Lifetime-Oriented Structural Design Concepts

Lifetime-Oriented Structural Design Concepts Friedhelm Stangenberg · Rolf Breitenbücher Otto T. Bruhns · Dietrich Har...

Author: Friedhelm Stangenberg

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Lifetime-Oriented Structural Design Concepts

Friedhelm Stangenberg · Rolf Breitenbücher Otto T. Bruhns · Dietrich Hartmann Rüdiger Höffer · Detlef Kuhl Günther Meschke (Eds.)

Lifetime-Oriented Structural Design Concepts

ABC

Prof. Dr.-Ing. Friedhelm Stangenberg Ruhr-University Bochum Institute for Reinforced and Prestressed Concrete Structures Universitätsstr. 150 44780 Bochum, Germany E-mail: sandra.krimpmann@ ruhr-uni-bochum.de, friedhelm.stangenberg@ ruhr-uni-bochum.de

Prof. Dr.-Ing. Dietrich Hartmann Ruhr-University Bochum Institute for Computational Engineering Universitätsstr. 150 44780 Bochum, Germany

Prof. Dr.-Ing. Rolf Breitenbücher Ruhr-University Bochum Institute for Building Materials Universitätsstr. 150 44780 Bochum, Germany

Prof. Dr.-Ing. Detlef Kuhl University of Kassel Institute of Mechanics and Dynamics Mönchebergstr. 7 34109 Kassel, Germany

Prof. Dr.-Ing. Otto T. Bruhns Ruhr-University Bochum Institute of Mechanics Universitätsstr. 150 44780 Bochum, Germany

Prof. Dr.-Ing. Günther Meschke Ruhr-University Bochum Institute for Structural Mechanics Universitätsstr. 150 44780 Bochum, Germany

ISBN 978-3-642-01461-1

e-ISBN 978-3-642-01462-8

Prof. Dr.-Ing. Rüdiger Höffer Ruhr-University Bochum Building Aerodynamics Laboratory Universitätsstr. 150 44780 Bochum, Germany

DOI 10.1007/978-3-642-01462-8 Library of Congress Control Number: Applied for c 2009 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting by the Author. Production: Scientific Publishing Services Pvt. Ltd., Chennai, India. Cover Design: WMX Design GmbH, Heidelberg. Printed in acid-free paper 30/3100/as 5 4 3 2 1 0 springer.com

For Our Students, Colleagues and Engineers in Industry and Academia

The Team of SFB 398 Mark Alexander Ahrens • Hussein Alawieh • Matthias Baitsch • Falko Bangert • Yavuz Ba¸sar • Christian Becker • Ivanka Bevanda • J¨ org Bockhold • Ndzi Christian Bongmba • Dietrich Braess • Rolf Breitenb¨ ucher • Otto T. Bruhns • Christian Duckheim • Andreas Eckstein • Frank Ensslen • Olaf Faber • M´ ozes Gálffy • Volkmar G¨ ornandt • Jaroslaw Gorski • Stefan Grasberger • Klaus Hackl • Ulrike Hansk¨ otter • Gerhard Hanswille • Dietrich Hartmann • Anne Hartmann • Gunnar Heibrock • Martin Heiderich • Jan Helm • Christa Hermichen • Erich Heymer • R¨ udiger H¨ offer • Norbert Hölscher • Jan-Hendrik Hommel • Wolfgang Hubert • Hur¸sit Ibuk • Mikhail Itskov • Hans-Ludwig Jessberger • Daniel Jun • Dirk Kamarys • Michael Kasperski • Christoph Kemblowski •Olaf Kintzel • Andreas S. Kompalka • Diethard K¨ onig • Karsten K¨ onke • Stefan Kopp • Wilfried B. Kr¨ atzig • Sandra Krimpmann • Jens Kruschwitz • Detlef Kuhl • Jan Laue • Armin Lenzen • Roland Littwin • Ludger Lohaus • Dimitar Mancevski • G¨ unther Meschke • Kianoush Molla-Abbassi • J¨ orn Mosler • Stephan M¨ uller • Thomas Nerzak • Hans-J¨ urgen Niemann • Andrzej Niemunis • Sam-Young Noh • Markus Peters • Lasse Petersen • Yuri Petryna • Daniel Pfanner • Tobias Pfister • Gero Pflanz • Igor Plazibat • Rainer P¨ olling • Markus Porsch • Thorsten Quent • Stefanie Reese • Christian Rickelt • Matthias Roik • Jan Saczuk • J¨ org Sahlmen • E. Scholz • Henning Sch¨ utte • Robert Schwetzke • Max J. Setzer • Bj¨ orn Siebert • Anne Spr¨ unken • Friedhelm Stangenberg • Zoran Stankovic • Sascha Stiehler • Mathias Strack • Helmut Stumpf • Theodoros ¨ undag • Heinz Waller • Claudia Walter • Heiner Triantafyllidis • Cenk Ust¨ Weber • Gisela Wegener • Andrés Wellmann Jelic • Torsten Wichtmann • Xuejin Xu • Natalia Yalovenko

Preface

At the beginning of 1996, the Cooperative Research Center SFB 398 financially supported by the German Science Foundation (DFG) was started at Ruhr-University Bochum (RUB). A scientists group representing the fields of structural engineering, structural mechanics, soil mechanics, material science, and numerical mathematics introduced a research program on “lifetimeoriented design concepts on the basis of damage and deterioration aspects”. Two scientists from neighbourhood universities, one from Wuppertal and the other one from Essen, joined the Bochum Research Center, after a few years. The SFB 398 was sponsored for 12 years, until the beginning of 2008 – this is the maximum possible duration of DFG financial support for an SFB. Safety and reliability are important for the whole expected service duration of an engineering structure. Therefore, prognostical solutions are needed and uncertainties have to be handled. A differentiation according to building types with different service life requirements is necessary. Life-cycle strategies to control future structural degradations by concepts of appropriate design have to be developed, in case including means of inspection, maintenance, and repair. Aspects of costs and sustainability also matter. The importance of structural life-cycle management is well recognized in the international science community. Therefore, parallel corresponding activities are proceeding in many countries. In Germany, two other related SFBs were established: SFB 524 “Materials and Structures in Revitalisation of Buildings” at Weimar University and the still running SFB 477 “LifeCycle Assessment of Structures via Innovative Monitoring” at Braunschweig University of Technology. With these two SFBs, a fruitful cooperation was developed. The Cooperative Research Center for Lifetime-Oriented Design Concepts (SFB 398) at Ruhr-University has carried out substantial work in many fields of structural lifetime management. Lifetime-related fundamentals are provided with respect to structural engineering, structural and soil mechanics, material science as well as computational methods and simulation techniques. Stochastic aspects and interactions between various influences are included.

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Thus, a solid basis is provided for future practical use and, e.g. also for standardization. The wide range of scientific topics among the specification and determination of external loading and the simulation based lifetime-oriented structural design concepts is presented in an extraordinary format. All scientists of the SFB 398, professors and Ph.D. students, have contributed with a great effort in matchless team work to the present book. As a result of this, the present work is not only a collection of project reports, in fact it is almost written in the style of a monograph, whereby several authors fruitfully interact in all sections from the highest to the deepest level. Within this philosophy of joint authorship, authors are denoted in chapters and sections down to the third level. In special cases, where authors have contributed to a selected deeper section level, they are denoted beside the standard procedure in the regarding text episode. All members of SFB 398, with sincere thanks, acknowledge the financial support of DFG over more than 12 years. The dedicated research work of all participating colleagues and of many guest scientists from diverse countries also is gratefully mentioned. Finally, the great efforts of Springer-Verlag, Heidelberg, to produce this attractive volume is appreciated very much.

Bochum, March 26th, 2009

Friedhelm Stangenberg, Chairman of SFB 398 Otto T. Bruhns, Vice-chairman of SFB 398

Contents

1

Lifetime-Oriented Design Concepts . . . . . . . . . . . . . . . . . . . . . . 1.1 Lifetime-Related Structural Damage Evolution . . . . . . . . . . . . 1.2 Time-Dependent Reliability of Ageing Structures . . . . . . . . . . 1.3 Idea of Working-Life Related Building Classes . . . . . . . . . . . . . 1.4 Economic and Further Aspects of Service-Life Control . . . . . . 1.5 Fundamentals of Lifetime-Oriented Design . . . . . . . . . . . . . . . .

1 1 3 4 5 7

2

Damage-Oriented Actions and Environmental Impact . . . . 2.1 Wind Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Wind Buffeting with Relation to Fatigue . . . . . . . . . . . 2.1.1.1 Gust Response Factor . . . . . . . . . . . . . . . . . . . . 2.1.1.2 Number of Gust Effects . . . . . . . . . . . . . . . . . . . 2.1.2 Influence of Wind Direction on Cycles of Gust Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2.1 Wind Data in the Sectors of the Wind Rosette . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2.2 Structural Safety Considering the Occurrence Probability of the Wind Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2.3 Advanced Directional Factors . . . . . . . . . . . . . 2.1.3 Vortex Excitation Including Lock-In . . . . . . . . . . . . . . . 2.1.3.1 Relevant Wind Load Models . . . . . . . . . . . . . . 2.1.3.2 Wind Load Model for the Fatigue Analysis of Bridge Hangers . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Micro and Macro Time Domain . . . . . . . . . . . . . . . . . . . 2.1.4.1 Renewal Processes and Pulse Processes . . . . . 2.2 Thermal Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 General Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Thermal Impacts on Structures . . . . . . . . . . . . . . . . . . .

9 9 10 11 14 18 19

22 23 25 27 29 33 34 35 35 35

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2.2.3 Test Stand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Modelling of Short Term Thermal Impacts and Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Application: Thermal Actions on a Cooling Tower Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Transport and Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Traffic Loads on Road Bridges . . . . . . . . . . . . . . . . . . . . 2.3.1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1.2 Basic European Traffic Data . . . . . . . . . . . . . . 2.3.1.3 Basic Assumptions of the Load Models for Ultimate and Serviceability Limit States in Eurocode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1.4 Principles for the Development of Fatigue Load Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1.5 Actual Traffic Trends and Required Future Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Aerodynamic Loads along High-Speed Railway Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2.1 Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2.2 Dynamic Load Parameters . . . . . . . . . . . . . . . . 2.3.2.3 Load Pattern for Static and Dynamic Design Calculations . . . . . . . . . . . . . . . . . . . . . . 2.3.2.4 Dynamic Response . . . . . . . . . . . . . . . . . . . . . . . 2.4 Load-Independent Environmental Impact . . . . . . . . . . . . . . . . . 2.4.1 Interactions of External Factors Influencing Durability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Frost Attack (with and without Deicing Agents) . . . . . 2.4.2.1 The ”Frost Environment”: External Factors and Frost Attack . . . . . . . . . . . . . . . . . 2.4.2.2 Damage Due to Frost Attack . . . . . . . . . . . . . . 2.4.3 External Chemical Attack . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3.1 Sulfate Attack . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3.2 Calcium Leaching . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Geotechnical Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Settlement Due to Cyclic Loading . . . . . . . . . . . . . . . . . 2.5.2 Multidimensional Amplitude for Soils under Cyclic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Deterioration of Materials and Structures . . . . . . . . . . . . . . . 3.1 Phenomena of Material Degradation on Various Scales . . . . . 3.1.1 Load Induced Degradation . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1.1 Quasi Static Loading in Cementitious Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 40 43 46 46 46 47

52 62 73 79 80 82 87 90 92 93 95 96 103 106 107 107 109 109 114 123 124 124 124

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3.1.1.1.1 Fracture Mechanism of Concrete Subjected to Uniaxial Compression Loading . . . . . . . . . . . 3.1.1.1.2 Fracture Mechanism of Concrete Subjected to Uniaxial Tension Loadings . . . . . . . . . . . . . . . . . . . . . . 3.1.1.1.3 Concrete under Multiaxial Loadings . . . . . . . . . . . . . . . . . . . . . . 3.1.1.2 Cyclic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1.2.1 Ductile Mode of Degradation in Metals . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1.2.2 Quasi-Brittle Damage . . . . . . . . . . . 3.1.1.2.2.1 Cementitious Materials . . . . . . . . . . . . 3.1.1.2.2.2 Metallic Materials . . . . 3.1.2 Non-mechanical Loading . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2.1 Thermal Loading . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2.1.1 Degradation of Concrete Due to Thermal Incompatibility of Its Components . . . . . . . . . . . . . . . . . . . 3.1.2.1.2 Stresses Due to Thermal Loading . . . . . . . . . . . . . . . . . . . . . . . 3.1.2.1.3 Temperature and Stress Development in Concrete at the Early Age Due to Heat of Hydration . . . . . . . . . . . . . . . . . . . . . 3.1.2.2 Thermo-Hygral Loading . . . . . . . . . . . . . . . . . . 3.1.2.2.1 Hygral Behaviour of Hardened Cement Paste . . . . . . . . . . . . . . . . . . 3.1.2.2.2 Influence of Cracks on the Moisture Transport . . . . . . . . . . . . . 3.1.2.2.3 Freeze Thaw . . . . . . . . . . . . . . . . . . . 3.1.2.3 Chemical Loading . . . . . . . . . . . . . . . . . . . . . . . 3.1.2.3.1 Microstructure of Cementitious Materials . . . . . . . . . . . . . . . . . . . . . . 3.1.2.3.2 Dissolution . . . . . . . . . . . . . . . . . . . . . 3.1.2.3.3 Expansion . . . . . . . . . . . . . . . . . . . . . 3.1.2.3.3.1 Sulphate Attack on Concrete and Mortar . . . . . . . . . . . . . . 3.1.2.3.3.2 Alkali-Aggregate Reaction in Concrete . . . . . . . . . . . . 3.1.3 Accumulation in Soils Due to Cyclic Loading: A Deterioration Phenomenon? . . . . . . . . . . . . . . . . . . . . . .

XI

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125 126 129 129 131 131 137 140 140

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3.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Laboratory Testing of Structural Materials . . . . . . . . . 3.2.1.1 Micro-macrocrack Detection in Metals . . . . . . 3.2.1.1.1 Electric Resistance Measurements . . . . . . . . . . . . . . . . . . 3.2.1.1.1.1 Introduction . . . . . . . . . 3.2.1.1.1.2 Measurement of the Electrical Resistance . . . . . . . . . . . 3.2.1.1.1.3 Calculation of the Electrical Resistance . . 3.2.1.1.1.4 Experiments . . . . . . . . . 3.2.1.1.1.5 Experimental Results . . . . . . . . . . . . . 3.2.1.1.2 Acoustic Emission . . . . . . . . . . . . . . 3.2.1.1.2.1 Location of Acoustic Emission Sources . . . . . . . . . . . . . 3.2.1.1.2.2 Linear Location of Acoustic Emission Sources . . . . . . . . . . . . . 3.2.1.1.2.3 Location of Sources in Two Dimensions . . . 3.2.1.1.2.4 Kaiser Effect . . . . . . . . 3.2.1.1.2.5 Experimental Procedures . . . . . . . . . . 3.2.1.1.2.6 Experimental Results . . . . . . . . . . . . . 3.2.1.2 Degradation of Concrete Subjected to Cyclic Compressive Loading . . . . . . . . . . . . . . . 3.2.1.2.1 Test Series and Experimental Strategy . . . . . . . . . . . . . . . . . . . . . . . 3.2.1.2.2 Degradation Determined by Decrease of Stiffness . . . . . . . . . . . . . 3.2.1.2.3 Degradation Determined by Changes in Stress-Strain Relation . . . . . . . . . . . . . . . . . . . . . . . 3.2.1.2.4 Adequate Description of Degradation by Fatigue Strain . . . . 3.2.1.2.5 Behaviour of High Strength Concrete and Air-Entrained Concrete . . . . . . . . . . . . . . . . . . . . . . . 3.2.1.2.6 Influence of Various Coarse Aggregates and Different Grading Curves . . . . . . . . . . . . . . . .

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165 166 166 167 169

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3.2.1.2.7 Cracking in the Microstructure Due to Cyclic Loading . . . . . . . . . . . 3.2.1.2.8 Influence of Single Rest Periods . . . 3.2.1.2.9 Sequence Effect Determined by Two-Stage Tests . . . . . . . . . . . . . . . . 3.2.1.3 Degradation of Concrete Subjected to Freeze Thaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 High-Cycle Laboratory Tests on Soils . . . . . . . . . . . . . . 3.2.3 Structural Testing of Composite Structures of Steel and Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3.2 Basic Tests for the Fatigue Resistance of Shear Connectors . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3.2.1 Test Program . . . . . . . . . . . . . . . . . . 3.2.3.2.2 Test Specimens . . . . . . . . . . . . . . . . . 3.2.3.2.3 Test Setup and Loading Procedure . . . . . . . . . . . . . . . . . . . . . 3.2.3.2.4 Material Properties . . . . . . . . . . . . . 3.2.3.2.5 Results of the Push-Out Tests . . . . 3.2.3.2.5.1 General . . . . . . . . . . . . . 3.2.3.2.5.2 Results of the Constant Amplitude Tests . . . . . . . . . . . . . . . 3.2.3.2.6 Results of the Tests with Multiple Blocks of Loading . . . . . . . 3.2.3.2.7 Results of the Tests Regarding the Mode Control and the Effect of Low Temperature . . . . . . . . . . . . 3.2.3.2.8 Results of the Tests Regarding Crack Initiation and Crack Propagation . . . . . . . . . . . . . . . . . . . . 3.2.3.3 Fatigue Tests of Full-Scale Composite Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3.3.2 Test Program . . . . . . . . . . . . . . . . . . 3.2.3.4 Test Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3.5 Test Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3.6 Material Properties . . . . . . . . . . . . . . . . . . . . . . 3.2.3.7 Main Results of the Beam Tests . . . . . . . . . . . 3.3 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Load Induced Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1.1 Damage in Cementitious Materials Subjected to Quasi Static Loading . . . . . . . . . 3.3.1.1.1 Continuum-Based Models . . . . . . . .

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3.3.1.1.1.1 Damage MechanicsBased Models . . . . . . . . 3.3.1.1.1.2 Elastoplastic Models . . 3.3.1.1.1.3 Coupled ElastoplasticDamage Models . . . . . . 3.3.1.1.1.4 Multisurface ElastoplasticDamage Model for Concrete . . . . . . . . . . . . 3.3.1.1.2 Embedded Crack Models . . . . . . . . 3.3.1.2 Cyclic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1.2.1 Mechanism-Oriented Simulation of Low Cycle Fatigue of Metallic Structures . . . . . . . . . . . . . . . . . . . . . 3.3.1.2.1.1 Macroscopic Elasto-Plastic Damage Model for Cyclic Loading . . . . . . . 3.3.1.2.1.2 Model Validation . . . . . 3.3.1.2.2 Quasi-Brittle Damage in Materials . . . . . . . . . . . . . . . . . . . . . . 3.3.1.2.2.1 Cementitious Materials . . . . . . . . . . . . 3.3.1.2.2.2 Metallic Materials . . . . 3.3.2 Non-mechanical Loading and Interactions . . . . . . . . . . 3.3.2.1 Thermo-Hygro-Mechanical Modelling of Cementitious Materials - Shrinkage and Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2.1.1 Introductory Remarks . . . . . . . . . . . 3.3.2.1.2 State Equations . . . . . . . . . . . . . . . . 3.3.2.1.3 Identification of Coupling Coefficients . . . . . . . . . . . . . . . . . . . . 3.3.2.1.4 Effective Stresses . . . . . . . . . . . . . . . 3.3.2.1.5 Multisurface Damage-Plasticity Model for Partially Saturated Concrete . . . . . . . . . . . . . . . . . . . . . . . 3.3.2.1.6 Long-Term Creep . . . . . . . . . . . . . . . 3.3.2.1.7 Moisture and Heat Transport . . . . 3.3.2.1.7.1 Freeze Thaw . . . . . . . . . 3.3.2.2 Chemo-Mechanical Modelling of Cementitious Materials . . . . . . . . . . . . . . . . . . . 3.3.2.2.1 Models for Ion Transport and Dissolution Processes . . . . . . . . . . . .

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3.3.2.2.1.1 Introductory Remarks . . . . . . . . . . . . 3.3.2.2.1.2 Initial Boundary Value Problem . . . . . . . 3.3.2.2.1.3 Constitutive Laws . . . . 3.3.2.2.1.4 Migration of Calcium Ions in Water and Electrolyte Solutions . . . . . . . . . . . . 3.3.2.2.1.5 Evolution Laws . . . . . . 3.3.2.2.2 Models for Expansive Processes . . . 3.3.2.2.2.1 Introductory Remarks . . . . . . . . . . . . 3.3.2.2.2.2 Balance Equations . . . 3.3.2.2.2.3 Constitutive Laws . . . . 3.3.2.2.2.4 Model Calibration . . . . 3.3.3 A High-Cycle Model for Soils . . . . . . . . . . . . . . . . . . . . . 3.3.4 Models for the Fatigue Resistance of Composite Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4.2 Modelling of the Local Behaviour of Shear Connectors in the Case of Cyclic Loading . . . 3.3.4.2.1 Static Strength of Headed Shear Studs without Any Pre-damage . . . 3.3.4.2.2 Failure Modes of Headed Shear Studs Subjected to High-Cycle Loading . . . . . . . . . . . . . . . . . . . . . . . 3.3.4.2.3 Correlation between the Reduced Static Strength and the Geometrical Property of the Fatigue Fracture Area . . . . . . . . . . . 3.3.4.2.4 Lifetime - Number of Cycles to Failure Based on Force Controlled Fatigue Tests . . . . . . . . . 3.3.4.2.5 Reduced Static Strength over Lifetime . . . . . . . . . . . . . . . . . . . . . . . 3.3.4.2.6 Load-Slip Behaviour . . . . . . . . . . . . 3.3.4.2.7 Crack Initiation and Crack Development . . . . . . . . . . . . . . . . . . . 3.3.4.2.8 Improved Damage Accumulation Model . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4.2.9 Ductility and Crack Formation . . .

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3.3.4.2.10 Finite Element Calculations of the (Reduced) Static Strength of Headed Shear Studs in Push-Out Specimens . . . . . . . . . . . . 3.3.4.2.11 Effect of the Control Mode Effect of Low Temperatures . . . . . . 3.3.4.3 Modelling of the Global Behaviour of Composite Beams Subjected to Cyclic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4.3.1 Material Model for the Concrete Behaviour . . . . . . . . . . . . . . . . . . . . . 3.3.4.3.2 Effect of High-Cycle Loading on Load Bearing Capacity of Composite Beams . . . . . . . . . . . . . . . 3.3.4.3.3 Cyclic Behaviour of Composite Beams - Development of Slip . . . . . 3.3.4.3.4 Effect of Cyclic Loading on Beams with Tension Flanges . . . . . 3.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Durability Analysis of a Concrete Tunnel Shell . . . . . . 3.4.2 Durability Analysis of a Cementitious Beam Exposed to Calcium Leaching and External Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Durability Analysis of a Sealed Panel with a Leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Numerical Simulation of a Concrete Beam Affected by Alkali-Silica Reaction . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.5 Lifetime Assessment of a Spherical Metallic Container . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Methodological Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Classification of Deterioration Problems . . . . . . . . . . . . 4.1.2 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Generalization of Single- and Multi-field Models . . . . . 4.2.1.1 Integral Format of Balance Equations . . . . . . 4.2.1.2 Strong Form of Individual Balance Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Strategy of Numerical Solution . . . . . . . . . . . . . . . . . . . . 4.2.3 Weak Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3.1 Weak Form of Coupled Balance Equations . .

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4.2.3.2 Linearized Weak Form of Coupled Balance Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Spatial Discretization Methods . . . . . . . . . . . . . . . . . . . . 4.2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4.2 Generalized Finite Element Discretization of Multifield Problems . . . . . . . . . . . . . . . . . . . . 4.2.4.2.1 Approximations . . . . . . . . . . . . . . . . 4.2.4.2.2 Non-Linear Semidiscrete Balance . . . . . . . . . . . . . . . . . . . . . . . 4.2.4.2.3 Linearized Semidiscrete Balance . . 4.2.4.2.4 Generation of Element and Structural Quantities . . . . . . . . . . . . 4.2.4.3 p-Finite Element Method . . . . . . . . . . . . . . . . . 4.2.4.3.1 Onedimensional Higher-Order Shape Function Concepts . . . . . . . . 4.2.4.3.1.1 Shape Functions of the Legendre-Type . . . 4.2.4.3.1.2 Comparison of Both Shape Function Concepts . . . . . . . . . . . . 4.2.4.3.2 3D-p-Finite Element Method Based on Hierarchical Legendre Polynomials . . . . . . . . . . . . . . . . . . . . 4.2.4.3.2.1 Generation of 3D-p-Shape Functions . . . . . . . . . . . 4.2.4.3.2.2 Spatially Anisotropic Approximation Orders . . . . . . . . . . . . . . 4.2.4.3.2.3 Field-wise Choice of the Approximation Order . . . . . . . . . . . . . . . 4.2.4.3.2.4 Geometry Approximation . . . . . . . 4.2.5 Solution of Stationary Problems . . . . . . . . . . . . . . . . . . . 4.2.5.1 Numerical Solution Technique . . . . . . . . . . . . . 4.2.5.2 Iteration Methods . . . . . . . . . . . . . . . . . . . . . . . 4.2.5.3 Arc-Length Controlled Analysis . . . . . . . . . . . 4.2.6 Temporal Discretization Methods . . . . . . . . . . . . . . . . . . 4.2.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.6.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . 4.2.6.1.2 Newmark-α Time Integration Schemes . . . . . . . . . . . . . . . . . . . . . . .

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4.2.6.1.3 Galerkin Time Integration Schemes . . . . . . . . . . . . . . . . . . . . . . . 4.2.6.2 Newmark-α Time Integration Schemes . . . . . 4.2.6.2.1 Non-linear Semidiscrete Initial Value Problem . . . . . . . . . . . . . . . . . 4.2.6.2.2 Numerical Concept of Newmark-α Time Integration Schemes . . . . . . . . . . . . . . . . . . . . . . . 4.2.6.2.3 Time Discretization . . . . . . . . . . . . . 4.2.6.2.4 Approximation of State Variables . . . . . . . . . . . . . . . . . . . . . . 4.2.6.2.5 Algorithmic Semidiscrete Balance Equation . . . . . . . . . . . . . . . 4.2.6.2.6 Effective Balance Equation . . . . . . . 4.2.6.2.7 Newmark-α Algorithm . . . . . . . . . . 4.2.6.3 Discontinuous and Continuous Galerkin Time Integration Schemes . . . . . . . . . . . . . . . . 4.2.6.3.1 Time Discretization . . . . . . . . . . . . . 4.2.6.3.2 Continuity Condition . . . . . . . . . . . . 4.2.6.3.3 Temporal Weak Form . . . . . . . . . . . 4.2.6.3.4 Linearization . . . . . . . . . . . . . . . . . . . 4.2.6.3.5 Temporal Galerkin Approximation . . . . . . . . . . . . . . . . . 4.2.6.3.6 Discontinuous Bubnov-Galerkin Schemes dG(p) . . . . . . . . . . . . . . . . . 4.2.6.3.7 Continuous Petrov-Galerkin Schemes cG(p) . . . . . . . . . . . . . . . . . 4.2.6.3.8 Newton-Raphson Iteration . . . . . . . 4.2.6.3.9 Algorithmic Set-Up of Galerkin Schemes . . . . . . . . . . . . . . . . . . . . . . . 4.2.7 Generalized Computational Durabilty Mechanics . . . . 4.2.8 Adaptivity in Space and Time . . . . . . . . . . . . . . . . . . . . 4.2.8.1 Error-Controlled Spatial Adaptivity . . . . . . . . 4.2.8.1.1 Variational Functional . . . . . . . . . . . 4.2.8.1.2 Interpolation . . . . . . . . . . . . . . . . . . . 4.2.8.1.3 Stress Computation . . . . . . . . . . . . . 4.2.8.1.4 Discretized Weak Form . . . . . . . . . . 4.2.8.1.5 Summary . . . . . . . . . . . . . . . . . . . . . . 4.2.8.1.6 Hanging Node Concept . . . . . . . . . . 4.2.8.1.7 Error Criteria . . . . . . . . . . . . . . . . . . 4.2.8.1.7.1 Warping-Based Error Criterion . . . . . . 4.2.8.1.7.2 Residual-Based Error Criterion . . . . . . 4.2.8.1.8 Program Flow . . . . . . . . . . . . . . . . . .

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4.2.8.1.9 Transfer of History Variables . . . . . 4.2.8.1.10 Examples . . . . . . . . . . . . . . . . . . . . . . 4.2.8.1.10.1 Uniaxial Bending (Beam of Uniform Thickness) . . . . . . . . . . 4.2.8.1.10.2 Uniaxial Bending (Beam of Variable Thickness) . . . . . . . . . . 4.2.8.1.10.3 Biaxial Bending (Thick Plate of Uniform Thickness) . . 4.2.8.2 Error-Controlled Temporal Adaptivity . . . . . . 4.2.8.2.1 Local a Posteriori h- and p-Method Error Estimates . . . . . . . 4.2.8.2.2 Local a Posteriori h- and p-Method Error Indicators . . . . . . . 4.2.8.2.3 Local Zienkiewicz a Posteriori Error Indicators . . . . . . . . . . . . . . . . 4.2.8.2.4 Adaptive Time Stepping Procedure . . . . . . . . . . . . . . . . . . . . . 4.2.8.2.5 Algorithmic Set-Up . . . . . . . . . . . . . 4.2.9 Discontinuous Finite Elements . . . . . . . . . . . . . . . . . . . . 4.2.9.1 Overview and Motivation . . . . . . . . . . . . . . . . . 4.2.9.2 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.9.2.1 Extended Finite Element Method (XFEM) . . . . . . . . . . . . . . . 4.2.9.2.1.1 Partition of Unity . . . . 4.2.9.2.1.2 XFEM Displacement Field . . . . . . . . . . . . . . . 4.2.9.2.1.3 Integrating Discontinuous Functions . . . . . . . . . . . 4.2.9.2.1.4 p-Version of the XFEM . . . . . . . . . . . . . . 4.2.9.2.1.5 3D XFEM . . . . . . . . . . 4.2.9.2.1.6 XFEM for Cohesive Cracks . . . . . . . . . . . . . . 4.2.9.2.2 Strong Discontinuity Approach and Enhanced Assumed Strain . . . 4.2.9.2.2.1 Kinematics: Modeling Embedded Strong Discontinuities . . . . . . .

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4.2.9.2.2.2 Numerical Implementation . . . . . . 4.2.9.2.2.3 Numerical Example: 3-Point Bending Problem . . . . . . . . . . . . 4.2.9.3 Crackgrowth Criteria . . . . . . . . . . . . . . . . . . . . . 4.2.9.3.1 Hoop Stresses . . . . . . . . . . . . . . . . . . 4.2.9.3.2 Mode-I-Crack Extension . . . . . . . . . 4.2.9.3.3 Minimum Energy . . . . . . . . . . . . . . . 4.2.9.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.9.4.1 Double Notched Slab . . . . . . . . . . . . 4.2.9.4.2 Anchor Pull-Out . . . . . . . . . . . . . . . . 4.2.10 Substructuring and Model Reduction of Partially Damaged Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.10.1 Motivation and Overview . . . . . . . . . . . . . . . . . 4.2.10.2 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.10.3 Derivation of a Substructure Technique for Nonlinear Dynamics . . . . . . . . . . . . . . . . . . . . . . 4.2.10.3.1 Craig-Bampton Method . . . . . . . . . 4.2.10.3.2 Model Reduction of Linear Dynamic Structures . . . . . . . . . . . . . 4.2.10.3.2.1 Modal Reduction . . . . . 4.2.10.3.2.2 Proper Orthogonal Decomposition . . . . . . . 4.2.10.3.2.3 Padé-Via-Lanczos Algorithm . . . . . . . . . . . 4.2.10.3.2.4 Load-Dependent Ritz Vectors . . . . . . . . . 4.2.10.3.3 Substructuring in the Framework of Nonlinear Dynamics . . . . . . . . . . . . . . . . . . . . . . 4.2.10.3.3.1 Discretisation and Linearisation . . . . . . . . 4.2.10.3.3.2 Primal Assembly . . . . . 4.2.10.3.3.3 Solution of the Decomposed Structure . . . . . . . . . . . 4.2.10.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.11 Strategy for Polycyclic Loading of Soil . . . . . . . . . . . . . 4.3 System Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Covariance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Subspace Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2.1 State Space Model . . . . . . . . . . . . . . . . . . . . . . . 4.3.2.2 Subspace Identification . . . . . . . . . . . . . . . . . . . 4.3.2.3 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.4 Reliability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 General Problem Definition . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Time-Invariant Problems . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2.1 Approximation Methods . . . . . . . . . . . . . . . . . . 4.4.2.2 Simulation Methods . . . . . . . . . . . . . . . . . . . . . . 4.4.2.2.1 Importance Sampling . . . . . . . . . . . . 4.4.2.2.2 Latin Hypercube Sampling . . . . . . . 4.4.2.2.3 Subset Methods . . . . . . . . . . . . . . . . 4.4.2.3 Response Surface Methods . . . . . . . . . . . . . . . . 4.4.2.4 Evaluation of Uncertainties and Choice of Random Variables . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Time-Variant Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3.1 Time-Integrated Approach . . . . . . . . . . . . . . . . 4.4.3.2 Time Discretization Approach . . . . . . . . . . . . . 4.4.3.3 Outcrossing Methods . . . . . . . . . . . . . . . . . . . . . 4.4.4 Parallelization of Reliability Analyses . . . . . . . . . . . . . . 4.4.4.1 Reliability Analysis of Fatigue Processes . . . . 4.4.4.2 Parallelization Example . . . . . . . . . . . . . . . . . . 4.5 Optimization and Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Classification of Optimization Problems . . . . . . . . . . . . 4.5.2 Design as an Optimization Problem . . . . . . . . . . . . . . . . 4.5.3 Numerical Optimization Methods . . . . . . . . . . . . . . . . . 4.5.3.1 Derivative-Based Methods . . . . . . . . . . . . . . . . 4.5.3.2 Derivative-Free Strategies . . . . . . . . . . . . . . . . . 4.5.4 Parallelization of Optimization Strategies . . . . . . . . . . . 4.5.4.1 Parallelization with Gradient-Based Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.4.2 Parallelization Using Evolution Strategies . . . 4.5.4.3 Distributed and Parallel Software Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Application of Lifetime-Oriented Analysis and Design . . . . . . 4.6.1 Testing of Beam-Like Structures . . . . . . . . . . . . . . . . . . . 4.6.1.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . 4.6.1.2 Identification of Modal Data . . . . . . . . . . . . . . 4.6.1.3 Updating of the Finite Element Model . . . . . . 4.6.2 Lifetime Analysis for Dynamically Loaded Structures at BMW AG . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2.1 Works for the New 3-Series Convertible . . . . . 4.6.2.2 The Shaker Test . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2.3 Approach 1: Time History Calculation and Amplitude Counting . . . . . . . . . . . . . . . . . . . . . 4.6.2.3.1 Structural Analysis Using Time Integration . . . . . . . . . . . . . . . . . . . . 4.6.2.3.2 Cycle Counting Using the Rainflow Method . . . . . . . . . . . . . . .

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4.6.2.3.3 Damage Calculation . . . . . . . . . . . . . 4.6.2.4 Approach 2: Power Spectral Density Functions and Calculation of Spectral Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2.4.1 Structural Analysis Using Power Spectral Density (PSD) Functions . . . . . . . . . . . . . . . . . . . . . 4.6.2.4.2 Analytical Counting Method . . . . . 4.6.2.4.3 Damage Accumulation for the Analytical Case . . . . . . . . . . . . . . . . . 4.6.2.5 Comparison of the Results . . . . . . . . . . . . . . . . 4.6.2.6 Summary and Outlook . . . . . . . . . . . . . . . . . . . 4.6.3 Lifetime-Oriented Analysis of Concrete Structures Subjected to Environmental Attack . . . . . . . . . . . . . . . . 4.6.3.1 Hygro-Mechanical Analysis of a Concrete Shell Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3.1.1 Conclusive Remarks on the Hygro-Mechanical Analysis . . . . . . 4.6.3.2 Calcium Leaching of Cementitious Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3.2.1 Calcium Leaching of a Cementitious Bar . . . . . . . . . . . . . . . 4.6.3.2.1.1 Analysis of the Numerical Results . . . . 4.6.3.2.1.2 Adaptive Newmark Solution . . . . . . . . . . . . 4.6.3.2.1.3 Robustness of Galerkin Solutions . . . . 4.6.3.2.1.4 Error Estimates for Newmark Solutions . . . 4.6.3.2.1.5 Error Estimates for Galerkin Solutions . . . . 4.6.3.2.1.6 Order of Accuracy of Galerkin Schemes . . 4.6.3.2.2 Calcium Leaching of a Cementitious Beam . . . . . . . . . . . . . 4.6.3.2.2.1 Analysis of the Numerical Results . . . . 4.6.3.2.2.2 Robustness of Continuous Galerkin Solutions . . . . 4.6.4 Arched Steel Bridge Under Wind Loading . . . . . . . . . . 4.6.4.1 Definition of Structural Problem . . . . . . . . . . . 4.6.4.2 Probabilistic Lifetime Assessment . . . . . . . . . . 4.6.4.2.1 Micro Time Scale . . . . . . . . . . . . . . .

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4.6.4.2.2 Macro Time Scale . . . . . . . . . . . . . . Results of Structural Optimization . . . . . . . . . Parallelization of Analyses . . . . . . . . . . . . . . . . Final Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . Reinforced Concrete Bridge . . . . . . . . . . . . . . . . Numerical Simulation . . . . . . . . . . . . . . . . . . . . 4.6.5.1.1 Experimental Investigation on Mechanical Concrete Properties . . 4.6.5.1.1.1 Non-destructive Tests . . . . . . . . . . . . . . . 4.6.5.1.1.2 Destructive Tests . . . . . 4.6.5.1.1.3 Microscopic Analysis . . . . . . . . . . . . 4.6.5.1.1.4 Cyclic Tests . . . . . . . . . 4.6.5.1.2 Finite Element Model . . . . . . . . . . . 4.6.5.1.3 Material Model . . . . . . . . . . . . . . . . . 4.6.5.1.4 Damage Mechanisms . . . . . . . . . . . . 4.6.5.1.4.1 Corrosion of the Reinforcement Steel Bars . . . . . . . . . . . . . . . . 4.6.5.1.4.2 Fatigue of the Prestressing Tendons . . . . . . . . . . . . 4.6.5.1.5 Modelling of Uncertainties . . . . . . . 4.6.5.1.5.1 Long-Term Developement of Concrete Strength . . . . 4.6.5.1.5.2 Determination of Material Properties . . . 4.6.5.1.5.3 Modelling of Spatial Scatter by Random Fields . . . . . . . . . . . . . . 4.6.5.1.6 Lifetime Simulation . . . . . . . . . . . . . 4.6.5.1.7 Conclusions . . . . . . . . . . . . . . . . . . . . 4.6.5.2 Experimental Verification . . . . . . . . . . . . . . . . . 4.6.5.2.1 State Space Model for Mechanical Structures . . . . . . . . . . . 4.6.5.2.2 White Box Model - Physical Interpretable Parameters . . . . . . . . 4.6.5.2.3 Identification of Measured Mechanical Structures . . . . . . . . . . . 4.6.5.2.3.1 Black Box Model Deterministic System Identification . . . . . . . .

4.6.4.3 4.6.4.4 4.6.4.5 4.6.5 Arched 4.6.5.1

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XXIV

Contents

4.6.5.2.3.2 Differences between Theory and Experiment . . . . . . . . . 4.6.5.2.4 Experiments . . . . . . . . . . . . . . . . . . . 4.6.5.2.4.1 Cantilever Bending Beam . . . . . . . . . . . . . . . 4.6.5.2.4.2 Tied-Arch Bridge near H¨ unxe - Germany . . . . 4.6.5.2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . 4.6.6 Examples for the Prediction of Settlement Due to Polycyclic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Future Life Time Oriented Design Concepts . . . . . . . . . . . . . 5.1 Exemplary Realization of Lifetime Control Using Concepts as Presented Here . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Reinforced Concrete Column under Fatigue Load . . . . 5.1.2 Connection Plates of an Arched Steel Bridge . . . . . . . . 5.1.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Lifetime-Control Provisions in Current Standardization . . . . . 5.3 Incorporation into Structural Engineering Standards . . . . . . .

638 641 641

642 645 646 653 653 653 655 658 658 659

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711

List of Figures

1.1 1.2 1.3 1.4 1.5 1.6 1.7 2.1 2.2

2.3 2.4 2.5

2.6 2.7 2.8 2.9 2.10

Lifetime-related aspects of structural concrete . . . . . . . . . . . . . . . Evolution of degradation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time-dependent reliability of structures . . . . . . . . . . . . . . . . . . . . . Time-dependent reliability of structures with upgrading by repairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Working-life related building classes . . . . . . . . . . . . . . . . . . . . . . . . Service Life control and economic aspects . . . . . . . . . . . . . . . . . . . Related Collaborative Research Centers . . . . . . . . . . . . . . . . . . . . . Typical wind load process (a), and related low frequency (b) and high frequency (c) response of a structure [572] . . . . . . . . . . Curve of the total variance of the base bending moment of a cantilever due to buffeting excitation plotted over frequencies [572] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of the occurence of repeated wind effects at different locations in Germany and a codified representation . . . Distribution of absolute frequencies of normalized gust responses into subsequent classes of different levels of effect . . . . Comparison of the distribution of cyclic stress amplitudes with the S-N curve (Wöhler curve) of stress concentration category 36* after [30] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rosettes of wind quantities at Hannover (12 sectors, 50 years return period) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Roughness lengths of the terrain in the farther vicinity of the building location [771] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sketch of a building contour and fa¸cade element exposed to a pressure coefficient cp = −1.4 [32] . . . . . . . . . . . . . . . . . . . . . . . . . . ´rma ń vortex trail formed by vortex shedding . . . . . . . . Von Ka Dependence of the vortex shedding frequency fv on the wind velocity u ¯. fn is the natural frequency of the structure . . . . . . . .

2 2 3 4 5 6 6

10

12 15 17

17 18 21 21 26 27

XXVI

2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23

2.24 2.25 2.26 2.27 2.28 2.29 2.30 2.31 2.32 2.33 2.34 2.35

List of Figures

Wind velocity, measured and simulated deflection vs. time for the bridge hanger 1 (left) and 2 (right) . . . . . . . . . . . . . . . . . . Width of the lock-in range for bridge tie rods . . . . . . . . . . . . . . . . Measured and simulated amplitude of the displacement within and outside of the lock-in range . . . . . . . . . . . . . . . . . . . . . Sample realizations of a renewal process (left) and of a pulse-process (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wavelength of the visible light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Climatic load on a structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test stand for the analysis of thermal actions on concrete specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measured temperature profile during a summer day . . . . . . . . . . Rainflow analysis of the macroscopic temperature behaviour . . . Temperature behaviour due to a sudden change in solar radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature distributions determined at 16 layers within a cooling tower shell under constant external load actions . . . . . . . Effect of the mean wind speed on the development of the temperature difference of a cooling tower shell . . . . . . . . . . . . . . . Frequency distribution of the total weight G of the representative lorries per 24 hours based on traffic data of Auxerre in France (1986) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gross vehicle and axle weight distribution of recorded traffic data from England, France and Germany . . . . . . . . . . . . . . . . . . . Histogram of vehicle Type 3 and approximation by two separate distribution functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of measured and theoretical values for the density function of intervehicle distances . . . . . . . . . . . . . . . . . . . . Model for the vehicles and local irregularities and power spectral density of the pavement . . . . . . . . . . . . . . . . . . . . . . . . . . . Eigenvalues of the first mode of steel and concrete Bridges [169] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cumulative frequency of the action effects for different vehicle speeds [530] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Influence of the quality of the pavement on the dynamic amplification factor ϕ [530] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Influence of the span length and the number of loaded lanes on the dynamic amplification factor ϕ . . . . . . . . . . . . . . . . . . . . . . Additional dynamic factor Δϕ taking into account irregularities of the pavement [9] . . . . . . . . . . . . . . . . . . . . . . . . . . . Determination of the characteristic values of the action effects from the random generations of loads . . . . . . . . . . . . . . . . . Load Model 1 according to Eurocode 1-2 . . . . . . . . . . . . . . . . . . . . Comparison of the Load Model 1 in Eurocode -2 with the characteristic values obtained from real traffic simulations . . . . .

30 32 33 34 38 38 40 41 42 43 44 45

48 48 49 51 53 54 55 56 56 57 58 59 59

List of Figures

2.36 2.37 2.38 2.39 2.40 2.41 2.42 2.43 2.44 2.45 2.46 2.47 2.48 2.49 2.50 2.51 2.52 2.53 2.54

2.55 2.56 2.57 2.58 2.59 2.60 2.61

XXVII

Determination of the representative values and the corresponding dynamic factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Factors ψT R for frequent design situations acc. to [37] for average pavement quality with Φ(Ωh ) = 16 . . . . . . . . . . . . . . . . . . Influence of the pavement quality on the factor ΨT R for frequent design situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Determination of stress spectra and damage accumulation due to fatigue loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fatigue strength curves for structural steel and reinforcement . . Typical examples for fatigue strength categories . . . . . . . . . . . . . . Set of lorries of Fatigue Load Model 4 in Eurocode -2 and contact surfaces of the wheels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distribution of transverse location of centre line of vehicles and dynamic load amplification factor near expansion joints . . . Linear damage accumulation and damage equivalent dynamic amplification factor ϕf at . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Influence of the pavement quality on the damage equivalent dynamic amplification factor [530] . . . . . . . . . . . . . . . . . . . . . . . . . . Fatigue Load model 3 in Eurocode 1-2 and fatigue verification for steel structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example for the damage equivalent factor λe [530] . . . . . . . . . . . Determination of the damage equivalent factor λ1 . . . . . . . . . . . . Factors λ1 for steel bridges given in Eurocode 3-2 . . . . . . . . . . . . Assumptions for the factor λ4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Damage equivalent factor λmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . Development of the freight traffic on roads, railways and ships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Development of the number of heavy vehicles per day and relative frequency of the gross weight for articulated vehicles . . Development of the number of permits of heavy transports in Bavaria and North-Rhine Westphalia and examples for vehicles for heavy transports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Traffic records from the Netherlands recorded in 2006 . . . . . . . . Heavy vehicles on the basis of the modular concept (Giga-Liners) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axle spacing and allowable axle weights of ”Giga-Liners” . . . . . Pressure time history at the track-side face of a 8 m high wall; at a fixed position; V = 234.3 km/h, [573] . . . . . . . . . . . . . . Pressure distribution along the track-side face of a wall at two different train speeds [573] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Full scale tests performed along the high speed line Cologne-Rhine/Main . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Effect of train speed stagnation pressure on the head pulse acting at the track-side face of a wall . . . . . . . . . . . . . . . . . . . . . . .

61 61 62 64 64 65 67 67 68 68 69 70 70 71 73 73 74 75

76 77 78 78 80 81 81 83

XXVIII

2.62 2.63 2.64 2.65 2.66 2.67 2.68 2.69 2.70 2.71 2.72 2.73 2.74 2.75 2.76 2.77 2.78 2.79 2.80

2.81 2.82 2.83 2.84

2.85 2.86 2.87 2.88

List of Figures

Pressure coefficients of the head pulse from 34 passages (at the track-side wall face) at 1.65 m above track level . . . . . . . Distance between the pulse peaks and the zero crossing (ΔL1 = pressure maximum, ΔL2 = pressure minimum) . . . . . . . . . . . . Head pulse in a free flow at various distances from the track axis [98] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Head pulse in the presence of a wall . . . . . . . . . . . . . . . . . . . . . . . . Load pattern over the height of the wall . . . . . . . . . . . . . . . . . . . . Variation of the time lag between maxima and minima of the head pulse over the wall height transformed to V = 300 m/s . . . Load factor for the load distribution over the height of the wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pattern of pressure coefficients cp for the ICE-3 train . . . . . . . . . Noise protection wall and mode shape of the 1st mode . . . . . . . . Time history of post top displacement calculated for a post in the middle of the wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resonant amplification of the displacement maximum vs. the natural frequency at train speeds between 200 and 300 km/h . . Resonant amplification of the displacement minimum vs. the natural frequency at train speeds between 200 and 300 km/h . . Schematic diagram - Interaction of climate, environmental attack and damage process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of reinforcement corrosion and concrete corrosion . . . Attacks on concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Surface of frost damaged concrete in situ . . . . . . . . . . . . . . . . . . . . Microcracking of cement paste(left); ESEM image of frost damaged concrete (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Field exposure (left); Modified multi-ring electrode (right) . . . . Effects at specific depths of water penetration(left); Dependence of Arrhenius factor b on moisture content (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Air temperature and rainfall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Freeze-thaw cycle illustrated by example (left); Temperature curve during thaw phase on November 26 (right) . . . . . . . . . . . . . Exemplary illustration of the change in resistance at depth level 6.6 cm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequency of freeze-thaw cycles depending on minimum temperature (left) and maximum cooling and thawing rates (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . External damage of concrete specimens after one winter . . . . . . Correlation between surface scaling and degree of visual damage on field exposed specimens . . . . . . . . . . . . . . . . . . . . . . . . . Development of external damage . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of the surface scaling obtained in laboratory and in field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84 84 85 85 87 88 88 89 90 91 92 92 93 94 95 96 97 97

98 99 100 101

103 103 104 105 106

List of Figures

2.89 2.90 2.91 2.92 2.93 2.94 2.95 2.96 2.97 2.98 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19

XXIX

Concrete damage caused by thaumasite . . . . . . . . . . . . . . . . . . . . . Corrosion on mortar coatings in two drinking water reservoirs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sources of cyclic loading of soils . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cyclic stresses in a soil element a) due to a passing wheel load and b) due to an earthquake loading . . . . . . . . . . . . . . . . . . . Accumulation of settlement due to cyclic loading . . . . . . . . . . . . . Decomposition of a signal with varying amplitudes into packages of cycles with constant amplitude . . . . . . . . . . . . . . . . . . Distinction between uniaxial IP-, multiaxial IP- and OOP-cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hodograph for detrending of a strain path . . . . . . . . . . . . . . . . . . Multiaxial amplitude definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex strain loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic stress-strain diagram of cementitous materials subjected to uniaxial compression [867] . . . . . . . . . . . . . . . . . . . . . Schematic stress-strain diagram of cementitous respectively geological materials due to tension [538] . . . . . . . . . . . . . . . . . . . . Stress-displacement diagram of a concrete specimen subjected to cyclic tensile loading [381] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Biaxial failure envelope for concrete [467, 567] . . . . . . . . . . . . . . . Stress-displacement diagrams obtained from triaxial compression tests for three levels of confining pressure σ2 . . . . . Failure surface of concrete in principal stress space and crack patterns corresponding to different triaxial loading conditions . . Ductile fracture surfaces of a round notched bar after 30 cycles with notch radius 2mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) Void nucleation due to fracture of inclusions, (b) partition of inclusion-matrix-area, (c) void coalescence . . . . . . . . . . . . . . . . Schematic S-N curves for concrete (W¨ ohler curves) . . . . . . . . . . . Fatigue fracture of concrete specimens due to cyclic compression load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Number of cycles to failure Nf for different load levels and their variation [627] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stress-strain relation of concrete measured after different number of cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Development of total longitudinal strain with the cycle ratio (N/Nf ) [383] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Change of secant modulus of elasticity [383] . . . . . . . . . . . . . . . . . Development of the value of the residual strength [70] . . . . . . . . W¨ ohler curves for tensile loads [207] . . . . . . . . . . . . . . . . . . . . . . . . W¨ ohler curves for flexural loads [865] . . . . . . . . . . . . . . . . . . . . . . . Development of strains in tensile loading [207] . . . . . . . . . . . . . . . Development of strains in bending [662] . . . . . . . . . . . . . . . . . . . . .

108 108 110 111 111 113 115 116 117 119

125 127 127 128 128 129 130 130 131 132 132 133 134 135 135 136 137 137 138

XXX

3.20 3.21 3.22 3.23 3.24 3.25

3.26 3.27 3.28 3.29 3.30 3.31 3.32 3.33 3.34 3.35 3.36

3.37 3.38 3.39 3.40 3.41 3.42 3.43 3.44 3.45 3.46

List of Figures

Degradation process of relevant concrete properties due to tensile loadings [429] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Degradation process of relevant concrete properties due to flexural loadings [866] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stiffness reduction by high cycle fatigue . . . . . . . . . . . . . . . . . . . . . Model for brittle damage by microcrack growth . . . . . . . . . . . . . . Stresses in a concrete slab at one-sided, non-linear cooling from the top [145] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature and stress development during the first hydration phase in restrained concrete elements [763, 145, 466] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hygric strains vs. relative humidity . . . . . . . . . . . . . . . . . . . . . . . . . Hygric strains vs. relative humidity & vs. water content . . . . . . . Hygric strains vs. surface free energy change . . . . . . . . . . . . . . . . . Hygric strains vs. surface free energy change & comparison between measured and calculated hygric strains . . . . . . . . . . . . . . Sorption isotherms vs. relative humidity . . . . . . . . . . . . . . . . . . . . Solid density vs. relative humidity . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic diagram of hygric mechanisms and properties of hardened cement paste . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of macroscopic and microscopic situation of the micro-ice-lens model during the heating and cooling phase . . . . Volume fractions of constituents of hardened cement paste as a function of the water cement ratio [448] . . . . . . . . . . . . . . . . . . . Schematic illustration of the dissolution- and loading induced long-term deterioration of concrete . . . . . . . . . . . . . . . . . . . . . . . . . Equilibrium states between the calcium concentration s and the ratio c/s: experimental [114, 115] and analytical [307, 308] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decrease of compressive strength as a function of the increase in porosity resulting from calcium leaching [172] . . . . . . . . . . . . . Expansion behaviour of flat mortar prisms with Portland cement during storage in sodium sulfate solution [502] . . . . . . . . Alkali-silica reaction damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Accumulation of stress or strain, illustrated for the two-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolution of the electrical resistance vs. number of cycles during fatigue - plain and circular specimen . . . . . . . . . . . . . . . . . Evolution of the electrical resistance during fatigue - plain specimen block-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrical potential - plane specimen . . . . . . . . . . . . . . . . . . . . . . . . Electrical potential - circular specimen . . . . . . . . . . . . . . . . . . . . . . Evolution of electrical resistance vs. crack length during fatigue - plain and circular specimen . . . . . . . . . . . . . . . . . . . . . . . Waveform parameters for a burst-signal . . . . . . . . . . . . . . . . . . . . .

138 139 139 140 141

142 144 144 145 146 146 147 148 149 151 153

154 155 158 159 161 167 167 168 168 169 170

List of Figures

3.47 3.48 3.49 3.50 3.51 3.52 3.53 3.54 3.55 3.56 3.57 3.58 3.59 3.60 3.61 3.62 3.63 3.64 3.65 3.66 3.67 3.68 3.69 3.70 3.71 3.72 3.73 3.74 3.75 3.76 3.77

XXXI

Location of the source in two dimensions . . . . . . . . . . . . . . . . . . . . Geometry of the plain specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometry of the circular specimen . . . . . . . . . . . . . . . . . . . . . . . . . The position of AE-transducers on the plain specimen . . . . . . . . The position of AE-transducers on the circular specimen . . . . . . Rate of event counts during fatigue - plain specimen . . . . . . . . . . Rate of event counts during fatigue - circular specimen . . . . . . . Total event counts during fatigue - plain specimen . . . . . . . . . . . Total event counts during fatigue - circular specimen . . . . . . . . . Origin of acoustic emission - plain specimen . . . . . . . . . . . . . . . . . Origin of acoustic emission - circular specimen . . . . . . . . . . . . . . . Acoustic emission event counts vs. amplitude - plain specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acoustic emission event counts vs. amplitude - circular specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acoustic emission event counts vs. frequency - plain specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acoustic emission event counts vs. frequency - circular specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Execution of the single-stage and two-stage test . . . . . . . . . . . . . Decrease and scatter of Estat at Smax /Smin = 0.675/0.10 (single-state-tests) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decrease and scatter of Edyn at Smax /Smin = 0.675/0.10 (single-state-tests) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variation of maximal bearable number of load cycles to failure Nf [627] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measured longitudinal strain at Smax (Smax /Smin = 0.60/0.10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stress-strain curves at different number of cycles (Smax /Smin = 0.60/0.10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total strain at Smax /Smin = 0.675/0.10 . . . . . . . . . . . . . . . . . . . . Calculation of fatigue strain at Smax . . . . . . . . . . . . . . . . . . . . . . . Formation of fatigue strain (schematically) . . . . . . . . . . . . . . . . . . Fatigue strain at Smax /Smin = 0.675/0.10 . . . . . . . . . . . . . . . . . . . Correlation between the fatigue strain and the residual stiffness for different load levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . Correlation between the fatigue strain and the residual stiffness of normal and high strength concrete . . . . . . . . . . . . . . . Correlation between the fatigue strain and the residual stiffness of normal and air-entrained concrete . . . . . . . . . . . . . . . . Correlation between the fatigue strain and the residual stiffness subjected to different aggregates in concrete . . . . . . . . . Correlation between the fatigue strain and the residual stiffness subjected to different grading curves in concrete . . . . . . Light microscopy micrographs . . . . . . . . . . . . . . . . . . . . . . . . . . . .

172 173 173 174 174 175 175 176 176 177 177 178 178 179 179 181 182 183 183 184 184 186 186 187 187 188 188 189 189 190 191

XXXII

3.78 3.79 3.80 3.81 3.82 3.83 3.84

3.85 3.86 3.87 3.88 3.89 3.90 3.91 3.92 3.93 3.94 3.95 3.96 3.97 3.98 3.99 3.100 3.101 3.102 3.103 3.104 3.105 3.106 3.107

List of Figures

Load history with various rest periods [150] . . . . . . . . . . . . . . . . . Behaviour of the longitudinal strain at Smax /Smin = 0.675/0.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Related longitudinal strain at Smax /Smin = 0.675/0.10 . . . . . . . Correlation between the fatigue strain and the residual stiffness subjected to different sequences of cyclic loading . . . . . Steps of exposure and measuring during CDF/CIF test [731] . . Example relationship between RDM and relative moisture uptake - concrete type 2 [610] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Internal damage due to freeze-thaw cycles at several depths of the specimen (left), Moisture uptake vs. number of freeze-thaw cycles (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test devices and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cyclic flow rule (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cyclic flow rule (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intensity of accumulation in drained cyclic element tests on soils (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intensity of accumulation in drained cyclic element tests on soils (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Influence of the grain size distribution curve on D acc . . . . . . . . . . Undrained cyclic tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of cycles at σ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Application of headed shear studs in composite bridges . . . . . . . Load-deflection behaviour of headed shear studs embedded in solid concrete slabs under static loading . . . . . . . . . . . . . . . . . . Fatigue strength curve for cyclic loaded headed shear studs according [685] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Safety concept to determine the lifetime of composite structures subjected to high cycle loading . . . . . . . . . . . . . . . . . . . Tests with multiple blocks of loading . . . . . . . . . . . . . . . . . . . . . . . Tests to compare the effect of the mode control - force control vs. displacement control - and the effect of low temperature . . . Duration of the crack initiation phase and crack growth velocity due to very low cyclic loads [685] . . . . . . . . . . . . . . . . . . . Details of the push-out test specimen . . . . . . . . . . . . . . . . . . . . . . . Servo hydraulic actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Position of transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Development of plastic slip over the fatigue life in series S1 - S4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decrease of static strength vs. lifetime due to high cycle loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test programme and loading parameters of the composite beam tests VT1 and VT2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Details of test beam VT1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Details of test beam VT2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

192 192 193 194 195 195

197 199 200 201 202 204 205 206 207 208 209 210 211 213 215 216 216 217 218 220 221 226 228 229

List of Figures

XXXIII

3.108 Test setup of test beams VT1 and VT2 . . . . . . . . . . . . . . . . . . . . . 3.109 Electric circuit to detect complete shear failure of headed studs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.110 Change of initial deflections due to cyclic loading . . . . . . . . . . . . 3.111 Load-deflection behaviour of test beams VT1 and VT2 in the static tests after cyclic loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.112 Experimental determination of the reduced static strength of the steel section near midspan after high cycle pre-loading . . . . 3.113 Slip along the interfaces of steel and concrete after first loading and after cyclic loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.114 Crack lengths at the stud feet after the cyclic loading phase Preparation stages for examination purposes . . . . . . . . . . . . . . . . 3.115 Representation of different failure surfaces in the principal strain space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.116 Stress-strain diagrams for uniaxial compressive and tensile loading obtained from the damage model by Mazars . . . . . . . . 3.117 Anisotropic damage model by [604]: Illustration of the failure surface in the principal stress space, see eq. (3.29) . . . . . . . . . . . . 3.118 Definition of a local coordinate system and decomposition of the traction vector t = into the normal part tn and the tangential part tm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.119 Anisotropic elastoplastic damage model by [534]: Influence of the scalar coupling parameter β on the stress-strain diagram . . 3.120 Yield conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.121 Stress-strain relation of concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.122 Discrete representation of cracks: Traction separation law of the format t = t( u ) across the crack surface . . . . . . . . . . . . . . 3.123 Strong Discontinuity Approach: Additive decomposition of the displacement field u (equation (3.84)) . . . . . . . . . . . . . . . . . . . 3.124 Strong Discontinuity Approach: Strain field resulting from ¯ (x) + u ˆ (x) . . . . . . . . . . . . . . . . . . . the displacement field u(x) = u 3.125 Model-based concept for life time assessment of metallic structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.126 Numerical and experimental data for (a) material softening and (b) ratcheting effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.127 Low Cycle Fatigue in metals: Numerical and experimental results for cyclically loaded round notched bar . . . . . . . . . . . . . . . 3.128 Low Cycle Fatigue in metals: Damage accumulation and predicted damage in a cyclically loaded round notched bar . . . . 3.129 S-N -approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.130 Degradation of compressive strength and sequence effects . . . . . 3.131 Evaluation of the approach for sequence effects . . . . . . . . . . . . . . 3.132 Rheological element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.133 Fatigue strain evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.134 Split of fatigue strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

230 231 234 234 235 236 237 239 240 242

243 246 247 249 253 254 254 257 259 260 261 263 263 264 265 267 268

XXXIV

List of Figures

3.135 Evaluation of the split variable β fat . . . . . . . . . . . . . . . . . . . . . . . . 3.136 Kinked crack and its equivalent elliptical crack . . . . . . . . . . . . . . . 3.137 Growth of the circular crack and its equivalent elliptical crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.138 Order of the considered sequential loading . . . . . . . . . . . . . . . . . . . 3.139 Evolution of the geometry and the orientations of the equivalent elliptical crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.140 Evolution of the stiffness components in the principle directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.141 Specimen geometry and different mesh patterns . . . . . . . . . . . . . . 3.142 Load-cycle curves for different mesh patterns . . . . . . . . . . . . . . . . 3.143 Chemo-mechanical damage of porous materials within the Theory of Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.144 Conductivity of the pore fluid D0 and macroscopic conductivity of non-reactive porous media φD0 . . . . . . . . . . . . . . 3.145 Chemical equilibrium function by G´ erard [307, 308] and Delagrave et al. [232] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.146 Microstructure, constituents and volume fractions of concrete as a partially saturated porous media . . . . . . . . . . . . . . . . . . . . . . . 3.147 Chemical material parameters k and u / r − 1 and of their dependence on the liquid saturation sl . . . . . . . . . . . . . . . . . . . . . . 3.148 Theoretical model for the prediction of the mean value of the ultimate shear resistance according [684] . . . . . . . . . . . . . . . . . . . . 3.149 Result of the statistical analysis of the results of 101 statically loaded push-out tests according to EN 1990 [16] . . . . . . . . . . . . . 3.150 Comparison of the result of the statistical analysis with the rules in current German and European standards . . . . . . . . . . . . 3.151 Preparation stages for examination purposes . . . . . . . . . . . . . . . . 3.152 Failure modes A and B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.153 Weld collar - Close-up view of the crack shown in Figure 3.152 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.154 Correlation between reduced static strength and damage at the stud feet based on the fatigue fracture area . . . . . . . . . . . . . . 3.155 Correlation between reduced static strength and damage at the stud feet based on crack lengths . . . . . . . . . . . . . . . . . . . . . . . . 3.156 Comparison of fatigue test results with the prediction in Eurocode 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.157 Model for the prediction of the fatigue life of a headed shear stud in a push-out test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.158 (a) Reduced static strength over lifetime, (b) Comparison of the reduced static strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.159 Load-slip curve of headed shear studs - load deflection behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.160 Effect of high-cycle loading on the load-slip behaviour . . . . . . . . 3.161 Elastic stiffness and accumulated plastic slip . . . . . . . . . . . . . . . . .

268 277 279 280 281 282 283 284 295 299 302 303 312 317 322 324 324 325 326 327 328 329 331 331 332 333 334

List of Figures

XXXV

3.162 Relationship between crack velocity, crack propagation and reduction of static strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.163 Fatigue strength and lifetime of cyclic loaded shear studs . . . . . 3.164 Comparison between the test results with the lifetime prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.165 Damage accumulation considering the load sequence effects . . . . 3.166 Damage accumulation in the case of multiple block loading tests with decreasing peak loads . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.167 Comparison between the test results with the results of the lifetime prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.168 Ductility after high cycle loading . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.169 Comparison between test results and finite element calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.170 Comparison between test results and finite element calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.171 Test series S9 - Effect of control mode - Effect of low temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.172 Failure surface of the improved material model CONCRETE . . 3.173 Comparison between the results of numerical simulations and test results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.174 Test beam VT1 - Effect of high cycle loading on load bearing capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.175 Cyclic behaviour of test beam VT1 . . . . . . . . . . . . . . . . . . . . . . . . . 3.176 Test beam VT2 - Effect of high cycle loading - Typical crack formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.177 Geometry of a tunnel lining subjected to cyclic hygral and thermal loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.178 Evolution of the crack width w of a tunnel lining subjected to cyclic hygral and thermal loading . . . . . . . . . . . . . . . . . . . . . . . . 3.179 Scalar damage measure d at the crown of a tunnel lining subjected to cyclic hygral and thermal loading . . . . . . . . . . . . . . . 3.180 Liquid saturation Sl at the crown of a tunnel lining subjected to cyclic hygral and thermal loading . . . . . . . . . . . . . . . . . . . . . . . . 3.181 Simulation of a cementitious beam exposed to calcium leaching and mechanical loading . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.182 Temporal evolution of the vertical displacement us of the cementitious beam and prediction of the collapse . . . . . . . . . . . . . 3.183 Chemo-mechanical analysis of a concrete panel: Conditions . . . . 3.184 Chemo-mechanical analysis of a concrete panel: Results I . . . . . 3.185 Chemo-mechanical analysis of a concrete panel: Results II . . . . . 3.186 Numerical simulation of a concrete beam affected by alkali-silica reaction: Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.187 Numerical simulation of a concrete beam affected by alkali-silica reaction: Results I . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

335 336 338 339 340 340 341 342 343 345 347 348 348 350 351 352 352 353 354 355 355 356 358 359 360 361

XXXVI

List of Figures

3.188 Numerical simulation of a concrete beam affected by alkali-silica reaction: Results II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.189 Numerical simulation of a concrete beam affected by alkali-silica reaction: Results III . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.190 Low Cycle Fatigue Model: (a) Spherical pressure vessel, (b) Vertical displacement-time plot of the El Centro earthquake . . . 3.191 Low Cycle Fatigue Model: (a) Damage accumulation (El Centro earthquake), (b) Temporal evolution of the maximal void volume fraction f . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13

4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21

Overview of the methodological implementation of lifetime oriented design concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical modeling and general multiphysics problem . . . . . . . . Modeling and numerical analysis of multiphysics problems . . . . Illustration of isotropic Lagrange shape functions . . . . . . . . . . . Illustration of anisotropic Lagrange shape functions . . . . . . . . Computation of generalized element tensors of multiphysics p-finite elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sinusoidial loading of a truss member and rel. error of internal energy plotted over the number of dof . . . . . . . . . . . . . . . Modified Legendre-polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of high order shape function concepts . . . . . . . . . . . . Comparison of the structure of element vectors and matrices for the Legendre- and Lagrange-concept . . . . . . . . . . . . . . . . . 3D-p-element: definition and numbering of element vertices (Ni ), edges (Ei ) and faces (Fi ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3D-p-shape functions: nodal, edge, face and internal modes for different polynomial degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure types, corresponding classical finite element models and 3D-p finite element models with spatially anisotropic approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hygro-thermo-mechanical loading of a structural segment, Fieldwise anisotropic discretization using the p-FEM . . . . . . . . . Discretization of the standard structures (truss, slab, shell) into an infinite numbers of elements . . . . . . . . . . . . . . . . . . . . . . . . Relative reduction of system nodes/dof for different structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strategy for solving non-linear vector equation ri (u) = r . . . . . . Control of load factor and Newton-Raphson iteration . . . . . . Algorithmic set-up of the load controlled Newton-Raphson scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Illustration of arc-length methods and predictor step calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithmic set-up of the arc-length controlled Newton-Raphson scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

362 363 363

364

366 375 376 381 382 387 388 390 391 392 393 395

396 398 399 402 404 404 406 407 410

List of Figures

4.22 4.23 4.24 4.25 4.26 4.27 4.28 4.29 4.30 4.31 4.32 4.33 4.34 4.35 4.36 4.37 4.38 4.39 4.40 4.41

4.42

4.43 4.44 4.45

XXXVII

Design of Newmark type time integration schemes . . . . . . . . . . Illustration of Newmark and generalized mid-point approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithmic set-up of Newmark-α schemes including error controlled adaptive time stepping . . . . . . . . . . . . . . . . . . . . . . . . . . Galerkin time integration schemes . . . . . . . . . . . . . . . . . . . . . . . . Algorithmic set-up of discontinuous and continuous Galerkin time integration schemes . . . . . . . . . . . . . . . . . . . . . . . . Modular concept for multiphysics finite element programs . . . . . Example geometry and warping-based error criterion . . . . . . . . . Two-element example with two hanging nodes . . . . . . . . . . . . . . . Beam 1: Geometry and boundary conditions . . . . . . . . . . . . . . . . . Beam 1: Load-displacement curve for tolerr = 10−5 and crit1 (various nGP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beam 1: Different states of mesh refinement (Q1SPs/o, 16El.), contours: accumulated plastic strain . . . . . . . . . . . . . . . . . . . . . . . . Beam 1: Load-displacement curve and number of elements for tolerr = 10−7 and crit1 (various nGP0) . . . . . . . . . . . . . . . . . . . Beam 1: Load-displacement curve and number of elements for different tolerances and crit2 (Q1SPs/o, nGP0 = 16) . . . . . . . Beam 2: Load-displacement curve and number of elements for different tolerances and crit2 (Q1SPs/o, nGP = 16) . . . . . . . . Beam 2: Different states of mesh refinement (Q1SPs/o, 16 El.), contours: accumulated plastic strain . . . . . . . . . . . . . . . . . Plate 1: Geometry and boundary conditions . . . . . . . . . . . . . . . . . Plate 1: Load-displacement curve and number of elements for different tolerances and crit2 (Q1SPs, nGP = 8) . . . . . . . . . . . . . . Plate 1: Load-displacement curve for different tolerances and crit2 (Q1SPs, nGP = 8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plate 1: Different states of mesh refinement (Q1SPs/o, 16 El.), contours: accumulated plastic strain . . . . . . . . . . . . . . . . . Plate 1: Load-displacement curve and number of elements for different load steps and crit2 (Q1SPs/o, nGP = 8, tolerr = 0.01) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plate 1: Load-displacement curve and number of elements for different load steps and crit2 (Q1SPs, nGP = 8, tolerr = 0.0001) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Illustration of h- and p-method error estimates and indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithmic set-up for the error controlled adaptive time integration by Newmark-α schemes . . . . . . . . . . . . . . . . . . . . . . . Algorithmic set-up for the error controlled adaptive time integration by Newmark-α or p-Galerkin methods and h-method error estimates/indicators . . . . . . . . . . . . . . . . . . . . . . . .

413 414 417 418 423 425 432 434 435 435 436 437 438 438 439 440 440 441 441

442

442 443 447

447

XXXVIII

4.46

4.47 4.48 4.49 4.50 4.51 4.52 4.53 4.54 4.55 4.56 4.57 4.58 4.59 4.60 4.61 4.62 4.63 4.64 4.65 4.66 4.67 4.68 4.69 4.70

4.71 4.72 4.73 4.74 4.75 4.76 4.77 4.78

List of Figures

Algorithmic set-up for the error controlled adaptive time integration by p-Galerkin methods and p-method error estimates/indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Function to be approximated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approximation of equation (4.147) . . . . . . . . . . . . . . . . . . . . . . . . . Normal and tangential vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Four crack tip functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crack with one kink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crack after mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple kinked crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple kinked crack after the first mapping . . . . . . . . . . . . . . . . ˆ.............................. Point x and mirrored point x Strain ε from equation (4.173) for the integral (4.175) . . . . . . . . Number of integration points used in the numerical integration of (4.174) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strain ε from equation (4.173) for the integral (4.177) . . . . . . . . Number of integration points used in the numerical integration of (4.176) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strain ε from equation (4.173) for the integral (4.179) . . . . . . . . Number of integration points used in the numerical integration of (4.178) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strain ε from equation (4.173) for the integral (4.181) . . . . . . . . Number of integration points used in the numerical integration of (4.180) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strain ε from equation (4.173) for the integral (4.183) . . . . . . . . Number of integration points used in the numerical integration of (4.182) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strain ε from equation (4.173) for the integral (4.185) . . . . . . . . Number of integration points used in the numerical integration of (4.184) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tension test configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Displacements ux for the deformed system using bilinear shape functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Displacements ux for the deformed system, left: using bi-quadratic shape functions, right: using quadratic hierarchical shape functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Differences of displacements inside the 1st blending element . . . Differences of displacements inside the 2nd blending element . . Differences of displacements inside the 3rd blending element . . . Differences of displacements inside the 4th blending element . . . Differences of displacements inside the 5th blending element . . . Numerical integration in the context of X-FEM: Subdivision of the continuum element into six sub-tetrahedrons . . . . . . . . . . . Separation of a sub-tetrahedron by a plane crack segment . . . . . C0 -crack plane evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

448 450 451 452 453 454 456 456 457 458 462 462 463 463 465 465 466 466 467 467 468 468 469 470

470 471 471 472 472 473 475 475 476

List of Figures

4.79 4.80

XXXIX

Definition of the crack plane by point P and normal vector n . . Constant strain triangular element cut by means of a planar internal boundary ∂s Ω; see [745] . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.81 Enhanced discontinuous displacement field ru (Hs − ϕ): (a) bi-linear approximation (2 nodes in Ω + ); (b) bi-quadratic approximation (1 node in Ω + ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.82 Numerical study of a notched concrete beam: dimensions (in [cm]) and material parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.83 Numerical study of a notched concrete beam using the proposed multiple crack concept and the rotating crack approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.84 Sketch for the computation of the SIF for a kinking crack with r → 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.85 Schematic figure for the calculation of the SIF with constant radius for kinking cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.86 Sketch of KII (left) and |KII | (right) depending on the angle θ for a three point bending test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.87 Energy function Πtot for a three point bending test . . . . . . . . . . 4.88 Crack simulation of a double notched slab: System, material data and finite element mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.89 Crack simulation of a double notched slab: Visualization of the crack topology by the φ = 0-level set . . . . . . . . . . . . . . . . . . . . 4.90 Crack simulation of a double notched slab: Comparison of crack topology and of load-displacement curves . . . . . . . . . . . . . . 4.91 Bumerical investigation of crack propagation of an anchor pull-out test: System and finite element mesh (N E = 996) . . . . 4.92 Numerical investigation of crack propagation of an anchor pull-out test: Crack topology and displacement u3 in pull-out direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.93 Numerical investigation of crack propagation of an anchor pull-out test: Stress σ 33 at the beginning and the end of the crack process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.94 Numerical investigation of crack propagation of an anchor pull-out test: Load-displacement curve . . . . . . . . . . . . . . . . . . . . . . 4.95 Concept for the efficient simulation of dynamic, partially damaged structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.96 Decomposition of the structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.97 Geometry and loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.98 Exploded view of the bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.99 Damage evolution in the largest two hangers . . . . . . . . . . . . . . . . 4.100 Displacement in X2 -direction in point B . . . . . . . . . . . . . . . . . . . . 4.101 Mean relative displacement-based error in point B . . . . . . . . . . . 4.102 Comparison of a pure implicit and an explicit calculation of accumulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

477 481

482 486

488 491 491 492 493 494 495 495 496

497

497 498 501 507 513 514 515 516 516 518

XL

List of Figures

4.103 General definition of the failure domain depending on scattering resistance (R) and stress (S) values . . . . . . . . . . . . . . . 4.104 Standardization of an exemplary 2D joint distribution function for a subsequent FORM/SORM analysis . . . . . . . . . . . . 4.105 Comparison of Latin Hypercube Sampling and Monte-Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.106 Parallel execution of stochastically independent DC-MCS of fatigue analyses on a distributed memory architecture [824] . . . 4.107 Parallel software framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.108 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.109 Damage equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.110 Singular values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.111 1’st eigenfrequency and mode shape . . . . . . . . . . . . . . . . . . . . . . . . 4.112 2’nd eigenfrequency and mode shape . . . . . . . . . . . . . . . . . . . . . . . 4.113 3’rd eigenfrequency and mode shape . . . . . . . . . . . . . . . . . . . . . . . . 4.114 4’th eigenfrequency and mode shape . . . . . . . . . . . . . . . . . . . . . . . . 4.115 Cut modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.116 Optimization topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.117 The new 3-series convertible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.118 3-series convertible with battery . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.119 Battery as vibration absorber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.120 FE model of the shaker test arrangement . . . . . . . . . . . . . . . . . . . 4.121 Measured acceleration data for the y-direction . . . . . . . . . . . . . . . 4.122 Power spectral density function of the resulting von Mises stress for the elements of Figure 4.119, load direction y . . . . . . . 4.123 Dirlik distribution function of the stress amplitudes . . . . . . . . 4.124 Typical stress picture for load in y-direction (Time History Analysis) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.125 Expected life time in arbitrary time units for the Time History calculation (acceleration load in y-direction) . . . . . . . . . . 4.126 Hygro-mechanically loaded concrete shell structure: System geometry and material data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.127 Hygro-mechanically loaded concrete shell structure: Hygral boundary conditions of the inner and outer surface of the shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.128 Hygro-mechanically loaded concrete shell structure: Finite element mesh of the numerical analysis . . . . . . . . . . . . . . . . . . . . . 4.129 Hygro-mechanically loaded concrete shell structure: Deformation and stresses due to dead load . . . . . . . . . . . . . . . . . . 4.130 Hygro-mechanically loaded concrete shell structure: Distribution of the saturation Sl . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.131 Hygro-mechanically loaded concrete shell structure: Damage evolution at the support area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.132 Hygro-mechanically loaded concrete shell structure: Damage zone and accelerated transport process in the area of cracks . . .

529 532 536 545 561 562 563 564 564 565 565 566 566 570 573 574 574 575 576 577 579 581 582 584

584 585 586 587 588 588

List of Figures

4.133 Hygro-mechanically loaded concrete shell structure: Distribution of saturation Sl and damage variable d across the shell thickness (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.134 Hygro-mechanically loaded concrete shell structure: Distribution of saturation Sl and damage variable d across the shell thickness (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.135 Calcium leaching of a cementitious bar and a cementitious beam: Geometry, FE mesh and chemical loading history . . . . . . 4.136 Calcium leaching of a cementitious bar: Numerical results obtained from the cG(1) method . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.137 Calcium leaching of a cementitious bar: Numerical results and time integration error obtained from adaptive Newmark integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.138 Calcium leaching of a cementitious bar: Time histories c(t, X1 )/c0 obtained from dG(p)-integration (t [108 s], X1 [mm]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.139 Calcium leaching of a cementitious bar: Time histories c(t, X1 )/c0 obtained from cG(p)-integration (t [108 s], X1 [mm]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.140 Calcium leaching of a cementitious bar: Spatial local and global error estimates for Newmark time integrations . . . . . . . . 4.141 Calcium leaching of a cementitious bar: Logarithm of error estimates eΔt/5 for dG-methods with different time steps Δt . . . 4.142 Calcium leaching of a cementitious bar: Logarithm of error estimates ep/p+1 for dG-methods with different time steps Δt . . 4.143 Calcium leaching of a cementitious bar: Logarithm of error estimates ep/p+1 and eΔt/5 for cG-methods with different time steps Δt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.144 Calcium leaching of a cementitious bar: Average relative errors of the Newmark method and Galerkin methods . . . . . 4.145 Calcium leaching of a cementitious beam: Numerical results obtained from cG(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.146 Calcium leaching of a cementitious beam: Investigation of the oscillations in the results of cG(1)- and cG(2)-solutions . . . . 4.147 Calcium leaching of a cementitious beam: Investigation of the robustness of the cG(1)-solution for small Tc . . . . . . . . . . . . 4.148 Pictures of damaged road bridge in M¨ unster (Germany) and correspondent FE models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.149 Refined FE models of a connecting plate and the correspondent welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.150 Effective stress values of a connecting plate under a constant rod deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.151 Representative surface of partial damage values for varying wind and initial displacements at the critical tie rod . . . . . . . . . .

XLI

589

590 591 593

595

596

597 598 599 600

601 603 604 605 606 607 608 610 611

XLII

List of Figures

4.152 Time-dependent evolution of the failure probability of critical material points in the welding region and the bulk material . . . 4.153 Optimization model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.154 Optimization results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.155 The road bridge at H¨ unxe (Germany) shortly before its deconstruction in 2006 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.156 Location of prestressing tendons and crack pattern observed on the bridges main girders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.157 Location of drilling cores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.158 Comparison of stress-strain curves between bridge concrete and laboratory concretes with different strengths [193] . . . . . . . . 4.159 LM-micrograph of in-situ concrete . . . . . . . . . . . . . . . . . . . . . . . . . 4.160 Total longitudinal (left) and fatigue strain (right) at Smax . . . . . 4.161 Correlation between fatigue strain and the residual stiffness for Smax /Smin = 0.675/0.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.162 Three dimensional Finite Element model of the road bridge at H¨ unxe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.163 Applied corrosion model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.164 Modified S-N curves for steel and fatigue damage evolution function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.165 Higher order statistical moments . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.166 Validation of input data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.167 Evolution of compressive strength and histogram of concrete strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.168 Random field dependency on correlation length and eigenvalues used for reconstruction of correlation matrix . . . . . . 4.169 Load deflection diagram and time deflection diagram 3D . . . . . . 4.170 Load deflection curves and lifetime distribution and estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.171 State space model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.172 Impulse excitation in laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.173 Comparison between measured signals and signals from identified model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.174 Cantilever bending beam used for experiments in laboratory . . . 4.175 Drawing from the cantilever bending beam with the location of saw cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.176 Markov parameters for damage detection . . . . . . . . . . . . . . . . . . . 4.177 Bridge near H¨ unxe / Germany (span: 62.5m) . . . . . . . . . . . . . . . . 4.178 System modification: hanger cut through . . . . . . . . . . . . . . . . . . . . 4.179 Torsional mode from reference system and after cut hanger . . . . 4.180 Recalculation of a centrifuge model test of Helm et al. [365] . . . 4.181 Parametric studies of shallow strip foundations under cyclic loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.182 FE calculations with stochastically fluctuating void ratio . . . . . .

612 613 614 616 617 618 620 621 622 623 624 626 627 628 628 629 632 633 633 636 639 639 641 641 642 643 643 644 646 647 648

List of Figures

4.183 FE calculation of vibratory compaction and bridge settlements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.184 Calculation of pore water pressure accumulation due to earthquake loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.185 FE calculation of a geogrid-reinforced embankment . . . . . . . . . . . 4.186 FE calculation of a monopile foundation of an offshore wind power plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 5.2 5.3 5.4 5.5 5.6

Reinforced concrete column under fatigue loading . . . . . . . . . . . . Load-carrying-capacity and response surface method . . . . . . . . . Time-dependent hazard function and time-dependent reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multi-level system approach followed during the lifetime analysis of the arched steel bridge [826] . . . . . . . . . . . . . . . . . . . . . Multi-scale modeling and analysis of fatigue-related structural problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of resulting time-dependent failure probabilities of the researched connection plate . . . . . . . . . . . . . . . . . . . . . . . . . .

XLIII

649 650 651 651 654 654 655 656 657 658

List of Tables

2.1 2.2 2.3 2.4

2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14

3.1 3.2 3.3

Conversion of the wind data of the observation station at the airport of Hannover into data for the building location . . . . . . . . 20 Determination of a reduced characteristic suction force on the fa¸cade element after Figure 2.8 . . . . . . . . . . . . . . . . . . . . . . . . . 24 Statistical parameters of the traffic records of Auxerre (1986) . . 49 Relation between gross weight of the heavy vehicles and the axle weights of the lorries of types 1 to 4 in % (mean values and standard deviation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Distance of axles in [m] of the different types of vehicles (mean values and standard deviation) . . . . . . . . . . . . . . . . . . . . . . 50 Statistical parameters of the corrected static traffic records of Auxerre (1986) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Different cross-sections and traffic types for the random generations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Traffic data of different locations and characteristic values of gross and axle weight [720] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Different design situations and corresponding return periods and fractiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Factors Ψ for the determination of the representative values for serviceability limit states acc. to [9] . . . . . . . . . . . . . . . . . . . . . 63 Traffic categories acc. to Eurocode 1-2 . . . . . . . . . . . . . . . . . . . . . . 66 Statistical parameters of the traffic records at highway A61 (2004) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Relation between gross weight of the heavy vehicles and the axle weights of the lorries of types 1 to 5 (mean values) . . . . . . . 76 Readings: winter 05/06 and winter 06/07; field station Meißen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Classification of pore sizes in concrete according to [724] . . . . . . 151 Influences on the degree of chemical attack . . . . . . . . . . . . . . . . . . 152 Changes of concrete properties due to cyclic loading . . . . . . . . . . 185

XLVI

3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 3.25 3.26 3.27 3.28

4.1 4.2 4.3

List of Tables

Crack characteristics at certain number of cycles Smax /Smin = 0.675/0.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Correlation between frost suction and internal damage due to freeze-thaw testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of the single level tests . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of the tests with multiple blocks of loading . . . . . . . . . Mean values of material properties of concrete according to EN 206-1 [12] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mean values of material properties of steel members . . . . . . . . . . Average test results per stud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Loading parameters and results of the tests with two blocks of loading (series S5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Loading parameters and results of the tests with four blocks of loading (series S6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Average test results per stud in series S9 . . . . . . . . . . . . . . . . . . . . Measured mean values of the peak load and the load range at discrete number of load cycles in tests S9 4 . . . . . . . . . . . . . . . Loading parameters and block lengths in tests S9 5 . . . . . . . . . . . Average test results per stud in series S11 and S13 . . . . . . . . . . . Mean values of material properties of concrete according to EN 206-1 [12] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mean values of material properties of steel members . . . . . . . . . . Main test results of beams VT1 and VT2 . . . . . . . . . . . . . . . . . . . Parameter of the elasto-plastic micropore damage model for 20MnMoNi55 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low Cycle Fatigue in metals: Number of load cycles until failure obtained . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characteristics of the applied sequential loading . . . . . . . . . . . . . Summary of the functions, material constants and reference quantities of the high-cycle model . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of the statically loaded push-out tests with decisive criterion “failure of the concrete” (tests 1 - 27) . . . . . . . . . . . . . . Summary of the statically loaded push-out tests with decisive criterion “failure of the concrete” (tests 28 - 58) . . . . . . . . . . . . . Summary of the statically loaded push-out tests with decisive criterion “shear failure of the stud” . . . . . . . . . . . . . . . . . . . . . . . . . Result of the statistical analysis according EN 1990, Annex D [16] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mean values of the crack length ah (see Figure 3.155) in test series S11 and S13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

191 197 213 214 218 219 219 222 222 223 223 224 225 231 232 233 260 261 281 314 319 320 321 323 337

Multi-dimensional Lagrange shape functions . . . . . . . . . . . . . . . 382 Total number of geometric entities (vertices, edges, faces) of the discretizations with an infinite number of elements . . . . . . . . 399 Convergence criteria of iterative solution methods . . . . . . . . . . . . 405

List of Tables

4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14

4.15

4.16 4.17

4.18

4.19 4.20 4.21 4.22

XLVII

Comparison of iteration methods . . . . . . . . . . . . . . . . . . . . . . . . . . . Constraints and load factor increments of selected arc-length methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Error indicators for Newmark type time integration schemes for non-linear second order initial value problems . . . . . . . . . . . . Error indicators for Newmark type time integration schemes for non-linear first order initial value problems . . . . . . . . . . . . . . . Equivalent square sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modal Assurance Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gauss-Newton (cp /cd =800/1smm) . . . . . . . . . . . . . . . . . . . . . . . Gauss-Newton iteration (cp /cd =1400/1mm) . . . . . . . . . . . . . . . Results for an early design proposal . . . . . . . . . . . . . . . . . . . . . . . . Standard parameter set [307, 454, 457] . . . . . . . . . . . . . . . . . . . . . . Calcium leaching of a cementitious bar: Average relative errors of the Newmark method, discontinuous Galerkin methods and continuous Galerkin methods . . . . . . . . . . . . . . . . Type of random variables (RV) included in the reliability problem used to describe the scatter of wind load parameters as well as material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of resulting runtime values analyzing the connecting plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic elastic moduli Edyn (mean) and their standard deviations (SD) of the concrete after a service life of 50 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relevant mechanical concrete properties Estat , u and fc (mean values) as well as their standard deviations (SD) after a service life of 50 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Number of elements of structural members . . . . . . . . . . . . . . . . . . Determination of compressive strength at time of construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concrete strength grades according to German standards . . . . . Summary of the results of the FE calculations of strip foundations under cyclic loading . . . . . . . . . . . . . . . . . . . . . . . . . . .

406 409 445 446 514 569 571 571 582 592

602

609 615

619

619 624 629 630 647

1 Lifetime-Oriented Design Concepts

Authored by Friedhelm Stangenberg

1.1 Lifetime-Related Structural Damage Evolution Authored by Friedhelm Stangenberg Structures deteriorate during their lifetimes, e.g. their original quality decreases. In terms of structural safety, this reduces the original safety margin, a process, which also can be described as an increase of structural damage. If, in such deterioration, the safety parameter decreases below the admissible safety limit, or the structural damage parameter increases beyond the admissible damage limit, then the structural service life will be terminated. If the failure safety value or the structural damage parameter both reach unity, the structure (theoretically) will fail. The initial structural properties must have sufficient reserves, in order to compensate future reductions of safety against failure and of safety against reaching the limits of serviceability. The final structural properties, at the end of the service life or at the end of the relevant inspection interval, respectively, must include a minimum resistance safety, a minimum serviceability level, and other minimum qualities. Lifetime-related deteriorations can happen in various forms and can consist of various components. For example for structural concrete, the lifetimerelated aspects with influence on the long time structural behavior are listed in Figure 1.1. Deteriorations therein can be induced mechanically, e.g. by load cycles leading to fatigue effects. They can also be induced non-mechanically, e.g. by corrosion or other chemical processes. In combination, deterioration effects can be superimposed by addition, if there is no interaction between them. In case of interactions, the superposition can be more than additive because of amplification effects due to influences

2

1 Lifetime-Oriented Design Concepts

Appropriate Quality Assurance for structural design, detailing and execution QA is a very important overhead necessity, also for lifetime-oriented design concepts, in order to eliminate big mistakes and big errors a priori, as well as to make sure that certain tolerable deviations of structural qualities are not exceeded. This is a fundamental requirement for service-life control.

Durability Integrated from outside into SFB 398 models and concepts Resistance against abrasion

Chemical durability against substances penetrating from outside

Own basic SFB 398 research

particularly: corrosion protection of reinforcing steel in concrete: • carbonation • penetration of chlorides • further deteriorations

Resistance against frost (-thaw cycles)

Resistance against fatigue (cyclic loading)

Simulations for combined mechanical and chemical processes and propagations

Integrated design concepts

Fig. 1.1. Lifetime-related aspects of structural concrete

between each other. E.g. the longtime increase of concrete crack widths leads to increased penetration of chemically aggressive media, through the cracks

damage phenomena d

mechanically induced degradation

d

Ti

d1

T

Te

Ti d

Te

interactions included d2

d2

Te

Ti : intermediate time Te : end of observation time Fig. 1.2. Evolution of degradation

d1

T Ti

T

Te

interaction

non-mechanically induced degradation, e.g. corrosion

Ti

without interactions d2

d1

d

superimposed

T

1.2

Time-Dependent Reliability of Ageing Structures

3

into the structure, causing increased chemical deteriorations, e.g. increased corrosion in the reinforcing steel. On the other hand, this leads to increased steel weakening and, thus, vice-versa to a higher amplification of concrete crack widths, and so on. Therefore, combinations of interactive effects can lead to additional deteriorations, which would be underestimated by additive superposition only. This is illustrated, in principle, in Figure 1.2.

1.2 Time-Dependent Reliability of Ageing Structures and Methodological Requirements Authored by Friedhelm Stangenberg In the beginning of the structural lifetime, the safety against failure and against losses of other important structural qualities must have sufficient reserves. In the early lifetime, maybe e.g. concrete post-hardening may improve the safety situation for a while, but later on, deteriorations lead to safety reductions. At the end of the planned service period, a remaining minimum safety is still required. This time variant safety problem or reliability problem, respectively, is presented in Figure 1.3, in principle, where the time histories of resistances and actions, together with their statistical distributions, are plotted in relation to each other. Degradations of the resistances and maybe certain increases of the actions effect time-dependent safety losses. For analytical predictions of these developments, methods for time-dependent stochastic calculations are needed. resistance R

R, S

actions S

R(t1 ) R(td )

Safety margin (fractile-based)

S(t1 )

S(td )

time t t1

td = planned service period

Fig. 1.3. Time-dependent reliability of structures

mean lifetime (50% probability)

4

1

Lifetime-Oriented Design Concepts

R, S R(t1 )

upgrading by repairs

resistance R actions S

¯ 1) R(t safety margin (fractile-based)

S(t1 )

t1

R(td )

S(td )

time t td = planned service period

Fig. 1.4. Time-dependent reliability of structures with upgrading by repairs

In Figure 1.4, two alternatives of service-life control are compared. The planned service period td can be obtained, by starting on a high level of structural reserves R(t1 ), high enough to reach td , with enough remaining safety, and without repairs during the full service life. The other alternative is: to ¯ 1 ), but to upgrade by restart at a lower level of initially invested reserves R(t pairs, i.e. by additional investments, before td , before the minimum acceptable safety is undergone. The success at time td may be the same for both alternatives. The second alternative means initial savings, but additional investments later, maybe combined with temporary restrictions or losses of use during the repairs. It depends on the special conditions, what alternative should have preference.

1.3 Idea of Working-Life Related Building Classes Authored by Friedhelm Stangenberg Current design standards do not provide a satisfactory basis or procedure to ensure expected structural lifetimes. These may vary from only a few years— for temporary structures—to more than a century for tunnels, dams of water reservoirs, or nuclear repositories. There is an urgent demand for handling this wide spectrum of lifetimes, in structural design and maintenance. An appropriate differentiation of the design service lives of different building classes is necessary. A proposal for such a differentiation of expected working lives is given in Figure 1.5.

1.4 Economic and Further Aspects of Service-Life Control

5

expected working life [years] ≥ 150

dams of water reservoirs

≥ 80

bridges

≥ 60

residential and buisness buildings

30–40

industrial buildings

≈ 10

temporary buildings

Fig. 1.5. Working-life related building classes

1.4 Economic and Further Aspects of Service-Life Control Authored by Friedhelm Stangenberg Service-life control is successful, if all initial and later investment costs are minimized, and if all aspects of sustainability are respected, from the beginning of the construction, over all the lifetime with perhaps intervals of repair, including final removal or perhaps recycling, revitalization etc. Lifetime-oriented strategies aim at successful investment economy combined with sustainability. Building investment costs comprise • • • •

costs of initial construction, maybe costs of periodic inspection, of maintenance and strengthening, costs consequent to temporary losses or restrictions of use (in cases of interruptions for maintenance), costs of financing for the initial construction and, in case, for later maintenance and repair.

6

1 P

Lifetime-Oriented Design Concepts

costs constant

designed for no maintenance

service life (without maintenance measures)

P

costs

savings at the beginnig

inspections and maintenance (eventually temporary loss of use)

Fig. 1.6. Service Life control and economic aspects

Life Cycle Assessment of Structures via Innovative Monitoring

Materials and Structures in Revitalization of Buildings Fig. 1.7. Related Collaborative Research Centers

Aspects of sustainability comprise • • •

saving of natural resources, e.g. by economizing raw materials, energy etc., prevention of future negative consequences and remains as well as harmony with nature and human life qualities, now and in future.

For the partial aspects of economic service-life control see the Figure 1.6. Service-life control with high probability of success, in future will also bring up new ideas in the field of warranty. Duration time of warranty for structural qualities is different in diverse countries (according to warranty laws in European countries: 5 or 10 years or other). New law aspects will perhaps follow in new design concepts making a successful service life more reliable. Duration time of warranty according to law should be in correlation with the degree of realizability of structural

1.5 Fundamentals of Lifetime-Oriented Design

7

lifetimes. The progress in lifetime-oriented design concepts can contribute to an international harmonization of warranty law.

1.5 Fundamentals of Lifetime-Oriented Design Authored by Friedhelm Stangenberg Current structural design concepts are oriented towards serviceability as well as towards safety against failure. They are based on structural virgin states, largely excluding pre-damage as well as later damage accumulation. Design standards use limit state formats in terms of load scenarios, material behavior, and required (partial) safeties. With respect to service-life control, such formats should include quantified damage predictions and assumptions for future tendencies of relevant action scenarios. Such anticipations are necessary for realistic simulations of future lifetime influences. Thus, expected or monitored structural safety and serviceability evolutions can be described properly. Such concepts are created presently and named life-cycle design. Methods for describing damage evolutions, resistance alterations, increases of actions etc. are needed as well as simulations techniques and methods for the estimation of the reliability of structures with respect to these phenomena. This book provides contributions to these topics, which form the basis for lifetimeoriented design.

2 Damage-Oriented Actions and Environmental Impact on Materials and Structures

Authored by R¨ udiger H¨ offer Mechanical loading and ambient actions on civil engineering structures and components cause lifetime-related deteriorations. Not the rare extreme loading events are in the first place responsible for the evolution of structural degradation but the ensemble of load effects during the life-time of the structure. It is of major importance to have models at hand which adequately reflect the experienced time histories of impacts, and which can include justified predictions of future trends. Leading types of loading and load-effects with relation to mechanical fatigue as well as damages due to hygro-thermal and chemical impacts are considered in this chapter. Selected contributions from wind and temperature effects with certain meteorological characteristics as well as from traffic loads on roads and railway lines are modeled as typical examples of contributions to mechanically induced degradations of structures. A specific aspect is the permanent settlement of soil due to high-cyclic, longterm loading, for which novel representations are developed. The attack of freeze-thaw circles in different environments and of chemical impacts leading to solving, swelling and leaching processes in concrete including principle interactions are discussed as examples for the main types of non-mechanically induced degradations.

2.1 Wind Actions Authored by R¨ udiger H¨ offer Wind-induced cyclic motions of structures can generate deterioration of constructions and materials. It is therefore required to check the exposure of structures and structural components regarding the probability and accumulation of such damages. Cyclic responses due to wind buffeting as one of the aerodynamic loading mechanisms are evaluated based on the concept of gust response factors. In general gust-response factors relate equivalent static loads

10

2 Damage-Oriented Actions and Environmental Impact

12

4.0

W [kN]

a[m]

a[m] 0.002

8

t[s]

4 0

20

40

60

2.0

0.001

80

20 0

-4

20

40

60

40

60

80

80

-8 -2.0

-12

f=0.18 Hz

-16

(a)

(b)

f=3.56 Hz

(c)

Fig. 2.1. Typical wind load process (a), and related low frequency (b) and high frequency (c) response of a structure [572]

to maximal dynamic effects. In the presented approach frequently repeated gust effects of lower levels than the extreme effect with a return period of 50 years are modeled using the statistical distributions of ensembles of registered wind speeds. The influence of the wind direction on a structural failure probability is included via advanced directional factors. Resonant vortex excitation of slender cylindrical structures is a classical aerodynamic interaction mechanism which can cause high-cyclic responses with large amplitudes. A time domain model is applied and validated using data from wind tunnel investigations and from an experiment in full scale. A novel representation in the micro and the macro time domain is developed. The succession of damage events is modeled through the adaption of renewal processes and pulse processes. 2.1.1 Wind Buffeting with Relation to Fatigue Authored by R¨ udiger H¨ offer Buffeting loading on structures consist of mean and fluctuating components. The mean wind load is the static load component, which is related to the mean wind speed in a deterministic manner. The second, fluctuating load component is primarily due to wind gustiness. Both parts are required for the evaluation of aerodynamic forces. The time histories of forces are derived from a superposition of the effect of wind buffeting and of body-induced turbulence, from vortex forces and from aeroelastic interaction forces (s. e.g. [200]). The forces vary randomly in space and time. Time domain or frequency domain methods are available to calculate the stochastic response. However, they are cumbersome in view of the input data required and the computer time needed. For practical design it is sufficient to apply equivalent static gust wind loads.

2.1 Wind Actions

11

They are based on the so-called gust response factor G that incorporates the most adverse gust effect on a structural response, which dominates in the design, the so-called leading response. Aeroelastic oscillations, such as galloping or flutter, are usually not object of a fatigue analysis as the associated, often continuously growing oscillation amplitudes can lead rapidly to a structural failure from an overload breakage. Structures which are prone to such type of excitations are dynamically detuned, damping devices or other dissipative mechanism are arranged. An important exception is the structural oscillation due to vortex resonance which - depending on the damping ration - can generate resonant amplitudes of different levels and also the lock-in phenomenon due to synchronized vortex seperations (s. e.g. [740]). Such fatigue behaviour must always be analysed. Often scruton coils against regular vortex separations are installed. In contrast, gust-induced oscillations of structures are unavoidable if the structure exhibits eigenfrequencies in the limits of up to ca. 2 Hz, Fig. 2.1 (b). The reason is that higher energy levels below 2 Hz are introduced into the flexible structure and are amplified due to the resonant behavior of the structure [571]. Above 2 Hz wind-induced oscillations are often marginal because the spectral energy of wind turbulence is minor. Here, the excitation process is not amplified but uniformly transferred into structural responses. However, in such case the structure follows the wind gusts quasi-statically, Fig. 2.1 (a), which alone by itself acts as a fatigue loading, Fig. 2.1 (c). 2.1.1.1 Gust Response Factor Authored by R¨ udiger H¨ offer and Norbert H¨ olscher The gust response factor G is the magnification of the static, mean reaction Ym of a structure to the fluctuations of the wind load. Stresses, internal forces, and displacements are the responses of interest for which a specific gust response factor is evaluated. The factor is applied to generate an equivalent static force FE which refers to the maximal dynamical effect. The maximal effect YP is composed from the mean response Ym , which is defined as the mean value of responses in a time window of 10 minutes, and the standard deviation of the reaction σY with the peak factor g as a weight. Yp = Ym ± g · σY

(2.1)

The total standard deviation of response σY results from quasi-static variance 2 2 σYQ and resonant contributions σYR which can originate from one or more resonant contributions (Figure 2.2). The assumed maximum reaction is given by 2 + σ2 = Y · Yp = Ym ± g · σYQ m YR

σYQ 1+g· · Ym

σ2 1 + YR 2 σYQ

= Ym · G (2.2)

12

2 Damage-Oriented Actions and Environmental Impact SM ⋅ f σ 2M

um

0.8

1.2

1.6

2.0

2.4

M

resonance 2 MR

σ

quasi-static

0.4

σ 2MQ

0.005

0.01

0.02

0.05

0.1

0.2

0.3

0.5

1.0

frequency in Hz

Fig. 2.2. Curve of the total variance of the base bending moment of a cantilever due to buffeting excitation plotted over frequencies [572]

which also yields the gust response factor. The equivalent quasi-static load is then defined as FE = G · Fm = G · (caero · qm · Aref )

(2.3)

The gust response factor G raises the original static wind load Fm which is calculated from the aerodynamic coefficient caero , the stagnation pressure qm due to the mean velocity vm , and the reference area Aref , which is in general the projection area of the component or structure in direction of the attack of the wind. The distribution of qm over the building height depends on the profile of the mean wind. For stiff building structures the low-frequency components of the excitation are not amplified but quasi-statically transferred into responses. The quantity of the quasi-static component is controlled through two sets of parameters of wind turbulence, the three components Ii , i = u, v, w, of turbulence intensity, and the integral length scales. The turbulence intensity Ii is a measure for the fluctuations of wind speed due to turbulence at a fixed point in space. Realizations of the turbulent fluctuations exist in longitudinal (Iu ), lateral (Iv ), and vertical (Iw ) directions. In a boundary layer flow the turbulence intensity is different for each of the three components, and it decays with increasing height above ground. For each component the intensity is estimated from the r.m.s. value of the associated statistically stationary process of fluctuations and the mean wind speed in longitudinal direction, Iu = σu /¯ u, Iv = σv /¯ u, Iw = σw /¯ u. The intensity in longitudinal directions is the largest of the three measures. Turbulence intensities of a wind flow over rough terrain are higher than those of a wind flow over a smoother terrain. The integral length scales Lij are statistical measures of turbulence which describe the mean spatial extent of the quantity of air which is homogenously moved

2.1 Wind Actions

13

by a wind gust. The length scales depend on the direction of observation x, y and z and on the observed wind speed components u, v and w. For example the integral length scale Lux describes the correlation of the longitudinal wind speed components in x-directions. In an atmospheric boundary layer flow Lux increases with height above ground. It results smaller in a wind flow over a rough terrain compared to a wind flow over a smoother terrain. Such behavior is also characteristic for the other components of length scales. Equivalent static forces are usually determined assuming a linear load bearing behavior of structures. The quasi-static reaction is then defined by σYQ ∼ σF = ¯ · Q0 Ym F

(2.4)

σF and F¯ are local wind loads applied at a characteristic point of the structure. Q0 is a proportionality factor which quantifies the inhomogeneity of the gust body along the surface of the structure. The r.m.s. value of a quasi-static structural response can directly be derived from the turbulence intensity and the proportionality factor Q0 (s. [572]). Simplifying it can be assumed after [32] that Q0 describes the effect of the longitudinal component u(t) = um + u (t) of the vector of wind speeds by Q20 =

1 1 + 0.9 · ((b + h)/Lux )0.63

(2.5)

b is the width and h is the height of the wind-exposed area. Dominant resonant gust effects are generated if a flexible structure is prone to oscillatory excitation. A required condition for that is the vicinity of both, eigenfrequencies of bending or torsion modes of the structure, and the highenergy range of the excitation, and low or moderate damping. It is often sufficient to consider the mechanical impedance of the structure at the fundamental mode only. In a modal analysis of wind-induced oscillations the transfer function Gν (f ) is formulated in terms of the modal amplification function Vν (f ). 2 2

kν kν 1 2 2 (2.6) · |Vν (f )| = · |Gν (f )| = mν mν [ων2 − ω 2 ]2 + 4 ξ 2 ω 2 ων2 kν mν |Gν (f )|2

active stiffness in the ν-th eigenform oscillating mass in the ν-th eigenform square of the modulus of the mechanical transfer function in the ν-th eigenform ω = 2 π · f circular frequency ν-th circular eigenfrequency ων

14


It can be assumed

σYR σYQ

2

≤

ν σYR ν σYQ

2

∞ = 0 ∞

SFν (f ) · |Gν (f )|2 df SFν (f )

· |Gν

(0)|2

∼ =

df

0

SFν (fν ) ∞

π 2 · fν 2·δ

(2.7)

SFν (f ) df

0

2

ν (σYR )

variance of the resonant component of the structural reaction in the ν-th eigenform

ν 2 σYQ variance of the quasi-static component of the structural reaction in the ν-th eigenform SFν (f ) modal power spectral density function of the modal wind load SFν (fν ) power spectral density function of the modal wind load at fν δ logarithmic damping decrement The resonant wind effects are superimposed from one or more modal effects which result from the mechanical transfer of the respective power spectral density function of the modal wind loads. The latter function results from SFν (f ) = ηiν · ηkν · SFi Fk (f ) (2.8) i

k

ηiν and ηkν are the modal ordinates of the eigenform ν at the nodal points i and k of the oscillating structure. The correlation of the wind loading in physical coordinates is introduced through the cross spectral density function SFi Fk (f ). The complete construction of the cross spectral density function follows sophisticated concepts (s. e.g. [379] and [385]). A simplified approach for vertical structures is realized and implemented in [32]. 2.1.1.2 Number of Gust Effects Authored by R¨ udiger H¨ offer and Volkmar G¨ ornandt The effect Y of actions due to gust loading reaches or exceeds the characteristic value of the action Yk once in 50 years in the statistical mean. The number Ng of exceedances of a lower level Y (Ng ) < Yk is higher. The curve drawn with a fat line in Figure 2.3 is implemented into the European standards, s. [32]. The value Y (Ng ) of the gust effect under consideration is related to the peak effect Yk due to a gust in a storm of a return period of 50 years, and represents the general behaviour of Y (Ng )/Yk through the equation Y (Ng ) 2 = 1 − 0.174 · log(Ng ) + 0.007 · (log(Ng )) Yk

(2.9)

Setting Equation 2.9 equal to (c1 − c2 log(Ng ))2 = c21 − 2 c1 c2 log(Ng ) + c22 (log(Ng ))2

(2.10)

(a)

Y(Ng) / Yk

2.1 Wind Actions

15

1.00 Y(Ng) / Yk after European standards Berlin-Tempelhof Berus Braunschweig Bremgarten

0.75

Frankfurt-Flughafen Hamburg-Fuhlsbüttel Hannover-Langenhagen Laupheim Memmingen

0.50

Nürnberg-Flughafen Saarbrücken-Ensheim Stötten Wasserkuppe

0.25

0.00 101

1

(b)

10 2

10 3

10 4

10 5

10 6

10 7

10 8

10 9

Ng

0.30 Y(Ng) / Yk after European standards Berlin-Tempelhof Bremgarten Hannover-Langenhagen Nürnberg-Flughafen

0.25

0.20

0.15

0.10

0.05

0.00 0.0

2.0

4.0

6.0

8.0 v (m/s)

10.0

12.0

14.0

Fig. 2.3. Comparison of the occurence of repeated wind effects at different locations in Germany and a codified representation. (a) Number Ng of exceedances of an effect Y (Ng ) ≤ Yk . (b) Probability density functions of ensembles of mean wind speeds

and adding a correction term Y (Ng ) 2 = (c1 − c2 log(Ng ))2 − c23 (log(Ng )) Yk

(2.11)

- where c1 , c2 and c3 are identification parameters - yields the form Y (Ng ) = Yk

2 0.174 2 (log(Ng )) − 0.000569 (log(Ng )) 1− 2

(2.12)

The statistical property of the quotient Y (Ng )/Yk is dominated from the probabilistic distribution of the square of the mean wind speeds vi (10-minutesmean). The return period of the mean velocity vi = vm is smaller or equal

16


than the return period of 50 years which applies for vm , the characteristic mean wind speed. The variable Ng can be interpreted as a measure for an exceedance probability of vi . Typically, a Weibull-distribution is applied to describe the probabilistic behavior of the complete ensemble of mean wind speeds. The cumulative distribution is v k W (v) = exp − (2.13) A with the probability density function k v k−1 v k w(v) = exp − A A A

for v ≥ 0

(2.14)

in which k is called the Weibull shape parameter, and A is the Weibull shape parameter. Troen and Petersen comment in [795] that extreme values are insufficiently represented in such distribution type. In alternative, a Gumbel-distribution or a logistic distribution is proposed. A logistic distribution with two parameters [263] which is comparatively referred to yields as the cumulative distribution L(v) =

1 v−a 1 + exp − b

The inverse function results from this as 1 −1 v(L) = a − b · ln L 2 1 v 2 (L) 1 − 1 = · a − b · ln 2 2 vm vm L

(2.15)

(2.16)

(2.17)

The two parameters a,b are extracted from meteorological data bases, e.g. [196]. Equation 2.17 is normalized with respect to v(L = 0.98) = vm and represents a first approximation of Y (Ng )/Yk . Equations 2.12 and 2.17 can directly be compared. Figure 2.3(a) shows the relation between Y (Ng ) as a fraction of Yk and the number Ng of exceedances at various locations in Germany. Figure 2.3(b) compares the density functions after Equation 2.14 of some of the locations to the assumed density function which corresponds to Y (Ng )/Yk after [32]. Equations 2.12 and 2.17 are applicable for the analysis of fatigue. A possible resonant contribution is included if Yk contains a resonant contribution as well. Figure 2.4 represents the absolute frequency Ni of normalized gust responses Yi which are partitioned into i classes of different response level and equal class widths of 0.1 times Yk . The double amplitude of the extreme stress

2.1 Wind Actions

0.95

2.9

0.85

12

0.75

59

0.65

2

3.1x10

0.55

Y(Ni)/Yk

17

3

1.9x10

0.45

4

1.4x10

0.35

5

1.4x10

0.25

6

2.1x10

0.15

7x10

7

11

3.1x10

0.05 10

0

10

1

10

2

10

3

10

4

10

5

10

6

N

10

7

10

8

10

9

10

10

10

11

10

12

10

13

Fig. 2.4. Distribution of absolute frequencies of normalized gust responses into subsequent classes of different levels of effect

Δσ = 2 · (

Nk Mk + ) W A

(2.18)

is the reference value, Mk an Nk are characteristic internal forces of a construction component, W is the elastic section modulus, A is the loaded area. Stress levels between 0.9 · Δσ and 1.0 · Δσ can occure 2.9 times in 50 years in the statistical mean. A damage accumulation after Palmgren-Miner D = i (Ni /Nci ) is performed in order to assess resistance of the considered component with respect to fatigue. Figure 2.5 shows an example taken from a fatigue analysis of the

S -N c u rv e (W ö h le r c u rv e ) o f s tre s s c o n c e n tra tio n c a te g o ry 3 6 *

Fig. 2.5. Comparison of the distribution of cyclic stress amplitudes with the S-N curve (W¨ ohler curve) of stress concentration category 36* after [30]

18


gust responses of steel archs of a road bridge. The considered cerb is sufficient to resist the repeated gust impacts. The application of the Equations 2.12 or 2.17 permits a detailed and safe method for the fatigue analysis of gust-induced effects at building structures. 2.1.2 Influence of Wind Direction on Cycles of Gust Responses Authored by R¨ udiger H¨ offer and Hans-J¨ urgen Niemann Meteorological observations document that the intensity of a storm is strongly related to its wind direction. Figure 2.6(a) shows the wind rosette of the airport Hannover, Germany, as an example. The probability of the first passage of the same threshold value can strongly vary for different sectors of wind direction. That means that the risk of a high wind induced stressing of a structural component is different between the wind directions. The failure risk

(a)

0◦

(b)

35 m/s

25 m/s

25 m/s

15 m/s

270◦

5 m/s

15 m/s

90◦

5 m/s

0

90◦

270◦

180◦

(c)

0◦ 35 m/s

180◦

◦

(d)

35 m/s

0◦

25 m/s

15 m/s

270◦

5 m/s

180◦

90◦

270◦

0.25

0.50

90◦ 0.75

1.00

180◦

Fig. 2.6. Rosettes of wind quantities at Hannover (12 sectors, 50 years return period) (a) extremes of 10-minutes means of wind velocities at the airport of Hannover at reference height of 10 m above ground (b) extremes of 10-minutes means of wind velocities at a building location at building height of 35 m above ground (c) extremes of gust wind speeds at a building location at building height of 35 m above ground (d) comparison of the load factors of the sectors; the largest load factor is valid for the design of the fa¸cade element after Figure 2.8

2.1 Wind Actions

19

of the structure or structural components is determined by the superposition of all probability fractions originating from the sectors of wind direction. Usually, codes follow the conservative approach to assume the same probability of an extreme wind speed for all wind directions. In general, more realistic and very often also more economic results can be achieved if the effect of wind direction is considered. This can be done by employing wind speeds for the structural loading which are adjusted in each sector with a directional factor. Such procedure is in principle permitted by the Eurocode [32]. It is left to the national application documents to regulate the procedures. The wind load is a non-permanent load; within statical proofs of the load bearing capacity it is employed using a characteristic value, which is defined as a 98% fractile, and an associated safety factor of 1.5. A load level is required which is exceeded not more than 0.02 times a year in a statistical sense. Such value is statistically evaluated from the collective of yearly extremes of the wind speeds. The intensity of the wind load is deduced from the level of the wind speed, or more exact, from its dynamic pressure. The related statistical parameters are used to determine the characteristic value of the load. The wind load depends on the wind direction as the wind speed is differently distributed regarding their compass, and as the aerodynamic coefficients varies with respect to the angle of flow attack. Taking this into account the most unfavourable load can originate from combining a lower characteristic value of the wind speed, which might be associated to a directional sector, and the related aerodynamic coefficient for this sector. In order to evaluate completely the effect of the influence of the wind direction it is required to take the structural response into account, e.g. after [227]. In such procedure a response quantity, which is a representative value of the wind action, is evaluated with the restriction to limit its exceedance probability of its yearly extremes to a value lower than 0.02 instead of focussing on loads. Using this requirement the characteristic wind velocities related to the different sectors can be deduced. 2.1.2.1 Wind Data in the Sectors of the Wind Rosette The maximum wind load effect on a structural component is resulting from the most unfavourable superposition of the function of the aerodynamic coefficient and the dynamic pressure. Both variables are independent and functions of the direction of mean wind. The usual zoning in statistical meteorology into twelve sectors of 30◦ each is a sufficient resolution in order to include distribution effects. The prediction of the risk requires an analysis of the extreme wind velocities for each sector at the building location. If available a complete set of data is taken from a local station for meteorological observations near the considered building location. The wind statistics of a considered building location in the city of Hannover in Germany is shown in Figure 2.6(a) as an example. The wind rosette is evaluated from data collected at the observation station at the airport of Hannover. The terrain in the environment of the station is plain with a relatively homogeneous surface represented by a roughness

20


Table 2.1. Conversion of the wind data of the observation station at the airport of Hannover into data for the building location Sectors of wind directions airport: 1 z0 = 0.05 m: vm (z = 10 m) in m/s arena: 2 z0 in m 3 kr · ln(

z

)

z0 4 vm (z = 35 m) 5 Iu (z = 35 m) v 6 gust factor vm 7 v(z = 35 m)

0◦

30◦

60◦

12.1

11.7

17.4

90◦ 120◦ 150◦ 180◦ 210◦ 240◦ 270◦ 300◦ 330◦

13.0

15.2

15.9

17.1

20.5

23.0

20.6

16.7

12.5

0.44

0.27

0.31

0.24

0.24

0.08

0.10

0.11

0.36

0.36

0.36

0.35

0.96

1.03

1.02

1.05

1.05

1.20

1.16

1.15

1.00

1.00

1.00

1.00

11.7 12.1 17.7 13.7 16.0 19.0 19.9 23.7 22.9 20.5 16.6 12.5 0.229 0.206 0.212 0.201 0.201 0.164 0.171 0.174 0.218 0.218 0.218 0.217 1.540 1.494 1.506 1.485 1.485 1.409 1.423 1.429 1.520 1.520 1.520 1.518 18.0

18.1

26.7

20.3

23.7

26.8

28.3

33.8

34.8

31.2

25.3

19.0

of ca. z0 = 0.05 m in all of the sectors. The measurements have been conducted in a standard height of 10 m above ground level, cf. J. Christoffer and M. Ulbricht-Eissing [196]. N yearly extremes of the mean wind velocity vm are ranked in each sector F , and respective probability distributions are identified. In the presented example distributions of Gumbel-type were adapted. The occurrence probability of an extreme value in a year, which is lower than a reference value vm,ref , is calculated from −a(vm,ref −U )

P (vm ≤ vm,ref ) = F (vm,ref ) = e−e

(2.19)

In Equation 2.19 U is the modal parameter, and the parameter a describes the diffusion. The wind velocities with return periods of 50 years for all sectors are listed in Table 2.1, line 1. In opposite to the conditions at the observation station, the building location is surrounded by a terrain with strongly non-homogeneous surface roughnesses. The effect of the varying roughnesses superpose the undisturbed conditions evaluated for the location of the observation station. These additional effects influence the wind velocity in reference height, its profile and the profile of gustiness over height, which vary between the directions according to the respective roughness conditions of a sector. The surface roughnesses for each sector are required. The local roughness lengths z0 of the surface roughness is analysed from aerial photographs over a radius of 50 to 100 times the height of the considered building, e.g. ca. 5 km in case of the considered stadium, Figure 2.7. Mixed profiles are evaluated for those sectors with significantly changing surface roughnesses; for approximation an equivalent roughness length is adapted. The results are shown in line 2 of Table 2.1; the conditions within each sector are described by conversion factors related to the undisturbed wind rosette. The factor in line 3 of Table 2.1 relates the mean wind speeds with a return period of 50 years at the building

2.1 Wind Actions

21

cp=-1.4

0° b/5

90°

b

Fig. 2.7. Roughness lengths of the terrain in the farther vicinity of the building location [771]

Fig. 2.8. Sketch of a building contour (top view) with b < 2 h and fa¸cade element exposed to a pressure coefficient cp = −1.4 [32] at the eastern fa¸cade in the case of winds from 0◦

location at a building height of 35 m of the stadium and the reference wind speed of the same return period at the location of the observation station in reference height of 10 m. The logarithmic law for the profile of the mean wind velocities is applied (Equation 2.20). The terrain factor kr is evaluated using an empirical relation (Equation 2.21). vm (z, z0 ) z = kr · ln( ) vm (zref , z0ref ) z0

kr = (

z0 0,07 1 ) · z0ref ln(zref /z0ref )

(2.20)

(2.21)

The wind velocities at the building location with a return period of 50 years are evaluated for each sector and are listed in line 4 of Table 2.1. As shown before, mean and gust wind speeds and the respective dynamic pressures are applied to determine equivalent loads which represent the resulting wind loading for design procedures. The dynamic gust pressure is calculated from the mean dynamic pressure qm and the turbulence intensity Iu . q = (1 + 2g · Iu · Q0 ) · qm

(2.22)

The gust velocity in the last row of Table 2.1 is calculated from Equation 2.23, where g is the peak factor and Q0 is the quasi-static gust reaction. Q20 is also called background response factor after [32].

22


v=

1 + 2 g Q0 Iu · vm

(2.23)

For simplicity Q0 can consistently be determined from 2 g Q0 = 6 assigning to Q0 its maximum value 1. It has to be pointed out that the surface roughness is also affecting the turbulence intensity, as shown in line 5 of Table 2.1. The statistical evaluation for all sectors leads to a mean wind of 50 years return period of 23.8 m/s at the building location. Figure 2.6(b) represents the rosette of mean wind speeds at the building location. In comparison of both wind rosettes, representing the building location and the location of the observation station, it can be concluded that the main character of the local wind climate is preserved but relevant changes due to the terrain roughness are introduced. 2.1.2.2 Structural Safety Considering the Occurrence Probability of the Wind Loading The wind load effect on a structure can be expressed in terms of a response quantity Y . For a linear, stiff structure without dynamic amplification, Y is calculated from: 1 2 Y (Φ) = ρvΦ · ηp (r) · cp (r, Φ) · dA (2.24) 2 A in which: ηp influence factor for the pressure p acting at the point on the surface of the structure; r - local vector; cp pressure coefficient at a point of the surface of the structure for a given wind direction Φ; ρ - mass density of air; A - pressure exposed influence area. A certain response force Y forms the basis for the determination of a characteristic wind velocity vik , which is valid over the sector with the central wind direction Φi . The starting point is vi,lim : 1 2 Yi,lim (Φi ) = CY (Φi ) · ρ · vi,lim 2

(2.25)

In Equation 2.25 the response Yi,lim is determined as an equivalent wind effect by use of the gust velocity v. The wind effect admittance depending on the wind direction Φ, CY = CY (Φ), is identical to the integral in Equation 2.24. It covers the distribution and the value of the aerodynamic coefficient within the influence area of the load as well as the mechanical admittance, which is the transfer from the dynamic pressure into the response quantity. This operation is conducted for a selected wind direction Φi . In a second step the complete risk is evaluated as the exceedance probability of the response quantity Y , which adds up from the contributions from each sector. The safety requirements are met if the total risk has a value smaller than 0.02. In case of a risk larger 0.02 an increased value of the vi,lim enters into the iteration until a value smaller 0.02 is achieved. In an analogeous manner a

2.1 Wind Actions

23

decreased value of vi,lim is introduced aiming on an economical optimization if the first iteration yields a value much smaller than 0.02. The total risk of exceeding the bearable response quantity Yi,lim , or as complementary formulation, the probability of non-exceedance of Yi,lim , is proved within the following steps. The main idea of the procedure is to make 2 use of combinations CY (Φ) · 12 ρ · vΦ,lim instead of a global CY · 12 ρ · v 2 . A probability of non-exceedance of 0.98 of the applied force must be guaranteed for both in the sectors and in total. 1 1 2 2 CY (Φi ) · ρ · vi,lim = CY (Φ) · ρ · vΦ,lim 2 2 The velocity limit vΦ,lim for a sector Φ results as CY (Φi ) 1 · vi,lim = · vi,lim vΦ,lim = CY (Φ) a(Φ)

(2.26)

(2.27)

The effect of the direction of the wind on the wind effect is expressed through a directional wind effect factor: a(Φ) =

CY (Φ) CY (Φi )

(2.28)

The probability P (v ≤ vΦ,lim ) = FΦ (vΦ,lim ) of the non-exceedance of vΦ,lim within the sector Φ also applies for the response Y ≤ Yi,lim . F (vΦ,lim ) can be calculated from the probability distribution of the mean wind velocity in the sector as given by Equation 2.19. The probability of the non-exceedance of the limit Yi,lim after Equation 2.25 under the condition of a certain vi,lim in sector Φi is satisfied from a product (Equation 2.29) of all non-exceedance probabilities under the condition that the yearly extremes in the different sectors are statistically independent. P (Y ≤ Yi,lim ) = P ((v ≤ v1,lim ) (v ≤ v2,lim ) · · · (v ≤ v12,lim )) 12 = 1 FΦ (vΦ,lim ) ≥ 0.98 (2.29) The considered value of the gust speed is adequate if the exceedance probability P (Y > Yi,lim ) is less or equal 0.02 which corresponds to the probability of non-exceedance of (1 − 0.02) = 0.98, Equation 2.29. Obviously, the condition P (Y = Yi,lim ) ≥ 0.98 must be observed in any sector. 2.1.2.3 Advanced Directional Factors The responses of a structure must be taken into consideration for the determination of the relevant wind speeds and wind loads for each sector. This

24


Table 2.2. Determination of a reduced characteristic suction force on the fa¸cade element after Figure 2.8 through the consideration of the effect of wind direction on loading. line 1: extreme gust speed at a building location at Hannover at building height of 35 m; line 2: cp,10 -values at the considered fa¸cade element for wind flow from the respective directions; line 3: directional wind effect factor after Equation 2.8; line 4: iterative determination of applicable wind speeds in sectors and associated non-exceedance probabilities in sectors; line 5: applicable fraction of codified standard load after the proposed method Sectors of wind directions 0◦ 1 2 3 4

30◦

18.0 18.1 -1.4 -1.4 1 1 18.0 18.1 0.98 0.98 18.2 18.3 0.9985 0.9985

5 0.194

0.196

60◦

90◦

120◦

26.7 – 0 ∞ 1.0 ∞ 1.0

20.3 – 0 ∞ 1.0 ∞ 1.0

23.7 – 0 ∞ 1.0 ∞ 1.0

–

–

–

150◦

180◦

210◦

240◦

270◦

300◦

330◦

26.8 28.3 33.8 34.8 31.2 25.3 19.0 -0.8 -0.8 -0.8 -0.6 -0.6 -0.6 -1.4 0.57 0.57 0.57 0.36 0.36 0.36 1 35.5 37.5 44.8 58.0 52.0 42.2 19.0 0.98 0.98 0.98 0.98 0.98 0.98 0.98 36.0 38.1 45.5 59.0 52.8 42.9 19.2 0.9985 0.9985 0.9985 0.9985 0.9985 0.9985 0.9985 0.434 0.486 0.694 0.874 0.701 0.462 0.216

can be achieved using the values of the wind effect admittance CY (Φ) for the respective sectors. The procedure of calculating the characteristic wind speed in the sectors is exemplified in Table 2.2 for a building located at Hannover, Germany. The fixing forces of fa¸cade claddings due to suction is considered. Figure 2.6 shows a topview sketch of a building cubus of 35 m height with fa¸cades oriented in northern, eastern, southern and western directions. The question is if reduced values of the suction forces at the cladding elements at the edge of the eastern fa¸cade can be adopted as the wind rosettes clearly indicate different wind extremes when comparing the sectors, cf. line 1 in Table 2.2. Wind from eastern directions generate pressure forces at the element, whereas suction forces at the same element are generated through winds from all other sectors. Suction coefficients from [26], Table 3, are used to describe the aerodynamic admittance in simplified terms. An element size of more than 10 m2 is assumed. The pressure minimum — or maximum suction — occurs for northern directions and is described through the pressure coefficient cp = −1.4 for h/b ≥ 5, h = 35 m. Southern wind directions generate a coefficient of cp = −0.8, cp = −0.6 is inserted for western wind directions (cf. line 2 in Table 2.2). The directional wind effect factor a(φ) in line 3 after Equation 2.28 is calculated refering the sectorial pressure coefficents to the minimum pressure coefficient cp = cp,min = −1.4. The results of two iterations are listed in line 4. The first two rows represent vΦ,lim = vi,lim and the corresponding probability of non-exceedance FΦ,lim (vΦ,lim ) which remains 0.98 according to the probability of non-exceedance of the values given in line 1, or it is 1 in sectors 0◦ , 60◦ and 90◦ as only pressure instead of suction can occur here. The application of Equation 2.29 leads to P = 0.8171 < 0.98. In a second iteration the extreme wind speeds are increased in such a way that the total probability

2.1 Wind Actions

25

of non-exceedance after Equation 2.29 results to be larger or equal to 0.98. The third and fourth row in line 4 of Table 2.2 represent a valid solution for which P = 0.9866 and results larger than the required value of P = 0.98. The codified standard design procedure requires a reference wind speed of vref = 25 m/s irrespective the wind direction. The calculation of a gust speed after the wind profile for midlands ([26], Table B.3) leads to a characteristic gust speed of v = 41.3 m/s at building height of 35 m. The standard suction force for the considered element — without any consideration of the influence of wind directions — must be calculated as Y = 12 ρ · v 2 · cp · A. The applicable characteristic suction force after Equation 2.25 — with consideration of the influence of wind directions — can be calculated as a fraction 2 2 (cp (φ) · vφ,lim )/(cp,min · vref ) of the standardized characteristic value. The quotient is listed in line 5 of Table 2.2, and it is represented in Figure 2.6, (d). The largest factor in line 5 must be applied. The respective characteristic velocity is ca. 59 m/s but the associated characteristic suction force after Equation 2.26 is lower than the standard suction force after the code. The reason is in the application of the much higher pressure coefficent — or lower suction coefficient — of cp = −0.5 for wind in the sector 240◦ instead of cp = −1.4. The procedure can also be adopted for a fatigue analysis after Equation 2.9. 2.1.3 Vortex Excitation Including Lock-In Authored by J¨ org Sahlmen and M´ ozes G´ alffy Vortex excitations represent an aerodynamic load type which can cause vibrations leading to fatigue, especially for slender bluff cylindrical structures — bridge hangers, towers or chimneys. The nature of air flow around the structure depends strongly on the wind velocity and on the dimensions of the structure. Accordingly, different wind velocity ranges can be defined, depending on the value of a non-dimensional parameter called the Reynolds-number Re =

u ¯D . ν

(2.30)

Here, u ¯ represents the mean wind velocity, D is the significant dimension of the body in the across-wind direction — for cylindrical structures, the diameter — and ν = 1.5 · 10−5 m2/s is the kinematic viscosity of air. In the Reynolds-number range between 30 and ca. 3 · 105 , vortices are formed and alternately shed in the wake of the cylinder creating the von ´rma ń vortex trail (Figure 2.9) and giving rise to the lift force — an alterKa nating force which acts on the structure in the across-wind direction. The nature of the vortex shedding and of the lift force is considerably influenced by the wind turbulence σu Iu = , (2.31) u ¯

26


´ rma ´ n vortex trail formed by vortex shedding Fig. 2.9. Von Ka

where σu denotes the standard deviation of the stochastically fluctuating wind velocity u. In a smooth wind flow, i. e. if the wind turbulence is low (Iu ≤ 0.03), the across-wind force is a harmonic function of the time t: Fl (t) =

ρ¯ u2 DCl sin 2πfs t. 2

(2.32)

Here, Fl denotes the lift force per unit span, ρ = 1.25 kg/m3 is the density of air, Cl is the dimensionless lift coefficient and fs = S

u ¯ D

(2.33)

is the frequency of the vortex shedding, also called the Strouhal-frequency. The non-dimensional coefficient S in (2.33) is the Strouhal-number which depends on the shape of the structure; its value for cylinders is S ≈ 0.2. In a turbulent flow, the excitation frequencies are distributed in an interval around the mean frequency, the width of the interval depending on the turbulence. When the Strouhal-frequency approaches one of the natural frequencies fn of the structure1 and the structure begins to oscillate at higher amplitudes because the resonance, an aeroelastic phenomenon, the so-called lock-in effect occurs. This results in the synchronization of the vortex shedding process to the motion of the excited structure (Figure 2.10), acting as a negative aerodynamic damping, and can lead to very large oscillation amplitudes. Consequently, the lock-in effect can play an essential role in the evolution of the fatigue processes in the damage-sensitive parts of the structure. The width of the lock-in range is zero for a fixed system and increases with increasing oscillation amplitude. As the amplitude depends on mass and damping, these system-parameters have a large influence on the lock-in effect. This influence can be numerically catched by introducing the dimensionless Scruton-number Sc =

2μδ , ρD2

(2.34)

where μ denotes the mass of the structure per unit length, and δ is the structural logarithmic damping decrement. The width of the lock-in range is 1

Generally only the first natural frequency is of practical importance.

2.1 Wind Actions

27

Fig. 2.10. Dependence of the vortex shedding frequency fv on the wind velocity u ¯. fn is the natural frequency of the structure

reduced with increasing Scruton-number, and for very large values of Sc, no lock-in effect occurs at all. In the case of a uniform smooth flow, the lift force per unit span acting on a circular cylinder fixed in both the along-wind and across-wind directions is given by (2.32). However, the force is not fully correlated along the cylinder span. If the cylinder is allowed to oscillate, the magnitude of the lift force and also the correlation increases. The equation of motion of the cylinder is given by m¨ y + cy˙ + ky = Fl (u, D, y, y, ˙ y¨, t),

(2.35)

where y denotes the across-wind displacement, m, c and k are the mass, the damping coefficient and the stiffness of the cylinder per unit span. As the lift force per unit span Fl depends not only on the wind velocity, on the cylinder diameter and on time, but also on the displacement, on the velocity and on the acceleration of the structure2 , it is not a trivial task to establish its explicite expression. Furthermore, the wind velocity u(t) is a stochastic variable which generally describes a turbulent wind process, and consequently a suitable wind load model must also correctly describe the oscillations in turbulent flow. Much effort has been done in order to find an expression for the across-wind force which fits the experimentally observed facts. However, all of the windload models developed up to the present can only describe the experimentally observed oscillations correctly if some limiting conditions are fulfilled. 2.1.3.1 Relevant Wind Load Models The Ruscheweyh-model [695], which is implemented in the German Codes DIN 4131 (Steel radio towers and masts) and DIN 4133 (Steel stacks), describes the across-wind oscillations in the time domain. The lift force per unit 2

The lift force also depends on the roughness of the cylinder surface, which is here not explicitely shown.

28


span is given by (2.32). The lift coefficient is Cl = 0.7 for Re ≤ 3 × 105 , for higher Reynolds-numbers, Cl decreases. It is assumed that the lift force acts over the correlation length Lc , which is the length-scale of the synchronized vortex shedding along the cylinder span. The increase of correlation with increasing oscillation amplitude Ay is described by the function ⎧ ⎪ for Ay ≤ 0.1D ⎨6D Lc = 4.8D + 12Ay for 0.1D < Ay < 0.6D (2.36) ⎪ ⎩ 12D for Ay ≥ 0.6D The width of the lock-in range is set to ±15 % around the critical velocity uc =

Dfn , S

(2.37)

which leads to a Strouhal-frequency equal to the natural frequency: fs = fn . This model predicts the oscillation amplitudes of slender cylindrical structures in a smooth wind flow for constant mean wind velocities within and outside of the lock-in range with a remarkable accuracy. Large estimation errors occur, however, in the case of high turbulence, or if the mean wind velocity considerably varies in time — especially in the case of entering or exiting the lock-in range. The Vickery-model [811] uses a frequency-domain-approach to describe the across-wind vibrations. Assuming a Gaussian distribution for the spectral density of the lift force, the standard deviation (rms-value or effective value) of the across-wind deflection is obtained as √ πLc h fs3 −( 1−fBn/fs )2 CLσ ρ D3 σy = e . (2.38) 2 2 8π S me 2Bξ fn3 Here, CLσ is the lift coefficient expressed as rms-value, me and ξ are the effective mass and damping ratio of the structure, h is the height of the cylinder and B is a dimensionless parameter which describes the relative width of the Gaussian spectral peak of the lift force. The parameters CLσ ≈ 0.1, Lc ≈ 0.6 D and B are obtained from fits to experimental data; obviously, B depends on the wind turbulence. The model is suitable for predicting the oscillation amplitudes, both in smooth and turbulent flow, but it is limited to the case of stationary flow, i. e. to constant mean wind velocities, and it doesn’t take the lock-in effect into consideration. The model of Vickery and Basu [810] describes the across-wind oscillations in smooth or turbulent flow, with mean wind velocities outside or within the lock-in range. The lift force is written as the sum of two forces: a narrowband stochastic term with a normal distribution of the spectral density and a motion dependent term — negative aerodynamic damping — which describes the lock-in effect. For the lock-in range, the rms-value of the displacement is obtained as

2.1 Wind Actions

σy = 2.5

Cl ρD3 Lc 16π 2 S 2

π me (μe ξ + μξa )

h 0

ψ 2 (z) dz

,

29

(2.39)

where μ and μe are mass and effective mass of the structure per unit span, ψ(z) is the value of the normalized mode shape at height z, and ξa is the aerodynamic damping ratio. The aerodynamic damping is negative in the lock-in range, and it depends on the ratio u ¯/uc, on the turbulence and on the Reynolds-number. Additionally, a dependence on the oscillation amplitude is defined in such a way that it limits the amplitude to a predefined value. The most exhaustive model of vortex-induced across-wind vibrations has been developed by ESDU [262], mainly based on the work of Vickery and Basu [810]. The response equations give the standard deviation of the oscillation amplitude and incorporate the influences of turbulence and of the lock-in effect. The system response is obtained from the superposition of a broadand of a narrow-band term. A very large variety of parameters, such as the surface roughness or the integral length of the turbulent wind, is included in the calculation. Also, the dependence of the lock-in range width on the oscillation amplitude is taken into consideration. Because of their complexity, the response equations will not be presented here. Like all the models presented above, also this model is only suitable to describe the across-wind vibrations in a stationary or quasi-stationary flow, i. e. if the mean wind velocity doesn’t change too rapidly and if there is no transition into or from the lock-in range. Based on the normal distribution of the lift force spectral density SF , suggested by Vickery and Clark [811], Lou has developed a convolution model [507] which describes the lift force in the time-domain, for a stationary turbulent flow, outside of the lock-in range: t ρ ¯¯ ) Fl (t) = DCl βu2 (τ ) e−ξ ω(t−τ cos ω ¯ (t − τ ) dτ, (2.40) 2 0 ω ¯ = 2πS u ¯/D denoting the Strouhal circular frequency corresponding to the mean wind velocity u ¯. From the assumption of the normal distribution for SF , the parameters β and ξ¯ can be determined as √ √ 2π ln 2 Iu ω ¯ (2 + 2 ln 2 Iu2 ) , ξ¯ = ln 4 Iu , β=u ¯ (2.41) 2 Su (¯ ω )(1 + 2 ln 2 Iu ) where Su is the spectral density of the wind velocity. 2.1.3.2 Wind Load Model for the Fatigue Analysis of Bridge Hangers In the project C5 of the Collaborative Research Center (SFB) 398, the vortexinduced across-wind vibrations of the vertical tie rods of an arched steel bridge in M¨ unster-Hiltrup have been analysed for the purpose of a fatigue analysis of

30


Fig. 2.11. Wind velocity, measured and simulated deflection vs. time for the bridge hanger 1 (left) and 2 (right). The horizontal lines in the upper panels show the mean width of the lock-in range

their extremely damage-sensitive welded connections. Therefor, the vibrations of two hangers have been filmed by digital cameras, and the time histories of the deflections have been extracted from the videos by means of a Java program. Simultaneously, the fluctuating wind velocity has been recorded with an ultrasonic 3D-anemometer. The mean wind velocity varied with time in such a way, that one of the hangers entered and exited the lock-in range several times during the measurement, while the other one stayed outside of the lockin range, see Figure 2.11. Because of the low oscillation amplitude, the lock-in range of the second hanger was very narrow; it lies within the width of the horizontal line in the upper right panel. In order to check the validity of the previously presented wind load models for bridge hangers, the amplitudes measured on hanger 1 in the lock-in range, in the time-interval between 8.5–12.5 min have been compared to the predictions of the Ruscheweyh- [695] and ESDU-models [262]. The experiment shows a peak amplitude of ca. 9 mm and an rms-amplitude of ca. 6 mm, while the Ruscheweyh-model predicts peak amplitude of about 5 mm and the ESDU-model an rms-amplitude of ca. 30 mm. As both models show a substantial discrepancy compared to the measured values, a new wind load model for the across-wind vibrations of bridge tie-rods in non-stationary, turbulent flow, including the lock-in effect, has been developed [296], based on the model by Lou [507]. For this purpose, a

2.1 Wind Actions

31

power-function dependence of the parameter β in (2.40) on the fluctuating wind velocity u has been supposed: β = Kun−2 ,

(2.42)

with the fit-parameters K and n. Furthermore, in order to describe the nonstationary wind process, the mean values in (2.40) have been replaced by the corresponding time-dependent quantities; only the wind turbulence Iu is supposed to be constant. The lift force per unit span obtained this way is: t ρ Fl (t) = DCl K un (τ ) eα(t,τ ) cos ϕ(t, τ ) dτ, (2.43) 2 0 with

√ α(t, τ ) = ln 4 Iu

τ

ω(θ) dθ,

t

ϕ(t, τ ) =

τ

ω(θ) dθ + ϕ0 (t).

(2.44)

t

ω(θ) = 2πSu(θ)/D is the Strouhal circular frequency corresponding to the fluctuating wind velocity u at the time moment θ. It is supposed that the lift force acts over the correlation length Lc which can be determined from equation (2.36). The phase angle ϕ0 in (2.44) describes the lock-in effect. For wind velocities in the lock-in range, it is set in phase with the rod motion: ϕ0 (t) = π + arctan

y(t) ˙ , ωn y(t)

(2.45)

outside of the lock-in range, it is set to 0. ωn = 2πfn denotes the angular natural frequency of the rod. The increase of the force amplitude caused by the phase-synchronization is compensated by the reduction of the multiplicative parameter K in equation (2.43) for the lock in range. It has been assumed that the lock-in range is symmetric with respect to the critical wind velocity (2.37) with a half width Δu depending on the oscillation amplitude Ay according to a simple parabolic function (Figure 2.12). The parabola is defined by three points, P1 , P2 and P3 , obtained from fits to the experimental data. The fit of the model parameters to the experimental data has been performed by simulating the vortex-induced vibrations in the time domain, on a finite-element model of the hanger, which has been excited by the force calculated using equation (2.43) applied to the experimental wind data u(τ ). The time dependent deflections have been calculated using the NewmarkWilson time-step method, applying Cl = 0.5 and Lc = 6D. The time histories obtained for the fitted values of the model parameters, K = 175 m−1 and n = 3, are shown in the lower panels of Figure 2.11. For the lock-in range, the multiplicative parameter K has been reduced by a factor 4.

32


Fig. 2.12. Width of the lock-in range for bridge tie rods

The time history of the measured and simulated oscillation amplitudes shows a remarkable similarity for both hangers (Figure 2.11). Furthermore, the averaged rms-amplitudes of the simulated deflections are very close to the values determined from the experiment: for hanger 1, 3.81 mm is obtained for both the measured and simulated data, while for hanger 2, measurement and simulation yield 0.133 mm and 0.130 mm respectively. The wind load model has also been validated by wind tunnel measurements, carried out on a rigid cylinder, elastically suspended in such a way that it could oscillate only in the across-wind direction. Wind velocity and displacement have been simultaneously recorded for 17 fixed values of the mean wind velocity. The displacements of both ends have been averaged in order to eliminate the rotational vibration of the cylinder around the axis parallel to the wind direction. The fit of the model parameters has been performed analogously to the full scale case, applying the same values for the parameters Cl and Lc , obtaining K = 23 m−1 and n = 3. The values for the full scale and the wind tunnel experiments differ because K obviously depends on the wind turbulence (see eq. (2.41) and (2.42)). Again, for the lock-in range, the parameter K has been reduced by the factor 4. The measured and simulated time histories of the amplitudes are shown in Figure 2.13 for a representative measurement within and another outside of the lock-in range. In both cases, the measured and the simulated data show time-dependent amplitudes with qualitatively and quantitatively similar characteristics. The ratio of the simulated to measured rms-amplitudes of the displacement varies between 0.47 and 1.95 for the different fixed mean wind velocities, which can be considered as a good agreement between model and experiment, in comparison to other models: The amplitudes are overestimated by a factor of ca. 7 by the Ruscheweyh- and by a factor of ca. 11 by the ESDU-model.

2.1 Wind Actions

33

Fig. 2.13. Measured and simulated amplitude of the displacement within and outside of the lock-in range

2.1.4 Micro and Macro Time Domain Authored by M´ ozes G´ alffy and Andrés Wellmann Jelic In modeling stochastic, especially time-variant fatigue processes, commonly the time scale is split into a micro and a macro time domain. In the micro time domain, loading events and resulting fatigue events are simulated. Theoretically, the loading and fatigue process can be considered as continuous in the micro time domain, but for practical calculations discrete realizations of these processes are used, which are separated in time by a constant increment called time step. The macro time domain is used for estimating the lifetime of the structure, taking into consideration the succession of fatigue events in time. The splitting procedure is applicable to any stochastic loading which causes fatigue — e. g. wind, traffic, sea-waves, etc. The reasons for splitting the time scale are: •

Within the micro time domain, the system properties, and in most cases also the excitation process, can be considered time-independent. Consequently, the simulation of a fatigue process in this time domain — a fatigue event — can be performed using time-independent stiffness, damping and

34


Fig. 2.14. Sample realizations of a renewal process (left) and of a pulse-process (right)

•

mass matrices and an excitation force derived from a stationary random function (generally white noise). The numerical simulation of the fatigue process over the macro time domain would result in unacceptably large computation times, especially for complex structures, where the solution of the equation of motion implies a laborious finite-element calculation at every time-step.

The advantage of the time scale splitting is that the fatigue results obtained for a load event in the micro time domain (e. g. using the rainflow cycle counting method) can be used in the macro time domain several times, without the need of recalculating the time histories of the loads and of the system responses. Generally, the length of the micro time domain is chosen btw. 1 ms and 1 s, depending on the properties of the structure and the loading. For some applications, however, considerably larger durations are needed, e. g. for the lifetime analysis of bridge hangers, performed in the project C5 of the Collaborative Research Center (SFB) 398. Because of the large mass and small damping of the tie rods (logarithmic damping decrement δ ≈ 6 × 10−4 ), the system answer to changes in the nature of the excitation force (e. g. on entering or exiting the lock-in range, see Section 2.1.3) is very slow and consequently it was necessary to choose a duration of ca. 1.5 hours for the micro time domain. Another uncommon feature of this application is that because of the lock-in effect, the stochastic excitation force cannot be considered stationary, even in the micro time domain [295]. The macro time domain spans the whole lifetime of the structure, implying an order of magnitude of several years. 2.1.4.1 Renewal Processes and Pulse Processes In the macro time domain, the succession of the fatigue events is numerically represented by discrete processes which occur at certain moments of time

2.2 Thermal Actions

35

ti , called renewal points. Each process causes a jump in the fatigue function, between the renewal points the function remains constant. The processes with constant height are called renewal processes, and those with variable height are called pulse processes. Renewal processes can be characterized by one single stochastic variable representing the length of the renewal period (the period between two successive renewal points). For pulse processes, a second stochastic variable is needed for the full description: the pulse height. Figure 2.14 presents the time dependence of the state function (e. g. fatigue) for a renewal process, represented by the number N of the occured processes, and of a pulse process, characterized by the pulse height X.

2.2 Thermal Actions Authored by J¨ org Sahlmen and Anne Spr¨ unken Climatic conditions (e.g. air temperature, solar radiation, wind velocity) cause a non-linear temperature profile within a structure or a structural component and stress due to thermal actions is induced. For the design and lifetime analysis of many engineering structures (e.g. bridges, cooling towers, tall buildings, etc.) thermal effects, in combination with moisture and chemical actions, remain an important issue. 2.2.1 General Comments Authored by J¨ org Sahlmen and Anne Spr¨ unken Temperature changes generate expansions or contractions, hence considerable stress may occur. The amount of stress is depending on the magnitude of loading. In the elastic range of deformation the material returns to its original dimension or shape when the load is removed. When subjected to sustained or long-term loading, many building materials experience additional deformation, which does not fully disappear when the loading is removed. Due to this special load cracks may occur and deterioration starts or proceeds. As a consequence the deterioration over time leads to a reduction of stiffness of the structure. The implementation of affected non-linearities due to thermal loads in the design process and lifetime analysis is still part of ongoing research. The numerical modelling of the temperature effects on structures based on experimental results are in the focus of this chapter. 2.2.2 Thermal Impacts on Structures Authored by J¨ org Sahlmen and Anne Spr¨ unken Permanent change of meteorological conditions (e.g. cloudiness, rain, sunny periods, etc.) leads to non-stationary und locale site-dominated loads on a structure. For the optimization of lifetime analysis a numerical algorithm is

36


needed to describe the physical thermal load scenario on an observed structure or structural component. A realistic temperature field, based on experimental data, has to be modelled to simulate the thermal transmission and moisture flux within a material with the final aim to determine the time dependent stress acting. Parameters like heat transfer and heat storage as well as the content of moisture have to be considered [517, 74, 704, 463]. Further more material and site conditions of the observed structure (location, climate, orientation, surrounding properties, etc.) have to be implemented in a numerical optimization model of thermal actions [518]. The process of heat transmission in materials is elementary controlled by three phenomena [807]: • • •

heat conduction natural convection thermal radiation

In the following the physical fundamentals of heat transmission are briefly described. Material properties and structure dimensions are affecting directly the heat conduction and the storage capacity. The rate at which heat is conducted through a material is proportional to the area normal to the heat flow and the temperature gradient along the heat flow path. For a one dimensional, steady-state heat flow the rate is expressed by Fourier’s differential equation: Q = −λ dT /dh = −λ grad T

(2.46)

with: T = T (x = h) − T (x = 0) and assuming stationary heat transfer the formula rearranges to: Q = −λ A(δT /h)

(2.47)

where: λ = thermal conductivity [W/mK] Q = rate of heat flow [W] δT = temperature difference [K] A = contact area [m2 ] h = thickness layer [m] Thermal conductivity λ is an intrinsic property of a homogeneous material which describes the material ability to conduct heat. This property is independent of material size, shape or orientation. For non-homogeneous materials, those having glass mesh or polymer film reinforcement, the term relative thermal conductivity is used because the thermal conductivity of these materials depends on the relative thickness of the layers and their orientation with respect to the heat flow direction. The thermal resistance R is another material property which describes the measure of how a material of a specific thickness resists to the flow of heat. This parameter is defined as follows:

2.2 Thermal Actions

R = A(δT /Q)

37

(2.48)

Hence, the relationship between λ and R is shown by the substitution of 2.47 and 2.48 and rearranging to the form: λ = h/R

(2.49)

Equation 2.49 reflects that for homogeneous materials, thermal resistance is directly proportional to the thickness. For non-homogeneous materials, the resistance generally increases with thickness but the relationship is maybe non-linear. Following this relation Eurocode 1 [19] is using a concept for the determination of the total resistance value as follows: Rtot = Rin + (hi /λi ) + Rout (2.50) where: Rin = thermal resistance at inner surface [m2 K/W] Rout = thermal resistance at outer surface [m2 K/W] λi = thermal conductivity of layer i [W/m K] hi = thickness of layer i [W/m K] The process of convection is dominated by the climatic conditions like wind, temperature, humidity, etc. Convection describes the transfer of heat energy by circulation and diffusion of the heated material. The fluid motion of the surrounding air is caused only by buoyancy forces set up by the temperature differences between the outer surface of the structure and the air temperature. The basic equation for the convective heat transfer is given as follows: Qconv = αconv (Tair − Tsurf ace )

(2.51)

where: = convection heat transfer coefficient [W/m2 K] αconv = air temperature [K] Tair Tsurf ace = surface temperature of the structure [K] Thermal radiation, essentially induced by the visible and non-visible light of the sun, consists of electromagnetic waves with different wavelengths (see Figure 2.15). The energy which a wave is able to transport is related to its wavelength. Shorter wavelengths carry more energy than longer wavelengths. The transported energy is released when these waves are absorbed by an object or structure. Due to solar radiation thermal actions on structures could be subdivided into two general types of solar impact depending on the wavelength: •

Short wave radiation with the highest heat energy content is described as global radiation. It includes the direct and the diffuse part of the thermal action on a structure as well as the reflected solar radiation from the immediate vicinity (see Figure 2.16) of the observed object.

38


Fig. 2.15. Wavelength of the visible light diffuse

direct wind atmospheric anti-radiation

reflection air-temperature

reflection of radiation of atmospheric immediate anti-radiation vicinity

reflection of global solar radiation

Fig. 2.16. Climatic load on a structure

•

Long wave radiation contains the atmospheric anti-radiation with its reflection to the surrounding area and to the atmosphere.

Additionally to the described external actions, the reflection of radiation at the structure is influencing the thermal stress. Figure 2.16 shows all types of radiation having a part on the thermal impact of a structure [517, 74, 286]. Heat transfer due to solar radiation is expressed by Boltzmann’s equation as follows: Qrad = αrad (Temitter − Tabsorber )

(2.52)

2.2 Thermal Actions

39

where: = heat transfer coefficient due to radiation [W/m2 K] αrad Temitter = absolute temperature of the emitter [K] Tabsorber = absolute temperature of the absorber [K] All parts of thermal radiation are directly affected by external interference effects. The local climatic conditions at the site (e.g. air-temperature, surface temperature, humidity, cloudiness, etc.) as well as the properties of the observed structural component control the intensity of the total thermal action. Surface colour and characteristic (colour, roughness, layer thickness of the wall, etc.) for example control absorption, reflection and transmission process. In addition to that the complete mechanism of heat transmission is considerably in dependency on the moisture content in the material of the structure and from other parameters like evaporation or condensation as well as special weather conditions like rain, snow and frost (see Section 2.4). Against this background long-term experiments are helpful to understand the complicated nature of the mechanisms involved. To give more precise recommendations for the reduction or elimination of cracking and failure of building materials better numerical models are needed where the interaction of all discussed parameters are implemented and non-stationary effects are taken into account.

2.2.3 Test Stand Authored by J¨ org Sahlmen and Anne Spr¨ unken For the analysis of thermal actions on structural elements under free atmospheric conditions a test stand with different test objects is performed. On the roof of the IA-Building of the Ruhr-University Bochum three different test plates, made of concrete, are installed (see Figure 2.17). Each test plate spans an area of 0.7 × 0.7 m2 (thickness: 0.1 m). Plate 1 is made of pure concrete whereas plate 2 and 3 contain two layers of reinforcement. The plates are mounted in the centre of the flat building roof to provide an undisturbed and direct solar radiation for the test bodies. Plate 1 and 2 are situated horizontally and parallel to the building roof in a height of 0.3 m above the ground. Whereas test object 3 is positioned in a height of 0.1 m above the building roof in vertical direction. The front side of this test plate is oriented to the south to get the maximal solar radiation impact at noon time. All test plates are equipped with thermo sensors on the front and the back side of the bodies to observe the outside surface temperature. Further more, simultaneous to this temperature measurement the basic atmospheric conditions are monitored. The wind speed and direction is measured next to the plates by an ultra-sonic anemometer. The global radiation is recorded with a CM3-pyranometer which is connected to the top side of plate 3 and the atmospheric temperature is measured by a thermo sensor (type k, class 2) at the feet of the ultra-sonic anemometer.

40


CM3

@

Usonic-anemometer data logger plate 1

thermo sensor Ts,pl3 plate 3 plate 2

thermo sensor Tair

Fig. 2.17. Test stand for the analysis of thermal actions on concrete specimen

A data logger in the centre of the test stand is used to collect all measured data in terms of time histories. For the measurements a sampling rate of one Hz is used for all sensors and the total time period of measurements is scheduled for one year. 2.2.4 Modelling of Short Term Thermal Impacts and Experimental Results Authored by J¨ org Sahlmen and Anne Spr¨ unken Seasonal and daily fluctuations in solar radiation, cloudiness and spacious air exchange due to global weather conditions cause a permanent change in the air temperature. Hence, in a first step of analysis the basic load of the thermal impact is subdivided in short term (daily) and long term (annually) actions. For the assessment of the short term action of the temperature on structures the field experiment provides a fundamental data base and is helpful to understand the physical causal relations between atmospheric conditions and surface temperature at the test plates. The measurements at the Ruhr-University Bochum have shown that the extreme values for the daily air temperatures can be found close before sunrise (minimum) and two to four hours after high noon

2.2 Thermal Actions

41

Fig. 2.18. Measured temperature profile during a summer day

(maximum). Thereby the amplitude-frequency characteristic in general is sinusoidal over the day and the daily extremes are characterized by the location and the season. Figure 2.18 shows the measured daily characteristic of the surface temperature for the three test plates. The surface temperatures, measured every second, are plotted against a 24-h period. The documented temperature distributions represent the typical behaviour of the air-temperature versus surface temperature on a structure during a summer day. Alternatively to the measurements the daily profile of the air temperature can be approximately described with the following idealized approach [286]: t1 ≤ t ≤ t 2 : ϑair (t) = 0.5 · (ϑair,max + ϑair,min ) + 0.5 · (ϑair,max − ϑair,min ) · sin(π · (

2t − (t1 + t2 ) ) 2(t2 − t1 )

(2.53)

t2 ≤ t ≤ t3 : ϑair (t) = 0.5 · (ϑair,max + ϑair,min ) + 0.5 · (ϑair,max − ϑair,min ) · sin(−π · (

2t − (t2 + t3 ) ) 2(t3 − t2 )

(2.54)

42


Fig. 2.19. Rainflow analysis of the macroscopic temperature behaviour

where ϑair,max and ϑair,min describe the maximum and minimum air temperature to the time t1 and t2 , t3 is to calculate with t3 = t1 + 24. This model is useful for the assessment of the macroscopic temperature behaviour of the air temperature. Sudden temperature changes in a short time period (microscopic temperature behaviour) occur due to special weather events (e.g. strong rain, thunderstorm, etc.). The time duration for a sudden temperature jump induced by such weather phenomena is in the frame of 0.5 to 1 hour. In contrast to that long term drops in temperature need one or two days to increase visible the air temperature. Figure 2.19 gives an example for the macroscopic temperature behaviour in the Northern part of Germany based on a rainflow analysis. For the investigation a time period of 30 years, based on DWD data-set, is used. The temperature is plotted in dependency on mean value and amplitude. Output of the analysis is the number of events for each temperature class. Additionally a time history of the temperature for a 2-years period is plotted. The figure shows a scattering for the mean values combined with relatively low amplitudes or rather low temperature differences. The daily rate is marginal whereas the annually behaviour is characterized by some single events with higher amplitudes. For the thermal loads on structures it can be presumed that the occurrence of low-cycle fatigue is low due to the fact that amplitude and number of events are not very significant. Based on this result the numerical model is concentrated on the shortterm thermal actions. Especially, the non-linear temperature impacts due to sudden changes in solar radiation are in the focus of the modelling. In Figure 2.20 the influence of the solar radiation on the air and surface temperatures for a summer period (2-days term) is shown. Additional to the measured temperatures the corresponding calculated values are plotted for verification.

2.2 Thermal Actions

43

Fig. 2.20. Temperature behaviour due to a sudden change in solar radiation

The strong correlation between the air and surface temperatures and the intensity of the solar radiation can be seen in Figure the plot. The temperature profiles follow with a time delay of about 3 to 4 hours the time history of the global solar radiation. For the calculated temperature trends over the observed time period a good estimation is found for the outer surface temperature. The numerical plot matches nearly exactly the measured distribution. In comparison to that the inner surface temperature is slightly over-estimated by the numerical analysis, but the general trend is captured. 2.2.5 Application: Thermal Actions on a Cooling Tower Shell Authored by J¨ org Sahlmen and Anne Spr¨ unken As an application of thermal loads on structures under atmospheric conditions a cooling tower with a shell thickness of 0.15 m is analysed for an extreme winter situation, where the temperature difference between inner and outer shell could rise to 40 − 50 ◦ C. Main issue for the calculations in this step is to find out how long it needs to get stationary temperature conditions in the shell for constant external actions. For the numerical analysis a constant temperature difference of ΔT = 45 K and a thermal inner transmission coefficient αi = 20 W/(m2 K) [235] are assumed. At the outer surface the thermal transmission coefficient is a function of wind speed. To minimise the calculation effort a mean wind speed of vmean = 4 m/s is selected where both coefficients have the same absolute value in terms of αi = αa = 20 W/(m2 K) [235]. Figure 2.21 presents as a result from the calculations the temperature distribution across the cooling tower shell (subdivided into 16 layers). The plotted shell temperature distributions for the layers is imaged with the inner temperature of the cooling tower and the outer atmospheric temperature as well as

44


Fig. 2.21. Temperature distributions determined at 16 layers within a cooling tower shell under constant external load actions

the intensity of the global radiation for a 4-days period with a constant wind speed of 4 m/s. It can be seen that the given external impacts need about 40 to 60 hours to reach a stationary temperature distribution within the shell. For shorter time periods a non-stationary thermal reaction can be observed. In a further step the variation of the wind speed and the induced thermal distribution within the cooling tower shell is in the focus of the numerical studies. The surface temperature difference ΔT is calculated under the action of ten different wind velocities, again for a 4-day time period. Figure 2.22 shows the results of the calculations in comparison to the value given by the VGB-guideline BTR [235]. Long-time observations of the surface temperatures of cooling towers have shown that the influence of the solar radiation in winter periods reduces the thermal actions due to the fact of a lower temperature difference between inner and outer shell in comparison to summer periods. The results in Figure 2.22 show for the variation of the wind speed a reverse trend for the temperature differences. It seems that an increase of the mean wind speed leads to an increase of the temperature difference ΔT . The reason for the rising is the cooling effect at the outer surface of the cooling tower. This effect is obviously visible for wind speeds between 0 and 10 m/s. For higher velocity steps the trend shows a lower significance. In comparison with the constant temperature

2.2 Thermal Actions

45

Fig. 2.22. Effect of the mean wind speed on the development of the temperature difference at the inner and outer surface of a cooling tower shell under external thermal load actions (4-days load period [235])

difference given by the VGB-guideline BTR [235] for a stationary status an underestimation of the temperature behaviour for higher wind velocities can be clearly seen. For lower wind speeds the guideline approach tends to be conservative. The results of the numerical analysis show for a strong intensity of solar radiation a reduction of the non-stationary behaviour of the temperature within a structure. The non-stationarities increase with lower global radiation action. This relation may lead to an increase of the thermal load action like it is shown in the comparison in Figure 2.22. Many load models used in the codes don’t include non-stationarities. In the current Eurocode [19] non-stationary thermal actions are included in chapter 6 for bridge decks, only. For buildings just the common rules are established. For advanced lifetime analysis concepts it is recommended to integrate non-linear and time-dependent temperature effects to precise the calculation of the stresses in the structure and hence, to optimize the detection of deterioration effects and lifetime estimates. Uncertainties of representative values of temperature, of initial conditions and of material parameters as well as uncertainties of the numerical model have to be included for the definition of safety factors and combination coefficients.

46


Recommendations for the implementation of such factors into Euro- and national codes for bridges have been done by Lichte [495] and Mangerig [518].

2.3 Transport and Mobility Authored by Gerhard Hanswille and Hans-J¨ urgen Niemann Highways, federal roads and high speed railway lines will in the future remain to be the most important parts of traffic infrastructure in Europe. All forecasts show that especially the amount of freight traffic on roads will continue to increase extremely over the next years. For the transport of persons over long distances especially the number of high speed railway lines will increase also significantly. These developments have to be considered for realistic lifetime oriented design concepts especially with regard to the fatigue damage of structures and structural members. The basis of such design concepts is the realistic modelling of actions. The clauses 2.3.1 and 2.3.2 exemplify the modelling of actions with regard to the static resistance and to the realistic lifetime oriented fatigue resistance of structures. Clause 2.3.1 gives the basis of the development of load models for traffic loads on road bridges and clause 2.3.2 shows in principal the procedure of the development of the load models for noise barrier walls to take into account the aerodynamic loads caused by high speed trains. 2.3.1 Traffic Loads on Road Bridges Authored by Gerhard Hanswille For a realistic lifetime oriented design especially with regard to the fatigue damage of structures and structural members realistic models for traffic loads are needed. These models have to cover several special aspects, because long time prognoses for the whole design life of a structure are necessary, e.g. 100 years for bridges. Traffic loads on road bridges are a good example, where several aspects must be considered for the development of lifetime oriented design concepts. 2.3.1.1 General In this case it should be pointed out, that especially for actions on bridges the models must cover current national and European traffic data and future developments due to the cross border trade. The main inputs required for the development of realistic traffic load models for bridges are: •

the currently available traffic data in Europe with information about the axle and vehicle weights, the different types of lorries and information of the European traffic composition,

2.3 Transport and Mobility

• • • • •

47

information about the influence of the dynamic behaviour of the vehicles and the bridge structures including information about the pavement quality, information about the different types of bridge structures and the corresponding influence surfaces, principles for the model calibration for ultimate limit and fatigue limit states and the damage accumulation under consideration of different materials, methods for the exploitation of the currently available traffic data, development of large capacity and heavy load transports not covered by the normal traffic models, the influence of future political decisions with regard to new traffic concepts.

2.3.1.2 Basic European Traffic Data With regard to the cross border trade, load models must be based on traffic data which are representative for the European traffic. For example the development of the models in Eurocode 1-2 [9] is based on data collected from 1977 to 1990 in several European countries [487, 720, 530, 37, 157, 361, 158]. The main data basis with information about the axle weights of heavy vehicles, about the spacing between axles and between vehicles and about the length of the vehicles came from France, Germany, Italy, United Kingdom and Spain. Most of the data relate to the slow lane of motorways and main roads and the duration of records varied from a few hours to more than 800 hours. Another important point is the medium flow of heavy vehicles per day on the slow lane. In order to analyse the composition of the traffic for the development of the load model in [9] four types of vehicles were defined for the European load model for bridges. Type 1 is a double-axle vehicle, Type 2 covers rigid vehicles with more than two axles, Type 3 articulated vehicles and Type 4 draw bar vehicles. Figure 2.23 shows the typical frequency distribution of these four types resulting from traffic records of the Auxerre traffic in France. The data base of different countries shows that the traffic composition is not identical in various European countries. The most frequent types of heavy vehicles are 1 and 3. Especially in Germany the traffic records in 1984 show that lorries with trailers (Type 4) dominated the traffic composition at that time. The traffic records of the Auxerre traffic (Motorway A6 between Paris and Lyon) gave a full set of the required information for the development of an European load model. In addition the Auxerre traffic includes a high percentage of heavy vehicles and gives a representative data base for the development of a realistic European load model. Figure 2.23 shows the distribution of the above explained types of heavy vehicles based on the Auxerre traffic records. Figure 2.24 shows the gross vehicle weight and the axle load distributions for the representative traffic in Auxerre and Brohltal (Germany) where n30 is the number of lorries with G ≥ 30 kN and n10 the number of axles with PA ≥ 10 kN. Especially for the development of models for the fatigue resistance

48

2 Damage-Oriented Actions and Environmental Impact N

N

800 700 600 500 400 300 200 100

80 70 60 50 40 30 20 10

Type 1 G(kN) 120

240

360

480

600

Type 2 G(kN)

720

120

N

240

360

480

600

720

N

1400

160 140 120 100 80 60

1200 1000 800 600 400 200

Type 3

G(kN) 120

240

360

480

600

Type 4

40 20

G(kN)

720

120

240

360

480

600

720

Fig. 2.23. Frequency distribution of the total weight G of the representative lorries per 24 hours based on traffic data of Auxerre in France (1986)

of structures further traffic records regarding the number of heavy vehicles per day are needed. These data were taken for the load model in [9] from several traffic records in Europe. From all the traffic records only the record locations

G[kN] 750 600 450

PA[kN]

total weight of heavy vehicles

Périphérique Brohltal Doxey

300

150

Brohltal Forth

100

Doxey

Auxerre

Forth

50

150 10-4

axle loads

200

Auxerre

10-3

10-2

10-1

1,0

n n30

10-4

10-3

10-2

10-1

n 1,0 n10

Fig. 2.24. Gross vehicle and axle weight distribution of recorded traffic data from England, France and Germany


49

Table 2.3. Statistical parameters of the traffic records of Auxerre (1986) mean value P of the total vehicle weight kN

standard deviation V kN

relative frequency %

Lane 1

Lane 2

Lane 1

Lane 2

Lane 1

Lane 2

Type 1

Go Gl

74 183

64 195

35 28

33 34

13,3 9,4

17,2 10,4

Type 2

Go Gl

123 251

107 257

46 38

45 43

0,3 1,0

1,3 2,2

Type 3

Go Gl

265 440

220 463

60 54

78 79

17,1 48,1

28,0 30,4

Type 4

Go Gl

254 429

196 443

45 68

69 78

3,6 7,2

4,1 6,4

N

Type 4

1500

Go G1 10,8%

10,5%

65,2 %

58,4%

1,3 %

3,5 %

22,7 %

27,6 %

lane 1

lane 2

Type 3 1000 Type 3 Type 2

500

G1

Go 120

240

360

G(kN) 480

600

Type 1

720

Fig. 2.25. Histogram of vehicle Type 3 and approximation by two separate distribution functions based on traffic data of Auxerre in France (1986 ) and frequency of the different vehicle types in the lanes 1 and 2

with a high rate of heavy vehicle in the total traffic are of interest, for example the traffic records of Brohltal and Auxerre in Figure 2.24. The histograms acc. to Figure 2.23 can be subdivided into two separated density functions, where the mean values correspond to loaded and unloaded vehicles. The statistical parameters of these distribution functions are given in Table 3.6. For the vehicle of Type 3 the distributions are shown examplarily in Figure 2.25. Furthermore for the development of the load model the frequency of the different vehicle types in the lanes 1 and 2 is needed. The records based on the Auxerre traffic are given in Figure 2.25. The number of axles per vehicle varies widely depending on the different vehicle manufactures. Nevertheless the frequency distributions of the axle

50


Table 2.4. Relation between gross weight of the heavy vehicles and the axle weights of the lorries of types 1 to 4 in % (mean values and standard deviation) Type of vehicle Type 1

Go Gl Go Gl Go Gl Go Gl

Type 2 Type 3 Type 4

Axle 1 m V 50,0 8,0 35,0 7,0 40,5 8,4 29,4 5,7 30,6 5,8 17,1 2,4 31,7 5,7 18,5 4,1

Axle 2 m V 50,0 8,0 65,0 7,0 36,2 8,8 42,8 4,2 27,5 4,4 26,9 4,4 31,3 5,8 29,1 4,2

Axle 3 m V

23,7 27,8 16,2 19,9 13,4 18,9

7,3 5,3 3,6 3,0 4,1 3,6

Axle 4 m V

13,6 19,0 13,7 18,3

Axle 5 m V

3,1 12,1 2,8 16,7 3,5 9,9 3,4 15,2

3,1 3,8 3,3 4,3

Table 2.5. Distance of axles in [m] of the different types of vehicles (mean values and standard deviation) Type of vehicle Type 1 Type 2 Type 3 Type 4

Axle 1-2 m V

Axle 2-3 m V

Axle 3-4 m V

Axle 4-5 m V

3,71

1,1

3,78

0,71

1,25

0,03

3,30

0,26

4,71

0,78 1,22 0,13 1,23

0,14

4,27

0,40

4,12

0,31 4,00 0,42 1,25

0,03

pacings show three cases with peak values nearly constant and very small standard deviations (vehicles of types 2, 3 and 4 with a space of 1.3 m corresponding to double and triple axles and with a space of 3.2 m corresponding to tractor axles of the articulated lorries). For the other spacings widely scattered distributions were recorded resulting from the different construction types of vehicles. As mentioned before, the traffic data given in Figures 2.23 and 2.24 are based on the traffic records of the Auxerre traffic in France. These data gave no sufficient information about the distribution of gross vehicle weight G on the single axles. Additional information from the traffic records of the Brohltal -Traffic in Germany (Highway A61) was used to define single axles weights and the spacing of the axles. These data (mean values of axle weight and axle spacing and corresponding standard deviations) are given in Tables 3.7 and 3.8. A further important parameter is the description of different traffic situations. For the development of load models the normal free flowing traffic as


f(a)

51

f(a)

0,005 D 90

0,004 0,003

O(1 D )

0,002 0,001

a[m]

a[m] 200

400

600

20

100

Fig. 2.26. Comparison of measured and theoretical values for the density function of intervehicle distances

well as condensed traffic and traffic jam have to be distinguished. The main parameters of the probability density functions for the distance are the lorry traffic density per lane (lorries per hour), the ratio between lorries and motorcars, the mean speed and the probability of occurrence of lorry distances less than 100 m to cover the development of convoys. A typical example for the distribution of distances measured at motorway A7 near Hamburg is given in Figure 2.26 and compared with an analytical function for high traffic densities given in [720]. The density function is approximated by a linear increase up to 20 m due to the minimum distance, a constant part up to a distance of 100 m because of convoys and an exponentially decreasing part for distances greater than 100 m for covering free flowing traffic. Another possibility is the approximation of the intervehicle distance by a log-normal distribution [305] which is based on new traffic data [314]. In Figure 2.26 the value α of the constant part between 20 and 100 m, giving the probability of occurrence for lorry distances less than 100 m, and the value λ were obtained from traffic records of 24 representative traffics in Germany. Additional information regarding the probability of occurrence of convoys are given in [267]. These accurate models apply mainly to the development of fatigue load models. Regarding load models for ultimate and serviceability limit states simplified models for the vehicle distances can be used on the safe side. In case of flowing traffic the distance between lorries is given by a minimum distance required, which results from a minimum reaction time of a driver to avoid a collision with the front vehicle in case of braking. On the safe side a minimum braking reaction time Ts of the driver of one second is assumed. Then the minimum distance a is given by a = v · (Ts ) where v is the mean speed of the vehicles. With this assumption also convoys are covered. The distance is limited to a minimum value of 5 m in case of jam situations.

52


2.3.1.3 Basic Assumptions of the Load Models for Ultimate and Serviceability Limit States in Eurocode As mentioned before, the load model in Eurocode 1 is mainly based on the traffic records of the A6 motorway near Auxerre with 2 × 2 lanes because these measurements were performed over long time periods in both lanes of the Highway and because these data represent approximately the current and future European traffic with a high rate of heavy vehicles related to the total traffic amount and also with a high percentage of loaded heavy vehicles (see also Figure 2.24). The European traffic records had been made on various locations and at various time periods. For the definition of the characteristic values of the load model therefore the target values of the traffic effects have to be determined. For Eurocode 1-2 it was decided, that these values correspond to a probability p = 5% of exceeding in a reference period RT = 50 years which leads to a mean return period of 1000 years. For the determination of target values of the traffic effects additional aspects have to be considered. The measurements of the moving traffic (e.g. by piezoelectric sensors) include some dynamic effect depending on the roughness profile of the pavement and the dynamic behaviour of the vehicles which has to be taken into account for modelling the traffic. The dynamic effects of the vehicles can be modelled acc. to Figure 2.27 taking into account the mass distribution of the vehicle, the number and spacing of axles, the axle characteristic (laminated spring, hydraulic or pneumatic axle suspension), the damping characteristics and the type of tires [720, 530, 238, 99, 330, 331]. The normal surface roughness can be modelled by a normally distributed stationary ergodic random process. The roughness is a spatial function h(x) and the relation between the spatial frequency Ω and the wave length L is given by Ω = 2π/L [1/m]. In the literature many surfaces have been classified by power spectral densities Φh (Ω) acc. to Figure 2.27. Increasing exponent w results in a larger number of wave length and increasing Φh (Ω) results in larger amplitudes of h(x). For modelling the surface roughness of road bridges w = 2 can be assumed. The quality of the pavement of German roads can be classified for motorways as ”very good”, for federal road as ”good” and for local roads as ”average”. While for the global effects of bridge structures an average roughness profile can be assumed, for shorter spans up to 15 m local irregularities (e.g. located default of the carriageway surface, special characteristics at expansion joints and differences of vertical deformation between end cross girders and the abutment) have to be taken into account. These irregularities were modelled in Eurocode 1-2 by a 30 mm thick plank as shown in Figure 2.27. As mentioned above, the axle and gross weights of the vehicles of the Auxerre traffic were measured by piezoelectric sensors. The calculations with fixed base and the vehicle model acc. to Figure 2.27 showed for good pavement quality, that the characteristic values determined from the measured gross and axle weights include a dynamic amplification of approximately 15% of


ª:º ) h ( : ) ) h ( : o )« » ¬ :o ¼

10-1

100

101

w

102

102

spatial frequency :=2S/L [m-1]

z

30 200 300

10-3 10-2 10-1 100 101

power spectral density )h(: ) [cm-3]

10-2

Model for irregularities

6 )=1 (: o 4 )h )= nt (: o me )h ave ent ep )=1 em rag (: o pav )h od ent go em pav

unevenness of the carriageway

ave

spring and damper of the tyre

:o=1 m-1 w=2

ood

h(x)

spring and damper of the vehicle body mass of the axle

yg

M

v er

mA,TA

S

103

PSD- spectras acc. to ISO-TC 108

Modelling of the vehicles

x

53

L

200

+h

+x[m] -h

Fig. 2.27. Model for the vehicles and local irregularities and power spectral density of the pavement

the axles weights and 10% of the vehicle gross weight. The filtering of the dynamic effects leads in comparison to the measured values to a reduced standard deviation. The corrected data of the static vehicle weights are given in Table 3.9. The dynamic behaviour of the bridge structure is mainly influenced by the span length and the dynamic characteristics of the structure [169] (eigenvalues acc. to Figure 2.28 and the damping characteristics). With the vehicle model and the modelling of the roughness of pavement surface acc. Table 2.6. Statistical parameters of the corrected static traffic records of Auxerre (1986)

Type 1 Type 2 Type 3 Type 4

Go Gl Go Gl Go Gl Go Gl

mean value P of the total standard deviation V vehicle weight [kN] [kN] lane 1 lane 2 lane1 lane 2 74 64 31 29 183 195 23 28 123 107 40 39 251 257 31 35 265 220 51 68 440 463 42 65 254 196 37 60 429 443 55 64


10

f

Comparison of calculated and measured dynamic amplification

Eigenvalues (1. mode)

f [Hz]

95,4

1 L0,933

70

V r 0,81 Hz

dynamic amplification in [%]

54

8

6

4

2

10

20

30

40 50 60

70 80

span length in [m]

90

60

36,95

calculated values

41,0m 32,35

50 measured values

40 30 20 10

10

20

30

40

50

60

70 80

vehicle speed [km/h]

Fig. 2.28. Measurements of the eigenvalues of the first mode of steel and concrete Bridges [169], and comparison of theoretically determined dynamic amplifications with measurements

to Figure 2.27 results can be obtained by dynamic calculations of the bridge and be compared with measurements at bridges. Figure 2.28 shows an example of the calculated and measured dynamic amplification of the Deibel-Bridge [720]. With the assumptions and models explained above, a realistic determination of the dynamic and static action effects due to traffic loads is possible. In a first step random generations of load files and roughness profiles of the pavement surface can be produced. Each load file consists of lorries with distances based on constant speed per lane. The main input parameters are the number and types of lorries, the probability of occurrence of each lorry type, the histogram of the static lorry weights of each type, the distribution of lorries to several lanes. For the load files simply supported and continuous bridges with one, two and four lanes and different span lengths between 1 and 200 m with a representative dynamic behaviour (mass, flexural rigidity, mean frequency acc. to Figure 2.28 and damping) have to be investigated in order to get results which are representative for the dynamic amplification of action effects of common bridges. Three different types of bridges with cross-sections with one, two and four lanes were investigated for the load model in Eurocode 1-2. For the different lanes the traffic types acc. to 3.10 were assumed, where traffic type 1 is a heavy lorry traffic for which motorcars were eliminated from the measured Auxerre traffic. The traffic type 2 is the measured traffic of lane


55

Table 2.7. Different cross-sections and traffic types for the random generations

number of lanes

type of cross section

traffic types of the different lanes

3,0 m

1

Type 1 3,0 m 3,0 m

Lane 1: Type 1 Lane 2: Type 2

2

3,0 m 3,0 m

3,0 m

Lane 1: Type 1 Lane 2: Type 3 Lane 3:Type 3 Lane 4: Type 2

3,0 m

4

1 in Auxerre, including motorcars and traffic type 3 is the measured traffic of lane two in Auxerre. Detailed information about the generation of these load files are given in [720, 530]. With random load files the static and the dynamic action effects of the different bridge types can be determined. The comparison of the static and dynamic action effects gives information about the dynamic amplification and the dynamic factor Φ, influenced by the dynamic behaviour of the lorries, the bridge structure and by the quality of the pavement. The results of the simulations can be plotted in diagrams which give the cumulative frequency of the action effects. A typical example is given in Figure 2.29 for a bridge with

cumulative frequency [%]

99,9 97

traffic jam

50 convoy v= 80 km/h

ME

convoy v= 60 km/h convoy v= 40 km/h

action effect 700

1000

1300

ME [kNm]

Fig. 2.29. Cumulative frequency of the action effects for different vehicle speeds [530]

56

2 Damage-Oriented Actions and Environmental Impact M 2,2 2,0 1,8

pavement irregularities (30 mm thick plank)

1,6

flowing traffic and average pavement quality

1,4 1,2

flowing traffic and good pavement quality

1,0 0,8

L [m] 10

20

30

40

50

60

70

80

Fig. 2.30. Influence of the quality of the pavement on the dynamic amplification factor ϕ[530]

one lane, good pavement quality and a span of 20 m. It can be seen that for this example the increase of the vehicle speed leads also to an increase of the dynamic action effects. Furthermore the dynamic amplification is extremely influenced by the roughness of the pavement and also by the span of the bridge. The influences of the pavement quality and traffic in more than one lane are shown in Figures 2.30 and 2.31. The results of the simulations show for condensed traffic no significant influence of the span length and the number of loaded lanes on the dynamic amplification. In case of flowing traffic the dynamic amplification of action effects depends significantly on the quality of the pavement, the number of loaded lanes, the span length and the type of the influence line of the action effect considered.

M

M

1,8

1,8

1,6

1,6

bending moment

1,4

1,4

vertical shear

1,2

bending moment

1,2 L [m]

L [m] 5 10

15

20

25

30 35

10

20 30 40 50 60

70 80

Fig. 2.31. Influence of the span length and the number of loaded lanes on the dynamic amplification factor ϕ


57

'M 1,3 1,2 1,1

30

1,0 L[m] 5

10 15

20

25

30

200 300

200

Model for irregularities

Fig. 2.32. Additional dynamic factor Δϕ taking into account irregularities of the pavement [9]

Figure 2.31 shows the envelope of the calculated dynamic factors ϕ for flowing traffic as a function of the span length. For the development of the load model in Eurocode 1-2 it was decided, that the dynamic amplification of the action effects should be included in the load model because otherwise different parameters like the traffic situation (flowing traffic or traffic jam, the quality of the pavement, the number of loaded lanes and the type of the influence line) had to be considered separately. The calculations show additionally, that the dynamic amplification due to flowing traffic is only relevant for shorter span length up to 50 m because for greater span length the condensed traffic with low vehicle spacings or the traffic jam lead to extreme action effects. As explained above the dynamic effects due to local irregularities were modelled by a 30 mm thick plank, which leads especially for shorter spans to a significant additional dynamic amplification factor. Figure 2.32 gives the additional dynamic factor Δϕ due to irregularities which has to be considered especially for fatigue verifications for short spans, e.g. for end cross girders and members near expansion joints (see Figure 2.32). With the random load files the static and the dynamic action effects and the characteristic values of the action effects can be determined. As mentioned above, the characteristic values in Eurocode 1-2 correspond to a probability p = 5% of exceeding in a reference period R = 50 years which leads to a return period of TR = 1000 years. The procedure for the determination is shown in Figure 2.33. The simulation of different bridge types gives a cumulative frequency of the considered action effects. The characteristic values can be determined by extrapolation. Finally these characteristic values can be compared with a simplified characteristic load model. The load model for global effects in Eurocode 1-2 [9] consists of uniformly distributed loads and simultaneously acting concentrated loads, so that global effects in large spans and the local effects in short spans can be covered by


99,90 99,00

50,00

extrapolation for the determination of the characteristic values

dynamic amplification factor: I

static values of simulations

E k ,dyn E k ,stat.

ME

dynamic values of simulations

Ek,dyn


99,9999

Ek,stat.

58

action effect

ME

influence line for ME

Fig. 2.33. Determination of the characteristic values of the action effects from the random generations of loads

the same model taking into account the dynamic amplification, where average pavement quality is expected. The carriageway with the width w is measured between kerbs or between the inner limits of vehicle restraint systems. For the notional lanes a width of wl = 3,0 m is assumed, and the greatest possible number nl of such lanes on the carriageway has to be considered. The locations of the notional lanes are not be necessarily related to their numbering. The lane giving the most unfavourable effect is numbered as Lane Number 1, the lane giving the second most unfavourable effect is numbered as Lane Number 2 and so on. For each individual verification the load models on each notional lane and on the remaining area outside the notional lanes have to be applied on such a length and longitudinally located so that the most adverse effect is obtained. The Load Model 1 in Eurocode 1-2 is shown in Figure 2.34. It consists of a double axle as concentrated loads (Tandem System TS) and uniformly distributed loads (UDL-System). For the verification of global effects it can be assumed that each tandem system travels centrally along the axes of notional lanes. For local effects the tandem system has to be located at the most unfavourable location and in case of two neighbouring tandem systems they have to be taken closer, with a distance between wheel axles not smaller than 0,5 m. With the adjustment factors αQi and αqi the expected traffic on different routes can be taken into account. The last step in the development of the load model is the comparison of the characteristic action effects caused by the normative load model with the characteristic values of the dynamic values of the real traffic simulations. Figure 2.35 shows this comparison for a three span bridge girder with one, two and four lanes. For the verification of local effects a Load Model 2 is given in Eurocode 1-2. This model consists of a single axle load equal to 400 kN, where the


Application of the Tandem System for local verifications

Application of the Tandem System for global verifications DQi Qik

1,20m

Dqi qik

2,00m 2,00m > 0,50m

DQi Qik

0,50

w1

Lane number 1: Q1k = 300 kN aQ1q1k = 9 KN/m²

2,00 0,50 0,50

w2

Lane number 2: Q2k= 200 aQ2 q2k = 2,5 KN/m²

2,00 0,50 0,50

w3

59

contact area of the wheel loads

Lane number 3: Q3k= 100 aQ2 q2k = 2,5 KN/m²

2,00 0,50

0,4 m Lane number 4 and further lanes as well as remaining areas: aQ3 q3k = 2,5 KN/m²

wi

0,4 m

Fig. 2.34. Load Model 1 according to Eurocode 1-2

ME/L 500

Load Model 1 acc. to Eurocode 1 400

simulation 300

200

ME

100

L

L 20

40

span length

60

80

L

L

Fig. 2.35. Comparison of the Load Model 1 in Eurocode -2 with the characteristic values obtained from real traffic simulations

dynamic amplification for average pavement quality is included. In the vicinity of expansion joints an additional dynamic amplification has to be applied for

60


Table 2.8. Traffic data of different locations and characteristic values of gross and axle weight [720] number nl of lorries per day

weight of one axle kN

tandem axles kN

tridem axles kN

gross weight of vehicle kN

1984

4793

211

357

434

853

Chamonix

1987

1204

192

355

480

724

Auxerre

1986

2630

245

397

527

811

France

Angers

1987

1272

192

340

456

670

France

Lyon

1987

1232

267

450

475

930

country

location

year

Germany

Brohltal

Belgium France

Table 2.9. Different design situations and corresponding return periods and fractiles

Design situation

Return period TR

infrequent frequent quasi - permanent

1 year 1 week 1 day

Fractile of the distribution of action effects in % 99,997 99,891 99,240

taking into account the local irregularities at expansion joints. The contact surface of each wheel can be taken into account as a rectangle of sides 0,35 m and 0,6 m. The evaluation of the traffic data of different locations lead to static characteristic axle values Qk given in Table 3.11, where the characteristic values relate to a return period TR of 1000 years (probability p of 5% in 50 years). It can be seen that the characteristic values are depending on the location. Taking into account the dynamic amplification for short spans (see Figure 2.31), this leads to the axle weight given in Eurocode 1-2. For serviceability limit states like limitation of deflections, crack width control and limitation of stresses to avoid inelastic behaviour, different design situations have to be distinguished. The Eurocodes distinguish between infrequent, frequent and quasi permanent design situations characterised by different return periods. The return periods and the corresponding fractile of the distribution of the dynamic action effects are given in Table 3.12. A change of the return period is equivalent with a change of the fractile of the distribution (see Figure 2.36). The representative values Frep of the action effects can then be written as Frep = ψ Fk , where Fk is the characteristic value. As explained above, the characteristic values were determined with adverse assumptions regarding the quality of the pavement Φ(Ωh ) = 16 acc. to


representative valuesErep=\ Ek

characteristic values:

static values of simulations

I

Ek,dyn

Ek,stat.

50,00

Erep,dyn.

dynamic values of simulations Erep,stat.


99,90 99,00

dynamic amplification factor

characteristic values Ek

99,9999

61

E k ,dyn E k ,stat.

representative values: E rep,dyn I E rep,stat. action effect E

ME Fig. 2.36. Determination of the representative values and the corresponding dynamic factors 10mm

2 3

3

L

'V

L Category 56 'Vc= 56 N/mm2 for L>100mm

1

'V

Fig. 2.41. Typical examples for fatigue strength categories

accumulation according to Miner [543] is used (Figure 2.40). Based on this assumption a realistic fatigue load model must fulfil the condition, that the cumulative damage produced by the real traffic must be equal to the cumulative damage caused by the load model. The main parameters which have to be considered are the design fatigue life, the type and number of lorries crossing the bridge, the traffic composition and the number of lanes with heavy traffic and in addition the quality of the pavement and the dynamic behaviour of the vehicles and the bridge. For fatigue problems of bridges only the traffic situation of flowing traffic has to be considered because the number of traffic jams is negligible during the design life. Furthermore the influence of motorcars can be neglected, because the stress ranges caused by motorcars do not reach the cut off limit of the fatigue strength curves. For the development of fatigue load models further considerations are necessary. For Eurocode 1-2 e.g. it was decided that the load model should include the dynamic amplification of the real traffic. Regarding the modelling several strategies are possible. One possibility is to consider only one type of vehicle in verifications and to take into account all other effects resulting from the real traffic by damage equivalent factors. This is the basis of the Load Model 3 in Eurocode 1-2. An other possibility is the definition of a set of lorries which together produce effects equivalent to those of typical traffic on European roads. An example for such a model is the Load Model 4 in Eurocode 1-2 (Figure 2.42). The fatigue models 3 and 4 are intended to be used for fatigue life verifications by reference to a fatigue strength curve. For the fatigue life verification it has to be distinguished between different traffic categories. The category is defined by the number of slow lanes, the number Nobs of heavy vehicles with a maximum gross weight more than 100 kN which was observed

66


Table 2.11. Traffic categories acc. to Eurocode 1-2

Traffic category

Nobs per year and per slow lane

1

Roads and motorways with 2 or more lanes per direction with high flow rates of lorries

2,0 x 106

2

Roads and motorways with medium flow rates of lorries

0,5 x 106

3

Main roads with low flow rates of lorries

0,125 x 106

4

Local roads with low flow rates of lorries

0,05 x 106

or estimated per year and per slow lane. Typical traffic categories acc. to Eurocode 1-2 are given in Table 3.14. Fatigue Load Model 4 in Eurocode 1-2 consists of a set of standard lorries (Figure 2.42) which together produce effects equivalent to those of typical traffic on European roads. This model is intended to determine stress range spectra resulting from the passage of lorries on bridge. The equivalent lorries are defined by the number of axles and the axle spacing, the equivalent load of each axle, the contact surface of the wheels, the transverse distance of the wheels and the percentage of each standard lorry in the traffic flow. For the verification of global action effects the model can be placed centrally on the notional lanes acc. to Figure 2.34. For local members (e.g. concrete slabs or orthotropic decks) the model has to be centred on notional lanes assumed to be located anywhere on the carriageway. Where the transverse location of the fatigue load model is significant for the action effects e.g. in orthotropic decks, a statistical distribution of this transverse location acc. to Figure 2.43 has to be taken into account. As mentioned above, the fatigue load models in Eurocode 1-2 include a dynamic load amplification ϕf at . An additional dynamic load amplification factor Δϕf at acc. to Figure 2.43 has to be taken into account near expansion joints to allow for the effects of local irregularities in this regions. For the other regions of the bridge the dynamic load amplification factor must take into account the high number of relative small load cycles. This can be achieved by introducing a damage equivalent dynamic amplification factor acc. to Figure 2.44 which results from the comparison of the cumulative damage calculated with and without dynamic amplification of the Auxerre traffic. The procedure for the determination of ϕf at is shown in Figure 2.44. Because most of the stress ranges are below the fatigue strength limit ΔσD , the dynamic factor can be determined with a constant value of m = 5 for the slope of the fatigue strength curve. In Eurocode 1-2 good pavement quality acc. to Figure 2.27 was assumed. The influence of the pavement quality on the dynamic amplification factor ϕf at can be seen from Figure 2.45. For good pavement qualities the dynamic

vehicle type Lorry

long distance

medium distance

local traffic

wheel type


20

40

80

A B

traffic type and lorry percentage axle spacing [m]

axle loads [kN]

4,50

70 130

4,20 1,30

70 120 120

5

10

5

A B B

3,20 5,20 1,30 1,30

70 150 90 90 90

50

30

5

A B C C C

3,40 6,00 1,80

70 140 90 90

15

15

5

A B B B

4,80 3,60 4,40 1,30

70 130 90 80 80

10

5

5

A B C C C

67

wheel types and dimensions of the wheel contact surface in mm 320

320

2,0 m

220

x

Type A

320 220

270

220

Type B

Type C

Fig. 2.42. Set of lorries of Fatigue Load Model 4 in Eurocode -2 and contact surfaces of the wheels

Dynamic load amplification factor near expansion joints

Distribution of transverse location of centre line of vehicle 50% 18%

D

7%

'Mfat 1,3 1,2 1,1 1,0

5 x 0,1 m

D [m] 2,0

4,0

6,0

Fig. 2.43. Distribution of transverse location of centre line of vehicles and dynamic load amplification factor near expansion joints

68

2 Damage-Oriented Actions and Environmental Impact 'V(log)

damage:

1

'V Ri

ªN ºm 'V D « D » ¬ Ni ¼

100

i

category 160

1 NC

ª 'Vi º « 'V » ¬ C¼

Di

ni N Ri

1 ND

ª 'Vi º « 'V » ¬ D¼

category 36

10

NL

NC ND 105

ni N Ri

Di

'Vi

106

107

108

m1

Di

for 'Vi t 'V D m2

for 'V D t 'Vi t 'V L

0

for 'Vi V L

Linear damage accumulation: D

N(log)

¦ Di

D Auxerre d 1,0

109

Damage equivalent dynamic amplification factor: i

m ¦ n i,dyn 'Vi,dyn i

m m ¦ n i,dyn 'Vi,dyn ¦ n i,stat Mfat 'Vstat Mfat

mi

i

¦

i n i,stat 'Vim,stat

mi

D Auxerre,dyn D Auxerre,stat

Fig. 2.44. Linear damage accumulation and damage equivalent dynamic amplification factor ϕf at

Mfat 1,8 1,6

average pavement quality )h(:o)=16

good pavement quality )h(:o)=4

flowing traffic with v= 80 km/h

ME

1,4 1,2 L [m]

1,0 10

20

30

40

50

60

70

L

L

L

80

Fig. 2.45. Influence of the pavement quality on the damage equivalent dynamic amplification factor [530]

factor ϕf at is in the range of 1.2, which is included in the load model in Figure 2.44. For average pavement qualities a mean increase in the range of 20% was obtained which leads to an increase of the damage D by a factor of 2.5 and a decrease of the fatigue life to 0.4 when for the slope of the fatigue strength curve m = 5 is assumed. This demonstrates that the authorities have the responsibility for a careful maintenance of the roads. As mentioned above, Fatigue Load Model 3 (Figure 2.46) consists of a single vehicle with four axles, each of them having two identical wheels with a squared surface contact area of each wheel with the side lenght of 0.4 m. The weight of the axles is equal to 120 kN and includes the dynamic amplification factor ϕf at . The damage of the real traffic is taken into account by a damage

2.3 Transport and Mobility Fatigue verification

Axle loads of Fatigue Load Model 3 120 kN 120 kN

69

fatigue strength curve

'VC

120 kN 120 kN

O 'VLM 1,20m

'VLM 0,4 m

0,4 m

3,00 m

6,00 m

lane width

2,00 m

1,20m

'Vi(ni) NC J F,fat O 'V LM d

ND 'VC J M ,fat

Fig. 2.46. Fatigue Load model 3 in Eurocode 1-2 and fatigue verification for steel structures

equivalent stress range λ · ΔσLM [31, 327] with λ = λ1 · λ2 · λ3 · λ4 ≤ λmax The factor λ1 takes into account the damage effect of traffic depending on the length of the critical influence length, λ2 is a factor for the traffic volume, λ3 allows for different design life and λ4 takes into account the traffic on other lanes. The fatigue verification can then be performed according to Figure 2.46, where ΔσLM is the stress range caused by the load model, ΔσC is the reference strength at 2 million load cycles and γF,f at and γM,f at are the partial safety factors for the equivalent constant amplitude stress λ · ΔσLM and the fatigue strength ΔσC . The damage equivalent factors must be determined from the real traffic, where for Eurocode 1-2 the Auxerre traffic was used. In a first step it is assumed that for the determination of λ1 the factor λ2 is equal to 1.0 for No = 0.5 × 106 lorries per year and slow lane and that a design life Tso = 100 years corresponds to a factor λ3 = 1.0. Furthermore only one slow lane is investigated which gives λ4 = 1.0. Then the factor λ1 must fulfil the condition, that the damage of the load model DLM is equal to the damage of the Auxerre traffic DAuxerre = Σ Di . For the calculation of λ1 at first random load files based on the Auxerre traffic and the corresponding stress range spectra acc. to Figure 2.44 have to be determined. The corresponding accumulative damage caused by the number nLs of simulated lorries results in DAuxerre = Σ Di . As the damage caused by the load model has to be equal to the cumulative damage DAuxerre , a correction factor λe for the stress range ΔσLM of the load model has to be introduced (see Figure 2.48). For the same number of lorries nLs the equivalent damage of the load model DLM and the factor λe is given by m nLs λe · ΔσLM ΔσD m ND · DAuxerre λe = (2.55) DLM = ND ΔσD ΔσLM nLS

70

2 Damage-Oriented Actions and Environmental Impact flowing traffic with v= 80 km/h

Oe

Oe static action effects

1,4 1,2

1,4

dynamic action effects

1,0

1,2 1,0

0,8

0,8

0,6

0,6

ME

0,4

L

L

0,2

L [m] 10

20

30

40

50

ME

0,4

L

60

L

L

L

0,2

70 80

L [m] 10

20

30

40

50

60 70 80

Fig. 2.47. Example for the damage equivalent factor λe [530]

'V(log)

m1=3 D

'VC O1 'VLM

¦ Di

D Auxerre

m2=5

'Ve=Oe'VLM 'VLM

'Vi(ni) N(log) NC

N D nS

NL

Fig. 2.48. Determination of the damage equivalent factor λ1

A typical example for the damage equivalent factor for a three span bridge is given in Figure 2.47. It can be seen that the dynamic amplification leads to a significant increase of the factor λe . Furthermore the factor depends on the type of the influence line and the assumption for the quality of the pavement. The values in Figure 2.47 were determined for a good pavement quality. For the fatigue verification acc. to Figure 2.46 it has to be taken into account that the verification is based on the fatigue strength ΔσC at NC = 2×106 load cycles and that in addition the relevant number of lorries during the design life Tso is given by NT O = No ·Tdo . This leads to a further transformation for the damage equivalent stress range Δσe = λe · ΔσLM (see Figure 2.48).

2.3 Transport and Mobility O1 2,8 2,6

O1 midspan regions 2,55

2,4

2,2

2,2

2,0

2,0 1,85

1,4

L [m] 10

20

30

L1

40

50 60

70

80

L2 2,2

2,0

1,8 1,6

1,70

1,4

L

1,2

internal supports

2,8 2,6

2,4

1,8 1,6

71

L= ½ (L1+L2)

1,2 10

20

30

40

50 60

70

L [m] 80

Fig. 2.49. Factors λ1 for steel bridges given in Eurocode 3-2

Because NT o is greater than ND in the first step a correction factor α for the damage equivalent stress related to ND is determined using the slope of the fatigue strength curve m2 = 5. NT o 5 5 (2.56) NT o [λe · ΔσLM ] = ND [α · λe · ΔσLM ] ⇒ α = 5 ND In the second step the transformation of the equivalent stress range related to NC follows using the slope m1 = 3 (see Figure 2.48) ND 3 3 (2.57) ND [α · λe · ΔσLM ] = NC [α · β · λe · ΔσLM ] = 3 NC The damage equivalent factor λ1 is then given by: ND 5 NT o · 3 λ1 = λe · α · β = λe · ND NC

(2.58)

The equivalent damage factor λ1 depends on the damage equivalent factor λe , the type of the fatigue strength curve (slopes m1 and m2 and the fatigue strength ΔσD and ΔσD respectively) and the relevant numbers NT o of lorries during the design life assumed for λ2 = 1.0. Therefore the factor differs for structures and structural members with different materials (e.g. structural steel, reinforcement, shear connectors). Figure 2.49 shows the λ1 values for steel bridges which are an envelope of the most adverse values determined for different types of influence lines. For concrete and composite bridges corresponding values are given in Eurocode 2-2 [29] and Eurocode 4-2 [31], respectively. As explained above, the factor λ1 was determined for the reference value No = 0.5 × 106 , where No corresponds to the traffic category 2 in Table 3.14. Furthermore for the design life a reference value Tso = 100 years was assumed. In case of another traffic category or design life the damage equivalent factor

72


has to be modified with the factors λ2 for the traffic category and λ3 for the design life. Regarding the traffic category it also has to be considered, that on special routes the mean gross weight of the lorries can be higher or less than the average gross weight of the Auxerre traffic. For the factor λ2 results Qml 5 Nobs (2.59) λ2 = Qo N0 where Nobs is the relevant number of lorries per year for the relevant traffic category given in Table 3.14, Qo = 480 kN is the reference value for the gross weight of the heavy vehicles and Qm1 is the damage equivalent gross weight of the lorries in the slow lane specified by the competent authority by the number ni of lorries and the corresponding gross weight Qi in the slow lane. Qml =

Σni · Qi Σni

1/5 (2.60)

With the reference value NT O = No ·Tdo the factor λ3 is given by equation 2.61 Td λ3 = 5 (2.61) Tdo The factor λ1 in Figure 2.49 is determined for lorries only in the slow lane of the bridge. In case of more than one heavy lane on the bridge the effect is taken into account by the factor λ4

N2 λ4 = 1 + N1

η2 · Qm2 η1 · Qm1

m

N3 + N1

η3 · Qm3 η1 · Qm1

m

Nk + ... + N1

ηk · Qmk η1 · Qm1

m m1 (2.62)

where Qmi

is the average gross weight of lorries in lane j

Nj

is the number of lorries per year in lane j

k

is the number of lanes with heavy traffic

m

is the slope of the fatigue strength curve (e.g. m = 5 for structural steel, m = 9 for reinforcement (straight bars) and m = 8 for headed stud shear connectors)

ηj

is the value of the transverse influence line for the internal force that produces the stress range in the middle of lane j acc. to Figure 2.50 and to be inserted in equation 8 with positive sign.


73

Lane 2

Lane 1

K1

K2

transverse influence line

1,0

Fig. 2.50. Assumptions for the factor λ4

Omax

Omax 2,8 2,6

internal supports

2,8

midspan regions 2,55

2,4

2,4

2,2

2,2

2,05

2,0

2,0

1,8

1,80

L= ½ (L1+L2)

1,8 1,6

1,6 1,4

1,4

L

1,2

L [m] 10

2,70

2,6

20

30

40

50 60

70

L1

L2

1,2

80

L [m] 10

20

30

40

50 60

70

80

Fig. 2.51. Damage equivalent factor λmax

For materials (e.g. structural steel) with a fatigue limit for constant amplitude stress ranges the damage equivalent factor λ is limited to a value λmax . Where all stress ranges caused by the real traffic do not exceed the fatigue limit (Δσmax ≤ ΔσD ) the fatigue life is unlimited. In this case results from the condition Δσmax = λmax · ΔσLM λmax =

Δσmax ΔσLM

(2.63)

where Δσmax can be determined from the traffic simulations of the Auxerre traffic. Figure 2.51 shows the values λmax given in [31] for steel bridges. 2.3.1.5 Actual Traffic Trends and Required Future Investigations For the transport of persons and goods bridges are an important part of the infrastructures in Europe. As explained above the load models for bridges in Eurocode 1-2 cover the European traffic of the year 2000. Contrary to all forecasts the amount of heavy traffic on motorways has increased in the last

74

2 Damage-Oriented Actions and Environmental Impact billion to km 600 500

forecast

400 300 200 100

2010

2000

1998

1996

1994

1992

1988

1990

1986

1984

1982

1980

year

Fig. 2.52. Development of the freight traffic on roads, railways and ships

years and it is expected that this increase will continue in future. Figure 2.52 shows the development of the freight traffic on roads, railways and ships in Germany [563]. In the last 20 years the total freight traffic on roads has increased significantly after the German reunification in comparison with traffic on rails and ships. It can be expected that a further increase of traffic will take place due to the increasing transit trade and the affiliated cross border traffic. This can also be seen from Figure 2.53 which gives the recorded and expected number of heavy vehicles per day [563]. The comparison with Table 3.14 shows that with regard to fatigue at present the number of heavy vehicles exceeds the values assumed for category 1 in Eurocode 1-2. In order to optimize the transport capacities and minimize the transport costs there is a strong tendency to produce vehicles with higher gross weights. This results from Figure 2.53 giving the relative frequency of articulated vehicle with two driving axles and triple axle semi-trailer, which is the most frequent type on German roads at present. Table 3.15 shows the results of actual traffic records (2004) at the Highway A61 near Brohltal [314]. The table demonstrates that the articulated vehicles (Type 5) dominate the traffic composition with a percentage of nearly 60%. The comparison with traffic data of the Auxerre traffic recorded in 1986 (see Table 1) shows, that presently the mean values of the gross vehicle weights in Germany are nearly conform with the values of the Auxerre traffic. For loaded articulated vehicles Table 3.15 gives a mean value of the gross weight of 405 kN which corresponds to a mean value of 463 kN of the Auxerre traffic, but with the addition that the standard deviation of the actual records is higher. The increasing of the standard deviation is mainly based on the fact that there is an increasing number of overloaded lorries.

2.3 Transport and Mobility number of heavy vehicles per day 7

12000

6

10000

5

8000

4 3

forecast

14000

6000 4000

2015

1998

1994

1996

1990

year 1992

f[%]

2

2000 1988

75

1

G[kN] 100

200

300

400

500

Fig. 2.53. Development of the number of heavy vehicles per day on highways and relative frequency of the gross weight for articulated vehicles with two driving axles and triple axle semi-trailer

Table 2.12. Statistical parameters of the traffic records at highway A61 (2004)

type of vehicle Type 1 Type 2 Type 3 Type 4 Type 5

Go Gl Go Gl Go Gl Go Gl Go Gl

mean value P of the standard deviation relative frequency % V total vehicle weight kN kN 59,6 14,6 5 91,7 44,0 6 190,3 23,2 1 208,4 73,9 4 276,8 59,5 12 414,5 32,5 5 156,7 18,8 3 211,4 52,8 5 259,6 92 37 405,3 24,8 22

The distribution of the gross weight to the single axles as recorded in 2004 is shown in Table 3.17. These new data are comparable with the values measured in Auxerre in 1986. Furthermore new data of the density function of the intervehicle distances [305] show in comparison with old traffic records that there is a trend to lower intervehicle distances mainly based on the fact that at present the number of convoys increases conditioned by modern breaking systems. This is especially important for the fatigue resistance of bridges with longer spans. In summary it can be stated, that the Auxerre traffic which was the basis of the load models in Eurocode 1-2 covers presently the actual traffic in Germany.

76


Table 2.13. Relation between gross weight of the heavy vehicles and the axle weights of the lorries of types 1 to 5 (mean values) Axle 1

Axle 2

Type 1

Type of vehicle

44,9 %

55,1 %

Type 2

25,8 %

Type 3 Type 4 Type 5

Axle 3

Axle 4

37,2 %

18,9 %

18,1 %

20,9 %

25,8 %

16,1 %

19,5 %

30,6 %

30,9 %

19,1 %

19,4 %

20,8 %

28,1 %

17,0 %

17,0 %

Axle 5

17,7 %

17,1 %

Nevertheless Figure 2.53 and Table 3.15 indicate that in the near future an adjustment of the load models in the codes is necessary. The data given in Tables 3.15 and 3.17 cover the normal traffic on highways, which do not include abnormal and heavy load transports. For such transports normally a special permit by the authorities is necessary. Figure 2.54 shows the development of the number of applications to authorisation for heavy load transports in Bavaria and North-Rhine Westphalia. The diagram demonstrates the significant increase in the number of such transports. At present in Germany it is considered to permit heavy load transports with gross weights up to 720 kN (see Figure 2.54) for defined routes over limited

n

(number of permits of heavy transports per year)

heavy vehicle with G= 520 kN

80000 70000 90 kN

60000

North Rhine Westphalia

50000

130 kN

3 x 100 kN

heavy vehicle with G= 550 kN

40000 90 kN

30000

North Bavaria

20000

2 x 130 kN

2 x 100 kN

crane with G= 720 KN

10000

2004

2002

1998

2000

1994

1996

1992

1988

1990

1984

1986

1982

1980

year

6 x 120 kN

Fig. 2.54. Development of the number of permits of heavy transports in Bavaria and North-Rhine Westphalia and examples for vehicles for heavy transports


77

G [kN] 1270 kN

1300 1200 1100 1000 900

vehicle weight

800 700 600 500 400 axle weight

300 200 100

n

102

103

104

105

106

107

number of vehivles and axles per year

Fig. 2.55. Traffic records from the Netherlands recorded in 2006

periods of one or two years. In this case a further increase of these transports can be expected and it cannot be excluded that a significant percentage of these transports is overloaded. A possible increase in the number of such vehicles in combination with a possible overloading has especially to be considered for the development of future fatigue load models. A comparable development takes place in other European countries. Figure 2.55 shows the vehicle weight and axle load distributions recorded in 2006 near the harbour of Rotterdam in the Netherlands. It can be seen that the extreme values of the gross weight and also the extreme values of the axle loads are significant higher than the values of the Auxerre traffic (see Figure 2.24). The shape of the distribution shows that the heavy load transports lead in comparison with the Auxerre traffic to a new shape of the distribution which could be taken into account by splitting the distribution into a distribution for normal traffic and a distribution for heavy load transports. Additionally the transport industry is extremely interested in new transport concepts at present. In some European countries and also in some German federal states field trials take place with modular vehicle concepts, the so called Giga-Liners with gross weight up to 600 kN and a total length of 25.25 m [314]. Typical vehicles and the corresponding allowable axle loads are shown in Figures 2.56 and 2.57. These types of vehicles have significant higher transport capacities and can reduce the transport cost. At present it cannot be foreseen how the future traffic composition will change. Some people argue that the new modular concept will reduce the total number of lorries on roads due to the higher transport capacity. On the other hand it has to be

78

2 Damage-Oriented Actions and Environmental Impact Current trucks in Germany (gross weight 400kN)

Vehicles acc. to the modular concept (gross weight up to 600kN)

25,25 m

16,50-18,25 m

Fig. 2.56. Heavy vehicles on the basis of the modular concept (Giga-Liners)

1,475

5,10 m

1,35

4,65

1,35

78 kN

78 kN 5,965 m

78 kN

54 kN

54 kN

92 kN

92 kN

74 kN

Giga – Liner with gross weight of 600 kN

1,36 1,36

2,64

1,475 3,215 1,36

5,965 m

1,36 1,36

90 kN 90 kN

65 kN 65 kN 65 kN

74 kN

74 kN

57 kN

Giga – Liner with gross weight of 580 kN

6,27 m

1,36

2,88

25,25 m

Fig. 2.57. Axle spacing and allowable axle weights of ”Giga-Liners”

considered that this new type of vehicle can not be loaded on trains, so that it can be expected that no significant reduction of the total road traffic will occur. First investigations [201] show that especially for bridges with longer spans the current European load model has to be modified, when the percentage of the new vehicles reaches 20% to 40% related to the total heavy traffic. Furthermore at present no information is available regarding the driving of such vehicles in convoys, especially on routes with acclivities, and the possible overloading and wrong loading which can lead to higher axle weights.


79

The new traffic concepts and development regarding heavy transports need new technologies to get more detailed information about the actual traffic situation and also a more close cooperation between the car industry and the authorities and experts for the development of realistic traffic models. The Weight in Motion (WIM) is a technology [407, 588] for the determination of the weight of vehicles without requiring it to stop for weighting. The system uses automated vehicle identification to classify the type of the vehicle and measures the dynamic tyre force of the moving vehicle when the vehicle drives over a sensor. From the dynamic tyre load then the corresponding tyre load of a static vehicle is estimated. The most common WIM device is a piezoelectric sensor embedded in the pavement which produces a charge that is equivalent to the deformation induced by the tyre loads on the pavements surface. Normally two inductive loops and two piezoelectric sensors in each monitoring lane are used. The system can be used in combination with an automatic vehicle classification system (AVC). Vehicles which do not meet the gross weight and axle weight requirements are notified with dynamic message signs. While in the USA this systems are used in some states all over the country, in Europe only in some countries these systems are used on special routes. First field trials with combined WIM and AVC methods take place presently in the Netherlands. The records demonstrate that besides the problem that the total weight of the vehicles exceed the permissible total weight there are also cases where the permissible total weight is not exceeded, but due to wrong loading of the vehicles the weight of single axles is significantly higher than the permissible axle weight. This can lead to excessive fatigue damage especially in orthotropic decks of steel bridges and also in concrete decks. These new traffic records demonstrate that in the future a better cooperation between bridge designers and truck producers is necessary. Strategies to avoid such overloading of single axles could be the implementation of immobiliser systems in trucks if single axles or the total gross weight of the truck are exceeded. 2.3.2 Aerodynamic Loads along High-Speed Railway Lines Authored by Hans-J¨ urgen Niemann Shelter walls often accompany high-speed railway lines for noise protection or to provide wind shelter for the trains. The walls consist of vertical cantilevered beams connected by horizontal panels. The pressure pulses from head and tail of the train induce a pressure load on the walls, which is in general smaller than the wind load. However, the load is dynamic which may cause resonant amplification. The load is furthermore frequent which may require design for fatigue. These issues are the topic of the following chapter.

80


Fig. 2.58. Pressure time history at the track-side face of a 8 m high wall; at a fixed position; V = 234.3 km/h, [573]

2.3.2.1 Phenomena As a train passes, a sudden rise and drop of the static pressure occurs. Structures at the trackside, such as noise barrier or wind shelter walls, in turn experience a time variant aerodynamic load [777]. It is caused by the pressure difference over the wall sides facing the track and the rear face. The load intensity of this aerodynamic loading is proportional to the square of the train speed. Figure 2.58 shows a pressure time history measured at a fixed position at the trackside surface of a wall, 1.65 m above rail level. The wall distance to the track axis is ag = 3.80 m. Typically, the head pulse starts with a positive pressure which is followed by a negative pressure approximately identical in magnitude. The subsequent tail pulse is reversed and its amplitudes are smaller unless the train is short. For short vehicles, head and tail pulse may merge and the negative pressure may dominate. Additional pulses occur at inter-car gaps with amplitudes much smaller than head and tail pulses. The measured time history clearly depends on the train speed. If instead of the time history the load pattern along the wall is considered, it becomes independent of the train speed. Figure 2.59 gives an example. The pattern of the pulse sequence travels along the wall at the train speed. It provides a dynamic load on the wall structure within a narrow bandwidth of frequencies determined by the train speed V . Furthermore, a spectral decomposition shows that the distance Δx of the positive and negative pulses is related to the prevailing frequency. Figure 2.59 gives two values of Δx measured at a track distance of ag = 3.80 m at two different train speeds. The effect of the train speed is within the scatter of the experimental results.


81

Fig. 2.59. Pressure distribution along the track-side face of a wall at two different train speeds [573]

(a)

(b)

Fig. 2.60. Full scale tests performed along the high speed line Cologne-Rhine/Main: view of the trough; (a) measuring the train speed, (b) with measurement set-up at the eastern wall

A spectral decomposition shows that the prevailing frequency fp is in the order of fp ≈

V 2.7Δx

(2.64)

Depending on the natural frequencies fn of the wall or any other trackside structure resonance may occur at a critical train speed Vres ≈ 2.7Δxfn , which in turn may cause considerable fatigue at rather few train passages. The

82


maximal pressure amplitude measured at a train speed of 304 km/h = 84.6 m/s is ca. 0.550 kN/m2. Typical wind loads are larger by a factor of 2 to 4. It has been argued that the load effect will become important only at very high speeds beyond 300 km/h (see [617]). In fact, the aerodynamic load does not dominate the design as long as the train speed is sufficiently below the critical. If however the critical speed is lower than the maximal track speed, resonant amplification will provide the dominant design situation. Fatigue damage occurred at protection walls along a high speed railway line in 2003. Previous investigations e.g. [36] had dealt with the static effect of the pulse and developed simplified design loads which cover the static action effect. However, they did not consider to model the loading process in view of the dynamic load effects. Therefore, additional investigations became necessary with a focus on the dynamic nature of the load. One issue concerned full-scale measurements of the aerodynamic load patterns along the wall and over the wall height, and the relation of natural wall frequency to the critical train speed. The following findings rely on the results of a campaign performed in 2003, see [573]. The measurements were performed along a concrete wall in order to avoid disturbances coming from the strong deformations of some of the walls. 2.3.2.2 Dynamic Load Parameters The streamlined shape of nose and tail, as well as the frontal area do not only determine the drag of the train but also the pulse amplitudes. As well, the nose length affects the distance between the pressure peaks. The ERRI-report [36] identifies three typical train nose shapes and gives load reduction factors as follows: freight trains express trains with Vmax = 220 km/h high speed trains (TGV, ICE, ETR)

k1 = 1, 00; k1 = 0, 85; k1 = 0, 60.

The dynamic stagnation pressure of the train speed clearly governs the aerodynamic pressures. Figure 2.61 is based on the pressures at the track-side wall surface. The diagram relates the measured pressure peaks of the head pulse, positive and negative, to the dynamic head of the train speed: q=

1 ρV2 2

(2.65)

The relation is linear with a high degree of correlation, and it follows that pressure coefficients may be introduced as cp =

p q

(2.66)


83

(a)

(b)

Fig. 2.61. 3 Effect of train speed stagnation pressure on the head pulse acting at the track-side face of a wall; (a) positive pressure; (b) negative pressure

Figure 2.62 shows the pattern of the head pulse in terms of pressure coefficients. The peak coefficients of ±0.15 are typical for the well shaped, slender nose of the ICE 3 train. The mean values are somewhat smaller. The detailed coefficients cp obtained for 152 train passages are: peak pressure maximum mean pressure maximum lowest pressure maximum

cp = 0, 1499 cp = 0, 1380 cp = 0, 1049

peak pressure minimum mean pressure minimum highest pressure minimum

cp = −0, 1520 cp = −0, 1419 cp = −0, 1041

84


Fig. 2.62. Pressure coefficients of the head pulse from 34 passages (at the track-side wall face) at 1.65 m above track level

Fig. 2.63. Distance between the pulse peaks and the zero crossing (ΔL1 = pressure maximum, ΔL2 = pressure minimum)

The dynamic effect is related to the distance between the pulse peaks. As is seen in Figure 2.63 a mean distance of Δx = 6.9 m is typical for the ICE 3 passing at a track distance of 3.80 m. At a train speed of 300 km/h, the related frequency is fp = 4.5 Hz. Natural frequencies of light protection walls are in the same order of magnitude. Obviously, the critical train speed may happen and its dynamic effect may become important.


85

Fig. 2.64. Head pulse in a free flow at various distances from the track axis [98]

Fig. 2.65. Head pulse in the presence of a wall

The results refer to a distance between the wall and the track axis of ag = 3.80 m. This parameter plays an important role both for the amplitude of and the distance between peaks. Figure 2.64 shows the result obtained theoretically regarding the pressure pulse in a free flow. As the track distance ag increases, the peak amplitudes max p and min p decrease whereas the separation Δx between the pulse peaks increases. Theory predicts that in free flow without walls, the separation Δx depends linearly on the track distance ag , see e.g. [98] √ (2.67) Δx = 2 ag

86


Experimental results can best be fitted by a slight modification: Δx = 1.424 a1.029 g

(2.68)

Figure 2.65 shows the head pulse in the presence of a wall for two different distances. The measurements at a track distance of 3.80 m and 8.30 m were performed simultaneously i.e. at identical train speeds at different walls, both 8 m high. The distance of the peaks at the wall decreases similar to the free flow case. However, the results indicate that the effect of the track distance becomes non-proportional in the presence of a wall. An analogous approximation matches the test results 0.653 ag Δx(ag ) = 6.9 (2.69) ag,ref in which ag,ref = 3.8 m is used as reference. The pressure amplitudes decrease with the inverse of the square of the track distance. Various empirical expressions take account of this theoretical result. The following formula developed in [36] is widely accepted: 2.5 cp,max = k1 + 0.025 (2.70) (ag + 0.25)2 Introducing the pressure at ag = 3.80 m as a reference, the peak pressure amplitude at any distance becomes 14.1 cp,max (ag ) = ca · cp,max (3.8) = + 0.14 cp,max (3.8) (2.71) (ag + 0.25)2 For ag = 8.3 m, the formula gives a wall distance factor of ca = 0.333. The experimental result is in this case a decrease by a mean factor of 0.3. The formula presented is a conservative estimate. The pressure varies over the wall height. Figure 2.66 is an example of a pressure pattern measured at a wall, 8 m high. The pressure intensity decreases at the upper end. This end effect coincides with a shift of the pulse peaks between wall foot and top, meaning that they do not occur simultaneously at each level. Figure 2.67 shows the time lag between head pulse maximum and minimum as it varies over the height of a 3.5 m wall. The measurements include various train speeds, the time lag has been transformed to V = 300 km/h. The maxima occur simultaneously at each level, whereas the minimum is not simultaneous but lags increasingly at higher levels. This will in general diminish the dynamic load effect. A conservative approximation is to assume identical and simultaneous pulse patterns at each level. Finally, the pressure magnitudes depend on the wall height. The experiments show that the pressures measured at low levels are higher in magnitude at high walls compared to lower walls. The pulse between the walls apparently levels out more rapidly when the walls are low. A convenient wall height factor is:


87

Fig. 2.66. Load pattern over the height of the wall

cWH =

1 − 0.03 HW ref , 1 − 0.03 HW

2 m < HW ≤ 5 m

(2.72)

where HW is the height of the wall above the track level in m is, and HW ref the reference wall height, for which the pressure coefficients have been determined. The results refer here to HW ref = 3.50 m. 2.3.2.3 Load Pattern for Static and Dynamic Design Calculations The following expression summarizes the observed effects and may be applied to static and in particular to dynamic design calculations: q1k (x, z, ag ) = cWH (HW ) ca (ag ) cz (z) cp (x) ρ

V2 2

(2.73)

where: q the pressure at a distance x from the train nose, at a level z above track height; cWH factor accounting for the wall height; cp pattern of the pressure coefficient at low levels acc. to Figure 2.69;

88


Fig. 2.67. Variation of the time lag between maxima and minima of the head pulse over the wall height transformed to V = 300 m/s

Fig. 2.68. Load factor for the load distribution over the height of the wall

cz ca ρ V ag x z

load factor accounting for the pressure variation over the wall height acc. to Figure 2.68; load factor accounting for the wall distance from the track axle; mass density of air; train speed in m/s; track axle distance; distance from zero-crossing of the head pulse; height above rail level.


89

(a)

(b)

Fig. 2.69. Pattern of pressure coefficients cp for the ICE-3 train: (a) pressure difference between track-side and rear-side faces of the wall; (b) pressure at the track-side face

The speed of an adverse wind has to be added to the train speed where required. The load factor cz in fig 2.68 neglects the phase shift occurring towards the top and is valid for any wall height. Figure 2.69 shows the reference load pattern. The stochastic component superimposed on the pressures by the boundary layer turbulence has been smoothed out by averaging. The head pulse at the track-side face (b) is symmetric. Considering the net pressure, the rear-side pressure has to be included. The measurements in ref. [229] include the required data. They show that the pressure maximum on the rear side precedes the track-side maximum. Therefore, regarding the net pressure the pulse maximum increases whereas the minimum decreases. The effect on the remaining load pattern is not noticeable.

90


(a)

(b)

Fig. 2.70. Noise protection wall (a): height 3.50 m above track level; post distance 5.00 m; lightweight panels (b) Mode shape of the 1st mode; natural frequency f1 = 4.67 Hz

The formula includes the wall distance effect on the pressure amplitude as a constant factor. It does not include the increasing distance between pressure maximum and minimum. In general, calculations of the dynamic load effect may be restricted to the head pulse. It governs the dynamic amplification of the response. A simple and sufficient approximation applicable to the symmetric load pattern is 2x |x| cp (x) = cp,max exp 1 − (2.74) Δx Δx The expression includes the effect of the track distance as well with regard to the pressure amplitude as to the distance of positive and negative peaks. 2.3.2.4 Dynamic Response A typical wall structure consists of concrete panels or lightweight metal panels filled with mineral wool. The panels are supported by steel posts at a distance of 2.00 or 5.00 m. Figure 2.70 (a) shows an example. It is rather laborious to model the dynamic behaviour of the structure. The transient response involves large parts of the wall between recesses. The attempt was misleading to identify the dynamic response at a single pole in a 1-D model. Similarly, the natural frequencies and the relevant mode shapes cannot be identified realistically in a simplified model: as an example, the panels have to be included as 2-D plates since their torsional stiffness contributes considerably to the system stiffness. Figure 2.70 (b) shows the 1st mode shape which is excited dominantly by the pulse load.


91

displacement in m

×10−1

time Fig. 2.71. Time history of post top displacement calculated for a post in the middle of the wall; displacement in m, positive direction outward

The natural frequencies are not well separated. For the wall shown above, the first 4 modes range from 4.67 Hz to 4.90 Hz, the 12th mode shape has a natural frequency of 6.04 Hz which is still rather close to the first one. The post top displacement from time history calculations, s. Figure 2.71 indicates that the wall moves outward at the pulse maximum. As it swings back, the negative pulse amplifies the movement: the 1st inward amplitude is ca. twice the 1st outward. This is a consequence of resonance. The effect of natural frequencies on the resonant amplification of the displacement may be studied in a simplified manner using modal decomposition. The response time history is calculated for a static behaviour and for various natural frequencies. A critical damping ratio of D = 0.05 was adopted independent of the natural frequency. The dynamic amplification of the response r is characterized by two resonant amplification factors: max ϕdyn =

max r rstat

min ϕdyn =

min r rstat

(2.75)

The Figures 2.72 and 2.73 show how the resonance factors depend on the natural frequency and the train speed, i.e. the pulse time lag. Both factors display identically that the maximal amplification is independent of the natural frequency with a value of max ϕdyn = 2.0 and min ϕdyn = 2.6. The range of natural frequencies where peak resonance occurs is however not identical in the two cases. At a train speed of 300 km/h, a natural frequency of 3.8 Hz provides the highest amplification of the outward displacement whereas the inward displacement is amplified most strongly at a natural frequency of 4.6 Hz. The wall considered suffers strong resonant vibrations.

92


Fig. 2.72. Resonant amplification of the displacement maximum vs. the natural frequency at train speeds between 200 and 300 km/h

Fig. 2.73. Resonant amplification of the displacement minimum vs. the natural frequency at train speeds between 200 and 300 km/h

2.4 Load-Independent Environmental Impact Authored by Ivanka Bevanda and Max J. Setzer During their serviceable life, concrete structures are exposed to various environmental influences which affect their durability to differing degrees. Ensuring durability is understood to mean that the load-independent influences which occur in the course of its serviceable life do not reduce the useful properties and the load-bearing capacity of the concrete structure. This means that a structure is sufficiently stable to be able to absorb the expected loads

2.4 Load-Independent Environmental Impact

93

(e.g.traffic, wind) on the one hand and at the same time that the load-bearing capacity is not reduced by environmental influences. An overview of the practical observations for the frost attack and a first introduction into external chemical attack are given in the following sections. 2.4.1 Interactions of External Factors Influencing Durability Authored by Ivanka Bevanda and Max J. Setzer The DIN EN 206-1 [1] introduces mechanism-related exposure classes which describe and account for environmental influences which are not directly taken into account as loads for constructional measurement (Figure 2.74). From a technological point of view, durability is determined by minimum concrete composition requirements (water/cement ratio, cement content). The design concept was derived from current knowledge of deterioration mechanisms and correlations between exposure and resistance. This simple approach does, however, have the major disadvantage that the application of new materials and concrete types for which there are as yet no empirical values is limited. Furthermore, it is not possible to evaluate existing structures whose composition is not known. Chronological changes in resistance to a different behavior compared with the original exposure are also not recorded. A durability prognosis of a concrete structure requires that the expected environmental conditions to which the structure will be exposed can be reasonably reliably predicted. The causes and correlations which lead to damage must be clearly recognized and understand. Knowledge of damage mechanisms and the complex interactions of external influences, transport and degradation process is necessary for forecasting durability and serviceable life (Figure 2.74).

Performance Concept

intensity

Environmental Impact (classification of EN 206-1) reinforcement corrosion

concrete corrosion

carbonation

chloride

frost attack with/

chemical

process

penetration

without de-icing agent

attack effect

Incubation Time

temperature and moisture 㩳 transport and/or reaction parameters

Serviceable Time

Climatic Conditions

Degradation Process Damage

damage

limit

criterion

Fig. 2.74. Schematic diagram - Interaction of climate, environmental attack and damage process - basis for the perfomance concept

94


(a)

(b)

(c)

Fig. 2.75. Reinforcement corrosion (above): (a) due to the influence of chloride (b) due to carbonation [400]; Concrete corrosion (below): (c) combined attack - AAR intensified by alternating frost and thawing [529]

The damage process depends on the transport process. The efficiency of the transport mechanisms is in turn dependent on moisture and/or temperature. Moisture is necessary as a transport and reaction medium and the external temperature works as a reaction accelerator. For example, the maximum carbonation speed occurs at humidities between 60% and 80% and the extent of sulfate corrosion rises with sinking temperatures. In case of frost attack the damage mechanisms only become active after the concrete texture is critically saturated through frost suction (transport mechanism). At the same time, the ”real” environmental attack is a complex strain, the sum total of several, sometimes simultaneous partial attacks which mutually influence one another. For example, weathering with deep craters can lead to increased chloride penetration of the concrete by deicing salt or a deeper carbonation of the concrete. This causes faster depassivation of the reinforcement, which causes more rapid corrosion of the outer reinforcement (Figure 2.75 (b)). A further example is the additional strain caused by temperature cycles, especially the alternation of frost and thawing of the alkali-aggregate reaction (AAR). These aid the development of the AAR by either leading to cracks in the concrete so that it can be better penetrated by moisture and an AAR can be initiated, or they lead to the expansion of existing AAR-related cracks (Figure 2.75 (c)).


Concrete Corrosion

95

Physical Action

thermal (e.g. freeze-thaw, freeze-deicing salt)

Chemical Action

dissolution (e.g. leaching, acid) expansion (e.g. sulfates, alkali-aggregate reaction)

Combined Action

e.g. alkali-aggregate reaction + freeze-thaw

Fig. 2.76. Attacks on concrete (in imitation of [872])

Figure 2.76 shows examples of physical and chemical environmental influences which cause concrete corrosion . The frost attack, the calcium leaching, the sulfate attack and the alkali-aggregate reaction were processed as part of SFB 398. It should be noted that in SFB 398 no practical examination of the listed chemical attacks was performed and a summary of the practical examinations in the literature can be found in Chapter 3. The laboratory tests are accordingly also listed in Chapter 3. In addition, more detailed summaries of the relevant aspects of durability in concrete structures can be found in e.g. [770],[702]. 2.4.2 Frost Attack (with and without Deicing Agents) Authored by Ivanka Bevanda and Max J. Setzer Frost and deicing salt attack are under the most detrimental environmental phenomena to be taken into account for durability design of concrete. Frost attack with and without the presence of deicing salt is a dynamic effect that involves both a transport mechanism and a damage mechanism. Setzer coined the term frost suction for the transport mechanism, and explained this phenomenon by surface physics described by the micro-ice-lens model (see Subsection 3.1.2.2.3). During the freeze-thaw cycle, external water is sucked inward by the action of the micro-ice-lens pump; the pore structure becomes saturated. Only once the critical degree of saturation is exceeded does ice expansion cause damage. Since there is not enough space in the concrete microstructure for lateral yield, critical internal stresses build up during the process of ice formation, and then abate again as micro-cracks form. The result of this is internal and/or external damage to the concrete structure. External damage known as scaling (Figure 2.77) can be recognized as (1) sandy decay and (2) local scaling of the hardened cement paste, and in the case of aggregate-related damage as (3) popouts and (4) D-cracking.

96


Fig. 2.77. Surface of frost damaged concrete in situ [20]

External damage is the most frequently observed frost damage. It starts off as an aesthetic fault, but then the surface destruction can lead to limitation and loss of the function of the component, although structural stability is still assured (e.g. in the case of airport taxiways). Internal damage is characterized by microstructural damage arising from microcracks (Figure 2.78), which influence the mechanical and physical properties of the concrete structure, and its structural integrity as a result. While both types of damage go hand-inhand with the critical degree of saturation and ice expansion, they still must be treated as separate phenomena, since they appear not to be strictly related. In the case of external damage, dissolved substances (salts) add their own damaging effect to the equation. This is an especially important factor in the case of deicing salt attack, and is discussed at length in literature. New findings, including those from SFB 398/ Project A11, show that even the influence of commonly ignored salt concentrations increases weathering in what is regarded as ”pure” frost attack. References in literature and our own investigations [21] show how diverse the possible variations of alternating frost and deicing salt stressing of concrete components can be. One actual overview is given in the progress report DAfStb3 [737]. The progression of damage following pure frost attack was also investigated under real climatic conditions (in situ) and under laboratory conditions in SFB 398/ Project A11. The essential results and their significance are summarized below. 2.4.2.1 The ”Frost Environment”: External Factors and Frost Attack Details on the composition and properties of the tested concretes are given in [119],[120]. In order to emulate the conditions as realistically as possible, the field samples were sealed and insulated on the sides, since moisture and 3

German Committee for Reinforced Concrete.


97

0,125 mm

Fig. 2.78. Microcracking of cement paste(left); ESEM image of frost damaged concrete (right) [20]

Fig. 2.79. Field exposure (left); Modified multi-ring electrode (right)

heat transport through components is typically one-dimensional in real applications. A side overlapping edge for catching rainwater was attached onto the test surfaces Figure 2.79. This way, a persistent water layer was simulated. In real situations, this type of frost attack typically occurs on horizontal components directly exposed to weathering, which are classified as exposure class XF3 according to DIN EN 206-1 (frost attack without deicing salt, high water saturation) [1]. Under the climatic conditions, there were alternating periods of wetness and dryness, i.e. periods with dynamic moisture entry and redistribution inside the specimen. Climatically induced humidity and temperature stressing of the component is the most important factor to consider when investigating frost damage. As such, it was decided to obtain information on the changes in moisture content and concrete temperature using a modified multi-ring electrode4 (MRE) Figure 2.79. The modified MRE is a humidity/temperature sensor. Detailed information on its construction and function are given in [660],[762]. 4

Humidity sensor with integrated thermometers, pursuant to the Aachen patent.

98

2 Damage-Oriented Actions and Environmental Impact 9,0 o

T >0 C

8,5 8,0

8,0

moisture pentetration

drying

7,0

7,5

6,5 6,0 5,5

o

T >0 C

8,5

ln (R)

ln (R)

7,5

9,0

o

T 104 ). This effect may be explained (at least partly) by abrasion and fragmentation of particle corners, i.e. by the non-permanency of grains [426]. The linear dependence εacc ∼ Nc is well-known from abrasion experiments on ballast material. Thus, the assumption of a permanent material usually made by theories for monotonic loading (plasticity, hypoplasticity) does not apply to cyclic loading, at least for very high numbers of cycles. However, in recent experiments on well-graded granular material the accumulation has been observed to run significantly faster than according to εacc ∼ ln(N ) already from the beginning of the tests [841]. This cannot be explained only by abrasion. The term ”deterioration” can be applied to stress relaxation observed in saturated soils. Under nearly undrained conditions (e.g. during an earthquake) the strength and the stiffness of the material decrease because the effective stress does. The pore pressure build-up is equivalent to a reduction of the effective stress (Figure 3.40b). In the extreme case σ av = 0 the stiffness and strength may completely vanish. The soil is then said to be ”liquefied” and the soil skeleton is temporarily ”deteriorated” until the pore water is squeezed out and volume changes occur. The contact loss of the grains gives rise to phenomena like cyclic mobility (Section 3.2.2), phase separation between soil layers and spontaneous densification during re-consolidation. Usually a few fast and strong cycles are necessary to reach a liquefaction and a collapse of a foundation. The number of cycles leading to liquefaction (Nc = 10 - 100) is much smaller than the Nc values usually considered in studies on the life time of structures. Therefore, this ”deterioration” phenomenon is irrelevant for lifetime oriented design concepts. The liquefaction phenomenon may also be utilized for passive isolation of structures or for soil improvement techniques (Section 2.5.1). Under drained conditions a ”deterioration” may take place if large amplitudes are applied, i.e. if the cyclic stress path significantly surpasses the

3.2 Experiments

163

critical state lines, or if small cycles are applied with an average stress ratio above the critical one. In both cases dilatancy occurs leading to a reduction of the strength and the stiffness of the soil. However, such stress paths should be avoided by an appropriate design of foundations considering also the cyclic part of the loading. While the extensive experimental study presented in Section 3.2.2 has significantly improved the understanding of the accumulation phenomenon during a drained cyclic loading, the post-cyclic behaviour is not well understood yet. The undrained displacement-controlled cyclic triaxial tests presented in Section 3.2.2 (Figure 3.92) reveal that a latent accumulation in the soil skeleton takes place even if the cycles are applied at σ av = 0 (i.e. after liquefaction). This latent accumulation becomes visible as a permanent deformation during re-consolidation. Similar observations have been reported also e.g. by Shamoto et al. [734] and Sento et al. [721]. Sento et al. have performed strain-controlled cyclic tests and could correlate the volume change during re-consolidation with the length of the shear strain path |γ| ˙ dt. Despite these recent findings, the post-cyclic behaviour of soils needs further experimental studies.

3.2 Experiments Authored by Otto T. Bruhns and G¨ unther Meschke This Section contains results from laboratory investigations of damage evolution in materials and structures performed in the conetxt of the Joint Collaborative Center SFB 398 at Ruhr University Bochum. It comprises non destructive investigations of crack propagation in metallic materials (Subsection 3.2.1.1), laboratory tests of concrete and soil subjected to cyclic loading (Subsections 3.2.1.2 and (Subsections 3.2.2) and structural experiments of steel-concrete composite structures (Subsection 3.2.3). 3.2.1 Laboratory Testing of Structural Materials Authored by Otto T. Bruhns and G¨ unther Meschke 3.2.1.1 Micro-macrocrack Detection in Metals Authored by Henning Sch¨ utte and Otto T. Bruhns 3.2.1.1.1 Electric Resistance Measurements 3.2.1.1.1.1 Introduction Crack growth due to fatigue and stress corrosion is normally a slow degradation process up to a point, beyond which failure may be sudden and catastrophic. For this reason early detection of crack growth during this initial

164

3 Deterioration of Materials and Structures

period is essential for the prevention of failure, especially for metallic structures subjected to cyclic loading. In fracture mechanics, experiments involving stable crack growth and measurement of the instantaneous crack length are essential. That is a reason why many experimental methods have been developed to monitor such crack growth. One of the most powerful nondestructive methods for monitoring crack initiation and measurement of crack growth is the electrical resistance method, in which a constant direct current is passed through the specimen and electrical resistance between two measurement points located on opposite sides of the crack is monitored. As the crack propagates, the uncracked cross-sectional area of the test specimen decreases, and electrical resistance between two points on either side of the crack rises. The advantage of the electrical resistance method lies in the simplicity of its system compared with other non-destructive techniques such as acoustic emission, vortex current, ultrasonic crack detection, etc. Also, the electrical resistance method has many advantages over the optical measurement of the crack propagation. It provides a total measurement, and, because it does not require visual accessibility, tests may be conducted in any sealed environment. The output is continuous which permits automated data collection and processing together with all day usage of testing machine capacity. The technique is capable to detect small increments of crack growth, which cannot be resolved optically. In addition to the majority of fatigue testing methods, which concentrate on intermediate and high stress intensities it is particularly useful in the study of fatigue thresholds and very slow crack propagation rates, which occur by high cycle fatigue tests. For a test specimen under cyclic fatigue loading the electrical resistance increases (Ra ), and comparing it with a reference electrical resistance (R0 ) measured on the test specimen before the beginning of the test, the crack length (a) or the crack length-to-specimen width ratio (a/w) may be determined. This determination is possible by making use of suitable calibration curves. In practice, the accuracy of an electrical resistance measurement of the crack length may be limited by a number of factors: • • • •

The electrical stability and resolution of the electrical resistance measurement system. Electrical contact between crack surfaces where the fracture morphology is rough or where significant crack closures are present. Changes in electrical resistance with plastic deformation. Changes in electrical resistance with temperature change.

This accuracy limitations are present in particular by the determination of calibration curves relating changes in electrical resistance across crack length (Ra ) to the crack length (a). In most cases, experimental calibration curves have been obtained by measuring the electrical resistance across:

3.2 Experiments

• • •

165

Machined slots of increasing length in a single test specimen. A growing fatigue crack, where the length of the crack at each point of measurement is marked on the fracture surface by a single overload cycle or by a change in the mean stress. A growing fatigue crack in thin specimens where the length of the crack is measured by surface observation.

3.2.1.1.1.2 Measurement of the Electrical Resistance The measurement of the electrical resistance was performed using an industry device Buster Resistomat 2304 with a maximal resolution of 10nΩ in the testing range up to 200μΩ and a constant direct current of 10A through the specimen. The specimen form and the distance between two points in which the electrical potential was measured were adjusted to reach this high resolution. The measurement of the electrical resistance was performed on two types of specimen: plain specimen and circular specimen, shown in the Figures 3.48 and 3.49 respectively. The measurement of the electrical resistance Rx in the observed area of the specimen was achieved using a Kelvin-procedure. In this procedure, a regulated constant direct electrical current of 10A is driven through the specimen and an internal reference resistance (Rk ) is measured. A difference in the electrical potential at any two points of the specimen can be measured, and the electrical resistance can be calculated. The potential drop over the testing range of the specimen can be related to the potential drop over the high-precision internal reference resistance and can be calculated using the known value of this reference resistance. The advantage of this procedure is the independence of the measured values of the parasitic resistance in the measuring system, such as contact resistance and transition resistance in the measurement connection. The measured values depend only on the quality of the internal reference resistance. To eliminate possible electrostatic effects or differences of the two amplifiers, the direction of the driven current and the polarity of the potential measurement are chanced within one period of the resistance measurement. For this reason the measuring value is the middle value from 4 different measurements. For all tests, the measuring time for one value was set to 1s. One value of the electrical resistance represents an average value of the electrical resistance over 9 loading cycles for the plain specimen and over 6 cycles for the circular specimen. This way, effects of elastic elongation of the specimen during the measuring time for one value of the electrical resistance are eliminated. The measurement of the difference in electrical potential was obtained using bypassed thermocouples, which are welded onto the specimen. Welding the spherical thermocouples onto the specimen surface also prevents the friction noise in the acoustic emission measurement. The temperature of the specimen is measured using the same bypassed thermocouple, which is welded onto the specimen between two thermocouples

166


for the measuring of the difference in the electrical potential. The measured temperature is used for the elimination of the temperature based effects in the electrical resistance, which are a result of the change of the specific resistance (resistivity) and the cross-section. 3.2.1.1.1.3 Calculation of the Electrical Resistance The measured values of the electrical resistance are resolved in the postprocessing using the influence of the temperature measured at the same time. A variation of the electric resistance can be calculated as a function of the possible temperature change of the specimen, using the conditional temperature change of the specific resistance ρ(Θ) = ρ20 (1 + αp ΔΘ20 ),

(3.2)

the temperature dependence of the length of the specimen l(Θ) = l20 (1 + α ΔΘ20 ),

(3.3)

the temperature dependence of the cross sectional area of the specimen A(Θ) = A20 (1 + α ΔΘ20 )2

(3.4)

and relation R(Θ) = R20

1 + αp ΔΘ20 . 1 + α ΔΘ20

(3.5)

Here, R20 is the electrical resistance at 20◦ C, αp is the temperature coefficient of specific electrical resistance, α is the linear thermal expansions coefficient and ΔΘ20 is the temperature difference between the current temperature and the reference temperature of 20◦ C. 3.2.1.1.1.4 Experiments Experiments were performed by a hydro-dynamic tension-torsion testing system (Schenck/Instron Fast Track 8800). The displacement was measured by a real time analog-built mean of 3 displacement transducers (HBMW5TK), and the load was measured by a 160kN load cell. The fatigue load was defined as cyclic sinusoidal load in tension range. Tests were performed either with constant amplitude for the entire duration of the test, or as a block-test with the amplitude which is constant inside the block and differs between the blocks. Stress-ratio range was R=0.05 and R=0.25. Test frequency was 9Hz for plain specimen and 6Hz for circular specimen. The electrical resistance is measured using an industry device Buster Resistomat 2304 with a constant direct current of 10A through the specimen, maximal resolution of 10nΩ and the testing range until 200μΩ.

3.2 Experiments

1.2

PS16 PS17 PS20 PS22 PS31

1.15 1.1

elect. Resistance R/Ro


1.2

1.05 1 0.95

167

CS02 CS05

1.15 1.1 1.05 1 0.95

0

150000

300000 Cycles

450000

600000

0

50000

100000 150000 Cycles

200000

Fig. 3.41. Evolution of the electrical resistance vs. number of cycles during fatigue - plain and circular specimen

1.2

PS24



1.2 1.15 1.1 1.05 1 0.95

PS29

1.15 1.1 1.05 1 0.95

0

200000

400000 Cycles

600000

800000

0

300000

600000 900000 Cycles

1.2e+06

Fig. 3.42. Evolution of the electrical resistance during fatigue - plain specimen block-test

3.2.1.1.1.5 Experimental Results The electrical resistance is normalised using average electrical resistance for the service life between 30% and 70%. The evolution of the normalised electrical resistance over the service life is shown in Figures 3.41 and 3.42. As can be seen in Figure 3.41, after a stabilisation during the first couple of load cycles, the electrical resistance is constant over a long time of the service life. To the end of the service life, an exponential growth of the electrical resistance is to be observed. Depending on the level of the cyclic load amplitude, an exponential evolution of the electrical resistance happens between 60% 80% of the service life. Figure 3.42, in which results of the block-test are represented shows that the normalised electrical resistance is not constant between the blocks. Although one value of the electrical resistance represents an average value of electrical resistance over 9 respectively 6 loading cycles, the influence of the different load amplitude level between the blocks on the electrical resistance can be seen clearly. The dependency between the evolution of the electric resistance and the crack propagation is calculated using finite element analysis software ANSYS.

168


Fig. 3.43. Electrical potential - plain specimen

Fig. 3.44. Electrical potential - circular specimen

The distribution of the electrical potential across the plain and circular specimen for different crack length is given in Figures 3.43 and 3.44. The results of both the finite element analysis and experiments for both types of specimen are represented in Figure 3.45 as a dimensionless plot of electrical resistance ratio R/R0 against a/w where R0 is the electrical resistance at the initial crack length a0 , R is the electrical resistance at the crack length a and w is half width of the specimen at the crack height. A good agreement between experimental result and the numerical solution of evolution of the electrical resistance with the crack propagation can be observed. The obtained evolution of the electrical resistance shows at the same time a high level of similarity to the measured crack propagation behaviour under cyclic fatigue load. Based on this results, it can be concluded that the method of the resistance measurement detects the appearance of the damage in the early phase, and it confirms the development of a damage evolution on the basis of microscopic crack incubation and initiation.

3.2 Experiments

1.5

ANSYS PS Exper PS



1.6

1.4 1.3 1.2 1.1 1 0.1

0.2

0.3

0.4

0.5

0.6

0.7

Crack length a/w

0.8

0.9

1

169

1.8 ANSYS CS Exper CS 1.6 1.4 1.2 1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Crack length a/w

Fig. 3.45. Evolution of electrical resistance vs. crack length during fatigue - plain and circular specimen

3.2.1.1.2 Acoustic Emission The mechanisms by which metals absorb and release strain energy under stress, the modelling of which is the basis of fracture mechanics analysis, can be different and complicated. Acoustic emission is the elastic energy that is spontaneously released by materials when they undergo deformation. The stress waves which result from this sudden release of elastic energy due to micro-fracture events are of most interest to the structural engineer. These events are typically 10μm to 100μm in linear dimension. Sources of acoustic emission include many different mechanisms of deformation and fracture. Sources that have been identified in metals include crack growth, moving dislocations, slip, twinning, grain boundary sliding and the fracture and decohesion of inclusion. Other mechanisms fall within the definition and are detectable with acoustic emission equipment. These include leaks and cavitation, friction (as in rotating bearings), liquefaction and solidification, solid-solid phase transformation. Sometimes these sources are called secondary sources to distinguish them from the classic acoustic emission due to mechanical deformation of stressed materials. Acoustic emission examination is a nondestructive testing method with demonstrated capabilities for monitoring structural integrity, detecting leaks and incipient failures in mechanical equipment. Acoustic emission differs from most other nondestructive methods in two significant respects: • •

The detected energy is released from within the test object rather than being supplied by the nondestructive method, as in ultrasonics or radiography. The acoustic emission method is capable of detecting the dynamic processes associated with the degradation of structural integrity.

Acoustic emission expected in fatigue studies is primarily of the burst type. Burst type emission signals originate from sources such as intermittent

170


Rise Time Decay Time Volts Amplitude

Energy

Threshold

Threshold Crossing

Time

Counts

Duration

Time

Fig. 3.46. Definition of simple waveform parameters for a burst-signal

dislocation motion and crack growth in metals. A Burst signal, given in the Figure 3.46, has the following characteristic parameters: • • • • • • • • •

•

threshold: A preset voltage level that has to be exceeded before an acoustic emission signal is detected and processed. This threshold is independent for every sensor, and must be chosen depending on the background noise. burst: A signal whose oscillations have a rapid increase in amplitude from an initial reference level, followed by a decrease to a value close to the initial value. hit: Total signal from the first to the last threshold crossing. amplitude: Maximum signal amplitude within duration of the burst. duration: The interval between the first and the last time the threshold was exceeded by the burst. counts: The number of times the signal amplitude exceeds the preset threshold. rise time: The time interval between the first threshold crossing and maximum amplitude of the burst. decay time: The time interval between the maximum amplitude of the burst and the last threshold crossing. event: A microstructural displacement that produces elastic waves in material under load or stress, which are detected by several AE-transducer. Using time analysis the origin of acoustic emission signal can then be detected. event counts: Counts which belong to an event.

3.2 Experiments

171

3.2.1.1.2.1 Location of Acoustic Emission Sources The ability to locate the sources of acoustic emission is one of the most important functions of the multichannel instrumentation system used in field application. One of the methods for detecting the emission source is the measurement of the time differences in reception of the stress waves at a number of sensors in an array. Depending on the sensor location linear (1D), two and three dimensional problems can be defined. 3.2.1.1.2.2 Linear Location of Acoustic Emission Sources Consider the situation where two sensors are mounted on a linear structure. Assume that an acoustic emission event occurs somewhere on the structure, and that the resulting stress waves propagate in both directions at the same velocity. Using the measurement of the time differences between hits it is possible to locate position of acoustic emission source. If the time difference between the hits of both sensors is zero, it would indicate a site precisely midway between the sensors. In general, for the case of constant velocity, the source location is given by: d=

1 (D − V Δt) 2

(3.6)

where D is the distance between sensors, V is the constant wave velocity, Δt is the time deference and d is the distance from the first hit sensor. If the source is outside the sensor array, the time difference measurement corresponds to the time of flight between outer sensor pair and remains constant. 3.2.1.1.2.3 Location of Sources in Two Dimensions The case of location of sources in two dimensions requires a minimum of three sensors. The input data now include a sequence of three hits and two time difference measurements (between the first and second hit sensors and the first and third hit sensors), as can be seen in the Figure (3.47). Then: Δt1 V = r1 − R

Δt2 V = r2 − R

(3.7)

which yields R=

D12 − Δt21 V 2 1 2 2 Δt1 V + D1 cos(Θ − Θ1 )

(3.8)

D22 − Δt22 V 2 1 2 2 Δt2 V + D2 cos(Θ3 − Θ)

(3.9)

and R=

Equations (3.8) and (3.9) can be solved simultaneously to provide the location of a source in two dimensions.

172


Sensor 3 X3 , Y 3

r2 D2 Z2 R

Source XS , Y S Z1

Θ Sensor 1 X1 , Y 1

Sensor 2 X2 , Y 2

r1

Θ1

D1 Θ3

Reference

Fig. 3.47. Location of the source in two dimensions

3.2.1.1.2.4 Kaiser Effect Kaiser effect is the phenomenon that a material under load emits acoustic waves only after a primary load level is exceeded. During reloading these materials behave elastically and little or no acoustic emission will be recorded before the previous maximum stress level is achieved. This is true only for materials in which no change in microstructure, such as dislocation movement or crack initiation, can be observed. The case when acoustic emission is recorded before the previous maximum load is reached is known as felicity effect and describes the breakdown of the Kaiser effect. If we define the ratio between the load level at which the acoustic emission appears and previous maximum load level as felicity ratio, it can be used as the associated quantitative measure of the felicity effect. In the case of the Kaiser effect the value of the felicity ratio is 1. 3.2.1.1.2.5 Experimental Procedures Experiments were performed by a hydro-dynamic tension-torsion testing system (Schenck/Instron Fast Track 8800. The displacement was measured by a real time analog-built mean of 3 displacement transducers (HBM-W5TK), and the load was measured by a 160kN load cell. The fatigue load was defined as cyclic sinusoidal load in tension range. Tests were performed either with constants amplitude for the entire duration of the test, or as a block-test with the amplitude which is constant inside the block and differs between the blocks. Stress-ratio range was R=0.05 and R=0.25. Test frequency was 9Hz for plain specimen and 6Hz for circular specimen. Acoustic emission was detected on two types of specimens. The first type is a plain specimen with thickness of 5mm shown in Figure 3.48. The second type is a circular specimen with inconstant thickness and outer diameter

3.2 Experiments

173

R5

330 280 230 225,5

50

26,2

10

22

R3

85

40 30

R1 0

t=5

Fig. 3.48. Geometry of the plain specimen (dimension in mm)

4,847

4,668

R757,25

0,8

R2 0,9

R40

19 9,701

166 150 146 111,883 108,169

Fig. 3.49. Geometry of the circular specimen (dimension in mm)

166mm shown in Figure 3.49. All specimens were made from heat-treatable steel 42CrMo4 (No. 1.7225). Two types of AE-transducers with appropriate preamplifiers were used for the detection of acoustic emission. The first set was 4 piezoelectric transducers R15 with resonant frequency 150kHz and 4 single In-Line 40dB preamplifiers with a 100-300kHz bandpass filter. The second set was 4 wideband piezoelectric transducers WD with operating frequency range 100-1000kHz and 4 voltage preamplifiers with 20/40/60dB selectable gain and 100-1200kHz bandpass filter. All acoustic emission signals were amplified with 40dB and recorded using Physical Acoustics software on a two Two-Chanel-AE-Boards. The sample rate for all measurements was 10MHz. All AE-transducers were clamped to the specimen with a spring clamp. The coupling between the specimen and the transducers was made with a silicon gel. The acoustic emission threshold was set to 35dB, as a compromise between effectively avoiding background noise and cutting off low level signals during damage evolution. The threshold setting depended on the experimental conditions.

174


117 x

3

4

1

2

60

67

40

Fig. 3.50. The position of AE-transducers on the plain specimen

4

1

2

90

3

98

Fig. 3.51. The position of AE-transducers on the circular specimen

The position of AE-transducers on the plain and circular specimen is given in Figures 3.50 and 3.51 respectively. The position of AE-transducers on the plain specimen was given in such a way, that transducers 1 and 4 are socalled guard transducers with the function to eliminate signals originating outside the specimen test section from the recorded data. Thus, extraneous signals such as those emanating from load-chain noise or from servo-valves and hydraulic pump were avoided without loss of data. The AE-transducer 2 and 3 were used as measuring sensors. In the case of the circular specimen all 4 transducers are measuring sensors. 3.2.1.1.2.6 Experimental Results Acoustic emission recorded during the test is represented using acoustic emission events counts per cycle over the whole stress range and cumulative acoustic emission event counts during fatigue damage. For all experiments, the load was in the range, which leads to high cycle fatigue with brittle damage. This type of load and the brittle damage behaviour lead to the significant

3.2 Experiments

175

1400 PS16 PS17 PS20 PS22 PS24 PS29 PS31

1200

Event Counts

1000

800

600

400

200

0 0

200000

400000

600000 800000 Cycles

1e+06

1.2e+06

Fig. 3.52. Acoustic emission event count rate during fatigue - plain specimen

8000 CS02 CS05 7000

Event Counts

6000 5000 4000 3000 2000 1000 0 0

50000

100000 Cycles

150000

200000

Fig. 3.53. Acoustic emission event count rate during fatigue - circular specimen

increase in the acoustic emission output, as the crack advances towards final failure. The rate of acoustic emission in the form of acoustic emission events counts per cycle over the whole stress range is given in the Figures 3.52 and 3.53. When the load was in the elastic range, the low acoustic emission output was evident during initial cycles, due mostly to microscopic dislocation dynamics. This stage was followed by a dead period with almost no acoustic emission. In this stage of fatigue damage accumulation results in the long crack-initiation lifetime. Low energy dislocation motion, which generates acoustic emission

176


PS16 PS17 PS20 PS22 PS24 PS29 PS31

50000

Total Event Counts

40000

30000

20000

10000

0 0

200000

400000

600000 800000 Cycles

1e+06

1.2e+06

Fig. 3.54. Acoustic emission total event counts during fatigue - plain specimen

1.1e+07 CS02 PS05

1e+07 9e+06

Total Event Counts

8e+06 7e+06 6e+06 5e+06 4e+06 3e+06 2e+06 1e+06 0 0

50000

100000 Cycles

150000

200000

Fig. 3.55. Acoustic emission total event counts during fatigue - circular specimen

waves, is frequently under background noise and is relatively hard to detect. Only discrete acoustic emission events counts represent the existence of acoustic emission and consequential evolution of fatigue damage. The crack propagation, which is connected with the high rate of the acoustic emission, occurs in the third stage. The release of the elastic energy due to the crack propagation has a significant level and the detection of the acoustic emission is not influenced by the background noise as in the second stage of the fatigue. Evolution of cumulative acoustic emission event counts during fatigue damage is given in the Figures 3.54 and 3.55 and represent a cumulative fatigue

3.2 Experiments

Total Event Counts

Total Event Counts

2500

PS16

2500 2000 1500 1000 500

PS20

2000 1500 1000 500

0

0 0

60 Location [mm]

2500

128

168

0

60 Location [mm]

400

PS29 Total Event Counts

Total Event Counts

177

2000 1500 1000 500 0

128

168

PS31

300 200 100 0

0

60 Location [mm]

128

168

0

60 Location [mm]

128

168

Fig. 3.56. The location of the origin of acoustic emission for the plain specimen

Fig. 3.57. The location of the origin of acoustic emission for the circular specimen

damage process. This process can be divided into the same three stages as indicated for the rate of acoustic emission: (i) microscopic dislocation dynamics; (ii) microscopic crack incubation and initiation; (iii) macroscopic crack propagation. The location of the origin of acoustic emission was computed using time difference measurement methods described earlier. The obtained distance represents the distance between origin of acoustic emission and AE-transducers. The location of the origin of acoustic emission for the plain specimen is given

178



Total Event Counts

200 150 100 50

6000 4000 2000

0

0 0

30

60 Amplitude [dB]

5000

90

120

0

30

60 Amplitude [dB]

90

250


Total Event Counts

PS22

8000

3750 2500 1250 0

120

PS31

200 150 100 50 0

0

30

60 Amplitude [dB]

90

120

0

30

60 Amplitude [dB]

90

120

Fig. 3.58. Acoustic emission event counts against amplitude for the plain specimen

500000

CS02 Total Event Counts

Total Event Counts

1.2e+06 900000 600000 300000 0

CS05

375000 250000 125000 0

0

30

60 Amplitude [dB]

90

120

0

30

60 Amplitude [dB]

90

120

Fig. 3.59. Acoustic emission event counts against amplitude for the circular specimen

in the Figure 3.56 and for the circular specimen in Figure 3.57. As can be seen in the Figures, the computed location of the origin is in good agreement with the real position of the fatigue damage. Most events are located at the position where the crack is present, confirming that the acoustic emission is from the crack. As a consequence of the specimen size, there are inherent problems with the location due to the size of the AE-transducers compared to their spacing. Namely, spacing between AE-transducers was 68mm for the plain specimen and between 90-98mm for the circular specimen and the diameter of transducers is 18mm, which excludes a point representation of AE-transducers. For this reason, signals which are located within 10mm of the real crack position are regarded as well located. Acoustic emission event counts with respect to the amplitude are given in the Figures 3.58 and 3.59. As can be seen most of acoustic emission

3.2 Experiments


Total Event Counts

300 225 150 75

PS22

10000 7500 5000 2500

0

0 0

200

400 600 Frequency [kHz]

5000

800

1000

0

200


PS24

800

1000


Total Event Counts

179

3750 2500 1250 0

300 200 100 0

0

200


800

1000

0

200


800

1000

Fig. 3.60. Acoustic emission event counts against frequency for the plain specimen

event counts occur with the amplitude between 40 and 70dB. This behaviour could be used as an additional condition in the elimination of the extraneous noise. Using wideband AE-transducers gives a possibility to investigate the frequency at which the acoustic emission occurs. In the Figures 3.60 and 3.61 acoustic emission event counts versus the frequency are given. Since almost all acoustic emission counts have a frequency between 50-300kHz, resonant piezoelectric transducers R15 with resonant frequency 150kHz can be used.

Total Event Counts

Total Event Counts

750000

CS02

1e+06 750000 500000 250000 0

CS05

600000 450000 300000 150000 0

0

200


800

1000

0

200


800

1000

Fig. 3.61. Acoustic emission event counts against frequency for the circular specimen

180


3.2.1.2 Degradation of Concrete Subjected to Cyclic Compressive Loading Authored by Rolf Breitenb¨ ucher and Hursit Ibuk A great number of concrete structures is exposed to cyclic mechanical loading scenarios. Therefore, the reliability of such structures depends among other influences also on the degree of structural degradation due to fatigue loading. In order to estimate the state of a structure it is necessary to know the development of the degradation of the material properties during its lifetime. However, up to now a statistically based description of the degradation processes and their effects on the compressive strength, stiffness and fracture energy referring to pure cyclic compression loads of plain concrete are still missing. For this purpose, within the large joint research project (Collaborative Research Center 398) extensive experimental investigations were carried out at the Ruhr-University in Bochum. In order to get information about the degradation processes with sufficient reliability a large number of specimens were tested by measuring ultrasonic transmission time, longitudinal strains and stress-strain curves during cycling loadings [148, 149, 402]. 3.2.1.2.1 Test Series and Experimental Strategy Most of the extensive cyclic tests were performed on normal-weight concrete of grade C 30/37. Furthermore also high strength concrete of grade C 70/85 as well as air-entrained concrete (grade C 30/37) were investigated. Within the C 30/37 series the types of the coarse aggregates (quartzite, basalt and sandstone) were varied. Additionally concretes with different coarse grading curves were tested, whereby in these cases the matrix of mortar was kept constant. For all the tests cylindrical specimens with a diameter d of 100 mm and a height h of 350 mm were used. These specimens were taken as cores, drilled from concrete blocks at an age of about 25 days. In comparison to specimens made in separate formworks, the drilled specimens have no accumulations of fine-grained mortar along the surface and represent a part of a real concrete structure in a better way. Thereby additional impairment of the peripheral zone can be prevented. The cyclic tests normally started at a concrete age of about 40 days. Previous the specimens remained on air; within these about two weeks between core-drilling and test-start nearly constant hygral conditions in the concrete specimens (equilibrium moisture content) could be reached. Most of the cyclic tests were performed as single-stage tests at constant stress levels. In some few series also two-stage tests with an alternating upper stress level were carried out. For all tests a hydraulic cylinder pulsators were used; the frequency of the cyclic loading was constantly f = 7 Hz.

3.2 Experiments Single-stage tests

Two-stage tests Variation of load sequence

S Smax= const.

N1 N2 N3 N... emax at Smax

fc, Estat., Edyn., etc.

181

S S1,max

Smin= const. N

S2,max Smin Ntotal N

S N

S2,max

S1,max

Sequence 2 Smin Ntotal N

N Aim: Revealing the degradation process (by US-measurement + s-e-relation)

Sequence 1

Aim: Revealing the influence of different sequences at Ntotal on the changes in fc, Estat., Edyn (by US-measurement + s-e-relation)

Fig. 3.62. Execution of the single-stage and two-stage test

In the single-stage test series the specimens were loaded within a defined stress-range Smax /Smin (Figure 3.62, left). In all of these tests the lower stress level Smin was adjusted constantly at 0.10 (i.e 10 per cent of fc ), while the upper stress level Smax was varied from series to series between 0.75, 0.675 and 0.60, however, within one series the Smax was kept also constant. Furthermore interruptions within these single-stage tests were investigated, mainly with the aim to check, to which extent degradations can recover in such rest periods between various load scenarios. Concerning the two-stage tests the lower stress level was also kept constant at Smin = 0.10, whereas the upper stress levels were varied within one series after a defined number of load cycles (Figure 3.62, right). In these series various sequences for the upper stress level were considered (Smax = 0.60 → Smax = 0.675 and v.v.; Smax = 0.75 → Smax = 0.675 and v.v.). So in both cases the tests were started with a higher Smax and then reduced to lower one as well as they started at a lower Smax and afterwards the upper stress level was increased. During these cyclic tests the longitudinal strains were measured continuously by two strain gauges (50 mm in length) which were applied in axial direction on opposite sides of each specimen. Additionally microdefects and their development were investigated by nondestructive ultrasonic (US) measurements perpendicular to the main direction of stress after applying a defined number of load cycles. Furthermore at the same stages also the static Young’s modulus was determined, however, in this

182


case in the main load-direction, by carrying out an additional load step at a Smax /Smin -ratio of 0.30 / 0.10 during a short interruption of the cyclic test. In addition to these non-destructive tests, the development of the mechanical properties - especially the changes in the stress-strain curve and strength - were investigated by destructive tests on specimens taken out of the setup after certain defined numbers of cycles. By comparison of the so obtained various stress-strain curves the development of the strength, stiffness and fracture energy of the respective concrete could be described. 3.2.1.2.2 Degradation Determined by Decrease of Stiffness The proceeding in the degradation process is mainly described by the changes in the stiffness. This can be demonstrated also within the performed testseries by means of the Young’s modulus Estat (Figure 3.63) as well as the dynamic E-modulus (3.64). For all considered stress levels a typical sharp decrease could be observed after only a few number of cycles. Exemplarily at the stress regime Smax /Smin = 0.675/0.10 the Young’s modulus Estat as well as the dynamic elastic modulus Edyn decreased within the first 180,000 load cycles by about 12.5 per cent (averaged value). A following steady decline with a significant lower slope was – also typically – observed between 180,000 and 400,000 cycles. After applying about 400,000 cycles some specimens showed an accelerated decrease in these characteristic values. Additionally an increasing scatter of the measured Estat and Edyn by increasing number of cycles was determined. So, e.g., the coefficient of variation in the Edyn grows up from 40.1 per cent after 10,000 cycles to 79.8 per cent after 600,000 cycles. Especially after about 400,000 cycles a significant increment in the standard deviation was observed.

Fig. 3.63. Decrease and scatter of Estat at Smax /Smin = 0.675/0.10 (single-statetests)

3.2 Experiments

183

Fig. 3.64. Decrease and scatter of Edyn at Smax /Smin = 0.675/0.10 (single-statetests)

Stress level

1.0 -smax/fc 0.9 0.8

Assimacopoulos 1959, Antrim 1959, Bennet 1967, Do 1991, Gaede 1962, Galloway 1979, Graf 1936, Gray 1961, Holmen 1979, Kessler 1958, Kim 1996, Oh 1991, Ople 1966, Weigler 1981, Williams 1943 (783 tests)

0.7 0.6 2 4 6 Number of cycles to failure

8 log Nf

Fig. 3.65. Variation of maximal bearable number of load cycles to failure Nf [627]

As already mentioned the specimens were not loaded until failure. Therefore the results in Figure 3.63 and 3.64 could not be referred to their specific failure cycles (Nf ). Moreover it is shown by many of W¨ ohler tests that the maximal bearable number of load cycles up to failure (Nf ) vary widely ([627]). Therefore, it is not suitable to predict the specific state of failure (Nf ) of the obtained specimens with an adequate accuracy (Figure 3.70). 3.2.1.2.3 Degradation Determined by Changes in Stress-Strain Relation The evolution of deformations due to the cyclic loadings was observed simultaneously by continuous measuring the longitudinal strain. The characteristic development of longitudinal strain at the upper stress level Smax is representatively illustrated for the stress regime Smax /Smin = 0.60/0.10 in

184


Fig. 3.66. Measured longitudinal strain at Smax (Smax /Smin = 0.60/0.10)

Figure 3.66. Under these conditions until the first 2.0 million cycles the longitudinal strain in general increased faster than in the phase after 2.0 million cycles. Here also up to 25.55 million cycles no failure due to cyclic loading could be observed. For the same test series (Smax /Smin = 0.60/0.10) the characteristic averaged results of the stress-strain relations after defined load cycles, determined in the destructive tests, are illustrated in Figure 3.67. Herein also the residual irreversible strains - caused by the cyclic loading itself, determined after unloading - are also considered at the beginning of the stress-strain curves. These residual deformations increased significantly with increasing numbers of load cycles. The ascending branches of the stress-strain curves themselves also changed with increasing cycle numbers from concave shape (towards the strain axis)

Fig. 3.67. Stress-strain curves at different number of cycles (Smax /Smin = 0.60/0.10)

3.2 Experiments

185

Table 3.3. Changes of concrete properties due to cyclic loading determined by changes in stress-strain relation at Smax /Smin = 0.60/0.10 Number of cycles N [Mio.] 0 1.8 4.15 25.5

Young’s modulus Estat kN/mm2 28.4 26.8 25.6 24.4

Compressive strength fc N/mm2 40.2 41.6 42.0 39.6

Compressive strain

u 0/00 2.3 2.0 1.9 1.8

Fracture energy gc kJ/m3 63.8 47.5 43.8 36.8

to a straight line and further to a convex shape in Figure 3.67. A similar development also was already observed in orientating investigations by e.g. Holmen [383]. For such a development mainly microcracking is responsible. This obviously correlates with the decreasing Young’s modulus (Estat ) – at least at lower stress-levels – with increasing number of cycles and parallel a reduction in the fracture energy (gc ) and the ultimate compressive strain (u ) (Table 3.3). According to the increase in the longitudinal strain the concrete properties Estat , gc and u decreased also faster up to the first 2.0 million cycles than afterwards, whereas the compressive strength (fc ) almost remained constant. 3.2.1.2.4 Adequate Description of Degradation by Fatigue Strain Investigations by Holmen [383] demonstrated the difficulties to formulate the state of concrete degradation and damage resp. depending on the ratio of already applied number of cycles N related to the maximal bearable number of load cycles Nf . So for example, at a defined ratio of Smax /Smin(0.675/0.050) some specimens failed already after only 105 cycles, whereas other ones under the same conditions exceeded 3 x 106 cycles [383]. This quite different behaviour indicates that each examined specimen has its own specific Nf -value. Hence an alternative approach was used to characterise the state of damage more precisely by a differentiation of the measured total longitudinal strain max into one part caused by the pure static loading 0 and in a second part derived by the cyclic loadings (Figure 3.68). In the following the second part is defined as fatigue strain f at,max (Figure 3.69). In Figure 3.71 these fatigue strains are illustrated for the same tests as in Figure 3.68. The increase in strain due to the cyclic loading, which takes place only within the fatigue strains, can also be explained in a stress-strain diagram of the complete cyclic test (Figure 3.70). (Within a simplification the stressstrain relation are linearised). Already the first single load (N1 ) leads directly to the initial strain 0,max at σmax . The values of the total strain max at σmax increase by increasing number of cycles (Ni ), i.e. that the cyclic loading cause further strains – namely the fatigue strain f at,max – exceeding the initial strain significantly. During the unloading at the end of the cyclic test (Ni )

186


Fig. 3.68. Total strain at Smax /Smin = 0.675/0.10

Fig. 3.69. Calculation of fatigue strain at Smax

from σmax to σmin and further to the unstressed state, the strain decreases on a significant lower rate as expected according to the deformation at the first loading 0,max . In Figure 3.72 the averaged values of the residual Young’s moduli are plotted against the corresponding fatigue strains f at,max at Smax for all 3 investigated Smax -levels. In general for all series a linear relationship between these two parameters could be proved. It became obvious that the approximation lines shift together. The trendlines for upper stress levels Smax = 0.60 and Smax = 0.675 almost were identical, while at Smax = 0.75 the residual Young’s modulus decreases somewhat more than expected ahead the achieved fatigue strain. So the process of fatigue degradation seems to be mainly coupled with the evolution of fatigue strain. Thus the parameters stress level and number of cycles seem to be negligible to describe the degradation process, especially at lower stress levels. In a global evaluation the residual Young’s

3.2 Experiments

emax efat,max cyclic loading

e0,max smax

s

N1

187

Ni

smin e0,max

eplastic

e

efat,max Fig. 3.70. Formation of fatigue strain (schematically)

Fig. 3.71. Fatigue strain at Smax /Smin = 0.675/0.10

modulus of all investigated stress levels of normal strength concrete can be resumed to just one trendline (Figure 3.72). 3.2.1.2.5 Behaviour of High Strength Concrete and Air-Entrained Concrete In comparison to the investigated normal strength concrete, the results of the high strength concrete followed in general almost the same trendline (Figure 3.73). In opposite to this air-entrained concrete showed a quite different relation between fatigue strain f at,max and the residual Young’s modulus in comparison with (non-air-entrained) normal strength concrete (Figure 3.74). After an accelerated drop at the beginning up to about f at,max = −0.20/00 ,

188


Fig. 3.72. Correlation between the fatigue strain and the residual stiffness for different load levels

the air-voids reduce the crack propagation so that the decline of the residual Young’s modulus continued only with a significant lower slope. This specific behaviour can be explained by the entrained micro air-voids into the concrete microstructure. Due to the so-called ”button-hole-effect”, raised by the micro-voids, micro-cracks, which penetrate up to such voids, require a much

Fig. 3.73. Correlation between the fatigue strain and the residual stiffness of normal and high strength concrete

3.2 Experiments

189

Fig. 3.74. Correlation between the fatigue strain and the residual stiffness of normal and air-entrained concrete

higher fracture energy for their further propagation than in a concrete matrix without micro-voids. 3.2.1.2.6 Influence of Various Coarse Aggregates and Different Grading Curves The influence of different aggregates and different grading curves on the degradation process and the residual Young’s modulus with increasing fatigue strain is illustrated in Figure 3.75 and 3.76. For the specimens containing sandstone Fatigue strain ?fat,max [‰] -1.5

-1.0

-0.5

0

Residual Young`s modulus [%]

100 90 80 70 60 50 basalt

quartzite

sandstone

Fig. 3.75. Correlation between the fatigue strain and the residual stiffness subjected to different aggregates in concrete (C 30/37)

190


-1.0

Fatigue strain ?fat,max [‰] -0.8 -0.6 -0.4 -0.2

0

Residual Young`s modulus [%]

100 90 80 70 60 50

high

middle

low

Fig. 3.76. Correlation between the fatigue strain and the residual stiffness subjected to different grading curves in concrete (C 30/37)

as coarse aggregate the changes in Young’s modulus (E˙ = dE/df at,max ) is about three times lower than that of specimens containing basalt or quartzite (Figure 3.75). This mainly can be explained by the ratio of the Young’s moduli of the aggregates and the hardened cement paste. The Young’s modulus of sandstone can be assumed between 21,000 N/mm2 until 58,000 N/mm2 and is therefore closer to the Young’s modulus of the hardened cement paste (about 20,000 N/mm2 ), whereas the corresponding values of basalt or quartzite exceed 60,000 N/mm2 significantly. Thus, in case of sandstone the concrete’s stiffness is a much more uniform within the cross-section. Therefore lower and less stress peaks within the cross-section, especially in the contact-zone coarse aggregate – cement-matrix, lead to less degradation in the concrete microstructure compared to the concrete with basalt or quartzite. The influence of different grading curves on the development of the Young’s modulus with increasing fatigue strain is shown in Figure 3.76. A high fraction of coarse grains (high) leads almost to a decisive higher velocity in the degradation process compared to the others with lower fraction of coarse grain (middle, low). Analogue to the influence of the type of aggregate, the stresses are spread more uniformly in the cross-section of concrete with middle or low fraction of coarse grains than in the specimens with high ones. 3.2.1.2.7 Cracking in the Microstructure Due to Cyclic Loading With the intention to determine the crack formation due to cyclic loading from some specimens, cyclic tested at the stress regime of Smax /Smin = 0.675/0.10, after certain cycles, sub-specimens were prepared for LMMicroscopy (light microscopy) analyses. In order to get representative results the microcrack characteristics were determined within a test area of 48 by 48 mm2 (polished surface) in the cross-section of the sub-specimens.

3.2 Experiments

191

Fig. 3.77. Light microscopy micrographs

Table 3.4. Crack characteristics at certain number of cycles Smax /Smin = 0.675/0.10 Number of cycles Crack width ∗ Crack amount Crack area ∗ averaged values

0 [μm] 4.0 [–] 2 [μm2 ] 3,400

1 8.0 1 2,900

180,000 7.0 5 28,400

600,000 11.0 14 179,200

The left photo in Figure 3.77 illustrates exemplarily the situation at an unloaded specimen (N = 0) without any cracks, while in the right photo after 600,000 cycles already a characteristic formation of microcracks with width of only a few μm could be proved. In order to quantify the crack formation and to enable a comparison between various concretes or load scenarios, the widths, lengths and number of cracks within the test area were determined as crack characteristics. From these values the crack area, i.e. the integrative product of crack width and length, was calculated (Table 3.4). It became obvious that the observed increase of the crack area is mainly governed by the increase of the crack amount. Further, it could be proved that the first load cycle (N = 1) - which corresponds to a static load - normally leads only to marginal changes in the microstructure in comparison with the further cyclic loads at the same stress level. Furthermore, this shows that the microcracking and therefore the degradation is mainly caused by the cyclic loading. 3.2.1.2.8 Influence of Single Rest Periods As already mentioned before, cyclic loading leads to some degradations and in consequence to an increase in strain and to a reduction in stiffness. The influence of single rest periods within periods of cyclic loading on the

192


Stress level Smax

N1

Rest period duration tp

N2

Smin

Investigated durations of the rest periods:

t respectively N 1. tp = 20 minutes 2. tp = 2 hours 3. tp = 72 hours

Fig. 3.78. Load history with various rest periods [150]

development of strains and stiffness has not been clarified up to now. Thus, the influence of single rest periods (RP) within the cyclic compression loadings at the stress regime of Smax /Smin = 0.675/0.10 on strain- and stiffnessbehaviour has been investigated also within this research project [150] (Figure 3.78). In these investigations also the durations of rest periods were modified. During the whole test periode the total longitudinal strains as well as the changes in stiffness by ultrasonic methods were determined in the same manner as described before. Initially, an increase in the total longitudinal strains up to 600,000 cycles at Smax = 0.675 (Figure 3.79) was observed (averaged: about +40.5 per cent). Because of the large scattering in the strain values already in the forefront of the rest periods, the development of the longitudinal strains up to N1 =

Fig. 3.79. Behaviour of the longitudinal strain at Smax /Smin = 0.675/0.10

3.2 Experiments

193

Fig. 3.80. Related longitudinal strain at Smax /Smin = 0.675/0.10

600,000 are related on the adequate value of the longitudinal strain at N1 to obtain a comparable basis (Figure 3.80). Immediately after unloading at the end of the first cyclic loading N1 resiliences were observed which correspond more or less to the elastic part of the strains at the beginning of the tests (by 50 to 60 per cent). These instantaneous deformations are followed by a gradual decrease in strain within the rest period. The longer the rest period, the higher the increase in strain. A similar behaviour has been observed also already by other scientists [693]. After starting the second cyclic loading N2 the strains increase more or less immediately up to the state at the end of the first loading period N1 . The further development obviously is not influenced by the rest period. Thus from these results it can be concluded, that under such conditions as investigated here the load histories and rest periods do not influence the strain behaviour of concrete under cyclic load significantly. Minor recoveries due to the gradual decrease in the strain-behaviour of the concrete within the rest periods were quickly used up after restarting the load regime. A similar behaviour was observed for the stiffness of the concrete [150]. 3.2.1.2.9 Sequence Effect Determined by Two-Stage Tests In order to simulate more practical load scenarios two-stage tests additionally were carried out. Within these two-stage tests the lower stress level was also kept constant on 0.10. The upper stress level was varied once after several numbers of cycles (Figure 3.81). Sequence 1 means that the cyclic loading started with the higher upper stress level, in sequence 2 the tests were started with the lower one. For both sequences a linear relationship of the fatigue strains at Ntotal against the residual Young’s modulus in case of the load parameters S1,max /S2,max = 0.75/0.675 could be observed (Figure 3.81). Additionally these trendlines are nearly identical. Thus it can be assumed in this case

194


Fig. 3.81. Correlation between the fatigue strain and the residual stiffness subjected to different sequences of cyclic loading for S1,max /S2,max = 0.75/0.675

that the load sequence did not affect the behaviour of changes in the residual Young’s modulus against the fatigue strain. 3.2.1.3 Degradation of Concrete Subjected to Freeze Thaw Authored by Max J. Setzer and Ivanka Bevanda Damage caused to constructions due to frost and deicing salt attack led to intensive research on development of reproducible practical test procedures. The durability of concrete against a frost and deicing salt attack is linked to both damage effects the surface scaling and the internal damage inside the material. An overview of the existing test procedures for determining frost and deicing salt resistance of concrete can be found in Auberg [69]. RILEM TC 117 FDC recommends the CDF test1 [732] for determining deicing salt resistance and RILEM TC 119 recommends the CIF test2 [729] for frost resistance. The CDF test is also one procedure in prEN 12390-9 [17]. The CIF test is subject to the standard CEN committee TC 51 and the BAW3 specification [24]. Both test methods are based on the micro-ice-lens model (refer to Subsection 3.1.2.2.3). The test procedure is described in detail in [732],[729]. With the CDF/CIF test, it is possible to simultaneously measure external and internal damage as well as moisture uptake (see Figure 3.82). The following data have been investigated using the CIF test in accordance with the RILEM Recommendations. In [69] was shown that the internal damage of concrete and moisture uptake correlate directly with each other. In his experiments with normal concrete, 1 2 3

C apillary suction of Deicing solution and F reeze-thaw test. C apillary suction, I nternal damage and F reeze-thaw test. German Federal Waterways Engineering and Research Institute.

3.2 Experiments

Isothermal Isothermal Isothermes suction +20 °C suction +20 °C

195

Freeze-thaw cycles Freeze-thaw cycles

-

Tau Zyklen

CIF/CDF test chest CDF/CIF test chest

0

Time Time[h] [h] 4 7 12

Temperature

Fros t

Saugen 20°C

20 °C 0 °C -20 °C

Tempering (+20°C°C...-20°C) ...-20°C) Tempering bath bath (+20

Ultrasonic transit time Ultrasonic timedynamic modulustransit of elasticity

Ultrasonic cleaning Ultrasonic cleaning bath bath

Mass Mass gain gain

Surfacescaling scaling Surface

dynamic modulus of elasticity

Mass Mass gain gain -

Internal damage Internal damage

Fig. 3.82. Steps of exposure and measuring during CDF/CIF test [731]

Auberg determined the dependence between frost suction and internal damage as well as length changes. Studies performed by Heine [363] and Palecki [610] show a direct material-specific connection between pore structure, moisture uptake and internal damage. Palecki discovered that there is a correlation between the rate of saturation, pore radius ratio and frost resistance, and divided concrete into damage types depending on moisture uptake behaviour. For example, the Figure 3.83 demonstrates the correlation between internal damage (calculated from ultrasonic pulse transit time) and moisture uptake by high performance concrete (hpc). The moisture uptake (frost Phases I

II

1

2

2,5%

III critical saturation in a depth of 35 mm - start of damage -

100%

2,0%

rapid decrease of RDM

1,5%

80%

0,30+7%SF+20%FA SERVICE LIFE

60%

1,0%

exponential moisture uptake

Incubation time

40%

0,5% damage has reached a depth of 35 mm

20%

0,0%

Rel.moisture uptake in M.-%

Rel. dyn. Modulus of elasticity in %

120%

first micro-cracking at surface layer - damage induction -

freezing at the surface layer

0%

-0,5% -7

0

7

14

21

28

35

42

Test duration in days [1d=2ftc]

Fig. 3.83. Example relationship between RDM and relative moisture uptake - concrete type 2 [610]

196


suction) rises continuously with an increasing number of freeze-thaw cycles (ftc). The non-saturated pores which cannot be sucked in capillary will be filled up to approx. 95 % of the original value of the relative dynamic modulus of elasticity (RDM=100%); no irreversible microcracks are observed, despite a very much increased moisture uptake [69]. Only after a critical saturation of these pores does the damage of the concrete structure start as a result of ice expansion. The critical saturation degree marks a nick point in the moisture curve i.e. in the RDM curve. The ”incubation period” up until the concrete does not show any damage is very striking. It effectively prolongs the service life. The last phase is distinguished by a rapid drop in the RDM and increased moisture uptake within a few ftc. Not only are the pores filled with moisture, but also the cracks will be filled due to frost attack. For the service life of a concrete structure under frost attack, this means that the saturation speed is the decisive parameter and not the amount of moisture [69],[610]. The aim of the tests in SFB 398/Project A11 was the non-destructive measurement of moisture and damage progress, working from the frost-exposed concrete surface. By applying multi-spherical electrodes (MSE), the moisture distribution was determined at several depths of specimen during the ftc; and the moisture volume calculated from the amount of moisture uptake (Figure 3.84, Table 3.5). The multi-spherical electrode is a moisture/ temperature sensor according to [660] improved by Xu [861] and Bevanda (see Subsection 2.4.2) during the SFB 398 project. Internal damage progress was detected by measuring the change of RDM (calculated from ultrasonic pulse transit time) also at several depths of the specimen. Figure 3.84 shows the internal damage as a function of moisture at corresponding ftc. Comparing the damage profile to the moisture profile, it follows that at an MSE signal of 80 %, the RDM is 95 % at all measured depths. If one considers the volume of moisture uptake at that time (Table 3.5), the volume does not exceed the concrete capillary porosity (rh >30 nm) of 3.5 % as measured by mercury intrusion porosimetry. Comparing the results found at a later time (Figure 3.84, 90 % of MSE signal), the volume of moisture uptake exceeds the capillary porosity. At the same time, the RDM decreases to e 3

e3

0.62

0.2 0 0 10

101

102

103

104

0.61

105

2 104

0

150

2 q

20 40

150

20

150

20 40

150

q 20

80

40 60

t

5 q60 150

20 40

t

6 q 80 60

80

150 t

105

t

4

80 60

8 104

2.0

t

3 q

6 104

all tests: pav = 200 kPa, ηav = 0.75, ID0 = 0.58 - 0.63, f = 0.25 Hz

2.4

60 40 80

εacc [%]

qampl = 60 80

1 q

4 104

Number of cycles Nc [-]


e)

0.6

Average stress ratio Y

c)

0.6

0.4

av

Average mean pressure pav [kPa]

0.8

all tests: pav = 200 kPa, ζ = 0.3, ID0 = 0.57 - 0.67, f = 1 Hz

1.6 1.2 qampl [kPa] = 80 60 40 20

0.8 0.4

40 20 t

0

0

25,000

50,000

75,000

100,000


Fig. 3.89. Results of drained cyclic triaxial tests: a) Dependence of Dacc on average mean pressure pav , b) on average stress ratio Y¯ av ≈ η av /Mc (ϕc ), c) on the number of cycles Nc and d) on cyclic preloading. e) Tests with packages of cycles with different amplitudes

with the liquefaction resistance, however, could be formulated [844] but its practical application has still to be proven. A correlation of cyclic preloading with acoustic emissions seems to be rather insufficient [581]. As an alternative, the cyclic preloading could be determined by cyclic test loadings in situ (some ideas are explained in [835]). In many practical problems the amplitude of the cycles is not constant but varies with the cycles. Such random cyclic loadings could be replaced by

Gravel Sand fine med. coarse fine

80

6

60

1.2

40

1.0

20 0 0.06

2

0.2

2 3 4 5

1

0.2 0.6 2 Grain diameter [mm]

6

3 5

0.1

0.8

205

Gravel Sand fine med. coarse fine 7 8 3

100 80 60 40 20

8

0 0.06

0.2 0.6 2 Grain diameter [mm]

6

0.6 7

0.4

4

after Nc = 105 0

Finer by weight [%]

1

100

εacc / fampl [%]

εacc / fampl [%]

0.3

Finer by weight [%]

3.2 Experiments

0.1

0.2

0.5

1

2

5

Mean grain diameter d50 [mm]

3

0.2

6

10

0

1

after Nc = 105 2

3

4

5

Uniformity coefficient Cu = d60/d10 [-]

Fig. 3.90. Influence of the grain size distribution curve on Dacc

packages of cycles each with a constant amplitude if the sequence of application would not affect the residual strain that means if Miner’s rule [543] were applicable to soil. In order to examine the influence of the order of packages cyclic triaxial tests were performed [839, 835]. In each test four packages each with 25,000 cycles were applied. The amplitudes q ampl = 20, 40, 60 and 80 kPa were applied in different sequences. Figure 3.89e presents the accumulation curves. Irrespectively of the sequence the residual strains at the end of the tests are quite similar. Thus, for a constant polarization of the cycles and as a first approximation Miner’s rule can be assumed to be valid for sand. In cyclic triaxial tests with different frequencies 0.05 ≤ f ≤ 2 Hz no influence of the loading frequency could be detected [835]. Thus, in this range the loading frequency does not need to be considered in a high-cycle model. In [835] also the influence of a static preloading was studied and found small. The grain size distribution curve has also a significant influence on the accumulation rate. In order to develop a simplified procedure for the determination of the material constants of the high-cycle model presented in Section 3.3.3, approx. 200 cyclic triaxial tests have been performed on eight different grain size distribution curves (Figure 3.85a) of a natural quartz sand. The results have been documented in detail in [841]. Figure 3.90 compares the strain remaining in the eight sands for similar test conditions (similar values of εampl , ID0 , σ av ). The accumulation rate increases with decreasing mean grain size d50 and grows significantly with increasing coefficient of uniformity Cu . In the accumulation model (Section 3.3.3) the influence of the grain size distribution curve has to be considered by different sets of material constants which enter the f -functions. Correlations of these constants with index properties (d50 , Cu , emin) are discussed in [841]. In contrast to drained cyclic tests the pore water pressure u accumulates in tests without drainage. Results of a typical test with an isotropic initial

206


a)

b)

500

Vertical strain ε1 [%]

Stresses σ3, u [kPa]

400 300 u 200

σ3 100 0

0

10 Nc,preload = 10

σ3 + u = const.

5

ID = 0.66 qampl = 45 kPa

-5

0

1,000 2,000 3,000 4,000 5,000 6,000

Time [s]

Time [s]

d)

60 Nc = 1-42 43 44 45 46

40

60 47

40

20

q [kPa]

Deviatoric stress q [kPa]

c)

47 46 45 44 Nc = 43

0

-10

1,000 2,000 3,000 4,000 5,000 6,000

ampl = 50 kPa qpreload

0

20 0 -20

-20

-40

-40 Nc = 46 45 44 43 42 1-41

-60 -10 -8

-6

-4

-2

0

2

-60 4

Vertical strain ε1 [%]

6

8

10

0

20

40

60

80

100

p [kPa]

Fig. 3.91. Results of an undrained cyclic triaxial test (after a drained cyclic preloadampl , see [844]): a) excess pore pressure ing with Nc,preload cycles at amplitude qpreload accumulation u(t), b) vertical strain ε1 (t), c) stress-strain hysteresis, d) effective stress path in the p-q-plane

stress are presented in Figure 3.91. The axial cyclic loading was applied stresscontrolled. The excess pore water pressure increased with each cycle (Figure 3.91a). When after some cycles the condition u ≈ −σ1 = −σ3 was approached (i.e. the effective stress was σ ≈ 0, so-called ”initial liquefaction”), the strain amplitude εampl started to grow rapidly (Figure 3.91b). During the subse1 quent cycles the stress-strain-hystereses (Figure 3.91c) showed no shearing resistance over a wide range of ε1 and the stress path in the p-q-plane followed a butterfly-like curve (Figure 3.91d). After several such ”cyclic mobility loops” the specimens failed during triaxial extension. A ”full liquefaction” is often quantified with a double amplitude 2εampl above approx. 10 %. 1 The stiffness E of a high-cycle model (Section 3.3.3) may be developed by comparing e.g. the accumulation of strain in drained cyclic triaxial tests and the relaxation of stress in undrained cyclic triaxial tests with similar initial conditions. From the rate of pore water pressure u˙ in undrained cyclic triaxial tests and the rate of volumetric strain accumulation Dvacc in drained cyclic triaxial tests one can derive the bulk modulus K = u/D ˙ vacc . A study of the stiffness E is documented in [841].

3.2 Experiments

207

undrained cycles package No. 2, Nc = 100 at σav = 0

380 340 0.64

-60 -80

0

40

80

p [kPa]

120

0.62

~ ~ ~ ~

~ ~

Δe1 = 0.010 Δe2 = 0.021

0.60 Δe3 = 0.024

0.58 0

500

1000

~ ~

-40

~ ~

0 -20


Re-cons. 3

420

20

Void ratio e [-]

q [kPa]

40

~ ~

u [kPa]

60

Re-cons. 2

Re-cons. 1

~ ~

460 80

~ ~


1500 3500

4000

4500 23000 23500

Time t [s]

Fig. 3.92. Change of void ratio Δe during re-consolidation after different numbers of cycles in the liqefied state (σ = 0)

An unsolved problem in connection with the post-cyclic behaviour is depicted in Figure 3.92 [579]. It presents displacement-controlled tests with three test phases. In the first phase the specimen was re-consolidated directly after a zero effective stress (σ = 0) was reached. In the second and third phase the specimen was monotonically re-consolidated after different numbers of cycles at σ = 0. Evidently the compaction during re-consolidation increases with the number of cycles at σ = 0. Similar observations were made by Shamoto et al. [734]. A latent accumulation in the grain skeleton seem to take place during the undrained cycles and it becomes visible during re-consolidation. Future investigations on this phenomenon are necessary. 3.2.3 Structural Testing of Composite Structures of Steel and Concrete Authored by Gerhard Hanswille and Markus Porsch 3.2.3.1 General In recent decades as a result of the benefits of combining the advantages of its components, steel-concrete composite beams have seen widespread use in buildings and bridges. The composite action of the components steel and concrete is realized by the shear connectors welded on the steel flange. Because of its economic and fast application headed shear studs are the most common used type of shear connectors in steel-concrete composite constructions. Typical examples of applications of headed studs in composite bridges are given in Figure 3.93.

208


Fig. 3.93. Application of headed shear studs in composite bridges

Due to the high initial stiffness headed shear studs effect nearly rigid composite action under service loads. At ultimate loads they allow for redistribution of longitudinal shear forces in the interface of steel and concrete due to their high ductility, which leads to robust composite structures. Especially in bridges and in industrial buildings with traffic actions or vibrations of machines these studs are subjected to high-cycle fatigue loading. In order to ensure high lifetimes and high reliability indices over lifetime it must be made sure, that the stiffness properties and the shear resistance of the studs remain sufficiently high. The behaviour of studs in solid concrete slabs under a static shear force is illustrated in Figure 3.94 [682, 686, 684]. At the beginning the shear force P acts mainly on the weld collar (component PW ). Under further increase of the load the concrete compressions at the weld collar lead to local destruction of the concrete in this region and thus to redistribution of the shear force into the stud shank. The stud shank is increasingly subjected to bending (component PB ) causing plastic deformations in its lower part. Due to the restraint of the stud head by the surrounding concrete further loading leads to tensile forces (Z) in the stud shank and a respective compressive force (D) in the concrete between the stud head and the flange of the steel beam. The horizontal component of the tensile force of the stud causes component PZ . The compressive force in the concrete activates additional frictional forces in the interface between concrete and steel flange, resulting in component PR . Because of the complex load deflection behaviour of headed studs embedded in solid slabs so far no design formula exists, which describes the ultimate shear resistance and the amount of each component by means of a mechanical model. The lifetime of cyclic loaded headed shear studs is effected by the load range, by its load bearing capacity and by the peak load. In current German

3.2 Experiments

209

P PR P

D PZ

D Z PR

PB PR

P G

PZ

PB PW

PW G

Fig. 3.94. Load-deflection behaviour of headed shear studs embedded in solid concrete slabs under static loading

design codes for bridges (DIN-Fachbericht 104) [18], which is based on the design rules of the Eurocode 4 [22, 23] the peak load level under service loads is limited to 60 % of the design value of the shear resistance of a stud. Due to this limitation mainly the components PW and PB are activated at studs of cyclic loaded composite structures. As the component PB leads to bending in the stud shank and thus to tension stresses the magnitude of this component has a significant influence on the development of cracks through the steel at the stud feet. In case of static loading even large bending moments can be resisted sufficiently by plastification of the shank cross section. Repeated loading causes a constant increase of the damage of the concrete in front of the stud, which leads to a steadily increase of the bending moments in the stud shank and to a gradual change of the stress state at the restraint during lifetime. Because of its costs and difficulties arising in the interpretation of the results of the full-scale beam tests, the evaluation of the behaviour of the shear studs generally takes place with standard push-out test specimen. Since 1960s various researchers conducted a great number of cyclic push-out tests under force control to determine the fatigue life of shear connectors. Some of them are given in [756, 513, 336, 681, 40, 591, 793, 687, 683]. As shown in Figure 3.95 by means of these investigations a fatigue strength curve for headed shear studs embedded in normal weight concrete was derived [685] based on the nominal stress concept. The scope of application is limited to studs in concrete flanges under compressive stressing and to peak loads less than 60 % of the ultimate stud shear resistance. In regions subjected to tensile stressing additionally the interaction between the shear stresses Δτ and the tensile stresses Δσ in the steel flange has to be considered [341, 594]. In the double logarithmic scale this model represents a linear relationship between the stress range Δτ in the stud shank and the number of cycles to failure N with the slope 1/m. It was taken as the basis for the design rules of

210


'WR (log) 1000

'W R

§N · ¨ ç ¨ N¸ © ¹

1m

'Wc N

§ 'W ¨ c ¨ 'W R ©

m

· ¸ N c ¸ ¹

test results: m = 8.658 Eurocode 4: m = 8

'Wcm = 110 N/mm²

100

5%-fractile

'P

d

'P 'W

'Wck = 90 N/mm²

4 'P Sd2

N (log)

10 104

105

106 Nc = 2x106

107

Fig. 3.95. Fatigue strength curve for cyclic loaded headed shear studs according [685]

cyclic loaded headed shear studs in Eurocode 4 in association with the linear damage accumulation rule of Palmgren-Miner [611, 543] in order to allow for load sequence effects. The magnitude of the value m (test results: m = 8.658) of the fatigue strength curve lies significantly above the values of m for details in steel structures (m = 5 or m = 3). This results in considerable deviations from the real lifetime, if the shear forces are already calculated slightly inexactly. Thus, particularly in hogging moment regions due to the influence of the axial stiffness of the concrete slab on the magnitude of the longitudinal shear forces the effect of the tension stiffening of the concrete between cracks, is of main interest [341]. In all before mentioned push-out tests the fatigue resistance was based on the total fracture of the studs but not on crack initiation being the main reason for the small slope of the fatigue strength curve. On this background one of the remarkable points in earlier investigations was that the repeated loading causes a reduction of the static strength of the shear studs not only at the end of their lifetime but within [593]. This indicates that the design concepts in the current codes [22, 23] does not describe sufficiently the real behaviour, because the determination of the ultimate load bearing capacity and the fatigue resistance takes place with separate and independent verifications at the ultimate limit state and the fatigue limit state.

3.2 Experiments

211

slip G P

G

P number of load cycles N

R, E ultimate load bearing capacity

R, E P

R resistance G

E

fatigue resistance 'W

effect of actions

N

a)

lifetime

b)

td

fatigue life tm

Fig. 3.96. (a) Safety concept to determine the lifetime of composite structures subjected to high cycle loading according to present codes, (b) Actual influence of high cycle loading on lifetime

In Figure 3.96 the design concept of the present codes is compared with the actual influence of high cycle loading on the lifetime of composite structures. So far the deterioration in strength of stud shear connectors and the change of the deformation behaviour due to cyclic loading remain unconsidered. In general this may result in a reduction of the reliability index of the composite structure so that it may fall below the target values in codes. The effect of the reduction of the static strength of the headed shear studs on the global load bearing capacity of a composite member depends on the magnitude of the loading and the capability of the redistribution of the shear forces. Further more it has to be considered that in flanges under tension the damage process at the stud welds can lead to a damage of the flange, on which the studs are welded. Except the test conducted by Oehlers [592] there were no further tests where the reduced static strength after high-cycle pre-loading was investigated. In the light of the information gained from previous researches a comprehensive program of more than 90 standard EC4-push-out test specimens and two full-scale beam tests were developed [349, 352, 347] which consider the crack propagation through the stud foot and the local damage of concrete surrounding the studs as relevant consequences of high cycle loading. In detail the research on the push-out tests were mainly focused on

212

• • • • • • • • •


the static strength Pu,0 of headed shear studs without any pre-damage the number of cycles to failure Nf taking into account the relevant (loading) parameters given as static strength Pu,0 , peak load Pmax and load range ΔP the reduced static strength Pu,N of headed shear studs after high cycle loading crack initiation and crack propagation through the stud feet as the main reason for the decrease of the static strength over lifetime load sequence effects regarding nonlinearities caused by cracking of the steel and crushing of the concrete static load deflection behaviour of undamaged and damaged shear studs as well as the cyclic load deflection behaviour the ductility behaviour of studs after pre-damage effect of the mode control on the results of cyclic loaded push-out tests (force control versus displacement control) effect of low temperature on the load deflection behaviour of damaged headed shear studs

Considering the interaction between the local damage and the behaviour of the global structure, these research results were subsequently taken as the basis to simulate the cyclic behaviour of composite beams by means of damage accumulation method. In order to verify the theoretical models derived from the push-out tests additionally two full-scale beam tests were performed, in which the effect of the deterioration of the properties of the interface between steel and concrete on the load-deflection behaviour, on the redistribution of the inner forces and on the reduced static strength after high cyclic loading were investigated. Regarding typical loading conditions in composite structures it was distinguished between a beam subjected to a sagging moment and a beam subjected to a hogging moment. 3.2.3.2 Basic Tests for the Fatigue Resistance of Shear Connectors 3.2.3.2.1 Test Program The experimental program of the push-out tests reported in the following chapters consists of a total of 9 series (S1 - S6, S9, S11 and S13) [352, 353]. The first four series S1-S4 deal with constant amplitude tests where the effect of unidirectional cyclic loading on the static strength and the fatigue life of the push-out specimen were investigated with the varying loading parameters peak load Pmax and the loading range ΔP . In each series initially three static tests were performed to determine the mean value of the ultimate static load P u,0 of the push-out specimen. The mean value of the ultimate static load represents the reference parameter for the relative values of loading parameter required for cyclic tests. Using the relative loading parameters three load controlled cyclic tests were performed to determine the mean fatigue life N f

3.2 Experiments

213

Table 3.6. Summary of the single level tests

series

'P Pu ,0

Pmax Pu ,0

S1 S2 S3 S4 S5E

0.20 0.25 0.25 0.20 0.25

0.44 0.71 0.44 0.71 0.30

Pu ,0

Nf

3 3 3 3 3

3 3 3 3 1

Pu,0

P

number of tests N ~ 0.3 N f 3 3 3 3 1

Pu,N

Pmax,4

Pmax,2 Pmax,1

3 3 3 3 1

Pu,0

P

Pu,N1 + N2 = Pmax,2

N ~ 0.7 N f

1 + N2 + N3 + N4

= Pmax,4

Pmax,1

'P

'P N1

N2

'P

N1 N2 N3

N4

load cycles load cycles

'P = constant

a)

Gu,N

1 + N2

G

'P = constant

b)

G

Gu,N

1 + N2 + N3 + N4

Fig. 3.97. Tests with multiple blocks of loading: (a) two blocks, (b) four blocks (increasing peak load)

of the push-out specimen. Subsequently six cyclic tests were conducted for approximately 30 and 70 percent of the mean fatigue life N f . After reaching the corresponding number of cycles each of these six test specimens was statically loaded to failure under displacement control to obtain the reduced static strength after high cycle pre-loading. The chosen loading parameters and number of performed tests for the first four series are summarized in Table 3.6. Based on the results of the constant amplitude tests in the series S5 and S6 tests with two and four blocks loading sequence were performed in order to check the validity of the linear damage accumulation according to the Palmgren-Miner rule in the case of headed shear studs embedded in solid slabs. In these tests the peak load was increased or decreased subsequently while the loading range was held constant (see Figure 3.97). Like the constant amplitude tests the series S5 and S6 initiate with shorttime static tests to determine the ultimate static load. During the tests with multiple blocks of loading the necessity of performing further constant amplitude tests with not investigated low peak load (Pmax / P u,0 = 0.3) was arisen.

214


Table 3.7. Summary of the tests with multiple blocks of loading series

number of tests

S5 - 2 S5 - 3 S5 - 4 S5 - 6 S6 - 3 S6 - 4

3 1 4 4 3 3

'P Pu ,0

0.25

0.20

Pmax Pu ,0 block 1 0.71 0.44 0.44 0.30 0.44 0.74

block 2 0.44 0.71 0.30 0.44 0.54 0.64

block 3 0.64 0.54

block 4 0.74 0.44

Thus, 3 complementary constant amplitude cyclic tests were additionally performed in these series, referred as series S5E. The chosen loading parameters and number of performed tests for tests with multiple blocks of loading are summarized in Table 3.7. The loading parameters of series S5E are also listed in Table 3.6. Regarding the simulation of the cyclic behaviour of composite beams by means of the results gained from force controlled push-out tests, in series S9 nine tests were performed in which the influence of the control mode was investigated (Figure 3.98). After experimental determination of the mean static strength P u,0 (test specimen S9 1b and S9 1c) three cyclic displacement controlled push-out tests with constant values of the maximum slip smax and the slip range Δs (initial peak load approximately 0.65 P u,0 ) subjected to 5 × 106 load cycles were conducted (S9 4). The resulting loading history of the varying peak load Pmax (N) and the load range ΔP (N) were classified in eight blocks with constant loading parameters and subsequently taken as input values for four additional cyclic force controlled push-out tests. The number of load cycles Ni in a successive block was chosen twice the number of load cycles in the former block. With the exception of test specimen S9 5d after each cyclic loading phase the reduced static strength was experimentally determined at a temperature of T = 20◦ C. In order to investigate the influence of cracks through the stud feet on both, the reduced static strength and the ductility test specimen S9 5d was tested after cooling down to T = −40◦ C. During the cooling process and subsequently in the test rig, the temperature of the concrete slabs were measured permanently by means of 2 temperature sensors PT 100, each embedded in the middle of one slab. By using thermal insulation it could be realized, that the core temperature of the concrete did not fall below a value of T = −38◦ C at the end of the static test procedure. Due to the notch effect of the stud welds in all test specimens in series S1 to S9 subjected to fatigue loading damage occurred at the stud feet in form of cracks through the steel leading to a reduction of the static shear resistance of the shear studs. On the background of decoupling of the ultimate limit state and the fatigue limit state in test series S11 and S13 7 additional

3.2 Experiments

215

P [kN] 1000

7 tests (S9_4, S9_5)

load applied displacement controlled

temperature sensor PT 100 's=0.08mm

smax

800

smin

P

P

600

s 400 Pmax

'P(N)

200

Pmin N [x 106]

load applied force controlled

0

2

1 specimen parameter Pu,0 S9_1 b-c -

-

S9_4 c–e

's [mm] block

S9_5 a–d

-

4

3

5

static strength T = 20°C

S9_5d during the cooling process in the climate chamber and in the test rig

cyclic loading - N = 5x 106 0.08 (initial load ~ 0.65 Pu,0 ) (displacement control)

Pmax Pu,0

1 0.54

2 0.51

3 0.48

4 0.44

5 0.41

6 0.36

7 0.27

8 0.21

'P Pu,0

0.23

0.23

0.22

0.21

0.20

0.17

0.15

0.12

load cycles 'N1 = 2 'N1 2 'N2 2 'N3 2 'N4 2 'N5 2 'N6 ~2 'N7 19608 reduced static strength -

S9_4 c–e S9_5 a–c

Pu,N

T = 20°C

S9_5 d

Pu,N

T = -40°C

Fig. 3.98. Tests to compare the effect of the mode control - force control vs. displacement control - and the effect of low temperature

tests in accordance with Figure 3.99 were carried out, in which the effect of extreme low peak loads on the crack initiation and the crack propagation at the stud feet was investigated. Due to the low cyclic loading parameters the cyclic loading phases were finished in each case after a certain number of load cycles, before determining the residual static strength. In most cases these numbers were only a small fraction of the mean number of cycles to failure according to the current design concept [685], taking into account a slope m = 8. 3.2.3.2.2 Test Specimens The specimen used in the push-out tests consists of a 650 mm high HEB260 profile and two 650 mm high, 600 mm wide and 150 mm thick concrete slabs. The slabs are connected to the steel beam by means of four headed shear studs of 22 mm diameter welded on each side of the beam. The weld collars complied with the requirements of EN ISO 13918 [10]. The mean height of the weld collars was 8.4 mm and the mean diameter 28.9 mm. The height of the welded stud was 125 mm. For casting the steel beams were cut into two halves and the concrete slabs were casted horizontally. The steel flanges were greased

216


Pu,0

P

Pu,N

test

Pmax / Pu,0 'P / Pu,0

Pmax

'P

[kN]

[kN]

S11-1a

Pmax 'P N

G

N / Nf,cal

(m = 8.0)

[-]

[-]

[-]

Nf,cal

N

[-]

[-]

static strength Pu,0 (valid for S11 and S13)

S11-4a

240

200

0.16

0.13

7.00 x 106

1.22 x 108

1 / 17.4

S11-4b

360

320

0.24

0.21

4.27 x 106

2.86 x 106

1 / 0.67

S11-4c

120

80

0.08

0.05

9.41 x 106

1.87 x 1011 1 / 19872

S13-2a

250

200

0.16

0.13

3.80 x 106

1.22 x 108

1 / 32.1

S13-2b

350

200

0.23

0.13

4.10 x 106

1.22 x 108

1 / 29.8

S13-2c

450

200

0.30

0.13

4.08 x 106

1.22 x 108

1 / 29.9

Gu,0 Gu,N

Fig. 3.99. Tests to investigate the effect on the duration of the crack initiation phase and crack growth velocity due to very low cyclic loads (Nf,cal according [685])

section B-B

section C-C

B

mortar

B

L120x120x12

mortar

130

150

A

B

L120x120x12

A

200

50

100

50

820

560

260

A

150

730

650

C

12 10

C

stud 22/125

108

C

HEB 260

30 20 200

C

HEB 260 (S235 J2G3)

250

150

50

HEB 260

80

A

10 12

30

108

B

head plate

130

section A-A

200

600

Fig. 3.100. Details of the push-out test specimen

prior to casting to remove friction between the concrete and steel. The two halves were subsequently welded together. This test specimen complies with the standard push-out specimen according to Eurocode 4 with an exception of lateral restraints of the concrete slabs at the specimen bottom. The lateral restraints avoid especially in the lower row of shear studs the introduction of additional tensile forces resulting from the moment of eccentricity and enable a better simulation of the real behaviour in composite beams. Details of the push-out specimens are given in Figure 3.100. 3.2.3.2.3 Test Setup and Loading Procedure Generally cyclic and monotonic loading was applied by a 2500 kN servo hydraulic actuator. Only in particular cases, for instance in order to determine effects of relaxation on the load bearing capacity, additionally a 2000 kN and a 10000 kN servo hydraulic actuator were used to apply static loading. The load

3.2 Experiments

2000 kN

2500 kN

217

10000 kN

Fig. 3.101. Servo hydraulic actuators

was introduced into the steel beam by means of a head plate welded on the cross section of the beam and an additional centring bar. All actuators were controlled by an Instron 8800 controller which allowed load and displacement control. For the cycling testing sinusoidal control waveforms were utilized. With the exception of test specimen S9 4c - S9 4e displacement control was used for the monotonic tests and load control was used for the cyclic phases. The actuators used in the experiments are shown in Figure 3.101. The monotonic tests were conducted at a displacement rate of 0.004 mm/sec. The time taken to reach the ultimate load was typically of the order of 50 minutes. After reaching the ultimate load the displacement rate was increased up to 0.008 mm/sec. Cyclic tests were conducted with a load frequency of 3 - 5.5 Hz. In order to collect data about the stiffness and plastic deformation, cyclic tests were held after specific number of cycles and the specimens were released and reloaded monotonically. During the tests the time, load from the actuator load cell, ram displacement from the built-in transducer in the actuator, vertical slip between the concrete slabs and steel beam (WV1a, WV1b, WV2a and WV2b) and horizontal displacement of the slabs (WHa and WHb) were measured (see Figure 3.102). The ram displacement included movement due to the compliance of the test rig, and therefore it was not used in any subsequent data analysis. In the case of specimen S9 4c - S9 4e also the cyclic test phase was performed under displacement control. For this purpose the measuring equipment was supplemented by two additional LVDT’s (WA20/1a and WA20/1b) in parallel connection placed directly above the before mentioned transducer (one at each flange) in the interface between the steel beam and the concrete slabs in order to control the actuator. 3.2.3.2.4 Material Properties At various intervals during the curing process at 28 days as well as at the beginning, in the middle and at the end of each series cylinder compression

218


Section A-A

Section C-C

Section B-B B

A

side b

side a

B side b

side a

WV20/1a WA20/1b

WV2a

WA20/1a

WV1b WV2b

WV2b

WV1a

80 45 45 80

WHa

WV2a WHb

WV2b WHa

WHb

B

WV1b

side b

A

WV20/1b

B

Fig. 3.102. Position of transducers

Table 3.8. Mean values of material properties of concrete according to EN 206-1 [12]

property fc [N/mm2] Ecm [N/mm2]

series S1

S2

S3

44.2 –

42.8 –

53.9 –

51.7

45.0

56.2

36400

33800

39000

S4

S5

S6

43.4

42.9

45.8

33900

33050

33700

S9 47.2 – 54.8 32250

S11, S13 38.5 30635

tests were carried out to determine the compressive cylinder strength and elastic stiffness of the concrete. Standard cylinders of 150 mm in diameter and 300 mm in length were used. Table 3.8 presents the interval of the mean concrete strength and the mean modulus of elasticity (taken as secant modulus Ecm according to EC 2[33]) associated with the test begin for each series according to EN 206-1 [12]. In test series S1 - S6 on one side and in series S9, S11 and S13 otherwise structural steel beams, headed shear studs and reinforcing bars were from the same batch. Structural steel beams of HEB 260 section with the material quality S235 J2G3 were used in each test. Stud shear connectors, which were welded automatically onto the steel beam flange, had a material quality of S235 J2G3+C450. As reinforcing steel standard deformed bars with diameters of 10 mm and 12 mm were used in the concrete slabs. In order to obtain detailed data about material properties tensile tests of all steel members were conducted according to the requirements of DIN EN 10002 [15]. The results of the corresponding values of each yield strength, tensile strength and modulus of elasticity are summarized in Table 3.9.

3.2 Experiments

219

Table 3.9. Mean values of material properties of steel members member yield strength, tensile strength and modulus of elasticity

steel beams

headed studs

reinforcement Ø 10

reinforcement Ø 12

property

S1 – S6

S9, S11, S13

fy fu Ea fy fu Es Rp0.2 Rm Es Rp0.2 Rm Es

337 448 210000 440 528 215000 549 606 197800 501 561 204500

413 516 204000 448 538 207800 543 586 192300 545 588 196500

[N/mm2]

3.2.3.2.5 Results of the Push-Out Tests 3.2.3.2.5.1 General In the following the main results of the push-out tests of series S1 - S6, S9, S11 and S13 are reported. They mainly concern load bearing capacities and number of load cycles. Further information especially results regarding simulation of cyclic loaded beams are presented in the associated Chapter 3.3.4. 3.2.3.2.5.2 Results of the Constant Amplitude Tests (S1 - S4, S5E) Table 3.10 shows the results of the static strength, the fatigue life N f and the reduced static strength after high cycle pre-loading for series S1 - S5E. The limit state of fatigue is given, when the reduced strength has reached the value of the peak load. Because of different static strength within each test Table 3.10. Average test results per stud series

Pu ,0

Nf

[-]

[kN]

[-]

S1 S2 S3

205 184 201

Pu ,N1

Pu ,N1 Pu ,0

N1 Nf

Pu ,N 2

Pu ,N 2 Pu ,0

N2 Nf

[kN]

[-]

[-]

[kN]

[-]

[-]

6.2x10

6

154

0.75

0.32

129

0.63

0.90

1.2x10

6

174

0.95

0.32

154

0.84

0.70

5.1x10

6

133

0.66

0.24

123

0.61

0.69

6

181

1.00

0.29

156

0.86

0.72

111

0.59

0.19

114

0.60

0.73

S4

181

3.5x10

S5E

189

6.4x106

220


series the absolute values of Pmax and ΔP differ slightly. All data given in this Table represent generally the mean values of three comparable tests. The strength data are based on short time behaviour. From the statically loaded push-out test specimens without any predamage it could be found out, that if the position of the hydraulic actuator is held constant during a test phase the test specimen typically relaxes and withdraws the applied loading. Near to failure the loss of loading amounts up to 10% of the applied load. On this background it is necessary to reduce the values given in Table 3.10 by 10% [345] when relaxation must be taken into account. In contrast to series S2 and S4 the low peak loads in series S1, S3 and S5E led to very high fatigue lives N f . This shows the significant influence of the relative peak load Pmax / P u,0 on the lifetime. On the background of a redistribution of inner forces due to a deterioration of the mechanical properties of the connectors under service loads not only the reduction of the static strength of the headed shear studs is of main interest but also the development of the deformation under repeated loads. In the case of high cyclic loading the load-deformation behaviour is characterized by an increasing plastic slip δ and a decreasing elastic stiffness Kel . In Figure 3.103 the inelastic displacement δ i related to the plastic slip in the first cycle δ 1 in the steel-concrete interface is plotted against the number of cycles over the fatigue life N i / N f for series S1 - S4. The beginning and the end of the lifetime are associated with a steep increase in the plastic slip with the number of cycles while in the remaining part of the lifetime a nearly linear increase of the plastic slip occurs with the number of cycles. Due to the

series S1

Gi G1

20

series S2

Gi G1

25

25 G1 0.06 mm

G1 0.52 mm

20

15

15

10

10 5

5 Ni Nf 0

0 0.0

0.2

0.4

1.0

0.8

1.0

Ni Nf

0.0

0.2

0.4

1.0

1.0

0.8

Pmax

P 25 20

Gi G1

series S3

G1 0.06 mm

25 20

15

15

10

10

5 0 0.0

Gi G1

series S4 c w,

G1 0.55 mm

w,1

Kel,1 G1

d0

Kel,i

G

Gi

5 Ni Nf

0.2

0.4

1.0

0.8

1.0

0 0.0

Ni Nf

0.2

0.4

1.0

0.8

1.0

Fig. 3.103. Development of plastic slip over the fatigue life in series S1 - S4

3.2 Experiments

Pu / Pu,0

221

Pu,0

P

Pu VX

1.0

0.03

VX

0 .03

Pmax

2 0.8

0 .09

Pmin

'P G

N

1 3

0.6

VX

0 .10

5E Stages of the strength reduction: I no reduction II non linear - declining III linear (stable) IV non linear - progressive

0.4

0.2

I

0

VX

4

0

static tests

3

4

5E

ǻP Pu,0

series

0.20 0.25

0.25

0.20

0.25

Pmax Pu,0

0.44 0.71

0.44

0.71

0.30

N f (x106)

6.2

5.1

3.5

6.4

2

1.2

III

II

0.2

1

0.4

0.6

tests with cyclic pre-loading

IV

0.8

N / Nf 1.0

fatigue tests

Fig. 3.104. Decrease of static strength vs. lifetime due to high cycle loading

increasing compliance of headed shear studs with higher load levels the mean value of the initial plastic slip δ 1 in the first cycle is in the test series with high peak loads approximately 8 times greater than in series with low peak loads. The influence of the cyclic loading becomes evident, when the static strengths are plotted versus number of load cycles. This is shown in Figure 3.104, where the results are related to the mean static strength and to the mean fatigue life of each series, respectively. Especially in series S1, S3 and S5E with low peak loads the rapid decrease of the static strength within the first 20 percent of the fatigue life is noteworthy. The reduction of the static strength over the lifetime is considered in four stages. In stage I there is no significant damage in form of cracks through the stud feet and therefore no noticeable reduction of the shear resistance. In stage II and IV in each case a more or less nonlinear reduction of the static strength could be observed. In the case of lower peak loads the decrease is disproportionately high. The range within stage II and stage IV shows a linear reduction of the static strength with the number of cycles. In the cases of low peak loads the decrease of static strength is disproportionately low. In Figure 3.104 the coefficient of variation Vx of the static strengths gained from three similar tests is marked out exemplarily for the series S4. The scatter of the results increases with the degradation of the strength of the shear studs. The

222


Table 3.11. Loading parameters and results of the tests with two blocks of loading (series S5) block 1

test

block 2

N1 + N2

Pmax,1

'P

N1

Pmax,2

'P

N2

[-]

[kN]

[kN]

[ u106 ]

[kN]

[kN]

[ u106 ]

[ u106 ]

S5-2a

133

0.204

83

0.792

0.996

S5-2c

133

0.198

83

1.440

1.638

S5-3a

83

1.099

133

-

1.099

S5-4a

83

0.473

56

1.365

1.838

S5-4b

83

0.517

56

0.772

1.289

S5-4c

83

0.544

56

S5-4d

83

0.542

S5-6a

56

S5-6b S5-6c S5-6d

47

47

0.735

1.279

56

3.396

3.938

0.537

83

5.821

6.358

56

1.223

83

0.761

1.984

56

1.295

83

1.744

3.039

56

1.277

83

3.206

4.483

comparison of the series with the same related load range makes clear that the static strength and the fatigue life are affected not only from the load range but also from the peak load of the cyclic loading. Furthermore it is also obvious that for the same amount of change the effect of the load range on the static strength is much greater than of the peak load. 3.2.3.2.6 Results of the Tests with Multiple Blocks of Loading (S5 and S6) The results of the two and four blocks loading sequence (see Figure 3.97) are given in Table 3.11 and 3.12 respectively. The mean value of the reference Table 3.12. Loading parameters and results of the tests with four blocks of loading (series S6) test

'P

block 1 Pmax,1

block 2

N1

Pmax,2 6

block 3

N2

Pmax,3 6

4

block 4

N3

Pmax,4 6

¦ Ni

N4

i 1

6

[-]

[kN]

[kN]

[ u10 ]

[kN]

[ u10 ]

[kN]

[ u10 ]

[kN]

[ u10 ]

[ u106 ]

S6-3a

38

83

0.756

101

0.768

120

0.770

139

0.868

3.162

S6-3b

38

83

0.765

101

0.804

120

0.785

139

0.324

2.678

S6-3c

38

83

0.754

101

0.759

120

0.750

139

0.449

2.712

S6-4a

38

139

0.550

120

0.763

101

0.754

83

0.583

2.650

S6-4b

38

139

0.550

120

0.758

101

0.750

83

0.756

2.815

S6-4c

38

139

0.540

120

0.753

101

0.753

83

1.208

3.254

3.2 Experiments

223

ultimate static strength P u,0 of the shear studs is 186 kN for the tests with two blocks of loading in series S5 and 196 kN for the tests with four blocks of loading in series S6. It is to be noticed that in tests with multiple blocks of loading the failure of the shear studs can occur on one hand during the cyclic loading by the decrease of static strength to the peak load and on the other hand during switching to the next block with a higher peak load by exceeding the reduced static strength. A typical example for the last case occurred in the test S5 3a during switching to the second block with a peak load of 133 kN by exceeding the reduced static strength of 124 kN per stud. 3.2.3.2.7 Results of the Tests Regarding the Mode Control and the Effect of Low Temperature (Series S9) The results of the tests of series S9 are given in Table 3.13, the corresponding loading histories are shown in Table 3.14 and Table 3.15. The mean value Table 3.13. Average test results per stud in series S9

N

Pu,0

Pu,N

T

[-]

[kN]

[kN]

°C

S9_1b

-

197

-

S9_1c

-

208

-

S9_4c

5019542

-

112

S9_4d

4966493

-

121

S9_4e

5031757

-

123

S9_5a

4964001

-

119

S9_5b

4999716

-

117

S9_5c

5457200

-

111

S9_5d

5000000

-

150

test

20

-40

Table 3.14. Measured mean values of the peak load and the load range at discrete number of load cycles in tests S9 4 test S9_4a-c

test S9_4a-c

Pmax,1

'P 1

N1

Pmax,2

'P 2

[kN] 873

N2

Pmax,3

'P3

[kN]

[-]

[kN]

375

9804

822

N3

Pmax,4

'P 4

[kN]

[-]

[kN]

366

39216

777

[kN]

[-]

[kN]

[kN]

[-]

356

98040

719

340

215688

N4

Pmax,5

'P 5

N5

Pmax,6

'P 6

N6

Pmax,7

'P7

N7

Pmax,8

'P 8

N8

[kN]

[kN]

[-]

[kN]

[kN]

[-]

[kN]

[kN]

[-]

[kN]

[kN]

[-]

660

318

450984

584

279

921576

443

236

1862760

339

190

3745108

224


Table 3.15. Loading parameters and block lengths in tests S9 5 cyclic phase 1

cyclic phase 2

cyclic phase 3

cyclic phase 4

test -

Pmax,1

'P 1

N1

Pmax,2

'P 2

N2

Pmax,3

'P3

N3

Pmax,4

'P 4

N4

[kN]

[kN]

[-]

[kN]

[kN]

[-]

[kN]

[kN]

[-]

[kN]

[kN]

[-]

S9_5a

19607

41408

76683

S9_5b

20209

37928

79566

S9_5c

873

375

S9_5d

19609

822

366

19616 cyclic phase 5

39209

777

356


78437

156421 719

340


156024 157160 156922

cyclic phase 8

test -

Pmax,5

'P 5

N5

Pmax,6

'P 6

N6

Pmax,7

'P7

N7

Pmax,8

'P 8

N8

[kN]

[kN]

[-]

[kN]

[kN]

[-]

[kN]

[kN]

[-]

[kN]

[kN]

[-]

S9_5a

313724

627452

1255227

2473479

S9_5b

313433

627744

1259583

2505229

S9_5c S9_5d

660

318

313429 313949

584

279

627452 627167

443

236

1257995 1256006

339

190

2963909 2508700

of the reference ultimate strength P u,0 of the shear studs is 202 kN. After 5 × 106 load cycles the load bearing capacity of a stud decreases to a mean value P u,N of 118 kN due to the displacement controlled high cycle loading and to an only slightly different mean value P u,N of 116 kN in the case of the force controlled cycle loading phase, respectively. As it can be seen from the test results of specimen S9 5d the lower temperature causes a remarkable increase of the reduced static strength compared to tests at a temperature of T = 20◦ C. Metallurgical investigation revealed that due to the loading history nearly the same size of crack lengths at the stud feet as in all other cases of series S9 were induced. In order to find the main reason for the higher reduced static strength further concrete cylinder compression tests of the same batch were carried out to determine the material properties of the concrete at a temperature T = −40◦ C. In these tests the compressive cylinder strength and the modulus of elasticity (taken as secant modulus) of specimens stored according to the requirements of EN 206 [12] were determined as fc = 67 N/mm2 and Ecm = 39250 N/mm2 . The cyclic loading parameters in the phases 1 to 8 of the force controlled tests were determined on the basis of the loading history of the displacement controlled tests. As shown in Table 3.14 and Table 3.15 for this purpose the measured values Pmax,i and ΔPi (i = 1 to 8) at 8 discrete numbers of load cycles of the displacement controlled tests were taken as the mean values for the corresponding blocks of loading for the force controlled tests.

3.2 Experiments

225

Table 3.16. Average test results per stud in series S11 and S13

N

Pu,0

Pu,N

[-]

[kN]

[kN]

S11_1a

-

190

-

S11_4a

7000600

-

197

S11_4b

4272816

-

157

S11_4c

9416000

-

205

S13_2a

3801147

-

202

S13_2b

4100300

-

206

S13_2c

4086509

-

189

test

3.2.3.2.8 Results of the Tests Regarding Crack Initiation and Crack Propagation in the Case of Low Peak Loads (Series S11 and S13) The results of the tests of series S11 and S13 are given in Table 3.16. The test specimens of both test series were casted at the same time using the same concrete mix, so that the reference ultimate strength Pu,0 of test S11 1a is also valid for test series S13. Although in most cases the static strength after high cyclic pre-loading were in the same order or even higher than the reference static strength without pre-damage Pu,0 the stud welds showed partly considerable fatigue fracture areas due to crack propagation during the cycling loading phase. 3.2.3.3 Fatigue Tests of Full-Scale Composite Beams 3.2.3.3.1 General Regarding the transition of results gained from push-out tests to the global behaviour of composite structures tests of composite beam are an essential part of experimental research. Within the framework of the project ”Modelling of damage mechanism to describe the fatigue life of composite steel-concrete structures”[348, 350] the test program was extended by 2 full-scale composite beams subjected to unidirectional high cyclic loading. In order to investigate the interaction between the local damage at the studs and the global behaviour of beams at the serviceability limit state and at the ultimate limit state these tests were mainly focused on • •

the increase of vertical deflections, the development of the elastic stiffness of the composite beams,

226

• • • •


the development of the plastic slip in the interfaces between steel and concrete, the crack propagation at the stud feet and the weld collars, the redistribution of the sectional forces of the concrete slabs and the steel cross sections, the reduced static strength of each beam after high cyclic pre-loading.

3.2.3.3.2 Test Program The test program consists of 2 simply supported beams, each subjected to a point load at midspan causing a sagging moment in the case of beam VT1 and a hogging moment in the case of beam VT2. As shown in Figure 3.105 the beams were loaded by 1.38 × 106 and 2.10 × 106 load cycles before testing their reduced static strengths. This procedure is in accordance with the principles of the constant amplitude tests of series S1-S4 of the test program of the push-out tests, in which also the effect of the high cyclic loading on the static strength was investigated. The cyclic loading phases should lead to noticeable reductions of the ultimate load bearing capacities but in no case to a complete

P

loading parameters

Pu,0 Pu,N 0.9 Pu,N

Gu2

Gu1

Pmax 'P

test

Pmax

'P

[-]

[kN]

[kN]

[-]

VT1

450

265

1.37x106

VT2

250

100

2.10x106

1. unloading 1. reloading

P N

N

VT1 and VT2 during testing

6m G Gu Gu,0

0 Pmax cyc G res Gcyc G LC Pmax , lc N res G ini

54 studs HEA300

load distribution plates (280x280) placed on mortar

b/h = 1500/150

VT1 VT1

(rb) HEA300

roller bearing (rb) b/h = 1500/150

0 G LC res

54 studs

load distribution plates (280x280)

(rb) 300

3000

3000

[mm]

VT2

(rb) 300

VT2

Fig. 3.105. Test programme and loading parameters of the composite beam tests VT1 and VT2

3.2 Experiments

227

fatigue shear failure of all studs in order to obtain clearly defined damage states between states with nearly no damage and states, in which the concrete slabs makes no contribution to ultimate load bearing capacity. During the cyclic loading phase the test beams were released and reloaded monotonically at specific number of load cycles, in order to collect data about accumulated plastic deformations and irreversible strains. 3.2.3.4 Test Specimen The specimens consist of a 6600 mm long HEA 300 profile and a likewise 6600 mm long, 1500 mm wide and a 150 mm thick concrete slab. In both cases the slabs were connected to the steel beam by means of 54 studs (in 2 rows with 27 studs per row) of 22 mm diameter and a height of 125 mm after welding. The weld collars complied with the requirements of EN ISO 13918 [10]. The mean height as well as the mean diameter of the weld collars were in the same magnitude as in the push-out tests reported in Chapter 3.2.3.2. Prior to casting the steel flanges were greased in order to remove friction in the interface between steel and concrete. Moreover thin sliding layers composed of steel plates and PTFE-foils were placed at midspan (VT1) and at each support (VT2) in order to avoid additional restraints caused by the introduction of the vertical forces directly into the concrete slabs at these sections. The longitudinal reinforcement was chosen to 18 bars with diameters of 12 mm (VT1) and 30 bars with diameters of 16 mm (VT2). As transversal reinforcement in both cases 52 closed stirrups with diameters of 12 mm were used. Like bridges the beams were casted in horizontal position with the concrete slabs at the upper side. Details of the test beams can be taken from Figure 3.106 and Figure 3.107. 3.2.3.5 Test Setup Cyclic and monotonic loading was applied by a 2000 kN servo hydraulic actuator, controlled by an Instron 8800 controller. The load from the actuator was introduced by means of 4 load distribution plates, placed directly on the steel flange (VT1) and on a mortar layer (VT2), respectively, at midspan of each test beam. The peak load and the load range during the cyclic loading phase was applied under force control with a frequency of 0.4 Hz. For all other loadings - first loading and first release, releases and re-loadings during the cyclic loading phase and loading up to failure - displacement control was used. The time taken to release and to re-load the test beams was typically of the order of 8 (VT1) and 4 (VT2) minutes. Regarding the evaluation of the tests extensive monitoring systems consisting of 62 measuring points at each beam according to Figure 3.108 were set up. By means of these systems data of the vertical deformations, of the slip


125

146

2

290

10

18.8 12.5 656

12.5

10

18.8 5 6 x 18

18 3

material properties: reinforcement: S500 structural steel: S460 sliding layer: PTFE (greased), S235

concrete: C35/45

8.6 cm

[cm] 15

3 18 [mm]

2 11 2

75 150 75

1 18 Ø 10 – spacing 18 cm – length 693.2 cm

47 x 12.5

concrete slab – reinforcement (sliding layer not shown)

6600

300

2650

700

2650

HEA 300

300

concrete slab – 15/150/660 [cm]

5

11 52 Ø 12 – spacing 12.5 and 18.8 cm – length 334 cm (alternately) 8.6

262x140x10 (stiffener)

100

sliding plate (700x300) on PTFE-foil (each 1mm thick – PTFE double sided greased – cut-out Ø 40 mm)

200 54 headed studs Ø22 - h/d = 125mm / 22mm (1a -27a, 1b-27b) 200

250

22 x 250

11 4 1 side b

side a

steel beam – headed studs – sliding layer

17

24

250

27

100

290

10

11

228

studs: S235 J2G3 + C450

Fig. 3.106. Details of test beam VT1

in the interfaces between steel and concrete and data of the strain states of the steel beams and the concrete slabs were collected continuously during all test phases.

125

146

2

290

10

18.8 12.5

675

concrete slab – 15/150/660 [cm]

5

11 52 Ø 12 – spacing 12.5 and 18.8 cm – length 334 cm (alternately) 8.6

656

12.5

10

18.8 5 5x9

18 18

5x9

material properties: reinforcement: S500 structural steel: S460 sliding layer: PTFE (greased), S235

concrete: C35/45

8.6 cm

[cm] 2 11 2

93

15

39 [mm] 262x140x10 (stiffener)

75 150 75

1 30 Ø 16 – spacing 9 cm and 18 cm – length 693.2 cm

47 x 12.5

concrete slab – reinforcement (sliding layer not shown)

500 675

2375

6600

2375

HEA 300

200 250 54 headed studs Ø22 - h/d = 125mm / 22mm (1a -27a, 1b-27b) 200 100

side a

side b

1

250

22 x 250

24 17 4

11

steel beam – headed studs – sliding layer

sliding plate (675x300) on PTFE-foil (each 1mm thick – PTFE double sided greased – cut-out Ø 40 mm)

27

100

290

229

10

11

3.2 Experiments

studs: S235 J2G3 + C450

Fig. 3.107. Details of test beam VT2

Without additional measurements or detailed monitoring it is not possible to determine the failure of studs. As known from the push-out tests the damage process in the interface between steel and concrete proceeds continuously and

230


300

1500

[mm]

145

50 (QS2, QS6: 20)

side A

50

290

50 (QS2, QS6: 20)

440

150

300

VT1

transducers

strain gauges (oriented in longitudinal direction / QS1 – QS7 ) 1500

side A

50

transducers (side A )

1125

675 QS0

QS1

1125 QS2

375 QS3

375

QS4

QS5

strain gauges (oriented in longitudinal direction / QS1 – QS7)

1125 QS6

QS8

QS7

transducers

VT2

side A 440

290

50 (QS2, QS6: 20)

675

50

145

side A

50

1125

150

50 (QS2, QS6: 20)

1500

300

[mm]

300

horizontal transducers (side A)

400

400

400

400

400

400

400

vertical transducers (side A)

1125

675 QS0

QS1

1125 QS2

375 QS3

375

QS4

1125

QS5

1125 QS6

675 QS7

QS8

Fig. 3.108. Test setup of test beams VT1 and VT2

so no significant change in properties of a beam can be observed after single stud failure. In order to avoid a complete shear failure of studs the studs of one row of each beam were coupled in an electric circuit. According to the circuit shown in Figure 3.109 shear failure during a cyclic loading phase can be detected, when the corresponding LED starts to flicker or extinguishes.

3.2 Experiments

231

VT1

LED with series resistance

3B


2B


1B 0V




+12V

25B

26B

27B

HEA 300

Fig. 3.109. Electric circuit to detect complete shear failure of headed studs

3.2.3.6 Material Properties At the beginning, in the middle and at the end of the test procedure of the test beams cylinder compression tests at standard cylinders according to EN 206 [12] (height 300 mm, diameter 150 mm, cured 28 days in water) were carried out. The results are shown in Table 3.17. Due to the high age of the test beams during the test procedure no increase of the concrete strength and the modulus of elasticity (taken as secant modulus according to EC 2 [33]) could be observed. In both tests structural steel beams of HEA 300 section with the material quality S460 were used. The stud shear connectors welded automatically onto the steel beam flanges had a material quality of S235 J2G3+C450. As reinforcing steel standard deformed bars with diameters of 10 mm, 12 mm and Table 3.17. Mean values of material properties of concrete according to EN 206-1 [12]

test beam

VT1, VT2

fc [N/mm2] 2

Ecm [N/mm ]

45.0 32040

232


Table 3.18. Mean values of material properties of steel members

steel beams

headed studs

longitudinal reinforcement

Ø 10 VT1 - Ø 16 VT2

transverse reinforcement

Ø 12 VT1 and VT2

yield strength, tensile strength and modulus of elasticity

member

property

VT1

VT2

fy

460

461

fu

527

531

Ea

203500

203500

fy

448

448

fu

538

538

207800

207800

Rp0.2

620

536

Rm

707

607

Es

216000

200000

Rp0.2

614

613

Rm

653

653

Es

203500

206500

Es

[N/mm2]

16 mm were used in the concrete slabs. In order to obtain detailed data about material properties tensile tests of all steel members were conducted according to the requirements of DIN EN 10002 [15]. The results of the corresponding values of each yield strength, tensile strength and modulus of elasticity are summarized in Table 3.18. 3.2.3.7 Main Results of the Beam Tests Table 3.19 gives an overview about the cyclic loading parameters, the number of load cycles, the reduced static strengths (as short time load bearing capacities) and of main deflections measured at midspan. Denotations are explained in Figure 3.105. During the static test phases the loss of the load bearing capacities near the ultimate loads was in the order of 5 % while holding the position of the actuators constant for visual inspection and checking the effects of relaxation. In the case of test beam VT1 cyclic loading caused an increase of the irreversible vertical deflections at midspan from 1.0 mm to 4.0 mm. Over the same period of time the vertical deflections at the peak load level rose from 18.9 mm to 25.4 mm. Despite these very high increments no apparent damage in the interface of steel and concrete could be observed. This changed when the load was increased up to the ultimate load. At a level of 650 kN the slab lifted from the steel flange by 0.75 mm on both sides of the load introduction area. This clearly indicated a high damage level of the studs which was also noticed near to fatigue failure in the case of the cyclic loaded push-out tests.

3.2 Experiments

233

Table 3.19. Main test results of beams VT1 and VT2 loading parameter test

Pmax

'P

[kN] [kN]

number of load cycles

reduced static strength

N

Pu,N

G iniPmax

[-]

[kN]

[mm] [mm]

[mm]

[mm]

18.9

25.4

4.0

VT1

450

265

1372194

756

VT2

250

100

2100000

625

deflections at midspan LC 0 G Pcyc G res max ,lc N

1.0

' = 17.9 18.9

5.1

' = 13.8

cyc G res

' = 21.4 22.5

7.3

' = 15.2

Gu

Gu1

Gu2

[mm] [mm] [mm] 80.0

38.6

10.1

90.0

27.9

> 27

Up to this time the maximum crack width at the bottom of the slab was 0.15 mm and the average distance between the cracks measured 12 cm. Crack formation at the bottom side of the concrete flange was finished to almost 80 % after initial loading to the peak load level. After development of a plastic hinge at midspan the beam failed at a maximum deflection of 80 mm at a load level of 756 kN caused by crushing of the concrete. After applying the initial peak load to test beam VT2 the slab was cracked nearly over the whole length between cross section 1 (QS1) and cross section 7 (QS7). The maximum crack width was 0.2 mm. The distance between two cracks measured 10 cm. Due to these cracks the subsequent reloading lead to very high irreversible vertical deflection at midspan of 5.1 mm, slightly increasing during the cyclic loading phase to 7.3 mm. In this period of time the vertical deflections at the peak load level grew from 18.9 mm to 22.5 mm. Unlike test beam VT1 the interface of steel and concrete showed no visible damage up to the end of the static test after cyclic pre-loading. The effect of repeated loading on the vertical deflections during the cyclic loading phases and the load-slip behaviour of both test beams in the subsequently performed static tests are shown in Figure 3.110 and Figure 3.111. By comparing the size of grey coloured areas surrounded by two related deflection curves in Figure 3.110 it becomes clear that the increase of the vertical deflections under the peak load level is significantly higher than the increase of the irreversible deflections. Consequently repeated loading not only causes an increase of plastic deformations but additionally a reduction in each elastic beam stiffness. In the case of test beam VT1 the reduction is in the order of approximately 20 %, in the case of test beam VT2 of approximately 10 %. This indicates that a remarkable redistribution of the inner forces had occurred. In order to allow for plastic deformations of the steel section near to midspan during the static test after cyclic pre-loading 4 transverse stiffeners were provided in a distance of 25 cm from the centre. The top flange was additionally welded to the lowest load introduction plate. At a load level of 580 kN one of the connection on side A between the top flange of the steel

234


-2.4

-3.0

-1.8

-1.2

-0.6

0.6

0.0

distance from midspan [m] 2.4 3.0 1.8

1.2

0 3.0

5

15

unloading level (10 kN) 13.8 peak load level

20 3.6

15.2 (+10%)

10

21.4 (+20%)

17.9

2.2

6.5

VT2

25

VT1

P 30

increase of deflection due to cycling loading - VT1

w 6 m

w [mm]

increase of deflection due to cycling loading - VT2

Fig. 3.110. Change of initial deflections due to cyclic loading

P [kN] 800

buckling of the top flange

756 kN

700 580 kN

600

VT 2

VT 1

625 kN

state after testing

500 VT 2

400

crushing of concrete

lifting of the slab

300 200 650 kN

100

w [mm]

0 0

20

40

60

80

100

120

756 kN

VT 1

140

Fig. 3.111. Load-deflection behaviour of test beams VT1 and VT2 in the static tests after cyclic loading

section and the load introduction plate were torn off unintentionally when the top flange began to buckle. This situation is shown in Figure 3.112 a). After this failure the composite beam was unloaded. As it can be seen in Figure 3.112 b) the top flange was subsequently straightened and the steel beam was stiffened by 4 additional massive round bars adjusted between the flanges. Although it must be mentioned that the top flange was not completely even after repairing the ultimate load bearing capacity could be significantly increased in the following static test phase. After reloading the beam failed at

3.2 Experiments

a)

235

b)

side A

(1) (2)

(1) side A

buckling of the top flange ( P ~ 580 kN)

straightened top flange and strengthened load introduction area by four massive round bars (state after first unloading)

c)

d)

side B

(1)

side A

two-sided buckling of the top flange (state after finishing the static test)

side B

buckling of the web in the load introduction area (state after finishing the static test)

Fig. 3.112. Steel section near midspan at different point of times during experimental determination of the reduced static strength after high cycle pre-loading

a maximum deflection of 90 mm at a load level of 625 kN. At this time the failure was primarily caused by local buckling of the top flange on side A between stiffener (1) and the adjacent round bar (2) followed by buckling of the top flange on the opposite side B and by buckling of the web beneath the load introduction plates (Figure 3.112 c) and d)). It cannot be excluded, that the experimental observed ultimate load was slightly affected by the first buckling at a load level of 580 kN. Because of the interaction between local stud behaviour and global beam behaviour the change of the deflections of the test beams during the cyclic loading phases decisively depends on the deterioration of the properties of the interface of steel and concrete. Analogous to the effect of cyclic loading on the behaviour of headed studs in push-out test specimens the repeated longitudinal shear forces lead to irreversible deformations at each stud and to a reduction of their elastic stiffness due to local crushing of the concrete and due to crack initiation at each stud foot. Thus the experimental observed load-bearing capacities given in Figure 3.111 are significantly affected by the stud damage and lie below corresponding ultimate load bearing capacities without any damage caused by cyclic pre-loading. The measured values of the irreversible part of the slip as well as the slip at the peak load level along each interface between the steel flange and the concrete slab at the beginning and at the end of the cyclic loading phases

236


-3.0 1.0 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1.0

-2.4

-1.8

-1.2

-0.6

0.0

0.6

1.2

distance from midspan [m] 2.4 3.0 1.8 VT 1

G

unloading level (10 kN) (f)

(c)

G (c)

(f)

peak load level (450 kN) (f): peak load level after first loading / unloading level after first loading

G [mm] 0.6 0.4 0.2 0

(c): peak load level at the end of the cyclic loading phase / unloading level after cyclic loading

VT 2

G

unloading level (10 kN) (f)

(c)

0.2 0.4 0.6

(c)

(f)

G

peak load level (250 kN)

Fig. 3.113. Slip along the interfaces of steel and concrete after first loading and after cyclic loading

can be taken from Figure 3.113. Comparable to the observations regarding the vertical deflections the increase of the slip under the peak load levels due to cyclic loading is significantly higher than the increase of the plastic slip at each unloading level. In Figure 3.114 the mean values of the crack lengths of two adjacent studs caused by the cyclic loading phases are given.

3.3 Modelling Authored by Otto T. Bruhns and G¨ unther Meschke This section contains numerical models for the description of long- and short-term damage in metallic and cementitious materials as well as in soil, developed within the Collaborative Reseacrch Center SFB 398 at Ruhr University Bochum. Following the classification of damage phenomena in Section 3.1 the structure of the section is differentiated into quasi-static and cyclic loading, in load-induced and environmentally induced damage and into ductile and brittle damage of metallic and cementitious materials as well as of soils. In Section 3.4 selected models are applied to life-time oriented finite element simulations of structures subjected to short and longterm degradation.

3.3 Modelling

237

a [mm] 35 30 stud B4 (VT2)

25

a

25

av

15

a = ah

stud B25 (VT1)

10 5

(1)

(27)

(14)

0 -2.4 -3.0 -1.8 (..) stud pair number

-1.2

-0.6

0.0

0.6

1.2

2.4 3.0 1.8 distance from midspan [m]

stu

VT1

ds

1 2, 3-1

5 4-2

VT1

Fig. 3.114. Crack lengths at the stud feet after the cyclic loading phase - Preparation stages for examination purposes

3.3.1 Load Induced Damage Authored by Otto T. Bruhns and G¨ unther Meschke 3.3.1.1 Damage in Cementitious Materials Subjected to Quasi Static Loading 3.3.1.1.1 Continuum-Based Models Authored by Tobias Pfister and G¨ unther Meschke This Subchapter provides a concise summary of continuum-based models for brittle damage of concrete subjected primarily to tensile stresses. After a short review of scalar damage models, anisotropic damage models are described. Although plasticity theory is a versatile concept for describing ductile material behavior, it is also frequently used for the modeling of the more or less ductile behavior of concrete subjected to uni- and triaxial compressive states of stresses. Hence, a concise overview over multisurface plasticity and combined plastic-damage models for concrete is provided in Subsections 3.3.1.1.1.2

238


and 3.3.1.1.1.3. Without use of regularization techniques, results from the finite element analyses exhibit a mesh dependency. For a review of existing regularization methods, the reader is referred to [141]. 3.3.1.1.1.1 Damage Mechanics-Based Models On the basis of one-dimensional damage models first proposed by [422] and [652], three-dimensional damage models were developed by [488, 489, 439] and [183]. Because of the conceptual simplicity and algorithmic robustness of these models, they are widely applied to numerical analyses of concrete despite the fact that cracks induce a significant material anisotropy. Starting with the strain energy density Ψ0 of the uncracked material, the free energy of the cracked material can be formulated as Ψ (ε, d) = (1 − d) Ψ0 (ε) = (1 − d)

1 ε : C0 : ε 2

(3.10)

with the scalar damage parameter d and the elastic constitutive tensor C 0 of the virgin material. From (3.10), using standard thermodynamic arguments, the stress tensor σ is obtained as σ = ∂ε Ψ (ε, d) = (1 − d) C 0 : ε. ! =: C

(3.11)

Frequently, the space of admissible states is controlled by the strain-like internal variable κ ≥ 0 defined in the strain space E ε := {(ε, κ) ∈ S × R + | f (ε, κ) ≤ 0} ,

(3.12)

where S is the space of symmetric second order tensors and R is the space of positive rational numbers. In Equation (3.12) f represents a failure surface. The evolution of the admissible strain space E ε is controlled by the strain-like internal variable κ. From the Kuhn-Tucker conditions, f (ε, κ) ≤ 0

κ˙ ≥ 0

κf ˙ = 0,

(3.13)

the current threshold of the equivalent strain κ is defined by the failure criterion f (ε, κ) = η(ε) − κ ⇒ κ = max {κ0 , max η(ε)} .

(3.14)

In general, the numerical implementation of material models within the finite element codes is strain driven. Hence, the algorithmic formulation of scalar damage models is relatively simple. The constitutive model is completed with the definition of a damage law d(κ) relating the equivalent strain κ to the damage parameter d: d(κ) = 0 ∀ κ ≤ κ0 , d(κ) > 0 ∀ κ > κ0 .

(3.15)

3.3 Modelling

239

If the fracture energy concept is used to avoid mesh-dependent results, d(κ) has to be related to the fracture energy Gf of concrete and to the size of the finite element [596]. For the approximation of brittle material characteristics of concrete under tensile loadings an equivalent strain corresponding to the Rankine criterion 1 max σ ˜i (ε) H(˜ σi (ε)), E

η(ε) =

(3.16)

with ˜ = C0 : ε = σ σ ˜i = 2μεi + γ

3 i=1 3

σ ˜i ni ⊗ ni ,

(3.17)

εk ,

(3.18)

i=k

making use of the Heaviside function H, can be applied (H(x) = 1 ∀x > 0, H(x) = 0 ∀x ≤ 0). The strain space illustration of Equation (3.16) is given in Figure 3.115b. A scalar damage model for the numerical analysis of concrete structures was proposed by [522]. Mazars introduced two damage parameters dt and dc , corresponding to tension and compression, respectively, to account for the different material behavior under compressive and tensile loadings. One of the key assumptions of this model is the additive decomposition of the damage variable d = dt αt + dc αc

(3.19)

1

ε2 τ

ν

2

√

ε:ε

ε2 τ2

Eτ 3κ

ε1

ε1

a) η(ε) =

ε2

b) η(ε) =

1 E

max σ ˜i H(˜ σi )

ε1

c) η(ε) =

3 i=1

ε2i H(εi)

Fig. 3.115. Representation of different failure surfaces f (η, κ) = η(ε) − κ = 0 in the principal strain space

240


2

σ11 [N/mm ]

0.0

−10.0

−20.0

−30.0 −0.005

−0.003

−0.001

ε11

0.001

Fig. 3.116. Stress-strain diagrams for uniaxial compressive and tensile loading obtained from the damage model by Mazars (Material parameters: E = 35000 N/mm2 , ν = 0.2, κ0 = 10−4 , At = 0.81, B − t = 1.045 · 104 , Ac = 1.34, B − c = 2.537 · 103 )

into a part (•)t corresponding to tensile loading and one (•)c associated with compressive states. In Equation (3.19) αt and αc represent weighting functions. The equivalent strain is defined in the format " # 3 # H(εi ) ε2i . (3.20) η(ε) = $ i=1

Figure 3.115c illustrates Equation (3.20) in the ε1 − ε2 -space. The weighting functions αt , αc are assumed to depend upon the state of the strain. The model is completed by the definition of the damage laws for dt (κ) and dt (κ) [522]. The stress-strain diagrams obtained from the analysis of concrete subjected to uniaxial tensile and compressive loading are illustrated in Figure 3.116. Several models have been proposed to extend the isotropic damage theory to capture anisotropic failure mechanisms. These models can be subdivided into formulations based on damage vectors (see [441]), formulations based on second-order damage tensors (see [523, 116, 187]) and formulations based on fourth-order damage tensors (see [604, 744, 178, 318, 60, 179]), respectively. In what follows, attention is restricted to models considering the fourth-order compliance tensor or the stiffness tensor as the fundamental internal variable. In an attempt to represent the anisotropic character of brittle failure of concrete within a continuum damage model formulated in the stress-space, [604] considered the complementary energy Ψ (σ, D , χ) =

1 σ : D : σ − Ψin (χ) 2

(3.21)

3.3 Modelling

241

D := with ζ(χ)2 = 2 ∂χ Ψin . In Equation (3.21) D is the compliance tensor (D C −1 ) and ζ(χ) a stress-like internal variable. From standard arguments of thermomechanics follows ε = ∂σ Ψ = D : σ.

(3.22)

From the rate form of Equation (3.22), an additive split of the strain rate ε˙ into an elastic part ε˙ e and an inelastic part ε˙ i results in ε˙ = D : σ˙ + D˙ : σ =: ε˙ e + ε˙ i .

(3.23)

One key point of this concrete model is the assumption of an additive split of the compliance tensor D D = D0 + D c

(3.24)

into the compliance tensor D 0 of the virgin material and the damage tensor D c associated with additional flexibility corresponding to active microcracks. Consequently, the total strains ε can be re-written into the format

ε = D 0 + D c : σ =: ε0 + εc . (3.25) Crack closure is taken into account by the restriction that the eigenvalues of εc must be positive. Tensile and compressive portions of the stresses can be re-written in the format σ+ = P+ : σ

and σ − = P − : σ.

(3.26)

using the projection tensors P+ = −

P =

3 k=1 3

H(σk ) nk ⊗ nk ⊗ nk ⊗ nk , (3.27) H(−σk ) nk ⊗ nk ⊗ nk ⊗ nk .

k=1

The space of admissible states is the stress space E σ = {(σ, χ) ∈ S × R | f (σ, χ) ≤ 0} ,

(3.28)

where the failure surface is defined as [604] f (σ, χ) =

1 1 1 + σ : σ + + c σ − : σ − − ζ(χ)2 . 2 2 2

(3.29)

In Equation (3.29), the parameter c represents a coefficient accounting for the cross-effect between compression and tension. The failure function (3.29)

242


σ2 q(α) σ1

elastic region √ q(α)/ c

Fig. 3.117. Anisotropic damage model by [604]: Illustration of the failure surface in the principal stress space, see eq. (3.29)

is illustrated in Figure 3.117. The dependence of the stress like internal variable ζ(χ) can be defined on the basis of uniaxial tensile tests [604]. From Equation (3.29) the evolution of the compliance tensor is derived exploiting the postulate of maximum dissipation. Adopting the idea to directly include the stiffness tensor (or as in [604] the compliance tensor) as arguments within the function of free energy, an anisotropic damage model was proposed by [318]. The model is based on the free energy Ψ (ε, C , χ) =

1 ε : C : ε + Ψin (χ). 2

(3.30)

The admissible stress space E σ is defined by a set of damage functions fk of the form fk = σkeq (σ) − fk + ζk (χ)

(3.31)

with ∂σ σkeq : σ = σkeq (σ) k ∈ {1, 2, · · · M }.

(3.32)

The evolution equations for the compliance tensor D and the internal variable χ are obtained in an associated format as D˙ =

M k=1

γk

∂σ fk ⊗ ∂σ fk , ∂σ fk : σ

χ˙ =

M k=1

γk ∂ζk fk .

(3.33)

3.3 Modelling

tm

243

t

m

tn n

crack Fig. 3.118. Concrete specimen with a crack: Definition of a local coordinate system by means of the crack normal vector n and the tangential vector m. Decomposition of the traction vector t = σ · n into the normal part tn = (n ⊗ n) : σ and the tangential part tm = (n ⊗ m) : σ

In Equation (3.31), the failure surfaces are defined as f1 (σ, χ) =

(n ⊗ n) : σ

−fn + kn ζ(χ),

f2 (σ, χ) =|(n ⊗ m1 )sym : σ| −fs + ks ζ(χ),

(3.34)

f3 (σ, χ) =|(n ⊗ m2 )sym : σ| −fs + ks ζ(χ), with n representing the normal vector of the crack surface and mα two vectors orthogonal to n (m1 × m2 = n and mα · n = 0) (see Fig 3.118). fn and fs denote the tensile and the shear strength, kn a parameter which defines the residual tensile strength and ks the residual shear strength, respectively. Since the vectors n and mα are assumed as constant during the deformation, the model falls within the class of fixed crack approaches (see [381]). Since t = σ·n, the failure surfaces (3.34) control all components of the traction vector t. For a detailed comparison between different interface models, reference is made to [434]. The integration of the rate of the stress tensor %

σ˙ = C : ε˙ −

&

M

γk ∂σ fk

(3.35)

k=1

within a time interval [tn , tn+1 ] by means of an implicit backward Euler scheme yields an algorithmic formulation σ n+1 = σ trial n+1 −

M

Δγk

C n : ∂σ fk ,

(3.36)

k=1

which is formally identical to standard plasticity. Hence, the standard return mapping algorithm can be used for the integration of the differential equations. In Equation (3.36) the definition of the trial stress tensor

244


σ trial n+1 = C n : εn+1

(3.37)

at the pseudo time tn+1 is used. The update of the compliance tensor results in ' M ∂σ fk ⊗ ∂σ fk '' D n+1 = D n + Δγk . (3.38) ∂σ fk : σ 'n+1 k=1

3.3.1.1.1.2 Elastoplastic Models In plasticity theory, cracking of hydrates is described by means of inelastic strains at the macroscale, denoted as εp . In addition, internal variables χ are employed to monitor evolution of inelastic mechanisms occurring at the microlevel. They represent microstructural changes caused by cracking of concrete. The energetically conjugated thermodynamic quantities are the hardening/softening forces ζ. They are related to the internal variables via the state equation ζ = ζ(χ). The hardening forces represent the actual strength of the material, defining the space of admissible stress states, C E . In case of multisurface plasticity, C E is defined by M yield surfaces: σ ∈ C E ⇔ fk = fk (σ, ζ(χ)) ≤ 0

(3.39)

for k ∈ [1, 2, . . . , M ], where fk denotes the k-th yield function. The internal variables and the tensor of plastic strains are obtained by means of evolution equations, reading ˙ = χ

k∈Jact

γ˙ k

∂hk , ∂ζ

ε˙ p =

k∈Jact

γ˙ k

∂gk , ∂σ

(3.40)

where γk denotes the plastic multiplier of the k-th yield function. gk and hk are potentials which generally depend on σ and ζ. In the Equations (3.40), Jact denotes the set of active yield surfaces. It is defined as Jact := {k ∈ [1, 2, . . . , M ]|fk (σ, ζ) = 0} .

(3.41)

A wide range of plasticity models for concrete (see, e.g., [339, 648, 276, 531, 166, 381] exists and the reader is referred to the respective references. A model representative for the class of single surface models applicable to the simulation of failure of concrete under triaxial stress states is the Extended Leon Model (ELM) [264]. Detailed information on the model and its algorithmic treatment is contained in [639], [516]. 3.3.1.1.1.3 Coupled Elastoplastic-Damage Models Failure of concrete in tension as well as in compression for low levels of confinement is associated with stiffness degradation as well as with inelastic deformations. To take account of both phenomena models characterized by a coupling between damage and plasticity theory have been developed.

3.3 Modelling

245

The simplest mode of coupling between damage and plasticity is a scalar damage elastoplastic model [420] based on the effective stress concept. Considering plastic strains in Equation (3.10) yields the stress-strain relation σ = (1 − d(κ)) C e : (ε − εp ) .

(3.42)

According to the effective stress concept, the plastic yield function fp is forˆ mulated in terms of effective stresses σ: σ ˆ − ζ(χ) with σ ˆ= fp = σeq (σ) . (3.43) 1−d The evolution of the damage parameter is controlled by a damage surface formulated in the strain space fd = η(ε) − κ.

(3.44)

An anisotropic elastoplastic-damage model for plain concrete using the principle of maximum dissipation has been proposed by Meschke, Lackner & Mang [534]. In contrast to [420], the stiffness degradation is taken into account by means of additional cracking strains. Conceptually, this model is an extension of the anisotropic damage model proposed by [318]. In contrast to [318], a rotating crack formulation is considered in [534]. The model is based on a function of free energy of the form C, ε, εp , χ) = Ψ (C

1 (ε − εp ) : C : (ε − εp ) − Ψin (χ). 2

(3.45)

From the postulate of maximum dissipation, the evolution equations are obtained as ε˙ p + D˙ : σ =

M

γk ∂σ fk (σ, χ)

(3.46)

k=1

and χ˙ =

M

γk ∂qk fk (σ, χ),

(3.47)

k=1

respectively. With the introduction of the rate of damage strains, ε˙ d := D˙ : σ, and a scalar coupling parameter β ∈ [0, 1], damage-induced and plastic strains are additively decomposed into ε˙ p = (1 − β) d

ε˙ = β

M k=1

M

γk ∂σ fk (σ, χ),

k=1

γk ∂σ fk (σ, χ),

(3.48)

246


σ

σ

ε

β = 0.5

β = 1.0

ε

Fig. 3.119. Anisotropic elastoplastic damage model by [534]: Influence of the scalar coupling parameter β on the stress-strain diagram obtained from cyclic uniaxial tensile loading

with β ∈ [0, 1]. Since the governing equations of the coupled model are formally identical with plasticity models, the standard return mapping can be used for the algorithmic formulation. Figure 3.119 illustrates the influence of the coupling parameter β on the un- and re-loading paths. The limit case (β = 1.0) corresponds to a damage model, ignoring plastic deformations. For β = 0.5, permanent deformation as well as stiffness degradation are taken into account. 3.3.1.1.1.4 Multisurface Elastoplastic-Damage Model for Concrete To account for brittle material response under tension and for the more ductile mode of damage under compression a 3D elastoplastic multisurface model for concrete is described in this Subsection. The model is designed such that the parameters can be derived from standard uniaxial tests or estimated on basis of the compressive strength [641, 444]. Applications and enhancements can be found in [133, 622, 628]. The model is characterized by a Drucker-Prager potential active in case of predominantely compressive states of stresses combined with Rankine damage surfaces governing the softening response in tension. The evolution laws for both loading states are derived from the free Helmholtz energies. In accordance with the proposal made in [318] the free energy ψc is defined for compression as ρ0 ψc (ε, εp , D d , χc ) =

−1 1 [ε − εp,c ] : D 0 + D d : [ε − εp,c ] + ρ0 ψc,in (χc ) 2 (3.49)

and for tension as ρ0 ψt (ε, D d,t , χt ) =

−1 1 ε : D 0 + D d,t : ε + ρ0 ψt,in (χt ). 2

(3.50)

3.3 Modelling

Drucker-Prager

247

−σII σIII Rankine Drucker-Prager

−σI

−σIII Rankine

σI

Fig. 3.120. Multisurface model for concrete: Yield and failure surfaces in the stress space

The compliance relation reads −1 σ = D 0 + D d + D d,t : [ε − εp,c ] .

(3.51)

The evolution laws for inelastic strains, for the damage compliance tensor and for the internal variables will be introduced in the following, separately for concrete under compression and under tension. The Drucker-Prager potential (see Fig. 3.120) and its derivatives are given as ( ) 1 φc (σ, ζ) = 1 μ I1 + J2 − ζc (χc ) , (3.52) √ −μ 3 ∂φc = ∂σ

1 s + sT √ , μ 1 + √1 − μ 4 J2 3

∂φc = −1 . ∂ζc

(3.53)

Using an associated flow rule and the split of inelastic strains into damaging and plastic parts according to [534], the evolution laws for the plastic and the damaging strains result in ε˙ p,c = β γ˙ c

∂ φc , ∂σ

ε˙ d,c = (1 − β) γ˙ c

d,c ∂ φc = D˙ : σ. ∂σ

(3.54)

The latter relation is the basis for the formulation of the rate of the damage d,c compliance tensor D˙ : d,c μ 1 β 1 ¯ ˙ 1⊗1+ √ I + I γ˙ c , − √ (3.55) D = 1 √ I1 6 J2 4 J2 3−μ

248


where isotropic damage evolution is assumed. Exploiting the symmetry of this tensor, which results from the postulation of the free Helmholtz energy according to Eq. (3.49), this fourth order tensor can be expressed by means of two independent variables: d,c d,c d,c D˙ I + ¯I . = D˙ s1 1 ⊗ 1 + D˙ s2

(3.56)

The evolution laws of the scalar variables are found by comparison of the coefficients of Eq. (3.55) and Eq. (3.56) as μ 1 β 1 β d,c d,c √ D˙ s1 γ˙ c , D˙ s2 γ˙ c . = 1 − √ = 1 (3.57) √ − μ I1 √ − μ 4 J2 6 J2 3 3 The rate of the internal variable χc is redefined in the format √1 − μ 2 2β 3 . χ˙ ∗c = γ˙ c with ψ(σ) = μ 1 √ ψ(σ) + I1 3 J

(3.58)

2

This definition enables the interpretation of the rate of the internal variable χ˙ ∗c as the rate of the uniaxial compliance χ˙ ∗c = D˙ d,c,1d ,

(3.59)

which is the basis for the derivation of the evolution law for the internal variable from the uniaxial stress-strain relation. Using the parameter b, the uniaxial inelastic strains may be decomposed into plastic and damaging parts: εp,c = b εin ,

εd,c = (1 − b) εin .

(3.60)

Together with Eq.3.59 the uniaxial stress-strain relation can be expressed as 1 1 ∗ ε= + χ σ. Ec 1−b c

(3.61)

In this equation, σ can be replaced by any stress-strain relation σ(ε) for concrete and the resulting expression can then be re-arranged to obtain a relation ε(χ∗c ). Inserted this relation into the given expression σ(ε), the relation ζc (χ∗c ) = σ(ε(χ∗c ))

(3.62)

can be found. The softening/hardening modulus then reads Hc =

dζc dσ dσ dε = = = dχ∗c dχ∗c dε dχ∗c

(1 − b) 1 −

ζc . dσ 1 χ∗c + Ec 1 − b dε

(3.63)

3.3 Modelling

2

1

3

2

stress σ

stress −σ

1

249

εp,c

εd,c

εel

εd,t

εel

strain −ε

strain ε

Fig. 3.121. Stress-strain relation of concrete in compression and in tension

The relation for the scalar variable β, which divides the inelastic strain rate into a rate for the plastic and the damaging strains, respectively, reads β(χ∗c ) = 1 +

−1 χ∗ dζc 1 1+ c . 1−b ζc dχ∗c

(3.64)

A particular stress-strain relation will be introduced in the following. It is subdivided into a linear elastic part for stresses below 13 of the compressive strength, a hardening part, which is a slight modification of the relation suggested in [182] and a softening part. This is illustrated in the left diagram in Fig. 3.121. The equations σ(ε) for these three domains read as follows:

σc1 = Ec ε ,

σc2

with Eci =

1 2 Ec

fc εc

σc3

2

−

2 Eci fεc + εεc = ε c 1 − Eci fcε−2 εc fc

fc 3 + Ec , εc 2

2 + γc f c ε c γc = + γc ε + 2 γc 2 εc ε2

with γc =

( 2 gcl,e −

(3.65)

(3.66)

−1

π 2 fc εc )2 . fc 1 ε (1 − b) + b c 2 Ec

(3.67)

(3.68)

250


In the softening domain, γc adjusts the area under the stress-strain relation. It is a function of the localised crushing energy gcl,e , which has to be adjusted to the element size in order to avoid ill-posedness and mesh dependency of the simulation results due to localisation. The equation for the Rankine damage potential for the tensile domain reads φt,i (σ, ζ t ) = ξi (σ, ζ t ) − fct ≤ 0 ,

(3.69)

as illustrated in Fig. 3.120. ζ t is called the back-stress tensor, that quantifies the internal (softening) state of the material. ξ is the difference between the stress and the back-stress tensor: ξ = σ − ζt ,

(3.70)

ξi is its projection into the ith crack direction: ξi = ξ : M i .

(3.71)

The crack direction is, different from the original description of the model in [641, 444], assumed to be determined from the strain tensor ε [628]. Thus, the eigenvalue basis M i is derived as the dyadic product of the eigenvectors xi of ε: M i = xi ⊗ xi .

(3.72)

The derivatives of the Rankine potential with respect to the stress and the back-stress tensor result in ∂φt,i = Mi , ∂σ

∂φt,i = −M i . ∂ζ t

(3.73)

In tension, the inelastic strains are assumed only to be associated with damage processes. The evolution law for the damaging strains εda and the internal variable χt read: ε˙ d,t =

3

γ˙ t,i

i=1

˙t = χ

3 i=1

γ˙ t,i

d,t ∂φt,i = D˙ : σ, ∂σ

∂φt,i . ∂ζ t

(3.74)

(3.75)

The evolution law of the back-stress tensor ζ t is derived from its total differential dζ t ∂φt,i dζt ˙t = ζ˙ t = :χ : γ˙ t,i . dχt dχt ∂ζ t i=1 3

(3.76)

3.3 Modelling

251

As the evolution of the softening behaviour is formulated by means of a kinematic softening law independently for each crack, the fourth order tensor dζ t /dχt can be replaced by three scalar softening moduli Ht,i , which will be derived from a uniaxial stress-strain relation. The evolution law for the back-stress tensor than simplifies to ζ˙ t =

3

∂φt,i γ˙ t,i = −Ht,i M i γt,i . ∂ζ t i=1 3

Ht,i

i=1

(3.77)

This relation can be transformed into the crack directions, so that the scalar values of ζ t result in ζ˙t,i = −Ht,i γ˙ t,i = Ht,i χ˙ t,i .

(3.78)

Hence, the evolution of ζ t can be evaluated separately for each crack direction with scalar variables, which then are assembled to the tensor ζ t using Eq. (3.77). The evolution law for the damage compliance tensor reads d,t D˙ =

3 1 γt,i M i ⊗ M i σ i=1 i

with σi = σ : M i .

(3.79)

This relation has also been used in [318]. It satisfies Eq. (3.74) as well the major symmetries of D d,t . The softening behaviour is again derived from uniaxial stress-strain relations. To this end, the uniaxial strain rate is expressed as 1 ε 1 ε dε ε˙ = − = − σ˙ ⇔ . (3.80) σ Ht dσ σ Ht In this formulation, an arbitrary stress-strain relation σ(ε) for concrete under tension is applied and the expression is rearranged for Ht . A specific stressstrain relation will be introduced in the following. It is decomposed into a linear elastic part before and an exponential softening part after the peak (see the diagram on the right hand side of Fig. 3.121): σt1 = Ec ε ,

1

σt2 = fct e γt

(εct −ε)

.

(3.81)

In analogy to the formulation of the softening branch in compression, the variable γt adjusts the area under the curve according to the fracture energy gfl,e and the element size in order to avoid mesh dependent solutions: γt =

gfl,e 1 fct − . fvt 2 Ec

(3.82)

252


3.3.1.1.2 Embedded Crack Models Since the early 1990s, attempts were made to represent the discontinuous character of cracks by incorporating the failure kinematics directly into the finite element formulation. From a mathematical point of view, the crack width can be interpreted as a jump of the displacement field. This idea has been pioneered by Dvorkin, Cuitino & Gioia [248], who enriched a two-dimensional finite element by a kinematics representing a discontinuous displacement field. These approaches account for the multiscale character of the problem (the failure zone is several dimensions smaller than the structure). They allow to use relatively large elements compared to the width of the localization zone. Hence, these methods are suitable for large scale structural applications. Enhanced finite element models considering discrete crack propagation can generally be categorized into element-based formulations, generally denoted as Embedded Crack Models (see [746, 598, 413, 59, 244], among others) and nodal-based formulations, e.g. the Extended Finite Element Method (X-FEM) (see [545, 544, 828, 243, 532]). For a comparative assessment of both approaches we refer to [412, 244]. In the Strong Discontinuity Approach (SDA), suggested by Simo, Oliver & Armero [746] and further elaborated in [597, 58, 671, 413, 559], among many others, the failure kinematics of solids, i.e. crack opening in brittle materials or sliding along shear failure zones in metallic materials, is approximated by means of discontinuous displacement fields locally embedded within the finite elements undergoing localization. With the exception of the “rotating” crack formulation of the Strong Discontinuity Approach [559], the topology of crack segments is held fixed once they are signaled to open. Furthermore in the framework of the Extended Finite Element Method crack path continuity is required. Hence, the correct prediction of the direction of new crack segments is crucial for the reliability as well as for the robustness of the numerical analysis. If the predicted propagation direction is incorrect, locking occurs, which generally leads to unreasonable results and eventually may cause failure of the analysis. A discrete representation of cohesive cracks based on a relation between the crack opening w and the normal component σn of the traction vector t was formulated in the context of the fictitious crack concept proposed by [369]. Starting with the definition of the fracture energy w max

Gf =

σn dw

(3.83)

w0 =0

in terms of σn of the traction vector t of a crack surface and the crack width w, a linear softening law of the form σn (w) was postulated. The underlying idea is illustrated in Figure 3.122. Since the softening law is formulated in terms of the traction vector and the crack displacement, the resulting load displacement diagram obtained from a computation is independent of the refinement of the respective finite element mesh.

3.3 Modelling

253

t

u

Gf

crack

Fig. 3.122. Discrete representation of cracks: Traction separation law of the format t = t( u ) across the crack surface

The so-called Strong Discontinuity Approach (SDA) was proposed by [746]. In this concept, the localization zone is represented as a surface of discontinuous displacements within the respective finite elements. This method accounts for the multiscale character of the problem. In the SDA, displacement jumps are embedded locally in the respective finite elements without affecting neighboring elements. In accordance with the underlying enhanced assumed strain (EAS) concept [747], only the (enhanced) strains resulting from the discontinuous displacement field appear explicitely in the formulation. The idea to enhance standard finite element models by additional modes to capture displacement jumps has been already suggested in the early work [417]. Since then, several variants of this concept have been proposed (see [105, 248, 435]). A more detailed insight into the incorporation of strong displacement discontinuities into classical (local) elastoplastic and elasto-damage continuum models was provided by [746]. This formulation corresponds to a Petrov-Galerkin method in the sense that the weak form involves test functions different from the variations of the enhanced strains. Details of the corresponding algorithmic formulations are given in [598, 57, 112, 757, 413] for constant strain elements and in [671, 599, 559] for the more general case. Based on the assumption of a jump in the displacement field across the crack surface, an additive decomposition of the displacement field ¯ (x) + u ˆ (x), u(x) = u

∀ x ∈ Ω,

(3.84)

with

ˆ (x) = u MS (x). u

(3.85)

254


u¯

u

uˆ u

x

x

x

Fig. 3.123. Strong Discontinuity Approach: Additive decompositionofthe dis¯ and a part u ˆ containing a jump u (Equaplacement field into a smooth part u tion (3.84))

u

¯ u

∞ x

ˆ u

x

∞ x

Fig. 3.124. Strong Discontinuity Approach: Strain field resulting from the displace¯ (x) + u ˆ (x) ment field u(x) = u

¯ (x) and a jump term u ˆ (x) is motivated. The function into a regular part u MS (x) can be decomposed into a Heaviside function HS (x) and a smooth function ϕ(x) MS (x) = HS (x) − ϕ(x),

∀ x ∈ Ω.

(3.86)

Figure 3.123 illustrates the additive decomposition of the displacement field according to Equation (3.84). Restricting the model to the geometrically linear theory and assuming ∇ u = 0 (see [139]), the strain field is obtained as ¯ + ∇sym u ˆ = ∇sym u ¯ − ˜ε, ε = ∇sym u = ∇sym u

(3.87)

with ˜ε =

sym sym u ⊗ ∇ϕ − u ⊗n δs ,

(3.88)

see Figure 3.124. In Equation (3.88) n represents the normal vector of the crack surface and δs is the Dirac-delta distribution. The displacement discontinuities u follow from a traction separation law t|∂s Ω = ts ( u ) to be defined along the crack surface ∂s Ω. From traction continuity follows the compatibility condition

3.3 Modelling

ts ( u ) = t+ = σ + · n.

255

(3.89)

The finite element formulation of the SDA according to [745] is based on the weak form of the equilibrium equations divσ + b = 0 ⇒ BT σ dV = r (3.90) Ω

as well as on the weak form of the traction equilibrium GT σ dV = 0. σ + · n = ts ( u ) ⇒

(3.91)

Ω

According to Equation (3.91), the softening behavior is controlled by means of a traction separation law ts ( u ). From Equations (3.90) and (3.91) the analogy with the enhanced assumed strain concept becomes obvious. An alternative implementation was proposed by [139] and [559], where the amplitude of the displacement jump is condensed out on the material level leaving the element routines of the 3D formulation unaffected. Consequently, the standard return mapping algorithms can be applied for the integration of the governing equations [560]. 3.3.1.2 Cyclic Loading 3.3.1.2.1 Mechanism-Oriented Simulation of Low Cycle Fatigue of Metallic Structures Authored by Jan-Hendrik Hommel and G¨ unther Meschke Classical phenomenological approaches for an assessment of damage accumulation of metallic structures subjected to fatigue loading are based on ¨ hler-type relations between the number of cycles to failure and quantiWo ties such as stress, strain amplitudes or amplitudes of dissipated energy (see [257] for an overview). In contrast, model-based approaches are based on constitutive models accounting for the accumulation of damage on a macroscopic level by means of appropriate internal variables representing the growth of damage at the micro-level (e.g. [186]). In [389, 388, 390] a mechanism-oriented concept for life-time prediction of metallic structures under (ultra) low cycle fatigue loading is proposed. It is characterized by combining two different strategies for the modelling of damage accumulation in highly plasticized zones of metallic materials related to two different levels of observation: One approach is based on unit-cell analyses representing, in a simplified manner, the microstructure characterized by a regular distribution of microvoids embedded within the matrix material. The matrix material is modelled by means of a sufficiently sophisticated elasto-plastic material model suitable for the representation of large plastic deformations under cyclic loading conditions as described in Section 3.3.1.2.1.1.

256


This model has to allow for the simulation of all relevant phenomena associated with cyclic plasticity such as the Bauschinger-effect, ratcheting or mean stress relaxation, cyclic hardening or softening. In addition to a nonlinear isotropic hardening rule for the modelling of cyclic plasticity a BariHassan-type kinematic hardening law [87], which represents a superposition of several kinematic hardening laws according to Armstrong-Frederick [62], is implemented in a finite deformation continuum mechanics framework. At the scale of the microstructure the growth of the initially spherical micropore represents the accumulated damage. On a macroscopic level, a Gursontype micropore damage model is employed to represent evolving damage. This micropore damage model is extended to cyclic loading and combined with isotropic and kinematic hardening similar to [63]. Within the presented life-time concept for metallic structures failure analyses are based upon the micropore damage model at the macro-level, which is calibrated to the specific characteristics of the metallic alloy by means of unit cell analyses performed on the micro-level [386]. The computational concept can be divided into three parts as shown in Figure 3.125. •

•

•

At first, numerical analyses of the material under investigation subjected to monotonic, constant stress triaxiality loading are performed both by means of a unit-cell model as well as by means of the macroscopic model to calibrate the parameters of the Gurson-type micropore model. This procedure is demonstrated in [386] for low alloy steel typically used for structures such as pressure vessels and is not part of this paper. After the calibration procedure, the micropore damage model is validated by means of typical fatigue tests, e.g. on notched specimens. To this end, results from cyclic tests on two different round notched bars performed by [638] are re-analyzed in Section 3.3.1.2.1.2. To study the damage evolution due to low cycle fatigue, the results are compared in terms of the degradation of the peak reaction force. Finally, the macroscopic micropore damage model is used for finite element analyses of a spherical pressure vessel supported by cylindrical columns subjected to earthquake loading (Section 3.4), which represents a typical loading scenario that may lead to structural failure induced by low cycle fatigue at highly stressed locations of the structures.

3.3.1.2.1.1 Macroscopic Elasto-Plastic Damage Model for Cyclic Loading The finite strain elasto-plastic damage model is based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts F=Fe Fp . The elastic material response is described by means of the compressible Neo-Hooke-model [750]. The Gurson-type yield condition, introduced by [332] and modified by [482], is formulated in the intermediate configuration ˆ2 ˆm Σ Σ 3 ln(f ∗ qA ) eq ∗ F = 2 + 2 f qA cosh qB − 1 − (f ∗ qA )2 ≤ 0 , (3.92) 2 ln(f ) qˆ ¯ qˆ ¯

3.3 Modelling

257

(I ) m ic r o -s tr u c tu r e : d e te r m in a tio n o f m o d e l p a r a m e te r s (i) e la s tic m o d e l p a r a m e te r s fro m

(ii) p la s tic m o d e l p a r a m e te r s e x p e rim e n ta l a n a ly s is

d a ta ta k e n lite ra tu re

s

s

e e

e la s to -p la s tic c a lc u la tio n c a lib ra tio n

s e

(iii) d a m a g e m o d e l p a r a m e te r s e la s to -p la s tic u n it c e ll a n a ly s is h o m o g e n is e d d a m a g e a n a ly s is v o id v o lu m e fra c tio n f(t)

c a lib ra tio n

v o id v o lu m e fra c tio n f(t)

(I I ) m a c r o -s tr u c tu r e : v a lid a tio n o f m o d e l p a r a m e te r s s p e c im e n te s t

e x p e rim e n ta l a n a ly s is lo a d c y c le s to fa ilu re s -e c u rv e

v a lid a tio n

n u m e ric a l a n a ly s is lo a d c y c le s to fa ilu re s -e c u rv e

(I I I ) e n g in e e r in g -s tr u c tu r e : life tim e a s s e s s m e n t g e o m e try &

lo a d in g

lo a d

s tr u c tu r a l a n a ly s is n u m e ric a l a n a ly s is

tim e

lo a d c y c le s to fa ilu re

Fig. 3.125. Model-based concept for life time assessment of metallic structures

ˆ eq = (G( ˆ Σ ˆ −κ ˆm = (BG( ˆ Σ ˆ −κ ˆ ˆ¯ ))m denotes the equivalent where Σ ¯ ))eq and Σ and the mean part of the relative stress, respectively. In (3.92) qA and qB represent the material parameters and the scalar damage variable f ∗ describes ˆ G ˆ −1 BC ˆ is a symmetric Mandel-type stress the void volume fraction. Σ= ˆS eT e ˆ tensor where BC ˆ =GF F is the elastic right Cauchy-Green-tensor and p ˆ = Fp G (C ) denotes the metric of the intermediate configuration. Note that the deviatoric part of (3.92) reduces for vanishing f ∗ to a von Mises-yield condition. The additional term ln(f ∗ qA )/ln(f ) introduced by

258


[386] within the mean part of the yield condition is a scaling factor, that leads to total material softening for f ∗ qA =1. Nonlinear isotropic hardening is considered by the following relation ˆ + β (σY∞ − σY0 )(1 − exp(−δiso α))] ˆ , qˆ ¯ = J p J e [σY0 + β Hiso α

(3.93)

ˆ ]/[(1 − f ∗ qA ) qˆ ˆ−κ ˆ ¯) : N ¯] describes the evolution of the scalar where α= ˆ˙ γ[( ˙ Σ isotropic hardening parameter α ˆ and the determinants J e and J p transform from Cauchy- to Kirchhoff-stresses. Cyclic loading is accounted for by an advanced kinematic hardening model [87] using a superposition of at most four 4 ˆ ˆ i with the following kinematic hardening tensors κ ¯ =J p J e (1 − f ∗ qA ) i=1 κ assumption for the material time derivative of the back stress tensor ˜ˆ D ˆ p − b ζ δ γ˙ κ ˆp ˆ + (1 − δ )(ˆ κ ˆ˙ = (1 − β) c D , (3.94) κ : N) i

i

i

kin

i

kin

i

κ ¯ ˆ˜ is the ˆ p =γ˙ F, ˆ > for i=4, D =γ˙ N ˆ) (Σ −κ ||ˆ κ4 || ˆ˜ describes the symmetric ˆ p =G ˆ −1 D ˆ pG ˆ −1 , N symmetric plastic strain rate, D ˆ ) is the norm of the back gradient of the yield surface, ||ˆ κ||= 3/2 tr(ˆ κκ ˆ ˆ κ ˆ and κ ˆ =Gˆ κG, ¯ , δkin are model parameters and β controls stress tensor κ the decomposition of isotropic and kinematic hardening. The evolution of the void volume fraction f is described by

where ζ=1 for i=1,2,3 or < 1 −

ˆ pG ˆ −1 ) + f˙nucl , f˙ = f˙growth + f˙nucl = qC (1 − f ) tr(D

(3.95)

∗

which is related to f in (3.92) according to (3.98). Note that an additional material coefficient qC is introduced in (3.95), which is necessary to calibrate the Gurson model according to the results from unit cell analyses [387]. f˙nucl in (3.95) represents a nucleation law according to [197] with % p 2 & − 1 f n n ˙p , exp − (3.96) f˙nucl = √ 2 sn sn 2 π where fn , sn and n are model parameters and ˙p is given by p ˆ pG ˆ −1 )2 : Î . ˙ = 3/2 (D

(3.97)

To describe the physical process of void nucleation adequately, the evolution of ˙p is only defined for loading. In case of unloading no nucleation of micropores is considered. For the consideration of the coalescence of the micropores the phenomenological law according to [797] is used, ⎧ ⎪ ⎨ f for f ≤ fc f∗ = ⎪ ⎩ fc + K(f − fc ) for f > fc (3.98) with

K=

fu∗ − fc 1 and fu∗ = , ff − fc qA

3.3 Modelling

259

Fig. 3.126. Numerical and experimental data for (a) material softening and (b) ratcheting effect

wherein f is transformed into f ∗ and an acceleration of the evolution of f is driven by a scalar factor K defined by fc and ff according to (3.98). In addition to the void volume fraction f ∗ , an additional variable S is used to characterize the void shape. The evolution of S is described by an equation proposed by [258]. It is implemented into the Gurson-model by a modification of the material parameter qB [386]. The implementation of the model is based upon the return-map algorithm and a consistent linearization procedure [743]. Because of the anisotropy induced by the kinematic hardening, the iterative solution involves 8 unknowns ˆ ˜ the plastic multiplier γ˙ and (the components of the symmetric gradient N, the void volume fraction f ). The proposed macroscopic elasto-plastic damage model has the ability to replicate all typical phenomena of cyclic plasticity such as the Bauschingereffect, ratcheting or mean stress relaxation, cyclic hardening or softening [386]. In the following, a comparison of numerical and experimental data shows the efficiency in case of simulating the effects of material softening and ratcheting. To this end, a cyclically loaded hollow cylindrical specimen of CS 1026 [86] is re-analysed numerically. Using the isotropic hardening law according to (3.93) the stress amplitude of the first 25 load cycles can be simulated in good agreement to the experimental results, as Figure 3.126(a) shows. The use of the Bari-Hassan-type of kinematic hardening rule allows for the simulation of the ratcheting effect, which is demonstrated by the evolution of the radial strain in case of biaxial loading in Figure 3.126(b). 3.3.1.2.1.2 Model Validation The following analysis are performed for 20MnMoNi55, a low alloy steel typically used for structures such as pressure vessels. For this special type of material a calibration leads to the model parameters presented in Table 3.20 [386]. After the calibration procedure, the micropore damage model is validated according to Figure 3.125 by means of fatigue tests. Therefore, results from

260


Table 3.20. Parameter of the elasto-plastic micropore damage model for 20MnMoNi55 E=204 [GPa]

β=0.5

ν=0.3 [-] b1 =25000 [-]

σY0 =220 [MPa] δiso =25 [-] σY∞ =410 [MPa] Hiso =0 [MPa]

b2 =500 [-]

b3 =5 [-]

b4 =5000 [-]

δkin =0.18 [-]

c1 =500000 [MPa] c2 =60000 [MPa] c3 =3000 [MPa] c4 =100000 [MPa] κ ˘ =0 [MPa] f0 =0.01 [-]

S0 =0.0 [-]

qA =1.85 [-]

qB =0.48 [-]

qC =1.4 [-]

fn =0.08 [-]

n =3.0 [-]

sn =1.0 [-]

fkrit =0.09 [-]

fBruch =0.14 [-]

Fig. 3.127. Low Cycle Fatigue in metals: Numerical and experimental results for cyclically loaded round notched bar with (a) 2mm notch radius and (b) 10mm notch radius

cyclically loaded round notched bars with two different notch radii (2mm and 10mm) performed by [638] are re-analyzed. To study the damage evolution due to low cycle fatigue, the results are compared in terms of the degradation of the peak reaction force in Figure 3.127. The chosen set of material specific model parameters result in a good agreement of experimental and numerical results. In particular the strong change of the slope of the reaction force curve, when coalescence becomes the dominant damage mechanism, is simulated by the micropore damage model in a good manner for both cases. This final change state can be correlated to the life time of the structure, which is reasonably well predicted (see Table 3.21). It should be noted that he numerical simulations also allow a localization of the position of damage accumulation in accordance with the experimental observations. For the smaller notch radius the micropore damage initiates and starts to accumulate from the notch root (Fig.3.128(a,b)). In contrast, for the specimen with the larger notch radius a nearly homogeneous damage accumulation initiating from the interior of the specimen is observed(Fig.3.128(c,d)).

3.3 Modelling

261

Table 3.21. Low Cycle Fatigue in metals: Number of load cycles until failure obtained from numerical simulations and experiments 2 mm notch radius

10 mm notch radius

Experiment

Num. Model

Experiment

Num. Model

Failure initiation

17 cycles

24 cycles

31 cycles

35 cycles

Life-time

23 cycles

28 cycles

43 cycles

40 cycles

(a )

(b )

(c )

(d )

Fig. 3.128. Low Cycle Fatigue in metals: Damage accumulation and numerically predicted damage in a cyclically loaded round notched bar: (a,b) 2 mm notch radius, (c,d) 10 mm notch radius

3.3.1.2.2 Quasi-Brittle Damage in Materials 3.3.1.2.2.1 Cementitious Materials Authored by Tobias Pfister and Friedhelm Stangenberg Concept In high-cycle fatigue processes, a large number of load cycles under moderate stress level leads to increase of strains and degradation of material properties. Different from approaches for low-cycle fatigue, the simulation of every single load cycle is too time-consuming for practical application. Therefore, degradation-, damage- and strain-evolutions are modelled indirectly, depending on the applied increment of load-cycles ΔN . For a standardised evaluation and formulation, the load cycles are related to the ultimate number of loadcycles Nf according to the S-N -approach. This leads to the standardised time scale n ∈ [0, 1] with increments Δn = ΔN Nf . This standardised time scale is

262


related to the time scale via Nf and the frequency f by: Δn =

ΔN Δt · f = Nf Nf

⇔

Δt =

Nf · Δn . f

(3.99)

This concept has already been applied in [627, 628, 629, 630]. S-N -Approach Thus, one basic quantity for the fatigue model is the fatigue lifetime Nf of concrete. Generally, any S-N -curve can be applied for its evaluation. In the following the approach presented in [392] will be re-used. It has been compared to other approaches and to a large number of experiments in [627] and proved to be well suitable. It takes the loading frequency into account and distinguishes between high-cycle and low-cylce fatigue: ⎧ tfat ⎪ ⎪ 1.0 − 0.0294 log + smax ⎪ ⎪ tref ⎪ ⎨ 0.062 (1 − 0.556 rfat ) . log Nf = max tfat ⎪ ⎪ 1.2 − 0.2 r + s − 0.053 (1 − 0.445 r ) log ⎪ fat max fat ⎪ ⎪ tref ⎩ 0.133 (1 − 0.778 rfat) (3.100) In this expression, tfat is the duration of one load cycle, which is the inverse of the frequency: tfat = 1/f , tref is a reference time of the same unit as tfat : tref = 1[tfat ], smax and smin are the related upper and lower stress limits, respectively: smax = σmax /fc , smin = σmin /fc and rfat is the relationship of lower to upper fatigue stress: rfat = σmin /σmax . The approach, together with a large number of experiments from the literature, is illustrated in Figure 3.129. The evaluation of the 0.05- and of the 0.95-quantile of Nf , which is also shown in this diagram, will be introduced later in this section. Degradation of the Compressive Strength The degradation of the compressive strength is formulated empirically with a direct approach in the time scale of the related number of load cycles n, as introduced above. To quantify the degradation, the variable dfc is introduced and the resulting compressive strength reads as follows: fc (n) = fc,28 · (1 − dfc (n))

(3.101)

According to the experimental results presented in [70, 374], the degradation process starts very slowly. Nevertheless, fatigue failure is associated with a drop of the compressive strength onto the level of the upper fatigue stress. Thus, the value of dfc results in dfc ,fail = 1 − |smax |

(3.102)

3.3 Modelling

263

related stress smax

1.0 0.9 0.8 0.7 0.6 0.5 100

102

104

106

108

1010

lifetime Nf Fig. 3.129. S-N approach (0.05-, 0.50-, and 0.95-quantiles) with experimental results

for the state of fatigue failure, n = 1. Based on an exponential approach, suggested in [370] for the description of sequence effects, dfc is sub-structured into dfc = nafc · dfc ,fail

with afc = 26.5 − 25.0

|σmax | . fc

(3.103)

This implies a faster degradation (in the time scale n = N/Nf ) of the compressive strength for higher stresses. The evaluation of the degradation is shown exemplarily for three different load levels in the left diagram in Figure 3.130.

1.0

relative strength fc /fc,28

relative strength fc /fc,28

1.2

1.0

0.8

0.6

0.4 0.0

0.2

0.4

0.6

0.8

related cycles n = N/Nf

1.0

0.6

0.7

0.8


Fig. 3.130. Degradation of compressive strength and sequence effects

0.9 0.9

264


2.0

} }

N2,res /Nf,2

1.5

1.0

0.5

0.0 0.0

0.2

0.4

0.6

0.8

1.0

N1 /Nf ,1

Fig. 3.131. Evaluation of the approach for sequence effects an comparison with single simulation results from [383]

Sequence Effects This direct formulation of the degradation of the compressive strength includes indirectly the formulation of sequence effects. As introduced, the degradation process depends on the applied upper fatigue-load level. When this load level is changed, the degradation curve changes, too. As illustrated in the right diagram of Figure 3.130, this requires a modification of the related number of load cycles n = N/Nf . Keeping n constant would imply a sudden drop or increase of the compressive strength, which is physically nonsensical. Thus, n has to be changed. The updated value of n can be evaluated from the approach for the description of the degradation process: !

dfc ,n = dfc ,n+1

⇒

n=

dfc ,n dfc ,fail,n+1

1/afc ,n (3.104)

Fig. 3.131 shows an evaluation of the presented approach together with the results of two stage tests from [383]. The mean values of the single results are plotted by solid lines. It can be seen, that the approach delivers qualitatively reasonable results, for quantitative evaluations, the data basis is too small. Strain Evolution As introduced in Section 3.1.1.2.2.1, the strain evolution in concrete under fatigue loading can be interpreted as the sum of creep and cyclic strain evolution. This interpretation is picked up for the modelling approach. In the following, the approaches for the creep strain evolution as well as for the evolution of cyclic strains is introduced.

3.3 Modelling

spring

265

frictional element

σ

σ

dashpot Fig. 3.132. Rheological element for the description of nonlinear creep processes

Creep Strain Evolution The creep strain evolution is modelled with rheological elements. The model is based on an approach presented in [735], which has been enhanced in [133, 134]. To account for the nonlinear relation of stress and creep strain rate, especially for concrete under higher stresses than approximately 0.4 fc , nonlinear rheological elements are used. They consist of a nonlinear spring with a friction element (to describe plastic deformations) and a nonlinear dashpot. Fig. 3.132 shows such elements. The nonlinear behaviour of the spring is described with the stress-strain relation for concrete under compression given in [182], with the compressive strength fc replaced by fc,T = 0.8 fc, taking long-term effects into account. Thus, the stress-strain relation for the spring reads: cr 2 cr Ec fεc,T + εεc fc,T . σs = εcr c 1 − Ec fεc,T −2 εc

(3.105)

In incremental formulation, this equation can be approximated as s σn+1 ≈ σns + Etan Δεcr n+1 ,

(3.106)

where Etan is the tangent d σ s /d εcr . Analogue to the smeared crack model described earlier in this chapter, strains which result from the nonlinearity of the spring are regarded dissipative. For the modelling of creep they are regarded as plastic strains, indicated by the friction element in Figure 3.132. For linear descriptions of dashpots, the viscosity is coupled linearly via the retardation time τ with the stiffness: η = τ Ec . This relation is now enhanced and is reformulated dependent on time and on the applied stress: η = τ Ec

t − t0 τ

12

σd 1− fc,T

ncr .

(3.107)

Herein, σ d is the stress in the dashpot and ncr a material parameter. According to [133, 134], for concrete it takes values between 1.5 and 2.0.

266


The classical relation between stress and strain rate, σ d = η ε˙cr ,

(3.108)

is now replaced by an incremental formulation: d d σn+1 ≈ σnd + Δt σ˙ n+1 .

(3.109)

The stress rate can be found as time derivative of eq. (3.108) to σ˙ d =

d (η ε˙cr ) = η˙ ε˙cr + η ε¨ , dt

(3.110)

the time derivative of eq. (3.107) results in the rate of viscosity: η˙ =

Ec 2

t + t0 τ

12 ncr σd 1− . fc,T

(3.111)

The equation for the resulting stress σ cr = σ s + σ d

(3.112)

yields a differential equation, which can be solved numerically. In [133] the Newmark method according to [569, 198, 409] is suggested. Cyclic Strain Evolution The rate of cyclic fatigue strains, in the time scale of related load cycles n:

ε˙fat,∗ =

∂εfat , ∂n

(3.113)

is formulated empirically on the basis of the experiments documented in [383]. Therefore, the typical S-shaped evolution curve of fatigue strains is devided into three parts, where a constant strain rate is assumed within each domain. The strain rates are evaluated from the experiments by linear regression, like illustrated in the left diagram of Figure 3.133 together with the experimental results from [383]. The borders between these domains are assumed at n = 0.1 and n = 0.9. In order to approximate the fatigue strain rates as functions of the applied load level, a scalar measure for the fatigue loading, taking upper and lower fatigue stress into account, is introduced as the product of mean stress and stress difference: s=

smax + smin · [smax − smin ] . 2

(3.114)

The evaluation of the experimental results of [383, 70] are shown in the right diagram of Figure 3.133. The experiments, that exhibit significant creep

3.3 Modelling

2.0

¯ smax /smin = 0.80/0.20 ¯ smax /smin = 0.75/0.05 ¯ smax /smin = 0.95/0.05

¯ ¯

4.0

Holmen Awad & Hilsdorf approach

3.0

1.5

2.0 1.0 dom. 3 0.5

1.0

domain 2 domain 1

0.0 0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.1


0.2

0.3

0.4

strain rate −ε˙fat,∗ [10−3 ]

fatigue strain −εfat [10−3 ]

2.5

267

0.0 0.5

stress measure s

Fig. 3.133. Fatigue strain evolution (stress measure vs strain rate by [383, 70])

strains due the test duration are plotted in white. These experiments have not been used for the evolution of cyclic strain evolution. Those ones that exhibit pure cyclic strain are plotted in black. By least square fitting, second order polynomials are evaluated to approximate the (pure cyclic) fatigue strain rate as function of the stress measure s: 2 ε˙fat,∗ 1,3 = −113.189 s + 67.5492 s − 4.50913 ,

(3.115)

ε˙fat,∗ = −6.54818 s2 + 4.55811 s − 0.268655 . 2

(3.116)

The right diagram in Figure 3.133 shows the polynomial as well as experimental results, exemplarily for domain 2. The results of [383] have been used for the evaluation of the polynomial. For additional proof, the results of [70] are plotted in the diagram. These values have been evaluated graphically. They are therefore regarded as too imprecise and where not taken into account for the evaluation. Fatigue Damage For the evaluation of fatigue damage, again the tests of [383] deliver the experimental basis. In these experiments, not only the evolution of the maximum, but also of the minimum4 fatigue strains is reported. That can be utilised for the sub-division of the total fatigue strains into damaging and plastic parts. At first, the measured total strains are reduced by the initial ones. Assuming linear unloading and reloading, according to the damage theory, the fatigue strains corresponding to σ = 0 can be extrapolated, like illustrated in Figure 3.134. This yields reversible and irreversible parts of the total fatigue strains, which are interpreted as damaging and plastic, respectively. 4

That means the strains, that correspond to the lower fatigue stress level.

268


σ

σ

stress

ε

0

ε − ε0 εfat

ε εir

εrev

strain Fig. 3.134. Split of total fatigue strains into reversible and irrversible parts

To distinguish the damaging part from the total fatigue strains, the variable β fat is introduced as β fat =

εda,fat . εfat

(3.117)

The evaluation of β fat , corresponding to n, is shown in Figure 3.135. It is evident, that β fat = 0.35 is a reasonable assumption. The curves of β fat for the series 0.675/0.05 (with |smax |/|smin | = 0.675/0.05) and 0.80/0.05 exhibit significant smaller values. The reason for these values for the series 0.80/0.20 is to be found in the relative high lower stress limit. The linear extrapolation leads,

split parameter β fat

0.8 smax /smin = 0.675/0.05 smax /smin = 0.80/0.20

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

related cycles n = N/Nf Fig. 3.135. Evaluation of the split parameter β fat

1.0

3.3 Modelling

269

due to the convex curvature of the unloading path, to an overestimation of the irreversible strains. The series 0.675/0.05 exhibits significant creep strains, which are regarded as plastic. Thus, the reduction of the damaging part of total measured time-dependent strains proves the assumption, that the total fatigue strains can be interpreted as the sum of time- and cycle-dependent parts, see Section 3.1.1.2.2.1. Multi-Axial Stress States To enhance the uniaxial approach for fatigue and creep strain evolution, a potential and a flow rule, analogous to the classical time-independent description of the behaviour of concrete according to the smeared crack approach, which has been introduced earlier in this section, is assumed: ε˙ fat,∗ =

∂φc ˙ fat,∗ . λ ∂σ c

(3.118)

In this equation, the derivative ∂φc /∂σ describes the direction of the strain evolution, λ˙ fat,∗ the norm. The latter one is derived from the uniaxial strain c rate and the assumption of energetic equality of uniaxial and multi-axial stress states: σeq ε˙fat,∗ 1d λ˙ fat,∗ = , c ∂φc σ: ∂σ

(3.119)

where the scalar equivalent stress σV is evaluated from the Drucker-Prager potential according to eq. (3.52): φc (σ, α) =

( ) 1 μ I + J2 −αc (qc ) . 1 √1 − μ 3 ! σeq

(3.120)

Fatigue damage is quantified by a fourth order compliance tensor, analogously to the smeared crack model described earlier in this section. The evolution laws of the two independent variables read μ β 1 da,fat,∗ ˙ Ds1 λ˙ fat,∗ = 1 − √ , c √ − μ I1 6 J 2 3

da,fat,∗ D˙ s2 =

1 β √ λ˙ fat,∗ , c √1 − μ 4 J2 3

(3.121)

which is similar to eq. (3.57). Analogously to eq. (3.56), the relation for the rate of the fatigue damage compliance tensor reads

270


da,fat,∗ da,fat,∗ da,fat,∗ D˙ I +¯ I , = D˙ s1 1 ⊗ 1 + D˙ s2

(3.122)

to evaluate increments of this tensor, it has to multiplied with the increment Δn of the number of related load cycles: da,fat,∗ Dda,fat = D˙ ΔD Δn .

(3.123)

Thus, the final compliance relation, which takes damage and plastic strains due to time-independent and time-dependent modelling of the material behaviour of concrete into account, reads −1 σ = D 0 + D da,c + D da,t + D da,fat : ε − εpl,c − εpl,fat − εcr .

(3.124)

Scatter of Basic Model Properties In order to account for the scatter, which is inherent to fatigue processes, the scatter of the basic quantity Nf is investigated. According to the observations in e.g. [738, 821], the scatter of the fatigue lifetime is related to the scatter of the compressive strength fc of concrete. An approach presented in [738] takes this into account; it describes the standard deviation of log Nf as a function of the standard deviation of fc and of the applied fatigue load: s(fc ) m(fc ) s(log Nf ) = √ −σmax . 1 − rfat s(fc ) 5.714

(3.125)

In order to avoid physically nonsensical solution, the values for the upper fatigue load and for the relation of lower to upper fatigue load are limited to −σmax ≥ 0.5 fc ,

rfat ≥ 0.75 .

(3.126)

This is a reasonable suggestion that is hardly to proof experimentally: At load levels below 0.5 fc, very high fatigue lifetimes Nf occur, which makes the realisation of a large number of experiments, the basis for the evaluation of the scatter, difficult. 3.3.1.2.2.2 Metallic Materials Authored by Henning Sch¨ utte It is known from experiments that most materials, and in particular brittle and quasi-brittle materials, under general loading conditions develop anisotropic damage [441]. For a given stress state, materials damaged by microcracks in general accumulate additional damage through the growth of these microcracks. Considering this and the mentioned points, the concern of this paper is to provide a consistent, continuum damage model based on the micromechanical framework and the local anisotropy induced by kinking

3.3 Modelling

271

and growing elliptical and/or circular microcracks. The reason for considering elliptical and circular cracks is that these geometries are good approximations for the shape of a flaw in many engineering materials. For clarity purposes and to explain the main issues of the proposed model in a more clear mathematical way, the complexity of the proposed damage model is reduced here by leaving out the thermal effects and other non-mechanical phenomena. Strains and rotations are assumed to be small, hence the framework of linear elastic fracture mechanics can be applied. Furthermore, viscous effects and permanent deformations are neglected and the material behavior is assumed to be linear elastic in its pristine state. The small strain assumption, and the lack of permanent deformations in this model makes it suitable to show the evolution of damage in structures with brittle and quasi-brittle fracture behavior experiencing high-cycle fatigue. Effective Continuum Elastic Properties of Damaged Media The micromechanical models are commonly referred to a class of analytical models which give the relation between the macroscopic state of a specimen and its micro-structure [164]. One of the goals of the micromechanical models is to provide relatively simple constitutive laws. Within this approach, the effective elastic properties are derived by using the pertinent results of microconstituent analysis, such as that of a planar crack embedded in an infinite medium. Using the concept of micromechanics, continuum damage models based on the framework of fracture mechanics and elasticity can give the local details of the damage response within a representative volume element. These class of models are based on the hypothesis of statistical homogeneity and weak interaction of defects, which are justifiable for reasonably modest concentration of heterogeneities [565]. In this respect, the first step in the formulation of the proposed continuum damage model requires the formulation of the change of continuum elastic properties due to the presence, kinking and growth of elliptical and/or circular microcracks. Applying the approach of micromechanics, the components of the effective compliance tensor of an infinite, homogeneous, isotropic (in its pristine state) and elastic continuum damaged by a single internal circular crack is given by [440]. Here, applying the same method, the components of the tensor for the change of compliance due to the presence of a single internal elliptical crack are derived, from which the results corresponding to a single circular crack can be reproduced (see also [163]). Within the approach of micromechanics, the effective elastic properties of a solid damaged by a planar internal elliptical crack are derived from the contribution to the complementary strain energy corresponding to the quasi-static, selfsimilar growth of the crack. For this, the stress intensity factors suffice to give the energy released during the quasi-static, selfsimilar growth of the crack. However, for the formulation of the complementary strain energy corresponding to the kinking of a crack, the analytical expressions for the so called T-stresses are required as well. The complete set for the T-stresses for internal elliptical and circular cracks

272


embedded in a homogenous isotropic infinite solid have been addressed by [549] and [716], respectively. Kinking of an Internal Elliptical Crack To study the degradation of the elastic material properties due to the kinking and growth of elliptical and/or circular microcracks, consider a single elliptical crack in an infinite, homogeneous, isotropic and elastic continuum subjected to mechanical loads applied at infinity. For a stress state outside the damage surface, the considered crack will kink and propagate to a new geometry, and the local kinking angle and the local crack extension length can be calculated from the considered fracture criterion coupled with a fatigue crack evolution law. In an analogous manner to the previous section, this problem can be decomposed into two sub-problems: that of the continuum without a crack subjected to the remote traction field, and that of the same continuum, where only the crack faces are subjected to the traction field. In the framework of linear elasticity, the compliance tensor of a material containing a kinked crack can be decomposed into three parts [715] * = S Matrix + ΔS SCrack + ΔS S Kink , S

(3.127)

SCrack refer respectively to the compliance tensor of the where S Matrix and ΔS matrix material in its pristine state and the change of compliance due to the SKink is the change of compliance due to presence of the microcrack, and ΔS the kinking and growth of the microcrack. The analytical expression for the tensor of the change of compliance due to the presence of a single active elliptical or circular microcrack was given by [163]. In a similar way, the tensor of the change of compliance due to the kinking of a crack can be calculated from the contribution to the complementary strain energy corresponding to the kinking of the crack, which is the energy released during the kinking growth of the crack. The rate of the change of the compliance tensor for a volume element V of elastic material, attributable to the extension rate s, ˙ through which a point on the perimeter of a single crack kinks to a new position and integrating this along the crack perimeter, the rate of the change of compliance due to the growth of an internal crack is resulting + 2 Kink 1 ∂ G(s) S˙ ijmn = s˙ dl , (3.128) V c ∂σij ∂σmn where ψ ∗∗ is the complementary energy associated with the kinking of the crack and G(s) is the energy released during the kinking of crack with a local extension of s, and is given by 1 − ν2 G(s) = E

2 KI2 (s) + KII (s) +

1 2 K (s) = Mαβ Kα (s) Kβ (s) , 1 − ν III (3.129)

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273

with Kα (s) being the stress intensity factors at the propagated crack front, given by (see [45, 481]) √ Kα (s) = Kα + Kα(1/2) s + O(s) , (3.130) with Kα = Fαβ (φ) Kβ ,

Kα(1/2) = Gαβ (φ) Tβ + a Hαβ (φ) Kβ ,

(3.131)

where Fαβ , Gαβ , and Hαβ are universal functions of the kinking angle φ, Kβ and Tβ are the stress intensity factors and the T-stresses of the crack prior to kinking, and a is the curvature parameter of the crack extension which for a straight extension vanishes (a = 0). Unlike the case of a single elliptical crack, the stress intensity factors alone would not suffice to give the complementary strain energy corresponding to the kinking of the crack, and the analytical expressions for the T-stresses of the elliptical crack prior to kinking are required as well. [549] and [716] derived the asymptotic solutions for the stress components, based on the potential method and a transformation technique, from which the T-stresses for mixedmode internal elliptical and circular cracks in homogeneous, isotropic linear elastic solids are resulting. Considering the expansion of the stress intensity factors in terms of the extension length and the crack tip parameters prior to kinking, Eq. (3.128) can be rewritten as + Kink ∂Kα (s) ∂Kβ (s) 1 S˙ ijmn = Mαβ s˙ dl , (3.132) V c ∂σij ∂σmn with √ ∂Kα (s) ∂ (Fαλ (φ) Kλ + Gαλ (φ) Tλ s) = ∂σij ∂σij ∂Kλ ∂Tλ √ = Fαλ (φ) + Gαλ (φ) s. ∂σij ∂σij

(3.133)

The local propagation rate s, ˙ measured in the direction normal to the crack front at a considered point, can be calculated using a fatigue crack evolution law coupled with the selected fracture criterion. For example considering the following modified Paris’ law [490, 713], which is combined with the fracture criterion of maximum driving force [476], and setting a threshold value for crack growth, it results [713] s˙ =

√ η ds =C G∗ − Gth , dN

(3.134)

where C and η are constants for the considered fatigue crack evolution law, N represents the number of load cycles, G∗ is the maximum driving force acting at the (infinitesimal) kinked crack tip

274


G = max lim G(s) = φ

s→0

' 1 − ν2 1 2 [(KI )2 + (KII (KIII ) + )2 ]'φ=φ E 1−ν (3.135)

and Gth is the threshold value, below which there is no damage growth. Here, the threshold value Gth is considered to take into account the load history and is varying by crack growth. Indeed, its value is decreasing by damage growth for brittle and quasi-brittle materials. The expression for the fourth-rank tensor of the rate of the change of compliance due to the kinking of an elliptical crack with a local growth rate s˙ is derived by substituting relations (3.133) and (3.134) into (3.132), and performing the requisite integration along the crack front in the following form Kink 1 2π S˙ ijmn = Mαβ α2 sin2 ϕ + β 2 cos2 ϕ 0 V , ∂Kλ ∂Kμ ∂Kλ ∂Tμ Fαλ Fβμ + Fαλ Gβμ + ∂σij ∂σmn ∂σij ∂σmn ∂Tλ ∂Kμ √ ∂Tλ ∂Tμ Gαλ Fβμ s + Gαλ Gβμ s s˙ dϕ , ∂σij ∂σmn ∂σij ∂σmn (3.136) where the Tα and Kα are given by [549]. A Fatigue Fracture Based Anisotropic Continuum Damage Model Generally, it is impossible to formulate a damage model covering the mixedmode propagation of microcracks in a fully traceable way. Considering microcracks in the form of elliptical and circular cracks, one may calculate the kinking of the initial cracks analytically only for the first growth increment. After the kinking of the initial cracks, however, the mathematical formulation of the next kinking steps is no longer possible. To overcome this difficulty, some researchers have introduced models based on simplifying assumption such as fixing the plane of crack growth, so that microcracks may only grow in a self-similar and/or co-planar manner [612, 860, 101]. This assumption may be acceptable for the case of monotonic loading and proportional loading, but for loads with changing direction and amplitude, as is the case for non-proportional loads or even sequential loads, this assumption leads to underestimating the damage, since it does not allow for crack kinking. The assumption of self-similar growth of mixed-mode cracks, in general, results in a smaller damage accumulation than what the real mixed-mode kinking results in. To overcome the difficulties in the formulation of a damage model, which accounts for the kinking and growth of microcracks in a mathematical traceable manner, the objective of this section is to propose a micromechanical based continuum damage model, which is based on the reduction of stiffness due to kinking elliptical microcracks. To be able to formulate the model in a

3.3 Modelling

275

fully mathematical traceable way, the concept of an equivalent elliptical crack is introduced in the sense that a kinked crack is replaced with an equivalent elliptical crack, resulting in an equivalent dissipation of energy. Basically, eight degrees of freedom can be considered for each equivalent elliptical microcrack replacing the kinking one. These are the major and minor axes of the ellipse (2 unknowns), orientation of the microcrack given by three Eulerian rotation angles (3 unknowns), and the position of the crack in the space (3 unknowns). Considering the concept of unit cell and assuming that the microcrack is located in the center of the cell, the position of the microcrack can be fixed and may be left out of the formulation. This is because in the case of noninteracting cracks, the position of the crack does not have an impact on the elastic properties of the material. To calculate the other five unknowns characteristics of the equivalent crack, the following postulates are proposed • •

equivalent type of damage induced anisotropy, and .Crack = ΔS SCrack + ΔS SKink , equivalent change of compliance tensors ΔS ijmn

ijmn

ijmn

where quantities with a hat indicate the ones corresponding to the equivalent replacement crack. With this, the geometry and the orientation of the equivalent elliptical crack replacing the kinked one then result by considering two optimization problems. The first postulate takes into account the fact that the local damage associated with a single planar elliptical crack results in an orthotropic material symmetry [163], thus it can be argued that changing the type of material symmetry from isotropy to orthotropy may imply the existence of local damage due to an elliptical crack. [714], have shown that the damage associated with a growing mixed-mode elliptical or circular crack also changes the virgin isotropic material into approximately orthotropic one. Considering the mentioned points, it is deduced that the orientation of the equivalent crack replacing the kinked one is such that the resulting orthotropy axes are aligned with the ones due to the damage associated with the kinked crack. It is reminded that for a single elliptical crack in an initially isotropic material, the axes of orthotropy are aligned with the axes of the ellipse, i.e. two axes of orthotropy are aligned with the major and minor axes of the ellipse and the third one is the normal to the plane of the ellipse. The second postulate results in the geometry of the equivalent elliptical crack in the sense that the components of the change of compliance due to the kinked crack and the equivalent crack are approximately identical. Such a formulation of the dissipative damage process due to kinking elliptical microcracks, taking into account the damage induced orthotropy of an elliptical crack in a local sense, results in a consistent damage model capturing the load history through the local orthotropic degradation of the mechanical properties.

276


Optimization Subroutines As mentioned in the previous section, based on the assumption of equivalent type of damage induced anisotropy, here orthotropy, the orientations of the equivalent crack are resulting. For this purpose the following optimization procedure is considered. Considering the fact that the damage associated with a growing elliptical or circular crack changes the virgin isotropic material properties into approximately orthotropic one [714], it is possible to find the orthotropy axes using an optimization method. The idea of the optimization procedure is to find the coordinate transformation that yields the best representation of a tensor with a known type of symmetry. Thus, the optimization algorithm consists of finding the appropriate coordinate transformation for the measured compliance tensor in a known coordinate system that yields the best orthotropic representation. Considering the transformation law for fourth-rank tensors, one possible way is considering the following optimization procedure which looks for the best Eulerian rotation angles minimizing the non-orthotropic components of the corresponding compliance tensor F1

=

non.ORT i,j,m,n

2 S ijmn (R(γx , γy , γz ))

S1213 )2 + (S S1223 )2 + (S S1323 )2 + = (S Sii12 )2 + (S S ii23 )2 + (S Sii13 )2 , + 3i=1 (S 0 / ' RORT (. γx , γ .y , . γz ) = arg min F1 (R) ' det R = 1, RT = R−1 , ⇒ where

(3.137)

0 / ORT S ORT , ijmn = ORT S ijmn R

S ijmn = Rir Rjs Rmt Rnu S rstu ,

where R is an orthogonal transformation including all rotations about the three cartesian axes (R(γx , γy , γz )), Figure 3.136, ORT (•) is the operator that nulls out the non-orthotropic components of S ijmn deviating from zero, and S and S are the compliance tensors in the considered global coordinate system and the local orthotropic coordinate system, respectively. The geometry of the equivalent crack is resulting from the postulate of equivalent tensors for the change of compliance. This is done by a direct comparison of the change of compliance tensors associated with the damages due to the kinked crack and due the equivalent elliptical crack, given with respect to a known coordinate system (local coordinates of the equivalent crack may be a good choice). In this regard, the following optimization problem should be solved

3.3 Modelling

y

da

y

γy

γ ^y

2 α

2 α ^

γx β

3 z

x γz

1

277

γ ^x

^ β 3 z

x γ ^z

φ

1

Fig. 3.136. Kinked crack and its equivalent elliptical crack

F2 =

2 3 .ijmn α S ., β. − S ORT , ijmn

i,j,m,n

/ 0 / ' 0 α ., β. = arg min F2 'α . ≥ αo , β. ≥ βo ,

(3.138)

. is the tensor of the change of compliance due to the equivalent elwhere S . = β/. . α, the components of which are liptical crack with an aspect ratio of λ given in the local crack coordinates in Eq. (3.136), and αo and βo are the semi-major and semi-minor axes of the elliptical crack prior to kinking. The proposed continuum damage model based on the reduction of stiffness due to the kinking of equivalent elliptical microcracks results in the effective elastic properties of a damaged material volume element in a consistent way. Based on the incremental analysis of the effective elasticity tensor for the given current values of the local stress and strain tensors, and taking the load history into account by introducing the concept of an equivalent elliptical crack, the propagation of microcrack is calculated by considering a crack evolution law. In this study, the propagation of microcracks is governed by the modified Paris’ law given by Eq. (3.134) coupled with the fracture criterion of maximum driving force [476]. In the incremental formulation, to have a more stable algorithm and for a faster convergence, the tangent stiffness of the damage material, the so called algorithmic tangent [749], should be introduced. The proposed damage model can also capture the unilateral effect observed in tension-compression tests, observed for a certain classes of materials including ceramics and concrete [156, 391], provided that for a passive crack the corresponding components of the compliance components are recovered, and they return to their degraded state upon the activation of the crack.

278


Numerical Examples The proposed continuum damage model, based on the reduction of stiffness due to kinking elliptical microcracks can be easily implemented in a finite element code. This has been performed here in the commercial finite element analysis software Ansys as a user material subroutine. In the following sections, a variety of numerical examples is presented. The first two examples are illustrative examples, which explain the proposed model in a fully analytical manner. The first example addresses the determination of the equivalent crack after a single kink step. As the second example, a unit cell damaged by a single mixed-mode circular crack is subjected to four stages of cyclic loadings with changing directions. With this, the degradation of the material properties and the evolution of the considered damage parameters are presented. This example is especially important, since it provides the reader with the details of the irreversible damage process due to growing microcracks subjected to cyclic loading with changing directions. The objective of the other examples is to show the mesh sensitivity of the model. All experiments are conducted on AISI 4130 steel with the mechanical properties and the chemical composition given in [394], except the first illustrative example. Example-1 The objective of this section is to explain the proposed continuum damage model by giving an illustrative example in a fully analytical manner. The considered crack problem is a circular crack with γz = 45◦ and γx = γy = 0◦ as the initial orientations. The loading level is σ ∞ /E = 1/1000, the Poisson’s ratio ν = 1/3, and the initial crack size is considered to be αo /L = 1/150 with L being the characteristic length of the volume element. To avoid more complexity, it is assumed that the stress state is outside the damage surface, thus the considered circular microcrack will propagate, and at this step due to simplicity purposes, the Wöhler’s limit stress (i.e. also the threshold for crack growth) is assumed to be negligible. The first step in the model is calculating the propagation of the crack under the given local stress state. Considering Eq. (3.134), the rate of crack growth is √ η s˙ = C G∗ , (3.139) where C and η are the Paris’ parameters. For clarity purposes, the propagation of the crack is exaggerated by selecting these parameters as C = 10−2 and η = 2. This and the considered loading level and material parameters result in crack extension length of approximately 12% of initial crack radius αo . The propagation parameters, i.e. the extension length and the kinking angle are resulting as functions of the geometrical angle ϕ along the crack front. With these the new crack geometry is resulting (Figs.3.137-left). The next step is to calculate the influence of crack growth on the compliance tensor through relations (3.136). These result in the effective compliance tensor modified by

3.3 Modelling

σ∞

σ∞

y 45

y

◦

z

1

2.65◦

^ 2

2

2

x 3

279

x 3 z

1

Fig. 3.137. Growth of the circular crack and its equivalent elliptical crack

the damage due to the propagated circular crack. In six-dimensional tensorial notation, normalizing the components of the compliance tensor with the tensile compliance component of the pristine material 1/E, and the factor (αo /L)3 , where αo and L are respectively the characteristic initial size of the crack and the length of the unit cell, results in: ⎞ 0.0527 0.0185 −0.0034 0.1884 0 0 ⎟ ⎜ ⎟ ⎜ ⎜ 0.0185 5.5727 0.0092 0.2988 0 0 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ 0 ⎟ 1 αo 3 ⎜−0.0034 0.0092 0.0193 0.0074 0 ⎟ SC = ΔS ⎟. ⎜ ⎟ ⎜ E L ⎜ 0.1884 0.2988 0.0074 3.7958 0 0 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ 0 0 0 0 3.5995 0.1220⎟ ⎟ ⎜ ⎠ ⎝ 0 0 0 0.1220 0.0396 ⎛

(3.140) To find the equivalent elliptical crack which replaces the propagated crack, it suffices to perform two further steps. The first step is finding the axes of orthotropy due to the damage induced by crack growth which can be deduced as the local coordinates of the equivalent crack. The second step is to find the geometry of the equivalent crack, the local axes of which are aligned with the axes of orthotropy calculated at the previous step. These two steps are performed by solving the optimization problems, given by Eqs. (3.137) and (3.138), respectively The application of this procedure to the considered

280


example results in the following parameters for the equivalent elliptical crack replacing the current propagated crack γ .x 0◦ ,

γ .y 0◦ ,

γ .z −2.65◦ ,

β. 1.00 , βo

α . = 1.14755 , αo

where γ .’s are given with respect to the local coordinates of the crack prior to propagation. The resulting local Eulerian angles result in the best approximate orthotropic representation of tensor (3.139), which is indeed given in the local coordinate system of the equivalent elliptical crack replacing the propagated one ⎞ ⎛ 0.0441 0.0071 −0.003 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎜ 0.0071 5.6042 0.0097 0 0 0 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ α 3 −0.003 0.0097 0.0193 0 0 0 ⎟ ORT 1 ⎟ ⎜ o . ΔS = ⎟. ⎜ C ⎟ ⎜ E L ⎜ 0 0 0 3.7729 0 0 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ 0 0 0 0 3.6032 0 ⎟ ⎟ ⎜ ⎠ ⎝ 0 0 0 0 0.0359 (3.141) Example-2 The objective of this example is to provide a better insight to the irreversible process of brittle damage in a local sense. For this a representative volume element of AISI 4130 steel is considered which embeds a single circular crack. The circular crack in its initial configuration has an inclination angle of 45◦ with respect to the 2-axis (Figure 3.138). The initial size of the crack is considered to

σ1

σ4

2 3

2 1

3

2 1

3

2 1

σ3

3

σ2 stage 1

stage 2

stage 3

Fig. 3.138. Order of the considered sequential loading

stage 4

1

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281

Table 3.22. Characteristics of the applied sequential loading stage 1

stage 2

stage 3

stage 4

stress σ1 cycles (MPa) N1




1500

1400

1350

1450

100,000

Α Αo

250

100

100,000

Γz Γy

crack orientation

150

100,000

40

Β Βo

200 crack size

100,000

Γx

30

20

10

50 0 0 0

100000 200000 300000 400000 number of loadcycle N

0

100000 200000 300000 number of loadcycle N

400000

Fig. 3.139. Evolution of the geometry and the orientations of the equivalent elliptical crack

be L/500, L being the characteristic size of the unit cell. The constants in the modified Paris’ equation (3.134) are chosen as C = 10−5 and η = 2. To study the local degradation of the considered material under fatigue conditions with the help of the proposed model, the unit cell is subjected to four stages of cyclic loading as presented in Table 3.22 and Figure 3.138. In the load stage 1, a stress of 1500 (MPa) is applied in the direction of 2-axis for 100, 000 cycles, and in the subsequent stages the direction of the applied stress is parallel to the 3-axis, 1-axis, and 2-axis, respectively. The applied stresses are 1400 (MPa), 1350 (MPa), 1450 (MPa) and the corresponding number of cycles are the same as for the stage 1. The proposed optimization subroutines are solved at each load increment, in order to calculate the geometry and the rotation of the equivalent elliptical crack replacing the kinked one. Figure 3.139 shows the evolution of the geometrical parameters and the orientation of the equivalent elliptical crack with respect to the number of cycles. It is observed that up to approximately 300, 000 loading cycles, the

282


evolution of the corresponding damage is relatively smooth, and at this moment application of the load stage 4 accelerates the accumulation of damage. To explain this, lets review each load stage separately. During load stage 1, it is observed that by crack growth the equivalent elliptical crack rotates to become perpendicular to the direction of σ1 . This is more evident in Figure 3.139-right, where at N = 100, 000 the equivalent crack has an inclination of approximately γz = 10◦ with direction-1. Changing the load to the load stage 2 results in no observable damage accumulation in the material, because at this stage, the equivalent crack is parallel to σ2 . Hence, very small changes in the presented parameters are induced, which due to the scaling of the curves is not observable. The load stage 3, similar to the load stage 2 has small contribution to the process of damage growth, since at this stage, plane of the crack is inclined nearly 10◦ with respect to the direction of σ3 (identical to the direction-1). This small angle induces a relatively small normal stress on the plane of the equivalent crack, leading to a relatively small driving force along the crack front, which consequently results in a slow propagation of the crack. This is observed between cycles 200, 000 and 300, 000 in Figure 3.139, as γz grows again bigger. At the end of this stage, γz has reached a value of approximately 15◦ . Application of the load stage 4, however, accelerates the propagation of the crack, since at this stage the orientation of the equivalent crack is such that the local normal stress acting on the plane of the crack induces a strong driving force on the crack front. As can be deduced from Figure 3.139, the dimensions of the crack evolve very fast to damage the whole material volume. Figure 3.140 shows the degradation of the stiffness components in the principle loading directions. It is observed that at the end of the load stage 4, the

1 0.95 0.9 E Eo 0.85

principle direction 1

0.8


0.75


0.7 0

100000 200000 300000 number of loadcycle N

400000

Fig. 3.140. Evolution of the stiffness components in the principle directions

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283

u, F

r2 h2 R h1

r1

mesh A

mesh B

mesh C

Fig. 3.141. Specimen geometry and different mesh patterns

stiffness in direction-3 does not show a considerable change, while in directions 1 and 2, the degradation of stiffness is obvious. Mesh Sensitivity The objective of this example is to show the degree of the mesh dependency for the proposed damage model. For this, the specimen of the form given in Figure 3.141 is considered, where r1 = 10 mm , r2 = 20 mm , h1 = 100 mm , h2 = 20 mm , R = 505 mm . Displacement controlled analyses have been performed for different discretization levels (Figure 3.141), where a constant displacement with the magnitude of 2% of the specimen’s initial length h1 + h2 is applied for 2, 000, 000 cycles. All models are meshed with the help of hexahedral elements with quadratic displacement behavior. Considering mesh pattern A, 48 hexahedral elements are generated, and mesh patterns B and C result in 384 and 1728 elements, respectively. The corresponding constants of the damage model are chosen as η = 2 , C = 7.39 × 10−9. The resulting force-cycle curves for these experiments are given in Figure 3.142, where FN and Fo are the resultant forces at the end cross section of the specimen (r2 ), which correspond respectively to the current load-cycle and the initial load-cycle. The good agreement between the results corresponding to different discretization levels demonstrate the mesh independency of the model. This is due to the fact that in the considered fatigue microcrack evolution law, the rate of the driving force is not appearing, as the process is parametrized with the number of cycles as a time-like parameter.

284


1 0.95 FN Fo

0.9 mesh A mesh B

0.85

mesh C

0

0.5 1 1.5 number of loadcycle N 106

2

Fig. 3.142. Load-cycle curves for different mesh patterns

Summary and Conclusions A micromechanical based continuum damage model based on the reduction of stiffness due to kinking elliptical microcracks has been proposed to show the anisotropic irreversible process of damage accumulation due to microcrack kinking and growth in brittle and quasi-brittle materials. The model is formulated consistently in a fully analytical way and the degradation of the elastic properties is associated with the irreversible process of crack kinking and growth. In order to make the formulation of the model mathematically traceable, the concept of an equivalent elliptical crack is proposed. The geometry and the orientation of the equivalent cracks are resulting from the postulates of equivalent dissipation and equivalent type of damage induced anisotropy. The proposed formulation yields a consistent damage model suitable for predicting the failure of structures and mechanical components subjected to fatigue conditions, independent of the type of loading. Accounting for the kinking and growth of microcracks and the type of damage induced anisotropy in the formulation of damage models is especially important in the case of non-proportional loads or even sequential loads, since the assumption of self-similar growth of mixed-mode cracks may underestimate the accumulated damage. The small strain assumption, and the lack of permanent deformations in this model makes it suitable to show the evolution of damage in structures with brittle and quasi-brittle fracture behavior experiencing highcycle fatigue. The proposed damage model can also capture the unilateral effect observed in tension-compression tests, observed for a certain classes of materials including ceramics and concrete, provided that for a passive crack the corresponding components of the compliance components are recovered, and they return to their degraded state upon the activation of the crack.

3.3 Modelling

285

3.3.2 Non-mechanical Loading and Interactions Authored by Otto T. Bruhns and G¨ unther Meschke 3.3.2.1 Thermo-Hygro-Mechanical Modelling of Cementitious Materials - Shrinkage and Creep Authored by Max J. Setzer and G¨ unther Meschke 3.3.2.1.1 Introductory Remarks While in engineering practice drying shrinkage is accounted for by means of shrinkage strains εs depending on an empirically determined shrinkage coefficient [182], recent progress in computational durability mechanics (see, e.g. [800, 142, 82]), together with appropriate numerical methods open the perspective of a more fundamental approach to obtain not only a better insight into the degradation mechanisms resulting from the interaction between mechanical, thermal and hygral loading but also to provide more reliable estimates for the (residual) life-time of concrete structures. In particular, thermodynamics of deformable porous media according to the Biot-Coussy-theory [122, 211] provides a suitable framework for the homogenization of microscopic or submicroscopic quantities to describe coupled hygro-mechanical mechanisms and damage on a macroscopic level. Within this framework, a non-linear poro-elastic model was employed in [212] for the modeling of drying shrinkage of concrete structures. In recent extensions of the theory of porous media, damage mechanics [89, 167, 177, 144, 533, 320, 533] and the viscous behavior of brittle materials [176] were taken into account. For saturated porous media the concept of plastic effective stress has been introduced at a macroscopic level as the thermodynamic force associated with the plastic strain of the skeleton [211]. This concept has been extended to partially saturated materials by [167] and, using micromechanical considerations, by [320, 321, 533]. In attempts to describe drying creep processes in concrete structures, the coupling between spatially and temporarily varying moisture distributions and the extent of long term creep have been considered by [110] and [320]. Cracks, irrespective of their origin, have a considerable influence on the moisture permeability of cementitious materials. As a consequence, the transport of aggressive substances may be promoted and the degradation process is further accelerated (see Subsection 3.1.2.3). In this Subchapter, a 3D coupled thermo-hygro-mechanical elastoplastic damage model for durability-oriented finite element analyses of concrete structures, formulated within the framework of the Biot-Coussy-theory [122, 211] that has been developed in [321, 533, 320] is presented. In accordance with the hygral processes acting on the various levels of the nano-porous skeleton (nano, micro and capillary pores) (see Subchapter 3.1.2.2.1 the effect of shrinkage is

286


taken into consideration by means of a macroscopic capillary pressure which represents, on a macroscopic level, hygrally induced stresses of various sources [533]. In addition to deformations and cracking resulting from drying shrinkage, the effect of cracks on the moisture transport, the moisture-dependence of the strength and stiffness of concrete and deformations resulting from longterm creep are considered in this model. Since long-term creep is associated with dislocation-like processes in the nano-pores of the cement gel which are prestressed by hygrally induced stresses (disjoining pressure) [93], creep depends on the moisture content. This coupling between moisture transport and creep deformations is also considered in the model. The specific coupled thermo-hygro-mechanical material behaviour of concrete described in the Subsection if formulated within the context of thermodynamics of deformable porous media and based on the Biot-Coussy-theory [122, 211]. In the model, concrete is assumed to consist of the matrix material (subscript s) – a mixture of cement paste and the aggregates – and the pores, which are partially saturated by liquid water (subscript l) and an ideal mixture of water vapour and dry air (subscript g). Provided that there is thermodynamic equilibrium between the mixture of water vapour and dry air and the external atmosphere, it is often assumed that the gaseous phase is at constant atmospheric pressure, taken as zero [97]. Therefore, for the sake of simplicity, the capillary pressure is expressed as pc = −pl in what follows. 3.3.2.1.2 State Equations Coupled phenomena on the microlevel of cementitious materials are described in a macroscopic framework using state variables. For the present 3D model, the function of free energy Ψ Ψ = W(ε − εp − εf , ml − ρl φpl , ψ, T ) + U(αR , αDP ),

(3.142)

depends on three external variables (ε, ml , T ) and six internal variables (εp , εf , φpl , ψ, αR , αDP ) [211, 213]. In Equation (3.142) ε denotes the linearised strain tensor, εp is the tensor of plastic strains, εf are the flow strains corresponding to long-term creep effects, ml denotes the liquid mass content variation, ρl is the mass density of the liquid phase, φpl stands for the nonrecoverable portion of the porosity occupied by the liquid phase, ψ = 1 − d is the integrity with d denoting the isotropic damage parameter 0 ≤ d ≤ 1, T denotes the absolute temperature and αR and αDP characterise the inelastic pre- and postfailure behaviour of concrete in tension (subscript R) and compression (subscript DP). From the entropy inequality, the state equations are obtained as ∂W ∂W σ= ; p l = ml ; ∂(ε − εp − εf ) ∂( − φpl ) (3.143) ρl ∂U ∂U ∂W ; qR = − ; qDP = − , S=− ∂T ∂αR ∂αDP

3.3 Modelling

287

where σ is the total stress tensor, pl is the liquid pressure and S is the entropy. qR and qDP are the thermodynamic forces conjugate to αR and αDP , which determine the damage-dependent size of the damage (fR ) and loading (fDP ) surface in the stress space. The differentiatial form of (3.143)) is obtained as dml p ed p f dσ = C u : (dε − dε − dε ) − ψM B (3.144) − dφl ρl + Λu dψ − Au dT, dpl = ψM

dml − dφpl ρl

− ψM B : (dε − dεp − dεf )

(3.145)

+ Ξdψ + LdT, Cu dT + Au : (dε − dεp − dεf ) + sl (dml − ρl dφpl ) T0 dml p −L − dφl + Πdψ. ρl

dS =

(3.146)

The symbols introduced in (3.144) – (3.146) represent the mixed partial derivatives of the free energy and can be interpreted as follows: Cu denotes the undrained fourth-order stiffness tensor, • C ed u = ψC • the term ψM B represents the hygro-mechanical couplings with M as the isotropic Biot modulus and B = b1 as the second-order tensor of tangential Biot coefficients b, • Λu is the undrained second-order tensor describing the coupling mechanisms between damage evolution and the total stress increment, • Au = C ed u : 1αt,u denotes the undrained second-order thermo-mechanical coupling tensor with αt,u as the undrained thermic dilatation coefficient, • Ξ is a coupling coefficient connected with the change of the liquid pressure due to damage evolution, • L = 3ψM αt,u characterises the thermo-hygral coupling mechanisms, • Cu denotes the undrained volume heat capacity and T0 the reference temperature, • sl is the internal entropy of the liquid phase and • T0 Π represents the latent heat due to damage evolution. Inserting (3.145) into (3.144) yields an alternative drained formulation for the differential stress tensor as dσ = C ed : (dε − dεp − dεf ) − Bdpl + Λdψ − AdT, with the drained stiffness tensor C, C ed = ψ C u − M b2 (1 ⊗ 1) = ψC

(3.147)

(3.148)

288


the drained thermo-mechanical coupling tensor A = Au − 3αt,u M B = C ed : 1αt ,

(3.149)

and the drained tensor Λ = Λu + ΞB,

(3.150)

respectively [321]. 3.3.2.1.3 Identification of Coupling Coefficients According to [321] the poroelastic hygro-mechanical coefficients b and M can be determined by relating differential stress and differential strain quantities defined on the meso-level to respective homogenised quantities on the macrolevel. The so-obtained tangential Biot coefficient is determined as K b = Sl 1 − ψ , (3.151) Ks which includes the expression b = Sl suggested by [211] for the special case of poroelastic materials with incompressible matrix behaviour. An expression for the Biot modulus M = ψM is obtained as −1 Sl pl ∂Sl φSl Sl (b − φSl ) M = φ 1− + + (3.152) Ks ∂pl Kl Ks see [705, 493] for a similar formulation. For cementitious materials, expression (3.152) can be replaced by −1 ∂Sl M≈ φ . (3.153) ∂pl In the special case of a fully saturated material (Sl = 1), (3.152) yields the classical relation [211, 493] −1 φ (b − φ) M Sl =1 = + . (3.154) Kl Ks The coefficients related to damage phenomena Λ and Ξ are identified by exploiting the symmetry relations that are connected to the existence of a macroscopic potential. Using the Maxwell symmetries, the drained tensor Λ can be expressed as [533] K p f Λ = C : (ε − ε − ε ) + Sl dpl 1 − C : 1αt T, (3.155) Ks pl and the coupling coefficient Ξ is obtained as M Sl K Ξ= Sl dpl ≈ 0. Ks2 pl

(3.156)

3.3 Modelling

289

3.3.2.1.4 Effective Stresses The concept of effective stress [281, 791] is a generally accepted approach in soil mechanics for the determination of stresses in the skeleton of fully saturated soils. In addition to the original proposal of [791], several alternative suggestions for the definition of effective stresses exist, taking the compressibility of the matrix material or the porosity into account (see e.g. [123, 587, 128]). Based on the relevance of the concept of effective stress for the analysis of fully saturated soils, this concept has also been adapted for the description of partially saturated soils. Early formulations introduced the capillary pressure in the (elastic) effective stress definition [127]. However, difficulties to obtain satisfactory agreements with experimental results have motivated the use of two independent stress fields for the constitutive modelling of unsaturated soils (see e.g. [129, 44]). As far as the numerical modelling of partially saturated cement-based materials is concerned, the assumption of (elastic) effective stresses seems not to be well suited for the description of shrinkage-induced cracks using stress-based crack-models. However, the concept of plastic effective stress first introduced at a macroscopic level by [210] for saturated porous media (see [211] for details), allows to overcome these difficulties in the framework of poroplasticity – porodamage models. The proposed form of the plastic effective stress is the same as the classical Biot-type, however, a plastic effective stress coefficient is used. A similar form has been derived from micromechanical considerations by [510]. This concept has been recently extended to partially saturated materials [167, 533], and is also adopted in the present formulation. From the coupled relations between total stresses, strains, liquid saturation and temperature C : (ε − εp − εf ) σ = ψC

K + 1−ψ Ks

(3.157) Sl (pc )dpc 1 − AT,

pc

the following definition of the elastic effective stress tensor C : (ε − εp − εf ) − AT, σ e = ψC

(3.158)

with

K Sl (pc )dpc 1. σ = σ e + 1 − ψ Ks pc

(3.159)

is obtained. The plastic effective stress tensor σ p = σ , defined as σ = σ − bp pc 1,

(3.160)

290


characterises the thermodynamic force associated with the plastic strain rate [211]. In contrast to the elastic effective stress tensor, σ represents the macroscopic counterpart to matrix-related micro-stresses with the coefficient bp as the plastic counterpart of the Biot coefficient b. By relating stress quantities on the meso-scale to respective macroscopic quantities, a possible identification of bp as a function of the integrity ψ, the porosity φ and the liquid saturation Sl can be accomplished as bp = ψφSl (pc ) ,

(3.161)

see [321] for details. 3.3.2.1.5 Multisurface Damage-Plasticity Model for Partially Saturated Concrete According to the concept of multisurface damage-plasticity theory, mechanisms characterised by the degradation of stiffness and inelastic deformations are controlled by four threshold functions defining a region of admissible stress states in the space of plastic effective stresses σ E = {(σ , qk )| fk (σ , qk (αk )) ≤ 0, k = 1, ..., 4} .

(3.162)

In (3.162), the index k = 1, 2, 3 stands for an active cracking mechanism associated with the damage function fR,k (σ , qR ) and k = 4 represents an active hardening/softening mechanism in compression associated with the loading function fDP (σ , qDP ). Cracking of concrete is accounted for by means of the Rankine criterion, employing three failure surfaces perpendicular to the axes of principal stresses fR,A (σ , qR ) = A − qR (αR ) ≤ 0,

A = 1, 2, 3.

(3.163)

In (3.163), the subscript A refers to one of the three principal directions and qR (αR ) = −∂U/∂αR denotes the softening parameter. The ductile behaviour of concrete subjected to compressive loading is described by a hardening/softening Drucker-Prager plasticity model qDP (αDP ) fDP (σ , qDP ) = J2 − κDP I1 − ≤ 0, (3.164) βDP with qDP (αDP ) = −∂U/∂αDP as the hardening/softening parameter. The determination of the model parameters κDP and βDP is based on the ratio of the biaxial and the uniaxial compressive strength of concrete fcb /fcu as [534] fcb /fcu − 1 1 , (3.165) κDP = √ 3 2fcb /fcu − 1 √ 2fcb /fcu − 1 βDP = 3 , (3.166) fcb /fcu

3.3 Modelling

291

whereby fcb /fcu is approximately equal to 1.16. The fracture energy concept is employed to ensure mesh-objective results in the post-peak regime. Details of the material model are found in [534]. For an efficient implementation of the multisurface model based on an algorithmic formulation in the principal stress space reference is made to [531]. The evolution equations of the tensor of plastic strains ε˙ p , of the reciprocal value of the integrity (ψ −1 )˙, of the plastic porosity occupied by the liquid phase φ˙ pl and of the internal variables α˙ R and α˙ DP are obtained from the postulate of stationarity of the dissipation functional [318] as ε˙ = (1 − β) p

4 k=1

γ˙ k

∂fk , ∂σ

(3.167)

∂fk ∂fk : Cu : ∂σ ∂σ γ˙ k , (ψ )˙ = β ∂fk k=1 : σ ∂σ 4 ∂fk γ˙ k : 1bp , φ˙ pl = ∂σ −1

4

(3.168)

(3.169)

k=1

α˙ R =

3 A=1

γ˙ R,A

∂fR,A , ∂qR

α˙ DP = γ˙ DP

∂fDP , ∂qDP

(3.170)

together with the loading/unloading conditions fk (σ , qk ) ≤ 0; γ˙ k ≥ 0; γ˙ k fk (σ , qk ) = 0.

(3.171)

The parameter 0 ≤ β ≤ 1 contained in (3.167) and (3.168) allows a simple partitioning of effects associated with inelastic deformations due to the crackinduced misalignment of the asperities of the crack surfaces, resulting in an increase of inelastic strains εp , and deterioration of the microstructure, resulting in a decrease of the integrity ψ. An elastoplastic model ((ψ −1 )˙ = 0, ε˙ p = 0) and a damage model ((ψ −1 )˙ = 0, ε˙ p = 0) are recovered as special cases by setting β = 0 and β = 1, respectively. 3.3.2.1.6 Long-Term Creep Consideration of long-term or flow creep effects is accomplished in the framework of the microprestress-solidification theory [93]. The evolution law of the flow strains is based on a linear relation between the rate ε˙ f and the stress tensor σ as 1 ε˙ f = G ed : σ, (3.172) ηf (Sf ) −1 with the fourth-order tensor G ed = E C ed and Young’s modulus E. The viscosity ηf is a decreasing function of the microprestress Sf and can be written as [93]

292


1 = cpSfp−1 , ηf (Sf )

(3.173)

where c and p > 1 are positive constants. According to [93], the microprestress relaxation is connected to changes of the disjoining pressure. Consequently, variations of the internal pore humidity h due to drying, which entail a changing disjoining pressure, lead to a change of the microprestress Sf . This mechanism partially explains the Pickett effect [631], also called drying creep. 3.3.2.1.7 Moisture and Heat Transport Starting with a simplified nonlinear diffusion approach, in which the different moisture transport mechanisms in liquid and in vapour form are represented by means of a single macroscopic moisture-dependent diffusivity [94], the relation between the moisture flux q l and the spatial gradient of the capillary pressure ∇pc is given by ql =

k · ∇pc . μl

(3.174)

In (3.174), k denotes the intrinsic liquid permeability tensor and μl is the viscosity of water. According to the hypothesis of dissipation decoupling [212], possible couplings between heat and moisture transport are disregarded in the present formulation. In order to account for the dependence of the moisture transport properties on the nonlinear material behaviour of concrete, k is additively decomposed into two portions as k = kr (Sl ) [kt (T ) kφ (φ) k 0 + kd (αR )] ,

(3.175)

one related to the moisture flow through the partially saturated pore space and one related to the flow within a crack, respectively [758]. This approach is consistent with the smeared crack concept. In (3.175), k0 denotes the initial isothermal permeability tensor, kr is the relative permeability, kt accounts for the dependence of the isothermal moisture transport properties on the temperature and kφ describes the relationship between the permeability and the porosity. Furthermore, kd is the permeability tensor relating plane Poiseuille flow through discrete fracture zones to the degree of damage in the continuum model, see [533, 319] for details. Using again the hypothesis of dissipation decoupling, the relation between the heat flux q t and the gradient of the temperature ∇T can be described by a linear heat conduction law reading q t = −Dt ∇T, whereby Dt (T, Sl , φ) denotes the effective thermal conductivity.

(3.176)

3.3 Modelling

293

3.3.2.1.7.1 Freeze Thaw Authored by Max J. Setzer and Jens Kruschwitz The main reason for frost damage in porous materials is the expansion by 9 Vol.-% in the transition from water to ice, if a critical degree of saturation in the pores is exceeded. This artificial saturation, e.g. observed by Auberg & Setzer [69], is as well a multi scaling as a coupled phenomenon. The scaling problem is characterised by the existence of two scales, which should be separated when modelling frost processes in hardened cement paste. Most relevant for the distinction between these scales are of course the macroscopic temperature changes and their typical time constants compared to the time necessary to obtain equilibrium within a certain scale. On the macroscopic scale transient conditions have to be modeled, i.e. mass transport due to viscous fluid flow is slow. On this scale the model deals with bigger volumes than on the microscale. In the big macroscopic volumes thermodynamic processes need a large time span to obtain equilibrium. This can be observed in practise as well as in standard experiments. The second part of the theory in this contribution is restricted to the nanoscopic CSH gel system consisting of solid CSH, pore water and air filled gel-pores with adsorbed water films. The liquid water film is an essential part of the Setzers model [726], which was determined by [812] experimentally. By going down in length scales it adopts primarily surface thermodynamics and the theory of disjoining pressure. At least thermal or thermodynamic equilibrium is established under normal conditions. This can be assumed for cubes of length up to 120 μm [731]. At constant temperature, the non-freezing interlayers and films are in equilibrium with ice and vapour. The temperature of the bulk ice governs the pressure and by this the equilibrium. Experiments have shown that the ice freezes in situ, referring to [778]. That means on the submicroscopic scale the motion of the pore water to the ice is highly dynamic. However, the response time for movement from gel to ice and the flow distance is rather small. Nevertheless, the pressure gradient is extremely high. By a combination of the Theory of porous Media (TPM), mainly influenced by de Boer [135], Ehlers [252], Bluhm [130], etc., and a micromechanical theory of surface forces developed by Setzer [723] the artificial saturation phenomenon can be described [448]. Basis of this model is the work of Kruschwitz & Setzer [450] and Kruschwitz & Bluhm [449] respectively. Last describe the frost heave of a critical filled cementitious matrix. In the mentioned combination the macroscopic, thermodynamic aspects of the model base on the Theory of Porous Media. This theory is a combination of the mixture theory and the concept of volume fractions. The interactions of the nanostructure of the hardened cement paste are modelled by a smeared micromechanical model. This part of the model is characterised by the properties of the two phase system solid and pore liquid. The transport on the micro structure and the unfrozen, adsorbed water film between matrix and ice are included.

294


3.3.2.2 Chemo-Mechanical Modelling of Cementitious Materials It has been shown in Subsections 3.1.2.3, 3.1.2.3.2, 3.1.2.3.3, 3.1.2.2.2 that the main microstructural mechanisms of environamentally induced corrosion and deterioration processes are by now fairly well understood. There seems to exist, however, a gap between research focused on the material level and durability oriented computational analysis of concrete structures. Although considerable progress has been achieved in the modeling of the mechanical behavior of concrete subjected to various loading conditions (see Subsection 3.1.1.1), environmental influences affecting the durability of concrete structures are stilc l accounted for by more or less heuristic evaluations of the degradation process and its influence on the residual structural safety. Recent progress in computational durability mechanics (see e.g. [75, 211, 800, 798, 214]), together with appropriate numerical discretization methods in space and time [460, 453] (see also Chapter 4) open the perspective of a more fundamental approach to obtain not only estimates for the life-time, but also to provide insight into the degradation mechanisms as a result of the interaction between mechanical and environmental loading. Using a continuum mechanics-based mode of description, concrete subjected to mechanical and non-mechanical loading is generally described as a multi-phase material whose behaviour is influenced by the interaction of the solid skeleton containing the cementititious matrix and the aggregates and the liquid and gaseous pore fluids. To this end, the scale of observation may either take the micro-scale or macro-scale as a point of departure. In the framework of a micro-scale approach the individual constituents are described by means of classical continuum mechanics for one-phase materials, formulating appropriately the interactions between the constituents and the contact conditions, respectively. To this end, the exact knowledge of the morphology of the material, in particular of the geometry of the pore space, is required. This is, however, not available in general. This difficulty motivates the description of porous materials on the basis of a macroscopic approach. The Theory of Mixtures (see e.g. [254] for more details) has been established as a suitable homogenisation procedure, which allows to treat multi-phase materials as a continuum while each constituent may be described by its own kinematics and balance equations. The interactions between the constituents are included by production terms within the balance equations. Since the Theory of Mixtures contains no microscopic information of the mixture it need to be complemented by the concept of volume. This leads to the well established concept of the Theory of Porous Media (TPM). It defines the volume fraction of each constituent dv α and the volume of the mixture dv, which provides a representation of the local microscopic composition of multi-phase materials: φα = dv α /dv. The sum of the volume fractions of all constituents has to be equal to one α φα = 1. The TPM provides a general continuum mechanically and thermo dynamically established concept for the macroscopic description of multi-phase materials like concrete.

3.3 Modelling

φ0 1 − φ0

mech. damage ← virgin material chem. dissolution ← virgin material

chem. dissolution ← virgin material

virgin material → mech. damage

295

dm φm

s˙ φc

φm φ0 φc 1−φ

Representative Elementary Volume (REV)

Theory of Mixture - material point

Fig. 3.143. Chemo-mechanical damage of porous materials within the Theory of Mixtures. Three types of deterioration are illustrated: virgin material, mechanically damaged material, chemically damaged material and chemo-mechanically damaged material

3.3.2.2.1 Models for Ion Transport and Dissolution Processes Authored by Detlef Kuhl and G¨ unther Meschke 3.3.2.2.1.1 Introductory Remarks Based on insights and data obtained from experimental investigations on calcium dissolution and coupled chemo-mechanical damage processes (see Subesection 3.1.2.3.2) constitutive models formulated on a macroscopic level of observation have been developed for the analysis of the time dependent dissolution process of concrete and concrete structures. One class of models is based on a phenomenological chemical equilibrium model relating the calcium concentration of the skeleton and the pore solution s(c) in conjunction with the concept of isotropic damage mechanics [422], as proposed by G´ erard [307] and subsequent publications (G´ erard [308], G´ erard et al. [311], Pijaudier-Cabot et al. [635, 634, 636] and Le Bell´ ego et al. [477, 479, 478]). Ulm et al. [801] and Ulm et al. [799] have proposed a chemo-plasticity model formulated within the Biot-Coussy-Theory of porous media [211]. This model is also based on a chemical equilibrium model, using empirical relations for the conductivity and aging. In both models, the irreversible character of skeleton dissolution is not accounted for. Hence, chemical unloading or cyclic chemical loading processes cannot be described. From the experiments the key-role of the porosity for the changing material and transport properties of chemo-mechanically loaded cementitious materials becomes obvious. Based on this observation and in order to consider

296


the interaction phenomena of chemical and mechanical material degradation described in Subsection 3.1.2.3.2 a fully coupled chemo-mechanical damage model has been developed in [454, 455] within the framework of the Theory of Porous Media. The material is described as ideal mixture of the fully saturated pore space and the matrix. In this model, the pore fluid acts as a transport medium for calcium ions. The pore pressure, however, is not accounted for in the present version of the model. The changing mechanical and transport properties are related to the total porosity defined as the sum of the initial porosity, the chemically induced porosity and the apparent mechanical porosity. Together with the assumptions of chemical and mechanical potentials the need for further assumptions or empirical models is circumvented. Micro-cracks are interpreted according to Kachanov [422] as equivalent pores affecting, on a macroscopic level, the conductivity and stiffness but not the mass balance. The evolution of the mechanically and chemically induced porosities are both controlled by internal parameters. This enables the modeling of cyclic loading conditions and allows a consistent thermodynamic formulation of the coupled field problems [454]. The link between the mechanical and the chemical field equations is accomplished by the definition of the total porosity φ as the sum of the initial porosity φ0 , the porosity due to matrix dissolution φc and the apparent mechanical porosity φm : φ = φ0 + φc + φm .

(3.177)

The chemically induced porosity φc can be calculated by multiplying the amount of dissolved calcium of the skeleton s0 − s by the averaged molar volume of the skeleton constituents M/ρ φc =

M [s0 − s] , ρ

(3.178)

where s0 denotes the initial skeleton concentration. The apparent mechanically induced porosity φm considers the influence of mechanically induced micro pores and micro cracks on the macroscopic material properties of the porous material. It is obtained by multiplying the scalar damage parameter dm by the current volume fraction of the skeleton 1 − φ0 − φc φm = [1 − φ0 − φc ] dm .

(3.179)

This definition of the mechanical porosity φm takes into account that microcracking is restricted to the solid matrix material. 3.3.2.2.1.2 Initial Boundary Value Problem The coupled system of calcium diffusion-dissolution, mechanical deformation and damage is characterized by the concentration field c of calcium ions

3.3 Modelling

297

in the pore solution and the displacement field u as external variables and a set of internal variables concerning the irreversible material behavior. The macroscopic balance of linear momentum is given by: div σ = 0 .

(3.180)

The matrix dissolution-diffusion problem is governed by the macroscopic balance of the calcium ion mass in the representative elementary volume div qc + [[φ0 + φc ] c ]· + s˙ = 0 ,

(3.181) ·

whereby qc is the mass flux of the solute. The term [[φ0 + φc ] c ] accounts for the change of the calcium mass due to the temporal change of the porosity and the concentration, which is up to one dimension smaller compared to the calcium mass production resulting from the dissolution of the skeleton s˙ [452]. The system of differential equations (3.180)-(3.181) is completed by boundary conditions on the boundary Γ given by σ · n = t , qc · n = qc , u = u , c = c

(3.182)

and initial conditions in the domain Ω given by u(t = 0) = u0 , c(t = 0) = c0 ,

(3.183)

where qc is the calcium ion mass flux across the boundary and c is the prescribed concentration. 3.3.2.2.1.3 Constitutive Laws The elasto-damage constitutive law is characterized by the free energy function Ψm : Ψm =

1−φ ε : Cs : ε . 2

(3.184)

Herein, ε denotes the linearized strain tensor and C s is the fourth order elasticity tensor of the the skeleton. The derivative of the free energy function Ψm with respect to the strain tensor ε yields the stress tensor σ: σ=

∂Ψm = [1 − φ] C s : ε . ∂ε

(3.185)

The diffusion-dissolution problem is defined by the dissipation potential Ψc of the calcium ions in the representative elementary volume Ψc =

φ Dl γ·γ, 2

(3.186)

298


where γ = −∇c is the negative gradient of the concentration field. The derivative of the dissipation potential Ψc with respect to the negative concentration gradient γ results in the calcium ion mass flux vector qc qc =

∂Ψc = φ Dl γ ∂γ

(3.187)

of the pore fluid, which is discussed in the next subsection. In eqs. (3.186) and (3.187) D 0 denotes the second order conductivity tensor of the pore fluid. Consequently, the macroscopic conductivity of the porous material is given by D = φD 0 . φ = φ0 defines the chemical and mechanical sound macroscopic material (φc = φm = 0), characterized by the subscript s. Hence, the macroscopic conductivity of the virgin material is given by D s = φ0 D 0 . In contrast to existing reaction-diffusion models describing calcium leaching, the dependence of D 0 on the square root of the calcium concentration within the pore fluid is considered. This dependency follows from Kohlrausch’s law, describing the molar conductivity of strong electrolytes (see Atkins [66] and Section 3.3.2.2.1.4), using Nernst-Einstein’s relation. In the isotropic case, the conductivity tensor D 0 is given in terms of the second order identity tensor 1, the calcium ion conductivity for the infinitely diluted solution D00 ≥ 0 and the constant D0c ≥ 0. √ D0 = D0 1 = D00 − D0c c 1 (3.188) It can be observed, that the conductivity decreases with an increasing calcium concentration. This follows from the interaction of moving cations Ca2+ and anions OH− by electrostatic forces and viscous forces. For D0c = 0 Kohlrausch’s law (3.188) degenerates to Fick’s law [280] of independent diffusing particles. 3.3.2.2.1.4 Migration of Calcium Ions in Water and Electrolyte Solutions The molar conductivity Λ of a strongly electrolyte solution is given as function of the calcium ion concentration in the pore fluid c by the empirical Kohlrausch law, see Kohlrausch [436]. D 0 = D0 1 =

RT Λ1 z2 F 2

Λ = Λ0 − Λc

√ c

(3.189)

Herein, R = 8.31451J/Kmol is the universal gas constant, T is the total temperature chosen as T = 298K, z = 2 is the number of elementary charges of a cation Ca2+ , F = 9.64853 ·104C/mol is the Faraday constant, Λ0 = 11.9 · 10−3Sm2/mol is the molar conductivity at infinite dilution and Λc is the Kohlrausch constant of the molar conductivity. Based upon the ¨ckel-Onsager theory (Debye & model of ionic clouds, the Debye-Hu ¨ckel [230] and Onsager [602]) verifies Kohlrausch’s law and allows to Hu determine the Kohlrausch constant Λc ,

3.3 Modelling fluid conductivity D0

macroscopic conductivity φD0

8

8

6

6 κc c0

323K φD0

D0

2

D0c = 0, T = 298 D0c = 0, T = 298 T = 273K, T = 323K

0 0

5

10 c 15

299

0.0

κc /c0 0.0 0.2 0.4 0.6 0.8 1.0

φ 0.725 0.497 0.457 0.430 0.414 0.200

0.2 273K

20

2 0 0

25

0.8 5

1.0 10 c 15

20

25

Fig. 3.144. Conductivity of the pore fluid D0 [10−10 m2 /s] as function of the calcium concentration c [mol/m3 ] and the total temperature T [K]. Macroscopic conductivity of non-reactive porous media φD0 [10−10 m2 /s] as function of the calcium concentration c [mol/m3 ] and the porosity φ [−] with φ = φ(κc , dm = 0)

2 z2 e F 2 A= 3πη R T 2 q z3 e F 2 B= 24 π η R T RT

Λ c = A + B Λ0

(3.190)

where the constants A and B account for electrophoretic and relaxation effects associated with the ion-ion interactions. These constants are given in terms of the universal gas constant, the total temperature, the elementary charge e = 1.602177 ·10−19C, the constant q = 0.586, the electric permittivity = 6.954·10−10C 2/Jm and the viscosity η = 0.891·10−3kg/ms of water (see e.g. Atkins [66]). From comparing equations (3.188) and (3.189) the macroscopic diffusion constants D00 and D0c can be determined. D00 =

2 RT −12 m Λ = 791.8·10 0 s z2F 2

(3.191)

D0c

m2 RT = 2 2 Λc = 96.85·10−12 s z F

m3 mol

(3.192)

In the present model the macroscopic diffusion coefficient φD0 can be determined for any state of chemo-mechanical degradation characterized by the history variables κc and κm and for any concentration c. Figure 3.144 contains plots of the conductivity D0 in the pore fluid and the macroscopic conductivity φD0 vs. the calcium concentration within the

300


range c = 0 corresponding to pure water and c = c0 corresponding to the concentrated pore solution of the virgin material. The diagrams on the left hand side of Figure 3.144 underline the relevance of using higher order ion transport models, considering electrophoretic and relaxation effects, as a basis for realistic calcium leaching models. Standard and higher order transport models are characterized by D0c = 0 and D0c = 0, respectively. A pronounced change of the conductivity D0 proportional to the square root of the concentration c can be observed within the considered concentration range. The ratio of the conductivities related to the fully degraded material D0 (0) and the virgin material D0 (c0 ) is approximately 9 : 4. The average value of the conductivity D0 is approximately D0 ≈ 4 ·10−10 m/s2 . This is in accordance with the suggestion by Delagrave et al. [232], that in numerical analyses D0 /2 should be used as the macroscopic conductivity in order to fit experimental results. Standard transport models are not capable to capture the significant increase of the conductivity with a decreasing calcium ion concentration corresponding to propagating chemical damage in reactive porous media. As expected, the results of the standard model and the present model are identical in the case of infinitely diluted solutions (c = 0). The sensitivity of the ion transport with regards to temperature changes is studied by including plots of the conductivity for T = 273 K, corresponding to the freezing point of water (no calcium ion transport occurs below this temperature), and for T = 323 K, representing approximately a desert climate, in Figure 3.144. According to equations (3.188), (3.191) and (3.192), the conductivity D0 depends linearly on the total temperature T . Within the considered temperature interval D0 is only changed by approximately 16%. Compared to the influence of the concentration the influence of the temperature plays a minor role in the transport process of ions within the pore water of cementitious materials. On the right hand side of Figure 3.144, the macroscopic conductivity is plotted for various values of the threshold calcium concentration κc and the corresponding values of the porosity, respectively, assuming a non-reactive porous material. 3.3.2.2.1.5 Evolution Laws According to Simo & Ju [744] the evolution of the damage parameter dm (κm ) is described by the damage criterion Φm = η(ε) − κm ≤ 0 ,

(3.193)

where η and κm are the equivalent strain function and the internal variable defining the current damage threshold. From the Kuhn-Tucker loading/unloading conditions and the consistency condition Φm ≤ 0 , κ˙ m ≥ 0 , Φm κ˙ m = 0 , Φ˙ m κ˙ m = 0 ,

(3.194)

3.3 Modelling

301

follows, that κm is unchanged for Φm < 0 and calculated as κm = η otherwise. The description of the elasto-damage material model is completed by the definition of the equivalent strain η and the damage function dm . Here the equivalent strain measure proposed by de Vree et al. [814] is used ks − 1 [ks − 1]2 1 12ks η= I1 + I2 + J2 , (3.195) 2 2ks [1 − νs ] 2ks [1 − 2νs ] [1 + νs ]2 in which I1 = tr[ε], I2 = [tr2 [ε] − ε : ε]/2 and J2 = [εdev : εdev ]/2 are the first and the second invariant of the strain tensor ε and the second invariant of the strain deviator εdev , respectively. The parameter ks denotes the ratio of tensile to compressive strength and νs the Poisson’s ratio of the skeleton. The exponential damage function is given by dm = 1−

κ0m 1−αm +αm exp[βm [κ0m −κm ]] , κm

(3.196)

where κ0m is the initial damage threshold and αm , βm are material parameters. The state of the chemically induced degradation of the porous material is characterized by the chemical porosity φc (s). Starting from a chemical equilibrium state between the calcium solved in the pore fluid and the calcium bound in the skeleton, the dissolution process requires a decreasing concentration c in the pore fluid. Otherwise, if c is increased, the structure of the skeleton is unchanged. In order to describe chemically induced degradation similarly to the elasto-damage problem, an internal variable κc is introduced, which corresponds to the current chemical equilibrium state. Based on this internal variable κc , the chemical reaction criterion Φc is formulated as Φc = κc − c ≤ 0 .

(3.197)

According to the Kuhn-Tucker conditions and the consistency condition Φc ≤ 0 ,

κ˙ c ≤ 0 ,

Φc κ˙ c = 0 ,

Φ˙ c κ˙ c = 0 ,

(3.198)

the process of matrix dissolution is associated with a decreasing chemical equilibrium calcium concentration (κ˙ c ≤ 0). The dissolution threshold κc is unchanged for Φc < 0 and equal to the current calcium concentration of the pore fluid (κc = c) otherwise. The conditions (3.197) and (3.198) for the occurence of chemical reactions are identical to those given by Mainguy & Coussy [512]. This identity is shown in Kuhl et al. [454]. As already mentioned, the current state of the calcium concentration in the skeleton s is controlled by the spontaneous calcium dissolution. It can be described as a function of the chemical equilibrium threshold κc given by G´ erard [307, 308] and Delagrave et al. [232])

302


Skeleton Concentration s [kmol/m3 ]

16 s0

14 12 10

sh

8 6 4 ccsh

2

dissolution cp

c0

0 0

5

10

15

20

25

Porefluid Concentration c [mol/m3 ]

Fig. 3.145. Chemical equilibrium function by G´ erard [307, 308] and Delagrave et al. [232]

1 1 2 κc + κc − s = s0 − [1 − αc ] sh 1 − 10 400

s0 − sh αc sh m n − κc κc 1+ 1+ cp ccsh (3.199)

for 0 < κc < c0 and s = s0 for κc ≥ c0 . αc , n and m are model parameters. c0 and s0 are the initial equilibrium concentrations of the sound material, cp and ccsh are material constants related to the averaged fluid calcium concentration of the progressive dissolution of the portlandite and the CSH phases, sh is the solid calcium concentration related to the portlandite-free cement matrix. A plot of function (3.199) and an illustration of the material parameters are given in Figure 3.145. 3.3.2.2.2 Models for Expansive Processes Authored by Falko Bangert and G¨ unther Meschke 3.3.2.2.2.1 Introductory Remarks Several numerical models have been developed in order to characterize the observed behavior of concrete affected by the Alkali-Silica Reaction (ASR) on a material level or even a structural level. Depending on the level of observation these models follow either a mesoscopic or a macroscopic approach. A mesoscopic approach involves the analysis of a single representative aggregate particle and its vicinity, whereby the kinetics of the chemical and diffusional processes involved are described on the scale of the aggregates, see e.g. Baˇzant & Steffens [96]. On the other hand, in a macroscopic approach concrete is described at the scale of laboratory specimens, see e.g. Larive &

Volume fractions

Microstructure

3.3 Modelling t=0

t>0

303

t→∞

ϕr g

ϕ ϕl ϕu

φs = φu

⎫ ⎪ φr ⎪ ⎬ ⎪ ⎪ φ ⎭

φs = φ r + φu

φs = φr

u

φg

φg

φg

φl

φl

φl

Fig. 3.146. Microstructure, constituents and volume fractions of concrete as a partially saturated porous media: light gray → unreacted part of the skeleton, dark gray → reacted part of the skeleton, white → pore gas, black → pore liquid

Coussy [470]. In these models, the main characteristics of ASR are incorporated phenomenologically on the macroscopic level. Hence, they can directly be used for numerical analysis of concrete structures [798]. Concluding from Subsection 3.1.2.3.3, there are two main mechanisms that have to be taken into account for a computational model which allows for realistic predictions of concrete deterioration caused by ASR. Firstly, the gel formation by the non-instantaneous dissolution of silica and secondly the swelling of the gel by the instantaneous imbibition of water. Both processes strongly depend on the moisture content within the concrete since water acts as a transport medium of ions and as a necessary compound for the formation of the swollen gel. Only very limited information on the properties of the individual constituents on the microscale, in particular of the gel, is available. In a chemo-hygro-mechanical damage model for the simulation of damage induced by the Alkali-Silica Reaction of concrete developed by [83, 81] the Theory of Porous Media (see e.g. Ehlers [254], Lewis & Schrefler [493]) together with a geometrically linear kinematics is used as the macroscopic continuum mechanics framework for the numerical simulation of concrete structures affected by the Alkali-Silica Reaction. Concrete is modeled as a partially saturated porous material consisting of a mixture of three main superimposed and interacting constituents ϕα , namely the non-porous skeleton (index α = s), the pore liquid (index α = l) and the pore gas (index α = g). When the alkali-silica reaction has not yet started (t=0), the skeleton represents a mixture of the unreacted aggregates and the hydration products. During the alkali-silica reaction, mass of the aggregates passes non-instantaneously into mass of the gel. The model formulation is based on the idea, that the gel formation is initiated at the surface of the aggregate particles and progresses

304


from the surface inward the particles. It is assumed, that the gel, which is responsible for the pressure build-up and the macroscopic expansion, is trapped at the reaction sites inside the reacting aggregate particles. The possibility, that the expansive gel may permeate in pores and cracks in the cement paste located near the surface of the aggregate particles or even may diffuse away through the connected pores space according to a through-solution mechanism is not explicitly considered in this model [83, 81]. Thus, for an instant t > 0 the skeleton ϕs is regarded as a mixture of the unreacted portion of the aggregates, the gel and the hydratation products. ϕs = ϕu + ϕr

(3.200)

At the same time, the pore space is solely saturated by the pore liquid ϕl and the pore gas ϕg . The unreacted phase ϕu represents the unreacted, unswollen skeleton material before it was affected by the alkali-silica reaction. On the other hand the reacted phase ϕr represents the reacted, swollen skeleton material after completion of ASR. The reactive aggregates of the reacted phase ϕr are completely converted into a gel. During the alkali-silica reaction, mass of the unreacted phase ϕr passes non-instantaneously into mass of the reacted phase ϕr . The mass exchange, which phenomenologically represents the gel formation by the dissolution of silica, is illustrated in Figure 3.146. At time t = 0 the skeleton is not affected by ASR as is indicated by the light gray color corresponding to unreacted material. At an instant t > 0 the dark gray part of the skeleton has already been affected by ASR. Finally, for t → ∞ the entire skeleton is affected by ASR. Following the standard concepts of the Theory of Porous Media, it is assumed, that the constituents ϕα are homogenized over a representative volume element, which is occupied by the mixture ϕ = ϕs + ϕl + ϕg . Therefore, material points of each constituent ϕα exist at each geometrical point x. Hence, the local composition of the mixture ϕ is described by the volume fraction φα , which is defined as the ratio of the volume element dv α occupied by the individual constituent ϕα and the volume element dv occupied by the mixture ϕ (see Figure 3.146): φα =

dv α . dv

(3.201)

Since the solid skeleton is regarded as a binary mixture, the respective volume fraction φs is given as the sum of the volume fraction of the unreacted volume fraction φu and the reacted volume fraction φr : φs = φu + φr .

(3.202)

It follows from definition (3.201), that the saturation condition must hold: φs + φl + φg = 1 .

(3.203)

3.3 Modelling

305

The material density $α and the partial density ρα of the constituent ϕα are introduced as $α =

dmα , dv α

ρα =

dv α dmα dmα = = φα $α . dv dv dv α

(3.204)

Herein, dmα denotes the local mass of the volume element dv α . The partial density of the skeleton is assumed to be composed by an unreacted and a reacted part: ρs = φs $s = φu $u + φr $r .

(3.205)

For the material densities of the unreacted and the reacted material the relationship $u > $ r

(3.206)

is assumed. By means of equation (3.206) it is considered, that during the noninstantaneous gel formation represented by the mass exchange between ϕu and ϕr the gel swells instantaneously. In other words, the ratio $u /$r represents phenomenologically the volume increase of the gel by the imbibition of water. The amount of water imbibed by the gel and consequently the ratio $u /$r strongly depend on the moisture content of the concrete. Since according to equation (3.206) the material densities of the unreacted and the reacted material are different, a variation of the volume fractions φu and φr due to the aforementioned mass exchange results in a variation of the material density of the skeleton $s , see equations (3.202) and (3.205). Thus, the ASRinduced swelling of the skeleton is associated with the variation of the material density $s . 3.3.2.2.2.2 Balance Equations Investigations on the role of water in the alkali-silica reaction have shown, that reactive concrete specimens do not absorb significantly more water than non-reactive ones, when they are stored under the same hygral conditions [469, 471]. Thus, no specific model needs to be developed to predict water movement in ASR affected concrete and it is reasonable to neglect any mass exchange between the skeleton and the pore fluids. In doing so, the mass balance equation of the skeleton ϕs as binary mixture reads [254] (φs $s )s + φs $s div(xs ) =

∂[φs $s ] + div(φs $s xs ) = 0, ∂t

(3.207)

where (•)α = ∂(•)/∂t + grad(•) · xα denotes the material time derivative of the quantity (•) following the individual motion of the respective constituent ϕα .

306


Assuming incompressible constituents ϕu and ϕr of the skeleton ϕs (→ $ =const., $r =const.), the associated partial mass balance equations of the unreacted and reacted phase result in the following volume balance equations: u

∂φu ∂φu→r + div(φu xs ) = , ∂t ∂t

∂φr ∂φr←u + div(φr xs ) = . ∂t ∂t

(3.208)

The terms $u ∂φu→r /∂t and $r ∂φr←u /∂t represent the mass exchange between the phases ϕu and ϕr due to the dissolution process. Since the summation of the partial balances (3.208)1 and (3.208)2 must result in the mixture balance equation (3.207), the following constraint must hold: $u

∂φu→r ∂φr←u + $r = 0. ∂t ∂t

(3.209)

Proceeding with the assumption, that the kinetics of the dissolution of silica and consequently the mass exchange between the constituents ϕu and ϕr follow a first order kinetic law, one may write (e.g. Atkins [66]) $u ∂φu→r $u ∂φr←u =− r = r k φu , ∂t $ ∂t $

∂φu→r = −k φu , ∂t

(3.210)

whereby the parameter k is the reaction velocity. Inserting the equations (3.210) into the volume balance equations (3.208) and neglecting the skeleton velocity xs ≈ 0 ,

(3.211)

yields: ∂φu = −k φu , ∂t

$u ∂φr = r k φu . ∂t $

(3.212)

For constant environmental conditions the volume balance equations (3.212) can be integrated analytically with the initial value φu0 = φs0 leading to φu = φu0 [1 − ξ] ,

φr =

$u u φ ξ, $r 0

(3.213)

where ane overall reaction extent ξ has been used. Finally, inserting (3.213) into (3.205) yields the material density of the skeleton $s as a function of the reaction extent ξ: $s =

$u $ r . $r + ξ [$u − $r ]

(3.214)

Thus, expression (3.214) reflects the swelling state of the skeleton ranging from an unswollen state (ξ = 0 ⇒ $s = $u ), if the alkali-silica reaction has

3.3 Modelling

307

not yet started, to a fully swollen state (ξ = 1 ⇒ $s = $r ) after the ASR process has come to an end. In analogy to equation (3.207), the mass balance equations of the pore fluids ϕβ (index β = l → liquid phase, index β = g → gas phase) are given by: ∂[φβ $β ] + div(φβ $β xβ ) = 0 . ∂t

(3.215)

Neglecting the material compressibility of the pore liquid in comparison to the material compressibility of the pore gas (→ $l = const.), and using the assumption (3.211) one obtains from equation (3.215): ∂φl + div(φl wl ) = 0 , ∂t

∂[φg $g ] + div(φg $g wg ) = 0 . ∂t

(3.216)

The partial momentum balances for the quasi-static case with the body forces neglected are given by: ˆα = 0 . div(σ α ) + p

(3.217)

ˆ α the momentum production, Herein, σ α is the partial stress tensor and p which can be interpreted as the local interaction force per unit volume between ϕα and the other constituents. Thereby the following constraint ˆs + p ˆl + p ˆg = 0 p

(3.218)

must hold due to the overall conservation of momentum div(σ) = 0 ,

(3.219)

with the overall stress tensor σ = σ s + σ l + σ g . 3.3.2.2.2.3 Constitutive Laws The pore space of concrete φl + φg is partially saturated with liquid and partially with gas, see Figure 3.146. The degree of liquid and gas saturation sβ , respectively, is given by: sβ =

φβ . φl + φg

(3.220)

The pore liquid and the pore gas are separated by a curved interface (meniscus) because of surface tensions. The radius of curvature of this interface depends on the pressure jump across the interface expressed by the so-called capillary pressure pc : pc = pg − pl .

(3.221)

308


In what follows, however, the capillary pressure pc will be interpreted as a macroscopic pressure representing all hygrally induced stresses acting on various scales of the nano-porous cementitious material, see e.g. [97]. There exists a relationship between the water content of the porous medium expressed by the liquid saturation sl and the capillary pressure pc . In [83] the following expression for the capillary pressure pc as a function of the liquid saturation sl is used pc = pr

( − 1 ) n1 sl m − 1 ,

(3.222)

according to van Genuchten [306]. In equation (3.222), pr , n, m denote material parameters, which have to be determined experimentally. The relation (3.222) has been originally proposed for soils. However, the pc (sl )-relations determined experimentally by Baroghel-Bouny et al. [88] for different types of cementitious materials by means of water vapor sorption isotherms are well fitted by expression (3.222). From thermodynamical considerations follows that the stress state of the skeleton and the fluid constituents is separated into two parts, where the first part is governed by the skeleton deformation and the pore fluid flow, respectively, while the second part is governed by the pore pressures (see e.g. [254]):

σ s = σ s − φs p 1 ,

σ β = σ β − φβ pβ 1 .

(3.223)

Therein, the pore pressure p is given by Dalton’s law p = sl p l + s g p g ,

(3.224)

where pl denotes the unspecified liquid pressure, whereas the gas pressure pg is related to the material density $g by the following constitutive law for an ideal gas: $g =

Mg g p . RT

(3.225)

In equation (3.225), Mg denotes the molar mass of the pore gas, R the universal gas constant and T the absolute temperature. The overall stress tensor σ of the porous material is given by the sum of the partial stress tensors σ α according to (3.223):

σ = σs + σl + σg − p 1 .

(3.226)

In the Theory of Porous Media the fluid frictional stresses σ β are usually neglected (σ β ≈ 0), yielding the well known concept of effective stress (see Bishop [127]): σ = σ s − p 1 .

(3.227)

3.3 Modelling

309

For the modeling of brittle failure of the skeleton (reduction of stiffness and strength) the continuum damage theory proposed by Kachanov [422] is employed. According to the effective area concept by Kachanov [422], the scalar damage parameter d can be interpreted as the ratio of the damaged cross section and the initial cross section. Thus, the undamaged material is characterized by d = 0, while d = 1 corresponds to the complete loss of integrity. Since the stresses in the skeleton are transferred by the intact, undamaged cross section, the effective stress reads

σ s = [1 − d] φs0 C s : [εs − εas 1] ,

(3.228)

with the effective elasticity tensor of the skeleton C s = E s [II + ν s /[1 − 2ν s]1 ⊗ 1]/[1 + ν s ], defined in terms of the Young’s modulus E s and the Poisson’s ratio ν s . In equation (3.228), the volumetric expansion resulting from ASR is considered by the volumetric strain εas . As mentioned above, the ASR swelling of the skeleton results from the variation of the material density $s of the skeleton, compare equation (3.214). Therfore, the volumetric expansion εas is defined as εas =

$s0 −1, $s

(3.229)

where $s0 denotes the initial material density of the skeleton. There is an ongoing debate whether the gel formed by the dissolution of silica initially saturates the pores in the cement paste located near the surface of the aggregates before a expansive pressure builds up (see Section 3.1.2.3.3). The definition 3.229 of the ASR-expansion implies, that the gel is trapped at the reaction sites inside the aggregates, thus representing a part of the skeleton. Consequently, the local ASR-progress directly results in the deformation of the skeleton. However, an initiation period due to a filling process can be considered by the model by introducing an initiation threshold for the ASR-expansion as e.g. suggested by Steffens et al. [772]. In the model proposed by [83], an isotropic damage model characterized by a single damage parameter d and a strain based description of the damage evolution in the sense of Simo & Ju [744] is used (see Section 3.3.1.2.2). Since the local continuum description of material degeneration suffers from the loss of well-posedness beyond a certain level of accumulated damage resulting in unphysical numerical results, a gradient enhanced damage formulation as proposed by Peerlings et al. [614] is used as a means of regularization. The evolution of damage is governed by the deformation of the skeleton. According to Simo & Ju [744] an internal variable κ is introduced, which represents the most severe deformation the skeleton material has experienced in the previous loading history and which acts as a threshold below which there is no further damage evolution. The damage parameter d is an explicit function of the internal variable κ. The evolution of κ is governed by the damage criterion

310


Φ = η¯ − κ ≤ 0 ,

(3.230)

where η¯ denotes the non-local equivalent strain. From the Kuhn-Tucker loading / unloading conditions and the consistency condition ∂κ ≥ 0, ∂t

Φ ≤ 0,

Φ

∂κ = 0, ∂t

∂Φ ∂κ =0 ∂t ∂t

(3.231)

follows, that κ is unchanged for Φ < 0 and calculated by κ = η¯ otherwise. The non-local equivalent strain η¯ in equation (3.230) is calculated on the basis of the following partial differential equation: η = η¯ − div(g grad(¯ η )) .

(3.232)

In this equation, η denotes the (local) equivalent strain representing a scalar measure of the local deformation state. Due to the gradient parameter g with the dimension of length squared an internal length scale is present in the formulation, which avoids the loss of well-posedness mentioned above. Finally, the equivalent strain η and the damage parameter d must be specified. Here, the equivalent strain measure corresponding to the Rankine criterion of maximal principal stress is used [83]: η=

1 max < σ ˜is > , s E

i = 1, 2, 3 ,

(3.233)

with max < σ ˜is > denoting the positive part of the largest eigenvalue of the ˜s . undamaged effective stress tensor σ The definition of the effective stress tensor σ s according to equation (3.228) is based on the assumption, that deterioration due to ASR only takes place if the ASR expansion εas is hindered. If the concrete can expand freely (σ s = 0 → εs = εas 1), the stiffness and strength are not affected by the alkali-silica reaction. This assumption, which is used for most model formulations in the literature [632, 798, 772], implies, that the degradation of concrete caused by ASR is mainly induced by structural effects. These structural effects may result from hindered deformations due to geometrical constraints or from gradients in the ASR expansion following from a non-uniform moisture distribu tion. It should be mentioned, that even under stress-free conditions (σ s = 0) microcracks can develop in the vicinity of the aggregate particles e.g. due to geometrical incompatibilities. However, on the macroscopic level the structural effects have a much more severe influence on the deterioration of concrete structures than these microcracks on the level of the aggregate particles. Although the fluid frictional stresses σ β are neglected, the fluid viscosity ˆ β in the partial momentum is included via the momentum production terms p balance equations (3.217). These are chosen as 2 μβ ˆ β = pβ grad(φβ ) − φβ p wβ , kβ

(3.234)

3.3 Modelling

311

with the dynamic viscosity μβ and the permeability k β [254]. In turn, the permeability k β depends on the intrinsic permeability k0 and on the nondimensional scaling factor krβ , which takes the dependence of the permeability k β on the saturation into account: k β = krβ k0 .

(3.235)

The intrinsic permeability k0 represents the permeability of the fully saturated porous material, which is independent of the saturating fluid phase. The influence of the saturation is considered according to van Genuchten [306] krl

( √ l m1 )m 2 l = s 1− 1− s ,

krg =

( 1 )2m 1 − sl 1 − sl m , (3.236)

where m is the same material parameter as used in the capillary pressure relation (3.222). ˆ β (3.234) into the related Finally, inserting the momentum productions p momentum balance equations (3.217) yields Darcy’s law: φβ wβ = −

kβ grad(pβ ) . μβ

(3.237)

By inserting the result into the partial mass balance equations of the pore fluids (3.216), the seepage velocities wβ can be eliminated as primary variables. 3.3.2.2.2.4 Model Calibration In this paragraph, the calibration of the chemical material parameters $u , $ and k, which control the deterioration caused by the alkali-silica reaction is described. First, a stress-free expansion test (σ = 0) of a reactive concrete specimen carried out at a certain temperature and humidity is considered. Inserting equation (3.228) into equation (3.227) yields after re-arrangement: r

εs = εas 1 +

1 Cs ]−1 : p 1 . [C [1 − d] φs0

(3.238)

For iso-hydro-thermal laboratory conditions the second part on the right hand side of equation (3.238) is almost constant since the pore pressure p does not change significantly in the course of the alkali-silica reaction. Hence, in laboratory tests on ASR affected concrete only the part εas of the strain tensor εs is measured, which is governed by the chemical reactions. Inserting the material density of the skeleton $s according to equation (3.214) and the initial value $s0 = $u into the ASR expansion εas defined in equation (3.229) yields for constant environmental conditions: u $ a εs = r − 1 ξ . (3.239) $

312


Differentiation of equation (3.239) with respect to time results in: u ∂εas ∂ξ $ ∂εas = = k [1 − ξ] r − 1 . ∂t ∂ξ ∂t $

(3.240)

From equations (3.239) and (3.240) the following values are obtained for the onset (t = 0 ⇒ ξ = 0) and for the completion of the alkali-silica reaction (t → ∞ ⇒ ξ = 1): u $ ∂εas ξ = 0 ⇒ εas = 0 , = k − 1 , ∂t $r u (3.241) $ ∂εas ξ = 1 ⇒ εas = r − 1 , = 0. ∂t $

0.4

Expansion εas [%]

1/k 0.3

0.2 u

r

/ − 1 0.1 Test results 0 0

100

200 Time t [d]

300

400

0.6

120 1/k

0.45

90

0.3

60 u

r

/ − 1 0.15

30 Test results Model results

0 0.7

Inverse velocity 1/k [d]

Asymptotic expasion u /r − 1 [%]

From equations (3.241) together, with the left diagram in Figure 3.147, which shows a typical strain evolution in a stress-free expansion test, the meaning of the chemical material parameters becomes clear: The parameter $u /$r − 1 represents the asymptotic strain in a stress-free expansion test. Furthermore, the parameter k controls the slope of the respective expansion-time-relation at the onset of ASR. Hence, the chemical material parameters $u /$r − 1 and k are well-defined and can be easily determined by means of macroscopic strain measurements on reactive concrete specimens. Both chemical material parameters ($u /$r − 1 and k) depend on the concrete mix design, the type of aggregates, the temperature and the moisture content. In particular, the moisture dependence plays a dominant role in the ASR deterioration. The role of moisture within the alkali-silica reaction has been studied in detail in an extensive test campaign at the Laboratoire Central des Ponts et Chaussées by Larive [469]. In these tests, cylindrical concrete

0 0.8

0.9

1

Liquid saturation sl [-]

Fig. 3.147. Illustration of the chemical material parameters k and u /r − 1 and of their dependence on the liquid saturation sl according to experimental results by Larive [469] and to model results by Steffens et al. [772]

3.3 Modelling

313

specimens of a certain mix design were stored under different hygral conditions (immersed in water, exposed to different relative humidities, wrapped in aluminum foil), whereby the temperature was kept constant at 38◦ C [471]. For each specimen, the macroscopic expansion and the weight change have been measured. The left diagram in Figure 3.147 shows a typical result of the expansion measurements. It turns out from the test results, that the asymptotic expansion $u /$r − 1 and the reaction velocity k increase with an increasing moisture content. In a recent study based on Larive’s results by Steffens et al. [772], the weight change of the specimens has been converted into an averaged liquid saturation sl within the specimens. An almost linear relation between the liquid saturation sl and the asymptotic expansion $u /$r − 1 as well as the inverse of the reaction velocity 1/k has been observed, which can be expressed by (Figure 3.147, right): $u − 1 = 1.27 sl − 0.754 [%], $r

1 = −300 sl + 326 [d]. k

(3.242)

Hence, the chemical material parameters of the proposed chemo-hygromechanical model controlling the deterioration caused by the alkali-silica reaction are known. 3.3.3 A High-Cycle Model for Soils Authored by Andrzej and Theodoros Triantafyllidis

Niemunis,

Torsten

Wichtmann

Based on the tests presented in Section 3.2.2 a high-cycle accumulation model [578] has been developed. The basic assumption of the model is that the strain path and the stress path that result from a cyclic loading can be decomposed into an oscillating part and a trend (accumulation) which can be treated separately. The oscillating part is described by the strain amplitude (Section 2.5.2). The model follows the trend using an empirical expression for the rate of strain accumulation D acc which depends (among others) on the oscillating portion of strain and enters the main constitutive equation σ˙ = E : (D − D acc − D pl )

(3.243)

with the Jaumann stress rate σ˙ of the effective stress σ, the strain rate D, the plastic strain rate D pl and the (barotropic) elastic stiffness E . In the high-cyclic context ”rate” means the derivative with respect to the number of cycles Nc , i.e. ˙ = ∂ /∂Nc , or the increment per cycle. In cyclic element tests D pl = 0 holds and we obtain pseudo-creep D = D acc at σ˙ = 0 or pseudoE : D acc at D = 0. In FE-calculations, apart from the highrelaxation σ˙ = −E cyclic loading, the elements are subjected to monotonic loading which requires a conventional plastic strain rate Dpl with the yield condition of Matsuoka and Nakai [520] (trσ tr(σ −1 ) < const) in order to restrict the admissible stress

314


ratios. For instance, such monotonic loading may be caused by accumulation effects in neighbouring elements. The direction m (a unit tensor) of D acc Dacc = Dacc m

(3.244)

turns out to be almost independent of the amplitude or the polarization of the cycles and independent of the density of the soil. The flow rule m is a function of the stress ratio only. The experiments (Section 3.2.2) show that the flow rule for monotonic loading proposed in the modified Cam clay (MCC) model → 1 q2 3 m= − p− 2 1 + 2 σ∗ 3 M p M

(3.245)

can be directly adopted to predict a sufficiently exact direction of accumulation, at least for the triaxial (axisymmetric) compression and extension. The asterisk ∗ denotes the deviatoric part of , the superposed arrow → denotes Euclidean normalization, and M (σ) describes in (3.245) the presumed (could not be fully tested in the conventional apparatus) dependence of m √ →∗ triaxial → → on the Lode angle θ = arccos(− 6 − σ ·− σ∗ : − σ ∗ ). The intensity of strain accumulation Dacc in Eq. (3.244) is calculated as a product of six empirical functions: Dacc = fampl f˙N fe fp fY fπ

(3.246)

The multiplicative combination of these functions has been also found empirically. Each function (see Table 3.23) considers separately the influence of a different parameter. The function fampl describes the proportionality between Dacc and the square of the strain amplitude (εampl )2 . The stress-dependence Table 3.23. Summary of the functions, material constants and reference quantities of the high-cycle model Influencing parameter Strain amplitude

Function

fampl = min

εampl ampl εref

2

Void ratio

fe =

(Ce −e)2 1+eref 1+e (Ce −eref )2

Reference quantities εampl = 10−4 ref

; 100

A B f˙N = f˙N + f˙N ) ( A A f˙N = CN1 CN2 exp − CN 1g f ampl B = CN1(CN3 f˙N ) av Average mean pressure fp = exp −Cp ppref − 1

Average stress ratio fY = exp CY Y¯ av

Cyclic preloading

Material constants

CN1 CN2 CN3 Cp

pref = 100 kPa

CY Ce

eref = emax

3.3 Modelling

315

(increase of Dacc with decreasing pav and with increasing stress ratio σ∗ /trσ) is captured by the functions fp and fY while fe expresses the increase of the rate with increasing void ratio. The increase of the accumulation rate due to changes in polarization is → − described by the factor fπ which compares the current polarization A of the amplitude with the so-called ’back polarization’ π , see [578]. If a package of cycles is directly followed by another package with the same polarization, i.e. →(1) − − →(2) A :: A = 1, then no correction of the accumulation rate is needed and fπ = 1. However, if the polarization has changed then the above product may become significantly smaller or even zero and then the rate of accumulation is increased up to fπ = 1 + Cπ1 . The 4-th rank back polarization tensor π represents the polarization in the recent history of cyclic deformation. The factor fπ entering Eq. (3.246) fπ = 1 + Cπ1 (1 − cos α)

(3.247)

→ − is proposed to be a function of the angle α = arccos( A :: π ) between the → − → − current polarization A and π . During cycles with A = const the tensor π → − is evolving (rotating) towards the current polarization, π → A . The angle α is proposed to evolve according to α˙ = −Cπ2 α (εampl )2

(3.248)

The constant Cπ2 > 0 influences the rate of this rotation. In order to perform π = R :: π by the angle Δα = αΔN the rotation π + Δπ ˙ we need to introduce the rotation operator (8-th rank tensor) → → → → → → → → R = (cos Δα − 1)(− μ ⊗− μ +− ν ⊗− ν ) + sin Δα(− ν ⊗− μ −− μ ⊗− ν ) +JJ (3.249) → − → − where μ = A + π and ν = A − π denote mutually orthogonal tensors constructed on the hyperplane perpendicular to the rotation axis. J denotes the 8-th rank identity tensor. The function f˙N describes the subtle problem of the dependence of Dacc on cyclic preloading (historiotropy). The cycles in the past are represented by their number Nc weighted by their amplitude [578]. For the sand samples in the laboratory the cycles can be counted from the beginning i.e. from their preparation, e.g. by pluviation. In situ the loading history is unknown and we must conclude the preloading from the behaviour of the soil under a cyclic test loading or from some empirical correlations. The cyclic loading in the past can be termed ”fatigue” which in case of soil means that preloaded soils respond with less accumulation (say of deformation) to the current cyclic loading than the freshly pluviated ones (at the same stress and density, see Figure 3.89d). The preloading can be quantified with g A defined via the following evolution equation gA A (3.250) g˙ = fampl CN 1 CN 2 exp − CN 1 fampl

316


The physical meaning (on the granular level) of this variable is not clear. Presumably it is related to the coordination number, to the statistical distribution of contact normals or to the spatial fluctuation of stress. The recent tests [581] could not definitively answer this question. The tensorial definition of the amplitude A ε for multidimensional strain loops is explained in Section 2.5.2. The determination of the material constants of the model has been discussed in detail in [842]. A simplified procedure proposed in [841] uses correlations of the constants with index properties (e.g. mean grain size d50 , uniformity coefficient Cu , minimum void ratio emin ). These correlations are based on approx. 200 cyclic triaxial tests. 3.3.4 Models for the Fatigue Resistance of Composite Structures Authored by Gerhard Hanswille and Markus Porsch 3.3.4.1 General Most design codes consider the static and fatigue resistance of composite steelconcrete structures with separate verifications for the ultimate limit state and the limit state of fatigue. For headed shear studs both verifications are based on world wide performed experimental investigations with push-out specimens. The determination of the fatigue life of headed shear studs in recent European codes is based on the S-N curve developed from the statistical analysis of a great number of cyclic push-out tests. The S-N curve considers the shear stress range only. In addition the maximum shear force is limited at serviceability limit states in order to avoid significant inelastic behaviour. However, recent researches showed that beside the shear stress range also the peak load and the static strength of the shear studs influence the fatigue life. On this background a comprehensive test program of more than 90 standard EC4-push-out test specimens and two full-scale beam tests was developed, in order to investigate the interaction between high cyclic loading and static strength as well as the effect of cyclic loading on the load-deflection behaviour of studs. This test program and the results are described in Chapter 3.2.3. More detailed information are given in [352, 349, 347]. The results clearly show that cyclic loading of headed shear studs leads to a substantial decrease of static strength of stud connectors during their lifetime, so that the assumption for independent limit states under service loads and under ultimate loads is not given[351, 354, 355, 353, 346, 643, 805]. In the following the results of the test program are evaluated and new models are presented considering the interaction between cyclic loading and a reduced static resistance in ultimate limit state. These models are not only focused on the local behaviour of headed shear studs, but also on the effect of the local damage on the global behaviour of composite beams.

3.3 Modelling

317

concrete failure

steel failure

concrete failure:

Pt ,c k c,m D d 2

steel failure:

Pt ,s

d fcm fu Ecm D kc,m , ks,m

E cm f cm

k s,m f u (S d 2 4)

diameter of the shank (16 d d d 22mm) mean value of the cylinder compressive strength mean value of the tensile strength of the stud shank mean value of the modulus of elasticity for concrete (secant modulus) = 0.2 [(h/d) + 1] for 3 d h/d d 4; = 1.0 for h/d > 4 coefficients to fit the theoretical model kc,m = 0.374, ks,m = 1.0 (mean prediction)

push-out test mean prediction: Pt = min (Pt,c , Pt,s)

Fig. 3.148. Theoretical model for the prediction of the mean value of the ultimate shear resistance according [684]

3.3.4.2 Modelling of the Local Behaviour of Shear Connectors in the Case of Cyclic Loading 3.3.4.2.1 Static Strength of Headed Shear Studs without Any Pre-damage Figure 3.148 shows the semi-empirical model for the prediction of the mean value of the ultimate shear resistance of headed studs [684], which was taken as the basis for the design rules in current national and international codes. It was derived for headed studs with a diameter of 16 mm to 22 mm embedded in solid slabs of normal weight concrete on the basis of the results of 76 statically loaded push-out test specimen, which were already in accordance to the version of the Eurocode 4 of the year 1994 [8]. The resistance is given by the minimum value of two equations, which describe the the failure mode ”shear failure of the stud” and ”concrete failure”, respectively. The model is based on the assumption, that in the case of low concrete strength the shear resistance is determined only by the failure of concrete in the lower part of the shank. In the case of high concrete strength it is assumed, that the shear resistance is determined by the shear resistance of the stud shank. It is known that this model is a simplified empirical approximation, but so far due to the complex interaction between the stud and the local concrete surrounding the stud it was not possible to find a verified mechanical model. The comparison of the static tests results (Pe ) for series S1 - S6 with the corresponding predictions according to Figure 3.149 shows that this model can also be applied on the new tests. Because of the relative high concrete strengths for all these cases the prediction according to the case ”shear failure of the stud” was decisive. Due to an erroneous specification of the secant modulus of elasticity Ecm in the earlier national and international concrete codes [5, 13] the evaluation of the statically loaded push-out tests was unintentionally based on both, the secant modulus Ecm and the tangent modulus Ec0m , although it was only

318


intended to use the secant modulus of elasticity. On this background the statistical analysis of [684] was repeated when specifying the national parameters of the German Annex of Eurocode 4 [34]. In this analysis additionally the new results of the static tests of series S1 - S6 and the results of larger headed studs with a diameter of 25 mm [343] were considered taking into account the revised secant modulus of elasticity Ecm according to the edited version of DIN 1045 [25]. In total 101 push-out tests could be included, which are summarized for the different failure modes in Table 3.24, Table 3.25 and Table 3.26. In these tables n means the number of studs per test specimen and h/d the ratio of the height of each stud (after welding) to its shank diameter. In 58 cases the criterion ”failure of the concrete” and in 43 cases the criterion ”shear failure of the stud” was relevant. Further information regarding specimen geometry and determination of the material properties are given in [345]. The result of the reanalysis according to EN 1990 [16] are shown in Table 3.27 and Figure 3.149. In accordance with the background report [684] the following coefficients of variation Vx were chosen. • • • •

Vx Vx Vx Vx

= = = =

3 % for the stud diameter d, 20 % for the modulus of elasticity (secant modulus) Ecm , 15 % for the cylinder compressive strength fcm , 5 % for the tensile strength of the headed stud fu .

In the case of relation of the equations of the theoretical model (Pt,c and Pt,s ) to the characteristic values (Xk ) of the cylinder compressive strength fck and the tensile strength of the headed studs fuk instead of each mean value (Xm ) the required partial safety factors γR shown in Table 3.27 can be reduced by the correction factors Δkc and Δks according equation 3.251. In the case of ”failure of the concrete” Δkc lies between 0.84 and 0.94 for a compressive strength range 20 ≤ fck ≤ 60 N/mm2 , thus a value of Δkc = 0.94 can be applied on the safe side. In the case of ”shear failure of the stud” Δk can be assumed constant equal to 0.92 for tensile strengths fuk between 400 and 620 N/mm2 . Δkc =

Pt,c (Xk ) Pt,c (Xm )

Δks =

Pt,s (Xk ) Pt,s (Xm )

(3.251)

Because of ∗ γR = Δkc · γR = 0.94 · 1.318 = 1.239

(3.252)

( γR according Table 3.27, column 3 ) and ∗ γR = Δks · γR = 0.92 · 1.198 = 1.102

(3.253)

( γR according Table 3.27, column 4 ) the design value of the shear resistance of a headed stud in concrete slabs with normal weight concrete as a short time static strength is given to:

3.3 Modelling

319

Table 3.24. Summary of the statically loaded push-out tests with decisive criterion ”failure of the concrete” (tests 1 - 27) reference [-]

[601]

[590]

PRd =

≤

test

no.

Pe

n

fcm

Ecm

fu

d

h/d

Pt,c

[-]

[-]

[kN]

[-]

[N/mm²]

[N/mm²]

[N/mm²]

[mm]

[-]

[kN]

SA1

1

88.5

8

28.2

25200

493

16

4.75

80.7

SA2

2

94.4

8

28.2

25200

493

16

4.75

80.7

SA3

3

90.3

8

28.2

25200

493

16

4.75

80.7

SB1

4

82.6

8

28.3

22300

493

16

4.75

76.1

SB2

5

76.7

8

28.3

22300

493

16

4.75

76.1

SB3

6

85.3

8

28.3

22300

493

16

4.75

76.1

A1

7

132.9

8

35.7

26300

499

19

4.00

130.8

A2

8

147.4

8

35.7

26300

499

19

4.00

130.8

A3

9

138.8

8

35.7

26300

499

19

4.00

130.8

LA1

10

111.1

8

25.6

24700

499

19

4.00

107.4

LA2

11

120.2

8

25.6

24700

499

19

4.00

107.4

LA3

12

112.0

8

25.6

24700

499

19

4.00

107.4

B1

13

124.3

8

33.6

22400

499

19

4.00

117.1

B2

14

115.2

8

33.6

22400

499

19

4.00

117.1

B3

15

115.2

8

33.6

22400

499

19

4.00

117.1

LB1

16

83.0

8

18.8

15400

499

19

4.00

72.6

LB2

17

82.1

8

18.8

15400

499

19

4.00

72.6

LB3

18

78.5

8

18.8

15400

499

19

4.00

72.6

2B1

19

118.4

8

33.6

22400

499

19

4.00

117.1

2B2

20

115.7

8

33.6

22400

499

19

4.00

117.1

2B3

21

113.4

8

33.6

22400

499

19

4.00

117.1

RSs1

22

135.0

2

27.0

24549

620

19

5.26

109.9

RSs2

23

133.0

2

27.0

24549

620

19

5.26

109.9

RSs3

24

122.0

2

21.8

22546

620

19

5.26

94.7

RSs4

25

131.0

2

21.8

22546

620

19

5.26

94.7

RSs5

26

133.0

2

25.5

23990

620

19

5.26

105.6

RSs6

27

142.0

2

25.5

23990

620

19

5.26

105.6

0.721 0.374 d2 α Ecm fck = 0.218 d2 α Ecm fck 1.239

(3.254)

d2 d2 0.811 1.000 π fuk = 0.736 π fuk 1.239 4 4

(3.255)

Due to short time relaxation effects in static tests under displacement control with structural composite members of steel and concrete a partly significant

320


Table 3.25. Summary of the statically loaded push-out tests with decisive criterion ”failure of the concrete” (tests 28 - 58) reference

test

no.

Pe

n

fcm

Ecm

fu

d

h/d

Pt,c

[-]

[-]

[-]

[kN]

[-]

[N/mm²]

[N/mm²]

[N/mm²]

[mm]

[-]

[kN]

[513]

S3 S4 S5 S6 S8 S11 S16 S19 S22 S26 S29

28 29 30 31 32 33 34 35 36 37 38

96.2 100.1 106.7 126.2 121.4 112.7 115.0 115.0 106.9 99.1 104.1

4 4 4 4 4 4 4 4 4 4 4

29.0 28.3 27.7 29.1 30.7 29.6 31.3 32.0 34.7 24.9 27.1

25273 25022 24805 25309 25873 25486 26081 26322 27233 23763 24586

600 600 600 600 600 600 600 600 600 600 600

19 19 19 19 19 19 19 19 19 19 19

5.33 5.33 5.33 5.33 5.33 5.33 5.33 5.33 5.33 5.33 5.33

115.6 113.6 111.9 115.9 120.3 117.3 122.0 123.9 131.2 103.9 110.2

[528]

P1 P2 P3 P4 P5 P6

39 40 41 42 43 44

97.5 96.5 97.0 127.0 127.0 127.0

4 4 4 4 4 4

16.6 16.6 16.6 40.8 40.8 40.8

20302 20302 20302 29196 29196 29196

600 600 600 600 600 600

19 19 19 19 19 19

5.33 5.33 5.33 5.33 5.33 5.33

78.4 78.4 78.4 147.4 147.4 147.4

[862]

D1/1 D1/2 D2/1 D2/2 D2/3 D3/1 D3/2 D3/3

45 46 47 48 49 50 51 52

99.0 94.0 123.0 128.8 126.5 148.5 148.0 146.8

4 4 4 4 4 4 4 4

30.2 30.2 30.2 30.2 30.2 30.2 30.2 30.2

25698 25698 25698 25698 25698 25698 25698 25698

580 580 500 500 500 548 548 548

16 16 19 19 19 22 22 22

6.25 6.25 5.26 5.26 5.26 4.54 4.54 4.54

84.3 84.3 118.9 118.9 118.9 159.5 159.5 159.5

[371]

2A

53

141.0

4

40.3

29040

485

19

3.68

136.7

[343]

I/1 I/2 I/3 I/4 I/5

54 55 56 57 58

179.5 183.0 180.4 183.1 178.6

8 8 8 8 8

23.7 23.7 23.7 23.7 23.7

29445 29445 29445 29445 29445

468 468 468 468 468

25 25 25 25 25

5.00 5.00 5.00 5.00 5.00

195.3 195.3 195.3 195.3 195.3

loss of load bearing capacity can be observed, when the actuator is held in constant position. In push-out tests near ultimate load this loss amounts approximately 10% [345], even if the tests are carried out with a very low displacement rate as in the present cases. In order to allow for these effects as a result of the test procedure the short time static strengths according equation (3.254) and (3.255) have to be reduced by an additional reduction factor in the order of 0.9. Thus on the basis a uniform partial safety factor γv = 1.25 for both failure modes the design value of the shear resistance of a single

3.3 Modelling

321

Table 3.26. Summary of the statically loaded push-out tests with decisive criterion ”shear failure of the stud” reference

test

no.

Pe

n

fcm

Ecm

fu

d

h/d

Pt,s

[-]

[-]

[-]

[kN]

[-]

[N/mm²]

[N/mm²]

[N/mm²]

[mm]

[-]

[kN]

T1/1 T1/2 T1/3 T1/4 T1/5 T3/1 T3/2 T4/1 T4/2 T4/3 T2/1 T2/2 T2/3 T2/4 T2/5 T5/1 T5/2 T6/1 T6/2 T6/3 3A 4A 5A II/1 II/2 II/3 II/4 II/5 S1-1a S1-1b S1-1c S2-1a S2-1b S2-1c S3-1a S3-1b S3-1c S4-1a S4-1b S4-1c S5-1a S5-1b S6-1a

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

144.5 147.8 135.5 148.9 137.8 140.1 145.1 137.3 133.7 137.7 170.1 168.1 165.9 170.6 168.8 176.3 177.5 166.1 159.9 177.9 166.0 160.0 172.0 233.0 238.0 234.9 243.5 232.8 191.3 211.3 213.0 201.3 173.3 175.3 216.0 200.6 201.0 186.8 176.5 179.1 184.6 186.8 196.0

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 4 4 4 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

36.7 36.7 36.7 38.3 38.3 44.7 44.7 44.7 44.7 44.7 36.3 36.3 36.3 36.3 36.3 59.0 59.0 57.3 57.3 57.3 39.1 47.1 57.5 41.3 41.3 41.3 41.3 41.3 44.2 49.0 49.7 44.7 42.8 42.8 56.2 53.9 53.9 43.4 43.4 43.4 42.9 42.9 45.8

27890 27890 27890 28405 28405 30397 30397 30397 30397 30397 27759 27759 27759 27759 27759 34546 34546 34069 34069 34069 28661 31119 34126 34687 34687 34687 34687 34687 36400 36400 36400 33800 33800 33800 39000 39000 39000 33900 33900 33900 33050 33050 33700

460 460 460 460 460 460 460 460 460 460 471 471 471 471 471 471 471 471 471 471 485 485 485 468 468 468 468 468 528 528 528 528 528 528 528 528 528 528 528 528 528 528 528

19 19 19 19 19 19 19 19 19 19 22 22 22 22 22 22 22 22 22 22 19 19 19 25 25 25 25 25 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22

5.26 5.26 5.26 5.26 5.26 5.26 5.26 5.26 5.26 5.26 4.50 4.50 4.50 4.50 4.50 4.50 4.50 4.50 4.50 4.50 5.26 5.26 5.26 5.00 5.00 5.00 5.00 5.00 5.68 5.68 5.68 5.68 5.68 5.68 5.68 5.68 5.68 5.68 5.68 5.68 5.68 5.68 5.68

130.4 130.4 130.4 130.4 130.4 130.4 130.4 130.4 130.4 130.4 179.0 179.0 179.0 179.0 179.0 179.0 179.0 179.0 179.0 179.0 137.5 137.5 137.5 229.8 229.8 229.8 229.8 229.8 200.7 200.7 200.7 200.7 200.7 200.7 200.7 200.7 200.7 200.7 200.7 200.7 200.7 200.7 200.7

[682]

[371]

[343]

[352]

stud connector considering time dependent effects due to high local concrete pressure in front of the studs is finally given by the minimum of equation (3.256) and (3.257).

322


Pe [kN]

Pe [kN] Pt,c

250

Pt ,c 0.374 d

2

200.7 Pt,s

250

E cm f cm

Pt ,s

200

PRk

VR = 0.19

150

PRd

200

Sd2 fu 4

PRk

VR = 0.12

PRd

150

100

100

50

50

S1-S6 PRk = 0.721 Pt,c

0 0

50

100

150

PRk = 0.811 Pt,s

Pt,c [kN]

PRd = 0.547 Pt,c

200

250

Pe experimental shear resistance

Pt,s [kN]

PRd = 0.678 Pt,s

0 0

50

100

150

200

250

PRk characteristic value of the shear resistance according

Pt,c mechanical model (concrete failure) (mean value)

EN 1990 (5%-fractile)

Pt,s mechanical model (steel failure) (mean value)

PRd design value of the shear resistance according EN 1990

Fig. 3.149. Result of the statistical analysis of the results of 101 statically loaded push-out tests according to EN 1990 [16]

PRd = 0.245 d2 α ≤ 0.83 π fuk

Ecm fck

d2 1 4 γv

1 γv

(γv = 1.25)

(3.256)

(γv = 1.25)

(3.257)

This result is nearly coincident to the original evaluation [684] and it confirms the use of the secant modulus of elasticity Ecm [33, 25] as one of the main material properties of the concrete in equation (3.256). In Figure 3.150 the result of the statistical re-analysis according EN 1990 is compared to the design rules of the German and the European rules. The design rules of DIN 18800-5 [27] are nearly identical to the result of the statistical re-analysis, whereas in the Eurocode 4 [22, 23] a significant higher shear resistance can be taken into account. In order to compensate this lower safety level in the German Annex of Eurocode 4 [34] a partial safety factor γv,c = 1.5 for the mode ”failure of the concrete” was introduced. 3.3.4.2.2 Failure Modes of Headed Shear Studs Subjected to High-Cycle Loading The test results given in Chapter 3.2.3 clearly indicate, that the mechanical properties of headed shear studs under static loading can not be applied without restrictions on the properties of headed shear studs subjected to

3.3 Modelling

323

Table 3.27. Result of the statistical analysis according EN 1990, Annex D [16] Table 3.24, Table 3.25 Pt,c

Table 3.26

58

43

¦ (P P ) ¦P

1.0

1.0

Pei b Pti

-

-

ln (G i )

-

-

0.035

0.012

0.124

0.087

0.124

0.088

0.139

0.078

VG2 Vrt2

0.187

0.117

test according theoretical model ("failure" mode) n

number of tests ei

b

b

Gi

Gi

'i

'i

1 n

'

'

1 n 1

s '2

s'

V G2

VG Vrt2

Vrt

6

V r2

Vr

¦'

¦ ('

i

i

') 2

exp (s 2' ) 1

ª 1 « « Pt ( X 1 ¬«

n i

ti

2 ti

º wPt Vi » » m ) wX i ¼»

Pt,s

2

QG

QG

ln (VG2

1)

0.124

0.087

Qrt

Qrt

ln ( Vrt2 1)

0.138

0.078

Q

Q

ln(Vr2 1)

0.185

0.117

Qrt2 Q2 k n G 0. 5 Q 2 ) Q Q

0.721 Pt,c

0.811 Pt,s

Q2 Qrt2 k d ,n G 0.5 Q 2 ) Q Q

0.547 Pt,c

0.678 Pt,s

1.318

1.198

PRk

PRk

b Pt (X m ) exp (1.645

PRd

PRd

b Pt (X m ) exp ( 3.04

JR

JR

PRk / PRd

kn

Vx unknown – 5%-fractile – (n)

1.694

1.713

k d ,n

Vx unknown – (n)

3.28

3.366

high-cyclic preloading. High cyclic loading leads to a reduction of the stiffness of the interface between steel and concrete due to the irreversible slip and moreover it results in an early reduction of the static strength. In order to find the reasons for the significant effect of high-cyclic loading, the concrete slabs were separated from the steel beams and the fractured surfaces at the

324


concrete failure: PRd ,c k c,d D d 2 steel failure: d fuk fck Ecm D kc,d , ks,d Jv,c , Jv,s

PRd,s

design value: PRd = min (PRd,c , PRd,s)

E cm f ck J v,c

kc,d / ks,d [-]

k s,d f uk (S d 2 4) J v,s

diameter of the shank (16 d d d 25mm) characteristic value of the ultimate tensile strength of the stud shank characteristic value of the compressive cylinder strength (according EN 206) mean value of the modulus of elasticity for concrete (secant modulus) (according EN 206) = 0.2 [(h/d) + 1] for 3 d h/d d 4; = 1.0 for h/d > 4 coefficients to fit the theoretical model partial safety factors for the design shear resistance

statistical analysis 0.245 / 0.83 (EN 1990)

Jv,c / Jv,s [-]

fu [N/mm²]

1.25 / 1.25

460 - 620

DIN 18800-5

0.25 / 0.80

1.25 / 1.25

< 450

EN 1994-1-1

0.29 / 0.80

1.25 / 1.25

< 500

EN 1994-1-1 incl. National Annex

0.29 / 0.80

1.50 / 1.25

< 500

Fig. 3.150. Comparison of the result of the statistical analysis with the rules in current German and European standards

metallurgical investigations

microstructure

forced fracture area and fatigue fracture area

Fig. 3.151. Preparation stages for examination purposes

foot of each headed stud of each test specimen were examined. Figure 3.151 shows in detail the stages of preparation of the test specimens after the test phases for examination purposes. In two specific cases additional metallurgical investigations of the microstructure were carried out. The exposed fracture surfaces at each stud foot consisted of a typical smooth fatigue fracture zone and a partly coarse forced fracture zone as shown in Figure 3.152. In nearly all cases these zones could be clearly distinguished

3.3 Modelling Mode A

Mode B

stud shank

fatigue fracture (with arrest lines)

325

weld collar

fatigue fracture P1

crack tip P2 forced fracture

mode B

forced fracture

Mode A: crack initiation at point P1 followed by a horizontal crack propagation through the shank

P1: transition between the stud shank and the weld collar P2: transition between the weld collar and the flange

Mode B: crack initiation at point P1 or at P2 followed by a crack propagation headed through the flange

Fig. 3.152. Failure modes A and B

from each other because of the different surface structures, so that it was possible to determine clearly the size and the geometry of the exposed fatigue fracture areas. The fatigue fracture area was in all cases caused by cracks at the stud foot, initiated at the points P1 or P2 and then propagating horizontal through the shank or headed through the flange. The corresponding forced fracture area was caused by a combination of a bending-shear failure of the residual cross section. This kind of failure occurred at the end of a fatigue test at which due to crack propagation the static strength was reduced to the applied peak load or during the static loading phase after high cyclic preloading, which was carried out in order to determine the residual strength. The failure modes were closely correlated with the peak load Pmax . For high peak loads such in series S2 and S4 only mode A occurred. For lower peak loads such in series S1, S3, S5E in most cases mode B occurred. Nevertheless in some cases two cracks of mode A and mode B were detected at the same time at a stud foot, which means, that two cracks grew directly above each other and both could initiate forced fracture. The investigations of the microstructure revealed that both points, P1 and P2, show exceptionally high geometrical and metallurgical notch effect due to welding technique. This is in no case in agreement with the requirement of common arc-welded joints in structural steelwork regarding the quality level according to [35]. Both sharp transitions are typical results of the drawn arc stud welding process. The process begins with pre-setting the current time and the welding time and placing the stud on the flange. Upon triggering a pilot arc occurs after lifting the stud to a pre-set height. Subsequently the main arc is ignited which melts the end of the stud and the flange on the opposite side. By means of a spring force finally the stud is forged into the molten flange.

326


crack initiation point (P2)

proper stud weld

200:1 crack propagation inter- and transcristallin

voids with rough surfaces and transitions

200:1

corresponding crack tip

Fig. 3.153. Weld collar (exterior appearance and inner state) - Close-up view of the crack shown in Figure 3.152 at the starting point (P2) and at the corresponding crack tip

This forces excessive material out into the ceramic ferrule shaping the weld collar. Due to the different aggregate states this does not lead to a fusion between the inside of the weld collar and the outside of the stud base and results in sharp edged transitions in P1 and P2. These two points coincide with the points of the highest stress levels and the crack growth consequently starts at these notches. Moreover Figure 3.153 (left) illustrates, that the drawn arc welding process leads to an apparent faultless weld collar on the outside, but on the inside it may contain voids due to the degassing process during welding. So contrary to the outside appearance the weld collar is generally not homogeneous and of lower strength compared to the stud and the base material. Figure 3.153 (right) shows the crack initiation point P2 and the corresponding crack tip of the crack in Figure 3.152 enlarged 200 times. It illustrates, that the transition between the weld collar and the flange is not smooth but undercut, being an ideal condition for early crack initiation in the case of high cycle loading. In the present case the crack propagated both transcrystalline and intercrystalline. Beginning near the line of fusion at the transition between the collar and the flange the crack grew through the fine grained structure of the heat affected zone, working its way through the coarse grained structure

3.3 Modelling

weld collar

327

crack orientation

shank

fatigue fracture area (AD)

crack front

mode B

forced fracture area (AG)

mode A

point C

circular

Pu / Pu,0

1.0

Pu AD | 1 ( Eq . B) Pu,0 AD AG

0.8

62 tests Æ 496 studs

Pu AD (Eq. A) | 1 0 . 6 Pu,0 AD AG

0.6

0.4

0.2

failure

cyclic loading

series

static

constant

S2, S4

fatigue static

constant constant

S2, S4 S1, S3, S5E

fatigue

constant

S1, S3, S5E

fatigue fatigue static

variable variable variable

S5 S6 S9

only mode A within a specimen

mode A and mode B within a specimen

AD/(AD+AG)

0.0

0.0

0.2

0.4

0.6

0.8

1.0

Fig. 3.154. Correlation between reduced static strength and damage at the stud feet for failure modes A and B based on the fatigue fracture area for av < ah

of the heat affected zone and ending at the non-affected base material of the flange. 3.3.4.2.3 Correlation between the Reduced Static Strength and the Geometrical Property of the Fatigue Fracture Area In order to detail the crack development, the test specimen were released and reloaded periodically during the cyclic loading phases. As shown in Figure 3.154 in the case of mode A it was possible to produce arrest line by means of this test procedure, which could be used for information about the number of load cycles causing crack initiation and about the crack propagation. Probably due to different microstructure no usable stop marks could be observed in the case of mode B although the testing procedure was always the same. However, in all cases geometrical properties of each fatigue fracture area (such as outline, size (area AD ) , extension in the direction of the loading (crack length ah ), extension into the base material (crack depths av )) can be used for evaluation purposes. The relationship between the reduced static strength and the relative size of the fatigue fracture zone can be assumed to be linear as a good approximation independently of the modes. This is illustrated in Figure 3.154, which shows the result of an evaluation of 496 studs of 62 push-out tests.

328


In Figure 3.154 AD is the area of the fatigue cracking zone and AG the area of the forced shear fracture, both taken as the horizontal projections. In the case of mode A the whole fracture area (AD + AG ) corresponds to the stud area, which is for this reason clearly defined. In the case of mode B due to the crack propagation into the flange the whole fracture area can be much larger than the stud area. In order to interpret the test results in a definite way and additionally allow for situations, in which only the fatigue fracture area AD (e.g. from non-destructive measurements) is known, it is necessary to make reasonable assumptions concerning the definition of the shape of the forced fracture area. Based on the observations of the failure modes the size of the forced fracture area was determined by assuming, that this area is bounded by the crack front and by a circular border passing through the outer diameter of the weld collar on the opposite side (given as point C in Figure 3.154). The coefficient of correlation of the linear relationship is 0.96 for series S2 and S4, in which due to the high peak loads of 0.70 Pu,0 exclusively mode A occurred. Except for very high degrees of damage of more than 90 % it can be deduced that the crack propagation in the shear stud independently of the modes has approximately 60 percent attribution in the reduction of the static strength. Regarding the reduced static strength the loading history during the cyclic loading phase (force controlled, displacement controlled, one block of loading and multiple blocks of loading) has only a minor influence. In the case of mode B (test series S1, S3 and S5E) the reduction of the static strength is very small for damage grades AD / (AD + AG ) between 35 % and 80 %. For estimations on the safe side the dotted relationship according equation (B) in Figure3.154 can be applied. For practical applications in which (e.g. from non-destructive inspection like ultrasonic) only the crack initiation point and the crack length at a stud foot instead of the whole outline of the fatigue fracture area is known, the relationships Eq. C and Eq. D according to Figure 3.155 can be used. If crack initiation starts at the outer edge of the weld collar (mode B) the horizontal crack length ah should be referred to the diameter dW of the stud weld. If crack initiation starts at the transition between the stud shank and the inner Pmax, Pu crack initiation at Point P2

dW

av

P2

Pmax, Pu crack initiation at Point P1

a ah

ah

d

av § 0 P1

Pmax, Pu

Eq . C

Pu Pu,0 | 1 0 .6 a h d W

ah

ah

Pmax, Pu

Eq . D

Pu Pu,0 | 1 0.6 a h d

Fig. 3.155. Correlation between reduced static strength and damage at the stud feet for failure modes A and B based on crack lengths and crack initiation points for av < ah

3.3 Modelling

329

edge of the stud weld (mode B) the crack length a (a ∼ ah ) should be referred to the stud shank diameter d. According to Figure 3.154 on the safe side the coefficient 0.6 can be substituted by 1.0. 3.3.4.2.4 Lifetime - Number of Cycles to Failure Based on Force Controlled Fatigue Tests In Figure 3.156 the results of the fatigue tests of series S1 to S4 and S5E are compared with the corresponding test results, from which the fatigue strength curve in Eurocode 4 was derived [685]. In this concept the prediction of the number of cycles to failure depends on the nominal shear stress in the shank of the studs, provided that the peak load Pmax is smaller than 0.6 Pu,0 [685]. It can be seen, that the lifetimes of the fatigue tests of series S1, S3 and S5E, which lie in the scope of application of the fatigue strength curve, are predicted very well. One of the reason is obviously the additional lateral supporting of the concrete slabs shown in Figure 3.100 of Chapter 3.2.3, which was not used in the tests on which the fatigue curve is based. However, the results of the fatigue tests clearly show the influence of the peak load Pmax on the life time. In the case of an identical relative load range ΔP / P u,0 it can be observed that if the relative peak load Pmax / P u,0 is increased the number of cycles to failure decreases from 6.2 × 106 to 3.5 × 106 load cycles (series S1 and S4) and from 6.4×106 over 5.1×106 to 1.2×106 load cycles (series S5E, S3 and S2), respectively. In order to develop a theoretical model for the prediction of the fatigue life, in which not only the effect of the load range ΔP can be taken into account, but also the effects of the static

'WR (log) 1000

'W R

§N · ¨ ç ¨ N ¸ © ¹

test results: m = 8.658 Eurocode 4: m = 8

1m

'Wc

S2, S4

(Pmax = 0.71 Pu,0 )

S1, S3, S5E (Pmax 0.44 Pu,0 ) 'Wcm = 110 N/mm²

100

5%-fractile 'Wck = 90 N/mm²

'P

d

'P 'W

10 104

105

4 'P Sd2

N (log) 106 Nc = 2x106

107

Fig. 3.156. Comparison of fatigue test results with the prediction in Eurocode 4

330


strength Pu,0 and the peak load Pmax , national and international fatigue tests of push-out test specimens subjected to unidirectional cyclic loading were reanalysed in the view of these parameters. To achieve comparable results great importance was attached to the geometry of the specimen, the number of welded studs and the lateral supporting condition of the concrete slabs. In this analysis only those tests were included, in which the requirements of the Eurocode 4 regarding geometry and test conditions were met. Thus the specimen had to consist of one steel beam and two lateral concrete slabs with four headed studs on each flange. The slabs had to be casted in horizontal position and the studs had to be welded with an adequate welding procedure ensuring the formation of weld collar in accordance with EN13918 [10] and EN14555 [11]. These requirements were fulfilled by 26 tests. In the case of 13 specimen the concrete slabs were additionally laterally supported. Among the group of test specimen without lateral supporting count the tests of Oehlers [591] and Hanswille [342] of 1989 and 1999. Among the other group count the fatigue tests listed in Table 3.6 (Chapter 3.2.3) and a fatigue test of Velkovic et al. [809] of 2003. In the case of [342] short time static tests were not carried out, so the reference value of the static resistance was calculated with the model given in Figure 3.148. In the 26 tests the concrete cylinder compressive strength fc according to EN 206 [12] varied between 31.0 N/mm2 and 54.3 N/mm2 . The range of the diameter d of the stud shanks was 13 mm to 25 mm and the tensile strength fu of the studs lied between 450 N/mm2 and 528 N/mm2 . For evaluation purposes the test were sorted in two groups each with identical supporting condition and evaluated by means of a common theoretical model according to Figure 3.157 giving the value of the fatigue life of a headed shear stud embedded in solid concrete slabs subjected to unidirectional cyclic loading. The free parameters K1 and K2 are to be chosen in dependence of the lateral supporting condition. In the case of additional lateral support, the parameters can be chosen to K1 = 0.1267 and K2 = 0.1344. In the case of no lateral support the parameters are to be chosen to and to K1 = 0.1483 and K2 = 0.1680. 3.3.4.2.5 Reduced Static Strength over Lifetime As it can be seen from the tests the static strength reduces with increasing number of cycles. The failure envelope, i.e. static strength over the number of cycles, is characterized by a sigmoidal shape as shown in Figure 3.158 (a). The results of the five more tests given in [809] with exact the same specimen geometry and supporting condition and a different relative peak load show the same characteristics and are also illustrated in Figure 3.158 (a). The sigmoidal relationship between the relative values of the static strength and the load cycles can be described with the equation given in Figure 3.158 (b). This equation is the result of a parametric study of totally 60 tests. It is to mention that the relative load range chosen in the tests was between 0.2 and 0.25. If further tests with different values of the load range are available

3.3 Modelling

331

Nf : number of load cycles to failure in a force-controlled push-out fatigue test

1 max Pu,0

10

theoretical model

Nf

P 0.5 'P K1 K 2 max Pu,0

with lateral restraint

Nf,t

108

P

without lateral restraint

107 26 tests 106

105 K1 = 0,1267

K1 = 0,1483

K2 = 0,1344

K2 = 0,1680

104 104

with without lateral restraint

105

106

107

108

Nf,e

experimental results

Fig. 3.157. Theoretical model for the prediction of the fatigue life of a headed shear stud in a push-out test - relationship between experimental and theoretical fatigue life a)

b) Pu Pu,0

1.2

(Pu Pu,0 ) theor

1.0 60 tests 0.8

1.0

0.6 0.8 0.4 0.6

0.2 series

0.4

0.2

0

0

1 2 3

0.20 0.25 0.25

0.44 0.71 0.44

4

0.20

0.71

5E

0.25

0.30

Vel

0.20

0.60

0.2

(Pu Pu,0 ) exp

ǻP Pu,0 Pmax Pu,0

0.4

0.6

0

Pu

N Nf 0.8

Pu,0

0

0.2

0.74

Pmax 'P Pu,0

0.4

0.6

0.8

1.0

d 1 § · ° 1 ¨ ¸ 0.54 - 0.04 ln ¨ 1¸ °® P ¨1 N N f ¸ °t max © ¹ °¯ Pu,0

1.0

Fig. 3.158. Analytical description of the reduced static strength over lifetime (a) Comparison of the theoretical and experimental values of the reduced static strength (b)

332


the parametric study should be repeated in order to extend the scope of application. 3.3.4.2.6 Load-Slip Behaviour Regarding the numerical simulation of composite beams the load deflection behaviour of headed shear studs under static and cyclic loading is of main interest. These results should not be neglected but be comprised as fundamental research results. According to different stages of the test procedure it was possible for the tests reported in Chapter 3.2.3 to deduce the load-deflection behaviour of headed studs embedded in normal weight concrete during initially static loading, during cyclic loading (including phases of releasing and reloading) and during static loading after high cycle pre-loading. As already known the initial static load-slip behaviour of headed shear studs embedded in normal weight concrete is characterized by a high initial stiffness and high ductility. Based on a statistical analysis of 15 comparable static push-out tests of the series S1-S6 and S9 the mean behaviour can be described by the exponential function, given in Figure 3.159, which can be applied up to mean value of the slip at ultimate load δu of 7.5 mm. The associated coefficient of variation Vx of the slip depends on the load level and P/Pu,0 [-]

P/Pu,0 [-]

scatter band

1.0

1.0 0.8

P Pu,0 | (1 e1.22 į

PRd = 0.68 Pu,0 (su)

0.6

)

mean behaviour

0.6

PRd = 0.55 Pu,0 (cu) 0.6 PRd = 0.41 Pu,0 (sf) 0.6 PRd = 0.33 Pu,0 (cf)

0.4 0.2

0.2

Gu = 7.5 mm

G [mm]

Gcf = 0.15 mm

0

1

2

3

G

0.4

Gsf = 0.24 mm

0

P

0.8

0.59

Vx [-]

0 4

5

6

7

8

10

9

11

12

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7

a)

abbreviations: s: "steel failure" - c: "concrete failure" - u: ultimate limit state - f: fatigue limit state

1.0

P/Pu,0 [-]

P/Pu,0 [-]

0.8

1. loading

ln (1 P Pu ,0 )1/ 0.59 1.22 ª1 º G K1 K1 « (P Pu ,0 )1.5 » «¬1.1 »¼

0.6

G

0.4

G pl

0.2

G 2.loading

1. unloading

2. loading

G K2 K 2

ª º 1 (P Pu ,0 ) 0.5 » «1 ¬« 7.5 ¼»

4

6

G [mm]

0 0

1

2

3

5

7

8

9

Gpl G G2. loading

G [mm]

b)

Fig. 3.159. Standardised load-slip curve of headed shear studs in normal weight concrete - load deflection behaviour after first unloading and successive reloading

3.3 Modelling

P / Pu,0

P/ Pu,0

(1 e 1.22 G

0.59

333

) (mean value) 7.5 mm

1.0 Pmax / Pu,0 = 0.71

0.8

P

0.6

G

0.4 0.2

Pmax / Pu,0 = 0.44

G [mm] 0

1

2

3

4

5

6

7 Gu 8

9

10

11

12

Fig. 3.160. Effect of high-cycle loading on the load-slip behaviour

varies between 0.65 at low levels and 0.25 at higher levels. In addition to the initial static behaviour the large number of tests under the same conditions were used for the evaluation of both, the magnitude of the plastic slip after first unloading and the magnitude of the slip after successive reloading to the preceding load. As shown in Figure 3.159 these values can be calculated by multiplication the initial static slip δ taken as a reference value with two simple functions η1 and η2 . Provided that the initial static load is less than the ultimate load first unloading and successive reloading only leads to an increase of the elastic and the accumulated plastic slip. However, high initial loading to the ultimate load followed by a hysteresis may additionally result in a lower load bearing capacity at the end of the reloading. The functions η1 and η2 illustrate that the secant stiffness during unloading and reloading are naturally slightly different and that their magnitudes decrease with increasing load level because of disproportionate increase of plastic slip. One additional important question in the concept of the numerical simulation of cyclic loaded beams is the influence of the cyclic loading on the load-slip behaviour of the studs after cyclic preloading. In Figure 3.160 the grey shaded area again shows the range of the load-slip curves of all statically loaded push-out tests without any pre-damage. The mean value of this range is given by the marked continuous blue line within the shaded area. All other curves show the static behaviour after different numbers of load cycles. The load-slip behaviour of a stud without any pre-damage can be interpreted as an envelope for all other cases, as all other curves lie within or significantly below the shaded area.

334


elastic stiffness

P

Pu,N

0.8Pu,N

P = Kel,N Pu,N (G – Gpl,N ) for P 0.8 Pu,N Kel,N =1.4 [1/mm]

G pl, N

Kel,N Gpl,N

G

§ N/N f D1 ¨ ¨ 1 N / Nf ©

1/ D 2

· ¸ ¸ ¹

5.04 ( Pmax / Pu , 0 )

D1

0.049 e

D2

§ P 0.5 'P · § · ¸ 11.522 ¨ Pmax 0.5 'P ¸ 3.054 24.865 ¨¨ max Pu , 0 Pu , 0 ¨ ¸¸ ¨¨ ¸¸ © ¹ © ¹

2

plastic slip

Fig. 3.161. Elastic stiffness and accumulated plastic slip after N number of load cycles - each based on the results of test series S1 - S4 and S5E

High-cyclic loading results in a linearization of the static load-slip behaviour up to approximately 80% of each corresponding reduced static strength. The elastic stiffness after high cyclic preloading can be assumed as being constant if the stiffness is determined on the basis of the reduced static strength Pu,N . The mathematical functions for both, the elastic stiffness and the accumulated plastic slip δpl,N after N numbers of load cycles are given in Figure 3.161. 3.3.4.2.7 Crack Initiation and Crack Development The effect of force controlled high cyclic pre-loading on the static strength as shown in Figure 3.104 in Chapter 3.2.3 is mainly caused by an early crack initiation followed by a long phase of crack propagation. Due to the reduction of the static strength under cyclic loading, the mechanical properties of headed shear studs under static loading cannot be adopted on the behaviour of studs under fatigue loading. In order to assess existing design concepts of current national and international codes and in order to develop new concepts based on based on crack propagation the knowledge of the exact time of crack initiation is of main interest. Another important question is the question about the cut-off limit. In comparison with typical welding details in steel structures, the sharp notches in the welding area lead to the conclusion that there is only a very low load limit or no load limit, where a crack initiation can be excluded. In the case of horizontal cracks of type A it is possible to produce systematically arrest lines (visible to the naked eye) on the fatigue fracture areas by releasing and reloading the test specimens during the cycle loading phases. As shown in Figure 3.162 for a representative stud of test specimen S2-4b (fatigue test in series S2) the arrest lines provide important details of crack initiation and crack growth velocity. By means of the correlation between the reduced static strength and the damage at the stud feet given in Figure 3.154 the reduced static strength can be determined if the fatigue fracture area AD is known. In the present case the crack velocity function shows a nearly

3.3 Modelling

Pu / Pu,0

reduction of static strength 2

1.0

AD AG S d 4

0.8 0.6

crack propagation

AD(N/Nf) [mm²] 200 test S2-4b stud D2b

150 Pu AD ~ 1 0.60 Pu,0 AD AG

335

100

0.4

50

0.2

crack type A

AD/(AD+AG)

0 0

0.1

0.2

0.3

0.4

0.5

0.6

N / Nf

0 0

0.7

0.2

0.4

0.6

0.8

1.0

dAD/dNi [10-3 mm²/load cycle] 0.4 fatigue fracture area (AD)

0.3

crack velocity

test S2-4b stud D2b

0.2 force fracture area (AG)

P Pmax

arrest lines caused by unloadings and reloadings during the cyclic loading phase load cycle

0.1 0 0

N / Nf 0.2

0.4

0.6

0.8

1.0

Fig. 3.162. Relationship between crack velocity, crack propagation and reduction of static strength for test series S2

sigmoidal characteristic. Similar to the development of plastic slip the crack velocity increases disproportionately at the end of the fatigue life reaching a critical crack length depending on the peak load level. The arrest lines clearly show that the start of the crack growth nearly from the beginning of the cyclic loading is possible. This observation led to the question whether in real composite structures significant cracking at the stud feet due to cyclic loading can be avoided if they are designed on the basis of current national and international codes. Based on the results explained above [351, 354, 355], in current German codes [27, 18, 34] the safety level for headed shear studs subjected to cyclic loading was increased compared to the safety level in other international codes based on Eurocode 4. The partial safety factor γMf,v was changed from 1.0 to 1.25 in the design model for the fatigue resistance of headed shear studs by means of the characteristic value of the fatigue resistance curve (5%fractile) shown in Figure 3.156. The effect of this approach is summarized in Figure 3.163. Due to the slope m = 8 an increase of the partial safety factor from 1.0 to 1.25 results in a decrease of the design value of the fatigue life Nf m of cyclic loaded headed shear studs by factor 6 (γMf,v = 1.258 ). On the other hand the characteristic value (5%-fractile) of the fatigue resistance curve used in the codes leads to a theoretical life time which is 5-times lower than the lifetime according to the mean value of the fatigue strength derived in [685]. Hence in current German codes in a design only 1/30 of the mean value of the

336


curve 1: fatigue strength curve – mean value (test results m = 8.658 ~ 8) curve 2: fatigue strength curve – characteristic value (5%-fractile) (JMf,v = 1.0) curve 3: fatigue strength curve – design value (JMf,v = 1.25)

'WR (log) 103 §N · ¨ ç ¨ N ¸ © ¹

'W R

1m

'W c N

§ 'W ¨ c ¨ 'W © R

m

'P

'P

· ¸ N c ¸ ¹

4 'P S d2

'W 'Wc = 110 N/mm²

1/ N f ,13 ~ (1/ 30 )

d

1/ N f ,12 ~ (1/ 5 )

1/ N f ,23 ~ (1/ 6)

curve 1 curve 2 curve 3

65.76 (tests S13_2)

101

'Wc = 72 N/mm² N f ,12 N f ( 'Wc

110 N / mm²) / N f ('W c

N f , 23 N f ( 'W c

90 N / mm ²) / N f ( 'Wc

N f ,13 N f ( 'Wc

110 N / mm²) / N f ('W c

104

105

90 N / mm²) 72 N / mm²) 72 N / mm²)

(110 / 90) 8 (90 / 72) 8 (110 / 72) 8

}

design codes m = 8

102

'Wc = 90 N/mm²

4.98 ~ 5 5.96 ~ 6

N (log)

29.7 ~ 30

106 Nc = 2x106

4.1x106

107 (tests S13_2)

Fig. 3.163. Fatigue strength and lifetime of cyclic loaded shear studs according different design concepts depending on the safety levels - curve 1 [685] - curve 2 [22, 23] (European codes) - curve 3 (German codes) [27, 18, 34]

lifetime given in [685] can be adopted, whereas in international codes based on Eurocode 4 1/5 can be adopted. In order to investigate, if these safety margins are sufficiently high to avoid significant cracking at the stud feet during lifetime test series S11 and S13 were performed as reported in Chapter 3.2.3. In the case of test series S13 the cyclic loading phases were aborted just after subjecting 1/30 (N = 4.1 × 106 load cycles) of the mean value of the number of cycles to failure according to [685] (Δτ = 65.76 N/mm2 - m = 8 - Nf = 1.22 × 108 load cycles) before determining the reduced static strength. In the cases of test specimens S11-4a and S11-4c the cyclic loading phases were completed after applying 1/17.4 and 1/19872 of the corresponding values according [685] (m = 8). In all cases the peak loads Pmax were lower than 0.6 PRd , so that the requirement of Eurocode 4 regarding the peak load level under service loads were fulfilled. In Table 3.28 the experimental observed crack lengths in test series S11 and S13 are listed. In the cases of test specimens S13-2b and S13-2c, in which the cyclic loading phase was aborted just after subjecting 1/30 of the mean value of the number of cycles to failure according to [685] the crack length was of remarkable size. According to Figure 3.155 the observed cracks result in a reduction of the static strength of approximately 10-15%. This can be accepted

3.3 Modelling

337

Table 3.28. Mean values of the crack length ah (see Figure 3.155) in test series S11 and S13 test

S11-4a

S11-4b

S11-4c

S13-2a

S13-2b

S13-2c

ah [mm]

~ 0.8

20.7

~ 0.7

~ 0.8

5.3

5.4

at the end of the numerical design life. However, it must be stated, that with the current design concepts cracks at the stud feet cannot be avoided. 3.3.4.2.8 Improved Damage Accumulation Model Palmgren-Miner cumulative linear damage rule [611, 543] provides a simple criterion for predicting the extent of fatigue damage induced by a particular block of constant amplitude cyclic loads in a loading sequence with different stress amplitudes. This linear damage rule assumes that the number of cycles imposed on a component, expressed as a percentage of the total number of cycles of the same amplitude to cause failure, gives the part of damage and the order of the loading blocks does not influence the fatigue life. If Ni is the number of cycles corresponding to the ith block of constant loading amplitude in a sequence of m blocks with Nf,i as the number of cycles to failure, the failure occurs, if condition 3.258 is fulfilled. m Ni =1 N f,i i=1

(3.258)

Evaluation of the tests with multiple blocks of loading on the basis of the linear damage accumulation hypothesis of Palmgren and Miner, on which the present design codes are based, is shown in Figure 3.164. The fatigue life Nf,i corresponding to each block of cyclic loading is gained from the results of the constant amplitude tests of series S1 to S4 and S5E. The missing values of number of cycles to fatigue for the blocks 2 and 3 in the test with four blocks of loading are determined by means of a linear interpolation from the results of series S1 and S4. Thus, for the peak loads of 101 kN and 120 kN per stud the fatigue life Nf is determined as 5.3 × 106 and 4.4 × 106 number of cycles, respectively. It is obvious that except for one test in Figure 3.164 all results of the lifetime prediction according to Palmgren and Miner lie on the unsafe side. Main reason for this is neglecting of the effects due to crack propagation in the shank of the stud and the increasing local crushing of concrete surrounding the stud weld. An improvement of the prediction is succeeded by the introduction of an additional damage term Δnf i in equation (3.259), which considers the effects resulting from crack propagation in the stud and steady increasing of crushing of concrete due to cyclic loading;

338


Nfe [x 106] 7 Ni

¦N

6

test results

1.0

fi

Ni

¦N

5

!

Ș N ft

fi

1 ¦ Ni Ș

4 3 2

test

K

test

K

S5-2a S5-2c

0.21

S5-6c

0.52

0.33

S5-6d

0.75

S5-3a

0.21

S6-3a

0.686

S5-4a

0.31

S6-3b

0.549

S5-4b

0.22

S6-3c

0.564

S5-4c

0.22

S6-4a

0.566

S5-4d

0.64

S6-4b

0.592

S5-6a

1.01

S6-4c

0.670

S5-6b

0.36

1 Nft [x 106]

damage accumulation according to Palmgren and Miner

0

0

1

2

4 5 3 lifetime prediction

6

7

Fig. 3.164. Comparison between the test results with the results of the lifetime prediction according to Palmgren-Miner m m−1 Ni + Δnf,i = 1 Nf,i i=1 i=1

(3.259)

Figure 3.165 explains this method by means of a cyclic test with two blocks of loading where the peak load of the first block is increased in the second block while the load range was held constant. The two curves 1 and 2 give the relationship between the relative static strength and relative number of load cycles and corresponding to the cyclic loading parameter of each block. After applying N1 number of cycles the static strength reduces to the value Pu,N 1 on curve 1 (Point B). The relative damage until this point can be expressed with the term N1 / Nf 1 based on the Palmgren-Miner rule. The point C on curve 2 corresponds to the same damage state, i.e. the same reduced static strength for the loading parameters of the second block and thus points up the starting value for the subsequent course of the reduction of the static strength along the path of curve 2. The horizontal offset Δnf between the damage equivalent points B and C can be interpreted as the loss of the lifetime and is introduced to the damage sum in the new model. The remaining lifetime is then governed by the value of N2 / Nf 2 until the failure of the specimen due to the decrease of the static strength to the value of peak load Pmax,2 . In Figure 3.165 the fatigue fracture zones corresponding to reduced static strength are depicted for different states from points A to E. As a consequence

3.3 Modelling

339

fatigue fracture area AD

P / Pu,0

A

1.0

1

AD,A = 0

0.8

Ni

¦N

D

f,i

¦ ǻn f,i d 1

C

B AD,B

AD,C ~ AD,B

2 Pmax,2 / Pu,0

D

0.6 Pmax,1 / Pu,0

'P / Pu,0

0.4

E

'P / Pu,0

'nf,1

N1 /Nf,1

0.2

'AD,E-D

AD,D

Pmax,1 / Pu,0 AD,E

N2 /Nf,2 Ni / Nf,i

0

0

0.2

0.4

0.6

0.8

1.0

Fig. 3.165. Damage accumulation considering the load sequence effects

of the correlation between the fatigue fracture area AD and the relative value of the reduced static strength Pu,N / Pu,0 the fracture areas in states D and E differ from each other due to different peak loads. Due to the raising of the peak load in the second block of loading, fatigue fracture area at the end of the lifetime can not be shaped corresponding to the fatigue fracture area AD,E of the first block of loading. The fatigue failure occurs by a rather smaller fatigue fracture area AD,D . In this case the damage term Δnf,1 considers the shortening of the fatigue life in consequence of the reduction of the fatigue fracture area to an extent of ΔAD,E−D . The results of the tests S5-2, S5-3 and series S6 showed that the loading sequence (i.e. increasing of decreasing the peak load) has a subsidiary effect on the fatigue life of a cyclic test with multiple blocks of loading. For an improved damage accumulation hypothesis before the analysis the load collectives with decreasing loadings must be resorted to collectives with increasing loadings. This procedure is shown exemplarily for the tests S6-4 in Figure 3.166. In such case the additional damage terms Δnf,i can be interpreted as the effect of concrete damage on the extent of the crack velocity. In test with increasing peak load the failure occurs by a fatigue fracture area AD,A and in the case of decreasing peak load by a fatigue fracture area AD,B which is greater then the AD,A by an extent of ΔAD,B−A . Considering identical lifetimes due to bending stresses in the stud shank resulting from local concrete damage and consequential high notch stresses at the crack tip, in the case of decreasing peak load the crack velocity is greater than in the case of increasing peak load.

340


Pu / Pu,0 N1 / Nf,1 'nf,1

'nf,2

N2 / Nf,2

'nf,3

N3 / Nf,3

N4 / Nf,4

1.0

4

AD,A

A

0.8

0.74 ~ 0.71

1

0.64 0.6

tserVe i M

:.1 6 3 0 8 5 7 .1 0 8 5 7

Pmax,4

0.4

P

4

Pmax,4

Pmax,4

2 . 1 .

0

0

20 + .1 7 + 5 0 4 .1 2 7 + 5 0 4

4 .1 0 2

6

.2 0

8 3

.1 2 0

0 .3

0 5 .4

.+ 4 0 6 7 4 9 6 4 7 . 0 + 9 6

1 7 + 6 0 + 5 3 8 6 7 1 + 0 5 3 8 6 7

.1 5 2 = 5 6

8 2 0 5

3 = 8 2

8 5 1

4 .1 0

5 )(c l a

8 3 2 1 5

0 5 .1

8 7 4 5 +

5(e ) p x

. 9 0 8

. 1 0

.+ 8 4 2 1 5

1 S 6 7 .1 0

2 "S 3 "S 4 S

0 4 7 .1

0 .6

.7 0

/ .2 (0

x p (e

)

x p (e

)

(l ca

)

). 4

/ .2 (0

). 4 5

/ .2 (0

.6 ) 4

(.2 /0

1 .7 )~

4 1 9 6 3

6

9 6 0 3 4 15

1 5

f,1 N =

N f,2 N = N = 4 ,f3

0 5 3

0

(.2 /0

4 7 )

fN

0.54 0.44

4

B

Pmax,1

AD,B

1 N1 N2 N3

0.2

2 0 .1 9 . 8 . 7 . 6 . 5 . 4 . 3 .

0 0

Pu,0

1 N4 N3 N2 N1

N4 'P konst.

0

rtueShw l ce

u P 1 0 0 0 0 0 0 0

Pu,0

P

0

0.2

'AD,B-A

G

Ni / Nf,i

G

0.4

0.6

0.8

1.0

Fig. 3.166. Damage accumulation in the case of multiple block loading tests with decreasing peak loads

Nfe [x 106]

7new damage accumulation rule

7

6

6

5

5

4 3 2 1

4 3 2 Nft [x

0 0

D



Nfe [x 106]

4 1 2 5 6 7 3 predicted lifetimes – single values

106]

(2)

1

¦

Ni ¦ ' n f, i d 1 N f, i

(3)

(3) (3) (4)

(1) ( ) number of tests

0 0

(1)

Nft [x 106]

4 1 2 5 6 7 3 predicted lifetimes – mean values

Fig. 3.167. Comparison between the test results with the results of the lifetime prediction according to the improved damage accumulation hypothesis

In Figure 3.167 the comparison of the results of multiple block loading tests with the theoretical values according the improved damage accumulation method is illustrated. With the exception of two tests nearly all single values

3.3 Modelling

341

Gu, 0.9 Pu [mm] 35

Pu 0.9 Pu

30

Pmax / Pu,0 = 0.71

P

type B

Gu, 0.9Pu

25

type A

d

20

3

15

1 2

10

Pmax / Pu,0 0.44

4 type A

5 0

0

0.2

0.4

0.6

0.8

1.0

type B 1.2

N / Nf

Fig. 3.168. Ductility after high cycle loading

of the test results are well predicted. In these two tests presumably due to worse compaction of the concrete a larger slip development at the stud feet was noticed. This shows the importance of the detailing of the shear connection as given in Eurocode 4 [22, 23]. Looking at similar tests it can be observed that the prediction matches the mean values leading to a significant improvement compared with the Palmgren-Miner rule. 3.3.4.2.9 Ductility and Crack Formation High initial stiffness and high ductility are main advantages of headed studs embedded in normal weight concrete. From the static tests carried out after cyclic preloading it could be found, that the load deflection behaviour is significantly affected by the crack formation, which is itself closely correlated to the peak load level. Very high peak loads cause horizontal cracks through the stud foot like crack type A shown in Figure 3.154. As shown in Figure 3.168 this formation results in a gradual decrease of ductility during lifetime and the values may fall below the target values of the codes. In the case of lower peak loads the cracks propagate into the flange like crack type B and ductility increases. This behaviour is of great importance regarding the capability of redistribution of shear forces in the interface between steel and concrete of composite beams subjected to fatigue loading. 3.3.4.2.10 Finite Element Calculations of the (Reduced) Static Strength of Headed Shear Studs in Push-Out Specimens The experimental results of the push-out tests can be taken as the basis for further theoretical investigations regarding the effect of cracks at the stud feet on the static strength of headed shear studs given in Figure 3.154. As shown

342


without pre-damage

with pre-damage (here: crack of type B at each stud foot)

P = 0.2

FE-model of a push-out test specimen (quarter)

P = 0.1

local deformation of concrete and steel at the stud feet at ultimate limit state P [kN] fc = 30 N/mm² Ecm = 27960 N/mm²

Pu = 1440 kN

1400 1200

test

'Pu,FEM ~ 0.26 Pu

FEM

fu = 528 N/mm E = 210000 N/mm²

1600

fu = 235 N/mm

1000 fu = 528 N/mm

800 600 400 200 0

crack pattern (here: type B) fy = 337 N/mm - fu = 448 N/mm

AD | 0.5 'Pu , exp | 0.30 | 'Pu , FEM AD AG 1

2

3

4

5

6

7

G [mm]

crack pattern (here: type B)

Distribution of main material properties at the stud feet

8

Fig. 3.169. Comparison between test results and finite element calculations of statically loaded push-out test specimens

in Figure 3.169 for this purpose a comprehensive three-dimensional FE-Model - using the finite element programme ANSYS - of a statically loaded push-out test specimen with lateral support of the concrete slabs according to Figure 3.100 of Chapter 3.2.3 has been built up in order to simulate he load-deflection behaviour of headed shear studs embedded in solid slabs without any pre-damage. Concerning the numerical simulation the material properties of the steel members and the concrete members are of main interest. So far no detailed information about the precise material properties of the steel in the heated affected zones, in the weld collar and in the melted zone at each stud are available. For this reason the material properties of the steel beam and the studs (determined by means of tensile tests) were taken as the basis of the material properties of the steel affected by the welding process. Microscopic examinations of the steel structure at the stud feet given in Figures 3.151 - 3.153 were performed in order to consider sufficiently the weld formations regarding the assignment of the main material properties in the FE-model. Metal plasticity behaviour of the steel was simulated by using the von-Mises criterion. The concrete behaviour was modelled elastic - perfectly plastic taking into account a yielding surface according to Drucker-Prager (DP) with an associated flow rule. The two parameters of the DP-yield surface were adjusted to the uniaxial (1.0 fc ) and to the biaxial compressive strength (taken as 1.2 fc ) of the concrete, obtained from concrete cylinders cured at air as the corresponding test specimens. As shown in Figure 3.153 the weld collar is commonly non

3.3 Modelling

343

homogeneous and of low strength, therefore the tensile strength of the collars was only chosen to fu = 235 N/mm2 . Figure 3.169 shows the results of a numerical simulation of a push-out test, considering concrete strength properties fc = 30 N/mm2 and Ecm = 27960 N/mm2 (cured at air), which are typical values like e.g. in test series S2 or S4. For the coefficient of friction in the interface between steel and concrete a very low value of μ = 0.2 was chosen, because the steel flanges were greased before casting. The FE-model leads to good agreement between the calculated load deflection curve, the ultimate static strength and the deformation of the studs with the corresponding experimental results of test series S1 to S9. After calibrating the FE-model of a statically loaded push-out specimen without damage subsequently cracks of different geometries were implemented in the FE-model at each stud foot and the calculations were repeated in order to check the influence of the steel damage on the reduced static strength. One typical crack pattern and the corresponding results are shown as an example in Figure 3.169. The coefficient of friction in the interface between steel and concrete was slightly reduced to μ = 0.1 in comparison to the non-damaged state in order to take the sliding in the interface due to cyclic preloading into account. The effect of three crack patterns with nearly half the stud foot size on the reduction of the static strength is shown in Figure 3.170. For the investigated sizes of the cracks, the results of the numerical simulations are in the same order as observed in the tests (see Figure 3.154). Regarding the shear resistance of pre-damaged studs the local crushing of the concrete due to cyclic loading is insignificant compared to the pre-damage in form of cracks in the steel at the stud feet.

Pu / Pu,0

crack pattern (FEM)

1.0 type B

0.8

FEM - type B

Pu / Pu,0

FEM - type A

0.6

experiment type A

0.4

Pu AD | 1 0 .6 Pu,0 all series A D A G

0.2 0.0

0.0

0.2

AD /(AD+AG ) 0.4

0.6

0.8

1.0

Fig. 3.170. Comparison between test results and finite element calculations of statically loaded push-out test specimens damaged by cracks in the steel at the stud feet

344


3.3.4.2.11 Effect of the Control Mode - Effect of Low Temperatures In force controlled push-out tests monotonic high cyclic loading leads to a successive increase of the accumulated plastic slip at the stud feet and to a gradual decrease of the elastic stiffness. In force controlled composite beams these effects cause successive redistribution of the longitudinal shear forces in the interface between steel and concrete and reduction of the normal forces in the steel and the concrete members. The stiffness of the interface between steel and concrete gradually decreases and the composite action deteriorates. In general the peak load Pmax and the load ranges ΔP of each stud decreases and the value of the maximum slip at the peak load level smax and the slip range Δs within a load cycle increases. Only in the cases with a very low load level the increase of the slip values smax and Δs at each stud foot over lifetime can be neglected. The deterioration of the mechanical properties of each stud in the interface between steel and concrete can then be predicted directly by means of displacement controlled push-out tests [527, 312, 278, 483]. In this context it is of main interest, which control mode should be considered in order to simulate cyclic behaviour of composite beams by means of results gained by push-out tests. In order to answer this question in series S9 nine tests according Figure 3.98 of Chapter 3.2.3 were performed in which the influence of the control mode on the results of cyclic loaded push-out test specimen was investigated. As shown in Figure 3.171 after 5 × 106 load cycles the load-slip behaviour and the mean values of the reduced static strength P u,N of the compared cyclic loaded push-out test specimens were nearly coincident. Moreover the mean values of the crack lengths acr through the stud feet were almost identical. Independent from the control mode in both cases the static shear resistance was reduced from P u,0 = 1620 kN to approximate P u,N = 930 kN. This shows that displacement controlled behaviour of headed studs can also be simulated by the results of force controlled push-out tests with multiple blocks of loading, taking into account an appropriate damage accumulation hypothesis in order to consider sequence effects. Regarding the numerical simulation of composite beams it appears, that the effect of the cyclic loading can be simulated by the results of force controlled push-out tests as gained from test series S1 - S6. Notched-bar impact tests carried out with test specimens taken from welded joints of headed shear studs show that the notched-bar impact values significantly fall below commonly accepted values of steel structures. As given in Figure 3.171 for a temperature of T = −20◦C in the tests the notched bar impact values decreased to 9 J near the melted line adjacent to the material of the steel flange. In order to investigate the interaction between the predamage and the toughness the reduced static strength of test specimen S9-5d (beforehand also being subjected to 5×106 load cycles under force control with multiple blocks of loadings) was tested after cooling down to T = −40◦ C. In comparison with the before mentioned cyclic loaded test specimens, in which the reduced static strength was tested at a temperature of T = 20◦ C, the

3.3 Modelling

P [kN]

T = -40°C (S9_5d)

1200

'Pu, N | 'f c ('T) 'E c ('T)

'Pu,N

1000

T = 20°C

800 600

Cyclic loading applied: displacement controlled force controlled

9J

type B reduced static strength after 5x106 load cycles 5

10

24J 19 J

acr

200

KV 150 / T = -20°C

notched-bar impact-bending test

Pu,0 1620kN

400

0

345

15

6J

G [mm]

Fig. 3.171. Test series S9 - Effect of control mode - Effect of low temperature on the load-slip behaviour of pre-damaged studs with cracks of type B

reduced static strength increases in accordance with the increase of the relevant material properties (fc , Ecm ) of the concrete at lower temperatures. The ductility decreases as a matter of course but compared to the requirements of the codes the ductility remains sufficiently high. In the present case the cracks propagated slightly into the steel flanges. For cracks of type A it must be presumed, that in the case of low temperatures the ductility lies below the corresponding values shown in Figure 3.168. This is again an import reason, why cracks of type A must be strongly avoided by means of limiting the peak load in the limit state of serviceability. 3.3.4.3 Modelling of the Global Behaviour of Composite Beams Subjected to Cyclic Loading 3.3.4.3.1 Material Model for the Concrete Behaviour Considering the interaction between the local damage of headed shear studs and the behaviour of the global structure, the research results based on the push-out tests were taken as the basis for numerical simulations of the static behaviour (with and without any pre-damage effects) and the cyclic behaviour of steel-composite beams subjected to fatigue loads. For this purpose at first the commercial finite element programme ANSYS, R7.1 was improved by implementing a modified version of the three-dimensional structural solid element SOLID65 [643]. In original ANSYS versions this element is not implemented satisfactorily, because the corresponding material model CONCRETE shows some severe errors. Due to numerical instabilities and inaccurate calculation results this material model is not applicable for typical problems of structural concrete members. The element behaviour has been updated in this manner that the primarily requested material failure modes of brittle materials subjected to static loading are enabled [48]. The element is defined by eight nodes having three

346


degrees of freedom at each node (translations in the nodal x, y, and z directions) based on linear shape functions. The solid is capable of cracking (in three orthogonal directions) in tension and crushing in compression. It can be adjusted to the behaviour of concrete by giving typical concrete material data like the shear transfer coefficient, ultimate uniaxial tensile strength, ultimate uniaxial compressive strength and ultimate compressive strength for states of multiaxial compression. Typical shear transfer coefficients range from 0.0 to 1.0, with 0.0 representing a smooth crack (complete loss of shear transfer) and 1.0 representing a rough crack (no loss of shear transfer). These specifications are possible for both a closed and an open crack. The new implemented failure surface in the compression domain (compression - compression - compression) is oriented at the 5-parameter failure criterion of Willam and Warnke [849], but having an open failure surface [190, 189, 191]. Moreover nonlinear material properties such as creep and plastic deformation can be treated by incorporating additional creep and plasticity options. In the case of the numerical simulation of structural composite members of steel and concrete, like composite beams and composite columns subjected to static loading up to failure, it is reasonable to incorporate the Multilinear Isotropic Hardening (MISO) option with yield surface according to von Mises in order to consider sufficiently the typical nonlinear stress-strain relationship of the concrete under compression. Figure 3.172 shows the implemented failure surface of the material model CONRETE corresponding to the structural element SOLID65 and a combination of this failure surface with a yield surface according to von Mises. As shown in Figure 3.173 it is possible to predict the load-deflection behaviour and the load bearing capacity of typical composite members of steel and concrete [342] under static loading up to failure by means of the improved material model. The properties of the studs were modelled by means of discrete non-linear spring elements, taking into account the analytical expressions developed from the statically loaded push-out tests of series S1 - S9. Similar good predictions were achieved in the case of numerical simulations of the behaviour of statically loaded composite columns of high strength steel and concrete within the framework of a research project at the University of Wuppertal, Germany [344, 501]. 3.3.4.3.2 Effect of High-Cycle Loading on Load Bearing Capacity of Composite Beams For real structures the effect of the local damage of the headed shear studs on the global behaviour and the resistance of beams subjected to fatigue loading is of main interest. The deterioration of the mechanical properties of the studs due to cyclic loading leads to a successive decrease of the composite action, resulting in a decrease of load bearing capacity and in a successive loss of elastic stiffness. The extent mainly depends the contribution of the steal beam to the load bearing capacity and to the stiffness of the composite beam.

347

improved failure surface of the material model CONCRETE incorporated by a yield surface according to von Mises

improved failure surface of the material model CONCRETE

3.3 Modelling

Fig. 3.172. Failure surface of the improved material model CONCRETE - Failure surface incorporated by a yield surface according to von Mises

For this reason the push-out test program was extended by two simply supported full-scale cyclic loaded composite beam tests (VT1 and VT2). In these tests the effect of the deterioration of the properties of the interface between steel and concrete on the load-deflection behaviour, on the redistribution of the inner forces and on the reduced static strength for two typical cyclic loading conditions (VT1: sagging moment - VT2: hogging moment) were investigated. The main results are given in Chapter 3.2.3. In both cases the test beams were statically loaded up to failure after the cyclic loading phase and subsequently checked for cracks in the steel at the stud feet. Figure 3.174 summarizes the main results of test beam VT1 regarding the interaction between the local damage of the studs and the load-deflection behaviour after high cyclic loading. The residual static strength of the beam was determined after 1.37 × 106 loadings. Based on the theoretical model given by Equation A in Figure 3.154 due to crack growth at the stud feet during the cyclic loading phase the static strengths of the studs were partly decreased by up to 65% of its original value. The value of the reduced static strength of all studs averages approximately 52% of the undamaged state. Numerical

348


Maximum concrete stresses at ultimate load (beam test 1)

min Vx = 43.0 MPa

HEA300-S460

failure surface (CONCRETE)

max Vx = 1.4MPa

F [kN]

700

beam test 1

600 test

500

VII

Von Mises

fc

HEA300-S460

FEM

400 300

ft

200

VI

fc

ft

beam test 4

uz

F 100

uz [mm]

6m

CONCRETE

0

0

10

20

30

40

50

60

70

80

90

100

Fig. 3.173. Comparison between the results of numerical simulations and test results of typical composite members of steel and concrete

shear resistance of the studs after 1.370.000 load cycles

1000

1.0

F [kN]

0.8

'F = 60 kN (8%)

0.65

0.6

800

Fu,N = 756 kN

0.4 0.2

0

600

Pu,N /Pu,0 54 studs

450 kN

1.370.000 cycles

FEM

F

400 200

reduced static strength first loading

3.0 [m] 1.0 0.5 0

0.0 0.52

3.0 0.47

0.48 Pu,N /Pu,0

0 1.0 0.89

0.53

uz

6.0 m

185 kN

20

uz [mm]

experiment (VT1)

60

40

80

100

M/Mu ' = 11%

0.8 (VT1)

0.6 0.4 b/h = 1500/150

0.2 HEA300 fy = 452N/mm² - fc = 36N/mm²

K

0.52

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Fig. 3.174. Test beam VT1 - Effect of high cycle loading on load bearing capacity

3.3 Modelling

349

investigations considering the experimental observed values of the accumulated plastic slip and the damage of each stud indicate that the deterioration of the strength of the interface between steel and concrete causes a loss of the load bearing capacity of the composite beam of nearly 8% compared to the undamaged beam. This result is in good agreement with the result obtained by applying partial-interaction theory taking into account the average damage of the studs along the interface instead of discrete local damage of each stud. In the case of test beam VT1 the reduction of the global load bearing capacity due to cyclic loading is considerably low compared to the reduced static strength of the studs. However, the effect of the local damage of the studs increases if the contribution of the steel beam on the load bearing capacity of the composite beam is low. Typical examples are composite girders with very small or even no steel top flanges, in which the steel flange is almost completely substituted by the concrete flange. Other typical structural details are cyclic loaded load introduction areas with headed shear studs, where no redistribution of the shear forces is possible. In these cases the reduction of the static strength of the studs directly affects the shear carrying capacity and with this the load bearing capacity of the load introduction area. On this background the design rules in current national and international design codes must be reconsidered. 3.3.4.3.3 Cyclic Behaviour of Composite Beams - Development of Slip In general the redistribution of the inner forces in the concrete slab and in the steel beam due to the decrease of the elastic stiffness and the plastic slip in the interface between concrete and steel is of main interest, where local buckling of the steel beam governs the design. In order to simulate the behaviour of cyclic loaded composite beams the first generation of incremental algorithms was developed, which are based on the damage accumulation method according to Figure 3.165. These algorithms are incremental processes in which the characteristics of each stud such as ”plastic slip”, ”elastic stiffness” and ”reduced static strength” are updated step by step regarding the accumulated damage D, the current peak load Pmax and current load range ΔP after each step. In each increment it is assumed that the loading parameters of each stud remain constant and thus the stud behaviour during the increment can directly be taken from appropriate force-controlled push-out test results as reported in Chapter 3.3.4.2. In Figure 3.175 for test beam VT1 the development of the plastic slip during the cyclic loading phase is given. Due to the high loading parameters the test beam shows a considerable increase of plastic slip during the cyclic loading phase. In the present case for the recalculation of the plastic slip the total number of load cycles Nk was split in 20 increments and after each FE-analysis, representing an increase of Nk / 20 numbers of load cycles, the relevant mechanical properties of each headed shear stud (plastic slip, elastic stiffness and reduced static strength) were updated. Although further research is needed the so far achieved numerical results are in agreement with the experimental observations in the beam test.

350


450 kN

Pu,N

1.370.000 cycles

P

185 kN

Kel,N

3.0 [m]

Gpl,N

3.0

0.0

G

G [mm] 1.0

D

0.8 0.6

1.370.000 load cycles

Pu,N

1. loading

Gpl

0.4 0.2 0

simulation

-0.2

20 increments with 68.500 load cycles

-0.4

P

-0.6

G

-0.8 -1.0

Nf

-3.0

-2.0

-1.0

0

1.0

2.0

3.0

[m]

Pmax, 'P

Fig. 3.175. Cyclic behaviour of test beam VT1 - Verification of the concept

3.3.4.3.4 Effect of Cyclic Loading on Beams with Tension Flanges The occurrence of cracks at the stud feet and the early crack initiation has to be assessed a in different way, if the flange, on which the studs are welded, is not in compression but in tension. For this purpose beam test VT2 in hogging bending was subjected to 2.1 million load cycles before the residual static strength of the beam was determined. In flanges under compression the cracks typically grow horizontal leading to a deterioration of the properties of the interface between steel and concrete. However, in tension flanges the direction of the cracks at the stud feet is additionally influenced by the tensile stresses in the steel flange. In test beam VT2 it could be observed, that the cracks partly propagate nearly vertical through the flange. In this case not only the properties of the interface are affected, but also the load bearing capacity of the cross sections. This is pointed out in Figure 3.176, in which the influence of pre-damage on the static load-deflection behaviour after high cyclic loading and a typical crack formation at a highly stressed stud (stud B9) is shown. The test shows that also in hogging bending the significant local damage causes only a small global reduction of the ultimate load, provided that the bending stiffness of the steel member is sufficiently high compared to the bending stiffness of the composite section. With regard to fatigue cracks growing in vertical direction through the top flange of the steel girder further research is needed.

3.4 Numerical Examples

351

tension flange

250 kN

2.100.000 cycles 150 kN

54 studs

stud B9

3.0 [m]

3.0

0.0

VT 2 – tension flange

F [kN] 800 'P = 47 kN

700

Pu,N = 625 kN

FEM, undamaged

600

a stud B9

500 400

6.0 m

200 100 0

VT 1 – compression flange

P

300

experiment (VT2)

0

20

40

60

80

uz

a

VT 2

100

120

140

uz [mm]

stud A20

Fig. 3.176. Test beam VT2 - Effect of high cycle loading on the load bearing capacity - typical crack formation in tension flanges and in compression flanges

3.4 Numerical Examples Authored by G¨ unther Meschke 3.4.1 Durability Analysis of a Concrete Tunnel Shell Authored by Stefan Grasberger and G¨ unther Meschke Large tunnels often exhibit crown cracks in the inner linings caused by restrained stresses due to cooling and drying of the surface. Using the hygromechanical model described in Subchapter 3.3.2.1 a durability analysis of the long-term degradation of an inner tunnel lining is presented in this section. The effect of creep is disregarded in this analysis. Figure 3.177 contains the dimensions of a cross section of the analysed tunnel. Exploiting symmetryconditions, only one half of the tunnel lining is discretised by means of 1832 finite volume elements. Details on the material and model parameters are contained in [322, 692]. The initial temperature is specified as T0 = 17.0◦ C, the initial pore humidity as h0 = 0.93 and the initial porosity as φ0 = 0.25. After application of the self-weight of the construction (g = 25 kN/m3 ), the inner tunnel lining is exposed to cyclic changes of the hygral and thermal environmental conditions based on meteorological data according to [680]. Elastic support conditions are assumed a long the interface of the tunnel shell to the surrounding soil.


9 5

8 3 5

9 3 0

352

5 4 0

Fig. 3.177. Numerical simulation of a tunnel lining subjected to cyclic hygral and thermal loading: Geometry of the investigated tunnel (dimensions in [cm])

1.2

Crack width Z [mm]

1 0.8 0.6 0.4 0.2 0

Model prediction

1

2

3

4

5

Time t [a]

Fig. 3.178. Numerical simulation of a tunnel lining subjected to cyclic hygral and thermal loading: Comparison of the calculated evolution of the crack width w at Point A at the crown and measurements by [764]

Figure 3.179 illustrates the computed evolution of the crown cracks represented by the scalar damage parameter d after 1, 2 and 5 years, respectively.


353

b A

a) t = 1 a

b) t = 2 a

c) t = 5 a 0.000

d [-]

1.000

Fig. 3.179. Numerical simulation of a tunnel lining subjected to cyclic hygral and thermal loading: Computed distribution of the scalar damage measure d at the crown

During the first year some cracks start to open at the top of the inner lining due to the arising temperature and moisture gradients. As the drying proceeds, these cracks are further opening, whereby the predicted crack spacing is about 20 ∼ 25 cm. According to the stress distribution owing to the self-weight of the construction, the lower part of the tunnel lining remains undamaged. Figure 3.178 contains a comparison of the predicted evolution of the crack width at the top of the inner lining (Point A in Figure 3.179) with reported in-situ measurements by [764]. The numerical results demonstrate that the finite element model is not only capable of reproducing shrinkage-induced cracks but also to predict the time of opening of the cracks, the location of the cracks and the width of the cracks in accordance with in-situ observations.

354


a) t = 1 a

b) t = 2 a

c) t = 5 a 0.900

Sl [-]

0.400

Fig. 3.180. Numerical simulation of a tunnel lining subjected to cyclic hygral and thermal loading: Computed distribution of the liquid saturation Sl at the crown after 1, 2, and 5 years

Figure 3.180 illustrates the predicted evolution of the drying process by means of the liquid saturation Sl . According to the strong interdependencies between moisture transport and the nonlinear material behaviour of concrete, the drying process at the top of the inner tunnel lining is clearly dominated by the influence of the growing crown cracks. 3.4.2 Durability Analysis of a Cementitious Beam Exposed to Calcium Leaching and External Loading Authored by G¨ unther Meschke The chemo-mechanical model described in Subchapter 3.3.2.2.1 is applied for durability analyses of a cement paste beam (Figures 3.181 and 3.182)

Ca 2+ -Concentration s h [kmol/m 3 ]


16 14 12 10 8 6 4 2 0

355

moving front of portlandite dissolution

P

moving front of CSH dissolution

z c-attack

t = 1600d, 3200d,... t = 9120d 0

10

20

30

40

50

Coordinate z [mm]

Displacement u s [mm]

Fig. 3.181. Simulation of a cementitious beam exposed to calcium leaching and mechanical loading. Distribution of the calcium concentration sh of the solid skeleton along the symmetry axis z

0.075 0.07 0.065 0.06 0.055 0.05 0.045 0.04

P us A

Collapse

c-attack

0

2000

4000

6000

8000

10000

Time t [d]

Fig. 3.182. Temporal evolution of the vertical displacement us in point A (P = const.). Collapse is predicted after approx. 25 years

exposed simultaneously to chemical dissolution and external loading. In the central part of the bottom face of the beam, the calcium concentration ci of the pore solution is decreased from an initial value to zero in order to simulate the contact with deionized water. Simultaneously, a constant load P (75% of ultimate load) is applied. The diagram in Figure 3.181 shows the distribution of the calcium concentration sh in the solid skeleton along the symmetry axis z at different time intervals. A decrease of sh below the initial value sh0 = 15 kmol/m3 is associated with both an increase of the porosity and a decrease of the stiffness and strength. The two fronts propagating from the bottom to the top are associated with the dissolution of portlandite and CSH. In diagram in Figure 3.182 the degrading structural behavior of the beam is represented by the evolution of the vertical displacement us in point A

356


square panel

mechanical load bearing behaviour

loading history

4 t 2 [N/mm]

point A

loading

40 mm

d = 1 mm

t 2,max

3 2 1

x2

Ω x1

40 mm

0 0 0.01 0.02 0.03 0.04 displacement u s, 2 in A [mm]

c /c i0 und t 2 /t 2,max

t2

c

c /c i0 t 2 /t 2,max

1

0 0

100 200 300 400 · · · ratio t/Δt

Fig. 3.183. Chemo-mechanical analysis of a concrete panel: a) Geometry, finite element mesh and boundary conditions, b) load-displacement curve when subjected to external loading only, c) chemo-mechanical loading history

along time. Since the leaching process results in a drastic reduction of the local and global stiffness, the displacement us increases in spite of a constant mechanical load P . The collapse of the structure is indicated at t = 9120 d ≈ 25 years, by a sudden increase of the displacement us . It is worth noting that in a purely mechanical analysis no collapse would be predicted. 3.4.3 Durability Analysis of a Sealed Panel with a Leakage Authored by Falko Bangert and G¨ unther Meschke As a second numerical application of the coupled chemo-mechanical model for concrete described in Subchapter 3.3.2.2.1 a panel made of cement paste, which is simultaneously subjected to external loading and exposed to calcium dissolution is investigated. Details of the material and model parameters and of the discretiztion are contained in [81]. Figure 3.183a illustrates the geometry, the finite element discretization, the boundary conditions of the concrete slab, which is discretized by 40 × 40 bi-quadratic finite elements Elements for the interpolation of the displacment field us and of the calcium concentration ci . For the nonlocal equivalent strains η¯ a bi-linear interpolation is used (see Subchapter 3.3.2.2.2). Plane stress conditions are assumed, using the algorithm proposed in [140]. The temporal evolution of the the external and chemical loading is shown in Figure 3.183c. In the upper left corner the calcium concentration c in the pore fluid is decreased from the initial value c = c0 corresponding to a state of chemical equilibrium to zero in order to simulate contact with deionized water. The rest of the surface of the panel is assumed to be sealed. This chemical loading scenario shall represent a very localized leakage of a sealing of an underground structure permanently exposed to deionized water. Simultaneousely, the panel is subjected to a time invariant external distributed loading


357

t∗ =2.5 N/mm. This level corresponds to 75 % of the ultimate load carrying capacity (tu = 3.4 N/mm) if this external loading would be monotonously increased up to structural failure (see the load-displacement curve in Figure 3.183b). Figure 3.184 illustrates the distribution of the calcium concentration c in the pore fluid, of the concentration s of the calcium bound in the skeleton, the rate of calcium dissolution s˙ and of the scalar damage parameter dm at four stages of the loading history. The results at t/Δt = 100 correspond to the end of the chemical loading at the top left corner of the panel (see Figure 3.183c), inducing a strong gradient of the calcium ion concentration grad(c) in the vicinity of this corner. This concentration gradient activates the calcium dissolution of the hydration products of the cement paste and the diffusion of the dissolved calcium ions towards the top left corner. The temporal evolution of the dissolution process is well represented by the contour plots showing the step-wise distribution of calcium-concentration sh of the hydratation products emanating from the top left corner, where the leakage is assumed. The third row in Figure 3.184 shows a dissolution front associated with the dissolution of the calciumhydroxide moving from the left top corner into the structure. The speed of propagation of the Ca(OH)2 dissolution front is shown in Figure 3.185 In contrast to the dissolution of the portlandite, the solution front associated with the dissolution of the CSHphases starts at a much later stage and only moves marginally. It is more or less restricted to the vicinity of the top left corner and propagates slowly into the direction of the opening vertical crack (see Figure 3.184 at t/Δt = 400 and t/Δt = 428). The sequence of contour plots on the right hand side of Figure 3.184 illustrates that the external loading in conjunction with the propagating chemical dissolution initiates a crack at the top left corner which propagates vertically along the fixed support. Although the external load is kept constant, the chemical attack continues to contribute to the weakening of the structure, resulting in an increase of the width of the crack and its propagation into the structure. The location of the crack tip propagates more or less simultaneousely with the dissoultion front. The presence of the crack accerlerates the dissolution process considerably. This is illustrated by the contour plots in Figure 3.184 at t/Δt = 300, 400), where the Ca(OH)2 -dissolution progresses faster along the vertical crack compared to the horizontal direction. This chemo-mechanical interaction is also demonstrated by the results shown in Figure 3.185a which contains the temporal evolution of the propagating Ca(OH)2 dissolution front along the vertical direction as obtained from a fully chemo-mechanically coupled analysis compared to results from an uncoupled chemical analysis, in which the effect of the damage is disregarded. While for a purely chemical load√ ing the Ca(OH)2 -dissolution front moves √ with a velocity of ac = 3.0 mm/ a, the speed increases to acm = 4.5 mm/ a if chemo-mechanical interactions are considered. This is an increase of 50%.

358


t Ca-Concentration ci Ca-Concentration sh rate log(−∂sh/∂t) Δt [1] 21

[mol/m3 ]

015

[kmol/m3 ]

0-2[kmol/[m3 d]] ≤−61

Damage dm [1]

0

100

200

300

400

428

Fig. 3.184. Chemo-mechanical analysis of a concrete panel: Distribution of the calcium concentration c (pore fluid), the calcium concentration s (skeleton), the dissolution rate s˙ and the scalar damage parameter dm at different states of exposure

40

displacement us,2 in point A [10−3 mm]

position of the Ca(OH)2 front x2 [mm]


chemo-mechanical chemical

35 30 25 20 15

acm

10

ac

5 failure 0 0

50

100

150

200 250 √ √ square root of time t [ d]

300

359

-4 chemo-mechanical mechanical

-6 -8 -10 -12 -14

failure

-16 0

200

400

600

800

1000

ratio t/Δt [1]

Fig. 3.185. Chemo-mechanical analysis of a concrete panel: a) Temporal propagation of the Ca(OH)2 dissolution front along the clamped support, b) Deformationtime relation

As a consequence of this (long-term) chemo-mechanical material degradation and the continuous propagation of the crack the structural stiffness decreases. This connected with a progressive growth of the vertical displacments u of the right edge of the panel (Figure 3.185b). 94 years after the start of the exposure to de-ionized water the resistance of the structure is finally exhausted. Without consideration of chemo-mechanical couplings no structural failure would have been predicted. 3.4.4 Numerical Simulation of a Concrete Beam Affected by Alkali-Silica Reaction Authored by Falko Bangert and G¨ unther Meschke The chemo-hygro-mechanical model for chemically expansive processes presented in Subchapter 3.3.2.2.2 is applied to the durability analysis of an ASRaffected concrete beam to predict the temporal deterioration of this structure. Details of the analysis including the chosen model parameters are contained in [83]. The left hand side of Figure 3.186 contains the geometry, the support conditions, the hygral Dirichlet boundaries Γp1l and Γp2l and the spatial discretization with finite elements. To avoid stress oscillations, the interpolation for the displacements us is chosen as one order higher than the interpolation of the non-local equivalent strain η¯, see Peerlings et al. [614]. The displacements us and the liquid pressure pl are interpolated with quadratic serendipity shape functions [782], while linear interpolations are used for the non-local equivalent strain η¯.

360

3 Deterioration of Materials and Structures Loading history

u

R

Γp2l

pl

pl on Γp1l

-5

p

l

on Γp2l

-10

16 cm

d = 0.1 cm

Liquid pressure pl [N/mm2]

0

Ω Γ 64 cm

Γp1l

pl

-15 -20 -25 -30 -35 0

50

100 150 200 250 300 Time t [d]

Fig. 3.186. Numerical simulation of a concrete beam affected by alkali-silica reaction: Geometry, mechanical and hygral boundary conditions, finite element mesh and hygral loading history

The diagram on the right hand side of Figure 3.186 illustrates the hygral loading history. The initial conditions are given by us0 = 0 for the displacements and pl0 = −15 N/mm2 for the liquid pressure corresponding to an initial liquid saturation of sl0 = 0.8. In an initial phase (Phase I), drying is simu lated by decreasing the liquid pressure from the initial value pl = pl0 to 2 pl = −29 N/mm corresponding to a liquid saturation of sl = 0.6. After 20 days of drying, the liquid pressure pl at the bottom of the beam is raised to 2 pl = −9 N/mm corresponding to a liquid saturation of sl = 0.9 (Phase II). In the third loading phase starting at t = 130 d, all hygral boundary conditions are reset to the initial value pl = pl0 and held constant afterwards. Figure 3.187 illustrates, from the left to the right, the distribution of the liquid saturation sl , the ASR expansion rate ∂εas /∂ t, the ASR expansion εas and the damage parameter d at selected stages of the hygral loading history. The first row of Figure 3.187 (t = 20 d) corresponds to the end of the initial drying process. Due to drying of the surface, the more humid inner part is characterized by larger values of the ASR expansion εas and the respective expansion rate ∂εas /∂t. Since the ASR-induced expansion of the core is at least partially hindered, the inner part of the beam is subjected to compression while the outer part is subjected to tension. These tensile stresses cause cracking along the surface. The opening of such surface cracks is frequently observed in ASR-affected concrete structures, see e.g. Poole [642] and Hobbs [373]. The contour plots in Figure 3.187 for t = 75 d and t = 130 d correspond to the wetting process (Phase II) characterized by a moisture transport oriented from the bottom towards the top. Accordingly, the ASR-induced expansion accelerates at the bottom face of the beam. At this stage, the ASR expansion


t [d]

Saturation sl

361

Expansion rate ε˙as Damage parameter d

20

0.6 75

ASR damage front

sl [-]

0.9

0.0 ε˙as [10−5 /d] 4.5 0.0

d [-]

1.0

130

Fig. 3.187. Numerical simulation of a concrete beam affected by alkali-silica reaction: Distribution of the liquid saturation sl , the expansion rate ∂εas /∂t, the expansion εas and the damage parameter d at different stages of the loading history

εas within the inner part of the beam is still larger compared to the expansion within the outer parts of the beam. The simultaneous moisture diffusion into the beam and the activation of reaction kinetics within the more humid zone near the surface leads to the formation of an ASR front which starts to propagate from the bottom to the top In front of the highly affected lower part of the beam the concrete is subjected to tensile stresses, which exhaust the tensile strength and lead to substantial cracking in the vicinity of the ASR front. At the top of the beam, cracks associated with the spatial gradient of the expansion εas due to drying of the upper part of the beam has penetrated about 5 cm into the beam. It should be noted, that in a purely hygro-mechanical analysis (i.e. εas = 0) corresponding to a beam made of concrete containing no reactive silica within the aggregates, no damage would be observed (i.e. d = 0) for the hygral loading scenario considered in this analysis. The left diagram in Figure 3.188 shows the evolution of the expansion εas in three points located along the axis of symmetry. The results clearly show, that the onset of moisture transport from the bottom into the beam (first vertical line at t = 20 d) considerably accelerates the ASR-expansion at the bottom side. At the beginning of Phase III (second vertical line at t = 130 d), the ASR reaction is accelerated at the top and decelerated at the bottom of the beam, respectively. However, at this relatively late stage the influence of varying hygral conditions on the kinetics of the ASR expansion is comparatively small since the sensitivity of the reaction kinetics on the moisture content reduces with an increasing reaction extent. Although the moisture content is uniformly distributed within the beam at the end of the numerical simulation (see Figure 3.187, t = 300 d), the expansion εas within

362


Evolution of ASR expansion

Load displacement diagrams 180

point A point B point C

0.3

C B A

0.2

loading at t = 0 d loading at t = 300 d

160 Reaction force R [N]

Expansion εas [%]

0.4

0.1

140

R0u

120 u R300

100 80 E0

60 40

E300

20 0

0 0

50

100

150

200

Time t [d]

250

300

0

0.2

0.4

0.6

0.8

1

Load factor λ [1]

Fig. 3.188. Numerical simulation of a concrete beam affected by alkali-silica reaction: Evolution of the ASR expansion εas in three points along the axis of symmetry and load displacement diagrams R(λ) of the virgin concrete beam (ultimate load analysis at t = 0 d) and the ASR-affected concrete beam (ultimate load analysis at t = 300 d)

the three points A, B and C differs considerable. This observation reflects the fact, that the ASR-induced expansion and, consequently, the deterioration of concrete structures caused by the alkali-silica reaction strongly depends not only on the moisture content but also on the hygral loading history. The right diagram in Figure 3.188 contains the load displacement relations R(λ) from ultimate load analyses performed for the ASR-affected beam and, for comparison, for an identical beam without being affected by the alkalisilica reaction. During 300 days of ASR-expansion, a considerable structural degradation is observed: the structural stiffness is reduced by [1 − E300 /E0 ] = 55% due to the alkali-silica reaction and the ultimate load is decreased by u [1−R300 /R0u ] = 27%. Furthermore, at time t = 300 d a more brittle structural response is observed in the very early post-peak regime compared to t = 0 d. 3.4.5 Lifetime Assessment of a Spherical Metallic Container Authored by Jan-Hendrik Hommel and G¨ unther Meschke This Subsection contains an application of the concept for life-time predictions of metallic structues subjected to cyclic loading [390] and described in Subsection 3.3.1.2.1. Figure 3.190(a) shows the spherical pressure vessel of the ICIWilhelmshaven supported by cylindrical columns. This structure is assumed to be subjected to earthquake loading, which causes settlements of the columns. In this prototype application of the life time assessment concept proposed in Section 3.3.1.2.1 the base displacements according to the El Centro


363

Damage parameter d

t [d]

0

[1]

1

0

300

λ=0

λ = 1/3

λ = 2/3

λ=1

Fig. 3.189. Numerical simulation of a concrete beam affected by alkali-silica reaction: Distribution of the damage parameter d at different stages of the mechanical loading test of the virgin concrete beam (ultimate load analysis at t = 0 d) and the ASR-affected concrete beam (ultimate load analysis at t = 300 d)

v e rtic a l d is p la c e m e n t [c m ]

2 0 1 0 0 -1 0 -2 0

(a )

0

1 0

2 0 tim e t [s ]

3 0

4 0

(b )

Fig. 3.190. Application of Low Cycle Fatigue Model: (a) Spherical pressure vessel of ICI-Wilhelmshaven, (b) vertical displacement-time plot of the El Centro earthquake

earthquake in 1940 (magnitude 6.9) is assumed as the relevant quasi-static loading scenario. The vertical displacement is illustrated for the first 32 seconds in Figure 3.190(b) [113]. Further details of the analysis are given in [386]. In the analysis, the structure is discretized by 7-parametric shell elements using a quadratic Mindlin-Reissner-type kinematics. A special updatedrotational formulation allows the simulation of the intersection of the spherical


v o id v o lu m e fra c tio n f [-]

0 .0 1 6 0 .0 1 5 0 .0 1 4 0 .0 1 3 0 .0 1 2 0 .0 1 1 0 .0 1 0 0 .0 0 9 (a )

m a x . v o id v o lu m e fra c tio n f [-]

364

0 .0 1 6 0 .0 1 4 0 .0 1 2 0 .0 1 0 0 .0 0 8 0

1 0

2 0 tim e t [s ]

3 0

4 0

(b )

Fig. 3.191. Application of Low Cycle Fatigue Model: (a) Damage accumulation resulting from the El Centro earthquake, (b) Temporal evolution of the maximal void volume fraction f

vessel and the cylindrical columms. To avoid locking phenomena an Enhanced Assumed Strain as well as an Assumed Natural Strain concept is implemented in the finite element procedure. Assuming the structue consistes of the steel alloy 20MnMoNi55 [390], the GTN-model with the calibrated model is used for the following numerical studies. Further details of the analysis are given in [386]. Figure 3.191(a) shows the damage accumulation in the vicinity of the vesselcolumn intersections. The maximum value of damage is observed at the highest point of the vessel-column intersection. An evolution of this maximum damage is plotted for the complete loading history in Figure 3.191(b). It is noted, that the maximum value of damage is not of critical level. It can be concluded, that this engineering structure would have survived the El Centro earthquake without failure.

4 Methodological Implementation

Authored by Dietrich Hartmann and Detlef Kuhl Based on the modeling of external as well as internal actions with respect to lifetime-oriented design (see Chapter 2), and by either applying or adapting various deterioration models of materials and structures (see Chapter 3), a methodological approach and implementation of appropriate concepts, numerical methods and solution paradigms are required to facilitate practical design to reality tasks. For this purpose, having addressed fundamental aspects of problem classification, numerical concepts, uncertainty and design methodology, details of specified solution methods for structural design are introduced. In this context, single and multi-field numerical methods, reliability analysis methods and optimization methods are dealt with, where particularly time-invariant behavior is taken into account. Finally, sophisticated practical applications are to demonstrate the capabilities of the established lifetime-oriented design concepts.

4.1 Fundamentals Authored by Dietrich Hartmann and Detlef Kuhl The previous chapters are dealing with the scientific bases required to establish lifetime-oriented design models for a solution by means of modern computer facilities. For that, the external impacts on materials and structures due to deterioration phenomena on micro-macro levels, to some extent even on a nano-micro level, have been scrutinized. As a result, an appropriate theoretical fortification of lifetime-oriented design concepts on the structural system level has been achieved, based on solid verification and, where required, on validation through experiments.

366

4 Methodological Implementation optimization and design reliability analysis numerical methods modeling of deterioration materials and structures

external attack

Chapter 3 • spatial discretization • stationary problems • temporal discretization • adaptivity • discontinuities • model reduction • reliability analysis • 1. and 2. order reliability • ν (t)-approximation • covariance analysis • Monte Carlo • response surface • parallelization • optimization problems • derivative based numerical solution methods • evolution strategies • parallelization Fig. 4.1. Overview of the methodological implementation of lifetime oriented design concepts

4.1.1 Classification of Deterioration Problems Authored by Dietrich Hartmann and Detlef Kuhl Analyzing and evaluating the evolution of the mechanical behavior of projected structures over short and long time spaces are the contemporary concern of research worldwide. To structure the current research endeavors, and to integrate the own work carried out in this field, a classification of the potential deterioration problems is taken with respect to the structural system level, in particular. Given that structures in civil engineering have to be designed, analyzed and constructed such that they are satisfying all requirements, imposed to them during their service performance, deteriorations and the following damages have to be scheduled within design concepts in a systemic fashion. According to the definitions introduced in the previous chapters, deteriorations are understood as irreversible microscopic alterations in a structure leading to macroscopic degradations on the structure level. In this connection, the characteristic mechanisms inducing the deteriorations and subsequent damages allow the classification as follows: • • • •

brittle deteriorations without creation of significant plastic strains ductile deteriorations due to pores during plastic effects creep damage caused by intergranular decohesion along with high temperature loading fatigue damage excited by means of transgranular fracture phenomena due to high as well as low load-cycles

4.1 Fundamentals

• •

367

chemo-mechanical, hygro-mechanical and thermo-mechanical damage chemically caused aging

Mathematical models describing these phenomena on the continuum mechanical level can be classified based on the character of the underlying differential equations: • • • • •

single field problems multi field problems stationary second order partial differential equations second order partial differential equations including derivatives with respect to spatial coordinates and time non-linear differential equations

Derivatives with respect to spatial coordinates are solved numerically by using the finite element method. As a result of this discretization process the deterioration mechanisms are described on the structural level. The associated semidiscrete structural equations are strongly non-linear vector equations depending on nodal values of time independent as well as time dependent state variables. Applying time integration schemes and consistent linearization, the approximated nodal solution of deterioration processes can be generated by the recurring solution of linear systems of equations. To allow for a goal-oriented and straightforward design procedure on the structural system level it is customary to represent the intricate internal deterioration and damage processes in terms of a set of appropriate limit states, either ultimate limit states or serviceability limit states. Both can be formulated as constraints to a structural design problem that must be satisfied necessarily. Also, dependent on the view a structural designer wants to take on the realistic representation of his design problem, the limit states may be defined in a deterministic or stochastic fashion. Obviously, the stochastic mapping of quantities as well as processes, describing the individual limit states, leads to more accurate, reliable and robust design results, however, the complexity and the computational effort increase drastically. In most of the cases, the approaches based upon stochastic problem representations require a multi-level solution philosophy with respect to space and time scales. Common ultimate limit states applied for civil engineering structures are: • • • • • •

loss of equilibrium change of position in the total structure or its parts (overtuning, sliding, lifting, etc.) fracture excessive displacement or twists passage to a kinematic mechanism instability

While ultimate limit states are serious threats to life or physical conditions of human beings the serviceability limit states constitute a danger to the

368


utilization of structures or buildings, respectively, and are described by means of limits for • • •

displacements which have to be below practical bounds natural frequencies whose exceedance avoid an usage of the structure stresses or strains which would jeopardize the durability of structures

In the particular cases, the nature of the design problem determines how many and which limit states are mandatory and have to be included in the lifetime-oriented design approach. As a matter of course, the computation of the quantities as well as processes incorporated in the individual limit states, according to the recent advances in computer technologies, are based on or aided by powerful numerical methods. 4.1.2 Numerical Methods Authored by Detlef Kuhl According to the character of mathematical models describing deterioration processes of concrete material and concrete structures several numerical methods are combined for their approximative solution and the prognosis of long term behavior. Spatial derivatives of single and multi field problems are approximated by using the spatial finite element method. The time variant formulation of durability models requires additionally the numerical integration by time integration methods. Adaptive methods in space and time supplement the basic discretization methods such that the quality of the numerical results can be controlled and the numerical effort is reduced. Finally, the numerical solution of non-linear differential equations includes the Newton-Raphson iteration based on the consistent linearization of partial differential equations, semidiscrete ordinary differential equations and fully discrete algebraic vector equations. The broad variety of mathematical models describing deterioration processes is numerically discretized by the finite element method. In order to simplify the development and coding process of numerical methods they are expressed within the framework of a generalized multi field problem containing all investigated models as particular examples. This generalized multifield problem is prepared for the finite element procedure by the weak formulation and the consistent linearization. Within the next development step a general finite element method for multifield processes is generated using a multi-dimensional spatial finite element concept. Approximations of state variables and test functions are evolved from one-dimensional Lagrange and Legendre polynomials of arbitrary polynomial degree, respectively. Both types of ansatz functions allow for the structured progress of two- and three-dimensional shape functions of arbitrary polynomial degree. Legendre and Lagrange polynomial based hierarchical shape functions provide additionally the generation of anisotropic ansatz functions. Furthermore, effectively formulated structural finite elements as special kinds of volume elements as for example beams, plates and shells are used for advanced structural calculations. The numerical simulation of fracture

4.1 Fundamentals

369

processes, as one of the main aspects of deterioration mechanics, can only be described with the standard finite element methods in combination with interface elements and remeshing in every load or time step. In order to avoid this inefficient computational technique, finite element methods including discontinuous displacement approximations are used. In particular, the embedded discontinuity approach and the extended finite element method are used for the computation of crack propagations. After the finite element discretization, semidiscrete ordinary differential equations are obtaind. The integration of these vector valued differential equations is realized by time integration schemes originally developed for structural dynamics. Furthermore continuous Galerkin methods, discontinuous Galerkin methods and generalized Newmark methods are used. Advantages of Newmark type integration schemes are the moderate numerical effort and the controllable numerical dissipation. Galerkin integration schemes are developed for arbitrary polynomial degree in time. Consequently, the order of accuracy overcomes the upper bound two of Newmark type integration schemes. Controlled by the polynomial degree, Galerkin integration schemes possessing an arbitrary order of accuracy, can be designed. Continuous and discontinuous Galerkin integration schemes are energy conserving and energy decaying methods, respectively. As a result of this, discontinuous Galerkin schemes are well suited for the calculation of transport processes including strong variations of source terms and boundary conditions. Both, Newmark and Galerkin type integrations schemes are supplemented by error based adaptive time stepping schemes. Because almost all present deterioration processes of material and structures in this book are highly non-linear, iterative schemes complete the collection of numerical methods solving these kind of problems. Exclusively, the pure Newton-Raphson scheme based on the consistent linearization, is adopted. The applied numerical methods are prepared for the iterative procedure in every development step. In particular, the underlying differential equations, weak forms, semidiscrete differential equations and algebraic equations are linearized. 4.1.3 Uncertainty Authored by Dietrich Hartmann Uncertainties, in general, have been of central interest in the solution of civil engineering problems over the past decades. Since the quantification of uncertainties in engineering is extremely relied on computing methods, it is not surprising that structural engineering, i. e. structural analysis as well as design, being the pathfinders for traditional deterministic computing in engineering, also substantially promoted the quantification of uncertainties using various computational approaches. In this context uncertainty should be understood according to the definition of Bothe [143] as the gradual assessment of the truth content of a proposition. Consequently, three categories of uncertainty can be

370


identified in engineering problems that need to have a computational representation if not only qualitative evaluation are to be carried out. These are • • •

stochastic uncertainty informal uncertainty lexical uncertainty

Depending on the nature of a problem considered the individual categories of uncertainty may occur separately from each other or in hybrid fashion. Stochastic uncertainty which are exclusively dealt with in the frame of this book is applicable when the facts of cases (either in terms of systems or processes) show a clear random behavior. This means that the principles, laws as well as rules of randomness, in harmony with the theory of probability, are completely satisfied. Characteristic assumptions that should theoretically apply (but are not always met in reality!) contain requirements such as (i) a sufficiently large universe (sample sizes), (ii) to specify the relevant stochastic parameters or functions correctly, (iii) the applicability of the describing parameters from reference cases to a new problem given (reproduction conditions) and (iv) the appropriateness of probability density (pdf) or distributed functions. Also, the often assumed identical independently distributed paradigm must apply accordingly. If the aforementioned premises do not or can not be satisfied because of lacking or unavailable information, e. g. if only undervalued sample sizes exist, then, strictly speaking, informal uncertainty models should or must be created. In such cases the fuzzy set theory, based upon pure mathematics, serves as a relatively new and powerful methodology to cope with such information deficits. The fuzzy set theory can also be applied if the third category of uncertainty occurs where uncertainty is quantified in terms of linguistic variables which are mapped to an adequate numerical scale. Hence, uncertainties stemming from information deficits or linked to linguistic variables are characterized by virtue of nonstatistical quantities which, to a large extend, exhibit subjective instead of of objective views on uncertainty. It should be mentioned that there are further mathematical quantification concepts for describing uncertainty. Although, in the following, the focus is placed solely on stochastic uncertainties some potential main concepts are to be named briefly; they include the method of subjective probability [231], [857], convex modelling [256] interval algebra [42] and chaos theory [425]. 4.1.4 Design Authored by Dietrich Hartmann The design of systems, in particular, the process of designing and erecting structures with respect to all the demands during their lifetime is a grand challenge problem. Complex structures, such as buildings, bridges, industrial plants and many others, are an excellent testimonial for the engineering design performance. However, if failures and casualties are reported one gets aware of the dangers and risks associated with stability, deteriorations damages, failures or inserviceability of structures. Then, the question arises if a

4.1 Fundamentals

371

structure could not have been designed better than the existing one to avoid the observed damages occurred or occurring. It is, therefore, obvious that the design process in civil engineering is a fairly complex process. Many assumptions must be made to develop structural models that can then be subjected to analysis by the available methods which are today, according to the revolution in computer technology and numerical computations, computer-based methods. Many possibilities and factors with respect to material properties, load actions and construction techniques must be considered during the design formulation phase. The larger the scheduled lifetime of a structure must be, the more is the increase in complexity and the more uncertainties have to be scrutinized and integrated into the design. Of course, also economic considerations play an important role in designing cost-effective structures. Despite the importance of costs, the lifetime-oriented design is assumed here to be represented by means of cost equivalent physical of structural quantities. This premise allows for a crisp structural mechanicsoriented formulation of the design process and avoids the consideration of further uncertain cost models. The design process begins by analyzing various options. Subsystems and their components are identified, preliminarily designed, evaluated, re-designed in more detail and then realized (fabricated). As a result, design represents an iterative process based upon the designer’s intuition and ingenuity as well as engineering competence and experience, in association with intensive structural analysis using computer facilities. In this context, the difference between structural analysis and structural design should be recognized. The analysis problem clearly addresses determining the static and dynamic behavior of a structural system designed for a given task. Hereby, the structural response due to specified actions (loadings) in terms of displacements, stresses, cracks, kinematic quantities (velocities, accelerations, frequencies, etc.) plays the dominant role. To compute these quantities the topology, geometry and sizes of the structural data of the various parts of a structure are totally known. By contrast, the design process calculates the sizes, shapes and, where applicable, even the topology of a structure to meet given performance requirements. Therefore, designs are estimated and then analyzed to check if they perform according to given specifications. If these specifications are satisfied, an acceptable or a feasible design is created; although it is attempted to change the design to improve its performance. If the design fails a re-design is mandatory to come up with a feasible solution. Both cases intensively rely on (computational) analysis capabilities, but, also represent more than pure analysis. According to the nature inherent in design problems, design processes must be understood as a ”synthesis (composition) approach”. Evidently, synthesis problems are more complex and cause greater efforts than analysis problems alone. Scarcity and efficiency in today’s competitive world force engineers to streamline and accelerate their design processes. Here, the computer-aided design provides means and ways to master complexity and time efforts. In

372


particular, those design problems either in preliminary or detailed design, where the conceptual decisions have been come already and an abstract formulation of design in terms of design variables, constraints and assessment (objective) criterion is available, can be transferred to an equivalent structural optimization problem (SOP). Based on such an optimization problem or model, numerical optimization strategies can be applied to solve design problems rapidly, within acceptable response times. This rigorous computeroriented design approach also helps a designer to gain a in-depth insight and understanding of the design problem, which is of particular use in lifetimeoriented design. To accomplish this, however, the structural optimum design process forces to identify explicitly a set of reasonable design variables, relevant constraints and significant and powerful optimization criterions.

4.2 Numerical Methods Authored by Detlef Kuhl The description of numerical methods, solving mathematical models for the prognosis of deterioration processes of materials and structures, contains the following aspects: • • • • • • • •

Deterioration models described in Chapter 3 are generalized in an abstract multi field problem. The solution strategy of highly non-linear and time dependent coupled field problems is outlined. General weak forms of coupled problems are generated and consistently linearized with respect to the field variables. Spatial finite element discretization techniques, using Lagrange or Legendre polynomial based multidimensional ansatz functions, are presented and discussed. Solution methods for stationary problems are briefly reviewed. Newmark and Galerkin type time integration schemes are presented. Adaptive methods in space and time as well as several underlying error measures and solution strategies are compared with attention to the algorithmic formulation. Standard finite elements methods are enriched for the description of discontinuities.

4.2.1 Generalization of Single- and Multi-Field Models Authored by Detlef Kuhl Mathematical models of single field and coupled field problems discussed in Chapter 3 are constituted by general balance equations, constitutive equations including history variables, driving forces and boundary conditions. In general, coupled processes in materials can be modeled by an abstract set of

4.2 Numerical Methods

373

interacting balance equations in local or, alternatively, global formulations. The local balances represent coupled partial differential equations which can be decomposed in the following parts: 1. 2. 3. 4.

time derivatives of the primary variables, divergenz of the balance quantity fluxes, constitutive laws connecting flux and driving forces and, finally, the driving forces expressed in terms of spatial gradients of primary variables. 5. Furthermore, interactions of the balance equations can be manifested in all parts of the differential equations. The modeling of generalized multiphysics problems is based on the generalized formulation of local and global balance equations by [253]. In order to prepare the general multiphysics model for its numerical solution, the strong form is weakly formulated and linearized. Before going into the technical aspects of this section, it is worth to embed the modeling of multiphysics proplems between the real physical problem and its virtual counterpart, compare Figure 4.2. This collection of the two step procedure from the original physical problem to its numerical simulation by means of modeling and numerical solution is denoted as numerical modeling. It is essential for reliable simulation of physical processes to develop adequate mathematical models capturing the main character of real phenomena and to apply numerical methods which solve the mathematical model preferably accurate and efficient. As a result of this statement, the modeling of multiphysics problems yielding partial differential equations and numerical methods, namely finite element spatial discretization methods and time integration methods, are discussed separately in the following sections. 4.2.1.1 Integral Format of Balance Equations The integral format of the balance equation related to the tensor field f ∈ [1, NF ] ∂ ˆ f dV Θ f dV = Φf · ndA + Σ f dV + Θ (4.1) ∂t Ω

ΓΦf

Ω

Ω

is described in terms of the volume specific density of a sf th order tensor valued balance quantity Θ f , the [sf + 1]th order tensor valued flux density Φf , the sf th order tensor valued production term Σ f and the balance quantity ˆ f . For the explanation of the exchange between the involved tensor fields Θ domain Ω and the boundaries ΓΦf see the example of a two field problem given in Figure 4.2. Since the model for ion transport, dissolution processes and mechanically caused material failure in Section 3.3.2.2 and [454] is one of the simplest

374


multifield problems investigated in the present book, it is intensively used as illustrative example for the abstract theory of numerical methods in computational durability mechanics. The chemo-mechanical damage model constitutes a two field multiphysics problem NF = 2 with tensor valued primary variables u1 = u and u2 = c of order s1 = 1 and s2 = 0, respectively. The associated balance equations are defined in terms of the stress tensor Φ1 = σ, ˙ 1 = 0, the calvanishing volume forces Σ 1 = 0 and linear momentum flux Θ cium ion mass flux tensor Φ2 = q, a vanishing production term Σ2 = 0, the storage/production term Θ2 = [[φ0 +φ2 ]c]˙ + s˙ and vanishing exchange terms ˆ 2 = 0. ˆ 1 = 0 and Θ Θ 4.2.1.2 Strong Form of Individual Balance Equations The local format of balance equations according to tensor fields f ˆf ˙ f = divΦf + Σ f + Θ Θ

˙ f = ∂Θf Θ ∂t

(4.2)

is given in terms of sf th order tensor valued field variables uf and associated internal variables κf . The time derivative of balance quantity Θ is in general defined in terms of all field variables uNF , time derivatives of field variables u˙ NF and internal variables κNF . ˙ f (u˙ 1 , . . . , uNF , u1 , . . . uNF , κ1 , . . . , κNF ) = Θ ˙ f (u˙ NF , uNF , κNF ) ˙f =Θ Θ (4.3) The constitutive law defining the flux density Φf is formulated as a function of the driving force γ f and the potential Ψf .

γ f = γ f (∇uf ), Φf (γ NF , uNF , κNF ) =

∂Ψf (γ NF , uNF , κNF ) ∂γ f

(4.4)

Dirichlet boundary conditions, Neumann boundary conditions uf = uf ∀ X ∈ Γuf

Φf · n = φf ∀ X ∈ ΓΦf

(4.5)

u˙ f (t0 ) = u˙ 0f

(4.6)

and initial conditions uf (t0 ) = u0f

complete the generalized initial boundary value problem for multiphysics analyses with Γ = Γuf ∪ΓΦf and Γuf ∩ΓΦf = ∅ (compare Figure 4.2).

4.2 Numerical Methods physical problem

375

abstract model of coupled processes φ1 n ΓΦ1 ,ΓΦ2

modeling

mathematical model

•

Γu 2

Γu 1 ˆ 1 = −Θ ˆ2 Θ

˙1 Θ Ω

•

numerical solution

˙2 Θ

φ2 Γ

simulation of physical problem

Fig. 4.2. Numerical modeling via modeling and numerical solution. General multiphysics problem with field variables u1 and u2

Applied to the representative example of coupled mechanically and chemically induced damage [454], the associated balance equations DIV[σ] = 0

DIV[q] + [[φ0 +φ2 ]c]˙+ s˙ = 0

(4.7)

and constitutive laws Ψ1 =

1−φ ε : C0 : ε 2

Ψ2 =

φ γ · D0 · γ 2

(4.8)

are based on the definition of the driving forces γ 1 = ε = ∇sym u

γ 2 = γ = −∇c

(4.9)

in terms of the elasticity tensor of the skeleton C 0 , the linear strain measure ε, the conductivity tensor of the pore fluid D 0 , the concentration gradient γ and the total porosity φ = φ0 + φ1 + φ2 . φ includes the initial porosity φ0 , the mechanically induced porosity φ1 = [1 − φ0 − φ2 ]d and the chemical porosity φ2 . Consequently, the stress tensor and the calcium ion flux tensor are obtained as follows: σ = [1 − φ] C 0 : ε

q = φ D0 · γ

(4.10)

Internal variables κ1 and κ2 , evolution equations and phenomenological models for mechanically

376


φ1 = φ1 (d, φ2 )

d = d(κ1 (η(ε)))

(4.11)

and chemically caused damage s = s(κ2 (c)), s˙ =

φ2 = φ2 (s)

∂s ∂κc c˙ ∂κc ∂c

(4.12)

complete the continuum mechanical description of chemo-mechanical damage. 4.2.2 Strategy of Numerical Solution Authored by Detlef Kuhl The strategy of the numerical solution of generalized multifield problems is designed by the character of the underlying differential equations. In general the coupled field problems are described by highly non-linear partial differential equations. This differential equations include second order spatial derivatives and up to second order temporal derivatives. The resulting concept for the spatial and temporal discretization of non-linear multifield problems is illustrated in Figure 4.3. 1. For the preparation of the numerical solution the highly non-linear and time dependend differential equations are weakly formulated and linearized. 2. The spatial finite element discretization is realized by multi-dimensional multifield elements of arbitrary polynomial degree.

a. Ψf = divΦf +Σf

1. b. δW = 0 2. M, D K

ri = r

c. semdiscrete balance d. discrete balance

3.

δW(◦) = 0

d. ri (◦) = rn+1−αf f. Newton correction e. linearized balance

ΔδW

c.

a. strong form b. weak form

1. weak formulation 2. spatial discretization 3. temporal discretization 4. linearization

ΔδW(◦)

e. K Δu dG dG 4. = rdG

5.

5. iterative solution

f. uk+1 n+1 = ukn+1+Δu

Fig. 4.3. Modeling and numerical analysis of multiphysics problems


377

3. The time derivatives within the underlying differential equations are numerically solved either by Newmark or Galerkin type time integration schemes. 4. Finally, the non-linear format of resulting algebraic vector equations is solved by the Newton-Raphson procedure. Spatial and temporal discretization methods are the main tasks of the numerical solution strategy. They are enriched by adaptive methods in order to control the numerical effort and the solution quality of structural reliability simulations. 4.2.3 Weak Formulation Authored by Detlef Kuhl Within the present section the weak formulation of generalized multiphysics problems is presented. In order to prepare the weak form also for iterative solution methods, the linearized weak form is additionally generated. 4.2.3.1 Weak Form of Coupled Balance Equations The weak form of individual balance equations and Neumann boundary conditions is generated by the contraction with the test function δuf , integration over the domain Ω and the boundary ΓΦf

˙ f dV − δuf ◦ Θ

δWf (•) =

δuf ◦ divΦf Ω

Ω

δuf ◦ Σ dV +

− Ω

dV

δuf ◦ Φf · n−φf dA = 0

(4.13)

ΓΦf

and application of the divergence theorem.

˙ f dV + δuf ◦ Θ

δWf (•)=

Ω

Ω

δuf ◦ Σ dV −

− Ω

◦

δ∇uf ◦ Φf dV δuf

◦ φf dA =0

(4.14)

ΓΦf

According to the multiphysics problems considered in the present book ˆ f = 0 is assumed for the sake of simplicity. In equation (4.14) ◦ and ◦◦ Θ represent the sf th and [sf + 1] tensor contraction, respectively, and argument

378


(•) represents all state variables u˙ NF , uNF and spatial gradients ∇uNF . On the basis of equation (4.14) the weak form of the coupled multiphysics system δW is generated in equation (4.15) by the weighted summation of the individual weak forms δWf , whereby Af is introduced to adapt physical units and dimensions of coupled fields.

δW (•) =

NF

Af δWf (•) = 0

(4.15)

f =1

4.2.3.2 Linearized Weak Form of Coupled Balance Equations In order to prepare the weak form for the numerical solution with the Newton-Raphson scheme, the weak form is expanded in a Taylor series about the trial solution of all state variables δW (•k+1 ) = δW (•k ) + ΔδW (•k ) = 0

(4.16)

and spatial gradients characterized by •k .

¨ NF , u˙ NF , uNF ) = ΔδW (•) = ΔδW (∇uNF , u

NF

Af ΔδWf (•) = 0

f =1

(4.17)

The increment of weak forms is generated by summation of individual portions in terms of the increments of field variables Δug and gradients of field variables ∇Δug with g ∈ [1, NF ].

ΔδWf (•) =

NF ∂δWf g=1

∂∇ug

◦

◦ Δ∇ug +

∂δWf ∂δWf ◦ Δu˙ g + ◦ Δug ∂ u˙ g ∂ug

(4.18)

It is worth to mention that the derivative with respect to gradient ∇ug is performed explicitly in oder to obtain an advantageous format for


379

the finite element discretization discussed in Section 4.2.4.2. The individual terms in equation (4.18) are expressed as follows: ∂δWf ◦ ◦ Δ∇ug = ∂∇ug ∂δWf ◦ Δu˙ g = ∂ u˙ g ∂δWf ◦ Δug = ∂ug

δuf ◦ Ω

Ω

˙f ◦ ∂Θ ◦Δ∇ug dV + ∂∇ug

˙f ∂Θ δuf ◦ ∂ u˙ g δuf ◦

˙f ∂Θ ∂ug

◦

∂Φf ◦ ◦Δ∇ug dV ∂∇ug

◦

∂Φf ∂ug

δ∇uf ◦ Ω

◦ Δu˙ g dV ◦ Δug dV +

Ω

δ∇uf ◦

◦ Δug dV

Ω

(4.19)

4.2.4 Spatial Discretization Methods Authored by Detlef Kuhl and Christian Becker Within the framework of the semdiscretization technique applied to solve durability single- and multiphysics problems, the spatial discretization is realized by the finite element method (see e.g. [90, 106, 223, 224, 870, 871]). The scientific and industrial oriented literature documents the broad range of applications of this method for the spatial discretization of differential equations. Highly non-linear problems, stationary and transient problems as well as single- and multifield problems can by solved adopting the finite element method. 4.2.4.1 Introduction Standard finite elements for one-, two- and three-dimensional problems are using low order approximations of the primary variables. The great advantage of these finite element methods is the effective formulation and implementation in finite element codes, see as a particular example the popular constant strain triangle [796]. The drawback of such kind of finite elements is the low computation accuracy because of only linear ansatz functions and locking phenomena. Basically finite element solutions can be improved by reducing the element size (h-method) and increasing the polynomial degree of ansatz functions (p-method), respectively. Furthermore, special element techniques can be used to overcome the well known locking problem. In summary, two main philosophies are used in the context of structural mechanics, to obtain high quality numerical results applying the finite element method. •

Low order finite element methods combined with methods to prevent locking [637, 790]:

380

•


◦ selective reduced integration [397, 649] and hourglass control [283] ◦ assumed natural strain concept [398, 247] ◦ B-bar methods [850, 789] ◦ enhanced assumed strain concept [747, 742] Higher order finite element methods using different kind of higher order ansatz polynomials: ◦ multidimensional Lagrange polynomials [870, 452] ◦ Legendre based hierarchical one-, two- and three-dimensional shape functions [72, 717, 246]

A literature review about high quality, low order finite element methods makes clear that the related element techniques are separately developed for selected applications in structural mechanics. But computational durability mechanics is characterized by manifold various underlying differential equations. Therfore, the more general higher order finite element method is presented in the following. 4.2.4.2 Generalized Finite Element Discretization of Multifield Problems The numerical analysis of non-linear multiphysics problems can be subdivided in the spatial finite element discretization, the temporal discretization and the iterative solution of the resulting non-linear algebraic equation. In the present section a detailed description of the spatial p-finite element discretization of generalized multiphysics problems is given. 4.2.4.2.1 Approximations The finite element formulation of the original and the linearized weak forms of multiphysics problems is based on the approximation of test functions, state variables and their gradients by shape functions and nodal values of state variables. Therefore, ND-dimensional anisotropic shape functions of arbitrary polynomial degrees pd for the ND spatial directions d are designed based on one-dimensional Lagrange shape functions. Furthermore, the approximation of state variables is given and the transformation between natural and physical element coordinates is performed by the Jacobi transformation. One-dimensional Lagrange shape functions of polynomial degrees pd can be generated for every spatial direction d ∈ [1, ND] by the product N i (ξd ) =

p: d +1 k=1 k=i

ξdk − ξd ξdk − ξdi

ξdi =

2[i − 1] −1 pd

(4.20)

in terms of the natural coordinate ξd and the natural nodal positions ξdi and ξdk with i, k ∈ [1, pd + 1]. Consequently, the derivatives of shape functions are also calculated for arbitrary polynomial degrees pd .


i N;d (ξd ) =

p: p d +1 d +1 −1 ξdk − ξd ∂N i (ξd ) = ∂ξd ξdl − ξdi k =1 ξdk − ξdi l= 1

381

(4.21)

k = i k = l

l= i

ND-dimensional isotropic or anisotropic shape functions are generated by the product of one-dimensional shape functions N i (ξ) =

ND :

ND d−1 : i = k1 + [kd −1] [pl +1]

N kd (ξd )

d=1

d=2

(4.22)

l=1

and their derivatives with respect to natural coordinates ξm with m ∈ [1, ND] kd analog by the product of the derivatives N;m . ND :

i kd N;m (ξ) = N;m (ξm )

N kd (ξd )

(4.23)

d=1 d=m

In equations (4.22) and (4.23) the counter kd ∈ [1, pd + 1] for one-dimensional shape functions is used. Representative examples for two-dimensional isotropic and anisotropic Lagrange shape functions as well as partial derivatives of Lagrange shape functions are given in Figures 4.4 and 4.5. Furthermore, the general design of one-, two- and three-dimensional shape functions is illustrated in Table 4.1. Shape functions (4.22) and and spatial derivatives of shape functions (4.23) are applied for the approximation of state variables

N24 (ξ2 )

N12 (ξ1 )

ξ2

N 11 (ξ1 , ξ2 )

ξ1 shape function N 11

N24 (ξ2 )

11 N;1 (ξ1 , ξ2 )

ξ2

ξ2

ξ1

ξ1 2 (ξ1 ) N1;1

ξ1 11 derivative N;1

Fig. 4.4. Illustration of isotropic Lagrange shape functions and derivatives of Lagrange shape functions by means of a cubic planar finite element

382


N22 (ξ2 )

N 8 (ξ1 , ξ2 )

ξ2

N12 (ξ1 )

2 N2;2 (ξ2 )

8 (ξ1 , ξ2 ) N;2

ξ2

ξ2

ξ1

ξ1 N12 (ξ1 )

ξ1

ξ1

shape function N 8

8 derivative N;2

Fig. 4.5. Illustration of anisotropic Lagrange shape functions and derivatives of Lagrange shape functions by means of a cubic-linear planar finite element Table 4.1. Multi-dimensional Lagrange shape functions and specialization to one-, two- and three-dimensional finite elements element type

position vector

field variable

X

u

integral • dV Ωe

ND = 1

( ) X = X1

( ) u = u1

1 −1

dVξ

truss element %

ND = 2

X1 X2

X=

ND = 3

X1 ⎢ ⎥ X = ⎣ X2 ⎦ X3

⎡

volume element

NN

%

&

plane element

uf ≈

• dξ1 A !

u= ⎤

N i uei f

i=1

u1 u2

⎡

&

−1−1

⎤

u1 ⎢ ⎥ u = ⎣ u2 ⎦ u3

u˙ f ≈

NN

1 1

• dξ1 dξ2 H ! dVξ

1 1 1 −1−1−1

• dξ1 dξ2 dξ3 ! dVξ

N i u˙ ei f

(4.24)

∂N i ∂X

(4.25)

i=1

and the gradient of state variables ∇uf ≈

NN i=1

i uei f ⊗∇N

∇N i (ξ) =


383

˙ ei in terms of nodal state variables uei f . Furthermore, equations (4.24) f and u and (4.25) are used for the approximation of δuf , Δuf , δ∇uf and Δ∇uf δuf (ξ) ≈

NN

N i (ξ) δuei f

δ∇uf (ξ) ≈

i=1

Δuf (ξ) ≈

NN

NN

i δuei f ⊗∇N (ξ)

i=1

N i (ξ) Δuei f

Δ∇uf (ξ) ≈

i=1

NN

i Δuei f ⊗∇N (ξ)

i=1

(4.26) ˙ f , Φf and their as well as for the implicit approximation of resulting terms Θ derivatives. Finally, the Jacobi tensor J(ξ) ≈

NN

X ei ⊗ ∇ξ N i (ξ)

∇ξ N i (ξ) =

i=1

∂N i (ξ) ∂ξ

(4.27)

allows for the transformation of the gradient of shape functions ∇N i (ξ) =

∂N i (ξ) ∂ξ · = J −T (ξ) · ∇ξ N i (ξ) ∂ξ ∂X

and the differential volume element. ⎧ ⎨ dξ1 dξ2 dξ3 for ND = 3 dV = |J | dVξ , dVξ = dξ1 dξ2 H for ND = 2 ⎩ for ND = 1 dξ1 A

(4.28)

(4.29)

4.2.4.2.2 Non-Linear Semidiscrete Balance Inserting the approximations discussed above into the weak form of multiphysics problems (4.14) and (4.15) for individual finite elements e ∈ [1, NE] yields the discretized weak form on the element level

δW e ≈

NF NN

ei ei δuei f ◦ r if − r f = 0

(4.30)

f =1 i=1

˙ eNF , ueNF ) and the in terms of the generalized internal force tensor r ei if (u ei generalized external force tensor r f according to the nodal values of the test function δuei f .

384


Ω e

r ei f

Φf · ∇N i dV

˙ f dV + Af Ni Θ

r ei if = Af

e Ω

N i φf

i

= Af

N Σ dV + Af

(4.31) dA

e ΓΦ

Ωe

f

After assembling the element quantities over element nodes, elements and tensor fields, application of the fundamental lemma of variational calculus and consideration of initial values the semidiscrete initial value problem ˙ u) = r ri (¨ u, u,

¨0 u(t0 ) = u0 , u˙ 0 , u

(4.32)

is obtained. In the non-linear second order vector differential equation ¨ represent the generalized vectors of (4.32) the vectors ri , r, u, u˙ and u internal forces, external forces, primary variables, first and second temporal rates of primary variables, respectively, whereby every structural vector contains the nodal values of all contributing fields f . As particular example the assembling of the generalized vector of internal forces ri based on the nodal element internal force tensors r ei i is given. ˙ u) = ri (¨ u, u,

NE,N AN

˙ eNF , ueNF ) rei i (u

(4.33)

e,i

It is worth to mention that the second time derivative only exists, if the ˙ f is identified with a second order temporal change of the balance quantity Θ time derivative as for example for the modeling of structural dynamics or wave propagation problems (see e.g [452]). The application of the present generalized multiphysics finite element concept for the discretization of the chemo-mechanical damage model yields the ei element tensors of internal and external forces (r ei i1 and r 1 ) and internal and ei ei external calcium ion mass fluxes (r i2 and r2 ). ei ∇N i · σ dV ri1 = Ω ei ri2

=− ∇N · q dV + i

Ω

rei 1 =

Ω i

N t dA Γσ

r2ei

N i q dA

= Γq

N i [[φ0 +φ2 ]c+s]˙ dV (4.34)


385

4.2.4.2.3 Linearized Semidiscrete Balance Applying the approximation procedure to the linearized weak form (4.17-4.19) on the element level yields its discrete counterpart

ΔδW e ≈

NN NF NF NN

eij eij ej ej ˙ δuei ◦ d ◦ Δ u + k ◦ Δu f g g fg fg

(4.35)

f =1 g=1 i=1 j=1

˙ eNF , ueNF ) and in terms of generalized tangent stiffness tensors keij f g (u ˙ eNF , ueNF ) according to the test generalized tangent damping tensors deij f g (u ej functions δuei f and increments Δug . keij f g = Af

Ni Ω

+ Af

N

i

˙f ∂Θ ∂ug

N

i

˙f ∂Θ ∂ u˙ g

Ω

deij fg

= Af

˙f ∂Θ · ∇N j dV + Af ∂∇ug

∇N i ·

∂Φf · ∇N j dV ∂∇ug

∇N i ·

∂Φf ∂ug

Ω

j

N dV + Af

N j dV

Ω j

N dV

Ω

(4.36) On the structural level the linearized discrete balance equation is obtained accordingly: ˙ u)Δ¨ ˙ u)Δu˙ + K(¨ ˙ u)Δu = r − ri (¨ ˙ u) (4.37) M(¨ u, u, u + D(¨ u, u, u, u, u, u, ˙ u), D(¨ ˙ u) and K(¨ ˙ u) are the generalized tanHerein M(¨ u, u, u, u, u, u, ˆteaux derivative of the generalized gents which can be defined by the Ga ¨. internal force vector with respect to the state variables u, u˙ and u

˙ u) = K(¨ u, u,

˙ u) u, u, ∂ri (¨ ∂u

D=

∂ri ∂ri , M= ¨ ∂ u˙ ∂u

(4.38)

As particular example the generalized tangent tensors associated with the chemo-mechanical model of coupled mechanical damage and calcium leaching are given. The generalized tangent stiffness tensors are given in terms of the second order mechanical tangent stiffness tensor,

386


keij 11

∇N i ·

=

∂σ · ∇N j dV ∂ε

(4.39)

Ω

the scalar valued chemical tangent c ∂q eij = − ∇N i · N j dV + ∇N i · D0 φ ·∇N j dV k22 ∂c Ω Ω (4.40) ∂ 2s ∂κc i ∂s ∂φ2 j j 2 cN ˙ φ N dV + N i c ˙ N dV + s ∂c ∂s ∂κ22 ∂c Ω

Ω

and the first order chemo-mechanical tangent coupling tensors. ∂q eij i ∂σ j keij N · ∇N j dV = ∇N · dV, k = − ∇N i · 12 21 ∂c ∂ε Ω

(4.41)

Ω

Furthemore, the damping tensor of the chemo-mechanical model of calcium leaching is computed. i ∂s j φ deij = N N dV + N i φ0 + φ2 N j dV (4.42) s 22 ∂c Ω

Ω

eij , Because of the time independent balance of momentum (4.7) the tensors d11 eij eij d12 and d21 vanish. In equations (4.40) and (4.42) the following abbreviations are used: c ∂q ∂φ2 ∂D 0 ∂φ2 D0 · γ + φ · γ, φs = 1 + c (4.43) = [1−d] ∂c ∂c ∂c ∂s

4.2.4.2.4 Generation of Element and Structural Quantities The integrals in equations (4.31) and (4.36) are computed by the GaußND Legendre quadrature with NG = d=1 NGd integration points ξdld , ld ∈ [1, NGd ] contained in vectors ξ l and the weights αl based on the onedimensional integration rule, see e.g. [870]. f (ξ)|J (ξ)|dVξ ≈ Ωξ

NG 1 NG 2 NG 3

αl f (ξ l )|J (ξ l )|, αl =

l1 =1 l2 =1 l3 =1

ND :

αld

(4.44)

ld =1

Figure 4.6 illustrates the final calculation of element quantities of generalized multiphysics p-finite elements. It is obvious that specific multiphysics problems as described in Chapter 3 can be implemented on the model or Gauß point level marked within the algorithmic set-up. Internal variables κf are just managed on the finite element level. Manipulations of internal variables take exclusively part on the material point level.

4.2 Numerical Methods select element nodal values for i ∈ [1, NN ]

387

˙ ei X ei , uei f , u f

ND loops over Gauss points ld ∈ [1, NGd ] coordinates and weight of Gauß point

ξ = [ξ1l1 ξ2l2 ξ3l3 ]T , α = αl1 αl2 αl3

loop over element nodes i ∈ [1, NN ] N i (ξ), ∇ξ N i (ξ)

shape functions and natural derivatives

J (ξ)

Jacobi transformation tensor next node i

|J (ξ)|, J −1 (ξ)

Jacobi determinant and inverse Jacobi tensor loop over element nodes i ∈ [1, NN ]

X(ξ), uf (ξ), u˙ f (ξ)

state variables physical gradient of shape functions physical gradient of tensor fields

∇N (ξ) = J −T(ξ)·∇ξ N i (ξ) i

∇uf

next node i model level

˙ f ,Φf ,∂ Θ ˙ f/∂∇ug ,∂ Θ ˙ f/∂ug ,∂ Θ ˙ f/∂ u˙ g ,∂Φf/∂∇ug , . . . Θ

loop over element nodes i ∈ [1, NN ] generalized external load vector

r ei f

generalized internal force vector

r ei if

summation and assembly

ei α|J |r ei f → r, α|J |r if → ri

loop over element nodes j ∈ [1, NN ] generalized tangent damping tensor

deij fg

generalized tangent stiffness tensor

k eij fg

summation and assembly

eij α|J |deij f g → D, α|J |k f g → K

next node j next node i next Gauß point ld Fig. 4.6. Computation of generalized element tensors of external and internal forces and generalized tangent damping and stiffness tensors of multiphysics p-finite elements

4.2.4.3 p-Finite Element Method The p-finite element method (p-FEM) is the exact counterpart to the h-finite element method (h-FEM). Whereas in the h-FEM a mesh of low-order elements is refined by increasing the number of elements that are naturally

388


1

p(X ) = sin(−8 X )

X

1

L = 1, 0

log rel. error [−]

0 1

-1 -2 -3 -4

p-mesh h-mesh

-5 -6 0

5

10

15

20

25

30

DOF [−]

Fig. 4.7. Sinusoidal loading of a truss member (left), Rel. error of internal energy plotted over the number of degrees of freedom (right)

smaller than the original ones, the p-FEM approach is similiar to the well-known Ritz-method. The basic idea of the approach is to use a relative coarse finite element mesh. If the quality of the numerical solution has to be improved the mesh remains unchanged but the polynomial degree of the shape functions, and therefore the polynomial degree of the approximation, is increased subsequently. In the limit case of p → ∞ the exact solution is gained. A considerable advantage of the p-finite element method in the case of a smooth solution is the exponential convergency compared to the almost linear convergence rate for the h-method. This is illustrated by an example that was analyzed in [245] and deals with a truss member that is loaded by a sinusoidal force that is acting along the member’s length (see Figure 4.7). The internal energy and the relativ error are defined as: 1 FE REF ||Wint − Wint || Wint = σ : ε dΩ E= . (4.45) 2 REF ||W || int Ω A standard h-approach and p-approach is used for the numerical analysis. The reference solution is the solution of the p-discretization with p = 11. In the figure the convergency rate with respect to the relative error in energy norm is ploted logarithmical over the degrees of freedom. The exponential convergence of the p-method becomes obvious in contrast to the almost linear convergency of the h-method. Concluding, if a certain degree of accuracy is required, the p-FEM is reaching this accuracy with less degrees of freedom than the h-FEM. Basically, there are two concepts of higher order shape function concepts within the p-FEM: the non-hierarchical Lagrange concept and the hierarchical concept based on the Legendre polynomials.


389

4.2.4.3.1 Onedimensional Higher-Order Shape Function Concepts Before dealing with details on threedimensional shape function concepts the basic concept of higher order approximations is illustrated in the onedimensional case. The generation of shape functions of the Lagrange-Type has already been introduced in Section 4.2.4.2.1. 4.2.4.3.1.1 Shape Functions of the Legendre-Type Legendre polynomials are the solutions Pn (ξ) of the homonymous differential equation [1 − ξ 2 ] Pn (ξ)− 2 ξ Pn (ξ)+ n [n+ 1] Pn(ξ) = 0 ,

ξ ∈ R,

n ∈ {0, 1, 2, 3...}. (4.46)

The solutions Pn (ξ) can be generated by the recursive formula by Bonnet Pn (ξ) =

1 [[2n − 1] ξ Pn−1 (ξ) − [n − 1] Pn−2 (ξ)] , n ≥ 2 n (4.47)

chosing P0 (ξ) = 1 and P1 (ξ) = ξ as starting polynomials. Before the polynomials (4.47) are applied within a shape function concept, they are modified such that their value vanish at the outer vertices of an onedimensional element: 1 Φi (ξ) = [Pi (ξ) − Pi−2 (ξ)] , 2 (2 i − 1)

i ≥ 2.

(4.48)

Figure 4.8 shows a selection of some modified higher-order Legendre polynomials. Within an onedimensional hierarchical shape function concept the first two shape functions are the standard linear Lagrange shape functions N1 (ξ) =

1 [1 − ξ] 2

N2 (ξ) =

1 [1 + ξ], 2

(4.49)

representing the linear approximation of the respective field variable, compare Section 4.2.4.2.1. The higher-order shape functions are taken from the modified Legendre polynomials (4.48): Nj (ξ) = Φj−1 (ξ) ,

3 ≤ j ≤ p + 1.

(4.50)

390


Φj [ξ i ] [−]

0.2 0 -0.2 -0.4 -0.6 -1

0

1

i

ξ [−] Fig. 4.8. Modified Legendre-polynomials for polynomial degrees p = 2, 3, 4, ..., 7

4.2.4.3.1.2 Comparison of Both Shape Function Concepts In general the numerical solutions of both concepts are identical if the same approximation degree is used. This is because hierarchical and nonhierarchical approximation describe the same polynomial Lp of degree p Lp (ξ) = a0 + a1 ξ 1 + a2 ξ 2 + ... + ap ξ p .

(4.51)

Solely the meaning of the discrete degrees of freedom differ from each other. This topic will be dealt with in a subsequent paragraph. Figure 4.9 shows the sets of shape functions for the Lagrange and for the Legendre concept respectively for the approximations p = 1, 2, 3. As it can be seen easily, the shape functions of the Lagrange type fullfill the interpolation property Ni (ξk1 ) = δik whereas this is only partially the case for the shape functions of the Legendre type (just for the linear ones). Within the Lagrange concept - contrary to the Legendre concept - the interpolation property can be used for an isoparametric approximation of the geometry, because all nodes are uniquely defined in space. A drawback of this concept is that all the shape functions have to be of the same degree of approximation and therefore have to be newly generated when the approximation order is increased. Regarding the hierarchical concept it can be seen that an increase of the approximation order from p to p + 1 results in generating only one new shape function of degree p+1 whereas the existing shape functions remain unchanged. This is the main characteristic of the hierarchical concept. Figure 4.10 illustrates the structure of the element stiffness matrices K and of the internal load vector r of both concepts for different polynomial degrees within an arbitrary linear problem. It can be seen that the bandwith of the stiffness matrix is smaller in the Legendre concept. Furthermore, the hierarchical structure of the matrices and vectors within the Legendre concept is obvious. In linear problems the

4.2 Numerical Methods 1 Nj [ξ i ] [−]

p=1

Nj [ξ i ] [−]

1 0.5 0 -0.5 -1

0

0.5 0 -0.5

1

-1

ξ i [−]

Nj [ξ i ] [−]

Nj [ξ i ] [−]

0.5 0 -0.5 0

0.5 0 -0.5

1

-1

ξ i [−]

0

1

ξ i [−]

1

1 Nj [ξ i ] [−]

Nj [ξ i ] [−]

1

1

-1

p=3

0 ξ i [−]

1

p=2

391

0.5 0 -0.5 -1

0 i

ξ [−] Legendre concept

1

0.5 0 -0.5 -1

0

1

i

ξ [−] Lagrange concept

Fig. 4.9. Set of hierarchically organized shape functions of the Legendre type for polynomial degrees p = 1, 2, 3 (left), Shape functions of the Lagrange type holding for the interpolation property for polynomial degrees p = 1, 2, 3 (right)

hierarchical concept can be used in adaptive p-refinements where the quality of the approximation is controlled by an error-estimator or indicator. If a p-refinement is needed most parts of the existing matrices and vectors can be used and only few entries have to be newly calculated. Furthermore it can be seen that within an onedimensional hierarchical concept the higher order degrees of freedom are decoupled from the other ones, because there are only entries on the main diagonal of the element stiffness matrix. In general, in a hierarchical concept the resulting equation system has a better conditioning [245]. The meaning of the nodal degrees of freedom within both concepts is as follows: In the Lagrange concept the nodal values represent the value of the

392


p=1 p1 p1

0

0

p1

p1 p1

0

0

p1

0

0 p2

0

p2

0

0

0 p3

p3

p=2

p=3

Legendre discretization

Lagrange discretization

Fig. 4.10. Comparison of the structure of element vectors and matrices for the Legendre- and Lagrange-concept for polyomial degrees p = 1, 2, 3

field variable at the corresponding node of the element. Because of the lack of the interpolation property the nodal values of the nodes within the Legendre concept - except the values at the vertices of a brick element - can not be interpreted that way. The higher-order degrees of freedom represent - multiplied with the corresponding shape function - a p-fraction of the resulting distribution of the solution. 4.2.4.3.2 3D-p-Finite Element Method Based on Hierarchical Legendre Polynomials After illustrating the main properties of the onedimensional hierarchical shape function concept in comparison to the Lagrange concept, the approximation techniques of the threedimensional hierarchical shape function concept are described subsequently. The basis of the threedimensional implementation is the brick element illustrated in Figure 4.11. Therein the definition and numbering of element vertices, edges and faces is given. 4.2.4.3.2.1 Generation of 3D-p-Shape Functions The threedimensional shape functions result from a spatial mutliplication of the onedimensional ones (4.49,4.50), leading to: i ∈ {1, 2, 3, ..., pξ1 + 1} Nl3D (ξ) = Ni1D (ξ 1 )Nj1D (ξ 2 )Nk1D (ξ 3 )

j ∈ {1, 2, 3, ..., pξ2 + 1} k ∈ {1, 2, 3, ..., pξ3 + 1} l ∈ {1, 2, 3, ..., NN 3D }.

(4.52)

4.2 Numerical Methods E3

N8 E6

F6 E4

N5

N6

E12 ξ 3 E9

F1 N4 E5 F5

N1

N7 E7

ξ2 F4 ξ1 E10E 2

E11

N3

F2 E 8

F3 E1

393

N2

Fig. 4.11. 3D-p-element: definition and numbering of element vertices (Ni ), edges (Ei ) and faces (Fi )

The pξi represent the maximal approximation degree in the direction of the natural element coordinates ξ i . The total number of element nodes NN 3D can be calculated as NN 3D = [pξ1 + 1][pξ2 + 1][pξ3 + 1].

(4.53)

According to the number and type of involved onedimensional shape functions, the threedimensional shape functions can be classified into four different shape function mode groups. Firstly, nodal modes are identical to the standard trilinear Lagrange shape functions

Ni (ξ) =

1 1 + ξi1 ξ 1 1 + ξi2 ξ 2 1 + ξi3 ξ 3 , 8

(4.54)

with ξij = ±1 corresponding to the natural coordinate ξ j of vertice node i, and their value correspond to the value of the field variable at the element vertices. Edge modes are a product of two linear and one higher-order polynomial, describing a polynomial approximation at the corresponding edges 2 2 3 3 NpE,1−4 (ξ) = 14 Np1 +1 (ξ 1 ) 1 + ξE ξ 1 + ξE ξ 1 −1 1 1 3 3 NpE,5−8 (ξ) = 14 Np2 +1 (ξ 2 ) 1 + ξE ξ 1 + ξE ξ 2 −1 1 1 2 2 NpE,9−12 (ξ) = 14 Np3 +1 (ξ 3 ) 1 + ξE ξ 1 + ξE ξ , 3 −1

(4.55)

394


with pi ∈ {2, 3, ..., pξi }. Furthermore face modes result from a multiplication of one linear function and two higher-order polynomials. Therewith a bipolynomial approximation is realized on the respective faces N F,1−2 (ξ) = 12 Np2 +1 (ξ 2 )Np3 +1 (ξ 3 ) 1 + ξF1 ξ 1 [pξ2 −1][p3 −2]+[p2 −1] (ξ) = 12 Np1 +1 (ξ 1 )Np3 +1 (ξ 3 ) 1 + ξF2 ξ 2 N F,3−4 [pξ1 −1][p3 −2]+[p1 −1] (ξ) = 12 Np1 +1 (ξ 1 )Np2 +1 (ξ 2 ) 1 + ξF3 ξ 3 . N F,5−6 [pξ1 −1][p2 −2]+[p1 −1]

(4.56)

Finally, functional values of internal modes I NNN (ξ) = Np1 +1 (ξ 1 ) Np2 +1 (ξ 2 ) Np3 +1 (ξ 3 ) I

(4.57)

with NNI = pξ1 − 1 pξ2 − 1 [p3 − 2] + pξ1 − 1 [p2 − 2] + [p1 − 1]

(4.58)

vanish at the boundaries of the finite brick element because they are a product of higher-order polynomials only. In some formulations the latter property is used to eliminate these nodes at the structural level by a static condensation technique to achieve a better condition of the matrices and a better performance at the structural level [245]. The number of nodes within each mode class depends on the order of polynomial degree in the natural coordinates and is calculated by: nodal modes

8

edge modes

4 · [pξ1 − 1] + 4 · [pξ2 − 1] + 4 · [pξ3 − 1]

face modes

2 · [pξ2 − 1] · [pξ3 − 1] + 2 · [pξ1 − 1] · [pξ3 − 1]+

(4.59)

2 · [pξ1 − 1] · [pξ2 − 1] internal modes [pξ1 − 1] · [pξ2 − 1] · [pξ3 − 1]. Figure 4.12 shows several higher-order modes for different polynomial degrees. 4.2.4.3.2.2 Spatially Anisotropic Approximation Orders In the subsequent paragraphs approximation orders are written as (•)i,j,k . Therein (•) represents an arbitrary field variable, whereas the ith index shows the degree of approximation in the direction of the natural coordinate ξ i .

395

nodal modes


N6

N7

N8

p2

p3

p4

p5

p2,2

p3,2

p3,3

p4,3

p2,2,2

p3,2,2

p3,3,3

p4,3,2

internal modes

face modes

edge modes

N5

Fig. 4.12. 3D-p-shape functions: nodal, edge, face and internal modes for different polynomial degrees

The 3D-p-formulation should allow for the numerical simulation of a wide range of structures, in particular shell structures. This is realized by applying the anisotropic shape functions for the approximation of the displacement field. Figure 4.13 illustrates some basic types of structures and how they are modelled by means of the classical finite element method and by a 3D-pformulation.

396


l >> (b, h)

ξ1 up,1,1

truss ξ2 ξ1 h > 1 and mostly even q > p to capture the fronts of the environmental variables adequately which play an important role in the transient transport process. At this point it can be seen - and it will be shown subsequently - that by using fieldwise anisotropic approximations a considerable reduction of computing time can be achieved compared to equally high spatially isotropic approximations for all fields. Up to this point only the individual requirements of all incorporated fields were taken into account. But concerning the numerical simulation of multiphysics of structures by using an approximative method like the finite element method, it is required that the FE-formulation converges to an unique solution. This requires to fullfill the well-known inf-sub- or rather Babuˇ ska-Brezzi-conditions [71, 154]. In accordance with [124], proving the fullfilling of these conditions mathematically is a major task and regarding the numerical level it can only be tested if the numerical model does not fullfill the conditions [188]. Consequently the fieldwise anisotropic approximation order has to be well-balanced. A suitable approach for well-balanced discretizations are Taylor-Hood-like approximations [255, 847]. Within this concept the approximations of environmental variables like temperature or capillary pressure are chosen one degree lower than the approximation of the displacement field: pθ = ppc = pu − 1.

(4.62)

399

n

→

∞


n→∞

n→∞

n→

∞

n→∞

Fig. 4.15. Discretization of the standard structures (truss, slab, shell) into an infinite numbers of elements Table 4.2. Total number of geometric entities (vertices, edges, faces) of the discretizations with an infinite number of elements

NNV Eξ1 Eξ2 Eξ3 Fξ1 ξ2 Fξ2 ξ3 Fξ1 ξ3

truss-discretization 4(n + 1) ≈ 4n 4n 2(n + 1) ≈ 2n 2(n + 1) ≈ 2n 2n n+1≈n 2n

slab-/plate-/shell-discretization 2(n + 1)2 ≈ 2n2 2n(n + 1) ≈ 2n2 2n(n + 1) ≈ 2n2 (n + 1)2 ≈ n2 2n2 n(n + 1) ≈ n2 n(n + 1) ≈ n2

At this point some considerations are made concerning the efficieny of the fieldwise spatially anisotropic approximation technique. As a qualitativ indicator of the efficiency, the relation between the number of nodes or rather degrees of freedom of a structure is investigated for the proposed discretization method and the standard fully isotropic approximation technique. Therefore, archetypes of structures (truss, slab, plate/shell structures) are considered which are meshed with an infinite numbers of p-elements in the dominating spatial dimensions. Figure 4.15 illustrates these archetypes. For slab-like-, plate-like- or slab-like-structures only one element is used in thickness direction. For plate and shell problems this would be compensated by using a higher-order kinematic in thickness direction. An infinite number is chosen to make the relative values independent of the number of elements. Table 4.2 gives the total numbers of vertice nodes NNV , edges in ξ i -direction Eξi and faces in ξ i ξ j -plane Fξi ξj for the archetype discretizations. These geometric entities are used to calculate the total number of nodes in the (•) respective mesh. In the following, NNp1,p2,p3 represents the total number of nodes when dealing with the field (•) with the underlying approximation order p1, p2, p3.When dealing with the effect of a fieldwise choice of the

400


approximation order only one environmental field variable will be added (temperature θ), leading to a maximum of 4 degrees of freedom (dof). Truss-Like Solid Discretizations Here, the polynomial degree is chosen to be p > 2. The total number of nodes for the spatially anisotropic approximation is u NNp,1,1 = 4n + 4n[p − 1]

(4.63)

whereas the full ansatz space leads to u NNp,p,p = 4n + 8n[p − 1] + 5n[p − 1]2 + n[p − 1]3

(4.64)

nodes. Because the quality of the numerical solution is not changed when using a spatially anisotropic approximation the efficiency of this method is indicated by the ratio of the total number of system nodes/dof. The structural efficiency and the combined fieldwise and structural efficiency for a truss structure is Effu struc =

u NNp,1,1 u NNp,p,p

Effu,θ TH =

u θ 3NNp,1,1 + NNp−1,1,1 , u 4NNp,p,p

(4.65)

whereas Taylor-Hood-like approximations were used for the coupled discretization. Slab-Like Solid Discretizations Also in this case the approximation order is chosen as p > 1. The total number of nodes for the anisotropic approximation is u NNp,p,1 = 2n2 + 4n2 [p − 1] + 2n2 [p − 1]2

(4.66)

and for the isotropic case: u NNp,p,p = 2n2 + 5n2 [p − 1] + 4n2 [p − 1]2 + n2 [p − 1]3 .

(4.67)

Once more the efficiency can be indicated by: Effu struc =

u NNp,p,1 u NNp,p,p

Effu,θ TH =

u θ 3NNp,p,1 + NNp−1,p−1,1 . u 4NNp,p,p

(4.68)

Shell-Like Solid Discretizations Concerning shell-structures, possible approximations in thickness direction are ps = 2 or ps = 3. The approximation degree in plane is p > 1 or rather p > ps . As mentioned in the previous section the approximation degree of the environmental variable pe should be chosen relatively high in thickness direction. So, pe >> ps and pe >> p. For simplicity, pe = pθ = 10 subsequently. The total number of nodes for the shell-like solid formulation is:

4.2 Numerical Methods u 2 2 2 s NNp,p,p s = 2n + 4n [p − 1] + n [p − 1]

401

(4.69)

+ 2n [p − 1] + 2n [p − 1][p − 1] + n [p − 1] [p − 1], 2

2

2

s

2

2

s

whereas for the full ansatz space it is: u = 2n2 + 5n2 [p − 1] + 4n2 [p − 1]2 + n2 [p − 1]3 . NNp,p,p

(4.70)

The anisotropic approximation for the thermal field needs only Θ u NN1,1,p θ = NN1,1,pθ

(4.71)

nodes. The pure structural efficiency can be calculated to Effu struc =

u NNp,p,p s . u NNp,p,p

(4.72)

The Taylor-Hood-discretizations are investigated for an approximation degree of ps in thickness direction and the necessary Taylor-Hood approximation in thickness direction of pθ + 1: Effu,θ =

u θ 3NNp,p,p s + NN 1,1,pθ

4NNpuθ ,pθ ,pθ

Effu,θ TH =

u θ 3NNp,p,p θ +1 + NN1,1,pθ

4NNpuθ +1,pθ +1,pθ +1

.

(4.73) Figure 4.16 shows the aforementioned relations of system nodes/dof for different types of structural discretizations. Regarding aspects of efficiency, it is stressed here, that the number of system nodes or dof is just an indicator for the real reduction of computing time. In fact, the reduction of nodes is at least proportional to the squared number of nodes because of the quadratic composition of the stiffness matrix. Obviously, the largest reduction of computational efforts can be gained with the truss-like solid or slab-like solid formulations, because truss-like solid discretization only lead to nodes at the vertices and edges of the element and slab-like solid formulations do not contain any internal modes. Furthermore, there is an additional reduction of computing time regarding multiphysics discretizations, which however is getting smaller with increasing p. A considerable reduction of computing time can be gained in analyses of shell-structures when using shell-like solid formulations with a quadratic ps = 2 or cubic ps = 3 approximation of the displacements in thickness direction. This is mainly caused by the considerable reduction of internal modes. Regarding the multiphysics approximation degrees it can be seen in Figure 4.16 d) u,θ that the function of the efficiencies EffTH1 and Effu,θ TH2 (4.73) in the range θ of p < p is increasing with increasing p, but we also see in this case a considerable reduction of system nodes/dof. For the uncommon case that p > pe = pθ the curves would look similiar to the curves a) to c). Considering these results it can be stated that using a fieldwise, spatially anisotropic

402


u,θ Effu struc , EffTH

u,θ Effu struc , EffTH

0.5 Effu struc Effu,θ TH

0.4 0.3 0.2 0.1

a)

Effu struc Effu,θ TH

0.4 0.3 0.2 0.1

2

3

4 p

5

6

b)

2

3

4 p

5

6

4 p

5

6

0.8 ps = 2 ps = 3 TH

0.2 Effu,θ TH

Effu struc

ps = 2 ps = 3 0.6

0.1

0.4 c)

4

5

6 p

7

8

d)

2

3

Fig. 4.16. Relative reduction of system nodes/dof for a) truss-like solid, b) slab-like solid, c) mechanical shell-like solid and d) multiphysics shell-like solid formulation

approximation technique is an adequate way of simulating coupled multiphysics of structures. 4.2.4.3.2.4 Geometry Approximation Inherently meshes for the p-finite element method appear to be relatively coarse. Therefore an adequate method for the description of the geometry is needed. An exact representation of the element geometry can be realized by using the blending function method [317, 782]. If this method is coupled with a professional CAD-environment a powerful tool is developed in conjunction with the p-finite element method [246]. Despite all the efficiency there is a lot of extra work to be done concerning the generation of the connectivities between structural and geometrical entities. Therefore in the presented formulation a subparametric concept is used. The geometry is described by a twenty-noded continuum element which allows for a quadratic Serendipity approximation of the element geometry. The data of the geometry approximation is independent of the approximation of the field variables. Consequently nodes used for the geometry are not identical to the nodes of the p-finite element formulation. This approximation in general holds for accurate results.


403

4.2.5 Solution of Stationary Problems Authored by Detlef Kuhl In the following section the solution of non-linear time independent durability mechanics is discussed. In particular, iteration methods and control strategies are combined in order to solve the non-linear vector equation obtained as result of the spatial discretization process. Since this procedures are intensively discussed in textbooks, see e.g. [106, 224, 225, 855], the present section is restricted to a brief summary. 4.2.5.1 Numerical Solution Technique Non-linear static systems can be expressed in terms of the generalized internal force vector ri , the generalized external force vector r and generalized vector valued variables or displacements u. ri (u) = r

(4.74)

As a basis for the numerical solution of equation (4.74), the consistent linearization of the generalized internal force vector, defining the generalized tangent stiffness matrix K, is given. ∂ri (u) = K(u) ∂u

(4.75)

In general, the numerical solution of non-linear vector equation (4.74) is realized by the combination of two algorithms: • •

The first one controls the application of generalized loads r, whereas the second one solves the resulting non-linear equation ri (u) = r of the particular load step.

In order to control the application of loads, the load factor λ is induced.

ri (un+1 ) = λn+1 r

[0, λ] =

NT −1 A

λn+1 − λn

(4.76)

n=0

Within every load step the non-linear vector equation (4.76) is solved iteratively. Figures 4.17 and 4.18 illustrate the resulting combination of computational methods to control the application or generalized external loads and the iterative solution of single load steps. 4.2.5.2 Iteration Methods For the explanation of iteration methods it is assumed that the load factor λn+1 is prescribed and un+1 is demanded. This represents the classical

404


non-linear vector equations ri (u) = r parameterized non-linear vector equations - load factor λ ri (u) = λ r control of load parameter λ NTB −1 [0, λ] = λn+1 − λn n=0

control of

iterative solution of non-linear vector equations, if λn is changed to λn+1

• • •

• • •

load displacement arc-length

Newton-Raphson modified Newton-Raphson quasi Newton

Fig. 4.17. Strategies for solving non-linear vector equations ri (u) = r λ λn+1

λ λn+1

iteration

λn

λ2 control parameter load displacement λn arc-length

λ1

λ0 u0 u1

u2

un

un+1

u

un = u0n+1

u1n+1

k+1

ukn+1un+1 un+1u

Fig. 4.18. Control of load factor and Newton-Raphson iteration

load controlled analysis. The consideration of variable load factors within the framework of arc-length methods is discussed in Section 4.2.5.3. In general, iterative solution methods are based on the Taylor expansion of the vector of generalized internal forces about the trial solution ukn+1 . k ri (uk+1 n+1 ) = ri (un+1 ) +

∂ri (ukn+1 ) k+1 un+1 − ukn+1 + · · · = λn+1 r k ∂un+1

(4.77)

Herein k counts the number of iterations. If the Taylor expansion is truncated after the linear term, the Newton correction k Δu = uk+1 n+1 − un+1

(4.78)


405

Table 4.3. Convergence criteria of iterative solution methods method

residuum

displacement

norm absolute

η¯rk+1 = λn+1 r − ri (uk+1 ¯r η¯uk+1 = n+1 ) ≤ η

norm relative

ηrk+1 =

Δu

≤ η¯u

λn+1 r − ri (uk+1

Δu n+1 ) ≤ ηr ηuk+1 = ≤ ηu

ri (ukn+1 )

uk+1 n+1 − un

components j k+1 η¯r = |λn+1 r j − rij (uk+1 ¯r η¯uj k+1 = n+1 )| ≤ η absolute

|Δuj |

≤ η¯u

components j k+1 |λn+1 r j − rij (uk+1 |Δuj | n+1 )| ≤ ηr ηuj k+1 = ≤ ηu ηr = j k j k+1 relative |ri (un+1 )| |un+1 − ujn |

can be calculated as result of a linear system of equations. ∂ri (ukn+1 ) ∂ukn+1

Δu = K(ukn+1 ) Δu = λn+1 r − ri (ukn+1 )

(4.79)

Finally, the sequence of solutions constitutes an iterative solution of the nonlinear vector equation (4.76). The quality of the approximative solution is checked by several alternative convergence criteria collected in Table 4.3. In summary the iterative solution of a single load step consists of the successive solution of linear systems of equation (4.79) and the convergence check. Several iteration methods are distinguishable by means of the used coefficient matrix K. If K is identified, as already derived in equation (4.79), by the current generalized tangent stiffness matrix, the pure Newton-Raphson method is obtained. If, in contrast to this, K is substituted by the initial tangent stiffness matrix of the current load step and an approximated tangent stiffness matrix, the modified Newton-Raphson method and a quasi Newton method are obtained, respectively. Alternative iteration methods fitting equation (4.79) are summarized in Table 4.4. It is obvious that the NewtonRaphson scheme is advantageous with respect to the number of iterations and the modified Newton-Raphson method is more effective within every iteration step. Since robust and reliable numerical methods are essential to solve highly non-linear multiphysics problems of the present book, the original Newton-Raphson scheme is preferred. In Figure 4.19 the algorithmic set-up of the load controlled NewtonRaphson scheme distinguishing between predictor and corrector iterations is given. According to the solution of the last load step ri (u0n+1 ) = λn r the implementation of the algorithm can be simplyfied by using identical predictor and corrector step. Therefore, in Figure 4.19 the frames ri (u0n+1 ) = λn r to u1n+1 = u0n+1 + Δu have to be deleted and the initial iteration counter k = 1 should be substituted by k = 0.

406


Table 4.4. Comparison of iteration methods based on the Taylor expansion of non-linear vector equations ri (un+1 ) = λn+1 r method

iteration with convergence

pure NewtonRaphson

current tangent K(ukn+1 )

quadratic

modified NewtonRaphson

initial tangent K(un )

non quadratic

quasi Newton

approx. inverse tangent ˆk K

non quadratic

effort of iteration

number of iterations

high generation K(ukn+1 ), ri (ukn+1 ) solution KΔu =λn+1 r−ri low generation ri (ukn+1 ) back substitution KΔu = λn+1 r−ri moderate generation ˆ k , ri (ukn+1 ) K back substitution ˆ k [λn+1 r−ri ] Δu = K

low

high

moderate

loop over load steps n = 0, N T − 1 external load vector and trial solution predictor internal force vector predictor tangent stiffness matrix solution of displacement increment update displacement vector

λn+1 r and u0n+1 = un ri (u0n+1 ) = λn r K(u0n+1 ) K(u0n+1 ) Δu = λn+1 r − ri (u0n+1 ) u1n+1 = u0n+1 + Δu

loop over iteration steps k = 1, · · · internal force vector

ri (ukn+1 )

tangent stiffness matrix

K(ukn+1 )

solution of displacement increment update displacements check for convergence, e.g.

K(ukn+1 ) Δu = λn+1 r − ri (ukn+1 ) k uk+1 n+1 = un+1 + Δu

ηuk+1 ≤ ηu

k + 1 −→ k n + 1 −→ n Fig. 4.19. Algorithmic set-up of the load controlled Newton-Raphson scheme distinguishing between predictor and corrector iterations


407

λ λn +1

λ

pre dic tor

λn+1

constraint f (un+1 , λn+1 ) = 0 equilibrium path ri (u) − λr = 0

s0 s

1 Δλ K(un )

λn

λn Δu Δuλ un+1

un

u

un

u

u1n+1 un + Δuλ

Fig. 4.20. Illustration of arc-length methods and predictor step calculation

4.2.5.3 Arc-Length Controlled Analysis Arc-length controlled iteration methods include the standard displacement and load controlled analyses as special cases. Consequently, only a brief summary of textbooks [832, 817, 224, 673, 855] and papers [91, 219, 222, 220, 221, 303, 304, 438, 655, 656, 677, 678, 679, 719, 829] is given. As a basis of arc-length methods the equilibrium path ri (un+1 ) = λn+1 r is enriched by a constraint f in terms of the displacements un+1 and the load factor λn+1 , compare Figure 4.20. ri (un+1 ) − λn+1 r = 0

f (un+1 , λn+1 ) = 0

(4.80)

This means that the load factor is also a variable and the solution should fulfill both non-linear equations. In Figure 4.20 the solution of the extended system (4.80) is given by the intersection of the equilibrium path and the constraint. In consequence of the consistent linearization of equation (4.80) the linear system of equations ⎡ ⎣

K(ukn+1 )

−r

T f,u (ukn+1 , λkn+1 )

f,λ (ukn+1 , λkn+1 )

⎤⎡ ⎦⎣

⎤ Δu

⎡

⎦=⎣

Δλ

λkn+1 r − ri (ukn+1 )

⎤ ⎦

(4.81)

−f (ukn+1 , λkn+1 )

for the solution of the Newton corrections k Δu = uk+1 n+1 − un+1

k Δλ = λk+1 n+1 − λn+1

(4.82)

408


is obtained. Since the linear system of equations (4.81) is non-symmetric and looses, furthermore, the band structure of the generalized tangent matrix, it is solved by applying the partitioning technique [91]. Therefore, the partial incremental solutions Δur and Δuλ are calculated in advance. K−1 (ukn+1 ) Δur = λkn+1 r − ri (ukn+1 ), K−1 (ukn+1 ) Δuλ = r

(4.83)

Afterwards the increments Δu = Δur + Δuλ Δλ

(4.84)

and Δλ = −

f (ukn+1 , λkn+1 ) + f,u (ukn+1 , λkn+1 ) · Δur f,u (ukn+1 , λkn+1 ) · Δuλ + f,λ (ukn+1 , λkn+1 )

(4.85)

are computed. Since this procedure is restricted to the corrector iteration, a specialized predictor step, adopting an user defined step length s, is implemented. As shown in Figure 4.20, the load factor is increased by one and the resulting displacement increment Δuλ and step length s0 are calculated. Δuλ = K−1 (un ) r, s0 = Δuλ · Δuλ + 1 (4.86) Afterwards the increments of the displacement vector and the load factor are scaled, such that the user defined step length s is obtained. Δλ =

s 1 s0

Δu =

s Δuλ s0

(4.87)

Selected constraints within the framework of the present generalized arc-length method are summarized in Table 4.5. It is worth to mention that the standard control algorithms used in the present book, namely the displacement and load controlled analyses, are also included in Table 4.5 and the load controlled Newton-Raphson scheme has already been discussed in Section 4.2.5.2. As a particular example of the generalized path following method, the algorithmic set-up of the arc-length controlled Newton-Raphson scheme is given in Figure 4.21. 4.2.6 Temporal Discretization Methods Authored by Detlef Kuhl and Sandra Krimpmann The present section is concerned with the numerical methods for the time integration of non-linear multiphysics problems by means of Newmarkα methods as well as discontinuous and continuous Galerkin schemes. Newmark-α time integration methods are using the semidiscrete balance equation evaluated at one selected time instant within a time step and finite


409

Table 4.5. Constraints and load factor increments of selected arc-length methods (f = f (ukn+1 , λkn+1 ), [817])

method

k+1 constraint f (uk+1 n+1 , λn+1 )

increment Δλ(ukn+1 , λkn+1 )

load control

f = λk+1 n+1 −c

displacement control

k+1 l f =un+1 −c k+1 k+1 l un+1 =[· · · un+1

initial normal plane

k f =[u1n+1 −un ]·[uk+1 n+1 −un+1 ] k+1 1 k +[λn+1 −λn ] [λn+1 −λn+1 ]

−[u1n+1 −un ] · Δur 1 [un+1 −un ] · Δuλ +λ1n+1 −λn

current normal plane

k f =[ukn+1 −un ]·[uk+1 n+1 −un+1 ] k+1 k k +[λn+1 −λn ] [λn+1 −λn+1 ]

−[ukn+1 −un ] · Δur k [un+1 −un ] · Δuλ +λkn+1 −λn

closest point projection

k f =Δuλ ·[uk+1 n+1 −un+1 ] k +1 [λk+1 −λ n+1 ] n+1

−Δuλ · Δur Δuλ · Δuλ + 1

sphere radius s

f+ s = sk+1 = 2 2 k+1 2

uk+1 n+1−un +ψ [λn+1−λn ]

−f [f +s]−[ukn+1−un ]·Δur k [un+1−un ]·Δuλ +ψ 2 [λkn+1−λn ]

f = uk+1 n+1−un − s

−

cylinder radius s

−f

···]

T

−

f +Δulr Δulλ

f [f +s]+[ukn+1−un ]·Δur [ukn+1−un ]·Δuλ

difference approximations of the state variables within a typical time interval. These classical methods are second order accurate and exhibit controllable numerical dissipation. In contrast to this, for discontinuous Galerkin methods the semidiscrete balance and the continuity of the primary variables are weakly formulated within time steps and between time steps, respectively. Continuous Galerkin methods are obtained by the strong enforcement of the continuity condition as special case. The introduction of a natural time coordinate allows for the application of standard higher order temporal shape functions of the p-Lagrange type and the well known Gauß-Legendre quadrature of associated time integrals. It is shown that arbitrary order accurate integration schemes can be developed within the framework of the proposed temporal p-Galerkin methods. 4.2.6.1 Introduction For the integration of time dependent durability problems either Newmark type finite difference methods or Galerkin type finite element methods are

410


loop over arc-length steps n = 0, N T − 1 predictor tangent stiffness matrix predictor increment scaling displacement and load factor increment update displacements and load factor

K(un ) −1

Δuλ = K

(un ) r

Δu and Δλ u1n+1

and λ1n+1

loop over iteration steps k = 1, · · · internal force vector

ri (ukn+1 )

tangent stiffness

K(ukn+1 )

partial incremental solutions

Δur and Δuλ

constraint

f (ukn+1 , λkn+1 )

increment of load factor

Δλ

displacement increment

Δu = Δur + Δuλ Δλ

update displacements and load factor check for convergence, e.g.

k+1 uk+1 n+1 and λn+1

ηuk+1 ≤ ηu

k + 1 −→ k n + 1 −→ n Fig. 4.21. Algorithmic set-up of the arc-length controlled Newton-Raphson scheme

used. Galerkin integration schemes can be furthermore classified in discontinuous and continuous formulations. Main advantages of Newmark schemes are the small numerical effort, the second order accuracy and the controllable numerical dissipation. Continuous Galerkin schemes are decorated by an arbitrary order of accuracy combined with a moderate numerical effort. Discontinuous Galerkin schemes allow for integrations with an arbitrary order of accuracy and numerically dissipative integrations. 4.2.6.1.1 Motivation Durability of concrete structures is limited by damage caused by external loading and its interaction with environmentally induced deterioration mechanisms (see Sections 3.1.2 and 3.3.2). Model based prognoses of the degradation of such structures are, in general, adapted from coupled damage models, accounting for the transport of moisture, heat and aggressive substances and the various interactions with diffuse or localized damage. Accurate numerical methods for the time integration of these kind of processes are indispensable for successful and reliable simulation based predictions of environmentally induced aging of structures. Standard time integration schemes of the finite difference or Newmark type (see e.g. [569, 198] and [409]) are not well suited


411

for non-smooth Dirichlet boundary conditions and pronounced changes of source terms typically arising in this class of parabolic differential equations (see e.g. [260, 415]). Since the order of accuracy of these algorithms according to the Dahlquist theorem [226] is restricted by two, adaptively controlled Newmark schemes or alternative time integration schemes are important ingredients of an efficient numerical strategy for the solution of multifield problems arising in durability oriented structural analyses. 4.2.6.1.2 Newmark-α Time Integration Schemes Newmark-α time stepping schemes represent a family of algorithms using finite difference based classical Newmark approximations of velocities and displacements and the strongly fulfilled semidiscrete algorithmic balance equation. The algorithmic balance equation is characterized by two time instants within a typical time step where selected terms of the balance equation are evaluated. This generalized family of Newmark type integration schemes collects the most popular integration schemes in industrial applications and engineering science: The classical Newmark method [568, 569], the Hilber-α method [368] and the Bossak-α method [854]. The generalized Newmark-α method is identical to the combination of the famous Hilber-α and Bossak-α methods. In the paper [198] this method is developed, the numerical properties are investigated and the algorithm is denoted as generalized-α method. In the present book we are using the denotation Newmark-α in honor of the great idea of Nathan Mortimore Newmark which represents still the main ingredient of the modern algorithm. The Newmark-α method is characterized by second order accurate integrations and controllable numerical dissipation. For linear applications it is unconditionally stable and in the non-linear case it can be simply modified to an energy conserving/decaying integration scheme [748, 61, 456, 461]. These positive features of the algorithm combined with its incomplex implementation give reasons for its popularity in science and engineering. However, the Dahlquist theorem [226] anticipates the boundless achievement of the Newmark-α scheme. 4.2.6.1.3 Galerkin Time Integration Schemes Galerkin time integration schemes are based on the temporal weak formulation of the ordinary differential equation and the finite element approximations of the state variables and the weight function. According to the weak and strong fulfillment of the continuity of the primary variable in-between two time steps, Galerkin methods are distinguished in their discontinuous and continuous versions, respectively. Historically, first ideas of Galerkin time integration schemes have been published at the end of the 1960’s. In particular, [55, 292, 589] have proposed the temporal weak formulation of semidiscrete balance laws. [399, 56] have presented the continuous Galerkin method for

412


the discretization of systems of first order differential equations. The accuracy of these methods has been improved by [616, 375] by using higher order polynomials analogous to the spatial p-finite element method [72]. Discontinuous Galerkin methods have been introduced as spatial discretization techniques by [663] and [480]. Later, the idea of the weak formulation of the continuity condition of primary variables has been applied by [408, 261] for the development of discontinuous Galerkin time integration schemes. In [204] a review on the development of discontinuous Galerkin methods is presented. Furthermore, the textbooks by [415, 260] include numerous applications of discontinuous and continuous Galerkin methods. In the present section discontinuous and continuous Galerkin time integration schemes for the solution of non-linear semidiscrete multiphysics problems are developed within a generalized framework. This generalized formulation is specialized to the discontinuous Bubnov-Galerkin method and the continuous Petrov-Galerkin method. For the temporal approximation of the state variables and the weight function Lagrange shape functions of arbitrary polynomial degree p in terms of the natural time coordinate ξt ∈ [−1, 1] are used. Furthermore, in Section 4.2.8.2 the Galerkin time integration schemes are enriched by error estimates of the h- and p-method in order to obtain information on the accuracy of the investigated methods. Adaptive time stepping schemes, presented in Section 4.2.8.2, complete the collection of numerical methods for the efficient numerical analysis of highly non-linear initial value problems. 4.2.6.2 Newmark-α Time Integration Schemes Originally the present Newmark methods have been designed for the integration of linear structural dynamics [569, 368, 854, 198]. However, the advantageous properties of these methods can also be transfered to first order semidiscrete balance equations of durability mechanics. For the sake of generality the Newmark-α method is presented for second order non-linear balance equations. A version for first order differential equations is simply obtained as special case by cancelation of the second time derivatives and the associated generalized tangent mass matrix [409, 455]. In the present section a brief summary of the Newmark-α method based on the papers [568, 569, 368, 854, 377, 378, 376, 198] and the textbooks [53, 54, 90, 102, 106, 225, 395, 396, 853, 855, 870] in the context of linear and non-linear structural dynamics is given. 4.2.6.2.1 Non-linear Semidiscrete Initial Value Problem The starting point for the development of Newmark-α integration schemes is the non-linear semidiscrete initial boundary value problem which is given in terms of the semidiscrete balance equation and initial conditions


413

˙ u) = r(t), u ¨ (t = t0 ) = u ¨ 0 , u(t ˙ = t0 ) = u˙ 0 , u(t = t0 ) = u0 ri (¨ u, u, (4.88) ¨ , u˙ and u. ri and r are the generalized internal and of the state variables u external force vectors, respectively. Linearization of equation (4.88) with respect to the state variables defines the generalized tangent matrices ˙ u) ∂ri (¨ u, u, ˙ u) = M(¨ u, u, ¨ ∂u ˙ u) u, u, ∂ri (¨ ˙ u) = D (¨ u, u, ∂ u˙

˙ u) ∂ri (¨ u, u, ˙ u) = K(¨ u, u, ∂u

(4.89)

denoted as generalized tangent mass matrix M, generalized tangent damping matrix D and generalized tangent stiffness matrix K. 4.2.6.2.2 Numerical Concept of Newmark-α Time Integration Schemes Figure 4.22 summarizes the main development steps of Newmark-α schemes for integrating non-linear first and second order initial value problems: 1. Subdivision of the time interval of interest [t0 , T ] in time steps Δt and consideration of a representative time step [tn , tn+1 ]. ¨ using Newmark approxi2. Approximation of state variables u, u˙ and u mations [569] and generalized mid-point approximations [198]. 3. Evaluation of the semidiscrete balance equation (4.88) at generalized midpoints tn+1−α of the representative time interval [198]. 4. Iterative Newton-Raphson solution of the resulting effective balance equation. 5. Repetition of steps 2.-4. for the successive solution of durability mechanics within the time interval [t0 , T ]. 1. subdivision of time interval

2. time approximations

t0 t0

tn

tn+1 tNT Δt T

in time steps Δt t 4. N-R-iteration

3.time tn+1−α

rn+1

tn+1 n+1

un u˙ n u ¨n tn

u(t) u(t) ˙ u ¨(t)

tn+1−α n+1−α

un+1 u˙n+1 u ¨n+1

tnn

K rn

α ∈ [0, 1] tn+1 t

tn

tn+1−α tn+1 t

t0

tn tn+1 tNT

un

5. repetition of steps 2.-4. t

Fig. 4.22. Design of Newmark type time integration schemes

un+1

414


4.2.6.2.3 Time Discretization As a basis of the numerical integration, the time interval of interest [t0 , T ] is subdivided in NT time steps Δt. [t0 , T ] =

NT −1 A

Δt = tn+1 − tn

[tn , tn+1 ]

(4.90)

n=0

˙ n) The state variables at the beginning of the time step un = u(tn ), u˙ n = u(t ¨n = u ¨ (tn ) are given and the state variables at the end of the time step and u ¨ n+1 should be determined by the time stepping scheme. un+1 , u˙ n+1 and u 4.2.6.2.4 Approximation of State Variables The approximation of state variables is realized by the combination of Newmark [569] and generalized mid-point approximations [198]. Newmark ap¨ proximations are based on the assumption of linear varying accelerations u and the inclusion of Newmark time integration parameters γ and β. 2γ ¨ n ] [t − tn ] [¨ un+1 − u Δt 6β ¨ (τ ) = u ¨n + ¨ n ] [t − tn ] [¨ un+1 − u u Δt

¨ (τ ) = u ¨n + u

(4.91)

Single and double integrations of the linear acceleration ansatz, evaluation of the resulting velocity, displacement approximations at the time t = tn+1 and solution of the resulting equations for u˙ n+1 and un+1 yields the well known Newmark approximations, compare Figure 4.23.

mid-point approximation

Newmark approximation

second derivative u ¨

first derivative u˙

u ¨n+1 u ¨n tn

u ¨(t) tn+1 t

u ¨n+1−α n+1−αm m tn+1−αmtn+1 t

un+1

u˙n+1 u˙ n tn

u(t) ˙

un tn+1 t

u ¨n+1 u ¨n tn

primary variable u

tn

u(t) tn+1 t

un+1

u˙n+1 u˙ n tn

u˙ n+1−α n+1−αd d tn+1−αd tn+1 t

un tn

un+1−α n+1−αff tn+1−αf tn+1 t

Fig. 4.23. Illustration of Newmark and generalized mid-point approximations of the Newmark-α method


γ γ−β γ − 2β ¨n [un+1 − un ] − u˙ n − Δt u βΔt β 2β 1 1 1 − 2β ¨n ¨ n+1 (un+1 ) = u˙ n − u u [un+1 − un ] − βΔt2 βΔt 2β

415

u˙ n+1 (un+1 ) =

(4.92)

Generalized mid-point approximations, expressed in terms of state variables at times tn and tn+1 , external loads and the time integration parameters αm and αf complete the set of approximations, compare Figure 4.23. ¨ n+1−αm (¨ ¨n ¨ n+1 (un+1 ) + αm u u un+1 (un+1 )) = [1−αm ] u u˙ n+1−αf (u˙ n+1 (un+1 )) = [1−αf ] u˙ n+1 (un+1 ) + αf u˙ n un+1−αf (un+1 ) = [1−αf ] un+1 + αf un rn+1−αf = [1−αf ] rn+1 + αf rn

(4.93)

4.2.6.2.5 Algorithmic Semidiscrete Balance Equation The algorithmic balance equation is obtained by applying the state variables at different time instants within the time interval [tn , tn+1 ] characterized by time integration parameters αm and αf [368, 854, 198]. ri (¨ un+1−αm , u˙ n+1−αf , un+1−αf ) = rn+1−αf

(4.94)

Equation (4.94) represents a non-linear algebraic equation for the solution of the end-point displacements un+1 , compare equations (4.92) and (4.93). 4.2.6.2.6 Effective Balance Equation The consistent linearization of equation (4.94), including approximations (4.92) and (4.93), with respect to the end-point displacements un+1 yields the effective balance equation for the iterative Newton-Raphson solution. Keff (ukn+1 ) Δu = reff (ukn+1 )

(4.95)

416


Herein the effective tangent matrix, 1−αm βΔt2 γ[1−αf ] + D(¨ un+1−αm , u˙ n+1−αf , un+1−αf ) βΔt

Keff (ukn+1 ) = M(¨ un+1−αm , u˙ n+1−αf , un+1−αf )

(4.96)

+ K(¨ un+1−αm , u˙ n+1−αf , un+1−αf ) [1−αf ] the effective right hand side reff (ukn+1 ) = rn+1−αf − ri (¨ un+1−αm , u˙ n+1−αf , un+1−αf )

(4.97)

and the Newton correction are used. k uk+1 n+1 = un+1 + Δu

(4.98)

Convergence criteria discussed in Section 4.2.5.2 for static analyses can also be applied for dynamics in order to judge the quality of the iterative solution. 4.2.6.2.7 Newmark-α Algorithm Figure 4.24 shows the algorithmic set-up of the Newmark-α method for non-linear analyses of first and second order durability problems. It is worth to mention that this algorithmic set-up already includes parts for the error based time step control explained in Section 4.2.8.2.4. Links to the calculation of element quantities and the update of history variables are marked on the right hand side by small rectangles. 4.2.6.3 Discontinuous and Continuous Galerkin Time Integration Schemes For the higher order accurate time integration of non-linear semidiscrete initial value problems continuous and discontinuous Galerkin methods in time have been developed (see e.g. [399, 56, 408, 261, 415, 260]). Since the discontinuous version of Galerkin integration schemes includes the continuous Galerkin method as a special case, the development of both methods will be described by means of the discontinuous Galerkin method of arbitrary polynomial degree p. Subsequently, the generalized method will be specialized to the continuous Galerkin method. The application of discontinuous and continuous Galerkin time integration schemes is investigated by means of the non-linear first order semidiscrete initial value problem.


417

¨0 u0 , u˙ 0 , u

generate initial conditions loop over time steps n

rn+1−αf

generate (or read) external loads loop over iteration steps steps k incremental update velocities and accelerations

¨ n+1 (ukn+1 ) u˙ n+1 (ukn+1 ), u

¨ n+1−αm , u˙ n+1−αf , un+1−αf u

state variables at tn+1−α

ri (◦), κf

calculation internal forces calculation tangential stiffness matrix

M(◦), D(◦), K(◦) reff (ukn+1 )

calculation effective right hand side

Keff (ukn+1 )

calculation effective tangent

Keff Δun+1 = reff

solution effective iterative structural equation Newton correction current displacements

k uk+1 n+1 = un+1 + Δun+1

ηuk+1 ≤ ηu

convergence check k+1→k

¨ n+1 (un+1 ) u˙ n+1 (un+1 ), u

final update velocities and accelerations

κf

final update internal variables

e

estimation/indication error

ν1 η ≤ e ≤ ν2 η

error check retain Δt

calculation time step

ν1 η > e

check (y|n) next time interval

Δtnew

[tn+1 , tn+2 ] −→ [tn , tn+1 ] retry

[tn , tn+1 ]

n+1 → n Fig. 4.24. Algorithmic set-up of Newmark-α schemes including error controlled adaptive time stepping [461]

˙ u) = r ri (u,

¨0 u(t0 ) = u0 , u˙ 0 , u

(4.99)

It is worth to mention that non-linear second order semidiscrete initial value problems of type (4.32) can be transformed into first order semidiscrete initial boundary value problems and additional constraints.

˙ v, u) ri (v, r = u˙ − v 0

ri (u˙ , u ) = r

(4.100)

418


It is obvious that this coupled system can also be expressed by the symbolic equation of a non-linear vector equation. Consequently, the present discontinuous and continuous Galerkin time integrations schemes can also be applied to second oder semidiscrete initial value problems, for details see [460]. 4.2.6.3.1 Time Discretization As basis of the time integration the time interval of interest t ∈ [t0 , T ] is subdivided in NT constant or adaptively controlled time intervals [tn , tn+1 ] with the time step size Δt. [t0 , T ] =

NT −1 A

Δt = tn+1 − tn

[tn , tn+1 ]

(4.101)

n=0

The state variables at the time instant tn are known and the state variables at the time instant tn+1 are to be computed by means of a time integration scheme. 4.2.6.3.2 Continuity Condition The continuity of the primary variable u at the boundaries of the individual time intervals [tn , tn+1 ] can be enforced by the continuity or the jump condition, compare Figure 4.25. − 1 0 un = u+ (4.102) n − un = u − u = 0 discontinuous Galerkin method dG(3) n

un

time step n + 1 n+2 un+1

u0 u1

u2 u3

+ u− n un

t1

t2

u4

−1

u0 u1

t4

t1

tn+1 ξt2

t

1

ξt

u2 u3

un+1 t2

t3

t4 tn+1

ξt1 −1

n+2

u4

tn

GP

×

0

time step n + 1

un

tn ×

n

+ u− n+1 un+1

t3

ξt1

continuous Galerkin method cG(3)

ξt2

×

t GP

×

0

1

ξt

Fig. 4.25. Galerkin time integration schemes illustrated by means of polynomial degree p = 3: Approximation of the primary variable u, defintion of the jump un , illustration of the physical time t, definition of the natural time coordinate ξt and the position of Gauß points (GP)


419

In equation (4.102) the primary variables at the end of the previous time step and the beginning of the current time step are defined by 0 u− n = u = lim u(tn − ε) ε→0

1 u+ n = u = lim u(tn + ε) ε→0

(4.103)

respectively. u0 and u1 are introduced according to the temporal finite element approximation of the primary variable in Section 4.2.6.3.5. 4.2.6.3.3 Temporal Weak Form The semidiscrete equilibrium equation is transformed into the temporal weak form by multiplication with an arbitrary weight function w(t) and integration over the time interval [tn , tn+1 ]. Furthermore, the continuity condition (4.102) is only weakly enforced using the weight w1 = w(tn ). The sum of these integrals defines the weak form of discontinuous Galerkin methods. t n+1

˙ u) dt + w · A un (u1 ) − w · ri (u,

t n+1

w · r dt = 0

1

tn

(4.104)

tn

The matrix A is introduced to balance the summarized weak forms and to adapt their physical units. 4.2.6.3.4 Linearization ˙ The linearization of the weak form (4.104) with respect to the unknowns u(t), u(t) and u1 leads to the linearized weak form of discontinuous Galerkin methods t n+1

˙ u) Δu˙ + K(u, ˙ u) Δu] dt + w1 · A Δu1 w · [D(u, tn t n+1

˙ u)] dt w · [r − ri (u,

=

− w1 · A un (u1 )

(4.105)

tn

˙ Δu and Δu1 are the increments of the unknowns and the genwhereby Δu, eralized tangent matrices D and K are identified as the generalized tangent damping matrix and the generalized tangent stiffness matrix. 4.2.6.3.5 Temporal Galerkin Approximation The state variables, the weight function and the increments are temporally approximated by shape functions N i according to temporal nodes i and associated nodal values (e.g. ui ). Standard Lagrange polynomials of

420


the polynomial degree p (see equation (4.20)) as function of the natural coordinate ξt ∈ [−1, 1] are used as shape functions. p+1 :

N i (ξt ) =

k=1 k=i

ξtk − ξt ξtk − ξti

ξti =

2 [i − 1] −1 p

(4.106)

In accordance with the isoparametric concept, the physical time t and the displacement vector are approximated by using the shape functions N i of the polynomial degree p. Independent approximations of the weight function ¯ i with the polynomial degree p¯ are assumed. w by shape functions N t(ξt ) ≈

p+1

j

N (ξt ) t

u(ξt ) ≈

j

w(ξt ) ≈

N j (ξt )

uj

j=1

j=1 p+1 ¯

p+1

¯ i (ξt ) wi N

Δu(ξt ) ≈

p+1

(4.107) j

j

N (ξt ) Δu

j=1

i=1

The vector of concentration rates u˙ and its increment are obtained by the time derivation of equation (4.107). ˙ t) ≈ u(ξ

p+1

N,tj (ξt ) uj

˙ t) ≈ Δu(ξ

j=1

p+1

N,tj (ξt ) Δuj

(4.108)

j=1

Herein N,tj (ξt (t)) = ∂N j (ξt (t))/∂t represent the time derivative of the shape function N j with respect to the physical time t. For the computation of these time derivatives N,tj (ξt ) =

N;tj (ξt ) ∂N j (ξt ) ∂ξt ∂N j (ξt ) = = N;tj (ξt )Jt−1 (ξt ) = ∂t ∂ξt ∂t Jt (ξt )

(4.109)

and of the time integrals t n+1

1 f (ξt ) |Jt (ξt )| dξt

f (t) dt = tn

(4.110)

−1

the temporal Jacobi transformer Jt is required. ∂t(ξt ) ∂N j (ξt ) j j ≈ t = N;t (ξt ) tj ∂ξt ∂ξ t j=1 j=1 p+1

Jt (ξt ) =

p+1

(4.111)

For equidistant nodal times tj − tj−1 = Δt/p with j ∈ [2, p + 1] the constant Jacobi transformer Jt = Δt/2 is obtained. The derivatives of the shape


421

functions with respect to the natural time coordinate ∂N i (ξt )/∂ξt = N;ti (ξt ) are obtained according to equation (4.21). Substitution of the approximations (4.107) and (4.108) in the linearized weak form (4.105) yields the discretized linearized temporal weak form of discontinuous Galerkin methods. p+1 p+1 ¯

ij Δuj + w1 · A Δu1 wi · Dij t + Kt

i=1 j=1

=

p+1 ¯

w · rit − riti − w1 · A un

(4.112)

i

i=1

In equation (4.112) the time integrals 1 Dij t

=

1 ¯i

N

N,tj

D |Jt | dξt

riti

−1 1

Kij t =

=

¯ i ri |Jt | dξt N

−1 1

¯ i N j K |Jt | dξt N

rit =

−1

(4.113) ¯ i r |Jt | dξt N

−1

are used. These integrals are computed numerically by the Gauß-Legendre integration scheme using standard Gauß points ξtl with l ∈ [1, NGt ] and weights αl (see e.g. [870]). 1 f (ξt ) dξt ≈ −1

NG t

αl f (ξtl )

(4.114)

l=1

4.2.6.3.6 Discontinuous Bubnov-Galerkin Schemes dG(p) For arbitrary weight functions wi equation (4.112) can be transformed into a linear system of equations (i ∈ [2, p¯ + 1] and j ∈ [2, p + 1]) %

1j 1j 11 D11 t + Kt + A D t + Kt ij i1 i1 D t + Kt Dt + Kij t

&

1 Δu1 rt − r1ti − A un = Δuj rit − riti (4.115)

which can be formally written as an effective linearized system of equations k k )ΔudG = rdG (udG ) (4.116) KdG (udG and solved for the increments of the primary variables ΔudG . For p¯ = p ¯ i = N i equation (4.116) leads to non-singular soand, consequently, for N lutions. Accordingly, discontinuous Galerkin time integrations schemes of

422


the polynomial degree p, denoted as dG(p)-methods, are Bubnov-Galerkin methods. 4.2.6.3.7 Continuous Petrov-Galerkin Schemes cG(p) For continuous Galerkin time integration schemes the continuity condition + (4.102) is strongly fulfilled and the primary variable u1 = u0 = u− n = un is known. Consequently, the derivations of Sections 4.2.6.3.3 to 4.2.6.3.6 are adapted to continuous Galerkin methods by substituting: Δu1 = 0

A=0

(4.117)

Hence, the effective linearized system of equations (

ij Dij t + Kt

)

Δuj = rit − riti

(4.118)

with i ∈ [1, p¯ + 1] and j ∈ [2, p + 1] can be briefly written as: k k )ΔucG = rcG (ucG ) KcG (ucG

(4.119)

¯ i = N i For the solution of equation (4.119) p¯ = p − 1 and, consequently, N are required. Accordingly, continuous Galerkin time integration schemes of the polynomial degree p, denoted as cG(p)-methods, are Petrov-Galerkin methods. 4.2.6.3.8 Newton-Raphson Iteration Based on the solution of equation (4.115) and equation (4.118), respectively, the improved iterative solution is calculated by using the Newton correction (j ∈ [1, p + 1] for dG-methods and j ∈ [2, p + 1] for cG-methods) uj k+1 = uj k + Δuj

(4.120)

where the initial values of each time step are given by uj 0 = u0 = u− n . The check of convergence is restricted to the primary variable up+1 = u(tn+1 ).

Δup+1

≤ ηu

up+1 k − u0

(4.121)

For alternative formulations of convergence criteria see Table 4.3. 4.2.6.3.9 Algorithmic Set-Up of Galerkin Schemes Figure 4.26 shows the algorithmic set-up of continuous and discontinuous Galerkin schemes for the time integration of non-linear first order semidiscrete initial value problems. Links to the calculation of element quantities and the update of history variables are marked on the right hand side by small rectangles.


423

loop over time steps n ∈ [1, NT ] initial calculations j ∈ [1, p + 1]

tj , t1 = tn , tp+1= tn+1 , u0 = uj 0 = u− n , Au

loop over iteration steps k

KdG (Au ), rdG (Au un )

initialization dG-matrix and dG-vector loop over Gauß points l ∈ [1, NGt ]

ξt = ξ l , αl

coordinate and weight of Gauß point loop over time nodes j ∈ [1, p + 1] and i ∈ [1, p¯ + 1]

¯ i (ξt ), N j (ξt ), N;tj (ξt ) N

shape functions, derivatives respective ξt

Jt (ξt )

Jacobian next nodes j and i

|Jt (ξt )|, Jt−1 (ξt )

Jacobi determinant and inverse Jacobian loop over time nodes j ∈ [1, p + 1]

N,tj (ξt )

derivatives of shape functions respective t

˙ t) t(ξt ), u(ξt ), u(ξ

local time and state variables next time node j Neumann- and Dirichlet boundary conditions system matrices, system vectors, temporary variables

r(t(ξt )), uu (t(ξt )) D, K, ri , r, κ

loop over time nodes i ∈ [1, p¯ + 1] terms of external, internal load vectors

¯ i (ξt )r, ri = N ¯ i (ξt )ri rit = N it

αl |Jt |rit → rdG , αl |Jt |riit → rdG loop over time nodes j ∈ [1 or 2, p + 1] summation & assembly

algorithmic matrices summation & assembly

ij ¯i j ¯i j Dij t = N N,t D, Kt = N N K ij αl |Jt |[Dij t +Kt ] → KdG

next time node i next time node j next Gauß point l = ukdG +ΔudG , ηdG solution, convergence check KdG ΔudG = rdG , uk+1 dG next iteration step k postprocessing and update internal variables

un+1 , u˙ n+1 , κ

next time step n Fig. 4.26. Algorithmic set-up of discontinuous and continuous Galerkin time integration schemes for first order non-linear systems

424


4.2.7 Generalized Computational Durabilty Mechanics Authored by Detlef Kuhl After the previous sections about the generalized modeling of deterioration problems of concrete materials and structures, spatial and temporal discretization methods and iterative solution methods, a modular numerical solution method is designed. The strict separation of the algorithmic, element and model levels allows for an effective and reliable computational implementation of time dependent and stationary durability mechanics models. The investigated numerical methods and durability models are collected within this generalized concept as follows: •

•

•

algorithmic level ◦ static solution methods · load control · arc-length control ◦ time integration methods · Newmark-α method · discontinuous p-Galerkin methods · continuous p-Galerkin methods ◦ iteration methods · Newton-Raphson method · modified Newton-Raphson method element level ◦ finite element method · Lagrange finite elements · Legendre finite elements ◦ discontinuous finite element methods · embedded discontinuity models · extended finite element method model level ◦ single field durability mechanics models ◦ multiphysics durability models

This basic concept may be enriched by adaptive methods in space and time. Time adaptive methods will take place in the algorithmic level. Spatial adaptive methods are mainly located in the element level and controlled by the algorithmic level. Figure 4.27 illustrates the proposed generalized computational durability mechanics by means of discontinuous and continuous p-Galerkin time integration schemes including the Newton-Raphson method, spatial finite element methods and several examples of multiphysics problems. The interactions of algorithmic, element and model levels are indicated by an arrow.

4.2 Numerical Methods algorithmic level

element level

model level

Sections 4.2.5, 4.2.6, 4.2.8.2

Sections 4.2.4, 4.2.9

Chapter 3

time loop n ∈ [0, NT ]

element loop e ∈ [1, NE]

iteration loop k Gauß point loop l

2 element node loops i geometry, Jacobi

time node loop i

variables, gradients

time node loop j tangent

model quantities, κ element node loop i right hand sides element node loop j

solution, convergence

mechanical damage model

ND Gauß point loops

variables, κ right hand side

425

tangents

postprocessing, κ

chemical damage model chemo-mechanical damage model .. . asr chemo-mechanical damage model hygro-thermo-mechanical damage model generalized multiphysics model, Section 4.2.1

Fig. 4.27. Modular concept for multiphysics finite element programs strictly separated in the algorithmic, element and model levels

4.2.8 Adaptivity in Space and Time Authored by Stefanie Reese and Detlef Kuhl Adaptive methods of the spatial and temporal discretization are applied to guarantee an user defined accuracy of the computation. The present adaptive strategies are controlled by error measures. As a consequence of the calculated error measures the spatial finite element meshes and the time step size are adapted. Furthermore, the underlying error measures can be used to judge the quality of numerical solutions. Recent research in error analysis and adaptive strategies is published in [775]. 4.2.8.1 Error-Controlled Spatial Adaptivity Authored by Stefanie Reese In the following section we concentrate on various concepts for continuumstructure transitions. This leads us to the notion of model-adaptivity which is certainly strongly connected to general spatial adaptivity. The literature offers a large variety of concepts for continuum-structure transitions. Most of them are restricted to linear elasticity ([447], [299], [736], [785], [497]). Suitable methods for large deformations are proposed by [873] (beam-3D transitions in concrete), [46] (adhesive joints) and [818] (shell-3D transitions). In none of these papers an adaptive algorithm is developed. In

426


the opinion of the authors this is, however, a crucial aspect because the optimal point of the continuum-structure transition varies in dependence on the loading and the material behaviour. This idea is followed in the context of p-adaptive concepts based on a hierarchical polynomial interpolation in thickness direction. Methods of this kind can be either formulated as a complete p-approach, i.e. with an additional p-interpolation in the shell midplane ([657], [73], [658]), as a hp-method (see [608], [773]) or in combination with shell or plate formulations ([282], [294], [442]), respectively. To our knowledge the notion model adaptivity has been coined by [773], see also the more recent papers [774], [570], [776] and further references therein. Their idea is to use a hierarchically expandable model in combination with an adaptive-controled switch between a 2 1/2D plate and a 3D continuum theory. The model-adaptive algorithm is based on a powerful residual a posteriori error estimate. We follow here a different approach in the regard that we intend to use only low-order elements, precisely speaking an eight-node solid-shell based on the concept of reduced integration and hourglass stabilization. It has been proven to be computationally very efficient and extremely robust. It is to be expected that these advantageous properties carry over to the more complex case of adaptive mesh refinement. In order to avoid mesh distortion we favour the hanging node (HN) concept. A similar argumentation is followed in the context of multigrid methods, see [100], [337], [468], [315] and [423]. One of the few engineering-oriented applications of the HN concept is found in a publication by [431]. In a strict sense we do not switch between different models. However, the element formulation is constructed in such a way that it includes both the continuum and the structural level. This can be explained as follows. Using one element over the thickness practically means to work with a one-directior shell theory. We are then at the structural level. The division of such an element into several elements over the thickness (to be performed in the context of the hanging node context) provides a classical continuum modelling of the considered domain. In contrast to many earlier approaches we do not have to neglect the stress in thickness direction at the structural level. [773], [774] and [776] propose an implicit residual error estimate of equilibrated local Neumann type, see in this context also the contributions of [416] and [39]. A similar view point, though in a more intuitive manner, is followed in the present paper. Inserting the solution of the discretized model into the strong form of the balance of linear momentum yields a residual-based error criterion which is well suited for the present purpose of adaptive thickness discretization. An hierarchichal error estimation has been suggested by [84]. Very common are also error indicators based on projection procedures (e.g. [869]) or dual methods (e.g. [659]). Finally we like to mention the group of gradient based error indicators where the L2 -norm of a stress- or strain-based quantity is


427

scaled by some reference value, for instance the maximum within a certain domain. The warping-based criterion proposed in this paper can be assigned to this group. To the knowledge of the authors, the HN-based thickness refinement in shell structures has been carried out only by [442], however, without error indicator. The astonishing small number of publications in this research domain can be explained by the fact that the element type “eight-node solid-shell”, upon which the HN concept for shells crucially relies, has been developed only recently. Earlier shell concepts can be split into classical four-node ([161], [117], [125], [539], [697], [118], [249]) and solid-shell four-node elements ([194], [427]). The mentioned formulations have in common that the shell geometry is projected on the shell midplane. In contrast, using solid-shell eight-node elements (see [357], [433], [358], [175], [783], [784], [484], [816] the extent of the structure in thickness direction is modelled accurately (in the framework of the isoparametric tri-linear geometry approximation). Such elements are ideally suited for an adaptive refinement in thickness direction since they include only displacement degrees-of-freedom and are as such very similar to classical 3D elements. For a more detailed survey of recent shell formulations see the review paper of [864]. For the current state of the art concerning the h-adaptive refinement of shell element meshes refer to [338], [467], [566]. The keypoint of the present formulation is the Taylor expansion of the relevant stress quantity (usually the first Piola-Kirchhoff stress tensor) with respect to the normal through the centre of the element. In this way the original nine internal element variables (related to the enhanced strain tensor) can be reduced to three. Further the integration over the element volume reduces to an integration in thickness direction. Therefore only two Gauss points are needed in total which makes the element very efficient from the computational point of view. 4.2.8.1.1 Variational Functional The present stabilization technique is strongly related to the enhanced strain method (EAS method) of [742] whose formulation is based on the HuWashizu variational functional. As starting point we state the three equations of strong form (I)

balance of linear momentum : Div P + ρ0 bv = 0 ∂W ∂H

(II) constitutive equation

: P−

(III) kinematical relation

: H − Grad u

=0

(4.122)

=0

where the strain energy function W = W (H, (Xi , i = 1, ..., n)) is a function of the strain tensor H and n internal variables Xi (i = 1, ..., n) to model

428


inelasticity. To determine Xi (i = 1, ..., n) suitable evolution equations have to be formulated. The vector u is the displacement vector. The tensor P represents the first Piola-Kirchhoff stress tensor. The quantity ρ0 bv denotes a volume force (e. g. gravity), ρ0 is the mass density in the undeformed configuration. The present element technology is based on the idea to fulfill these three equations only in weak form, i. e. not pointwise. In this way so-called incompatible modes, expressed by the so-called “enhanced” strain Henh = H − Grad u, can be introduced. They are constructed in such a way that the undesirable defect of low order finite elements (“locking”) is avoided. In the following sections of the paper the compatible strain Grad u is alternatively denoted as Hcomp. The equations (4.122) are multiplied with test functions δu, δH and δP, respectively, followed by an integration over the domain under investigation (B0 ). Assuming further that P is constant within the element, we can finally eliminate the independent stress field and arrive at two equations of weak form: ∂W g1 (u, Henh ) = · Grad δu dV − gext (4.123) B ∂H 0 ∂W · δHenh dV = 0 g2 (u, Henh ) = (4.124) B0 ∂H where gext is a short hand notation for the virtual work of the external loading and the dot characterizes the scalar product of two tensors. Obviously the stress-like strain-dependent quantity ¯ = ∂W P ∂H

(4.125)

takes over the role of the originally introduced stress P. Due to the fact that P has been eliminated, we can simplify the notation by omitting the bar in ¯ what follows: P = P. 4.2.8.1.2 Interpolation In the following we differentiate between tensor and matrix (Voigt) notation. Quantities written in Voigt notation are indicated by slanted bold letters whereas tensors are denoted by straight bold letters. The interpolation of H h does not differ from the one chosen for the hexahedral element formulation proposed by [670] (restricted to elasticity). The 24x1 vector U e (U Te = {U T1 e , ..., U TI e , ..., U T8 e }, I = 1, ..., 8) and the 9x1 vector W e include the nodal displacement degrees-of-freedom and the internal degreesof-freedom, respectively. 4.2.8.1.3 Stress Computation In order to transfer the enhanced strain approach derived up to this point into a powerful reduced integration concept for solid-shells further steps are


429

necessary. [664], [665] suggests to carry out a Taylor expansion of the first Piola-Kirchhoff stress tensor with respect to the element centre ξ = 0. In this way a stress-strain relation is obtained which is linear in the local coordinates ξ, η and ζ. The linear dependence on ζ (if ζ is the thickness direction) is, however, here not useful because the solid-shell concept should be able to capture any non-linear stress-strain behaviour over the thickness. For this reason the Taylor expansion is done with respect to the point ξ (ξ T = {0, 0, ζ}) such that the non-linear dependence on ζ is retained in the constitutive quantities. 4.2.8.1.4 Discretized Weak Form The stress relation and the strain interpolation are inserted into the second equation of weak form (4.124). After a longer analysis which is in detail presented in [666], we arrive at the equation T δW ζe T Rw + δW ξη (K wu U e + K ww W ξη e e )= 0

(4.126)

to be solved at the element level. The vectors W ζe and W ξη e include the internal element degrees-of-freedom. Rw is the residual vector for this equation, the matrices K wu and K ww define various submatrices of the stiffness matrix. The relation (4.126) must hold for arbitrary δW ζe and δW ξη e and results into one non-linear equation Rw = 0

(4.127)

for W ζe and one linear equation for W ξη e : −1 W ξη e = −K ww K wu U e

(4.128)

Using (4.128) and again the stress and strain interpolation e h e the element contributions of the global weak form g1 ≈ g*1h = ne=1 g1 = 0 (ne number of * elements) are finally summarized in the equation T lin hg hg −1 he g1 = δU e Ru + Ru + (K uu − K uw K ww K wu ) U e * (4.129) ! := K stab hg hg T Rlin u and Ru are both residual force vectors, K uu and K uw = K wu further stiffness submatrices. Important is the so-called hourglass stabilization matrix K stab . It should be noted that the efficiency of the element crucially depends on the fact that the thickness direction is treated differently than the two other coordinate directions. It can be said that in contrast to earlier 3D reduced integration concepts (see e.g. [103], [765], [136], [650], [484]) an “anisotropic” ansatz has been made. Therefore it is important to correctly identify the thickness direction in each element. This is suitably done by computing the

430


element distortion on the basis of the Jacobi matrix. The thickness direction is the direction where the element has the smallest extent. In general the orientation of the coordinate ζ varies from element to element. Using this procedure the solid-shell element can be arbitrarily oriented in space. 4.2.8.1.5 Summary To conclude we summarize the differences of the present stabilized reduced integration technique with respect to standard displacement-based approaches as well as alternative locking-free element formulations. The present element will from now on be denoted as Q1SPs. Implementation ◦ ◦ ◦ ◦

Instead of a full (2x2x2) integration the present formulation requires only two Gauss points (located on the line ξ = {0, 0, ζ}). We have to solve the non-linear equation Rw = 0 at the element level. These are three scalar equations. Obviously the computational effort caused by this additional step is very small. The hourglass stabilization matrix K stab can be evaluated analytically. It has to be computed once per time step in an extra subroutine. The use of the modified constitutive equations means to work with an “anisotropic” ansatz. For this reason in each element a coordinate transformation has to be carried out which clearly identifies the thickness direction. For details see [666]. Advantages

◦ ◦

◦ ◦ ◦

The element is free of volumetric, shear, membrane and thickness locking. This property is usually only achieved in the framework of highly sophisticated shell formulations. Due to the absence of shear locking, thin shell computations can be carried out with only one element over the thickness. First of all this reduces the computational effort enormously. Secondly, the ratio of the element side lengths is less extreme so that the critical time step for an explicit time integration is significantly increased. Thus both the number of elements and the number of time steps can be reduced! The element possesses eight nodes. The extension of the structure in thickness direction is correctly displayed. This property proves to be in particular advantageous e.g. for contact modelling. It is trivial to take several elements over the thickness. Usually this is necessary for the modelling of thick structures. The elements can be easily coupled to classical 3D elements.


431

4.2.8.1.6 Hanging Node Concept The last two points on the previous list are our motivation to establish an adaptive procedure for the automatical refinement in thickness direction. The idea is to hold the number of elements in thickness direction as small as possible in order to save CPU time. On the other hand the bending behaviour of the structure should be correctly modelled. In thick or e.g. strongly damaged parts of a structure the strain distribution over the thickness is far from being linear. Due to the fact that the displacement distribution is only linear (or at most quadratic if one takes into account the effect of the enhanced degreesof-freedom) such a situation can usually not be modelled with only one (loworder) element over the thickness. On the other hand the assumption of a linear strain distribution holds approximately quite well for thin structures where a refinement in thickness direction should not be needed. The varying thickness discretization is here realized in the context of the hanging node concept. 4.2.8.1.7 Error Criteria 4.2.8.1.7.1 Warping-Based Error Criterion Obviously we are in need of a suitable error criterion which mirrors the previous circumstances. Very common in the literature about adaptive mesh refinement are residual error estimators (e.g. [416]), error indicators based on projection procedures (e.g. [869]) and finally error estimators using equilibrium relations on element patches (e.g. [39]) or dual methods (e.g. [659]). However, the use of a criterion of these kinds would not let us distinguish between the overall discretization error, i.e. in all three directions, and the error in the thickness discretization. For this reason we investigate in addition a criterion based on the coefficients be 11 and be 22 of the left elastic CauchyGreen tensor T be = Fe FTe = F C−1 p F

(4.130)

In the latter relation the multiplicative decomposition F = Fe Fp (Cp = FTp Fp ) of the deformation gradient has been exploited. It should be emphasized again that due to the element-wise coordinate transformation to detect the thickness direction the two coefficients be 11 and be 22 characterize the bending deformation in the structure. A plane cross-section means that the bending components be 11 and be 22 are linearly distributed over the height. The more strongly the warping of the cross-section develops the larger is the deviation of be 11 and be 22 from this linear function. Accordingly the error is defined as the sum of the deviations from the linear function defined by the values of be 11 and be 22 in the outer Gauss points, see Figure 4.28. The parameter lint refers to the number of Gauss points. We work with the relative measure (abbreviated from now on by crit1)

432


e

Element e:

linear deformation distribution

err1 err2

Fig. 4.28. Example geometry and warping-based error criterion

errorerel =

lint−1 ( i=2 erri ) with errmax = err21 + err2lint lint errmax

(4.131)

Applications have shown that the criterion crit1 is very well suited to control the mesh refinement in a physically reasonable way. However, it does not judge the quality of the solution (e.g. measurable as the deviation of the weak solution from its strong counterpart). In other words, using the same tolerance firstly in combination with a locking-free element formulation and then with a less sophisticated one (which exhibits locking) about the same level of refinement is finally obtained. 4.2.8.1.7.2 Residual-Based Error Criterion For this reason we additionally suggest a residual-based error criterion given as the norm of the vector r = div σ h = 0 (volume force omitted). If the strong form of the balance of linear momentum is exactly fulfilled with the solution uh of the discretized model we have found the analytical solution (r = 0). In general we obtain a deviation r = 0 the norm of which represents a good measure for the discretization error. The vector r reads in index notation ri =

h ∂σij

∂xhj

=

h h h ∂σij ∂σij ∂σij ∂ξ ∂η ∂ζ + + h h ∂ξ ∂xj ∂η ∂xj ∂ζ ∂xhj

(4.132)

ˆ h of the Cauchy stress tensor σ h is in consistency The Voigt notation σ with the element formulation approximated by ˆh = σ

1 ˆh = σ ˆ h + σ ˆ hhg fT P det Fh0 0

(4.133)

ˆ depends non-linearly on ζ, the remainder, the hourglass The summand σ part, is linear in ξ, η and ζ.


433

ˆ h with respect to ξ and η requires the To determine the derivatives of σ knowledge of W e . The three coefficients included in W ζe are anyway determined in the code. The vector W ξη e is easily computed by exploiting the linear ˆ h with respect to ζ varies within relation (4.128). The partial derivative of σ the element. It could be quite easily calculated by measuring the increments from Gauss point to Gauss point. The present analysis is, however, based on the idea to have one error per element. For this reason we divide the Gauss points into two groups (upper and lower half of the element), calculate the stress average within one of these groups and finally compute the slope on the basis of these average values. The residual-based error (denoted from now on as crit2) is calculated by means of errore = r12 + r22 + r32 (4.134) 4.2.8.1.8 Program Flow At the beginning of a computation the structure will be discretized with only one solid-shell element over the thickness. To obtain a preferably good quantitative estimation of the error it is reasonable to work with a relatively large number of Gauss points (from now on abbreviated as GP), e.g. 16 GP. The chosen error criterion is checked after each load step. If the error exceeds a prescribed tolerance the element is divided. The lower half keeps the element number (“old” element) the upper one obtains a new one (“new” element). In the division procedure the number of GP is retained, i.e. the divided elements have only half as much GP as the original element. Up to four hanging nodes are introduced (see Figure 4.29). The hanging node concept serves to combine regions with different numbers of elements over the thickness. The degrees-of-freedom of the hanging nodes stored in the vector U HN depend on the displacements U UN and U ON of the neighbouring nodes UN and ON: U HN =

1 (U UN + U ON ) 2

(4.135)

The nodes UN and ON share the same element edge as HN. As such they have the same ξ- and η-coordinates as HN. UN is located at ζ = −1, ON at ζ = +1 if the local coordinates of the undivided element are taken as reference. Employing the coupling condition (4.135) the elements in the residual force vector and the tangential stiffness matrix which are connected to HN have to be moved to the corresponding vector or matrix elements of UN and ON. Load steps where elements are added are repeated until the error criterion is fulfilled in all old and new elements. If one starts with 32 GP 4 divisions are possible before the elements with the smallest thickness have only 2 GP. The distribution of the bending strains is then by definition linear, i.e. the error criterion crit1 is no longer effective. Using crit2 it is theoretically possible that

434


ON

HN

UN

Fig. 4.29. Two-element example with two hanging nodes

further divisions are required. According to the experience of the authors this case does not have practical relevance because it is always possible to start with more Gauss points. Other possibilities are to hold the number of GP constant within each element (when 2 GP are reached) or to switch to alternative refinement procedures. 4.2.8.1.9 Transfer of History Variables An important issue concerns the transfer of the history variables from the original to the divided elements. One part of the history data, e.g. the internal variables to model inelasticity, is computed in the Gauss points. The second group of history variables acts in the context of finite element technology and is as such associated with the element directly (independent of the Gauss point). Fortunately the transfer of the second group is not necessary because these variables are newly computed at the beginning of each Newton iteration. On the other hand the GP-oriented history data have an important influence on the solution. Large changes of the GP positions during the process of element division should be avoided. A good way to achieve this goal is to retain the total number of GP during the division. Exploiting the linear interpolation of the original history data the history variables at the GP locations of the divided elements are easily computed. 4.2.8.1.10 Examples These results are also documented in [668]. 4.2.8.1.10.1 Uniaxial Bending (Beam of Uniform Thickness) The first example is a clamped thick beam of uniform thickness (beam 1). Geometry and boundary conditons are given in Figure 4.30. At the boundary X = 12000 mm the displacements in X-direction are constrained. The displacements in Y -direction are in general set equal to zero. The load is


435

Z

X

Y

2000 mm

F F 1200 mm

w 12000 mm

Fig. 4.30. Beam 1: Geometry and boundary conditions

applied according to the function F = ν F0 (F0 = 10000 N) where ν represents the load level. The maximum load level νmax = 75 is reached in 30 equidistant steps. We work with a Neo-Hooke based material model of finite elasto-plasticity (see [669]). The material parameters are Λ = 1153.85 N/mm2 , μ = 1730.77 N/mm2 , τY = 2.43 N/mm2 , H = 400 N/mm2 . The stabilization parameter μ is set equal to μ = 105 N/mm2 . In the first computation we investigate how the number of Gauss points (nGP) influences the deformation behaviour (see Figure 4.31). In Figure 4.32 different states of the mesh refinement are plotted (Q1SPs/o, 16El.). Two element formulations are compared. The first is the solid-shell formulation presented in the first part of the paper (Q1SPs). The second one is a

Beam 1 (tolerr = 10^-5, crit1) 80 70

load level [-]

60 50 40 30 20 10

Q1SPs 2 El. Q1SPs 4 El. Q1SPs 8 El. Q1SPs/o 2 El. Q1SPs/o 4 El. Q1SPs/o 8 El. Q1SPs/o 16 El.

0 -900 -800 -700 -600 -500 -400 -300 -200 -100 w [mm]

0

Fig. 4.31. Beam 1: Load-displacement curve for tolerr = 10−5 and crit1 (various nGP)

436


E2:

3E-05

0.00372608

0.00742217

0.0111183

0.0148143

0.0185104

E2:

3E-05

0.00372608

0.00742217

0.0111183

0.0148143

0.0185104

Fig. 4.32. Beam 1: Different states of mesh refinement (Q1SPs/o, 16El.), contours: accumulated plastic strain

modification of Q1SPs where the three enhanced degrees-of-freedom W ζe are set equal to zero (Q1SPs/o). The element is then only suitable for thin structures, i.e. it exhibits noticeable thickness locking for thick structures. As expected the value of nGP influences the results of Q1SPs/o noticeably. Due to the thickness locking the element requires a rather fine discretization over the thickness. With nGP = 4 (2 El.) the algorithm refines up to 2 elements over the thickness, the deformation behaviour is much too stiff. With increasing nGP the results significantly improve and approach the curves obtained with Q1SPs which are almost independent on nGP. It is striking that the main refinement in the calculation with Q1SPs/o occurs at about a load level of 30. This can be explained by the choice of the error tolerance. It is rather large so that the main mesh refinement happens after many parts of the structure have already plastified. However, the locking is certainly also eminent in the elastic range where we observe a large deviation between the curves of Q1SPs and Q1SPs/o. The situation improves immediately when the error tolerance is chosen to be much smaller, e.g. tolerr = 10−7 (Figure 4.33a). Again the discretization with nGP = 4 and nGP = 8 (Q1SPs/o) yields too stiff results. However, the refinement starts already in the elastic region and gives a much better


Beam 1 (tolerr = 10^-7, crit1)

Beam 1 (tolerr = 10^-7, crit1) 80

120 number of elements [-]

load level [-]

60 50 40

20 10

Q1SPs, nGP=4 Q1SPs, nGP=8 Q1SPs, nGP=16 Q1SPs/o, nGP=4 Q1SPs/o, nGP=8 Q1SPs/o, nGP=16 nGP=4 maximum nGP=8 maximum nGP=16 maximum

140

70

30

437

Q1SPs 2 El. Q1SPs 4 El. Q1SPs 8 El. Q1SPs/o 2 El. Q1SPs/o 4 El. Q1SPs/o 8 El.

100 80 60 40 20

0 -900 -800 -700 -600 -500 -400 -300 -200 -100 w [mm]

0

0 -900

-800

-700

-600

-500

-400

-300

-200

-100

0

w [mm]

Fig. 4.33. Beam 1: Load-displacement curve and number of elements for tolerr = 10−7 and crit1 (various nGP0)

approximation for load levels below 30. For nGP = 16 (8 El.) the results of Q1SPs and Q1SPs/o are almost equal. As in the previous study the curves for Q1SPs are very close together. The latter observation indicates that Q1SPs should require much less elements to obtain a solution of the same quality as the one of Q1SPs/o. It must be admitted that the number of elements rather depends on the error tolerance than on the choice of the element formulation (see Figure 4.33b). This points to the fact that although the use of the error criterion crit1 (based on the elastic left Cauchy-Green tensor) yields a physically reasonable mesh refinement it is not sensitive with regard to the quality of the solution. For this reason we use now the same example in combination with the residual-based error criterion crit2 (Figure 4.34a). We obtain the very interesting result that in fact this criterion yields much smaller values for Q1SPs than for Q1SPs/o. Obviously the enhanced strains controlled by the internal element degrees-of-freedom W ζe are crucial to model the deformation behaviour of thick structures with very few elements. At a tolerance of 0.001 Q1SPs not even needs a mesh refinement whereas Q1SPs/o refines almost up to the here prescribed maximum (94 elements, see Figure 4.34b). 4.2.8.1.10.2 Uniaxial Bending (Beam of Variable Thickness) In order to investigate the efficiency of the algorithm in a structure of variable thickness we increase the thickness at X = 0 up to 3600 mm. It is assumed that it linearly varies until the end value of 1200 mm is reached at X = 12000 mm. The reference load F0 is given as 500000 N. We load up to a load level of νmax = 100 in 40 equidistant steps. The hardening parameter reads H = 600 N/mm2 . We work with the stabilization parameter μ = 420 N/mm2 . All other material constants as well as the boundary conditions are left unchanged. The behaviour (see Figure 4.35a) resembles the previous observations. For tolerr = 0.05 Q1SPs does not create any new elements whereas Q1SPs/o must

438


Beam 1 (nGP=16, crit2)

Beam 1 (nGP=16, crit2, Q1SPs/o) 100

70


80

load level [-]

60 50 40 30 20 10 0 -700

tolerr=0.1 tolerr=0.01 tolerr=0.005 tolerr=0.001 Q1SPs -600

-500

80 70 60 50 40 30 20

-400

-300

w [mm]

-200

-100

0

10 -700

tolerr=0.1 tolerr=0.01 tolerr=0.005 tolerr=0.001 -600

-500

-400

-300

-200

-100

0

w [mm]

Fig. 4.34. Beam 1: Load-displacement curve and number of elements for different tolerances and crit2 (Q1SPs/o, nGP0 = 16)

Fig. 4.35. Beam 2: Load-displacement curve and number of elements for different tolerances and crit2 (Q1SPs/o, nGP = 16)

increase the number of elements up to 67 (Figure 4.35b). The curve of Q1SPs computed with a very small tolerance (see the curve s : tolerr = 0.00001 in Figure 4.35a) hardly differs from the result achieved with a rather large tolerance (Figure 4.35a, s : tolerr = 0.05). Different states of mesh refinement for a computation with Q1SPs/o are plotted in Figure 4.36. As expected refinement starts in the loading area and in the parts of the structure where the plastic strain begins to accumulate (X ≈ 4500 mm). In conclusion it should be emphasized that also the error criterion crit2 provides a physically reasonable mesh refinement. Additionally, in contrast to crit1, it measures the quality of the solution accurately enough, i.e. in such a way that computations based on a too stiff element formulation (e.g. Q1SPs/o) require a much higher mesh density for the same quality of solution. Therefore


E2: 0.000648603

0.0705965

0.140544

E2: 0.000648603

0.0705965

0.140544

E2: 0.000648603

0.0705965

0.140544

E2: 0.000648603

0.0705965

0.140544

439

Fig. 4.36. Beam 2: Different states of mesh refinement (Q1SPs/o, 16 El.), contours: accumulated plastic strain

the mesh refinement in Figure 4.36 is here shown for Q1SPs/o. Q1SPs shows the same mesh refinement when the error tolerance is set equal to an extremely small value. 4.2.8.1.10.3 Biaxial Bending (Thick Plate of Uniform Thickness) Geometry and boundary conditions are depicted in Figure 4.37. At the boundary X = 12000 mm the displacement in X-direction is constrained. Analogously at the boundary Y = 12000 mm we apply constraints in Y direction. The material parameters and the reference load are the same as in the previous example. The maximum load level is νmax = 60 (applied in 24 equidistant steps). For this example also Q1SPs requires a refinement of the mesh in thickness direction to yield a converged result (see Figure 4.38a). The solution for tolerr = 0.0001 (refinement up to 140 elements, see Figure 4.38b) is very close to the one computed with a mesh of 7x7x4 elements. The curves for the larger tolerances (tolerr ≤ 0.0002) bring up the problem that at the beginning of the refinement (one to two new elements) the solution stiffens, i.e. the displacement w becomes smaller, see the detail A in Figure 4.39. At a larger load the solution softens again (detail B in Figure 4.39). However, the solution for tolerr = 0.001 must be judged as worse as the result for tolerr = 0.01 where

440

4 Methodological Implementation Z

X

Y

F

F

1200 mm 12000 mm

w

12000 mm

Fig. 4.37. Plate 1: Geometry and boundary conditions

Plate 1 (nGP=8, crit2, Q1SPs) 60

40


50 load level [-]

Plate 1 (nGP=8, crit2, Q1SPs) 200

tolerr=0.01 tolerr=0.001 tolerr=0.0005 tolerr=0.0002 tolerr=0.0001 tolerr=0.00001 nGP=8 maximum

30 20 10 0 -3500

160 140 120 100 80 60

-3000

-2500

-2000 -1500 w [mm]

-1000

-500

0

40 -3500

-3000

-2500

-2000 -1500 w [mm]

-1000

-500

0

Fig. 4.38. Plate 1: Load-displacement curve and number of elements for different tolerances and crit2 (Q1SPs, nGP = 8)

no mesh refinement takes place. We assume that the problem is due to the transfer of the history variables which can never be exact. The element formulation Q1SPs reacts much more sensitively to this transaction than Q1SPs/o. The deficiency is easily overcome by working with a slightly smaller tolerance, where right at the beginning several new elements are inserted. In summary it can be, however, stated that the differences between the curves for rather large and very small tolerances are again small. The picture is as expected very different if one uses Q1SPs/o (see Figure 4.39). Whereas Q1SPs does not need additional elements to undercut the error tolerance tolerr = 0.01, Q1SPs/o requires for a solution of the same quality a refinement up to 148 elements. Applying the tolerance tolerr = 0.001 leads already to the maximum refinement (196 elements), i.e. with nGP = 8 the solution cannot be further improved. It has still not completely converged at this point.


Plate 1 (nGP=8, crit2, Q1SPs)

Plate 1 (nGP=8, crit2, Q1SPs/o)

60

60

tolerr=1.0 tolerr=0.1 tolerr=0.01 tolerr=0.001

40 30 20 10 0 -3500

B

50 load level [-]

load level [-]

50

441

tolerr=0.01 tolerr=0.001 nGP=8 maximum

40 30

A

20 10

-3000

-2500

-2000

-1500

w [mm]

-1000

-500

0

0 -3500

-3000

-2500

-2000 -1500 w [mm]

-1000

-500

0

Fig. 4.39. Plate 1: Load-displacement curve (left) and details of it (right) for different tolerances and crit2 (Q1SPs, nGP = 8)

Fig. 4.40. Plate 1: Different states of mesh refinement (Q1SPs/o, 16 El.), contours: accumulated plastic strain

The adaptive increase of the number of elements is visualized in Figure 4.40 (computation with Q1SPs/o). The mesh refines strongly in the loading domain where a severe element distortion is detected. At this point a simultaneous refinement in the plate plane is necessary. It should be made available in the future.

442


Fig. 4.41. Plate 1: Load-displacement curve and number of elements for different load steps and crit2 (Q1SPs/o, nGP = 8, tolerr = 0.01)

Plate 1 (nGP=8, crit2, Q1SPs, tolerr=0.0001) 60

40

number of elements [-]

50 load level [-]

Plate 1 (nGP=8, crit2, Q1SPs, tolerr=0.0001)

dF=2.5 dF=5.0 dF=10.0 dF=20.0 dF=30.0 dF=60.0

30 20 10 0 -3500

-3000

-2500

-2000 -1500 w [mm]

-1000

-500

0

150 140 130 120 110 100 90 80 70 60 50 40 -3500

dF=2.5 dF=5.0 dF=10.0 dF=20.0 dF=30.0 dF=60.0

-3000

-2500

-2000 -1500 w [mm]

-1000

-500

0

Fig. 4.42. Plate 1: Load-displacement curve and number of elements for different load steps and crit2 (Q1SPs, nGP = 8, tolerr = 0.0001)

A point of further interest is the robustness of the algorithm. For this purpose we compare first for Q1SPs/o and tolerr = 0.01 the load-displacement curves computed with different load steps (see Figure 4.41). We obtain the quite impressive result that the total load (ν = 60) can be applied in one step which shows the robustness of the proposed algorithm. The resulting displacement is only marginally smaller than the correct value (for tolerr = 0.01). The new elements appear in almost the same order as for much smaller load steps. The same robust behaviour is observed for the computation with Q1SPs and tolerr = 0.0001. However, the number of new elements is in in this case influenced by the load step. Fewer new elements are generated if the load step is chosen to be very large. Moreover the solution shows the peculiar stiffening again, i.e. the insertion of a few new elements does not lead as expected to a softer (in this case better) solution.


443

4.2.8.2 Error-Controlled Temporal Adaptivity Authored by Detlef Kuhl and Sandra Krimpmann Strategies for the error controlled adaptive time integration of durability problems consist of to main parts: The computation of error measures and the control algorithm of the time step size. Local error estimates and different local error indicators can be used as controlling parameter in adaptive time stepping schemes. 4.2.8.2.1 Local a Posteriori h- and p-Method Error Estimates In order to obtain an estimate of the local time integration error, error measures based on the comparison of the primary integration results un+1 with simultaneously performed time integrations of higher accuracy are used. As illustrated in Figure 4.43 for Newmark and Galerkin type integration schemes the integration quality can be improved by reduction of the time step size Δt. Reasoned by the fact that the time integration error e ∼ Δto is proportional to the time step size Δt to the power of the order of accuracy o, the error e can be significantly reduced if the time step size is divided by m. Δt/m The resulting improved solution un+1 allows for the estimation of the local time integration error by the h-method. Δt/m

eΔt/m = un+1 − un+1

(4.136)

error estimates error indicators

The order of accuracy of Galerkin type integration schemes is controlled by the polynomial degree p whereby the higher the polynomial degree the higher h-method error measures

p-method error measures

enlarged time step size

polynomial degree p1 = p − m < p

umΔt n+2

mΔt uΔt n−1

basic time steps mΔt e Δt uΔt uΔt n+1 n+2 Δt un t eΔt/m

eΔt/m

eΔt/m

Δt/m Δt/m Δt uΔt/m un+1 un+2 m n decreased time step size

1 upn+1

upn1 uΔt n−1

ep/p1 Δt

ep/p1 uΔt n

ep/p2 upn2

1 upn+2

ep/p1 uΔt n+1

ep/p2 2 upn+1

uΔt n+2 t

ep/p2 2 upn+2

polynomial degree p2 = p + m > p

Fig. 4.43. Illustration of h-method error estimates and indicators associated with Newmark and Galerkin time integration schemes as well as p-method error estimates and indicators for p-Galerkin time integration schemes

444


the order of accuracy. For p-Galerkin type integration schemes a higher order accurate comparison solution can be alternatively generated by Galerkin integration schemes of a higher polynomial degree p + m or a Galerkin integration with a smaller time step size Δt/m, compare Figure 4.43. The Δt/m resulting improved solution, denoted by un+1 , allows for the estimation of the local time integration error by the h-method. The resulting improved solution up+m n+1 allows for the estimation of the local time integration error by the p-method. ep/p+m = upn+1 − up+m n+1

(4.137)

Both present error estimates of the h- and p-method are excellently appropriate to estimate the real time integration error. This exceeding quality of the error measures enforces, of course, a very high numerical effort. As a consequence of this, these error estimates are applied if highly robust adaptive integrations and absolutely reliable calculations of multiphysics problems are necessary. Since unreliable prognoses of the long term behavior of concrete structures are worthless, it is highly recommended to invest the additional computational time for the error estimates of the h- and p-method. If the solution behavior of the durability problem is completely understood by the engineer he may switch to the error indicators of the h- and p-method for further parametric studies. Furthermore, these error estimates are applied to study the numerical properties of the present time integration schemes in the context on non-linear durability problems. 4.2.8.2.2 Local a Posteriori h- and p-Method Error Indicators Error indicators of the h- and p-method are only applied, if the numerical effort for the error estimates discussed in the previous section is significantly to high. These kind of error indicators are characterized by comparison solutions of lower quality. The present error indicators are either based on the h-method emΔt = un+1 − umΔt n+1

(4.138)

or the p-method, compare Figure 4.43. ep/p−m = upn+1 − up−m n+1

(4.139)

4.2.8.2.3 Local Zienkiewicz a Posteriori Error Indicators Motivated by the high numerical effort of h- and p-method error estimators as well as indicators, alternative error measures using the Taylor expansion of the solution un+1 for the comparison with the Newmark solution were developed. This basic idea was firstly published by [868] and later enriched by several extensions by [494, 675, 833, 834, 566]. So called Zienkiewicz error


445

Table 4.6. Error indicators for Newmark type time integration schemes (eZX : [868], eLZW : [494], eRS : [675], eR : [676]) for non-linear second order initial value ˙ u) = r u, u, problems ri (¨

¨˙ n derivative u eZX

1 [¨ ¨n] u −u Δt n+1

2 [¨ ¨ n ]− u ¨˙ n eLZW Δt un+1 − u

¨¨ n derivative u 0 1 ¨˙ ¨˙ Δt [un+1 − un ]

error indicator ein 6β −1 ¨ n ]Δt2 un+1 − u 6 [¨ 12β −1 ¨ n ]Δt2 un+1 − u 12 [¨ 3 1 ¨˙ n Δt − 12 u

eRS

1 u ¨ n−1 ] n+1 − u 2Δt [¨

1 [¨ ¨ n−1 ] 8β−1 ¨ n+1 + 1−12β ¨n un+1 −2¨ un + u 8 u 12 u Δt2 1 ¨ n−1 Δt2 + 24 u

eR

0

0

¨ n ]Δt2 β[¨ un+1 − u

indicators compare the Taylor expansion of the primary variable at the end of the time step ˙n + uZX n+1 = un + Δtu

Δt2 Δt3 ˙ Δt4 ¨ ¨n + ¨n + ¨n + · · · u u u 2 6 24

(4.140)

and the Newmark approximation. eZX = un+1 − uZX n+1 β 1 1˙ 1 ¨ ¨ n Δt2 − u ¨ n Δt3 − u ¨ n Δt4 − · · · = [u˙ n+1 − u˙ n ]Δt − u γ 2 6 24

(4.141)

Table 4.6 summarizes various existing error indicators developed for second order initial value problems on basis of equation (4.141). They are distinguished ¨¨ n . Namely, ¨˙ n and u by the estimation of the higher order time derivatives u • • • •

the the the the

Zienkiewicz Xie error indicator, Li Zeng error indicator, Riccius error indicator, and Rickelt error indicator

are special cases of the error indicator defined by equation (4.141). Since this error indicators are defined in terms of accelerations different derivations for first order initial value problems compared to second order initial value problems on ¨ n and u ¨˙ n by adethe basis of equation (4.141) are required. They substitute u quate difference approximations in terms of velocities, see Table 4.7.

446


Table 4.7. Error indicators for Newmark type time integration schemes (eZX : [868], eLZW : [494], eRS : [675], eR : [676]) for non-linear first order initial value prob˙ u) = r lems ri (u,

¨˙ n derivative u

¨n derivative u eZX

1 ˙ ˙ Δt [un+1 − un ]

2 [u˙ ˙ n] − u ¨n eLZW Δt n+1 − u

eRS

1 ˙ ˙ 2Δt [un+1 − un−1 ]

eR

0

error indicator ein 2β − γ ˙ 2γ [un+1 − u˙ n ]Δt

0 1 u ¨ n] n+1 − u Δt [¨

3β − γ ˙ 3γ [un+1 − u˙ n ]Δt 1u ¨ n Δt2 −6

−5γ 1 [u˙ ˙ n + u˙ n−1 ] Δt 12β n+1 −2u 2 12γ u˙ n+1 Δt γ −3β 1 u˙ + 3γ u˙ n + 12 n−1 β ˙ γ [un+1 − u˙ n ]Δt

0

4.2.8.2.4 Adaptive Time Stepping Procedure As a basis of adaptive time stepping procedures the error vectors (4.136,4.137, 4.141) are transformed to a scalar valued relative error measure by using various alternative reference values uref :

e

e= uref

uref = un+1 − un

uref = un+1

uref = u0

(4.142)

The error measure e is compared with the user defined error bounds ν1 η and ν2 η. ν1 η ≤ e ≤ ν2 η

(4.143)

If equation (4.143) is fulfilled the time step remains unchanged. Otherwise, the time step will be adapted: η (4.144) Δtnew = Δtold o e o represents the order of accuracy of the basis time stepping scheme. For e > ν2 η the last time step is repeated with Δtnew and for e < ν1 η the next time step is solved with Δtnew .


447

initial conditions loop over time steps n Newmark-α time integration (Figure 4.24)

¨ n+1 , u˙ n+1 , un+1 u

indication of error

ein = un+1 − uin n+1 e = ein /uref

scalar valued relative error measure

ν1 η ≤ e ≤ ν 2 η

error check (y|n) retain time step size Δt

calculation of time step size

ν1 η > e

error check (y|n) next time interval

Δtnew

[tn+1 , tn+2 ] −→ [tn , tn+1 ] retry time interval κ

final update of internal variables

[tn , tn+1 ]

n+1 → n Fig. 4.44. Algorithmic set-up for the error controlled adaptive time integration by Newmark-α integration schemes combined with error indicators and the adaptive time step control by [868], compare Figure 4.24

initial conditions loop over time steps n standard integration by Newmark-α un+1 or p-Galerkin methods (Figures 4.24 and 4.26) indication/estimation of error

loop over time steps m integration

Δt/m

umΔt m+1 , um+1

m+1 →m mΔt

e

Δt/m

Δt/m = un+1 − umΔt = un+1 − un+1 n+1 , e


e = e /uref

error based adaptive time step control (Figure 4.44) n+1 → n Fig. 4.45. Algorithmic set-up for the error controlled adaptive time integration by Newmark-α or p-Galerkin methods and h-method error estimates/indicators

4.2.8.2.5 Algorithmic Set-Up A typical algorithmic set-up for the adaptively controlled time integration of non-linear second order semidiscrete initial value problems is shown in Figure 4.44. In this overview the Newmark type integration scheme by [569, 198] is combined with error indicators and the adaptive time stepping procedure by [868] as a representative example. The boxes in Figure 4.44 illustrate links with the element and material levels of multifield durability finite element programs (compare Section 4.2.7). As illustrated by Figures 4.45 and 4.46

448


initial conditions loop over time steps n integration by p-Galerkin methods

[p ± m]-Galerkin methods

upn+1 (Figure 4.26)

up±m n+1 (Figure 4.26)

p/p+m indication/estimation of error ep/p−m = upn+1 − up−m = upn+1 − up+m n+1 , e n+1


e = e /uref

error based adaptive time step control (Figure 4.44) n+1 → n Fig. 4.46. Algorithmic set-up for the error controlled adaptive time integration by p-Galerkin methods and p-method error estimates/indicators

the error analysis using improved comparison solutions performed by the hor p-method causes high numerical expenses. In order to avoid this, error indicatiors of the h- or p-method can be used as basis for the adaptive time stepping scheme (compare [452, 459]). 4.2.9 Discontinuous Finite Elements Authored by Klaus Hackl 4.2.9.1 Overview and Motivation Authored by Markus Peters and Klaus Hackl To simulate the displacements in the vicinity of a crack there are several numerical possibilities. In this section we focus on the finite element method using elements with embedded discontinuous fields — on the one hand the Strong Discontinuity Approach (SDA) and the Enhanced Assumed Strain (EAS) which is derived from the SDA, on the other hand the eXtended Finite Element Method (XFEM). All these methods have in common that the strain or displacement field is enhanced to allow specific discontinuities independent of element edges. The SDA and the EAS concepts are based on the same idea of enhancing the strain field so that a discontinuous strain field is obtained across the crack. For this reason the stress field is discontinuous while the displacement field is still continuous. In contrast to that by using the XFEM concept enhanced displacements are used. In addition this enhancement contains functions which span the asymptotic near tip displacement field. So the displacements and stresses are discontinuous across the crack.


449

4.2.9.2 Concepts Authored by Markus Peters and Klaus Hackl The discontinuous concepts which are presented in this book are the Strong Discontinuity Approach (SDA) and the Enhanced Assumed Strain (EAS) which are described in Section 4.2.9.2.2 and the eXtended Finite Element Method (XFEM) which is discussed in more detail in Section 4.2.9.2.1. 4.2.9.2.1 Extended Finite Element Method (XFEM) The eXtended Finite Element Method (XFEM) is an efficient way to calculate the displacements and stresses of the near tip field. The ansatz for the displacements in the finite element approximation are enhanced by using functions which span the asymptotic near tip displacement field. This improves the accuracy of the approximation results and the mesh dependency can be reduced significantly. The concept of the XFEM is based on the Partition of Unity which is described in the following section. By using it the finite element approach is enhanced and results in the XFEM that is described in Section 4.2.9.2.1.2. The functions which are used for this enhancement are those functions that span the function space of the near tip field which is derived in Section 4.2.9.2.1.2. After deriving the XFEM displacement field it is necessary to develop a new integration method which converges accurately for singular functions (see 4.2.9.2.1.3). To improve the accuracy of the XFEM results a p-version using hierarchical higher order Legendre polynomials 4.2.9.2.1.4 is recommended. At last a three dimensional implementation of the XFEM is introduced in Section 4.2.9.2.1.5 and the differences between the XFEM for linear elastic fracture mechanics and cohesive cracks are discussed in Section 4.2.9.2.1.6. 4.2.9.2.1.1 Partition of Unity The idea of the XFEM is based on the principle of the Partition of Unity published in [526]. The keynote of the Partition of Unity is to approximate a function Ψ (x) not alone by using the shape functions, but to implement known functions into the approximation procedure. The Lagrange polynomials which were used as standard shape functions have the following property inside one element Ni = 1 (4.145) i

Multiplying both sides of equation 4.145 above with the functions Ψ delivers i N i Ψ = Ψ, (4.146) =Ψ N Ψ i

i

450

4 Methodological Implementation 60

40

20

0

1

2

3

-20

-40

-60

Fig. 4.47. Function to be approximated, equation (4.147)

which means that an arbitrary function can be approximated by a sum of the function multiplied with the shape functions. The possibility arising from equation 4.146 becomes more clear using an example. The following function has to be approximated: 5

f (x) =

10 (−0.9 + x) + 10 (−0.9 + x) + 0.5 cos(0.9 − x) | − 0.9 + x|

3

(4.147)

The function is divided into three parts (cp. figure 4.47). In each part it is evaluated at p + 1 equidistant points to compute an interpolation polynomial whose polynomial order is varied with p = 1, . . . , 8. In the middle partition the function is approximated by using a sum of polynomials of order p using g1 (x) = f (xi )Lin (x) (4.148) i

as well as using g2 (x) =

i

f (xi )|xi − 0.9|

Lin (x) |x − 0.9|

(4.149)

In equation 4.149 the knowledge of the position and the kind of the singularity is implemented. The functions Lin (x) are the Lagrange polynomials which are defined as Lin (x) =

(x − x0 ) . . . (x − xi−1 )(x − xi+1 ) . . . (xn ) . (xi − x0 ) . . . (xi − xi−1 )(xi − xi+1 ) . . . (xi − xn )

(4.150)

The results of the approximations are shown in figure 4.48. It can easily be discovered that by using equation 4.149 the function f (x) is approximated much better for lower polynomial order than by using equation 4.148. The right and left part are approximated without singularity by using equation 4.148. We also refer to [526]. In [192] the Partition of Unity Method is investigated concerning the XFEM and the included blending elements. The accuracy in these blending elements of the displacements and stresses using the XFEM is also illustrated in Section 4.2.9.2.1.4.


451

polynomial order 1 200 75

150 50 100 25 50

-1

1

2

3 -1

1

2

3

1

2

3

1

2

3

1

2

3

-25 -50

-50 -100

-75

polynomial order 3 75

60

50

40

25

-1

20

1

2

3

-1

-25

-20

-50 -40

-75 -60

polynomial order 4 80 75

60 50

40 25 20

-1

1

2

3 -1

-25 -20

-50 -40

-75

-60

polynomial order 8 80 60 60 40 40

20 20

-1

1

2

3 -1

-20 -20

-40

-60

-40

-60

Fig. 4.48. Function to be approximated from equation (4.147) together with the approximations, left from equation (4.148), right from equation (4.149)

452


s

Et x∗ En x

Fig. 4.49. Definition of normal and tangential vector

4.2.9.2.1.2 XFEM Displacement Field It was shown in Section 4.2.9.2.1.1 that the Partition of Unity Method offers the possibility to enrich the finite element approximation by analytic functions and to implement them into the displacement field. By using the finite element method the continuity of the displacement field can be ensured. The XFEM displacement field can be written as u ˆ = ui Ni (x) + bj Nj (x)H(x) + ckl Nk (x)Fl (x)

(4.151)

It contains • •

•

the standard FEM approximation (ui Ni ) a term to model the crack opening (bj Nj H), with bj representing the half of the crack width and H as the Heaviside function ⎧ ⎪ ⎨ 1 for (x − x∗ ) · En > 0 H(x) = ⎪ ⎩ −1 else x∗ in this definition means the point on the crack surface which has the smallest distance to the point x, En is the normal vector to the crack with its orientation deemed to be Et × En = Ez . The vector Et is the tangential vector to the crack (cp. Figure 4.49). the near tip displacement field (ckl Nk Fl ). The functions Fl herein span the asymptotic near tip displacement field (see Figure 4.50). 0 /√ ϕ √ ϕ √ ϕ ϕ √ , r cos , r sin sin (ϕ) , r cos sin (ϕ) Fl = r sin 2 2 2 2 (4.152)

The element stiffness matrix can now be written as ⎛ ⎡ ⎤⎞T ⎛ ⎡ ⎤⎞ N N F F ⎜ ⎢ 1 ⎥⎟ ⎜ ⎢ 1 ⎥⎟ ⎜ ⎢ ⎥⎟ ⎥⎟ ⎜ ⎢ ⎟ ⎢ ⎜ ⎥ ⎥⎟ ⎜ ⎢ N N F F ⎟ ⎢ ⎢ ⎜ ⎥ ⎥⎟ ⎜ 2 2 e ⎟ ⎢ ⎢ ⎜ ⎥ ⎥⎟ ⎜ KXF EM = ⎜D ⎢ . ⎥⎟ C ⎜D ⎢ . ⎥⎟ dΩ. Ω⎜ ⎢ .. ⎥⎟ ⎜ ⎢ .. ⎥⎟ ⎜ ⎢ ⎥⎟ ⎥⎟ ⎜ ⎢ ⎝ ⎣ ⎦⎠ ⎦⎠ ⎝ ⎣ Nn F Nn F

(4.153)

4.2 Numerical Methods √

r sin

ϕ

453

ϕ √ r cos 2

2

1 1 0.5

1

0

1

0.75 0.5

0.5

-0.5

0.5 0.25

-1

0

-1

-1

0 -0.5

0 -0.5

0

0

-0.5

0.5

1

√

r sin

ϕ 2

-0.5

0.5

-1

1

√

sin (ϕ)

r cos

ϕ 2

-1

sin (ϕ)

0.8 0.5 0.6

1

1 0

0.4

0.5

0.5

0.2

-0.5

0 -1

0

-1

0 -0.5

-0.5

0

-0.5

0

-0.5

0.5

0.5

1

-1

1

-1

Fig. 4.50. The four crack tip functions Fl of equation 4.152

The included vector F is defined as F = [1, H, F1 , F2 , F3 , F4 ]T

(4.154)

and D represents the matrix of derivatives. ⎡

∂ ⎢ ∂x ⎢ ⎢ D=⎢ 0 ⎢ ⎣ ∂ ∂y

⎤ 0 ⎥ ∂ ⎥ ⎥ ⎥ ∂y ⎥ ∂ ⎦ ∂x

The functions Fl are valid for straight cracks. Kinking cracks have to be aligned to straight cracks. In [104, 545] the following mapping is used: The

454


ytip

α θR xn

xtip xtip

r xn−1

x

Fig. 4.51. Crack with one kink

coordinates of one point under consideration can be used to calculate an angle α. The kinking angle is defined as θR (cp. Figure 4.51). The distance r of the considered point to xn can also be computed. If a line perpendicular onto the crack segment containing the crack tip can be drawn through the point (−π/2 ≤ α ≤ π/2 — right of the thin dotted line in Figure 4.51) the point can be used by its polar coordinates directly in the functions Fl . Otherwise a mapping algorithm has to be defined as follows (see also Figure 4.51). ⎛ ⎞ ⎛ ⎞ ¯ x −l − r cos( θ) ⎜ tip ⎟ ⎜ ⎟ (4.155) ⎝ ⎠=⎝ ⎠ ¯ −r sin(θ) ytip This definition contains θ¯ = c(α − θR ) ⎧ π/2 ⎪ ⎪ for α > θR ⎨ 3/2 π − θR with c = π/2 ⎪ ⎪ ⎩ for α < θR θR − π/2

(4.156)

l = |xtip − xn | r = |xn − x| Because of this mapping the matrix of equation 4.157 has to be implemented into the derivatives at mapped points.


⎛

455

⎞

∂

⎜ ∂xxyloc ⎟ ⎜ ⎟= ⎝ ⎠ ∂ ∂yxyloc

⎞⎛ ∂ ¯ ¯ ¯ ¯ ⎜ − cos θ cos α − c sin θ sin α − sin θ cos α − c cos θ sin α ⎟ ⎜ ∂xtip ⎝ ⎠⎜ ⎝ ∂ − cos θ¯ sin α + c sin θ¯ cos α − sin θ¯ sin α − c cos θ¯ cos α ∂ytip ⎛

⎞ ⎟ ⎟ ⎠ (4.157)

The included values c are those from equation 4.156. ∂x∂tip means the derivative in x direction in the local crack tip coordinate system after the mapping ∂ and ∂xxyloc represents the derivative in x direction of the local crack tip coordinate system with the kinked crack. The same notation is used for the local y direction. The matrix in equation 4.157 is calculated as follows ∂ ∂ ∂ f (xtip (x)) = xtip f ∂x ∂xtip ∂x Using the definition of equation 4.155 yields ∂ ∂ ∂ xtip = − r cos θ¯ − r cos θ¯ ∂x ∂x ∂x

¯ ¯ ∂ = − 1+x r cos θ − r −ck sin θ ∂x α = − cos α cos θ¯ + ck sin θ¯ −y r = − cos α cos θ¯ − ck sin θ¯ sin α ∂ ∂ ¯ ¯ r cos θ − r cos θ − ∂y ∂y y ∂ ¯ ¯ = − cos θ − r −ck sin θ α r ∂y x = − sin α cos θ¯ + ck sin θ¯ r ¯ ¯ = − sin α cos θ − ck sin θ cos α

∂ xtip = ∂y

∂ −y ytip = − cos α sin θ¯ − ck cos θ¯ ∂x r ¯ ¯ = − cos α sin θ + ck cos θ sin α ∂ x ytip = − sin α sin θ¯ − ck cos θ¯ ∂y r = − sin α sin θ¯ − ck cos θ¯ cos α.

456


ytip xtip θ¯

x∗n−1

x

r

xn

xtip

∗

Fig. 4.52. Crack with one kink after mapping

If the crack is kinked multiple times the mapping algorithm of equations 4.155– 4.157 has to be evaluated for every kink starting with the kink that is close to the crack tip (cp. Figures 4.53 and 4.54). Finally the derivatives have to be transfered from the local crack tip coordinate system to the global system which uses a simple rotation matrix of equation 4.158.

ytip

α

xtip

θR xn r xn−1 x xn−2

Fig. 4.53. Multiple kinked crack

xtip


457

ytip x∗n−1

xtip θ¯

x∗n−2

x∗

r

xn

xtip

Fig. 4.54. Multiple kinked crack after the first mapping

⎛

∂

⎜ ∂xxyglo ⎜ ⎝ ∂ ∂yxyglo

⎞

⎞⎛ ∂ ⎟ ⎜ cos ϕ0 − sin ϕ0 ⎟ ⎜ ∂xxyloc ⎟=⎝ ⎠⎜ ⎠ ⎝ ∂ sin ϕ0 cos ϕ0 ∂yxyloc ⎛

⎞ ⎟ ⎟ ⎠

(4.158)

ϕ0 means the angle between local and global coordinate system. The displacements calculated by the XFEM can now be used to compute the stress intensity factors. The integral formulation emerges to be more robust against numerical errors when using the integration which is introduced in Section 4.2.9.2.1.3. For the integration of the J-integral for every mode separately a mode separation procedure is used [404]. It yields for the displacements ⎫ ⎫ ⎧ ⎧ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ ⎨ ˆ1 ⎬ 1 u1 − u 1 u1 + uˆ1 u = uI + uII = + (4.159) ⎪ 2⎪ ⎪ 2⎪ ⎩ u2 − u ⎩ u2 + u ˆ2 ⎭ ˆ2 ⎭ and for the stresses

σ = σ I + σ II =

⎫ ⎧ ⎪ ⎪ ⎬ ⎨ σ + σ ˆ σ − σ ˆ 11 11 12 12 1 2⎪ ⎩

σ21 − σ ˆ21

⎪ σ22 + σ ˆ22 ⎭

+

⎫ ⎧ ⎪ ⎪ ⎬ ⎨ σ − σ ˆ σ + σ ˆ 11 11 12 12 1 2⎪ ⎩

⎪ σ21 + σ ˆ21 σ22 − σ ˆ22 ⎭ (4.160)

ˆ and stresses σ ˆ are evaluated at point x ˆ . This point is The displacements u mirrored from x at the crack (see Figure 4.55).

458


x

ˆ x ˆ Fig. 4.55. Point x and mirrored point x

4.2.9.2.1.3 Integrating Discontinuous Functions A Combination of 5 Point Newton-Cotes Integration and 3 Point Gaussian Quadrature It is necessary to integrate the element stiffness matrix and the J-integrals accurately because of the fact that they contain singular functions which have to be integrated. Therefore a new integration routine is suggested. It is an adaptive combination of a 5 point Newton-Cotes integration scheme and a 3 point Gaußian quadrature. In this case for both integration schemes the analytical error depends on the sixth derivative of the function to be integrated. In the following derivation of this integration scheme the index N C and G represents parts of the Newton-Cotes integration and the Gaußian quadrature respectively. The Newton-Cotes-Formulas deliver the result a+h h (7f1 + 32f2 + 12f3 + 32f4 + 7f5 ) f (x) dx ≈ IN C = (4.161) 90 a The analytical error is N C =

h7 f (6) (ξN C ) 1935360

(4.162)

The Gaußian quadrature provides

a+h

f (x) dx ≈ IG =

a

h (5f1 + 8f2 + 5f3 ) 18

(4.163)

with its error G =

h7 f (6) (ξG ) 2016000

(4.164)

The sum of the integration results and errors have to be equal for both integration schemes. IG + G = IN C + N C

(4.165)


459

Using equations 4.162 and 4.164 in equation 4.165 and assuming that f (6) (ξG ) = f (6) (ξN C )

(4.166)

the sixth derivative can be estimated as −987429 (IG + IN C ) (4.167) f (6) (ξ) = h7 Equation 4.167 can now be used to express the error of the Newton-Cotes integration in terms of the integration results 49 (IG − IN C ) (4.168) 25 With this value the integration result of the Newton-Cotes integration can be improved. a+h f (x) dx ≈ IG (4.169) N C =

a

IˆN C = IN C + N C

G+N C ≈ IG − IˆN C

(4.170)

The derivation was shown for one dimension but it is similar for two dimensions where integrals over the sixth derivative have to be estimated (cp. equation 4.166). % & 7 6 7 6 b y − ay d d 8 b x − ax IN C − f dx + f dy 945 4 dy 4 dx x y % & 6 6 d d 1 7 7 (by − ay ) f dx + (bx − ax ) f dy = IG − 2016000 dy dx x y (4.171) By using the assumption that [. . . ]N C ≈ [. . . ]G the error of the NewtonCotes integration can be estimated as 49 (IG − IN C ) (4.172) 25 Again the integration result of the Newton-Cotes integration can be improved similar to the procedure in one dimension (see equations 4.169 and 4.170). This method is used in an adaptive scheme which means that in a first step the total domain is integrated. If the estimated error is bigger than a given error limit the domain is divided into four parts and in every part the integration is repeated. Now that part is decomposed and integrated which has the biggest predicted error. This is repeated until the sum of all predicted errors are smaller than the given error limit. N C =

460


Example Functions and Integration The following comparison of some integration methods is done here for scalar functions only. The integration of vector functions is not pictured here. Because a vector function has to be integrated in the computation of the element stiffness matrices we refer to the comparison of the introduced combination method with an integration routine that is based on an extrapolation method in [619]. In the definitions of the functions which are integrated in the following the functions Ni are the bi-linear shape functions which belong to node i in one element. Furthermore the singularity is lying in the origin so we have to consider r(x, y) = x2 + y 2 and ϕ(x, y) = arctan(x, y) The regions are chosen in such a way that the position of the singularity is not in the middle of the integration range because during the XFEM computation this only happens in exceptional cases. Only for the first example function 1 with f (x) = r(x,y) the integration range is chosen to (x, y) ∈ [−1; 1] to show that the combination method is able to integrate a function with singular value lying at one integration point, too. The numerical integration of every function is calculated using 1. 2. 3. 4.

the adaptive Simpson integration the adaptive 3 point Gauss integration the adaptive 4 point Gauss integration the adaptive combination of 5 point Newton-Cotes integration and 3 point Gaussian quadrature

This computation is done for different values of the maximal relative error εrel = 10−i , i = 1, . . . , 6. In figures 4.56-4.66 the relation of the errors ε = max(|

εgeschtzt εvorh |, | |) εgeschtzt εrel

(4.173)

and the number of integration points which were used are shown. The maximal admissible error εrel is a value which given by the user. The beams which show the relation of the errors are filled with different colors. This color is white if the condition ε≤1 is fulfilled which means that εrel ≥ εestimated ≥ εpresent . The beam is filled with gray color if the deviation is 3% at maximum which is true for


461

1 < ε ≤ 1.03. If the error of the integration routine is bigger the color of the beam is black. In the following computations the number of integration points is limited to 1.25 · 106 points. The first function to be integrated is 1 r(x, y)

f1 (x, y) =

(4.174)

The analytical integration in the region (x, y) ∈ [−1; 1]2 yields

1

1

f1 (x, y) dx = 7.0509887. x=−1

(4.175)

y=−1

The results of the numerical integration is shown in Figures 4.56 und 4.57. The second function is the derivative in x direction of the first crack tip function 2 ∂ f2 (x, y) = (F1 (x, y)) . ∂x By simplifying the function is r(x, y) − x . 8 r(x, y)2

f2 (x, y) =

(4.176)

The analytic integration of f2 (x, y) in the range (x, y) ∈ [−1.01; 1]2

1

1

f2 (x, y) dx = 0.88773515. x=−1.01

(4.177)

y=−1.01

The numerical results are shown in Figures 4.58 und 4.59. The function to be integrated is build by multiplication of one bi-linear shape function and the first crack tip function f3 (x, y) =

∂ (N1 (x, y)F1 (x, y)) . ∂x

By inserting the functions N1 and F1 and simplifying the function can be written as

(−1 + y) − (−1 + x) y cos( ϕ2 ) + x (−1 + 3 x) + 2 y 2 sin( ϕ2 ) f3 (x, y) = . 3 8 (r2 ) 4 (4.178) The analytic integration of function f3 (x, y) in the range (x, y) ∈ [−1.01; 1]2 gives

462 ε

4 Methodological Implementation adapt. 3-Pt-Gauss-Integration

ε

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2 1

ε

2

3

4

5

εrel =10−i

6

1

adapt. Simpson-Integration

ε

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2 1

2

3

4

5

εrel =10−i

6

adapt. 4-Pt-Gauss-Integration

2

3

4

5

εrel =10−i

6

adapt. Combination NC–Gauss

1

2

3

4

5

εrel =10−i

6

Fig. 4.56. Strain ε from equation (4.173) for the integral (4.175) εrel = 10−1

n

εrel = 10−2

n 7000

6000

6000

5000

5000 4000 4000 3000

3000

2000

2000

1000 1

2

3

4

Int.Meth.

εrel = 10−3

n

1000 1

2

3

4

Int.Meth.

4

Int.Meth.

4

Int.Meth.

εrel = 10−4

n

8000 8000 6000 6000 4000 4000 2000

2000

1

2

3

4

Int.Meth.

εrel = 10−5

n

1

2

3

εrel = 10−6

n 60000

20000

50000 15000

40000 30000

10000

20000 5000

1

2

3

4

Int.Meth.

10000 1

2

3

Fig. 4.57. Number of integration points used in the numerical integration of (4.174)

4.2 Numerical Methods ε


ε

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2 1

ε

2

3

4

5

εrel =10−i

6

ε

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2 1

2

3

4

5

εrel =10−i

6


1


463

2

3

4

5

εrel =10−i

6


1

2

3

4

5

εrel =10−i

6


n

εrel = 10−2

n

600

1000

500

800

400

600

300 400 200 100 1

2

3

4

Int.Meth.

εrel = 10−3

n

200 1

2

3

4

Int.Meth.

4

Int.Meth.

4

Int.Meth.

εrel = 10−4

n 7000

2000

6000 1500

5000 4000

1000

3000 2000

500

1

2

3

4

Int.Meth.

εrel = 10−5

n

1000 1

2

3

εrel = 10−6

n 70000

20000 60000 50000

15000

40000 10000

30000 20000

5000

1

2

3

4

Int.Meth.

10000 1

2

3


464


1

1

f3 (x, y) dx = 0.54239955.

(4.179)

y=−1.01

x=−1.01

The numerical results are shown in Figures 4.60 and 4.61. The next example function is (f3 (x, y))2 ; or ∂ 2 (N1 (x, y)F1 (x, y)) . ∂x

f4 (x, y) =

By simplifying the function it can be written as 2

(−1 + y)

f4 (x, y) =

2 (−1 + x) y cos( ϕ2 ) + x − 3 x2 − 2 y 2 sin( ϕ2 ) 3

.

64 (r2 ) 2 (4.180)

The analytic integration of f4 (x, y) in the range (x, y) ∈ [−1.01; 1]2 results in

1

1

f4 (x, y) dx = 0.49637119

(4.181)

y=−1.01

x=−1.01

The numerical results of the integration are given in Figures 4.62 and 4.63. The function which is integrated in this example is formed by the bi-linear shape functions which are defined for nodes 1 and 3 and the first crack tip function in the following form: ∂ ∂ (N1 (x, y)F1 (x, y)) (N3 (x, y)F1 (x, y)) . ∂x ∂x

f5 (x, y) =

Computing the derivatives and simplifying the function it can be written as f5 (x, y) =

−

−1 + y 2

2 − x + y 2 −1 + 9 x2 + 4 y 2 + r x −1 + 9 x2 + 8 y 2 3

.

128 (r2 ) 2

(4.182) Integrating this function analytically in the range (x, y) ∈ [−1.01; 1]2 the integration value is 1 1 f5 (x, y) dx = −0.058125164. (4.183) x=−1.01

y=−1.01

The results of the numerical integrations are shown in Figures 4.64 and 4.65. The function to be integrated is defined as f6 (x, y) =

∂ ∂x

∂ (N3 (x, y)F3 (x, y)) ∂x (N2 (x, y)F4 (x, y))

∂ ∂ + ∂y (N3 (x, y)F3 (x, y)) ∂y (N2 (x, y)F4 (x, y)) .



ε

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2 1

ε

2

3

4

5

6

εrel =10−i ε

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2 1

2

3

4

5

6

εrel =10−i


1


465

2

3

4

5

6

εrel =10−i


1

2

3

4

5

6

εrel =10−i

Fig. 4.60. Strain ε from equation (4.173) for the integral (4.179)


466 ε


ε

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2 1

ε

2

3

4

5

εrel =10−i

6

1


ε

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2 1

2

3

4

5

εrel =10−i

6


2

3

4

5

εrel =10−i

6


1

2

3

4

5

εrel =10−i

6

Fig. 4.62. Strain ε from equation (4.173) for the integral (4.181)

350

εrel = 10−1

n

εrel = 10−2

n 700

300

600

250

500

200

400

150

300

100

200

50 1

2

3

4

Int.Meth.

εrel = 10−3

n

100 1

2500

800

2000

600

1500

400

1000

200 2

3

4

Int.Meth.

εrel = 10−5

n

3

4

Int.Meth.

4

Int.Meth.

4

Int.Meth.

εrel = 10−4

n

1000

1

2

500 1

2

3

εrel = 10−6

n 30000

8000

25000 20000

6000

15000 4000 10000 2000

1

2

3

4

Int.Meth.

5000 1

2

3




ε

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2 1

ε

2

3

4

5

εrel =10−i

6

ε

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2 1

2

3

4

5

εrel =10−i

6


1


467

2

3

4

5

εrel =10−i

6


1

2

3

4

5

εrel =10−i

6


n

εrel = 10−2

n

600

1000

500

800

400

600

300 400 200 100 1

2

3

4

Int.Meth.

εrel = 10−3

n

200 1

2

3

4

Int.Meth.

4

Int.Meth.

4

Int.Meth.

εrel = 10−4

n 7000

2000

6000 5000

1500

4000 1000

3000 2000

500

1

2

3

4

Int.Meth.

εrel = 10−5

n

1000 1

2

3

εrel = 10−6

n 70000

20000

60000 50000

15000

40000 10000

30000 20000

5000

1

2

3

4

Int.Meth.

10000 1

2

3


468


ε

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2 1

2

3

4

5

εrel =10−i

6

1


ε 1

1 0.8

0.6

0.6

0.4

0.4

0.2

0.2 2

3

4

5

εrel =10−i

6

2

3

4

5

εrel =10−i

6


ε

0.8

1


ε

1

1

2

3

4

5

εrel =10−i

6


n

εrel = 10−2

n 3000

800 2500 600

2000

400

1500 1000

200

1

2

3

4

Int.Meth.

εrel = 10−3

n

500 1

2

30000

8000

25000

4

Int.Meth.

4

Int.Meth.

4

Int.Meth.

εrel = 10−4

n

10000

3

20000

6000

15000 4000 10000 2000 1

2

3

4

Int.Meth.

εrel = 10−5

n

5000 1

3

εrel = 10−6

n

100000

2

300000

80000

250000

60000

200000 150000

40000 100000 20000 1

2

3

4

Int.Meth.

50000 1

2

3



469

Fig. 4.68. Configuration of a tension test

which can be rewritten as 2 2 f6 (x, y) = 32yr4 2 x2 (1 + x) − r x (1 + x) y

2 −x (2 + x (7 + 6 x (2 + x))) y 2 − (1 + x) y 4 − y 6 .

(4.184)

By integrating the function f6 (x, y) analytically in the integration range x ∈ [−1.02; 1] and y ∈ [−1.01; 1] the integration value 1 1 f6 (x, y) dx = −0.027713772 (4.185) x=−1.02

y=−1.01

is obtained. The numerical results are shown in Figures 4.66 and 4.67. 4.2.9.2.1.4 p-Version of the XFEM By using the XFEM with bi-linear shape functions to compute the displacements of the tension test shown in Figure 4.68 the resulting displacements ux around the crack tip are shown in Figure 4.69 for the deformed system. The black line indicates the asymptotic displacements of the crack surfaces in the vicinity of the crack tip. It can be seen from the figure that the approximation of the displacement field results in big differences to the analytical solution. Furthermore a constriction is noticeable which occurs at the position where the crack cuts the edge of the crack tip element. The results can be improved by using a refinement of the element mesh (h-version or h-extension) as well as higher order polynomials for the shape functions (p-version or p-extension). Using a h-extension will improve the results but the constriction is still visible but closer to the crack tip while using a p-version for the shape functions results in the displacements

470


Fig. 4.69. Displacements ux for the deformed system using bilinear shape functions

Fig. 4.70. Displacements ux for the deformed system, left: using bi-quadratic shape functions, right: using quadratic hierarchical shape functions

shown in figure 4.70. In both cases the accuracy of the results is improved, but using the hierarchical shape functions the displacement field results in smaller differences and furthermore the constriction is not visible here. To find out more about this phenomenon we examine a single element and compute the coefficients in the XFEM displacement field using the minimizing process of Equation 4.186 n coll i=1

2

(uanalyt (xi ) − uXFEM (xi )) → min

(4.186)


471

The displacements in this equation are the functions of the analytical crack tip field and 4.151. The computed coefficients are used to calculate the integral of the error as defined in Equation 4.187. ε= |uanalyt (xi ) − uXFEM (xi ) | dΩ (4.187) Ω

Figures 4.71-4.75 show the results of equation 4.187 for different blending elements. Because the condition of the Partition of Unity is satisfied inside the crack tip element the integrated error is zero and not shown here. The errors

|Δu| dΩ

2 lO2 5 2 lO3 5 2 lO4 5 2 lO5 5

Δux Δuy Δux Δuy

2 lO6 5 2 lO7 5

hierarchic hierarchic standard standard

2 1

2

3

4

5

6

7

polynomial order

Fig. 4.71. Differences of displacements inside the 1st blending element

|Δu| dΩ

2 lO2 5 2 lO3 5 2 lO4 5 2 lO5 5

Δux Δuy Δux Δuy

2 lO6 5 2 lO7 5 1

2

hierarchic hierarchic standard standard 3

4

5

6

7

polynomial order

Fig. 4.72. Differences of displacements inside the 2nd blending element

472

4 Methodological Implementation |Δu| dΩ

0.1 5 2 lO2 5 2 lO3 5 2 lO4 5 2 lO5 5 2 lO6 5 2 lO7 5 2 lO8 5 2

Δux Δuy Δux Δuy 1

2


4

5

6

7

polynomial order

Fig. 4.73. Differences of displacements inside the 3rd blending element

|Δu| dΩ 2

lO2 5 2 lO3 5 2 lO4 5 2 lO5

Δux Δuy Δux Δuy

5 2 lO6 5 2 1

2


4

5

6

7

polynomial order

Fig. 4.74. Differences of displacements inside the 4th blending element

for the elements that are not enriched are equal for using Lagrange polynomials (the standard shape functions) or Legendre polynomials (hierarchical shape functions). Because of the fact that the shape functions defined for the blending elements span different function spaces, differences for the error of Equation 4.187 exist. The results of the blending elements which are pictured here indicate the enormous influence of the selection of the basis to the approximation of the near tip field. The results show that the blending elements cause a big error of the finite element analysis, because the conditions of the Partition of Unity are not satisfied in the blending elements. The fact that only some of the nodes or


473

|Δu| dΩ

lO2 5 2 lO3 5 2 lO4 5 2 lO5

Δux Δuy Δux Δuy

5 2 lO6 5 2 1

2


4

5

6

7

polynomial order

Fig. 4.75. Differences of displacements inside the 5th blending element

modes inside these elements are enriched by the crack tip functions, produces this difficulty. Using the hierarchical shape functions result in smaller errors because the influenced region of the crack tip functions become bigger. For more details on this phenomenon we refer to [618, 334]. 4.2.9.2.1.5 3D XFEM Authored by Christian Becker and G¨ unther Meschke Because of the variety and complexity of mechanical engineering crackpropagation problems the powerful two-dimensional X-FEM simulation tools have to be enhanced for three-dimensional problems. In recent years the extended finite element method was succesfully applied to fully threedimensional problems [781, 547, 780, 301]. The description of the crack topology and crack propagation is more frequently described implicitly by the level set method [606, 605]. Considering the three-dimensional extended finite element method, there is mostly a conflict between the accuracy, applicability, realistic results and the complexity of the numerical implementation, in particular regarding crack propagation criteria. As an example, in a two-dimensional analysis of a single crack, the crack front is represented by a single point P . Consequently, the crack propagation is simply reduced to the determination of the crack propagation angle Θc and the crack propagation length lc . Even with a crack propagation criterion like the minimization of the total energy, this problem is solved quite easily by introducing two additional system degrees of freedom Θc and lc . On the contrary, in a full three-dimensional analysis of crack propagation of a single crack, the crack front is represented by a number of line segments bounding the crack surface. It can be seen, that this problem is of much more complexity than the two-dimensional problem.

474


In most of the three-dimensional X-FEM implementations tetrahedral elements with linear approximations of the displacement field are used in combination with elementwise plane crack propagation (see e.g. [52]). This represents the classical h-finite element approach to capture the crack propagation process. A more complex simulation concept applies the well-known level-set method (see e.g. [546, 324]). Within this method arbitrary crack growth is accounted for by solving Hamilton-Jacobi-like equations. A crucial point in this concept is that the velocity of the crack surface evolution has to be known to capture the propagation process. Alternatively, the proposed combination of the p-FEM and the X-FEM that was stressed in the last section (Section 4.2.9.2.1.4) and holding for high accuracy, is applied to the numerical simulation of three-dimensional crack propagation. In this concept, because of the complexity of the fully three-dimensional problem, some assumptions are introduced. The underlying finite element formulation is a continuum brick element holding for arbitrary higher-order shape functions (see Section 4.2.4.3). The proposed higher order X-FEM simulation strategy holds for: • •

element-wise crack propagation plane crack surface at the element level

With the proposed assumptions a row of questions arise concerning numerical integration and the determination of crack propagation. Because of the jump in the enhanced strain field, standard GaussIntegration cannot be applied. Hence, both parts of a cracked element have to be integrated separately. In two dimensional analyses the well-known Delauney-triangulation is used. A division into sub-tetrahedra in the sense of the Delauney-triangulation is way too complicated, therefore a fixed subdivision into six subtetrahedra is used (see Figure 4.76). Each of these six tetrahedra may be cut by the crack plane into the sub-domains of a pentahedron, tetrahedron or a pyramide by either a triangular or quadrilateral crack plane (see Figure 4.77). The element quantities (•) like stiffness matrices or internal load vectors are generated by summing over all obtained sub-domains that are numerically integrated with a Gauss-integration according to the geometrical shape of the sub-domain Vηi : ∂X || dVξ (•) = f (X) dΩ = f (X(ξ)) || ∂ξ Ω Vξ (4.188) n ∂X ∂ξ || dVηi . = f (X(ξ(η))) || ∂ξ ∂η i=1 Vηi

Besides numerical integration, the proposed assumptions lead to a problem concerning crack propagation, that is adressed in the following paragraph. In general, the crack path continuity condition is fullfilled in the context of the


475

Fig. 4.76. Numerical integration in the context of the X-FEM: Subdivision of the continuum element into six sub-tetrahedrons

n

Ω−

*

n

∂S Ω

Ω+

Ω− ∂ Ω S

Ω+

Fig. 4.77. Separation of a sub-tetrahedron by a plane crack segment: left: separation into two pentahedra by a quadrilateral, right: separation into pentahedron and tetrahedron by a triangle

X-FEM. In conjunction with elementwise, plane crack propagation this leads to strong restrictions concerning the kinematics of the evolving crack surface. These restrictions are addressed in Figure 4.78. For a start, the crack front is represented by line segments at those element faces where the crack surface ended so far. Sound elements that are neighbours to that faces are crack candidates and are investigated in the subsequent load

476


P2 nmod n P3

X3

P1 Ä

P1

P2

P2 X

P1

1

Fig. 4.78. C0 -crack plane evolution: Left: fixed position of the potential crack segment by two neighbouring crack tip line segments, middle: restricted position of the crack segment by one neighbouring crack tip, right: modification of the computed normal vector considering one neighbouring crack tip segment

steps if the crack propagates through them. In Figure 4.78 the element in the middle is neighboured by only one crack segment whereas the candidate at the left is bounded by two segments of the crack front. Because of the already existing crack line of the middle element, the position of the new crack segment is not fully arbitrary but can only be positioned by a rotation around the existing crack line segment. Therefore the normal vector resulting from the crack propagation criterion has to be modified, like it is illustrated at the right hand side of Figure 4.78. Regarding the left element, the position of a crack segment is already predefined by the two linear independent line segments. Therefore, it is only possible to decide when the new crack plane is inserted. The preceding thoughts occur only in the context of a C0 -continuous description of the crack surface, using an algorithm according to e.g. [52]. To gain more flexibility concerning the kinematics of the evolving crack surface, the C0 -continuity of the crack surface is neglected and a C0 -discontinuous algorithm according to [301] can be used. In this algorithm the new crack segment introduced within a crack candidate is defined by a point P and the normal vector n obtained through the crack propagation criterion. The point P is the geometrical mean of all existing mid-points of existing crack segments at the element boundaries (see Figure 4.79). With this method the crack propagation gains a lot of flexibility by keeping the numerical implementation very simple. In cases where the gaps between neighboured crack segments are getting to wide, a smoothing algorithm for the crack surface is provided by [302]. 4.2.9.2.1.6 XFEM for Cohesive Cracks Authored by Christian Becker and G¨ unther Meschke First of all, the X-FEM was applicated to problems of Linear Elastic Fracture Mechanics (LEFM). Therein, the general balance of momentum


477

P1 P P2

Fig. 4.79. Definition of the crack plane by point P and normal vector n. P is the geometric mean of all midpoints of the crack front lines

at the discontinuity, requiring the jump of the traction to vanish at the discontinuity [[td ]] = 0,

(4.189)

was accounted for by stating vanishing traction td = 0.

(4.190)

LEFM is primarily used in the context of numerical simulations of brittle materials like ceramics. Numerical simulations of quasi-brittle material behaviour of concrete require the consideration of a cohesive zone by a cohesive zone modell [240, 85, 369]. Within the cohesive zone the traction vector is not vanishing, but is dependend on the energetically conjugated variable of the displacement jump [[u]]: td = td ([[u]]).

(4.191)

Because of the non-vanishing traction vector, there is an additional term of the internal virtual work within the weak form of balance of momentum, here, for brevity without volume or external loads: δ¯ ε : σ dΩ + δ[[u]] · td dΓd = 0. (4.192) Ω

Γd

δW int

!

Exemplary, an anisotropic softening law according to [828] is investigated. The hyperbolic softening is realized by application of a scalar damage criterion

478


similiar to continuum damage mechanics [422]. Here, the traction vector is supposed to point only in normal direction of the crack: td = tn ([[u]]) n.

(4.193)

The normal component of the traction vector is a function of the equivalent displacement jump, which in this case is simply the absolute value of the normal component of the displacement jump: tn = tn ([[ueq ]]), [[ueq ]] = [[u]]2n = ||[[u]]n ||. (4.194) The normal traction is calculated with the help of an initial stiffness T and the damage compliance tensor T da which is also a function of the equivalent displacement jump: tn = T − T da([[ueq ]]) [[ueq ]], (4.195) with the hyperbolical softening law ftu α0 da T =T 1− exp{− [α − α0 ]} , α Gf

(4.196)

where Gf is the fracture energy, ftu is the ultimate tension stress, α is an internal history variable and α0 the corresponding initial value. The history variable α is related to the equivalent displacement jump by the damage criterion: Φ([[ueq ]], α) = [[ueq ]] − α ≤ 0

(4.197)

In case of a violation of the damage criterion, the internal variable α is updated to the current value of the equivalent displacement jump. This criterion is completed by the well-known Kuhn-Tucker conditions Φ ≤ 0,

α˙ ≥ 0 ,

Φ α˙ = 0

(4.198)

and the consistency condition Φ˙ α˙ = 0. The linearization of the normal traction component is derived as ∂T da eq tn,[[u]]eq = T − T da ([[u]]eq ) − · [[u]] ∂[[u]]eq with the partial derivative of the damage compliance tensor ∂T da α0 exp{•} α0 ftu exp{•} . = T + ∂[[u]]eq [[u]]eq2 Gf [[u]]eq

(4.199)

(4.200)

(4.201)


479

Therewith, the complete linearization of the traction vector with respect to the displacement jump [[u]] is obtained as: ∂tS [[u]]eq eq ∂[[u]] ∂tS ∂[[u]]eq · [[u]] = ⊗ eq ∂[[u]] ∂[[u]] ∂[[u]]eq = tn,[[u]]eq n ⊗ · [[u]] ∂[[u]] [[u]]n = tn,[[u]]eq n ⊗ n · [[u]] eq [[u]] [[u]]n eq = tn,[[u]] n ⊗ n ·[[u]] [[u]]eq !

tS =

(4.202)

Ttan

4.2.9.2.2 Strong Discontinuity Approach and Enhanced Assumed Strain Authored by J¨ orn Mosler An overview of the Strong Discontinuity Approach (SDA) is given in this section. The SDA is characterized by the incorporation of strong discontinuities, i.e. discontinuous displacement fields, into standard displacement-based finite elements by means of the Enhanced Assumed Strain (EAS) concept. The fundamentals of the SDA are illustrated and compared to those of other models based on discontinuous deformation mappings. The main part of this contribution deals with the numerical implementation of the SDA. Besides the original finite element formulation of the SDA, a more recently proposed algorithmic framework which avoids the use of the static condensation technique is presented as well. This section follows to a large extent [554]. 4.2.9.2.2.1 Kinematics: Modeling Embedded Strong Discontinuities In this section, a review of the kinematics associated with the Strong Discontinuity Approach (SDA) is given. For the sake of simplicity, attention is restricted to a geometrically linearized framework. Further details on the geometrically exact SDA can be found in [555]. Fundamentals According to Simo & Oliver [745, 598], the SDA is characterized by a displacement field of the type ¯ + ru (Hs − ϕ), u=u

¯ ∈ C ∞ (Ω, R3 ), ϕ ∈ C ∞ (Ω, R). with u

(4.203)

Here, Hs is the Heaviside function, ru the displacement discontinuity and ϕ is a smooth ramp function which allows to prescribe the Dirichlet ¯ (see [745, 598]). This will be described later. boundary conditions in terms of u

480


Applying the generalized derivative D to the Heaviside function (see [766, 767]) which results in the identity DHs = N δs , the linearized strains are computed from Equation (4.203) as sym ¯ ∂u ∂ru ∂ϕ ε= + (Hs − ϕ) + ru ⊗ N δs − ru ⊗ . (4.204) ∂X ∂X ∂X In Equation (4.204), δs denotes the Dirac-delta distribution with respect to the surface of discontinuities. It is noteworthy that assuming a displacement field of the type (4.203) already implies some kinematical restrictions. Focusing on the one-dimensional case for now, the linearized strains ε in Ω ± := Ω + ∪ Ω − are computed as ε=

∂ϕ ∂u ¯ − ru . ∂X ∂X

(4.205)

Consequently, the limits of the strain at the surface of discontinuities yield ∂ϕ ∂u ¯ ± ε = lim − ru , with X0 = ∂s Ω. (4.206) ∂X ∂X X ± →X0 Since u ¯, ϕ ∈ C ∞ , the equivalence ' ∂ϕ '' ∂u ¯ − ru ε+ = ε− = ∂X ∂X 'X0

(4.207)

holds. Hence, in contrast to the displacement field associated with the X-FEM, the kinematics corresponding to Ω − and Ω + are not completely independent of one another. For further details, refer to [412, 556]. It is straightforward to show that identity (4.207) is also fulfilled in the three-dimensional case. Numerical Implementation Referring to the finite element method and focusing on constant strain triangular elements which are cut by means of a planar surface ∂s Ω for now, Simo & Oliver [745] proposed an approximation of the displacement field (4.203) of the type ¯ + ru (Hs − ϕ), with u ¯= u=u

n node

¯ ei , ϕ(X) = 1 − (X − X e∗ ) · mi /he , Ni u

i=1

(4.208) cf. [58, 59, 298]. In Equation (4.208), Ni denotes the standard interpolation ¯ ei the nodal displacements at node i, nnode the number of nodes of functions, u the respective finite element, X e∗ the node that connects the two sides of the element which are cut by ∂s Ω and he represents the distance from X e∗ to the opposite side, with unit vector mi (see Figure 4.80). According to Figure 4.80,


481

he

X e∗

N

mi

∂s Ω

Fig. 4.80. Constant strain triangular element cut by means of a planar internal boundary ∂s Ω; see [745]

ϕ(X = X e∗ ) = 1,

ϕ(X = X ei ) = 0, ∀X ei = X e∗ (1 ≤ i ≤ 3),

(4.209)

with the nodal coordinates X ei . Hence, by combining Eqs. (4.208)3 and Equation (4.203), the displacements at node i with coordinates X ei are computed as ¯ (X ei ) ∀X ei . u(X ei ) = u

(4.210)

Consequently, as mentioned before, the function ϕ allows to prescribe the ¯ . Clearly, Ni and ϕ are linear and Dirichlet boundary conditions in terms of u continuous functions and Conditions (4.209) are fulfilled for the interpolation function N∗ associated with node X e∗ . Thus, the identity ϕ(X) = N∗ (X)

(4.211)

holds (see [598]). Originally, the ramp function (4.211) was proposed for the design of a numerical length scale, cf. [596]. The extension to higher order elements is straightforward. Applying the interpolation conditions and considering the most general case, ϕ is designed according to

nΩ ¯+

ϕ=

Ni .

(4.212)

i=1

Here and henceforth, the notation

n ¯+ Ω i=1

denotes the summation over all nodes

¯ + . For bi-linear and bi-quadratic of the respective finite element belonging to Ω shape functions see [559] (Figure 4.81). According to [746, 745], the incorporation of the discontinuous displacement field into the finite element formulation is achieved by adopting the EAS concept, cf. [747, 742]. Hence, the enriched displacement field is modeled in an incompatible fashion. Consequently, it is admissible to neglect the gradient of

482


(a)

(b)

Fig. 4.81. Enhanced discontinuous displacement field ru (Hs − ϕ): (a) bi-linear approximation (2 nodes in Ω + ); (b) bi-quadratic approximation (1 node in Ω + )

the displacement discontinuity, i.e. ∂ru/∂X = 0, and to consider a linearized strain tensor of the type ¯ − (ru ⊗ ∇ϕ)sym +(ru ⊗ N )sym δs . ε := ∇sym u = ∇sym u ! := ˜ε

(4.213)

Interestingly, the local decomposition (4.213) is similar to the additive split ε = εe + εp used in standard plasticity models, cf. [556]. It it noteworthy that although most SDA models in the literature are based on the assumption ∂ru/∂X = 0, the more general case can be derived in a relatively straightforward manner, cf. [499]. 4.2.9.2.2.2 Numerical Implementation This section contains different algorithmic formulations of the strong discontinuity approach. In Subsection 4.2.9.2.2.2, the original finite element model proposed by Simo et al. [746, 745] is summarized first (see also [598]). This implementation is based on the nowadays classical EAS concept and the static condensation technique. Starting from this SDA implementation, a recently suggested algorithmic formulations is discussed next. This method avoids the static condensation technique and results in linearized constitutive equations formally identical to those of classical continuum models such as standard plasticity theory. Numerical Implementation Based on Static Condensation Technique The additive decomposition of the strain tensor (4.213) is formally identical to the corresponding additive split of the by now standard EAS concept (see [747, 742]). Hence, the original implementation of the strong discontinuity approach as proposed by Simo and co-workers was based on the algorithmic formulation of the EAS concept, cf. [746, 745, 600]. According to Simo & Rifai [747], the starting point of the EAS method is represented by the two-field functional


∇¯ η : σ dV =

Ω e

¯ dV + f ·η

¯ dΓ t∗ · η

Γσ

Ωe

γˆ : σ dV

483

(4.214)

= 0.

Ωe

¯ , f , t∗ , Ω e and γˆ denote a compatible, continuous test funcIn Eqs. (4.214), η tion, body forces, prescribed traction vectors acting on the Neumann boundary Γσ , the domain of the considered finite element e and the variations of the enhanced, i.e. incompatible, strains, respectively. Clearly, particularly for constant stress fields, Equation (4.214)2 has to be fulfilled. As a consequence,

!

γˆ dV = 0.

(4.215)

Ωe

However, by adopting a Galerkin-type approximation of the enhanced strains, γˆ is computed as (see Equation (4.213)) γˆ = − (β ⊗ ∇ϕ)

sym

sym

+ (β ⊗ N )

δs

(4.216)

where β denotes the variation of the displacement discontinuity. By applying the definition of δs and restricting attention to constant gradients ∇ϕ for now, Equation (4.215) yields sym sym γˆ dV = 0 ⇔ V e (β ⊗ ∇ϕ) = As (β ⊗ N ) , (4.217) Ωe

with As := ∂s Ω dΓ , i.e. the area of the surface ∂s Ω. Hence, the variation of the enhanced strains has to comply with the restriction ∇ϕ ∈ span(N ).

(4.218)

It is obvious that this condition is, in general, not fulfilled. Probably based on this observation, Simo & Oliver [745] applied, in contrast the classical EAS concept, a Petrov-Galerkin formulation to the enhanced strains. More precisely, γˆ was specified by (see also [59, 472]) γˆ = −

As sym sym (N ⊗ β) + (N ⊗ β) δs . Ve

(4.219)

Inserting Equation (4.219) into Equation (4.215), the identity Ω e γˆ dV = 0 is obtained. It should be noted that alternatively to Equation (4.214)2 and (4.219), some authors, e.g. [597, 598], use the equivalent condition γˆ ˆ · σ dV = 0, (4.220) Ωe

484


with As (4.221) γˆ ˆ = − N + N δs . V So far, the special choice (4.219) seems to be only mathematically justified. However, by combining Equation (4.214)2 and Equation (4.219), the equivalence 1 1 γˆ : σ dV = 0 ⇔ σ · N dV = ts dΓ (4.222) Ve As Ωe

Ωe

∂s Ω

is derived, cf. [745, 598]. Hence, the L2 orthogonality condition (4.214)2 is equivalent to the weak form of traction continuity between Ω ± and ∂s Ω. As a consequence, the special choice (4.219) is not only mathematically but also physically sound. The approximation is completed by employing a standard Bubnov-Galerkin method to the continuous (globally conforming) part of the displacement field, i.e., ¯≈ u

n node i=1

¯ ei , Ni u

¯= η

n node

¯ ei . Ni η

(4.223)

i=1

In line with the X-FEM discussed before, a cohesive model can be incorporated by postulating a traction-separation law of the type ts = ts (ru). Clearly, a fully open macro crack is characterized by vanishing tractions, i.e., ts = 0. ¯ and ru follow from solving Eqs. (4.214). The unknown displacement fields u Usually, this is done simultaneously by employing the concept of static condensation. Hence, only the conforming part of the displacement field, namely ¯ , is present at the global level. u Numerical Technique

Implementations

without

Employing

Static

Condensation

A new algorithmic formulation for the numerical implementation of locally (incompatible) embedded strong discontinuities was proposed in [557, 558]. A similar approach was recently presented in [139]. However, the formulation [139] is restricted to Constant Strain Triangle (CST) elements. In contrast to the strong discontinuity models [745, 598, 59, 472, 57, 828, 413], the numerical formulations suggested in [139, 557, 558] are not based on the static condensation technique. More precisely, the parameters defining the displacement jump within the finite element are condensed out at the material point level. These approaches result in linearized constitutive equations formally identical to those of standard (local) continuum models. Therefore, the by now well-known return mapping algorithm [743, 741] can be applied to solve the resulting nonlinear set of differential equations. Only minor modifications of material subroutines designed for standard plasticity models are required. Clearly, this represents an important advantage of the finite element formulations proposed in [139, 557, 558] compared to the original one presented in [746, 745, 598].


485

Fundamentals One of the key ideas of the novel implementation of the SDA is the reformulation of the weak form of traction continuity (4.222), cf. [552, 551, 555]. More precisely, introducing the notation 1 (•) := e (•) dV (4.224) V Ωe

Equation (4.222), can be re-written as φ := ||t − ts || = 0.

(4.225)

Here, t = σ · N . Accordingly, the weak form of traction continuity is fully equivalent to the necessary condition of yielding known from standard plasticity models. Hence, it is possible to derive a traction-separation based constitutive model fully analogously to classical plasticity theory. For that purpose, an admissible stress space Et in terms of the average traction vector t can be defined, i.e., C D Et := (t, q) ∈ R3 × Rn | φ(t, q) ≤ 0 . (4.226) In contrast to classical plasticity theory, this space depends now on the average stress vector t, instead of the local stress tensor σ. Et is defined uniquely by means of a yield (failure) function φ(t, q), which depends on the traction vector t and a vector of stress-like hardening/softening parameters q = q(α) conjugated to a displacement-like internal variable α. For the special choice α = ru, φ(t, q) = ||t − q|| and q = ts , φ = 0 is equivalent to the condition of traction continuity. Hence, the condition of traction continuity is included in the more general space of admissible stresses (4.226). It is noteworthy that Definition (4.226) allows for the modeling of isotropic hardening/softening as well as the kinematical counterpart (see [550, 553]). The model is completed by postulating physically sound evolution equations. In line with standard plasticity theory, they may be derived by maximizing the dissipation resulting in normality rules, i.e. ˙ = λ ∂q φ ru˙ = λ ∂t φ and α

(4.227)

with λ being the plastic multiplier. The analogy to classical plasticity theory characterized by ε˙ p = λ ∂σ φ

˙ = λ ∂q φ and α

(4.228)

is remarkable. The evolution Eqs. (4.227) can be generalized by introducing a hardening potential h and a plastic potential g leading to ru˙ = λ ∂t g

˙ = λ ∂q h. and α

(4.229)

486


Numerical Implementation Since the re-formulated SDA as described in the previous subsection is formally identical to classical plasticity theory, algorithms originally designed for standard plasticity models can be applied with only minor modifications necessary. In this section, an adapted return-mapping algorithms is briefly presented, cf. [743, 741]. Further details may be found in [552, 551]. Starting with the analogy between Eqs. (4.227) and Eqs. (4.228) and in line with the classical return-mapping algorithm, a backward Euler integration is considered and the residuals ' ' ' Rru := −run+1 +run +Δλ ∂t g ' , Rα := −αn+1 +αn +Δλ ∂q h'n+1 n+1

(4.230) are introduced. Here, the supscript (•)n is associated with time step n. Following the solution procedure of the return mapping algorithm, the unknown displacement jump run+1 , the internal variables αn+1 , together with the plastic mul t tiplier Δλ = tnn+1 λ dt are obtained by solving the set of nonlinear equations Rru = 0,

Rα = 0,

φn+1 = 0.

(4.231)

For that purpose, Newton’s method can conveniently be adopted. Once the solution is computed within the prescribed tolerance, a linearization of the residuals yields the algorithmic tangent required for an asymptotically quadratic convergence. Further details may be found in [743, 741, 552, 551]. 4.2.9.2.2.3 Numerical Example: 3-Point Bending Problem In this paragraph, the performance of the SDA is demonstrated by means of a finite element analysis of a notched concrete beam, cf. [559]. The geometry, the loading and boundary conditions of the beam and the material parameters

48.45

4.7

48.45

u,F t

= 0.127

[m]

11 00

E = 4.36 · 10

3

[KN/cm2 ]

25.4 7.8 50.55

0.5 101.6

50.55

ν

= 0.2

ftu = 0.4

[KN/cm2 ]

Gf = 1.195 · 10−3 [KN cm/cm2 ]

Fig. 4.82. Numerical study of a notched concrete beam: dimensions (in [cm]) and material parameters


487

are depicted in Figure 4.82. In this figure, t, E, ν, ftu and Gf denote the thickness of the slab, the Young’s modulus, the Poisson’s ratio, the maximal uniaxial tensile strength and the fracture energy (in tension), respectively. This problem represents a standard benchmark for cracking models, cf. [248, 506, 534, 827, 413, 550]. For the analysis of cracking in brittle materials the normal vector N is computed from the direction of the maximum principal stress. After crack initiation the traction-separation law is defined by the failure function φ(t, q(α)) = tn + κ2 t2m + −q(α).

(4.232)

with tn and tm being the normal and the tangential component of the traction vector t. The softening behavior after onset of cracking is assumed to follow the exponential law ftu q(α) = ftu exp −α , (4.233) Gf with the internal variable α := ru·N representing the crack width. The model is completed by associative evolution equations. Three different numerical implementations are analyzed and compared to each other in what follows: • • •

Single fixed crack model: one crack per element having a fixed direction, ˙ =0 i.e., N Rotating crack model: one crack per element which is allowed to rotated, ˙ = 0 i.e., N Multiple fixed crack model: two orthogonal cracks per element having a ˙ =0 fixed direction, i.e., N

First, a numerical analysis based on the single fixed crack approach is performed. During the first loading stage and independently of κ2 , a vertically oriented crack propagating along the symmetry line starts to open. However, at a later stage of deformation locking effects resulting in convergence problems occur. The smaller κ2 , the earlier these problems arise. Even in the case κ2 → ∞ this stress-locking is not eliminated, cf. [552]. The same phenomenon can be observed if constant strain elements are used. It is noteworthy that the numerical problems reported are not restricted to the specific strong discontinuity model. According to [244], the Extended Finite Element Method (X-FEM) (see [545]) shows the same problem. Furthermore, this locking effect is also present if quadratic elements (displacement approximation) are used, cf. [559]. The reason for the stress-locking effect is a wrong prediction of the orientation of the macro-crack. Independently of κ2 , the direction of the maximum principal stress in the uncracked part of the beam in the vicinity of the crack tip is computed to be orthogonal to the line of symmetry at an early stage of the analysis. As a consequence, the resulting crack surface is vertically oriented. However, at a later stage of deformation, this direction rotates about

488


Force F [kN]

15

10

5 multiple SDA rotating SDA 0 0

(a)

0.01

0.02

Displacement u [cm]

(b)

Fig. 4.83. Numerical study of a notched concrete beam using the proposed multiple crack concept and the rotating crack approach: (a) load-displacement digram, (b) deformed structure at u = 0.025 cm (1000-fold magnification of displacements)

90◦ . Hence, a horizontally oriented micro-crack starts to open. Assuming a fixed crack concept, this direction is held constant during subsequent loading. Consequently, a wrong orientation of the resulting macro-crack is predicted. Next, the notched concrete beam is re-analyzed using a rotating crack approach. Since in this case, N is parallel to the maximum principal stress, the shear component of the traction vector vanishes and hence, without loss of generality, κ2 (see Equation (4.232) can be considered as κ2 = 0. Thus, a Rankine model is adopted. In contrast to the analyses based on the fixed crack concept, no convergence problems occur now even in the post-peak regime. The resulting structural response is given in Figure 4.83. Finally, a numerical analysis based on a multiple fixed crack model is performed, i.e., two orthogonal cracks per element are allowed which do not rotate ˙ = 0). Analogously to the rotating SDA, no convergence problems occur (N if intersecting cracks are allowed. However, as illustrated in Figure 4.83, the structural response computed from the multiple SDA is slightly stiffer compared to the results obtained from the rotating SDA. This stems from the fact that in the multiple SDA both cracks are assumed to be uncoupled, i.e. q (1) = q (2) . Consequently, the total fracture energy predicted by the multiple SDA is greater than that of the rotating SDA, cf. [552]. 4.2.9.3 Crackgrowth Criteria Authored by Klaus Hackl and Markus Peters The consideration of crack growth is the main problem which has to be taken into account for the estimation of the durability of a part or the whole construction. After Griffith has published a energy based criterion in [325] for brittle materials a criterion was published in [405] that was able to capture


489

the properties of ductile materials by using the concept of the strain energy release rate. In this chapter we focus on the linear elastic fracture mechanics. So mainly the following crack growth criteria are standard: • • • • •

the the the the the

maximum hoop stress [259], maximum principle stress [514], minimum of the elastic energy density [739], crack direction criterion using the assumption KII = 0 [209] maximum driving force [476]

All of them disregard the SIF KIII . So they are basically practical for two dimensional problems. In the following sections three classical criteria and their implementations are presented because the are of specific importance for the linear elastic fracture mechanics in two dimensions: 1. the criterion of maximum hoop stresses 2. the crack direction criterion using the assumption KII = 0 3. the Griffith criterion which is based on the principle of minimum potential energy These criteria are used to simulate some examples in Section 4.2.9.4. 4.2.9.3.1 Hoop Stresses The stresses can be computed very easily at a point x in the following way ˆ B (x) u ˆ. σ(x) = C Herein these definitions are used ⎧ ⎪ ⎪ ⎪ DN (x) for the FEM ⎪ ⎪ ⎡ ⎤ ⎪ ⎪ ⎪ ⎪ ⎪ N1 (x) F(x) ⎥ ⎪ ⎪ ⎨ ⎢ ⎢ ⎥ ⎢ ⎥ B (x) = . N (x) F(x) ⎢ ⎥ 2 ⎪ ⎢ ⎥ ⎪ D for the XFEM, and F from equation 4.154 ⎪ ⎢ ⎥ .. ⎪ ⎪ ⎢ ⎥ ⎪ . ⎪ ⎢ ⎥ ⎪ ⎪ ⎣ ⎦ ⎪ ⎪ ⎪ ⎩ Nn (x) F(x) The hoop stress can now be calculated by using σT =

σxx − σyy σxx + σyy − cos (2 θ) − τxy sin (2 θ) 2 2

at points with constant distance to the crack tip.

(4.234)

490


From these stresses the angle θ has to be found so that the stresses become a maximum. If the maximum stress σT,max is bigger than a threshold the crack will grow into the calculated direction. To improve the results the stresses can be integrated over a region σm = w σ dΩ Ω

using a weighting function 1 r2 w= exp − 3 2 3 2rm (2π) 2 rm This computation was described in [241] (cp. [827, 414]). The parameter rm defines how rapidly the weighting function decreases with increasing distance r to the crack tip. Typically values of 1.5 times the characteristic element length are used for rm . Using this criterion the crack growth direction can be calculated but the change of the crack length is not part of the criterion. Here a constant value for Δr as material constant is used. The same criterion can be defined depending on the SIF KI and KII ⎛ ⎛ ⎞⎞ 2 1 K K I I Δθ = 2 arctan ⎝ ⎝ ± + 8⎠⎠ . (4.235) 4 KII KII This equation implements the same criterion of the maximum hoop stresses (cp. [544, 242]). In contrast to the first possible implementation here the change of crack length can be defined in terms of the SIF KI Δr = Δr(KI ). 4.2.9.3.2 Mode-I-Crack Extension This criterion is based on the assupmtion that the crack will grow in a direction so that after crack growth a Mode-I-Crack occurs. It means that the value of the SIF KII has to vanish. This can be implemented in two different ways: 1. The SIF can be computed at a kink of the crack by using the limit of a crack extension which goes to zero (cp. [45] and Figure 4.84) with ⎛ ⎞ ∗ K ⎜ I ⎟ lim ⎝ ⎠. r→0 ∗ KII In this case (cp. [476]) the SIFs are computed by using


491

∗ KI∗ , KII

lim

r→0

θ KI , KII

Fig. 4.84. Sketch for the computation of the SIF for a kinking crack with r → 0

r

θmax

KI∗ (θ) ∗ KII (θ)

θmax

Fig. 4.85. Schematic figure for the calculation of the SIF with constant radius for kinking cracks

⎛ ⎜ ⎝

⎞

⎛

⎞⎛

⎞

KI∗

⎟ ⎜ FI,I (θ) FI,II (θ) ⎟ ⎜ KI ⎟ ⎠=⎝ ⎠. ⎠⎝ ∗ FII,I (θ) FII,II (θ) KII KII

The functions Fi,j (θ) which were used are taken from [45]. 2. The SIF can be computed at a constant value for r with varying the angle in a range ±θmax . So the local minimum of the SIF |KII | can be found. In Figure 4.85 the region is pictured. At the circular arc the values of KII can be computed at different virtual crack tip positions. By using an interpolation polynomial all positions can be approximated and the position where KII = 0 can be obtained. As an example the progress of KII is shown in Figure 4.86 for a three point bending test and the kinking angle −25 ≤ θmax ≤ 25

(4.236)

492

4 Methodological Implementation 8 8

∗ KII (θ)

∗ |KII (θ)|

6

6

4

4

2

-20

-10

10

20

θ

2

-2

-4 -20

-10

10

20

θ

Fig. 4.86. Sketch of KII (left) and |KII | (right) depending on the angle θ for a three point bending test

and a radius of r = 0.5. Because of the fact that the values of KII are evaluated at some points inside the interval 4.236 the position of KII = 0 is not found in general. So it has to be interpolated. Again the change of crack length can be defined in terms of the SIF KI Δr = Δr(KI ). 4.2.9.3.3 Minimum Energy Criteria which are based on the principle of minimal energy are already discussed in [325, 613, 364, 475, 859]. Here it is assumed that crack growth will occur to that point which describes the minimum of the total energy for a certain crack growth. For linear elastic fracture mechanics in homogeneous materials the strain energy is 1 Πint = − σ : dΩ 2 Ω and can be implemented in a FE program using the nodal force and nodal displacement vector Πint = −

1 f u. 2

Using a virtual crack extension the energy can be defined as function of the crack changes. The energy which is necessary to let the crack grow to a specific position is added to the determined energy Πint . In [325] it is outlined that for crack growth energy in form of Πc (r, ϕ) = f1 r

(4.237)

is necessary. Because of the fact that using Griffith theory only unstable crack growth or no crack growth can be simulated the criterion is extended to


493

Πges

r

Fig. 4.87. Energy function Πtot for a three point bending test, left: as 3D-Plot, right: as 2D-Plot plotted for constant angles depending on the radius

Πc (r, ϕ) =

f1 r 2 + f3 ϕ2 1 + f2 r

(4.238)

so that also stable crack growth can be simulated. This criterion reduces to the Griffith criterion for big values of Δr and if f3 is chosen to be zero. This phenomenological extension of the Griffith potential makes it possible to capture local effects and dynamic processes in a very easy manner which were unconsidered in linear elastic fracture mechanics otherwise. In addition to that using the extended potential the simulation is numerically more stable. The energies are added to the total energy Πtot = Πint + Πc .

(4.239)

In Figure 4.87 an energy function is plotted for a three point bending test. The virtual crack extension is from the range −25 ≤ ϕ ≤ 25

Δ rmax = 0.95.

(4.240)

4.2.9.4 Examples Authored by Christian Becker 4.2.9.4.1 Double Notched Slab A double notched slab subjected to a combined loading of shear and normal load is analyzed numerically. The corresponding experiment was realized by [584]. The numerical results, in particular crack topology and load-deflection curve will be compared to the experimental test results. The geometry, relvant material data as well as the finite element discretization are illustrated in Figure 4.88. Because of the quasi two-dimensional problem the approximation of the displacement field is chosen as:

494


25 [mm]

Fn , un N Young’s modulus E = 30000 [ mm 2]

Poisson’s ratio ν = 0.2 [−] 200 [mm]

FS

5

N ] fracture energy Gf = 0.11 [ mm N tensile strength ftu = 3.0 [ mm 2]

thickness t = 50 [mm]

FS

Fn , un 200 [mm] Fig. 4.88. Crack simulation of a double notched slab: System, material data and finite element mesh

u ≈ u3,3,1 .

(4.241)

This represents a slab-like solid formulation (see Section 4.2.4.3) with a high inplane approximation degree and the minimum degree in thickness direction. Hence, computational effort is saved here compared to full cubic approximation u3,3,3 . Firstly, the slab is loaded with the aforementioned shear load of Fs = 10 kN is applied. By keeping the load level of the shear load, the normal load Fn is applied displacement driven subsequently. Figure 4.88 presents the visualization of both obtained curved cracks, resulting from the combined loading scenario. The visualization of the crack path is realized implicitly by plotting the φ = 0-level set. This scalar value represents the shortest distance of a continuum point towards the crack surface. Hence, a point where φ = 0 is a point of the crack surface. Figure (4.89,left) shows a comparison of the determined crack topology of the slab-like solid formulation and of a formulation by [241] as well as an experimentally determined region of obtained cracks (grey). The determined crack topology fits very well with the results of [241] and the experiment. Figure (4.89,right) shows the corresponding load reflection curves of the numerical simulations of [277, 241], the presented formulation and of the experiment. All numerical analyses show the same result but differ significantly from the experimental tests. In [241] a possible reason for that is an overestimation of the proposed fracture energy. 4.2.9.4.2 Anchor Pull-Out To illustrate the capability of the proposed three-dimensional implementation of the extended finite element method, a relatively coarse discretized anchor


495

φ=0

φ=0

Fig. 4.89. Crack simulation of a double notched slab: Visualization of the crack topology by the φ = 0-level set

18 X-FEM p3,3,1 [Dumstorff 05]

16 14 F [kN ]

12

X-FEM p3,3,1 [Dumstorff 05] [Feist 04] experiment

6 4 2 0.02 0.04 u [mm] 0.08

0.1

Fig. 4.90. Crack simulation of a double notched slab: Comparison of crack topology (left), Comparison of load-displacement curves (right)

pull-out test should be investigated numerically. Figure 4.91 shows the analyzed structure and the corresponding finite element mesh (NE=996). The relevant material parameters ,e.g. the tensile strength or the fracture energy,

496


400 [mm]

z = 0

Z Y m 0[ 40

m

]

400 [m m]

X

[m 80

m

]

Fig. 4.91. Bumerical investigation of crack propagation of an anchor pull-out test: System and finite element mesh (N E = 996)

are the same as for the numerical example of the double-notched slab. For symmetry reasons only a quarter of the structure is discretized. To represent the anchor plate, elements are removed from the mesh. The pull-out of the anchor is simulated by subsequently applying a constant displacement at the bottom side of the elements being atop of the removed ones. For simplicity the pole of the anchor is not discretized. The simulation is realized by applying a softening traction-separation law according to [828]. The approximation degree of the displacement field for the compact structure is chosen to: u = u2,2,2

(4.242)

The concrete structure is completely clamped at the bottom side. At the top surface of the concrete structure there is an area which is supposed to not move vertically to simulate a counter pressure area. Figure (4.92,left) shows the resulting crack surface of the pull-out test. It is clear from the figure that the crack surface has a three-dimensional character and is expanding conically from the anchor plate towards the outer boundaries of the structure. Furthermore it can be seen that the crack surface is C0 -discontinuous. But in this example there are no too excessively big gaps between the crack surface segments. Figure (4.92,right) illustrates the displacement component uz in the pullout direction. The jump in the displacement component uz is clearly identified at the position of the determined crack surface, indicating a complete separation of the upper and lower part of the concrete structure in the vicinity of the crack surface. Also the distribution of the stress component σ 33 (see Figure (4.93)) indicates this separation by the propagation of the stress concentration towards the boundaries of the concrete structure. Figure 4.94 shows the load displacement curve of the numerical test. The discrete steps in the curve represent the cracking of single elements. These jump depend on the size of

4.2 Numerical Methods −0.01

uz [mm]

497 0.32

φ=0

8.5

8.0

σ 33 [N/mm2 ]

σ 33 [N/mm2 ]

Fig. 4.92. Numerical investigation of crack propagation of an anchor pull-out test: Crack topology (left), Displacement u3 in pull-out direction (right)

−40

−80

Fig. 4.93. Numerical investigation of crack propagation of an anchor pull-out test: Stress component σ 33 at the beginning of the test (left), Stress component σ 33 at the end of crack process (right)

the finite elements due to the assumption of elementwise crack propagation. At first, the curve describes a linear elastic behaviour of the structure up to a prescribed displacement of uz ≈ 0.04 [mm]. From this point, a clearly nonlinear relation between load and displacment is observed without showing a global softening behaviour. This nonlinear relation continues up to a displacement of uz ≈ 0.21 [mm]. From this point the response of the structure is linear. The reason for this response is, that in the beginning of the loading process, the load is primarly carried by the material around the anchor plate and from there distributed in the neighbouring continuum. When the

498


350 300

F [kN ]

250 200 150 100 50 0 0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 u [mm]

Fig. 4.94. Numerical investigation of crack propagation of an anchor pull-out test: Load-displacement curve

displacement reaches uz ≈ 0.21 [mm] the carck propagation nearly stops due to the fact that most of the load is carried by the area of counter pressure. This is evident from Figure(4.93,right) that shows the strongly increased value of the stress component σ 33 at the counter pressure support. On the contrary in the beginning of the cracking process (figure(4.93,left)) there is nearly no compression at the area of counter pressure because of the load carrying of the concrete in the vicinity of the anchor plate. When the displacement reaches uz ≈ 0.21 [mm], the load is primarily induced to the counter pressure supports by the nearly fully separated upper part of the concrete structure. This bearing behaviour of the system is primarly characterized by the position of the crack surface that is running underneath the pressure supports. In the case that the crack reaches the upper surface of the concrete structure being in front of the supports, there is going to be a structural softening of the concrete block. This interpretation is similiary to the one in [301] where a similiar numerical test is performed. As can be further deduced from Figure 4.93 there is a high compression atop of the anchor plate, which would also lead to compression failure of concrete. This does not occur here, because this analysis only takes failure of concrete in tension into account. 4.2.10 Substructuring and Model Reduction of Partially Damaged Structures Authored by Christian Rickelt and Stefanie Reese The objective of this section is to present an efficient strategy to cope with demanding dynamical simulations of complex structures. We are in particular interested in long term calculations like life cycle investigations. Our


499

ansatz emanates from the idea that a structure only comprises locally distributed damaged zones. The proposed method is an evolution of the classical Craig-Bampton approach to partially damaged problems and incorporates an advantageous decomposition of the entire structure into linear and nonlinear segments. The former are reduced by model reduction techniques. In the nonlinear components the deterioration of the material is simulated by a continuum damage model for ductile damage phenomena of metals. This material model is included into a new solid beam finite element formulation Q1SPb based on reduced integration with hourglass stabilisation. The section closes with an example to validate the proposed strategy and to determine the approximation error caused by the model reduction of the linear components. 4.2.10.1 Motivation and Overview The importance of simulation tools for the design and lifetime estimation of complex structures increases noticeably. Highly demanding tasks and requirements on the accuracy of numerical computations necessitate powerful simulation techniques and more elaborate models. Since in general these models cannot be solved analytically the underlying ordinary or partial differential equations are often discretised and solved by the finite element method. Dynamic problems require beside the spatial discretisation as well a discretisation in time. This can be accomplished by implicit or explicit time integration schemes. While the former lead to large, sparse equation systems, whose solution is memory and time consuming, the disadvantage of the explicit methods is that they are only conditionally stable. Accordingly its time step size is limited by a critical time step. As a consequence the latter schemes are not suitable for long-term computations. Strategies to solve such complex long-term calculation numerically efficient have been a topic of research in the field of engineering technologies and mathematics. More related to the former one are the condensation methods and the staggered schemes. The domain decomposition methods are mostly researched in mathematics while the model reduction techniques are established in both communities. The idea of model reduction is to transform the large equation system into a small, dynamically equivalent substitute model. Powerful model reduction methods exist for the solution of linear and weakly nonlinear systems. Highly nonlinear problems like the evolution of damage cannot be solved. [761] and [808] employ nonlinear model reduction techniques to reduce geometrically nonlinear problems and fluid-structure interactions, respectively. For an overview of model reduction methods in general refer to the publications of [111, 761, 49, 50, 651, 288, 76, 51, 451, 287, 236, 583, 485, 813]. Linear model reduction methods can be classified into three different groups: the singular value decomposition (SVD)-based methods, the Krylovbased methods and the condensation methods. Out of the group of SVDbased model reduction two well-known methods are the Modal Reduction

500


(MODRED) (see the standard text books [195, 203]) and the Proper Orthogonal Decomposition (POD) (see [121, 384, 755]). Two important members of the Krylov-based methods are the Load-Dependent Ritz Vectors (LDRV) (see [328, 203, 585]) as well as the Pad´ e-Via-Lanczos algorithm (PVL). In this publication a symmetric version for undamped second order systems, called symmPVL, is used (see e.g. [288, 76]). The above mentioned group of condensation methods comprise model reduction methods which condense the inner degrees-of-freedom (dofs) and conserve the physical interface dofs. This idea can be structured into static as well as dynamic condensation methods. According to [620] the latter yield exactly reduced dynamical systems which even for linear problems result in nonlinear functions which depend on the frequency. Hence they are not numerically efficient. In contrast the static condensation of [333] is suitable to only a limited extent for the model reduction of linear dynamic systems. Corresponding to the text book of [651] component mode synthesis techniques (CMS techniques) establish a significant extension of Guyan’s reduction to methods with hybrid transformation matrices. Initialised by the classical papers of [401] and [216] (see also the review article of [215]), CMS techniques with fixed interfaces have been developed within the scope of reduction methods for large structural dynamic models and the design of single dynamical components of challenging structures. These methods are based on the superposition of different contribution of the deformations. Hence they are only valid for linear problems. For further information on CMS techniques see the overview publications of [217, 206, 498, 722, 511]. Alternatively the objective of staggered algorithms (see for example [548]) is to solve multi-field or dynamic problems which are discretised by different time integration schemes (implicit/explicit) and varying time step lengths (subcycling) within each subdomain (see the classical papers of [108] and [107]). An interesting approach of an explicit-implicit multi time step method for nonlinear structural dynamics which prescribes the continuity of velocities at the interface and uses a dual Schur formulation has been published by [323]. [274] and [275] extend this ansatz successively to nonmatching meshes and linear as well as nonlinear model reduction techniques. The primary interest of domain decomposition methods, grouped into overlapping (”Schwarz methods”) and non-overlapping (”iterative substructuring methods”) methods, is the development of highly efficient parallelised iterative solution techniques. These usually result in conjugate gradient schemes. The latter are subdivided into the Neumann-Neumann- (Balanced Domain Decomposition (BDD), [515]), Dirichlet-Neumann- and Dirichlet-Dirichlet-Algorithm (Finite Element Tearing and Interconnecting Method (FETI), [273]). Most of the publications deal with linear problems. Applications to geometrically nonlinear problems can be found in the papers of [218] and [272].


501

Structural Dynamics

Modelling (FEM)

Linear Components

Model Reduction

Nonlinear Components

Substructure Technique

Damage Modelling

Fig. 4.95. Concept for the efficient simulation of dynamic, partially damaged structures by means of model reduction and substructuring

4.2.10.2 Concept In this section an efficient strategy for dynamic long-term simulations of complex structures with local nonlinearities by means of the finite element method is presented. Our approach emanates from the idea that engineering structures are usually designed in such a way that most components of a structure behave linearly elastically and under the assumption of small deformations. Thus undesired effects such as the evolution of damage or possibly occurring large deformations are localised in small parts of a structure. This ansatz, depicted in Figure 4.95, enables to decompose any discretised structure strictly into its linear and nonlinear components. In the latter the evolution of ductile damage of metals is considered. The damage model is based on the void growth model of [691, 690]. Further we assume that the evolution of damage is only influenced indirectly by the dominating linear subsystems. Hence, to increase the efficiency of the strategy, the latter are reduced by model reduction methods in conjunction with the Craig-Bampton method – one of the widely used CMS substructure techniques. Beside the common Modal Reduction, linear model reduction methods of superior accuracy – the Pad´ e-Via-Lanczos algorithm (PVL), the Load-Dependent Ritz Vectors (LDRV) and the Proper Orthogonal Decomposition (POD) – are employed. Finally the substructuring of the total structure into reduced linear and nonlinear components is exploited. Instead of solving a large monolithic equation system the system response of the redundant interfaces of the components is computed. Subsequently the inner dofs of all components are calculated. One advantage of the presented concept rests upon the beneficial combination of well-known and robust linear model reduction methods, the CMS

502


technique, the material as well as the efficient element formulation. Another important aspect is the numerical implementation. For this purpose the mathematical development environment Matlab and the finite element program Feap are linked by the interface Feapmex1 . 4.2.10.3 Derivation of a Substructure Technique for Nonlinear Dynamics In this section the Craig-Bampton method is summarised. Afterwards the linear model reduction methods and the employed substructure technique are discussed. 4.2.10.3.1 Craig-Bampton Method The concept of CMS techniques has been developed [401] and later rewritten and simplified by [216] to analyse complex structural systems decomposed into interconnected components with fixed interfaces. The Craig-Bampton method superposes two different fractions of the motion: the static or constraint modes Ψic of the matrix ΨT := ΨTic ITcc are defined as the static deformation of a structure when a unit displacement is applied to one interface dof while the remaining interface dofs are restrained. The matrix Icc herein is an identity matrix of dimension c × c. The indices i and c indicate the inner and the interface dofs, respectively. The second fraction are the k (k i) remaining inner dynamical or so-called normal modes φj , j = 1, . . . , k of the fixed subsystem s. They are stored column by column into the matrix ⎡ ⎤ ⎡ ⎤ φ Φ , . . . , φ k⎥ ⎢ 1 ⎢ ik ⎥ Φr := ⎣ (4.243) ⎦=⎣ ⎦ . Ock Ock The matrix Ock is a zero matrix of dimension c × k. Usually the modes Φik are represented by eigenvectors. In this contribution the LDRV-, POD- or symmPVL-vectors of alternative model reduction methods of superior accuracy, as presented in section 4.2.10.3.2, are utilised. With these modes the physical coordinates u can be transformed by the relation ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎢ ui ⎥ ⎢ Φik Ψic ⎥ ⎢ qk ⎥ (4.244) ⎣ ⎦=⎣ ⎦⎣ ⎦ uc Ock Icc qc ! ! ! (s) u(s) q (s) VCB into generalised coordinates q. VCB symbolises the well-known Craig-Bampton transformation matrix of one single component s. Transforming with 1

For further information see the following homepage: www.cims.nyu.edu/ dbindel/feapmex/feapmex/doc/feapmex.html


503

the relation (4.244) the linear equation of motion of one component s by an orthogonal projection (s) T

V CB

(s)

(s) T

(s)

(s) T

¨ (s) (t) + VCB K(s) VCB q (s) (t) = VCB b(s) f (t)(s) M(s) VCB q ! ! ! (s) (s) Mrr Krr b(s) r (4.245)

leads for linear undamped systems to a component of smaller dimension in which the interface dofs uc = q c are conserved in physical coordinates. In the latter equation M, K, b and f (t) are the mass matrix, the stiffness matrix, the load distribution and the loading function. The index r denotes the dimension of the reduced component. 4.2.10.3.2 Model Reduction of Linear Dynamic Structures Besides our objective to save computational effort by decomposing a structure into its linear and non-linear parts the linear substructures are approximated by dynamically equivalent subsystems. Within the Craig-Bampton method the fixed interface normal modes of each component are defined as a reduced set of modes by restraining all boundary dofs. These modes Φik form part of the Craig-Bampton transformation matrix VCB (see section 4.2.10.3.1). In this contribution we will derive the model reduction of linear dynamic systems within a general framework. For simplicity we leave out the superscript s. We start from the ansatz ui = Φik q k

(4.246)

in which ui is the displacement vector, q k is the vector of the reduced system and Φik is a rectangular projection matrix of the dimension (i × k). This approach is inserted into the linear equation of motion ¨ i = bi f (t) . Kii ui + Mii u

(4.247)

This leads to a set of linear equations of reduced dimension ¨ = (Φik )T bi f (t) . (Φik )T Kii Φik q + (Φik )T Mii Φik q ! ! ! Kkk Mkk bk

(4.248)

In the following four different projection-based model reduction methods are summarised. These methods are the Modal Reduction, the Proper Orthogonal Decomposition (POD), a symmetric Pad´ e-Via-Lanczos algorithm (symmPVL) and the Load-Dependent Ritz Vectors (LDRV). 4.2.10.3.2.1 Modal Reduction Modal Reduction, also known as Modal Truncation, is the most simple and popular model reduction method. The idea is to solve a subset of the

504


generalised eigenvalue problem in which Φik = [φ1, φ2 , · · · , φk ] is the reduced modal matrix and Λkk = diag ω12 , ω22 , · · · , ωk2 is the reduced eigenvalue matrix. After a mass normalisation (Φik )T Kii Φik = Λkk ,

(Φik )T Mii Φik = Ikk

(4.249)

a reduced decoupled differential equation system is obtained: ¨ k = bk f (t) . Λkk q k + Ikk q

(4.250)

4.2.10.3.2.2 Proper Orthogonal Decomposition A second possibility is the POD method. This method is also known as empirical eigenvectors, Karhunen-Lo` eve expansion, principle component analysis, empirical orthogonal eigenvectors, etc. An overview of nomenclatures used in the literature and areas of application are given e.g. in [121]. The mathematical basis for the POD method is the spectral theory of compact, selfadjoint operators which is explained e.g. in the standard text book of [384]. One problem of this ansatz is that even for small systems the eigenvectors of a large spatial covariance matrix have to be calculated. One approach to lower the computational costs is known as the “method of snapshots” ([755]). In this case each POD basis vector φl =

m

βj w j

l = 1, . . . , k

(4.251)

j=1

is generated out of m uncorrelated zero-mean “snapshots” wj . In the latter ¯ describes the deviation of the “snapshot” uj from their equation wj = uj − u ¯ . βj are unknown coefficients which have to be determined. temporal mean u After some derivations and using the assumption that the investigated process is ergodic (see e.g. [536, 384]) only a reduced eigenvalue problem of dimension m × m 1 W T W β l = λl β l m

with

W = [w 1 , · · · , wm ] ,

(4.252)

in which W contains the m zero-mean “snapshots” has to be solved. The k basis vectors of the POD φl = W βl

,

(4.253)

corresponding to the eigenvalues λ1 > λ2 > · · · > λl > · · · > λk , result from a linear combination of the zero-mean “snapshots”. 4.2.10.3.2.3 Padé-Via-Lanczos Algorithm The Pad´ e-Via-Lanczos algorithm and the Dual Rational Arnoldi method belong to the Krylov-based model reduction methods. This


505

system-theoretical approach for first order differential equations can also be applied to second order systems. A differential-algebraic equation system ¨ i (t) = bi f (t) Kii ui (t) + Mii u

y(t) = ci ui (t)

(4.254)

is converted by the Laplace transformation to the transfer function H(s) = ci [s2 Mii + Kii ]−1 bi

(4.255)

Here the equations are given for a single input single output (SISO) systems. The measurement vector ci of the dimension (1 × i) relates the displacement vector ui (t) to the measured output y(t) of the system. The transfer function (4.255) is re-written and expanded around an expansion point σ 2 into a power series (Laurent or Taylor series) H(s) = ci [(Kii − σ 2 Mii ) + (σ 2 + s2 ) Mii ]−1 bi = ci [Iii − (Kii − σ 2 Mii )−1 (−σ 2 − s2 )Mii ]−1 (Kii − σ 2 Mii )−1 bi ∞ = μj (−σ 2 − s2 )j j=0

(4.256) The coefficients μj = ci ((Kii − σ 2 Mii )−1 Mii )j (Kii − σ 2 Mii )−1 bi are termed “moments”. Thus the method is also called “moment matching” in the literature. An important observation for the Pad´ e approximation presents the fact that the moments can be computed in a numerically stable fashion by Krylov subspace methods like the Lanczos or the Arnoldi method. For the special case cTi = bi and symmetric, positive definite matrices Mii and Kii [290] show that the reduced systems are stable. According to [486] for mechanical problems purely imaginary expansion points σ = jωc are chosen (ωc is the angular frequency in the centre of the interesting frequency range). Employing a Cholesky decomposition Kii − σ 2 Mii = Nii NTii and −1 −T the relations ri = N−1 the transfer function is ii b and Gii = Nii Mii Nii transformed into H(s) = rTi [Iii + Gii (σ 2 + s2 )]−1 r i

.

(4.257)

Using the Lanczos algorithm the matrix Gii is approximated by a tridiagonalised matrix Tkk of the dimension k × k. In the time domain the reduced system results in ¨ k (t) + [Ikk + σ 2 Tkk ]q k (t) = rk f (t) Tkk q y(t) =

rTk q k (t)

(4.258)

506


The reduced vector r k = (Φik )T r i is computed according to the projection (4.248). The proposed algorithm for symmetric positive definite system is termed in the following symmPVL. 4.2.10.3.2.4 Load-Dependent Ritz Vectors The method of Load-Dependent Ritz Vectors (LDRV) is an approach of structural dynamics. In the special case that the matrices Mii and Kii are symmetric positive definite matrices, the expansion point is zero (σ = 0), the basis vectors are mass normalised and the input and measuring vectors are identical (cTi = bi ) this algorithm is similar to the SyPVL algorithm proposed by [289]. The LDRV are based on the Lanczos algorithm together with a special start vector. Here the static deflection is used as the first Ritz vector so that all following Ritz vectors may be regarded as the balancing of this initial deflection (see [851]). The advantage of this method is that no eigenvalue problem has to be solved. According to [585] the method delivers the following reduced coupled differential equation system: T

Tkk q¨k + Ikk q k = {β1 , 0, · · · , 0} f (t)

(4.259)

Herein the stiffness matrix and the mass matrix are degenerated to an identity matrix Ikk and a tridiagonal matrix Tkk in generalised coordinates, respectively. If we assume that the load distribution on the structure is constant during the simulation, the projected external load vector reduces to T bk = {β1 , 0, · · · , 0} f (t). The scalar value β1 = ϕT1 M ϕ1 is given by the first not mass normalised Ritz vector ϕ1 . 4.2.10.3.3 Substructuring in the Framework of Nonlinear Dynamics The derivation is based, as displayed in Figure 4.96, on a decomposition of the structure into two arbitrary components. Only with the assembly and the solution of the equation system one of the two components is limited to a reduced linear subsystem. 4.2.10.3.3.1 Discretisation and Linearisation Starting point is the balance of linear momentum of a subsystem s in the reference configuration ¨ (s) + F c(s) = 0 Div P (s) + ρ0 bv(s) − ρ0 u

s = 1, 2 .

(4.260) (s)

Herein is P (s) the first Piola-Kirchhoff stress tensor, bv the volume force ¨ (s) the acceleration vector and ρ0 the mass density in the reference vector, u (s) configuration. Additionally interface forces F c have to be introduced. These internal forces only possess non-zero components at the redundant interfaces (s) Γc . As constraints the equilibrium of the interface forces


Component (1)

(1)

Γc

(1)

Ω0

Component (2)

(2)

Γc

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507

(2)

Ω0

Internal node Interface node

Fig. 4.96. Decomposition of the structure into two components (1) F (2) c − Fc = 0

(4.261)

and the compatibility condition of the deformed configuration (1) g = x(2) c − xc = 0

(4.262)

(s)

(s)

(s)

has to be fulfilled. xc = uc + X c is the position vector of the deformed (s) configuration at the interface. It is composed of the displacement vector uc (s) and the position vector of the undeformed reference configuration X c . The interface forces may be replaced by the Lagrange-multipliers (2) λ(1) c = λc = λc

(4.263)

since the interface area is of identical size. In the field of contact simulations λc is named contact pressure. If equation (4.263) is inserted into the strong forms (4.260) and (4.262) in accordance with [473, 856] each subsystem s can be transformed into the weak form 2 ( (s) (s) (s) (s) (s) ¨ (s) · δu(s) dΩ0 g1 (u , λ) = P · Grad δu dΩ0 + ρ0 u (s) (s) Ω Ω s=1 0 0 ) (s) (s) (s) − ρ0 bv(s) · δu(s) dΩ0 − T¯ · δu(s) dΓT (s) (s) ΓT Ω0 + λ · (δu(2) − δu(1) ) dΓc Γc

=0 (x(2) − x(1) ) · δλ dΓc = 0

g2 (u(s) , λ) = Γc

(4.264)

508

4 Methodological Implementation (s)

which refers to the initial configuration Ω0 . The last integral in Equation (4.264a) specifies the virtual work of the Lagrange multipliers λ at the interface. Equation (4.264b) enforces the constraint condition in a weak sense. δu(s) and δλ are the test functions of the independent variables u(s) and λ. (s) (s) (s) The external boundary of each component Γ (s) = Γu ∪ ΓT ∪ Γc consists of (s) (s) the Dirichlet boundary Γu , the Neumann boundary ΓT and the interface (s) (s) represents the external tensions. boundary Γc . The vector T¯ Beside a spatial discretisation according to the isoparametric element concept the constraints at the interface are fulfilled in a strong manner. Hence the virtual work of the Lagrange multiplier at the interface (4.264a) and the compatibility constraint (4.264b) λ · (δu(2) − δu(1) ) dΓc = Γc

nnc

(2)

Λi (δui

(1)

− δui ) ⎡

i=1 (1) T

= [δui

(2) T

δui

(1) T

⎤T

⎢− C ⎥ ]⎣ ⎦ Λ T d (2) C d

T

= δuT d Cc Λ (4.265) (x(2) − x(1) ) · δλ dΓc Γc

=

nnc

(2)

(ui

(1)

− ui ) δΛi

i=1

= δΛT [−d C(1)

d

C(2) ]u

= δΛT d Cc u = δΛT d Gc are transformed into a summation over all interface nodes nnc . The interface force Λi = λi · Ai at node i is the product of the Lagrange multiplier λi and (s) the corresponding area Ai of node i. δui and δΛi are the test functions of node i. Additionally the position vectors at the corresponding interface nodes X (1) ≡ X (2) are identical. As a result in Equation (4.265b) the difference between the position vectors of the deformed configuration x(s) is replaced by the the difference of the displacements u(s) . The matrix d C reduces in the discrete case to a Boolean allocation matrix. At this the index d indicates that the interface constraints are fulfilled in a strong manner. Finally incorporating any time integration like e.g. the Newmark method we result in the fully discretised nonlinear equation system subjected to a constraint:

4.2 Numerical Methods T

R(un+1 ) + M¨ un+1 (un+1 )− P ext + d Cc Λn+1

509

:=

T

G(un+1 ) + d Cc Λn+1 ≈ 0 ! d G (u g n+1 ,Λn+1 ) d

Gc (un+1 )

=0

(4.266) .

n + 1 denotes the current time step which is omitted below. R(u) and P ext are the inner and the external force vectors, respectively. d Gg (un+1 ) and d Gc (un+1 ) are the residual vectors. To solve Equation (4.266) by the Newton-Raphson method a consistent linearisation with respect to the independent variables u and Λ leads to the linearised and decoupled system ⎡ ⎤ ⎤⎡ ⎤ ⎡ m d T m+1 m d T m ⎢ KT eff (u ) Cc ⎥ ⎢ Δu ⎥ ⎢ G(u ) + Cc Λ ⎥ (4.267) ⎣ ⎦ ⎦⎣ ⎦ = −⎣ d d Cc O ΔΛm+1 . Cc u m in matrix notation. The indices eff and m denote that in the tangential stiffness matrix KT eff the time discretisation is already included as well as m signifies the number of iterations. Both indices are omitted below in order to improve the readability. 4.2.10.3.3.2 Primal Assembly The objective of this strategy is to solve partially reduced systems with local nonlinearities such as material damage behaviour. Thus in the following derivation the components (1) and (2) are regarded to be the nonlinear subsystem (nl) and subsystem (2) the reduced, but linear subsystem (lin). The ( * )-symbol is used to indicate reduced components. The transformation of the reduced linear subsystem (2) results from the presented Craig(2) Bampton transformation u(2) = VCB q (2) . The vectors u(2) and q (2) are the physical and the generalised coordinates of the linear component (2). In the following synthesis of the components, which is published by [217] for linear systems, the interface forces are regarded as Lagrange multipliers. At first the compatibility condition (4.266b) has to be transformed into generalised coordinates d

Cq q = 0

(4.268)

and is split d

Cdd Cde

⎡

⎤

⎢ qd ⎥ ⎣ ⎦=0 q qe

(4.269)

510


into e coordinates which have to be kept and d coordinates which are deleted or assembled. The transformation of the not assembled generalised coordinates q to assembled generalised coordinates p is carried out by the projection ⎡ ⎤ ⎡ ⎤ % & −1 ⎢ q d ⎥ ⎢ −Cdd Cde ⎥ . ⎣ ⎦=⎣ ⎦ qe (4.270) qe Iee ! ! ! p q R The latter equation also holds in incremental form: Δq = R Δp

.

(4.271)

Considering the two relations (4.269) and (4.270) the identity d

Cq R = O

(4.272)

holds. For the considered case of a nonlinear and a linear reduced subsystem with physical interface dofs the relation (4.270) transforms to: ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ (1) nl (1) nl ui I O O ⎡ (1) nl ⎤ qe ⎥ ⎢ (1) nl ⎥ ⎢ (1) nl ⎥ ⎢ ⎢q ⎥ ⎢u ⎥ ⎢ O O I ⎥ ⎢ ui ⎥ ⎢ (2) lin ⎥ ⎢ d ⎥ ⎢ c ⎥ ⎢ ⎥ . q = ⎢ (2) lin ⎥ = ⎢ (2) lin ⎥ = ⎢ (4.273) ⎥ q ⎦ ⎢ qe ⎥ ⎢q ⎥ ⎢O I O⎥⎣ k k ⎦ u ⎣ ⎦ ⎣ ⎦ ⎣ c (2) lin (2) lin OO I qe uc Substituting (4.273) in incremental form into the first equation of (4.267) ⎡ ⎤⎡ ⎤ ⎤ ⎡ (1) nl (1) nl (1) nl (1) nl Δui Gi Kii Kic ⎢ ⎥⎢ ⎥ ⎥ ⎢ ⎢ (1) nl ⎥ ⎥⎢ ⎢ (1) nl (1) nl nl ⎥ ⎢ Gc ⎥ ⎢ Δu(1) ⎥ ⎥ d T m+1 ⎢ Kci Kcc c ⎥⎢ ⎥ = −⎢ ⎥ ⎢ (2) lin (2) lin ⎥ ⎢ (2) lin ⎥ + Cq Λ ⎢ ⎥ ⎢ (2) lin * * * ⎢ ⎢ ⎥ ⎥ ⎥ ⎢ Δq K K G kk kc k ⎣ k(2) lin ⎦ ⎦⎣ ⎦ ⎣ (2) lin (2) lin (2) lin * * * Δuc K K G ck cc c (4.274) which is further multiplied by RT , we arrive at the direct assembled global system ⎡ ⎡ (1) nl ⎤⎡ ⎤ ⎤ (1) nl (1) nl (1) nl Gi Kii Kic Δui ⎢ ⎢ ⎥⎢ ⎥ ⎥ ⎢ ⎥ ⎢ Δq (2) lin ⎥ = − ⎢ ⎥ * (2) lin * (2) lin * (2) lin K K G kk kc k k ⎣ ⎣ ⎦⎣ ⎦ ⎦ (2) lin (2) lin (2) lin (1) nl * (1) nl * * Δuc K Kcc + K G(1) nl + G K ci

ck

cc

c

c

(4.275)


511

The iteration index m + 1 in Equation (4.274) is only applied to underline that the latter equation depends on the current Lagrange multipliers. 4.2.10.3.3.3 Solution of the Decomposed Structure The most common solution in the literature is the monolithic solution of the overall equation system. In this work the existing decomposition of the structure into large linear and small nonlinear components and the model reduction of the linear subsystems is exploited to substitute the solution of one monolithic equation system by the efficient solution of a number of small equation systems. On the basis of Equation (4.273) the individual linear and nonlinear components are transformed by means of static condensation into the local Schur complement systems −1 (s) lin (s) lin (s) lin (s) lin (s) lin * * * * K K −K Δuc K cc

ck

kk

kc

! (s) lin S −1 T (s) lin (s) lin (s) lin * * * * (s) lin +d Cq(s) Λ(s) Kkk = Gc − Kck G k ! (s) lin G

(4.276)

and −1 (s) nl (s) nl (s) nl (s) nl (s) nl Kcc − Kci Kii Kic Δuc ! (s) nl S −1 (s) nl (s) nl (s) nl d (s) T (s) (s) nl Kii = Gc − Kci Gi + Cc Λ ! G(s) nl

(4.277)

S(s) lin and S(s) nl represent the local Schur complements as well as G(s) lin and G(s) nl the modified residual vectors. According to the previous section the Lagrange multipliers cancel out during the assembly process by means of direct elimination. As a result the global Schur complement system Sg Δuc = Gg

(4.278)

is independent of the Lagrange multipliers. The global Schur complement Sg and the modified global residual Gg are divided into its linear and nonlinear components:

512


nl Sg = Slin g + Sg

=

N lin

(R(s) )T S(s) lin R(s) +

s=1 nl Gg = Glin g + Gg =

Nnl s=1

Nnl

(R(s) )T S(s) nl R(s)

s=1

(R(s) )T G(s) lin

+

Nnl

(R(s) )T G(s) nl

s=1

(4.279) R(s) represent rectangular Boolean submatrices of the global assembly ma(s) trix R. They link the local interface dofs uc with the global interface uc . The complete number of subsystems N = Nlin + Nnl consists of Nlin linear components and Nnl nonlinear components. Finally, depending on the solution of the global Schur complement system (4.278) the internal dofs −1 (s) lin (s) lin (s) lin (s) lin * k(s) lin − K * kk * kc Δq k G = K Δuc (s) nl

Δuk

−1 (s) nl (s) nl (s) nl (s) nl Gi = Kii − Kic Δuc

(4.280)

are to be determined. 4.2.10.4 Example: M¨ unster-Hiltruper Road Bridge This example serves to validate the overall strategy. At first the solution of the decomposed but unreduced structure is compared to a monolithic solution. Subsequently the influence of the different employed model reduction methods on the accuracy of the computation is investigated. In this example the bar-like structure of an arched steel bridge is regarded. Its geometry is based on the road bridge in M¨ unster-Hiltrup (federal road B54), which is one of the reference buildings of the Collaboratory Research Centre 398. The dimensions, cross sections and parameters are chosen according to existing mechanical drawings. In accordance with the proposed strategy the structure is subdivided into linear and nonlinear components. The former are additionally reduced to increase the numerical efficiency. The results of the simulation are compared to the solution of a monolithic transient analysis. The road bridge, depicted in Figure 4.97, is l = 87.37 m long, b = 17.85 m wide and h = 13.68 m high. In the nonlinear substructures we model the evolution of material deterioration for ductile damage behaviour of metals. For this purpose we extend the material model of Rousselier (see [691]). Compare also [251] where an alternative approach has been chosen. The material parameters read E = 210 000 N/mm2 , ν = 0.3, ρ0 = 7.85 kg/dm3 , σy0 = 400 N/mm2 , H = 2100 N/mm2 , D = 2.0, σk = 400 N/mm2 , f0 = 0.01, fN = 0.25, εN = 0.2 and s = 0.4.


513

q1

P

h l Load Function X1

b

X2 q2

X3

t1 t2 t3

Fig. 4.97. Discretised quarter of the bridge - geometry and loading

To simulate the deterioration within the hangers, a horizontal spatial force of q1 = 2.2 N/mm2 is applied to the largest hanger. Additionally a vertical force of q2 = 0.25 N/mm2 and two single forces of P = 600 kN are loaded on the cross girders of the girder and the cross girders of the arch, respectively. The loading process is carried out linearly in t1 = 1.2 · 10−1 s. After a constant period of 1 · 10−2 s the structure is completely unloaded. The loading function is plotted in Figure 4.97. The bridge structure is investigated for a symmetric loading case. As a result only one quarter of the bridge is discretised. Besides the loading conditions which result from the symmetric configuration the bridge is supported vertically on lbearing = 1.189 m long bearings at the ends of the girder. The spatial discretisation of the bridge is accomplished by solid beam elements Q1SPb of the finite element family Q1SP. These eight node solid beam elements are an extension of the solid shell element formulation of [666]. They possess 2 × 2 Gauss points on their mid plane (perpendicular to the beam axis). As a result Q1SPb elements are suitable to capture bending phenomena of slender structures in a numerically efficient way with only one element over the height. A further advantage of this finite element formulation is that any inelastic material behaviour can be implemented without further kinematic restrictions. These solid beam elements demonstrate their efficiency in the paper of [667]. Therein they are used to simulate the behaviour of medical stents made of shape memory alloys. For an appropriate discretisation of the bridge some modifications are required. To reduce the numerical effort the cross sections of the arch, the girder and the hanger have to be transformed into equivalent rectangular cross sections. In Table 4.8 the corresponding cross sections and the associated material parameter are listed. Another simplification concerns the hanger of the bridge: Since they hold only small-sized cross sections compared to the

514


Table 4.8. Equivalent square sections and corresponding material parameters Section Girder (lin) Arch (lin) Hanger (nl)

Square Section Young’s Modulus (b×h) [ mm × mm ] [ N/mm2 ] 240 × 1400 205000 700 × 600 198000 100 × 100 181000

Density [ kg/dm3 ] 1.81 1.88 7.46

C B

A

Fig. 4.98. Exploded view of the bridge (complete system)

remaining bridge, they are approximated and connected to the structure by (solid) beam elements. The time integration of the example is carried out by the Newmark method and a time step length of Δt = 10−3 s. The same length ΔtPOD = Δt is utilised in the precalculation step to collect the data sets for the POD method. In this example a data basis composed of 400 data sets has shown to be advantageous. A transient analysis is accomplished under the assumption that large deformations and an accompanying evolution of damage only occur in the slender hangers. Hence the girders, the arches as well as the cross girders are discretised by linear elastic Q1SPb solid beam elements, while the damage model is only considered within the hangers and their junctions.


515

f 0,018

0,014

0,010

Fig. 4.99. Damage evolution in the largest two hangers at the end of the simulation (initial damage f0 = 0.01)

This enables to decompose the discretised quarter of the bridge into its linear and nonlinear components according to the proposed strategy. In this example these are the linear girder, the linear arch as well as five nonlinear hangers of the discretised quarter of the bridge. The single components of the entire structure and the points of interest are visualised in Figure 4.98 by an exploded view. In point B (centre of the largest hanger) the system response of a nonlinear subsystem is evaluated. Point C (upper junction of the largest hanger) is the place of maximal damage evolution (cp. Figure 4.99). The maximal deterioration at the end of the simulation is fmax = 2.73 · 10−2 . In the following investigation the dimension of the linear subsystems is reduced via the presented model reduction techniques. The solution is compared to the system response of the unreduced structure. In the chosen discretisation the arch and the girder consist of 2226 and 4898 inner dofs, respectively. Therewith the reduction of the linear subsystems from 1 up to 100 basis vectors corresponds to a reduction of the original size of the linear inner equation systems from 0.02 % up to 4 %. The dimension of the Schur complement is 138 dofs. In Figure 4.100 the system response concerning the largest examined reduction basis of 100 basis vectors is regarded. Between the unreduced and the partially reduced solution of the MODRED, the LDRV and the

516


Complete system 100 MODRED 100 POD 400 100 symmPVL 100 LDRV

1000 in point B [mm]

Displacement in X2 − direction

1200

800

600

400

200 0.2

0.4

0.6

0.8

1

1.2

Time [s]

Fig. 4.100. Displacement in X2 -direction in point B and a reduction of the linear subsystems to 100 basis vectors

in X2 − direction in point B [−]

Error in the displacement

10+00 MODRED POD 400 symmPVL LDRV

10−01

10−02

10−03

10

20

30 40 50 60 70 Number of basis vectors

80

90

100

Fig. 4.101. Mean relative displacement-based error u,Σ in X2 -direction in point B and a reduction of the linear subsystems

symmPVL no difference is distinguishable. Solely the solution of the POD exhibits a significant deviation to the unreduced solution. This result shows the slight influence which has an exchange of the local element shape functions towards a small number of unphysical global functions. For a more detailed quantitative evaluation of the reduction error the mean relative displacement-based error u,Σ =

N ¯A (ti )| 1 |* uA (ti ) − u ; 1.0) min( N i=1 (|¯ uA (ti )|)


517

is defined. Herein N is the number of discrete time steps. The estimated error over an interval of 1 up to 100 basis vectors is depicted in Figure 4.101. Obviously all reduction basis exhibit quantitatively up to a size of 30 basis vectors comparable reduction errors. Subsequently the LDRV, the symmPVL and the POD reveal up to a basis of 50 or 60 basis vectors a reduction period in which the model reduction errors do not improve or even increases with the increasing dimension of the reduced components. Aside from Modal Reduction it becomes evident that an increase of the number of basis vectors does not automatically lead to a better approximation of the dynamical behaviour of the reduced subsystems. The LDRV and the symmPVL, which both dispose the Lanczos-Algorithm, show an almost identical reduction error. For reduced subsystems with more than 60 basis vectors both methods show a slightly larger reduction error than the Modal Reduction. The latter yields in conjunction with the Craig-Bampton method the lowest reduction error. In contrast the POD only captures the principle dynamic characteristics of the systems. Higher modes do not induce an improvement of the accuracy. As a consequence the method remains on an error level which has already been reached with only 10 or 20 basis vectors. The reason for the good performance of the Modal Reduction is based on the fact that the reduced linear parts of this example are only slightly loaded. Thus the main disadvantage of the Modal Reduction - to approximate external loads inaccurate - has less influence on the result. Accordingly the Modal Reduction yields for the reduction of unloaded subsystems, as proposed by [313], accurate results. Additionally [111] mentions that the difference in the reduction quality of reduced linear total systems between the Modal Reduction and reduction methods of higher accuracy (like the LDRV, symmPVL or POD) is in the time domain of lower importance than in the frequency domain. Finally the error level of the Craig-Bampton substructure technique and the reduction methods superpose each other. All these reasons lead to results which differ from the model reduction of total systems. There the minor accuracy of the Modal Reduction is not competitive to the other model reduction methods. Altogether the substructuring of complex systems into a number of smaller components and the strong reduction of the dominating linear subsystems results in a numerically efficient simulation strategy for partially damaged systems. The additionally introduced reduction error is small. 4.2.11 Strategy for Polycyclic Loading of Soil Authored by Andrzej and Theodoros Triantafyllidis

Niemunis,

Torsten

Wichtmann

In FE calculations of the accumulation of settlement due to high-cyclic loading two different numerical strategies are combined. They are termed the implicit and the explicit mode of operation.

518


˙ In the implicit mode each cycle is calculated with small increments σ(D)Δt. The accumulation results as a by-product due to the not perfectly closed stress or strain loops. Elastoplastic multi-surface models [561, 184, 185], endochronic models [806] or the hypoplastic model with intergranular strain [576] can be used for implicit calculations. The applicability of the pure implicit method is restricted to a low number of cycles (Nc < 50) because with each increment systematic errors of the constitutive model or the integration scheme are accumulated too [574]. Even small systematic errors may become significant after large Nc (e.g. multiplied with a factor 106 in the case of 104 cycles with 100 increments each). Thus, a constitutive model of an unreachable perfection would be necessary, let alone the large calculation effort. The explicit strategy is a time integration dedicated to high-cyclic loading only. It requires a special constitutive formulation (Section 3.3.3) which accepts packages of cycles as input. The accumulation D acc ΔNc due to a package of ΔNc cycles of a given amplitude A (Section 2.5.2) is treated similarly as a creep deformation due to time increments Δt in viscoplastic models. The number of cycles Nc just replaces the time t. In FE-calculations the strain amplitude A (strictly speaking one needs its spatial field) is usually unknown and therefore one uses a combination of an implicit calculation (in order to evaluate the amplitude) and of an explicit calculation in order to evaluate the accumulation. A few first cycles are calculated implicitly with strain increments (Figure 4.102) using a conventional constitutive model (quasi-static or dynamic analysis). The first cycle is irregular so we use rather the second or the third one to record the strain path ε(t) for the evaluation of the amplitude for the first package of cycles. The path ε(t) is stored as a series of discrete strain states. This should be done for each Gauss integration point. A smart recording algorithm has been developed to economize on the computer memory, e.g. intermediate strain states along a

F

F

ε

t

t F

F

ε cyclic "pseudo-creep"

t

εampl

εampl

t

Fig. 4.102. Comparison of a pure implicit and an explicit calculation of accumulation

4.3 System Identification

519

straight line would not be recorded. The A strain amplitude is determined from this strain path as described in Section 2.5.2. The first ’irregular’ cycle is not suitable for the determination of A , since the deformations in the first cycle can significantly differ from those in the subsequent cycles, cf. Figure 4.102 or the discussion in [578]. The amplitudes from the second or third cycle are more representative for the amplitudes during the following packages of cycles with constant excitation. The explicit mode calculates directly the accumulation rate D acc which enters the constitutive equation (3.243). During this explicit calculation the strain amplitude A is assumed constant. After several thousand cycles this assumption is not realistic anymore. Due to compaction and re-distribution of stress the stiffness and thus also the strain amplitude may significantly change. The explicit calculation should be therefore interrupted after definite numbers of cycles and εampl should be recalculated using the implicit mode. Such sporadic control cycles (Figure 4.102) are recommended in particular during the early cycles (in a so-called conditioning phase). In a control cycle also the static admissibility of the state of stress and the overall stability can be checked. The latter one can get lost e.g. in the undrained case due to excess pore water pressures. The plastic stretching Dpl in (3.243) is necessary for a monotonic loading applied simultaneously with the cyclic loading. A monotonic deformation can be caused either by the monotonic changes at boundaries or by a strong accumulation in neighbouring elements of the one under consideration. The plastic stretching Dpl is treated separately from the cumulative stretching D acc although from the physical point of view they cannot be distinguished. The decomposition of the irreversible strain is enforced by the explicit strategy of calculation. Implicit models need not such separation. The plastic deformation is necessary to restrict the stress paths due to monotonic loading not to surpass the yield surface. A similar effect cannot happen due to the pseudoE : D acc which always tends inwards the yield surface. In relaxation σ˙ acc = −E other words, the plastic rate D pl is indispensable in an element under small cyclic loading to make it compliant with the large deformation of surrounding elements under strong cyclic loading. Another difficulty of the presented high-cycle model is the mesh locking. This difficulty is typical for initial stress BVPs. Using full Gauss spatial integration we have much more constraints (= prescribed strains at the Gauss points) than degrees of freedom (= nodal displacements). Reduced integration algorithms are therefore recommended.

4.3 System Identification Authored by Stefanie Reese, Heinz Waller and Armin Lenzen System identification is a methodology developed mainly in the area of automatic control, by which we can choose the best model from a given model

520


set based on the observed input-output data from a given physical system. The input and output data are measured by appropriate experiments. 4.3.1 Covariance Analysis Authored by Stefanie Reese, Heinz Waller and Armin Lenzen Conventionally, the dynamic computation is performed by means of the time history method, which applies a direct integration of the differential equation of motion. This approach is very inefficient because it has to be evaluated numerous times in the case of stochastic loadings. There exist many strategies to include the time dependent influences in reliability problems, but a totally dynamic computation is always avoided. So an efficient method to solve this problem is needed. A useful alternative is to apply a covariance analysis using a finite element method along with a shaping filter. Within the shaping filter, the wind process is represented through a black box parameter model which is realized in terms of linear algebra techniques by measured data. 4.3.2 Subspace Methods Authored by Stefanie Reese and Andreas S. Kompalka In [462] the subspace identification procedure was published first time. In this section the data-driven subspace identification method is presented with attention to the estimation of modal data (frequencies and mode shapes) of mechanical structures. The prefix ’data-driven’ expresses that the measurements are analyzed without prefiltering. In the following subsections the three most common state space models are derived and the identification procedure are explained. Finally, the calculation of the modal data is illustrated in an appropriate subsection. 4.3.2.1 State Space Model The mechanical basis of the state space model is given by the equation of motion ˙ p(t) = M¨ q (t) + Dq(t) + Sq(t) = Gu(t)

(4.281)

¨ (t), where the products of the mass matrix M with the acceleration vector q ˙ the damping matrix D with the velocity vector q(t) and the stiffness matrix ¨ (t) are equal to the load vector p(t). The S with the displacement vector q load vector p(t) is factorized into the input location influence matrix G and the input force vector u(t). In system or control theory u(t) is labeled as input vector. Using the equation of motion (4.281) and the trivial statement ˙ ˙ q(t) = q(t) we obtain the state space equation


⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ˙ q(t) 0 I q(t) 0 ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦=⎣ ⎦⎣ ⎦+⎣ ⎦ u(t) −1 −1 −1 ˙ q¨ (t) −M S −M D q(t) M G

521

⎡

(4.282)

T ˙ , the derivative of the state with the state space vector x(t) = [q(t) q(t)] ˙ ˙ space vector x(t) = [q(t) q¨(t)]T , the system matrix ⎡ ⎤ 0 I ⎥ ¯ =⎢ A (4.283) ⎣ ⎦ −1 −1 −M S −M D

and the input matrix ⎤ ⎡ 0 ⎥ ¯ =⎢ B ⎦. ⎣ M−1 G

(4.284)

A second order linear differential equation ˙ + Ca q¨(t) + D u(t) y(t) = Cd q(t) + Cv q(t)

(4.285)

is used to describe the relation between the measurements y(t) and the me˙ ¨ (t). In system or control theory y(t) is chanical state variables q(t), q(t) and q labeled as an output vector. Cd , Cv and Ca denote the displacement, velocity and acceleration calibration matrices, respectively. The matrix D is the direct transmission matrix. Inserting the reshaped equation of motion (4.281) into the second order linear differential equation (4.285) yields the output equation

y(t) =

⎡

⎤

⎢ q(t) ⎥ ⎦ Cd −Ca M−1 S Cv −Ca M−1 D ⎣ ˙ q(t) + Ca M−1 G + D u(t)

(4.286)

with the output matrix ¯ −1 −1 C = Cd −Ca M S Cv −Ca M D

(4.287)

and the direct transmission matrix ¯ −1 D = Ca M G + D .

(4.288)

Using the state space equation (4.282) and the output equation (4.286) with the matrix definitions (4.283), (4.284), (4.287) and (4.288) we obtain the continuous-time state space model

522


⎧ ⎪ ⎨ x(t) ¯ ¯ ˙ = Ax(t) + Bu(t) Mt :=

⎪ ⎩ y(t) = Cx(t) ¯ ¯ + Du(t)

.

(4.289)

If the measurements y(t) are taken at discrete times tk we can state the distrete-time deterministic state space model ⎧ ⎪ ⎨ xk+1 = Axk + Buk Md := (4.290) ⎪ ⎩ y = Cxk + Duk k equivalent to the continuous-time state space model (4.289). A, B, C and D are the discrete-time system, input, output and direct transmission matricies. The experimental measurements are usually contaminated by noise. Introducing the process noise wk and the measurement noise v k we obtain the combined deterministic-stochastic state space model ⎧ ⎪ ⎨ xk+1 = Axk + Buk + wk Mc := . (4.291) ⎪ ⎩ y = Cxk + Duk + v k k For many experimental applications it is difficult or impossible to measure the load terms uk . If the input terms are included in the noise terms, we finally arrive at the stochastic state space model ⎧ ⎪ ⎨ xk+1 = Axk + wk Ms := (4.292) ⎪ ⎩ y = Cxk + v k k which is also known as output-only. Further state space models with other extentions exist in the literature (see e.g. [359]). In this contribution the focus lies on the estimation of the modal data (frequencies and mode shapes) of mechanical structures. As shown in subsection 4.3.3.3, only the discrete-time system matrix A and the discrete-time output matrix C are needed to calculate the modal data. 4.3.2.2 Subspace Identification In this subsection the subspace identification method is explained to estimate the discrete-time system matrix A and the discrete-time output matrix C of the deterministic (4.290), the deterministic-stochastic (4.291) and the stochastic state space model (4.292).


523

The deterministic subspace identification (dsi) estimates the deterministic state space model (4.290) without noise terms. The input and output vectors are measured and sorted into the input block Hankel matrix ⎤ ⎡ u u ·· u 0 1 N −1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ u1 u2 ·· uN ⎥ ⎢ (4.293) U0|j|N = ⎢ . ⎥ . . .. .. ⎥ ⎢ .. ⎥ ⎢ ⎦ ⎣ uj−1 uj ·· uj+N −2 and the output block Hankel matrix ⎤ ⎡ ⎢ y 0 y 1 ·· y N −1 ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ y 1 y 2 ·· y N ⎥. Y0|j|N = ⎢ ⎥ ⎢. .. .. ⎥ ⎢ .. . . ⎥ ⎢ ⎦ ⎣ y j−1 y j ·· y j+N −2

(4.294)

The block Hankel matrix is characterized by the shape where each antidiagonal term is a constant measurement vector (block). This shape is an essential property to separate the state space vectors xk from the input vectors uk and, if present, from the noise terms w k and v k (see e.g. [609]). In a first step the input block Hankel matrix (4.293) and the output block Hankel matrix (4.294) are assembled in a matrix ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ Q1 ⎢ ⎥ ⎥ ⎢ U0|j|N ⎥ ⎢ L11 0 0 ⎥ ⎢ ⎢ (4.295) ⎣ ⎦=⎣ ⎦ ⎢ Q2 ⎥ ⎥ ⎦ L21 L22 0 ⎣ Y 0|j|N ! Q3 ! L Q and factorized by the LQ-factorization into a left lower matrix L and an orthogonal matrix Q. The left lower submatrix L22 is equal to the transformed observability matrix which includes the distrete-time system matrix A and the distrete-time output matrix C (see e.g. [505]). The key point of the subspace identification is the singular value decomposition ⎡ ⎤ ⎤⎡ T ⎢ Σ1 0 ⎥ ⎢ V1 ⎥ L22 = U1 U2 ⎣ (4.296) ⎦ ⎦⎣ 0 VT2 ! 0 * ! ! U S VT

524


of the submatrix L22 into a left singular vector matrix U, a right singular vector matrix VT and the singular values matrix S. The singular values different from zero signalize the excited mode shapes of the experiment and are sorted in the diagonal matrix Σ1 . The reduced singular value decomposition L22 = U1 Σ1 VT1

(4.297)

separates the subspace which includes the information of the measurement. This is the reason for the denotation subspace identification for this kind of algorithms. Finally, the observability matrix Γj = U1 Σ1

(4.298)

includes the discrete-time output matrix C = Γj (1 : n, s)

(4.299)

in the first n lines. With the upper observability matrix Γu = Γj (1 : end − n, s)

(4.300)

and lower observability matrix Γl = Γj (1 + n : end, s)

(4.301)

which are equivalent to the observability matrix Γj (4.298) without the last and the first n lines, it is possible to calculate discrete-time system matrix A = Γ†u Γl

(4.302)

where (•)† represents the pseudo-inverse of the matrix (•). Here, the ’:’ denotes the MATLAB notation which is common in the literature for system and control theory. The discrete-time system matrix A and the discrete-time output matrix C were identified by using the observability matrix Γj Eq. (4.298). Alternatively, the states can be used for identification (see e.g. [609]). Furthermore, the observability matrix Γj Eq. (4.298) was calculated with the singular √ value matrix Σ1 . [462] and [609] use the root of the singular value matrix Σ1 . In [359] the singular value matrix is dropped and [505] states that the multiplication with the singular value matrix is equivalent to an arbitrary state space transformation. The authors agree with [505] with the exception of the case without the singular value matrix where important information is lost. Some applications use weighting matrices to improve the estimates (see e.g. [609] or [505]). In this contribution, this aspect is dropped because for the investigated mechanical structures the authors realize no improvements in the identification of the modal data. The deterministic-stochastic subspace identification (dssi) estimates the discrete-time system matrix A and the discrete-time output matrix C of the combined deterministic-stochastic state space model (4.291). Equal to the


525

deterministic subspace identification (dsi) the measurements are sorted into the input block Hankel matrix ⎤ ⎡

U0|2j|N

⎢ u0 ⎢ ⎢ ⎢ u1 ⎢ ⎢. ⎢ .. ⎢ ⎢ ⎢ ⎢ uj−1 ⎢ =⎢ ⎢ ⎢ uj ⎢ ⎢ ⎢u ⎢ j+1 ⎢ ⎢ .. ⎢. ⎢ ⎣ u2j−1

u1

·· uN −1

u2 .. .

·· uN .. .

uj

·· uj+N −2

uj+1 ·· uj+N −1 uj+2 ·· uj+N .. .. . . u2j

·· u2j+N −2

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎡ ⎤ ⎥ ⎥ U ⎥ ⎢ 0|j|N ⎥ ⎥=⎣ ⎦ ⎥ ⎥ Uj|j|N ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(4.303)

and the output block Hankel matrix ⎤ ⎡

Y0|2j|N

⎢ y0 ⎢ ⎢ ⎢ y1 ⎢ ⎢. ⎢ .. ⎢ ⎢ ⎢ ⎢ y j−1 ⎢ =⎢ ⎢ ⎢ yj ⎢ ⎢ ⎢y ⎢ j+1 ⎢ ⎢ .. ⎢. ⎢ ⎣ y 2j−1

y1

·· y N −1

y2 .. .

·· y N .. .

yj

·· y j+N −2

y j+1 ·· y j+N −1 y j+2 ·· y j+N .. .. . . y 2j

·· y 2j+N −2

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎡ ⎤ ⎥ ⎥ Y 0|j|N ⎥ ⎢ ⎥ ⎥=⎣ ⎦. ⎥ ⎥ Y j|j|N ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(4.304)

Then, the block Hankel matricies (4.303) and (4.304) are assembled in a matrix ⎡ ⎤ ⎡ ⎤⎡ ⎤ L11 0 0 0 U0|j|N Q1 ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ Uj|j|N ⎥ ⎢ L21 L22 0 0 ⎥ ⎢ Q2 ⎥ ⎢ ⎥=⎢ ⎥⎢ ⎥ (4.305) ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ Y 0|j|N ⎥ ⎢ L31 L32 L33 0 ⎥ ⎢ Q3 ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦ Yj|j|N Q4 L41 L42 L43 L44 ! ! L Q

526


and factorized by the LQ-factorization. Unlike the deterministic subspace identification (dsi) the singular value decomposition is applied on the submatrix ⎡ ⎤ ⎤⎡ ( ) ( ) Σ1 0 VT1 ⎦ ⎦⎣ (4.306) L42 L43 = U1 U2 ⎣ * VT2 ! 0 0 ! ! U S VT where the input vectors uk and noise terms wk and v k are eliminated (proof see [359]). The segmentation of the input and output block Hankel matrix (4.303) and (4.304) into a lower and upper part defines the size of the submatrix [L42 L43 ]. Finally, the reduced singular value decomposition T (4.307) L42 L43 = U1 Σ1 V1 enables the calculation of the discrete-time output matrix C and the discretetime system matrix A with Eq. (4.298)-(4.302) similar to the deterministic subspace identification (dsi). To estimate the discrete-time system matrix A and the discrete-time output matrix C of the stochastic state space model (4.292) only the system responce is available. For the stochastic subspace indentification procedure (ssi) only the output measurements are sorted in the output block Hankel matrix ⎤ ⎡

Y0|2j|N

⎢ y0 ⎢ ⎢ ⎢ y1 ⎢ ⎢. ⎢ .. ⎢ ⎢ ⎢ ⎢ y j−1 ⎢ =⎢ ⎢ ⎢ yj ⎢ ⎢ ⎢y ⎢ j+1 ⎢ ⎢ .. ⎢. ⎢ ⎣ y 2j−1

y1

·· y N −1

y2 .. .

·· y N .. .

yj

·· y j+N −2

y j+1 ·· y j+N −1 y j+2 ·· y j+N .. .. . . y 2j

·· y 2j+N −2

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎡ ⎤ ⎥ ⎥ Y ⎥ ⎢ 0|j|N ⎥ ⎥=⎣ ⎦ ⎥ ⎥ Y j|j|N ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

and factorized by the LQ-factorization ⎡ ⎤ ⎤ ⎡ ⎤ Q1 ⎡ ⎢ ⎥ ⎥ ⎢ Y 0|j|N ⎥ ⎢ L11 0 0 ⎥ ⎢ ⎢ ⎦=⎣ ⎦ ⎢ Q2 ⎥ ⎣ ⎥. ⎦ L21 L22 0 ⎣ Yj|j|N ! Q3 ! L Q

(4.308)

(4.309)


527

Assuming a sufficient number of measurement information the lower submatrix L21 is noise-free. The singular value decomposition ⎡ ⎤ ⎤⎡ T ⎢ Σ1 0 ⎥ ⎢ V1 ⎥ (4.310) L21 = U1 U2 ⎣ ⎦ ⎦⎣ T * 0 0 V ! 2 ! ! U S VT of the submatrix L21 leads to the reduced singular value decomposition L21 = U1 Σ1 VT1 .

(4.311)

Similar to the deterministic subspace identification (dsi) the discrete-time system matrix A and the discrete-time output matrix C are obtained from the reduced singular value decomposition (4.311) and Eq. (4.298)-(4.302). Summarizing we can state, the deterministic, the deterministic-stochastic and the stochastic identification methods differ only in the available information for the LQ-factorization and the used submatrix for the singular value decomposition. 4.3.2.3 Modal Analysis In chapter 4.3.3.2 the subspace identification was explained to estimate the system matrix A and the output matrix C. The modal analysis utilizes this information to calculate the modal data (frequencies and mode shapes). Using the special ansatz q(t) = φi eωi t

(4.312)

with the complex circular frequency ωi = −δi − ωi i, the complex eigenvector φi = φri + φii i and the state space equation (4.282) without load (u(t) = 0) yields the special eigenvalue problem ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎤⎡ ∗ ∗ Φ Φ Ω 0 0 I Φ Φ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎥⎢ (4.313) ⎣ ⎦⎣ ⎦=⎣ ⎦ ⎦⎣ 0 Ω∗ −M−1 S −M−1 D ΦΩ Φ∗ Ω∗ ΦΩ Φ∗ Ω∗ with the complex conjugate circular frequency matrix ⎤ ⎡ ⎢Ω 0 ⎥ Λ=⎣ ⎦ 0 Ω∗ including the complex circular frequency matrix

(4.314)

528


⎡ ⎢ ⎢ Ω=⎢ ⎢ ⎣

..

⎤ .

⎥ ⎥ ⎥ ⎥ ⎦

−δi − ωi i ..

(4.315)

.

and the complex conjugate eigenvector matrix ⎡ ⎤ ∗ ⎢ Φ Φ ⎥ Ψ=⎣ ⎦ ΦΩ Φ∗ Ω∗ including the complex eigenvector matrix .. .. r i Φ = . φi − φi i . .

(4.316)

(4.317)

Unfortunately, the solution of the subspace identification for the system matrix A (4.283) is discrete-time and includes an arbitrary state space transformation. The complex circular frequency matrix Λ (4.314) obtained from the solution of the special eigenvalue problem (4.313) is a diagonal matrix and independent from the state space transformation. The continuous-time complex conjugate circular frequency matrix ¯ = 1 ln (Λ) Λ Δt

(4.318)

can be calculated with the matrix logarithm ln(•) (see [228]). The complex eigenvector matrix Φ (4.317) is similar in discrete-time and continuous-time but includes an abritrary state space transformation. Therefore, the transformed complex eigenvector matrix ⎡ ⎤ Φ ⎥ −2 ˆ = C⎢ Φ (4.319) ⎣ ⎦Ω ΦΩ is calculated with the output matrix C which includes the same state space transformation (see e.g. [615]).

4.4 Reliability Analysis Authored by Dietrich Hartmann, Yuri Petryna and Andrés Wellmann Jelic The computer-based determination and analysis of structural reliability aims at the realistic spatiotemporal assessment of the probability that given structural systems will adequately perform their intended functions subject to

4.4 Reliability Analysis

529

existing environmental conditions. Thus, the computation of time-variant, but also time-invariant failure probabilities of a structure, in total or in parts, governs the reliability analysis approach. In addition, based on the success gained in reliability analysis in the past, it becomes more and more popular to extend the reliability analysis to a reliability-based optimum design. By that, structural optimization of structures, which often leads to infeasible structural layouts in sensitive cases, naturally incorporates probabilistic effects in the optimization variables, criteria and constraints making real world optimization models. According to the fact that various modes of failure are customarily possible, a formidable task, in particular for large and complex structural systems, is to be solved. In the following sections the currently most powerful approaches in reliability analysis are described to demonstrate their possible potentials. 4.4.1 General Problem Definition Authored by Dietrich Hartmann, Yuri Petryna and Andrés Wellmann Jelic A reliability analysis aims at quantifying the reliability of a structure accounting for uncertainties inherent to the model properties (material parameters, geometry) as well as environmental data (loads), respectively. The quantification is achieved by estimating the failure probability Pf of the structure. For problems in the scope of civil engineering, this probability depends on the random nature of the stress values S as well as the resistance values R, as depicted in Figure 4.103. The structural failure at the particular time T is defined by the event R < S, leading to a failure probability Pf = P (R < S) = P (R < S|S = s)P (S = s) . (4.320) all s

fS (s), fR (r) Resistance R Stresses S

r, s μr

μs failure domain

Fig. 4.103. General definition of the failure domain depending on scattering resistance (R) and stress (S) values

530


where the quantities r, s are realizations of the random values R and S, respectively. By assuming an uncorrelated relationship and continuous distribution functions of the random values R and S the formulation in eq. (4.320) can be simplified to ∞ FR (s) · fS (s) ds

Pf =

(4.321)

0

where FR (s), fS (s) represent the general cumulative distribution function (cdf) of the resistance values and the probability density function (pdf) of the stress values, respectively. Mathematically, the random nature of the values R and S is modelled T in terms of a vector of random variables X = {X1 , X2 , . . . , Xd } and the correspondent joint probability density function fx (x) =

P (xd < Xd ≤ xd + dx) . dxd

(4.322)

In this context, the parameter d quantifies the total number of random variables which corresponds to the stochastic dimension of the researched structural reliability problem. By using the joint probability density function in eq. (4.322), the failure probability in eq. (4.321) is reformulated to Pf = P [g(X) ≤ 0] =

fx (x)dx .

(4.323)

g(x,T )=0

g(x, T ) = 0 is the relevant time-dependent limit state function for a prescribed failure criterion and divides the safe state from the unsafe state as follows

≤0 failure g (x, T ) = (4.324) >0 survival . In general, multiple distinct limit state functions may be defined. In the following, however, only one limit state function is considered for a better readability. By solving the multidimensional integral in Eq.(4.323) an estimate for the failure probability at the point in time T is quantified. Additionally to the time-dependent formulation of the limit state function in eq. (4.324), also the stress values S as well as the resistance values R of certain structural problems may exhibit a significant time dependency. As a consequence, the formulation of the resulting failure probability has to incorporate this time dependency as follows


Pf (t) = P (R(t) < S(t)) = FT (t) .

531

(4.325)

Thus, by evaluating this failure probability for multiple discrete time points ti the evolution of Pf (t) can be estimated. Following the above explained differentiation between time-variant and time-invariant modelling of reliability problems, consequently, also the corresponding solution methods for solving these reliability problems are presented separately in the following. 4.4.2 Time-Invariant Problems Authored by Dietrich Hartmann, Yuri Petryna and Andrés Wellmann Jelic Existing methods for solving time-invariant reliability problems can be mainly separated into three different groups: analytical solutions, approximate methods and simulation methods. In the initially named group an analytical, closed-form solution of the multidimensional integral in eq. (4.323) is researched. However, this approach is only realizable for low-dimensional reliability problems with a small number of random variables. As structural reliability problems in civil engineering primarily comprise several random variables together with nonlinear limit state functions this analytical approach can not be applied. For the analysis and solution of structural reliability problems the approximate and simulation methods are most favorable so that they are to be explained more detailed. 4.4.2.1 Approximation Methods Well developed methods for approximating the failure probability are FORM and SORM (First-Order and Second-Order Reliability Methods). These are analytical solutions converting the integration into an optimization problem. In order to simplify the calculation the distribution functions of the random variables and the limit state function are transformed into a standardized Gaussian space, as outlined in Figure 4.104. This transformation is defined via the Cumulative Distribution Function FXi (xi ) = Φ(yi )

(4.326)

where yi are the transformed and standardized Gaussian variables, leading to yi = Φ−1 (FXi (yi )) . This transformation leads to a nonlinear limit state function

−1 −1 (Φ(x1 )) , . . . , FX (Φ(xm )) h(y) = g FX 1 m

(4.327)

(4.328)

532


Y1

X1

g(X) h(Y)

μX

Y2

X2

Fig. 4.104. Standardization of an exemplary 2D joint distribution function for a subsequent FORM/SORM analysis

in almost all cases. The FORM and SORM now simplify these functions calculating linear and quadratic tangential surfaces respectively. These surfaces are adapted in the so-called design point y ∗ . This point of the limit state function h(y) is defined via the shortest distance (e.g. in FORM) h(y) − δ=

m j=1

∂h yj ∂y j

2 m ∂h j=1

1/2

(4.329)

∂yj

between h(y) and the coordinate origin of the standardized Gaussian space. From this distance measure the safety index ⎧ ⎪ ⎨ +δ, h(0) > 0 (4.330) β= ⎪ ⎩ −δ, h(0) < 0 is derived. This leads to a simplified formulation of the failure probability Pf ≈ Φ(−β)

(4.331)

in FORM and to Pf = Φ(−β)

m−1 : i=1

in SORM.

(1 − βκi )−1/2

(4.332)


533

A main computational task in these methods is finding the design point by means of suitable search algorithms. Conceptually simple analytical algorithms – like the Hasofer-Lind-algorithm [356] or the derived Rackwitz-Fiessleralgorithm [654] – have been developed initially and are still used for well-behaved reliability problems. As the search of the design point y ∗ can be formulated in terms of an optimization problem, alternatively, also gradient-based optimization strategies like the Sequential Programming method (SQP) [64] are heavily employed. More detailed information on FORM/SORM and, particularly, on further developed derivatives of these methods are presented in [653]. A critical view on this approximate methods conclude to the following statements. In general, these methods only approximate the researched failure probability by ignoring existing non-linearities, e.g. in the limit state function, such that an significant error may be introduced providing only poor information about this possible error. Furthermore, the above explained identification of the design point y ∗ by means of optimization strategies may only deliver a local minimum of the limit state function, possibly ignoring a global minimum. Another disadvantage is the low numerical efficiency when solving high-dimensional structural reliability problems (high-dimensional in terms of number d of random variables). This low efficiency results from the computation of gradients in multiple point of the limit state function as this computation itself comprise – in most cases – multiple FE analyses. In this context, the authors in [711] state a lack of robustness, accuracy and competitiveness compared to simulation methods for d > 30. An exemplary analysis of the influence of the number d on the results is given in [653]. Conversely, the approximate methods are well suited for the computation of small values of failure probability, say Pf ≤ 106 , when reliability problems with a small number d of random variables are to be analyzed. Also problems with multiple limit state functions (union or intersection of failure domains) can be analyzed very efficiently when extended version of FORM/SORM – like summarized in [653] – are employed. Due to this high efficiency for lowdimensional problems (in terms of random variables) these approximate methods are widely used in the scope of Reliability-Based Design Optimization (RBDO, see Section 4.5), as stated in [300, 519]. 4.4.2.2 Simulation Methods Authored by Dietrich Hartmann, Andrés Wellmann Jelic and Yuri Petryna In contrast to the approximation methods named above the class of Monte Carlo Simulations (MCS) has to be mentioned. These methods use the given density functions to create multiple sets of realizations of all random variables. For each set of realizations, a deterministic analysis of the researched limit state function g(x) is performed, in civil engineering dominantly a structural analysis using the Finite Element Method. Afterwards, the results are evaluated concerning failure or survival. In order to simplify the description of the analysis results an Indicator function

534


⎧ ⎪ ⎨ 1, f¨ ur g(x) < 0 I(g(x)) =

⎪ ⎩ 0, f¨ ur g(x) ≥ 0,

(4.333)

is used. This leads to an alternative formulation of the failure probability in eq. (4.323) ∞ I(g(x)) · fX (x)dx.

Pf =

(4.334)

−∞

In a discrete simulation this can be reduced to the finite sum 1 I[g(xi ) < 0] n i=1 n

Pf =

(4.335)

with n is describing the number of simulations and xi is the ith set of generated realizations. The big disadvantage of the classical Monte Carlo Simulation is that the √ accuracy of the estimated results are proportional to 1/ n. Therefore, an increase of accuracy by one order of magnitude demands an increased execution of discrete simulations by around two orders of magnitude. The main reason is the clustered generation of realizations of the random variables near their mean values. As the demanded failure probabilities in structural engineering are very small, an uneconomic number of simulations have to be performed intending to get good estimations. Consequently, the class of variance reducing methods have been developed based on the classic Monte Carlo Simulations. Some variations are e.g Importance Sampling, Stratified Sampling or Adaptive Sampling, more details can be found in [159, 706]. 4.4.2.2.1 Importance Sampling Representative for the above named variance-reducing simulation methods the main principles of Importance Sampling will be explained shortly. The Importance Sampling method moves the main generation point for realizations near the design point y ∗ , shown in eq. (4.329), and then defines a new simulation density function h(v) in y ∗ . This expands the integral in Eq.(4.323) to fx (v) hV (v) dv. Pf = · · · I(v) (4.336) hV (v) Hence, the failure probability can be estimated with Pf =

m 1 fx (vn ) I(vn ) m n=1 hV (vn )

(4.337)


535

using m simulation runs and the sample vn defined by h. In order to calculate approximate estimates for the failure probability a good choice of the sampling density h(v) is essential. The variance of eq. (4.337) is % & 2 m 1 1 fx (vn ) 2 Var[Pf ] = (4.338) I(vn ) − Pf , m − 1 m n=1 hV (vn ) leading to a coefficient of variance υPf =

(Var[Pf ])1/2 . Pf

(4.339)

The exact solution for Pf is obtained for a proportional definition of hV (v) to the real density function fX (v), which, however, implies the knowledge of the searched probability. Instead, [760] proposes the use of the original density function of fV (v), a normal or a uniform distribution. The design point y ∗ can be determined from a pre-executed FORM or SORM calculation, respectively. 4.4.2.2.2 Latin Hypercube Sampling Stochastic modelling of random values for reliability analysis using direct Monte-Carlo simulations requires a huge number of samples, if the failure probability is small. In other cases, one requires solely statistical characteristics of structural response such as displacements or stresses estimated over a certain range of input values. Sometimes, critical parameter combinations corresponding to structural failure conditions, i.e. to limit states, are of interest. For those purposes, the number of the required simulations can be significantly reduced by special simulation techniques such as the Latin Hypercube Sampling (LHS) [285]. According to the direct Monte-Carlo approach, uniformly distributed random values xk , k = 1, 2, ... are generated within the probability range [0, 1] and then transformed into the actual random samples of a certain variable Xk by −1 means of its inverse probability function Xk = FX (xk ). In such a way, a uniform distribution xk can be ”mapped” onto an arbitrary distribution function of interest. For most practically important statistical distributions, like the normal distribution, the probability density function (PDF) is concentrated more or less around the mean value. Thus, rare values Xk corresponding to the tales of the PDF can be reliably generated only within a large number of Monte-Carlo simulations. The main idea of the Latin Hypercube Sampling is to divide the probability range [0, 1] in a Nsim number of equal intervals and take their centroids, randomly permutated, for the mapping onto the probability function of interest (Figure 4.105). At that, Nsim denotes the number of simulations resp. the size of the set of samples. It is evident that LHS covers the entire range of values much better than the direct MCS for the same, relatively small number Nsim . The applications of LHS to stochastic structural analysis in [474, 624] confirm that already ten to hundred LHS simulations provide acceptable results.

536

4 Methodological Implementation M o n te - C a r lo S im u la tio n

F X (x )

v a lu e s

1 .0

x

L a tin H y p e r c u b e S a m p lin g

x

F X (x )

1 .0 k

1 /N

s im

ra n d o m

N

s im

k

X

0 .0 H is to g r a m

X

0 .0

k

M C S

R a n d o m

6

5 4 .5

0 .1

3 2 .5

ra n d o m

0

2

2

-0 .0 5

1

0

3 .5

0 .0 5

3

1 .5

-0 .0 8

-0 .0 6

-0 .0 4

-0 .0 2

0

0 .0 2

0 .0 4

0 .0 6

0 .0 8

0

0 .1

1

L H S M C S

-0 .1

-0 .1

L H S

L H S

4

v a lu e X 4

k

H is to g r a m

s e ts

0 .1 5

M C S 5

X X

X

5

1 0

1 5

2 0

2 5

3 0

3 5

s im u la tio n n u m b e r

4 0

0 .5 4 5

5 0

0

-0 .2

-0 .1 5

-0 .1

-0 .0 5

0

0 .0 5

0 .1

0 .1 5

X

Fig. 4.105. Comparison of Latin Hypercube Sampling and Monte-Carlo Simulation

Figure 4.105 illustrates the difference between LHS and MCS by means of Nsim = 50 simulations: The histogram of the random set of a Gaussian variable generated by LHS is almost ideal compared to that generated by MCS using the same Nsim . Latin Hypercube Sampling has been successfully applied in [624] to stochastic sensitivity analysis of a reinforced/prestressed concrete bridge and to calculation of the limit state points in the framework of the Response Surface Method (Chapter 4.4.2.3). 4.4.2.2.3 Subset Methods A novel and very promising simulation method called Subset simulation (SubSim) has been proposed by Au & Beck in [67] for estimating small Pf values. This method reduces the numerical effort compared to direct MCS by expressing small failure probabilities as a product of larger, conditional probabilities. These conditional probabilities are estimated for decreasing intermediate failure events (subsets) {Fi }m i=1 such that F1 ⊃ F2 ⊃ . . . Fm = F

(4.340)

with F defined as the main failure event. Consequently, the researched failure probability Pf = P (Fm ) = P (F1 )

m−1 :

P (Fi+1 |Fi ) ,

(4.341)

i=1

is defined as the product of all conditional failure probabilities P (Fi+1 |Fi ). By selecting the intermediate failure events Fi appropriately, large corresponding


537

failure probability values are achieved such that they can be computed efficiently by direct Monte Carlo estimators. Three variants of Subset Simulation have been developed so far namely SubSim/MCMC, SubSim/Splitting and SubSim/Hybrid. All variants are based on the same adaptive simulation procedure, however, the differ in the generation of the next conditional sample when reaching an intermediate failure event. A general summary of these variants together with their application in the context of a benchmark study is given in [68]. This benchmark study on reliability estimation in higher dimension of structural systems is organized since 2004 by the Institute of Engineering Mechanics, University of Innsbruck, Austria (Prof. Schuëller, Prof. Pradlwater). The intermediate results of this benchmark, presented in [710], attest a general applicability together with a very high computational efficiency to almost all Subset Simulation variants. 4.4.2.3 Response Surface Methods The reliability assessment of structures is usually focused on the evaluation of the failure probability: ( ) pf = P g(X) ≤ 0 = fx (X)dX. (4.342) g(X )≤0 This task includes, on one hand, an expensive structural analysis for determination of the limit state function g(X) and, on the other hand, the solution of the multi-dimensional integral (4.342). The reliability analysis of large structures imposes a number of typical difficulties influencing the choice of an appropriate approach to calculate failure probabilities: • • •

the need of multiple and expensive nonlinear analyses of the entire structure; generally implicit limit state functions, which can be determined only for discrete values of governing parameters; the vagueness about the parameters dominating the failure probability.

The Response Surface Method (RSM) [271, 160, 712], combined with efficient Monte-Carlo simulations [712], helps to avoid at least the first two difficulties. Its application includes in general the following steps. First, the limit state points are determined by means of deterministic nonlinear structural simulations, as described above. The corresponding critical values of the random variables X (k) belong to the implicit limit state function g(X) = 0 and satisfy global equilibrium of the system, for example in static case: F I (u, X (k) ) = λP (X (k) )

→

X (k) g(X (k) )=0 .

(4.343)

538


In the second step, the actual limit state function g(X) - the response surface - is approximated by an analytical function g ∗ (X) providing the minimum least square error for a set of limit state points X (k) , k = 1, . . . , np : S = min

% np

&1/2 ∗

(g (X (k) ) − g(X (k) ))

2

.

(4.344)

k=1

Polynomials of the second order g ∗ (X) = a + X T b + X T cX

(4.345)

are most frequently used approximations [709]. For nV random variables one needs to find at least np = nV + 12 (nV (nV + 1)) limit state points in order to determine the polynomial coefficients a, b, c uniquely. In the last step, the failure probability is calculated by means of statistical simulation of the analytical limit state function g ∗ (X) (4.345): ⎧ ⎪ N sim ⎨ 0 : g ∗ (X i ) > 0 1 . (4.346) pf = I(X i ) with I(X i ) = ⎪ Nsim i=1 ⎩ 1 : g ∗ (X i ) ≤ 0 The efficiency is generally improved by use of variance-reducing or weighted simulation techniques [707]. The response surface method provides a satisfactory ratio of accuracy and efficiency especially for large structures for a relatively small number of random variables [626, 625, 624]. Further advantages consist in its applicability to arbitrary limit states and failure criteria, for linear as well as nonlinear, static and dynamic problems in similar manner. A clear separation of deterministic structural analysis, of the response surface approximation a and statistical simulations through simple interfaces in form of the limit state points X (k) and the coefficients a, b, c, allows for independent use of the best currently available state-of-the-art solutions at each step. Consistent nonlinear structural analysis of arbitrary sophistication and probabilistic reliability assessment without theoretical limitations become possible. One of the practical advantages of the RSM is its availability in form of commercial software like COSSAN [708]. The disadvantages of the RSM are mainly related to the accuracy of the approximation g ∗ (X) in cases of complex, fold-like or only piecewise-smooth limit state functions [688]. Such situations occur, for example, due to different possible failure modes in dynamic systems. For static or long-term problems, each limit state function is typically defined with respect to the associated load cases and, therefore, can be mostly distinguished from other limit states. Due to the fundamental advantages of the RSM, new types of approximations and algorithms are currently under development, see XX for example. According to [709], the maximum number of random variables acceptable due to efficiency reasons is currently about nV < 20. Therefore, the choice


539

of representative sets of random parameters is a challenging task for reliability analysis as well, but especially by use of the Response Surface Methods. Therefore, a preliminary sensitivity analysis is required in order to weight the impact of each random parameter on the failure probability. 4.4.2.4 Evaluation of Uncertainties and Choice of Random Variables Many parameters in structural analysis are not known exactly and thus introduce uncertainties. Those of the input information can be generally classified into load, material and structural uncertainties. Additionally, we must account for model uncertainties and output uncertainties mirrored in structural response variables. We consider in this contribution only stochastic approaches to handle uncertainties in structural and reliability analysis. An overview of uncertainty models with respect to stochastic finite elements is given in [521]. Stochastic uncertainties and models are already a part of the Probabilistic Model Code developed by the Joint Committee for Structural Safety [815]. For reasonable computer expenses, only the most important variables shall be treated statistically. On the other hand, the set of selected random variables shall reflect all principal sources of uncertainty for realistic response predictions. The importance of each uncertain parameter Xi , independently of its origin, can be quantified by its contribution σi2 to the variance σg2 of the limit state function g(X) approximated at point X in the space of random variables: ' 2 nV nV ∂g '' 2 σg2 = σi2 = · σX . (4.347) ' i ∂X i X i=1 i=1 ' ∂g ' 2 stays denotes the variance of the random variable X and Herein, σX i ∂Xi 'X i for the gradient of the limit state function on Xi at point X computed as follows: ' g(X1 , . . . , Xi + ΔXi , . . . , XnV ) − g(X1 , . . . , Xi , . . . , XnV ) ∂g '' ≈ . ∂Xi 'X ΔXi (4.348) As limit state functions are generally nonlinear, the gradients (4.348) are different for different points X. Therefore, a special sensitivity analysis shall be performed by calculating the gradients (4.348) on a grid of points within a physically meaningful range of values of the considered random variables Xi . Usually, it is sufficient to consider the gradients at the boundaries and in the center of the domain of interest. If the importance measure of Xi , estimated by σi2 , exceeds a given threshold value TOLX :

540


σi2

=

' 2 ∂g '' 2 · σX ≥ TOLX , i ∂Xi 'X

(4.349)

the parameter Xi must be considered as random. The relevant uncertainties are then simulated as individual random variables, random fields or random processes. 4.4.3 Time-Variant Problems Authored by Dietrich Hartmann, Yuri Petryna and Andrés Wellmann Jelic The time-dependent formulation of Pf in eq. (4.325) is equivalent to the distribution of the first passage point of the time-dependent state function into the failure domain. As this distribution is rarely known explicitly, correspondent numerical algorithms have to be employed. The existing algorithms in the literature can be assigned to one of the three following categories. 4.4.3.1 Time-Integrated Approach Basic idea of this approach is the transformation of the time-variant problem eq. (4.325) to a time-invariant problem. This transformation is accomplished by means of an integration of the stress values S as well as the resistance values R over the targeted lifetime TL,D . The failure probability Pf (TL,D ) = P [g (Rmin (TL,D ), Smax (TL,D )) ≤ 0]

(4.350)

is estimated based on the extreme values distributions of the {R, S} values, like proposed by Wen & Chen in [831]. A main disadvantage of this approach is a probable overestimation of the failure probability value Pf as the considered extreme values {Rmin , Smax } rarely occur simultaneously. As a consequence, equivalent time-independent load combinations, like published by Wen in [830], have to be defined for each researched structural reliability problem in order to estimate realistic results of Pf . 4.4.3.2 Time Discretization Approach Basically, this approach overrides the above named drawback of the timeintegrated approach by defining the extreme values {Rmin , Smax } for a time period ΔTi shorter than the demanded lifetime TL,D , e.g. a duration of a single storm or, alternatively, one year. At first, the failure probability Pf (ΔTi ) = P [g (X (ΔTi )) ≤ 0]

(4.351)


541

within the time period ΔTi is computed based on the extreme values X corresponding to this period. Subsequently, by solving the hazard function hTL,D (T ) =

fTL,D (T ) 1 − FTL,D (T )

(4.352)

as defined in the literature ([47, 525]) for the time T the researched failure probability can be estimated as follows: T Pf (T ) =

hTL,D (t)dt .

(4.353)

0

Further information about the theoretical aspect of this approach together with representative applications is given in [622]. This time discretization approach allows the approximate computation of time-variant Pf values. Thereby, based on the fact that variance-reducing simulation techniques for time-invariant reliability analyses can be used for the discrete time period ΔTi the overall computation can be accomplished within a relatively short runtime. A restriction to be stated in this context is the inefficiency of this method for the solution of dynamical structural problems like the following example analyzed in Section 4.6.4. 4.4.3.3 Outcrossing Methods Within this approach the time-variant reliability problem in eq. (4.325) is formulated as an outcrossing or first passage problem, respectively, based on the positive outcrossing rate νξ+ and the failure threshold ξ. Analogously to the time-invariant solution methods, also, approximation as well as simulation methods have been developed in order to solve this first passage problem. The existing approximation methods have drawbacks with respect to their applicability. Rackwitz states in [653] that only specific categories of problems have been solved by employing these methods, e.g. problems with Gaussian vector processes or, alternatively, with rectangular wave renewal processes [152]. Conversely, the existing simulation methods exhibit a general applicability allowing the runtime-efficient estimation of the time-variant failure probability. For that, the demanded evolution of the failure probability over the time can be computed by estimating the first passage probability Pf (τ ) = P (T ≤ t) = FT (t) = Ie (t, x0 , ω)dF (ω) (4.354) Ω

in a time interval [0, t]. The vector x0 contains the initial conditions of all stochastic processes and ω is a probability parameter in the probability space Ω, respectively. Furthermore, the first excursion indicator function

542


Ie (t, x0 , ω) =

1 for ∃τ ∈ [0, t] : yω (τ ) ≥ ξ 0 for ∀τ ∈ [0, t] : yω (τ ) < ξ

failure domain safe domain

(4.355)

is evaluated at a given time t and indicates whether the resulting stochastic process yω (τ ) exceeds a predefined threshold level ξ. For the solution of the time-variant reliability problems dealt with in the Collaborative Research Center 398 the Distance-Controlled Monte Carlo Simulation (DC-MCS) has been adapted. The DC-MCS method was proposed by [647] and facilitates a runtime-efficient solution of complex dynamical systems. It is based on a time-variant MCS including a vector wt (ω) of weight values for each of the nsim realizations in the generated ensemble. Initially, all vector elements are set to a value wt (ω) =

1 w ω0 nsim t

where wtω0 = 1 .

(4.356)

At predefined time steps of the simulation, this weight vector is modified dynamically aiming at an approximately uniform density function of the correspondent realizations in the sample space. This targeted uniform density function leads to an increasing number of realizations in the researched sample state area of rare failure events. The modification is accomplished in terms of the Russian Roulette & Splitting (RR&S) technique which doubles ’important’ realizations (S) and deletes ’unimportant’ ones (RR). Thereby, a priori undefined importance is quantified during the runtime by means of an evolutionary distance criterion presented in [647]. 4.4.4 Parallelization of Reliability Analyses Authored by Dietrich Hartmann and Andrés Wellmann Jelic Processing uncertainties by means of probabilistic methods to determine structural reliability results in exeptionally increased computational effort even if only moderately complex structures are to be considered. This obvious difficulty, known already for a long time, forms the main obstacle for a rapid and enthusiastic acceptance of probabilistic methodologies in practical engineering. Needless to say that nevertheless the application of uncertainty methods is getting mandatory in the time to come. In this context, parallel processing or distributed computing, including modern methods of autonomous computing, e.g. agent-based parallelization, appear to be an appropriate way of overcoming the dilemma and existing drawbacks because parallelization of reliability analysis enables drastic cuts of the computer time required for a given task. In addition, cost barriers placed by expensive special parallel computer systems in the past have become obsolete as clusters made out of customary personal computers are available as well as affordable for civil engineering institutions. From the viewpoint of algorithms or software implementation, reliability analysis methods furthermore allow for


543

a vast parallelization because numerous tasks and subtasks can be executed independently; a representative example for this statement is discussed in the next two sections. 4.4.4.1 Reliability Analysis of Fatigue Processes Evaluations of fatigue tests like [205] have proven a considerable scatter of experimental lifetime results. Consequently, the material properties in fatigue simulations have to be modelled in terms of random variables or stochastic processes, respectively, if realistic lifetime values are to be strived. The inclusion of random properties in the numerical model, of course, leads to a highly complex sample space to be analyzed. In order to solve this complex task a hybrid simulation concept has been developed [268] and evaluated positively. This hybrid concept proposes a decoupled sample space, distinguishing between a sample space ΩR0 of material parameters R0 as well as a sample space ΩDt of loading-induced fatigue processes. Both sample spaces are coupled based on the conditional probabilistic formulation Dt |r0 where r0 describes the realization of initial material parameters which are deteriorating over the simulated lifetime. The above-named vector ΩR0 may contain different quantities depending on the numerical model employed for simulating the fatigue of materials. As an representative example the vector ΩR0 = {ΔσD , k, DLim }T is to be defined when using a S-N approach based on Wöhler curves. A more detailed explanation of this stochastic S-N approach together with a typical application is given in one of the following examples (s. Section 4.6.4). Based on this theoretical decoupling the general integral in Eq.(4.323) can be written into Pf (t) = ΩR0

⎡ ⎣

⎤ I [g(D(t)|r0 )] · dFD|R0 (D(t)|r0 )⎦ fR0 (r0 )dr0 (4.357)

ΩD

where FD|R0 defines the distribution function of the above explained conditional formulation. The inner integral represents a time-variant conditional reliability problem for R0 = r0 such that Eq.(4.357) takes the following form Pf (t) = (Pf (t)|R0 = r0 ) fR0 (r0 )dr0 . (4.358) ΩR0

The time-variant reliability problem Eq.(4.358) is solved by using the aforementioned DC-MCS using a predefined number of realizations. The outer integral forms a time-invariant reliability problem which is currently solved by means of a direct MCS. Finally, the researched time-variant failure probability

544


Pf (t) = ER0 [Pf (t)|R0 = r0 ]

(4.359)

can be estimated by computing the expected value of the conditional failure probability in sample space ΩR0 of the initial material properties. 4.4.4.2 Parallelization Example In the scope of the above named hybrid simulation concept, several levels of parallelization can be identified. One possible level is the fine-grained parallelization of the time-variant DC-MCS conditionally defined for R0 = r0 . For that, the ensemble of generated damage process realizations can be split up into smaller ensemble groups which could be processed each on a single computing node of a distributed memory system. Due to the already explained dynamical modification of the weight vector wt (ω) the parallel simulation processes have to be synchronized at prescribed time steps based on an explicit message passing technique. Such a fine-grained level of parallelization has been confirmed in [418] as very efficient for reliability problems containing complex structures. However, this approach will not allow an efficient parallelization of the fatigue-related reliability analyses researched in this contribution. One reason for that is the relatively low computational effort at each time step of the simulated fatigue process. Instead of starting each time a structural analysis (as done in [418]), already computed stress values are extracted (in the so-called macro time scale) from a database, then mapped to correspondent partial damage values and, finally, accumulated until a limit damage value is reached. The database is filled in before in a separate computation (within the micro time scale) by starting structural analyses for a predefined parameter range (more information on this multi-scale approach in Section 2.1.4 and in Section 5.1.2). Hence, the computational effort of the simple accumulation is very small compared to the needed communication between all parallel processes and, therefore, leads to a very poor parallel efficiency. This poor efficiency has been confirmed by [418] for low dimensional structural problems. Based on the explained low computational effort, a more coarse-grained level of parallelization is used in the work presented here. More precisely, the time-invariant MCS for solving the Eq.(4.359) is parallelized on a parallel system. This is achieved first by generating a vector r0 of realizations of initial material properties. Subsequently, a DC-MCS of an ensemble of damage processes Dt |r0 is simulated on each available computing node based on the predefined realization r0 . When the demanded number of DC-MCS ensembles has been computed all resulting estimates of the time-variant failure probability are collected in the master process and combined to estimate the expected value of the researched failure probability Pf (t). A simplified overview of this parallelization approach is shown in Figure 4.106. A major advantage of this coarse-grained parallelization is the comparatively low communication overhead needed to synchronize the parallel

4.5 Optimization and Design

545

Total damage d ωi Limit damage D Limit

¢di

Workstation

Workstation

Workstation

: Damage sum Σ di = D Lim

Time T 2. Simulation with r 0,2 3. Simulation with r 0,3

Workstation

n. Simulation with r 0,n Workstation

Realisation r0 of material properties

Fig. 4.106. Parallel execution of stochastically independent DC-MCS of fatigue analyses on a distributed memory architecture [824]

processes. Discrete message passing is basically needed only at the beginning as well as the end of a child process. This advantage is highly favorable, in particular, on a distributed memory system with a relatively slow interconnect between the computing nodes as used in the research project reported here.

4.5 Optimization and Design Authored by Dietrich Hartmann As demonstrated in Section 4.1.4 structural design problems can be transformed into equivalent structural optimization problems because ‘optimizing something’ is always inherent in design. Naturally, only the numerically representable aspects of a design problem can be captured by such a conversion. It is customary to divide structural optimization into subcategories, in dependence of what types of design variables are to be optimized. Further characteristics are then catenated to the methodology applied within the iterative optimization as structural analysis kernel. From the previous remarks it is known that different finite computational methods may be chosen. Details of the canonical classification in optimum design are, therefore, resolved in the next subsection. Once a structural optimization model is established, i.e. the optimization criterion, the design variables and the set of constraints have been defined, the numerical optimization strategy needs to be determined. The dilemma is that a myriad of rival optimization methods have been developed since the past decades, and new methods are still underway. Hence, it is not an easy taste to select the best suitable optimization method compatible to the problem given. Obviously, this selection is never unique. In the subsection after next,

546


therefore, representative optimization methods are to be introduced provided that they are applicable with respect to lifetime-oriented design problems. At the end of this section, the aspects of parallelization of optimization strategies are addressed. The parallel solution of optimum design is always necessary if a large scale optimization problem has to be dealt with where the response time of a computer becomes unacceptable (e.g. months or even years). 4.5.1 Classification of Optimization Problems Authored by Dietrich Hartmann The required numerical formalization of an optimization problem for computer-based solutions leads to the following description of the aforementioned design variables, optimization criterion and constraints: • •

•

design variables xi , i = 1, 2, 3, . . . , n representing the vital parameters of a structural system, concentrated in the design vector x optimization criterion or objective function introduced as f (x) and, as a rule, being a non-linear function of the design variables xi , i = 1, 2, 3, . . . , n. In some cases of optimum design the function f (x) turns into a general sequence of instructions (algorithm) instead of a ‘facile’ mathematical function such that a algorithmic non-linearity appears constraints where three categories are possible: ◦ side constraints for upper and lower limits of specified design variables xi ∈ R xi ≤ xi ≤ xi or in vector form x≤x≤x ◦ equality constraints hk (x) = 0, k = 1, 2, 3, . . . , p

or

h(x) = 0

or

g(x) ≤ 0

◦ inequality constraints gj (x) ≤ 0, j = 1, 2, 3, . . . , m

using the relational operator of the ≤-type to declare the inequality. Again, in some case of optimum design specified components in the set g(x) ≤ 0 may be described in terms of an algorithm to express practicable limits to the choice of design variables. According to the notation mostly used in structural optimization the optimization is defined as a minimization problem. This, the optimization problem takes the general format


' ⎫ ' 'x ≤ x ≤ x ⎪ ⎪ ⎪ ' ⎪ ⎬ ' ' min f (x) ' h(x) = 0 x ⎪ ⎪ ' ⎪ ⎪ ⎪ ⎪ ' ⎪ ⎪ ⎩ ' g(x) ≤ 0 ⎭

547

⎧ ⎪ ⎪ ⎪ ⎪ ⎨

(4.360)

If algorithmic definitions have to be applied, either in the optimization criterion or the set of the use qualities, this can be expressed by means of the ‘alg’-operator ' ⎫ ' ⎪ 'x ≤ x ≤ x ⎪ ⎪ ' ⎪ ⎬ ' ' min alg → f (x) ' alg → h(x) = 0 x ⎪ ⎪ ' ⎪ ⎪ ⎪ ⎪ ' ⎪ ⎪ ⎩ ' alg → g(x) ≤ 0 ⎭ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨

(4.361)

In cases where competing optimization criteria are reasonable, because one criterion alone cannot fully cover the desired goals of the optimization approach, the function f (x) becomes a vector as well and the term f (x) has to be rewritten f (x) → fl (x), l = 1, 2, 3, . . . , lmax or in vector form f (x) → f (x) The multi-criteria optimization is also called vector optimization. In this book, however, only single-criterion optimization problems are to be considered without exception. 4.5.2 Design as an Optimization Problem Authored by Gerhard Hanswille and Yuri Petryna From the discussion of the algorithmic nature of quantities it should be made clear that structural optimization distinguishes from ‘pure’ mathematical optimization. This fact becomes also visible if the optimization variables, i.e. the design variables are looked at. While mathematical optimization variables have neutral character, in structural optimization, the variables are linked to a physical meaning associated with consequences to how to solve the optimization problem numerically. If design variables are describing the topology of a structure they are usually discrete, integer or binary variables such that a discrete non-linear optimization problem is created. Topology optimization is again not in the focus of this book. Here, of prime interest are, however, sizing and shaping of structural systems and their structural parts

548


assuming that the choice of the structural material (e.g. concrete, steel, etc.) is fixed already and must remain unaltered. As a result, material optimization is disregarded such that only size and shape optimization is dealt with. In lifetime-oriented design, size as well as shape optimization can appear apart to each other but also jointly. The design variable in both cases are continuously varying and/or of discrete nature, e.g. if specified profile tables must be met. Accordingly, the optimization problem is continuously non-linear or discretely non-linear. Compared to the mathematical view on optimization problems, much more incisive is the implicit nature of structural optimization problems. Implicitness emerges through the necessity to embed structural analysis approaches into the optimization. As known in large scale problems, including complicated structural systems subjected to complicated actions, comprehensive numerical methods and program systems (finite element programs as already mentioned) are needed to compute the desired structural response (kinematic quantities, stresses, etc.). Depending on the optimization problem the response quantities are incorporated either in the optimization criterion or within the equality constraints. Implicit in this context, therefore, means that the relevant components of the constraint vector g(x) ≤ 0 only indirectly depend from their design variables. Vice versa, to be able to compute an implicit constraint, say the j-th constraint gj (x) ≤ 0, induces a complete and probably computationally extensive structural analysis of the total structure for the current design vector x. It is readily identifiable that this forms a serious obstacle because of computational effort. This effort is affected by the computational methodology that is needed to obtain an appropriate but also realistic structural response r. With respect to the computations utilized in the lifetime assessment various problem dependent computing methods come into play. If linear structural behavior can be assumed in specified time intervals during the lifetime of a structure, because non-linear response r is negligible, the system equations of a structure, according to the finite element method, have the following forms: a. Linear stiffness equations if motions are not present: K(x) · u(x) = F (x)

(4.362)

where K(x) = stiffness matrix of the structure whose components are a function of the design variables x generalized u(x) = displacement vector, also being a function of the vector x and describing the basic constituent of the structural response r F(x) = generalized load vector as a function of the design variables x b. Systems equations of motion if dynamic effects need to be include into the design


˙ M(x)¨ u(x) + D(x)u(x) + K(x)u(x) = F (t)

549

(4.363)

where M(x) D(x) F(t) ¨ (x), u(x), ˙ u u(x)

= = = =

mass matrix depending on the design variables x damping matrix excitation load as a function of time t kinematic quantities, i.e. accelerations, velocities and displacements, representing structural response r

c. System equations for bifurcation problems to compute critical loads K(x) − λ(x)Kg (x) u(x) = 0 (4.364) where Kg (x) = geometric stiffness matrix of the structure λ(x) = vector of eigenvalues from which critical load can be determined If non-linear structural behavior is dominant, then, the finite element equations for the static elastic-plastic problem can be written in an incremental form leading to

K x, u(x), σ(x) · Δu(x) = ΔF (x) (4.365) where σ(x) = stress vector computed from the displacements u(x) Δu(x) = vector of displacement increments ΔF (x)= load increment vector The incremental equation has to be solved in association with the governing yielding function Φ(x) = 0. Based upon the discussion of the potential governing system equations for a total structure, the optimum design problem can be substantiated more target-oriented. The structural response r(x) contains displacements u(x), stresses σ(x), etc. which are computed from the individual finite element equations as indicated above. These equations can be recapitulated by the generalized system equation

S x, u(x), σ(x), . . . = 0 (4.366)

from which the generalized structural response r x, u(x), σ(x), . . . is determined. This view on the design problem allows the reformulation of the optimization problem introduced above in eq. (4.360). The equality equations are replaced by the generalized system equation. Also, those parts of the inequality constraints that have to be evaluated by

550


the aid of quantities in the response vector r and are, therefore, implicit constraints are rewritten. This, the relations ' ⎧ ⎫ ' ⎪ ⎪ ' ⎪ ⎪ ⎪ ⎪ 'x ≤ x ≤ x ⎪ ⎪ ⎪ ⎪ ' ⎪ ⎪ ⎪ ⎪

' ⎪ ⎪ ⎨ ⎬ ' S x, u(x), σ(x), . . . = 0 ' (4.367) min f (x) ' x ⎪ ' g impl (x, r(x, u(x), . . . )) ≤ 0, j ∈ J impl ⎪ ⎪ ⎪ ⎪ ⎪ j ' ⎪ ⎪ ⎪ ⎪ ' ⎪ ⎪ ⎪ ⎪ ' expl ⎪ ⎪ ⎩ ⎭ expl ' gj (x) ≤ 0, j ∈ J are obtained where where gjimpl gjexpl mimpl mexpl m

= = = = =

implicit constraints j = 1, 2, 3, . . . , mimpl , j ∈ J impl explicit constraints j = 1, 2, 3, . . . , mexpl, j ∈ J expl total number of implicit constraints total number of explicit constraints total number of inequality constraints, i.e. m = mimpl + mexpl .

Hereby, it is understood that the objective function f (x) is formulated in an explicit fashion, e.g. in terms of a cost equivalent expression such as weight depending directly from the variable x. If the objective function needs to have a parts of the response vector r too, e.g. specified maximal stresses to be minimized, the objective function f (x) becomes implicit as well. Hence, in this case f (x) → f (x, r(x), σ(x), . . . ) .

(4.368)

The fact that some of the constituent parts g(x) ≤ 0 or f (x) may have an algorithmic nature and are not ordinary mathematical functions is mentioned only and not put into the above relations. The advantage of structural optimization to create the best possible design, although highly desirable, bears potentially serious risks. Since the optimization method applied is inherently adjusted to bring out the best subject to the active constraints, the smallest exceedance of these constraints through imperfections during the erection of a real world structure, may cause infeasible designs. This fact can put optimization into question because failures, collapses and damages can happen. To avid unacceptable and infeasible designs in practise and to achieve robust optimum designs, structural reliability must be incorporated into the optimization model, taking into account the governing uncertainties with respect to structural parameters, geometry of the structural system and loading. As expedited in the previous section on uncertainties, here, solely stochastic structural uncertainties are to be examined. This necessitates a further revision of the optimization model and a further amplification by means of stochastic quantities leading to stochastic structural optimization. Of course, the


551

computational effort increases considerable, however, in particular in lifetimeoriented design the enhancements are clearly indispensable. Modification of the deterministic structural optimization model requires to introduce stochastic parameters (basic variables) Y , e.g. stochastic material parameters using given probability distributions, as well as actions or loadings which have to be represented in terms of time-variant stochastic processes Z(t), e.g. wind actions or traffic loads. The transition to a stochastic nonlinear stochastic optimization problem applied to structural systems leads to the following formulation ' ⎧ ' ⎪ 'x ≤ x ≤ x ⎪ ⎪ ' ⎪ ⎪ ' ⎪ ⎪ ' ⎪ ⎨ ' S x, u(x), σ(x), . . . , Y , Z(t) = 0 min f x, Y , Z(t) '' x ⎪ ' g impl (x, r(x, u(x), . . . , Y , Z(t))) ≤ 0, j ∈ J impl ⎪ ⎪ ' j ⎪ ⎪ ' ⎪ ⎪ ' expl ⎪ ⎩ ' gj (x, Y , Z(t)) ≤ 0, j ∈ J expl

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(4.369) where the newly introduced quantities denote Y = vector of the stochastic basic variables Y α , α = 1, 2, 3, . . . , αmax Z(t) = vector of the stochastic processes including loadings and damage processes Z β (t), β = 1, 2, 3, . . . , β max t = time. As a consequence of this, the optimization problem is representing a multidimensional highly non-linear stochastic problem. Since the solution must be satisfied in each point of time and space, a semi-infinite structural optimization problem is encountered. 4.5.3 Numerical Optimization Methods Authored by Dietrich Hartmann The various categories of nonlinear optimization problems have led to an enormous plenitude of different optimization concepts, methods and strategies. From an engineering viewpoint, the determination on which method is better than others primarily depends on the targets that govern the structural optimization problem considered. In the cases where the optimization quantities represent smooth optimization domains perhaps even in association with uni-modality or convexity (such that definitely only one global optimum exists), it is obvious to apply derivative-based methods. Since the first partial derivatives of the objective function and constraints provide significant topological information on the behavior of the optimization domain if a small change with respect to the independent variables takes place, derivative-based

552


optimization methods tend to be extra-ordinarily fast. The convergence towards the optimal solution is even more accelerated if second order information in terms of the second partial derivatives of functions is exploited, leading to Newton- oder quasi-Newton methods. However, derivative-based methods become obsolete if smoothness, convexity, uni-modality, continuity etc. are no longer valid, when more realistic (and complex) structural optimization problem are to be solved. Then, robustness, general applicability and the chance to find the global optimum solution amongst many local optima, of course without negligence of the computational effort, are the key features of interest. As a consequence, derivative-free methodologies are indispensable. Although, tremendous progress has been made in the improvement of derivative-free optimization in the past years, research in this area continues to develop even better techniques. An extract of the state-of-the-art with respect to both, derivative-based and derivativefree optimization, is to be outlined in the next subsections. Hereby, the focus is particulary placed on those aspects that are relevant to lifetime-oriented design approaches. 4.5.3.1 Derivative-Based Methods The governing constraints in complex structural optimization models, in particular the implicit nature of the mechanics-determined components in the set of constraints, prevents the application of the well-known transformation methods which transform the original constrained problem into a sequence of unconstrained subproblems. The solution of the unconstrained subproblems are hereby tailored such that a rapid and safe convergence of the approximated optima in the sequence towards the actual optimum can be obtained. To this end, a problem-dependent sophisticated composite function is constructed from the given objective function and the constraint function along with certain controlling (or penalty) parameters. Prohibitive is the modeling effort, but also the navigation of the optimization iteration forms a serious obstacle. As a consequence, transformation methods are of only vanishing relevance in complex lifetime-oriented optimization. Therefore, popular variants of the transformation methods, like Sequential Unconstrained Minimization Techniques SUMT, e.g. the various Barrier Function Methods and Penalty Methods, and also the Multiplier (Augmented Lagrangian) Method have not been come into operation in the lifetime-oriented design. Detailed reasons why these methods become inappropriate may be consulted from the references [674], [335] or [64] which also indicate the already long history of these methods. In contrast to transformation methods, optimization methods - often called primal methods - to directly solve the original constraint optimization problem are more suitable for complex engineering optimization problems. Two reasons can be named for that: First, the modeling of the respective optimization problem is less time consuming. Second, the navigation of the iteration process


553

to find the searched optimum can be mastered with less interactions due to the better condition of the optimization problem compared to transformation methods. It is not surprising that primal methods are therefore favored if derivativeoriented methods can be applied to complex nonlinear optimization subject to nonlinear constraints. They are applicable if first and/or second order, or approximations of them, are available, either in terms of explicit derivatives (manually determined) or derivatives created by symbolic manipulators (Maple, Mathematica, etc.). As a result, so called Sequential Programming Techniques emerged leading to the three variants Sequential Linear Programming (SLP), Sequential Quadratic Programming (SQP) and Sequential Convex Programming (SCP). Most recently, SQP has proved to be an efficient and reliable method. Properties like sufficient smoothness of the functions and limitations to only small and medium-sized optimization problems are no longer required because recent enhancements and improvements, respectively, have overcome such difficulties (see [562], [410] and [504]). Hence, SQP methods represent the state of the art in nonlinear derivative-oriented solutions methods that outperforms many other methods in terms of efficiency, accuracy, applicability and handling. Nevertheless, SQP cannot be used simply as black box tool because expertise and human interaction is still mandatory in specific cases. Without going to much into details, the SQP approach is to described briefly: The general nonlinear constrained problem is transformed into a sequence of also constrained quadratic subproblems that stepwise approximate the real optimum. In analogy to the transformation methods in the unconstrained case, the sequence of approximations converge against the constrained optimum solution. All in all, the SQP mimics Newton’s method for constrained optimization. The basic idea is the formulation of a quadratic programming subproblem based on a quadratic approximation of the Lagrangian function L(x, λ) = f (x) +

m

λj gj (x)

(4.370)

j=1

where λj = Lagrange multipliers for inequality constraints gj (x) (to be considered only in structural optimization because equality constraints are incorporated into the implicit inequality constraints) The quadratic programming subproblem is based on quadratic objective function and linear constraints. To obtain such an approximate subproblem a Taylor series expansion of the original optimization about the current point of interest x(k) is established. By that the quadratic programming subproblem (QPSP) takes the following form assuming minimization as the standard optimization approach:

554


, min

1 (Δx(k) )T · ∇2 L(x(k) ) · Δx(k) + ∇f (x(k) )Δx(k) 2

(4.371)

subject to ∇gj (x(k) )T Δx(k) + gj (x(k) ) ≤ 0,

j = 1, 2, 3, . . . , m

(4.372)

where direction vector Δx(k) = ∇2 L(x(k) ) = positive definite approximation of the Hessian matrix of the Lagrangian function L (see eq. (4.371)), also designated as H(k) , which contains the ordered second partial derivatives of the function L The subproblem is solved by using any QP algorithm as available in many software libraries and yields Δx(k+1) as the direction for the new iterate x(k+1) = x(k) + α(k) Δx(k)

(4.373)

The step length α(k) needs to be computed as well and can be determined through an appropriate line search procedure. Of particular importance is updating of the Hessian matrix H(k) at each major iteration using a quasi-Newton updating formula. Customarily, a so called Broyden-Fletcher-Goldfarb-Shannon (BFGS) update is carried out according to H(k+1) = H(k) +

q (k) · q (k)

T

T

q (k) · s(k)

T

−

H(k) · H(k) T

s(k) · H(k) s(k)

(4.374)

where s(k) =x(k+1) − x(k) m q (k) =∇f x(k+1) + λj ∇gj x(k+1) ⎛

j=1

− ⎝∇f x(k) +

m

⎞

(4.375)

λj ∇gj x(k) ⎠

j=1

and λj = estimates of the Lagrange multipliers Using the SQP approach gives rapidly an optimum design vector for structural design, provided that continuous functions can be assumed.


555

4.5.3.2 Derivative-Free Strategies In practical engineering problems, where the design optimization must be based on nonlinear mechanics simulation methods, e.g. nonlinear transient finite element computations, the assumptions or preconditions of first and second differentiability can often no longer be perpetuated. The more complex a structural optimization becomes, the more intricate gets the topology of the multi-dimensional optimization domain. Multimodal optimization criteria (objective functions), discontinuities and (algorithmically) nonlinear constraints, inducing “jagged” and loopy boundaries of the optimization domain, enforce the application of derivative-free optimization strategies. This all the more, if multi-scale or multi-level optimization has to be employed, as is the case in lifetime-oriented design problems to which the key attention is turned here. Derivative-free strategies unexceptionally apply zero order information of the optimization model, i.e. “simple” function evaluations to search for the global optimum. Hence, such strategies are also named search strategies. Since derivative-free strategies abstain from derivative information’s (gradients and Hessian matrices), of course, they tend to have a slower convergence in all the cases which are tailored to derivative-oriented methods. On the other hand, they are by far superior if derivatives are unstable or not available at all. Furthermore, search strategies by far outperform derivative-bounded methods with respect to robustness, general applicability and global convergence (which is significant in multimodal optimization). Historically, the development and research of derivative-free methods has created a high opulence of miscellaneous strategies which can roughly distributed into • • • •

enumeration strategies, Monte-Carlo-Strategies, direct search algorithms, evolutionary algorithms.

With respect to performance and popularity, only the last category needs to be considered here. Evolutionary algorithms represent approaches which mimic natural optimization strategies inherent in biological processes, e. g. the interplay between mutation, variation and selection of the fittest. Transferred to mathematical optimization models, it has been figured out, in the past two decades, that biologically motivated optimization paradigms demonstrate surprisingly powerful sophistication and successes in formerly hopeless problem cases. Nearly isochronally, two main currents of evolutionary algorithms have emerged; (i) genetic algorithms and (ii) evolution strategies (compare references [382, 419, 316, 661, 718]). Although the stochastic rationale of genetic algorithms and evolution strategies has much in common, because both strategies adopt similar

556


optimization mechanisms (mutation, recombination, selection, etc.) from the biological ideal, evolution strategies are preferred in lifetime-oriented reliability-based structural optimization: Evolution strategies show advantages in this area because the real-valued design variables are directly implemented, instead of bit-string codings used in genetic algorithms. Moreover, in the past two decades a comprehensive theoretical body of knowledge could be acquired that has led to an independent consistent philosophy of evolution strategies, along with a huge family of strategy variants or alternatives, respectively. The research in this field is still in progress, e.g. the most modern development are so-called meta-evolution strategies representing nested levels of population-based evolution strategies (evolution-evolution methods like function-functions or functionals in the variation calculus). Hereby “population” means that many optimization vectors are examined simultaneously as point clouds instead of single candidates x(k) , e.g. in the aforementioned SQP method. Further details on the common and different behavior of both approaches can be learned from the publication [380]. In the following, the internal operational steps as well as the underlying principles of a medium-sophisticated evolution strategy (ES) is to be explained as brief as possible. A good representative for the family of evolution strategies epitomizes the so called (μ/$+, λ)-ES. This strategy is the result of long-term research and unifies various branches of strategy variants in one generalized concept. The parameter μ, $ and λ indicate that a population-based strategy is considered. According to optimization mechanisms in the biological evolution process, entire populations,in the mathematical optimization sets S of candidate optimization vectors, are created to trace a (global) optimum within the solution domain. Hereby, the parameter μ determines the number of input or parent vectors within the current iteration/generation which creates in total λ offsprings or “child” vectors. This generation entilities various stochastic optimizing mechanisms, like different mutation and recombination schemes. In particular, the stochastic recombination is navigated by means of the parameter $ which defines the number of individuals of the μ parents participating in a specified recombination (discrete or intermedial) process to create a new child vectors. The $ instances (2 ≤ $ ≤ μ) are chosen randomly from the μ parents. The (+)- or (,)-symbol represents two selection options; while (+) symbol means that both the parents and childs are the μ+ λ competitors with respect to the selection of the μ parents in the next iteration step of the optimization process, the (,)-symbol indicates that only the λ childs are evaluated regarding their fitness to generate the next μ parents. The operating sequence in the g-th iteration/generation starts from a parent generation (point cloud) 0 / SPg = xgP1 , xgP2 , xgP3 , . . . , xgPμ (4.376)


557

If a recombination schema, described subsequently, is applied as an option, g then, an intermediate set SˆC of $ child vectors is created leading to 0 / g (4.377) = x ˆgC1 , x ˆgC2 , x ˆgC3 , . . . , x ˆgC , $ ≤ μ SˆC Using a mutation, i.e. a Gauss-normally distributed step in the n-dimensional g optimization space about a point given by the vectors in the sets SPg or SˆC (if g recombination is active), the actual set SC of λ child vectors is established. Hence D C g SC (4.378) = xgC1 , xgC2 , xgC3 , . . . , xgCλ , λ μ This set forms the basis for the next, hopefully ameliorative parent generation SPg+1 . To accomplish this, according to the principle of the survival of the fittest, a selection mechanism is activated. In the case of a (+)-selection, the g union of SPg and SC defines the candidates for selecting the best μ vectors with respect to the optimization model (optimization criterion and constraints). Thus, μ best of (μ+λ)

g SPg ∪ SC −−−−−−−−−−→ SPg+1 .

(4.379)

In the (−)-selection, the parent set SPg is no longer racing (“lethality” pring ciple) and only the child generation SC represents the selection basis. Hence, μ best of (λ)

g g+1 SC −−−−−−−−→ SC .

(4.380)

The recombination mechanism, represented by the parameter $, acts on $ parents of the μ candidates in the set SPg , where 1 < $ ≤ μ, but $ = μ is always a good choice. To choose the recombination participants, $ equally distributed random numbers Rd from the interval [1, 2, 3, . . . , μ], i.e. Rd1 , Rd2 , . . . , Rd , are extracted which are then subjected to a recombination (crossing over) mechanism. Two distinct options are eligible which, for exemplification, are explained only for a single optimization variable, designated as xβC , in one of g the different child vectors in the set SˆC . The discrete recombination results in the following assignments: ⎧ ⎪ ⎪ xβP,Rd1 , ∀ 0 < R < 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ xβ , ∀ 1 < R < 2 P,Rd2 xβC . (4.381) ⎪ .. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩x , ∀ −1 < R < 1 βP,Rd

where R = random[0, 1]. Therefore, the probability that a parent variable is chosen is equal to 1/$. The average recombination adds up to a shorter expression:

558


1 = xβ . $ j=1 P,Rdj

xβC

(4.382)

Correspondingly, the recombination pattern has to be repeatedly used on g all $ parent vectors until the set SˆC containing the individuals xˆgCα , α = 1, 2, 3, . . . , $, is determined. The mutation mechanisms in the (μ/$+, λ)-ES is the heart of the strategy and the most vigorous optimization force. At this, a specific strength is the property that besides the original optimization/design variables, also the step lengthes during the iteration of the optimization are becoming part of the continuous adjustment and adaptation of variables towards optimal quantities. In the most general case, the population-based model allows for an adaptive adjustment of the step lengthes of each corresponding optimization/design variable (called anisotropic mutative step length control). This necessitates to expand the original optimization vector x by the step lengths or strategy parameters, concentrated in the vector Δ, leading to the new vector ⎡

⎤ x1 ⎢ x2 ⎥ ⎢ ⎥ ⎢. ⎥ ⎢ .. ⎥ ⎥ ⎢ ⎢ ⎥ x ⎢ xn ⎥ x ˜= =⎢ ⎥ ⎢ δ1 ⎥ Δ ⎢ ⎥ ⎢ δ2 ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎣. ⎦ δn

(4.383)

where the components δi , i = 1, 2, 3, . . . , n are the step lengths associated to the variables xi , i = 1, 2, 3, . . . , n. The mutation schema to create new child vectors from parent vectors takes the following form if the i-th optimization variable of all λ child vectors in the g set SC are contemplated exclusively: xgiC = xgiP,R + δiP,R1 · ξi1 · [∼ N (0, 1)] 1

1

xgiC = xgiP,R + δiP,R2 · ξi2 · [∼ N (0, 1)] 2

2

.. .

(4.384)

xgiC = xgiP,R + δiP,Rλ · ξiλ · [∼ N (0, 1)] . λ

λ

Of course, these instructions of generations have to be carried out for all indices i, i = 1, 2, 3, . . . , n. The following explanations clarify the effects of the equations above. Parent variables are the originators of the generation where at the parent individuals are again randomly drawn from the set SPg according to


Rj = random of{1, 2, 3, . . . , μ} ,

j = 1, 2, 3, . . . , λ ,

λμ.

559

(4.385)

The increments added to the parent components are also random quantities. According to the nature of a mutation they are mainly driven by Gaussdistributed values (large changes are rare, small changes are more frequent!). This behavior is assured by computing a new Gauss-normally distributed random number from the Intervall [0,1], indicated by [∼ N (0, 1)], for each equation of the above generation instruction. Multiplying such random numbers by appropriate scalars yield the standard deviation of the Gauss distribution which can be interpreted as a step length to navigate the optimization process. The two factors in front of [∼ N (0, 1)] both together represent the standard deviation without going into the details (for an accurate derivation see e.g. [607, 340]), It should be only mentioned here that the ξ-quantities are providing that the step lengthes or standard deviations are adapted continuously due to the current topology of the optimization domain. The adaptation is handled by means of a so called multiplicative mutation ansatz avoiding negative values and adequate scaling tailored to the convergence needed. Recapitulatorily, the (μ/$+, λ)-ES exhibits a plethora of powerful mechanisms and concepts to circumnavigate the most difficult optimization scenarios at reasonable convergence speed. The main benefits can be seen in the robust behavior compared to other competitive methods, in the ability to find a global optimum with a good chance and in the general applicability, particularly, in algorithmically nonlinear structural optimization problems (as mentioned already in Section 4.5.1). A further significant advantage is the fact that the population-based evolution strategies are inherently parallel in their behavior and, therefore, contain numerous opportunities for parallelization. 4.5.4 Parallelization of Optimization Strategies Authored by Dietrich Hartmann and Matthias Baitsch Since numerical optimization algorithms rely on the repeated evaluation of objective and constraint functions, the process of numerical optimization can be very time consuming when function evaluations are costly. Typically, the number of function evaluations using gradient based algorithms is of order of magnitude of 102 where evolution strategies typically require up to 104 or even more function evaluations. The potential for parallelization and the associated strategies are determined by the type of analysis involved and the type of the optimization methods used. For instance, in multilevel structural optimization, the original optimization problem is decomposed into a number of smaller non-interacting subproblems coupled on a coordination level [802]. In contrast to such highly specialized schemes, the following two sections cover generally applicable techniques feasible for a wide range of structural optimization problems.

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4.5.4.1 Parallelization with Gradient-Based Algorithms As outlined in Section 4.5.3.1, many gradient based algorithms repeatedly determine a descent direction using the gradients of the objective function and constraints (see e.g. problem (4.371)) and carry out a line search search along this direction to solve the one dimensional problem (4.373). Hence, there are mainly two possibilities for parallelization: The computation of derivatives and the line-search step. For many problems involving numerical simulation, derivatives can only be approximated numerically using either forward differences ∂f f (x + Δei ) − f (x) (x) ≈ ∂xi Δ

(4.386)

or central differences ∂f f (x + Δei ) − f (x − Δei ) , (x) ≈ ∂xi 2Δ

(4.387)

i = 1, . . . , n, where n is the number of design variables and ei is the i-th unit vector. Obviously, either n + 1 or 2n independent function evaluations are required which can easily be carried out in parallel. In the line-search step, several points on the one-dimensional search direction can be evaluated in parallel which can yield a substantial parallel speed-up. For example, in [703] Schittkowski proposes a sequential quadratic programming algorithm with distributed and non-monotone line search. Combining the parallel approximation of gradients and a parallel line search, gradient based optimization requires in the ideal case two computational steps per iteration: One for the gradients and one for the line search. However, both techniques do not involve enough parallel processes to make full use of modern cluster computers with more than 150 CPUs. Therefore, the described techniques can been combined with a parallel structural analysis in order to save more computing time (see e.g. [78] for an application with high-order finite element methods). 4.5.4.2 Parallelization Using Evolution Strategies Population-based evolution strategies as introduced in Section 4.5.3.2 require λ function evaluations in each optimization step where λ is the population size (number of children in one generation). The population size is chosen according to the type and size of problem at hand and typically ranges from 50 to 200. Taking into account that up to 400 iteration steps might be required, it becomes obvious that parallelization is mandatory when evolution strategies are applied to complex engineering problems. On the other side, the large number of designs to be evaluated in each iteration step allows for an efficient parallelization since the required computations do not depend on each other. Although a straightforward parallelization

4.6 Application of Lifetime-Oriented Analysis and Design

Server

Linux-Cluster

MPI

Workstation

x1

Problem

x2

Problem

fi(x1) fi(x2)

GUI

LAN CORBA

561

Optimizer

MProblem

.. xn

fi(xn)

Problem

{x1, x2, ... , xm} { fi(x1), ... , fi(xm)}

Fig. 4.107. Parallel software framework

scheme such as the manager-worker approach often renders good performance, further improvements can be achieved if the communication overhead is reduced by applying packeting or load balancing mechanisms [326]. 4.5.4.3 Distributed and Parallel Software Architecture There are basically two demands for a parallel optimization software: (i) a wide variety of optimization algorithms have to be readily available in order to enable the designer to choose a suitable method for the problem at hand and (ii) the parallel part of the software should be isolated as much as possible in order to facilitate software development. These requirements are accomplished by the software framework shown in Fig. 4.107. Here, the optimizer software component provides a wide variety of optimization algorithms such as evolution strategies and different variants of gradient-based algorithms in a unified fashion [79]. This software component is implemented as a CORBA server such that it can be used remotely over the Internet. The second part is the multi-problem parallelization component which preferably runs on a cluster of Linux computers. This component receives a set of design vectors from the optimizer and dispatches them to the individual instances of the actual optimization problem running on the compute nodes. The overall optimization process is driven from a GUI application running on the user’s workstation or laptop computer.

4.6 Application of Lifetime-Oriented Analysis and Design Authored by Dietrich Hartmann and Detlef Kuhl The successful application as well as the practical implementation of results based on sophisticated long-term research in lifetime-oriented analysis and design is the most essential achievement and the best possible evidence for work

562


performed. For that reason, a wide variety of highly different application examples are shown in the following chapters ranging from the lifetime-oriented analysis and design of beam-like structures over structural components used in the automobile industry up to concrete as well as steel structures, where particularly bridge systems are dealt with. According to the specific nature of the structural systems considered with respect to material aspects and/or structural behaviors, all relevant concepts and methodologies uncovered in the recent years of research are elucidated. Hereby, eminent importance is put on the verification and the validation of theoretical findings.

4.6.1 Testing of Beam-Like Structures Authored by Stefanie Reese and Andreas S. Kompalka In the literature a couple of publications focus on the identification of a damage in beam-like structures. The publications from [424] and [858] localize a cut damage in a simple beam made of steel. The localization and quantification of a cut damage in a cantilever beam made of aluminum are announced in [535] and [751, 752, 754, 753]. In the following sections a subspace method (see chapter 4.3.2) is combined with a derivative-based optimization method

Fig. 4.108. Experimental setup


563

Fig. 4.109. Damage equipment

(see chapter 4.5.3.1) to identify a cut damage in a cantilever beam made of steel. 4.6.1.1 Experimental Setup The experimental setup is a clamped cantilever beam with a length of 1.62m and a rectangular cross-section of 40 × 15mm made of steel. The cantilever beam is fixed with several clamps and a steel bar (HEB-100) at a massive steel plate (1000 × 800 × 100mm) on a vibration decoupled foundation (see Figure (4.108)). The used measurement technology from Hottinger & Baldwin consists of 16 micro-mechanic accelerometers and two amplifiers. The structural damage is a cut with a rectangular cross-section of 10 × 5mm. The cut is realized by a milling machine and a cross-support (see Figure (4.109)). The central position of the cut is 450.00mm from the clamping. The system is excited by a static displacement. Three measurements of the excited structure are recorded in the undamaged and damaged state. 4.6.1.2 Identification of Modal Data To obtain the modal data (frequencies and mode shapes) of the experimental setup, the accelerations of the 16 channels are analyzed with the data-driven stochastic subspace identification of chapter 4.3.2 (with (4.308)-(4.311)

564


500

400

300

200

100

0

5

10

15

20

Fig. 4.110. Singular values

0.20

0.15

0.10

0.05

0.00

-0.05

-0.10

-0.15

f1 = 4.74, hr = 15.0mm f1 = 4.60, hr = 7.5mm

-0.20

-0.20

0.00

0.20

0.40

Fig. 4.111. 1’st eigenfrequency and mode shape

and (4.298)-(4.302)). The first twenty singular values are visualized in Figure (4.110). In Figure (4.111)-(4.114) the frequencies and mode shapes in the undamaged and damaged state are visualized. The standard deviations of the mode shapes are smaller than of the frequencies. Comparing the undamaged and damaged state, the relative changes in the coordinates of the mode shapes are


565

0.20

0.15

0.10

0.05

0.00

-0.05

-0.10

-0.15

f2 = 29.68, hr = 15.0mm f2 = 28.45, hr = 7.5mm

-0.20

-0.20

0.00

0.20

0.40

Fig. 4.112. 2’nd eigenfrequency and mode shape

0.20

0.15

0.10

0.05

0.00

-0.05

-0.10

-0.15

f3 = 83.04, hr = 15.0mm f3 = 80.99, hr = 7.5mm

-0.20

-0.20

0.00

0.20

0.40

Fig. 4.113. 3’rd eigenfrequency and mode shape

much smaller than the relative frequency changes. In the damaged state, the coordinates of the first mode shape almost do not change. The coordinates of the higher mode shapes show only small changes in the damaged state.

566


0.20

0.15

0.10

0.05

0.00

-0.05

-0.10

-0.15

f4 = 162.58, hr = 15.0mm f4 = 160.52, hr = 7.5mm

-0.20

0.00

-0.20

0.20

0.40

Fig. 4.114. 4’th eigenfrequency and mode shape

f1 = 4.6414Hz Y-Coordinate [m]

f2 = 29.5777Hz 0.0075

0

-0.0075 0.43

0.44

0.45

0.46

0.47

X-Coordinate [m]

Fig. 4.115. Cut modelling

4.6.1.3 Updating of the Finite Element Model The experimental setup is discretized by means of a finite element model. A two-dimensional four-node shell element with bilinear ansatz functions and a two-dimensional nine-node shell element with biquadratic ansatz functions are compared by a convergency study. The nine-node shell element with biquadratic ansatz functions enables a better approximation of the bending


567

modes especially in the damaged state with the modeled cut (see Figure (4.115)). Based on the convergency study, six nine-node shell elements over the cross-sectional height and 1296 elements in length direction are used to discretize the cantilever beam structure in the undamaged and damaged state. In Chapter 4.5.3.1, derivative-based methods like the Newton method are explained. In the context of this chapter, the Gauss-Newton method is derived to solve the least squares problem. The sum of squares, which have to be minimized, are the residuals or differences between the experimental measures und numerical calculated modal data (frequencies and mode shapes). Finding the minimum of the objective function f (x) =

1 r(x)T r(x) 2

(4.388)

of the sum of squares with the residual vector ⎞ ⎛ r (x , ... , x ) 1 1 n ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ r2 (x1 , ... , xn ) ⎟ ⎟ ⎜ r(x) = ⎜ ⎟ .. ⎟ ⎜ . ⎟ ⎜ ⎠ ⎝ rm (x1 , ... , xn )

(4.389)

is equal to finding the zero point of the first partial derivatives of the object function ∇f (x) = J(x)T r(x)

(4.390)

with the Jacobian matrix ⎡ ∂r1 (x) ∂r1 (x) ∂x2 ⎢ ∂x1 ⎢ ⎢ ⎢ ∂r2 (x) ∂r2 (x) ⎢ ∂x1 ∂x2 ⎢ J(x) = ⎢ ⎢ .. .. ⎢ . . ⎢ ⎢ ⎣ ∂rm (x) ∂rm (x) ∂x1

∂x2

∂r1 (x) ∂xn

⎤

⎡

⎤

∇r1 (x) ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ T ⎥ 2 (x) ⎥ ⎢ ∇r (x) . . . ∂r∂x ⎥ 2 ⎢ ⎥ n ⎥ ⎢ ⎥. ⎥=⎢ ⎥ ⎥ ⎢ .. . .. ⎥ . ⎥ ⎢ . . ⎥ . ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎦ ⎣ ⎦ ∂rm (x) . . . ∂xn ∇rm (x)T ...

T

(4.391)

The Jacobian matrix includes the transformed gradients of the residuals ∇rj (x)T = ∂rj (x) ∂rj (x) . . . ∂rj (x) (4.392) ∂x1

∂x2

∂xn

in each row. Using a second-order Taylor series to approximate the first partial derivatives of the object function

568


∇f (x) = ∇f (x0 ) + ∇2 f (x0 )(x − x0 ) leads to the Newton method C D−1 x = x0 − ∇2 f (x0 ) ∇f (x0 )

(4.393)

(4.394)

with the second partial derivatives of the object function ∇2 f (x) = J(x)T J(x) +

m

rj (x) Hj (x)

(4.395)

j=1

and the Hessian matrix ⎡ ∂rj (x) ∂rj (x) ⎢ ∂x1 ∂x1 ∂x2 ∂x1 ⎢ ⎢ ⎢ ∂rj (x) ∂rj (x) ⎢ ∂x1 ∂x1 ∂x2 ∂x1 ⎢ Hj (x) = ⎢ ⎢ .. .. ⎢ . . ⎢ ⎢ ⎣ ∂rj (x) ∂rj (x) ∂x1 ∂x1 ∂x2 ∂x1

...

∂rj (x) ∂xn ∂x1

⎤

⎥ ⎥ ⎥ ∂rj (x) ⎥ . . . ∂xn ∂x1 ⎥ ⎥ ⎥. .. ⎥ .. . . ⎥ ⎥ ⎥ ⎦ ∂rj (x) . . . ∂xn ∂x1

(4.396)

Due to the linearization in Eq. (4.393) with the second-order Taylor series, the calculation of the zero point with Eq. (4.394) is only an approximation. Using the Eq. (4.390) and (4.395) in Eq. (4.394) and substituting the model parameter x and x0 by the incremental sizes xk+1 and xk leads to the Newton iteration xk+1 = xk −

⎧ ⎨ ⎩

J(xk )T J(xk ) +

m j=1

⎫−1 ⎬ rj (xk ) Hj (xk ) J(xk )T r(xk ) ⎭ (4.397)

for the least squares problem with the Newton search direction sN k =−

⎧ ⎨ ⎩

J(xk )T J(xk ) +

m j=1

⎫−1 ⎬ rj (xk ) Hj (xk ) J(xk )T r(xk ). ⎭

(4.398)

In practice, the computation effort for the Hessian matrix is very high. Furthermore, the residuals close to the minimum are small if the model is a good approximation of the problem. Therefore, the Gauss-Newton search direction C D−1 sGN = − J(xk )T J(xk ) J(xk )T r(xk ) (4.399) k


569

Table 4.9. Modal Assurance Criterion

φf e1 φf e2 φf e3 φf e4

φex1 0.9992 0.0037 0.0077 0.0065

φex2 0.0043 0.9981 0.0013 0.0050

φex3 0.0069 0.0040 0.9974 0.0025

φex1 0.0053 0.0031 0.0072 0.9960

neglects the term with the Hessian matrix in Eq. (4.398). The methods of Newton and Gauss-Newton are summarized in the iterative procedure xk+1 = xk + αk sk

(4.400)

with the search direction sk and the stepsize parameter αk . The control of the stepsize is important for the updating algorithm. One of the first publications of Gauss-Newton iteration with line search approach was given by [233]. A simple choice for the stepsize is the exponential ansatz αk = 0.5mk . Depending on the sum of squares, the integer mk reduces or enlarges the stepsize. The method uses one information of the object function to control the stepsize. A stepsize control with a quadratic-cubic ansatz is published by [284] or [582]. The method uses a quadratic or cubic extrapolation based on three evaluations of the object function to estimate the optimum stepsize. Here, the iteration stops if the changes in the model parameter (cut position and cut deepness) are smaller than the realization of the cut which is 1/10mm. Prerequisite for a successfull updating of the finite element model is the mode pairing and the mode scaling. The “Modal Assurance Criterion” M ACij =

(φTexi φf ej )2 (φTexi φexi )(φTfej φf ej )

(4.401)

was introduced by [43] and compares the experimental identified and the numerical calculated eigenvectors φexi and φf ei . The MAC values are sorted in a table where values close to 1 denote a good agreement of the measured and modeled data and values close to 0 denote a bad agreement. In Tab. (4.9) the correct pairing of the modal data is verified. MAC values close to 100 percentage indicate a good approximation of the experimental setup with the finite element model. The “Coordinate Modal Assurance Criterion” m ( i=1 φf eji φexji )2 COM ACj = m (4.402) 2 ( i=1 φ2exji )( m i=1 φf eji ) was published first by [496] and compares the eigenvector coordinates of the finite element model and the experimental setup. The COMAC values are sorted in a vector and values are interpreted similar to the MAC values. In [293] and [511] other mode pairing criterions are mentioned as e.g. the

570


7

6

Cut deepness cd [mm]

5

4

3

2

1

200

400

600

800

1000

1200

1400

1600

Cut position cp [mm]

Fig. 4.116. Optimization topology

“Orthogonality Check” or the “Normalized Cross Orthogonality”. The disadvantge of the mentioned methods is that the mass matrix has to be assumed and the results depend on the accuracy of these assumptions. After pairing of the modes it is important to scale the eigenvectors before calculating the residuals. The “Modal Scale Factor” M SFi =

φTfei φexi φTexi φexi

.

(4.403)

multiplied with the numerically calculated eigenvetor ensures the correct scaling and the correct orientation of the mode shape. In Figure (4.116) the sums of squares with the first four frequencies are plotted in an optimization topology. The plot is cut off at a value of 0.0015 visualized with white arrays. The optimization topology shows a global minimum with the sum of squares of 3.2 · 10−7 at the cut position cp = 450.60mm and the cut deepness cd = 4.95mm (see Figure (4.116) circle symbol). This is very close to the damage in the experimental setup with the cut position cp = 450.00mm and the cut deepness cd = 5.00mm. There is a deep local minimum with the sum of squares round about 1.6 · 10−4 at the cut position cp = 13.35mm and the cut deepness cd = 2.50mm (see Figure (4.116) diamond symbol). Another high local minimum with the sum of squares round


571

Table 4.10. Gauss-Newton (cp /cd =800/1smm) Cut position cp [mm] 1 800.0000 2 326.2682 3 345.2333 4 382.9806 5 428.5175 6 448.3661 7 452.0016 8 450.9110 9 450.6433 10 450.5973

Cut deepness cd [mm] 1.0000 1.4511 1.5350 1.8797 2.5789 3.5447 4.3401 4.9733 4.9523 4.9514

Residual rk2 [-] 0.00057917 0.00056076 0.00049416 0.00036085 0.00017330 0.00004369 0.00000039 0.00000032 0.00000032 -

Gradient 2 2 rk,1 [-] rk,2 [-] -0.00039927 0.00183427 0.00155315 0.00153707 0.00282315 0.00135235 0.00468388 0.00133469 0.01511816 0.00135926 0.05704360 0.00114078 -0.06352743 0.00047868 -0.00036151 -0.00002018 -0.00000388 -0.00000088 -

Stepsize αk [-] 0.03125 0.03125 0.06250 0.12500 0.25000 0.50000 1.00000 1.00000 0.25000 -

Table 4.11. Gauss-Newton iteration (cp /cd =1400/1mm) Cut position cp [mm] 1 1400.0000 2 1333.2877 3 1350.0403 4 1275.6526 5 1222.3885 6 1135.5092 7 1189.3673 8 1209.9656 9 1186.0790 10 1198.9238 11 1198.5966 12 1198.3513

Cut deepness cd [mm] 1.0000 0.6470 0.8431 0.8014 1.0727 1.3560 2.2344 3.0453 4.2576 4.0723 3.8351 3.8333

Residual rk2 [-] 0.00071567 0.00070936 0.00069332 0.00066246 0.00064613 0.00053004 0.00043665 0.00040122 0.00039218 0.00038984 0.00038984 -

Gradient 2 2 rk,1 [-] rk,2 [-] -0.00016494 -0.00013640 0.00060276 0.00022530 -0.00034495 -0.00269144 -0.00092701 0.00079634 -0.00030087 0.00040366 0.00344872 0.00092509 0.00725386 0.00080615 -0.00237344 0.00020460 0.00112548 -0.00034139 -0.01146584 -0.00006919 -0.00000549 -0.00000322 -

Stepsize αk [-] 0.01563 0.01563 0.03125 0.06250 0.12500 0.25000 0.50000 1.00000 1.00000 1.00000 0.01563 -

about 3.9 · 10−4 lies at the cut position cp = 1198.35mm and the cut deepness cd = 3.83mm (see Figure (4.116) square symbol). The Gauss-Newton iteration with the start values cp /cd =800/1mm and cp /cd =1400/1mm are visualized in Figure (4.116) and summarized in Tab. (4.10) and (4.11). In the neighborhood of small gradients the Gauss-Newton search direction is not rectangular to the contour line and the stepsize control is required. Finally we can state, minimizing the sum of squares of the modal data with the Gauss-Newton method it is possible to identify the cut position and the cut deepness in a cantilever beam (length 1.62m) with an accuracy less than 1mm.

572


4.6.2 Lifetime Analysis for Dynamically Loaded Structures at BMW AG Authored by Dietrich Hartmann, Heiner Weber and Gero Pflanz The design of the car body-in-white using CAE-technology is an iterative process which can be described as an extensive optimization process. A desirable target of such an optimization can be the minimum weight of the resulting structure taking into account prescribed boundary conditions. Boundary conditions may include cost, geometrical guidelines, production engineering demands and functional specifications. For instance, geometrical guidelines are the length of the car, the height or width of the door sill. Production engineering demands to consider the deep drawing process or the thickness. Characteristic functional specifications comprise crash behaviour, static and dynamic stiffness and acoustics as well as strength and durability requirements. Static stiffness requirements include both the torsional stiffness of the bodyin-white as well as several bending and transverse stiffness cases. For each of these load cases target values are defined for the body-in-white component which have to comply with certain handling performance or further desired properties of the whole car. Dynamic stiffness requirements include the frequency range of certain global eigenmodes of the structure, for example, to avoid the excitation of an eigenmode of the car body while the engine is running in idle-speed. Additional dynamic stiffness targets are necessary to guarantee the desired vibration comfort. Strength requirements define the maximum tolerable plastic deformation for a particular load case, e.g. towing the car onto a tow-track. Durability requirements are set up to guarantee that no cracks in sheet-metal parts or failure in weld-spots occur within a given mileage under certain statistical loading conditions. Plenty of months before the first prototypes of a car are built, the functional requirements are analysed and iteratively improved in virtual prototypes by means of the finite element method. The finite element model is hereby generated from early CAD data. Typically, those virtual prototypes have a size of a few million degrees of freedom. 4.6.2.1 Works for the New 3-Series Convertible The 3-series convertible (see Figure 4.117) is the fourth member of the current BMW 3-series along with sedan, station wagon and coupé. While the frontal, middle and rear part of the body-in-white are similar or identical to the other model members, the side frame is completely different. The missing roof load-path has to be compensated by the reinforced side-frame design and additional torsion bars, which are packaged at several locations in the convertible structure.


573

Fig. 4.117. The new 3-series convertible

Low-frequency vibration comfort in convertibles is significantly dominated by the excitation of the car-body at the strut towers through the wheeldamper-chain. In particular, the run-flat tires of the 3-series cars result in higher coupling forces at the strut towers compared to the normal tires of the other variants. The first wheel eigenmode has a resonant frequency just below the first torsion eigenfrequency of the car body. In order to further improve the customer relevant vibration level of the car body, therefore, for this kind of excitation a tuned mass damper is implemented. A tuned-mass damper is a secondary mass which is attached to the primary mass (the car body) via a spring and a damper such that the resonant frequency of the second mass equals the problem-frequency of the primary mass. As a result, the forces acting between the tuned-mass damper and the primary mass reduce the vibration amplitudes of the primary mass. To achieve the best effect, the tuned mass damper has to be attached at that position of the car where the damper extinguishes large oscillation amplitudes when the relevant mode is excited. Since additional masses are not desired for the car, the idea is to use an existing mass as a tuned-mass damper. In the case of the 3-series convertible, the battery of the car is used because its mass is sufficient and its location in the rear side part of the trunk floor is well suited for damping (see Figure 4.118). Instead of a tight bolt connection the battery is connected to the car body by a hinge-joint at the one end and mounted on a hydro mount at the other end (see Figure 4.119). By calibrating the hydro mount the desired resonant frequency and damping characteristic of the tuned-mass damper system is reached.

574


Fig. 4.118. 3-series convertible with bat- Fig. 4.119. Battery as vibration absorber tery

4.6.2.2 The Shaker Test The electrical components of a car have to pass a shaker test according to a Group Standard [132] developed by BMW which describes the load profiles and the test conditions. Generally, the test set-up consists of a part of the car body and the relevant attachment parts. The component of the car body is excited to random vibrations. Hereby, the loads are defined by their power spectral density (PSD) functions. All tests are performed for the three spatial dimensions, where each test has a duration of 8 hours. The shape of the PSD functions depend on the location of the electrical components in the car and the spatial direction tested. The application of the battery as a tuned-mass damper leads to considerably higher loading of the battery and the electrical components attached on top of it. Furthermore, the forces acting in the area of the hydro mount and the hinge joint will be considerably higher than in a tight bolt connection. For the virtual and the real shaker test, a part of the car body has been cut out of the complete body-in-white and the battery has been attached to the relevant electrical components as depicted in Figure 4.120. A typical load can be seen in Figure 4.121. While the durability performance of the electrical components has been only analysed in the real shaker test, the sheet metal parts connecting the battery rack with the welded body-in-white have also been tested with regard to durability by two numerical approaches. The first approach using a time domain integration is shown in the following Section 4.6.2.3, the second approach using a stochastic load in the frequency domain is presented in Section 4.6.2.4. 4.6.2.3 Approach 1: Time History Calculation and Amplitude Counting As mentioned above, the customary way to calculate a structure which is loaded by a stochastic dynamic acceleration is a time history integration described in the following subchapter.


575

Fig. 4.120. FE model of the shaker test arrangement

4.6.2.3.1 Structural Analysis Using Time Integration Time integration means that the loading must be given in a discrete fashion for being entered into the computation. For example, the Nastran solution routine 112 provides this approach. The computation result is a time series of the structural response, either displacements, section forces or stresses. In particluar, the integration is a time consuming task. To obtain stable results, the time series of the loading must have an appropriate, not too short length. Furthermore, the time series is not directly available, only its power spectral density is given in the Group Standard. As a consequence, the time series of the loading is created by a routine when the shaker test is being performed. For the test, the accelerations of the structures have been measured for all three directions. By this, representative 130 seconds of the loading have been given for x, y and z accelerations which can be used for finite element calculations. Then, the results can be extrapolated from 130 seconds to the test duration of 8 hours for each direction. 4.6.2.3.2 Cycle Counting Using the Rainflow Method A fatigue analysis is based on the stress time series for elements of interest. In a second step, the counting of stress amplitudes has to follow the FE

576


Fig. 4.121. Measured acceleration data for the y-direction

calculation. To this end, the Rainflow method [202] is used which takes into account the hysteresis loops. By comparing the number of stress amplitudes with the allowed number which is given by the Δσ-N -curve, a damage accumulation rule is applied. By that, the safety against fatigue can be determined. For this second step, it is necessary to prepare the results of the structural analysis for a separate program. Here, the program Falancs has been used. This two program approach is a disadvantage of the two-step-procedure. Furthermore, the counting algorithm is time-consuming, too. 4.6.2.3.3 Damage Calculation For the damage calculation the accumulation rule of Palmgren-Miner [543, 611] is applied. Due to this rule all partial damages ΔDi consist of the sum of single damages Δdi , each of them belonging to one amplitude level. The single damage Δdi represents the relation of the applied number of amplitudes ni and the allowed number Ni given by means of the Δσ-N -curve: ni Δdi = (4.404) ΔDi = Ni All partial damages are summed up and a total damage Dtot is resulting, for example for loading in different directions x, y, z: ΔDi = ΔDi + ΔDi + ΔDi (4.405) Dtot = x

y

z


Power spectral density SXX(f)

10000

577

Shell 7672094 vonMises-1 Shell 7672094 vonMises-2 Shell 7672154 vonMises-1 Shell 7672154 vonMises-2 Shell 7672171 vonMises-1 Shell 7672171 vonMises-2 Shell 7672467 vonMises-1 Shell 7672467 vonMises-2 Shell 7672488 vonMises-1 Shell 7672488 vonMises-2

1000 100 10 1 0,1 0,01 1E-3 0

50

100 Frequency f

150

200

Fig. 4.122. Power spectral density function of the resulting von Mises stress for the elements of Figure 4.119, load direction y

For a design, the total damage must be smaller than a limit damage which is usually equated with 1: Dtot < DLim

(4.406)

4.6.2.4 Approach 2: Power Spectral Density Functions and Calculation of Spectral Moments An alternative way to the approach above uses the power spectral density functions of the load directly. The full particulars of this alternative are outlined in the next subchapter. 4.6.2.4.1 Structural Analysis Using Power Spectral Density (PSD) Functions The popular program system Nastran offers a module called “RANDOM” which allows power spectral density functions as input. Here, a frequency response calculation - Nastran solution 111 - is performed followed by the calculation of the Random module. The results are again power spectral density functions, either for displacements, section forces or stresses (see Figure 4.122). Then, it is possible to determine characteristic statistical numbers of the resulting functions, the so-called spectral moments: 1 λi = · π

∞

∞ i

ω · SXX (ω) dω = 2 ·

(2πf) · SXX (f ) df

i

0

0

(4.407)

578


These moments represent the variances of the stress process and its derivatives. The zeroth spectral moment is the variance of the process X: 1 λ0 = · π

∞ 2 SXX (ω) dω = σXX

(4.408)

0

The second spectral moment represents the variance of the derivative process x˙ 1 λ2 = · π

∞ 2 ω 2 · SXX (ω) dω = σX ˙X ˙

(4.409)

0

and the fourth spectral moment reflects the variance of the second derivative process x ¨: 1 λ4 = · π

∞ 2 ω 4 · SXX (ω) dω = σX ¨X ¨

(4.410)

0

All three spectral moments can be calculated easily by integrating the PSD functions of the results [621]. This alternative approach is much faster than the time history calculation. The only disadvantage is that one must know which elements are decisive for the fatigue analysis, because the PSD function is calculated for each element separately. It is possible to compute numerous result functions, but calculating a large number of functions nullifies the time advantage. 4.6.2.4.2 Analytical Counting Method Having calculated the spectral moments, the Rainflow counting method can be replaced by an analytical calculation [109]. The distribution function of the stress amplitudes has been already examined by several researchers. Decades ago, Dirlik found a formula consisting of three parts which very well fits the Rainflow amplitudes [237]: ϑ1 Δσ √ exp − √ f (Δσ) = 2Q λ0 2Q λ0 2 Δσ 1 ϑ2 √ · Δσ · exp − + (4.411) 4λ0 R2 2 2R λ0 2 Δσ ϑ3 1 √ + · Δσ · exp − 4λ0 2 2 λ0 Here, the following parameters are introduced:

Dirlik probability density function f(¢¾)


579

0,006 Shell 7672094 vonMises-1 Shell 7672094 vonMises-2 Shell 7672154 vonMises-1 Shell 7672154 vonMises-2 Shell 7672171 vonMises-1 Shell 7672171 vonMises-2 Shell 7672467 vonMises-1 Shell 7672467 vonMises-2 Shell 7672488 vonMises-1 Shell 7672488 vonMises-2

0,005 0,004 0,003 0,002 0,001 0,000 0

200

400 600 800 1000 von Mises stress ranges ¢¾ [N/mm2]

1200

Fig. 4.123. Dirlik distribution function of the stress amplitudes

λ1 λ2 α − xm − ϑ21 xm = R= λ0 λ4 1 − α − ϑ1 + ϑ21

2 xm − α2 1 − α − ϑ1 + ϑ21 ϑ1 = ϑ = 2 1 + α2 1−R

Q=

1.25 (α − ϑ3 − ϑ2 R) ϑ1 ϑ3 = 1 − ϑ1 − ϑ2 (4.412)

It can be seen that all parameters only depend on the spectral moments λ1 , λ2 , λ3 and λ4 . Fig. 4.123 shows the distribution function for the stress PDF presented in Figure 4.122. 4.6.2.4.3 Damage Accumulation for the Analytical Case The damage accumulation rule eq. (4.404) is identical for both approaches. In the case of Rainflow counting, the summation has to be performed explicitly. In the analytical case the summation can be replaced and an analytical formulation is obtained which can be applied rapidly and easily by using the formula for the Δσ-N -curve: ni ΔDi = Δdi = Ni ∞ (4.413) 1 N · E[Δσ ϕ ] = (Δσ ϕ · f (Δσ))dΔσ = ϕ · ϕ ND · ΔσD ND · ΔσD 0

580


Here, f (Δσ) is the distribution function of the stress amplitudes according to the formula of Dirlik in eq. (4.411). N is the number of amplitudes during a damage event: t λ4 · N= , (4.414) 2π λ2 where t is the length of the time interval, which is 8 hours for each direction, according to the Group Standard. The value of expectation of Δσ ϕ can be calculated analytically using Dirlik’s formula ϕ E [Δσ ϕ ] = λ0 · ϑ1 · Qϕ · Γ (1 + ϕ) (4.415) ϕ √ ϕ + λ0 · ( 2)ϕ · (ϑ2 · |R| + ϑ3 ) · Γ (1 + 0.5 · ϕ) The above formulas are valid for a Δσ-N -curve without fatigue endurance limit (i. e. the “elementary case”). In the case of using a curve with fatigue endurance limit (i. e. Miner’s “original case”) the stress amplitudes beneath the fatigue endurance limit are neglected. Then, it can be written: √ ϕ λ0 ΔσD ϕ ϕ √ ) E [Δσ ] =ϑ1 · Q · · Γ (1 + ϕ; 2 2 · Q · λ0 √ 2 ϕ ϕ ΔσD (4.416) 2 · R · λ0 · Γ (1 + ; ) + ϑ2 2 8 · R2 · λ0 √ ϕ 2 ϕ ΔσD + ϑ3 · 2 · λ0 · Γ (1 + ; ) 2 8 · λ0 Consequently, the amplitude counting can be replaced by analytical formulas. The parameters needed are only the spectral moments λ1 , λ2 , λ3 and λ4 , the duration time t and the material parameters of the Δσ-N -curve ΔσD , ND and φ. 4.6.2.5 Comparison of the Results The first approach provides the time history series of the desired results, for example the von Mises stress. Fig. 4.124 shows a typical stress distribution for the loading with respect to the y-direction. The stress information can be established easily for all necessary elements and all time steps with the aid of the FE solver, i. e. Nastran. The von Mises stress can be calculated as 2 + σ2 − σ σ 2 σV M = σxx (4.417) xx yy + 3τxy yy Subsequent to the structural analysis, the Rainflow counting and the damage accumulation calculation is carried out, here by means of Falancs. Results are the total damages as defined in eq. (4.404). These damages for the calculated time can be extrapolated to damages for the whole test duration of 8


581

Fig. 4.124. Typical stress picture for load in y-direction (Time History Analysis)

hours. Inverting the damages, the expected lifetime of the critical elements can be determined (see Figure 4.125). Damages less than 1 or expected lifetimes greater than 8 h mean survival of the sheet plate. The results of the PSD analysis are power spectral density functions of the stresses (see Figure 4.122). In this case, the FE solver outputs the quadratic values for each of the normal and shear stresses. Unfortunately, for the von Mises stress also mixed values for E[σxx σyy ] are required which are not given and cannot be calculated: 2 2 2 E σV2 M = E σxx + E σyy − E [σxx σyy ] + 3E τxy (4.418) This is a well-known problem [137]. To overcome this problem, an estimation has to be done for the correlation between the stresses in the two directions of the shell elements: 2 ] E[σ 2 ] means total correlation and same orientaa) E[σxx σyy ] = + E[σxx yy tion 2 ] E[σ 2 ] means total correlation and opposed orib) E[σxx σyy ] = − E[σxx yy entation c) E[σxx σyy ] = 0 means no correlation between the two directions To play safe, alternative b) is chosen. For the given problem the influence of the mixed value is small. From the PSD functions, all necessary spectral moments can be calculated and the damages are determined analytically. Tab. 4.12 shows a comparison of the results for an early design proposal. The first crack was predicted at location A (see Figure 4.125). This failure prediction was confirmed by the shaker test. In Table 4.12 also two Falancs results for different equivalent stress hypotheses and the PSD results are shown. The last two columns can be compared directly: For all critical elements both

582


A Fig. 4.125. Expected life time in arbitrary time units for the Time History calculation (acceleration load in y-direction) Table 4.12. Results for an early design proposal Element No. Time History + Rainflow Time History + Rainflow PSD + Dirlik 7672094 1.11 1.51 1.67 7672154 0.88 0.94 1.10 7672171 0.61 0.65 0.83 7672467 1.23 1.31 1.70 7672488 39.78 700.11 81.67

methods are in good accordance. The last row contains an element with low stress where many cycles are below the endurance limit. 4.6.2.6 Summary and Outlook The example described above describes how the fatigue analysis is being performed in practice. An alternative way using power spectral density functions directly is presented. For this approach, spectral moments are calculated and applied in a damage accumulation. This approach has been employed in the work of the SFB project C5, too, where the spectral moments were calculated by a covariance analysis using correlation functions instead of PSD functions [820]. The presented method avoids the time consuming time history integration. Furthermore, it is more precise in the case of flat spectra because no


583

additional random phenomena appear in measuring the accelerations. The method can be applied if natural stochastic processes like wind are considered. Rough roads can be another excitation. The method is always suitable if the distribution function of the load process approximately equals the Gaussian type. In particular, this means that the method can be chosen for artificially created signals like shaker test accelerations. Today, the use of the PSD method is applicable in practice, but it is inappropriately supported by the FE solver program. Particularly the calculation of the spectral values has to be included. A disadvantage of the PSD method is the limitation of the number of elements due to the necessity to perform one calculation for each value of the spectral function. Here, the covariance analysis – already explained in Section 4.3.1 – has a great time advantage. By using this method, it is possible to calculate the damages for all elements at once. Furthermore it is possible to consider the von Mises stresses without having accuracy problems because the covariance method provides all values needed. 4.6.3 Lifetime-Oriented Analysis of Concrete Structures Subjected to Environmental Attack Authored by Detlef Kuhl, Christian Becker and Sandra Krimpmann 4.6.3.1 Hygro-Mechanical Analysis of a Concrete Shell Structure The following simulation should prove and illustrate the effects of coupled hygro-mechanical attacks on structures as well as the capability of the higherorder spatial discretization concept proposed in Section 4.2.4.3.2. The hygromechanical material model for concrete [319] is implemented in the previously discussed higher-order spatial discretization platform. This numerical example should illustrate the neccessity of incorporating additional environmental influences as well as the interaction between the participating field variables. In particular the potential damage, that is a result of environmental loading, and the increase of moisture transport due to an increase of macroscopic permeability caused by cracking of concrete will be investigated. Figure 4.126 gives an overview of the investigated shell structure including all relevant geometrical as well as material data. The sphere-like structure is suspended at the upper part to give an additional support to the clamping at the footprint of the structure. Firstly, the structure is loaded by its deadweight only. After that, the capillary pressure boundary conditions at the inner and outer surface are changed according to Figure 4.127 to simulate environmental moisture changes. It is started from an initial moisture of h ≈ 93%. This is about the initial moisture of a freshly cast concrete shell. Both, inner and outer surface are dried equally down to moisture of h ≈ 80% (phase I). After that, the outer surface of the shell is re-humidified up to h ≈ 91% to simulate an extrem

584

4 Methodological Implementation 868 [mm]

suspension

5000 [mm]

d = 100 [mm] clamping Young’s modulus:E = 36.700 [N/mm2 ] tensile strength: Poission’s ratio: ν = 0.2 [−] initial porosity: density: ρ = 2300 [kg/m3 ] fracture energy: compr. strength: fcu = 64.5 [N/mm2 ]

ftu = 3.8 [N/mm2 ] φ0 = 0.25[−] Gf = 0.169 [N/mm]

Fig. 4.126. Hygro-mechanically loaded concrete shell structure: System geometry and material data

h [%] outside inside 100 % 91 %

93 %

81 %

69 % t [d]

I 713

II 1313

III

2738

Fig. 4.127. Hygro-mechanically loaded concrete shell structure: Hygral boundary conditions of the inner and outer surface of the shell

wet environment (phase II). Finally, the outer surface is dried again down to h ≈ 69% (phase III). The embrasure is isolated against moisture transport. Like illustrated in Figure 4.128 the shell is discretized with the help of 3D-p-elements according to Section 4.2.4.3.2 into 10 × 10 elements in-plane of the shell and into four elements in thickness direction. Obviously, the mesh is chosen relatively coarse. Therefore, to come to adequate results, the polynomial degrees of the approximation of the displacements u and the


585

Fig. 4.128. Hygro-mechanically loaded concrete shell structure: Finite element mesh of the numerical analysis

capillary pressure pc , which is representing the moisture in the structure, have to be chosen appropriately high. In this example the following statements concerning the approximation should hold: Firstly, the discretization should use a considerably high approximation in-plane of the shell for the displacement field. For the thickness direction a quadratic approximation is a suitable choice. Considering the moisture transport and an adequate degree of its approximation, the primary transport direction is of vital importance. Because of the changes of humidity at the outer and inner surface of the structure and of the isolation of the embrasure the primary transport direction is the thickness direction. Consequently, the approximation of the capillary pressure pc has to be high enough in thickness direction to capture the developing moisture fronts. Also in-plane of the structure, a moderately high approximation degree of pc should be chosen. Finally, such approximations are applied, that fullfill the Babuˇ ska-Brezzi-conditions in the sense of the Taylor-Hood-elements to guarantee the uniqueness u ≈ u3,3,4

pc ≈ pc,2,2,3 .

(4.419)

Figure 4.129 shows the deformed configuration of the shell structure after applying dead weight as well as the stress components in equatorial σ ˜ 22 11 and meridional direction σ ˜ . The purely mechanical state does obviously not lead to any kind of structural damage. The distributions of the stress components show some tensile stress areas at the clamped area of the shell which so far do not reach the tensile strength of concrete. In Figure 4.130 representative states of the moisture content, with the help of the saturation Sl , are illustrated over the thickness direction at the peak of the structure. Obviously, in phase I the outer and inner surface become drier than the inner core of the structure, that is still very moist. In phase II it

586


0, 99

0, 00

||u|| [mm]

2, 30

1, 47

−2, 00 σ ˜ 11 [N/mm2 ]

−1, 15

σ ˜ 22 [N/mm2 ]

Fig. 4.129. Hygro-mechanically loaded concrete shell structure: Deformation, meridional stress σ ˜ 11 and equatorial stress σ ˜ 22 in case of dead load

can be seen that the drying process has continued whereas the outer surface shows a considerable degree of moisture saturation. The gradient between the states at the inner and outer surface can be seen clearly. Figure 4.130 (bottom) shows the state in phase III, which is characterized by a very low humidity throughout the structure. Only the domain around the middle surface shows a relatively high level of humidity. As mentioned before, the loading with dead load only does not lead to any kind of structural damage. That is changing when drying phase I is applied. Because of the combined loading of dead load and drying, structural damage in the form of cracks, represented by the scalar damage variable d, appears in the domain of the clamped edge. This damage zone is spreading across the whole width of the structure, like it is illustrated in Figure 4.131 for different time steps in the loading scenario. The opening of cracks is initiated by the drying process. This process normally leads to a shrinkage of the dried parts. Because of the constraint of the clamped edge this shrinkage movement cannot be realized and results in additional stresses due to this constraint. The superposition of the pure mechanical state and the constrained stress state leads to an exceeding of the tensile strength of the material at the outer surface and consequently to macroscopic cracks. As can be seen here, a neglect of environmental influences, like moisture or heat effects, may result in an overestimation of the structural safety, because structural damage due to additional environmental influences can not be detected. Figure 4.131 shows, how the damage zone is widening from the middle domain of the clamped support to the boundaries. After that the damage zone is processing slightly in meridional direction. The damage zone also leads to an increase of the moisture transport in the plane of the shell structure, like it is illustrated in Figure 4.132. In this figure the damage zone at one point in phase III is plotted as well as the


100

587

0.72 outside

x[mm]

75 50

Sl [−]

25 0

inside 0.4

100

0.6

0.8

0.42 phase I, t = 713 d

Sl [−]

0.72

outside

x[mm]

75 50 Sl [−] 25 0

inside 0.4

0.6

0.8

Sl [−] 100

0.42 phase II, t = 1313 d 0.72

outside

x[mm]

75 50 Sl [−] 25 0

inside 0.4

0.6 Sl [−]

0.8 0.42

phase III, t = 2438 d

Fig. 4.130. Hygro-mechanically loaded concrete shell structure: Distribution of the saturation Sl across shell thickness at the peak at different points in time

degree of saturation. As can be seen, the damage zone has reached the outer boundaries of the structure and starts to propagate in meridional direction. It is obvious from the saturation degree at the clamped edge, that the cracks in the support zone also lead to an increase of permeability in the plane of the structure resulting in an accelerated in-plane transport process. For a detailed description of the coupled transport-damage process, the distributions of the scalar damage variable d and of the saturation Sl are considered at the damage zone at the supports. Figure 4.133 (left) shows these distributions for selected points in time over the thickness direction.

588


1

0

t = 638d

t = 713d

t = 2583d

t = 2740d

Fig. 4.131. Hygro-mechanically loaded concrete shell structure: Damage evolution at the support area. Plot of scalar damage variable d[−]

1

0.84

d [−]

Sl [−]

0

0.42

Fig. 4.132. Hygro-mechanically loaded concrete shell structure: Damage zone and accelerated transport process in the area of cracks (t = 2590 d)

Obviously, the saturation decreases equally at the inner and outer surface during the drying process in phase I. Despite the little number of elements in thickness direction it can be seen that the p-finite element method captures the distribution of the saturation Sl very accurately. Starting from the time point t = 638 d the saturation decreases rapidly at the outer surface. This accelerated transport process is due to the already existing cracks in the support zone. The saturation at t = 938 d shows a point in phase II, the moisturizing phase of the outer surface. It can be seen, that because of the high crack induced permeability the saturation increases nearly equally (at one level) in the area of cracks. The next paragraphs will deal with each of the phases I,II,III in detail.


inside 0.9

outside

inside

0.85

1

0.8

0.8

589

outside

Sl

Sl [−]

0.75 0.6

0.7

t=37d t=375d 0.4 t=187d t=562d 0.6 t=375d t=637d d 0.2 t=637d t=675d 0.55 t=712d t=712d 0.5 0 t=937d 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 0.65

x[mm]

x[mm]

Fig. 4.133. Hygro-mechanically loaded concrete shell structure: Distribution of saturation Sl and damage variable d across the shell thickness in the middle of the footprint. Left: distribution of saturation for different points in time, right: distribution of saturation in the drying phase I

Figure 4.133 (right) illustrates the distribution of the saturation Sl and the scalar damage variable d for different points in time within phase I, which is the drying phase for the inner and outer surface of the shell. As long as no damage occurs, both inner and outer surface are dried equally. From t = 562 d cracks start to open at the outer surface of the shell which lead to an accelerated drying process in the form of a drying front proceeding from the outside to the inside of the shell. This accelerated process becomes obvious by regarding the almost equal decrease of moisture in the damaged part of the shell thickness and the decrease of moisture at the outer surface of the shell. Damage is increasing because of the constraint shrinkage which is proceeding through the thickness of the shell. By this accumulative damage process the coupling of both participating field becomes evident. Considering phases II and III, Figure 4.134 shows the distributions of saturation Sl and damage variable d. In the left part the distributions for the re-moisturizing phase II of the outer surface are depicted. Firstly, it once again becomes clear that the damage zone leads to an increase of the transport process. This results in an accelerated moisturing of the damaged part of the shell thickness in which the saturation level increases almost equally. Secondly, cracks continue to propagate through the thickness of the shell. But it should be stated here that the moisturizing phase is for the benefit of the shell, because the shrinkage process (and therefore the stresses due to constraints) is almost reversed and additional damage is propagating very slowly. Finally, Figure 4.134 (right) shows the drying phase III. Once again, the

590


inside

outside

1

inside

outside

1 d

0.8

d 0.8

Sl

Sl 0.6

t=712d t=937d 0.4 t=1012d t=1125d 0.2 t=1237d t=1312d 0 0 10 20 30 40 50 60 70 80 90 100 x[mm]

0.6 0.4 0.2 0

[t] = [103 d]

t=1.3 t=1.5 t=1.7 t=2.1 t=2.3 t=2.7

0 10 20 30 40 50 60 70 80 90 100 x[mm]

Fig. 4.134. Hygro-mechanically loaded concrete shell structure: Distribution of saturation Sl and damage variable d across the shell thickness in the middle of the footprint. Left: distribution of saturation in moisturizing phase II, right: distribution of saturation in dryig phase III

existing cracks accelerate the drying process. In the end, the additional drying of the outer surface of the shell leads to fully opened cracks over the complete thickness of the shell. 4.6.3.1.1 Conclusive Remarks on the Hygro-Mechanical Analysis The numerical analysis of a concrete shell subjected to combined hygromechanical loading has proven the capabilities of the hygro-mechanical material model by [319] for durability analyses of engineering structures. The interactions of both participating fields (the displacement field u and the capillary pressure pc ) are represented accurately. This holds for the incorporation of the pressure field within the macroscopic stress tensor as well as the accelerating influence of macroscopic cracks on the macroscopic permeability of concrete. As presented the combined loading of a structure with mechanical and environmental loads, like moisture attack in this example, leads to structural damage which, in turn, results in an increase of moisture transport. This may lead to an accumulative loading scenario. Considering lifetime-oriented durability analyses of concrete structures and in particular structural safety it is concluded that an incorporation of environmental loadings is of vital importance, because otherwise possible damage effects may not be detected and the structural safety would be overrated. The simulation of the hygro-mechanical loading of a concrete shell with the 3D-p-finite element platform leads to adequate results in conjunction with a reduced discretization effort. Despite the coarse mesh, primary and secondary variables show a good resolution. This


591

is clear from the plots of the stress distributions and of the saturation over the thickness at the top of the shell. In particular, the moisture/drying fronts were very well captured by using only four higher-order elements in thickness direction. From that we can conclude that for threedimensional analyses the discretization effort can be strongly reduced. The subparametric concept for the geometry description enables, as can be concluded from the results, an adequate approximation of the spatially curved geometry of the shell structure. Without being illustrated, the fieldwise choice of spatially anisotropic shape functions leads to a considerable reduction of assembly time, by maintaining the accuracy of the numerical results at the same time. 4.6.3.2 Calcium Leaching of Cementitious Materials Cementitious specimens are investigated as prototype examples solving the semidiscrete reaction-diffusion equation of calcium leaching as shown in Figure 4.135. In the initial state the cementitious specimens are in chemical equilibrium with the environment. According to the Dirichlet boundary conditions (Figure 4.135) the calcium concentration is reduced and the degradation of the specimens by calcium leaching is initiated. As the calcium ion concentration c in the pore fluid falls below the equilibrium concentration, calcium is dissolved (characterized by the rate s) ˙ from the cementitious skeleton. Consequently, the pore space of this skeleton increases which results in a weakening of the material. Propagating chemical deterioration is characterized by the calcium concentration of the pore fluid c, the remaining calcium concentration of the skeleton s and the calcium ion production s. ˙ These model problems have been investigated in detail by [445, 446, 455, 457, 452]. In particular, the

geometry and finite element mesh c c˙

Ω c00

NE = 40 X1

Γc

geometry and finite element mesh X2 Ω c00 H NE = 40 × 20 Γc X1 c , c˙ L

L

2L

Dirichlet boundary condition c˙ , c˙ = c0 1 c˙0 0 Tc c 0 c0 0 Tc T t

Dirichlet boundary condition c˙ , c˙ = c0 1 c˙0 0 Tc c 0 c0 0 T t Tc

Fig. 4.135. Calcium leaching of a cementitious bar (left) and a cementitious beam (right): Geometry, finite element discretization and chemical loading history

592


Table 4.13. Standard parameter set [307, 454, 457]

DN =

1.83 · 10−9

A1 = −2.10 · 10−19 A2 =

1.80 · 108

m2 s

c0 = 20.7378

mol m3

m3 s

cp = 19

mol m3

p

A3 = −3.57 · 10−10 m a=

4.25 · 10−10 m

m mol

αc = 0.565

L = 0.16 m

n = 85

H = 0.08 m

mol m3

m=5

Tc = 109 s

s0 = 15

kmol m3

φ0 = 0.2

Δt = 107 s

sh = 9

kmol M ρ m3

ccsh = 1.5

= 3.5 · 10−5

m3 mol

Au = I

phenomenological dissolution model by [307, 308] and the electrolyte diffusion model by [445, 446, 457, 452] are applied. Model parameters, the dimensions of the specimens and algorithmic data are summarized in Table 4.13. Initial conditions are given by the stationary state (c˙0 = 0) characterized by the equilibrium concentration between the pore fluid and the skeleton of the virgin material c0 . For the analyses the chemical loading time Tc = 109 s (see Figure 4.135) is chosen which complies with almost 32 years. For the spatial discretization quadratic one- and two-dimensional Lagrange elements are used. 4.6.3.2.1 Calcium Leaching of a Cementitious Bar Calcium leaching of the cementitious bar in Figure 4.135 is simulated for T = 1.2 · 1010 s. For the simulations both Newmark and Galerkin time integration schemes are applied. The robustness of the integration schemes for problems characterized by non-smooth Dirichlet boundary conditions c and pronounced changes of the reaction rate s˙ as well as the accuracy of time stepping schemes are investigated. 4.6.3.2.1.1 Analysis of the Numerical Results Figure 4.136 shows the contour plots of the calcium concentration of the pore fluid c and the production s˙ obtained from the continuous Galerkin scheme with linear approximations in time as function of the time t/T and the position X1 /L. The variables c and s˙ are normalized by the initial calcium concentration c0 and the maximum production s˙ max , respectively. log s˙ n =

log[−s˙ 109 m3 s/mol] − 1 , log[s˙ max 109 m3 s/mol] − 1

s˙ max = 1.2019·10−4

mol s m3

(4.420)

The production s˙ shows two pronounced reaction zones which can be distinguished by their propagation velocity and the value of the reaction rate. The strong and fast dissolution zone corresponds to the dissolution of calcium

4.6 Application of Lifetime-Oriented Analysis and Design normalized production log s˙ n

normalized concentration c/c0 1.0 c/c00 = 0.0

1.0 0.0

0.8

0.8

0.6 t T 0.4

0.6 t T 0.4

0.2

0.2

0.0

c/c0 = 1.0 0.0 0.2 0.4 X1 /L 0.6 0.8 1.0

593

0.0

1.0 0.0 0.2 0.4 X1 /L 0.6 0.8 1.0

Fig. 4.136. Calcium leaching of a cementitious bar: Numerical results obtained from the cG(1) method in terms of the calcium concentration of the pore solution c/c0 and the calcium ion production log s˙ n resulting from dissociation

hydroxide and the significantly slower propagating dissolution zone with an approximately three orders of magnitude smaller rate s˙ (compare [454, 455]) represents the decalcification of calcium silicate hydrates (CSH). After the calcium hydroxide dissolution zone has reached the right boundary the reaction term s˙ within the domain Ω is significantly reduced and the propagation of the CSH-dissolution front is accelerated. During the dissolution of calcium hydroxide the calcium concentration of the pore fluid c is reduced only slowly within the domain Ω. After the complete dissolution of calcium hydroxide the concentration c decreases at a larger rate. Furthermore, the horizontally aligned kinks in the s-contours ˙ at t/T ≈ 0.1 and t/T ≈ 0.6 demonstrate, that non-smooth changes of the Dirichlet boundary condition and the reaction term result in pronounced changes of the system characteristics. The c-plot allows for the differentiation between the chemical loading range t/T ≤ Tc /T = 1/12, the calcium hydroxide dissolution phase t/T ≤ 0.6 and the decalcification of CSH-phases within the remaining time interval of the simulation. It should be noted, that the results in Figure 4.136 can be obtained by all investigated time integration schemes using constant time steps Δt = 107 s. In ¨ = 0, dG(p) particular, the Newmark [569] method with 2β = γ = 0.5 and u and cG(p) time integration schemes are applied without any numerical problems. Even the computation of non-smooth Dirichlet boundary conditions and significant changes of the reaction term yields no numerical difficulty. However, if the chemical loading time Tc is chosen very small, Newmark time integration schemes fail during the Newton-Raphson iteration.

594


4.6.3.2.1.2 Adaptive Newmark Solution For the solution of calcium leaching using realistic chemical loading times Tc the Newmark time integration is enriched by an error-based adaptive time step control (for details see Kuhl &Meschke [458]). Therefore, the error estimate eΔt/2 (compare Section 4.2.8.2.1) is applied for the check of the admissible error range 0.8η ≤ e ≤ 1.2η with η = 10−6 and for the adaption of the time step Δtnew = Δtold [η/e]1/2 . For this study the total time T of the numerical analysis is reduced to 2 · 109 s. With the described adaptive Newmark scheme the dissolution process can be analyzed for chemical loading times Tc set to 1.0 · 109 s, 5.0 · 108 s, 2.5 · 108 s and 1.25 · 108 s by using 328, 481, 598 and 717 adaptively controlled time steps. The results of these analyses are given in Figure 4.137 (t [108 s], Δt [s], X1 [mm], e [1011 ]). The diagrams show the profiles of the concentration c for selected times t, the time histories of the concentration c for selected positions X1 as well as the time step Δt obtained from the time step control and the respective error estimate eΔt/2 ≤ 1.2η. In order to plot the profiles c(X1 , t)/c0 for selected time instants t set to 2 · 108 s, 4 · 108 s, 6 · 108 s, 8 · 108 s, 1 · 109 s, 1.2 · 109 s, 1.4 · 109 s, 1.6 · 109 s, 1.8 · 109 s and 2 · 109 s, the time step is controlled such that this times are exactly approached by the times tn+1 of the integration scheme. Consequently, the errors at the selected times are smaller compared to the remaining integration time. 4.6.3.2.1.3 Robustness of Galerkin Solutions Using Galerkin time integration schemes of the polynomial degree p with the time step Δt = 107 s allows for the simulation of calcium leaching even for short chemical loading times Tc . Figures 4.138 and 4.139 show the results of discontinuous and continuous Galerkin integration schemes with polynomial degrees up to p = 3. Only the cubic discontinuous and continuous Galerkin schemes failed to converge during the Newton-Raphson iteration. This failure is caused by oscillations of the primary variable within the time finite element resulting from the higher order polynomial shape functions (compare [452]). Qualitative differences between discontinuous and continuous Galerkin integrations are not observed. 4.6.3.2.1.4 Error Estimates for Newmark Solutions For comparison reasons the proposed error estimate eΔt/m is applied to the Newmark integration scheme used for the analysis of calcium leaching. Figure 4.140 shows the normalized temporal and spatial local error estimate Δt/10

Δt/10

el

=

Δt/10 |cn+1 − cΔt log[el 1012 ] − 1 n+1 | , log en = c0 log[emax 1012 ] − 1

(4.421)

with emax = 2.38 ·10−3 for constant time steps Δt = 107 s and the temporal local and spatial global error estimate eΔt/5 (4.136) for different constant time steps Δt. The light gray bars on the right hand side of Figure 4.140 characterize two time intervals during chemical loading and free reaction-diffusion

4.6 Application of Lifetime-Oriented Analysis and Design profiles c(X1 , t)/c0

Tc = 1.0·109 s

1.0 0.8 c c0

Tc = 5.0·108 s

1.0

108

0.8

7

8 8

50 50 10 10 12 12 14 14 20 20

0.2

c c0

0 0 10 10 20 20 30 30

40 40

X1

Δt e

t 0.0 X1 0 4 40 50

8 t 12 16 20

104

1.0

108

0.8

0.8

107

c c0

c c0

Δt e

0.2

0.2

105

0 10

X1

40 50

0.0

0

4

8 t 12 16 20

104

1.0

1.0

108

0.8

0.8

107

c c0

c c0

Δt e

0.2

0.2

105

0.0

0 10

X1

40 50

0.0

0

4

8 t 12 16 20

104

1.0

1.0

108

0.8

0.8

107

c c0

c c0

Δt e

0.2

0.2

105

0.0

0 10

X1

40 50

0.0

e

105

0.2 0 10

Δt

10

1.0

0.0

Tc = 2.5·108 s

4 4

time histories c(t, X1 )/c0 time step Δt, error eΔt/2

6 6

0.0

Tc = 1.25·108 s

2 2

595

0

4

8 t 12 16 20

104

0

4

8 t 12 16 20

0

4

8 t 12 16 20

0

4

8 t 12 16 20

0

4

8 t 12 16 20

Fig. 4.137. Calcium leaching of a cementitious bar: Numerical results and time integration error obtained from adaptive Newmark integration

which will be used for the analysis of averaged error estimates discussed in Δt/10 Section 4.6.3.2.1.6. From the error el -contour plot on the left hand side of Figure 4.140 space-time regions where large time integration errors occur can be identified: Region a) where non-smooth boundary conditions are

596


Tc = 5.0·108 s

Tc = 1.0·109 s

dG(1), NGT = 4 1.0

1.0

0.8

0.8

Tc = 2.5·108 s

dG(3), NGT = 10 1.0 0.8

50 50

c c0

c c0

0.2

0.2

0.0

0.0

0

4

8 t 12 16 20

0 0 10 10 20 20 30 30

40 40

c c0 0.2

X1 0 4

8 t 12 16 20

0.0

1.0

1.0

1.0

0.8

0.8

0.8

c c0

c c0

c c0

0.2

0.2

0.2

0.0

Tc = 1.25·108 s

dG(2), NGT = 6

0

4

8 t 12 16 20

0.0

0

4

8 t 12 16 20

0.0

1.0

1.0

1.0

0.8

0.8

0.8

c c0

c c0

c c0

0.2

0.2

0.2

0.0

0.0

0

4

8 t 12 16 20

0

4

8 t 12 16 20

0.0

0

4

8 t 12 16 20

0

4

8 t 12 16 20

0

4

8 t 12 16 20

1.0

1.0

1.0

0.8

0.8

0.8

c c0

c c0

c c0 no convergence

0.2

0.2

0.2

0.0

0.0

0

4

8 t 12 16 20

0

4

8 t 12 16 20

0.0

0

4

8 t 12 16 20

Fig. 4.138. Calcium leaching of a cementitious bar: Time histories c(t, X1 )/c0 obtained from dG(p)-integration (t [108 s], X1 [mm])

applied, the dissolution zone of calcium hydroxide b), the zone c) characterized by a significant change of the reaction term s˙ and zone d) in which an initial error is identified. These maxima of the time integration error Δt/10 el are also shown on the right hand side of Figure 4.140 by means of the


Tc = 5.0·108 s

Tc = 1.0·109 s

cG(1), NGT = 4 1.0

1.0

0.8

0.8

Tc = 2.5·108 s

cG(3), NGT = 10 1.0 0.8

50 50

c c0

c c0

0.2

0.2

0.0

0.0

0

4

8 t 12 16 20

0 0 10 10 20 20 30 30

40 40

c c0 0.2

X1 0 4

8 t 12 16 20

0.0

1.0

1.0

1.0

0.8

0.8

0.8

c c0

c c0

c c0

0.2

0.2

0.2

0.0

Tc = 1.25·108 s

cG(2), NGT = 6

597

0

4

8 t 12 16 20

0.0

0

4

8 t 12 16 20

0.0

1.0

1.0

1.0

0.8

0.8

0.8

c c0

c c0

c c0

0.2

0.2

0.2

0.0

0.0

0

4

8 t 12 16 20

0

4

8 t 12 16 20

0.0

0

4

8 t 12 16 20

0

4

8 t 12 16 20

0

4

8 t 12 16 20

1.0

1.0

1.0

0.8

0.8

0.8

c c0

c c0

c c0 no convergence

0.2

0.2

0.2

0.0

0.0

0

4

8 t 12 16 20

0

4

8 t 12 16 20

0.0

0

4

8 t 12 16 20

Fig. 4.139. Calcium leaching of a cementitious bar: Time histories c(t, X1 )/c0 obtained from cG(p)-integration (t [108 s], X1 [mm])

spatial global error estimate eΔt/5 . In the first few time steps an extremely large error followed by a moderately large error during the chemical loading range can be observed. At the end of the chemical loading a peak of the eΔt/5-curve indicates the discontinuity of the Dirichlet boundary condition.

598

4 Methodological Implementation Δt/10

Newmark, eΔt/5

Newmark, Δt = 107 s, log en 1.0

−1

0.8

−2

0.6 t T 0.4

d

−3 log e

c

0.2 b 0.0

0.2 0.5 1.0 2.0 Δt [107 s]

a 0.0 0.2 0.4 X1 /L 0.6 0.8 1.0

−5 −6 −7

0

2

4

t

8

10 12

Fig. 4.140. Calcium leaching of a cementitious bar: Spatial local error estimate Δt/10 (4.421) and spatial global error estimate eΔt/5 (4.136) for Newmark time el integrations. Sources for large errors and characteristics of the numerical solution are identified in the left hand figure: Zone a: Initial error, zone b: Non-smooth Dirichlet boundary condition, zone c: Dissolution front of calcium hydroxide, zone d: Change of the reaction term s˙

During the free reaction-diffusion phase the error is reduced until the calcium hydroxide solution stops. This phase is followed by a significant change of the reaction rate s˙ which causes an enormous increase of the error by more than three decades. After this peak the error is decreased to the previous level of the free reaction-diffusion. As shown in Figure 4.140, the error eΔt/5 increases with increasing time steps. The distances between the error curves seem to be constant during the integration time. 4.6.3.2.1.5 Error Estimates for Galerkin Solutions In Figure 4.141 the error eΔt/5 is plotted for discontinuous Galerkin time integration schemes with polynomial degrees p = 1, p = 2 and p = 3. Just like for Newmark integrations the error estimates applied to dG-methods indicate the different phases of the dissolution process. The bandwidth of the estimated errors increases considerably with increasing polynomial degrees. Furthermore, the distances between the error curves for different time steps increase with increasing p. Compared to the Newmark integration the error for dG(1) is larger, for dG(2) slightly smaller and for dG(3) significantly smaller. The comparison of dG(1) and dG(3) within the time interval [0, 2.5 · 108 s] illustrates the initial error of the time integrations. Figure 4.142 contains the plots of the error ep/p+1 for dG-methods of polynomial degrees p ∈ [1, 3]. For this study the number of temporal Gauß points is chosen according to the polynomial degree p + 1 of the comparison

4.6 Application of Lifetime-Oriented Analysis and Design dG(2), NGT = 6

dG(1), NGT = 4 1

0.2 0.5 1.0 2.0 Δt [107 s]

0

−1 −2

log e

−3

log e

−1

−3

−5

−4

−6

−5

0

2

4

t

8

10 12

−7

−4

−2

−5

−3 log e

−1

log e

−3

−7

−5

−8

−6 0

2

4

t

8

0

2

4

t

8

10 12

dG(1) vs. dG(3)

dG(3), NGT = 10

−9

599

10 12

−7

dG(1)

dG(3)

0.000.05 0.10 t 0.15 0.20 0.25

Fig. 4.141. Calcium leaching of a cementitious bar: Logarithm of error estimates eΔt/5 for dG-methods with different time steps Δt (t [109 s])

integration. The estimated errors eΔt/5 (Figure 4.141) and ep/p+1 (Figure 4.142) are almost identical. This comparison verifies the error estimates of the h- and p-method. In Figure 4.142 the comparison of error estimates ep/p+1 illustrates the considerable improvement of the quality of the numerical results with an increasing polynomial degree p. Figure 4.143 shows the analogous study for continuous Galerkin schemes. Qualitatively, the same observations as for the dG-methods are made. However, the error obtained from the cG-methods is significantly smaller. In other words, compared to dG-methods cG-methods lead to more accurate solutions with minor numerical expense (compare equations (4.115) and (4.118)).

600

4 Methodological Implementation dG(1), NGT = 6, e1/2 1

0.2 0.5 1.0 2.0 Δt [107 s]

0

−1 −2

log e

−3

log e

−1

dG(2), NGT = 10, e2/3

−3

−5

−4

−6

−5

0

2

4

t

8

10 12

−7

4

t

−1 −2

−4

8

10 12

e1/2 e2/3 e3/4

−3

−5

log e

log e

−4 −6

−7

−7

−8 −9

2

comparison ep/p+1 , Δt = 107 s

dG(3), NGT = 12, e3/4 −3

0

0

2

4

t

8

10 12

−8 −9

0

2

4

t

8

10 12

Fig. 4.142. Calcium leaching of a cementitious bar: Logarithm of error estimates ep/p+1 for dG-methods with different time steps Δt (t [109 s])

4.6.3.2.1.6 Order of Accuracy of Galerkin Schemes For the purpose of determining the order of accuracy of Newmark and Galerkin time integration schemes the average errors within the time intervals I1 = [575, 625] · 107 s and I2 = [50, 100] · 107 s are summarized in Table 4.14. The time intervals I1 and I2 are chosen as representative intervals for free reaction-diffusion and for the chemical loading range. Table 4.14 confirms again the equivalence of h- and p-error estimates. The errors eΔt/5 of the Newmark time integration and the errors ep/p+1 of the Galerkin integration schemes are plotted in Figure 4.144 as functions of the time step Δt. The slope of the error curves log e = a+b log Δt defines the proportionality of the error e ∼ Δtb and the order of accuracy O(b). For Newmark, discontinuous Galerkin and continuous Galerkin methods the following proportionality holds:

4.6 Application of Lifetime-Oriented Analysis and Design cG(1), NGT = 6, e1/2 −1

cG(2), NGT = 10, e2/3

0.2 0.5 1.0 2.0 Δt [107 s]

−2

−3 −4

log e

−5

log e

−3

−5

−7

−6

−8

−7

0

2

4

t

8

10 12

−9

0

2

4

t

8

10 12

error estimate eΔt/5 , Δt = 107 s

cG(3), NGT = 12, e3/4 −5

−2 −3

−6

cG(1) cG(2) cG(3) cG(4)

−4

−7

log e

log e

−5 −7

−9

−8

−10 −11

601

0

2

4

t

8

10 12

−9 −10

0

2

4

t

8

10 12

Fig. 4.143. Calcium leaching of a cementitious bar: Logarithm of error estimates ep/p+1 and eΔt/5 for cG-methods with different time steps Δt (t [109 s])

eN ∼ Δt2 ,

edG ∼ Δtp ,

ecG ∼ Δtp+1

(4.422)

As expected, the order of accuracy of Newmark schemes is two (see e.g. Hughes [396]). In contrast, Galerkin time integration schemes allow for arbitrary order of accuracy which is controlled by the temporal polynomial degree p. From Figure 4.144 follows for discontinuous Galerkin methods the order of accuracy p and for continuous Galerkin methods the order of accuracy p + 1. 4.6.3.2.2 Calcium Leaching of a Cementitious Beam In this section calcium leaching is further investigated by means of a twodimensional discretization of a cementitious beam shown in Figure 4.135. The investigated integration time is T = 5 · 109 s. Again, the initial conditions

602


Table 4.14. Calcium leaching of a cementitious bar: Average relative errors of the Newmark method (N), discontinuous Galerkin methods (dG) and continuous Galerkin methods (cG) within the time intervals I1 = [575, 625] · 107 s (reactiondiffusion phase) and I2 = [50, 100]·107 s (chemical loading phase) Δt[107 s] N dG(1) dG(2) dG(3) dG(1) dG(2) dG(3) cG(1) cG(2) cG(3)

I1 I2 I1 I2 I1 I2 I1 I2 I1 I2 I1 I2 I1 I2 I1 I2 I1 I2 I1 I2

0.2

0.5 −7

4.244·10 8.072·10−5 1.262·10−4 1.014·10−3 8.405·10−8 1.368·10−5 8.034·10−11 4.010·10−7 1.574·10−4 1.262·10−3 8.759·10−8 1.435·10−5 8.133·10−11 4.056·10−7 3.592·10−6 2.323·10−4 3.777·10−10 1.191·10−6 2.623·10−12 1.077·10−8

1.0 −6

2.628·10 4.248·10−4 3.113·10−4 2.895·10−3 5.184·10−7 6.626·10−5 1.165·10−9 4.954·10−6 3.886·10−4 3.553·10−3 5.400·10−7 7.028·10−5 1.175·10−9 5.009·10−6 2.159·10−5 1.038·10−3 9.169·10−9 1.844·10−5 8.644·10−12 4.782·10−7

2.0 −5

1.040·10 1.461·10−3 6.150·10−4 6.888·10−3 2.096·10−6 2.203·10−4 8.596·10−9 3.176·10−5 7.655·10−4 8.445·10−3 2.181·10−6 2.342·10−4 8.869·10−9 3.348·10−5 8.156·10−5 2.928·10−3 9.811·10−8 1.140·10−4 2.257·10−10 8.843·10−6

4.0 −5

4.169·10 2.945·10−3 1.238·10−3 1.648·10−2 8.752·10−6 6.703·10−4 5.823·10−8 1.621·10−4 1.523·10−3 2.058·10−2 9.086·10−6 7.400·10−4 5.877·10−8 1.732·10−4 2.949·10−4 7.504·10−3 9.873·10−7 7.336·10−4 5.425·10−9 1.218·10−4

no convergence 2.714·10−3 3.872·10−2 3.558·10−5 1.643·10−3 3.648·10−7 5.968·10−4 3.200·10−3 4.713·10−2 3.700·10−5 1.954·10−3 3.690·10−7 6.970·10−4 1.011·10−3 1.630·10−2 9.076·10−6 3.133·10−3 1.111·10−7 1.180·10−3

e e

Δt/5

eΔt/5

ep/p+1

ep/p+1

are characterized by the stationary chemical equilibrium state of the virgin material. 4.6.3.2.2.1 Analysis of the Numerical Results Figure 4.145 shows the evolution of the calcium concentration of the pore fluid c, the calcium concentration of the skeleton s and the reaction rate s. ˙ According to equation (4.420), the reaction rate is normalized with s˙ max = 1.8324 10−4 mol/s m3 . The peaks in the reaction rate s˙ and the steps in the concentration field s are indicators for the propagation of the calcium hydroxide dissolution and the CSH decalcification zones. As discussed in [454, 455] the reaction rate s˙ for calcium hydroxide dissolution is orders of magnitude larger than for CSH. Also, the calcium hydroxide dissolution zone propagates much faster than the region of CSH-dissolution. At time t = 3·109 s oscillations of the reaction rate s˙ are visible. The reason for these oscillations and the effect of linear and quadratic continuous Galerkin time integration schemes on this solution behavior are inΔt/2 vestigated in Figure 4.146. The left column shows the spatial local error el

4.6 Application of Lifetime-Oriented Analysis and Design chemical loading I2 = [50, 100]·107 s

N dG cG p=1 p=2 p=3 1

3

10 Δt [106 s] 100

log e

log e −7 −8 −9 −10

reaction-diffusion I1 = [575, 625]·107 s −2 −3 −4 1 : 1 −5

0 −1 −2 −3

603

1 : 1.9 1 : 3.3

1:2 1:2

−9 −10 1 : 2.8 −11 −12 1 3

1 : 4.5

10 Δt [106 s] 100

Fig. 4.144. Calcium leaching of a cementitious bar: Average relative errors of the Newmark method (N), discontinuous and continuous Galerkin methods (dG und cG, ep/p+1 ) within the time intervals I1 = [575, 625]·107 s (reaction-diffusion phase) and I2 = [50, 100]·107 s (chemical loading phase)

(compare equation (4.421)) of the cG(1)-solution. In the middle and right columns cG(1)- and cG(2)-solutions are compared by means of the reaction rate s. ˙ The s-plots ˙ show, that oscillations start after the calcium hydroxide dissolution front has arrived at the top of the beam (X2 = H). Since the dissolution front approaches the end of the beam parallel to the boundary, the horizontal speed of the front grows to infinity. Consequently, a singularity of the rate s˙ causes the oscillations. The spatial position of the resulting oscillations is fixed (compare also Figure 4.145). Only a slow decay of the oscillations in the time domain can be observed. In conclusion, the error plots in Figures 4.146 and 4.147 demonstrate two characteristic properties of the numerical solution: Firstly, a local error maximum follows the dissolution front of calcium hydroxide. Secondly, the aforementioned singularity of s˙ introduces a large time integration error. The region of the singularity induced error is fixed and the amplitude is only slowly reduced in the time domain. 4.6.3.2.2.2 Robustness of Continuous Galerkin Solutions In Figure 4.147 the robustness of continuous Galerkin schemes is investigated if realistically small chemical loading times Tc are applied. The middle column shows again the spatial distribution of the calcium concentration s of the skeleton obtained from using the standard chemical loading time Tc = 109 s. This solution represents the minimal chemical loading time which can be employed if non-adaptive Newmark time integration schemes are adopted (compare [452]). As illustrated in the right column of Figure 4.147, the cG(1)-integration allows a robust numerical simulation of this

604


t = 109 s

concentration c/c0 1

1

1

0

0

0

X1

t = 2 · 109 s

X2

X1 1

X2

t = 3 · 109 s

X2

t = 4 · 109 s

X1 1

X2

1

0

X1

0

X1 1

1

0

X1 X2

0

t = 5 · 109 s

X2

0

X1

X2

X1 1

0

X2

1

0

X1 X2

X1 1

0

1

1

0

X1 1

X2

X2

0

X1

X2

X1 1

0

X2

reaction rate log s˙ n

concentration s/s0

0

X1 X2

X1 X2

Fig. 4.145. Calcium leaching of a cementitious beam: Numerical results obtained from cG(1)

initial boundary value problem, even for a very small value for the chemical loading time taken as Tc = 6.25 · 107 s. The full chemical loading can be applied within seven time steps.


cG(1), log s˙ n

Δt/2

cG(1), − log el

t = 2.1 · 109 s

2 4 6 8 X1

t = 2.2 · 109 s

X2

cG(2), log s˙ n 1

1

0

0

X1 2 4 6 8

X2

X1 1

X2

1

0 X1

t = 2.3 · 109 s

X2

0

X1 2 4 6 8

X2

X1 1

X2

1

0 X1

t = 2.4 · 109 s

X2

0

X1 2 4 6 8

X2

X1 1

X2

1

0 X1

t = 2.5 · 109 s

X2

0

X1 2 4 6 8

X2

X1 1

X2

1

0 X1 X2

0

X1 X2

605

X1 X2

Fig. 4.146. Calcium leaching of a cementitious beam: Investigation of the oscillations appearing in the numerical results obtained from cG(1)- and cG(2)-solutions

606


t = 7 · 107 s

2 4 6 8 X1

t = 109 s

X2

Tc = 6.25 · 107 s, s/s0

Tc = 109 s, s/s0

Δt/2

Tc = 109 s, − log el

1

1

0

0

X1 2 4 6 8

X2

X1 1

X2

1

0 X1

t = 2 · 109 s

X2

0

X1 2 4 6 8

X2

X1 1

X2

1

0 X1

t = 3 · 109 s

X2

0

X1 2 4 6 8

X2

X1 1

X2

1

0 X1

t = 4 · 109 s

X2

0

X1 2 4 6 8

X2

X1 1

X2

1

0 X1 X2

0

X1 X2

X1 X2

Fig. 4.147. Calcium leaching of a cementitious beam: Investigation of the robustness of the cG(1)-solution for small values of the chemical loading time Tc


(a)

(b)

607

(c)

Fig. 4.148. Damaged road bridge in M¨ unster-Hiltrup (Germany) (a), correspondent Finite Element model of the bridge (b) as well as of a singular tie rod with connecting plates(c)[297]

4.6.4 Arched Steel Bridge Under Wind Loading Authored by Dietrich Hartmann, and Andrés Wellmann Jelic

Matthias

Baitsch,

M´ ozes

G´ alffy

In the following example a representative application of the beforeexplained lifetime-oriented design concepts for steel structures is explained in detail. This application combines numerical methods like spatial as well as temporal discretization methods (Section 4.2) with methods for solving timevariant reliability problems (Section 4.4) together with suitable optimization strategies (Section 4.5) such that robust structural design candidates are identified for predefined lifetime values. Robustness implies the minimization of sensitivities of the researched structural response quantities to the inherent scatter of the incorporated design parameters like actions or material properties, respectively. 4.6.4.1 Definition of Structural Problem The example to be analyzed is an arched steel bridge or, more precisely, a connecting plate between a vertical tie rod and the main horizontal girder. The referenced full-scale structure in M¨ unster, Germany (see Figure 4.148(a)) already showed significant cracks in the connecting plates two years after the construction. This high sensitivity to fatigue was caused by sharp notches (like welding seams and manufactured holes) in combination with high-frequent oscillations of the vertical tie rods causing displacements y ≤ 10 mm in the mid length of the tie rod. Main cause for these strong oscillations was the aeroelastic phenomena of vortex shedding, often dominated by the so-called Lock-In effect, on the circular rod cross section. More information about this phenomena is already given in Section 2.1.1. Fundamental dimensions of the referenced bridge are the length l = 87.37 m, width w = 17.85 m and the total height h = 13.68 m. All vertical tie rods have

608


(a)

(b)

(c)

Fig. 4.149. Refined FE model of the connection plate for the original geometry (a) and for the optimized geometry (c) together with the discretization of the welding (b)

a circular cross section with a diameter d = 110 mm and length values varying in the range lH = 4.03 − 11.03 m. The structural response values are computed by means of Finite Element analyses on three different structural levels. At first, a coarse model of the total bridge structure, as pictured in Figure 4.148(b), is generated to identify the decisive connecting plate exhibiting the highest stress values. In the second step, a realistic, but still rather coarse Finite Element model of the bridge tie rod and its two connecting plates is generated, using bar- and shell-elements (Figure 4.148(c)). Additionally, on the upper end of the system also a portion of the bridge arch is generated in order to correctly map the clamping conditions of the plate edge welded to the arch profile. For the last-named finite element model, the eigenshape corresponding to the lowest eigenfrequency is identified. Thereby, also the geometric stiffness values of all bars and shell elements resulting from the tension force under self-weight are taken into account. Based on the computed eigenshape, scaled to a deflection-amplitude of 1 mm, the displacements and rotations of the nodes located in the connecting plate can be determined. The third level comprises the generation of a refined FE model of the upper connection plate at the bridge arch using 3D-solid elements (Figure 4.149). At this, the notches of the weldings are to be modelled as a cylindrical surface with a radius r = 1 mm according to the R1MS-concept (also known as notch stress approach) adopted by the IIW-guidelines [372]. The displacements and rotations of the nodes on the contours, obtained in the second step, are applied as support displacements and rotations on the corresponding nodes of the refined model.


609

Table 4.15. Type of random variables (RV) included in the reliability problem used to describe the scatter of wind load parameters as well as material properties

Load parameters

Material properties

wind veloc. wind direct. initial displ. wind process ID [m/s]

[-]

[mm]

user-defined user-defined

ΔσD

k

DLim

[-]

[N/mm2 ]

[-]

[-]

Uniform

Logn.

RV

Weibull

Logn. Logn.

μX

2.06

-

-

(nsim + 1)/2

160.0

3.0

1.0

V (X)

1.09

-

-

(nsim − 1)/6

0.21

0.15

0.65

As stated before, the vortex-induced across-wind vibrations represent the dominant load factor in this example. A lifetime-oriented and realistic modelling of this wind load is accomplished by means of non-intermittent stochastic pulse processes based on a multi-scale load modell. The load event researched within the micro time scale is a local wind process with a constant pulse duration of t = 60 min. To this end, a continuous stochastic process is modelled following the definition in Section 2.1.1. For the computation of the correspondent wind loading on the researched structure the load model for vortex-induced cross vibrations in Section 2.1.3 is applied. Subsequently, the long-term nature of the natural wind is represented by the abovementioned pulse processes where each pulse represents a wind process with a 60 min duration. The magnitude of each pulse is described by a set of random variables summarized in Table 4.15. Additionally, Table 4.15 lists the parameters quantifying the scatter of material properties which are used to estimate the fatigue damage induced by the external loading. Hereby, the phenomenological, stochastically enhanced notch stress concept is used. As already stated before in Section 4.4.4.1 the parameters in Table 4.15 related to the material properties can be summarized in the vector R0 . In contrast to the standardized phenomenological proving methods for fatigue - such as the notch stress concept or the structural stress concept proposed in standards (e.g. the Eurocode 3 [7]) – the enhanced method describes two central parameters of the damage model by means of density functions fX (x) which are to be adapted to experimental results related to fatigue (e.g. [205]). The two parameters are the slope k and, respectively, the fatigue limit ΔσD of the fatigue curves employed. Additionally, the limit damage DLim is introduced as a third random variable, based on the results of experiments under variable cyclic loading published in [205]. As a consequence of this stochastic formulation, a damage analysis of each single wind series

610


realization leads to different estimates of damage values. In order to get reliable statistical estimates of the damage values multiple fatigue analyses have to be performed implying a time-expensive computational effort. With respect to this effort, the selected fatigue proving concept is highly favorable as it enables the approximate computation of lifetime values within a comparatively short response time. More information on this proving method for fatigue is already given in Section 4.6.2.3. 4.6.4.2 Probabilistic Lifetime Assessment As already mentioned before, the probabilistic lifetime assessment of the researched structural detail is accomplished in two conceptual steps. Therefore, the correspondent numerical results are presented separately in more detail. 4.6.4.2.1 Micro Time Scale The computed stresses in the nodes of the refined model with 3D-solid elements are shown in Figure 4.150, the position of the maximal values being marked with arrows. Maximal effective stresses, caused by bending in the connecting plate, are computed in two different regions – the bulk material without any notch as well as the weld material – as the resistance values in terms of S-N curves differ for these regions. A maximal stress value σe = 12.0 N/mm2 is identified in the welding parallel to the plate and σe = 20.7 N/mm2 in the welding perpendicular to the plate, in the re-entrant corner. For the bulk material of the plate, the maximal stress cannot be unambiguously defined, because the stress on the edge of the semi-circular cut increases on approaching the welding. For the lifetime

[N/mm2]

[N/mm2] 20.0

10.0

4.74 N/mm2

12.0 N/mm2 5.0

15.0

20.7 N/mm2

10.0 5.0

0.0

0.0

Fig. 4.150. Effective stresses σe caused by a rod deflection of 1 mm in the welding parallel to the plate (middle) and perpendicular to the plate (right)


611

evaluation, the value σe = 4.74 N/mm2 is considered, which represents a local maximum at a height of ca. 32 mm above the re-entrant corner. All named values refer to the nodes on the structural surface. They are used as stress concentration factors for determining the stress from the amplitudes computed on the secondly named FE model, consisting of bar elements. By using this simplified model the stress-time histories σ(t) can be computed efficiently for varying values of the mean wind velocity u ¯x,60 and the initial displacement y0 of the tie rod, respectively. In the next computational step, the stress ranges relevant for crack evolution are counted in each resulting stress-time history σ(t) by means of the enhanced Rainflow counting method proposed by Clormann/Seeger in [202]. Finally, the information about identified stress ranges is stored as representative empirical density functions hΔσ (Δσ) which are to be used in the macro time scale as follows. 4.6.4.2.2 Macro Time Scale

Partial damage value di [-]

Within this time scale the main probabilistic lifetime analysis is executed. For that, the above-named density functions hΔσ (Δσ) are transposed to corresponding partial damage values by using the S-N curves in combination with damage accumulation rules. Figure 4.151 shows a representative computation of partial damage values based on one generated set of the random material property values R0 . The partial damage values plotted there are defined

0.016 0.016

0.012

0.012 0.008

0.008

0.004

0.004 0

0

12.5 7.5

0

2.0

10.0

y0 ent em c a pl dis

] [mm

5.0 2.5

4.0

6.0 8.0 Mean wind velocity u[m/s]

10.0

0

l tia I ni

Fig. 4.151. Representative surface of partial damage values for varying mean wind velocities and initial displacement values at the critical tie rod [824]

612


Failure probability Pf(t) [-]

1 0.1 0.01 0.001 0.0001 1e-05 1e-06

Welding perpendicular to plate Welding parallel to plate Bulk material

1e-07 0.001

0.01

0.1 1 Service lifetime TS [a]

10

100

Fig. 4.152. Time-dependent evolution of the failure probability of critical material points in the welding region as well as the bulk material

as a percentage of the total structural lifetime. In this 3D-plot a significant structural impact for a range of very small and, therefore, very frequent mean wind velocities u ¯x,60 can be identified. Based on this fact it is to be stated that the existing connecting plate, as built in the reference structure, exhibits an unfavorably high sensitivity to fatigue damage. In a subsequent step, fatigue damage processes are simulated following the definition of the before-explained stochastic pulse processes. Final results of these simulations are time-dependent evolution functions Pf (t) of the failure probability of the researched structural detail as depicted in Figure 4.152. This plot shows the evolution of two different failure probability functions each corresponding to one of the material points with the maximum effective stress values identified in the micro time scale. Assuming a failure probability Pf = 2.3 % which corresponds to the 97.7 % fractile given in the Eurocode and in the IIW-guidelines, the calculation predicts a lifetime of 23.4 years for the bulk plate in the original geometry. The estimated lifetime of the weldings for the same Pf value is equal to 0.025 years (welding perpendicular to the plate), respectively 0.14 years (welding parallel to the plate). By this results the welding in the manufactured hole is clearly identified as the most critical region of the connecting plate, with an extremely short lifetime of 0.025 years. Therefore, the structural optimization, following as a next computational step, has to place a special focus on the shape optimization of this sensitive part.


613

1100 x9

140

x1

160

x10 x11

t = 25 1115

x12 x13 x14 x15

x8

x7

3000

x4

x6 x5 x3 x2

Ø110 w F

(a) Dimensions of initial design

(b) Discretization and geometry model

Fig. 4.153. Optimization model

4.6.4.3 Results of Structural Optimization In this section, the upper connection plate is optimized with respect to a maximum-stress criterion. The initial dimensions of the plate are shown in Figure 4.153(a). Two loading cases are taken into account: Loading A is a in-plane tension force of F = 975 kN acting on the hanger and loading B is a prescribed out-of-plane displacement w = 57 mm, in the middle of the hanger. The material is steel where E = 210000 N/mm2 and ν = 0.3. Linear elastic material behavior is assumed. The system is modelled by means of high-order two-dimensional elements using an object-oriented software system [80, 77]. Fig. 4.153(b) shows the discretization by means of 39 elements and utilizing the system’s symmetry. The connection plate is optimized with respect to the maximum stress criterion. The objective function taken is the the sum of the maximum stresses of both loading cases where loading B is weighted by a factor of 0.75. The shape of the plate is described by 15 design variables being coupled to the control points of the NURBS representing the outer curve as well as the shape of the void (see Figure 4.153(b)). Bounds are imposed on the design variables such that the overall dimensions are limited to the initial ones. The resulting optimization problem is therefore unconstrained. The objective function is not continuously differentiable because of the max-operator involved in the evaluation. Therefore, parallel evolution strategies have been used for the optimization of the connection plate. A comparison of the initial and the optimized shape of the plate is shown in Fig. 4.154. It

614

4 Methodological Implementation 0.0

Initial shape

400.0 [N/mm²]

Improved shape

Fig. 4.154. Optimization results

can be seen that the void moves towards the boundary of the plate and its shape changes substantially. Together with the adaption of the outer shape the optimized geometry shows a significant reduction of the stress peak present in the initial geometry. This reduction has a direct impact on the estimated lifetime. Comparing the lifetimes obtained for the original and for the optimized geometry, it can be concluded that the lifetime specially of the welding has been enormously increased by the optimization from originally 0.025 to more than 1000 years. Furthermore, the structural optimization leads to a substantial increase of the overall lifetime of the connection plate from 0.02 to 5.6 years. For the optimized geometry, the overall lifetime of the structure is solely governed by the bulk plate, not the welding, as in the case of the original geometry. 4.6.4.4 Parallelization of Analyses As already mentioned before, a significant drawback of an sophisticated timevariant reliability analysis is the high computational runtime needed to finish the calculations. For example, the sequential execution of the analysis in the micro and macro time scale on a modern computer hardware2 leads to a total theoretical runtime of 194 days. Obviously, this runtime has to be minimized significantly to enable a practical application of the theoretical concept presented here. The same problem is to be dealt with at the methodological level of structural optimization. As the structural optimization strategies employed here comprise discrete FE analyses, again, high theoretical runtime values are to be expected. A strong reduction can be achieved by means of a parallelization of the structural optimization shown in Section 4.5.4. In the scope of this example 2

PC with AMD Opteron (2.2 GHz), 2 GB RAM, 80 GB HDD running under SuSE Linux.


615

Table 4.16. Comparison of resulting runtime values analyzing the connecting plate Sequential execution runtime [h] efficiency ep [-]

1680.0 -

Parallel execution 1 Process/node 2 Processes/node 31.5 16.3 0.97 0.94

different parallelization techniques are used in order to test and compare their practical robustness and parallel efficiency. In the following these techniques are explained shortly. The parallelization of the reliability analysis has been accomplished by means of an agent-based parallelization technique developed by Bilek,Wellmann and Hartmann presented in [825]. Main advantages of this technique to be named here are the high robustness as well as flexibility during the runtime of the parallel analysis. Robustness indicates that the crash of a running parallel task will not lead to the total crash of the complete analysis as it occurs when concurrent parallelization techniques like MPI or OpenMP are used; Flexibility means that new computing nodes can be added during the runtime in order to enhance the total number of computing nodes. Based on this technique the parallel analysis has been accomplished on a dedicated PC cluster with 55 computing nodes each equipped with two AMD Opterons. Because of occasional network problems some node were unreachable and had to be restarted, afterwards. However, due to the dynamical runtime behavior of the agent-based technique the restarted nodes could be integrated again within the group of computing nodes. A list of resulting runtime values of the parallel execution as well as parallel efficiency values is given in Table 4.16. The values given there demonstrate the strongly reduced runtime values of the different computations named before as well as a very good efficiency of the parallelization near to the value 1. In this context, the theoretical nature of the given efficiency values has to be emphasized as the corresponding sequential runtime can only be estimated theoretically. 4.6.4.5 Final Conclusion Finally, it is to be summarized that by combining the multi-scale modelling of external loads with enhanced simulation techniques like the DC-MCS an runtime-efficient computation of realistic lifetime values is enabled. A wide applicability of this concept employed here has already been substantiated in the scope of further probabilistic lifetime analyses like free-standing chimneys [268] as well as framed structures under crane loading [824]. Furthermore, an evidence for the good runtime performance of the variance-reducing simulation technique has been given empirically by Faber in [268]. However, the strongest runtime reduction has been achieved by employing innovative parallelization techniques which facilitate a very robust and fast execution

616


Fig. 4.155. The road bridge at H¨ unxe (Germany) shortly before its deconstruction in 2006

of the numerical analyses. By using these parallelization methods technically feasible computational response times can be attained. 4.6.5 Arched Reinforced Concrete Bridge Authored by Yuri Petryna, Rolf Breitenb¨ ucher and Alexander Ahrens A remarkable number of bridges in Germany has been build after the Second World War. These structures will reach their scheduled lifetime within the next years. Over the intervening years a wide variety of mechanical, physical and chemical damage processes has weakened their structural resistance. In this context the fundamental question arises, if it is economically advantageous to strengthen those old structures to maintain their bearing capacity, or to replace them sumptuously by new ones. Additionally, the intended lifetime of new structures should be regarded right from the beginning of their planning processes by the involved design engineers. Therefore, it is essential to provide concepts and numerical tools to assist their work. In the following example a rational approach to simulation and assessment of structural damage and lifetime is applied to a complex engineering structure, a degraded road bridge at H¨ unxe in Germany (Figure 4.155). The proposed approach bases on a finite set of deterministic nonlinear simulations up to structural failure. Spatial variability of mechanical and geometrical properties of the structural system has been taken into account by the use of random fields (Section 4.4.2.4). Mean values and standard deviations of relevant material properties, e.g. compressive strength and Young’s modulus, have been determined by testing of drilling cores extracted form different parts of the structure. To account for the inherent non-deterministic chararacter (uncertainty) of these values stochastic sampling methods have been applied. The creation of random fields by means of Latin Hypercube Sampling (Section 4.4.2.2.2) instead of pure Monte Carlo Simulations enables to signifcantly decrease the number of finite element simulations required. Each simulation results in a discrete lifetime prediction regarding to the currently active parameter set. The total number of prediction points delivers on the


617

Fig. 4.156. Location of prestressing tendons and crack pattern observed on the bridges main girders

one hand a direct estimation of structural lifetime distribution and allows on the other hand for an approximation by second order polynomials to obtain an analytical function of structural lifetime. In a further step, this function can serve as a basis of an design optimization task using for example derivative based methods (Section 4.5.3.1) if crucial design variables are among the chosen input random varibales. 4.6.5.1 Numerical Simulation Authored by Yuri and Hursit Ibuk

Petryna,

Rolf

Breitenb¨ ucher,

Alexander

Ahrens

The investigated road bridge on the Wesel-Datteln channel located at H¨ unxe in the Ruhr area in Germany was built in 1951 as a reinforced, prestressed concrete arch bridge (Figure 4.155). The bridge has been designed for two traffic lanes according to the former German standard for traffic loads DIN 1072 [2]. The structural system consists of two centrically prestressed main girders and twelve prestressed cross girders that carry the deck slab which is prestressed in both directions. The girders and the deck slab are suspended by ten pairs of prestressed concrete hangers to the reinforced concrete arches. Height and width of the arches vary linearly from h = 1.50 m, b = 0.60 m at the foot to h = 0.58 m, b = 1.45 m at the vertex, respectively. The bridge with a total length of 62.50 m is divided by those concrete hangers into eleven sub-spans of equal length. It is simply supported at both ends using roller bearings (Figure 4.156, left). Since in 1988 unexpected displacements and first concrete cracks occurred in the main girders, the bridge has continuously been under observation (Figure 4.156, right). Due to progressive damage the Waterways and Shipping administration Duisburg-Meiderich commissioned an extensive expertise in 2001. This expertise identified loss of prestress due to fatigue corrosion of the tendons as main cause for a critically reduced bearing capacity. Hence, in 2006 the bridge was replaced by a new one. In this context before deconstruction various investigations could be performed for studying the actual material properties and the structural behaviour after more than 50 years of traffic use. For this purpose also numerous concrete cores (h/d = 200/100 mm) were

618


side view: eastern arch 8,57

western arch

5,68

5,68

5,68

5,68

5,68

13,30

top view: main- and cross girder

5,68

5,68

5,68

5,68

5,68

6

7

8

9

10

11

5,68

5,68

5,68

5,68

5,68

5,68

west

1

2

3

4

5

5,68

5,68

5,68

5,68

5,68

64,20

east points

Fig. 4.157. Location of drilling cores

drilled out from various sections of the structure for further laboratory testings (Figure 4.157). The total number of drilling cores was limited by logistical reasons within the deconstruction process and in order to ensure the structural stability also during and after the core-drilling. In order to get information on the actual concrete properties, laboratory tests were performed according to the same methods described in Section 3.2.1.2. 4.6.5.1.1 Experimental Investigation on Mechanical Concrete Properties According to the available construction drawings the concrete used for all structural members tested now was conform to the formerly strength grade B 450 [3], which correlates at time of construction more or less with a C 30/37 according to the current standards. Due to lack of more detailed information it was assumed that the concrete composition in all parts of the bridge was the same. The maximum size of aggregates could be determined on the drilling cores to 32 mm. 4.6.5.1.1.1 Non-Destructive Tests In order to determine the concrete’s stiffness, resp. the dynamic elastic modulus Edyn non-destructive ultrasonic tests (US-measurements) were performed. Comparing these results, the mean values of Edyn are on a similar level of about


619

Table 4.17. Dynamic elastic moduli Edyn (mean) and their standard deviations (SD) of the concrete after a service life of 50 years for the different members of the bridge structural member main girder cross girder arches carriageway slab cantilever section

number of specimens Edyn [−] [N/mm2 ] 12 49,700 8 47,600 17 50,600 5 50,300 18 52,900

SD [N/mm2 ] 4,200 2,700 3,300 2,300 5,300

50, 000 N/mm2 within all investigated parts of the structure (Table 4.17). The standard deviations (SD) in Edyn vary from 2, 700 N/mm2 to 5, 300 N/mm2. This scatter of in-situ concretes exceeds significantly the experiences on Edyn of laboratory concretes tested at an age of 28 days without former loadings. In the latter typical standard deviations of about 2, 000 N/mm2 were observed. Thus, it can be assumed, that a main part of the determined standard deviations in Edyn from the structural members is affected – besides the deviations due to testing and materials inhomogeneities – by deviations due to concrete’s post-hydratation, environmental impacts and mechanical effects raised by the cyclic loading itself. 4.6.5.1.1.2 Destructive Tests In addition, static compression tests were performed on a couple of specimens from the main structural members, mainly to determine the stress-strain relation. The results reveal that the mean values of the Young’s Modulus Estat , the ultimate strain u as well as the compressive strength fc are approximately on the same level for the different structural members (Table 4.18). (This would confirm the assumption, that in all investigated members nearly the same concrete had been used). On the other hand the scatters, e.g. a threeto sixfold standard deviation in fc of the in-situ concrete could be observed in Table 4.18. Relevant mechanical concrete properties Estat , u and fc (mean values) as well as their standard deviations (SD) after a service life of 50 years for the different structural members of the bridge structural member main girder cross girder arches

number of specimens [-] 7 6 5

Estat

SD

εu

SD

fc

SD

[N/mm2 ] 38,800 39,400 41,000

[N/mm2 ] 6,700 6,300 8,100

[%0 ] 2.17 2.02 2.28

[%0 ] 0.33 0.27 0.13

[N/mm2 ] 72.4 70.6 86.2

[N/mm2 ] 18.5 13.7 15.4

620


Cylinder compressive strength [N/mm²]

comparison to results of common static compression 28 days-tests on common separately fabricated concrete specimens. Although fc scattered within a wide range, it became also obvious, that the compressive strength fc has increased due to the post-hardening during 50 years. Assuming, that the concrete has fulfilled the requirements to a B 450, resp. C 30/37, at construction time, a post-hardening of 80 ... 100 % can be stated. On the other hand, the ultimate strain u remained more or less on the value of a C 30/37 at an age of 28 days (Figure 4.158). At first, these results seem to be in contrast to the typical stress-strain relations of laboratory concrete at an age of 28 days (Figure 4.158). In the latter, the ultimate strain u also increases with increasing compressive strength fc . Although the posthardening effect on the one hand leads to a significant increase in strength in the bridge’s in-situ concrete, on the other hand the cyclic loadings reduced – analogue to investigations on laboratory test concretes – the ultimate strain u of the in-situ concrete significantly (Section 3.2.1.2). (The strength fc is impaired by cyclic loadings only barely). Additionally also the shape of the stress-strain relation diverges significantly between laboratory concretes (at 28 days) and the in-situ concrete of the 50 year old bridge (Figure 4.158). The typical concave shape towards the strain axis was not observable at the insitu concrete. Thus, it could be proved also by these tests, that cyclic loadings

Bridge concrete

C 80/95

80

60

C 50/60 C 35/45

40

C 20/25

20

Ultimate strains eu

0

0

-1

-2 -3 Strain e [‰]

-4

Fig. 4.158. Comparison of stress-strain curves between bridge concrete (dashed line) and laboratory concretes with different strengths at the age of 28 days (solid lines) [193]


621

300 µm Fig. 4.159. LM-micrograph of in-situ concrete

change the stress-strain curve from a concave form towards the strain axis to a straight line, as it was also observed in cyclic tests on laboratory concretes (Section 3.2.1.2). 4.6.5.1.1.3 Microscopic Analysis Furthermore, microscopic analyses partly proved the existence of microcracks within the concrete microstructure caused by cyclic loading (Figure 4.159). The path of these microcracks are similar to those of laboratory concretes (Section 3.2.1.2), which were subjected to about 600,000 load cycles at a stress regime of Smax /Smin = 0.675/0.10. In both cases microcracking starts in the transition zone between cement paste and coarse aggregate grains. By the continuous cyclic loadings a prolongation of these microcracks through the cement paste was raised. However, it must be emphasised that the existence of microcracks significantly was depending on the extraction point within the respective drilling core, i.e. samples without any microcracks were observed as well. 4.6.5.1.1.4 Cyclic Tests The further changes in the concrete properties were also investigated by applying further cyclic loads on specimens taken from the bridge. Initially, the upper and lower stress levels Smax (= σmax · fc ) and Smin (= σmin · fc ) for the cyclic test regime were determined on the basis of actual determined concrete strength. For all cyclic tests the upper and lower stress levels were adjusted to 0.675 fc and 0.10 fc, respectively. Altogether, sixteen specimens were subjected to cyclic loading using the test setup described in Section 3.2.1.2.

622


0.0

Fatigue strain [‰]

Total strain [‰]

0.0 -0.5 -1.0

QT3-1

-1.5

HT11

-2.0 Smax/Smin = 0.675/0.10

O3

-0.1

-0.2 Smax/Smin = 0.675/0.10

-0.3

-2.5

0

10.0

20.0

30.0

0

200,000

arch main and cross girder

400,000

600,000

Number of cycles N [ - ]

Number of cycles N [in million] O1

O2

O3

O5

QT3-1

HT3

HT8

HT11

W3

Fig. 4.160. Total longitudinal (left) and fatigue strain (right) at Smax

Seven of these in-situ specimens failed already within a comparatively low number of load cycles between 652 and 307,000. Other ones, however, resisted millions of load cycles without any occurrence of failure. In comparison to these in-situ specimens, none of the laboratory concrete specimens of grade C 30/37 failed before applying about 800,000 load cycles at the same test regime. Hence, it can be revealed that the bridge’s in-situ concrete has a more sensitive behaviour to further cyclic loading in comparison to laboratory concrete of C 30/37 at an age of 28 days. The development in total longitudinal strain at Smax of all nine (unfailed) specimens during the cyclic tests is illustrated (Figure 4.160, left). On three of these specimens the cyclic tests were carried out as long-term tests for 12.0 millions to 27.8 millions load cycles. (The others were tested only up to 600,000 load cycles). At first, it became evident that the initial strains scatter within a wide range. In order to reveal these differences it is more suitable to take into account only the increase in strains during the cyclic loadings (”fatigue strains”, see also Section 3.2.1.2). For this purpose the fatigue strains of the nine in-situ specimens up to 600,000 cycles are separately illustrated in Figure 4.160, right. The development of the fatigue strains within the first 600,000 cycles are quite differently shaped for each specimen, which indicates also a wide scattering in the maximal bearable number of cycles up to failure Nf . Before the cyclic tests were started the mean value of the compressive strength fc had been determined. This averaged strength was taken as the reference value to adjust the stress levels Smax and Smin . Since the compressive strength fc of the bridge’s concrete varied significant (Table 4.18), the parameters Smax and Smin of the cyclic stress regime could not precisely be adjusted to 0.675 and 0.10, respectively.


t = 50 a Fatigue strain e fat,max [‰] -0.6 -0.4 -0.2

0

-0.8 100

Residual Young’s modulus [%]

95 90 85 80 75 70 Smax*/Smin* = 0.55 - 0.64/0.10

t = 50 a

65

Fatigue strain e fat,max [‰] -0.6 -0.4 -0.2

95 90 85 80 75 70 Smax*/Smin* = 0.55 - 0.64/0.10

60 O3

QT3-1

HT11

0 100

Residual dynamic elastic modulus [%]

-0.8

623

65 60

O3

QT3-1

HT11

Fig. 4.161. Correlation between fatigue strain and the residual stiffness for Smax /Smin = 0.675/0.10

Since the compressive strength fc remained nearly constant during the cyclic tests (Section 3.2.1.2), it was possible to calculate almost the actual ∗ ∗ ∗ specific values for both stress levels Smax (Smax = 0.675 fc/fc∗ ) and Smin ∗ ∗ ∗ (Smin = 0.10 fc/fc ) by determining the specific compressive strength fc after the cyclic test. The specific strengths fc∗ of the three long-term cyclic loaded specimens amount to 86.6 N/mm2 for QT 3-1, 88.9 N/mm2 for HT 11 and ∗ 90.9 N/mm2 for O 3. Thus, the real values of the upper stress level Smax amount to 0.55 (QT 3-1, HT 11) as well as 0.64 (O 3). Cyclic loadings lead to degradation processes combined with changes in the mechanical concrete properties. An adequate description of the changes in the Young’s modulus, referred to the fatigue strain, is given in Section 3.2.1.2. Following this approach here, the Young’s modulus as well as the dynamic elastic modulus versus the fatigue strain are illustrated in Figure 4.161 for the three long-term cyclic tests. Thereby, it has to be considered, that the concrete specimens in this case are already about 50 years old and had imprinted already a certain amount of fatigue strain within this period. However, this amount of the accumulated fatigue strains remains unknown in value and is surely different for each specimen. Nevertheless, at first roughly an almost linear relationship between the residual Young’s modulus/dynamic elastic modulus resp. and the fatigue strain has been observed. Although, the actual stress levels of the applied loads on each specimen are not equal, the changes in the stiffness can be approximated adequately by a common trendline. This underlines again that the linear relationship between the residual stiffness and the fatigue strain ∗ at Smax is also valid for lower load levels as observed in other tests (Section 3.2.1.2). Furthermore, it could be proved that quite different accumulated fatigue strains – as it can be assumed within the 50 years of service lifetime

624


Fig. 4.162. Three dimensional Finite Element model of the road bridge at H¨ unxe

Table 4.19. Number of elements of structural members structural deck slab main member girders number of 1632 816 elements

cantilevers arches 1428

204

cross girders 736

hangers

sum

224

5040

of the bridge – have no significant influence on further development of the ratios between residual stiffness and fatigue strain. Additionally, it could be observed that the dynamic elastic modulus is quite more influenced by the cyclic loading than the Young’s one. In comparison to investigated normal and high strength laboratory concretes without any pre-loadings (Section 3.2.1.2), the development of the residual Young’s modulus versus fatigue strain of the bridge’s concrete follows nearly the same trendline. 4.6.5.1.2 Finite Element Model A three dimensional finite element model of the bridge has been developed for numerical analysis of the structural state after 50 years of service (Figure 4.162). To match the geometrical shape of the bridge as well as possible and to model the connections of all structural members correctly, a quite large number of elements according to Table 4.19 has been required. The size of the resulting stiffness matrix is about one billion entries, which only could be handled using bandwidth optimization and sparse storage schemes offered by the finite element program [788]. A three dimensional shell element suitable for geometrically and physically nonlinear analyses has been implemented for calculation purposes (see e.g. [421, 443]). This element employs a layered approach to combine the both composites of reinforced concrete. The formulation of the finite element allows for up to four uniaxial steel layers to model reinforcement bars as well as the prestressing tendons in an accordant position.


625

4.6.5.1.3 Material Model To mirror the complex material behaviour of concrete correctly, a three ¨tzig and Po ¨ lling (see e.g. dimensional material model developed by Kra [444]) was used for the nonlinear finite element simulations of the structure. To avoid mesh-dependencies, the crack band and fracture energy approach has been incorporated into the material model [95]. Both, reinforcement steel bars as well as the prestressing tendons are predominantly subjected to tension. Therefore, they are modeled, according to the layered element concept, as dimensionless steel layers, using an uniaxial elasto-plastic material law with a damage component d. Hence, the resulting stress-strain relations for reinforcement bars and tendons read: σs = Es (1 − d)s

σs = Es (1 − d)(s + ps )

(4.423)

In the upper eq. (4.423) σs , s denote stress and strain of the steel due to loading. The prestrains of the tendons are termed ps , whereas Es stands for the Young’s modulus of steel. 4.6.5.1.4 Damage Mechanisms According to the expertise two damage mechanisms are considered as relevant for the time-dependent degradation of the structure, namely fatigue of the prestressing tendons and corrosion of the reinforcement steel bars. For both appropriate numerical models are incorporated into the basic material model of reinforced/prestressed concrete for structural simulations. 4.6.5.1.4.1 Corrosion of the Reinforcement Steel Bars A first impact of corrosion on structural response is the reduction of the reinforcement bars’ cross-section during time. Assuming a constant corrosion rate ks for the entire perimeter of the reinforcement bars according to Figure 4.163, the cross-section area As of each steel bar at time t reads: As =

π(D0 − 2ks (t − ti ))2 4

(4.424)

In eq. (4.424) D0 denotes the initial undamaged diameter of the steel bars cross-section and ti the initiation time. For structural elements without concrete cracks corrosion is assumed to start after initiation time ti is passed. Within that time the corrosion attack front is presumed to permeate through the concrete cover to the steel bars. If concrete cracks appear in structural elements due to mechanical loading, corrosion initiates immediately thereafter (ti = 0). The second effect of corrosion concerns the damage of the bond between concrete and steel bars due to expanding rust products, which can reach up to the ninefold of the original steel volume. The impact of bond damage on


g lo b a l c o rro s io n

lo c a l c o rro s io n

a tta c k fro n t

n o c ra c k s

, d , d

w ith c ra c k s

@

d

k s

, d = 2 k st 0

t

c o n c re te c o v e r

in itia tio n t0

d a m a g e

b o n d d a m a g e

626

1 ,0 0 ,9 d 0 ,8 0 ,7 0 ,6 0 ,5 0 ,4 0 ,3 0 ,2 0 ,1 0 0 b

, A S/A 0 ,1

0 ,2

0 ,3

0 ,4

0 ,5

0 ,6

0 ,7

0 ,8

0 ,9

S 0

1 ,0

Fig. 4.163. Applied corrosion model

the structural response is a reduction of the part of the concrete strains c which is transferred to the steel strains s . Within our approach it is modeled as follows: s = (1 − db )c

(4.425)

Further, the evolution of bond damage is guided by a variable db taken from experiments of [181] and depicted here on the right hand side of Figure 4.163:

db =

0 1−

1 33·ΔAs

for ΔAs /As < 0.03 for ΔAs /As ≥ 0.03

(4.426)

4.6.5.1.4.2 Fatigue of the Prestressing Tendons The second long-term damage mechanism, namely fatigue of the steel rein¨ hler-approach anchored in strucforcement bars, is modeled within the Wo tural design codes [182]. The failure criterion is defined by the bilinear S-N curve (Figure 4.164), which relates the stress amplitudes ΔσRsk resulting from each truck crossing, to the number of load cycles to fatigue failure Nf :

1 6 ∗ k1 ΔσRsk1 [ 10 for Nf ≤ 106 Nf ] (4.427) ΔσRsk = 1 8 ∗ k2 ΔσRsk2 [ 10 for Nf > 106 Nf ] ∗ ∗ Herein, ΔσRsk1 , ΔσRsk2 denote the limit values of the stress amplitude for 6 8 10 and 10 load cycles, respectively, depending on the diameter of steel bars; (k1 = 5) and (k2 = 9) are parameters defining the logarithmic slope of the S-N curves. This relationship has been modified to account for uncertainties of the fatigue life Nf by a parameter κs affecting the slope of both sections of the fatigue curve [627]. The evolution of the fatigue damage variable dfs at is described by a nonlinear function: ⎡ ⎤ m Nj (Δσs,j ) ⎦ 1 dfs at = − · ⎣1 − (1 − e−ϑs ) (4.428) ϑs N f j (Δσs,j ) j=1


R s k

1 0 0 0 d

s tre s s m a g n itu d e

c

3 0 0 2 0 0 1 0 0 5 0 3 0 2 0

M o d ific a tio n , c M o d e l C o d e , c

1 0 4

5

6

s s

¹ 5 .0 = 5 .0 7

n u m b e r o f c y c le s

s

fa t

0 .8

= 1 .0 2 .0 3 .0 4 .0 5 .0 6 .0 7 .0 c s 8 .0 = 9 .0 s

0 .6

d a m a g e

D s

[M N /m ²] 5 0 0

J

0 .4

5 0 .2

1 0 0

8

9

lo g N

627

1 0 f

0

0 .2

0 .4

0 .6

0 .8

2 0 m

j= 1

5 0 N j( d s , j ) N fj( d s , j )

Fig. 4.164. Modified S-N curves for steel and fatigue damage evolution function

where ϑs defines the degree of nonlinearity (Figure 4.164). The sum argument in the brackets reflects a normalized fatigue life accumulated at different stress amplitudes Δσs,j . Furthermore, the impact of fatigue damage is taken into account by a reduction of the material stiffness as follows: Esf at = (1 − dfs at )Es

(4.429)

4.6.5.1.5 Modelling of Uncertainties During an ordinary design process, all input parameters are usually treated in a deterministic way using just mean values as input. Such an approach denies the stochastic character of material properties and damage driving forces ab initio. In the context of generating input data for numerical simulations two important questions arise. The first one is, how many data sets have to be generated to ensure a good representation of the population characteristics, namely mean value, standard deviation and type of distribution. Thereby, it should be considered, that the higher the number of sets is chosen, the more expensive – in terms of computation time – the presented approach will be. The second question concerns the method to be used for this purpose. Therefore the statistical moments mean, standard deviation, skewness and kurtosis have been regarded (Figure 4.165). Obviously, an impressive small number of simulations seems to be sufficient for generation of input data with the postulated characteristics taking Latin Hypercube sampling instead of pure Monte Carlo method. This even holds for the higher order statistical moments skewness and kurtosis. Further, the generated data sets have been compared to the expected values in Figure 4.166. The Gaussian shape of the distributions of expected and generated values are in great accordance. Just little differences in mean and standard deviation of the material parameters compressive strength and elastic modulus were found. It is assumed that all material parameters obtained by testing reflect the bridge’s structural state after 50 years of service, shortly before its deconstruction. Therefore, the properties have to be transformed back to the structural virgin state, to serve as realistic input data for the lifetime simulations.

628

4 Methodological Implementation m e a n v a lu e

4 0 3 9

s ta n d a r d d e v ia tio n 6

e x p e c ta tio n M C S L H S


5 .5

3 8 5 3 7 4 .5

3 6

n s im

3 5 0

1 0

3 0

1 0 0

5 0 0

n s im 4

5 0 0 0

0

s k e w n e s s 1

3 0

1 0 0

5 0 0

5 0 0 0

k u r to s is 4


0 .5

1 0


3 .5 3

0 2 .5 -0 .5 2

n s im

-1 0

1 0

3 0

1 0 0

5 0 0

n s im

1 .5

5 0 0 0

0

1 0

3 0

1 0 0

5 0 0

5 0 0 0

Fig. 4.165. Higher order statistical moments

Y o u n g ' s m o d u lu s

c o m p r e s s iv e s tr e n g th e x p e c te d g e n e ra te d

0

1 0

2 0

e x p e c te d g e n e ra te d

m e a n

m e a n

s ta n d a r d d e v ia tio n

s ta n d a r d d e v ia tio n

3 0

4 0

5 0

6 0

7 0

0

1 0 0 0 0

2 0 0 0 0

3 0 0 0 0

4 0 0 0 0

5 0 0 0 0

6 0 0 0 0

Fig. 4.166. Validation of input data

4.6.5.1.5.1 Long-Term Developement of Concrete Strength The long-term evolution of the compressive strength fc depends on the cement type, the curing conditions and the ambient temperature. For a


629

fc5% 100

50

80

~1.60

60

compression strenght f

c

[N/mm²]

120

40

fcm

fc95%

60

30

Washa, Seamann, Cramer fc,50a ~ 1.60 x f c

20 CEB-FIP Modelcode 90

10

fc,50a ~ 1.45 x f c fc,50a ~ 1.28 x f c fc,50a ~ 1.22 x f c

0 0

10

20

30

40

drilling cores

20

time [a]

40

50

0

20

25

30

35

40

45

50

55

Fig. 4.167. Evolution of compressive strength and histogram of concrete strength

Table 4.20. Determination of compressive strength at time of construction structural member unit fc,cyl,50a fc,cyl,0a = fc,cyl,50a /1.60

cross girder [N/mm2 ] 70.6 43.8

main girder [N/mm2 ] 71.8 44.8

arches [N/mm2 ] 85.5 53.4

mean temperature of 20 ◦ C the Model Code 1990 provides a factor βcc (t) in eq. (4.430) to estimate the developement of compressive strength in time [182]. fcm (t) = βcc (t) · fcm

fcm = fck + 8[N/mm2 ]

(4.430)

Additionally, some data on long-term evolution of the compressive strength is available in the literature. Three long-time test series initiated about 100 years ago at the University of Wisconsin-Madison provide data of various concrete mixtures. A wide range of cement types, mix proportions and ambient conditions has been investigated [819]. An increase of the compressive strength occurs predominantly in the first ten years. Thereafter, the concrete strength remains nearly at a constant level of about 160% with respect to its initial value. This corresponds to the model code prediction, which is contrasted to the mean value of the experimental data stemming from the long-time test series C in Figure 4.167. Due to lack of information on the cement type of the bridge’s structural concrete, the measured concrete strength given in Table 4.20, is reduced by a factor of 1.60 to account for its evolution in time. Differences between geometric shape of specimens and standardised cylinders have been regarded as well.

630


Table 4.21. Concrete strength grades according to German standards DIN 1045 [1959] B 300 B 450

DIN 1045 [1978] B 25 B 35

DIN 1045-1 [2001] C 20/25 C 30/37

4.6.5.1.5.2 Determination of Material Properties According to the expertise and declarations on the available construction drawings, concrete conform to the formerly strength grades B 300 and B 450 has been used to build the bridge. As a side note, the lower grade B 300 has only been used for the concrete hangers. These strength grades were defined as mean values of at least three test specimen in the former German standard DIN 1045 [3]. To determine a realistic mean value for material parameter concrete strength, the notation there has to be transformed to the recent definition of concrete strength classes. The cube strength according to the six defined concrete strength classes given in the old standard represent a mean value of minimal three test cubes. Since the fourth edition of the German standard released in 1959 the classification of concrete classes has changed twice [4, 13]. By contrast, the classification of concrete today uses the 5%-fractile value of the concrete strength to set the crucial limits. The classification of both used concrete types according to the different definitions in German standards through time can be taken from Table 4.21. Assuming the compressive strength to be Gaussian distributed, the 5%fractile value allows the calculation of the corresponding expected mean value using eq. (4.431): fc,cyl,5% ≡

fc,cyl

= 30 [N/mm2]

fcm,cyl = 30 + 1 · 1.645 · s ≈ 38 [N/mm2]

(4.431)

fc,cyl,95% = 30 + 2 · 1.645 · s ≈ 46 [N/mm2] wherein s denotes the standard deviation of the compressive strength which is stated to 5 N/mm2 according to an evaluation of international results of castin-place concrete structures [694]. In Figure 4.167 a histogram of the resulting concrete strength population is given. Obviously, the results gained by testing the drilling cores fit to this population quite well. Further, the material properties Young’s modulus and tension strength are modeled as fully correlated to the compressive strength. Again the Model Code 1990 provides eq. (4.432) and eq. (4.433) to estimate those quantities with respect to fck . This is reasonable to keep the number of uncertain independent parameters as small as possible.


fck + 8 E(fck ) = 21500 · 10 fct (fck ) = 1.40 ·

fck 10

631

13 (4.432)

23 (4.433)

4.6.5.1.5.3 Modelling of Spatial Scatter by Random Fields The spatial variability of relevant material properties and damage driving forces can be described by the use of random fields. To create the random fields of those input parameters the midpoint approach, a type of the point discretization methods, has been used to determine the input values for all elements in all simulations [863, 521]. This procedure leads to an element-wise constant representation of each parameter in one simulation. The isotropic exponential autocorrelation function RHH which depends on the correlation 2 length lH and the variance of the random field σHH is evaluated by pairs for all element midpoints xi , xj of a chosen discretization of the structure. RHH =

2 σHH

(xi,1 − xj,1 )2 + (xi,2 − xj,2 )2 + (xi,3 − xj,3 )2 · exp − lH (4.434)

The correlation length lH describes the inter-elemental fluctuation rate of the properties under consideration. The upper limit of the correlation length lH → ∞ is also known as linear dependency of values. The other limit lH → 0 stands for uncorrelated or stochastically independent behaviour. A visualization of the effect of different correlation lengths on the bridges element mesh is given in Figure 4.168. In this study the calculation of all random fields has been performed assuming the correlation length equal to the largest dimension of the bridge - its length of 62.50 m. Additionally, all parameters were assumed to be independent and Gaussian distributed. An evaluation of all possible combinations of the element midpoints xi , xj following this procedure leads to the correlation matrix CH . CH = Φ · Λ · ΦT

(4.435)

CH is then transformed to the orthogonal uncorrelated space by means of its eigenvalues Λ = diag{λ1 , ..., λn } and eigenvectors Φ. The basic features ¯ Λ ¯ of each random field can be well approximated by a small subset Φ, of 20 eigenvalues only as depicted in Figure 4.168 [586]. Thereby, the use of even a limited number of Latin Hypercube samples grants an accurate representation of the required statistical distribution. ¯ ·Λ ¯ ΔH(x, y, z) = Φ

¯ = diag{λ1 , ..., λn } Λ

(4.436)

632


Fig. 4.168. Random field dependency on correlation length and eigenvalues used for reconstruction of correlation matrix

Each random field is finally represented as a sum of the mean value H0 , independent of location, and the fluctuation ΔH: H = H0 + ΔH(x, y, z)

(4.437)

4.6.5.1.6 Lifetime Simulation In general, structural degradation modelling in our concept follows a two step procedure, displayed in Figure 4.169. First, the structure is stepwise subjected to a design load combination consisting of dead (G) and traffic (Q) load as well as the prestressing (V ) of the tendons. At that load level, the external forces are kept constant. A further augmentation of the external load would lead to structural failure due to static overloading. The corresponding tangential stiffness relation in the first domain reads: KT (u, d)Δu(n+1) = G + V + λ(n+1) Q − FI (u, d)

(4.438)

where u, Δu denote the vectors of the system node displacement and its increment, respectively; KT indicates the system tangential matrix and FI the vector of internal forces, both depending on the current system u and damage d states. The low indices (n + 1) indicate values incremented on step (n + 1). A nonlinear structural simulation over the lifetime T , under the fixed load combination G + V + Q and the long-term degradation mechanisms described above, follows in the second step. The corresponding time-dependent system governing eq. (4.439) reads [622]: KT (u, d)Δu(Tm+1 ) = G + V + Q − (FI (Tm ) + ΔFI (Tm+1 ))

(4.439)


633

Fig. 4.169. Load deflection diagram and time deflection diagram 3D

tim e - d e fle c tio n d ia g r a m

life tim e h is to g r a m

d is p la c e m e n t [m ]

-0 .0 1 0

T

fa ilu r e

-0 .0 1 5

L ,fre q

T T

L ,m e a n L ,d e t

= 2 4 = 3 0 = 3 5

-0 .0 2 0 -0 .0 2 5 -0 .0 3 0 0 .0

1 0 .0

2 0 .0

life tim e T

3 0 .0 L

4 0 .0

[y e a rs ]

5 0 .0

6 0 .0 0

1 0

2 0

3 0

life tim e T

4 0 L

5 0

6 0

[y e a rs ]

Fig. 4.170. Load deflection curves and lifetime distribution and estimation

Herein, the system matrices and vectors depend on the time increments ΔTm+1 = Tm+1 − Tm instead of the load increment λ. The change of the system state is now solely caused by an increment of internal forces ΔFI (Tm+1 ) due to damage. The structural degradation in time is reflected in the left part of Figure 4.169 as an increase of the deflections with time, solely caused by action of damage mechanisms introduced above. All long-term simulations are performed until a computational limit state is reached. Each corresponding time instant provides a discrete estimation of structural lifetime under given conditions (Figure 4.170, right). On the one hand, the entirety of all limit state points provides an impression of the lifetime distribution of the structure and on the other hand allows for approximation by a second order polynomial to obtain an analytical function of structural lifetime [623]. This function enables for reliability analysis of the structure not requiring any further time consuming nonlinear finite element calculations [622].

634


4.6.5.1.7 Conclusions The presented approach allows a combination of stochastic modelling and nonlinear damage-oriented structural analyses by the means of the finite element method considering relevant scattering properties. A method to determine from samples, reflecting the structural state after 50 years of service, to the corresponding material properties at the structural virgin state is presented. The relevant damage mechanisms and the corresponding implementation into numerical models have been introduced. Thereby, the interface of random fields - generated by Latin Hypercube Sampling - provides an efficient and simple way to treat uncertainties within long-term simulations of real structures. 4.6.5.2 Experimental Verification Authored by Heinz Waller and Armin Lenzen In the following text it is proposed to identify the dynamic characteristic of a structure by vibration measurements. In the case of damage this characteristic will be alternated. At the present time the experimental modal analysis is often used to determine eigenfrequencies, modal damping and mode shapes. For damage detection finite element models will be optimized by experimental data through model updating. The correct choice of parameters which will be adapted in the finite-element model is crucial for this procedure. Here another method is to be presented for damage detection and localization. Black-box state space models can be identified by subspace method from measurement data. These identified state space models represent the transfer function between input and output. The black box model for the intact system is compared with the black box model of the damaged monitored system. Variations of the structure can be detected by evaluating special damage indicators for instance by the static or dynamic influence coefficients ([492],[500]). Additionally a first step to transfer a black-box state-space models into a white-box model is presented. White-box models are physically interpretable and permit direct damage localization. The possibility of extracting mechanical properties like mass, stiffness or damping direct from identified state space models will be shown by theoretical mechanical equations. Because of differences between theory, simulation and experiment this is more difficult by real measurements. Nevertheless the identified model parameters (e.g. MarkovBlocks) are able to detect and localize variances of mechanical properties. Results from experimental measurements in our laboratory on a cantilever bending beam will show that the presented methods are able to localize changes of stiffness and mass. A rectangular steel pipe (80x40x2.9mm) with a length of 2.45m was used as test object. For vibration measurements eight one-dimensional acceleration sensors were attached equidistant. The mechanical structure was excited by impulse loads. Furthermore experiments on a


635

prestressed concrete tied-arch bridge in H¨ unxe (Germany) will be presented. The bridge (built in 1952) had a span of 62.5 meters and was deconstructed in 2005. Main- and cross-girder, track-slab and the hanger consisted of prestressed concrete, the arch was built in reinforced concrete. On the verge of deconstruction it was possible to accomplish numerous vibration measurements. For the experiments two states as a variation of the structure were induced. First an additional support near the bridge bearing of one main girder was set-up. In a second experiment one hanger from one tied arch was cut through. 4.6.5.2.1 State Space Model for Mechanical Structures There are two main concepts for the modelling of mechanical systems. Here it will be called: I. analytical physical or white-box modelling II. black-box modelling III. hybrid or grey-box modelling (combination of methods I / II) Mechanical systems can be analyzed by principles of mechanics to get a mathematical model on an analytical physical basis. This can be called white-box modelling. As an alternative the black-box modelling can be used to describe the input - output relation of the system. Mechanical structures are characterized through continuous physical parameters as mass and stiffness. To describe the dynamic properties of such structures partial differential equations can be used. For the numeric processing by digital computers a discretization is necessary. This can be done for example by finite element method with the following time continuous equation. M¨ xt + Dx˙ t + Kxt = ut

(4.440)

This second order differential equation can be transferred in a system of first order differential equations by introducing the velocities as derivation of the displacements. Both mechanical parameters will be merged in the state space vector z, x˙ t = v t ¨ t = v˙ t = −M−1 Dv t − M−1 Kxt + M−1 ut x or summarized in matrix notation: ⎡ ⎤ 0 I x˙ t 0 ⎢ ⎥ xt =⎣ + ⎦ v˙ t vt M−1 ut −M−1 K −M−1 D

(4.441) (4.442)

(4.443)

Written as well known state space formulation follows: Z˙ t = AZ t + But y t = CZ t + Dut

(4.444)

636


ut

-c- B

z˙ t - c- 6

zt

- c- C

- c - yt 6

A

- D Fig. 4.171. State space model

The state matrices for linear mechanical systems are then: ⎡ ⎤ 0 I ⎢ ⎥ A=⎣ ⎦ −M−1 K −M−1 D ⎡ ⎤ 0 ⎢ ⎥ B=⎣ ⎦ M−1 C= I0 ; D=0

(4.445)

This state space system (eq. (4.444)) describes the transfer behavior of a linear mechanical system. For linear models the system matrices are constant over time. In Figure 4.171 you can see that the input will be transferred to the output by the state space system. Remind that the transfer function can be described by the state space matrices after a Laplace-transformation of the state space system as following. y(s) = H(s)u(s)

(4.446) −1

= C(sI − A) Bu(s) CA0 B CA1 B CAi−1 B + u(s) = + . . . + s s2 si

(4.447) (4.448)

4.6.5.2.2 White Box Model - Physical Interpretable Parameters State space matrices identified by real experiments don’t show the physical interpretable structure from eq. (4.445) because the found state space matrices are ambiguous. The identified state space matrices C, A and B can be transformed by arbitrarily selectable transformation matrices T to a new ˜ A ˜ and B. ˜ equivalent state space system characterized trough the matrices C,


Ã ˜ nB ˜ = CT T−1 An T T−1 B = CAn B C

637

(4.449)

Because of the possibility of transformation identified state space matrices do not contain the mechanical interpretable structure from eq. (4.445) and the separated state space matrices are not qualified to extract mechanical information directly. In opposite to this the products of the state space matrices C, A and B (the so-called Markov parameters) are unique and physically interpretable as you can see in eq. (4.449). If one has a state space system build by a linear mechanical system (eq. (4.445)) and calculate the matrix product CAB it leads to the inverse mass matrix: ⎡ ⎤ ⎤⎡ 0 I ⎢ ⎥⎢ 0 ⎥ −1 CAB = I 0 ⎣ (4.450) ⎦=M ⎦⎣ −1 −1 −1 −M K −M D M Further more one can extract the flexibility or stiffness matrix by using an other power of the system matrix A: ⎤ ⎤⎡ ⎡ −1 −1 D −K M 0 −K ⎥ ⎥⎢ ⎢ −1 CA−1 B = I 0 ⎣ (4.451) ⎦ = −F = −K ⎦⎣ −1 I 0 M In theory the equations (4.450) and (4.451) describe the possibility to extract mass, flexibility and stiffness matrices from a state space model build from a linear mechanical model. Because of this the original mass, flexibility and stiffness matrices can be extracted by the Markov parameters if a state space system was identified by simulated vibration data from a finite element model. 4.6.5.2.3 Identification of Measured Mechanical Structures 4.6.5.2.3.1 Black Box Model - Deterministic System Identification To determine the real dynamic properties of mechanical structures experiments are often accomplished. With the measured signals of cause and effect for example the impulse response by the mechanical quantities displacement, velocity or acceleration - black-box state space models can be identified. Most basic theoretical concepts and methods for black-box identification follow on the basis of the system theory. According to the system theory and to the principle cause - effect, technical systems can be formulated as transfer systems, whereas the cause is assigned to the input and the effect to the output of a system. Above we have shown the possibility to build a state space model from the mechanical equation of motion. By experiments the measured values are simply available at discrete locations and time steps. For mathematical modelling a time discrete state space formulation is necessary:

638


¯ k + Bu ¯ k z k+1 = Az ¯ k + Du ¯ k y k = Cz

(4.452) (4.453)

The measured values y k can be arranged in a so-called Hankel matrix by the well known classical procedure: ⎡ ⎢ Yt1 Yt2 Yt3 ⎢ ⎢ ⎢ Yt 2 Yt 3 Yt 4 ⎢ ⎢ ¯ H=⎢ ⎢ Yt3 Yt4 ⎢ ⎢ ⎢ Yt4 ⎢ ⎣ . ..

⎤

⎡

¯A ¯ 0B ¯ Yt4 · · · ⎥ ⎢ C ⎥ ⎢ ⎥ ⎢ ¯ ¯1¯ ⎥ ⎢ CA B ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢C ¯ 2B ¯ ⎥ ⎢ ¯A ⎥ ⎢ ⎥ ⎢ ¯ ¯3¯ ⎥ ⎢C ⎥ ⎢ A B ⎦ ⎣ . ..

⎤ 1¯ ¯ ¯2¯ ¯ ¯4¯ ¯ ¯ CA B CA B CA B · · · ⎥ ⎥ ⎥ ¯A ¯ 2B ¯ C ¯A ¯ 3B ¯ ⎥ C ⎥ ⎥ 3 ⎥ ¯A ¯ B ¯ C ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (4.454)

¯ is indefinite it can be shown Although the dimension of the Hankel matrix H ¯ that the rank of H is finite. The rank can be estimated by Singular Value Decomposition (SVD). By SVD the Hankel matrix can be decomposed into two matrices, the so-called controllable matrix and observable matrix. From this ¯ A ¯ and B ¯ can be detertwo matrices the discrete state space parameters C, mined. For further information for identifying parameters from deterministic and stochastic excited structures see [491]. Additional algorithms can be found in [293] or [609]. For mechanical systems the identified time-discrete state space parameters have to be transformed into time-continuous parameters. For this transformation there exist different algorithms for example the zero order hold: ¯ ln A A= (4.455) Δt

−1 ¯ −I ¯ B= A ·A·B (4.456) ¯ C=C (4.457) ¯ D=D (4.458) In Section 4.6.5.2.2 we have shown that Markov parameters of state space systems are unique and physically interpretable. If the state space system is build by the linear equation of motion (eq. (4.440)) one can extract theoretical and in simulations the mechanical matrices mass, flexibility and stiffness from Markov parameters as shown Section 4.6.5.2.2. At real experiments this is more difficult. Some causes are described in the next section. 4.6.5.2.3.2 Differences between Theory and Experiment In Section 4.6.5.2.3.1 we have shortly presented an algorithm for identification of measured structures excited by impulse loads. The presented algorithm is based on the ideal theoretic impulse function with infinitesimal duration.


639

20

acceleration in [m/s2]

0 −20 −40 −60 −80 0

(a) experimental setup

100 200 300 400 discrete time steps − fs=10kHz

500

600

(b) measured impulse and system reaction (8 channels)

Fig. 4.172. Impulse excitation in laboratory

In a numerical time discrete description the impulse function is characterized through an amplitude unequal to zero at one time step. In opposite to this the impulse excitation at experiments has a finite duration. The presented impulse excitation of a laboratory experiment has a duration of about one hundred time steps for example (Figure 4.172). Additionally the response of the structure is delayed regarding the excitation by the impulse load. First the response of the structure near the impulse excitation took place and in the end the farthest locations. The structure needs a short time for transient vibration then free vibration took place. At theoretic derivations and simulations based on the equation of motion this effect is neglected. In Figure 4.173 the differences between a measured signal and a signal created by the identified state space model are shown exemplary

8 free vibration

transient vibration

acceleration in m/s

2

6 4 2 0 −2 −4

channel 1 measured channel 3 measured channel 1 identified channel 3 identified

−6 −8 0

ideally starting point

50

100 150 discrete time steps − f =10kHz

200

250

s

Fig. 4.173. Comparison between measured signals and signals from identified model

640


on two measured channels. One can clearly locate the transient vibration time characterized through bad consistence between measured signal and the signal created by the identified state space model. After this short time (here about 10ms) at the beginning of the vibration the consistence is good. To get mechanical interpretable Markov parameters the following important constraint must be fulfilled (see eq. (4.445)): CB = 0

(4.459)

This constraint is equivalent to the postulation that the measured values at the time t = 0 (impulse excitation) have to be zero. This theoretic formulated starting point can not be defined clearly at real experiments as you can see in Figure 4.172(b). One possibility to overcome this problem is to start the identification with measured data from the free vibration after the impulse load and then count back the identified discrete state space system to an ideally starting point characterized by a value near zero of the Markov block ¯ B. ¯ C Another difference between theory and experiment is the symmetry of Markov parameters. If one has a look at the theoretical mechanical formulation it becomes clear that in theory the physical system matrices (mass, stiffness and damping) are symmetric. This symmetry is produced from the Maxwell’s reciprocal theorem that describes the permutability between cause and effect. Because of this the Markov parameters CAk B have to be symmetric. But even on experiments in laboratory with nearly perfect conditions it is difficult to get approximately this symmetry. Furthermore for mechanical interpretation the identified time-discrete parameters must be transformed to time-continuous parameters. This transformation can possibly cause numerical problems. In the text above some essential differences between theory and real measurements are demonstrated. In succession the identified Markov parameters are nonsymmetric and the equations (4.451) and (4.451) don’t apply exactly. Nevertheless the Markov parameters are physically interpretable. For damage detection we define the following Markov parameters as corresponding mass and flexibility matrices: ˜ −1 (CAB)id = M −1

(CA

B)id

˜ = −K ˜ −1 = −F

(4.460) (4.461)

It must be pointed out that this equations are valid for displacement measurements. If velocities or accelerations are measured the identified state space model can be converted through differentiation. In the next section it will be shown that changes of the structure stiffness or mass can be identified and localized by this corresponding mass and flexibility matrices.


641

Fig. 4.174. Cantilever bending beam used for experiments in laboratory

saw cut

30 cm

1

2

3

4

5

6

7

8

u1

u2

u3

u4

u5

u6

u7

u8

2,40 m

Fig. 4.175. Drawing from the cantilever bending beam with the location of saw cut

4.6.5.2.4 Experiments 4.6.5.2.4.1 Cantilever Bending Beam A rectangular steel pipe was used as test object. The rectangular pipe (80x40x2.9mm) with a length of 2.45m was restrained at one end and free on the other end. Eight acceleration sensors were attached equidistant and have measured the vertical vibrations of the bending beam. The mechanical structure was excited by impulse loads. To generate a nearly perfect impulse with very short impact time a weight fixed on a spring was used. The weight was lifted and then dropped on the structure. The impact was measured by an acceleration sensor and was used to normalize all measurements to an uniform excitation level. To get a complete state-space model with the same numbers at inputs and outputs the structure was excited on all sensor locations. This excitation on all degrees of freedom is essential to get mechanical interpretable quadratic system matrices. First the reference system was measured in undamaged state. To detect changes of the system two induced damages were created. First an additional

642


differences of inverse markov blocks : (CA−1B)−1

reference

− (CA−1B)−1

damage

−3

−6

0.4

−3

differences of markov blocks: (CA B)reference − (CA B)damage

differences of corresponding mass matrices 8

x 10

differences of corresponding fexibility matrices

0.3 6 0.2 4

0.1 0

2

−0.1 0

−0.2 −0.3 0

1

2

3

4 DOF

5

6

7

8

(a) Additional mass (1kg) at DOF 2

−2 1

2

3

4

DOF

5

6

7

8

(b) Saw cut (20mm) at DOF 6

Fig. 4.176. Markov parameters for damage detection

mass was attached at one sensor location. The changed structure was excited on all sensor positions again and the vibration responds were measured in analogous to the undamaged reference system. From the reference and the damaged measurements two state space models were identified. In figure 4.176a the additional mass (1kg) on sensor position two can be located clearly. Displayed is the difference between the inverted continuous Markov parameters (Cd AB)−1 from the reference and the damaged system. The weight of the undamaged reference structure including the eight acceleration sensors was about 20.8kg. On further experiments additional weights at other sensor locations could also be identified by the described method. To detect changes of stiffness the cantilever beam was damaged through a saw cut between sensor position five and six (1.70m far from the fixed support). The saw cut has a depth of 20mm. Analogous to the reference measurement the damaged system was excited by impulse loads on all sensor locations and the accelerations of the structure were recorded. On the basis of the recorded data a state space model was identified. In figure 4.176b the differences between the inverted continuous Markov parameters (Cd A−1 B)−1 from the reference and the damaged system are displayed. The saw cut near sensor position six can be identified clearly. 4.6.5.2.4.2 Tied-Arch Bridge near H¨ unxe - Germany Near H¨ unxe (Germany) a tied-arch bridge with a span of 62.5m (Fig. 4.177) was deconstructed in 2005 because of corrosion. The bridge was built in 1952 in order to lead a country road across the Wesel-Datteln-Canal. Main- and crossgirder, track-slab and the hanger consisted of prestressed concrete, the arch was built in reinforced concrete. On the verge of deconstruction it was possible to accomplish numerous vibration measurements. For the experiments two damaged states were induced. First an additional support near the bridge bearing of one main girder was set-up. In a second experiment one hanger from one tied arch was cut through. For the different experiments the bridge was excited through


643

Fig. 4.177. Bridge near H¨ unxe / Germany (span: 62.5m)

deterministic (by impulse hammer) and stochastic (by traffic, wind, etc.) loads. The used special constructed impulse hammer has a moveable weight of 150kg and was fixed on the bridge near the sensor points. The moveable weight was advanced by a pneumatic system. At impact a acceleration of 1000 m/s2 was measured.The available measurement system could handle sixteen acceleration sensors. Only the vertical acceleration of the two main girders were measured. First the actual condition of the bridge was measured individual for all excitations. With this data state space models of the reference condition were identified. Afterwards the damaged conditions were measured and identified. System Variation through Cut Hanger Before the deconstruction of the bridge took place one of the twenty hangers was cut through. The third hanger on south-west side was selected. The cross-section of the prestressed concrete hanger varies over the height from about 55x50cm down to 35x30cm upside. After this induced damage vibration measurements with deterministic (impulse) and stochastic (traffic, wind) excitation took place analogously to the reference measurements.

Fig. 4.178. System modification: hanger cut through

644


reference system − mode 10, 16.11 Hz

damaged hanger − mode 10a, 15.66 Hz 3rd hanger main girder west cut through

main girder east

main girder west

(a) reference system - mode 10

curvature reference system − mode 10a

main girder east

main girder west

(b) damaged hanger - mode 10a

curvature damaged hanger − mode 10a 3rd hanger main girder west cut through

main girder east

main girder west

(c) reference system

main girder east

main girder west

(d) damaged hanger - mode 10a

difference of curvature − mode 10a 3rd hanger main girder west cut through

main girder east

main girder west

(e) difference curvature Fig. 4.179. Torsional mode from reference system and after cut hanger

The measured modes identified by the system transfer function (eq. (4.448)) should be shown as first step for damage localization here. For lack of space only one mode should be introduced here. At the example of the 10th mode (figure 4.179 (a) and (b)) it is possible to see the change of the system by the generated damage clearly. The undamaged reference system has one torsional


645

mode with five maxima (16.11 Hz). After the failure of the hanger the belonging torsional mode can be identified with clearly changed frequency (15.66). The mode has his most essential change of amplitude in the area of the damage. Other modes show also significant changes in frequencies and shapes. An further damage indicator are the curvatures of the mode shapes displayed for the 10th mode in figure 4.179 (c) and (d). In figure 4.179 (e) the difference of the curvature from the reference and the damaged system is shown. Visible is that the undamaged main girder has no differences and the damaged main girder has a significant differences between damaged and undamaged state. Especially higher modes can identify local damages as you can see in the experiment with cut hanger. In a further experiment an additional support near the bridge bearing of one main girder was set-up. For this system variance lower modes (especially the first one) are adapted to identify the variation (further informations in [250]). Intense research is ongoing to extract further mechanical interpretable parameters (e.g. Markov parameters) for damage localization on this real measured tied-arch bridge. 4.6.5.2.5 Conclusion In this text a method was described that can identify changes of mechanical structures on the basis of vibration measurements. Initially a mathematical state space model will be identified by subspace method that describe the transfer behavior of the measured mechanical structure. It can be shown theoretically how characteristic system matrices like mass and stiffness can be extracted from such state space models. The shown differences between theory and real measured experiments produce problems that prevent the direct estimation of mass and stiffness matrices. From the theory divergent effects arises in particular from not ideal impulse loads, absence of symmetry (Maxwell’s reciprocal theorem), possible nonlinearities and measurement noise. Although special Markov parameters can be used for estimation of corresponding mass and flexibility matrices that are adapted for damage localization from vibration experiments. On a cantilever bending beam we could accomplish numerous vibration measurement for undamaged and damaged states in our laboratory. Different damage states were measured such as additional weights on single sensors positions or saw cuts between two sensors. The structure - a steel rectangle pipe - was excited by impulse loads on all degrees of freedom. For these real experiments it could be shown that products of the identified state space matrices C, A and B are able to locate discrete changes of masses and stiffness clearly. During further experimental vibration tests on a wide-span prestressedconcrete tied arch bridge methods of system identification were used to estimate state space models from measured data. Two system modifications should show the potential of the presented algorithms to identify and locate damages. Intense research is ongoing to extract mechanical interpretable parameters direct from the identified black-box model.

646 a) σ

4 Methodological Implementation σ

b)

f = 0.44 Hz σampl = 75 kPa σav = 89 kPa

t

Sand, ID0 = 0.9 K0 = 0.5 g0A = 0

7.3 m

b/2 = 0.5 m

Settlement [cm]

8

6

4 centrifuge model test Helm et al.

2

0 9.1 m

FE calculation

100

101

102

103

104

105


Fig. 4.180. Recalculation of a centrifuge model test of Helm et al. [365]

4.6.6 Examples for the Prediction of Settlement Due to Polycyclic Loading Authored by Torsten and Theodoros Triantafyllidis

Wichtmann,

Andrzej

Niemunis

Several boundary value problems have been calculated with the high-cyclemodel (Section 3.3.3). In order to confirm the prediction of the model a centrifuge model test of Helm et al. [365] was re-calculated using the FEM. In the model test (acceleration level 30g) a strip foundation was placed without embedding on a freshly pluviated dense fine sand and afterwards loaded by Nc = 105 cycles. The dimensions in the prototype scale and the loading are given in Figure 4.180a. In order to re-calculate the model test, the material constants of the fine sand were determined [835]. Figure 4.180a presents the FE discretisation with CPE4 elements. Figure 4.180b compares the settlement s(Nc ) in the model test and in the FE calculation. A good congruence of the curves can be observed. Thus, the high-cycle model was confirmed to deliver realistic settlement predictions. Parametric studies of strip foundations under cyclic loading were performed by Wichtmann et al. [836]. The variables of the soil (initial density ID0 , coefficient of lateral earth pressure K0 = σh /σv , historiotropic variable g0A ), the loading of the foundation (average value σ av , amplitude σ ampl ) and the geometry of the foundation (depth t of embedding, width b) were varied. Beside the strip foundation also the influence of the shape of the foundation was studied in calculations of shallow foundations with a quadratic or a circular cross section. Two examples showing the increase of settlement with the number of load cycles are presented in Figure 4.181. They demonstrate that the settlement increases with increasing load amplitude and decreasing depth of embedding. Table 4.22 summarizes qualitatively a parametric study of the settlement

4.6 Application of Lifetime-Oriented Analysis and Design σ

a)

σ

b)

σ

σ

σ

σampl = 150 kPa σav = 200 kPa

max

s

t

σmin = 0

t

24

10 8 6 4

l mp

a

[mm

]) =

20

] (s x Pa .50) ma [k (1 σ 300 6) (1.2 250 .06) 200 (1 .84) 150 (0 100 (0.58)

Settlement s [cm]

b = 1 m, t = 0, ID0 = 0.9, K0 = 0.38, g0A = 0

12

Settlement s [cm]

t

s

14

16

101

102

103

104

4

105

l

amp

t=

8

0 100

b = 1 m, ID0 = 0.8, K0 = 0.38, g0A = 0

12

50 (0.31)

2 0

647

100

101

0

s m(

.99

=1

)

mm

(1.36) t=1m (1.13) m 2 t= 7) t = 3 m (0.9 t = 4 m (0.85)

102

103

104

105



Fig. 4.181. Parametric studies of shallow strip foundations under cyclic loading: variation of a) load amplitude and b) depth of embedding

Table 4.22. Summary of the results of the FE calculations of strip foundations under cyclic loading: influence (↓ = decrease, ↑ = increase) of several parameters on the settlements (σ BC = bearing capacity) Varied parameter

Constant parameter

ID0 ↑ σ av , σ ampl , b, t, K0 , g0A max σ ↑ σ min = 0, b, t, ID0 , K0 , g0A ampl av σ /σ ↑ σ av , b, t, ID0 , K0 , g0A av σ ↑ σ ampl /σ av , b, t, ID0 , K0 , g0A σ av ↑ σ ampl , b, t, ID0 , K0 , g0A b↑ σ av , σ ampl , t, ID0 , K0 , g0A b↑ F av , F ampl , t, ID0 , K0 , g0A t↑ σ av , σ ampl , b, ID0 , K0 , g0A t↑ σ av /σ BC , σ ampl /σ av , b, ID0 , K0 , g0A K0 ↑ σ av , σ ampl , b, t, ID0 , g0A A g0 ↑ σ av , σ ampl , b, t, ID0 , K0

sstat sampl s(Nc = 105 ) s(Nc = 105 ) +s1 −(sstat +s1 ) ↓ ↑ ↑ ↑ ↑ ↑ ↓ ↓ ↑ ↓ -

↓ ↑ ↑ ↑ ↓ ↑ ↓ ↓ ↑ ↓ -

↓ ↑ ↑ ↑ ↓ ↑ ↓ ↓ ↑ ↓ ↓

↓ ↑ ↑ ↑ ↑ ↑ ↓ ↓ ↑ ↓ ↓

sstat + s1 after the application of the static load and the first cycle, the amplitude sampl of settlement, the settlement sacc = s(Nc = 105 ) − (sstat + s1 ) accumulated during the subsequent 105 cycles and the total settlement s(Nc = 105 ) after 105 cycles. Niemunis et al. [577] calculated the differential settlements of two neighboured foundations. The spatial distribution of the void ratio e(x) was stochastically generated (with three different spatial correlation lengths). 30

648


a)

σav = 100 kPa σampl = 50 kPa

Nc = 105 cycles

b)

50

σ

sl

sr

s

σ

σ

sl

sr 5.0

40

1.0

1.0

26.0 m

(Δs/s)cyc (Δs/s)stat a = 3.1

s

sstat scyc

13.0 m

1.0

σav

(Δs/s)cyc [%]

Δs Void ratio e

a=

a = 2.9

30

a = 2.6

20

Correlation length 0.5 m 2.0 m 20.0 m

10 0 0

5

10

15

20

(Δs/s)stat [%]

Fig. 4.182. FE calculations with stochastically fluctuating fields of the initial void ratio: a) dimensions and an example of a field e(x), b) differential settlement due to cyclic loading as a function of the differential settlement due to static loading

different fields e(x) (see an example in Figure 4.182a) were tested. Let sl and sr be the settlements of the left and the right foundation, respectively (Figure 4.182a). The differential settlement Δs = |sl − sr | was divided by the mean value s¯ = (sl + sr )/2. The ratio (Δs/¯ s)stat due to static loading up to σ av was compared to the ratio (Δs/¯ s)cyc describing the additional differential settlement accumulated during the subsequent 105 cycles. Independently of the correlation length the differential settlement (Δs/¯ s)cyc resulting from cyclic loading was observed to be approximately three times larger than (Δs/¯ s)stat caused by static loading (Figure 4.182b). This finding can be attributed to the fact that the settlement due to monotonic loading is proportional to the load, while the accumulation rate under cyclic loading is proportional to the square of the strain amplitude, i.e. approximately proportional to the square of the load. Therefore, a cyclic loading has a smaller range of influence than a monotonic loading and inhomogeneities of the field e(x) near the foundations have a larger effect (i.e. the differential settlements are larger due to averaging over a smaller region). Keßler [428] used the high-cycle model to simulate a vibratory compaction in a certain depth (Figure 4.183a, the pulling-out of the vibrator was not modelled yet). The initial densitiy and the frequency were varied. In that case, the implicit steps of the calculation were performed dynamically. Canbolat [168] determined the settlements of the abutment of a bridge (”H¨ unxer Br¨ ucke”) under 53 years of traffic loading. The geometry of the problem and the FE mesh are given in Figure 4.183b. The profile of void ratio with depth was chosen in accordance with in-situ CPT measurements using correlations. A special calculation strategy was used in order to apply the initial stress within the embankment [168]. The traffic loading was estimated based on the general development of traffic in the period 1951 - 2004, with traffic measurements for similar streets and with an information about


b)

σ 10

t void ratio

s [cm]

a)

649

15 m

σ

Pack. large

small

8 6

Pack. small

4 0

2

4

large 6

8

10

Number of cycles Nc [106] Settlement s [m]

vibrator

s

infinite elements

e0 = 0.715 ID0 = 0.4 Nc = 4,000

Fig. 4.183. a) FE calculation of a vibratory compaction see Keßler [428], b) FE calculation of the settlements of a bridge cp. Canbolat [168]

the percentage of the different classes of vehicles. The varying amplitudes due to different classes of vehicles were collected in packages of a constant amplitude. The accumulation model was also used by Niemunis et al. [580] for the calculation of excess pore water pressures and settlements in a water-saturated sand layer under earthquake loading. This problem was studied using the Finite Difference Method. A special numerical strategy was tested (Figure 4.184a). The fast processes (propagation of shear wave) were decoupled from the slow processes (accumulation of the mean values, e.g. excess pore water pressure) for one period T of the harmonic excitation of the rock bed. The dynamic calculation of the shear wave propagation in the sand layer during the first period T of excitation was performed with fixed values of σ av (average effective stress), uav (average excess pore water pressure) and eav (average void ratio). At the end of the period, the change of σ av , uav and eav during T was calculated by means of the accumulation model. For this purpose the strain amplitude εampl was obtained from the dynamic calculation. The pore water dissipation was also calculated in a separate step, i.e. decoupled from the ”dynamic” and the ”cumulative” mode. The values uav and σ av were modified over a period T (consolidation). The dynamic calculation of the wave propagation during the second period of excitation followed using the modified values of σ av , uav and eav , and so on. The introduction of special boundary conditions lead to a reflection of the shear wave at liquefied layers. Figure 4.184b presents an example of a calculation, i.e. the distributions of shear strain γ, shear strain amplitude γ ampl and excess pore water pressure uav with depth z for 15 calculated periods T (N = 15). It has to be critically remarked, that the shear strain amplitudes mostly exceed γ ampl = 10−3 (Figure 4.184b) and thus lay in a range, which was scarcely covered by laboratory tests up to now.

650


a)

time increments Δt 5 m). The FE mesh is given in Figure 4.186b. Since an uni-directional cyclic loading was studied the symmetry of the system could be utilized. The idealized loading resulting from wind and waves was grouped into packages with similar average value and amplitude (Figure 4.186c). Figure 4.186d exhibits that the high-cycle model predicts an increase of the horizontal deformations, i.e. an increase of the tilting of the OWPP with the number of cycles Nc . The aim of future research will be to exploit the limits of foundation design for extreme load events.

5 Future Life Time Oriented Design Concepts

Authored by Friedhelm Stangenberg, Dietrich Hartmann, Tobias Pfister and Andrés Wellmann Jelic

5.1 Exemplary Realization of Lifetime Control Using Concepts as Presented Here Authored by Friedhelm Stangenberg, Dietrich Hartmann, Tobias Pfister and Andrés Wellmann Jelic In the following two possible applications of the lifetime control concepts proposed within this book are examplarily presented. 5.1.1 Reinforced Concrete Column under Fatigue Load Authored by Friedhelm Stangenberg and Tobias Pfister In this first example, a reinforced concrete column under static and fatigue load is investigated. It is subjected to a static load case, which is assumed to appear once a year and a fatigue load case with one million cycles per year. The reliability of the structure as the major design matter is investigated in the initial state and during the scheduled lifetime of 80 years. The column is shown in Figure 5.1. A basic quantity for the estimation of the reliability is the scatter of material and model properties and of the load: the compressive strength, the tensile strength, the stiffness, and fracture energy are assumed normally distributed and fully correlated. The scatter of the lifetime Nf according the S-N -approach, as the basic quantity for the fatigue model, is correlated to the scatter of the compressive strength like described in Section 3.3.1.2.2.1, see e.g. eq. (3.125). The static load is assumed normally distributed.

654

5

Future Life Time Oriented Design Concepts

h = 4.00 m

e

6 elments, each 10 concrete layers

P

40 cm

2×3 d 16

static

P

f

cyclic static

T

40 cm

Fig. 5.1. Reinforced concrete column under fatigue loading

P

2.0

f

1.5

1.5

1.0

1.0

0.5

0.5

0.0

0

2

4

6

8

10

displacement f [cm]

12

20

30

40

50

60

load P [MN]

load P [MN]

2.0

0.0 70

compressive stregth fc [MPa]

Fig. 5.2. Degradation of the load-carrying-capacity and response surfaces at T = 0 a and T = 80 a together with Monte Carlo simulation points

The reliability in the initial state is estimated with the response surface method according to Section 4.4.2.3. The original design with 3 bars of diameter 16 mm on each side results in Pf = 0.532 × 10−6 and could thus be accepted. The time-dependent reliability is estimated with the time-discretization approach according to Section 4.4.3.2. The failure rate is evaluated with the response surface method for each time instant and integrated over the number of load events. Figure 5.2 (left diagram) shows the simulated degradation of the load-carrying capacity of the column due to increasing deformation and damage. The right diagram shows the resulting response surfaces in the

Exemplary Realization of Lifetime Control

-3

-3

-4

-4

-5

-5

-6

-6

-7

-7

-8

hP (t)

-9 -10

-8

Pf (t)

0

20

40

time T [a]

655

log hP (t), log Pf (t)

log hP (t), log Pf (t)

5.1

-9 60

80

0

20

40

60

-10 80

time T [a]

Fig. 5.3. Time-dependent hazard function and time-dependent reliability: original design (left) and improved design (right)

initial state and after 80 years lifetime, together with a cloud of Monte Carlo simulation points. The developing of the values of the hazard function as well as of the time-dependent reliability are shown in the left diagram in Figure 5.3. After 50 years, EC1 demands a safety index of β = 3.8. Under assumption of a normal distributed limit state function, this corresponds to a failure probability of Pf = 7.24×10−5. Like indicated in the diagram, this failure probability is missed, so the design has to be changed. As one possible alternative, the number of reinforcing bars has been changed from 3 to 4 on each side. The resulting values of the hazard function and of the reliability are shown in the right diagram in Figure 5.3. This design could be accepted. 5.1.2 Connection Plates of an Arched Steel Bridge Authored by Dietrich Hartmann and Andrés Wellmann Jelic The lifetime-oriented design of an arched steel bridge has been discussed already in Section 4.6.4, at full length. Here, therefore only the general approach for the lifetime analysis is recapitulated with respect to an implementation into the practice. The bridge contemplated in Section 4.6.4 is an arched steel bridge erected in M¨ unster, Germany, in 2001. Structural details of this structure, which are sensitive to fatigue, are the plates connecting the vertical tie rods with the main girders. During the design phase of this bridge, the checking methods for fatigue according to the German standard EC3 have been applied indicating that the stresses in the connecting plates are not exceeding the corresponding limit values. However, several connecting plates of the real structure showed macro cracks, already two years after the construction. Hence, more sophisticated

656

5

Future Life Time Oriented Design Concepts

e rall Pa

l / dis

tributed software s

Level

of optimization

yst em

Level of reliability

EA F t

Strucural detail Load model

EI

Loading / Load capacity

Damage model

Total structure

M i c r o ti m e s c a l e Damage evolution

M a c r o ti m e s c a l e

Fig. 5.4. Multi-level system approach followed during the lifetime analysis of the arched steel bridge [826]

lifetime-oriented design concept methods have been developed to estimate reliable lifetime values and, furthermore, investigate possible structural improvements. The design concept, suggested in Subsection 4.6.4, comprises two main approaches (Figure 5.4 and 5.5) which are to be explained more detailed. As depicted in Figure 5.4, a multi-level system approach is chosen having the following sublevels: • • • • • •

Level of load model where external loads are described analytically Level of total structure where those structural members are identified that are most sensitive with respect to fatigue Level of structural member for which a structural analysis and a computation of stress-time histories is carried out Level of fatigue in the critical structural components of the bridge identified according to used verification concepts for fatigue Reliability level for the time-variant limit state of fatigue Optimization level of the total structure as well as the identified weak structural components

All sublevels are embedded in a parallel and distributed software system as illustrated by the external shell in the diagram (see Figure 5.4). Another key idea of the design concept is the multi-scale resolution of load actions with respect to time allowing the separation of the two computing

5.1

Exemplary Realization of Lifetime Control

657

Micro time [t] = s Load F(t)

Structure

Response

X2 X1 t

Macro time [t] =h Load process X(t)

Fatigue process

!

d(t)

!

!=3 ! = 2 ! = 1

X3

t

!=3 ! = 2 ! = 1

t

Fig. 5.5. Multi-scale modeling and analysis of fatigue-related structural problems

tasks structural analysis and reliability analysis. For that, different time scales in the micro and macro scale are introduced and analyzed in an interlocked fashion, as demonstrated in Figure 5.5. Within the micro time scale single load events, i.e. 10-min wind processes or vehicle crossing, are analyzed with regard to their structural impact. The numerical results of the structural analyses, carried out by means of a Finite Element Analysis for different parameter sets of the corresponding load event, are stored in a file-based lookup table. Subsequently, the random sequence of these single load events is modelled in the macro time scale represented by stochastic pulse processes. Here, partial damage values induced by each load event are estimated using stochastically defined S-N-curves. Finally, the partial damages are accumulated analogously to the pulse process until a predefined damage limit state is reached. The numerical methods, used in the above-named sublevels, have already been explained in section 4, together with exemplary results in Subsection 4.6.4. In the given context, only the achieved lifetime increase of the exemplarily researched structural problem is highlighted. For that, Figure 5.6 shows the time-dependent evolution of computed failure probabilities of the connection plate. The comparison of the two plotted curves substantiates the drastically increased lifetime of the optimized plate shape. E.g. at a reliability level of Pf = 2.3%, the lifetime of the original shape (TL = 0.025 a) has been improved to TL = 5.6 a for the optimized plate shape. Finally, the main benefits of the proposed multi-scale and multi-level approach can be summarized as follows: According to the multi-level system approach, a well-organized and simplified model of the initially complex structural design problem is provided. By that, suitable analytical solution methods

658

5

Future Life Time Oriented Design Concepts 1

Failure probability Pf (t) [-]

0.1 Initial shape

Improved shape

0.01 0.001 0.0001 1e-05 1e-06

Welding perpendicular to plate Bulk material

1e-07 0.001

0.01

0.1 1 Service lifetime TS [a]

10

100

Fig. 5.6. Comparison of resulting time-dependent failure probabilities of the researched connection plate

for each different system level can be developed and considered efficiently. A benefit resulting from the multi-scale approach is a lifetime-oriented modelling of actions and, additionally, a runtime-efficient analysis of the correspondent lifetime of the structure. Particularly, the combination of this multi-scale approach with parallelization techniques, as explained in Subsection 4.5.4, allow the numerical analysis in a comparatively short response time. 5.1.3 Conclusion Authored by Friedhelm Stangenberg, Andrés Wellmann Jelic and Tobias Pfister

Dietrich

Hartmann,

These short examples show, how the models and methods presented in this book can be applied to extend the design process according to standard design codes. These applications are time-consuming and need extensive knowledge. Thus, they can be applied only selectively for specific problems. They can only be established in engineering practice by degrees going along with enhancement and improvement of the applied models and methods.

5.2 Lifetime-Control Provisions in Current Standardization Authored by Friedhelm Stangenberg In the Eurocodes (ECs) and other codes, links are implemented for providing later detailing of regulations concerning lifetime control.

5.3 Incorporation into Structural Engineering Standards

659

E.g. EC1 mentions five “building classes” distinguishing different “design working life”: • • • • •

temporary buildings, renewable structural components (e.g. bearing elements), agricultural or similar structures, residential and business buildings, monumental and engineering structures (particularly bridges).

EC2 has chapters about “durability of reinforced concrete”, particularly with respect to “environmental conditions” as well as to “chemical and physical attacks”. For steel reinforcement, EC2 mentions, in context with reinforced concrete structural longtime resistance: “Where required, the products shall have adequate fatigue strength”. EC3, for steel structures, gives regulations for taking into account degradation effects due to “fatigue”. Further references to structural lifetime control can be found in other modern building codes. However, these hints are only regulatory frames, which must be filled by detailed precisions, in the next future. The research work documented in this book aims to contribute to a scientific basis for lifetime control and to derivations of regulations and specifications for later practical use. There is a common understanding that lifetime control—of course, combined with quality assurance on a high level—must be introduced into structural design processes and into the management of existing structures. A reasonable and sufficiently simplified handling of the resulting tools of these lifetime related structural control concepts will be the next step in a continuing development.

5.3 Incorporation into Structural Engineering Standards Authored by Friedhelm Stangenberg The realization of lifetime control in structural engineering practise will be followed by: • • • •

effecting a change in structural engineering mentality; pointing out the significance of a reliable service-life control to owners, users, licensing authorities, insurance companies, designing and controlling engineers; integrating service-life control aspects in quality assurance systems; transfer into codes, model codes and international regulatory principles of structural engineering.

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1. DIN EN 206-1. Festlegung, Eigenschaften, Herstellung und Konformit¨ at 2. DIN 1072. Straßen- und Wegbr¨ ucken, Belastungsannahmen (1952) 3. DIN 1045. Bestimmungen f¨ ur die Ausf¨ uhrung von Bauwerken aus Stahlbeton - Teil A (1959) 4. DIN 1045. Beton- und Stahlbeton: Bemessung und Ausf¨ uhrung (1978) 5. DIN V ENV 1992-1-1. EC 2: Planung von Stahlbeton- und Spannbetontragwerken, Teil 1-1: Grundlagen und Anwendungsregeln f¨ ur den Hochbau, Entwurf (1989) 6. DIN 4030-1. Beurteilung betonangreifender W¨ asser, B¨ oden und Gase - Grundlagen und Grenzwerte (1991) 7. EC 3: Design of steel structures. Part 1.9: Fatigue, CEN (1993) 8. DIN V ENV 1994-1-1. EC 4: Bemessung und Konstruktion von Verbundtragwerken aus Stahl und Beton, Teil 1-1: Allgemeine Bemessungsregeln und Anwendungsregeln f¨ ur den Hochbau (1994) 9. EN 1991. Actions on structures - Part 2: Traffic loads on bridges (1995) 10. EN ISO 13918. Welding - Studs and ceramic ferrules for arc stud welding (1998) 11. EN ISO 14555. Welding - Arc stud welding of metallic materials (1998) 12. EN 206-1. Concrete - Part 1: Specification, performance, production and conformity (2000) 13. DIN 1045-1. Tragwerke aus Beton, Stahlbeton und Spannbeton - Teil 1: Bemessung und Konstruktion (2001) 14. DIN 1045-2. Tragwerke aus Beton, Stahlbeton und Spannbetonn - Teil 2: Beton - Festlegung, Eigenschaften, Herstellung und Konformit¨ at - Anwendungsregeln zu DIN EN 206-1 (2001) 15. DIN EN 10002-1. Werkstoffe - Zugversuch - Teil 1: Pr¨ ufverfahren bei Raumtemperatur (2001) 16. EN 1990. Grundlagen der Tragwerksplanung (2002) 17. prEN 12390. Testing hardened concrete - Part 9: Freeze-thaw resistance, Scaling (2002) 18. DIN-Fachbericht 104. Verbundbr¨ ucken. Beuth-Verlag (2003) 19. ENV 1991-5. EC: Thermal actions, part 2-5: Thermal Actions, Steering Panel Draft (2003) 20. Conlife deliverable report 10: Recommendations for application of highperfomance concrete. Technical report, Project Co-Ordinator: Institut f¨ ur Bauphysik und Materialwissenschaft (2004)

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Subject Index

3D-p-finite element method

392

abrasion 1, 162 abstract multi field problem 372 abutment of a bridge 649 accumulation 111, 160, 198, 313, 626 acid attack 152 acidic fluids 152 acoustic emissions 169, 204 across-wind force 26 dimensionless lift coefficient, 26 Strouhal-number, 26 across-wind vibrations 29 action 4 adaptive method, 368, 372, 424, 425 time step control, 594 stepping procedure, 446 stepping scheme, 369, 412, 443 adaptivity error-controlled temporal, 443 in space and time, 425 advanced directional factors 23 aerodynamic admittance, 24 applicable characteristic suction force, 25 directional wind effect, 24 iteration, 24 suction force on the fa¸cade element, 24 ageing 3 chemically caused, 367 aggregate 189, 302, 307

algebraic equation 369 algorithmic balance equation 415 alkali-aggregate reaction 158 alkali-carbonate reaction 160 alkali-silica reaction 158, 302, 305, 311, 359 amplitude counting 574 analytical counting method 578 anchor pull-out test 495 angular frequency 115 anisotropic shape function, 380 softening law, 478 apodization 120 approximation of state variable 381, 414 approximativ solution 368 arbitrary order of accuracy 410 arbitrary polynomial degree 380 arc-length controlled iteration method 407 arched steel bridge 512, 607 assembling 384 assumed natural strain 362, 380 asymptotic displacement 469 asymptotic state 112 Babuˇska-Brezzi-condition 398, 585 back polarization 315 back stress tensor 246, 256 balance equation 373–375 balance of linear momentum 296, 426, 427 band-pass 120

712

Subject Index

Bauschinger effect 255, 256 beam finite elements 369 beat 118 Bernoulli formulation 397 biaxial strength 126 Biot coefficients, 286, 288 modulus, 286 Biot-Coussy theory 285 black-box modelling 635 blending function method 402 BMW 3-serie 572 Bossak-α method 411 boundary condition 373 brittle failure 307 Bubnov-Galerkin discontinuous method, 412, 421 building class 4, 659 C0 -continous 476 C0 -discontinuous 476 calcium diffusion, 296 dissolution, 295, 296, 300, 356 ion concentration, 297, 298, 300, 354 leaching, 107, 150, 152, 297, 591, 592 cantilever beam 563 capillary pressure, 285, 289, 292, 307 tension, 143 carbonation 1 Cauchy-Green tensor 437 cementitious beam, 601 material, 124, 236, 591 cement paste 124 centrifuge model test 646 change of position 367 chaos theory 370 chemical dissolution, 354 equilibrium, 295 expansive process, 359 loading, 592 range, 600 porosity, 300 chemo-mechanical damage model, 374 tangent coupling tensor, 386

chloride 1 CMDSS device 200 coalescence 256 coaxial states 116 complementary energy, 238 strain energy, 271 compliance tensor 238, 269, 278 composite structure 207 compressive strength 246, 262, 265, 270, 290, 622, 629 computational durability mechanics 285 computer-aided design 372 concrete 180, 237, 244, 265 corrosion, 95 cover, 625 high strength, 187 permeability, 590 properties, 618 shell structure, 583 strength, 628 conductivity tensor 297 connection plate 613 consistent linearization 367, 369, 377, 415 constitutive equation, 373, 427 law, 374 continuity 412 condition, 409, 418 continuum damage theory 307 continuum mechanics 294 control cycle 519 controllable numerical dissipation 409–411 convection 37 convergence 422 convex modelling 370 cooling 351 rate, 101 coordinate modal assurance criterion 569 corrector iteration 405, 408 correlation length 631 corrosion 1 attack, 625 rate, 625 reinforcement, 625

Subject Index COSSAN 538 coupled balance equation 377 coupled field problem 372, 376 covariance analysis 520 crack band, 625 elliptical, 270, 272, 274, 278, 280 extension, 492 formation, 190 growth, 489 growth criteria, 488 initiation, 207 kinked, 274, 276 kinking, 453 microcrack, 190 elliptical, 284 mixed mode, 272, 274, 278 propagation, 174, 225, 369, 474 simulation, 493 surface, 476 tip, 454, 469 topology, 493 cracking strains 244 Craig-Bampton approach 499 creep 143, 285, 291 effects, 133 long-term, 286 strain, 265–267 critical degree of saturation 95 crown cracks 351 crushing energy 246 CSH 293, 300, 356 cumulative distribution function 531 cycle amplitude double amplitude, 120 scalar amplitude, 118 tensorial amplitude, 114, 316 average, 114 circulation, 117 counting, 575 in-phase (=IP) cycles, 114 irregular cycle, 116, 198, 518 multiaxial, 115 out-of-phase (=OOP) cycles, 115 ovality, 114 packages of cycles, 205 polarization, 114, 202, 315 polarization changes, 203

713

regular cycles, 198 shape, 203 span, 117 strain cycles, 114 uniaxial, 115 cyclic flow rule, 161, 201 loading, 123, 129, 256, 519, 623, 647, 650 mobility, 113, 162, 206 multidimensional simple shear tests, 198 preloading, 112, 203, 315 tensile and flexural load, 136 test, 180, 621 triaxial tests, 198 Dahlquist theorem 411 Daltons law 307 damage 237, 366 accumulation, 7, 579, 581 chemical, 298 chemo-mechanical, 295, 367 creep, 367 criterion, 300, 478 detection, 634, 640 driving forces, 627 evolution, 167, 626 fatigue, 367 gradient enhanced, 307 hygro-mechanical, 367 isotropic, 238 mechanisms, 625 model anisotropic, 244 anisotropic elastoplastic, 244 continuum, 270, 274, 284 micropore, 255, 259 parameter, 238, 295, 300, 307 thermo-mechanical, 367 damage equivalent factors 69 damaging process 139 Darcys law 307 data windowing 120 Debye-H¨ uckel-Onsager theory 298 degeneration of a trajectory 118 degradation 3, 262, 411, 654 macroscopic, 367 process, 131, 182, 623

714

Subject Index

stiffness, 134 structural, 180, 359 degraded road bridge 617 deionized water 356 derivative-based method 552 derivative-free strategy 555 design concept, 93 process, 371 working life, 659 destructive test 619 deterioration 1, 307, 366 brittle, 367 ductile, 367 mechanic, 369 mechanism, 411 model, 372 deterministic subspace identification 527 detrended strain path 118 detrending 116 differential settlements 109 diffusion 292 direction of accumulation 161, 200, 201, 314 direct search algorithm 555 Dirichlet boundary condition 374, 411, 591, 598 Dirlik distribution function 579 discontinuous displacement approximation, 369 finite elements, 448 discrete-time system matrix 526 discrete Fourier transform (DFT) 119 discrete inverse Fourier transform (DIFT) 120 discretization 414 method, 294 disjoining pressure 143, 291, 293 dissipation potential 297 dissolution 152, 305 front, 356 process, 150, 152, 374 drilling core 618 driving force 373 Drucker-Prager plasticity model, 290 potential, 246, 269

drying 351 creep, 291 process, 590 shrinkage, 285 ductile mode 129 durability 92, 150, 294, 411, 424, 574, 659 analysis, 354 mechanics model, 424 structural, 368 dynamic influence coefficient, 634 load parameters, 82 distance between the pulse peaks, 84 pressure coefficients, 82 pulse amplitudes, 82 wall distance factor, 86 wall height factor, 86 response, 90 critical damping ratio, 91 modal decomposition, 91 resonant amplification factors, 91 stiffness requirement, 572 E-Modulus 134 earthquake 113, 162, 362, 649 edge modes 393 effective balance equation, 415 elastic properties, 271, 276 linearized system of equation, 421, 422 right hand side, 416 stress, 244, 285, 289, 307 elastic, 289 plastic, 289 tangent matrix, 416 eigenfrequency 634 elastoplastic models 244 electrical resistance method 163 electrophoretic effect 298 element 379 embedded crack models, 252 discontinuity approach, 369 endochronic models 518 energy complementary, 272

Subject Index conserving, 369 decaying method, 369 enhanced assumed strain 252, 362, 380, 427, 448 enriched finite elements 372 entropy 286 enumeration strategy 555 environmental attack, 583 humidity, 143 equivalent elliptical crack, 278 elliptical microcrack, 276 strain, 238, 307 error criterion, 431, 432 estimate, 412, 444, 594 h-method, 443 local, 443 p-method, 444 estimator, 391, 445 indicator, 391, 444 h-method, 444 local, 443 p-method, 444 measure, 372, 425 ettringite formation 157 Eurocode 659 evolution algorithm, 555 long-term, 629 strategy, 556 excessive displacement 367 exchangeable salt 152 expansion 157, 307, 311 experimental results 40 macroscopic temperature behaviour, 42 explicit mode 518 exponential convergency 388 exposure classes 93 extended finite element method 369, 448 external damage 95 fabric 109, 162 face modes 394 failure function, 238 probability, 529, 530, 534, 541, 612

715

Falancs 581 Faraday constant 298 fatigue 1, 163, 169, 180, 626, 653, 655, 659 analysis of bridge hangers, 29 crack, 272 damage, 267, 367 process, 612 failure, 262 high-cycle, 262, 270 high cycle, 129, 174, 207 life, 627 lifetime, 270 load, 270 low-cycle, 259, 262 low cycle, 129, 255 Newmark-Wilson time-step method, 31 strain, 185, 266 test, 259 ultra low cycle, 255 Ficks law 297 filtering 119 finite difference method 649 finite element method 367–369 first-order reliability method 531 first order kinetic law 305 first passage 540 probability, 541 flattened strain trajectory 117 foundations 109, 646 Fourier transform 119 fracture energy, 187, 238, 252, 290, 478 approach, 625 mechanism, 124 process, 369 fragmentation 162 free energy 238, 286, 297 freeze thaw attack 148, 293 frost attack, 194 damage, 293 deicing salt attack, 148 suction, 95 full scale experiments 32 Galerkin 377 continuous method, 369, 408, 412, 416

716

Subject Index

discontinuous method, 369, 408, 412, 416 solution, 594 temporal approximation, 419 time integration scheme, 372, 411, 416, 594, 598, 600 type finite element method, 410 Gˆ ateaux derivative 385 Gauss-Legendre integration, 421, 460 quadrature, 386, 458 generalized external force vector, 413 internal force tensor, 383 vector, 403, 413 mid-point approximations, 414, 415 tangent damping matrix, 413 damping tensor, 385 mass matrix, 413 stiffness matrix, 403, 413 stiffness tensor, 385 genetic algorithm 556 geogrids 650 global balance equation 373 gradient-based algorithm 561 grading curve 189 grain size distribution curve 205 grey-box modelling 635 Griffith criterion 493 gust response factor 11 equivalent quasi-static load, 12 fatigue analysis of gust-induced effects, 18 h-finite element method 379, 388 h-method 412 H¨ unxer Br¨ ucke 617, 648 hanging node concept 431 Hankel matrix 523, 638 hardening isotropic, 256 kinematic, 256 modulus, 246 harmonic excitation, 649 oscillation, 115 harmony with nature 6

Hasofer-Lind-algorithm 533 hazard 654 headed stud 207 heat capacity, 286 conduction, 36, 292 of hydration, 142 transmission, 39 Helmholtz energy 246 Hessian matrix 568 hierarchical concept, 388 polynomial interpolation, 426 high-cycle model for soils 517, 646 high-speed railway lines 79 aerodynamic loads, 79 critical train speed, 81 noise barrier, 80 prevailing frequency fp , 81 wind shelter walls, 80 higher order finite elements 380 Hilber-α method 411 historiotropy 203, 315 history integration, 583 variable, 434, 440, 478 hodograph 116 hoop stress 489 hourglass control 380 hydration products 302 hydraulic pressure 148 hygral behaviour, 143 conditions, 359 hygro-mechanical analysis, 583, 590 damage, 367 identification 637 implicit backward Euler scheme, 238 mode, 518 Importance Sampling 534 indicator function 533 inf-sub-condition 398 initial condition 412 in situ test stand 39 inspection 6 instability 367

Subject Index integration 377 scheme energy conserving, 411 scheme energy decaying, 411 integrity 289, 290 intensity of accumulation 161, 314 interaction 3 interactive effect 3 intermediate configuration 256 internal damage, 96 modes, 394 interpolation property 390 interval algebra 370 intrinsic permeability 307 inverse Fourier transform 120 investment cost 5 ion transport 295, 374 isomorphic variables 198 isoparametric approximation 390 isotropic damage parameter 286 iteration methods 403 Jacobi temporal transformer, 420 tensor, 383 transformation, 383 jump condition 418 kinematical relation 427 kinematic hardening law 255 kinking angle 272 Kirchhoff-Love formulation 397 Kohlrauschs law 297, 298 Krylov-based model reduction methods 500, 504 Kuhn-Tucker conditions 238, 300, 478 Lagrange 390 multipliers, 507, 511 non-hierarchical concept, 388 polynomials, 369, 380 shape function, 380, 381, 412 latent heat 286 Latin hypercube sampling 535, 536, 617, 627, 631 leaching 354 leakage 356

least squares problem 567 Legendre 380, 389, 390 modified polynomials, 389 polynomials, 369, 389 shape function, 389 type, 392 Leon model 244 life-cycle design, 7 investigation, 499 lifetime 207 analysis of bridge hangers, 34 lock-in effect, 34 stochastic excitation force, 34 assessment, 610 control, 659 lifetime-oriented analysis, 562, 583 design, 1, 371, 548, 562 concept, 365, 607 model, 365 limit state 367 serviceability, 367 ultimate, 367 linear convergence rate 388 linearized discrete balance equation, 385 weak form, 419 liquefaction 113, 162, 206 liquid saturation 289 load -deflection curve, 493 -independent influences, 92 cycles, 266, 269 pattern, 87 static and dynamic design calculations, 87 symmetric load pattern, 90 scenario, 7 loading biaxial, 256 cycles, 280 cyclic, 278 local balance equation 373 localization 634 lock-in 25 effect, 26 aerodynamic damping, 26 large oscillation amplitudes, 26

717

718

Subject Index

Scruton number, 26 range, 28 lift force, 28 locking-free element formulation locking phenomena 379 Lode angle 314 long-term calculation, 499 creep, 285 degradation mechanisms, 632 experiments, 39 loss of equilibrium 367 low-frequency vibration comfort low order approximation, 379 finite elements, 379

430

573

macro time 657 maintenance 6 Markov parameters 638, 640 material degradation, 124 failure, 374 model, 625 matrix dissolution 295 Matsuoka and Nakai condition 313 Maxwell symmetry 288 mean wind speeds 15 Gumbel-distribution, 16 logistic distribution, 16 occurrence probability of an extreme value, 20 rosettes of 10-minutes means of wind velocities, 18 terrain factor, 21 Weibull-distribution, 16 wind direction, 18 wind effect admittance, 22 wind rosette, 18 mesh dependency, 246 locking, 519 metallic materials 137 micro-ice-lens model 95, 148, 194 micro and macro time domain 33 microcrack 124, 133 formation, 194 micromechanical model 293

micromechanics 271 microplasticity 137 microprestress 291 microscopic alteration, 367 analysis, 621 microstructure 133, 150 micro time 657 Mindlin-Reissner formulation, 397 kinematics, 362 Miner’s rule 205 minimum energy 492 modal analysis, 527, 634 of wind-induced oscillations, 13 assurance criterion, 569 damping, 634 reduction, 503 mode-I-crack extension 490 model adaptivity, 426 reduction, 498, 499, 503 mode shapes 634 moisture permeability, 147 transport, 147, 285, 359, 586 uptake, 194 molar conductivity 298 monopile foundation 650 Monte Carlo method, 627 Simulation, 533, 534, 617, 654 distance-controlled, 542 strategy, 555 multi-axial stress state 269 multi-dimensional multi-field element 377 multi-field numerical method, 365 problem, 367, 369, 380 multi-level solution philosophy 367 multi-scale 658 modeling, 616, 657 multi-stage cyclic loading 136 multi-surface models 518 multiaxial loading 126 multiphysics problems 373, 379, 380 multisurface

Subject Index damage-plasticity model, 290 plasticity, 237

nanostructure 293 natural convection 36 natural resource 6 Neo-Hooke model 256 Nernst-Einstein relation 297 Neumann boundary condition 374 Newmark 377, 414 adaptive solution, 594 approximations, 414 generalized α method, 411 integration, 598 integration scheme, 594 method, 369, 411, 412, 514 time integration, 372, 594 time integration scheme, 600 type finite difference, 410 Newmark-α time intergration method 408, 411, 412, 416 Newton correction, 407, 416 quasi Newton method, 405 Newton-Cotes integration 458–460 Newton-Raphson iteration, 368, 422 method, 405, 424 modified method, 405 procedure, 377 scheme, 378, 408 non-destructive measurement, 194 test, 618 non-linear algebraic equation, 415 differential equations, 367 first order semidiscrete initial value problem, 416 vector equation, 403 non-local equivalent strain 359 normalization 117 normalized cross orthogonality 570 notch filter 121 nucleation law 256 numerical method, 368 modeling, 373

719

models, 39 numerically dissipative integration 410 NURBS 613 Nyquist frequency 119 optimization 656 method, 365, 551 problem, 546 result, 614 strategy, 372 optimum design 372 order of accuracy 369 orthogonality check 570 oscillation 115 P- and S-wave velocities 203 p-finite element method 379, 387, 388, 402, 588 p-method 412 p-refinement 391 Padé-Via-Lanczos algorithm 500 Palmgren-Miner’s hypothesis 136 parallel optimization software, 561 processing, 543 parallelization 542–544, 559, 615 Paris law 272, 276, 280 partial density, 302 differential equations, 367 partially saturated material 289, 302 partition of unity 449, 471 passage to a kinematic mechanism 367 PC cluster 615 perfomance concept 93 period T 116 permeability 292, 307, 590 Petrov-Galerkin continuous method, 252, 412, 422 phase shift 118 Pickett effect 291 Piola-Kirchhoff stress tensor 427, 428, 506 plastic -damage model, 237 deformation, 216 stretching, 519 plasticity theorie 237

720

Subject Index

plate finite elements 369 polarization 314 changes, 162 polycyclic loading 198, 646 population-based evolution strategy 560 pore pressure, 307 size distribution, 150 water pressure accumulation, 113, 205, 649 porosity 289, 292, 295 porous media 285 post-cyclic behaviour 163, 207 power spectral density function 577 practical design 365 predictor iteration, 405 step, 408 pressure vessel 259, 362 prestressing tendon 625, 626 primal method 553 principle of maximum dissipation 244 probabilistic lifetime analysis 611 probability 657 density function, 535 subjective, 370 process 543 prognosis of deterioration 368, 372 projection 117 proper orthogonal decomposition 504 pseudo -creep, 116 -relaxation, 116, 160, 519 pulse processes 34 push-out test 207 quality assurance 1, 659 quasi-brittle damage 131 Rackwitz-Fiessler-algorithm rainflow counting, 581 method, 575 random field, 631 variable, 609 Rankine criterion, 238, 290, 307

533

damage surface, 246 ratcheting effect 256 reaction-diffusion 297, 600 reaction kinetics 359 reactive aggregates, 302 porous media, 298 recycling 5 reduced integration technique 430 regularization 237, 307 relaxation effect 298 reliability 3, 4, 653, 656 analysis, 528, 529, 542, 543 time-variant, 614 analysis method, 365 problem, 520, 531, 541 time-invariant, 531 reliability-based optimum design 529 renewal processes 34 fatigue events, 34 macro time domain, 34 repair 4, 6 representative elementary volume 297 residual -based error criterion, 426 deformation, 183 strain, 116, 160, 198 resilient strain 116, 198 resistance 4 Response Surface Method 537, 538 rest period 191 restraint stresses 141 return mapping algorithm 252 revitalization 5 right stretch tensor 114 Ritz load-dependent Ritz vectors, 500, 506 road bridge 617 robust structural design 607 Roscoe’s invariants 198 rotating crack 244 rotation operator 315 Russian Roulette & Splitting 542 S-N curve safety 1 salt attack saturation

626 194 588, 589

Subject Index scaling 293 Schur complement 511 second-order reliability method 531 second order accuracy 409–411 seepage velocity 307 selective reduced integration 380 semidiscrete balance equation, 412 differential equation, 369 structural equation, 367 sequential convex programming, 553 linear programming, 553 quadratic programming, 533, 553 serendipity approximation 402 service -life control, 4, 7 life, 1 performance, 366 serviceability 1 settlement 109, 646 shakedown 112 shaker test 574 shape function 3-D, 392 anisotropic, 369, 381, 395, 397 hierarchical, 369 isotropic, 381 shaping filter 520 shell finite elements, 369 structure, 583 short term thermal impacts 40 shrinkage 143, 285, 289, 351 side frame 572 Simpson integration 460 single-field numerical method, 365 problem, 367 singular value decomposition 500, 526 smeared crack approach, 269 concept, 292 model, 265 softened water 152 softening modulus 246 soil 109, 114, 160, 198, 517, 646 -structure interaction, 109

721

solid discretization shell-like, 400 slab-like, 400 truss-like, 400 solid formulation shell-like, 397 slab-like, 397 truss-like, 397 sorption behaviour 143 spatial discretization, 379 method, 377, 379 finite element discretization, 377, 380 scatter, 631 special eigenvalue problem 527 spectral analysis, 119 moment, 577, 578 spherical cavaties 129 state space matrix, 636 model, 520, 634 system, 636 variable, 413 stationary problem 372, 403 steel bars, 625 pipe, 641 structure, 607 stiffness 216 degradation, 134, 139 reduction, 137 stochastic sampling method, 617 void ratio, 647 strain energy function, 428 release rate, 489 strategy for polycyclic loading of soil 517 strengthening 6 stress Cauchy, 256 fluctuation, 316 intensity factor, 137, 272 Kirchhoff, 256 von Mises, 581 stress-strain

722

Subject Index

curve, 124 relation, 265 strong discontinuity approach 252, 448 structural analysis, 371 degradation, 180, 359 design, 365, 371 failure, 356 optimization, 529, 545, 546, 548, 555, 612 problem, 372, 545 optimization problem implicit nature, 548 semi-infinite, 551 subset method, 536 simulation, 536, 537 subspace identification, 522, 524, 525 methods, 520 substructure technique 502 sulfate 107 attack, 157 surface energy 143 sustainability 5, 6 swelling 302 synthesis (composition) approach 371 system identification 519 T-stress 271 tangential stiffness matrix 632 Taylor expansion, 427 series, 378 Taylor-Hood 400 approximation, 398 element, 585 temperature profile 141 temporal discretization, 380 method, 408 discretization method, 377 weak form, 419 weak formulation, 412 test function 377 thaumasite formation 157 thawing rate 101 theory of mixtures 294

theory of porous media 293–295, 302, 307 theory of probability 370 thermal actions, 35 design process, 35 lifetime analysis, 35 behaviour, 140 conductivity, 292 cracking, 142 expansion, 140 coefficient, 140 incompatibility, 140 loading, 141 loads on structures, 42 low-cycle fatigue, 42 properties, 140 radiation, 36, 37 transmission, 36 thermic dilatation coefficient 286 thickness refinement 427 tied-arch bridge 642 time derivatives, 377 discretisation, 499 history calculation, 574 method, 520 integration, 514, 518, 595 error, 443 method, 368 parameter, 414 scheme, 367, 369 variant formulation, 368 time-invariant behavior, 365 problem, 531 time-variant fatigue processes 33 rainflow cycle counting, 34 Timoshenko formulation 397 traction vector 477, 478 traffic load 617 fatigue load models, 62 models, 46 on road bridges, 46 fatigue load model 3, 69 load model 1, 58 load model 2, 60 transformation method 552, 553

Subject Index transient vibration time 640 transport-damage process 588 triaxial stresses 126 tuned-mass damper 573, 574 tunnel lining 351 ultimate tension stress 478 ultrasonic transmission time 180 uncertainty 369, 370, 539, 627 categories, 370 informal, 370 lexical, 370 stochastic, 370 uniaxial compression, 124 cyclic load, 133 elasto-plastic material law, 625 stress state, 269 tension, 125 unit-cell analyses 255 upgrading 4 variance-reducing simulation method 534 verification 634 vertical tie rod 607 vibration comfort, 572 measurement, 645 vibratory compaction 648 viscosity 265 visual degree of (external)damage 103 void volume fraction 256 Voigt notation 428 volume fraction 295 vortex -induced across-wind vibration, 609 excitation, 25

723

shedding, 25 wind turbulence, 25 warranty 7 weak form, 372 formulation, 369, 377 welding 610 white-box modelling 635 wind buffeting, 10 equivalent roughness length, 20 gustiness, 10 gust loading, 14 integral length scales, 12 turbulence intensity, 12 load, 27, 607 ESDU model, 29 Ruscheweyh-model, 27 Vickery-model, 28 power plant, 650–652 offshore, 650 tunnel experiment, 32 windowing 120 W¨ ohler approach, 626 limit stress, 278 test, 182 working-life 4 XFEM 473, 474 displacement field, 452 p-version, 469 yield condition Gurson, 256 von Mises, 256 yield strength 137 Zienkiewicz error indicator

444, 445

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Programming Language Design Concepts

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Earthquake engineering for structural design

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Earthquake Engineering for Structural Design

Earthquake Engineering for Structural Design

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Advanced Design Concepts for Engineers

Design Concepts in Programming Languages

Distributed Systems: Concepts and Design

Structural Steel Design: A Practice Oriented Approach

Computer-Aided Materials Selection During Structural Design

Computer-Aided Materials Selection During Structural Design

Limit States Design of Structural Steelwork

Structural Steel Design: A Practice Oriented Approach

Thermostable Proteins: Structural Stability and Design

Lifetime-Oriented Structural Design Concepts

Masonry Structural Design

Design of Structural Elements

Structural Timber Design

Reliability-based Structural Design

Structural Design for Architecture

Marine Structural Design

Principles of Structural Design

Structural Timber Design

SOI Circuit Design Concepts

Programming Language Design Concepts

SOI Circuit Design Concepts

Programming Language Design Concepts

Innovative Shear Design (Structural Engineering)

Earthquake engineering for structural design

Structural Stability of Steel: Concepts and Applications for Structural Engineers

Structural Concrete: Theory and Design

Introduction to Structural Aluminium Design

Earthquake Engineering for Structural Design

Earthquake Engineering for Structural Design

Structural Design Optimization Considering Uncertainties

Structural Equation Modeling: Concepts, Issues, and Applications

Advanced Design Concepts for Engineers

Design Concepts in Programming Languages

Distributed Systems: Concepts and Design

Structural Steel Design: A Practice Oriented Approach

Computer-Aided Materials Selection During Structural Design

Computer-Aided Materials Selection During Structural Design

Limit States Design of Structural Steelwork

Structural Steel Design: A Practice Oriented Approach

Thermostable Proteins: Structural Stability and Design

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