Stability of rock structures: proceedings of the 5th International Conference on Analysis of Discontinuous Deformation, ICADD-5, Ben-Gurion University of the Negev, Beer Sheva, Israel, 6-10 October 2002

STABILITY OF ROCK STRUCTURES Copyright © 2002 Swets & Zeitlinger B.V., Lisse, The Netherlands PROCEEDINGS OF THE FIF...

Author: Yossef H. Hatzor

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STABILITY OF ROCK STRUCTURES

Copyright © 2002 Swets & Zeitlinger B.V., Lisse, The Netherlands

PROCEEDINGS OF THE FIFTH INTERNATIONAL CONFERENCE ON ANALYSIS OF DISCONTINUOUS DEFORMATION, ICADD-5/BEN-GURION UNIVERSITY OF THE NEGEV/BEER SHEVA/ISRAEL/6–10 OCTOBER 2002

Stability of Rock Structures

Edited by Yossef H. Hatzor Ben-Gurion University of the Negev, Beer Sheva, Israel

A.A. BALKEMA PUBLISHERS

LISSE/ABINGDON/EXTON (PA)/TOKYO


Cover: Upper terrace of King Herod’s Palace, Masada/photo by Yael Ilan Curtesy of Israel Nature and Parks Authority Copyright © 2002 Swets & Zeitlinger B.V., Lisse, The Netherlands All rights reserved. No part of this publication or the information contained herein may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, by photocopying, recording or otherwise, without written prior permission from the publisher. Although all care is taken to ensure the integrity and quality of this publication and the information herein, no responsibility is assumed by the publishers nor the author for any damage to property or persons as a result of operation or use of this publication and/or the information contained herein. Published by: A.A. Balkema, a member of Swets & Zeitlinger Publishers www.balkema.nl and www.szp.swets.nl

ISBN 90 5809 519 3

Printed in The Netherlands


Stability of Rock Structures, Hatzor (ed) © 2002 Swets & Zeitlinger, Lisse, ISBN 90 5809 519 3

Table of contents

Preface

VII

Organisation

IX

Sponsorship

XI

Keynote lecture Single and multiple block limit equilibrium of key block method and discontinuous deformation analysis Gen-Hua Shi

3

Plenary lectures Realistic dynamic analysis of jointed rock slopes using DDA Y.H. Hatzor, A.A. Arzi & M. Tsesarsky Numerical models for coupled thermo-hydro-mechanical processes in fractured rocks-continuum and discrete approaches L. Jing, K.-B. Min & O. Stephansson

47

57

Grand challenge of discontinuous deformation analysis A. Munjiza & J.P. Latham

69

High-order manifold method with simplex integration M. Lu

75

Case studies in rock slope and underground openings in discontinuous rock Experimental investigations into floor bearing strength of jointed and layered rock mass D. Kumar & S.K. Das

87

Stability analysis for rock blocks in Three Gorges Project W. Aiqing & H. Zhengjia

95

Some approaches on the prediction of hillsides stability in karstic massif E. Rocamora Alvarez

101

Analysis of displacement and stress around a tunnel S. Chen, Y.-N. Oh, D.-S. Jeng & L.-K. Chien

107

Analysis, response, prediction and monitoring of existing rock and stone monuments A parametric study using discontinuous deformation analysis to model wave-induced seabed response Y.-N. Oh, D.-S. Jeng, S. Chen & L.-K. Chien

V


113

Simulations of underground structures subjected to dynamic loading using the distinct element method J.P. Morris, L.A. Glenn, F.E. Heuzé & S.C. Blair

121

Numerical analysis of Gjøvik olympic cavern: a comparison of continuous and discontinuous results by using Phase2 and DDA T. Scheldt, M. Lu & A. Myrvang

125

Earthquake site response on hard rock – empirical study Y. Zaslavsky, A. Shapira & A.A. Arzi

133

Numerical simulation of shear sliding effects at the connecting interface of two megalithic column drums N.L. Ninis, A.K. Kakaliagos, H. Mouzakis & P. Carydis

145

On determining appropriate parameters of mechanical strength for numeric simulation of building stones N.L. Ninis & S.K. Kourkoulis

153

Validation of theoretical models Experimental validation of combined FEM/DEM simulation of R.C. beams under impact induced failure T. Bangash & A. Munjiza

165

A study of wedge stability using physical models, block theory and three-dimensional discontinuous deformation analysis M.R. Yeung, N. Sun & Q.H. Jiang

171

Shaking table tests of coarse granular materials with discontinuous analysis T. Ishikawa, E. Sekine & Y. Ohnishi

181

Pre-failure damage, time-dependent creep and strength variations of a brittle granite O. Katz & Z. Reches

189

Dynamic block displacement prediction – validation of DDA using analytical solutions and shaking table experiments M. Tsesarsky, Y.H. Hatzor & N. Sitar

195

Theoretical developments in modelling discontinuous deformation Crack propagation modelling by numerical manifold method S. Wang & M. Lu

207

Continuum models with microstructure for discontinuous rock mass J. Sulem, V. de Gennaro & M. Cerrolaza

215

Development of a three-dimensional discontinuous deformation analysis technique and its application to toppling failure H.I. Jang & C.I. Lee Three-dimensional discontinuity network analysis (TDNA) on rock mass X.-C. Peng & H.-B. Tang

VI


225 231


Preface

Rock is typically discontinuous due to the presence of joints, faults, shears, bedding planes, and foliation planes. Stability analysis of structures in rocks must therefore address deformation along block boundaries and interaction between rock blocks, which may be impossible to address properly by imposing a continuum framework model. Over the past decade we have seen significant progress in development of methods of analysis for discontinuous media. Of note in particular are the DEM (distinct element method), DDA (discontinuous deformation analysis), block theory, combined FEM/DEM, and the new manifold method. Four international meetings on analysis of discontinuous deformation have been held, in Taipei in 1995, Kyoto in 1997, Vail in 1999, and Glasgow in 2001. The objective of this ICADD-5 is to focus on the application potential of discontinuous analysis methods to the stability evaluation of structures in rock, including both modern engineered rock slopes and underground openings, as well as ancient monuments in fractured rock. Nevertheless, this proceedings volume also contains original, high quality theoretical papers which explore issues such as fracture mechanics modeling in the new manifold method, continuum models with microstructure for discontinuous rock mass, coupled thermo-hydromechanical processes in fractured rocks, and recent developments in three dimensional DDA. It is believed that the collection of papers in this volume demonstrates the directions in which theoretical developments in analysis of discontinuous deformation should proceed, the validity and limitations of existing codes, and the range of engineering problems to which discontinuous analysis can be applied. Yossef H. Hatzor Chair, ICADD-5 Organizing Committee President, Israel Rock Mechanics Association

VII



Organisation

INTERNATIONAL ADVISORY PANEL Bernard Amadei (USA) Nenad Bicanic (UK) Yossef H. Hatzor (ISRAEL) Ante Munjiza (UK) Yuzo Ohnishi (JAPAN) Friedrich Scheele (SOUTH AFRICA) Gen-Hua Shi (USA) Nicholas Sitar (USA) Chung-Yue Wang (TAIWAN, ROC) Aiqing Wu (CHINA) Man-chu Ronald Yeung (HONG KONG, CHINA)

ORGANISING COMMITTEE Yossef H. Hatzor Chairman John P. Tinucci Website and peer-review site manager Michael Tsesarsky Secretary Avner A. Arzi, Chaim Benjamini, Tsvika Tzuk Members

IX



Sponsorship

The conference is sponspored by The President of Ben-Gurion University The Rector of Ben-Gurion University The Kreitman Foundation Fellowships – Ben-Gurion University The Faculty of Natural Sciences – Ben-Gurion University

XI


Keynote lecture



Single and multiple block limit equilibrium of key block method and discontinuous deformation analysis Gen-hua Shi Belmont, CA, USA

ABSTRACT: Limit equilibrium is still the basic method to do the stability analysis including slope stability, tunnel stability and dam foundation stability. Key blocks are the blocks very possible to reach limit equilibrium first. Newmark key block method is a dynamic limit equilibrium method. The latest version of delimiting key blocks on the unrolled tunnel joint trace map is also used. Two and three dimensional dynamic DDA is used to compute multiple block multiple step dynamic limit equilibrium. The recorded seismic loads are input. Block system statics with large displacements is the results of stabilized dynamic DDA method. Statics is infinite long time and stabilized dynamics therefore is much more difficult. In the following, key block limit equilibrium and limit equilibrium related to the multiple block and time depending multiple step DDA computations are studied: one step static computation, non-stabilized dynamics computation, and stabilized dynamics computation.

1

LIMIT EQUILIBRIUM ANALYSIS OF BLOCKS

For small displacement and continuous computation, the normal computation solve the equation only once. It is one time step computation. Its assumption is the single time step is very large. Due to the long time and the small displacements in this single step, the velocity and the acceleration are nearly null. In this case, the inertia is neglected. Most traditional limit equilibrium, key block and structure mechanics computations belong to this case. However this kind of simple statics computation cannot compute the large displacements, large deformation and discontinuous cases. It even has substantial difficulties for the majority of material non-linear computation. For the large displacement, large deformation or discontinuous cases, both statics and dynamics use time steps. Statics is infinite long time and stabilized dynamics. Therefore the general statics is much more difficult than dynamics. Based upon the two or three dimensional block kinematics, the loading conditions and the friction law, limit equilibrium analyses were performed. The following results are required at the end of each time step from the analyses: 1. 2. 3. 4. 5. 6. 7. 8.

Normal forces perpendicular to the sliding interface Resisting force from the shear strength at sliding interface Sliding forces along with the sliding interface Dynamic or static equilibrium Satisfying friction law at all interfaces Inertia forces for multiple step dynamic computation Convergence of contact forces for multiple step static computation Modes of failure including sliding surfaces and sliding direction

3


2

LIMIT EQUILIBRIUM OF SLOPE KEY BLOCKS

In the following case, the slope has double free planes. Four joint sets and two slope planes are listed in Table 2.1. Figure 2.1 and 2.2 are the upper hemisphere stereographic projection of dominant joint sets A, B, C and D. The two dashed circle are the side slope and top slope. From Figure 2.1 and 2.2, the region entirely included in the union of two dashed circles (projections of two slope planes) are regions 0000, 0010, 0110 and 0100. Also, regions 0020, 0200, 0120 and 0220 are in the same union of two dashed circles. In the field, only one key block with JP code 0220 has been found. Each joint and slope plane is located by one point: • • • • •

the key blocks of JP code 0220 is on the upper side of joint set A on the upper side of joint set D on the lower side of slope side plane on the lower side of slope top plane

In order to compute the block shape and block volume, one point is chosen in each face of this specific block. There are four faces in this block. In the following table, the intersection point of first three faces can be chosen as the coordinate system original (0,0,0). A point (−2.30, 6.77, 14.00) is chosen in the fourth face (Table 2.2). Table 2.1.

Figure 2.1.

Input data.

Joint set

Dip angle

Dip direction

Friction angle

Joint set A Joint set B Joint set C Joint set D Side slope Top slope

73◦ 58◦ 70◦ 32◦ 67◦ 10◦

108◦ 20◦ 219◦ 225◦ 195◦ 195◦

33◦ 25◦ 25◦ 33◦

Key blocks and sliding forces.

4


Figure 2.3 shows the block shape. From Figure 2.1, 2.2 and 2.3 the following results can be read from Table 2.3. This block is stable under gravity load. This slope is stable under gravity load. Figure 2.4 is the upper hemisphere equal angle projection of the joint sets and slope planes. The following section 3 and 4 are for earth quake stability analysis of this block 0220. 3

STEREOGRAPHIC PROJECTION SOLUTIONS OF LIMIT EQUILIBRIUM WITH EARTH QUAKE LOADS

P. Londe stereographic projection solution is well known for dam foundation and dam abutment stability analysis. This method is suitable for earth quake loads. As the limitation of this method, cohesion c = 0 is assumed. The input data are given in Tables 3.1 and 3.2.

Figure 2.2.

Sliding planes and factor of safety. Table 2.2.

X Y Z of a point on each face of the block.

Coordinates

X (feet)

Y (feet)

Z (feet)

Joint set A Joint set D Slope Top slope

0 0 0 −2.30

0 0 0 6.77

0 0 0 14.00

Table 2.3.

Block stability.

Block JP code Block volume Sliding force per unit weight Factor of safety (under gravity only)

0220 2742 cubic feet −0.28 1.61

5


Figure 2.3.

Key block JP = 0220.

Figure 2.4.

Equal angle projection of joint sets and slopes. Table 3.1.

Joint data.

Joint set

Dip angle

Dip direction

Joint set A Joint set D

73◦ 32◦

108◦ 225◦

Table 3.2.

Stereographic projection.

Contour step 5◦ Stereographic projection lower hemisphere Resultants weight and earth quake accelerations

P. Londe pictures are contours of friction angles. The contour value of the projection point of a resultant force is the required friction angle for limit equilibrium. This friction angle corresponds factor of safety 1.0. Figure 3.1 shows the required friction angles of different time intervals from the whole earth quake time history loads.

6


Figure 3.1.

Friction angle contours of key block limit equilibrium.

Results Figure

Time interval

Stable friction angle

Figure 3.1 Figure 3.1 Figure 3.1 Figure 3.1 Figure 3.1 Figure 3.1

0–50 seconds 0–10 seconds 10–20 seconds 20–30 seconds 30–40 seconds 40–50 seconds

70◦ 53◦ 70◦ 52◦ 43◦ 23◦

The Figures show 70 degrees of friction angle makes the block stable under the given time depending earth quake loads. The solid circle in the center of the picture represents the loads corresponding all possible 0.6 g earth quake loads. It can be found from the Figures, 55 degrees of friction angle is stable for 0.6 g earth quake. 4

NEWMARK DISPLACEMENT SOLUTION OF LIMIT EQUILIBRIUM WITH EARTH QUAKE LOADS

Since the block 0220 has two sliding joint faces of dominant joint sets A and D respectively, double face sliding Newmark displacement algorithm is needed. A C-language code including PostScript graphics is written for

7


this special purpose. Due to the variant earth quake load, the changing of sliding planes and sliding modes is considered. The results and the formulations are consistent with the previous P. Londe graphic solutions. The input data of Newmark approach is simply the joint information, the block geometry and the earth quake acceleration data are given in Table 4.1 and 4.2. The earth quake data are given by Figure 4.1 to 4.2. The curves of Figure 4.3 gives the accumulated displacements and the instant velocity during the 50 second earth quake process. This computation is single block three dimensional dynamic limit equilibrium analysis without rotations. The final displacement is 2.2 feet under 33 degrees friction angle and 0 cohesion. The all-time maximum sliding velocity is 2.10 feet per second. The cohesion is sensitive relative to the results. If cohesion 4.0 psi or 0.576 kips/ft2 is applied to the joint set D, using 25◦ friction angle for both sliding joint sets A and D, the cohesion of joint set A remain to zero, there are no permanent displacement in the entire 50 second earth quake process.

Table 4.1.

Joint information.

Joint set

Dip angle

Dip direction

Friction angle

Joint set A Joint set D

73◦ 32◦

108◦ 225◦

33◦ 33◦

Table 4.2.

Block geometry.

Block volume Sliding force per unit weight Rock unit weight Area of side slope face Maximum distance to side slope Sliding force on side slope Area of top slope face Maximum distance to top slope Sliding force on side slope

2742 cubic feet 0.5528 0.15 kips per cubic feet 293 square feet 28.1 feet 0.7754 kips per square feet 645 square feet 12.8 feet 0.3474 kips per square feet

Table 4.3. Time depending accelerations.

Figure

Time interval seconds

Direction g

Direction g

Figure 4.1 Figure 4.1 Figure 4.2 Figure 4.2

0–25 25–50 0–25 25–50

X X Z Z

Y Y resultant resultant

Table 4.4.

Results.

Figure

Time interval seconds

Displacement Maximum velocity feet feet/second

Figure 4.3 Figure 4.3 Figure 4.3 Figure 4.3 Figure 4.3

0–10 10–20 20–30 30–40 40–50

0.5444 1.7895 2.2131 2.2369 2.2369

8


2.10 1.15 1.75 0.22 0.00

Figure 4.1.

X and Y components of earth quake forces.

9


Figure 4.2.

Z component and resultant of earth quake forces.

10


Figure 4.3.

Displacements and velocity of the key block.

11


5

LIMIT EQUILIBRIUM OF TUNNEL KEY BLOCKS

The joint sets are given in Table 5.1. Figure 5.1 is the upper hemisphere equal angle stereographic projection of the joint sets. Table 5.1.

Figure 5.1.

Joint set data of joint system.

Joint set

Dip angle

Dip d.

Friction angle

Cohesion

Joint set 1 Joint set 2 Joint set 3

79◦ 81◦ 5◦

270◦ 230◦ 45◦

39◦ 39◦ 39◦

0 ton/m2 0 ton/m2 0 ton/m2

Equal angle stereographic projection of joint sets.

Figure 5.2. Total key block sliding force for all tunnel directions.

12


The equal angle stereographic projection has the following three advantages: 1. The projection of any plane is a arc or a segment of a circle. 2. The intersection angle between two projection arcs is the true intersection angle between the two joint set planes. 3. This simple diagram shows all of the angular relationship between joint sets. The volume and sliding forces of key blocks are sensitive to the tunnel directions. The following is the study: If the given tunnel direction is safe in terms of the key block sliding force. This tunnel direction study only based on the tunnel directions and the joint set directions. The assumptions are: 1. The joints are very long. 2. The joint spacing are very small. 3. The key blocks in between two parallel joints are not considered. Figure 5.2 shows the contours of the sliding forces of maximum key blocks for all tunnel directions. The contours is equal area projection of the tunnel axis inside of the reference circle. In this case, the tunnel direction N 75◦ E is marked as a small circle in the drawing. The maximum sliding force of this direction is about 20% to 30% of the over all largest sliding force for all tunnel directions. Due to the two vertical joint sets, the tunnels with nearly 90 degrees rise angle or shafts have relatively large key block sliding force. 6

KEY BLOCK ZONES REACHES LIMIT EQUILIBRIUM FIRST

For a given tunnel direction, Figure 6.1 shows the zones of maximum key blocks for each joint pyramid (JP). The maximum key block zones are the projections of the maximum three dimensional key block on the tunnel

Figure 6.1.

Key block zones with JP codes and sliding forces.

13


Figure 6.2.

Key block zones with sliding planes and factor of safety.

section plane. Also all the key blocks of the same joint pyramid (JP) are in the corresponding key block zone. The numbers under the JP codes are the sliding forces with unit g. The zone marked “011” means JP = 011, where “0” means upper side of the joint set and “1” means lower side of the joint set. All key blocks of JP = 011 are in the upper side of joint set 1, lower side of joint set 2 and lower side of joint set 3. Under the number “011” is “+0.83”. The sliding force of the JP = 011 key blocks is 0.83 times the key block weight. Figure 6.2 shows the sliding joint sets of the maximum key blocks. The zone marked “1” means all of the key blocks of this JP slide along joint set 1. The zone marked “13” means all of the key blocks of this JP slide along the intersection line of joint set 1 and joint set 3. The second number under the sliding joint set number is the factor of safety of all the key blocks of the corresponding JP. The factor of safety of JP = 011 key blocks is 0.16. If the factor of safety is greater than 9.99, 9.99 is printed. All of the computations of sliding forces and factor of safety are based on limit equilibrium. The key blocks are these blocks which very likely reach limit equilibrium first. As the joints have limited lengths and wide spacing, the key blocks can occupy only a part of the maximum key block zone. Most of the real key blocks will lie near the tunnel surface. Therefore, relatively smaller key block region can be considered in the stability analysis. From Figure 6.1 to 6.2, only key blocks of JP = 011 and JP = 101 can fall. The blocks of JP = 100 and JP = 010 are only removable. 7 THREE DIMENSIONAL VIEW OF MAXIMUM KEY BLOCKS Figure 7.1 shows the three dimensional view of the maximum key block JP = 011. Here the key block can be in between two parallel joints of joint set 1 or JP = 311 (Table 7.1). Figure 7.2 shows the three dimensional view of the maximum key block JP = 011. Here the key block can be in between two parallel joints of joint set 2 or JP = 031 (Table 7.2).

14


Figure 7.1.

Maximum key block JP = 311.

15


Figure 7.3 shows the three dimensional view of the maximum key block JP = 101. Here the key block can be in between two parallel joints of joint set 1 or JP = 301 (Table 7.3). Figure 7.4 shows the three dimensional view of the maximum key block JP = 101. Here the key block can be in between two parallel joints of joint set 2 or JP = 131 (Table 7.4). For the maximum three dimensional key blocks, the assumptions are: the joint length in each joint set is sufficiently large and the joint spacing in each joint set is sufficiently small. Under these extreme assumptions, the maximum key blocks are drawn. Any actual key block can not be larger than these maximum key blocks. The actual key blocks could be much smaller due to the limited length and substantial spacing.

8

KEY BLOCK AREAS ON TUNNEL SURFACE JOINT TRACE MAP

The joint sets are given by Table 5.1. Based upon the joint statistics, the joint geometric parameters are given by the following: From the joint direction, joint spacing and joint length of Table 5.1 and Table 8.1, the joint traces on the curved tunnel surface are produced statistically. Here for a practical reason, the joint bridges are all assumed to be 0.1 m. It has been assumed here that the joints having traces in the tunnel surface extend sufficiently far behind the tunnel surface as to form blocks by their mutual intersections. It has been proved that if the joints do thus extend behind the tunnel surface, the three dimensional key blocks of the tunnel can be delimited by operating only with the joint traces exposed on the tunnel surface. Then using key block theory, the key block zones are delimited from the curved polygons of the unrolled joint trace map. Figure 8.1 is the diagram which shows the way the tunnel, including the joint sets, is unrolled. Figure 8.2 is the statistically produced unrolled joint trace map of the whole tunnel. Figure 8.3 is the key blocks on the unrolled joint trace map of the whole tunnel.

Table 7.1.


Key block JP code Key block volume Area in tunnel surface

Table 7.2.

011 or 311 0.77 m3 2.16 m2



Table 7.3.

011 or 031 0.64 m3 1.84 m2



Table 7.4.

101 or 301 0.39 m3 1.47 m2



101 or 131 0.47 m3 1.68 m2

16


Figure 7.2.


17


Figure 7.3.


18


Figure 7.4.


19


Table 8.1.

Statistical joint set data.

Joint set

Spacing (m)

Length (m)

Bridge (m)


0.30 0.30 0.50

1.8 2.4 1.8

0.30 0.30 0.50

Table 8.2.

Key block zones.

JP code

Key block area (m2 )

Sliding joint set

311 031 301 131

9.81 7.70 11.19 5.88

1 1 2 2

Figure 8.1. Tunnel wall unroll.

Figure 8.4 is the three dimensional far side view of statistically produced joint traces. Figure 8.5 is the three dimensional far side view of key block. Figure 8.6 is the three dimensional near side view of statistically produced joint traces. Figure 8.7 is the three dimensional near side view of key block.

20


Figure 8.2.

Joint trace unroll map.

Figure 8.3.

Key blocks on unroll map.

21


Figure 8.4.

Far side view of joint traces.

Figure 8.5.

Far side view of key blocks.

22


Figure 8.6.

Near side view of joint traces.

Figure 8.7.

Near side view of key blocks.

23


9

DYNAMIC LIMIT EQUILIBRIUM OF DDA

The discontinuous deformation analysis (DDA) computes deformable block systems. In the current version, the block displacements are complete linear functions of the coordinates. Each block is assumed to have constant stresses and strains. In spite of the complex shape of DDA blocks, DDA method uses analytic integrations for all of its matrices. This analytic integration is simplex integration. The simplex integration can compute ordinary integrations without subdividing domains to simple elements. Using simplex integration, the integration of any polynomials can be represented by the coordinates of boundary vertices of generally shaped two or three dimensional blocks. DDAcomputation offers the movements, stresses and strains of each block. The computed block displacements are often large enough to be visible, the modes of failure and the final damage can be seen directly. On the other side, the DDA codes can perform traditional limit equilibrium analysis for whole block systems. When large deformation are involved, the static solution is the stabilized state from the dynamic solution due to friction or real damping. The current DDA program treats the damping in a simple manner: the dynamic computation inherits the full velocity at the end of the previous time step. The static computation inherits only a part of the velocity at the end of the previous time step as the initial velocity at the beginning of this time step. The DDA computation must satisfy the following conditions at the end of each time step: 1. Each degree of freedom of each block has an equilibrium equation. The simultaneous equilibrium equations are derived by minimizing the total potential energy at the end of each time step. All external forces acting on each block, including loads and contact forces with other blocks, reach equilibrium in all directions and reach moment equilibrium for all rotations. Equilibrium is also achieved between block stresses and external forces on the block. 2. Entrance theory is used to identify all possible first entrance positions. Contacts occur only on the first entrance position, interpenetrations are prevented on the first entrance positions and sliding is controlled by the friction law. 3. Within each time step, if the tensile force from the normal contact spring exceeds the limit, this normal spring will be removed. If interpenetration occurs in a entrance position, a normal spring is applied. The global equations have to be solved repeatedly while selecting the closed entrance positions. This procedure for adding or removing springs and solving equilibrium equations is referred to as an open-close iteration. The open-close iteration will continue until all tensile force and all interpenetrations are within set limits over all the entrances.

10 THE GEOMETRY AND MECHANICAL DATA OF DYNAMIC DDA COMPUTATION The joint sets are given by Table 5.1. Based upon the statistics, the joint geometric parameters are given by Table 8.1. Based on the geometric data of Table 5.1 and Table 8.1, DDA-DL program produces the joints and tunnel boundary lines. From the joint and tunnel boundary lines, DDA-DC program produces the block system. The block system is the geometric input of DDA-DF program. The mechanical parameters of both rock masses and joints are the following: Based on the mechanical data of Table 10.1, the program DDA-DF computes the time depending block movements and block stresses. The process of block falling can be shown. Table 10.1.

Mechanical data. 2.27 ton /m3 3000000 ton /m3 0.21 20000 0.0010 second 20 second 39◦ 0 ton /m3

Unit weight E of rock mass ν of rock mass Number of time steps Time step Earth quake duration Joint friction angle Cohesion

24


For this computation, the earth quake acceleration data are from California Department of Transportation. The original data are 50 seconds, our computation only uses from 10 second to 30 second. However these 20 second data are the main part of the strong earth quake. In DDA computation, as an extension of Newmark method from one block to multiple blocks, the earth quake accelerations are applied as body forces. Figure 4.1 show X and Y components of the time depending earth quake acceleration data. Figure 4.2 also show Z components and the resultants of the time depending earth quake acceleration data.

11

CASES OF ROCK FALLING DDA COMPUTATION WITH EARTH QUAKE LOAD

Figure 11.1 and 11.5 show the joint maps statistically produced on the tunnel section plane based upon the joint direction, joint length, joint spacing and joint bridge on Table 5.1 and Table 8.1. Figure 11.2 and 11.6 show the block meshes on the tunnel section plane produced from statistically produced joint maps Figure 11.1 and 11.5 respectively. Figure 11.3, 11.4, 11.7 and 11.8 show the rock falling of the meshes Figure 11.2 and 11.6. In the computation, the earth quake load is applied.

12

DYNAMIC THREE DIMENSIONAL DISCONTINUOUS DEFORMATION ANALYSES

The joint polygons and block systems are three-dimensional. Most of joints are not perpendicular to a given two dimensional cross section. Therefore, the two-dimensional computations of jointed rocks or block systems are of limited reliability and accuracy. The three-dimensional analyses of block systems are important. Threedimensional discontinuous deformation analysis (3-D DDA) forms blocks directly from general polygons. The blocks can be convex or concave. Also, the blocks can have any numbers of polygonal faces. The 3-D DDA program computes three-dimensional deformable block systems. In the current version, there are 12 degrees of freedom per block: displacements on X, Y, Z directions, rotations around axis X, Y, Z and six 3-D strains. The block displacements are complete linear functions of the coordinates. Each block is assumed to have constant stresses and strains. The discontinuous contacts between 3-D blocks are the main part of 3-D DDA algorithms. There are more ways for blocks to contact in three-dimensions compared with two-dimensional block contacts. For the friction law, the two-dimensional sliding directions form a line, while the three-dimensional sliding directions form a plane. The example of Figure 12.1 is a free falling block. g = 9.8 m/s2 , step time = 0.009035 second, total 50 steps. Theoretical falling distance is: 1 s = gt 2 = 1.0000 m. 2 Result of 3-D DDA is: s = 1.0000 m Figure 12.2 A block slides on one plane Friction angle of the sliding plane is 0. The sliding plane has 45◦ friction angle. g = 9.8 m/s2 Step time = 0.0080812 second, total 50 steps t = 100 × 0.0080812 second = 0.80812 second Vertical sliding distance is 1 sv = gt 2 = 1.6000 m 4 Result of 3-D DDA is: s = 1.5998 m

25


Figure 11.1.

Statistically produced joints of case 1.

Figure 11.2.

Block mesh formed by joints of case 1.

26


Figure 11.3a.

Case 1 rock falling after 0 steps.

Figure 11.3b.


27


Figure 11.4a.


Figure 11.4b.


28


Figure 11.5.

Statistically produced joints of case 2.

Figure 11.6.

Block mesh formed by joints of case 2.

29


Figure 11.7a.


Figure 11.7b.


30


Figure 11.8a.


Figure 11.8b.


31


Figure 12.1a.

Block free falling after 0 steps.

Figure 12.1b.

Block free falling after 50 steps.

Figure 12.3 A block slides on two planes The force along the sliding line is 1 mg cos (45◦ ) = √ mg = ma, 2 1 a = √ g, 2 where a is the acceleration along the sliding intersection line.

32


Figure 12.2a. A block slides on one plane after 0 steps.

Figure 12.2b. A block slides on one plane after 200 steps.

33


Figure 12.3a. A block slides on two planes after 0 steps.

Figure 12.3b. A block slides on two planes after 55 steps.

The sliding distance along the intersection line is: 1 1 s = at 2 = √ gt 2 . 2 2 2 Theoretical sliding distances sh and sv along horizontal and vertical directions are: 1 1 sv = sh = √ s = gt 2 = 0.4000 m. 4 2

34


Figure 12.4a. Three block simultaneously sliding after 0 steps.

Figure 12.4b. Three block simultaneously sliding after 70 steps.

Results of 3-D DDA are: sh = 0.3998m, sv = 0.4007m. Figure 12.4 is the dynamic displacements of four blocks. In the process of three block sliding, the middle block separate the other two blocks.

35


Figure 12.5a. Arch block displacements after 0 steps.

Figure 12.5b. Arch block displacements after 110 steps.

The sliding modes are the same with two dimensional results and the base friction machine test. Figure 12.5 is the dynamic displacements of seven arch blocks. It shows the three dimensional block mesh of the arch and the three dimensional block movements after 105 time steps. The movements keep symmetries in all directions and consistent with two dimensional DDA computations. Figure 12.6 is a 30000 cubic meter key block computation. From key block theory, this is a single face sliding. However, three dimensional DDA shows rotations.

13 THE STATIC STABILITY ANALYSIS OF DAM FOUNDATIONS Figure 13.1 to 13.4 are dam foundation cases. The mechanical parameters of rock masses are given in Table 13.1. Figure 13.1 and 13.2 are the dam foundation with resisting blocks. If the two major horizontal joints have friction angle 45◦ and 29◦ respectively, the dam is stable. This is statics case. The statics is computed by time depending dynamics. In the end of each time step, set the velocity to zero. In case the friction angles are 0◦ and 17◦ respectively, the dam will slide together with the resisting blocks as shown in Figure 13.2.

36


Figure 12.6a.

One block complex sliding after 0 steps.

Figure 12.6b.

One block complex sliding after 700 steps.

Table 13.1.

Mechanical data.

Reduce step velocity E of rock mass Contact spring ν of rock mass Number of time steps Cohesion

1.00 432000 ton /m 3 9216000 ton /m 0.20 100 0 ton/m 3

Figure 13.1 The dam and rock foundation with resisting blocks. Figure 13.2 Dam, foundation and resisting blocks movement. Figure 13.3 and 13.4 are the dam foundation. If the two major horizontal joints have friction angle 32◦ , the dam is stable. This is statics case. The statics is computed by time depending dynamics. In the end of each time step, set the velocity to zero. In case major horizontal joints have friction angles 29◦ and 17◦ respectively, the dam will slide as shown in Figure 13.4.

37


Figure 13.1.

Dam and foundation movement case 1 after 0 steps.

Figure 13.2.


Figure 13.3 The dam and rock foundation block mesh. Figure 13.4 Dam and foundation movement.

14 THE GEOMETRY AND MECHANICAL DATA OF TUNNEL STATICS DDA COMPUTATION The rock block movement near the tunnel is basically controlled by existing joints. The joint sets of joint system and mechanical parameters of the tunnel given in Table 14.1. Based upon the statistics, the joint geometric parameters are given in Table 14.2. The geometry of the tunnels are given in Table 14.3. Based on the geometric data of Table 14.1, 14.2 and 14.3, program produces the joints and tunnel boundary lines.

38


Figure 13.3.


Figure 13.4.


Table 14.1. Angle data of joint system. Joint set

Dip angle

Dip d.

Friction angle

Cohesion


82◦ 82◦ 14◦

288◦ 229◦ 40◦

39◦ 39◦ 39◦

0 ton/m2 0 ton/m2 0 ton/m2

From the joints and tunnel boundary lines, program produces the block system. The block system is the geometric input of the mechanical analysis program. The total block number is 3784.

39


Table 14.2.

Statistical length data of joint system.

Joint set

Spacing: m

Length: m

Bridge: m


2.6 m 2.6 m 2.6 m

16.45 m 14.9 m 25.8 m

−1.0 m −1.0 m 0.4 m

Table 14.3. Tunnel data. 75◦ 0◦ 5.5 meter circular

bearing angle of tunnel axis rise angle of tunnel axis tunnel diameter tunnel shape

Figure 15.1.

Rock displacements after 2000 steps.

The mechanical parameters of both rock masses and joints are given in Table 14.4. Based on the mechanical data of Table 14.4, the program computes the time depending block movements and block stresses. The process of block falling can be shown.

40


Table 14.4.

Mechanical data.

Reduce step velocity E of rock mass Contact spring ν of rock mass Number of time steps Time step Computation duration Joint friction angle Cohesion

Figure 15.2.

15

0.01 3000000 ton /m3 6000000 ton /m 0.21 20000 0.0010 second 2 second 39◦ 0 ton /m3

Rock block stresses after 2000 steps.

STATIC DDA TUNNEL COMPUTATION USING DYNAMICS WITH UNIT MASS DAMPING

Here in this following case, the dynamics with unit mass damping is used. The relative displacements are reduced near zero following the time steps. When the step time is 0.001 seconds, the next step uses 0.99 (normally 0.95– 0.99) of the velocity from the end of the previous time step. If this number is 0.97, the relative displacement reduces much faster. Figure 15.1 The block boundary after 2000 time steps (2.0 seconds). Figure 15.2 The principle stresses of the blocks after 2000 time steps (2.0 seconds).

41


Table 15.1. Average relative displacement. Time step

Relative displacement

Open-close iteration

1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

0.00083 9 0.00041 5 0.00032 8 0.00030 5 0.00016 5 0.00028 5 0.00038 5 0.00047 5 0.00055 5 0.00062 5 0.00103 5 0.00106 5 0.00086 4 0.00054 4 0.00023 3 0.00025 3 0.00046 2 0.00049 2 0.00039 1 0.00014 1 0.00001 1 0.00001 1 0.00000 1 0.00000 1 0.00000 1 0.00000 1 0.00000 1 0.00000 1 0.00000 1 0.00000 1 0.00000 1 0.00000 1 0.00000 1 0.00000 1 0.00000 1 0.00000 1 0.00000 1 0.00000 1 Equation solving iteration: 40–50 Factor of SOR: 1.25

The following table is the computation of the statics using dynamics. The relative displacements are reduced following the progress of the time steps. The damping is made by reducing 0.01 times of velocity after each 0.001 seconds (time interval). It is also can be noticed, the open-close iterations are all 1 after time step 100. This means all contacts keep the same after time step 100. The computation is stable. REFERENCES [1] Gen-hua Shi, “Applications of Discontinuous Deformation Analysis (DDA) and Manifold Method” The Third International Conference on Analysis of Discontinuous Deformation, pp. 3–15 Vail, Colorado (1999) [2] Gen-hua Shi, “Block System Modeling by Discontinuous Deformation Analysis” Computational Mechanics Publications, Southampton UK and Boston USA (1993) [3] Gen-hua Shi and Richard E. Goodman, “Generalization of Two Dimensional Discontinuous Deformation Analysis for Forward Modeling,” International Journal for Numerical and Analytical Methods in Geomechanics. Vol. 13, pp. 131–158 (1989)

42


[4] Gen-hua Shi and Richard E. Goodman, “The Key Blocks of Unrolled Joint Traces in Developed Maps of Tunnel Walls,” International Journal for Numerical and Analytical Methods in Geomechanics. Vol. 13, pp. 359–380 (1989) [5] Gen-hua Shi and Richard E. Goodman, “ Stability analysis of infinite block systems using block theory,” Proc. Analytical and computational methods in engineering rock mechanics, E.T. Brown, London: Allen and Unwin, pp. 205–245 (1987) [6] Richard E. Goodman and Gen-hua Shi, “The Application of Block Theory to the Design of Rock Bolt Supports for Tunnels,” Felsbau 5 Nr. 2, pp. 79–86 (1987) [7] Gen-hua Shi and Richard E. Goodman, “Two Dimensional Discontinuous Deformation Analysis,” International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 9, pp. 541–556 (1985) [8] Richard E. Goodman and Gen-hua Shi, “Block theory and its application to rock engineering,” Englewood Cliffs, NJ: Prentice-Hall (1985) [9] Gen-hua Shi and Richard E. Goodman, “Keyblock Bolting,” Proc. of International Symposium on Rock Bolting, pp. 143–167 Sweden (1983) [10] Gen-hua Shi and Richard E. Goodman, “Underground Support Design Using Block Theory to Determine Keyblock Bolting Requirements,” Proc. of the Symposium on Rock Mechanics in the Design of Tunnels, South Africa (1983) [11] Richard E. Goodman and Gen-hua Shi, “Geology and Rock Slope Stability – Application of a Keyblock Concept for Rock Slopes,” Proc. of Third International Conference on Stability in Surface Mining, pp. 347–373, (SME) (1983) [12] Gen-hua Shi, “A Geometric Method of Stability Analysis of Discontinuous Rocks,” Scientia Sinica, Vol. 25, No. 1, pp. 125–148 Peking, China (1982) [13] Gen-hua Shi and Richard E. Goodman, 1981. “A New Concept for Support of Underground and Surface Excavation in Discontinuous Rocks Based on a Keystone Principle,” Proc. 22th U. S. Symposium on Rock Mechanics, pp. 290–296 MIT (1981) [14] Gen-hua Shi, 1977. “The Stereographic Projection Method of Stability Analysis of Rock Mass,” Scientia Sinica, Vol. 3, pp. 260–271 Peking, China (1977)

43


Plenary lectures



Realistic dynamic analysis of jointed rock slopes using DDA Y.H. Hatzor, A.A. Arzi, and M. Tsesarsky Department of Geological and Environmental Sciences, Ben Gurion University of the Negev, Beer–Sheva, Israel

ABSTRACT: A fully dynamic, two dimensional, stability analysis of a highly discontinuous rock slope is demonstrated in this paper using DDA. The analytically determined failure modes of critical keyblocks are clearly predicted by DDA. However, application of a fully dynamic analysis with no damping results in unrealistically large displacements that cannot be confirmed by field studies. With introduction of dynamic damping the calculated results can be made to match historic evidence. Our study shows that introduction of at least 5% dynamic damping is necessary to predict realistically the earthquake damage in a highly discontinuous rock slope with about 400 individual blocks. The introduction of dynamic damping is necessary to account for 2D limitations as well as for various energy loss mechanisms, which are not modeled in DDA.

1

INTRODUCTION

Mount Masada is a table mountain, having a comparatively flat summit surrounded by steep slopes, rising about 480 meters above the nearby Dead Sea. The uppermost tens of meters of the slopes consist of nearly vertical cliffs. About two thousand years ago, King Herod fortified the mountain and built a major palace based on three natural rock terraces at the northern tip of the summit (Figure 1). Mount Masada was the site of heroic Jewish resistance against the Romans. It is a national historic monument. The Israel Nature and Parks Authority commissioned this study of the stability of the upper rock terrace of Herod’s Palace under earthquake loading, as part of its preservation work. We carried out the stability evaluation using a fully dynamic version of DDA, with inputs based on a comprehensive field and laboratory study.

2

GEOLOGICAL – SEISMOLOGICAL SETTING

The upper portion of Mount Masada consists of essentially bare hard rock. The rock is mainly bedded limestone and dolomite, with near vertical jointing. Structurally, the entire mountain is an uplifted block within the band of faults which forms the western boundary of the Dead Sea Rift. The Dead Sea Rift is a seismically active transform (Garfunkel et al., 1981; Garfunkel and Ben-Avraham,

Figure 1. A photo of the North Face of Masada showing the upper terrace of King Herod’s Palace.

47


Locally it was even founded slightly beyond the rim, on a somewhat lower edge of rock. On the aforementioned three palace terraces, jutting at the northern tip of the mountain top, construction was again carried out up to the rim and beyond in order to achieve architectural effects and utilize fully the limited space. Thus, the remaining foundations effectively serve to delineate the position of the natural rim of the flat mountain top and associated northern terraces about 2000 years ago. Missing portions along such foundation lines indicate locations in which the rim has most probably receded due to rockfalls, unless the portions are missing due to other obvious reasons such as local erosion of the flat top by water or an apparent location of the foundation on fill beyond the rim. Our inspection of the entire rim of the top of Masada, aided by Hebrew University archeologist Guy Stiebel, reveals that over almost the entire length of the casemate wall, which is about 1400 m, the rock rim has not receded during the past two thousand years more than a few decimeters, if at all. Only over a cumulative total of less than 40 m, i.e. about 3% of the wall length, there are indications of rockfalls involving rim recessions exceeding 1.5 m, but not exceeding 4.0 m. Since the height of the nearly-vertical cliffs below the rim is in the order of tens of meters, these observations attest to remarkable overall stability in the face of the recurring earthquakes. On King Herod’s palace terraces there has been apparent widespread destruction, mostly of walls and fills which were somehow founded on the steep slopes. However, in the natural cliffs themselves there are few indications of rockfalls involving rim recessions of more than a few decimeters. Remarkably, most of the high retaining walls surrounding the middle and lower terraces are still standing, attesting to the stability of the rock behind them. In the upper terrace, on which this study is focused, there appears to be only one rockfall with depth exceeding several decimeters. It is a local rockfall near the top of the 22 m cliff, in the northeast, causing a rim recession of about 2.0 m. It is notable that this particular section of the terrace cliff was substantially modified by the palace builders, perhaps de-stabilizing the preexisting natural cliff. We have also inspected rare aerial photographs of Mount Masada dated December 29th 1924, i.e. predating the 1927 earthquake. Our comparison with recent aerial photographs would have been capable of detecting rim recessions exceeding about one meter, if any had occurred in the northern part of the mountain. None were found, suggesting that the 1927 earthquake did not cause any significant rockfalls there (the southern part was less clear in the old photographs). The information presented above essentially constitutes results of a rare rock-mechanics field-scale “experiment”. Two thousand years ago the Masada cliff top was marked by construction. The mountain was later shaken by several major earthquakes, with

1996). According to the Israel building code – Israel Standard 413, based on research by the Geophysical Institute of Israel Seismology Division, under the direction of Dr. A. Shapira, the Dead Sea region has been classified as a region in which an earthquakeinduced peak ground acceleration. (PGA) exceeding 0.2 g at the deep bedrock level is expected with a 10% probability within any 50 year window. This is analogous to a 475 year average recurrence interval for such acceleration. In this paper we repeatedly refer to PGA for simplicity, which is adequate in the present context, although PGA is not generally the best measure of destructive potential (Shapira, 1983; Shapira and van Eck, 1993). Inspection of the historic earthquake record (BenMenahem, 1979; Turcotte and Arieh, 1988; Amiran et al., 1994) suggests that the strongest shaking events which have actually affected Mount Masada within the past two thousand years were due to about ten identified earthquakes with estimated magnitudes in the range of 6.0 ± 0.4 and focal distances probably in the order of several kilometers to a few tens of kilometers from the site. With these parameters, it is highly likely that some of these earthquakes have caused at Mount Masada bedrock PGA’s reaching and even exceeding 0.2 g, in general agreement with predictions for a 2000 year period based on the aforementioned building code assumptions. One of the most notable historic earthquakes in this region occurred probably in the year 362 or 363, with a magnitude estimated at 6.4 (Ben-Menahem, 1979) or even 7.0 (Turcotte and Arieh, 1988). Reported effects included a tsunami in the Dead Sea and destruction in cities tens of kilometers from the Dead Sea both east and west. This is probably the earthquake identified by archeologists as “the great earthquake which destroyed most of the walls on Masada sometime during the second to the fourth centuries” (Netzer, 1991). The most recent of the major historic earthquakes near Mount Masada occurred on July 11th, 1927. This earthquake was recorded by tens of seismographs, yielding a magnitude determination of 6.2 and an epicenter location 30 ± 10 km north of Masada. It also caused a tsunami in the Dead Sea and destruction in cities tens of kilometers away (Shapira et al., 1993) 3

OBSERVED HISTORICAL STABILITY

The fortifications built by King Herod on Mount Masada about two thousands years ago (Netzer, 1991) included a casemate wall surrounding the relatively flat top of the mountain. Clearly, because of its defensive function, the outer face of this wall was built so as to continue upward the face of the natural cliff, as much as possible. The outer wall was therefore founded typically on the flat top within several decimeters from its rim.

48


Table 1. Discontinuity data for the foundations of King Herod’s Palace – Masada. Set #

Type

Dip

s m

C MPa

φ◦

1 2 3

Bedding Joints Joints

5/N 80/ESE 80/NNE

0.60 0.14 0.17

0 0 0

41◦ 41◦ 41◦

J1 Spacing

Frequency

25 15 10 5 0

MECHANICAL BEHAVIOR OF THE ROCK AT MASADA

40

0

80 120 160 200 Spacing (cm)

10 20 30 Spacing (cm)

12

40

Length

N = 69 Mean = 16.8 cm

8 4 0

50 Frequency

Rock mass structure

Herod’s Palace, also known as the North Palace, is built on three terraces at the north face of Masada. The rock mass structure at the foundations consists of two orthogonal, sub vertical, joint sets striking roughly parallel and normal to the NE trending axis of the mountain, and a set of well developed bedding planes gently dipping to the north (Table 1). The joints are persistent, with mean length of 2.7 m. The bedding planes, designated here as J1 , dip gently to the north with mean spacing of 60 cm. The two joint sets, J2 and J3 , are closely spaced with mean spacing of 14 cm and 17 cm respectively (Figure 2). 4.2

10

J3 Spacing 16 Frequency

4.1

N = 80 Mean = 14 cm

20

0 0

4

J2 Spacing

30

N = 59 Mean = 60 cm

20

Frequency

deep bedrock accelerations certainly exceeding 0.1 g and probably even exceeding 0.2 g. Due to the topography affect, motions at the top are substantially amplified at frequencies about 1.3 Hz (Zaslavsky et al., 2002). Observations at the present stage of the “experiment” show that all the cliffs surrounding the top of Mount Masada essentially withstood the shaking, with some relatively minor rockfalls at the top of the cliffs. The above is a substantial result of a full-scale “experiment” on the real rock structure. Therefore, a fundamental test of any model of this structure is that it must essentially duplicate the above “experiment”. As shown in the sequel, we subjected our DDA model to this test, obtaining instructive results.

N = 100 Mean = 2.7 m

40 30 20 10 0

0

10 20 30 Spacing (cm)

40

0

2

4 6 8 10 12 14 Spacing (cm)

Physical properties

The rock mass consists of bedded dolomites with local karstic voids between beds. The bulk porosity of intact samples ranges between 3% – 12% and the dry unit weight is 25 kN/m3 . The bedding planes are generally clean and tight. 4.3

Strength and elasticity of intact rock

The elastic behavior of the rock was studied using a stiff, hydraulic, closed-loop servo controlled load frame with maximum axial force of 1.4 MN, and stiffness of 5 ∗ 109 N/m (Terra-Tek model FX-S-33090). Testing procedures are described by Hatzor and Palchik, 1997, 1998) and Palchik and Hatzor (2002). Tests were performed at a constant strain rate of 5 ∗ 10−6 s−1 . In Figure 3 the result of a load – unload loop of uniaxial compression performed on a solid cylinder from the dolomite at Masada is shown. This result indicates that the uniaxial compressive strength of the tested dolomite sample is greater than 315 MPa, that the elastic modulus is 43 GPa, and that Poisson’s ratio is 0.18 (radial strains are not shown). These data

Figure 2. Discontinuity length, spacing, and orientation distribution at the foundations of King Herod’s Palace, Masada. Upper Hemisphere projection of poles.

fall within the range of strength and elasticity values determined experimentally for other dolomites in Israel (Hatzor and Palchik, 1997, 1998; Palchik and Hatzor, 2002). 4.4 Residual shear strength of discontinuities The residual friction angle of joints was determined using tilt tests performed on saw-cut and ground surfaces of dolomite, assuming the joint planes are

49


1.2

MNP3 - Natural bedding plane 1.38 MPa

300 200 100

0.8

1.2 MPa

0.4

1.03 MPa 0.86 MPa 0.7 MPa 0.52 MPa

Shear Stress (MPa)

E = 42,909 MPa v = 0.1844 Shear Stress (MPa)

Axial Stress (MPa)

400

1.2

Failure Envelope

0.8 0.4 0 0

0.4 0.8 1.2 Normal Stress (MPa)

0.35 MPa

0

Normal Stress 0.17 MPa

0

0.1

0.2

0.3

0.4

0.5 0

Axial Strain (%)

0

Figure 3. Result of a load – unload cycle under uniaxial compression performed on an NX size solid cylinder of the dolomite in Masada.

Shear strength of filled bedding planes

2

0 < σn < 0.5 MPa : τ = 0.88σn (R2 = 0.999) 0.5 < σn ≤ 12 MPa : τ = 0.083 + 0.71σn (R2 = 0.998)

Shear strength of rough bedding planes

These results indicate that for low normal load (up to 0.5 MPa) the peak friction angle for the bedding planes at Masada is 41.3◦ . For higher normal loads the peak friction angle is 35.3◦ . The residual friction angle may be taken from the triaxial tests of the filled discontinuities as 23◦ . The maximum height of the terrace at the North Palace is 25 m and therefore the normal stress acting on bedding planes at the North

The shear strength of rough bedding planes was determined using real bedding plane samples from the foundations of the North Palace. The upper and lower sides of the mating planes were kept in contact with no disturbance and were transported to the lab at natural water content. The two samples were cast inside two 200 mm ∗ 200 mm ∗ 150 mm shear boxes while the mating surfaces were kept intact. The gap between

50


1.6

the rock and the box frame was filled with Portland cement. Direct shear tests were performed using a hydraulic, close loop servo-controlled, direct shear system with normal force capacity of 1000 kN and horizontal force capacity of 300 kN (Product of TerraTek Systems Inc.). The stiffness of the normal and shear load frames is 7.0 and 3.5 MN/m respectively. Normal and horizontal displacement during shear were measured using four and two 50 mm LVDT’s with 0.25% linearity full scale. Axial load was measured using a 1000 kN capacity load cell with 0.5% linearity full scale. Shear load was measured using a 300 kN load cell with 0.5% linearity full scale. The direct shear tests were performed for two samples (MNP2, MNP3) under constant normal stress and shear displacement rate of 1mil/sec (0.025 mm/sec). In Figure 4 shear stress vs. shear displacement is shown for sample MNP-3 that was loaded, unloaded, and reloaded in eight cycles of increasing normal stress from 0.17 MPa to 1.38 MPa. In each cycle the sample was sheared forward, in the first cycle a distance of 1.3 mm, and then additional 0.5 mm of forward shear displacement in each consecutive segment. Plotting the peak shear stress vs. normal load for the two segment tests (Figure 5) reveals a bilinear failure envelope with the following failure criterion:

The shear strength of filled bedding planes was estimated using a segment triaxial test performed on a right cylinder containing an inclined saw cut plane at 55◦ to the axis of the cylinder, filled with crushed dolomite. Seven different segments were performed, with confining pressure values ranging between 2.2 and 16.2 MPa. A linear Coulomb – Mohr failure criterion was found, with zero cohesion and a residual friction angle of 22.7◦ . The test procedures and results are discussed by Hatzor (in press). The similarity between the result of tilt tests on ground surfaces (23◦ ) and the segment triaxial test on a filled saw-cut plane (22.7◦ ) suggests that during shear the infilling material crushed all remaining asperities in the saw-cut sample resulting in a failure envelope representing residual conditions. The residual friction angle value of 23◦ may therefore be applicable for very large blocks where some initial shear displacements have already taken place due to historic cycles of seismic loading (see Hatzor, in press). However, for dynamic analysis of smaller blocks with high static factor of safety the strengthening effect of initial asperities ought to be considered. 4.6

0.8 1.2 Shear Displacement (mm)

Figure 4. Shear stress vs. shear displacement for a natural bedding plane sample from the foundations of Herod’s Palace in Masada.

clean and tight. 20 tilt tests performed on saw-cut and ground surfaces provided a mean friction angle of 28◦ and 23◦ respectively. The 5◦ difference is attributed to roughness resulting from saw-cutting. 4.5

0.4

Direct Shear of Natural Bedding Planes Triaxial Shear of Filled Saw-cut

Shear Stress (MPa)

12

8

4

0 0

5

10 15 Normal Stress (MPa)

20

25

Figure 5. Failure envelope for rough bedding planes – direct shear tests.

Figure 7. A photogeological trace map of the northern face of Herod’s Palace upper terrace.

Figure 8. A deterministic joint trace map of the terrace prepared using the photogeological map (Figure 7) and the block cutting algorithm (DC) of Shi (1993). Figure 6. A synthetic joint trace map of the upper rock terrace of Herod’s Palace in Masada using the statistical joint trace generation code (DL) of Shi (1993).

individual blocks. Kinematic, mode, and removability analyses confirm these intuitive expectations.

Palace cannot be greater than 682 kPa. Therefore, in light of the experimental results, the low normal load criterion should be used for dynamic analysis.

5 5.1

5.2 Deterministic joint trace generation While it is quite convenient to use mean joint set attitudes and spacings in order to generate statistically a synthetic mesh, the resulting mesh is quite unrealistic and bears little resemblance to the actual slope. The contact between blocks obtained this way is unrealistically planar, thus interlocking between blocks is not modeled. Consequently the results of dynamic calculations will be overly conservative. In order to analyze the dynamic response of the slope realistically a photo-geological trace map of the face was prepared using aerial photographs (Figure 7), and the trace lines were then digitized. Thus, the block-cutting (DC code) algorithm of Shi (1993) could be utilized in order to generate a trace map which represents more closely the reality in the field (Figure 8).

MESH GENERATION METHODS AND EXPECTED FAILURE MODES Synthetic joint trace generation

Two principal joint sets and a systematic set of bedding planes comprise the structure of the foundations of Herod’s Palace (Figure 2). An E-W cross section of the upper terrace is shown in Figure 6, computed using the statistical joint trace generation code (DL) of Shi (1993). It can be seen intuitively that while the East face of the rock terrace is prone to sliding of wedges, the West face is more likely to fail by toppling of

51


7

The same failure modes would be anticipated in the deterministic mesh shown in Figure 8. However, the interlocking between blocks within the slope is much higher and therefore the results of the dynamic analysis should be less conservative and more realistic. 6

7.1 The numerical discontinuous deformation analysis The DDA method (Shi, 1993) incorporates dynamics, kinematics, and elastic deformability of geological materials, and models actual displacements of individual blocks in the rock mass using a time-step marching scheme. The formulation is based on minimization of potential energy and uses a “penalty” method to prevent penetration or tension between blocks. Numerical penalties in the form of stiff springs are applied at the contacts to prevent either penetration or tension between blocks. Since tension or penetration at the contacts will result in expansion or contraction of the springs, a process that requires energy, the minimum energy solution is one with no tension or penetration. When the system converges to an equilibrium state however, there are inevitable penetration energies at each contact, which balance the contact forces. Thus, the energy of the penetration (the deformation of the springs) can be used to calculate the normal and shear contact forces. Shear displacement along boundaries is modeled in DDA using the Coulomb-Mohr failure criterion. The fixed boundaries are implemented using the same penalty method formulation: stiff springs are applied at the fixed points. Since displacement of the fixed points requires great energy, the minimum energy solution will not permit fixed-point displacement. The blocks are simply deformable: stresses and strains within a block are constant across the whole region of the block. In this research a new C/PC version of DDA, recently developed by Shi (1999), is used. In this new version earthquake acceleration can be input directly in every time step. A necessary condition for direct input of earthquake acceleration is that the numerical computation has no artificial damping, because damping may reduce the earthquake dynamic energy and the damage may thus be underestimated (Shi, 1999). In DDA the solution of the equilibrium equations is performed without damping (Shi, 1999) for the purpose of a fully dynamic analysis in jointed rock masses. As we shall see below however, application of dynamic DDA with no damping returns unrealistically high displacements and is therefore overly conservative.

INPUT MOTION FOR DYNAMIC ANALYSIS

In order to perform a realistic dynamic analysis for the rock slopes at Masada, we prefer modeled input motion representing ground motions for the Dead Sea rift system. In this research we chose to use the recorded time history of the Mw = 7.1. Nuweiba earthquake which occurred in November 1995 in the Gulf of Eilat (Aqaba) with an epicenter near the village of Nuweiba, Egypt. The main shock was recorded at the city of Eilat where the tremor was felt by people, and structural damage was detected in houses and buildings. The city of Eilat is located 91 km north from the Nuweiba earthquake epicenter and 186 km south of Masada, on the northern coast of the gulf of Eilat (Aqaba). Figure 9 shows the vertical and EW components of the accelerogram that were recorded in Eilat. The horizontal Peak Ground Acceleration (PGA) of the Nuweiba record was 0.08 g. The Eilat accelerograph station was on a thick fill of Pleistocene alluvial fan deposits. The recorded accelerogram therefore represents the response of a site situated on deep fill rather than on sound bedrock. However, we regard this as a secondary issue in the present context. As shown in the sequel, we utilized as input both the 0.08 g PGA accelerogram as well as the same accelerogram normalized to a 0.18 g PGA, so as to explore a range of PGA’s. As explained in chapters 2 and 3 above, the Masada cliffs, including Herod’s Palace upper terrace, have withstood historic earthquakes in this PGArange with only minor rockfalls. 0.12

Vertical

Accl. (g)

0.08 0.04 0 -0.04 -0.08 0.12

FULLY DYNAMIC ANALYSIS USING DDA

E-W

Accl. (g)

0.08

7.2

0.04 0 -0.04

Hatzor and Feintuch (2001) demonstrated the validity of DDA results for fully dynamic analysis of a single block on an incline subjected to dynamic loading. First the dynamic solution for a single block on an incline subjected to gravitational load (constant acceleration), a case which was investigated originally by MacLaughlin (1997), was repeated using the new

-0.08 0

10

20

30

40

50

60

Time (sec)

Figure 9. Time history of the Mw = 7.1 Nuweiba earthquake (Nov. 22, 1995) as recorded at the city of Eilat.

52


Validation of dynamic displacement prediction by DDA using analytical solutions

dynamic code (Shi, 1999). For a slope inclination of 22.6◦ , four dynamic displacement tests were performed for interface friction angle values of 5◦ , 10◦ , 15◦ , and 20◦ . The agreement between the analytical and DDA solution was within 1–2%. Next, Hatzor and Feintuch investigated three different sinusoidal functions of increasing complexity for the dynamic load input function, and checked the agreement between DDA and the derived analytical solutions. A very good agreement between analytical solution and DDA was obtained in all cases, with errors between 5% to 10% (Figure 10). It was found by Hatzor and Feintuch that in order to get a good agreement between DDA and the analytical solution, the maximum size of the time step (g1) had to be properly conditioned. For example, with increasing block velocity the maximum time step size had to be reduced in order to obtain good agreement with the analytical solution. The best method to estimate the proper time step size would be to check the ratio between the assumed maximum displacement per time step (g2) and the actual displacement per time step calculated by DDA. In order to get good agreement between DDA and the analytical solution that ratio should be as close as possible to 1.0. Such an optimization procedure however is only possible for single block cases. 20

The dynamic displacement problem of a block on an incline was studied by Wartman (1999) using shaking table experiments that were performed at the U.C. Berkeley Earthquake Engineering Laboratory. The same tests were repeated numerically by Tsesarsky et al. (2002) using dynamic DDA, and some results are shown in Figure 11 (see complete paper in this volume). The results of Tsesarsky et al. suggest that with zero dynamic damping DDA overestimates the physical displacements by as much as 80%. However, with as little as 2% dynamic damping the results of DDA match the experimental results within 5% accuracy. This result suggests that realistic application of dynamic DDA must introduce some measure of damping in order to account for energy loss mechanisms that are not modeled by DDA, the first of which is energy consumption due to irreversible deformation during block interactions. The results of Tsesarsky et al. (2002) pertain to a single block on an incline. A multi-block problem was studied by McBride and Scheele (2001), using a slope with a stepped base consisting of 50 blocks that undergo sliding failure under gravitational load. Their conclusion was that as much as 20% dynamic damping was necessary in order to obtain realistic agreement between the physical model and DDA. Perhaps better conditioning of the control parameters would have reduced the required dynamic damping by a significant amount.

Acceleration Velocity Displacement DDA displacement

15 a (m/s2), v (m/s), s (m)

7.3 Validation of dynamic displacement prediction by DDA using shaking table experiments

10

5

8

8.1 Numerical details

0 0

2

1

3

4

5

In all DDA simulations the complete record was computed for the entire 50 seconds of earthquake duration (see Figure 9). The numerical input parameters used in this work are listed in Tables 2 and 3 (for explanation of each control parameter see Shi, 1993). DDA computations were performed on a P41.5 GHz processor with 128 Mb RAM. To complete the required 25000 time steps (earthquake duration of 50 seconds) approximately 42 hours of processor time were requires, namely a computation rate of approximately 600 time steps per hour. The mesh consists of 344 blocks.

a(t) = 2sint + 3sin2t

-5

Time (sec)

Figure 10. Validation of dynamic DDA using analytical solutions (after Hatzor and Feintuch, 2001). 0.1 Measured Block Displacement DDA (k01 = 1)

0.08 Displ. (m)

RESULTS OF DYNAMIC ANALYSIS

DDA (k01 = 0.98)

DDA solution for g0 = 500*106 N/m g1 = 0.0025 sec g2 = 0.0075

0.06 0.04 0.02

8.2 Realistic damage prediction by DDA

0 0

1

2

3

4

In Figure 12 the computed response of the upper terrace is shown with various amounts of dynamic damping. Figure 12A shows the computed damage with zero dynamic damping, namely the initial velocity in every time step is inherited from the previous

5

Time (sec)

Figure 11. Validation of dynamic DDA using shaking tabel experiments from Berkeley (after Tsesarsky et al., 2002).

53


Table 2.

Numeric Control Parameters.

Total number of time steps: Time step size (g1): Assumed maximum displacement ratio (g2): Contact spring stiffness (g0): Factor of over-relaxation:

Table 3.

25,000 0.002 0.0015 5 ∗ 106 kN/m 1.3

Material Properties.

Unit weight of rock (γ ): Elastic Modulus (E): Poisson’s Ratio (v): Friction angle of all discontinuities (φ): Cohesion of all discontinuities (C): Tensile strength of all discontinuities (σt ):

25 kN/m3 43 ∗ 106 kN/m2 0.184 41◦ 0 0

time step. After 50 seconds of shaking, with 0.08 g PGA, the upper terrace seems to disintegrate completely. In light of the historic observations this result is clearly unrealistic. In Figure 12B the response of the terrace is shown again after 50 seconds of shaking but with 5% dynamic damping, namely the initial velocity of each block at the beginning of a time step is reduced by 5% with respect to its terminal velocity at the end of the previous time step. The model predicts onset of toppling failure at the foot of the west slope and minor sliding deformations at the east slope. The failure modes predicted by the model are similar to the expected modes from both field and analytical studies. The extent of damage in the terrace and the depth of the loosened zone in the west slope are reduced significantly with comparison to the undamped analysis. The performance of the slope with 10% dynamic damping (Figure 12C) is roughly the same as with 5% dynamic damping and therefore the justification for more than 5% dynamic damping seems questionable. Assuming that 5% damping is the correct amount necessary to account for energy loss mechanisms ignored by DDA, we studied the response of the terrace slopes to the same accelerogram when normalized to a 0.18 g PGA, which is still within the range of historic earthquakes as discussed in chapters 2,3 and 6 above. As shown in Figure 13, after 50 seconds of shaking the damage is not much different than that which was modeled for the original time history (Figure 12B). Both Figures 12B and 13 indicate the expected depth of the loosened zone in the slope due to the seismic loading. We believe that the graphical output in Figure 13 is still very conservative because it does not compensate fully for: a) various real energy dissipation mechanisms, b) reinforcing potential of the third, in slope, dimension.

Figure 12. Results of dynamic DDA calculation of the original Nuweiba record (PGA = 0.08 g) after 50 seconds of shaking. A) No dynamic damping in DDA, B) 5% damping, C) 10% damping.

Nevertheless, a graphical result such as the one presented in Figure 13 can be used as an aid for support design. Both the spacing and length of the support elements (anchors or rock bolts) can be dimensioned using the graphical output. The required capacity of the anchors may be estimated using a pseudo-static analysis for a representative block with the peak horizontal acceleration taken for the pseudo-static inertia force.

54


slope for which exists a good historic record of stability under recurring strong earthquake shaking. We found that in order to obtain realistic predictions for this multi-block analysis, at least 5% dynamic damping was required. • Only when proven to be realistic, the graphical output of such an analysis may be used to estimate the depth of the loosening zone following the earthquake, and the spacing, length and capacity of support elements – if required.

ACKNOWLEDGEMENTS Figure 13. Results of dynamic DDA calculation (with 5% dynamic damping) after 50 seconds of shaking using the Nuweiba record normalized to PGA = 0.18 g.

9

This research was funded by Israel Nature and Parks authority and partially by the Bi-national Science Foundation (BSF) through grant 98-399. The support of the two agencies is hereby acknowledged. A. Shapira and Y. Zaslavsky from the Geophysical Institute of Israel are thanked for the Nuweiba record. Gen-Hua Shi is thanked for making his dynamic DDA code available for this study. Finally, Guy Stiebel is thanked for guiding us on Masada with archeological insights and for providing the 1924 aerial photographs.

SUMMARY AND CONCLUSIONS

In this paper a highly jointed rock slope, which withstood many events of strong seismic ground motions in historic times, is modeled using dynamic DDA. The field observations are compared with the results of the numerical model. The following are tentative conclusions:

REFERENCES

•

We find that for a realistic calculation of dynamic response the modeled joint trace map (the mesh) must be as similar as possible to the geological reality. We recommend using digitized photogeological trace maps in conjunction with the DC block-cutting algorithm of Shi (1993) in order to generate the realistic mesh, rather than a statistical joint trace generation algorithm such as the DL code of Shi (1993). Such a deterministic approach will capture some block-interlocking mechanisms active in the modeled slope because of dissimilarities in joint attitudes and variations in dip angle along the surface of the discontinuities. • Previous studies have shown that dynamic DDA with zero dynamic damping will match analytical solutions for a single block on an incline with great accuracy. However, when results of dynamic DDA are compared with shaking table experiments for a single block it is found that at least 2.5% of dynamic damping is necessary for accurate displacement predictions. We believe that the dynamic damping is necessary in order to account for energy loss mechanisms, which are abundant in the physical reality but are ignored by the linear – elastic approach taken by DDA. Also, the dynamic damping may partially compensate for the two dimensional formulation which does not allow modeling the strengthening effect of the third, in slope, dimension. • Using a real time history from the Dead Sea rift system we modeled the response of the jointed rock

Amiran, D.H.K., Arieh, E. and Turcotte, T. 1994. Earthquakes in Israel and adjacent areas: Macro – seismicity observations since 100 B.C.E. Israel Explor. J., 41, 261–305. Ben – Menahem, A. 1979. Earthquake Catalog for the Middle East. Bollettino di Geofisica Teorica e Applicata, v. XXI, pp. 245–313. Garfunkel, Z. and Ben-Avraham, Z. 1996. The structure of the Dead Sea basin. In : Dynamics of extensional basins and inversion tectonics. Tectonophysics, 266, 155–176. Garfunkel, Z., Zak, I. and Freund, R., 1981. Active faulting in the Dead Sea Rift. Tectonophysics, 80, 1–26. Hatzor, Y.H. and V. Palchik, 1997. The influence of grain size and porosity on crack initiation stress and critical flaw length in dolomites. International Journal of Rock Mechanics and Mining Sciences, Vol. 34, No. 5, pp. 805–816. Hatzor, Y.H. and Palchik V., 1998. A microstructure-based failure criterion forAminadav dolomites – Technical Note. International Journal of Rock Mechanics and Mining Sciences, Vol. 35, No. 6, pp. 797–805. Hatzor, Y.H. and Feintuch, A. 2001. The validity of dynamic displacement prediction using DDA. International Journal of Rock Mechanics and Mining Sciences, Vol. 38, No. 4, pp. 599–606. Hatzor, Y.H. in press. Keyblock stability in seismically active rock slopes – the Snake Path cliff – Masada. Journal of Geotechnical and Geoenvironmental Engineering,ASCE. MacLaughlin, M.M. Discontinuous Deformation Analysis of the kinematics of landslides 1997. Ph.D. Dissertation, Dept. of Civil and Env. Engrg., University of California, Berkeley.

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McBride, A. and Scheele, F. Investigation of discontinuous deformation analysis using physical laboratory models. Proceedings of ICADD-4, 4th International Conference of Analysis of Discontinuous Deformation (N. Bicanic Ed) Glasgow, Scotland, UK. Netzer, E. 1991. Masada III – the Yigael Yadin Excavations 1963–1965. Final Reports – The Buildings Stratigraphy and Architecture. Israel Exploration Society, The Hebrew University, Jerusalem, Israel, 665p. Palchik, V. and Y.H. Hatzor, 2002. Crack damage stress as a composite function of porosity and elastic matrix stiffness in dolomites and limestones. Engineering Geology. Vol. 63, pp. 233–245. Shapira, A. 1983, A probabilistic approach for evaluating earthquake risk with application to the Afro-Eurasian junction, Tectonophysics, 95:75–89. Shapira, A., Avni, R. and Nur, A. 1993. A new estimate for the epicenter of the Jericho earthquake of 11th July 1927. Israel Journal of Earth Science, Vol. 42, No. 2, pp. 93–96. Shapira, A. and van Eck, T. 1993. Synthetic uniform hazard site specific response spectrum, Natural Hazard, 8: 201–205 Shi, G.-H. 1993. Block System Modeling by Discontinuous Deformation Analysis, Computational Mechanics Publications, Southampton UK, p. 209.

Shi, Gen-Hua, 1999. Application of Discontinuous Deformation Analysis and Manifold Method. Proceedings of ICADD-3, Third International Conference of Analysis of Discontinuous Deformation (B. Amadei, Ed) Vail, Colorado, pp. 3–15. Tsesarsy, M., Hatzor, Y.H. and Sitar, N. 2002. Dynamic block displacement prediction – validation of DDA using analytical solutions and shaking table experiments. Proceedings of ICADD-5, 5th International Conference of Analysis of Discontinuous Deformation (Y.H. Hatzor, Ed) Beer-Sheva, Israel. Published by Balkema Rotterdam. Turcotte, T. and Arieh, E. 1988. Catalog of earthquakes in and around Israel, Appendix 2.5A in: Shivta site Preliminary Safety Analysis Report, Israel Electric Corp. LTD. Wartman, J. 1999. Physical model studies of seismically induced deformation in slopes. Ph.D. thesis, Department of Civil Engineering, University of California, Berkley. Zaslavsky, Y., Shapira, A. and Arzi, A.A. 2002. Earthquake site response on hard rock – empirical study. Proceedings of ICADD-5, 5th International Conference on Analysis of Discontinuous Deformation (Y.H. Hatzor, Ed) Beer Sheva, Israel. Published by Balkema, Rotterdam.

56



Numerical models for coupled thermo-hydro-mechanical processes in fractured rocks-continuum and discrete approaches L. Jing, K.-B. Min & O. Stephansson Department of Land and Water Resources Engineering, Royal Institute of Technology, Stockholm, Sweden

ABSTRACT: This article reviews the two main approaches of numerical modeling for the coupled thermohydro-mechanical (THM) processes in fractured rocks – the discrete and the equivalent continuum models, respectively. The presentation covers the governing equations derived from the conservation laws of mass, momentum and energy of the continuum mechanics, with focus on the FEM formulations and explicit blocfracture system representations for discrete models. Two applications related to nuclear waste repository design and performance assessment are presented as examples. Special attention is given to the DEM approach of homogenisation and upscaling for deriving equivalent continuum properties of fractured rocks based on the REV concept, with realistic representations of fracture systems. A brief summary of the trends, characteristics and outstanding issues in numerical modeling of fractured rocks is given at the end to highlight the advances and remaining difficulties.

1

INTRODUCTION

be crossed so that their interactions be fully expressed in the resultant mathematical models and computer codes. The coupled THM process is mainly described by mechanics of porous geological media, such as soils, sands, clays and fractured rocks. The first theory is von Terzaghi’s 1-D consolidation theory of soils (Terzaghi, 1923), followed later by Biot’s phenomenological approach of poroelasticity (Biot, 1941, 1956), which was further enriched by the mixture theory (Morland, 1972, Bowen, 1982). Non-isothermal consolidation of deformable porous media is the basis of coupled THM models of geological media, using either the averaging approach as proposed by Hassanizadeh & Gray (1979a, b, 1980, 1990) and Achanta et al. (1994), or an extension to the Biot’s theory with a thermal component (de Boer, 1998). The former is more suitable for understanding the microscopic behavior of porous media and the latter is better suited for macroscopic description and computer implementation. The subject has attracted very active research activities because of its wide reaching impacts in the fields of both mechanics and engineering, and generated extensive publications. The fundamentals are systematically presented in many volumes, e.g. Whitaker (1977), Domenico and Schwartz (1990), Charlez (1991), Charlez and Keramsi (1995), Coussy (1995), Sahimi (1995), Selvadurai (1996), Lewis and Schrefler

Many rock engineering projects, such as radioactive waste disposal in underground repositories, Hot-DryRock geothermal energy extraction, oil/gas reservoir exploitation and oil/gas underground storage caverns, require understanding of interactions among different physio-chemical processes in various geological media. In most of the cases, mainly the mechanical process of rock stress, deformation, strength and failure (M), hydraulic processes of fluid flow and pressure (H), thermal processes of heat transfer (T) and chemical processes (C) of contaminant transport related to different fluid-rock interaction mechanisms are important. These processes are coupled, meaning that one process affects the initiation and progress of others. Therefore, the rock mass response to natural (such as in-situ stresses and groundwater flow) and engineering (such as excavation, fluid injection and extraction, etc.) perturbations cannot be predicted with acceptable confidence by considering each process independently. The requirement to include coupling of these processes depends on the specifics of the engineering design and performance/safety requirements. To gain a proper understanding of coupled behavior of rock masses, the boundaries of traditional fields of research, e.g. rock/soil mechanics, heat transfer, hydrogeology and geochemistry must

57


In this review, only the models and applications for THM coupling in fractured rocks are presented, using FEM and DEM approaches.

(1987, 1998) and Bai & Elsworth (2000), with focus on multiphase fluid flow and transport in deformable porous continua under isothermal or non-isothermal conditions. The volume edited by Tsang (1987) is more focused on fractured rocks, especially for nuclear waste disposal applications. Extension to poroplasticity and rock fractures are reported in Pariseau (1999) and Selvadurai and Nguyen (1999), respectively, and comprehensive reviews on applications for geothermal reservoir simulations are given by WillisRichards & Wallroth (1995) and Kohl et al. (1995). The physics of the coupled THM processes in continuous porous media are well understood, and the mathematical theory, numerical methods, computer implementation, code verification and applications for practical soil mechanics and reservoir engineering problems have become widely accepted. On the other hand, coupled THM phenomena in fractured rocks are more complex in almost all aspects, from processes, material models, properties, parameters, geometry, initial/boundary conditions, computer methods, down to code verifications and validations. Comprehensive studies, with both continuum and discrete approaches, were conducted in the international DECOVALEX projects for coupled THM processes in fractured rocks and buffer materials for underground radioactive waste disposal. The results were published in a series of reports (Jing et al., 1996, 1999), an edited book by Stephansson et al. (1996), and two special issues of Int. J. Rock Mech. Min. Sci. (32(5), 1995 & 38(1), 2000). Applications in other areas have also been reported, such as for reservoir simulations and non-Darcian flow (Sasaki & Morikawa, 1996, Nithiarasu et al., 2000), mechanics of soils and clays (Gawin & Schrefler, 1996, Thomas & Missoum, 1999, Masters et al., 2000) and tunneling in cold regions (Lai et al., 1998). The interactions between chemical and THM processes for geological media, however, are not well understood and less reported. Tsang (1991) summarized some fundamental THMC issues regarding applications for nuclear waste disposal. Some recent developments are reported by Zhao et al. (2000) for numerical modeling of fluid-water interaction for transport in porous media, Yang (2001) for reservoir compaction with mineral reaction, Yeh et al. (2001) for reactive flow and transport, and Sausse et al. (2001) for change of fracture surface and permeability due to fluid-rock interaction process. Consideration for chemical alteration on mechanical properties of geological materials is reported in Renard et al. (1997), Loret et al. (2002) and Hueckel (2002), respectively. The most well known code for THC coupling is perhaps the TOUGH2 code (Pruess, 1991), with wide applications in geothermal reservoir simulations and nuclear waste repository design and performance assessments.

2

The equivalent continuum approach means that the macroscopic properties of the fractured media have their corresponding supporting volumes, or representative elementary volume (REV). The governing equations and FEM formulations given below are perhaps the most basic and common for porous continua, but they vary with specific requirements for processes, properties and parameters. 2.1

The governing equations

Assuming that the porous medium is a mixture of solid phase of homogeneous, isotropic and linear elasticity (characterized by Lame’s constant µ and λ), and fluid phase (water and gas) with saturation degree S (0 ≤ S ≤ 1) and porosity φ, the primary variables are the displacement vector u (relative) temperature T and fluid pressure P, the governing equations are derived based on the basic laws of momentum, mass and energy conservations of the mixture and the individual phases (s-solid, l-liquid and g-gas), as given below. a) Linear momentum conservation equation of the mixture: ∂ Dijkl εkl − αl Pl + αg Pg + γ Ts δij ∂xj + (1 − φ)ρs + φS(ρl + ρg ) gi = 0

(1)

where Dijkl is the elasticity tensor of solid phase, εkl = (uk,l + ul,k )/2 the solid strain tensor, gi the acceleration vector by gravity, απ the Biot parameter for phase π(π = s, g, l) of density ρπ , γ = (2µ + 3λ)β the thermo-elastic constant, and β the linear thermal expansion coefficient, respectively. b) Gas (dry air) mass conservation equation: φ

∂ ∂ (1 − S)ρga + (1 − S)ρga (∇ · u) ∂t ∂t ∂ d + =0 ρg vg + ρg vˆ gw ∂xi

(2)

where ρga is the mass concentration of dry air in the gas phase, vg is the velocity vector of the gas phase, d and vˆ gw is the average diffusion velocity of the dry air species, respectively.

58


FEM SOLUTION OF COUPLE THM PROCESSES IN FRACTURED ROCKS – THE EQUIVALENT CONTINUUM APPROACH

c) Water species (liquid and vapor) mass conservation equation: φ

∂S ∂ [(1 − S)ρgw ] + φl ∂t ∂t

∂ + [(1 − S)ρgw + Sρl ] (∇ · u) ∂t d + ∇ · ρgw vg + ρg vˆ gw − ρl vl = 0

where dots over primary variables indicates their rate of change with time. The explicit expressions of the integrals for the coefficient matrices in (5) are given in Lewis and Schrefler (1987, 1998), Schrefler (2001), and Schrefler et al. (1997), with small variations. The equation can be symbolically written as ˙ + CX = F X

(3)

(6)

Then a standard staggered scheme for temporal integration with time-marching leads to

where ρgw is the mass concentration of water vapour in the gas phase, and vl is the velocity of the liquid water phase, respectively. d) Energy conservation (enthalpy balance):

(I + tC(k+1) )X(k+1) = tF(k+1) + X(k) = (x based on mid-point rule x˙ an arbitrary scalar variable x. (k+1)

∂T + (ρw Cpw vl + ρgw Cpg vg ) · ∇T ∂t ∂S ∂ −∇ · (KmT ∇T ) = hvap φρl + Sρl (∇ · u) ∂t ∂t ∂εkk −∇ · (ρl vl ) − T0 γ (4) ∂t

k+1

(7)

− x )/t for (k)

(ρCp )m

2.3 An example of application-Kamaishi in-situ experiments One of the problems studied in the DECOVALEX II project is the numerical simulation of an in-situ THM experiment carried out in Kamaishi Mine, Japan, where field scale tests of a heating-and-cooling period of more than one year was conducted to verify numerical modeling techniques for coupled THM analysis. Measured results of temperature, water content, stress, strain and displacements at numerous monitoring points were used for prediction and calibration of FEM models and material parameters. This work is continued in DECOVALEX III project for better understanding of the physics of the rock-buffer system, based on a simplified axi-symmetric psuedo-2D model as shown in Figure 1. Temperature, water contents, radial strains and radial stresses are calculated at four points at positions r = 0.52 m, 0.685 m, 0.85 m and 1.45 m, respectively, where values of these parameters were measured during the experiments. Four FEM codes were applied to investigate the problem: ROCMAS (KTH, Sweden), FRACON (CSNC, Canada), THAMES (JNC, Japan), and CASTEM 2000

where (ρCp )m = (1 − φ)ρs Cps + φρl Cpl + φρg Cpg is the composite heat capacity of the medium, Cpπ , (π = s, g, l) are the heat capacity of the phase π, KmT = (1−φ)KsT +φKlT +φKgT is the effective thermal conductivity of the medium, and hvap is the enthalpy of vaporization per unit mass. 2.2

FEM formulation and solution

Applying standard Galerkin FEM spatial discretization approach leads to the following set of matrix equations:  Kuu u + Cuw Pw + Cug Pg + CuT T = Fu    C u˙ + H P + P P˙ + C P˙ + C T˙ = F wu ww w ww w wg g wT w  Cgu u˙ + Cgw P˙ w + Hgg Pg + Pgg P˙ g + CwT T˙ = Fg    CTu u˙ + CTw P˙ w + CTG P˙ g + PTT T˙ + HTT T = FT

(5)

Test cavern Heater

Steel bars

Steel Bentonite

r = 0.47 m Concrete

r=0

r = 0.85 m

Rock

r = 1.0 m

r = 0.52 m

1-D Axi-symmetric model Bentonite

Figure 1.

Heater

Geometry of the simplified axisymmetric model-BMT1A-DECOVALEX III project (Jing and Nguyen, 2001).

59


(CEA, France). The governing equations and constitutive models used in these codes are similar but different to that presented in section 2, especially regarding thermo-hydraulic behavior of the bentonite used as buffer for the experiments. The details of the

Temperature at Point 1

Temperature (degree)

120

Water content (%)

Measured JNC KTH CNSC CEA INERIS

100 80 60 40 20 0

0

50

100

150

200 250 Time (days)

Water content at Point 1

18 16 14 12 10 8 6 4 2 0 0

Figure 2.

background, code formulations, initial/boundary conditions and material properties can be seen in Jing and Nguyen (2001). The results at Point 1 are shown in Figures 2 and 3. It illustrates that the FEM codes applied can predict very accurately the temperature

50

100

150

200 250 Time (days)

300

350

Measured KTH CEA

300

400

450

JNC CSNC INERIS

350

400

450

Calculated and measured temperature and water content at Point 1 (r = 0.52 m) (Jing and Nguyen, 2001).

Radial stress at Point 1 (tensile stress negative) DDA

1000

Stress (KPa)

800

BBC

JNC

KTH

CNSC

CEA

600 400 200 0

Figure 3.

0

50

100

150

200 250 300 Time (days)

350

400

Calculated and measured radial stress at Point 1 (r = 0.52 m) (Jing and Nguyen, 2001).

60


450

evolution, in general. The models can also predict reasonably well about fluid flow and saturation evolutions. The prediction to stress behavior, on the other hand, is much less reliable, due mainly to the unknown effects of the fractures and rock-bentonite interfaces presented in the test area and the changes the fractures caused in the initial hydro-mechanical conditions.

3

each “domain” (fracture sections for flow calculations determined by contact locations) is given by

t V P = P 0 + Kf Q − Kf (9a) V Vm V = V − V 0 ,

The most representative discrete numerical method for coupled THM processes in fractured rocks is perhaps the UDEC code (ITASCA, 1993), with the assumption that fluid flow is conducted only through fractures. The THM coupling algorithm is therefore split so that in the rock matrix (blocks), only the one-way TM coupling, e.g. the impact of thermal stress increments and thermal expansion on stress and displacement of rock matrix by heat conduction, is considered, and in fractures, the effective stress (in full saturation sense) and convective heat transfer by fluid flow are considered. Therefore, the equations are partially uncoupled and can be solved separately with updated primary variables (displacement, fluid pressure and temperature) as the results of solution of independent equations for deformation/stress, flow and heat transfer, through a time-marching process. No special coupling parameters (such as Biot’s coefficients απ ) are needed as for the continuum case where coupling parameters must form an integral part of the constitutive laws. This is possible since the fractures are explicitly expressed in the model and flow is limited only in fractures. Therefore the constitutive laws are much simplified.

or in discrete form

t t ∂qxt ≈− T = − cp ρ ∂xi 2cp ρ 3 qit,a + qit,b ni S k × qit = −kit

e = e0 + un

∂T ∂xi

σij = −δij Ks βT

(11b)

(12)

where Ks is the bulk modulus of the solid matrix. Heat convection due to fluid flow along fractures can be considered (Abdaliah, et al., 1995), but partial saturation and fluid phase change have not been incorporated yet since no fluid in matrix is assumed. 3.2 HM coupling of block systems-implicit DDA approach The coupling of rock block deformation and fluid flow (through connected fracture systems) was incorporated in DDA by Kim et al. (1999) and Jing et al. (2000), respectively, with the more general block deformation and discretization compatibility considered in the latter. The fluid flow in fractures are governed by the Cubic law

ρf ge3 ∂(h − bx x) (13) q= − 12µ ∂x

(8)

where e0 is the residual hydraulic aperture of fracture of length L, and un is the normal displacement of the fracture, respectively. The fluid pressure at

61


(11a)

where the heat flux follows the Fourier’s law with kit being the thermal conductivity of the rocks and qit,a , qit,b are the heat fluxes at two nodes (a and b) along a side of triangle elements in a block. The thermal stress increment by a temperature increment T is given by

Partial THM coupling by the UDEC code-explicit approach

e3 (P/L) , 12µ

(9b)

k=1

For the explicit DEM approach such as the UDEC code, the equations of motion for blocks and fracture deformation are solved by the standard dynamic relaxation approach through the contact-deformationmotion loop scheme (Hart, 1993). The flow through connected fractures are simulated by parallel plate model (or Cubic law) based on the fracture aperture, e, with the flow-rate given by q=

V + V0 2

where P 0 is the domain pressure at previous time step, Q the sum of flow rate into the domain from all surrounding contacts, Kf the bulk modulus of the fluid, t the time step, and V 0 and V the domain volumes at previous and current time step, respectively. The heat conduction equation for rock blocks is written as

1 ∂T ∂T ∂ ∂T ∂ = kx + ky (10) ∂t cp ρ ∂x ∂x ∂y ∂y

DEM SOLUTION OF COUPLED THM PROCESSES IN FRACTURED ROCKS – THE DISCRETE APPROACH

3.1

Vm =

the pressure information. For unconfined flow field, an iterative procedure with initial pressure is used to determine the final geometric locations of the free piezometric surface (water table). The standard DDA equation for deformable block systems can be written simply as (Jing, 1998) kij dj = fj (21) where kij is the global stiffness matrix of the block system, consisting of contributions from elastic deformation and block contacts, etc., dj is the global nodal displacement vector and fj is the resultant global RHS vector, with contributions from mainly the boundary conditions. The combination of the equations (18) and (21) then leads to the coupled HM equation for DDA T d p = qˆ ij j j j (22) kij dj = fj pj

where a correction for equivalent aperture for unparallel fractures is given by (Iwai, 1976) 1/3 em 16r 2 = em (1 + r)4 F

e=

em = (ea + eb )/2,

r = ea /eb

(14) (15)

where ea and eb are the hydraulic apertures at the two ends of the wedge-shaped fracture, respectively. Assuming that there are ni fracture segments connected at intersection i, where there exists also an external resultant recharge (or discharge) rate qis . From the law of mass conservation, the sum of total inflow and outflow rates should be equal to the recharge (positive) or discharge (negative) rate, i.e. 3 ni ρf g eij pi − pj = qis 12µ Lij j=1

(16a)

or ni j=1

3 pi − pj 12µf s = q eij Lij gρf i

The solution of this equation is through a time marching process using properly selected time step t. The coupled analysis requires to perform two tasks at the end of each time step: 1. updating the conductivity matrix Tij dj according to current values of nodal displacements, by re-calculating equivalent aperture eij and length Lij of the fracture connecting i and j. intersections 2. updating the load vector fj pj according to the pressure distribution along the boundaries of blocks or boundary edges of elements.

(16b)

where pi and pj are the pressures at intersections i and j (j = 1, 2, …, ni ), eij and Lij the equivalent hydraulic aperture and length of the fracture segment between intersection i and j (j = 1, 2, …). The collection of all similar equations at all intersections (including the ones at boundaries with known values of pressure or flux) results in a simultaneous set of algebraic equation Tij pj = qˆ j (17)

The full THM coupling for DDAhas not been developed at present. However, since FEM is the basis of DDA formulation, incorporation of THM coupling using the standard FEM algorithms presented above inside each block is then much more straightforward without pausing any additional difficulty, and the heat convection along fracture due to fluid velocity can be considered using the same model as developed in Abdaliah et al. (1995). Special attention, however, is needed to matrix-fracture interaction in terms of fluid flow.

after moving the terms with known pressures into the right-hand-side of the equation. The matrix Tij is called the conductivity matrix, with its elements defined by ni 3 eij Tii = Lij j=1

(18)

3 e Tij = − ( Lijij) , i = j, but are adjacent Tij = 0, i = j, and not adjacent i 12µ s (eik )3 qj + pˆ k ρf g Lik

3.3 An example of application – derivation of equivalent properties of fractured rocks using DEM approach

(19)

When FEM models are used for simulating THM behavior of fractured rocks, equivalent deformability and permeability of the rock masses often need to be established with their supporting REVs, especially for large scale practical problems. Closed-form solutions exist rarely except for problems with regular fracture patterns and numerical solution must be used for general cases. Because of the fact the effect of fracture systems is the central issue of homogenisation/upscaling

n

qˆ j =

(20)

k=1

where pˆ k (k = 1, 2, …, ni ) is the known pressure at intersection k adjacent to intersection i. Solution of equation (17) will lead to values of pressures at all intersections, and the rest of unknowns (piezometric heads and flow velocity, etc.) can be obtained with

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process for deriving equivalent properties, discrete models are more natural choices for such works. The example given below contains some results from the ongoing DECOVALEX III project for deriving equivalent hydro-mechanical properties of three geological formations (Formations 1 and 2 and a fault zone) (Fig. 4) of fracture rocks using UDEC code, with data from Sellafield, UK, for fracture statistics and hydro-mechanical properties. The derived equivalent properties will then be used by the FEM code ROCMAS to investigate the impact of THM processes on transport of nuclides from a hypothetic repository to

the sea or ground surface (Fig. 4). The details of the background, material properties, fracture geometry statistics, and their treatment and results can be seen in Mas-Ivars et al. (1999), Min et al. (1999) and Min et al. (2002). In this section we only report the approach of the homogenization and results of hydraulic permeability for Formation 1. Figure 5 shows the stochastic generation of fracture system realizations for the problem, with increasing domain side lengths from 1 m to 15 m, and 10 realizations for each size according to the fracture statistics. The boundary conditions are that two constant pressures P1

Recharge 100 m asl Sea Vertical fault zone Detailed model area

500 m

Formation 2 50 m

100 m 1 km

100m Formation 1

10 m 50 m 100 m 5m

20 m Repository block

Not to Scale!

5 km

Figure 4. The global model for investigating impact of THM processes on performance assessment of a hypothetical nuclear waste repository.

Figure 5.

Stochastic generations of fracture system realisations for Formation 1.

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permeability,kxx (m2)

3.0E-13 1 2 3 4 5 6 7 8 9 10

2.5E-13 2.0E-13 1.5E-13 1.0E-13 5.0E-14 0.0E+00 0

1

2

3

5

4

6

7

9

8

10

side length of square model (m) Figure 6. Variation of permeability component Kxx with side length of the test domains.

permeability (m2)

1.5E-13 1.0E-13 5.0E-14

kxy

0.0E+00

kyx -5.0E-14 -1.0E-13 -1.5E-13 0

1

2

3

4

5

6

7

8

9

10

side length of square model (m) Figure 7. Variations permeability components Kxy and Kyx with side length of the test domains.

Probability Density

side length10 m

5m

1m

0.5 m 0.25 m

0.0E+00

5.0E-14

1.0E-13

1.5E-13

2.0E-13

2.5E-13

3.0E-13

directional permeability, kyy (m2) Figure 8.

PDFs of the Kyy with increasing side length of testing domain.

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3.5E-13

4.0E-13

5.E+06 0 330

5.E+06 0 330

30

300

60

270

0.E+00

300

90

240

120 210

240 210

300

150

5.E+06

90

0.E+00

240

5.E+06 0

120

270

0.E+00

90

240

120 150

210

30

330 60

300

side length 2 m

Figure 9.

180

30

150 180

150

side length 1 m

0

330 60

210

120 210

side length 0.5 m

30

90

0.E+00

270

180

side length 0.25 m

270

60

240

120

180

5.E+06 0 330

90

0.E+00

30

300

60

270

150

5.E+06 0 330

30

60

300

270

0.E+00

90

240

120 150

210 180

180

side length 5 m

side length 10 m

√ Elliptical form of permeability (values are expressed as 1/ Kyy).

and P2 (P1 > P2) are described for the two opposing side of the test domain and the other two sides have linearly variable pressure conditions. Thereafter the model is rotated clockwise by a 30◦ interval to investigate the anisotropy of the permeability. The resultant values of permeability parameters, their statistics and their homogenization process with test domain size are shown in Figures 6–8, respectively. It should be noted that the permeability parameters (Figs. 6, 7) do not converge to single values but a band after reaching an approximate REV size. Therefore, distributions instead of single values of the permeability components need to be established for flow analysis by the large global model, using stochastic FEM approach. Figure 8 shows the probabilistic density functions (PDF) thus established for Kyy, assuming a Gaussian distribution. The evolution of the directional permeability, plotted as inverse of the square roots of Kyy in Figure 9, confirms that a permeability tensor can be determined with a supporting REV of a side length of 5 m, thus justifying applicability of continuum mechanics for fluid flow analysis. The work is continued currently for obtaining equivalent mechanical properties and relationships between the permeability tensor and stresses.

been impressive – especially in numerical methods, based on both continuum and discrete approaches. The progress is especially significant in the areas of representation of fracture systems, comprehensive constitutive models of fractures and interfaces, discrete element methods and coupled THM or THMC models. It appears that continuum and discrete model are more linked than before, especially when homogenization/upscaling processes are needed for characterization of fractured rock masses. Many well-verified FEM and DEM codes are developed and applied to practical problems where full or partial THM coupling is required, often with reliable results. Despite all the advances, our computer methods and codes can still be inadequate when facing the challenge of some practical problems, especially when adequate representation of rock fracture systems and fracture behavior are a pre-condition for successful modeling. Some of the issues of special difficulty and importance are: •

• • •

4 ADVANCES, TRENDS AND OUTSTANDING ISSUES

• •

Over the last three decades, advances in the use of computational methods in rock mechanics have

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Systematic evaluation of geological and engineering uncertainties and how to represent them in numerical models. Understanding and mathematical representation of large rock fractures (e.g. fracture zones). More realistic constitutive models of rock fractures with special attention to roughness effects. Quantification of fracture shape, size, connectivity and effect of fracture intersections for DEM models. Time effects (e.g. fracture creeping). Scale effects, and representation of rock mass properties and behavior as an equivalent continuum and

De Boer R. The thermodynamic structure and constitutive equations for fluid-saturated compressible and incompressible elastic porous solids. Int. J. Solids & Structures, 1998; 35(34–35): 4557–4573. Domenico PA, Schwartz FW. Physical and chemical hydrogeology. John Wiley & Sons. New York. 1990. Gawin D, Schrefler BA. Thermo-hydro-mechanical analysis of partially saturated materials. Engineering Computations, 1996; 13(7): 113–143. Hart RD. An introduction to distinct element modeling for rock engineering. In: Comprehensive Rock Engineering, J. A. Hudson (Ed. inchief), Vol. 2, Pergamon Press, Oxford, 1993, 245–261. Hassanizadeh M, Gray WG. General conservation equations for multiphase systems: 1. Averaging procedures. Adv. Water Res., 2 (1979), 131–144. Hassanizadeh M, Gray WG. General conservation equations for multiphase systems: 2. Mass momenta, energy and entropy equations. Adv. Water Res., 2 (1979), 191–203. Hassanizadeh M, Gray WG. General conservation equations for multiphase systems: 3. Constitutive theory for porous media flow. Adv. Water Res., 3 (1980), 25–40. Hassanizadeh M, Gray WG. Mechanics and theormodynamics of multiphase flow in porous media including interphase transport. Adv. Water Res., 13 (1990), 169–186. Hueckel T. Reactive plasticity for clays during dehydration and rehydration. Part 1: concepts and options. Int. J. Plasticity, 18 (2002), 281–312. ITASCA Consulting Group Lt. UDEC Manual. 1993. Iwai K. Fluid flow in simulated fractures. American Institute of Chemical Engineering Journal, 2 (1976), 259–263. Kim Y, Amadei B, Pan E. Modeling the effect of water, excavation sequence and rock reinforcement with discontinuous deformation analysis. Int. J. Rock Mech. Min. Sci., 1999; 36(7): 949–970. Kohl T, Evans KF, Hopkirk RJ, Rybach L. Coupled hydraulic, thermal and mechanical considerations for the simulation of hot dry rock reservoirs. Geothermics, 1995; 24(3): 345–359. Jing L. Formulations of discontinuous deformation analysis for block systems. Int. J. Engineering Geology, 49 (1998), 371–381. Jing L, Stephansson O, Tsang CF, Kautsky F. DECOVALEX – mathematical models of coupled T-H-M processes for nuclear waste repositories. Executive summary for Phases I, II and III. SKI Report 96:58. Swedish Nuclear Power Inspectorate, Stockholm, Sweden. 1996. Jing L, Stephansson O, Tsang CF, Knight LJ, Kautsky F. DECOVALEX II project, executive summary. SKI Report 99:24. Swedish Nuclear Power Inspectorate, Stockholm, Sweden. 1999. Jing L, Ma Y, Fang Z. Modeling of Fluid Flow and Solid Deformation for Fractured Rocks with discontinuous Deformation Analysis (DDA) Method. Int. J. Rock Mech. Min. Sci., 2000; 38(3): 343–356. Jing L, Nguyen ST (eds.). Technical Report of BMT1A. DECOVALEX III Project. 2001. Lai YM, Wu ZW, Zhu YL, Zhu LN. Nonlinear analysis for the coupled problem of temperature, seepage and stress fields in cold-region tunnels. Tunneling and Underground Space Technology, 1998; 13(4): 435–440. Lewis RW, Schrefler BA. The finite element method in the static and dynamic deformation and consolidation of

existence of the REV with complexity in aperture, width, size and shape behaviors. • Representation of interfaces (contact zones of different materials or system components, such as rock-reinforcements, rock-buffer, rock-soil, etc). • Numerical representation of engineering processes, such as excavation sequence, grouting and reinforcement. • Large-scale computational capacities. The numerical modeling for coupled THM and THMC processes played a very significant role in extending rock mechanics from an art of design and analysis of rock construction works based on “empirical” concepts of stress, failure and strength to a more “scientific” branch of engineering mechanics based on conservation laws, with integrated understanding and treatment of diverse information about geology, physics, construction technique, the environment and their interactions. Linking up with geo-chemical processes will further enhance the field of rock mechanics and rock engineering, with numerical modeling as the basic platform of development. Further extension to include biochemical, electrical, acoustic and magnetic processes have also started to appear in the literature and are an indication of future research directions.

ACKNOWLEDGEMENT The funding organizations of the DECOVALEX III project and EC supported the example works presented in this article.

REFERENCES Abdaliah G, Thoraval A, Sfeir A, Piguet JP. Thermal convection of fluid in fractured media. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 1995; 32(5): 481–490. Achanta S, Cushman JH, Okos MR. On multicomponent, multiphase thermomechanics with interfaces. Int. J. Engng. Sci., 32 (1994), 1717–1738. Bai M, Elsworth D. Coupled processes in subsurface deformation, flow and transport.ASCE Press. Reston, VA. 2000. Biot MA. General theory of three-dimensional consolidation. J. Appl. Phy., 12 (1941), 155–164. Biot MA. General solution of the equation of elasticity and consolidation for a porous material. J. Appl. Mech., 23 (1956), 91–96. Bowen RM. Compressible porous media models by use of theories of mixtures. Int. J. Engng. Sci., 20 (1982), 697–735. Charlez PA. Rock mechanics. Vol. 1-Theoretical fundamentals. Editions Technip. Paris. 1991. Charlez P, Keramsi D (eds.). Mechanics of porous media (Lecture notes of the Mechanics of Porous Media summer school, June 1994). Balkema, Rotterdam. 1995. Coussy O. Mechanics of porous media. John Wiley & Sons. Chichester. 1995.

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porous media. 2nd edition. John Wiley & Sons. Chichester. 1998. Lewis RW, Schrefler BA. The finite element method in the deformation and consolidation of porous media. John Wiley & Sons. Chichester. 1987. Loret B, Hueckel T, Gajo A. Chemo-mechanical coupling in saturated porous media: elastic-plastic behavior of homoionic expansive clays. Int. J. Solids and Structures. 2002 (in press). Mas-Ivas D, Min KB, Jing L. Homogenization of mechanical properties of fracture rocks by DEM modeling. In: Wang S, Fu B, Li Z (eds.), Frontiers of rock mechanics and sustainable development in 21st century (Proc. Of the 2nd Asia Rock Mechanics Symp. Sep. 11–14, 2001, Beijing, China). Balkema, Rotterdam 322–314. 2001. Masters I, Pao WKS, Lewis RW. Coupling temperature to a double-porosity model of deformable porous media. Int. J. Numer. Anal. Meth. Geomech., 49 (2000), 421–438. Min KB, Mas-Ivars D, Jing L. Numerical derivation of the equivalent hydro-mechanical properties of fractured rock masses using distinct element method. In: Rock Mechanics in the National Interest. Elsworth, Tinucci & Heasley (eds.), Swets & Zeitlinger Lisse, 1469–1476. 2001. Min KB, Jing, Stephansson O. Determination of the permeability tensor of fractured rock masses based on stochastic REV approach, ISRM regional symposium, 3rd Korea-Japan Joint symposium on rock engineering, Seoul, Korea, 2002. (in press) Morland LW. A simple constitutive theory for fluid saturated porous solids. J. Geophys. Res., 77 (1972), 890–900. Nithiarasu P, Sujatha KS, Ravindran K, Sundararajan T, Seetharamu KN. Non-Darcy natural convection in a hydrodynamically and thermally anisotropic porous medium. Comput. Methods Appl. Mech. Engng., 188 (2000), 413–430. Pariseau WG. Poroelastic-plastic consolidation. analytical solution. Int. J. Numer. Anal. Meth. Geomech., 23 (1999), 577–594. Renard F, Ortoleva P, Gratier JP. Pressure solution in sandstones: influence of clays and dependence on temperature and stress. Tectonophysics, 280 (1997), 257–266. Pruess K. TOUGH2 – A general purpose numerical simulator for multiphase fluid and heat flow. Lawrence Berkeley Laboratory Report LBL-29400, Berkeley, CA. 1991. Sahimi M. Flow and transport in porous media and fractured rock: from classical methods to modern approaches. VCH Verlagsgesellschaft mbH. Weinheim. 1995. Sasaki T, Morikawa S. Thermo-mechanical consolidation coupling analysis on jointed rock mass by the finite element method. Engineering. Computations, 1996; 13(7): 70–86.

Sausse J, Jacquot E, Fritz B, Leroy J, Lespinasse M. Evolution of crack permeability during fluidrock interaction. Example of the Brézouard granite (Vosges, France). Tectonophysics, 336 (2001), 199–214. Schrefler BA. Computer modelling in environmental geomechanics. Computers and Structures, 79 (2001), 2209–2223. Schrefler BA, Simoni L, Turska E. Standard staggered and staggered Newton schemes in thermo-hydro-mechanical problems. Compt. Methods Appl. Mech. Engng. 144 (1997), 93–109. Selvadurai APS (ed.). Mechanics of poroelastic media. Kluwer Academic Publishers. Dordrecht. 1996. Selvadurai APS, Nguyen TS. Mechanics and fluid transport in a degradable discontinuity. Engineering Geology, 53 (1999), 243–249. Stephansson O, Jing L, Tsang CF (eds.). Mathematical models for coupled thermo-hydro-mechanical processes in fractured media. Elsevier, Rotterdam. 1996. Thomas HR, Missoum H. Three-dimensional coupled heat, moisture and air transfer in a deformable unsaturated soil. Int. J. Numer. Meth. Engng., 44 (1999), 919–943. Tsang CF(ed.). Coupled processes associated with nuclear waste repositories. Academic Press Inc. 1987. Tsang CF. Coupled thermomechanical hydrochemical processes in rock fractures. Rev. of Geophys., 29 (1991), 537–551. von Terzaghi, K. Die berechnug der Durchlässigkeitsziffer des Tones aus dem Verlauf der hydrodynamischen Spannungserscheinungen. Sitzungsber. Akad. Wiss., Math.Naturwiss, Section IIa, 1923; 132(3/4): 125–138. Whitaker S. Simultaneous heat, mass and momentum transfer in porous media: a theory of drying. Academic Press, New York. 1977. Willis-Richards J, Wallroth T. Approaches to the modelling of HDR reservoirs: a review. Geothermics, 1995; 24(3): 307–332. Yang XS, 2001. A unified approach to mechanical compaction, pressure solution, mineral reactions and the temperature distribution in hydrocarbon basins. Tectonophysics, 330 (2001), 141–151. Yeh GT, Siegel MD, Li MH. Numerical modeling of coupled variably saturated fluid flow and reactive transport with fast and slow chemical reactions. J. of Contaminant Hydrology, 47 (2001), 379–390. Zhao C, Hobbs BE, Mühlhaus HB, Ord A. Numercal modeling of double diffusion driven reactive flow transport in deformable fluid-saturated porous media with particular consideration of temperature-dependent chemical reaction rates. Engineering Computations, 2000; 17(4): 367–385.

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Grand challenge of discontinuous deformation analysis A. Munjiza Department of Engineering, Queen Mary, University of London, UK

J.P. Latham Department of Earth Sciences and Engineering, Imperial College of Science Technology and Medicine, London, UK

ABSTRACT: It is now accepted that many processes in nature and industry, and problems of science and engineering, cannot be modeled using the assumption of continuum properties. In recent decades a number of formulations have been developed, based on the a priori assumption that these processes are best modeled by considering interacting discontinua. These have been accompanied with approximate computational methods such as the discrete element method (DEM) combined discrete-finite element methods (FEM/DEM) and discontinuous deformation analysis (DDA). In practice, the applicability and success of these methods, up until recently, has been greatly limited by the CPU power available. In this article the feasibility of large-scale computer simulations of discontinuum problems is investigated in the light of our recent algorithmic developments and ever-decreasing hardware constraints. We also report on the implications of our latest algorithm, which in principle enables us to model a whole range of particulate processes using real particles. A virtual game of 3D snooker (or pool) in which we can introduce a million arbitrary-shaped snooker “balls” of all sizes and as many simultaneous players as one likes is now within our reach. 1

INTRODUCTION

size of the sample. From the constitutive law and the conservation laws (for energy, momentum, mass, etc.) a set of governing equations defining the physical problem is derived. Solution of these equations was first sought in analytical form. Later, approximate numerical techniques were developed and in the last few decades, computational techniques have been applied to almost all conceivable physical systems for which continuum models have the slightest chance of success. Continuum models can only be an approximation of the real physical process or problem, and thus are only as valid as the underlying assumptions on which they are based. The most important assumption necessary for the validity of a continuum model is the assumption that the scale of the problem or the scale of the part of the problem of practical significance is much larger than some characteristic length defined by the microstructure of the material. For elastic analysis of isotropic and homogeneous solids this characteristic length is a few orders of magnitude greater than the size of molecules making up the material. For example, practical problems invoking the theory of elasticity are well represented by the continuum model. On the other hand, there exists a completely different class of problems, an example of which is the

Early in 2001, an algorithmic breakthrough was achieved which now sits on top of a decade of computational developments, allowing 3D transient dynamic modelling of particulate systems of realshaped particles1 . In short, we now have the essential algorithms to be able to model a game of 3D snooker (or pool) with a million arbitrary-shaped real particles as snooker “balls” of any size and as many simultaneous players as one likes. Whereas with spheres it is relatively simple to establish whether particles are in contact from the position of their centres and their radii, and to establish forces and trajectory paths associated with collisions, considerable algorithmic sophistication is required for collisions of rock fragment-shaped particles. 2

CONTINUA VERSUS DISCONTINUA

Formulations for solutions to problems of continuous media (for simplicity, called continuum models) are based on what are termed constitutive laws. These are that the physical properties of matter are described from the premise that the underlying microstructure of the physical matter is the same irrespective of the

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such as DEM, FEM/DEM, DDA, etc. A diverse research community is now spawning leading-edge particulate modelling tools based on these methods. The teams include collaborations with physicists, powder technologists2 , geotechnical engineers3 , mineral processing engineers4 , and petroleum reservoir engineers5 . A broad cross-section of particulate modelling applications in powder technology is given by Thornton6 .

process of filling a cube-shaped container with pebbles. One might pose the question: how many pebbles can fit into a single container? (The detailed solution will of course need to know whether the filling is by pouring, by dynamic dumping or involving vibrocompaction). In the text that follows, this problem is referred to as the container problem. The point to note about the container problem is that the size of the container (edge length) might be, say, only five times larger than the size (diameter) of a single pebble. It is therefore evident that the scale of the problem is similar to the characteristic length of the microstructure of the material. Indeed, this is also the case if the container size is 500 times the pebble size because details of the granular micromechanics between pebbles would reveal localised shear behaviour and flow cells extending perhaps over hundreds of particle diameters. 3

4 THE CHALLENGE AND THE OPPORTUNITIES In here we now introduce our approach and examine the opportunities for many such teams facing the challenge of how to model systems with many millions of particles, while at the same time accommodating particle characteristics such as realistic shapes. Up to the present time, numerical models of particulate systems that track motions have mainly used discs and superquadrics in 2D or spheres, bonded spheres and ellipsoids in 3D. These discrete approaches have struggled to approximate real behaviour for the more angular particulate systems normally encountered in nature or during mineral processing. (There will be notable examples of particulate materials such as flint beach shingle and other very rounded granular media for which reported methods have proven effective.) Simulations like those in Figure 1 are now feasible only because of a series of recent algorithmic breakthroughs which include linear search algorithms7,8 , potential contact force interaction strategies9,10,11 , discretised contact solution strategies9 , and crucially, 3D irregular body transient motion solutions1 . It is emphasised that the animation shown in Figure 1 is meant to

MODELLING DISCONTINUA

Particulate problems requiring microstructural examination similar to the container problem are of practical importance in many branches of science and engineering. The common feature of these process phenomena is that the representative volume of the physical matter whose behaviour is being modelled is either much larger or of a similar order of magnitude to the physical problem to be analysed. Thus, the necessary conditions for successful modelling using a continuum-based governing set of equations are not satisfied. Researchers and engineers have long recognised this fact. Historically the solution to problems characterised by the container problem was sought through either experimental investigations or phenomenological analytical approximations. In recent decades, a set of modelling approaches based on the a priori assumption of discontinua, has also been developed. The common feature of all these discontinua-based approaches is that no constitutive law is formulated. Instead, they take into account the physical characteristics of the building blocks of the material which may include for instance: shape of individual particles, interaction among particles, friction among particles, adhesion between particles, transient dynamics of individual particles, deformability of individual particles, etc. Sets of governing equations from such discontinuabased formulations are then solved. For very simple microstructures and shapes a set of analytical solutions is readily available (for instance some problems of packing of spheres). Unfortunately, these are only special cases and in general numerical (spatial and temporal) discretisations of the physical problem are necessary. These discretisations usually necessitate use of digital computers. The set of computational methods developed for this purpose include methods

Figure 1. Numerical simulation of initial motion, collision, bouncing off container walls, pirouetting, rocking and final rest state of real-shaped particles (3D laser-scanned pieces of rock aggregate) propelled into a container.

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era. This is not the case with the problems of discontinua because in most cases the scale of the problem was computationally prohibitive in terms of the CPU time. This is best illustrated by comparing a linear elastic analysis of a 3D elastic block of edge size measuring 1 m. The total number of finite elements needed ranges from 1000 to 30000 elements. In contrast discontinua-based analysis of a similar size container problem (container measuring 1 m filled with particles with an average size of 1 mm) would involve over one billion separate particles. In terms of CPU time many of the problems of this type are therefore in the realm of what are often described by numerical modellers as grand-challenge problems. There are two necessary conditions to make grand-challenge problems solvable in practical timescales: (1) algorithmic breakthroughs must be made and (2) large CPU power at low cost must be available. Before leaving algorithms, it is worth dwelling on the contact search problem. Up until recently the most CPU efficient search algorithms required total CPU time proportional to N log2 N , where N is the total number of elements comprising the grid. The new generation of no binary search, or “NBS-based” search algorithms7,8 requires CPU time proportional to N (i.e. are as fast as is theoretically possible). These contact search algorithms together with those described earlier for dealing with complex shapes combined with limited hardware of today has enabled us to address real problems involving granular material. This is demonstrated in Figure 2, where transient dynamic simulation of a container problem comprising over 3 million degrees of freedom is shown. Again the transient motion and final state of rest are as a result of motion and interaction of all individual particles making up the system. Cube packing experiments were also performed and these provided an ideal

illustrate the combined potential of this latest generation of algorithmic solutions. To deal with the irregular shapes, the discretised contact solution strategy we use is based on discretisation of the complex surface shape of irregular bodies into sets of simplex contact geometries. In the context of the combined finite-discrete element method, these discretisations are also used for deformability analysis, while in the case of rigid particles it is enough to discretise the surface layer only. It is often assumed implicitly that such discretisations increase CPU time. In fact the opposite is true. The number of contacts, and thus solutions per time step, is a function of the geometry of the discrete system and is not a function of contact discretisation, which is in any case meant only to simplify contact geometry. The actual situation is that discretisation speeds up contact procedures and this is the reason why systems comprising millions of degrees of freedom can be handled effectively1,7 . Algorithmic solutions for dynamically interacting discontinua1 , even ones incorporating fracturing elements, e.g. by combined discrete-finite elements12,13 have been developed. As discussed at ICADD-41,8,14,15 in June of 2001,the problems of how to detect16,7,8 and how to represent the contact interactions17,18,9 between sufficient numbers of bodies and how to include the effects of realisticshaped particles1 and display results in a manageable CPU time3,15 remain outstanding challenges for future research. The common feature of both continuum-based and discontinuum-based computational methods is the spatial grid. The size of the spatial grid for continuumbased problems (say linear elasticity) is in essence governed by the geometry of the problem. For instance, an elastic beam of 5 m span is analysed using the same number of elements as a similar beam of 0.5 m span. In contrast, the size of the spatial grid for discontinuumbased problems is governed by the microstructure of the material. For the container problem, the size of the spatial grid is largely defined by the total number of individual pebbles and their shapes. This means that if the size of the container in the container problem is increased by 10 times, the size of the grid increases by at least 1000 times. Grid size is the most important and the most challenging aspect of discontinua representation and modelling. Problems of continua do not in general involve length scale as the defining factor in representation of the physical matter and usually result in spatial grids comprising a relatively small number of elements. Such coarse grids still provide approximations of sufficient accuracy. The problems of discontinua, however, do involve this length scale and in most cases result in grids comprising millions or even billions of elements. Computational solutions for problems of continua have been developed since the dawn of the computing

Figure 2. A 3-million degrees of freedom dynamic deposition simulation showing a collapsing “cloud” of variously coloured particles and their final rest state. Animation and analysis of key diagnostics indicate the pulsating or bouncing nature of the dumping and settling process as the bulk packing density oscillates before stabilising.

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required depends on the information to be collected. In general there is a limit to what information one can collect and what one cannot. For instance it is not feasible to measure interaction forces between all the particles. In short, there is a limit to what can be observed and how much data can be recorded. In this light it would be beneficial to be able to perform a numerical experiment instead. With numerical or “virtual” experimentation, any key diagnostic information required is readily available and new perhaps simplified theories can then be developed to more efficiently approximate the physics involved.

benchmark problem from which initial confidence in the modelling technology was established. In 1998 the Royal Society held a discussion meeting on the mechanics of granular materials in engineering and earth sciences19 . Researchers with interests as diverse as pyroclastic flows on Montserrat to flow in cement and grain storage silos were assembled. One objective was to bring researchers a step closer to developing strategies that can deal with the dual solidlike and fluid-like rheologies of granular systems. We were struck by a conclusion in the summing up, that much of the particulate behaviour is related to effects occurring as a result of grain shape and that more work was required in this area if scientists were to accommodate the effects of real particle shapes within new computational tools for granular media. Scientists and engineers can now look towards building on these powerful algorithmic tools in the search for a deeper understanding of the vexing and often unpredictable behaviour of granular systems. Hitherto has been difficult because of the limitations in our ability to understand and model the complex graingrain interactions. The numerical experimentation that now seems possible will be invaluable in advancing subjects such as packing, mixing and segregation, avalanching, and spontaneous stratification20 . These subjects are considered vital for precision mixing in the pharmaceutical industry, for explaining particulate behaviour under mechanised sieving in the minerals industry and a host of natural phenomena such as sedimentation and avalanches including the enigmatic long-runout rockslides21 . Many researchers are working on simple theoretical models of these processes that would be greatly enhanced by an ability to model realistic systems and to test the validity of the more simple models. Improving an understanding of particulate behaviour and especially concepts in particulate packing has in fact triggered breakthroughs in many disciplines e.g. aeronautics, agriculture, biology, ceramics, chemical engineering, chemistry, civil engineering, composites, electrical engineering, foods, geology, mechanical engineering, medicine, metallurgy, nuclear, paint technology, pharmaceuticals, physics, polymers22 . The international research community is witnessing an explosion in the development of computer modelling for particulate systems designed to tackle their complexity. The activity is a consequence of developments in algorithmic solutions for handling systems of discrete elements and increasingly affordable CPU/RAM power, which in turn have resulted in an increasing range of science and engineering problems that appear amenable to discontinuum numerical modelling approaches (DEM, FEM/DEM, DDA). At present, problems exemplified by the container problem mentioned above can be addressed most readily by experimental means. The instrumentation

5

For many engineering and scientific applications involving discontinua, the problems at present of going down the numerical route are the massive CPU times required. For instance, only 1 CPU second per 1 mm particle in the one-metre cube container problem translates into about 32 CPU years. When these massive CPU times are translated into cost, the sums obtained are unreasonable. The encouraging fact is that the cost of CPU is going down and performance is going up. With future computer technologies and future computer architectures one can easily estimate that problems of the above scale may require very short CPU times on a 2020 generation PC. On a “1 kg ultimate laptop”23 the above mentioned container problem would only take 32 · 365 · 24 · 3600 · 10−40 = 10−31 seconds. If it is possible to extrapolate Moore’s law into the future, it would take 250 years for such a laptop to become available. It may just become available in a shorter time than it would take to solve the above problem on a present day PC set running today. The point is that, although at present some of these problems appear to be of a grand-challenge type, it is evident that in the near future some of these problems will become relatively small-scale in terms of both CPU time and computational cost.

ACKNOWLEDGEMENTS We gratefully acknowledge the Engineering and Physical Sciences Research Council of Great Britain, for their support under GR/L93454. REFERENCES [1] A. Munjiza, J.P. Latham, N.W.M. John. Transient motion of irregular 3D discrete elements. Proceedings of the Fourth International Conference on Analysis of Discontinuous Deformation (ICADD-4), 23–33,

72


CONCLUSION

[2] [3]

[4] [5] [6] [7] [8] [9] [10]

[11]

Ed. Nenad Bicanic, Glasgow, Scotland UK, 6–8 June 2001 (Full paper: A. Munjiza, J.P. Latham, N.W.M. John. Transient motion of irregular 3D discrete elements. Int. J. Num. M. Eng. submitted June 2001.) Z.P. Zhang, A.B. Yu, et al. A simulation study of the effects of dynamic variables on the packing of spheres. Powder Tech. 116 (2001) 23–32. P.A. Cundall. A discontinuous future for numerical modelling in geomechanics? Proceedings of the Institution of Civil Engineers, Geotechnical Engineering. 149 (2001) 41–47. P.W. Cleary. DEM simulation of industrial particulate flows: dragline excavators, mixing and centrifugal mills. Powder Tech. 109 (2000) 83–112. P.-E. Oren, S. Bakke, O.J. Arntzen. Extending predictive capabilities to network models. SPE J. 3, (1998) 324–336. C. Thornton (Editor) Special Issue: Numerical simulations of discrete particle systems, Powder Tech. 109(1–3) (2000) 1–298. A. Munjiza, K.R.F. Andrews. NBS contact detection algorithm for bodies of similar size. Int. J. Num. M. Eng. 43 (1998) 131–149. P. Perkins, J.R. Williams. Cgrid: Neighbor searching for many body simulation. (ICADD-4, p427–438, see ref 1) A. Munjiza, K.R.F. Andrews. Discretised penalty function method in combined FEM/DEM analysis. Int. J. Num. M. Eng. 49 (2000) 1495–1520. A. Munjiza, D.R.J. Owen, N. Bicanic. A combined finite-discrete element method in transient dynamics of fracturing solids. Int. J. Engineering Computations, 12 (1995) 145–174. A. Munjiza, D.R.J. Owen, A.J.L. Crook. Energy and momentum preserving contact algorithm for general 2D and 3D contact problems. Proc. Third Intnl. Conf.

[12] [13]

[14] [15] [16] [17]

[18] [19] [20] [21] [22] [23]

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on Computational Plasticity: Fundamentals and Applications, 829–841, Barcelona, April (1995). A. Munjiza, et al. Combined single and smeared crack model in combined FEM/DEM. Int. J. Num. M. Eng. 44(1999) 41–57. A. Munjiza. Fracture, fragmentation and rock blasting models in the combined finite-discrete element method, Chapter in Fracture of Rock, Computational Mechanics Publications 1999. G.-H. Shi. Three-dimensional discontinuum deformation analysis. (ICADD-4, p1–21, see ref 1) D.R.J. Owen, Y.T. Feng. Parallel processing strategies for particulates and multi-fracturing solids. (ICADD-4, p299–313, see ref 1) Z.H. Zong, L. Nilsson. A contact search algorithm for general 3-D contact-impact problems, Comp. Struct. 34 (1990) 327–335. G.-H. Shi. Discontinuum deformation analysis – a new numerical method for the statics and dynamics of block systems, PhD Thesis, Dept. Civil Engng., Univ. of California, Berkeley (1988). J.R. Williams, G. Mustoe. Proc. 2nd U.S. Conference on Discrete Element Methods. MIT, MA (1993). D. Muir Wood, G.S. Boulton J.M. Rotter. Mechanics of granular materials in engineering and earth sciences. Phil. Trans. R. Soc. Lond. A 356 (1998) 2451–2452. H.A. Makse, S. Havlin, P.R. King, H.E. Stanley. Spontaneous stratification in granular mixtures. Nature 386 (1997) 379–382. Nature Editorial, News & Views editorial by J. Fineberg, Nature 386 (1997) 323. R.M. German, Particle Packing Characteristics, Metal Powder Industries Federation, Princetown, NJ (1989). S. Lloyd. Ultimate physical limits to computation, Nature 406 (2000) 1047–1054.


High-order manifold method with simplex integration Ming Lu SINTEF Civil and Environmental Engineering, Trondheim, Norway

ABSTRACT: The original MM program developed by Genhua Shi uses a constant cover function, leading to constant strains and stresses for the triangle elements. Experience indicates such a cover function is not adequate for applications in which accurate computation of stresses is required. It is particularly the case for simulating hydraulic fracturing tests. This paper presents formulation of a complete N-order manifold method and verification examples. Also given in the paper is a potential problem associated with high-order manifold method as well as a possible solution.

1

INTRODUCTION

identical, particularly for high order MM. For instance, the basic unknowns of FEM are nodal displacements, whilst for the high order MM they are coefficients of the polynomial cover function. The nature of the right-hand “loading” vector is also different. In MM it is no longer “nodal force”, rather than the “loading terms” corresponding to the relevant displacement coefficients. Consequently, the way of handling initial stress, boundary conditions and various types of loading is different. Various terms have been used in publications for describing geometry in MM, such as mathematical cover, physical cover, manifold element, node, vertex and etc. In this paper four terms are used and defined as follows:

The original numerical manifold method (MM) invented by Shi (1997) and corresponding computer programs use a constant cover function, leading to constant strains and stresses for the triangle elements. The programs give satisfactory results for problems with crack dominating failure mode. However, experience indicates such a constant cover function is not adequate for applications in which accurate computation of stresses is required. It is particularly the case for simulating hydraulic fracturing, in which very accurate evaluation of displacements and stresses at the crack tip is absolutely essential. Chen et al (1998) proposed a high-order manifold method. In order to develop a MM program for simulating hydraulic fracturing tests, following the ideas of Shi and Chen et al, numerical manifold formulation with a complete N-order cover function is worked out at SINTEF Civil and Environmental Engineering. A computer program in Fortran is written to implement the computation and the code is verified by means of comparing to the closed form solutions. During the development, however, some problems associated with small blocks at the model boundaries are also revealed. Attempts are made to solve the problem and it is found that at least one of the alternative solutions may work in certain conditions. This paper presents the formulae of the complete N-order manifold method and a verification example. Also given in the paper is the potential problem associated with the high-order manifold method as well as a possible solution. The governing equations of the MM are similar to those of FEM. However, the basic formulations are not

•

MM element: Basic geometry generated by the code (triangle); occupied with material either fully or partly; may be cut by joint(s) such containing more than one element blocks; similar to FEM. • Element block: A part of an element fully occupied by material and cut by joints; may also occupy an entire element if the element is not cut by any joints; may be triangle or polygon. • Node: Connection points between elements; same as FEM; a geometrical point may be associated with more than one node if one of the elements connecting to the node is cut by any joint; also called physical cover in other publications. • Vertex: Apex of element block. Figure 1 illustrates the definitions. The N-order MM formulation presented in this paper is based on triangle elements. It should be mentioned that numerical integration is commonly used for FEM, whilst simplex integration

75


Figure 2. Illustration of weighting function for triangle element (after Chen et al, 1998).

 a1 = x2 y3 − x3 y2 a2 = x3 y1 − x1 y3 a = x y − x y 2 1 3 1 2  b1 = y2 − y3 b2 = y3 − y1 b = y − y 3 1 2  c1 = x3 − x2 c2 = x1 − x3 c = x − x 3 2 1

Figure 1. Illustration of MM element, node, element block and vertex.

is used for Shi’s original MM program. It is true that the simplex integration gives analytical solution and is more accurate than the numerical integration. All formulae presented in this paper are based on the simplex integration. The difficulty involved in high-order MM is greatly enhanced by adopting simplex integration since all integrands have to be expressed explicitly in the polynomial form.

2 2.1

Element displacement

 u = ui1 + ui2 x + ui3 y + ui4 x2 + ui5 xy   i + ui6 y2 + · · · + uim yN v = vi1 + vi2 x + vi3 y + vi4 x2 + vi5 xy   i + vi6 y2 + · · · + vim yN

In each MM element, the displacements of a point (x, y) are computed from the weighting function wi and the displacements of three nodes of the element ui and vi :   3    w (x, y)u (x, y)     i i   u(x, y) (1) = i=1 3 v(x, y)        wi (x, y)vi (x, y)   

m=

Weighting functions are smaller than or equal to unity and their summation is always equal to unity. For the triangle elements the weight function at the nodal points is 1 and it is zero at the outer edges, with linear variation, see Figure 2. Having such defined, the weight functions for the triangle elements become      a1 b1 c1    w1  1  1   w2 = (2)  a2 b2 c2  x   w  y   a b c  3 3 3 3

(4)

(N + 1)(N + 2) 2

(5)

Coefficients of the cover function, uij and vij , are the basic unknowns. Eqn (1) can be rewritten as: U = TD u(x, y) U= v(x, y) T D = D1 D2 D3   ui1       vi1       ui1         ui2      ui2  Di = vi2 = .. ..           .  .      u   im    uim   vim

(3.1)

76


(3.4)

m is the number of terms of the cover function which is related to the order of the cover function N as follows:

i=1

" y1 "" " y2 " " y3 "

(3.3)

xi , yi are coordinates of the three nodes of the element. For a complete N-order polynomial cover function the nodal displacement can be expressed as:

BASIC FORMULATIONS

where " "1 x1 " " = "1 x2 " "1 x3

(3.2)

(6) (7) (8)

(9)

uik =

uik vik

Then the element stiffness matrix for the elastic material can be computed as follows:

(10)

T = T1 T2 T3

Kij = ∫ BTi EBj dA

(11)

1 0 x 0 y 0 x 0 xy 0 y 0 · · · y 0 Ti = wi 0 1 0 x 0 y 0 x2 0 xy 0 y2 · · · 0 yN 2

2

N

E is the elastic matrix, the same as FEM. The global displacement vector can be expressed as: T U = U1 U2 U3 · · · Un (19)

(12) 2.2

T Ui = ui1 ui2 ui3 · · · uim

Strain matrix

∂ ∂x        ε x   εy = 0    γ   xy ∂ ∂y

 0    ∂   u(x, y) ∂y   v(x, y)  ∂  ∂x 



∂  ∂x   0 =   ∂ ∂y

0

  ∂   TD = BD ∂y   ∂ 

(21) Its location in the global stiffness matrix is: Row – (i − 1) m + j; Column – (k − 1) m + l. Here index i and k are in the global system. For the explicit expressions of the integrand of elements of [Kij,kl ] and their coefficients, see Lu (2001).

Strain matrices B can be written as: B = B1 B2 B3

(14)

Bi = Bi1 Bi2 Bi3 · · · Bim

(15)

2.4 Initial stress matrix In MM the equivalent force of stress at the end of a time step is transferred to the next time step. This includes the first time step in which the initial stress is the in-situ stress. The stress is computed as follows:



σ = Eε = ENL = SL (16)

  N11 N12 N13 · · · N1m N = N21 N22 N23 · · · N2m  N31 N32 N33 · · · N3m

Element stiffness matrix

 S11 S12 S13 · · · S1m S = S21 S22 S23 · · · S2m  S31 S32 S33 · · · S3m

(17)

77


(24)



Similar to the FEM the global governing equilibrium equation is KU = F

(22)

where stress vector σ , strain distribution matrix N, stress distribution matrix S and coordinate vector L are defined as: T σ = σx σy τxy (23)

Exponents mj and nj are evaluated from the location index j. For explicit expressions of general terms see Lu (2001). 2.3

(13)

∂x

 ∂(xmj ynj wi ) 0   ∂x     mj nj ∂(x y wi )     0 Bij =   ∂y      ∂(xmj ynj w ) ∂(xmj ynj w )  i i ∂y ∂x

(20)

n is the total number of nodes and m is the total number of DOFs of each node in one dimension. So the total number of DOFs for a 2-D problem is 2nm. The basic sub-matrix [Kij,kl ], which is the stiffness of jth DOF of ith node to lth DOF of kth node, is a 2 × 2 matrix and computed as follows: # [Kij, kl ] = BTij EBkl d #  # kij,kl (1, 1) dx dy kij,kl (1, 2) dx dy    # =  # kij,kl (2, 1) dx dy kij,kl (2, 2) dx dy

The strain vector can be expressed as: 

(18)

(25)

T L = 1 x y · · · yN

2.5 Point loading matrix

(26)

Different from FEM, a point load can be applied on any location of the material in MM, not necessarily $ Px on a node. When a point load is acting on the Py location (x, y), it’s contribution to the load vector is: P fp = T T x (34) Py

After the displacement coefficients D have been computed matrix N can be computed from Eqn (22), and then matrix S is obtained. Derivation of matrix N is tedious and interested readers are referred to Lu (2001). The stress-equivalent load vector fs is computed as follows: # (27) fS = BT σ dA

where TT is computed from Eqn (12). On the element level the force vector is corresponding to the displacement vector defined in Eqns (9) and (10) T fS = fS1 fS2 fS3 (28)   fsxi1      fsyi1           fsi1   fsx  i2        fsi2  fSi = fsyi2 = .. .       ..    .        fs   im   fsx   im   fsyim Substituting Eqn (22) into (27) leads to # # fS = BT SL dA = GL dA

2.6 Body force matrix

$

gx acts on an element gy block, the equivalent loading vector fg is # g TT x dA (35) fg = gy A

When constant body force

(29)

Since T(1,2) = T(2,1) = 0, Eqn (35) can also be written as: # f Tij (1, 1) gx fg,ij = gij, x = dA (36) Tij (2, 2) gy fgij, y A Explicit expressions are needed that are given in Lu (2001).

(30)

where G = BT S

2.7 (31)

The contributions of the inertia force to the global stiffness matrix and load vector are given below: # 2ρ TT Tkl dA (37) [Kij,kl ] = t 2 A ij

Elements of S are constants, but elements of BT are functions of (xm yn ). The final form of matrix G can be written as Eqn (32). The explicit expressions of Gij,k can be found in Lu (2001). For the first time step, the in-situ stress is used, which is assumed to be linearly distributed, as given in Eqn (33).   G11,1 G11,2 · · · G11,m  G12,1 G12,2 · · · G12,m     ··· ··· ··· ···   G G · · · G 1m,m   1m,1 1m,2  G  21,1 G21,2 · · · G21,m   G22,1 G22,2 · · · G22,m   G= (32)  ··· ··· ··· ···    G2m,m G2m,2 · · · G2m,m     G31,1 G31,2 · · · G31,m    G · · · G G  32,1 32,2 32,m   ··· ··· ··· ···  G3m,1 G3m,2 · · · G3m,m  0 σx = σ0 (1, 1) + σ0 (1, 2)x + σ0 (1, 3)y σ 0 = σ0 (2, 1) + σ0 (2, 2)x + σ0 (2, 3)y  y0 τx = σ0 (3, 1) + σ0 (3, 2)x + σ0 (3, 3)y

fρ =

# 2ρ fρij, x TijT Tkl dx dy Vkl = fρij, y t A

(38)

where ρ is mass density of the material, t is the time step and Vkl is the “velocity” term at the end of previous time step. Explicit expressions required by the simplex integration are given in Lu (2001). 2.8 Fixed point matrix The fixed points are handled by applying hard springs. The same as the concentrated forces, the fixed points are not necessarily at the nodal points. The contribution of the springs to the element stiffness matrix of jth DOF of ith node to lth DOF of kth node is: [Kij,kl ] = kTijT (x0 , y0 )Tkl (x0 , y0 )I

(33)

(39)

where k is the stiffness of the spring(s) and (x0 , y0 ) is the location of the fixed point. The matrix I is

Array σ0 is input data.

78


Inertial force matrix

 1     0    1  0  

    0 0

0 1

0 0

if fixed in

x&y x direction(s) y

Element I

(40)

0 1

P1(x1,y1)

The corresponding load term of jth DOF of ith node is: f ffptij = fptij, x = kTijT (x0 , y0 )ufptij (41) ffptij, y

Figure 3.

ufpt, x is computed displacement of the ufpt, y

Illustration of block contact.

where i,k = 1,2,3 and j,l = 1–m. HIij and HIkl are for element I and J, respectively and computed as:

fixed point. 2.9

P3(x3,y3) Element J

$

where ufpt =

P2(x2,y2)

1 y − y3 Tij (x1 , y1 ) 2 (48) x3 − x2 l Tkl (x3 , y3 ) y1 − y2 Tkl (x2 , y2 ) y3 − y1 + HJkl = x1 − x3 x2 − x1 l l (49) % (50) l = (x2 − x3 )2 + (y2 − y3 )2 " " "1 x1 y1 " " " S0 = ""1 x2 y2 "" (51) "1 x y " 3 3

Normal contact matrix

HIij =

Assume the stiffness of the normal spring is kn , the contributions of the spring to the global stiffness matrix are: T [Kij,kl ]ii = kn HIij HIkl

(42)

T [Kij,kl ]ij = kn HIij HJkl

(43)

T [Kij,kl ]ji = kn HJij HIkl

(44)

T [Kij,kl ]jj = kn HJij HJkl

(45) 2.10 Shear contact matrix

where [Kij,kl ]ii

Sub-stiffness matrix (2 × 2) of jth DOF of ith node of element I to lth DOF of kth node of element I

Similar to the normal contact matrices, the shear contact matrices and forces are computed as follows:

[Kij,kl ]ij

Sub-stiffness matrix (2 × 2) of jth DOF of ith node of element J to lth DOF of kth node of element I

T [Kij,kl ]ii = ks HIij HIkl

(52)

T [Kij,kl ]ij = ks HIij HJkl

(53)

T [Kij,kl ]ji = ks HJij HIkl

(54)

T [Kij,kl ]jj = ks HJij HJkl

S0 I HIij fsij = −ks l

S0 J fsjk = −ks HJjk l

(55)

[Kij,kl ]ji

[Kij,kl ]jj

Sub-stiffness matrix (2 × 2) of jth DOF of ith node of element I to lth DOF of kth node of element J Sub-stiffness matrix (2 × 2) of jth DOF of ith node of element J to lth DOF of kth node of element J.

The normal contact force terms for nodes of element I and J are computed from Eqns (46) and (47)

S0 I fnij HIij = −kn (46) l

J = −kn fnjk

S0 l

(47)

79


(57)

where ks is the stiffness of the shear spring. HIij and HIkl are for element I and J respectively and computed as: 1 x − x2 HIij = Tij (x1 , y1 ) 3 (58) y3 − y2 l

HJjk

(56)

Element I

P1(x1,y1)

P2(x2,y2)

P0(x0,y0)

P3(x3,y3)

Element J Figure 4.

HJkl = l=

Illustration of shear contact.

1 x − x3 Tkl (x0 , y0 ) 2 y2 − y3 l %

(x2 − x3 )2 + (y2 − y3 )2

(59) (60)

S0 = (x1 − x0 )(x3 − x2 ) − (y1 − y0 )(y3 − y2 ) (61) 2.11

Figure 5. Two MM models: (a) coarse mesh and (b) fine mesh.

Contact friction force

polynomial integrand are derived and a computer subroutine is written to implement the integration. Detailed description can be found in Lu (2001).

The force terms resulting from the sliding of the boundary contact for nodes of element I and J are computed from following equations (see Figure 4): I = −HIij kn ds tan φ ffij

(62)

= −HJij kn ds tan φ

(63)

J ffij

4 VERIFICATION The high order MM formulation and the corresponding computer program have been tested with a number of examples and verified by means of comparing to the closed form solutions. Given below is one of the testing examples, which is a cylinder subjected to internal pressure. As shown in Figure 5, two MM models are used, one has a coarse mesh and the other has a finer one, consisting of 116 and 478 triangle elements, respectively. The basic geometrical and mechanical data are:

where kn is the stiffness of the normal contact spring, d is the penetration distance, φ is friction angle,  1 x>0 if x = 0 s = sgn(x) = 0 −1 x 0

(2)

Where S is slide direction, R is the exterior force and Vk is the normal vector toward block. Eq (1) represents slide direction is at the same direction with the exterior force. Eq (1) represents the block must be departed or take off the all joint surfaces. (2) Single slide Figure 1.

When block slides along joint surface i, the slide direction

Convex and concave block.

S=

(ni × R) × ni |ni × R|

(3)

and R Vi 0 S Vk > 0

(4) ki

(5)

where ni is the up normal of slide joint surface i. (3) Double slide

Figure 2. Water pressure distribution model.

If a block slides along joint surface i with joint surface j, the slide direction

or studies is only restricted to the underground water distribution model for tetrahedral blocks. Since the shape of many block is nontetrahedral, below is a new underground water distribution model for general shape block (Fig 2). The polygon in figure 2 represents a joint planar surface of one block. Because lines A5A1 and A5A4 are free surfaces, the pressures of apex A1 , A4 and A5 are zero. The water pressure is according to static water pressure distribution in the line section A1A2 , increasely linear distribution in the line section A2A3 and decreasely linear distribution in the line section A3A4 . The pressure of apex A2 and A3 is calculated by above pressure distribution. With linear interpolating function the pressure force along the joint surface is exactly defined by the pressure of apex. Then the water press on the joint surface is calculated and the direction perpendicular to the planar surface is towards the block. 2.3

S=

ni × nj sign[(ni × nj ) · R] |ni × nj |

(6)

and Si Vj 0

(7)

Sj Vi 0

(8)

S Vk > 0

kij

(9)

(4) Self lockage Each removable block must be in one of the three slide modes under one exterior force. If a block is not satisfied the one of the three slide modes, the slide mode is called self lockage. When one block is in self lockage slide, it is stable even if the friction and cohesion coefficient of joint surface are zero. Based on the results of slide mode, the stability safety factor for block is taken different formula. The stability safety factor is zero if the block is sloughage off and a block is stable if it is self lockage. The stability safety factor formula for single and double slide are found out in reference [2].

Slide mode and stability analysis

The first step for block stability analysis is judging the block slide mode and then calculating the safety stability factor. The block slide mode is divided into 4 types, that is sloughage slide, single slide, double slide and self lockage.

3

(1) Sloughage slide

ROCK BLOCKS IN ROCK WALLS OF THE PERMANENT SHIPLOCK IN TGP

One block is sloughage mode, the slide direction S = R/|R|

The excavation for main shiplock begins in spring 1996 and basically ends in spring 2000. Although

(1)

96


45.6% 140

30

120

25

N Slope S Slope

20 15

Mound

10

Block Num

Block Num

40 35

80

17.0%

60

17.7% 11.1%

8.5%

40

5 0

100

20

1

2

4 Chamber No. 3

5

0 100-300

300-500

500-700

700-1000

>1000

Volume

Figure 3. The block distribution along chamber in shiplock slope.

Figure 5. Volume distribution in shiplock slope.

250

Block num

200 Gravity

150

Cable 100

water

50 0

Figure 4.

Concave block in shiplock slope.

Figure 6.

784 blocks are found out under construction, there are only 317 blocks whose volumes are greater than 100 m3 . In these blocks, there exist 171 blocks in the left shiplock, 146 ones in right and 167 ones in isolated middle mound. Because the shiplock are five stage lock, each lock chamber has different block and figure 3 is the block distribution according to lock chamber. Since there are 12 blocks which have been slided off or excavated during the construction periods, the remaining 305 blocks are studied and calculated with block theory and its code on detail in the papers. The shape of blocks which are made up of faults or joints and excavation face are mainly tetrahedron, nontetrahedron with more than 4 surfaces which consist of them and concave. Most blocks are nontetrahedrons in the entire 305 blocks. There are only 111 tetrahedron blocks with 36.4 percent and one is concave block (Fig 4). The geometric characteristic of block is mainly considered the volume and depth of block. Figure 5 is the statistic distribution diagram for volume. In the permanent shiplock block, most block volumes are distributed between 100 m3 and 500 m3 , having 62.0 percent of the entire block, and there are 54 huge blocks with a volume greater than 1,000 m3 , among which the largest volume is up to 29,658.6 m3 . The depth for 49.2 percent block is 5–10 m and there are 185 blocks whose depth is less than 10 m. It is

double

sloughage self lockage Slide

Statistic slide mode in shiplock slope.

explained that the length for the design bolt with 8–10 m is reasonable. In the gravity case, most blocks are double slide modes with 70.5 percent of the entire ones. There are 59 single slide mode blocks with 19.3 percent, and 10.2 percent blocks are in self lockage. Because of the pre-stressed cable and different exterior forces, the block slide mode is changed, for example, the slide mode from single to double and from double to self lockage. In pre-stressed cable case, the block number of single slide mode is decreased from 59 to 32 and the self lockage one is increased from 31 to 64. With the change of block slide mode, the safety stability factor of block is enhanced. In the water pressure with prestressed cable case, there are 3 blocks which take in sloughage mode due to the water pressure. Figure 6 is the statistic diagram of block slide mode. The block stability factor Kc considering the slide joints cohesion is different in 3 cases that is in the gravity, prestressed cable and water pressure with cable case. First the block is basically stable in the gravity case because the slope with stability factor greater than 1.3 is stable according to design standard value of stability factor in the high slope of shiplock in TGP. Second, nearly all block is stable under the prestressed case because the stability factor Kc of 99 percent of block is greater than 1.3. It is favourable for

97


single

Table 1.

300

Joint dip dip dir. in underground house. Strike◦

Block num

250 200

Gravity cable water

150 100 50

Type

steep

0

Kc " Kij "" (9) " j=1 " " " j=i Larger inertia terms on the diagonal of the global stiffness matrix increase the stability of the computation. A small time step size is needed to increase the inertia terms, which are inversely proportional to the square of time step. This effect can be seen in Figure 6. For small time steps (0.0025 sec) the numeric error does not exceed 10% for increasing penalty values up to 5 ∗ 1010 N/m, higher values introduce significant error as the off diagonal sub-matrices become larger, resulting in loss of diagonal dominance. Enlarging the time step results in reduction of the inertia term in the diagonal sub-matrices. Thus, for a given value of time step size the loss of diagonal dominance will occur at lower penalty values. Figure 8 shows the accuracy of the DDA solution for different penalty values, for a given values of g1 and g2. When the penalty is lower than 5 ∗ 106 N/m inter-block penetration occurs. For penalty values of 5 ∗ 106 N/m and up to 600 ∗ 106 N/m the accuracy of

201


DISCUSSION

10000

EN(%)

1000

100

k01 = 1

10

All DDA simulations for: g1 = 0.0025, g2 = 0.005

0.985 0.975

g0 = 500*106 N/m, φ = 16˚

k0.98

1

0

0.2

0.4 0.6 Elapsed time/Total time

0.8

1

Figure 10. Numeric error evolution of DDA solution for sinusoidal input function at 2.66 Hz input frequency.

the solution is well confined between relatively narrow error margins. With φav = 17◦ the error is reduced from 110% to 20% over the studied range of penalties. Similarly, with φav = 16◦ the error is reduced from 120% to 80% over the same penalty range. When the penalty is exceedingly high an abrupt accumulation of error occurs due to loss of diagonal dominance of the global stiffness matrix, or due to matrix ill conditioning. Most of the error is accumulated at the beginning of the analysis and it declines with time, a phenomenon known as algorithmic damping (Figure 10). Similar observations were reported by Doolin and Sitar (2001) for the case of a gravity driven block. The maximum error value is a test artifact associated with the transition from a ramped motion to a steady sine input motion in the shaking table experiment and can be ignored. This trend is maintained here for all values of k01 selected; greater accuracy is attained when k01 is optimized. Algorithmic damping is typical to implicit time integration schemes. In DDA, a Newmark type implicit, time integration scheme (collocation parameters are β = 0.5, δ = 1) assures unconditional stability of integration and high algorithmic damping (Wang et al., 1996). Thus, damping is performed without the introduction of energy consuming devices. The amount of algorithmic damping depends on the time integration method, the time step size, and the natural period of the system. In this study we have limited the duration of the analysis to 5 seconds, in conjunction with the physical model. It has been shown that algorithmic damping reduces the numeric error as calculation evolves. Doolin and Sitar (2001) showed that error reduction is evident for sliding distances of up to 250 m over 16 sec. Thus, for larger time spans the error will decline with calculation progress to a certain minimum value, further improving solution accuracy. Dynamic formulation of DDA is essentially undamped, thus for evolving systems the only way to

dissipate energy is by frictional resistance. The physical model is however more complicated, energy losses through structural vibrations, heat radiation, drag, and other physical mechanisms are present, and not accounted for by DDA. Reduction of the transferred velocity at each time step reduces the overall dynamic behavior of the discrete system without imposing illconditioning of the stiffness matrix (Wang et al., 1996). In a similar manner a quasy-static analysis is performed by setting k01 = 0. Thus we recommend that for full-scale simulations a certain amount of kinetic damping should be applied. McBride and Scheele (2001) showed similar effect for a gravity driven multi–block structure, showing that optimal results were achieved for k01 = 0.8. It is reasonable to assume that higher kinetic damping is required for multi – block structures, to account for a large number of contacts and block interactions. However, this estimate should be examined in conjunction with the time step size and the penalty value.

6 •

The results of the validation study show that DDA solution of an idealized system for which an analytical solution exists, is accurate. The DDA intra-block contact algorithm is therefore a true replication of the analytical model for frictional sliding. • The accuracy of DDA is governed by the conditioning of the stiffness matrix. DDA solution is accurate provided that the chosen time step is small enough to assure diagonal dominance of the global stiffness matrix. • Numeric spring stiffness should be optimized in conjunction with the chosen time step size to assure accurate solution and to preclude ill conditioning of the global stiffness matrix. • Comparison between a shaking table model and DDA calculation shows that the DDA solution is conservative. For accurate prediction of dynamic displacement of single block on an incline a reduction of the dynamic control parameter (k01) by 2% is recommended.

ACKNOWLEDGMENTS This research is funded by the US–Israel Binational Science Foundation through grant 98–399. The authors wish to express their gratitude to Gen-hua Shi who kindly provided his new dynamic version of DDA. Shaking table data were provided by J. Wartman of Drexel University, R. Seed, and J. Bray of University of California, Berkeley, and their cooperation is greatly appreciated.

202


SUMMARY AND CONCLUSIONS

REFERENCES Doolin, D. & Sitar, N. 2001. Accuracy of the DDA method with respect to a single sliding block. In: Rock Mechanics in the National Interest, proceedings of the 38th U.S. Rock Mechanics Symposium, Washington D.C., July 5–7, 2001. Balkema, Rotterdam. Goodman, R. E. & Seed, H. B. 1965. Earthquake induced displacements in sand embankments. J. of Soil Mech. and Foundations Div. ASCE. 92(SM2): 125–146. Hatzor, Y. H. & Feintuch, A. 2001. The validity of dynamic block displacement prediction using DDA. Int. J. of Rock Mech. and Min. Sci. 38: 599–606. Kim, J., Bray, J. D., Reimer, M. F. & Seed, R. B. 1999. Dynamic interface friction properties of geosyntetics. Unpublished report, University of California at Berkeley, Department of Civil Engineering. MacLaughlin, M. 1997. Discontinuous Deformation Analysis of the kinematics of rock slopes. Ph.D. thesis, Department of Civil Engineering, University of California, Berkley. McBride, A., Scheele, F. 2001. Investigation of discontinuous deformation analysis using physical laboratory models. In: Bicanic, N. (ed.). Proc. of the Fourth International Conference on Discontinuous Deformation Analysis. 73–82. Glasgow, 6–8 June. Newmark, N. M. 1959. A method of computation for structural dynamics. J. of the Eng. Mech. Div. ASCE. 85(EM3) O’Sullivan, C. & Bray, J. D. 2001. A comparative evaluation of two approaches to discrete element modeling to

particulate media. In: Bicanic, N. (ed.). Proceedings of the Fourth International Conference on Discontinuous Deformation Analysis. 97–110. Glasgow, 6–8 June. Shi, G-h. 1988. Discontinuous Deformation Analysis – A new model for the statics and dynamics of block systems. Ph.D. thesis, Department of Civil Engineering, University of California, Berkley. Shi, G-h. 1993. Block system modeling by discontinuous deformation analysis. In: Brebbia, C. A. & Connor, J. J. (eds). Topics in Engineering, Vol. 11. Computational Mechanics Publication. Shi, G-h. 1999. Applications of Discontinuous Deformation Analysis and Manifold method. In: Amadei, B (ed.). Third International Conference on Analysis of Discontinuous Deformation. 3–16. Vail, Colorado, 3–4 June. Wang, C-Y., Chuang C-C. & Sheng, J. 1996. Time integration theories for the DDA method with Finite Element meshes. In: Reza Salami, M. & Banks, M. (eds.). Proceedings of the Fifth International Forum on Discontinuous Deformation Analysis (DDA) and Simulation of Discontinuous Media. 97–110. Berkeley, 12–14 June. TSI Press: Albuquerque. Wartman, J. 1999. Physical model studies of seismically induced deformation in slopes. Ph.D. thesis, Department of Civil Engineering, University of California, Berkley. Yeung, M. R. 1991. Application of Shi’s DDA to the study of rock behavior. Ph.D. thesis, Department of Civil Engineering, University of California, Berkley.

203


Theoretical developments in modelling discontinuous deformation



Crack propagation modeling by numerical manifold method Shuilin Wang Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan, P. R. China

Ming Lu SINTEF Civil and Environmental Engineering, Trondheim, Norway

ABSTRACT: Manifold method, proposed by Dr. Genhua Shi about 10 years ago, catches the attention of many scholars around the world. It employs cover systems to form elements similar to the finite element method using meshes. In this paper, manifold method is employed to simulate crack propagation for the reason that the mathematical meshes can be kept unchanged during the simulation processes. The theory of linear elastic fracture mechanics (LEFM) is chosen to judge whether a crack extends or not. Stress intensity factors in LEFM are calculated by a path independent contour integral to avoid singularity at crack tip. High order cover functions are used on physical covers to improve the accuracy of the crack propagation simulation. An algorithm to deal with the crack tip’s stopping at any place of an element is presented. Finally, test examples are given to validate the method and the corresponding program.

1

INTRODUCTION

Crack propagation is encountered in many engineering problems and simulation of crack propagation has been a challenging problem. As far as the authors’ knowledge, there are 4 kinds of numerical methods that can be used to model crack growth. They are finite element method (FEM) (Lu & Bostrøm 1999), boundary element method (BEM) (Scivia 1995), element free Galerkin method (EFGM) (Belytschko et al 1996) and manifold method (Wang 1998). Some FEM programs (Bittencourt et al 1996, Lu & Bostrøm 1999) for simulating crack propagation have been developed. But remeshing is an overburden for problems in which the crack path is not previously known, especially when problem is extended from 2-D to 3-D. BEM has advantages in crack propagation simulation for simplifying the problems (reducing the dimension by 1). But it also has its own limitations and requires simple material property and geometry. EFGM is proposed recently and produced on the basis of moving least square interpolation. No mesh is needed in this method except that a background rectangular mesh is sometimes added during numerical integration. However, according to our experiences, further studies are needed on the numerical stability of the method. Numerical manifold method, a newly proposed method, is similar to FEM in some respects. But the element shape in this method is not as restricted as in FEM. It can be irregular. The initial meshes (i.e.

mathematical meshes) can be kept unchanged during the simulation process. Therefore, manifold method is especially suitable for simulating crack propagation in solids. In this paper, MM is chosen to model crack propagation. Numerical procedure about simulating crack propagation by the method is described. In the procedure, theory of linear elastic fracture mechanics (LEFM) is used for dealing with crack propagation. Related equations for computing stress intensity factors (SIF) and criterion for judging crack propagation are given. An algorithm to make the crack tip stops at any place is presented. Meanwhile, corresponding program was written as a part of the hydraulic fracturing simulation code (Lu et al 2001). The programs are tested with two numerical examples.

2 THEORETICAL BACKGROUND 2.1 Criterion of judging crack propagation Only 2-D problems are discussed in this paper and the crack propagation model is based on LEFM theory. Generally speaking, once relative displacements occur on the crack plane with cracks, stress singularity will appear at the crack tip. As shown in Figure 1, local Cartesian coordinate XoY is located at the crack tip with X-axis in the crack plane.

207


Thereby, the classical strength theory is not applicable to judge whether failure will occur at the crack tip. In this situation, LEFM theory, in which SIFs KI and KII are the key parameters, is utilized to evaluate the potential of crack extension. In mixed mode problems, the maximum circumferential tensile stress criterion is simplified to compare the equivalent SIF Keq , which is a function of KI and KII and θ, with the material toughness. When the following condition is met 3 θ θ Keq = cos [KI cos2 − KII sin θ] = KIc 2 2 2 Figure 1. Crack tip, integration contour 2 and its corresponding encompassed domain.

Stress and displacement fields around the tip can be written in the following analytical form (Whittaker et al 1992):

3θ KI θ θ 1 − sin sin σx = √ cos 2 2 2 2πr

3θ θ θ −KII 2 + cos cos sin +√ 2 2 2 2πr

3θ KI θ θ 1 + sin sin cos σy = √ 2 2 2 2πr θ 3θ KII θ +√ sin cos cos 2 2 2 2πr θ 3θ KI θ τxy = √ cos sin cos 2 2 2 2πr

3θ θ KII θ 1 − sin sin +√ cos 2 2 2 2πr / 3θ θ KI 2r (2k − 1) cos − cos u= 8G π 2 2 / 3θ θ KII 2r (2k + 3) sin + sin + 8G π 2 2 / 3θ θ KI 2r (2k + 1) sin − sin v= 8G π 2 2 / 3θ θ KII 2r −(2k − 3) cos − cos + 8G π 2 2

(1)

where KI and KII are mode I and II stress intensity factors, respectively. G is the shear modulus. k = (3 − v)/(1 + v) and k = 3 − 4v for plane stress and plane strain conditions, respectively. v is Poisson’s ratio. r and θ are local polar coordinates originating from the crack tip. Equation (1) shows that the stresses will be infinite at crack tip under loading, even if the load is small.

the crack will be considered to extend in the direction θ0 , in which Keq takes its maximum value. θ0 satisfies the following equation ∂Keq = 0; ⇒ KI sin θ0 + KII (3 cos θ0 − 1) = 0 ∂θ θ = θ0 (3) Now the problem is to evaluate SIFs KI and KII . 2.2 Computation of stress intensity factors In LEFM, stress singularities exist at crack tip. Because SIFs are used to evaluate crack, it is important to evaluate them accurately. In practice, analytical solutions are available only for few simple problems. Numerical methods are usually needed. SIFs can be computed directly from the displacements on the crack plane near the crack tip. Usually fine meshes are required due to the high stress gradient around the tip. Alternatively, a kind of singular element can be constructed to model the stress singularities at crack tip. One can also use contour integration away from the crack tip. Then SIFs are computed from the displacements and stresses along the contour. In such a way, stress singularities can be avoided. In this paper, the contour integration method derived based on Betti’s work reciprocal theorem is used. In the method, an auxiliary displacement and its corresponding stress fields represented by u, v, σx , σy , τxy are constructed. They can be written in analytical expressions. Equation (4) is a closed form solution derived from the complex functions proposed by Muskhelishvili (Yang 1996). cI and cII are constants similar to KI and KII . We utilize Betti’s work reciprocal theorem. i.e., the work the true stress fields σx , σy , τxy do on auxiliary displacements uˆ , vˆ equals to what auxiliary stresses σˆ x , σˆ y , τˆxy do on true displacements u, v. 1 σˆ x = √ 2πr 3

208


(2)

cos

+

3 5θ 3θ − sin θ sin 2 2 2

−2 sin

cI

3 5θ 3θ − sin θ cos 2 2 2

cII

σˆ y = √

1 2πr 3

cos

3 5θ 3θ + sin θ sin 2 2 2

cI

5θ 3 sin θ cos cII 2 2 5θ 3 1 sin θ cos cI τˆxy = √ 2 2 2πr 3

5θ 3θ 3 − sin θ sin cII + cos (4) 2 2 2 3θ θ 1 cI (1 − k) cos + sin θ sin uˆ = √ 2 2 2G 2πr 3θ θ + (1 + k) sin + sin θ cos cII 2 2 3θ θ 1 cI (1 + k) sin − sin θ cos vˆ = √ 2 2 2G 2πr 3θ θ cII + (k − 1) cos + sin θ sin 2 2 +

As illustrated in Figure 1, domain encompassed by contour = 2 + AB + (−1 ) + CD is considered. 2 = DEFGA, and 1 = CE1 F1 G1 B. Betti’s work reciprocal theorem is applied on the domain and we can have # (ui ˆti − uˆ i ti ) d = 0 (5)

i is a dummy index and i = 1, 2. Subscripts 1 and 2 denote axis x and y in local coordinates. u1 and u2 represent the displacements in x and y directions, and uˆ 1 and uˆ 2 represent the auxiliary field displacements of x and y directions. t1 = σx • nx + τxy • ny . t2 = τxy • nx + σy • ny . (6) ˆt2 = τˆxy • nx + σˆ y • ny (7)

ˆt1 = σˆ x • nx + τˆxy • ny ,

where nx and ny are unit outer normal along the contour in local coordinate XoY. Equation (5) can be further written as # # (ui ˆti − uˆ i ti ) d = (ui ˆti − uˆ i ti ) d (8) 1

2

for the reason that the sum of integration along AB and CD will disappear. After a tedious manipulation on the integration along path 1 , the left side of Equation (8) is simplified as an analytical expression. # k +1 (KI cI + KII cII ) (ui ˆti − uˆ i ti ) d = (9) 2G 1 Integration along path 2 , is evaluated by numerical means. Substituting numerical solutions of σx , σy , τxy ,

u and v into the right side of Equation (8) results in # (10) (ui ˆti − uˆ i ti )d = m1 cI + m2 cII 2

m1 and m2 are coefficients obtained by numerical computation. Equating (9) with (10) leads to k +1 (KI cI + KII cII ) = m1 cI + m2 cII 2G

By comparing the coefficients of cI and cII , we can get KI and KII . Even if there is pressure acting on the plane, the above equation still holds. See Wang (1998) for a detailed explanation. 2.3 Numerical integration As discussed in section 2.2, the computation of SIFs comes down to the integration along a contour starting from a point on one crack plane and ending at a point on the other crack plane. In our program, a nonclosed circle similar to path 2 in Figure 1 is chosen as the integration contour. The center of the circle is located at crack tip. The arc of the circle is divided into N sections. Numerical integration is performed section by section. The sum of the integration of all sections gives the coefficients m1 and m2 . Then KI and KII are obtained from Equation (11). During the computation, displacements u, v and stresses σx , σy , τxy at the end points of each section are obtained by numerical analysis. The auxiliary displacements uˆ , vˆ and stresses σˆ x , σˆ y , τˆxy along the contour are calculated from Equation (4). 2.4 An algorithm for managing crack tip ending at any places In MM, only when an element or edge of an element is penetrated completely by a crack, the crack will be considered to exist in that part of the element. In reality, the crack tip may stop at any place of the domain after extension. In order to handle the possible termination of crack tip in the element, a penalty method is adopted. In dealing with crack propagation, two cases will occur as showed in Figures 2 and 3. For case 1, crack tip may end within an element. Here we suppose that crack tip ends at node T as shown in Figure 2(a); for case 2, crack extends along the boundary of two elements and stops at any point of the boundary. Here we assume that crack tip ends at node T as shown in Figure 3(a). If two penalty springs with stiffness ks and kn in tangential and normal directions are applied at the “true” crack tip T with coordinate (xt , yt ) as illustrated in Figures 2 and 3, then the potential of the springs is 1 k 0 uu − ul w = {uu − ul vu − vl } s (12) 0 kn vu − vl (x ,y ) 2 t t

209


(11)

kn

displacement function of physical cover (12 , 21 , 31 ). Displacement function of manifold element (11 21 31 ) is

ing

ing

lty

a pen

spr

spr

alty

pen ks

Yt

X t,

3 31 31 31 21 12 T 12 31

Y X

Yt

X t,

3 31 31 31 21 21 2

12

11 11

21

12

21 21 2

31

(14)

11 11 1

1

(a)

(b)

Figure 2. Crack propagating across an element edge and ending at a point within an element (Case 1). ing

ing

alty

spr

alty

spr

pen

pen kn

ks Yt

Yt

X t,

X t,

3

3 31

31 21 2 α 21

Y

where wi (x, y) (i = 1,2,3) is the weight function of physical cover (11 , 21 , 31 ). They are the same as those of manifold element (12 21 31 ) because both manifold elements have the same mathematical covers. u11 , v11 , u21 , v21 and u31 , v31 displacement function of physical cover (11 ,21 ,31 ). Therefore,

12 1 11

T

1

12 11

w=

4

4

(a)

Equation (12) can be rewritten as 41

41

X

(15)

21 2 21 T

(b)

Figure 3. Crack propagating along the boundary of two elements and ending at a point on the boundary (Case 2).

uu

vu

where and are tangential and normal displacements along the crack plane on the upper element; ul and vl are tangential and normal displacements along the crack plane on the lower element. ks and kn are tangential and normal penalty stiffness, respectively. (xt , yt ) are the coordinates of the current crack tip where the penalty springs are inserted and displacements take values at that node. For case 1, displacement function on manifold element (12 21 31 ) is

(13) where wi (x, y) (i = 1,2,3) is the weight function of physical cover (12 , 21 , 31 ). ui and vi (i = 12 , 21 , 31 ) are

t

t

Equation (16) can be further written in the following form 1 T T T T T D11 w= D11 D12 (F C W KWCF)(xt ,yt ) (17) D12 2 where K =

ks 0 , 0 kn

 cos α sin α 0 0 0  −sin α cos α 0 C= 0 0 cos α sin α 0 0 −sin α cos α 

T11 0 , F = 0 T12 0 w1 (x, y) 0 −w1 (x, y) W = , 0 −w1 (x, y) 0 w1 (x, y)

Di (i = 11 , 12 ) are general unknown variable vectors of physical cover 11 and 12 . Ti (i = 11 , 12 ) has the same meaning as in report (Lu 2001) and is the function of coordinates. C is a transform matrix, α is the angle between crack plane and x-axis as shown in Figures 2

210


1 w1 (x, y)(u12 − u11 )w1 (x, y)(v1 2 − v1 1 ) 2 w1 (x, y)(u12 − u11 ) ks 0 × (16) 0 kn w1 (x, y)(v1 2 − v1 1 ) (x ,y )

and 3, and K is the local stiffness matrix. For case 2, we can write the same potential of springs as Equation (12) and get an equation similar to Equation (17). If we want to let crack tip stop at any place of the crack plane within the extending element or along the boundary, a penalty stiffness matrix Kp = [FT CT WT KWCF](xt,yt) must be added in the global stiffness matrix. Note that this algorithm will work only when high order (≥1) cover functions on physical cover are used. 2.5

σY0 10 m

1m

σx0

10 m

P

σx0 0.6 m

Simulation of crack propagation

At each step, the general procedure for the numerical modeling process is as following: (a) The mixed mode SIFs KI and KII are computed from Equation (11) after displacement and stress fields are obtained. (b) The maximum equivalent Keq and its corresponding angle θ0 are determined from Equations (2) and (3). (c) If Keq is greater than or equal to toughness KIc of the material, the crack will propagate. The crack tip extends in the direction of θ0 and ends temporarily at a node or at a boundary of an element. The crack length is measured from the tip. If the crack length is less than the length specified by the user, the crack will continue extending and end at another node or at a boundary of another element. This process goes on until the extending length is greater than or equal to the inputted crack length. Then the true crack tip is determined. Meanwhile, physical covers and manifold elements are added and updated. (d) If Keq is less than the toughness KIc , increase the load until Keq is equal to fracture toughness KIc . Then the crack starts to propagate as what is described above. (e) The procedures are repeated from (a) to (d) for the subsequent steps.

σY0

Figure 4. A plate with a circular hole and an initial crack subjected to internal pressure and outside uniform compression.

Given below are two test examples, in which the first order cover function is used and crack propagation is modeled following the procedure.

Figure 5.

3

Table 1. Relationship between internal pressure and crack extension (case 1, σx0 = 0.0 and σy0 = 0.0).

NUMERICAL EXAMPLES

Firstly, the model is applied to simulate propagation of an initial crack (0.6 m) in a plate with a circular hole. The geometry and loading conditions are shown in Figure 4. Figure 5 presents the finite element mesh used as mathematical covers. Plane stress condition is adopted. The material properties are E = 10 GPa, V = 0.23 and KIc = 0.52 MPam1/2 . Three loading cases are discussed and their results are given in Tables 1, 2 and 3. The size of crack increment is set to be 0.8 m, meaning that the new displacement and

Internal pressure P (MPa) Crack extension (m)

0.67 0

0.68 0.8

0.71 1.6

0.70 2.4

Table 2. Relationship between internal pressure and crack extension (case 2, σx0 = 0.375 MPa and σy0 = 0.25 MPa). Internal pressure P (MPa) Crack extension (m)

211


Initial finite element mesh.

1.27 0

1.58 0.8

1.85 1.6

1.95 2.4

Table 3. Relationship between internal pressure and crack extension (case 3, σx0 = 0.25 MPa and σy0 = 0.25 MPa). Internal pressure P (MPa) Crack extension (m)

1.35 0

1.63 0.8

1.88 1.6

2.05 2.4

Figure 8.

Initial finite element mesh.

Table 4. Relationship between the applied forces and crack extension. Px = Py (103 KN) Crack extension (m)

Figure 6.

0.60 0.0

0.22 0.8

0.30 1.6

0.17 2.4

Predicted crack trajectories. 54˚

P x

3m

44˚

Py

7m

2˚

34˚

2m

1m

45

8m 6m

Figure 9.

Crack path.

10 m

Figure 7.

Structure geometry, cutting force and restriction.

stress field will be computed after the crack extends 0.8 m at each step. Three steps are computed for each loading case. The development of crack propagation at the last step is shown in Figure 6, in which the deformation is scaled up a little so that crack path can be seen clearly. Following are observations from the results. (1) Crack extension is stable for all of the 3 cases. After the crack extends a certain length, it will stop if the internal pressure doesn’t increase. (2) Compared with cases 2 and 3, it is easy for the crack to extend in case 1 because no outside

compression is applied. In case 2, the horizontal compression σx0 is greater than the one in case 3. It becomes easier for the crack to extend in case 2 than in case 3. The results sound reasonable. Additionally, in order to make the crack keep going, higher internal pressure is needed in cases 2 and 3 than in case 1. (3) Theoretically, the crack will extend in a straightline due to the symmetry in geometry and loading. Trivial computation error makes the crack route deviate from the horizontal line. This will be corrected in the next step. The other example is to simulate the rock cutting process due to drag picks. The geometry, loadings and boundary conditions of the model are shown in Figure 7. There is an initial crack of 1 m long in

212


processes. This will get rid of the overburden of remeshing as in FEM. An algorithm to deal with the crack tip’s stopping at any place of an element is presented so that the crack extension can be controlled. Test examples are given and their results look reasonable. REFERENCES

Figure 10. 2.4 m.

Deformed geometry at crack extension length

the structure. The material properties are E = 10 GPa, V = 0.25 and KIc = 1.2 MPam1/2 . Plane stress condition is assumed. Figure 8 presents the finite element mesh used as mathematical covers. Relationship between the applied forces and crack extension is presented in Table 4. The crack path and the deformed geometry are shown in Figures 9 and 10, respectively.

4

DISCUSSION AND CONCLUSION

The presented methodology gives an effective approach to simulate crack extension in solids. Its advantage is that the mathematical meshes (initial mesh) can be kept unchanged during the simulation

Belytschko, T., Krongauz, Y., Organ, D., Fleming, M. & Krysl, P. 1996. “Meshless methods: An overview and recent developments” Comput. Meth. Appl. Mech. Engng., 139: 3–47. Bittencourt, T., Wawrzynek, P., Ingraffea, A. & Sousa, J. 1996. Quasi-automatic simulation of crack propagation for 2D LEFM problems. Engineering Fracture Mechanics. 55(2): 321–334. Lu, M. 2001. Complete N-order polynomial cover function for numerical manifold method. SINTEF report 2001 F01139. Lu, M. & Bostrøm, B. 1999. Investigation of capacity of existing computer programs for simulating crack propagation. STF22 A99105. Lu, M., Bostrøm, B. & Svanø, G. 2001. Hydraulic fracturing simulation with numerical manifold method. ICADD-4: 391–401, Univ. of Glasgow, Scotland, UK Scavia, C. 1995. A method for the study of crack propagation in rock structures. Geotechnique. 45(3): 447–463. Wang, S. 1998. Numerical manifold method and simulation of crack propagation. [Ph.D. dissertation]. Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Whittaker, B., Singh, R. & Sun, G. (ed.) 1992. Rock fracture mechanics: principles, design and applications. Amsterdam: Elsevier. Yang, X., Fan, J. & Kuang, Z. 1996.Acontour integral method for stress intensity factors of mixed-mode crack. Chinese Journal of Computational Mechanics. 13(1): 84–89.

213



Continuum models with microstructure for discontinuous rock mass J. Sulem & V. De Gennaro CERMES, Ecole Nationale des Ponts et Chaussées-LCPC, Paris, France

M. Cerrolaza IMME, Facultad de Ingeniería, Universidad Central de Venezuela, Caracas, Venezuela

ABSTRACT: The discontinuous structure of layered or blocky rock is approximated by an equivalent Cosserat continuum. This implies the introduction of couple stresses and internal rotations which model the relative rotations between blocks and the bending stiffness of the layers. The advantage of the Cosserat homogenisation is also that various failure modes such as inter-block slip, block tilting, layer bending can be easily described through a multi-criteria plasticity model. This model is illustrated by two applications to geotechnical problems such as foundation in blocky rock and slope stability in foliated rock.

1

INTRODUCTION

The numerical analysis of discontinuous rock mass can be dealt with (in most cases) using off the shelf discrete or finite element codes. In the latter case, special interface elements are needed in order to account for the unilateral kinematics of the rock joints. The major drawback of these discrete analyses is that they are very computer time intensive when the number of layers is large. Moreover, detailed information on the geometry and the properties of the individual layers is needed for such models. The interest of developing continuous models for discontinuous rock mass or discrete structures is that for practical applications, a homogenized continuum model would provide a large-scale (average) response of the medium. However the validity of the approximation is restricted to the case where the characteristic size of the recurrent cell of the periodic medium (e.g. layer thickness or block size) is small as compared to the characteristic size of the problem (e.g. the wavelength of the deformation field). An other important limitation of the homogenization of layered or blocky structures with classical continuum theories is that they cannot account for elementary bending due to inter-layer or inter-block slip and may thus considerably overestimate the deformation. In order to overcome these limitations and to expand the domain of validity of the continuum approach one has to consider the salient features of the discontinuum within the frame of continuum theories with microstructure (Vardoulakis & Sulem 1995). The Cosserat theory has been used with some success in the recent years for analyzing blocky

and laminated systems (e.g. Mühlhaus, 1993, 1995, Adhikary & Dyskin, 1996, Sulem & Mühlhaus 1997). The enriched kinematics of the Cosserat continuum allows to model systems of microelements undergoing rotations which are different from the local rotations of the continuum. For blocky rock various failure modes such as inter-block slip and block tilting can then be easily described. In this paper we present several geotechnical applications of Cosserat continuum for layered and blocky rock as encountered in slope stability and foundations problems. It is shown that for toppling failure of rock slopes a Cosserat model provides the necessary link between the slipping mechanism along the layers and the subsequent bending of the rock columns that may lead to tensile breakage. The Cosserat continuum modeling can then be coupled to a discrete approach of block stability. Zones of bending failure are identified in the material to determine the failure surface of the slope that will intersect the foliation discontinuities. This allows to overcome one of the major shortcomings of the limit equilibrium approach: the arbitrary and sometimes unrealistic assumption of a pre-defined failure plane roughly normal to the set of discontinuities of the foliation. 2 A COSSERAT CONTINUUM FOR BLOCKY ROCK In a rock mass the continuity of the material is generally interrupted by a system of bedding planes, faults or joints. Among discontinuous rocks with “regular”

215


network of discontinuities one can mention stratified rock mass where the rock mass is divided into parallel bedding planes and blocky rock for which the rock mass is jointed in a collection of separate blocks in such a way that from the macroscopic it gives the impression of a dry brickwork. In Cosserat theory a material point of the continuum has three additional rotational degrees of freedom as well as the three translations of a classical continuum. In a regular block structure, one can consider the influence of relative rotations between blocks by means of additional Cosserat rotations. The relative rotations cause moments and consequently, material parameters with dimension of length (here the block dimensions) appear in the constitutive relationship. 2.1

The Cosserat elastic model

In a two-dimensional Cosserat continuum each material point has two translational degree of freedom (u1 , u2 ) and one rotational degree of freedom ωc . The index is used to distinguish the Cosserat rotation from the rotation ω=

1 u2,1 − u1,2 ; 2

(·),i =

∂(·) ∂xi

i = 1, 2

(1)

The state of deformation is described by the four components of the rate of the so-called “relative deformation” (Schaefer 1962) γ11 = ∂u1 /∂x1 ;

γ12 = ∂u1 /∂x2 + ωc

γ22 = ∂u2 /∂x2 ;

γ21 = ∂u2 /∂x1 − ω

c

(2)

and the two components of the gradient of the Cosserat rotation which is called the curvature of the deformation κ1 = ∂ωc /∂x1 ;

κ2 = ∂ωc /∂x2

(3)

The six deformation quantities (equations 3 and 4) are conjugate in energy to six stress quantities. First we have the four components of the non symmetric stress tensor σij which is conjugate to the non symmetric deformation tensor γij and second we have two moment stresses (moment per unit area) m1 and m2 , which are conjugate to the two curvatures κ1 and κ2 . Force and moment equilibrium at the element (dx1 ,dx2 ) lead to

relationships for a 2D anisotropic Cosserat continuum are σ11 = C11 γ11 + C12 γ22 σ22 = C21 γ11 + C22 γ22 σ12 = [G + Gc (1 − α)] γ12 + [G − Gc ] γ21

where α is a parameter of anisotropy. We consider here a simple model for blocky rock (Fig. 1). Each block is surrounded by six others. We are mainly concerned with the accuracy with which the continuum model reflects the domain of rigidity set by the size of the blocks. The elasticity of the blocks and the joints elasticities are lumped at the block edges for simplicity. We assume fully elastic joint behavior. We assume that the interaction between the blocks is concentrated in six points of the edges as shown on Fig. 1. Normal and shear forces are written as Qkl = cQ ukl

(6)

Nkl = cN vkl

where cQ and cN are the elastic shear and normal stiffness respectively and u and v at various contact points are given by ui±1, j±1 = ui±1, j±1 − ui, j ±

where 2a and b are the dimensions of the block, ui,j , vi,j and ϕi,j are the displacements and rotation of the block number (i,j). The continuum model is derived by identifying the elastic energy of the equivalent Cosserat continuum with that of the discrete structure leading to

N

i-2, j

Q

i+1, j+1 i, j

i-1, j-1

i+2, j i+1, j-1

Figure 1. The blocky structure.

216


(8)

a ϕi±1, j±1 + ϕi, j (9) vi±1, j±1 = vi±1, j±1 − vi, j ∓ 2 (10) vi±2, j = vi+2, j − vi, j ∓ a ϕi±2, j + ϕi, j

(4)

In the above equations dynamic effects are included through inertial forces and moment. The stress-strain

b ϕi±1, j±1 + ϕi, j (7) 2

ui±2, j = ui±2, j − ui, j

i-1, j+1

σ11,1 + σ12,2 − ρ u¨ 1 = 0 σ21,1 + σ22,2 − ρ u¨ 2 = 0 m1,1 + m2,2 + σ21 − σ12 − I ω¨ c = 0

(5)

σ21 = [G − Gc ] γ12 + [G + Gc (1 + α)] γ21 m1 = M1 κ1 m2 = M2 κ2

b 2a

the following expressions for the elastic constitutive parameters of the Cosserat continuum (Sulem and Mühlhaus, 1997) a b C11 = cQ + 2cN ; C22 = cN ; C12 = C21 = 0 b a a 1 b cQ + cN + 2cQ G = Gc = 4 a b 2 2 a cN + 2cQ − cQ b (11) α = 2 2 a cN + 2cQ + cQ b2

2 a2 b a + 2a2 + cQ cN M1 = 4 4 b 2 2 a b b + cQ cN M2 = a 4 4 The domain of validity of the above representation of a blocky structure by a Cosserat continuum is evaluated by comparing the dynamic response of the discrete and the homogenized structures. The dynamic response of a structure is characterized by its dispersion function which relates the wave propagation velocity to the wave length of the input signal. For elastic behavior it is possible to derive analytical solutions for the dispersion function of the discrete and the continuous systems by using 2D discrete and continuous Fourier transform (Sulem and Mühlhaus, 1997). It was obtained that the Cosserat model is appropriate for wave-lengths greater than 5 times the size of the block (Fig. 2). 1.0

2.2 Extension to elasto-plastic joints The above elastic Cosserat continuum can be extended to an elasto-plastic Cosserat continuum (Mühlhaus 1993). Two different plastic mechanisms can in a blocky structure: block sliding along the joints and/or block tilting. Consequently several yield conditions have to be examined simultaneously. The state of joint slip is defined by a simple Mohr-Coulomb yield condition (compression is assumed to be negative) F 1 = |σ12 | + tan φσ22 − c ≤ 0

(12)

where c and φ are the joint cohesion and friction angle respectively. It is physically acceptable to assume zero dilatancy for friction mechanism so that the corresponding plastic potential is expressed as Q1 = |σ12 |

(13)

Statically admissible force/moment states of a volume element of the block structure are characterized by the tilting conditions (Fig. 3) F (1,2) = −N (1,2) +

2 "" (1,2) "" M ≤0 a

(14)

where for incipient gap opening we have F = 0. The tilting yield criterion for the corresponding Cosserat continuum is expressed as " " b 2" b " F (2,3) = σ22 ± σ21 − ""m2 ± m1 "" ≤ 0 a a a

(15)

with normality flow rule V-Cosserat/V-discrete

0.8

Q

(2,3)

=F

(2,3)

" " b 2 "" b "" = σ22 ± σ21 − "m2 ± m1 " a a a

(16)

0.6

N (2) N (1)

0.4 wave in x-direction wave in y-direction

(2)

M

(2)

M

(1)

(1)

∆␻c

(1)

0.2

∆u2

(0) 0.0 2

1

3

4 5 6 7 89

2

3

(3)

4 5 6 7 89

10

(4)

100

w1/2a ; w2 /b a/2 Figure 2. Dispersion function for continuous and discrete approach.

Figure 3. Tilting conditions.

217


(0)

2.3

Example of application: foundation on blocky rock

We consider here the 2D-example of a half-space under uniform normal load over an extent L. This simple configuration can represent the classical problem

Figure 4.

of a strip-footing on a blocky rock mass (Fig. 4). For numerical applications the following set of geometrical and mechanical data is considered: – Block characteristics: length: a = 1 m, width: b = 0.5 m – Joint characteristics: joint stiffness: cN = cQ = 1 GPa friction angle: φ = 20◦ cohesion: c = 1 KPa dilatancy angle ψ = 0◦ – Length of the footing: L = 10 m. For symmetry reasons, only half of the domain is discretized. On Fig. 5a (respectively 5b) the plastic zones for the sliding criterion (respectively the tilting criterion) are represented in dark color. These results show that the tilting criterion is reached at the surface of the half-space on a limited extent at the vicinity of the side of the footing. The sliding criterion is reached deep inside the rock mass with an orientation of about 30◦ with respect to the horizontal axis. These results are compared to those obtained with a classical isotropic elastic-plastic Mohr-Coulomb yield surface with the geomechanical characteristics of a gravel (Young’s modulus = 25 MPa, Poisson’s ratio = 0.3, friction angle = 40◦ , zero cohesion). In the latter case, the classical result of standard soil mechanics is retrieved: an elastic cone is formed under the footing and plastic yield occurs under it (Fig. 6).

Strip-footing of a periodic block structure.

Figure 5. Uniform normal loading on a blocky structure, (a) sliding zones (p = 8.4 KPa), (b) tilting zones (p = 4.5 KPa).

Figure 6. Plastic zones for a uniform normal loading on an isotropic Mohr-Coulomb half-space.

218


3 ANALYSIS OF TOPPLING FAILURES IN JOINTED ROCK SLOPES: AN EXAMPLE OF TRANSITION FROM CONTINUOUS TO DISCRETE APPROACH Toppling failures can develop in natural slopes as well as in artificial cuts (e.g. open pit walls) provided that the spatial distribution of fractures gives rise to a layered (foliated) rock mass with a principal system of parallel discontinuities that dip deeply into the slope at a given angle. According to existing published researches two main toppling failure mechanisms are usually considered: (a) flexural toppling, (b) block toppling (among others: Hoek & Bray 1977). Both type of failures have a common characteristic: slender blocks may tend to topple thrusting forwards downhill elements. In case (a) the triggering mechanism inducing slope instability is often the flexion of slender rock columns due to joint slipping and gravity force. At a critical depth the induced tensile stress due to bending exceeds the tensile strength of rock material, and fractures progressively develop in the cross section of blocks leading to the progressive ruin of columnar array. In case (b) rock mass has a well defined structural pattern, resulting from the system of pseudo-vertical discontinuities intersecting at right angles a system of bedding planes dipping towards the slope surface. In this configuration, even if the inclination of bedding planes is lower than the angle of friction mobilized at the interface between the base of block and the basal surface, toppling of slender columns causes, in turn, toppling and sliding of blocks below.

Figure 7.

In a reference system with directions 1 and 2 as in Figure 6, for this layered rock, the elastic constitutive parameters of equivalent Cosserat continuum can be expressed as follows (Zvolinskii and Shkhinek, 1984, Mühlhaus, 1995, Adhikary and Dyskin, 1996) C11 = C22 =

1−

−

ν 2 (1+ν)2 1−ν 2 +E/(kn )

(1 − ν)E (1 + ν)(1 − 2ν) + (1 − ν)E/(kn )

νE (1 + ν)(1 − 2ν) + (1 − ν)E/(kn ) (17) E 5ks + E/(2(1 + ν)) G= 8(1 + ν) ks + E/(2(1 + ν))

Gc =

E ; 8(1 + ν)

α=2

E/(2(1 + ν)) Eh2 ; M1 = 12(1 − ν) ks + E/(2(1 + ν))

M2 = 0

A simplified analysis is developed along the following lines: (a) when no sliding occurs the homogenized continuum reduces to a classical anisotropic elastic one; (b) when sliding occurs (i.e. when the frictional resistance of joints is reached), the corresponding parts of material fall in the Cosserat state with zero-joints stiffness. We emphasize the fact that in that case the sliding can be restrained by the bending rigidity and thus, the Cosserat model can address also the situation of zero-joints stiffness as opposed to the conventional homogenization which breaks down. In the parts of material in Cosserat state, the distribution of bending moments can be computed. Bending moments induce a “microscopically” non-uniform distribution of normal stress in the individual layers which may reach

219


E ν2

C12 = C21 =

3.1 A Cosserat model for foliated rock In the models based on classical continuum approach, the layered material is replaced with a homogeneous anisotropic medium characterized by “effective” elastic moduli (e.g. Salamon 1968). If sliding between the layers can occur, the equivalent continuum is viewed as elasto-plastic as for example in the “multi-laminate” model proposed by Zienkiewicz & Pande (1977). Such models provide a good approximation of discontinuous material behavior when the shear stiffness along the joints is comparable to the individual layer shear modulus. In that case, joint slip is small enough to neglect layer bending. However if it is not the case, bending rigidity has to be included in the model otherwise the homogenized model may considerably overestimate the deformation. Bending stiffness of layers can be accounted for by using Cosserat theory where bending moments are considered in addition to conventional stresses. In order to verify the reliability of such an approach we present on Figure 7 the geometry of a slope in a foliated rock mass susceptible of toppling failure.

Slope geometry

the material tensile resistance. Zones of bending failure are thus identified in the material to determine the failure slope surface that will intersect the foliation discontinuities. This allows to overcome the major shortcomings of limit equilibrium approach: the arbitrary and sometimes unrealistic assumption of a pre-defined failure plane roughly normal to the set of discontinuities of foliation. 3.2

Finite element analysis

For the numerical application the following set of geometrical data has been considered: – slope characteristics: height, H = 80 m; slope face inclination, α = 45◦ ; – foliation characteristics: orientation with respect to the horizontal axis, β = 70◦ , thickness = 2 m.

Figure 8.

Sliding zones for layered rock slope.

Figure 9.

Distribution of bending moment.

Material data are as follows: – rock mass characteristics: Young’s modulus = 20 GPa, Poisson’s ratio = 0.2, friction angle = 30◦ , cohesion = 1 MPa; – joints characteristics: shear stiffness = 0.1 GPa/m, normal stiffness = 1 GPa/m, friction angle = 22◦ , cohesion = 10 kPa. The finite element analysis for Cosserat material is performed, using the code COSSBLPL (Cerrolaza et al. 1999). A first computation is performed to determine the zones where the sliding criterion is reached (Fig. 8). A second computation is then performed for which, it is assumed that the shear stiffness along the joints is zero for the part of model where sliding occurs. In this zone, the effect of bending stiffness as introduced by the Cosserat model is thus more important. The results for the bending moment are presented on Figure 9. As mentioned above, bending moments will result in a microscopically non-uniform distribution of normal stress in the layers. In a first approximation the microscopic stress distribution is assumed to be linear within the layer. From elementary beam theory, the normal stress can be evaluated as σN = −

M N + y A I

(18)

where N is the axial force in the layer, M is the bending moment, A is the cross-sectional area and I is the second moment of inertia. The maximum value of tensile stresses acting in the layer can be thus estimated as σtensile = 6

m1 + σ11 b

(19)

where m1 is the Cosserat couple stress, b is the layer thickness and σ11 is the microscopic stress.

Figure 10.

If one assumes that the rock tensile strength is 1 MPa, the zones of possible tensile failure are represented on Figure 10. 3.3 From flexural toppling to block toppling As depicted in Figure 10, if one assume that 1 MPa is the rock tensile strength, a well defined pattern gives the failure surface direction expected. Once that this surface is localized, the system of parallel discontinuities located inside the wedge shaped zone delimited by the slope face, the ground level (Fig. 11) and the failure surface, gives rise to a system of interacting unstable blocks.

220


Zones of tensile failure.

Figure 12. Limit equilibrium analysis: (a) slope geometry, (b) forces and geometry of i-th block (Mongiovì et al. 1995).

Figure 11. analysis.

Slope geometry used for limit equilibrium

The problem of flexural toppling failure is then reconducted to a block toppling failure. For the latter, the basal failure plane, usually assumed as pre-existing, is obtained from a preliminary computation based on an equivalent continuum approach, accounting for microstructure. The analysis of block toppling is usually performed in a more conventional way, adopting the well known limit equilibrium approach (Goodman & Bray 1976). It seems then licit to investigate whether the wedge obtained in Figure 10 is unstable also in the context of limit equilibrium analysis. In this respect a theoretical 2D model based on the original method proposed by Goodman & Bray (1976) is used. The slope response is analyzed with reference to the geometry at the onset of failure induced by flexural toppling (Fig. 10). The new slope geometry is presented in Figure 11. All blocks have the same width b and form a regular system on a basal surface stepped upwards, as schematically represented in Figure 12a. The system is delimited by the slope face (α3 ), the ground level (α1 ) and the basal surface (α2 ). The inclination of bedding planes is αo . The inclination of basal surface has been depicted from the pattern of tensile stress shown in the same figure. The value retained is α2 = 30.2◦ . This value corresponds to the basal surface inclination obtained joining the point at the slope toe and the point on the ground level where the tensile stress reaches the rock tensile strength. Assuming that blocks have constant width (b = 2 m), the basal surface inclination involves about 80 blocks in the calculation. The other angles (all calculated versus the direction of X-axis) are as follows: αo = 20◦ , α1 = 0, α3 = 45◦ . The block i is subjected to gravity force (Pi ), side forces (Si ,Ti , Si−1 , Ti−1 ) and basal forces (Qi , Ri ) (Fig. 12b). It is assumed that Mohr-Coulomb cohesionless failure criterion holds true for the basal and

side forces, with a friction angle ϕ = 22◦ as in previous finite element analysis. The force distribution on each element is determined by solving simultaneously the three equations of equilibrium and assuming the less favorable configuration kinematically compatible among the following conditions: stability, downhill sliding, downhill toppling, simultaneous uphill sliding and downhill toppling. Four compatibility conditions are introduced, related to the failure frictional criterion given at the interfaces between blocks and to the points of application of base and side forces. Considering the scheme in Figure 11b, they are as follows: |Ri | − Qi tan ϕ ≤ 0

(20)

|Ti | − Si tan ϕ ≤ 0 " " " " " fi − di " ≤ di " 2" 2 " " " " "ei − b " ≤ b " 2" 2

(21)

(23)

where b is the block width and di is the block height. The condition of simultaneous uphill sliding and downhill toppling was proposed by Mongiovì et al. (1995), who verified that often no solutions of equilibrium equations exist and satisfy the kinematic compatibility for downhill toppling. The analysis starts from the block at the top, progressing down to the toe. The response of each block is determined selecting among the solutions related to the four modes of behavior considered the only acceptable one that: (a) yields the highest value for the force Si , transmitted by block i to block i+1 below, and (b) fulfills the compatibility conditions (20)–(23). Obviously, being So the force acting at the left side face of block at the slope toe, the slope is considered:

221


(22)

– stable if: So < 0; – at limit equilibrium if: So = 0; – unstable if: So > 0.

3.4

Numerical results

The results of calculations performed in order to verify the consistency of equivalent continuum approach and the limit equilibrium analysis, are presented on Figure 13. The dimensionless value S on Figure 13 is the ratio between the forces Si , transmitted by block i to block i + 1, and the total weight of the unstable region defined in Figure 11. As it can be observed, S increases moving from the top (Lo /L = 0) to the toe (Lo /L = 1), and for Lo /L = 1 is So > 0. Consequently, the inclination α2 considered, issued from the finite element calculation for Cosserat material, leads to an unstable block system. We present in the same figure the evolution along the slope of the kinematics of blocks. As expected, the small blocks at the top of the slope are stables. In fact, being ϕ > αo , sliding on the surface of bedding planes will never occur. Moreover, neither downhill toppling nor downhill toppling and simultaneous uphill sliding are possible due to their reduced slenderness. Progressing towards the toe the height of blocks increases and blocks start topple. Finally, when approaching the toe of the slope complex failure mechanisms are activated. The occurrence of complex mechanisms is analyzed in Figure 14. If only downhill toppling or sliding are accounted in the calculations (see for instance: Goodman & Bray 1976, Zanbak 1983), by inspecting the evolution of the normal force Qi and the tangential force Ri at block base is easy to verify that condition (20) is violated at Lo /L = 0.65. Moreover, being Ri < 0, this implies simultaneous block uphill sliding. If this incompatibility is ignored, further calculations will lead also to negative values of Qi forces (dot line

in Figure 14), corresponding to an hypothetical and rather unrealistic block lifting. The problem can be regularized assuming that a new mechanism is developing, where the downhill toppling is accompanied by a simultaneous uphill sliding (Mongiovì et al. 1995). In this case, due to uphill sliding, the unstable block lower corner may knock against the stepped base riser, and extra forces are generated at the uphill side of the block in order to restore the equilibrium. The implications related to the evaluation of the action at the toe of the slope are quite significant. Indeed, as presented in Figure 15, the computed So in the hypothesis of simple mechanisms is of about 0.04

Q, R 0.02

0

-0.02

-0.04

Dimensionless forces Q and R on bedding planes Q simple mechanisms R simple mechanisms Q complex mechanisms R complex mechanisms

-0.06

-0.08 0

0.2

0.4

0.6

0.8

1

Lo

/L

Figure 14. 20

Dimensionless forces Q and R at block bases.

25 250 Dimensionless force S ␣2 = 30.2° (80 blocks)

S

Dimensionless force S S

16

simple mechanisms complex mechanisms

20 200

3

3

12

15 150

2

MECHANISM 0 = stable

1 = downhill sliding 2 = downhill toppling 3 = downhill toppling & uphill sliding

4

1

MECHANISM

2 8

10 100

50

0 0

0 0

0.2

0.4

0.6

0.8

0

1

0

Lo

Figure 13. blocks.

0.4 0

0.6 0

0.8 0

1

/L

Dimensionless force S and mechanisms on the

Figure 15. Influence of block kinematics on the dimensionless forces S.

222


0.2 0

Lo

/L

10 orders of magnitude higher than the force at the toe of the slope as obtained admitting complex mechanisms. 4

CONCLUSION

In this paper we have discussed the representation of a rock mass with a regular network of fractures, using a Cosserat continuum. For layered and blocky rock the enriched kinematics of the Cosserat continuum allows to account for individual block rotation and bending of individual layer. Geometrical characteristics of the discontinuities such as block size and orientation, layer thickness are thus introduced directly in the constitutive relationships of the equivalent continuum. It is shown that this representation is valid even for a structure with a relatively small number of blocks or layers. Various failure mechanisms such as sliding or tilting can be considered with appropriate multi-criteria plasticity model. The practical relevance of using continuum models for discontinuous structures is that it is extremely flexible when used with numerical methods since no interface elements are needed and since the topology of the finite element is independent of block size and geometry (one mesh can be used to study several different structures). The homogenization procedure is however restricted to the case of a fixed and regular structure of orthogonal joints and it is the major limitation of the approach. It is shown that for the analysis of toppling failure of layered rock slopes, the Cosserat continuum approach provides the link between the slipping mechanism along the joints and the formation of a tensile failure surface which intersects the foliation discontinuities. The continuous approach is thus coupled with subsequent discrete approach of block stability.

Cerrolaza, M., Sulem, J. & El Bied, A. 1999. A Cosserat non-linear finite element analysis software for blocky structures. Int. J. of Advances in Eng. Soft. (30): 69–83. Goodman, R.E. & Bray, J.W. 1976. Toppling of rock slopes. In ASCE (ed.), Proc. Specialty Conf. on Rock Engineering for Found. and Slopes. Boulder, Colorado: 201–234. Hoek, E. & Bray, J.W. 1981. Rock slope engineering. The Institution of Mining and Metallurgy. Cambridge. Mongiovì, L., Bosco, G. & De Gennaro, V. 1995. Analysis of complex rotational and translational failure mechanisms in jointed rock slopes. In Rossmanith (ed.), Proc. Mech. Of Jointed and Faulted Rock. Wien, Austria: 617–622. Rotterdam: Balkema. Mühlhaus, H.-B. 1993. Continuum models for layered for layered and blocky rock. In: Comprehensive Rock Engng., Vol. 2 (Charles Fairhurst ed.) Pergamon Press: 209–230. Mühlhaus, H.B. 1995. A relative gradient model for laminated materials. In H.B. Mühlhaus (ed.), Continuum Models for Materials with Micro-Structure: 450–482, J. Wiley. Salamon, M.D.G. 1968. Elastic moduli of stratified rock mass. Int. J. Rock Mech. Min. Sci. and Geomech. Abstr. (5): 519–527. Schaefer, H. 1962. Versuch einer Elastizitätstheorie des zweidimensionalen ebenen Cosserat-Kontinuums. In: Miszellannenn der Angewandten Mekanik, Akademie Verlag, Berlin: 277–292. Sulem, J. & Mühlhaus, H.B. 1997. A continuum model for periodic two-dimensional block structures. Mechanics of cohesive-frictional materials. (2): 31–46. Vardoulakis, I. & Sulem, J. 1995. Bifurcation analysis in geomechanics. Blackie Academic & Professional. Zanbak, C. 1983. Design chart for rock slopes susceptible to toppling. ASCE, Journal of Geotechnical Engineering. (109, 8): 1039–1062. Zienkiewicz, O.C. & Pande, G.N. 1977. Time dependent multilaminate model of rocks – a numerical study of deformation and failure of rock masses. Int. J. Numer. Anal. Meth. Geomech.(1): 219–247. Zvolinskii, N.V. & Shkhinek, K.N. 1984. Continual model of laminar elastic medium. Mechanics of Solids. (19): 1–9.

REFERENCES Adhikary & D.P., Dyskin. 1996. A Cosserat continuum model for layered materials. Computers and Geotechnics. (20): 15–45.

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Development of a three-dimensional discontinuous deformation analysis technique and its application to toppling failure H.I. Jang Research Institute of Engineering Science, Seoul National University, Korea

C.I. Lee School of Civil, Urban & Geo-system Engineering, Seoul National University, Korea

ABSTRACT: We developed a three-dimensional discontinuous deformation analysis (DDA) theory and computer program to analyze the deformation of rock blocks influenced by discontinuous planes in rock masses. In this, a first order deformation function was made and the potential energy and sub-matrices of a single block and contacts were obtained using this. Contacts between the blocks were classified into 4 categories (vertex-vertex, vertex-edge, vertex-triangle, edge-edge contact) according to their distance and position criteria. Using input parameters such as center, normal vectors and the radius of the discontinuous plane the three-dimensional block generation program was developed. In the verification test, two cases were analyzed using the three-dimensional DDA program: two-block sliding (one sliding face), and wedge sliding (two sliding faces). The results showed a good match when compared with those derived from an alternative theoretical analysis. The toppling mechanism (domino effect) was also analyzed and similarly gave good results.

1

INTRODUCTION

Since the development of Discontinuous Deformation Analysis (DDA) by Shi (1984), there has been much improvement in the theory and programs. These, however, are all based on the assumption of a twodimensional plane strain or plane stress state; and because a rock block system is a three-dimensional problem a two-dimensional analysis has limited application. A three-dimensional analysis required in the design of rock slopes and underground spaces where three-dimensional discontinuities dominate stability. In this paper, Shi’s two-dimensional DDA theory is extended to encompass three-dimensional theory. The three-dimensional DDA program, thus developed, was verified against three cases that had been solved by alternate means.

2 THREE-DIMENSIONAL DDA THEORY 2.1

Block deformation function and simultaneous equations

DDA calculates the equilibrium equations by minimization of the potential energies of single blocks and contacts between two blocks. To calculate the

simultaneous equilibrium equations, deformation functions must be defined. The deformation function calculates the deformation of all the blocks using the displacement of each block centroid. This function is similar to the shape function of Finite Element Method and can represent the potential energy of the blocks and the simultaneous equilibrium equations simply. Assuming all displacements are small and each block has constant stress and constant strain throughout, the displacement (u, v, w) of any point (x, y, z) of a block can be represented by 12 displacement variables. In the 12 variables, (u0 , v0 , w0 ) is the rigid body translation of a specific point (x0 , y0 , z0 ), r1 is the rotation angle (radians) of block around z axis, r2 is the rotation angle of block around x axis, r3 is the rotation angle of block around y axis, εx , εy , εz , γxy , γyz , γzx are the normal and shear strains in the block. The displacement of any point (x, y, z) in the block can be represented by Eq. (1). In DDA, the equilibrium equation is established by differentiation of the potential energy of the block as with FEM. The stiffness matrix is constructed using the potential energy of a single block and the contacts between two blocks. As DDA is a displacement method like FEM, the equilibrium equations are established by transposition of the constant to the right side, which is calculated by the differentiation of the

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total potential energy with respect to the displacement variables. Assuming there are n blocks in the defined block system, the simultaneous equilibrium equation has the same form as Eq. (2). In Eq. (2).

3.1 Two blocks with a single sliding face Two blocks with a single sliding face were analyzed. Table 1 shows the input variables of the blocks and

  0 0 0 (z − z0 ) (x − x0 ) 1 0 0 −(y − y0 ) u  v  = 0 1 0 (x − x0 ) −(z − z0 ) 0 0 (y − y0 ) 0  w 0 0 1 0 (y − y0 ) −(x − x0 ) 0 0 (z − z0 )

Kij (i = 1, . . ., n, j = 1, . . ., n) is a stiffness matrix which is a 12 × 12 matrix calculated for a single block and its contact with two other blocks; Di represents the displacement variables; Fi is the loading on the blocks, distributed to the 12 displacement variables.      D1 K11 K12 K13 · · · K1n F1 K K K · · · K  21 22 23 F2  2n  D2  K K K · · · K  D  F  3n   3  =  3   31 32 33 (2)      . . . .  . . . . . ..   ..   ..  . . . . . . Kn1 Kn2 Kn3 · · · Knn Dn Fn

2.2 Analysis algorithm of DDA

0

(x−x0 ) 2

(z−z0 ) 2

0

(y−y0 ) 2

the discontinuity properties used in this verification. Fig. 1 shows the initial state, before sliding of the two blocks. The slope angle between the bottom and upper block is arctan(1/2), 26.57 degrees. These two blocks have 4 contacts, which are divided into two categories; the upper two vertex-edge and lower two vertex-edge contacts. Fig. 2 shows the blocks after 100 time steps. Once the centroid of the upper block passed the edge of the bottom block, the upper block fell down while rotating in a counterclockwise direction. There were only two edge-edge contacts in Fig. 2. Fig. 3 shows the z-axis value (height) of the point “a” on block A Table 1.

The algorithm of the three-dimensional DDA is the same as that of the two-dimensional DDA. First, the block generation program generated the block data based on the discontinuities, whether they were fixed or sliding, and the loading point data. The block data, loading data, properties of the blocks, and discontinuity data were then saved. The next step was the contact finding process, which is crucial to discontinuous analysis; for this, a sub-matrix for each single block was calculated by adding or subtracting a normal spring or shear spring according to the contact condition (sliding or fixed). Subsequently, all the contacts were reviewed. If the no penetration and the no tension conditions between each block were not satisfied, the sub-matrices were recalculated until they were satisfied, and the results then saved.

(y−y0 ) 2

  u0 v0    w0 r  1 (z−z0 )   r  2  2   r3    0   · εx  (1)   (x−x0 ) εy    2  εz    γxy γ  yz γzx

Properties of blocks and discontinuities in 3.1.

Spring stiffness Block stiffness Unit mass

2 GN/m 1 GPa 2.7 t/m3

Time step Total steps Poisson’s ratio

0.1 (second) 200 (step) 0.24

3 VERIFICATION Using the three-dimensional DDA program, two cases were used for verification. The first had two blocks with a single sliding face, and the other, a wedge analysis. The unit of length in this chapter is meter (m).

Figure 1.

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Sliding of block A over block B (initial state).

Figure 2.

Sliding of block A over block B (after 100 steps). Figure 4. Deformed block B of wedge sliding analysis (initial state).

Figure 3. Position of block (z-coordinate of point a) as a function of friction angle (after 100 steps).

(shown in Fig. 1) at different friction angles after 50 time steps. For this slope, the z value of “a” was at the initial value of 7.5; this value changed with different friction angles. There was however, no movement when the friction angle was greater than the slope angle. Conversely when the friction angle was smaller than slope angle, the measured point moved down (i.e. z got smaller), and the smaller the friction angle the greater the movement. This is illustrated in Fig. 3 the measured point does not move with friction angle of 26.57◦ or larger but remains constant at a value of 7.5. Thus, this shows that analysis program calculates the sliding between two blocks exactly. 3.2

Wedge sliding

This analysis was performed to determine whether the three-dimensional DDA program could calculate an

Figure 5. Deformed block B of wedge sliding analysis (after 200 steps).

exact value when sliding occurs along two sliding planes. The mechanical properties were the same as those given in table 1, chapter 3.1. Fig. 4 shows the blocks before analysis. The overall block was 10 m on each side and was divided into 4 blocks by two planes whose center points were (0, 5, 5), (0, 5, −1.25) with dips of 40◦ and 60◦ , and dip directions of 130◦ and 200◦ , respectively. The cases in Figs 1–3 were analyzed for a single slope angle whereas the cases in Figs 4–6 were analyzed for two different slope angles. In Fig. 4 all the blocks were fixed except the wedge shaped block B, so only block B could move as the friction angle was changed. This case was also modeled for comparison using 3DEC, the three-dimensional

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Figure 7. Three-dimensional blocks used in toppling analysis. Figure 6. Position of block (z-coordinate of point b) as a function of friction angle (after 200 steps).

discrete element code; this model has the same geometry and properties as the “sliding wedge model” in the manual “Verification Problem” 3DEC version 2.0 (Itasca, 1998). Fig. 5 shows the displaced position of block B after 200 steps with a friction angle of 10◦ . In the initial state there were a total 6 contacts (two sets of 3 vertex-vertex contacts) between block B and the other side-blocks; after sliding down, the contact state changed to a different 6 contacts, including two sets of two vertex-triangle contacts and two edge-edge contacts. In the analysis of Fig. 6, the z-axis value of point “b” of block B is measured against a changing friction angle varied from 31◦ to 36◦ , in order to calculate the lowest friction angle when the wedge begins to slide. The critical friction angle of wedge block calculated theoretically by Hoek & Bray (1979) was 33.36◦ and the calculated value by 3DEC was 33.19◦ , which is about 0.5% smaller than theoretical value. The calculated value (Fig. 6) using the three-dimensional DDA was 33.36◦ , exactly the same as the theoretical value.

Figure 8. Deformed blocks in toppling analysis after 30 seconds.

4 TOPPLING FAILURE SIMULATION To simulate toppling failure the simplified model in Fig. 7 was tested. There were ten blocks on the plate and two 0.1 MN forces were applied at the upper corners of the No. 10 block as in Fig. 7. The forces were applied for 90 seconds out of the total time of 120 seconds using a friction angle of 10◦ . Fig. 8 shows the results after 30 seconds. All the blocks rotated in a counterclockwise direction. Fig. 9 shows the results after 50 seconds. No. 1 and No. 2 blocks began to separate after more rotation. After 80 seconds, No. 1 block was contacting the lower plate. Because the forces were loaded at two points only up until 90 seconds,


Fig. 11, which is after 120 seconds, has no external force associated with it. Accordingly No. 2 and No. 3 blocks have separated and No. 10 block is shown sliding down the face of the No. 9 block. The state of ten blocks in Fig. 7 was analyzed changing on angle between the blocks from 20◦ to 23◦ , the

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all contacts change to sliding state and all blocks start moving. This means that 21.8◦ is the critical friction angle for this toppling failure analysis. 5



Figure 12.

Shi’s discontinuous deformation analysis (DDA) has been developed in many parts, but until now DDA analysis has been based on two-dimensional plain strain or plain stress. Two-dimensional analyses are limited because discontinuities are basically a three-dimensional problem. We developed a three-dimensional discontinuous deformation analysis theory and computer program, to analyse the deformation of blocks made by discontinuous planes in the rock mass. To develop the three-dimensional DDA theory, a first order deformation function was created and the potential energy and sub-matrices of a single block and contacts were obtained using this. In the verification test, the program calculated the two exact theoretical friction angles at the onset of sliding, which were 26.57◦ for the two-block sliding case and 33.36◦ for the wedge sliding case. In the toppling mechanism analysis, the program calculated the block deformation after 30, 50, 80, 120 seconds and critical friction angle and gave good results. REFERENCES Hoek, E. and Bray, J.W., 1979, Rock slope engineering, Institute of Mining and Metallurgy, London. Shi, G.-H. and Goodman, R.E., 1984, Discontinuous deformation analysis, Proceedings of the 25th U.S. Symposium on Rock Mechanics, pp. 269–277. Shi, G.-H. 1988, Discontinuous deformation analysis: a new numerical model for the static and dynamics of block systems, PhD thesis, Civil Eng., University of California, Berkeley. Yeung, M.R., 1991, Application of Shi’s discontinuous deformation analysis to the study of rock behavior, PhD. Dissertation, Civil Eng., U.C. Berkeley Ohnishi, Y., Chen, G. and Miki, S., 1995, Recent development of DDA in rock mechanics, Proceedings of the First International Conference on Analysis of Discontinuous Deformation, Chunghi, Taiwan, pp. 26–47. Cundall, 1998, 3DEC User’s manual, Itasca consulting group.

Stability analysis of toppling blocks.

result is showed in Fig. 12. Fig. 12 shows the change of the contact number which can be fixed or sliding according to the change of the friction angle. Below the friction angle 21.8◦ , all contacts remain fixed state and blocks show no movement, but above that angle,

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CONCLUSION


Three-dimensional Discontinuity Network Analysis (TDNA) on rock mass Xiao-Chu Peng & Hong-bo Tang Mid-south Institute of Investigation Design Research, Changsha, Hunan Province, PRC, China

ABSTRACT: TDNA is of the discontinuity mechanics. It is a new three-dimensional discontinuity numerical analysis method for analyzing the structural features of rock mass. TDNA will firstly vividly show the space attitude, location, size and mechanical characteristics of the actual structured planes system (which is fully random but with regularity) in the analyzed zone by using distinct multi-dimensional Montre-Carlo method; secondly, search out the network system of the structural planes; thirdly, analyze the topologic, geometric, kinetic and mechanical characteristics of the system; and finally, display the moving tendency of the rock mass by stepped animation frames. Based on this, the process and the final formation of the failure of the rock mass will be inferred, and the specific zone and depth of the failure of the rock mass will be defined. So, the fairly accurate parameters such as the depth, orientation and anchoring force of the anchorages can be obtained.

1

INTRODUCTION

Structural planes such as joints, fissures, beddings and faults intersecting in the rock mass in versatile direction is an important structural characteristics of rock mass. While existing of a lot of structural planes make the deformation mechanism and the mechanic mechanism of the rock mass have substantial difference from those of the continuous media, the pattern of and the principles for the rock disruption also have large difference form those of the continuous media, and mainly depend on the density, the combination pattern and the mechanical characteristics of the structural planes. Therefore, numerical analysis on the stability of the rock mass requires to accurately express the structural planes in density, dimensions, attitude, location and mechanical characteristics. In this sector, experts as Priest, S.D. & Hudson, J.A. (1)(2)(3) have done much work and comparatively mature conclusions have been made. This paper will set forth the way to establish the mathematical model of the discontinuity system closer to the actual conditions, and judge the stability of the rock mass and provide the supporting parameters for the unstable blocks on the basis of calculation of the internal forces in the system after obtaining the discontinuity system. TDNA method, as described in this paper, supplements the Block Theory and has overcome the following disadvantages of “Block Theory”: 1. As a numerical method of discontinuity mechanics, the accuracy of the formation of the discontinuity

system directly determines the reliability of the analysis results. During the simulation of the discontinuity system by “Block Theory”, the structural planes are attitude-simplified in groups and then are simulated, that undoubtedly results in the inconsistency between the simulation result and the actual random discontinuity system; so, it is naturally difficult to gain final analysis results approximate to the actual results. 2. The Block Theory of “Complete Three Dimensional Analysis” established by Dr. Shi Genhua is, in fact, a combined analysis on trace lines on the three-dimensional structural planes (or polygonal planes). Hence, it is quasi-three-dimensional. Therefore, the Block Theory can only consider the self gravity of each block but not any external force (however, the external forces such as, crustal stress, seepage pressure and ex-system agent, are the decisive factors for stability of rock mass), and neither the internal force of the structural planes system, so that many dangerous factors are ignored and cases such as stability of anti-dip structural planes can not be distinguished. All these result in distortion of the analysis results. 3. Block Theory can not provide the process, extent and degree of the failure as well as the final pattern and the total volume. It can only distinguish the key block, but can not estimate the impacts of the failure of the key block in the rock mass. 4. Block Theory can not provide the supporting forces required for engineering stability, since it only

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considers the self-gravity of each block but can not consider the impacts on the whole system, therefore, the supporting forces provided by Block Theory can not reflect the supporting forces actually required by the rock slope engineering. TDNA has resolved the above problems in a relatively good way. TDNA has stronger pre-processing and postprocessing functions and less input data, showing the designer in vivid three dimensional graphs the output results that make the analysis results visibly clear. The principles of TDNA are introduced as follows. 2 2.1

NETWORK SIMULATION OF STRUCTURAL PLANES Simulation of the attitudes of random structural planes

If the attitudes of structural planes are expressed in pole density graph, its density distribution can hardly be expressed by a simple function. Figure 1 shows the statistical density diagraph of poles of small faults in an adit of a slope works. It is obvious that the distribution of pole points is discontinuous and it’s impossible to get the bivariate function expressing the attitude density distribution by means of fitting. Therefore, the author suggests to gain the subsamples, which are for a relatively accurate simulation of attitude of each fault, by means of the distinct two-dimensional Montre-Carlo method. The distinct two-dimensional Montre-Carlo method is described as follows: Suppose that there is two dimensional variable (x, y) and its probability density function is f(x, y), a ≤ x ≤ b, c ≤ y ≤ d, and its accumulative distribution is F(x, y). In accordance with the definition of probability, there shall be F(b,d) = 1 Step one: calculate the total distribution of x, F1 (x): # d f (x, y)dy F1 (x) = c

inf

F1(t)≥r2n−1

t

where, ζxn is the independent subsample sampled in the nth turn and in accordance with the distribution function F1 (x), and r2n−1 is the (2n − 1)th pseudo-uniform random number, 0 ≤ r2n − 1 ≤ 1 Step three: calculate subsample of y, ζyn ζyn =

inf

F(ζxn,t )≥r2n

Contour diagram of joint poles of insite.

Figure 2. Contour diagram of joint poles by twodimensional sampling.

Step two: calculate the subsample of x, ζxn ζxn =

Figure 1.

t

Subsamples gained in this way form a point couple (ζ xi, ζ yi). The point couples gained from several sampling, i = 1, 2, . . ., n, are independent from each other

and their density distribution follows the distribution function, f(x, y). On the basis of the statistical data as shown in Figure 1, Figure 3 is the density distribution of pole points in accordance with the independent samples and Figure 2 the density distribution of pole points sampled by means of two-dimensional Montre-Carlo method. From comparison of these three diagraphs, it can be concluded that the general distribution by means of two-dimensional Montre-Carlo sampling method has much higher accuracy than that by means of independent sampling.

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structural plane as a plane or a polygonal plane and adjusts the mechanical parameters of the structural plane to compensate the impacts in form. Suppose that structural plane P is a plane disk with a radius of R, a normal vector of n(A, B, C) and a disk center of PC (xC , yC , zC ), then the disk can be expressed as:  M(x, y, z) ∈ P : Ax + By + Cz = D   (x − xC )2 + (y − yC )2 + (z − zC )2 = R 2 D = AxC + ByC + CzC  A2 + B2 + C2 = 1

Figure 3. Contour diagram of joint poles by independent sampling.

After the attitude of the structural plane is illustrated, there yet exist the following factors affecting the structural characteristics of the rock mass: a) density, b) dimensions, c) opening, d) characteristics of the filling and e) location of the structural plane. In the process of numerical analysis, characteristics of the opening and the filling of the fault can be expressed together in coefficients to show its characteristic parameters such as deformation, strength, seepage etc. The above mentioned five parameters have close correlation between each other. If they are sampled in an independent way, undoubtedly the results gained will have great difference from those of the actual cases. Therefore, multi-dimensional MontreCarlo method shall be adopted for sampling. The structural plane system simulated in this way is closer to the actual conditions in density, size, location and mechanical characteristics etc. The determined large faults or the discontinuities that has significant importance for an engineering analysis can be accurately defined in this method. 2.2

Expression of the structural plane

Barton and Long (4) think that the boundary of a structural plane can be deemed as a circle or an oval. When a structural plane develops to intersect with an earlier structural plane, it will not develop further more since the strain energy is be fully released. Therefore, the simulated structural plane is a circle at the very beginning but will not have a final boundary as a circle. A structural plane is three-dimensional in form, but its thickness is very small in comparison with its plane dimensions. In simulation, the author deems the

For convenience of calculation, the periphery of the said disk can be simplified in straight-line segments. For example, if a line segment is corresponding to a center angle of π/6, the start point of the segment is pr−1 (xr−1 , yr−1 , zr−1 ), and the terminal point is pr (xr , yr , zr ), r = 1, 2, . . ., 12, then the equation for this line can be written as x − xr−1 y − yr−1 z − zr−1 Lr : = = = tr xr − xr−1 yr − yr−1 zr − zr−1 (0 ≤ tr ≤ 1) The equations of 12 straight-line segments are calculated. In turn, the boundary of the disk can be approximately determined. After the random structural plane being calculated by the said method, the intersection with each other of the structural planes can be judged. 2.3 Judging the correlation of the structural planes Suppose that the coordinates of the center points of structural plane Pi and Pj are Pci (xci , yci , zci ) and Pcj (xcj , ycj , zcj ) respectively, the normal vectors Ni (Ai , Bi , Ci ) and Nj (Aj , Bj , Cj ), the radii Ri and Rj . The vector of the intersection line of the two planes is   i j k Nij = Ni × Nj = Ai Bi Ci  Aj Bj Cj = Aij i + Bij j + Cij k where Aij = Bi Cj − Bj Ci , Bij = Ci Aj − Cj Ai , Cij = Ai Bj − Aj Bi . On the basis of a point, Dij (xij , yij , zij ), on the intersection line of the two planes Pi and Pj , the equation of the intersection line of planes (or extended planes) Pi and Pj is gained as follows: x − xij y − yij z − zij = = = t0 Aij Bij Cij (−s∞ < t0 < ∞)

Lij :

Lij is an infinite long line. First use Lij to make the judge with the periphery straight line segments of plane Pi , then the segment Lij (of Lij ) within Pi can

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be gained. Consequently, make the judge with line Lij and the periphery straight lines segments of plane Pj . There can be three cases which follows: 1. With no intersection point: It illustrates that Lij can’t intersect the periphery line segments of Pj and there are two possibilities: the first is that Lij is completely out of Pj , planes Pi and Pj can’t intersect; the second is that Lij is completely in the domain of Pj and the intersected part of the planes Pi and Pj is Lij . 2. With two intersection points: it illustrates that the two planes must intersect with two possibilities: the first is that Lij is within the very scope of the periphery lines of Pj and the actual intersect line is Lij ; the second is that Lij has two ends beyond the domain of Pj and middle part is within the domain of Pj , and the actual intersected line of the two planes is the middle part. 3. With only one intersection point: It illustrates that the planes must intersect but the intersection line is only a part of Lij .

Figure 5.

Zone cut by one discontinuity.

Figure 6.

Zone cut by two discontinuities.

Figure 7.

Zone cut by three discontinuities.

Thus, it can be determined whether the two planes intersect or not. If they intersect, the specific location and scope of the intersection line can also be gained. Cap all the planes with one another in the system, for each structural plane, the following information can be gained: 1. Quantity, Nos. and the intersecting scope of the other planes intersecting with it. 2. Trace lines of all the planes which intersect with it, and the corresponding network relationship. 3. Mechanical characteristics and seepage characteristics of the structural planes intersecting with it. After these steps, the correlations of the structural plane system are clear and structural plane network system is formed, and, the slope outlook can be clearly expressed. The following figures (Figures 4–9) show the situation of a slope cut by structural planes.

Figure 4.

Model of analyzed zone.

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of information as follows: 1. Geometrical information such as space location, dimensions, direction etc. of each subset constituting the block. 2. Topological information expressing the relationship of the subsets.

Figure 8.

Zone cut by four discontinuities.

Figure 9.

Zone cut by five discontinuities.

3

Geometrical information and topological information are necessary for complete expression of a polyhedron. In the past, the method usually used for gaining the geometrical information was lower hemisphere equal area projection in combination with scaled solid project. For a simple polyhedron, this method is simple and effective. However, for any arbitrary polyhedron with over five boundary planes, the geometrical and topological information is pretty hard to gain. Furthermore, such an arbitrary polyhedron often occurs in an actual project. Therefore, the author initiates the searching method of a polyhedron as from the plane to the edges and then from the side to the planes. The method can make geometrical and topological information of any polyhedron be easily gained. Each plane has many trace lines intersected with other planes. Some of the trace lines can form closed loops but others can not. While the finite space plane enclosed by a closed loop must be a boundary plane of the polyhedron, an edge of the closed loop must be an edge of the polyhedron and nodes of the closed loop must be vertexes of the polyhedron. Therefore, after setting a closed loop of a structural plane Pi , and a point Q, in a given semi-space of this loop, the another plane Pj , which intersects Pi , can be sought out on the basis of edges of the loop, sand the Pj ’s semi-space in which the block is can be defined by point Q, so on and so forth, all the information of the block can be gained.

SEARCHING OF STRUCTURAL BODIES

A structural body in actual projects is a polyhedron with very complicated pattern. A polyhedron correctly defined shall not be self-intersecting and shall be directed and shall have no gap between each two intersected planes. Let H express finite number of three-dimensional structural planes, semi-space subdivision can be done by H. The set of the subdivided semi-spaces is a compound of the polyhedron, let’s say, A(H). Obviously, subsets ofA(H) are the points, the edges, planes and the polyhedron. A polyhedron is the sum of the directed boundary planes set and the inbody points set. The subsets of a polyhedron are points, sides and planes. Therefore, a polyhedron can be expressed by two parts

1. Coordinates of the point (on one side of the block) which all the polygonal planes in the block and constituting the block point to. 2. Array of the plane nos. 3. Number of the semi-space (of the plane) in which the block is. 4. Quantity of the edges of the plane. 5. Coordinates of each acnode. To avoid any omission in the block searching, a check matrix, ITEST(i, j, k), shall be established. The matrix expresses the searching case of the blocks in No. k semi-space of No. j closed loop of No. i plane (k = 1 expresses the upper semi-space of the loop and k = 2 the lower one). Before searching, let all array elements of ITEST be 0. If the blocks in No. k semispace of No. j closed loop of No. i plane have been sought out. Let ITEST(i, j, k) = 1. And do one check on ITEST after each block being sought out. If it is found that elements of an array are 0, using the plane,

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Figure 10. The main blocks in the structural system.

loop and semi-space expressed by this array element as the first data of the new block to conduct the searching. When all ITEST elements are equal to 1, it means that the block searching has been finished and the block system is formed. Figure 10 shows the main blocks in a slope cut by five structural planes. 4

CALCULATION OF INTERNAL FORCES OF THE STRUCTURAL PLANES NETWORK SYSTEM

After formation of the block system, circulating iteration is done on the system under external forces and self gravities so as to obtain the force-field of the whole system, and then force balancing analysis is made for individual blocks in the system. If any block can not be naturally balanced, a stability failure occurs. After the block falls, a new system will form, consequently

the force-field will redistribute to form a new balance, so on so forth, balance and failure of new system, rebalance and refailure, until a new fully-stable and balanced status forms. The force transmission calculation by TDNA method is greatly different from that by means of distinct element method. TDNA method deems that the system under study is transiently stable. The deformation of the system has been settled in the long geological history and the system is in stable conditions. Therefore, the contact relationship of all the blocks keeps unchangeable and the contact forces between in the blocks are static. The block is a rigid body that is unbreakable. Hence, the unbalanced forces acted on each block are balanced by the counter forces on the boundary planes. Magnitude and distribution of the counter forces can be gained by means of establishing three-axis force balancing and force couple balancing. Suppose that block A contacts elastically with blocks B, C, D, . . . surrounding it, block A must shift and rotate while it is acted by force F and force couple M. The shift value and the rotating angle can be gained by establishing balancing equations on the basis of the contact cases of block A with its surrounding blocks as well as the normal and tangential stiffness of the contact surfaces. On the basis of the gained shift value of the block and rotating angle of the block round its gravity center, the displacement increment of each acnode of the polyhedron, δ, can be solved. Superimpose δ on the locations of each acnode of block A before movement, the updated location of the form center, boundary planes and acnodes of block can be solved. Block A at the updated location will produce unbalancing forces and force couples on the surrounding blocks B, C, D, . . . By the same way for block A, the new locations after balancing of the block B, C, D, . . . can be solved. The calculation is done so on and so forth in such a pattern, until the locations of all the blocks in the system are in the balanced location and unbalanced forces will no longer exist. If the force system of one block can not be balanced, the block fails and consequently the systematic structure and form change and a new regulation process of unbalanced forces occurs. Calculation is done round and round in this way until new failure no longer occurs in the damaged system. The process of the block failure will be showed in three-dimensional blanking graphs by means of normal axonometric projection. The graphs have strong sense of three dimensions and clear dimensional scales. The drawings also show the location, shape, volume, supporting force to be provided and the failure pattern, which can be directly employed for the project site. The following figure (Figures 11–12) shows the slope shape after failure of two blocks.

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method, the probability distribution of the critical blocks in the slope can be sought out. Provided the statistical information is reliable, the method can predict and judge, in a microscopic view, the slope stability and the failure pattern as well as the chain reaction process of the collapse in a significant accuracy. This method can describe the form of any complicated block, including the concave blocks, as well as make accurate analysis on single dangerous block at the project site and provide corresponding supporting parameters.

REFERENCES Figure 11.

Figure 12. failure.

5

First falling block and zone outline after failure.

Second falling block and zone outline after

CONCLUSIONS

In the random three-dimensional structural plane network (of the rock slope) generated by statistical

Priest S.D. & Hudson J.A. Estimation of discontinuity spacing and trace length using scanline surveys. Int. J. Rock Mech. Mine Sci. and Geomech. Abstra. 18, 183–197 (1981). Sen Z. & Kazi A. Discontinuity Spacing and RQD Estimates from Finite Length Scanline. Int. J. Rock Mech. Mine Sci. and Geomech. Abstra. 21, 203–212 (1984). Pahl P.J. Estimation the Mean Length of Discontinuity Traces. Int. J. Rock Mech. Mine Sci. and Geomech. Abstra. 18, 221–228 (1981). Goodman R.E. & Shi, GenHua. Block Theory and its Application to Rock Engineering Prentice Hall INC 1985. Shi GenHua & Goodman R.E. The Key Blocks of Unroolled Joint Trace in Developed Maps of Tunnel Walls. Int. J. for Num. and And. Methods Geo. Vol. 13, No. 2, Mar–Apr. 1989. Hoek E. Methods for the Rapid Assessment of the Stability of Three-dimensional Rock Slopes. Q. JL Engeng. Geol. 6(3) (1973). Lin D. & Fairhurst C. Static Analysis of the Stability of Three-dimensional Blocky Systems Around Excavations in Rock. Int. J. Rock Mech. Mine Sci. and Geomech. Abstra. Vol. 25, No. 3, 139–147 (1988).

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