N U M E R I C A L M O D E L L I N G OF R O C K M O V E M E N T S A R O U N D

M I N E O P E N I N G S

A thesis submitted to the University of London

by

Xiao—Dong Pan

In partial fulfilment of the requirements for the degree of

Doctor of Philosophy and for the Diploma of Membership

of Imperial College

November 1988

ABSTRACT

A primary concern in mining operations, be they surface or underground, is the

control of the displacements in the rock surrounding the excavations. In the cases

of weak or jointed rock strata, the displacements are often large, so that they are

referred to as 'strata movement'.

The existing numerical methods currently used in geomechanics are examined

in order to provide the basis for the development of an efficient tool for modelling

the strata movement. It is concluded that the full nonlinear analysis of ground

response is still fax beyond the capability of any existing numerical programs.

The work in the thesis is intended to apply updated numerical techniques for this

specific rock engineering problem.

In the modelling of a rock mass as a continuum, the two-dimensional elasto-

viscoplastic finite element method is used. A rock mass response model is incor-

porated with a modified Hoek-Brown yield criterion. A geometric nonlinearity

analysis, which is essential for modelling large displacements of the rock masses,

is implemented in the program using the updated Lagrangian formulation. The

influence of both the time-dependent behaviour of rock masses and the axztual

excavation sequence are also considered.

A numerical analysis to investigate the validation and to highlight any discrep-

ancies of plane strain analysis in modelling three-dimensional tunnel excavations

is presented. Some of the relations between the results given by these 2-D and 3-D

analyses are clarified.

In the modelling of a rock mass as a discontinuum, the two-dimensional dis-

tinct element method using the static relaxation procedure is adopted. For a

more general case, a hybrid computational scheme is proposed, in which the finite

element method is used in conjunction with distinct elements to model the inter-

action of blocky rock with a softer rock mass. The hybrid program COUPLE is

described.

The practical implications of the research are demonstrated by analysing typ-

ical gate roadways in Coventry Colliery in the U.K.. The factors influencing the

roadway deformations and stabilities are investigated. It is shown that the hybrid

program is capable of dealing with the most general analysis of strata movements

and assisting in the design of mining excavations.

ACKNOWLEDGEMENTS

The author wishes to express his sincere appreciation to the Royal School of Mines

of Imperial College for the grant of the 'Stephen & Anna Hui Fellowship' and to

the HQTD of British Coal for financial support and providing in situ data for this

research project.

He is also grateful to many people who assisted in the completion of this

thesis. Special thanks are given to the following:

Dr. J. A. Hudson for supervision of this work in many ways, for providing

a very good research environment, and for the proofreading of this thesis.

Dr. J. O. Watson for his initial supervision of this work and for stimulating

the author's ideas.

Professor E. T. Brown for the arrangement and encouragement of the re-

search and for enlightening discussions.

Dr. M. B. Reed of Brunei University for co-operation throughout this re-

search programme and for his enthusiasm, assessment and advice of this work.

Professors Z. H. Zhao and D. F. Hu at Tongji University, Shanghai, China

for arrangement and support of the author's leaving to undertake this re-

search.

The postgraduate students with whom the author has worked very enjoyably

and hopes has developed lasting friendships. They are: Ming Lii, Chris

Dyke, Ian Harrison, Andy Hyett , Bailin Wu and Paul Carton.

The staff of the Engineering Rock Mechanics Research Group, notably. Miss

M . J. Knox, Dr. D . H. Spencer, Mr. J. P. Harrison and Miss E. M.

Diskin for help of many kinds.

Finally, the author wishes to take this opportunity to thank his wife Min,

whose understanding, patience and assistance enabled him to complete this thesis,

and his daughter Qin, to whom the author feels deeply in debt for her long-time

separation from her parents during childhood.

CONTENTS

Contents Page

A B S T R A C T 1

ACKNOWLEDGEMENTS 2

CONTENTS 3

LIST OF FIGURES 7

LIST OF TABLES 15

LIST OF SYMBOLS 16

CHAPTER 1

INTRODUCTION 19

1.1 Conceptual models of rock mass response to excavation 21

1.2 Strata movement problems in rock engineering 24

1.3 Roadway stability analysis in longwall coal mining 26

1.4 Purpose and structure of this thesis .33

CHAPTER 2

REVIEW OF THE NUMERICAL METHODS AVAILABLE

FOR STRATA MOVEMENT ANALYSIS 36

2.1 Introduction 36

2.2 The finite difference method 40

2.3 The finite element method 44

2.4 The boundary element method 51

2.5 The distinct element method 56

2.6 Hybrid methods 62

2.7 Other methods 67

2.8 Conclusions and discussion 69

CHAPTER 3

TWO-DIMENSIONAL NONLINEAR FINITE

ELEMENT ANALYSIS OF ROCK MASSES 71

3.1 Introduction 71

3.2 Nonlinear finite element analysis 72

3.2.1 The basic expressions and algorithm 72

3.2.2 Description of the program 77

3.3 The rock mass behaviour model 80

3.3.1 Constitutive relation and plasticity theory 80

3.3.2 Yield criteria for rocks 82

3.3.3 The Hoek-Brown criterion 88

3.3.4 A simplified 3-D criterion 99

3.4 Material nonlinear analysis 103

3.4.1 Introduction 103

3.4.2 Yield function and plastic potential function 103

3.4.3 The flow vectors and the matrix H"" 106

3.4.4 Modelling of time-dependent behaviour of rock masses 110

3.4.5 Validation examples 112

3.5 Geometric nonlinear analysis 126

3.5.1 Introduction 126

3.5.2 Some definitions and assumptions 127

3.5.3 Basic formulations 128

3.5.4 Incremental procedure 132

3.5.5 Validation examples 133

3.6 Excavation sequence analysis 143

3.7 Summary 147

CHAPTER 4

P L A N E STRAIN ANALYSIS IN MODELLING THREE-

DIMENSIONAL TUNNEL EXCAVATIONS 148

4.1 Introduction 148

4.2 An assessment of the previous opproaches 149

4.2.1 Plane strain solutions in Rock Mechanics 149

4.2.2 Advancing tunnel analysis 151

4.2.3 Influence of axial stress and loading history 153

4.2.4 Concluding remarks 155

4.3 A numerical procedure for investigation 156

4.3.1 Preliminary considerations 156

4.3.2 Numerical simulations 157

4.4 Computational results 162

4

4.4.1 Elastic solutions 162

4.4.2 Elasto-viscoplastic solutions 164

4.5 Application of the results 176

4.6 Conclusions and discussion 177

CHAPTER 5

A HYBRID COMPUTATIONAL SCHEME 179

5.1 Introduction 179

5.2 The distinct element method 180

5.2.1 Description of the method 180

5.2.2 Validation of the program 188

5.2.3 Some modifications 197

5.2.4 The input parameters for contacts 201

5.2.5 Summary and discussion 205

5.3 The hybrid finite element - distinct element method 207

5.3.1 The coupling principles 207

5.3.2 The algorithm and compatibility conditions 208

5.3.3 The program structure and convergence criteria 212

5.4 Test of the hybrid program 217

5.4.1 Consolidation of blocks on a deformable body 217

5.4.2 A tunnel roof undergoing support deformation 220

5.4.3 Excavation of a square tunnel 222

5.5 Conclusions 230

CHAPTER 6

A NUMERICAL CASE STUDY

— Roadway deformation at Coventry Colliery 232

6.1 Introduction 232

6.2 Descriptions of the roadways at Coventry Colliery 232

6.2.1 Overview 232

6.2.2 Roadway details 233

6.2.3 In situ measurements 237

6.2.4 Conclusions and discussion 242

6.3 A simplified 2-D roadway model 244

6.3.1 Problem idealisation 244

6.3.2 Numerical idealisation 246

6.3.3 Simulation of 3-D excavations 252

6.3.4 Summary and discussion 254

6.4 Computational results 255

6.4.1 A representative simulation —simulation 1 255

6.4.2 Weak floor strata —simulation 2 272

6.4.3 Weak roof strata —simulation 3 275

6.4.4 Weak coal seam —simulation 4 280

6.4.5 Conclusions concerning the numerical simulation 284

6.5 Comparisons of results with measurements 285

6.5.1 Comparison of closure 285

6.5.2 Comparison of floor lift 288

6.5.3 Comparison of pack load 289

6.5.4 Beick analysis of in situ rock mass properties 290

6.6 Conclusions and practical implications 290

CHAPTER 7

S U M M A R Y A N D CONCLUSIONS 293

REFERENCES 299

APPENDICES

APPENDDC 1 DESCRIPTIONS OF PROGRAM 'VISCO' 318

APPENDIX 2 SOME OTHER PROPERTIES OF THE EXTENDED

H-B CRITERION 320

APPENDIX 3 INSTRUCTIONS OF THE PROGRAM 'COAL' 326

APPENDDC 4 DERIVATION OF LARGE DEFORMATION MATRIX

337

APPENDIX 5 DESCRIPTIONS OF PROGRAM 'BLOCK' 341

APPENDDC 6 DESCRIPTION OF NEW SUBROUTINES IN

PROGRAM'COUPLE' 354

APPENDDC 7 STRATA PROPERTIES AND MEASUREMENTS

FOR CASE STUDY 356

APPENDIX 8 REST OF THE FIGURES 367

LIST OF FIGURES

Figure Title Page

1.1 Engineering rock mechanics problems 20

1.2 Conceptual models relating rock structure ajid

rock response 22

1.3 An example of slope excavation in a

complex medium 25

1.4 Roadways in longwall coal mining 29

1.5 Three main systems of roadway formation 29

1.6 An example of roadway support system in

cross sectional view 30

1.7 Roadway convergence measured along the gate

axis 30

1.8 Large roadway deformations at British coalfields 32

2.1 The static problem of a tunnel excavation 37

2.2 Analysis of slope stability using the explicit

finite difference method 43

2.3 Analysis of underground excavation using the

implicit finite difference method 43

2.4 Finite element analysis of tunnel excavation 48

2.5 Finite element analysis of mine subsidence 48

2.6 3-D finite element analysis of longwall coal mining 50

2.7 Example of the boundary element model for

underground excavations 54

2.8 A boundary element analysis for mining excavation 55

2.9 Modelling surface subsidence using program

BLOCKS (rigid block model) 59

2.10 Modelling surface subsidence using program UDEC 61

2.11 Example of the hybrid FE - BE analysis of

underground excavations 63

2.12 Illustration example of the hybrid DE - BE method 65

3.1 Basic one-dimensional elasto-viscoplastic

model 72

3.2 Isoparametric quadrilateral finite elements 78

3.3 Stress - strain curves for rocks 80

3.4 Mohr-Coulomb and Drucker-Prager yield surfaces

in 3-D principal stress space 83

3.5 General nonlinear relation of stress components

of rock at failure 84

3.6 Serata et al.'s theory (1972) on yielding and

failure surface charaxzteristics of rocks 87

3.7 Influence of the principal stress erg

on rock failure 89

3.8 An assumption associated with the Hoek-Brown yield

surface (before 1987) 92

3.9 Characteristics of a stress point on a general

yield surface 92

3.10 The Hoek-Brown yield surface in 3-D principal

stress space 94

3.11 An interpretation of test data in the octahedral plane

using the Hoek-Brown criterion 95

3.12 The construction of the Hoek-Brown yield surface

in 3-D principal stress space 97

3.13 Johnston's comparison of the two criteria 98

3.14 The alternative interpretations of the data

using the Hoek-Brown criterion 98

3.15a The Hoek-Brown yield surface for weak rock masses 101

3.15b The extended Hoek-Brown yield surface 101

3.16 The average surface between the 'outer apices'

and 'iimer apices' of the Hoek-Brown criterion 102

3.17 Singularities on the Hoek-Brown surface 105

3.18 A flow rule using the Hoek-Brown yield surface

and its extended form 105

3.19 A confined block under uniaxial compression 113

3.20 Computational results of the block problem 115

8

3.21 Thick cylinder analysis 116

3.22 Load - displacement relation 117

3.23 Stresses distribution of the thick cylinder 118

3.24 Axisymmetric finite element mesh for circular

tunnel excavation problems 120

3.25 Displacement distributions calculated by

different flow rules 124

3.26 Stress distribution and its relaxation with time 125

3.27 Influence of the parameter r on the solution

convergence and accuracy 125

3.28 Colunm buckling analysis 135

3.29 Computed response of the column 136

3.30 Nodal displacements and element shapes

before column collapse 137

3.31 Cantilever under distributed load 138

3.32 Large displacement analysis of the cantilever,

- comparison of computed and analytical results 140

3.33 The Influence of iteration steps on solution

accuracy 140

3.34 Mesh deformations calculated using the

large deformation theory 141

3.35 Mesh deformations calculated using the

infinitesimal deformation theory 141

3.36 Large displacement analysis considering

material yielding 142

3.37 Influence of different simulations of excavation

on prediction of plastic zone 145

4.1 Advancing tunnel analysis 152

4.2 Comparisons of stresses and displacements for

two- and three- dimensional analyses 154

4.3 Axisymmetric three-dimensional model for tunnel

excavation 158

4.4 Axisymmetric finite element meshes used for three-

and two-dimensional tunnel excavation analyses 160

4.5 A reference cross section in the three-dimensional

model for compaxison with a two-dimensional analysis 161

4.6 Elastic solutions for radial displacements behind

the advancing face 163

4.7 2-D simulations of the advancing tunnel problem

using the proposed numerical excavation procedure 165

4.8 Elastic solutions for stress distributions in

a tunnel section 166

4.9 Stress distributions in a section near the

tunnel face 167

4.10 Some of the yield surfaces used in the

nonlinear analysis 168

4.11 Elasto-viscoplastic solutions for radial

displacements behind the advancing face 170

4.12 Elasto-viscoplastic solutions for radial

displacements in the tunnel wall 171

4.13 Elasto-viscoplastic solutions for stress

distributions in the tunnel wall 172

4.14 Development of the plastic zones in the 3-D

axisymmetric model 174

4.15 The path of axial stress Oz as the

tunnel face advances 175

5.1 Calculation procedure of the distinct element

method 181

5.2 Normal and shear interactions for static relaxation 183

5.3 Typical force - deformation relation for joints 183

5.4 Block relaxation and calculation procedure 184

5.5 Block geometry and property for comparison with

dynamic relaxation 189

5.6 Table for comparison of results 189

5.7 A tunnel roof analysis 190

5.8a Block displacement of the tunnel roof 191

5.8b Force and displacement distribution 192

5.9 Modelling the tunnel roof problem with the base

friction model 194

5.10 A physical model simulating a rectangular opening

10

in a jointed rock mass 195

5.11 Numerical simulation of the rectangular opening

in a jointed rock mass 196

5.12 The hyperbolic normal pressure - normal displacement

relation for a discontinuity 198

5.13 Comparison of the measured normal stiflFhess

and the constant stiffness assumed in the calculation 198

5.14 Modification of the program using the nonlinear

normal stiffness - overlap relation 199

5.15 Derivation of edge stiffness for block interaction

in an elastic discontinuous body 202

5.16 Determination of normal stiffness Kn 202

5.17 Interaction of a continuum and a discontinuum 209

5.18 Shape functions of the 8-node serendipity

quadrilateral element 210

5.19 Introduction of a 7-node finite element for the

compatibility conditions at the interface 210

5.20 Definition of the 7-node finite element 211

5.21 An outline of the hybrid program structure and

its developments 213

5.22 The fiow chart of program COUPLE 215

5.23 Consolidation of blocks on a deformable body 218

5.24 Influence of iteration number on displacement

distribution at the interface 219

5.25 Test of the program on the tunnel roof problem 221

5.26 Calculated roof block movements as the results

of the support medium deformation 221

5.27 Finite element and distinct element meshes for

modelling a rock mass characterised as a combination

of discontinuum and equivalent continuum 223

5.28 Plot of principal stresses in the finite element

region showing a hydrostatic stress field 223

5.29 Plot of pre-mining stresses by the hybrid

computation 224

5.30 Excavation of the tunnel by reducing the

11

stress and stiffness of the elements 226

5.31 Force and stress plot 227

5.32 Force and stress plot after further excavation and

iteration 228

5.33 Plot of displacement of the roof blocks 229

6.1 Location of districts where measurements were taken 234

6.2 An outline of geological strata section of Coventry

Colliery with the associated uniaxial compressive strength 235

6.3 Roadway measuring section in Coventry Colliery 236

6.4 Roadway deformations at Coventry Colliery 238

6.5 Measured roadway closure versus distance of face

advance 239

6.6 Measured floor-lift in two roadway sections of

Coventry Colliery 239

6.7 Measured pack load and closure versus distance of

face advance 240

6.8 Measured roof bed separation versus distance of

face advance 240

6.9 Measured angle change of support arch leg 243

6.10 Measured penetration of support arch leg 243

6.11 Measured arch support load versus distance of

face advance 243

6.12 Idealised typical roadway section for numerical

modelling 245

6.13 Finite element representation of the roadway problem 250

6.14 Influence of lamination of strata on calculation

of pre-mining stress field 251

6.15a Calculated pre-mining stress field in the

bedded strata 253

6.15b Simulation of progressive excavation of roadway

and face 253

6.16 Plots of principal stresses around the roadway 257-261

6.17 Stresses distributions inside the crown, fioor and

ribside of the roadway 262

6.18 Plot of principal stresses calculated by elastic

12

analysis

6.19 Stresses distributions inside crown, floor and ribside

of the roadway predicted by elastic analysis 264 x

6.20 Plot of mesh deformations and plastic zones

6.21 Roadway deformation versus excavation steps

— comparisons of large deformation predictions

and small strain analysis

6.22 Calculated strata deformation and plastic zone in a

weak floor case — simulation 2

6.23 Roadway deformation versus excavation steps

in a weak floor strata case — simulation 2

6.24 Stress distributions in the floor and roof

— simulation 3

6.25 Stresses versus excavation steps at two fixed

points — simulation 3

6.26 Calculated mesh deformation and plastic zone for

a weak roof case — simulation 3

6.27 Roadway deformation versus excavation steps for a

weak roof case — simulation 3

6.28 Stresses in the ribside after excavation step 7

— simulation 4

6.29 Calculated mesh deformation and plastic zone for a

weak coal seam case — simulation 4

6.30 Roadway deformation versus excavation steps for a

weak coal seam case — simulation 4

6.31 Comparison of measured and computed roadway

closures — simulation 1

6.32 Comparison of measured and computed roadway

closures — using modified input data for the case

of simulation 2

6.33 Comparison of measured and computed roadway

closures — simulation 1 (for another section)

6.34 Comparison of measured and computed floor-lift

6.35 Comparison of measured and computed pack load

7.1 An outline of conclusions of each chapter

263

266-268

270-271

273

274

276

277

278

279

281

282

283

286

286

287

288

289

295

13

A.1.1 The flow chart of program VISCO 319

A.2.1 The extended H-B criterion in the / i and (J2) ' space 321

A,2.2 The influence of empirical parameters m, s, 0^ 323

A.2.3 The concept of the 'average space' of the

extended H-B criterion 325

A.3.1 The flow chart of the program COAL 327

A.4.1 Body deformation and displacement notation 337

A.5.1 Calculation procedure of the program BLOCK 342

A.5.2 Routine structures of the program BLOCK 343

A.7.1 Seam section of the Warwickshire Thick Seam 357

A.7.2 Strata section of Coventry Colliery with test

results 358

A.7.3 Road cell results at Coventry Colliery 363

A.7.4 Measurements of arch leg penetrations 364

A.7.5 Roadway closures at Coventry Colliery 365

A.7.6 Measured floor-lift at Coventry Colliery 366

A.8.1 The Hoek-Brown yield surface for various

rock masses 368-370

A.8.2 The Mohr-Coulomb and the extended H-B yield

surface 370

A.8.3 Plots of principal stresses for a case of

strong pack support 371-373

A.8.4 Plots of mesh deformation for a case of weak

floor strata 374-375

14

LIST OF TABLES

Table Title Page

3.1 A review of rock strength criteria developed

since the 1960's 86

3.2 The flow vector components 107

3.3 Comparison of the computed results 121-123

5.1 Comparison of displacements predicted by

physical model and numerical analysis 196

6.1 Material constants used in the computations 248-249

6.2 Excavation factors used in simulating face

advance 252

6.3 Computed time with reference to each excavation

steps — simulation 1 271

6.4 Computed time with reference to each excavation

steps — simulation 2 274

A.7.1 Coal sample strength parameters by laboratory

tests 359

A.7.2 Elasticity constants of the coal samples by

laboratory tests 360

A.7.3 Strength parameters of other rock samples by

laboratory tests 361

A.7.4 Strength of rock samples around the roadway

and pack samples 362

15

LIST OF SYMBOLS

Symbols Definition

[A] CoefRcient matrix relating forces and displacements

[A ] The unit stiffness matrix of the block

equilibrium equation

€bi,aii,aiii Flow vector components

a, b The flow vectors

B The strain matrix

Bo Linear component of strain matrix

Bffi, Nonlinear component of strain matrix

P The empirical parameter for the extended

Hoek-Brown criterion

CjjCajCs Parameters of flow vector components

c, (j) Cohesion and frictional angle

D The stress matrix

d Vector of nodal displacements

AF" Vector of incremental pseudo loads

6 A small increment (or variation)

E Young's modulus of intact rock

c strain

€e Elastic component of strain

e„p Viscoplastic component of strain

F Yield function

FQ Equivalent yield stress

Fi{ ) Equivalent stress level

Fn , Fs Normal and shear forces at a contact

Vector of equivalent nodal loads

Fi Specific boundaries

7 Fluidity parameter

Ixy^lyz Shear strains

16

H"' The matrix for calculation of viscoplastic

strain rate increment

A J/z 5 3 The first, second and third stress invariants

Ji , 2? « 3 The first, second and third deviatoric

stress invariants

[JD] The deformation Jacobian matrix

i fy Stiffness matrix

i f j Stiffness matrix for small deformation

iiCjj Stress dependent stiffness matrix

K n , Ks Normal and shear stiffness of block contacts

K Strain hardening or softening parameter

M,N Exponential constants in $ function

m, 5 The empirical parameters of Hoek-Brown criterion

m',s' Dilation parameters of the Hoek-Brown

yield surface

Ni Shape function for node i

n — 1, n, n + 1 At time step n. — 1, n, n + 1 respectively

1/ Poisson's ratio

w,w' Infinitesimal numbers for convergence

and equilibrium check

Pa Atmospheric pressure

Px, Py 5 Pz situ stresses

d Partial derivatives

$ Function of yield function

^ Vector of residual forces

Q Plastic potential function

{g} Force vector at interfaces

{JR} Applied forces on blocks

Rep Plastic radius around the tunnel

p Unit weight of rock masses

R Tunnel radius

a Stress

<T„ Normal stress on a discontinuity

am Normal stress on a boundary

Oe Elastic component of stress

17

t 5

J 0 5 z

O'ci f^t

<^ii

*3 ^oct J 'oct

E

Superscript ^

t

To

T

r»

6"

6

0

Ur

U,V,U}

^mc V

x , y

Xc,} ;

X, y,z

^nt tin

i,ri [ ] { }

Applied in situ stresses in numerical analysis

Principal stresses

Stress components in Cartentian coordinates

Stress components in cylindrical polar

coordinates

The uniaxial compressive and tensile strengths

Stress tensor

Deviatoric stress tensor

Octaliedral normal and shear stresses

Summation

Transform of a matrix (or vector)

Time

Uniaxial tensile strngth in Murrell's criterion

Shear stress

Shear stress on a boundary

Rotation angle of a distinct element

The 'Rode' angle

Constant determining integration algorithm

Tunnel face displacement

Displacements in x, y, z directions

Maximum closure of a joint

Joint closure

Components of body force per unit volume

Coordinates of contact point i

Coordinates of centroid point of a block

Cartentian coordinates

Coordinates at time step n

Local coordinates

Matrix

Vector

18

CHAPTER 1 INTRODUCTION

Computer-based numerical methods are a vital part of the modern

discipline of rock mechanics.

E. T. Brown (1988)

The purpose of engineering rock mechanics is to provide effective tools and regu-

lations for engineering design related to rock mass stability. Many of the problems

encountered in engineering geology, civil engineering and mining engineering re-

quire the prediction of the response of the rock strata. This is well illustrated

by Figure 1.1 (from Hudson 1988) in which all rock engineering problems and

subjects are abstraxztly indicated. The diagram also shows that many of the rock

engineering activities are related to excavations.

A primary concern in the excavation of rock, be it surface or underground, is

the control of displacements in the rock surrounding the excavation. In each exca-

vation, the design objective is to ensure that displacements of the rock around the

excavation are compatible with the performance of specified engineering activities.

There are generally two approaches to excavation aad support design. One is

based on observation and case records of existing practice and attempts to give a

simple answer by correlating information about the rock mass conditions with the

supports that were used. This has led to several rock mass classification systems

designed solely to elucidate support requirements (Bieniawski 1973, 1984; Barton

et al. 1974). This approach perpetuates existing practice, and does not necessarily

distinguish conservative work, or even perhaps unsatisfactory practice. It takes no

account of excavation and construction procedures and of rock mass post-failure

behaviour.

19

COMPLETE ROCK ENGINEERING PROBLEMS'

— ANALYSIS OF COUPLED MECHANISMS —

ANALYSIS OF INDIVIDUAL SUBJECTS

Foundation

Is5 o

/WW

BOUNDARY CONDITIONS:

In Situ St ress, Hydrogeologlcal Regime

Borehole/ Shaft

Underground Excavat ion

e.g. BLOCK ANALYSIS OR STRESS ANALYSIS

-ROCK MECHANICS INTERACTION MATRICES —

-KNOWLEDGE-BASED EXPERT SYSTEMS

Figure 1.1 Engineering rock mechanics problems (from Hudson, 1988).

The second approach involves obtaining a more complete knowledge of strata

deformation mechanics by means of continuum and discontinuum analysis of rock

mass behaviour, considering support systems, their interactions with the rock mass

and the effects of the progressive excavation and construction.

Numerical modelling, based on the techniques of mathematics and mechanics,

is a basic tool in the second approach and is nowadays popularly used in different

stages of excavation design, including the initial design, the monitoring programs,

the interpretation of measurements and the retrospective analysis and redesign. It

should be noted that development of a numerical model is a process of mathemat-

ical and physical abstraction. Difficulty arises in applying the existing numerical

models in rock mechanics because of the complexity and measurement of rock

mass properties. Some of these difficulties and analysis of specific problems of

large rock movements induced by mining excavations, with a detailed example of

roadway closure in long wall coal mines, are discussed in this chapter. Based on

these introductory discussions, the purpose of this research is described.

1.1 Conceptual models of rock mass response to excavation

A rock mass is defined as a medium which is comprised of intact rock, its net-

work of discontinuities and weathering profile (Goodman 1976). Generally, a rock

mass under an initial state of stress responds differently to excavation according

to all these components. In choosing a correct analytical or numerical method

for modelling rock mass response to excavation, Brady (1987) distinguished four

conceptual models of rock masses on the basis of the displacements induced by the

excavation, as illustrated by the models a, b, c, d in Figure 1.2 (the model e will be

described later). For a massive good quality rock mass, the displacement field may

be continuous throughout the near field of the excavation and this is illustrated

in Figure 1.2a. For the rock mass structure shown in Figure 1.26, within a large

discrete region of the near field rock the displacement field is continuous. These

21

I

LZL: z L/:_; J

Continuum

Continuum and joints

Discontinuum

Pseudo-continuum

Discontinuum and pseudo-continuum

Linear elastic, small deformation problem.

Nonlinear large deformation and strata movement problems (purpose of this research)

Figure 1.2 Conceptual models relating rock structure and rock response

(extended from Brady, 1987).

22

regions are separated by planes of weakness on which slip or separation can occur.

The conceptual model c is defined as a frequently jointed rock mass with sets of

discontinuities on the scale of the excavation, as shown in Figure 1.2c. For this

case, slip and separation on joints may occur so that the near field displacements

are determined by the rigid body translation and rotation of rock blocks. When a

rock mass is frequently and randomly fractured, as illustrated in Figure 1.2d, the

joints ubiquitously exist in the rock mass so that the displacement field around

the excavation is pseudo-continuous.

The theories and analyses used to study these conceptual models of under-

ground excavations are of three general types: continuum approach, discontinuum

approach and limit equilibrium method. For conceptual models a and d, contin-

uum approaches such as numerical modelling with the finite element and boundary

element methods are used to calculate the stress and displacements induced in the

initially stressed rock mass following excavation. However, there is a significant

difference between conceptual models a and d. Because of the complexities and un-

certainties involved in the conceptual model d, the theory and numerical approach

for the model d have not been well validated. As for the resultant displacements

induced in these two types of rock masses, model d usually gives much larger values

than model a, often of one to several orders of magnitude. In search of solutions

for conceptual models b and c, discontinuum approaches such as the distinct el-

ement method and the finite element method incorporating joint elements have

been used to calculate the stress (or forces) and displacements. An alternative

method for model b is the limiting equilibrium approach based on block theory or

stereonet projection (Goodman and Shi 1985; Priest 1985), by which the stability

of the excavation can be assessed.

The displacements for the models a and b are usually of elastic orders of mag-

nitude, hence they have been well analysed by the methods using conventional

small strain theory. However, the rock masses in models c and d may experience

relatively large displacements resulting from slip and separation of joints, so that

23

small strain theory cannot be applied in the analysis of these models. The differ-

ence between models c and d is in the manner of their large displacements: the

former is dominated by discrete translations and rotations of rigid rock blocks,

while the latter usually exhibits an equivalent continuous yielding with plastic di-

lation of the rock masses. Analysis of these two models is still currently an active

research axea in rock mechanics.

For a practical mining excavation, rock strata in the near field of the exca-

vation can actually be a combination of any of these four models of Figure 1.2

(Lorig and Brady 1984). One of the cases is illustrated in Figure 1.2e where the

upper part of the strata can be modelled using conceptual model c whereas the

lower part has to be treated as a pseudo-continuous medium as in model d. This

is referred to as the conceptual model e throughout the following discussions, and

practical examples of the model are given.

1.2 Strata movement problems in rock engineering

In the case of excavations in weak or jointed rock strata, considering conceptual

models c,d and e often results in rock displacements so large that they are re-

garded as 'rock (or strata) movements'. These rock movements can either be

pseudo-continuous or discontinuous, resulting from local failures, such as material

yield and shear, fracture propagation, joint slip and separation in the near field of

the excavation. One of the important aspects of rock mechanics is ground control

in mining engineering, which involves controlling the displcicement of rock sur-

rounding the excavations generated by mining activity. An example is the surface

subsidence induced by various underground mining extractions (Brady and Brown

1985; Peng 1986). There are two types of subsidence - continuous and discontinu-

ous; both of them are consequences of large strata movements due to excavations,

which are inevitable for the rock mass in transition from one state of equilibrium

to another. Another example is the near field strata deformation of underground

24

excavations, such as tunnels, underground power plants and iongwall mining ex-

cavations. Ward (1978) described the Girl tunnel (5.5m diameter), the lining of

which underwent up to 1.0m of diametrical convergence in passing through a weak

thrust zone at a depth of 200 — 300m. More examples can be found in mining

engineering, and they will be further discussed in detail in the next section of this

chapter.

With the advances of modern engineering rock mechanics and numerical

techniques, rock movements have been investigated by using quantitative anal-

ysis rather than qualitative analysis and empirical evaluation. However, existing

quantitative analyses are usually unsatisfactory If the rock strata movements are

relatively large and at the same time the material properties and geometries of the

problem are relatively complex. It is even more difficult when, in special cases,

the conceptual model e in Figure 1.2 has to be considered. For example, it often

occurs in the analysis of slope deformation that a slope material is composed of

more than two types of rock masses as shown in Figure 1.3. The analysis of road-

way closure as described in the next section has also to face complex material and

mechanism problems.

Jointed Sandstone

Severely Weathered Granite dyke

Figure 1.3 An example of slope excavation in a complex medium (discontlnuum and pseudo-continuum).

25

There are numerous other examples of difficult mining conditions resulting

from large strata movements, often leading to increased expenditure and losses

in production. Many of the conditions, however, could be improved at little cost

by an understanding of the mechanics of strata deformation and prediction of the

extent of strata movement,

1.3 Roadway stability analysis in longwall coal mining

In the further discussions of numerical modelling of rock movements, the specific

problem of gate roadway stability in Coal Measures strata will be considered. The

problem is described in detail in order to provide an insight into the nature of the

analysis and because it provided the basic motivation for this study.

Behaviour of Coal Measures strata:

In general, Coal Measures primarily comprise mudstone, sandstone, seat earth

and coals, arranged in a series of rhythmic or cyclic sequences varying throughout

coalfields and through successions. Coal Measures rocks are relatively soft rocks.

In British coal fields, the range of uniaxial compressive strength of the Coal Mea-

sures rocks (intact samples) is about 10 — 120MPa (Whittaker and Singh 1981;

Farmer 1985). This compares with the other relatively hard rocks which may give

a range of 100 — 550MPa of the uniaxial compressive strength as that listed by

Hoek and Brown (1980).

A general tendency is that sandstones are stronger than seat earth and mud-

stone; the latter two are usually the weaJcest materials in British coalfields. The

uniaxial compressive strength of the seat earth is generally in the range 10 —

30MPa. The roof strata are often composed of relatively hard rock contain-

ing numerous discontinuities, whilst the coal seam and floor strata are usually of

weaker rock. In British coalfields the roof strata are sandstones, siltstone, etc.

and the floor strata normally seat earth and mudstone.

26

It was found that, for soft rocks, the compressive strength is only an index

and is not an indication of the strength of the in situ rock mass (Wilson 1983).

For example, the strength of the coal in situ may only be about one-fifth of the

strength determined from a small sample in the laboratory.

Deformation characteristics of Coal Measures rocks have been studied by

many investigators (Hobbs 1966; Bieniawski and Vogler 1970; Farmer 1982,1983;

Wilson 1983). The nonlinear deformation behaviour of the Coal Measures rocks

includes strain softening or strain hardening deformations, resulting from brittle

or ductile failures of the rock, rock block movements due to joint slips and sep-

arations and time-dependent or water-dependent rock deformations (viscous and

swelling phenomena).

In situ stress

The coal seams in the United Kingdom axe often at depths of 500 to 1000

meters. The pre-mining in situ stress at depth is derived from several sources,

including gravity loading of the overburden, the tectonic stresses, the local stress

concentrations and the residual stresses (Peng 1986). The most important factor

is probably the gravitational loading which, according to simple statics, makes the

vertical stress at depth equal to the overburden load per unit area.

In the relatively soft rocks of the Coal Measures, it is probable that creep over

geological time (a time span measurable in millions of years) will have caused near

equalisation of horizontal and vertical stresses and so the pre-mining stress field

in this case may be approximately hydrostatic. In the absence of other factors

influencing the in situ state of stress such as the surface topography, tectonic

loading and local inhomogeneity, it is often supposed that the principal directions

of stress are vertical and horizontal, and the vertical stress is given by:

Pz = pZ. (1.1)

where p is the rock mass unit weight, and Z is the depth below ground surface. In

27

the United Kingdom, back analyses based on observed excavation performances

suggest that the stress field, in these weak, sedimentary rocks, is approximately

hydrostatic and can be calculated using the above equation.

However, in some cases, the harder rock overlying the coal is essentially elastic

and may sustain shear stresses over a long period of time. Therefore horizontal

principal stresses could, at depth of around 800 ~ 1000 meters, be anywhere in the

range 0.5 to 2 times the vertical stress in the relatively harder rock strata (Brown

and Hoek 1978).

Longwall mining excavation and roadway formation

The longwall mining method is used in coal mining because coal seams are

generally flat-dipping and of large area extent. Details of the longwall mining

method can be found in the books by Brady and Brown (1985) and Peng (1986).

Figure 1.4 illustrates briefly the typical longwall coal mining process. It is seen

from the figure that the primary objective of the design is to achieve pseudo-

continuous deformation of the main upper strata overlying the seam.

In longwall mining, special gate roads are constructed to service a longwall

face. According to the method and location or timing of the formation, there

are different kinds of roadways. In general, the roadway for an advancing face is

situated in the waste area behind the face, and at either side of it there may be

ribside packs. The roadway excavation sequence is of a three dimensional nature.

The packs are usually constructed after the longwall face has advanced to a certain

distance away. Figure 1.5 illustrates three main systems of roadway formation in

longwall advance mining. After formation, the roadway is subjected to stress,

induced as the face advances further away and the surrounding waste caves.

Roadway structure and deformation

A typical roadway structure in plane view is shown in Figure 1.6. Since the

load capacity of the roadway support and the near field rock is always small com-

28

roof collapses behind

supported area

longwall ^ f a c e

drift supported by yieldable steel arches

\

chain conveyer ^^^^sport drift

belt conveyer

Figure 1.4 Roadways in longwall coal mining (from Hamrin, 1982).

(a)

~XA

n

(b) (c)

Figure 1.5 Three main systems of roadway formation: a) advancing heading; b) conventional rip; and c) half heading (from Wittaker and Pye, 1977).

29

Rib side

ripping lip lagging

coal rib.

steel arch

With ribside pack

Hp supports

mm

Figure 1.6 An example of roadway support system in cross section view (from tftomas, 1978).

<U (U -r;

(a)

(b)

(c)

1 1 40 60 80 100 120

Distance along gate, A: (m) 140

Figure 1.7 Roadway convergence measured along the gate axis (c.f. Figure 1.5) (from Wittaker and Pye, 1977).

30

pared with the overburden load, it is possible for a roadway to undergo as much as

1 ~ 1.5 meters of vertical closure when the depth is in excess of about 150 meters.

Therefore, a roadway support system, using yieldable steel arches or permitting

the arch legs to punch into the floor, is usually introduced. These steel arches

are installed at intervals along the roadway and often in conjunction with wooden

lagging. It is clear that the formation of roadways is a kind of excavation lying

between the extremes of fully supported and completely unsupported excavation.

The stress distribution and associated deformation of an advancing longwall

roadway at a given location depend greatly on how far the mining face has ad-

vanced beyond that location. Figure 1.7 illustrates the roadway convergence along

its axis for three different systems of roadway formation (cf. Figure 1.5). Even

when the face has advanced a large distance compared with the roadway dimen-

sions, it is still possible for the roadway to undergo further deformation with time.

It is found that the rock mass around roadway openings will fracture and

yield because of the low strength of the rocks and the high in situ stress field at

depth. This leads to unavoidable large displacements of the strata, such as floor

heave and squeeze of the side wall, often resulting in roadway closure. Figure 1.8

shows two photographs of large roadway deformations in British coalfields.

Conclusions

From the literature (Farmer 1982,1983; Peng 1986; Wilson 1983) and the

descriptions of the above points, it can be concluded that the main factors deter-

mining the magnitude of roadway deformation at depth are:

a. the stresses resulting from redistribution of pre-mining stresses during exca-

vation of coal from the longwall working or from adjax:ent workings in the

same or other seams,

b. the seam height and the compressibility of the face side park,

c. the strength of roof and floor rock.

31

3R

4:

b)

I igi irc 1.8 L a r g e r o a d w a y d e f o r m a t i o n s at Br i t i sh c o a l f i e l d s ( p l i o t o g r a g h s p r o v i d e d

by Bri t ish C o a l , a) ins tab i l i ty ; b) f loo r - l i f t .

3 2

d. geological structure including the presence of major discontinuities, the direc-

tion of leating and the presence of strong layers in the roof strata, resisting

caving,

e. the method of forming the roadway, particularly the method of excavation,

the degree of roadway support and the treatment of the ribside, and

f. the layout of faces and the presence of ribside pillars.

One of the aims of mining engineering design is to protect major service

openings until they are no longer required, so the roadways in longwall coal min-

ing must be formed and supported in such a way that displacements are kept

within operationally acceptable limits in a certain work period. Thus, an effective

means of predicting the distribution of stress and displacement of the strata is

needed. Unfortunately, however, no satisfactory means of dealing with the prob-

lem of roadways with such complex material and geometrical conditions has been

developed, and the improvement of existing numerical methods or establishment

of new numerical model is necessary to provide quantitative analytical results.

1.4 Purpose and structure of this thesis

Purpose of the thesis

Practical rock engineering design requires a more complete knowledge of rock

mass response to excavations and the mechanics of support - rock interaction.

Jointed or weak rock mass response to various types of excavation can be analysed

using the continuum and discontinuum models (Figure 1.2), which may be char-

acterised by large rock movement. In mining engineering, large rock movements

often lead to excessive expenditure and losses in production. Understanding of

the mechanics of strata deformation and correct prediction of the extent of strata

movement could improve mining excavation and support design.

Numerical modelling is a basic tool to achieve this complete knowledge, in

33

particular, for a complicated problem like roadway deformation analysis. However,

the full nonlinear analysis of ground response is still far beyond the capability of

any existing numerical programs.

The objective of this research is to apply updated numerical techniques and

to develop complementary methods to this specific rock engineering problem. This

required the following work during a three year period:

a. an assessment of the existing numerical methods currently used in geome-

chanics,

b. choosing the correct numerical methods for the rock movement analysis ac-

cording to the conceptual models,

c. improving the numerical techniques for the specific problems such as roadway

analysis, and

d. validation and applications of the developed program.

Structure of the thesis

Apart from this introduction, there are six chapters covering the above indi-

vidual studies.

In Chapter 2, the existing numerical methods currently available in geome-

chanics are examined in order to provide the basis for the development of an efii-

cient tool for the modelling of strata movement. The basic principles, the current

state of the development, the application and limitations of the various methods

with reference to strata movement analysis are assessed.

Chapter 3 describes a two-dimensional nonlinear finite element method for

continuous or pseudo-continuous rock masses. Main improvements of the method

include provision of a more suitable rock mass response model, implementation of

the large deformation theory and simulation of excavation sequences. Validations

of the developed program are described.

34

In order to ajiswer a key question in analytical methods in rock mechanics,

Chapter 4 presents a numerical investigation procedure to study the validation and

discrepancies of two- dimensional (2-D) analysis in modelling three-dimensional (3-

D) excavation problems. The application of the developed finite element program

is thereby described, and the results clarify some of the relations between the 2-D

and 3-D analyses.

Chapter 5 proposes a new hybrid computational scheme in which the finite

element method is used in conjunction with distinct elements to model the inter-

action of blocky rock with a softer rock mass. The 2-D distinct element program

using the static relaxation procedure is described and validation examples are

given.

The practical implications of the research are demonstrated in Chapter 6, in

which the roadway deformation problem in Coventry Colliery, United Kingdom

is analysed. It is shown that the hybrid program is capable of dealing with the

most general analysis of strata movements and assisting in the design of mining

excavations.

The last chapter. Chapter 7, gives conclusions of the research and suggestions

for useful future study.

35

CHAPTER 2 REVIEW OF THE NUMERICAL METHODS

AVAILABLE FOR STRATA MOVEMENT ANALYSIS

2.1 Introduction

Analysis of stress and displacement within rock mechanics problems usually re-

quires the solution of partial differential equations for the given boundary con-

ditions. Once the mathematical equations governing a rock engineering problem

have been established, then, with the exception of very simple cases, they must

be solved using a computer-based numerical (approximate) method. The type

of numerical method selected is important as it determines the accuracy of the

solution.

In choosing a suitable method for various rock engineering problems, con-

ceptual models as described in section 1.2 have to be considered. This leads to

two alternative approaches in the modelling of rock mass response. One is the

continuum approach. Historically, the numerical modelling of rock masses origi-

nated from methods which already existed for the modelling of continua. Later,

as the understanding of the influence of discontinuities on rock mass properties

improved, some special techniques to model the behaviour of these discontinuities

were developed.

The aim of a general numerical method is to reduce a governing equation (or

a set of equations) plus boundary conditions to a system of algebraic equations

which cajx be easily solved by a digital computer. Consider, for instance, a tun-

nel excavated in a pre-stressed linear elastic continuous body under plane strain

36

conditions (Figure 2.1).

Gy: Horizontal

and vertical stresses;

P i : The tunnel

surface; r2 . The outer

boundary of the body.

Figure 2.1 The static problem of a horizontal tunnel excavated in a continuous,

homogeneous, isotropic, linear elastic body.

The displacements in the body are governed by the following set of partial differ-

ential equations (Xu 1979):

E {l + u){l-2u)

E ( l + i / ) ( l - 2 i / )

. .d'^u l — lud^u 1 (1 - W — + _ . o + r-dx^ 2 dy"^ 2 dxdy

. . d'^v l — 2i/ d^v 1 d'^u

+ x = o,

+ r = o , (2.1)

2 2 dxdy

where u and v are the exact solution of horizontal and vertical displacements, X

and Y denote the components of body force per unit volume, and E and i/ are the

Young's modulus and Poisson's ratio. The boundary conditions of the problem

are:

stresses : a^n = 7 , = 0, on Ti,

displacements : u = t; = 0, on Tg oo) (2.2)

where Orn,Tt are the normal and shear stresses at the tunnel surface, Ti is the

tunnel boundary, Fg is the outer boundary of the body and R is the distance. The

37

stresses are related to displacements by the assumptions of Hooke's law and the

infinitesimal strain theory (Timoshenko and Goodier 1970).*

If the exact solution of u and v can be found by solving equation (2.1) with the

defined boundary conditions, the displacements at any point in the body, including

that on the tunnel wall, are known. Unfortunately, it is difficult to find the exact

solution for this simple problem. In general, the solution has to be approximated

using numerical methods in which the continuum is often subdivided into a number

of cells or elements. The whole problem (equations (2.1) and (2.2)) can then be

re-expressed as

[A]{u} = {F}, (2.3)

where [A] is a coefficient matrix, {u} is a vector of the approximate displacements

at certain points in each discrete region, and { f } is a vector of the equivalent force

components at those points. Thus equation (2.3) can be easily solved by using

computers, and the rest of the unknowns, such as stresses and displacements at

other points, can be determined as secondary variables.

For a preictical rock mass which is governed by the conceptual models b,c,d,e

of Figure 1.2, the representative mathematical model (in which non-homogeneous,

anisotropic, inelastic, discontinuous and/or large deformations have to be taken

into account) will be so complex that only numerical methods can be applied to

find any of these solutions. When discontinuum behaviour dominates the rock

mass, special mathematical models and numerical techniques are required.

To date, available numerical methods which can be used for the analysis of

stress and displacement of rock strata are primarily the finite difference method,

* An infinitesimal strain analysis (small deformation) is also called geometric

linear analysis in which the relation between strain and displacement is linear. For

a large deformation problem, the strain may be finite and the relation between

the strain and displacement is no longer linear. This geometric nonlinear problem

is to be further described in Chapter 3.

38

the finite element method, the boundary element method, the distinct element

method and several hybrid techniques. In recent years, many computer programs

based on these methods have been developed and used not only in the field of re-

search, but also in engineering practice. A list of those programs available in vari-

ous countries, including a brief explanation of their capabilities, has been produced

very recently by the Commission on Computer Programs of the International So-

ciety for Rock Mechanics (Plischke 1988). From the list of all 198 programs,

some statistical figures on the percentages of a particular method used in Rock

Mechanics have been drawn as shown below:

the finite element method: 64%,

the boundary element method: 7%,

the finite difference method (including

the distinct element method): 6%,

hybrid methods; 4%,

other methods (limit equilibrium,

test data processor, etc.): 19%.

A similar feature can also be seen through the series of proceedings of the

International Conference on Numerical Methods in Geomechanics (1972, 1976,

1979, 1982, 1985, 1988). In all these conferences, most of the papers presented

are related to finite element analysis; however recently, other methods such as the

boundary element and distinct element methods are gaining popularity.

In order to provide the basis for the development of an efficient tool for the

purpose of this research, a brief review of the above methods is given in the

following sections. The detailed theories and formulations of the various methods

are not discussed in the review, but the related key references on these methods

are given. Assessments are made of the basic principles, the current state of

development, and the applications and limitations of the various methods for strata

movement analysis.

39

2.2 The finite difference method

The methodology

The finite difference method is a technique to solve the various types of partial

differential equations encountered in engineering problems (Mithchell and Grif-

fiths 1980; Smith 1985). It is based on the simple theorem that any derivatives

(or partial derivatives) can be approximated by using certain finite difference for-

mulations. In the method, the region under consideration is usually divided into

a number of equally spaced rectangular grids. The original problem of deter-

mining unknown continuous functions, such as the u and t; in equation (2.1), is

replaced by the problem of solving a matrix equation for the discrete set of values

Ui,vi ,u2,v2,-" ,Un,Vn at the points where the grid lines intersect. Thus, the

method will give information about the function values at the grid points, but it

usually gives no information about the function values between these points.

According to the formulas adopted in the approximation of a real problem,

the finite difference method can be classed as either explicit or implicit. The

formulas which express each unknown value directly in terms of known values are

called explicit formulas, while the method where the calculation of unknown values

necessitates the solution of a set of simultaneous equations is described as implicit.

Generally, an explicit method has the advantage of simplicity in computation, but

has a serious drawback of poor stability. Both explicit and implicit finite difference

methods have been used with success in diverse areas of geomechanics.

The current state of development

For continuum problems, both material and geometric nonlineaxities can be

treated and have been coded in programs to solve quasi-static problems (Payne

and Isaac 1985; Coulthard and Perkings 1987; Plischke 1988). Among the available

finite difference programs, the explicit formulation is preferably used in modelling

rock mass nonlinear behaviour. It is reported that the benefits of an explicit

40

finite difference algorithm lie in the direct way in which nonlinearities are treated,

without recourse to devices such as equivalent stiffness, initial stresses or initial

strains, which need to be introduced into matrix oriented programs to preserve the

linearity dictated by matrix formulation (Cundall 1976), It is also reported that

the updated explicit finite difference programs, such as the 2-D program FLAG

(see Goulthard and Perkings 1987) and the 3-D program RMS (see Plischke 1988),

are as flexible as the finite element method because they allow irregular-shaped

zones and the capability to have different material properties for each zone. The

yield criteria used in FLAG and RMS are the Mohr-Goulomb and Drucker-Prager

criteria with non-associated flow rule and limited tensile strength. The large strain

plasticity problem can also be analysed using FLAG (Gundall and Board 1988),

One more advantage of the explicit finite difference method (such as dynamic

relaxation or fast Lagrangian analysis) is that it is well-suited to dynamic calcu-

lations, particularly in nonlinear situations including dynamic loading and wave

propagation in the ground. Modelling of discontinuities such as joints and faults

by the finite difference method is not described in the literature. However, an

alternative procedure based on the finite difference principle was developed by

Gundall (1971, 1974, 1979) for the analysis of a heavily jointed rock mass where

large movements can occur on discontinuities. The method became a new numer-

ical model, the distinct element method, which will be discussed further in section

2.5.

Applications

Applications of the finite difference method in rock mechanics problems are

not as popular as those of other numerical methods, such as the finite element

or boundary element methods. Several examples of applying dynamic relaxation

or the finite difference method in geomechanics were described by Scott (1987).

To illustrate a typical finite difference analysis, an example of soil slope stabil-

ity analysis by using the method, taking into account large strain and plastic

41

flow (Mohr-Coulomb yield criterion and associated flow rule), is shown in Fig.2.2

(Cundall 1976). Figure 2.2a shows the deformed flnite diflerence grid after failure

occurred and some movements had taken place. Figure 2.2b shows the velocity

vectors corresponding to the state shown in figure 2.2a. It can be seen that the

relative large deformation of the slope was well modelled, and the solution is a

continuum approach. A more recent application of the method was described by

Johansson et al. (1988).

Another example is the analysis of coal rib pillar deformation, as shown in

figure 2.3. A finite difference numerical model using the implicit method of Gauss-

Seidel iteration (Smith 1985) was adopted to predict support system deformation

at an underground excavation in soft coal measures strata (Mayer et al. 1984).

In the program, the plasticity and tensile failure were simulated by altering the

Young's modulus and Poisson's ratio of the individual grid points, hence they

were not properly simulated in a real 'failure' sense. Large deformation or finite

strain analysis was not formulated in the program. Figure 2.3a is the calculated

displacement plot and figure 2.3b shows the predicted failure zone.

Conclusion

The advantage of the finite difference method is that for the explicit approach

such as in the dynamic relaxation procedure, it does not include matrix operations

and this significantly benefits the modelling of nonlinear problems involving unsta-

ble material behaviour. For the implicit method, the above advantages diminish

because of the difficulties involved in the matrix operation. One of the limitations

in dynamic relaxation is the artificial damping factor which has to be introduced

for static analysis which may influence the solution. The other drawbacks of the

method, particularly in comparison with the finite element method, include diffi-

culties in representing complicated geometric forms, in modelling joint behaviour,

and in calculating the values in zones between the grid points.

42

Figure 2.2 Analysis of slope stability using the explicit finite difference method, a)

deformed finite difference grids; b) velocity vectors corresponding to a)

(from Hammett 1974).

Relative displacement

Failure pattern

Figure 2.3 Analysis of underground excavation using the implicit finite difference

method (from Payne and Isaac 1985).

43

2.3 The finite element method

The methodology

The finite element method is a general numerical technique for the solution of

partial differential equation systems subject to appropriate boundary conditions

and initial conditions (Zienkiewicz 1977; Hinton and Owen 1979). In many aspects,

the formulations of the finite element and finite difference approximation are iden-

tical and it was shown that the finite difference approximation is only a special

case of the finite element method (Zienkiewicz and Morgan 1983). The theoretical

base of the finite element method has been verified by the classical approximation

techniques known as weighted residual methods; the method always involves solv-

ing a matrix equation. In the dynamic analyses such as propagation and vibration

analyses, however, it can also be described as explicit or implicit method according

to the time integration procedure (Bathe 1982; Owen and Hinton 1980).

When the finite element approximation is applied to equations (2.1) and (2.2)

in which the unknown function is chosen as the displacements in the body, it is

called the 'displacement method'. In the approximation, the continuum is first

divided into discrete non-overlapping regions known as elements, over which the

main variables are interpolated. These elements are connected at a certain number

of points along their edges known as 'nodal points'. In the finite element displace-

ment method, the displacement values at these nodal points are the unknowns to

be calculated, and the other values such as stresses and strains at or between the

nodal points are approximated using the interpolation functions, which are one of

the important components of the method and whose use distinguishes itself from

the finite difference method. The simultaneous equations are derived using the

fact that the potential energy of the system is minimised when the mathematical

model, such as equations (2.1) and (2.2), is satisfied.

44

The current state of development

For the nonlinear analysis of geotechnical problems, the finite element pro-

cedure based on incremental theory is the most powerful and well established

method. A historically thorough review of the development and application of the

technique in geotechnical engineering in the early days (before 1972) was given

by Desai (1972). Since then, the technique itself has reached a higher plateau of

development; a great number of improvements have been made to the efficiency

of the technique and to the realistic modelling of geological media. Among these

improvements, an 'infinite' finite element has been developed and applied success-

fully to overcome difficulties which used to exist in using the finite element method

to model an 'infinite domain' of a geological problem (Beer 1983).

However, the nonlinear analysis of many geotechnical engineering problems,

especially rock mechanics problems, is still rather intractable and a number of

aspects of the finite element method still need investigation. This is because the

related nonlinear problems in rock strata analysis, such as roadway analysis, are

coupled field problems in which the following three factors are usually considered

(Kawai 1980):

a) material nonlinearity (plasticity, viscosity, strain softening, fracture, etc.),

b) geometrical nonlinearity (large deformations, buckling phenomena, etc.), and

c) relative movements of the discontinuities.

For the material nonlinearity analysis, Owen and Hinton (1980) and Naylor

et al. (1981) presented some well established theory and finite element computer

algorithms for geotechnical engineering. The work being carried out by many

other researchers involves consideration of the flow rule (Desai and Zhang 1987;

Reed 1988), softening law (Lee et al. 1985; Gumusoglu et al. 1986), yield criteria

(Michelis 1987; Pan and Hudson 1988), fractures (Crook et al. 1987), viscosity

(Akagi 1985; Swoboda and Mertz 1987), etc. based on the concept of the equivalent

continuum.

45

Geometrical nonlineaxity in the finite element analysis has also been consid-

ered by Zienkiewicz (1977) and described in detail by Bathe (1982), and it has

been applied in soil mechanics (Yamada and Wifi 1977; Kiousis et al. 1986).

However, the application of large deformation theory in rock strata analysis, espe-

cially in conjunction with material nonlinearity, is rarely described in the existing

literature.

Within the discipline of rock mechanics, discontinua have been analysed using

Zienkiewicz's 'no tension' model (1968, 1970) and later Goodman's joint element

(1968, 1976). Many attempts to develop new joint elements have been made

(Ghaboussi et al. 1973; Pande and Sharma 1979; Carol and Alonso 1983; Beer

1985; Pande 1985). However, existing types of models incorporating discontinu-

ities in finite element analysis cannot treat large displacements on discontinuities.

This is because such discontinuity of the displacement field cannot be treated eas-

ily by the existing theories of structural mechanics based on the concept of the

continuum. Adjacent elements in the conventional finite element method share

common nodes, and therefore separation or relatively large displacement at con-

necting nodes cannot be represented easily, even with the joint/interface element

in which node pairs are defined (Beer 1985; Swoboda et al. 1988).

Finite element modelling of frictional contacts has been extensively studied,

notably in mechanical engineering analysis, but almost invariably only infinitesimal

slip is considered (Okamota and Nataaama 1979; Sachdeva and Ramakrishman

1981). An alternative method of contact analysis in which the finite element mesh

is updated during solution was described by Haber and Hariandia (1985). A similar

method, of allowing the finite element mesh to change with time, was proposed

by Miller and Miller (1981) for simulating shock fronts in fluid flow (Barnes 1985).

These methods are in an early stage of development and appears not practical in

geotechnical engineering.

Recently, a concept of the representative volume element (RVE) in modelling

46

jointed rock using the finite element method has been assessed by Pariseau (1988).

The influence of the joints on the rock mass is modelled by averaging the overall

properties of intact rock and joints within the RVE. However, the validity of the

concept is still in question, and needs to be investigated (Cundall 1988).

Most of the existing finite element programs can deal with nonlinear analysis,

but only some aspects of the above factors are considered in those examples. For

instance, the well developed nonlinear stress analysis program ADINA (U.S.A)

is capable of modelling static and dynamic problems considering material and

geometric nonlinearities, but is not efficient in modelling yield and strain softening

of rock masses and joint behaviour. Other well known programs, such as BMINE

(U.S.A.) and NOLINA (U.K.), are more suitable for modelling rock material, but

often fail to model large deformation and weak rock mass behaviour.

Applications

Very many applications of finite element analysis have been reported in the

literature. Most of the analyses were carried out to investigate the nonlinear

behaviour of rock masses for various geotechnical problems. As an illustrative

example. Figure 2.4 shows a case study using finite element analysis, in which the

behaviour of the Mont-Blanc tunnel (France - Italy) was modelled (Yuritzinn et al.

1982). In the analysis, the rock mass strain softening behaviour (Drucker - Prager

criterion with associated flow rule), the function of the support system and the

influence of the tunnel face advance (Panet and Guellec 1974) were considered.

The finite element model predicted the different failure zones of the rock mass

around the tunnel as shown in Figure 2.4b, and the ground response curves were

also calculated. A similar application using the program ADINA was described

by Danger and Stockman (1985).

The method has also been used to model weak rock strata behaviour. Mikula

and Holt (1983) used a finite element model to predict the subsidence due to

coal extraction in Eastern Australia, as shown in Figure 2.5. In their model

47

a)

\N

Elastic r / J

Residual

Softenir

b)

Figure 2.4 Finite element analysis of tunnel excavation, a) the finite element meshes; b) the calculated plastic zones (firom Yuritzinn et al. 1982).

a)

O O o

o o o to

I— 0 . 0 0

I r 100 .00

I r

2 0 0 . 0 0 300.00 4 0 0 . 0 0 I

5 0 0 - 0 0

b)

o o o -

o o o r>

o o 0 r) 1

I 0.00

—I 60-00 120.00

—I r 180.00 2 4 0 . 0 0

I 3 0 0 . 0 0

Figure 2.5 Finite element analysis of mine subsidence, a) strata discretizations;

b) predicted failure zone and subsidence (from Mikula and Holt 1983).

48

(using constant strain elements), the anisotropy of the Coal Measures rocks and

the behaviour of joints ajid bedding planes were taken into account, and plastic

yielding was also simulated. Calculated large subsidence using this conventional

finite element analysis (small strain) is indicated in Figure 2.5b. Strata behaviour

of longwall coal mining problems has also been analysed using 3-D finite element

programs. Figure 2.6 shows such an example given by Ash and Park (1987),

in which the response of roof and pillar in a coal mine was simulated using a

progressive failure concept.

Conclusion

The finite element method has a special significance in rock mechanics, and

forms an intrinsic part of the numerical modelling technique. As has been men-

tioned in the preceding section, more than sixty per cent of existing computer

programs in rock mechanics axe finite element programs. Likewise, most of the

research and applications of numerical methods in rock mechanics have been based

on finite element analysis. Consideration of rock material properties and disconti-

nuities in the finite element approach mean the method can be treated as a mature

branch of engineering science in geomechanics. Geometric nonlinear analysis has

not been well investigated in rock mechanics, especially for weak rock masses in

which large deformation usually occurs. The existing finite element programs

usually concentrate on particular aspects of the nonlinear behaviour, so they are

usually not suitable for complex problems such as the roadway deformations.

49

Unmlned p a n e l

a)

Chain p i l l a r

Head Entry

P a n e l

b)

Figure 2.6 3-D finite element analysis of longwall coal mining, a) calculated initial

stage of stress distribution; b) final stage of stress distribution (from Ash

and Park 1987).

50

2.4 The boundary element method

Methodology

The boundary element method is a relatively new numerical technique in

which the differential equation for the region under study is transformed into

an integral equation relating unknowns on the boundary (Watson 1979; Brebbia

1978). In the process of transformation, two general approaches are used: one is

the indirect formulation method and the other is the direct method (see later).

In both methods, the boundaries of the problems, such as that in Figure 2.1, are

divided into elements, each of which has a number of 'nodes'. The unknowns of the

problem are taken as these boundary nodal values, and displacement and traction

between the nodes are interpolated using certain type of functions, similar to the

shape function of the finite elements. After the integral equation is derived, simul-

taneous equations can be written, which express the feict that the integral equation

is satisfied at each node of the boundary elements. The interior unknowns are then

evaluated using the boundary values solved from the simultaneous equations.

Compared with the finite element method, the well known advantages of the

boundary element method are: a) less input data to prepare; b) smaller system of

equations to solve; and c) the partial difi"erential equation can be exactly satisfied

at every point of the interior region.

If the indirect boundary element method is applied for the problem of Figure

2.1, the related integral equation is often derived from the fictitious stress concept

(Brady and Brown 1985), which is based on the principle of superposition and the

known boundary value problem solution. Another indirect method popularly used

in rock mechanics is the displacement discontinuity method (Crouch and Starfield

1983) which is based on the analytical solution to the problem of a constant dis-

continuity in displacement over a finite line segment in a 2-D elastic solid (Crouch

1976). When the alternative direct formulation is used, the partial differential

equations of (2.1) can be transformed into integral equations by direct integra-

51

tion of (2.1) or, more conveniently, by using the Betti reciprocal work theorem

(Butterfield 1979).

Current state of development

A brief review of the development of the boundary element technique in rock

mechanics has been given by Vargas (1982), and much work on the research and

application of the method for rock engineering problems has been done within the

Royal School of Mines, Imperial College (Bray 1976; Hocking 1977; Brady 1979;

Watson 1979; Eissa 1980; Sofianos 1984). Nevertheless, it appears that in all

these studies and applications, it was assumed that the rock mass is an elastic ally

homogeneous continuum.

In the investigation of nonlinear problems using the boundary el-

ement method, Nishimura and Kobayshi (1984) described elastoplastic analyses

using indirect boundary element methods. Mukherjee and Chandra (1984) coded

large deformation formulations into their viscoplasticity boundary element pro-

gram. Recent progress in nonlinear analysis using the method in geomechanics

was described by Banerjee and Dargush (1988). Unfortunately, however, some of

the advantages of the boundary element method diminish in these nonlinear analy-

ses, because interior meshes (cells) are usually introduced to model the progressive

yield of the material. Other advances in the method include introducing higher

order boundary elements, such as the Hermitian cubic elements, to model com-

plex boundary geometry (Watson 1986) and dynamic analysis to model vibration

of structures.

In rock mechanics, the indirect boundary element method is often used. The

fictitious stress method has a physical significance, so is easy to understand. The

displacement discontinuity method has been devised to calculate the stress distri-

bution around cracks and joints of rock masses, and can model some aspects of

the rock mass discontinuous behaviour. However, the current application of either

indirect or direct boundary element methods in geotechnical engineering is still in

52

general restricted to relatively simple linear elastic problems, so it is only applied

for predicting hard massive rock mass response.

Most existing boundary element programs for rock mechanics cannot deal

with material nonlinear analysis, but can calculate stress and displacement for

piecewise homogeneous media containing discontinuities. For example, the pro-

grams BITEMJ (see Coulthard and Perkings 1987), THREE (see Watson 1987)

and BEM (see Plischke 1988) are all well developed 2-D or 3-D codes, but caji

only predict the elastic behaviour of rock materials.

Applications

The method has become very popular in the solution of rock mechanics prob-

lems, particularly in hard rock excavations, because it is an efficient numerical

method for stress analysis with the boundaries of regions extending to infinity.

Figure 2.7a shows an example of a boundary element model of an underground

excavation intersected by a joint. The elastic principal stresses and directions near

the excavations and joints can be easily calculated using the method, as shown in

Figure 2.7b. A similar application of the method to mining excavation in a faulted

zone is shown in Figure 2.8, in which the stresses and displacements around the

fault and the mine-out are plotted. Both these examples axe only suitable for

analysis of strong hard rock strata. Some examples of nonlinear analysis using

the interior cell were described by other authors (Banerjee and Dargush 1988;

Chen and Yuan 1988; Li 1988). However, the efficiency of these analyses and

their practical implications have not been verified, and further investigations are

needed.

Conclusions

The boundary element method is an increasingly popular numerical technique

for rock mechanics. It has obvious advantages in the analysis of a linear elastic,

homogeneous and isotropic medium, and is superior for modelling the behaviour of

53

a)

0 5 0 0 0 kPc

1 stress scale 1

Figure 2.7 Example of the boundary element model for underground excavations,

a) the boundary elements for the excavation intersected by a

joint; b) predicted principal stresses and directions around excavations

and joints (from Crouch and Starfield 1983).

54

an infinite region in rock mechanics. The analysis of material and geometric non-

linearities using the method is at an early stage of development, so the application

to complex problems such as strata movement analysis has not been found in the

literature. Existing boundary element programs in rock mechanics are restricted

to relatively simple, linear elastic problems with or without discontinuities.

a)

S C A L E S :

to m L(near;

100 mm

D A T A : D e p t h « 2 5 0 0 m E - 7 0 GPo

" 0,20 0 I spIacement:

b)

S C A L E S }

1 0 0 H P a

D A T A : D e p t h - 2 5 0 0 m E - 7 0 GPe ' - 0 ,20

Figure 2.8 A boundary element analysis for mining excavation, a) computed prin-

cipal stresses around fault and excavations; b) computed displacements

(from Brummer 1988).

55

2.5 The distinct element method

The distinct element method is a discontinuum modelling approach for simulating

the behaviour of jointed rock masses (Lemos et al. 1985). The origin and devel-

opment of the method have been reviewed in detail by Stewart (1981) and Vargas

(1982). The method was first proposed by Cundall (1971) as an application of the

explicit finite difference method and dynamic relaxation technique to the problem

of the discrete block model of rock slopes. It was afterwards improved and applied

in engineering rock mechanics by Cundall (1974, 1979) and many others, such as

Voegele (1978), Maini et al. (1978), Stewart (1981), Vargas (1982), Lemos et al.

(1985) and Hart et al. (1988).

Methodology

In the distinct element method, the region of interest is a discontinuous

medium, with 'real' joints intersecting the body to form a system of various shaped

individual 'blocks', as in the conceptual model c (see Figure 1.2). The stresses and

displacements are continuous within each block, but are generally discontinuous

between these blocks. Therefore, the general partial differential equations for the

static equilibrium problem of Figure 2.1 (Equation (2.1)) are usually not satis-

fied for the whole region, because overlaps and cavitation often occur. Other

equations governing the constitutive relations between the joints have to be intro-

duced. This also means that unlike the other methods, the compatibility condition

(Timoshenko and Goodier 1970) is no longer satisfied in the governing equations

of the distinct element method so that more assumptions are usually involved in

the solution procedure.

In the original and basic form of the distinct element method, the blocks are

taken to be rigid, and deformations are associated with the surface of contact

between blocks; physically this means that the deformability of the surface mate-

rial, such as asperities of the joints, is far greater than that of the solid rock. A

56

universal law, Newton's second law of motion, was then applied as the primary

governing equation for the problem, and the behaviour of contacts between blocks

is simulated using other assumptions concerning the geometry, the contact points

and the force - displacement relations at the contacts.

The solution procedure is by use of a modification of the dynamic relaxation

technique of the finite difference method (Southwell 1940,1946, Otter et al. 1967),

so that in the procedure one block is equivalent to a nodal point of these classical

dynamic relaxations. Like other explicit methods, it is not necessary to solve global

simultaneous equations, and the numerical iteration is stable only if the time step

is taken as very small.

Current state of development

Since the method was first proposed, several forms of distinct element codes

have been developed to cover a variety of in situ conditions. The main research

and development of the method has been at Imperial College, U.K., and the Uni-

versity of Minnesota, U.S.A. By 1980, the distinct element program had been

developed from the original modelling of rigid blocks to the analysis of cracking

of blocks, the elastic deformability of intact rock and the fully deformable block,

using internal finite differences or finite elements (Cundall and Marti 1979; Vargas

1982). Before that time, however, various versions of distinct element programs

were written for these individual problems and no particular emphasis was made

on efficiency or practical applicability.

A new version of the distinct element program, UDEC (the universal distinct

element code), has been developed to include all the previous capabilities for mod-

elling various rock mass features (Lemos et al. 1985). Fluid pressure and flow

may also be analysed by the program. UDEC and its various versions appear to

be becoming the primary distinct element programs used in rock mechanics.

It is also noted that, among investigations carried out in the last ten years,

57

Stewart (1981) presented a different version of the relaxation procedure, called

static relaxation, applied to a system of interacting rigid blocks (Stewart and

Brown 1984). The benefits of this procedure were reported as:

a) the rate of convergence for static relaxation was shown to be higher than for

dynamic relaxation,

b) the requirement of very small time steps in dynamic relaxation was eliminated

because of the omission of the inertia term.

Nevertheless, some numerical ill-conditioning will arise in using the existing static

relaxation procedure for analysis involving loose block assemblage and/or having

widely varying joint stiffness. The method is to be further discussed in Chapter 5.

Current work lies in the development of three - dimensional distinct element

programs. The original version, 3DEC, has been described by Hart et al. (1988).

The capabilities of the updated distinct element programs include modelling elasto-

plastic behaviour of intact blocks, the various joint responses and other influences

such as structure elements, fluid interactions, infinite domains, etc.

Applications

Most of the earlier examples (before 1980) of applying the distinct element

method had been for the purpose of illustrating the applicability of the new tech-

nique. As well as Cundall (1971, 1974) and Maini et al. (1978), Voegele (1978)

described a series of such applications ranging in complexity from the modelling of

a jointed rock slope, to examining the behaviour of a discontinuous rock mass be-

ing mined by caving techniques. The applicability and correctness of the method

have been verified by physical models. More efforts have been made recently to

apply the distinct element method in engineering practice. Two examples are

shown below to illustrate distinct element analysis in rock mechanics.

Figure 2.9 shows an applied example in underground excavation design using

program BLOCKS (Sutherland et al. 1984). The jointed rock blocks were sim-

58

a)

•

b)

Figure 2.9 Modelling surface subsidence using program BLOCKS (rigid block

model), a) the distinct element meshes; b) simulated strata movements

(from Sutherland et al. 1984).

59

ulated by various shapes of rigid distinct elements, as illustrated in Figure 2.9a.

Modelling of excavations was carried out by removing the support blocks at the

bottom, and Figure 2.9b shows the strata deformation after 7 support blocks had

been removed.

Coulthard and Button (1988) used the distinct element program UDEC to

predict the subsidence induced by single panel coal extractions. Unlike the example

of Figure 2.9, the rock blocks were modelled by using fully deformable elastic

distinct elements. The calculated subsidence is shown in Figure 2.10a, and block

deformation due to the roof collapse and subsidence is shown in Figure 2.10b.

Conclusions

To date, the distinct element method is the only available method of analysis

of blocky media subject to large displacements. It is a newly developed special

numerical technique for the analysis of problems characterised by the conceptual

model c (Figure 1.2c). Although the theoretical basis of the method and the

parameters involved in the constitutive relations have not been fully clarified or

verified, its developments and applications have shown distinct advantages over

other numerical methods, and show great potential for numerical analysis in rock

mechanics.

60

Surface subsidence - UDEC calculations

00

25.0 50.0 75.0 100.0 125.0 150.0 175.0

Distance from centre of excavation (m) 200.0

Fine upper strata: W = 120 m Fine uppter strata: W = 160 m Fine upper strata: W = 200 m

Fine upper strata: W = 240 m Coarse upper strata: W = 250 m

CSIRO Division of Geomechanics

a)

b)

Figure 2.10 Modelling surface subsidence using program UDEC (fully deformable

block model), a) calculated subsidence values; b) calculated large block

deformation (from Coulthard and Dulton 1988).

61

2.6 Hybrid methods

The advantages and limitations of the various numerical methods of analysis in

engineering rock mechanics have been briefly discussed in the preceding sections.

In the application of these numerical methods, it has been found that none of

the methods is ideally suitable for all practical problems. This is because of the

limitations involved in each of the methods. A hybrid method is a combined (or

coupled) computational scheme in which two or three different methods are used to

preserve the advantages of each numerical procedure and eliminate disadvantages.

The existing hybrid methods used in rock mechanics include the finite element -

boundary element method, the boundary element - distinct element method and

the finite element - distinct element method.

The finite element - boundary element method

The procedures of coupling the finite element method with boundary integral

solutions were first described by Zienkiewicz et al. (1977). Brady and Wassying

(1981) implemented a combination of the boundary element method for modelling

a homogeneous elastic region and the finite element method for an inhomogeneous

region. The procedure has since been further developed and widely applied (Dun-

bar 1982; Yeung and Brady 1982; Meek and Beer 1982, 1984; Sofianos 1984).

In the coupling procedure, the partial differential equation and boundary

conditions of the problem are reduced to a system of equations involving both

finite element and boundary element discretizations (Beer 1985). The solution

procedure is implicit. Recent investigations in development and application of this

hybrid method have been focused on the nonlinear analysis of rock mass behaviour

(Swoboda et al. 1987) and three - dimensional excavation problems (Beer et al.

1987). An example of underground excavation analysis using the coupled finite

element and boundary element methods is shown in Figure 2.11.

62

a)

Bedding planes

Finite elements

Boundary elements

50 m

b)

scale

0 80 160 mm displ . L -scale

Figure 2.11 Example of the hybrid finite element - boundary element analysis of un-

derground excavations, a) finite element and boundary element meshes;

b) displaced shape showing slip and separation on bedding planes (from

Meek and Beer 1984).

63

The distinct element - boundary element method

In order to analyse the stress distribution and displacement in a jointed and

fractured region of a rock mass, a hybrid distinct element - boundary element

model has been developed (Lorig and Brady 1982, 1984; Lemos and Brady 1983).

When representing the rock which constitutes the near field of an excavation with

distinct elements, and the far field with boundary elements, the problem has gen-

erally to be solved in an iterative way. This is unlike the linkage of the finite

element and the boundary element method, because the distinct element method

is a specially developed explicit technique while the boundary element solution is

an implicit procedure. The calculation proceeds by considering the satisfaction of

the displacement continuity and equilibrium conditions at the interface between

the two solution domains (Brady 1985). In other words, it was assumed that no

slip or separation could occur at the interface.

Verification of the performance of this hybrid method appears very difficult,

particularly for the occurrence of large movement of distinct elements. Practical

applications of the method have not been found in the literature. Figure 2.12a

illustrates a demonstration example of the method, simulating an underground

opening.

The finite element - distinct element method

Dowding et al. (1983) presented a coupled finite element - rigid block (distinct

element) model to analyse lined openings in a jointed rock mass, as illustrated in

Figure 2.12b. In their model, an explicit finite element formulation (dynamic

analysis) was used to coincide with the relaxation algorithm of distinct element

analysis. Finite element nodes at the interface were assumed to be fixed on the

neighbouring rigid blocks, and only linear elastic analysis examples were given. No

existing program of this type was found in the literature (Plischke 1988; Coulthard

and Perkings 1987).

As has been described, the hybrid scheme is usually carried out by coupling

64

\ . \ \ far- f ie ld

d.0. domain

beam element structural

elements

distinct element

boundary element normal contact

element

transverse contact element

a)

Rigid Blocks

Si lent Boundaries

Area Shown in Figure 1 Continuum

Elements

Source

b)

Figure 2.12 Illustration example of the hybrid distinct element - boundary element

method for underground excavations, a) the coupling of distinct elements

and boundary elements; b) the coupling of distinct elements and finite

elements (from Brady 1985).

65

the different codes to model different parts of the region. One variation is the

deformable block model of a jointed rock mass developed by Maini et al. (1978)

and Vargas (1982) in which finite differences and finite elements were used to

represent the deformable rocks within the distinct elements. This technique has

been used in the development of updated distinct element programs (Lemos et al.

1985),

Conclusion

The purpose of przictical hybrid computational schemes has been to model

the far field rock mass with boundary elements (as a continuum) and the more

complex constitutive behaviour with the appropriate finite elements (as a pseudo-

continuum) or distinct elements (as a discontinuum). The coupling and applica-

tion of these hybrid schemes have proved effective and successful, and appear to

have great potential for future numerical analysis in rock mechanics. Although

relatively limited attention has been paid to the coupling of the finite element

and distinct element methods, and its development is only at an early stage, the

method proposed by Dowding et al. (1983) has shown the potential to model the

near field complex rock mass behaviour such as that of conceptual model e (Figure

2.2).

66

2.7 Other methods

A number of other computer aided numerical and analytical tools are available for

rock strata analysis involving excavations. These include the conventional limiting

equilibrium analysis, block theory, discontinuity deformation analysis, the rigid

block - spring model, and the rigid body - joint element method. Apart from

the limiting equilibrium analysis, the others are all relatively new methods and

are mainly developed for analysis of discontinuous rock mass behaviour. In this

section, these methods are briefly reviewed and instructive comparisons of the

methods are given.

The block theory

Based on the stereographic and geometric methods of stability analysis of rock

masses, Goodman and Shi (1985) presented a complete theory of block analysis in

hard rock excavations. Recent development and application of the method, with

particular emphasis on using Lnterax:tive computer graphics, was reported by Lin

and Fairhurst (1988). The theory provides a technique for describing the geometric

relations of intersecting discontinuities penetrating a three - dimensional solid.

The system of joints and other discontinuities is then analysed to find the critical

blocks of the rock mass when excavated along defined surfaces. Calculations of

stability are only required for these critical blocks, or key blocks, using limiting

equilibrium theory to give a fcictor of safety. The advantages of the method include

ease of use and the direct way of determining instability, so it is a basic tool for

practical engineering design. However, the method is limited only to the analysis

of a rock mass characterised by the conceptual model b (Figure 2.2), and stresses

and displacements of the rock mass cannot be calculated by the method.

The discontinuous deformation analysis (DDA)

Being aware of the limitations of the block theory in analysis of rock mass

response to excavation, Shi and Goodman (1988) developed a new method to

67

compute stress, strain, sliding and opening of rock blocks. In this method, all the

rigid block movements and deformations occur simultaneously in one step of the

analysis. It was reported that the method parallels the finite element method, but

it is more general because it solves a finite element type of mesh where all the

elements are real isolated blocks, bounded by pre-existing discontinuities. Each

block was assumed to possess constant stresses and strains with global equilibrium

equations derived by the minimum work theory. However, the method is only at

its early stage of development and, some mechanisms and procedure, especially

for nonlinear analysis, are still not clear.

The rigid body - spring model (RBSM)

Kawai et al. (1981) described a rigid body - spring model to simulate discrete

weak rock mass behaviour. In the model, arbitrary shaped blocks are connected

by distributed elastic springs, and strain energy due to the relative displacements

of the spring system is then used to derive the system equations. Because of the

assumption that the springs are fixed to each block, the discontinuum behaviour

cannot be modelled well by this method, especially when relatively large move-

ments of blocks occur. Suzuki et al. (1985) used this model to simulate the elasto -

plastic, dynamic response of blocky media, but the analysis was limited to a small

deformation assumption.

The rigid-body joint-element method (RJM)

In order to overcome disadvantages of methods such as RBSM and the finite

element or distinct element methods, an alternative approach was presented by

Asai et al. (1981). The method is called the rigid-body joint-element method,

and is based on a combination of Goodman's joint element and Cundall's distinct

rigid element. Unfortunately, however, a small displacement between blocks was

again assumed, and it is not suitable for analysis of the nonlinear behaviour of

discontinuous rock masses.

68

2.8 Conclusions and discussions

Critical assessments of the existing numerical methods in rock mechanics have been

made, and it is seen that although various numerical techniques have been devel-

oped to simulate the majority of rock mechanics problems, the presently available

programs cannot be used effectively in strata movement analysis, such as road-

way stability analysis. This results from the complexity of strata properties and

geometries. Therefore, in order to evaluate ultimate strength and displacement

in strata stability problems, a practical computer program is required by which

the previously mentioned coupled problems can be hajidled effectively. Such a

program can possibly be established based on the existing numerical techniques

with necessary modifications.

For the purpose of this research, the following additional conclusions are

drawn;

a. The finite element method appears the most mature and versatile technique

in modelling weak rock mass behaviour. However, more investigations have

to be made in applying the technique to strata movement analysis, including:

implementation of suitable yield criteria for rock masses, introduction of geo-

metric nonlinear formulations for large deformation analysis, and simulation

of 3-D excavation and time-dependent effects.

b. Most existing programs and their applications of various numerical meth-

ods are two-dimensional, but practical rock engineering problems require 3-D

analyses. Although in the current state of development, 2-D numerical analy-

sis appears more practical for a complex problem like roadway analysis, more

attention must be paid to the interpretation of 2-D computation results in

explaining 3-D phenomena.

c. A number of numerical models have been proposed to simulate the behaviour

of discontinuities (joints or bedding planes). Nevertheless, it appears that the

69

distinct element model is the only one which can deal with relatively large

movements of joints. There is no satisfactory program available for complex

strata movement analysis. Consequently, a hybrid computational scheme is

potentially an effective method for general analyses of strata movement, but

further investigations are required.

Based on the above considerations, a series of studies is suggested and pre-

sented. Firstly, the finite element technique is adopted in this research, with

necessary improvements to the numerical procedure to model the fully nonlinear

behaviour of an equivalent continuous rock mass. Secondly, the developed finite

element program is used to investigate the validity of a 2-D plane strain analysis

in modelling 3-D tunnelling. Thirdly, a new coupling procedure to link finite el-

ements with distinct elements is proposed , which is believed to be currently the

only method that can be used in the analysis of conceptual model e, and finally,

the application of the developed program is demonstrated.

A main weakness of the proposed technique is expected to be that realistic

problems require very large amounts of computation. However, this appears to be

a practical and unavoidable feature of a fully nonlinear process of rock mass defor-

mation and failure. One of the solutions is to use the power of the supercomputer,

as has been done in the aerospace and petroleum industries (Janakiraman and

Das 1987). On the other hand, the development of the PC or desktop computer

has advanced to such a level that they could replace the mainframe (or perhaps

even supercomputer) in the near future. The research described in the following

chapters is therefore believed to have great potential in rock engineering design.

70

CHAPTER 3 TWO-DIMENSIONAL NONLINEAR FINITE ELEMENT ANALYSIS OF ROCK MASSES

3.1 Introduction

In the modelling of a rock mass as a continuum (e.g. the coal seam or the seat

earth in the Coal Measures), the finite element displacement method is used. It

is assumed for the current stage that the rock strata under consideration are

homogeneous, isotropic, elastoplastic or elasto-viscoplastic media.

Although it is known that the finite element method is superior to any other

numerical methods in modelling nonlinear behaviour of a continuum (Section 2.2

and 2.7), there are still a number of facets which need to be investigated in applying

the method for the purpose of rock engineering. Furthermore, however powerful

it is, a numerical model can never give an accurate prediction unless the input

parameters reflect the real properties of the rock mass and in situ stress.

In this chapter, the finite element formulations used in this research are briefly

presented and the developed finite element program COAL is described (Section

3.2). In order to incorporate into the program a suitable yield criterion for the

analysis, a rock mass response model is proposed in Section 3.3. Incorporation

of the nonlinear material model and some validation examples are presented in

Section 3.4. Implementation of the geometrically nonlinear analysis is described

and validated in Section 3.5. A method of simulating the excavations in this elasto-

viscoplastic analysis is developed in Section 3.6. Some conclusions ajid discussions

of the above work are given in Section 3.7,

71

3.2 Nonlinear finite element analysis

3.2.1 The basic expressions and algorithm

The program for plane strain elasto-viscoplastic analysis described by Owen and

Hinton (1980) is taken as the original version of the finite element analysis in

the numerical model. The analysis is based upon the one dimensional rheological

model which was originally proposed by Zienkiewicz and Cormeau (1974) and

is illustrated in Figure 3.1. The friction slider component develops a stress <Tp,

becoming active only if a > Y, where a is the total applied stress and Y is some

limiting yield value. The excess stress cr = a—a^ is carried by the viscous dashpot.

Instantaneous elastic response is provided by the linear spring. The presence of

daahpot allows the stress level to instantaneously exceed the value predicted by

plasticity theory, the solution tending to this equilibrium level as steady state

conditions are achieved in the system.

crj = <r - o-p

A

i

A

V

Inactive if

//////////

Figure 3.1 Basic one-dimensional elasto-viscoplastic model

(from Owen and Hinton 1980).

72

Stress, strain and flow rule:

In this model, the total strain is given by the sum of the elastic and viscoplastic

components as (see Figure 3.1)

£ — €e €%p. (3.1)

The total stress is equal to the stress in the linear spring and is related to the

elastic strain by

a = Og = Ese, (3.2)

where E is the elastic modulus of the linear spring. In figure 3.1, Y stands for the

yield stress which can vary with the viscoplastic strain (it increases with strain

hardening and decreases with strain softening).

For 2-D or 3-D problems, the above a ajid e are vectors, and the E is usually

replaced by an elastic matrix D. In that case, differentiate (3.1) and (3.2) with

respect to time t to give the incremental relations

de dse dt dt

1

dt ' (3.3)

da due dt dt

(3.4)

The onset of viscoplastic behaviour is governed by a scalar condition of the

form

F = Fi{(t,€„p) - Fo = 0, (3.5)

in which FQ is the equivalent yield stress similar to the above Y which may itself

be a function of a hardening or softening parameter, Ft is the equivalent stress

level and function F usually defines a yield surface in the stress (or strain) space

(see Section 3.3).

The viscoplastic flow rule is taken in the explicit form:

(3.6)

73

where Q is the plastic potential and 7 is a fluidity parameter controlling the plastic

flow rate. The function is defined as

In general, $(JF) can be any positive monotonically increasing function of F, such

as

and

« ( f ) = - I, (3.8)

(3.9)

in which M, N are arbitrary prescribed constants, but (3.7) has given the most

satisfactory results.

The viscoplastic strain increment

With the viscoplastic flow rule defined by (3.6) the viscoplastic strain incre-

ment during the time interval is

( 1 _ 0 ) 4 ^ + 0 . dt dt

(3.10)

By a truncated Taylor series approximation,

de: "p vp dt dt

(3.11)

and using (3.6)

F " = d^i vp

dtda ^ dF da I dF\

a * / . (3.12)

In Equation (3.10), 0 is chosen within the range 0 to 1.0. If 0 = 0, the

algorithm is an explicit method of integration with respect to time; if © = 1.0, it

is a fully implicit method.

Thus, (3.10) can be rewritten as:

dt ^^'"'6tr, + e6tr,n"-6a' (3.13)

74

The stress increments

The stress increment during time [in,in+i] is:

6a^ = D6e^ = D{6e^ - (3.14)

in which

5c" = (3.15)

where jB" is the strain matrix and 6d^ is the incremental displacements vector.

Substituting (3.13) and (3.15) into (3.14), the stress increments become

= IKp (B-'Sd" - ^ S t „ \ , (3.16)

where

D':^ = {D-^ + Q6t^H^)- \ (3.17)

and

= Bo + (3.18)

where BQ represents the standard linear strain matrix and B^j^ is the nonlinear

component. A detailed derivation of the matrix B"" is given in Appendix 4 and its

implementation in the program is presented in Section 3.5.

Equilibrium equations and their incremental forms

The equations of equilibrium can be derived either by application of the virtual

work theorem or by minimising the total potential energy of the structure, and

can be written as

[Bfa'^dn + r = 0, (3.19) L 'n

which are to be satisfied at any instant of time, tn. Here f"' is the vector of

equivalent nodal loads and the integration is proceeded in the region f].

During a time increment, the equilibrium equations which must be satisfied

are given by the incremental form of (3.19) to be (see Section 3.5.2)

[ [BfSa'^dn + f [SB^^far^dn + 6^ = 0, (3.20) Ja Jq

75

in which 5 /" represents the change in loads during the time interval 6tn and

is related to the current displacements.

For small strain analysis, the incremental form of (3.19) is

+ S r = 0. (3.21) f [Bof Jn

From the above equations, the incremental stiffness equation can be written

in the form:

K^Sdr- = AV", (3.22)

where

AV"- = j j B " - f D ' : ^ ^ S t , , d n + S r , (3.23)

K}=K'/- + KYi= f + K^j, (3.24) Jn

in which Kf is the conventional stiffness matrix for small strain analysis and Kfj

is the stress dependent stiffiiess matrix for large strain analysis which is discussed

in detail in Section 3.5. The incremental displacements are, therefore, given as

SdT- = [K^]-^AV'^. (3.25)

Computational procedure

From the above formulations, the incremental displacements at time tn

can be used to determine the incremental stress Sa"", and the stress and displace-

ment at time t = are

= + (3.26)

<r+^=<r + 5<r. (3.27)

The strain is calculated by

fc" = (3.28)

= Cp + (3-29)

76

The new equilibrium state should be:

/ = 0. (3.30) Jn

and, if the equilibrium state has not been reached (because of the approximation

in equation (3.11)), the residual forces are

^"+1 = f ^ 0. (3.31) Jn

^n+i jg then added to the applied force increment at the next time step. The

procedure in each time step can be summarised as:

1) calculation of viscoplastic strain rate from current stress state by using (3.6),

2) using (3.16), (3.21) and (3.22) to find and A V ,

3) finding by (3.24) and (3.25),

4) evaluation of by (3.28) to use in next time step.

A flow chart of the original program (VISCO) to fulfil the above algorithm,

with its outlined features, is given in Appendix 1.

3.2.2 Description of the program

a. The original program (Owen and Hinton 1980)

In the original program, three different types of finite elements are used, namely,

the 4-node, 8-node and 9-node isoparametric quadrilateral elements (Figure 3.2).

The equation solution is by the frontal solver (Hinton and Owen 1977), which is

a very efficient method. The spatial discretization reduces the problem to that

of the integration of a system of simultaneous ordinary differential equations in

which time is the independent variable and nodal displacements are the dependent

variables, subject to initial conditions. The user of the program is given a choice of

either explicit or implicit linear methods of solution of these equations

by choice of 0 .

77

The program as presented in their book is based on the theory of infinitesimal

strain and therefore is incapable of predicting geometrically nonlinear behaviour

such as buckling. The yield function may be chosen to be that of the Tresca, Von

Mises, Mohr-Coulomb or Drucker-Prager criteria, none of which is likely to model

satisfactorily the behaviour of the weak rock strata like coal or seat earth (see sec-

tion 3.3). The inelastic strain rate is computed according to the relevant associated

flow rule, which is not satisfactory for the rock strata under consideration as it will

overestimate the inelastic dilation. The program can only allow an incremental

loading sequence so that the excavation cannot be simulated. The program does

not incorporate any special elements, such as joint element or contact element, for

discontinuities.

4-noded element 8-noded element 9-noded element

Figure 3.2 Isoparametric quadrilateral finite elements

There are some errors existing in the original program. One of them is the

calculation of axial stress in plane strain analysis in which the stress cr was

calculated based on the cissumption = e® = = 0. Unfortunately, this is not

correct because the strain in the z direction should be 6 = + = 0 (Reed 1988).

78

Another error occurs in the incremental form of the equilibrium equation in which

the stress dependent stiffness Kf'j is ignored. When geometrical nonlinearity is

taken into account in an analysis, this stiffness plays an important part,

b. The modified program

Into the program has been incorporated the nonlinear Hoek-Brown yield function

which is believed to be the most suitable criterion currently available for rocks. A

modified Hoek-Brown yield criterion (the extended 3-D H-B yield surface) is also

provided in the program, which is shown to have several significant advantages,

especially for weak rock masses. A choice is given for the user to apply either

the associated flow rule or the non-associated flow rule which uses the modified

Hoek-Brown yield function as the plastic potential Q (Section 3.4).

The large displacement theory has been implemented into the program using

the Updated Lagrangian formulations (Section 3.5). It can be determined by

a controlling parameter to take account of the geometric nonlineaxities in the

analysis, and if so, the loading (or excavation) must be carried out in a stepwise

manner. The program has also been provided with a procedure to remove material

progressively (reducing stress and stiffness) to simulate the excavation sequence.

Some other features of the program including the provision of 7-node isopara-

metric elements and contact loading for the hybrid computation are outlined in

Section 5.3. In Appendix 3, a flow chart of the modified program COAL and

descriptions of the new subroutines are given, together with the user instructions

of the program.

79

3.3 T h e rock mass behaviour mode l

3.3.1 Const i tut ive relation and plasticity theory

An important phenomenon manifested by rock strata in the vicinity of under-

ground openings is their nonlinear response to induced stress. In situ obser-

vations and experimental results show that this is caused by a combination of

nonlinear elasticity and post peak behaviour. After failure, a rock mass may ex-

hibit strain softening, perfect plastic or strain hardening behaviour depending on

confining pressure, rate of loading and rheologic characteristics of the material as

shown in Figure 3.3 (Hudson at al. 1972a,b; Farmer 1983). For the mechanical

description of these nonlinear post failure behaviour modes of a rock mass, a yield

criterion based on the theory of plasticity is usually used.

Axial stress

Strain-softening

(Uncontrolled)

Controlled)

Axial strain

Axial stress

Strain-hardening

\ Perfect-plastic

Axial strain

Figure 3.3 Stress - strain curves for: a) brittle fracture in uniaxial compression; b) dectile behaviour in compression (after Hoek and Brown, 1980).

80

The theory of plasticity was initially developed for ductile metals which show

strain hardening or perfect plastic behaviour. Although for rock material ductility

is of much less importance, the theory has been of great value in rock mechanics.

This is because the study of yield and flow has been very highly developed, and

the phenomena are essentially simpler than those involved in fracture. Theoretical

study and large scale tests on jointed rock masses have shown that it appears

reasonable to use plasticity theory to describe some aspects of the constitutive

behaviour of rocks and rock masses (Muller-Salzburg and Ge 1983; Brown 1987).

With the recent advance of numerical techniques, the strain softening be-

haviour has also been widely analysed using the theory of plasticity(e.g. Gumu-

soglu et al. 1986; Reed 1987; Crook et al. 1987). In this research, therefore, the

rock mass behaviour model will be based on the theory of plasticity. In particular,

using the finite element algorithm presented in section 3.2.1, the time dependent

plastic deformation can be modelled.

In plasticity theory, the actual rock material is replaced by an idealistic one

that behaves elastically up to some limiting state of stress at which failure occurs.

The essential elements of the plasticity theory are equations of equilibrium, the

geometry of strain, a stress-strain relation of some kind, and a yield condition(Hill

1950; Prager 1959). The former two have been defined in the finite element expres-

sions of section 3.2.1. In practice, the elasto-viscoplastic algorithm can treat either

strain hardening or strain softening problem effectively (Reed 1987; Crook et al.

1987). In this analysis of rock strata movements, however, the elastic-perfectly

plastic stress-strain relation for the rock mass is assumed and the yield criterion,

in this case, is the same as the peak strength criterion. This is the main topic

discussed below.

81

3.3.2 Yield criteria for rocks

Yield criteria in 3-D principal stress space

To determine the conditions which govern the failure or yielding of a rock

mass considered as an equivalent continuum, a 3-D strength criterion is generally

required. In plasticity theory, this criterion is defined as a function of stress

components satisfied at the onset of plastic deformation. It is usually written

as

F{a ,K)=0, (3.32)

where a is the stress components and /c is a parameter related to strain hardening

or softening. In the numerical analysis of geomechanical engineering, two of the

classical yield criteria for rocks are the Mohr-Coulomb (1773-1882) criterion and

the Drucker-Prager (1952) criterion (Zienkiewicz and Cormeau 1974; Yuritizinn

et al. 1982). In 3-D principal stress space, they define the simple yield surfaces

of Figure 3.4. Any state of stress may be defined in terms of principal stress

components ai, <73 and 0-3, so that a stress may be represented graphically as

a point in stress spare whose three axes are those of principal stress. A change

in stress may be represented as a stress path, the locus of the stress point as it

moves through stress space. The definitions of a general yield surface (e.g. Mohr-

Coulomb's or Drucker-Prager's yield surface) are:

a) if a stress state point (o'i,<T2>< 3) is inside the domain, the material behaves

elastically and F[a, k) < 0,

b) if the stress point is located on the surface, the material behaves elasto-

plastically and F{a, k) = 0,

c) the stress state outside the surface (i.e. F{a, A;) > 0) is not possible because

of the material strength.

82

Mohr-Coulomb yield surface

Drucker-Prager yield surface

a| = 0 2 = CT3

Figure 3.4 Mohr-Coulomb and Drucker-Prager yield surfaces in 3-D principal stress

space.

Yield criteria for isotropic rock

Usually, a similar form of rock strength criteria is used as a yield surface in

practice. Of the classical strength criteria, the Mohr-Coulomb criterion is generally

considered the most acceptable for describing rock behaviour. However, through

considering the strength criterion in 3-D principal stress space, the main discrep-

ancies between the experimental results and those predicted by the Mohr-Coulomb

criterion are the curvature of its yield surface. The 3-D Mohr-Coulomb yield sur-

face is composed of six planes, whereas true triaxial test data for rocks show that

83

the surface is significantly curved. This discrepancy in two dimensions is shown in

Figure 3.5. The well known Griffith criterion (see Brady and Brown 1985) predicts

a parabolic relation between the stress components in plane compression but, for

a number of reasons, such ajs its prediction of rock tensile strength, often gives an

inadequate fit to experimental data (Brady and Brown 1985).

••

, 1 • •

•

1 t*'

r 1

0 }

von-iVlises

a) ( after Hoek and

Brown. 1980 )

M o h r -Coulomb

b)

(a f te r Akai and M o r i . 1967)

Figure 3.5 General nonlinear relation of stress components of rock at failure, a)

CTi ~ 0-3 plane view; b) octahedral plane view.

Since the 1960's, numerous investigations have been performed concerned with

correcting this discrepancy and trying to fit a nonlinear criterion with observed

84

data (Murrell 1963; Fairhurst 1964; Hobbs 1966; Hoek 1968; Franklin 1971; Bi-

eniawski 1974; Yoshinaka and Yamabe 1980; Hoek and Brown 1980a; Kim and

Lade 1984; Johnston 1985; Desai and Salami 1987; Michelis 1987). A brief review

of the available criteria is given in Table 3.1 in which the development of each

criterion and the constants involved are outlined. Many of these criteria provide

good explanations of some aspects of rock behaviour, but fail to explain others.

The ideal shape of yield surfaces for rock materials has been well accepted as a

curved, pointed bullet with curved cross sections in the octahedral planes similar to

those of Mohr-Coulomb's but without sharp intersection points or corners (Serata

et al. 1967; Akai and Mori 1970; Franklin 1971; Kim and Lade 1984).

Serata et al. (1967) suggested that the shape of the yield surface in the

octahedral plane should be different from that of the failure surface, the former

is a circular whereas the latter is a triangular with curved convex sides as shown

in Figure 3.6. Recently, Michelis (1987) interpreted his truly triaxial test data in

the meridian and deviatoric sections of yield surfaces and verified that the general

characteristics of a strength surface are:

1. In the meridian plane: a) the yield curves are smooth, curved, and convex,

b) the yield curves depend on the hydrostatic pressure.

2. In the deviatoric plane (for compressive stress): a) the yield curves are smooth,

curved, and convex, b) the value of Jg is a function of 6 and for the same

yield state and material the curve shapes are similar.

Taking into account the practical application of a strength criterion in exca-

vation design, not only the intact rock, but also the rock mass behaviour has to be

considered in the development of a criterion(Hoek and Brown 1980). On the other

hand, a rock strength criterion is at best only an approximate fit to the observed

data, and thus it should be expressed in a simple way and involving the minimum

number of parameters for practical excavation design.

85

No. Authors The criteria Constants Involved

Development of the criteria

Murrell(1963) 7" pet — 8To<7 Oct- or: J2 — ATQII. One constant (3D criterion)

Extended 3D Griffith theory.

Fairhurst (1964)

if m(2m — l)ci + cj > 0; A\ ^ K,

if m{2m — l)<7i + cTj < 0; 7K

Two constants (2D criterion)

Empirical genera lisatlon of 20 Griffith theory for Intact rock.

Hobbs (1966) ai = Bo^ + 03, or; Two constants (2D crtterin)

Empirical test data fitting for Intact rocks.

Hoek (1968)

or;

oI — ~ 2C + A{oi + <73) 1

^max ~ ^maio

Three para-meters (2D criterion)

Empirical curve fitting for Intact rock.

Franklin (1971)

- 03 = crj (cTi + 03)^-Two constants (20 criterion)

Empirical curve fitting for 500 rock

specimens.

Bienlawski (1974)

<71 = K'o^ + CTci

or; ^ — 5'ct^ + O.IcTc.

Three constant (20 criterion)

Empirical curve fitting for700 rock specimens. (5 types)

Yoshinaka & Yamabe

(1980)

aj — 02 = aK{q){oi + 02 +

Three para-meters (30 criterion)

Empirical test data analysis for soft rocks (mudstone, etc).

Hoek and Brown (1980)

<71 — <73 = V rUTc< 5 -f S0C .

or: r = A(0„ + B)^.

Three para-meters (20 cri. for rocks and rock masses)

Appl. of Griffith theory and empi-rical curve fitting for rock and rock mass.

Kim and Lade (1984)

M3 - 2 7

Pa J 'U-Three para-meters (3D criterion)

Analytical exami-nation on test data (originally for soil

and concrete).

10 Johnston (1985)

01 0«

M 03 Is Z + 1

Three para-meters (20 criterion)

Empirical curve fitting for soft rock specimens.

11 Desai and Salami

(1987)

4 = More than six parameters (30 criterion)

Polynomlnal ex-pansion In terms of stress Invariants to curve fitting.

12 Micheiis (1987)

/ q in + aiPy + j — Four constants (30 criterion)

Analytical and ex-perimental exami nation on yield sur face (true triaxlal test).

Table 3.1 A review of rock strength criteria developed since the 1960's.

86

Failure surface

Yield surface

Figure 3.6 Serata et al.'s theory (1972) on yielding and failure surface characteristics of rocks in principal stress space.

87

Effect of the intermediate principal stress on yield

It can be seen from Table 3.1 that most of the criteria were proposed on

the assumption that rock strength is independent of intermediate principal stress.

This assumption implies that the yield surface is generated by lines parallel to

the principal stress axes. Following experiments on hollow cylinders with external

pressure and axial load, Obert and Stephenson (1965) suggested that the failure

stresses are independent of However, Hajidin et al. (1967) demonstrated

quantitatively the influence of on shear strength of the Solenhofen limestone.

Mogi (1967) pointed out that the effect of the intermediate principal stress did

exist, but it was relatively small. Hoskins' (1969) hollow cylinder experiment on

trachyte shows that <72 has a significant influence on the rock strength. However,

Akai and Mori's (1967) test data on sandstone manifests no such an influence at

all (see Figure 3.7). Likewise, yield criteria for isotropic rock material have usually

been based on empirical criteria independent of the intermediate principal stress.

It is more acceptable to assume that the intermediate principal stress will have a

negligible influence on yield conditions for intact rock than it is for jointed rock

(Reik and Zacas 1978). Desai (1980,1987) and Kim and Lade (1984) proposed

some yield surfaces which are three dimensional and erg dependent (see Table 3.1).

These surfaces may be used in theoretical analysis, but will be very difficult, if not

impossible, to apply in laboratory and practical engineering design because of the

complexity and because too many constants are involved.

3.3.3 The Hoek-Brown criterion

The assumptions

Among the available criteria, the only one which takes account of the strength

of intact rock as well as jointed rock masses (excluding a regularly jointed rock

mass such as that discussed by Amadei, 1988) is the Hoek-Brown criterion (Hoek

and Brown 1980a). In developing their empirical criterion, Hoek and Brown as-

sumed that intermediate principal stress has a negligible influence on rock failure

88

a )

b )

cr (Kb) 0-69<")

0 52(A)

0-35(o)

M = l-2

0 9<t - 0 . 2

( kb)

M = 7 0

1-4 or 4/(1+ 3 8

o (kb)

Figure 3.7 Influence of the intermediate principal stress <J2 on rock

failure, a) Hosking's results; O2" dependent; b) Akai

and Mori's results: (72 " independent.

89

conditions and a rock or a rock mass failure is dominated by brittle behaviour

because the parameters used in the criterion are derived from the test data under

relatively low confining pressure. It has been noticed that for a jointed rock mass

the former assumption could be less acceptable so that in applying the criterion to

multiply jointed rock, the rock mass is assumed to contain more than four random

sets of discontinuities having similar properties. If ductile failure occurs, more

attempts should be made to determine the required empirical parameters. Some

other assumptions such as the influence of pore fluid, loading rate, scale ,etc. were

discussed in detail by Hoek (1983).

The formulation

The criterion was based on classical Grifiith theory and the examining of a

wide range of experimental data for intact rocks and rock masses. It assumes the

following relation between the principal stresses* at failure;

=<^3 + y/macffa + sa^. (3.33)

where (Xc is the uniaxial compressive strength of the intact rock material and m

and 5 are empirical parameters, which depend on the properties of the rock mass

and on the extent to which it had been broken before being subject to any load.

An updated discussion on how to determine the parameters m and s was recently

given by Hoek and Brown (1988). The uniaxial tensile strength as given by the

criterion is

(Xt = acTc, (3.34)

where

a = + (3.35)

For intact rock and m > > 0 as that listed in Hoek and Brown's table (Hoek

and Brown 1988), a « — and therefore, Hoek-Brown criterion predicts a ratio

* In the existence of pore pressure, these should be eff ective principal stresses

(see Hoek, 1983).

90

1 ; 7 ~ 25 of uniaxial tensile strength to uniaxial compressive strength, which

is much more realistic for rock than the classic Griffith theory. It appears that

the yield surface of Hoek-Brown criterion in 3-D principal stress space has not

been well understood, and it was assumed that the only diSerence between Hoek-

Brown's and Mohr-Coulomb's criteria is that the sides of the conical hexagon yield

surface illustrated in Figure 3.8 are curved. However, an investigation by means

of computer graphics reveals that the Hoek-Brown yield surface satisfies another

of the attributes of the ideal shape of a yield criterion for rock materials, namely,

its cross section in the octahedral planes shows a curved hexagonal rather than a

Mohr-Coulomb type cross section.

The Hoek-Brown yield surface in 3-D principal stress space

If compressive stress is taken positive to coincide with rock mechanics con-

ventions, the yield function can be written from equation (3.33) as

F = ai — as — y/mOcOs+s^ = 0. (3.36)

The relations between stress invariants and the principal stresses are:

h = = O"! + 0-2 +

h = + crsai, (3.37)

^3 = = aia2(Ts. and deviatoric stress invariants:

Ji = 0,

J2 = + {02 - <73) + (< 3 - < i)% (3.38)

• 3 =

In order to use a convenient form of the yield surface equation, the cubic

equation

<7 — J2(T — Jz = 0 (3.39)

is solved by introducing the 'Lode' angle 6: (Lode,1926)

sin 30 = (3.40) 2Ji

91

Curved conical hexagon sides.

Assumed Hoek-Brown surface with the linear Mohr-Coulomb cross sections

Figure 3.8 An assumption of the Hoek-Brown yield surface (before 1987).

P: ( 0 1 , 0 2 , 0 3 )

(r , h, (j))

Figure 3.9 Characteristics of a stress point on a general yield surface.

92

to give the principal stresses as (see Nayak et al. 1972)

\ /3 ^

'5m(tf + |7r) I J sinO > + -7i

. 5m {0 + ITT) (3.41)

with <7i > <T2 > < 3 and — f < ^ < f • Substituting (3.41) into (3.37), the Hoek-

Brown's yield function can be written as

F = 2J2 cos 6 — rruTcJ^ (sin 0 + VScos 0J + IirruTc + aaj = 0. (3.42)

The advantages of using a criterion defined in the form F = ( / i , J2 , ) = 0 are

its convenience in the numerical analysis and its physical significance which are

discussed as below.

Considering a general yield function F{ax,(72,<^z) = 0, one can obtain a sur-

face in 3-D principal stress space as shown in Figure 3.9. A stress point p on this

surface is characterised by h,r and <f>. After the above coordinate transformation

and noticing that

f^oct — g A, (3.43)

/2 a ' Toct ~ Y 3

where Toct and (Xoct are the octahedral shear and normal stresses (Nayak et al.

1972), the following relations yield:

k -

(3.44) r = ,

(f) = 0.

With (3.42) and (3.44) the Hoek-Brown yield surface in principal stress space

can be easily drawn by using the computer graphics facility as shown in Figure

3.10. Other yield surfaces, such as Mohr-Coulomb's can also be drawn in the same

way, so that different yield criteria with variant material constants or empirical

parameters can be compared with each other easily. It is seen that the Hoek-

Brown yield surface gives an acceptable approximation to the observed data in

Figure 3.11 (Akai 1967; Serata et al. 1972; Kim and Lade 1984).

93

HOEK AND BROWN S U R F A C E M = 0 - 3 0 0 S = 0 - 0 0 0 1 S I G M A - C = 3 5 - 0

H O E K AND BROWN S U R F A C E M = 7 - 0 0 0 S = 1 . 0 0 0 0 S I G M R - C z 3 5 - 0

Figure 3.10 The Hoek-Brown yield surface in 3-D principal stress space, a) for rock masses; b) for intact rocks.

94

HOEK-BROWN

KIM-LADE

(72

• Experimental results from Mogi's tests on trachyte.

Figure 3.11 An interpretation of test data in octahedral plane using the Hoek-Brown criterion (data from Mogi's (1967) results and Kim & Lade's interpretation).

95

In Figure 3.12, the 3-D Hoek-Brown yield surface is shown to be a combination

of the components of six 'single curved' parabolic surfaces which explains why the

intermediate principal stress 02 has no influence on the yield criterion. As the

stress point moves from point 1 to 3, passing through 2 on the yield surface, the

locus is a straight line and the variable remains unchanged at points 1 and 3.

This means that an increase of the intermediate principal stress 02 will lead to an

increase of the first stress invariant / i , but may not change the second deviatoric

stress invariant Jg. This may be a common feature of most nonlinear empirical

criteria as long as they are intermediate principal stress independent.

The criterion in predicting soft rock strength

It is generally accepted that the Hoek-Brown criterion is suitable for most

types of rock including soft rock. In applying the criterion, however, one must be

very careful in choosing these material constants. As Hoek and Brown said: "It

must be strongly emphasised that the relationships set out in Table 3 are based

on very sparse data and therefore very approximate; • • • every attempt should be

made to determine the required strength parameters by laboratory and in situ

testing • • •".

Johnston (1986) compared Hoek-Brown criterion with his own criterion devel-

oped for soft rock by analysing triaxial test data for mudstone (Figure 3.13). He

found a large discrepancy and concluded that Hoek-Brown criterion would seem

to be restricted to relatively hard rocks. Nevertheless, this comparison can be in-

terpreted in an alternative way through consideration of the choice of parameters.

In spite of one material parameter less, the curve would have been as well fitted by

the Hoek-Brown criterion as by his own criterion if the parameter m were calcu-

lated as suggested by Hoek (1983). For example, in Figure 3.14 is shown the curve

fitting using m = 20. Other investigations on soft rock strength, such as Hobbs

(1966) and Yoshinaka and Yamabe (1980), resulted in similar criteria as shown in

Table 3.1. After conducting tests on a soft rock material, Yudhbir et al, (1983)

96

a,= 02=03

Figure 3.12 The construction of the Hoek-Brown yield surface in 3-D principal stress space ini,%2 and 713 are the octahedral planes which are

perpendicular to the axis aj=0'2=<T3)-

97

10

14

12

10

Gin 8

6

4

2

0

I r

M-4j,8 - B - 0 . 8 1

# / /

/ - I /

/

G;n-G3n+(mG^+1)«^ m "10

• R«su(U of triaxlal le t t s on Melbourne -

rtudftone.0c - 2MP«

_L _L _L _L 1

^3n

Figure 3.13 Johnston's comparison of the two criteria (after Johnston, 1986).

/ / / J r —y

M: r

M: •2( > / -

• M: :1( ) r

/ I

'P: 0 . 6 5 vi; 1 (

1

/ f

Figure 3.14 The alternative interpretations to the data using the Hoek-Brown criterion (m=20, or (3=0.65).

98

also concluded that the Hoek-Brown criterion is suitable for soft rocks and predicts

test results very well in the brittle rock behaviour range. The discrepancies of the

Hoek-Brown curve fitting to observed data are usually in the curvature of the

yield surface because the Hoek-Brown criterion gives a quadratic approximation

to this yield surface; however, many other criteria use more complicated empirical

curved surfaces which usually involve more parameters. In fact, the Hoek-Brown

criterion can be easily modified to take account of a more precise curve fitting by

introducing a third empirical parameter /3 and, consequently, the criterion (3.33)

becomes:

(3.45) "o "„ \ "c ]

For instance, if /? = 0.65 is chosen for the Melbourne mudstone of Johnston (1986),

the Hoek-Brown criterion fits the test data very well using the original parameter

m = 10 (see Figure 3.14). Equation (3.45) can be used for applications La which a

specific curve fitting or a higher precision is required. For modelling of a rock mass,

however, the quadratic approximation seems more realistic and suitable because

of its simplicity and relative accuracy.

3.3.4 A simplified 3-D criterion

Although it appears that the Hoek-Brown empirical strength criterion is still the

most suitable and realistic criterion for predicting the strength of a rock mass,

difficulties have often occurred in. applying the criterion in practical analysis or

numerical modelling. These difficulties include the following points:

1. The consideration of the influence of the intermediate principal stress on the

rock mass,

2. The treatment of the singularities on the Hoek-Brown yield surface,

3. Differentiation of the yield function or the plastic potential, and

4. Determination of the empirical parameters m and s.

In the modelling of soft rock mass behaviour, it was noticed that the Hoek-

99

Brown yield surface is very close to a paraboloid surface if either the uniaxial

compressive strength or the parameters m and s are small. In fact, as m

and s decrease, the shape of the cross section in the octahedral planes changes

from a curved triangular shape to a curved hexagonal shape (see Figure 3.15a),

which approximates to a circle. The criterion derived specifically for soft rocks by

Yoshinaka and Yamabe (1980) exhibits a similar characteristic. It is, therefore,

possible to extend the original Hoek-Brown criterion into a paraboloid surface

with a central axis of ai = <72 = as (Figure 3.15b.). Using this simplified (or

extended) yield criterion, some of the above difficulties are overcome and the

resulting discrepancies are found to be negligible see Appendix 2. The proposed

criterion is derived and verified as follows. In equation (3.41), let <7i = erg to obtain

Substituting ^ f into (3.42) gives

i*" = ^—- rruXcJ^ 4" —mli + , (3.46)

which is a 'inner apices' surface as shown in Figure 3.16b. In the same way, if

(T2 = 03 is substituted into (3.41), an 'outer apices' surface is derived:

F — y/sJ^ — ^ + —mil ~t~ • (3.47)

For a closer approximation, a mean surface (between the 'inner' and 'outer' apices

surface) can be derived by rearranging the parameter using an approximately

average of (3.46) and (3.47) as:

— J2 + (3.48) Of. i O

where m,s and Oc are the empirical Hoek- Brown parameters (Hoek and Brown

1988). Thus, criterion (3.48) takes into account the influence of the intermediate

principal stress in a similar way to the criteria of Von-Mises, Drucker-Prager and

the extended Griffith criteria (Murrell 1963) (see Figure 3.16b and Appendix 2).

100

HOEK ANO BROWN SURFACE M =0 .140 5 z 0-0001 5IGMA-C = 35.0

Figure 3.15a The Hoek-Brown yield surface for weak rock masses.

SlnPLIFIEO H-B SURFACE M =0.080 S = 0-00001 SIGMA-C = 35.0

0-,

Figure 3.15b The extended (or simplified) Hoek-Brown yield surface.

101

Hoek-Brown surface (with comers)

The extended (or simplified) 'average' surface

Figure 3.16 The average surface between the 'outer apices' and 'inner apices' of the Hoek-Brown criterion.

102

3.4 Material nonlinear analysis

3.4.1 Introduction

In the modelling of nonlinear material behaviour of a rock mass, the finite element

elasto-viscoplastic analysis as presented in Section 3.2 has been adopted. In ap-

plying the procedure, the decisions that must be made concern the finite element

type, the yield surface (strain hardening or softening), the flow rule, and the choice

of an explicit or implicit algorithm.

The eight-noded quadrilateral element is used, allowing a parabolic variation

of stress across the element as well as accurate modelling of curved tunnel bound-

aries. The associated seven-node element (see section 5.3.2) is available for use on

the mesh interface to model the coupling behaviour of finite elements and distinct

elements. The choice of yield criteria for the rock mass has been discussed in

previous sections. In this section, the incorporation of yield surface and flow rules

in the program COAL is presented, and the discussion will be limited to the per-

fect plastic situation. The time integration algorithm is discussed together with

modelling creep behaviour of rock masses, and finally, some validation examples

are given.

3.4.2 Yield function and plastic potential function

Generally, a yield function in numerical analysis can be written in the form

F = f [a) — k{K) = 0, (3.49)

where / is some function of stress components and A: is a material parameter to

be determined experimentally. The term k may be a function of a hardening or

softening parameter K. The Hoek-Brown criterion (3.42) can thus be rearranged

into

F — —-mil H J2 cos"^ 6 171 (cos 0 -\—sin 0 j y/ J2 — — 0* (3. 3 o"c \ v 3 /

50)

103

The yield function for soft or weak rock masses is taken to be the extended form

of (3.50) as (see section 3.3.4):

tw 3 \/3 f— ——Ii H J2 + — m y J g = sac- (3.51)

Classical plasticity theory is based on the concept of associated flow; in that

case, the plastic potential function is given as the same as the yield function or

Q = F. It has been recognised that this usually over-estimates the degree of

plastic dilation of rock masses (Brown 1987). A plastic potential often used in

conjunction with the Mohr-Coulomb surface is obtained by replacing the friction

angle <}> by an dilation angle tf). It seems that for the Hoek-Brown surface, a

similar approach can be adopted to give the required plastic potential (Reed 1986,

Gumusoglu 1986). This will, however, give rise to a number of shortcomings for

the current numerical model. The sharp comers on the Hoek-Brown surface must

be 'rounded off' in a certain way to avoid singularities in the flow vector (see

Figure 3.17). If the algorithm of 'rounding -off' suggested by Sloan (1986) is used,

the new parameters have to be introduced and the solution will be sensitive to

this because the flow rule tends to direct the stress state precisely towards these

corners (Reed 1987). On the other hand, it will be a very difficult task to form

the matrix for the implicit algorithm (see section 3.2.1) because this requires

derivation of the second partial derivatives of Q{o).

The above consideration has lead to a suggestion of using the simple form of

the simplified Hoek-Brown surface as the plastic potential:

+ — J2 + V ^ , (3.52) * O ^

where m' is the dilation parameter of the Hoek-Brown criterion. This flow rule is

shown in Figure 3.18 and the advantages are obvious:

i) dilatancy reduces continuously (nonlinear) with increasing of confining pres-

sure,

ii) the numerical difficulties in dealing with singularities are avoided,

104

Plastic potential surface (or yield surface for associated flow) in pr^ipal stress space.

Undetermined plastic flow direction at the comer points.

Figure 3.17 Singularities on the Hoek-Brown surface.

Df latancy reduces

Plastic flow direction

Hoek-Brown yield function

Plastic potential using Equation (3.52)

Figure 3.18 A flow rule using the Hoek-Brown yield surface and its extended form as the the yield function and the plastic potential function.

105

iii) it is relatively easy to obtain the second partial derivatives of Q (a).

In the analysis of weak rock masses, the yield function can also be taken as

Equation (3.51) and if the parameter m' = m, the associate flow is achieved.

3.4.3 The flow vectors and the matrix [5 "]

The flow vectors

In order to calculate the strain rate in (3.6) the flow vector {o} can be ex-

pressed explicitly as (see Owen and Hinton,1980):

W = = Ciioi} + C2{aii} + C3{aiii}, (3.53)

where for plane strain:

{c} = {''a;, Cg, Taig, <Tz}, (3.54)

and for axisymmetric problems:

{cr}^ = {(^r,<rz,'rrz,ce}, (3.55)

the vectors:

H"}'

{%} = 1 ^ . (3-56) a{*}' dJs

a{*} '

are explicitly given in Table 3.2 and the constants:

dQ Cx =

a i l '

dQ tan dQ \

Ca =

'2 ^ 2

- \ / 3 1 dQ 2 cos 30 TJ do '

''2

can be chosen according to different plastic potential functions. For associated

106

flow using the Hoek-Brown criterion, these constants are: ( | # | < 29°)

_ 1 < i =

8 1 C-i = COS 0[—Jg cos d{\. + tan 6 tan Z0)

1 1 (3.58) + m(l -\—•= tanO + tan 0 tan SO •= tanZO)],

y/3 Vs B Sv S -1 —^[ Jg sinO — m(l — -^tanO)]. 2 J2 cos Z0 <T,

dJi = {1, 1, 1, 0, 0, 0}

8(7

aCJgyvs 1 "2 — - — •> 2rzx, 2Txy}

oa 2( J2 J '

, \^x <^z —'^xz

^x ~Txy^ J '^{jxz'^xy (^X ^yz),

^i^xyTyz ^y '^xz)^ '^ijyz'^'xz <^z

Table 3.2 The flow vector components

For those points at the corners, i.e. | # | ^ 29°, the constants are derived as;

if 6 = +30°

Ci =

% = - 4 + (3M) cr,

C3 = 0,

107

and if ^ = —30°

Ci = - - m ,

6 i 1 C2 — —J2 H—

Vs" (3.60)

C3 = 0.

For non-associated flow using the extended Hoek-Brown surface as the plastic

potential, the constants can be derived from (3.52) and (3.57) as:

( 1 — —gfM,

C2 = — + ^ m , -Oc "

(3.61)

C3 = 0

These are much simpler and easier to use.

The evaluation of [H"-]

For a general case of non-associated flow, the matrix [H"] in section 3.2.1 is

(3.62)

where {6} = dF/d{a} and {a} = dQ/d{a}. For plane strain and stress, or

axisymmetric problems:

{a} = {ai, 02, as, 04}^, {6} = {61,62,63,64}^.

and thus ®i6i < 261 &361 #461 &162 0262 #362 <1462 0163 0263 A363 #463

. 164 0264 0364 0464.

which is usually nonsymmetric. For example, {a} and {6} can be determined by

= (3.63)

using (3.61) and (3.58) for the plastic potential and yield functions respectively.

In (3.62), the only term unknown at this stage is the derivative of the flow

vector {a}, that is d{a}^/d{a}. For the extended Hoek-Brown plastic potential

(3.52), it can be derived from (3.61) as;

• dai/da'^ 5^2 da^jdo'^ da4,/da'^

_ da^jda'y da^ldo'y da^jda'y da^/da'y d{<j} daxjdr'^y da-ifdT'^y das/dr'^y da^/dr'^y

.datldo'^ da^jdo'^ das/da'^ daj^jdo'^

(3.64)

108

Calculating the elements in the matrix, it can be written as:

c u

where

d{a}

Cl2 Cl3 C22 C23 C24

Symm. C33 C34 C44 J

cii = H

C22 3

— +

C33 = 1-Oc 3

C44 = 1-

C12 = C21

Cl3 — C31 =

Ci4 - C41

C23 = C32 =

C24 — C42 —

C34 = C43

\ / 3 m ' v /Sm' i 1

AJi 8J2'

y/Zm' y/Zm' 1 8Ji

\ / 3 m ' y/Zm' 1 2J^

y/Zm' \ / 3 m ' X

4 4 ' 8 J ^

K r ,

8J2'

V i m -O'T! 3 ^x'xyi

•v /3m' , ,

y/Zm' , , aj! ^

\/3m' , ,

W ' - ' "

(3.65)

(3.66)

With equations (3.62) to (3.66) the matrix [.ff"] can be easily evaluated. For

plane stress and strain problems, only the upper 3 x 3 partition is employed; while,

for axisymmetric situations, the complete matrices are utilised with x and y being

replaced by r and z respectively.

109

3.4.4 Modelling of time-dependent behaviour rocks

It has been well accepted that the Theological behaviour of rock masses should be

taken into account in a numerical model as it has a great influence on the tunnel de-

formation and support system (Panet 1979; Ladanyi 1984). The viscoplastic model

presented by Zienkiewicz and Cormeau (1974) has proved useful to describe time

dependent phenomena and to simulate the real behaviour of rocks (Katsoulidis

1986). For a more precise description of the time-dependent behaviour of rocks,

many other viscoplastic rheological models have been proposed and compared to

some in situ measurements (e.g. Lombarti 1977; Gioda 1981; Sun and Lee 1985).

In the viscoplastic model adopted in this research, the rock mass plastic strain

is defined as:

where

= ,y$(F)^^ dt da

and 6t is the time step used in each iteration. Calculation of de„p/dt has been dis-

cussed in the preceding section (Equation 3.6) and the rock mass time-dependent

property can be modelled by choosing the correct value of the fluidity parameter

7 which should agree with in situ measurements. Another factor which may

have an influence on the time-dependent behaviour of rock masses is the time step

length At adopted in the computation.

As can be seen in section 3.2.1, the whole computational process is based

on the time iteration procedure in. which the magnitude of St can significantly

influence the numerical solution. If too large a time step 6t is taken, numerical

oscillations may be induced which can lead to incorrect or even divergent solutions;

whereas, too small a time step may be unrealistic for the computation due to

the extremely high computation cost. For explicit time stepping schemes, the

integration process is only conditionally stable so that St must be less than a

critical value. Cormeau (1975) provided a theoretical restriction on the time step

110

length for some specific forms of the viscoplastic flow rule. In particular, for the

Mohr-Coulomb criterion with associated viscoplasticity, Q = F, and = F,

the time step length limit is:

where ^ is fluidity parameter, (j> is frictional angle, c is cohesion and E is Young's

modulus.

Similar formulations for the Von-Mises and Tresca criteria have been derived

by Cormeau and they all show a common property of the limit value, that is,

^tmax is directly proportional to material strength and inversely proportional to

parameters E and 7. Although there is no theoretical formulation available for

determining the time step length in using the Hoek-Brown criterion and the sim-

plified Hoek-Brown criterion, the above properties can be considered in empirically

estimating a suitable time step.

Two empirical criteria (Owen and Hinton 1980) for determining the time step

length are used in this research and they were found to be useful and reasonable

in the numerical tests. The first one is:

STRI < T (3.68)

where is the first total strain invariant, d{e^-)^pldt is the first viscoplastic strain

rate invariant and the minimum is that taken over all integration points in the

rock mass. The constant r is a empirical input parameter and for an explicit time

stepping scheme it was suggested to be in the range 0.01 < r < 0.15 to obtain

accurate results. However, in this research, it was found that the range

0.03 < r < 0.08, (3.69)

can be acceptable for both Mohr-Coulomb and Hoek-Brown criteria(a detailed dis-

cussion of the influence of this parameter in the program was given by Katsoulidis,

1986). The second empirical criterion is for the variable time stepping scheme; the

111

change in time step length between any two intervals should be limited according

to:

< kStr,, (3.70)

where A; is a specified constant and the empirical value of A: = 1.5 has been sug-

gested from experience.

3.4.5 Validation examples

Modelling of nonlinear material behaviour as discussed in the preceding section

using program COAL has been verified against theoretical solutions in the litera-

ture. The results of four tests of the finite element analysis are shown below. The

first is an elastic-perfectly plastic analysis of a rectangular block under uniform

compression; the second is a thick cylinder analysis described by many investi-

gators; the third is an axisymmetric tunnel example; and the last is the tunnel

problem considering the time dependent behaviour of a rock mass.

3.4.5.1 A confined block under uniform compression

The test problem is designed to validate the basic formulations of the elasto-

viscoplastic algorithm. As shown in Figure 3.19(a), a block is subjected to dis-

tributed load on its surface and all the other directions axe constrained to move

normally. This simple problem was modelled using two plane strain 8-node finite

elements shown in Figure 3.19(b). The material is assumed to be continuous,

homogeneous, isotropic and elastic perfectly plastic complying with the Mohr-

Coulomb yield criterion and associated flow rule. The material constants are

taken as Young's modulus 1000 MPa, Poisson's ratio 0.2, cohesion 0.325 MPa and

a friction angle of zero. Considering that

= ( I = = 0,

112

l a

1

1 - -

a

i U i i U U l

i v w - ' i 23 m

5.0 tn

a )

Surrounding material acts as a rigid body

E = 1000.0 MPa,

C = 0.325 MPa,

(j) = 0.00,

V = 0.2,

Mohr-Coulomb criterion.

Associated flow.

b )

Figure 3.19 A confined block under uniaxial compression.

113

where superscripts T, e, p represent total, elastic and plastic respectively, and the

directions of Oxi< y,< z those of principal stresses, the analytical solution can

be easily obtained. Under elastic condition, the stress components are related by

the well known equation 1 — 1/

(Ty = — (Xx — Py

The plastic yield condition is

Cfy (Tx 2C Py •

The displacement at the block surface is (associated flow)

Uy = — (<7y + 2crx]h.

The failure is predicted to occur at the uniform pressure a = 0.87 MPa. If the

strength parameter c is reduced to zero after failure, a hydrostatic stress state in

the block can be expected. The computed results and the theoretical solutions are

identical as shown in Figure 3.20.

3.4.5.2 Thick cylinder subject to internal pressure

The problem has been described by many investigators (e.g. Zienkiewicz and

Nayak 1971; Owen and Hinton 1980). The material is taken to be elastic perfectly

plastic, with Young's modulus 2.1 X 10^ MPa, Poisson's ratio 0.3 and uniaxial

yield stress 240 MPa, and is taken to behave according to the Von Mises yield

criterion and associated flow rule. Such properties approximate to those of Grade

43 structure steel. The fluidity parameter in equation(3.12) of section 3.2.1 is taken

to be O.OOl/day. The outer diameter is taken to be twice the inner diameter, and

one quarter of the cross section is modelled by 12 elements as shown in Figure

3.21.

Computed displacements of the inner surface of the tube for large t (i.e. once

the viscoplastic strain rate has decayed to a negligible value) compare well with

114

2.000

% 1.000

a 0.867 a. <

0.000

Analytical solution

Cbmputed results.

0.0 0.2 0.4 0.6 0.8

Stress Ox ( ( ^ z )

1.0

a)

10.00

Analytical solution

E 4.00

0.0

Computed results.

I • I • 0.4 0.6

Stress Cf X ( Og )

-T— 0.8 1.0

b)

Figure 3.20 Computational results of the block problem, a) stresses — load; b) stresses — displacement.

115

100 M M I

E = 2 . 1 x 1 0 ^ MPa

V = 0 . 3

^ T = 2 4 0 MPa

H'= 0 . 0

Von M i ses yield c r i t e r i o n

/ = O.OOI/day

2 0 0 m m

Figure 3.21 Thick cylinder suialysis (2 by 2 Gauss point integration).

116

p.

(MP

20

16

12

8

/ /

/

Theoretical Solution

V Elasto-Plastic Analysis

o Elasto Visco-Plastic Analysis

A 'Visco' Analysis Considering Geo-metric Nonlinearity

_i I ) I 1 1 1 1 ^ . 4 8 12 16 20 24 28 32(x10"^mm) 0 (INNER FACE)

Figure 3.22 Load - displacement relation (a point on the inner face).

1.00 -\

\D

m a> U) (0

2 CO

0.00 -

-1.00

Plastic

— =0.76 OT

Elastic

1 1 1 1 1 1 ' 1 ' 1 1.0 1.2 1.4 1.6 1.8 2.0

Distance (r/a)

-g- Oj- —Radial stress

• Gq -Tangential stress

"B" —Axial stress

Results of Hodge and White (1950) for plastic zone up to 1.6a.

Uniaxial yield strength of the material.

Figure 3.23 Stresses-distribution of the thick cylinder (VonMises criterion and elasto- perfectly plastic analysis).

118

the exact solution and those predicted by Owen and Hinton's program for elasto-

plasticity (Figure 3.22), computed stress distribution including the axial stress

along the radial axis is presented in Figure 3.23 in which the theoretical solution

given by Hodge and White (1950) is also shown; the failure is predicted to occur

at an internal pressure of about 170 MPa.

The geometric nonlinearity analysis (see section 3.5) is also considered in this

test and the effect of the large strain is computed to be negligible, as would be

expected in this case.

3.4.5.3 Axisymmetr ic tunnel analysis

A 4.0m diameter circular tunnel excavated in a homogeneous, isotropic, elastic

perfectly plastic rock mass, complying with the Mohr-Coulomb yield criterion and

associated flow rule, is considered. The in situ stress is assumed to be hydrostatic

Po = 5.0 MPa and the body forces across the modelled region are ignored. The

following material property constants are assumed to give the input data: E =

7.5 X 10^ MPa, V = 0.25, c = 1.0 MPa, <j) = 30.0, 7 = O.OOl/rfay, T — 0.05. The

finite element mesh is shown in Figure 3.24. The mesh consists of 10 8-noded

isoparametric elements with the ratio of length in radial direction: 2 : 2 : 1 : 1 :

2 : 4 : 8 : 16 : 28 : 56. When the excavation sequence is not modelled, the first two

elements are not included in the finite element formulations. The radius of the

external boundary is 30 times the tunnel radius, i.e. R = 30r. Displacements in

the z direction are prescribed to be zero as demanded by the plane strain conditions

and the rock mass is free to move in the radial direction. The influence of this

boundary condition on the analysis of tunnel excavation is to be further discussed

in chapter 4.

The theoretical results of this problem were given by Gumusoglu (1986) and

a radius of the elastic-plastic zone interface Rep = 2.79m is predicted by both

analytical solution and elastic-plastic finite element analysis. Table 3.3 gives a

119

/

2m. 58 m

.Elements to be excavated Normally fixed boundary conditions

58 m

Element dimension:

1:1 :0.5 :0.5 :1; 2:4 : 8 :14 : 28

Figure 3.24 Axisymmetric finite element mesh for circular tunnel excavation problems.

120

Gauss points (7r (MPa)

r (m), {0 = 0°) Anal, solution Elasto-plastic Visco-plastic V.P. (excav.)

2.100 0.188 0,176 0.188 0.196

2.394 0.750 0.751 0.757 0.765

2.606 1.208 1.215 1.225 1.233

2.894 1.877 1.853 1.864 1.872

3.211 2.462 2.450 2.458 2.464

3.789 3.177 3.174 3.180 3.185

4.123 3.662 3.656 3.660 3.664

5.577 4.159 4.163 4.166 4.168

6.822 4.438 4.437 4.443 4.445

9.118 4.685 4.692 4.696 4.698

11.660 4.808 4.812 4.813 4.814

16.280 4.901 4.908 4.910 4.910

20.954 4.940 4.947 4.947 4.947

29.042 4.969 4.976 4.976 4.976

37.917 4.982 4.989 4.989 4.989

54.034 4.991 4.998 4.998 4.998

Notes c — Z.QMPa, <f> = 30°, Mohr-Coulomb criterion

E = 75000A^Pa, i/ = 0.25

(a)

Table 3.3 Comparisons of results of various analyses,

(a) The radial stress distribution.

121

(Continued:)

Gauss points T0 (MPa)

r (m), {0 = 0°) Anal, solution Elas to-plastic Visco-plastic V.P. (excav.)

2.100 3.997 3.992 4.061 4.065

2.394 5.715 5.717 5.756 5.769

2.606 7.088 7.110 7.155 7.172

2.894 8.123 8.111 8.102 8.095

3.211 7.538 7.565 7.557 7.550

3.789 6,823 6.841 6.835 6.830

4.123 6.338 6.339 6.354 6.351

5.577 5.841 5.852 5.849 5.847

6.822 5.562 5.577 5.571 5.570

9.118 5.315 5.323 5.319 5.318

11.660 5.192 5.203 5.201 5.201

16.280 5.099 5.106 5.105 5.105

20.954 5.060 5.068 5.068 5.067

29.042 5.031 5.038 5.038 5.038

37.917 5.018 5.025 5.026 5.026

54.034 5.009 5.016 5.016 5.016

Notes c = Z.OMPa, (f) = 30°, Mohx-Coulomb criterion

E = 75000MPa, u = 0.25

(b)

Table 3.3 Comparisons of results of various analyses,

b) The tangential stress distribution.

122

(Continued:)

Nodal points Ur (mm)

r (m), {e = 0°) Anal, solution Elasto-plastic Visco-plastic V.P. (excav.)

2.00 -0.369 -0.371 -0.373 -0.362

2.25 -0.253 -0.255 -0.256 -0.241

2.50 -0.190 -0.191 -0.192 -0.195

2.75 -0.159 -0.160 -0.160 -0.162

3.00 -0.145 -0.146 -0.146 -0.145

3.50 -0.125 -0.126 -0.125 -0.125

4.00 -0.109 -0.110 -0.109 -0.109

6.00 -0.073 -0.074 -0.073 -0.073

10.00 -0.044 -0.045 -0.044 -0.044

18.00 -0.024 -0.025 -0.025 -0.025

32.00 -0.015 -0.016 -0.016 -0.016

60.00 -0.007 -0.008 -0.011 -0.011

Notes c = S.OMPa, (j) = 30°, Mohr-Coulomb criterion

E 75000MPa, v = 0.25

(c)

Table 3.3 Comparisons of results of various analyses,

(c) The displaxrement distribution.

123

comparison of computed elasto-viscoplzistic analysis results with those existing

solutions. As cam. be seen from the table, steady state elasto-viscoplastic solutions

agree well with the prediction of the theoretical analysis and the elatstic-plastic

analysis, although relatively large errors occur for stress values around the position

of Rep and for displacements at the tunnel wall respectively. The maximum error

is found in the displacement at tunnel wall which is about 0.8% and the elastic-

plastic analysis using viscoplastic algorithm is shown to be satisfactory.

The non-associated flow using the simplified Hoek-Brown yield criterion as the

plastic potential function Q is also considered in this problem, and the results axe

shown in Figure 3.25. It is as expected that the analysis using the non-aissociated

flow rule (zero dilation with m' = 0 in equation 3.52) predicts smaller displacement

at the tunnel wall.

The stress relaxations with the increase of time are shown in Figure 3.26,

from which it can be seen that increase of plastic zone in the tunnel wall with time

can be modelled by the numerical algorithm. Figure 3.27 is a plot of displacement

against iteration number showing the influence of the parameter r on the solution

convergence and accuracy.

Associated f low

Non-associated flow X = 0.0025/day, tg = 1.0.

Figure 3.25 Displacement distributions calculated by different flow rules.

124

Stress (MPa)

tj = 3 days

l epi Rep2

Figure 3.26 Stress distribution and its relaxation with time.

0.50

+ Ur (0.1m)

0.45

0.40

0.35

0.30

Divergent

Convergent solution: 0.457.

X = 0.0025 / day, to= 1.0.

0 10 20 30 40 50 6 0 70 8 0 9 0 100

Iteration number

Figure 3.27 Influence of the parameter x (time step length control parameter) on the solution convergence and accuracy.

125

3.5 Geometric nonlinear analysis

3.5.1 Introduction

In the study of rock mass behaviour, a great number of incremental plasticity

models from strain hardening to strain softening have been proposed (see Section

2.4 and 3.2). Most of these models are generally based on the assumption that

the strains, both elastic and plastic, are infinitesimal and that the initial geometry

of a deformed body is not appreciably altered during the deformation process. A

numerical analysis with this small deformation assumption is called a geometric

linear analysis, A geometric linear analysis may be justified for a hard intact

rock mass but not for soft or weak rock masses. This is just like a recently well

accepted concept that the small strain can be acceptable for a material such as

steel or concrete, but has very limited applicability in soil deformation (Desai and

Phan 1980; Kiousis et al. 1986). The floor heave in a roadway of coal mine is

a very good example manifesting the inadequacy of small strain theory. Another

example is that the buckling phenomena of steel arches in the roadway cannot be

predicted by a small strain theory due to its relatively high strength.

Modelling of geometric nonlinearity in geomechanics by using the finite ele-

ment method started in early 1970's and has been a subject of study by many

investigators (Zienkiewicz and Nayak 1973; Carter et al. 1977; Yamada and Wifi

1977; Desai and Phan 1980; Kiousis et al. 1986). Almost all of their work is

related to soil deformation and the examples of soil ground subject to footing load

are usually considered. In comparison with the work done in material nonlinear

analysis in geomechanics, significantly less effort seems to have been spent on this

equally important problem. In the Rock Mechanics literature it appears that there

is no finite element program available which can deal with large deformation (finite

strain) problems.

There are several different approaches to geometric nonlinear problems and

126

they are basically belong to two types of formulations. One is the Eulerian formu-

lation (Banerjee and Fathallah 1979) and the other is the Lagrangian formulation

(Bathe et al. 1975). In both types of formulations, specific stress and strain

tensors as well as their derivatives are usually introduced because the stress and

strain in these formulations are no longer defined in an engineering sense. This

requires a reader to be familiar with all those precise definitions such as the second

Piola-Kirchhoff stress and its Jaumann rate, etc. (Fung 1965) in order to follow

the deduction of any finite element equilibrium equations and their incremental

formulations.

In this section, however, a straightforward deduction similar to that of Kanchi

et al. (1978) and based on the elasto viscoplastic incremental formulations of

Section 3.2 is adopted to demonstrate the provision of large deformation analysis

in this project. The method is equivalent to the Updated Lagrangian approach as

described by Bathe (1976, 1982) and adopted by Desai and Phan (1980).

3.5.2 Some definitions and assumptions

In the following presentation of the finite element formulations for finite strain

analysis, two dimensional plane problems with specialisation to the axisymmetric

case are considered for the purpose of this study and it is easy to extend all the

formulations to a fully three dimensional solid. -

127

The finite strain is defined as (Lagrangian formulation):

^xy dUr. , , dUr^ duru ,

dx dy dx dy dx dy .

(3.71)

where €x,ey,exp are the sheaj components in x,y plane and Un,Vn are the dis-

placements at time tn.

Matrix notations will be used throughout the following discussion.

3.5.3 Basic formulations

In section 3.2, the equilibrium equation was (equation 3.19):

f [B^'f + { / " } = 0,

Jq

where [£"•] is the nonlinear strain matrix at time t = tn- It can be written

(Appendix 4): I B - l = |Bol + i B J i l = [Bol [ J c l , (3.72)

where [5^^] contains the nonlinear quadratic terms and [Jp] is called the defor-

mation Jacobian matrix defined at a point in a body as

where Xn,yn are the new coordinates at time and for the Updated Lagrangian

formulation x,y are the coordinates at time (*-!-

In order to obtain the incremental equilibrium equation, (3.71) is differenti-

ated to give

d Q [B'^f a^dn^ + d r = j a^dn + j [B^'f da^dU + d f . (3.74)

Substitute (3.72) in to (3.74) and note that d[Bo] = 0 to give

f [B^'f dondU + d r + f [ d B ^ ^ f <^r.dn = 0, (3.75) Ja Jo

128

JD = (3.73)

which is the same as (3.20), Equation (3.75) can also be verified by using incre-

mental formulation from time n to n+1 as follows.

Since (3.71) is the equilibrium equation at time step n, the new equilibrium

after a time step 6t is

/ = 0, (3.76) Ja

which is the equilibrium condition at time step n+1.

During this time step,

Sp = - / » ,

SOn = Ofi+l O'n,

therefore (3.76) can be written as

/ (B" + (( » + f<;») dn + r + S r = 0. (3.77) Ja

Rearrange (3.77) by using (3.19) to give

f { S B ^ L f o ^ d n + f [B"f 6a„dn+ / + = 0 , (3.78) Ja Jn Ja

where matrix is a linear function of incremental displcicements 6Un and

6vn.

Since 5o"„ in (3.78) is also a linear function of 6un and5t;„, the third term is

infinitesimal compared to the other two terms. Therefore, it can be neglected to

give the incremental form of (3.71):

/ [B^'f Sa^dn + [ [SB^^f ^ndn + s r = 0, (3.79) Ja Jn

which is the same as (3.74).

129

In Appendix 4, the nonlinear strain matrix SB^j^ at node point i has been

derived as:

dUn dNj dx dx*

dur^ dNi

dvn dNj dx dx

dvn dNi dy dy dy dy

dun dNi I dun dNi dvn dNi • dvn dNi - 5 ^ - ^ + - ^ - ^ + dx dy

(3.80)

Therefore, its incremental form can written as:

dSun dNi dx 8a:

^ U n dNi

dy dy

dSvn dNi dx dx

dSvn dNi dy dy

dSun dNi , dSun dNi dSvn d N j , dSvn dNi

+ dy dx + dx dy J

(3.81a)

and for the axisymmetric problems (3.81a) becomes

SB"- NL

dSvn dNi dr r

dSfn dNi

dz dz

dSzn dNi dr dr

dSzr. dNi

dz dz

dSrn dNi , dSrn dNi dSz^ dNi . dSzn d N j

dz dr + dr dz

0

(3.816)

(3.81a) and (3.81b) can be further written as

[SB[ n iT NL

dNi

dx

0

0

dNi

dx

dy

0

— [^»]2X4 [-^'14x3

0

r dSu

dx

dSv„ dx

0

0

IL 0

0

dSu a. dy

dSvn

dSUn 1 dy

dSv„.

dy

dSu.

dx

dSv.

IL

Ik dx J

(3.82a)

, and

0

0 ^

^ 0

0 ^

— [^ilaxs [ »]5X4

r

0

dSr.

d^ dr

0

0

0

IL 0

0 dSr, n. 9z

dSz„

dz

0

d^

dSz,

dSz,

a.

dr

0

U.

0

0

0

0 5

Srn r J

(3.826)

130

It can be found from equation (3.23) of Section 3.2.1 that the first term of

(3.79) has a form of

b = K f , (3.83)

in which the stiffness Kf is related to the the current material property and

geometry. The second term of (3.79) is also to be written in a similar form to

(3.23).

Using (3.81) to (3.82) for a point associated with node i, it is easy to write

{<T„} = [Gi]^ lAf K } dSu

= [Gif

dx

DSUR, .

'scy dSv^^ dSv,

Txyl

(3.84)

= [Gif

= [Gd'' [M\ [Gil { « * } ,

where

and

[M| = O'x I

Txyl O'yl (3.85)

1 = 1 0 0 1

If the axisymmetric problem is considered, the [M] will be a 5 x 5 matrix and [G]

should also be changed accordingly. For an 8-node element,

[G®] — [Gi Ga . . . Gs > (3.86)

so the element stiffness matrix is

M i e x i e - l G ' f M M , (3.87)

and for a 7-node element as is to be used in Chapter 5,

m = {G, G2 . . . (37)4x14,

1 3 1

(3.88)

and

= [G'f |M| |G'| . (3.89)

Thus, the component of global stiffness matrix depending on the stress state at

time step n can be derived by integration on a whole body:

/ \BNLf ".(fn = \K?,] { < 4 , (3.90) Jn

where

[ % l = f IGf |M| [G] rfn. (3.91) Ja

If the problem is axisymmetric, formulations of (3.81b) and (3.82b) can be used

and, in the element stiffness matrix, only the relevant matrix [G®] is to be changed.

Substituting (3.90) into (3.74), the displacement increment during time step

tn to tn+i can be calculated as

AF", (3.92)

where

[ir-l = |jir?| + |irp,|,

as in (3.83) and (3.91), AV„ is the same as that in (3.24) of section 3.2.1.

3.5.4 Incremental procedure

The computational procedure including large deformation is similar to that de-

scribed in section 3.2 except that the extra term in the equilibrium equation and

the updated strain matrix [£"•] are calculated.

A very important aspect in evaluation of the strain matrix is its definition.

The deformation Jacobian matrix [Jd] in (3.18) includes the terms like dxn/dx,

d x n j d y , etc. and x,y caji be either the original coordinates or the updated ones

before time if they are not defined.

132

In principle, for the Updated Lagrangian formulation of large deformation

analysis, the coordinates of the system under consideration at time should be

referred to as the updated one, so that incremental values of displacements,

strains, stresses and their derivatives are kept small enough to guarantee the val-

idation of the adopted constitutive laws and the accuracy of the results. In this

respect, x,y in the formulation are the updated coordinates at time i„_i.

For this visco-plastic approach, however, the plasticity and creep usually lead

to hundreds of time step iterations for a convergent solution, especially when

the explicit time integration scheme is adopted. The above manipulation is time

consuming and not necessary. Therefore, an alternative incremental procedure is

used. During the calculation, the x,y are updated either when an incremental load

is applied or if a certain number of time steps has been iterated (say 15 steps).

The deformation matrix [JD] in this case is defined as

Jjj —

dXr^ dtJn dxi dxi

DXR^ DVN

. dyi 'dm .

(3.93)

where

current load increment number < / < n — 1.

3.5.5 Validation examples

The solutions presented in the following examples have been obtained using the

algorithm presented in Section 3.2 in which the additional nonlinear stress depen-

dent stiffness matrix Kji and the updated strain matrix 5 " are considered.

For the finite element discretization, 8-noded two-dimensional elements have

been employed. Since the results are to be compared to theoretical or analytical

solutions, the material properties are assumed linear elastic and isotropic for the

column and the cantilever. The plastic solution of the cantilever is compared with

the existing numerical results.

133

a) Buckling of a column

Consider a slender pin jointed column fixed at one end and free to move in

the axial direction on the other end with a compressive axial force P, as shown

in Figure 3.28(a). For convenience in imposing constraints on displacement, only

half the column is modelled (Figure 3.28(c)). The finite element mesh used is as

shown in Figure 3.28(d). The yield stress is set to a relatively high value compared

with Young's modulus so that the column remains elastic when subject to large

strain, to permit comparison of the computed buckling load with that predicted

by Euler's theory, which can be found in many text books on elasticity theory. In

the finite element analysis, it is necessary to apply in addition to the axial load P

a small lateral load fiP where = 0.001 or 0.0005, to initiate nonlinear behaviour.

The values of /? are so small that this lateral load does not significantly affect the

stress distribution according to the theory of infinitesimal strain; if =0.001, then

the line of thrust is only 0.01m from the centerline of the column at its base.

In Figure 3.29 is shown lateral displacement 5 as a function of applied load

P, computed by the program and according to infinitesimal strain beam bending

theory. The program using the small strain theory, in this case, cannot predict

the buckling phenomenon as indicated in the figure. However, the program incor-

porating a large strain theory predicts collapse at about 0.97 Pcr where Pcr is

the value given by Euler's theory. In Figure 3.30, the computed displacements are

shown, to scale, at axial loads of 0.96Pcr and 0.97Pc'r.

The above example has verified that the adopted large strain analysis proce-

dure can predict possible buckling phenomena, but this is still a small displacement

problem because after the critical load was applied the problem became unstable.

Therefore the following large displacement problem was considered.

134

(a)

L = 2 t

(b)

P=PCr

Pr.r = TT^EI

c r - ( E u l e r s o l u t i o n )

(c)

P = TT^E I 4 £2

( d ) 1 m

y — O - O O—V -O O O-' O" • Q——Q- O I

10 m

V T ^

Figure 3.28 Column buckling analysis, a) pin-jointed column; b) buckling of the column for P = Per, c) problem simplification; d) the finite element meshes.

135

P/PCr

L • 6 = 0.001

A B= 0.0005

O 6=0.001

( i n f i n i t e s i m a l a n a l y s i s )

Figure 3.29 Computed response of the column.

136

BP

P = 0.9 6 Per

7

P = 0.97Pcr

6 = 0.001

-•X

Figure 3.30 Nodal displacements and element shapes before the column collapse.

137

b) Large displacement of a cantilever

The cantilever in Figure 3.31 under uniformly distributed load was analysed

using the program as another validation example. The cantilever was idealised

using five 8-noded plane stress elements which was the same as that in Bathe's

example (Bathe and Ozdemir 1976). The material of the cantilever is assumed to

be homogeneous, isotropic and linear elastic so that, in praxztice, the yield stress

of the material was selected sufficiently high to prevent plastic response. The

problem was chosen to compare the results with Bathe's solution and also with

the analytical solution given by Holden (1972).

L / h = 10.0, E= 12000 MPa, v = 0.2.

Figure 3.31 Cantilever under uniformly distributed load (data taken from Bathe and

Ozdemir, 1976).

138

Elastic solution

The response of the cantilever predicted by using the procedure presented in

3.3.2 and 3.3.3 is shown in Figure 3.32 in which Holden's solution and Bathe's

results are also given. The solution was obtained by applying the incremental load

in 35 steps. The accuracy can be improved by increasing the incremental load

number as shown in Figure 3.33.

The mesh deformations predicted by using the large deformation theory are

shown in Figure 3.34. For comparison, the results predicted by infinitesimal strain

theory are also shown to scale in Figure 3.35, and it reveals that a significant

error appears when the large deformation was modelled by a conventional finite

element analysis (small strain). In this analysis of cantilever deformation, the

small strain theory over-predicts the vertical displacements and cannot calculate

the horizontal displacement (central node) and consequently, the mesh geometry

change is incorrectly predicted.

Elastic-plastic solution

The plastic behaviour of the cantilever is considered using the Von Mises

criterion. The material is assumed to be elastic perfectly plastic with a tensile

strength cry = 500.0 MPa. The small strain finite element analysis in this case

predicts a typical load ~ displacement curve as shown in Figure 3.36. However,

it is known that the large displacement effects will stiffen the material, thus op-

posing the effects of plasticity (Kanchi et al. 1978). This is verified by the large

displacement analysis and the computed curve is shown in the same figure (Figure

3.36). A different phase of behaviour after yielding is clearly shown in the figure

and the results are satisfactory compared to that of Kanchi et al. (1978).

139

Deflec-tion ratio (W/L)

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

/ /

i

I

Small defor-Tiation analysis

r

y — Analytical solution

(Holden), O Bathe's results, • Current analysis (Updated

Lagrangian formulation, with no equilibrium iterations).

— Analytical solution (Holden),

O Bathe's results, • Current analysis (Updated

Lagrangian formulation, with no equilibrium iterations).

— Analytical solution (Holden),

O Bathe's results, • Current analysis (Updated

Lagrangian formulation, with no equilibrium iterations).

— Analytical solution (Holden),

O Bathe's results, • Current analysis (Updated

Lagrangian formulation, with no equilibrium iterations).

0 1 8 10 2 3 4 5 6 7

Load parameter k = PL^ / (EI)

Figure 3.32 Large displacement analysis of the cantilever, comparison of computed results with existing solutions.

Deflec-tion ratio (W/L)

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

A

Analytical solution (Holden),

® Bathe's results (1(X) steps),

^ Bathe's results (5 steps),

• Current analysis (35steps).

J /

Analytical solution (Holden),

® Bathe's results (1(X) steps),

^ Bathe's results (5 steps),

• Current analysis (35steps). 9 /

Analytical solution (Holden),

® Bathe's results (1(X) steps),

^ Bathe's results (5 steps),

• Current analysis (35steps).

Analytical solution (Holden),

® Bathe's results (1(X) steps),

^ Bathe's results (5 steps),

• Current analysis (35steps).

0 1 8 2 3 4 5 6 7

Load parameter k = PL^ / (EI)

Figure 3.33 The influence of iteration steps on solution accuracies.

10

140

5 . 4 8

P = 10.(

Figure 3.34 Mesh deformations calculated by using the large deformation theory (to scale).

> r- n

\ \

X

N X

\ / ^ \ P = 6 . 5

X \ \

\

\ \

8 . 1 9

\ \ \

\ ZL_ V "

Figure 3.35 Mesh deformations calculated by using the infinitesimal deformation theory (to scale), results showing a significant error as is clear from the prediction of displacements and the element shapes.

141

Deflec- 0 .7 tion ratio (W/L)

iSmall displacement lelastic analysis

Gy = 500.0 MPa,

Von -Mises criterion.

Large displacement lelastic analysis

" Analytical solution (Holden), O Small displacement elasto-plastic

analysis. • Large displacement elasto-plastic

analysis (Updated Lagrangian formulation, with equilibrium iterations for material nonlinearities).

2 3 4 5 6 7

Load parameter k = PL^ / (EI)

Figure 3.36 Large displacement analysis considering material yielding.

142

3.6 Excavation sequence analysis

Another problem that has to be considered in the numerical model of underground

excavation is the influence of loading history and/or stress path (including 3-D ef-

fects) on rock mass strength and deformations (Kaiser 1981; Amadei et al. 1986).

This requires modelling the excavation sequence as realistically as possible. There

are two kinds of excavation sequence to be simulated, one is for the large under-

ground caverns in which the whole cavern section is divided into sub- divisions

and excavated in several stages, and the other is for the tunnel face advance in

which a full section of tunnel material is progressively removed as the feuce passes

through. In numerical analysis, the former can be simulated by removing the re-

lated elements in a similar way to the real multiple stage excavation process and

the latter is a more complex problem which will be discussed in detail in Chapter

4.

For the excavation in an elastic-plastic or viscoplastic rock mass, Dolezalova

(1977) and Akagi et al. (1984) found that the amount of plastic deformation from

the multiple stage excavation sequence is less than that by the full section one,

although it seems this is not always the case because Sun and Lee (1985) obtained

the opposite result from their viscous elasto-plastic tunnel excavation analysis.

Conventionally, the excavation process can be numerically simulated by apply-

ing the equivalent nodal forces in opposite directions on the tunnel boundary after

removing the elements inside the tunnel (Christian and Wong 1974) or by reducing

incrementally the stiffness of the material inside the tunnel (Chandrasekaran and

King 1974). In the former method, the unassembled nodal forces, which can be

printed at the end of the initial step of analysis, are taken and assembled by hand

to obtain the forces which will reduce the stresses on the new boundary to zero.

It is a tedious task for fine element meshes. In the latter method, the elements in

the excavation area are retained as 'ghost' elements and their stiffness is simply

reduced to a negligible (but finite) value. This removes restrictions on element

143

numbering or in the matching of element numbers between analysis. The use of

'ghost' elements requires coding to be introduced to prevent the assembly of the

nodal forces from these elements (including body weight and in situ stress). This

probably outweights the advantages of the 'ghost' elements.

For elastic-plastic analysis of tunnel excavations, the above two simulations

may give different results as shown in Figure 3.37. In the numerical incremental

calculation, the former method leads to a necessity of recalculation of the global

stiffness matrix excluding the elements inside the tunnel boundary, and for the

latter, the global stiffness matrix is recalculated at each incremental step using

the same element meshes.

The stiffness reduction method takes some account of the effect of the pro-

gressive advance of the excavation face, so that it may be more reasonable than

the "opposite nodal force " method. For nonlinear material analysis problems in

which the stiffness of the elements has to be gradually reduced to zero, however, it

is reported that difficulties arise in determining the stress-dependent stiffness for

each step due to their nonlinear variation with the stress components (Lu, 1987).

In this research, a modified stiffness reduction method for the elasto- viscoplas-

tic algorithm, developed by the author and called 'stress and stiffness reduction'

method, is applied to simulate the excavation process and is proved effective and

convenient.

Assuming that at step t — tn, the equilibrium state under initial stress field

and applied loading has been established, so that equation (3.19) yields:

f + r = 0. Ja

For one step excavation without consideration of geometrical nonlinearity, the sim-

ulation can be carried out by reducing the stresses of the elements to be excavated

to a negligible (but finite) value. This disturbs the equilibrium state of the system

and (3.19) will no longer valid. The "out-of-balance" forces at nodal points due

144

Ow = 5.0 MPa

Ox = 1.5 MPa

E = 750.0 MPa

V =0.25

Oq = 25.0 MPa

m = 0.7

s = 0.004

Stress concentration zone predicted

by elastic analysis.

Simulation 1: The tunnel exists before

applying the loads.

Simulation 2: Excavation by reducing

the stiffness of 'ghost' elements.

Simulation 3: Excavation by applying

the opposite equivalent load at the

tunnel face.

Figure 3.37 Influence of different simulations of excavation on prediction of

plastic zone (after Giimiisoglu, 1987).

145

to this 'stress release' are

= / {B-? JQ

+ r 9^0, (3.94)

JU

where

(3.95)

and a is a constant for the stress reduction. After this manipulation, the numerical

iteration process can be continued corresponding to the incremental form of (3.19)

and (3.94): / + r " = 0, (3.96) Jci

from which the incremental displacements induced by the excavation axe:

SdT = (3.97)

where \K^\ is a recalculated reduced stiffness matrix and can be written as:

\Kf\ = f (3.98) Ja

= ccD-. (3.99)

Here the oc is the same as that in (3.95).

The incremental process presented in Section 3.2 can then be carried on until

a new equilibrium state is reax;hed. It is not necessary to introduce the code to

prevent the assembly of the nodal forc^ from the 'ghost elements', even at the

final excavation stage.

For multiple-step excavation which is particularly required for geometrically

nonlinear analysis and modelling of the progressive advance of the tunnel face, the

above T is usually given in a stepwise manner corresponding to an incremental

reduction of stresses of the elements to be excavated (say in 20 steps). The manip-

ulation and iteration process are identical to the above method. It is found that

the excavation sequence can be modelled accurately and realistically by using the

above procedure. The elastic solution of excavation is identical to that of loading

146

analysis (namely, the tunnel exists before loading) and are independent of excava-

tion steps. The elastic-plastic solution is compared to the theoretical solution for

axisymmetric tunnel analysis as presented in Table 3.3.

3.7 Summary

The application and improvements of the 2-D finite element method for analysis

of weak rock mass response to the mining excavations have been presented. The

features and some validations of the program COAL using the elasto visco-plastic

theory have been described.

The review and investigation of the existing strength (yield) criteria for rocks

during the development of a rock mass behaviour model highlighted the need for

a new yield function in the analysis. A modified Hoek-Brown yield criterion has

been proposed and shown to have significant advantages. It has been incorporated

into the program and can be used either as the yield function, or as the plastic

potential function if the non-associated flow rule is applied. A large displacement

theory using the Updated Lagrangian formulations is described and implemented

in the program. The influence of time-dependent behaviour of a rock mass and

the excavation sequence are also modelled.

The program COAL has been verified against existing theoretical solutions

and other numerical solutions, including those of material nonlinear analysis and

geometrically nonlinear analysis. It is concluded that the developed program

COAL is capable of modelling the nonlinear behaviour of equivalent continuum

rock masses which exhibit plasticity, creep, buckling, large displacements, etc.

147

CHAPTER 4 PLANE STRAIN ANALYSIS IN MODELLING

3-D TUNNEL EXCAVATIONS

4.1 Introduction

In the preceding chapter, a 2-D nonlinear finite element model for underground

excavations was described. Stresses and displacements in the rock surrounding

tunnels and in the support systems are usually analysed using plane strain solu-

tions and 2-D numerical modelling due to the complexity of the rock material and

the limitations of existing computational tools (see, e.g., Detounay and Fairhurst

1987; Sulem et al. 1987). It appears that plane strain analysis has been considered

accurate for a tunnel section which is far away from the face because the state of

stress there satisfies essentially two-dimensional plane strain conditions. However,

it is well known that plane strain analysis cannot be used near the tunnel face or

near branches of the tunnel. When the rock mass behaves in a nonlinear manner,

the loading history and stress path will also have a great influence on the 3-D

stresses and deformation of the tunnel.

It was discussed in chapter 1 that one of the purposes of this study is to

simulate the roadway deformation in a coal mine which has been shown to be a

fully nonlinear, three dimensional excavation problem. In this regard, the same

key question as that raised recently by Cundall (1988) is: "Are 2-D numerical

simulations of any use for practical rock engineering problems?". Moreover, if

plane strain analysis is used in modelling 3-D tunnel excavations, what is the

possible error? In this chapter, therefore, understanding of the problem is firstly

assessed in Section 4.2. Then an investigation of the relation between the 2-D

simulation and 3-D tunnelling behaviour is undertaken. The procedure and results

148

are described and presented in Section 4.3 and Section 4.4. Potential application

of the results is discussed in Section 4.5. Finally, in Section 4.6, conclusions are

drawn from the investigation.

4.2 A n assessment of previous approaches

4.2.1 P lane strain solutions in Rock Mechanics

The definition of a plane strain problem can be found in most text books on

elasticity theory such as that of Timoshenko and Goodier (1970). The main feature

of the plane strain condition is that the deformations are restricted to a plane with

coordinates, say, (a;,y). That is:

Czz = Ixz = lyz = 0. (4.1)

In classical stress analysis using elasticity theory, a tunnel excavated in an isotropic,

homogeneous, elastic body with a constant load condition has often been taken as

an example of the plane strain condition. In applying the principles of classical

stress analysis in Rock Mechanics, these plane strain conditions have been adopted

for tunnel problems involving plastic rock deformations (Jaeger and Cook 1979,

Brady and Brown 1985). Most existing solutions used in underground excavation

design are plane strain solutions. For the excavation cross section near the tunnel

face or an intersection, the stress field in the section is of a 3-D nature. However,

it has been concluded that if the location of the section is at a distance of two

excavation 'diameters' away from the face or intersection, the elastic plane strain

solution usually approximates the correct 3-D solution to 90 ~ 95% (Panet and

Guellec 1974, Hocking 1978). For an elastic analysis, therefore, the plane strain

solutions often provide more widely applicable results than might be expected and

accordingly, the need to carry out fully 3-D analyses arises only rarely. Another

feature of the elastic plane strain problem in homogeneous, isotropic materials is

that the stresses induced around an excavation in the plane of the cross section

149

axe independent of the elastic constants, but the displacements are not.

A number of plastic solutions for the axisymmetric (plane strain) tunnel prob-

lems have also been proposed (e.g. Florence and Schwer 1978; Brown et al. 1983;

Fritz 1984; Detournay ajid Fairhurst 1987; Reed 1988). A more exhaustive review

of the available solutions for various material behaviour was given by Brown et

al. (1983). Some of these analytical solutions allow for considerations of strain

softening, viscosity, non-hydrostatic loading, influence of the tunnel face, etc. In

deriving all those solutions, the axis of the axisymmetric tunnel was assumed to

be coincident with one of the in situ principal stress directions.

In order to determine stress and displacement around long openings in a

general triaxial stress field, an alternative plane strain analysis - the complete

plane strain model was proposed by Brady and Bray (1978). In the model, the

displacement in the tunnel axial direction (z) is not assumed to be zero but a

function of x and y, or

w = w{x,y), (4.2)

thus the general condition of plane strain (Equation 4.1) is changed to:

dw 0, 0,

du dw dw

dx dx dw dv dw

^ 0, (4.3)

A major advantage of the model is that in the analysis the tunnel axis is not re-

quired to be coincident with one of the in situ principal directions and, therefore,

some aspects of the 3-D nature can be incorporated (Brady 1979).

150

4.2.2 Advancing tunnel analysis

For a practical 3-D tunnel excavation, the stability and convergence of the tunnel

behind an advancing face are of primary interest. Panet and Guellec (1974) pro-

posed that the problem can possibly be approached by an equivalent plane strain

problem in which the radial stress Cr is progressively decreased from an initial

value equal to the initial stress to zero in the case of no support. This approx-

imation process is shown in Figure 4.1. A simple formulation is used to simulate

the face effect and it is given by

(Xr = [1 — A)(7°. ( 4 . 4 )

where the parameter A varies from 0 to 1. In the elastic case, it is defined as:

in which Ur (x) is the radial displacement at a distance x behind the face and C/j?°

is the radial displacement at infinity which is given by plane strain solution

u r = (4.6)

The influence of tunnel face advance was further investigated and discussed by

many others (Ranken et al. 1978; Panet 1979; Hanafy and Emery 1980; Panet

and Guenot 1982).

Ohnishi et al. (1982) developed a multiple element method in a conventional

plane strain finite element analysis to model the tunnel face advance, but the

results appear still of a 2-D nature.

A general conclusion has been drawn that the tunnel convergence behind

an advancing fa^e is due to the process of excavation and the time-dependent

behaviour of the rock mass respectively. Recently, Sulem et al. (1987) provided a

2-D solution taking into account both of these effects. In those 2-D approximations,

the calculated tunnel deformation with the support stress = 0 represents the

151

I

\ (j» -4 \ /

x«=o

X ( I

y / / / / \ / / / / / / / / / / / / / / / / ^ / I

0 < X < 1 X —1

cr =cr° or=(i - A) a" c r - o

X •« • ov 2G

tlr

Figure 4.1 Advancing tunnel analysis, a) 2-D model of 3-D face advancement; b) radial displacements behind advancing face (from Panet, 1982).

152

ultimate tumiel convergence far away from the face. However, for specific values of

Poisson's ratio and in situ stress, this assumption may produce significant error

when the material behaves in a nonlinear manner.

4.2.3 Influence of axial stress and loading history

It has also been accepted that another important factor in the analysis of tunnel

excavation is the influence of stress path and loading history (Kaiser 1980; Amadei

et al. 1986), but the effect of the intermediate principal stress (<72) on the stress

history is more intractable and cannot be analysed by a plane strain model. Hence,

the axial stress of the tunnel is seldom investigated and considered in tunnelling

design.

Bamdis et al. (1986,1987) contributed to the understanding of the failure

behaviour of 3-D physical models and compared the results with the relevant 2-D

plane strain analyses. The effect of the axial stress (or the intermediate principal

stress) on the failure mode in their physical model was discussed. Two of the

conclusions drawn from their 3-D physical model and 2-D theoretical analyses

were that 2-D plane strain solutions over-predicted the failure zone, and the axial

stress ((T ) stabilised the fractured rock near the tunnel wall. Unfortunately, these

conclusions do not appear to be supported by existing theoretical studies and the

mechanisms of the phenomena cannot be explained yet.

Beer et al. (1987) applied an efficient coupling procedure using nonlinear

finite element and infinite boundary elements to investigate the 3-D nature of the

displacements and stresses near the tunnel face. A significant difference was found

between 2-D and 3-D predictions. In contrast with the conclusions of Bandis et

al. (1986, 1987), Beer et al. (1987) showed that their 2-D model greatly under-

predicted tunnel convergence and, accordingly, under-predicted the lining stresses.

Some of their results are shown in Figure 4.2.

153

a )

ro X w Siu

till liiWim It' 1 5 ! f g S : Cfl !o

b)

1 1

I |CJ

ro :«o Sl«

a

s Loading 12

p

Figure 4.2 Comparisons of stresses and displacements for two- and three-dimensional analyses, a) comparison of lining stresses; b) comparison of vertical displacements (from Beer et al.,1987).

154

4.2.4 Concluding remarks

Use of 2-D plane strain analysis in rock mechanics ajid rock engineering started

from the application of classical elasticity theory. Since then, the plane strain

analysis has been developed to a more mature and advanced level. Based upon

secure theoretical foundations, a number of 2-D solutions for 3-D excavations,

with real rock mass properties, have been proposed. The available plane strain

solutions include those considering an advancing face, time dependent behaviour,

rock material failure, support linings, non-axial stress field, etc. Plane strain

finite element analysis for various nonlinear problems has been more versatile

and has a special significance in analysing tunnel excavations. It appears that

only limited fully three-dimensional numerical analyses have been carried out in

studying tunnel problems.

It has been noted that there is a significant lack of information on validity of

various 2-D models in approaching fully 3-D tunnel excavation problems. More

surprisingly, among those unknowns are a number of key questions such as:

- 'Will a pleine strain analysis over-predict or under-predict tunnel convergence

if 3-D nonlinear behaviour of rock mass is taken into account?',

- 'What is the influence of the tunnel axial stress?',

- 'What is the influence of stress history on tunnel convergence and stability?'.

Before these questions can be answered, doubts will always exist on the va-

lidity of a 2-D nonlinear analysis, even if a rock mass does behave as a continuum.

155

4.3 A numerical procedure for investigation

4.3.1. Preliminary considerations

The influence of the face advance on the behaviour of a tunnel has been studied

by many investigators, using axisymmetric finite element models. It was noted

that the full simulation of the advancing face sequence and its 3-D effects can

be accommodated by these models (Panet and Guellec 1974; Hanafy and Emery

1980; Cunha 1983), However, most of those investigations have concentrated on

the rock material properties and time- dependent behaviour. Few of them have

been related to the study of the effects of loading history and stress path or, in

particular, the discrepancies between 2-D and 3-D models.

The conventional plane strain analysis of tunnel excavation assumes that a

tunnel section far away from the face satisfies plane strain conditions. In that case,

U^ = { U ^ U . (4.7)

where is the tunnel convergence predicted by a plane strain model and (C/^)oo

is the true convergence far away from the face in the three dimensional excava-

tion. Unfortunately, any tunnel section excavated in a three dimensional space

can hardly satisfy the plane strain assumptions, especially, when the material is

not linearly elastic and when the face advancement sequence is taken into account.

If the plane strain conditions are assumed in an axisymmetric tunnel problem,

the induced tangential stress ae and radial stress Or (polar coordinates system)

in the rock mass surrounding the tunnel will be the principal stresses. In an

elastic rock mass, the axial stress cr is generally the intermediate principal stress.

After failure occurs in the rock mass, the axial stress (or out-of- plane stress) in the

plastic zone may not remain the intermediate principal stress (Florence and Schwer

1978; Reed 1988). Therefore, ag and cr cannot be assumed to be major and minor

principal stresses respectively. Furthermore, the general constitutive relation does

156

not apply because the axial stress predicted by plane strain formulations is no

longer correct in the plastic zone if the in situ stress field and the process of

stress release cannot be modelled. It is this discrepancy in the axial stress (or

strain) that makes the plane strain modelling unrealistic.

The above considerations suggest that equation (4.7) could not be satisfied in

a nonlinear analysis of a tunnelling problem. It is therefore necessary to investigate

quantitatively the possible errors introduced by the plane strain assumptions. In

order to get a better understanding of the problem, comparisons of the stresses,

displacements and their path-dependent effects between the predictions by 2-D

and 3-D analyses are particularly required.

4.3.2 Numerical simulations

A numerical modelling procedure has been carried out to investigate quantitatively

the errors usually introduced by the 2-D assumptions in various situations. A

circular tunnel driven in a homogeneous, isotropic, elasto-viscoplastic medium is

considered. This 3-D tunnelling problem can be simulated by the axisymmetric

finite element model as shown in Figure 4.3. Axisymmetric finite element models

for studies of 3-D tunnel excavation have been developed as an extension of the

plane strain finite element simulation, and are still the simplest and most efficient

in the modelling of nonlinear 3-D incremental tunnelling. The model shown in

Figure 4.3 is used in conjunction with the plane strain model to investigate the

behaviour and errors introduced by the plane strain assumptions.

In a conventional 2-D finite element analysis with plane strain assumptions,

the in situ stress field is generally simulated by specifying a horizontal initial

stress <7° and a vertical initial stress cr° (they may be applied at boundary). If

a® is taken the same as cr®, the tunnel initial axial stress is usually calculated as;

cr® = 2i/cr2 (using the plane strain condition: = l /E[az — v{ax + cTj,)] = 0).

In the 3-D simulations, therefore, the initial stress field is modelled in two ways:

157

y

Stress field I: o r ^ = ( T y =

Stress field I I : = c r y , a , = 2v<j.

Figure 4.3 Axisymmetric three-dimensional model for tunnel excavation.

158

hydrostatic stress (a® = = <t°) and plane strain conditional stress field (<7® =

<7°, a® = 2i/a®). Comparisons can thus be made by the assumed stress conditions

within the same model. The advancement of the tunnel face of length 9i2 {R is

the radius of the tunnel) is modelled by 12 excavation steps and, for comparison,

an instantaneous excavation of such a distance is also considered. Taking account

of the progressive advance of a tunnel, the axisymmetric finite element mesh and

the boundary conditions are designed as shown in Figure 4.4. The plan.e strain

representation of the problem is also shown in the figure.

The program described in Chapter 3 is used for the investigation. The nonlin-

ear rock mass behaviour is modelled by the elasto-viscoplastic analysis, with which

the standard creep phenomena can be reproduced and the general plasticity solu-

tions are approached when stationary conditions are reached. The yield criteria

used in the analyses are the Mohr-Coulomb criterion and the Hoek-Brown crite-

rion. The influence of the non-associated plastic flow is investigated using the 3-D

Hoek-Brown yield surface as the plastic potential function (Equation 3.52). The

influence of large strain analysis in this 3-D model is also checked by the available

formulations in the program. The excavation process is modelled by the method

described in 3.6. In each excavation step, the stress and stiffness of either two or

three elements are reduced by a factor of 1.0 X 10~® representing a 0.75i2 or 0.5i2

advancement of the tunnel face (jR-tunnel radius). Thus the length of the 9R or

6R face advancement is simulated in 12 steps. A reference cross section is taken

as shown in Figure 4.5, in which the stresses and displacements are investigated

in detail as the face advances and the results are compared with the related 2-D

analysis. The results presented below were obtained by running the program on

the Cray computer at ULCC (University of London Computer Centre) with an av-

erage 100 to 150 iterations for each excavation step when the material is modelled

via viscoplasticity.

159

§

Of O

21 3 2 5 3 6 4 8 5 9 6 117 128 149 160 181 1 9 2 213 224 2 4 5 2 5 6 277 2 8 8 3 0 9 320 341 3 5 2 373 3 8 4 4 0 5 416 437 4 4 8 4 6 9 4 8 0 501 5 1 2 5 3 3

ar

2 0 52 84 116 148 180 212 2 4 4 276 3 0 8 3 4 0 372 4 0 4 4 3 6 4 6 8 5 0 0 5 3 2 5 6 4 5 9 6 10 20 3 0 4 0 5 0 6 0 7 0 8 0 9 0 t o o 110 1 2 0 1 3 0 1 4 0 150 160 1 7 0 1 8 0

19 31 51 6 3 6 3 95 115 127 147 159 179 191 211 2 2 3 2 4 3 255 275 2 8 7 3 0 7 3 ( 9 3 3 9 351 371 3 8 3 4 0 3 415 435 4 4 7 4 6 7 4 7 9 4 9 9 511 531 5 4 3 5 6 3 5 7 5 5 9 5

18 5 0 82 114 14 6 178 210 2 4 2 274 3 0 6 3 3 8 3 7 0 4 0 2 4 3 4 4 6 6 4 9 8 5 3 0 5 6 2 5 9 4

9 19 2 9 3 9 4 9 5 9 6 9 7 9 8 9 9 9 109 119 1 2 9 1 3 9 149 159 1 6 9 179

17 3 0 4 9 6 2 81 9 4 113 1 2 6 143 158 177 1 9 0 2 0 9 2 2 2 241 254 2 7 3 2 8 6 3 0 5 318 337 3 5 0 3 6 9 382 401 414 433 4 4 6 4 6 5 4 7 8 497 510 5 2 9 5 4 2 561 5 7 4 5 9 3

16 4fl fiO 11? 144 176 ?Ofl 2 4 0 2 7 ? 3 0 4 336 36fl 4 0 0 432 4 6 4 4 9 6 5 2 8 5 6 0 5 9 2 16 8 16 2 8 3 8 4 8 5 8 6 8 78 8 8 9 8 108 118 128 138 148 158 168 1 7 8

16 2 9 47 61 7 9 9 3 111 1 2 5 143 157 175 189 2 0 7 221 2 3 9 2 5 3 271 2 8 5 3 0 3 317 3 3 5 3 4 9 3 6 7 3 8 1 3 9 9 413 431 4 4 5 4 6 3 4 7 7 4 9 5 5 0 9 5 2 7 541 5 5 9 5 7 3 5 9 1

14 4 6 7 8 MO 142 174 2 0 6 238 270 3 0 2 334 366 3 9 8 4 3 0 4 6 2 4 9 4 5 2 6 5 5 8 5 9 0 7 17 27 37 4 7 5 7 6 7 7 7 8 7 9 7 1 0 7 117 127 1 3 7 1 4 7 157 1 6 7 1 7 7

13 2 8 4 5 6 0 77 9 2 109 124 141 156 173 188 2 0 5 2 2 0 2 3 7 2 5 2 2 6 9 2 8 4 301 316 333 3 4 8 3 6 5 3 8 0 3 9 7 412 4 2 9 444 461 476 4 9 3 5 0 8 5 2 5 5 4 0 557 572 5 8 9

12 6 4 4 1 6 7 6 2 6 108 3 6 140 4 6 172 5 6 2 0 4 6 6 2 3 6 76 2 6 8 8 6 3 0 0 9 6 3 3 2 106 3 6 4 116 3 9 6 ^26 426 136 4 6 0 1 4 6 4 9 2 156 5 2 4 166 9 9 6 176 5 8 8 11 27 4 3 5 9 7 5 91 107; 123 139 155 i n 187 2 0 3 219 2 3 5 251 2 6 7 2 8 3 2 9 9 315 331 3 4 7 3 6 3 3 7 9 3 9 5 411 4 2 7 4 4 3 4 5 9 4 7 5 4 9 1 507 9 2 3 9 3 9 9 9 9 571 5 8 7

10 5 4 2 15 74 ?5

106 35

138 4 5 170 5 5 2 0 2 6 5

2 3 4 75 2 6 6 fln 2 9 8 9 5 105 3 6 2 115 3®"* 125 4 2 6 , 3 5 4 5 8 145 4 5 0 155 5 2 2 168 5 5 4 175 5 8 6

9 28 41 5 8 73 9 0 105 122 137 154 169 186 2 0 1 218 2 3 3 2 5 0 2 6 5 2 8 2 297 314 3 2 9 3 4 6 361 378 3 9 3 410 4 2 5 4 4 2 457 4 7 4 4 8 9 5 0 6 5 2 1 5 3 8 5 5 3 5 7 0 5 8 5

e 4 4 0 14 72 2 4 104 3 4 136 44 168 5 4 2 0 0 6 4 2 3 2 74 264 84 296 9 4 3 2 8 104 3 6 0 114 3 9 2 , 2 4 424 , 3 4 456 144 4 8 8 , 5 4 5 2 0 164 5 9 2 4 7 4 5 8 4

7 25 3 9 57 71 8 9 103 121 135 153 167 185 199 217 231 2 4 9 2 6 3 281 295 313 327 345 3 5 9 377 391 4 0 9 4 2 3 4 41 4 5 5 4 7 3 4 8 7 9 0 5 5 1 9 5 3 7 991 9 6 9 5 8 3

6 3 3 8 13 7 0

23 102

3 3 134 4 3 166 53 198 6 3 2 3 0 73 2 6 2 8 3 294 9 3 326 , 0 3 3 5 8 113 3 9 0 123 4 2 2 , 3 3 454

143 " " 1 5 3 518 1 6 3 5 5 0 1 7 3 5 8 2

5 24 37 56 6 9 8 8 101 120 133 152 165 184 197 216 2 2 9 2 4 8 261 280 2 9 3 312 3 2 5 3 4 4 357 376 389 4 0 8 421 4 4 0 4 5 3 4 7 2 4 6 5 5 0 4 917 9 3 6 5 4 9 5 6 8 5 8 1 4 2 3 6 12 6 8 22 100 32 132 4 2 164 5 2 196 6 2 2 2 8 7 2 2 6 0 8 2 2 9 2 9 2 3 2 4 ,Qg 3 5 6 112 3 6 8 , 2 2 4 2 0 , 3 2 45& 142 4 8 4 , 5 2 516 162 5 4 8 1 7 2 5 8 0

3 23 3 5 5 5 67 87 9 9 119 131 151 163 183 195 215 227 247 2 5 9 279 291 311 3 2 3 3 4 3 3 5 5 375 3 8 7 4 0 7 4 1 9 4 3 9 4 51 471 4 8 3 5 0 3 515 535 5 4 7 5 6 7 5 7 9

2 1 34

11 6 6

?1 9 8

31 130

41 162 51 194 61 226 71 2 5 8 81 2 9 0 91 322 ,01 3 5 4 111 3 8 6 ,21 41fi 131 4 5 0 141 4 8 2 ,51 614 161 5 4 6 171 5 7 8

22 3 3 54 6 5 8 6 97 118 129 150 161 182 193 214 2 2 5 2 4 6 2 5 7 278 2 8 9 310 321 3 4 2 353 374 3 8 5 4 0 6 4 17 4 3 8 4 4 9 4 7 0 4 8 1 5 0 2 513 534 5 4 5 5 6 6 9 7 7

L o a d i n g i : a , < /- Loading Ii: o> = Zva,

Figure 4.4 Axisymmetric finite element meshes used for three- and two-dimensional tunnel excavation analyses.

Advancing . Face /

Reference Section

Figure 4.5 A reference cross section in the three-dimensional model for

comparison with a two-dimensional analysis.

161

4.4 Computation results

4.4.1 Elastic solutions

The elastic constants used in the computations are ^ = 10 GPa, v — 0.25. This

is chosen as a typical hard rock mass (Hoek and Brown 1980). In the three

dimensional modelling, the in situ stress equivalent to a ground depth of about

400 meters is applied (Brown and Hoek 1978): Oro = cr o = 10 MPa (loading I).

The plane strain conditional in situ stress in this case is applied as: Cro = 10

MPa, <7 0 = 2i/(7rO = 5 MPa (loading II). Tunnel face advance of 6iE is simulated

in twelve steps and one step respectively.

The results are identical for these different simulations. The radial displace-

ments at the tunnel wall of the reference section are shown in Figure 4.6. Curve B

in the figure is the normalised displacement computed under loading condition I,

namely, the excavation is proceeding in a real hydrostatic stress field. Curve A is

the displacement under loading condition II which is equivalent to a plane strain

loading condition. The dashed line shows the plane strain solution (one step exca-

vation). The results indicate that for elastic media with plane strain conditional

loading, the ultimate tunnel convergence can be well modelled by an equivalent

plane strain analysis. However, the hydrostatic stress field in this case results in a

slightly smaller tunnel convergence than that predicted by the plane strain model.

This is because the plane strain analysis assumes an axial stress smaller than the

required hydrostatic value. It appears that in an elastic medium, increasing of tun-

nel axial stress will decrease the tunnel convergence and thus benefit the stability

of the tunnel.

The influence of the face advance is also investigated in the 2-D analysis using

the excavation simulation procedure described in Section 3.6. The 2-D tunnel

profile was excavated in twelve steps corresponding to the 3-D excavations. The

3-D tunnel convergence is well traced by choosing a set of suitable factors for the

162

a : l o a d i n g n .

2 - d s o l u t i o n

b : l o a d i n g i

X / The distance from the calculation section to the tunnel face \

R The radius of the tunnel

Figure 4.6 Elastic solutions for radial displacements behind the advancing face.

163

excavation. The results are presented in Figure 4.7 where the influence of the

excavation factors is also shown. In the present case, the following factors give a

perfect simulation of the 3-D face advance:

0.8,0.8,0.2,0.4,0.5,0.5, . ,0.5

The stress distribution in the tunnel wall is shown in Figure 4.8 and there is

no difference between the 2-D and 3-D models for a section fax from the face. In

both cases, the shear stress is zero. Figure 4.8c shows the stresses predicted

by the 3-D model under the hydrostatic load condition, and the results are again

identical to those of Figure 4.8a and b except for the axial stress Figure 4.9a

shows the stress redistribution in a section when the tunnel fax:e is very close to it,

while Figure 4.9b shows the 2-D representation of the stresses in this location. It

is seen that ar,Ot and Os in the 3-D analysis are not principal stresses because of

the presence of a non-zero shear stress tv- . This may greatly influence the plastic

flow when the material yields and thus it cannot be well modelled by a plane strain

analysis.

4.4.2 Elasto-viscoplastic solutions

The same elastic constants are used for most computations of the elasto-plastic

analysis. The strength constants for the Mohr-Coulomb's criterion are chosen

as c — 2.0, and 1.0 MPa, ^ = 40° respectively. The strength constants and

the empirical parameters for the Hoek-Brown criterion and the simplified plastic

potential are: m = 7.0, s = 1.0, m' — 0.0, <Tc = 8.6 MPa and 4.3 Mpa respectively

(these were chosen in comparison with the Mohr Coulomb yield surface by using

the 3-D computer graphics, so that the practical rock mass properties m, a and

<Jc were not considered). All these yield surfaces in three dimensional principal

stress space, for the above parameters, are shown in Figure 4.10. In situ stress

conditions were the same as for the elastic analysis. In the modelling of stepwise

excavations, stationary conditions axe always reached after each excavation.

164

Ur u P

1 • 0 —

05 Cn

A /

/

/

— —11 1

- 1

(A t face)

12 steps excavations Excavation factors

- A - — Simulation 1: 0.3: 0.3; 0.4; 0.4; 0 .4 :0 .5 - : 0.5,

Simulation 2: 0.5: 0.5; 0.5; , 0.5.

Simulation 3: 0.8; 0.8; 0.2; 0.4; 0.5; — ; 0.5.

- - 0 - Simulation 4: 0.9; 0.8; 0.2; 0.4; 0.5; •••; 0.5.

Three-dimensional simulation results.

R

Figure 4.7 Two-dimensional simulations of the advancing tunnel problem using the proposed numerical excavation procedure.

MtC74t JO (C«3<IC MMltSIS Aflito - O.N

i s »!!

s " * 2 I . . .

i«.«

I ••

• -00 'CO I.W Tlw "T,

ROOIUS Of ClCUlAtlON (HETEOSI

S07IJ41 JO ciasiic mntsis

RADIUS Of OLCULnilOH CHEIERSI

a) 2-D analysis b) 3-D analysis (loading IL)

S0Je24X M CLASTIC'AMACtSlS roi&OM

«-00 f .00 J .00 4'00 RfiOlOS OP CALCULATION I METERS I

c) 3-D analysis (loading L )

Figure 4.8 Elastic solutions for stress distributions in a tunnel section.

166

Stress (MPa)

1 5 . 0

10.0

5 . 0 -

0.0

Stress plot at a section very close to face, hydrostatic stress field: = 10.0 MPa.

y O - ^

o-o—o—o

2 3

Radius of calculation (m)

I 4

T " 5

Stress (MPa)

1 5 . 0 -

10.0-

5 . 0 -

0.0

o - Radial stress:

• - Tangential stress; og

A-Axia l stress:

• - Shear stress:

Stress plot of the 2-D analysis by step-wise excavations, stress field: gO = 10.0 MPa.

— • — » - # «

or sy o-o-o-

# -—o=- -o=-

4 4 A A A A A A — A - A • -

2 3

Radius of calculation (m)

Figure 4.9 Stress distributions in a section near the tunnel face, a) three-dimensional solution; b) two-dimensional solution.

167

M - C

a 00

4 (7 Modified H - B

Figure 4.10 Some of the yield surfaces used in the nonlinear analysis.

Tunnel convergence

The influence of the tunnel face and its excavation sequence on tunnel wall

displacements are shown in Figure 4.11. Both the Mohr-Coulomb and Hoek-Brown

yield criteria were applied for 3-D and 2-D models and the results did not show

much diff'erence because both the yield surfaces were chosen for an equivalent

strength value (see Figure 4.10). Curve A in Figure 4.11 was predicted by twelve

step excavations under loading condition I and shows the maximum tunnel conver-

gence. Excavation under loading condition II, in this case, produced much smaller

radial displacement. However, it was still larger than that of the 2-D solution. The

results given by one step excavation under loading condition II are almost identical

to those of the plane strain solution. The distribution of the displacements in the

tunnel wall is shown in Figure 4.12 where the influence of the loading condition

on the results is again well displayed.

The above results show that the 3-D model predicts a larger tunnel conver-

gence than the 2-D model, especially, when the real in situ stress and excavation

sequence are taken into account. The influence of the stress path on tunnel con-

vergence is obvious, because compared with a multi-step excavation process, one

step simulation of tunnel advance also gave a smaller prediction of those values.

The progressive tunnel plastic deformation cannot be well modelled by the instan-

taneous excavation of a full length of the tunnel. The 2-D numerical solution in

modelling this one step excavation seems more acceptable because the plane strain

condition for a tunnel section far from the face can be satisfied.

Stresses and plastic zone

Some typical stress distributions predicted by the nonlinear models are given

in Figure 4.13 in which the influence of stress history is obvious, particularly, in

the plastic zone. Figure 4.13c shows the stress distribution predicted by the 2-D

analysis using the Hoek-Brown criterion. An inner plastic zone in which Or <

Oz Pi ae, as discussed by Reed (1988), is clearly seen. The related 3-D predictions

169

s

A: 3-D MULTI-STEP EXCAVATION (Loading I )

C 3 -D MULTI-STEP EXCAVATION (Loading I I )

6 JD n o

2-0 SOLUTION

.ly B: 3-D ONE STEP EXCAVATION

3 4 5

The distance from the calculation section to the tunnel face

The radius of the tunnel

Figure 4.11 Elasto-viscoplastic solutions for radial displacements behind the advancing face.

2 . 0 -

A : 3 -D M U L T I - S T E P EXCAVATION

(Loading I )

B; 3 - D ONE STEP EXCAVATION

(Loading I I )

^ ^ k k —I— 3r 4 r —I—

5 r R

Figure 4.12 Elasto-viscoplastic solutions for radial displacements in the tunnel wall (a section far from the face).

171

STRESS OISTRieUTIOH IN IME lONNCL WALL

7-0 flXT. AWAITS IS USING M-O CRITERION

IS.O 14.0

O 12.0

a)

00 1.00 2.00 3.00 4.00 S.'OO

R A T I O OF C A L C U L A T I O N P O I N T TO TUNNEL R A D I U S

STRESS OlSTRieUTION IN THE TUNNEL WALL

3-0 AXT. ANALTSIS USING N-8 CRITERION

DROSTATIC STRESS LOAOtNO

ZO.O 19.0

10.0

or. n . o 0 .

C IQ .0

IS.O

14.0

O 17.0

c 11.0 o o 10.0 (O 9.0

1.0

6 . 0

5.0

4.0

3.0

2.0

I .0

b )

1.00 t .00 2.00 3J00 4.'00 S.'OO

R A T I O OF C A L C U L A T I O N P O I N T TO TUNNEL R A D I U S

STRESS OISTR18UTION IN THE TUNNEL WALL

3-0 AXl. ANALTSIS USING M-8 CRITERION

A SECTION VERT CLOSE TO FACE

C)

>•00 UQO 2.00 3.00 4 Zoo sJoO

R A T I O O f C A L C U L A T I O N P O I N T TO TUNNEL R A D I U S

Figure 4.13 Elasto-viscoplastic solutions for stress distributions in the tunnel

wall, a) 2-D analysis; b) 3-D analysis (Loading I); c) 3-D

analysis of a section close to the face (Loading U).

172

are presented in Figure 4.13b where the induced stresses exhibit a more disturbed

manner, and shear stresses still exist within the plastic zone. Figure 4.13b and c

also show that significant shear stress will be generated and remain in the plastic

zone even after the face has advanced to some distance away and consequently,

ar,<Tz and ae axe no longer principal stresses as assumed and predicted in the

plane strain model of Figure 4.13a.

A significant difference between the axial stress distributions for 2-D and 3-D

analyses is obvious and it can also be seen that the 3-D model predicts a larger

plastic zone than that of the 2-D model. The development of the plastic zone in

the 3-D model is shown in Figure 3.14. It should be noted that the failure zone in

front of the face has a significant influence on the final tunnel convergence. In the

analyses, the difference in the predicted radius of the plastic zone, Rgp, between

3-D and 2-D models is found to be up to 13% depending on Poisson's ratio, in

situ stress, excavation sequence and tunnel dimension, etc. Figure 4.15 shows the

distribution of axial stress at a fixed point close to the tunnel wall as the tunnel

passes by. Both the maximum and minimum values of axial stress occur when the

face is in close to this point.

The above results indicate that the plane strain modelling of tunnel excavation

usually predicts a smaller plastic zone and smaller displacements than a fully 3-D

analysis. This error arises within the 2-D model because of

a) the inadequate modelling of the stress path and

b) inadequate representation of the axial in situ stress.

A compromise value of Poisson's ratio which provides an approximation to

incorporate both in situ stresses and elastic mass behaviour must be chosen for

plane strain analysis. For instance, Poisson's ratio of the rock mass can be chosen

as 0.499 in the plane strain model to simulate the hydrostatic stress field, but

this will give rise to an over estimation of volumetric strain of the rock mass

and, consequently an error will still exist. An alternative approach to solve this

173

a "

o '

4 >

Hoek-Brown criterion

0(5=8.6 MPa, m=7.0,

s=1.0, (Loading I) 12 step excavation.

r E I

( V p*

Plastic Zone I

-Face advance

O a °

a

a °

Hoek-Brown criterion Og=4.3 MPa, m=7.0,

s=1.0, (Loading I) 12 step excavation.

s Plastic Zone \

Face advance

a '

Figure 4.14 Development of the plastic zones in the 3-D axisymmetric model.

174

problem is to use a different Poisson's ratio at different stage of calculation, which

is described and adopted in Chapter 6.

The axial stress in the plastic zone of a rock mass can be the maYimiiTn or

minimum principal stress under certain conditions (This usually occurs when the

plastic zone is very large so that an 'inner plastic zone' is created), however, for

most other situations, the axial stress is the intermediate principal stress, as in the

plane strain analysis. Unlike the plane strain analysis, an axial plastic flow near

the tunnel wall can possibly arise in 3-D modelling due to the high stress gradients

and this can never be simulated by a general plane strain model (Figure 4.15).

10.0

5.0<>

(MPa)

Face advance

0 .0

• Reference po in t

Figure 4.15 The path of axial stress (a^) as the tunnel face advances (curve

a: Loading I; curve b: Loading II).

175

4.5 Appl icat ion of the results

Among a number of solutions for tunnel convergence analysis, a unique conver-

gence law has been obtained by Sulem et al. (1987). It is based on numerical

solutions and convergence measurements and can be written in the form

2" C{x,t) = C, 1 -

1 + 1 + m 1 -

1 + y 0*8)

C(x, t) is the tunnel closure which depends on the distance to the face x and time

t. Apart from x and t, the function depends on four main parameters:

Repi plastic radius around the tunnel far behind the face,

T: characteristic parameter of the time-dependent properties of the ground,

Coox- instantaneous closure of the tunnel and

m: a constant related to the final closure.

The turmel convergence calculated by equation (4.8) was found to agree well

with the in situ measurements for two different tunnels (Frejus and Las Planas).

Unfortunately, however, the determination of these parameters based on data from

in situ investigation is very delicate. It is even more difficult to apply this law

in predicting the tunnel convergence in a design stage, so that practical use of the

equation is still doubtful.

The results obtained in this study appear to have a great potential for solving

this problem. According to the numerical investigations, it is clear that there is a

relation between the plastic radius estimated by 2-D analysis and 3-D modelling,

which can be expressed as:

= + (4.9)

where

( = / ( ! ' , ( 4 . 1 0 )

176

and / is a correction function, u is Poisson's ratio, Z is a ratio of radial stress

to axial stress, e is a excavation constant and r is the tunnel radius. In this

study, 0 < f < 0.13. Likewise, an under-estimation of the final convergence by

a 2-D numerical model would be expected to be up to 70% (see Figure 4.2 and

Figure 4.11). By understanding of the above relations between a 2-D model and

a 3-D excavation problem, some of the parameters involved in equation (4.8) can

be estimated by a plane strain analysis. Equation (4.8) can thus be applied in

coordination with a simple 2-D numerical analysis in which the plastic radius, the

tunnel convergence and time-dependent behaviour, as that required by equation

(4.8), can all be investigated. A factor of safety can also be considered in evaluating

these parameters.

4.6 Conclusions and discussions

A nonlinear axisymmetric finite element simulation of tunnelling in a rock mass has

been carried out to investigate the stress and displacements behind an advancing

tunnel face. All results obtained from the three-dimensional modelling have been

compared with the related two-dimensional model of similar mesh and material

properties. Although it is often assumed that 2-D numerical analysis over-predicts

tunnel convergence, computation results reveal this not to be generally true. It

can lead to a significant error (under-estimation) when a 2-D modelling of a tunnel

excavation problem is used, especially in the prediction of ultimate unlined tunnel

convergence.

The yield criteria used in the nonlinear numerical models are the Mohr-

Coulomb ajid the Hoek-Brown criteria, both of which are intermediate principal

stress independent; however, if an intermediate principal stress dependent yield

criterion, such as Drucker Prager's or modified Hoek-Brown's, is adopted in the

analysis, an even bigger difference between the 2-D and 3-D predictions will be ex-

177

pected. It is possible to correct this error by understjoiding the mechanism of the

stress redistribution and the influence of excavation sequence in three dimensions.

Such a correction has been suggested by the author (Equation 4.9 and 4.10) and

further investigation of the function and parameters is needed.

For a time dependent rock mass, it is possible that a certain length of a

tunnel is formed before the final induced stress and displacement in the tunnel

wall occur (namely, the excavation can be considered as infinitely rapid due to

the time dependent response of the rock mass). In that case, the modelling of

one step excavation along the whole length of the tunnel can be correct and a

2-D plane strain model can be acceptable; otherwise, either the real excavation

sequence should be modelled by using three dimensional numerical analysis or a

correction of the results is necessary.

Application of the results is briefly discussed with an example of the useful

convergence law proposed recently by Sulem et al. (1987). In that respect, the

investigations carried out in this study are not only of theoretical significance but

also of practical importance.

178

CHAPTER 5 A HYBRID COMPUTATIONAL SCHEME

5.1 Introduction

As has been discussed in the preceding chapters, a rock mass is distinguished

from other engineering materials by the presence of discontinuities within it. For

the conceptual models (b), (c) in Figure 1.2, the discontinuities must be allowed

for, either implicitly or explicitly, in a numerical or an analytical model. The

development of methods of modelling discontinuities and their effects has been

the distinctive feature of the adaptation to engineering rock mechanics of methods

used in other branches of engineering mechanics.

It has also been concluded in chapter 2 that for a block-jointed rock mass,

the distinct element method is currently the only one suited to the analysis of its

behaviour; while for an ubiquitously jointed rock mass, as the conceptual model

(d) in Figure 1.2, the nonlinear finite element method is appropriate. A logical

answer has been drawn in choosing a method to model the rock mass response

which can be represented by the conceptual model shown in Figure 1.2 (e). The

answer is to develop a hybrid computational scheme in which the finite element

method is used in conjunction with distinct elements to model the interaction of

blocky rock with a softer rock mass. This is the main topic of this chapter.

The conventional distinct element method in rock mechanics (dynamic relax-

ation) can be used in the hybrid computation. Nevertheless, the calculation of

block displacements in the dynamic relaxation involves an extremely small time

step 6t, so that it is hardly possible for this small time step to coincide with the real

time steps in a viscoplastic finite element analysis. It has, therefore, been deter-

179

mined to adopt an alternative relaxation procedure, the static relaxation (Stewart

and Brown 1984) for the hybrid computation. This relatively new distinct element

method is described with an assessment of its theoretical basis, and before being

used in the coupling analysis, is further validated with some modifications (Sec-

tion 5.2). The principles and algorithms of the coupling procedure are presented

in Section 5.3. Validation examples of the hybrid program COUPLE are given in

Section 5.4. In the end of the chapter, some conclusions and discussions are drawn

in Section 5.5.

5.2 The dist inct e lement m e t h o d

5.2.1 Descr ipt ion of the m e t h o d

The relaxation theory

As was described by Stewart (1981), the method of static relaxation analy-

sis is based on the general relaxation principles developed by Cross (1932) and

Southwell (1940,1946) in order to eliminate the necessity of solving large numbers

of simultajieous equations in the analysis of engineering structures. In the method,

each node (or block) is given a displacement that would bring it to equilibrium if

other nodes were to remain fixed at their positions at the start of the iteration. As

each node is moved in turn, the equilibrium of previous nodes is destroyed, but,

after several passes through the system of equations, the out- of-balance forces

become small and all nodes are approximately in equilibrium. The method is sim-

ilar to that of Jacobi as compared to Gauss-Seidel so that the solution is path

independent (Otter et al. 1967; Cundall 1987).

The calculation procedure

The same basic calculation procedure as that in the dynamic relaxation dis-

tinct element program is adopted, and the frame chart of the program is shown in

180

Figure A.5.1 of Appendix 5. The complete calculation cycle can be summarised

as below:

SYSTEM FORCE DISTRIBUTION

^CONSTITUTIVE RELATION li

BLOCK SYSTEM DISPLACEMENTS

"BLOCK RELAXATION

LAW OF A SINGLE BLOCK f DISPLACEMENT:

f = m a, (dynamic) or:

Z Fj (Ax, Ay, AG) = 0.

(static)_

Figure 5.1 Calculation procedure of the distinct element method.

The program computes force and moment balance for ea^h block at each itera-

tion within a complete cycle. After all blocks have moved to their new positions

(relaxed), a new force distribution of the system can be calculated according to a

certain constitutive relation. If the imbalance still exists, a cycle begins in which

the blocks move in accordance with the net force until new equilibrium is estab-

lished, or simply continues to move if a true collapse condition hats been reached.

The boundary conditions and loading (or excavation) can be applied within each

cycle.

The constitutive relation

Block interactions are governed by realistic friction and stiffness laws which

are the relations of force and displacement at contacts. The deformation properties

181

of both the intact material and the joint are modelled collectively at the block

interfaces. Contraction and shear movements between blocks are simulated by

overlapping and sliding, respectively, along the interfaces. Contacts and associated

forces between blocks are assumed to occur at discrete points where block comers

interact with neighbouring block edges and vice versa. Each contact between

blocks is modelled as a normal interaction and shear interax:tion which are assumed

to be the models shown in Figure 5.2. Typical load - deformation characteristics

for rock joints are shown in Figure 5.3.

In the numerical calculation using the constitutive relation assumed, omission

of a viscous element in the model maJses the block assemblies unable to rearh

exact equilibrium, and small continuing vibrations of blocks relative to each other

usually occur. Convergence can be assured by controlling the rate at which loading

is applied. It is also possible to incorporate some form of damping at the contacts

to obtain better equilibrium, but comparisons with dynamic relaxation indicated

that the effect of not including viscous damping has little effect on the final results

(Stewart 1981).

The solution of block movements

Assuming that load and boundary conditions have been applied and specified

on the block system, the solution is begun by calculating the force distributions

in the system according to the block positions and the constitutive relations. The

solution of a single block displacement is illustrated with reference to the following

example.*

For a single block in the system, it may be surrounded by six other blocks as

shown in Figure 5.4 and, therefore, this gives rise to 8 corner-edge contact forces.

* The problem can be, or ideally should be, three dimensional. Formulations of

the 3-D distinct element analysis have been described recently by Cundall (1988)

and Hart et al. (1988). As aji illustration example here, the discussions are

restricted to 2-D.

182

Normal interaction Shear interaction

m Q)

Overlap Shear displacement

Figure 5.2 Normal and shear interactions for static relaxation (after Stewart and Brown 1984).

/ / / / - \ V 1

e

X / ^ \ ' ' 1 - ' 1

a) cr compression

TTJ / / M / / // / A v

'mc

Unfilled Joints

Filled Joints

Normal Displacement

Shear Displacement

Figure 5.3 Typical force - deformation relation of joints, a) normal deformation

of joints; , b) shear deformation of joints (after Goodman 1976).

183

a)

Block e is to be relaxed while the other blocks are tempararily fixed at their positions calculated in the last iteration.

b) Before block displacement and rotation:

c ) After block displacement and rotation:

P f } ( n + 1 ) + { Z R j = } = 0,

({^i )(n+1) = { } n + { ^ } )'

Figure 5.4 Block relaxation and calculation procedure.

184

Now, for the particular block relaxation in the iteration, all the other blocks except

block e are fixed at their positions calculated in the last iteration. The resultant

force is usually not in equilibrium, that is:

{S /?} + {SR}} + 0. (6.1)

where Rj is the applied load including body force and

m (5.2)

in which and are the force and moment components at ith contact point

of block e. Subscripts x, y are the global coordinates, and 6 is the rotation angle

around the centroid of the block. The block e must now move to a new position

under the action of these out-of-balance forces and moment.

Since the blocks are considered completely rigid, the incremental displace-

ments of the block contact points can be defined in terms of the incremental

displaxzements of the block centroid. The displacement vector is defined as:

{6u^} = {6x\6y^,6e^}^. (5.3)

After this displax:ement, the block n should be in a temporary equilibrium state

due to the contact force changes and the equilibrium condition is (Figure 5.4 c):

{ E / f } + { m n + { S B ; } = 0 . (5.4)

where

{£</?} = lA'\{Su'}. (5.5)

and \A \ is the unit stiffness matrix which can be written as: *

ail fli2 ®13

&21 <122 ®23

<131 < 32 < 33 (5.6)

* In the dynamic relaxation, (5.5) is replaced by the Newton's Law of motion

and the incremental displacement vector is derived from the integration of the

Newton's Law (see Cundall 1974).

185

The elements of the matrix [A®] are related to the positions and geometries of

block e and its neighbouring blocks. They also depend on the constitutive law

used for the block contacts and can be generally given as:

NOP

= E (GLk (KsuKm,ai, (X? - X|), (!;• - Y;)) . (5.7) i=l

where NCP is the number of contact points. The definitions of the other symbols

are explicitly given in Appendix 5 for the constitutive relation defined in Figure

5.2.

It is thus easy to solve equation (5.4) as:

where

{ f } = - ( {Ef iJ} + {E/ / } ) .

(5.8)

(6.9)

Assuming that the system consists of n blocks which are temporarily relaxed

at the same time during one iteration, a global stiffness equation with a specific

explicit nature can also be established. The displacements of the n blocks within

this relaxation are:

{*%} = [K]-XF}. (5.10)

where

and

m =

Ai 0 . . 0 . .. 0 " 0 A2 . . 0 . .. 0

0 0 . . A' . .. 0

0 0 . . 0 . .. A» -1 3n.x3n

{6u} = . ,6%° ^''"}Tx3n'

M = F»}L3„-

(511)

(5.12)

(5.13)

The submatrix A® and vector are defined by (5.6) and (5.9) which depend on the

current block positions and are recalculated after each iteration. The incremental

186

displacements are used to evaluate the incremental forces and the residual forces

are thus updated for the next iteration.

The program and its numerical convergence

From the above discussion, the complete calculation cycle has been traced and

the calculation is continued until the difference between incremental displacements

determined from successive iterations falls below some pre-determined threshold

or when a mode of failure results. It was shown by Stewart (1981) that the rate

of convergence using static relaxation can be faster than the dynamic relaxation

method. However, a disadvantage of the method is that the computation cannot

be carried on if any block lacks sufficient contact points to approach the static

equilibrium. This requires the method to be applied for the analysis of relatively

dense assemblages of blocks . Therefore, the current calculation procedure is not

capable of dealing with the problems in which the separated block movements have

to be included.

Although the program procedure seems very simple, the operations and rou-

tines necessary to allow for virtually unrestricted large scale displacements and

rotations of the rock blocks result in a computer program of considerable com-

plexity. Another feature of the program developed by Stewart is the introduc-

tion of dynamic memory allocation utilising linked lists for the one dimensional

data storage array, which makes the program almost unreadable. The program is

supported by the graphics facilities available on the Imperial College Computing

System.

187

5.2.2 Validation of the program

The program for two dimensional analysis of underground excavations in discon-

tinuous rock masses described in the previous section is taken as the means to

model the discontinuous rock movements in the present work. After establishing

the code on the main frame (the CDC at Imperial College and the Amdahl at

ULCC), the program was tested on several examples which axe related either to

the closed form solution and other numerical solution or the physical model tests,

A) Consolidation of five blocks under self weight

The block geometry shown in Figure 5.5 was modelled as in Stewart's example

(Stewart 1981). It was used to compare the rate of convergence towards the final

solution with that of dynamic relaxation. The blocks were subjected to self weight

and the iteration continued until equilibrium was attained. The properties of the

blocks are also shown in Figure 5.5, The results , presented in Figure 5.6, are the

same as those predicted by Stewart. It is seen that the static relaxation method

in this example converges fax more rapidly thaxi dynamic relaxation. After 30

iterations, the total force at the block base converged to 99.9% of the final value.

B) Tunnel roof stability under self weight

A tunnel roof with geometry similar to that considered by Voegele (1978) is

chosen as the second test example (Figure 5,7). The roof strata are divided into

blocks by two main sets of joints and the model has the properties: p — 2.5 t/m,

iiCjv = 200 MN/m, Ks = 10 MN/m, c - - 0.001 MPa and / = tan<f) = 0.3.

Three cases were considered:

1) gravity load is applied after excavation;

2) gravity load is applied before excavation; and

3) as case 2) but the excavation was modelled by removing blocks gradually.

Some of these results are shown in Figure 5.8 and the displacements predicted

188

C H

A B

0 0

Normal stiffness: K|\| = 1000 MPa,

Shear stiffness; Kg = 0.0,

Frictional coeficient: f = 0.01,

Gravitation: g = 9.81 m/(sec)

Block size: L = 5.0 m,

H = 2.5 m.

Figure 5.5 Block geometry and property for comparison with dynamic relaxation (data from Stewart, 1981).

Iteration number

Force by static relaxation (0.1 MN)

force by dynamic relaxation (0.1 MN) Iteration

number

A B c A B c

10 1.918 2.547 0.806

20 1.968 2.914 0.871

30 1.971 2.952 0.876 1.791 2.468 0.739

40 1.971 2.955 0.877

50 1.971 2.956 0.877

60 1.949 2.882 0.849

90 1.964 2.946 0.862

100 1.971 2.956 0.877 1.965 2.958 0.863

Figure 5.6 Table for comparison of results (data for dynamic relaxation are from Stewart, 1981).

189

SLOCK PLOT

ITI Rfl ION NUMBER : 0

NunSER OF PROBLEM UNITS PER INCH = 0 . 8 0

a]

NunSCK or Moailm UNITS "t* l<(CH : O-dC N EA ION MUNSEN S 20

••ufBC* or MoeiEM UNITS fc* INCH t o.«c 10 101 NUFLFLC* T M O

NUtiBCR OF Moaien UNITS PC* INCH % c-«c t' 101 T 400

b)

Figure 5.7 A tunnel roof analysis: a) the proposed excavation geometry; b) removing blocks to model excavation.

190

BLOCK M 0 1 NUMBER OF PROBLEM UNITS PER INCH s 0.60

ITl RA ION NUMBER s 400

TOTAL OlSPL.PLOT NUMBER Of PROBLEM UNITS PER INCH s 0.12784860

TERRTION NUMBER s 400

\

1 1

SCALE . I INCH s

I

1

\

\

\

1

\ I

\

1 1

1 \

i I

1

I

I

1 /

^ / /

Figure 5.8a Block displacement of the tunnel roof (gravity was applied after

excavation).

191

Comer Force Plot, Units Per Inch - 232664.165 Incremental Number - 50 Scale; 1 Inch - —

* '

I

J

J

Total Displ. Plot. Units Per Inch = 0.11623214 Iteration Number = 50 Scale: 1 Inch = —

I I

I

/

(i)

Corner Force Plot, Units Per Inch = 208577.070 Incremental Number = 50 Scale 1 Inch = —

1 1 * I '

(

1 \ \ \

\ r ' \

\ ( \

\ - r \

Total Displ. Plot, Units Per Inch = 0.02712792 Iteration Number = 50 Scale: 1 Inch = —

I I

/

(ii)

Figure 5.8b Force and displacement distribution, (i) linear normal contact force;

(ii) nonlinear normal contact force.

192

in cases 2) and 3) axe obviously smaller than that in case 1). The block movements

and the possible failure mode were also verified visually by the base friction test

as shown in Figure 5.9.

C) A rectangular opening in a jo inted rock m a s s

As a study to compare the results of numerical analyses and physical models,

Stewart (1981) conducted a physical model test in which a rectangular excavation

in a prestressed jointed rock mass was analysed. The geometry of the physical

model with the block size is shown in Figure 10. The elastic modulus was measured

as: E = 1.088 x 10^ MP a, so that the normal stiffness for the joints can be

calculated as (see section 5.2.4):

JpTk Kff = Ks = . = 1.36 X lO'^MN/m.

AH

The in situ stress field was modelled as: ah = 0.155 MPa, <7„ = 1.395 MPa, that

is, the ratio of the forces applied in the vertical (v) and horizontal (h) directions

is 9.0.

A numerical simulation of the above physical model test was analysed using

the existing distinct element program. The model geometry and the unit block

size in the analysis is shown in Figure 5.11. The block stiffness in the analysis can

thus be taken as

KN =Ks= 2.72 X lO^MN/m.

which is equivalent to the measurements in the physical model. A comparison of

the displacements measured in the physical model test and predicted in the nu-

merical analysis is listed in Table 5.1 which shows a reasonable agreement between

the two analyses.

193

'Jr<

1

Figure 5.9 Modelling of the tunnel roof problem by base friction model.

194

" ' 4

i i Y i Y u L _ U U

m • m

• • • •

• n n n

a

Gv = 9.0.

H 1

L

A single plaster block in the physical model.

H : L : b = 1 : 1 : 2.

Figure 5.10 A physical model simulating a rectangular opening in a jointed rock mass (tested by Stewart, 1981).

195

4

168

145

104 106

61 64

2 0 22 24

H ; L : b = 1 :1 :1.

0y / Of, = 9.0.

A single distinct element in the numerical model.

Figure 5.11 Numerical simulation of the rectangular opening in the jointed rock mass.

Block number Physical model (mm) Numerical model (mm)

20 0.59 0.122

22 1.90 1.058

24 1.97 0.982

61 0.43 0.332

64 1.77 0.874

103 0.39 0.482

106 1.38 0.844

145 0.51 0.701

168 0.75 0.820

Table 5.1 Comparison of displacements predicted by physical model and numerical analysis.

196

5.2.3 Some modifications

A) Nonl inear normal contact force

The typical relation of the normal contact force and displacement between

a rock joint is generally known as shown in Figure 5.12. A measurement of the

normal stiffness between the blocks under different pressures was taJcen in the

physical model described in the last section. Figure 5.13 shows the measured

nonlinear experimental stiffness and the assumed constant stiffness. In the testing

of Stewart's program, it was found that the constant normal stiffness assumption

in the constitutive model may be one of the major reasons why a discrepancy

exists in those displacement values as shown in Table 5.1. Another shortcoming of

the assumption is that the constant Kjj has to be pressure dependent in order to

avoid too large or too small an overlap between the blocks. From Figure 5.13, it is

clear that with the constant too small or too large an overlap at the contact

point may result and this may give rise to a further error on force distribution.

It appears, therefore, necessary to use a nonlinear normal stiffness law which

has been chosen as Goodman's hyperbolic relation of stress and deformation for a

joint. The relation can be rewritten (Figure 5.12);

where v and Umc are indicated on the diagram. For ease of computation,

A = t = 1.0 is assumed [A > 1.0,( < 1.0, in fact).

Considering the initial state of the distinct elements, there is neither stress

nor overlap between the blocks, so the relation can be assumed by setting

CTn - C = (5.15)

to give

197

o,

Kn"

V

a) mc

b)

mc

Figure 5.12 The hyperbolic normal pressure - nomal displacement relation for a discontinuity, a) definition of the initial pressure (seating pressure) ^ and the maximum closure (cf. Figure 5.3);

b) definition of the initial stiffness of the joint.

Load (MPa)

0.6

0.5

0.4

0.3

0.2

0.1

Experimental measurements.

0.0 V (mm) 0.0 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200

Figure 5.13 Comparison of the measured normal stiffness and the constant stiffness assumed in the calculation (data from Stewart, 1981).

198

Differentiating (5.16):

and letting

— ^/^mc

(5.17)

(5.18)

where is the initial stiffness, (5.17) becomes

da'„ = K^{v)dv. (5.19)

where

(5.20) (Wmc - U) •

is the nonlinear stiffness between the contacts. For each contact point, the incre-

mental form of (5.19) used in the program is

6/^+"- = K^{v'')6v^+^ = K^- (5.21) (1 - V"/Vm.c)

where n is the iteration number, N is the normal direction of contact, 6 / is the

incremental contact force and v is the overlap. This assumed nonlinear relation

and its iteration process are illustrated in Figure 5.14.

O QC O

non i i nea r

near

CJ DC O theory curve

app rox ima t i on

" V m c

Figure 5.14 Modification of the program using the nonlinear normal stiffness

overlap relation.

199

The incorporation of (5.21) into the program requires that normal stiffness

values be associated with comers of blocks rather than edges, which means that

each block corner can have a different normal stiffness at step n. A new

subroutine JSTIF has been written for this purpose (see Appendix 5).

B ) Boundary loading and constrained displacements

In the original program, only concentrated external forces can be applied to

the boundaries, which makes consideration of the in situ stress field somewhat

difficult. A modification to the program is therefore needed to make it possible to

apply distributed boundary forces in any direction. This is done by introducing

codes to transform linearly distributed edge loads into centroid force and moment

acting on the boundary blocks. The requirements of input data for distributed

loads can then be in a similar form of the finite element analysis.

Constrained block displacements at boundaries also need to be incorporated

into the program as an application of the boundary displacement condition which

is to occur in coupled problems. These modifications have also been carried out

and successfully tested.

200

5.2.4 T h e input parameters for contacts

The parameters for block contact properties in the distinct element method are

generally: 1) The normal and shear stiffnesses ,Kg, 2) the normal and shear

dashpot (damping) constants , rjs and 3) joint strength parameters c, <f>.

In the static relaxation procedure, the damping dashpots axe not introduced

and the parameters axe reduced to four, in which the and Ks are the most

important constants dominating the joint behaviour and the block displacements.

However, the method of determining the joint parameters Ki^ and Ks in the

distinct element analysis has not yet been established.

Normal stiffness

In modelling an elastic continuum using the distinct element method, the

normal stiffness can usually be derived by considering a rectangular array of blocks

as shown in Figure 5.15, and in that case:

KN. =

KNy = 26

where E is the Young's modulus of the body and a,b, and c are the element

length, height and thickness, respectively. (5.22) is usually applied in the validation

examples of the distinct element analysis (Stewart 1981; Lorig and Brady 1982).

It must be pointed out that (5.18) must be changed if the distinct element

meshes are taken in a different way such as in Figure 5.15(b). In that case, the

normal stiffnesses become:

KN. =

^ _ L

- 46 •

It is obvious that in the large deformation analysis of a jointed rock mass, some

of the contact points between blocks change throughout the process of calculation.

Moreover, the global behaviour of a discrete system cannot be represented by

201

/

/

b a

r

a)

b)

Figure 5.15 Derivation of edge stiffness for block interaction in an elastic discontinuous body.

Overlap

5m * 5m

Hq = 20; 60 m.

Fixed block

a) b)

Figure 5.16 Detemunation of normal stiffness K , a) by consideration of

cylinders; b) by free drop test (after Ohnishi et al., 1985).

202

an elastic continutim, so that the above simplified constant normal stiffness may

produce a significant error when applied to a practical jointed media.

Difficulties have always existed in choosing the stiffness Ki^ and it was often

used as a parameter for ensuring convergence rather than a constant reflecting any

physical meaning. Ohnishi et al. (1985) reviewed the previous work on how to

determine the iiCjv and reported that for two cylinder elements in contact (Figure

5.16a), ifjv had been previously derived as:

TT E

" 2{1 - + 2 In {4r/by

where i/ is Poisson's ratio, r is the radius of the cylinders and b is the width of

contact. They also concluded that Kff could be determined by measuring the

rebound coefiicient in a block free drop test (Figure 5.16b).

In this study, a modification of the program to use the corner related non-

linear stiffness parameter has been carried out (see the last section), and this

may overcome some of the difficulties involved in evaluating the parameter. In

that case, only the initial stiffness is required and it caji be determined by

equation (5.18).

Shear stiffness Ks

If the same assumptions are made as shown in Figure 5.15 a., the shear stiff-

ness can be similarly derived as (Stewart 1981):

+ = 4 - (5.25) Ksxf> KsyO, E

One of the simple solutions is

In most of the previous work the normal stiffness was set as

Knx = = KN,

203

(5.26)

so that

Ksx = Ksy = Ks.

and if slip is not allowed,

Ks = Kif

The errors possibly involved in these formulations can also be shown by considering

Figure 5.15 b. in which the shear stiffness should be written as

. A ~ 2a •

If slip and separation of the blocks are allowed, the above assumptions cannot

be realistic, hence some suggestions have been made in the literature, one of which

is (see Ohnishi 1985):

Ks = sKjf. (5.28)

where 0 < 5 < 1, and it was reported that good results have been obtained using

5 = 0.25. Another investigation (Belytschko et al. 1984) suggested a ratio of

normal and shear stiffness re (= K^/Ks) which should be in the range of

2 < re < 100. (5.29)

In fact, however, the stiffness KN and Ks of joints are quite different properties

and above estimations are only for solution convergence and computational conve-

nience. When the nonlinear normal contact model is used in this study, the shear

stiffness can be taken as

Ks = aK^. (5.30)

where K^ is the initial normal stiffness defined in (5.18).

The joint strength parameters c, <f)

In Figure 5.2 it has been shown that the criterion of plastic slip at a contact

is:

Fs > c + Fff tan <}>. (5.31)

204

where c is the cohesion and <l> is the frictional angle. If (5,31) is satisfied, the shear

force is set to the sheax strength and slip continues until elastic conditions can be

re- established. However, if the shear stiffness is set to a low value (e.g. s = 0.01 or

K = 99, etc.), a large slip will occur from the beginning and hence the parameters

c and <f>, in this case, are not important.

Typical values of c and <l> for various joints can be found in the existing

literature and can also be obtained by performing various shear tests.

5.2.5 Summary and discussions

The presentation of the distinct element method in this chapter indicates that the

static relaxation procedure is an alternative method in analysing jointed rock mass

behaviour in which large scale translational and rotational movements can occur.

The basis of the method is assessed and the explicit formulations of the distinct

element method using static relaxation procedure have been expressed in a matrix

form.

The program was tested against analytical solutions, other numerical solutions

and physical models. Good agreement was found in all cases. As a modification

of the constitutive law, a nonlinear normal stiffness relation at contact points has

been implemented in the program. The evaluation of the input parameters for the

static relaxation program in analysing joint contact behaviour was described.

Of particular engineering interest is that this method may simulate the be-

haviour of an aggregation of joint-defined blocks in a manner conceptually and

physically consistent with the observed performance of jointed rock masses.

In applying the program to realistic, practical problems involving complex

material properties and large block movements, three possible limitations of the

method follow:

205

1) incapable of modelling far field boundary conditions;

2) unable to model separated block movements due to the algorithm adopted in

the static relaxation procedure; and

3) difficult to evaluate the input parameters for different joints.

The first limitation is caused by the 'infinite domain' problem because the

rock excavation problem usually involves a domain which is much larger than the

region to be analysed. Similar to the conventional finite element or finite differ-

ence analysis, the distinct element method can only apply a zero displacement or

prescribed force boundary condition at a finite large distance away from the region

of interest. These two 'artificial' boundary conditions may produce significantly

different results. A suitable method of solving this problem is to adopt a hybrid

method in which boundary elements are used in coupling with the distinct ele-

ments to model the infinite domain (Lorig and Brady 1982). Otherwise, like some

of the examples in the remaining chapters, both boundary conditions should be

applied in different tests and their influence on the results have to be considered.

Since the numerical algorithm in static relaxation is of an explicit nature

whereby incremental displacements are calculated from the static equilibrium of

forces acting on the blocks, a separated block can no longer satisfy the equilibrium

equation as the matrix [A ] is singular. As described in section 5.2.2, the program

may stop executing when any block has no contact force (or too few contact

points) at its edges or corners. This limitation might be overcome by introducing a

damping parameter similar to that of dynamic relaxation to establish a temporarily

pseudo equilibrium. In this study, however, the separated block is either to be

temporarily fixed at its current position or deleted so that calculation caji continue.

Due to the lack of practical joint property parameters, the computation should

be carried out using trial values evaluated through the literature and experience.

206

5.3 The hybrid finite element - distinct element method

5.3.1 The coupling principles

By representing the rock strata, which constitutes the jointed hard rock around

an excavation, with distinct elements, and the relatively weak and soft rock mass

with finite elements, the conceptual advantages of the distinct element method are

retained, and the full nonlinear problem can be solved.

Like other hybrid analysis methods such as the distinct element - boundary

element method and the finite element - boundary element method, the basic prin-

ciple of the coupling is that the different numerical techniques suited to different

material responses can be applied in one analysis if the boundary conditions at

the interfcice between the two zones can be satisfied in both types of analysis. A

specific feature of the hybrid analysis in this research is that both domains of rock

mass may experience relatively large deformation including slip at the interface.

The interface between the two domains contains points common to both re-

gions. Determination of induced contact forces permits direct determination of the

induced total nodal forces of finite elements. These induced nodal forces are then

added to the residual force vector ^ from equation (3.31) and applied as the incre-

mental load at the current iteration step of finite element analysis. The induced

nodal displacements of the finite elements at the interface are thus calculated using

equation (3.25). These displacements are then fed into the distinct element do-

main analysis as displacement boundary conditions. Thus the relaxation process

continues with nodal displacements and forces being updated and used in the sub-

sequent hybrid computation cycles. The iteration can continue until equilibrium

is approximately reached. In principle, the distinct element program should be

run at each time step of the finite element analysis. However, in the current stage

of the program, it appears not realistic due to the high computing cost.

207

5.3.2 The algorithm and compatibility conditions

The hybrid distinct element and finite element computation is performed by sat-

isfying the conditions of displacement compatibility (usually not continuous) and

force or traction equilibrium at the interface between the two solution domains.

The algorithm used in this research is different from that of the linkage of dis-

tinct element - boundary element described by Lorig and Brady (1984). In their

coupling procedure (see Section 2.6), a displacement continuity condition at the

interface, which is an elastic body surface was assumed. In this full nonlinear

analysis, however, a discontinuous displacement phenomenon including slip and

separation at the interface is required to be modelled.

In considering the nonlinear continuum domain interacting with the set of dis-

tinct elements, shown in Figure 5.17, the most convenient procedure is to develop

a contact force - nodal force relation for the finite element interface represented

by Figure 5.17b. Treated in the way as illustrated in Figure 5.17, the two different

domains of material behaviour can be analysed using the different methods with

some compatible updated boundary conditions at the interface.

It was described in Chapter 3 that in the 2-D plane strain analysis, the 8-

node isoparametric finite element is used which is usually with curved sides and

a quadratic variation of the displacement field within the element (Figure 5.18a).

The induced nodal forces and displacements are related through the incremental

stiffness equation (3.22) and it may be recast as (i.e. equation 3.25);

where AF" involves the induced nodal forces 6f' and the calculated Sd"" is added

to the total displacement vector expressed in equation 3.27, i.e.

cr+^ = dr + 6cr.

After a number of time step iterations when either the specified maximum step

number has been reached or the residual force becomes zero, the induced displace-

208

a)

b) c)

Figure 5.17 Interaction of a continuum and a discontinuum, a) finite elements and distinct elements; b) distinct element calculation; c) finite element calculation.

209

Figure 5.18 Shape functions of the 8-node Serendipity quadrilateral element.

After deformation:

a) b)

Figure 5.19 Introduction of a 7-node finite element for the compatibility conditions at the interface.

210

ments at the finite element interface can be determined The interface sides

of the finite elements are generally curved as shown in Figure 19a rather than the

initial meshes as in Figure 5.17a.

X

Figure 5.20 Definition of the 7-node finite element (in accordance with equation (5.32)).

Difficulties exist in applying this curved displacement boundary condition to

the subsequent distinct element analysis because the distinct elements are defined

as rigid blocks with linear edges. The best solution for this problem is to introduce

a 7-node isoparametric finite element at the interface. Figure 5.20 illustrates the

element with the order of node numbering. In that case, the shape functions of

the element are derived as;

= ^(1 - 0 ( 1 +

= ^(1 - £)(i -

= ^(1 — 6)(i —'?)(—^ — u — 1),

»4(C, f?) = g ( l — ^^)(l — *?)> (5.32)

^5((,n) = ^(1 + 0 ( 1 ~ ^) (~ i + C-v),

n6( >*7) = 2(1 + 0 ( 1 ~

nriCtV) = ^(1 + 0(1 + %)%)

where rj are the element local coordinates. With this element, the displacements

at the interface solved from the finite element analysis can coincide with

211

the required boundary constraints for the distinct element analysis as illustrated

in Figure 5.19b. Thus, the distinct element solution caji proceed and the force

components {g^} at the interface can be solved using equation (5.5) to give

{,•<} = [ i f ' K i u ' } + { , g } . (6.33)

in which [jK"'] is the explicit stiffness submatrix related to the contact force at

interface, and {^q} represents a set of initial contact forces.

To satisfy force equilibrium at the interface, the force components expected

by the distinct element domain on the nodes of the abutting finite elements are

given by:

and {gf} becomes an input force bounda/ry condition for the computation of the

finite element domain. Thus, the compatibility of interface boundary conditions of

the two regions can be guaranteed and the accuracy of the finite element analysis

is maintained.

5.3.3 The program structure and convergence criteria

A hybrid computation program, COUPLE, has been developed following the al-

gorithm described in the previous section. The program involves two basic sub-

routines, one is based on the finite element program COAL described in chapter

3 and the other is taken from the distinct element program BLOCK presented in

Section 5.2. The aim of the program design is to provide a basic means for the

numerical analysis of general problems of rock mechanics including all five concep-

tual models shown in Figure 1.2. The program structure and the work involved in

the program development can be summarised as in Figure 5.21.

It is shown in Figure 5.21 that the full nonlinear modelling of rock mass be-

haviour around mine excavations can be analysed by the proposed hybrid computa-

tional scheme. In its current state, the program does not contain any joint elements

212

Answer

Boundary conditions.

Constitutive relation of contacts

Simulation of coupled behaviour.

Interaction of continuum and discontinuum.

Simulation of 3-D excavation.

Geometric non-linear analysis for large deformation.

Material nonlinear analysis for yield,

Convergence of the hybrid solution.

Program COUPLE for hybrid problem analyses.

Numerical modelling of rock movements around mine openings (near field nonlinear problems).

Finite element analysis using subroutine COAL for conceptual models a), b), and d).

Distinct element analysis using subroutine BLOCK for conceptual mode

Hybrid analysis to model coupled behaviour of complex rock masses (such as that characterised by the conceptual model

Figure 5.21 An outline of the hybrid program structure and its developments.

2 1 3

STraigh*-

in the finite element analysis. These could be included, however, by forward im-

plementation of the existing formulations, such as Goodman's zero thickness joint

element (Goodman 1976), for analysis of the rock mass structure represented by

conceptual model b (Figure 1.2b).

The computation sequence for the program is shown in Figure 5.22. and

the program structure follows the design chart of Figure 5.21. The finite element

program COAL and the distinct element program BLOCK are utilised as two

main subroutines with several modifications of the related codes. The majority of

the subroutines utilised in the two programs remain unchanged and all modified

routines and the additional subroutines required are described in Appendix 6.

For viscoplastic analysis of the continuum region, the time stepping loop is

temporarily stopped to update the boundary condition by the distinct element

analysis in which only the static equilibrium of the rock blocks is considered.

It usually takes several hundred iterations for the distinct elements to reach a

new equilibrium after changing the displacement boundary conditions. Here, two

criteria are necessary for the iterations from calculations of COAL to BLOCK and

vice versa.

In subroutine COAL, the criterion for updating the interface boundary con-

dition is the attainment of steady state conditions which are monitored by accu-

mulating some measure of the viscoplastic strain rate in the finite elements, that

r{e„p}] ' « 0. (5.34)

In most nonlinear hybrid analyses, the loading (or excavation) proceeds in a

small incremental manner so that the deformation at the interface is not large.

In this case, equation (5.34) can be satisfactorily used within the finite element

iterations. However, it is possible for large displacement to occur at the interface

due to viscoplastic yielding which results in a separation of the two regions. In this

214

(START }

Yes

Yes INDEX = 2 ?

No

No

No

L Yes

INDEX = 1: F£.M. only, INDEX = 2: D£.M.only,

( Etro )

INDEX = 3? Ym~~I

Change of load anc excava. condition? or converged ?

Outout

D.E.M. analysis by calUng BLOCK

F.E.M. analysis by calling COAL

Input control para -meters & choose analysis method

Store contact forces of interfaces for F.E.M. analysis: FORFC

Loading and excavation control, or convergence check using (5.34)-(5.36).

Update coordinates of interface elements for feeding back to D.E.M. analysis in next cycle, by calling BACK

Figure 5.22 The flow chart of program COUPLE.

2 1 5

case, the criterion can be alternatively given by a limit number of time iterations

to avoid such a separation. It is then necessary to apply the residual stress of

the finite element together with the interface boundary loading change in the next

calculation cycle since equation (5.34) is no longer satisfied.

In subroutine BLOCK, the criterion for the attainment of an approximate

static equilibrium state can be monitored by a measure of either the maximum in-

cremental force changes or the maximum incremental displacement changes within

the distinct elements. In this analysis, the maximum incremental displacement of

all the blocks is recorded during each iteration and used as the criterion for ter-

minating the current relaxations. It is expressed as:

^Stt^Ynax ^ (5.35)

where w is an infinitesimal number, say 10~®. In the practical test of the hybrid

program, another simple criterion which is based directly on the static equilibrium

was found more effective. It can be written as:

|{E/i}7 - { E / a ; | < w'. (6.36)

where {S/^}" is the total contact force at the interface in the nth cycle,

is the initial total contact force at the interface and w' is a small value controlling

the approximate equilibrium. Equation (5.36) has been adopted in the program

in combination with (5.35) to preserve a convergent solution.

216

5.4 Test of the hybrid program

5.4.1 Consolidation of blocks on a deformable body

The problem illustrated in Figure 5.5 of section 5.2.2 has been chosen as one of

the validation tests for the hybrid code COUPLE, but the fixed base in this case

is taken as a finite element domain composed of four 7-node elements (see Figure

5.23a).

In the test, only the self weight of five blocks is considered. The yield strength

of the finite elements is chosen such that these elements remain elastic. Various

methods of iteration were tested, all with the same material properties:

finite element domain:

E = 4000 MPa, u = 0.36, ay = 3.7 MPa,

H' = 0, Von Mises yield criterion.

distinct element domain:

p — 0.005 MN/m^, KN = EL/AH — 2000 MN/m,

Ks = 0, c = 0, / , = 0.

In accordance with the principles of the relaxation procedure, the number of

iterations in each distinct element analysis cycle has to be large, especially at the

start of analysis (i.e. the first distinct element calculation cycle). Figure 5.23b

shows the influence of the number of iterations of eax:h cycle on the solution con-

vergence. In this simple problem, the total self weight of the blocks is 15.32SMN.

When the analysis is carried out with 10 iterations of distinct element analysis

per cycle, the total reaction at the base of the finite elements is 14.617MN, so the

error is about 5%. This error is only 0.1% if the number of iterations is 20 per cy-

cle. The computed force and displacement distribution along the interface is also

dependent on the number of iterations per cycle because the more iterations per

distinct element cycle, the more accurate equilibrium is approached. Figure 5.24

illustrates this path dependency and it shows that the real solution may be satis-

217

i T

D I S T I N C T E L E M E N T S a)

7 - N O O E F I N I T E ELMENTS

1 X & It ji X %

o < IXI

cc 15.3 >-

cc < a 2 o 15.1 CO

< (-o

15.9

14.7

14.5

TOTAL BLOCK SELFWEIGHT = 1 5 . 3 2 8 b)

-m—

ONE COMPLETE CYCLE = D E M + F E M

ANALYSES

10 20 30 40 50 60

NUMBER OF DEM ITERATIONS PER CYCLE

Figure 5.23 Consolidation of blocks on a deformable body, a) blocks and deformable body geometry (D.E. and F.E.); b) calculation solutions.

218

o < O-CO

.56

.52

.48

.44

.40

- A - -

^ • POINT A

^ A / A POINT B

% y

A A

^ 20 30 40 50 60 NUMBER OF D E M ITERATIONS PER CYCLE

Figure 5.24 Influence of iteration number on displacement distribution at the interface.

219

factorily approximated by adopting a large enough number of iterations within a

calculation cycle to maintain the equilibrium state.

5.4.2 A tunnel roof undergoing support deformation

The same geometry for the tunnel roof as in example B of section 5.2.2 was con-

sidered, but in this case the support system was taken as a soft elastic body which

can be modelled by finite elements as illustrated in Figure 5.25. A hydrostatic

stress field of magnitude about ten times greater than the stress at the interface

due to self weight was applied at the boundaries. The simulation was carried out

by applying both boundary load and gravity after excavation, and then estimating

the lower soft body deformations which gave rise to a downwards displacement of

the interface during the process of calculation. In order to investigate the dis-

continuous behaviour of the distinct elements, only the tunnel roof response is of

primary interest in this example.

The material properties of the roof strata are assumed as: p = 2.5 t/m^,

Kj^ = 2000 MN/m, Kg = 100 MN/m, c = 0.001 MPa, / = tan4> = 0.3 and in

situ stress <to = 2.5 MPa. A maximum vertical displacement at the interface was

evaluated as about 0.2 m and the deformed tunnel roof geometry, in this case, is

shown in Figure 5.26. It is seen from the figure that the distinct element method in

the hybrid calculation is capable of modelling the large movements of roof blocks,

and in comparison with the example in section 5.2.2, the significajnt effects of the

support strata deformations on the roof block stability are obvious.

220

Figure 5.25 Test of the program, on the tunnel roof problem.

BLOCK PLOT NUMBER OF PROBLEM U N I T S PER INCH = 0 . 8 0

I K "lUN'NUMHtK = IVUU S C f i C t

Figure 5.26 Calculated roof block movements as the results of the support medium deformation

221

5.4.3 Excavat ion of a square tunnel

The main purpose of this test example is to verify the applicability of the program

for a fairly large practical problem. Before the test, the dimensions of the arrays in

the program were expanded to allow realistic problem geometries to be considered

and subsequently, the program had to be run on the CRAY of ULCC. The potential

of the hybrid program for application to mine design is illustrated by the example.

Problem geometry and material properties

The rock mass geometry under consideration shown in Figure 5.27 was chosen

as being representative of a particular mining environment. As shown in Figure

1,2, the upper layer of the rock mass is characterised as a discontinuum and lower

layer as an equivalent continuum. The excavation proceeds in the continuum

close to the interface at a depth of 600 m and a hydrostatic in situ stress of 15

MPa and gravitational load were applied. A total of 234 rectangular rock blocks

and 126 finite elements were included in the model. The material parameters

for the blocky rock are: KNQ = 3000 MN/m, Kso — 1333 MN/m, c = 0.1,

/ = 0.364, p = 0.025 MN/m^, and the input rock mass constants for finite

elements are: E = 8000 MPa, u = 0.25, p = 0.025 MN/m^, c = 20 MPa,

^ = 40, 7 = 0.002, T — 0.05 and k = 1.5. It should be noted that the idealised

geometry and material properties were chosen for testing the program and were

not necessarily representative of particular rock mass prototypes. The problem

may also be assumed as an excavation in a brick-soil media, hence, the results are

primarily qualitative but are useful in demonstrating the method and the influence

of various parameters.

Loading and excavation

A pre-mining stress field of 15 MPa was applied before the excavation. The

gravitational load, compared to this stress level, has little influence. Figure 5.28

shows a stress distribution in the finite element domain obtained from applying the

222

B L O C K P L O T

I T E R A T I O N N U M B E R =

N U M B E R O F P R O B L E M U N I T S P E R I N C H

5 0 . 0 0 0 0 0 S C A L E . 1 I N C H = _

2 . 2 5 0 0 0

r n

3 3

Figure 5.27 Finite element and distinct element meshes for modelling a rock

mass characterised as a combination of discontinuum and equivalent

continuum.

F.E. REGION TO MODEL SOFTER ROCK MASS AS AN EQUIVALENT flEDlU

THE LENGTH OF PLOT = 3 0 . 0 MPA. BOUNOflRY LOfiO = I S . 0 flPfl

t t t t ^

~ ¥

# # # # # # ? ? f -F -f- + 4-t t t t $ $ $$ t t t $ tttt

$ $ $ $ $$ ##### t $ t t $ $ tttt $$#### t $ + + + + + + + + + + 4- + + + + +

4- +

4- 4-

4- 4-

4- 4-

t $ t $

$ t $ $ $ $ $ - — + •

t $ $ $ $ 4- 4- 4- 4- 4--f- 4- 4- 4- 4-4- 4- 4- 4- 4-

4- 4- 4- 4- 4-

+ 4- + 4- 4-

4- + 4- 4- 4-

+ + + 4- 4-

+ 4- 4- 4- 4-

4- 4- 4- 4- 4-

+ 4-4-

4-

4-

4-4-

4-

4-

4-

4-

+ 4-

4- 4-

+ -f-

+ 4-

4- 4- 4 - 4 - + + 4-4- 4- +

Figure 5.28 Plot of principal stresses in the finite element region showing a

hydrostatic stress field (without considering the influence of discon-

tinuous structure).

223

a)

C O R N E R F O R C E P L O T N U M B E R O F P R O B L E M U N I T S P E R I N C H =

I T E R A T I O N N U M B E R = 5 8 2 . 0 0 0 0 0 S C A L E . 1 I N C H =

12.81800

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 , 1 ' 1 1 1 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 l i l t I l l 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 I I 1 1 1 1 1

I I 1 1 1 1 I I I I I I 1 I I

1 1 1

STRESS D ISTRIBUTION BEFORE EXCAVATION

I N S I T U STRESS = I S . G flPA. CONTACT LOAD BY 0 . E .

LENGTH OF STRESS PLOT = 3 0 . 0 flPA

b )

4 -A-- f -V X A - .

t %%%# * * * < •

yjc y. +

• -v -v

-T-*

t n :zc •V -V-

Jf

Jr

$ % y-y. y-y-y-

-yr t t -f •f

% +

! +

+ + ++ ++++++++++ + + ++ ++-H-H-+f++ +

+ + + +

+ +

+ +

+ + ++ ++-H--H-++++ + + + + ++ ++-H-+++f++ + +

+ + ++ ++++++++++ + +

+ + + + + + + + - + + 4 4 - + + + +

$

-V •V

-V +

+

Jr -V

-V

•V

X X X X X X

%

TL y.

f .

Y.

f.

+ + + + + + ++ +t44"H-H-++ + +

Figure 5.29 Plot of pre-mining stresses (forces) by the hybrid computation, a)

corner force plot of the distinct element region; b) stress plot of the

finite element region.

224

in situ stress at its boundary, and here, the gravitational load is also considered.

This 'hydrostatic' stress field will be disturbed by the presence of a discontinuous

rock mass. Figure 5.29 shows the plots of forces and stresses under the practical

loading condition. It is seen that the stress field in the finite element domain is no

longer hydrostatic due to the boundary load condition at the interface where a non-

distributed pressure was transferred from the block region. However, the resulting

stress field in the finite element domain is still approximately hydrostatic. The

force distribution at the interface of the distinct element domain was calculated

from the consolidation of blocks under the applied loads and the converged solution

was achieved after 582 iterations of relaxation. From the plot of block corner

forces (Figure 5.29a), the difference between the stress (or force) distributions of

a continuum and a discontinuum, particularly at the interface, is well displayed.

The excavation of the tunnel was modelled by reducing the stress and stiffness

of the elements, representing the tunnel profile, in ten steps. The meshes after

excavation are shown in Figure 5.30. In each excavation step, the iterations were

carried out until a temporary equilibrium was reached ajid the interaction between

the block movements and continuum deformation was thus simulated.

Induced stress and displacement

The induced forces and stresses due to the excavation are shown in Figure

5.31 and Figure 5.32. Figure 5.31 was plotted after 99% of the stress and stiffness

of the tunnel had been removed and small arch effects are seen in both roof and

floor. At this stage, the rock mass still behave elastically and no slip occurred

between rock blocks. With further iteration and excavation, a relatively large

plastic deformation occurred in the continuum so that slip between the interfaces

continued and stress (or force) in the blocks was further released. As a result,

as shown in Figure 5.32, a larger influence zone and a much wider arch shaped

distribution of stress and force formed. Corresponding with the above stress and

force plots, displax;ements of roof blocks are plotted in Figure 5.33.

225

B L O C K P L O T N U M B E R O F P R O B L E M U N I T S P E R I N C H

I T E R A T I O N N U M B E R = 5 8 3 . 0 0 0 0 S C A L E . 1 I N C H = _

2 . 2 5 0 0 0

E Z O T l

in in.

n I I I I I m

1- 1- 1- y 1-

Figure 5.30 Excavation of the tunnel by reducing the stress and stiffness of the

elements.

226

C O R N E R F O R C E P L O T

I T E R A T I O N N U M B E R =

N U M B E R O F P R O B L E M U N I T S P E R I N C H =

1 4 0 0 . 0 0 0 0 0 S C A L E . 1 I N C H =

16.20868

V V V

a)

STRESS OlSTRteUTIOK AFTER EXCAVATION

ONE PER CENT SUPPORT LOAD iNSlOE THE TUNNEL

LENGTH OF STRESS PLOT = 3 0 . 0 flPfl

T { -TC-$ + t

: $ - $ — ^

5 X H

y. %

y -y-$ A" 4-4-y-

+ X

+ -V + +

4 -

V. % f.

"f-

X X X

+ "f. + Jr + + +

X WY+AXX/.-H- -f f. X + + + -VJr-V++f+H-V -V X

-H-H-++ -V -V

-H-+t-++ + +

+ + ++ ++++-H-H++ + + + + ++ +f-H-H-H-++ + +

X X X

Jr

A-

X Af Jk" Jr

+

$

4 -+

4 "

4 -

+ 4- ++++44'a(^#-H'+++ 4-

b)

Figure 5.31 Force and stress plot, a) Induced force distribution showing a small

arch in the roof; b) induced stress distribution showing a small arch

in the floor.

227

C O R N E R F O R C E P L O T

I T E R A T I O N N U M B E R =

N U M B E R O F P R O B L E M U N I T S P E R I N C H =

1 8 0 0 . 0 0 0 0 0 S C A L E . 1 I N C H =

40.93729

•>v t"-s s S s S S 4 4

n "-T -J "7 "7 -7 --7 ' s s s 4 4 4 4 4

-1 ^ 1 ^ n 1 -7 -7 -7 -7 '•

-

-v-v—r-A'

I

H-

'-I

' « -V

^ r' v--^ x-^ fr^ r-'r-^r-'r i

fJJJ/Xs'X n 7 ^ S ^ S / —^

7 7 / 7

a) f 1

STRESS O lSTR lBUr iON AFTER OEFORMAriON ANO BLOCK MOVEMENf

0 . 1 PER CENT SUPPORT INSIOE THE TUNNEL

THE LENGTH OF STRESS PLOT = 3 0 . 0 flPA

n If I

X X

X X

Sk. SL

V V

X X.

X ^ X

^

X ^ \ .

X s. Y. "A

:::

"Ik

S. •+-

"f- -f.

T i

/ / / X

X

: : z # # r r r r r r i T : r I I I M I H-

s- -K-W-4'/ ! I H I I f I 4-~f- ~i~~h -H- -f+4-HH- -f-^ - f - - f - f . - f -

^ - f . - f -

^ ^ YJC XWiL"A(.Y."A ^

- 4 - > r

4- Jr

4 A'

+

+ + + + + + 4-4-+4-~W::)(\:fAr)f;r;(XY. +

4-4-

-V

-V

4 - 4 -

n I X X

X

X

X X

i X

X

X

X

X

X

b)

Figure 5.32 Force and stress plot after further excavation and iteration, a) force

distribution showing a wider arch shape in the tunnel roof; b) stress

distribution showing a stress release zone near tunnel roof.

228

TOTAL. DISPLPLDT NUMBER OF PROBLEM UNITS PER INCH = 0.00220

ITERATION NUMBER = 1400.00000 SCALE. 1 INCH =

^ ^ \ \ \ \ — ^ — ~ ^ \ \ \ \ I / X _ . ^

MmimM a)

TOTAL OISPL.PLOT NUMBER OF PROBLEM UNITS PER INCH = 0.00285

ITERATION NUMBER = 1800.00000 SCALE . 1 INCH = —

N \ \ \ / / / ^ - ^ \ A \ ; /

X \ \ / / / / "

b)

Figure 5.33 Plot of displacement of the roof blocks, a) at iteration number 1400;

b) at iteration number 1800.

229

5.5 Conclusions

The computational scheme described allows for consideration of particular prob-

lems such as the conceptual model e (Figure 1.2). For a hard rock material and a

jointed rock mass, the dominant modes of local rock mass response to excavation

involve slip and separation along weakness planes. Any acceptable model of this

behaviour must accommodate rigid body translation and rotation of constituent

blocks. For a soft rock material and a ubiquitously jointed rock mass, a nonlinear

continuum numerical analysis adopting a proper constitutive model is acceptable.

Therefore, for the rock masses with the above hybrid characteristics, the hybrid

computational scheme proves to be computationally efficient and conceptually ap-

propriate. In particular, unlike the other hybrid computational schemes such as

the boundary element - distinct element and the finite element - boundary ele-

ment, the proposed method is to model the large displacements induced by the

excavation in both regions. Moreover, large slips between interfaces are also al-

lowed. The developed program has proven to be able to deal with such a fully

nonlinear problem.

At the current stage, however, there are still a number of limitations involved

in the program and they are mainly the following points:

1. The distinct element representation requires that all rock blocks be consid-

ered as rigid entities. This may appear to place a severe restriction on the

applicability of the method, especially, for problems at depth where a high

confining pressure usually exists.

2. The static relaxation procedure does not appear capable of modelling sepa-

rated block movements and difficulties always occurred whenever such a block

separation took place in the simulation process.

3. The criterion of determining the computing iterations between the two mod-

elling regions was still chosen in an artificial way and hence the solution may

230

be path dependent and sometimes the model may not reflect a reality.

To solve these problems, further study and development of the scheme is

needed and possible solutions are discussed in chapter 7.

2 3 1

CHAPTER 6 A NUMERICAL CASE STUDY

Roadway deformation at Coventry Colliery

6.1 Introduction

This chapter presents a case study using the developed program for analysis of

roadway deformation and stability in underground longwall mining. Typical road-

way large deformation problems at Coventry Colliery in U.K. are chosen as the

demonstration example. A number of laboratory tests of the rock properties and

various in situ measurements have been conducted by the Rock Mechanics Branch

of British Coal at Burton on Trent and also by the Mining Department of Not-

tingham University (Baxter 1987), which provide excellent quantitative data for

comparison with numerical predictions and parametric back analysis. Typical re-

sults of these tests and measurements are presented in Section 6.2, from which

some conclusions are drawn. In Section 6.3, a simplified 2-D numerical model for

the roadway problems at Coventry Colliery is described. Results of the numeri-

cal simulation are presented in Section 6.4 and 6.5, through which the practical

implications of the research axe demonstrated. Finally, conclusions and practical

implications are discussed.

6.2 Descriptions of the roadways at Coventry Colliery

6.2.1 Overview

Investigations undertaken at Coventry Colliery include the roadways in the dis-

tricts of 21's, Si 's and S3's as shown in Figure 6.1. The main and tail gate

232

roadways of these districts axe situated in the middle section of the Warwickshire

Thick Seam. The seam at 21's district is about 5.5 m thick and lies at a depth of

about 750 m; whereas, that at S3's district is thicker (7.3 m) and the depth is over

800 m. The coal seams are overlain by several layers of mudstone and sandstone

and below the coal floor are situated bands of weak seatearth mudstone. Figure

6.2 illustrates a typical geological section of the thick seam which is abstracted

from the strata sections of the colliery included in Appendix 7. The representative

strength parameters of the strata are also shown in the Figure 6.2; the original

data are presented in Appendix 7.

The longwall face is about 3.5 m in height and 230 m in length. Coal was

left in both floor and roof to help stabilize the weak strata in these areas. For

example, the face and roadway floor at 21's district was normally very strong, so

that the entire Two Yard Seam was left to support the incompetent roof mudstone.

However, at S3's district, both Two Yard and Nine Feet coal seams were left due

to the existence of the weak fireclay mudstone below the coal floor (Figure 6.2).

6.2.2 Roadway details

The roadways were formed in-line by the end ranging drum shearer, and had a

height of 3.5 m and a width of 4.4 m width, as shown in Figure 6.3. They were

supported by three piece steel arches set at about 0.76 m intervals. Pack formation

was by the Aquapak packing system using 1.6 m high by 3 m wide bags placed

one on top of the other (see Figure 6.3).

General conditions in the roadways were very poor and considerable arch dis-

tortion and roadway closure occurred. The photo in Figure 6.4a gives an indication

of the floor instability encountered in S3's tail gate. The original roadway height

of 3.5 m was reduced to 1 m in some areas resulting in severe restriction in access

to the face. However, conditions were much improved by the better standard of

packing and the yield offered by stilts.

233

NO.l Pit Road

Road

South

Retttfn

Figure 6.1 Location of districts where measurements were taken, Conventry

Colliery (after Baxter, 1987), plane view.

234

Notes; Strata: Roadway position

Uniaxial compressive strength Og (MPa)

Mudstone, sandstone, etc.

^ 4 0 ^ 4 0 ^ 4 0 ^ 4 0

^30 ^30 ^30 ^30 ^30

Two Yard & others 1.8 - 2.5 m

Ryder & others 1.8 - 2.4 m

Nine Feet & High Main 1.9-2.5 m

Extracted coal, longwall face Might, and roadway position.

£^40

e ^ 3 5

Seatearth, (Fireclay), Mudstone, etc.

c/ 20

Mudstone, sandstone, etc.

40

Figure 6.2 An outline of the geological strata section at Coventry Colliery with

the associated uniaxial compressive strength.

235

Sandstone & mudstone

Extensometer

1.6 m 3.0 m

4.4 m Aquapak packing

Seatearth & mudstone

Figure 6.3 Roadway measuring section in Coventry Colliery.

2 3 6

In situ measurements with yielding steel arches were carried out in the road-

ways of Si 's and SB's districts. The conclusion relating to the use of yielding

arches was that they provided extra resistance and maintained roadway profile

and strength, but created much greater floor lift than that had occurred with

the rigid arches. This can be explained by the 'ground characteristic or required

support curve' described by Brady and Brown (1985).

Floor heave continued to affect the roadway and to date repair has been

mainly by dinting*. Figure 6.4b shows an area where ripping** and dinting took

place, but the floorlift and side squeeze can still be clearly seen. In conclusion,

the deformation of roadways at the colliery is very large and has the character of

significant strata movement of the relatively weak strata around the underground

excavations.

6.2.3 In situ measurements

In situ investigations of roadway deformation and stability were undertaken at

Coventry Colliery. The results obtained are briefly described as follows.

1. Roadway vertical and horizontal closure

The roadway closure was determined by steel tape measurements of the ver-

tical distance between the roof mark and the practical floor and the horizontal

distance between marks on the two sidewalls. The measurements were taken in

different sections of roadway in the three districts. A general of the mea-

surements was that vertical closure was much greater than the horizontal closure.

In some of the sections, the maximum vertical closure was up to 2.5 m. Figure

6.5 shows a plot of roadway closure versus distance of face advance measured in

one of the roadway sections of the colliery. A conclusion was drawn by in situ

observations that the magnitude of the roadway closure at the colliery was mainly

* dinting: re-excavation of floor to keep level constant. ** ripping: re-excavation of roof and adjacent area to maintain height clearance.

237

a)

b)

Figure 6.4 Roadway deformations at Coventry Colliery, a) floor-lift and support

deformation; b) ripping and dinting in keeping the access of the

roadway profile (from Baxter, 1987).

238

Vertical closure (section 11)

, ' - ' X \ Vertical closure

(section 12)

Horizontal closure (section 11) Horizontal closure

(section 12)

Face Distance (m)

Figure 6.5 Measured roadway closure versus distance of face advance, (after Mallary, 1982).

Measured after dinting.

Floor-lift (yieldable arch)

Floor-lift (rigid arch)

50

Face Distance (m)

Figure 6.6 Measured floor-lift in two roadway sections of Coventry Colliery (data

from Baxter, 1987).

239

2.0 -

S. s •o §1.0

o a

0.0

Pack load (section 11) Pack closure

(section 12)

Pack closure (section 11)

o 0.5

100

Face Distance (m)

Pack load (section 12)

Figure 6.7 Measured pack load and closure versus distance of face advance

(after Mallory, 1982).

60

50

c 40

s CO

i " 30 T3 O m

20 -

10

0 10 20 r

40

Anchor point 1

Q. .O cf

Extensometer results

Anchor point 2 "O O- o—

T

50 T

70 80 90

Face distance (m)

Figure 6.8 Measured roof bed separation versus distance of face advance

(after Baxter, 1987).

240

due to the local geology rather than other factors, such as the method of forma-

tion, the system of support and packing, because these factors were identical in

the different measurement sections.

2. Floor heave

The vertical closure of roadways at Coventry Colliery was mainly due to floor-

lift, even in the 21's district where the coal seam was overlain by an incompetent

mudstone strata extending up to over 9 m. A high rate of floorlift often occurred

when the face advance was within 32 m, and the rate decreased after further ad-

vance of the face but showed no signs of stopping completely. This is illustrated

in Figure 6.6 which was replotted from the original data shown in Appendix 7.2.

S. Pack load and closure

Pack load was monitored by the load cells installed in the packs and their

convergence was found by direct measurement from a wire on a strut , that in-

creased in length as the pack closed up. One of the main conclusions from the

measurements is that pack loads along the roadway were not high, the greatest

being 2 MPa. The peak loads were recorded at a distance of between 18 m and

26 m from the face. Figure 6.7 illustrates the relations between face advance and

pack load as well as paxJt closure. It also shows that the maximum closure of the

packs was about 25 %.

4. Roof bed separation

Roof bed separation was monitored by installing aji extensometer in the roof

to a depth of 2.5 m in the S3's district. The results show that the majority of the

bed separation took place between the face plate and 1 meter (or from the roof

surface to 1 m inside the roof strata, see Figure 6,3), whereas less separation was

monitored between 1 and 2.5 m. This is shown in. Figure 6.8 in which the point

1 was at the depth of 2.5 m from roadway face plate and point 2 is at distance of

1.0 m. A decrease in the amount of separation would be expected if the measuring

241

depth was increased beyond 2.5 m.

5. Change of support leg angle

As the face advanced, the legs of steel arches moved in a 3-D nature. The

changes of leg angles were recorded in both planes parallel and perpendicular to

the roadway axis. Some of the results are shown in Figure 6.9.

6. Leg penetration

Leg penetration into the floor was measured by grouting a 1.5 m rock bolt,

which acted as a datum line, into the ribside 1 m above the foot of the support.

The penetration was recorded as shown in Figure 6.10.

7. Arch support load

Loading on the crown of the arch supports was measured by hydraulic load

cells placed around the crown. The measurement results showed that in the centre

of the crown, the loading magnitude was greater than 6 tonnes (points 2,3 in Figure

6.11), whereas it was less than 2 tonnes in the outermost positions (points 1,4).

6.2.4 Conclusions and discussions

Results from the in situ measurements at the colliery are of great use to this study.

They give a good description of the characteristics of roadway deformation and

provided a better understanding of the relations between various factors. Attention

is drawn to the following points:

i. roadway closures were significantly different at the same colliery, due to the

strata properties,

ii. floor heave is the main feature of roadway vertical closure at the colliery,

regardless of the properties of the roof strata,

iii. pack loads were not high, the peak loads of about 2 MPa were recorded at a

face distance of between 18m ~ 26m, and

242

goatSKfe

Angle change (") 8 ••

30 40 50 60 7Q 80

Face advance (m)

Figure 6.9 Measured angle change of support arch leg (from Baxter, 1987).

400 T

300 r-

Penetration (mm)

2 0 0 ••

100

30 40 50

Face advance (m)

Figure 6.10 Measured penetration of support arch leg (from Baxter, 1987).

Load (kN)

70 -

0 10 20 30 40 50 60

Face advance (m)

Figure 6.11 Measured arch support load (cf. Figure 6.3) versus distance of face

advance (from Baxter, 1987).

243

iv. roadway closure continued during the whole process of face advance with no

sign of stopping in most cases.

Although the behaviour of the roadway is clear through the in situ observa-

tions, the mechanism of phenomena such as floor lift and long term deformation at

different conditions is still obscure. Hence, in research or engineering design, it is

difiicult to explain and predict roadway deformation and stability by using these

data. Consequently, as discussed in Section 1.3, a numerical model is required and

in this case, the measurements described previously, provide an excellent base for

checking hypotheses, parametric back analyses and numerical simulation.

6.3 A simplified 2-D roadway model

The first stage in numerical simulation of a rock mass response to mining excava-

tion is the development of an accurate geotechnical model or conceptual model,

since the results obtained from any mathematical modelling are no more reliable

than the data on which the model has been based. If the established model does

not maJce a fair attempt to represent reality, doubts concerning the validity of

subsequent analysis always remain. In this section, therefore, a simplified 2-D

roadway model which is abstracted from the practical information at Coventry

Colliery is presented. The finite element meshes, boundary conditions and mate-

rial properties are described. Modelling of the in situ stress field and practical

excavation sequences is also described and, finally, a discussion of the assumptions

involved in the model is given.

6.3.1 Problem idealisation

Considering the roadway details described in the previous section (Section 6.2), a

typical 2-D cross-section perpendicular to the axis of the gate roadway is chosen

as shown in Figure 6.12. The dimension of the domain to be modelled is taken as

about 11 times of the roadway diameter. The components to be considered in the

244

Mining advance

Subsidence above worked out area

Roof Strata

Support

Floor heave

^ Floor Strata

m 11:111 III III i i m ^ ^

mmmm

Floor heave below worked " oul area

Figure 6.12 Idealised 2-D typical roadway section for numerical modelling.

245

model are roof strata, coal seam, seat earth, goaf, pack and yieldable steel arch.

The goaf consists of voids and rocks detached from the roof and the arch may yield

or sometimes punch into the floor. These two are difficult to model numerically

so that assumptions are made in order to take into account of their effects in the

analysis.

As the longwall face advances beyond the cross section, the support afforded

by the coal face rapidly diminishes to zero and then slowly returns to a certain

pressure as the broken zone around the excavation extends. In the numerical

model, therefore, excavation is simulated by progressive removal or reduction in

modulus of the material to be excavated (see section 6.3.3).

It is assumed that a constant 'reduced in situ stress ' remains in the goaf

(waste area) after the longwall face advances. This remaining stress is taken to be

equivalent to the stress rise in the area, but it is simplified from a probable triangle

distribution (see Wilson 1983) to a constant one. The effect of the yieldable arch

on the roadway deformation is also simplified by maintaining a constant stress

within the roadway profile, which is equivalent to the measured average load on

the arch.

The applied loads are the field stresses which exist before mining takes place

and, because of the absence of field data, it has been assumed that hydrostatic

stresses exist with magnitude p = pH where H is the depth of the roadway (see

Section 1.3). However, the influence or sensitivity to the variation of the horizontal

field stress can be easily determined using the model, and back analysis may also

be carried out if it is required.

6.3.2 Numerical idealisation

A conclusion is drawn from the description of roadway strata behaviour in Section

6.2 that the problem can be modelled by the finite element analysis rather than

246

the hybrid finite element and distinct element analysis. Figure 6.13 gives the finite

element representation of the problem using 231 8-node quadrilateral elements and

754 nodal points. The strata layers with different material properties are assumed

to be horizontal with perfect contact between these layers. No symmetry can be

allowed due to the geometry and properties of the problem. The grading of the

finite element meshes was chosen to be reasonable in terms of reproducing the

main characteristics of the rock mass nonlinear behaviour and being economic to

run as a test example.

An initial hydrostatic stress field of 20MPa is applied in all simulation cases.

Seven layers of different strata plus pack materials are considered in the modelling,

and parameters for different materials used in a series of simulations axe shown

in Table 6.1. This lamination of the calculation domain gives rise to difficulties

of applying a uniform hydrostatic stress field. If the initial loading is applied at

all the boundaries, the calculated stress field will be no longer hydrostatic (Figure

6.14). Therefore, an alternative way of applying the initial loading is adopted.

The modelled region is confined horizontally and the initial load is applied only

in the vertical direction with the Poisson's ratios of all rock masses 0.499*. The

calculated principal initial stresses are plotted in Figure 6.15 and an advantage of

this approach is that a practical stress field in the domain, namely, Oy

can be established. After loading, the proper values of Poisson's ratio, as shown

in Table 6.1, are re-specified for those particular layers of strata and they are used

for the rest of the calculations.

The failure criterion used in the computations is the Hoek-Brown criterion

with the assumptions of associated flow and perfectly viscoplastic post failure

behaviour. Hence, the minimum amount of material constants are involved. The

time dependent behaviour of rock mass is modelled by parameters 7 and r (see

Section 3.4 and Table 6.1), and it is also related to the specific excavation process.

* According to Hooke's law and plane strain condition, Cz = 2uax (see Chapter

4).

2 4 7

Simulation 1: Representative data from the colliery

Strata No. E (MPa) V Oc (MPa) m 5 7 ( l / d a y )

1 2000.0 0.25 30,0 3.0 0.04 0.0002

2 2000.0 0.25 25.0 3.0 0.04 0.0002

3 1500.0 0.25 40.0 3.0 0.04 0.0002

4 1500.0 0.25 40.0 3.0 0.04 0.0002

5 1200.0 0.25 20.0 3.0 0.04 0.0002

6 1500.0 0.25 40.0 3.0 0.04 0.0002

7 1200.0 0.25 20.0 3.0 0.04 0.0002

Pack 2000.0 0.25 2.0 7.0 1.0 0.0002

Simulation 2: Weak floor

Strata No. E (MPa) V Oo (MPa) m 5 '^{Ifday]

1 3000.0 0.25 60.0 3.0 0.04 0.0002

2 2500.0 0.25 50.0 3.0 0.04 0.0002

3 2500.0 0.25 50.0 3.0 0.04 0.0002

4 1200.0 0.25 40.0 3.0 0.04 0.0002

5 1000.0 0.25 20.0 2.0 0.004 0.0002

6 1000.0 0.25 20.0 2.0 0.004 0.0002

7 1000.0 0.25 20.0 2.0 0.004 0.0002

Pack 2000.0 0.25 2.0 7.0 1.0 0.0002

Table 6.1 Material constants used in the computations

248

(continued from the last page)

Simulation 3: Weak roof

Strata No. E (MPa) V <Tc (MPa) m 6 7 ( l / d a y )

1 3000.0 0.25 60.0 0.5 0.0003 0.0002

2 2500.0 0.25 50.0 0.3 0.0001 0.0002

3 2500.0 0.25 50.0 0.3 0.0001 0.0002

4 1200.0 0.25 40.0 3.0 0.04 0.0002

5 1500.0 0.25 20.0 3.0 0.04 0.0002

6 1500.0 0.25 20.0 3.0 0.04 0.0002

7 1500.0 0.25 20.0 3.0 0.04 0.0002

Pack 2000.0 0.25 2.0 7.0 1.0 0.0002

Simulation 4; Weak coal seam

Strata No. E (MPa) V cTe (MPa) m 5 ^ ( l / d a y )

1 2000.0 0.25 60.0 3.0 0.04 0.0002

2 2000.0 0.25 60.0 3.0 0.04 0.0002

3 2000.0 0.25 60.0 3.0 0.04 0.0002

4 1200.0 0.25 20.0 0.3 0.0001 0.0002

5 2000.0 0.25 60.0 3.0 0.04 0.0002

6 2000.0 0.25 60.0 3.0 0.04 0.0002

7 2000.0 0.25 60.0 3.0 0.04 0.0002

Pack 2000.0 0.25 2.0 7.0 1.0 0.0002

Table 6.1 Material constants used in the computations (continued)

249

Node Number

Gauss-Point

Element Number

to o\ o

565 573

.566218

-1567 5 7 4

579219

1580 587

' 568 204 581205

5 8 2 586

570 190 583

584 S89 .a2l_| dSS

20

22 160

a i

578 586

5 9 3 6 0 0

5 9 4

5 9 5 6 0 1

191 5 9 6

597 602 177 598

M AM

'I • " _il_

113 _6S_

52 129

53 6 i :

145

5g 67 '

56 161

57 68'

,591 5 9 9

5 9 2 220

206 6 0 7

5 0 8 6 1 4

192 6 0 9 ,

3 1 0 S I S

178 sn

114 is. aa_ 86

130

87 10(3

146

,89 ,101

90 1 6 2

9 1 102

6 0 4 6 1 2

SOS 2 2 1

506 .613

2 0 7

9 3

179

1 0,, ' 11 , l i t

i» J,

MS 19 133-

120 131

121 134

1 2 2 , 4 7

123 135

124: 6 3

1 2 5 1 3 6

6aft.AU.

aiMt

A

lM,,g ( 3 18 ' )

M no

aw-UB m-..703-708,

U kL-iia.

Am'

r>os 229

71^

201

719 '

" ie« m 711V

|7I5 ,87

• 124 JIL

716

215

.lib

m LUU

h" 141 • 471

Hs. 157

M' 472

173

lint ,473

7 2 1 729 734 742 747

748 722 2 3 0

723 7 3 0

724 216

725 .731

726 202

7?7 73? 7 2 8

Ui-l 188 ,1M

13 ««?

m 1 2 6

<90- SQ< •

491 142 49'2 5 0 5

493 1 5 8

4 94 .506

495 1 7 4

4 9 6 5 0 7

7 3 5 2 3 1

7 3 6 7 4 3

7 3 7 2 1 7

738 744

7 3 9 2 0 3

740, , .745

7 4 1

tOL

323 iU

189 —2ii-

n

12? M8

tit M* M4 {4S

5 2 5 1 4 3

5 2 6 5 3 9

5 2 7 1 5 9

528 540

5 2 9 1 7 5

5 3 0 5 4 1

7 4 9

7 5 0

7 5 1

7 5 2

753 754

367 558 5 5 9

5 6 0

5 6 1

5 6 2

5 6 3

5 6 4

Figure 6.13 Finite element representation of the roadway problem (231 elements and 754 nodal points).

u m i i m m m m a y

a)

Hydrostatic stress loading:

CJH = Gy.

Normally fixed boundary

b)

E i i i i i i i i n i H i i i i

J X

Ov

c)

0 3 O i > a ^

+ +

stiff layer

Intermediate

Soft layer

+ +

Ar +

+ +

The

maximum principal stress,

(Tg. The

minimum principal stress.

^3 < - ^v.

Figure 6.14 Influence of lamination of strata on calculation of pre-mining stress

field, a) hydrostatic stress loading on a bedded material (a vertical

cross section); b) numerical analysis of the problem using the sym-

metric condition; c) indication of calculated sprincipal stresses.

251

6.3.3 Simulation of 3-D excavations

Excavations are simulated using the numerical procedure proposed in Section 3.6.

The gate roads at Coventry Colliery are for advancing longwall faces, so that

after formation they are aflFected by further excavation of the waste area and, as

the face advances, the pack is constructed to support the nearby strata. These

roadway formation and support sequences and their 3-D effects should be taken

into account as much as possible because it has been accepted that the stress path

has a great influence on the nonlinear behaviour of a rock mass. Therefore, the

following excavation sequence is adopted in the analysis:

1. excavate the roadway profile incrementally (say, in 6 steps) to model the gate

road advance (Figure 6.15b(i)),

2. excavate the goaf and pack area incrementally to a certain value to model the

longwall face advance to the section under consideration (Figure 6.15b(ii)),

and

3. excavate the goaf area incrementally to model the longwall face advance as-

suming that the pack has been constructed (Figure 6.15b(iii)).

The excavation factors for different simulations are shown in Table 6.2.

Table 6.2 Excavation factors used in simulating face advance

L o a d o r E x c a . N o . Load o r E x c a . F a c t o r E l e m e n t s E x c a v a t e d

1 1.0 —

2 0.20 52 ~ 60

3 0.04 52 ~ 60

4 0.012 52 ~ 60

5 0.0036 30 ~ 35; 46 ~ 51; 68 ~ 73

6 0.2 30 ~ 35; 46 ~ 51; 68 73

7 0.06 30 ~ 35; 46 ~ 51; 68 ~ 73

8 0.018 30 ~ 33; 46 ~ 49; 68 ~ 71

9 0.009 30 ~ 33; 46 ~ 49; 68 ~ 71

10 0.0036 30 ~ 33; 46 ~ 49; 68 ~ 71

11 0.0011 30 ~ 33; 46 ~ 49; 68 ~ 71

252

= 30

4- 4- 4- 4- + 4- 4-

4- 4- 4- 4- 4- 4- 4-

4- 4- 4- 4- 4- 4- 4-

4- 4- 4- 4- 4- 4- 4-

4- 4- 4- 4- 4- 4- 4-

4- 4- 4- 4- 4- 4- 4-4- 4- 4- + 4- 4- 4-

+ + ++ + +

+ + 4-4- 44-

4- 4- -h4- H-4-

- 4 - 4 - 4 -

-4 -4 -4 -

4-4-4-4-4-4-

4-4-4-4-4-4-

- 4 - 4 - 4 -

- 4 - 4 - 4 -

4 - 4 - 4 - 4 - 4 4-4-4-4-4-4

4 - 4 - 4 -

4- 4- 4- 4- 4-

4- 4- 4- 4- 4-

4- 4- 4- 4- 4-

4- 4- 4- 4- 4-

4- 4- 4- 4- 4-

4- 4- 4- 4- 4-4- 4- 4- 4- 4-

4- 4- 4- 4- 4- 4- 4- 4- 4- t 1 II 1144 [-44-444+ff 4- 4- 4- 4- 4- 4- 4-4- 4- 4- 4- 4- 4- 4- 4- 4- 1 1 II 1144 1 1 mil 1 4- 4- 4- 4- 4- 4- 4-

4- 4- 4- 4- 4- 4- 4- + 4- 1 1 II 1144 II mil 1 4- 4- 4- 4- 4- 4- 4-4- 4- 4- 4- 4- 4- 4- 4- 4- 1 HI H44 -44-444144- 4- 4- 4- 4- 4- 4- 4-

4- 4- 4- 4- 4- 4- 4- 4- 4- 1 1 II 1144 h44~Wt44- 4- 4- 4- 4- 4- 4- 4-

4- 4- 4- 4- 4- 4- 4- 4- 4- 1 1 II 1144 -44-44444- 4- 4- 4- 4- 4- 4- 4-

4- 4- 4- 4- 4- 4- 4- 4- 4- 444444-44 -44-444444- 4- 4- 4- 4- 4- 4- 4-

Figure 6.15a Calculated pre-mining stress field (hydrostatic) in the bedded strata (seven layers).

Roadway Longwall Face

r a c K

R

i-J 4

Figure 6.15b Simulation of progressive excavation of roadway and face, i) gateroad

advance; ii) longwall face advance; iii) further advance of the face

(after construction of the pack).

253

6.3.4 Summary and discussion

A 2-D plane strain numerical model for the roadway problem at Coventry Colliery

has been described. The model takes account of the laminated s trata of the Coal

Measures and their post failure behaviour. The roadway support, including pack

and arch is also considered and simplifications have been made in modelling the

goaf area and the yieldable arches. The initial stress field, which is assumed to be

hydrostatic, is applied and a 3-D excavation sequence is simulated using a flexible

new approach in the 2-D analysis.

Limitations of this 2-D model consist chiefly of the following points:

a. the fully 3-D problem cannot be well modelled by a plane strain analysis,

b. the region considered in the model is much smaller than that in a practical

longwall mining situation, so that the boundary conditions used in the 2-D

model axe only a rough approximation,

c. some of the post failure behaviour such as fracture and strain softening are

not considered in the model.

However, an estimation of the possible errors introduced by the above limi-

tations can be made for guidance. As investigated in chapter 4, a 2-D model may

under-predict the displacement of a 3-D nonlinear problem by more than 10%.

Nevertheless, the artificial stress boundary conditions in the model would possibly

over-predict the displacement to the same extent. This is also the case when mod-

elling the rock mass post failure behaviour. The assumption of a perfect plastic

stress- strain relation will give an under-estimation of the brittle failure of rock

masses, while the associated fiow rule assumed in this model is known to over-

estimate the dilation and the displacement of rock masses. Therefore, the global

effect of all these assumptions may only produce a few discrepancies between the

model and real problems.

Although there are questions concerning the validity of a 2-D numerical anal-

254

ysis in modelling 3-D fully nonlinear problems, the above considerations and the

investigations undertaken so far in this study have lead to the conclusion that the

proposed case study model should be able to simulate and analyse the character-

istics of strata movements around roadways.

6.4 Computation results

In this section, computed results of several numerical simulations are presented.

Typical results computed using representative strata parameters are discussed in

detail, and then different material parameters are used in a series of simulations

to model various strata conditions. The influence of geometric nonlinearity on

roadway deformation is investigated, and the discrepancies between the predictions

using infinitesimal theory and using large deformation theory are described. The

results are presented and interpreted in the forms of graphics, plots and tables.

Finally, conclusions concerning the work are presented.

6.4.1 A representative simulation —simulation 1

The material constants used in this simulation are those representative values of

the colliery shown in Table 6.1. Roof strata axe the remaining coal seam and weak

sandstones, the floor consists of coal seam, weak sandstones and seat earth. The

strength of the coal seam is generally greater than that of the roof and floor. For

the small strain analysis [NLAPS = 1), eleven excavation steps are used and

the excavation factors are shown in Table 6.2. For the analysis considering large

strain and large displacement {NLAPS = 2), the excavation must be undertaken

using much smaller steps (Section 3.5), so that the above eleven excavation steps

are further divided into 117 steps. The results are interpreted as follows:

255

Stress analysis

In predicting stress redistribution during or after excavation, the analyses

using small or large displacement theory {NLAPS = 1 or NLAPS = 2) give very

similar results. As the roadway and longwall face are excavated progressively, the

computed principal stress redistribution is plotted as shown in Figure 6.16a~ i.

The region of interest is the rock mass around the roadway; the left side pillar is

considered for the purpose of numerical stability and convenience.

Figure 6.16a shows the principal stress distribution before the roadway face

reached the section, whereas the stresses shown in Figure 6.16c are that after the

face had passed away from the section. Approximately zero normal stresses around

the roadway surface have been established by simulation of the excavation. Figure

6.16d,e,f are the stress plots when the longwall face including the pack region has

advanced, and the remaining stresses in the excavated region in Figure 6.16f are

assumed to exist due to the longwall face support. Figure 6.16g is of the situation

when the support near the roadway is removed to construct the paick and from

this stage the longwall face is to pass away from the section. As the longwall face

advances, the stress in the pack is increased to its maximum value, which is about

2 MPa (Figure 6.16i).

Taking three reference lines inside the crown, ribside and floor of the roadway,

the stress distributions {a^^Oy and <7 ) along these lines are plotted in Figure 6.17

in which the nonlinear nature of these stresses is clearly shown. The stresses far

from the roadway are approximately equal to the premining stress field, namely,

20,0MPa. It can be seen from Figure 6.17 that the stress concentration in the

ribside is much higher than that in the crown or floor. The maximum stress in

the ribside is about SOMPa {oy) which is at the elastic and plastic interface, at a

distance of about 2.2m. At the Gauss integration points near the roadway surface,

the normal stress in the crown and sidewalls is higher than 2MPa (points a ajid

b) whereas it is much lower at the floor due to the geometry (point c).

256

+ + + + + + + + + + + +++4-++-H-H-+ + + + 4- 4- +

+ + + + + + + + + ++ ++++-H-++-H- + + + + + + + + + + + + + + + •f ++ ++++++++++ + + + + + + +

+ + + + + + + Y-V. ¥-V-+-f-H-VJrJr-V -v •V + + + + + + + + + + + Y- y.xxx-fxxjc-vrir Jc -V -V + + +

+ + + + + + + •f Y- y-X x x A p W + y x x x •>c A- + + + + +

+ + + + + + + Jc A- + + + + + V- Y- / X X J ( - J t + - f + K - * X X Jc -V + Tt XX->c+-hfT XX k -V

+ + + + + + + -V •V\)«w.-W-Wr WW/ - 4 + + + + + + + + + + + + + + -V J C ^ > 0 0 0 < . T I - + 4 ^ - > < W X X i- V- + + + + +

+ + + + + + + +" -v % + + + + + + + + + + + + V -v J c - V ^ X K x-)<.+>c Y- + + + +

+ + + + + + + + -V i- + + + + + + + + + + + + -f-Y- + + + +

+ + + + + + + + + + + + + + + +

a) Roadway face position 1: before reaching the section.

+ + + + + + + + + + + + + - H - + + - H - + + + + + + + + +

+ + + + + + - f f - y . V . - H - H - W - A r J f J r -V + + +

+ + + + + + Y- Y- •/ . ><.><. y . T i . - t O f j f j f W J f -V -V -V + + +

+ + + + + + V- y - y . s x x x ^ + + y . x x x x -V + + + + + + + + + + y - / X X x j f - \ r + + f y . x x x X A- + + +

+ + + + + + + Y- y . X X X J c - H - ^ + Y - y - X X X + + + 4- + + + + V- / X X X J r - H - + + - / - - > < . x x X •>r + + + +

+ / X X ) ( -v

•f -f ^ -M-W-SXXX ->c -V X > ' - M - M - y X X X -!r

+ i

t

yxxx -V Jr

4 - + + + + + + + y - + + + + + + + + + + + + + Jr > c X > o o y . - M - W - x « i » < X / • f + + + +

+ + + + • + + - v JC X X X K y / . + f + t ' A » ( W ( X X y - i- + + + + + + + + + + + j ( x x w y y . - w - H r - ' c ^ f x x X y + + + +

+ + + + + + - v J f X x x w x x ) o » » # ( x X X • f + + + + 4 . + + + + + -V -V X x x > o o < x x y - v - x >o«««<x X X V- + + + +

4- + + + + + + + -V x x > « < . y - + + + - H - w « < x y - - f + + + + +

b) Roadway face position 2: at the section.

Figure 6.16 Plots of principal stresses around the roadway, a), b), c): Stress distributions as

the roadway advances; d), e), f): stress distributions as the longwall face advance

(excavation includes pack area); and g), h), i): stress distributions as the further

advance of longwall face (after construction of the pack).

2 5 7

(Continued)

+ + + + + + + f- i- y-x •>fJr-H-f+-Mxx Ar + + + + + +

+ + + + + y. X XX xxxxyxxxxx X X A- + + + + + + + + + % X XX xxJfV+x-xxxx X X A A- + + +

+ + + + + y- X X XX XJr-H-f+XXXX X X A A- + + + + + + + + + ¥ • y X XX xA--H-H-y-y-xx X X A Ar + + +

+ + + + + + y- X XX A-Jr-H-W-itX-XX X X A- + + + + + + + + + + •f-/ X X X-Jr-H-W-YaiXX X A- + + + + +

Y- y- X X A-Jr-H-f-- iCJ<.XX \ -V + y- XX XX < Jc ' 4:

+ + + + + + + + + + + + + +

+ + + + + + A-+ + + + 4- A- A-+ + + + + A- A-+ + + + + A- Ar + + + + + + +

-V Jc X XXXAL-H-W-A%)WX /

f -l-+ + + + + + -f + + + + +

•f + + + +

X V- + + + +

X V- ¥ • + + +

X y- + + +

y- + + + + +

c) Roadway face position 3: in front of the section.

+ % Jr + + + ++ -f-f-H-N-Aixx A- + + +

+ + + + + +

+ f- i- % X

X X

X X

X X

X

X

X

X y. % $ X

$

Ar + ++ i-TL-Atyxxxxx X A-A" Jr ++ •fy-%x.xxxxxx X -X X Jr ++ +Y-sy.xx.xxxx x \ X X- ->r+ ++'Aiyx.xxxx X -V X X ->r-v 4-4--w-yx.xxxX X \ X X XJr -M-W-AXXXXX ^ -V •i- y. XX -t-+-(--«x>xxXH \ -V y. / X X Jr-I—M--<x.XXX\ -V 4

-v •v

•V -V

4-+

4-+

+ + + + + +

+ A X y. X X A- Ar A X X W A k - - f + + + + + + + X X X X X A- X X x x v y ^ H — i - -h + + + + +

+ A X X X X X X X y.y<+H+A4rX3fW»XX / i- + + + + + + A X X X X X X -x. y--/-f-H+A-A-xxx>«»<X X + + + +

+ A X X X X X X y. -H-HAA-XXXXOWX X X •f + + +

4- A- A X X X X y- y- •f++HA-A-xxxw««<x X X + + + + +

+ + + A- X y- + + + -H-+H-H+A4-AW(X y- + + + + + +

d) Position of longwall face 1: 10 m in front of the section.

Figure 6.16 Plots of principal stresses around the roadway, a), b), c): Stress distributions as the roadway advances; d), e), f): stress distributions as the longwall face advance (excavation includes pack area); and g), h), i): stress distributions as the further advance of longwall face (after construction of the pack).

258

(Continued)

X Jr 4- +

X X X X -V

X X X X -V

X X X X A-

X X X X 4r

X X X X X X X 4r / -V X X X 4r X X

$ X X $

< 4 ? f t

I ^ X \ X \ X

X

X X

I I i i

v. y. V.

X v.

X X

X

X

+ + ++ -V +

+ +T^y-yxxxxxxx \ \ + + V-V- XXXXKXXXX 5 \

4- + -f-V- nx-xxxXXXHX \ >C + + -Y-V- -!<-•*. XX XX XX XH ^ \

+ +-+- XXXXXk + -H-H -#.'*.-*XVXXXH\

A- + •V \

\

-V

+ +

+ + + + % *

I i

+ +

+ +

+ + + +

+ y.

1 ^

+ + + + + +

Tt + -f- + + + I t H f +

f + + S. + 4-f + + -V S +

y- + + + 4- 4-

/ + + 4- 4- 4-

/ Y- + 4- 4- 4-

f + 4- + +

+ + + 4- + +

e) Position of longwall face 2: 23 m in front of the face.

•/• X - ¥ • 4- - + -

X X X •Ar 4-/ X X Jr Jr

/ X X .V 4-/ X / X .4-/ X

1 X

*

\ \ \ X \ X

\ \

V X X X

I I

X

•K

4- + -Y-S- Tt-AXXXXXXXX X + + Y-V. XXXXXXXXHX X

+ s- v xxxxxXXXX \ 4- -+- -k-*. xxxxxXXWK \ + -f- t-Y. - .xvxxxWW \ -I- •*- - xvxxxXXW \

I 1111 t 4 44-1

-1 -+-

-V- /

• 4- .wAveyAy/yxfWX /

4- + +

+ + 4-•V + +

-V + 4-•V + +

+ + X + + % + Jr •f X -V t

$

+ + + y.

-V +

i:

+

4-+

Jr

4-4-

y. y- -h-h 4—t-1 \ + +

+ + X +• + 4-+ + + V- + 4-•f + + •X •X. + / + + + 4-

/ + + 4- 4- 4-+ + 4- + +

+ + + + + +

f) Position of longwall face 3; 36 m in front of the section.

Figure 6.16 Plots of principal stresses around the roadway, a), b), c): Stress distributions as

the roadway advances; d), e), f): stress distributions as the longwall face advance

(excavation includes pack area); and g), h), i): stress distributions as the further

advance of longwall face (after construction of the pack).

259

(Continued)

; f I + ; +

i +

il

I +

! 4-

Jr +

+ X 4-

/ X X X

/ X X X

/ / X X

/ / X X

/ / X

/ / X

X X

K -

\ \ \ X

H \ \ X

•X- -V 4- 4- 4-H—M- x >( -V +

A- -A- 4- 4- -f--A IW.'XX.XXXXXX X ^

i t

-V Jr-Jf A-

+ •+- X.XXXXKX\X\ \ \ + + f-'A. xxxxxxXXXH \ \ + •*- •»-xxx\XX\H\ \ \ + + \ -V

1 i ^ S.

X X Ik s.

X X Ys -f.

y. -f- ~h -+- H—t WWW-^^x^i- -h f

+ +

+ + +

v-•v +

1:

+ + X + + + f + -V + 4-

•/• + + X % +

•f + + + 4-

/ + + + + +

/ + + + + +

+ i + + + +

g) Position of longwall face 4: 50 m in front of the section.

— • (THE PACK MAS BEEN CONSTRUCTED 1

-V -V 4-

A-+

f X 4-

/ X X X'

/ / X X

/ / X X

/ / / X

1 / / X

X"

X -

I H X \ \

X \ \ \

V ^

h)

4 (- 4—(-4--f--f+ -Ttj<->ocXJc -V +

Jr 4-.y 4r-- r -V-

4- -f- -rL-iL YCX.'WXXXX X X X \ + -f- v^xXxvXXXXXX X X

•+- -i- -K x xxX* ,XWX\X X \

+ xxXXWWW \ \ +- -»- xxXsVXWW \ \

-vxxxxwxw \ 4

X X ^ X X X X 4-

X X -f_

4—I-4-HWPW^XWT' -f

+ -f- + 4 - 4 -

i- + X + 4 -

+ 4- •A +

/ + + X X X

/ + + + + +

/ + + + + +

f t + + + +

Position of longwall face 5: 70 m in front of the section.

Figure 6.16 Plots of principal stresses around the roadway, a), b), c): Stress distributions as

the roadway advances; d), e), f): stress distributions as the longwall face advance

(excavation includes pack area); and g), h), i): stress distributions as the further

advance of longwall face (after construction of the pack).

260

(Continued)

THE nnxinun PACK loho of 2 npn is thus hooelleo

(STRESS PLOTTED AT THEIR ORIGlONflL POINTS (UNDEFORHEOl

• f : f ; f

: i i f

% •V

•V

I «

f / Jr 4- 4 (- - ) — \ + / / X A" -v- 4- -+. s- -xacocxx W X X k \ / / X ^ -Jr "Ik"*. xxXx XXWW k \

/ -t- ^ -x-X XXX\W\\\\ \ \ X ^ -- -- ^x xxxWWXW \ \

'^x\x\\W\\ \ ^ ^ -Y

\ \

\ \ \ X X X, X -f-

-+- -*- • « ^ / ' y / / / / / /

I -t • H44rVPWXX)«AWf i f

+ + + + +

+ + • + + +

+ + + +

-V + + Y- •f + + Jf +

+ •A -V -V + + X + +

1 1 1 4: 1 1

- f -h 4 - 4 -

+ + + + 4 -

+ •V y - + +

+ + X X

+ + •V + +

+ + + +

t 4 +

i) Position of the longwall face 6: 100 m in front of the section.

Figure 6.16 Plots of principal stresses around the roadway, a), b),c); Stress distributions as the roadway advances; d), e), £): stress distributions as the longwall face advance (excavation includes pack area); and g), h), i): stress distributions as the further advance of longwall face (after construction of the pack).

261

i g L ia ^

Wr -vr - . __'__.T_TTfer_"_a • 0 ^

^ » n <

-0-—

Vertical stress, Oy.

Horizontal stress,

Axial stress (out-of-plane stress),

The sections where stresses values are plotted.

Figure 6.17 Stresses distributions inside the crown, floor and ribside of the roadway (simulation 1).

262

As a comparison, a prediction of the stresses by linear elastic analysis, con-

sidering the same excavation sequences is shown in Figure 6.18. Also in Figure

6.19 the elastic stress distribution at the reference lines of Figure 6.17 are plotted.

A tensile stress zone occurs near the roadway floor where the axial stress becomes

the maximum principal stress (shadowed part in Figure 6.19). These results give

obvious indications that significant discrepancies would exist if the stresses of a

nonlineeir problem are predicted by elasticity theory.

ELASTIC ANALYSIS OF THE STRESS OtSTRlBUnON

+ y ^ -Jr 4 - 4 - -V -V

X X X

X X

X

X ^ X

-V- •+- \ \

^ •*r -h-

•j-

\ \

\

X X -f-

X -1k -K X ^ -h

- t - / "/•

4 — / /

+ + + +

-V + + +

A- + + +

+ + •f + + X

t + X -V T % A- 4-

1 § 1 4 - -

f + + +

-f -A -f X % A- Jr

+ + + +

+ +

- f +

+ 4-

+ +

-V +

+

+ i +

Figure 6.18 Plot of principal stresses calculated by elastic analysis (same excavation

procedure is used).

263

.

- Vertical stress, a,

Horizontal stress, a

-B— Axial stress (out-of-plane stress),

The sections where stresses values are plotted.

Figure 6.19 Stresses distributions inside crown, floor and ribside of the roadway predicted by elastic analysis (c.f. Figure 6.17).

264

Deformation analysis

The roadway deformation process is well simulated using the proposed model.

The computed progressive deformations are plotted by the program during the

excavation sequencing and they are shown in Figure 6.20a ~ i in which development

of plastic zones around the roadway and the longwall face are also illustrated. It is

noted that failure did not initiate at zones of the highest stress concentration such

as those near foot corners, but where large deviatoric stresses exist (see Figure

6.20a). In some places, the 'outer layer' weaJcer rock may fail first when the 'inner

layer' strata is still elastic, so that local elastic zones may exist in the plastic zone

(see Figure 6,20e ~ 6.20i). After the final stage of excavation, the diameter of

the plastic zone has extended to more than 6 times that of the original roadway

'diameter', and the vertical closure is 36% of the original roadway height. The

process of roadway deformation also shows that most of the displacement is due

to the advance of the longwall face, and that the advance roadway, ahead of the

longwall face, created only about 10% of the final deformation.

The displacements of the roadway at the crown, sidewalls and floor are plot-

ted against excavation step in Figure 6.21a,b,c where the process of deformation

is illustrated. The analysis of displacement considering the geometric nonlinear-

ity {NLAPS = 2) gives significantly lower values compared with the geometric

linear analysis {NLAPS = 1). Like the plastic analysis of the cantilever problem

described in Section 3.5, as the rock deformation becomes larger, the nonlinear

behaviour of the roadway geometry has a 'stiffening' effect on the rock mass and,

therefore, reduces its deformability. In the analysis using large deformation the-

ory, although only 55 excavation steps out of a total of 117 were executed, this

'stiffening' effect is shown well in Figure 6.21a, and 6.21b. When only the roadway

is excavated (steps 1,2,3 and 4), the deformation is relatively small, hence, both

analyses {NLAPS = 1 and NLAPS = 2) give similar results. However, a rela-

tively large deformation builds up when the longwall face is excavated (start from

step 5) and in that stage, a significant discrepancy between the analyses occurs

265

a)

b)

c) m

Figure 6.20 Plot of mesh deformations and plastic zones, a), b), c): deformation and plastic zone as the process of excavation of roadway profile.

266

(Continued)

d)

e) uz

m L

1

uz

m % > —

m m

f)

1 L m n

wJ//I//A TZ : —m

— —T z 0 I —

1 1

. __

Figure 6.20 Plot of mesh deformations and plastic zones, d), e), f): deformation and plastic zone as the process of longwall face advance.

267

(Continued)

g) % 12^97// K

—

1 1 1 i 1 i

h)

f

i)

mm/mm/M Ui////Ah7f///A^hr/n

Figure 6.20 Plot of mesh deformations and plastic zones, g), h), i); deformation and plastic zone as the process of further face advance (after construction of the pack).

268

(see the diflFerence between dashed lines and solid lines in Figure 6.21).

An interesting phenomenon is noted in Figure 6.21b for excavation step 6,

whereby the floor lift predicted by large deformation theory is greater than that

predicted by small deformation theory. This is in contrast with the results shown

in Figure 6.21a and 6.21c. In fact, it is a good indication of the buckling effect

on the results because only large deformation theory can predict this phenomenon

(see Section 3.5). It can be explained from Figure 6.21b that after excavation step

5, the floor lift is affected by a buckling factor due to the high tangential stress

and a further release of normal stress. After excavation step 7 (about excavation

step 50 in the large deformation analysis), the 'stiffening' effect dominated again

and the dashed line in Figure 6.21b gives a lower value. In this case, therefore, a

smaller difference is found in predicting the floor lift using the two theories. If the

floor is much softer (or weaker) than the roof and seam, it is expected that there

will be more buckling effect in the analysis of large deformation theory.

Time dependent behaviour

In the analysis, plastic deformation of a rock mass is assumed to be time

dependent, and after each excavation step a certain amount of time must be given

to allow the rock mass to deform plastically. These time intervals are controlled by

the parameters 7, k and T as discussed previously, and they are also controlled by

the maximum number of iterations specified for each excavation step. Therefore,

computed time dependent behaviour in the analysis involves both the excavation

effect and viscous rock mass property. Table 6.3 lists the computed time at each

excavation step predicted by the analysis and the total time needed for the flnal

deformation is 167.7 days. The computation stopped when the time dependent

deformation was still occurring in a convergent manner (convergence code = 1).

269

Calculation (plot) point

E u ^ 2.5

tS 2.0 g ©

8 1-5 (0

S E s m 0.5 Q. w ° 0.0

Face advance

NLAPS = 1

NLAPS = 2

Pack is - -a' constructed

I-

Lonwall advance

3 4 5 6 7 8 9 10 11

Excavation steps (Roadway and longwall face advance)

Calculation (plot) point NLAPS = 2

Face advance

NLAPS = 1

Pack IS constructed

Lonwall advance

3 4 5 6 7 8 9 10 11

Excavation steps (Roadway and longwall face advance)

Figure 6.21 Roadway deformation versus excavation steps (face advances) — comparisons of large deformation predictions and small strain analysis , a) crown displacement vs. excavations; b) floor-lift vs. excavation; c) ribside displacement vs. excavations.

270

(Continued) c)

4.0

I 3.5

" 3.0 L. o o •2 2.5 o 8 (2. 2.0

c 0 1 U a w 1.0

0.5

0.0

/ \ Calculation I 8 (plot) point

NLAPS = 1

Face advance

, ^ t i _ , . Pack IS constructed

x " »-

^ NLAPS = 2

Lonwall advance

3 4 5 6 7 8 9 10 11

Excavation steps (Roadway and longwall face advance)

— o — Small strain analysis: NLAPS = 1

- -Q - - Large deformation analysis: NLAPS = 2

Figure 6.21 Roadway deformation versus excavation steps (face advances) — comparisons of large deformation predictions and small strain analysis., a) crown displacement vs. excavations; b) floor-lift vs. excavation; c) ribside displacement vs. excavations.

Exca. step 1 2 3 4 5 6 7 8 9 10 11

Exca. factor - 0.2 0.04 0.012 0.0036 0.2 0.06 0.018 0.009 0.0036 0.0011

Time (Days) - 0.3 1.0 2.4 6.3 24.5 39.8 60.7 90.1 151.2 167.7

Table 6.3 Computed time with reference to each excavation steps — simulation 1

271

6.4.2 Weak floor strata —simulation 2

Roadway deformation, assuming that the roof and coal seam are fairly strong rock

masses ajid that he floor strata are sequence of weaker strata, such as seat earth

underlain by weak mudstones and heavily fractured weak rock, was investigated.

The strata parameters are listed in Table 6.1b and it is seen that the in situ

uniaxial compressive strength of the floor strata given by the Hoek-Brown criterion

is only about 1.2 MPa.

Computed stress redistributions are similar to that of Figure 6.16 except for

the magnitudes near the roof and floor. Significant differences are found in the

deformation and plastic zones. The roadway closure and the development of the

plastic zone after the 11th excavation is plotted in Figure 6.22a,b. A large plastic

zone occurred in the floor while only a few shallow local plastic zones are seen in

the roof. The floor heave is well simulated in the model and the vertical roadway

closure is about 55% of the original roadway height. The deformation of roadway

surfaces is plotted against the excavation steps in Figure 6.23. As indicated in

the figure, the displacements at three points o, p, q on the surface were recorded

as the roadway and longwall face advanced. The floor lift is shown dominating

the roadway closure. Crown displacement is only 13% of the vertical closure and

horizontal closure of the roadway is not large. It is also well indicated by the curves

that floor lift increases continuously with no sign of stopping, which corresponds

to an extending yield zone in the strata (Figure 6.22a,b), but the displacements

of crown and ribside decrease gradually if there is no further development of the

failure zone. The large deformation theory, in this case, gives an obvious prediction

of a buckling phenomenon of the weak floor as shown by the dashed line in the

figure. The time period of plastic deformation that is predicted corresponds to

each excavation step and some of the values are listed in Table 6.4. Compared

with simulation 1, a longer time period is needed to obtain the predicted roadway

closure.

272

a) mrm J —

r r v TTTT m y I — h —

Vr-nr-. r m f / f / t — 1 'A I 3

Excavation step 8, time: t = 113 days.

b)

Excavation step 11, time; t = 320 days.

Figure 6.22 Calculated strata deformation and plastic zone in a weak floor case simulation 2, a) at excavation step 8; b) at excavation step 11.

273

8.0

7.0

E 6.0 o M T— II

o 5.0 u C8 O s <0

c 0) E s m Q. w h

4.0

3.0

2.0

1.0

0.0

NLAPS = 2 (point q)

NLAPS = 1 (Point q)

> Point p

.xy-

A -

2 3 4 5 6 7 8 9 10 11

Excavation steps (Roadway and longwall face advance)

Figure 6.23 Roadway deformation versus excavations in a weak floor strata case, — simulation 2.

Exca. step 1 2 3 4 5 6 7 8 9 10 11

Exca. factor - 0.2 0.04 0.012 0.0036 0.2 0.06 0.018 0.009 0.0036 0.0011

Time (Days) - 6.0 11.0 20.0 70.0 95.0 105.0 113.0 143.0 177.0 320.0

Table 6.4 Computed time with reference to eaxzh excavation steps — simulation 2

274

6.4.3 Weak roof strata —simulation 3

In this simulation, the roof strata are considered as a relatively hard but broken

rock mass. Hence its uniaxial compressive strength is assumed as 50 ~ 60MPa

and the parameter m and s are taken as 0.3 and 0.0001 respectively. The In situ

rock mass compressive strength of roof strata in this case is only O.SMPa according

to the Hoek-Brown criterion. Material constants for different strata are listed in

Table 6.1c.

The process of stress redistributions is again very similar to the previous two

simulations. Taking a vertical reference line in the roof and floor, the computed

values of stresses axe plotted in Figure 6.24. It is noted that a high stress zone

(stress concentration) exists within the plastic zone near the surface of the crown.

This is considered to be caused by the increase of a support load within the

roadway, and also probably by the arch shaped boundary. However, the stresses

in the roof and floor are not high compared with that in the rib side (cf. Figure

6.17). In order to understand the stress path as the excavation proceeds, stresses

near the surfaces of the crown and floor are plotted versus the excavation steps

in Figure 6.25. In the figure, the eff'ects of longwall face advance (start from step

4) is clearly illustrated and it is revealed that the induced stress changes, at these

positions, in a significantly different manner.

The calculated roadway deformation and plastic zone are shown in Figure

6.26, where only two computer plots are presented. Displacements of the crown,

ribside and floor surface are illustrated in Figure 6.27 in the same form of Figure

6.21 and Figure 6.23. Although the roof strata are much weaker than the floor and

a much larger plastic zone has been seen in Figure 6.26, the floor lift is still larger

than the crown displacement. This explains very well the general phenomena of

roadway deformations in coal mines and the influence of the caving shapes.

275

&

(MPa)

- Horizontal stress

- O — Vertical stress Oy;

- A — Axial stress (out-of-plane)

Calculation section.

Figure 6.24 Stress distributions in the floor and roof, — simulation 3.

276

vertica stress o

Axial stress a Point of observation O 15.0 (out-of-plane)

Honzontal stress a

Excavation steps (Roadway and longwall face advance)

b)

CD 20.0 Vertica stress a

OT 15.0

5510.0

Point of observation

Axial stress o.

/ (out-of-plane)

Horizontal stress a.

3 4 5 6 7 8 9 10 11

Excavation steps (Roadway and longwall face advance)

Figure 6.25 Stresses versus excavation steps at two fixed points, — simulation 3, a) the stress paths at a point in the roof; b) the stress paths at a point in the floor.

277

a)

At excavation step 8

Km

• 'WM mm m » VLLUJ #

% 4 T/ ^ 1

m f

b)

At excavation step 11

Figure 6.26 Calculated mesh deformation and plastic zone for a weak roof case, — simulation 3.

278

5 *

4.0

"e ^ ^ u

II 3.0

O o « 2.5 w s W 2.0

c E 1.5

8 OT

0.5 f-

0.0

Floor-lift (point q)

Crown displacement, (point o)

° ^ Diho; Ribside displacement (point p)

2 3 4 5 6 7 8 9 10 11

Excavation steps (Roadway and longwall face advance)

Figure 6.27 Roadway deformation versus excavation steps for a weak roof case, — simulation 3.

279

6.4.4 Weak coal seam —simulation 4

E the coal seam is the weakest strata, and contains discontinuities, how will the

roadway deform? This simulation is designed to answer the question. Material

parameters are assumed as listed in Table 6.1d. The uniaxial compressive strength

of the intact coal is 20MPa, so that with the assumed Hoek-Brown empirical

parameters, the in situ compressive strength is only about 0.2MPa, which can

be considered as the average strength of the jointed rock strata. The roof consists

of stronger rocks, whereas the floor has 1.7 m of the remaining weak coal seam.

Excavations were identical to those in the preceding studies and in each step,

the principal stresses and updated meshes are plotted. The results show that a

much larger horizontal stress reduction area is developed corresponding to the

plastic zone. Conferring with Figure 6.17, the stresses in the ribside are plotted in

Figure 6,28 and they show a much different distribution. The maximum vertical

stress is only about 40MPa, which is about half of that in Figure 6.17. A local stress

concentration exists close to the ribside surface, and this is due to the provision

of a support load within the roadway. Two of the mesh deformation plots are

presented in Figure 6.29a,b from which the development of a horizontal yield zone

is clearly shown. After excavation step 10, the roadway closure is almost 80% and

the roof and floor in the goaf area deform to such an extent that part of the floor

touches the roof. Although the results are affected by the boundary conditions,

the trends of the development of the failure zone are well manifested. The lateral

ribside displacement is predicted to be larger than the floor lift, ajid crown closure

is small. This is also demonstrated by Figure 6.30 in which the displacements at

three reference points are plotted versus excavation steps.

280

S. 40 s

/ \ ^ y

30-

9a Distance to tlie ribside wall (Gauss points)

horizontal stress.

Oy! vertical stress.

Gz: axial stress

(out-of-plane).

Figure 6.28 Stresses in the ribside after excavation step 7 (in a weak coal seam case) — simulation 4.

281

a) urmm

m m m m

// y m W// \ r i

At excavation step 7

b)

At excavation step 10

Figure 6.29 Calculated mesh deformation and plastic zone for a weak coal seam case, — simulation 4.

282

i

18.0

E o N

15.0

OB 12.0

o 8

I 9.0 O o (0 Q. OT Q

6.0 h

3.0

Ribside displacement (point p)

0.0

Floor-lift (point q)

Crown displacement (point o)

2 3 4 5 6 7 8 9 10

Excavation steps (Roadway and longwail face advance)

Figure 6.30 Roadway deformation versus excavation steps for a weak coal seam case, — simulation 4.

283

6.4.5 Conclusions concerning the numerical s imulation

A number of numerical simulations of roadway deformation at Coventry Colliery

were conducted. The first simulation used representative rock mass parameters

abstracted from test data. The stress redistribution in the process of excavation

was presented. Failure of the rock mass and its post failure behaviour was pre-

dicted. The calculated roadway deformation and the plastic zone were plotted in

accordance with each excavation step. Time dependent behaviour was also de-

scribed. The effect of geometrical nonlineaxity in the large deformation problem

was investigated and results were compared and discussed. The rest of the simu-

lations were based on the assumption that one layer of the strata among the roof,

coal seam and floor was significantly weaker due to either intact rock property

or presence of random discontinuities. The simulations were performed in a ex-

perimental way and show great advantages compared with physical models. Only

some of the results were presented. Some mechanisms of the large deformation of

the roadway, as well as the influence of the strata properties, are clarified. Several

conclusions are drawn from the analysis;

1. An elasto-plastic finite element analysis using infinitesimal theory may over-

predict displacements if the deformation associated with the problem are

large. However, when a buckling effect exists in some part of the problem,

infinitesimal theory may also under-predict the displacement in this part. In

general, therefore, the effect of geometrical nonlinearity must be considered

in a large deformation analysis.

2. Stress concentrations may exist in the failure zone near the roadway boundary

due to the provision of a support load at its surfaxre (arches), and this high

stress zone contributes to the stability of the yielded (or fractured) rock mass

near the surface.

3. It has been verified that in most cases, floor heave is the dominant behaviour

of roadway deformation in coal measures and this is sometimes still the case

284

when the roof strata are weaker than those of the floor.

4. It appears that large lateral deformation occurs only if the coal seam is par-

ticularly weak.

6.5 Comparisons between calculation and measured values

In order to compare the calculated results with field measurements, it is firstly

necessary to correlate the simulated excavation steps in the 2-D model with the

practical 3-D longwall face advance. Unfortunately, there is no existing equation or

function for such a relation. In this study, therefore, numerical explorations have

been made to establish this correlation. Because the field measurements were

taken after the roadway was formed, while the excavations in the model start with

the roadway advance, only those results computed after excavation of the roadway

(from step 5) are used in the comparisons. Comparisons are made between the

measured and calculated roadway closure, floor lift and pack load; however, it has

not been possible to take into account other aspects such as computed stresses,

measured bed separation and leg penetration.

6.5.1 Comparison of roadway closure

Two sections of measured roadway closure, one is in 21S' and the other in SB's

district, were taken for comparison. The vertical closure in the section of 21S'

district was up to 2.5m (Figure 6.5), but the analysis using the representative

rock mass parameters only predicted a closure of 1.1m, as shown in Figure 6.31.

Moreover, the calculation of simulation 1 predicted a much larger lateral closure

than that measured in situ. A probable explanation is that the parameters

assumed in simulation 1 were not suitable for the rock mass at this roadway section.

An alternative set of parameters, similar to simulation 2, but using reduced values

of m and s (m = 2.0,1.0 and « = 0.002,0.0002) was found to give satisfactory

results and is presented in Figure 6.32.

285

Measured vertical tosure (section 11)

Measured vertical closure (section 12)

Computed horizontal closure Computed

vertical closure

Measured horizontal closure

(4) (5) (6) (7) (8) (9)

Face Distance (m) & Excavation steps

Figure 6.31 Comparison of measured and computed roadway closures, —simulation 1 (for measured sections 11 and 12).

Measured vertical losure (section 11) Measured vertical

closure (section 12)

Computed vertical closure

Computed horizontal closure

Measured horizontal closure

0 50

(4) (5) (6) (7) (8) (9)

Face Distance (m) & Excavation steps

Figure 6.32 Comparison of measured and computed roadway closures, —using

modified input data for the case of simulation 2.

286

However, it was found that the results predicted by simulation 1 axe in much

better agreement with the measurements taken in the other section. Figure 6.33

shows the computed vertical and lateral closure in comparison with the measured

values. It is thus concluded on the basis of this modelling that in the location where

very large vertical closure occurs, the floor strata is likely to be much weaker than

that of the roof, while in the other location the properties of the floor strata are

probably very similar. The results predicted by the 2-D numerical model are thus

seen to be very encouraging.

Computed horizontal closure

Computed vertical closure

Measured vertical closure.

O 0.6

Measured horizontal closure

(7) (8) (9)

Face Distance (m) & Excavation steps

Figure 6.33 Comparison of measured and computed roadway closures, —simulation 1 (for another roadway section in S3's district).

287

6.5.2 Comparison of floor lift

The magnitude of floor lift is more directly related to the properties of the floor

strata. Figure 6.34 shows the computed floor lifts of simulation 1,2 and 4, as well

as the values measured at two sections mentioned previously. Although in Figure

6.33 the vertical closure calculated by simulation 1 is fairly close to the measured

value, the plot of its floor lift in Figure 6.34 shows a much bigger difference.

This means that the overall material constants used in the calculation must be

adjusted if better agreement is to be obtained. Simulation 4 in Figure 6.34 gives

the maximum floor lift, but it is still less than the field data. The floor lift predicted

by simulation 2 appears to be in very good agreement with one of the field curves,

and in this case, the parameters assumed in the computation are probably a good

reflection of the actual rock mass.

Measured after dinting.

Calculated floor-lift

Measured floor-lift

Simulation 4

Simulation 2

-r.rr.-S^'

Simulation 1

10 20 30 40

(5) (6) (7)

-r 60

„o..

70

(9) 80 90 100

(10)

Face Distance (m) & excavation steps

Figure 6.34 Comparison of measured and computed floor-lift —simulations l,2,and 4) (of. Figure 6.6).

288

6.5.3 Comparison of pack load

As was concluded in §6.2, in practice, pack load is not high and its maximum

value is about 2 MPa. However, in the excavation simulation described in 6.3.3,

the real path of pack load is very difficult to trace. The computed results show

that after the 6th step of the excavation, which is equivalent to 30~40 metres of

longwall face advance, the peak load is still fairly high. This is because of the

existence of stresses in the goaf which have not been fully reduced. These stresses

apply a confining pressure on the pack and thus make the strength of the pack

relatively high. With further simulations of longwall face advance, this pack stress

is reduced to a required value of less than 2 MPa and the process is presented in

Figure 6.35.

2.0 -

<0 OL 2 TJ §1.0

o <a 0.

0.0

Simulated pack load

(section 11)

100 (8) (9) (10)

Face Distance (m) & excavation steps

Pack load (section 12)

Figure 6.35 Comparison of measured and computed pack load, —simulation 1 (of. Figure 6.7).

289

6.5.4 Back analysis of in situ rock m a s s propert ies

Simulations using the input parameters shown in Table 6.1 have been found to be

in general agreement with some of the field measurements. For those simulations

which do not fit the field data, the related rock mass parameters must be modified

according to the in situ deformation behaviour. If after several adjustments, the

numerical simulation can well model the in situ s trata behaviour, the parameters

used in the simulation can be considered as a true representative of the in situ

rock mass. In this regard, the practical in situ rock mass properties can be

back analysed well by the finite element model. Thus, a better understanding is

acquired for the relation between the properties of an in situ rock mass and the

intact rock samples.

For example, in simulating the roadway deformation characterised by Figure

6.5 and Figure 6.6, simulation 1 was found to predict too small a vertical closure

and too large a lateral closure. Therefore, parameters have to be modified and after

several calculation trials, the required parameters are acquired. Good agreement

of the results is then achieved and thus, the actual in situ rock s trata properties,

in the specific location where measurements were taken, are correctly evaluated

in the context of this modelling. In this example, the strength of the in situ

floor strata is calculated as trj w 0 and ctJ. w 0 .28MPa, which is equivalent to a

moderately weathered rock mass with CSIR rating RMR fn 40 (Hoek and Brown

1988).

6.6 Conclusions and practical implicat ions

By analysing a practical roadway closure problem using the developed numer-

ical model, comparisons between calculations and field measurements and back

analyses have been conducted. From this has ensued better understanding of the

behaviour of the Coal Measures strata, particularly at depth.

290

No doubt some improvement could be achieved by adjusting the input data

and running more analyses. A decrease in modulus would increase the deformation.

A very large number of combinations of strata properties could be tried in an

attempt to fit the field data more closely. However, a number of main conclusions

have been drawn from the study and they merit emphasis as follows:

1. As input parameters, rock mass properties represent an important source of

analysis error, while laboratory intact rock properties contribute less to the

errors.

2. Consideration of geometric non-linearities in the deformation analysis is par-

ticularly important in order to understand an important source of error which

has to date been ignored.

3. In the numerical analysis, the adopted constitutive model of the rock mass

has a great advantage in simulating the strata deformation.

4. The proposed excavation procedure is very flexible and successful in simulat-

ing a real sequence of 3-D mining face advajice. Finally,

5. The fully nonlinear and 3-D roadway problem can be approximately modelled

well by a proper simplified 2-D model based on an understanding of a number

of sources of error.

These conclusions have significant practical implications.

- For evaluating rock and rock mass properties, the accuracy of laboratory

intact rock tests is not so important as a correct classification of the rock

mass, ff the Hoek-Brown criterion or its simplified form is used in analysis,

effort has to be made to determine the rock mass type, its nature and the

joint occurrence, etc. This is supported by the recent remarks of Hoek and

Brown (1988) on using the updated empirical criterion.

- Large deformation theory, in predicting large rock mass deformation, is at this

time a pioneering research area in Rock Mechanics, and the author believes

that the results and conclusions from this study will be of help in answering

291

some of the classical key questions of Rock Mechanics and also for further

research.

As long as the rock mass is correctly classified, the numerical analysis using

the proposed constitutive model is believed applicable to a number of practical

problems and a reasonable prediction can be expected.

The excavation procedure used in the study has a very clear physical meaning

and is easy to conduct. It will be applicable in a truly 3-D finite element

analysis and more advantages in that case are expected.

The doubt in the applicability of a 2-D numerical analysis in Rock Mechanics

is further clarified by this study, and as regards economy and engineering sim-

plicity, a 2-D analysis still has superior advantages over a fully 3-D nonlinear

finite element analysis.

292

CHAPTER 7 SUMMARY AND CONCLUSIONS

A brief summary

Numerical modelling is a basic tool to achieve a more complete knowledge of

the rock mass response to excavations. The philosophy adopted for this research

of modelling rock movement around mine openings is to apply available numerical

methods and to develop complementary techniques of analysis. To this end, avail-

able numerical methods in rock mechanics are critically assessed. Based on this

assessment, the finite element visco-plastic analysis is chosen for the continuum

approach to simulate weak rock mass behaviour. Improvements and developments

of the finite element program are described and systematically validated against

theoretical or other numerical solutions.

A nonlinear axisymmetric finite element analysis is carried out to investigate

the validity of a plane strain analysis in simulating 3-D tunnelling in a rock mass.

All results obtained from the 3-D modelling are compared with the related plane

strain analysis, from which some relations between the 2-D and 3-D analyses are

clarified.

The distinct element method using the static relaxation procedure is further

investigated, with rewriting of the formulations in a matrix form. The method

is tested to compare it with physical models. For the complex behaviour of rock

masses (combination of continuum and discontinuum), a new coupling procedure

to use the nonlinear finite element method in conjunction with distinct elements

is proposed. The developed hybrid program is described, and validation examples

are given.

293

Practical implications of the research are demonstrated by analysing the road-

way deformation of Coventry Colliery using the developed numerical model, in

which comparisons between calculations and field measurements as well as back

analyses are presented.

The above investigations have been described in the previous six chapters

in a relatively independent way as illustrated in Figure 7.1, in which the major

conclusions of each chapter are outlined.

Main conclusions

Complete conclusions of the research of each chapter are presented at the end

of that chapter, as indicated in Figure 7.1; thus, only some main conclusions of

the results are presented here.

a) In the continuum approach to rock mass behaviour, the Hoek-Brown yield

criterion currently appears to be the most suitable one of those considered.

In the finite element analysis, however, it is shown that the modified Hoek-

Brown criterion has significant advantages, particularly for simulating weak

rock mass behaviour. The smooth surface of the modified Hoek-Brown cri-

terion is also suitable for the plastic potential function if the non-associated

flow rule is applied.

b) The implementation of large deformation theory using the Updated La-

grangian formulations is shown to be successful in either elastic analysis or

viscoplastic analysis. The influence of the geometric nonlinearity on the com-

putation results, especially for the weak rock mass behaviour, is investigated

in detail and shown to be significant. It is therefore possible to correct the

errors which may arise in the application of a conventional infinitesimal defor-

mation analysis. For simulating the excavation process, the proposed 'stress

and stiffness reduction' method is proved to be effective and convenient, and

thus it is recommended for future analysis.

c) 2-D nonlinear numerical analysis (plane strain) usually under-estimates the

294

THESIS STRUCTURE

CONCLUSIONS ON OUTCOME OF EACH CHAPTER

What? Why? How? (Section 1.4)

Possible tools? (Section 2.8)

Improvements & developments (Section 3.7)

Is plane strain analysis of any use? (Section 4.6)

A general purpose program (Section 5.5)

A numerical case study — Practical implications of the research (Section 6.6)

Past, present and future.

Practical applications

Numerical analysis

Theoretical investigations

Validation examples

Coupled procedure of the hybrid analysis

Distinct element analysis (static relaxation)

Rock mass res-ponse model & its numerical procedure

2-D simulation of stepwise and 3-D excavation

Implementation of large deform-tion theory

CHAPTER 1 Introduction

CHAPTER 7 Conclusions

CHAPTER 6 Applications

CHAPTER 5 2-D Hybrid analysis

CHAPTER 2 Assessment of the methods

CHAPTER 4 Plane strain analysis for 3-D problems

CHAPTER 3 2-D Nonlinear F.E. analysis

Figure 7.1 An outline of conclusions of each chapter.

295

failure zone and convergence of a 3-D tunnel excavation, because, as the face

advances, the axial stress path and the yield process cannot be modelled

well in plane strain analysis. However, the errors may be corrected by un-

derstanding of the relations between the 2-D model and a 3-D excavation

problem. Such a correction function is suggested, but further investigation of

the relevant parameters is necessary.

d) A complementary hybrid conceptual model and its practical examples are

described . A new coupling procedure of the finite elements and distinct

elements is proposed to take account of both continuum and discontinuum

behaviour in one analysis. The developed method is capable of modelling the

large displacements of continuum and discontinuum induced by an excavation.

It proves to be computationally efficient and conceptually appropriate.

e) The research is supported by application of the developed program to roadway

deformation analysis. The results of the case study indicate that :

i. For evaluating rock mass properties, the accuracy of laboratory intact

rock tests is not as important as a correct classification of the rock mass.

In other words, the most sensitive input parameters are the empirical

constants for rock masses (Hoek-Brown or the modified criterion).

ii. Large deformation theory for predicting large rock mass deformation is

at this time a pioneering research area in rock mechanics, but the results

show that it is particularly important because it represents a significant

source of error which has to date been ignored.

iii. As long as the rock mass is correctly classified and the relevant strength

parameters can be determined, numerical analysis using any of the Mohr-

Coulomb, the Hoek-Brown and the modified Hoek-Brown criteria is found

to be satisfactory, or a reasonable prediction can be expected. In this

respect, the proposed rock mass response model also shows advantages

because of its ease in reflecting rock mass type and its simplicity of math-

ematical formulations.

296

iv, 2-D numerical simulations are well verified by the 3-D in situ measure-

ments. The excavation procedure used in the program is thus further

proved and doubt concerning the validity of a 2-D numerical analysis is

clarified.

Suggestions for further research

Limitations of the current program are discussed in previous chapters, and

further developments and investigations are required. The following suggestions

axe made to improve the efiiciency of the proposed model and to enhance the

generality of the program.

1. In the finite element analysis, some straightforward implementations of exist-

ing techniques are suggested. As is described in Chapter 2, these include: to

extend the program to 3-D analysis, to introduce infinite finite elements, to

incorporate strain softening modelling procedure, and to adopt appropriate

joint elements.

2. Considering practical applications of the program, the computation time, or

the convergence rate, of the nonlinear iterations is of primary concern, so that

further development should take account of the improvement of the compu-

tation algorithm. When strain softening or a more complicated rheological

model is considered, the validation of the large deformation theory in the

finite element analysis needs to be further investigated.

3. Further investigation is necessary for applying a two- dimensional numerical

model in simulating a tunnel face advance problem in which a rock mass ex-

hibits nonlinear behaviour. Detailed correction functions based on equations

(4.9) and (4.10) are required.

4. For the distinct element representation of discontinua, deformable blocks us-

ing internal finite elements or finite diff^erence should be introduced. It is also

suggested that other available techniques such as modelling cracks in blocks

and simulating tunnel lining using structure elements, etc., are incorporated.

297

In view of the difficulties involved in the static relaxation procedure for mod-

elling separated block movements, more investigations into the static relax-

ation algorithm are required. For each separated or ill-conditioned block, the

possible solutions to overcome the difficulties include development of a pseudo

dynamic analysis procedure, or introduction of an incremental pseudo oppo-

site force for the static equilibrium of the block. Those approaches do not

appear fundamentally difficult.

5. In simulating the coupled nonlinear behaviour, a better iteration procedure

may be that the hybrid calculation is carried out during each viscous time step.

This can eliminate the necessity of choosing the criterion for determining the

computing iterations between the two modelling regions and thus guarantee

a path independent solution.

6. The effect of ground water on the stability of surface or underground exca-

vations is important, so further research should also take account of the fluid

interaction and the effects of pore pressure, etc.

298

REFERENCES

Akagi, T., Ichikawa, Y., Kuroda, H. & Kawamoto, T. (1984) A nonlinear rheolog-

ical analysis of deeply located tunnels. Int. J. Num. & Anal. Meth. in

Geomtch., Vol. 8, No. 5, pp. 457- 471.

Akagi, T. (1985) Application of rheological models to nonlinear creep properties

of rock. Proc. 5th Conf. on Num. Meth. in Geomech., Nagoya, pp.

415-420.

Akai, K. and Mori, H. (1967) Study on the failure mechajiism of a sandstone under

combined compressive stress. Proc. 2nd Congr. ISRM, Vol. 11, Paper

No. 3-30, Belgrade, Yugoslavia.

Amadei B., Robinson M.J. and Yassin Y.Y. (1986) Rock strength and design of

underground excavations. In Large Rock Caverns, K. Saari (ed.), Proc.

Int. Sym., Vol. 2, pp. 1135-1147. Helsinki.

Amadei, B. (1988) Strength of a regularly jointed rock mass under biaxial and

axisymmetric loading conditions. Int. J. Rock Mech. Min. Sci. &

Geomech. Abstr., Vol.25, No. 1, pp.3-13.

Ash, N.F. and Park, D.W. (1987) 3-D finite element modelling of longwall mining

using progressive failure concept. S8th U.S. Sym. on Rock Mech., pp.

725-733, Tucson.

Asi, T., Nishimura, M. and Saito, T. (1981) A numerical modelling technique for

analysing the behaviour of discontinuous rock masses. Proc. Int. Sym.

on weak rock, pp.737-742, Tokyo.

Baines, M. J. (1985) On approximate solutions of time- dependent partial differ-

ential equations by the moving Finite Element Method. UCINA Report,

Number 85/1.

Bandis, B.C., Nadim, F., Lindmann, J. and Barton, N. (1986) Stability investiga-

tion of modelled rock caverns at great depth. In Large Rock Caverns,

299

Proc. Int. Sym., Vol. 2, pp. 1161-1170. Helsinki.

Bandis, S.C., Lindmann, J . and Barton, N. (1987) Three-dimensional stress state

and fracturing around cavities in overstressed weak rock. In Proc. of

6th Int. Congr. on Rock Mechanics, pp. 769-775. Montreal.

Banerjee, P.K., Cathie, D.N. and Davis, T.G. (1979) Two- and three dimensional

problems of elasto - plasticity. Developments in Boundary Element

Methods - 1, P.K.Banerjee and R.Butterfield (eds.). Applied Science

Publishers, pp. 65-95.

Banerjee, P.K. and Dargush, G.F. (1988) Progress in BEM applications in geome-

chanics via examples. Proc. 6th Conf. on Num. Meth. in Geomech. ,

pp. 13-22, Innsbruck.

Barton, N., Lien, R. and Lunde, J . (1974) Engineering classification of rock masses

for the design of tunnel support. Rock Mechanics, Vol. 6, pp. 189-236.

Bathe, K.J. and Ozdemir, H. (1976) Elastic - plastic large deformation static and

dynamic analysis. Ird. J. of Solids and Structures, Vol. 6, 81-92.

Bathe, K.J. (1982) Finite Element Procedures in Engineering Analysis, Prentice -

Hall Inc.

Baxter, N.G. (1987) Design and stability of steel support systems with special

reference to gate roadways in the Warwickshire coalfield. PhD thesis,

Nottingham University.

Beer, G. and Meek, J.L. (1982) Coupled finite element - boundary element analysis

in elasticity and plasticity. Proc. Int. Conf. Finite Element Meth., pp.

686-692. Shanghai, China.

Beer, G. (1983) 'Infinite domain' elements in finite element analysis of underground

excavations. Int. J. Num. Anal. Meth. in Geomech., Vol. 7, pp. 1-7.

Beer, G. (1985) An isoparametric joint / interface element for finite element anal-

ysis. Int. J. Num. Meth. in Eng., Vol. 21, pp. 585-600.

Beer, G. (1985) Combined boundary element - finite element analysis. In De-

velopments in Boundary Element Methods - 4, P.K. Banerjee and J.O.

Watson (eds). Applied Science Publishers, pp. 191-225.

Beer G., Watson J.O. and Swoboda G. (1987) Three-dimensional analysis of tun-

300

nels using infinite boundary elements. Computers and Geomechanics,

Vol. 3, pp. 37-58.

Bieniawski,Z.T. (1974) Estimating the strength of rock materials. J . of South

African Institute of Min. & MetalL, Vol. 74, pp. 312-320.

Bieniawski, Z.T. (1984) Rock mechanics design in mining and tunnelling, Balkema,

Rotterdam.

Bishop A.W. (1966)The strength of soils as engineering materials. 6th Rankine

Lecture. Geotecnique, Vol. 16, No. 2, pp. 91-130.

Bohatier, C. and Chenot, J.L. (1985) Finite element formulations for non-steady-

state large viscoplastic deformation. Int. J. Num. Meth. in Eng., Vol.

21, pp. 1697-1708.

Brady, B.H.G. and Bray, J.W. (1978) The boundary element method for deter-

mining stress and displacement around long openings in a triaxial stress

field. Int. J. Rock Mech. Min. Sci. Geomech. Abstr., Vol. 15, pp.

21-28.

Brady, B.H.G. (1979) Boundary Element Methods for Mine Design. PhD Thesis,

University of London (Imperial College).

Brady, B.H.G. and Wassying, A. (1981) A coupled Finite Element-Boundary El-

ement Method of Stress Analysis. Int. J. Rock Mech. Min. Sci., Vol.

18, pp. 475-485.

Brady, B.H.G. and Brown, E.T. (1985) Rock Mechanics for Underground Mining,

London: George Allen & Unwin.

Bray, J.W. (1976) A program for two dimensional stress analysis using the bound-

ary element method. Imperial College, Rock Mechanics Progress Report

no.l6.

Brebbia, C.A. (1978) The Boundary Element Method for Engineering, Pentech

Press, London.

Brockman, R.A. (1984) Explicit forms for the tangent modulus tensor in visco

plastic stress analysis. Int. J. for Num. Meth. in Eng., Vol.20, pp.

315-319.

Brown, E.T. and Hoek, E. (1978) Trends in relationships between measured rock

301

in-situ stresses and depth. Int. J. Rock Mech. Min. Sci., vol.15, pp.

211-215.

Brown E.T., Bray, J.W. Ladanyi, B. and Hoek E. (1983) Characteristic line calcu-

lations for rock tunnels. J. Geotech. Eng. Am. Soc. Civ. Engrs., Vol.

109, pp. 15-39.

Brown, E.T. (1987) Introduction in Analytical and Computational Methods in En-

gineering Rock Mechanics, Allen & Unwin, pp. 1-31.

Brown, E.T. (1987) Personal communication.

Brown, E.T. (1988) Foreword. Int. J. Rock. Mech. Min. Sci. & Geomech. Abstr.,

Vol. 25, No. 4, pp. iii.

Brummer, R.K. (1988) Active methods to combat the rockburst hazard in South

African gold mines. Proc. of CARE'88 (Conference on Applied Rock

Engineering), IMM, pp. 35-43.

Butterfield, R. (1979) New concept illustrated by old problems. In Developments

in Boundary Element Methods - 1, P.K. Banerjee and R. Butterfield

(eds.). Applied Science Publishers, pp. 1-20.

Carol, I. and Alonso, E.E. (1983) A new joint element for the analysis of fractured

rock. Fifth Conf. Sym. of ISRM, pp. F147-F151, Australia.

Carter, J.P., Booker, J.R. and Davis, E.H. (1977) Finite deformation of an elasto-

plastic Soil. Int. J. Num. Anal. Meth. Geomech., Vol. 1, pp. 25-43.

Chandrasekaran, V.S. & King, G. J.W. (1974) Simulation of excavation using finite

elements. Technical Notes. J. Geotech. Eng. Div., Proc. ASCE., Vol.

100, No. GT9, pp. 1086-1089.

Chen, J. and Yuan, W. (1988) The progressive failure of the rock mass surrounding

an opening by boundary element method. Proc. 6th Conf. on Num.

Meth. in Geomech. , pp. 997-1001, Innsbruck.

Christiaji, J .T. & Wong, I.E. (1974) Errors in simulating excavation in elastic

media by finite elements. Soils and Foundations, Vol. 13, No. 1, pp.

1-10 .

Cormean, I. (1975) Numerical stability in quasi-static elasto/visco-plasticity. Int.

J. for Num. Meth. in Eng., Vol. 9, pp. 109-127.

302

Coulthaxd, M. A. and Perkings, B. R. (1987) Computer programs abstracts, (4th

edition), CSRIO, Institute of energy and earth resources, Div. of Ge-

omech.. Technical Report No. 99.

Crook, A.J.L., Hinton, E. and Stagg, K.G. (1987) Proc. CARE' 88 (Conference

on Applied Rock Engineering), pp. 45-54, Newcastle, England.

Crouch, S.I. (1976) Solution of plane elasticity problems by the displacement dis-

continuity method. Parts 1 and 2. Int. J. Rock Mech. Min. Set., Vol.

10, pp. 301-343.

Crouch, S.L. and Starfield, A.M. (1983) Boundary Element Method in Solid Me-

chanics, Allen & Unwin, London.

Cundall, P.A. (1971) The measurement and analysis of accelerations in rock slopes.

PhD Thesis, University of London (Imperial College).

Cundall, P.A. (1971a) A computer model for simulating progressive large scale

movements in blocky rock systems. Proc. Int. Sym. on Rock Fracture,

Nancy (ISRM), paper 2-8.

Cundall, P.A. (1974) A computer model for rock mass behaviour using interactive

graphics. U.S. Army, Corps of Engineers, Technical Report MRP2-74,

Missouri River Division.

Cundall, P.A. (1976) Explicit finite difference methods of geomechanics. proc. of

ASCE Eng. Foundation Conf. on Num. Meth. in Geomechanics, pp.

132-150, Virginia.

Cundall, P.A. and Marti, J. (1979) Some new developments in discrete numerical

methods for dynamic modelling of jointed rock masses. Rapid Excava-

tion and Tunnelling Conference, Atlanta, Geogia.

Cundall, P.A. and Hart, R.D. (1985) Development of generalised 2-D and 3-D

distinct element programs for modelling jointed rock. Itasca Consulting

Group, Misc. Paper SL-85-1, U.S. Army Corps, of Engineers.

Cundall, P.A. (1988) Conceptual, analytical and numerical modelling. Keynote

address in 29th U.S. Sym. on Rock Mech. , Minneapolis, Minnesota.

Cundall, P. A. and Board, M. (1988) A microcomputer program for modelling

large-strain plasticity problems. Proc. 6th Conf. on Num. Meth. in

303

Geomech., pp. 2101-2108, Innsbruck.

Cunha, A.P. (1983) Analysis of advancing tunnels in rock. 5th Con}. Sym. of

ISRM, pp. 153-157, Australia.

Desai, C.S. (1972) Overview, Trends and Projections, Theory and Applications of

the Finite Element Method in Geotechnical Engineering. In Applications

o} the Finite Element Method in Geotechnical Engineering, pp. 3-90,

Proc. Sym. held at Vicksburg, Mississippi.

Desai, C.S. (1980) A general basis for yield, failure and potential functions in

plasticity. Int. J. for Num. Meth. in Geomech., Vol. 4, pp. 361-375.

Desai C.S. and Phan H.V. (1980) Three dimensional finite element analysis includ-

ing material and geometrical non-linearities. In Computational Methods

in Nonlinear Mechanics J.T.Oden (ed.), North-Holland Publishing Com-

pany, pp. 205-224.

Desai, C.S. and Salami, M.R. (1987) A constitutive model and associated testing

for soft rock. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., Vol.

24, No. 5, pp. 299-307.

Desai, C.S. and Zhang, D. (1987) Viscoplastic model for geologic materials with

generalised flow rule. Int. J. for Num. and Anal. Meth. in Geomech. ,

Vol. 11, pp. 603-620.

Detournay E. and Fairhurst C. (1987) Two-dimensional elaatoplastic analysis of

a long cylindrical cavity under non- hydrostatic loading. Int. J. Rock

Mech. Min. Sci. & Geomech. Abstr., Vol. 24, No. 4, pp. 197-211.

Dillen, D.E.V. (1981) A finite element code for Rock Mechanics applications. Proc.

22nd U.S. Sym. on Rock Mech., pp. 373-378.

Dolezalova, M. (1977) A contribution to the common application of field mea-

surements and the finite element method. Proc. Int, Sym. on Field

Measurements in Rock Mech., Vol. 2, pp. 873-888, Zurich.

Dowding, C.H., Belytshko, T.B. and Yen, H.J. (1983) A coupled finite element -

rigid block method for transient analysis of rock caverns. Int. J. Num.

and Anal. Meth. in Geomech., Vol. 7, pp. 117-127.

Dunbar, W.S. (1982) Implementation of finite element - boundary integral linkage

304

algorithms for rock mechanics applications. Issues in Rock Mechanics,

23rd U.S. Sym. on Rock Mech., pp. 604-611, Berkeley.

Eissa, E.A. (1980) Stress analysis of underground excavations in isotropic and

stratified rock using the boundary element method. PhD Thesis, Uni-

versity of London (Imperial College).

Fairhurst, C. (1964) On the validity of the 'Brazilian' test for brittle materials.

Int. J. Rock Mech. Min. Sci., Vol. 1, pp. 535-546.

Farmer, I.W. (1982) Deformation of access roadways and roadside packs in coal

mines. In Strata Mechanics, Developments in Geotechnical Engineering,

Vol. 32, pp. 207-212.

Farmer, I.W. (1983) Engineering behaviour of rocks, Chapman & Hall, London.

Farmer, I.W. (1985) Coal Mine Structures, Chapman & Hall, London.

Florence A.L. and Schwer L.E. (1978) Axisymmetric compression of a Mohr-

Coulomb Medium around a Circular Hole. Int. J. Num. Anal. Meth.

Geomech., Vol.2, pp. 367-379.

Franklin, J.A. (1971) Triaxial strength of rock materials. Rock Mechanics, Vol. 3,

pp. 86-98.

Fritz, P. (1984) An analytical solution for axisymmetric tunnel problems in elasto-

viscoplastic media. Int. J. Num. Anal. Meth. Geomech. , Vol. 8, pp.

325-342.

Fun, Y.C. (1965) Foundation of Solid Mechanics, Prentice-Hall, Inc.

Ghaboussi, J. et al (1973) Finite Element for Rock Joints and Interfaces. J. of soil

mech. and found, division, ASCE, Vol. 99, No. SMIO, pp. 833-848.

Goodman, R.E. et al (1968) A model for the Mechanics of Jointed Rock. J . of soil

mech. and found, division, ASCE, VOL. 94, NO. SMS, pp. 637-659.

Goodman, R.E. (1976) Methods of Geological Engineering in Discontinuous Rock,

West Publishing Company.

Goodman, R.E. and Shi, G.H. (1985) Block Theory and its Application to Rock

Engineering, Prentice - Hall, New Jersey.

Gumusoglu, M.C., Bray, J.W. and Watson, J O. (1986) Analysis of underground

305

excavations incorporating the strain softening of rock masses. Proc. 6th

Congr. of ISRM, pp. 923-928, Montreal, Canada.

Haber, R.B. and Hariandia, B.H. (1985) An Eulerian - Lagrangian finite element

approach to large deformation frictional contact. Computers and Struc-

tures, Vol. 20, No.1-3, pp. 193-201.

Hammet, R.D. (1974) A study of the behaviour of discontinuous rock. PhD Thesis,

James Cook University, North Queensland, Australia.

Hanafy E.A. and Emery J.J. (1980) Advancing face simulation of tunnel exca-

vations and lining placement. In Underground rock engineering: 13th

Canadian rock mechanics symposium (the H.R. Rice Memorial sym-

posium), pp. 119-125. Montreal: Canadian Institute of Mining and

Metallurgy.

Handin, J. , Heaxd, H.C. and Magouirk, J.N. (1967) Effects of the intermediate

principal stress on the failure of limestone, Dolomite, and Glass at dif-

ferent temperatures and strain rates. J. Geophys. Res., Vol. 72, pp.

611-640.

Hart, R., Cundall, P.A. and Lemos, J. (1988) Formulation of a three dimensional

distinct element model - part II. Mechanical calculations for motion and

interaction of a system composed of many polyhedral blocks. Int. J.

Rock Mech. Min. Sci. & Geomech. Abstr. , Vol. 25, No.3, pp. 117-125.

Hill, R. (1959) The mathematical theory of plasticity. University Press, Oxford.

Hinton, E. and Owen, D.R.J. (1977) Finite Element Programming, Academic

Press.

Hinton, E. and Owen, D.R.J. (1979) An Introduction to Finite Element Compu-

tations, Pineridge Press, Swansea.

Hobbs, D.W. (1966) A study of the behaviour of broken rock under triaxial com-

pression and its application to mine roadways. Int. J. Rock Mech. Min.

Sci., Vol. 3, No. 1, pp. 11-43.

Hocking, G. (1977) Development and application of the boundary integral and

rigid block methods for geotechnics. PhD Thesis, University of London

(Imperial College).

306

Hocking, G. (1978) Stresses around tunnel intersections. In Computer methods in

tunnel design, pp. 41-60, London: Instn. of Civil Engrs.

Hodge, P.G. ajid White, G.N. (1950) A quantitative comparison of flow and de-

formation theories of plasticity. J. of Appl. Mech. Vol. 17, pp. 180-184.

Hoek, E. (1968) Brittle failure of rock. In Rock Mech. in Eng. Practice, Stagg,

K.G. (ed.), John Wiley, pp. 99-124.

Hoek E. and Brown E.T. (1980) Underground Excavations in Rock. The Institute

of Mining and Metallurgy, London.

Hoek, E. and Brown, E.T. (1980a) Empirical strength criterion for rock masses. J.

of Geotech. Eng. Division, ASCE, Vol. 106, No. GT9, pp. 1013-1035.

Hoek, E. (1983) Strength of jointed rock masses, the twenty third Rankine Lecture.

Geotechnique, Vol. 33, pp. 187-223.

Hoek, E. (1987) Practical Rock Mechanics - developments over the past 25 years.

Trans. Instn. Min. Metall. (sect. A: Min. industry). Vol. 96, pp.

Al-6 .

Hoek, E. and Brown, E.T. (1988) The Hoek-Brown failure criterion - a 1988

update. To be published in Proc. 18th Canadian Rock Mech. Sym.,

Toronto.

Holden, J .T. (1972) On the finite deflection of thin beams. Int. J. Solids and

Struct., Vol. 8, pp. 1051-1055.

Hoskins, E.R. (1967) The failure of thick-walled hollow cylinders of isotropic rock.

Int. J. Rock Mech. Min. Sci., Vol. 6, pp. 99-125.

Hudson J.A. (1987) The understanding of measured changes in rock structure, in

situ stress and water flow caused by underground excavation. (2nd Int.

Sym. on Field Measurements in Geomech.), pp. 467-474. Kobe.

Hudson, J.A. (1988). Rock Mechanics Principles in Engineering Practice, Butter-

worths on behalf of CIRIA, (in press).

Hudson, J.A., Brown, E.T. and Fairhurst, C. (1972) Shape of the complete stress

strain curve for rock. In Stability of Rock Slopes. Proc. ISth Sym. Rock

Mech., pp. 773-795, Illinois, ASCE, N.Y.

307

Jaeger, J.C. and Cook, N.G. (1979) Fundamentals of rock mechanics^ 3rd edn.

London, Chapman and Hall.

Janakiraman, C. and Das, B. (1987) Numerical modelling applications to rock

mechanics on a supercomputer. Proc. 28th U.S. Sym. on Rock Mech. ,

pp. 735-743, Tucson.

Johansson, E., Riekkola, R. and Lorig, L. (1988) Design analysis of multiple paral-

lel caverns using explicit finite different methods. Key Questions in Rock

Mechanics, 29th U.S. Sym. on Rock Mechanics, pp. 325-333, Minnesota.

Johnston, I.W. (1985a) Strength of intact geomechanical materials. J . Geotechni-

cal Engineering, ASCE, Vol. I l l , No. 6, pp. 730-749.

Johnston, I.W. (1985b) Comparison of two strength criteria for intact rock. J.

Geotechnical Engineering, Vol. I l l , No. 12, pp. 1449-1454.

Kaiser P.K. (1980) Effect of stress-history on the deformation behaviour of un-

derground openings. In Underground Rock Engineering'. 13th Canadian

rock mechanics symposium (the H.R. Rice Memorial symposium), pp.

133-140. Montreal: Canadian Institute of Mining and Metallurgy.

Kaiser, P.K. (1981) A new concept to evaluate tunnel performance-influence of

excavation procedure. Proc. 22nd U.S. Sym. Rock Mech., pp. 264-271,

Massachusetts.

Kanchi, M.B., Zienkiewicz, O.C. and Owen, D.R.J.(1978) The visco-plastic ap-

proach to problems of plasticity and creep involving geometric non-linear

effects. Int. J. Num. Meth. in Eng., Vol. 12, pp. 169-181.

Katsoulidis, C. (1986) An analysis of the viscoplastic behaviour of rocks, MSc

Thesis, London University (Imperial College).

Kawai,T.G. (1980) Some considerations on the finite element method. Int. J.

Num. Meth. in Eng., Vol. 16, pp. 81-120.

Kawai, T., Takeuchi, N. and Kumeta, T. (1981) New discrete models and their

application to rock mechanics. Proc. Int. Sym. on Weak Rock , pp.

725-730, Tokyo.

Kim, M.K. and Lade, P.V. (1984) Modelling rock strength in three dimensions.

Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., Vol. 21, No. 1, pp.

308

21-33.

Kiousis, P.D., Voyiadjis, G,Z. and Tumay, M.T. (1986) A large strain theory for

the two dimensional problems in geomechanics. Int. J. Num. Anal.

Meth. Geomech., Vol. 10, pp. 17-39.

Ladajiyi, B. (1984) Tunnel lining design in a creeping rock. In Design and Per-

formance of Underground Excavations, (Proc. Sym. ISRM B.G.S.), pp.

19-26, Cambridge.

Lade P.V. (1984) Failure criterion for frictional materials. In Mechanics of Engi-

neering Materials. (Edited by C.S.Desai and R.H.Gallagher, John Wiley

& Sons Ltd, pp. 385-402.

Langer, F.T. and Stockmann, K. (1985) Stability analysis of tunnels using the

program ADINA. Computers and Structures, Vol. 21, No. 1/2, pp.

341-351.

Lee, H.K., Yang, H.S. and Mizuda, Y. (1985) An elasoplastic analysis for post

failure behaviours of strain softening rocks. Fifth Int. Conf. Num.

Meth. in Geomechanics, pp. 165-171. Nagoya.

Lemos, J.V. and Brady, B.H.G. (1983) Stress distribution in a jointed and frac-

tured medium. Proc. 24th U.S. Sym. on Rock Mech., pp. 53-59.

Lemos, J.V., Hart, R.D. and Cundall, P.A. (1985) A generalised distinct element

program for modelling jointed rock mass, A keynote lecture. Proc. Int.

Sym. on Fundamentals of Rock Joints, pp. 335-343, Bjorkliden.

Li, L.H. (1988) Elastoplastic boundary element analysis of well- uniform self -

supporting structure. Proc. 6th Conf. on Num. Meth. in Geomech. ,

pp. 991-996, Innsbruck.

Lin, D. and Fair hurst, C. (1988) Static analysis of the stability of three-dimensional

blocky systems around excavations in rock. Int. J. Rock Mech. Min.

Sci. & Geomech. Abstr., Vol. 25, No. 3, pp. 139-147.

Lode, W. (1926) Versuche under den Einfluss der mitt leren Hauptspannung auf

das Fliessen der Metalle Eisen Kupfer und Nickel. Zeitschrift fuer Physik,

Vol. 36, pp. 913-939.

Lombardi, G. (1977) Long term measurements in underground openings and their

309

interpretation with special consideration to the Theological behaviour of

the rock. Int. Sym. on field measurements in rock mechanics, Vol. 2,

pp. 839-858. Zurich.

Lorig, L.J. and Brady, B.H.G. (1982) A hybrid discrete element-boundary element

method of stress analysis. Proc. SSrd U.S. Sym. on Rock Mech., pp.

628-636

Lorig, L.J. and Brady, B.H.G, (1984) A hybrid computational Scheme for excava-

tion and support design in jointed rock media, inDesign and Performance

of Underground Excavations, E.T.Brown and J.A.Hudson (eds), London:

British Geotechnical Society, pp. 105-112.

Maini, T., Cundall, P. and Marti, J. (1978) Computer modelling of jointed rock

masses, U.S. Army Corps of Engineers, Technical Report N-78-4.

Mallory, R. (1982) Roadway investigation in 21's district, Warwickshire thick seam,

Coventry Colliery. First Interim Report, National Coal Board.

Mayer, A.D., Isaac, A.K. and Neve, P. (1984) Validation and error analysis of a

strata modelling technique. Int. J. of Mining Engineering, Vol. 2, pp.

305-321.

Meek, J.L. and Beer, G. (1984) A review of analysis techniques for the determina-

tion of stresses around and performance of excavations in hard rock, in

Design and Performance of Underground Excavations, E.T.Brown and

J.A.Hudson (eds), London: British Geotechnical Society, pp. 1-10.

Michelis, P. (1987) True triaxial yielding and hardening of rock. J. Geotech. Eng.

Div. ASCE, Vol. 113, No. 6, pp. 616-635.

Mikula, P.A. and Holt, G.E. (1983) Prediction of mine subsidence in eastern

australia by mathematical modelling. Proc. 5th Conf. of ISRM, pp.

E119-126, Australia. Miller, K. and Miller, R.N. (1981) Moving Finite

Elements, Part I. SIAM J. Numer. Anal., Vol. 18, pp. 1019-1032.

Mitchell, A.R. and Griffiths, D.F. (1980) The Finite Difference Method in Partial

Differential Equations, Wiley, Chichester.

Mogi, K. (1967) Effect of the intermediate principal stress on rock failure. J.

geophys., Res., Vol. 72, pp. 5117-5131.

310

Mukherjee, S. and Chandra, A. (1984) Boundary element formulations for large

strain - large deformation problems of plasticity and visco-plasticity. In

Developments in Boundary Element Methods — S, P.K.Banerjee and S.

Mukherjee (eds), pp. 27-58.

Muller-Salzburg, L. and Ge Xiu Run (1983) Untersuchungen mechanichen ver-

halten geklufteten gebirges unter Wechsellasten. Proc. 5th Int. Congr.

Rock Mech. , Melboum, Vol.1, pp. A43-49, Rotterdam Balkema.

Murrel, S.A.F. (1963) A criterion for brittle fracture of rocks and concrete under

stress and the effect of pore pressure on the criterion. Proc. 5th Rock

Mech. Sym., University of Minnesota, Pergamon, pp. 563- 577.

Naylor, D.J., Pande, G.N., Simpson, B. and Tabb, R. (1981) Finite Elements in

Geotechnical Engineering, Pineridge Press, Swansea.

Nayak, G.C, and Zienkiewicz, O.C. (1972) Convenient form of stress invariants

for plasticity. J. Structural Div., Proc. ASCE., Vol. 98, No. ST4, pp.

949-954.

Nishimura, N. and Kobayshi, S. (1984) Elastoplastic analysis by indirect methods.

In Developments of Boundary Element Methods - S, P.K.Banerjee and

T.G.Davies (eds.). Applied Science Publishers, pp. 59-85.

Obert, L. and Stephenson, D.E. (1965) Stress conditions under which core discing

occurs. Soc. Min. Engrs. Trans., pp. 227-234.

Ohnishi, Y., Mimuro, T., Takewaki, N. and Yoshida, I. (1985) Verification of input

parameters for distinct element analysis of jointed rock mass. Proc. Int.

Sym. on Fundamentals of Rock Joints, pp. 205-214, Bjokliden.

Ohnishi, Y., Kishimoto, H., Nishigaki, Y. and Tanaka, Y, (1987) Analysis of ad-

vancing tunnel by two-dimensional F.E.M.. Proc. Int. Conf. on

Num. Meth. in Geomech., Vol. 2, pp. 571-578, Edmonton.

Okamoto, N. and Natazama, M. (1979) Finite element incremental contact analysis

with various frictional conditions. Int. J. Num. Meth. Eng., Vol. 14,

pp. 337-357.

Otter, J.R.H., Cassell, A.C. and Hobbs, R.E. (1967) Dynamic relaxation. Proc.

Instn. Civ. Engrs. , Vol. 35, pp. 633- 656.

311

Owen, D.R.J, and Hinton, E. (1980) Finite Elements in Plasticity, — Theory and

Practice ^ Pineridge Press Ltd, Swansea.

Pan, X.D. and Hudson, J.A. (1988) A simplified three dimensional Hoek and

Brown yield criterion. In Rock Mechanics and Power Plants, M. Romana

(ed.), ISRM Sym., pp. 95-103, Madrid, Spain.

Pan, X. D. and Hudson, J. A. (1988) Plane strain analysis in modelling of 3-D

tunnel excavations. Int. J. Rock Mech. & Min. Sci. & Geomech. Abstr.,

to be printed in October.

Pande, N.G. and Sharma, K.G. (1979) On joint / interface element and associated

problems of numerical ill conditioning. Int. J. Num. Anal. Meth. in

Geomechanics, Vol. 3, pp. 293-300.

Pande, G.N. and Pietruszczak, S. (1982) Symmetric tangential stiffness formu-

lation for non-associated plasticity. C/R/405, U. College of Swansea,

U.K.

Pande, G.N. (1985) A constitutive model of rock joints. Proc. of Int. Sym. on

Fundamentals of Rock Joints, pp. 429-439, Bjorkliden.

Panet M. and Guellec P. (1974) Contribution to the problem of the design of

tunnel support behind the face. In Advances in Rock Mechanics, volume

II, part B: (proceedings 3rd congress ISRM.), pp. 1163-1168. Denver.

Panet, M. (1979) Time-dependent deformations in underground works. Proc. 4th

Congr. ISRM., Vol. 3, pp. 279-290. Montreux.

Panet M. ajid Guenot A. (1982) Analysis of convergence behind the face of a

tunnel. Proc. of Tunnelling 82, pp. 197-204. Brighton.

Pariseau, W.G. (1988) On the concept of rock mass plasticity. Key Questions in

Rock Mechanics, 29th U.S. Sym. on Rock Mech., pp. 291-302, Min-

nesota.

Payne, A. R. and Isaac, A. K. (1985) The application of numerical models in coal

rib pillar design at longwall panels. 26th U.S. Sym. on Rock Mech. ,

pp. 685-692, Rapid City.

Plischke, LB. (1988) A report compiled by the ISRM Commission on Computer

Programs. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., Vol. 25,

312

No. 4, pp. 183-252.

Prager, W. (1959) An Introduction to Plasticity, Addison-Wesley Publ. Co., Am-

sterdam & London.

Priest, S.D. (1985) Hemispherical Projection Methods in Rock Mechanics, Allen

and Unwin, London.

Ranken R.E., Ghaboussi J . and Hendron A.J. Jr. (1978) Analysis of Ground-liner

Interaction for Tunnels, UMTA-IL-0043-78-3, NTIS, Springfield.

Reed, M.B. (1986) Stress and displacements around a cylindrical cavity in soft

rock. IMA J. of Applie, Oxford University Press, pp. 223-245.

Reed, M.B. (1987) A viscoplastic model for soft rock. Int. Conf. on Computational

Plasticity, - Models, Software & applications, Barcelona, Spain.

Reed, M.B. (1988) The influence of out-of-plane stress on a plane strain problem

in Rock Mechanics. Int. J. Num. Anal. Meth. in Geomech., Vol. 12,

pp. 171-181.

Reed, M.B. (1988a), Non-associated flow rules in computational plasticity. Proc.

6th Sym. on Num. Meth. in Geomech., pp. 481-488, Innsbruck.

Reik, G. and Zacas, M. (1978) Strength and deformation characteristics of jointed

media in true triaxial compression. Int. J. Rock Mech. Min. Sci. &

Geomech. Abstr., Vol. 15, pp. 295-303.

Sachdeva, T.D. and Ramakrishnan, C.V, (1981) A finite element solution for the

two-dimensional elastic contact problems with friction. Int. J. Num.

Meth Eng., Vol. 17, pp. 1257-1271.

Sakurai, S. (1978) Approximate time-dependent analysis of tunnel support struc-

ture considering progress of tunnel face. Int. J. Num. Anal. Meth. in

Geomech. , Vol. 2, pp. 159-175.

Serata, S., Sakurai, S. and Adachi, T. (1972) Theory of aggregate rock behaviour

based on absolute three dimensional testing (ATT) of rock salt. Basic

and Applied Rock Mechanics, K.E. Gray (ed.), (Proc. 10th Sym. on

Rock Mech.), pp. 431-473, Texas.

Scott, R.F. (1987) Failure, the twenty-seventh Rankine lecture, Geotechnique, Vol.

37, No.4, pp. 423-466.

313

Sharma, K.G., Varadarajan, A. and Srivastava, R.K. (1985) Elasto-viscoplastic

finite element analysis of tunnels. Fifth Int. Conf. Num. Meth. in

Geomechanics, pp. 1141-1148, Nagoya.

Shi, G.H. and Goodman, R.E. (1988) Discontinuous deformation analysis - A new

method for computing stress, strain and sliding of blocky systems. Key

Questions in Rock Mechanics, 29th U.S. Sym. on Rock Mech., pp. 381-

393, Minnesota.

Sloan, S.W. and Booker, J.R. (1986) Removal of singularities in Tresca and Mohr-

Coulomb yield functions. Comm. Appl. Num. Methods, Vol. 2, pp.

173-179.

Smith, G.D. (1985) Numerical Solution of Partial Differential Equations: Finite

Difference Methods, Third edition. Clarendon Press.

Sofianos, A. (1984) Numerical Simulation of Excavations within Jointed Rock of

Infinite Extent. PhD Thesis, University of London (Imperial College).

Southwell, R.V. (1940) Relaxation Methods in Engineering Science, Oxford Uni-

versity, New York.

Southwell, R.V. (1946) Relaxation Methods in Theoretical Physics, Oxford at the

Clarendon Press.

Stewart, I.J. (1981) Numerical and physical modelling of underground excavations

in discontinuous rock. PhD Thesis, University of London (Imperial Col-

lege).

Stewart, I.J. and Brown, E.T. (1984) A static relaxation method for the analysis

of excavations in discontinuous rock. In Design and performance of un-

derground excavations, E.T. Brown and J.A. Hudson (eds.), ISRM Sym.,

pp. 149-155, Cambridge,

Stricklin, J.A., Haisler, W.E. and Riesemann, W.A.V. (1973) Evaluation of solu-

tion procedures for material and / or geometrical nonlinear structural

analysis. AIAA J., Vol. 11, pp. 292-299.

Sulem J., Panet M. and Guenot A. C(1987) Closure Analysis in deep tunnels.

Int. J. Rock Mech. Min. Sci. & Geomech, Abstr., Vol. 24, No. 3, pp.

145-154.

314

Sulem J., Panet M. and Guenot A. (1987) An analytical solution for time-

dependent displacements in a circular tunnel. Int. J. Rock Mech. Min.

Set. & Geomech. Abstr., Vol. 24, No. 4, pp. 155- 164.

Sutherland, H.J., Heckes, A.A. and Taylor, L.M. (1984) Physical and numerical

simulations of subsidence above high extraction coal mines. Design and

performance of underground excavations, E.T.Brown and J.A.Hudson

(eds.), ISRM Sym., pp. 65-72, Cambridge.

Suzuki, K., Sugimoto, M., Kakurai, M., Yamashita, K., Ueda, M. and Kawai,

T. (1985) Elasto-plastic, dynamic response analysis based on the rigid-

body-spring-model for earth structure. Proc. 5th Int. Conf. on Num.

Meth. in Geomech. , pp. 215-220, Nagoya.

Swoboda, G. and Mertz, W. (1987) Rheological analysis of tunnel excavations

by means of coupled finite element (FEM) - boundary element (BEM)

analyses. Int. J. Num. and Anal. Meth. in Geomech., Vol. 11, pp.

115-129.

Swoboda, G., Edner, H., Wang, S.J. and Zhang, J.M. (1988) Application of the

'decoupled finite element analysis' in tunnelling. Proc. 6th Int. Conf.

on Num. Meth. in Geomech. , pp. 1465-1472, Innsbruck.

Tax>, Z.Y. and Yu, Q.H. (1985) The model of accelerating creep and the finite

element analysis of the process of rheological failure of rock. Fifth Int.

Conf. Num. Meth. in Geomechanics, pp. 445-450. Nagoya.

Timoshenko, S. P. and Goodier, J . N. (1970) Theory of Elasticity, London, Wiley.

Vargas Jr, E. (1982) Development and application of numerical methods to simu-

late the behaviour of fractured rock masses. PhD Thesis, University of

London (Imperial College).

Voegele, M.D. (1978) Rational design of supports: An interactive graphics

based analysis of the support requirements of excavations in jointed

rock masses, Final Report, Dept. of the Army Contract no. DACW

45-74-C0066.

Ward, W.H. (1978) Ground supports for tunnels in weak rocks, eighteenth Rankine

lecture, Geotechnique, Vol. 28, No. 2, pp. 133- 171.

315

Watson, J.O. (1979) Advanced implementation of the boundary element method

for two-and three-dimensional elastostatics, in Development in Boundary

Element Methods, P. Banerjee and R. Butterfield (eds). Applied Science

Publishers Ltd.

Watson, J.O. (1979) Boundary integral equations in Rock Mechanics. J. of the

Royal School of Mines (Imperial College) No. 29.

Watson, J.O. (1986) Hermitian cubic and singular elements for plane strain. In

Development in Boundary Element Methods - 4 , P.K.Banerjee and

J.O. Watson (eds.), pp. 1-28.

Watson, J.O. (1987) User Instructions for program THREE, Imperial College,

London.

WhittaJcer, B.N. and Singh, R.N. (1981) Technical Note, Stability of Longwall

Mining Gate Roadways in Relation to Rib Pillar Size. Int. J. Rock

Mech. Min. Sci. & Geo. Abstr. Vol. 18, pp. 331-334.

Wilson, A.H. (1983) The stability of underground workings in the soft rocks of the

coal measures. Int. J. Min. Eng., Vol. 1, No. 2, pp. 91-187.

Wold, M.B. (1985) A blocky physical model of longwall caving under strong roof

conditions. 26th U.S. Sym. on Rock Mech., pp. 1007-1014, Rapid City.

Xiong, W. (1985) Symmetric formulation of tangential stiffness for non-associated

plasticity. Fifth Int. Conf. Num. Meth. in Geomechanics, pp. 341-347,

Nagoya.

Xu, Z.R. (1979) Mechanics of Elasticity, People's Education Publishing Ltd. (in

Chinese).

Yamada, Y. and Wifi, A.S.(1977) Large strain analysis of some geomechanics

problems by the finite element methods. Int. J. Num. Anal. Meth.

Geomech., Vol. 1, pp. 299-318.

Yeung, D. and Brady, B.H.G. (1982) A quadratic, isoparametric, hybrid FE-BE

code for underground excavation design. Proc. 2Srd U.S. Sym. on Rock

Mech. , pp. 692-703, Berkeley, California.

Yoshinaka, R. and Yamabe, T. (1980) Strength criterion of rocks. Soils and foun-

dations, Vol. 20, No. 4, pp. 113-126.

316

Yudhbir, Lemanza, W. and Prinzl, F. (1983) An empirical failure criterion for rock

masses. Proc. 5th Congr. of ISRM, Vol. 1, pp. B1-B8.

Yuritzinn, T., Panet, M. & Guenot, A. (1982) Analysis of tunnels in strain soften-

ing grounds. Proc. 4th Int. Conf. Num. Meth. Geomech., pp. 635-644.

Edmonton, Canada.

Zienkiewicz, O.C., Valliapan, S. and King, LP. (1968) Stress analysis of rock as a

'no tension' material. Geoteehnique, Vol. 18, pp. 56-66.

Zienkiewicz, O.C., Best, B., Dullage, C. ajid Stagg, K.G. (1970) Analysis of non-

linear problems in rock mechanics with particular reference to jointed

rock systems. Proc. Snd Congr. of ISRM, pp. 501-509, Belgrade.

Zienkiewicz,0.C. and Nayak, G.C. (1973) A general approach to problems of large

deformation and plasticity using isoparametric elements. Srd Conf. Ma-

trix Meth. Struc. Meth., pp. 881-928, Wright-Patterson Air Force Base,

Ohio.

Zienkiewicz, O.C. and Cormean I.C. (1974) Visco-plasticity- plasticity and creep

in elastic solids - a unified numerical approach. Int. J. Num. Meth.

Eng., Vol.8, pp. 821-845.

Zienkiewicz, O.C. (1977) Finite Element Method in Engineering, McGraw-Hill

Book Company Ltd., U.K.

Zienkiewicz, O.C., Kelly, D.W. and Bettess, P. (1977) The coupling of the finite

element method and boundary solution procedures. Int. J. Num. Meth.

Eng., Vol. 11, pp. 355- 375.

Zienkiewicz, O.C. and Morgan, K. (1983) Finite Element and Approximation,

John Wiley & Sons.

317

APPENDIX 1 DESCRIPTIONS OF PROGRAM 'VISCO'

The flow chart of program VISCO is shown in Figure A l . l . The main facilities of

the program are summarised below:

a. two dimensional plane stress / strain and axisymmetric elasto-viscoplastic

analysis, with the solution scheme being explicit, implicit or implicit - explicit;

b. three element types (4, 8, and 9 noded elements) and two or three point Gauss

integration rules;

c. dynamic dimensioning in subroutines and frontal solver for solution procedure;

d. strain - hardening or perfect plastic constitutive model with four yield criteria:

Tresca, Von-Mises, Mohr-Coulomb and Drucker-Prager;

e. incremental loading with four types of loads: point loads, initial stresses,

gravity loads and distributed edge loads;

f. nonlinear iteration algorithms including initial stress, tangential and mixed

methods.

The program structure and all its subroutines are described by Owen and

Hinton (1980).

318

START

END

INVAR Evaluates the effective stress level

TANGVP Evaluates Z)" according to (8.18)

OUTPUT Prints the results for the current timestep

STIFVP Calculates the element stiffnesses as Kr^tr") (Eq. (8.24))

YIELDF & FLOWVP

Determines;— a) The flow

vector, a b) =

Y<<t>)a"+i

INPUT Inputs data defining geometry, boundary conditions and material properties

ZERO Sets to zero arrays required for accumulation of data

LOADPS Evaluates the equivalent nodal forces for pressure loading, gravity loading, etc.

DIMEN Presets the variables associated with the dynamic dimensioning process

INCREM Increments the applied loads according to specified load factors

FRONT Solves the simultaneous equation system by the frontal method, i.e. Ad" = AV" < / » + ! =

STEPVP Evaluates quantities at the end of the timestep a) A<r"=d''(i?"Ad—€„p"A/n) b) 0-"+'- =(r" +Ao-" c) e»p"+'-=e„p"+€Bp"A^

Calculate residual forces and pseudo loads for next time step a) = Jo dv +/"+! b) A «+i Atn idu

Figure A. 1.1 The flow chart of program VIS CO.

319

APPENDIX 2 SOME OTHER PROPERTIES OF THE

EXTENDED H-B CRITERION

a) Influence of parameters

The parameters involved in the criterion

J2 + —II = SOc (A.2.1)

are m, s and <Tc

In investigating the influences of these constants on the yield surface, a two

- dimensional parabolic curve in coordinates Ix and (^2)^ is considered, as illus-

trated in Figure A.2.1.

It is seen from Figure A.2.1b that the yield surface is characterised by two

variables A and B which are functions of three vparameters m, 5 and <TC- They

can be calculated from equation (A.2.1) as:

A = 3 — , (A2.2) m

B = -^^rruxc ^ \ / l + Qsjm? — i j , (A.2.3)

The influence of changing a specific parameter can then be easily investigated

by fixing the other two constants and examing the values of A and B which

determine the location and shape of the parabolic curve in the stress space.

For example, if s is fixed to 1.0 and <7c is assumed to be 50.0 MPa, A and B

320

~ ^2 ~ ^3

Figure A.2.1 The extended Hoek-Brown criterion in the I i and space.

321

are functions of m only. For sandstone, m = 7.0, so

7 X 50.0

sVs

For granite, m = 25.0 and

In general, the influence of parameter m is shown in Figure A.2.2a. In a

similar way, the influence of parameters s and Cc can be investigated and Figures

A.2.2b,c illustrate the change of yield surface as changing of the parameters.

The conclusions are:

1. Change of parameter m will change the shape of the yield surface, and it is

sensitive for predicting the rock tensile strength,

2. Parameter « controls the isotropic expansion and contraction of the yield

surface, and

3. Uniaxial compressive strength Uc determines the basic dimension (or size) of

the yield surface.

b) Approximation of the 'average' surface

The average surface

The extended formula of the Hoek-Brown criterion can be either:

—J2 H—y=m-\/j2 — ~ I i — sOc = 0, (A.2.4) V 3 j

322

a)

(J2) 0.5

b)

s1

(J2) 0.5

0

c)

ml>m2>m3.

I l

0

Figure A.2.2 The influence of empirical parameters m, s, Cc

323

or

—Jg H—y=my/j2——/i — S0C — 0" (A.2.5) V 3 3

In the 3-D pricipal stress space, the surface expressed by (A.2.4) contains the

original Hoek-Brown surface and is called 'outer apices' surface which intersects the

three 'outer' corner sides of the Hoek-Brown surface. Equation (A.2.5) expresses

an 'inner apices' surface which intersects the three 'inner' sides of the Hoek-Brown

surfcLce (see Figure A.2.3).

An approximately average surface is derived by taking the average parameters

of (A.2.4) and (A.2.5) as:

— J2 + ——/i — sCc = 0, (A.2.6). 0(. Z o

Verification

It is found that the approximation of (A.2.6) is reasonable only for rock masses

because for intact rock the difference between the 'outer' and 'inner' apices surfaces

is large. This is demonstrated by the following example.

For a hard intact sandstone, typical parameters can be assumed as: m =

7.0,5 = 1.0, <Tc = lOO.OMPa. Considering the stress level Ii = SO.OMPa, the

deviatoric shear strength of the rock (equivalent to the radius of the strength

surface at Ji) can be calculated as:

ri = 33.57, — using (A.2.4),

rg = 19.58, — using (A.2.5),

f = 25.0, — using (A.2.6).

Therefore, the maximum discrepancy between the results predicted by (A.2.6)

and the Hoek-Brown criterion may be (for stress points located near the corners):

— TtlQfX |ri - r| |r2 - r|

324

0.34 = 34%.

For weak rock masses, however, the discrepancies are small and negligible.

For example, if the typical values of m, 5 and cr for weak rock masses are taken

as: m < 0.1, 5 < 0.0001 and ac < SO.OMPa (Hoek and Brown 1988), and the

stress level is again considered as Ii = SO.OMPa, the shear strengths (the radius

of yield strength surface) are calculated as:

ri = 2.89, — using (A.2.4),

rg = 2.64, — using (A.2.5),

f = 2.76, — using (A.2.6).

(It is noted here that the average radius is 2.765).

Then the maximum discrepancies of the approximation is

^MAAJ — TTXCLX |ri - r| |r2 - 0.047 = 4.7%.

A conclusion of the above calculations is that the proposed yield surface can

provide a good approximation to the Hoek-Brown criterion in modelling weak rock

masses behaviour.

Outer apex sur-face of the ex-tended criferion.

The Hoek-Brown surface.

mner apex sur-face of the ex-tended criterion.

Figure A.2.3 The concept of the 'average space' of the extended H-B criterion.

325

APPENDIX 3 INSTRUCTIONS OF PROGRAM 'COAL'

A3.1 General descriptions

The program COAL is a two-dimensional finite element code capable of analysing

plane stress / plane strain and axisymmetric problems in elasto - viscoplastic ma-

terials. It is developed from the program VIS CO (Owen and Hinton 1980). In

comparison with the flow chart in appendix 1, Figure A.3.1 illustrates its calcula-

tion procedures and new features, which include following points;

a. A new constitutive model for rock mass is incorporated using the Hoek-Brown

and the extended Hoek-Brown yield function (or plastic potential function),

b. Excavation process is simulated using the stress and stiffness reduction pro-

cedure, which is proved to be efficient and convenient in modelling various

excavations, and

c. The large deformation theory using the updated Lagrangian formulations is

incorporated in the program to model the geometrically nonlinear behaviour

of weak rock masses. A choice can be made for the conventional small defor-

mation analysis or the large deformation analysis.

New subroutines are written and the related subroutines are modified to im-

plement the computations. These include:

COAL: the main program to control the new computing procedure,

EXCAVT: deal with incremental excavations,

GMAT: evaluate the nonlinear stiffness due to existing stresses,

(G-matrix),

MM AT: calculate a special stress matrix used for evaluating nonlinear

stiffness (M-matrix),

MGMAT: form the MG matrix,

BLARGE: form the large displacement B-matrix,

JACOBD: deformation Jacobian matrix.

326

(START^

INPUT

LOADPS

Loading Excavation Loading or excavation?

4 (by reducing stress and INCREM gsdmiess)

ALGOR

TANVP STIFVP GMAT MMAT

MGMAT FRONT

j(New sub-trouUnes for

JACOBD large deforma-ition analysis

STEPVP

FLOWVP

STEADY INVAR

OUTPUT

(or the maximum time step reached?)

Converged?

Plot specified graphs Plot?

ncrementJZ

Figure A.3.1 The flow chart of program COAL.

327

In accordance with the above routines, following existing subroutines

are further developed and modified: STIFVP, STEP VP, INVAR, YIELDF,

FLOWVP and STEADY.

A 3 . 2 Input instructions for the program COAL:

Card set 1: Title card (12A6)

— one card

cols.

1 - 8 0 TITLE Title of the analysis (maximum number

of 72 characters).

Card set 2: Control card (1215)

— one card

cols.

I - 5 NPOIN Total number of nodal points.

6 - 1 0 NELEM Total number of elements.

I I - 1 5 NVFDC Total number of restrained boundary nodes.

16 - 20 NTYPE Problem type: -

1 - plane stress,

2 - plane strain,

3 - axisymmetric.

2 1 - 2 5 NNODE Maximum number of nodes per element,

4 - linear quadrilateral element,

7 - seven node quadratic element,

8 - quadratic Serendipity element,

9 - quadratic Lagrangian element.

26 - 30 NMATS Total number of different materials.

31 - 35 NGAUS Order of Gauss quadrature for numerical

integration.

2 - two point Gauss quadrature rule,

3 - three point Gauss quaxirature rule.

328

3 6 - 4 0 NALGO

41 - 45

4 6 - 5 0

51 - 55

56 - 60

NCRIT

NINCS

NSTRE

NLAPS

Time integration parameter: -

1 - explicit scheme,

2 - implicit scheme,

3 - implicit - explicit scheme.

Yield criterion parameter:

1 - Tresca,

2 - Von Mises,

3 - Mohr - Coulomb,

4 - Drucker - Prager,

5 - Hoek - Brown,

6 - the extended Hoek - Brown.

Total number of proportional load increments

and excavation increments.

Number of stress components at a point.

3 - plane stress or plane strain,

4 - axisymmetric.

Deformation type: -

1 - small deformation theory,

2 - large deformation theory.

Card set 3: Element connection cards (1215)

- One card for each element (total NELEM cards)

cols.

1 - 5 NUMEL Element number.

6 - 1 0 MATNO(numel) Material property number.

11 - 15 LNODS(l,numel) 1st nodal connection number.

16 - 20 LNODS(2,numel) 2nd nodal connection number.

51 - 55 LN0DS(9,numel) 9th nodal connection number.

56 - 60 KNODE Number of nodes on the element.

Note:

a. Columns 3 1 - 5 5 remain blank for linear 4 - nodal elements,

b. Columns 46 - 55 remain blank for 7 - nodal elements.

329

c. Column 51 - 55 remain blank for 8 - noded element,

d. The nodal connection numbers must be listed in an anti - clockwise sequence,

starting from any corner node.

Card set 4- Nodal coordinate cards (15, 2F10.5)

One card for each nodal coordinate input.

cols.

1 - 5 IPOIN nodal point number.

6 - 1 5 COORD(l,ipoui) x (or r) coordinate of the node.

16 - 25 COORD(2,ipoin) y (or z) coordinate of the node.

N o t e :

a. The program will automatically compute:

i) midside nodes on a straight edge, and

ii) central nodes of 9 noded Lagrangian elements,

b. The total number of cards in this set will generally differ from NPOIN due to

the above point,

c. The last card must be the highest numbered node card regardless of whether

it is a midside node or not.

Card set 5: Restrained boundary node cards

(IX, 14, 5x, 15, 5X, 2F10.6)

One card for each restrained node (total = NVFIX).

cols.

1 - 5 NOFIX(ivfix) Restrained node number.

11 - 15 IFPRE Restraint code:

10 - nodal displacement prescribed

in X (or r) direction,

01 - nodal displacement prescribed

in y (or z) direction,

11 - nodal displacement prescribed

in both coordinate directions.

21 - 30 PRESC(l,ivfix) The prescribed value of the x (or r)

component of nodal displaxzement.

3 1 - 4 0 PRESC(2,ivfix) The prescribed value of the y (or z)

330

component of nodal displacement.

Card set 6: Material cards

Three cards for each different material.

Ist card: control card (15)

cols.

1 - 5 NUMAT Material identification number,

£nd card: properties card (8F10.5)

cols.

1 - 1 0 PROPS (numat,1) Elastic modulus, E.

11 - 20 PR0PS(numat,2) Poisson's ratio, u.

21 - 30 PROPS (numat,3) Material thickness, t (leave blank for

plane strain and axisymmetric problems).

31 - 40 PROPS (numat ,4) Mass density, p.

41 - 50 PR0PS(numat,5) Uniaxial yield stress, (7y - for

Von - Mises and Tresca material.

Cohesion, c - for Mohr - Coulomb

or Drucker - Prager material.

Uniaxial compressive strength, <Tc - for

Hoek-Brown and the extended H-B

material.

51 - 60 PRO PS (numat ,6) strain hardening parameter, H'

61 - 70 PR0PS(numat,7) Friction angle, (f> (measured in degrees)

for Mohr-Coulomb and Drucker-Prager

materials only.

71 - 80 PR0PS(numat,8) Fluidity parameter, 7.

3rd card: Properties card (continued), (4F10.5)

cols.

1 - 1 0 PROPS (numat ,9) The constant M or N for flow rule.

11 - 20 PROPS (numat, 10) Parameter defining flow rule: -

0 — exponential function,

1 — power function.

21 - 30 PROPS (numat,11) H-B empirical parameter, m.

31 - 40 PROPS(numat,12) H-B empirical parameter, s.

331

Card set 7: Load case title card (12A6)

One card

cols.

1 - 8 0 TITLE Title for load case (maximum of

72 characters).

Card set 8: Load control card (415)

One card

cols.

I - 5 IPLOD 1 - Applied point load (0 - otherwise).

6 - 1 0 IGRAV 1 - Gravity loading (0 - otherwise).

I I - 1 5 lEDGE 1 - Distributed edge load (0 - otherwise).

Card set 9: Point load card (15, 2F10.S)

One caxd for each nodal point

cols.

1 - 5 LODPT Node number.

6 - 1 5 POINT(l) Load component in x (or r) direction.

16 - 25 P0INT(2) Load component in y (or z) direction.

Note :

a. The last card should be the highest node number in mesh.

b. For NTYPE = 3, the load component should be the total load acting on the

circumferential ring passing through the nodal point.

c. If IPLOD = 0, omit this card set.

Card set 10: Gravity loading card (2F10.S)

One card

cols.

I - 1 0 THETA Angle at which gravity acts from the positive

y axis (anticlockwise positive).

I I - 2 0 GRAVY Gravitational constant expressed as a multiple

of the acceleration due to gravity, g.

Note: If IGRAV = 0, omit this card set.

Card set 11: Distributed edge load cards

332

Ist card: control card (15)

cols.

1 - 5 NEDGE Number of element edges on which distributed

loads are to be applied.

From the 2nd card, edge topology and load - two cards for each edge (total

number NEDGE).

i) Element edge topology card (415)

cols.

1 - 5 NEASS The element number, which the element

edge is associated.

6 - 1 0 NOPRS(l) First nodal number of the edge.

11 - 15 N0PRS(2) Second nodal number of the edge.

16 - 20 N0PRS(3) Third nodal number of the edge.

Note :

a. Nodal numbers of the loaded edge is counted in an anticlockwise sequence,

b. For linear 4 - noded elements, cols. 16 - 20 remain blank.

ii) Distributed load card (6F10.S)

cols.

PRESS{1,1) 1 - 1 0

11 - 2 0

21 - 30

Value of normal component of distributed

load at node NOPRS(l).

PRESS (1,2) Value of tangential component of distributed

load at node NOPRS(l).

PRESS(2,1) Value of normal component of distributed

load at node N0PRS(2).

3 1 - 4 0 PRESS(2,2)

4 1 - 5 0 PRESS(3,1)

51 - 60 PRESS(3,2)

Note:

a. If NNODE = KNODE = 4, columns 41 - 60 remain blank.

b. The element edge to be loaded can be considered in any order.

c. If lEDGE = 0 in card set 12, omit this card set.

Card set 12: Time stepping parameter card (4F10.3, 15)

333

One card

cols.

1 - 1 0

11 - 20

21 - 30

31 - 40

41 - 50

TIMEX Time stepping parameter, 0 .

TAUFT Time step length control parameter, T.

DTINT Initial time step length, SIQ.

FTIME Time step length control parameter, R.

NTICR Time step length control criterion:

1- 6t — mtn[6t{T),6t{k)],

2-6t = Min{6t{T),6t{k),6t{T)] ,

— where T is the critical length obtained

by Cormeau's formulae.

Card set 13: Loading or excavation control card (15)

One card

cols.

1 - 5 ILOAD 1 - Incremental loading,

2 - incremental excavation.

Card set 14: Incremental loading card (3F10.5, 715)

(If ILOAD = 1).

One card for eawrh proportional load increment.

cols.

1 - 1 0 FACTO

11 - 20

2 1 - 3 0

TOLER

TOLEV

31 - 35

3 6 - 4 0

MITER

NOUTP(l)

Proportional load factor for the current

increment specified as a feictor of the loading

read in card set 7 - 1 1 .

Convergence tolerance factor for the criterion

based on the ratio of the norms of the residuals.

Convergence tolerajice factor for the criterion

based on the sum of the effective viscoplastic

strain rate over all Gauss points at the first

stop of the load increment.

Maximum number of time steps allowed for

the load increment.

Results output every NOUTP(l) steps.

334

41 - 45 N0UTP(2) Parameter controlling the output of results at

steady state: -

0 - selective output,

1 - complete output.

46 — 50 NOUTP(3) Parameter controlling the output of

displacements, (l - yes, 0 - no).

5 1 - 5 5 NOUTP(4) Parameter controlling the output of

reactions (1 - yes, 0 - no).

56 - 60 NOUTP(5) Parameter controlling the output of

stress, (1 - yes, 0 - no).

61 - 65 IPLOT Parameter controlling plotting output,

1 — output produced,

2 — no output.

Note:

a. If ILOAD = 2, or for excavation steps, omit this card set,

b. The output facilities can be complete, selective or combined.

Card set 15: Incremental excavation card set

(Three cards for each proportional excavation increment)

Ist card: (SF10.5, 715)

cols.

1 - 1 0

11 - 20

21 - 30

31 - 35

3 6 - 4 0

41 - 45

46 - 50

51 - 55

56 - 60

61 - 65

FACTO

TOLER

TOLEV

MITER

NOUTP(l)

N0UTP(2)

NOUTP(3)

N0UTP(4)

N0UTP(5)

IPLOT

Incremental excavation factor for the

current step, specified as a proportional

factor referring to the remaining stress

and stiffness of the excavated elements.

Identical to the card set 14.

335

2nd card: (215)

cols.

1 - 5 NEXCA Number of elements to be excavated.

6 - 1 0 NEWLD Parameter controlling the excavation sequence,

1 — starting to excavate new elements,

2 — continuing to excavate the

current elements,

3 — continuing to excavate the

previously excavated elements.

Note; Excavations can be carried out in different orders for different groups of

elements.

Srd card: (* - free format input)

cols.

1 - 8 0 LEXCA(nexca) Excavating element numbers (in any order).

Note:

a. If ILOAD = 1, or for loading steps, omit this card set.

b. This caxd may not be enough when the excavation element number is greater

than 20. Changes of the input statement is then necessary.

336

APPENDIX 4 DERIVATION OF LARGE DEFORMATION

MATRIX [Bnl]

In the derivation of the nonlinear deformation matrix [BnL\t the explicit forms

of Lagrangian formulation is used (Zienkiewicz and Nayak 1971). Considering a

body in a Cartesian coordinate system as shown in Figure A.4.1, the coordinates

of a particle in the body in vector notation are:

{a:} = {a;o,yo}^. (A.4.1)

If the body moves to a new location under certain conditions and undergoes some

changes of configuration (deforms), the displacements and new coordinates are:

{«}« = (A.4.2)

= {xn, = {xo + yo + (A.4.3)

0 X

Figure A.4.1 Body deformation and displacement notation

337

The Green's strain can be written using the engineering definitions as

or:

{c}* =

r 2

{^}» — Cj/i fxy}„ —

(A.4.4)

(A.4.5)

where {ex,} is the usually linear, infinitesimal strain vector and {cArx,} can be

further written as:

r 2

dUrt dUrt , dVr^ dVn , dx dy dx dy ,

1 2

dx dx 0 0

0 0

dUn dVn dy dy

d^ dx

du„. dx

dx d^

dVn K dy )

(A.4.6)

By using isoparametric elements, the displacement vector can be written as

{«n} = [ N \ { 6 } , (A.4.7)

where [N] is the element shape function and {^} is the nodal displacement vector.

The incremental strain is given by differentiating (A.4.5):

d{e}n = d{€L}n + d{€NL}n, (A.4.8)

where

d{eL}n = d % d ^ , d ^ dy dx ,

1 ddu„. \

s . w ddUn I ddVn

dy dx (A.4.9)

Ur [BL]d j y" [ = [BL]d{6}.

338

Using (A.4.6), the differentiation of the second term of equation (A.4.8) is

(6.4.10)

and a transformation can be done by using the following property of [A] and {#}:

d{eNL}n = ^d[A\{e} +

d[A]{e} =

0 0

0 0

dvr,.

dx dx 0 0

0 0 d^ dy

dur^ . dy ^y

d^ dx

d^ dx .

r jdUr.

= [A]d{e}.

Thus

d{eNL}n =

The vector can be further written as:

w = dUr^ dy dVrL

y. dy )

— []{}>

where

[G] ^ 0 # 0

L 0 0 S g i J

Substituting (A.4.13) into (A.4.12):

d{eiiL} = [A\d{0} = [A]<i[G]{^}

= [A][G]rf{5} = [BNL\d{6}.

From equation (A.4.8) we now have:

d{€}n = {[BLU + [-B/\ri,]n)<i{^} =

(A.4.11)

(A.4.12)

(A.4.13)

(A.4.14)

(A.4.15)

(A.4.16)

339

where

[B\n = [BL\ +

For each node t, the matrix can be written in the explicit form:

[-Bfjn = [BLI] + rdNi

dx 0

0 dNi V +

dNi . dy

dNi St.

dUn dNi dx dUn dNi dy dy

dvn dNi dx dx dvn dNi dy dy

dun dNj . dun dNi dvn dNi , dvn dNi

Thus, the explicit form of the nonlinear B matrix is:

dur^ dNi dx

dy dy

d vr,. dNi dx dx dvn dNi dy dy

du„. dNi I dur^ dNi dy„. dNi , dy„. dNi , dy dx dx dy dy dx dx dy J

(A.4.17)

(^.6.18)

As we know,

^0 J

Vn = yn- yo, equation (A.4.17) can be written in an alternative simple form as:

[^t] =

^ 0 0 #

dXr^ dyn dx dx

I d ^ ^ I L dy dy J

The matrix [Jp] is the deformation Jacob ian matrix.

(A.4.19)

(A.4.20)

The matrix of Green's strain can also be expressed by [JD]'

€ —

L :{[JD]n[JD]n " [ - ^ ] ) >

and here.

[I] = 1 0 0 1

(A.4.21)

(AA.22)

Equation (A.4. 21) can be derived by substituting (A.4.19) into (A.4.4) and noting

that dxp _ dyo dx dy dxp _ dyo dy dx

= 1,

= 0.

340

APPENDIX 5 DESCRIPTIONS OF PROGRAM 'BLOCK'

A5.1 Program structure

Computer code BLOCK is a distinct element program using the static relaxation

formulation for rigid block system analysis. The program was described in detail

by Stewart (1981). The calculation procedure and its routine structure are shown

in Figure A.5.1 and A.5.2.

A brief description of eaxzh subroutine is given below:

BLOCK: The main program.

BEGIN: Starts a job, the job can be a start run or restart

run, a) if start run, it reads title; Initial problem definition,

plotting and printing intervals, updating and memory data

and initialises data, b) if restart run, it reads saved

information from tape 1 which was written by a previous job.

Reads in keywords and jumps to appropriate code. Returns control

to BLOCK when ready for job execution.

Once blocks have been created and before job execution,

it is called to perform the initial classification of block

corners into boxes.

CYCLE: This is the driving routine for the relaxation algorithm.

controls calls to most of the remaining routines.

MOTION: Calculates incremental rigid body displacements for relaxation

of a single block.

LOAD: Routine to calculate resultant surface forces on block.

ELIMIN: The equation solver solves the simultaneous linear equations

by Gaussian elimination with partial pivoting. Writes out appropriate

message if equations singular or ill-conditioned.

DATA:

BOX:

341

START

Search all blocks for contacts with their neighbours: UPDATE

Has the prog run for the specified no.

cycles ?

Yes

Relax a block and solve the force equilibrium equations for the unknown displacements.

Check if all the blocks have been

relaxed ?

Stop the program and print out the results.

Yes

Update all the contact forces between the neighbouring blocks.

Has any block moved more than the threshold

distance ?

Figure A.5.1 Static relaxation calculation procedure (after Stewart, 1981).

342

BEGIN

BLOCK

BLPLOT

DATA BOX DATA BOX

FINISH

LOAD

CYCLE MOTION

ELIMLN

FORD

UPDATE

REBOX

DUMP

HSTIF

JSTIF

Plotting routines:

BLPLOT FORPLT DISPLT TDISLP

VECTOR

TITLES

Figure A.5.2 Connection tree of the subroutines of program BLOCK.

343

FORD:

REBOX:

UPDATE:

DUMP:

HSTIFF:

JSTIF:

BLPLOT:

FORPLT:

DISPLT:

VECTOR:

TITLES:

FINISH:

Applies the constitutive laws for a single block.

Contact forces between blocks are determined from their positions

in space.

Routine to re-box a single block's corners. Relaxing is triggered

when a block centroid exceeds specified displacements.

Updates all contacts between blocks. UPDATE is

called either at regular intervals or when the maximum

cumulative block displax:ement exceeds a specified limit.

Prints out information in memory.

Routine to calculate displacements on boundary blocks carrying

external loads with external half - springs of double stiffness.

Routine to calculate nonlinear normal corner stiffness

after each system relaxation.

Plots block geometry.

Plots inter - block contact forces.

Plots incremental block displacements.

Draws vector.

Prints heading on plots.

Terminates the program and writes restart file on tape 2.

The program has general features of rigid block dynamic relaxation programs

except for the relaxation algorithm used in subroutine MOTION. The allocation

of memory is via linked list arrays stored in one large, single - subscripted array

Q, all storage locations being held simultaneously in core. By use of memory

pointers occupying fixed locations throughout the array, any selected information

can be economically referenced. (For further information of the memory partition

of array Q, see page 430 - 433, Stewart, 1981).

In the original program, arbitrary physical properties, such as normal and

shear stiffness, are assigned to any edge of any block. In order to incorporate a

nonlinear normal contact force - overlap relation, an alternative way of storage

is developed. Subroutine JSTIF is written to calculate the nonlinear stiffness of

each block corner according to the overlap which the corner has penetrated into

an edge. The locations for storing edge stiffness are then specified for storing the

related corner stiffness, arranged in the same order of the local corner coordinates.

344

The number of storage locations of array Q required by a problem depends on

the number of blocks and the nature of properties for each block. The maximum

amount of storage locations required can be estimated by the following formula:

M = (2 X NBLOKS) + (16 + 6 x NC) x NBLOKSl

+ (16 + 10 x NC) X NBLOKS2 + (16 + 14 x NC) x NBLOKSZ NBLOKS NBLOKS

(A5.1)

+ {NBOXES+ Y , {NCx3)+ Y , {NCONT X 15). i = l t=l

where

M: -the

NBLOKS: -the

NC: -the

NBLOKSl: -the

NBLOKS2: -the

NBLOKS3: -the

NBOXES: -the

NCONT: -the

A5.2 Input instructions :

Card set 1: Start and title card (12A6)

— Two cards

1st card (A4)

cols.

1 - 4 word STAR - start a new run.

2nd card Title card (8A10)

cols.

1 - 8 0 HED(80) Title of problem.

Card set 2: Dimension control card

— one card (4110)

345

cols.

1 - 1 0

1 1 - 2 0

21 - 30

31 - 40

NBLOKS Maximum number of blocks.

IBOXES Number of boxes in X - direction.

JBOXES Number of boxes in Y - direction.

IBSIZE Size of single box (boxes are square).

NOTE: If WORD = REST, omit this card.

Card set 3: Plotting control card

- One card (8110)

cols.

1 - 10 IBPLT Block plot interval.

11 - 20 IFPLT Contact force plot interval.

21 - 30 IDPLT Incremental displacement plot interval.

31 - 40 ITDPLT Cumulative displacement plot interval.

41 - 50 ICONDA Contact data arrays print interval.

51 - 60 IBLKDA Block data arrays print interval.

61 - 70 IBOXDA Box data arrays print interval.

71 - 80 IRFPO Resultant force print interval.

NOTE: If WORD = REST, omit this card.

Card set 4' Contact updating control card

- One card (2F10.0)

cols.

1 - 1 0

11 - 20

XYL

THL

Translation threshold.

Rotation threshold.

NOTE: If WORD = REST, omit this card.

Card set 5: Dimension card

- One card (110)

cols.

346

1 - 1 0 M7D Number of storage locations.

Card set 6: Controlling keyword

- One card (A4) (20 controlling keywords for choose)

cols.

1 - 4 CARD CREA - Reads in block properties.

DELE - Removes blocks.

DUMP - Prints contact forces.

CYCL - Commences job execution.

STOP - Stops the program.

PLOT - Plots block geometry.

INTF - Applies internal forces.

RSET - Sets real data.

ISET - Set integer data.

EXTF - Applies external concentrated forces.

EXTS - Applies external distributed forces.

STIF - Sets normal and shear stiflFness.

COHN - Sets joint cohesion.

FRIC - Sets friction coefficient.

RINV - Resets intervals for plotting and

resets size of memory.

MDFY - Resets thresholds for updating.

UPDA - Calls routine UPDATE.

DCON - Prints contact data.

ZTDP - Zeros total block displacements.

STCH - Reads in interface information

(only for hybrid analysis).

NOTE: After card set 6, the program jumps to the appropriate segments of code

to read in the relevant data or to call relevant subroutines.

Card set for key word CREA

The 1st card (15)

347

cols.

1 — 5 NBLOK Number of blocks to be considered.

Following two cards for each block are required, totally 2 X NBLOK cards.

2nd card: Block property card (110, FIO.O, 3110)

cols.

I — 10 NC Number of comers (< 50).

I I -20 RHO Block density.

2 1 - 3 0 JFIX 0 - block free to move,

1 - block fixed.

3 1 - 4 0 JBOUND External load flag,

0 - no external loads on block edges

1 - external loads applied to edges.

41 - 50 JINT Internal load flag.

0 - no internal loads on block edges,

1 - internal loads applied.

The 3rd card: Block corner coordinates card (8F10.0)

cols.

1 - 8 0 X(I), Y(I) 1=1, NC, (X and Y - global coordinates)

NOTES:

1. Block corner is numbered in a clockwise direction (starting from any corner).

2. If NC > 4, more than one card is needed.

Card set for keyword DELE

- Two cards

The 1st card (15)

cols.

1 - 5 NREM Number of blocks removed.

The 2nd card: (free format)

348

cols.

1 — 80 BNREM(k) Number of blocks to be removed

(maximum 19 blocks).

NOTE: If NREM > 19, extra caxd is needed.

Card set for keyword CYGL

- One card (IlO)

cols.

1 - 1 0 NCYC Number of iterations.

Card set for keyword INTF

- One card (2110, 4F10.0)

cols.

1 - 10 NB Block number.

11 - 20 NE Edge number.

21 - 30 INFL Force applied to lower corner.

31 - 40 INANGL Angle of lower corner force.

41 - 50 INFU Force applied to upper corner.

51 - 60 INANGU Angle of upper comer force.

NOTE: For the sign convention and definition, see Stewart, 1981).

Card set for keyword RSET

— One card for each set of data (110, FIO.O)

cols.

1 - 1 0 lADR Storage location in array Q.

11 - 20 VAL Value set, i.e. Q(IADR) = VAL.

Card set for keyword ISET

- One card for each set of data (2110)

cols.

349

1 - 1 0

1 1 - 2 0

lADR

IVAL

Storage location in array Q.

Value set, i.e. IQ(IADR) = IVAL.

Card set for each external force

- One card for each external force (2110, 4F10.0)

cols.

1 - 1 0

11 - 20

21 - 30

31 - 40

41 - 50

51 - 60

NB

NE

EXFL

EXANGL

EXFU

EXANGU

Block number.

Edge number.

Same as card set for INTF.

Card set for keyword EXTS

- One card (2110, 2F10.4)

cols.

1 - 1 0

11 - 20

21 - 30

31 - 40

NB

NE

EXSN

EXST

Block number.

Edge number.

Value of normal component of distributed

load.

Value of tangential component of

distributed load.

Card set for keyword STIF

- One card (110, 2E10.4)

cols.

1 - 1 0 NON

11 - 2 0

21 - 30

STIFN

STIFS

0 - linear normal stiffness,

1 - nonlinear normal stiffness.

Normal stiffness (if NON = 1,

it is the initial normal stiffness).

Shear stiffness.

350

NOTE: Different values of STIFN, STIFS may be assigned by using the RSET

command.

Card set for keyword COHN

- One card (F10.4)

cols.

1 - 1 0 COHES Joint cohesion.

NOTE: See above note.

Card set for keyword FRIC

- One card (F10.4)

cols.

1 - 1 0 FRIC Joint friction coefficient.

NOTE: See above note.

Card set for keyword RINV

— Two cards

The 1st card (8110)

cols.

1 - 8 0 (The same as card set 3).

The 2nd card (110)

cols.

1 - 1 0 (the same as card set 5).

Card set for keyword MDFY

— One caxd (2F10.0) cols.

1 - 2 0 (The same as card set 4).

Card set for keyword DCON

351

- Two cards

The Ist card (15)

cols.

1 - 5 NBCODD Number of blocks for dumping of

contact data.

The 2nd card (19F4'0)

cols.

1 - 8 0 (The same as the 2nd card of card set for

keyword DELE).

NOTE: All the other keywords do not require input cards.

A.5 .3 Components of the unit stiffness m a t r i x

In section 5.2.1, the unit stiffness matrix is expressed as:

[ ^ 1 = ®11 ®12 ®13 021 ®22 ®23 «31 O32 fl33

(A.5.2)

According to Stewart (1981), [A] matrix is symmetric and its elements can be

written as follows: NOP

Oil = ^ 2 (^11)*' t=l

NOP

®12 — ®21 — ^ ] (^12)*) t - 1

NCP

ai3 = 031 — ^ ] (^13)* > t=i

NOP

®22 — (^22)1' i=l

NCP

O23 — ®32 — ^ ] (^23)1' t=l

NCP

®33 — y ] ((^33)*'

(A.5.3)

*=1

352

in which the function can be explicitly written as:

= —Ksi cos^ a — Km sin^ a,

(^12)* = (^2i)» — ~^Si cos a sin a + Km cos a sin a,

(G!s).' = = (GSi)i(xi - x'J - (G!i)((k - y%

((^22)* ~ ~Ksi sin^ a. — K^i cos^ a,

- < ) - ( % ) ( ( % -1 / : ) ,

((^33)* = (((^23)1 — Fxi){xi — X®) - ((G^a),- + Fyi){yi — y®).

(A.5.4)

where

Km,Ksi'.

OL\

Xi,yi :

K^yl-

NCP:

are the normal and shear stiffness at the

ith contact point of block e.

is the angle between the horizontal axis x and the

block edge on which the contact point i lies.

are the contact point coordinates.

are the centroid coordinates of block e.

are the contact forces in global coordinates.

is the total number of contact points on block e.

353

APPENDIX 6 DESCRIPTIONS OF NEW SUBROUTINES

IN PROGRAM 'COUPLE'

The structure and flow chart of the hybrid program COUPLE have been described

in Chapter 5 (see Figure 5.21 - 5.22). As was mentioned in Section 5.2, the majority

of the subroutines utilised in the programs COAL and BLOCK. Following six

additional subroutines have been written for the coupling analysis:

COUPLE:

The main program controlling the type of analysis such as the finite

element analysis, the distinct element analysis or the hybrid analysis. It

calls subroutines COAL, BLOCK and BACK, as well as various plotting

routines.

BACK:

The subroutine for hybrid analysis. It starts a new cycle by updating the

displacement conditions after the finite element analysis and applying

these conditions for the distinct element analysis.

LFORC:

The subroutine for hybrid analysis. It calculates and stores contact forces

for the finite element ajialysis (called by CYCLE in BLOCK).

LOADFC:

The subroutine for hybrid analysis. It applies contact loads on the edges

and nodes of finite elements (called by LOADPS in COAL).

354

FEPLOT:

The routine for finite element mesh geometry plot, (called by COUPLE

or COAL).

STPLOT:

The routine for principal stresses plot in the finite element region (called

by COUPLE or COAL).

In accordance with the above new subroutines, a number of related routines

are modified. These include: BLOCK, COAL, DATA, CYCLE, INPUT, SFR2,

STEPVP, STEADY and so on.

355

APPENDIX 7 STRATA PROPERTIES AND MASUREMENTS

FOR CASE STUDY

The geological data, strata properties (strength parameters) and in situ mea-

surements presented in this appendix are from Baxter (1987), Malliory (1982) and

other internal reports of NCB. Only are some of the raw data included. Rep-

resentative in situ measurement results have been presented and interpreted in

Chapter 6. Some of their original data are shown here.

Geological data

The Warwickshire Thick Seam lies at a depth of 740 ~ 800 meters and is

about 5.5 ~ 7.5 meters thick. It is overlain by layers of mudstones, sajidstones

siltstones, etc. Below the seam is the seatearths, sandstones, mudstones, etc.

Typical geological sections are shown in Figure A.7.1 and A.7.2. The average

strength parameters for various layers of strata are also indicated in Figure A.7.2.

Rock strength parameters

A number of laboratory tests have been carried out to specify the strength

parameters of rocks at Coventry Colliery. Table A.7.1 shows the laboratory test

results on coal samples. The samples (number 1 to 31) were taken from the top to

the bottom of the thick coal seam. Table A.7.2 shows the elastic constants of the

coal given by the laboratory tests. Table A.7.3 shows the strength parameters of

other rocks, and Table A.7.4 also shows the strength parameter for pack material.

356

Muds tone

Two Y a r d

F i r e c l a y

B a r e C o a l

S h a l e

Ryde r

B l a c k S h a l e

E l l C o a l

N i n e F e e t and

H i g h M a i n

F i r e c l a y A ^ A

1 . 8 3 0 m

0 . 0 7 6 m

0 . 6 0 9 m

0 . 0 2 5 m

1 . 8 3 0 m

TOTAL

7 . 2 3 9 m

0 . 0 5 0 m

0 . 8 3 8 m

1.981 m

E x t r a c t e d C o a l

Figure A.7.1 Seam section of the Warwickshire Thick Coal.

357

SECTION OF STRATA IN THE NORTH ROCK HEAD WITH TEST RESULTS

"fe

1 40 —I

35 I 30

20 -

•s -

TTT

a

11 3€ I :i

hll

76-5 9-9 3 5

78-6 144 3 5

75 3 155 3 5

773 17-4 35

569 156 25

%3 84 2

575 125 25

505 135 25

673 141 3

573 128 25

m 137 4

730 13-5 35

509 131 25 1360 20-4 4-5 755 113 3-5

733 B-5 35

528 151 2-5

491 127 25 629 S I 3

TWO YARD

Seafearth mudsfone

Coarse Sandstone

Fine Sandstone

Silts tone

- • — Mud; tone silty

Mudstene

Coal

Figure A.7.2 Strata section of Coventry Colliery with test results.

358

SAMPLE COMPRESSIVE TRIAXIAL YOUNG'S POISSON'S TENSILE NO STRENGTH STRESS MODULAS RATIO STRENGTH

(MPa) FACTOR (k) (GPa) (MPa)

1 Coal 39.4 3.2 2.5 0.40

2 " 22.9 4.1 2.7 0.39 -

3 24.7 4.5 4.0 0.32 5.20

4 26.0 4.7 2.1 • 0.38 -

5 31.8 3.4 2.2 0.33 -

6 20.7 3.6 - - -

7 Mudstone 3.3 - - - -

8 Coal 31.4 4.0 2.0 0.32 -

9 " 37.2 2.7 3.4 0.43 -

10 " 48.3 2.4 4.1 0.46 - •

11 " 12.5 3.9 2.4 0.50 -

12 " 34.3 3.6 2.6 0.31 -

13 " 29.4 4.1 2.4 0.39 6.90

14 " 23.6 3.7 2.3 0.37 -

15 " 19.2 4.3 3.0 0.37 -

16 " 10.9 4.7 2.4 0.51 -

17 " 14.6 4.0 2.2 0.48 -

18 " 29.3 4.5 - - -

19 " 22.8 4.5 - - -

20 " 36.7 3.4 1.7 0.47 -

21 " 67.5 3.0 3.3 0.48 -

22 " 45.7 3.4 2.9 0.34 9.8

23 << 26.0 4.5 2.6 0.32 -

24 " 46.4 4.1 2.5 0.35 -

25 " 22.8 3.6 1.8 0.31 -

26 " 36.5 - - - -

27 " 26.1 4.5 2.8 0.40 -

28 " 20.5 3.7 3.0 0.37 3.6

29 " 24.5 4.5 - - -

30 " 33.3 3.5 2-8 0.41 -

31 << 17.0 3.3

Table A.7.1 Coal sample strength parameters by laboratory tests.

359

ELASTICITY

SAMPLE YOUNG'S POISSON'S LOAD SPECIMEN NO MODULUS RATIO RANGE DIAMETER

(GP ) (kN) (mm)

1 2.5 0.40 2 - 5 28

2 2.7 0.39 2 - 5 28

3 4.0 0.32 2 - 5 28

4 2.1 0.38 2 - 5 28

5 2.2 0-33 2 - 5 28

8 2.0 0,32 2 - 5 28

9 3.4 0-43 2 - 5 28

10 4.1 0.46 2 - 5 28

11 2.4 0.50 2 - 5 28

12 2.6 0.31 2 - 5 25

13 2.4 0.39 2 - 5 28

14 2.3 0.37 2 - 5 28

15 3.0 0.37 2 - 5 25

16 2.4 0.51 2 - 5 25

17 2.2 0.48 2 - 5 28

20 1.7 0.47 2 - 5 28

21 3.3 0.48 2 - 5 28

22 2.9 0.34 2 - 5 25

23 2.6 0.32 2 - 5 25

24 2.5 0.35 2 - 5 28

25 1.8 0.31 2 - 5 28

27 2.8 0.40 2 - 5 28

28 3.0 0.37 2 - 5 28

30 2.8 0.41 2 - 5 28

Table A.7.2 Elasticity constants of the coal samples by laboratory tests.

360

Conpreasion, Indirect Tensile

Sample No.

1061

Compreasive Strength - MPa Tenaile Strength UPa Sample

No. 1061

No. of teata made

Actual Equivalent K C B Standard H/D Ratio = 2:1

Tenaile Strength UPa Sample

No. 1061

No. of teata made

Actual Equivalent K C B Standard H/D Ratio = 2:1 No. of

Teata Average Range

Sample No.

1061 No. of teata made Average Range Average Range

No. of Teata Average Range

l. Mudatone 2 66.9 65.7-60.1 62.9 62.4-63.3 2 1,5.1 ll.'Mti.b l. Mudatone 2 2

B. Mudstone 4 50.5 46.7-55.8 49.1 45.3-53.9 3 12.7 12.2-13.3 B. Mudstone 4 SD - 3.8 SB = 4.4

3 SB - 0.6

D. Muddy siltstone 5 56,1 47.1-64.9 52.8 44.1-61.2 5 15.1 13.3-15.7 D. Muddy siltstone 5 SB - 7.0 SB = 6.1

5 SD - 1.0

E. Mudstone seatearth 6 78.0 73.3-80.2 73.3 67.4-77.2 5 13.5 13.0-1/.0 E. Mudstone seatearth 6

SB - 2.6 SB = 3.3 5

SB - 0.4 F. Silty sandstone 4 78.8 1 72.1-90.5 7 5 . 5 1 70.2-86.9 3 11.3 1 9.9-12.7 F. Silty sandstone 4

SB - 8.3 SB = 7.8 3

SB - 1.4 3. Sandstone with dark

laminae 6 139.7 104.3-199.0 136.0 96.8-189.6 5 20.9 1 17.6-23.9 3. Sandstone with dark

laminae 6

SB - 32.7 SB - 31.1 5

SB - 2.6 H, Muddy siltstone 6 52.9 48.3-63.0 50.9 47.2-61.5 5 13.1 1 11.9-14.0 H, Muddy siltstone 6

SB - 5.3, SB = 5.3 5

SD . 0.9 JK. Silty sandstone 6 78.9 1 72.9-86.7 75 .2 69.6-83.1 5 13.9 ' 12.9-14.8 JK. Silty sandstone 6

SB = 5.4 SB . 0.8

L. Sandstone 6 98.5 1 86.5-116.8 95.2 1 84.5-113.6 5 13.7 13.1-14.3 L. Sandstone 6

SB . 10.3 SB = 10.2 SB - 0.5

Note: 10 UPa = 1450 Ibf/in^ SD denotes Standard Deviation

Compression, Indirect Tensile

Sample No, 1061

Compressive Strength - MPa Tensile Strength UPa Sample

No, 1061

No. of tests made

Actual Equivalent N C B Standard

H/d Ratio = 2:1

Tensile Strength UPa Sample

No, 1061

No. of tests made

Actual Equivalent N C B Standard

H/d Ratio = 2:1 No. of Tests Average Range

Sample No, 1061

No. of tests made Average Range Average Range

No. of Tests Average Range

M. Muddy siltstone 6 63.4 60.4-66.1 57.3 52.8-62.6 4 12.8 12.3-13.1 M. Muddy siltstone 6

SD = 2.4 SB = 4.0

4 SD - 0.3

N. Siltstone 4 70.5 1 68.7-72.7 6 7 . 3 1 65.6-69.6 3 14.1 1 13.9-14.4 N. Siltstone 4 SB = 1.7 SD = 1.9

3 SD . 0.3

0. Mudstone 4 51.4 1 47.9-54.8 50.5 1 47.2-54.1 5 13 .5 1 12.5-15.6 0. Mudstone 4 SD - 2.9 SB = 2.9

5 SD - 1.2

P. Muddy alltstone 5 63.2 1 58.6-66.5 57.7 1 54.2-62.4 5 12.5 1 11.9-13.6 P. Muddy alltstone 5 SB - 3.1 SD = 3.6

5 SD - 0.6

Q. Mudstone seatearth 4 32.9 1 26.3^1.0 31.3 1 25.4-39.2 2 8.4 1 8.2-8.6 Q. Mudstone seatearth 4 SB - 6.1 SB = 5.9

2

R. Slialy muds tone R. Slialy muds tone Too degraded to test

5. Black mudstone 4 61.2 1 57.6-64.4 56.9 1 53.9-61.2 3 15.6 13.3-16.9 5. Black mudstone 4 SB - 3.5 SB . 3.5

3 SD . 2.0

J T. Mudstone 6 81.8 74.8-89.1 77.3 1 71.3-^.5 6 17.4 15.3-19.4 J T. Mudstone 6 SB - 5.0 SD = 5*6

6 SB - 1.5

' n, Mudstone/sandstone 6 60.9 1 76.6-86.1 78.8 1 74.6-84.0 5 14.4 1 14.3-14.5 ' n, Mudstone/sandstone 6 SB - 3.7 ^ •" 3*6

5 SB - 0.1

" 7. Sandstone 6 7 8 . 8 1 66.5-87.1 76.5 1 64.6-85.0 6 9.9 8.0-11.3 " 7. Sandstone 6 SD - 8.0 SD - 7.4

6 SB - 1.2

SD denotes Standard Deviation

Table A.7.3 Stength parameters of other rock samples by laboratory tests.

361

Compression, Indlreot Tensile

Note: 10 MPa . 1/1)0 Itf/in^ ST denotes Standard Deviation

Coomressive Strength - UPa Tensile Strength

Sample Mo.

1061

No. of tests

Actual Equivalent N C B Standard

H/b Ratio = 2:1

•tra Sample Mo.

1061

No. of tests

Actual Equivalent N C B Standard

H/b Ratio = 2:1 No. of Tests made

Average Range

Sample Mo.

1061

No. of tests

Average Range Average Range

No. of Tests made

Average Range

(A 1 4 33.3 32.1-35.0 31.6 28.9-34.0 No tensiles

o(. 2 4 27.1 20.2-33.5 26.0 19.6-26.9

No tensiles

2« Sandstone seateartl from 'beneath thick coal in BHj at 117 m

6 56.9 43.6-66.3 55.9 42.7-65.1 5 12.3 11.1-13.8 2« Sandstone seateartl from 'beneath thick coal in BHj at 117 m

6

SD - 8.5 SD - 8.4

5

SD - 1.0

3. Sandstone with coaly plant remains SH5s EHO at 11? m

2 68.0 66.7-69.3 60.0 57.5-62.4 1 No tensiles

3. Sandstone with coaly plant remains SH5s EHO at 11? m

2 1 No tensiles

A. (Q) Seatearth, siltj mudstone slightly ferruginous

6 28.6 17.2-39.6 27.5 16.8-37.5 5 8.6 7.1-10.1 A. (Q) Seatearth, siltj mudstone slightly ferruginous

6

SD - 7.5 SD = 7.0

5

SD - 1,2

B- Washout sandstone at 530 m along the South Main Return

6 77.2 1 66.3-84.5 73.7 1 63.6-80.2 5 14.2 1 12.8-15.9 B- Washout sandstone at 530 m along the South Main Return

6

SB = 6.7 SD = 6.5

5

SD . 1.3

C, Siltstone 56O m along the South Main Return '

6 79.3 71.1-83.8 76.9 67.7-81.6 5 13.3 12.7-14.1 C, Siltstone 56O m along the South Main Return '

6

SD - 4.7 SD - 5.1 SD - 0.7

D. Mudstone laminated with sandstone from beneath the Seven Feel Seam in SW$s new coal-Kate

6 62.6 55.0-67.1 59.3 51.4-64.3 5 10.1 9.5-10.4 D. Mudstone laminated with sandstone from beneath the Seven Feel Seam in SW$s new coal-Kate

6

SD = 4.3 SD = 4.8 SB « 0.4

E. Mudstone with spha«rosiderite

6 48.4 42.4-52.7 46.9 41.2-51.3 - -E. Mudstone with spha«rosiderite

6

SB = 4.1 SD = 4.0

Table A.7.3 (Continued) Stength parameters of other rock samples by laboratory tests.

Sample

Uniaxial Compressive Strength (MPa)

Stress at Failure (MPa) Triaxial Stress Factor (k)

Sample Actual Equivalent

NCB Standard 2:1 H/D Ratio

Confin

21 MPa

ing Press

14 MPa

ure

7 MPa

Triaxial Stress Factor (k)

Aquapak 19.8 19.8 51.3 38.7 31.0 1.5

Mudstone 35.1 35.0 87.0 - - 2.5

Two Yard 31.3 29.8 102.9 - - 3.5

. 2 Note: 10 MPa = 1450 Ibf/in

Table A.7.4 Strength of rock samples around the roadway and Aquapack pack samples.

362

Load cell readings

Load (kN) 40

30 40 50

Face advance (m)

a)

Load cell readings

Load (kN)

20 +

10 20 30 40 50

Face advance (m)

60 70 80

b)

Figure A.7.3 Road cell results at Coventry Colliery, a) yieldable arches; b) rigid arches.

363

120 T

1 0 0 -•

Penetration (mm)

Leg penetration

10 20 30 40 50

Face advance (m)

a)

Leg penetration

60 70 80

Penetration (mm)

400 T

300 • •

2 0 0 -•

1 0 0 ••

10 20 30 40 50

Face advance (m)

60 70 80

b)

Figure A.7.4 Measurements of arch leg penetrations, a) Yieldable arches; b) rigid

arches.

364

Support profile

2500 T

2000

1500 -•

Closure (mm)

1000

500

0 10 20 30 40 50

Face advance (m)

60

floorlift

lateral convergence

^ roof convergence

i 1 70 80

Closure (mm)

- 1 0 0

a)

Profile

800 T

700 ••

600

500 ••

400

300 ••

200

100

=1,1— 1 1 I 1 0 \ ^ 2 0 ^ ^ 3 0 4 0

lateral convergence

floorlift

' 50——60, .—70, 80 roof convergence

Face advance (m)

b)

Figure A.7.5 Roadway closures at Coventry Colliery, a) yieldable arches; b) rigid

arches.

365

Measured roadway deformations

An average roadway support pressure can be estimated by the measurements

shown in Figure A.7.3 in which the loadcell readings are plotted versus the face

advance. Figure A.7.4 shows a measurement of arch leg penetration as the face

advances. The measurements of roadway closures for different support systems

are shown in Figure A.7.5. Floor-lift of the roadway are plotted in Figure A.7.6.

Floor l i f t

2500 T

2 0 0 0 -

Floorlift (mm)

1500 --

1 0 0 0 ••

500 ••

3 0 4 0 5 0

Face advance (m)

rigid

Figure A.7.6 Measurements of floor-lift at Coventiy Colliery.

366

APPENDIX 8 REST OF THE FIGURES

In this appendix, several more figures for Chapter 3 and 6 are given. Figure A.8.1

shows the plots of Hoek-Brown surface for various of rock masses. These are drawn

by using the different values of m and s according to Hoek and Brown (1980). The

influence of cr on the surface shape is also shown. Figure A.8.2 plots the Mohr-

Coulomb yield surfaces and the extended H-B yield surface for comparison with

the related Hoek- Brown surfaces. Figure A.8.3 shows plots of results from another

numerical simulation, in which principal stresses were calculated by assuming that

the pack strength is high enough to support the roof strata. A relatively high

stress concentration is clearly seen in the pack whereas the stresses in the ribside

are much lower than those predicted by the simulations mentioned in Section 6.4.

Figure A.8.4 shows mesh deformations as the excavation proceeds for a case that

floor s trata axe relatively weak. A muck large deformation of the floor strata is

seen.

367

HOEK ANO GROWN SURFACE n = 2 5 . 0 0 0 5 : 1 . 0 0 0 0 S I O M A - C = 3 5 . 0

HOEK ANO GROWN SURFACE M = 1 5 . 0 0 0 5 = I . 0 0 0 0 S I C n A - C = 3 5 . 0

HOEK ANO BROWN SURFACE n = 7 . 0 0 0 S = I . 0 0 0 0 S I C H A - C = 3 5 . 0

HOEK ANO BROWN SURFACE M = 1 2 . 5 0 0 5 = 0 . 1 0 0 0 S I O f l A - C = 3 5 . 0

HOEK AND GROWN SURFACE M = 7 . 5 0 0 S = 0 . 1 0 0 0 S l O n f l - C = 3 5 . 0

HOEK ANO BROWN SURFACE n = 3 . 5 0 0 S = 0 1 0 0 0 S I O M A - C = 3 5 . 0

Figure A.8.1 The Hoek-Brown yield surface for various rock masses.

368

HOEK AND BROWN SURFACE

HOEK AND BROWN SURFACE f1 5 0 . 7 0 0 S = 0 . 0 0 4 0 S l C n A - C = 3 5 . 0

HOEK ANO GROWN SURFACE n = 0 . 5 0 0 S = 0 . 0 0 0 1 SIGMA-C = 3 5 . 0

HOEK ANO BROWN SURFACE ft = 0 . 3 0 0 S = 0 . 0 0 0 1 S IGMA-C = 3 5 . 0

HOEK ANO BROWN SURFACE « = 0 . 1 WO 5 = 0 . 0 0 0 1 SICMA-C z 3 5 . 0

Figure A.8.1 (Continued) The Hoek-Brown yield surface for various rock masses.

369

HO£K ANO GROWN SURFACE M = 0 . 0 6 0 S r 0 . 0 0 0 0 S I G M A - C = 3 5 . 0

HOEK ANO BROWN SURFACE M = 0 . 0 8 0 S = 0 . 0 0 0 0 S I C M A - C = 1 0 0 . 0

HOEK ANO BROWN SURFACE M = 1 0 . 0 0 0 5 = 1 . 0 0 0 0 S I G M A - C = 2 . 0

HOCK ANO BROHN SURFACE O = 0 . 0 0 7 S = 0 . 0 0 0 0 S I G M A - C = 3 5 . 0

Figure A.8.1 (Continued) The Hoek-Brown yield surface for various rock masses.

S I M P L E F I E O H - 6 SURFACE M = 0 - 0 6 0 S = 0.00001 S I G M A - C = 3 5 . 0

Figure A.8.2 The Mohr-Coloumb and the extended H-B yield surface.

370

= 3 0 t iPf l

+ + + + + + + + + ++ ++4+4 +++++ + + + + + + +

+ + + + + + + + + ++ +++++ +4+4-f + + + + + + + + + + + + + + + + ++ ++4++ +++++ + + + + + + +

+ + + + + + + + + ++ +++++ +++++ + + + + + + + + + + + + + + + + ++ ++4++ +4+++ + + + + + + + + + + + + + + + + ++ +++f+ +4+++ + + + + + + + + + + + + + + + + ++ 4++++ +4+4-4- + + + + + + +

+ + + + + + + + + nil 1144 1 1 Mill 1 + + + + + + + + + + + + + + + + 44+H++4 11 mil 1 + + + + + + + + + + + + + + + + 1 m 1144 11 mil 1 + + + + + + + + + + + + + + + + 1 1 111144 11 iimi + + + + + + +

+ + + + + + + + + 1 1 111144 -+++HH+ + + + + + + + + + + + + + + + + II H 1144 II mill + + + + + + +

+ + + + + + + + + -1 1 II 1144 II mil 1 + + + + + + +

a)

+ + + + + + + + + ++ ++4+4+4+++ + + + + + + +

+ + + + + + + + + ++ ++4+4+4+++ + + + + + + +

+ + + + + + + •f + ++++4+++++ + + + + + + +

+ + + + + + •f Tt -V Jr + + + + +

+ + + + + f /X XXJfWY-XXXX Jr Jr + + + + +

+ + + + + + y. XX Xjr-H-+T s.xxx Jc Jr + + + + + + + + + + + Y- Tt XX xJr++++-)t>txx Jr Jr + + + + +

f /X XJf-M-W-SXXX -V Ar # 4:

1:

+ + + + + + + + + JrJrXXxx.T-f-VJrJo<>4<Tt + + + + + + + + + + + + + + + Jr JrJcX>oo<.-/-f-+VJo«««<X •f + + + + +

+ + + + + + + Jr Jr )rxxxXA-H-++J»x»(X y- Tt + + + + + + + + + + + + Jr Jr JrXxo V++JrJ w<x y- + + +

+ + + + + + + Jr Jr JfJrWxx xww y y- •f + + +

+ + + + + + + Jr Jr JrJrJtkJOr Jr+yy. -A4W/. Y- •f + + + +

+ + + + + + + + + ++J*VrJr+-A<--AW+ + + + + + + +

b)

Figure A.8.3 Plots of principal stresses for a case of strong pack support. (for geometry being considered, see Figure 6.15b, page 253.

371

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375