Computational Methods and Experimental Measurements XIII

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Computational Methods and Experimental Measurements XIII

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WITeLibrary Home of the Transactions of the Wessex Institute. Papers presented at Computational Methods and Experimental Measurements XIII are archived in the WIT eLibrary in volume 46 of WIT Transactions on Modelling and Simulation (ISSN 1743-355X). The WIT eLibrary provides the international scientific community with immediate and permanent access to individual papers presented at WIT conferences. Visit the WIT eLibrary at www.witpress.com.

THIRTEENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL METHODS AND EXPERIMENTAL MEASUREMENTS

CMEM XIII CONFERENCE CHAIRMEN C.A. Brebbia Wessex Institute of Technology, UK G. M. Carlomagno University of Naples Federico II, Italy

INTERNATIONAL SCIENTIFIC ADVISORY COMMITTEE M. Attard Z. Bielecki R. Cerny J. Everett L. Fryba C. Golia W. Graf S. Hernandez

C. Karayannis R. Khanbilvardi G. Lorenzini O. Manca R.A. Pitarma P. Prochazka H. Sakamoto

F. Seiler P. Stehlik K. Takayama M. Trajkovic M. Tsutahara F. Viadero Rueda M. Wnuk

Organised by Wessex Institute of Technology, UK,and University of Naples Federico II, Italy Sponsored by WIT Transactions on Modelling and Simulation

WIT Transactions on Modelling and Simulation Transactions Editor Carlos Brebbia Wessex Institute of Technology Ashurst Lodge, Ashurst Southampton SO40 7AA, UK Email: [email protected]

Editorial Board C Alessandri Universita di Ferrara Italy J Baish Bucknell University USA D E Beskos University of Patras Greece J A Bryant University of Exeter UK M A Celia Princeton University USA J J Connor Massachusetts Institute of Technology USA D F Cutler Royal Botanic Gardens UK G De Mey Ghent State University Belgium Q H Du Tsinghua University China A El-Zafrany Cranfield University UK S Finger Carnegie Mellon University USA M J Fritzler University of Calgary Canada G S Gipson Oklahoma State University USA

M A Atherton South Bank University UK C D Bertram The University of New South Wales Australia M Bonnet Ecole Polytechnique France M B Bush The University of Western Australia Australia A H-D Cheng University of Mississippi USA D E Cormack University of Toronto Canada E R de Arantes e Oliveira Insituto Superior Tecnico Portugal J Dominguez University of Seville Spain S Elghobashi University of California Irvine USA P Fedelinski Silesian Technical University Poland J I Frankel University of Tennessee USA L Gaul Universitat Stuttgart Germany S Grilli University of Rhode Island USA

K Hayami National Institute of Informatics Japan D B Ingham The University of Leeds UK D L Karabalis University of Patras Greece H Lui State Seismological Bureau Harbin China R A Meric Research Institute for Basic Sciences Turkey K Onishi Ibaraki University Japan M Predeleanu University Paris VI France S Rinaldi Politecnico di Milano Italy G Schmid Ruhr-Universitat Bochum Germany X Shixiong Fudan University China V Sladek Slovak Academy of Sciences Slovakia J Stasiek Technical University of Gdansk Poland M Tanaka Shinshu University Japan T Tran-Cong University of Southern Queensland Australia J F V Vincent The University of Bath UK Z-Y Yan Peking University China G Zharkova Institute of Theoretical and Applied Mechanics Russia

J A C Humphrey Bucknell University USA N Kamiya Nagoya University Japan J T Katsikadelis National Technical University of Athens Greece W J Mansur COPPE/UFRJ Brazil J Mikielewicz Polish Academy of Sciences Poland E L Ortiz Imperial College London UK D Qinghua Tsinghua University China T J Rudolphi Iowa State University USA A P S Selvadurai McGill University Canada P Skerget University of Maribor Slovenia T Speck Albert-Ludwigs-Universitaet Freiburg Germany S Syngellakis University of Southampton UK N Tosaka Nihon University Japan W S Venturini University of Sao Paulo Brazil J R Whiteman Brunel University UK K Yoshizato Hiroshima University Japan

Computational Methods and Experimental Measurements XIII EDITORS C.A. Brebbia Wessex Institute of Technology, UK G.M. Carlomagno University of Naples Federico II, Italy

Editors: C.A. Brebbia Wessex Institute of Technology, UK G.M. Carlomagno University of Naples Federico II, Italy Published by WIT Press Ashurst Lodge, Ashurst, Southampton, SO40 7AA, UK Tel: 44 (0) 238 029 3223; Fax: 44 (0) 238 029 2853 E-Mail: [email protected] http://www.witpress.com For USA, Canada and Mexico Computational Mechanics Inc 25 Bridge Street, Billerica, MA 01821, USA Tel: 978 667 5841; Fax: 978 667 7582 E-Mail: [email protected] http://www.witpress.com British Library Cataloguing-in-Publication Data A Catalogue record for this book is available from the British Library ISBN: 978-1-84564-084-2 ISSN: 1746-4064 (print) ISSN: 1743-355X (on-line) The texts of the papers in this volume were set individually by the authors or under their supervision. Only minor corrections to the text may have been carried out by the publisher. No responsibility is assumed by the Publisher, the Editors and Authors for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. © WIT Press 2007 Printed in Great Britain by Athenaeum Press Ltd. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the Publisher.

Preface

This book contains most of the papers presented at the 13 th International Conference on Computational Methods and Experimental Measurements (CMEM/07) held in Prague in 2007. This series of conferences started in Washington DC at the beginning of the 1980s and has been reconvened every two years with continuous success. The primary aim of the meeting is to offer to the international scientific community an arena for the presentation and exchange of advanced approaches and applications in the fields of numerical methods and experimental determinations, with the principal attention and emphasis being devoted to their beneficial interaction and mutual influence. Recent advances in the speed and efficiency of computers, and in computational techniques, have been major factors in the growth of numerical methods that now affect not only engineering, but our everyday lives. However, even if computational codes have been increasingly successful in simulating engineering problems, there still exists a wide range of them that need a comprehensive validation that can be accomplished only by means of experimental analysis. In the meantime, experimental measurements have become so sophisticated that experiments must very often be carried out with the help of computers and the data obtained have to be processed by means of numerical methods. This volume contains a substantial number of excellent scientific papers, which have been grouped in the following sections: • Computational and experimental methods • Experimental and computational analysis • Fluid flow • Salts in porous materials • Heat transfer • Structural and stress analysis • Experiments and analysis of reinforced concrete members • Structural dynamics • Dynamics and vibrations • Detection and signal • Advances in measurements and experiments • Industrial applications

The Editors are grateful to all the authors for their valuable contributions and to the Members of the International Scientific Advisory Committee, as well as other colleagues, for their help in reviewing the papers published in this book and are most especially indebted to Prof. R. Èerný, C. G. Karayannis and Z. Bielecki who organised special sessions within the conference. The Editors Prague, 2007

Contents Section 1: Computational and experimental methods Direct simulations of fluid dynamic sounds by the finite difference lattice Boltzmann method M. Tsutahara, A. Tamura, S. Tajiri & W. Long ....................................................3 Spectral analysis of a transitional separated-reattached flow using Fourier and wavelet transforms I. E. Abdalla & M. J. Cook..................................................................................13 Observation of solid fuel in a supersonic flowfield J. M. Char & U. K. Hsu ......................................................................................25 Temperature field at the twin-roll casting of aluminium alloys: computational model and measurements H. Gjerkeš, S. Hartman, R. Vertnik & B. Šarler .................................................35 A newly developed test rig for the study of thermal fatigue M. Fazarinc, R. Turk, G. Kugler & M. Terčelj ...................................................45 Numerical simulation with flow feature extraction of a propeller turbine unsteady rotor-stator interaction J.-M. Gagnon & C. Deschênes............................................................................55 Computer and experimental study of the gate dielectric in a memory transistor R. Avichail-Bibi, D. Fuks, A. Kiv & Ya. Roizin...................................................65 Electron band structure and properties of disordered semiconductor compound alloys D. Alexandrov, K. S. A. Butcher & T. L. Tansley................................................75

Fast calculation of the dimensioning factors of the railway power supply system L. Abrahamsson & L. Söder................................................................................85 A formulation of a multi-wave elastodynamic infinite element K. Kazakov ..........................................................................................................97 Section 2: Experimental and computational analysis Influence of the collision speed and angle of a bullet: experimental reconstruction of bullet configuration and FE-analysis H. Sakamoto, T. Hiwatashi, T. Yamaguchi & M. Yamamoto............................109 Influence of the cross-section shape on the lateral torsional buckling capacity D. Djurić-Mijović & M. Trajković ....................................................................117 Computational fluid dynamic modelling and simulation evaluation of the plume evacuation device efficiencies F. Farshad, H. Rieke, L. C. LaHaye & S. C. Nulu............................................127 Flow estimations through spillways under submerged tidal conditions P. D. Scarlatos, M. Ansar & Z. Chen................................................................137 Analysis of the wave-flow interaction with submerged breakwaters A. C. Neves, F. Veloso Gomes & F. Taveira Pinto ...........................................147 An application of an edge effect based vacuum blower to a lyophilizer M. Kitamura, M. Tsutahara & H. Sasaki..........................................................155 The wavelength reconstruction from toroidal spectrometer image data J. Blazej, M. Tamas, L. Pina, A. Jancarek, S. Palinek, P. Vrba & M. Vrbova .....................................................................................................165 Numerical noise in satellite laser ranging data processing J. Blazej & I. Prochazka ...................................................................................171 Expanding the definition of multivariate correlation W. Conley ..........................................................................................................177

3D analysis of solid reinforced concrete beams subjected to combined load of bending, torsion and shear A. S. Alnuaimi....................................................................................................185 Use of correlation of iron loss and copper loss for improving the efficiency of three phase squirrel cage induction motors B. B. Saanane, A. H. Nzali & D. J. Chambega .................................................195 Effect of Zr addition on the fatigue strength of Cu-6Ni-2Mn-2Sn-2Al alloy M. Goto, S.-Z. Han, C.-J. Kim & N. Kawagoishi..............................................205 An analysis of superplastic free forming at constant pressure G. Giuliano & S. Franchitti ..............................................................................215 A mathematical model approach to a glycerolysis reaction for monoacylglycerol production B. Cheirsilp & A. H-Kittikul..............................................................................225 Evaluation of experimental procedures for confined concrete columns using 3D finite element analyses H. O. Köksal, C. Karakoç, Z. Polat, T. Turgay & Ş. Akgün .............................233 Geometrically nonlinear static analysis of 3D trusses using the arc-length method G. A. Hrinda......................................................................................................243 Section 3: Fluid flow On the accuracy of integral representation of differential operators in Lagrangian blob mesh-less methods C. Golia & B. Buonomo ....................................................................................255 Construction of a non-Newtonian fluid model based on the finite difference lattice Boltzmann method S. Tajiri & M. Tsutahara...................................................................................267 Experimental measurements for the control of a vortex shaft theoretical model G. Ciaravino, L. Ciaravino & G. Pulci Doria ..................................................277 Stability of stratified spin-up flows S. A. Smirnov .....................................................................................................287

On the relation between geometric and flow properties of a miniaturized fluid oscillator J. O. Sotero-Esteva, R. Furlan & J. J. Santiago-Avilés ....................................297 On-stream floodplain storage: experimental research G. De Martino, F. De Paola, G. Marini & A. Ranucci.....................................307 Analytical and hydraulic model study of highway culvert sand-blockages M. Kamaka, E. Cheng, M. Teng & C. Matsuda ................................................319 Incorporating computational fluid dynamics in the design/build of a single family residence J. Chang & N. Rosemann..................................................................................329 Section 4: Salts in porous materials (Special session organised by Professor R. Černý) Mathematical modeling of water and salt transport in porous materials R. Černý ............................................................................................................339 Determination of water and salt transport parameters of porous materials using methods of inverse modelling L. Fiala, Z. Pavlík, M. Pavlíková & R. Černý...................................................349 Effect of metakaolin on chloride binding in lime-based composites R. Pernicová, M. Pavlíková & R. Černý ...........................................................357 Computational simulation of the effect of crystallization inhibitors on salt transport and crystallization in porous materials J. Kelnar, J. Maděra & R. Černý ......................................................................367 Desalination of historical masonry using hydrophilic mineral wool boards P. Michálek, V. Tydlitát, M. Jerman & R. Černý..............................................377 Section 5: Heat transfer Numerical investigation on natural convection in asymmetric channel-chimney systems A. Andreozzi, B. Buonomo & O. Manca ...........................................................389

Plate distance effect on mixed convection in horizontal channels heated from below G. Foglia, O. Manca & S. Nardini....................................................................399 Numerical analysis of mixed convection in air in an inclined channel with a moving plate A. Andreozzi, N. Bianco, G. Lacasa & V. Naso ................................................411 A numerical method for studying impulsively generated convection from heated tubes S. J. D. D'Alessio...............................................................................................425 Heat transfer in a ribbed square duct by Large-Eddy-Simulation O. Labbé............................................................................................................437 Effect of thermal boundary conditions on conjugate natural convection flow in vertical eccentric annuli A. Jamal, M. A. I. El-Shaarawi & E. M. A. Mokheimer....................................447 Foam flow turn influence on the in-line tube bundle heat transfer intensity J. Gylys, S. Sinkunas, T. Zdankus, V. Giedraitis & A. Balcius .........................457 Importance of experimental measurements and simulations for “sludge-to-energy” systems L. Houdkova, J. Boran, T. Elsäßer & P. Stehlik ...............................................465 Simulation of thermal barrier coating behaviour during dynamic thermal loading by the Exodus method J. Sroub, M. Honner & Z. Vesely ......................................................................475 Forced convection in a variable section axisymmetric channel with different porous layers and heat generation E. Pilevne & A. Misirlioglu...............................................................................485 Comparison of h-, p- and hp-adaptation for convective heat transfer D. W. Pepper & X. Wang ..................................................................................495 Modified spherical harmonics method for one-speed transport equation with anisotropic scattering M. S. Li & B. Yang ............................................................................................505

Section 6: Structural and stress analysis Inverse variational principle based coupled modeling of underground structures P. Procházka .....................................................................................................517 Elastic-plastic simulation of plate with a blunt slit subjected to uni-axial tension S. Ohtaki, S. Kobayashi & T. Yamamoto ..........................................................527 A simplified shear strength evaluation model for reinforced concrete corbels J. K. Lu, S. Y. Kuo, J. Y. Lin & S. H. Hsu .........................................................537 Post-buckling behaviour of a slender beam in a circular tube, under axial load M. Gh. Munteanu & A. Barraco .......................................................................547 Structural properties of a new material made of waste paper J. Santamaria, B. Fuller & A. Fafitis................................................................557 Non-Hertzian rolling contact stress analysis C. H. Liu & W.-E. Hsu ......................................................................................569 The areolar strain concept applied to elasticity I. D. Kotchergenko ............................................................................................579 Methodology for the manufacture of smart composites with thermoplastic matrix L. Elsoufi, K. Khalil, R. Lachat, W. Charon & M. Zoaeter...............................589 Experimental investigation on the folding of axially crushed hexagonal tubes M. R. Said, A. A. Mokhtar, A. Alias & A. Ibrahim ............................................601 Application of the shakedown analysis in the elastic: plastic assessment of cracked plates M. A. Belouchrani .............................................................................................611 Section 7: Experiments and analysis of reinforced concrete members (Special session organised by Professor C. G. Karayannis) Cyclic testing of reinforced concrete beam-column joints with crossed inclined bars C. E. Chalioris, C. G. Karayannis & M. I. Favvata .........................................623

Tests and analysis of reinforced concrete beams under torsion retrofitted with FRP strips C. E. Chalioris ..................................................................................................633 Influence of masonry strength and rectangular spiral shear reinforcement on infilled RC frames under cyclic loading D. J. Kakaletsis .................................................................................................643 Application of the Cement Hydration Equation in self-compacting concrete’s compressive strength N. Anagnostopoulos, A. Gergiadis & K. K. Sideris ..........................................655 Section 8: Structural dynamics Response of a double system beam and string with an elastic layer to the dynamic excitations L. Frýba, C. Fischer & Sh. Urushadze .............................................................671 Excessive accelerations in bridges for Korea high-speed railway J. W. Kwark, J. R. Cho, W. J. Chin, B. S. Kim & E. K. Cho .............................681 Parameter identification for multiple modes of cable-stayed bridge cables using ambient vibration measurements from a single station W.-H. Wu, C.-A. Liao & C.-C. Chen ........................................................ 691 Experimental evaluation of dynamic properties of an incrementally prestressed concrete girder railway bridge S. I. Kim, I. H. Yeo, N. S. Kim, J. W. Kwark & J. S. Lee...................................701 Section 9: Dynamics and vibrations A model of spur gears supported by ball bearings F. Viadero, A. Fernandez del Rincon, R. Sancibrian, P. Garcia Fernandez & A. de Juan...................................................................711 On a numerical model of a complete washing machine T. Argentini, M. Belloli, N. Gaudiano, G. Fraternale, F. Panetta, D. Sabato & M. Vanali......................................................................................723

Section 10: Detection and signal processing (Special session organised by Professor Z. Bielecki) EUV detection system with calibrated responsivity J. Mikolajczyk & Z. Bielecki .............................................................................737 Analysis of radiating structures placed on multilayer dielectric M. Wnuk & M. Bugaj ........................................................................................747 Method of signal processing in passive infrared detectors for security systems H. Madura .........................................................................................................757 Influence of the displacement effect on compressed LFM signal parameters A. Kawalec, Cz. Leśnik, W. Komorniczak, W. Czarnecki & J. Pietrasiński................................................................................................769 A comparison of estimation accuracy by the use of KF, EKF & UKF filters S. Konatowski & A. T. Pieniężny.......................................................................779 Novel method for watermarking system operating on the HF and VHF radio links Z. Piotrowski & P. Gajewski.............................................................................791 Optoelectronic system for phase array antenna beam steering E. Sędek, Z. Bielecki, M. Muszkowski, W. Kołosowski, G. Różański & M. Wnuk....................................................................................801 Nitrogen dioxide detection using an optoelectronic sensor Z. Bielecki, W. Kołosowski, G. Różański, E. Sędek & J. Wojtas.......................809 Automatic processing analysis of infrared images for monitoring pantograph catenary interactions A. Balestrino, O. Bruno, A. Landi & L. Sani.....................................................819 Architectural hot-swap in NURBS surfaces: versioning of case studies, work in progress A. Prichard-Schmitzberger ...............................................................................829

Section 11: Advances in measurements and experiments Development of advanced instrumentation for operational oceanography G. Zappalà ........................................................................................................841 In-situ measurement of formwork pressures generated by Self-Compacting Concrete M. M. Giammatteo, A. Gregori & G. Totani.....................................................851 Evaluation of phenolic resins from one-pot microwave synthesis A. Britten, M. M. MacIntyre & A. Miadonye ....................................................861 Modern techniques of measure and control of deformations – an experimental test: Senerchia landslide M. Caprioli & G. Strisciuglio ...........................................................................871 Section 12: Industrial applications Calculating the dilution between successive trainloads of iron ore during processing J. E. Everett .......................................................................................................883 Rock bumps due to the creation of a dislocation during deep mining V. Doležel & P. Procházka ...............................................................................893 Author Index ...................................................................................................903

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Section 1 Computational and experimental methods



3

Direct simulations of fluid dynamic sounds by the finite difference lattice Boltzmann method M. Tsutahara, A. Tamura, S. Tajiri & W. Long Graduate School of Science and Technology, Kobe University, Rokko, Nada, Kobe, Japan

Abstract In this paper we present some applications of the finite difference lattice Boltzmann method (FDLBM) to direct simulations of fluid dynamic sound. The Arbitrary Lagrangian Eulerian formulation is introduced to FDLBM and the sounds emitted from moving bodies are successfully simulated. The two-particle model is used to simulate two-phase flows, and introducing a fluid elasticity the sound propagation inside the liquid is simulated. The sounds generated on the interface between the liquid and gas are also successfully simulated. Keywords: fluid dynamic sound, lattice Boltzmann method, Arbitrary Lagrangian Eulerian formulation, two-phase flow, under water sound.

1

Introduction

The lattice Boltzmann method [1–6] is now a very powerful tool of computational fluid dynamics (CFD). This method is different from ordinary Navier-Stokes equations based CFD methods, and is based on the particle motions. However, mostly successful model so far is for incompressible fluids, but several models for thermal compressible models have been proposed including our model [7–13]. On the other hand, this method has great advantage to simulate multi-phase flows, because the interface is automatically determined in this method without special treatment [14–17]. We use a compressible fluid model of LBM and perform direct simulations of aerodynamic sound emitted from moving bodies using the arbitrary Lagrangian Eulerian formulation, especially the sound sources are detected. We also propose a new model for liquids considering the elasticity of liquid, and the sound speed propagating inside the liquid is correctly realized. A simulation of a water drop colliding the water surface and sound emission is performed. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070011

4 Computational Methods and Experimental Measurements XIII

2

Discrete BGK equation

In this paper, we apply the discrete BGK equation as a governing equation of the FDLBM. The discrete BGK equation represents evolution of a velocity distribution function of particles fi (x, t ) as ∂fi ( x, t )

∂fi ( x, t )

1 (1) { f ( x, t ) − fi (0) ( x, t )} τ i where t and x indicate the time and the space, respectively. Subscript i and α denote directions of particles' motion and the space directions in Cartesian coordinates, and ciα represents a particle velocity. τ is called a relaxation time and fi (0) (x, t ) is the equilibrium distribution function, that is determined so as to ∂t

+ ciα

∂xα

=−

recover the corresponding fluid dynamic equations. The details are shown later. The right hand side is a collision term, and represents that a particle distribution approaches an equilibrium state by the collisions among particles. We introduce a particle model into equation (1) in order to determine the particle velocity and the equilibrium distribution function. In this paper, we apply the D2Q21 model for two-dimensional flows and the D3Q39 model for three-dimensional flows proposed by Takada and Tsutahara [9]. The particle velocity vector is shown in Tables 1 and 2. The equilibrium distribution function is given by a following polynomial of the flow velocity up to the third order.

f i ( 0 ) = F i ρ (1 − 2 Bc iα u α + 2 B 2 c i α c iβ u α u β + Bu

2

(2) 4 3 B c iα c iβ c iγ u α u β u γ ) 3 where ρ, u and e denote the density, the flow velocity and the internal energy, respectively. They are defined by the particle velocity and the distribution function as follows. (3) ρ (x, t ) = ∑ f (x, t )

− 2 B 2 c iα u α u 2 −

i

i

ρ (x, t )uα (x, t ) = ∑ f i (x, t )ciα

(4)

i



u (x, t ) 2

 c2 (5)  = ∑ f i (x, t ) i  2 i   The coefficients Fi and B in equation (2) depend on the internal energy. They are determined to satisfy following constraints in order to recover the compressible Navier-Stokes equations. The coefficients Fi and B are presented in Tsutahara et al [13]. (6) ∑ f i (0) = ρ

ρ (x, t ) e(x, t ) +

2

i

∑

∑f

f i c iα = ρ u α (0)

(7)

i

(0) i

c iα c iβ = ρ (eδ αβ + u α u β )

i

WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

(8)


∑f

i

 c i2 u2   = ρ  e + 2 2  

c iα

 c i2 u2 = ρu  2e + 2 2 

( 0)

i

∑f i

Table 1:

2-7 8-13 14-19 20-31 32-39

3

(9)   

(10)

Velocity set in D2Q21 model.

i 1 2-5 6-9 10-13

Velocity vector (0, 0) (1, 0), (0, 1), (-1, 0), (0,-1) (2, 0), (0, 2), (-2, 0), (0,-2) (3, 0), (0, 3), (-3, 0), (0,-3)

|c| 0 1 2 3

14-17

(1, 1), (-1, 1), (-1,-1), (1,-1)

2

18-21

(2, 2), (-2, 2), (-2,-2), (2,-2)

2 2

Table 2: i 1

( 0) i

5

Velocity set in 3D39V model.

Velocity vector (0,0,0) (1, 0, 0), (-1, 0, 0), (0, 1, 0), (0,-1, 0), (0, 0, 1), (0, 0,-1) (2, 0, 0), (-2, 0, 0), (0, 2, 0), (0,-2, 0), (0, 0, 2), (0, 0,-2) (3, 0, 0), (-3, 0, 0), (0, 3, 0), (0,-3, 0), (0, 0, 3), (0, 0,-3) (2, 2, 0), (-2, 2, 0), (-2,-2, 0), (2,-2, 0), (0, 2, 2), (0,-2, 2), (0,-2,-2), (0, 2,-2), (2, 0, 2), (-2, 0, 2), (-2, 0,-2), (2, 0,-2) (1, 1, 1), (-1, 1, 1), (-1,-1, 1), (1,-1, 1), (1, 1,-1), (-1, 1,-1), (-1,-1,-1), (1,-1,-1)

|c| 0 1 2 3

2 2 3

Arbitrary Lagrangian Eulerian (ALE) formulation

The ALE method is generally used to simulate flows around moving bodies or to calculate the problem that contains interface of multi-phase flows. In the ALE method, a flow-field is described by the Eulerian formulation, and a grid moves together with moving boundaries. In this paper, a grid deformation is not considered and the grid only translates or rotates as a solid body. Thus, we can easily present a formulation of the ALE method, where the convection velocity of the equation is replaced with the relative velocity between the convection velocity and the grid velocity. On the surface of moving boundary, a flow velocity is given by that of the moving body as the boundary condition.


6 Computational Methods and Experimental Measurements XIII If the discrete BGK equation is regarded as a convection equation, the particle velocity corresponds to a convection velocity. Thus, the discrete BGK equation is reformulated as follows [18]. ∂fi ( x, t ) ∂t =−

1

φ

+ {ciα − Vα ( x, t )}

∂f i ( x, t ) ∂xα

−

A

φ

ciα

{

}

∂ fi ( x, t ) − f i (0) ( x, t ) ∂xα

(11)

{ f ( x, t ) − f ( x, t )} (0)

i

i

where V is the grid velocity vector. The third term of the left hand side means negative viscosity. We can set large time increment in high Reynolds number flows due to this additional term [19]. As a result, computational time becomes shorter. This term is not replaced although it includes a particle velocity, because this term represents the negative viscosity and is not a convection term. By applying the Chapman-Enskog expansion, we obtain the Navier-Stokes equations

∂ρ ∂ + {ρ (uα − Vα )} = 0 ∂t ∂xα ∂ (ρuα ) + ∂ {ρuα (u β − Vβ ) + pδ αβ } ∂t ∂x β ∂   ∂u β ∂uα + − µ ∂x β   ∂xα ∂x β

(13)

  ∂u  + λ γ δ αβ  = 0  ∂xγ  

   + puα     ∂u β ∂uα  ∂u  ∂  ∂e  + λuα β  = 0 + − + µu β  κ ′   ∂x β  ∂xα  ∂xα  ∂xα ∂x β  ∂   u2  ρ  e + ∂t   2

(12)

  ∂  u2  +  ρ (uα − Vα ) e + 2   ∂xα 

(14)

where the pressure, the viscosity, the second viscosity, and the thermal diffusivity are expressed as follows p = ρ e , µ = ρ e (φ − A ) , λ = − ρ e (φ − A) = − µ , κ ' = 2ρe(φ − A) (15) The sound speed is given as

c s = 2e

(16)

3.1 Sound emitted from a rapidly rotating elliptic cylinder and the grid system

The grid system consists of a boundary fitted co-ordinate system near the elliptic cylinder which rotates with the cylinder and outside of which is a cylindrical coordinates at rest as shown in Fig.1. Between the two grid systems, we set a buffer region and variables are connected through the region.



Figure 1:

7

Grid for computation.

3.2 Sound pressure field and sound source

The Reynolds number, the Mach number and the aspect ratio are nondimensional parameters of this problem and they are defined as follows. Re =

L ρUL , U , Ar = Ma = l µ cs

(17)

where U is the peripheral velocity of the edge of the elliptic cylinder, L is the chord length, l is the thickness. We show the result for Ar= 10.0, Re=10000 and Ma=0.2. The number of the grid is (r×θ)=(421×301). The streamlines are shown in Fig. 2(a) and two vortices appear and rotate slowly in the same direction as the cylinder rotation. The pressure field in the vicinity of the cylinder at the same time is shown in Fig.2(b). When the edge of the cylinder passes through the vortices and goes into the potential region, the pressure on the front side of the edge rise sharply and on the other side the pressure drops and this pressure change is the sound source. The sound pressure fields when the sound is emitted are shown in Fig.3.

(a) Figure 2:

(b)

Stream-lines and pressure distributions at non-dimensional time t* = 6.24.



(a) t* = 6.24 Figure 3:

4

(b) t* = 6.40

Sound pressure field near the elliptic cylinder.

Two-phase flow model

As a two-phase, liquid and gas, model, Swift et al have presented models employing free energy and they are widely used. But it is difficult to extend these models to a real liquid and gas model. We use a two-particle model in this paper. Two particle distribution functions are defined for, say, blue and red particles, and are written as fb and f r , respectively. Macroscopic variables are defined in (3) (4) and (5) for each fluid, but the fluid velocity is common to both fluids and the internal energy for liquid is neglected. 4.1 A model for liquid

The liquid has density much larger than that of the gas and also has larger elasticity, but it is difficult to realize these properties by the LBM because the LBM model is for an ideal gas. For large elasticity model, we introduce a new definition of pressure, which has a relation with the bulk elasticity as (18) P ' = P + β ( ρb − ρb ,ref ) where ρb ,ref is the reference density defined appropriately. Then the sound speed in liquid is given as

cs =

∆P ' ∼ β ∆ρ

(19)

and this sound speed is shown to be realized properly. 4.2 Two-phase flow model with large density difference

For two-phase, liquid and gas, flow models with large density, He et al. [20] introduce a model with the density difference up to 20 or so. They introduce two distribution functions, and one is only used as an index function that is detected the position of the interface. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


9

Inamuro et al. [21] also present a model with the density difference up to 1000, but in their model the velocity field and the pressure field must be recalculated because the densities are uniformly defined in liquid and gas phases. We present novel model including the density difference. The idea is very simple, that is the effect of the density difference appears the difference of the acceleration of the fluid acted some force. Therefore, we can consider the density difference by changing the acceleration. If the density of liquid, corresponding to the blue particles, is m times larger than that of gas, corresponding the red particles, the decelerations of the gas and liquid are

µ 2 1 1 µ ( − ∇P + ∇ 2 u) + τ g ∇ u) + ρ ρ mr ρ 1 µ 1 1 µ (− ∇P '+ ∇ 2u) + τ g = −(− ∇P + ∇ 2u) + mb ρ ρ ρ ρ

a mod,r = −(− a mod,b

1

ρ

∇P +

(20) (21)

in which mr and mb are the masses of the gas and the liquid particles, respectively, and the last terms in (20) and (21) are effect of the gravitational force. 4.3 Phase separation and surface tension 4.3.1 Shang-Chen model The surface tension is defined by the gradient of blue particles, and the velocity for determining the local equilibrium distribution function is changed by the fluid velocity due to the impulse of the surface tension force [16,17] as (22) ub = u + τ × a mod,b + τ × κρ r ∇ρ b

(23) u r = u + τ × a mod,r − τ × κρ b∇ρ b where κ represent the strength of the surface tension. By substituting the above mentioned velocities to the local equilibrium distribution function, a two-phase fluid model with large density difference is obtained. 4.3.2 Latva-Kokko and Rothman model In order to separate the liquid and the gas phases, the surface tension force is commonly used as the above mentioned Shang-Chen model, but this technique gives large spurious velocities. Then we use a method proposed by Latva-Kokko and Rothman [22], in which the distribution of particles is separated on the liquid side and the gas side on the interface.

ρ L ρG f eq (0) + f iGeq (0) ) cos ϕ 2 ( iL ρ L + ρG ( ρ L + ρG ) ρG ρ ρ fiG′ = ( fiL + fiG ) + κ1 L G 2 ( fiLeq (0) + fiGeq (0) ) cos ϕ ρ L + ρG ( ρ L + ρG ) fiL′ =

ρL

( fiL + fiG ) − κ1

(24) (25)

where κ1 is a factor of particle separation on the interface, f iLeq ( 0 ) is a local equilibrium distribution function for velocity 0, and ϕ represents the angle of the interface. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

10 Computational Methods and Experimental Measurements XIII The surface tension is defined as a normal force to the interface, and the force is proportional to the radius of curvature as (26) Fsα = σκnα where σ is the radius of curvature,

κ

is the surface tension, and nα is a normal

unit vector on the interface. By substituting the above mentioned velocities to the local equilibrium distribution function, a two-phase fluid model with large density difference is obtained.

Figure 4:

Phase separation model of Latova-Kokko and Rothman.

Figure 5:

Surface tension model.

4.4 Water drop colliding the water surface and generated sound

Figure 6 shows the sound generated by the collision of a 3D water drop with a water surface. Calculation parameters are as follows: the factor of particle separation κ1 =2.0, the surface tension κ = 0.004, the bulk elasticity β =16000, and the acceleration of gravity g =0.001. In (a) the shape of the drop at the collision and the shadow graph of the sound generated on the interface are shown. In (b) the shape of the splash and the its shadow graph are shown. The sound propagating in the air and also the under water sound are clearly seen.

5

Conclusions

Direct simulations of fluid dynamic sound are performed by the finite difference lattice Boltzmann method. The Arbitrary Lagrangian Eulerian formulation is WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


11

applied to the simulation of sound emitted from a rapidly rotating elliptic cylinder and the sound source is detected. A two-phase flow model with large density difference and including the elasticity of liquid is proposed and a simulation of sound generated when a water drop collides with a water surface is also performed.

(a) Figure 6:

(b)

Sound generated at the collision of a water drop with a water surface.

References [1] [2] [3]

[4] [5] [6] [7] [8] [9]

Qian, Y.H., Succi, S. and Orszag, S.A., Recent Advances in Lattice Boltzmann Computing, Ann. Rev. of Comp. Phy. III, D. Stauffer ed. World Scientific, pp.195-242, 1995. Rothman, D.H. and Zalenski, S., Lattice-Gas Cellular Automata, Cambridge U.P., 1997. Chopard, B. and Droz, M., Cellular Automata Modeling of Physical Systems, Cambridge University Press, 1998. Chen, S. and Doolen, G.D., Lattice Boltzmann method for fluid flows, Ann. Rev. Fluid Mech., Ann. Rev. Inc. pp.329-364, 1998. Wolf-Gladrow, D.A., Lattice-Gas Cellular Automata and Lattice Boltzmann Models, Lecture Notes in Mathematics, Springer, 2000. Succi, S., The lattice Boltzmann Equation for Fluid Dynamics and Beyond, Oxford, 2001. Alexander, F.J. et al., Lattice Boltzmann thermodynamics, Phys. Rev. E, 47 R2249-R2252, 1993. Chen, Y., et al., Thermal lattice Bhatanagar Gross Knook model without nonlinear deviations in macrodynamic equations, Phys. Rev. E, 50, pp.2776-2283, 1994. Takada, N. and Tsutahara M., Proposal of Lattice BGK model with internal degrees of freedom in lattice Boltzmann method, Transaction of JSME B， 65-629, pp.92-99, 1999 (in Japanese). WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

12 Computational Methods and Experimental Measurements XIII [10] [11] [12] [13] [14] [15] [16] [17] [18]

[19] [20]

[21] [22]

McNamara, G.R., et al., Stabilization of thermal lattice Boltzmann models, J. Stat. Phys., 81(1/2), pp. 395-408, 1995. Kataoka, T. and Tsutahara, T., Lattice Boltzmann model for the compressible Navier-Stokes equations with flexible specific-heat ratio, Phys. Rev. E, 67, pp.036306-1-4, 2004. Watari, M. and Tsutahara, T., Two-dimensional thermal model of the finite-difference lattice Boltzmann method with high spatial isotropy, Phys. Rev. E, 69, pp.035701-1-7, 2004. Tsutahara M, Takada N, Kataoka T, Lattice gas and lattice Boltzmann methods, Corona-sha, (in Japanese) 1999. Swift, M.R., Orlandini, E., Osborn, W.R., and Yeomans, J.M., Lattice Boltzmann simulations of liquid-gas and binary-fluid systems. Phys. Rev. E 54, pp.5041-5052, 1996. Swift, M.R., Osborn, W.R., and Yeomans, J.M., Lattice Boltzmann simulation of non-ideal fluids. Phys. Rev. Lett. 75, pp.830-833, 1995. Shan, X. and Chen, H., Lattice Boltzmann model for simulating flows with multiple phases and components. Phys. Rev. E 47, pp.1815-1819, 1993. Shan, X. and Chen, H. Simulation of non-ideal gases and liquid-gas phase-transitions by the lattice Boltzmann-equation. Phys. Rev. E 49, pp.2941-2948, 1994. Tamura, A., Tsutahara, M., Matsuoka. H. Direct simulation of bladevortex interaction by the finite difference lattice Boltzmann method, WESPAC 2006 CD-ROM, 2006. Tsutahara, M., Kurita, M., and Kataoka, T., Direct simulation of Aeolian tone by the finite difference Lattice Boltzmann method, Computational Fluid Dynamics 2002, pp.508-513, 2003. He X., Chen S., and Zhang R., A lattice Boltzmann Scheme for Incompressible Multiphase Flow and Its Application in Simulation of Rayleigh-Taylor Instability, J. Computational Physics 152, pp.642-663, 1999. Inamuro T., Ogata T., Tajima S. and Konishi N., A lattice Boltzmann method for incompressible two-phase flows with large density differences, J. Computational Physics 198, pp.628-644, 2004. Latva-Kokko, M. and Rothman D.H., Diffusion Properties of Gradientbased Lattice Boltzmann Models of Immiscible Fluids, Physical Review E, 71 pp. 056702 1-8, 2005.



13

Spectral analysis of a transitional separated-reattached flow using Fourier and wavelet transforms I. E. Abdalla & M. J. Cook Institute of Energy & Sustainable Development, De Montfort University, Leicester, UK

Abstract Large-eddy simulations (LES) of transitional separating-reattaching flow over a square surface mounted obstacle (SSMO) and a forward-facing step (FFS) have been performed. The Reynolds number based on the uniform inlet velocity and the obstacle height is 4.5 × 103 . A dynamic subgrid-scale model is employed in this work. The mean LES results compare favourably with the available experimental and DNS data. This paper addresses the characteristic shedding modes associated with the separated-reattached flows on the SSMO and the FFS and sheds light on the use of the wavelet transform (WT) in extracting the content of a time history of a (velocity and/or pressure) signal compared to the traditional Fourier transform (FT). The turbulence spectra for the geometries revealed amplified frequency modes both upstream and downstream of the separation edge with those associated with the SSMO showing more clearly compared to the FFS. A frequency peak was detected at a location upstream of the separation line and immediately above the SSMO. The value of this frequency suggests that the upstream separated region is unstable via the Kelvin–Helmholtz instability and the peak can not be attributed to the flapping of the separated shear layer which is a phenomenon commonly associated with this class of flows. The WT captured events that are characterised by narrow periods (scales) and which happened over shorter times. Such events are smoothed out by the Fourier transform indicating the superiority of the WT over the FT. Keywords: large-eddy simulation, wavelet and Fourier transform, spectral analysis, transitional to turbulence.

WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070021


1 Introduction Turbulent and transitional flows over obstruction such as the SSMO and the FFS are an important group of separated-reattached flows that occur in many industrial and environmental applications. For example, control of boundary layers, river flows, wind loads on structures, and the spread of pollutants in the vicinity of buildings. Separated-reattached flows over an obstruction is quite complicated when compared with other bluff body geometries such as the backward-facing step and the blunt leading edge plate aligned to a flow field. The complication comes as a result of an additional separation in the upstream region displayed by the obstruction leading to a closed or open bubble (Satür et al [1]). For the SSMO, there exist few studies which are focused on the very basic features of this class of flows such as the variation of the mean reattachment length with the Reynolds number (Tropea and Gackstatter [2]), the influence of the obstacle aspect ratio ( hl ) on the mean reattachment length (Bergeles and Athanassiadis [3]), the effect of varying the blockage ratio (Durst and Rastogi [4]). Little work has been published on either laminar or turbulent separated flows over the FFS flow. Recently, Satür et al [1] and Wilhelm et al [5] performed experimental and computational work respectively to study the laminar separation on a forward facing step for Reynolds numbers as low as 30. This work is mainly focused on the flow structure and the instability causing three-dimensionality of the flow downstream of the leading edge. Previous studies for separated-reattached flows have identified specific frequency modes associated with some physical phenomena of the separatedreattached flows, such as the shedding frequency, in addition to a lower one which is attributed to flapping of the shear layer (Kiya and Sasaki [6]). The main technique used by the researchers to identify these range of frequencies is through FT to a time series of the velocity and pressure field with the aid of flow visualization. Kaiser [7] suggests that the FT method is inaccurate and inefficient for time-frequency localisation. The spectra of transitional/turbulent flow usually contain a range of frequencies. For analysis where a predetermined scaling may not be appropriate because of a wide range of dominant frequencies (such as signals resulting from a transitional/turbulent flows), a method of timefrequency localization that is scale dependent, such as wavelet analysis, might be more successful. The aim of this paper is to shed light on the use of the WT as a method of extracting the content of a time series of the velocity and pressure fields obtained from a large-eddy simulation on separated-reattached flow on a SSMO and a FFS and discuss the processes responsible for the content of the spectra (shedding and interaction of the coherent structures). The WT method is not commonly used in this field and its strengths/weaknesses are not well known when applied to data from unsteady turbulent flows. The objectives are: (i) to identify the amplified



15

frequencies associated with the SSMO and FFS, and (ii) to compare the spectra details using the FT and WT and highlight the strengths and weaknesses of the two methods.

2 Details of numerical computation The governing equations are discretised on a staggered grid using the finite volume method. Any small-scale (smaller than the control volume) motions are averaged out and have to be accounted for by a subgrid-scale model. A standard dynamic subgrid-scale model in cartesian co-ordinates has been employed in the present study. The ratio νs /ν is zero in the laminar region before transition occurs and starts to increase shortly after the separation line reaching a maximum value of about 9 around the mean reattachment location and dropping down to about 5 after reattachment. The explicit second order AdamsBashforth scheme is used for the momentum advancement. The Poisson equation for pressure is solved using an efficient hybrid Fourier multigrid method. The spatial discretisation is secondorder central differencing which is widely used in LES owing to its non-dissipative and conservative properties. More details of the mathematical formulation and numerical methods have been reported elsewhere by Yang and Voke [8]. 8

y/h

6 4 2 0 -5

-2.5

0

2.5

5

7.5

10

12.5

x/h

15

17.5

20

22.5

25

27.5

30

Figure 1: The computational domain and mesh used for the first simulation of the SSMO.

Two simulations were performed for the obstacle case. In the first simulation (figure 1) 288 × 128 × 64 cells along the streamwise, wall-normal and spanwise directions respectively were employed. The blockage ratio is 8 and the spanwise dimension of the domain is 4h. A free-slip but impermeable boundary is applied at the lateral boundary, periodic boundary along the spanwise and no-slip boundary conditions are used at all other walls. At the inflow boundary, a uniform velocity profile is applied and the Reynolds number based on the inflow velocity and obstacle height is 4500. At the outflow boundary, a convective boundary condition is applied. In terms of wall units based on the friction velocity downstream of reattachment at x/h = 27, the streamwise mesh sizes vary from ∆x+ = 6.77 to ∆x+ = 43.04 , while ∆z + = 10.625 and at the wall ∆y + = 1.28. The time step used in this simulation is 4.75 × 10−6 second (0.001425 Uh0 ). The simulation ran for 129, 000 time steps, equivalent to more than 5 flow passes through the WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

16 Computational Methods and Experimental Measurements XIII domain (or residence times) to allow the transition and turbulent boundary layer to be established, i.e. the flow to have reached a statistically stationary state. The averaged results were gathered over a further 249, 900 time steps, with a sample taken every 10 time steps (24, 990 samples) averaged over the spanwise direction, corresponding to more than 10 flow passes or residence times. The second simulation used 320 × 220 × 64 cells for 0.25 m × 0.15 m × 0.04 m resulting in a blockage ratio of 15. The streamwise mesh sizes vary from ∆x+ = 6.09 to ∆x+ = 19.988 , while ∆z + = 10.96 and at the wall ∆y + = 1.14. The averaged results gathered from this simulation show no significant changes in the mean reattachment length from the first simulation. The same computational domain used in the second SSMO simulation was adopted for the FFS (with the step leading edge again located at x/h=5). The FFS case ran for a total of 404,000 time steps with the sampling for the mean field started 100,000 after the start of the run.

3 Mean flow field An important parameter characterising a separated-reattached flow is the time mean position of the reattachment. The mean streamlines and the mean axial velocity profile at the first cell from the solid boundary (a method used to determine the mean reattachment location) are shown in figure 2(a) and b respectively and indicate that the mean reattachment length is ≈ 15.5h. The predicted mean reattachment length compares well with the experimental measurements of 15.5h reported by Tropea and Gackstatter [2] which is the benchmark used in this work for the current LES of the SSMO. Other values include Bergeles and Athanassiadis [3] (xR /h = 11), Durst and Rastogi [4] (xR /h = 16), all of which are under a turbulent condition with high free-stream turbulence. Similar scatter was reported for the fence geometry including Tropea and Gackstatter [2] (xR /h = 17), the DNS study of Orellano and Wengle [9] (xR /h = 13.2) and the experiment of Larsen [10] (xR /h = 11.7). Overall, the LES simulation has predicted the line mean position of the reattachment well. Similarly, figures 3(a), and (b) are the corresponding mean stream line and the mean streamline velocity profiles at the first cell from the solid surface for the FFS flow. The predicted mean reattachment length downstream the separation line read from the two figures is 8.1h. Experimental and computational studies for the FFS are few and those which exist have predicted a shorter length than 8.1h. Ko [11] simulation predicted this length as 5.5h and the measured value from Moss and Baker [12] experiment is 4.8h. Similarly, the work of Bergeles and Athanassiadis [3] showed that the mean reattachment length downstream an extended obstacle is of order 4h. In contrast with the current simulation it appears that the LES has over-predicted this parameter. But once again the difference is thought to be due to the high Reynolds number and the nature of turbulent flows in the work cited here. For laminar separation, as in the current case, the transition could be delayed leading to a longer bubble than the case for turbulent separation. A relevant type of flow to the FFS is the blunt plate experiment of Castro and WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


y/h

2 1.5 1 0.5 0 -0.5 -1

(a)

-2

0

2

4

6x/h

8

10

8

10

12

14

17

16

0.5

y/h

0.25 0 -0.25 (b) -0.5

2

4

6

x/h

12

14

16

Figure 2: LES prediction of flow over the SSMO: (a) Mean streamlines, (b) profile of mean axial velocity at the first cell from the solid surface along the streamwise direction. 1.5

(a)

y/h

1 0.5 0 -0.5 -1 -3

-2

-1

0

1

2

3

4

x/h

5

6

7

8

1

(b)

y/h

0.5

0

-0.5

-1

1

2

3

4

x/h

5

6

7

8

9

Figure 3: LES prediction of flow over the FFS: (a) Mean streamlines, (b) profile of mean axial velocity at the first cell from the solid surface along the streamwise direction.

Epik [13] (ReD = 6500) in which the reattachment is reported to be 7.7D, where D is the plate thickness. This is comparable to the mean reattachment length for the current simulation. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

y/h

18 Computational Methods and Experimental Measurements XIII 9 8 7 6 5 4 3 2 1 0

0

1

2

3

4

5

Um/U0

6

7

8

9

10

y/h

Figure 4: The SSMO flow: profiles of mean streamwise velocity Um /U0 at six streamwise locations measured from the separation line (leading edge). Left to right x/xR =0.05, 0.2, 0.4, 0.6, 0.8, 1.025. Also shown are measurements by Tropea and Gackstatter [2] (triangle), Larsen [10] (square) and the DNS data of Orellano and Wengle [9] (circle symbol) at Re=3,000. 9 8 7 6 5 4 3 2 1 0

0

0.4

0.8

1.2

1.6

2

urms/U0

Figure 5: The SSMO flow: profiles of mean streamwise turbulent intensity urms /U0 at six streamwise locations measured from the separation line (leading edge). Left to right x/xR =0.05, 0.2, 0.4, 0.6, 0.8, 1.025. Also shown are measurements by Larsen [10] (square) and the DNS data of Orellano and Wengle [9] (circle) at Re=3,000. 8 7 6 y/h

5 4 3 2 1 0

0

1

2

3 Um/U0

4

5

6

Figure 6: The FFS flow: profiles of mean streamwise velocity Um /U0 at four streamwise locations measured from the separation line (leading edge). Left to right x/xR =0.208, 0.625, 1.04, and 1.25. Also shown are measurements by Moss and Baker [12] (circle) at Re=46,000.



19

Figure 4 compares the mean streamwise velocity distribution U /U0 at 6 locations downstream of the separation line with the experimental data of Tropea and Gackstatter [2] (available only at 3 locations), Larsen [10] and the DNS data of Orellano and Wengle [9]. The results show good agreement with the data of Larsen [10] and the DNS data of Orellano and Wengle [9]. The freestream velocities of the data from Tropea and Gackstatter [2] are bigger than those predicted by the LES and the other two results, and peak at lower y-values. This is attributed to the difference in blockage ratio used by Tropea and Gackstatter [2] which is very low (= 2), 5 in the case of Orellano and Wengle [9] and 8 for the current LES. Profiles of the rms streamwise velocity, urms , normalised by U0 , at the same six stations are shown in figure 5. The agreement between the LES results and the data of Larsen [10] and the DNS data of Orellano and Wengle [9] is encouraging. No measured data were presented by Tropea and Gackstatter [2]. Data for the mean velocity profiles for the FFS flow at low Reynolds number does not exist. Therefore, the results were compared with experimental data by Moss and Baker [12] at higher Reynolds number (Reh = 46 000). This is thought not to be detrimental to the comparison and any discrepancy will be discussed taking this fact into consideration. Figure 6 compares the mean streamwise velocity distribution U /U0 at 4 locations downstream of the separation line with the experimental data of Baker [12]. The agreement between the experimental and LES results is reasonably good. At the location x/xR = 0.625 the peak negative value is slightly higher than the experimental value. This could be due to the low Reynolds number for the LES case leading to a stronger back flow and low mixing at this specific region of the recirculation region downstream the step. The profiles of the rms streamwise velocity, urms , normalised by U0 (not shown here), at the same four locations presented in figure 6 also display good agreement when compared with the data of Moss and Baker [12].

4 Spectra using Fourier and wavelet transforms For the current work, a well tested code that uses the FT methods for autocorrelation is used to process the data. For the WT, the Morlet wavelet is used and a code developed by Torrence and Compo ([14]) was modified to perform the analysis for the time series signals shown in this section. The WT analysis produces a 2D picture showing wavelet power concentration in time (x-axis) vs scales or period (y-axis) (which is an approximate measure to the Fourier period of the signal). For more details on wavelet transform techniques, the reader is advised to consult Torrence and Compo [14]. A total of 24990 samples at each point taken every 10 time-steps with time step = 4.75 × 10−6 seconds (sampling frequency 21.053 kHz) were collected for the SSMO case. This corresponds to a total period of 1.187025 seconds. For the FFS simulation, 32,312 samples were collected with each sample taken every 10 time-steps with time step = 1.5 × 10−6 seconds (sampling frequency 66.67 kHz). This corresponds to a total period of 0.484680 seconds. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

20 Computational Methods and Experimental Measurements XIII It is essential to choose a set of scaling parameters s, such that the wavelet transform adequately samples all the frequencies present in the time series. The smallest resolvable scale, s0 , is usually selected as a multiple of the time resolution dt and for the current studies is chosen as 2dt for the SSMO and FFS data. Torrence and Compo [14] recommended that the largest scale chosen should be less than 1/2 the length of the entire time series. For the two cases presented here, the largest scale chosen is of order one third of the total time span and hence, no interest in event with long periods is sought. However, larger periods are also investigated to shed light on the difference of choosing this parameter associated with WT. Shown in figures 7(a) and (c) are the time history for the velocity u and v at the position described by the co-ordinates (h=-0.375, y/h=0.04, z/h=2), immediately upstream and slightly above the leading edge of the SSMO. The WT spectra based, on scales of order one third of the total sampling time, is shown in figures 7(b) and (d) and the spectra based on the FT for these velocities appear in figure 7(e). The FT spectra clearly show a sharp frequency peak (band) centered at approximately 105 Hz for both velocity components (the normalised value is 5.425 xUR0 ). The wavelet spectra for the velocity U (figure 7(c)) reveal power concentration associated with four instances of time. The major power is centered around t ≈ 0.26s with a band scale ≤ 0.2. Linked to this event, there is another event which is characterised by less power compared the major event, having a lower band (0.1 ≤ scale ≤ 0.2) and centered around t ≈ 0.5s (the two events are thought to be associated with the peak noticed in the FT spectra). The wavelet spectra also indicates that an event with lower power concentration and much narrower band (0.09 ≤ scale ≤ 0.11) has taken place between 0.85 ≤ t ≤ 1.2. Also, towards the end of the sampling period, the WT spectra indicates that the major event (or velocity peak) could possibly reoccur. The last point is clear from the WT spectra for the velocity v (figure 7(d)) which shows power concentration shortly after the start of the sampling and towards the end of the time history. Comparing the events shown by the WT spectra with the time series, it is clear that the WT is able to interpret exactly the content of the event displayed in the history for each velocity component. Whenever the signal shows a sharp positive or negative peak which is most likely an indication of an event (shedding, or pairing of large-scale structures), it was captured in the WT spectra with its representative magnitude and at the exact time where the process happened. All these features represent the benefits of using WT to perform spectral analysis for unsteady turbulent flow. One of the critical features of the WT spectra is the fact that it provides only qualitative results (many criticise the method for this feature). However, it does give a clear picture of the extent of the event (amplitude) and any smaller events associated with it and the time of occurrence and possibly the cyclic behaviour of the major event. This information could be useful in controlling such events (damping or exciting as required for specific applications) It is essential to set an appropriate range of scales for the WT spectra to be realistic. As an example, the largest scale for the velocity u in figure 7(b) has been increased in figure 7(f) in order to seek events with higher period. It is clear that the content shown by this figure is a little misleading when compared with the WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


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corresponding time history of the signal where events with lower concentration and those occurring at shorter times have vanished. This confirms that care should be taken when using WT spectra for turbulent flow data analysis. For the FFS flow and for a point similar to the location of the SSMO discussed above (x/h=-0.225, y/h=0.2, z/h=2), the time series for the velocities u and w are shown in figures 7(g) and (i) respectively with the WT spectra shown in figures 7(h) and (j) respectively. The FT spectra is displayed in figure 7(k). The interesting point here is the fact that the FT spectra does not show any particularly amplified frequency. However, the WT spectra for the velocity u shows power concentrations both at the beginning of the sampling period and towards the end while that for the velocity w shows a clear and significant concentration shortly after the start of the sampling process. The reason for these events not being shown in the FT spectra is probably due to the short total time (≈ 0.45 sec compared to 1.25 sec for the SSMO) for the samples collected in the FFS which may be too short to resolve this particular frequency. However, the WT spectra shows that there are significant events at the location mentioned above for the SSMO flow. It is worth pointing out that this detected frequency is most likely due to Kelvin-Helmholtz (K-H) instability rather than being attributed to the flapping of the shear layer since its reduced value is much higher than the later which is of order 0.15 xUR0 . For the SSMO and downstream the separation line (x/h=1.65, y/h=0.85, z/h=2), figures 7(l) shows a fading frequency peak (centered at f=200 Hz) which most likely represents vortex shedding/pairing of vortices rolling-up at the fundamental instability frequency of the separating shear layer. A similar value is reported for the FFS downstream the leading edge (not shown here).

5 Conclusion Analysis of the velocities and pressure fields signals predicted by LES for a transitional separated-reattached flow over a SSMO and FFS was performed employing the FT and the WT. The turbulence spectra for the geometries revealed amplified frequency modes both upstream and downstream of the separation edge with those associated with the SSMO more apparent compared to the FFS. The value of amplified frequency upstream strongly suggest that the upstream separated region is unstable via the K-H instability and the frequency could not be attributed to the flapping of the shear layer as commonly reported for such class of flows. Those which appear downstream the separation line are attributed to the shedding and pairing activities of large-scale motions dominating the separated boundary layer. The WT correctly interpreted the content of the time signal for the presented data. Whenever the time signal indicates the occurrence of a certain event, it was marked in the WT spectra with its representative amplitude. The WT was also able to capture events that are characterised by narrow periods (scales) and which happened over shorter times. Such events are smoothed out by the Fourier transform indicating the superiority of the WT over the FT in revealing a signal WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Figure 7: Time signal, FT and WT spectra for the SSMO (a), (b), (c), (d), (e), (f) and (l) and FFS (g), (h), (i), and (j).



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contents. However, it was found that appropriate range of scales must be selected for the WT in order to adequately sample the existing frequencies within a signal and avoid misleading interpretation.

References [1] Satür, H., Gyr, A. and Kinzelbach, W. Laminar separation on a forward facing step. Eur. J. Mechj. B/Fluids 18, 675-692, 1999. [2] Tropea CD, Gackstatter R. The flow over two-dimensional surface-mounted obstacles at low Reynolds number, Journal of Fluids Engineering; 107:489494, 1984. [3] Bergeles G, and Athanassiadis N. The flow pass a surface mounted obstacle. ASME Journal of Fluid Engineering; 105:461-463, 1983. [4] Durst F, Rastogi AK. Turbulent flow over two-dimensional fences, in Turb. Shear Flows 2. Springer Verlag: Berlin, 218-231, 1980. [5] Wilhelm, D., Härtel, C., and Kleiser, L. Computational analysis of the two-dimensional-three-dimensional transition in forward-facing step flow. J. Fluid Mech; 179:439-468, 1987. [6] Kiya M, Sasaki K. Structure of a turbulent separation bubble, J. Fluid Mech; 137:83-113, 1983. [7] Kaiser G. A Friendly Guide to Wavelets. Cambridge, MA: Birkhuser, 300pp, 1994. [8] Yang, Z. and Voke, R.P. Large-eddy simulation of separated leading-edge flow in general co-ordinates. International J. Numer. Meth. Engng. 49, pp. 681696, 2000. [9] Orellano A, Wengle H. Numerical simulation (DNS and LES) of manipulated turbulent boundary layer flow over a surface-mounted fence. Er. J. B - Fluids, 19:765-788, 2000. [10] Larsen, P. S. Database on tc-2C and tc-2D fence-on-wall and obstacle-onwall test case. Report AFM-ETMA 95-01, ISSN 0590-8809; TU Denmark, 1995. [11] Ko, S. H. Computation of turbulent flows over backward and forward-facing steps using a near-wall Reynolds stress model. CTR Annual Research Briefs, Stanford University/NASA Ames, 75-90, 1993. [12] Moss W. D., Baker S. Re-circulating Flows Associated with Twodimensional Steps, Aero Quart., 151-172, 1980. [13] Castro IP, Epik E. Boundary layer development after a separated region. Journal of Fluid Mechanics, Vol.374(), pp.91-116, 1998. [14] Torrence C, Compo GP. A practical guide to wavelet analysis. Bulletin of the American Meteorological Society, 61-77, 1998.




25

Observation of solid fuel in a supersonic flowfield J. M. Char1 & U. K. Hsu2 1

Air Force Institute of Technology, Gangshan, Kaohsiung, Taiwan, Republic of China 2 Department of Aircraft Engineering, Air Force Institute of Technology, Gangshan, Kaohsiung, Taiwan, Republic of China

Abstract Hypersonic vehicles will be the new-generation of aerial transport. Hence, supersonic combustor design becomes important. Many investigations have been conducted on this subject, however, they are focused on gaseous or liquid fuels combustor. The use of a solid fuel combustor can substantially decrease complexity and cost, so for certain purposes, solid fuel supersonic combustors show advantages over other fuel systems. This research adopts a shock tube, 16 meter long and with a bore of 9 cm to create a supersonic, high-temperature, and high-pressure flowfield to observe the gasification and ignition of HTPB solid fuel under different environments. Also, full-scale 3D numerical simulation is executed to enhance the comprehension of this complex phenomenon. The CFD code is based on the control volume method and the pre-conditioning method for solving the Navier-Stokes equations to simulate the compressible and incompressible coupling problem. In the tests, a HTPB slab is placed in the windowed-test section. Various test conditions generate different supersonic Mach numbers and environmental temperatures, meanwhile the HTPB slab changes its incident angles relative to the coming shock wave. Results show that when the Mach number around the slab section is beyond 1.25, the flowfield temperature can achieve above 1100K, which is higher than the HTPB gasification temperature (930K~1090K), then the gasification happens and a short-period ignition can be observed. In particular, as the slab angle is 7°, the phenomenon is more visible. This is due to the flowfield temperature increasing when the slab angle is at 7°. The comparison between test results and CFD simulation show good agreement, so the CFD results help the understanding and analysis of this complicated test event. Several pictures demonstrating the research results are shown below. Keywords: ignition, shock tube, HTPB, scramjet, finite volume. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070031


1

Introduction

With the development of the space shuttle and solar system exploration, the hypersonic high technology in aviation will play an important role in the nextgeneration frontier [1]. However, the engine of the hypersonic vehicle is a kind of scamjet, but the combustor inlet air is supersonic and will be much more useful and powerful than the Ramjet [2]. In the 1960’s, researchers indicated that regression ratio is the key in mixer-rocket studies. For this reason, many models are developed in different combustion conditions. Marxman and Gilbert [3] consider that the optimal position of the flame should be in the top of the fuel surface, and the regression rate is the minimum in the turbulence layer of about 10-20%. Muzzy [4] also verify that the heat convection effect is very important in fuel consumption. Smoot and Price [5] indicated the fuel regression ratio is proportional to the oxidizer flow rate in 0.8 order in lower oxidizer flow rate. During the 1990s, there were lots of basic studies conducted. Greiner and Grederick [6] expressed the fuel regression ratio as proportional to the oxidizer flow rate, and the pressure fluctuation as decreasing in raising the mixed region length. Chiaverini et al. [7] indicated that the vapor temperature of the fuel surface is between 930K and 1190K according to several different HTPB composites. In order to investigate the ignition and combustion efficiency of a supersonic combustion ramjet and simplify the components, there is a 16 meters shock tube established as shown in Fig.1. The device consists of a long tube divided into a high pressure and a low pressure section by an aluminium diaphragm. When the diaphragm is rupturing in the high pressure driver section, a series of compression waves coalesce into a single shock front which compresses and heats the high pressure region test gas to low pressure region, and creates the supersonic gas flow condition. In the shock tube, there are lots of complex phenomena including the normal shock, contact surface, expansion wave etc. We investigate the flowfield of the supersonic flow through the plate-like HTPB solid fuel under this unsteady condition. The difficult problem, however, is that it is not easy to create a supersonic condition over a length of time. The best test period is about 10 ms. Therefore, we must be carefully in experiments and operation. There are lots of shock tube wind tunnels established for research from 1950, for example: Glass and Hall [8], Lukasiewicz [9], Nagamatsu [10], Bradley [11], Soloukhin [12], etc. In fact, the wind tunnel test is very important in classical aerodynamics. However, there is not a complete understanding of the full phenomena because of the limit of the experiment. The Computational Fluid Dynamics, CFD, is a good tool to deal with the problems. In this study, both of these two methods are used. Using CFD simulates supersonic flow through the HTPB slab, and compares with the experimental data.

2

Experimental apparatus

In Fig.1, the length of a shock tube is 16m. The high pressure region is higher than 147 psi, and low pressure region is below 1 psi. A HTPB slab, 15cm(length)×3cm(spread) ×0.5cm (thickness), is placed in 14.55m from the WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


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start point of the high pressure region. There are two pressure transducer placed in 14.55m (#1) and 14.65m (#2), separately. The initial tube temperature is 300K. To capture the reaction, a 10W pulse laser (t=50µs) and high speed CCD camera (t=200µs) is set up.

(a) high pressure section (length: 290 cm, diameter: 28.5cm)

(c) test section (window: 25cm×3cm)

(b) convergent nozzle (length: 10cm, internal diameter in divergent section: 28.5cm, internal diameter in convergent section: 9cm)

(d) dimension

Figure 1:

3

Shock tube dimension.

Numerical model

Algebraically spaced grids are used to cover the flowfield, and the stretching transformation clusters using the Roberts generalized stretching transformation technique are made near the boundary layer. The Shock tube is symmetrical about the centre-plane and, therefore, only the right half of the Shock tube and plate-like model needs to be modelled. The multi-block grid approach is used in the present study. The total number of cells is 2094750 with respect to the half 3D shock tube as shown in Fig.2. The instantaneous solution was obtained by WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

28 Computational Methods and Experimental Measurements XIII solving the time-dependent governing equations, and the residual is measured by the order of magnitude of the decay. The convergent solution was achieved when the residual had decayed by about 3 orders of magnitudes. Computation was performed on finer and coarser grids for a grid resolution steady; it was found that the total grid sizes, especially in the y direction, depend on the turbulence model used. According to present study, the average value of y+ closest to the surface is 0.2 with the exact solution for turbulence models [13]. The DELL OPTIPLX GX270 workstation is used for the computation.

Figure 2:

Longitudinal cut view of the grid system.

The numerical scheme, using the preconditioning finite volume method, is introduced to solve the governing flow equations. A 2nd-order scheme is initially applied, so the left and right states are chosen to be the cell average values on the left and right of the cell faces. In a high-resolution scheme, in order to raise the order of accuracy of upwind differencing, all one needs to do is to raise the order of accuracy of the initial-value interpolation that yields the zone-boundary data. Such schemes are labelled as high resolution schemes as opposed to Total Variation Diminishing (TVD) schemes, which completely eliminate any of those spurious oscillations when applied to one dimensional nonlinear hyperbolic conservation laws and linear hyperbolic systems. The van Leer kappa-scheme, in which the kappa number is one-third, was selected to obtain a high-resolution upwind differencing [14–16]. An optimal multi-stage scheme is used for the time integration, the multistage coefficients are modified by Tai et al [17] and redefined using the Courant number for multi-dimensional use. Also, a residual smoothing method is imposed to accelerate convergence and to improve numerical stability.

4

Results and discussion

4.1 Initial conditions In order to understand the velocity and temperature statute in a shock tube, 1-D shock tube theorem is applied to determine the shock speed, temperature, and action time as following:



29

P4 147 ( psi ) = = 147 P1 1( psi ) 2γ

 γ −1   P2   − 1   P P  γ −1   P1  ∵ 1 = 1 1 −  P4 P2 2γ  γ + 1  P2   − 1  1+  2γ  P1   

P2 = 7.0 P1

so,

from

normal

shock

table,

it

shows

M s = 2.48, and

hence

M 2 = 0.5149, T2 = 635.4 K . The history of the action time is shown in Fig.3. The main shock wave has reflected to the test model before the contact surface arrived. Therefore, the period of the test time is only 3.2 ms.

Figure 3: 1-D theorem determined result.

Figure 4:

Pressure and temperature profile at point #1& #2.

4.2 Full-scale shock tube simulation We can set the time step in unsteady simulation from the above 1-D theorem determined. In Fig.4, P1 and P2 are presented the position of the pressure transducer (#1 and #2) individually. The result shows the shock speed is 787.4 m/s in test section. The main shock arrived point #1 at t=11.08ms, and is faster by 2.32ms than theorem determined. The numerical model simulates the real case that the calibre is reduced to 1/3 when section is in the low pressure region from high pressure region. For this reason, the shock speed is rapidly increased. The calculation of the theory regards, in terms of the main straight tube as, WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

30 Computational Methods and Experimental Measurements XIII neglects the heat effect and boundary layer effect, because their response time is relatively slow if comparing with the test event interval. Fig.5 and Fig.6 show pressure and temperature change in different time steps: (a) shows the initial state after the diaphragm broken; (b) and (c) show the incident shock wave propagation phenomena. Fig.6(b) shows the temperature distribution, and a contact surface can clearly find. Fig.6(d) and 6(e) show the detail of the reflection shock wave. Because the wind tunnel is a close type, the pressure is dropped naturally in the high pressure section after the main shock released as shown in Fig.5.

Figure 5: Snapshots of the pressure distribution in a tube.

Figure 6:

Snapshots of the temperature distribution in a tube.

4.3 Different fuel slab angle analysis After understanding the phenomena of a full scale shock tube, a fuel slab is placed in test section (X=14.55~14.70M), to investigate the physical phenomena of the fuel slab surface in different angles of attack (AOA=0°, 7°, and 10°). Figs 7–9 show the slab surface pressure, temperature, and Mach number. Due to the symmetric shape across the upper and lower area of the slab at AOA=0°, the distribution curves are merged as one line. But at AOA=7° and 10°, the temperature and pressure curve across the upper and lower surface show differently in Figs 7–9. In Fig. 10, at AOA=0°, the blunt shape of the leading edge causes the supersonic bow shock, and one more shock wave is followed to make the temperature rise once again near the leading edge afterwards. The channel flow seems to be through a convergent nozzle between the upper and lower passage because of the boundary layer effect. Also, because of the passage flow wall effect, the shock wave is reflected, which makes the speed rapidly reduced. Therefore, the Mach number is decreasing from 1.25 to M=0.5 when air flow attached a reflection shock at X/C=0.1. Following on, the flow speed is raised at X/C=0.15~0.4 because of the convergent passage that the boundary layer affects. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Figure 7: Pressure profile of HTPB surface in different AOA.

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Figure 8: Temp. profile of HTPB surface in different AOA.

(a) AOA=0 deg

(b) AOA=7 deg.

(c) AOA=10 deg.

Figure 9: Mach No. profile of HTPB surface in different AOA.

Figure 10:

Iso-density contour.

However, the energy after the reflecting shock wave is reduced too fast, so the Mach number tends towards stability after X/C=0.5. As 7 degrees of angles of attack, the impact effect on the upper surface increases so temperature is higher than 0 degrees case. The supersonic flow will reduce speed and raise pressure because of a geometric convergent passage. The shock wave which occurred in leading edge of upper surface will reattach at the trailing edge through the reflection shock. The main shock wave has the deflection characteristic due to the decline of the slab so the air flow oppresses the boundary layer. For this WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

32 Computational Methods and Experimental Measurements XIII reason, the temperature is increasing on the upper surface. The supersonic flow speed raised and pressure reduced when the air flow over the lower surface which is like in a divergent nozzle. Therefore, at AOA=7° and HTPB fuel is mounted on the upper surface of the slab; high temperature helps the gasification and ignition in HTPB plate. In Fig.9, we found the air flow speed still above M=1 either on upper or low surface in AOA=7°. Therefore, this situation can satisfy the ignition in supersonic flow. As AOA=10°, the average temperature is not so good as in AOA=7°, but the speed is still above M=1. Because of the reflection of the main shock wave in the leading edge, the pressure is reduced, the temperature is raised, and Mach number is reduced after the flow through the reflection shock (Fig.7 to 9, X/C=0.3). 4.4 Experiment visualization Fig.11 shows the time history snapshot of the shock wave using a CCD camera. The shock wave arrived to slab (AOA=7deg) at t=11.1 ms. The oblique shock has occurred because of the angle shape configuration. The secondary oblique shock wave is induced at t=11.7ms when the flow over the HTPB. Comparing with CFD results, in Fig.12, the flow speed behind the shock is kept in transonic on the upper side, and the speed of the lower side increased to M=2. Although the speed is reduced behind the oblique shock on the upper side, the temperature is also proportional to the length as shown in Fig.8. The reflection shock is induced in the trailing edge at t=14.1ms as shown in Fig.11(d). The period is about 3 ms between the shock arrived and reflection shock attached the test section. Fig.13 shows the time history of pressure transducer record at 14.55M (#1) and 14.65M (#2). Because the HTPB fuel is not burned, the change of pressure transducer record is not clear until the hit of main shock and reflection shock. We can find the pressure fluctuation after a rapid peak as shown in Fig.13, because of the interaction of shocks in the tube. The total pressure is not reduced until the pressure goes to stable after the peak value. It is observed that gasification exits when the shock wave goes across the HTPB. From the recovered fuel, there is a melt in the leading edge as shown in Fig.14. This region is a stagnation zone for a bow shock, where flow speed is slow and temperature is high. The temperature can reach 1100K as shown in Fig.8. The temperature has already been higher than the surface gasification temperature (930K 1190K) of HTPB, and there is enough time in the district for burning. The flow speed in other places is too fast, so the temperature has not reached the gasification criterion, and it is unable to burn.

5

Conclusion

From the numerical simulation result, the HTPB slab at an angle of attack of 7 degrees has a higher temperature and pressure in the upper surface than 0 degrees and 10 degrees, and the flow speed of the upper and lower surface keeps in supersonic flow and contributing to gasification ignition. According to the WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


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experimental data and numerical result, the test periods are both about 3 ms, and it had the melt in the leading edge of the tested HTPB slab. In other areas, although they reach the gasification criterion, it is unable to burn because the flow speed is too fast and the test time is limited by 3ms of shock tube facility.

(a) before event (t=0 sec) Figure 12:

Mach No. distribution.

Figure 13:

Pressure measurement (point#1).

(b) shock wave arrived (dir.:Î, t=11.1ms)

(c) shock wave through (dir.:Î, t=11.7ms)

(d) ref. Shock (dir.:Í, t=14.1ms)

Figure 11:

Snapshots of shock wave.

(a)

(b)

Figure 14: Comparison before and after the action: (a) without action, (b) with action.

Acknowledgement The authors are grateful to the National Science Council of the Republic of China for financial support under contract number NSC 93-2212-E- 013-007.



References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

[15] [16] [17]

Waltrup, P. J., White, M.E., Zarlingo, F., and Gravlin, E. S., “History of U.S. Navy Ramjet, Scramjet, and Mixed-Cycle Propulsion Development,” Journal of Propulsion and Power, Vol.18, No.1, pp.14-27, 2000. Jones, R. A. and Huber, P.W. “Toward Scramjet Aircraft,” AIAA Journal, Vol.16, pp.38-48, 1978. Marxman, C. A., and Gilbert, M., “Turbulent Boundary Layer Combustion in the Hybrid Rocket,” 9th International Symposium on Combustion, Academic Press, Inc., New York, 1963, pp.371-383. Muzzy, R. J., “Applied Hybrid Combustion Theory,” AIAA Paper No. 72 1143, 1972. Smoot, L. D., and Price, C. F., “Regression Rates of Nonmetalized Hybrid Fuel Systems,” AIAA Journal, Vol. 3, No. 8, August 1965, pp. 1408-1413. Greiner, B. and Grederick, R. A. Jr., “Results of Labscale Hybrid Rocket Motor Investigation,” AIAA Paper No. 92-3301, 1992. Chiaverini, M. J. et al., ”Fuel Decomposition and Boundary Layer Combustion Processes of Hybrid Rocket Motors,” AIAA Paper 95-2686, 1995. Glass, I. I., and Hall, J. G., “Shock Tubes, Handbook of Supersonic Aerodynamics,” NAVORD Report 1488, Vol. 6, Section 18. (1958) Lukasiewicz, J., “Shock Tube Theory and Application,” National Aeronautical Establishment, Rept. 15, Ottawa, Canada. (1952) Nagamatsu, H. T., “Shock Tube Technology and Design,” Fundamental Data Obtained From a Shock-Tube Experiments,” Edited by A. Feri, pp. 86-136, Pergamon Press. (1961) Bradley, J. N., “Shock Waves in Chemistry and Physics,” Methuen & Co. (London), J. Wiley & Sons (New York). (1962) Soloukhin, R. I., “Shock Waves and Detonation in Gases,” State Publishing House of Physical-Mathematical Literature, Moscow; English Translation Published by Mono Book Corp., Baltimore. (1966) Wilcox, D. C., “Comparison of Two-Equation Turbulence Models for Boundary Layers with Pressure Gradient”, J of AIAA, Vol.31, No.8, pp.1414-1421 (1993) Van Leer, B., "Upwind-Difference Methods for Aerodynamic Problems Governed by the Euler Equations, in Large-Scale Computations in Fluid Mechanics," Lectures in Applied Mathematics, Vol. 22, pp. 327336(1985). Roe, P. L., "Approximate Riemann Solvers, Parameter Vector, and Difference Schemes," Journal of Computational Physics, Vol. 43, pp.357372(1981). Edwards J.R. and Liou M. S. “Low-Diffusion Flux-Splitting Methods for Flows at All Speeds” AIAA J., Vol.36. No.9, 1610-1617, (1998). Tai, C. H., Sheu, J.H, and van Leer, B., "Optimally Multi-Stage Schemes for the Euler Equations with Residual Smoothing," Journal of AIAA, Vol.33, No.6, pp.1008-1016 (1995). WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


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Temperature field at the twin-roll casting of aluminium alloys: computational model and measurements H. Gjerkeš, S. Hartman, R. Vertnik & B. Šarler Laboratory for Multiphase Processes, University of Nova Gorica, Slovenia

Abstract Industrial twin-roll casting of aluminium alloys requires optimisation of connections between process parameters and product properties. To a large extent, the cast strip properties depend on the spatial and temporal characteristics of heat transfer from the cast strip, which needs estimation of the influence of process parameters on the temperature field and temperature gradients in a strip for various alloys, which are cast on an industrial machine. For this purpose, the recently developed meshless Local Radial Basis Function Collocation Method (LRBFCM) was used in the simulation of heat transfer from the strip. The solution of the nonlinear convection-diffusion equation is based on a mixture continuum formulation of the energy transport in solid-liquid phase change systems. Simulation results were compared with experimental values, which were obtained during the industrial production process. They were measured in situ by the specially designed apparatus with fast response thermocouples, which can measure the time-dependent temperature of the cast strip in several locations on the moving strip surface simultaneously. The time-dependent measurement was coupled with a digital camera recording to gather the spatial domain measurement of temperature. The differences between the numerical simulation and experimental results were smaller than 6 K. An explanation for the discrepancy is given. Keywords: twin-roll casting, aluminium alloys, measurements, surface temperature, meshless method, radial basis functions.



1

Introduction

The twin-roll continuous casting is a promising commercial material processing technique, however it is extremely difficult to control [1]. Process modelling [2] has been introduced in order to improve quality, reduce production cost, and improve safety. Like most commercial material processes, the twin roll casting involves many interacting phenomena of great complexity, which makes it difficult to include them all in the model. The principal aim of the presented model is to predict the product properties as a function of the process parameters. A broad class of meshfree methods in development today are based on Radial Basis Functions (RBFs). The RBF collocation method or Kansa method is the simplest of them. This paper describes the numerical solution of the convectivediffusive solidification problem in twin-roll casting process by the recently developed meshless local radial basis function collocation approach [3–5]. The method is structured on multiquadrics radial basis functions. Instead of global, the collocation is made locally over a set of overlapping domains of influence and the time-stepping is performed in an explicit way. Only small systems of linear equations with the dimension of the number of nodes included in the domain of influence have to be solved for each node. The computational effort grows roughly linearly with the number of the nodes. The developed approach overcomes the principal large-scale problem bottleneck of the original Kansa method. Simulation results were compared with the measured temperatures on the cast strip surface after exiting the rolls. Presented experimental technique enables temperature measurements during the industrial continuous twin-roll casting without disturbing the production process. The technique is based on fast response thermocouples, mounted on the cast strip using the specially designed apparatus. The time-dependent temperature measurement on the moving cast strip was coupled with digital camera recording to obtain the measurement of surface temperature over a substantial part of the strip exiting the cooling rolls. Influence of the experimental set-up on the measurement results is considered. Based on experimental results, the improvements of the numerical model boundary conditions are proposed.

2

Model

The twin-roll casting process, fig. 1, consists of two oppositely rotating internally water cooled rolls which are independently driven. The molten metal is fed between the rolls through a pouring nozzle. The strip thickness is defined with the separation distance between rolls, which is typical few millimetres at strip width of a meter or more. The heat transfer in twin-roll casting can be reasonably represented in the framework of the mixture continuum formulation which assumes local thermodynamic equilibrium between the phases. This formulation can in solidification context involve quite complicated constitutive relations. Presented model focuses on a convective-diffusive heat transport as a first step. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Figure 1:

37

Twin-roll casting process.

2.1 Governing equations Consider a connected fixed domain Ω with boundary Γ occupied by a phase change material described with the temperature dependent density ρ℘ of the phase ℘ , temperature dependent specific heat at constant pressure c p , effective thermal conductivity k , and the specific latent heat of the solid-liquid phase change hm . The mixture continuum formulation of the enthalpy conservation for the assumed system is ∂ G G G G ( ρ h ) + ∇ ⋅ ( ρ vh ) = ∇ ⋅ ( k ∇T ) + ∇ ⋅ ( ρ vh − f SV ρ S vS hS − f LV ρ L vL hL ) (1) ∂t with subscripts S and L denoting the solid and the liquid phase, respectively. The mixture density is defined as ρ = f SV ρ S + f LV ρ L , the mixture velocity is G G G defined as ρ v = f SV ρ S vS + f LV ρ L vL , and the mixture enthalpy is defined as h = f SV hS + f LV hL . The constitutive mixture temperature - mixture enthalpy relationships are hS = ∫

T

Tref

cS dT , hL = hS (T ) + ∫

T

TS

( cL − cS )dT + hm

(2,3)

with Tref and TS standing for the reference temperature and solidus temperature, respectively. Thermal conductivity and specific heat of the phases can arbitrarily depend on temperature. The liquid volume fraction f LV is assumed to vary from 0 to 1 between solidus TS and liquidus temperature TL . We seek for mixture temperature at time t0 + ∆t by assuming known temperature and velocity fields at time t0 , and boundary conditions.


38 Computational Methods and Experimental Measurements XIII 2.2 Solution procedure The solution of the problem is demonstrated on the general transport equation defined on a fixed domain Ω with boundary Γ , standing for a reasonably broad spectra of mass, energy, momentum and species transfer problems (and includes also eqn. (1) as a special case): ∂ G (4)  ρ C ( Φ )  + ∇ ⋅  ρ v C ( Φ )  = −∇ ⋅ ( − D∇Φ ) + S ∂t  G with ρ , Φ, t , v , D, and S standing for density, transport variable, time, velocity, diffusion matrix and source, respectively. Scalar function C stands for possible more involved constitutive relations between conserved and diffused quantities. The solution of the governing equation for the transport variable at the final time t0 + ∆t is sought, where t0 represents the initial time and ∆t the positive time increment. The solution is constructed by the initial and boundary conditions that G follow. The initial value of the transport variable Φ ( p, t ) at point with position G vector p and time t0 is defined through the known function Φ 0 : G G Φ ( p, t ) = Φ 0 ( p ) ; p ∈ Ω + Γ . (5) The boundary Γ is divided into not necessarily connected parts Γ = Γ D ∪ Γ N ∪ Γ R with Dirichlet, Neumann and Robin type boundary G conditions, respectively. These boundary conditions are at the boundary point p G with normal nΓ and time t0 < t ≤ t0 + ∆t defined through known functions Φ ΓD , Φ ΓR , Φ ΓRref : ∂ ∂ G Φ = Φ ΓD ; p ∈ Γ D , Φ = Φ ΓR ( Φ − Φ ΓRref ) ; p ∈ Γ R . Φ = Φ ΓN ; p ∈ Γ N , ∂nΓ ∂nΓ

(6,7,8) The involved parameters of the governing equation and boundary conditions are assumed to depend on the transport variable, space and time. The solution procedure is in this paper based on the combined explicit-implicit scheme. The discretisation in time can be written as dC ρ C− ρ 0 C0 ρ C + ρ d Φ ( Φ − Φ ) − ρ 0 C 0 ∂ (9) ( ρ C ( Φ ) ) ≈ ∆t ≈ ∂t ∆t by using the two-level time discretisation and Taylor expansion of the function C ( Φ ) . The known quantities are denoted with overbar. The source term can be

expanded as dS ( Φ−Φ ) . dΦ The unknown Φ can be calculated from the equation S (Φ) ≈ S +


(10)


ρ0

G ρ dC dS Φ + ∇ ⋅ ( D0 ∇Φ 0 ) − ∇ ⋅ ( ρ0 v0 C0 ) + S − Φ ∆t d Φ d Φ . (11) ρ dC dS − ∆t d Φ d Φ The value of the transport variable Φ n is solved in as set of nodes G pn ; n = 1, 2,..., N of which N Ω belong to the domain and N Γ to the boundary. The iterations over one timestep are completed when the equation (12) is satisfied, and the steady-state is achieved when the eqn. (13) is achieved max Φ n − Φ n ≤ Φ itr , max Φ n − Φ 0 ≤ Φ ste . (12,13) G The value of the unknown derivatives of the variable Φ n in point pn is Φ = ∆t

C0 −

ρ

39

∆t

C+

approximated by the collocation method which uses the values of Φ i at I G G points pi ; i = 1, 2,..., I , situated in the vicinity of and including pn . One can write the following approximation of the function and its first and second order partial derivatives ∂ ∂ G G G K K G G G Φ ( p ) ≈ ∑ k =1 n α kψ k ( p − pn ) , ψ k ( p − pn ) (14,15) Φ ( p ) ≈ ∑ k =1 n α k ∂pς ∂pς ∂2 ∂2 K G G G ψ k ( p − pn ) ; Φ ( p ) ≈ ∑ k =1 n α k ∂pςξ ∂pςξ

ς , ξ = x, y .

(16)

The scaled multiquadrics are used for representation functions 1/ 2 G G G G 2 ψ k ( p − pn ) = ( p − pn ) − c 2 r02  (17)   where r0 represents the maximum distance between points in a subdomain. Other details of the solution (particularly implementation of boundary conditions) are elaborated in [3–5]. 2.3 Model results The aluminium alloy composition and the nominal process parameters, used as simulation input parameters, are given in tables 1 and 2, respectively. Table 1: El. Wt [%]

Fe 0.85

Si 0.6

Mn 0.022

Table 2:

Aluminium alloy composition. Mg 0.05

Cu 0.05

Zn Ti 0.05 0.0375

Cr 0.002

Process and input parameters.

Casting Speed [m/min] Strip thickness [mm] Roll temperature [°C] Setback [mm] Heat transfer coefficient strip-cooling rolls [W/(m2K)] Convection heat transfer coefficient strip-air [W/(m2K)] WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

1.22 6.00 35.1 69.22 3600 10.7

B 0.01


Figure 2:

Predicted temperature field in cast strip cross-section at nominal process parameters.

Fig. 2 shows prediction of the temperature field in cast aluminium alloy strip cross-section at the process parameters, which were used at the time of the measurements, presented below. More simulation results for temperature field in the cast strip as a function of various process parameters: casting speed, strip thickness, roll temperature, setback, etc., can be found in [3–5].

3

Experimental set up

A new measurement technique was developed for surface temperature measurements of twin-roll cast aluminium alloys. The fast response Omega CO1 Type K thermocouples with butt welded junction of thickness 0.013 mm were used. The thermocouple is embedded between two thin, glass reinforced high temperature polymer laminates for support and electrical insulation. The laminated sensor has dimensions 19 x 9.4 x 0.13 mm. Declared response time is between 10 and 20 ms, when grounded to surface. The sensor was installed on a small, 12 x 8 x 5 mm graphite block with a 2 mm thick layer of temperature resistant silicone putty. The latter was used to reduce the influence of the transient heat conduction in the thermocouple support on the dynamic temperature measurements, i.e. to improve the response of the sensor. Thermocouples were fastened via graphite blocks to a specially constructed holder. Gravity of the holder assured good contact between thermocouples and measured surface, whereby it could be placed down and removed from the strip surface without disturbing or influencing the production process. The detail of the thermocouple fixation and the set up, placed on the cast strip during the measurement are shown in fig. 3. Measurement procedure started with placing down the holder with thermocouples on the upper cast aluminium alloy strip surface as near as possible to the cast rolls, typically 100 mm from the rolls' outlet (point 0 in fig. 1). Thermocouples travelled together with cast strip without relative moving. The temperatures were sampled using the high-speed data acquisition system Agilent 34970A with multiplexer module 34902A. Simultaneously, the position of the thermocouples was recorded with a digital camera, whereby clocks in the camera, the data acquisition system and the process control system were synchronised. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Figure 3:

41

The set up during measurement (left bottom) and thermocouple installation detail (right top).

Information from digital camera recording enabled conversion of the timedependent into spatial-dependent measurement of the strip surface temperature. The specially designed apparatus gives possibility to measure strip surface temperature during the industrial continuous twin-roll casting of aluminium alloys, where no influences of the measurement procedure on the production process are allowed. On the other hand, the thermocouple installation on the graphite block deteriorates the sensor dynamic response because of the additional thermal inertia, which increases the delay time of the sensor. The influence of the thermocouple installation support was considered with the heat flow balance of the sensor: dT (TA − Tm )α Am A = ( ρ cV ) m m + Q1 (18) dt where TA is the surface temperature of the aluminium strip, Tm is the measured temperature, αAm is the coefficient describing the heat transfer between the aluminium surface and sensor, A is the sensor contact area, (ρcV)m is the thermal capacitance of the sensor, and Q1 is heat flow from the sensor to the silicone layer and graphite installation support. Defining the thermocouple time constant ( ρ cV ) m as t m = , eqn. (18) can be rearranged to express surface temperature α Am A measurement error: dT q (TA − Tm ) = t m m + 1 . (19) dt α Am Time derivatives of the measured temperature were calculated with the first order central difference scheme. The thermocouple time constant tm was determined in separated test runs to be 3 s and should be the same for the all measurement runs, as well as the fitting coefficient αAm. The heat flux from measurement sensor to the silicone layer q1 was calculated using the equations describing the response of a semi-infinite solid to a step input of heat flux q1 at the surface of the silicone layer [6]. Subtracting the WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

42 Computational Methods and Experimental Measurements XIII temperature variation at the surface between the sensor and silicone layer from temperature variation at silicone layer depth lSl with temperature TG gives the expression for variation of q1: (Tm − TG ) k Si q1 = (20) 1 −   τ   1  τ lSi    1 − e  + erfc 

 π



 τ 

where TG is measured temperature at the back of the graphite support block with 4κ t the same kind of the thermocouple as Tm, τ = 2Sl is the dimensionless time for lSl the silicone layer of thickness lSl and of thermal diffusivity κ Sl =

k Sl

ρ Sl cSl

. ρSl, c Sl,

and V Sl are density, specific heat and volume of the silicone layer, respectively. 3.1 Experimental results The difference between the actual and measured temperature of the cast aluminium strip surface, TA-Tm, is calculated with eqn. (19) for four measurement runs at the same process parameters and shown in fig. 4 in dependence of the sensor installation support starting temperature (runs #30 and #32: 50°C; runs #31 and #33: 210 °C). At the beginning, the differences are the largest, from 10 to 22 K, and are falling to 2.5 K in all runs at the end of the measurement, which was limited to about 1.9 m or 90 s. In fig. 5, the measured, Tm, and the actual, TA, time dependent strip surface temperatures are shown for two runs. It is evident, that TA, calculated with eqn. (19), is independent of Tm starting temperature. 25 run #30 run #31 run #32 run #33

TA-Tm [K]

20

15

10

5

0 0

Figure 4:

0.2

0.4

0.6

0.8

1 x [m]

1.2

1.4

1.6

1.8

Difference between measured and actual strip surface temperature, spatial coordinate.



43

350 300

T [°C]

250 200 150

Tm run #32 Tm run #33 TA run #32

100

TA run #33

50 -20 -10

Figure 5:

0

10

20

30 40 t [s]

50

60

70

80

Measured and actual strip surface temperature, time coordinate. 350 run #30 run #31 run #32 run #33 model

340

T [°C]

330

320

310

300 0

Figure 6:

4

0.2

0.4

x [m]

0.6

0.8

1

Comparison of the simulated and measured strip surface temperature in four runs.

Experimental validation of the model

Results of the predicted temperature field of the cast aluminium strip crosssection were compared with the measured temperature on the upper side in the middle of the cast strip surface. Standard deviation of TA for all runs is 1.3 K. Fig. 6 shows good agreement between simulation and measurements with maximum discrepancy of 6 K near the cooling rolls. Comparison indicates overestimation of the convection heat transfer between strip and surrounding air, WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

44 Computational Methods and Experimental Measurements XIII and need for the radiation heat transfer consideration between cold rolls and strip surface, which has large influence on strip cooling in the vicinity of the rolls.

5

Conclusion

The meshless Local Radial Basis Function Collocation Method (LRBFCM) was used for estimation of the process parameters influence on the temperature field in the industrial twin-roll continuous casting aluminium alloy strip in the simulation of heat transfer from the strip. The model can consider influence of various process parameters on the temperature of the cast strip, which determines its properties. Simulation results were in good agreement with in situ measurements during the production process with the developed measurement technique. In addition, the experimental results gave important information for the model improvement, especially at the definition of the boundary conditions. Mounting three or five temperature sensors in a row and positioning the thermocouples on the same spots on the upper and the lower cast strip surface will represent next steps in development of the presented measurement technique, which will enable measurement of the possible temperature gradient in the width and the transverse sections of the cast strip, which could be responsible for inhomogeneous material structure and consecutive, difficulties at further strip processing.


[4] [5]

[6]

D. G. Altenpohl, Aluminum: Technology, Applications, and Environment: a Profile of a Modern Metal, Aluminium Association & TMS, 1998. T. Saitoh, H. Hojo, H. Yaguchi and C.K. Kang, Two dimensional model for twin roll continuous casting”, Metallurgical Transactions, 20B, pp.381-390 1989. B. Šarler, R. Vertnik & S. Šaletić, Solution of the thermal model of the twin-roll casting process by the meshless local radial basis function collocation technique. Computational methods for coupled problems in science and engineering, eds. M. Papadrakakis, E. Onate & B. Schrefler, Barcelona: International Center for Numerical Methods in Engineering (CIMNE), 2005. B. Šarler & R. Vertnik, Meshfree explicit radial basis function collocation method for diffusion problems, Computers and Mathematics with Applications, 51, pp. 1269-1282, 2006. R. Vertnik & B. Šarler, Meshless local radial basis function collocation method for convective-diffusive solid-liquid phase change problems, International Journal of Numerical Methods for Heat and Fluid Flow, 16, pp. 617-640, 2006. B. Lawton & G. Klingerberg, Transient Temperature in Engineering and Science, Oxford University Press, 1996.



45

A newly developed test rig for the study of thermal fatigue M. Fazarinc, R. Turk, G. Kugler & M. Terčelj Department of Materials and Metallurgy, Faculty of Natural Sciences and Engineering, University of Ljubljana, Aškerčeva cesta 12, 1000 Ljubljana, Slovenia

Abstract For simulation and better study of thermal fatigue, a new test rig has been developed. The rig has computer guided heating and quenching of the specimen that enables constant thermal loading and gathering of reliable experimental data. At the same maximal testing temperature it is possible to generate different and higher temperature gradients in comparison to other tests. The cooling rates that were measured were almost 4000 °C/s, which is considerably higher than the known rates so far measured (calculated) in cases of thermal fatigue cases (500– 1000 °C) and thus fewer cycles to crack nucleation were needed. Verification of the abilities of the rig was carried out with specimens (AISI H11 tool steel) which had different wall thicknesses (2.75–4 mm) and different surface qualities (heat treated and gas nitrided). The gas nitrided specimens exhibited lower thermal fatigue resistance. The shape of cracks was a grid, which is a typical characteristic of tools subjected to thermal fatigue (tools for hot forming of materials, etc.). Some specimens were additionally mechanically loaded and cracks that were at right-angles to the direction of the compression force were essentially detained; their nucleation and growth were suppressed. The measured temperatures in the surface layer were used to calculate the initial stress field using the Finite Element Method (FEM). The computed results matched well with the experimental data on the number of cycles needed for crack initiation. Keywords: test rig, thermal fatigue, tool steel AISI H11, FEM analysis.



1

Introduction

Thermal shock and/or fatigue resistance are very important characteristics of a material. The laboratory test rig (equipment) for their evaluation must be capable of carrying out such tests at various temperatures and have the capability, at the same testing temperature, to generate various temperature (stress) fields in order to better evaluate the usage of materials for a specific application. The tests that can be found in the literature cannot fulfil the above mentioned tasks satisfactorily. Poor knowledge of the temperature field within the specimen and consequently also poor knowledge of the stress/strain field results in an inaccurate estimation of the influence of parameters that are responsible for the nucleation and growth of cracks. In the past years an increased demand for improvement experimental data for better numerical modelling of crack growth can be found in the literature. Quenching in water is a very popular method to measure of thermal shock or thermal fatigue resistance but it has some weaknesses due to the unknown (unstable) heat transfer coefficient as a consequence of the vaporisation of water in the vicinity of the tested specimen surface [1–4]. Some authors designed special nozzles in order to minimize the effect of water vaporisation [5]. Marsh [6], Amiabe et al. [4] and Hadder and Fissolo [7] applied the so-called SPLASH test that also utilises water quenching of samples to generate temperature fields. Although they made great progress in modelling of crack growth on the basis of the experimental data further experimental improvements in this research area are desired. Therefore the goal of this paper was to present a new test rig for better evaluation of thermal shock or thermal fatigue resistance of materials. This test rig enables generation of high thermal stress with well defined thermal boundary conditions which allows the study of the temperature and thermal stress distribution in order to better evaluate the thermal shock or fatigue resistance of the tested material. An especially desirable characteristic of the test should have be the ability to generate various stress fields in the tested sample at the same maximal temperature.

2

Experimental set-up

2.1 Test rig and testing parameters For obtaining reliable test date it is of a great importance that the test is computer controlled and thus highly repeatable. Thus it was found helpful to carry out the tests on a thermo-mechanical simulator of metallurgical states, the Gleeble 1500D, in order to utilize the possibility of computer guided resistance heating of specimens and movement of the working jaws, simultaneously. The main idea was that the specimens would be rapidly heated to the maximum holding temperature and then the surface would be quenched with water. It was estimated that the cracks would form earlier than in 1000 cycles (for tool steel); this would shorten the testing time of a given specimen.



47

Circular and hollow shaped specimens were applied for testing (Figure 1). The thermocouple (type K) needed for temperature control and guidance was welded inside the specimen in the middle of its testing area, as shown on Figure 1. The outer testing part of the specimen was placed in a cooling chamber (Figure 2). The cooling and emptying process was optimized by a pair of magnetic computer controlled valves. One valve controlled the water quench and the other controlled the air compression to empty the cooling chamber. The valves were guided using the Gleeble 1500D control computer that was programmed simultaneously with the program for thermal and mechanical loading of the specimen.

Figure 1:

Figure 2:

Cross section of the testing specimen with welded wires.

The cooling chamber with water cooled clamps.

The setup of the test rig, without working jaws, is shown in Figure 3a and it’s positioning in the Gleeble load cell is shown in Figure 3b. For proving the test’s abilities of the test rig, nine different specimens (made from AISI H11 tool steel) with various characteristics given in Table 1 were tested. Two different types of specimens were compared, i.e. five were heat treated (I - IV and IX, initial microstructure on Figure 4a) and four were heat treated plus gas nitrided (V - VIII, initial microstructure on Figure 4b). Further, the specimens differed also in specimen thickness (2.75 - 4 mm). Two specimens of each mentioned types were also mechanically loaded during testing by applying a compression force (0-19kN) equal to 90% of the yield stress at 650°C corresponding to a value around 150MPa. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

48 Computational Methods and Experimental Measurements XIII The program used for thermal and mechanical loading (see Figure 5 a-b) of the specimen was written in the applicative program QuickSim and was composed in following way: - resistance heating of the sample to the holding temperature of 650°C with increasing mechanical loading (0 to 19kN, see Figure 5 a-b), - water quenching (0-0.50s) during maintenance of the programmed temperature and mechanical loading, - interruption of heating and quenching (and mechanical unloading), - blowing of compressed air into the chamber (3s). After 500 cycles the tested surface was examined visually and after 1000 cycles the specimens were cut and metallographically prepared for microscopic analysis. Table 1: Surface treatment Specimen Thickness Tmax (°C) Fmax (kN)

a)

Characteristics of specimens (AISI H11) and loadings (thickness [mm]). Heat treated I II 2.75 3.25 650 0

Heat treated III IV 2.75 3.25 650 -19.0

H.T. + nitrided V VI 2.75 3.25 650 0

H.T. + nitrided VII VIII 2.75 3.25 650 -19.0

Heat treated IX 4 650 0

b)

Figure 3:

a)

a) Testing device with specimen, b) Inserted test rig in testing cell.

b)

Figure 4:

Microstructure a) Heat treated specimens, b) Nitrided specimens.


Computational Methods and Experimental Measurements XIII Temp. Force

600

-12

heating strat cycle period

400

700

-16 -14

quenching finish

500

-18

-10

300

-8

200

-6

100

-4

Temperature [°C]

Temperature [°C]

600

Force [kN]

Quenching start

700

49

TC1(C)

500 400 300 200 100

-2

0

0 0

180

185

190

195

200

205

Time [s]

a) Figure 5:

quench strart

600

800

1000

1200

1400

500

quench finish

1,0 mm

450 400 350

0,7 mm

300

0

Cooling rate [°C/s]

Temperature [°C]

400

Time [s]

0.3 mm 0.7 mm 1.0 mm 2.75 mm

600

2,75 mm

250 200

200

a) The thermal and mechanical loading in one cycle, b) 50 temperature cycles.

650

550

0

b)

0,3 mm

150

-1000

-2000

-3000

-4000

100 87.5

88.0

88.5

a)

89.0

89.5

90.0

90.5

91.0

91.5

92.0

Figure 6:

86

b)

Time [s]

87

88

89

90

91

92

93

Time [s]

a) Measured temperatures at different depths during the quenching sequence, b) Cooling rate at the depth of 0.30 mm. 700

500

Temperature gradient 600

Temperature [°C]

480

∆T [°C]

460

440

420

Temperature profile 500

400

300

200

400 100

0.1

0.2

a)

0.4

Quenching time [s]

Figure 7:

3

0.3

0.5

0.0

0.6

0.5

b)

1.0

1.5

2.0

2.5

3.0

Thickness [mm]

a) Measured minimal temperature (temperature profile) in various depths at quenching time of 0.5s, b) Measured temperature gradient in the specimen at various quenching times.

Results

3.1 Measurements of temperature in the specimen A typical example of a thermal and mechanical cycle is shown in Figure 5a. The simultaneous increasing of temperature and mechanical loading of the tested WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

50 Computational Methods and Experimental Measurements XIII specimen is visible, while during quenching both of them maintain the programmed (same) value. The repetitiveness of the testing cycles obtained is presented on Figure 5b. The temperature of the specimen was controlled by the thermocouple, as presented in Figure 1. In order to determine (measure) the temperature field on the tested surface, a specially made specimen was applied in which inside had welded three additional highly responsive thermocouples (wire diameter 0.2 mm, type K) on various distances (0.30 mm, 0.70 mm, 1 mm) bellow quenched surface, while the fourth thermocouple (Figure 1) served for computer guidance and control of temperature. Their measured values are shown in Figure 6a. It is apparent that the temperature of the fourth and computer guided thermocouple remained at the programmed value of 650 °C also during quenching. After the quenching period the temperature fall of the fourth thermocouple was higher in comparison to the third thermocouple, since the heat transfer on the outer surface in the hollow specimen is also higher. The highest temperature gradient was registered at the quenching time of 0.5s where the measured temperature difference between the first and the fourth thermocouple was 473°C. The cooling rate was determined by differentiating the temperature/time curve at the depth of 0.30 mm from the quenched surface. Using this technique the maximum cooling rate was determined (3918 °C/s, Figure 6b). At the specimen surface these rates were certainly higher. Thus with a combination of simultaneously controlled cooling and resistance heating of the specimen, greater thermal gradients in relatively thin specimen surface layer were achieved, while the remaining depth of the specimen kept approximately the same value (see Figures 5a and 6a). This reduced the number of cycles needed for crack nucleation and resulted in faster crack growth. For comparison Hadder and Fissolo [7] and Marsh [6] report cooling rates between 500 and 1000 °C/s. The minimal measured temperature for the first thermocouple and various quenching times (0.1 - 0.55 s) are shown in Figure 7a while Figure 7b shows the temperature profile (field) in the tested specimen at a quenching time of 0.5s. With the possibility of generation of various thermal gradient, the test can also simulate various thermal loading conditions in an applicative environment. 3.2 Estimation of initial stress field by FEM When the entire temperature profile was known the FEM was applied in order to estimate initial stress field (thermal stress) for different quenching times and specimen thicknesses. The MSC Super Forge 2005 code was used for this. The results of these analyses are shown in Figure 8 a-b. The analyses show that the maximum stresses are obtained after 0.5s of quenching and are roughly 140 MPa for a specimen thickness of 2.75 mm. At 0.12s of quench time the stresses reached the values around 125 MPa. This shows that the temperature and consequently the stress field can be varied by changing the quenching time. The stress obtained also depends on specimen thickness; thus they reached values of cca 160 MPa and 175 MPa for specimen thicknesses of 3.25 mm and 4 mm, respectively. Furthermore the temperature can be varied by changing the quenching liquid (nitrogen, air, etc) and changing its pressure. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


a)

51

b)

Figure 8:

Estimated stress field simulated by FEM method, a) t = 0.12 s, b) t = 0.5s; wall thickness 2.75 mm, T = 650 °C.

3.3 Appearance of surface cracking From the estimated stress field it could be predicted that cracks would appear in thicker specimens sooner than in thinner specimens. The next batch of pictures were taken by an optical microscope after 1000 cycles of testing. The results are given in sequences according to thickness. 3.3.1 Wall thickness 2.75 mm Estimated initial maximal stresses for the wall thickness of 2.75 mm were namely around 140 MPa that is lower as yield stress of AISI H11 tool steel at 650 °C (cca 150 MPa). This fact explain why on heat treated no cracks occurred but on contrary on gas nitrided specimens network of cracks was observed after 1000 cycles; the depth of obtained crack was in range 20 - 60 µm (Figure 9).

Figure 9:

Appearance of surfaces of nitrided specimens after 1000 cycles, axially unloaded specimen; quenching time 0.5s, thickness 2.55 mm, radial cross-section.

3.3.2 Wall thickness 3.25 mm The next batch of pictures was taken on nitrided specimens after 1000 cycles where the thickness of the specimens was 3.25mm. Estimated maximal stress using the FEM wall thickness of the specimen 3.25 mm amounted around WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

52 Computational Methods and Experimental Measurements XIII 160MPa. Consequently the cracks appearing were deeper and denser. Figure 10a shows the surface of nitrided specimen after 1000 cycles. The cracks here are clearly visible and they are appearing in both directions (radial and axial). As expected, the axial loading of the specimen restricted the nucleation of cracks in the radial direction. The maximal compression force during cooling was 19 MN corresponding cca 140 MPa. Thus it means that maximal stresses in axial direction of the specimen were close to zero and consequently no cracks appeared in the radial direction (Figure 10b). The depths of the obtained cracks in the radial direction for an axially unloaded specimen are presented on Figure 11a and for an axially loaded specimen on Figure 11b. The crack depth obtained was cca 180 µm in the radial direction and cca 120 µm in the axial direction. The same phenomenon also occurred on heat treated specimen, only the depth of the cracks were lower (cca 20-30 µm).

a)

b)

Figure 10:

Appearance of surface of nitrided specimens after 1000 cycles, a) Axially unloaded specimen, b) Axially loaded, ← direction of axial loading; quenching time 0.3 s, and thickness 3.25 mm.

b)

a) Figure 11:

Cross-section of nitrided specimen of axially loaded nitrided specimens, a) Axial section, b) Radial section; quenching time 0.3s, wall thickness 3.25 mm.

3.3.3 Wall thickness 4 mm As was expected denser cracks were obtained in thicker specimen and at a quenching time of 0.5s since the estimated maximal stress (for the specimen thicknesses of 4.0 mm) amounted around 170 MPa. On Figure 12 a-b the cracks WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


53

network on heat treated specimen is visible. Mentioned cracks were visible still after 500 quenching cycles and after 1000 quenching cycles the crack net obtained was denser. The depth of the cracks was around 40 µm. 3.3.4 Discussion The test enables us to vary all important parameters which influence the experimental results. These are the heating rate, maximum specimen temperature, quenching time, quenching medium, pressure of the quenching medium, specimen wall thickness, and the external specimen loading (tensile compressive, cyclic - constant). The appearance of cracks on tested specimen was very similar to those obtained on a hot working surface of rolls as presented on Figure 13. Namely, after its contact with heated specimen the roll surface is subjected to rapid water cooling leading to surface cracking. In the literature we can find contradictory result regarding to the thermal fatigue resistance of some nitrided surfaces. Thus Pellizari et al. [8] claim that nitrided surfaces decreased the fatigue resistance on the contrary Spies et al. [9] claim that the nitrided surface increased fatigue resistance. The authors did not present the temperature field and consequently they also could not calculate (asses) the stress field (predominately tensile or compression stresses) on the tested specimens from which the reason for the appearance of cracks based. Therefore, different results should be carefully compared according to the specific test conditions.

a)

b)

Figure 12:

Appearance of the surface of heat treated specimens, a) After 500 cycles, b) After 1000; axially unloaded specimen, quenching time 0.5s, wall thickness 4 mm, heat treated.

Figure 13:

Appearance of cracks on the surface of hot roll.



4

Conclusions

A new thermal fatigue test was developed for testing materials. It is based on computer guided heating and quenching of the specimen. The specific shape of the specimen and special execution of the program enables the achievement of very high temperature gradients that are greater than those found in the literature. This reduces the number of cycles needed for crack nucleation and enables shorter testing times. In combination with mechanical loading different stress states on specimens can be achieved (mechanical pre-loading of forging tools, etc.). The test can generate a wide range of conditions to which the real parts can be subjected. Calculation of the stress field using the FEM showed the highest stresses in the thicker walls of the specimen. Thus on specimens with thicker walls and longer cooling times, the cracks nucleated earlier and are denser. A comparison of two differently treated materials (heat treated or heat treated + gas nitrided) showed that the nitrided layer is brittle and rapidly cracks under tensile thermal stresses. The newly developed test with its proven characteristics will contribute to better understanding of crack nucleation and their growth. This is the basis for better modelling of the processes involved.

References [1] [2] [3] [4] [5] [6] [7] [8] [9]

A. Weroński, T. Hejwowski, Thermal fatigue of metals, Marcel Dekker 1991. N. Hadder, A. Fissolo, V. Maillot, Thermal fatigue crack networks: a computational study, International Journal of Solids and Structures 42 (2005) 771-788. A. Persson, S. Hogmark, J. Bergström, Simulation and evaluation of thermal fatigue cracking of hot work tool steel, International Journal of Fatigue 26 (2004) 1095-1107. S. Amiable, S. Chapuliot, A. Constantinescu, A. Fissolo, A comparison of life time prediction methods for a thermal fatigue experiment, International Journal of Fatigue, 28 (2006) 692-706. J. Absi, J.C. Glandus, Improved methods for severe thermal shocks testing of ceramics by water quenching, Journal of European Ceramic society 24 (2004) 2835-2838. D.J. Marsh, A thermal shock fatigue study of type 304 and 316 stainless steels, Fatigue of Engineering Materials and Structures, 4/2 (1981) 179195. N. Hadder, A. Fissolo, 2D simulation of the initiation and propagation of crack array under thermal fatigue, Nuclear Engineering and Design 235 (2005) 945-964. M. Pellizzari, A. Molinari, G. Straffelini, Thermal fatigue resistance of gas and plasma nitrided 41CrAlMo7 steel, Materials Science & Engineering A352 (2003) 186-194. H.J. Spies, F. Vogt, M. Svennson, Neue Hütte 8 (1983) 281-287.



55

Numerical simulation with flow feature extraction of a propeller turbine unsteady rotor-stator interaction J.-M. Gagnon & C. Deschênes Hydraulic Machinery Laboratory, Laval University, Canada

Abstract We have seen many papers in the past few years reporting research on Francis hydraulic turbine flow. Interesting papers assessing the accuracy of the CFD numerical simulation were also presented for axial turbine draft tubes, as in the three Turbine 99 workshops. However, little knowledge is available on the flow taking place inside an axial turbine. This paper focuses on the unsteady rotor-stator interaction in a propeller axial turbine. The flow behaviour is analysed at different rotor and stator relative locations with numerical simulations using a commercial code and k-ε turbulence model. The main goal is to study unsteady flow phenomena such as wake, separation, forces and pressure fluctuations in the propeller turbine. This investigation will help to design a series of flow measurements used in turn to improve future CFD simulations with realistic velocity profiles as boundary conditions. Keywords: propeller turbine, numerical simulation, blade torque, partial load.

1

Introduction

Low head power plants are expected to be implemented increasingly in the future for economical, geographical and environmental purposes. Propeller turbines are well suited for these types of applications. They operate at higher flow rate, smaller head and faster rotational speed, thus being more compact than other types of machines. The US Department of Energy (DOE) is anticipating major growth for low head power plants [1] and studies such as Turbine 99 [2] or the work of Roussopoulos and Muntean [3, 4] show research trends on axial hydraulic machine. On a global perspective, optimization of size and weight WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070061

56 Computational Methods and Experimental Measurements XIII towards highly compact system has led researchers to seek for a better understanding of fluid dynamic and unsteady effects. In turbomachinery applications, the interaction of stationary guide vanes and rotating runner may be considered as the source of unsteady phenomena. Rotor-stator interactions taking place in turbines may be harmful for blades and surrounding systems as they induce pressure pulsations and torque variations on runner blades [5, 6]. Fluctuations of the velocity field between the rotating and stationary parts also contribute to unsteadiness. Extensive literature is available on this subject for Francis turbines, pumps and pump-turbines. In [7], Page et al. analyse the time-averaged relative velocity profile and pressure fluctuations on a pump impeller outlet and a Francis turbine. Vu and Nennemann [8] have shown a method to analyse unsteady interactions in a Francis turbine using torque fluctuations and pressure spectrum. The unsteady torque on guide vanes pitching axis of a pump turbine in pump mode was predicted by Lipej et al. [9]. Recently, blade cracking and power plant failure were reported and due to the high dynamic stress of wicket gates and runner interaction [10]. In numerical simulation, Ruprecht et al. [11, 12] showed parallel computation capabilities to simulate a complete Francis Turbine including rotor-stator unsteady hydrodynamic effect. Performance of an in-house code to study the flow in an axial turbine with unequal pitch ratio was also addressed. We can also gather great amounts of information on rotor-stator interaction and flow features inside axial turbines or compressors from the aerodynamic field. Basic phenomena such as separation, wake and secondary flows are similar when there is no shock and Mach number stay below 0.3. In this avenue, Zaccaria and Lakshminaryana [13] have exposed an extensive review of phenomena taking place in axial air turbine with an experimental investigation of the flow. The present paper focuses on 3-D Navier-Stokes simulations of the flow in a propeller turbine using the commercial code ANSYS CFX. Three operating regimes near the best efficiency point are studied. Numerical methods are first detailed and then results from both steady and unsteady computations are analysed. Relevant quantities as torque on blades and pressure fluctuations are investigated, as well as wake characteristics behind guide vanes. This research is carried out within the new Canadian Consortium in Axial Hydraulic Turbine taking place at Laval University, Quebec. Figure 1 illustrates a cut plane of the hydraulic propeller turbine along with future measurement access.

2

Experimental overview

Numerical simulations begin with the definition of different geometries of the domain and interfaces. This step allows us to look forward for the experimental setup that will be used to validate numerical results. Optical accesses for PIV and LDV measurements will be installed on critical regions, where the velocity and pressure fields need to be known to improve numerical simulations. CAD drawings are also used to help locating the calibration plate for PIV inside the distributor and for inter-blade PIV. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


57

Optical access 1

PIV/LDV

Optical access 2

Optical access 3,4

PIV/LDV Figure 1:

3

Cut plane of the hydraulic turbine with planned measurement concept.

Computational methods

We consider an incompressible unsteady 3-D turbulent flow in a single rotorstator passage of the model scale propeller turbine. RANS equations are used in combination with the k-ε turbulence model and scalable wall functions accounting for unequal wall cells sizes and y+ in the boundary layer calculation. The main advantage is to allow uses of coarser or finer mesh sizes and different turbulence models without having to remesh the whole boundary layer around runner blades and guide vanes. We use an advection scheme switching locally from a first order upwind differencing scheme to a second order scheme when necessary. Solver robustness is thus increased and convergence accelerated. 3.1 Interface definition and analysis Several interfaces were used to connect stationary and moving meshes for steady and unsteady calculations. First, we estimated mesh performance and torque on the runner blade with a steady simulation averaging the circumferential velocity at the interface (Stage simulation). Since the effect of circumferential velocity is filtered through the averaging operation, it makes possible the use of a partial stator passage having only one guide vane and one stay vane instead of four. Total numbers of blades and guide vanes is respectively 6 and 24 for the whole machine, giving even pitch ratios. Other types of simulations were performed using a Frozen rotor for steady calculations and a Transient Rotor-Stator interface for unsteady calculations. Frozen rotor simulation cases were used as initial conditions for unsteady calculations. Figure 2 below shows interfaces definition: on the left with a stator pitch angle of 15 degrees in the azimutal plane for stage calculation and on the right with 60 degrees rotor-stator passage.


58 Computational Methods and Experimental Measurements XIII 60

15q Rotor interface

60q rotor periodicity

60q

Stay vanes

Rotor blade

Guide vanes

Axis b)

a)

Simulation domain and interface a) 15° stator - 60° rotor b) 60° rotor/stator.

Figure 2:

coarser

+

+ coarse

Figure 3:

medium

+

fine

+

Torque in function of cells number (left), y+ distribution on blade and guide vanes walls (top and bottom right).

3.2 Mesh independence We performed a mesh independence test to evaluate what size of mesh is needed for the type of flow phenomena which is analysed. In our case, the runner blade torque is the quantity investigated since it is directly linked to unsteady pressure fluctuations. Four mesh sizes were compared against computed torque values to ensure mesh independence. Figure 3 shows normalized runner blade torque in WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


59

function of number of mesh elements (on left). Comparison of y+ distribution for different mesh sizes was also done on guide vanes and runner blade walls. The lowest blade normalized torque value of 1.05 corresponding to the coarse mesh computation gives the best computed torque compared to experimental data. However, having a look at the right figures, this mesh exceeds the recommended maximum y+ value of 200 on the runner blade. This may lead to poor boundary layer calculations. Therefore the medium grid of 1000k elements has been chosen for all calculations at peak, partial and overload conditions to fulfil the solver boundary requirement. Note that no calculations got closer to 5% of the experimental torque value, leaving place for numerical simulation improvement.

4

Rotor-Stator interaction analysis

Unsteady rotor-stator interactions may be attributed to instantaneous pressure fluctuations. Wakes building along the boundary layer of stay vanes and guide vanes of the distributor are other unsteady phenomena which may be considered. These flow phenomena are weaker in intensity and reflect higher harmonics on the frequency spectrum. Results from steady state simulation using a Frozen rotor interface are presented in this section for wake interaction. Dissipation of the wake is investigated with the following indicators: the turbulent kinetic energy, the vorticity and the velocity profile. Figure 4 shows contours of turbulent kinetic energy and vorticity in a blade-to-blade view for three operating regimes. Here, α* is the normalized opening guide vane angle equal to the ratio of the actual opening angle to the angle corresponding to the best efficiency point, α/αpeak. Note that the flow velocity, U∞, changes direction across interfaces accounting for rotation terms in N.-S. equations. Figure 4 presents two periodic profiles having a total of eight guide vanes and two runner blades. The blade-to-blade planes of cases shown in Figure 4a) are located at low span, near the shroud, to capture the small gap between guide vanes and runner blades. From the contours, we can distinguish a slight increase in k across the interface with increasing guide vane opening (from left to right, light grey contour). At overload condition (α* = 1.15, top right), the contour of k almost reaches the blades and may indicate the possibility of wake interaction with runner blades. The level of wake kinetic energy is low for all cases and around 0.03 m2/s2. In figure 4b), the slice is at mid span and one can see that wakes behind guide vanes and runner blades induces regions of high radial vorticity (dark contour). The vorticity intensity is kept high across the interface and propagates until it reaches runner blades surfaces for all cases. A further investigation of these steady state calculations gives insight about how far gradients of velocity are convected into the rotor domain. In Figure 5, one can see a typical case of how velocity profiles on circular lines in the rotor-stator passage are gradually dissipated. Velocity gradients are dissipated up to 99% before reaching the blade at peak condition.


60 Computational Methods and Experimental Measurements XIII D* =0.94

D* = 1

D* = 1.15

Interface position

Blade

Interface position

a)

k increase

G.v.

Uf

Uf

Uf

b)

Z

Figure 4:

Z

Z

Blade-to-blade contours of a) turbulent kinetic energy b) radial vorticity.

Line 1

Line 2 Line 3 Line 4

Figure 5:

Meridional velocity profiles along different streamwise stations in the rotor-stator passage.



5

61

Unsteady rotor-stator analysis

Unsteady interactions of runner blades with stationary guide vanes have been studied by looking at the frequency spectrum of instantaneous force, torque and pressure signals over time at various locations. Figure 6 illustrates the time and frequency signal of drag force on each of the four guide vanes in the distributor.

Figure 6:

Guide vanes force for α*=1 with signal in time (top) and frequency spectra (bottom).

Fluctuations of instantaneous forces on the guide vanes are about 5% of the average force. The peak on the frequency spectrum corresponds to the blade passing frequency, fb = Nb⋅ fω , where fω is the rotor frequency and Nb the number of blades. The phase shift between two adjacent force signals equals 2π/Ngv, where Ngv is the number of guide vanes. We investigated the instantaneous blade torque on the rotor side to assess the effect of transient interactions on the runner in the rotating frame of reference. In figure 7, the three operating regimes are shown with a FFT transform applied on the same amount of cycles for each torque signal. Amplitude peaks for all regimes correspond to the wicket gate passing frequency, fgv = Ngv⋅fω. Wake interactions have very small effects on the unsteady torque since there is no other peak at higher frequencies. Also, we see that the amplitude of the signal at overload condition (α* = 1.15) is higher than in the other cases confirming the fact that pressure fluctuations effects on blades and on other unsteady phenomena are increased as we move away from peak condition. Fluctuations of the torque are small and estimated about 0.3% of the time-averaged torque. To complete this analysis, we have evaluated the pressure signals from two points located very close to the interface at mid span. Figure 8 shows the time and frequency signals for these two points. Similar conclusions applied as for the torque and forces analysis and we see that pressure frequencies match with fb for the stator point and fgv for the rotor point. One can also find additional higher harmonics around 7 and 14 times fb (top axis, bottom left figure). These WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

62 Computational Methods and Experimental Measurements XIII harmonics are likely due to numerical noise rather than to unsteady interactions (wakes, von Karman vortices). Further investigations are needed to better understand the phenomenon. Pressure fluctuations around the time-averaged signals are about 2.4% and 0.5% respectively for the stator and the rotor points.

Figure 7:

Frequency spectra of torque on runner blades.

Higher harmonics

Figure 8:

6

Pressure time and frequency signal of two points situated close to the interface.

Conclusion

In this paper, we studied wakes and unsteady rotor-stator interactions for different operating regimes of a propeller turbine. The wake behind guide vanes is dissipating very fast. It is still unsure whether it has an effect on the rotorWIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


63

stator unsteady interactions since higher harmonics of the spectrum were not well defined. The main interaction is therefore attributed to the pressure field fluctuations. Peak amplitudes correlate with the runner frequency for all pressure related signals. Although it was found that unsteady forces, torque and pressure fluctuations were weak in amplitude compared to time-averaged values (below 5%), one must keep in mind that current computations were done at model scale. Same types of study on Francis turbine indicate that fluctuations might double at prototype scale. Finally, we are really looking forward the upcoming experimental phase to increase our knowledge and understanding of the propeller turbine flow.

Acknowledgements The authors would like to thank Hydro-Quebec and Alstom Hydro Canada Inc. for their financial and technical supports. Their contribution for turbine geometries and experimental model is of great importance to this research project. Richard Fraser, Maryse Page, Sebastien Houde and Michel Sabourin are also gratefully acknowledged for their time and devotion to the project.

References [1]

[2] [3]

[4] [5] [6] [7] [8]

Douglas, G. H., Kelly, S.R. (2006), Feasibility Assessment of the Water Energy Resources of the United States for New Low Power and Small Hydro Classes of Hydroelectric Plants, U.S. Department of Energy, DOEID-11263. Andersson U., Karlsson R. (1999), Quality aspects of the Turbine 99 draft tube experiments, Proceeding of Turbine 99 – workshop on draft tube flow, Technical report, Lulea University of Technology, Sweden. Roussopoulos K., Monkewitz P.A. (2000), Measurements of Tip Vortex Characteristics and the Effect of an Anti-Cavitation Lip on a Model Kaplan Turbine Blade, Flow, Turbulence and Combustion, Netherlands, vol. 64, p. 119-144. Muntean, S., Balint, D. (2006), Analytical representation of the swirling flow upstream the Kaplan turbine for variable guide vane opening, XXIII IAHR Symposium, Yokohama. Nennemann, B., Vu, T.C., Farhat, M. (2005), CFD prediction of unsteady wicket gate-runner interaction in Francis turbines: A new standard hydraulic procedure, Hydro 2005. Ciocan, G.D., Kueny, J.L. (2006), Experimental Analysis of Rotor-Stator Interaction in a Pump-Turbine, XXIII IAHR Symposium, Yokohama. Page, M. Théroux, E., Trépanier, J.-Y. (2004), Unsteady rotor-stator analysis of a Francis turbine, XXII IAHR Symposium on Hydraulic Machinery and System, Stockholm, Sweden. Vu, T.C., Nennemann, B. (2006), Modern trend of CFD application for hydraulic design procedure, XXIII IAHR Symposium, Yokohama.


64 Computational Methods and Experimental Measurements XIII [9] [10] [11] [12] [13]

Lipej, A., Jost, D., Meznar, P. (2006), Numerical analysis of rotor-stator interaction in a reversible pump-turbine – pump mode, XXIII IAHR Symposium, Yokohama. Coutu, A., Proulx, D., Coulson, S., Demers, A. (2004), Dynamic Assessment of Hydraulic Turbines, Proceedings of HydroVision 2004, Montreal, Quebec, Canada, August 16-20. Ruprecht, A., Heitele, M., Helmrich, T. (2000), Numerical Simulation of a Complete Francis Turbine including unsteady rotor/stator interactions, XX IAHR Symposium, Charlotte, North Carolina. Ruprecht, A., Bauer, C., Gentner, C., Lein, G. (1999), Parallel Computation of Stator-Rotor Interaction in an Axial Turbine, ASME PVP Conference, CFD Symposium, Boston. Zaccaria, M. and Lakshminarayana B. (1997), An experimental investigation of steady and unsteady flow field in an axial flow turbine, NASA contractor report; 4778, National Aeronautics and Space Administration.



65

Computer and experimental study of the gate dielectric in a memory transistor R. Avichail-Bibi1, D. Fuks1, A. Kiv1 & Ya. Roizin2 1 2

Ben-Gurion University of the Negev, Israel Tower Semiconductor Ltd., Israel

Abstract We demonstrate a novel approach that enables combining microscopic studies of the behaviour of the injected charge (IC) in the gate dielectric (GD) of the memory transistor and the description of kinetics of memory device service parameters. To study the microscopic processes of the redistribution of the IC in the GD a special package of programs was developed that allows the modelling the migration of injected electrons and holes in the GD. The model accounts real properties of dielectric (spatial distribution of local centres and their characteristics, the dielectric constant and its changes on the microscopic distances, a complex composition of dielectric, temperature conditions and the geometry of the GD). The results of the computer simulation of microscopic characteristics of the IC were used as input data for the commercial Device simulation program "Medici". We found a correlation between microscopic characteristics of IC in GD and the service parameters of the memory device and realized the feedback procedure changing the GD characteristics in the simulation model. Keywords: memory transistor, gate dielectric, trapping mechanisms, molecular dynamics, computer simulation.

1

Introduction

The NROM (nitride read only memories) are non-volatile memories with local storage of charge at the edges of the memory transistor channel and a thick (>35 Å) bottom oxide (BOX) [1] that became popular in the nonvolatile semiconductor memory market. A ONO (SiO2-Si3N4-SiO2) stack with nontunnel bottom oxide is the GD in a two-bit per cell memory transistor. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070071

66 Computational Methods and Experimental Measurements XIII Information is stored as the charges injected into the Nitride at the channel edges of the memory transistor. Programming is performed by channel hot electrons. Holes, created by band-to-band tunneling in the drain region erase the programmed bit [1–3]. The device is read out in the “reverse” direction compared to programming. These memories are free of the limited retention and high read disturb of the previous SONOS generations having thin (~20 Å) BOX. Properties of traps are especially important in NROM devices because the reliability performance of these memories is to a great extent determined by the lateral migration of the charge trapped in the silicon nitride layer [2]. Reliable values of trap parameters in silicon nitride, in particular the trap activation energy, can be found from device measurements, as well as the effects related to BOX can be distinguished from those connected with silicon nitride properties. Despite numerous experimental data on traps collected for different types of silicon nitride, the chemical nature of traps in a Si3N4 remains unclear. This is why the phenomenological description and computer simulation of trapping processes in Si3N4 stack is of great interest. In [4] a new approach in Molecular Dynamics (MD) simulation is described. It allowed the study of real physical processes in the GD of memory device and the prediction of its retentionendurance characteristics. Retention of microFLASH® memory transistors is characterized by the stability of the programmed and erased state threshold voltages (Vt). The Vt shift of the programmed state of the device after cycling (a series of programmingerase (P/E) operations) is experimentally controlled after high temperature bakes. The Vt shift of the programmed state is sometimes called a high temperature, HT Vt shift. The second type of Vt shift characterizes the erased state of the device after cycling. This shift has weak temperature dependence and is called room temperature, RT Vt shift. HT and RT Vt shifts determine the operation margins of the memory cell [5]. Analysis of time dependences of HT Vt shift (∆Vt) shows two stages in its kinetics: “fast” Vt shift and “slow” Vt shift. The “fast” Vt shift depends on the programming window ∆V = Vt high – Vt initial and is typically 50 mV – 300 mV for ∆V=1-3 V. At a temperature ~2000C the “fast” period of Vt decrease lasts several hours [14–16]. The “fast” relaxation period is followed by a “slow” Vt decrease process with high activation energy (~1.8eV) [6]. At the beginning of the relaxation process there is a limited decrease of Vt (~50-200 mV) even for the one-time programmed cell. Additional “fast” memory window loss (~250500mV) is observed after 1k-100k program-erase cycles [5–7]. It was found in [8] that at least for a small ( N*traps was manifested by the significant increase of activation energy for carrier migration [15]. A strong dependence of the PR transformation on the degree of the overlapping of electron wave functions for the neighbor traps and on the value of potential wells of the individual traps was revealed. The overlapping of wave functions was modeled by changing the probability of tunneling between the nearest traps [4].

U=2.0eV U=2.5eV

0

1000

2000

3000

4000

Number of steps

Figure 4:

The kinetics of charge leaving the IR for the large number of traps: Ntraps > N*traps.

The increase of the depth of potential wells in PR of GD can be explained by formation of a strongly disordered PR as a result of the random overlapping of wave functions of neighboring trapped electrons. Thus we can suppose that in the case when Ntraps>N*traps the Anderson localization [16] occurs decreasing sharply the mobility of carriers. This effect leads to significant growth of the activation energy for carrier migration. Such a situation reveals itself in the fact that the kinetics of redistribution of the IC in the GD becomes independent on the activation energy of individual traps. Fig. 4 illustrates these results. One can WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


71

see that at the first steps of the simulation procedure the escape of some number of the IC from the IR occurs. After that we do not observe the redistribution of spreading IC. We see almost the same kinetics of the IC spreading for two different activation energies, U of individual traps (2 eV and 2.5 eV). These results show the way of the modification of the PR of the GD to achieve a stable distribution of the IC in the GD for a long time: it is necessary to create the disordered potential relief for electron migration with deep enough wells for Anderson localization. The next example of the influence of the features of PR on the kinetics of the IC in the GD is linked to the creation of high-k layers (HKL) in the boundary regions of the GD [10]. In the case of ONO with HKL at the late stage of baking (in simulation procedure after ~ 10000 steps) we observed a sharp increase of the fraction of spreading electrons that results in a large increase of Vt shift. At the interface between the HKL and the nitride a large density of traps exists. These local centres trap electrons that lead to formation of additional charge at the interface. The newly injected carriers scatter on these charges and begin to move in a lateral direction. Later the number of spreading electrons can decrease due to the effect of competition between the formation of located charges in PP and those in the interface traps of the nitride. In Fig. 5 we demonstrate the behaviour of retention parameters of the memory cell with HKL that hinders the penetration of 90% of the IC outward from the ONO stack. One can see that at the first stages the existence of HKL improved retention characteristics of the memory cell. At some stage of baking the non-monotonous behaviour of the retention characteristics of the ONO stack with HKL is seen. Analysis of simulation results leads to the conclusion that the properties of the traps in the bulk and in the interface region are really responsible for this effect. Consequently in the process of memory device exploitation the effect of the nonmonotonous kinetics of the retention characteristics of ONO memories with HKL can be expected. This effect depends also on the thickness of ONO (Fig. 6). with HKL without HKL

0

5000

10000

15000

20000

Number of steps

Figure 5:

Two kinetics of IC in ONO without HKL and with HKL.


72 Computational Methods and Experimental Measurements XIII The vertical axis in Fig. 6 presents the number of steps preceding the jump on the kinetic curve shown in Fig. 5 for the case of ONO with HKL. The number of steps that corresponds to an increase of charge loss from the IR for the given depth of the ONO stack is larger for the HKL with smaller "k". This means that two limits of scaling down a memory cell exist. The upper one is caused by intensive scattering of injected carriers on trapping centers located at the interface between HKL and the nitride. The lower one is linked to charge leakage caused by tunneling current. The problem that should be solved is how to decrease the interval between these two limits. The vertical axis in Fig. 6 presents the number of steps preceding the jump on the kinetic curve shown in Fig. 5 for the case of ONO with HKL. The number of steps that corresponds to an increase of charge loss from the IR for the given depth of ONO stack is larger for the HKL with smaller "k". This means that two limits of scaling down a memory cell exist. The upper one is caused by intensive scattering of injected carriers on trapping centers located at the interface between HKL and the nitride. The lower one is linked to charge leakage caused by the tunneling current. The problem that should be solved is how to decrease the interval between these two limits.

0000

1 2

8000

6000

4000

2000

0 80

100

120

140

160

Thikness of ONO stack (Angstrom)

Figure 6:

4

Influence of the quality of HKL on the possibility of scaling down of ONO stack. Graphs 1 and 2 correspond to HKL that hinders the ejection of 90% and 50% of IC from the ONO stack.

Conclusions

Novel software was used for simulation of physical processes responsible for Vt shift in a memory device. The proposed model of the GD accounts for both classical and quantum properties of the system. It was found that Coulomb repulsion determines the redistribution of the IC in the GD. As a result the nature of the Vt shift in cycled microFLASH® memory transistors is explained and the ways for improvement of their parameters are indicated. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


73

References [1] [2]

[3]

[4] [5]

[6] [7]

[8] [9] [10] [11] [12] [13] [14]

Eitan B., Pavan P., Bloom I., Aloni E., Frommer A. & Finzi D., NROM: A novel localized trapping, 2-bit nonvolatile memory cell IEEE Elec. Dev. Lett., 21, pp. 543-545, 2000. Roizin Y., Aloni E., Gutman M., Kairys V. & Zisman P., Retention characteristics of microFLASH® Memory, IEEE 2001 NVSMW, Monterey, CA, 2001. Gritsenko V. A., Svitasheva S. N., Petrenko I. P., Wong H., Xu J. B. & Wilson I. H., Study of Excess Silicon at Si3N4/Thermal SiO2 Interface Using EELS and Ellipsometric Measurements J. Electrochem. Soc. 146(2), pp. 780-785, 1999. Fuks D., Kiv A., Maximova T., Bibi R., Roizin Ya. & Gutman M., Computer model of the trapping media in microFLASH memory cells, Journal of Computer-Aided Materials Design, 9, pp. 21-32, 2002. Yen C., Tsai W. J. & Lu T. C., Novel operation Schemes to Improve Device Reliability in a Localized Trapping Storage SONOS-type FLASH Memory, Electron Devices Meeting, IEDM '03 Technical Digest. IEEE International, pp. 7.5.1-7.5.4, 2003. Roizin Y., Yankelevich A. & Netzer Y., Novel technique for data retention and Leff measurements in two bit microFLASH memory cells, AIP Proceedings, USA, pp. 181-185, 2001. Wang T., Tsai W. J., Gu S. H., Chan C. T., Yeh C. C., Zous N. K., Lu T. C., Pan S. & Lu C. Y., Reliability models of Data Retention and readDisturb in 2-bit Nitride Storage FLASH Memory Cells, Electron Devices Meeting, 2003, IEDM '03 Technical Digest. IEEE International, pp.7.4.17.4.4, 2003. Fuks D., Kiv A., Roizin Ya., Gutman M., Bibi R. & Maximova T., The nature of HT Vt shift in NROM memory transistors, IEEE Transactions on Electron Device, 53, pp. 304-313, 2006. Fuks D., Kiv A., Roizin Ya., & Gutman M., Computer simulation and experimental study of retention of SONOS device, Computational Electronics, 5, pp. 49-52, 2006. Avichail-Bibi R., Kiv A., Maximova T., Roizin Y. & Fuks D., Behavior of injected electrons in high-k dielectric layers, Materials Science in Semiconductor Processing, 9, pp. 985-988, 2006. MEDICI. User Guide, Version 2003.12. Synopsys, December 2003. Janai M., Data Retention, Endurance and Acceleration Factors of NROM Devices, IEEE International Reliability Physics Seminar, USA, pp. 502503, 2003. Zisman P., Roizin Ya. & Gutman M., Vt drift of cycled two bit per cell microFLASH cells, SSDM Proceedings, Tokyo, Japan, pp. 228-230, 2003. Rudnikov T., Ostrovsky N., Fuks D., Kiv A., Bibi R., Roizin Ya. & Gutman M., Behavior of injected charges in dielectric layers of memory devices. Proceedings of the 2nd International Conference on Information Technologies and Management, Riga, Latvia, pp.69-74, 2004. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

74 Computational Methods and Experimental Measurements XIII [15]

[16]

Avichail-Bibi R., Fuks D., Kiv A., Roizin Y. Maximova T., Gouternakht A. & Shtermer V., Software for simulation of retention loss in memory transistor, Computer Modeling & New Technologies, 10(2), pp.30-39, 2006. Mott N. F. & Devis E. A., Electron processes in non-crystal materials, Clarendon Press, Oxford, 1979.



75

Electron band structure and properties of disordered semiconductor compound alloys D. Alexandrov1, K. S. A. Butcher2 & T. L. Tansley2 1 2

Department of Electrical Engineering, Lakehead University, Canada Physics Department, Macquarie University, Australia

Abstract A new metric system called the electron metric system, having a basic metric constant, is introduced. The connection between the electron metric system and the external metric system is defined. The symmetry relationships of the multinary semiconductor compound alloys are defined according to the electron metric system. The basic metric constant is found on the basis of a diatomic tetrahedral cell. The electron wave vector in the new system is found and the electron energy states are determined. Correlation is shown between the electron energy and the electron wave vector in the multinary crystal. The LCAO electron band structures of InxAl1-xN are presented. The phenomenon tunnel optical absorption is investigated in InxAl1-xN, in InxGa1-xN, in InN containing oxygen and in non-stoichiometric InN. It is found the optical absorption edges begin in energies much lower than the energy band gaps due to this phenomenon. Existence of excitons of the structure is shown in these semiconductors and it is found that the peaks of the PL spectra correspond to annihilation energies of these excitons. Keywords: semiconductor compound alloys, optical properties.

1

Introduction

The relatively recent observation of 0.7 eV photoluminescence for InN, and of absorption features near this energy have been the subject of a number of recent papers [1–3]. It has been proposed that the low energy features indicate a 0.7 eV band-gap. However, the material had long been held to have a much higher band-gap of 1.9 eV. The large difference between these values is not presently understood. The Moss-Burstein effect does not explain the variation seen for WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070081

76 Computational Methods and Experimental Measurements XIII material of low carrier concentration, and Vegard’s law indicates that oxygen levels in the higher band-gap material are insufficient to account for the difference [4]. Sample inhomogeneity offers a strong possible explanation. The grown conditions are known to form non-stoichiometric indium nitride during the growth of nitride films. One now seeks to investigate the possibility that these low energy features arise as growth artifacts, due to an alloy formed by non-stoichiometric InN containing both single In substitutions on N sites and single N substitutions on In sites. A new metric system called the electron metric system is introduced in this paper. The connection between the electron metric system and the external metric system is defined. The symmetry relationships of the multinary semiconductor compound alloys are defined according to the electron metric system. The electron wave vector in the electron metric system is found and the electron energy states are determined. Correlation is shown between the electron energy and the electron wave vector in the multinary crystal. LCAO electron band structure of InxAl1-xN is presented. The phenomenon tunnel optical absorption is investigated in InxAl1-xN, in InxGa1-xN, in InN containing oxygen and in non-stoichiometric InN. It is found the optical absorption edges in these semiconductors begin in energies much lower than the energy band gaps due to this phenomenon. Existence of excitons of the structure is shown in these semiconductors and it is found that the peaks of the PL spectra correspond to annihilation energies of these excitons.

2

Metric system and symmetries in multinary crystal

The existing metric system in the solid state physics is defined on the basis of positions of the ions building the crystal lattice. One can call it the metric system of the external observer or simpler external metric system. It does not account for the electron interactions. We will define the metric system in the multinary crystal in terms of electron interaction because it is the basis of the determination of the electronic and optical properties of solid state. One can call it the electron metric system. The definition will be done on the basis of the following assumptions: i) Every quasi-elementary cell (defined in [5] for multinary crystal) is built by points, which are identical with the corresponding points in all other quasi-elementary cells of the multinary crystal in term of electron propagator (The quasi-elementary cell doesn’t contain any other points.); ii) The quasi-elementary cell is electro-neutral; iii) The atomic substitutions in multinary crystal save the valences of the corresponding atoms; iv) The electrical charges are of point type and they are concentrated exclusively in the nodes of the crystal lattice; v) One electron approximation has place. Weyl’s metric system will be used [6, 7], i.e. the length is defined by



l = l0 exp(α∫Φi dxi)

77 (1)

where l0 is metric constant, α is proportional factor, Φi is component of the electric field, and xi is the coordinate in the external metric system (i=1,2,3). The integration takes place between two points of the external system and l is the corresponding distance in the electron metric system. One will define the metric length in the multinary crystal in term of change of electron energy when the electron moves in the electrical field of the crystal lattice formed by the electrical fields of the nuclei, i.e. l = l0 exp(α∫Єi dxi)

(2)

where Єi = Єi(xi) is the corresponding component of the electrical field strength. The coordinate li (i = 1, 2, 3) in the electron metric system can be defined using (2). The integration in (2) between two nodes of the crystal lattice having equal electron charges is equal to zero, i.e. l = l0. Using this result one will define l0 to be the distance in the electron metric system between nearest neighbouring atoms of the crystal lattice having equal electron charges. Let us consider that we have tetrahedral quasi-elementary cell of a multinary semiconductor compound alloy and that this cell is built by two sorts of nuclei – cationic and anionic. Also let us consider at this point that the cationic sub-lattice is built by atoms of one chemical element, and to consider the same for anionic sub-lattice (however both cationic element and anionic element are different atoms). Due to the distribution of the valence electrons the cationic atom has charge +|Z|, and the charge of the anionic atom is -|Z|. According to the assumption iv) given above these charges are concentrated exclusively in the nodes of the corresponding sub-lattices. The tetrahedral quasi-elementary cell contains cationic atom having charge +|Z| and anionic atom having charge -|Z|. The determination of the metric constant l0 for this tetrahedral cell will be done and one-electron approximation will be considered (Fig.1). (Only part of the diatomic tetrahedral cell is given in Fig.1. However this part is enough to represent the interactions and the corresponding lengths due to the symmetry of the tetrahedral cell. The electro-neutrality of the tetrahedral cell is saved.) The one-electron Schrödinger equation is given by [-ћ2∆/2m + e2( |Z|2/(4 rAB ) - |Z|2/(2 rAC) -|Z|2/(2 rBC) - |Z|/(2 rA) - |Z|/(2 rB) + |Z|/rC)] ψ = E ψ (3) where e is electron charge, m is electron mass, ψ is wave function in one-electron approximation, and E is electron energy. This equation is invariant in term of the metric system. Let’s consider that the electron metric system takes place, i.e. rAB = l0 Because the following equality is valid


78 Computational Methods and Experimental Measurements XIII cationic nucleus A

cationic nucleus B

electron rAB

|Z|/2

|Z|/2 rB

rA rAC

rC

rBC

-|Z| anionic nucleus C Figure 1:

Part of the tetrahedral cell containing all possible distances. ξB

∫Є dξ = 0

(4)

ξA where ξA and ξB are the positions of the atoms A and B in the external metric system, and the integration in (4) takes place in the external metric system as well. Let’s consider the positions of the electron and of the atom C in the external metric system to be ξe and ξC respectively. One has found ξC rAC = l0 exp(α∫Є dξ) = l0 exp(-3αe|Z|/2) ξA Using (4) and (5) one has found rBC = l0 exp(3αe|Z|/2) Let’s designate

ξe

∫Є dξ = β ξA Using this designation and expressions (4) and (5) one has found rA = l0 exp(αβ) rB = l0 exp(-αβ) rC = l0 exp(-αβ) exp(-3αe|Z|/2) WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

(5)


79

Considering the expressions for the distances found above the Schrödinger equation becomes {-ћ2∆/2m + e2 [ |Z|2/(4 l0) - |Z|2/(2 l0 exp(-3αe|Z|/2) ) - |Z|2/(2 l0 exp(3αe|Z|/2)) – |Z|/(2 l0 exp(αβ) ) - |Z|/(2 l0 exp(-αβ) ) + |Z|/ (l0 exp(-αβ) exp(-3αe|Z|/2) ) ]} ψ =Eψ (6) The metric constant l0 is found as solution of (6) in the electron metric system assuming that ψ is ortho-normal wave function l0 = e2 /(Emin – H0) [( |Z|2/4 - |Z|2/(2 exp(-3αe|Z|/2) ) - |Z|2/(2 exp(3αe|Z|/2) ) - |Z| H-β /2 - |Z| Hβ /2 + |Z| Hβ / ( exp(-3αe|Z|/2) )]

(7)

Where the matrix elements H0, H-β and Hβ are as follows H0 = H-β = and Hβ =

(8)

It is important to mention that integration for finding β has a place in the external metric system, but the matrix elements are determined in the electron metric system. β has continuous values within the tetrahedral cell and these values are equivalent for all tetrahedral cells having equal |Z|, i.e. the matrix elements H-β and Hβ depend only on |Z|. Emin is the minimum of the electron energy in tetrahedral cell having |Z|. The authors believe there are other methods for determination of the metric constant l0. The distances between two nearest neighbouring cationic is l0, also the distances between two nearest neighbouring anionic is l0 as well. Important conclusions can be made: i) Tetrahedral cells containing two atoms and having the same values of |Z| have equal l0 or in a multinary semiconductor compound alloy the different quasielementary tetrahedral cells containing two atoms and having equal values of |Z| have equal values of l0, i.e. the lengths of the tetrahedral edges are equal in the electron metric system. ii) The electron wave function ψ(l) and the potential function U(l) of the multinary crystal have symmetry in the electron metric system that is the same as the symmetries of both the electron wave functions ψ(x) and the potential functions U(x) of the binary constituents in the external metric system if |Z| remains constant for every quasi-elementary tetrahedral cell. (The binary constituents are building the multinary crystal.) The binary alloys InN, GaN and AlN have values of |Z| - 1.56, 1.48 and 1.36 respectively. (The calculations are performed on the basis of the polarities of these alloys given in [8]). The average value is |Zav| = 1.47. It means that it can WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

80 Computational Methods and Experimental Measurements XIII be considered that |Z| = 1.47 for the quasi-elementary tetrahedral cells of InxGa1-xN, of InxAl1-xN and of GaxAl1-xN with error not greater than 7.5%, and the corresponding tetrahedral edge l0 in the electron metric system. However this conclusion is made on the basis of the independent binary alloys that are possible binary constituents. The authors assume that a detail investigation may give constant value of |Z| without theoretical error. The purpose of the electron metric system in this paper is to define the symmetry of the multinary crystal in term of crystal lattice and the corresponding symmetries of both the electron wave function and the potential function. Further development of this metric system and its application in the quantum mechanics of solids goes beyond the scope of this paper and it is subject of other author’s papers.

3

Electron wave vector and electron states in multinary crystal

The change of the metric system requires a new approach in the determination of the electron wave vector because the electron wave length has to be determined in the corresponding metric system. This problem will be solved on the basis of the following assumptions: i) The quasi-elementary tetrahedral cell is electro-neutral; ii) The electrons belonging to both the conduction band and the valence band have energies E greater than the potential energy U of the nuclei of certain quasi-elementary tetrahedral cell; Using these assumptions and the result in [9] one can write (kxi and kl are the electron wave vectors in the external metric system and in the electron metric system respectively, and n is positive integer number): - for the external metric system xB (1/ћ ) ∫ {2m [E – U(xi)]}1/2 dxi = |xB – xA| kxi = n π xA (|xB – xA| is tetrahedral edge) - for the electron metric system l0 (1/ћ ) ∫ {2m [E – U(l)]}1/2 d l = l0 kl = n π 0 The equations (9) and (10) give |xB – xA| kxi = l0 kl = n π

(9)

(10)

(11)

As a matter of fact the distance |xB – xA| determines the length of the tetrahedral edge of certain quasi-elementary cell in the external metric system (the tetrahedral edges of different quasi-elementary cells having |Z| have different lengths in the external metric system, and equal lengths in the electron metric system). The equalities (11) have important meaning – they are basis of the following conclusions: WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


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1) The electron wave length λl in the electron metric system stays unchanged when the electron is moving through different quasi-elementary tetrahedral cells and λl = 2 l0 / n

(12)

2) The electron wave length λxi in the external metric system changes when the electron is moving through different quasi-elementary tetrahedral cells. 3) The Bloch’s theorem is satisfied in the electron metric system. 4) The number of electron states remains unchanged in different metric systems. (It means one can determine the electron states in the electron metric system and they give the corresponding states in the external metric system.) 5) Considering that a sub-lattice of the multinary crystal is built by different sort of atoms one must expect that the electron energy corresponding to certain electron state depends on the quasi-elementary cell, i.e. the formula derived in [10] is valid. E(r) = ∑q δ (r – Rq) E(q)

(13)

where E(q) is electron energy in quasi-elementary cell having radius-vector Rq, and r is radius-vector. Both Rq and r can be determined in both metric systems, however the application of the electron metric system in calculation of the Hamiltonian matrix elements goes beyond the scope of this paper and the calculations of the corresponding matrix elements will be done in the external metric system furthermore according to the conclusion 4), which is given above, the determination of the electron states can be made in the electron metric system, and the calculation of the corresponding electron energies can be done in the external metric system (conclusion 5). Calculations of LCAO electron energies for the electron state k=0 (i.e. point Γ of the electron band structure) will be presented for InxAl1-xN. (Details of these calculations are given in [5, 10].) Each type of quasi-elementary cell forms sector υ of the corresponding electron band structure (υ = 1, 2, 3, 4, 5). In terms of both the optical absorption and the photoluminescence the energy band gap of InxAl1-xN has to be determined as energy differences between Γυc1 (the bottom of the conduction band of sector υ) and Γυv15 (the top of the valence band of sector υ). The results of the calculations are given in Fig.2 for InxAl1-xN. The energy levels Γυc1 and Γυv15 are determined by taking the energy of the vacuum as being equal to zero. The energy difference Eυg = (Γυc1 - Γυv15) gives the energy band gap of sector υ. The shifts of the boundaries of the energy band gaps in Fig.2, and the corresponding energy intervals are due to defects in the crystal lattices of InxAl1-xN. The nature of these shifts is different from the nature of the shifts of the boundaries of the energy band gap described in [11, 12]. The same approach is used for determination of the energy states in InxGa1-xN [10], in InOyN1-y [5] and in non-stoichiometric InN [13].


82 Computational Methods and Experimental Measurements XIII Sector 5

Sector 4

Sector3

Sector2

Sector 1

*2c1 = -4.93

*1c1 = -4.92

*2v15 = -11.21

*1v15 = -11.22

Evac = 0

_f! *5c1 = -5.05

*5v15 = -7.04

*4c1 = -5.07

*4v15 = -7.02

_i!

Figure 2:

4

*3c1 = -5.43

*3v15 = -8.80

The energy band gap of InxAl1-xN. The energies Γυc1 and Γυv15 are shown (the sectors are υ=1, 2, 3, 4, 5). The shapes of parts of the electron wave functions corresponding to both the initial i > and final statef > are given and the allowed optical absorption transition is shown.

Electron and optical phenomena in semiconductor compound alloys related to InN

4.1 Excitons of the structure in InxAl1-xN, in InxGa1-xN, and in InOyN1-y An exciton of the structure in InxAl1-xN is formed by an electron occupying state Γ3c1 and a hole occupying state Γ4v15 (Fig.2). This exciton state Γ3c1 ↔ Γ4v15 is defined as the ground exciton state for this type of exciton in InxAl1-xN. Using the method given in [10] the hydrogen like energy level En for the ground state (n = 1) is found to depend on the ratio between neighbouring Al cationic and neighbouring In cationic surrounding the quasi-elementary cells of sectors 3 and 4 of Fig.2. It is found En varies in the interval 0.765 –0.778 eV and these energies are in agreement with the optical transitions corresponding to the experimental photoluminescence spectra of samples containing interface layers InxAl1-xN - ~ 0.77 eV reported in [2] and ~ 0.8 eV given in [3]. As a mater of fact En determines the optical transitions connected with photon radiation due to annihilations between the electrons and the holes belonging to the exciton Γ3c1 ↔ Γ4v15 in InxAl1-xN. Excitons of the structure in InxGa1-xN are investigated in WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


83

details in [10]. It is found that the hydrogen like energy level En varies over the interval 0.50 – 0.82 eV and these energies are close to the experimental results about the photoluminescence spectra of In-rich regions of InxGa1-xN reported in [14–16]. The same type of exciton in InOyN1-y is investigated in details in [5]. It is found that the hydrogen like energy level En varies over the interval 0.84 – 1.01 eV, and that it determines the optical transitions connected with photon radiation due to annihilations between the electrons and the holes belonging to these excitons in InOyN1-y. The photoluminescence rates in InxAl1-xN, in InxGa1-xN and in InOyN1-y depend on the concentration of the corresponding excitons of the structure, and this in turn depends on the technological circumstances forming this alloys. 4.2 Tunnel optical absorptions in semiconductors related to InN The tunnel optical absorption in InxAl1-xN, in InOyN1-y, in InxGa1-xN is investigated in details in [5]. The basis of this phenomenon is the overlapping between the electron wave function | i > of the initial electron state and the electron wave function of the final electron state | f > in term of optical transition. Part of the graphics of figure 2 provides the electron wave functions of the initial state i > and of the final state f > for InxAl1-xN (it is important to note that the state i > is an electron state, it is not a hole state). It is found [5] the lengths (in the one-dimensional case) of the regions where the matrix element of the optical absorption has non-zero values due to the penetrations of the electron wave functions into the barriers. The length for InxAl1-xN is determined to vary in the interval 4.70 – 4.72 Angstrom depending on the number of In and Al atoms surrounding the sectors 3 and 4 (Fig.2), the length for InxGa1-xN is determined to vary in the interval 4.61 – 4.64 Angstrom, and the length for InOyN1-y is found to vary in the interval 1.98 – 2.00 Angstrom. The conclusion is made [5] that the optical absorption transitions Γ4v15→ Γ3c1 for InxAl1-xN and Γ3v15→ Γ4c1 for InxGa1-xN are allowed and they give the optical absorption edges and the corresponding energy band gaps (for InxAl1-xN Eg=Γ3c1-Γ4v15 and for InxGa1-xN Eg=Γ4c1-Γ3v15). The similar conclusions are made for InOyN1-y as well. It is important to note that if the distances between the corresponding quasielementary cells (forming the sectors) are longer than the lengths of the regions determined above the optical absorption transitions given above will not be allowed). The optical absorption rates depend on the number of Γυv15→ Γξc1 pairs for InxAl1-xN, for InOyN1-y and for InxGa1-xN in the corresponding primitive super-cells that are connected with the technological circumstances (growth conditions) forming the layers InxAl1-xN, InOyN1-y and InxGa1-xN. The energy Eg for InxAl1-xN is found to vary in the interval 1.58 – 1.62 eV, the energy Eg for InOyN1-y is found to have a small variation around 1.19 eV, and the energy Eg for InxGa1-xN is found to vary in the interval 1.40 – 1.56 eV. These energies (especially the energy Eg ~ 1.19 eV for InOyN1-y) are close to the optical absorption edge that shows the optical transmission data for the Ioffe sample W431. The tunnel optical absorption in non-stoichiometric InN:In is investigated in details in [13]. It determines energy band gap Eg = 0.2 eV. (However the single substitutions In atom on N site do not reduce the energy band gap to zero.) WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


5

Conclusion

The development of the electron metric system and its application in calculations of the electron band structure of multinary crystal has an important impact over the investigation of the disordered atomic systems. In fact, the calculations of the Hamiltonian matrix elements can be done in this metric system and the authors believe these calculations to become easier and more accurate. The observed phenomena excitons of the structure and tunnel optical absorption can be used in design of semiconductor devices on InN and related alloys.


[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

K.S.A. Butcher, M. Wintrebert-Fouquet, K.E. Prince and T.L. Tansley, Materials Research Society Symposium Proceeding 743, 707 (2003). J. Wu, W. Walukiewicz, K. M. Yu, J. W. Ager III, E. E. Haller, H. Lu, W. J. Schaff, Y. Saito, Y. Nanishi, Applied Physics Letters 80, 3967 (2002). V. Yu. Davydov, A. A. Klochikhin, V. V. Emtsev, S. V. Ivanov, V. A. Vekchin, F. Bechsted, J. Furthmuller, J. Aderhold, J. Graul, A. V. Mudryi, H. Harima, A. Hashimoto, Y. Yamamoto, E. E. Haller, Physica Status Solidi. (b) 234, 787 (2002). T.L. Tansley, presentation at the First Internat. InN Workshop, Fremantle, Australia, Nov. 2003. D. Alexandrov, K.S.A. Butcher, M. Wintrebert-Fouquet, Journal of Crystal Growth 269, 77 (2004) F. London, Zeit. F. Phys. 42, 375 (1927) L. O’Raifeartaigh, The Dawning of Gauge Theory, (Princeton university press, Princeton, 1997), part I W.A. Harrison, Electronic Structure and the Properties of Solids, Dover Publ. Inc, (1989) A.S. Davydov, Quantum Mechanics (Pergamon press, Don Mills, 1965) D. Alexandrov, Journal of Crystal Growth, 246, 325 (2002) H. Fritzsche, Journal of Non-Crystal Solids, 6, 49 (1971) A. Efros, B. Shklovskii, Electronic Properties of Doped Semiconductors (Springer-Verlag, Berlin – Heidelberg 1984) D. Alexandrov, K.S.A. Butcher, T. Tansley, Journal of Crystal Growth 288, 261 (2006). V. Yu. Davydov, A. A. Klochikhin et al., Phys. Stat. Sol. (b), 229,1 (2002) V. Yu. Davydov, A. A. Klochikhin et al., Phys. Stat. Sol. (b), 230, 4 (2002) V. Yu. Davydov, A. A. Klochikhin et al., International Workshop on Nitride Semiconductors, 22 – 25 July 2002, Aachen, Germany, p.133 (2002)



85

Fast calculation of the dimensioning factors of the railway power supply system L. Abrahamsson & L. Söder Electric Power Systems, Royal Institute of Technology (KTH), Sweden

Abstract Because of environmental and economical reasons, in Sweden and the rest of Europe, both personal and goods transports on railway are increasing. Therefore great railway infrastructure investments are expected to come. An important part of this infrastructure is the railway power supply system. Exactly how much, when and where the traffic will increase is not known for sure. This means investment planning for an uncertain future. The more uncertain parameters, such as traffic density and weight of trains, and the further future considered, the greater the inevitable amount of cases that have to be considered. When doing simulations concerning a tremendous amount of cases, each part of the simulation model has to be computationally fast – in real life this means approximations. The two most important issues to estimate given a certain power system configuration, when planning for an electric traction system, are the energy consumption of the grid and the train delays that a too weak system would cause. In this paper, some modeling suggestions of the energy consumption and the maximal train velocities are presented. Two linear, and one nonlinear model are presented and compared. The comparisons regard both computer speed and representability. The independent variables of these models are a selection of parameters describing the power system, i.e.: power system technology used on each section, and traffic intensity. Keywords: railway, traction system, power supply, energy consumption.

1 Introduction During the last decades, the railway has in many countries experienced a renaissance. The main reasons for the expansion of the railway are environmental and economical. This, in turn, has increased the interest in railway grid research. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070091


Figure 1: Overview of the main purpose of the paper.

When making a decision about the future railway power supply system, possible under-investments or over-investments need to be estimated in an appropriate manner. The costs of over-investments are simply put: the price difference between the investment cost of the “too strong” and the “sufficiently strong” power system configurations. Costs that are related to under-dimensioning are somewhat more subtle, though. Some examples will be mentioned in the following. While a voltage drop in the ordinary power system would cause occasional disconnections of customers, it would in the power supply system of the railway simply cause the trains to run slower. The slowing down of trains do, however, immediately lead to costs – either by lower incomes due to reduced competitiveness on the transportation market – or for greater voltage drops, disturbed, or even modified time tables. With the trend of increasing energy prices in mind, power losses might be as important to study as delays caused by low voltage when looking into the future. Therefore, the initial focus on underinvestment costs will be set to train velocities limited by the power system, as well as differences in energy use – losses will vary, as also train power demand – between different power system solutions. The objective of this paper is to present an idea of how to pick out relevant information of the outputs of a basic simulation method (TTS), and by presenting the input variables to an approximator (TTSA) estimate these relevant outputs, see Figure 1 (Main results). Relevant outputs are here chosen to be the maximal train throughput velocity, as well as the corresponding energy consumption of the system, for a given train traffic and electric power supply system. The main ideas behind TTS, its model, as well as TTSA, and its accuracy and ability to generalize the results, are presented. A further developed TTSA (box C in Figure 1) approximating main results (box E) from TTS (box B), is planned to be used in a future investment planning tool. Planning will be done for several years ahead, allowing investments to be done stepwise. This planning tool should in the end be able to manage a huge amount of uncertain variables, such as: train types, train weights, locomotive types, energy prices, increasing or decreasing demand, taxes, the economical situation, and so forth. All the possible combinations of realizations of variables like these cannot be simulated (box B), because it would demand too much time. The aim is to use results from a limited number of TTS simulations to determine parameters for the WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


87

TTSA model in box C. With this approach, the much faster TTSA model can be used for a large amount of scenarios.

2 Train traffic simulator (TTS) The aim of TTS is to, as accurate as possible, simulate a certain traffic and a certain infrastructure. 2.1 Modeling 2.1.1 Electrical and mechanical power, for each time step In this part of the paper, the system of equations to be solved for each time step is presented. A great part of the modeling is the same as in [1, 2], and therefore only additional and updated equations will be presented here. The maximal tractive force of an Rc locomotive is a function of the catenary voltage, U , and the velocity of the train that it is hauling, v. The function can be expressed as a polynomial Fmotor,max = c1 + c2 U + c3 v + c4 U v + c5 v 2 + c6 v 2 U + c7 v 3 + + c8 v 3 U + c9 v 4 + c10 v 4 U + c11 v 5 + c12 v 5 U + c13 U 2 + 2

2 3

(1)

2 4

+ c14 U v + c15 U v + c16 U v , where the parameters can be obtained from data sheets [3] using least squares fitting. The motor force can be modeled as Fmotor,max − KJ · 4 · a · (1 + ζ) if Braking = 0 Fmotor = (2) 0 if Braking = 0 where ζ is the slippage ratio [4], KJ is related to rotational inertia [4], a is the acceleration, and Braking is a variable that will be described in the method part. The adhesive tractive force between train and rail,   madh,drive · g · 0.161 + 7.5 for dry rail 44+3.6v (3) Ftract,adh = 3.78  madh,drive · g · 23.6+v for wet rail where g is the gravitational constant, and madh,drive is the mass on the driving axles of the train [4]. The effective tractive force Ftract = min {Fmotor , Ftract,adh }

(4)

because it is indifferent how strong the engine is if there is no grip [4]. The train resistive force due to mechanical and air resistances, Fair,mech = A + B · v + C · v 2 WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

(5)

88 Computational Methods and Experimental Measurements XIII where A, B, and C are train dependent [4]. The resistive force due to grades Fgrades = m · g · incl

(6)

where incl is the inclination of the track. The total resistive force is simply Fres = Fair,mech + Fgrades . And

a=

(7)

Ftract −Fres madh,drive +(m−madh,drive )(1+H)

if Braking = 0

abrake

if Braking = 0

(8)

where m is the total train mass [4], H is the relative factor accounting for rotational inertia of the unbraked wheel sets [4], and abrake will be described in the braking part. The mechanical power of the motor Pmotor = Fmotor · v · (1 + ζ)

(9)

[4]. The electrical power demand PD = Pmotor

(10)

which implicates an assumption of a lossless motor. The DC voltage of the motor, Udiα

= Emax · min 1,

v vbase

· min 1,

U

(11)

U14kV

which is a wiser modeling when allowing greater voltage drops [5]. In converter stations with several converters of the same kind, PG , QG , and Q50 (the total active, reactive, and 50 Hz side reactive power generations, respectively) can simply be divided by the number of converters, #conv , in order to give the proper U and ψ values [6]. G G Xqm · #Pconv Xqg · #Pconv 1 ψ = − arctan − arctan 2 2 3 (U m ) + Xqm · #Q50 (U g ) + Xqg · #QG conv

(12)

conv

Both Um and Ug are assumed to be at nominal voltage levels constantly. The phase shift on the 50 Hz side of the converter due to train power consumption is, according to [6], θ0 = θ50 −

X50 · PG 1 · arctan . m 3 (U )2 − X50 · Q50


(13)


89

2.1.2 The braking The problem would have grown tremendously if the driving behavior would have been subject for optimization with respect to time. In order to avoid the time dimension of the problem, the driver is assumed to be aggressive. When accelerating, he/she does it as hard as ever possible. Braking, on the other hand, demands a more sophisticated modeling in order to be able to stop at the train stations. In order to determine the shortest braking distance for a given initial velocity, a small optimization problem was set up. The braking acceleration was constrained to 0 a −0.85 m/s2 , while v was constrained to be nonnegative. The figure 0.85 m/s2 is due to comfortability reasons and according to [7], there is never any problem achieving that retardation level. The position, p, was constrained to lie within 0 p pstation . The remaining constraints were as follows pt = pt−1 + vt ∆t + at

∆2t 2

pt pt−1 pt − pstation −vt · M vstart + at ∆t , t = 1 vt = vt−1 + at ∆t , t ∈ {2, 3, ..., tmax } z= pstation − pt

(14)

∀t

where index t ∈ {1, 2, ..., tmax } is time step index, M is a large number (in this case 1000), and ∆t is the time step length. The value of tmax must like pstation be big enough for the train to have time and place to stop. The second constraint remedies the phenomenon in discretized time that the position might be reduced when traveling forward if the retardation turns bigger than suitable for the problem. The third constraint ensures that the train stops at the station. The objective z is minimized. This LP problem is solved for all integer velocities vstart between 61 and 160 km/h, 61 because it is the lowest that gives feasible solutions, 160 because Rc locomotives rarely go faster. The braking accelerations are stored as a discrete function abrake [vstart , t] to be used in TTS later on, paired with the critical braking distance, dbrake . However, these critical distances does not really form a smooth function of vstart because of the time discretization. Therefore, the trend is extracted by least squares fitting into a sixth grade polynomial of vstart . 2.2 The method The Newton Raphson method of [1, 2] did soon turn out to be too weak for these nonlinear models. In TTS, whose working idea is illustrated in Figure 2, Matlab mainly does the bookkeeping. GAMS is a powerful optimization program that is WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Figure 2: Main ideas of the work flow of TTS. used for solving the system of nonlinear equations for each time step. The objective function of the GAMS program is the sum of the squared two-norms of the power flow error vector and the reactive power flow error vector, but could be chosen differently. In order to reduce the computation complexity of the system of nonlinear equations, GAMS is programmed to use vt−1 , the prior time step velocity, rather than the in the presented models assumed vt . Since vt−1 is a parameter, and vt would, if introduced, be a variable depending on at that in turn depends intricately on several other variables, the simplification is obvious. Moreover, v will normally not change that much between two small consecutive time steps. Doing the same with, e.g., U would be harder to justify – especially for weak BT power systems. The consecutive time steps are thus connected by at = SOE (pt−1 , vt−1 ) vt = vt−1 + at ∆t ∆2 pt = pt−1 + vt ∆t + at t 2

(15)

where SOE denotes the system of equations in section 2.1.1. The TTS time table (unidirectional traffic intensity) remains as in [1, 2], i.e., a train is let loose in the start every nth minute, and once the entire track is filled up with trains the forthcoming train to let loose gets a label. When the labeled train reaches its final destination, the TTS simulation halts. The precalculated braking schedules that are described in the model section are used as follows. Before solving the equations, TTS checks if there is time to brake for any train. It is considered time to brake when 0 < pstation − pt < dbrake (vt ) + vt ∆t , where vt ∆t is a sort of insecurity factor due to the discrete time model. If there is time for a train to brake, then the parameter Braking = 0 is raised one step and vt is stored as vbrake . This is done for bookkeeping of train braking time and choosing an appropriate braking schedule. Matlab thereafter checks trains with parameter Braking > 0. The parameter is raised one step for the forthcoming time step. The braking acceleration is then determined by at =

1 (abrake [vbrake , Brakingt ] + abrake [vbrake , Brakingt ]) 2


(16)


91

Figure 3: The assumed dependencies of a piece of the railway grid.

because the braking schedules are only computed for integer velocities. Five minutes after that at has reached zero, Braking is set back to zero and the train can start accelerating, aiming for the next station on the track.

3 The approximator (TTSA) The aim of the TTSA is to construct a method of retrieving reliable results fast, much faster than would be possible in TTS. 3.1 The dependency models The inputs (Box A in Figure 1) chosen to be variables in this paper were: the catenary type (BT or AT [1, 2]), the option of having an HV transmission line (130 kV 2-phase line [1, 2]), and the traffic intensities quantified into three measures. The choice of catenary type, AT , as well as the HV line option, HV , are binary variables that tells whether the power supply system has AT catenaries or not, and HV transmission line or not, respectively. There is one such AT and HV pair for each section in the railway power system. The traffic intensity is described by ET (Et (vt,T )), the average velocity, and VT (Et (vt,T )), the variance of the mean velocities, both of them taken over all trains (subscript T ) during a certain time window (subscript t). The third traffic intensity measure is N oT r, the number of trains. There is one such E (v), V (v), and N oT r triplet for: each train type, each track section, and both traveling directions. Apart from the electromechanical properties, also the classification of trains as either “accelerating” or “speedmaintaining” is a train type demarcation. A train is classified as “accelerating” on a specific section if it is stopped on the section border before entering the section. This is indeed a crude measure, and a future TTSA should be able to handle trains stopping several times in each section. Track section borders are defined by the converter station locations in the non-HV cases, e.g., in Figure 4 there are two sections. In the simple example of this paper, mixed traffic is not studied, and all trains have the same Et (vt,T ) such that the variances can be neglected. Moreover, all trains are “accelerating” so no separation between “accelerating” and “speedWIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

92 Computational Methods and Experimental Measurements XIII maintaining” trains is needed. The general dependency model assumed for TTSA in this paper is illustrated in Figure 3. There are two kinds of outputs of TTSA, see Figure 3. First, the average energy consumption of the power system. Second, vmax , the maximal attainable velocity for each section, direction, and train type. In this paper, all outputs are assumed to depend on all inputs. Since the power flows are not local in the power system, especially not when using AT and/or HV, the energy consumption is hard to separate into section components. It is however tempting to, in a future improved TTSA, model at least the vmax s as functions merely of the traffic in the concerned section. Three different methods of modeling how the assumed outputs depend upon the assumed inputs are proposed. The input and output assumptions are illustrated

in Figure 3. The first model, M 1, assumes a linear dependency outi = bi + in w , where i is output index, k is input index, and w k i,k i,k and bi are k

i − parameters to be determined by minimizing the mean square of the error, out outi , in an optimization program. The second model, M 2, is also linear; with the same inputs, outputs, and parameters. In M 2, however, the parameters are determined by the Matlab Neural Network (NN) Toolbox algorithm trainb (batch learning). In other words, M 2 is a single layered neural network with inputs ink , outputs outi , and have |i| neurons with linear transfer functions. The third model, M 3, is a nonlinear NN with two layers. The first, “hidden”, layer has tanh transfer functions, and the second (output) layer has linear transfer functions. According to the theory [8, 9], this kind of network can be used as a general function approximator, given sufficient neurons in the hidden layer. The hidden layer was chosen to have 3 neurons, the linear (output) layer naturally has |i| neurons, and the network is trained using the trainbr (Bayesian regularization backpropagation) algorithm with an error goal of 10−5 . Both the ink and outi data are normalized to lie in the interval [−1, 1] before training and testing the approximators. Furthermore, the 128 TTS results are separated into one randomly chosen training set of size 32, and one remaining test set. The figure 128 comes from four different power system configurations and 32 different train departure periodicities n leaping from 6 to 20, from 21 by 1.5 to 30, and from 33 by 3 to 60 (minutes). The main difference by minimizing the mean square error (MSE) by an optimization algorithm compared to a NNs algorithm is that one can perform a fewer amount of iterations in a wiser way in the latter case. Of course, that leads to a non optimal MSE, but hopefully a model that better generalizes the behavior of the system studied.

4 Numerical example on a test system 4.1 System configuration The system that the TTS has simulated is a three city test system (Figure 4), mainly using the same ideas as in [1, 2]. In the test system the converters are of type WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


93

Q48/Q49 [10]. In the HV case, there are converter stations situated in City1 and City2 with 6 converters in each. In the non-HV case, there are three converter stations – one in each city – with 4 converters in each. In the test system, trains denoted “F1 Mixed” in [4] are used. ∆t was set to 0.1 minutes in TTS. The stretch “City Distance” was 50 km.

Figure 4: An illustration of the test system. The inclinations on the test system are inspired by the stretch between Lule˚a and Bastuträsk. The hight curve is measured from a graph [11] every 6.25th km whereafter the inclinations are calculated. The rail is assumed to be dry. The slippage ratio, ζ, is for simplicity set to be zero, and 4 · KJ is assumed to be 10750 N, a typical figure [4]. Moreover, the 50 Hz sides of the converters stations have no load angle θ50 = 0◦ like in [1, 2]. Finally, H is assumed to equal zero. 4.2 Results and conclusions In Figure 5 there are two plots of selected TTS data: one for the strongest system with the lightest load, and one for the weakest system with heaviest load simulated. The variables v, PD , QD , and U are shown for the labeled train while driving the first 100 km. v is a part of box E in Figure 1, the others of box D. v is normalized by 150 km/h, PD and QD are normalized by 5 MW/MVAr, and U by 16.5 kV. The inclination of the track (a part of box A) is included to show its influence, incl is scaled so that −1, 1 corresponds to −10, 10 per mill. The remainder of the section is devoted to TTSA. As one would have expected, the linear NN tends to coincide with the GAMS solution when training it for WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

94 Computational Methods and Experimental Measurements XIII AT=0,HV=0,n=6 1

0.8

0.8

0.6

v P

D

0.4

Q

D

U incl incl=0

0.2 0

0

20

40

60

80

100

normalized data

normalized data

AT=1,HV=1,n=60 1

0.6

v P

D

0.4

Q

D

U incl incl=0

0.2 0

0

20

40

km

60

80

100

km

Figure 5: A selection of TTS data. thousands of iterations and when the MSE goal is set small. The errors of the mean energy of such an approximator were about 10−5 for the training set and 10−4 for the test set. The errors of the maximal velocities were negligible compared to the approximation errors of the energy. For a less trained linear NN, the errors on the output vector are more equally spread and the generalization is slightly better. The approximation errors for the nonlinear NN are evenly spread, with norms similar to the linear cases. In a minor modification of the GAMS program, certain w and b were set to zero in order to determine whether the maximal velocity of a section could be modeled as depending only on the traffic on that very section. This assumption would be reasonable because of the voltage control on the section borders. Simulations shown just a slight decrease in approximator performance, so one could conclude that the traffic of neighboring sections do not affect each other much. The computation times for making one estimation might be of interest. Since it is unfair comparing different programs, the both NNs are compared. By the usage of tic and toc in Matlab, the approximation calculation times turned out to be less than 3 · 10−2 in the linear network and less than 1.6 · 10−2 seconds on average on an IBM X40 portable computer.

References [1] Abrahamsson L., Basic Modeling for Electric Traction Systems under Uncertainty, UPEC (Universities Power Engineering Conference) 2006, 2006. [2] Abrahamsson L., Operation Simulation of Traction Systems, to be published in the Comprail 2008 preceedings, presented orally at Comprail 2006, 2006. [3] Jansson, N., Electrical Data for the Locomotive Types Rc4 and Rc6 (original title in Swedish), TrainTech, Solna, 2004. [4] Lukaszewicz, P., Energy Consumption and Running Time for Trains, Ph.D. Thesis, Division of Railway Technology, KTH, Stockholm, 2001. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


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¨ [5] Ostlund S., Personal communication, 2 February 2007, Professor at the division of Electrical Machines and Power Electronics, KTH, Stockholm. [6] Olofsson M., Optimal Operation of the Swedish Railway Electrical System, Ph.D. Thesis, Electric Power Systems, KTH, 1996. [7] Friman E., Personal communication, 10 January 2007, M.Sc. E.E. at the Swedish Railway Administration (Banverket), Borlänge. [8] Gurney, K., An Introduction to Neural Networks, CRC Press, p. 78, 2003. [9] Matlab online help, Neural Network Toolbox, www.mathworks.com / access / helpdesk / help / toolbox / nnet / backpro4.html, 9 March 2007. [10] Kols, H., Frequency Converters for Railway Feeding (original title in Swedish), BVH 543.17000, Banverket, 2004. [11] Banverket, Track profile Lule˚a-Borlänge (original title in Swedish), 2007.




97

A formulation of a multi-wave elastodynamic infinite element K. Kazakov Department of Structural Mechanics, VSU “Luben Karavelov”, Sofia, Bulgaria

Abstract In this paper, a formulation of a multi-wave elastodynamic four and eight node infinite element is proposed. Such a kind of element is appropriate for multi-wave soil-structure interaction problems. The formulation follows the standard infinite element formulation steps which are the same as for the Finite element method after mapping the infinite element domain to a finite element domain. It is shown that if only one wave function is used (only one frequency) the proposed multi-wave elastodynamic infinite element is reduced to a singlewave elastodynamic infinite element. The mapping and the Lagrange isoparametric shape functions for a 2D axisymmetric four and eight node multi-wave elastodynamic quadrilateral infinite element are also given. The basic aspects of the continuity along the finite/infinite element (artificial boundary) line are discussed in brief. In this type of model such a line marks the boundary between the near and the far field of the model. The formulation is appropriate for wave propagation problems only. Keywords: wave propagation, infinite elements, finite element method, soil-structure interaction.

1

Introduction

This section is devoted to the review of the historical background of infinite elements from the original works to the latest contribution. Exterior domain scattering problems appear in many engineering fields such as electrodynamics, magnetics, fluid and thermal analyses and so on. Wave propagation in an elastic infinite media and scattering of waves on bodies in a fluid which extends WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070101

98 Computational Methods and Experimental Measurements XIII infinitely are of particular interest. The difficulty in such problems when numerical methods are used arises from the unbounded domain that has to be discretized. Many suggestions and ideas for the treatment of the exterior domain have been presented and discussed in a number of research papers for the period of three decades. The exterior (infinite) domain cannot be discretized with standard finite elements. A lot of efforts have been spent in the development of new infinite element formulations and techniques, based on the changes of the polynomial shape functions with trigonometric or exponential forms. In soil-structure interaction problems one possible approach is just to truncate the computational domain at some distance (line) away and to impose some “appropriate” boundary conditions. Such a line is called an “artificial” boundary. In this case viscous, absorbing or transmitting boundary conditions can be used. It is evident that the computational efficiency depends then on the localization of the “artificial” boundary and the type of the boundary conditions. In many cases such techniques give acceptable results. In soil-structure interaction problems that approach is known as the Substructural approach.

2

Earlier infinite element method works

The infinite element method was introduced about three decades ago in the original work of Bettess [5]. Then this method have been developed and refined in many works. Between them are the works of Pissanetzky on the magnetostatics and Kim on the magnetic field problems. The original Bettess formulation is similar to the finite element concept except the element domain. In this formulation the domain extends toward infinity in one direction. The corresponding shape functions are analytically integrable over the element. Such an infinite element is directly applicable to the Finite element method. The mapped infinite elements were developed by Zienkiewicz et al. [18]. These elements are based on polynomial shape functions, attenuating in the infinity. The mapping technique assures direct integration. A mathematically precise variational formulation of infinite elements has only recently been proposed [15].

3

Practical classification of infinite elements

From a practical point of view infinite elements can be classified into five classes: • classical infinite elements, • decay infinite elements, • mapped infinite elements, • elastodynamic infinite elements and • Wave envelope infinite elements. The origin of the idea and the development of every one of the above classes are difficult to be dated. The first class is based on the original so-called “classical” formulation of the infinite elements. In the decay, infinite element WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


99

decay functions from different mathematical types are used. The mapped infinite elements are developed by usage of mapping techniques. These techniques map the infinite domain of the element into a finite. The obtained element is similar to the classical finite element. The latest researches of infinite elements are devoted to the development of the elastodynamic infinite elements and the wave envelope infinite elements. In some cases the last two classes can be considered as a special combination of the mapped and decay infinite elements.

4

Multi-wave elastodynamic infinite element formulation

The displacement field in the multi-wave elastodynamic infinite element can be described in the standard form by a finite number of shape functions based on wave propagation functions [6] as n

m

u( x, z ,ω ) = ∑∑ N iq ( x, z ,ω )p iq (ω ) i =1 q =1

or

u(x, z, ω ) = N p (x, z, ω )p(ω )

(1)

where N iq ( x, z ,ω ) are the standard shape displacement functions, p iq (ω ) are the generalized coordinates associated with N iq ( x, z ,ω ) , n is the number of

nodes for the element and m is the number of wave functions included in the formulation of the infinite element. For horizontal wave propagation the shape displacement functions can be expressed as:

N iq (x, z, ω ) = L(η )Wq (ξ , ω ) where

(2)

Wq (ξ , ω ) are wave functions related to a horizontal propagation (in ξ

()

direction) and L η is a Lagrange polynomial. The infinite element domain is shown in fig.1. By taking into account only the real parts of the wave functions, the equations of the wave propagation can be written as

 iω  −αξ Re Wq (ξ , ω ) = cos ξ e  cs  or

 iω  −αξ Re Wq (ξ , ω ) = cos ξ e c   p  where c s and

c p are the velocities of the S-waves and P-waves respectively.


(3)


z

η ξ Ωi

x Figure 1:

The infinite element domain.

If the number m is known these functions can be collected preliminary as

ReW (ξ ) =

 iϖq  −αξ 1 m ξ e Aq cos ∑ m q =1  cs 

ReW (ξ ) =

 iϖq  −αξ 1 m Aq cos ξ e ∑  c m q =1  p 

or

where

ϖ

is the lowest frequency and

ω = ϖq .

(4)

The coefficients Aq can be

written as:

Тq Aq =

∫ 0

 iϖ q  Re W (ξ , ω ) cos  ξ dt c s  

(5)

Now so-called united shape function can be written as

Ni (x, z ) =

m

∑

N iq (x, z, ω ) = L(η ) Re W (ξ )

q =1

(6)

Finally equation (1) can be expressed as n

u( x, z , t ) = ∑ N i ( x, z , t )p i (t )

(7)

i =1

or

u (x, z ) = N p (x, z )p WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

(8)


101

The procedure described by the equations (4)–(6) can be treated as a superposing procedure based on a finite number of terms, where the real components of the wave functions ReWq (ξ , ω ) are called “preliminary” shape functions. The coefficients

Aq are generalized coordinates with only one component,

corresponding to the node i. It is easy to be shown that in the case of only one wave function used in the computational model, only one frequency, the proposed multi-wave elastodynamic infinite element is reduced to a single-wave elastodynamic infinite element. It can be treated as a special case. Then or

5

N i (x, z ) = N iq (x, z, ω ) = L(η ) Re W (ξ , ω )

(9)

N i (x, z ) = Niq (x, z ) = L(η ) Re W (ξ ) .

(10)

Two dimensional mapped infinite element

The next step is to generate mapping to map the infinite element domain to a finite domain and vice versa. Mapping functions and the Lagrange isoparametric shape functions for a 2D axisymmetric four node quadrilateral mapping infinite element and for a 2D axisymmetric eight node quadrilateral mapping infinite element can be written as follows. 5.1 2D axisymmetric four node quadrilateral mapping infinite element 5.1.1 Mapping functions

x = xI

(1 −η )(− ξ ) + x (1 + η )(− ξ ) + 1 x (1 +η )(1 + ξ ) +

1− ξ 1 (1 −η )(1 + ξ ) + xL 2 1− ξ y = yI

1− ξ

J

2

K

1− ξ

(11)

(1 −η )(− ξ ) + y (1 + η )(− ξ ) + 1 y (1 + η )(1 + ξ ) +

1− ξ 1 (1 −η )(1 + ξ ) + yL 2 1− ξ

J

1−ξ

2

K

1− ξ

5.1.2 Lagrange isoparametric shape functions (displacement field) 1 1 u = u I (1 − η ) ξ 2 − ξ  + u J (1 + η ) ξ 2 − ξ  +    4  4 1 1 2 2     + u K (1 + η )1 − ξ  + u L (1 − η )1 − ξ    4   4 WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

(12)

(13)

102 Computational Methods and Experimental Measurements XIII 5.2 2D axisymmetric eight node quadrilateral mapping infinite element 5.2.1 Mapping functions

x = xI

(1 −η )(− 1 − ξ −η ) + 2x

y = yI

(1 −η )(− 1 − ξ −η ) + 2 y

(1 −η ) + x (1 +η )(−1 − ξ +η ) + 2

1− ξ 1− ξ ( )( ) (1 + η )(1 + ξ ) 1 1 +η 1 + ξ 1 + xM + xL 2 1− ξ 2 1−ξ J

1− ξ

K

(14)

(1 −η ) + y (1 +η )(−1 − ξ +η ) + 2

1− ξ 1−ξ (1 + η )(1 + ξ ) 1 (1 + η )(1 + ξ ) 1 + yL + yM 2 1−ξ 2 1− ξ J

1− ξ

K

(15)

5.2.2 Lagrange isoparametric shape functions (displacement field)

(

)

1 1 u = uI (1 −η )(1 − ξ )(− 1 − η − ξ ) + uJ 1 − η 2 (1 − ξ ) + 2 4 1 1 + uK (1 + η )(1 − ξ )(− 1 + η − ξ ) + uL (1 + η ) 1 − ξ 2 + 2 4 1 + uM (1 − η ) 1 − ξ 2 2

(

(

)

(16)

)

5.3 Mass and stiffness matrices The stiffness and mass matrices can be given in a standard of the Finite element method form as     [k e ] = [B ]T [D ][B ]dΩ e [me ] =  [N ]T [N ]dΩ e  I and (17)   Ωe Ω  e 

∫

∫

where [N ] are the shape functions and the vectors {Bi } in the matrix [B ] are written as

 ∂N i  {Bi } =  ∂∂Nx  i  ∂y

    

or

 ∂N i   ∂ξ 1 Bi = [J ]   ∂N i  ∂η 

{ }


      

(18)


103

[ ]1

where J is the Jacobian matrix which defines the geometrical mapping and can be written as

 ∂ξ  [J ]1 =  ∂∂ξx   ∂y

∂η   ∂x  ∂η  ∂y 

(19)

The domain differential dΩe must also be written in terms of the local coordinates as

dΩe = dxdy = [J ]dηdξ

(20)

{Bi }

and dΩe , which involves the mapping functions, the element stiffness and mass matrices may not be computed with standard Gaussian procedure.

Subject to the evaluation of

6 Continuity through finite and infinite elements The continuity through finite and infinite elements can be enforced in exactly the same way as between two finite elements in the case they have the same degrees of freedom and the degree of approximation. A sketch of the boundary between finite and infinite elements is given in fig. 2.

ηf

ηi

Ω ie

ξi Ω

fe

Figure 2:

7

ξf Sketch of the boundary between finite and infinite elements.

Conclusion

This paper proposes a formulation of a multi-wave elastodynamic infinite element, appropriate for multi-wave soil-structure interaction problems. In the case of only one included wave function, the proposed multi-wave elastodynamic infinite element is reduced to a single-wave elastodynamic infinite element. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

104 Computational Methods and Experimental Measurements XIII The formulation follows the standard infinite element formulation steps which are the same as in the Finite element method after the mapping the infinite domain to a finite domain of the element. Also the mapping and the Lagrange isoparametric shape functions for a 2D axisymmetric four node multi-wave elastodynamic quadrilateral infinite element and for a 2D axisymmetric eight node multi-wave elastodynamic quadrilateral infinite element are given.

References [1] [2] [3] [4] [5] [6] [7] [8] [9]

[10]

[11]

[12] [13] [14]

Wolf J.P, Song C, Finite-element modeling of unbounded media. Willey, 1996. Bathe K.J., Finite element procedures in engineering analysis. PrenticeHill, 1982. Bancov B. P., Palova J. B., “The Finite Element Method in Structural Mechanics”, UACG, 1996, (in Bulgarian). Ungless R.F., “Un infinite elements”, M.A. Sc. Dissertation, University of British Columbia, 1973. Bettess P., “Infinite elements”, International Journal for Numerical Methods in Engineering, 1978; 11:54-64. Yan Ch. B., Kim D.K., Kim J.N., “Analytical frequency-dependent infinite elements for soil-structure interaction analysis in two-dimensional medium”, Engineering Structures 22 (2000); 258-271. Wolf J.P., “Soil-Structure Interaction Analysis in Time Domain”, Englewood Cliffs, N.J.: Prentice-Hill, 1988. Kazakov K, “A model of one-dimensional wave propagation in homogeneous media”, Journal Stroitelstvo, 6/2004, 12-14. (in Bulgarian) Kazakov K, “An adequate computational model of the infinite soil for Soil-Structure Interaction Problems”, Proceedings of X Congress of applied mechanics, Bulgarian Academy of Science, Varna, Bulgaria, 2005. Kazakov K, “On the model of elastodynamic infinite element for the far field in Soil-Structure Interaction problems”, Proceedings of National conference with international participation of VSU “Liuben Karavelov”, Sofia, 2005, (in Bulgarian). Kazakov K, “A model if FEM type elastodynamic infinite element for Soil-Structure Interaction”, Proceedings of the 4th International Conference on New trends in Static and Dynamics of structure 20-21 October 2005. Bratislava, Slovakia. Kazakov K., “Continuity between Finite and Infinite Elements, Along Artificial Boundary in Soil-Structure Interaction Problems”, Proceedings of the Jubilee Conference in UACG 2007, Sofia, Bulgaria Madabhushi S. P. G., “Modeling of deformations in Dynamic SoilStructure Interaction problems”, VELACS, 1996. Park K. L., Watanabe E., Utsunomiya T., “Development of 3D elastodynamic infinite elements for Soil-Structure Interaction Problems”, WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


[15] [16] [17] [18]

105

International Journal of Structural Stability and Dynamics, Vol. 4, No. 3 (2004) 423-441 Cecot W., Demkowicz L., Rachowicz W., “A three-dimensional infinite element for Maxwell’s equations”, TICAM Report 00.20 Gerdes K., “A review of Infinite Element Method”, Journal of Computational Acoustics Zhao Ch., Valliappan S., “A Dynamic Infinite Element for Threedimensional Infinite Domain Wave Problems”, International Journal for Numerical Methods in Engineering, Vol. 36, (1993), 2567-2580 Zienkievicz O. C., Bando K., Bettess P., Emson C., Chiam T. C., “Mapped Infinite Elements for Exterior Wave Problems”, International Journal for Numerical Methods in Engineering, Vol. 21, (1985), 12291251



Section 2 Experimental and computational analysis



109

Influence of the collision speed and angle of a bullet: experimental reconstruction of bullet configuration and FE-analysis H. Sakamoto1, T. Hiwatashi1, T. Yamaguchi2 & M. Yamamoto3 1

Mechanical System Division, Graduate School of Science and Technology, Kumamoto University, Japan 2 Forensic Science Laboratory, Kumamoto Pref. Police H.Q., Japan 3 Faculty of Engineering, Kumamoto University, Japan

Abstract In this research, the bullet-firing test was carried out at different velocities and collision angles of the bullet. The influence of its velocity and angle on the deformation shape after firing was discussed by comparison with experiment and simulation. The velocity of the bullet was chosen in several kinds of speed ranges at 80m/s–250m/s. As for the angle of incidence with the collision object, four kinds of angles 90° (head-on collision), 45°, 30° and 60° were set. The deformation shape was measured by a 3D coordinate measurement machine and reconstructed with 3D-CAD based on the 3D digital data. The bullet hole and deformation of the polycarbonate board caused by the bullet's collision with the object was also examined. In addition, the collision simulation of the bullet was carried out using LS-DYNA, and these analytical results were compared with the 3D digital data of the bullet. The comparison of the FE-simulation analysis results of the experiment enabled the quantitative evaluation of the collision deformation. Keywords: collision, bullet, deformation analysis, LS-DYNA, 3D-CAD.

1

Introduction

Gun crime has been increasing year on year. Collecting the bullet after firing is needed to solve these affairs [1]. It is often that a bullet is the only evidence in the affair. Therefore, studies about a bullet deformed by a collision are very WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070111

110 Computational Methods and Experimental Measurements XIII important. For example, rifling marks on a bullet after firing give information about the type of gun used; in the same way, deformation of the target will indicate the firing position and firing angle. The purpose of this research is to clarify the influence of collision velocity and collision angle on deformed shapes of bullets and targets. In addition, we simulated the collision of the bullet by FEM (Finite Element Method) and attempted a comparison between experimental and simulated results.

2

Experiment

2.1 Materials Three types of bullet with full metal were used. The external appearances are shown in Figure 1. These are called “round nose bullet” and the lead core is coated with copper film. Bullets were shot with 38Auto and Wheel (S&W) gun (Figure 2) at a polycarbonate plate (PC), which was used as a shield. PC board of 10mm in thickness was used. 2.2 Experimental procedure The distance from gun to target was about 6.4m shown in Figure 3. The velocity of the bullet was measured by a laser ballistic chronograph. The bullet velocity was controlled by adjusting the amount of explosive powder in the cartridge case. The 25AUTO and 380AUTO shown in Figure 1 (b)(c) were used in headon collision tests. The 38SPL shown in Figure 1(a) was used in oblique collision tests. The collision angles are θ=30°, 45° and 60°to the target shown in Figure 4.

3


Gun

3.1 Test firing The upper part in Figure 5 shows the bullet’s original shape and deformed shapes in the case of head-on collision after firing. The corresponding bullet holes of the target plate were shown in the lower part. These configurations were precisely measured by a 3D coordinate measurement machine and were reconstructed by using a 3D-CAD system as digital data as shown in Figure 6. The reconstructed models of the deformed bullet were done to compare experimental and analytical results quantitatively. Moreover, by using the digital data, the deformed bullet’s dimensions, for example, cross sectional areas and volumes, can be evaluated precisely. Figure 7 shows the relationship between the bullet velocity and deformation rate. This indicates that the deformation rate’s peak occurs at 150 m/s (corresponding to a kinetic energy of 100J). The reason comes from the fact that the compressive strength of the bullet overtakes the compressive strength of polycarbonate in the case of the low collision velocity (less than 150 m/s). The relationship between velocity and bullet hole are given in Table 1. The depth of WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


111

11.35

11.25

17.3

the bullet hole becomes bigger with increasing collision velocity. The diameter of the bullet also shows a similar trend. The depth of the bullet hole is comparatively small due to energy absorption by bending deformation of the polycarbonate. However, in the case of high velocity (more over 150 m/s), the plastic deformation region was limited to the contact region only. Therefore, both bullet hole diameter and compressive plastic deformation of the bullet have been decreased at 251 m/s [2,3].

φ9

φ6.3

φ9

38SPL

25AUTO

380AUTO

Figure 1:

Geometry and dimension of bullets.

Figure 2: 38Auto and Wheel (S&W) gun.

Target

Laser ballistic chronograph Gun

Figure 3:

gun

Firing experimental layout.

Polycarbonate board

θ

center

Overhead view Figure 4:

Overhead view of firing experiment system.



Original 発射前 shape

79.2

103.5

126.4

150.6

228.1

holes

250.9 m /s

Figure 5:

Bullet’s shapes and bullet’s 380AUTO,V=79.2~250.9m/s).

(head-on

collision,

Figure 6:

Reconstruction bullet shape by using 3D measurement and 3DCAD.

Deformation rate (％)

30

20

10 ●　380AUTO 0 0

Figure 7:

100 200 Velocity of bullet (m/s)

300

Relationship between bullet velocity and deformation rate (head-on collision).

3.2 Numerical simulation Finite Element models for the bullet and polycarbonate plate are shown in Figure 8. As analysis conditions, the collision velocities and the angles in experimental conditions were used. In addition, the spinning of the bullet, material’s strain rate WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


113

dependence, Mach number and friction were considered [3–6]. The main analytical constants of the material properties used in the simulation experiment are shown in table 2. Table 1:

The dimensions of bullet holes.

Collision velocity (m/s) 79 104 126 151 228 251

Type of bullet

380AUTO (head-on collision)

Diameter of bullet Depth of bullet hole hole (mm) (mm) 7.2 0 8.3 0.1 10.3 0.3 11.4 0.4 12.2 3.9 11.7 4.0

(a) Target model (polycarbonate board) Figure 8:

(b) Bullet model (380AUTO)

An example of the analysis model using an FEM simulation.

Table 2:

Analytical constants of material properties.

Cu 3

Mass density (kg/m ) Young’s modulus (GPa) Yield stress (MPa) Poisson’s ratio

8.93×10 132 196 0.34

Pb 3

11.337×10 17.2 60 0.44

PC 3

1.2×103 2.06 100 0.3

Comparison between analysis results and experimental ones in oblique collision (collision angle 30°) are shown in Figure 9. In the case of not considering the strain rate dependence, as the bullet deformation for high velocity is overestimated, the simulation was performed by using the strain-strain relation obtained under different velocity compression tests. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

114 Computational Methods and Experimental Measurements XIII The deformation shapes obtained from simulation were in good agreement with experimental results for the collision angle of 30°. However, for the angle of 60-degrees, the calculated model and the experimental results were a little different. Although the contact between the bullet bottom part and PC-plate in the experiments was recognized, this contact mark on the PC plate was not observed in the simulation. The same tendency is observed for the collision angle of 45-degrees. This is probably caused by the definition of friction between the bullet and polycarbonate and the bonding condition between lead and copper of the bullet used in the simulation. Results of head-on collisions are not presented here, however experimental results and simulation results show very good correlation.

(a) Analysis results in collision angle 30° (V = 79～218m/s)

(b) Experiment results in angle 30° (V = 79～218m/s) Figure 9:

Comparison of analysis results with experiment ones (38SPL, collision angle: 30°).

Cracks on copper coated bullets were recognized in oblique collision experiments. These cracks caused by the rifling mark in spirals have little influence on the deformed shape of the bullets shown in Figure 9. However, the non-coated lead bullet breaks with cracks cased along rifling marks in the headon collision experiment (Figure 10). Figure 11 shows an example of the simulation result for the non-coated lead bullet [7,8]. From comparison with both figures, it is found that the deformation and fracture behaviour can be evaluated by simulation quantitatively.

4

Conclusion

The effect of collision velocity and collision angle on bullet deformation was discussed and performance of a protection board made of polycarbonate was estimated by experiment and FE Analysis. The results obtained are summarized as follows. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Figure 10: Fracture under head-on collision for non-coated bullet.

115

Figure 11: Fracture analysis under head-on collision for noncoated bullet.

(1) For the low collision velocity region, polycarbonate plate undergoes bending deformation. In the case of high collision velocity, the plastic deformation region of polycarbonate plate was limited to the bullet’s contact region only. (2) For a collision angle smaller than 30o, the simulation results show good agreement with experimental ones under consideration of the train rate dependence. (3) When the collision angle is bigger than 45°, the simulation results and the experimental ones are a little different because of the friction between the bullet and polycarbonate, and the bonding condition between lead and copper. (4) In collision simulations, the rifling marks on full metal shield bullets of firing have little influence on the deformed shape of bullets. However, the crash and fracture phenomenon occur along the cracks ignited by the rifling marks.

References [1] [2] [3] [4] [5] [6]

K. Kashiwatani, Forensic science for Investigation, 1983. N. Cristescu: Dynamic Plasticity, North-Holland Publishing Company, 1967. T. Borvik, O.S. Hopperstad, T. Berstad, M. Langseth, A computational model of viscoplasticity and suctile damage for impact and penetration, Eur. J. Mech. A/Solids 20, pp.685-712, 2001 T. Negishi, T. Ogura, T. Masumoto, T. Goto, K. Fukuoka, and J. Syono: Mat. Sci. 20,1985. R. Barauskas & A. Abraitiene, Computational analysis of impact of impact of a bullet against the multiplayer fabrics in LS-DYNA, International Journal of Impact Engineering, 2005 K. Sakino, Strain Rate Dependence of Dynamic Flow Stress of Aluminum Alloy 6061 at Very High Straom Rates, Original Paper, vol.54 No.12 pp.1301-1306 WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

116 Computational Methods and Experimental Measurements XIII [7] [8]

H. Kurtaran, M. Buyuk & A. Eskandarian, Ballistic impact simulation of GT model vehicle door using finite element method, Theoretical and Applied Fracture Mechanics 40,pp.113-121, 2003 P.A. Vityaz, V. Roman: Proc.13th Int. J. Mach. Tool Des. & Res., 1972.



117

Influence of the cross-section shape on the lateral torsional buckling capacity D. Djurić-Mijović & M. Trajković Faculty of Civil Engineering and Architecture, University of Niš, Republic of Serbia

Abstract The bifurcation-load of a bending girder, the ideal lateral torsional buckling moment MKI, is of great importance for the verification of stability. The solutions in the literature exist almost exclusively for double-symmetrical cross sections. The whole problem considering the lateral torsional buckling is quite complex as the analytical solutions for determining the lateral torsional buckling strength exist in closed-form only for the most simple cases, e.g. for the case of a simply supported double symmetric beam of a constant cross-section under uniform moment. However, for most cases, in order to obtain the buckling load, numerical or approximate solutions are required. This article examines the problem of elastic lateral torsional bucking of simply supported, monosymmetric I and T-beams under two different transverse load cases. Correct and approximate Euro Code 3 (EC3) approaches for obtaining elastic lateral torsional buckling capacities of monosymmetric I and T-beams were investigated for each load case. Solutions were obtained in terms of the easily evaluated degree of beam monosymmetry, β f , beam parameter, K , and monosymmetry parameter, rz. The results obtained are graphically presented and compared. It was found that approximation formulae for rz given by Kitipornchai and Trahair are much more accurate compared to the correct formulae than the approximation formulae proposed by EC3. T-beams are considered as a special case of monosymmetric I cross sections. In conclusion, the authors recommend utilisation of different types of cross sections for different types of loading and length of the beams. The result can be significant savings in material as well as increased stability of the structure regarding lateral torsional buckling of monosymmetric cross section girders. Keywords: lateral torsional buckling, monosymmetric cross section, EC3, thinwalled open cross section, monosymmetry parameter. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070121


1

Lateral torsional buckling

Lateral torsional buckling is a mode of structural failure in which one or more members (beams, trusses….) of a frame suddenly deflect and twist out of the plane of loading (Trahair [2]). If it is not prevented in the right way, lateral torsional buckling may reduce the load-carrying-capacity of the structure because members subjected to a flexure have much greater strength and stiffness in the plane of the loading (major principal axis) than in the minor principal axis. The implication is that structural members are subjected to a failure by lateral torsional buckling before they even reach their full in-plane capacity. This problem is frequent and it is perceived at slender members or structures, and so, of great importance in the design of steel structures. Also, it often occurs in the construction phase. This kind of structure deformation does not concern only individual members, but also occurs in rigid-jointed structures, where continuity of rotations between adjacent members causes them to interact during buckling. Lateral torsional buckling of a member, as a combination of lateral buckling and torsional buckling, is a case in which transverse displacements of a member, out-of-plane deflection v and in-plane deflection w, occur in combination with rotation ϑ, around its major axis.

Figure 1:

Elastic bending and buckling.

A beam, which is bent in its stiffer principal plane, may buckle out of that plane by deflecting laterally out-of-plane v and rotating (twisting) ϑ, as shown in the Figure 1(b). These deformations are interdependent. For example, a twist rotation ϑ of the beam will cause the in-plane bending moment My to have an out-plane component Myϑ as shown in Figure 2(a), which will cause lateral deflections v. Conversely, lateral deflections v will cause the moment My to have a torque component Myv' as shown in Figure 2(b), which will cause twist rotations ϑ. Lateral torsional buckling is resisted by combinations of the bending resistances EIzd2v/dx2 and -EIyd2w/dx2 and the torsional resistances GJtdϑ/dx and -EIwd3ϑ/dx3 (Trahair [2]). WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Figure 2:

2

119

Interdependence of v and ϑ .

Buckling capacities of monosymmetric I-cross sections

2.1 Description of the system, boundary conditions and load cases In this paper the single span beams, simply supported in-plane ( w 0 = w L = 0 , w ′0 , w ′L ≠ 0 ,) and simply supported out-of-plane are considered. The ends of the beams, that are simply supported out-of-plane, are fixed against out-of-plane deflections and twist rotations v 0 = v L = ϑ 0 = ϑ L = 0 , but are unrestrained against minor axis rotations v ′0 , v ′L (so that v ′0′ = v ′L′ = 0 ) and against warping displacements proportional to ϑ′0 , ϑ′L (so that ϑ′0′ = ϑ′L′ = 0 ), Figure 3(c). The beams are assumed to be perfectly straight and untwisted before loading and are exposed to the loads that initially cause deflections only in the plane of loading. It is also assumed that the direction of the load remains unchanged during buckling. Within this paper the central concentrated load, Figure 3(a) and uniformly distributed load case, Figure 3(b) were examined.

(c) Figure 3:

System, boundary conditions and load cases.

2.2 Differential equations Differential equilibrium equations (Trahair [2]) for a simply supported beam with monosymmetric cross-section under a uniform moment induced by equal and opposite end moments M, are shown in Figure 4. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

120 Computational Methods and Experimental Measurements XIII (EI z v′′)′′ + (M y ϑ)′′ = 0 (EI W ϑ′′) ′′ − (GI t ϑ′) ′ + M y v ′′ − (M y rz ϑ′) ′ = 0

with the boundary conditions: v 0,L = v′0′,L = ϑ 0,L = ϑ′0′, L = 0

Figure 4:

(1) (2) (3)

Simply supported beam under end moments.

Terms (M y rz ϑ′)′ are associated with the monosymmetry property rz, called the monosymmetry parameter of the cross section or “Wagner effect”, which is given by 1 z( y 2 + z 2 )dA − 2z S (4) rz = Iy

∫ A

zS is the distance between the centroid and shear centre and Iy is the major axis moment of inertia. Many authors have developed various approximate formulas for rz. One of them was developed by Trahair and Kitipornchai (Kitipornchai and Trahair [4]) rZ ≅ 0.9h S (2β f − 1)[1 − (I z / I y ) 2 ] (5) in which Iz/Iy is the ratio of the second moment of area of the section about minor and major axes respectively, hs is the distance between shear centres of the flanges. This formula has very good accuracy and is used for practical purposes. 2.3 Buckling capacities of monosymmetric I cross-sections In the case of double symmetric cross sections the tensile and compressive bending stresses are equal and the flanges are at the same distance from the shear centre. That leads to increasing buckling resistance caused by tensile stresses balanced by increasing buckling action caused by compressive stresses. However, this balance is upset in monosymmetric beams. When such a beam twists during buckling, the longitudinal bending stresses exert a torque around the axis of twist of the member. This torque causes an effective change in the torsional stiffness from GIt to (GIt+Myrz), in which My is the major axis moment and rz is the monosymmetry parameter. As a consequence, when the larger flange is in compression the buckling resistance is increased, and opposite (Anderson and Trahair [5]). The elastic critical moment, MKI, of the monosymmetric I-beam under uniform moment is a solution of eqns (1) and (2) and is given as: 2    rZ K  2 rZ K  π    + (6) M KI = EI Z GI t 1 + 4β f (1 − β f )K +  h   hS  L  S    WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Figure 5:

121

Monosymmetric I-cross sections.

where EIz is the minor axis flexural rigidity, GIt is the torsional rigidity, L is the length of the beam. The next parameters are K , the beam parameter, also called the torsion parameter and βf, the measure of monosymmetry or degree of monosymmetry: K=

π 2 EI Z h S 4GI t L2

2

;

βf =

I fc ; 0 ≤ β f ≤ 1.0 I fc + I ft

(7)

where hs is the distance between shear centres of the flanges. The values of K for practical beams are in the range between 0.1 and 2.5 with low values representing long beams and/or compact cross-sections and high values corresponding to short beams and/or slender cross-sections.

Various cross sections depending on βf.

Figure 6:

These two parameters, βf and K , enable easily visualization of the monosymmetric I-beam. 2.4 Euro Code 3 - buckling formulae The general formula according to Euro Code 3 for a beam of uniform cross section, elastic critical moment for lateral torsional buckling, MKI is given by M KI

 π 2 EI Z  k = C1  (kL)2  k W 

 I W (kL )2 GI t   I + π 2 EI + C 2 z g − C 3 z j  Z Z 2

(

)

2

   

1/ 2


  − C 2 z g − C 3 z j  (8)  

(

)

122 Computational Methods and Experimental Measurements XIII in which C1,C2,C3 are factors depending on the loading and boundary conditions, k and kw are effective length factors and these coefficients are given in Annex F of Euro Code 3. 1 zg = za − zs ; z y 2 + z 2 dA (9) z j = zs − 2I y

∫(

)

A

where za is the coordinate of the load application point and zs is the coordinate of the shear centre. For an I-section with unequal flanges it is recommended to calculate the warping section constant, Iw by the following formula 2 I W = β f (1 − β f )I Z h S (10) where hs is the distance between the shear centres of the flanges. Using eqn (8) and zj calculated by eqn (9)2 the correct values of MKI were obtained. Also, the approximate formula for zj was proposed in Annex F of Euro Code 3, for βf >0.5 z j = 0.8(2β f − 1)h S / 2 (11) for βf 1.5 even the double symmetric cross section (βf =0.5) has greater capacity than the T-beam (βf =1.0). Figure 7 shows elastic critical loads, MKI, of monosymmetric I beams under central concentrated load applied at top flange, shear centre and bottom flange. Figure 9 shows MKI of the monosymmetric I beams under uniformly distributed load applied at the top flange, shear centre and bottom flange. The Kitipornchai and Trahair approximate formulae for calculation of rz are reported to be of good accuracy in a range between 0.1 ≤ βf ≤ 0.9 (deviation from correct formulae up to 6%). In the case of highly monosymmetric cross sections for which β f approaches 0 or 1,0 the deviation is up to 18%, as opposed to the EC3 approximate formulae for which the deviation is up to 50%, Figure 8.



Bf=0 Bf=0,1

30

Non-Dimensional Mk i

Bf=0,2 25

Bf=0,3 Bf=0,4

20

Bf=0,5 Bf=0,6

15

Bf=0,7 10

Bf=0,8 Bf=0,9

5

Bf=1,0 0 0.0

0.5

1.0

1.5

2.0

2.5

35

Bf=0 Bf=0,1

30

Non-Dimensional Mki

Bf=0,2 25

Bf=0,3 Bf=0,4

20

Bf=0,5 Bf=0,6

15

Bf=0,7 10

Bf=0,8 Bf=0,9

5

Bf=1,0

0 0.0

0.5

1.0

1.5

2.0

2.5

60

Bf=0 Bf=0,1 Bf=0,2 Bf=0,3

40

Bf=0,4 Bf=0,5

30

Bf=0,6 Bf=0,7

20

Bf=0,8 Bf=0,9

10

Bf=1,0

0 0.0

0.5

1.0

1.5

2.0

2.5

K (Beam Parameter)

Figure 7:

Central concentrated load.

60 50 40 EC3 correct

30

EC3 app.

rz

Non-Dimensional Mki

50

20

Kitip.&Wang app.

10 0 0.0

-10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Bf

Figure 8:

Deviations [%] of rz.



10

Bf=0

9

Non-Dimensional Mki

125

Bf=0,1

8

Bf=0,2

7

Bf=0,3 Bf=0,4

6

Bf=0,5

5

Bf=0.6

4

Bf=0,7

3

Bf=0,8 Bf=0,9

2

Bf=1,0

1 0 0.0

0.5

1.0

1.5

2.0

2.5

10

Bf=0

Non-Dimensional Mki

9

Bf=0,1

8

Bf=0,2

7

Bf=0,3 Bf=0,4

6

Bf=0,5

5

Bf=0.6

4

Bf=0,7

3

Bf=0,8 Bf=0,9

2

Bf=1,0

1 0 0.0

0.5

1.0

1.5

2.0

2.5

10

Bf=0

Non-Dimensional Mki

9

Bf=0,1

8

Bf=0,2

7

Bf=0,3 Bf=0,4

6

Bf=0,5

5

Bf=0.6

4

Bf=0,7

3

Bf=0,8 Bf=0,9

2

Bf=1,0

1 0 0.0

0.5

1.0

Figure 9:

4

1.5

2.0

2.5

Uniformly distributed load.

Conclusions

Influence of the monosymmetric cross section shape on lateral torsional buckling capacity has been investigated in this paper. These results based on EC3, where it was found that the best capacities have cross sections with βf =1.0 (0.9), are different in comparison with previously investigated approaches of RUBSTAHL-KSTAB 2000 (Djurić [1]) and Kitipornchai and Trahair (Kitipornchai and Trahair [4]) where for the same load cases the capacity of the T-beam is much lower then for beams with βf = 0.9 and 0.7.


126 Computational Methods and Experimental Measurements XIII Also it was found that the approximate formulae for the monosymmetry parameter rz given by Kitipornchai and Trahair is more accurate in comparison to the rz correct formulae than the EC3 approximation formulae. After this extensive theoretical and numerical research, it appears necessary to experimentally verify these approaches in order to determine which is the most accurate in describing the behaviour of real models. Using the approximation in respect to the accurate formula for calculation of rz, the calculation time would be significantly shortened, which would be very important for the design of demanding structures. It has been theoretically demonstrated that certain monosymmetric I cross sections possess higher bearing capacity to lateral torsional buckling than the double symmetrical I cross sections. Using the monosymmetric cross sections, considerable savings could be accomplished, and the steel structure weight would be reduced. This is very important for contemporary engineering practice. Experimental investigation is a present occupation of the authors.

Acknowledgement The support from the Ministry of Science and Environmental Protection, Republic of Serbia, through the projects ON144002 and ON144027 is gratefully acknowledged.

References [1] [2] [3] [4] [5] [6] [7]

Djurić D., Influence of the Cross-Section-Shape on the Bifurcation-Loads of Bending Girders, Master thesis, Ruhr University Bochum, Germany, 2002. Trahair, N.S., Flexural-Torsional Buckling of Structures, 1st edn, E&FN Spon (Chapman & Hall) London, 1993. Galambos T.V. Guide to Stability Design Criteria for Metal Structures. 4th edn, John Wiley and Sons, New York, 1988. Kitipornchai, S. and Trahair, N.S. Buckling properties of monosymmetric I-beams. Journal of the Structural Division, ASCE, 106 (941-57), 1980. Anderson, J.M. and Trahair N.S. Stability of monosymmetric I-beams and cantilevers. Journal of Structural Division, ASCE, 98 (269-86), 1972 Wang, C.M. and Kitipornchai, S. Buckling capacities of monosymmetric Ibeams. Journal of Structural Engineering, ASCE 112 (2373-91), 1986. R. Kindmann, Computed-oriented Design of Steel Structures. RuhrUniversity, Bochum, 2001.



127

Computational fluid dynamic modelling and simulation evaluation of the plume evacuation device efficiencies F. Farshad1, H. Rieke1, L. C. LaHaye2 & S. C. Nulu1 1 2

University of Louisiana at Lafayette, USA Vision Pro LLC, USA

Abstract The purpose of our work has been to evaluate the fluid flow dynamics of distal and proximal handheld plume evacuation devices used during LASIK eye surgery using Computational Fluid Dynamic (CFD) Modelling. Fluid flow dynamics studies using CFD simulations were conducted on a proximal plume evacuator, LAHayeSIK™ surgical device, and on the VISX Star S3, which is a distal large volume plume evacuation device. The resulting data was compared and analyzed with experimental data. CFD results show that the proximal plume evacuation system generated a uniform laminar airflow velocity of 0.94 m/s across the corneal surface as compared to 1.3 m/s reported by the distal evacuation system. Flow profiles indicate high shear regions resulting in vortex formations, for the large volume distal evacuator. The CFD simulations conducted to determine the airflow profiles generated by the two surgical plume evacuation devices concur with data obtained from experiments. Flow patterns simulated by the CFD modeling, indicate that the proximal plume evacuation devices generate a gentle laminar airflow profiles over the stromal surface. On the other hand, the distal large volume plume evacuators generate multiple regions of varying air flow velocities contributing to ineffective plume capture. Keywords: computational fluid dynamics, fluent, LAHayeSIKTM, LASIK, plume evacuation, CFD simulation.



1

Introduction

One of the ultimate goals in performing any surgical procedure is to minimize less than desirable outcomes arising from both infectious and noninfectious contaminants entering the surgical field (LaHaye et al., [1, 2]). Although not normally thought of as such, plume smoke is a by-product contaminate of excimer laser surgery. The complexities of plume formation and its rapid dynamic movements both vertically and laterally impose problems that present undesirable outcomes and present health related issues for the surgeon, patient, and nursing staff. In excimer refractive surgery a laser’s accuracy, effectiveness, and reproducibility can be directly affected by how well the surgical operating environment is managed. The authors emphasize that managing the microclimate of the stromal bed during the excimer refractive procedures is the only important avenue where substantial improvements can be made through advancements in design for better outcomes and fewer health risks. Air flow dynamics generated by plume evacuation systems can have direct and indirect influences on refractive outcome. A direct effect of LASIK plume smoke is the masking effect created as the plume hangs just over the ablating stromal bed, blocking subsequent excimer pulses which can cause a measurable difference in the resulting ablation (Duffey, [3]). Researchers contend that the plume particles falling back onto the on the ablating stromal bed creates additional beam masking (Noack et al., [4]) and may be a contributing factor to the “Sands of the Sahara” syndrome (Dell, [5]). Research has demonstrated that plume vapor condensation with precipitation contributes to visible fluid accumulation on the surface of the stromal bed during ablation. This additional regional accumulation of fluid can interfere with beam etching to cause a decrease in transmission of energy to the stroma through increased reflection and absorption of incident laser energy. This resulted in undesirable ablation, such as central islands, “hot and cold” spots, and under corrections (Oshika et al., [7]). Most corporate and physician-based nomograms are based on a certain portion of the excimer beam being blocked by plume particles (Maguen and Machat, [8], Duffey, [3]). Laser manufacturers add pulses to the nomograms based on outcome averages to “compensate” for plume masking attributes of their systems. The “blanket” method of re-mediating the numerous problems associated with plume appears to have little logic. The many potential problems associated with plume should direct one to solving the underlying issue by removing the cause and therefore the effect as opposed to simply adding additional laser pulses in an attempt to compensate. Since the adoption of excimer refractive procedures some 15 years ago, the industry has several alternatives for plume management. The options include; (1) no plume management; (2) operating room ventilation fan generated room air blown across the surgical field; (3) laser integrated distal plume evacuation; (4) laser integrated devices which combine blown air and distal plume evacuation; and (5) handheld proximal plume evacuation devices. Because of the potential health hazards associated with plume, we have chosen to compare systems that



129

are, in theory, designed to only remove plume. Experimental analysis and Computational Fluid Dynamics were used to comprehend the dynamics of the plume generated during the LASIK surgery. Two types of plume evacuation devices, the proximal plume evacuation device – LAHayeSIK™ Surgical System and the distal plume evacuation system – VISX Star S3 are compared for their effectiveness in design to remove plume generated during LASIK surgery. Our CFD results of this study emphasize modern computational techniques like CFD, which can be used with a great effect in determining the best design technique for medical equipment design.

2

CFD simulation of the proximal LAHayeSIK™ plume evacuation system

The flow domain for the proximal plume evacuation system is identified as the path inside and outside the handpiece in the vicinity of the stromal surface, where the plume particles travel under the influence of the plume evacuation force. Figure 1 shows the plume flow domain of the proximal plume evacuation system that is to be modeled.

Figure 1:

Solid model (domain) of the LAHayeSIK™ surgical device.

The solid model of the domain is created in FLUENT’s GAMBIT using the basic geometrical tools such as edges, faces, and volumes. The design of the solid model incorporates all the nuances in the model and the exact measurements of angles and distances. The model is then meshed using various meshing strategies to come up with the best quality mesh. The mesh can be used by FLUENT to solve the numerical equations without any divergence problems. Figure 2 shows the meshed model of the plume evacuation function of the LAHayeSIKTM surgical device. The meshed model from GAMBIT is then set up inside the FLUENT console after performing a grid check for negative volumes and inconsistent meshing. Thus, the solution is setup with the required parameters and boundary conditions and is iterated for convergence with a constant monitoring of the solution using the residuals. The convergence criteria are set to be 10-6 and once the residuals reach this prescribed value, the solution is said to have converged and the data is post processed and the results analyzed. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Figure 2:

Meshed model of the plume evacuation flow domain of the LAHayeSIK™ surgical device.

Figure 3:

Solid model of the plume evacuation flow domain of the distal plume evacuation surgical device.

3

CFD simulation of the distal VISX Star S3 plume evacuation system

The flow domain for the distal plume evacuation system is identified as the path in the vicinity of the stromal surface from the evacuation tube, where the plume particles travel under the influence of the suction force. Figure 3 presents the plume flow domain that has been modeled. To better capture the effects of facial features such as the nose, setting of the eyes, and the evacuation tube’s influence on the flow of room air. A modeled face is in the surgical position as the subject that under goes surgery. Notice the considerably sharp features of the nose and reasonably deep-set eyes in figure 3. The model is then meshed using various meshing strategies to come up with the best quality mesh, which can be used by FLUENT to solve the numerical equations without any divergence problems. The plume evacuation function of the VISX surgical device was constructed as a mesh model. The meshed model from GAMBIT is then exported to the solver, which is FLUENT in the present CFD analysis.

4 CFD simulation results The CFD results obtained in this simulation indicate some important flow aspects of the plume under the evacuation field generated by the plume WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


131

evacuation devices. The proximal plume evacuation system shows a more effective evacuation influence on the microclimate over the corneal surface and thus assists in the onsite plume removal without giving the plume particles a longer time of travel. It is noted that the longer the plume particles are in the vicinity of the corneal surface, the more chances for plume masking and other complications. The proximal plume evacuation technique allows a quick 360o on site plume removal technique along the circumference of the cornea with high velocity gradients created just above the corneal surface. The design of the ports is such that low and uniform velocity fields are generated over the corneal surface itself, thereby reducing the risk of over dehydrating the cornea.

Figure 4:

Flow path of the plume particles which are situated at height of the plume channel of the LAHayeSIK™ surgical device. Simulation results show that all the plume is captured by the seven strategically placed ports.

Figure 4 shows the trajectories of the plume particles that are generated under the influence of the evacuation force of the LAHayeSIK™ plume evacuation function. The particles are created at a distance away from the corneal surface and the seven ports at a height of 2 cm from the corneal surface. The simulation shows that the entire plume is being captured by the seven strategically placed ports on the circumference of the handpiece. Simulating the particle paths to evaluate the velocity functions with a certain direction over the surface is a common practice to either create a set of points and plot their velocities or simulate the paths of the particle trajectories. In FLUENT, rakes are used to serve the same purpose. Rakes are a predetermined number of points between two specified endpoints. Figures 5 and 6 simulate the actual flow path of the plume particles that are generated during the actual surgery. These simulations show that not only the entire plume is effectively captured by the seven ports but also the flow paths indicate that there are no vortices or turbulent behavior during the plume travel. Contour maps show the velocity profiles over specified cross sections of a fluid domain (Figs. 7 and 8). These maps play an important role in indicating the change in velocities along the radial direction of the device. Figure 7 presents a contour map of simulated velocities generated by the plume evacuation function of the LAHayeSIK™ surgical device at the corneal height. It shows low WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

132 Computational Methods and Experimental Measurements XIII velocities with uniform profiles, which is important to minimize uneven dehydration of the cornea. Figure 8 shows the contour map of simulated velocities generated by the plume evacuation function of the LAHayeSIK™ surgical device at the plume channel height, which is about 3 cm from the cornea. This indicates that there is a high velocity field generated centimeters above the corneal surface that assists in the complete removal of the plume particles from the microclimate with no escape into the operating room, as shown by the particle paths. Figure 9 shows the plot of velocities on the rakes. This plot is an indication of the directional velocity changes as one goes away from the plume evacuation ports. The flatness of the curves indicates that there are minimal changes in the velocities in the radial direction and that an overall uniform evacuation is achieved by the combined effect of the seven evacuation ports.

Figure 5:

Flow path of the plume particles situated at different places on the cornea under the influence of the LAHayeSIK™ plume evacuation function.

Figure 6:

Flow path of the plume particles situated at different places on the cornea under the influence of the LAHayeSIK™ plume evacuation function. A complete capture is shown of all the plume particles generated at the corneal surface.

CFD simulations used the distal large volume VISX Star S3 plume evacuator to compare the CFD results with the experimental results and to delve deeper into the flow dynamics of distal plume evacuators. Figure 10 shows the velocity vectors generated over the facial contours by the VISX Star S3 plume evacuation WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


133

function achieved by employing the 22 mm diameter tube. The vector field indicates uneven velocity gradients with high and low velocity fields in the vertical plane, which create shear and cause vortices resulting in a high degree of turbulence. These vectors support the high turbulence and vortices observed during experiments with the artificial plume (Figure 11).

Figure 7:

Contour map of velocities generated by the plume evacuation function of the LAHayeSIK™ surgical device at the corneal height.

Figure 8:

Contour map of velocities generated by the plume evacuation function of the LAHayeSIK™ surgical device at the plume channel height which is about 3 cm from the cornea.

Figure 9:

Plot of velocities with respect to position generated over the cornea in the direction of a rake.



Figure 10:

Vector plot of the velocities generated by the VISX Star S3 distal plume evacuation device. The vertical plane of interest where the experimental velocities are measured.

Figure 11:

Photographs of the formation of vortices and plume escape in the field of a large volume distal plume evacuator documenting inefficient plume capture during LASIK surgery.

Figure 12:

Contour map of the velocity field generated by VISX Star S3 large volume plume evacuator in a plane perpendicular to corneal surface.

Figure 12 is the contour map of the velocities on the plane perpendicular to the corneal surface. These contour maps show similar profiles as obtained by WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


135

experimental studies, thereby validating the CFD simulation results. Figure 13 shows the trajectories of the plume particle paths under the influence of the distal plume evacuation velocity field generated by the distal large volume VISX Star S3 plume evacuator. The high shear generated by the high and low velocity fields in close proximity of the corneal surface cause turbulence and vortices. The non-uniform flow patterns cause escape of plume as shown in Figure 13.

Figure 13:

5

Path lines of trajectories of plume particles in the influence of the plume evacuation force field of the VISX Star S3 device.

Conclusions

Both the experimental and CFD simulation results indicate that the proximal plume evacuation systems are designed to better handle the plume evacuation as compared to the distal large volume plume evacuators. The proximal plume evacuators, owing to their proximity of the plume evacuation ports from the stromal surface have a greater effect in removing the generated plume. On the other hand, the particle trajectory simulations of the large volume evacuators indicate escape of particles and an inefficient plume removal. Also the presence of varying velocity fields just above the stromal surface results in high shear and thus there is a good possibility of vortex formations due to turbulence. This study concludes that the proximal plume evacuation systems such as LAHayeSIKTM are better designed to remove the plume generated during LASIK surgery as opposed to existing large volume evacuation technology.

References [1] [2]

LaHaye, L.C., Rieke, H.H., and Farshad, F.F., Is REFRACTIVE PROCEDURES Possible? Part Management, Oct 2005, pp. 97-99. LaHaye, L.C., Rieke, H.H., and Farshad, F.F., Is REFRACTIVE PROCEDURES Possible? Part Management, Jan 2006, pp. 45-46. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

Cleaner EXCIMER I, Ophthalmology Cleaner EXCIMER 2, Ophthalmology

136 Computational Methods and Experimental Measurements XIII [3] [4] [5] [6] [7] [8]

Duffey, R., Central Islands and Decentered Ablations After Excimer Refractive Procedures, International Ophthalmology Clinics 2000, 40, pp. 93-101. Noack et al., Influence of Ablation Plume Dynamics on the Formation of Islands in Excimer Laser Photorefractive Keratectomy, Ophthalmology 1997, 104(5), pp. 823-830. Dell, New System permits safe, more effective plume evacuation, Ophthalmology Times, April 2003. Charle, K., Effects of Laser Plume Evacuation on Laser in situ Keratomileusis Outcomes, Journal of Refractive Surgery, June 2002 (Suppl), pp. 340-341. Oshika et al., Corneal hydration and central islands after excimer laser photorefractive keratectomy, J Cataract Refractive Surgery, Dec 1998, 24, pp. 1575-1579. Maguen, E., Machat, J.J., Complications of photorefractive keratectomy, primarily with the VISX excimer laser, Corneal Laser Surgery 1995, pp. 143-158.



137

Flow estimations through spillways under submerged tidal conditions P. D. Scarlatos1, M. Ansar2 & Z. Chen2 1

Department of Civil Engineering & Center for Intermodal Transportation Safety and Security, Florida Atlantic University, USA 2 Operations & Hydro Data Management Division, South Florida Water Management District, USA

Abstract The South Florida Water Management District (SFWMD), Florida, USA, operates and maintains about twenty-five coastal spillways that discharge excess runoff water directly into the environmentally sensitive area of Biscayne Bay, south of Miami. Over the past decade serious concerns have been raised regarding the large fluctuations of salinity levels in Biscayne Bay caused by the freshwater releases. At the SFWMD, discharges at gated spillways are computed from an instantaneous stage and operational control information by using basic formulas developed for the estimation of orifice and weir type of flows. However, the accuracy of discharge estimates is compromised whenever flow occurs under submerged conditions and particularly at tidally affected structures. This paper is focused on the estimation of discharges through gated spillways under tidal submerged flow conditions. Data are analyzed using dimensional analysis and an empirical model is developed based on data from two coastal spillways. The model relates the discharge as a function of the tail-water head and the low tide elevation during each tidal cycle. All of the other parameters are treated as constants and lumped into the empirical coefficients. Keywords: dimensional analysis, empirical formulas, flow measurements, spillways, submerged flows, tidal effects.

1

Introduction

The South Florida Water Management District (SFWMD) operates and maintains over four hundred hydraulic structures including spillways, weirs, culverts and pump stations. The main driving force for the development of water WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070141

138 Computational Methods and Experimental Measurements XIII controls in southern Florida is to maintain adequate water supplies for the rapidly growing population along the lower east coast, to sustain agricultural activities and to restore and protect the Everglades National Park and other delicate wetland and coastal ecosystems. Approximately one hundred and twenty of the hydraulic structures are gated spillways and about twenty-five of them are coastal spillways that discharge excess runoff water directly into the environmentally sensitive area of Biscayne Bay, south of the Miami-Dade metropolitan area. Over the past decade serious concerns have been raised regarding the large fluctuations of salinity levels in Biscayne Bay caused by the freshwater releases and the negative impact that they have on the quality and biodiversity of the native environment. Thus, for an effective and efficient management of the regional water resources system there is a substantial need for accurate estimates of those freshwater discharges. The purpose of this study is to improve the existing state-of-the-art for estimation of discharges at coastal spillways under tidally-affected and submerged flow conditions.

2

Spillway gate operations

Depending on the gate operating conditions and the water elevation on either side of the spillway flow at coastal spillways is classified into five different flow categories [1, 5]. More specifically those categories are: • Free orifice-flow (partially opened gate – i.e. gate is in the water) • Submerged orifice-flow (partially opened gate) • Free weir-flow (fully opened gate – i.e. gate is out of the water) • Submerged weir-flow (fully opened gate) • Submerged tidally-affected weir-flow (fully or partially opened gate) At the SFWMD flows at gated spillways are generally computed from instantaneous stage and operational control information using an in-house developed program based on orifice and weir type discharge formulas [6]. However, the accuracy of discharge estimates is compromised in cases where flow occurs under submerged tidal conditions. An improvement on the discharge calculations was done by using dimensional analysis and field measurements from coastal spillway S-26 [2]. That model incorporated tidal effects in lumped terms of the tidal range and period. This study improves the Ansar and Raymond model by incorporating and analyzing additional available data from spillway S21.

3

Submerged tidal spillway flows

Generally, there are many parameters that control flow through tidally-affected submerged spillways. Those parameters can be categorized as being related to flow, fluid and geometric features of the structure [3]. Traditionally submerged flow is estimated as [6]: Q = Cs BH t 2g(H − H t ) WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

(1)


139

where B is the gate width, H is the upstream head, Ht is the tail-water elevation, g is the gravitational acceleration, and Cs is an empirical coefficient expressed as:

 Ht    H 

β

Cs = α

(2)

where α and β are experimental constants [5]. Another formula proposed by Skogerboe and Hyatt [4] estimates submerged flow as:

Q=

α C1 (H - H t ) 1

  Ht   −  log H + C 2    

(3)

β1

where both exponents α1 and β1 are empirical constants. However, application of the above formulas may lead to substantial inaccuracies particularly when the head difference, H – Ht, is too small or even a negative number. In certain occasion flow estimations for spillway S-26 using the above formulas provided negative flows, while field measurements indicated a positive flow. In order to improve the discharge computational formulas under submerged tidal conditions, Ansar and Raymond [2] assumed that the flow rate, Q, at a tidal gate is a function of eight variables (fig. 1), i.e.

(

Q = f H t , Pw , B, g, µ, ρ, A, T

)

(4)

where Pw is the height of the spillway weir, µ is the dynamic viscosity, ρ is the water density, A is the tidal range and T is the tidal half-period (fig. 1). By applying dimensional analysis and π-theorem the eight variables are combined into four dimensional groups: 2 3 Q 2 1 B 3 g 3 Pw

H ρQ  A , = Φ t ,   Pw T gPw µB 

(5)

Due to the high turbulent flow regime, the last dimensionless group within the parenthesis (Reynolds Number) was dropped from further consideration since the viscous effects are negligible. Thus, the expression (5) was re-written as:


140 Computational Methods and Experimental Measurements XIII 2 Q3 2 1 B 3 g 3 Pw

=

H Yc A = Φ t , Pw  Pw T gPw

  

(6)

By using actual data from a coastal spillway in Southeast Florida (Structure S26) the above relation was calibrated and verified as: 2 Q3

ξ

2 1 B 3 g 3 Pw

 A 6  H  = λ 10   t    PW   T gPw

ζ

(7)

The values of the calibration coefficients that satisfy data for spillway S-26 are different for the falling and rising tide (Table 1). Table 1:

Calibration coefficients for S-26 discharge formula.

Tidal Conditions Flood Ebb

ξ -0.44 -0.10

λ 2,179.35 111.77

ζ -2.38 -1.48

Gate

H-Ht Ht

L

P

H

Flow

Pw Lw LT

Figure 1:

Standard geometric features of spillways.

The fact that in equation (7) only the tail-water was included is justifiable due to the fact that under submerged flow conditions H is approximately equal to Ht.



4

141

Data analysis and model development

The South Florida Water Management District (SFWMD) and the US Geological Survey (USGS) operate and maintain a number of Ultrasonic Velocity Meters (UVMs) upstream of various gated structures. UVMs are utilized to estimate flows through the structures. However, since UVMs are “point” instruments their discharge estimates are adjusted using flow data collected by Acoustic Doppler Current Profiler (ADCP) which integrates velocities throughout the entire cross-sectional area of the channel. Then the UVM and/or ADCP discharge data are used to calibrate the empirical formulas for discharge estimation which are based on real-time telemetry measurements of upstream and tail-water elevations at the gate. The Ansar and Raymond model was calibrated and verified using discharge and water elevation data obtained at structure S-26 [2]. In the present study, the proposed model utilizes data from spillway S-26 (events of June 1997 and February 1998) and also spillway S-21 (event of September 2005). 4.1 Discussion of the Ansar and Raymond model The Ansar and Raymond model was tested under a variety of different conditions using data from spillway S-26 and the following conclusions were derived: • The model is extremely sensitive to the value of the total weir height, Pw (fig. 1). By using instead the top weir height, P, flow estimations are improved, but in certain cases that height has still to be reduced in order to produce adequate discharge estimations. • Both the tidal range, A, and the half-tidal period, T, may slightly affect the discharge estimation but not as drastically as the weir height. • In cases that the upstream head, H, is noticeably different that the tail-water, Ht, discharge estimates improve by using H instead of Ht. • In certain occasions, the ebb cycle is estimated better by using the calibration coefficients obtained for flood flow conditions. • Adopting the value of P instead of Pw in the Ansar and Raymond model, the value of the parameter λ may be need to be reduced accordingly in order to improve flow calculations. 4.2 Critical review of the spillway S-26 and S-21 data Since the focus of this study was on submerged tidal flows, only those data sets that the upstream and downstream heads were almost equal were considered. The data were analyzed separately for the flood (rising tide) and the ebb (falling tide) cycles. 4.2.1 Flood tide All of the data regarding the flood part of the tidal cycle showed a distinct linear, inversely proportional correlation between the discharge, Q, and the tail-water head, Ht, (as measured from the MSL) (figs. 2a,b,c). Thus, during the rising tide WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

142 Computational Methods and Experimental Measurements XIII discharge decreased due to the decreasing energy gradient along the discharging canal. The slope of that linear relation remained almost constant for each discharge event and varied only slightly among different gate operation events. S-26 Flood Feb 98

S-26 Flood Sept 97 800

1050

700 Discharge (cfs)

Discharge (cfs)

950 850 750 650 550 450

500 400 300 200 100

350 250 0.50

600

1.00

1.50

2.00

0 0.00

2.50

0.50

1.00

1.50

2.00

2.50

Tailw ater MSL (ft)

Tailwater MSL (ft)

(a)

(b)

Discharge (cfs)

S-21 Flood Sept 05 4500 4000 3500 3000 2500 2000 1500 1000 500 0 -500 0

1

2

3

4

Tailwater MSL (ft)

(c) Figure 2:

Discharge versus tail-water elevation during rising tide.

Another interesting observation was that no identifiable correlation appeared to exist between the discharge and either the tidal range, A, or half-period, T, as defined in the Ansar and Raymond model [2]. On the other hand, from a close scrutiny of figs. 2a,b,c it is suggested that the discharge is correlated to the lowest tail-water stage, (Htmin) during each tidal cycle. The discharge increases exponentially for increasing minimum tail-water stage (fig. 3). The exponential correlation that matches the data for spillway S-26 is plotted also in figure 3. Q = 6.633 exp(3.623Htmin) + 675.1

(8)

where Q is in cubic feet per second (cfs) and Htmin is measured in feet (in reference to the mean sea level (MSL)). For very high values of the tail-water head (Ht > 2.5 ft) the discharge becomes negligible. This is an indication that due to the high tidal stage the energy gradient within the discharging canal is diminished.



143

S-26 Discharge vs Minimum Tailwater Discharge (cfs)

1100 1000 900

Measured

800

Estimated

700 600 0.20

0.70

1.20

Minimum Tailwater (ft)

Figure 3:

Discharge versus low tail-water during each tidal cycle. S-26 Ebb June 97

S-26 Ebb Feb 98

1200

900 800 700 Discharge (cfs)

Discharge (cfs)

1000 800 600 400 200

600 500 400 300 200 100

0 0.00

0.50

1.00

1.50

2.00

2.50

0 0.00

3.00

0.50

Tailwater MSL (ft)

1.00

1.50

2.00

2.50

Tailwater MSL (ft)

(a)

(b) S-21 Ebb Sept 05 6000

Discharge (cfs)

5000 4000 3000 2000 1000 0 -1000 0

1

2

3

4

Tailwater MSL (ft)

(c) Figure 4:

Discharge versus tail-water elevation during falling tide.

4.2.2 Ebb tide The relationship between the discharge and tail-water for the falling water part of the tidal cycle is not linear. The discharge increases with the falling tail-water to a certain level but then before the end of the ebb cycle it levels or decreases even though the tide is still falling (fig. 4a,b,c). Since the high’s and low’s of tidal elevations are common points for both ebb and flood cycles the preceding remarks on the relationship between Htmin and Q, and diminishing Q for Ht > 2.5 ft are also observed. In addition, among the various expressions that were tried, the one that best fitted the nonlinear behavior for all of the data for spillway S-26 was a sinusoidal equation: WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

144 Computational Methods and Experimental Measurements XIII Q = 867 sin(Ht + 260) + 102

(9)

where Q is the discharge in cfs and Ht is the tail-water in ft (above MSL). The results are presented in fig. 5 (QF is the observed and QSIN are the simulated data). 1200

1100

1000

900

QF QSIN

800

700

600

500

400 11.4

11.6

11.8

12

12.2

12.4

12.6

12.8

XF

Tail-water (ft) Figure 5:

Measured versus simulated values of ebb tide discharges.

Once the trend (shape of the curve) was captured, the simulation was further improved by adjusting the discharge in terms of the minimum tidal water elevation, Htmin, as was demonstrated by equation 6.

5

Conclusions

Based on the analysis of this study involving data from spillways S-21 and S-26 the following conclusions were derived: • •

The weir height, Pw, (or the top weir height, P), do not have any direct significance in the discharge formula and can be treated as just another calibration constant. The variation in the tidal half-cycle period, T, in the Ansar Raymond model [2] is artificially introduced due to minor tailwater stage fluctuations and the WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


•

•

•

•

•

•

145

15 minutes data recording step. In reality the tide is astronomically driven with a fixed semi-diurnal period of 12.6 hours. Therefore, T should be treated as a constant. From the recorded data the tidal range, A, varies from 1.27 ft to 1.97 ft. However, no correlation was found between the tidal range and the flow rate. The tidal range showed a periodicity ranging from 2 to 3 tidal cycles (i.e., approximately 24 to 36 hours). This may be due to astronomical diurnal tidal effects and some prevailing weather conditions during the recorded event. During rising tide (flood), the discharge, Q, is inversely proportional to the tailwater elevation, Ht. In addition, the rate of flow reduction due to increasing tailwater stage is almost constant (linear relationship). Some inaccuracies to the above rule may occur in the neighbourhood of high tailwater stages (Ht > 2.5 ft). During falling tide (ebb), the discharge, Q, decreases with increasing tailwater elevation. The relationship although not linear as in the case of the rising tide, it follows a distinct pattern that can be described with a sinusoidal curve. The maximum flow rate, Qmax, occurring during the minimum tailwater elevation, Htmin, depends on the Htmin. As Htmin increases, the maximum flow increases in an exponential manner. This explains the “overshooting” of the Ansar and Raymond model that was calibrated using the high values of Htmin (S-26 June 1997 data) and was verified for lower values of Htmin (S-26 February 1998 data). The discharge conditions approaching the minimum tailwater elevation are more close to “tidal-free” regime since the flow was moving unopposed during the entire ebb cycle. The energy gradient decreases with the rising tide until it reaches its minimum value at high tide. At that time the energy gradient is from one to three orders of magnitude less than the one during the low tide. The amount of energy gradient decrease appears to be correlated to the tidal range but no particular pattern was identifiable. The flow behaviour occurring under fully opened gate conditions (S-26) was very similar to that occurring under partially opened gates (S-21) as long as the upstream and tail-water stages were about the same.

Acknowledgement This study was made possible through the support of the Operations and Hydro Data Management Division, South Florida Water Management District, West Palm Beach, Florida, USA.

References [1]

Ansar, M., Alexis, A. & Damisse, E., 2002. “Atlas of Flow Computations at District Hydraulic Structures”, Technical Report, Hydrology and WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


[2] [3] [4] [5]

[6]

Hydraulics Division, South Florida Water Management District, West Palm Beach, Florida. Ansar, M. & Raymond J.H., 2003. “Tidal Flow Computations at a Coastal Prototype Spillway”, Submitted to Journal of Hydraulic Engineering, ASCE. Scarlatos, P.D., 2005. “Tidal Hydraulics at Coastal Spillways”, Report submitted to SFWMD, West Palm Beach, Florida. Skogerboe, G.V. & Hyatt, L., 1967. “Analysis of Submergence in Flow Measuring Flumes”, J. Hydraulic Engineering, ASCE, 93(HY4): 183-200. Tillis, G.M. & Swain, E.D., 1998. “Determining Discharge-Coefficient Ratings for Selected Coastal Control Structures in Broward and Palm Beach Counties, Florida”, US Geological Survey, Water-Resources Investigations Report 98-4007, Tallahassee, Florida. US Army Corps of Engineers, 1963. “Typical Spillway Structure for Central and Southern Florida Water-Control Project: Hydraulic Model Investigation”, Technical Report 2-633. Vicksburg, Mississippi.



147

Analysis of the wave-flow interaction with submerged breakwaters A. C. Neves, F. Veloso Gomes & F. Taveira Pinto Institute of Hydraulics and Water Resources, Faculty of Engineering of the University of Porto, Porto, Portugal

Abstract The use of submerged breakwaters against coastal erosion problems has some advantages when compared, for example, with the use of similar emerged coastal protection structures and for that reason their use in coastal protection is becoming popular all over the world. Their effect in the wave field and especially in the wave-induced velocities has been analysed by several authors though there is still a lack of knowledge on the wave-flow interaction. The aim of this study was to analyse the behaviour of submerged breakwaters and especially the wave induced velocity field, which can have great impact in the stability of the structures and in the sediment circulation. Two-dimensional scaled physical tests were carried out in the Hydraulics Laboratory wave tank of the University of Porto, in order to understand with more accuracy the influence of the permeability and of the submergence of the breakwaters in the wave-structure interaction. For that reason, tests were performed with the same wave conditions (regular waves), with two different models with the same geometry, one with permeable and one with impermeable rough slopes, and with two water depths (leading to two different freeboards of the structure). Keywords: submerged breakwater, wave-flow interaction.

1

Introduction

Several investigations have been performed in relation to submerged breakwaters. Although much is already known in what concerns to the stability of these structures, it is believed that there is a lack of information about the hydrodynamics in their vicinity, namely in the velocity field. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070151

148 Computational Methods and Experimental Measurements XIII Losada et al. [1], in DELOS research report, have referred that the knowledge of the velocity field in the immediacy of submerged breakwaters is essential in what respects to the stability of the blocks. The authors have developed a numerical model, calibrated by several experimental results, to the prediction of the behaviour of submerged breakwaters and their influence in the wave propagation and in the water particle velocities. A high degree of agreement between th numerical and experimental data relatively to the free surface displacement in the vicinity of the structure and the velocities in the seaward slope was found. Tirindelli and Lamberti [3], also analysed velocities in the proximity of the stones of the armour later of submerged breakwaters. Through Morrison-type equations, the authors have calculated the wave forces in armour stones. They also pointed out the importance of knowing the velocity and acceleration of the water particles in the proximity of the blocks, as the wave-induced force applied to an armour unit depends on them. Saitoh and Ishida [2] analysed the wave-flow interaction with submerged breakwaters, namely velocity field in their proximity. Experiments with an impermeable model using PIV measurements and Laser Dopper Anemometry have been performed with several wave conditions. It was found that the maximum velocity offshore occurred in the upper part of the slope. Mass transport velocity and acceleration, among other parameters were also calculated, once they were considered to strongly affect the stability of the blocks and erosion at the toe of these structures. Taveira-Pinto [4] also analysed the velocity field near submerged breakwaters, through experimental modelling. The author has found high values of the turbulence intensity near the bottom and near the surface, in sections where an inversion of the flow velocity occurred and where the flow was passing over the obstacle. This last observation could be explained by the flow constriction occurring due to the water depth reduction in this section. It is considered that much of the existent literature has not paid significant attention to the influence of the roughness and of the permeability of the submerged breakwaters in the velocity near and far field and, thus, the aim of this study was to analyse the behaviour of submerged breakwaters and, especially the wave induced velocities near them and the parameters influencing them

2

Experimental set-up

The experiments were conducted in the Hydraulics Laboratory of the Faculty of Engineering of the University of Porto. The wave tank is 12 m wide, 28 m long and has a maximum water depth of 0.6 m. A thin dividing wall was used to isolate the measuring section from the rest of the tank, avoiding threedimensional effects and wave diffraction during the tests. A gently inclined absorbing beach was also constructed, diminishing wave reflection by causing sufficient dissipation of wave energy. A piston-type wave maker was used.



149

Two models of submerged breakwaters were used: a rough impermeable model and a rough permeable one, in order to analyse the effect of the permeability in the wave-velocity field. The models had similar cross-section, with a 0.30 m crest width, a 0.40 m crest height and 1/1 slopes. Figure 1 shows the rough impermeable model, where stones (Dn50=4.4 cm) were glued to a perspex base, simulating an armour layer. The permeable model had an armour layer of 2 Dn50 width and a nucleus.

Figure 1:

Rough impermeable model.

The velocity measurements were done in different vertical profiles, at different distances from the bottom and, successively nearer to the models, using Laser Doppler Anemometry technique. The optical system was formed by an Argon-Ion Laser, modular LDA optics based on a one-component fibre optic system and a 60 mm probe, working in a backscatter configuration. A capacitance wave probe was placed outside the testing section in order to verify the incident wave and in the vertical alignment where the velocity component was being measured. A Burst Spectrum Analyser allowed the simultaneous record of the analogical signals relative to the instantaneous water surface elevations and the Doppler signal relative to the velocity, in a way that each validated velocity corresponded to a water surface elevation value. The testes were performed with regular waves, with the same characteristics: H=0.065 m and T=1.5 s. Two water depths were tested, allowing the analysis of the effect of the submergence of the model in the velocity-field, 0.4 m and 0.45 m, which is equivalent to a 0 and a +5 cm freeboard.

3


As referred before the velocity measurements were done in different vertical profiles, at different distances from the bottom. The location of the measured profiles is indicated in Figure 2. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

150 Computational Methods and Experimental Measurements XIII Wave probe 17 19 21 23 25 27

18

15 16

13 14

11 12 10

20

(Hi, Ti)

9 8

7 6

22

5 4

24

3 2

26

Figure 2:

z/d=0.257

0.2

0.12

0.1

0.04

v (m/s)

u (m/s)

0

Location of the measured profiles.

z/d=0.257

0 -0.1

-0.04 -0.12

-0.2 0

0

60 120 180 240 300 360

60 120 180 240 300 360

Wave phase (º)

Wave phase (º)

z/d=0.561

z/d=0.561

0.2

0.12

0.1

0.04

v (m/s)

u (m/s)

1

0 -0.1

-0.04 -0.12

-0.2 0

60 120 180 240 300 360 Wave phase (º)

rough impermeable model

Figure 3:

0

60 120 180 240 300 360 Wave phase (º)

rough permeable model

Mean values of the horizontal (a) and vertical (b) component of the velocity (H=0.065m, T=1.5s, d=0.45 m, z/d=0.181 and z/d=0.257, profile 1).

Each one of the measured points generated a file, with the instantaneous water surface elevation and the respective velocity component values. A specific program was then used for the wave analysis and the calculation of the mean values of the variables for 50 different phases of the wave period. Figures 3 and 4 show some of the velocity measurement results. The results respect to the WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


151

rough impermeable and the rough permeable model and were obtained in profile 1 and profile 27, for two of the measurement levels and for one of the water depths. z/d=0.333 0.12

0.06

0.06

v (m/s)

u (m/s)

z/d=0.333 0.12

0 -0.06

0 -0.06 -0.12

-0.12 0

0

60 120 180 240 300 360 Wave phase (º)

60 120 180 240 300 360 Wave phase (º)

z/d=0.409 0.12

0.06

0.06

v (m/s)

u (m/s)

z/d=0.409 0.12

0 -0.06 -0.12

-0.06 -0.12

0

60 120 180 240 300 360 Wave phase (º) rough impermeable model

Figure 4:

0

0

60 120 180 240 300 360 Wave phase (º)

rough permeable model

Mean values of the horizontal (a) and vertical (b) component of the velocity (H=0.065m, T=1.5s, d=0.45 m, z/d=0.181 and z/d=0.257, profile 27)

The results proved that the horizontal component of the velocity is almost always greater than the vertical one. This fact is more obvious seaward of the model than in the leeward of the model, where the vertical velocity has more or less the same magnitude than the horizontal one. It can also be seen that the permeability induces in a greater attenuation of the velocities. Both velocity components were clearly attenuated by the obstacle, as it can be seen in the comparison between figures 3 and 4. A great area in the vicinity of the submerged breakwater models was covered and in total, 212 points were measured with the rough impermeable model and 169 points with the rough permeable one. Phase-averaged values of the velocities were calculated and analysed as a function of the relative distance to the bottom, allowing a global analysis of the velocity field.


152 Computational Methods and Experimental Measurements XIII 3.1 Permeability effect

z/d

PC

z/d

1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.45 -0.23

0.00

0.23

0.45

phase-averaged u (m/s)

1.2 1 0.8 0.6 0.4 0.2 0 -0.45 -0.225

PC

0

0.225 0.45

phase-averaged v (m/s)

1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.12 -0.06

P27

z/d

z/d

Figure 5 illustrates some of the results, where the values obtained with the rough impermeable model and the permeable one were placed in the same graphic in order to easily compare the permeability effect in the wave-induced flow velocity field.

1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.12 -0.06 0.00

P27

0.06

0.12


0.06

0.12


impermeable model

Figure 5:

0.00

permeable model

Variation along relative distance to the bottom of the mean, maximum and minimum values of both velocities components for permeable and impermeable models, profiles C and 27.

The graphics clearly illustrate that permeability has an important role in the attenuation of the flow velocities. In the permeable model, the water percolates inside the structure, which allows the dissipation of velocities through it, instead of concentrating them only in the surface, conducting the mean horizontal velocities in the permeable model to be most of the time lower than the ones obtained in the impermeable one, in profiles located seaward of the model or in the seaward slope. This can be explained by the effect of the structure’s permeability in the decreasing of the reflection properties of the breakwater model, which attenuates the seaward wave-induced velocities. On the other hand, it was found that the percolation of the flow through the structure caused an increment of the horizontal component of the velocity in the bottom region till the crest of the structure, as it can be seen in the graphic that respects to phaseaveraged horizontal velocity results calculated for profile 27. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


153

It was found that the most critical areas, in terms of velocities were located along the crown and in the upper part of the seaward slope. The upper part of the leeward slopes also revealed large mean horizontal velocities. The results proved that the permeable breakwater had greater impact in terms of wave energy dissipation than the other model, allowing a significant reduction of the horizontal velocities in the upper sections. 3.2 Freeboard of the structure effect

z/d

PC 1.2 1 0.8 0.6 0.4 0.2 0 -0.1

PC

z/d

1.2 1 0.8 0.6 0.4 0.2 0 -0.4

-0.2

0

0.2

0.4


-0.05

0

0.05


1.2 1 0.8 0.6 0.4 0.2 0 -0.4

P27

z/d

z/d

A similar analysis was performed for the results obtained with the two water depths. Figure 6 represents some of the results. The used water depths, 0.40 and 0.45 m conducted to submergences of 0 cm (mean water level coincident with the crest of the model) and 5 cm, respectively. In total, 169 points were measured with the 0.45 m water depth and 111 points with the 0.40m water depth.

0.1

1.2 1 0.8 0.6 0.4 0.2 0 -0.1

P27

-0.2

-0.05

0.2

0.4

0

0.05

0.1


F=-5 cm Figure 6:

0


F=0 cm

Comparison of phase-average velocities near submerged breakwaters, for different freeboards, profiles C and 27.

The results have proved that the submergence of the structure affects the wave-induced flow characteristics. The results obtained with the zero freeboard were almost always smaller than the ones found in the +5 cm submergence case. These differences were especially significant in the profiles located leeward of the model, in the sections above the structure, due to the (in)existence of WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

154 Computational Methods and Experimental Measurements XIII overtopping. When d=0.45 m, for example, it can be observed the enlargement of the mean velocities in the leeward of the model, directed onshore, while for d=0. 40 m, the respective velocities are much closer to zero.

4

Conclusions and final remarks

Experimental tests were conducted with two models of submerged breakwaters with different permeability characteristics for the same wave conditions. The results have confirmed that permeability of these structures can strongly affect the wave-induced flow characteristics, seawards and leewards of the structure. Most of the time, the mean horizontal velocities registered in the permeable model were lower than the ones obtained in the impermeable model, due to the effect of the structure’s permeability that decreases the reflection properties of the breakwater model and consequentially attenuates the seaward flow velocities. Different structure’s freeboards were also tested and the results of the measurements proved that the level of the crest of structure in relation to the mean water level has great influence in the wave-induced velocities, as it influences the passage of water over the structure. It is intended to extend this analysis to other characteristics of the models, such as the roughness of the surface, in the wave-induced flow. Other wave conditions are also being tested, in other to investigate their influence in the flow field. Tests with irregular waves are also intended to perform in order to prove the validity of the conclusions.

References [1]

[2]

[3]

[4]

Losada, I., Garcia, N., Lara, J., 2005. Report on turbulent flow velocities in the surface region of low-crested structures. DELOS (Environmental Design of Low Crested Coastal Defence Structures) Project, Deliverable 23 and 44. Saitoh, T., Ishida, H., 2001. Kinematics and Transformation of New Type Wave Front Breaker Over Submerged Breakwater. Proceedings of the 4th International Symposium Waves 2001, San Francisco, USA, ASCE, volume II, pp. 1032-1041, ISBN 0-7844-0604-9. Tirindelli, M., Lamberti, A., 2004. Wave-induced forces on Structural and Biotic Elements of Low Crested Structures, Proceedings of the 29th International Conference Coastal Engineering 2004, Lisbon, Portugal, pp. 4228-4239. Taveira-Pinto, F. 2001. Velocities Fields in the Vicinity of Submerged Breakwaters, PhD Thesis, Faculty of Engineering of University of Porto (in Portuguese), Porto, Portugal



155

An application of an edge effect based vacuum blower to a lyophilizer M. Kitamura, M. Tsutahara & H. Sasaki Graduate School of Science and Technology, Kobe University, Japan

Abstract In the rarefied gas, the edge effect flow is induced at a heated edge of a flat plate and its mechanism is well known. Usually edge effect flow is generated at both sides of the flat plate and the effects cancel each other out because the directions of the flows are completely opposite. By rounding the edge of a flat plate at one side we can obtain one-way flow and can use this flow for a vacuum blower. By confirming the characteristic of the effect of this edge effect flow we try to use this flow as an edge effect blower for lyophilizers. We confirmed that there is a pressure difference between the inlet and outlet of this blower caused by this edge effect flow and also this pressure difference is very obvious in the 200–400 Pa range of the surrounding pressure. We set this edge effect blower in a vacuum tank and examined whether this blower could promote the sublimation phenomenon of water. We confirmed that the sublimation was increased by using this blower. We also performed a numerical analysis using the finite difference lattice Boltzmann method to simulate the sublimation phenomena. Gas flows caused by evaporation and condensation phases are simulated. Keywords: rarefied gas, edge effect, lattice Boltzmann method, finite difference method, evaporation, condensation.

1

Introduction

We can usually consider the gas to be a continuum and describe a phenomenon by applying the Navier-Stokes equations as the governing equations. However, for low pressure gases, the Navier-Stokes equations and the no-slip boundary condition cannot be applied, because the mean free path of molecules cannot be neglected. We can describe such a gas phenomenon by the Boltzman equation.


156 Computational Methods and Experimental Measurements XIII In the rarefied gas, the thermal transpiration flow is induced when a solid wall has a temperature gradient [1–3]. The edge effect flow is also induced at the edge of a flat plate and its mechanism is the same as thermal transpiration flow. As an engineering application of this edge effect flow, we take up the lyophilizer this time. There is no system of the ventilation effect in the current lyophilizers that is necessary for the sublimation process. By applying an edge effect flow for the ventilation of the lyophilizer the rise of water sublimation speed is examined by experiments. The effectiveness of numerical simulation by the FDLBM is also checked by comparing it with the results by a molecular gas dynamics based calculation.

2

Experiments

We made an edge effect based vacuum blower (edge effect blower) using the edge effect flow. The edge effect flow usually occurs at both edges of a plate and it is offsetting. But by rounding one edge we can obtain a flow only for one direction [4]. We confirmed the pressure at the outlet of the blower was raised. We also confirmed that this blower is effective for speeding up the process of l sublimation. 2.1 Experiments by edge effect blower 2.1.1 Experimental device and method We use the accumulation unit that incorporated edge plates in that unit. A schematic view of the experimental device is shown in Fig 1. The accumulation unit of edge effect blower is settled in a vacuum chamber. The pressure in the chamber and the accumulator are measured by Pirani gauges. Each edge plate is equipped with heaters to be heated.

Pirani gauge Accumulation chamber Edge plate

Edge flow

Heater

Figure 1:

Schematic view of the edge effect blower.



157

The value of the vacuum gauge which is installed in a vacuum tank shows a back pressure of an edge effect blower, and a difference between a value in the edge effect blower and that of the vacuum tank is understood as a rise of pressure by an edge effect flow. After confirming at arrival the pressure in a vacuum tank at about 90Pa, the pressure measurement is started. 2.1.2 Test results and discussion The test results are shown in Fig. 2. The measurements are done in two cases. One case (case 1) is tested without heating the edge plate dotted square marks. The second case (case 2) is measured with heating the edge plate dotted diamond marks. We can confirm that the pressure rises in case 2 by heating an edge plate in the back pressure of 200–400 Pa region. It is thought that the edge effect is most effective in this region. Drops of pressure in case 2 around 400 Pa is caused by the characteristics of gauges.

Rising Pressure (Pa)

20 15 10 5 0 -5

0

100

200

300

400

500

600

Pressure of Chamber (Pa)

Figure 2:

Relation between pressure rise and back pressure in vacuum chamber.

2.2 An application of edge effect blower to a lyophilizer An edge effect blower is installed in a vacuum tank and the speeding up the process of the sublimation is examined. 2.2.1 Experimental device and method A view of an edge effect blower is shown in Fig 3. This blower consists of three plates bent at one end comprising heater in one row, and has two rows of plates. The experimental apparatus is shown in Fig. 4. The performance of an edge effect blower is tested by the sponge soaked with water and frozen up. The difference of quantity of water left in the sponge is measured by the following three cases. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Figure 3:

Edge effect blower.

Figure 4:

View of apparatus.

experimental

Water after Experiment(kg)e-3

Non-heater Edge effect fan

0.006

Non-edge effect fan

0.005 0.004 0.003 0.002 0.001 0 0.080

0.090

0.100

0.110

0.120

Water before experiment(kg)e-3 Figure 5:

Relation between contained water before and after experiment.

Case 1: Experiment with edge effect plate of no heat. Case 2: Experiment with edge effect plate of heat Case 3: Experiment without edge effect plate 2.2.2 Test results and discussion The test result is shown in Fig. 5 in the case of a start pressure 200 Pa in the vacuum chamber. It takes about 4 minutes before arriving at the experimental start pressure 200 Pa. During the experiment the pressure rises about 50Pa in the WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


159

vacuum chamber. Some dispersion of the data is seen, but, as for the case without the source of heat, the quantity of water sublimation decreases as the pressure rises. Cases 1 and 3: The test results of case 3 (without edge effect plate), as the pressure rises, the quantity of water sublimation decreases as well as case 1 (without the heat). This cause is similar to the case without the source of heat. Case 2: In the case of the edge effect blower, quantity of water sublimation becomes greatest at the starting pressure 200 Pa in the vacuum chamber. Therefore, it seems that sublimation is promoted by the ventilation effect of the edge effect blower. Although the dispersion is rather large, it seems that a ventilation effect by the edge effect flow is effective on lyophilization in the range of 100 – 400 Pa.

3

Numerical analysis

The lattice Boltzmann method (LBM) is the computational tool to analyse the continuous thermal viscous fluid. The flow in the vacuum chamber is governed by the Navier-Stokes equations except the Knudsen layer near the sublimation surface. The LBM model is a strongly discretized version of the Boltzmann equation. Therefore the LBM can be considered to simulate the flow except the Knudsen layer without any special treatment for the boundary conditions. In this paper, the finite difference lattice Boltzmann method (FDLBM) is used. The two-dimensional thermal model (the D2Q21model) [5, 6] of FDLBM is presented briefly in section 3.1. The numerical analysis method is presented in section 3.2. 3.1 Finite difference lattice Boltzmann method The discrete BGK equation for the FDLBM is written as follows with the distribution function fi k (x, t ) having the particle velocity ci

where

φ

∂fi ∂f 1 + ci i = −  fi − fi (0)  ∂t ∂x φ

(1)

is the collision parameter (the relaxation time) and f i 0 is the local equilibrium distribution function chosen to satisfy the Navier-Stokes equation. The local equilibrium distribution function is defined as follows in the case of the thermal model. f i 0 = ω i ρ (1 − 2 Buα ci ,α + 2 B 2 (uα ci ,α ) 2 + Bu 2 (2) 4 − B 3 (uα ci ,α )3 − 2 B 2uα ci ,α u 2 ) 3 The macroscopic variables for continuous fluids are defined as

ρ = ∑ f i = ∑ f i (0) i

i


(3)


ρ u = ∑ fi ci = ∑ f i (0)ci i

(4)

i

1 2 1 1 ρu + ρe = ∑ f i ci 2 = ∑ f i ( 0) ci2 2 i 2 i 2

(5)

for the density, the momentum and energy, respectively, where e is the internal energy per unit mass. For the finite difference lattice Boltzmann method, the governing differential equation is discretized in finite difference schemes, and the following calculation procedures are employed. The time integration is performed by the second-order Runge-Kutta method and the third-order upwind scheme is employed for space differential. Evaporated area （eA ,ρA） X2 r

θ

X1

L Condensed area （eB ,ρB） (a) Parallel condensed phases Figure 6:

(b) Cylindrical condensed phase Schematic diagram.

3.2 Method of analysis 3.2.1 Evaporation and condensation between two parallel plates One-dimensional behaviours of gas between two parallel condensed phases is considered as shown in Fig. 6(a). Both condensed phases spread infinitely and are kept at different temperatures. We only consider the condensed gas. Evaporation occurs at the high temperature condensed phase and condensation in the low temperature condensed phase and flow of the gas occurs towards the low temperature side from the high temperature side. It is assumed that the temperature is TA (internal energy eA) at the high temperature side and TB (internal energy eB) at the low temperature side, and the saturated vapour pressure of gas at each side is given pA and pB. (Fig. 6(a)) The local equilibrium distribution functions defined by these temperatures and the pressures (densities), and the zero flow velocity are used. We compare the flows in the steady state to those given by the molecular dynamics based calculations. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


161

3.2.2 Evaporation from cylindrical condensed phase Two-dimensional evaporation from a cylindrical surface in an infinitely expanded domain as shown in Fig. 6(b) is also considered. We assume the temperature of the cylinder surface is Tw (internal energy e) and assume saturated vapour pressure of gas is pw. The temperature and the pressure at infinity are T∞ and p∞, respectively. The local equilibrium distribution function defined by the temperature, the pressure（ density） and the zero flow velocity is used. 1.2

1

e/eB

0.8

0.6 0.4

pA/pB=10 pA/pB=3

0.2 0

X

(a) T/TB

12

10

pA/pB=10 pA/pB=3

p/pB

8

6

4

2

0 X (b) p/pB Figure 7: Temperature (internal energy) and pressure distributions in case of pA/pB =10, TA/TB =1.1 and pA/pB =3.0, TA/TB=1.1. Left side (A side) is the evaporation phase and right side (B side) is the condensation phase. (a) e/eB (T/TB): Temperature distribution as a function of X, distance from A for pA/pB=10 and pA/pB =3. (b) p/pB: Pressure distribution as a function of X for pA/pB =10 and pA/pB =3. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

162 Computational Methods and Experimental Measurements XIII 3.3 Results and discussion 3.3.1 Evaporation and condensation between two parallel plates The distributions of the temperature and the saturated pressure between both phases in case of pA/pB =3.0, TA/TB =1.1, and pA/pB =10.0, TA/TB =1.1 are shown in Fig. 7. The rapid changes of the pressure and the temperature in both cases arise near the condensation and the evaporation phases, but except for these layers the pressure and temperature are uniform. This shows the same tendency with the molecular gas dynamics result [7] based on the BGK equation with KnB=0. Comparing the results for pA/pB=10 with those for pA/pB=3, a temperature descent of the gas is more remarkable in the case of pA/pB=10 because the flow from the condensed side to the evaporated side becomes fast caused by the large difference of saturated vapour pressures. The reverse temperature incline phenomenon that appeared in the results of molecular gas dynamics is not detected. 1.00

T∞/ Tw

0.95

0.90

0.85

0.80 0.0

0.2

0.4

0.6

0.8

1.0

ｐ∞/ pw

Figure 8:

The temperature ratio versus the pressure ratio.

3.3.2 Evaporation from cylindrical condensed phase The relation of T∞/Tw vs. p∞/pw is shown in Fig. 8. The x marks show calculation results and the line shows the results by molecule gas dynamics. The calculation results by FDLBM become closer to the latter when p∞/pw approaches to unity. The pressure distributions p/pw for p∞/pw (0.82,0.70,0.61 and 0.55) are shown in figure 9. When the pressure ratio p∞/pw becomes smaller, the pressure profile p/pw near the cylinder surface becomes inclined and to be smaller. These results show the same tendency as those of the molecule gas dynamic results near KnB=0 [8]. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


163

0.85 0.80

p∞/pw =0.82 p∞/pw =0.70 p∞/pw =0.61 p∞/pw =0.55

p/pw

0.75 0.70 0.65 0.60 0.55 0.50 1

2

3

6

11

19

34

62

111 200

r/L

Figure 9:

4

Distribution of non-dimensional pressure from the cylinder surface.

Conclusions

The experiment of applying an edge effect blower to lyophilization confirms that the sublimation is promoted by the presence of the edge effect blower in the pressure area of 200 ~ 400 Pa. However, in higher vacuum domain of 1~10Pa, the remarkable influence is not confirmed. We need more detailed experimental work around these pressure regions and have to confirm the usefulness of the edge effect blower to a lyophilizer. By the numerical analysis, comparison with a calculation result based on molecular gas dynamics gives a good agreement. It is confirmed that the FDLBM model (the D2Q21 model) for compressible fluid with boundary condition given by the local equilibrium distribution function is very useful for simulation of flows caused by the condensations and/or evaporations. This method is also powerful for complex shaped boundaries.

References [1] [2] [3] [4]

Y. Sone, K Aoki, Molecular Gas Dynamics, Asakura-syoten 1994,in Japanese Y. Sone, Y. Waniguchi, K. Aoki, One-way flow of rarefied gas induced in a channel with a periodic temperature distribution, Phys.Fluids, 8,pp22272235, 1996 Y Sone, Kinetic Theory and Fluid Dynamics, BIRKHAUSER 2002 M Tutahara, K Ogawa, T Kataoka, M Shoji, Y Sakai and M Kirimuro, Study on engineering application of the Knudsen pump in rarefied gas, Transactions of the Japan Society of Mechanical Engineers NO.044-1, 2004, in Japanese. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

164 Computational Methods and Experimental Measurements XIII [5] [6]

[7]

[8]

M Tsutahara, N Takada and T Kataoka, Lattice gas and lattice Boltzmann methods, Corona-sha 1999; in Japanese. M. Tsutahara, T. Kataoka, K. Shikata, N. Takada, New Model and Scheme for Compressible Fluids of the Finite Difference Lattice Boltzmann Method and Direct Simulations of Aerodynamic Sound” Computers and Fluids, (to be published) Kazuo Aoki and Noboru Masukawa “Gas flows caused by evaporation and condensation on two parallel condensed phases and the negative temperature gradient: Numerical analysis by using a nonlinear kinetic equation” Phys. Fluids 6(3), p1379-1395 March 1994 Hiroshi Sugimoto, Yoshio Sone “Numerical analysis of steady flows of a gas evaporating from its cylindrical condensed phase on the basis of kinetic theory” Phys. Fluids A 4(2), p419-440 Feb.1992



165

The wavelength reconstruction from toroidal spectrometer image data J. Blazej1, M. Tamas1, L. Pina1, A. Jancarek1, S. Palinek1, P. Vrba2 & M. Vrbova1 1 2

Czech Technical University in Prague, Czech Republic Institute of Plasma Physics, Academy of Science, Czech Republic

Abstract We are reporting spectroscopy studies of Nitrogen filled capillary discharge. The identification procedure of spectra was carried out with the use of an extreme ultraviolet grazing incidence spectrometer with toroidal grating. The source of radiation and the object under study is a pinching alumina capillary discharge in Nitrogen, where stimulated emission in the 13.4 nm wavelength region is expected. To simplify the alignment of the spectrometer during the experiment the off Rowland circle registration scheme is used. In this scheme, spectra are recorded in a single plane and thus exact focusing of the input slit takes place only for one single wavelength, which corresponds to the intersection of the plane of registration with the Rowland circle. Appropriate image processing must be applied to reconstruct the spectra profile and to calibrate wavelength positions. The cooled extreme ultraviolet sensitive CCD camera was used in the role of planar detector. It has a 512×512 elements matrix with 24 µm square pixels. The 16-bit dynamic range together with very low-noise provide a good data source for image post-processing. The experimental impossibility to record simultaneously zeroth order maximum and the first diffraction order spectrum was a crucial problem in the presented process. The presented methodology can be applied to any experiment using toroidal grating spectrometer. Keywords: spectrometer, image distortion, calibration.

1

Introduction

We are reporting spectroscopy studies of Nitrogen filled capillary discharge. The goal of our work is to create a technique for routine use of a spectrometer with WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070171

166 Computational Methods and Experimental Measurements XIII toroidal grating with relatively small image detector in the off Rowland circle registration scheme. The usual type of detector for this scheme is a linear CCD that allowing images simultaneously zeroth and first order maximum and that in principle eliminate geometrical distortion of lines aside the circle plane. Spectra acquired by a planar CCD are distorted by non-ideal imaging in the spectrograph and non-linear dispersion must be eliminated to reconstruct wavelength information. The first problem is that geometrical distortion of acquired spectra origins from imaging of input slit to a plane. In standard spectrograph configuration the output slit is used to overcome this problem. This is an on Rowland circle registration scheme. The other possibility is to use a line (linear) CCD without output slit in the plane of the Rowland circle. In this case the distortion is converted into modification of dispersion. To improve the signal to noise ratio and overcome experimental difficulties with low detection yield and energy budget we decided to use a planar CCD.

2

Distortion elimination

The spatial configuration of the spectrometer with toroidal grating is schematically drawn in figure 1. For the theoretical review see Haber [1]. The cross section of the screen with the Rowland circle defines on preferred wavelength. Its line is imaged without any astigmatism. All others line are distorted in correlation with the distance between the screen and the Rowland circle. φ0

O [0,0]

α D φ x F S

A

scree

n

R

C

Figure 1:

The spatial configuration of the spectrometer with toroidal grating. The input slit is imaging to the screen. The cross section of the screen with the Rowland circle defines on preferred wavelength. It is only one line imaged without any astigmatism. φ is the angle of virtual scanning line p, the coordinate origin is point O. R is the Rowland circle with diameter R, center C. The angle φ0 is the zeroth order angle, α is the screen angle. The origin of coordinates at the screen is the point F, the cross-section with the Rowland circle. The distance between grid and screen is called D = D-O.



167

The analyzed radiation is coming from the left and is mirrored by the grating under grazing angle φ0. The zero order maximum angle φ0 is 4º and the angle of screen is 10º. The coordinate origin is the point O – the grid center. X axis is the grid tangent and y axis is going through point C – the center of the Rowland circle The screen is at distance D from the origin, containing the α angle from the x-axis. To eliminate geometrical distortion of spectra the following method is used. Several horizontal profiles (raw cuts) are taken on the image with good contrast line pattern. From the line pattern several well-defined lines are selected. Each horizontal profile is separately fitted by multiple Gaussian peaks at selected lines. Positions of their centers define a matrix of distortion. Each column of this matrix is fitted by a parabola. The radii of curvature of these parabolas are changing for different columns and this dependence has a minimum. The position of this minimum can be used for absolute calibration. The generated matrix can be used for distortion elimination after its smoothing by the parabola fit. For distortion elimination – image warping – we are using open source program xmorph [2] with sophisticated wavelet amplitude correction to ensure the photometric data from original spectra. The result of these steps is an image with parallel spectral lines over the entire screen, see figure 2.

a) Figure 2:

3

b)

The negative of the primary picture taken by CCD (a) and final warped image (b). All its lines are straight and parallel. Optionally the flat field and background correction can be applied to enhance the signal to noise ratio and recognize and eliminate the dead pixels.

Calibration

The spectrometer dispersion curve is well known, but the problems of calibration of exposed spectra are two: 1) in real experimental conditions it is impossible to determine the distance D 2) the origin of x axis on screen is unknown, the screen is too small to include zeroth order maximum


168 Computational Methods and Experimental Measurements XIII Therefore the experiment has two degree of freedom from the absolute calibration point of view; the position of zeroth order maximum on detector plate and the distance of detector from grating center. Both are experimentally unattainable for us. The parameters for calibration can be derived from following sources: 3.1 Calibration filter For these wavelengths the L-edge of aluminium can be used. The problem is the absence of continuous source. Some experiment configuration generating a lot of lines can be used, but the final resolution of edge position identification depends on actual spectra. Nevertheless, only one parameter can be evaluated from one edge and filtered spectra were not included in a particular analyzed spectra set. 3.2 Width of spectral lines The width is proportional to focusing. The imaging of spectra to the Rowland circle can be modelled as an imaging by a lens with variable focus length. Assuming a hypothesis about the quasi-Gaussian character of the input signal, the model of dependence of line width on wavelength can be derived. But we are not succeeding to find any algorithm for a reliable line width calculation from spectra with unknown spectral line intensities. 3.3 Position of non-distorted spectral line on screen This invariant reduces the number of degrees of freedom to one defining a relation between searched parameters. The position of the non-distorted line can be estimated with a precision of 10 pixels. 3.4 Assignment of known wavelengths to bright lines The fourth source is the assignment of known wavelengths to bright lines. It is dependent on several presumptions about observed plasma source or on an independent spectral analysis of similar plasma source. 3.5 Solution We decide to use the fourth method. A spectrum identified in this work is generated by a capillary discharge in nitrogen pre-filled alumina capillary (nitrogen pre-fill pressure 0,49 torr), capillary dimensions were 297 mm long, 3.0 mm in diameter and the current form is damped sinus with quarter period t1=87.5 ns, damping time t2=600 ns and I0=22 kA. For comparison we have two independently calibrated spectra, see Vrbova et al. [3]. The first one is a spectrum of capillary discharge in nitrogen pre-filled alumina capillary (nitrogen pre-fill pressure 0.9 torr). Capillary dimensions were 56 mm length and 3 mm diameter, the current form is damped sinus with quarter period t1=150 ns, damping time t2=600 ns and I0=15.5 kA. And the second one is a spectrum of ablative capillary discharge in polyoxymethylene (POM) capillary. Capillary WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


169

dimensions were 56 mm length and 1.1 mm initial diameter, the current form is damped sinus with quarter period t1=65 ns, damping time t2=150 ns and I0=12 kA. We have identified a strong resemblance between three spectra. Considering that the only common element present in every discharge is oxygen, there is a strong indication to assign the most intense lines to oxygen ions. By comparing the database [4] and the simulated spectra of different ions made by Sapphire software [5] with measured values we decided to identify the most intense lines around 13 nm, 15 nm and 17.4 nm as lines belonging to helium like oxygen. Modifying the spectra accordingly we could further identify minor peaks from their absolute wavelength position. By assigning the most intense lines to oxygen ions a very good accordance of measurement with simulated lines of He-like oxygen (O VI) was found. Further very good accordance can be seen with simulated lines of He-like nitrogen (N V), especially in experiments with higher nitrogen pre-fill pressure.

4

Conclusion

The experimental impossibility to record simultaneously zeroth order maximum and the first diffraction order spectrum was a crucial problem in the presented process. We succeeded in eliminating an image distortion and verified its successful calibration. Spectral lines belonging to excited Nitrogen ions N4+ and N5+ were identified. We may note that we were not able to identify as many lines in the interval above 22 nm as we did in the interval 7-22 nm. This may be caused by lower resolution as we are far away off the Rowland circle. The presented methodology can be applied to any similar experiment using a toroidal grating spectrometer.

Acknowledgements This research has been supported by the research framework MSM6840770022, section 3 of Ministry of Education of Czech Republic and experimental data has been obtained thanks to support of grant 1P04LA235.

References [1] [2] [3] [4] [5]

Haber, H.: The Torus Grating, In: Journal of Optical Society of America vol. 40, number 3, Optical Society of America, Washington DC, USA, March 1950, pp. 153–165. Mennucci, A.C.G.: Xmorph, [online] , 2004, compilation gtk2.2 July 17th 2004. Vrbova, M., Jancarek, A., Vrba, P., Fojtik, A., Scholzova, L., Havlikova, R., Palinek, S., Soft X-ray Emission from Nitrogen Capillary Discharge, ICXRL Peking 2004. NIST Atomic Spectra Database Lines Form, [online], . Sapphire professional, version 1.0.3.3 (Cavendish Instruments Ltd. 2003). WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)



171

Numerical noise in satellite laser ranging data processing J. Blazej & I. Prochazka Czech Technical University in Prague, Czech Republic

Abstract Satellite laser ranging is a highly accurate measuring technique providing the accurate range to the retroreflector equipped Earth satellites. It operates on a classical radar principle. Measuring the time interval between the pulse transmittal and reception, and considering the speed of light and the way of its propagation, the target distance may be evaluated. The picosecond laser pulses together with highly precise timing systems and optical detectors enable resolution and accuracy of the entire ranging system to an accuracy of several millimetres. To demonstrate the ultimate precision of satellite laser ranging two different and independent algorithms were used for ranging data processing. The internal consistency and the numerical noise of the data reduction, fitting and normal point forming procedure have been tested. The completed experiment demonstrates the ultimate normal point precision at the level of 2.5 picoseconds, that corresponds to 0.37 mm in the range of the retroreflector equipped satellite in space. The key contributor to this value is the interpolation used in the satellite orbit modelling procedure. Keywords: picosecond ranging, orbit modelling.

1

Introduction

The testing of precision limits of algorithms used is interesting for any technique whose precision during recent years is improving by technology reasons without changing of the data processing algorithm. One of these is the Satellite Laser Ranging (SLR). The SLR is a highly accurate measuring technique providing accurate range to retroreflector equipped Earth satellites. It operates on a classical radar principle. Measuring the time interval between the pulse transmission and reception and considering the speed of light and the way of its WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070181

172 Computational Methods and Experimental Measurements XIII propagation, the target distance may be evaluated. The picosecond laser pulses together with highly precise timing systems and optical detectors enable resolution and accuracy of the entire ranging system to a range of several millimetres. The results of SLR are used, among others, for the determination of terrestrial reference frame and the product of the universal gravitational constant and the Earth mass, which represent one of the fundamental constants in physics, for details see Pearlman et al [1]. In this view, the SLR serves as a fundamental technique to calibrate other measurements. That is why the precision and accuracy of the SLR data itself is a critical issue.

2

Data processing

The SLR is a technique developed over more than thirty years. One unpropitious consequence of this fact is the squeezing by standard data interchanging formats of a data processing algorithm. To reduce the requirements on data transfer and archiving, to simplify the data analysis procedures, and to simplify the data use, the individual (so-called single shot) SLR measurements are compressed into socalled normal points. The data time series are divided into time slots – time bins of common predefined size of 5÷300 seconds, depending on the satellite orbit altitude. The individual ranging data within one bin are fitted and averaged and at the end are represented by a single range called the normal point. The goal of this paper is to compare two independently implemented algorithms to help identify error source limitations of the normal point precision.

Figure 1:

The schematic diagram of SLR principle, time interval between laser pulse transmission and detection is measured together with time of measurement – epoch.



3

173

Experiment

The main factors limiting the normal point precision are: the overall stability of the laser ranging chain and the precision of the data reduction and fitting procedure. The laser ranging hardware and its stability were characterized by means of the Portable Calibration Standard in numerous trials, see Hamal et al [2]. The best laser ranging stations achieve long term system bias stability of the order of picoseconds in time that corresponds to the 0.1 mm in range. The internal consistency and the numerical noise of the data reduction, fitting and normal point forming procedure were tested by the following experiment. The SLR raw ranging data were processed, fitted, and normal points were formed using two different and independent algorithms. 3.1 Graz SLR data reduction The orbit is modelled by a IRVINT Sinclair [3] integrator providing 1 minute x,y,z coordinates, 8-points Lagrange interpolation and topocentric conversion, and optional manual range and time bias tuning. The generated orbit is subtracted from range data and the residuum series is fitted by a polynomial fitting (standard scheme, degree 5-10, optionally 20) data screening and manual editing (validation). All interchange data files are in MERIT2 data format, i.e. with 1 picosecond granularity. 3.2 Portable Calibration Standard data reduction This software package, see Hamal et al [2], consists of orbit integration using the Herstmonceux RGO scheme providing 1 minute x,y,z coordinates, 8-points Lagrange interpolation and topocentric conversion, automated range bias, time bias, and device universal time tuning procedure. The generated orbit is subtracted from the range data and residua are iterative polynomial fitting with fully automated data editing. In this data reduction package, the processing consists of a sequence of individual programs, the data are passed from one program to another via a formatted data file with the least significant digit of 1 picosecond, and thus the rounding is implemented 3 to 5 times consequently. 3.3 Results The ranging data, described in Prochazka and Kirchner [4], from SLR station in Graz, Austria, were used. Segments of the ERS-2 satellite ranging with the data rate of about 750 echoes per second, acquired from October 2003 to January 2004 were selected. The echoes from the closest corner retroreflector were selected to eliminate the influence of the target completely from the process. The single shot precision achieved in the tested data series was 18 to 20 picoseconds rms, that corresponds to a 2.5 to 3.0 mm range precision of a single shot measurement.


174 Computational Methods and Experimental Measurements XIII The range residuals of individual ranges (single shot) were compared for all measurements. The difference between the computed ranging residuals on a shot by shot basis indicates the estimate of the data processing procedure. The computed differences have the characteristics displayed in figure 2. One can see a slowly varying component with a varying period of about 15 seconds and an amplitude of 2.0 picoseconds, and random spread within 1.5 picoseconds. The slowly varying component was attributed to the interpolation, the random component to the numerical noise of the computation and to the rounding process. The slowly varying component including absolute shift of the order of 2 picoseconds is the dominant contributor. 10

Range difference [ps]

5

0

-5 # points/pixel densitogram, 5 pixels/s -10 74770

Figure 2:

0

75

74780 74790 Time epoch, seconds in current day [s]

150 74800

Data residuals comparison of two described algorithms. The difference is computed for each shot, the integral value for 0.2 second is plotted.

From range residuals, results were calculated. The numerical experiments of Prochazka and Kirchner [4] showed the deviation from the ideal model in the case when about 75 individual range measurements are compressed into one normal point with the corresponding normal point precision of 2.5 ps. The second break point occurred when compressing more than 2000 points with the corresponding normal point precision of 1 ps. These two break points correspond perfectly to the two limiting factors identified as the interpolation and rounding errors.



4

175

Conclusion

The completed experiment demonstrates the ultimate normal point precision at the level of 2.5 picoseconds, that corresponds to 0.37 mm in the range of the retroreflector equipped satellite in space. This limit value is negligible in comparison to the signatures of actually used satellites. The key contributor to this value is the interpolation used in the orbit modelling procedure. The last significant digit in order of one picosecond is acceptable in today’s SLR data processing formats.

Acknowledgements The authors are grateful to Georg Kirchner and Franz Koidl, Graz SLR station. This research was supported by the research framework MSM6840770015 of the Ministry of Education of the Czech Republic.

Reference [1] [2]

[3] [4]

Pearlman, M.R., Degnan, J.J., and Bosworth, J.M., "The International Laser Ranging Service", Advances in Space Research, Vol. 30, No. 2, pp. 135-143, July 2002. Hamal, K., Prochazka, I., Blazej, J., Kirchner, G., Schreiber, U., Riepl, S., Sperber, P., Gurtner, W., Appleby, G., Gibbs, P., Yang Fumin, Neubert, R., and Grunwaldt, L., Satelite Laser Ranging Portable Calibration Standard Missions 1997-2002, In Abstract Book 'Geophysical Research Abstracts' Volume 5, 2003, EGS - AGU - EUG Joint Assembly 2003, Nice, France, p. 243. Sinclair, A, TIRV Reference System, [online], [cit. 02/03/07], http://ilrs.gsfc.nasa.gov/products_formats_procedures/predictions/tirv.htm l. Prochazka, I., Kirchner, G., Numerical noise in satellite laser ranging data processing, Boletin ROA No. 4/2004, ed. J.M.Davila, San Fernando: Real Instituto Observatorion de la Armada, ISSN 1131-5040, 2004.




177

Expanding the definition of multivariate correlation W. Conley Departments of Mathematics and Business Administration, University of Wisconsin at Green Bay, USA

Abstract The complexities of large scale data analysis, in our computer age, invite the development of new sophisticated statistics to help in this task. One entry into this arena is the CTSP multivariate correlation statistic. A five variable 49 line spreadsheet of data was analyzed using CTSP by Conley and the relationship was found to be linear. Presented here is a much larger example involving nine variables and 89 lines of data, where CTSP reveals a correlation that is nonlinear (exponential in this case). Specifically, nine columns (representing nine variables) of 89 lines of data are being analyzed to see if the variables they represent are correlated in some fashion (linear or nonlinear). Therefore, the data is read into the CTSP statistical analysis simulation program which is adjusted for nine variables and 89 lines of data. Then, using a generalization of the Pythagorean distance measure to nine dimensions, a shortest route connecting the 89 points in a closed loop tour is calculated. Then several random data sets of 9 x 89 size (in similar ranges to the original data) are generated and the shortest routes are calculated for them. In this case, the actual data had a much shorter shortest route than the random data’s shortest routes. Therefore, statistically speaking it can be argued that the actual data is correlated in some fashion because it is following a pattern and hence the points are more compact (closer together in nine dimensional space) leading to a shorter shortest route. The relationship is exponential in this case. This expanded view of correlation (linear or nonlinear) can complement the standard linear analysis Anderson currently uses. Additionally, a second example involving eight variables is presented for comparison purposes. Keywords: multivariate correlation, CTSP statistic, shortest route statistical test, linear and nonlinear analysis. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070191


1

Introduction

The new CTSP correlation statistic was developed in 2002 to help discover multivariate relationships whether they be linear or nonlinear. The CTSP is short for (an acronym) correlation using the travelling salesman problem. The idea in two dimensions can be illustrated by thinking of X, Y pairs of points that are following the shape of a parabola when graphed. These points will have a shorter shortest route connecting them, than the shortest route connecting the same number of random points (in the same range as the parabola points). A three dimensional example would be X, Y, Z triples on a flat plane going through the points (0, 0, 0), (100, 100, 100), (100, 0, 100), and (0, 100, 0) in the region 0 to 100 for all three variables. The shortest route connecting these X, Y, Z triples (say 75 of them, for example) would be much shorter than the shortest route connecting 75 random triples (in the same range) that are not following a pattern. This can be exploited statistically to show a correlation (or lack thereof) when analyzing multivariate data. The name CTSP statistic comes from the mathematically famous TSP problems (so called travelling salesman problem) of finding a shortest route to connect n points in a closed loop tour. This analysis presented here uses the multi stage Monte Carlo optimization MSMCO TSP algorithm for the statistical analysis. Let us look at an example.

2

Numeric example one

Researches believe that the nine variables represented by the nine columns of data in Table 1 are correlated in a linear or nonlinear fashion. They think that the first eight variables are perhaps driving the ninth variable (X9 whose data is in Column 9). Therefore, to test the null hypothesis of no correlation between the nine variables (X1, X2…X9) versus the alternative hypothesis of correlation, the 89 lines of 9 columns of data are read into the MSMSCO TSP algorithm program adjusted for 89 lines of data and nine variables. The Pythagorean theorem distance measure is used after expanding it to nine dimensional distance calculations. A few seconds to a minute of computer run time (on a desktop PC) yielded a shortest route of total distance 5,776.308 (see Table 2 for the route read left to right) connecting the 89 nine dimensional points in a closed loop tour. Then four sets of 89 lines of nine columns of random data (in similar 0-100 ranges as the actual data) were read into the MSMCO TSP algorithm. Their shortest routes were calculated to be 6786.314, 6518.579, 6600.060 and 6410.455. Therefore, CTSP = A / B = 5776.308 /(6518.579 + 6600.060) / 2 = .88

(1)

where A is the shortest route distance for the actual data and B is the median of the four random data shortest routes. Now taking the 3 x 4 = 12 Ai/Bi quotients using all combinations of the four random shortest routes, we see that their range WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Table 1: Point 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

X1 73 66 9 40 40 5 0 63 61 28 31 50 65 32 100 40 52 68 17 71 78 70 46 58 65 88 51 57 12 33 100 4 13 38 77 52 55 11 36 65 50 14 84

X2 12 5 17 41 70 51 50 99 56 70 100 60 23 13 10 94 20 89 90 48 56 47 0 39 61 90 28 74 77 3 97 19 23 26 57 63 19 41 20 66 58 79 44

X3 46 69 18 95 58 49 89 29 100 57 11 16 77 100 95 94 67 29 72 66 57 75 37 22 93 62 20 36 41 32 98 69 25 21 72 10 35 42 10 15 3 29 35

X4 59 49 58 91 36 44 77 13 14 30 99 79 39 90 60 30 14 2 40 71 44 25 70 97 5 90 77 71 72 63 32 33 42 100 19 18 41 89 87 23 23 88 21

179

The data. X5 41 94 85 8 8 37 83 27 91 86 73 11 65 18 24 70 3 62 16 36 62 100 88 68 31 86 75 100 63 88 45 100 27 1 97 22 32 46 4 0 100 56 24

X6 24 18 85 68 35 33 13 46 1 31 76 86 36 33 24 46 99 38 49 1 82 70 100 57 33 17 64 74 48 57 85 9 96 26 91 68 37 5 86 3 97 72 54


X7 95 100 62 18 30 72 64 30 25 47 5 100 51 38 33 65 41 92 40 99 39 20 34 42 29 69 29 57 36 81 0 63 21 40 91 100 35 35 58 16 73 34 2

X8 96 21 29 23 72 23 52 82 31 100 82 43 87 29 88 82 94 100 4 53 19 20 20 52 81 42 13 91 92 73 70 88 92 5 44 17 82 57 95 67 92 75 57

X9 3.6 4.9 3 3.6 2.7 2 5.1 3.7 3.4 6 7.5 5.8 5.7 2.8 5.3 10.7 3.8 7.7 2.3 5.8 5.5 5.1 5.4 5.4 4 12.9 2.9 14.6 5.7 5.2 11.2 3.6 2.5 1.3 13.3 2.7 2.4 2.3 3.9 1.3 8.8 5.9 2.2

180 Computational Methods and Experimental Measurements XIII Table 1: Point 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86

X1 28 51 51 56 35 83 40 90 33 55 45 43 96 13 60 18 29 57 43 57 29 81 41 93 45 7 31 11 95 62 66 88 8 66 31 62 29 35 17 44 1 2 98

X2 32 40 94 2 88 95 4 43 23 100 51 100 7 43 58 28 29 63 57 95 100 100 31 47 27 95 72 20 7 23 74 53 10 43 60 40 83 36 40 56 20 56 77

X3 44 100 2 31 5 27 85 73 53 9 3 31 13 37 14 6 15 75 60 50 45 1 33 79 70 77 51 19 31 3 60 46 92 81 79 100 24 9 81 46 100 0 100

X4 80 22 4 60 79 46 50 52 2 79 83 87 16 41 90 97 32 91 53 82 66 18 16 48 65 30 73 43 15 17 10 9 39 93 37 48 28 39 23 84 20 20 19

Continued. X5 88 79 6 74 1 32 56 76 61 70 48 31 81 19 93 1 60 81 90 35 92 99 13 100 79 12 88 82 26 71 41 65 37 39 33 91 56 49 36 91 34 15 48

X6 13 61 100 12 36 90 99 92 34 35 63 34 27 59 65 76 65 83 62 100 63 6 35 76 92 10 75 100 12 60 52 94 83 27 75 6 80 43 11 100 16 12 8


X7 21 74 27 37 98 40 85 74 74 0 49 25 66 77 75 22 64 86 90 90 78 90 90 60 56 56 82 95 51 42 46 82 55 21 46 55 77 28 55 19 2 35 35

X8 10 99 88 1 48 59 42 36 30 3 37 55 44 35 74 81 6 80 86 33 100 62 8 91 26 36 25 58 89 20 17 68 3 95 25 66 63 96 14 10 74 100 35

X9 2.1 11.2 3.3 1.5 3.8 7.2 6.6 12.1 2 2.7 3.4 4.3 2.7 2.2 11.4 2.3 1.8 22.9 12.6 12.7 16.3 6.4 1.4 19.2 6.6 2.2 8.8 5.1 2.3 1.8 3.1 9.4 2.3 6.8 3.6 7 5.6 2.4 1.5 6.1 1.4 1.1 4.8


Table 1: Point 87 88 89

X1 72 39 97

X2 56 4 16

X3 4 94 59

X4 39 98 100

Table 2: 43 29 76 12 1 47 67 72 41

17 42 78 63 20 44 89 37 75

33 11 19 70 88 38 61 81 35

39 55 69 26 7 34 62 85 51

59 53 82 79 32 4 30 40 21

181

Continued. X5 97 60 81

X6 15 2 95

X7 60 100 52

X8 8 74 89

X9 2.8 7.2 18.5

58 23 66 74 18 15 60 46 25

28 68 36 52 65 13 73 49 85

64 50 48 2 87 45 56 80 43

The route. 54 27 6 9 10 14 71 5 22

24 83 57 86 16 77 3 8 31

is 6410.455/6786.314 = .945 to 6786.314/6410.455 = 1.059. This range contains the vast majority of the sampling distribution of CTSP under the null hypothesis of no correlation. However, from eqn (1) our CTSP = .88, which is less than this range. Therefore, the null hypothesis of no correlation can be confidently rejected. The nine variables are correlated and the equation X 9 = .1666 exp(( X 1 + X 2 + X 3 + X 4 + X 5 + X 6 + X 7 + X 8 ) / 125)

(2)

fits the data fairly well. It should be noted that the range of the actual X9 variable values is somewhat smaller (0-25) rather than 0-100 for the other eight variables. However, if one evaluates eqn (2) with 100 for X1 through X8, X9 is 100. Therefore, the restricted range of X9 was a function of the correlation, so random sampling all nine ranges in the area 0-100 seems appropriate for the randomness comparison. However, if the researcher or engineer believes that there is scientific justification for having different ranges for the variables, they can be used for the random data sets generation and subsequent testing. The engineer or scientist will know what ranges are believable and appropriate for the application at hand. The central point is that a shorter shortest route (than randomly generated data’s shortest routes) will indicate that the data is following a pattern and is hence indicating that the variables are correlated.

3

Example two

Researchers studying the eight columns of n = 49 lines of data in Table 3 want to test the hypothesis of no correlation between the eight variables represented by the eight columns of data. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

182 Computational Methods and Experimental Measurements XIII Therefore, the 8 x 49 array is read into the MSMCO TSP algorithm. A less than one minute computer run produces a shortest route of total distance A = 3571.170 (see Table 4) for analysis. Table 3: Point 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

X1 55 27 82 72 37 4 94 92 46 0 85 74 38 57 54 40 71 88 100 47 16 36 21 29 59 54 65 10 23 48 55 66 20 74 80 26 24 9 64

X2 10 38 98 74 30 100 50 99 100 18 61 70 84 22 26 52 56 57 37 8 45 63 100 51 85 72 40 45 65 15 42 7 8 63 64 58 40 54 88

X3 89 86 61 4 57 50 82 83 82 7 21 70 59 36 92 60 10 75 45 41 20 46 13 41 83 49 71 73 76 80 70 82 71 26 81 5 26 31 54

Example two data. X4 17 67 13 93 70 6 39 23 1 85 100 100 17 19 3 97 82 42 42 67 25 28 94 100 41 62 65 30 80 28 40 100 37 9 15 35 2 18 56

X5 33 50 51 100 15 3 19 21 4 24 100 100 98 2 26 38 79 2 55 36 71 67 22 39 27 32 69 49 5 80 28 98 7 33 99 56 90 33 61

X6 50 12 81 83 90 66 79 27 12 12 17 94 9 47 25 96 52 47 52 29 99 40 20 36 1 58 34 21 44 11 17 73 20 4 38 17 100 41 68


X7 100 43 27 12 86 47 84 77 38 64 23 57 11 49 19 20 22 56 26 59 53 61 21 75 79 55 3 98 6 84 56 80 55 44 72 21 3 49 69

X8 45 47 61 56 17 4 3 4 10 62 89 18 87 82 99 93 87 15 3 87 62 3 52 21 40 15 27 84 57 89 76 33 39 14 86 75 53 26 8


Table 3: Point 40 41 42 43 44 45 46 47 48 49

X1 78 32 61 89 61 98 32 100 85 13

X2 21 53 88 44 48 100 96 93 47 88

45 5 46 6 27

35 44 25 9 19

43 16 8 42 34

Continued.

X3 56 34 82 92 54 55 70 47 39 96

Table 4: 30 29 18 3 48

183

X4 37 39 10 48 100 29 87 73 25 98

X5 35 28 61 91 8 97 95 17 5 39

X6 86 29 61 6 88 1 0 75 14 88

X7 92 25 23 58 34 83 89 81 33 68

X8 92 99 7 83 44 62 8 17 71 37

Example two route. 28 23 7 37 14

31 11 47 21 15

20 17 26 40 13

10 4 39 1 41

24 12 22 33 36

49 32 38 2

Then four sets of random data of 8 x 49 are read into the MSMCO TSP algorithm. Their shortest routes turn out to be 3346.265, 3473.206, 3564.802 and 3363.000. Note that the real data shortest route distance (3571.170) is comparable to these and will not produce a relatively smaller CTSP. Therefore, the null hypothesis of no correlation between the variables cannot be rejected.

4 Expanded definition of correlation We are now prepared with the CTSP multivariate statistic (for the computer age) to expand our definition of multivariate correlation to the discovery of any type of pattern, whether it be linear, nonlinear and even non-functional (like a mathematical relation). Additionally, if one wishes to assume and/or test for various underlying distributions, CTSP can help in that accommodation also.

5

Conclusion

Much of the standard multivariate analysis used today assumes sampling from multivariate normal distributions and looking for linear relationships. However, with CTSP we can go beyond this and look for any type of relationship that is revealed by a relatively shorter shortest route through the k dimensional space (for k variables) when compared with similar random data. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

184 Computational Methods and Experimental Measurements XIII The various TSP algorithms can help trucking companies and transportation entities in their deliveries to their valued customers. However, the TSP analysis when combined with the CTSP multivariate correlation statistic, can do much more than deliver products and services to customers. It can also deliver sophisticated multivariate statistical correlation analysis on spreadsheets of data that researcher and engineers encounter on an almost daily basis.

References [1] [2]

Conley, W.C., Multi stage Monte Carlo optimization applied to systems of integral equations. Proc. of the 15th Int. Conf. on Boundary Element Technology, ed. C.A. Brebbia-WIT Press, Southampton, pp 75-86, 2003. Anderson, T.W. Multivariate Statistical Analysis, 3rd edition, John Wiley Inc, New York, 2003.



185

3D analysis of solid reinforced concrete beams subjected to combined load of bending, torsion and shear A. S. Alnuaimi Civil and Architectural Engineering, Sultan Qaboos University, Sultanate of Oman

Abstract This paper presents a comparison between experimental and an in house 3-D finite element analysis results of three reinforced concrete solid beams subjected to combined loadings of bending, shear and torsion. The finite element program adopted was based on a 20 node isoparametric element. A non-linear elastic isotropic model, proposed by Kotsovos, was used to model concrete behaviour, while steel was modelled as an embedded element exhibiting elastic-perfectly plastic response. Allowance was made for shear retention and for tension stiffening in concrete after cracking. Only fixed direction, smeared cracking modelling was adopted. The beam dimensions were 300x300mm cross section, 3800mm length. Experimental results were compared with the non-linear predictions. The comparison was judged by load displacement relationship, steel strain, and load and mode of failure. Good agreement was observed between predicted ultimate and experimentally measured loads. It was concluded that the present program can confidently be used to predict the behaviour and failure load of reinforced concrete solid beams subjected to combined load of bending, torsion and shear. Keywords: beam, solid beam, bending, shear, torsion, direct design, concrete, reinforced concrete, stress analysis, combined loading.

1

Introduction

The behaviour of solid beams when subjected to combined loading is very complex. A detailed analysis would normally require a three-dimensional finite element model. Rahal and Collins [1] developed a three-dimensional analytical WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070201

186 Computational Methods and Experimental Measurements XIII model capable of analysing rectangular sections subjected to combined loading of biaxial bending, biaxial shear, torsion and axial load. The model takes into account the shear-torsion interaction and concrete spalling. It idealizes the rectangular cross-section resisting shear and torsion as made of four transversely reinforced walls with varying thickness and varying angle of principal compressive strains. The vertical shear stress due to shear force is uniformly resisted by the vertical walls and the lateral shear stress is resisted by the horizontal walls. They tested their model and concluded that the model predicts very close results to experimental behavioural and ultimate load results. Ibell et al [2] used an upper-bound plasticity analysis in a 2D model for the assessment of shear in reinforced concrete beams. The results from this model were compared with experimental results. It was recommended that the 2D analysis to be extended to more general 3D collapse analysis. Rabczuk and Eibl [3] presented a model using a mesh free 2D Galerkin/ finite element approach. The concrete was modelled with particles and reinforcement with beam element. For steel, an elastoplastic constitutive law with isotropic hardening and tension cut-off was used. The concrete was modelled via a continuum damage model, where an anisotropic tensile damage variable was used to capture the behaviour of concrete in tension. They used a displacement controlled approach for testing their model. They compared the results from their model with experimental results from one rectangular and one I-section pre-stressed beams. They concluded that a full 3D simulation would be more appropriate. In this research an in-house 3-D finite element program was used for nonlinear analysis of this computational study. The program was developed by ElNuonu [4] using Kotsovs’ concrete model. This model was based on experimental data obtained at Imperial College London from tests on the behaviour of concrete under complex stress states (Kotsovos and Newman [5] and Kotsovos [6]). The testing techniques used to obtain this data were validated by comparing them with those obtained in an international co-operative programme of research into the effect of different test methods on the behaviour of concrete. This model is capable of describing the behaviour of concrete under uniaxial, biaxial and triaxial stress conditions. It requires only the concrete cube compressive strength f cu to define the behaviour of concrete under different stress states. More information about this model is given in Kotsovos and Pavlovic [7].

2

A 3D finite element program

In the 3D program, a 150x150x150mm iso-parametric solid element with twenty node and twenty seven Gauss points was used. The concrete cylinder '

compressive strength f c is taken as f’c= 0.8fcu N/mm2, the Young’s modulus ' ' ' f c N/mm2, the split cylinder tensile strength f t = 0.54 f c N/mm2 and the Poisson’s ratio was set at a constant value of 0.15. Before cracking or crushing, the concrete behaviour is assumed to be non-linear elastic isotropic.

Ec = 5000



187

Crushing occurs at a point when all the three principal stresses are compressive and the state of stress is on the ‘yield’ surface. In the case of concrete crushing, complete loss of strength is assumed i.e. no compression softening is allowed for. When the concrete cracks in any direction, concrete ceases to be isotropic and cracking can occur if the minimum principal strain (compressive) reaches a value taken as equal to 0.003. After cracking, smeared crack approach with simple tension stiffening and shear retention equations are employed to represent the post cracking behaviour of concrete. Cracks are assumed to be orthogonal and once formed remain in their direction. The stress-strain relationship in '

tension was assumed to be linear up to f t and immediately after cracking the '

tensile stress f t is reduced to 0.8 f t . Thereafter, f t decreases linearly with strain and is zero at the maximum strain of 0.003 which roughly corresponds to yield strain of steel of 0.0025. Transfer of shear stresses across cracks is modelled by means of the 'shear retention' factor β which defines the shear modulus of cracked concrete as βG, where G is the elastic shear modulus of the un-cracked concrete. The shear retention factor β= 1.0 if εn ≤ εcr and β = 0.25 εcr/εn if εn > εcr, where εcr = cracking strain ( ε cr = f t ' Ec ) and εn = average of the three principal strains at any cracked point ( ε n = (ε1 + ε 2 + ε 3 ) 3 ). The reinforcement is modelled as one dimensional element embedded in the solid concrete elements. Elastic-plastic stress-strain behaviour without strain hardening was used in this research. Only uniaxial resistance is considered with no provision for kinking or dowel action of bars. Standard incremental-iterative procedure was adopted for solution. The load increments were equal to 10% of the design load for the first three increments and 5% for the remaining increments. The maximum number of increments was 50 and the maximum number of iterations in each increment was 200. The convergence being deemed satisfactory if the ratio of the square roots of the sum of the squares of the residual forces to that of the applied loads did not exceed 5%. The stresses in the cross-section nearest to mid-span were analysed. The stress distribution at the last converged increment was used for the analysis. In deciding on the predicted mode of failure, the load-deflection relationship, steel strain and ultimate load were taken into consideration. The program was extensively used by Bhatt and Lim for the analysis of slabs, internal column-flat slab junctions and punching shear failure of flat slabs (Bhatt and Lim [8,9]). Good agreement between predicted and experimental results was found.

3

Tested beams

Three reinforced concrete beams tested by Alnuaimi and Bhatt [10] were analysed. All beams were 300x300mm cross section and 3.8m length. They were subjected to combined load of bending, torsion and shear (Table 1). The main variables studied were the ratio of the shear stress due to torsion to shear stress


188 Computational Methods and Experimental Measurements XIII due to shear force τtor/τshr which varied between 0.69 and 3.04 and the ratio of the torsion to bending moment Td/Md which varied between 0.26 and 1.19. The concrete mix consisted of cement, uncrushed 10mm gravel and sand with water/ cement ratio of 0.55. Three cubes, 100x100x100mm, and six cylinders, 150x300mm, for each beam were cast from the same concrete used for casting each beam. The specimen and the samples were kept under damp Hessian for about four days and then under room condition. The samples were tested on the day the beam was tested to determine the cube and cylinder compressive strengths and split cylinder tensile strength of concrete. Table 1: Beam No. BTV13 BTV14 BTV15

Td

Md

kNm 26 13 39

kNm 50.89 50.89 32.89

Load combination. Vd

τtor

kN 61.08 61.08 41.08

τshr 2

2

N/mm 4.16 2.08 6.24

N/mm 3.00 3.00 2.05

Td/Md

τtor / τshr

Ratio 0.51 0.26 1.19

Ratio 1.39 0.69 3.04

Table 2 shows the average yield strengths of reinforcement and compressive and tensile strengths of concrete. The concrete cube and cylinder compressive strengths shown for each beam in Table 2 are the measured average strengths of the three cubes and three cylinders respectively and the concrete tensile strength shown is the measured average strength of three cylinders tested for split test. All results were obtained from samples cured along side each beam. Table 2: Beam No. BTV13 BTV14 BTV15

fcu

Average material properties. f'c

2

f’t 2

fy 2

fyv 2

N/mm

N/mm

N/mm

N/mm

N/mm2

40 37 61

28.5 25.7 38.2

3.45 2.92 4.38

500 500 500

500 500 500

Figure 1 shows the provided reinforcement and arrangement of longitudinal bars for each beam. The solid circles in Figure 1 represent the longitudinal bars in which strain was measured nearest to mid-span. Strains in the stirrups nearest to mid-span are also reported. 2Y8

2Y8

2Y8

2Y8

2Y8 2Y10 4Y12

2Y10 4Y12

BTV13

2Y10 4Y12

BTV15

Y8@120 mm

BTV14

Figure 1:

Provided reinforcement at test-span.

Y8@170 mm


Y8@82 mm


4

189

Test setup and instrumentation

Figure 2 shows a testing rig with typical beam installed. The test rig is a threedimensional frame designed to allow application of torsion, bending moment and shear force. The model was mounted on two steel stools fixed to the concrete floor at a distance of 1.8m a part. The test span was 1.2m long centred at midspan. The beam was simply supported by a set of two perpendicular rollers at each support and a system of pin-and-roller at the mid-span of the top face. At the support, the lower roller allows axial displacement and the upper one allows rotation about a horizontal axis at the soffit level of the beam. Torsion was applied by means of a torsion arm fixed to each end of the beam while bending moment and shear force were a result of applied load at mid-span across the beam width at the top face. This support and loading arrangement allowed full rotation about the centre line of the beam soffit and displacement in the beam axial direction. It produced constant torsional shear stress over the entire length of the beam and maximum normal stress due to bending and shear stress due to shear force occurred near the mid-span. The load was measured using a data logger for data acquisition. Linear voltage displacement transducers (LVDT) were used to measure the vertical displacement at the bottom face of the beam. To measure strain in the bars, a pair of strain gauges, 6mm long, was fixed on directly opposite faces of the bar and connected to a data logger. Accordingly, the axial strain recorded at each load stage was taken as the average reading of both gauges. Crack width and crack development were measured by means of a crack width measuring microscope.

Figure 2:

Test rig with a typical beam installation.



5

Comparison between experimental and computational results

5.1 Load displacement relationship Figure 3 shows the vertical measured and computed displacements at the centre of the bottom face of each beam. It is clear from this figure that, good agreement was achieved between experimental and computational results for beam BTV14. In the case of beams BTV13 the program predicted stiffer behaviour than the measured. The predicted and measured values from beams BTV15 were small and difficult to judge. The measured and computed displacements of beams BTV13 and BTV14 with bending dominance (Td/Md1) experienced measured and computed relatively smaller displacements and did not reach the span/250 limit. BTV13 1.2 1

L.F.

0.8 0.6

EXP

0.4

Comput

0.2 0 -0.2

0.0

2.0

4.0

6.0

8.0

Disp.(m m )

BTV14 1.4 1.2 1

L.F.

0.8 0.6 0.4

EXP

0.2

Comput.

0 -0.2 0.0

2.0

4.0

6.0

8.0

Disp.(m m )

BTV15 1 0.8

L.F.

0.6 0.4

EXP

0.2 0 -0.1 -0.2

Comput 0.1

0.3

0.5

0.7

0.9

1.1

1.3

1.5

1.7

1.9

Disp.(m m )

Figure 3:

Vertical displacement at mid-span.



191

5.2 Strain in the longitudinal steel Figure 4 shows good agreement between the measured and computed strain ratios in the longitudinal steel. In beams BTV13 and BTV14 with Td/Md1. 5.3 Strain in the transverse steel Figure 5 shows strain ratios in the transverse steel; in general, a very good agreement between measured and computed results was achieved. The transverse steel yielded or reached near yield strain. Longitudinal steel BTV13 1.2 1

L.F.

0.8 0.6

EXP

0.4

Comput

0.2 0 -0.2 0

0.2

0.4

0.6

0.8

1

ε/ εy

Longitudinal steel BTV14 1.4 1.2 1

L.F.

0.8 0.6 0.4

EXP

0.2

Comput

0 -0.2 0

0.2

0.4

0.6

0.8

1

1.2

ε/ εy

Longitudinal steel BTV15

1 0.8

L.F.

0.6 0.4 EXP

0.2

Comput

0 -0.2

-0.2

0

Figure 4:

0.2

0.4

0.6

0.8

ε/ εy

Strain ratios in the longitudinal steel.


1

192 Computational Methods and Experimental Measurements XIII Stirrup BTV13 1.2 1

L.F.

0.8 0.6

EXP

0.4

Comput

0.2 0 -0.2 0

0.1

0.2

0.3

0.4

ε/ εy

Stirrup BTV14 1.4 1.2 1 L.F.

0.8 0.6

-0.1

0.4

EXP

0.2

Comput

0 -0.2 0

0.1

0.2

0.3

0.4

0.5

ε/ εy

Stirrups BTV15

1 0.8

L.F.

0.6 0.4

EXP

0.2

Comput

0 -0.2

0

0.2

0.4

0.6

0.8

1

ε/ εy

Figure 5:

Strain ratios in the transverse steel.

5.4 Failure load and mode of failure Column 4 of Table 3 shows very good agreement between the measured Le and computed Lc failure loads. All beams failed near their design loads. Both the computed and measured results showed that in the case of beams in which bending was dominant (Td/Md1), inclined cracks extended into the bottom face one increment after they were formed in the front and rear sides. In both groups, the smaller the ratio Td/Md, the closer is the angle of crack to vertical. In beams WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


193

BTV13, BTV14 the mode of failure was mostly flexural where the beam experienced relatively large displacement and the flexural steel yielded. A small number of large cracks caused failure at the time of flexural steel yielding. Beam BTV15 failed by diagonal cracking due to high torsional shear stress and the failure mode was less ductile with small displacement, less longitudinal steel strain and larger transverse steel strain than the bending dominant beams. Table 3: 1

BTV13

2 Td/Md Ratio 0.51

3 τtor/τshr Ratio 1.39

4 Le/Lc Ratio 0.98

BTV14

0.26

0.69

0.96

BTV15

1.19

3.04

0.93

Beam No.

6

Ratios of measured and predicted failure loads.

Conclusion

From the results presented in this paper it can be concluded that the 3-D finite element program was shown to be in a good agreement with the experimental results and therefore, proven to be a good tool for the prediction of beam behaviour and ultimate load of solid reinforced concrete beams subjected to combined load of bending, torsion and shear.


[4] [5] [6] [7]

Rahal K. N. and Collins M. P., Analysis of Sections Subjected to Combined Shear and Torsion – A Theoretical Model. ACI Structural Journal, Vol. 92, No. 4, July-August 1995, pp. 459-469. Ibell T. J., Morley C.T. and Middleton C.R., An Upper-bound Plastic Analysis for Shear. Magazine of Concrete Research, 50, No.1, March 1998, pp. 67-73. Rabczuk T. and Eibl J., Numerical analysis of prestressed concrete beams using a coupled element free Galerkin/finite approach, International Journal of Solid and Structures, v.41, n.3-4, February 2004, pp 10611080. El-Nuonu G. F. R., Design of Shear Wall-Floor Slab Connections. Ph.D. thesis, University of Glasgow, 1985. Kotsovos M. D. and Newman J. B., A mathematical Description of the Deformation Behaviour of Concrete under Complex Loading. Magazine of Concrete Research, Vol. 31, No. 107, June 1979, pp.77-90. Kotsovos M. D., A Mathematical Description of the Strength Properties of Concrete under Generalized Stress. Magazine of Concrete Research, Vol. 31, No. 108, Sep. 1979, pp. 151-158. Kotsovos M. D. and Pavlovic M. N., Structural Concrete, Finite element analysis for limit-state design, Thomas Telford Publications, 1 Heron Quey, London E14 4JD, 1995. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


[9]

[10]

Bhatt P. and Lim B. T., Flat Slab-Column Junctions with Shear and Moment Transfer: A Comparison between Finite Element Predictions and Experiments. Proc. of 7th ACME Conference, University of Durham, (Ed. Bettes, P.) pp. 11 – 14, 1999. Bhatt P. and Lim B. T., Punching Shear Capacity of Internal Column-Flat Slab Junction with In-Plane Restraint: A Comparison between Finite Element Predictions and Experiments. Developments in Analysis and Design Using Finite Element Methods, Civil-Comp Press, (Ed. B.H.V. Topping), pp. 141 – 147, 1999. Alnuaimi A. S. and Bhatt P., Design of Reinforced Concrete Solid Beams, Structures and Buildings Journal, Thomas Telford Limited, v.159, n.4, August 2006, pp 197-216.



195

Use of correlation of iron loss and copper loss for improving the efficiency of three phase squirrel cage induction motors B. B. Saanane, A. H. Nzali & D. J. Chambega Department of Electrical Power Engineering, University of Dares Salaam, Tanzania

Abstract So far computation of iron (core) losses in induction motors cannot be performed through exact analytical methods but is dependent mainly on empirical formulae and experience of motor designers and manufacturers. In comparison to copper losses, which are to a larger extent easier to calculate, iron losses are mostly associated with some practical parameters, for example the type of material and manufacturing conditions. This paper proposes a new correlation between these two losses with the aim of getting minimized total machine loss in order to improve the efficiency. A total loss prediction model is developed on a matlab 6.5 platform such that the optimal airgap magnetic flux density and airgap length points are established which offer minimized total loss, min(Pfe + Pcu). These points are then used to reconfigure a new motor geometry with a minimized total loss. The new motor design approach was simulated on a 2D-FEM to analyse the new motor response. Experimental results which agree with the results of the design show an improvement of motor efficiency. Also, empirical formulae are developed and validated which can greatly assist motor designers. Keywords: total loss model, optimization, analysis, design formulae, motor efficiency.

1

Introduction

Electrical motors in general, and industry’s “workhorse ” AC induction motors in particular, represent a great potential and realizable energy savings. Motors account for approximately 64% of electricity consumption in the US, at a yearly WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070211

196 Computational Methods and Experimental Measurements XIII cost of US $ 112 billion. Every 1% reduction in motor demand therefore cuts 0.64 % or US $ 716, 800,000 off the industry-wide bill [1]. Motors lose energy in several ways. The difference between the power input and power output make up the motor losses, which are generally defined as noload and load-dependent losses. The no-load losses consist of the friction and windage loss which come from bearing friction and air resistance of the spinning fan/rotor respectively. Also, included are the iron losses which are a result of a combination of hysteresis and eddy current effects due to the changing magnetic fields in the motor’s steel core. The load dependent losses consist of the stator losses (product of stator input current squared and stator resistance at operating temperature), the rotor losses (product of induced rotor current squared and rotor resistance) and stray losses which come from additional harmonics due to the supply and circulating current losses in the magnetic steel and windings.

2

Statement of the problem

All losses with the exception of the core losses, can be expressed analytically. But core losses can only be expressed through empirical formulae, as many authors until to-date have found out [2]. This situation is due to the fact that the material used for magnetic circuits of the machines is non-linear. So in reality, it has not been possible to get analytical expressions to fully describe the phenomena of hysteresis and eddy currents in the machine cores. Therefore, in three phase induction motors, about 16% of total losses are core losses, while about 48% are copper losses. Hence, it is still important to continue finding how to reduce iron and copper losses so as to raise the motor efficiency. This paper proposes a new correlation between these two losses with the aim of getting minimized total machine loss in order to improve the efficiency.

3

The employed methodology

The parameters of any motor frame including its geometry form the initial data for developing the total loss optimization model. Through the developed model, the airgap magnetic flux density B δ and airgap diameter (stator bore) D are both varied. As a consequence, also varies the geometry of the stator and rotor teeth, slots and backs so as to achieve minimized total loss min(Pfe + Pcu) in the same motor frame. From this procedure, it is then possible to locate the optimal points of B and D respectively, for the airgap magnetic flux density and airgap diameter. Using these optimal points a new motor geometry is reconfigured which assures a minimized total loss of the motor. This approach is summarized in a flowchart which is given in Figure 1. The new motor design according to the above approach is simulated on a twodimensional finite element method (2D-FEM) to analyze the new motor response. Experimental results are also compared with the results of the design in



197

order to check for any improvement of motor efficiency. Also, empirical formulae are developed and validated which can greatly assist motor designers.

4

Motor prediction model

The optimisation was carried out through a model incorporating also, the complete stator and rotor geometry. The model was then implemented on a matlab 6.5 platform. 4.1 Assumptions for the approximate iron loss prediction model Prediction and subsequent optimisation of iron loss and copper loss was achieved by minimizing the losses in different sections of the core through minimization of their magnetic flux densities by influencing on the air gap flux density. The limit of the air gap flux density B for a given maximum induction in the stator and rotor teeth and backs, depended on the thickness of the teeth and the backs. In this model, the parasitic effects were not considered, with the exception of the surface and teeth reluctance losses in the rotor. Hence, for a given motor geometry and by varying the air gap induction B and air gap diameter D, the motor loss prediction model was developed with consideration of the following assumptions: (1) The non-linear magnetic behaviour of the iron material was taken into consideration by allocating a maximum flux density in different iron regions of the machine; (2) The leakage fluxes in the air gap and slots were neglected, such that, all magnetic flux crossing the air gap was assumed to flow radially through the teeth; (3) The overhang effects were neglected; (4) Stray losses were not included in the model; (5) A sinusoidally distributed air gap flux density was assumed; and (6) The current loadings in the stator and rotor were determined by the cooling capacity and the available slot areas in the motor cross-section. 4.2 Flowchart of optimization for the approximate iron loss prediction model The flowchart as shown in Figure1 for the approximate motor loss prediction model was developed on the basis of the motor geometry including the simple thermal model of a motor and the empirical iron loss formulae. The initial conditions were considered to be the nominal values of the original motor frame type 160 L-4 for 15 kW. The idea therefore, was to try the motor loss optimisation method on this original motor geometry, in order to get a new geometry with lowered total loss. So, this therefore, could lead to efficiency improvement of the same motor frame size, but with a re-configured geometry. ^

The iteration was conducted on varying the air gap induction, B δ and air gap diameter, D and also through the logic loops for the model to be able to compute the minimized value of motor loss, Ptot ( B, D ) , of the motor.



Start

No

Srtop(n,m) > = Sr(n,m)

Set initial motor parameters

Yes

Set maximum values of flux densities in stator and rotor sections

Srtop(n,m) > = Sr(n,m) Vary the air gap diameter (bore), D(n) : n = n+1

Compute stator and rotor winding copper loss:Psw, Prw, Pcu(n,m)=Psw+Prw

Vary the air gap induction, B(m): m = m+1

Compute maximum torque, maxTq(n,m) Compute geometrical stator and rotor dimensions No

maxTq(n,m) > Tr rated torque

Compute stator and rotor available slot area and current loadings

Yes Tq(n,m)=Tr

Compute Iron loss components in the stator and rotor

No

Tq(n,m) =Tr and modulus Pcu(n,m) Yes

Minimize sum, min(Pfe(n,m)+Pcu(n,m) as function of D and B

3-D plot, Ptot(B,D) and minPtot

Get optD and optB from minPtot(n,m)

Compute new geometry for stator and rotor with optD and optB for minPfe

END

Figure 1: Flowchart of optimisation for the motor loss prediction model. 4.3 Optimization of motor loss prediction model In this approximate motor loss prediction model, optimization of motor loss was achieved by minimizing the losses in different sections of the core by influencing the airgap flux density Bˆ δ and air gap diameter D. The limit of the air gap WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


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induction Bˆ δ for a given maximum induction in the stator and rotor teeth and back depended on the thickness of the teeth bts, btr and backs hrs, hrr. These geometrical dimensions depended also on the airgap diameter D. Therefore, by increasing Bˆ δ the available space areas for slots A ss , A sr decreased and as a consequence also, the current loadings S rs , S rr decreased. Conversely, by increasing the current loadings brought about an increase in the slot areas and a reduction in the widths of the teeth and the backs. As a result, the airgap induction Bˆ δ decreased. Also, the iron losses were considered together with the copper losses as in a real motor. Implementation of the model was done on the matlab 6.5 platform on the basis of the flowchart shown in Figure 1. Thereafter, the motor loss curve as a function of Bˆ δ and D was minimized as shown in Figure 3. In this model, however, the motor for every shaft power was dimensioned by maintaining the same nominal (rated) torque, the same outer diameter of the stator core and the same airgap thickness. This was done so in order to keep the same frame size. Through minimization of Ptot (B , D ) the optimal point was located and the corresponding values for B and D were determined as optB and optD. These new values optB and optD, then facilitated to compute the new motor geometry for stator and rotor cores with a result of reduced motor losses.

Ptot, Total Loss (Watts)

1500 1000 500 0

-500 0.4 1.5

0.3 0.2 A irgap Diam eter, (m eters )

Figure 2:

0.1

1 A irgap Induc tion, B , (Tes la) 0.5

Motor Loss, Ptot, (Watts) for a 15kW motor type M3AP 160 L-4 as a function of Air gap Induction, B, and Air gap Diameter, D.

4.4 Optimization of motor loss prediction model The developed approximate motor loss prediction model was applied to the frame size of a three phase squirrel cage motor, 15 kW 4-pole. The iron loss correction factors were introduced in the iron loss empirical expressions, in order to account for other loss making mechanisms, like the stray losses which were WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

200 Computational Methods and Experimental Measurements XIII not included in the model. This motor loss prediction model was able to give theoretical results as shown in Figure 3 and Figure 4. From Figure 3, it was possible to locate the point with minimum motor loss, minPtot, and the corresponding values, optB and optD. Then, values of randomly selected seventeen options around the optimal points optB and optD were computed and plotted as shown in Figure 4. They were generated through various combinations of peak values of magnetic flux densities in different motor sections for both stator and rotor. Through this procedure could generate a new motor geometry inside the same frame with minimized total loss min(Pfe+Pcu).

min(Pf e+Pcu), [ W]

1200 1100 1000 900 800 I

IV

VII

X

XIII XVI

Option Nos. I-XVII

Figure 3:

Seventeen options for motor loss optimization on a 15 kW, 160 L-4 motor.

4.5 Formulation of the correlation empirical formula between the iron loss and copper loss The statistical analysis of the model results was performed using a software [3]. So, the optimised parameters B, D, Pcu and Pfe were then statistically analysed. Empirical correlation relationships were therefore formulated as given in Equations (1), (2), (3) and the curve relationship shown in Figure 5: Following below are the developed novel empirical formulae: (1) D = a + b B + c B 2 + d B 3 , where: a = 4 .8 1 2 1 0 5 1, b = − 2 2 .1 1 2 4 7 6 , c = 3 1 .9 3 5 4 5 5 , d = − 1 8 .7 7 3 0 3 5 ; D[m] and B[T]. The limits are: 0 . 5700 < B < 0 . 6900 T and 0 . 1580 < D < 0 . 1760 m . 2 3 , (2) P fe = a + bB + cB + dB where: a = 1 8 6 3 6 3 .0 5 , b = − 8 3 4 4 5 6 .0 3 , c = 1 2 4 8 9 1 2 .5 , d = − 6 2 2 9 8 4 .3 9

;

and P fe [W] and B[T]. The limits are: 0 .6 3 0 0 < B < 0 .6 9 0 0

P

c u

T

=

and

5 3 9 .9 3 0 0 < P

a + b P

fe

+ c P

2 fe

fe

< 5 7 3 .8 0 0

+ d P


3 fe

,

W

. (3)


201

where: a = −4049416.8, b = 22282.868, c = −40.851615, d = 0.024954481; and P cu [W] and P fe [W]. The limits are: 5 3 6 .1 1 0 0 < P

fe

< 5 4 6 .1 5 0 0

W

a nd

3 8 2 .5 0 0 < Pcu < 4 5 1 .5 5 0 0

W

.

4 .5 8 e + 0 0 2

Y Ax is (P cu, W)

4 .4 5 e + 0 0 2

4 .3 1 e + 0 0 2

4 .1 7 e + 0 0 2

4 .0 3 e + 0 0 2

3 .8 9 e + 0 0 2

3 .7 6 e + 0 052.4 e + 0 0 2

5 .4 e + 0 0 2

5 .4 e + 0 0 2

5 .4 e + 0 0 2

5 .4 e + 0 0 2

5 .5 e + 0 0 2

5 .5 e + 0 0 2

X A x is (P fe , W )

Figure 4:

Relationship Pcu vs. Pfe for 15 kW motor M3AP 160 L-4.

Figure 5: Air gap flux density and the higher harmonics present including the fundamental one.

Figure 6:

Chart of magnetic field lines for no-load motor operation.

4.6 The Finite Element Method Analysis of new motor geometry The evaluation of electromagnetic field in all the simulations was based on the finite element computation of the unknown represented by the magnetic vector potential, a vector normally oriented to the computation domain, [5]. Two magneto-harmonic models of no-load operation for rated source voltage and frequency were employed: (1) simulation with a value lower than the rated WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

202 Computational Methods and Experimental Measurements XIII slip; and (2) simulation with rated slip and with a value of rotor bar resistivity 5 much larger than the real value; a value 10 times greater was used. Both options could give practically the same results, which are shown in Figure 5 and Figure 6. The main numerical results of no-load simulation were: (1) the value of no-load current for each phase was, I10 = 18.7A, 19A, 20.9A; (2) stator and rotor iron loss was 343 W. Table 1: Motor type for ABB frames

Computed data from original geometry

Simulated data for new geometry with developed model

Simulated data for new geometry with FEM

Experimental data on original motor

M3AP 160 L-4

A) At unity iron loss correction factors: Kbts=1, Kbrs=1, Kbtr=1, Kbrr=1 Pcu=346.74 W Pfet=68.18 W, Pfer=170.35 W Ptar=5.87 W Pytr=77.65 W Pexc=0.71 W Pfe=322.76 W Ptot=669.50 W

A) At unity iron loss correction factors: Kbts=1, Kbrs=1, Kbtr=1,

A) At unity iron loss correction factors: Kbts=1, Kbrs=1, Kbtr=1,

A) Standard efficiency type Pcu=325.24 W

B) At non-unity iron loss correction factors: Kbts=1.51, Kbrs=2.234 Kbrr=1.52, Kbtr=1.20 Pcu=346.74 W Pfet=102.95 W Pfer=380.55 W Ptar=7.05 W Pytr=118.03 W Pexc=0.71 W Pfe=609.29 W Ptot=956.03 W

5

Comparison of results.

Kbrr=1 Pcu=383.48 W Pfet=57.86 W Pfer=139.06 W Ptar=10.72 W Pytr=46.47 W Pexc=0.58 W Pfe=254.68 W Ptot=638.16 W B) At non-unity ironloss correction factors: Kbts=1.51, Kbrs=2.234 Kbrr=1.52, Kbtr=1.20 Pcu=383.48 W Pfet=87.37 W Pfer=310.66 W Ptar=12.86 W Pytr=70.63 W Pexc=0.58 W Pfe=482.09 W Ptot=928.65 W

Kbrr=1 Stator iron loss , 207.20 W Rotor iron loss, 136 W Total iron loss, 343.2 W

Pfet=57.9 W Pfer=180.4 W Ptar=4.60 W Pytr=45.9 W Field factor=1.08 Pfe=337.09 W Ptot=662.33 W No load Iron loss Calculated=310 W Tested=262 W B) High Efficiency type (Eff.1) Pcu=325.24 W Pfet=55.1 W Pfer=171.6 W Ptar=2.00 W Pytr=18.8 W Field factor=2.15 Pfe=532.05 W Ptot= 857.29 W

Discussion of results

The winding materials are assumed to be copper for the stator winding and aluminium for the rotor winding and the steel sheet material is used for the magnetic circuit. For a given airgap induction and maximum flux densities in the teeth and backs, the copper loss is calculated using the formulae according to Chandur [2]. Through the methodology outlined in sec.3, the developed total loss prediction model made possible to initially locate the optimal points of airgap magnetic induction, airgap diameter and the minimized total loss, min(Pfe+Pcu).



203

With these points, a new motor geometry was re-configured and verified its performance including comparison with experimental results. The analytical and experimental results are summarized in Table 1.

6

Conclusion

In comparing the analytical model results with the experimental data there is a good and acceptable agreement. The model gives a lower value of the total loss than with the results from the original motor frame. So through this motor loss model the motor efficiency can be improved. Therefore, this model and the developed empirical formulae given as equations (1), (2) and (3) are very useful tools to motor designers.

List of symbols Ass

=

Asr

= available space area for rotor slots,

available space area for stator slots,

B, Bˆ δ = airgap magnetic flux density, bts = width of stator teeth, btr = width of rotor teeth, D= airgap diameter or bore of stator core, hrs = thickness of back of stator core, hrr = thickness of back of rotor core, n = matrix row in iteration of a model parameter, m = matrix column in iteration of a model parameter, S rs = stator current loading,

S rr =

rotor current loading,

Srtop = the biggest value between the two current loadings, Tq = computed developed torque by the motor, Tr = rated motor torque, optB= optimal point of airgap magnetic flux density, optD= optimal point of airgap diameter, Psw = stator winding copper loss, Prw = rotor winding copper loss, Pcu = total copper loss, Pfet = stator teeth loss, Pfer = stator back loss, Ptar = rotor teeth reluctance loss, Pytr = rotor surface reluctance loss, Pexc = excess or anomalous loss of stator back, Pfe = total iron loss, Ptot = total motor loss, minPtot= minimized total motor loss.

References [1]

Application Note, “An In-Depth Examination of an Energy Efficiency Technology”, Pacific Gas and Electric Company May 1997. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


[3] [4] [5] [6] [7]

[8]

Chandur, S. “Electrical machine design and analysis of induction and permanent magnet motors”, Department of Electric Power Engineering, Electrical Machines and Power Electronics, Royal Institute of Technology, Stockholm, August 2000. Chapra S. & Canale R. “Numerical Methods for Engineers”, McGraw 1998. Electrical Technology, “http://www.very last page of the internet.com/elctromagnetic dv/muller/electrical_gene…16/1/02”. Masato E. and Kenji O. “Designing a low-loss induction motor considering the vector magnetic properties”, IEEE Transactions on Magnetics, Vol.38, No.2, pp 877-880, March 2002. Oriano B., Aldo C., Mario C. and Maurizio R. “Iron losses in Electrical Machines: Influence of different material models”, IEEE Transactions on Magnetics, Vol.38, No.2, pp 805-808, March 2002. Saanane B.B, Nzali A.H and Chambega D.J. “Design Approach of Squirrel Cage Induction Motors by Use of Iron Loss Optimization Method for Improving Efficiency”, Electrocomp 2005, Wessex Institute of Technology, WIT Transactions on Modeling and Simulation, Vol. 41, pp 621-629, © 2005 WIT Press, UK, March 2005 Stumberger B., Gorican V., Stumberger G., Hamler A., Trlep M. and Jesenik M. “Accuracy of iron loss calculation in electrical machines by using different iron loss models”, Journal of Magnetism and Magnetic Materials, pp 254-255, 2003.



205

Effect of Zr addition on the fatigue strength of Cu-6Ni-2Mn-2Sn-2Al alloy M. Goto1, S.-Z. Han2, C.-J. Kim2 & N. Kawagoishi3 1

Department of Mechanical Engineering, Oita University, Japan Korea Institute of Machinery & Materials, Korea 3 Department of Mechanical Engineering, Kagoshima University, Japan 2

Abstract Fatigue tests of Cu-6Ni-2Mn-2Sn-2Al alloy smooth specimens were carried out to clarify the effect of trace Zr on the fatigue strength. The growth behaviour of a major crack, which led to the final fracture of the specimen, was monitored to study the physical basis of fatigue damage. When stress amplitude was less than σa = 350 MPa, the fatigue life of Zr-containing alloys was about two times larger than that of alloys without Zr. When σa > 350 MPa, increments of fatigue life due to Zr decrease with an increase in σa and the increments were negligible at σa = 400 MPa. Increased fatigue life due to Zr addition resulted from an increase in crack initiation life and microcrack growth life. The growth rate of a small crack was determined by a term, σanl, independent of Zr addition. The effects of trace Zr on fatigue strength were discussed with relation to the initiation and propagation behaviour of a major crack. Keywords: fatigue strength, small crack, crack growth rate, plastic replication technique, Cu-Ni-Sn alloy systems.

1

Introduction

The substitutes for Cu-Be alloys with a high production cost have been developed. Cu-Ni-Sn alloy systems have been regarded as potential substitutes for Cu-Be alloys. Among the Cu-Ni-Sn alloy systems, Cu-9Ni-6Sn alloys [1–4] are considered to have the best strength/ductility combination. When hot rolling is applied to produce sheet products, hot cracking problems occur [5] as a result of Sn-rich segregates formed during the solidification in castings. Cu-6Ni-2MnWIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070221

206 Computational Methods and Experimental Measurements XIII 2Sn-2Al alloys with a lower production cost and improved hot workability [6] have been recently developed as possible substitutes for Cu-9Ni-6Sn alloys. Mn is isomorphous compared to Cu and can partially replace the high-cost Ni which is beneficial to castability and tensile strength. The low Sn content makes it possible to be hot-rolled to produce sheet products. The addition of Al, which is a strong solid solution strengthener in Cu, gives this alloy a proper tensile strength without sacrificing the tensile ductility. The normal cast process for alloy production gives low production cost. Thus, Cu-6Ni-2Mn-2Sn-2Al alloys are applicable for various components in the electrical industry. However, further increase in the strength of alloys is required. It is known that the strength of pure copper is improved by the addition of a small amount of Zr, e.g., the precipitation hardening due to aging tends to saturate with the addition of only 0.15 % Zr. This indicates that the effects of small amounts of Zr addition to Cu6Ni-2Mn- 2Sn-2Al alloys on the strength should be studied. On the other hand, when alloys are used for their actual components, it is important to estimate the fatigue damage of the material. To estimate fatigue damage precisely, the physical basis of fatigue damage must be clarified. However, there are only a few studies on fatigue of Cu-6Ni-2Mn-2Al alloys [7]. In the present study, fatigue tests of Cu-6Ni-2Mn-2Al alloys without and with Zr (0.1 and 0.3 %) have been carried out. The physical basis of fatigue damage and the effects of Zr addition on fatigue strength were discussed based on the initiation and growth behaviour of a crack monitored by the plastic replication technique.

Figure 1:

2

Microstructure of the materials; (a) NZ, (b) Z1 and (c) Z3.

Experimental procedures

Cu-6Ni-2Mn-2Sn-2Al alloys without and with Zr (0.1 and 0.3 %) were prepared using a vertical continuous casting of high purity elemental Cu, Ni, Mn, Sn, Al and Zr in air. After casting, solution treatment was conducted at 850 oC for 1 h, followed by swaging with a total swaging amount of 80 %. The diameter of swaged bars was 11 mm. Prior to machining the specimens, materials were aged at 400 oC for 3 h. From here on, alloys without Zr, alloys with 0.1 and 0.3 % Zr are referred to NZ, Z1 and Z3, respectively. Fig.1 shows the microstructure of WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


207

the materials. Mechanical properties after aging were 876, 915 and 902 MPa tensile strength, 6.0, 7.0 and 6.3 % elongation, and 275, 288 and 287 Vickers hardness (load: 9.81 N) for NZ, Z1 and Z3, respectively. Fig. 2 shows the shape and dimensions of the specimen. The round bar specimens with 5 mm diameter were machined from the bars. Although the specimens have a shallow circumferential notch (depth: t = 0.25 mm, radius: ρ = 20 mm), the strength reduction factor for this geometry is close to unity, so that the specimens can be considered as plain specimens. Before testing, all specimens were electro-polished to remove about 20 µm from the surface layer, in order to facilitate changes in the surface state. All tests were carried out using a rotating bending fatigue machine with a constant bending moment type of a capacity of 14.7 Nm operating at 50 Hz. Specimens were fatigued at ambient air under constant stress amplitudes. The observation of surface fatigue damage and the measurements of crack length were made via plastic replicas using an optical microscope at a magnification of x 400. The stress value used in the present study is that of the nominal stress amplitude, σa, at the minimal cross section. The crack length, l, is the length along the circumferential direction on the surface.

Figure 2:

Shape of the specimen.

3 Experimental results and discussion 3.1 Mechanical properties and S-N curve Fig. 3 shows the S-N curve. The S-N curve of the Zr-containing alloys shifted toward the long life field, and showed a change in slope at around σa = 350 MPa. Namely, when stress amplitude is less than σa = 350 MPa, fatigue life of Zrcontaining alloys is about 2 times larger than without Zr. When σa > 350 MPa, differences in fatigue life between alloys with and without Zr decrease with an increase in stress amplitude, and no increase in fatigue life at σa = 400 MPa is observed. In addition, the value of fatigue limit stress at 107 cycles, σw, was about 5 % larger in alloys with Zr than in alloys without Zr. With regard to the differences in Zr content between Z1 and Z3, Zr effects on fatigue strength were negligibly small.



Figure 3:

Figure 4:

S-N curve.

Typical growth paths of a major crack.

3.2 Crack initiation and growth behaviour Fig. 4 illustrates the typical growth paths of a major crack, which led to the final fracture of the specimen. The initiation site of a major crack was studied from the direct observation of surface etched after the detection of a 0.2mm length crack [7]. The results showed that, at σa = 300 and 350 MPa, the crack initiation site of NZ was grain boundaries (GBs), whereas the sites of Z1 and Z3 were the slip bands inside the grains. At σa = 400 MPa, both the GBs and slip bands were the initiation site of major cracks independent of the existence of Zr. With regard to other small cracks initiated after the major crack initiation, the initiation sites of those cracks were both the GBs and slip bands independent of stress amplitude and Zr addition. Crack growth behaviour was monitored successively by the plastic replication technique, showing that, after initiation of a crack from GBs in NZ alloy, although the crack propagated along the initiation direction for a small WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


209

distance, it changed growth direction and propagated with a shear mode along the slip orientation within the adjacent grains. In Zr-containing alloys, a shearmode crack from slip bands propagated along a slip orientation to GBs. A change in propagation direction occurred at GBs, followed by crack growth with shear-mode within the adjacent grains. In all alloys without and with Zr, when adjacent grains did not have suitable slip orientations for shear crack growth, crack growth with tensile mode was observed. Although growth behaviour of a small crack is influenced by the inhomogeneity of its microstructure, the influence of the microstructure on growth behaviour can be negligible for cracks larger than 0.3 mm. The propagation of relatively large cracks (l > 0.3 mm) was principally controlled by tensile mode. Similar growth behaviours have been observed in other cyclic-softening alloys such as age-hardened aluminium alloys [8–10]. As for the differences in Zr content, there were no significant differences in crack initiation and growth behaviour between the Z1 and Z3 alloys. Fig. 5 shows the growth curve, the lnl versus N relation, of a major crack. At σa = 300 and 350 MPa, the relation shifted toward the long life field due to the existence of Zr, but negligible effects of Zr addition on the growth curve were observed at σa = 400 MPa. Namely, at σa = 400 MPa, no significant differences in both the initiation life of a grain size order crack (e.g., l ≑ 30 µm) and the slope of crack growth curves were observed. Conversely, at σa = 300 and 350 MPa, addition of Zr increases the crack initiation life and makes the slope of the growth curves decrease. Strictly speaking, at σa = 350 MPa, crack growth behaviour in the range l < 0.1 mm is strongly retarded due to Zr addition. For l > 0.1 mm, although the slope of the growth curve for Zr-containing alloys tends to be slightly smaller than in alloys without Zr, its effect on growth life is practically negligible. Thus, Zr addition affects the behaviour of a crack propagating with low driving force (in other words, the smaller the crack length and stress amplitude, the larger the retardation of crack growth). Retardation of a 10

σa = 400M P a

Crack length l mm

350M P a

300M P a

1

0.1

0.01 0.0

○,●:N Z △,▲:Z1 □,■:Z3

5.0x105

1.0x106

1.5x106

2.0x106

2.5x106

Number of cycles N

Figure 5: Crack growth curve (the lnl, logarithm of crack length, vs. N relation).


210 Computational Methods and Experimental Measurements XIII crack growth may be related to the strengthening of the matrix due to Zr addition. Since strengthening of the matrix is not relatively large (an increase in Vickers hardness was about 5 %), significant retardation of growth behaviour might occur for a small crack propagating under a low stress amplitude. On the other hand, the effects of difference in Zr content on crack initiation and crack growth behaviour are not clear. Fig. 6 illustrates σa versus Ni and Ni→f. Ni and Ni→f refer to the initiation life of a 0.05 mm crack and to the crack growth life from l = 0.05 mm to the fracture, respectively. When σa ≲ 350 MPa, Ni for Z1 and Z3 is about 2 to 3 times larger than NZ. However, at σa = 400 MPa, a negligibly small increase in Ni due to Zr addition is observed. On the other hand, increments of Ni→f due to Zr addition gradually decrease with an increase in σa, and there is no difference in Ni→f at σa = 400 MPa. It has been shown in the previous report [7] that the significant large enhancement of crack initiation life in Zr-containing alloys was resulted from the strengthened GBs caused by the participation of Zr compounds. In order to determine the compositions of Zr compounds precipitated at GB regions, energy-dispersive X-ray diffractometry (EDX) was used. However, accurate analysis failed because of the too small compound sizes. From the binary phase diagram, Zr never exists, taking the form of single particles in CuNi-Mn-Sn-Al alloys because Zr tends to form compounds in Cu alloys. Table 1

Figure 6:

Effects of Zr on crack initiation life, Ni, and crack growth life, Ni→f.

Table 1:

Enthalpy of formation (∆Hfor) of the binary zirconium compounds. Compound ∆ H for Cu9 Zr2 -34 Ni3 Zr2 -73 Mn5 Zr3 -23 SnZr -86 AlZr -83 ∆Η for : kJ/mole of atoms

Element Cu Ni Mn Sn Al



211

shows the formation enthalpy of Zr-related compounds. SnZr and AlZr compounds show large formation enthalpy, while Al tends to solute in Cu matrix. Moreover, Sn tends to segregate at the GB regions when alloys were cast. Thus, we concluded that particles precipitated at GB regions are SnZr compounds. 3.3 Effect of Zr on the growth rate of a small crack

Crack growth rate dl/dN mm/cycle

Fig.7 shows the dl/dN versus l relation of major cracks at σa = 300 MPa. Here, the crack growth rate dl/dN is calculated from the growth curve approximated by a smoothed curve. The range of crack length specified by "SMCG" indicates a crack propagating with shear mode. When the crack length was small (e.g., l < 0.2 mm), a crack tended to propagate with shear mode. For the crack length in excess of 0.2 mm, the propagation mechanism was principally dominant by tensile mode, and dl/dN is nearly proportional to l. Since a shear microcrack is strongly influenced by the inhomogeneity of its microstructure [11–15], the fluctuation of dl/dN is relatively larger than that of large crack propagating with tensile mode. In addition, the growth rate of SMCG is higher than that of a tensile-mode crack with corresponding crack length, evaluated from the relation holds for l > 0.3 mm.

10-5

σa = 300M P a

○：N Z △：Z1 □：Z3

1

10-6

10-7

SM C G

1

S M C G :S hear m ode crack grow th

SM C G

0.1

1

Crack length l mm Figure 7:

dl/dN versus l relation (σa = 300 MPa).

The growth rate of fatigue crack was usually evaluated in terms of a stress intensity factor range ∆K. Here, ∆K is the effective parameter describing the stress field in the vicinity of a crack when the condition of small scale yielding at a crack tip is satisfied. The value of ∆K for an infinite plate with a through thickness crack is given by the equation ∆K = ∆σ (πa)1/2. This equation indicates that the stress range has to be higher for a small crack in order to get the same WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

212 Computational Methods and Experimental Measurements XIII growth rate as for a large crack. However, when a sufficiently small crack propagates at finite growth rate (e.g., 10-6 - 10-3 mm/cycle), the condition of small scale yielding is not usually satisfied. Thus, the growth rate of a small surface crack cannot be determined uniquely by ∆K. Nisitani and Goto [16–20] have studied small crack growth behaviour using carbon steels, low alloy steels, aluminium alloys, Ni-base superalloys and indicated that the growth rate of a small crack in which the condition of small scale yielding does not hold can be uniquely determined by a term σanl, but, not by the stress intensity factor range, i.e. (1)

1

dl/dN mm/cycle

(a)

NZ l = 0.3 - 1.5 mm n = 8.0

Crack growth rate

Crack growth rate

dl/dN mm/cycle

In eqn (1), C1 and n are material constants. Furthermore, they have proposed a convenient method for predicting the fatigue life based on the small crack growth law. The validity of the method has been confirmed by its application to the other researchers fatigue data [18]. The expression σanl (n = 3) was first proposed by Frost and Dugdale [21]. They applied it to comparatively large cracks in which the condition of small scale yielding nearly holds.σa3l can be considered as an application for ∆K, whereas σanl in the present study is a parameter for crack propagating under large scale yielding. Fig. 7 suggested that the dl/dN is proportional to l for a crack larger than l = 0.3 mm under the constant stress amplitude. The dependency of dl/dN on stress amplitude was also investigated and it was found that the dl/dN was proportional to σan for a constant crack length. The value of n for the present alloys was 8.0 independent of Zr addition. Putting these results together, we obtained the small crack growth law, eqn (1) with n = 8. Fig. 8 shows the dl/dN vs. σa8l relation. The growth rate of a small crack (l > 0.3 mm) can be uniquely determined by eqn (1). With regard to the effect of Zr, trace Zr makes the growth rate slightly decrease.

10-5

1

10-6 1019

σanl

1020

(MPa)nmm

1021

(b)

▲： Z1 ■： Z3

10-5

NZ

1 -6

10

1019

1

σanl

l = 0.3 - 1.5 mm n = 8.0

1020

(MPa)nmm

1021

Figure 8: Crack growth data (dl/dN vs. σanl relation); (a) relation for NZ alloy, (b) relation for Zr-containing alloys and comparison of the relation between non-Zr and Zr-containing alloys. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


4

213

Conclusions

In order to study the fatigue behaviour of Cu-6Ni-2Mn-2Sn-2Al alloys without Zr, or with 0.1 and 0.3 % Zr, rotating bending fatigue tests of smooth specimens have been carried out. The plastic replication technique was used for monitoring crack initiation and growth behaviour. Here, alloys with no Zr, or containing 0.1 % Zr and 0.3% Zr were identified by the terms NZ, Z1 and Z3, respectively. The main conclusions can be summarized as follows: (1) When the stress amplitude is less than σa = 350 MPa, fatigue life of Z1 and Z3 is about 2 to 2.5 times larger than that of NZ. When σa > 350 MPa, increments of fatigue life due to Zr addition decrease with an increase in σa and the increments are negligible at σa = 400 MPa. Thus, the S-N curve for Z1 and Z3 alloys shows a change in slope at around σa = 350 MPa. The increase in fatigue life in the range of σa ≲ 350 MPa results from increases in crack initiation life and micro-crack growth life. With regard to differences in Zr content between Z1 and Z3, their effects on fatigue strength are negligibly small. (2) The preferential initiation sites of a major crack, which led to the final fracture of specimens, in the range of σa ≲ 350 MPa are grain boundaries (GBs) and slip bands for alloys with and without Zr, respectively. At σa = 400 MPa, both GBs and slip bands are initiation sites of the major cracks independent of the existence of Zr. (3) Zr addition generated strengthened GBs resulting from precipitation of SnZr compounds. Strengthened GBs contributed to the increase in crack initiation life. (4) The increase in microcrack growth life is related to the strengthening of matrix due to Zr addition. However, since the strengthening of the matrix is not large (e.g., the increase in HV is about 5 %), significant retardation of growth behaviour can occur for a small crack propagating under a low stress amplitude less than σa = 350 MPa. (5) Growth rate of a small crack can be determined by a term σanl. n is a material constant. The value of n was about 8 independent of Zr addition.

Acknowledgement This research was partially supported by a grant (code #: 06K1501-00230) from 'Center for Nanostructured Materials Technology' under '21st Century Frontier R&D Programs' of the Ministry of Science and Technology, Korea.

References [1] [2] [3] [4]

Plewes, J.T., Metal Trans., 6A, pp. 537-544, 1975. Lefevre, B.G., Dannessa, A.T., Kalish, D., Metall. Trans., 9A, pp. 577586, 1978. Ray, R.K. & Narayanan, S.C., Metall. Trans., 13A, pp. 565-570, 1982. Sato, A., Katsuta, S. & Kato, M., Acta Metall Trans., 13A, pp. 633-640, 1988. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

214 Computational Methods and Experimental Measurements XIII [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

Han, S.Z., Kim, H.I, Lee, J.M. & Kim, C.J., J. Kor. Inst. Machinery Mater., 37, pp. 882-886, 1999. Rhu, C.J., Kim, S.S., Han, S.Z, Jung, Y.C. & Kim, C.J, Scripta Mater., 42 pp. 83-89, 2000. Goto, M., Han, S.Z., Kim, C.J. & Kawagoishi, N., Mater. Letters, to be published, 2007. Goto, M., Nisitani, H., Kawagoishi, N., Miyagawa, H. & Chujoh, N., Trans. Jpn Soc. Mech. Eng. (in Japanese), A-59, pp. 205-211, 1993. Goto, M., Kawagoishi, N., J. Soc. Mater. Sci. Jpn (in Japanese), 45, pp. 675-680, 1996. Goto, M. & DuQuesnay, D.L., SAE Technical Paper Series, No.970703, Soc. Automotive Engng-USA, pp. 1-7, 1997. Suh, C.M., Yuuki, R. & Kitagawa, H., Fatigue Fract. Engng Mater. Struct., 8, pp.193-203, 1985. Ochi, Y., Ishii, A. & Sasaki, S.K., Fatigue Fract. Engng Mater. Struct., 8, pp. 327-339, 1985. Goto, M., Fatigue Fract. Engng Mater. Struct., 14, pp. 833-845, 1991. Goto, M., Fatigue Fract. Engng Mater. Struct., 16, pp. 795-809, 1993. Goto, M., Fatigue Fract. Engng Mater. Struct., 17, pp. 635-649, 1994. Nisitani, H. & Goto, M., The Behaviour of Short Cracks, EGF 1, eds. K.J. Miller & E.R. de los Rios, Mech. Eng. Publications, London. pp.461-478, 1987. Goto, M. & Nisitani, H., Trans. Jpn. Soc. Mech. Eng. (in Japanese), A-56, pp.1938-1944, 1990. Nisitani, H., Goto, M. & Kawagoishi, N., Eng. Fract. Mech., 41, pp. 499513, 1992. Goto, M., Fatigue Fract. Engng Mater. Struct., 17, pp. 171-185, 1994. Goto, M. & Knowles, D.M., Eng. Fract. Mech., 60, pp. 1-18, 1998. Frost, N.E. & Dugdale, D.S., J. Mech. Phys. Solids, 6, pp. 92-110, 1958.



215

An analysis of superplastic free forming at constant pressure G. Giuliano & S. Franchitti Department of Industrial Engineering, Cassino University, Italy

Abstract Superplastic materials are characterised by very low strength during forming and by large plastic deformations. Superplasticity is observed in some metal alloys when deformed under particular conditions, namely: • very fine grain size (10µm); • high temperature (greater than about one-half the absolute melting point); • a controlled strain rate (10-4–10-2 s-1). In recent years, there has been a considerable interest in the aircraft and automotive industries using superplastic forming to obtain complex parts. In these industrial sectors superplastic forming (SPF) of sheet metal has been used to produce, with a low number of mechanical steps, several sheet metal components that are lighter and stronger than conventional components. The increasing spread of superplastic forming processes has focussed the attention of technologists onto problems of optimising process parameters. The design of these processes is more difficult than conventional manufacturing operations. In fact, to successfully produce an SPF component it is essential to control a variety of different parameters during the forming process: temperature, material grain size, strain-rate distribution due to the pressure forming and the final distribution of thicknesses. In this study the superplastic behaviour of PbSn60, an alloy that is superplastic at room temperature, is evaluated during constant pressure free bulging. A set of experimental tests were carried out in order to determine the characteristic parameters of the PbSn60. Keywords: superplastic forming, constitutive equation, finite element method.



1

Introduction

An ever-increasing number of fields in mechanical industry need to be able to produce light parts of complex shape and elevated mechanical strength. In particular, the automotive and aeronautical industries require innovatory technologies in the field of sheet metal forming processes: being able to have materials and technological processes capable of obtaining elevated plastic strain, is very attractive to designers. A decisive innovation in the field of sheet metal forming processes, was the discovery of some metallic alloys that in given microstructural conditions of temperature and strain rate, exhibit a behaviour called superplasticity. This is characterised by an extraordinary ductility, as can be seen from percentage elongations that are one or two orders of magnitudes greater than those observed for conventional metals and alloys [1,2]. The superplastic materials of industrial interest are aluminium, titanium and magnesium-based alloys. The need to master process parameters (particularly the pressure-time load curve) during superplastic forming requires more and more sophisticated simulation techniques. Nowadays, the finite element method represents the most common tool for planning sheet metal forming processes [3–13]. The starting point for a reliable simulation of the process is to know the characteristics of the material. Being able to accurately model the deformation conditions is critical in superplasticity; in fact the constitutive equation, used to define the relationship in a superplastic material between the flow stress σ, the strain ε and the strain rate, ε , considers the effect of multiple factors, namely temperature, grain size, fraction of cavities, strain hardening/softening and strain-rate. Various researchers have proposed equations to describe the behaviour of superplastic materials using models based on three different levels: macrocospic, mesoscopic and atomic levels [14]. On the basis of the phenomenological aspects of superplastic behaviour it is possible to note a strong correlation between the flow stress and the strain rate and a weak one with the strain and the grain size: The equation can be written as: σ = Kε m ε n d p (1) Often the dependence of the flow stress on the grain size is neglected, and hence the constitutive equation is simplified: (2) σ = Kε m ε n The method usually used to characterise superplastic material is the tensile test in which the specimen is subjected to monoaxial stress. It has been noted, however, that the free forming tests enable the characteristic parameters to be determined more reliably since the material is subject to stress conditions (biaxial stress) closer to those of the real forming process [15]. This study analyses the free forming process, at constant pressure, of PbSn60 an alloy that is superplastic at room temperature. It aims to verify whether eqn WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


217

(2) is easy to implement in a computational model and whether it represents the behaviour of superplastic material to a good degree of accuracy.

2

Experimental work

The constant pressure free forming experiments were performed using the equipment designed by the authors and shown in Figure 1 [16]. TRANSDUCER OF PRESSURE

PROPORTIONAL VALVE

COMPARATOR VIDEOCAMERA

FORMING DIE

Figure 1:

AIR COMPRESSOR

VOLTAGE GENERATOR

The equipment used to carry out the experimental tests.

It comprises a compressor that sends pressurised air to a proportional valve whose function is to regulate the pressure within the circuit; a pressure transducer; and an open die that includes the upper and lower parts, both of which made of steel. The upper plate measures 180x100x100 mm and it has a 60mm diameter central hole of through which the sheet is formed and four holes for linking it up to the lower plate. The latter is of similar dimensions to the former and has a central hole too to allow the compressed air to be fed through (figure 2). The material used for SPF was 0.3mm thick Pb/Sn alloy composed of (in wt. %) 60%Pb and 40%Sn (non eutectic composition). This material displays superplastic characteristics at room temperature and it therefore does not require the use of a furnace and expensive apparatus. The superplastic PbSn60 alloy has mechanical properties that are too poor to be used to produce real industrial components, but it proves to be advantageous for laboratory activity. This material is commercially available in the form of PbSn60 welding bars; in order to drastically reduce grain size, and therefore to respect one of the conditions necessary for exhibiting superplastic behaviour, it is subjected to bending and rolling cycles [16]. Two series of tests were carried out: the first at a forming pressure of 0.10MPa and the second at a pressure of 0.18MPa. For each single pressure value 5 tests were carried out for a total of 10 tests, and the time step taken for the sheet to pass through the normalized height H=0 to H=1 was measured for each single test. The normalized height is defined as H=h/a, where h is the polar dome height and a is the die radius.



Figure 2:

Shape of the die.

Since it was not possible by means of an analogic comparator to memorize the heights reached by the sheet, the trend of height versus time was obtained using a video camera. The videos of the forming test were uploaded to a PC by means of an image acquisition system, thereby enabling the data to be read. Figure 3 shows the results obtained. The bars showing the mean standard deviation are highlighted. The results obtained were analysed using the Grubbs test to check for the presence or not of anomalous values. The tests predict that for each value of H, a shift in the time measured compared to the mean value is determined. The measured shift in relation to the standard deviation of the population under examination is compared to a critical value linked to the distribution of the measured data. The constant of the material m was determined as shown by various authors [17], by means of the expression: ln(p 2 / p1 ) (3) m= ln(t1 / t 2 ) where, in this paper, t1 and t2 are the forming time values necessary to produce the same dome geometry (that is the same value of H) at constant pressures p1 and p2, respectively. Table 1 summarizes the m values measured during the forming process. Figure 4 shows how this parameter is dependent on the deformation obtained (H). In particular it proves to decrease slightly during deformation. A means value of m between H=0.2 and H=1 will be considered for the numerical analysis. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


400

219

p=0.10MPa

350 300

t[s]

250 200 150 100 50 0 0

0.2

0.4

0.6

0.8

1

1.2

0.8

1

1.2

H

p=0.18MPa

120 100

t[s]

80 60 40 20 0 0

0.2

0.4

0.6 H

Figure 3:

3

Experimental results.

Numerical analysis

Using commercial finite element software, a constant pressure free bulging process of the superplastic PbSn60 alloy was simulated. The element type used in the analysis is a four-node, isoparametric, arbitrary quadrilateral written for axisymmetric applications [18]. Due to the symmetry of the geometry, load and WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

220 Computational Methods and Experimental Measurements XIII constraint conditions, only half of the cross-section of the sheet metal was analysed. Moreover, it was necessary to impose constraint conditions on the periphery of the sheet, in order to simulate the action of a blank holder. The die has a circular geometry with an aperture diameter of 60.0 mm and a die entry radius of 2.0 mm. Rigid-plastic flow formulation was applied to the superplastic forming analysis [19]. Table 1:

Experimental values of the strain-rate sensitivity index. H

m

0.200 0.267 0.333 0.400 0.467 0.500 0.533 0.600 0.667 0.733 0.800 0.867 0.933 1.000

0.467 0.482 0.481 0.478 0.476 0.473 0.470 0.468 0.465 0.462 0.458 0.454 0.449 0.467

1 0.8

m

0.6 0.4 0.2 0 0.2

0.4

0.6

0.8

H

Figure 4:

Experimental trend of m value.


1.0


400

221

p=0.10MPa

350 300

t[s]

250 200 150 100 50 0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.80

1.00

1.20

H

p=0.18MPa

120 100

t[s]

80 60 40 20 0 0.00

0.20

0.40

0.60 H

Figure 5:

Comparison between the numerical and experimental results.

The constants of the material, introduced in the code, for the process simulation, were determined starting from an original characterisation methodology proposed by authors in [17]. The material constants were determined as: m=0.468 n=-0.029 K=146.515 MPa*s WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

222 Computational Methods and Experimental Measurements XIII From the results of the numerical analysis it is possible to plot the trend of time versus the normalized polar height H. It can be seen that this curve lies between the bars of the standard deviation of the mean experimentally obtained results (figure 5).

4

Conclusions

This study analysed the behaviour of the superplastic Pb-Sn60 alloy as it underwent a superplastic free forming process at a constant pressure. A comparison between the experimental results and the ones from a numerical analysis of the forming process justifies the approximation introduced to determine the constant m. In fact to consider a simplified constitutive equation (eq.2), the numerical analysis uses a mean value of m in relation to the deformation (H).

References [1] [2] [3] [4] [5] [6]

[7]

[8]

[9]

Hamilton C.H. and Ghosh A.K., Superplastic sheet forming, Metals Handbook, 1988. Pilling J. and Ridley N., Superplasticity in Crystalline Solids, The Institute of Metals, London, 1989. Sadeghi R. and Pursell Z., Finite element modeling of superplastic forming with precise dies, Superplasticity and Superplastic Forming, The Minerals, Metals & Materials Society, Warrendale, 1995. Chandra N., 1988, Analysis of superplastic metal forming by a finite element method, International Journal for Numerical Methods in Engineering, vol. 26, pp. 1925–1944. Chandra N. and Rama S.C., 1992, Application of finite element method to the design of superplastic forming processes, Journal of Engineering for Industry, vol. 114, pp. 452–458. Bonet J., Bhargava P. and Wood R.D., 1997, Finite element analysis of the superplastic forming of thick sheet using the incremental flow formulation, International Journal for Numerical Methods in Engineering, vol. 40, n°17, pp. 3205–3228. Bellet M., Massoni E. and Chenot J.L., 1987, A viscoplastic membrane formulation for the 3-D analysis of superplastic forming of thin sheet, Proceedings of the International Conference on Computational Plasticity, pp. 917–926. Huh H., Han S.S., Lee J.S. and Hong S.S., 1995, Experimental verification of superplastic sheet-metal forming analysis by the finiteelement method, Journal of Materials Processing Technology, vol. 49, pp. 355–369. Kim Yong H., Hong S.S., Lee J.S. and Wagoner R.H., 1996, Analysis of superplastic forming processes using a finite-element method, Journal of Materials Processing Technology vol. 62, pp. 90–99.



[10] [11] [12]

[13] [14] [15] [16] [17] [18] [19]

223

Xing H.L. and Wang Z.R., 1997, Finite-element analysis and design of thin sheet superplastic forming, Journal of Materials Processing Technology, vol. 68, pp. 1–7. Doltsinis J. St., Loginsland J. and Nolting S., 1987, Some developments in the numerical simulation of metal forming processes, Proceedings of the International Conference on Computational Plasticity, Barcelona. Carrino L., Giuliano G. and Napolitano G., 2003, A posteriori optimisation of the forming pressure in superplastic forming processes by the finite element method, Finite Elements in Analysis and Design, vol. 39, n° 11, pp. 1083–1093. Carrino L., Giuliano G. and Palmieri C., 2003, On the optimization of superplastic forming processes by the finite element method, Journal of Materials Processing Technology, vol. 143–144, pp. 373–377. Chandra N., 2002, Constitutive behavior of superplastic materials, International Journal of Non-Linear Mechanics, vol. 37, pp. 461–484. Carrino L., Giuliano G. and Polini W., 2003, A method to characterise superplastic materials in comparison with alternative methods, Journal of Materials Processing Technology, vol. 138, pp. 417–422. Carrino L. and Giuliano G., 1999, Finite element modelling and the experimental verification of superplastic forming, Advanced Performance Materials, vol. 6, n° 2, pp. 159–169. Giuliano G. and Franchitti S., 2006, On the Evaluation of Superplastic Characteristics using the Finite Element Method, International Journal of Machine Tools and Manufacture, vol. 47, pp. 471–476. MSC. Marc 2005 Vol. B, Element library. Zienkiewicz O.C., Flow formulation for numerical solution of forming processes, Numerical Analysis of Forming Processes, Wiley, New York, 1984.




225

A mathematical model approach to a glycerolysis reaction for monoacylglycerol production B. Cheirsilp & A. H-Kittikul Department of Industrial Biotechnology, Faculty of Agro-Industry, Prince of Songkla University, Hat Yai, Thailand

Abstract Monoacylglycerol emulsifiers commonly employed in the food, cosmetic and pharmaceutical industries can be produced by a glycerolysis reaction using lipase. The concentrations of two substrates (palm olein and glycerol) in the glycerolysis reaction were strictly interdependent. Therefore, it was impossible to perform a classic kinetic study by fixing the concentration of one substrate and changing the concentration of the other substrate. Here, a mathematical model approach is a useful tool to independently assess the effects of hypothetical changes in the concentrations of each. In essence, this analysis permits one to examine various aspects associated with the dynamics and equilibrium of the glycerolysis reaction that cannot be investigated experimentally. In this study a mathematical model, taking into account the mechanism of the glycerolysis reaction for monoacylglycerol production using immobilized lipase, has been developed. From the proposed model, the effects of varying the initial concentrations of substrates on the initial production rate and yield of monoacylglycerol were simulated. The most significant finding from simulation was that an increase in the initial concentration of triacylglycerol leads to an increase in the initial production rate of monoacylglycerol, but there is a limit beyond which increasing the initial concentration of triacylglycerol results in a low yield of monoacylglycerol. The simulation results show that mostly glycerol reacts with fatty acid of triacylglycerol to produce monoacylglycerol in a glycerolysis reaction. From a thermodynamic standpoint, a greater incorporation of glycerol is expected because a higher concentration of this acyl acceptor should shift the equilibrium towards a greater glycerolysis reaction. Keywords: glycerolysis, lipase, mathematical model, monoacylglycerol. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070241


1

Introduction

The applications of glycerolysis reaction using lipase have been carried out to produce monoacylglycerol due to its mild reaction conditions and position specific products [1–3]. The concentrations of two substrates, triacylglycerol and glycerol, in glycerolysis reaction were strictly interdependent. To identify the optimal conditions for lipase catalyzed glycerolysis reaction, it is essential to understand the kinetics of this reaction. Until now all kinetic mechanisms on lipase catalytic reactions are only based on hydrolysis of triacylglycerol [4,5] or esterification of fatty acid [6,7]. Only a limited number of kinetic studies for glycerolysis using glycerol as acyl acceptor have been found in literature [1]. However, the reported model is rather complicated and narrow range of application. Here, in this paper the kinetics of glycerolysis of palm olein for monoacylglycerol production was studied. First, a simple model based on PingPong Bi Bi was proposed to describe the kinetics of hydrolysis and esterification steps involved in glycerolysis reaction. Then, the mathematical model for glycerolysis reaction was considered. The effects of enzyme, water, glycerol and palm olein concentrations on monoacylglycerol production were contributed in the model. The constructed model was used to obtain a better understanding of the effects of two substrates, triacylglycerol and glycerol in glycerolysis reaction. Finally, optimal condition was determined by simulation study using the model.

2

Materials and methods

2.1 Materials Lipase PS (Pseudomonas sp.) was a gift from Amano Pharmaceutical Co. Ltd., Japan. Microporous polypropylene powder; Accurel EP-100 (particle size 200400 µm) was a gift from Akzo Nobel (Obermburg, Germany). Palm olein was purchased from Morakot Industry Co. Ltd., Thailand. All other chemicals were analytical grade reagents obtained from commercial sources. 2.2 Immobilization To immobilize lipase, Accurel EP-100 (10g) was added to 100 ml of 0.1 M phosphate buffer (pH 7) containing lipase PS approximately 100 U/ml and the reaction mixture was stirred with a magnetic bar at 100 rpm for 30 min. Afterward, 100 ml of 0.1 M phosphate buffer (pH 7) was added and the suspension was filtered through a filter paper by vacuum. The immobilized lipase PS on Accurel EP-100 (330 U/g) was stored at 4 °C for further studies. 2.3 Glycerolysis reaction The glycerolysis experiments were carried out in batch system. The reaction mixture consisted of various enzyme, water, glycerol and palm olein concentrations in organic solvent (acetone/isooctane mixture 3:1, v/v). The WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


227

temperature was controlled at 45 °C. The reaction mixture was mixed by magnetic stirrer at 300 rpm. Samples of the reaction mixture were centrifuged to remove immobilized lipase before analysis. 2.4 Analytical method The components of oil phase were analyzed for triacylglycerol (TAG), diacylglycerol (DAG), monoacylglycerol (MAG) and fatty acid (FA) using a thin-layer chromatography with flame ionization detection (TLC/FID) (IATROSCAN MK5, Iatron Laboratories Inc. Tokyo, Japan). In this experiment, percent of peak area was assumed as percent content of the corresponding compound. Activity of lipase was determined by the modified cupric acetate method. One unit of hydrolytic activity was defined as the amount of the enzyme, which liberates 1 µmol equivalent of palmitic acid from palm olein in 1 min at 30 °C. Ordinary differential equations were solved by the Runge-Kutta single-step fourth-order method [8]. The programs were coded in the Visual Basic program ver. 6.0 (Microsoft Inc., USA). (a) W

DAG

G

ExTAGxW

ExDAGxFA

ExFAxG

W

MAG

G

ExDAGxW

ExMAGxFA

Ex FAxG

TAG E (b) DAG E

Figure 1:

3

W MAG E W MAG E

Schematic diagram of Ping Pong Bi Bi mechanisms for glycerolysis reaction of triacylglycerol (a) and diacylglycerol (b). TAG, DAG, MAG, W, G and E denote triacylglycerol, diacylglycerol, monoacylglycerol, water, glycerol and enzyme, respectively. ExTAGxW, ExDAGxW, Ex DAGxFA, ExMAGxFA and ExFAxG are different complexes between enzyme and the species defined above.


3.1 Modeling of glycerolysis reaction Mechanisms of glycerolysis reaction for monoacylglycerol production were interesterifications of tri- or di-acylglycerol with excess glycerol. The interesterification reaction involves sequential execution of the hydrolysis and reesterification steps, and thus requires multiple entrances and exits of reactant and product species in such a manner as to render the overall mechanism of the WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

228 Computational Methods and Experimental Measurements XIII Ping-Pong type as in Fig. 1. Theoretically glycerolysis of one mole of triacylglycerol and two moles of glycerol could produce 3 moles of monoacylglycerol, however the yields of monoacylglycerol depend on flavored equilibrium in various conditions [9]. In kinetic study of immobilized enzyme, the reaction rate might be controlled by an internal mass transfer. However, it was also reported that mass transfer limitation in porous supports could be neglected [10,11]. Therefore, to describe a simple mathematical model for glycerolysis reaction, mass transfer limitation in reaction system was first neglected. The appearance rates of triacylglycerol (TAG), diacylglycerol (DAG), monoacylglycerol (MAG), glycerol (G), fatty acid (FA) and water (W) from hydrolysis and reesterification steps were derived as in Table 1. Table 1:

A mathematical model of glycerolysis reaction.

Differential equations d [TAG ] dt

=

( −VmDAG [TAG ][W ] + VrDAG [ DAG ][ FA]) ET ([W ] + K mDAG [ DAG ] + K mMAG [ MAG ] + K mG [G ])[ FA]

d [ DAG] (VmDAG [TAG][W ] + VrMAG [MAG][FA] − VmMAG [DAG][W ] −VrDAG [DAG][FA]) ET = dt ([W ] + KmDAG [DAG] + KmMAG [MAG] + KmG [G])[FA] d [ MAG ] (VmMAG [ DAG ][W ] + VrG [G ][ FA] − VmG [ MAG ][W ] − VrMAG [ MAG ][ FA]) ET = dt ([W ] + K mDAG [ DAG ] + K mMAG [ MAG ] + K mG [G ])[ FA] d [G ] VmG [MAG][W ] − VrG [G][ FA]) ET ( = dt ([W ] + KmDAG [ DAG] + KmMAG [MAG] + KmG [G])[ FA] d [ FA] (VmDAG [TAG ][W ] + VmMAG [ DAG ][W ] + VmG [ MAG ][W ]) ET = ([W ] + K mDAG [ DAG ] + K mMAG [ MAG ] + K mG [G ])[ FA] dt (V [ DAG ][ FA] + VrMAG [ MAG ][ FA] + VrG [G ][ FA]) ET − rDAG ([W ] + K mDAG [ DAG ] + K mMAG [ MAG ] + K mG [G ])[ FA]

d [W ] (VrDAG [ DAG ][ FA] + VrMAG [ MAG ][ FA] + VrG [G ][ FA]) ET = dt ([W ] + K mDAG [ DAG ] + K mMAG [ MAG ] + K mG [G ])[ FA] −

(1) (2) (3) (4) (5)

(6)

(VmDAG [TAG ][W ] + VmMAG [ DAG ][W ] + VmG [ MAG ][W ]) ET

([W ] + K mDAG [ DAG ] + K mMAG [ MAG ] + K mG [G ])[ FA] FA: fatty acid concentration, W: water concentration, ET: total enzyme concentration. VmDAG and VrDAG are maximum initial reaction rates for hydrolysis of TAG and reesterification of DAG defined as: k1k3 and k 4 k6 , respectively. KmDAG, KmMAG and KmG are equilibrium constants

VmDAG =

k 2 + k3

VrDAG =

k 4 + k5

k5 , k , respectively. VmMAG and k and K mG = 17 K mMAG = 11 k6 k12 k18 VrMAG are maximum initial reaction rates for hydrolysis of DAG (production of MAG) and k k , respectively. VmG and VrG reesterification of MAG defined as: k k and VrMAG = 10 12 VmMAG = 7 9 k10 + k11 k8 + k 9 are maximum initial reaction rates for hydrolysis of MAG (production of G) and reesterification of G k k , respectively. k k defined as: and VrG = 16 18 VmG = 13 15 k16 + k17 k14 + k15 for DAG, MAG and G defined as:

K mDAG =



229

The concentrations of triacylglycerol, diacylglycerol, monoacylglycerol and fatty acid at different times were obtained experimentally when the substrate concentration of triacylglycerol and glycerol were 7.16 and 19.14 mM, respectively. The concentrations of enzyme and water were 0.6 g and 9.89 mM, respectively. The nonlinear curve fitting by Simplex’s method [12] was used for fitting the system of differential equations (1-6) (Table 1) into the experimental data. By fitting the above differential equations to the experimental data, the rate constants were estimated and listed in Table 2. Base upon the obtained rate constants and the kinetic scheme, the concentrations of each composition at different reactions could be calculated. Comparison between calculated and experimental data was presented in Fig. 2 and a good agreement was obtained. In terms of the reaction rate constants in Table 2, the forward reaction rate constants in the first and second reactions (VmDAG = 0.115 mM-1g-1h-1, VmMAG = 0.151 mM-1g-1h-1) were much lower than that in the third forward reactions (VmG = 0.811 mM-1g-1h-1). The results indicated that the first and second hydrolysis reactions (TAG to DAG and DAG to MAG) were the limiting steps during the overall reactions. The production rate of MAG by glycerolysis reaction (VrG = 1.23 mM-1g-1h-1) was higher than the production rate by hydrolysis reaction of DAG (VmMAG = 0.151 mM-1g-1h-1). These rate constants show that the intermediates (G and FA) were easily converted to MAG by glycerolysis reaction. The experimental results also showed that the concentration of the measurable intermediate (FA) was low during time course of glycerolysis (Fig. 2). Table 2: Maximum initial reaction rates (mM-1g-1h-1) VmDAG VmMAG VmG VrDAG VrMAG VrG

Rate constants in the model of glycerolysis reaction. Value

Equilibrium constants (mM-2)

0.115 0.151 0.811 0.109 0.320 1.23

KmDAG KmMAG KmG

Value 1.09 × 10-4 3.20 × 10-4 1.23 × 10-3

3.2 Effect of triacylglycerol and glycerol The effects of varying the initial concentrations of triacylglycerol and glycerol on the initial production rate and yield of monoacylglycerol were also simulated. Simulation of glycerolysis system under a variety of initial conditions provides a more complete picture of the dynamics and equilibrium behavior of this system (Fig. 3). Two of the most significant findings were (1) an increase in the initial



Concentration (mM)

12 10 8 6 4 2 0 0

5

10

15

20

25

Time (h) TAG MAG

Figure 2:

DAG FA

Comparison between calculated (lines) and experimental results (symbols) of glycerolysis reaction. Triacylglycerol 7.16 mM; glycerol 19.14 mM; water 9.89 mM; immobilized lipase 0.6 g.

1

0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 11

Figure 3:

0.9 0.8 0.7 0.6 Yield

rMAG

0.50 0.45

14 19 Glyc erol

21

25 1

2

5

7

10

G TA

12

0.5 0.4 0.3 0.2 0.1 0 11

14 19 Gl yc erol

21

25 1

2

5

7

10

12

G TA

Simulation results of effects of triacylglycerol (TAG) and glycerol concentrations on the initial production rate (rMAG) and yield of monoacylglycerol.

concentration of triacylglycerol leads to an increase in the initial production rate of monoacylglycerol, but there is a limit beyond which increasing the initial concentration of triacylglycerol results low yield of monoacylglycerol at each glycerol concentration and (2) increasing in glycerol concentration more than WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


231

19.14 mM the effect becomes less pronounced in initial production rate and yield of monoacylglycerol at equilibrium. The former result is expected from a thermodynamic standpoint, because a higher concentration of triacylglycerol should enhance the greater production rate of monoacylglycerol. The latter result might be due to the limitation of equilibrium comes from limited enzyme concentration. Therefore, it was concluded that glycerol concentration of 19.14 mM was the optimal concentration for glycerolysis reaction with enzyme concentration of 0.6 g. From all of the above, in the case of high reaction rate and acceptable yield of monoacylglycerol are set as target in process optimization, the glycerolysis reaction with high concentrations of triacylglycerol and glycerol are suitable to be carried out. On the other hand, in the case of high yield of monoacylglycerol is set as target in process optimization, low concentration of triacylglycerol and excess concentration of glycerol are suitable.

4

Conclusion

The kinetics of lipase-catalyzed glycerolysis reaction between triacylglycerol and glycerol were successfully modeled using rate expressions requiring adjustable parameters. The present model was effective for prediction the synergic effect of two substrates (triacylglycerol and glycerol) in glycerolysis reaction. The simulation results showed that mostly glycerol reacted with fatty acid of triacylglycerol to produce monoacylglycerol in glycerolysis reaction. From a thermodynamic standpoint, a greater incorporation of glycerol is expected because a higher concentration of this acyl acceptor should shift the equilibrium toward greater glycerolysis reaction. Another important observation was that the behavior of high triacylglycerol concentrations showed high initial production rates but low yields of monoacylglycerol.

References [1] [2] [3] [4] [5]

Tan, T. & Yin, C., The mechanism and kinetic model for glycerolysis by 1,3 position specific lipase from Rhizopus arrhizus. Biochemical Engineering Journal, 25, pp. 39-45, 2005. McNeill, G.P., Shimizu, S. & Yamane, T., High-yield enzymatic glycerolysis of fats and oils. Journal American Oil Chemical Society, 68(1), pp. 1-5, 1991. McNeill, G.P. & Yamane, T., Further improvements in the yield of monoglycerides during enzymatic glycerolysis of fats and oils. Journal American Oil Chemical Society, 68(1), pp. 6-10, 1991. Taylor, F., Kurantz, M.J. & Craig, J.C., Kinetics of continuous hydrolysis of tallow in a multi-layered flat-plate immobilized-lipase reactor. Journal American Oil Chemical Society, 69(6), pp. 591-594, 1992. Padmini, P., Rakshit, S.K. & Baradarajan, A., Kinetics of enzymatic hydrolysis of rice bran oil in organic system. Enzyme and Microbial Technology, 16, pp. 432-435, 1994. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


[7]

[8] [9] [10] [11] [12]

Zhang, T., Yang, L., Zhu, Z. & Wu, J., The kinetic study on lipasecatalyzed transesterification of α-cyano-3-phenoxybenzyl alcohol in organic media. Journal of Molecular Catalysis B: Enzymatic, 18, pp. 315323, 2002. Xu, Y., Du, X. & Liu, D., Study on the kinetics of enzymatic interesterification of triglycerides for biodiesel production with methyl acetate as the acyl acceptor. Journal of Molecular Catalysis B: Enzymatic, 32, pp. 241-245, 2005. Danby, J.M.A., Computer modeling, Willmann-Bell Inc Richmond Va 1997. Reyes H.R., Kinetic modeling of interesterification reactions catalyzed by immobilized lipase. Biotechnology and Bioengineering, 43, pp. 171-182, 1994. Chen, J.P. & Wang, H.Y., Improved properties of bilirubin oxidase by entrapment in alginate-silicate sol-gel matrix. Biotechnology Techniques, 12(11), pp. 851-853, 1998. Romero, M.D., Calvo, L., Alba, C. & Daneshfar, A., A kinetic study of isoamyl acetate synthesis by immobilized lipase-catalyzed acetylation in n-hexane. Journal of Biotechnology, 127, pp. 269–277, 2007. Nelder, J.A. & Mead, R., A simplex method for function minimization. Computational Journal, 7, pp. 308-313, 1964.



233

Evaluation of experimental procedures for confined concrete columns using 3D finite element analyses H. O. Köksal1, C. Karakoç2, Z. Polat1, T. Turgay1 & Ş. Akgün1 1 2

Civil Engineering Faculty, Yıldız Technical University, Turkey Civil Engineering Department, Boğaziçi University, Turkey

Abstract This paper presents the results of the early stages of both a continuing experimental work on the square confined concrete columns and their 3D finite element modeling based on the isotropic damage theory in order to establish a realistic approach for the confinement pressure. Based on the axial behavior of four RC columns with a 200 mm square cross section tested under concentric loading, existing experimental data and procedures in the literature are evaluated. As demonstrated from the comparison of the FE analysis and the test results, obtaining a uniform axial loading and deformation state is questionable. Keywords: confinement pressure, reinforced concrete column, finite element, isotropic damage theory, geometrical defects.

1

Introduction

The compressive strength of RC columns increase with increasing confining pressure. The confinement mechanisms are transverse reinforcements such as stirrups or spirals, FRP wraps, and steel jackets, etc. extensive research on the axial behavior of confined concrete has been carried out since the pioneering study of Richardt et al. [1]. The constitutive model for confined concrete based on the experimental findings plays an important role in the pushover analysis of RC structures. There are some frequently cited models (Hognestad [2], Kent and Park [3], Sheikh and Uzumeri [4], Mander et al. [5], Saatcioglu and Razvi [6]) to predict the peak stress or the stress–strain curve of confined concrete. Only the latest two WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070251

234 Computational Methods and Experimental Measurements XIII recommends a relation for finding the confining pressure. Mainly two experimental works conducted by Sheikh and Uzumeri [7], Mander et al. [8], are employed mainly for derivation of these relations. Techniques for measuring longitudinal deformations and the effects of possible small defects on the cross-section of the columns during the production of the specimens to the experimental results are evaluated.

2

Experimental work

The square specimens in the test program have 200x200 mm cross-section dimensions and 1000 mm height. The columns tested in the structural laboratory of Yıldız Technical University, are a first part of a Ph.D. study about confined RC columns. Fig. 1 shows the details of test setup and instrumentation.

Figure 1:

Test setup and the details of test specimens.

There is only one type of concrete mix for all test columns. C1, C2, and C3 type columns are tested respectively at 30 days, 60 days and 90 days. All longitudinal bars are 10mm in diameter and L4 and L8 shows the number of the bars in the cross-section of a column. The tie spacing is 100mm and the S8 and S12 represent tie diameters. In this paper, the results of the C1 type-columns are presented. For measurement of axial strains, four linear variable displacement transducers (LVDTs) are placed over the central 400mm gage length at each side of a column in a similar way used to assess any eccentricity of the applied load WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


235

as recommended in the study of Shrive et al. [9]. A pre-loading up to the onefourth of the predicted axial capacity is applied to maintain similar displacement readings at LVDTs so that in the linear elastic stage of the overall behavior any eccentricity can be eliminated. Similar procedures have been used for measuring the axial shortening in the literature. In the experimental study of Sheikh and Uzumeri [7], the load versus deformation behavior of the test region of the columns was recorded using two linear variable displacement transducers (LVDT), one on the west, and the other on the east side of the column. Strain curves were averaged along the strain axis to obtain the mean load versus average strain characteristics of the test region. Mander et al. [8] took several readings of longitudinal strains over the central 450mm gage length of each column using four linear potentiometers. The average of the four potentiometers around the circumference was used prior to reaching maximum load. After this load level, some critical potentiometers were defined by establishing that the failure region occurred wholly within the gage length of the potentiometers instead of averaging the readings of four potentiometers. 1000 C1L8S8

Axial load (kN)

800 600 400

LVDT LVDT LVDT LVDT

200

#2 #3 #4 #7

0 0

0.1

0.2

0.3

0.4

Axial shortening (mm)

Figure 2:

Four separate LVDT readings of axial load-shortening curves for the column specimen C1L8S8.

Although the LVDT readings were provided very close to each other, next to the maximum axial load there can be a significant variation between the minimum and maximum values of shortening reaching very high values.

3

3D Finite element modeling of RC columns

The concrete elements are modeled using eight-noded isoparametric solid continuum elements. Vertical and lateral reinforcements are meshed using twonoded 3D bar elements. Taking the advantage of symmetry and uniform loading WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

236 Computational Methods and Experimental Measurements XIII of system, 3D model of the concentrically loaded RC columns is created using one-fourth of the cross-sectional symmetry. The bottom surface is fully restrained while the lateral translations are only restricted in the top surface [10]. The Oliver’s damage model [11] primarily developed for concrete elements and now available within the LUSAS software package [12], are employed for the material modeling of concrete. There are initially three parameters, which have to be defined before the analysis: 1) The initial threshold, τ*, can be given as f τ* = t (1) E0 CEB-FIB [13] Equation is adopted for the initial elasticity modulus of elasticity, E0: 1/ 3

 f  E = α ∗ 21500 ∗  c  (2)  10  in which fc is the cylindrical compressive strength of concrete at 28 days. ft is taken as

f t = 0.35

fc

(3)

2) A material parameter, A, is a limiting factor of the maximum size of the element that used for a mesh size choice and is given by [11]  G f E0 1  − ≥0 (4) A=  hf2 2  t  in which h is the characteristic length of the finite element and Gf is the fracture energy of concrete. A simple analytical relation, derived from a semitheoretical approach [14] is used for the fracture energy of concrete: f2 (5) G f = 15.5 d max t E0 in which dmax is the maximum aggregate size in the concrete mix. A final form of A can be obtained as in a previous study [15] h (6) A= 310 Eq (6) clearly reflects only the mesh size effect into the analysis. A representative mesh size for nonlinear FEA can be used in this study as recommended by Bazant and Oh [16]: h= 3 h x h y hz

(7)

where hx, hy, and hz are sizes of an eight-noded solid element. An ideal elasto-plastic behavior is adopted with a Von Misses type of stress potential for steel elements. 3D-bar elements are used for the reinforcing steel bars and ties. 3) A damage ratio is the ratio of the stresses that cause initial damage in tension and compression. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


4

237

Model verification and evaluation of experimental results

3D-Finite element analyses of four RC columns tested in the structural laboratory of Yıldız Technical University are performed in this study. As shown in fig.3 and fig.4, the mesh sizes chosen for concrete elements are optimum sizes since the expression for material parameter A in eq (6) is derived for representative concrete meshes having approximately a size of three times the maximum aggregate diameter [16–18].

Figure 3:

(a) Cross-sectional dimensions and (b) FE meshing of RC columns with and without any defects.

Figure 4:

(a) Test setup; (b) Failure of a RC column.

It is important to model the confining action of the stirrups properly. If only the bar elements are used for ties, there is no possibility to reflect the confining action of ties between two nodes. Therefore, springs, acting on the lateral surface of concrete element are defined and the total spring stiffness acts on the nodes as shown in fig.3. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Axial load (kN)

1200 C1L4S8 800 Average reading 0.01 h defect 0.005 h defect Ideal column

400 0 0

0.2

0.4

0.6

0.8

1


Axial load (kN)


400 0 0

0.2

0.4 Axial shortening (mm)

0.6

0.8

Axial load (kN)

1000 C1L8S8 500

Average reading 0.01 h defect 0.005 h defect Ideal column

0 0

0.2 0.4 Axial shortening (mm)

0.6

Axial load (kN)


400 0 0

Figure 5:

0.2

0.4 Axial shortening (mm)

0.6

0.8

Influence of geometrical defects on axial load-shortening curves of finite element analysis and comparison with the experimental results obtained from averaging the readings of three LVDT’s.



239

Axial load (kN)

1000 C1L4S8

800 600 400

LVDT #2 Ideal column

200 0 0

0.1

0.2

0.3

0.4

0.5

0.6


Axial load (kN)

1200 1000 800 600 400 200 0

C1L4S12

LVDT #2 Ideal column 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7


Axial load (kN)

1000 C1L8S8

800 600 400

LVDT #7

200

Ideal column

0

Axial load (kN)

0

0.1

1200 1000 800 600 400 200 0

0.4

0.5

C1L8S12

LVDT #7 Ideal column 0

Figure 6:

0.2 0.3 Axial shortening (mm)

0.1

0.2

0.3 0.4 0.5 Axial shortening (mm)

0.6

0.7

Comparison of the readings of the critical LVDT located on the heavily damaged region and the results of FEA of an ideal column.


240 Computational Methods and Experimental Measurements XIII The assumption of existing of some specific geometrical defects is initially imposed on the cross-section of the column as shown in fig. 3(b). A perfect flat top surface is modified to a deformed one by imposing a height difference of 10mm (0.01h) and 5mm (0.005h) respectively to one edge of the column. These deformations are really small values, which cannot be observed easily. As can be seen in fig.5, the results obtained from finite element analysis of deformed specimens for the axial load-shortening curves get closer to the curve drawn averaging readings of three LVDTs. It is clearly stated that there is no ideal column subjected to a perfect concentric loading up to the failure load. Especially, when the plastic deformations are getting larger at approximately 75% of the maximum load, the cracks begin to grow unstably and macro-cracks arise pointing out the localization of the damage. The readings obtained from the critical LVDT and shown in fig.6 are also close to the results of the 3D finite element analysis of an ideal RC column. The critical LVDT represents the readings belonging to the heavily deformed side of the column, even the differential settlement of this side due to the localized damage can be observed by naked eye. While the test column continues to consolidate near to the failure load, volumetric expansion occurs by the opening of macro cracks on the adjacent side of the column as shown in fig.7. A similar approach is adopted by Mander et al. [8] defining some critical potentiometers on the failure region within the gage length of the potentiometers instead of averaging the readings of four potentiometers. However, it is clear that there is an uncertainty about the exact place of the localized damage region on which the critical LVDT to be placed for obtaining the maximum shortening values of the column.

Figure 7:

(a) Heavily consolidated side; (b) macro-crack at neighbouring side.



5

241

Conclusion

This paper has been concerned with the evaluation of the experimental techniques for obtaining the axial load-shortening curves of RC columns subjected to concentric loading. Comparing the experimental data and 3D finite element analyses of four RC columns tested in the structural laboratory of Yıldız Technical University, some uncertainties measuring the axial deformation is studied. For this purpose, while the average of LVDTs can be used near to the failure load, the critical LVDT readings showing the accumulated damage on heavily consolidated side of the column should be preferred after the beginning of the unstable crack propagation approximately taken as 75% of the failure load. The FEA results and experimental data show good agreement for this case. On the other hand, there is some uncertainty about the exact place for the replacement of the critical LVDT on the failure region. Alternative places for the LVDT on this damaged side of the column may be schemed for measuring the consolidation of the column. Thus, a need for a definition for the replacement of the critical LVDT to measure the axial deformations of RC columns is apparent. Even it is possible to use some averaging values for the determination of the axial shortening. If a very small height difference between two opposite sides on the top surface of the specimen is imposed on the 3D modelling of columns, the results of FEA and the average of LVDTs are close to each other. This kind of defects is possible during the preparation stage of the specimens and should be carefully checked.

Acknowledgement The support of B.U. Research Fund (ref: research project 05A403) for this paper is gratefully acknowledged.

References [1]

[2] [3] [4]

Richart, F.E., Bradtzaeg, A. & Brown, R. L., A study of the Failure of Concrete under Combined Compressive Stresses, Bulletin Np. 185, Engineering experimental station University of Illinois, Urbana, pp. 104, 1928. Hognestad, E., A Study of Combined Bending and Axial Load in Reinforced Concrete Members, Bulletin Series No.399, University of Illinois Eng. Exp. Station, Urbana. 1951. Kent, D.C. & Park, R., Flexural Members with Confined Concrete, Journal of the Structural Division, Proc. of the American Society of Civil Engineers, 97(ST7), pp.1969-1990, 1971. Sheikh, S.A. & Uzumeri, S.M., Analytical Model for Concrete Confinement in Tied Columns, Journal of the Structural Division, Proc. of WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


[5] [6] [7] [8] [9] [10]

[11]

[12] [13] [14] [15]

[16] [17] [18]

the American Society of Civil Engineers, 108(ST12), pp. 2703-2722, 1982. Mander, J.B., Priestly, M.J.N. & Park, R., Theoretical Stress-Strain Model for Confined Concrete, Journal of the Structural Engineering, ASCE, 114(8), pp.1804-1826, 1988. Saatcioglu, M. & Razvi, S.R., Strength and Ductility of Confined Concrete, Journal of the Structural Engineering, ASCE, 118(6), pp.1590-1607, 1992. Sheikh, S.A. & Uzumeri, S.M., Strength and Ductility of Tied Concrete Columns, Journal of the Structural Division, ASCE, 106(ST5), pp.10791102, 1980. Mander, J.B., Priestly, M.J.N. & Park, R., Observed Stress-Strain Behavior of Confined Concrete, Journal of the Structural Engineering, ASCE, 114(8), pp.1827-1849, 1988. Shrive, P.L., Azarnejad, A., Tadros, G., McWhinnie, C. & Shrive, N.G, Strength of Concrete Columns with Carbon Fibre Reinforcement Wrap, Canadian Journal of Civil Engineering, Volume 30, pp. 543-554, 2003. Karakoç, C., Köksal, H.O., & Özsoy, A.E., The Behaviour of Reinforced Block Masonry Columns Under Axial Compression, Earthquake Resistant Engineering Structures IV, WIT Press, Southampton, U.K., pp. 371-379, 2003. Oliver, J., Cervera, M., Oller, S. & Lubliner, J., Isotropic Damage Models and Smeared Crack Analysis of Concrete, In N. Bićanić et al. (ed) Proc. SCI-C Computer Aided Analysis and Design of Concrete Structures, pp. 945-957, 1990. LUSAS Modeller (v13.6) User Manual v13, FEA ltd., UK CEB-FIB. 1990 Model Code. Comité Euro-International du Béton (CEB), Lausanne, Switzerland. Köksal, H.O., Modelling of Concrete Fracture, Ph.D. thesis, Division of Civil Engineering, Boğaziçi University, İstanbul, Turkey, 1998. Köksal, H.O., Doran, B., Özsoy, A.E. & Alacali, S.N., Nonlinear Modeling of Concentrically Loaded Reinforced Blockwork Masonry Columns, Canadian Journal of Civil Engineering, Volume 31, Number 6, pp.1012-1023, 2004. Bažant, Z.P. & Oh, B., Crack Band Theory for Fracture of Concrete, Materiaux et Constructions, 16, 93, pp.155-177, 1983. Bedard, C. & Kostovos, M.D., Fracture Process of Concrete for NLFEA Methods, ASCE Journal of Structural Engineering, 112, 3, 573-586, 1986. Köksal, H.O. & Arslan, G., Damage Analysis of RC Beams without Web Reinforcement, Magazine of Concrete Research, Volume: 56, pp.231-241, 2004.



243

Geometrically nonlinear static analysis of 3D trusses using the arc-length method G. A. Hrinda NASA Langley Research Center, Hampton, Virginia, USA

Abstract Rigorous analysis of geometrically nonlinear structures demands the creation of mathematical models that accurately include loading and support conditions and, more importantly, model the stiffness and response of the structure. Nonlinear geometric structures often contain critical points with snap-through behavior during the response to large loads. Studying the post buckling behavior during a portion of a structure's unstable load history may be necessary. Primary structures made from ductile materials will stretch enough prior to failure for loads to redistribute producing sudden and often catastrophic collapses that are difficult to predict. The responses and redistribution of the internal loads during collapses and possible sharp snap-back of structures have frequently caused numerical difficulties in analysis procedures. The presence of critical stability points and unstable equilibrium paths are major difficulties that numerical solutions must pass to fully capture the nonlinear response. Some hurdles still exist in finding nonlinear responses of structures under large geometric changes. Predicting snap-through and snap-back of certain structures has been difficult and time consuming. Also difficult is finding how much load a structure may still carry safely. Highly geometrically nonlinear responses of structures exhibiting complex snap-back behavior are presented and analyzed with a finite element approach. The arc-length method will be reviewed and shown to predict the proper response and follow the nonlinear equilibrium path through limit points. Keywords: nonlinear, truss, arc-length, finite element, snap-back, Crisfield, tangent stiffness, equilibrium path, Nastran, bifurcation.



1

Introduction

Three-dimensional space trusses may experience loading conditions that cause large displacements that significantly change the geometry of the structure and require the equations of equilibrium to be formulated for the deformed structure. The large deflections are described by nonlinear differential equations that can be solved using incremental techniques. In nonlinear analysis the tangent stiffness matrix replaces the stiffness matrix used in linear analysis. Iterative time stepping is used to apply small incremental loads to the structure and find the corresponding incremental displacements. A plot of the results defines a curve of the equilibrium path of the structure under the applied loading. A truss structure undergoing large changes in geometry often exhibits critical points with an unstable snapping response during a static collapse. The solution to these structural instabilities is difficult to find with common nonlinear equation solvers such as the Newton-Raphson method. These methods often fail whenever snap-back behavior occurs along the loading path and they may not correctly define the response immediately after snap-through. Large gaps in the equilibrium path will occur with artificial results being plotted. Researchers have continually investigated these shortcomings and have offered improvements to the process that have been gradually introduced into commercial finite element analysis (FEA) programs. The techniques investigated bring the above FEA problems into focus. Geometric nonlinear FEA may be challenged to find all possible responses during large loading. A finite element computer program was created and tested by means of a number of examples exhibiting geometric nonlinearity. The sophistication of the finite element program presented in this work is measured by the path-following techniques enabling the fundamental path to be followed after bifurcation. The results are compared with nonlinear Nastran solutions and Crisfield [1,2].

2

Geometrically nonlinear finite element static analysis by the Riks-Wempner arc-length method

Passing through critical points during the geometrically nonlinear response is challenging. Two critical points encountered during this type of behavior are: load limit points that are reached whenever the response path has a local snapthrough; and control limit points that define a local snap-back. At a control limit point the loading may reverse as the deflections change directions and a local maximum is passed. An important family of nonlinear equations solvers called the arc-length method as developed by Riks-Wempner [6] can overcome the difficulties of passing critical points. The technique resembles the Newton-Raphson method described in Riks [5] except the applied load increment becomes an additional unknown. The Riks-Wempner method computes load magnitudes as part of the solution. The length of a vector tangent to the equilibrium path is used to find a new point that is the intersection of the plane normal to the tangent. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


245

A user-supplied load will estimate magnitudes of the initial load increment for a step. Termination of the method is done by the user specifying a maximum load proportionality factor or a maximum nodal displacement. The process also ends when the maximum number of increments for a step is reached. 2.1 Derivations of arc-length formulas The Riks-Wempner arc-length method traces the nonlinear equilibrium path using an iterative process that begins with computing initial displacements due to a user defined load increment. The method proceeds to find the next equilibrium point from the initial point i as shown in fig. 2 and detailed by Crisfield [1,2], Owen and Hinton [3], Owen et al [4] and Riks [6]. The figure shows the loaddisplacement curve for a single-degree-of-freedom system. A vector tangent to the curve at i can be drawn and written as G ti =

G

 ∆qi   ∆λ   i

(1) G

where ∆λi is the incremental applied load at i and ∆qi is the incremental displacement vector found from computing

G G K T ∆qi = ∆λi Q

(2)

i

G The normal vector, ni , is also shown in fig. 2 and can be written as G  ∆qk  G ni =    −∆λk 

(3)

The tangent stiffness matrix, KT , is assembled using the nonlinear truss shown i

in fig. 1 and derived in [1]. Trusses undergoing large deflections must be analyzed for the deformed geometry of the structure. The linear equations G G F = [K ] u (4) G

that relates the applied forces F with the truss element stiffness [ K ] and G displacements u must be modified to account for changes in nodal geometry as the load is applied. The sum of the linear elastic and nonlinear matrices produce the global tangent stiffness at point i along the load-displacement path of the single-degree-of-freedom system. The standard elastic stiffness matrix [ K ]E will be modified to give the tangent stiffness:

(

)

 3 (ux 2 −ux1 ) 2  1 + ( 3(ux 2 −ux1 ) ) + ( 3(uy 2 −uy1 ) ) + ( 3(uz 2 −uz1 ) ) + +  (5)   xl xl xl 2 xl 2 EA KT =  xl  2 2 3 (uz 2 −uz1 )  3 (uy 2 −uy1 )  +   2 2 2 xl 2 xl  

(

) (

)



Figure 1:

Figure 2:

Nonlinear 3d Truss variables.

Riks-Wempner arc-length method on a normal plane for a singledegree-of-freedom system.



Figure 3:

247

Start of Riks-Wempner normal to a plane method.

At the start of the Riks-Wempner arc-length method an initial load increment, G ∆λ0 , is used to compute the first displacement vector, ∆q0 , and the length of the

G

first tangent vector t0 . The variables are shown in the load displacement plot in G fig. 3. Similar triangles are used to find the initial displacements, ∆q0 . During this initial increment the tangent stiffness is the same as the linear stiffness. The load increment is a user-defined value that divides the total applied load into even increments. A given load increment starts the process and finds the displacements ∆q0 using the tangent stiffness matrix KT . The initial 0

displacements ∆q0 are found using ∆λ0 λ G = G ∆q0 ∆qtot

(6)

where λ = 1 and ∆qtot is found from the expression KT ∆qtot = λ Q (7) G The length of the tangent vector t0 along the equilibrium path can be calculated 0

as ∆s0 = t 0 ⋅ t 0 =

GT GT 2 ∆λ + ∆ q 0 ∆ q

(8)

Throughout the rest of the iterations the arc-length is constant or can be scaled by the user input into the following 1

 I 2 ∆si = ∆si − 1  des   I i −1 

(9)

The user decides on the required number of iterations, I i −1 , and on the number of desired iterations, I des . WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

248 Computational Methods and Experimental Measurements XIII The internal forces in the truss element are required for the tangent stiffness and will be defined using matrix notation. The element strain formulation uses a constant cross sectional area and assumes the length/area of the truss will remain large. The strain energy or work done is 1/2 the nodal forces multiplied by the corresponding deflections. The internal force in the truss will now be defined to include nonlinear effects. The force is axially directed and is needed in updating the nonlinear stiffness matrix. Now using the strain equation

ε

2

=

xli − xl

2 xl

the matrix notation can be written as GT G ε = x21 p21 +

2

(10)

2

1 2

GT G p 21 p21

(11)

 ( ux2 − ux1 )  GT G G 2 where p21 =  ( uy 2 − uy1 )  and xl = x21 x21   ( uz2



− uz1 )  

The first term on the right-hand side of (11) represents the linear strain and the second term is the nonlinear contribution. The internal force in the truss is then: AN = EAε (12)

3

Verification models

The proposed finite element program was tested and verified using several examples found in literature with nonlinear Nastran solutions. Of particular interest was the ability to reproduce snap-through and snap-back behavior found in some structures. The examples were chosen as a robust test of this unstable behavior and bring confidence to the computer coding and numerical techniques. 3.1 Single-degree-of-freedom nonlinear example The following single-degree-of-freedom example uses a truss that follows the National Agency of Finite Elements (NAFEMS) benchmark tests. The problem is used in [1] and by others. Fig. 4 shows the problem with the variables used in expressing the exact response equation.

Figure 4:

Single-degree-of-freedom truss snap through problem.



249

The exact load-displacement path is given in [1] as: 2

qG =

EA0 Z (2 Zw − w )

(13)

3

2lo

where Z = ( z + w) and E =Young's modulus, A0 =area The truss in fig. 4 was solved using the exact eq. (13), performing a nonlinear Nastran FEA and using the proposed Hrinda FEA computer program for static nonlinear trusses presented in this work. Letting EA0 = 5.e 7 , x = 2500. ,

Displacement (inches)

z = 25. , q1 = 1.0 , then the following plot of the solution points are: 60

Nastran

50

Hrinda

40

Exact

30 20 10 0 -0.2

-0.1

0

0.1

0.2

0.3

Load ( x 1e7 lbs.) Load/Deflections for Crisfield Sdof Figure 5:

Single-degree-of-freedom load/deflection plot.

3.2 Star dome truss The Crisfield shallow star dome model was taken from [2] and has been studied by others to demonstrate a complex equilibrium path. The 24 member 3d truss model, shown in fig. 6, has one concentrated load at the center and is solved in the arc-length computer program with results compared to a Nastran model. The load increment vs. vertical displacements of the center node are plotted in fig. 7 and compared. This model introduced a major difficulty following the load path at snap points. The proposed Hrinda FEA program was able to accurately follow the Nastran results through several snap-through and snap-back points. The structure snaps through just before -13" of vertical displacement and then snaps back to -3.7". The loading is reversed as shown by the horizontal axis in fig. 7. The maximum negative load increment is reached at -.443 and the displacements continue to increase.



Figure 6:

Crisfield three-dimensional star dome. -25


-20

Nastran -15

Hrinda

-10

-5 -1

-0.5

0

0.5

1

1.5

0

5

Load (x 1.e6 lbs.)

Load/Deflections for Star Dom e at Center Figure 7:

Star dome load increment vs. vertical displacement.



251

3.3 Crisfield arch truss The model is taken from [2] and tests multiple snap-through and snap-back behavior. The model has 101 elements with 42 nodes with a total of 126 degrees-of-freedom. Fig. 8 shows the test model and the applied load at the apex. Fig. 9 displays the Nastran results and the predicted response of the Hrinda arc-length FEA computer program. The first snap-through, shown in figure 9, occurs at a load increment of -.1 and -29.2 inches. A snap-back occurs at a load increment of .29 with a displacement of -3.33 inches. A second snapthrough begins and continues with increasing loads. The arc-length FEA program closely follows the equilibrium path found by a nonlinear Nastran solution.

Figure 8:

Crisfield large circular arch. -35


-30

-25

N astran

H rinda -20

-15

-10

-5 -0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0

Load ( x 1e7 lbs. ) Load/D eflections for C risfield Arch @ Apex Figure 9:

Arch load deflection apex comparisons of Nastran vs. Hrinda.



Figure 10:

4

Arch deflections at snap-through and snap-back.

Conclusion and future work

The arc-length method was used in a finite element program created to properly find the equilibrium path of highly geometrically nonlinear truss structures. Critical points along the path were found and passed to show snap-through and snap-back behavior of the truss structures. Future work will include the Newark method for solving nonlinear transient problems. The Hrinda finite element program for the nonlinear static load case will be revised to include nonlinear dynamics.

References [1] [2] [3] [4] [5] [6]

Crisfield, M. A., Non-Linear Finite Element Analysis of Solids and Structures: Volume I Essentials, John Wiley & Sons 1991. Crisfield, M. A., Non-Linear Finite Element Analysis of Solids and Structures: Volume 2 Advanced Topics, John Wiley & Sons 1997. Owen, D. R. J., Hinton, E., Finite Elements in Plasticity: Theory and Practice, Pineridge Press Ltd., 1980. Owen, D. R. J., Hinton, E., Taylor, C., Numerical Methods for Non-Linear Problems, Volume 1, Pineridge Press, 1980. Riks, E., The Application of Newton's Methods to the Problem of Elastic Stability, Journal of Applied Mechanics, Vol. 39 (1060-1065), 1972. Riks, E., An Incremental Approach to the Solution of Snapping and Buckling Problems, International Journal of Solids and Structures, Vol. 15 (529-551), 1979.


Section 3 Fluid flow



255

On the accuracy of integral representation of differential operators in Lagrangian blob mesh-less methods C. Golia & B. Buonomo Department of Aerospace and Mechanical Engineering, Second University of Naples, Italy

Abstract We explore novel ideas to improve the accuracy of the integral approximation of differential operators (Gradient and Laplacian) in the simulation of thermal viscous problems with Lagrangian Blob mesh-less methods. Basically we investigate and develop a novel convolution integral discretization of the differential operators by using 2D-Taylor series expansions and a Gaussian like kernel function defined on a compact support around the blob centre of a given particle. This allows us to overtake: • deficiency of cells in the compact domain due to irregular distribution of the particles around the given blob, • deficiency of cells in the compact domain caused by the presence of a boundary cutting the support of a nearby blob. The accuracy and order of approximation of such a discretization are determined in regular and randomly distorted grids of various sizes, and compared with the widely used PSE (Particle Strength Exchange) formulation. Results obtained in the solution of thermal buoyant problems at realistic values of the Grashoff number demonstrate validity and benefits of the novel findings. Keywords: integral definition of differential operators, lagrangian mesh-less methods, vortex/thermal blobs, thermal buoyant problems.



1

Introduction

Mesh-less methods (both 2 and 3 D) are of growing interest in the simulation of viscous unsteady fluid-dynamic problems. In particular, we focus on the analysis of the particle based Lagrangian approach to the Helmholtz formulation of the complete Navier-Stokes equations. Such analysis is made using individual discrete particles, blobs, that, as computational elements, transport (with the velocity induced by the vorticity field) momentum/vorticity and energy. In the regularized vortex blob method, the discretization of the equations is made by considering N-blobs problems where vorticity/energy is represented, for a general particle located in (x, t), by convolution integrals with kernel radial functions, W(r), r = |x|, satisfying given normalization properties. The accuracy of the solution depends on the approximation of the differential operators of the equations. Usually diffusive Laplacian operators are discretized according to the Particle Strength Exchange (PSE) method proposed by Degond and Mas-Gallic [1] and Gradient operators (needed in the buoyant term and in the 3D vorticity equation stretching term) according to a similar method proposed by Eldredge et al [2]. Both such discretizations are defined by convolution integrals with kernel radial functions derived according to the specific kernel function, W(r), used. The discretization of each term is substituted in the equations and integrated over the volume around each blob particle: this will result in a time integration of N-body problem. The discretization of the differential operators is quite accurate for blobs regularly distributed in the field and away from boundaries/discontinuities. Usually, to avoid inaccuracy due to distorted particle field, a regrid is used, after a number of integration steps, to project the field on a regular mesh. The interest of the authors lay in the modeling of Unsteady Free Convection Buoyant Flows (Helmholtz formulation) characterized by strong unsteady start up phenomena (usually mushroom rising domes) and by a timely growing convective field that requires vanishing asymptotic boundary conditions (pressure closure). Since regrid of the actual blob field on a regular symmetric grid will result in a somewhat viscous step, the authors are willing to search for formulations of the differential operators that are accurate also for distorted blob fields.

2

Taylor convolution formulation

We consider a function f(x) = f(x,y), on a 2D field, represented by blobs located on a non regular grid and we are looking to compute the values of Gradient and Laplacian of the field at a given blob located at xo = (xo,yo) by a convolution integral strategy on a compact domain around the blob, by using a kernel function W(x-x o, h). Following an idea developed by Liu and Lui [3], we consider a Taylor expansion of f(x,y) around the given point (xo,yo) truncated at the 3rd order:



257

f (x, y) = f (x o , yo ) + ∆x f x o + ∆y f y + 12 ∆x 2 f xx o + ∆x ∆y f xy + 12 ∆y 2 f yy + ... (1) o

o

o

where: ∆x = x - xo, ∆y = y - yo. We consider a kernel function W(x-xo,h) that is compact on a domain Ω of radius k times the grid size, h, around xo. The kernel vanish on the boundary of Ω and has unitary zero-order moment, whereas all other n-th order moments are zero: M 0 ≡ ∫ W ( x − x o , h ) dx = 1

(2)

Ω

M n ≡ ∫ ( x − x o ) W ( x − x o , h ) dx = 0 ∀n ≠ 0 n

Ω

Moreover we assume that the first partial derivatives of the kernel, Wx(x-xo, h) and Wy(x-xo, h), vanish on the boundary and have unitary first-order moments whereas all others vanish: M1 ≡ ∫ ( x − x o ) Wa ( x − x o , h ) dx = −1

(3)

Ω

M n ≡ ∫ ( x − x o ) Wa ( x − x o , h ) dx = 0 ; ∀n ≠ 1, a = ( x, y ) n

Ω

Finally we assume also that all second partial derivatives of the kernel, Wxx(x-xo, h), Wxy(x-xo, h) and Wyy(x-xo, h) vanish on the boundary and have unitary second-order moments whereas all others vanish: M 2 ≡ ∫ ( x − x o ) Wab ( x − x o , h ) dx = 2 2

(4)

Ω

M n ≡ ∫ ( x − x o ) Wab ( x − x o , h ) dx = 0 ; ∀n ≠ 2, ab = ( x, y ) ⊗ ( x, y ) n

Ω

Since we are looking for a convolution representation of operators, we multiply eq. (1), truncated to 2nd order, alternatively, by the two first partial derivatives of the kernel function W(x-xo, h) and integrate on the compact domain. We obtain then a system of two linear equations: f x   f x 

0

∫ ∆x W dΩ + f ∫ ∆y W dΩ ≈ ∫ [f − f ] W dΩ x

Ω

y 0

∆x Wy dΩ + f y 0 ∫ Ω

0

x

Ω

o

x

Ω

(5)

∫ ∆y Wy dΩ ≈ ∫ [f − fo ] WydΩ

Ω

Ω

where: f = f(x), f o = f( x o ). The solution of eq. (5) furnishes the representation of the components of the field gradient, fx fy, at xo. Similarly by multiplying alternatively eq. (1) by the three second partial derivatives of the kernel function W(x-xo, h), by truncating the expansion to the 3rd order and by integrating on the compact domain Ω, we can obtain the representation of the second derivatives needed for the Laplacian (fxx , fyy) by solving the system of 3 linear equations: WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

258 Computational Methods and Experimental Measurements XIII  f xx   f xx  f  xx 

1 o 2 1 o 2 1 o 2

∫ ∆x W

dΩ + f xy

∫ ∆x W

dΩ + f xy

∫ ∆x W

dΩ + f xy

2

xx

Ω

2

xy

Ω

2

yy

Ω

o

o

o

∫ ∆x∆yW

dΩ + f yy

∫ ∆x∆yW

dΩ + f yy

∫ ∆x∆yW

dΩ + f yy

xx

Ω

xy

Ω

yy

Ω

1 o 2 1 o 2 1 o 2

∫ ∆y W

dΩ ≈ ∫ ∆fWxx dΩ − f x o ∫ ∆xWxx dΩ − f y

o

∫ ∆y W

dΩ ≈ ∫ ∆fWxy dΩ − f x o ∫ ∆xWxy dΩ − f y

o

∫ ∆y W

dΩ ≈ ∫ ∆fWyy dΩ − f x o ∫ ∆xWyy dΩ − f y

o

2

xx

Ω

2

xy

Ω

2

yy

Ω

Ω

Ω

Ω

Ω

Ω

Ω

∫ ∆yW

dΩ

∫ ∆yW

dΩ

∫ ∆yW

dΩ

xx

Ω

xy

Ω

yy

Ω

(6) It is worthy to note that in case of symmetrical grid, many of the convolution integrals will vanish, and a much simple representations will result: for the Gradient:

∫ ∆f W dΩ x

fx o ≈

Ω

∫ ∆x W dΩ

∫ ∆f W dΩ

(7)

y

;

fy ≈ o

x

Ω

Ω

∫ ∆y W dΩ y

v

for the Laplacian:

∫ ∆f W

xx

f xx o ≈

1 2

Ω

∫ ∆x

2

Ω

f yy ≈ o

1 2

∫ ∆x

dΩ * ∫ ∆y 2 Wyy dΩ − ∫ ∆f Wyy dΩ * ∫ ∆y 2 Wxx dΩ Ω

Ω

Ω

Ω

Ω

Wxx dΩ * ∫ ∆y 2 Wyy dΩ − ∫ ∆x 2 Wyy dΩ * ∫ ∆y 2 Wxx dΩ 2

Ω

(8)

Ω

Wxx dΩ *∫ ∆f Wyy dΩ − ∫ ∆y 2 Wyy dΩ * ∫ ∆f Wxx dΩ Ω

Ω

Ω

Ω

Ω

2 2 2 2 ∫ ∆x Wxx dΩ * ∫ ∆y Wyy dΩ − ∫ ∆x Wyy dΩ * ∫ ∆y Wxx dΩ

Ω

Ω

In the following we shall denote the representations given by eqs. (5) and (6) as “TCFC” and the representations given by eqs. (7) and (8) as “TCFP”.

3

Kernel function

The kernel used is a modified (2D) Gaussian: W(x, y, h) =

(

 4 x 2 + y2 4 1 − exp  π h2 h2 

)   

(9)

It can be shown that such kernel, with its derivatives, satisfy the Moment closures as per eqs. (2)–(4). The kernel of eq. (9) is compact on r/h ≤ 2 with an approximation of 1.5E-7; its first derivatives are compact with an approximation of 1.5E-6, the second derivatives are compact with an approximation of 1.5E-5.

4

Strategy of the performance analysis

We assume a given known field on a regular or perturbed grid over a region and we proceed to compute the values of Gradient and Laplacian on a number of collocation points. Thereafter, we do compare the values computed with various methods with the exact ones and determine the statistics of the errors. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


259

Among others used, we report the analysis for the field: f(x,y) = cos(2πx) sin(2πy) / (2π) (x,y)∈[0,1]x[0,1]

(10)

that represents the pressure field of a well known test case for benchmarking of Navier Stokes equations [4].

1 0

0

1

0.25 0

0.5

0.2 0.4

Y

X

0.75

0.6 0.8 1

1

Figure 1. Grid Comparison: Igrid= 0 ; NRandom=0.5 1 0.8

Y

0.6 0.4 0.2 0 0

0.25

0.5

X

0.75

1

REGULAR GRID COLLOCATION POINTS DISTORTED GRID

Figure 2. We note that this field implies homogeneous Neumann conditions on the boundaries x=0 and x=1, and homogeneous Dirichlet conditions on the boundaries y=0 and y=1. The reference grid is regular or perturbed according to the value of a random parameter Nrandom. For Nrandom=0 the reference grid is symmetric.


260 Computational Methods and Experimental Measurements XIII The grid of the collocation points, where we are computing the values of the Gradient and Laplacian, is located in the mid points of the original regular grid. For various grid size and values of Nrandom, we will compare the performances of the methods: TCFC, TCFP, and PSE, in terms of: •

Field survey comparison maps,

•

Map of the error in the field.

•

Global RMS error,

For each formulation, the trend of the Global RMS error in terms of the grid size, h, will give the real value of the order of approximation of the methods. 4.1 Field survey comparison maps We show the Laplacian maps obtained with TCFC, TCFP, PSE, compared to the exact one for the case: Nx*Ny =21·21, h=0.05, Nrandom=0 in Figure 3. We note that all the three methods catch the general behavior of the Laplacian field, but the PSE lacks for magnitude, and the TCFP locally suffers the Dirichlet boundary conditions on y=0 and y=1. T.C.F.Complete 0.9

0.8

0.8

0.7

0.7

0.6

0.6

Yr

Yr

Exact 0.9

0.5

0.5 0.4

0.4

0.3

0.3

0.2

0.2

LAPLACIAN Maps Nx * Ny = 21 *21 h = 0.05 Nrandom = 0.0

0.1 0.25

0.5

Xr

0.75

1

0.1 0.25

0.5

0.75

Xr

1

P.S.E.

T.C.F.Partial 1 0.9

0.9 0.8

0.8 0.7

0.7

0.6

Yr

Yr

0.6 0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.5

Xr

0.2

1

0.4

Xr

0.6

0.8

1

Figure 3. The same case, obtained with a strongly distorted grid denoted by Nrandom=0.5, results in the maps in Figure 4. It can be seen that TCFC is still able to give quantitative and qualitative satisfactory results, it suffers only the intersection zones with change of sign. For the other two maps, TCFP is clearly failing and PSE is clearly unsatisfactory. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

Computational Methods and Experimental Measurements XIII T.C.F. Complete

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

Yr

Yr

Exact

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2 0.1

0.1 0.25

0.5

0.75

Xr

0.25

1

0.5

LAPLACIAN Maps Nx * NY = 21*21

T.C.F. Partial 1 0.9 0.8

0.75

Xr

1

P.S.E.

h = 0.5

0.9

Nrandom = 0.50

0.8 0.7

0.7

0.6

Yr

0.6

Yr

261

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.5

0.2

1

Xr

0.4

Xr

0.6

0.8

1

Figure 4. 4.2 Field maps of the error For conciseness we report solely the error maps for TCFC and PSE corresponding to the case Nx*Ny =41*11, h=0.025, Nrandom=0.25 in Figure 5. % Error Map on Laplacian: T.C.F. Complete Method Nx*Ny = 41*41; NRandon = 0.25

-85.7143

-42.85 71

-71. 4286

29 -57. 14

-57.14 -42.8571 -28.5714 -8529 .7143

-57.1429 -14.2857

-71.42 86

71 -42.85

28 6

1.4

-7

-71. 4286

-71. 4286

-2 5

Y

5 -7 50

Y

-25

-71.42 86

-5 7.1

-85.714

7

-57.1429

71

.2085 -14

85

42 9

3-100-71.428

6

0

25 -25 10 0

0.25

-71.4286

-57.1429

-100

3

0

0

-57.1429 -100 -85.7143

-71.4286

-42.

-100

14

10

25

7 85 -100 .2 14 14 .57 28

5.7

5

0.1

9 42

-8

-75

-2

0.1

6

7.1

29 -57.14

42 9

28

1 57 2.8 0 -4 -14.2857

-100

0.2

1.4

PSE-Lap 100 85.7143 71.4286 57.1429 42.8571 28.5714 14.2857 0 -14.2857 -28.5714 -42.8571 -57.1429 -71.4286 -85.7143 -100

7

-5

-57.1429 -71.42 86

-57.1429

-100 -71.4286 -5 7.1

85

-57.1429

-7

143 -57.1429

4.2

86 42 -71. 9 42 7.1

-85.7

0.3

0.2

0

0.5 0.4

0.3

-85.7143

-1

9 42 7.1 -5

-71. 4286

-100

25

00

-42.8571 -71.42 -71.428 6 86

-100

-1 00

0.6

1 -42.857 57 -14.28

0

0.4

50 50 75

5

-100 -57.1429

-85.7143 -71. 4286

1429 -57.

-100 -2

-1

10

0.5

0.7

8571 -42. 29 .14 -57

-25

-25

-25

0.8

6 1.4 28 -7-57.1429

0.7

0.9

-5

-2 5

Comp-Lap 100 75 50 25 -25 -50 -75 -100

0.8

-57.1429 -85.7143

-85.7143

25

00 -1

0.9

0.6

% Error Map on Laplacian: P.S.E. Nx*Ny = 41*41; NRandom=0.25.

1 00

50

-1

75

-5 0

1

0

5000 1

0.5

X

0.75

1

0.25

0.5

X

0.75

1

Figure 5. These maps confirm what said in the previous paragraph. It is clear the much better accuracy of the TCFC. 4.3 Error versus grid size We consider the error on the RMS norm of Gradient and Laplacian with varying grid size and distortion of the grid. Since we realize that such norms are affected by strong local errors nearby boundaries, we report the norm for the computations of the operators in collocation points within one grid size step away from the boundaries, that we call Inner grid. We consider firstly the Gradient of the three methods. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

262 Computational Methods and Experimental Measurements XIII Integral Computation of the Gradient Operator Distorted Grid: NRandom = 0.25

Integral Computation of the Gradient Operator Symmetric INNER Grid 1

1

O(h2)

0.1

0.01

Gradient RMS Error Norm

Gradient RMS Error Norm

O(h) 0.1

O(h2)

0.01

0.001

Partial Complete P.S.E.

0.001 0.01

h (grid size)

0.0001 0.01

h (grid size)

0.1

0.1

Figure 6. Integral Computation of the Laplace Operator Distorted INNER Grid, NRandom=0.25

Integral Computation of the Laplace Operator Symmetric INNER Grid Laplacian Operator RMS Error Norm

Laplacian Operator RMS Error Norm

1

O(h2)

0.1

0.01


0.001 0.01

h (grid size)

0.1

10

O(h2)

1

0.1


0.01

0.001 0.01

h (grid size)

0.1

Figure 7. The trends clearly show that TCFC and TCFP are almost equivalent, of 2nd order, and practically not sensible to grid distortion. PSE is almost zero order. TCFC and clearly inadequate. A comparison for the Laplacian shows that TCFC and TCFP are equivalent for symmetrical grid and 2nd order. For distorted grid TCFC maintains roughly the 2nd order accuracy, whereas TCFP and PSE are not well performing. In conclusion, the TCFC perform much better of the other formulations both for Laplacian and for Gradients, and it is second order.

5

Buoyant problem test

We are going to compare the results of running the same 2D code with PSE and TCFC formulations of Gradients and Laplacians. The code, detailed elsewhere in [5–7], is based upon Lagrangian particles (vortex and heat, independently treated) that, as computational elements, transport (with the velocity induced by the vorticity field) momentum/vorticity and energy. Vorticity is produced by the thermal gradient present in the flow field, in the context of validity of Boussinesq hypothesis. Heat is generated by the Thermal Boundary/Initial Conditions. Both vorticity and heat diffuse according to their transport coefficients and Laplacian terms. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


263

The formulation is based on a splitting technique of the convective and diffusive terms: it considers an hyperbolic equation for the trajectory of the blobs, and 2 parabolic equations for the diffusive processes, of thermal energy and vorticity, along the characteristics curves (trajectories). The evaluation of the velocity from the vorticity field, needed to integrate the trajectories of the N blobs, is a classical N-Body problem that requires O(N2) operations. The code uses a Fast Multipole Method (FMM) that is a O(N) algorithm, capable of self organize in order to warrant an imposed error level on the calculation of the velocity field. The resulting methodology has strong advantages, among the others: it is intrinsically unsteady, continuity is satisfied by definition, blobs move where they are needed (loci of strong gradients), asymptotic conditions are automatically satisfied, generates very robust codes (CFL condition removed). This will allow, with modern workstations and reasonable computing time, to simulate thermal free convection problems of engineering interest at realistic values of the Grashoff number. The code is organized to perform, after a given number of computational steps, a regrid process, i.e. project the field on a regular grid. At time of the regrid the code automatically creates if it is the case, new blob particles to take into account the natural evolution of the flow. T.C.F.Complete Gr=.454E8, Re=1. Regrid Steps=5

time =

P.S.E. Gr=.454E8, Re=1. Regrid Steps=5

1.000

0.6

0.4

0.4

1.000

Y

Y

0.6

time =

0.2

0.2

0

0

-0.2

-0.2 -0.2

0

X

0.2

-0.2

Figure 8. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

0

X

0.2

264 Computational Methods and Experimental Measurements XIII Since regrid steps result in a somewhat viscous step, object of this analysis is to compare the capacity of use a value of “Regrid Steps” as high as possible still preserving the main characteristics of the solutions. First we use the same low Regrid Steps=5, for both cases, with a low resolution grid. The figure shows the isotherm maps. We note that the solutions are very similar at initial stages, PSE slightly more varying at late times. This is caused, perhaps by the smoothing effect of the regrid processes. Then we run the same case, low resolution grid, with larger Regrid steps =20. The figure below shows isothermal maps and velocity plots. We can note that for the PSE, at late times the velocities grow with untidiness and this cause an abnormal break up of the mushroom rising cup and of its stem. Further analysis with a much finer grid reveals indeed shapes and mushroom rising cup very similar to the ones given by the TCFC. TCFC - Regrid steps = 20

time =

1.000

PSE - Regrid steps = 20

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

1.000

Y

Y

0.6

time =

0.1

0.1

0

0

-0.1

-0.1

-0.2

-0.2 -0.2

-0.1

0

X

0.1

0.2

-0.2

-0.1

0

X

0.1

0.2

Figure 9.

6

Conclusions

Analysis and comparisons made in this paper confirm definitely the superiority of the TCFC for the representation of differential operators in blob mesh less methods. Preliminary tests with 3-D problems fully confirm such findings; details are reported in a companion paper [8].

References [1] [2]

Degond, P. and Mas-Gallic, S., 1989, “The weighted Particle method for Convection-Diffusion Equations, part1: the case of an isotropic viscosity” Maths of Computation 53 (188), pp. 485-507 Eldredge, J.D., Leonard, A. and Colonius, T., 2002, “A General Deterministic Treatment of Derivatives in Particle Methods”, J. Comp. Phys. 180, pp. 686-709 WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


[3] [4] [5] [6] [7] [8]

265

Liu, G.R. and Liu, M.B., 2003, “Smoothed Particle Hydrodynamics”, World Scientific Pub. Co. Ross Ethier, C. and Streinman, D.A., 1994, ”Exact fully 3D Navier-Stokes solutions for benchmarking”, Int. J. Num. Meth. in Fluids 19, pp. 369375. Golia, C., Buonomo, B., Manca, O., and Viviani, A., 2004, “A VortexThermal Blobs Method For Unsteady Buoyancy Driven Flows”, ASMEIMECE. Anaheim, California. Golia, C., and Buonomo, B., 2005, “An Effective Blob Approach to Unsteady Thermal Buoyant Flows”, CMEM 2005, Malta Golia, C., and Buonomo, B., 2005, “Numerical Simulation of Unsteady Natural Convection by Blobs Methods “, 60th ATI Congress, Rome, Italy Golia, C., Buonomo, B., and Viviani, A., 2007, “A corrected Vortex Blob Method for 3D Thermal Buoyan Flows”, ICTEA Amman, Jordan




267

Construction of a non-Newtonian fluid model based on the finite difference lattice Boltzmann method S. Tajiri & M. Tsutahara Graduate School of Science and Technology, Kobe University, Japan

Abstract A model of the lattice Boltzmann method for non-Newtonian fluids was constructed. The shear stress of purely viscous but non-elastic non-Newtonian fluid is a function of shear rate only. For the power-law model, only two constant parameters can cover shear-thinning and shear-thickening fluids. Two power-law models are introduced to the finite difference lattice Boltzmann method. One is a model in which the collision parameter (the relaxation time) is determined as a function of the shear rate, and then the viscosity changes point by point according to the shear rate. For the other model, the effect of the variable viscosity is introduced as an external force which is determined by the local shear rate. Two-dimensional channel flow between two parallel plates was calculated by using the above two models. Both models are shown to give satisfactory results. However, some discontinuity appears in the calculation by the former model that is due to instability of the calculation. For the latter model, smooth velocity distributions are always obtained. The shear rate is estimated by the second order and fourth order central finite difference scheme, but the accuracy of the velocity distribution is to first order. A model in which the normal stress can be introduced by the shear is being constructed in the same manner. The normal stress was given by introducing the single mode Giesekus constitutive model to the finite difference lattice Boltzmann method. Keywords: finite difference lattice Boltzmann method (FDLBM), non-Newtonian fluids, power-law model, Giesekus model.



1

Introduction

The lattice Boltzmann method (LBM) is a computational tool to analyze a thermal viscous fluid [1–4]. In particular, calculations of the flow within complicated porous media and the multi-phase flow are effective [5]. Recently, some models of LBM for purely viscous but non-elastic non-Newtonian fluid are proposed. The flow between parallel plates and open cavity flow were analyzed by Gabbanelli et al. [6] with the constructed LB model for the non-Newtonian fluid, and the examination about the calculation accuracy of the model was performed. Sullivan et al. [7] analyzed the detailed behaviour of the nonNewtonian fluid in three-dimensional complicated porous media. Their LB models gave the viscosity locally by determining the collision parameter (the relaxation time). The collision parameter was determined according to the shear rate. However, the relaxation time has a direct relation to the calculation stability, and the stability of the calculation is not enough. Yoshino et al. [8] presented the LB model for a non-Newtonian fluid using the Lattice Kinetic Scheme (LKS), in order to improve the problem. They analyzed the Darcy law of a two-dimensional porous media, and obtained the appropriate result. In this paper, the non-Newtonian model of the finite difference lattice Boltzmann method (FDLBM) is presented. The viscosity of the non-Newtonian fluid was given using the relaxation time and the external force. The channel flow between two parallel plates was calculated by using the constructed models. Normal stress plays an important role in the non-Newtonian viscoelastic fluid. The model which can generate the normal stress was also constructed in the same manner using the external force.

2

Numerical method

The original two-dimensional and three-dimensional isothermal models (the D2Q9 and D3Q15 model [9]) are presented briefly in section 2.1. The introduced non-Newtonian model is presented in section 2.2. The power-law model is introduced for the non-Newtonian purely viscous fluid. The power-law model has only two constant parameters, and can cover shear-thinning and shearthickening fluids [10]. The single mode Giesekus constitutive model is used for the viscoelastic fluid [11, 12]. The Giesekus model is a popular choice for several flows [13, 14]. For example, it is known that the Giesekus model is useful for describing processing flows of polymer solutions. 2.1 Finite difference lattice Boltzmann method (multi particle model) A discrete BGK equation for the FDLBM is written as follows with the distribution function fi k (x, t ) having the particle velocity ci

∂fi k 1 k + ci ∇fi k = − f − fi eqk φ i ∂t

(

)


(1)


269

φ

is the collision parameter (the relaxation time) and f i eqk is the local equilibrium distribution function chosen to satisfy the Navier-Stokes equation (refer to [9]). Macroscopic variables on each lattice site are defined as

where

ρ k = ∑ fi k = ∑ fi eqk i ,k

∑ρ k

k

(2)

i ,k

u = ∑ f i k ci = ∑ f i eqk ci i ,k

(3)

i ,k

for, respectively, the density, the momentum. For the finite difference lattice Boltzmann method, the governing equation is discretized, so the corresponding calculation procedures are given as follows. The time integration is performed by the second-order Runge-Kutta method and the third-order upwind scheme is employed for space dispersion of ci ∇f i . 2.2 Non-Newtonian model 2.2.1 Power-law model The constitutive equation of the viscosity of the power-law model is given as

η = η0 γ

n −1

(4)

η0 and n are the parameters of the power-law fluid. n = 1 corresponds to the Newtonian fluid in eqn. (4). Then, η0 is the coefficient of the Newtonian where

viscosity. n > 1 corresponds to the shear-thickening fluid. n < 1 corresponds to the shear-thinning fluid which decreases the viscosity coefficient when the shear rate becomes larger. The shear rate γ has a relation to the symmetrical rate of strain tensor Dαβ as follows

γ = Dαβ Dαβ

(5)

∂u 1  ∂u Dαβ =  β + α  2  ∂xα ∂xβ

  

(6)

where the subscript α and β represent the Descartes coordinates and follow the summation convention. The velocity gradient ∂uα ∂xβ was derived by using the second order central difference scheme. The power-law model is introduced to the collision parameter φ and the external force Fαpower −law as follows

φ = 3η

( D2Q9 ) ,

3 2

φ= η

( D3Q15)


(7)

270 Computational Methods and Experimental Measurements XIII Fαpower −law = −

η1

∂ ∂xβ

 ∂u η1 α  ∂xβ

 ∂  +  ∂xβ

 ∂ uα η  ∂xβ

  

(8)

is the original viscosity as the Newtonian fluid in eqn. (8). The above force is

introduced to the local equilibrium distribution function in eqn. (3) by replacing

∑ρ k

k

uα → ∑ ρ k ( uα + φ Fα )

(9)

k

2.2.2 Single mode Giesekus constitutive model Here the viscoelastic fluid model proposed by Giesekus is shown briefly. The constitutive equation is ∇

τ αβ + λτ αβ + α

λ τ τ = 2η0 Dαβ η0 αχ χβ

(10)

where τ αβ represents the stress tensor. The non-linear term τ αχτ χβ is a key role in the Giesekus model.

α

is the mobility factor ( 0 ≤ α ≤ 1) .

λ

is the relaxation

∇

time of the viscoelastc fluid． τ αβ represents the upper convected derivative as ∇

τ αβ =

∂τ αβ ∂t

+ uχ

∂τ iαβ ∂xχ

−

∂u ∂ui τ χβ − τ αχ χ ∂xχ ∂xβ

(11)

The above equations correspond to the Giesekus model with non-considering the infinitesimal viscosity in the shear rate γ → 0 . The Giesekus stress is introduced to the external force in the same manner with section 2.2.1. The total stress σ αβ of the Giesekus fluid is σ αβ = − Pδ αβ + τ αβ . Hence, the external force can be written as follows:

Fαgiesekus

3

=−

∂ ∂xβ

 ∂u  − Pδαβ + η1 α ∂xβ 

 ∂σ αβ  +  ∂xβ

(12)

Result and discussion

3.1 Channel flow between two parallel plates The channel flow between two parallel plates was calculated by using the two introduced power-law models. The exact solution of normalized velocity WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

Normalized Velocity u/umean


271

0.5

0.4 0.3

n=0.50

0.2

exact solution Δy=1/20 (CP) Δy=1/40 (CP) Δy=1/80 (CP) Δy=1/20 (EF) Δy=1/40 (EF) Δy=1/80 (EF)

0.1

0 0

0.02

0.04

0.06

0.08

0.1

Normalized Distance x/L Figure 1:

Normalized velocity profile near one of two parallel plates. The power-law exponent of the fluid is n=0.50 (shear thinning). Full symbols correspond to the external force (EF) model, and blank symbols correspond to the collision parameter (CP) model not giving the extrapolation of non-equilibrium distribution functions on the plate boundary.

* uexa = uexa ( x ) uexa _ mean is written in eqn. (13). uexa _ mean represents the mean

velocity between two plates. n +1   n   L−x      2n + 1  2 *   uexa − + 1 ( x) =   n +1   L       2    

(13)

where L represents the distance between two parallel plates. The external force model could give us the satisfactory velocity distribution, but the collision parameter model made a discontinuity of the velocity distribution near boundary in fig.1. The velocity discontinuity on boundary is the conventional characteristic of the FDLBM. This problem is improved by the below-mentioned technique. The distribution functions fi k (x) on the plate boundary are given the extrapolation of non-equilibrium distribution functions as f i k (x) = f i keq (x) +  2 f i kneq ( x + ∆x) − f i kneq (x + 2∆x) 

(14)

The calculation result with the above technique agreed with the exact solution even if using the collision parameter model (fig.2). The evaluation about the external force model is described below. The satisfied velocity distribution was obtained in the parameter n changed from 0.5 (shear-thinning fluid) to 1.5 (shear-thickening fluid) as shown in fig.3. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)



1.5

1

n=0.50

0.5

exact solution Δx=1/80 (CP) Δx=1/80 (EF)

0 0

0.2

0.4

0.6

0.8

1

Normalized Distance x/L Figure 2:

Normalized velocity profiles between two parallel plates. The power-law exponent of the fluid is n=0.50 (shear thinning). Full symbols correspond to the external force (EF) model, and blank symbols correspond to the collision parameter (CP) model. The distribution functions on the plate boundary are given the extrapolation of non-equilibrium distribution functions.

The accuracy of the velocity distribution was checked. The error norm in the steady flow (the velocity fluctuation become smaller than 10-8) was defined as

∑ u ( x) − u ( x) error = ×100 ∑ u ( x) *

* exa

x

* exa

(15)

[%]

x

where u* ( x ) is the velocity normalized by the mean velocity. The shear rate is estimated by the second order and also fourth order central finite difference scheme. The error norm in the highest resolution of lattice was about 0.02%, but the accuracy of the velocity distribution is the first order as shown in fig.4. 3.2 Shear flow of two component fluids on gravity field The shear flow with the free surface is simulated by using the introduced Giesekus model. The moderate diffusion scheme proposed by Latva-Kokko and Rothman [15, 16] was introduced to the FDLBM in order to simulate the behaviour of two immiscible phases. The governing equation (1) becomes

∂fi k 1 k + ci ∇f i k = − f − f i eqk + f i k − fi ′k ∂t φ i

(

) (


)

(16)



273

2

1

exact solutions n=0.50 n=0.75 n=1.00 n=1.50

0 0

0.2

0.4

0.6

0.8

1

Normalized Distance x/L Normalized velocity profiles between parallel plates for different power indices by the external force model of the power-law fluid. The solid lines represent exact solutions.

Relative error [%]

Figure 3:

Figure 4:

101 nd

: 2th order center difference : 4 order center difference

th

4 order rd

3 order

0

10

2nd order

10-1 st

1 order

10-2 -2 10

Grid size ⊿x

10-1

Error norms of the channel flow between parallel plates with ∆x=1/20, 1/40, 1/80 by the external force model. ∆ represent results of the 2nd order central finite difference scheme for estimating the shear rate. ● represent results of the 4th order central finite difference scheme. The grey solid lines indicate inclines of the order.


274 Computational Methods and Experimental Measurements XIII where f i ′k is the redistribution function as follows: f i ′G = fi′N =

ρG

ρG ρ N f eqG (0) + f i eqN (0) ) cos ϕ 2 ( i ( ρG + ρ N )

i

ρN ( f G + fi N ) − κ ρ ρG+ρρN 2 ( fi eqG (0) + fi eqN (0) ) cos ϕ ρG + ρ N i ( G N)

i

ρG + ρ N

(f

G i

)

+ fi N + κ

(17)

where subscript N and G represent the components of the fluid. The parameter κ controls the thickness of interface. f i eqk (0) is the zero-vector local equilibrium

distribution function. ϕ represents the angle between the particle direction and normal direction of the interface. The angle was determined as follows:

cos ϕ i =

G ⋅ ci G ⋅ ci

G ( x ) = ∑ ci  ρ G ( x + ci ) − ρ N ( x + ci ) 

(18) (19)

i

Fig.5 shows the three-dimensional calculation domain. The calculation grid was 9×31×31. The periodic boundary was employed in the x-direction, and the other boundaries were non-slip condition. The right side boundary in fig.5 moved at the velocity U0=0.05. Fig.6(a) is the pressure profile at the parameter α=1.0 and λ=10000. It is shown that the pressure near a right-hand side boundary becomes large by the normal stress produced by the shear. Such pressure distribution in the Newtonian fluid is not seen (fig.6(b)). When the calculation time passed, the calculation became unstable.

4

Conclusion

The model of the finite difference lattice Boltzmann method for non-Newtonian fluids was presented. For purely viscous fluids, the power-law model is introduced to the collision parameter and the external force. Some discontinuity of velocity distributions appeared by using the collision parameter model in the two-dimensional channel flow between parallel plates. The stability of calculation was not enough. The external force model allowed us to obtain smooth velocity distributions. The single mode Giesekus constitutive model for viscoelastic fluids was also introduced in the same manner as the external force model. The normal stress appeared in the shear flow on the gravity field. However, the calculation was unstable. In the future work, the infinitesimal viscosity of the Giesekus model will have to be considered for improving the stability of calculation.



275

Newtonian fluid

g z

y

U0

Giesekus fluid

x Figure 5:

Calculation domain (grid number 9×31×31). The boundary condition of x-direction is the periodic. The other boundaries are non-slip solid walls. The velocity U0=0.05. The Giesekus fluid is given the gravity force g=0.0001.

Newtonian fluid

Giesekus fluid

(a) Giesekus-Newtonian Figure 6:

Newtonian fluid

Newtonian fluid

(b) Newtonian-Newtonian

Comparison of the pressure fields between the Giesekus fluid and the Newtonian fluid. The parameter α=1.0, λ=10000.


Qian, Y. H., Succi, S. & Orszag, S. A., Recent Advances in Lattice Boltzmann Computing, Ann. Rev. of Comp. Phy. (D. Stauffer ed.), World Scientific, pp. 195-242, 1995. Rothman, D. H. & Zalenski, S., Lattice-Gas Celluar Automata, Cambridge U. P., pp. 73-90, 1997. Chen, S. & Doolen, G. D., Lattice Boltzmann Method for Fluid Flows, Ann. Rev. Fluid Mech., Ann. Rev. Inc, pp. 329-364, 1998. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

276 Computational Methods and Experimental Measurements XIII [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

S. Succi, The lattice Boltzmann Equation for Fluid Dynamics and Beyond, Oxford, pp. 51-123, 2001. Inamuro, T., Ogata, T., Tajima, S. & Konishi, N., A lattice Boltzmann method for incompressible two-phase flows with large density differences, J. Computational Physics, 198, pp. 628-644, 2004 Gabbanelli, S., Drazer, G. & Koplik, J., Lattice Boltzmann Method for Non-Newtonian (Power-law) Fluids, Phys. Rev. E, 72, 2005. Sullivan, S. P., Gladden, L. F. & Johns, M. L., Simulation of Power-Law Fluid Flow through Porous Media using Lattice Boltzmann Techniques, J. Non-Newton. Fluid Mech., 133, pp.91-98, 2006. Yoshino, M., Hotta, Y., Hirozane, T. & Endo, M., A Lattice Boltzmann Method for Non-Newtonian Fluid Flows, Japan Society for Computational Methods in Engineering, 6(2), 2006. Tsutahara, M., Takada, N. & Kataoka, T., Lattice Gas Method & Lattice Boltzmann Method, Corona-sya, pp.101-112, 1999; in Japanese. Nakamura, K., Non-Newtonian Fluid Mechanics. Corona-sya, pp.68-71, 1997; in Japanese Giesekus, H., A Simple Constitutive Equation for Polymer Fluids Based on the Concept of Deformation-dependent Tensorial Mobility, J. NonNewton. Fluid Mech., 11, pp.69-109, 1982. Giesekus, H., Stressing behaviour in simple shear flow as predicted by a new constitutive model for polymer fluids, J. Non-Newton. Fluid Mech., 12, pp.376-374, 1983. Yoo, J. Y. & Choi, H. Ch., On the steady simple shear flows of the onemode Giesekus fluid, Rheologica Acta, 28, pp.13-24, 1989. Mostafaiyan, M., Khodabandehlou, K. & Sharif, F., Analysis of a viscoelastic fluid in an annulus using Giesekus model, J. Non-Newton. Fluid Mech., 118, pp.49-55, 2004. Latva-Kokko, M. & Rothman, D. H., Diffusion properties of gradientbased lattice Boltzmann models of immiscible fluids, Phys. Rev. E, 71, 2005. Latva-Kokko, M. & Rothman, D. H., Static contact angle in lattice Boltzmann models of immiscible fluids, Phys. Rev. E, 72, 2005.



277

Experimental measurements for the control of a vortex shaft theoretical model G. Ciaravino, L. Ciaravino & G. Pulci Doria Department of Hydraulic and Environmental Engineering, “G. Ippolito”, University of Naples Federico II, Italy

Abstract With reference to the hydrodynamic working of a dropshaft fitted with a vortex inlet, there are still two problems that are open to debate: the actual distribution of velocities at the inlet and the pressure distribution along the radius in the first cross section of the shaft. The majority of researchers assume that the velocity distribution is both irrotational and axially symmetrical, although some forty years ago Viparelli experimentally showed that such an assumption is actually incorrect, hypothesizing that the flow is symmetrical but not irrotational. Moreover, the determination of pressure distribution in the different proposed theories remains a debatable issue, with some claiming that the distribution of pressure is equal to zero while others maintain it is positive. In the present paper, two series of experimental measurements concerning the above-mentioned problems are analyzed. A first series of experimental tests performed with a Laser Doppler Anemometer confirms a different velocity distribution hypothesis: irrotational but not entirely symmetrical. A second series of experimental tests deals with pressure measurements made in the vertical shaft inlet. Contrary to what has been hypothesized by other researchers, these measurements indicate negative pressure values. Keywords: dropshaft, vortex flow, mathematical model, experimental measurements, Laser-Doppler Anemometer.

1

Introduction

Inflow in a dropshaft with vortex inlet is a rather complex phenomenon which still presents certain issues that have not been fully clarified. In spite of this such device is of great interest because of its undeniable technical importance. Its WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070291

278 Computational Methods and Experimental Measurements XIII inventor Drioli [1], and increasing numbers of authors thereafter (i.e. [2–9]) have investigated the vortex inlet with seemingly different procedures that can, however, be traced back to a single template. Nevertheless, the definition of the actual velocity distribution in the inlet chamber and the pressure distribution along the radius r in the first cross section of the shaft remain controversial issues. In particular the characteristic quantities of Drioli’s inlet are as follows (see fig. 1): - ∆: distance between the two axes of inflow channel and of vertical dropshaft; - h: stream depth in the inflow channel; - H: total energy (head) in the inflow channel; - b: width of the inflow channel; - δ: characteristic dimension of the inlet; - R: radius of the shaft; - ro: internal radius of the vortex in section 0-0 (see fig. 1); - Vt(r): tangential velocity in the section 0-0; - A or C: kinetic constant; - p(r): pressure in the section 0-0; - Q: flow rate discharged. A

Sec. A-A b

ro P1 h P4

P2

0

P3

P1

0

P3

R A

Figure 1: Typical Drioli vortex-flow inlet. Furthermore, the equations considered by the various authors are as follows: 1) Bernoulli’s equation for the definition of total head H; 2) the equation of flow rate:

(

)

Q = 2π 2g ∫r H − (p / γ ) − Vt2 / 2g ⋅ r ⋅ dr ; o R

(1)

in which Vt and the integral are determined in the section 0-0 while r is the generic radius in that section; WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


279

3) the equation needed for the calculation of ro determined either by means of experimental data or by means of the principle maximum flow rate which provides (i.e. Knapp [5]): ro = A / 2gH ;

(2)

4) the equation of continuity or the equation of momentum; the latter leads (i.e. Viparelli [3]) to the determination of the following formula: A=

Q∆ ; bh

(3)

5) the distribution of tangential velocity provided either by: Vt (r ) = C ;

(4)

which represents a constant distribution, or by: Vt (r ) = A r ;

(5)

which represents an irrotational distribution; 6) distribution of pressure p(r) along the radius ro in the section 0-0 provided either by: p( r ) γ = 0 ;

(6)

or by:

(

)(

)

p(r ) γ = A 2 2g ⋅ 1 ro2 − 1 r 2 .

(7)

As far as the velocity distribution is concerned, the overwhelming majority of authors hypothesizes that motion is irrotatational throughout the field and therefore assumes the characteristic expression of irrotatational flow shown in eqn. (5). Adopting eqn. (4) would contradict the hypothesis of irrotational flow. In actual fact, the velocity distribution in the curve of type Vt = constant mentioned by Ramponi [10] refers to secondary motion arising in the curve when the velocity before the curve is no longer constant in the straight section as a result of friction with the channel walls and, therefore, the motion is already no longer irrotataional. Viparelli [3] bases his reasoning both on eqn. (4) and on eqn. (5): even if Viparelli assumes eqn. (4) nevertheless continues to refer to the validity of Bernoulli’s equation which requires motion to be irrotational. Furthermore, the tangential velocity distribution of eqn. (5) combined with the equation of momentum (3) provides Viparelli with values of A that are so high as to make the flow rate expression imaginary, forcing Viparelli to arbitrarily WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

280 Computational Methods and Experimental Measurements XIII reduce the value of A (thus effectively negating the validity of the momentum equation). Nor it is possible to share Knapp’s hypothesis [5] which introduces ∆+b/2 instead of ∆ into eqn. (3). Given the uncertainty of the hypothesis, Pica [7] suggests using eqn. (4) as it makes the process simpler to deal with. As regards the pressure distribution along the radius in the first cross section of the shaft, the authors investigating this issue fall into two broad groups. On the basis of experimental measurements, Viparelli [3], Knapp [5] and Pica [7] hold eqn. (6) to be valid, theoretically justifying this position with the hypothesis of a balancing of the outward centrifugal forces (caused by the curvature of the trajectories in the horizontal plane 0-0, reported in fig.1) with the inward centrifugal forces (caused by the curvature of the trajectories in vertical planes). Binnie and Hookings [2], Ackers and Crump [4] and Adami [6], on the other hand, believe that only the curvature in the horizontal plane should be taken into account and, therefore, although different schemes are considered, they hypothesize a pressure distribution reported in eqn. (7).

2

Velocity distribution analysis

Almost all the models assume motion to be irrotataional and symmetrical to the vertical axis of the vortex shaft. Only Viparelli [3] and Pica [7] assume there is symmetry with respect to the so-called core of the vortex (the free space in the proximity of the shaft axis through which a continual supply of air is known to pass, thus preventing the closure of the vortex) whose axis turns out in their tests not to perfectly match the vertical axis of the shaft. The hypothesized symmetry generally requires the values of the three velocity components Vr, Vθ and Vz (in cylindrical coordinates r, θ and z, where the z axis is shared with the vertical axis of the vortex) to be constant along circumferences lying on horizontal planes or on arcs of these circumferences up to the point where they meet the solid wall: this makes it possible to define a symmetrical behaviour also for a system that is geometrically non-symmetrical. The three components of the rotation vector Ω along the three axes z, r, and t (the latter normal to z and r in the considered point) can be written as follows: Ωr =

1 ∂Vz ∂Vt − ; ∂z r ∂θ

(8)

∂Vz ∂Vr + ; ∂r ∂z

(9)

Ωt = −

Ωz =

∂Vt 1 ∂Vr Vt − + . ∂r r r ∂θ

(10)

If the hypotheses of symmetry and irrotationality are assumed to be simultaneously verified, then it must be true that:



281

∂Vt =0; ∂z

(11)

∂Vz ∂Vr = ; ∂r ∂z

(12)

∂Vt V =− t . ∂r r

(13)

The first of these relations tells us that Vt must be constant along every vertical. Since, by symmetry, Vt must be constant along every circumference (r=constant, z=constant) it follows that Vt must be constant along every cylinder of axis z. Furthermore, the third relation makes it possible to determine the law of velocity distribution of eqn. (5) by means of simple integration. However, the experimental measurements of Viparelli (who moreover did not take precise velocity measurements) do not verify this velocity distribution. From his observations, Viparelli therefore has inferred that a law of eqn. (5) could not be verified and Viparelli has completed his model after abandoning the hypothesis of irrotationality. Relinquishing the hypothesis of irrotational flow moreover seems to be a solution requiring suitable analyses as no other author (except, as already mentioned, Pica) has gone down this road. In actual fact, the other authors have assumed a velocity distribution of eqn. (5) leaving the contradiction of experimental data highlighted by Viparelli still unsolved. In their studies directed to formulate a new mathematical model on the hydraulic working of vortex shaft, Ciaravino et al. [8] have examined a different velocity distribution based on the renunciation to the symmetry hypothesis rather than on the renunciation to the irrotationality hypothesis. If the symmetry hypothesis is abandoned also the following position is abandoned: ∂Vr =0. ∂θ

(14)

Therefore a relation of the following type can be hypothesized in the whole flow field: ∂Vr = k ⋅ VΑ = cos t ≠ 0 ; ∂θ

(15)

where VA is the velocity in the inlet channel. Starting from the irrotational flow condition, eqn. (13) becomes: ∂Vt V 1 ∂Vr =− t + . ∂r r r ∂θ


(16)

282 Computational Methods and Experimental Measurements XIII Taking eqn. (15) into account, after the integration it is simple process to obtain: Vt =

A A + k ⋅ VΑ = + B . r r

(17)

It is interesting to observe that eqn. (17) is also obtained by Viparelli, not starting from eqn. (15) but by assuming non-irrotatational flow. In actual fact, Viparelli sees the term kVA as representing the motion rotation value rather than the contribution of Vr which contrasts the rotationality generated by Vt. Moreover, it is possible to hypothesize the existence of a second type of asymmetry on the basis of the fact that the inlet chamber wall gradually draws closer to the shaft axis: this determines ever smaller sections for the passage of the annular flow. This appears to bring about rises in the free surface (measured experimentally) and an increase in velocity, even when there is a decrease in the flow rate determined by the centripetal flow entering the shaft. The assumed increase in tangential velocity can be taken into account (in a first approximation) by adding a linear term in θ to the velocity distribution shown in eqn. (17), so that the tangential velocity Vt is expressed by: Vt =

A + B+ kθ ⋅θ . r

(18)

The decision to assume either the tangential velocity distribution of eqn. (5) or that of eqn. (18) requires considerable differences also in the definition of the distribution of radial velocity Vr. Thus introducing eqn. (5) or eqn. (18) into eqn. (17) yields, in the two different cases, by integrating with respect to Vr: Vr = Vro (r ) ; Vr = Vro (r ) + B ⋅ θ + k θ

(19) θ2 . 2

(20)

In the first case a symmetrical radial velocity distribution is verified (the θ terms are missing) dependent on r; in the second case, in addition to the dependence on r there are two extra θ terms, one linear and the other quadratic, which introduce the presumed asymmetry of motion. In order to verify which of these two velocity distributions is closer to reality, a series of preliminary experimental measurements of velocity have been taken inside the vortex (measurements which, as already mentioned, Viparelli did not have). Velocity measurements have been taken using a Laser Doppler Anemometer (LDA) on the experimental installation available at the Department of Hydraulic and Environmental Engineering of the University of Naples Federico II [9]. These tests have been conducted with a flow rate of 0.049 m3/s and with two alignments (orthogonal to the axis of the inlet channel) placed in a horizontal plane at a height of 0.06 m from the bottom of the shaft. This position WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


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has made it possible to achieve measurement points with conditions of movement that have been disturbance free both on the bottom and on the free surface. The distance from the shaft axis of the two alignments has been 0.115 m (for the inner one) and 0.150m (for the outer one). The first of these two distances has allowed the first alignment to be taken as close as possible to the shaft axis without interfering with the free surface. The second distance has allowed the second alignment to be tangential to the vertical cylinder that is the ideal continuation of the vertical shaft receiving the flow. Therefore, velocity measurements have been taken in different points of each of the two alignments. Under these conditions the measured velocity component is a combination of the Vr and Vt as a function of the considered alignment point. The measured component V(θ) (indicated by the value θ of the angular coordinate) is orthogonal at that point to the alignment and therefore: V(θ) = Vt sin θ − Vr cos θ .

(21)

For negative values of the angle θ it is alternatively possible to write: V(−θ) = − Vt sin θ − Vr cos θ .

(22)

Consistently with Viparelli’s observations [3], the simultaneous hypotheses of symmetry and irrotationality have given rise to incongruences in the interpretation of the results obtained from the experimental velocity measurements. In actual fact, introducing the expressions of Vt and Vr derived from eqn. (5) and eqn. (19) (valid for the simultaneous hypotheses of symmetry and irrotationality) into eqn. (21) and eqn. (22) yields: V(θ) =

A sin θ − Vro (r ) cos θ ; r

V(− θ ) = −

A sin θ − Vro (r ) cos θ . r

(23)

(24)

Adding these relations member by member, it is a simple process to obtain: Vro (r ) = −

V(θ) + V(−θ) . 2 cos θ

(25)

The latter expression makes it possible to determine the distribution of the radial velocity component orthogonally to the considered alignment. The results of the calculations performed with eqn. (25) turn out to be incongruent as – in the field where the two adopted alignments overlap – they return substantially different and non-matching radial velocity values that are such as to put in crisis the hypothesis of simultaneous symmetry and WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

284 Computational Methods and Experimental Measurements XIII irrotationality. If, on the other hand, the validity is assumed of eqns (18) and (20), which abandon the symmetry hypothesis, then the same methodology used above provides:  θ2  A  cos θ ; V(θ) =  + B + k θ ⋅ θ  sin θ −  Vro (r ) + B ⋅ θ + k θ  2   r   

(26)

 θ 2   A V(− θ ) = − + B − k θ ⋅ θ  sin θ −  Vro (r ) − B ⋅ θ + k θ cos θ ;  2    r  

(27)

Vro (r ) = −

 V(θ) + V(− θ ) θ2 + k ⋅  θ ⋅ tgθ −  2 cos θ 2 

 .  

(28)

The results of the calculations made using eqn. (28), with k = 0.236, point to a good coincidence of the radial velocity component values calculated in the field where the two chosen alignments overlap. At this level of investigation it therefore seems reasonable to assume a velocity distribution represented by eqns (18) and (20).

3

Pressure distribution analysis

Viparelli’s experimental measurements [3] indicate zero pressure values in the section 0-0 (fig.1) thus justifying eqn. (6) (as Knapp [5] and Pica [7] do), while the measurements of Binnie and Hookings [2] return positive values, thus justifying eqn. (7) (as Ackers and Crump [4] do). In order to account for this difference, it can be assumed that Viparelli’s section 0-0 does not match that of Binnie and Hookings. Viparelli’s section is probably very close to the edge of the inlet with the result that the curvatures in the vertical planes acquire greater value while the pressure can be evaluated using eqn. (6) only in that part of the shaft in which the radius of the vortex core no longer changes, i.e. just below the inlet channel (moreover, in this case the value of H to be inserted into the equations should be slightly higher than the one supplied by eqn. (1)). Moreover, as far as Adami [6], the distinction between eqn. (6) and eqn. (7) refers only to the theory for flow rates beyond critical value. The criteria adopted with eqn. (6) (i.e. Knapp [5]) result in the calculated flow rate values being systematically and significantly lower than the experimental ones. On the other hand, the adoption of eqn. (7) leads to even smaller flow rate values which are therefore even further from the experimental data. Then it has been considered that the section 0-0 (in which the edge effect must be felt) has a negative pressure distribution. Such pressure distribution results in even higher velocity values. Therefore, if the values of the tangential velocity components Vt WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


285

are fixed in the computation model, the increase in the vertical components Vz results in an increase in the calculated flow rate. Adequate negative pressure values can therefore account for the flow values measured experimentally. In particular, it has been decided to use a pressure distribution law which, when inserted into eqn. (2), facilitates its integration. In effect the trajectories are curved both in the plane of section 0-0 and in vertical planes: the two curvatures determine centrifugal forces which are opposing and, therefore, they have an opposite effect on the pressure value in the section 0-0. The experimental data (table 1) can be interpreted to show that the influence of curvatures in vertical planes can be prevalent and, therefore, negative pressure can be verified (even if not along the entire perimeter with the higher flow values). Consequently it has been assumed, in alternative to eqn. (6) or eqn. (7), that: p B2 = γ 2g

1 1   − ;  r 2 ro2 

(29)

where B is a constant value which has to be determined. In actual fact, if r = ro, eqn. (29) yields that p/γ is zero; if r > ro, p/γ is negative. In conclusion, it can also be noted that the pressure measurements (taken at the four points P indicated in fig. 1 and reported in table 1 in water column) constantly show a tendency towards an eminently asymmetrical distribution. Table 1: Q (m3/s) 0.0010 0.0052 0.0160 0.0260 0.0380 0.0490 0.0580 0.0690 0.0790

4

P1 (m) -0.0030 -0-0028 -0.0027 -0.0027 -0.0025 -0.0020 -0.0016 -0.0010 -0.0005

Experimental tests. P2 (m) - 0.0003 - 0.0012 - 0.0013 - 0.0014 - 0.0015 - 0.0017 - 0.0018 - 0.0024 - 0.0025

P3 (m) -0.0015 -0.0006 -0.0015 0.0005 0.0012 0.0021 0.0027 0.0041 0.0048

P4 (m) -0.0008 -0.0003 0.0000 0.0002 0.0004 0.0005 0.0010 0.0015 0.0018

Conclusions

The theoretical analysis, performed using experimental tests, shows how the flow in a vortex shaft, contrary to the hypotheses made by all the other authors, is asymmetrical and has negative pressures in the inlet section. Above all in the initial part of the vortex flow (which is the one examined with the reported preliminary tests), the tangential velocities are accelerated, a characteristic which is reflected in particular trends of radial velocities which are once again nonWIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

286 Computational Methods and Experimental Measurements XIII symmetrical. The asymmetry of the vortex motion and the negative pressure in the inlet section are able to justify certain experimental incongruences reported by some of the cited authors and make it possible to formulate a theory of hydraulic working (on the basis only of the geometric data of the outlet device and the water depth in the inflow channel) which leads to determine flow rates and discharge coefficients that are consistent and in good agreement with the experimental measurements. The results obtained through the analysis of a limited number of experimental tests certainly need to be verified by extending the measurements to the entire flow field and to a larger number of measurements regarding the pressure distribution in the inlet section. It can nevertheless be concluded that an increased accuracy in the quantitative assessment of current theoretical models on the hydraulic working of vortex shafts can only be assured by taking the asymmetry hypothesis into account.

References [1] [2] [3] [4] [5] [6] [7] [8] [9]

[10]

Drioli, C.,Su un particolare tipo di imbocco per pozzi di scaricoScaricatore idraulico a vortice, L’Energia Elettrica, pp. 447-463, ottobre 1947. Binnie, A.M. & Hookings, G.A., Laboratory experiments on whirlpools, Proc. Roy. Soc., Series A, Vol.194, London, 1948. Viparelli, M., Su un particolare tipo di imbocco e sull’efflusso con vortice, L’Energia Elettrica, Vol. XXVII(10), pp. 610-621, October 1950. Ackers, P. & Crump, E.S., The vortex drop, Institution of Civil Engineers, Vol.16 (4), pp. 433-442, august 1960. Knapp, F.H., Aufluss, Uberfall und Durchfluss im Wasserbau, Verlag G. Braun, pp. 502-517, Karlsruhe, 1960. Adami, A., Analisi del moto in uno scaricatore a vortice, L’Energia Elettrica, Vol. XLIV (7), pp. 406-410, luglio 1967. Pica, M., Scaricatori a vortice, L’Energia Elettrica, Vol XLVII (4), pp 217-234, aprile 1970. Ciaravino, G., Galasso, V., Mancini, P. & Pulci Doria, G., A mathematic model for vortex shaft. Theory and experimental control, Congress IAHRAIRH, Lausanne, 1987. Ciaravino, G. & Pulci Doria, G., Rilievi iniziali con LDA delle distribuzioni di velocità in un pozzo a vortice, II Simposio sull’anemometria Laser Doppler nella sperimentazione idraulica, Napoli, 28-29 marzo 1988. Ramponi, F., Sul moto dell’acqua nei canali aperti ad asse curvilineo, L’Energia Elettrica, aprile 1940.



287

Stability of stratified spin-up flows S. A. Smirnov Mechanical Engineering Department, Texas Tech University, USA

Abstract The stability of stratified rotating flows is investigated by means of laboratory experiments in axisymmetric cylindrical and annular containers with both horizontal and sloping bottoms. The baroclinic current is initiated via incremental spin-up/down of a linearly stratified fluid by an abrupt change in the rotation rate of the system (from Ω ± ∆Ω to Ω). The flow stability depends on the characteristic values of the Rossby number, ε = ∆Ω/Ω, and the Burger number, Bu = NH/fR, where f = 2Ω is the Coriolis parameter, R is the characteristic horizontal length scale of the flow, H is the depth of the fluid layer, and N is the buoyancy frequency. Particular attention is given to the nonlinear flow regime (finite Rossby numbers). It is found that axisymmetric spin-up current loses its azimuthal symmetry when Bu < 1, and breaks into a system of large-scale cyclonic and anticyclonic vortices with a predominantly vertical axis of rotation. The eddies always develop at the density fronts formed by the corner regions adjacent to the sidewalls of the container. The corner regions reach a quasi-equilibrium state at the characteristic time scale E-1/2Ω-1 (where E = ν/ΩH2 is the Ekman number and ν is the kinematic viscosity), which is also observed for homogenous fluids. It is also shown that the stability of the spin-up flow is affected by the bottom slope. In the presence of the latter the bottom boundary layer experiences a qualitatively different behavior. While the density field demonstrates a smooth monotonic behavior in the case of stratified spin-up at all times, it reveals high-frequency fluctuations in the spin-down case, suggesting the turbulent nature of the bottom boundary layer. The results of observations may be found useful in interpreting in-situ measurements of upwelling- and downwelling-favorable oceanic currents in the littoral zones. Keywords: geophysical systems, rotating stratified flows, spin-up, flow instability.



1

Introduction

Dynamics of the spin-up flow initiated by a sudden acceleration/deceleration of the fluid container presents a fundamental interest from the point of view of the flow stability. Spin-up problem addresses an important question of how the torque exerted by the sidewalls of the fluid container is transmitted throughout the bulk of the rotating fluid. Although, the final condition of the fluid system is obvious – a new solid-body rotation state – it is far less clear what “routes” lead to this state. Spin-up in a homogeneous fluid has been extensively studied in the past (e.g., Greenspan [1], Duck and Foster [2]). It was shown that flow dynamics depends on the relative value of the Rossby number, ε = ∆Ω/Ω. The flow evolution is governed by the Ekman layers formed on the top and bottom of the container. They initially convey the fluid towards the sidewalls into the Stewartson boundary layers. Passing through the boundary layers, fluid parcels roughly attain a new angular velocity of the container before moving into the interior. This secondary circulation is responsible for global spin-up of the fluid which is achieved at the homogeneous spin-up time scale E-1/2Ω-1. In a stratified fluid gravity force limits the vertical fluid velocity and influences the secondary circulation. Flow dynamics is characterized by an additional parameter, the Burger number, Bu = NH/fR. The initial phase of stratified spin-up is similar to that of homogeneous spin-up (Benton and Clark [3], Buzyna and Veronis [4], Holton [5], Pedlosky [6]). Near-bottom heavy fluid is transported radially outward causing a strong deformation of isopycnals (surfaces of constant density) near the outer wall. This process is called upwelling. Stratified spin-up problem has been examined for different flow regimes in ε-Bu parameter space, but the flow instability that breaks the axial symmetry was observed only for incremental spin-down (Hewitt et al [7, 8]) or highly nonlinear spin-up from rest, when ε = 1 (Flor et al [9, 10]). Nonaxisymmetric stage of incremental spin-up of a stratified fluid received special attention only recently (Kanda [11], Smirnov et al [12]). The formation of largescale eddies through the development of non-axisymmetric instabilities provides an alternative “route” in a stratified fluid for the transport of angular momentum from solid boundaries to the bulk of the fluid. This mechanism reduces the spinup time making it significantly less compared to the viscous time scale E-1Ω-1. In the present paper we report on the cases of flow instability during incremental stratified spin-up and the conditions under which a sloping bottom topography may serve to stabilize the flow. Analysis of the boundary layer formed by an upwelling-favorable flow shows that buoyancy forces are always important unless Bu = 0 (MacCready and Rhines [13]). In the presence of a sloping bottom buoyancy inhibits the cross-slope transport. The density field redistribution in the horizontal plane due to the cross-slope transport establishes vertical shear through the thermal-wind balance. The decrease of the upslope transport prevents the heavier fluid from reaching the corner regions and closing the secondary meridional circulation. Buoyancy forces weaken the stress at the bottom boundary (free-slip boundary condition) and, therefore, affect the spin-up characteristics. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


2

289

Experimental arrangements

Laboratory experiments were conducted in a cylindrical tank (radius R = 91 cm) positioned on a rotating table whose angular velocity was set within the frequency range (0.1 ≤ f ≤ 1.0 s-1) by a computer-controlled motor. The annular geometry was created by adding the second cylinder (radius R1 = 55 cm) concentric with the outer cylinder. The bottom slope topography was modeled by an inverted cone (slope angle α = 100), which was centered inside a cylindrical container. The tank was stratified by salt using the standard two-bucket technique, so that the buoyancy frequency varied in the range, 0.5 ≤ N ≤ 2.0 s-1. The tank was covered with a transparent screen in order to reduce shear stress exerted by the air on the water free surface. The actual values of the stratification (N) and rotation (f) parameters were chosen to satisfy the condition Wf/N > H, i.e. the height of a vertically sheared baroclinic current always exceeds the depth of the fluid layer at a given location. Continuous stratification in the conical geometry was produced by two different methods. One method employed five fluid layers with uniform density increments. In this case the tank was stratified whilst at rest. The initial density profile always had a staircase shape, which was transformed into a linear one after three-four experimental runs. The second method employed a standard twobucket technique with a constant rate of filling, so that the tank can be stratified whilst rotating to reduce mixing. Measurements of the vertical density profiles were conducted prior to the beginning of spin-up in the center of the cone and at r/R = 0.54 in order to estimate the effect of a sloping boundary on the background density profile (fig. 1). These two density profiles were almost identical from the free surface to the bottom of the cone at r/R = 0.54. The density profile in the center of the cone preserved the same gradient over the next few centimeters and then started deviating sharply towards a constant-density profile (see almost flat curve in fig. 1) at the apex. The flow was visualized using a neutrally buoyant dye-tracer (thymol blue pH indicator). Its evolution was recorded from above with a camera rotating with the tank. Particle tracking velocimetry (PTV) technique was employed to collect the quantitative data on the fluid velocity at different vertical levels. Small (mean diameter 100 microns) neutrally buoyant polystyrene particles were used as the flow tracers. They were distributed uniformly at various depths and illuminated with a light source from the side. The resulting motion was recorded from above and processed using DigImage software. The change in the rotation rate of the tank was performed during a five-second-time interval, ∆t, and may be considered as impulsive (∆t 1, until they reach a quasi-equilibrium height determined by the relative values of ε and Bu. The density structure of the corner regions was found to be the key factor in determining the stability of stratified spin-up flows. The evolution of the isopycnal displacements at a given location is shown in fig. 5 for the annular geometry. The tracking of the conductivity signal was conducted at three different depths and at the same distance form the center of rotation. Because all probes were located approximately in the middle of the channel, there is no pronounced increase or decrease of the density at early times. Undulations of the signals correspond to the developing instability. The WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


293

instability develops in the form of eddies, which are represented by regular lowfrequency (about 0.1f) oscillations in fig. 5. Note that oscillations at different levels are practically in phase and the largest isopycnal deformation is registered closer to the bottom of the annulus, while its manifestation close to the free surface is relatively weak.

2

t/T 1 η/H

3

Figure 5:

Time evolution of the normalized isopycnal displacements during stratified spin-up in the annulus. Conductivity probes are distributed vertically at z = (1) 2.5, (2) 6, and (3) 10 cm from the bottom of the annulus (H = 12.5 cm is the total depth of the fluid layer) and positioned at r = 74 cm from the center of rotation. Time t = 0 represents the beginning of measurements, while the end of the flat region corresponds to the initiation of spin-up. Parameters: ε = 0.12, Bu = 0.7.

Stratified spin-up/down in a conical geometry appeared to be fundamentally different from the above cases. The flow evolution visualized with a passive dye tracer is shown in fig. 6a (stable regime) and 6b (unstable regime). The absolute values of the flow regime parameters, ε and Bu, were kept the same in both cases. The passive dye tracer, introduced near the bottom, follows the boundary-layer current. The difference in the flow patterns becomes clear after a few rotation periods. A system of relatively small and densely packed anticyclonic eddies spawns out of the initially chaotic distribution of the dye tracer. These eddies merge and finally form a system of much larger anticyclonic vortices (fig. 6b). Contrary to that, in the spin-up case the dye tracer dispersed by the shear flow preserves its axisymmetry even at very late times of the flow evolution (fig. 6a). The observations suggest that stratified spin-up flows remain stable at all times. A comparison of the vertical density profiles for positive (spin-up) and negative (spin-down) Rossby numbers shows that in the former case the density gradient decreases (compared to the background value) in the upper part of the fluid column and increases near the bottom, while in the latter case this tendency is reversed. This observation indicates to the existence of intensive mixing regions localized in the vicinity of the bottom boundary in the spin-down case. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

294 Computational Methods and Experimental Measurements XIII (a)

(b)

Figure 6:

Stratified spin-up/down in the axisymmetric conical geometry. Top view. The block arrow shows the direction of the mean current. The radial distance between two concentric rings is 30 cm. Parameters: (a) spin-up ε = 0.3, Bu = 0.42, t/T = 41.5, (b) spin-down ε = − 0.3, Bu = 0.42, t/T = 22.6.

Figure 7:

Dye visualization of stratified spin-up in the shallow part of the conical geometry. Parameters: ε = 0.3, Bu = 0.46, t/T = 102. The distance between two marks along the radius is 3 cm. The dark band in the left bottom corner corresponds to the outer edge of the cone. The free surface intersects the sloping bottom at the location shown by the broken line.

The differences between spin-up and spin-down flow patterns may also be delineated through measurements of instantaneous vertical salinity profiles. It was found that the fundamental difference between stratified spin-up and spindown is manifested in the way the boundary layer behaves shortly after the change in the rotation rate of the container. In the case of spin-up the conductivity probe positioned inside the boundary layer registered a smooth and monotonic increase in density, while in the case of spin-down the registered signal experienced a large number of random high-frequency oscillations shortly after the beginning of spin-down. This observation suggests that the WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


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downwelling boundary-layer current becomes unstable and turbulent relatively fast, while upwelling boundary-layer remains stable at all times. This, in turn, leads to qualitatively different boundary conditions for the interior flow. The segment of the conical geometry shown in fig. 6 represents the mid-depth region of the flow domain, with the lower and upper circles being at 30 cm and 60 cm respectively from the center of the cone measured along the bottom of the tank. In general, the flow evolution is expected to be different in the deep and shallow parts of the flow domain and, therefore, the flow region where the sloping bottom intersects the free surface needs special attention as a potential source of flow instability. Experiments conducted with the dye tracer being introduced in the vicinity of the outer edge of the tank (fig. 7) showed that even after one hundred revolutions the stratified spin-up flow remains axisymmetric in this part of the conical geometry, which means that it is globally stable. It is also interesting to note, that in the case of spin-down the eddies do not form close to the outer rim of the cone, where the fluid reaches a new solid-body rotation relatively quickly because of the small depth of the fluid layer, but rather in the mid-depth section of the tank shown in fig. 6.

4

Conclusions

It was demonstrated that spin-up flows of rotating, continuously stratified fluid in axisymmetric geometries may become unstable at late times (several tens of rotation periods). The instability and formation of cyclonic/anticyclonic eddies may be suppressed by increasing Bu and decreasing ε. The eddy formation time decreases for larger ε and is independent of Bu (for the range of parameters investigated). Unquestionably the bottom topography plays a crucial role in setting up the density stratification and vertical shear in the bottom boundary layer that influences both the development of non-axisymmetric instabilities and global spin-up. If the characteristic “shutdown” time is relatively small, the heavier fluid may never reach the corner regions and the formation of the “corner jets”, which close the meridional circulation, does not occur. Thus, the slippery boundary condition should affect the global spin-up time of the interior fluid.

Acknowledgements The author is grateful to Professors H.J.S. Fernando and D.L. Boyer for the support of this research under ONR grant N00014-0-1-0626 and NSF grant OCE-0137197.

References [1] [2]

Greenspan, H.P., The Theory of Rotating Fluids, Cambridge Univ. Press: London and New York, 1968. Duck, P.W. & Foster, M.R., Spin-up of homogeneous and stratified fluids. Ann. Rev. Fluid Mech., 33, pp. 231, 2001. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

296 Computational Methods and Experimental Measurements XIII [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

Benton, E.R. & Clark, A., Spin-up. Ann. Rev. Fluid Mech., 6, pp. 257, 1974. Buzyna, G. & Veronis, G., Spin-up of a stratified fluid: theory and experiment. J. Fluid Mech., 50, pp. 579, 1971. Holton, J.R., The influence of viscous boundary layers on transient motions in a stratified rotating fluid: Part I. J. Atmos. Sci., 22, pp. 402, 1965. Pedlosky, J., The spin up of a stratified fluid. J. Fluid Mech., 28, pp. 463, 1967. Hewitt, R.E., Davies, P.A., Duck, P.W. & Foster, M.R., Spin-up of stratified rotating flows at large Schmidt number: experiment and theory. J. Fluid Mech., 389, pp. 169, 1999. Hewitt, R.E., Foster, M.R. & Davies, P.A., Spin-up of a two-layer rotating stratified fluid in a variable-depth container. J. Fluid Mech., 438, pp. 379, 2001. Flor, J.B., Ungarish, M. & Bush, J.W.M., Spin-up from rest in a stratified fluid: boundary flows. J. Fluid Mech., 472, pp. 51, 2002. Flor, J.B., Bush, J.W.M. & Ungarish, M., An experimental investigation of spin-up from rest of a stratified fluid. Geophys. Astrophys. Fluid Dyn., 98, pp. 277, 2004. Kanda, I., A laboratory study of columnar baroclinic vortices in a continuously stratified fluid. Dyn. Atmos. Oceans, 38, pp. 69, 2004. Smirnov, S.A., Baines, P.G., Boyer, D.L., Voropayev, S.I. & SrdicMitrovic, A.N., Long-time evolution of linearly stratified spin-up flows in axisymmetric geometries. Phys. Fluids, 17, 016601, 2005. MacCready, P. & Rhines, P., Buoyant inhibition of Ekman transport on a slope and its effect on stratified spin-up. J. Fluid Mech., 223, pp. 631, 1991.



297

On the relation between geometric and flow properties of a miniaturized fluid oscillator J. O. Sotero-Esteva1, R. Furlan2 & J. J. Santiago-Avilés3 1

Department of Mathematics, University of Puerto Rico at Humacao, Puerto Rico 2 Department of Physics and Electronics, University of Puerto Rico at Humacao, Puerto Rico 3 University of Pennsylvania, Philadelphia, USA

Abstract A miniaturized fluid oscillator with no movable parts composed by a switching cavity, one inlet, two outlets and two feedback channels was studied. Examples of these devices have been built before and had shown oscillation of fluid flows ranging from tens of Hz for liquids to thousand of Hz for gases. The present work consists of a study of the flow within the device by means of a computer model. The computer model was built using COMSOL 3.3. It uses conventional Navier-Stokes equations numerically solved by Finite Element Methods. A new level of detail of the qualitative description of the flow within the device is achieved which allows for a better understanding of why some geometries produce better devices than others. It shows that homogeneous oscillation, not only depend on the direct force exerted by the feedback flow on the inlet stream, but also on the disruption of the Coanda's effect that diverts the stream towards one of the output channels. On the other hand, the quantitative part of the study serves to validate the simulation as well as a basis for proposing new empirical models. The quantitative measurements are consistent with a previously known mathematical model that describes the relation between frequency, input velocity and geometrical and physical properties. Ongoing testing with actual devices also supports the proposed operating mechanisms and models. Keywords: fluidic, flow control, flow meter, fluidic oscillator, finite element method, computer simulation.



1

Introduction

The miniaturized oscillator studied here is a structure without movable parts composed of cavities and channels. It can be used for flow control, flow measurement, and substance identification. They find applications in the areas of medicine, aeronautics and automotive industries among others. The use of macro sensors and actuators based on several designs of these oscillators can be traced back to 1960 when the demands for reliable controls for space and marine environments stimulated research in this area. A patent for a substance identification device using oscillators very similar to the one studied here was filled in 1963 and issued in 1966 [9]. The use of oscillators for the mentioned applications depend on relating their frequency response to factors such as the composition of the fluid, temperature, volumetric flow and variations in geometrical and other physical properties of the device. Most of the characterizations of fluidic devices found in the literature are based on experimental methods where actual devices are made and tested by varying the desired parameters. Most frequently the factors that are studied are those that require building one or a few devices for all tests. To study of the effect of geometry variations in the laboratory gets complicated by the fact that the researcher would have to build and test dozens, or perhaps hundreds, of devices. Up until recently computational testing methods were difficult to use as well. Limited software tools for numerical computations and visualization, lack of processor speed, and limitations in the mathematical modeling severely restricted the use of the computer for the study of these devices. Software libraries and simulation environments for solving systems of differential equations based on Finite Element Methods (FEM) such as FreeFem, ANSYS and COMSOL, among others, are now mature enough for such complex simulations. They have excellent visualization capabilities, automatic initial and adaptive meshing, scripting for automatic geometry design and graphical user interface (GUI) generation, and even parallel processing capabilities. In a "V" shaped device without feedback arms in which the inlet is placed at the junction and the outlets at the end of the upper arms the fluid flow tends to stick to one of the lateral walls due to the Coanda's effect. Fluid emerging from the inlet creates a partial vacuum just after entering to the switching chamber. A fluid vortex is formed in this low pressure region. The miniaturized fluid oscillator with no movable parts studied in this work is composed by a switching cavity, one inlet, two outlets and two feedback channels (see figure 1). The feedback channels divert part of the flow back to the junction, switching the flow to the other arm where the same event is repeated, thus creating the oscillations. Experimental results indicate that the operation of this type of fluid oscillator is a direct function of the length of the feedback loops and of the velocity inside of the interaction region. The main oscillation frequency modes range from tens of Hz for liquids (water, isopropyl alcohol and



Figure 1:

299

A feedback fluid oscillator with one inlet nozzle, two feedback loops and two output channels.

acetone) to thousand of Hz for gases (nitrogen, argon, and carbon dioxide) [7]. Experimental evidence of internal flow oscillation with operation with water has also been reported in [5]. Simões et al. [8] analyzed a feedback oscillator numerically using ANSYS 5.7. They studied variations in fluid properties and proved the viability of using FEM based numerical methods for the simulation of fluid flow in these devices. This work is a computational study on how the positions and width of the feedback arms affect the oscillation patterns when water flows through the channels at low input speeds (< 200mm/s). All other geometry parameters are kept constant.

2

Analytical modelling

For subsonic or transonic flow, associated with quasi-laminar or turbulent regime, the frequency of oscillation is determined by: the time of interaction of the fluid in the feedback loop (τf), by the amplifier switching dynamics (τs), and by the flow-rate (u). In this case, the typical feedback oscillator can be designed to give a long linear range of frequency against velocity characteristics. The total oscillation time is described as in [8] as T = 2(τf + τs). The feedback transmission time is τf = l / c where, c is the wave propagation speed, l is the feedback loop length. The switching time is τS = ξ R / u where u is the jet velocity, R is the nozzle to splitter distance, and ξ is an empirical constant. If the duct is not small, the speed of wave propagation tends to the speed of sound. For liquids the speed of wave propagation is two to four orders of magnitude higher than the jet velocity in the nozzle-to-splitter path. Therefore, for a fixed geometry the following linear relation between frequency and input fluid speed holds:

f=

1 ξ = u. 2τ S 2 R


(1)

300 Computational Methods and Experimental Measurements XIII The oscillator frequency increases linearly with increasing volume flow. This behavior allows using of the feedback fluid oscillator for the measurement of the flow of Newtonian fluids. A feedback oscillator may exhibit many oscillation frequency modes. The conditions under which a device exhibits few modes is crucial for many applications. In this work the term properly designed device refers to a device in which the number of dominant oscillatory modes is low. Ideally, an oscillator should have only one dominant mode. Equation (1) describes a system with only one oscillation frequency; it implicitly assumes a properly designed device. Since the apparent flow in the switching cavity is far more complex than in the feedback arms, closely studying what happens inside the switching chamber is a crucial step in the design of properly crafted oscillators.

3

Simulation procedure

In order to simplify the computational effort a two dimensional model was used. The validity of the simplification is justified by the aspect ratio used in actual devices. The computer model was built using COMSOL 3.3 running on a Silicon Graphics Altix 350 with eight Itanium2 processors. Navier-Stokes equations were numerically solved by Finite Element Methods. 3.1 Mathematical model The incompressible Navier-Stokes equations [1, 11],

ρ

[

(

)]

∂u T + ρu ⋅ ∇u = ∇ ⋅ − pI + η ∇u + (∇u ) , ∂t ∇ ⋅u = 0 ,

where used. The density and the dynamic viscosity were set to ρ = 1000 kg/m³ and η = 0.001 kg/ms respectively, p is pleasure and I is the identity matrix. These are values commonly used in simulations of water when variations due to temperature and other factors are not to be considered. No force field and noslip boundaries were used. Zero pressure was initially set at the output boundaries and inward velocity uin at the input boundary. 3.2 Geometry The geometries are designed to resemble those used at previous experiments with actual devices (figure 2). The feedback channels have a constriction at the return joint that varies from 0.04mm to 2mm. The separation from the base of the switching chamber varies from 0mm to 4mm in 1mm increments. The rest of the dimensions are kept fixed. The input, output and feedback channels have a width of 2mm.



Figure 2:

301

Geometry, measurements, and distribution of triangles throughout the mesh.

The mesh used was generated by COMSOL 3.3. It is a triangular mesh with higher density of vertices around borders with high curvature. The distribution of vertices is neither uniform nor symmetric. The maximum length of a triangle edge in the mesh varied. A maximum of 1mm (shown in figure 2) was used for smaller flow rates. Denser meshes were used for higher flow rates. A Delaunay triangulation was adopted. 3.3 Simulation parameters A series of test runs were performed with a duration of three seconds of simulated time. From those tests it was determined that all devices that showed oscillations after one second were most likely to continue oscillating until the end of the test run. It was inferred that when a test case pass this point it is most likely to exhibit persistent oscillations, that is, it will keep oscillating during an arbitrary long period of time. It was also observed in the test runs that oscillatory behavior usually stabilized before that moment and that 1.5s of simulated time gives sufficient data to capture the main oscillation modes of the device. Simulated data was produced for 25 different geometries. Input fluid speed ranged from 50mm/s to 300mm/s in 50mm/s increases for a total of 150 runs of the simulation. The maximum fluid speed at the end of the output channels was measured every 0.05s. The transient analysis was performed using a default element type Lagrange P2P1, the state of the system is stored every 0.05s of simulated time. 3.4 Post-processing Two graphs were produced for each simulation. The first one is a plot of time versus the differences between maximal output fluid velocities at the end of the output channels at that time. The second was the Fourier Transform of the WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

302 Computational Methods and Experimental Measurements XIII previous graph that gives information of the different oscillations modes of the device. The frequencies of the three most conspicuous peaks of this graph were registered. 3.5 Computational tools COMSOL Script was adopted in order to speed up the simulation process, allowing a more complete analysis of the influence of geometry of the device. Modules for geometry generation, definition of the model, automatic calling of meshing functions, setting of the model, time-dependent simulation functions and post-processing were written (see [9] for details). Those scripts were also useful for batch processing. A Graphical User Interface (GUI) was built on top of the modules (figure 3). The GUI integrates all previous functions and made possible a faster setup and manipulation of the simulation, especially for closer inspection of interesting geometries. .

Figure 3:

4

A Graphical User Interface (GUI) for feedback oscillator simulations and analysis.


The discussion of results is organized in order of increasing detail. Each section considers a subset of the cases of the previous section. 4.1 Persistent oscillations All geometries exhibited persistent oscillations at the lowest volumetric flow tests (50mm/s). No persistent oscillations were found for input speeds over 200mm/s. In all cases in which the simulation stopped before the 1.5s mark a small and highly turbulent region that caused excessive numerical instability appeared when the simulation stopped. The position of that region does not seem to follow a pattern. Attempts were made to extend the simulation time on those cases for longer periods by refining and/or adapting the mesh but were WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


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unsuccessful. A mathematical model that accounts for turbulence, such as the κε turbulence model, could produce longer simulations. But turbulence and patterns sometimes appear together in the same physical system [3], sometimes do not. The failure to converge in some cases may also reflect that there exist some combinations of properties that separate the devices in which both turbulence (oscillation) and pattern (persistence) appear from those where they do not appear The count of cases that showed persistent oscillations shown in table 1 is evidence against designing devices with a combination of wide feedback channel close to the inlet channel. Table 1:

Number of devices with persistent oscillations.

separation feedback width (mm) (mm) 0.4 0.8 1.2 1.6 2 1 1 1 1 0 3 1 1 1 1 1 3 1 1 2 2 2 3 3 3 3 3 3 3 3 3 4 2 2 2 total 14 8 8 10 10

(a) Figure 4:

total 7 7 9 15 12 50

(b)

Oscillations and frequency analysis of a (a) non-persistent oscillator device and a (b) device classified as properly designed.

4.2 Oscillation frequency modes A variety of oscillation patterns were observed in cases with persistent oscillations (figure 4). For each case, the six highest values of discrete Fourier transform amplitudes and their corresponding frequencies were tabulated. The main oscillation frequency was determined by selecting the frequency corresponding to the highest amplitude after discarding very low frequency modes (less than 5Hz). Averages over all the different geometries were WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

304 Computational Methods and Experimental Measurements XIII calculated for each input velocity. The graph of input velocity versus frequency (figure 5) shows an excellent agreement with the linear model in eqn (1) and the experimental observations. The variance in the data captures experimental noise as well as small variations produced by the changes in geometric measures. 50

frequency (Hz)

40 30 20 10 0 50

100

150

200

input speed (mm/s)

Figure 5:

The average frequency in all devices with persistent oscillations. The bars cover one standard deviation from the averages.

Table 2:

Properly designed devices by feedback separation and width.

separation (mm) 0 1 2 3 4 totals

0.4 0 2 0 0 0 2

0.8 0 0 0 1 0 1

feedback width (mm) 1.2 1.6 2 0 0 0 0 0 0 0 0 0 1 2 2 1 1 0 2 3 2

totals 0 2 0 6 2 10

4.3 Proper designs As stated earlier, we define properly designed oscillator as an oscillator that has few dominant oscillatory modes; ideally only one such as the one in figure 4(b). The quantitative criterion used for classifying a geometry as a proper design was as follows: the value of the frequency for which the second highest discrete Fourier transform is less than half of the most dominant one. Ten devices satisfied the criterion (table 2) 4.4 Analysis of vorticity A closer inspection of the vorticity plots in the switching chamber reveals details of the physical mechanisms that determine why some combinations of geometries and input velocities perform better than others. As seen in figure 6, WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


305

when the feedback arms are positioned where the eddy responsible for the Coanda's effect is located, the feedback flow not only exerts a force on the inlet stream but also disrupts the effect that keeps the jet attached to that wall.

5

Conclusions

From the analyses presented above the following conclusions are drawn about a feedback fluid oscillator. Proper positioning and width of the feedback channel are important in the design of the devices. The proper adjustment of these parameters leads to devices with a well defined oscillating framework. Homogeneous oscillations, not only depend on the direct force exerted by the feedback flow on the inlet stream, but also on the disruption of the Coanda's effect. Linear dependence between frequency and volumetric flow was confirmed for simulated water.

Figure 6:

Vorticity in the switching chamber of a properly designed feedback oscillator. Light shades indicate counter-clockwise rotation, darker shades indicate clockwise rotation. The arrows show the direction and magnitude of the velocity of the fluid at the middle of the feedback arm when entering the switching chamber.

Acknowledgements This work was supported by the US National Science Foundation through the Penn-UPR Partnership for Research and Education in Materials project (NSFDMR-353730) and by the US National Security Agency through the Humacao Undergraduate Research in Mathematics to Promote Academic Achievement program (NSA-H98230-04-C-0486).

References [1] [2]

Chorin, A.J. & Marsden, J.E., A Mathematical Introduction to Fluid Dynamics, Second Edition, Springer-Verlag: New York, 1990. COMSOL 3.3 User’s Guide and Introduction, Comsol AB, Sweden, 2006. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

306 Computational Methods and Experimental Measurements XIII [3] [4] [5]

[6] [7]

[8]

[9] [10] [11]

Field, M, & Golubitsky, M., Symmetry in Chaos, A search for Pattern in Mathematics, Art and Nature, Oxford University Press: Oxford, 1992. Gebhardy, U., Hein, H. & Schmidt, U., Numerical investigation of fluidic micro-oscillators, J. Micromech. Microeng. 6 (1996) 115–117. Rogerio Furlan, Maria Lucia Pereira da Silva, Eliphas Wagner Simoes, Roberto Eduardo Bruzetti Leminski, Jorge J. Santiago Aviles, Visualization of internal liquid flow interactions in meso planar structures, Flow Measurement and Instrumentation, Vol. 17, No. 5, pp. 298-302, 2006. Gregory, J.W., Sullivan, J. P., Raman, G. & Raghu, S., Characterization of a micro fluidic oscillator for flow control, 2nd AIAA Flow Control Conference, Portland Oregon, June 28 – July 1, 2004, paper 2692. Simões, E.W., Furlan, R., Pereira, M.T., Numerical analysis of a microfluidic oscillator flowmeter operating with gases or liquids, Technical Proceedings of the Fifth International Conference on Modelling and Simulation of Microsystems, MSM 2002, ISBN: 0-9708275-7-1, pp. 36–39, 2002. Simoes, E.W., Furlan, R., Leminski, E.B., Gongora-Rubio, M.R., Pereira, M.P., Morimoto, N.I. & Santiago-Aviles, J. J., Microfluidic oscillator for gas flow control and measurement, Flow Measurement and Instrumentation, Vol. 16, Issue 1, pp. 7–12, 2005. Sotero Esteva, J.O., Furlan, R., Santiago Avilés & J.J., Simulation of miniaturized fluidic oscillators using COMSOL Script, Proceedings of the COMSOL Users Conference 2006, Boston, 2006. Testerman, M.K., McLeod Jr., P. C., Fluid oscillator analyzer and method, US Patent Office, patent #3273377, 1966. Zienkiewicz, O.C., Taylor, R.L. & Nithiarasu, P., The Finite Element Method for Fluid Dynamics, 6th edition, Elsevier Ed., Oxford, 2005.



307

On-stream floodplain storage: experimental research G. De Martino, F. De Paola, G. Marini & A. Ranucci University of Naples Federico II, Department of Hydraulic and Environmental Engineering G. Ippolito, Italy

Abstract In order to achieve a reduction of flood wave peak discharge in a stream plain zone, it is often possible to take advantage of structural active measures consisting of flood plain storage which can be on-stream or off-stream, whose task is the temporary storage of water flood volumes with the consequent outflow of a discharge that is compatible with the hydraulic characteristic of a stream. For preliminary sizing of these measures, the hypothesis of uniform storage is made, and it is also possible to show it in the technical literature. In order to verify the reliability of the simplified hypothesis of uniform storage, in the Laboratory of the Department of Hydraulic and Environmental Engineering G. Ippolito of the University of Naples Federico II, the experimental prototype of on-stream floodplain storage has been installed and the tests have been carried out, by using a suitable electrical level probe in the inflow and outflow hydrograph reconstruction. The tests, managed by both varying the dimension of the free bottom outlet of the floodplain storage and its surface, have confirmed the reliability of the uniform storage hypothesis. Tests in the presence of obstacles, made of synthetic grass, on the bottom of the floodplain zone, are in progress to simulate the presence of vegetation, and therefore the possible influence of hydraulic resistances on the reliability still of the uniform storage hypothesis. Keywords: hydraulic risk mitigation, structural measures, floodplain storage.



1

Introduction

The floodplain storage represents a structural active measure for the reduction of the flood wave peak discharge [1, 10, 16]. The storage and drainage phenomena of these infrastructures are of two-dimensional type and are, moreover, influenced by a lot of momentum exchanges that take place between the current in the main river bed and that in the expansion areas, as well as by the presence of possible obstacles and unevenness that induce localized resistances to the motion [21]. For a first hydraulic sizing of the floodplain storage, the uniform storage hypothesis is generally adopted, taking into account that the water surface maintains itself horizontal and parallel to the bottom [21, 17], without, therefore, considering the two-dimensional model. Such an assumption allows one to define a procedure of hydraulic sizing of the floodplain storage, based on the hypothesis of establishing in advance the hydrograph that try the storage [2, 8, 12, 18, 20, 22], and also on the use of a transformation inflows-outflows model for the definition of the critical rain duration. If the water level maintains itself horizontally during the filling of the floodplain storage it allows a meaningful simplification of the calculations, with the possibility to define sizing nomographs for a quick preliminary evaluation of the volume to assign for a fixed lamination of the flood. In a previous work [15], the Authors have reported the first results of test experiences run on a physical prototype of an on-stream floodplain storage with a surface of approximately 8 m2 and subsequently of approximately 30 m2, and it has been observed that, at least for a first dimensioning of the structure, the uniform storage hypothesis can be reliable. In the same job, besides, it was suggested to plan further experiences finalized to the evaluation of possible influence of marked resistances and patchiness on the storage-drainage phenomena. The possible presence of important resistances to motion and/or patchiness can in fact be present in a real situation because of the floodplain storage are infrastructures that don’t have a continuous functioning but these are tried by big water volumes just in a few occasions during their life. This means that we can have longer or shorter periods, but, however, in terms of years, in which the area destined to be invaded by water is the centre of morphologic transformations mainly due to the vegetation growth that can also be of high stem plants. The tendency of the last years, in fact, [24] is to realize the floodplain storages integrating themselves in the economic and social life of the area in which they are built; the huge areas destined to the flooding can be used for the most of their life for other purposes (cultivation, pasture, free time etc.). The next experimental research that will be presented in this paper is exactly aimed to verify the reliability of the uniform storage hypothesis in presence of marked obstacles due to the greater dimensions of the flooding surface and to the presence of important resistances.



309

The presence of the vegetation involves essentially two problems: - a difference between the geometric storage volume and the useful volume to be invaded; - an obstacle to the water outflow that moves in the storage zones. The effect of the resistances to the motion, in general, when the water speeds are low in the river bed, is negligible, but the effect on the reduction of the geometric volume is no doubt to be taken into account [21]. When the areas to be invaded are affected by the movement of the water in a lengthwise direction to the bottom, the vegetation does offer resistance to the motion that can be evaluated with various procedures [4–6, 21, 23] that could invalidate the uniform storage hypothesis used in the dimensioning models.

2

Experimental installation

The prototype of the on-stream floodplain storage has been realized in the Laboratory of the Department of Hydraulic and Environmental Engineering "G. Ippolito" of the University of the Studies of Naples Federico II, and it is shown in detail in fig. 1. Such prototypes allow one to reproduce the hydraulic behaviour of an on stream floodplain storage equipped with a support structure made of a floodgate: in such a way the outflow of the flows to the inside of the channel occurs undisturbed until the water level gets almost to the bottom side of the floodgate, while the further increase of the discharge causes a sudden rise of the water surface, with the transition to pouring outflow, that allows the filling of the floodplain storage with contemporary reduction of the flood wave peak discharge. Upstream resistive probe Tank charge

Downstream resistive probe (configuration 2) Weir outflow coefficient µb=0.465

Downstream resistive probe (configuration 1 and 3) Flood gate (configuration 1 and 3) outflow hole height: 0.05 m outflow coefficient µb=0.612

Quiet wall Gate valve

a Channel width: 0.45 m height: 0.30 m

b

Floodpalin storage Configuration 1 and 3: flooding area: 69.20 m² Configuration 2: flooding area: 29.12 m²

Flood gate (configuration 2) outflow hole height: 0.05 m outflow coefficient µb=0.640

Figure 1:

Experimental installation scheme.

The installation (fig. 1) is made of a little charge tank in which the capacity needed for the tests is conveyed by a steel pipe and is regulated by a gate valve; from this tank, through a calibrated weir (µs= 0.465), the water comes in to the channel with rectangular section along which there is the obstruction with a floodgate that leaves a free outflow on the bottom, calibrated as well (µb= 0.640; 0,612). The presence of the floodgate induces a pouring of the flows and WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

310 Computational Methods and Experimental Measurements XIII consequently the progressive storage of the water volumes in a properly embanked area. For the water level measurement, two resistive level probes are installed one in the tank charge and another one upstream of the floodgate. Further clarifications about the prototype description can be found in the previous work [15].

3

Experimental research

The tests dealt with the reconstruction of incoming and outgoing hydrographs from the storage floodplain, starting from the water levels measured by resistive level probes. The knowledge of the incoming flow to the system has also allowed an estimation of water levels, assuming the uniform storage hypothesis that have been subsequently compared to experimental values. The proposed numerical model for the estimation of flooding event is based on integration of the differential equation (1):

[

dh 1 = Qi − µblb 2 g (h − s 2) dt S ( h)

]

(1)

where S(h) is the flooding surface which is a function of water levels h reached in the storage area (uniform storage curve), lb and s are respectively the width and the height of the outflow in correspondence of the gate, g is the gravitational acceleration, Qi is the incoming flow calculated in the prototype by measuring water level up-flow the calibrated weir crest (with outflow coefficient µs). The first tests run on such prototype [15] show that the gap between experimentally measured water levels and the ones calculated with the proposed numerical model, based on integration of (1), are modest and therefore the uniform storage hypothesis seems reliable. The present note refers to the follow up of experimental studies on the same prototype described in paragraph 2 but modified in order to obtain three further configurations aimed to assign, possibly, a greater generality to the conclusions drawn previously: - configuration 1- The physical prototype is the one of fig. 1b essentially characterized by a flooding area (69,20 m2) more than twice the one used in the previous tests; - configuration 2- The physical prototype is the one of fig.1a with a flooding area of 29.12 m2 covered by synthetic grass to simulate the presence of vegetation on the flooding surface. - configuration 3- The physical prototype is the one of configuration 1 with the surface covered by synthetic grass. The configurations are numbered and shown following a logical and not chronological order, in fact: the first configuration is aimed to gain an understanding, in particular, which is the effect on flooding and draining events of the area, the second one, those due to the presence of high roughness and in the third configuration the two effects are concomitants. Hereafter we report the results about the examined configurations. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


311

3.1 Configuration 1 (the effect of the flooded area on the flooding and draining phenomena) To verify the possible influence, on the reliability of the uniform storage hypothesis, of the storage surface area on the modalities of flooding, some modifications were applied to the first experimental installation (fig. 1a) in order to increase, by repositioning some boundary banks, the flooding surface (fig.1b). Also, in this case the tests carried out have shown a good agreement among the experimental values and the ones obtained by numerical integration of (1); for example in figs 2 and 3 the results of one of numerous tests, in terms of hydrographs and water levels upstream of the floodgate, are shown. In the diagram of hydrographs is shown: with continuous line the incoming hydrograph to the storage (Qi), with broken line the outcoming one measured during the tests (Qu) and with dotted line the last one obtained by the numerical model proposed (Qu,num). In the water levels diagram the comparison is done between measured levels upstream the floodgate (hu) displayed with continuous line and the calculated with the numerical model one (hu, num) in broken line. The phase of filling is important for the sizing of the volume of the storage floodplain; but the emptying one is useless for understanding the flooding and draining events. The numerical model of uniform storage based on the integration of the (1) represents very well the filling phase of the box, but not the emptying one. Such a gap between numerical and experimental data in the final phase is to be referred to the increase of the surface plain-highness irregularities of the storage floodplain in comparison to the configuration of Fig. 1a.

Q [l/s]

80 70

Qi

60

Qu

50

Qu,num

40 30 20 10 0 0

100

Figure 2:

200

300

t [s]

400

500

600

Configuration 1: Incoming and outgoing hydrographs.

Such irregularities imply the formation of "backwater areas": a part of the water in the storage is held back and therefore the volume of water that flows to the prototype’s channel is reduced in comparison to the one actually held; this implies a more rapid decrement of the water levels in the channel with the final effect, that does not actually occur, of a more rapid emptying of the storage: in reality the floodplain storage, until the measurements can be acquired, is not completely empty but it keeps a certain amount of water in the "backwater WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

312 Computational Methods and Experimental Measurements XIII areas", that can be estimated with good approximation by the hydrographs integrating the inclusive area between the numerical curve and the experimental one considered in phase of draining. 50 hu hu,num

h[cm]

40

30

20 0

100

Figure 3:

200

300

t [s]

400

500

600

Configuration 1: water levels upstream the floodgate.

3.2 Configuration 2 (effect of the presence of vegetation on flooding and draining phenomena) In order to clarify the hydraulic aspects related to the presence of vegetation, tests have been carried out to estimate the effects of the same one on the modalities of filling the storage, aimed to verify if the hypothesis of uniform storage can be still reliable. The presence of the vegetation inside of the floodplain storage has been simulated by putting some synthetic grass, characterized by flexible leaves made of 100% poliolefine, type of fiber B-SOFT, height 55 mm, total height 57 mm, points/m2 11250, total weight 2665 g/ m2. In fig. 4 an overview of the prototype with covered surface and a detail of the synthetic grass used for the cover is shown.

Figure 4:

Overview of surface and particular of synthetic grass.



313

As an example we report the results only in terms of water course level upstream the floodgate (fig. 5) about one of the many tests carried out. In the phase of filling, the presence of the vegetation induces a modest increment of water level in the channel in the case of absence of grass; this is due to the need that the water mass has to exceed the obstacle represented by the grass and to fill the storage. This involves a small gap between the experimental curve (hu) and the numerical one (hu,num) obtained by the model that does not consider the presence of the vegetation. In the phase of draining, the presence of vegetation involves a slower draining because of two concomitant effects: - the first, and marginal one, is due to the obstacle represented by the presence of the grass to the motion of the water: the measured water level in the channel are reduced with respect to the ones estimated by the numerical method by integrating the (1) that does not considered the presence of the grass; such an event, by itself does not justify the remarkable gap between experimental curve and numerical one; - the second one is due to the formation of “backwater areas”: part of the stored water is held back and therefore the water volume that flows to the channel is reduced with respect to the one actually stored; this implies a quicker decrease of water level in the channel with the final effect, not actually occurring, of a quicker draining of the storage: in reality the storage, until the measurements can be acquired, is not completely empty but it maintains a water part in the “backwater areas”. The withheld volume is estimable with good approximation starting from the diagrams of the hydrographs integrating the area included between the numerical curve and the experimental one considered in phase of draining. The final phase of the draining is interpretable applying the theory of the storage, probably not linear. In fig. 5 two significant points are shown, found during the tests: first (A), represents the moment in which the water it begins to submerge the grass, second (B) represent the first moment in which all the grass is submerged. The positioning on the curve of water level of these two points allows one to define three flooding phases: the first included between the initial point and the point A; the second one, included between the point A and the point B, is the starting phase of filling of the storage in which progressively the grass is submerged; the third, that goes from the point B to the beginning of the drain, is the one in which the storage is flooded, with the grass completely submerged. The increment of water level in time (the gradient of the curve) in the first phase is greater in comparison to the one of following phases because the surface of the channel is much lower than that one of the storage; the two following phases have a gradient much similar, and this allows one to guess that the hydraulic behaviour in the two cases is the same one. In the final phase of the flooding, in fact, the vegetation is completely submerged and therefore the motion is mostly developed without interacting with it. In phase A-B the motion happens evidently interacting with the grass, but, however, by observing section A-B we do not find differences regarding the following section; this is justified WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

314 Computational Methods and Experimental Measurements XIII by the fact that also in this case the resistances to the motion applied by the grass are definitely negligible because the speeds with which the water moves in the storage are modest. 3.3 Configuration 3 (effect of area influence and vegetation presence) Also the prototype flooding surface with a bigger area (fig 1b) was covered by synthetic grass (fig. 4), as described in the former paragraph, in order to obtain a configuration which could take into account the combined influence of the surface widening and vegetation presence. The results of one of the many tests run are shown in fig. 6. In the filling phase we found a similar behaviour to the one related to configuration 2 tests: the numerical – experimental gap between water level curves is due to the presence of the grass, the draining hydrographs matching is still good. In the draining phase the behaviour is similar to the one related to the previous configurations and, in particular, it is still present the phenomena of “backwater areas” caused by some irregularities linked to the widening of flooding surface and to the presence of grass. Dealing with the considerations related to the flooding phase which can be read on a water level diagram by applying the points A and B, what we said in the previous paragraph about configuration 2 is still valid. 50 hu

45

hu,num

B (43,6 cm)

h[cm]

40 35

A (35,5 cm)

30 25 20 0

50

Figure 5:

100

150

200 t [s]

250

300

350

400


55

hu

50

h[cm]

45

hu,num

B (44,3 cm)

40 35

A (32,2 cm)

30 25 20 0

100

200

300

400

500

600

700

t [s]

Figure 6:



800


4

315

Conclusions

In a previous paper [15] the result was explained of an experimental research carried out on the physical prototype represented here in fig. 1a. In this paper it was emphasized of the good correspondence between experimental and numerical data obtained by the model based on the hypothesis of uniform storage and considering the flooding surface S of the storage not constant but function of water level h in the storage. The tests run dealt with three further different configurations: the first one in which the flooding surface of the storage is increased (fig.1b) with respect to the previous cases, the second one covered by synthetic grass to simulate the presence of vegetation on the surface of the storage and the last one in which the widening of the area and the presence of grass are concomitant. In the configuration with widened area and absence of grass the proposed numerical model approximates very well the experimental measures them during the phase of filling, not equally the phase of draining; in this last one, in fact, a marked theoretical-experimental gap is found that is explained by the creation of “backwater areas” in which the water temporary is hold back and then is given back slowly. In the configuration with small area and grassy covering the gap between the theoretical behaviour (based on the hypothesis of uniform storage) and that effective (experimental measures) is justified by two things: the obstacle created by the presence of the vegetation to the motion of the water and the presence of “backwater areas”. The first one justifies the modest mismatch that there is in phase of filling, the second, similar to what said before, is introduced in phase of draining. In the configuration with widened area and presence of grass, the deductions drawn for the other configurations were confirmed. In the cases with grass presence it is allowed to conclude that given the low speeds with which the water moves inside the storage, the existing hydraulic resistance to the motion produced by the vegetation is completely negligible: the hydraulic behaviour in the phase of filling in which the water progressively submerges the grass is very similar to what is developed in the case of completely submerged grass. In a few words, the filling of the storage is simulated by the numerical model with a higher precision rate in the configurations with absence of grass. This means that the numerical method, that considers the flooding surface S variable (stored curve), simulates well the presence of the patchiness in phase of filling; in fact the numerical-experimental correspondence is smaller in the configurations in which the stored curve is not enough to considered also the vegetation presence. In phase of draining, where the presence of or patchiness or vegetation is not contemplated by the stored curve, the numerical-experimental correspondence is lowered because of the macroscopic effect of the “backwater areas”. In conclusion the uniform storage hypothesis, on which the proposed numerical model is based, at least in the experimentations carried out, can be WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

316 Computational Methods and Experimental Measurements XIII used for the dimensioning of the storage since it allows one estimate values of the useful volume of the storage much closer than the effective ones.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

[12]

[13] [14]

[15]

Adami, “Casse di espansione fluviali - Aspetti idraulici” Atti del Corso di Aggiornamento del Politecnico di Milano: La difesa idraulica dei territori fortemente antropizzati, Milano, 1997. Editoriale Bios, (1998) A. O. Akan, “Detention pond sizing for multiple return periods”, ASCE J. of Hydr. Engrg., 115 (5), (1989), pp. 650-664 A. O. Akan, “Single outlet detention-pond analysis and design”, ASCE J. Irrig. and Drain Engrg., 115 (5), (1990), pp. 527-536 Armanini A.: Principi di idraulica fluviale. BIOS editore – Cosenza, 1999 Armanini A. Righetti M., Flow resistance in compound vegetated channel, ICHE 98 Advances in Hydro-Science and engineering, Cottbus, Berlin, Germany Sept. 1988 Armanini A. Righetti M., Flow resistance in open channel flows with sparsely distributed bushes, Journal of Hydrology, 2002, v. 269, p. 55-64 Armanini A. Righetti M. Grisenti P., Direct measurement of vegetation resistance in prototype scale, Journal of hydraulic research, 2005, v. 43, n. 5, p. 481-487 H. A. Basha, “Non linear reservoir routing: a particular analytical solution”, ASCE J. of Hydr. Engrg., 120 (5), (1994), pp. 624-632 H. A. Basha, “Routing Equations for Detention Reservoirs”, ASCE J. of Hydr. Engrg., 121 (12), (1995), pp. 885-888 L. Da Deppo, “Laminazione delle piene con casse di espansione”, Tecniche per la difesa dall’inquinamento, Ed. Bios, (1998) G. De Martino, D. Pianese, F. De Paola, N. Fontana, M. Giugni, “Considerazioni sulla redazione dei piani stralcio per la tutela del territorio dal rischio idrogeologico”, La difesa idraulica del territorio, Atti delle giornate di studio, Trieste, 1999 G. De Martino, F. De Paola, N. Fontana, M. Giugni, “Hydraulic design of on stream floodplain storages”, New Trends in Water and Environmental Engineering for Safety and Life, Maione, Majone Lehto & Monti (eds), Balkema, Rotterdam, (2000) G. De Martino, F. De Paola, N. Fontana, M. Giugni, “Sul dimensionamento di casse di espansione in linea”, 28° Convegno di Idraulica e Costruzioni Idrauliche, Potenza, (2004) F. De Paola, N. Fontana, “Alcune considerazioni sul dimensionamento idraulico di casse di espansione in linea”, L’efficienza e la vulnerabilità delle opere ed infrastrutture fluviali a seguito di eventi idrologici estremi, Attività svolta nell'ambito del PRIN 2000-2002, Castorani e De Martino (eds), (2005) F. De Paola, N. Fontana, A. Ranucci “Indagine sperimentale per la verifica del comportamento idraulico di casse di espansione in linea”, XXX Convegno di Idraulica e Costruzioni Idrauliche, Roma 2006 WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


[16]

[17] [18] [19] [20] [21] [22] [23] [24]

317

V. Ferro, M. Santoro, “Fondamenti teorici e recenti acquisizioni nel settore delle sistemazioni dei bacini idrografici” Università di Palermo – Dipartimento di Ingegneria e Tecnologie Agro-Forestali, Dipartimento di Ingegneria Idraulica e Applicazioni Ambientali, (1999) U. Maione, “Le casse di espansione”, Linee guida per la progettazione delle casse di laminazione. Autorità di Bacino del Fiume Arno, (2000) B. M. McEnroe, “Preliminary sizing of detention reservoirs to reduce peak discharges”, ASCE J. Hydr. Engrg., 118(11), (1992), 1540-1549 V. Marone, “Calcolo di massima di un serbatoio di laminazione”, L’Energia Elettrica, 9, (1971) V. Marone, “Calcolo di massima dell’effetto di laminazione di un serbatoio sulle piene”, L’Energia Elettrica, 10, (1964) A. Paoletti, “Resistenze al moto e processi di laminazione nei corsi d'acqua”, Atti del Corso di Aggiornamento del Politecnico di Milano: La difesa idraulica del territorio, Milano, 1996. Editoriale Bios, (1997) D. Pianese e F. Rossi, “Curve di possibilità di laminazione delle piene”, L'Energia Elettrica, 2, (1986), pp. 131-149 G. Pulci Doria, P. Gualtieri, R. Catapano, “Experimental observations through LDA of a current with almost rigid submerged vegetation”, IAHR, Venezia, (2007) E. Paris, Rischio idraulico: interventi per la protezione idraulica del territorio le casse di espansione. International centre for mechanical sciences monografie CISM. Udine, 2004




319

Analytical and hydraulic model study of highway culvert sand-blockages M. Kamaka1, E. Cheng2, M. Teng2 & C. Matsuda3 1

KFC Airport, Inc., Honolulu, HI, USA Department of Civil and Environmental Engineering, University of Hawaii at Manoa, Honolulu, HI, USA 3 Hawaii State Department of Transportation, Kapolei, HI, USA 2

Abstract Culverts are essential elements of a highway system. Currently, many hydraulic engineers are facing problems of sand-blockage inside culverts along coastal highways. To establish design criteria and a maintenance policy for Hawaii coastal highways, a research program is being carried out at the Department of Civil and Environmental Engineering, University of Hawaii at Manoa, USA. In establishing the design criteria for coastal culverts, we use overland flow theory to consider the design storm, drainage area, detention basin and culvert size as a system. A computational scheme is thus created. For predicting whether a sand-blocked culvert that may be opened by floodwater or it may need manual cleaning, we built hydraulic models for two existing culverts on the Island of Oahu, Hawaii in our Fluid Mechanics Laboratory. The preliminary results of our computer model and hydraulic model are very encouraging. Keywords: highway culverts, send-blockage, simulation model, direct runoff hydrograph, hydraulic model.

1

Introduction

Coastal highway flooding or “overtopping” by surface runoff is the concern. Mitigation measures considered herein are two-fold. The first management measure to minimize overtopping is attributed to the routine maintenance of a culvert, where the culvert is maintained free of debris to facilitate unimpeded surface runoff through a culvert. The second management measure is to assess the adequate nature of the existing culvert and detention pond combination as the WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070331

320 Computational Methods and Experimental Measurements XIII drainage area and design storm changes. It is important to recognize that maintaining an open or clear culvert does not necessarily ensure adequate drainage from surface runoff that may occur. In addition to maintaining an open culvert, minimizing overtopping by surface runoff over a roadway is also managed by a combination of culvert size and storage capacity of a detention pond. Various combinations of the culvert size and the storage capacity of the detention pond may be appropriate to facilitate proper drainage of surface runoff. To minimize overtopping of roadways for a given design storm, (a) the ability to keep drainage culverts clear of debris and/or sand blockage; and (b) having an adequate combination of culvert size and detention pond storage capacity for a specific drainage basin, are two key criteria for long-term management. In order to establish the management criteria, two tasks are undertaken. The first task is to create a flood routing based computational scheme. This simulation model considers the design storm, drainage area, detention pond, and culvert size as a system. For predicting whether a sand-blocked culvert may be opened by floodwater or it may require manual cleaning, the second task is to build hydraulic models for existing culverts on Windward Oahu, Hawaii in the Fluid Mechanics Laboratory at the University of Hawaii at Manoa. Therefore, physical phenomena may be observed from these models.

2

The simulation model

The computer simulation model developed in this study is a flood routing based computational scheme, which considers design storm, drainage area, detention pond and culvert size as an interactive system. As indicated in fig. 1, for the drainage area, a direct runoff hydrograph may be synthesized for a given design storm. This synthesized direct runoff hydrograph from the defined drainage area is actually the inflow hydrograph to the detention pond. The dynamic routing of the inflow hydrograph through the detention pond and culvert system will result in an outflow hydrograph. This routing process is carried off by using PondPack [1]. 2.1 Design storm In accordance with Hawaii Department of Transportation’s Design Criteria for Highway Drainage [2], this study uses an one-hour 50-year rainstorms or sixhour 50-year rainstorms for drainage areas less than 200 acres and drainage areas equal to or greater than 200 acres, respectively. 2.2 Synthesizing direct runoff hydrograph Direct runoff from a drainage basin may be simulated by the Nash-Muskingum method if one assumes that inflow to the basin is a known design storm; and the drainage basin storage may be represented by one or more reservoir type storage. The Nash-Muskingum method is used to compute direct runoff of the drainage basin given the area, rainfall intensity for a 50-year recurrence interval, and the WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


321

recession constant. The direct runoff hydrograph from the drainage basin is the inflow hydrograph to the detention pond. The outflow hydrograph of the drainage basin is calculated by the following relationship:

O2 = Co I 2 + C1I1 + C 2 I 2

(1)

Where O2, O1 are the outflow rates; I2, I1 are the inflow rates for time intervals 2 and 1 respectively; and C0, C1, and C2 are:

C2 = e

− dt 



(

)

(

)

K  ; C =  K  * 1 − C − C ; C = − K  * 1 − C + 1   1  dt  2 2 0 2  dt 

The Muskingum coefficient, K, is considered as a function of recession constant, K1, of a watershed [3]. 2.3 Inflow hydrograph rationale and assumptions The inflow hydrograph used in the generalized simulation model is actually the time-dependent outflow from the drainage basin for a given design storm. For drainage areas less than 200 acres, the 50-year 1-hour design storm is used. Detention Pond Drainage Area

Main stream Drainage Culvert Roadway Figure 1:

Conceptual simulation model.

2.4 Detention pond rationale and assumptions Three different conceptual detention ponds were considered in this study (fig. 2). An average slope of 1%, 2% or 3% of a detention basin was used to estimate the storage capacity prior to flood routing process through a culvert. As indicated in WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

322 Computational Methods and Experimental Measurements XIII fig. 3, the detention basin is assumed to be half of a cylindrical cone, where the center of the cone is the projection of the culvert, headwall, and roadway. The average slope represents the slope of the right cone.

Drainage Culvert

Figure 2:

Details of a culvert.

Culvert barrel

Detention pond

Culvert barrel

Figure 3:

3

Schematic diagram of a detention pond-culvert system.

Punaluu and Hauula culverts

The Punaluu and Hauula drainage areas on the windward of Oahu, Hawaii were selected for the study by using GIS software [4] and topological maps [5]. The watershed areas Punaluu and Hauula are estimated at 262 acres and 167.3 acres, WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


323

respectively. Hydraulic models of the Punaluu circular culvert and Hauula box culvert were fabricated in the Fluid Mechanics laboratory of the University of Hawaii at Manoa. The purpose of the model study is to observe the time for a culvert to self-open under various completely sand blocked conditions. The scaled models were blocked with beach sand, and experiments were conducted to evaluate the time the culvert would open as a function of water depths. 3.1 Laboratory culvert model and dynamic similitude The model scale was calculated using the concept of dynamic similitude for open channel flow. Dynamic similitude will provide a relationship between the laboratory results and actual conditions expected in the field. For free surface flow the parameter in dynamic similitude between the model and prototype is the Froude Number, Fr. By definition: V (2) Fr = g*y where V is the velocity; g is the gravitational constant; and y is the depth of the flow. Therefore, for free surface flow models, the dynamic similitude between the model and the prototype is: Frm = Frp Vp Vm (3) = ym yp where Vm and Vp are model velocity and prototype velocity, respectively; and ym and yp are the flow depths in the model and prototype. 1 1 2 2 Vm  y m  Since, = =L (4)  yp  r Vp   or,

 lm    Vm  t m  or, = Vp  lp   t   p Therefore,

where the model scale L r =

 tp Vm = L r  Vp  tm

   

y

(5)

m , and l and l are length dimensions in model m p y p

and prototype, respectively.


324 Computational Methods and Experimental Measurements XIII Substituting eqn. 4 into eqn. 5 yields: t m = t p * Lr

(6)

where tm = model time for culvert to self-open; and tp = actual time for the culvert to self-open. 3.2 Description of Laboratory Experiment The culverts were “packed” with beach sand to simulate complete blockages. The sand was placed in the model culvert with various compaction efforts. A constant volume of sand was placed in the model culvert for all experimental trails. The slope of the culverts was placed at 0 % for both the 9-inch by 12-inch model box culvert (fig. 4) and 6-in model circular culvert. The sand was placed in the middle of the model culvert barrel, simulating actual conditions along the coast of Windward Oahu, Hawaii.

Sluice Gate Water Tank

Box Culvert

Sand Blockage

Figure 4:

Photograph of model box culvert experiment for Hauula.

4 Results and discussions A block diagram of model runs is summarized in fig. 5. This figure indicates that a total of 12 series of model runs were performed. Forty different sizes of box culverts and fifteen different sizes of circular pipe culverts were modeled for each of the three different conceptual detention ponds of 1%, 2% or 3% slopes (fig. 2) under 17 different inflow hydrograph scenarios. Laboratory observations from model study of both the box culvert as wall as the circular culvert indicate that the time for the culvert to open varies as a function of the soil moisture and the degree of “compactness” of the sand in the model culvert. The time for the model box culvert to self-open ranged from 15 seconds to 8 minutes 39 seconds. The time for the circular culvert to self-open WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


325

Maximum Head Water Elevation

Inflow Hydrograph Watershed Area > 200 acres 50-year , 6-hour storm

Figure 5: Table 1:

Circular Culvert

Detention Basin 3% slope

Box Culvert


Circular Culvert

Circular Culvert


Box Culvert

Circular Culvert


Box Culvert

Circular Culvert


Box Culvert

Circular Culvert

Box Culvert


Box Culvert

Inflow Hydrograph Watershed Area < 200 acres 50-year , 1-hour storm

Block diagram of model runs.

Values for inflow hydrograph for 0.

When α , β ,b1 and c are specified we can solve critical thickness 2a . WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

(4)


3

507

Asymptotic analytic method and integral method

Following Inonu [7], we divide the ordinary part of the scattering function into one symmetric and one antisymmetric part. We make the substitutions [6] 1−αc − γ κ ( x , µ ) = ψ ( x, µ ) − ψ ( x, − µ ) (5) βc where c(1 − α − β ) c= (6) 1 − c(α + β ) (7) γ = (1 − α c) 2 − β 2 c 2 . Obviously, for γ to be real it is required that (α + β )c ≤ 1 . With these substitutions we find that κ ( x, µ ) is a solution of the equation ∂κ ( x, µ ) cγ 1 + κ ( x, µ ) = µ (8) ∫ (1 + 3b1µµ ')κ ( x, µ ')d µ ' ∂x

2

−1

where 1−αc − β c b1 . 1−αc + β c The boundary conditions for the transformed flux κ ( x, µ ) will then be b1 =

µ < 0; κ (a, µ ) = Rκ (a, − µ )  κ ( a , µ ) R κ ( a , µ ) µ > 0. − = − − 

(9)

(10)

where R=

βc . 1−αc + γ

(11)

Eliminating ψ (τ , − µ ) from eqn (5) with arguments µ and − µ we obtain 1−αc + γ βc κ ( x, µ ) + κ ( x, − µ ). (12) 2γ 2γ 1 1−αc + β c + γ 1 (13) ∫−1ψ ( x, µ )d µ = ∫−1κ ( x, µ )d µ. 2γ Utilizing the symmetry of the angular flux at x = 0 ψ (0, µ ) = ψ (0, − µ ) we obtain 1−αc + β c + γ ψ (0, µ ) = κ (0, µ ). (14) 2γ From eqns (13) and (14) we know the zero moments of the angular flux

ψ ( x, µ ) =

1

1

−1

−1

ψ 0 ( x) = ∫ ψ ( x, µ )d µ , κ 0 ( x) = ∫ κ ( x, µ )d µ as a function of x have a similar distribution and ψ (0, µ ) , κ (0, µ ) as a function of µ have a similar distribution. If we assume that b1 = 0 and κ 0 ( x) = cos( x / η ). (15) WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

508 Computational Methods and Experimental Measurements XIII When eqn (15) is inserted into eqn (8) we obtain 2 aγ

aγ

 a |µ| a  (e |µ | − R ) + ( R − 1) cos( ) + ( R + 1)( ) sin( )  e |µ | η γη η  c  k (0, µ ) = . 2 aγ 2 µ [1 + ( ) 2 ][e | µ | − R ]

(16)

γη

(a) For β = 0 from eqn (11) it follows that R = 0 k (0, µ ) =

c 1 − cos(a / η )e − aγ /| µ | + sin(a / η )e − aγ /| µ | 2 1 + ( µ / γη ) 2

(17)

When α → 1/ c and a / η → 0 if we assume the scale flux as a function of x is a constant we obtain c − aγ / µ k (0, µ ) = (1 − e ). (18) 2 κ 0 (0) = cγ 1 − e − aγ + aγ E1 (aγ )  (19) where E1 ( x) is exponential integral function. Neglecting terms proportional to γ and of higher order cγ a[1 − γ 0 − log(γ a )] 1 ca (1 − α ){1 − γ 0 − log[(1 − α c)a ]} 1 where γ 0 is Euler’s constant γ 0 = 0.577216 .

(20) (21)

(b) For β ≠ 0 when α + β → 1/ c from eqn (11) it follows R = −1, κ 0 (a) = 0 , we obtain cos(a / η ) ≈ 0 and sin(a / η ) ≈ 1 k (0, µ ) =

c 1 . 2 1 + ( µ / γη ) 2

k0 (0) = cγη arctg (1/ γη ). η To get we use Eqs. (15) and (23) cγη arctg (1/ γη ) = 1. This is close to the discrete eigenvalues of the Case spectrum [8]. It shows that the expression 2ca (1 − α − β )(1 − α c + β c) / γ

(22) (23) (24)

approaches a constant when α + β → 1/ c . The limiting value is 2 (1.954 in reference [6]). 2ca (1 − α − β )(1 − α c + β c) / γ = 2.0. (25) (c) As shown by Siewert and Williams [9] and Sahni et al [6] eqn (8) can be transformed into an integral equation for the zero moments of the angular flux κ 0 ( x) . If we assume that b1 = 0 and κ 0 ( x) is an even function of x it will read



κ 0 ( x) =

cγ 2 ×∫

∫

a

−a

1

κ 0 ( y ) E1 (γ | x − y |)dy +

exp(

−2aγ

µ

){exp[

( y − x)γ

cγ 2

∫

a

−a

κ 0 ( y )dy

] + exp[

( y + x)γ

µ µ µ[1 − R exp(−2aγ / µ )]

0

509

(26)

]}d µ .

For β = 0 with a constant scale flux assumption eqn (26) will be cγ a (27) E1 (γ | y |)dy. 2 ∫− a Expand E1 ( x) at x = 0 and neglecting terms proportional to γ and of higher order, we could obtain eqs. (20) and (21).

κ 0 (0) =

4

Spherical harmonics method

For the Spherical harmonics method solution, the angular flux is expanded in a series of Legendre polynomials as N  2n + 1  ψ ( x, µ ) = ∑  (28)  Φ n ( x ) Pn ( µ ). 2  n=0  This expansion (28) can now be substituted into eqn (3) in order to obtain the function Φ n ( x ) . Multiplying both sides of the resulting equation by Pm ( µ ) , integrating overt and utilizing the orthogonally properties and the recursion relations of the Legendre polynomials, after some rearrangement we have: n

d Φ n −1 ( x )

+ ( n + 1)

d Φ n +1 ( x )

+ ( 2n + 1) Φ n ( x ) dx dx = ( 2n + 1) [c (1 − α − β ) Φ 0 ( x ) δ n 0 b1c (1 − α − β ) Φ1 ( x ) δ n1 + α cΦ n ( x ) + β c ( −1) Φ n ( x )] n

(29)

n = 0,1, 2,… N .

One may employ the well-known procedure of seeking a solution of the homogeneous eqn (29) in the form Φ n ( x ) = Gn ( v ) exp ( − x v )

(30)

where the Gn (v) are some constants. Each of the Φ n ( x ) defined in eqn (30) will satisfy eqn (29) provided that the characteristic PN equations are satisfied：

( n + 1) Gn +1 ( v ) + nGn −1 ( v ) − ( 2n + 1) v{1 − c (1 − α − β ) n × (δ n 0 + b1δ n1 ) − c α + ( −1) β }Gn ( v ) = 0.  

(31)

Eqn (31) has a homogeneous matrix form:  M ( v )  G = 0.


(32)

510 Computational Methods and Experimental Measurements XIII The coefficient matrix is (N + 1) × (N + 1) , and G = [G0 , G1 . . . Gn ]T . The unknown constant vector G is determined from the normalization G−1 (ν ) = 0 and G0 (ν ) = 1 . The general solutions of eqn (3) with N odd can be expressed, for n = 0,1, N , as Φn ( x ) =

( N +1)

∑ j =1

2

n Gn ( v j )  Aj Fn ( − x v j ) + ( −1) B j Fn ( x v j )   

(33)

where Gn (−v) = (−1) n Gn (v) , Fn (± x / v j ) = exp(± x / v j ) and Aj and B j are coefficients to be determined from the boundary and symmetry conditions of the problem. The essential idea of the standard PN method is that Φ N +1 (v) = 0 , i.e. the permissible eigenvalueν j is the jth positive zero of GN +1 (v) . The determination of the roots is obtained using the Newton-Raphson iterative technique. For a finite slab of half-thickness a we apply the Marshak boundary condition

∫

1

0

P2 k −1 ( µ )ψ ( a, − µ ) d µ = 0

k = 1, 2,… ( N + 1) 2

(34)

to the general solution (31), and then obtain an eigenvalue equation which relates the critical dimensions a to the parameter c , or vice versa. This can be written in matrix form as (35)  M ( a )  A = [ 0] where A is a vector with elements A k , k = 1, 2,… ( N + 1) 2 and the [( N + 1) 2]2 elements of matrix M k , j (a ) are given by

Mk, j ( a) =

( N −1)

2

∑ ϕ ( n, k ) G ( v ) cosh ( a v ) + G ( v ) sinh ( a v ) n =0

2n

j

j

2 k −1

j

j

(36)

where

( 4n + 1)( −1) ( 2k − 1)!( 2n )! . ϕ ( n, k ) = 2 2 k + 2 n −1 2 ( 2n − 2k + 1)( k + n )  n !( k − 1)! n+ k

(37)

Clearly eqn (37) is a linear system of algebraic equations. This system has a non-trivial solution (for A k ) if determinant M k , j (a ) vanishes, and this condition yields the desired results.

5

Modified spherical harmonics method

The PN approximation ( N odd) consists of truncating the expansion (28) after N + 1 terms, i.e. at n = N . One also chooses a finite number of points of the ν spectrum, given by the roots of the eqn (31) and GN +1 (v) = 0 . Since c > 1 , the root ν 1 is imaginary (we letν 1 = iη ) and very close to the discrete eigenvalue ν 0 of the Case spectrum [8]. The eigenfunction corresponding imaginary



511

eigenvalueν 1 is dominant part of the angular flux when α + β approaches the value 1/ c . It is necessary to count for the high-order terms of eqn (28) as the angular flux is a very strongly peaked function along the direction parallel to the slab surface. The modified spherical harmonics ( δ PN ) method is that using the finite width delta functions analyse the high-order truncation error of the angular flux and the high-order part of the angular flux is calculated analytically. Gn (ν 1 ) = Dn for n ∈ even and n ≥ N + 1 (38) where 1

Dn = ∫ δ ( µ )Pn ( µ )d µ . −1

(39)

From the previous section we know the finite width delta functions δ ( µ ) are expressed by eqns (18) and (22) for β = 0 and β ≠ 0 separately. 1 1 − e − aγ /|µ | if β = 0;  − aγ  2 1 − e + aγ E1 (aγ ) δ (µ ) =  1 1  if β ≠ 0.  2γη arctg (1/ γη ) 1 + ( µ / γη ) 2

(40)

The value of GN +1 (ν 1 ) is not equal to zero at this time GN +1 (ν 1 ) = DN +1 . (41) To apply the Marshak boundary condition (34) we must take into account the high-order parts of angular flux so eqn (36) for j = 1 will be changed to

6

∞  ( N −1) 2  M k ,1 ( a ) =  ∑ ϕ ( n, k ) G2 n (v1 ) + ∑ ϕ ( n, k ) D2 n (η )   n =0  n = ( N +1) 2   × cos ( a η ) + G2 k −1 ( v1 ) sin ( a η ) .

(42)

 ( N −1) 2  1 M k ,1 ( a ) =  ∑ ϕ ( n, k )( G2 n (v1 ) − D2 n (η ) ) + ∫ P2 k −1 ( µ ) δ ( µ ) d µ  0  n=0  × cos ( a η ) + G2 k −1 ( v1 ) sin ( a η ) .

(43)

Numerical results and discussion

In order to examine the validity and accuracy of the present method two computer programs ( δ PN and integrate method) in FORTRAN were written to calculate the critical thickness. We took the results from integral equation method as reference values. The integral eqn (26) was solved iteratively by substituting improved flux approximation in to the right-hand side of the equation. When α + β → 1/ c the angular flux κ ( x, µ ) are high-order functions of µ for β ≠ 0 we use N = 24 to 520 double-Gauss quadrature sets for different combinations α , β and c .


512 Computational Methods and Experimental Measurements XIII Table 1:

The critical slab thickness 2a obtained in PN , δ PN approximation and integrate ( Int ) method with different value of c, α , β and degree of linearly anisotropic scattering comparison with results of C. Yildiz [1].

c, α , β

b1

P11

δ P3

δ P11

1.2 0.70, 0.0

-0.3 0 0.3

2.95114 3.11594 3.32199 2.60780 2.66236 -0.52586 0.53742 -0.52156 0.53247 -1.14702 1.15204 1.15715 0.95199 0.95384 0.95569 0.33843 0.34093 0.34349 0.33164 0.33389 0.33618 0.67499 0.67993 0.68501 0.73696 0.74475 0.75285 0.93023 0.95138 0.97453

2.95434 3.14486 3.34232 0.86736 0.87926 -0.31261 0.32049 -0.17348 0.17586 -0.97472 0.98166 0.98875 0.00987 0.00987 0.00988 0.08039 0.08114 0.08191 0.00204 0.00204 0.00204 0.16057 0.16207 0.16360 0.45872 0.46617 0.47388 0.79248 0.81434 0.83624

2.93843 3.10476 3.31169 0.87132 0.87928 -0.31780 0.32095 -0.17428 0.17587 -1.02865 1.03315 1.03772 0.01707 0.01707 0.01708 0.11128 0.11188 0.11250 0.00378 0.00378 0.00378 0.22206 0.22326 0.22449 0.55192 0.55693 0.56202 0.88245 0.90186 0.92250

1.2 0.8333, 0.0 2.0 0.49, 0.0 2.0 0.4999, 0.0 1.2 0.0, 0.80

1.2 0.0, 0.8333 2.0 0.0, 0.49 2.0 0.0, 0.4999 1.5 0.33, 0.33 1.5 0.33, 0.3 1.5 0.42, 0.15

-0.3 0 0.3 -0.3 0.0 0.3 -0.3 0.0 0.3 -0.3 0 0.3

-0.3 0 0.3 -0.3 0.0 0.3 -0.3 0.0 0.3 -0.3 0 0.3 -0.3 0 0.3 -0.3 0 0.3

Int

3.08555 0.89557 0.32932 0.17913 1.01267 0.04210 0.14402 0.01852 0.28622 0.56570 0.88135

P11 P13* [1] 2.95115 3.11595 3.32199 2.60780 2.66236 2.72127 0.52586 0.53742 0.54996 0.52156 0.53247 0.54425 1.14702 1.15204 1.15715 0.95199 0.95384 0.95569 0.33843 0.34093 0.34349 0.33164 0.33389 0.33618 0.66691* 0.67168* 0.67658* 0,73009* 0.73770* 0.74562* 0.92990* 0.95144* 0.97506*

For the specified α , β , c and b1 we first computed the eigenvalueν 1 = iη from eqn (24) and start the calculation with some assumed value of halfthickness a , then we could calculate Dn (ν 1 ) by eqn (39). Except GN +1 (ν 1 )



513

and M k ,1 ( a ) the calculation procedures in present work are the same as standard PN method. After we obtain an improved value of half-thickness a and continue this process until we get a convergence accurate result. In table 1, we present the critical slab thickness 2a obtained in PN method, δ PN method and integrate ( Int ) method with different value of c, α , β and degree of linearly anisotropic scattering b1 . In reference [1] Yildiz had compared the results calculated by standard PN method with the results of Sahni et al [6] and others. The conclusion was that the standard PN method could give reasonable results for thick slab. We directly compare our results obtained by standard PN method with those in reference [1]. The agreement is generally within five or six significant decimal places for N < 13 . For N = 13,15 the agreement is generally within three or four significant decimal places. We couldn’t obtain converged results for some combination of α , β when α + β → 1/ c with b1 = 0.3 . When γ ≈ 1 the value of GN +1 is nearly zero. The low-order PN approximations generally give accurate results for thick slabs [1,2]. For example when α = 0.7, β = 0.0, c = 1.2, b1 = 0.0 , γ = 0.16 and G12 = 0.0066 the PN , δ PN methods give closely results. When the values α + β approach the limit value 1/ c the angular distribution is very strongly peaked along the plane of the slab. The critical slab thickness 2a varies rapidly with c and approaches to zero but the standard PN method give an unchanged result. For example while for α = 0.0, β = 0.4999, c = 2.0, b1 = 0.0 , γ = 0.02 , GN +1 = 0.2252 the critical slab thickness 2a are 0.33389 for P11 . The δ PN method gives 2a = 0.00378 for δ P11 . Even δ P3 approximation could give a reasonable result 2a = 0.00204 . In table 1 we could see when α = 0.8333 , β = 0.0 , c = 1.2 , b1 = 0.0 , γ = 4 × 10−5 , GN +1 = 0.164 the critical slab thickness 2a are 2.6623,0.87928 for P11 , δ P11 respectively. It is seen from Table 1 that the δ PN method has more accuracy than standard PN method but has less accuracy than integral equation method. When α + β → 1/ c the critical slab thickness 2a as a function of α , β approaches to zero more rapidly than real. We think that the errors occur in the lack of taking account of the high-order moments of the angular flux in the boundary condition eqn (34).

7

Conclusion

In this paper we have considered the slab criticality problem of the linear transport equation with forward, backward and linearly anisotropic scattering in a homogeneous slab. Using the asymptotic analytic method we obtain the asymptotic angular flux which expressed by finite width delta functions at the central point of the slab when α + β approaches the value 1/ c . Using the finite WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

514 Computational Methods and Experimental Measurements XIII width delta functions to consider more terms in the expansion of angular flux we could obtain accurately results with low-order N and less computational effort. The present scheme converges quite well and enables us to obtain accurate results even with the low-order formulas. The present method is applicable to large system, and also to very small system. The slab thickness as a function of α , β are continuously and smoothly for various α + β ranging from 0 to the extreme case 1/ c . The present method can also be extended to more general problems with two or three dimension high-order anisotropy and to multiregional energy dependent problems. We have started looking into these aspects and the conclusion awaits further work.

Acknowledgement This work was partially supported by the National Key Laboratory of Computational Physics under Grant No.51479050105ZW0906.

References [1] [2] [3] [4] [5] [6] [7] [8] [9]

C. Yildiz, Variation of the critical slab thickness with the degree of strongly anisotropic scattering in one-speed neutron transport theory, Ann. Nucl. Energy, 1998; Vol. 25, No. 8, pp. 529 540, C. Yildiz. Influence of anisotropic scattering on the size of timedependent systems in monoenergetic neutron transport, Journal of Physics D 1999; 32: 317. R.C. Barros and E.W. Larsen. A numerical method for one group slab geometry discrete ordinates problems with no spatial truncation error. Nuclear Science and Engineering 1990; 104: 199. W. A. Fiveland. Three-dimensional radiative heat transfer solutions by the Discrete Ordinates Method. AIAA J. Thermophysics 1988; 2(4): 309-316. J. E. Morel, A hybrid collocation-Galerkin-Sn method for solving the Boltzmann transport equations. Nuclear Science and Engineering 1989; 101: 72-87. D. C. Sahni, N. G. Sjostrand and N. S. Garis, Criticality and time eigenvalues for one-speed neutrons in a slab with forward and backward scattering. Journal of Physics D 1992; 25:1381. E. Inonu, Transport Theory and Statistical Physics 1973; 3: 137 K. M. Case and P. F. Zweifel, Linear Transport Theory. Addison Wesley, Reading, MA, 1967. C. E. Siewert and M. M. R. Williams, The effect of anisotropic scattering on the critical slab problem in neutron transport theory using a synthetic kernel. Journal of Physics D 1977; 10: 2031-2040.


Section 6 Structural and stress analysis



517

Inverse variational principle based coupled modeling of underground structures P. Procházka Czech Technical University in Prague, Czech Republic

Abstract In some previous papers of the author coupled numerical–experimental modeling of tunnels embedded in surrounding rock was based on minimization of a certain functional describing the steepest descend mode of differences of measured and computed values of stresses or displacements at selected points. The idea consisted of a choice of subdomains (patches), the eigenstrains in which were introduced using the unit impulse technique. Influence matrices were generated and the linear hull of eigenstrain effects together with the optimization problem lead to the identification of plastic stresses inside the domain describing the surrounding rock. Consequently, a nonlinear model in numerical analysis can be improved using eigenparameters as design parameters in optimization. The only problem remaining is how to select the patches. In this paper inverse variational principles are applied to help solve this principal problem. The 2D problem is solved with moving patches (support subdomains) with uniformly introduced eigenstrains. Keywords: coupled modeling, tunnel face stability, Inverse variational principles, Desai’s model.

1

Introduction

Using a very powerful tool, a combination of Transformation field analysis (TFA) and a certain plasticity rule (possibly softening included), back analysis of structures can be regarded as seeking the optimal distribution of eigenparameters (eigenstrains or eigenstresses) in the domain describing undeformed rock in a certain sense. For the first time the TFA was applied to optimization of prestressing of composite structures, [1]. The eigenparameters can represent many phenomena (not only prestress), and also the influence of plastic WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070521

518 Computational Methods and Experimental Measurements XIII deformation, or the influence of the length of purchase in tunneling. A similar approach was applied to nonlinear problems in composite structures, [2]. In the first above mentioned publication the areas with possible eigenparameters were firmly given. In the second publication the areas (patches) were selected from experience and also were done. Unit impulses of subdomain-wise uniformly distributed eigenparameters (generally linear or higher order polynomials) enable one to find material properties from comparison of results from experimental studies and numerical analysis at selected points, as was done in other papers of the author of this paper on geomechanical problems, [3, 4], for example. The only problem still occurring is: find the optimal distribution of patches (subregions), where eigenstrains are introduced or considered. This is a problem of combined optimization, where the principal variables depend on subdomains (patches), the uniform distribution (this is one of possible approximations) of eigenparameters is assumed. This problem is not easy to solve, as the optimization of principal variables must be iterated, and a reasonable tool for it should be found. One such tool can serve Inverse variational principles, which hold the volume of the domain as constant, and design variables are subdomains, or their representatives, [5].

2

Transformation field analysis

In this section the general procedure for coupled modeling on the firm patches is briefly introduced using the TFA. It may be done in terms of many modern numerical methods. First, let us consider that the body Ω under consideration (part of a structure, element, and system of more elements, composite, in our case rock surrounding tunnel) behaves linearly, i.e. Hooke’s linear law is valid in the entire body. When the problem is correctly posed, the displacement vector, strain and stress tensors can be obtained from the Navier equations, kinematical equations, and linear Hooke’s law. In the second step we select points, where the measured values are available, either from experiments in laboratory, or from “in situ” measurements. We also select points Ar, or disjoint regions (subdomains) Ωr, r = 1,...,n, from the body under study, and apply there successively unit eigenparameter impulses (either eigenstresses or eigenstrains) to get an influence tensors (matrices). Moreover, let the set of points where the measured values are known be Bs, s = 1,...,m. Then the real stress (σ ) s at Bs is a linear hull of stress (σ ext ) s at Bs due to external loading and eigenstrains ( µ ) r and (ε pl ) r , or eigenstress (λ ) r and relaxation stress (σ rel ) r at Ar (similar relations are valid for overall strain field ε or displacements u): (σ i ) s = (σ iext ) s + ( Pikσ ) sr ( µk ) r + (Qikσ ) sr ( ε kpl ) r ,

or (σ i ) s = (σ iext ) s + ( Rikσ ) sr (λk ) r + (Tikσ ) sr (σ krel ) r WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

(1)


519

(ui ) s = (uiext ) s + ( Piku ) sr ( µ k ) r + (Qiku ) sr (ε kpl ) r ,

or (ui ) s = (uiext ) s + ( Riku ) sr (λk ) r + (Tiku ) sr (σ krel ) r

(2)

where the Einstein summation rule has been used and the influence tensors P, Q, and also R and T may be identical, as any eigenparameter may stand for the plastic or relaxation parameter (say, eigenstrain may stand for plastic strain, which is obvious from (1) and (2)). The strain and stress components are written in vector form. Note that λ = − Lµ holds, where L is the elastic stiffness tensor. The first relations in (1) and (2) describe the initial strain method while the second relations in those equations formulate the initial stress method. On the other hand measured stresses (σimeas)s, or measured displacements meas s (u i) are available in a discrete set of points. A natural requirement is formulated in terms of steepest descent type “error functionals” I, which express that the values of measured and computed values be as close as possible: I σ [( µk ) r ] =

6

m

∑∑

[(σ i ) s − (σ imeas ) s ]2 → minimum,

(3)

[( ui ) s − (uimeas ) s ]2 → minimum,

(4)

í =1 s =1

or I u [( µ k ) r ] =

6

m

∑∑ í =1 s =1

where the sum is taken over i and s. Differentiating I by (µα)β yields a linear system of equations for (µj)l: (Aαk)βr (µk)r = Yαβ, α = 1,...,6, β = 1,...,m, where ( Aαk ) βr =

6

m

∑∑ ( Pikσ ) sr ( Piασ ) sβ i =1 s =1

Yαβ =

6

m

∑∑ [(σ imeas ) s − (σ iext ) s − (Qikσ ) sr (ε kpl ) r ]( Piασ ) sβ i =1 s =1

in the case (3), and ( Aαk ) βr =

6

m

∑∑ ( Piku ) sr ( Piαu ) sβ i =1 s =1


(5)

520 Computational Methods and Experimental Measurements XIII Yαβ =

6

m

∑∑ [(uimeas ) s − (uiext ) s − (Qiku ) sr (ε kpl ) r ]( Piαu ) sβ i =1 s =1

in the case (4). Recall that the patches are selected as fixed, their position and shape are firmly given.

3

Nonlinear law given

The previous section was devoted to application of the TFA to coupled modeling assuming the starting (initial) stage of computation is linear elastic. This case is appropriate for large scale problems, involving plasticity, hereditary problems and some softening rules. On the other hand, probably a more precise approach would be to start with the plastic law, at least approximately. It is well known that some cases of softening lead to multivalued functions. The procedure described in the previous section cannot include this case. If the initial stage is described by a certain plastic rule, the larger scale of materially nonlinear problems can be solved in similar way as before. Let us concentrate on such problems. If a certain plasticity law is proposed we can write using the splitting of elastic and plastic influences the resulting stresses and displacements as: (σ i ) s = (σ iext ) s + (Qikσ ) sr ( ε kpl ) r ,

or (σ i ) s = (σ iext ) s + (Tikσ ) sr (σ krel ) r s

(ui ) =

(uiext ) s

+ (Qiku ) sr ( ε kpl ) r

(6)

,

or (ui ) s = (uiext ) s + (Tiku ) sr (σ krel ) r

(7)

From the above equations it immediately follows that, for example, (σ i ) s = ( S ikσ ) s + ( Pikσ ) sr ( µ k ) r , (ui ) s = ( Siku ) s + ( Piku ) sr ( µ k ) r ,

(8)

holds, i.e. the influence of elasticity and plasticity is hidden in the first terms of the right hand sides of (8). From (8) two possibilities obviously appear: Either plastic effects disappear in the first terms of the r.h.s. of (8) or they are considered there. Certain starting plastic rules involved in (8) are discussed in [4]. Using the expressions (8) and the minimum conditions (3) or (4), the conditions for a minimum are obtained from (5) with Yαβ =

6

m

∑∑ [(σ imeas ) s − (σ iext ) s ]( Piασ ) sβ i =1 s =1



Yαβ =

6

521

m

∑∑ [(uimeas ) s − (uiext ) s ]( Piαu ) sβ i =1 s =1

The left hand sides hold their former expression.

4

Inverse variational principles

Following extended primary variational principles one can write the energy functionals on the whole domain Ω as: Π=

1 a (u, u ) − [ p, u ] → minimum 2

(9)

where a(.,.) is an energetic norm and [.,.] is the scalar product on the boundary Γ of Ω, with p being prescribed tractions. Let us divide Ω into m disjoint subregions (patches) Ωr, union of them is Ω. Then (9) can be rewritten as Π=

m

∑ r =1

1 a r (u, u ) − [ p, u ] → minimum 2

(10)

where ar(.,.) is an energetic norm on each Ωr. Then the problem appears not to be properly defined. In order to ensure that the problem is correctly posed, the volumes (or areas in 2D) have to be bounded and their measure has to be given. The functional (9) then is improved as: Π (u, Ω ) =

∑

1 [ a r ( u , u ) + ωr ( 2

∫

ΩdΩr − C r )] − [ p, u ] →

→ stationary

(11)

where Cr is a measure of Ωr, see [5], and ωr are the lagrangian multipliers. The internal energy is a sum of integrals over appropriate domains Ωr of potentials (W ) r , which in our case reads as: (W ) r = (σ i ) r [(ε i ) r − ( µ i ) r ]

(12)

where ( µi ) r has been considered uniformly distributed in each Ωr . Hence, Euler’s equations follow as: 1. Variation by displacements yields equilibrium equations involving partially uniformly distributed eigenstrains ( µi ) r . They are given from (5) for a supposed distribution (shape) of Ωr .


522 Computational Methods and Experimental Measurements XIII 2. Approximating the problem in the sense of FEM with K, say, then the changes in the fields with respect to the subdomains can formally be written as: ∂Ω r 1 ∂K =0 U iU k + ωr 2 ∂pi ∂pi

(13)

where pr are internal parameters declaring the shape of the subdomain Ωr . From (10) it immediately follows that the lagrangian multipliers have to be constant for each r, i.e. on each subdomain Ωr all components of eigenstrain tensor remain uniform. 3. Partial differentiation of (8) by ω r ensures that the measures of the subdomains are unchanged. Some recommendations on how to introduce the internal shape parameters could be found in [5], for example.

5

Numerical procedure for two subdomains

To show the ability of the above submitted procedure consider a simply supported steel fiber reinforced concrete beam with the length of 30 cm, the height of 12 cm, the bending stiffness I = 1440 cm4, E = 2 GPa. The width is 10 cm. The problem is solved as a stretched plate, the shape of the external boundary of which remains constant and one patch is considered. Concentrated forces are applied at one third and two thirds of the span, i.e. four points test is studied. The symmetric case is solved, shear eigenstrains are neglected, and transversal eigenstrain disappears, too. The force F applied is 4 kN, distributed along the width, so that the stress in the elastic state is 1.67 MPa. Measured values of deflections are as follows: under the force at the lower edge of the beam it was 0.69 cm, at the middle of the span 0.828 cm. The initial stage is considered elastic, plasticity occurs due to eigenparameters, which stand here for plastic strains. For the optimized shape of the patch we use condition (4). Note that the number of unknown eigenparameters cannot exceed the number of measured values. This must be understood in such a way that one component of the eigenparameter tensor is one unknown. For example, if we take into consideration a full 2D tensor, than four components create the unknowns in one patch. Consequently, for one patch we need at least five measured values of either displacement, or stresses, or both, as obviously the conditions (3) and (4) can be combined. The strain at the middle of the span in the lower edge was calculated as 0.000835, while the resulting strain dropped to the value of 0.000738, i.e. the eigenstrain lowered the strain by approximately 0.0001. The movement of the boundary of the optimal patch undergoes criteria, which are described in the second example, which is more general. The final situation is described in Fig. 1, where the shape of the uniformly distributed eigenstrains is shown (we consider only one component of the eigenstrain tensor, namely the horizontal eigenstrain, according to our above mentioned assumption). WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Figure 1:

523

Four point test on a beam with one patch.

The second example is much more complicated and shows us how to use the procedure in underground structure assessment, namely for description of behavior of the rock surrounding the tunnel lining. Also here, one patch is considered for simplicity. As in the eigenparameters also the influence of purchase can be involved, the problem of loading of the lining is solved as a 2D problem.

a) Figure 2:

b)

a) Starting triangular mesh and b) highlighted domain describes the patch considered.

The height of the domain, in which the problem is solved, is 30 m, the diameter of the tunnel is 10 m, the distance of the axis of the tunnel from the lower boundary of the domain is 15 m, and the width of the model is 15 m. A triangular mesh is depicted in Fig. 2a), and the starting shape of the patch is displayed in Fig. 2b). In these pictures as well as in the next pictures no tunnel lining is depicted in order not to disturb the contour lines (hypsography). WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

524 Computational Methods and Experimental Measurements XIII The problem starts with the solution of the plastic stage obeying the generalized Mohr-Coulomb hypothesis with the following material parameters: Modulus of elasticity E = 1000 MPa Plastic E = 800 MPa Residual E = 500 MPa Poisson’s ratio ν = 0.25 Plastic equals residual ν = 0.46 Shear strength C = 0.08 MPa Plastic C = 0.06 MPa Residual C = 0.02 MPa tan ϕ = 240 Plastic tan ϕ = 300 Residual tan ϕ = 100 Volume weight ρ = 2100 kg/m3 where ϕ is the angle of internal friction. Using the above said parameters describing material properties, which correspond with the class R3 of the rock according to geological standards. The tunnel lining is made from concrete and no plastic behavior is assumed. After computing the plastic state with the above parameters, the initial stage is created and the eigenstrains can be introduced. The initial shape of the patch is depicted by the highlighted subdomain in Fig. 2b). In this figure also possible movement of one nodal point is illustrated by thick lines. Parameters pi are distances of the patch boundary nodal points from the center of the tunnel in our case. Movement of each point causes also a change in the mesh, as is obvious from Fig. 2b). The measured values were also vertical deflections on the contact of the lining and rock. The values were taken from a scale model built up in stands, as described in [3], for example. There were nine measurement points along the lining; the values at symmetric points were averaged. In Fig. 3a) hypsography of vertical stresses in the starting plastic stage is depicted and in Fig. 3b) hypsography of vertical stresses in the final stage after optimization of the shape of patch is displayed. The difference between these two pictures is not as distinct as supposed, the plastic law and the material properties were selected in a good way, and the influence of the purchase is not too important. From this assertion immediately follows the conclusion that the eigenstrains can be also considered as a measure of error of the plastic law selected. On the other hand, if the choice of the plastic law is wrong, the initial stage starts with elasticity, for instance, as was the case of the previous example, more accurate results can be expected only if more patches are selected, not only one, as in our examples. In Fig. 4a), vertical displacements, which were compared with the measurements on the scale model, are again drawn in the form of hypsography. Important is the fact that at one of the measured points, at point A, the measured vertical displacement was 4.12 cm, so that the results are very versatile. For completeness in Fig. 4b) the shape of the patch in the final stage is also shown. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


a) Figure 3:

6

b)

Vertical stresses in a) plastic stage and b) after optimization.

a) Figure 4:

525

b) a) final displacements and b) shape of the optimal patch.

Conclusions

In some previous papers of the author it has been shown that the coupled numerical and experimental (scale) modeling or the on site measurements can basically improve identification of a numerical mechanical model. The only problem appeared the choice of subdomains (patches). No solution has been proposed so far. This paper tries to improve this lack of information using Inverse variational principles. Although simple examples are presented here, the generalization to more patches is straightforward. It is worth noting that for subdomains of large extent an extensive number of measurements are necessary. In former papers three or at most four subregions have been considered in applications to underground structures, particularly for WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

526 Computational Methods and Experimental Measurements XIII assessment of tunnel face stability. The reason consists of the fact that (5) is generally created for 3 in 2D or 6 in 3D components of eigenstrains (or eigenstresses), which means it is necessary to determine at least 24 unknown eigenparameters in 3D for four subdomains in each iterative step. From the point of view of numerical analysis this does not cause any problem, but to feed (5) at least by 25 measurements the original domain Ω can lead to quite a complicated problem or an insufficient set of data.

Acknowledgements The financial support of GAČR, project number 103/06/1124 is appreciated. The research has also been supported by a grant of Ministry of Education of the Czech Republic number MSM6840770001.

References [1] [2]

[3] [4] [5]

G. J. Dvorak, P. P. Prochazka: Thick-walled composite cylinders with optimal fiber prestress, Composites Part B 27B, 643-649, 1996 P. P. Procházka, A. E. Yiakoumi: Nonlinear transformation field analysis of inelastic composite structures, In Proc. Conference on Composites/Nano Engineering, Colorado, Univ. of New Orleans, CD, 2.7.-8.7. 2006. P. P. Procházka, J. Trčková: Coupled Modeling of Concrete Tunnel Lining, In Proc. Our World in Concrete and Structures, Singapore, 2000, 125-132. P. P. Procházka, J. Trčková: Assessment and Control of Tunnel Structures based on coupled modeling. Submitted to Géotechnique. P. P. Procházka: Shape optimal design using Inverse Variational Principles. Submitted to EABE.



527

Elastic-plastic simulation of plate with a blunt slit subjected to uni-axial tension S. Ohtaki1, S. Kobayashi2 & T. Yamamoto3 1

Department of Mechanical Systems Engineering, Hokkaido Institute of Technology, Japan 2 Actis Co., Japan 3 J-tec Co., Japan

Abstract By the progress of a computer, the elastic-plastic simulation becomes easier under the mathematical assumption. However, results of the numerical analysis of the elastic-plastic problem depend on the assumption of yield conditions, definition of yield points and approximation of the stress-strain diagram of the material and other factors. The best way to testify the precision is to compare the numerical results with the experimental ones. In this paper, the authors study a numerical simulation of the elastic-plastic problem under plane stress by the finite element method. Calculation is executed by using FEM software LUSAS. The plastic theory is based on the strain incremental one, and Prandtl-Reuss equations are used. As a yield condition, von Mises’ one is adopted. The definition of the yield stress is determined by the proportional limit. Three approximations of stress-strain diagrams are selected. To verify the numerical results, an experiment is conducted under the same condition by the photoelastic coating method. However, it is not easy to evaluate the stress distributions from the isochromatic fringes. A numerical example of thin plate with a blunt slit subjected to uni-axial tension is presented. Stress distributions in the minimum cross-section and stress contours are presented and examined. Keywords: structural analysis, finite element method, elastic-plastic problem, stress concentration.



1

Introduction

Machines and structures are usually composed of various kinds of metals. Stress concentration due to the existence of cracks or notches in metals causes fracture or failure of the structures. Therefore, it is important to investigate the elasticplastic behavior of metals. The finite element method is one of the most effective tools to deal with the numerical simulation of elastic-plastic problem. Various mathematical methods in the theory of plasticity have been proposed by many researchers [1–3]. Also, Ellyin and Wu analyzed the stationary crack simulation under the cyclic loading containing the effect by the over- loading cycle using their constitutive equations [4]. In this paper, simulation of elastic-plastic problem is studied. To evaluate the validity of the numerical simulation of the elastic-plastic problem by the finite element method, it is desirable to compare the result with an experimental one by the photoelastic coating method (PCM) under the same dimensions and conditions [5,6]. Important factors in the elastic-plastic calculation are the definition of the yield points and the assumption of stress-strain curves [7]. For example, simple prediction of a stress-strain diagram is an approximation with two bi-linear lines, taking the work-hardening ratio constant value during the plastic deformation. There are several ways of the definition of yield points. Numerical simulations are conducted for three cases subjected to the tensile load. The software called LUSAS is used for the numerical calculation so as to consider the convenience and visualization [8]. The incremental step-by-step calculation is adopted in the convergent process of Newton-Raphson method. Numerical example is presented for a rectangular thin plate with a blunt slit subjected to uni-axial tensile load. Stresses and strains in the minimum crosssection and contour lines are shown. The contour lines called quasi-isochromatic fringes by the finite element model are compared with isochromatic ones obtained by the PCM experiment. And the reinforcing effect of the PCM is also simulated by using three-dimensional solid elements.

2

Elastic-plastic theory

2.1 Yield function Considering the second and the third invariants of the deviatoric stress tensor

J 2' and J 3' , the yield condition f for the isotropic materials is written as the next equation. f = F ( J 2' , J 3' )

(1)

Where those two invariants can be expressed using the tensor notation as, J 2' = σ ij' σ ij' / 2 J 3' = σ ij' σ ij' σ ij' / 3

(2)

Assuming that the third invariants J 3' does not affect the yielding behavior, the von Mises yield condition is adopted and it is given as, WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


f = 6 j2' − 2σ 2 2 2 = (σ x − σ y ) 2 + (σ y − σ z ) 2 + (σ z − σ x ) 2 + 6(τ yz + τ zy + τ xy ) − 2σ Y2

529 (3)

Under the plane stress, the yield surface is depicted with an elliptical form. 2.2 Constitutive equations The relations between stress and strain increments under the plane stress are written as In the elastic region

{dσ } = [D e ]{ε } where

(4)

{dσ } = {

σ x σ y τ xy

}

(5)

{dε } = {

ε x ε y γ xy

}

(6)

[D ] = 1 −Eν

T

T

1 ν

0 1 −ν 1 2

e

2

ν

(7)

In the plastic region

{dσ } = [D p ]{ε }

(8)

where

[D ] P

 E S2 − 1  2 S 1 − ν  =    SYM . 

SS νE − 1 2 S 1 −ν 2 S2 E − 2 2 S 1 −ν

 S1 S 6  S   S S − 2 6  S  2 S  E − 6 2(1 + ν ) S  −

(9)

where S=

4 2 ' σ H + S1σ x' + S 2σ y' + 2 S6τ xy 9 E S1 = σ x' + νσ 'y 1 −ν 2 E S2 = σ y' + νσ x' 1 −ν 2 E S6 = τ xy 1 +ν

(10)

(

)

(11)

(

)

(12) (13)

where, E, ν , H ' and σ i' are Young’s modulus, Poisson’s ratio, work-hardening ratio and deviatoric stresses, respectively.



3

Finite element calculation

3.1 Configuration Numerical example of a thin plate with a blunt slit under the plane stress has been calculated by the finite element method. Originally, the shape of the model is the same as the specimen conducted by the photoelastic coating method. The material used for the PCM experiment is made of aluminum alloy (A5052PH34) and the mechanical properties are shown in Table 1. Yield surfaces under the plane stress are depicted in Fig.1. The shape of the specimen and its dimensions are shown in Fig. 2. Without taking into account the geometrical symmetric condition, whole specimen is divided into the finite element model as shown in Fig. 3 using auto-mesh function. The total number of elements is 16176, and that of nodal points is 14680. Table 1:

H34

Mechanical properties and yield stress. YOUNG'S POISSON'S MODULUS RATIO GPa 72.7 0.29 σ ２/ σ

YIELD STRESS MPa 120

Y

σ １/ σ

Y

Tr esca von Mi ses

Figure 1:

von Mises and Tresca yield surfaces.

3.2 Yield points and stress-strain diagram There are many definitions of yield points. We used two kinds of definitions. One is the proportional limit and the other is the intersection between two straight lines. In the elastic-plastic calculation, there are some approximate ways to predict the stress-strain diagram. One of the simplest ways is to approximate with bi-linear lines. The precise prediction of the stress-strain diagram is to use multi-linear lines to express the gradual change of work-hardening ratio H’. In the calculation of the plastic region, we deal with three cases. In the Case 1, the WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


531

yield point is the proportional limit and work-hardening ratio is taken as a constant. In the Case 2, the yield point is defined with intersection of the two extrapolated lines and H’ is constant. In the Case 3, stress-strain curve is approximated with twelve straight lines. Stress-strain diagrams for three cases are shown in Fig. 4.

2

3 60

1

10

50

Figure 2:

Rectangular specimen with a blunt slit.

16176 elements, 14680 nodes Figure 3:

Finite element model.



Figure 4:

4

Stress-strain diagrams of the aluminum alloy.

Numerical results

The stress component σ y of the tensile direction in the minimum cross-section for the Case 1 is presented in Fig. 5 for several loading steps. Similarly,

σ y for

the Case 2 is shown in Fig. 6. In the Cases 1 and 2, as the work-hardening ratios H’ are assumed to be constant, the maximum stresses near the tip of the slit are constant too. 250

250 5880[N]

7840[N]

9800[N]

11760[N]

13720[N]

3920[N] 7840[N] 11760[N]

200

5880[N] 9800[N] 13720[N]

stress[MPa]

strese[MPa]

200

3920[N]

150

150

100

100

50

50

0

0 0

Figure 5:

5

σy

10 distance[mm]

15

20

in the x-axis (Case 1).

0

Figure 6:

5

σy

10 15 distance[mm]

20

in the x-axis (Case 2).



Figure 7:

Quasi-isochromatic fringe by FEM (Py=11760 N).

533

Figure 8: Isochromatic fringe by PCM (Py=11760 N).

The difference of definition of the yield points makes significant difference in the magnitude of stress distribution. In the case 3, the increase of the change of σ y in the plastic range is evident. From this fact, the numerical calculation had conducted using the stress-strain diagram of the case 3.

Figure 9:

Equivalent stress Mises).

σ

without coating in case of Py=8829Ｎ (Von



Figure 10:

Equivalent stress Mises).

σ

with coating in case of Py=8829Ｎ (Von

In the calculation with LUSAS, the von Mises yield criterion is adopted. Therefore, we can easily obtain the contour lines of equivalent stress. To compare the results by PCM experiment, Tresca’s quasi-isochromatic lines that are proportional to the difference of two principal stresses are calculated and shown in Fig. 7 for the case of Py=11760 N. The isochromatic fringes conducted by the PCM experiment are presented in Fig. 8 for the same loading step. The tendency of these results is roughly similar to each other, though quantitative analysis is not adequate yet due to the limitation of the maximum number of color values of the software. Thus, without complicated calculation of stress and strains of PCM experimental procedure, the validity of the results by the finite element can be assured by the comparison of these two Figs. 7 and 8. It is known that if the stiffness of the polymer coating which is adhered on the surface of the specimen in the PCM is larger compared with the surface metal, the reinforcing effect is not negligible. We have calculated the reinforcing effect with three-dimensional solid elements taking account of the stiffness in the thickness direction of the aluminum plate and polymer. The contour lines of equivalent stress σ without coating in case of Py=8829 N are shown in Fig. 9. Similarly, contour lines of σ with coating are shown in Fig.10. However, from these figures, the comparison of the quantities of the equivalent stress is not clear because of the difference of the color scale. By plotting the equivalent stress in the minimum cross-section, the reinforcing effect is calculated as 6 percent.

5 Conclusion In this paper, the finite element simulation has been conducted for the thin plate with a blunt slit subjected to uni-axial tension. Three cases had been calculated WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


535

with different definition of yield points and the approximation of stress-strain diagrams in the proportional tensile load. The stress distribution in the minimum cross-section and stress contour lines are obtained.

(1) The definition of yield point affects the magnitude of the distribution of stress state in plastic range greatly. The best approximation of stressstrain diagram is to use the proportional limit as the yield point and multi-linear work-hardening ratios of case 3. (2) The evolution of elastic-plastic boundaries is roughly similar to those obtained by the photoelastic coating experiment. However, quantitative analysis is not adequate in the present method. (3) The reinforcing effect of the polymer of the PCM experiment is simulated by three dimensional solid elements. It is found that the reinforced effect is about 6 percent which coincide with theoretical value.

References [1] [2] [3] [4] [5] [6] [7] [8]

Marcal, P.V. and King, I.P., Elastic-plastic analysis of two-dimensional stress by the finite element method, International Journal of Mechanical Science, 9-3, pp. 143-155, 1967. Zienkiwitcz, O.C., The finite element method, Third Ed. McGraw-Hill, 1977. Owen, D.R.J. and Hinton, E., Finite element in Plasticity (Theory and Practice), Pineridge Press Limited, Swansea, UK, 1980. Ellyin, F. and Wu, J., Elastic-plastic analysis of a stationary crack under cyclic loading and effect of over load, International Journal of Fracture, 56, pp. 189-208, 1992. Zandman, F. et al., Photoelastic Coating, ISU Press & SEM, 1977. Theocaris, P.S., Experimental Mechanics, 3-9, 1963. Ikegami, K., A Historical Perspective of the Experimental Study of Subsequent Yield Surface for Metal-Part 1.2, British Industrial & Scientific International Translation Service, pp.1-33, 1976. LUSAS theory manual Version 13， FEA Ltd., UK, 2000.




537

A simplified shear strength evaluation model for reinforced concrete corbels J. K. Lu1, S. Y. Kuo2, J. Y. Lin1 & S. H. Hsu3 1

Department of Civil Engineering, National Pingtung University of Science and Technology, Taiwan ROC 2 Department of Civil Engineering, De Lin Institute of Technology, Taiwan ROC 3 New Construction Office, Taipei City Government, Taiwan ROC

Abstract A shear strength model for reinforced concrete corbels is developed by modifying the softened strut-and-tie model. Using the concept of Lu and Wu, the constitutive equation is developed. Then, the constitutive equation is combined with the equilibrium equations of Hwang and Lee. The model is used to predict the shear strength of reinforced concrete corbels. The results of the model are compared with the experimental data and the results of ACI code. Keywords: softened strut-and-tie model, reinforced concrete corbels, shear strength.

1

Introduction

Corbels are cantilevers having shear span-to-depth ratios not greater than unity, which tend to control by shear as a deep beam, rather than flexural members. Reinforced concrete corbels are becoming a common feature in building construction. Simple design procedures to produce safe and economic corbels are therefore required. Much research is carried into investigation about the shear strength of corbels. In test, the most common modes of failure of corbels are crushing or splitting of the compression strut. ACI 318-95 [3] used the shear-friction method to calculate the shear capacities of corbels. It is shown that most of the shear capacities of corbels calculated by ACI code are underestimated [4–6]. ACI 318-02 [7] provides the strut-and-tie model as an alternate method to WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070541

538 Computational Methods and Experimental Measurements XIII calculate the shear capacities of corbels. A softened strut-and-tie model for predicting the shear strengths of reinforced concrete corbels is proposed by Hwang et al [4]. It is based on the strut-and-tie concept with satisfying equilibrium, compatibility and the constitutive law of concrete. Hwang and Lee [2] modified the softened strut-and-tie model to develop a simple predicting procedure. But, the procedure is still complicate. To further simplify the design procedure, the present paper developed a shear strength model for reinforced concrete corbels. Using the concept of Lu and Wu [1], the constitutive equation is developed. Then, the constitutive equation is applied with the equilibrium equations of Hwang and Lee [2]. The results of the present model are compared with the results of ACI code and the experimental data from published papers [6, 8–11].

2

The ACI model

The design procedure for corbels given by section 11.9 in ACI-02 [7] code is based on the shear-friction method. The shear capacity of the corbel Vn is obtained by:

Vn = µAvf f y

(1)

Vn ≤ 0.2 f c' bw d

(2)

Vn ≤ 5.6bw d

(3)

with

and

where Avf represents the area of shear-friction reinforcement; µ is coefficient of friction; bw is the web width and d is the distance from extreme compression fiber to centroid of tension reinforcement; f y represents the yield strength of '

reinforcement and f c represents the compressive strength of concrete.

3

The present model

In this paper, a simplified shear strength model for reinforced concrete corbels is developed by modifying the softened strut-and-tie model. The constitutive equation is developed by using the concept of Lu and Wu [1]. The diagonal compression stress is defined in terms of the yield strength of reinforced steel and the compression strength of concrete. Then, the constitutive equation is combined with the equilibrium equations of Hwang and Lee [2]. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


539

3.1 Equilibrium equations The force equilibrium of a diagonal compression using a strut-and-tie perspective is shown in figure 1.

(a)

(b) (Hwang et al [4]) Figure 1:

The softened strut-and-tie model of corbel.


540 Computational Methods and Experimental Measurements XIII The vertical shear force Vv can be obtained by:

Vv = C d sin θ

(4)

where Cd represents the diagonal compression and the θ denotes the angle of inclination of the diagonal with respect to the compression horizontal axis. Also, the horizontal shear force Vh can be obtained by:

Vh = C d cosθ

(5)

By using the equation (4) and (5), following relationship can be obtained by:

Vv A v = = tan θ Vh A h

(6)

Then, the angle of inclination θ can be obtained:

 Av Ah

θ = tan −1 

  

(7)

where lv and lh represent the internal level arms of the vertical and horizontal shear couples, respectively. According to linear bending theory, the lv and lh can be obtained as:

lh = a

(8)

l v = jd

(9)

and

with the lever arm jd can be estimated as:

jd = d −

kd 3

(10)

where a presents the shear span and the kd represents the depth of the compression zone at the section with the coefficient k.



541

The coefficient k is chosen as 0.375 [12] in this paper. Equation (10) can be expressed as:

jd = 0.875d

(11)

Substituting equations (8) and (11) into equation (7), we obtain

 0.875d    a 

θ = tan −1 

(12)

3.2 Constitutive laws The constitutive equation of concrete used by Hwang and Lee [2] is:

C d = Kζ f c' Astr

(13)

where K is the strut-and-tie index, ζ denotes softening coefficient, Astr is the effective area of the diagonal strut. Equation (13) is first suggested by Zhang and Jirsa [13]. The effective area of the diagonal strut Astr used by Hwang and Lee [2] is:

Astr = a s bs

(14)

with

a s = kd

(15)

where a s represents the depth of the diagonal strut, bs represents the width of the diagonal strut which can be taken as width of the corbel. The coefficient k is chosen as 0.375 [12] in this paper. Equation (15) can be expressed as:

a s = 0 .375 d

(16)

Substituting equation (17) into equation (15), we obtain:

Astr = 0.375dbs

(17)

Instead of equation (13), the compression strength Cd is redefined in present paper as:

C d = σ d Astr WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

(18)

542 Computational Methods and Experimental Measurements XIII where σd represents the diagonal compression stress. In this paper, the diagonal compression stress σd is defined in terms of the yield strength of reinforced steel and the compression strength of concrete. Based on the concept of Lu and Wu [1], the σd is defined by this paper as:

σd = f1 (ρh , f yh ) + f 2 (f 'c )

(19)

where ρ h represents the ratio of horizontal reinforced steel and f yh represents the yield strength of horizontal reinforced steel. The first terms of equation (18), f1 (ρh , f yh ) present the effect of reinforced steel and the second term, f 2 (f 'c ) presents the effect of the compressive strength of concrete. In this paper, f1 (ρh , f yh ) and f 2 (f 'c ) are determined by fitting the proposed model to the experimental data of Foster, Powell and Selim [9]. It is based on that the average of ratio of the calculated shear strength Vv to experimental result Vt is 1.00. There are:

f1 (ρh , f yh ) =0.59 ρh f yh

(20)

and

f 2 (f 'c ) = 0.53f c' − 0.00143(f c' ) 2 with

f c' in MPa. Then, equation (19) can be expressed as: σd = 0.59ρh f yh + 0.53f c' − 0.00143(f c' ) 2

4

(21)

(22)

The procedure of numerical calculation

The procedure of determining the shear force corresponding to the reinforced concrete corbels is now further described. In this application, the

ρ h , f c' , f yv

a , bs, d, ρ v ,

and f yh are known. The procedure of the calculation for the

shear force case is shown in figure 2 and described as follows: Step 1. Calculate Astr by use of equation (18), Astr = 0.375dbs Step 2. Calculate θ by use of equation (12),

 0.875d    a 

θ = tan −1 



543

Step 3. Calculate σd by use of equation (22),

σd = 0.59ρh f yh + 0.53f c' − 0.00143(f c' ) 2 Step 4. Calculate Cd by use of equation (14), C d = σ d Astr Step 5. Calculate Vv by use of equation (4), Vv = C d sin θ

Figure 2:

Comparison of the theoretical results with experimental data.



5

Comparison of theoretical and experimental results

A total of 105 experimental data from the literature are used to compare with the present model. The experimental data are from Foster et al [9], Lin [11], Fattuhi and Hughes [8], Yong and Balaguru [6], and Fattuhi [10]. The calculated shear strength of present model and ACI code Vv are compared with the experimental results Vt. Figure 2(a) shows the theoretical results and the experimental data of Lin [11], Fattuhi and Hughes [8], and Fattuhi [10]. Figure 2(b) shows the theoretical results and the experimental data of Foster et al [9] and Yong and Balaguru [6]. It is seen that the ACI code are under estimate and the present model is quite satisfactory. Table 1:

The theoretical results and experimental data.

Author

Number of specimen

Foster et al [9]

22

Lin [11]

24

Ali and White [5]

11

Yong and Balaguru [6]

11

Fattuhi and Hughes [8]

37

Total

Vv Vt

105

ACI code AVG COV AVG COV AVG COV AVG COV AVG COV AVG COV

0.52 0.31 0.46 0.22 0.48 0.12 0.21 0.23 0.84 0.20 0.580 0.457

Present model 1.00 0.23 0.90 0.20 1.02 0.07 1.05 0.12 0.87 0.18 0.917 0.180

The average (AVG) and the coefficient of variation (COV) of the ratio of the calculated shear strength Vv to experimental result Vt is calculated and shown in table 1. It is noted that the AVG of the present model and ACI code, with respected to total experimental data, are 0.917 and 0.580, respectively. It is shown that the shear strength calculated from present model is closer to the experimental data than the shear strength calculated from ACI code. The COV of present model and ACI code are 0.180 and 0.457, respectively. It is shown that the calculated results from present model are more stable than the results from ACI code.



6

545

Concluding remarks

In this paper, a shear strength model for reinforced concrete corbels is developed by modifying the softened strut-and-tie model. Using the concept of Lu and Wu [1], the constitutive equation is developed. Then, the constitutive equation is combined with the equilibrium equations of Hwang, and Lee [2]. Present model are used to predict the shear strength of reinforced concrete corbel. The results of present model are compared with the experimental data and the results of ACI code. It is seen that present model are compares quite well.

Acknowledgement The authors would like to thank the financial support provided by the National Science Council, Taiwan.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

Lu, J.K., & Wu, W.H., The application of soften truss model with plastic approach for reinforced concrete beams, Structural Engineering and Mechanics, An International Journal, 11(4), pp. 393-406, 2001. Hwang, S.J., & Lee, H.J., Strength prediction for discontinuity regions by softened strut-and-tie model, Journal of Structural Engineering, ASCE, 128(12), pp. 1519-1526, 2002. ACI Committee 318, Building Code Requirements for Structural Concrete (ACI 318-95) and Commentary (ACI 318R-95), American Concrete Institute, Farmington Hills, Michigan, USA, 1995. Hwang, S.J., Lu, W.Y., & Lee, H.J., Shear strength prediction for reinforced concrete corbels, ACI Structural Journal, 97(4), pp. 543-552, 2000. Ali, M. A. & White, R. N., Consideration of compression stress bulging and strut degradation in truss modeling of ductile and brittle corbels, Engineering Structures, 23, pp. 240–249, 2001. Yong, Y. K. & Balaguru, P., Behavior of reinforced high-strengthconcrete corbels, Journal of Structural Engineering, ASCE, 120(4), pp. 1182-1201, 1994. ACI Committee 318, Building Code Requirements for Structural Concrete (ACI 318-02) and Commentary (ACI 318R-02), American Concrete Institute, Farmington Hills, Michigan, USA, 2002. Fattuhi, N.I., & Hughes, B.P., Ductility of reinforced concrete corbels containing either steel fibers or stirrups, ACI Structural Journal, 86(6), pp. 644-651, 1989. Foster, S.J., Powell, R.E., & Selim, H.S., Performance of high-strength concrete corbels, ACI Structural Journal, 93(5), pp. 555-563, 1996. Fattuhi, N.I., Strength of FRP corbels in flexure, Journal of Structural Engineering, ASCE, 120(2), pp. 360-377, 1994. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Lin, R.-D., Behavior of Reinforced Concrete Corbels, Master thesis, Department of construction engineering, National Taiwan University of Science and Technology, Taiwan, 2002. Nilson, A.H., Design of Concrete Structure. 11th Ed, McGraw-Hill, NY, USA, pp. 146, 1997. Zhang, L. & Jirsa, J. O., A study of shear behavior of reinforced concrete beam-column joints, PMFSEL Rep. No. 82-1, Department of Civil Engineering, University of Texas, Austin, TX, USA, 1982.



547

Post-buckling behaviour of a slender beam in a circular tube, under axial load M. Gh. Munteanu1 & A. Barraco2 1 2

Transilvania University of Brasov, Romania Ecole Nationale Supérieure d'Arts et Métiers ENSAM), Paris, France

Abstract This paper deals with the study of the behaviour of a slender beam introduced in a cylindrical tube and subjected to an axial compressive force. The beam is very long compared to its transversal dimensions and therefore it will buckle to a very small axial force. The post-buckling behaviour is examined. The study has important applications in the petroleum industry, for coiled tubing in the case of drilling in horizontal or inclined wellbores. The slender beam has a constant cross-section that can have any form, although a circular cross-section is the most used in practice. The problem has a geometrical non-linearity to which the non-linearity caused by the friction has to be added. Rotations could be large and a special isoparametric 3D beam finite element is elaborated: the EulerRodrigues quaternion was preferred to describe the finite cross-section rotations. The paper presents only the static case, but extending the presented approach to dynamic analysis is quite natural. The method is very accurate and it is rapidly convergent due to the fact that the exact equations, written for the deformed configuration, are solved. The iterative Newton-Raphson method was used to solve the nonlinear differential equations. Keywords: coiled tubing, finite element method, post-buckling behaviour, Euler quaternion, geometrical non-linearity.

1

Introduction

A long slender initially straight beam compressed and constrained within a circular cylinder is studied in this work. This problem presents a great interest in rock engineering and petroleum production. The beam buckles within the narrow space of a drill hole under the action of axial force and its own weight. Due to its WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070551

548 Computational Methods and Experimental Measurements XIII length, the beam will first buckle according to the classic Euler formula, a sinusoidal buckling, when the displacements are considered infinitely small. The post-buckling behaviour is much more important because the beam lies on the cylinder and friction will occur that diminishes the axial load transmission. The problem is highly complicated because it involves a geometrical non-linearity to which the friction has to be added. Several theoretical models were proposed, all of them based on analytical models, some of them complicated, [1, 2], or based on simplified (or even very simple) models using in general a variational approach, [3–12]. Experimental measurements were performed as well: some works are based only on experiments, [13, 14], while in other works the experiments were meant to verify the theoretical models, [1, 2, 4, 11, 12]. Almost all theoretical models assume that post-buckling shape is a helix that is not always true. Some of them consider even a constant pitch helix that it is obviously not exact mainly when the friction is taken into consideration, [4, 12]. In this work a numerical method based on finite element method is proposed. A special type of finite element is presented, based on some previous works, [15–17]. This finite element type is specially conceived for geometrical nonlinear beam systems and uses Euler-Rodrigues quaternion and the axial strain as generalized coordinates. It allows a very exact and fast convergent numerical model to be elaborated. The beam is initially straight and lies along a generatrix; it lies always on the cylindrical surface even after deformation due to the axial force. Two boundary conditions were examined, clamped-free beam and double hinged beam; significantly different results were obtained for these two cases.

2

Beam theory

Because of beam length, the Euler-Bernoulli beam theory is applied. The beam is considered initially straight and it has a constant cross-section. The first variation of the total potential energy has the expression:

δΠ = δU + δV = EA∫ δε 0 Τ ε 0 ds + ∫ δκ Τ Dκ ds −  ∫ δu T pds + ∫ δϕ mds  = 0. l l

l l  δU

(1)

δV

where U is the deformation energy, V is the potential of the external load, u is the displacement vector, ϕ represents the small rotation vector, p and m are the distributed force vector and the distributed moment vector, respectively, acting on the beam. Efforts and strains are linked by the relations (Hooke’s low):

N = EA ε 0 , M = Dκ ,

(2)

where N is the axial effort, E is the Young modulus, A the cross-section area and ε0 is the axial deformation of the beam centreline; the moment vector M and the curvature κ have following expressions in the local reference frame:

M = (M 1

M2

M 3 ) , κ = (κ 1 κ 2 κ 3 ) T


(3)


549

Mt is the torque and M2, M3 are the bending moments; κ1 , κ 2 and κ 3 are the torsion and the two curvatures, that is the variation of angles between two local reference frames situated at the infinitesimal distance ds .The Hooke’s low matrix D is  M t  GI t   M = Dκ or  M 2   0 M   0  3 

0 EI 2 0

0  κ 1    0  κ 2  . EI 3  κ 3 

(4)

It, I2 and I3 are second order geometrical properties of the cross-section (I2 and I3 are the principal inertia moments); G is the transversal elasticity modulus. The main difficulty in 3D geometrical non-linear study, in classical incremental formulation, is that the rotations form no longer a vector. This is why in this work the Euler-Rodrigues parameters are used to describe crosssection rotation, [15–17]. They are defined for a finite rotation of angle θ around T the axis defined by the unit vector ν = (ν x ν y ν z ) . The rotation is

θ

θ

and by the vector sin ν grouped in the 2 2 following column matrix, the well-known Euler-Rodrigues quaternion: represented by the scalar l 0 = cos

T

θ θ θ θ  ν x sin ν y sin ν z sin  = (l0 , l1 , l 2 , l3 ) T . l =  cos 2 2 2 2 

(5)

In this work the finite rotation θ transforms the initial infinite small length element ds of the initial straight beam into the final position due to its deformation. The four Euler-Rodrigues parameters must verify the relation: lTl = 1 .

(6)

Knowing the Euler-Rodrigues parameters, the rotation matrix from the initial configuration to the actual one, for each cross-section, has the form:

(

)

2 l02 + l12 − 1 2(l1l2 − l0 l3 ) 2(l1l3 + l0 l2 )   ℜ =  2(l1l2 + l0 l3 ) 2 l02 + l22 − 1 2(l2 l3 − l0 l1 ) .  2(l1l3 − l0 l2 ) 2(l2 l3 + l0 l1 ) 2 l02 + l32 − 1  

(

)

(

)

(7)

To solve the problem, the total potential energy (1) has to be minimized and usually numerical methods are used, finite element method being the most effective. The classical large displacement finite element approach considers as nodal unknowns (nodal generalized coordinates) small increments of nodal displacements and rotations and this approach leads to an extremely timeconsuming incremental formulation. In this work the unknowns are axial deformation ε 0 and the four EulerRodrigues parameters l. They are grouped into the generalized coordinate vector: q = (ε 0

l0

l1 l 2

l3 ) . T


(8)

550 Computational Methods and Experimental Measurements XIII Curvature vector may be easily expressed in Euler-Rodrigues parameters as:

κ = Gl ′ ,

(9)

where l ′ is the space derivative that is with respect to the curvilinear coordinates s. The G matrix has the form:  − l1 G = 2  − l2  − l3

l0

l3

− l3

l0

l2

− l1

− l2  l1  . l0 

(10)

The equations are written in the actual reference frame of the current crosssection and therefore the spatial derivatives are performed with respect of actual curvilinear coordinate S different of the curvilinear coordinate s for the initial configuration. As the small strain hypothesis was considered, we may consider that the length of the beam keep its length during the deformation: dS = (1 + ε 0 )ds ≈ ds . Thus the following symbolic expression may be written: ∂ ∂ ≈ . ∂S ∂s In the local reference frame, virtual small rotation vector is expressed in EulerRodrigues parameters as:

δ Lϕ = Gδl

(11)

δϕ = ℜGδl .

(12)

and in the global reference frame: Finally the displacements vector u of the current point has the form:   2(l02 + l12 ) - 1 1       u( s ) = u0 + (ℜn(1 + ε 0 ) − n)ds = u0 +  2(l1l 2 + l0 l3 ) (1 + ε 0 ) − 0 ds , (13) 0  0 0  2(l1l3 − l0 l 2 )      s

∫

s

∫

where n is the unit vector perpendicular to cross-section. Now there are all the ingredients to solve numerically the problem using the variational formulation (1).

3

Kinematic equations

In the case of the beam in contact with the tube, the deformed shape of the beam is given by the equations:  x = x (s )   y = R cosϕ (s ) .  z = R sin ϕ (s ) 

(14)

The following relations exists between angles θ, ϕ (see figure 1) and the abscise x (for small axial strain ε 0 ): WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


dx = sin θ ds cosθds dϕ = R

or

dx ds , dϕ cosθ = ds R

551

sin θ =

or

(15)

R being the radius of the tube, figure 1. Therefore the director parameters of an infinite small segment ds of the beam centre-line are: dx = sin θ , ds

dy = − R cosθ sin ϕ , ds

dz = R cosθ cosϕ , ds

(16)

where the angles θ and ϕ result from figure 1.That must coincide to the first column of the matrix ℜ given by (7): 1 + sin θ  2 2  l0 + l1 = 2  cosθ sin ϕ . l l l l + = − 12 0 3 2   l1l3 − l0 l 2 = cosθ cosϕ  2

(17)

It is easy to show that the Euler-Rodrigues parameters satisfying equations (17), satisfy also equation (6). The pipe (beam)

θ

R

ds

z

x

ϕ y

dx The tube Figure 1:

Definition of angles θ and ϕ.

The system resulting from (1) is nonlinear and the iterative Newton-Raphson method will be applied, that is at each iteration the system is solved in unknown increments. Equations (17) become in increments:   l0   l3  - l 2 

l1

l2

l3

l2

l1

l0

l3

- l0

l1

0 1 cosθ cosϕ 2 1 cosθ sin ϕ 2

δl0  0     0  δl1  0  δl  0 1 − sin θ sin ϕ   2  =   , 2  δl3  0 1 sin θ cosϕ  δϕ  0 2      δθ  0

to which the incremental form of the second equation (15) is added: WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

(18)

552 Computational Methods and Experimental Measurements XIII s

s

sin θδθ (1 + ε 0 )ds + ∫ cosθ δε 0ds . R R 0 0

δϕ = − ∫

(19)

Equations (18) and (19) represent the relations between the increments of EulerRodrigues parameters and the increments of the two angles θ and ϕ. There are 4 equations, linking 7 unknown functions, ε 0 , l0 , l1 , l2 , l3 , θ , ϕ , imposing the deformed beam to lie on the internal surface of a tube of radius R. Therefore only 3 independent generalized coordinates were chosen: T qˆ = (ε 0 l1 θ ) . (20)

4

Friction forces

Between the beam and the lateral surface of the tube there are reaction forces. Moment equilibrium equations for a slice of the beam, written in the local reference frame of each cross-section, is: ∂M ∂ (κ ) + n × R = 0 or D + κ × (D κ ) + n × R = 0 . ∂s ∂s

(21)

After each iteration, the generalized coordinates qˆ are known and the EulerRodrigues parameters are known as well; that allow the curvatures κ to be found. The axial effort N and the two shear efforts Q2 and Q3 (the components of the vector R) are computed using equation (21). Finally, from the force equilibrium equation written in the cross section local reference frame: ∂R +κ × R + p = 0 ∂s

(22)

it results the contact pressure p between the beam and the tube. Distributed force p will be represented in the cylindrical reference frame. Axes of the cylindrical reference frame are: 1 – x direction of the global frame (see figure 1), 2- normal direction and 3 – tangent direction in the yz global plane. The normal distributed force is p2 . The friction force cannot exceed the value µ p2 , where µ is the friction coefficient. If the tangential force is smaller than µ p2 , there is no relative motion between the points in contact belonging to the beam and to the tube. In any point the following condition must be respected: ~ p12 + ~ p 32 ≤ µ~ p2 ,

(23)

otherwise the “link” is broken and a relative motion is allowed. At each load step few iterations are needed in order to establish which among all nodes are moving and which are not because of the friction. The tube is compressed by an axial force with the condition that the tube always lies on the surface of the tube. In the absence of the friction the method is not incremental and the solution may be find directly for the final load of the load. But in the presence of the friction the approach must be incremental, the result depending on the history of the force variation. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Figure 2:

5

553

Two-node and three-node finite elements.

Special finite element

To solve the problem, finite element method was applied. Isoparametric twonode and three-node finite elements were used, figure 2. This finite element is described in detail in [15–17]. Each node has five degrees of freedom per node accordingly to (8). For the three-node finite element the shape function is:

 s2 s2 s s  s2  qk,i-1 + ( − 2 + 1) qk ,i + ( 2 + ) qk ,i +1 , k = 1...5 . (24) qk ( s ) =  2 − 2h  2h 2h h  2h The beam is divided in several finite elements and the load is applied incrementally. Finally a non-linear differential system is obtained from functional (1) using a procedure similar to the standard one:

E (q ) = K (q )q − Q = 0 ,

(25)

where K is the secant stiffness matrix that is function of q, obtained by assembling elemental stiffness matrices; q is the vector of the generalized nodal unknown (8) for the whole structure and Q is the vector of nodal loads. System (25) includes the equation (6) for each node. It is interesting to note that all elements of secant stiffness matrix K are polynomials in Euler-Rodrigues l and strain ε 0 . To solve the non-linear system, the iterative Newton-Raphson method is applied. At each step, at each iteration, equations (18) and (19) are used in order to replace nodal degrees of freedom (8) with (20), to impose the beam to lie on the cylindrical surface of the tube. In the load step i and for the k-th iteration it can be written: ˆ (k )δqˆ (k +1) + Eˆ (k ) = 0 , K i t

(26)

where: ˆ C , E = CE ˆ, q = Cqˆ , K t = C T K t

(27)

C being a square matrix provided by applying of equations (18) and (19) for all nodes of the discretization and δqˆ i(k +1) = qˆ i(k +1) − qˆ i(k ) . Kt is the tangent stiffness modulus, in fact the Jacobean J: WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

554 Computational Methods and Experimental Measurements XIII J = Kt =

∂E . ∂q

(28)

The approximate evaluation of the Jacobean may be used, although in the paper the exact computing were performed: Kt ≈ K +

∂Q . ∂q

(29)

Often an under-relaxation factor β could improve the convergence: qˆ i(+k1+ 1 ) = qˆ i(+k1) + βδqˆ i(+k1+ 1 ) .

(30)

Angle θ [o]

z

18 16 14 12 10 8

θ

6 4 2 0

-8 -6 -4 -2

0

y

Axial force [N]

a) Rotation angle θ of the mid node versus axial b) Beam (pipe) view along x force. axis, the beam axis. Figure 3:

6

Post-buckling diagram.

Numerical example

To illustrate the method a MATLAB program was elaborate and the following example was analyzed, [6, 7, 13]: double hinged beam (pipe), length L=21340 mm, outer pipe diameter D=13.72 mm, inner pipe diameter d=6.35 mm, diameter of the tube D0=48.26 mm, Young modulus E=2.07e5 MPa, Poisson coefficient ν=0.3, 30 3-node finite elements, 61 nodes. Figure 3 shows the dependence angle θ versus axial force, dependence found by the computer code. The Euler buckling force is 7.45 N (see figure 3a) and much higher force values were studied. Figure 4 presents the deformation configuration of the pipe under an axial force of 800 N. One can see that the shape is not exactly a helix: the symmetry about the plane perpendicular on the beam, on the mid-node, has to be ensured. On the contrary, figure 5 shows the unsymmetrical deformed configuration of a clamped-free beam. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


a) Isometric view (view direction: [1; 0.006; 0.003]) Figure 4:

555

b) View along x axis, the hole axis.

Post-buckling shape of a double hinged beam; compression force 800 N. Clamped end

Free end

View direction: [1; 0.005; 0.005].

Figure 5:

7

Post buckling configuration for a compressed clamped-free beam.

Conclusions

In the case of complex structural problems, numerical methods, especially those based on finite element method, may describe much more exactly the real phenomenon than analytical methods. This is also the case for long and slender beam compressed in a cylinder tube, the topic of the present work. A special finite element type, mainly conceived for geometrical non-linearity of beam systems, was proposed which proved to be fast convergent and exact. Equilibrium equations are exact and written on the deformed configuration of the beam: this is possible only because the generalized nodal coordinates are the elements of Euler-Rodrigues quaternion and the axial strain. In this way the boundary conditions can be described better and a significant influence of boundary conditions was found even for quite long beams (pipes).

References [1]

Sorenson, K.G., Post buckling behavior of a circular rod constrained within circular cylinder, PhD thesis, Rice University, Houston, Texas, 1984. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

556 Computational Methods and Experimental Measurements XIII [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

Chen, Yu-Che, Post buckling behavior of a circular rod constrained within an inclined hole, PhD thesis, Rice University, Houston, Texas, 1988. Miska, A & Cunha, J.C., An analysis of helical Buckling of Tubulars Subjected to axial and torsional Loading in inclined wellbores, Society of Petroleum Engineers, SPE 29460, pp. 173-180, 1995. Tan, X.C. & Forsman, B., Buckling of slender string in cylindrical tube under axial load: experiments and theoretical analysis, Experimental Mechanics, 35(1), pp. 55-60, 1995. Jang Wu & Juvkam-Wold, H.C., Coiled tubing buckling implication in drilling and completing horizontal wells, SPE Drilling & Completion, March, pp. 16-21, 1995. Miska, S., Qiu, W., Volk L. & Cuhna, J.C., An improved analysis of axial force along coiled tubing in inclined, horizontal wellbores, Society of Petroleum Engineers, SPE 37056, pp. 207-214, 1996. Kuru, E., Martinez, A. & Miska, S., The buckling behavior of pipes and its influence on the axial force transfer in directional wells, Society of Petroleum Engineers, SPE 53840, pp. 1-9, 1999. Qiu, W, Force transmission of coiled tubing in horizontal wells, Society of Petroleum Engineers, SPE 54584, pp. 1-7, 1999. Mitchell, R.F., Exact analytic solutions for pipe buckling in vertical and horizontal wells, SPE Journal, December, pp. 35-51, 2002. Mitchell, R.F., Lateral Buckling of pipes with connectors in horizontal wells, SPE Journal, June, pp. 41-52, 2003. Gao, De-Li, Gao, Bao-Kui, A method for calculating tubing behavior in HPHT wells, Journal of Petroleum Science & Engineering, 41, pp. 183188, 2004. Sun, C. & Lukasiewicz, S., A new method on the buckling of a rod in tubing, Journal of Petroleum Science & Engineering, 50, pp.78-82, 2006. Martinez, A., Miska, S., Kuru, E. & Sorem, J., Experimental Evaluation of the Lateral Contact Force in Horizontal Wells, Journal of Energy Resources Technology, 122(3), pp. 123-128, 2000. Kuru, E. Martinez, A., Miska, S. & Qiu, W., The buckling behavior of pipes and its influence on the axial force transfer in directional wells, Journal of Energy Resources Technology, 122(3), pp. 129-135, 2000. Barraco, A. & Munteanu, M.Gh., A special finite element for static and dynamic study of mechanical systems under large motion”, part 1, Revue européenne des éléments finis, Vol. 11(6), pp.773-790, 2002. Munteanu, M.Gh. & Barraco, A., A special finite element for static and dynamic study of mechanical systems under large motion”, part 2, Revue européenne des éléments finis, Vol. 11(6), pp.791-814, 2002. Barraco, A. & Munteanu, M.Gh., Dynamic study of elastic beam systems under large motion, Proc. of European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS, Finland, 24 28 July, published in extensor on CD, 2004.



557

Structural properties of a new material made of waste paper J. Santamaria1, B. Fuller2 & A. Fafitis1 1

Department of Civil and Environmental Engineering, Arizona State University, USA 2 Center for Alternative Building Studies, USA

Abstract Papercrete is the name of a new construction material made basically of waste paper, cement and water. Contractors have started using papercrete to build low cost houses relying on empirical knowledge of its structural properties. The purpose of this study is to obtain some mechanical and physical parameters of papercrete by doing several laboratory tests. The samples tested were made following the most common procedures that papercrete makers are currently using. The experimental setup used to test the samples is briefly described and some test results are presented. Some conclusions and recommendations for the use of papercrete for building houses are also presented. Keywords: waste paper, recycling, sustainability, low-cost housing, greenconstruction.

1

Introduction

The necessity of low-cost housing has pushed people to look for alternative construction materials. “Papercrete” is a relatively new material basically made of waste paper, cement, and water. Papercrete is a slightly misleading name. It seems to imply a mix of paper and concrete, hence papercrete. But more accurately, only the Portland cement part of concrete is used in the mix – along with other admixtures. Although some sand and other additives to improve its behavior under compressive load may be used, the basic components are still the same. According to the Environmental Protection Agency (EPA), the United States recycles about 45% of discarded paper annually, U.S. EPA, 2000 [10]. This WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070561

558 Computational Methods and Experimental Measurements XIII means that about 55% or 48 million tons of paper is thrown away or goes into the landfills. Conservatively speaking, it takes about fifteen trees to make a ton of paper. That means that 720 million trees are used once and then buried in a landfill each year. The volume of the paper material wasted every year is equivalent to the volume of a wall 48 feet high and 1 foot wide, around the entire perimeter of the United States, Fig. 1.

Figure 1:

Waste paper wall around the U.S.A. made of waste paper.

At the same time, in Arizona and many other states, there is a pressing need for affordable, sustainable housing. On Native American lands in Arizona alone, there is an immediate and pressing need for tens of thousands of homes. According to a report by the Arizona Housing Commission, “The urgent overriding message is clear; housing affordability is an impending crisis in Arizona”, Arizona Housing Commission, 2000 [1]. The research and testing on papercrete is very limited or nonexistent. From a scientific standpoint papercrete is, for all practical purposes, invisible. To the best of our knowledge there is no systematic experimental or analytical study of this material. So there are few ways to reference papercrete because there are no antecedents in research - only locations, structures and people who have built them. Some parallel work has been done on fiber cement composites for roofing and siding, Mohr et al. 2004 [5], but to our knowledge, no formal research. The purpose of this research is to determine if papercrete has suitable mechanical and physical properties so that it can be used as a construction material for houses, to find out what anecdotal evidence about it is accurate, and to determine areas of further study, if any. The parameters to be studied are the Young’s Modulus (E), thermal conductivity (K), thermal resistance (R), bond characteristics, and creep behavior.

2

Anecdotal evidence

As part of the search for anecdotal evidence several papercrete structures were visited and practitioners were interviewed. It was found that the locations of the structures represent a cross-section of climate and geography. Many of these structures are standing for years and they are quite attractive, Fig. 2. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Figure 2:

559

Papercrete structures are attractive, and said to have a number of positive properties.

Homes and other buildings of up to 3200 square feet have been built with papercrete for $25 per square foot, not including labor. Finished designer homes made of papercrete have cost $70-$80 per square foot, with labor included. There seems to be no harmful by-products or excessive use of energy in the production of papercrete, Hart 2004 [3]. Current mixes can be produced using electric motors powered by solar energy, McCain 2004 [4]. Since the mortar, exterior stucco and interior plaster used with papercrete is made of the same paper material, much wood and other construction materials used in framing and "building out" the interior and exterior of the home is saved. With an R-Value between 2.0 and 3.0 per inch, U.S. DOE 2002 [9], in walls 12 to16 inches thick, the long-term energy savings of buildings with papercrete is considerable for the homeowner and the environment. Papercrete is reported to have good sound absorption quality. It's also reported to be flame and fungus retardant, as well as bug, rodent and pest resistant, Patterson 2004 [6]. Since it is very light, it would be an ideal interior wall material for high-rise buildings in active seismic areas. Using it in place of concrete would reduce the gauge of steel framework and the depth of foundations - cutting materials, labor and energy costs significantly. Papercrete wall and roof material is, in itself, insulation. It surrounds, supports and insulates the structure's climate control, plumbing, and electrical systems. Communication, security, fire, moisture and structural stress monitors can be run between courses of the material during construction or cut into the walls after completion, Terry 2004 [7]. Vaults and domes made of papercrete have been built, Fig. 3, on walls with no reinforcement of any kind, Curry 2004 [2]. Due to the fact that it is so lightweight and easy to install, labor cost is reduced. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Figure 3:

Papercrete domes and vaults built without interior reinforcement.

Experimental structures, which are still standing, have been built without footings, directly on the ground and on sand bags, McCain 2004 [4]. Others have been built more conventionally on piers and on concrete footings and slabs, Terry 2004 [7]. Walls have been dry stacked, pinned with rebar, and poured in forms on the ground and then tipped up. They also have been poured monolithically, slip formed or used as infill in post and beam construction with a wood or steel framework, McCain 2004 [4]. Papercrete blocks made with sufficient Portland cement are fire retardant. They will not burn with an open flame. The blocks can be made waterproof by using a concrete sealer. In rural communities, with no current recycling services, making papercrete blocks increases the recycling of waste paper and creates new rural jobs and economic opportunities Fig. 4). Small communities, with even a modest waste paper stream, can start a cottage industry based on this material and save waste transportation fees to regional landfills.

Figure 4:

Papercrete house built in a small community.

The economic bonus is that there is no waste. Sawdust, cuttings, spills and broken pieces can be returned to the mixer for re-use, McCain 2004 [4]. The environmental bonus is that there is a sizable savings in trees and landfill space, new uses for other waste materials such as glass, fly ash and Styrofoam, and an estimated 50% to 70% savings in energy to heat and cool a papercrete home during its lifetime. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


3

561

Experimental program

3.1 Compressive test In theses tests an increasing uniaxial compressive load was applied at constant speed, uniformly distributed in order to develop the stress-strain curve and determine the stiffness of the material. The following testing procedure was used for the compression test: Since some samples had irregular faces, they were made flat by using normal commercial mortar cap. The samples were tested under uniaxial compressive force using a 100ton-compression machine, Fig. 5. The loading rate at the displacement control mode was 0.35 in/min, and all samples were loaded up to approximately 10 kips, unloaded, and reloaded to approximately 15 kips.

Figure 5:

Compression test.

Failure was defined by deformation criteria rather than load because the material is not brittle, and it does not exhibit descending branch in the stressstrain curve. It was found that at 15 kips the deformation was excessive, rendering the material useless. 3.2 Pull-out test The purpose of this test was to measure the bond capacity of a material by applying an increasing force to extract a corrugated steel bar that was previously driven. The following testing procedure was used for the Pull-Out Test: The samples were prepared by driving the rebar in the middle of a block of Papercrete. Two different kinds of samples were tested. The sample is subjected to an increasing load in order to pull out the steel bar. The loading rate at the displacement control mode was 0.35 in/min. In the same way as the compression test, failure was defined by deformation criteria.


562 Computational Methods and Experimental Measurements XIII 3.3 Creep test The purpose of this test was to study the behavior of the material when it is subjected to a constant compressive load for a long period of time. The following testing procedure was used for the Creep Test: All creep samples were made by cutting papercrete blocks of approximately 3 in x 3 in x 9 in. The axial deformations were measured by a mechanical gage. Each sample was subjected to a constant load of approximately 300 lb for a period of time of four weeks approximately. It was found that practically all creep had taken place within approximately three weeks. The deformation was measured with 0.001 in. accuracy. PAPERCRETE Stress vs Strain Sample No2 300.00

Stress - psi

250.00

200.00

150.00

E=560 psi

100.00

50.00

E=1200 psi

0.00 0.00

0.10

0.20

0.30

0.40

0.50

Strain - %

Figure 6:

4

Stress vs. Strain graph.

Test results and analysis

4.1 Compressive tests The data collected from the compression tests were used to develop stress-strain curves for each sample. The stiffness or elastic modulus of the material (E) is the slope of the stress-strain curve. A trend line was obtained using Microsoft Excel in order to get the right value of the slope of the curve, Fig. 6. Note that the material is non-linear, and as a result there is no Elastic (Young’s) Modulus. A working Young’s Modulus is an approximate value obtained from the stressstrain curves, and which can be used as an index to characterize the compressive behavior up to some stress. In practice, the allowable compressive stress is WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


563

expected to be at about this level. The softer part of the curve, Fig. 6, is probably due to irregularities of the surfaces of the specimens. The compressive test results of all samples are shown in Table 1. 4.2 Pull-out tests The data collected from five pull-out tests were used to develop a Load vs. Deformation graph for each sample. A typical force-deformation curve is shown in Fig.7. The maximum pull-out force for each specimen is given in Table 2. Table 1: Group 1 1 1 2 2 2 3 3 3 4 4 4 5 5 6 6 6 7 8 8 9 9 9 10 10 10 11 12 13 14 14 15 16 17

Sample 1 2 3 4 5 6 7 8 9 13 14 15 16 17-18 19 20 21 22 23 24 25 26 27 30 33 35 36 37 60 38 39 40 41 42

Papercrete compressive tests.

Material

Proportions

Paper/Portland Paper/Portland Paper/Portland Paper/Port/Sand Paper/Port/Sand Paper/Port/Sand Paper/Port/Fly ash Paper/Port/Fly ash Paper/Port/Fly ash Paper/Port/Styrofoam Paper/Port/Styrofoam Paper/Port/Styrofoam Sludge/Port/Fly Sludge/Port/Fly Paper/Port/Glass Paper/Port/Glass Paper/Port/Glass Paper/Clay Paper/Port/Clay Paper/Port/Clay Paper/Port/Lime Paper/Port/Lime Paper/Port/Lime 1/8 inch grind 3/8 inch grind 5/8 inch grind Clyde T. Curry: Poured 2/13/05 Zach Rabon: Poured 2/05/05 Cardboard Pour 3/17/05 Paper/Port/Fly ash/Sand Paper/Port/Fly ash/Sand SRP Printing Paper Mixed Waste Paper Paper/Port/Fly Ash/Sand

1-1 1-2 1-3 1-1-5gal 1-1-10gal 1-1-15gal 1-.7-.25 1-.6-.30 1-.5-.35 15% Sty 20% Sty 25% Sty 1-.7-.25 1-.6-.30 1-1-5 gal 1-1-10 gal 1-1-15 gal 0 % Portland 1 bag mix 2 bag mix 1-.5-.5 1-1-1 1-1.5-1.5 1-2 1-2 1-2 Per yard Per yard .7 per yard 200 gal 200 gal 0.70 1/2 batch 0.70

Elastic Modulus 600 1200 2000 800 700 590 950 420 400 1200 1430 860 1390 2700 470 570 700 1394 855 1375 400 570 660 1550 2000 1200 1250 3000 220 1200 900 1500 1300 2100

4.3 Creep tests The data collected from the creep test were used to develop a Deformation vs. Time graph for each sample. Atypical curve is shown in Fig. 8. It is apparent that the deformation (creep) behavior under a constant load tends quickly to an WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

564 Computational Methods and Experimental Measurements XIII asymptote. To smoothen the curve, a trend line was applied using Microsoft Excel. Six samples were tested for creep and the results are tabulated in Table 3. Samples No2 and No4 were made using the corresponding mixtures in Table 1.

5

Conclusions and recommendations

Papercrete has wide-ranging implications for home construction and the environment. The challenge of papercrete is the lack of information about it. More research is needed in order to learn more about the material and to its properties. The results of this study indicate that papercrete, properly mixed and applied, should be safe and practical for two-story residential home construction. PULL OUT Load vs Deformation Sample No3 (double block)

700 650 600 550 500

Load - lb

450 400 350

Pmax = 285.3 lb

300 250 200 150 100 50 0 0.00

0.40

0.80

Deformation - in

Figure 7: Table 2:

Load – deformation. Pull out tests.

Group

Sample

Type

1 1 2 2 2

1 2 3 4 5

Single Single Double Double Double

Pmax (lb) 60.4 47.0 285.0 130.1 694.0

5.1 Compressive test When the samples were capped, all of them absorbed a lot of water very quickly. However, no apparent change in the samples after the seven day curing period was observed. During the compression test, the stress-strain curve is WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


565

monotonically increasing and the sample starts packing rather than disintegrating. For that reason, deformation is the criterion for failure. The stressstrain curves suggest that, papercrete is a ductile material that can sustain large deformations. Cement plays an important role in the compressive strength and behavior. Specimens with higher proportion of cement exhibit larger Young’s Modulus. As pointed out, the stress-strain curves exhibit a softer segment at the beginning. This is probably because of the inherent irregularities of the specimens due to shrinkage. It is believed that, in practice (for example in the construction of a wall), the self-weight of the structure will apply a moderate pressure which will bring the stress at the level of the working Young’s Modulus which can be used in design. PAPERCRETE Deformation vs Time Sample No4

0.1000 Dimensions: a=3.0in b=3.0in t=8.75in

Deformation - in

0.0800

Trendline

0.0600

0.0400

0.0200

0.0000 0

2

4

6

8

10

12

14

16

18

20

Time - Days

Figure 8:

Deformation vs. time curve.

Table 3:

Sample No 2.1 2.2 2.3 4.1 4.2 4.3

Contac Area (in²) 9.0 9.0 9.0 9.0 9.0 9.0

Load (lb) 307.5 307.5 307.5 307.5 307.5 307.5

Creep test results.

Stress (psi) 34.17 34.17 34.17 34.17 34.17 34.17

Deformation creep (inch) 0.0108 0.0282 0.0177 0.0774 0.0831 0.0471


due

to

566 Computational Methods and Experimental Measurements XIII 5.2 Pull out test The purpose of this test was to evaluate the pull-out capacity of a driven bar as a fastener. The load-deformation curve is monotonically increasing in a non-linear way until it reaches its maximum. Then it starts decreasing slowly due to the bar wrinkles or folds which prevent the opposite force (friction force) from decreasing drastically. All load-deformation curves exhibit a large number of peaks. This is because the material is broken and packed as the corrugated steel bar is driven by the pulling force. When the material is broken, the pulling load drops, and when the material starts packing, the load increases. The results exhibit considerable scatter. Since the corrugated steel bar was driven into the papercrete blocks by hammering, some factors such as perpendicularity, and packing produced by bar wrinkles or folds may contribute to the scatter. 5.3 Creep test During the creep test, the deformation-time curve is monotonically increasing in a non-linear way until it reaches its maximum. In general, the deformation due to a constant load stopped increasing in less than a month, which is a relative small period of time. It appears that creep depends on the amount of cement, increasing with increasing cement content. However the observed creep was rather small and it does not seem to pose significant structural problems.

References [1] [2] [3] [4] [5]

[6] [7] [8]

Arizona Housing Commission (2000) The State Of Housing in Arizona Retrieved October 6, 2005, from http://www.housingaz.com/UPLOAD/ HsgReprt.pdf Curry, C. Interview. Living In Paper (work in progress) [Videotape] Dir. Barry J. Fuller Bloomington, New Mexico October 4, 2004 Hart, K. Interview. Living In Paper (work in progress) [Videotape] Dir. Barry J. Fuller Crestone, Colorado June 9, 2004 McCain, M. (producer) (1991) Introduction to Fibercrete [Videotape] (Available from: RV Productions: Albuquerque, New Mexico) Mohr, B.J., El-Ashkar, N.H., and Kurtis, K.E. (2004) Fiber-cement composites for housing construction: state of the art review. Proceedings of the NSF Housing Research Agenda Workshop, Feb 12-14, 2004, Orlando, Florida. Eds. Syal, M., Hastak, M., and Mullens, M. Vol 2, Focus Group 2, 112-128 Patterson, E. Interview. Living In Paper (work in progress) [Videotape] Dir. Barry J. Fuller Silver City, New Mexico October 2, 2004 Terry, L. Interview. Living In Paper (work in progress) [Videotape] Dir. Barry J. Fuller Columbus, New Mexico June 7, 2004 Thomas, M. Interview. Living In Paper (work in progress) [Videotape] Dir. Barry J. Fuller Navajo Dam, New Mexico April 14, 2004



[9] [10]

567

U.S. DOE (2002) R-value of papercrete [Email: Arun Vohra to Lex Terry] April 9, 2002 U.S. EPA (2000) Jobs Through Recycling Retrieved March 4, 2005, from http://www.epa.gov/epaoswer/non-hw/recycle/jtr/comm/paper.htm




569

Non-Hertzian rolling contact stress analysis C. H. Liu & W.-E. Hsu Department of Mechanical & Electro-mechanical Engineering, Tamkang University, Tamsui, Taipei Shien, Taiwan, 251

Abstract In a previous study the three-dimensional rolling contact problem under Hertzian pressure has been dealt with. Two new initial guesses for the Newton-Raphson method were proposed, which always lead to convergent solutions for tangential stresses. However, non-Hertzian contact often appears in moderately used contact elements. The present study extends the previous study to cases with non-Hertzian contact. In particular the counter-formal case is treated. In this case the contact region can still be bounded by a plane, although the contact pressure is not Hertzian. The previous numerical algorithm is used to treat counter-formal contacts, and convergent results can always be obtained. Keywords: rolling contact, rail and wheel contact, non-Hertzian contact, computational stress analysis.

1

Introduction

Rolling contact occurs in many mechanical pairs, such as gear and pinion, cam and follower, wheel and rail, and also in ball to ball contacts in bearings. Many analytical as well as numerical techniques have been suggested to solve for rolling contact stresses. Basically these techniques can be grouped into the following three categories: the integral equation method [1–5], the method based on variational principles [6,7], and the mixed method [8–10] that makes use of both of the two preceding methods. Among the various techniques based upon integral equations, the numerical procedure developed by Liu and Paul [5] may determine tangential contact stress distribution for elastically similar bodies in rolling contact. Their iterative procedure showed fast convergence for cases with small spins, but might fail to converge for cases with even moderate values of spin. In a previous study by Liu and Hsu [11], Liu and Paul’s numerical scheme was improved by using two new initial guesses, so that it might also converge WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070571

570 Computational Methods and Experimental Measurements XIII under very large spins. In Liu and Paul’s original study, both Hertzian and nonHertzian contacts were treated, but only cases with Hertzian contact pressure were presented by Liu and Hsu. The purpose of this study, therefore, is to extend this previous study to cases with non-Hertzian contacts. The Hertzian pressure distribution was obtained under the following assumptions: 1. the contact region is very small so that it can be bounded in a plane; 2. in the vicinity of the contact region the two contacting bodies can be approximated by quadratic surfaces; and 3. the surfaces in contact are frictionless. While these are valid assumptions for new contacting surfaces, for moderately used contact elements, however, the above assumptions are generally not valid, and the corresponding contact pressures are non-Hertzian. In wheel and rail contact, although the pressure is non-Hertzian, but the contact region can still be confined on a plane [12]. In this study we assume that the loading is monotonically increasing, and that a fixed ratio of normal to tangential forces is maintained, so that only the final force values are considered. braking moment M

N

rolling direction

y

Mz upper body (body 1)

x

Fx z

: x

lower body (body 2)

rolling direction

Mx

(a)

Figure 1:

Fy

(b)

(a) two bodies in rolling contact; (b) tangential forces on contact region Ω.

2 Kinematic equation of rolling contact When two rolling bodies are pressed by a normal force N, the contact region Ω develops in the xy plane, as Fig. 1 shows. In addition to the normal force N, tangential force (Fx, Fy) and twisting moment Mz are also transmitted in the contact region. If the two bodies in contact are elastically identical, namely, they have the same elastic properties, then the problem can be separated into two parts: the normal problem and the tangential problem [13]. In the normal problem, the contact region Ω and the pressure distribution p(x,y) in the region are determined. Tangential tractions Tx(x,y) and Ty(x,y) in the contact region are then determined in the tangential problem. This separation is valid for elastically identical materials because the tangential tractions imposed later on the contact WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


571

region do not affect the conformity of the two bodies in contact. Hence the normal pressure and the size of the contact region remain the same. In this study we assume the two bodies rolling over each other are elastically identical. The non-Hertzian contact region Ω and the pressure p(x,y) in Ω are given. The purpose is to determine the tangential tractions Tx(x,y) and Ty(x,y) in the contact region Ω. When a braking moment M in the direction of −y is applied to body 1, a tangential force Fx develops in the contact region Ω to oppose the tendency of body 1 to slide over body 2. Likewise, a moment Mx along the x axis creates a tangential Fy in Ω. Figure 1(b) shows Fx and Fy in the contact region on body 2. As a result of Fx and Fy, certain points on the upper body (body 1) may slip over the contacting points on the lower body (body 2). The slippages sx and sy defined by ( s x , s y ) = ( v x1 − v x 2 , v y1 − v y 2 ) / V0 (1) represent the velocities at a point (x, y ) on body 1 relative to the point occupying the same position but on body 2. The normalization constant V0 is the rolling velocity when the two rolling bodies are rigid, that is, when they do not deform. The contact region hence reduces to a single contact point. In the case of gear and pinion pairs, rolling velocity V0 is the velocity of the contact point, and in the case of wheel-rail rolling contact, V0 is the velocity of the center of the rigid wheel. Johnson [1,2] derived the following kinematic equation for a state of steady rolling ∂ (2) ( s x , s y ) = (v x ,ν y ) + φ (− y, x) + 2 (u x , u y ) ∂x This equation shows that slippage (sx , s y) may be separated into three terms. The first term includes two constants, the longitudinal creepage νx and the lateral creepage νy. They represent the change in rolling speeds due to the previous mentioned moments M and Mx. Disregarding temporarily the tangential forces Fx and Fy developed in the contact region so that there becomes no resistance to M and Mx, then body 1’s rolling velocity deviates from its original value (V0 ,0 ). Note that equal but opposite moments −M and −Mx are applied to body 2 as well (not shown in Fig. 1), and a change in rolling speed of body 2 also occurs. Let (δVix , δ Viy) represents the change in rolling velocity of body i (i=1 or 2) from the value (V0 ,0 ), then longitudinal and lateral creepages νx and ν y are defined as (v x ,ν y ) = (δV1x − δV2 x , δV1 y − δV2 y ) V0 (3) The second term of (sx ,s y) is due to the moment Mz applied to body 1, and also the equal but opposite moment −Mz applied to body 2 (not shown in Fig. 1). At the point of application of Mz on body 1, which we assume to be far away from the contact region, body 1 rotates about the z axis with an angular velocity ω 1. Neglecting temporarily the resistant moment −Mz on body 1 so that body 1 can rotate freely, then the linear velocity at the point (x,y) in the contact region on body 1 is ω 1(−y, x). The velocity of this point relative to the same point on body 2 is then φ ( −y, x), where the constant φ is called the relative spin, or simply the spin, defined by the equation WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

572 Computational Methods and Experimental Measurements XIII φ = (ω1 − ω 2 ) V0 (4) In obtaining the first two slip components, namely (νx , ν y) and φ ( −y, x), the resisting forces (Fx , F y) and resisting moment M z were not taken into account. The elastic deformation caused by these forces (moment) was neglected. Let ux and uy denote the elastic displacements produced by (Fx , F y, M z) at a point (x ,y ) in the contact region on body 2. The relative slip velocity due to (ux ,u y) is given by the last term of equation (2).

3

Discretization

The elastic displacements (ux ,u y) in Eq. (2) are caused by the resisting forces Fx, Fy, and Mz, which are the resultants of tangential traction (Tx ,T y) in the contact region. Using principle of superposition, one may express (ux ,u y) as follows u x ( x, y )   f xx f xy  Tx (α , β )  (5)  dα dβ   = ∫∫  f f yy  Ty (α , β ) u y ( x, y ) Ω  yx where fij(x, y, α , β ) are Cerruti functions [14]. For example, fxy represents the displacement at the point (x ,y ), in the x direction, which is produced by a unit force at the point (α , β ), in the y direction. The contact region is first replaced by a number of horizontal strips, as Fig. 2(a) shows, and each strip is further divided into several rectangular cells. Hence the contact region is discretized into n rectangular cells. The sub-region of Ω where relative slip occurs is called the slip zone, denoted by the symbol ΩS. The remaining region of Ω where relative slip is prohibited is called the stick zone (or locked zone), and we denote it by ΩL. Equation (2) is valid at each cell center, but instead of dealing with this equation directly, Liu and Paul [5], Hsu [15], and also Liu and Hsu [11] integrated Eq. (2) with respect to x to remove the partial derivative. Following the procedure suggested by Liu and Hsu, we integrate both sides of Eq. (2) from the center of the i'th cell (xi , y i), to a point (xU ,y i), also making use of Eq. (5), obtaining v ( x − xi ) − φyi ( xU − xi ) x n Txj   hxx hxy  s xj   x U  2 2 2 + ∆x j   =   x x ( ) φ −  dαdβ   (6) h ∑ ∑ ∫∫ U i s h Tyj  j = xi j =1 Ω  yx yy   yj  v y ( xU − xi ) +  2   In the last equation (sxj , s y j) and (Txj , T y j) are slippage and traction at the center of the j'th cell, whose width is ∆xj; functions hij are defined by hij (xi , yi , α , β ) = f ij (xU , yi , α , β ) − f ij (xi , yi , α , β ) (7) U

j

and Ωj denotes the rectangular region of the j'th cell, over which the double integral is evaluated. Tractions and slippages are taken out of the integrals since we assume they take constant values in a cell. The upper limit of integration (xU ,y i) depends upon position of the cell center (xi , y i). Generally speaking (xU ,y i) is the closest boundary on the right of (xi ,y i). It can either be boundary that separates the slip-stick zone, or boundary of the contact region. For example, if (xi , y i) is point A, point B, or point C in Fig. 2(b), then xU is xS 1 , xS 2 , and xE , respectively. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


(xs1 , y)

573

(xs2 , y) (xE , y )

a cell strip i

strip 2 strip 1

xB

:S

:L y

trailing edge

x x x C

E

x leading edge

(b)

(a)

Figure 2:

xA x

(a) Discretized contact region; (b) integration limits.

Note that equation (6) can be written at each cell center (xi , y i), with i=1,2,…,n, and for each cell center there are components in the x and in the y directions. Hence equation (6) implies 2n equations. In addition, when (xi , y i) is in the slip zone, tangential tractions in this cell satisfy Coulomb’s law of friction, i.e. (TxiS ) 2 + (TyiS ) 2 − ( µpi ) 2 = 0 , for i = 1,2,..., n S (8) In the last equation the superscript S means the associated variable is for the slip zone, in particular, (TxiS , TyiS ) is the traction of a cell i in the slip zone, and n S is the number of cells in the slip zone. The traction vector should be collinear with the slip vector, hence s xiTyiS − s yiTxiS = 0, i = 1,2,..., n S (9) Note that the last equation only requires traction and slippage to be collinear. We also require that traction on the surface of body 2 is in exactly the same direction as the relative motion of body 1 to body 2, hence s ⋅ T S ≥ 0 , or s xiTxiS + s yiTyiS ≥ 0, i = 1,2,..., n S (10) Therefore we have 2n+2 nS equations, given by Eqs. (6), (8), and (9), for the same number of unknowns, which are the 2n tractions (Txi ,T y i) and the 2nS slippages (sxi ,s y i).

4

Numerical procedure

If there are nL cells in the stick zone ΩL, we may write Eq. (6) nL times, each time (xi , y i) is replaced by a different cell center coordinate in ΩL, then we obtain 2nL equations as follows [11,15]

0 = e − At L − Βt S , ( xi , yi ) ∈ Ω L

(11) In the last equation t and t are traction vectors of lengths 2n and 2nS, respectively, representing tractions in the stick and the slip zones; A and B are L

S


L

574 Computational Methods and Experimental Measurements XIII coefficient matrices obtained by integrating Cerruti functions over cells in ΩL and in ΩS, respectively; e is the vector containing the constants of the first term on the right hand side of Eq. (6). Zero vector at the left hand side is due to the fact that no relative slip occurs in stick cells. Similarly we may write 2nS equations for cells in ΩS, as follows s = q − Ct L − Dt S , ( xi , yi ) ∈ Ω S (12) S L where s is the slippage vector of length 2n . We may solve t from Eq. (11) and then substitute it into Eq. (12), obtaining s = q − CA −1e + (CA −1 B − D)t S (13) Thus the procedure is to solve for s and tS from equations (8), (9), and (13). After tS is obtained, then tL can be found from Eq. (11). Since equations (8) and (9) are nonlinear, the Newton-Raphson scheme is utilized by Liu and Paul [5], Hsu [15], and Liu and Hsu [11]. The initial guesses suggested by Liu and Hsu are given below Initial guess 1: (ν x − φ y i ,ν y + φ x i ) µ p( x i , y i ) (T xiS , T yiS ) = , ( xi , y i ) ∈ Ω S (14) (ν x − φ y i ) 2 + (ν y + φ x i ) 2 Initial guess 2: (TxiS , TyiS ) =

µ p ( xi , y i ) ( xi − x P ) 2 − ( y i − y P ) 2

[− ( y

i

− y P ), ( xi − x P )] , ( xi , y i ) ∈ Ω S (15)

where (xp , y p) is coordinate of a point called spin pole, whose location is to be estimated, and we will discuss this later. These initial guesses will be used in the subsequent analysis. In addition to the equation-solving procedure discussed in the last paragraph, another iterative procedure is to determine the sizes of the stick and slip zones [5,11,15]. This procedure starts by assuming the whole contact region is the stick zone. Tractions of the first iteration tL can be obtained by solving Eq. (11), but without the last term. In subsequent iterations a slip zone generally appears, then tractions tS, tL, and slippage s are obtained using the previous mentioned equation-solving procedure. At the end of the an iteration, if traction at a cell in the stick zone equals to or exceeds its limiting value µp, then in the next iteration this cell is in the slip zone. Also, if inequality (10) is violated within any cell in the slip zone, then in the next iteration this cell is in the stick zone. The procedure terminates when all tractions in the stick zone fall below their limiting values, and inequality (10) is satisfied throughout the slip zone.

5 Results and discussions In the following analysis the non-Hertzian contact region and the pressure on the region are taken from Paul and Hashemi [12]. By using initial guess 1, the Newton-Raphson procedure may always converge within 1%, provided that a



575

coarse mesh with less than 30 cells is used. In a case with very large spin, the stick zone may become very small, and a very fine mesh is necessary to expose this zone. In such a situation initial guess 1 may lead to divergence, and we can only turn to initial guess 2, which requires an estimation of the spin pole coordinate. We follow a two-stage analysis procedure suggested by Liu and Hsu [11] for Hertzian contact. In the first stage initial guess 1 with a coarse mesh is used, tractions so obtained may indicate the spin pole location. With this spin pole location, in the second analysis one may use initial guess 2 in a fine mesh. We found this procedure also work for non-Hertzian contact. Figure 3 shows slip-stick boundaries for various values of normalized spin Φ, defined by Φ=Ea3φ /[µN(1−σ2 ) ], when the normalized longitudinal creepage Ea2ν x /[µN(1−σ2 ) ]=0.6; E is modulus of elasticity, σ is Poisson’s ratio, and a is the half length of the longest strip in the contact region, as shown in the figure. Figure 4 shows both the slip and stick zones under pure spin. One may notice that the stick zone may reduce to separated poles. In figure 5 we show the resultant force Fx and moment Mz due to longitudinal creepage ν x . All these results show that the initial guesses 1 and 2 may give convergent results under non-Hertzian contact.

Figure 3:

6

Slip-stick boundaries under various Ea 2vx [ µN (1 − σ 2 )] =0.6, Φ=Ea3φ /[µN(1−σ2 ) ].

spin

when

Conclusions

In this study it is shown that the numerical technique suggested by Liu and Hsu for Hertzian contact may be extended to cases with non-Hertzian contact. The two initial guesses suggested by them may be used together and convergence may always be obtained. For cases with large spins, initial guess 1 under a coarse mesh may provide spin pole locations, which are used in the analysis using initial guess 2, with a much finer mesh.



Figure 4:

Slip and Stick zones under pure spin.



577

Fx /(4µP0a2) or Mz /(8µP0a3)

0.35 0.3

F

x

0.25 0.2 0.15

M

z

0.1 0.05 0 0

0.5

1

1.5

Ea2ν /[µN(1-σ2)] x

Figure 5:

Tangential force Fx and twisting moment Mz due to longitudinal creepage νx.

Acknowledgement The authors gratefully acknowledge that this study was partially supported by the National Science Council of ROC under grant no. NSC95-2212-E-032-015.

References [1] [2] [3] [4] [5] [6]

[7]

Johnson, K.L., The Effect of a Tangential Contact Force upon the Rolling Motion of an Elastic Sphere on a Plane, Journal of Applied Mechanics, 25, pp. 339−346, 1958. Johnson, K.L., The Effect of Spin upon the Rolling Motion of an Elastic Sphere on a Plane, Journal of Applied Mechanics, 25, pp. 332−338, 1958. Carter, F.W., On the Action of a Locomotive Driving Wheel, Proceedings of Royal Society, Vol. A112, pp. 151−157, 1926. Hills, D.A, Nowell, D., and Sackfield, A., Mechanics of Elastic Contact, Butterworth-Heinemann, 1993. Liu, C. and Paul, B., Rolling Contact With Friction and Non-Hertzian Pressure Distribution, Journal of Applied Mechanics, 56(4), pp. 814−820, 1989. Zastrau, B., Nackenhorst, U., & Jarewsky, J., On the Computation of Elastic-Elastic Rolling Contact Using Adaptive Finite Element Techniques, Proceeding of 3rd International Conference on Contact Mechanics, pp. 129−138, 1997. Wriggers, P., Computational Contact Mechanics, 2nd ed., Springer, 2006. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

578 Computational Methods and Experimental Measurements XIII [8] [9] [10] [11]

[12]

[13] [14] [15]

Kalker, J.J., On the Rolling Contact of Two Elastic Bodies in the Presence of Dry Friction, Ph. D. dissertation, Delft University of Technology, Delft, Netherlands, 1967. Kalker, J.J., Computation of Three-Dimensional Rolling Contact With Dry Friction, International Journal for Numerical Methods in Engineering, 14(9), pp. 1293−1307, 1979. Kalker, J.J., Wheel-rail wear calculations with the program CONTACT, Contact Mechanics and Wear of Rail-Wheel Systems II, University of Waterloo Press, Waterloo, Ontario, pp.3−26, 1987. Liu, C.H. and Hsu, W-E, “Three-Dimensional Rolling Contact Stress Analysis”, in Computer Methods and Experimental Measurements for Surface Effects and Contact Mechanics VII, editors J. T. M. de Hosson, C. A. Brebbia, and S-I Nishida. WIT press, pp. 269-278, 2005. Paul, B. and Hashemi, J. “User’s Manual for Program COUNTACT (COUNTerformal contACT stress programs),” Technical Report No. 4, FRA-ORD-78-72, (PB 286097/AS, available from National Technical Information Service, Springfield, Va.22151), 1977. Johnson, K.L., Contact Mechanics, Cambridge University Press, 1985. Love, A.E.H., A Treatise On The Mathematical Theory Of Elasticity, New York: Dover Publications, 1944. Hsu, W-E., Three-dimensional Rolling Contact Stress Analyses, Master thesis, Dept. of mechanical & electro-mechanical engineering, Tamkang University, 2005, in Chinese.



579

The areolar strain concept applied to elasticity I. D. Kotchergenko Instituto Militar de Engenharia, Rio de Janeiro, Brazil

Abstract In contrast to the books that start with solutions of Lamé-Navier equations using complex variables, the present article starts with a presentation of the fundamentals of the complex theory of two-dimensional elasticity. A new strain expression is derived and the compatibility conditions for these strains are given. The fundamental equations for the isotropic and orthotropic plane elasticity are also presented in somewhat new complex forms. The homogeneous equilibrium equation is presented in a complex form that proved to be easily solvable. Kolosov’s general solution for the isotropic case is obtained in a fairly straightforward fashion. New equilibrium equations and boundary conditions for finite rotation are also given. Keywords: areolar strain, compatibility equation, finite rotation, complex elasticity.

1

Introduction

The development of the theory of two-dimensional elasticity hereof is grounded on the concept of areolar strain. This concept was first presented by Kotchergenko [8], in 1983, though containing some errors. In this approach the strain is obtained by the division of two complex-valued quantities associated with 2D vectors and called areolar strain owing to the fact that its real part represents a radial strain and its imaginary part represents either a circumferential strain or a rotation. The quaternion concept can probably be used to extend the areolar strain concept to the 3D vectors case. Improvements on the linear theory were recently presented, Kotchergenko [9,10]. Equilibrium equations for finite rotation and several other improvements are now included. A detailed review of the main results obtained until now is provided in order to make this article self contained.



2

The areolar strain

Let a region in the plane of the variables z = x + iy and z = x − iy be mapped in a one-to-one manner onto the plane of the displacements u ( x, y ) and v( x, y ) by means of the transformation w( z, z ) = u ( x, y ) + iv ( x, y ) . Given the direction α of the vector z − z0 , where z and z 0 are two neighboring points of the plane, the areolar strain is defined as the gradient of the vector field w( z, z ) in the direction α , through the areolar derivative: ε = lim z → z0

∂w ∂w dz + dz w − w0 z ∂ ∂z = z − z0 dz

or ε = ∂w + ∂w e −i 2α , ∂z

∂z

(1)

−iα

where the polar form has been used for the ratio dz = | dz | e = e −i 2α . This iα dz

| dz | e

expression presupposes that z tends to z 0 , maintaining the direction α . Since 2

∂w ∂u ∂v ∂v ∂u = ( + ) + i ( − ) = θ + i 2ω , ∂z ∂x ∂y ∂x ∂y 2

∂w ∂u ∂v ∂v ∂u = ( − ) + i ( + ) = ξ + iγ , ∂z ∂x ∂y ∂x ∂y

(2)

the areolar strain can also be written in the form ε=

1

2 (θ

+ i 2ω ) + 1 2 (ξ + iγ )e −2iα .

(3)

When viewed in the polar form, the real part of the areolar strain represents a radial strain and the imaginary part represents either, a circumferential strain or a rotation. The second complex term is the complex shear strain. The components of the areolar strain are orthogonal as the quadratic form contained in the integrand of the work expression U = 1 / 2∫ C ij ε i ε j dV can be converted into its canonic form

U=

3

1

2

∫ [ 1 2(C11 + C12 )θ

2

+ 1 2 (C11 − C12 )ξ 2 + C13γ 2 ]dV .

(4)

Compatibility equations

If z0 and z are two points pertaining to the complex plane, their relative displacement is given by w − w0 = ∫ εdz = ∫ ( C

C

∂w ∂w dz )dz = + ∂z ∂z dz

∂w

∂w

∫ ∂z dz + ∂z dz .

(5)

C

Since w − w0 is independent of the path of integration C, dw =

∂w ∂w dz + dz ∂z ∂z


(6)


Figure 1:

581

Fundamental modes of the areolar strain.

is a total differential. Consequentially, the displacement field must comply with the condition of continuity ∂ ∂w ∂ ∂w ( )− ( )=0 ∂z ∂z ∂z ∂z . (7) Separating the real and imaginary parts of this equation, the following compatibility equations are obtained: ∂ ∂ (2ω − γ ) + (θ + ξ ) = 0 , ∂y ∂x ∂ ∂ (2ω + γ ) − (θ − ξ ) = 0 . ∂x ∂y

(8)

Saint-Venant’s compatibility equation is obtained from these equations through elimination of the mode ω , by applying a cross-differentiation followed by a subtraction. Saint-Venant’s compatibility equation will then be satisfied for any rotational field, which may thus violate the compatibility conditions (8).

4

Equilibrium equations

For isotropic material, 1 2 (C11 +C 12 ) = λ + µ , 1 2 (C11 −C 12 ) = µ and C 13 = µ , thus the work expression given by eqn. (4) reduces to


582 Computational Methods and Experimental Measurements XIII U=

1

2

∫ [(λ + µ )θ

2

+ µ (ξ 2 + γ 2 )]dV

(9)

where λ and µ are Lame’s elastic material constants. The Euler equations for this functional are ∂θ ∂ξ ∂γ + µ( + (λ + µ ) )=0 ∂x ∂x ∂y . (10) ∂θ ∂γ ∂ξ (λ + µ ) + µ( − )=0 ∂y ∂x ∂y Taking into account that ∂ξ ∂γ ∂γ ∂ξ ∂ (ξ + iγ ) = ( + ) + i ( − ) = ∆u ( x, y ) + i∆v( x, y ) , ∂z

∂x

∂y

∂x

∂y

where the symbol ∆ stands for the Laplacian operator, Lamé’s homogeneous equilibrium equations can be presented in the following complex form: 2

∂ ∂ [(λ + µ )θ ] + 2 [ µ (ξ + iγ )] = 0 ∂z ∂z .

(11)

Multiplying each term of eqn (7) by a different complex elastic constant will result in an equation in terms of stresses. For isotropic material, undergoing small strain and small rotations, the operations required for obtaining Lame’s homogeneous equilibrium equation are ∂w ∂ λ ∂w ∂ − iω )] − ( 2 ) = 0 . [(λ + µ )( 2 ∂z ∂z ∂z 2 ∂z

(12)

Observe that only one out of the two iω , which appear in the term i 2ω , was removed. The term iω represents a local rotation. Taking the derivatives of ω from eqns. (8) and substituting into eqn. (12) will give eqn. (11)

5

Boundary conditions

Applying Green’s formula (e.g. Courant and Hilbert [1]) to eqn. (11), in the complex form, 2 ∫∫{ Ω

∂ ∂ 1 [(λ + µ )θ ] + [ µ (ξ + iγ )]}dxdy = ∫ (λ + µ )θdz − µ (ξ + iγ )dz , ∂z ∂z i

(13)

C

the traction vector on a boundary curve C results T = (λ + µ )θ − µ (ξ + iγ )e −2iα ,

(14)

where α points towards the direction of the vector element of arc dz of the closed contour curve C. Observe that if n the unit outward vector, normal to the element of arc ds = dz , then nds = −idz = 1 i dz . WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


6

583

Other forms for the equilibrium equations

The compatibility equations (7) can also be rewritten in the following complex form ∂ ∂ (θ + i 2ω ) − (ξ + iγ ) = 0 ∂z ∂z .

(15)

Equation (11) can hence be reduced to the following holomorphic function (Love, [4]), ∂ [θ + iη 2ω ] = 0 ∂z ,

where η =

µ λ + 2µ

(16)

. The modes θ and ω are hence harmonic functions.

The writer succeeded in rewriting the equilibrium equation in another form that proved to be easily solvable. The elimination of the derivatives of θ between equations (8) and (16), gives ∂ω λ + 2 µ ∂γ ∂ξ , 2 ( − ) = λ + µ ∂x ∂y ∂x λ + 2 µ ∂ξ ∂γ . ∂ω (17) 2 ( + ) =− λ + µ ∂x ∂y ∂y The elimination of the derivatives of ω , from the same pair of equations, gives µ ∂ξ ∂γ ∂θ ( =− + ), λ + µ ∂x ∂y ∂x µ ∂γ ∂ξ ∂θ ( − ). (18) =− λ + µ ∂x ∂y ∂y Combination of these equations yields: 2

∂γ ∂ξ ∂ξ ∂γ ∂ (θ + i 2ω ) = − χ [( + ) − i ( − )] , ∂x ∂y ∂x ∂y ∂z

(19)

Or after eqns. (2), ∂2w ∂z 2

+χ

∂2w = 0. ∂z∂z

(20)

∗ where χ = λ + 3µ for plane strain and χ = λ + 3µ , for plane stress, observing ∗

λ+µ

λ +µ

that λ∗ = 2λµ . λ + 2µ

7

General solution

Differentiation of eqn. (11) in z results


584 Computational Methods and Experimental Measurements XIII ∂ 2θ ∂2 [(ξ + iγ )] = 0 + 2µ ∂z∂z ∂z 2

2( λ + µ )

Since 4

2

∂ ∂z∂z

(21)

is the Laplacian operator and θ is a harmonic function, this

equation reduces to 2µ

∂2 ∂z 2

[(ξ + iγ )] = 0

,

(22)

∂2w 1 =− ϕ ' ' ( z) , ∂z∂z 2µ

(23)

or after eqns. (2), to 2

∂ ∂2w ( )=0 ∂z ∂z∂z .

Integration in z gives

where ϕ ' ' ( z ) is the conjugate of an analytic function. Integrating eqn. (23) in z, results ∂w 1 1 = (ξ + iγ ) = − [ zϕ ' ' ( z ) + ψ ' ( z )] . ∂z 2 2µ

(24)

Equations (20) and (23) furnish ∂2w ∂z which after integration in z, results

2

=

χ 2µ

ϕ ' ' ( z) ,

χ ∂w = ϕ ' ( z) + ζ ' ( z ) . ∂z 2µ

(25)

Differentiation of this equation with respect to z yields ∂ ∂w ( ) = ζ ' ' (z) . ∂z dz

1 ϕ ' ' ( z ) and hence 2µ 1 ζ ' (z) = − ϕ ' ( z) + c , 2µ

(26)

From eqn. (23), results ζ ' ' ( z ) = −

(27)

where c is a complex constant. Substitution into eqn. (25) yields ∂w 1 1 1 = (θ + i 2ω ) = [ χϕ ' ( z ) − ϕ ' ( z )] + (θ 0 + i 2ω 0 ) . ∂z 2 2µ 2

(28)

Using eqns. (28) and (24), the integration of the total differential, eqn. (5), gives Kolosov-Muskhelishvili’s general solution, [2,3,5,7]: w = w0 +

1 1 (θ 0 + i 2ω 0 ) z + [ χϕ ( z ) − zϕ ' ( z ) −ψ ( z )] . 2 2µ


(29)


585

The addition of Eqn (28) to its conjugate gives θ=

χ −1 [ϕ ' ( z ) + ϕ ' ( z )] + θ 0 . 2µ

(30)

The subtraction of Eqn (28) from its conjugate gives ω=

χ + 1 ϕ ' ( z) − ϕ ' ( z) [ ] + ω0 . 2µ 2i

(31)

Kolosov´s formulas, for stresses are obtained directly from eqns. (30) and (24): σ 11 + σ 22 = 2[ϕ ' ( z ) + ϕ ' ( z )] + 2(λ + µ )θ 0 , (32) σ 11 − σ 22 + i 2σ 12 = −2[ zϕ ' ' ( z ) + ψ ' ( z )] .

(33)

The areolar strain, in terms of analytic functions assumes the form ε=

8

1 1 1 (θ 0 + i 2ω0 ) + [ χϕ ' ( z ) − ϕ ' ( z )] − [ zϕ ' ' ( z ) + ψ ' ( z )]e −i 2α . 2 2µ 2µ

(34)

Equilibrium equations for an orthotropic plane

The writer explored the possibility of applying to the orthotropic plane case the same approach used with eqn. (11). Taking the compatibility equation in the form of eqn. (15) and removing one out of the two iω present into the term i 2ω ; then multiplying each complex term of this equation by different complex elastic constants, will result in the following equation in terms of stresses: ∂ ∂ (θ + iω ) − (b1 + ib2 ) (ξ + iγ ) = 0 . ∂z ∂z

(35)

∂θ ∂ + [(a1 + ia 2 ) − 2(b1 + ib2 )] (ξ + iγ ) = 0 . ∂z ∂z

(36)

(a1 + ia 2 )

Substitution of the derivatives of the rotation, obtained from eqn. (8), gives (a1 + ia 2 )

Lamé’s equilibrium equations are obtained letting a 2 = 0 , b2 = 0 , a1 = λ + µ and b1 = λ 2 . Application of Green’s formula to eqn. (36), gives the following generalization of the traction vector formula, previously depicted in eqn. (14): T = (a1 + ia 2 )θ − [(a1 + ia 2 ) − 2(b1 + ib2 )](ξ + iγ )e −i 2α .

(37)

Using the same procedure as the one used for obtaining eqn. (16) leads to the same Love’s form of holomorphic function: ∂ [θ + iη 2ω ] = 0 , ∂z

where η=

(a1 − 2b1 ) + i( a 2 − 2b2 ) . 2[(a1 − b1 ) + i (a 2 − b2 )]

Observe that for the isotropic plane, η will be reduced to a real constant. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

(38)

(39)

586 Computational Methods and Experimental Measurements XIII Separating the real and imaginary parts yields: ∂θ ∂ω ∂ω , = η1 2 +η 2 2 ∂x ∂y ∂x ∂θ ∂ω ∂ω , = −η1 2 +η 2 2 ∂y ∂x ∂y

(40)

with η = η1 + iη2 , where η1 =

a12 + a 22 − 3( a1b1 + a 2 b2 ) + 2(b12 + b22 ) 2[(a1 − b1 ) 2 + (a 2 − b2 ) 2 ]

η2 =

a 2 b1 − a1b2

,

.

(41)

∂ 2ω . ) ∂y 2

(42)

2[(a1 − b1 ) 2 + ( a 2 − b2 ) 2 ]

Differentiations of eqns. (40), yields ∂ 2θ ∂x 2

+

∂ 2θ ∂y 2

= η 2 2(

∂ 2ω ∂x 2

+

Other differentiations of eqns. (40), yields additionally: ∂ 2ω ∂x 2

=

∂ 2 ω ∂ 2θ 1 − (η 2 2 ) ∂x∂y ∂x∂y 2η1

∂ 2ω

and

∂ 2ω ∂y 2

=

∂ 2 ω ∂ 2θ . 1 + ( −η 2 2 ) ∂x∂y ∂x∂y 2η1

∂ 2ω

= 0 , and after eqn. (42), both ω and θ remain harmonics ∂x 2 ∂y 2 also for the orthotropic plane case.

Hence

9

+

Finite rotations

The gain of area during a plane deformation is given by dA = θ +

θ2 4

+ ω2 −

2

2

1 2 ∂w ∂w , (ξ + γ 2 ) = θ + − 4 dz ∂z

(43)

which is obtained from the determinant ∂u   ∂u 1 + ∂x ∂y  1 + dA =  . ∂v  ∂v 1+   ∂x ∂y 

(44)

Observing fig. 1, it can be seen that a finite “rotation” ω has a significant influence into the change of area of a plane elastic body. In some bending and buckling problems, it is admitted to disregard strain terms squared, except for

ω 2 , reducing dA to dA = θ + ω 2 . WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

(45)

587


A rigid body rotation in this case can be approximated by the condition

θ + ω 2 = 0 . Then, for a point to describe a circular path of arc Ω = tan −1 ω , it is required that the expansion produced by the mode ω be neutralized by the shrinkage θ = −ω 2 . For the condition given by eqn. (45), the work expression assumes the form U = 1 / 2∫ (λ + µ )(θ + ω 2 ) 2 + µ (ξ 2 + γ 2 ) dxdy .

(46)

The Euler equations for this functional are (λ + µ )

∂ξ ∂γ ∂ω ∂ ∂ ∂θ (ωθ ) + (ω 2 ) − 3ω 2 ]=0, + ) + (λ + µ )[− + µ( ∂y ∂x ∂y ∂x ∂y ∂x

(λ + µ )

∂γ ∂ξ ∂ω ∂ ∂ ∂θ (ω 2 ) + 3ω 2 ]=0. + µ ( − ) + (λ + µ )[ (ωθ ) + ∂x ∂y ∂x ∂x ∂y ∂y

(47)

This system of equations can be written in the complex form (λ + µ )

∂ ∂ [(1 + iω )(θ + ω 2 ] + µ (ξ + iγ ) = 0 . ∂z ∂z

(48)

Applying Green’s formula, the traction vector on a closed curve C, in the undeformed reference frame, results T = (λ + µ )[(1 + iω )(θ + ω 2 )] − µ (ξ + iγ )e −i 2α . (49) Accordingly with eqns. (I.22) from Novozhilov [6], the shearing modes due to 1 1 finite strains are: ξˆ = ξ + θξ + ωγ and γˆ = γ + θγ − ωξ , hence discarding the 2 2 strains squared, other than those containing ω , results ξ = ξˆ − ωγ and

γ = γˆ + ωξ . If the following approximations are used accordingly: σx −σ y σx −σ y σ x +σ y τ τ θ +ω2 = , ξ= −ω and γ = + ω , 2(λ + µ ) µ 2µ µ 2µ

(50)

the equilibrium equations (47) in terms of stresses, referred to the undeformed reference frame, become ∂ ∂σ x ∂τ ∂ − (ωσ y ) − (ωτ ) = 0 , + ∂x ∂y ∂y ∂x ∂σ y ∂y

+

∂τ ∂ ∂ + (ωσ x ) + (ωτ ) = 0 . ∂x ∂x ∂y

(51)

These are exactly the equilibrium equations (II.49) given by Novozhilov, [6].

References [1]

Courant, R. & Hilbert, D., Methods of Mathematical Physics, Vol. II, Wiley: New York, pp. 350-351, 1989. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

588 Computational Methods and Experimental Measurements XIII [2] [3] [4] [5] [6] [7] [8]

[9] [10]

England, A. H., Complex Variable Methods in Elasticity, Wiley, pp. 2849, 1971. Kalandiya, A. I., Mathematical Methods of Two-Dimensional Elasticity, Mir Publishers, pp. 307-339, 1975. Love, A.E.H., A Treatise on the Mathematical Theory of Elasticity, Cambridge University Press, pp. 204-220, 1927. Muskhelishvili, N.I., Some Basic Problems of the Theory of Elasticity, Noordhoff, Groningen, 1963. Novozhilov, V.V., Foundations of the Nonlinear Theory of Elasticity, Dover Publications: Mineola, New York, pp. 83-84, 1999. Sokolnikoff, I.S., Mathematical Theory of Elasticity, McGraw-Hill, 1956. Kotchergenko, I.D., Unsymmetrical Plane Elasticity, Recent Advances in Engineering Mechanics, Proceedings of the Fourth Engineering Mechanics Division Specialty Conference, ASCE, eds. W.F. Chen & A.D.M. Lewis, Vol. I, pp. 385-388, 1983. Kotchergenko, I.D., Kolosov-Mushkhelishvili Formulas Revisited, 11th International Conference on Fracture, Turin, March 2005. Kotchergenko, I.D., Applications of Generalized Analytic Functions to Elasticity, CMM-2005-Computer Methods in Mechanics, Polish Academy of Sciences, June 2005.



589

Methodology for the manufacture of smart composites with thermoplastic matrix L. Elsoufi1,2, K. Khalil3, R. Lachat1, W. Charon1 & M. Zoaeter2 1

University of Technology of Belfort-Montbeliard (UTBM), M3M Laboratory, France 2 Lebanese University (UL), LPM laboratory, Lebanon 3 Centre universitaire de technologie (CUT), Lebanon

Abstract The adaptable mechanical structures in the form of Shell have found large developments and use in many applications, especially in the field of smart structures where the piezoelectric components are used as actuators and sensors. Many functional constraints prevent the control of a structure by elements reported on the surfaces of the object. Thus, the piezoelectric components necessary for the control of the structures will be integrated into this, i.e., in material, even the material of the wall. Certain work describes the manner and the performances of structures by using composite materials with thermoset matrices as structural support and piezoelectric components as the actuator or sensor elements. The current difficulties in recycling the thermoset materials are hindering the industrial development of such structures. For this reason, we propose to use composites with a thermoplastic matrix. Unfortunately, the current processes of achieving models in smart thermoplastic structures are not directly exploitable for the integration of components such as the piezoelectric actuators and sensors which are fragile and sensitive to temperature. This work evaluates the sizes typically reached in the process transformations of such composites with a thermoplastic matrix in order to be able to establish the behavior models for the realized structures. A thermo-mechanic testing method using dynamic mechanical analysis (DMA) is also proposed. Keywords: smart thermoplastic manufacturing, cooling rate, thermoplastic filling time, PZT sensor, piezoelectric properties degradation.



1

Introduction

Recent applications in smart structures and materials are limited by the manufacture of composites with a thermosetting matrix [2] such as: shape control of airplane wings [3], car bodies, reflector antennas [4], deformable mirrors, and actuators such as the so-called bimorph and C-block actuators [5, 6], which are widely used in the automation and aeronautics industries. This study is a contribution to modeling in smart thermoplastic structures manufacturing which have a great importance in recycling. The relatively high viscosity of melted thermoplastic materials leads the manufacturers of composites with thermoplastic matrix to raise the temperature of these materials during the manufacturing processes until temperatures can reach 1.5 to 2 times the melting point of these materials [1]. Among the processes we can enumerate: multi-material injection, multiphase injection, rolling, and thermoforming. The integration of components like memory-shape alloys such as the piezoelectric actuators or sensors in composite parts with thermoplastic matrix causes the thermal limit to exceed that of these components Tc/2 (Tc: Curie temperature of the piezoelectric component) which varies according to the type of piezoelectric. There are recent piezoelectric materials which have relatively high curie temperatures compared to those traditional ones [7], such as: 0.36BiScO30.64PbTiO3 and 0.3BiYbO3-0.7PbTiO3. However, these have not been commercialized yet because they are still under research. The aim of this study is to determine the relationship between the thermoplastic manufacturing conditions and those of commercial traditional piezoelectric materials. In this context, experimental tests have been carried out on piezoceramic plates by the dynamical-mechanical analysis test machine (DMA) which is called “Viscoanalyzer”. In this work, we will analyze the curvetype of temperature variation over time in different conditions of fabrication using simulation under ANSYS. The thermoplastic manufacturing process treated in this study is the injection molding of a rectangular model, in which we integrate a piezoelectric sensor. We vary the model thickness and the sensor position in order to study their thermal influence on the sensor and to minimize its exposure to heat. This introduction is followed by three sections which represent respectively: Modeling procedures, experimental results, and a conclusion.

2

Modeling procedures

In the injection process the thermal phenomena are often the slowest and they mainly control the time cycle as well as the degree of the damage of the piezoelectric component during the process. In practice, the cooling time of a molded part corresponds to the longest phase of the molding cycle as shown in figure 1. The thermal resistance of the piezoelectric sensor requires the knowledge of the thermal phenomena during the molding cycle. In this paper we present the various aspects of the thermal transfers during the thermoplastics injection process. We treat the cooling of the model and the heating of the WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


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piezoelectric sensor. Furthermore, we introduce concepts of compromise between the heating and cooling of the polymer, and the on-heating of the piezoelectric sensor during the melt plastic filling in the mould. The thermal analysis of the injection process requires the knowledge of three modes of heat transfers: conduction, convection and radiation. The conductive transfers take place in the materials (part and mould). The heat transfer between the wall of the cooling channels and the fluid of the regulation system is controlled by convective transfers.

Figure 1:

Cycle time.

2.1 Model description and properties The studied model is composed of a thin polypropylene plate (Borealis BD500P) which contains a piezoceramic sensor PZT made of two circular plates: yellow brass and ceramics as shown in figure 2. The polypropylene plate is molded in a steel mould of type AISI 1008. All properties are given by [9]. The properties of polypropylene (Borealis BD500P) are as follows: − density: ρ2 = 800 kg/m3 − thermal conductivity: k2 = 0.179 W/m.K − specific heat: c2 = 2800 J/kg.K − consistence (degree of viscosity): K = 11000 − index of pseudoplasticity: n = 0.308 The properties of the steel mould (AISI 1008) are as follows: − density: ρ1 = 7800 kg/m3 − the thermal conductivity (k1) of this type of steel is variable with temperature as shown in the table below: Table 1:

AISI 1008 thermal conductivity.

Temperature [°C] Conductivity [W/m.K]

0 65.2

100 60.2

200 54.7

400 45.2

− the specific heat (c1) is also variable with temperature as shown in the table below: Table 2: Temperature [°C] Specific heat [J/kg.K]

50-100 481

AISI 1008 specific heat. 150-200 519

200-250 536


250-300 553

350-400 595

592 Computational Methods and Experimental Measurements XIII The properties of the piezoelectric ceramic (PZT upper plate) are as follows: − density: ρ3 = 7500 kg/m3 − thermal conductivity: k3 = 6 W/m.K − specific heat: c3 = 670 J/kg.K The properties of the yellow brass (PZT lower plate) are as follows: − density: ρ4 = 8750 kg/m3 − thermal conductivity: k4 = 159 W/m.K − specific heat: c4 = 380 J/kg.K The necessary data to calculation are as follows: − the mould is controlled using a coolant fluid at the temperature of Tm = 40°C − the injection temperature of polypropylene part is TM = 240°C − the initial temperature of the piezoelectric sensor is Tp = 25°C

Figure 2:

Description of the mould for simulation.

2.2 Modeling results The modeling of the transient problem is built under ANSYS by studying the thermal distribution over time of the upper part of the molding plate (above the piezoceramic sensor: between sensor and mould) and the piezoceramic sensor during manufacturing at all point in z direction. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


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In order to study the several states of manufacturing influence of such composites with thermoplastic matrix on the integrated piezoelectric components, we analyze three sizes for the plate thickness and five positions for the sensor as follows: − h = 2mm with h1 = 0.5mm, 0.75mm and 1mm − h = 5mm and 10mm with h1 = 0.5mm, 0.75mm, 1mm, 1.75mm and 2.5mm 2.2.1 Part thickness influence To illustrate the part thickness influence on the injection thermal cycle and the piezoceramic sensor thermal resistance over time, three simulations were realized for three different part thicknesses h = 2mm, 5mm and 10 mm, to the same piezoceramic sensor position hp = 1mm as shown in figures 3 and 4. In the PP zone between mould and sensor, we notice an important increase in the temperature discard between centre and part surfaces with the increase of the global model thickness. In addition, this model thickness increasing causes an increment in the piezoceramic sensor temperature from 100°C to 140°C.

Figure 3:

Part thickness influence on plastic zone (hp = 1mm).

2.2.2 Sensor position influence Five simulations were realized on five various piezoceramic sensor positions with aim to study the thermal resistance of the sensor over time under various conditions of manufacture including its positions in model (figures 5 and 6). We notice a systematic increase in the sensor maximum temperature with its transverse penetration towards the centre of model to reach 160 °C as shown in figure 6. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Figure 4:

Figure 5:

Part thickness influence on piezoceramic sensor (hp = 1mm).

Sensor position influence on plastic zone (h = 5mm).

2.2.3 Liquid plastic filling effect During the mould filling, a solid layer is formed on the flow surfaces (mould and sensor surfaces), this solidification is able to reduce significantly the available flow passage distance (liquid vein hv(t), fig.7). The reduction of the liquid vein, related to an increase in the filling rate, causes an important rise in the filling pressure of the mould. However, a significant reduction of the filling time involves an increasing in the pressure due WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


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to flow increment. Finally, the evolution of the filling pressure related to the filling time, presents successively a decreasing in pressure profile passing by a minimum P* than increasing [8]: n

 2(2 n + 1)   2 n  P * ( z , hv , t * ) = 2 K     n    3n + 1 

−2 n

  2n   1 −    3n + 1  

− ( n +1)

z n +1

h −p (3 n +1)

α −n

(1)

The optimal filling time values are higher than the ones known in industrial practice. For evident reasons of productivity, it isn’t necessary to prolong the filling time until the minimum of the filling pressure. It is preferable to be located at the point that constitutes the best compromise between a significant reduction of the filling time and a minimum increase ∆P in the filling pressure. The optimal filling time is the average value of the filling times obtained by decreasing solidified hp – s(t) thickness and increasing solidified thickness z (t ) = αt * as follows [8]:

Figure 6:

Sensor position influence on piezoceramic sensor (h = 5mm).

Figure 7:

Flow section in the solidification course.



tf

  1    h p =    

   

n +1 n

 3n + 1   +  h p ( n + 1)    2

n +1 n

    ×    

(2)

  3 n +1 1    h p n  2 n  2   2 n   n +1  ∆P   +  1 −     n α  3n + 1    3n + 1     2 K  2(2 n + 1)  x n +1        n  

1 − −n  n

     

The filling speed is given by [8] as follows: 2 1    3n + 1  3  aL  3  − 1.384 2 n   2   h   V =  2n + 1 n  n +1 KαLh − n −1    2  n  k∆T 

3

 3n+ 2       

(3)

with ∆T = TM - Tm and L = model length. For a plate with 5 mm of thickness (PP Borealis BD500P), we obtain tf = 12.6 s and V = 0.2 m/s which allow us to calculate the convection heat transfer coefficient of liquid plastic on the sensor (equal 18 W/m2K). This coefficient is necessary to re-modeling the problem in filling conditions.

Figure 8:

Filling influence on plastic zone (h = 5mm, hp = 1mm).



Figure 9:

597

Filling influence on piezoceramic sensor (h = 5mm, hp = 1mm).

We notice that the filling process causes a big rise in the temperature maximum values of the different manufacture elements: 20% of temperature increment (fig.8) in plastic zone (sensor side) and 42% of sensor temperature increment (fig.9). The temperature increment and the cooling deceleration play an effective role in the reduction during the creation of solidified thermoplastic thickness on the cold surfaces (mould and sensor). However, the sudden and enormous temperature rise in piezoelectric sensor (figure 9) until exceeding the piezoelectric thermal limit causes the degradation of the sensor i.e. the degradation of these piezoelectric properties (polarization of the piezoelectric ceramic).

3

Experimental results

In this work, experimental tests were carried out in M3M laboratory on piezoceramic sensor by a dynamical-mechanical analysis test machine (DMA) which is called “Viscoanalyzer” of type METRAVIB RDS VA 2000, with an acquisition system (TEAC GX-1) used to measure the electric voltage produced by the tested piezoelectric sensor. The viscoanalyzer is a testing machine which makes it possible to determine the viscoelastic properties of materials. The dynamic analysis carried out by this machine allows the thermomechanical performance evaluation of materials and the choice of materials for a function of damping or sound-proofing, for the optimization of the formulation or manufacture processes for the forecast of the products durability. The operation principle consists on applying a sinusoidal force to a sample model and measuring the displacement results. A thermal enclosure makes it possible to cover an experimental window of temperature going from -150°C to +450°C. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

598 Computational Methods and Experimental Measurements XIII The capacity of the machine is about 100 N. The frequency of the test-sample excitation varies from 0.001 Hz to 200 Hz. The materials tested can be solid, liquid or pasty. Several samples support making it possible to make the following tests: traction, compression, plane or annular shearing, annular pumping, bending, creep or strain relaxation tests. The principle used in this test installation is based on the relation between the applied sinusoidal displacement and the measured sinusoidal force responding to this displacement (stiffness and loss angle). The model studied in the experimental test is a circular actuator piezoceramic PZT (same properties in §2.1), simply supported on the periphery; and applying at the center of the plate a sinusoidal displacement of amplitude 50 µm and frequencies: 10, 15.85, 25.12, 39.81, 63.1, 100 Hz, with temperature variation as the protocol illustrated in figure 10. This experimental protocol allows us to measure the different sensor parameters and the voltage emitted after each temperature crawl up (10°C/min) and down (-5°C/min) in points k, l, m and n as indicated in figure 10.

Figure 10:

Experimental protocol.

The aim of this experience is to determine the degree of the mechanical and piezoelectric characteristics change during or after temperature variation.

4

Conclusion

In conclusion, we can notice through the results that the systematic increase in the temperature during the manufacture of smart composites with thermoplastic matrix causes an exceeding of the piezoelectric thermal limit. This fact makes a degrading in the ceramic crystal polarization of the piezoelectric components, i.e. the loss in the piezoelectric properties.



Figure 11:

599

Variation curves of various parameters with the temperature increment measured by viscoanalyzer.

The empirical results obtained by viscoanalyzer show a beginning of this phenomenon from a temperature of approximately 75°C: until this temperature the voltage is reversible after each temperature crawl up and down. In addition, simulations show an important influence of the piezoelectric positioning component and the molded plate dimensions. In spite of these facts which are compatible in certain cases and incompatible in others, a comparison was made between modeling and experiment. This comparison shows the requirement of integrating the piezoelectric component in a position which takes into consideration two effects: On one hand the liquid plastic filling effect (solidification of the plastic on surfaces and the temperature increment of piezoelectric component). On the other hand the thickness of the manufactured model. These results constitute a beneficial contribution for the smart thermoplastic composite manufacture.

Acknowledgement The authors wish to acknowledge that this work was supported by the Francophone University Agency (AUF).

References [1] [2]

[3]

Cracknell, Dyson, Handbook Of Thermoplastics Injection Mould Design, Blackie Academic & Professional, pp. 10-80, 1993. Hori, Aoki, Ohira, Yano, New type of mechanical damping composites composed of piezoelectric ceramics, carbon black and epoxy resin. Composites Part A: Applied Science and Manufacturing, 32(2), pp. 287290, 2001. Tzou, Tseng, Distributed piezoelectric sensor/actuator design for dynamic measurement/control of distributed parameter systems: a piezoelectric finite element approach. J. Sound Vib., 138, pp. 17–34, 1990. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

600 Computational Methods and Experimental Measurements XIII [4] [5] [6] [7]

[8] [9]

Agrawal B. N., Treanor K. E., Shape control of a beam using piezoelectric actuators. Smart Mater. Struct., 8, pp. 729–40, 1999. Brei, Moskalik, Deflection performance of a bi-directional distributed polymeric piezoelectric micromotor. J. Microelectromech. Syst., 6, pp. 62–9, 1997. Ervin, Brei, Recurve piezoelectric-strain-amplifying actuator architecture. IEEE/ASME Trans. Mechatron., 3, pp. 293–301, 1998. Eitel, Randall, Shrout, Rehrig, Hackenberger, Park, New High Temperature Morphotropic Phase Boundary Piezoelectrics Based on Bi(Me)O3-PbTiO3 Ceramics. The Japan Society of Applied Physics, 40(1), pp. 5999-6002, 2001. Mousseau, Sarda, Deterre, Thermique de l’injection des thermoplastiques /Optimisation. Technique de l’ingénieur, AM 3 685, pp. 1-18, 2005. MATWEB, www.matweb.com



601

Experimental investigation on the folding of axially crushed hexagonal tubes M. R. Said1, A. A. Mokhtar1, A. Alias2 & A. Ibrahim3 1

Faculty of Mechanical Engineering, Kolej Universiti Teknikal, Kebangsaan, Malaysia, 2 Faculty of Mechanical Engineering, Universiti Teknologi, Malaysia 3 Universiti Utara, Malaysia

Abstract The objective of this paper is to investigate the folding mechanism of an empty hexagonal tube made of mild steel with respect to the energy absorption. This includes the determination of the plastic wavelength, type of deforming mode and the mean load of the tube subjected to an axial loading. The introduction of chamfer at one end of the tube is also examined and studied. The comparison load-displacement curves between experiment and Finite Element Analysis (FEA) are made. The experimental mean load is also compared with theoretical analysis. The response of the hexagonal tube subjected to axial loading is observed through experiments. The formation of plastic fold length is observed and measured. The mean load and area under the load-displacement curve are calculated to obtain the energy absorption. By introducing the chamfer at one end, the deforming pattern can be determined, i.e. in-out chamfer gives a diamond mode while all-out chamfer gives a concentina mode. The FEA loaddisplacement curve agrees well with the experiment. However, the predicted mean load is overestimated, while the equivalent circular cylinder underestimates the experimental result. Keywords: energy absorption, hexagonal tubes, triggering, and crushed.

1

Introduction

Energy absorption systems made of thin-walled circular tubes and tube nests have been given significant attention by researchers for the past few decades. In particular, the initiation progressive axisymmetric and non-axisymmetrical WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070601

602 Computational Methods and Experimental Measurements XIII folding in tubes has been a subject of intense investigation by many investigators. Alexander [1] was first to present an analysis for the prediction of a mean crushing load, for a tube subjected to axial compression. He managed to obtain an expression for the folding length and the mean crushing load by implementing global energy balance. However, he assumed the tube to be folding into axisymmetric rings, in which the tube wall moves out of its initial position. But in reality there is outward as well as inward movement. This anomaly was first addressed by Reddy and Reid [2] albeit by an over prescribed hinge (four) mechanism. Wierzbicki et al [3] presented a more realistic three hinges mechanism in the form of a concentina collapse mode. The mechanism allows for radially inward as well outward folding in a proportion relative to the tube generator. The outer portion of the folding length relative to the total (inward plus outward) length was defined by the eccentricity factor. The mean load deduced was independent of the eccentricity factor, which was indeterminate. Singace et al [4] further modified Wierzbicki’s analysis enabling the determination of the eccentricity factor, which was in good agreement with the experimental results. The absolute mean crushing load produced by Singace et al [4] was similar to that of Wierzbicki et al [3] except for an added factor 5.632. These and other analysis are applicable only on the second and subsequent folds. The formation of the first fold is unique and there appears to be no analysis yet to describe it. Pugsley and Macaulay [5] observed diamond type folding in thinner tubes and derived a semi-empirical expression for the mean load. Horton et al [6] reported the findings of an extensive experimental study of quasi-static axial buckling of thin-walled cylindrical tubes, concluding that the buckling mode changed from axisymmetric to non-symmetric patterns if the geometric parameter, radius over wall thickness, t increases. Johnson et al [7] developed, a mechanism of inextensional mode of deformation of thin-walled tube under axial loading which involved the stationary and travelling hinges in the diamond mode and derived an expression for the mean crushing load. However, they [7] did not correlate the change of concentina to diamond mode. Tvergaard [8] investigated the influence of buckling pattern localisation, which could cause the transition of diamond mode to axisymmetric mode. The dependence of folding mode on tube geometry has been investigated experimentally by Andrews et al [9]. Square, rectangular, hexagonal are more popular than circular tubes in automobile industry. Early investigations [10,11] were concerned with rectangular tubes made of sheet metal and were aimed to understand the behaviour of vehicle body shells. Wierzbicki and Abramowicz [12] formulated kinematically admissible global deformation mechanisms for thin-walled rectangular tubes comprised of flat plates. Reid and Reddy [13], Reid et al. [14] and Reddy and Al-Hassani [15] have shown that introducing different fillers like foam or wood increase the overall energy absorption of the tubes. Said [16] presented experimental measurement of elastic half wavelength and plastic fold length during the crushing of rectangular tubes. No work has been found to study the effect of trigger by introducing the chamfer in one end tube. Abramowicz and Wierzbicki [17] developed a mechanism for predicting the crush behaviour of WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


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multicorner columns with an arbitrary corner angle. This was applied to find crushing strength of hexagonal tube under axial compression and produce a simple expression for the mean crushing load. As the hexagonal consists of 6 sides and circular is infinity sides, the correlation may be made by assuming the hexagonal sides as an equivalent circular cylinder. The objective of this paper is to investigate the folding mechanism of empty hexagonal tube made of mild steel with respect to the energy absorption. The effect of triggering with mean load and plastic folding is also examined.

2

Experiment

Axial compression tests were carried out on as-received 170 mm and 200 mm long hexagonal tubes made from the same tube. Longer specimens were labelled as HEX1 to HEX12 and shorter (170 mm long) specimens as HEX170. Some of them were marked with 4 mm grid circles and some chamfered at one end in order to trigger the mode of deformation. Distance of 4 mm is measured from a point of arc intersection to the next point. All the axial compression tests were on specimens in the as-received state only. Tests were carried out on a 200 kN universal testing machine at a loading rate of 10 mm/min.

3

Observations and discussions

This section describes the influence of triggering, the characteristic of loaddisplacement curves, and the effect of chamfer. This includes the determination of the plastic fold length, mean load and energy absorbed and also compared with the circular tube under axial loading. 3.1 Triggering the mode of deformation Mode of deformation of hexagonal tube under axial loading can be triggered by introducing the chamfer at one end. Without chamfering, the tube deforms in uncertainty manner, i.e. may deform in concentina or diamond or mixed mode. Experiments have shown that by introducing the chamfer, the mode of collapse can be determined. However, the edge of the tubes under investigations, have to be suitably chamfered. The chamfer was made to half of the original thickness of the tube and with an angle of approximately 450 as shown in Figure 1. This results in the axial end forces applied being offset with respect to the centreline of the tube wall. The larger the chamfer is the larger the eccentricity. Experiments have shown that if the chamfering one side only of all end faces (either inner or outer ends), tube collapses into concentina mode. With external chamfers on all faces (referred to as all-out chamfers), the first buckle was outwards and vice versa. However, if adjacent faces were chamfered on opposites edges (i.e. one on the outside and the other on the inside, referred to as in-out chamfers), the tubes were seen to fold into the diamond mode. In the inout chamfered tubes, fractures were observed along the corners when the chamfers were extended over the whole face width. By chamfering only over the central region, such corner failures were prevented. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Figure 1:

A schematic view of the tube wall, showing the chamfered angle. (a)

Op

(i)

(ii)

(b)

Figure 2:

(a) A typical load-displacement curve for a single hexagonal tube (HEX170). Insert: A one-sixth segment of a deformed specimen. (b) Deformed unchamfered tubes (i) Diamond mode (ii) Concentina mode.

3.2 The load-displacement characteristic for unchamfered specimens Figure 2 shows a typical load-displacement trace for an unchamfered specimen marked as HEX170 and the deformed tubes. A cross-section of the crushed specimen is shown in the inset. During compression, just before the peak load (point 1 in Figure 2a), a series of elastic ripples were noticeable in the sides of WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


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the hexagonal tube. It is noticed that, the peak load is 184 kN. Ripples disappeared when the load dropped (point 1 to 2) and were not noticeable afterwards. They reflect the state of buckling and influenced mostly by strainhardening characteristics of these tubes. At the peak load the tube buckled locally at the top end and initiated a plastic fold. The faces were observed to buckle before the corners collapsed indicating the start of plastic fold. The first fold was followed by a series of load fluctuations. Each cycle in the fluctuations (e.g. 3-5, 5-7 etc) corresponds to a plastic fold that formed in the tube at section labelled as A, B, C, D, E, F and G shown in the inset. The number of peaks in the load-displacement curve indicates the number of folds formed. The mean load during compression was calculated as the ratio of the area under the curve and the displacement is almost the same for all folds. A secondary (minor) load fluctuation was observed between the first and second primary peak only. Such fluctuation was not seen beyond the second peak. A characteristic (plasticbuckling) wavelength can be identified as, λp (=2H) as shown in the inset in which mode of deformation seen is of a concentina type, in which all the sides of the tube buckle in phase i.e. inward or outwards simultaneously. The plastic wavelength can be estimated from experiment, as the average length per fold in the deformed specimens. It was found that, λp = 33 mm, this also being approximately the displacement difference between two adjacent peaks in the load-displacement characteristic (Figure 2a). A detailed examination of the deformed specimen also indicated that, the first fold was at about 16 mm from top end as shown in the inset. From the calculation, it is noticed that the mean load is about 84 kN. Some specimens exhibited progressive incomplete diamond and concentina type folding patterns in a fold. The diamond type is where in adjacent sides of the tube were buckling out of phase (i.e. one inward and the other outward etc). Incomplete mode is a mixture of half concentina and half diamond in one plastic fold. For example, specimen HEX1 deformed in four incomplete diamond and concentina folds as can be seen from photograph in Figure 2b(i-ii). In some specimens, concentina fold formation was seen in the initial few folds and then followed by incomplete diamond and concentina folding mechanism was seen. For example in specimen HEX7, Figure 2b(ii), three concentina folds are followed by two folds of incomplete diamond and concentina. None of the specimen was seen to fold in mixed mode. Mixed mode is a complete concentina and diamond folds in the same tube. This concludes that unchamfered specimens exhibit uncertainty of folding mechanisms. A summary of results that includes mean load and plastic fold length for the entire specimen tested is shown in Table 1. This includes the specimens in which triggers are introduced to induce specific mode, i.e. concentina or diamond type of deformation. 3.3 The load-displacement characteristic for chamfered specimens Figure 3a-b shows typical load-displacement curve for in-out and all-out chamfered hexagonal tubes subjected to axial loading. In-out chamfered WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

606 Computational Methods and Experimental Measurements XIII specimen exhibits diamond mode, while all-out in concentina mode. The sequence of deformation for the case of all-out chamfers (point 0 to 15) is shown in Figure 4 corresponding to Figure 3b. Point 1 of the same figure indicates the peak load and the initiation of first plastic fold. The peak load shows 165 kN for both in-out and all-out chamfers. Table 1: Spec. no.

Summary of results of hexagonal tubes subjected to axial crushing. average

λp at mid

mean Load,

Energy absorbed W (Nm) at δ=100 mm

Mode of deformation and length of specimen, L Concentina, L=170 mm (unchamfer) 4 folds incomplete L=200 mm (unchamfer) 4 folds incomplete L=200 mm (unchamfer) Concentina, L=200 mm (All-out chamfer)

face (mm)

Fm (kN)

HEX170

33

84

8401

HEX1

36

84

8376

HEX2

35

84

8372

HEX3, 8, 11, 12

36

85

8488

HEX4

-

-

-

HEX5

36

86

8592

HEX6

-

-

-

HEX7

34

85

8477

HEX10

44

80

7939

Broke halfway, L=200 mm (in-out chamfer) Concentina, L=200 mm (all-out chamfer) Broke halfway, L=200mm (in-out chamfer) 3 folds Concentina, 2 folds incomplete, L=200mm Diamond, L=200 mm (in-out chamfer)

This means that insignificant effect on peak load for in-out and all-out chamfers, as the initial compression areas are the same. However, the load fluctuation (amplitude) for in-out chamfer is smaller than all-out chamfer by about 25 kN. Referring to Figure 3b, two secondary peaks (point 3 and 5) exist in load-displacement curve between the first and second primary peak and one between second and third primary peak. But, the secondary peaks were not seen after the third primary peak (point 8). The existing of two secondary peaks may be due to the plastic fold in contact with the top platen. However, the secondary peaks were not seen for the case of in-out chamfered specimen. A summary of results that includes the mean load, and plastic wavelength for all the specimens also is included in Table 1. It shows no significant change on with respect to mean load and plastic folding length for concentina mode for the case of all-out chamfered. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Figure 3:

607

Load-displacement curve of hexagonal chamfered tubes under quasi-static compression (a) In-out chamfer (b) All-out chamfer.

However, in-out chamfer produces diamond mode, which gives the lower mean load, thus gives less energy absorption. Figures 5a-b show the typical cross-sectional views of the deformation patterns of all-out and in-out chamfered specimen crushed under nominally identical conditions. Figure 5a illustrates an all-out chamfered specimen, exhibiting concentina folds, while Figure 5b shows cross-section view of the diamond mode seen in the crushed specimen with inout chamfers. 3.4 Effect of chamfer The effect of chamfer is observed for both all-out and in-out chamfer. The significant effect of all-out chamfer is noticed in the load-displacement curve (Figure 3b), which shows the existing of two secondary peaks at point 3 and 5. Figure 3b also show the repeatability of the peaks. No secondary peak has been found in in-out chamfer. The reason of non-existing of the secondary peak has not been explored. However, they also appear in unchamfered tubes but only one secondary peak exists at random. In general, from energy absorption point of view, the all-out chamfered tubes give a higher value of energy absorption. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Figure 4:

Sequence of deformation of axially crushed tube with all-out chamfer.

(a)

(b)

Figure 5:

Typical cross-sectional views showing deformation pattern.

3.5 Comparison with FEA and equivalent circular cylinder Figure 6 is the comparison of experimental load-displacement curves and those predicted by FEA [18,19], which are seen to be in good agreement. The FEA mean load is almost the same value with experiment. However the plastic fold length, λp(=36 mm) derived from the FEA [18,19] overestimates the experiment by 10%. The prediction of mean loads by various upperbound solutions for equivalent circular cylinder, Alexander [1] and Singace et al [4] and for hexagonal tubes, Abramowics and Wierzbicki [17] are also shown in the figure 6. The radius of an equivalent circular cylinder is assumed to be equal side length, b of hexagonal tubes. The mean load for the equivalent circular cylinders underestimates the experiment by about 30%. This is not surprising because WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


609

these solutions underestimate the mean load for the crushing of cylinder by about the same margin. Their plastic fold length, λp [4] is also underestimated by about 30%. The mean load predicted by the analysis of Abramowics and Wierzbicki [17] is 105 kN which overestimates the experiment by about 25%.

Figure 6:

4

Load-displacement curves for axial compression quasi-static loading and comparison of the mean load, Fm with previous researchers.

Conclusion

With chamfering, the tube deforms in certainty manner, i.e. in-out chamfer gives a diamond mode while all-out in concentina mode. However, a slightly higher energy absorption is obtained from concentina mode. It also shows no significant change on with respect to mean load and plastic folding length for concentina mode for the case of all-out chamfered. However, in-out chamfer produces diamond mode, which gives the lower mean load, thus gives less energy absorption. The mean load for equivalent circular cylinder underestimates the experimental result, however FEA give excellent results with experiment.

References [1] [2]

Alexander, J. M. “An approximate analysis of the collapse of thin cylindrical shells under axial loading”, Quart. J. Mech. Appl. Math, Vol.13, pp 10-14,1960 Reddy, T. Y. and Reid, S.R., “Plastic folding of cylinders under axial compression” Presented at Euro Mech. 1st European Solid Mech Conference at Munich, 1990


610 Computational Methods and Experimental Measurements XIII [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

Wierzbicki, T., Bhat, S. U., Abramowicz, W. and Brodkin, D., “Alexander revisited-a two folding elements models of progressive crushing of tubes” Int. J. Solids Struct, Vol. 29, pp. 3269-3288, 1992 Singace, A. A., El-Sobky, H. and Reddy, T. Y. “On the eccentricity factor in the progressive crushing of tubes” Int. J. Solids Struct Vol. 32, pp 3589-3602,1995 Pugsley, A. and Macaulay, M. “The large scale crumpling of thin tubes”, Quart J. Mech. Appl Maths, part 1, vol. 13, 1960 Horton, W. H., Bailey, S. C. and Edwards, A. M. “Nonsymmteric buckle patterns in progressive plastic buckling”, Experimental Mechanics, Vol. 6, No. 9. pp 433-444, 1966 Johnson, W., Soden, P. D and Al-Hassani, S. T. S. “Inextensional collapse of thin-walled tube under axial compression”, J. Strain Analysis Vol.12, No.4, 1977 Tvergaard, V. “On the transition from a diamond mode to an axisymmetric mode of collapse in cylinder shells” Int. J. Solids Struct, Vol.19, No.10, pp 845-8561983 Andrews, K. R. F, England G. L. and Ghani, E. “Classification of the axial collapse of cylindrical tubes under quasi-static loading” Int. J. Mech. Sci. Vol. 25, No. 9-10, pp. 687-696, 1983 Macaulay, M. A. and Redwood, R. G. “Small scale model railway coaches under impact”, Engineer, 1041-1046, 1964 Postlethwaite, H. E. and Mills, B. “Use of collapsible structural elements as impact isolators with special reference to automotive application”, J. Strain Analysis, 5, pp 58-73, 1970 Wierzbicki, T. and Abramowicz, W. “On the crushing mechanics of thin walled structures”, J. Appl Mech, Vol. 50, pp 727-734, 1983 Reid, S. R. and Reddy, T.Y. “Axial crushing of foam-filled tapered sheet metal tubes”. Int. J. Mech. Sci., Vol.28, No.10, pp 643-656, 1986 ] Reid, S. R., Reddy, T.Y. and Gray, M. D. “Static and dynamics axial crushing of foam-filled sheet metal tubes”, Int. J. Mech. Sci., 28, pp. 295322, 1986 Reddy, T. Y. and Al-Hassani, S. T. S., “Axial crushing of wood-filled square metal tubes”. Int. J. Mech. Sci., Vol.35, No.3/4, pp 231-246,1993 Said, M. R. “Axial compression of empty and filled rectangular tubes”. MSc. Dissertation, UMIST, U.K., 1988 Abramowicz, W. and Wierzbicki, T. “Axial crushing of multicorners sheet metal columns” J. App. Mech., Trans ASME., 56, pp 113-120, 1989 Said, M. R. “Energy absorption in certain cellular structures under uniaxial and biaxial loading” PhD. Thesis, UMIST, U.K., 2000 Said, M. R. Ahmad R. and Alias A. “ Finite Element Analysis of hexagonal tubes structure under axial loading” Proceeding in 1st International Conference On Safety and Security Engineering, pp 83-91, Rome, Italy, 2005



611

Application of the shakedown analysis in the elastic: plastic assessment of cracked plates M. A. Belouchrani Materials laboratory, E.M.P., BP 17C Bordj El Bahri Alger, Algeria

Abstract In this work, a new method in the elastic-plastic calculation of cracked plates, based on shakedown analysis, is presented. This method presents an interesting alternative to classical methods in the structure design, especially when the loading is variable. It permits, via a simple elastic calculation and a mathematical optimisation, the determination of a load domain situated between the elastic and the limit load domains, within which the loading can evolve arbitrarily, while insuring the reliability of the cracked structure versus the plastic ruin and the unstable crack propagation. Also, by taking into account the microstructure of the material and the residual stresses, this method allows the determination of a stress intensity factor corresponding to the shakedown state. This factor is comparable to the fatigue threshold and can be used in the crack admissibility criteria. Keywords: shakedown, crack, notch, finite elements analysis, stress intensity factor, fatigue threshold.

1

Introduction

The structures used in the aeronautics, nuclear plants and naval yards are generally plane forms, originally, constituted of ductile material plates (metals and alloys) that could support irreversible plastic deformations before to break, and that even when they contain manufacture defects or cracks. Cracks constitute the major problem of designers, because they can falsify completely the prediction of the conceived structure behavior, by accelerating their ruin through brutal propagation of these cracks. When a crack is detected in a system, the designer undertakes an analysis, where he will have to explore the brutal rupture risk, the plastic ruin risk and the foreseeable evolution of the fissure. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070611

612 Computational Methods and Experimental Measurements XIII The most often, because of the geometrical singularity fathered by the crack, one uses the fracture mechanics, that has replaced the classical design methods. Nevertheless, one knows that the fracture mechanics alone cannot insure that a crack is admissible; it attends only what happens at the vicinity of the crack tip (front), where the strong constraints and deformations can cause plastic yield of the material and therefore the propagation of the crack. Criteria of defect admissibility of the fracture mechanics are naturally local, attached to event being able to happen at the crack tip. One knows also that even flawless, a structure cannot support unlimited loads, it finishes by bankrupting. To take into account these two modes of ruin and to anticipate the stability and the security of the cracked structure, Belouchrani and Weichert [1][2], have proposed a new approach by the application of the shakedown theory, that finds its preferential application framework when the loading is variable. This new approach consists of an extension of the static shakedown theorem to the cracked structures. The goal of this work is to present this new prediction method of the cracked plate security and reliability, by using the finite elements method and the nonlinear mathematical optimization. The crack is assimilated to a sharp notch according to Neubers material block concept [3].

2

Formulation of the shakedown theorem for a cracked body

It has been suggested [1,2], that a cracked body shakes down with respect to a given loading history, if a time-independent state of residual stress ρ D ( x ) exists, such that for all times t > 0 : ρ Dij, j = 0

in Ω

(1)

n j ρ Dij = 0

on Γσ

(2)

in Ω

(3)

(

F

σ ijc ( t ) + ρ Dij , σ y

)< 0

With a supplementary condition imposed on the admissible length of a typical micro-crack in the material a lim < a c (4) Here, F is the plastic yield surface assumed of Von-Mises type, convex by definition, σ y is the yield stress and a lim is the largest admissible crack length determined by means of shakedown analysis. In inequality (3), σ c ( x , t ) is the time-dependent stress state for a purely elastic comparison problem, differing from the original problem only by the fact that the material reacts purely elastically with the same elastic moduli as for the elastic part of the material law in the original problem. For the Lemaitre-Chaboche model [4] adopted for ductile fracture, a lim is given by [1,2] a lim

m +1 1 α = a0 +   m K α −1

∫ (

)

1 D  ρ ij L ijkl ρ Dkl dΩ Ω2 


m

( m +1)

(5)


613

Here, a 0 is the initial crack length, L is the positive definite and timeindependent tensor of elastic moduli, m and K are material constants characterizing the R-curve parameters and α is the shakedown safety factor. As mentioned before, the problem of stress singularity of the elastic stress field deserves special attention for the application of the shakedown theorem to cracked structures. In this case, no time-independent field of residual stresses ρ ° ( x ) satisfying inequality (3) can be found and classical shakedown theory does not deliver comprehensive results, even for loads for which limit states physically exist. We bypass this problem by assimilating the crack tip to a notch, following the concept of material block introduced by [3] and used in the same spirit as in the present work by [5]. So, following [6], the stress distribution in the neighborhood of the root of the notch given by rf = r + nε (6) with, according to figure 1, rf as the effective notch root radius, ε as the length of the Neuber material block (assumed to be a material constant), and n as the factor depending on the loading mode. The factor n is equal to 2 in mode I. Following this concept, the effective notch radius is equal to the original notch radius augmented of n times the dimension of the Neuber material block. ε

ε

r

ρf

rf

Figure 1:

Modified notch and crack.

In the case of a sharp crack, the radius at the root of the effective crack, denoted ρ f , can be obtained just by putting r = 0 in Eqn. (6) (7) ρ f = nε Eqn. (7) indicates implicitly that the crack can be treated as a notch with ρ f radius. Physically, Neubers material block may be explained as being the sum of the minimum number of individual microscopic material particles (such as grains in polycrystalline metals). The properties of which may differ from each other, but in average they should have the property of the macroscopic material. In [5], ρ f is put to be about ten times the size of a grain, for the mode I loading. Following this suggestion we write (8) ε ≈ 5ξ

3

Shakedown safety factor

We consider an elastic-plastic plate, subjected to uniaxial loads P(t). The values of P(t) vary arbitrarily with time t, but remain between prescribed loads Pmin WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

614 Computational Methods and Experimental Measurements XIII and Pmax . One then looks for the maximum value of the load factor α , such that the plate will shake down under the load αP( t ) . This load factor will be called the shakedown load factor α SD and can be determined as solution of the following optimization problem α SD = maxD α (9) α ,ρ

with the subsidiary conditions ρ Dij, j = 0

in Ω

(10)

n j ρ Dij = 0

on Γσ

(11)

F(ασ c (P) + ρ D , σ y ) < 0 in Ω ∀P ∈ [Pmin , Pmax ]

(

)

1 D  m +1 1 α  ρ ij L ijkl ρ Dkl dΩ  a lim = a 0 +  ∫ Ω α − m K 1 2  

4

m

( m +1)

(12) < ac

(13)

Fatigue threshold and influence of the microstructure

In what follows, the classical concept of existence of a threshold K S for the stress intensity factor K I in opening mode I is used. K I = σ πa (14) Therefore, it is assumed that the micro-crack propagation does not occur if K I < KS (15) and the material state is safe against failure by fatigue. Among other parameters, the fatigue threshold is influenced by the microstructure of the material. Generally, it is admitted that K S increases with the size of the grain as has been observed on ferritic steels by [7–9]. It has been suggested that the dependence between K S and ξ1 2 (ξ is the diameter of the grain) are related by a linear function [7–9] K S = a 1 + b1 ξ1 2 (16)

where the dimensions of K S and ξ are, respectively, [MPa.m1/2] and [m]. Here, a 1 and b1 are material constants.

5 Shakedown stress intensity factor KSD With the shakedown load factor α SD computed for a cracked plate loaded in mode I, we will compute the stress intensity factor (Eqn. (14)) corresponding to the shakedown state by [10] K SD = α SD P πa lim


(17)


6

615

Numerical examples

Here, the finite element method is used combined with mathematical programming method to maximize the load factor under the subsidiary conditions (10)–(13) such that the yield criterion F is not violated in any point of the plate and that the admissible crack length a lim remains inferior to its critical value ac. The resolution of this mathematical programming problem is performed using the code LPNLP [11] that is based on an augmented Lagrangian method and the BFGS-algorithm. For the solution one needs: − the solution of the purely elastic comparison problem in the sense of shakedown analysis, − the construction of a time-independent residual stress field. To this end, we use the finite element force method based on the principle of minimum complementary energy [12]. This approach uses stress functions for the construction of the complementary energy function and represents an algebraic dual to the finite element displacement method. This method has been used by [13] for the study of limit and shakedown analysis of two-dimensional structures. 6.1 Shakedown loads domain: We consider a rectangular cracked plate, subjected to uniform loading (fig.2):

Figure 2:

Rectangular plate with lateral crack.


616 Computational Methods and Experimental Measurements XIII The obtained results (fig. 3) show that the safety load is reduced in the presence of crack; this reduction depends on the crack length. The load P2 amends adversely the domain of load and reduces the limits of the load obtained. While under the load P1, the influence of the crack is lesser because the stress concentration is less important, and is situated on crack lips, not in front of the crack tip. Figure 4 shows well that in the case of a cracked plate, the dimensioning based on the limit analysis is not recommended if the applied loads could vary in an independent way. 0.4

0.32

0.24

P2 P0 0.16 Shakedown domain a = 0.2 W a = 0.3 W a = 0.4 W a = 0.5 W

0.08

0

0

0.2

Figure 3:

0.4

P1 P0

0.6

0.8

1

Shakedown domain.

0.75 a = 0.5 W Limit analysis Shakedown 0.6

0.45

P2 P0 0.3

0.15

0

0

Figure 4:

0.2

0.4

P1 P0 0.6

0.8

1

Shakedown and load limits domains.



617

1.25 Shakedown Limit analysis Limit analysis (analytic) 1

Load factor

0.75

0.5

0.25

0

0

0.1

0.2

0.3

0.4

0.5

a/W

Figure 5:

Influence of the ratio a/W on the load factor. 12.5

10

7.5

K SD 5

2.5

0

0

0.2

0.4

0.6

0.8

1

a/W

Figure 6:

Independence of K SD with a/W.

Figure 5 shows that for the cracked plate subjected to the load P2, the value of the limit load values agrees with the analytical ones. 6.2 Shakedown stress intensity factor KSD We have computed the stress intensity factor at the shakedown state for different crack lengths, the obtained results (fig 6), show that this factor K SD is crack length independent, and it can be considered as the safety parameter against the cracked structure ruin, by plasticity and crack propagation for the mode I of loading. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

618 Computational Methods and Experimental Measurements XIII 0.075

0.06

0.045

K SD σy 0.03

0.015

0

0

Figure 7:

2

4

1

ξ2

6

8

10

Relation between K SD and ξ1 2 .

We have then studied the influence of the grain size on the shakedown stress intensity factor K SD , figure 7 shows that the ratio K SD / σ y varies linearly with ξ1 2 . K SD = a + bξ1 2 . σy

(18)

By comparing this equation with the equation giving the fatigue threshold, one notes that the yield stress influences the shakedown stress intensity factor. This can be explained by the fact that the shakedown intensity factor is computed by taking into account the yield stress and the largest admissible crack length. 6.3 Comparison of KSD and the fatigue threshold values for some materials To validate the proposed approach, a comparison is made between the values of K SD computed in the case of a rectangular plate solicited in mode I, the fatigue threshold K S given by [7] and the shakedown stress intensity factor K sh given by [5] for some materials. The characteristics of these materials are given in the Table 1. According to the results given in the Table 2, one remarks that the values of K SD agree with the values of K sh given by [5]. On the other hand, one notices a disparity with the fatigue threshold K S given by [7] for materials A and E. However, the results indicate that indeed K SD can be considered as fatigue threshold. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Table 1:

Diameter of the grain and mechanical material data.

Material

Designation

ξ (µm)

σ y (MPa)

Docol 350 SS 141147 HP steel HP steel HP steel

A B C D E

8 15 29 45 82

260 185 210 160 120

Table 2:

619

Fatigue threshold and shakedown stress intensity factors.

Material A B C D E

KS 5.4 6.0 6.2 6.7 8.2

K sh 8.9 7.6 9.5 8.1 7.8

K SD 8.43 6.49 8.22 6.8 5.73

More, knowing that the relationship existing between the yield stress and the diameter of the grain has the following form [7,8]: σ y = c + dξ

−

1 2

,

(19)

σ y increases when the diameter of the grain decreases and since the residual

stress intensity increases with the increase of the elastic limit, one can say, considering the equation (16), that disparities observed between K SD and K S values are mainly caused by the residual stresses. One concludes that the computed stress intensity factor can be considered as a fatigue threshold taking into account the residual stresses.

7

Conclusion

In this work, we have presented a new method in the prediction of the inelastic cracked structures failure by the application of the shakedown analysis. This new method presents several advantages, among which, the facility of computation since one undertakes only an elastic calculation. Also, it allows going more far than the elastic limit in the structures design. We can also compute the fatigue threshold using this method.

References [1]

Belouchrani M. A., Contribution to the shakedown analysis of inelastic cracked structures, PhD Thesis, University of Lille, (1997).


620 Computational Methods and Experimental Measurements XIII [2] [3] [4] [5] [6] [7]

[8] [9] [10] [11] [12] [13]

Belouchrani M. A. and Weichert D., An extension of the static shakedown theorem to inelastic cracked structures, Int. J. Mech. Sci., 41, 163-177, (1998). Neuber H., Über die Berücksichtigug der Spannungskonzentration bei Festigkeitsberechnungen, Konstruktion 20 (7), 245-251, (1968). Lemaitre J. and Chaboche J. L., Mécanique des matériaux solides, Dunod, Paris, (1985). Huang Y. and Stein E., Shakedown of a cracked body consisting of kinematic hardening material, Eng. Frac. Mech, 54, 1, 107-112, (1996). Creager M., Master Thesis, Lehigh University, (1966). Wasen J., Hamberg K. and Karlsson B., The influence of Grain Size and Fracture Surface Geometry on the Near-threshold Fatigue Crack Growth in Ferritic Steels, Materials Science and Engineering A, 102, 217-226, (1988). Xu-Dong Li and Edwards L., Theoretical Modelling of Fatigue Threshold for Aluminium Alloys, Eng. Frac. Mech., 54, 35-48, (1996). Radaj D. and Zhang S., On the relation between notch stress and crack stress intensity in plane shear and mixed mode loading, Eng. Frac. Mech., 44, N°5, 691-704, (1993). Belouchrani M. A., Weichert D. and Hachemi A., Fatigue threshold computation by shakedown theory, Mech. Resea. Comm., vol. 27, N° 3, pp. 287-293, (2000). Pierre D. A. and Lowe M. J., Mathematical programming via Augmented Lagrangians, London: Addison-Wesley, (1975). Gallagher R. H. and Dhalla A. K., Direct flexibility finite element elastoplastic analysis, Englewood Cliffs, New Jersey, USA, (1975). Gross-Weege J. and Weichert D., Elastic-plastic shells under variable mechanical and thermal loads, Int. J. Mech. Sci., 34, 863-880, (1992).


Section 7 Experiments and analysis of reinforced concrete members (Special session organised by Professor C. G. Karayannis)



623

Cyclic testing of reinforced concrete beam-column joints with crossed inclined bars C. E. Chalioris, C. G. Karayannis & M. I. Favvata Department of Civil Engineering, Democritus University of Thrace, Greece

Abstract The use of crossed inclined bars in external beam-column connections under cyclic deformations is experimentally investigated. For this purpose, test results of four Reinforced Concrete (RC) joint subassemblages subjected to constantly increasing pseudo-seismic loading are presented. The shear reinforcement in the joint area for two specimens was two pairs of inclined bars that formed a pair of X-type reinforcement. The other two specimens were conventionally reinforced joints (control specimens). The effectiveness of this X-type, non-conventional reinforcement on the overall seismic performance of the tested joints is examined. The beam and the columns of all the specimens were designed according to the requirements of ACI 318-02 and the recommendations of ACI-ASCE 352-02 (Type 2 exterior connections). The design of the joint area for one control specimen was also carried out according the ACI Design Codes and the required amount of steel stirrups (5∅8) was added in the joint body. The other control specimen had no stirrup at the joint area. Comparisons between the test results of the examined specimens indicated that the cyclic behaviour of the joints with X-bars was ameliorated with respect to the response of the control specimen without stirrups. Further, load capacity and hysteretic energy dissipation values of the joint with 2X-bars ∅14 were slightly lower than the values of the control specimen which joint area had stirrups (5∅8) according to the specifications of ACI Design Codes. Keywords: beam-column joint, cyclic tests, reinforced concrete, inclined bars.



1

Introduction

The behaviour of beam-column joints has long been recognized as a significant factor that frequently becomes critical for the overall behaviour of Reinforced Concrete (RC) framed structures subjected to seismic excitations. The response of RC connections involves the influence of complex interacting phenomena such as shear, bond, confinement, fatigue, which even independently are not yet well understood [1]. Therefore, experimental research is very often the main method for the investigation of the parameters that influence and improve the joint performance [2]. It is also noted that detailed design recommendations for RC beam-column joints were first published in the last two decades in Europe and in the last three decades in USA. Attempts at any improvement of the seismic properties of these members are mainly focus on the use of non-conventional reinforcement, such as steel fibres, composite materials (FRPs), inclined bars and spiral reinforcement [3]. The first idea of the use of crossed inclined bars in RC joints was developed in 1984 [4]. Since then, further experimental and analytical studies indicated that joints with X-type reinforcement exhibited improved behaviour in respect to joints with conventional reinforcement [5–8]. The aim of the present work is to experimentally investigate the effectiveness of crossed inclined bars in external beam-column connections subjected to cyclic loading. Four RC joint specimens with different reinforcement arrangements in the joint area, that included X-bars and stirrups, were tested to the same constantly increasing cyclic loading sequence and useful concluding remarks were derived.

2


The experimental program of this paper includes 4 external beam-column joint specimens. The geometry of the specimens was the same; total columns height and cross-section dimensions 1800 mm and 300×200 mm, respectively, whereas beam length and cross-section dimensions were 1300 mm and 200/300 mm, respectively. The beam and the columns of all the specimens were designed according to the specifications of ACI 318-02 [9] and ACI-ASCE 352-02 [10] for Type 2 exterior connections (see Table 1 and Figure 1). The shear reinforcement of the joint area varied. First, control specimen JA-0 had no stirrups (2 vertical ∅10 column bars were placed through the joint area as in all the specimens). The joint area of control specimen JA-s5 was designed according the ACI Design Codes and 5∅8 steel stirrups were added (required amount of stirrups). Finally, specimens JA-2X12 and JA-2X14 had two pairs of inclined bars ∅12 and ∅14, respectively, which formed a pair of X-type reinforcement (Figure 1). Geometry and detailed reinforcement arrangement of the tested specimens are shown in Figure 1 and presented in Table 1. Concrete mean cylinder compressive strength at the age of 28 days was fcm = 34 MPa and steel yield strength was 580 MPa. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Table 1:

Reinforcement arrangement of the beam-column joint specimens.

Specimen

Columns

JA-0

(+ 2∅10 middle bars) Stirrups: ∅8/50 mm

JA-s5

JA-2X12

JA-2X14

625

4∅14 (corner bars)

4∅14 (corner bars) (+ 2∅10 middle bars) Stirrups: ∅8/50 mm

4∅14 (+ 4∅12)* (corner) (+ 2∅10 middle bars) Stirrups: ∅8/50 mm

4∅14 (+ 4∅14)* (corner) (+ 2∅10 middle bars) Stirrups: ∅8/50 mm

*

Beam 4∅12 top 4∅12 bottom Stir.: ∅8/65 mm

4∅12 top 4∅12 bottom Stir.: ∅8/65 mm



Joint area (2∅10 vertical bars)

5∅8 (∅8/50 mm) (+ 2∅10 vertical bars)

2X∅12 (+ 2∅10 vertical bars)

2X∅14 (+ 2∅10 vertical bars)

The inclined X-bars in joint area were extended to the entire height of both columns. Test setup and instrumentation details are shown in Figure 2. Supports that allow rotation were used to simulate the inflection points assumed to occur at a point of the columns in a laterally-loaded frame structure. Column axial load with value equal to Nc = 0.05×Ac×fcm was applied during the tests. All specimens were subjected to full cycle deformations imposed near the free end of the beam by a pinned-end actuator. The moment arm for the applied load was equal to 1.2 m. Tested specimens were suffered a loading history of five full loading steps with maximum displacements ± 6 mm, ± 20 mm, ± 40 mm, ± 60 mm and ± 80 mm at each step. Every loading step included two full load cycles, thus, the loading sequence was performed the way it is shown in Figure 3. The imposed load was measured by a load cell with accuracy equal to 0.025 kN and the displacements of the beam were measured by linear variable differential transducer (LVDT) with accuracy equal to 0.01 mm.

3 Test results The hysteretic responses of the tested specimens are compared in Figures 4 and 5. Further, the values of the maximum bending moment (load capacity) and the hysteretic energy dissipation in terms of the area of the response loading cycle for each loading cycle and for each joint are presented in Table 2.



1502

200

72

5∅8

65

130

4∅12

in joint

300

1800

JA-s5

4∅12

50

600

200

1300

80

260

∅8

JA-0 130

50

JA-s5 4∅14+2∅10 240

160

200

300

80

∅12 − ∅14

724

∅8

182

65

130

300

1800

36 0

352

4∅12

4∅12

50

600 1300

200

724

JA-2X12 50

JA-2X14

Figure 1:

Geometry and reinforcement arrangements of the tested specimens.



627

Load frame

actuator

Nc

load cell

actuator

LVDT load cell

P

1.20

+ 1.80 1.50

-

LVDT 1.30

LVDT

Figure 2:

Test setup.

80

Deformation (mm)

60 40 20 0 -20 -40 -60 -80

Figure 3:

Loading history.

In order to evaluate the effectiveness of the examined reinforcement in the joint area, the final cracking patterns of the tested joints are illustrated in the photos of Figure 6. Based on the damage modes and the observations of cracking propagation during the entire cyclic loading procedure of the specimens, it is deduced that crossed inclined bars inhibited the damage in the joint area and considerably improved the damage mode characteristics (distinct plastic hinges were formed). Further, the presence of stirrups (specimen JA-s5) proved to be


Moment (kNm)

628 Computational Methods and Experimental Measurements XIII 70 60 50 40 30 20 10 0 -80

-70

-60

-50

-40

-30

-20

-10 0 -10

10

20

30

40

50

60

70

80

Deformation (mm)

-20 -30 -40

JA-0

-50

JA-2X12

Moment (kNm)

-60 -70 70 60 50 40 30 20 10 0 -80

-70

-60

-50

-40

-30

-20

-10 0 -10

10

20

30

40

50

60

70

80

Deformation (mm)

-20 -30 -40

JA-0

-50

JA-2X14

-60 -70

Figure 4:

Comparison of the hysteretic responses of the tested specimens.


Moment (kNm)


629

70 60 50 40 30 20 10 0

-80

-70

-60

-50

-40

-30

-20

-10 0 -10

10

20

30

40

50

60

70

80

Deformation (mm)

-20 -30 -40

JA-s5

-50

JA-2X14

-60 -70

Figure 5:

Comparison of the hysteretic responses of joints JA-s5 and JA2X14. Table 2:

Test results.

JA-0 JA-s5 JA-2X12 JA-2X14 Cycle Moment Energy Moment Energy Moment Energy Moment Energy (kN⋅m)

6-1+ 6-16-2+ 6-220-1+ 20-120-2+ 20-240-1+ 40-140-2+ 40-260-1+ 60-160-2+ 60-280-1+ 80-180-2+ 80-2-

39.8 -35.6 38.9 -32.4 62.6 -62.8 56.7 -54.2 64.6 -58.5 44.5 -38.7 38.6 -30.8 22.5 -19.6 -

(kN⋅m⋅mm)

178.1 149.4 1267.8 535.3 3454.6 1454.4 2164.8 899.4 -

(kN⋅m)

36.7 -41.4 35.7 -38.7 60.4 -61.2 54.2 -60.0 60.9 -64.2 59.2 -63.3 63.0 -63.6 59.0 -58.0 36.7 -38.8 15.3 -25.0

(kN⋅m⋅mm)

215.2 158.1 1405.2 1095.3 4013.4 3917.2 6644.7 4613.3 5414.0 2975.3

(kN⋅m)

33.7 -34.6 31.1 -31.1 62.6 -61.1 58.3 -59.0 64.2 -60.0 56.4 -57.0 55.8 -49.2 30.6 -27.0 13.8 -22.2 10.2 -16.4

(kN⋅m⋅mm)

206.7 227.8 985.9 885.6 3207.2 2625.9 4385.0 2143.6 1913.5 1202.5


(kN⋅m)

33.8 -33.8 33.1 -32.9 63.0 -61.2 60.6 -58.9 65.4 -64.6 58.2 -64.3 59.4 -65.3 52.2 -50.3 36.6 -27.9 15.0 -13.0

(kN⋅m⋅mm)

275.5 196.3 969.1 767.3 3817.6 3479.9 5975.9 4417.7 3694.5 1731.7


JA-2X12 JA-0

JA-s5 Figure 6:

JA-2X14

Damage modes and crack patterns at the end of the loading sequence.

essential since stirrups increased the joint shear capacity and restrained the deformations of the bend anchorage of the beam’s bars that caused severe damages at the back of the joint area in the other specimens without stirrups.

4

Concluding remarks

Cyclic test results of four RC external beam-column joint specimens were presented herein, in order to investigate the effectiveness of crossed inclined bars as shear reinforcement in joint area. The specimens were designed according the specifications of ACI 318-02 and ACI-ASCE 352-02. Based on the hysteretic WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


631

responses and the cracking modes of the tested joints it is deduced that the overall seismic performance of the joints with X-type reinforcement was considerably improved with respect to the response of the control specimen without stirrups. Joints with X-bars, in comparison with the control specimen without stirrups, exhibited higher values of load capacity in most of the loading cycles and increased hysteretic energy dissipation practically in the entire loading sequence. This improvement was greater in the higher deformation loading cycles. Further, the hysteretic response of the specimen with 2X∅14 as shear reinforcement in joint area was slightly lower from the response of the control specimen with 5∅8 stirrups, which followed the guidelines of ACI Design Codes. Concerning the cracking patterns, the deformations of the bend anchorage of the beam’s bars (anchorage failure) in combination with the absence of stirrups in the joint area contributed to significant damage of the concrete cover at the back of the joint area. Stirrups restrained these deformations and kept the joint body quite intact. Specimens with X-bars performed enhanced damage mode since distinct flexural hinge was developed in the beam-joint interface.

References [1]

[2]

[3]

[4] [5] [6] [7]

Karayannis, C.G. & Sirkelis, G.M., Effectiveness of RC beam-column connections strengthening using carbon-FRP jackets, Proc. of the 12th European Conference on Earthquake Engineering, London, UK, PR 549, 2002. Karayannis, C.G., Sirkelis, G.M. & Chalioris, C.E., Seismic performance of RC beam-column joints retrofitted using light RC jacket - Experimental study, Proc. of the 1st European Conference on Earthquake Engineering and Seismology, Geneva, Switzerland, PN.136, 2006. Karayannis, C.G. & Sirkelis, G.M., Response of columns and joints with spiral shear reinforcement, Proc. of the 12th Int. Conference: Computational Methods and Experimental Measurements (CMEM), Malta, Wessex Institute of Technology Transactions on Modelling and Simulation, Vol. 41, 2005. Paulay, T. & Park, R., Joints of reinforced concrete frames designed for earthquake resistance. Research report 84-9, Dept. of Civil Engineering, Univ. of Canterbury, Christchurch, New Zealand, 1984. Tsonos, A.G., Tegos, I.A. & Penelis, G.G., Seismic resistance of Type 2 exterior beam - column joints reinforced with inclined bars, ACI Structural Journal, 89(1), pp. 3-12, 1992. Karayannis, C.G., Chalioris, C.E. & Sideris, K.K., Effectiveness of RC beam - column connection repair using epoxy resin injections. Journal of Earthquake Engineering, 2(2), pp. 217-240, 1998. Bakir, P.G., Seismic resistance and mechanical behaviour of exterior beam - column joints with crossed inclined bars. Journal of Structural Engineering and Mechanics, 16(4), pp. 493-517, 2003.



[9] [10]

Tsonos, A.G., Improvement of the earthquake resistance of R/C beam column joints under the influence of P-∆ effect and axial force variations using inclined bars, Journal of Structural Engineering and Mechanics, 18(4), pp. 389-410, 2004. ACI Committee 318. Building code requirements for structural concrete (ACI 318-02) and commentary (ACI 318R-02). American Concrete Institute, Farmington Hills, Mich., 2002. ACI-ASCE Committee 352. Recommendations for design of beamcolumn connections in monolithic reinforced concrete structures (ACI 352R-02). American Concrete Institute, Farmington Hills, Mich., 2002.



633

Tests and analysis of reinforced concrete beams under torsion retrofitted with FRP strips C. E. Chalioris Department of Civil Engineering, Democritus University of Thrace, Greece

Abstract The aim of the present study is twofold. First, the results of an experimental investigation on the torsional response of reinforced concrete (RC) beams retrofitted with epoxy-bonded carbon fibre-reinforced-polymer (FRP) strips as external transverse reinforcement are presented. Second, an analytical approach for the estimation of the torsional capacity and the entire behaviour of RC beams retrofitted with FRP fabrics is attempted. The experimental program comprises eight beams with a rectangular cross-section tested under pure torsion. Based on the measured values of the torsional moment at cracking and at ultimate and the behavioural curves of the beams, useful conclusions concerning the effectiveness of the FRP strengthening technique are indicated. Further, in order to describe the entire torsional behaviour of RC beams retrofitted with FRP, a method that employs the combination of two different analytical theories is used. The prediction of the elastic till the first cracking part is achieved using a smeared crack analysis for plain concrete in torsion, whereas for the description of the post-cracking response a modified softened truss model is proposed. The wellknown softened truss model is properly modified to include the influence of the FRP fabrics on the torsional response as external reinforcement. Analytical torsional behaviour curves are compared with experimental ones from the present study and the literature, and first concluding remarks are mentioned. Keywords: beams, reinforced concrete, fibre-reinforced polymers (FRP), retrofit, model, torsion, tests.



1

Introduction

Torsion occurs more frequently in modern structures but rarely occurs alone. However, torsion forms one of the basic structural actions besides flexure, shear and axial force. Torsional failure of concrete members is initiated by the tensile stress developed due to a state of pure shear, which arises due to torsion [1]. Epoxy-bonded Fibre-Reinforced Polymers (FRP) materials to the surface of Reinforced Concrete (RC) members are considered as external reinforcement that can bear the developed tensile stresses. The advanced mechanical properties and the easy-to-apply character of these composite materials inspired the researchers to study their use on strengthening and rehabilitation of deficit concrete elements. A growing number of studies and applications in this area has been carried out for the flexural and shear strengthening of RC subassemblages under monotonic and cyclic deformations [2–4]. Nevertheless, the conducted research on the torsional upgrading of RC beams using FRP materials is quite limited and has preliminary and exploratory character so far. Earlier investigations indicated that FRP materials caused a significant increase on the torsional capacity of the tested beams [5–7]. Torsion of plain, steel reinforced and fibre reinforced concrete beams has extensively been studied by several experimental and analytical researches [8–11]. The present study deals with the upgrading of torsional resistance of RC beams using FRP fabrics, which is still an open field of study. The recent increase interest for the use of these materials, the catastrophic character of the torsional failure and the lack of relative studies are the main motives behind this effort. The objective of this work has two parts; an experimental and an analytical one. The experimental program consists of 8 specimens and aims to evaluate the effectiveness of the use of epoxy-bonded carbon FRP strips as external transverse reinforcement to RC beams subjected to pure torsion. The analytical part of the study employs a method that has already been used for the prediction of the entire torsional response of RC beams [12, 13]. This model derives from the combination of two well-known theories [1, 14] and it is properly modified herein, in order to include the influence of the FRP materials on the torsional behaviour.

2


The experimental program comprises 8 beams of rectangular cross-section sorted in two groups (Ra and Rb) and tested under pure torsion action. The crosssection dimensions of the beams of group Ra was 100/200 mm, whereas the beams of group Rb had cross-section dimensions 150/300 mm. Each group comprises four specimens. Two of them were strengthened using epoxy-bonded carbon FRP strips that wrapped around the rectangular cross-section of the beams as external transverse reinforcement. The longitudinal reinforcement of all



635

the tested specimens was the same; four deformed bars of diameter 8 mm (4∅8) at the corners of the rectangular cross-section. The transverse reinforcement of the beams varies (none, steel stirrups or/and FRP strips) as shown in Table 1. Reinforcement arrangement of all the tested beams is also presented in Table 1. The concrete compressive and tensile strength values were measured from supplementary compression and splitting tests, respectively, and equal to 27.5 MPa and 2.8 MPa for the beams of group Ra and 28.8 MPa and 2.9 MPa for the beams of group Rb. Steel yield strength was 560 MPa for the ∅8 deformed longitudinal steel bars and 350 MPa for the ∅5.5 mild steel stirrups. Unidirectional carbon FRP fabrics with thickness tf = 0.11 mm per ply were used as external transverse reinforcement (SikaWrap-200C). The fibre direction oriented perpendicular to the longitudinal axis of the beam. According to the FRP supplier data, the elastic modulus, the ultimate tensile strength and the elongation at failure of the fibres were 230 GPa, 3900 MPa and 1.5% mm/mm, respectively. A two-component rubber-toughened cold-curing-construction epoxy adhesive with density 1310 kg/m3, elastic modulus 3800 MPa and tensile strength 30 MPa was used for bonding the FRP strips to concrete (Sikadur 330).

Table 1:

Reinforcement and concrete strength of tested beams.

Beam code name

Steel bars (mm)

Steel stirrups (mm)

nf

Ra-c Ra-Fs150(2) Ra-S Ra-SFs150(2) Rb-c Rb-Fs200(1) Rb-S Rb-SFs200(1)

4∅8 4∅8 4∅8 4∅8 4∅8 4∅8 4∅8 4∅8

∅5.5/130 ∅5.5/130 ∅5.5/100 ∅5.5/100

2 2 1 1

FRP strips wf (mm) 150 150 200 200

sf (mm) 300 300 400 400

Af/sf (mm) 0.110 0.110 0.055 0.055

ρ sAn

ρ stn

ρ nft

(%)

(%)

(%)

1.01 1.01 1.01 1.01 0.45 0.45 0.45 0.45

0.38 0.38 0.38 0.38

0.33 0.33 0.11 0.11

All tested specimens had common total length equal to 1.60 m and were tested monotonically under pure torsion. Details about the experimental apparatus and the test procedure can be found in references 7 and 13. The values of torque moment at cracking and ultimate torque moment, as measured from the tests, are presented in Table 2. Further, the entire torsional behaviour of the beams is also presented in Figure 1 in terms of torsional moment versus angle of twist per unit length experimental curves.



Torsional moment (kN.m)

5 Ra-SFs150(2) 4 3

Ra-Fs150(2) Ra-S

2 1 0 0.00

Ra-c

0.05

0.10

0.15

Angle of twist per length (rad/m)


10 Rb-SFs200(1) 8

Rb-Fs200(1) Rb-S

6 Rb-c

4 2 0 0.00

0.05

0.10

0.15

Angle of twist per length (rad/m) Figure 1:

3

Experimental curves of the tested beams.

Behavioural model and comparisons with test results

The prediction of the entire torsional behaviour of RC beams strengthened with FRP strips is achieved using a combined approach that has already been proposed for RC beams [12, 13]. In the present study this method is extended to include the influence of FRP fabrics. The method employs the combination of two different analytical models, since the torsional response of RC members consists of a pre-cracking and a post-cracking part and justifies this WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


637

simulation [13]. The elastic till the first cracking part is described by a smeared crack analysis for plain concrete in torsion [1] and the post-cracking part is described by a softened truss model [14]. It is justified that for the elastic till the first cracking region the percentage of reinforcement has a minor effect on the torsional response and RC elements behave, more or less, as plain concrete members. Therefore, the analytical smeared crack model for plain concrete in torsion proposed by Karayannis [1] is applicable to RC beams for the prediction of the first elastic part till the developing of concrete cracking. The model is based on an analytical technique that employs constitutive relations expressed in terms of normal stress and crack width, for the behaviour of the crack process zones. Detailed derivation of the equations and the solution technique of this theory can be found in reference 1. For the calculation of the torsional behaviour of a RC beam strengthened with FRP strips in the post-cracking region the basic equations and considerations of the softened truss model [14] are adopted and properly modified. This method relies on solving three equilibrium and three compatibility equations along with the constitutive laws of an element taken from a member subjected to pure torsion. The equations of equilibrium and compatibility are based on the assumption that the material is continuous. Especially for the concrete in compression, it is considered that concrete struts strength is greatly reduced by the diagonal cracking caused by tension in the perpendicular direction (concrete softening). The influence of the epoxy-bonded FRP fabrics as external reinforcement is implemented as an additional component that contributes to the torsional resistance along with the steel reinforcement. Particularly, considering the state of stress in a finite concrete element of the cracked beam, which is assumed to lie in the plane of the shear flow, it is subjected to a set of stresses in the plane represented by Mohr’s circle [14]. Thus, from the stresses equilibrium the following relationships that include the influence of FRP reinforcement are deduced:

σ A = σ d cos 2 α + σ r sin 2 α + ρ sA f sA + ρ fA f fA σ t = σ d sin 2 α + σ r cos 2 α + ρ st f st + ρ ft f ft τ At = (− σ d + σ r ) sin α cos α where σ A and σ t are the normal stresses of the element in longitudinal and transverse direction, respectively; σ d and σ r are the principal compressive and tensile stresses, respectively; α is the inclination angle of the diagonal compression struts; τ At is the shear stress; ρ sA and ρ st are the longitudinal and transverse steel reinforcement ratio, respectively; ρ fA and ρ ft are the ratio of the FRP fabrics in longitudinal and transverse direction, respectively; f sA , f st , f fA and f ft are the stresses of steel and FRP reinforcement in longitudinal and transverse direction. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

638 Computational Methods and Experimental Measurements XIII It is noted that the above mentioned ratios are not the normalised reinforcement ratios presented in Table 1, but they are calculated by the following expressions:

ρ sA =

ρ fA =

AsA po t d

A fA po t d

for the steel bars and =

(n fA ⋅ t fA ) p fA po t d

and ρ ft =

ρ st =

A ft p ft po t d s f

=

Ast p st po t d s

for the stirrups,

(n ft ⋅ t ft )w f p ft po t d s f

for the FRP

fabrics in longitudinal and transverse direction, respectively, where p o is the perimeter of the centreline of the shear flow; t d is the effective thickness of the compression zone in the diagonal compression struts; AsA is the total area of steel longitudinal bars; Ast is the area of one steel stirrup; p st is the perimeter of the steel stirrup; s is the spacing of steel stirrups; n fA , n ft and t fA , t ft are the numbers of plies and the thicknesses of one FRP ply of the epoxy-bonded FRP fabrics in longitudinal and transverse direction, respectively; p fA , p ft are the perimeters of the strengthened beam cross-section using FRP fabrics in longitudinal and transverse direction, respectively; w f is the width of the FRP strips; s f is the length along the beam over that FRP area is distributed, which is equal to the spacing between the centroid line of the FRP strips; A fA , A ft are the FRP fabrics area in longitudinal and transverse direction, respectively. Further, the developed tensile stress of the FRP fabrics is calculated by the strain of the fibres, ε f , using the following expression: f f = E f ε f ≤ f fu and

ε f ≤ ε fu ; where E f , f fu and ε fu are the elastic modulus, the ultimate tensile strength and the elongation at failure (ultimate strain) of the fibres, respectively. The shear flow derived from the Bredt equation has the form:

τ At =

T and therefore T = 2 Ao t d (− σ d + σ r ) sin α cos α 2 Ao t d

where T is the torsional moment; Ao is the area enclosed by the centreline of the shear flow. The compatibility equations relate the shear distortion in the wall, γ At , to the strains of the concrete and reinforcements:

ε A = ε d cos 2 α + ε r sin 2 α ε t = ε d sin 2 α + ε r cos 2 α

γ At = (− ε d + ε r ) sin 2α εA + εt = εd + εr WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


639

The inclination angle of the diagonal compression struts can be calculated by: tan α =

εA − εd εt − εd

The angle of twist per unit length, ϑ , is related with the shear distortion:

ϑ=

po γ At 2 Ao

Especially for the concrete in compression, it is taking into account the fact that concrete struts strength is reduced by the diagonal cracking caused by tension in the perpendicular direction (concrete softening) and appropriate constitutive laws with softening effect have been formulated [13, 14]. Analyses for the prediction of the torsional behaviour of the 8 tested beams and 5 more beams from the literature [5] using the proposed combined approach were performed. Analytical values of the torsional moment at cracking and the ultimate torsional moment are presented and compared with the measured ones in Table 2. Furthermore, full analytical torque curves for the behaviour of 4 beams are compared with the experimental ones in Figures 2 and 3. 10

5

4



9

3

2

1

Ra-SFs150(2)

8 7 6 5

Test results

4

Proposed model

3 2

Rb-SFs200(1)

1 0 0.00

0.05

0.10

0.15

0 0.00

0.05

0.10

0.15

Angle of twist per length (rad/m) Figure 2:

Comparisons between experimental and calculated curves.


18

18

16

16

14

14




12 10 8

C2

6 4

12 10 8

C4

6 4 Test results [Ghobarah et al (2002)]

2

2

0 0.00

0.10

0.20

0 0.00

Proposed model

0.10

0.20

Angle of twist per length (rad/m) Figure 3: Table 2:

Comparisons between experimental [5] and calculated curves. Test results and analytical predictions of the tested beams.

Cracking torsional moment Ultimate torsional moment (kN⋅m) (kN⋅m) exp . exp . exp. calc. exp. calc. calc. calc. Present experimental study & Chalioris [7] Ra-c 2.39 2.33 0.975 * Ra-Fs150(2) 2.22 2.33 1.050 3.02 2.75 0.911 Ra-S 2.25 2.33 1.036 2.41 2.31 0.959 Ra-SFs150(2) 2.35 2.33 0.991 4.33 4.22 0.975 Rb-c 6.95 6.85 0.985 * Rb-Fs200(1) 6.73 6.85 1.018 9.32 8.72 0.936 Rb-S 6.90 6.85 0.993 7.15 6.90 0.965 Rb-SFs200(1) 6.93 6.85 0.988 9.80 9.60 0.980 Ghobarah et al [5] N2 5.00 5.75 1.150 11.02 12.11 1.099 C2 5.53 5.75 1.039 13.96 15.41 1.104 C4 6.57 5.75 0.875 15.83 17.12 1.081 C5 5.87 5.75 0.979 13.42 15.08 1.124 G2 6.29 5.75 0.914 13.15 14.52 1.104 *:post-cracking behaviour of the beam did not exhibit increased torsional strength. Beam code name



4

641

Concluding remarks

Retrofitting with epoxy-bonded FRP strips is a feasible strengthening technique for existing under-reinforced beams subjected to torsion. Strengthened beams exhibited better overall torsional performance than the non-strengthened control specimens. Failure of wrapped beams with FRP strips was partially delayed in respect to the failure of the control specimens and fibres initially prevented cracking. However, torsional diagonal cracks eventually appeared and widened in the unwrapped concrete of the beams. Two distinct regions are observed in a typical experimental curve of a RC beam with FRP strips. The different character of the response in these regions reveals the different nature of the load resisting mechanism in each part. For the prediction of the entire torsional behaviour of RC beams strengthened with FRP strips a combined method is adopted that employs a smeared crack analysis for the elastic till the first cracking response and a softened truss model that has been extended to include the influence of the FRP fabrics for the post-cracking response. Comparisons between analytical and experimental torque curves showed promising results.

References [1] [2]

[3]

[4] [5] [6] [7]

[8]

Karayannis, C.G., Smeared Crack Analysis for Plain Concrete in Torsion, Structural Engineering, ASCE, 126(6), pp. 638-645, 2000. Karayannis, C.G. & Sirkelis, G.M., Effectiveness of RC Beam-column Connections Strengthening using carbon-FRP Jackets, Proc. of the 12th European Conference on Earthquake Engineering, London, UK, PR 549, 2002. Tsonos, A. & Papanikolaou, K., Post-Earthquake Repair and Strengthening of RC Beam-column Connections (Theoretical and Experimental Investigation), Bulletin of the New Zealand National Society for Earthquake Engineering, 36(2), pp. 73-93, 2003. Chalioris, C.E., Shear Performance of RC Beams using FRP Sheets covering Part of the Shear Span. Proc. of the 1st International Conference on Concrete Repair, St-Malo, Brittany, France, Vol. 2, pp. 809-816, 2003. Ghobarah, A., Ghorbel, M.N. & Chidiac, S.E., Upgrading Torsional Resistance of Reinforced Concrete Beams using Fiber-Reinforced Polymer, Composites for Construction, ASCE, 6(4), pp. 257-263, 2002. Salom, P.R., Gergely, J.M. & Young, D.T., Torsional Strengthening of Spandrel Beams with Fiber-Reinforced Polymer Laminates, Composites for Construction, ASCE, 8(2), pp. 157-162, 2004. Chalioris, C.E. Torsional Strengthening of Rectangular and Flanged Beams using Carbon Fibre-Reinforced-Polymers – Experimental Study, Construction and Building Materials, available on-line from 16 Nov. 2006 (to appear in 2007). Karayannis, C.G. & Chalioris, C.E., Experimental Validation of Smeared Analysis for Plain Concrete in Torsion, Structural Engineering, ASCE, 126(6), pp. 646-653, 2000. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


[12]

[13] [14]

Karayannis, C.G., Nonlinear Analysis and Tests of Steel-fiber Concrete Beams in Torsion, Structural Engineering and Mechanics, 9(4), pp. 323338, 2000. Karayannis, C.G. & Chalioris, C.E., Strength of Prestressed Concrete Beams in Torsion, Structural Engineering and Mechanics, 10(2), 165180, 2000. Chalioris, C.E., Cracking and Ultimate Torque Capacity of Reinforced Concrete Beams, Proc. of the Int. Symposia Celebrating Concrete: People and Practice, University of Dundee, Scotland, UK, Vol. Role of Concrete Bridges in Sustainable Development, pp. 109-118, 2003. Chalioris, C.E., Behaviour Model and Experimental Study for the Torsion of Reinforced Concrete Members, Proc. of the 3rd Int. Conference: High Performance Structures and Materials, Ostend, Belgium, Wessex Institute of Technology Transactions on The Built Environment, Vol. 85, pp. 459468, 2006. Chalioris, C.E. Experimental Study of the Torsion of Reinforced Concrete Members, Structural Engineering and Mechanics, 23(6), pp. 713-737, 2006. Hsu, T.C., Toward a Unified Nomenclature for Reinforced-Concrete Theory, Structural Engineering, ASCE, 122(3), pp. 275-283, 1996.



643

Influence of masonry strength and rectangular spiral shear reinforcement on infilled RC frames under cyclic loading D. J. Kakaletsis Technological Educational Institution of Serres, Greece

Abstract The effect of two types of shear reinforcement of the concrete members and two types of masonry infills on the seismic performance of reinforced concrete (RC) frames was experimentally investigated. Six single-story, one-bay, 1/3-scale frame specimens were tested under cyclic horizontal loading, up to a drift level of 40‰. Bare frames and infilled frames with weak and strong infills were sorted into two groups: Specimens of group A had stirrups while specimens of group B had spirals respectively, as shear reinforcement. The frames were designed in accordance with modern codes provisions. The types of masonry infills had different compressive strength but almost identical shear strength. Infills were designed so that the infill lateral cracking load is less than the available column shear resistance. The results from the specimens of group A were compared with the results from specimens of group B, in terms of hysteretic response, ductility and energy absorption. From the observed responses of the tested specimens it can be deduced that the use of rectangular spiral reinforcement in the beam and columns, even in the case of strong infills, improved the seismic capacity of the examined infilled RC frames. Keywords: infilled R/C frames, masonry strength, spiral shear reinforcement.

1

Introduction

From the experimental investigations that were carried out by several researchers, [1–4], it has been shown that the presence of infill panels improves the seismic performance of a frame. The stronger the infill and the frame is, the higher is the seismic resistance (Mehrabi et al [5]). However the brittle shear failure of columns which might jeopardise the stability and repairability of a WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070641

644 Computational Methods and Experimental Measurements XIII structure must be avoided. The results also suggest that infill panels can be used for retrofitting existing RC structures. In this case, new panels must be designed in such a way that their strength will be compatible with those of the columns (Klingner and Bertero [6]). Therefore, the infilled frame is suggested to be designed to prevent or delay shear failure of the frame members, by designing them for high resistance to cycles of shear reversal, and by examining closely the relationship between column shear resistance and infill panel strength. The above demands must be satisfied by high percentages of transverse steel used in the beams and columns in excess of the proper concrete section and panel thickness. Henceforth, it is obvious that the improvement of the concrete response in terms of the confinement of frame members would help to the improvement of the total seismic response of the infilled frame. On the other hand it is generally accepted that the use of continuous spiral reinforcement in concrete elements with cyclic cross section can substantial improve the strength and the ductility of the concrete and henceforth the total seismic response and capacity of the structural element (Park and Paulay [7]). International codes in these cases propose increased performance factors for the concrete confinement (ACI 318, EC8). The extension of the use of continuous spiral reinforcement in elements with rectangular cross sections is a new promising technology that is believed it can improve the seismic capacity of structures. Considering that the application of the Rectangular Spiral Reinforcement (RSR) could contribute to the improvement of the external beamcolumn joint properties (Karayannis et al [8]) it is expected to contribute to the total improvement of the response of infilled frames. In this paper the experimental results that are presented are a part from an experimental program that has the aim to investigate the performance of masonry – infilled RC frames under in-plane lateral cyclic loads. The objects of the present paper were mainly of: (a) Finding the effect of two types of shear reinforcement that is spirals and equally spaced stirrups, on the hysteretic characteristics of infilled frames. (b) Examining the behavior of two types of masonry infills that is weak and strong, under identical geometry and loading conditions.

2


2.1 Test specimens The experimental program as shown in table 1 consisted of testing six singlestory, one-bay, 1/3-scale specimens of reinforced concrete frames. Specimens B and BS were bare frames, one with transverse steel in the form of common stirrups and one with continuous rectangular spiral reinforcement of the same spacing. Specimens S and SS were infilled frames with a solid weak infill of clay bricks, one with transverse steel in the form of common stirrups and one with continuous rectangular spiral reinforcement of the same spacing. Specimens IS and ISS were infilled frames with a solid strong infill of vitrified ceramic bricks,



Table 1:

Test specimens.

Masonry type Group

A

B

Specimen B S IS ΒS SS ISS

Bare frame

645

Weak

Strong

Shear reinforcement type Common Rectangular stirrups spirals

(a)

(b) Figure 1:

(c)

Description of infilled frame specimens: (a) Reinforcement detailing of the RC frame models (mm); (b) Infilled frame and instrumentation (cm); (c) Weak and strong brick units (mm).

one with transverse steel in the form of common stirrups and one with continuous rectangular spiral reinforcement of the same spacing. The geometric characteristics of the RC frames were the same for all specimens. The elevation, the corresponding cross-sections of the members and the design details for the RC frame specimens are shown in figs. 1a, b. The reinforced concrete frame represented typical ductile concrete construction, particularly structures built in accordance to currently used codes and standards in Greece. Masonry infills had WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

646 Computational Methods and Experimental Measurements XIII a height/length ratio h/l = 1/1,5 and were constructed with two selected brick types cut into two halves for complete simulation to the test scale. Their configuration is shown in fig. 1c. The former “weak” common clay brick usually used in Greece had a thickness 60 mm, while the latter “strong” vitrified ceramic brick that felt to be important for the specimen behavior had a thickness 52 mm. A representative mortar mix was used for the two types of infills contained the portions 1:1:6 (cement: lime: sand) and produced mechanical properties similarly to type M1 mortar according to EN 998-2 standard. Masonry properties were chosen in such a way to produce the desired lateral strength of the two types in a magnitude Vw,u = 27.36 or 25.58 KN lower than that of the lateral strength of the frame Vf,u = 40.28 KN as presented in the following paragraph. This closely represents actual construction in Greece. 2.2 Material properties Material tests were conducted on concrete, reinforcing steel and masonry samples. The mean compressive strength of the frame concrete was 28.51 MPa. The yield stress of longitudinal and transverse steel was 390.47 and 212.2 MPa respectively. The main results of mortar, bricks and infill masonry tests are presented in table 2. It can be noted from the table that the compressive strength of the “weak” masonry prisms was considerably lower than those of the “strong” while the shear strength of the bed joints in the “weak” and “strong” specimens with the same to the full size infills length / height ratio (l/h = fv/fn = 1.5/1) was almost identical. Table 2:

Mechanical properties of the materials used (MPa).

Material Properties MORTAR Compressive Strength fm BRICK UNITS Compressive Strength fbc MASONRY Compressive Strength ⊥ to hollows fc Elastic Modulus ⊥ to hollows E Compressive strength // to hollows fc90 Elastic Modulus // to hollows E90 Friction Coefficient µ (rads) Shear Modulus G Shear Strength without normal stress fvo Shear Strength with normal stress fv/ fn * On full size infills

Masonry type Weak Strong t = 6 cm t = 5.2 cm 1.53

1.75

3.1

26.4

2.63 660.66 5.11 670.3 0.77 259.39 0.08 0.38*/0.25* 0.33/0.22 0.39/0.30 0.21/0.37 0.20/0.73

15.18 2837.14 17.68 540.19 0.957 351.37 0.12 0.41*/0.27* 0.26/0.17 0.60/0.61 0.39/0.72 0.41/1.55



Figure 2:

647

Test setup of specimen SS and loading program.

2.3 Test setup and instrumentation The test setup is shown in fig. 2. The lateral load was applied by means of a double action hydraulic actuator. The vertical loads were exerted by manually controlled hydraulic jacks that were tensioning four strands at the top of the column whose forces were maintained constant during each test. The level of this axial compressive load per column was set 50 KN (0.1 of the ultimate). One LVDT measured the lateral drift of the frame and a load cell measured the lateral force of the hydraulic actuator. The loading program included full reversals of gradually increasing displacements. Two reversals were applied for each displacement level. The cycles started from a ductility level 0.8 corresponding to an amplitude of about ±2 mm (the displacement of yield initiation to the system is considered as ductility level µ=1) and were followed gradually by ductility levels 2, 4, 6, 8, 10, 12 corresponding about to amplitudes 6, 12, 18, 24, 30, 36 mm (fig. 2).

3


The main output of the experimental investigation was a load – displacement curve for each frame (figs. 3, 4, 5a). The initial stiffnesses, critical loads, energy dissipation capacities and critical displacements attained during the tests of the six specimens were derived. It must be pointed out that the hysteretic characteristics of the weak masonry infill were some times larger because of the larger net bedded area for the weak masonry units. The appearance and propagation of cracking was also recorded for both infill and frame throughout each test (figs. 3, 4). Specimens “B” and “BS” were bare reference frames. Flexural cracks and corresponding plastic hinges occurred at predicted critical locations at the bottom and the top of the columns and the ends of the beam – at a drift 4-6‰ – (figs. 3a, 4a). Specimens “S”, “SS” and “IS”, “ISS” had solid weak and solid strong infill respectively. The nonlinear behavior was initiated by the cracking of the infill. Then developed plastic hinges at the top and the bottom of the columns – at a WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

648 Computational Methods and Experimental Measurements XIII drift 4-11‰ –. However, as shown by the damage patterns of specimens, the failure of the specimens “S” and “SS” with the weak solid infill (figs. 3d, 4d) was dominated by internal crushing in the infill – at a drift 19‰ – while the failure of the specimens “IS” and “ISS” with the strong solid infill (fig. 3f, 4f) was dominated by sliding of the infill along its bed joints – at a drift 14‰ –.

(a)

Figure 3:

(b)

(c)

(d)

(e)

(f)

Lateral load – displacement hysteresis curves and failure modes of specimens of group A with stirrups: (a), (b) Bare frame; (c), (d) Weak solid infill; (e), (f) Strong solid infill.



(a)

(c)

(e) Figure 4:

649

(b)

(d)

(f)

Lateral load – displacement hysteresis curves and failure modes of specimens of group B with spirals: (a), (b) Bare frame; (c), (d) Weak solid infill; (e), (f) Strong solid infill.

In all infilled specimens the cracking of the beam occurred far from the column face towards the mid – span vicinity of the beam. Plastic hinges were developed at drifts higher than 11‰ or they did not developed at all. Generally WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

650 Computational Methods and Experimental Measurements XIII the infills restrained the beams from bending and, there by, postpone the development of plastic hinges in the beams. In the case of the present project shear failure of the columns was not observed.

4

Interpretation of experimental results

From the data shown in Table 3 it can be concluded that: For all cases lateral resistance (v) of infilled frames was from 1.63 up to 1.94 times that of the corresponding bare frames. Spirals increased resistance as far strong infills only. The residual resistance (βres) was observed to be increased in the case of strong infills. Spirals did not influence very much residual resistance. The presence of strong infills increased considerably the initial stiffness (k) of the system. Spirals decreased the initial stiffness. It should be noted that the confinement type of the surrounding frame members, did not influence considerably the limit states which had been regarded corresponding to the drifts (γy) and (γu). Only the specimen with strong infill of group B had ultimate limit occurring at a much lower drift level than that of group A. The presence and behavior of spirals increased the ductility factor (µ0,85), corresponding to a lateral force response equal to 85% of the maximum, only in the bare frame while the specimens with infills exhibited higher ductility than that of the bare frames. The total energy dissipation capacity (ΣW) of the infilled frames was of order 1,44 up to 1,64 times the capacity of the corresponding bare frames. It must be pointed out that infill strength and type of shear reinforcement did not influence very much the values of dissipation ratio. Specimens with strong infills and spirals seemed to loose a larger amount of strength and energy during the second loading cycle. From fig. 5b, it can be concluded that in all specimen cases spirals lessened the loss of stiffness. Strong infills increased the loss of stiffness because of different failure mechanism respectively to that of weak infills. From fig. 5c and fig. 5f it can be concluded that strong infills resulted in higher values of average added strength and average added energy dissipation to the system due to infills, especially at low displacement ranges, because those infills developed a better distribution of cracking than weak infills. Almost in all specimen cases, spirals lessened the effectiveness of an infill in increasing the lateral strength and the energy dissipation capacity of a frame. From the cumulative energy dissipated per cycle shown in fig. 5d it can be concluded that the contribution of spirals to energy dissipation capacity of the system seems to be slightly greater than the contribution of stirrups only at very high distortions and only in bare frame and frame with strong infill. Infill strength did not influence very much the dissipation capacity. From fig. 5e it is evident that the energy dissipation during a given cycle normalized by the total peak-to-peak displacement variation for that cycle was greatest just prior to crushing of the critical equivalent compression struts at about γ=13‰-20‰ in weak infills and prior to shear sliding at about γ = 7‰ in strong infills coinciding with the formation of plastic hinges in frame elements. After this, dissipation dropped with a steeper branch in the case of weak infills or with a smoother branch in the case of strong infills and WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


651

continued to decrease gradually with increasing deflections, tending to reach the values of the corresponding bare frame. Spirals in bare frame at high distortion levels resulted in higher normalized energy dissipation capacity. Table 3:

Comparison of hysteretic characteristics for test specimens.

(a) Spec. Structural v γy γu k vlim µ0,85 βres V2/V1 W2/W1 ΣW/ΣWB Morphology (‰) (‰) (m. v.) (m. v.) Bare frame 1.00 stirrups 1.00 5.06 12.09 1.00 0.74 2.81 1.00 0.89 0.84 B

Weak infill stirrups 1.84 2.82 9.23 2.88 0.65 4.24 1.40 0.87

0.85

1.64

Strong infill stirrups 1.65 3.10 13.69 3.04 0.84 6.31 1.75 0.87

0.70

1.48

S

IS

(b) Spec. Structural v γy γu k vlim µ0,85 βres V2/V1 W2/W1 ΣW/ΣWBS Morphology (‰) (‰) (m. v.) (m. v.) Bare frame 1.00 spirals 1.00 3.44 15.50 1.00 0.54 3.97 1.00 0.90 0.70 BS

Weak infill spirals 1.63 2.77 13.33 1.92 0.51 4.09 1.47 0.87

0.79

1.46

Strong infill spirals 1.94 3.33 6.81 2.36 0.65 3.36 1.56 0.87

0.70

1.44

SS

ISS v: Lateral norm. resistance, βres: Residual nor. resistance, γy: Serviceability limit, γu: Ultimate limit, k: In. norm. stiffness, µ0,85: Ductility factor, ΣW: cumulative energy, V: max. Recorded force, W: Energy dissipation, 1/2: 1st/2nd cycle. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Figure 5:

5

Comparison of displacements.

hysteretic

characteristics

versus

imposed

Conclusions

The authors have carried out investigations on several bare frames and infilled frames with weak and strong infills that were sorted into two groups based on the shear reinforcement, providing data for a parametric evaluation of different shear reinforcement and different infill compressive strengths. The experimental results indicated that the presence, behavior and failure of the infills can significantly improve the performance of RC frames. As long as the infilled frames are designed so that the infill cracking resistance will be less than the combined available shear resistance of the columns, the use of infills does not cause a brittle frame failure. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


653

Furthermore specimens with strong infills exhibited a better performance than those with weak infills in terms of the load resistance, stiffness, ductility and energy – dissipation capacity. Strong infills exhibited a better distribution of cracking than weak infills and, thereby, a more drastic mechanism of energy dissipation. The extension of the use of continuous spiral reinforcement in elements with rectangular cross sections is a new promising technology that is believed it can improve the seismic capacity of structures. However, considering that the application of the Rectangular Spiral Reinforcement was not observed to offer a clear total improvement of the response of infilled frames, it is recommended that more refined experimental techniques be pursued in future research.

References [1]

[2] [3] [4] [5] [6] [7] [8]

Fiorato, A.E., Sozen, M.A. & Gambel, W.L., An investigation of the interaction of reinforced concrete frames with masonry filler walls, Civil Engineering Studies, University of Illinois: Urbana. IL, Struct. Res. Series No. 370, pp.117, 1970. Maghaddam, H.A. & Dowling, P.J., The State of the Art in Infilled Frames, Civil Engineering Department, Imperial College: London, ESEE Research Report No, 87(2), pp. 231–284, 1987 Valiasis, T. & Stylianidis, K., Masonry infilled R/C frames under horizontal loading. Experimental results. Europ. Earthq. Engng, III(3), pp. 10–20, 1989. Comite Euro - International du Beton, Reinforced Concrete Infilled frames (Chapter 5). RC Frames under Earthquake Loading – State of the art report, ed. Thomas Telford: London, pp. 231–303, 1996. Mehrabi, A.B., Shing, P.B., Schuller, M.P. & Noland, J.L., Experimental evaluation of masonry-infilled RC frames, Journal of Structural Engineering, Vol. 122, March, pp. 228–237, 1996. Klingner, R.E. & Bertero, V.V., Infilled frames in earthquake resistant construction, Earthquake Engineering Research Centre, University of California: Berkeley, Report No. EERC 76-32, 1976. Park R. & Paulay T., Reinforced Concrete Structures, John Wiley & Sons: New York, pp. 118–122, 1975. Karayannis, C.G., Kakaletsis, D.J. & Favvata, M.J., Improvement of seismic capacity of external beam-column joints using rectangular spiral shear reinforcement. Proc. of the Fifth Int. Conf. On Earthquake Resistant Engineering Structures, Wessex Institute of Technology, Un. of Patras, Aristotele Un. of Thessaloniki, National Technical Un. of Athens: Skiathos, pp. 429–438, 2005.




655

Application of the Cement Hydration Equation in self-compacting concrete’s compressive strength N. Anagnostopoulos, A. Gergiadis & K. K. Sideris Laboratory of Building Materials, Democritus University of Thrace, Greece

Abstract The development of the compressive strength of different self-compacting concretes is experimentally investigated in this research. The self compacting concretes belonged to different strength classes C20/25, C25/30, C30/37 and C35/45. A total of eight mixtures with different cements and different types of aggregates were produced. For comparison reasons additional conventionally vibrated concretes (NCC) of the same strength classes were also produced with the same cements and aggregates. The compressive strength of all mixtures was studied with the help of the cement hydration equation. The hydration number p was determined using the hydration criterion of mortar’s compressive strength. The compressive strength equations were therefore set up for all SCC and NCC produced and their compressive strength values at different hydration ages up to 15 years were calculated. The results indicate that in the case of SCC of all tested classes their compressive strength was significantly increased after the age of 7 days. This increase was much higher than the one measured on conventional concretes of the same strength class. It seems therefore that self compacting concretes produced with limestone filler have a significantly high safety coefficient, regarding the increase of their compressive strength at the late hydration ages. Keywords: self compacting concrete, cement hydration equation, compressive strength, limestone filler, aggregates.



1

Introduction

Self compacting concrete is the latest achievement in the concrete technology ground. It is about a concrete which can be self consolidated without the use of any mechanical means, by its own weight exclusively. In that way the use of energy spending and noisy consolidation mechanisms can be avoided, while at the same time, due to decreased noise, better personnel communication is established. The convenience of flow and self consolidation, that the new concrete has, leads to brief placing and setting (RILEM [1]). In conclusion due to thorough consolidation SCC mixtures appear to have increased durability and as a result better reinforced concrete quality (Sideris and Sideris [2, 3]). Despite the extensive researches that have taken place in the last decade in order to clarify the new material behavior, the growth of its strength in due time has not yet been searched. At the same time the use of greater fine material quantities which are added in the case of SCC, has led to different hydration rate, especially during the first days, resulting to complications as far as the empiric determination of the strength mixture class is concerned (fracture f7/f28). For the reasons mentioned above, on the present paper the compressive strength growth of SCC of different strength classes, which have been produced by Greek materials, is studied. At the same time for comparison matters conventional concretes of the same strength class were produced. The compressive strength growth was studied using the implementation of the Cement Hydration Equation that was used in all mixtures

2


There have been produced SCC mixtures of different strength classes such as C20/25, C25/30, C30/37 and C35/45. The coarse aggregates used for mixture production were limestone and siliceous, while only limestone filler was used. Two different cement types of the same class were used (CEM II 42.5N) originated from Athens and Thessalonica. The one originated from Athens was used for mixture production using limestone aggregates which have been sent from a quarry in Attica, whereas the cement from Thessalonica was used for mixture production using siliceous aggregates from a quarry in Xanthi. In that way the image of the local cement market can better be simulated. The mixture production and the check of their rheological characteristics took place according to the European Guidelines for Self-Compacting Concrete: Specification, Production and Use [4]. At the same time conventional concretes have been produced in all strength classes with the same cement and aggregate proportion as in SCC. A total of eight SCC mixtures and eight conventional concretes were studied. For each concrete 150mm and 100mm (edge) cubes were prepared. The first specimens were used for the compressive strength assessment in 28 days, whereas the others were used for the compressive strength assessment in 2, 7, 21, 28, 60, 90 and 180 days. The mix proportions, their rheological features and their compressive strength in the age of 28 days are respectively listed in the following tables. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Table 1:

657

(a) Mix design characteristics of conventional concretes (NCC) produced with limestone aggregates; (b) Mix design characteristics of conventional concretes (NCC) produced with siliceous aggregates; (c) Mix design characteristics of self-compacting concretes (SCC) produced with limestone aggregates; (d) Mix design characteristics of self-compacting concretes (SCC) produced with siliceous aggregates. (a)

Mixture Proportions Cement (IIΑ/Μ42.5N) Silica Fume Silicate natural Sand Limestone crushed Sand Aggregates Water W/Paste Superpl/zer Slump (cm) fc,28 (Mpa)

NCC 20/25 Limestone

NCC 25/30 Limestone

NCC 30/37 Limestone

NCC 35/45 Limestone

280

325

370

450

-

-

-

-

-

-

-

-

1022

940

870

805

880 186 0,66 1% 19 29,3

927 183 0,56 1% 19 36

955 185 0,50 1% 20 52,7

940 185 0,41 1% 20 56,7

(b) Mixture Proportions Cement (IIΑ/Μ42.5N) Silica Fume Silicate natural Sand Limestone crushed Sand Aggregates Water W/Paste Superpl/zer Slump (cm) fc,28 (Mpa)

NCC 20/25 Siliceous

NCC 25/30 Siliceous

NCC 30/37 Siliceous

NCC 35/45 Siliceous

330

350

430

430

-

-

-

20

430

280

530

510

655

610

385

385

760 212 0,64 1% 20 30,6

850 200 0,57 1% 19 41,6

760 200 0,47 1% 18 47,3

510 160 0,50 1% 17 53



Continued. (c)

Mixture SCC 20/25 SCC 25/30 SCC 30/37 SCC 35/45 Proportions Limestone Limestone Limestone Limestone Cement 301,6 336 374 435 (IIΑ/Μ42.5N) 20 Silica Fume 184,2 136 104 100 Limestone Filler Siliceous Sand 861,6 916 898 808 Limestone Sand 800 800 800 800 Aggregates 186,8 173,6 180,6 192,2 Water 0,62 0,52 0,48 0,42 W/Paste 1,27% 1,63% 1,88% 1,51% Superpl/zer 75,5 75,5 77 76 Slump Flow (cm) 0,92 0,88 0,88 0,86 L-Box (H2/H1) 6,5 10,5 10 13,16 V-funnel (sec) 0,5 0,5 0,5 0,5 J-ring (cm) 35,8 48,3 50 55,3 fc,28 (Mpa) (d) Mixture SCC 20/25 SCC 25/30 SCC 30/37 SCC 35/45 Proportions Siliceous Siliceous Siliceous Siliceous Cement 337,2 353,4 432 435,8 (IIΑ/Μ42.5N) 20 Silica Fume 206 144 120 100 Limestone Filler 808 897,6 808 807,2 Siliceous Sand Limestone Sand 800 800 800 800 Aggregates 187,6 171,7 189,4 192,2 Water 0,56 0,49 0,44 0,44 W/Paste 1,61% 1,85% 1,88% 1,51% Superpl/zer 77,5 71,5 77 70 Slump Flow (cm) 0,97 0,8 0,93 0,85 L-Box (H2/H1) 6 11,47 7,25 6,78 V-funnel (sec) 0,5 0,5 0,6 0,5 J-ring (cm) 36,5 47,3 52,9 58,3 fc,28 (Mpa)

3

Hydration equation of cements

The cement hydration equation came forward for the first time in 1993 by Sideris [5]. The two phases of hydration (first and second phase) expressed by a hydration criterion K, are represented by straight lines in the coordinate system WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


659

(K - (1/t) p). The quantitative expression of hydration in relation to time t is given by the equation: or

K = K ∞ ± b × (1 / t )

p

K = K ∞ ± b × (1 / t )

−p

(1)

where: Κ= Κ∞ = b= t=

Hydration Criterion. Constant variable (intersection of the line with the y-coordinate). Line slope. Hydration time (days), (t>>0). Final hydration time = 15 years for curing temperatures of 5 - 60 oC and adequate relative humidity >90 %. p= Hydration number of the cement used for mixture production. This coefficient depends only on the chemical cement composition. The term hydration criterion envelopes any given established hydration criterion, of cement or concrete hydration, the way it is analyzed by Sideris and Sideris [2]. In the case of the compressive concrete strength criterion (fc) the general equation (1) transforms to

fc = f∞ − b × t − p

(2)

The application of the hydration equation procedure is analytically presented by Sideris and Sideris [3]. According to this procedure the hydration equations of the produced self compacting and conventional mixtures were determined. The hydration number p was defined in mortar specimens. The values which came up from coefficient p were later on used for linear correlation among fcmeas-t-p pairs (where fcmeas is the measured values of the concretes under study and t-p the corresponding hydration age (in days) raised in minus p) in order to define the compressive strength equations of the sixteen concretes which are under study. The procedure is presented suggestively in table 2 for the case of self compacting concretes using cement originated from Athens and limestone aggregates. The hydration equations of all mixtures the way the came up after the process mentioned above are listed in table 3. The graph of these equations is a straight line in the diagram of fc-t-p, where fc stands for compressive strength (measured in Mpa) and t-p stands for the modified time scale (t in days and p the hydration number of cement and the other cementitious materials which were used for this specific concrete production). By using these equations, one can achieve the calculation of the compressive strength which concrete mixtures develop in any age until the hydration end (15 years or 5475 days). These values are named as measured compressive strength values (fc,calc) and do not really differ from the experimental fc,meas values.



(a) Hydration equations of self-compacting concretes with CEM II 42.5N and limestone aggregates-strength class C20/25; (b) Hydration equations of self-compacting concretes with CEM II 42.5N and limestone aggregates-strength class C25/30; (c) Hydration equations of self-compacting concretes with CEM II 42.5N and limestone aggregates-strength class C30/37; (d) Hydration equations of self-compacting concretes with CEM II 42.5N and limestone aggregates-strength class C35/45. (a)

SCC-C20/25 Limestone Hydration 2 7 21 Age t (days) fc(meas) 13,67 24,33 35,33 (MPa)

Linear correlation among the values of fcmeas – t-0.424

t-0.424

fc(calc)

28

90

180

36,2 36,5

38,27

0,745

0,438

0,275

0,24 0,148

0,111

13,7

24,3

35,3

36,2

36,5

38,3

0,745

0,438

0,275

0,24 0,148

0,111

365

730

5475

0,082

0,061

0,026

Hydration number: p = 0,424. Correlation coefficient: r = 0.980979.Standard deviation: s = 2,11145 (%) Hydrations equation: fc(calc) = 43.9982 – 40.6362*t-0.424 MPa, t≥2 13.70

26.19

32.82 34.1

37.97

39.5

40.67

41.52

365

730

42.94

(b)

Linear correlation among the values of fcmeas - t-0.381

SCC-C25/30 Limestone Hydration 2 7 Age t (days) fc(meas) 17 32,5 (MPa) 0,768 0,477 t-0.381

fc(calc)

21

28

90

180

----

47,2

50,2

52,4

0,281

0,18

0,138

17

32,5

----

47,2

50,2

52,4

0,768

0,477

0,314

0,281

0,18

0,138

0,106 0,0811

5475

0,0376

Hydration number: p = 0,381 Relation coefficient: r = 0.9959.Standard deviation s = 1,554476 (%) Hydrations equation: fc(calc) = 61,0937 – 57,6024*t-0.381 MPa, t≥2 16,86

33,65

43,03

44,91

50,7

53,13


55,0

56,4

58,93


Table 2:

661

Continued. (c)

SCC-C30/37 Limestone Hydration 2 Age t (days) fc(meas) 23,3 (MPa)

Linear correlation among the values of fcmeas- t-0.087

t-0.087

fc(calc)

7

21

28

90

45,6

49

58

61,8

0,942

0,844

0,767

0,748

23,3

45,6

49

58

180

365

0,676 0,637 0,599

730

5475

0,564

0,473

61,8

0,9415 0,8443 0,7673 0,7483 0,6761

Hydration number: p = 0,087. Correlation coefficient: r =0.999207. Standard deviation: s =0.691645 (%) Hydrations equation: fc(calc) =144.0134 – 127.9438*t-0.087 MPa, t≥2 23.55

35.99

45.84

48.27

57.5

62.6

67.4

71.9

83.5

(d) SCC-C35/45 Limestone Hydration 2 Age t (days) fc(meas) 29,5 (MPa)

Linear correlation among the values of fcmeas – t-238

t-238

fc(calc)

7

21

28

90

180

46,5

54,5

59

64

66,8

0,848 0,629

0,485

29,5

46,5

54,5

0,848 0,629

0,485

0,453 0,343 0,291 59

64

365

730

5475

0,246

0,208

0,129

66,8

0,453 0,343 0,291

Hydration number: p = 0,238. Correlation coefficient: r =0.995709. Standard deviation: s =1.421161(%) Hydrations equation: fc(calc) =87,4010 – 66,9769*t-0.238 MPa, t≥2 30,6

45,25

54,94

57,09

64,5 67,93

70,95


73,46

79,7


Hydration equations of self compacting (SCC) and conventional (NCC) concretes a) with CEM II 42.5N from Athens and limestone aggregates (L) and b) with CEM II 42.5N from Thessalonica and siliceous aggregates (S)

SCC-20/25 L SCC-25/30 L SCC-30/37 L SCC-35/45 L SCC-20/25 S SCC-25/30 S SCC-30/37 S SCC-35/45 S NCC-20/25 L NCC-25/30 L NCC-30/37 L NCC-35/45 L NCC-20/25 S NCC-25/30 S NCC-30/37 S NCC-35/45 S

4

fc(calc) = 43.9982 – 40.6362*t-0.424, r = 0.980979,s = 2,111445 fc(calc) =61,0937 – 57,6024*t-0.381, r = 0.9959,s = 1,554476 fc(calc) =144.0134 – 127.9438*t-0.087, r =0.999207,s =0.691645 fc(calc) =87,4010 – 66,9769*t-0.238, r =0.995709,s =1.421161 fc(calc) = 75,4647 – 62,4647*t-0.224, r = 0.994447,s = 1,471585 fc(calc) = 69,3884 – 44,0293*t-0.253, r = 0.986597,s = 1,700644 fc(calc) =82,9112 – 70,3783*t-0.298, r =0.991974,s =2.355236 fc(calc) =85,9980 – 64,8322*t-0.238, r =0.993301, s =1.716112 fc(calc) = 33,2357 – 27,6425*t-0.422, r = 0.986239,s = 1,213672 fc(calc) = 38,1373 – 15,88879*t-0.422, r = 0.975240,s = 1,58887 fc(calc) = 65,4272 – 61,6919*t-0.422, r =0.997443,s =1,331999 fc(calc) = 77,2356 – 57,7096*t-0.238, r =0.993134,s =1.774112 fc(calc) = 51,0120 – 42,4016*t-0.36, r = 0.989084,s = 1,626802 fc(calc) = 52,4366 – 34,8214*t-0.36, r = 0.999773,s = 0,127685 fc(calc) = 57,5746 – 26,8129*t-0.36, r =0.990374,s =1,077980 fc(calc) = 77,6585 – 64,6109*t-0.236, r =0.989448,s =2,152820

Results and analysis

The equation graphs from Table 3 are presented for all mixtures produced in Figure 1. The mixture compressive strength alteration is comparably presented for self compacting (SCC) and conventional (NCC) concretes of all strength classes in Figure 2. The diagrams were drawn using the measured compressive strength values (fccalc) as they were calculated from the corresponding hydration equations. While studying those diagrams one can realise the alternation in the compressive strength development which self compacting mixtures have. Even though these mixtures were produced with the same cement quantity and similar w/c number compared to conventional concretes of the same strength class (table 1), they appear to have different compressive strength development, especially after the age of 7 days. This is mainly attributed to better grain placement from which the paste consists of: The use of great quantities of fine material (limestone filler) has as a result the filling of the gaps which are created after the formation of calcium-siliceous hydrous and finally the decrease in the active porosity of self compacting concretes (De Schutter et al. [6], Audenaert and De Schutter [7], Träghård and De Schutter [8], Audenaert et al. [9], Popee and De Schutter [10], Audenaert et al. [11]). This phenomenon leads to increased compressive strength of self compacting concretes. This increase gains is significance after the age of 7 days. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


663

80

100

80

fc (MPa)

fc (MPa)

60

60

40 40

20

SCC 20/25 Limestone

20

SCC 20/25 Siliceous SCC 25/30 Siliceous

SCC 25/30 Limestone SCC 30/37 Limestone SCC 35/45 Limestone

SCC 30/37 Siliceous SCC 35/45 Siliceous

t-p

t-p

0

0 0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

(a)

0.6

1

(b)

80

NCC 20/25 Limestone

80 NCC 20/25 Siliceous

NCC 25/30 Limestone NCC 30/37 Limestone

NCC 25/30 Siliceous NCC 30/37 Siliceous NCC 35/45 Siliceous

NCC 35/45 Limestone

60

0.8

fc (MPa)

fc (MPa)

60

40

40

20

20

t-p

t-p

0

0

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

(c) Figure 1:

0.6

0.8

1

(d)

Hydration equations of a) SCC concretes using limestone aggregates, b) SCC using siliceous aggregates, c) NCC concretes using limestone aggregates and d) NCC using siliceous aggregates in the modified time scale t-p (p values from Table 3).

The ratio of the compressive strength in the age of 7 days to the strength in the age of 28 days and 15 years (5475 days) as well as the degree of hydration at the ages of 7, 28, 365 and 5475 days are presented for all mixtures in Table 4. The calculations took place with reference to measured values of the mixture compressive strength in every age (fc, calc), which were defined by the equations in Table 3.



100

SCC-C20/25 Limestone


NCC-C20/25 Limestone





80


fc (MPa)

fc (MPa)

60

60

40

40

20 20

0

0

28 days

15 years

1 year

28 days

(a)

1 year

15 years

(b)

80

80

60

fc (MPa)

fc (MPa)

60

40

40

20

20

SCC-C20/25 Siliceous


NCC-C20/25 Siliceous

NCC-C30/37 SIliceous





0

0

28 days

15 years

1 year

28 days

(c) Figure 2:

1 year

15 years

(d)

Comparable diagrams of the compressive strength development of a) SCC and NCC concretes C20/25 και C25/30 using limestone aggregates, b) SCC and NCC concretes C30/37 και C35/45 using limestone aggregates, c) SCC and ΝCC concretes C20/25 και C25/30 using siliceous aggregates and d) SCC and ΝCC concretes C30/37 και C35/45 using siliceous aggregates

After studying Table 4, the difference in the development of self compacting concretes’ strength among 7 and 28 days, as well as in later ages (365 days and 15 years), becomes more than clear. Ratios f7/f28 and f7/f5475 are lesser in SCCs compared to NCC of the same strength class, which as a fact indicates that the increase of the compressive strength after the age of 7 days is greater in SCCs compared to NCCs of the same strength class. Indeed the compressive strength of the SCCs in the age of 28 days appears to be greater from 26 to 61% WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


665

compared to the strength of 7 days when the corresponding increase percentage in NCCs is in the order of magnitude of 19-31%. The respective percentages for the age of one year vary among 55 and 74% for the SCCs and 36 to 57% for NCCs. For the final hydration age (15 years) the respective percentages are 63 to 110% for SCCs and only 41 to 70% for NCCs. Table 4:

Ratios f7/f28 and f7/f5475, degree of hydration at the ages of 7, 28, 365 and 5475 days for all produced mixtures.

SCC20/25 L SCC25/30 L SCC30/37 L SCC35/45 L SCC20/25 S SCC25/30 S SCC30/37 S SCC35/45 S NCC20/25 L NCC25/30 L NCC30/37 L NCC35/45 L NCC20/25 S NCC25/30 S NCC30/37 S NCC35/45 S

f7/f28

f7/f5475

α7

0,76833 0,62347 0,7456 0,79261 0,76489 0,84219 0,76351 0,79773 0,7963 0,83513 0,76143 0,80047 0,78394 0,83783 0,89434 0,7640

0,61015 0,47514 0,500 0,57446 0,52832 0,65963 0,56136 0,58109 0,64831 0,70505 0,60031 0,58625 0,61051 0,69014 0,78549 0,53252

0,61015 0,47514 0,500 0,57446 0,52832 0,65963 0,56136 0,58109 0,64831 0,70505 0,60031 0,58625 0,61051 0,69014 0,78549 0,53252

α28

α365

α5475

0,79413 0,76209 0,67116 0,72477 0,69072 0,78323 0,73351 0,72842 0,81415 0,84424 0,7884 0,73238 0,77877 0,82373 0,87828 0,69702

0,94714 0,93331 0,87013 0,9007 0,88581 0,92376 0,91341 0,90156 0,95231 0,95912 0,9453 0,90344 0,93583 0,94865 0,96451 0,89043

1,00 1,00 1.00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00

In every case SCCs appear to have greater increase in their compressive strength when compared with NCC of the same strength class. The greater increase percentages appear in strength classes such as C20/25 and C25/30 due to greater limestone filler quantity that is added to the mixture. On the other hand, when the case is self compacting concretes of high strength class such as C35/45 then no limestone filler is used. Even though these mixtures developed higher strength compared to the corresponding conventional C35/45, they didn’t really differ in the f7/f28 and f7/f5475 ratios or in the degree of hydration in the ages of 7, 28 and 365 days. More than significant is the increase in the mixture compressive strength after the age of 28 days. It is well known that the strength in that particular age is the one which is taken into consideration about the structural calculations. But in structures the sum of the planning loads are really taken into consideration in ages not less than one year. This variation (f365-f28) indicates a security coefficient for structures. Compressive strength in the age of one year increases in SCC mixtures by 20 to 30% compared to the 28 days strength, when the corresponding percentage for NCCs is among 13 and 23%. If the security coefficient (f365-f28) for the same strength class concretes is examined using the WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

666 Computational Methods and Experimental Measurements XIII equations of Table 4, it will become clear that it is every time by 25 to 40% greater for the case of SCC mixtures. It can then be said that the different development rate as far as strength is concerned, after the age of 28 days gives to self compacting concrete mixtures an extra security coefficient.

5

Conclusions

Self compacting concrete mixtures which are produced with limestone filler appear to have difference in compressive strength in the age of 7 days. Not only the f7/f28 and f7/f5475 ratios but also the degree of hydration at the ages of 7, 28 and 365 days are lesser compared to the same number in conventional concretes of the same strength class which were produced with the same cement amount and the same w/c ratio. This fact is independent of the strength class, the cement origin and the aggregates type (limestone or siliceous). The compressive strength development in later ages indicates an extra security coefficient for these mixtures. For the production of self compacting concretes of high strength class such as C35/45 no limestone filler amount was used. Even though these mixtures developed grater strength compared to the corresponding conventional concretes, they did not really differ in the f7/f28 and f7/f5475 ratios nor in their hydration degrees at the ages of 7, 28 and 365 days. After all it seems that the addition of limestone filler is the main reason which caused the different rate in the compressive strength development of self compacting concretes in later ages.

References [1] [2]

[3]

[4] [5] [6]

RILEM (1999): Self-Compacting Concrete”, State of the art Report of RILEM Technical Committee 174-SCC, Skarendahl Å. and Petersson Ö. Editors, p.154. Sideris K., Sideris K.K. (2003a): Ten Years Cement Hydration Equation and its Applications to Chemistry and Physics of cement paste, mortar and concrete, Xanthi 2003, ISBN 960-343-722-0. Chapter 1: Ten Years Cement Hydration Equation, p. 1-24. Sideris K., Sideris K.K. (2003b): Ten Years Cement Hydration Equation and its Applications to Chemistry and Physics of cement paste, mortar and concrete, Xanthi 2003, ISBN 960-343-722-0. Chapter 2: Application procedures of Cement Hydration Equation, p. 25-38. ΒIBM, CEMBUREAU, EFCA, EFNARC, ERMCO (2005): European Guidelines for Self-Compacting Concrete: Specification, Production and Use, May 2005, downloadable from www.efnarc.org Sideris K. (1993) The cement hydration equation, Zement-Kalk-Gips, 12 (1993), Edition B, pp. E337-E344. De Schutter, G., Audenaert, K., Boel, V., Vandewalle L., Dupont, D., Heirman, G., Vantomme, J., D’Hemricourt, J. (2003). Transport properties in self consolidating concrete and relation with durability: Overview of a Belgian research project”, Proc., Third Int. Symp. on SCC, WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


[7] [8] [9] [10]

[11]

667

RILEM, Reykjavik, Iceland, Wallenik O. and Nielsson I. Editors, 799807. Audenaert K., Boel, V., De Schutter, G. (2003). Water permeability of self-consolidating concrete, Proc., 11th Int. Congr. Chem. of Cem., Durban, South Africa, Grieve G. and Owens G. Editors, 1574-1584. Träghård, Jan. (1999): Microstructural features and related properties of self-consolidating concrete”, Proc. First Int. Symp. on SCC, RILEM, Stockholm, Sweden, Skarendahl Å. And Petersson ö. Editors, 175-186 Audenaert, K., De Schutter, G. (2003). Chloride penetration in self consolidating concrete, Proc., Third Int. Symp. on SCC, RILEM, Reykjavik, Iceland, 818-825. Popee, A-M., De Schutter, G. (2005). Creep and Shrinkage of selfconsolidating concrete”, Proc., Design Performance and Use of SCC, Changsha, Hunan, China, Yu Z., Shi C., Khayat K.H. and Xie Y. Editors, 329-336. Audenaert, K., Boel, V., De Schutter, G. (2005). Chloride penetration in self-consolidating concrete by cyclic immersion”, Proc., Design Performance and Use of SCC, Changsha, Hunan, China, Yu Z., Shi C., Khayat K.H. and Xie Y. Editors, 355-362.



Section 8 Structural dynamics



671

Response of a double system beam and string with an elastic layer to the dynamic excitations L. Frýba, C. Fischer & Sh. Urushadze Institute of Theoretical and Applied Mechanics, Academy of Sciences of the Czech Republic, Prague

Abstract The beam with an axial force is coupled with the pretensiled string by an elastic layer of Winkler type. It is subjected to a row of moving forces. The theoretical model corresponds to a prestressed bridge. The governing equations form a coupled set of partial differential equations which are solved using the Fourier and Laplace-Carson integral transformation methods. A simple experimental model was constructed in a form of a plexiglass beam and two strings bound together with several spring and damping elements. The beam was excited by an electro-magnetic exciter and its response was measured at several places. Three steps of stiffness and damping characteristics of the elastic layer were proved to show their effect on the dynamic response of the double system. Its low response is searched. Keywords: prestressed beam, string, dynamic excitation.

1

Introduction

Many years standing effort has been devoted to damp the dynamic effects of both the highway and railway vehicles when they cross a bridge. For that purpose, a lot of systems were developed, e.g. elastic supports of bridges, triangular falsework system with controlled damping, double systems with two beams or two strings connecting together with an elastic layer, etc. They are briefly described in [1] and cited in details in [2] and [3]. They are, especially, the papers of Kawanazoe et al. [4] and Oniszczuk [5], who firstly introduced the double beam and double string system, respectively. However, each of the systems mentioned above shows its technical or economic effectiveness only in some specific conditions. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070661

672 Computational Methods and Experimental Measurements XIII On the other hand, the prestressed bridges, world wide used for highway as well as railway bridges of small and medium spans, form naturally a double system with two elements: beam and pretensiled strings. That’s the reason, why an idea arose – to bind both the elements with an elastic layer and dampers to diminish the dynamic response. The aim of the present paper is to show the effect of the double system beam and string and put a question on the effectiveness of damping.

2 Theory The Fig. 1 represents the theoretical model of a beam, pretensiled string and an elastic layer subjected to a row of axle forces. The axle forces Fn , n = 1, 2,3,..., N , move with a constant speed c from the left hand side to the right one. The simple supported beam and string provide the span l. The beam is subjected to an axial force N1 (generally tension), while the string is tensed by a force N 2 (in practice, of course, N1 = − N 2 ). An elastic layer of Winkler type binds together both the carrying elements, its characteristic is k 2

[N/mm ] and its viscous damping

Figure 1:

ωd 1 or ωd 2 , respectively.

Theoretical model.

A system of partial differential equations describes the dynamic behaviour of the Bernoulli-Euler beam and string:



EI

673

∂ 4 v1 ( x, t ) ∂ 2v1 ( x, t ) ∂ 2 v1 ( x, t ) − N + + k [ v1 ( x, t ) − v2 ( x, t )] + µ 1 1 ∂x 4 ∂t 2 ∂x 2

 ∂v ( x, t ) ∂v2 ( x, t )  N +2 µ1ωd 1  1 − = ∑ Fnε n (t )δ ( x − xn ), ∂t  n =1  ∂t (1)

∂ v2 ( x , t ) ∂ v2 ( x , t ) + µ2 + k [ v2 ( x, t ) − v1 ( x, t ) ] + 2 ∂x ∂t 2  ∂v ( x, t ) ∂v1 ( x, t )  +2 µ2ωd 2  2 − = 0. ∂t   ∂t −N2

2

2

(2)

The following notations were applied: vi ( x, t ), i = 1, 2 – vertical deflection of the beam (i=1) and the string (i=2), respectively, at place x and time t, E, I – modulus of elasticity and inertial cross-section moment, respectively, of the beam, µi – mass of the beam (i=1) and of the string (i=2), respectively, per unit length,

δ ( x) – Dirac delta function, ε n (t ) = h(t − tn ) − h(t − Tn )

– the function describing the position of the

n-th force with respect to the beam, h(t ) = 0 for t 500 (Cole [1]). It was also pointed out by other studies that N = 1000 is the optimal number of superposition and N = 100 is the minimum acceptable number (Jeary [2]). Taking a single velocity signal x (t ) for example, its random decrement signature can be obtained with the following detailed steps: 1. Choose a fixed value of velocity xs as the cutting threshold such that x (t ) is with values of xs at N different time instants t1 , t2 , "" , t N . 2. Set the extracted signal duration Td such that the extracted signal can adequately reflect the dynamic characteristics of system. 3. Extract N different time histories, all with a duration Td , from the measured signal. Average all those time histories to yield the corresponding random decrement signature y (τ ) .

2.2 Ibrahim time domain method If the random decrement signature can be obtained from the ambient vibration measurements of cables and sufficiently represents the corresponding free vibration response, the ITD method is then usually applied to determine the associated modal parameters. Consider the free vibration system illustrated in eqn (3), but replace the time variable τ by t. Transformation of this equation into the state space further results in a first-order differential equation:


694 Computational Methods and Experimental Measurements XIII y   0n×n  = y   − M −1 K 

I n× n   y    −M −1C  y 

(4)

where I indicates an identity matrix with its dimension shown in the subscript. Solving eqn (4) as an eigenvalue problem, the displacement and velocity vectors can be expressed, respectively as: 2n

2n

2n

y (t ) = ∑ Γi e λ t ; y (t ) = ∑ λi Γi e λ t = ∑ Φi e λ t i

i

i =1

(5)

i

i =1

i =1

where λi ’s are the eigenvalues of the system matrix and Γi ’s are the corresponding n × 1 eigenvectors. Based on eqn (5), the velocity vectors at s different time instants can be assembled into a matrix as

[y 1 y 2 " y s ] = [Φ1 Φ 2

 e λ1t1  λ2t1 e " Φ2n ]   #  λ2 n t1  e

e λ1t2

e λ1ts   " e λ2ts  % #   " e λ2 n ts  "

e λ2t2 # e λ2 n t2

(6)

where y i = y (ti ), i = 1, 2, " , s . If eqn (6) is shifted forward in the time axis with m ∆ t and 2m ∆ t , respectively, then

Y = [ y 1 y 2 " y s ] = [Φ1 Φ 2

ˆ = [yˆ yˆ " yˆ ] = [Φ ˆ Φ ˆ Y 1 2 1 2 s

 e λ1t1  λ2t1 e " Φ2n ]   #  λ2 n t1  e  e λ1t1  λ2t1 e ˆ " Φ2n ]   #  λ2 n t1  e

(7)

e λ2 n t2

e λ1ts   " e λ2ts  = ΦΛ % #   " e λ2 n ts 

e λ1t2

"

e λ1ts   " e λ2ts  ˆ = ΦΛ % #   " e λ2 n ts 

(8)

e λ1t2

"

e λ2t2 #

e λ2t2 # e λ2 n t2

y i = y (ti + m∆t ) and yˆ i = y (ti + 2m∆t ), i = 1, 2, " , s. In addition, ˆ = Φ e λ 2 m∆t , i = 1, 2, " , 2n. Two combinations of eqns Φi = Φi e λ m∆t and Φ i i (6)–(8) can be obtained as

where

i

i

 Y  Φ   Y  Φ  ˆ U =   =   Λ = ΨΛ and V =   =   Λ = ΨΛ ˆ ˆ Y Φ Y Φ        

(9)

ˆ are 2n × 2n matrices. where U and V are 2n × s matrices and Ψ and Ψ Examination of eqn (9) reveals that if a system related matrix A is defined as WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


695

 eλ1m+t  0 ˆ AΨ = Ψ = Ψ   #   0

(10)

" 0   % # #  # % 0   λ m+t " 0 e  0

2n

then eqn (10) obviously indicates that eλ m+t , " , eλ m+t are the eigenvalues of A and directly related to the system eigenvalues: λ1 , " , λ2 n . Pre-multiplying the first part in eqn (9) by A and applying eqn (10) lead to: 2n

1

ˆ =V AU = AΨΛ = ΨΛ

(11)

Therefore, the least squares method can be conveniently adopted to obtain A based on eqn (11), followed by the previously mentioned eigenvalue analysis. It has been shown in the literature that the eigenvalues of the state space system in eqn (4) have to be in complex conjugate pairs and the eigenvalues λ2 j −1 and λ2 j corresponding to the j-th mode are related to the modal frequency

ω j and modal damping ratio ξ j by λ2 j −1 , λ2 j = −ξ j ω j ± iω j 1 − ξ j 2 = α j ± iβ j , j = 1, 2, " , n

(12)

Consequently, the eigenvalues of A also appear in complex conjugate pairs as e

λ2 j −1m+t

,e

λ2 j m +t

(

)

(

)

 cos β j m+t ± i sin β j m+t    = a j ± ib j , j = 1, 2, " , n α j m+ t

=e

(13)

With eqn (13), it is easy to obtain: αj =

 bj  1 1 tan −1   , j = 1, 2, " , n ln a j 2 + b j 2 and β j =  aj  2m ∆ t m∆t  

(

)

(14)

and then yield: ω j = λ2 j −1 = λ2 j = α j 2 + β j 2 and ξ j =

3

−α j ωj

, j = 1, 2, " , n

(15)

Parametric study for time shifting in the ITD method

The combination of RD and ITD methods, as described in the previous section, can usually provide meaningful results for several modal parameters if the ambient vibration measurements can be obtained for various stations of the target system and the frequency content of environmental excitations is not far away from that of a white noise. Unfortunately, the above two conditions are not always possible for certain civil structures, especially the cable-stayed bridge WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

696 Computational Methods and Experimental Measurements XIII cables. To overcome these critical difficulties in practical applications, a first step is to improve the accuracy of the ITD method if the input time history is not an ideally decaying function under free vibration. As described in Section 2.2, the matrix A utilized in the ITD method has to be constructed by shifting a certain amount of time and its eigenvalues are then solved to determine the modal frequencies and damping ratios of system. It is obvious that different adopted values for the number of shifted time steps m will certainly lead to different matrices of A and altered eigenvalues, especially for the case with a response not closed related to a free vibration function. The parametric study of m is consequently conducted in this section to determine the optimal selection. Table 1:

No. of shifted time steps m 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 150 200

Identified parameters of SDOF systems for various time shifting values. SDOF system with damping ratio 0.1% Natural Damping Eigenvalues Ratio Frequency of A ξ (%) ω (Hz)

SDOF system with damping ratio 1% Natural Damping Eigenvalues Frequency Ratio of A ξ (%) ω (Hz)

0.987 ± 0.156i 0.951 ± 0.309i 0.891 ± 0.454i 0.809 ± 0.587i 0.707 ± 0.707i 0.588 ± 0.809i 0.454 ± 0.890i 0.309 ± 0.950i 0.157 ± 0.987i 0.001 ± 0.999i

0.965 ± 0.155i 0.930 ± 0.306i 0.873 ± 0.450i 0.799 ± 0.584i 0.702 ± 0.705i 0.584 ± 0.806i 0.451 ± 0.887i 0.307 ± 0.946i 0.155 ± 0.982i

−0.156 ± 0.987i −0.308 ± 0.951i −0.453 ± 0.891i −0.586 ± 0.809i −0.706 ± 0.707i −0.808 ± 0.588i −0.890 ± 0.455i −0.950 ± 0.310i −0.986 ± 0.157i 0.236; −0.988 −0.002 ± 0.998i 0.189; 0.973

0.9999 0.9997 0.9997 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 •

1.000 0.103 0.067 0.047 0.038 0.036 0.036 0.036 0.036 0.036 0.036 0.036 0.036 0.036 0.036 0.037 0.038 0.041 0.062 •

0.9996 •

0.035 •

−0.001 ± 0.993i −0.156 ± 0.980i −0.307 ± 0.944i −0.449 ± 0.883i −0.581 ± 0.801i −0.697 ± 0.699i −0.796 ± 0.581i −0.872 ± 0.448i −0.919± 0.303i −0.889± 0.124i 0.760; −0.999 0.001 ± 0.981i 0.535; 0.998

1.0259 1.0149 1.0096 1.0056 1.0027 1.0010 1.0005 1.0005 1.0006 0.9994 1.0004 1.0001 0.9997 0.9995 0.9993 0.9991 0.9988 0.9984 1.0069 •

14.006 6.719 3.736 1.650 0.724 0.488 0.453 0.436 0.430 0.427 0.418 0.419 0.450 0.496 0.535 0.588 0.734 1.170 3.596 •

1.0002 •

0.414 •

The simulated response functions for a single-degree-of-freedom (SDOF) system with various parameter values (natural frequency ω = 1Hz and damping ratio ξ = 0.1, 1% ) are adopted to investigate the effects of the time shifting parameter in the ITD method. For these two different cases, the same initial displacement and ground acceleration reflecting the measured environmental WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


697

excitation are assumed to compute the corresponding displacement time histories of 300 seconds. A threshold of 80% of the standard deviation of each time history and an extracted duration of Td = 150sec are adopted to carry out the RD method. Those selected values for the RD parameters do not correspond to the optimal choice (Liao [3]) and are intended to result in the imperfect RD signatures where the excitation effects are not totally filtered out. In addition, the examined values of m are limited to the time shifting range within a period of system, especially focused on the shifting less than a half of the period. It should be noted that m = 200 corresponds to a time shifting of a period since ∆t = 0.005sec is taken in computation. The identified frequencies and damping ratios from the ITD method for all these different cases are listed in Table 1. From the results in Table 1, it is evidently observed that the identified natural frequencies and damping ratios from the ITD method converge to stable values as the shifting parameter m gradually increases from small values, especially when the time shifting reaches one quarter of the system period ( m = 50 ). Nevertheless, the identified results would get worse if m keeps increasing to larger values. The worse case is bumped as the time shifting grows up to a half of the system period ( m = 100 ) where real eigenvalues of A are obtained and no feasible system parameters can be identified. The above comparison clearly indicates that the value of m corresponding to a shifting of one quarter of the system period is the optimal selection while the value of m corresponding to a shifting of a half of the system period needs to be always avoided. The reason why the shifting parameter m is capable of inducing such a critical difference can be explained by considering the periodic characteristics for the free vibration of SDOF systems. If the ITD method is applied on an ideal time history of free vibration, the time shifting of one half of the system period would only find another time history that is different from the original time history in signs and with a scalar factor. Thus, the associated matrix A will not be full-ranked and it is impossible to solve for sufficient eigenvalues. In a more general sense, the same problem will be encountered if the time shifting is taken as a multiple of one half of the system period. Theoretically, the other values of time shifting are eligible for effective identifications. When the RD signature is not an ideal time history of free vibration, however, different values of time shifting in the application of the ITD method will include deviated errors into A and produce the identified results with a variety of accuracy. As the time shifting is close to one quarter of the system period, the least related original and shifted time histories will be obtained and consequently lead to the optimal identification. Similarly, the values of time shifting such as three or five quarters of the system period will also yield pretty good results. As for the values of time shifting close to a multiple of one half of the system period, the shifted time history will be nearly dependent on the original time history such that real eigenvalues of A will be obtained and no feasible identification is possible. It is also noteworthy that the identified natural frequencies in different cases are all in excellent accuracy while the errors for the identified damping ratios may not be negligible. This problem comes from the fact that the RD method cannot totally exclude the effects of external excitations and will be next discussed. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


4

Multiple random decrement method

In a very recent study by the authors (Wu [4]), a novel multiple random decrement (MRD) method was proposed to successfully exclude the effects from background excitation. It will be briefly reviewed and illustrate in this section for developing an effective method to identify the modal parameters of cable. The ambient vibration measurement for a cable numbered L33 (length = 126.4 m and inclined angle = 26D ) of the Chi-Lu cable-stayed bridge close to the epicenter of 1999 Ch-Chi earthquake occurred in Taiwan can be taken as an example for illustration. In Figures 1 and 2, respectively, the originally measured velocity signal and its corresponding RD signature (cutting threshold = 0.8 × standard deviation of the original measurement and Td = half of the original time history) are displayed in both forms of time history and Fourier amplitude spectrum. It is noteworthy in Figure 1 that all the cable frequencies are nearly in an arithmetic sequence, as predicted by the string theory. However, there exist two additional peak frequencies. These two extra peak frequencies have been pointed out to be corresponding to two significant modal frequencies of the bridge deck (Liao [3]). From the cable’s standpoint, these two frequencies can be regarded as the particularly concentrated parts in the frequency content of external excitation. Even in this case where the excitation is far away from a white noise, Figure 2 indicates that the contributions from these two non-cable frequencies have been considerably suppressed after conducting the RD method. Therefore, in the cases where the excitation force is not close to a white noise, the RD method can still greatly raise the contributions from all the modal frequencies and also relatively diminish those from the rest frequencies even though the resulted RD signature may not be a perfect free vibration time history. Based on this generalized concept, a novel multiple random decrement (MRD) method was proposed. More specifically, the conventional RD method can be repeatedly applied on each round of the resulted RD signature to exclusively filter out the initially substantial effects of the excitation frequencies such that the goal of extracting the free vibration time history can be practically attained. 10000 Fourier Amplitude (cm)

Velocity (cm/sec)

1 0.5 0 -0.5 -1 0

Figure 1:

100

Time (sec)

200

300

1000 100 10 1 0.1

0

2

4 Frequency (Hz)

6

8

Original measurement for Cable 33 of the Chi-Lu bridge.

Two associated problems have to be solved before the MRD can be effectively applied in the identification of cable parameters. First of all, the extracted duration Td in each round of RD is the length of time history for the WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


699

next round and thus needs to be taken as large as possible such that a sufficient superposition number N can be reached in the next round. But contradictorily, no adequate number N would be attained if a very large value of Td is selected in the current round. To balance this problem for mutually compensated N and Td in each round of RD, it was suggested (Wu [4]) to choose Td as a half length of the signal from the previous round. As for the second problem, the associated characteristic of the MRD method to also suppress the contributions from the secondary modes makes it impossible for the subsequent ITD identification to directly obtain meaningful parameters for several modes at one time. This obstacle will be overcome in the next section. Fourier Amplitude (cm)

Velocity (cm/sec)

10000 0.2

0

-0.2 0

40

Figure 2:

5

80 Time (sec)

120

1000 100 10 1 0.1 0.01 0

2

4 Frequency (Hz)

6

8

First round RD for Cable 33 of the Chi-Lu bridge.

Mode separation and example

Due to the inherited properties of uniformly separated modal frequencies and extremely small modal damping ratios, the interaction between any two modes of cable would be very limited. This feature was further exploited (Wu et al [4]) to develop a mode separation technique for effectively identifying the parameters of multiple modes. After Fourier transform is taken on the measured velocity time history, the modal frequencies of cable can be accurately determined from the associated Fourier amplitude spectrum. With these available modal frequencies to decide different frequency ranges for separation, the frequency response contributed by each mode of cable is then conveniently extracted in the frequency domain. In this research, the mid-points between any two adjacent modal frequencies are adopted to divide the frequency ranges for different modes. The corresponding time history for each separated mode is subsequently obtained by performing the inverse Fourier transform. Finally, the MRD method together with the ITD method can be independently applied on these individual modal responses to identify the dynamic parameters for each mode. Cable 33 of the Chi-Lu bridge is again taken as an example. The parameters for the first 10 modes of this cable are aimed for identification. After the mode separation is performed as previously described, the MRD technique is applied on each separated modal time history with a cutting threshold = 0.8 × standard deviation of the target time history and Td = half of the original time history for each round of RD. The MRD method is set to stop before the criterion of WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

700 Computational Methods and Experimental Measurements XIII N ≥ 100 cannot be guaranteed. In addition, the ITD technique is also utilized to identify the corresponding parameters after each round of RD. As discussed in Section 3, the time shifting for each mode is in consistent correspondence to 1/4 of each modal period. The identified parameter values for each mode in all the RD rounds are arranged and listed in Table 2. Based on these results, it is obvious that the conventionally ambiguous modal damping ratios all reach stably convergent values, usually with very few rounds of RD. This trend shows that not a lot of RD rounds are required to obtain satisfactory parameters from identification and also indicates the effective performance of the approach proposed in this study.

Table 2:

Identified modal parameters of Cable L33 in each round of MRD. Identified modal parameters

MRD

ω1

ξ1

ω2

ξ2

ω3

ξ3

ω4

ξ4

ω5

ξ5

(Hz)

1st Round

(Hz) (Hz) (%) (%) (%) (Hz) (Hz) (%) (%) 0.913 0.2061 1.808 0.0745 2.720 0.0994 3.627 0.1546 4.534 0.0630

2nd Round

0.914 0.2312 1.808 0.0559 2.720 0.0934 3.626 0.2007 4.534 0.0685

3rd Round

0.914 0.2349 1.807 0.0530 2.721 0.1003 3.625 0.2324 4.533 0.0717

MRD 1st Round

ω6

ξ6

ω7

ξ7

ω8

ξ8

ω9

ξ9

ω10

ξ 10

(Hz) (Hz) (Hz) (Hz) (Hz) (%) (%) (%) (%) (%) 5.474 0.1302 6.365 0.0461 7.278 0.1075 8.221 0.0881 9.141 0.0480

2nd Round

5.473 0.1260 6.365 0.0444 7.279 0.1108 8.220 0.0696 9.141 0.0533

3rd Round

5.475 0.1140 6.365 0.0426 7.278 0.1141 8.220 0.0669 9.141 0.0560

4th Round

•

•

6.365 0.0431

•

•

•

•

•

•

References [1] [2] [3] [4]

Cole, H.A. Jr., Methods and apparatus for measuring the damping characteristics of a structure. United States Patent No. 3,620,069, 1971. Jeary, A.P., The description and measurement of nonlinear damping in structures. Journal of Wind Engineering and Industrial Aerodynamics, 59(2-3), pp. 103-114, 1996. Liao, C.-A., Modal parameter identification for both the cables and girders of cable-stayed bridges based on ambient vibration measurements, Master Thesis, National Yunlin University of Science & Technology, 2007. Wu, W.-H., Liao, C.-A., and Chen, C.-C., A novel multiple random decrement method for modal parameter identification of cable-stayed bridge cables, International Operational Modal Analysis Conference 2007, paper no. 92, Copenhagen, Denmark, 2007.



701

Experimental evaluation of dynamic properties of an incrementally prestressed concrete girder railway bridge S. I. Kim1, I. H. Yeo1, N. S. Kim2, J. W. Kwark3 & J. S. Lee1 1

Korea Railroad Research Institute, Republic of Korea Pusan National University, Republic of Korea 3 DKorea Institute of Construction Technology, Republic of Korea 2

Abstract The estimation of exact dynamic behavior is very important in railway bridges which can undergo resonance state from repetitive moving axle forces with uniform intervals. The dynamic interaction between bridge superstructures and passing trains is one of the critical issues concerning newly developed bridges designed with more flexibility. Therefore, it is very important to evaluate the modal parameters of PSC girders that have been newly designed before doing dynamic analyses. In this paper, a full scale incrementally prestressed concrete girder of 25 meters long as a test specimen was fabricated and modal testing on it at every prestressing stage was carried out to evaluate the modal parameters, including the natural frequency and the modal damping ratio. Young’s modulus was also obtained from global stiffness of the test specimen. During the modal testing, a digitally controlled vibration exciter as well as an impact hammer is applied, in order to obtain precise frequency response functions, and the modal parameters were evaluated with various construction stages. Keywords: railway bridge, modal test, dynamic performance estimation, moving train analysis.

1

Introduction

Repetitive moving forces with uniform intervals induced from a passing train can cause some undesirable behavior of railway bridges. The resonance of a structure can be broken out when the natural frequency of the bridge coincides with the WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070691

702 Computational Methods and Experimental Measurements XIII exciting frequency of moving forces. For that reason, an exact application of the dynamic properties of a structure will lead to an exact understanding of the dynamic behavior of a structure under moving train loads. As an alternative of conventional prestressed concrete girders, various types of PSC girders are being developed for, and applied in, as roadway and railway bridges. Incrementally prestressed concrete girder (IPC) and concrete girders with encased steel I-shaped beams are two representative types of these newly developed girders. According to the design concept, these new types of PSC girders have considerable advantages in terms of weight, while presenting the capability of creating longer spans. However, the dynamic interaction between bridge superstructures and passing trains is one of the critical issues concerning these railway bridges that have been designed with more flexibility. Therefore, it is very important to evaluate the modal parameters of PSC girders that have been newly designed before doing dynamic analyses. The investigation of the dynamic behavior of bridges under moving loads dates back many years. During the past three decades, numerous experimental and theoretical study on moving loads problem of bridges has been carried out from early studies (Fryba, [3]) to recent studies (Yang et al. [6], Kwark et al. [4]). From simple concentrated force simulation to sophisticated interaction, various vehicle-bridge models and analysis methods have been implemented for more precise verification of the dynamic behavior of bridges under moving roadway or railway loads. However, most of experimental studies and theoretical studies are not related effectively. In the present study, dynamic properties from modal test of a full-scale girder are tried to be linked with the bridge-train interaction analysis for the estimation of dynamic performances. A 25-meter-long full scale IPC girder as a test specimen was fabricated and modal testing on it at every prestressing stage was carried out to evaluate the modal parameters including the natural frequency and modal damping ratio. Young’s modulus was also obtained from static loading tests. During the modal testing, a digitally controlled vibration exciter as well as an impact hammer is utilized, in order to obtain precise frequency response functions, and the modal parameters were evaluated with various construction stages. The sectional and material properties of design values can have a number of differences compared to what is actual constructed, and the damping ratio of a structure cannot but depend on assumption particularly. Thus, an understanding of the exact dynamic behavior of railway bridges should be started from the exact modal properties of an applied bridge.

2

Modal tests of an incrementally prestressed girder

2.1 General For the purpose of observing the modal properties of an IPC girder, modal experiments were performed sequentially, as shown in table 1. Experiments from Test I to Test IV were conducted in order to consider structural state changes according to the construction stages of an IPC girder. Test V was performed to WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


703

verify the mode shapes from the impact hammer. During the modal testing, a vibration exciter that was controlled digitally was utilized, as was the impact hammer, in order to obtain the frequency response functions more accurately. Figure 1 shows a complete view of a 25m full scale IPC girder model, a cross section of an IPC girder, and five channels of accelerometers and a LVDT used in experiments. Modal tests procedure.

Test Case

Structural Condition

Vibration Source

Test I

PSC girder only tensioned with bottom tendons

Hammer

Test II

PSC girder with slab composition

Hammer & Exciter

Test III

PSC girder tensioned with one of the upper tendons

Hammer

Test IV

IPC girder complete stages

Test V

Mode shape configuration

C

2000 Y1

Y0

Y2

Hammer & Exciter Hammer

A

B

Accelerometer Y3

Y4 1800

340 920 540

Table 1:

LVDT

200 400 1400

C'

Figure 1:

25000

B'

A'

A complete view and location of sensors.

In figure 1, Y0 and Y4 are used only in Test V, which is performed in order to examine the mode shapes resulting from the impact hammer test. The mass of the impact hammer was 5.5kg, and its frequency range was 0~250Hz. The vibration exciter was composed of two unbalanced masses of 100kg each, and had a frequency range of 0~12Hz. The sampling rate of the tests was set to be 500Hz for each channel. The time interval ∆t can be determined by sampling


704 Computational Methods and Experimental Measurements XIII frequency fA as shown in eq. (1), and the frequency resolution ∆f in the frequency domain can be acquired with the sampling data number N, as shown in eq. (2).

∆t = 1 / f A ∆f = 1 /( N × ∆t )

(1) (2)

2.2 Evaluation of modal properties A 25m full scale IPC girder for a railway bridge was fabricated and tested before the estimation of the dynamic performance under moving train loads. The natural frequency and damping ratio were evaluated from the modal experiments by the impact hammer and vibration exciter. The impacts by the impact hammer numbered 15~25 at the L/4 and L/2 points for each test. The vibration exciter is excited at the L/2 position of the model. Figure 2 shows APSD functions (Bendat and Piersol [1]) at each accelerometer, which were acquired from the impact hammer’s actions. The damping ratio can be evaluated by a half-power band width method from APSD functions. Figure 3 shows another response by the vibration exciter. In a resonance test by the vibration exciter, a sine sweeping was performed to cause a resonance state, and steady-state response could be acquired from the resonance frequency. The time history of each accelerometer from the resonance frequency was a free vibration response. The damping ratio was evaluated using the logarithmic decrement method. -8

-8

10

Power Spectrum of Acceleration


10

Mid hammer Y1 Average(N=15) 7.4463 -9

10

154.5410

-10

10

72.8760 124.3897

14.8926 23.4375 29.7852

-11

10

0

25

50

75

179.4434

7.4463

Mid hammer Y2 Average(N=15)

-9

10

-10

10

72.8760 125.6104 29.6631

-11

10

100 125 150 175 200 225 250

228.1494

181.1744 155.0293

15.0147

22.4609

0

25

50

75

Frequency(Hz)

100 125 150 175 200 225 250

Frequency(Hz)

-8


10

Mid hammer Y3 Average(N=15)

7.4463 -9

10

-10

72.8760

10

154.9072 124.3897

-11

10

23.5596 29.7852 14.8926

0

25

50

75

179.0772

228.7598

100 125 150 175 200 225 250

Frequency(Hz)

Figure 2:

APSD functions at Y1, Y2, Y3 (impact hammer tests).


Computational Methods and Experimental Measurements XIII 0.8

6.900Hz Y(1)

0.4

Power Spectrum

Acceleration(volt)

0.6

0.2 0.0 -0.2 -0.4 -0.6

3.0x10

-6

2.5x10

-6

2.0x10

-6

1.5x10

-6

1.0x10

-6

5.0x10

-7

2070-2RPM(6.900Hz) Y(1)

705

6.896Hz

0.0

-0.8 55

56

57

58

59

60

61

0

62

2

4

6

8

10

Frequency(Hz)

Time(sec) 0.06

6.900Hz Y(1)

Damping Ratio (ξ)

0.05 0.04 0.03

ξaverage=0.0252

0.02 0.01 0.00 5

10

15

20

25

30

35

Shifted Peak Number (m=4)

Figure 3:

Free vibration response, APSD function at Y2 and damping ratio (vibration exciter tests).

Table 2:

The 1st natural frequency and damping ratio from experiments and analysis.

Impact Hammer Resonance Tests by Vibration Tests Exciter 1st 1st Damping Bending Bending Damping Ratio Ratio Frequency Frequency TEST I 6.95 Hz 2.19 % TEST II 7.32 Hz 1.25 % 6.73 Hz 2.68 % TEST 7.45 Hz 2.70 % III TEST 7.45 Hz 1.43 % 6.90 Hz 2.57 % IV

Free Vibration Analysis 1st Bending Frequency 6.02Hz 6.79Hz

Table 2 summarizes the results of each experiment, and shows the fundamental frequencies in flexural mode and the damping ratio of the 25m full scale IPC girder model. It was found that the natural frequency from Test II had an increase of 5.32% before the composition of the deck. In addition, from a comparison of Test II and Test IV, through a tensioning of the upper tendon, an increase of 1.78% in the impact test and an increase of 2.53% in the resonance WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Normalized Magnitude of Power Spectrum

test were found. This result coincides with other theoretical and experimental studies concerning the effects of prestressing on the stiffness of structures (Dall’asta and Leoni [2], Saiidi et al. [5]). In the free vibration analysis that was done before the tests, the first bending frequency appeared at 6.01 Hz in Test I which was for a PSC beam-only, and 6.79Hz in Test IV which was for a complete IPC girder state. The result of the free vibration analysis agrees well with the vibration exciter tests, and the impact hammer test shows relatively higher values of natural frequencies. A compressive strength of 430MPa for a test specimen is similar to the design compressive strength of 400kgf/m2, and the additional weight by the vibration exciter is exceedingly small compared to the total weight of a 25m full-sized model. Therefore, it can be expected that the resonance test by vibration exciter is more credible than the impact hammer test. In order to verify these results, an impact hammer test (Test V) with additional accelerometers to acquire more detailed mode shapes of the model was conducted. From the result of Test V, it was found that the overall exciting force of impact hammer was insufficient for a massive 25m full scale model, and the first bending mode shape of the test model cannot be found accurately by the impact hammer tests. Therefore, the natural frequency from the impact hammer test results in higher values than the resonance tests by the vibration exciter. This can be explained in figure 4 which shows the effective length of the model. As expected, in the impact hammer test, a higher frequency must appear, as the first bending frequency is evaluated from an effective length of 22.07m. 1.2

Measured Mode Shape Polynomial Curve Fit

1.0 0.8 0.6 0.4 0.2

Effective Length = 22.073m

0.0 0

1.474

5

10

15

20

23.547

25

Location (m)

Figure 4:

Curve fitting of the effective length of test model from the impact hammer test.



707

In order to estimate the damping ratio, it can be said that the logarithmic decrement method is more reliable than the half-power bandwidth method. In the present study, the damping ratio from a resonance test can be accepted as producing more reliable values, as the damping ratio tends to increase as the deformation of structure increases. However, the damping ratio varies with the acceleration response, indicating that the acquired free vibration response does not show exponential decay accurately. Therefore, the average value of several damping ratio is selected to representative the damping ratio for the test model. Finally, it can be concluded from Test IV that an IPC girder in the final stages has a damping ratio of 2.5%, as shown in figure 3. Consequently, the natural frequency of an IPC girder from the experiments has error of 1.62% comparing with the analysis. From this result, it can be concluded that the sectional and material properties of design values can be used for estimating dynamic performance. Conversely, a damping ratio of 2.5% would be adequate. These results from the modal tests can be used as credible input data for a dynamic analysis under moving train loads.

3

Conclusion

An estimation of the dynamic performance of railway bridges which undergo a uniform interval of axles and repetitive moving forces induced from the passing train is very important and it is essential for a newly developed bridge that have been designed with more flexibility. The sectional and material properties of design values can have a number of differences from the actual constructed values, and the damping ratio of the structure cannot but particularly depend on such an assumption. An understanding of the exact dynamic behavior of a railway bridge should be undertaken from the exact modal properties of the bridge to be considered. In this paper, a full scale IPC girder of 25 meters long as a test specimen was fabricated, and modal testing on it at every construction stage was carried out in order to evaluate the modal parameters, including the natural frequency and the modal damping ratio. During the modal testing, a digitally controlled vibration exciter and the impact hammer were utilized in order to obtain the frequency response functions more precisely. In addition, the modal parameters were evaluated varying through various construction stages. Various parametric studies on the dynamic behavior under the passage of a moving train can be performed for an estimation of the dynamic performances of an IPC girder railway bridge with the modal properties from the experiments. a.

It can be known from the modal tests that the impact hammer test can overestimate the natural frequency of the structure. The effective length of the fundamental mode of the structure can be shortened if the vibrating force against the experimental model is insufficient. The damping ratio also appears to increase as the deformation of structure increases. Therefore, a damping ratio from a resonance test can be accepted as producing more reliable values. In the present study, the



b.

c.

average value from experimentally produced damping ratios was selected as a representative damping ratio for the test model. Consequently, it can be concluded that a resonance test by a vibration exciter is more credible than impact hammer tests, and that attention is required if using an impact hammer in modal tests. The natural frequency of an IPC girder from the experiments has an error of 1.62% compared to the analysis. From this result, it can be derived that the sectional and material properties of the design values can be adequate for the dynamic performance estimation. On the other hand, a damping ratio of 2.5% is adequate for the dynamic performance estimation. The effects of prestressing on the stiffness of structures can be founded at every prestressing stage. This result coincides with other theoretical and experimental studies concerning the effects of prestressing on the stiffness of structures. However, further theoretical and experimental studies referring to this problem should be conducted for a more detail discussion.

References [1] [2] [3] [4] [5] [6]

Bendat, J. S. and Piersol, A. G., Random Data: Analysis and Measurement Procedures (3rd Ed)., John Wiley & Sons, Inc., 2000. Dall'asta, A. and Leoni, G., Vibrations of Beams Prestressed by Internal Frictionless Cables, Journal of Sound & Vibration, 222(1), pp. 1-18, 1999. Fryba, L., Vibration of Solids and Structures under Moving Loads, Noordhoff International, 1972. Kwark, J.W., Choi, E.S., Kim, Y.J., Kim, B.S. and Kim, S.I., Dynamic Behavior of Two Span Continuous Concrete Bridges under Moving HighSpeed Train, Computers & Structures, 82(4-5), pp. 464-474, 2004. Saiidi, M., Douglas, B., and Feng, S., Prestress Force Effect on Vibration Frequency of Concrete Bridges, ASCE Journal of Structural Engineering, 120(7), pp.2233-2241, 1994. Yang Y.B., Yau J. D. and Hsu L.C., Vibration of Simple Beams due to Trains Moving at High Speeds, Engineering Structures, 19(11), pp.936944, 1997.


Section 9 Dynamics and vibrations



711

A model of spur gears supported by ball bearings F. Viadero, A. Fernandez del Rincon, R. Sancibrian, P. Garcia Fernandez & A. de Juan Department Structural and Mechanical Engineering, University of Cantabria, Santander, Spain

Abstract In this work a model of a 2D spur gear transmission is described for analysis of tooth contact forces and deformations. Assuming the position of each wheel is known, the contact points between gears are obtained taking into account the geometric description of the tooth profiles including profile errors and relief modifications. Then the deformation in each contact point is separated into a global and a local term combining a finite element model and an analytical formulation originating from Hertzian contact theory. The proposed procedure does not need new element meshing for each angular position thus obtaining an important computational advantage. Afterwards, a non-linear system of equations is obtained and solved for each gear position in order to calculate the meshing contact forces. The model can include the possibility of bidirectional single-flank or double-flank action as well as friction forces in the out-of-action line. Once the contact forces are known, it is possible to use the procedure in the calculation of loaded transmission error and meshing stiffness. Furthermore, each gear is supported by ball bearings that are included in the model taking into account their clearance and their variable stiffness due to the change in the number of balls supporting the load. This variable bearing compliance modifies the gear centre distance and as a consequence the transmission error during a turn. Using this methodology a numerical example is presented where the static behaviour of a spur gear transmission is described and analysed. Special attention is focused on the influence of load level on the final loaded transmission error. Keywords: gear, transmission error, bearings, tooth contact, load ratio.



1

Introduction

Gears are one of the most important mechanical components in many advanced machines. Ranging from industry to space, automotive or agricultural equipment, there are a wide range of machines that use gears. This fact justifies the interest in the study of gear dynamics with the aim of design, condition monitoring and vibration and noise control. In this sense it is possible to find works about topics related to gear dynamics such as profile modifications, surface generation kinematics, non-linear interactions, noise, friction, dynamic tooth loads, wear, contact and bending stresses, condition monitoring, efficiency, etc. All of these subjects are also closely related to the so-called Transmission Error (TE), which is defined as “the difference between the position that the output shaft of a drive would occupy if the drive were perfect and the actual position of the output” [1]. There are three main sources of TE: geometry, deflections and dynamics. Although transmission error is a common term used by the gearing community it is possible to add certain names depending on the source that produces the final TE. Thus, it is possible to distinguish between manufacturing, kinematic (sometimes confused with manufacturing), static and dynamic TE. In this work we are interested in the static transmission error also known as Loaded Transmission Error (LTE) as this could be used as an external input excitation for dynamic analysis [2] or as a measure directly related to the noise level of a certain gear transmission. There is also one other quantity of interest, the meshing stiffness, which governs the dynamic behaviour of geared systems acting as a parametric excitation of the gear pair. The periodic change of the number of contacting teeth pairs is one of the most important phenomena involved in gear dynamics. There are several approaches to include it in the dynamic models from the simple average throughout a meshing cycle to more accurate formulations taking into account the stiffness variation along the tooth profile for each pair [3,4]. Nevertheless, some aspects are normally neglected such as the elastic coupling between successive teeth under load, the non-linear variation with the load level or the influence of the support deflection in the final meshing stiffness. Many available dynamic models use a simple formulation for the gear stiffness as their interest is focused on dynamics neglecting all of the terms that could be very useful in certain cases. Special attention should be paid to the case of condition monitoring where a good description of the dynamic forces is crucial for good prediction of machinery condition [5]. The most accurate works [6,7] use the finite element method combined with contact algorithms for evaluating the forces in meshing teeth. Nevertheless, a new meshing is necessary for each angular position, thus making it a high consumption computational task. There are some proposals to avoid this such as the application of Artificial Neural Networks [8] or the combination of finite elements and analytical formulations for the contacts [9,10]. In this work a new model for analysis of contact forces between gear pairs including the interaction with rolling bearings is presented. The gear model is based on the formulation proposed in [9], while bearings include clearances and WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


713

the non linearity due to the changing number of rolling elements supporting the load, following the approach proposed in [11]. Bearing clearances and variable stiffness modify the operation distance between gear centres as well as the pressure angle of the transmission. As a consequence, LTE, meshing stiffness and load ratio should be modified. In this work the proposed model is used only for quasi-static calculations in order to obtain the teeth contact forces and deflections as well as their derived quantities such as LTE, load ratio and meshing stiffness.

2

Gear contact forces

The calculation of gear contact forces requires the solution of three different problems; the description of the gear body geometry, the determination of the contact points as a function of the gear position, and the calculation of the contact forces themselves. The first task will be the definition of the tooth profiles in order to be able to solve the other problems. In this work, the gear generation will be based on a rack-type tool following Litvin’s vector approach [12] taking into account the possibility of tool displacements and also undercutting conditions. Furthermore, a rounding profile was added in the tooth tip to handle corner contact. Contact points and their corresponding separation distances (δi) have been obtained taking advantage of the analytical properties of involute profiles and tip rounding arcs. Positive values for separation distance mean that the points should be in contact and negative values indicate a non-contact condition. Two types of contacting profiles have been considered: involute-involute and involute-circle arc. The number of potential contact points will depend on the contact ratio (ε) and can be expressed as N = 2 ( Ceil (ε ) + 1) where Ceil(x) is a function that rounds x to the nearest integer towards infinity. Contacts on both flanks of any contacting tooth have been considered. In figure 1 the potential contact points for a spur gear pair with a contact ratio between 1and 2, which means two contact teeth pairs, are shown. The calculation of gear forces requires a relationship between the forces and displacements of the contact points. This relation should take into account not only the elastic deflection of a pair of teeth in contact but also the local deflection in the vicinity of the load as well as the load sharing between more than one teeth pair. Due to the complexity of gear geometry, which is composed of several parts (involute, fillet, tip rounding), a common procedure for handling this problem is the development of a finite element model for two gears or only a portion of their teeth and applying a type of gap elements in order to simulate the contact [6,7,13]. Special attention should be paid to the definition of contact loads and boundary conditions. The resulting load, neglecting frictional forces, should be approximated to the elliptic distribution characteristic of the Hertzian contacts. Furthermore, as the contact width will be affected by the load, a small size meshing in the vicinity of contact points should be provided to be valid from WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

714 Computational Methods and Experimental Measurements XIII low to high load levels. Finally, a different meshing is desired for each position of the contact along the tooth profile. This approach is very time consuming and would not be practical if a dynamic simulation were desired.

6 6

1

3

5 2

52 3 4

4

1

Figure 1:

Potential contact points.

Another possibility is to consider the elastic deflections divided in two different contributions: local deflections of Hertzian type in the vicinity of contact points and global deflections, also called structural, that include all the deflections (bending, and shearing) except those due to the local contact. Local deflections are normally approximated using a non-linear formulation based on Herztian theory or any of its variants. In this work, the local deformation was obtained applying the following expression derived by WeberBanashek for bi-dimensional problems. The deformation between the surface and a line at a depth h is uL (q) =

2 2 (1 −ν 2 )   h h  ν q  Ln  + 1 +    − πE  L  L   1 −ν  

h   L

2

2   h  1 +   − 1     L  

where q is the load intensity along the thickness and 2L is the extension of the elliptic distribution of the pressure around the location of the load, which can be written as L=

2 2 ρ p ρg 4  1 −ν p 1 − ν g  +   qρ ; ρ = E g  π  Ep ρ p + ρg

On the other hand, structural deflections have been obtained by means of a plane strain (or plane stress) finite element model presented in figure 2. The model only contains a certain number of teeth depending of the contact ratio of the gear pair. A unitary load is applied in each node in the profile of the tooth located in the vertical position, which will be called loaded active flank, obtaining the displacement in the rest of the nodes of the other teeth both in the right flank but also in the left one in order to apply the procedure in case of two tooth flanks contacts. The displacements obtained will be the flexibility coefficients (βi,j) which represent the displacement of the node j due to a unitary load applied in the node i of the loaded active flank. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


βi, j

Fi

Figure 2:

715

Fi

Load and displacements of the finite element model.

This procedure is not valid for the point where the load is applied, as it is a stress concentration point. In this case, a local finite element model with the same meshing as the global one and a depth h is used to correct the displacements in the vicinity of the load point. This model has a unitary load acting in the opposite sense to the one applied in the global model, considering the nodes on the boundary with the global model as fixed. The displacement obtained with this model is removed from the original one avoiding the inaccuracy in the local displacement of the node where the concentrated load was applied. Then, when n contacts take place, the structural displacements uSj of node j will be defined by n

uSj = ∑ ( β i , j Fi ) i

The total displacement of node j is obtained by addition of local and structural components for both gears, wheel and pinion, which for n contacts is

uTj = u Ljp ( F j ) + u Ljw ( F j ) + u Ljp ({ F1 ,..., Fn }) + u Ljw ({ F1 ,..., Fn })

Once the initial separations {δi} of the contact points are known as a function of the pinion and wheel centre displacement (rp,rw), and angular position, the load distribution {Fi} of gear teeth can be defined on the basis of the following conditions: compatibility of initial separations and elastic deflections and complementary condition to avoid non realistic negative loads. Taking into account the profile errors for each gear the resulting forces should be obtained from the following non-linear system of equations for n contact points defined by a positive separation distance G G G G G G δ i ({rp ,θ p } , {rw ,θ w }) = uTj ({rp ,θ p }{rw ,θ w }{ F }) + eip {rp ,θ p } + eiw ({rw ,θ w })

(

)

under the condition that Fi ≥ 0; i = 1,..., n Here rp and rw are both vectors representing the location for pinion and wheel, θp, θw are the angular positions for each gear and eip, eiw are the profile errors for gear and wheel corresponding to the contact point i. The geometric separation distance will be used as a first step for calculation of contact forces, considering only those that are positive. Nevertheless, the solution algorithm verifies the deflection in the other contact points looking for a new possible contact. Following this procedure it is possible to calculate the forces and their application direction as a function of gear position. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


3

Ball bearing contact forces

Forces in ball bearings involve a similar problem to the gear contact. Again the changing number of contacting elements (see fig. 3) involves a variable stiffness. Furthermore, load increases the period with the maximum number of elements in contact and even the maximun number of contacts themselves. Fortunately, bearing geometry is simpler and the following assumptions can be made: • Planar movement is considered as in the case of gear forces. • Inner and outer races are rigidly fixed to the gear shaft and support. • Only deflections of Hertzian type are considered, neglecting bending and shearing of races and rolling elements. • The angular separation between rolling elements is constant (θb=2π/n). • Rolling elements roll on the surface races without slipping. The last assumption provides a simple expression to calculate the cage angular position, which controls the angular position for each rolling element, as  r  θ cage = θ shaft   r+R where R and r are the outer and inner race radii (see fig 3).

c/2

i c/2

Y

θi

X r R

θb

Figure 3:

Schema of rolling bearing.

Force deflection relationship for the local Hertzian contact is defined by Fθ i = k Bδθpi with p = 3 / 2 for ball bearings where kB is the contact stiffness, both inner and outer contacts connected in series are considered, and δθi the radial deformation (geometric overlapping) of the ith rolling element (located in angular position θi), which will depend on the inner race centre movement (x,y) and the bearing clearance c δθpi = x cos θi + y sin θi − c Only positive radial deformations should be considered in the calculation of resultant force as otherwise the ball will not be in contact. Therefore, the resulting force for the ith rolling element is obtained as



717

1 δ θ i ≥ 0 Fθ i = k B H (δθ i )δθpi ; H (δθ i )  0 δ θ i < 0 Finally the total force is obtained by summation of the n individual forces from each rolling element. The resultant force, knowing the angular positions θi, can be expressed by the x and y direction components, with the following equations, n  Fx = k B ∑ H (δθ i )δθpi cos θi   i =1  ; with θi = θ cage + θ b (i − 1) n Fy = k B ∑ H (δθ i )δθpi sin θi   i =1

4

Application example

The models described in the previous paragraphs have been applied to a pair of gear wheels with the same number of teeth, each one supported by two identical ball bearings. The main gear parameters are contained in table 1, while the support data bearings are defined in table 2 Table 1:

Gear data parameters.

Parameter Number of teeth (Z) Module (m) Modulus of elasticity, E Poisson’s ratio Pressure angle Rack addendum Rack deddendum Rack tip rounding Gear tip rounding Gear face width Gear shaft radius Table 2:

Pinion / Gear 23 3 (mm) 210 (GPa) 0.3 20 (degree) 1.25 m 1m 0.25 m 0.05 m 15 (mm) 9 (mm)

Bearing data parameters.

Parameter Contact Stiffness kB Number of ball bearings n Radial clearance c Outer race radius Inner race radius

Value 7.055 109 N/m3/2 9 20 (µm) 14.13 (mm) 9.37 (mm)

Before carrying out simulations, taking into account both force models, each one is tested individually to validate it. The bearing model was tested applying an increasing load in the x direction (corresponding to a torque from -10 to – WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

718 Computational Methods and Experimental Measurements XIII 100 Nm) obtaining the equilibrium position for a complete turn of the shaft (360 degree). The number of active contacts is shown in figure 4, while the corresponding orbits are shown in figure 5. It can be observed that a displacement in y direction appears even though there is no force applied in this direction. Moreover displacement in the y direction is greater than the displacement in x direction. Therefore, when bearings interact with gear forces a variation of the gear centre position should be expected and therefore a modification of the operation distance and pressure angle.

4 3.8 3.6 3.4

Elements

3.2 3 2.8 2.6 2.4 2.2 2 3000 2500 2000

300

Load (N)

350

250

1500

200

1000 500 0

Figure 4:

50

100

150 degree

0

Number of active elements.

-6

2

x 10

1.5

1

Y displacement (m)

0.5

0

-0.5

-1

LOAD -1.5

-2 2.5

3

Figure 5:

3.5

4 4.5 X displacement (m)

5

Orbits for several load values.


5.5

6 -5

x 10


719

The gear model was also tested taking into account a fixed position for gear centres (rigid support) neglecting profile errors. Then an increasing torque (from 10 to 100 Nm) was applied and the turn angle necessary to obtain the desired torque that will be the LTE was calculated for several angular positions and the torsional stiffness was deduced as in reference [6].

-4

x 10 8 7

L T E (ra d ia n s )

6 5 4 3 2 1 0 100 80 0.1

60 Torque (Nm)

0.05

40

0 -0.05

20 0

Figure 6:

Angle (radians)

-0.1

LTE (radians) for several load values (rigid support).

6

7

x 10

Torsional Stiffness (Nm/rad)

6.5

6

5.5

5

4.5

Figure 7:

-0.1

-0.05

0 Angle (radians)

0.05

0.1

Torsional Stiffness for several load values (rigid support).


720 Computational Methods and Experimental Measurements XIII Both magnitudes are shown in figure 6 and figure 7 for several torque levels. Reference angular position corresponds to the contact in the primitive point without any clearance. It is clear that the torque increases the period of double contact (higher stiffness in figure 7) and the corresponding transmission error. -5

x 10 3

2

1

0

-1

-2

-3 -4

-3

-2

-1

0

1

2

3

4 -5

x 10

Figure 8:

Orbits for several load values with support flexibility.

-3

3.6

x 10

3.4

3.2

3

LTE (radians)

2.8

2.6

2.4

2.2

2

1.8

1.6

0

Figure 9:

0.2

0.4

0.6

0.8 Angle (Degree)

1

1.2

1.4

1.6

LTE (radians) for several load values with support flexibility.



721

Finally the whole model was tested with the same torque levels. This time the interaction between gears and bearings is recorded as can be seen in figure 8 where the orbits for each gear centre are shown. Figure 9 shows the resulting LTE corresponding to the orbits presented in figure 8, assuming a flexible support due the bearing flexibility. In spite of the fact that, in figure 9 it is not appreciated due to the scale, the model is able to predict an oscillation wave in the resulting LTE due to the bearing’s variable compliance. It should also be noted that the LTE shown in the figure 9 should be corrected removing the angular clearance that appears as a consequence of the variable position of gears.

5

Conclusions

A quasi-static model for the study of interactions between gears and ball bearings is presented. The main features of the model developed are the approximation of the gear contact by decomposition of local and global deformations and the inclusion of bearing variable compliance with the angular position. An application example was presented where the interaction between elements can be observed taking into account the effect of the load, obtaining several parameters representative of the system behaviour such as the LTE, centre orbits and the number of active contacts in bearings and gears. In this work some quasi-static analyses were presented, nevertheless one of the aims of this model was its application to the dynamic analysis of gear transmission. Taking into account this fact, special attention was paid to achieving a good compromise between accuracy and computational load.

Acknowledgements This paper has been developed in the framework of the Projects DPI2003-1845 and DPI2006-14348 funded by the Spanish Ministry of Science and Technology.

References [1] [2] [3] [4] [5]

J. Dereck Smith, Gear Noise and Vibration, Marcel Dekker, Inc, 1999. P. Velex, M. Ajmi, On the modelling of excitations in geared systems by transmission errors, Journal of Sound and Vibration, 290(3-5), pp. 882909, 2006. J. H. Kuang, Y. T. Yang, An estimate of mesh stiffness and load sharing ratio of a spur gear pair, Proceedings of the ASME International Power Transmission and Gearing Conference 1992. Y. Cai, T. Hayashi, The linear approximated equation of vibration of a pair of spur gears (theory and experiment), Transactions of the ASME, Journal of Mechanical Design, 116, pp. 558-564, 1994. A. Parey, M. El Badaoui, F. Guillet, N. Tandon, Dynamic modelling of spur gear pair and application of empirical mode decomposition-based statistical analysis for early detection of localized tooth defect, Journal of Sound and Vibration, 294(3), pp. 547-561, 2006. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

722 Computational Methods and Experimental Measurements XIII [6] [7] [8] [9] [10] [11] [12] [13]

Ian Howard, Shengxiang Jia, Jiande Wang, The dynamic modelling of a spur gear in mesh including friction and a crack, Mechanical Systems and Signal Processing, 15(5), pp. 831-853, 2001. Shengxiang Jia, Ian Howard, Comparison of localised spalling and crack damage from dynamic modelling of spur gear vibrations, Mechanical Systems and Signal Processing, 20(2), pp. 332-349, 2006. L. D. MacLennan, An analytical method to determine the influence of shape deviation on load distribution and mesh stiffness for spur gears, Journal of Mechanical Science Part C, 216, pp. 1005-1016, 2002. A. Andersson, L. Vedmar, A method to determine dynamic loads on spur gear teeth and on bearings, Journal of Sound and Vibration, 267(5), pp. 1065-1084, 2003. R.G. Parker, S. M. Vijayakar, T.B. Imajo, Non-linear dynamic response of a spur gear pair: modelling and experimental comparisons, Journal of Sound and Vibration 237 (3), pp.435-455, 2000. M. Tiwari, K. Gupta, O. Prakash, Dynamic response of an unbalanced rotor supported on ball bearings, Journal of Sound and Vibration, 238(5), pp. 757-779, 2000. F.L. Litvin and A. Fuentes, Gear Geometry and Applied Theory, Cambridge University Press, 2004. G. Bonori, A. O. Andrisano, F. Pellicano, Stiffness Evaluation and Vibration in a tractor gear, Proceedings of ASME International Mechanical Engineering Congress and Exposition (IMECE) 2004.



723

On a numerical model of a complete washing machine T. Argentini1, M. Belloli1, N. Gaudiano2, G. Fraternale2, F. Panetta1, D. Sabato1 & M. Vanali1 1 2

Department of Mechanics, Politecnico di Milano, Milano, Italy Indesit Company Spa, Fabriano (AN), Italy

Abstract We present a numerical model of a complete horizontal-axis washing machine and its validation. Our purpose is to develop an effective mathematical model able to predict the machine dynamics during steady-state spinning cycles, with a specific focus on the structural vibrations of the cabinet. The complete model couples a parametric linearized multi-body model under unbalanced-mass forcing for the drum unit and a structural finite element model for the cabinet. The drum unit and cabinet are connected by means of a suspension system composed of three extension springs and two dry-friction dampers. We present two different models to simulate the dampers’ behaviour and we compute the response of the cabinet using modal superposition. Besides numerical modelling, we carried out an extensive experimental campaign in order to characterize every element of the washing machine, to determine every parameter of the model, and to collect a large database to validate the model. Experimental results are reported in Argentini et al. “Experimental characterization of the vibro-acoustic behaviour of a complete washing machine. XXV IMAC, 2007”. A satisfying reproduction of the machine dynamics emerges from the comparison between numerical and experimental responses. The numerical model will be used to design and test new solutions to improve the dynamical response and to reduce structural vibrations. Keywords: washing machine, numerical model, multi-body, FEM, friction dampers, vibrations.



1

Introduction

In a market that is more and more oriented towards customer satisfaction, high performances together with high comfort levels provide the competitive edge to gain market share. In the case of washing machines, very high spin speeds (up to 1600 rpm) are offered to end-users. However, high performances often drop comfort levels: indeed, a drawback of having high spinning speeds is that the cabinet of the washer is forced at higher frequencies than it was designed for and it vibrates excessively, generating noise. Therefore, reducing the noise emitted during washing cycles, and specifically during spinning cycles, is a very attractive target, because it permits the matching of performance and comfort. Up to now, research and development of washing machines have been mainly carried on adapting existing machines by means of experimental and heuristic approaches, with high development costs and long research time. Within this scenario, considering that computational speed is continuously improving, numerical models are gaining ground as design tools that allow to improve products’ time-to-market and to reduce design costs. In this perspective, we present a numerical model of a horizontal-axis washing machine able to predict its dynamics. In particular, we focus our modelling on the dynamics during spinning cycles since, as we showed in [1], at these forcing frequencies the noise due to structural vibrations is particularly unpleasant for end-users. To obtain the global equations of motion, we divided the model into two distinct parts. The drum unit is modelled with a parametrical linear multi-body approach, using lumped parameters to represent the inertial, elastic or dissipative contribution of each element. A parametrical approach, i.e. the possibility to modify one or more elements independently, is necessary to get a versatile design tool; a linear approach is adopted since we study the steady-state dynamics of the drum unit, that during spinning cycles is forced in its seismographic zone, and therefore we account for small displacements and linearized equations of motion. To model the cabinet we used a finite element approach [2], and we used a reduced modal superposition to evaluate the vibratory response of some significant points.

2

Drum unit model

The model we used to schematize the system is depicted in Figure 1. The drum unit is composed of four rigid bodies: a plastic tub (in which we include the inertia of the electrical motor), a rotating stainless steel basket, and two concrete ballasts constrained at the tub. The unit is linked to the cabinet by means of three springs, two dampers and a gasket. An unbalanced mass, constrained to the tub with a known eccentricity, forces the system. The model global reference system is placed in a corner of the cabinet: from it, we define the coordinate of the points of interest, such as the centre of gravity of each body and the hinges of springs and dampers, as shown in Fig. 1.



725

To write the equations of motion of the drum we consider six degrees of freedom (displacements and rotations of the tub) and the imposed angular velocity of the basket. Gballast1 tub’s local reference system Gtub Gmecc Gbasket

main reference system z x

Gballast2

Figure 1:

y

Idealization of the real washing machine: tub, gasket, suspensions, main and a local reference system.

The total kinetic energy of the drum is obtained through the sum of the kinetic energies of each body; N 1 [ M ] EkTOT = ∑ Eki , with Eki = yTi  2  [ 0] i =1

[ 0]  y [ J ] i

=

1 T y M i  y 2 i   i

(1)

where y is a vector that represents the dofs of the centre of gravity of the i-th body, [M] is the local mass matrix, and [J] is the principal inertia tensor: these inertial and geometric properties can be easily retrieved from 3D solid models of new prototypes. The projection of the local principal references onto the global reference is made in two steps: first, we rotate it using a direction cosine matrix [Λθ] onto a reference system parallel to the global one and centred in Gi, i.e.: Second, we project the [Mi*] matrix onto the global reference system using a linear transformation matrix, [ΛT], which contains opportunely the three distances between global and local reference system.

Eki =

T T 1 T 1 T q i  ΛTi   Λϑi   M i*   Λϑi   ΛTi  q i = q i  M i  q i 2 2

(2)

To reproduce the effect of the unbalanced mass we consider it as an external force applied to the centre of mass of the tub. We write the equation of motion of the mass in the coordinate system of the basket, computing its kinetic energy and applying Lagrange’s formalism. With reference to Figure 3, the velocity of the mass vm can be written as follows: ecc


726 Computational Methods and Experimental Measurements XIII vxc = xc + θ yc e cos(θ yc ) − θ zc ey  v yc = yc + θ zc e sin(θ yc ) +    − θ xc e cos(θ yc )  vzc = zc − θ yc e sin(θ yc ) − θ xc ey

vmecc = vx2c + v y2c + vz2c

(3)

y esin(θyc ) z mecc

ey

θy c

x x θz c z

ey

ecos(θyc ) y

θx c

Figure 2:

Position of the unbalanced mass in the basket local coordinates.

It can be noticed that the velocity of the unbalanced mass has a non-linear dependence upon the rotation and on the angular velocity of the basket. Deriving these terms according to the Lagrange’s formalism, their dependence on the six degrees of freedom of the system is highlighted. Organizing the terms according to their dependence upon the six dofs of the tub and their derivatives, we can write the contribute of the unbalanced mass as follows:

Qc = [ M ecc (t ) ] qc + [ Recc (t ) ] qc + f ecc (t )

(4)

The terms [Mecc(t)] and [Recc(t)] contain the dependence of these elements on the accelerations and velocities of the tub, the external force fecc(t) contains the terms that have no dependence upon the chosen degrees of freedom. The elastic potential is due to the three springs. The general formulation of the potential energy for each spring is: T

 ∆lx  1 T 1  Vs = ∆l [ K ] ∆l = ∆l y  2 2   ∆lz 

kx 0   0

0 ky 0

0   ∆l x    0  ∆l y  k z   ∆lz 


(5)


727

where ∆l is a vector containing the dynamical extension of the spring projected onto the local orthogonal coordinate system of the spring, shown in Figure 3. The stiffness kx is the value found experimentally [1], whereas ky and kz are the stiffness values due to the fact that the spring is statically pre-loaded [5]. These values can be computed as:

k y = k x ∆l0

∂ ∆l ∂z A2

= kx yA =0

2

= kx z A =0

ldef − l0 ldef

(6)

ldef − l0 ldef

x

z

k z = k x ∆l0

∂ 2 ∆l ∂y A2

Figure 3:

Spring local coordinate system (2D view).

where yA and zA are the displacement of the hinge on the drum in the spring local reference, while ldef and l0 are respectively the pre-loaded and unloaded length of the spring. A big part of the dissipating contribution to the system is due to the gasket, which also preserves elastic properties. This element is not easy to model (with non-linear dependence on the amplitude and the frequency of motion) but, at the effects of our work, it is sufficient to consider a system of linear elastic and dissipative elements, placed in an equivalent point (see Fig.1). This point is the center of the porthole and, knowing its coordinates, we can write the potential and dissipating terms of the gasket. We consider as well the linearized gyroscopic effect due to the spin speed of the basket [5].

3

Damper models

Beside the gasket, most part of the dissipative energy is due to the two dampers placed under the washing machine tub (see Fig.1). During the experimental campaign, we evidenced the non-linear characteristic of these elements: we now propose two different models to reproduce this behavior. The first approach uses a formula that fits the hysteresis loop of the damper [3]: we will call it the closed form formulation, in witch the damper force can be expressed as: WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

728 Computational Methods and Experimental Measurements XIII      ∆f xv (t ) − xb  f damper (t ) = f ab +  ∓ f ab   f 2      x 1 − 2 ab  ± ( xv (t ) − xb )  ∆f   s  

(7)

where xv(t) represents the motion of the connection point of the damper with the tub, xb and fab are the last value of xv(t) and fdamper(t) before the reversal of the motion, ∆f is the amplitude of the hysteresis loop, and xs a parameter proportional to the slope of the hysteresis loop (see Fig. 4). The operators ± and ∓ change their value if the damper is in a compression or traction phase. 40 20 [N]

slope 0 -20 -40

Figure 4:

-5

0 [m]

5 x 10

-3

Hysteresis loops: closed formulation (dotted) and rheologic model (solid).

The second model implemented is a rheologic one. This model can be schematized with a one degree of freedom damper-spring-mass system that slides with friction (see Fig. 5). yd , y˙ d , y¨d kd xv , x˙ v , x ¨v md fdamper rd T

Figure 5:

Rheologic model scheme.

The damper force (fdamper) can be easily computed by force balance, considering that kd is a linear stiffness, rd a viscous damping and T is a friction force whose amplitude is proportional to the mass and whose sign is function of its velocity. A typical hysteresis cycle is represented in Fig. 4. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


729

For both models the determination of the parameters comes from the minimization of the error between the numerical and the experimental dissipated energy. The main advantage of the closed formulation model is its simplicity and the little computational time that it requires; unfortunately this method does not contain any information about the excitation frequency. Experimentally, we evidenced that this is acceptable for the real damper in the range over 5 Hz, therefore this model is very useful especially for spinning cycles simulations. On the other hand, the peculiarity of the rheologic model is its frequency dependent response and the consequent easiness of performing a sensitivity analysis since each parameter has a physical meaning. Its major drawback is the substantial increase in the integration time due to the reduced time step necessary to simulate the friction force and to the addition of one dof for each damper. To test both models we made a comparison with the real damper, using the results of the experimental campaign. In particular, we focus our attention on the total dissipated energy per cycle and on the frequency content of the force. In Fig. 6 we show the hysteresis loop in a test at 10 Hz with 5 mm amplitude of sinusoidal imposed motion. 40

[N]

20 Experimental Closed form Rheologic

0 -20 -40

Figure 6:

-5

0 [m]

5 x 10

-3

Hysteresis loop comparison between experimental and numerical dampers (5 mm, 10 Hz).

It can be seen that the non linearity of the experimental force (red line) is better represented by the rheologic damper (dotted line), but its dependence on the frequency and the motion amplitude is distant from the experimental one (solid line). Yet the closed-form damper (dashed line) has always the same behavior in the frequency domain and it best fits the area of the loop, even if it introduce more energy at higher frequencies (see Fig. 7). It is possible for the closed-form model to fit better the high frequency forces using a low-pass filter. Coupling these systems to the drum unit model, we evaluated that the closed form damper (especially during spinning cycles) guarantees the best agreement between experimental and numerical results for all the drum angular velocities from 600 to 1600 rpm. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


4

Cabinet model

A model of the cabinet is necessary to integrate the drum unit model in order to get a complete washing machine dynamical model. The cabinet, whose solid geometric model was available from its designers, can be modeled using a FEM approach [2]. The resulting FE model leads to very large structural matrices ( n = 817218 degrees of freedom, in our case), consequently solving directly the equation of motion is computationally expensive and time consuming. Experimental

[N]

10 10

10

Figure 7:

Rheologic Closed formulation

0

-2

-4

0

100

200 [Hz]

300

400

Force spectrum comparison between experimental and numerical dampers (5 mm, 10 Hz).

Figure 8:

Example of FEM analysis result, 18° mode at 81Hz.

Therefore, to predict the behaviour of the cabinet we developed a model using a reduced modal superposition approach using the results of a FEM eigenvalue analysis (see Fig. 8). We approximate the modal response using k eigenmodes of the n we arrange, considering a defined bandwidth, in our case 200 Hz. For this reason we need to solve a system of k decoupled single-dof equations in modal coordinates qi . Each equation has the following form:

qi + 2hiω0i qi + ω02i qi = Qi (t )

i = 1,… , k


(8)


731

where i in the mode number, hi the modal damping, ω0i2 the eigenvalue associated with the i-th mode, and Qi the force vector projected onto the i-th eigenvector. Forces in Qi are the forces transmitted by the three springs and the two dampers.

5

Model validation

It is important to verify that the hypothesis made during its development is justified by a good correspondence between numerical and experimental results. First we validate the tub motion, then we proceed with the cabinet motion. One of the model results, at an imposed angular velocity excitation, is the time history of the tub’s centre of mass, while experimentally we can evaluate the acceleration of six points on its surface [1]. Through an appropriate transformation matrix the output of the model has been made consistent with the experimental one. In Fig.9 we show a comparison between steady-state responses at different excitation frequencies: the washing machine is tested with the two dampers disconnected, and with an unbalanced mass of 190g. The model represents the response of the vertical accelerometers (z-front, zback), but, while in the experimental results the vertical mode is coupled to a yaw one, in the numerical response this phenomenon does not appear. The rear and lateral accelerometers responses (y-right, y-left, x-front, x-back) are overestimated at 1.5-3 Hz, while they are underestimated in the range 4-5 Hz. z front

Simulation 0.02

z back y right

[m]

0.015

y left x front

0.01

x back 0.005 0

1

1.5

2

2.5

1

1.5

2

2.5

3 3.5 [Hz] Experimental

4

4.5

5

4

4.5

5

0.02

[m]

0.015 0.01 0.005 0

Figure 9:

3 [Hz]

3.5

Drum unit motion: numerical vs. experimental.


732 Computational Methods and Experimental Measurements XIII These discrepancies are due to the simplifications introduced in the gasket modeling as it has been evidenced by a comparison done excluding it. Since our objective is the simulation of the vibratory response of the cabinet at high rpms, it is more meaningful to see directly a comparison between the experimental and numerical responses of the cabinet for a high frequency excitation. The cabinet motion is measured with a number of accelerometers placed on the five panels. As an example, in Figure 9 we show the response of one point the cabinet during a spinning cycle with a 500g unbalanced mass at 1100rpm. As visible, the odd harmonics, which are the most relevant, are very close to the experimental one, whereas the even harmonics, whose contribution to the frequency content is minor, reflect the characteristics of the proposed models for the dampers.

2

[m/s ]

3

experimental

2 1 0

20

40

60

80

100 [Hz]

120

140

2

[m/s ]

3

180

200

simulation

2 1 0

Figure 10:

6

160

20

40

60

80

100 [Hz]

120

140

160

180

200

Spectra of the accelerometer placed in the centre of the cabinet left panel.

Conclusions

We summarized the passages taken to realize a numerical model of a complete washing machine. The linear approach adopted is justified by our initial hypothesis and by the results presented that validate the model for the simulation of the vibratory response of the cabinet during spinning cycles. The limits of the model are as well evidenced. This tool allows the prediction of the dynamic behaviour of the machine when it is modified by either moving the position of the concrete ballasts or adding other springs to the system. It is therefore a versatile design tool to test very easily many different solutions able to improve the system response.



733

References [1] [2] [3] [4] [5]

T. Argentini, M. Belloli, N. Gaudiano, G. Fraternale, D. Sabato, M. Vanali. Experimental characterization of the vibro-acoustic behaviour of a complete washing machine. XXV IMAC, 2007 F.Panetta, M.Belloli, S.Miccoli, A.Tosi, R.Viganò. Analisi numerico sperimentale del comportamento dinamico di una macchina lavatrice. In XXXIV Convegno Nazionale AIAS, 2005 B. Warner. An analytical and experimental investigation of racing car suspension system. Journal of commercial vehicle, 1996. A.A. Shabana, Dynamics of Multibody Systems, Third Edition, Cambridge University Press, 2005 G.Diana and F.Cheli: Dinamica e vibrazioni dei sistemi meccanici. Vol. 1 and 2. UTET Spiegel. Milano, Italy 1993.



Section 10 Detection and signal processing (Special session organised by Professor Z. Bielecki)



737

EUV detection system with calibrated responsivity J. Mikolajczyk & Z. Bielecki Institute of Optoelectronics, Military University of Technology, Poland

Abstract We describe a measuring system of EUV radiation energy using special calibration responsivity. This system is specified by responsivity of 0.043 A/W at the wavelength range of 13.5±0.5 nm (FWHM=7.2%). A determination of the calibration responsivity for a laser-plasma source with Xe/He gas puff target is also discussed. This parameter makes it possible to reference measurement results to the standard wavelengths band of 13.5±0.13 nm (FWHM=2%). The value of the calibration responsivity depends on the spectra of measured radiation and features of the system elements. In this paper, the analysis and investigation results of the source and the measuring system are demonstrated. Keywords: EUV radiation, optical sensors, energy meter.

1

Introduction

Accurate measurements of energy radiation are very important in nanolithography. The resists or masks can be damaged due to the measurements errors. Nowadays, energy meters of EUV radiation are also used for testing of lithography tools. The investigations are connected with characterization of EUV radiation sources, features of optical elements, etc. The responsivity and measurement accuracy are the main features of the measuring devices. The parameters are referenced to a standard wavelength range of 13.5±0.13 nm. The range corresponds with the FWHM=2%. The developed commercial meters are usually built from multilayer mirrors and semiconductor detectors [1]. The mirrors simplify construction of the instruments. Using photodiodes makes it possible to achieve a high efficiency and plain measurements procedures. The pulse energy of radiation is determined by the measured charge generated in a photodiode, the value of the detector WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070721

738 Computational Methods and Experimental Measurements XIII responsivity and the features of optical elements. Achievement of appropriate selectivity of measuring spectrum (FWHM=2%) is important issue of construction process of the meters. The special mirrors with wavelength band of 2% have been constructed. Analyses and investigations showed that the mirrors are characterized by low reflectivity [2]. As the amount of photons reaching detector is limited, the instrument responsivity is decreased. The described effect is avoided using the optical elements with broader wavelength range and a special calibration factor. This factor provides to refer measured data taken in the broader spectrum to the arbitrary one. The factor value depends on spectra of the measured radiation and some performances of the instrument elements. Basing on this factor and the meter responsivity, the calibrated responsivity is calculated by SS = Cλ Sλ . (1) The construction of the laser-plasma source with gas-puff target necessitated application of an appropriate system for energy measurement [3]. The features of the commercial meters (their dimensions and prices) forced to design a special measuring system (M-EUV system) [4]. The small size, the high-calibrated responsivity and the fully automated measuring process, all are the main system advantages. M-EUV system uses Mo/Si multilayer mirror (FWHM=7.3%), silicon photodiode with integrated absorption filter and the calibration factor. To the best of analysed knowledge, this work presents the experimental results of the first investigation of the system with calibrated responsivity used for measurements of energy emitted by the laser plasma source with Xe/He gas puff target.

2

Calibrated responsivity

The determined value and the measured range of wavelengths describe the energy of source radiation using M-EUV system. The spectrum influence on measurement results is taken into consideration by determination of the system responsivity. The responsivity is the most important parameter of absolute energy measurements and it is defined by

RD =

∆Iph Pλ

,

(2)

where Iph is the current signal generated in detector and Pλ is the power of the exposing radiation. The spectral characteristic of M-EUV system responsivity depends on the mirror reflectivity and the detector responsivity. This characteristic is given by S λ = SDet R Mir . (3) Theoretically determined characteristic of the system responsivity is shown in Fig. 1.



739

Figure 1: Theoretical characteristic of the system responsivity [5]. The maximum value is 0.043 A/W at the wavelength range of 13.5±0.5 nm. Analyses and investigations of the spectra generated from Xe/He gas-puff target gave an opportunity to calculate the calibrated responsivity of the M-EUV system. In Fig. 2, it is presented the spectrum of the source radiation in the two ranges of wavelengths 13.5±0.5 nm and 13.5±0.13 nm describing the system responsivity.

Figure 2: Radiation spectrum with analysed ranges of wavelength [5]. The marked areas specify the radiation energies in the chosen wavelength ranges. The energies are connected with the charge generated in the irradiated photodiode. The value of this charge is given by ∞

Q=Ω

∫ E(λ ) S

λ

dλ ,

(4)

0

where Ω is the solid angle of the source radiation. The energy value of radiation at the range of wavelengths of 13.5±0.13 nm is determined by

Q

E 2% (λ ) =

,

∞

Ω

∫ I(λ ) S

λ

dλ

0


(5)

740 Computational Methods and Experimental Measurements XIII where I(λ) is the normalized function characterising the spectrum of radiation and it is described by

∫ I(λ ) dλ = 1.

(6)

13.5 ± 0.13 nm

The expression given by ∞

S S( λ ) = ∫ I(λ ) S λ dλ ,

(7)

0

defines the calibration responsivity of the M-EUV system [1]. This responsivity depends on features of system elements and the source spectra. The spectra can be shifted by changing conditions of the plasma generation in the investigated source. The aforesaid conditions concern a delay of the nozzles valve opening with respect to a laser pulse, power density of Nd:YAG laser radiation on the target surface, position of a laser beam focus with respect to the gas target axis, pressure of the remaining gases in the source chamber, gases pressures in the valve nozzles [6].

3

Measurement method

Measurements of the calibration responsivity were taken using the laser plasma source with Xe/He gas puff target and the model instrument (E-Mon energy meter [1]). The main aim of the investigations was determination of energy radiation in two ranges of wavelengths 13.5±0.27 nm and 13.5±0.5 nm. The ranges are defined by spectral characteristics of mirrors reflectivity used in E-Mon meter and M-EUV system. The calibration responsivity (SS E-Mon) and measurement uncertainty of E-Mon meter referenced to the wavelength range of 13.5±0.13 nm is taken into consideration for determination of the M-EUV system responsivity. The calibration factor (CM-EUV) is determined by measured energies with EMon meter and M-EUV system. The value of the factor is given by

CM−EUV =

E 7% , E 3%

(8)

where E7% and E3% are the energy values in the wavelength ranges of 13.5±0.5 nm and 13.5±0.27 nm. This factor makes it possible to specify the calibrated responsivity of M-EUV system referenced to FWHM=2%. This responsivity is calculated by

S S M−EUV = CM−EUV S S E−Mon

4

.

(9)


The measurements of the source energy and determination of the energies ratio in two wavelength ranges were the main aim of the investigations. The researches defined the calibration responsivity and also the calibration factor of WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


741

M-EUV system. The spectra of the radiation emitted by the laser-plasma source with Xe/He gas puff target were also analysed. The measurements were taken for different conditions of plasma generation in the source. 4.1 Influence of delay of Xe/He nozzles valve opening The delay of the nozzles valve opening with respect to a laser pulse has direct impact on a dimension and density of the gas target. Measured characteristics of the energy and the calibration factor as a function of the delay of Xe and Xe gas nozzles are shown in Fig. 3.

Figure 3:

Energy radiation and the calibration factor at various delays of Xe and Xe gas nozzles.

The value changes of the calibration factor with delay of Xe gas nozzle are up to relative value of 10%. Its mean value is 1.85±0.06. The maximum energy emitted by the source was achieved for delay from 650 to 850 µs. For He gas, a threshold delay was observed. The energy drops off dramatically for the delay of 200 µs. The noticed threshold has no influence on the factor characteristic. 4.2 Influence of Xe/He gases pressures in the valve nozzles The gases pressure similar to nozzles delay forms the target performances. This influence concerns the source spectra. The characteristic changes of energy and calibration factor as a function of the pressures are shown in Fig. 4. The increase in the energy radiation is determined by simultaneous increase in the Xe gas pressure. The maximum value of energy is limited by strength of gas pipes. The calibration factor characteristic follows the energy characteristics. For He gas pressure below 0.5 MPa, the optimal range of energies is observed. In this case, the calibrated factor decreases negligibly with He gas pressure. 4.3 Influence of pressure of the remaining gases The EUV radiation is absorbed not only by a solid body but also by a gaseous one. The remaining gas target can be accumulated in the chamber of the laserplasma source. The pressure of the remaining gas depends on the source WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

742 Computational Methods and Experimental Measurements XIII repetition and the efficiency of the vacuum pump. The measured value of the pressure was shifted from 10-6 to 10-2 mbar at the source repetition of 10 Hz. The changes of energy and the calibration factor for the remaining gases are shown in Fig. 6.

Figure 4:

Energy radiation and the calibration factor vs. Xe/He gases pressures in the valve nozzles.

Figure 5:

Energy and the calibration factor as a function of pressure of the remaining gases in the source chamber.

The energy and the calibration factor vary significantly with the Xe remaining gas. The threshold pressure of the energy dropping is observed near the value of 10-2 mbar. The calibration factor increases with the pressure build-up. Its mean value is 1.82±0.06. For the He gas, the described influences are not so noticeable. The mean value of the calibrated factor is 1.83±0.03. 4.4 Influence of position of a laser beam focus The valve construction makes it possible to control a beam focus position on the gas target space. Figure 6 shows energy and calibration factor characteristics varied with a position of a laser beam focus. The measured energy and the calibration factor decrease with shifting the target position more than value of +10 µm. The mean value of the calibrated factor was 1.77±0.06. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Figure 6:

743

Energy and the calibration factor as a function of positions of a laser beam focus.

4.5 Influence of energy radiation of Nd:YAG laser The Nd:YAG laser with output energy adjustment was used during the investigation procedures. The energy control was executed by changing a duration time of the active laser pumping. This time can be adjusted in the range from 200 to 2000 µs. Figure 7 shows the source energies and the calibrated factor for changes of duration time of the active laser pumping.

Figure 7:

5

Energy and the calibration factor vs. duration time of the laser pumping.

Results verification

The results were verified by spectral measuring of the source radiation. The analysed spectra were taken by the spectrometer consisted of the reflecting grating (HITACHI 1200 l/mm) and CCD camera (Roper Scientists). The values of the calibrated factor were calculated using the processed graphs of the measured spectra. The graphs were given by multiplication of the spectral characteristics of multilayer mirrors (used in the E-Mon meter and M-EUV system) and the source spectra. The example of the analysed graph is shown in Fig. 8.



Figure 8:

Resultant graphs of the spectra detected by the E-Mon meter and M-EUV system.

The maximum differences between the data taken by the measuring instruments and the spectral analyses were observed for changes of position of a laser beam focus (12%), pressure of the remaining gases (9.5%), and delay of the Xe nozzle valve opening (8.8%). For other conditions of the plasma generations, the noticed differences were less than 4%.

6

Conclusions

The paper presents a unique system for energy measurement of radiation emitted by the laser-plasma source with Xe/He gas puff target. The experimental results specified not only the features of the system, but also characterized an influence of the conditions of the plasma generation on the source spectra. The measured responsivity of the system is 0.043 A/W in the range of wavelengths of 13.5±0.5 nm. For the maximum efficiency of the source, the calculated value of the calibrated factor is 1.79±0.03 in the wavelength range of 13.5 ±0.13 nm hence the system responsivity of 0.291±0.027 A/W. Using the mirrors with wider spectral characteristic of the reflectivity (worse selectivity) has improved the measuring performance of the system. Acceptance of the constant value of the calibration factor (independent on conditions of the plasma generation) is able to make the measurement error of 14%. The uncertainty of factor calculation was 7.4%. The relative value of the determined uncertainty of energy measurement in the spectrum of FWHM=2% is 10.1%. The presented system provides a simple, reliable solution to EUV energy measurements. That is why it is promising alternative for commercial meters.

Acknowledgements This work was financially supported by the Polish Ministry of Science and Higher Education in the frame of research project No 0004/T00/2005/29. The authors would like to thank prof. H. Fiedorowicz (Institute of Optoelectronics, MUT, Warsaw) for fruitful discussion and advice. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


745

References [1] [2] [3] [4] [5] [6]

Cross-Calibration of Extreme Ultraviolet (EUV) Energy Sensors International SEMATECH Technology Transfer #04024498A-TR http://www.sematech.org/docubase/document/4498atr.pdf. http://www.iof.fhg.de/departments/optical-coatings/vuv-euv-ray/projects /euv_schmalband_e.html. Kranzusch S. & Mann K. Spectral characterization of EUV radiation emitted from a laser-irradiated gas puff target, Optics Communications, vol. 200, Issue: 1-6, pp. 223-230, December 15, 2001. Mikolajczyk J., Bielecki Z. & Wojtas J. Testing system for extreme ultraviolet detectors, Processing of SPIE, vol. 5948, pp. 594822-(1-9), April, 2005. Bielecki Z. & Mikolajczyk J. Energy meter for Xe/He gas-puff laser plasma source, Optica Applicata, 2007 in print. Fiedorowicz H. & et al. Compact laser plasma EUV source based on a gas puff target for metrology applications, Journal of Alloys and Compounds, vol. 401, Issue: 1-2, pp. 99-103, September 29, 2005.




747

Analysis of radiating structures placed on multilayer dielectric M. Wnuk & M. Bugaj Faculty of Electronics, Military University of Technology, Poland

Abstract In this work, a solution to the problem of electromagnetic fields scattering on periodical, limited, planar antenna structures, placed on the border of two dielectric layers is formulated. One led out of scattering this kind of structure and defined in its generalization for the case of antenna with a multilayered dielectric basis. By using Galerkin’s spectral method the problem was brought to an arrangement of algebraical equations with disposition on coefficients of the current’s disposition on metal elements of an antenna arrangement, with regard to disposition of field on Floquet’s harmonics. The complete transverse dimension of an antenna was taken into account by introducing antenna’s window functioning solution of unlimited antenna. The formalism allows the modeling of periodical actions, dielectrics, and composite antenna arrays. Keywords: Antenna arrays, multilayer dielectric, Green function.

1

Introduction

Antenna arrays usually consist of periodical metal structures placed on, or set in, multilayer dielectric. Changes of size or shape of each of the antenna’s elements allows for effective modelling of spectral characteristics of parameters of antenna array. These changes can also be modelled by the selection of geometrical and physical parameters of dielectric layer surface and coverage. In the case of big transversal dimensions of antenna (100 lengths of wave) and several dozen or more single radiating metal elements, characteristics of antenna radiation in good approach respond to characteristics of an unlimited antenna structure. When the condition of fulfilling periodicality of such an antenna is fulfilled and it is excited by a flat wave with linear and homogenize module faze, the problem of diffraction of such a structure is brought to an analysis of a single WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070731

748 Computational Methods and Experimental Measurements XIII elementary antenna cell. For structures of smaller shapes and fewer numbers of elements, effects connected with their limitation, especially boundary effects occurring on external cells can have an influence on characteristics of antenna work and should be taken into account during modelling them. So far, a strict method of solving the problem of diffraction on periodical limited structure is not known. Attempts to directly solve the question, with direct usage of spectral moment method, are only formally strict, because their exactitude is limited by limitations of dimensions of matrixes used in procedures of solving them [L-3,4,5]. z

y

(-M,N)

(M,N)

θ

-r

(-1,1)

dy

(0,1) (1,1)

ρ -

φ

(-1,0) (0,0) (1,0) (-1,-1) (0,-1) (1,-1)

x

(M,-N)

(-M,-N)

ε0 µ 0

ε1 b1 ε2 b2 ε3 b3 ε n bn

Figure 1:

Planar periodic multilayer antenna structure.

Nevertheless, for antenna arrays consisting of several or several dozen of basic cells of regular shape, this method gives good results. Other known methods of solving problems of diffraction on larger periodical structures, such as modifications mentioned in the above method by limiting the number of input base functions and accurate selection of their course [L-1] or iteration method used in solution by sequenced usage of fast Fourier transform are approached, and can be used only in some simple, easy to analyse cases. The analysis presented is based on the idea of window function [L-2] used in seeking current distribution and composed in the structure of a problem of diffraction on periodical unlimited structure. Presented in the paper is the sequence of the next steps of solving leads to strict solution and takes into account only predicted difficulties with convergence of the used procedure.

2

Diffraction of electromagnetic field on limited periodical antenna array

We consider a case with limited antenna array periodical in plane z=0 in direction of x and y “axis”. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


749

Antenna array, as stated before, can be set in any unlimited multilayer dielectric in planes perpendicular to axis z or placed on border of two layers of dielectric. For a simpler analyse we divide the antenna surface in plane z=0 on identical cells (Fig. 2) numbered with integer numbers (m,n), where the central point ((x0,y0) of the central cell(0,0) is placed at the beginning of coordinate system x=0, y=0, and points x=xm, y=yn are central points of next cells(m,n).

y' y ksinΘ

basic cell

ϕ

x Ω

dy

x' dx antenna plane

Figure 2:

Partition of the antenna surface on cells.

Antenna cells with carrier function hmn cover the part of plane z=0 responding to carrier function HMN(x,y) where:

H MN ( x , y ) = 1, dla x ∈ [− Md x , Md x ]i y ∈ [− Nd y , Nd y ],

H MN ( x , y ) = 0, dla x ∉ [− Md x , Md x ]lub y ∉ [− Nd y , Nd y ],

(1)

Integer numbers m, n number cells in direction of axis x m=-M,M+1,...,0,...,M-1,M and towards axis y n=-N,-N+1,...,0,...,N-1,N. Metalized elements are placed periodically on the same limited area of plane z=0 in each cell of antenna array (m,n) and cover the area of carrier function AMN(x,y):

∑a

mn

( x , y ) = AMN ( x , y ) .

(2)

m ,n

Let’s notice that functions HMN(x,y), AMN(x,y) are decomposed on components dependent from x and y:

H MN ( x , y ) = H MNx ( x )H MNy ( y ) , WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

(3)


AMN ( x , y ) = AMNx ( x ) AMNy ( y ) .

(4)

Because of the limitation of the antenna in plane z=0 antenna’s cells are distinguished, by their placement towards the beginning of coordinate system (x,y)=(0,0). However, in this case the solution (current distribution) is changing from cell to cell. In a periodical unlimited system each cell (m,n) can be a central cell. In a limited periodical system the central cell is univocally defined by its placement in the centre area of the antenna array. Every other cell is univocally defined by placement of its centre towards the centre of the array area (x,y)=(0,0). For further consideration it is necessary to define window carrier function HMNmn(x,y) of the antenna array. HMNmn(x,y) functions change themselves depending on in which cell a hypothetical beginning of coordinate system has been placed and are defined by vector translation (xm,yn)=(mdx,ndy) of function HMN:

H mn ( x , y ) = H 00 ( x − x m , y − y m ) = H MN ( x − x m , y − y m ) . (5) Their Fourier transforms equal respectively:

F ( H 00 )( k x , k y ) =

MNd x d y sin Mk x d x sin Nk y d y , Md x Nd y π2

(6)

H mnk ≡ H MN 00 (k xk , k yk ) = (d x d y ) −1 F ( H 00 )(k xk , k yk ) = =

H mnk ≡ H mn (k xk , k yk ) = H 00 (k xk , k yk ) ⋅

[

(7)

MN sin Mk xk d x sin Nk yk d y , Md x Nd y π2

]

[

]

exp − i (mk xk d x + nk yk d y ) = H 00 (k xk , k yk ) exp − i (mϕ xk + nϕ ky ) . (8) So, as we see Fourier transforms of the following cells experience phase modulation which equals to respective Floquet harmonic.

3 Solving the problem of electromagnetic field diffraction on antenna array with limited dimensions Formulating the problem of diffraction by defining matrixes of diffraction antenna array in spectral space, allows for direct introduction of window function to make a solution for unlimited case. [L-11]. In such a case only equations containing current distribution are modified:



751

J ( x , y ) = H mn ( x , y )∑ j q ϕ q ( x , y ) = ∑ J k H mnk e k ( x , y ) , (9) q

(

k

)

E s ( x, y,0) = G ∗ J ∗ H mn ( x, y ) =

∑G J k

k

H mnk ek ( x, y )

(10)

k

It is obtained by multiplying amplitudes of their spectral representations by Fourier transform of window function H mnk responding to selected cell (m,n). Another form of window function Hmn(x,y) responds to each cell (m,n) because of its placement towards the central carrier of window function HMN(x,y). For each couple of numbers (m,n) another solution determining current distribution on the metalised part of cell (m,n) is calculated. . From the formal point of view, formulating and solving the problem of diffraction for an antenna array with limited dimensions should take into account the continuous spectrum of the value of the wave number k in spectral distribution of parameters field and currents. It means that one should use continuous Fourier transform instead of spectral distributions in those points which respond to discreet values of wave numbers of the following Floquet harmonics. The numerical solution of such a problem also in this case would bring the analysed problem to a problem with discreet spectrum used by numerical procedures such as fast Fourier transform (FFT). This is why it seems reasonable to use spectral distributions of analysed values with spectrum determined by harmonics responding to a case with unlimited periodical structure. So when concerning formal note of equations, except for modifications in the above equations by introducing a window function, formulating and solving presented problem of diffraction on unlimited periodical antenna structure is obtained by distribution of base functions, currents and fields on Floquet harmonics, in a way similar to unlimited antenna array:

ϕ q ( x , y ) = ∑ ϕ qk Ψ k ( x , y ) ,

(11)

J ( x , y ) = ∑ j q ϕ qk H mnk e k ( x , y ) ,

(12)

E t ( x , y ,0 ) = ∑ v ki Ψ k ( x , y ) ,

(13)

E t ( x , y ,0 ) = ∑ ( 1 ± Γk )v ki Ψ k ( x , y ) ,

(14)

k

q ,k

i

k

e

k

E t ( x, y,0) = ∑ v ks Ψ k ( x, y ) , s

(15)

k

by solving diffracted field with inducted currents, with usage of Green function WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

752 Computational Methods and Experimental Measurements XIII s ( x, y,0) = E tmn

∑G J k

k

H mnk ek ( x, y )

(16)

k

and throwing on, with usage of the Galerkin method, inducted currents condition of resetting total field on surface of metal antenna elements:

(

)

s E tmn ( x, y,0) AMN = E te ( x, y,0) + E tmn ( x, y,0) AMN = 0

(17)

As a result we obtain system with double matrix algebraically equations:

W j =V

i

NV = −U j

s

where:

[W ]kqj ≡ Wkqj =

[N ] pqij

* ki G kij ϕ qkj H mnk

(19)

i

*

qkj

[U ]

∑ψ

(18)

≡ N qkj = (1 + Γk )ϕ qkjψ kj

≡ U pqij =

∑

,

(20)

*

ϕ pki G kij ϕ qkj H mnk

(21)

k

[ j ]qj ≡

jqj ,

(22)

k

≡ ν ks ,

(23)

k

≡ ν ki .

(24)

[V ] s

[V ] i

Solution of these equations gives coordinates of current distribution:

[ j ]qxmn ≡ jqxmn , [ j ]qymn ≡ jqymn ,

(25) (26)

in a cell defined by indexes of window function Hmn Solving the above system of algebraical equations for the next window functions Hmn, we obtain current distribution Jmn on next cells (m,n) of limited periodical antenna array, so we obtain MxN solutions of the above system of equations for MxN courses of functions Hmn(x,y). Summarized current distribution on all cells at the same time can be presented in a following form:



J ( x, y ) =

∑J

mn ( x,

y )amn ( x, y ) ,

753 (27)

m, n

where:

J ( x , y ) = ∑ j q ϕ q ( x, y ) = q

= ∑ J k ek ( x, y ) = ∑∑ j q ϕ qk ek ( x, y ) k

k

(28)

q

However, the final solution needs univocal connection of current distribution with raining and diffracted field. s

V = −W U

−1

NV

i

(29)

equivalent to construction of matrix of diffraction of antenna structure:

V s + + Γ V i −   S S  V i −  11 12  s− =  i+  i+ S S  V − Γ V   21 22   V 

(30)

so it needs one univocal solution of above system of equations, which takes into account current distributions on all cells of antenna at the same time.

Figure 3:

Eight element antenna array.



field pattern [dB]

Calculations have been made for a structure presented in fig. 3. Obtained results of calculations with comparison to the made measurement are presented in fig.4 although the time of calculations is very long, they are encouraging to further work in the direction. 0

P/Pmax [dB]

Theory

-10

Measurement

-20 -30 -40 -50

0

20

Figure 4:

4

40

60

80

100

120

140

160 180 angle [deg]

Radiation characteristic of antenna array.

Conclusions

Construction of antenna arrays placed on multilayer dielectric allows for the wanted modelling parameters and characteristics of the antenna array. An analysis of such a type of antennas which take into account the structure of the surface, true for each bandwidth, is based on the green function and moment method. The problem has been analysed, taking into account the distribution of field on Floquet harmonics, with usage of vector distribution on base functions. The proposed solution also allows for an analysis of antenna arrays with variable dimensions of a basic cell. It is a proposition of a precise and effective solution of problem of diagnosis on periodical antenna array with limited dimensions. One should stress, that a very important problem during solution is optimising the computer program in order to obtain maximum exactitude of calculations with minimal time needed.


Hall P., James J.: “Handbook of microstrip antennas” Peter Peregrinus Ltd., London 1989 Oltman G.H. "Electromagnetically coupled microstrip dipoles", IEEE Transactions on Antennas and Propagation AP-29, No 1, 1981. Pozar D., Tsay W.: “Application of the FDTD technique to periodic problems in scattering and radiation”, IEEE Microwave and Guided Wave Letters, vol.3, August 1993. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


[4] [5] [6] [7] [8]

[9] [10] [11]

[12] [13]

755

Scott C.: „The spectral domain method in electromagnetics”, Artech House, London 1989. Hansen, Numerical Solution of Antennas in Layered Media, Wiley, New York, 1989. E. Munson "Conformal microstrip antennas and microstrip phased arrays ” IEEE Trans. Antennas and Propagation vol. 28 197 4 pp. 74- 78 K. S. Yee "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic medias”, IEEE Trans. Antennas and Propagation, vol. 14, 1966. pp. 302 - 307. C. J. Railton, E. M. Daniel, D. L. Paul, J. P. McGreen "Optimized absorbing boundary conditions handicap the analysis of planar circuits using the finite difference time domain method” IEEE Trans. Antennas and Propagation vol. 41, 1993 pp. 290 - 296 Z. Bi, K. Wu, Ch. Wu, J. Litva "And dispersive boundary condition handicap microstrip component analysis using the FDTD method”, IEEE Trans. Antennas and Propagation, vol. 40, 1992. pp. 774 - 777 A. Taflowe "Advance in computational electrodynamics The Finite Difference Time Domain ” Artech House Boston, 1995 D. M. Sheen, S. M. Ali, M. D. Abouzahra, J. A. Kong, "Application of the three - dimensional finite difference time domain Method this the analysis of planar microstrip circuits “ IEEE Trans. he Microwave Theory and Techniques, vol. 38, 1990 pp. 849-857 Z. Bi, K. Wu, Ch. Wu, J. Litva, "Accurate characterization of planar printed antennas using finite difference time domain method" IEEE Trans. he Antennas and Propagation vol. 40 pp. 526-533 A. Reinex, B. Jecko "Analysis of Microstrip Patch Anntenas Using Finite Difference Time Domain IEEE Trans. Antennas and Propagation, vol. 37, 1989. pp. 1361 - 1369.




757

Method of signal processing in passive infrared detectors for security systems H. Madura Institute of Optoelectronics, Military University of Technology, Warsaw, Poland

Abstract This article presents the construction and principle of operation of passive IR detectors (PIR detectors) of a large detection range. An important virtue of these detectors is highly efficient detection of slowly moving or crawling people. The PIR detector described here detects crawling people at a distance of 140 m. High signal-to-noise ratio was obtained by using a larger number of pyroelectric sensors, i.e., by using a larger number of detection zones (channels). The original electronic system for the PIR detector is presented in which DC signal amplifiers from pyroelectric signals are used. In order to ensure large detection ranges, a new method of signal analysis was used. Keywords: PIR detector, security systems.

1

Introduction

Main elements of security systems are PIR detectors. In general, detectors operating inside buildings have small detection range, small ranges of working temperature, and relatively simple algorithms of intruder detection. The detectors used for protection of objects or large areas (buildings, airports) have larger detection ranges and complex algorithms of signal processing on which significantly depend the efficiency of their operation. An essential drawback of currently available PIR detectors is low efficiency of detection of slowly moving or crawling people. It is because radiation from such objects is similar to background thermal noise. Moreover, to detect slowly moving or crawling people, the lower limit frequency of a transfer band of PIR detector should be near zero. By fulfilling this condition, increase in lowfrequency noise occurs causing a decrease in the next detector’s sensitivity. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070741

758 Computational Methods and Experimental Measurements XIII Algorithms of intruder detection have to be different than these for typical PIR detectors. To detect crawling people, larger number of sensors should be used (more detection zones) which will cause an increase in signal-to-noise ratio because each of the sensors will “see” the smaller field of view. PIR detectors used in security systems for people detection operate in far infrared spectral range (8÷14 µm). In these detectors, the most frequently used are pyroelectric sensors allowing the detection of temperature changes as small as 10-6K. Application of a single pyroelectric sensor does not ensure distinguishing the events of an alarm nature from, so-called, false alarms caused by, e.g., air turbulences or background temperature changes resulting from sun radiation. Therefore in PIR detectors, the most frequently used are pyroelectric sensors with two active elements (two sensors) and an alarm signal is determined on the basis of analysis of a difference (or a sum) of their output signals [1, 2]. Usually, pyroelectric sensors are mounted, together with a transistor and a resistor polarizing its gate, in standard hermetic housings. A value of this resistor can be even up to 1011Ω, depending on the preamplifier configuration. JFET or MOSFET transistors are used most frequently as the amplifying elements that are mounted near a detector. Electronic Units

Unit of sensors with optical collectors Lens (germanium, gasir)

Optical Diaphragms

Figure 1:

2

Simplified diagram of PIR detector.

Construction of PIR detector

Main elements of PIR detector are: objective (mirror or refraction one), set of pyroelectric sensors, and electronic systems (Fig. 1). The sensors convert an optical signal emitted from the “being observed” surface into an electrical signal. This signal is processed in the electronic systems (it is amplified, filtered, sampled) and next it is analysed in a microprocessor system. The presented PIR detector detects the crawling people at a distance of 140 m. High signal-to-noise ratio was obtained due to application of the larger numbers of pyroelectric sensors, i.e., larger number of detection zones WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


759

(channels). Application of larger number of sensors forces the necessity to develop a complex optical system (Fig. 2). The optical system of the PIR detector has to ensure such a position of the detection zones to the avoid presence of the areas which are “not seen” by a detector [3–5]. Channel 6

Channel 2

Channel 4

Channel 7

Channel 5 Channel 3

4,5m

Channel 1

1,3m 17m 25m

43m

94m

100m

140m

Dual pyroelectric sensor

Figure 2:

Detection zones of PIR detector in horizontal and vertical planes (top) and pyroelectric sensors with optical concentrators and general view of a detection set with seven two-segment sensors (bottom).

Radiation signals caused by slowly moving, especially crawling, people are characterized by similar luminance amplitudes and velocities of its change in time such as fluctuations of background radiation. The signal amplitude from pyroelectric sensors is directly proportional to the velocity of a change of radiation signal in time (i.e., to the velocity of a moving object). A disadvantageous property of pyroelectric sensors is a voltage drift (pyroelectric detector is equipped with a field transistor operating as a voltage follower) which at low velocities of intruder movement can have temporal characteristics of signals, comparable with the characteristics originating from a moving person. Thus, in order that the sensor could efficiently detect slowly moving person, it is necessary to develop an algorithm distinguishing both characteristic features of signal changes from a sensor caused by a photon noise and signal changes caused by a temperature drift of a pyroelectric sensor. A simplified diagram of the electronic circuit of PIR detector is shown in Fig. 3. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

760 Computational Methods and Experimental Measurements XIII Vcc R12

C2

PIR

C9

C8

U1A

C7

R6

C10

R4

C11

C14

R10

D2

R11

U3A U2A

U3B

U2B

Vout

C5

R5

SO SI SCK /CS

PWM

R9

A/D

Sunlight sensor

T1

D1

Heater

R14

U4

R15

Alarm

A/D

C6

Vcc C4

C15

R8

R7

C3

D3

C13

C12

R3

R2

9÷24V

DC/DC

Microprocessor

R1

C1

6,2V

Voltage Stabilizer

R13

A/D

D D

T2

RS485

U5

Temperature sensor

Figure 3:

Simplified diagram of electronic circuit of PIR detector (one channel).

A system of DC amplifier with regulation (by means of digital electronic potentiometers) of a level of a constant component consists of three operation amplifiers. The first U1A amplifier, because of significant output resistance of a pyroelectric sensor operates in a follower system and two consecutive ones in the systems reversing a signal phase. The first stage of amplification is realized by the operation amplifier U3A. It is characterized with the signal amplification, K1= - R4/R3. At its „+” input, a reference voltage is applied obtained from the resistance divider R5, R6, U2A (digital potentiometer) included between the supply voltage and the system ground. Reference voltage (regulated by means of a digital electronic potentiometer) is used for elimination of a constant component of a signal obtained from a pyroelectric sensor. This component is significantly changed together with temperature changes occurring near a pyroelectric sensor (because of a change of the output current change of a field transistor dependent on temperature). Next, the signal is delivered to the second amplifying stage of the U3B operation amplifier the amplification of which is K = - R8/R7. At its “+” input, it has the reference voltage obtained from the R9, R10, U2B resistance divider (digital potentiometer). This voltage (regulated by means of a digital electronic potentiometer) protects the amplifier against saturation state. The output signal is filtered in the filter made from R11 and C6 elements. The electronic potentiometers, used in a preamplifier, are connected through a SPI series bus with a controlling processor which on the basis of the output signal values makes current adequate regulations (change in the value of potentiometer resistances) of the values of reference voltages of particular amplifying stages. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


761

Knowing the NA and NB settings (number of a selected level) of particular digital potentiometers, their total resistances RA and RB and the values of amplifications of particular K1, K2 preamplifier stages, the signal value VOUT at the amplifier output can be calculated: VOUT = VIN K 1 K 2 − U pA K 1 K 2 − U pB K 2 (1) where VIN is the input voltage of the first amplifying stage, UpA is the reference voltage of the first amplifying stage, and UpB is the reference voltage of the second amplifying stage. The values of reference voltages of particular amplifying stages for 8-bit resolution of digital potentiometers settings can be calculated from the relationship:

U pA =

N Vcc    R 5 + A RA  R 6 + RA + R 5  256 

(2)

U pB =

N Vcc    R 9 + B RB  R10 + RB + R 9  256 

(3)

where Vcc is the supply voltage. The UpA and UpB values are currently matched by a microprocessor ensuring adequate dynamic of the amplifier and preventing the saturation state of particular amplifying stages. If the output voltage of the amplifier will be measured and analyzed by a microprocessor, which in its operation algorithm considers the shift of levels (potentiometer settings), it can be shown that the voltage calculated (by a microprocessor) can be significantly higher than the supply voltage. It is an important virtue of this system. Application of electronic potentiometers requires initial determination of their working ranges by means of external resistors. A better solution is application of digital-analogue converters instead of potentiometers. In a sensor’s electronic system, the system for measurement of sun radiation intensity was used. The information from this system is taken into account for in the intruder detection algorithm. The data from the temperature measurement system can be used for switching-on a miniature heater system mounted inside the PIR detector housing. A small increase in detector’s interior temperature (in relation to ambient temperature) protects against water vapour deposition on optical elements (lens, concentrator mirrors, optical windows of sensors).

3

Method of signal analysis

An object in the sensor’s “observation zone” (inspection zone) is detected when a conventional detection threshold is exceeded by a signal level at the detection system output, caused by IR emitted from an object. In order to minimize the probability of false alarms, an adaptation detection threshold should be determined “following up” the changing atmospheric conditions causing changes of “thermal scene” parameters [6].


762 Computational Methods and Experimental Measurements XIII N

a

1 N+M+K

i

Method of creation of “time windows” in which samples of signals from pyroelectric sensors are analyzed.


763


The analysis method is based on determination of an average moving in three “time windows” of a defined length [7]. The principle of formation of “time windows” is explained in Fig. 4. The windows are marked with N, M, and K letters while N > M >> K is assumed. Signal processing is carried out in the following way. The signal from a detection set is sampled and the voltages of successive signal samples of instantaneous values ai are added. The time window is formed, containing N samples, which in each consecutive cycle “shifts” by one sample. On the basis of an arithmetic average of voltage values of the samples in the window (Fig. 5), a value of the reference voltage level ϕ i is determined

ϕi =

1 (ai −N +1 + ai −N + ... + ai ) , for i > N N

(4)

where i is the sample number which takes the values i = 1, 2, 3, ...

N N

In following cycles of analysis N window is moving for one sample

N Signal

ϕi - reference level

5

Figure 5:

15

20 30 Number of sample

35

45

i

Sampled signal from a detection channel and the reference level ϕi calculated according to Eq. (4).

In the next window M, that is in the previously formed window N and containing M samples (for i ≥ N + M ), the instantaneous values of the signal β i (Fig. 6) are determined as a difference of instantaneous values of voltages of consecutive samples of the signal and the reference level voltage ϕ i

β i = a i − ϕ i , for i ≥ N + M . For further calculations, the absolute value βi is taken as

βi = ai − ϕi , for i ≥ N + M .


(5) (6)


Deviation of signal value from reference level

βi

5

Figure 6:

15


35

45

i

Instantaneous deviation of the signal βi from reference level calculated according to Eq. (5).

On the basis of such determined data, the instantaneous value of the detection threshold voltage D i which for i > N + M is calculated as:

Di = A

1 ( βi −M +1 + βi −M + ... + βi ) , M

(7)

where A is the coefficient considering detector design parameters. For typical solutions, the A coefficient takes the values 1÷5. For PIR detectors of several detection zones and extended operation range, the A coefficient of the higher values for the zone being near detector is taken and A of the lower values is assumed for distant zones. In order to diminish the influence of sun radiation, precipitation, and ambient temperature, the instantaneous corrected detection threshold Dki is determined:

Dki = Di ⋅ k s ⋅ kw ⋅ kt

i > N +M

(8)

where k s is the correction coefficient considering sun radiation, k w is the correction coefficient of precipitation influence, and k t is the correction coefficient of ambient temperature. The correction coefficient k s increases the level of a detection threshold in the case of sun illumination [8–10]. The values of changes in a detection threshold were determined experimentally from investigations on the influence of sun illumination level on the increase in both object temperature and background thermal noise. Because the object is also illuminated with sun radiation, which increases its equivalent temperature, increase in a detection threshold value does not cause a decrease in a sensor range and undesirable disturbances are eliminated. The value of the k s coefficient was taken as 1÷1.5.


765

Signal


5

Figure 7:

15


35

45

i

Instantaneous value of the detection threshold voltage Di and the absolute values of βi signal.

The k w coefficient causes a decrease in a detection threshold in the case of precipitation in the detection zone of the PIR detector. When intensity of precipitation (rain, snow) is higher, IR is much more attenuated which results in lower radiation reaching the detector, thus the signal at its output is weaker. When keeping the same value of a detection threshold, the sensor range is smaller. The k w coefficient is just to compensate the influence of precipitation in such a way as to keep the sensor range unchanged. The value of the k w coefficient was taken in the limits 0.7÷1. In the next step, the samples in the time window K are analysed in which the parameter γ i is determined as an arithmetic average of the values of the signal samples βi :

γi =

1 ( βi −K +1 + βi −K + ... + βi ) , for i ≥ N + M + K . K

(9)

An object is detected (Fig. 8) in the analysed signal when the parameter γ i is higher than the instantaneous value of the corrected detection threshold Dki , i.e., when

γ i − Dki > 0 .

(10)

In Fig. 8, also short lasting disturbances of high amplitudes are illustrated (samples 18 and 45) which are not interpreted as object detection.


Value of parameters γ i , D ki


8 6

Detection level Dki

Detection of object γi - average value of |βi | in window K

4 2

5

Figure 8:

15


35

45

i

Object is detected when a value of γ i parameter exceeds a value of corrected detection threshold (in figure for samples No 26÷No 31). Channel 1 500 450 400 350 Signal

300 250 200 150 100 50 0 65

Figure 9:

4

70

75

80

85 t [s]

90

95

100

110

Signal in channel 1 of PIR detector recorded during night (ambient temperature 25ºC, velocity of human target 1 m/s, distance 140 meters).

Results of range investigations

The PIR detector is equipped with a data transmission path in the standard RS 485. For measurement results registration, special software developed for detector diagnostics was used. The software allows for signal registration from particular detection zones. In Figs. 9–11, the signals from the PIR detector are shown which were caused by a moving person in an inspection zone.



767

Channel 1 200 180 160 140 Signal

120 100 80 60 40 20 0

Figure 10:

90

100

110

120

130 t [s]

140

150

160 170

Signal in channel 1 of PIR detector recorded during sunny day (ambient temperature 28ºC, velocity of human target 5 m/s, distance 140 meters). Channel 2 400 350 300

Signal

250 200 150 100 50 0 100

Figure 11:

150

200

250

300 t [s]

350

400

450

500

Signal in channel 2 of the PIR detector (ambient temperature 26ºC, velocity of crawling human target 0.1 m/s, distance 140 meters).

The investigation results, shown in Figs. 9–11, confirmed the proper operation of the PIR detector, in particular correctness and high efficiency of the signal processing method. Figure 10 shows that the detection threshold follows up (upper course) the thermal changes of a background which for this measurement case were caused by sun radiation.



Acknowledgement This work was financially supported by the Polish Ministry of Science and Higher Education in the frame of research project No 0005/T00/2005/28.

References [1] [2] [3] [4] [5]

[6] [7] [8]

[9] [10]

ELTEC Instruments Inc.: Catalogue of firm products, P.O. Box 9610, Central Business Park, Daytona Beach, Florida 32120-9610, USA, 1996. Madura H., Sikorski Z., Polakowski H., Kastek M. Long-range passive IR sensor, Quantitative Infrared Thermography 5 QIRT’2000, Reims, France, 18-21.07.2000. Kastek M.: Method of objects detection in IR system of elongated detection zone. Doctors Thesis, Warsaw, The Military University of Technology Library, 2002 (in Polish). Madura H., Sikorski Z., Polakowski H., Kastek M. Automated stand for measurement of parameters of long-range passive IR sensor. Quantitative Infrared Thermography, Reims France, pp. 118-121, 2000. Kastek M., Madura H., Morawski M., Piatkowski T., Powiada E., Polakowski H. Test bed for measurement of angular parameters of passive infrared sensors. Infrared Physics & Technology, Volume 49, No 3, pp. 198-201, 2007. Leach G. A single performance measure for perimeter intruder detection systems, European Conference on Security and Detection, Conference Publication No 437, pp. 114-119. 28-30.04. 1997. Madura H., Kastek M., Powiada E. Method of objects detection by means of IR sensors, Polish Patent Office, Patent Application P-360064, Warsaw 2003 (in Polish). Polakowski H., Piątkowski T., Madura H., Kastek M., Stojak Z., Powiada E., Chmielewski K., Kulewski R. Method of disturbances correction in IR receivers, Polish Patent Office, Patent Application P-352889, Warsaw 2002, (in Polish). Madura H., Kołodziejczyk M. Influence of sun radiation on results of noncontact temperature measurements in far infrared range. Opto-Electronics Review, No 13 (3), pp. 253-257, 2005. Madura H., Piątkowski T., Powiada E. Multispectral precise pyrometer for measurement of seawater surface temperature. Infrared Physics & Technology Vol. 46 pp. 69-73, 2004.



769

Influence of the displacement effect on compressed LFM signal parameters A. Kawalec, Cz. Leśnik, W. Komorniczak, W. Czarnecki & J. Pietrasiński Military University of Technology, Institute of Radioelectronics, Electronics Department, Warsaw, Poland

Abstract This paper deals with the digital signal processing in the application of a modern radar receiver. The influence of the signal discretization and its side effects on radar performance are considered. In particular, the paper presents the problem of the random displacement effect of LFM (linear frequency modulation) radar echo pulse beginning in accordance with the closest discrete time value. This effect has an impact on the characteristics of the pulse compression filter output signal. Its equivalence to the Doppler frequency shift has also been proven. The shift is responsible for additional, unwanted amplitude modulation of the signal samples on the output of the matched filter, as a result of the discretization process. The resulting losses of the signal level (LPG) can reach 4dB. The losses follow the function of the sampling frequency, whose relation has been developed. In addition, the displacement effect can significantly decrease the effectiveness of the MTI process. In the paper it has been shown that the influence of the displacement effect and discretization process on the relative level of the side lobes is not significant. The losses of PSL (Peak Side-lobe Level) do not exceed a fraction of dB. Characteristics evaluation of the discretization process has been performed on the basis of the proposed discrete time generation algorithm. The testing results are shown, as well as a discussion and some practical conclusions. Keywords: signal processing, linear frequency modulation, radar signal, ambiguity function, signal discretization.



1

Introduction

The time representation of the signal at the matched filter output can be drawn from Woodward’s general definition of the radar ambiguity function: χ(τ, f D ) =

where:

u (t )

∞

∫ u (t )u * (t − τ )e

j 2π fD t

dt

−∞

,

(1)

-

signal complex envelope, u * (t ) complex conjugate of u (t ) , fD Doppler frequency shift, τ relative time shift (related to time of maximum output signal). During further consideration the radar ambiguity function modulus χ(τ, f D ) will be taken into account. The most popular type of radar signal is a pulse with inner linear frequency modulation (LFM). Its normalized time representation is as follows [1,2]: s (t ) =

µt2  t   rect   cos ω0 t + 2  T T   ,

1

(2)

where: T ω0

- pulse duration, - carrier pulsation,

[

]

2 πB rad/s 2 , T B - total frequency deviation, t rect   - uniform rectangular pulse: T 

µ

- frequency slope, µ = ±

T T   t  1 for − < t < rect   =  2 2  T  0 for other t 

.

The complex signal envelope of (2) is represented by: t u (t ) = rect  T T 1

 j e 

µ t2 2

.

(3)

From (1) and (3) the equation describing the modulus of the LFM signal ambiguity function can be drawn [2]: WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


 τ  χ(τ, f D ) = rect   2T 

  τ  τ T  1 −  Sa (2πf D + µτ)1 −   ,  T  T2      

771 (4)

where:  sin x for x ≠ 0  Sa (x ) =  x  1 for x = 0  An expression (4) shows that a frequency axis position of the signal maximum at the matched filter output depends on Doppler frequency shift along with a modulation described by the function: (1 − τ T ) . LPG [dB]

0.0

-10.0 BT=50 BT=100 BT=200 BT=400

-20.0

-30.0

-40.0 -1.00

-0.50

0.00

0.50

1.00

fD / B

a. PSL [dB]

-3.0 BT=50 BT=100 BT=200 BT=400

-6.0

-9.0

-12.0

-15.0 -1.00

-0.50

0.00

0.50

b. Figure 1:

1.00

fD / B

LPG (a.) and PSL (b.) values as a function of relative Doppler frequency shift.

Two of the most common matched filter output signal characteristics are as follows: LPG (Loss in Process Gain, the level of the mainlobe in no-match case U ML related to its maximum value in the perfect match case U MLmx ) and PSL (Peak Sidelobe Level, relative level of the maximum sidelobe U SL related to the maximum of the mainlobe U ML ):


772 Computational Methods and Experimental Measurements XIII LPG = 20 log

U ML U MLmx

, PSL = 20 log

U SL

.

U ML

The LPG and PSL parameters as a function of the Doppler shift normalized to the deviation are derived according to equation (4). The characteristics for selected values of time-bandwidth product BT are presented on Fig. 1 [4]. Minimum sidelobe level for zero Doppler shift is not a constant value, but it depends on time-bandwidth product BT. PSL parameter value fluctuations for low real (met in practice) values of Doppler frequency shift are presented on Fig. 2. PSL [dB] -13.0

-13.2 BT=50 BT=100 BT=200 BT=400

-13.4

-13.6 -0.10

-0.05

0.00

0.05

0.10

fD / B

Figure 2:

PSL values as a function of small values of relative Doppler frequency shift. PSL [dB]

-13.0

-13.2

-13.4

-13.6

-13.8

-14.0 0

200

400

600

800

1000

BT

Figure 3:

Minimum PSL as a function of time-bandwidth product.

Sidelobe minimum level can be found as a result of looking for the second extreme of the function described by the expression (4) under assumption that f D = 0 . Precise solution leads to very complicated expressions. That is why it is interesting to find the solution using numerical methods. Results of the calculations are shown on Fig. 3.



2

773

Radar signal discretization

In the contemporary radar systems, the received echo signal is converted from its native analogue form to the digital representation. The conversion process is repeated with a sampling clock period t s . The same period characterizes the set of samples which forms the response of the matched filter used for signal matched filtering. In the perfect condition the beginning of the echo signal overlaps with the sampling period beginning of the analogue-digital converter. In the general case, due to the random position of the detected object (relating to radar location), those two time moments do not overlap. The first signal sample is delayed from the beginning of the signal of the time t j , which is shorter than t s . The problem is illustrated on Fig. 4. The mentioned phenomenon is called a displacement effect. s(t)

without displacement

with displacement effect

t ts

tj

ts

tj

f(t) fk´ fk

fp´ fp

Figure 4:

t tj

The displacement effect phenomenon.

The time shift fulfils the rule t j ∈ 〈 0, t s ) . In the case of LFM signal, the temporary signal frequency is a linear function of time. In the idealized case the first and last sample of the received echo signal correspond to the values of initial f p and final f k frequency. In reality the first and last sample corresponds to the other initial f p' and final f k' frequency. As a result the frequency relations are going to be as on Fig. 5. The displacement effect results are equivalent to the Doppler frequency shift whose maximum value is given by: µ ∆f mx = ts 2π . (5) According to equation (5), evaluation of the influence of the discretization process and displacement effect on the after compression signal characteristics can be reduced to examination of the Doppler frequency shift.


774 Computational Methods and Experimental Measurements XIII The carrier frequency shift of the LFM signal causes changes of both the main signal lobe level at the matched filter output and the signal location on the time axis. It is shown on the Fig. 6. For simplification, presented signals are limited only to the first sidelobe pair. f (t) fk´

displacement effe

fk f0´

∆f = f0

µ tj 2π

fp´ without displacement effect

t

fp

T

T /2

Figure 5:

Temporary frequency in the cases of the lack and presence of the displacement effect. |χ (τ, fDi)| 1

|χ (τ, 0)|

(1 + τ/T) |χ (τ, fD1)|

|χ (τ, -fD1)| (1- τ/T) τ 0

-T

Figure 6:

T

The output of the matched filter for chosen Doppler frequency shift values vs. time.

The time shift value can be found as an abscissa of the max of the function described by equation (4) for the constant f D value. Assuming f d = ∆f and t s α i+1 , and for logical bit “0”: α i < α i+1 . two binary values are assumed: α ∈ {0,1} . The

spectral line amplitude value rule:

In the described system only information mapper work principle for the OFDM modulator is shown in fig. 2. The quick brown fox jumps over a lazy dog

11011010101011010111110101010110101111011... Bit:

1

1

0

f [Hz] 1 2 FFT bins

Figure 2:

3

4

5

6

OFDM information mapper – work principle.

The six chosen spectral lines are considered as spectral phase “pilots” used on the receiver’s side for the output phase drift estimation Φ ∆ of the radio link. The WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

794 Computational Methods and Experimental Measurements XIII SSDS signal based on the preliminary BPSK modulator is used for synchronisation of the watermark sequence. The following parameters are assumed: conversion gain SNRSSDS = 24,77 dB , spread spectrum pass band:

Bs = 3kHz . A spreading signal is generated using a pseudorandom information sequence. Depending on the assumed block data format, a system can be considered as steganographic (OFDM sequence for data block precedes by SSDS sequence block) or watermarking (each OFDM block with the same data payload in the stream). A steganographic system has a dedicated bit rate, a watermarking system is characterised by a data payload parameter (16 bits in the case of the described system). The host signal is transformed to the frequency domain and psychoacoustic processing is performed using the Mpeg1 Layer1 algorithm to compute the Just Noticeable Difference threshold level (JND) as a difference between host level and the minimal masking threshold. The parameter Signal-to-Mask Ratio (SMR) is computed, separately for each critical sub-band (logarithmic Bark scale) for the watermark spectrum range. The SMR parameter is used for the watermark correction procedure (block: shape-level watermark equalizer) to ensure watermark signal perceptual transparency at the host speech signal presence. The corrected watermark in frequency shape and level is transformed back to the time domain using IFFT and summed together with the host signal. The developed system is based on direct host signal summing with corrected watermark, unlike the system described in [1] where the mean value of the host signal’s FFT magnitudes in the dedicated sub-band is computed to add the watermark spectrum line. An example of the watermarked signal spectrum is shown in fig. 3.

Figure 3:

Example of the watermarked signal spectrum (host speech signal) and hidden (JND level) watermark.



795

2.2 Watermark decoding The basic block scheme of the watermark decoder is shown in fig. 4. The watermark decoder consists of a synchronisation block. This block uses a SSDS demodulator. The demodulator generates the same PN binary sequence as on the transmitter side, a 2kHz carrier wave allows one to make a transformation from 500-3500Hz into the base pass band. The decoding process based on correlation analysis finds synchronisation points for correct data block bit decoding. Optional LPC analysis (carried out in synchronizer block) is useful for the effective decoding process (the decorrelation procedure generates a residual signal with small standard deviation). Buffer delay delays the start of the data blocks for the decoding process. An example of a BPSK DS signal spectrum taking part in the decoding process is shown in fig. 5. Drift Phase Scanner

FFT

Synchronizer

SS DS demodulator

Figure 4:

Phase equalizer

Spectrum bins comparator

Buffer (delay)

PN sequence

Coherent Averaging

Output: Correspondent’s PIN

PIN

Basic block scheme of the watermark decoder.

Figure 5:

BPSK SSDS signal spectrum.

The synchronisation process is optional and useful only in the steganographic system case. The main goal of the projected algorithm is a watermarking system based on a drift phase scanner, a phase angle correction and a coherent averaging process. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

796 Computational Methods and Experimental Measurements XIII The drift phase scanner method was developed and described in detail in the PhD dissertation [2] and then published in [3]. The decoded signal has ultimate frequency detuning (phase drift) caused by phase drift A/C and C/A converters’ (clock short-term stability) as well as output small stability heterodyne devices’ (in transceiver radios). Spectrum coherention as a key requirement for proper spectrum averaging must be met to decrease host speech signal standard deviation and receive proper coherent gain: SNRcoh = 10 ⋅ log10 (M ) (dB) (1) where: M - iterations number (signal block averaging) Example: signal duration equals 10 second, fs=48kHz and data block 512 samples gives: SNRcoh = 19,72 (dB). This result means that the watermark signal “hidden” under the host speech signal at value SMR < SNRcoh is still possible to be correctly detected, of course, if coherence is met. Spectrum averaging is based on the formula (2) for the chosen n spectral line: N

FFTcoh (n ) =

Figure 6:

∑ Re{FFTi (n )} i =1

N

N

+j

∑ Im{FFTi (n )} i =1

(2)

N

Example of searching radio link drift phase, result: rad/block.

φ=

-8.57e-4

The final result as a module for a complex number has a maximum value only in the case when the coherence requirement is met:

γ max = abs(FFTcoh (n )) WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

(3)


797

which means even a small drift value of the phase angle for dedicated spectral line during spectrum averaging will decrease the module value γ . The drift phase scanner searches the all angle range < −φ , φ > (limited for predicted output drift values and desired method resolution) to find the maximum value γ max . The searching process is based on virtual module computation. The virtual module is considered as a sum of modules for “pilots” spectral lines and is divided onto two stages: rough and precise searching (fig. 6, fig. 7). The output drift phase value found in the drift scanner is a base parameter in the block: phase equalizer. After the phase equalization process (phase value correction) the signal spectrum is averaged and spectrum bins are pairs’ compared to find the output PIN binary sequence. The watermarked host speech spectrum after coherent averaging after drift equalisation is shown in fig. 8. Input PIN signature on the transmitter side: [1111111111111111]. OFDM information mapper allocated energy to the frequency bins numbers: 7-38 (with DC component) in the following sequence: [10101010101010101010101010101010]. The PIN signature was decoded on the receiver’s side with 0 errors using spectrum lines comparator.

a)

b) Figure 7:

“Pilots” spectral lines before a) and after b) drift correction procedure Im-Re plots.



Figure 8:

3

Watermarked host spectrum after coherent averaging.


Experiment number 1 was dedicated for perceptual transparency assessment and based on standard ITU-R BS 1116-1 subjective fidelity test. The full procedure and the environmental conditions that were required to be met were described in detail in [4]. Results for subjective assessment by listeners are shown in fig. 9. 1,00

0,00 1

2

3

4

5

6

7

8

9

10

SDG

-1,00

-2,00 -3,00

-4,00 track number

Figure 9:

ITU-R BS 1116-1 results for 10 tracks.

Subjective fidelity assessment was done for 10 (male/female) speech tracks with an embedded watermark and degraded by HF/VHF noise. These tracks were compared with host speech tracks degraded also by HF/VHF noise. Every measurement point was characterized by 15 listeners with normal hearing (based on preliminary audiogram tests). Measurement results were taking into a statistical estimation procedure with a 95% confidence level. A SGD score above -1 SDG value shows that the embedded watermark is completely transparent at the host speech signal presence. A SDG positive score shows listeners better assessed the fidelity watermarked speech signal in comparison with the host speech signal. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


799

Experiment number 2 was dedicated for estimating of the decoding data payload efficiency. The computer simulated model of the HF/VHF radio channel degraded watermarked signal as well as the simulation of the radio simplex mode was taken into account. The result of the radio simplex mode simulation for the output received signal is a set of short signal fragments with various durations. These signal fragments are decoded separately (averaging complex spectrum and modules computation) and finally summed, as shown in table 1. Experiment number 3 was dedicated for estimation of the decoding data payload efficiency in the real radio channel using a standard FM/AM modulator and a standard radio FM/AM receiver. The modulated watermarked signal (duration: 10 second) was transmitted over the radio channel and demodulated on the receiver’s side and decoded correctly in PC software (0 errors / 16 bits correct decoded). Table 1:

PIN decoding efficiency with simulation of the radio simplex mode and HF/VHF channel. SNR* (dB)

0.877 0.93 0.922 1.181 1.256 0.934 2.034 1.187

Track Avg RMS Power (dB) -20.4 -22.87 -22.65 -29.34 -23.68 -24.97 -47.44 -24.38

1.92 5.46 6.54 4.30 6.59 6.02 2.76 9.25

Decoder error / total 4/16 3/16 5/16 4/16 3/16 5/16 7/16 1/16

9.321

-24.77

5.64

1/16

Duration (s)

Track Track - part 01 Track - part 02 Track - part 03 Track - part 04 Track - part 05 Track - part 06 Track - part 07 Track - part 08

∑ abs(Re+ j Im)

01−08

Track 10 -25.99 11.27 0/16 SNR* - denotes: power of the spectral line “1” to power of the spectral line “0” ratio. Assumption: logical bit “1” is pair spectral lines mapped as a sequence “1” and “0”, logical bit “0” is pair spectral line mapped as a sequence “0” and “1”.

4

Conclusions

The proposed system proved the usefulness of the computer simulated HF/VHF radio link as well as of the preliminary radio link tests with FM/AM modulation. Watermark perceptual transparency at the host speech signal presence was confirmed in the experiment number 1 using subjective fidelity test ITU-R BS 1116-1. The watermark binary signature (PIN) is decoded correctly depending on the signal track’s duration time and SMR value. Tracks with duration time of 10 seconds were decoded with 1 error in experiment number 2 (computer simulation), and 0 error in experiment number 3 (radio link). The steganographic system mode is used when the SSDS signal is embedded into the host signal. The WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

800 Computational Methods and Experimental Measurements XIII phase drift scanner is a key element of this watermarking system and can be useful for providing the signal’s coherency. The system is dedicated for radio correspondent authorisation. At present the described system is implemented on TI DSK 6713 for real-time processing.


[4] [5]

Nedeljko Cvejic, Tapo Seppänen, Spread spectrum audio watermarking using frequency hopping and attack characterization. Signal Processing, 84 (2004), pp.207-213, 2004. Zbigniew Piotrowski, Effective method of watermark embedding and decoding in the audio broadcast band. Military University of Technology, Warsaw, Poland, PhD dissertation, pp. 51-59, 2005 Piotr Gajewski, Jerzy Łopatka, Zbigniew Piotrowski, A New method of frequency offset correction using coherent averaging. Journal of Telecommunications and Information Technology, 1/2005, pp. 142-146, 2005 Methods for the Subjective Assessment of Small Impairments in Audio Systems Including Multichannel Sound Systems; Recommendation ITU-R BS.1116. Richard G. Lyons, Understanding Digital Signal Processing, Prentice Hall PTR Publication, 2004



801

Optoelectronic system for phase array antenna beam steering E. Sędek1, Z. Bielecki2, M. Muszkowski1, W. Kołosowski1, G. Różański2 & M. Wnuk2 1 2

Telecommunications Research Institute (PIT), Poland Military University of Technology (WAT), Poland

Abstract This paper presents a system for optoelectronic beam steering of a phase antenna array. The microwave signal from an RF generator controlled by a pulse generator is transmitted to the electrical input of an optical elevation control unit, which generates N microwave signals with independent amplitude and phase. The control unit utilizes an optical wave from a laser. All outputs of those control units are connected to inputs of N optical azimuth control units, which provide M linear element antennas control. Those units utilize an optical wave from a second laser. This way an M x N element of a planar antenna control is provided. The presented method is compared with the classical and electronic method, which consists of multi-bits microwave phase shifters for producing an electronic scanning effect. On the basis of this concept we realized 16-element linear antenna array printed on a dielectric substrate fully controlled by an optoelectronic system. A tuned wavelength laser in the range 1520–1600 nm and 10 mW optical output power has been used as an optical signal source. The optical signal is modulated by a microwave signal. The applied modulator operates in the third optical window, allowing optical signal modulation to 10GHz. Very high resolution and an excellent accuracy of the antenna beam positioning can be achieved. The optimal technique depends on the number of antenna elements, which implicates beam width. The presented method is preferred for very narrow antenna beams. Keywords: optoelectronic system, phase array antenna, beam steering.



1

Introduction

Active phase array antennas [1,2] have played an ever more important role in modern radar systems. This direction of antenna techniques has been possible due to electronic space scanning within a wide range without the need for mechanical rotation of the antenna as well as free options in the generated beam spatial characteristic shaping, achieved through relevant control of numerous transmission elements. The microwave control signals distribution for traditional systems take advantage of shared axis systems and metal wave-guides of considerable size; all these - due to a large number of active elements in an array antenna - make impossible a simple delivery of control signals and reception of output microwave signals. The rapid development of optoelectronic techniques, in particular development of numerous modern optic elements, used for fiber optic transmissions and optic signals processing, provide a practical opportunity to solve these problems [3–5].

2

Electronic scanning of phase array antenna beam

Active phased array antennas are composed of the matrix of independent microwave transmitting-receiving elements (TR-modules) to which separate control signals are being supplied by means of the independent regulated phase shifter systems or through delaying lines. The proper positioning in space of the radiating elements, as well as proper power supply, shape the resultant field distribution, making it possible to achieve the desired direction characteristic [6]. The generated beam spatial characteristics and propagation direction are controlled electronically, by changing an amplitude and phase of the elementary signals distributed to the individual antenna elements. The beam steering accuracy and spatial characteristic distortions depend upon the number of the applied antenna transmitting elements and accuracy in defining the phase delays and microwave signals power. Phased array antenna beam steering [7, 8] is based on supplying antenna radiating elements by microwave signals with gradually increasing phases. In the case of one-dimensional active antenna, the phase difference between antenna array consecutive elements required for shifting the antenna beam off an antenna axis by θ angle, can be presented as follows:

α = 2π

d

λ

cos θ

(1)

where: λ - microwave signal wavelength, d - distance between the radiating elements. Based on the dependence λ = c / f , where c is the vacuum light speed, and f - microwave signal frequency, we obtain:

α = 2π ⋅ f

d cosθ c


(2)


803

The above equation indicates that the phase of microwave signals connected to the antenna consecutive elements is dependent upon the microwave signal frequency. Thus, the antenna operational frequency change results in the beam propagation direction change; in turn, this indicates that varying frequencies are being positioned in different directions. This phenomenon is known in the literature as beam squinting. This disadvantage limits the antenna operation to monochromatic signals, thus disabling wide band systems implementation. Substituting in eqn (2), the dependence α = 2π ⋅ f ⋅ t , where t is a time needed for the wave to change phase by α and transforming the equation, we arrive at a simple dependence for the delay time of the microwave signal distributed to the antenna consecutive elements to propagate the beam under the θ angle:

t=

d cos θ c

(3)

The above equation shows that the signal delay time between the consecutive antenna elements - as opposite to the phase - does not depend on frequency, implicating a possibility of wide beam operation. Thus, the phase - array – in this case – should be replaced with the time-array type of steering. This method of steering is defined in the literature as TTD (true-time delay).

3

Optoelectronic units

A block scheme of the optoelectronic unit for 2-dimensional control of an antenna beam is presented in Fig. 1. Laser 2

Optical

Fibre Optic Splitter

Laser 1

Optical

RF

Generator

Optical Azimuth Control Unit 1xN

Optical

1 2

2

3 M

1

1 2

2 3

{

M

{ {

Optical

Figure 1:

1

Azimuth

Generator

PULSE

M

1

1 2

2

3

2-dimensional optoelectronic control unit of antenna beam.


804 Computational Methods and Experimental Measurements XIII Microwave signals from an RF generator controlled by a pulse generator are transmitted to an electrical input of an optical elevation control unit, which generates N microwave signals with independent amplitude and phase. The control unit utilizes an optical wave from a laser. All outputs of those control units are connected to inputs of N optical azimuth control units, which provide M linear element antennas control. Those units utilize an optical wave from a second laser. This way an MxN element of planar antenna control is provided. A block scheme of a 1-dimensional optoelectronic control unit of an antenna beam is presented in Fig. 2. Optical Input

Electrical Input Driver

MZM

Optical Receiver

Fibre

Optical Receiver

Optic Network

Optical Receiver

Figure 2:

1-dimensional optoelectronic control unit of antenna beam.

Optical input of the unit is supplied by the optical carrier signal from a laser. Electrical input is supplied by a microwave signal from the RF generator. A Mach-Zender optical modulator has been used to perform modulation of the optical carrier wave by a microwave signal. The modulated signal is routed to optical receivers via a fibre optic network, where the optical signal is split on N or M outputs and amplitude and phase control of each signal is provided. Optical to electrical conversion is done by optical receivers. There are many different optoelectronic approaches to arrange fibre optic network to amplitude and phase control, which can be divided into coherent and non-coherent systems. A coherent approach is based on optical signal phase modulation using coherent detection. Non-coherent systems are based on different lengths of optical path applications to provide various delay times of optical signals. In this case phase shift is proportional to length of optical fibre as follows:

∆α = 2π ⋅ n ⋅ f

L c

(4)

where: n - refractive index coefficient, f - microwave signal frequency, L fibre optic length, c - light velocity. This means that the optical signal phase change required to generate a microwave beam from a planar antenna propagated in the required direction needs to use a large number of different length fibre optic lines. For each beam position another set of different lengths of fibre optic



805

is required, then we need M x N x R fibres, where R – number of beam position. Thus systems like this are very large and complicated. Generally large numbers of different optic techniques were applied to achieve a variable phase shift. Most popular applications are based on: optical switches, piezo-electric crystals, liquid and aqustooptic crystals, high dispersion fibres and bragg gratings. Two types of fibre optic techniques based on binary fibre optic delay line and dispersion delay line will be considered and compared to classic microwave phase shifters. The microwave beam position generated from a linear antenna can be described as follows:

c α  d 2π ⋅

θ = arccos

  f 

(5)

where α is the phase difference of signals supplying antenna neighbour’s elements, and d is a space between antenna elements. Concerning classical microwave systems applying 5-bit microwave phase shifters, the maximum phase resolution is 11.25 degrees. Then the maximum beam position angle is 4.77 degrees. Using an optic delay line with optical switches and different lengths of fibres the maximum resolution of fibre length is equal to 5 mm so then the maximum beam position angle is 1.7 degrees for the 5GHz microwave signal frequency. Much higher resolution can be achieved using a dispersion delay line. In this case we need to apply a variable wavelength laser with 0.1nm resolution. Then the maximum phase resolution is equal to 0.25 degrees, and thus the maximum beam position angle is equal to 0.09 degrees for the 5GHz microwave signal frequency. The scheme of the control system applies a material dispersion phenomenon in single-mode optical fibres for a 16-element linear antenna as presented in Fig. 3. A tuned wavelength laser in the range 1520÷1600nm and an 10mW optical output power has been used as an optic signal source. The optic signal is routed to n electrooptic modulator.

MZM

Laser RF Generator

Driver LDO

1:2

LDZ

1:8

1:8

Attenuator

Optical Receiver Amplifier

Figure 3:

Antenna control system structure.


806 Computational Methods and Experimental Measurements XIII The applied modulator operates in a third optic window, allowing for 10GHz optical signal modulation. A specialised microwave amplifier has been applied as a modulator driver. The microwave input signal, following relevant amplification in a driver, is routed to the modulator electric input. Thus, an optic signal on the modulator output is amplitude modulated with an envelope according to the control microwave signal from the RF generator. Next, the modulated optic signal is distributed to a binary tree introducing the relevant microwave signal delays on individual outputs of the system. Optical signals are attenuated in fibre optic attenuators to obtain the required power distribution. Next, they are routed to the inputs of 16 optical receivers, performing optic electric signal conversion and finally distributed to the microwave amplifiers, thus additional amplifcation of 22dB is performed. The antenna beam control system requires selection of high dispersion fibre optic length LD and zero dispersion fibre optic length LZD, according to the operation laser wavelength range to obtain the desired antenna beam scanning range.

4

Control system measurements

Detail measurements of the developed system are based on measurements of the output signal power distribution and microwave signal delay between a modulating signal and 16 outputs signals of the control system. The measurements have been done on a microwave network analyser HP8720B. The power and phase of the microwave signal have been measured. The phase was converted into a time delay for a signal frequency equal to 5GHz. The antenna elevation characteristics have been calculated on the basis of the obtained results. Antenna beam scanning in the range of 0º ÷ 45º has been achieved. In figure 4 high compatibility between the position angle of a 16-element linear antenna characteristic in theory (black curve) and measurement (red curve) for a fibre optic dispersion delay line application can be observed.

Figure 4:

16-element linear antenna characteristics for 30º degree antenna beam position.



807

Elevation characteristics of an antenna, calculated theoretically and based on measurements for a distribution of output power like cos2x + 0.4 are presented in Figs 5 and 6. Radiation characteristics of the 16-element antenna row, performed for a non-uniform signal power distribution at the control system outputs proves considerable compatibility with theoretical simulations. The developed system ensures an opportunity to control the beam propagation direction with an excellent accuracy equal to 0.1º.

Figure 5:

16-element linear antenna characteristics for 0º, 10º, 30º and 45º degree antenna beam position (simulation).

Figure 6:

Antenna characteristics for 0º, 10º, 30º and 45º degree antenna beam position (measurements).

5

Conclusion

Using digital microwave phase shifters the calculated phase shift must be approximated according to phase shifter resolution, then the accuracy of beam WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

808 Computational Methods and Experimental Measurements XIII positioning is equal to a few degrees. Application of optical weigh units in phase array beam forming provides full antenna characteristic control. Very high resolution and an excellent accuracy of the antenna beam positioning can be achieved. The optimal technique depends on the number of antenna elements, which implicates beam width. For very narrow beams the best solution is optical delay line application in weigh units providing high angle resolution and positioning accuracy.

References [1] [2] [3] [4] [5] [6] [7] [8]

Mailloux R.J., Phased Array Antenna Handbook, Artch House Inc. 1994 Dufrêne R., Sędek E., Kołosowski W., Wnuk M., Lisowski J., Muszkowski M., Array Antenna Direction Characteristics Shaping Optic and Electronic Methods, PIT Works No. 127, Warsaw 2001 Zmuda H., Toughlian E.N., Photonic Aspect of Modern Radar, Artech House Inc. 1994 Kumar A., Antenna Design with Fiber Optics, Artech House Inc. 1996 Muszkowski M., Radar Signals Fiber Optic Transmission System, New Constructions and Technologies, Rościszów 1988 Seeds A.J., Application of Opto-Electronic Techniques in Phased Array Antenna Beamforming, Microwave Photonic, Technical Digest, MWP 1997 Frigyes I., Seeds A.J., Optically Generated True-Tine Delay in PhasedArray Antennas, IEEE Trans. Microwave theory & Tech., vol. 43. p. 2378-2386, Sept. 1995 Soref R., Optical Dispersion Techniques for Time-Delay Beam Steering, Appl. Opt. Vol. 31, No. 36, p. 7395-7397, Dec. 1992



809

Nitrogen dioxide detection using an optoelectronic sensor Z. Bielecki1, W. Kołosowski1, G. Różański1, E. Sędek2 & J. Wojtas1 1 2

Military University of Technology, Warsaw, Poland Telecommunications Research Institute, Warsaw, Poland

Abstract Sensitive laser absorption spectroscopy requires a long effective pathlength of the laser beam in the analyzed media. Traditionally, this requirement is satisfied using an optical multipass cell. We present the opportunities of detection of nitrogen dioxide using cavity enhanced spectroscopies. These techniques can be applied for construction of a fully optoelectronic NO2 sensor with a detection limit better than 1 ppb. The setup usually includes the resonance optical cavity, which is equipped with spherical and high reflectance mirrors, the pulsed or cw laser and a photomultiplier which is connected with the digital oscilloscope or with others digitizers. The optical signal leaving a cavity can be used to determine an absorption coefficient of the intracavity medium. Keywords: optoelectronic sensor, absorption spectroscopy.

1

Introduction

Nitrogen dioxide is one of the most important factors in atmosphere quality. At present, NO2 is commonly detected using the methods based on chemiluminescence. Their sensitivity reaches single ppb. Laser absorption spectroscopy is an effective tool in studies of gaseous species. The Beer-Lambert law for week absorption indicates that the minimum detectable concentration of absorbers is inversely proportional to the effective sample-path length, and directly proportional to the minimum intensity fluctuation detected by a photodetector. In order to increase the absorption sensitivity, the optical path through the absorber should be maximised and the relative-intensity variations should be minimised. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070791


Figure 1:

Example of commonly used optical multipass cell.

Long optical path lengths for absorption can be achieved by the use of White or Herriott cells (Fig. 1), a passive high-finesse cavity or laser intracavity technique [1–3]. An effective absorption path length up to several kilometres has been obtained using laser intracavity absorption spectroscopy. Use of a high-finesse cavity permits hundreds, or even thousands, of traverses through the absorber, and thus reaches long effective path lengths and provides high detection sensitivity. One of the methods of laser intracavity absorption spectroscopy is cavity ring-down spectroscopy (CRDS). The CRDS based on the measurement of the time for light to decay within the cavity, has been developed for use with a pulsed laser (P-CRDS) or with a continuous-wave laser (CW-CRDS) with acousto-optic modulator or mechanical beam chopper (Fig. 2) [4, 5]. There are others methods using a high-finesse optical cavity consisting of two highly reflective mirrors (R > 0.9999) such as cavity enhanced absorption spectroscopy (CEAS) with off-axis laser and cavity adjustment, integrated cavity output spectroscopy (ICOS) with piezoelement or ring-down spectral photography (RSP) [5–8].

2

Experimental design

The laser intracavity absorption spectroscopy techniques share a common instrument layout. A basic schematic layout of the instrument design is shown in Fig. 2. In all cases, a laser light is coupled into high-finesse optical cavity, which is equipped with gas valves. The valves are connected with the gas mixing system, which allows for precise gas mixing and establishing the ordered NO2 concentration. The laser radiation can be directed to the cavity using two mirrors. Due to pulsed character of the radiation used, one can avoid various problems with the light modulation and the laser – cavity mode coupling. A photodetector, often PMT, monitors the power escaping from the cavity, which is proportional to the intensity inside a cavity. PMT can be equipped with the interference filter, the bandpass of which was well matched to the laser line. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Figure 2:

811

Example of CRDS setup.

Figure 3:

Idea of CRDS.

Next, the signal is recorded with a fast digital oscilloscope and after digitization usually a computer is used to store and analyze the measurement data.

3

Analyses

In CRDS technique, the gas sample is placed inside a high-finesse optical cavity. It consists of two highly reflective mirrors. A short laser pulse is coupled into the cavity. The light is reflected back and forth inside the cavity and, every time that the light is reflected, a small fraction of this light leaks out of the cavity and is registered by a photodetector (Fig. 3).


812 Computational Methods and Experimental Measurements XIII The presence of absorbing species in the cavity gives an additional loss channel for the light inside the cavity. If the absorption follows Beer′s law, the light intensity inside the cavity will still decay exponentially, resulting in a decrease in the decay time as

I (t ) = I 0 e

−

[(1− R )+αL ]c t L

= I 0e

−

t

τ ,

(1) where I0 is the initial intensity, c denotes the light speed, α is the absorption coefficient, and L is the resonator length. The decay time of radiation is measured once when the cavity is empty (Fig. 4, dashed line B) and next when the cavity is filled with the absorber (Fig. 3, line A).

Figure 4:

Decay signal from a cavity.

In an empty cavity, this ring-down transient is a single-exponentially decaying function of time with a 1/e CRDS time which is solely determined by the reflectivity of the mirrors and the optical path length between the mirrors. Speed of the decay intensity I(t) of the pulse of the laser light is dependent on the mirrors reflectivity coefficient R, the resonator length L, diffraction losses, and extinction, that is absorption and scattering of a light in the absorber filled cavity. Therefore, by measuring the resonator quality, determination of the absorption coefficient is possible. The resonator quality can be determined by measurement of the radiation decay time constant τ

τ=

L . c ⋅ [(1 − R ) + αL]

(2)

By comparison of decay times for these two cases, a value of the absorber density N can be found α 1 1 1 , (3)  −  N= = σ σ c  τ τ 0  where σ denotes the absorption cross section, while τ0 and τ are the time constants of the exponential decay of the output signal for empty resonator and for the resonator filled with the absorber, respectively. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


813

When the high reflectivity mirrors (R > 0.9999) are applied, the absorption coefficients as low as 10-9 m-1 can be measured. There are several advantages to this approach. Since the absorption is determined from the time behaviour of the signal, it is independent of pulse-topulse fluctuation of the laser. Furthermore, the effective absorption path length, which depends on the reflectivity of the cavity mirrors, can be very long (several km), while the sample volume can be kept rather small. Another advantage is that the absorption is measured on the absolute scale. The CRDS technique can be applied with success when the molecule′s excited state does not fluoresce or can be ionized. In high-pressure samples, such as plasmas and flames, CRDS can be successfully used to extract quantitative absolute concentration data, which is nearly impossible by using other methods. As long as the mirrors with a sufficiently high reflectivity, detectors with a sufficiently sensitivity and fast time response, and tunable light sources are available, there is no intrinsic limitation to the spectral region in which CRDS can be applied. This method can be used from the ultraviolet part of the spectrum to the infrared spectral region.

4

Nitrogen dioxide concentration measurement

The nitrogen dioxide absorption cross section σ is shown in Fig. 5 [9]. The greatest σ values reached around 400 nm. There are several minima and maxima, but they varying about value of 6×10-19 cm2.

Figure 5:

NO2 absorption cross section.

In order to achieve a high value of limit detection of NO2 reflectivity, cavity mirrors and wavelength of a laser beam should be appropriately matched to the greatest values of nitrogen dioxide absorption cross section. Furthermore, to achieve long decay times and high precision of the decay time determination, the proper adjustment of the cavity is necessary. Moreover, the measurement with a good detection limit also requires appropriate filtration WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

814 Computational Methods and Experimental Measurements XIII of the investigated air. This is necessary in order to avoid the light scattering in the aerosol particles as well as the dust deposition on the mirror surfaces. When the concentration of aerosol particles in gas is similar to that in free atmosphere, the extinction of about 10-6 m-1 occurs, consequently at high sensitivity measurements, the losses due to the light scattering can be the same or larger than the absorption losses. Sensitivity of nitrogen dioxide optoelectronic sensor based on cavity enhanced spectroscopy technique is determined by detectable concentration limit NLimit, which is described by the formula

N Limit =

X , cσ (λ )τ 0

where X is the relative precision of time determination, and is equal to τ −τ X = 0 .

τ0

(4)

(5)

In Fig. 6, the dependence of sensitivity nitrogen dioxide sensor on cavity length and mirrors reflectivity is shown. The calculations were performed for 5% of decay time precision determination and absorption cross section equaled to 5.1 cm2. As it is shown in the picture, strong influence on sensor sensitivity has mirrors reflectivity.

Figure 6:

Dependence of NO2 sensor sensitivity on cavity length and mirrors reflectivity.



815

Detectable concentration limit is also strongly biased by decay time precision determination. For instance, if 1% precision is ensured, sensor with mirrors, the reflectivity of which reached a value of 99.995%, provides possibility of measurement of 0.5 ppb of nitrogen dioxide, what is presented in Fig. 7. The dependence of NO2 sensor sensitivity on decay time precision determination and cavity mirrors reflectivity is shown.

Figure 7:

Dependence of NO2 sensor sensitivity on decay time precision determination and cavity mirrors reflectivity.

It is possible to reach highly precise decay time determination using coherence averaging of samples of signal from a photodetector [10]. Experiment shows, that thanks to this precision better than 0.2% is attainable after the averaged 10 000 CRDS pulses (Fig. 8).

5

Conclusions

In this paper, we have presented cavity enhanced spectroscopic technique which can be applied for construction of a fully optoelectronic NO2 detection system. A manner of resonator quality determination by measuring the time of the radiation imprisonment is not sensitive for the laser power fluctuation and photodetector sensitivity fluctuation.



Figure 8:

Precision of decay time determination as a function of a pulses number.

The features of the described detection system show, that it is possible to construct a NO2 sensor, the sensitivity of which could be comparable with that of chemical detectors. Such a kind of system has several advantages such as: low price, small size and weight and possibility of detection of other gases. In the next stage of the experiment we are going to construct a fully optoelectronic system based on cavity enhanced spectroscopic technique for detection of trace concentration nitrogen dioxide.

References [1] [2] [3] [4] [5]

[6] [7]

J.U. White, Very long optical paths in air, J. Opt. Soc. Am. 66, 411, 1976. D. Herriott, H. Kogelnik, R. Kompfner, Off-axis paths in spherical mirror interferometers, Applied Optics, 3, p. 523, 1964. J. Doussin, R. Dominique, C. Patrick, Multiple-pass cell for very-longpath infrared spectrometry, Applied Optics, Vol. 38, No. 19, pp. 41454151, 1999. A. O’Keefe, D.A.G. Deacon, “Cavity ring-down optical spectrometer for absorption measurements using pulsed laser sources”, Rev. Sci. Instrum. 59, No. 12, pp.2544 – 2551, 1988. G.S. Engel, E.J. Moyer, F.N. Ketusch, J.G. Anderson, Innovations in Cavity Enhanced Laser Absorption Spectroscopy: Using in situ Measurements to Probe the Mechanisms Driving Climate Change, Earth Science Technology Conference, Laser Sensor Technologies, June, 2003. J.J. Scherer, Ringdown spectral photography, Chemical Physics Letters, No. 292, pp. 143–153, 1998. J.J. Scherer, J.B. Paul, H. Jiao, A. O’Keefe, Broadband ringdown spectral photography, Applied Optics, Vol. 40, No. 36, pp. 6725-6732, 2001.



[8] [9] [10]

817

R.F. Curl, F.K. Tittel, Tunable infrared laser spectroscopy, Annu. Rep. Prog. Chem., Sect. C, 98, pp. 219–272, 2002. M.F. Merienne, A Jenouvrier, B. Coquart, The NO2 absorption spectrum. I: absorption cross-sections at ambient temperature in the 300-500 nm region, J. Atmos. Chem., Vol. 20, No. 3, pp. 281-297, 1995. J. Wojtas, A. Czyżewski, T. Stacewicz, Z. Bielecki, J. Mikolajczyk, Cavity enhanced spectroscopy for NO2 detection, Proc. SPIE, Vol. 5954, pp. 174-178, 2005.




819

Automatic processing analysis of infrared images for monitoring pantograph catenary interactions A. Balestrino, O. Bruno, A. Landi & L. Sani Dipartimento di Sistemi Elettrici e Automazione, Università di Pisa, Italy

Abstract This paper shows a new sensor based on infrared images for monitoring pantograph catenary interaction. Due to the displacement of the contact point with respect to the reference position an image processing analysis of each frame is performed by using the Hough transformation in order to detect automatically monitoring variables of interest (e.g. temperature at the contact point and the position of the support towers along the railway line). Keywords: infrared image, pantograph-catenary interaction, Hough transformation, segment detection.

1

Introduction

In order to improve the maintenance activities, a relevant objective for railway companies is the development of new sensors for a continuous monitoring of the quality of the current transmission between the overhead line and the collector strips of the pantograph. This problem is grown crucial with the introduction of the European Standard 96/48/CE about the interoperability. It is well-known that a poor electric contact between is the origin of arcing between the overhead wire and the collector strips of the pantograph. Several investigations have shown that the main damages of the overhead contact line installations are caused by the short term thermal effect of these arcing. The process which leads to the deterioration of the contact wire is associated with a localised recrystallization of the copper (i.e. a transaction to a stable crystalline microstructure with loss of all physical characteristics typical of the cold-drawn) and the formation of pits and dents on the surface. Unfortunately high speeds worsen the problem and a monitoring system has to be set up and tested to plan maintenance activity. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070801

820 Computational Methods and Experimental Measurements XIII Previous research activity (Bruno et al [1], Barmada et al [2]) revealed that a new measurement system, based on a phototube sensor, could detect break arcs between the pantograph and the contact line. A different approach is a measurement of the quality of the overhead contact by using infrared cameras. Preliminary results based on the analysis of infrared images were proposed by Balestrino et al [3]. Measurements carried out during high speed test runs and a post processing of the data collected have revealed the effectiveness of the proposed method for detecting the losses of contact, in order to evaluate and test the performance of running pantographs and of the contact wires. The task of the proposed research is the application of tracking algorithm of predefined objects (e.g., the contact point on the pantograph strip) based on the Hough transform over a sequence of images acquired from thermo-camera, in order to monitoring variables of interest, (e.g., temperature at the contact point and the position of the support towers along the railway line).

2

Thermography for monitoring pantograph-catenary interactions

Thermo-cameras have not been used extensively in railway research. Main advantages offered by thermo vision are: a) it is a non-contact and non-destructive technique; b) it is suited to monitoring devices operating under high voltage or carrying high currents; c) is more informative than a standard camera image, because of its insensitivity to different weather conditions, to the daily-nightly runs or to the presence of tunnels. A critical aspect of thermo vision is that the exact temperature of the body under test cannot be directly revealed. Nevertheless infrared imaging is an excellent method for extracting a qualitative map of the superficial temperature. As a matter of fact thermal analysis gives a relative information on the temperature, following the reasoning of Runciman [4]: all objects at temperatures above absolute zero emit electromagnetic radiation. Radiation thermometry makes use of this fact to estimate the temperatures of objects by measuring the radiated energy from selected regions. Every physical process characterized by an increase or decrease in surface temperature is detectable with infrared thermography. The intensity of the emitted radiation depends on two factors: the body temperature and the factor emissivity of the surface, i.e. the ability of the object to radiate, defined by the Stefan-Boltzmann equation:

E = εσ T 4

(1)

where: E is the radiation in W/m2, ε the emissivity, σ the Boltzmann constant and T is the temperature in Kelvin.



821

The emissivity factor ε lies between zero and the unity, depending on the material nature and on the superficial roughness. Infrared imaging systems convert infrared heat emissions into a picture that shows the relative temperature differences in a range of grey tones, or in a series of colours, in such a way that the desired temperature information is easily interpreted by the user. In fig. 1.a a typical infrared image of pantograph-catenary interaction in the case of a burst of arcing (i.e. losses of contact between catenary and pantograph) is shown (the train speed was 200 km/h in this case). The maximum infrared emission is in the neighbourhood of the contact surface between strip and wire. Interaction region between the overhead feeder and the strip shows a high thermal gradient. A continuous monitoring of the contact region leads to interesting information on the quality of the current transmission.

Figure 1:

3

Thermal image in case of a burst of arcing.

The automatic image processing

It is well-known that the contact point between the pantograph strip and the overhead line is not fixed. In fact the wire is zigzagged relative to the centre line of the track, to even the wear on the train's pantograph as it runs underneath. Moreover, because of the no uniform elasticity of the catenary, the height of the contact wire may change (e.g. entering a tunnel). Therefore a relevant displacement of the contact point may complicate the image processing, unless considering the knowledge of the moving spatial coordinates of the contact area. A first step for an automatic image processing analysis is to follow frame-byframe the position of some critical features (pantograph strips, overhead electric line). All these elements can be characterized by linear contours. This observation allows the use of standard algorithms for detecting straight lines inside each image under examination. The Hough transform is well-suited for detecting straight-lines in the real image: its application can be extremely useful for monitoring the contact point, characterized from the intersection of a segment representing the strip and a straight line representing the overhead line contact. Furthermore it can identify different rectilinear structures, such as the poles along the railway line, helping in finding a more precise correlation between the overheating of the contact point (or a burst of arcing) and the position of the train. A detailed description of the Hough transform is reported by Hough [5] and Deans [6]. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

822 Computational Methods and Experimental Measurements XIII The object recognizing procedure is depicted in the followings steps. a) Independently from the electronic format of the recorded file (we considered outputs in .bmp, .mpg or .avi formats), single frames are extracted from the image sequence. Each image is first stored in a matrix, then converted from true colour (RGB) to grey scale. b) Each image is suitably filtered for reducing noises and sharpen objects, then edges are extracted and image is then converted from a grey scale to a binary image. Edge detection was performed using Canny algorithm with a preliminary noise reduction (Canny [7]). After this step, each frame is represented by a binary image with edges evidenced. It is an easy task to identify the pantograph, the overhead contact lines and the poles. c) Object locations in every frame is characterised from an analysis of edges and a detection of reference straight lines. All straight lines can be localised using the Hough transform. In order to increase algorithm efficiency and to reduce computational burden, we suppose to know, within a limited variation, the position and the possible movement of the features to be monitored, therefore two observation windows into the image are locked: a first one includes the pantograph structure and the second one reveals the passage of the support structures. The straight line search can be limited inside these rectangular regions of the image. The procedure allows a quick detection of the object position and a repetitive algorithm allows following the trajectory of an object inside successive frames. After a first detection of the trajectory, it is a good strategy to eliminate frames with discontinuous trends of the object and to filter the image for reducing noise and to sharpen the image. Following this logic a software tool was created: it allows the user to detect automatically: 1. pantograph strip position; 2. catenary wires; 3. position of the poles. This procedure can be repeated for each frame of the sequence. Since the pantograph moves slowly with respect the frame rate of the camera, we can use the position of the strip detected in the previous image to window next image appropriately. So the computation time can be greatly reduced. 3.1 Position of the pantograph inside an image The pantograph appears at each frame and its position moves slowly. Small displacements both in vertical and in horizontal direction are allowed: they are due to the movements of the main frame and of the contact arms. In thermo images are evidenced two different pantograph heads and two different strips: the goal of the following study is to follow only the strip nearest the IR camera. Contact analysis is applied to the squared window of fig. 2.a, named reference box: its size and position inside the image will remain unchanged for the limited displacement of the pantograph head. Several segments are easily recognised by Hough transform (fig. 2.a): only the upper WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Figure 2:

(a)

(b)

(c)

(d)

(e)

(f)

823

Application of the Hough transformation. (a) Horizontal detected segments into the reference box of the strip. (b) Upper edge of the nearest strip. (c) Detected segments into the reference boxes of the lateral supports. (d) Detected lateral supports. (e) Detected line by Hough transformation. (f) Reference shape.

one represents the contact line for monitoring (fig. 2.b). Reference box allows following pantograph displacements along a vertical direction. For a correct analysis also the horizontal displacement has to be measured. Lateral displacements can be then esteemed from an analysis of the two lateral arms sustaining the pantograph heads (fig. 2.c). The two segments representing the external profile of these lateral supports can be identified by repeating the Hough analysis inside two new reference boxes, as shown in (fig. 2.d). The three reference boxes can be joined together for localising a triangle as the reference shape representing the pantograph, as shown in fig. 2.e and fig. 2.f Upper side of the triangle is representing the line useful for monitoring the WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

824 Computational Methods and Experimental Measurements XIII contact with the catenary wires. Note that the representation of the pantograph head with a straight line is only an approximation of the real scenario. Nevertheless the maximum error accumulates in the lateral regions of the segment, where the pantograph head is not rectilinear: fortunately in such region an analysis of the contact is less relevant, since the local temperature is lower than in the middle zone of the segment. 3.2 Identification of the contact between pantograph and contact wire After the identification of the pantograph structure, the following step was devoted to a searching of the contact zone between pantograph and the overhead wire for monitoring the temperature and for checking a correct positioning of the contact wire in terms of its zigzagged movement, relative to the centre line of the track. In the case of the image sequences recorded along the railways from Florence to Rome, it can be observed that there are many straight lines, between them because two overhead wires (fig. 3.a) are in close proximity to the contact region.

Figure 3:

(a)

(b)

(c)

(d)

Identification of the contact between pantograph and overhead contact wire. (a) Image under examination (b) Detected straight lines representing catenary wires. (c) Detected boundaries (d) Identified contact region.

It must be highlighted that the contact points are not corresponding to the hottest points of the strip, usually in the middle of the contact strip. Hough transform can be applied again to detect overhead wires, as shown in fig. 3.b WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


825

many straight lines are detected. A selection of the correct contact wires was performed, based on the two following rules: contact wires are always adjacent and at the exterior of the set of the identified lines and contact wires are thicker than non-contact ones. The contact points are uniquely located via the following steps: 1) computing the intersection between the upper side of the triangle (pantograph head) and the two straight lines exterior to the pencil of lines (fig. 3.c); 2) analysing the colour distribution of the pixels inside the image, near the two points selected at step 1. The thickest line of the two is selected, for detecting the contact zone, whose extension is predefined by a fixed number of pixels in the direction of the middle of the pantograph head (fig. 3.d). 3.3 Passages under the portal structures of the railways In the case of a railway line with portal structures (e.g. the railway line connecting Florence to Rome) a similar method can be employed for detecting the poles. The horizontal structure is the easiest element to be recognized: it is the wider structure and it appears even if the poles are not included inside the image, as shown in fig. 4.a.

(a) Figure 4:

(b)

(a) Horizontal bridge sustaining catenary wires. (b) Observation box for counting the passages of portals.

Therefore counting passages of horizontal portals constitutes a safe method for localising the train position along the line: it can be performed using an observation box positioned in a suitable region of the image. An easy solution is to locate this box in a region where only the horizontal portals are passing, e.g., in the upper left side of each frame (fig 4.b). The Hough transform for straight lines recognizing can be applied inside the observation box and a reliable criterion of portal detection may be based on an accurate analysis of the number of identified lines. Its maximum value is very high in case of passages of portal structures: a threshold value is therefore useful for discriminating noisy images, or single straight lines passages.



4


A preliminary laboratory test was carried out in order to yield the emissivity of the pantograph strip. In Italian Railways (3 kV d.c.) the contact material strip is copper (or copper based alloys). By using the IR camera software, the average emissivity of the strip was determined to be 0.2. The measurement equipment was then installed on board of an ETR500, a high-speed train of Italian Railways. IR-camera (produced by the Flir Systems) was located on the top deck of the locomotive, in front of the pantograph. The block diagram of fig. 5 illustrates the measurement chain. Trial runs have been carried out travelling along the railway line connecting Florence to Rome and vice versa.

Figure 5:

Measurement chain.

4.1 Portal structure detection an positioning of the overhead contact line In fig. 6.a and 6.b two plots are shown, the first one consider a portal structure detection, the second one reports the position of the contact point with respect to the centre of the pantograph head (a positive value means the contact point is located on the right w.r.t. the median position). They are referred to an analysis of 200 frames. Plots are functions of a position relative to the first frame analysed. Train speed was 200 km/h.

Portal structure

Position of the contact point (pixels)

1

0

0

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 Space (km)

30 20 10 0 -10 -20 -30 0

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22

Space (km)

(a) Figure 6:

(b)

(a) Positioning of structural portal. (b) Contact point position respect to the centre of the pantograph head.



827

4.2 Example of break arc detection Consider now 100 consecutive frames with a break arc corresponding to frame 50, shown in fig.7.a. The observation box is representative of the contact point. The observation box is representative of the contact point. Temperature profile along the strip surface for frame 50 is shown in fig. 7.b. Figure 8 shows the infrared map of the strip along the railway line connecting Florence to Rome between km 60 and km 63. Temperature peaks are correlated with arcing occurrences. This elaboration has been obtained by processing 2700 frame of the video sequence with a computation time of 540 s (using an AMD ® Athlon processor at 1.2 GHz).

(a) Figure 7:

(a) Frame 50 with break arc detection. (b) Temperature profile on the strip surface.

Figure 8:

5

(b)

Infrared map of pantograph strip along the railways.

Conclusions

The use of an infrared camera for monitoring the pantograph-catenary status has been proposed and experimentally tested. The temperature profile of the pantograph strip was detected on infrared images by using the Hough WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

828 Computational Methods and Experimental Measurements XIII transformation. An analysis of the stored images could help maintenance operations, revealing, for example an irregular positioning of the line. New typologies of pantographs, new materials for collector strips, and catenary layout can be thermically analyzed.

Acknowledgement The financial support of the Italian Ministry of University and Research - MIUR (Project on Innovative Controls in High-Speed Transport Systems) is gratefully acknowledged.

References [1]

[2]

[3]

[4] [5] [6] [7]

Bruno, O., Landi, A., Papi, M. and Sani, L. Phototube sensor for monitoring the quality of current collection on overhead electrified railways. In Proc. IMECHE, Part F: Journal of Rail and Rapid Transit, vol.215, n. 3, pp. 231-241, 2001. Barmada, S., Landi, A., Papi, M. and Sani, L., Wavelet multi-resolution analysis for monitoring the occurrence of arcing on overhead electrified railways, In Proc. IMECHE, Part F: Journal of Rail and Rapid Transit, vol. 217, n 3, pp. 177-187, 2003. Balestrino, A., Bruno, O., Landi, A., Masini, P., Mingozzi, E., Papi, M. and Sani, L. Infrared camera for monitoring pantograph-catenary interactions. In Proc. of World Conference of Railway Research 2003, Edinburgh (UK), pp. 1-7, 2003. Runciman, H.M. Thermal imaging, in Measurement, Instrumentation, and Sensors Handbook CRCnetBase 1999, J.G. Webster Editor, CRC Press, 2000. Hough, P. Method and means for recognizing complex patterns, in U.S. Patent 3069654, 1962. Deans, S, R. Hough Transform from the Radon Transform, in IEEE Transactions on Pattern Analysis and Machine Intelligence Vol. PAMI-3, No 2, pp.185-188, (Mar.1981). Canny, J. A Computational Approach to Edge Detection, in IEEE Transactions on Pattern Analysis and Machine Intelligence Vol. PAMI-8, No 6, pp.679-698, (Nov.1986).



829

Architectural hot-swap in NURBS surfaces: versioning of case studies, work in progress A. Prichard-Schmitzberger California State Polytechnic University, Pomona, USA

Abstract This paper documents the testing of various applications of hot-swapping and precision modelling with minimal programming input reflecting the limited resources of small-scale practices. Hot-swapping identifies the principle of replacing components of a building during active design processes without altering its general appearance. The methodologies were tested through a series of courses on Digitally Enhanced Construction and Fabrication with architecture students at Calpoly Pomona. The teaching methodology allows not only the investigation of the required modelling accuracy but also the creation of prototypes and various versions of assembly alternatives. The paper covers the procedures developed to derive architectural form in a process of reverse engineering of specific architectural case. Flexibility or limitation of nonuniform surfaces for rapid prototyping and file-to-factory construction are investigated. Using versioning, a centralized design methodology with one source object for controlling and tracking 3d NURBS surfaces are detailed to material properties and using current building material which allows the implementation of pre-fabrication as well as mass-customization. Resulting projects allow assessment of time-consumption in the design-process; deviation of the proposed result from the original; overall complexity of assembly method; customization amounts of individual parts and manual effort. The procedures outlined in this paper would enable small architectural offices to enter competition with larger offices through integrating positive aspects of hotswapping in conjunction with prototyping without the typically necessary technical and fiscal casts. Keywords: hot-swapping in architecture, reverse engineering, versioning, computer integrated construction and manufacturing, design processes.



1

Introduction

The course Digitally Enhanced Construction and Fabrication establishes first contact with so-called “file to factory” [2] scenarios and rapid prototyping in architecture. A major aspect of the course, however, focuses on the practical application and the supposed seamless exchange of digital information in 3dmodelling, simulating a typical small-scale practice. Small-scale offices experience on one hand the financial strain of software and equipment costs and on the other hand affordability of qualified employees to be in competing range with larger office structures.

2

Objective

The advance of BIM – Building Information Modelling – and the integration thereof in the curriculum for accreditation purposes offers another challenge to digital architecture and its educators. BIM is associated with the relatively ‘smart’ structure of current software packages, among them so-called parametric modelers such as Autocad Revit and Archicad. However, these particular programs differ in application from potential software packages such as Generative Components, running on Microstation platforms, due to the fact that the major input follows industry standards rather than input derived from a designed architectural model, enabling definition and specification of customized detailing. Hot-swapping, a term borrowed from computer hardware, describes a potential method which allows a) quick development of a base 3d model with enough flexibility and accuracy for future manipulation and b) exchange of construction methods up through a late stage of design and production. 2.1 Course setup The initial 10 week, 3 unit course layout encompasses a lecture portion with a brief introduction to the history of prototyping and numeric machining. Differentiation between polygonal, Boolean and NURBS modelling is necessary to establish an understanding for the advantages and disadvantages in creation processes. NURBS models offer mostly more flexibility due to their parametric setup. Polygonal mesh models represent complex structures themselves and are therefore undesired. Much of contemporary architecture shows intention and interest in generating quasi-automatic forms and structures, genetic algorithms, and parametric design methodology. They engage programming/scripting, setting inputs, initial conditions, controls, and performance, all of which are criteria to evolve a form or structure or to produce a “population” of possible architectural phenotypes which are then selected according to extrinsic criteria or self-referencing elaborated. Manuel DeLanda once characterized this shift towards the genetic algorithm as the third wave of digital modelling software, one which would finally supplant all the previous identities of “designer” and intentionality in favor of diagrammatic processes [3]. This process is mostly WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


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formal and despite its aspiration a very often pseudo-scientific approach that appears removed from any material aspects. Hot-swapping follows a different premise. 2.1.1 Hot-swapping configuration Hot-swapping as a technique is detached from the architectural discussion on form and formal tendencies. The re-association with fabrication, construction and affiliated practices is tied to the computational procedures. The digital model serves as surface model and offers the conditions upon which material properties and structural methods can be exchanged in a timely fashion without the need for re-creating the geometries. In architectural education the tools are limited, and the understanding of architectural assembly is still a traditional one. Architectural construction methods are sequenced in accordance with the history of modernist, post industrial-revolution production, choosing pre-arranged certified techniques and using discrete building elements to form a hierarchical build-up sequence. These independent elements represent a simple nested hierarchy, easy to control in assembly and for its assembly schedule. 2.1.2 File-to-factory File-to-factory scenarios and the proclamation “one-building – one detail” [4] by K. Oosterhuis of ONL require a new production setting, reduction of material and the generation of assemblies at a pre-construction phase as well as a different structural fabrication scenario. While bypassing the rather linear, formgenerating relationship of architecture and technology, mostly represented through the software and its commanding algorithms, he further re-introduces the relevance of practical application, looking for a new position in the construction process. Utilizing the digital model not only for representational, diagrammatic or conditional purposes, but also for applying an entire material and assembly process as well as documentation method is core to Building Information Modelling. However closely related, his practical application does not submit to standardized detailing and building methods but offers via ‘hot-swapping’ and parametric design the ability to control the production process in conjunction with the digital model until a very late stage in the design-to-production process [5]. 2.2 Design methodology Throughout studies it appears that for an applicable generation of conditions which should allow a hot-swap of features, a parametric surface model provides the highest flexibility in a) establishing a controllable form and b) continually applying detailing features. For this investigation it is necessary that current discussions concerning performative aspects of surfaces strictly follow the practical application only to avoid unnecessary distractions from the objected goal. A reversed, traditional approach of developing a structural grid and forming the architecture after its constraints would limit any flexibility in the exchange of generation methods. Any parametric surface can be treated as a field WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

832 Computational Methods and Experimental Measurements XIII and therefore subdivided into elements, following a process similar to tessellation of digital surfaces first-hand. However, a structurally correct, generative tessellation would again require specific scripting effort. The engineering-design application mainly focuses on a realistic structural composite solution at this point. The maintenance of a relatively simple NURBS surface in Rhino3D offers a highly flexible base for any further manipulation, since the parametric values are reduced to few control-points in contrast to polygonal meshes, with a comparatively excessive amount of manipulation vertices. Structural application can be simply pulled, projected onto the surface or created via isocurves, fig. 1, and despite causing occasional undesirable, uncontrolled effects in varying degrees of the surface curvatures, the results remain largely flexible for manipulation. Further detailing is has a high level of time consumption due to the manual aspect of modelling; each elemental intersection is revisited with each alteration of the conditions and if necessarily corrected through either Boolean operations or remodelling.

Figure 1:

3

Example for pulling structure grids to a surface. (Student: D. Rogers).

Practical application

Given practical limitations the course setting intentionally abandons the introduction of scripting to investigate and manage architectural data in a more traditional sense. The importance of programming and respectively scripting is undisputed; programmability is a unique property of the digital medium. Design methodology might be opposed by the exactness inherent to programming itself; though this concern was not reflected by the objectives of this course.



833

3.1 Concrete modelling The case-studies chosen for reverse engineering are F. Candela’s Restaurant Mantiales in Xochimilco, Mexico, built in 1958–59, and the 1959 version of F. Kiesler’s Endless House. Candela’s Restaurant Mantiales represents an elaborate thin concrete shell structure; the main form, a rotated Hypar – hyperbolic paraboloid (a non-planar surface in which at any point are two intersecting straight lines) represents a challenge in reconstruction itself without the support of a mathematical formula; in addition its non-conventional selfsupporting structure ultimately removes it from the typical possibilities to be developed under use of regular BIM approaches. Similarly, Kiesler’s Endless House follows in its bio-morphic structure a non-regular architectural form, which demands not only strategic modelling consideration but also understanding of complex geometries in space. Both models were reduced to slightly simpler geometries in order to allow adequate time-management of the course, fig. 2. 3.1.1 Reverse engineering In order to offer best resolution in the surveillance practice, Reverse Engineering (RE) is brought into consideration. RE is a process describing recovery or discovery of technological principles of a mechanical, or in this case architectural, application through analysis of its structure, function and operation, in architecture becomes comparable to a detailed case study. An architectural example can be evaluated and understood through its geometric, structural and technological assets. The procedure further enables understanding of core-principles of generating ruled surfaces and their flexibility as well as applying those principles to fixed parameters derived from the studied object.

Figure 2:

Deriving the Hypar from defining geometries. (Student: D. Rogers).


834 Computational Methods and Experimental Measurements XIII 3.1.2 Versioning After assessment of scope and modelling technique the principle of versioning is introduced and applied in parts. Versioning describes a centralized design methodology with one source object for controlling and tracking and in chosen case studies students are assigned one successfully developed model to generate individual considerations of construction method. 3.2 Implementation Adopting the traditional method of tracing a digital line-drawing of the original architecture, a set of B-Splines and resulting 1/8 segments of NURBS surfaces for the Mantiales model are produced. To reconnect those after the rotational array is itself a transitional step, non-regular surfaces need to be inserted which transform a single a surface ultimately into a polysurface, reducing the flexibility of the system as a whole. Afterwards a cleaner surface can be derived through a lofting procedure from extracted sections and edges, fig. 3.

Figure 3:

Completing the full rotational structure for the Restaurant Mantiales (Student: M. Gonzales).

In the case of Kiesler’s Endless house the geometry is far more complex, tracing efforts to develop boundary curvatures and respectively cross-sectional shapes demand advanced modelling skills. A lofting method of interpolating equally spaced cross sections within the boundary trace-lines for individual pods provides satisfying results. A reduced geometric model of the exterior envelope of Kiesler’s project is developed. The blending of the pod-shapes to one joint shape adds a level of imprecision and deprives the NURBS model of its initial flexibility, fig. 3. Full parametric modelling as offered through Microstation GC might deliver a better, faster product; in current version of the House the time consumption is demanding to fall back on available tools with steep learning curve such as Rhino3D, fig. 4. In both Kiesler’s Endless House and Candela’s Restaurant Mantiales, various approaches in different complexities were repeatedly re-drawn to optimize the base model; these models received further manipulation through individual interpretations and considerations of structural systems through projections onto the surfaces themselves. This initial projections already allow first analysis and WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


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re-application of different structural systems, which then, through simple sequenced processes of lofting and extruding lead to rudimentary but applicable structural solutions, figs. 5(a), (b). Any additional, engineered input from this point on illustrates the limitations of modelling without scripted environments. Despite the maintained precision, manual input of eventually corrected, structural or material assumptions would require re-adoption in a manual, linear fashion and therefore would add up to potential human error or imprecision as well as consumed time. (a)

(b)

(d)

(c)

Figure 4:

Development of Kiesler’s surface model (a) mesh transformation delivers uncontrollable surface, (b) pod-method generates more accurate solution, (c) pods are merged, (d) an isoparm curve skeleton is derived) (Student: J. Cabreras).

3.3 Results Results vary in quality, but demonstrate initial success of the conceptual state of hot-swapping method in architectural design. The ability to offer a multiplicity of solutions – versioning – in a relatively short time allows affability in material and engineering negotiation, fig. 6.

4 Conclusion: programmer – an architectural specialization Schedule, availability of technical support and architectural challenge remain key-factors for competition of small architectural practices to large offices; displayed approach demonstrates that with some geometric constraints the small



(a)

Figure 5:

(b) (a) Clockwise: (i), (ii) intersections between crossing members are derived manually, a time-consuming process, lacking specific accuracy, (iii) CNC-prototype underplayed the original drawing and (iv) laser-cut structural model (production time incl. file preparation 50 hours) (Students: J. Leong, E. Liu, H. Ming, A. Hernandez); (b) Clockwise (i), (ii) finalization for production, creation of cut-sheets for laser-cutting (student: M. Gonzales), (iii), (iv) Tessellation approach and detailing including tagging of different elements (Student: E. Scott.)



Figure 6:

837

Clockwise from top left: different approaches with different Prototypes, shingled surface model with diagonally braced sheet metal frame (Student: T. Vincent), hexagonal structure (Student D. Rogers), cut-sheet for welding mock-up (Student T. Vincent), tessellation model (Student E. Scott) and grid-lattice (Student M. Gonzales).

office is in competition with the traditional large office. However, hot-swapping as technique remains, despite improvements in modelling skills, modelling accuracy and software advancement, a prerogative of designers with adequate programming skills and education. Real-time events, as Axel Kilian points out, require programming of these events so that the computational event takes place during its calculated execution [6]. Common affordable and available software packages currently does not provide this commodity in other form than indigenous scripting language; scripting itself supports customization of software which does not inherently support enough flexibility in its command structure and can therefore be seen as a reverse approach itself. Software designed for architects seems to gear more towards implementing industry standards rather than generating a desirable tool for architectural design. Education itself and its accreditation systems abandon vital elements for at least allowing students to customize the appropriated software; neither descriptive geometry nor programming skills are enforced. Therefore educators are forced to rather comply with standards initiated by software producers and industry, than to represent a contributing element to the industry itself. Respectively, a small practice cannot currently take full advantage of new trends in architecture, due to the budget and schedule challenges for employment of specialists and the rigid hierarchy in construction business. Given the current climate of academic education in architecture programs and the lack of substantial pre-education in computing skills, the course is partially used to examine success and failure of the system in parts. Building Information Modelling appears to offer desirable appeal to such structures, relying on the safe implementation of established construction methods and materials. Alternative routes, as demonstrated in examples by practices of Frank O. Gehry Associates and ONL become showcases and demonstrate the position of corporation versus academia; academia is currently coerced to follow the lead of few successful WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

838 Computational Methods and Experimental Measurements XIII large practices an their techniques. Knowledge and understanding of engineering, industrial production sequences, explicit computational and geometric training simplify the architectural and respectively digital production process and would further allow architectural academia structures to regain its position as informant educator.

References [1] [2] [3] [4] [5] [6]

Supported by a grant from the College of Environmental Design, California State Polytechnic University. Oosterhuis K, The Architect’s new Data-Driven Practice (Chapter 3). Hyperbodies, Towards an E-motive Architecture, Birkhaeuser: Basel, Boston, Berlin, pp7-10, 2003. De Landa, M.l, Philosophies of Design, The Case of Modelling Software, Verb Processing, Actar: Barcelona, 2001. Oosterhuis, K., Building Bodies (chapter 7), Hyperbodies, Towards an Emotive Architecture, Birkhaeuser: Basel, Boston, Berlin, pp 17-22,2003. Oosterhuis, K., Bier, H. Aalbers, C., Boer, S., Programmable Architecture, Proc. of ACADIA 23, Fabrication, pp 3-5, Cambridge and Toronto, Ontario, Canada, 2004. Kilian, A., The Importance of Programming, Defining Digital Space Through a Visual Language, p 13, Master of Sciences Thesis, Massachusetts Institute of Technology, Department of Architecture, 2000, http://destech.mit.edu/akilian/newscreens/thesis2000/pdfversion.PDF


Section 11 Advances in measurements and experiments



841

Development of advanced instrumentation for operational oceanography G. Zappalà CNR IAMC, Messina, Italy

Abstract Operational oceanography is an applied science requiring new instrumentation to perform cost effective surveys. The use of XBT probes from Voluntary Observing Ships is a good way to obtain temperature profiles, but it is limited by the cost of the probes and of the operators on board. An automatic multiple XBT launcher was developed, to work unattended and be recharged by a crew member, without the need for a technician. Keywords: operational oceanography, marine monitoring, XBT, voluntary observing ships.

1

Introduction

The assessment of environmental conditions requires a series of measurements, with a good spatial and temporal resolution. The high cost and limits of traditional oceanographic surveys stimulate the use of new techniques to obtain a proper coverage. So, traditional moorings and oceanographic cruises are complemented with autonomous devices (e.g. drifters and gliders) and with the use of "ships of opportunity". In the framework of the EU funded "Mediterranean ocean Forecasting System Toward Environmental Prediction" (MFSTEP), several experiences were performed both in the use and in the development of advanced instrumentation. Started in 1999 as a part of the "Mediterranean ocean Forecasting System Pilot Project" (MFSPP), a Voluntary Observing Ship (VOS) program is still collecting temperature XBT profiles along seven Mediterranean transects, designed to study, in each of the sub-basins (the Algero-Provençal, the Tyrrhenian, the south Adriatic, the Ionian and the Levantine), the variability of the main circulation features [1]. Although less expensive than dedicated oceanographic cruises, the WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070821

842 Computational Methods and Experimental Measurements XIII use of commercial ships to launch XBT probes requires however the presence of one or more technicians on board the ship, with the related costs. In order to obtain cheaper operations, an automatic device was designed, to launch unattended up to eight probes, easy enough to be recharged by a member of the crew.

2

Materials and methods

An XBT probe has the shape of a small missile, contained in a plastic canister with an overall length of 36 centimetres and external diameter of 7 centimetres; its weight is about 1200 grams. The temperature sensor is a precision thermistor enclosed in the lead nose; the electrical wire to connect it to the measuring system is contained in two spools, one in the plastic body of the probe and the other one in the canister, that also hosts on its back the electrical connections towards the data acquisition system. Traditional use of XBT (expendable bathy-thermograph) probes implies the presence of a technician on board the ship to manually perform the "launch". The plastic canister is inserted in a hand-held "gun"; pressing the trigger the probe is released; while the probe falls into water, the wire inside it starts unreeling. The wire unreels also from the spool in the canister, compensating the movement of the ship and allowing the probe to freefall from the sea surface down to several hundred meters depth. A data acquisition board connected to a computer controls the measurement; when the wire breaks, the profile is completed and the system can be prepared for another launch. The nominal accuracy of the probe is 0,1°C. The depth is estimated as a function of time, using a formula of the kind Z(t)=At-Bt2. Acquired data can be transmitted using satellite or GSM-GPRS modems.

3

The automatic multiple launcher

The new system is an integrated set of mechanical and electronic hardware and software programs offering the maximum of flexibility. 3.1 The mechanical hardware Heart of the mechanical hardware is the launch tube, built in AISI 316 steel, in which the probe is fitted with its envelope. An upper cap, holding the electrical connections of the probe, closes the launch tube; opening the lower door the probe is released and falls into seawater. Two pneumatic cylinders control the door: a small one keeps it closed acting as a safety lock also in case of pressure loss, a bigger one moves it. In the actual MFS-VOS design, the system assembles on a frame eight launch tubes with their pneumatic actuators (fig. 1). A small compressor supplies compressed air to the system, keeping a reserve of 5 litres at 6 bar; a filtered pressure regulator is mounted to prevent damages to the pneumatic system.



Figure 1:

843

A close-up view of two launch cylinders with the pneumatic actuators.

Two versions of the launcher were built: the first to work hanged outboard the ship, the second to stay inboard, on a deck. Fig. 2 (left) shows the "outboard" version, and (right) the inboard one, mounting a funnel with a pipe to drive probes outboard.



Figure 2:

Figure 3:

The two versions of the multiple launcher.

The watertight boxes containing the electro-pneumatic valves (left) and the control computer (right).

Two watertight boxes host respectively the electro-pneumatic valves feeding the cylinders and the computerized control system (fig. 3). 3.2 The control computer hardware and software The control computer hardware and software are enhanced versions of those formerly designed to be used in environmental monitoring networks, described by Zappalà [2]. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


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3.2.1 The electronic hardware All operations are coordinated by an industrial grade Pentium computer, based on IEEE 696 compliant boards, interfaced with GPS, data acquisition and communication devices (fig. 4). Both analog and digital interfaces are available, to collect data coming from the most various devices (passive and active expendable probes, meteo sensors...). GSM -GPRS

SATELLITE

GPS

MODEM(S)

MODEM

RECEIVER OPERATOR CONSOLE

CONTROL ELECTRONICS AND DATA ACQUISITION SYSTEM

LAPTOP PC

POWER ACTUATORS

MULTIPLE LAUNCHER

Figure 4:

ANALOG AND DIGITAL INPUTS FROM PROBES AND SHIP SENSORS

Block diagram of the control computer and of the data acquisition and transmission system.

Remote communications are performed using a GPRS modem with an embedded TCP-IP stack; a serial port is available to connect other communication devices (e.g. satellite modems), but also other communications systems could be used. A balanced source circuit (fig. 5) was designed to interface standard passive temperature probes with 12 or 16 bit Analog to Digital Converters (ADC). Two equal currents are injected in both the wires coming from the probe and the potentials VA and VB are measured; the circuit is closed by the sea water to which the circuit is "grounded" through the ship's hull; the switch SW1 (a relay contact) allows to test the continuity of the probe circuit before the launch. The use of sea water as a third wire is necessary because of the high variability in the wire resistance also in probes belonging to the same production lot. Applying Ohm's law, we obtain the voltage across the wires, the thermistor and the fictitious resistor constituted by the sea water multiplying their resistance by the flowing current. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Sw1

I1

I2

Rballast

Rballast

A

x1 x2

B

Rwire1

Vw1

A

Vth

Vw2

Rwire2

to ADC B

x1

Vsea Rthermistor Rsea

Figure 5:

The schematics of the probe interface board.

Assuming we can write:

I 1 = I2 = I Rwire1 = Rwire2 = Rwire Vsea = Rsea x 2 I Vw1 = Rwire x I Vw2 = Rwire x I Vw1 = Vw2 = Vw Vth = Rth x I VA = Vsea + Vw VB = Vsea + Vth + Vw VB = VA + Vth

So, VA is the sum of the voltage across Rwire1 and Rsea (the resistance of sea water), VB is the sum of the voltage across Rwire2, Rthermistor and Rsea. The voltage across the thermistor is obtained subtracting VA from VB. To avoid any perturbation to the measurement, a multiple stage amplifier was used to adapt the signal coming from the probe to the need of the ADC; the circuit was built using high quality precision instrumentation amplifier ICs. The first stage uses unity gain configuration, having high input and low output impedance; the second stage is a differential amplifier with gain = 2; the third stage is a low output impedance low-pass filter with gain= 1 and ft = 40 Hz. Ten times a second, the mean of 16 (VB-VA) measurements is calculated; the resistance of the thermistor is obtained using Ohm's law, or, better, using the regression coefficients obtained after calibration of the circuit against a set of standard high precision resistors. The measured temperature is finally obtained using the standard formula by Steinhart and Hart [3]: T = -273.15 + {1 / [A + (B x ln R) + C x (ln R)3]} where: A = 1.29502 × 10-3, B = 2.34546 × 10-4, and C = 9.9434 × 10-8. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


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The constants were determined empirically from laboratory tests of XBT thermistors [4]. 3.2.2 The software A local and a remote control programs were developed, able to manage 96 launch events. Launch options are: • northern than a defined latitude • southern than a defined latitude • eastern than a defined longitude • western than a defined longitude • far from a previous launch • at GPS time • at PC time Collected data are locally stored and can be transmitted as e-mails. The local control program was written in Microsoft Compiled Basic v. 7.1 and runs in Datalight DOS environment. Assembly language routines are used to manage the Analog to Digital Converter. This program, executed on the launcher control computer, is able to control all the launcher functions, i.e. real time and position acquisition, comparison against set points-times, launch, data acquisition and transmission, ancillary functions. Every hour, a “sequence manager” starts a macro-command sequence, that can be different for each time and is remotely reprogrammable; new releases of the software and of the sequences are uploadable to the station without suspending its normal activity. The macro-commands enable to manage the data acquisition and transmission, the mission programming, the station hardware and the measuring instruments. The entire system can be connected to a computer (local laptop or remote desktop), in order to use a remote control program (fig. 6), written in Microsoft Visual Basic, running in Windows environment. This program enables to set up all the launcher functionalities, prepare launch event sequences, transfer files to and from the launcher, and, if needed, to take control of all the launcher operations, including the time-position acquisition and comparison.

4

Tests

The automatic multiple launcher was tested in laboratory and during two short cruises. The in situ tests were performed in March 2006 using the Italian Hydrographic Institute vessel Magnaghi and in June 2006 with the ENEA boat S. Teresa, in the Ligurian Sea – North western Mediterranean. During these tests, data was recorded with the multiple launcher to verify the efficiency of the whole system (hardware and software), and, for comparison, XBT probes were launched with a standard Sippican system (hand launcher LM 3A, MK21 readout card). Figure 7 shows some profiles obtained by the multiple launcher. Temperature values measured with reference systems demonstrate the high quality of the multiple launcher profiles. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Figure 6: °C

13

15

A screenshot of the remote control software. 17

19

21

23

25

-10

-60

-110

-160

-210

-260

-310

Figure 7:

Profiles obtained with the multiple launcher.


27


5

849

Discussion and conclusions

Figure 8 shows the multiple launcher installed on the Magnaghi ship, assuring safety to operators and undisturbed operations. The use of a multiple launcher improves the cost effectiveness of an operational observing system. Working in the Mediterranean Sea with commercial ships, it is necessary to launch a probe every 20-30 minutes to obtain a good spatial resolution of the mesoscale phenomena on sections several hundreds of miles long, lasting up to 24-36 hours. So, using the manual launcher, at least two technicians are needed, with a cost up to 2000 USD for each travel; using the automatic launcher, it is possible to avoid this cost, or at least halve it, engaging one person only. The small number of electric and mechanic components reduces the risks of malfunctions; the remote control capability enhances the flexibility of the instrument and allows real time operations. The system proved its reliability and is open to further developments and improvements to be used with other expendable probes (XCTDs, T-FLAPs...).

Figure 8:

The multiple launcher installed on the deck of the Magnaghi ship.

Acknowledgements The activity was funded in the frame of the MFSTEP project, 5th EU FP. The Author acknowledges the support provided by N. Pinardi and G. Coppini. The work of technicians during these years has provided the necessary information for the design. The Author thanks G. Manzella and F. Reseghetti for the useful suggestions and A. Baldi and F. Conte from ENEA, for their technical support.



References [1]

[2]

[3] [4]

Pinardi, N., Allen, I., Demirov, E., DeMey, P., Korres, G., Lascaratos, A., LeTraon, P.Y. Maillard, C., Manzella, G., Tziavos, C., The Mediterranean ocean Forecasting System: first phase of implementation (1998 - 2001). Ann. Geophys., 21, pp. 3-20, 2003. Zappalà, G.: A software for environment monitoring networks, Proc. of the 10th Int. Conf. On the Development and Application of Computer Techniques to Environmental Studies, eds. G. Latini, G. Passerini & C.A. Brebbia, WIT Press: Southampton, pp. 3-12, 2004. Steinhart, J. S., Hart, S. R., Calibration curves for thermistors, Deep-Sea Research 15, pp. 497-503, 1968. Georgi, D. T., Dean, J. P., Chase, J. A., Temperature calibration of expendable bathy-thermographs, Ocean Engineering, 7, pp. 491-499, 1980.



851

In-situ measurement of formwork pressures generated by Self-Compacting Concrete M. M. Giammatteo, A. Gregori & G. Totani Department of Civil Engineering, L’Aquila University, Italy

Abstract Self-Compacting Concretes (SCC) are a special category of concretes and directions for designing formwork relative to ordinary concretes do not apply to them. In this paper an experimental study on formwork pressure exerted by SCC is reported. The behaviour of the SCC was investigated in instrumented formworks provided for two different types of pressure sensors: common diaphragm pressure transducers and dilatometer cells. In situ measurements of formwork pressure were carried out on walls and concrete columns 6m and 9m high, 0.30m thick and 2.50 or 0.50m wide respectively. Successfully, both types of sensor were in agreement describing the continuous time variation of the pressures. Keywords: in-situ measurements, self-compacting concrete, formwork pressures, dilatometer cells.

1

Introduction

The horizontal pressure exerted by ordinary plastic concretes is generally calculated as a function of many parameters: final height and casting rate, vibration method, temperature of the environment and of the concrete, nominal dimension and shape of the aggregate, consistency of fresh concrete, setting time, typology of the additives used, formwork’s shape. Currently, design purposes for ordinary concrete are various, but the resulting pressure diagrams along the casting height are relatively unvarying (CIRIA [1], ACI [2]). Unfortunately, recommendations for designing formworks relative to ordinary concretes do not apply to Self-Compacting Concrete (SCC) which represents a relatively new technology developed in Japan since the 1980s (Ozawa et al. [3]). Driven by their own weight, SCCs quickly flow in the formwork and reach an optimum compaction degree without the need for vibration, also in case of WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070831

852 Computational Methods and Experimental Measurements XIII heavily reinforced structures and/or of a narrow cross section. Saving in manpower is provided and noise pollution is reduced, improving working condition. Apart from the large use in the precast industry, a certain number of technical issues have slowed down its use in cast-in-place applications, in particular due to the lack of knowledge on the lateral pressure that such concrete can exert on formwork systems. In fact, directions for designing formworks relative to ordinary concretes do not apply to SCC, for which the design recommendation to design formwork for full hydrostatic pressure (fluid density of 2400kg/m3) drastically increases the cost of constructions made of SCC [4]. There are indications, however, confirming the possibility of saving considerable amounts of money when constructing formwork for SCC, because data of pressures measured both in the laboratory (Assaad and Khayat [5, 6]) and in the field (Skarendahl [7], Brameshuber and Uebachs [8], Billberg [9]) often resulted to be lower than hydrostatic. When measured SCC formwork pressure were found to be in agreement with the hydrostatic hypothesis, this behaviour from the concrete was mainly attributed to the effect of relatively high casting rates that, however, not always were higher than in other case with SCC behaving differently. The reason for so wide a range of literature results is still not clear, therefore extensive campaigns of experimental measurements are required for the validation of new predicting models. Various techniques have already been adopted, so far, to enquire the concrete behaviour into formwork. Pressure gauges and other alternative devices are often used for investigation curried out in form of experimental size tests and on reduced scale column of concrete cast and monitored into a laboratory. These approaches usually require delicate equipments conceptually more rigorous than other techniques that are more robust, maybe less sensitive but more appropriate for the field. In this second situation, monitoring the stress in the formwork anchors can already provide an indirect measure of the pressure exerted by the concrete (Brameshuber and Uebachs [8]). Although diaphragm pressure transducers have also been used in both the cases of laboratory and field measurements, the use of such kinds of devices has been criticized by Amziane et al. [10], Amziane [11] and Andriamantsilavo and Amziane [12]. They observed that, under the pressure of the fluid (liquid or gas), the membrane of a pressure transducer deforms and produces a variation in both the sensor wire resistance and the output voltage. This, however, makes unsure whether the measurement method suits hardening materials, like concrete, in the setting process. In fact, when the material is setting, the deformation of the transducers diaphragm due to the flow of freshly mixed materials, indeed, is not reversible, although no pressure is applied anymore. Consequently, a displacement controlled procedure can represent the most reliable technique for monitoring the concrete formwork pressures that, eventually, are influenced by the material setting. These authors, therefore, proposed an alternative method based on the balance of zero measurement procedure. Their apparatus, however, results to be as accurate as delicate, making the equipment not suitable for in situ measurement of formwork pressure. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


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On the other hand, the same working principle is adopted by dilatometer cells that are currently used, in the geotechnical field, to conduct measures of lateral pressures in the ground (Flat Dilatometer Tests, DMT) (Marchetti [13], Marchetti et al. [14]). The apparatus, as described ahead, is so robust that is usually pushed for many meters deep into ground, being also insensitive to difficulties related to electric power supply (no needed), unstable signals, humidity, dusty etc. This makes the use of dilatometer cells an interesting option for appropriate descriptions of how the lateral pressure on formwork increases during the concreting and decreases with the material setting. In this paper, data of formwork pressure recorded by using this special technique were compared with those recorded by diaphragm pressure transducers, showing that dilatometer cells represent a reliable and easier alternative to more expensive way to measure formwork pressure.

2

Experimental investigation

The behaviour of various commercial SCC mixtures in the formworks was investigated performing in situ measurements of formwork pressure. The cast of real concrete elements took place in formworks designed for casting heights of 6 and 9m respectively. Effect on pressure due to different structural shapes was investigate comparing the case of walls and column measuring 2.50m and 0.50m in wideness and 0.30m in thickness. Formworks were instrumented disposing several diaphragm pressure transducers along their height: closer each other at the bottom part of the structure and gradually more spaced going up along the middle span of the main vertical facade. A picture of the wall formwork system is given in Fig. 1 together with the precise position of the sensors. In addition to pressure transducers, also alternative devices (named dilatometer cells) were applied, for the first time, for measuring formwork pressure in the construction field. Their description is given in the next section. Four of these cells were attached to the instrumented wall formwork system, all aligned at the horizontal level of 1.10m from the bottom of the cast. This provided redundancy in the acquisitions, and the mean of four measurements was assumed as value of the concrete formwork pressure at that level. This value was than compared with the readings from the pressure transducers mounted immediately above and below the four cells alignment. Pore water pressure in concrete mass was also detected, and a specific measuring devise was properly designed for this task. Two of this pore water pressure measuring devices were mounted, one for each side of the formwork, at the level of 1.10m from the structure bottom. The mean of the two readings was assumed as value of pore water pressure at that depth. Lateral pressures exerted by the tested SCCs were monitored during the filling of the formwork and for several hours after the end of the casting. Effect of various casting rate on concrete formwork pressure was investigate by repeating tests at different speeds. The adopted filling rates ranged from 3m/h to 12m/h. Due to the limited amount of concrete required for a test, and according to the mixing and pumping truck limitations, low values of the casting rate were respected dividing the cast in layers and suspending the discharge of WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

854 Computational Methods and Experimental Measurements XIII the concrete between a layer and the next. Height of these layers was fixed in 1m. Measurements of formwork pressure were carried out right after the discharge of any layer and repeated right before the cast of the next.

775

75

300

75

850

75

700

75

625

75

550

75

475

Figure 1:

300 250 200 150 100 50

25

25 50 50 50 50 50 50 50

600

400 350

25

Wall casting and sensors positions.

2.1 The dilatometer cell In the geotechnical field, dilatometer cells are known for being used in Flat Dilatometer Tests (DMT). They provide measures of pressure according to a displacement control procedure (Marchetti [13], Marchetti et al. [14] and Totani et al. [15]). Bigger, more robust and easier to be used than diaphragm pressure transducers, this type of sensors (Fig. 2a) consist in a 60mm diameter circular steel membrane (0.2mm in thickness) that, as result of an applied internal gas pressure, expands into the material which pressure has to be measured. The cells were attached to the formwork with its membrane being flush with the inside formwork surface. The in-situ test layout is given in Fig. 2b: the cell is connected to a control unit by a pneumatic-electrical tube transmitting the gas pressure and the electrical continuity. A gas tank, connected to the control unit by a pneumatic cable, supplies the gas pressure required to expand the membrane. The working principle of the cell is illustrated in Fig. 3a. During the pressure measurement, the dilatometer cell works as an electric switch (on/off). First the external lateral pressure keeps the membrane touching the cell body and the control unit emits a sound (circuit close). Now the operator starts increasing the internal pressure and, for some time, the membrane does not move and remains in contact with its WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


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metal support (signal on). When the internal pressure counterbalance the external concrete pressure, the membrane starts moving and loses contact with its support (signal off). This causes the interruption of the signal and sound stops, so prompting the operator to read on the manometer (Fig. 3b) the “liftoff” lateral pressure.

steel membrane formwork system

Dilatometer cells (back side)

pneumatic cable cell body

control unit

gas tank

pneumatic cable

(a) Figure 2:

(a) Detail of a dilatometer pressure cell; (b) general layout of the in situ dilatometer test method.

(a) Figure 3:

(b)

(b)

(a) Dilatometer cell working principle; (b) control unit: manometers for pressure reading.


856 Computational Methods and Experimental Measurements XIII Due to the balance of zero pressure measurement method (null method), the dilatometer cell readings are highly accurate (1/20bar = 5kPa). Moreover, the entire equipment can measure values of pressure as high as 10MPa and over. 2.2 Self-Compacting Concrete mixtures Two SCC mixtures were tested. The basic combinations of ingredients, called Mixture A and B respectively, are indicated in table 1. Table 1: Materials Sand (0 – 6mm) Coarse aggregate (7 – 15mm) Portland Cement (32,5) Filler Superplasticizer Viscosity modifying agent Water/Cement Water/Binder

SCC mixtures dosages. kg kg kg kg lt lt

Mixture A 974 650 355 170 8.88 1.00 0.52 0.35

Mixture B 916 721 400 146 10.00 1.50 0.41 0.30

In order to maintain the desired condition of fluidity and prevent slump loss, use of different chemical admixtures where required (superplasticizers and set retarding). Several tests were first performed on Mixture A, and later on mixture B, this last designed to have similar fresh property than in case A with a reduced water/binder ratio from 0.35 to 0.30. Compared to mixture A, thixotropic behaviour of mixture B should be so increased and, consequently, lateral pressure exerted on formwork potentially reduced. An example of the SCC fluid behaviour is given in Fig. 4, were the concrete slump flow is being measured after the Abrams cone test.

Figure 4:

Abrams cone test: SCC slump flow.



3

857


Tests were performed on different days. Collected data of formwork pressure are exposed by graphs indicating the evolution of the lateral pressure during the filling of the formwork and after the casting end. Fig. 5 shows data of formwork pressures collected during the cast of a 6m high wall performed at the casting rate of 12m/h. Mixture A was used with a slump flow diameter of 620mm. SCC was found to behave in a perfect hydraulic way, maybe due to the combined effect of the set-retarding agent incorporated in the mix and the relatively high casting rate adopted. Broken lines in Fig. 5 highlight that also when DMT measures (star dots in the graph) did not overlap the pressure profile indicated by the transducers, data kept maintaining the alignment with the pressure transducers readings made immediately over and below the dilatometer cells level. This indicates that DMT technique can result in a reduced data scattering compared with use of common diaphragm pressure transducers.

C asting height [cm ]

700

DMT

600

1m

500

3m

2m

4m

400

5m

300

6m Hydraulic

200 100 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

Pressure [bar]

Figure 5:

Formwork pressure profiles during the cast of a 6m high wall adopting Mixture A. Casting rate 12m/h.

Formwork pressures were recorded for several minutes after the casting end as well. Pressure profile was found to reduce quite fast with time (Fig. 6). This suggests that adoption of lower casting rate values can help reducing the initial formwork pressure. In both Figs. 5 and 6, a perfect agreement can be seen between readings of formwork pressure collected by the diaphragm pressure transducers and the dilatometer cells respectively. DMT technique confirmed good performances and consistency also in different cases of in situ measurements of SCC formwork pressures. Also data collected on column shape structures instead of wall structures showed that readings of pressure curried out by dilatometer cells are fast and reliable and always in good agreement with those performed by the diaphragm pressure transducers. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

858 Computational Methods and Experimental Measurements XIII A last experiment was carried out monitoring the cast of a 9m high wall built at the casting rate of 4m/h. Mixture B was used adopting a slump flow diameter of 680mm. According to references Ferron et al. [16] and Gregori et al. [17], reduction in water and increase in superplasticizer dosages enhanced the thixotropyc behaviour of concrete. As expected, this provided some benefit on formwork pressures that, in fact, were found to deviate from the hydrostatic hypothesis already several meters before the casting end. Figure 7 shows that Mixture B maintained a perfect fluid behavior until the concrete level into formwork did not exceed 5m in height. hydrostatic pressures

C asting height [cm ]

700 600

pressures profile at the casting end 48 min after

500

80 min after 105 min after

400

DMT

300 200 100 0 0

0.2

0.4

0.6

0.8

1

1.4

1.2

Pressure [bar]

Figure 6:

Evolution of the formwork pressure profile with time after the end of the cast.

1000

Mixture B Formwork Pressure Profiles During the Cast of a 9 m High Wall (slump flow =680mm; casting rate v=4m/h)

900

DMT

height of the cast (cm)

800

1m 2m 3m 5m 6m 7m

700 600 500

8m 9m hydraulic pressure

400 300

pore water pressure

Dilatometer cells level

200 100 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

pressure (bar)

Figure 7:

Formwork pressures growth during the cast of a 9 m high wall using Mixture B. Adopted casting rate: 4 m/h.



859

Keep casting above this level, formwork pressures stopped to grow at the bottom of the structure where, at the end of the casting, only 50% of the hydraulic pressure was measured. Data in the graph highlight that formwork pressure measured by diaphragm pressure transducers and dilatometer cells were in good agreement confirming this particular concrete behaviour. Pore water pressures in the concrete mass were also measured. At any stage of the concrete casting, response of the measuring apparatus to the variations of pore water pressure resulted instantaneous. The mean of two readings, taken at 1m from the bottom on both sides of the formwork, is indicated in Fig. 7 with blue dot signs. It can be noted that, according with laboratory results concerning simple cement pastes (Andriamanantsilavo and Amziane [12]), also in situ measurements of pore water pressures in concrete were found to be identical to the total lateral stress on formworks.

4

Conclusions

In this paper, results from an experimental campaign of study carried out in the construction field were reported. Lateral pressures exerted by different SCC mixtures were monitored in situ by use of instrumented formworks. Two different methods for measuring concrete formwork pressure were compared. Pore water pressure measurements in concrete mass were also curried out: specific devices designed and built on purpose were applied. Although intense work still have to be done in order to provide complete models for predicting SCC formwork pressures, the following main conclusions can be drawn: Lateral pressures exerted by fresh concrete into formwork can be monitored according to several approaches. Use of diaphragm pressure transducers can provide accurate description of the total lateral pressures profile along the formwork height. Formwork pressures decay due to the concrete setting is also detectable by the transducers. Electro-working devices, however, are usually delicate and expensive, requiring special cares to be managed and used in situ. Dusty, humidity and mechanical shocks are all undesired factors representing hindrances to a their direct application in the construction field. In alternative to diaphragm pressure transducers, DMT system represents a robust, cheap and sensitive technique able to provide high reliable in situ measures of concrete formwork pressures. In addition, also building cost of pore water measuring devices is relatively low. They have been used already for a long time in the geotechnical field, and here it has been proved that they can provide correct measures of the concrete formwork pressure in situ.

References [1] CIRIA (Civil Industries Research and Information Association); The pressure of concrete on formwork, Research Report No. 1, London, 1965. [2] ACI Committee 347-01. Guide to Formwork for Concrete, American Concrete Institute, Farmington Hills, MI, 32 pp., 2001. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

860 Computational Methods and Experimental Measurements XIII [3] Ozawa, K., Naekawa, K., Kunishima, M. & Okamura, H., Development of high-performance concrete based on the durability design of concrete structures, Proc. 2nd East Asia and Pacific Conference on Structural Engineering and Construction (EASEC-2), pp. 445-450, 1989. [4] ACI Committee 347R-03. Guide to Formwork for Concrete, American Concrete Institute, Farmington Hills, MI, 32 pp., 2004. [5] Assaad, J. & Khayat, K.H., Effect of Coarse Aggregate Characteristics on Lateral pressure Exerted by Self-Consolidating Concrete, ACI Material Journal, V. 102, No. 3, pp. 145-153, 2005. [6] Assaad, J. & Khayat, K.H., Formwork Pressure of Self-Consolidating Concrete Made with Various Binder Types and Contents, ACI Material Journal, V. 102, No. 4, pp. 215-223, 2005. [7] Skarendahl, A., Self-Compacting Concrete for Improved Productivity, Working Environment, and Performance, IREX-Meeting, Paris, 12 pp, 1999. [8] Brameshuber, W. & Uebachs, S., Investigations on the Formwork Pressure Using Self-Compacting Concrete, 3rd International Symposium on SelfCompacting Concrete, Reykjavik, Iceland, pp. 281-287, 2003. [9] Billberg, P., Form Pressure Generated by Self-Compacting Concrete, 3rd International Symposium on Self-Compacting Concrete, 3rd International Symposium on Self-Compacting Concrete, Reykjavik, Iceland, pp. 271-280, 2003. [10] Amziane, S., Andriamanantsilavo, N.R. & Bandeau, P., An Experimental Study on the Pressure of Cement Based Materials Against Formwork, 15th ASCE Engineering Mechanics Conference, Columbia University, New York, NY, 8 pp., 2002. [11] Amziane, S., Setting Time Determination of Cementitious Materials Based on Measurements of the Hydraulic Pressure Variations, Cement and Concrete Research, 2005. [12] Andriamanantsilavo, N.R. & Amziane, S., Maturation of fresh Cement Paste Within 1 to 10m Large Formwork, Cement and Concrete Research 34, pp. 2141-2152, 2004. [13] Marchetti, S., In Situ Tests by Flat Dilatometer, ASCE Journal GED, Vol. 106, No. GT3, pp. 299-321, 1980. [14] Marchetti, S., Monaco, P., Totani, G. & Calabrese, M., The flat dilatometer test (DMT) in soil investigations, Report by the ISSMGE Committee TC16, Proc. of the 2nd Int. Conf. on the Flat Dilatometer, Washington, D.C., pp. 848, 2006. [15] Totani, G., Calabrese, M. & Monaco, P., Shaft resistance of piles driven in clay: field study of an instrumented full scale pile. Workshop in Pile foundations experimental investigations, analysis and design, Napoli, Italy, pp. 199-253, 1994. [16] Ferron, R., Gregori, A., Sun, Z. & Shah, S. P., Rheological method to evacuate structural build-up in SCC cement pastes, ACI Materials Journal, publication in press. [17] Gregori, A., Ferron, R., Sun, Z. & Shah, S. P., Experimental simulation of SCC formwork pressure, ACI Materials Journal, publication in progress. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


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Evaluation of phenolic resins from one-pot microwave synthesis A. Britten, M. M. MacIntyre & A. Miadonye Department of Chemistry, Cape Breton University, Sydney, NS, Canada

Abstract The characteristics of oxalic acid catalyzed phenolic resins, synthesized using the conventional heating method and by a one-pot microwave process, were evaluated and compared. Phenolic resins were produced in an autoclave with reflux times of 3, 5 and 7 hours, and by one-pot microwave synthesis at different power levels (176.364, 315.092, 453.820, 592.548W) for reaction times of 15–90 minutes. UV-vis spectroscopy and FT-IR analyses showed similarities in the characteristics of the resins synthesized by both methods. Concentration of residual phenol in the products decreased with reaction time. For the conventional method the decrease in phenol after an initial 3hrs of reaction were 4.3% (in 5hrs) and 22.3% (7hrs), while in microwave synthesis at a power level of 176.364W and an initial reaction time of 30mins the phenol decreased by 33.05% (in 60mins) and 91.61% (in 90mins). At a power level of 315.092W and an reaction time of 15mins it decreased by 36.3% (in 20mins), 16.5% (in 25mins) and 10.8% (in 30mins). For resins from the conventional method, melting points increased with increasing reaction time 83oC, 97oC and 109oC for respectively 3hrs, 5hrs and 7hrs, and between 95oC and 103oC for resins from the microwave method at various power levels and reaction times. The FT-IR spectra analyses for the products showed strong similarity in characteristics, while the melting point data supported the GC-MS results that for the conventional method mostly the degree of polymerization depends on reaction time. The substitution of phenol with cresol results in resins with improved hardness and melting points, but these differences were not obvious in the spectral characteristics. It was also observed that for the conventional heating method the melting points of the CF pre-polymers produced were higher than those of the PF prepolymers (119ºC and 127ºC for 3hr and 5hr CF). One-pot microwave synthesis was found to reduce the polymerization time significantly and produced resins with shorter gel times. Closest similarities in characteristics were found between resins heated in an autoclave for 7hrs and the resins made by one-pot microwave synthesis at reduced reaction times. Keywords: phenolic resins, microwave synthesis, GC-MS, FT-IR spectroscopy, prepolymers, condensation polymerization. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070841


1

Introduction

Phenolic resins, like other thermosetting resins, are very important because their use as engineering materials is increasing rapidly due to their good mechanical properties, high hardness, and excellent stability at high temperatures [1]. Traditionally, phenolic resins are synthesized by step-growth polymerization in an autoclave from phenol or substituted phenols with formaldehyde. The type of resin synthesized depends on the catalyst and the mole ratio of phenol to formaldehyde. The catalyst can be acid, base or neutral divalent metal such as Co, Mg, Zn, Cu or Ni. Figure 1 displays the reaction for the acid catalyzed formation of phenol formaldehyde (PF) which involves the hydrolysis of formaldehyde to methyl glycol. Phenolic resins prepared under acidic conditions are known as novolak, while resins prepared under basic conditions are called resoles [2–4]. Microwave chemistry applies microwave irradiation to chemical synthesis. Recent developments in microwave chemistry have provided scientists with a method of organic synthesis that is more efficient than traditional heating. Microwaves are a form of electromagnetic energy with wavelengths between 300-300,000 megahertz (MHz). Molecular rotation is a property that is affected within this region of electromagnetic spectrum however, molecular structure is not. Samples can be heated using microwave irradiation due to the wave-material interactions. These interactions transform electromagnetic energy from the microwave into heat through dipole rotation or ionic conduction. An important advantage that microwave-assisted organic synthesis has over conventional heating methods is that samples are heated rapidly in a homogeneous manner. Homogeneous heating creates higher yields and faster reaction times. This type of heating occurs in microwave synthesis because heat is generated within the reaction medium rather than being transferred from an external source [5–7]. OH

OH + 2n H2C O

n

H3O+

H2C

OH

CH2

(CH2)6N4

H2C

CH2

HMTL

n CH2

Phenol

Formaldehyde

Figure 1:

2

A-stage resin

n

C-stage resin

Typical reactions for formation of phenol-formaldehyde [2].

Experimental

2.1 Materials The reagents used for the synthesis were formaldehyde solution (36.5 – 38% v/v) from VWR International, loose phenol crystals, oxalic acid from Fisher Scientific Company and liquid m-cresol (99% purity) from Alfa Aesar. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


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2.2 Resin synthesis Acid catalyzed phenol-formaldehyde (PF) and cresol formaldehyde (CF) prepolymers were synthesized using molar ratios of phenol or cresol to formaldehyde of 1: 0.8. The PF prepolymers were produced in autoclave with constant stirring and reaction times of 3, 5 and 7 hours. They were also produced by one-pot microwave synthesis at different power levels (PL): 176.364W (PL1), 315.092W (PL3), 453.82W (PL5), and 592.548W (PL7), for reaction times ranging from 15-90 minutes. The CF prepolymers were synthesized in autoclave with constant stirring for reflux times of 3 and 5 hours. 2.3 Resins analysis The Fourier transformed infrared spectroscopy (FT-IR) of the resins was obtained with a Nicolet spectroscopy, model 6700. The analysis conditions were 500 – 4000 cm-1 spectral range, 10 scans, and a resolution of 4cm-1. The samples were ground and vacuum pressed on KBr pellets of approximately 1mm thick. The determination of residual phenol in the resins were made with uv-vis spectroscopy by phenol absorbance method. Figure 2 shows a plot of uv absorbance determined for the molar concentration of phenol in aqueous solution. The plot used as reference source for the evaluation of residual phenol in the resins. 2

Absorbance

1.6

1.2

0.8

0.4

0 0

0.2

0.4

0.6

0.8

Concentration (mol/L)

Figure 2:

UV Absorbance versus concentration of phenol in aqueous solution.


864 Computational Methods and Experimental Measurements XIII Gas Chromatography Mass Spectrometry (GC-MS) using an ICB-WAX 30m x 0.25mm ID x 0.25µm film column (J & K Scientific, Milton, Ont., Canada) was used to monitor phenol residues. GC conditions were: initial oven temperature 80oC, hold 1min, and then 20oC/min to 250oC; injector temperature 225oC; MS interface temperature 280oC. Phenol eluted at 6.6min. Chromatographic Total Ion Current (TIC) peak height and peak area were used for quantification. 250

Temperature (ºC)

200

150

100

50

Power level 1

Power Level 5

Power Level 3

Power Level 7

0 0

500

1000

1500

2000

Time (s)

Figure 3:

Relationship between time and temperature for 30min. microwave reactions of PF at different power levels.

3 Discussion of results The temperature was recorded at regular time intervals throughout the microwave synthesis processes. Figure 3 shows the relationship between temperature and time for PF synthesized in 30 minutes at different power levels. For each power level the temperature rapidly increased within the first 60 seconds to about 100oC. After this period of time the temperature of the reaction mixture increased steadily at different rates depending on the power level in use, except for PL 1 which remained at 100oC throughout the duration of the reaction. The pattern of increase in temperature of the reaction was similar for the other WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


865

three power levels, peaking in 20 minutes at different temperatures. The temperature difference observed for different power levels has also been reported by Liao et al [8], who showed that the higher the power, the higher the temperature and different equilibrium temperature is reached for each power level. Analyses from UV-Vis and GC-MS showed that the concentration of unreacted phenol in the products of PF prepolymers synthesized from one-pot microwave synthesis and conventional heating decreased with reaction time. Typical GC-MS chromatograms and spectra for phenolic resins using the conventional heating method for 7 hours and with microwave method for 30 minutes are compared in Figure 4. The decrease in phenol for the conventional method with an initial reaction time of 3hrs was by 4.3% (5hrs) and 22.3% (7hrs) as shown in Figure 5, while the decrease in phenol for the microwave synthesis of PF prepolymers at 176.364W after an initial reaction time of 30mins was by 33.05% (in 60mins) and 91.61% (in 90mins). At 315.092W after an initial reaction time of 15mins it decreased by 36.3% (in 20mins), 16.5% (in 25mins) and 10.8% (in 30mins). These trends are shown in Figure 6, and could account for the temperature profiles observed in Figure 3. The UV-Vis results are shown in Table 1. The FT-IR spectra analyses of the m-cresol modified PF resins showed strong similarity in characteristics between the resins. Figure 7 demonstrates the spectra of three different resins synthesized by microwave method at reaction times of 30 minutes. The modified resins were rusty brown colour and highly viscous, which formed very brittle solids. Conventional heating (7hrs)

Figure 4:

Microwave at PL3 (30min)

GC-MS chromatograms and spectra of residual phenol with two different synthesis techniques.



Method, reaction time, and absorbance at 297nm of residual phenol obtained from UV-Vis analyses.

Residual Microwave Energy Reaction Time Uv-vis Phenol absorbance (molar) 173.364W

315.092W

Conventional

30mins

1.139

0.122

60mins

1.117

0.115

15mins

1.222

0.147

20mins

1.205

0.140

3hrs

1.131

0.118

5hrs

1.118

0.115

7hrs

0.974

0.089

1.45

Conventional heating Power Level 1

Power Level 3

Absorbance

1.35

1.25

1.15

1.05

0.95 1

1.5

2

2.5

3

Log10 (time)

Figure 5:

Comparison of absorbance of residual phenol with time at different process conditions.



867

1x107 1.E+07

176.364W 315.092W

Abundance

6 1.E+06 1x10

5

1x10 1.E+05

4

1x10 1.E+04

10

40

70

100

Reaction Time (min)

Figure 6:

Relative abundance of phenol with reaction time for two power levels.

80:20 phenol: cresol



Figure 7:

FT-IR spectra for cresol modified PF resins synthesized by Microwave irradiation method at 30 minutes reaction time.


868 Computational Methods and Experimental Measurements XIII The melting points of the PF prepolymers produced by the conventional heating method increased with increasing reaction time (83ºC, 97ºC and 109ºC respectively for 3hrs, 5hrs and 7hrs). Melting points varied between 95ºC and 103ºC for the PF resins formed by microwave synthesis at various power levels and reaction time. It was also noted that the melting points for the CF prepolymers produced by the conventional heating method were higher than those for the PF prepolymers produced by the conventional method (119ºC and 127ºC for 3hr and 5hr CF). The melting point data supported the GC-MS results that for conventional method mostly, the degree of polymerization depends on reaction time. Finally, close similarity in characteristics were found between resins heated in autoclave for 7hrs and the resins made by one-pot microwave synthesis at reduced reaction time.

4

Conclusions

Prepolymers of CF and PF, produced by the conventional heating method, have similar characteristics although PF prepolymers seem to be more malleable and have lower melting points. The degree of polymerization of phenolic resins produced by the conventional method depend on the reaction time and the strongest similarities between prepolymers produced in autoclave and by microwave synthesis is found between those produced by long autoclave reaction times and those produced by short one-pot microwave reaction times.

Acknowledgements This work was funded by USRA grant from Natural Sciences and Engineering Research Council of Canada, and the RP grant from Cape Breton University, Canada. The authors wish to acknowledge the contribution of Dr. D. Irwin for the microwave set-up and experiment.

References [1] [2] [3] [4] [5]

Odian, G., Principles of Polymerization. New York: McGraw-Hill, 1970. Carraher, C.E., & Swymour, R.B., Polymer Chemistry: An Introduction. New York Marcel Dekker, Inc., 1992. Tigani, L.W., Pinhas, A.R., & Mark, J.E., Some Attempts at Introducing Flexibility into Phenolic Resins. Polym. Plast. Technol. Eng., 39(4), pp. 711-721, 2000. Poljansek, I. & Krajnc, M., Characterization of Phenol-Formaldehyde Prepolymer Resins by in Line FT-IR Spectroscopy. Acta Chim. Slov., 52, pp. 238-244, 2005. Hayes, B.L., Microwave Synthesis: Chemistry at the Speed of Light. NC: CEM Publishing, 2002



[6] [7] [8]

869

Gyorgy, I., Radiation Chemistry of Hydrocarbons (ed. G. Goldiak), Chapter 2, Elsevier, Amsterdam, 1981. Loupy, A., Solvent-free microwave organic synthesis as an efficient procedure for green chemistry, C.R. Chemie, 7, pp. 103-112, 2004. Liao, L.Q., Liu, L.J., Zhang, C., He, F., & Zhuo, R.X., Heating Characteristics and Polymerization of Caprolactone under Microwave Irradiation. J. Appl. Poly. Sc., 90, pp. 2657-2664, 2003.




871

Modern techniques of measure and control of deformations – an experimental test: Senerchia landslide M. Caprioli & G. Strisciuglio Dipartimento di Vie e Trasporti, Politechnic of Bari, Italy

Abstract Control of movements and deformations is based on the comparison between surveys repeated in different times and in particular on the determination of a certain number of points. The topographical method represents one of the possible solutions in order to reach such an aim. A control network is constituted by reference points (located in stable zones) and control points (located in unstable zones). The movements can be of two kinds: “relative” between the control points (network deformations) and “absolute” (control points respective to the reference network). Its necessary, therefore, to estimate, during the time, the stability of reference points. In this work the stability of reference network – located in Senerchia (Southern Italy) and constituted for the monitoring of a landslide area with static GPS – has been investigated. Both the classical and the Bayesian statistical analysis have been executed on the network in order to identify small movements in comparison with accuracy of the measurements. The application of the Bayes theory on the network has confirmed its stability. Keywords: deformation analysis, statistical inference, Bayesian approach, GPS.

1

Network in deformation analysis

Control of movements and deformations is based on the comparison between surveys repeated in different times and particularly on the determination of the variation between the position of the points at the beginning and at the end of the considered period. The topographical method represents one of the possible ways to reach this aim. It can synthetically be summarized in the following phases: choice of the technique and instrumentation to be used, planning of the network configuration, monumentation of the vertexes, execution of the measurements WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070851

872 Computational Methods and Experimental Measurements XIII (manuals/automatic), data processing, deformation analysis using statistical inference, optimization of geodetic network. Nowadays the development of technology allows a great range of solutions for survey execution (digital levelling, motorized TPS, laser scanners, SAR, GPS with RTK transmission): our survey has been performed with GPS instrumentation. Static GPS is the most adopted method, especially when high precision is requested: the standard precision in relative positioning is 10-6 D or 10-7 D in some experiences, with particular elaborations and satellite configurations.

Figure 1:

GPS acquisition.

In deformation analysis, the optimal network configuration has a strategic role to reach desired accuracies, to define an efficient program of measurements, to establish a proper mathematical model for the description of investigated phenomena. A topographical network must be planned keeping in mind that it is necessary to effect measurements in hyper determined number, in order to reach evaluation of measurements uncertainty and of standard deviations. The redundant scheme of the network allows to make a preliminary analysis, with the aim to determine the presence of gross errors, and, in second phase, the network adjustment with the aim to get 3D points position and the estimated value of the accuracies with which they has been determined. In this paper the adjustment method is based on Gauss-Markov model. At the end of the adjustment process, it can be determined the interval of confidence for the respect of the position of every unknown point of the network (error ellipse/ellipsdoids). The analysis of the residuals is not enough to individualize gross errors: Date Snooping method (Baarda, [12]) proposes the use of normalized residuals of the measurements and local redundancy (contribution of every single observation to the general redundancy of the network). Control network is constituted by reference vertexes (points situated in stable zones) and by control vertexes (points situated in not stable zones, inside the monitoring zone). There are two typologies of movements: relative among the control points (deformations of the control network) and absolute (control points in comparison to a stable structure or to the reference frame. It’s important, therefore, to appraise, during the time, the stability of the reference vertexes. In this paper, the aim is to verify the stability a control network, constituted for monitoring a landslide area. The investigation results interesting especially for the particular location of a point. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Figure 2:

2

873

Control network of Senerchia.

Experimental test

Experimental test has been effected using campaigns of measurements related to the landslide of Serra dell’ Acquara-Vadoncello, Senerchia (SW Italy). The aim is to verify the application of the theoretical studies to a series of observations of control network, obtained using modern techniques of survey. The area of study is located in a zone of the southern Appennino, characterized by catastrophic events in 1980 and 1996, due to unstable nature of the area. The choice of such site is tied up to the presence of database coming from research works effected in the past years: data have been compared and integrated with those coming from the surveys directly effected. In this context, it is inserted the topographical study of the zone, finalized to the geodetic monitoring of small control network constituted by five vertexes. The particular location of vertex T4 (outside the landslide area of 1993 but inside the 1980 surface) requests a closer examination during the analysis, justifying the chance to use innovative tools/theory in the analysis of the observations (Bayes theory). The objective is to verify the stability of all the vertexes retained fixed and used as stations of distantiometric measurements to control position of points belonging to the landslide mass. Reference vertexes have been materialized through the construction of specific 3D concrete monuments (circular section 30 cm, height 120 cm) for the stable positioning of GPS antenna. During survey campaigns the following instrumentation has been used: • n° 2 LEICA GPS1200 (GX1230, 12 L1s + 12 L2s / WAAS / EGNOS) with Smart Track antenna AX 1202, Ground incorporated plane; • n°1 GPS LEICA System 500- SR530 double frequency (L1 codes C/A; L2 code P), antenna AT502, controller TR500; WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Figure 3:

Vertex T4 of the network.

• n°2 GPS Trimble 4700 (L1 code C/A,) with controller TSC1; • n°1 Smartstation with Smart Antenna. Use of Smartstation has allowed one to integrate GPS survey, primary source of data in the experimental test, with classical topographical data: angles and distances. Every GPS survey has been conducted in 3 independent static sessions, duration of 45 minutes. The tools rate has been planned of 10 seconds. The satellite configuration and the climatic conditions are optimal during the whole execution of survey operations. Download and post-processing phase have been executed in Geomatics Laboratory of Polytechnic of Bari, using software furnished with the receivers (Leica Geo Office 2.1 and Trimble Geomatics Office 1.01). All the data have subsequently imported in the second application to perform treatment according to the "single base" approach. At the end, the test of ratio has performed: it has broadly been always overcome the limit of 1.5 for GPS measurements. The network adjustment has been executed with least square method.

3

Classical analysis of the network movements

It has been decided to conduct on control points both the classical and the Bayesian analysis. While classical analysis has the purpose to quantify the entity of movements, the Bayesian approach has the purpose to identify the zones interested by significant movements. The simplification of the statistic problem is a necessary condition to allow the comparison among the results: therefore, the coordinates (N, E, h) of every vertex Pj (j=1,... ,5) of the network have been considered separately . The first hypothesis consists of considering not correlated coordinates in space and time (among the different sessions of measurements).



Figure 4:

875

GPS network and error ellipses.

The hypothesis to be submitted to inference is that movements are not significant:

H 0 : δx = 0 It is possible to build the following statistic parameter:

Z~

∆x

σ + σ υ22 2 υ1

.

Fixed the level of α=5% (corresponding to Z=1.96), the hypothesis H0 is approved (so the movement of the point is not significant) if:

− Z α ⋅ σ υ21 + σ υ22 ≤ ∆x ≤ Z α ⋅ σ υ21 + σ υ22 . 2

2

The calculation has been effected considering all the sessions of measurements that are divided in: GPS data coming from the effected surveys (fifth and sixth session); GPS data coming from a series of campaigns of measurements (first 4 sessions) conducted starting from the 1996. Sessions of measurements are totally 6: all data are the result of network adjustment effected with least square method.



Figure 5:

GPS post-processed data.

In the following charts for every point of the network and for every coordinate (E, N, h), the observed values are the following:

Z oss = Z oss = Z oss =

∆E

σ + σ υ2i 2 υ1

∆N

σ + σ υ2i 2 υ1

∆h

σ υ21 + σ υ2i

As concerns Figure 6, on celestial background there are values of Zoss that underline the presence of a possible significant movement in comparison to the first campaign of measurements. Classical analysis furnishes unequivocal results on the planimetric coordinates, because the values of Zoss are inferior to the threshold value (1.96) denoting absence of movement. Some problem arises on the height, as it was possible to expect for GPS observations characteristics.

4

Bayesian analysis for the movements of the network

In Bayesian analysis, the only coordinate h got by the network adjustment in different times has been considered: classical analysis is not able to clearly WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


877

define the vertical behavior of control vertexes. The quantities to be considered therefore, are movement ∆h among the different sessions of all the not-fixed points:

(∆h )P

J

= (hi − h1 )Pj

with i = 2, 3, 4, 5, 6 and j = 1, 2, 3, 4, 5.

Figure 6:

ZOSS reported to the E, N and h coordinates.

We can suppose that ∆h follow a normal distribution:

(

)

∆h ~ N δh, σ h2 . Parameters of this distribution are average (unknown) and the known variance factor that is obtained by the effected least square adjustment. For every point of the network we have:

∆h = hi − h1 = δh + σ h .

δh

is an aleatory variable, distributed according to a normal density of

probability, of average µ and variance

δh

σ 02 : ~N

(µ ,σ ) . 2 0

Parameters of this distribution (prior function of Bayesian theory), are information of the problem (they will have been fixed during the numerical WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

878 Computational Methods and Experimental Measurements XIII analysis). Since we effect an analysis of vertical movements of points in landslide area, it is supposed that the movements are void or downward. Then, considering an axle opportunely directed, it is reasonable to impose previously a further condition: δh ≥ 0 . This condition becomes complicated to impose in classical inference. Bayesian theory, explicited all the terms, is:

f (δh / ∆h ) =

f (∆h / δh ) ⋅ f (δh ) = K

(δh − m )   − ∆h 2 q h ∆ h 2 + q 0 µ 2 − q ⋅m 2 ) ( − 2σ 2 −   P0 e 1 ϑ (δh ) 2σ h2 2 = ⋅ ⋅e ⋅ δ (δh ) + ⋅e ⋅ ⋅ σ . A + B  2π ⋅ σ h 2π ⋅ σ  2π σ h σ 0   The analysis of the movements through the Bayesian approach can reduce then to the comparison between the two quantities: 2

P(δh = 0 \ ∆h) =

A A+ B

P(δh > 0 \ ∆h ) =

and

B . A+ B

From the comparison between the two quantities above, it is to be verified which of the two alternatives has greater probabilities to verify:

P0 A≡ e 2π ⋅ σ h

−∆h

2

2σ h2

and B ≡

σ ⋅e

−

(q ∆h +q µ h

2

0

2

−qm2

)

2

2π σ hσ 0

  m  ⋅ 1 − erf  −  .  σ  

All the terms that appear in these expressions are numerically determinable; particularly, the known values of µ and of σ0.

Figure 7:

Bayesian analysis.

In classical analysis the statistic test has been performed considering a level of significance α =5%. The interpretation of the results in the Bayesian analysis happens as shown in Figure 7. Six different elaborations have been effected, according to the initial hypotheses, always considering the only h coordinate, using a simplified model WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


879

and selecting different values of prior data. They calculated the values of P(dh>0) consequential from the comparison of all the measurement sessions with the first assumed as reference. Later, we present the results deriving from the sixth elaboration (Figure 8), in which we consider the following data prior: σ0 = 0.05 m for every network vertex, µ equal to the average of the ∆h in correspondence of every vertex .

Figure 8:

Bayesian analysis.

Prior data has been select exploiting the accumulated experience and knowledge of the phenomenon coming from a first analysis of raw data. It immediately emerges that, in all the elaborations, the numerical value of P(dh>0) calculated for network vertexes underlines that the relative movement is "not significant". The doubts appeared after the application of classical analysis has been resolved: there is no vertical movement of the vertexes.

5

Conclusions

The application of Bayes theory on the network has confirmed its stability during the time: significant movements are not present. The vertexes are stable and therefore usable as stations of measurements to control points position in landslide area. The application is limited only to the vertical direction because of classical analysis has already clarified the absence of planimetric movements. Results coming from Bayesian analysis are characterized by great coherence: also varying some prior data they appear stable. It is clarified, with Bayesian analysis, the behaviour of T4 vertex but it appears a necessary constant monitoring on it, due to its particular positioning. The application of Bayes theory is efficient for the aim of this paper: information insertion (critical phase of the procedure) has allowed to conduct a statistical analysis finalized to detect small movements in comparison to the standard deviation of the residuals. This method must not be considered as substitutive of classical analysis but integrative, because it uses the values of the parameters got with least square method. The difficulty of the Bayesian approach is to fit probability distribution to data and to make inferences from parameter distributions.



References [1] [2] [3] [4] [5] [6] [7] [8]

[9] [10] [11]

[12]

Augutis, J., Zutautaite, I., Alzbutas, R., Uspuras, E., Iterative estimation of reliability parameters using the Bayesian approach, Risk Analysis IV, WIT Press 2004, ISBN 1-85312-736-1. Barbarella, M. & Radicioni, F., Kalman Filtering in levelling: an application, Global and Regional Geodynamics, Springer-Verlag, NY, 1990. Barbarella, M., Canuti, P., Fiani, M., & Vannocci, P., Controllo di movimenti gravitativi profondi, XIII GNGTS Proceedings, Rome, 551562. Bernstein, S., & Bernstein, R., Statistica Inferenziale, McGraw-Hill, 2003. Caprioli, M., Importance and role of the GPS permanent national network for topographic and professional application, ASI-CGS, Matera, 1997. Caprioli, M., Stability controls and deformations measurements in civil engineering works, Allied Publishers Ltd, ISBN 81-7764-245-6. Cina, A., GPS – Principi, modalità e tecniche di posizionamento, Celid, ISBN 88-7661-417-6, 2000. Erol, S., Erol, B., Ayan, T., A general review of the deformation monitoring techniques and a case study: analysing deformations using GPS/levelling, ISPRS-WG VII/5,XXth ISPRS Congress Geo-Imagery Bridging Continents, Istanbul, Turkey, 12-23 July 2004. Pinto, L., & al., Controllo delle deformazioni: un approccio statistico di tipo Bayesiano, VII ASITA National Conference, 2003. Sacerdote, F., Albertella, A., Cazzaniga, N., Applicazione di metodi bayesiani per il controllo delle deformazioni, VIII ASITA National Conference, 2004. Trizzino, R. & Sorgente, M., Ottimizzazione di una rete di controllo GPS per il monitoraggio di una frana, Proceedings of International Conference “Prevention of Hydrogeological Hazards: the Role of Scientific Research", 1996. Baarda, W., A testing procedure for use in the geodetic network, Neth. Geod. Comm., n°5, 1968.


Section 12 Industrial applications



883

Calculating the dilution between successive trainloads of iron ore during processing J. E. Everett Faculty of Economics and Commerce, The University of Western Australia, Australia

Abstract Iron ore is railed several hundred kilometres from inland mines to port, where it is crushed and separated into lump and fines products for shipment to customers. During crushing, each trainload is sampled and assayed to measure the grade, in iron, phosphorus, silica, alumina and a number of other minerals. Trainloads from different mines are of systematically different grade. Accurate product grades for each trainload, from each mine, are required for production planning and quality control. During crushing each trainload is contaminated with ore from the preceding and possibly from the following trainloads, so that the reported grade is biased by this dilution from the neighbouring trainloads. Although the dilution effect has long been recognised, it has not previously been quantified and therefore has not been corrected for. This paper describes the development, verification and use of a non-linear regression model enabling the amount of dilution to be estimated, so that the diluted grades can be corrected to undiluted grades. Keywords: quality control, decision support, mining, non-linear regression, weighted least squares.

1

Introduction

The quality of the ore product depends upon it closely matching target composition, not only in iron, but also in phosphorus, silica and alumina. The four critically important composition elements will be referred to as the vector {Fe, P, SiO2, Al2O3}. A number of other elements are also monitored. The customer blast furnaces are tuned to receive ore of the agreed target composition. Issues relating to quality control in iron ore production are discussed in [1,2]. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070861

884 Computational Methods and Experimental Measurements XIII Iron ore is railed from five mines of differing average grades. At the port, run of mine trainloads are processed sequentially and separated into lump and fines products. Each trainload is sampled during processing. The lump and fines products are stacked onto lump and fines stockpiles, blended from the five mines The mine grades vary around consistently different means. For example, average Fe grade is considerably higher for Mine A than for the others. It has long been observed that trainload assays sampled at the port are biased according to the mine sources of trainloads processed before and after it. For example, a Mine A trainload followed and preceded by trainloads not from Mine A has lower Fe content than if its neighbours are other Mine A trainloads. The effect is because ore from the neighbouring trainloads contaminates the beginning and end of each trainload sample, during dumping, screening and crushing. As each trainload is processed, some material mixes into the next trainload. Each trainload sample has a sharp beginning and end, but the sample contains some material from the following and/or preceding trainloads. The dilution does not affect the estimated composition of the final product, because dilution of each trainload is compensated by dilution of its neighbours. But it does affect the reported composition of ore from each contributing mine, and this systematic error is important because it interferes with the planning and quality control for each mine. For each mine, each day’s production is planned to satisfy composition targets, based on mine estimates of the currently available ore sources. Comparison between mine estimates and subsequent port assays are used to monitor and adjust the mine estimates: this adjustment is compromised if the port assays have error, depending on which type of trainload preceded or followed them. A non-linear regression model is described, to estimate the average dilution from the preceding and following trainloads. It is shown that the lump and fines samples for 12 kilotonnes trainloads are each contaminated by nearly one kilotonne from the preceding trainload and a median of about 450 tonnes from the following trainload. The results of the analysis are now being routinely used to provide more accurate trainload grade data.

2

The model

Consider a variable X, with mean M., E[Xn] = M

(1)

X n = M + dn

(2)

where dn is the deviation for the nth observation Xn. M can be estimated as the average of Xn. But it can equally well be estimated as the value that minimises ∑dn2. The two formulations are equivalent, giving the same estimate for M, but the second formulation is a simple regression model.



885

Now consider the grade Xn for a trainload portion, which may come from any one of the five mines. It has total tonnage Wn kt, of which p kt comes from the previous trainload portion, and q kt comes from the following trainload portion. The expected value of Xn is now: E[Xn] = (pMp + (Wn-p-q)Mc + qMq)/Wn

(3)

Xn = (pMp + (Wn-p-q)Mc + qMq)/Wn + dn

(4)

where Mp, Mc, Mq are the mean values for the previous, current and following trainload portions, each being MA, MB, MC, MD or ME, according to which of the five mines is its source. Define M as the vector {MA, MB, MC, MD, ME}, and unit vectors Sp, Sc, Sq to specify the source mines of the preceding, current and following trainload portions. For example, if the preceding trainload portion comes from Mine B, then Sp is {0, 1, 0, 0, 0}. Thus: Mp = Sp.M´; Mc = Sc.M´; Mq = Sq.M´

(5)

and equation (4) becomes: Xn = (pSp + (Wn-p-q)Sc + qSq).M´/Wn + dn

(6)

Equation (6) is a non-linear regression model. Ordinary least squares (OLS) regression minimises ∑dn2. Weighted least squares regression (WLS) weights the variance terms to minimise ∑Wndn2. This can be achieved by multiplying both sides of equation (6) by √Wn: Yn = Xn√Wn= (pSp + (Wn-p-q)Sc + qSq).M´/√Wn + en

(7)

The OLS solution to equation (7) is thus a WLS solution to equation (6), providing estimates of the dilution factors p and q and the mine means, M.

3

Data used and its preparation

The analysis to be reported here is based upon a total of 9,403 “Port Actual” records, for ore crushed during the period of about ten months. Each “Port Actual” record reported a trainload portion of lump or fines from a total of 2,468 trainloads, being 1,471 trainloads from Mine A, 548 from Mine B, 279 from Mine C, 108 from Mine D, and 62 from Mine E. Each trainload portion was processed through one of two buildings, “TCB1” or” TCB2”. Many trainloads were split between TCB1 and TCB2, with a trainload portion going to each. Successive trainload portions of the same


886 Computational Methods and Experimental Measurements XIII product (Lump or Fines) of the same trainload going through the same building were merged. For each building and product, the trainload portions were ordered by start time. For each trainload portion record, five fields were added for the each of the unit vectors Sp, Sc and Sq, to specify the preceding, current and following mine, as defined above. Each vector had four zero components, and a single unit component. Consider: Sp={0,1,0,0,0}; Sc={1,0,0,0,0}; Sq={0,0,0,0,1}

(8)

In the example of equation (8) the preceding, current and following trainloads come from Mines B, A and E respectively. Each trainload portion had a grade vector G, defining its assayed composition. Assays were available not only for the four control minerals {Fe, P, SiO2, Al2O3}, but also for {H2O, LOI, Mn, TiO2, CaO, MgO, S, K2O}. LOI is “loss on ignition”, a measure of moisture content. G = {Fe, P, SiO2, Al2O3, H2O, LOI, Mn, TiO2, CaO, MgO, S, K2O} The vector fields Prod = {L(ump), F(ines)} and Plant = {TCB1, TCB2} were added, specifying the product (lump or fines) and plant building for each trainload portion. For example, a lump trainload portion going through TCB2 has Prod = {1,0} and Plant = {0,1}. Fields Tp and Tq were calculated, as the time interval (in minutes) before the start and after the end of each trainload portion.

4

Discriminant analysis

Solution of the non-linear regression equation (7) requires that the mean values of X differ significantly between the five mine sources. The model could be run repeatedly, for each mineral, giving repeat estimates for the dilution. It is more efficient to let X be the linear composite of the minerals that best separates the sources. The “best” separation is defined as maximising the between-group variance divided by the within-group variance. The linear composite satisfying this condition is known as the first discriminant function. The second discriminant function again maximises the between-group variance divided by the within group variance, subject to the constraint that the second discriminant function is orthogonal (uncorrelated) to the first discriminant function. A useful description of discriminant analysis is supplied in the SPSS™ reference guide [3]. With five groups, four orthogonal discriminant functions can be extracted. We are interested only in the first discriminant function, since it has the greatest possible power to discriminate between the five groups. The statistical package



887

SPSS™ was used to apply discriminant analysis. The first discriminant function scores were added to the data file. The discriminant analysis was run separately for lump and for fines. The correlations between the mineral grades and a discriminant function are referred to as the “loadings” of the mineral grades on the discriminant function. The loadings for the first discriminant function, for lump and for fines, are tabulated below. Table 1:

Mineral

5

Fe

Loadings on the first discriminant function.

P SiO2 Al2O3 H2O LOI Mn TiO2 CaO MgO

S K2O

Lump -.53

.54

.09

.58 -.01

.87 -.04

.33 -.26 -.28

.29 -.17

Fines -.77

.47

.57

.60 -.01

.72 -.17

.22 -.29 -.33

.38 -.13

Model solution

The non-linear regression equation (7) can be solved to estimate p and q and the mine means M. The dilutants p and q are not necessarily constants, but may themselves be expressed as functions, of the trainload portion weight Wn, the crusher plant (TCB1 or TCB2), the product (Lump or Fines), time, et cetera. The model of equation (7) could be run separately for lump and fines, and it is to be expected that the mean value vector will be different for lump and fines. However, the two analyses can be combined. Letting the vector of mean values M be ML and MF respectively for lump and fines, equation (7) becomes the combined equation: Yn=Xn√Wn=(pSp+(Wn-p-q)Sc+qSq).(L.ML´+F.MF´)/√Wn+en

(9)

The statistical package SPSS™ was again used, to solve the non-linear regression model for the unknown parameters p, q, ML and MF, which could be expressed as functions. A range of physically believable models were tested, rejecting any terms not significant at the 5% significance level. The following model was found significant in all its terms: p = P1*Wn

(10)

q = Q1*Wn*exp(-H*Tq)

(11)

ML = {MAL, MBL, MCL+GCL*Time, MDL, MEL}

(12)

MF = {MAF+GAF*Time, MBF+GBF*Time, MCF, MDF, MEF}

(13)


888 Computational Methods and Experimental Measurements XIII Parameters defining p and q are of operational interest. All are significant at the 0.0001% level, as tabulated below. Parameters for ML and MF represent discriminant function score means and are of no further interest. The coefficients GCL, GAF and GBF, were all significantly negative, indicating that the means for Lump from Mine C Lump and for Fines from Mines A and B decrease with time. Slopes against time for the other means were tested, and none were significant at the 5% level. Table 2:

Dilution parameter estimates.

Parameter

6

StdErr

Sig.

P1

.083

.005

< 0.0001%

Q1

.119

.017

< 0.0001%

H/minute

.060

.012

< 0.0001%

Testing for further dilution models

The dilution model of equations (10) and (11) support the model for which: 1) The tonnage dilution from the previous trainload portion is proportional to the tonnage of the current trainload portion. 2) The tonnage dilution from the following trainload portion is proportional to the tonnage of the current trainload portion, declining exponentially according to the time gap between the current and following trainload portions. Further terms were added to the model of equations (10) to (13), one by one, to test if the model required significant augmentation. 6.1 Potential additions to the “p” model p = P0 + P1.Wn P0 = .034 +/- .023 Sig.(P0=0) =14%

(14)

p = P1*Wn.exp(-K*Tp) K = 1.92 +/- 1.11 Sig.(K=0) = 8.4%

(15)

p = P1.WnK

K = .808 +/- .091 Sig.(K=1) = 3.5%

(16)

p = (PB.TCB2+P1).Wn PB = -.0005 +/- .008 Sig.(PB=0) = 94%

(17)

p = (PF.Fines+P1).Wn PF = -.015 +/- .010 Sig.(PF=0) = 15%

(18)



889

6.2 Potential additions to the “q” model q = Q0+Q1.Wn.exp(-H*Tq) Q0 = -.016 +/- .019 Sig.(Q0=0) = 40%

(19)

q = Q1.WnK.exp(-H*Tq) K = .895 +/- .175 Sig.(K=1) = 55%

(20)

q = (QB.TCB2+Q1).Wn QB = -.042 +/- .020 Sig.(QB=0) = 3.7%

(21)

q = (QF.Fines+Q1).Wn

7

QF = -.019 +/- .024 Sig.(QF=0) = 43%

(22)

Results

None of the examined extensions to the model of equations (14) to (22) are significant at the 3% level. Two extensions, (16) and (21), significant at the 5% level, provide weak evidence that: 1) p may increase a little less than linearly with Wn. 2) q may be smaller for the second plant TCB2. It was concluded that the evidence for these effects was not strong enough to justify further complicating the model.

8

Undiluting the data

Having established that each trainload is diluted with a tonnage p from the previous rake, and q from the following rake, we can write the formula for calculating the trainload’s undiluted composition vector XU from its diluted composition XD, and the compositions XP and XQ of the preceding and following trainloads. XU = (XD – pXP – qXQ)/(1-p-q)

(23)

where p = 0.083, and q = 0.119*exp(-.06Tq) The dilution model of equations (10) and (11) was used to undilute the diluted data, in accord with equation (23). The compositions XP and XQ of the preceding and following trainloads should really be the undiluted values. For convenience, the diluted values were used instead, introducing a negligible second-order error. As an example of the undilution process, cumulative distributions for diluted and undiluted lump Fe values are plotted in Figure 1, and for lump Al2O3 values are plotted in Figure 2. The graphs show the results only for Mines A and B. Fe is consistently higher for Mine A, and undilution makes it higher still. Al2O3 is consistently lower for Mine A, and undilution makes it lower still. The reverse applies to Mine B. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Figure 1:

9

Cumulative distributions for lump Fe, Mines A and B.

Conclusion

The results plotted in Figures 1 and 2 show examples of the undilution process. Similar comparisons can be made for the other three mines, and for the other minerals of interest, P and SiO2. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Figure 2:

891

Cumulative distributions for lump Al2O3, Mines A and B.

It should be noted that the undilution applied to a particular trainload takes account of the particular trainloads that precede and follow it. If a trainload from a particular mine is preceded and followed by trainloads also from the same mine, then the undiluted composition will be close to the diluted composition. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

892 Computational Methods and Experimental Measurements XIII The procedure is now used routinely to improve the estimates of composition for trainloads from each mine. The greater accuracy has helped in improving the tracking of mine estimate errors and thus has improved the mine planning capability. The model of equations (9)–(13), leading to the undilution formula of equation (23) is somewhat counter-intuitive. It finds that the tonnage contaminated from the previous and following trainloads is proportional to the tonnage of the trainload being contaminated. Our expectation had been that the contamination should be a tonnage amount independent of the tonnage of the contaminated trainload. The significances of the model parameters in equations (10), (11), (14) and (19) support the counter-intuitive model. The model finds that the contamination from the following trainload decays exponentially, according to the time interval between the trainload being contaminated and the following trainload. The decay has a half-life of about twelve minutes. This makes sense. The sampling from a trainload stops when the operator judges that the trainload processing has finished. If the following trainload starts being fed in straight away, some of it will be picked up in the sample, but not if there is an appreciable time interval between the trainloads.


Everett, J.E. Iron ore handling procedures enhance export quality, Interfaces, 26(6), pp. 82-94, 1996. Everett J.E. Iron ore production scheduling to improve product quality, European Journal of Operational Research, 129, pp. 355-361, 2001. SPSS, Reference Guide, SPSS Inc: Chicago, 1990.



893

Rock bumps due to the creation of a dislocation during deep mining V. Doležel1 & P. Procházka2 1 2

University of Pardubice, Czech Republic Czech Technical University in Prague, Czech Republic

Abstract The safety of deep mines is sometimes threatened by sudden and hazardous failures of a coal face, as in the case of bumps, for example. It has been shown in many experimental studies that coal bumps are influenced either by stress, stiffness, and yield ability of surrounding rocks, or by dynamic effects associated with the failure of surrounding strata. In addition, bumps occurred in such parts of mines where there had been rapid stress changes over a short period of time. The dynamic effects associated with the failure of surrounding strata triggered bumps in these marginally stable seam structures. While it was not possible to evaluate the influence of mine stiffness directly, it was shown that coal bumps generally occurred in mines with uniaxial compressive strength and Young’s modulus ratios (roof to coal) exceeding 3 to 5. In addition, bump-prone coal exhibited the potential for storing high horizontal stresses. Yielding of the immediate roof and floor reduced horizontal stresses and enhanced gradual failure of coal. A method is proposed to assess coal bumps in which stress analyses, in situ strength data, stiffness and strength ratios of roof to coal, and affected wave magnitude resulting from strata failure and mining experience are incorporated. A rheological model is involved to fulfill the time dependence of the phenomenon of bumps. It has namely been shown in the literature that the time plays a decisive role in the possible occurrence of rock bursts. In our case the static (stability) problem is solved and the dynamical influence of inertia forces is initiated after occurrence of bumps. This is not of any importance for us, as we suppose that the mine is far enough from other openings (other mines). In this way the stability problem with possible rheology (or creep) is studied. Keywords: rock bursts (bumps), scale modeling, coupled modeling, natural existence of occurrence of bumps, numerical model of contact problem. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CMEM070871


1

Introduction

The paper deals with an application of coupled modeling in identification of dislocation occurring in coal mines threatening workers in the underground structures. The bumps can be induced for different reasons, one being the accumulation of energy in unpredictable dislocations. The measurement on site is very expensive and unreliable. One of the most reliable is physical modeling, which enables one to carry out parametric studies and after certain results from these models one can assess the most probable concentration of stresses. On the other hand, the stresses are measured in a very difficult way, so that numerical analysis should take place. In numerical analysis the contact problem in limit state estimation is based on data from physical modeling. The physical modeling seems to be the best for linear analysis. This is not the case of our study and large attempts have to be carried out to estimate the real behavior of the material. With support of physical modeling mathematical formulation and numerical treatment can lead to the danger of occurrence of bumps. In general, the large iteration should be used to solve the strongly nonlinear problem in both subdomains ranged in boundaries given by possible dislocations, and on the interfacial boundary (dislocations). In order to eliminate some principal directions of iteration the physical modeling is used and the numerical processes become bearable. A typical example from praxis verifies the theory based on back analysis. Extensive studies were done by Elices et al., Haramy and McDonnell [3] and Haramy et al., [4], who showed a methodology on how to estimate the possibility of bumps. In [2] the authors tried to explain reasons for bumps based on a couple of experiments. The conditions for bumps in coal mines under a strong roof are described in [3], where possible bumps are analyzed and explained. A methodology for assessment of pillars in longwall mining is suggested in [5]. Scale modeling of rock bursts is described in [6]. The method and equipment mentioned in [6] are very powerful for illustration of bump evaluation, as the extrusion of coal grains into free space can be seen in a natural way. Numerical modeling can be found in [7], for example, where the cracking is described by a cohesive zone method, which starts with Griffith and Barenblatt theories. A very promising method seems to be partition of unity, presented in [8], for example, where a theoretical explanation can be found, or in [9], where a direct application of the mentioned method to a cracking material is shown. Another method appropriate for description of cracking is called the manifold method; its advanced version is published in [10]. Discrete methods became very popular in describing nucleation of cracks and consequent occurrence of bumps. One of such method is denoted as free hexagons of statical particle flow code, [11]. Coupled modeling, consisting of implementation of results from experimental (scale) models into numerical computation, is proposed in [12]. It is based on involvement of eigenstrains or eigenstresses as design parameters in identification of material properties. In this paper the most dangerous crack in a rock overburden of a coal seam is determined. WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


2

895

Experimental study

The principal objective of the experimental and numerical model research was the clarification of mechanical behavior of the coal seam and a close and distant overburden of one selected seam in connection with the origin of rock bumps in the proximity of localization in the north Czech Republic. The first phase comprised the simulation of exploitation of the seam under consideration. The extraordinarily difficult and exacting character of the problem required an unconventional approach to its solution. For this reason the physical model of equivalent materials was supplemented with a mathematical and a scale model. This combined approach was selected to make the individual partial methods mutually link up and complement each other to advantage and so enable a more comprehensive insight into the whole problem. As some data required by mechanical properties, particularly those concerning the shear and triaxial strength, were not available (the required measurements had not been made in the given area), it was necessary to verify the strength characteristics of the rock mass in the environs of the principal crosscut by a contact problem. Experiments with models from physically equivalent materials provided a detailed stress distribution in rock environment (with the assumption of its ideally elastic behavior) and supplemented very appropriately the results of measurements made on physical models. Generally, two concepts of rock bursts are distinguished: The first assumes that a first bump occurs and then the dislocation appears as a consequent of weakened overburden. The second idea starts with the dislocation in the overburden and the bumps is an aftermath of the situation caused by the cracked dislocation. The latter case is considered in this study after obtaining the realistic material properties from measurements on site. Hence the appraisal of the relations and consequently the maintenance of the situation on site can be carried out. The geometry and possible failure in the neighborhood of the face of the long wall mining is depicted in Fig. 1, where also the source of dynamic propagation is shown.

Figure 1:

Static and seismic energy released after limit concentration at the face of the wall.



3

Verification of material parameters of rock environment

To determine the characteristics of equivalent material used for physical model construction it was necessary to comply with the requirements of physical similarity between the model and the rock environment. However, the measurements required for this purpose were not made in the full extent. For this reason it was decided to verify and/or refine the selected shear parameters of the rock mass by the DEM in such a way that the results for the free hexagons fed by data from laboratory experiments had to comply with the results from beards build up in the beginning of the construction of the mines. It appeared that there were deep differences in these results and the numerical approach had to be basically improved. Long term monitoring of the rock mass in crosscut environs has ascertained that its extraction does not impair ambient rock. Consequently the shear parameters not producing any extensive failure zone in the environs of the working may be considered as the basic shear strength of the rock mass. Under these assumptions the problems were formulated as follows: 1. The shear parameters were estimated on the basis of the rock strength (admissible values of the material parameters) in uniaxial compression and tension and literary data. Alternative A : c = 5.5 MPa φ = 35° σ + = 5.0 MPa where c is the cohesion (shear strength), φ is the angle of internal friction, and σ + is the tensile strength. 2. The non-linear problem, based on the same shear strength assumptions, was solved to refine the convergence of the working. 3. The problem solved as another alternative, was based on the following strength parameters: Alternative B: c = 10 MPa φ = 65.5° σ + = 4.1 MPa The difference in the extent of failure zones should show the minimum shear strength of the rock mass. The solution of all problems assumed the initial stress state due to overburden weight and characterized by the value of the coefficient of the side pressure Ke = 0.54. Physical non-linearity was concentrated on the dislocation. The program makes it possible to respect the different behavior of the materials in compression, in tension and after load relief. Compressive strength is controlled in accordance with the first failure theory, i.e., by a comparison of the maximum shear stress with the shear strength of the respective material. This control is performed only if the octahedral normal stress σoct is a shear stress. In the opposite case the control concerns shear according to the generalized Mohr theory of failure expressed in octahedral stresses.



4

897

Solution of underground continuum by the BEM

In this part we briefly describe an implementation of the boundary element method to the solution of specific problems of underground continuum, for which the numerical method appears to be extraordinary advantageous. The method, among others, reduces the problem by one. Further good application of the boundary element method is the optimization and/or contact problems which concern the boundary only. Then, in spite of the finite element method it suffices to study a change of location of boundary elements only. The problem is solved as two-dimensional, i.e. a possible dislocation is long enough, and a narrow seam is considered. Moreover, the nonlinear behavior is considered in the region, which is sufficiently close to the dislocation, according to Mises theory. Suspicious dislocation is given from the experimental model from physically equivalent materials. In our following consideration we will concentrate on the physically nonlinear problems (nonlinear evolution is also included in boundary conditions). Let us solve the problem in domain Ω. We originate from the Cauchy equations:

(α + µ )

∂σ ij0 ∂ div u + µ ∆ ui + X i + ∑ = 0, i = 1,...,3 ∂x j ∂x i j

(1)

where div u =

∂u1 ∂u 2 , + ∂x1 ∂u 2

∆=

∂ ∂ + ∂x1 ∂x 2

and u = (u1, u2) is the displacement field, (X1, X2) are components of the volume weight and σij0 are components of the tensor of initial stress. These equations will be solved in the coordinate system 0x1,x2. In the sense of BEM, (1) may be reformulated in an equivalent form:

(

)

[

] (

* c k 1 (ξ )u1 (ξ ) = ε ijk , σ ij + [ pik , ui ] − uik* , pi − uik* , X i

)

(2)

where [.] are boundary integrals, (.) are plane integrals, ck1 is the matrix of coefficients depending on a position of ξ, p is the vector of external forces, and quantities with an asterisk denote the relevant quantities of the fundamental solution. From the Cauchy equations (respecting shearing stresses to be zero) we have the well known relations for a virgin state to get the initial stresses:

u1o = −

1 − 2ν X 1 x12 + const., 4 µ(1 − ν )

σ1o,1 = − X 1 x1,σ1o,2 = 0, σ 2o,2 = −

u 2o = 0 ν X 1 x1 1− ν


(3)


5

Contact problem

Before we start the analysis preliminary considerations will be introduced. In order to explain the process of computation the two-dimensional problem will be treated. The three-dimensional problems are solved similarly. Let the problem be described from experimental study. Ωseam is the domain of the seam, for which the dislocation and the bearing capacity is to be assessed.

Figure 2:

Domain and denotation of the example under study.

After discretization of (2) in the sense of the boundary element method the problem leads to the system of algebraic equations:  H 11 H  21  H 31

H 12 H 22 H 32

H 13   u   G11 G12   A 23   u -c  − G 21 G 22   A 33  u c+  G 31 G 32  

g  F  G13       g  -  G 23 + G11  g c  =  Fc-   g +   F +  G 33   c   c 

(4)

where the upper index - denotes "from the left" and + denotes "from the right“, g is the vector of prescribed surface forces along the boundaries Г and Гp, pc is the vector of surface forces on fictitious contact Гc and F includes the effect of volume weight. As the vector g contains known quantities we can rearrange the previous equations to obtain:  H 11 H  21  H 31

H 12 H 22 H 32

g H 13   u   G 12 p -c + G 13 p c+   F + G 11           A 23   u -c  − G 21 p -c + G 23 p c+  =  Fc- + G g21   g  A 33  u c+  G 31 p -c + G 33 p c+   Fc+ + G 31      


(5)


899

Suppose now that for example u c− and u c+ is known. Then the problem is uniquely solvable, so that the matrix H 11 is regular. For a similar reason the matrices Hkk are regular, too. Also, the same assertion holds for the matrices G11,, i = 1,2. This is the general result of solvability of linear problems of elasticity by the boundary element method. We can conclude that the matrix H is singular, but the last submatrices are regular matrices. This is why it is possible to rearrange the system in the sense of matrix canonical transformations (in algorithm we use Gaussian elimination) to obtain:  H 11  0   0

H 12 A 22 A 32

H 13   u   B11  C11   -      A 23   u c  −  B 21 {Pc } = C 21  C  A 33  u c+   B 31   31 

(6)

where the balance condition pc = pc− = − pc+

(7)

was employed. The matrices are known while the vectors u and p remain unknown. From the last form the reducibility follows and we can employ the following system of equations: A 32 u -c + A 33 u c+ − B31p c = c 21 A 22 u -c + A 23u c+ − B21p c = c 22

(8)

Generally, along the contact line only balance condition holds and the compatibility is prescribed with the aid of more complicated relations. For example, suppose that at each nodal point along the contact line holds:

[u]n = u1n − un2 ≥ 0 pt ≤ Tpn + c pt ≤ Tpn + c ⇒ Ελ > 0, [u ]n = − λpt

(9)

where T and c are prescribed coefficients (they may very along the contact), pn and pt are projections of tractions to the normal and tangential direction with respect to the contact line, respectively. Then Uzawa’s algorithm can be applied to the contact problem – see, e.g., [12]. In the domain the rheological model is applied according to Fig. 3 with the coefficients: Elasticity modulus E = 10 GPa, Poisson’s ratio ν = 0.21, peak cohesion = 1.2 MPa, peak angle of internal fiction = 300, dilation angle = 00, peak tensile strength = 4 MPa (not used), residua cohesion = 0.2 MPa, cohesion softening rate = 0.001, residua fiction angle = 200, fiction softening rate = 100, rock viscosity = 10-19 MPa (average used). WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


Figure 3:

Schematic of the rheological model.

The critical shape of the searched dislocation is depicted in Fig. 2. The way of overall vertical stresses together with a description of typical longwall mining is illustrated in Fig. 4. Fig. 5 displays distribution of principal strain rates for two basic materials: the left describes a roof with plastic properties and the right material properties being “almost” brittle. The envelope of the peak stresses describes the most dangerous states for possible occurrence of bumps. The picture was obtained from numerical computation, based on tuned material properties from the scale model and in situ measurement of material parameters.

Figure 4:

6

Stress state in the ground after the excavation of the panel.

Conclusions

Geotechnical data, mining experience, and long-term underground observations were analyzed in an effort to better understand causes of violent failure in U.S. coal mines. There it was shown that coal bumps are influenced by the interaction WIT Transactions on Modelling and Simulation, Vol 46, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)


901

of geologic and stress conditions that govern the post failure behavior of coalmeasure rocks. Case studies provided new insight into the buildup of horizontal stresses, geometric factors that cause zones with high stress gradients, contrasts in stiffness and the mechanical properties of rock and seam, and failure of upper strata. In this paper a combination of experimental tests in scale models from physically equivalent materials provided input data for numerical models, the mechanical properties which were tuned to get a more realistic view of the real situation. The aim of the numerical model was to determine critical dislocation, which causes the critical state on the interface of the overburden and coal seam. The energy accumulated at the tip can be calculated from the numerical model after finishing the computation using the above described conditions. The decisive factors are principal strain rates. In any case, experience, numerical modeling, and engineering judgment should be used to assess the extent of yielding in mine roofs and floors; such controlled yielding may help reduce horizontal stresses and lead to less violent failures. Lack of yielding within the roof, coal, and seam promotes high bump potential where there is a high risk of a seismic event and a low factor of safety.

Figure 5:

Two types of material response in compression for varying principal strain rates.

Acknowledgements The first author has been financially supported by the Grant agency of the Czech Republic, project number 103/05/0679, and the second author was supported by the Grant agency of the Academy of the Czech Republic, project number IAA 2119402.

References [1]

Maleki, H., Jung, Y. & Hollberg, K: Case Study of Monitoring Changes in Roof Stability. Int. J. Rock Mech. Min. Sci. and Geomech. Abstr., V. 30, No. 7, 1395-1401, 1993.


902 Computational Methods and Experimental Measurements XIII [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

Elices, M., Guinea, G.V., Gomez, J. & Planas, J. The cohesive zone model: advantages, limitations and challenges. Engineering Fracture Mechanics 69, 2002, 137-163. Haramy, K.H., & McDonnell, J.P.: Causes and Control of Coal Mine Bumps. USBM RI 9225, 35 pp, 1988 Haramy, K.Y., Magers, J.A. & McDonnell, J.P. Mining under strong roof. 7th Int. Conf. on Ground Control in Mining, Bureau of Mines, Denver, USA, 1992, 179-194. Haramy, K.Y. & Brady, B.T. A methodology to determine in situ rock mass failure. Internal report of Bureau of Mines, Denver, CO, USA, 1995. Mark, C.: Pillar Design Methods for Longwall Mining. USBM IC 9247, 52 p., 1990. Kuch, R., Lippmann, H. & Zhang, J. Simulating coal mine bumps with model material. Rockbursts and seismicity in mines, Gibowitz & Lasocki (eds.), Balkema, Rotterdam, 1997, 23-25. Babuska, I., Melenk, J.M.: The partition of unity method, Int. J. Numer. Meth. Engrg. 40 (1997) 727-758 Lin, J.S.: A mesh-based partition of unity method for discontinuity modeling, Comput. Meth. Appl. Mech. Engrg. 192 (2003) 1515-1532 Chen, G., Ohnoshi, Y., Ito, T. Development of high-order manifold method, Int. J. Numer. Meth. Engrg. 43 (1998) 685-712 Procházka, P. Application of discrete element methods to fracture mechanics of rock bursts. Engng. Fract. Mech. 2003. Dolezel, V. & Prochazka, P.: Characterization of dislocations in underground mass using coupled modeling. CMEM, Portland, Maine, USA, 2005, 333-341



903

Author Index Abdalla I. E................................ 13 Abrahamsson L.......................... 85 Akgün Ş. .................................. 233 Alexandrov D............................. 75 Alias A..................................... 601 Alnuaimi A. S. ......................... 185 Anagnostopoulos N. ................ 655 Andreozzi A..................... 389, 411 Ansar M. .................................. 137 Argentini T. ............................. 723 Avichail-Bibi R.......................... 65 Balcius A. ................................ 457 Balestrino A............................. 819 Barraco A................................. 547 Belloli M.................................. 723 Belouchrani M. A. ................... 611 Bianco N. ................................. 411 Bielecki Z. ............... 737, 801, 809 Blazej J. ........................... 165, 171 Boran J..................................... 465 Britten A. ................................. 861 Bruno O. .................................. 819 Bugaj M. .................................. 747 Buonomo B...................... 255, 389 Butcher K. S. A.......................... 75 Caprioli M................................ 871 Černý R.... 339, 349, 357, 367, 377 Chalioris C. E. ................. 623, 633 Chambega D. J......................... 195 Chang J. ................................... 329 Char J. M. .................................. 25 Charon W................................. 589 Cheirsilp B............................... 225 Chen C.-C............................ 691 Chen Z. .................................... 137 Cheng E. .................................. 319 Chin W. J. ................................ 681 Cho E. K. ................................. 681 Cho J. R. .................................. 681 Ciaravino G.............................. 277 Ciaravino L. ............................. 277

Conley W................................. 177 Cook M. J. ................................. 13 Czarnecki W. ........................... 769 D'Alessio S. J. D...................... 425 de Juan A. ................................ 711 De Martino G........................... 307 De Paola F. .............................. 307 Deschênes C. ............................. 55 Djurić-Mijović D. .................... 117 Doležel V................................. 893 Elsäßer T.................................. 465 El-Shaarawi M. A. I................. 447 Elsoufi L. ................................. 589 Everett J. E............................... 883 Fafitis A. .................................. 557 Farshad F. ................................ 127 Favvata M. I............................. 623 Fazarinc M................................. 45 Fernandez del Rincon A. ......... 711 Fiala L...................................... 349 Fischer C.................................. 671 Foglia G. .................................. 399 Franchitti S. ............................. 215 Fraternale G. ............................ 723 Frýba L. ................................... 671 Fuks D. ...................................... 65 Fuller B.................................... 557 Furlan R. .................................. 297 Gagnon J.-M. ............................. 55 Gajewski P............................... 791 Garcia Fernandez P.................. 711 Gaudiano N.............................. 723 Gergiadis A.............................. 655 Giammatteo M. M. .................. 851 Giedraitis V. ............................ 457 Giuliano G. .............................. 215 Gjerkeš H................................... 35 Golia C..................................... 255 Goto M..................................... 205

904 Computational Methods and Experimental Measurements XIII Gregori A................................. 851 Gylys J. .................................... 457 Han S.-Z................................... 205 Hartman S. ................................. 35 Hiwatashi T.............................. 109 H-Kittikul A............................. 225 Honner M................................. 475 Houdkova L. ............................ 465 Hrinda G. A. ............................ 243 Hsu S. H................................... 537 Hsu U. K. ................................... 25 Hsu W.-E. ................................ 569 Ibrahim A................................. 601 Jamal A. ................................... 447 Jancarek A. .............................. 165 Jerman M. ................................ 377 Kakaletsis D. J. ........................ 643 Kamaka M. .............................. 319 Karakoç C. ............................... 233 Karayannis C. G....................... 623 Kawagoishi N. ......................... 205 Kawalec A. .............................. 769 Kazakov K. ................................ 97 Kelnar J.................................... 367 Khalil K. .................................. 589 Kim B. S. ................................. 681 Kim C.-J................................... 205 Kim N. S. ................................. 701 Kim S. I.................................... 701 Kitamura M.............................. 155 Kiv A. ........................................ 65 Kobayashi S............................. 527 Köksal H. O. ............................ 233 Kołosowski W. ................ 801, 809 Komorniczak W....................... 769 Konatowski S........................... 779 Kotchergenko I. D. .................. 579 Kugler G. ................................... 45 Kuo S. Y. ................................. 537 Kwark J. W. ..................... 681, 701 Labbé O. .................................. 437

Lacasa G. ................................. 411 Lachat R................................... 589 LaHaye L. C. ........................... 127 Landi A.................................... 819 Lee J. S. ................................... 701 Leśnik Cz................................. 769 Li M. S..................................... 505 Liao C.-A................................. 691 Lin J. Y. ................................... 537 Liu C. H. .................................. 569 Long W........................................ 3 Lu J. K. .................................... 537 MacIntyre M. M. ..................... 861 Maděra J. ................................. 367 Madura H................................. 757 Manca O. ......................... 389, 399 Marini G. ................................. 307 Matsuda C................................ 319 Miadonye A. ............................ 861 Michálek P............................... 377 Mikolajczyk J. ......................... 737 Misirlioglu A. .......................... 485 Mokheimer E. M. A................. 447 Mokhtar A. A........................... 601 Munteanu M. Gh...................... 547 Muszkowski M. ....................... 801 Nardini S.................................. 399 Naso V. .................................... 411 Neves A. C............................... 147 Nulu S. C. ................................ 127 Nzali A. H................................ 195 Ohtaki S. .................................. 527 Palinek S.................................. 165 Panetta F. ................................. 723 Pavlík Z. .................................. 349 Pavlíková M..................... 349, 357 Pepper D. W. ........................... 495 Pernicová R. ............................ 357 Pieniężny A. T. ........................ 779 Pietrasiński J............................ 769 Pilevne E.................................. 485 Pina L....................................... 165


Piotrowski Z. ........................... 791 Polat Z...................................... 233 Prichard-Schmitzberger A. ...... 829 Prochazka I. ............................. 171 Procházka P. .................... 517, 893 Pulci Doria G. .......................... 277 Ranucci A. ............................... 307 Rieke H. ................................... 127 Roizin Ya................................... 65 Rosemann N. ........................... 329 Różański G. ..................... 801, 809

905

Tamura A..................................... 3 Tansley T. L............................... 75 Taveira Pinto F. ....................... 147 Teng M. ................................... 319 Terčelj M. .................................. 45 Totani G................................... 851 Trajković M. ............................ 117 Tsutahara M................. 3, 155, 267 Turgay T. ................................. 233 Turk R........................................ 45 Tydlitát V................................. 377 Urushadze Sh........................... 671

Saanane B. B............................ 195 Sabato D. ................................. 723 Said M. R................................. 601 Sakamoto H. ............................ 109 Sancibrian R. ........................... 711 Sani L....................................... 819 Santamaria J............................. 557 Santiago-Avilés J. J. ................ 297 Šarler B. ..................................... 35 Sasaki H................................... 155 Scarlatos P. D. ......................... 137 Sędek E. ........................... 801, 809 Sideris K. K. ............................ 655 Sinkunas S. .............................. 457 Smirnov S. A. .......................... 287 Söder L. ..................................... 85 Sotero-Esteva J. O. .................. 297 Sroub J. .................................... 475 Stehlik P................................... 465 Strisciuglio G. .......................... 871 Tajiri S. ................................ 3, 267 Tamas M. ................................. 165

Vanali M.................................. 723 Veloso Gomes F. ..................... 147 Vertnik R. .................................. 35 Vesely Z................................... 475 Viadero F. ................................ 711 Vrba P...................................... 165 Vrbova M................................. 165 Wang X.................................... 495 Wnuk M........................... 747, 801 Wojtas J. .................................. 809 Wu W.-H. ................................ 691 Yamaguchi T. .......................... 109 Yamamoto M........................... 109 Yamamoto T. ........................... 527 Yang B..................................... 505 Yeo I. H. .................................. 701 Zappalà G. ............................... 841 Zdankus T................................ 457 Zoaeter M. ............................... 589


...for scientists by scientists

Computational The Wonderful World Methods and Experimental of Simon Stevin Edited by: J.T. DEVREESE, University of Measurements XII Antwerp, Belgium and G. VANDEN Magic is No Magic:

BERGHE, University of Ghent, Belgium This book gives a comprehensive picture of the activities and the creative heritage of Simon Stevin, who made outstanding contributions to various fields of science in particular, physics and mathematics and many more. Among the striking spectrum of his ingenious achievements, it is worth emphasizing, that Simon Stevin is rightly considered as the father of the system of decimal fractions as it is in use today. Stevin also urged the universal use of decimal fractions along with standardization in coinage, measures and weights.Stevin was the first since Archimedes to make a significant new contribution to statics and hydrostatics. The impact of the Stevin’s works has been multilateral and worldwide, including literature (William Shakespeare), science (from Christian Huygens to Richard Feynman), politics (Thomas Jefferson) and many other fields. Thomas Jefferson, together with Alexander Hamilton and Robert Morris, advocated introducing the decimal monetary units in the USA with reference to the book “De Thiende” by S. Stevin and in particular to the English translation of the book: “Disme: The Art of Tenths” by Robert Norton. In accordance with the title of this translation, the name of the first silver coin issued in the USA in 1792 was ‘disme’ (since 1837 the spelling changed to (‘dime’). It was considered as a symbol of national independence of the USA. ISBN: 978-1-84564-092-7 2007 apx 343pp apx £90.00/US$145.00/€135.00

Edited by: C.A. BREBBIA, Wessex Institute of Technology, UK and G.M. CARLOMAGNO, University of Naples di Napoli, Italy This volume contains the papers presented at the 12th International Conference on Computational Methods and Experimental Measurements (CMEM) held in Malta in 2005. These biannual conferences provide a forum to review the latest work on the interaction between computational methods and experimental measurements. New types of experiments allowing for more reliable interpretation of physical systems, result in a better virtual representation of reality. Experimental results are themselves increasingly dependent on specialised computer codes. It is only through the harmonious progressive development of the experimental and computational fields that engineering sciences will be able to progress. This volume contains over 80 papers grouped in the following sections. Computational and Analytical Methods; Experimental and Computational Analysis; Direct, indirect and In-Situ Measurements; Particle Methods; Structural and Stress Analysis; Structural Dynamics; Dynamics and Vibrations; Electrical and Electromagnetic Applications; Bioengineering Applications; Heat Transfer; Thermal Processes and Fluid Flow. WIT Transactions on Modelling and Simulation, Vol 41 ISBN: 1-84564-020-9 2005 944pp £330.00/US$528.00/€495.00 Find us at http://www.witpress.com


Elastic and Elastoplastic Contact Analysis

New Solutions in Contact Mechanics

Using Boundary Elements and Mathematical Programming

The result of around 20 years of research by the author, this book features some previously unpublished solutions that will be useful for scientific investigation and mechanical design. A boundary element algorithm for contact with friction is discussed and a demonstration version with 800 contact points is included on an accompanying CD-ROM. All of the chapters are more or less self-contained, while the derivations used are suitable for undergraduate students. Readers will also find new information which may aid further developments in this field. Contents: Introduction; Summary of the Basic Equations; Solutions for Plane Contact; Solutions for Axi-Symmetric Profiles; General Contact and Comparison with FEM and BEM Programs; Computation of Contact Problems; Gauss-Seidel Method for Frictional Contact Problems; Numerical Results for Incremental Load-Histories; Basic Equations and Solutions for Impact; Tangential and Torsional Impact of Spheres; Numerical Results for Impact; Applications.

A. FARAJI, University of Tehran, Iran This book presents a general elastic and elastoplastic analysis method for the treatment of two- and three-dimensional contact problems between two deformable bodies undergoing small displacements with and without friction. The author’s approach uses the Boundary Element Method (BEM) and Mathematical Programming (MP). This is applied to the contacting bodies pressed together by a normal force which is sufficient to expand the size of the contact area, mainly through plastic deformation, acted on subsequently by a tangential force less than that necessary to cause overall sliding. The formulated method described in this book is straightforward and applicable to practical engineering problems. Series: Topics in Engineering, Vol 45 ISBN: 1-85312-733-7 2005 144pp £48.00/US$86.00/€72.00

WIT eLibrary Home of the Transactions of the Wessex Institute, the WIT electronic-library provides the international scientific community with immediate and permanent access to individual papers presented at WIT conferences. Visitors to the WIT eLibrary can freely browse and search abstracts of all papers in the collection before progressing to download their full text. Visit the WIT eLibrary at http://library.witpress.com

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Computational Methods in Contact Mechanics VI Edited by: C.A. BREBBIA, Wessex Institute of Technology, UK Including discussions of mechanical models, numerical aspects, experimental measurements and engineering applications, this volume features the proceedings of the Sixth International Conference on Computational Methods and Experimental Measurements in Contact Mechanics. WIT Transactions on Engineering Sciences, Vol 38 ISBN: 1-85312-963-1 2003 232pp £89.00/US$142.00/€133.50

Computational Methods and Experimental Measurements XI Edited by: C.A. BREBBIA, Wessex Institute of Technology, UK, G.M. CARLOMAGNO, University of Naples, Italy and P. ANAGNOSTOPOULOS, University of Thessaloniki, Greece This book consists of over 70 papers from the eleventh international conference on this subject. WIT Transactions on Modelling and Simulation, Vol 33 ISBN: 1-85312-969-0 2003 776pp £232.00/US$371.00/€348.00

All prices correct at time of going to press but subject to change. WIT Press books are available through your bookseller or direct from the publisher.

Applied Numerical Analysis M. RAHMAN, Dalhousie University, Canada This book presents a clear and well-organised treatment of the concepts behind the development of mathematics and numerical techniques. The central topic is numerical methods and the calculus of variations to physical problems. Based on the author’s course taught at many universities around the world, the text is primarily intended for undergraduates in electrical, mechanical, chemical and civil engineering, physics, applied mathematics and computer science. Many sections are also directly relevant to graduate students in the mathematical and physical sciences. More than 100 solved problems and approximately 120 exercises are also featured. ISBN: 1-85312-891-0 2004 408pp+CD-ROM £149.00/US$238.00/€223.50

Moving Boundaries VII Computational Modelling of Free and Moving Boundary Problems Edited by: A.A. MAMMOLI, University of New Mexico, USA and C.A. BREBBIA, Wessex Institute of Technology, UK Features papers presented at the seventh international conference on this topic. WIT Transactions on Modelling and Simulation, Vol 36 ISBN: 1-85312-987-9 2004 392pp £129.00/US$206.00/€193.50 Find us at http://www.witpress.com


Mathematical Methods with Applications M. RAHMAN, Dalhousie University, Canada “This well-thought-out masterpiece has come from an author with considerable experience in teaching mathematical methods in many universities all over the world. His text will certainly appeal to a broader audience, including upper-level undergraduates and graduate students in engineering, mathematics, computer science, and the physical sciences. A fantastic addition to all college libraries.” CHOICE

Chosen by Choice as an “Outstanding Academic Title”, this book is a clear and well-organized description of the mathematical methods required for solving physical problems. The author focuses in particular on differential equations applied to physical problems. Many practical examples are used and an accompanying CD-ROM features exercises, selected answers and an appendix with short tables of Z-transforms, Fourier, Hankel and Laplace transforms. ISBN: 1-85312-847-3 2000 456pp + CD-ROM £175.00/US$269.00/€262.50

Introduction to Regression Analysis M.A. GOLBERG, Las Vegas, Nevada, USA and H.A. CHO, University of Nevada, USA In order to apply regression analysis effectively, it is necessary to understand both the underlying theory and its practical application. This book explores conventional topics as well as recent practical

developments, linking theory with application. Intended to continue from where most basic statistics texts end, it is designed primarily for advanced undergraduates, graduate students and researchers in various fields of engineering, chemical and physical sciences, mathematical sciences and statistics. Contents: Some Basic Results in Probability and Statistics; Simple Linear Regression; Random Vectors and Matrix Algebra; Multiple Regression; Residuals, Diagnostics and Transformations; Further Applications of Regression Techniques; Selection of a Regression Model; Multicollinearity: Diagnosis and Remedies; Appendix. ISBN: 1-85312-624-1 2004 452pp £122.00/US$195.00/€183.00

Introduction to Multi-Disciplinary Model-Building H.G. NATKE, Curt-Risch-Institute, Germany “.…a stimulating overview.”

MEASUREMENT + CONTROL

Designed for use by scientists and practitioners in a wide range of disciplines, this title highlights the advantages of model-building based on systems theory. It includes hints, methods and tools for obtaining reliable mathematical models and avoiding pitfalls ISBN: 1-85312-954-2 2002 396pp £119.00/US$184.00/€178.50 We are now able to supply you with details of new WIT Press titles via E-Mail. To subscribe to this free service, or for information on any of our titles, please contact the Marketing Department, WIT Press, Ashurst Lodge, Ashurst, Southampton, SO40 7AA, UK Tel: +44 (0) 238 029 3223 Fax: +44 (0) 238 029 2853 E-mail: [email protected]

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Inductive Reasoning: Experimental, Developmental, and Computational Approaches

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Computational Methods in Photochemistry

Molecular Physics: Theoretical Principles and Experimental Methods

Molecular Physics: Theoretical Principles and Experimental Methods

Computational Methods for Geodynamics

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Computational Methods and Experimental Measurements XIV (WIT Transactions on Modelling and Simulation) (Wit Transactions on Modeling and Simulation; Fourteenth ... Methods and Experimental Measurements)

Computational Methods and Experimental Measurements XV (Wit Transactions on Modelling and Simulation)

Geophysics: Field Measurements (Experimental Methods in the Physical Sciences)

Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods

Bridge and Potentiometer Methods of Electrical Measurements

Engineering Measurements Methods and Intrinsic Errors

Experimental Methods in Biology

Environmental Economics, Experimental Methods

Inductive Reasoning: Experimental, Developmental, and Computational Approaches

Electrocatalysis.. Computational, Experimental, and Industrial Aspects

Multiscale Modeling in Solid Mechanics: Computational Approaches (Computational and Experimental Methods in Structures)

Computer Methods and Experimental Measurements for Surface Effects and Contact Mechanics

Computational Intelligence: methods and techniques

Computational geometry: methods and applications

Computational Photography: Methods and Applications

Computational Intelligence: Methods and Techniques

Computational Neuroanatomy: Principles and Methods

Computational Neuroanatomy: Principles and Methods

Computational Methods And Function Theory

Transactions on Computational Systems Biology XIII

Methods of experimental physics. Classical methods

Physical Optics and Light Measurements, Volume 26 (Methods in Experimental Physics)

Computational Methods in Photochemistry

Molecular Physics: Theoretical Principles and Experimental Methods

Molecular Physics: Theoretical Principles and Experimental Methods

Computational Methods for Geodynamics

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