A Study on Deformation of Tunnels

Excavated in Fractured Rocks

MÉMOIRE

Amir Rahim Khoshboresh

Maîtrise en génie civil Maître ès sciences (M.Sc.)

Québec, Canada

© Amir Rahim Khoshboresh, 2013

iii

RÉSUMÉ

La déformation due au fluage d'un massif rocheux autour d'un tunnel a été rencontrée fréquemment. Ce

phénomène est évident où il y a des tunnels creusés dans la roche tendre, des masses rocheuses faible et

fortement cisaillées, ou des massifs rocheux soumis à des contraintes in-situ élevées. La déformation due au

fluage se produit fréquemment au moment d’excavation des tunnels longs où il y a des failles et des zones

fracturées et cisaillées. Ce phénomène peut causer différents dommages sur des systèmes de soutènement

en raison de la déformation excessive et des effondrements. La déformation excessive impose une ré-

excavation de la section du tunnel, qui monte le coût supplémentaire, la durée de la réalisation du projet et le

risque de la sécurité sur le projet. En plus, comme la stabilité de terrain est dans un état critique durant la ré-

excavation, une petite négligence peut conduire à une grande caverne. Bien que la déformation de fluage est

commune dans un massif rocheux à une faible résistance dans un tunnel très profond, mais ce phénomène a

été observé dans des tunnels peu profonds.

Une bonne compréhension des déformations causées par une excavation souterraine requiert la

connaissance de l'interaction roche-support et l'interprétation des données de terrain. Auparavant, l’objet

principal de la surveillance effectuée durant la construction du tunnel était des mesures de la pression au

terrain imposé sur le revêtement du tunnel. Mais aujourd’hui, les méthodes modernes de construction de

tunnel se concentrent sur la surveillance des déplacements pendant et après la construction.

Afin de déterminer des déformations dans les tunnels, Panet et Sulem ont supposé que "Le tunnel a une

section transversale circulaire et le milieu est homogène et isotrope, aussi le tunnel est suffisamment profond

pour considérer que la distribution des contraintes est homogène". Mais dans le cas quasi réel, la distribution

de la contrainte autour du tunnel est hétérogène et anisotrope. Dans cette étude, pour la modification des

équations Panet et Sulem, certaines équations sont proposées en cas de matériau hétérogène et anisotrope

pour généraliser le problème.

La galerie de force motrice Seymareh a été considérée comme l’étude de cas. Celle-ci est une partie du

conduit d’eau dans le projet de centrale électrique du barrage Seymareh. Ce projet est situé à l'ouest de l'Iran.

Les données de surveillance de la galerie de force motrice sont collectées au moment de l’excavation du

tunnel, et sont comparées avec les résultats de la modélisation numérique et de la solution analytique. Cette

comparaison montre que les résultats des données expérimentales obtenues par la surveillance sont très

proches des résultats de la solution analytique, mais il y a une différence entre les deux et la modélisation

numérique. Il était prévisible, car l’effet d’autres activités comme l’excavation des tunnels verticaux n’est pas

prise en compte dans l’analyse numérique et aussi dans la solution analytique. Il est évident que les autres

activités comme l’excavation des tunnels verticaux et l'excavation du tunnel principal vers deux directions

opposées, peuvent affecter sur les résultats de la surveillance. D'autre part, les données initiales utilisées

dans l'analyse numérique et la solution analytique ne sont pas tout à fait exactes, car elles sont obtenues en

tant que représentatives du massif rocheux de la région, mais pas pour une section particulière. Toutefois, le

but de cette étude est le développement d'une solution analytique de la déformation dans les tunnels sur les

conditions générales et la poursuite de cette étude pourra être plus développée.

iv

v

ABSTRACT

The creep deformation of a rock mass around a tunnel has been encountered frequently. It is particularly

common in tunnels excavated in soft rock, heavily sheared weak rock masses or rock masses subjected to

high in-situ stresses. Creep deformation in fault and shear fractured zones are one of the frequently

encountered difficulties in long tunnel construction, which tend to cause failure of supporting systems due to

excessive deformation and cavern. Excessive deformation would necessitate re-mining of the tunnel cross

section, thus imposing impacts such as extra cost, extended time schedule and safety risk on the project.

Furthermore, as the ground stability is in critical condition during re-mining, the slightest negligence would lead

to major cavern. Although creep deformation is common to extremely poor rock mass under high overburden

in a tunnel alignment, but however this phenomenon is not limited to tunnels with high overburden.

A good understanding of the deformations caused by an underground excavation requires simultaneously

knowledge of the rock-support interaction and interpretation of field data. Formerly, the main purpose of the

monitoring carried out during tunnel construction was to measure the ground pressures acting on the tunnel

lining. Modern tunneling practice emphasizes the monitoring of the displacements occurring during and after

the construction.

Panet and Sulem for determining of deformations in tunnels have assumed that "The tunnel has a circular

cross section and around the tunnel, the rock is homogeneous and isotropic and also the tunnel is deep

enough to consider that the stress distribution is homogenous". But in almost real cases, the stresses

distribution around the tunnel is not homogeneous and isotropic. In this study, for modification of the Panet and

Sulem equations, some equations are proposed in case of nonhomogeneous and anisotropic for generalizing

of the problem.

Seymareh power tunnel which is considered as a case study is a part of the powerhouse waterways

system of the Seymareh dam and hydroelectric power plant project. The project is located in west of Iran. The

monitoring data of power tunnel which are collected during excavation of tunnel is compared with the results of

numerical modelling and analytical solution results as well as. The results obtained from comparison show

although the field data, which are collected through the monitoring, are very close to the analytical solution

results (approximately), but there is a significant difference between both of them and numerical modelling

results. It was predictable; because the influence of the other activities such as excavation of shaft and surge

tank in the numerical analysis and also analytical solution are not considered. It is obvious that other activities

such as excavation of shaft and surge tank and also excavation of mean tunnel from other direction which

were under operation at the same time can effect on the results of monitoring. On the other hand, the initial

data which are used in numerical analysis and analytical solution are not quite accurate; because they are

extracted as a representative of the rock mass of region, not for a particular section. However the goal of this

study is development of analytical solution of deformation in tunnels on general conditions and pursuit of the

study could be leaded to more development in this field.

vi

vii

CONTENTS

Résumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Scope of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Chapter 2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1 Theoretical Background, Time-dependent behavior Models . . . . . . . . . . . . . . . . . . 9 2.1.1 Visco-elastic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.2 Visco-elasto-plastic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.3 Empirical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Previous Researches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Chapter 3 Theories of Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1 Rheological models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.1.1 Maxwell rheological model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.1.2 Kelvin and Voight rheological model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1.3 Bingham rheological model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Phenomenological models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2.1 Application to stress-strain behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2.2 Application to strain-time behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Chapter 4 Study of Creep phenomenon in intact rocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.2 Influence of deviator stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.3 Influence of confining pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.4 Effects of water content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.5 Effect of the rate of crack formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.6 Effect of dilatancy hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Chapter 5 Study of Creep phenomenon in fractured rocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.2 Fracture shear strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.2.1 Unfilled discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.2.2 Filled discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.2.3 Time-dependent shear strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.3 Creep tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.3.1 Unfilled discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

viii

5.3.2 Filled discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Chapter 6 Deformation analysis in tunnels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.2 Convergence measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 6.3 Models used in rock tunnel engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.4 Deformations caused by a tunnel driven in an elastic medium . . . . . . . . . . . . . . . . . 59 6.4.1 The axisymmetric case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.4.2 Anisotropy of the initial state of stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 6.4.3 Anisotropic elastic rock mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 6.5 Deformations caused by a tunnel driven in a yielding rock mass . . . . . . . . . . . . . . . 68 6.5.1 The development of a plastic zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.5.2 The ground-support interaction for an elasto perfectly plastic medium . . . . . . . . . . . 71 6.5.3 The ground-support interaction for a tunnel driven in an elastoplastic rock mass with a strain-softening or brittle behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 6.5.4 Allowance for the weight of the yielded rock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.6 Time-dependent deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.6.2 The viscoelastic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.6.3 The viscoplastic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Chapter 7 An Analytical Solution for deformations of tunnels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.1 Kirsch equations for estimation of induced stresses around a tunnel . . . . . . . . . . . . . 89

7.2 Modification of Panet fictitious support pressure coefficient () . . . . . . . . . . . . . . . . . . 90

7.3 Generalized Hoek-Brown failure criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 7.4 Modification of the Panet equations using non-linear Generalized Hoek-Brown

failure criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Chapter 8 Case study, Seymareh power tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 8.2 Geotechnical & Geological Features of the Region . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 8.3 Specifications of Powerhouse Waterways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 8.4 Excavation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 8.5 Tunnel Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 8.6 Data obtained by Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 8.7 Deformation Analysis using Numerical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 8.8 Analytical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 8.8 Comparison of the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Summary, Conclusions & Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

9.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 9.2 Recommendations for future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Apendix-A COMSOL Multiphysics software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Apendix-B Hoek-Brown failure criterion-2002 edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

ix

LIST OF TABLES

Table 2.1. Visco-Elastic Creep Models 11

Table 2.2. Empirical Models (After Farmer ,1983) 14

Table 5.1. Basic friction angles of various un-weathered rocks obtained from flat surfaces (Modified after Barton and Choubey 1977) 39

Table 5.2. Shear strength of filled discontinuities and filling material (Modified after Barton1974) 41

Table 5.3. Friction angle of mineral as a function of dry and wet surface (Modified after Rummel1982) 41

Table 5.4. Values for coefficients

and for granite and quartzite at a normal stress of approximately

10 MPa (After Dieterich 1972) 43

Table 8.1. Characteristics of rock mass discontinuities in the Seymareh dam site 102

Table 8.2. Geomechanical parameters of rock mass in the Seymareh dam site 103

Table 8.3. Some of the geotechnical features of the Asmari rock formation in direction of the Seymareh power tunnel 103

Table 8.4. Maximum values of the convergence in four sections of the tunnel (mm) 109

Table 8.5. Some of the material properties in the sections under study 110

Table 8.6. the Norton-Bailey creep parameters 112

Table 8.7. the Hoek-Brown parameters for Seymareh intact rocks 113

Table 8.8. the ratio of horizontal applied stress to vertical applied stress ( ) 114

Table 8.9. Value of in the roof, floor and sidewalls of the tunnel in four sections 114

Table 8.10. Value of (radius of the plastic zone) in the roof and floor and in the sidewalls of tunnel in four sections (m) 114

Table B.1 Guidelines for estimating disturbance factor D 138

x

LIST OF FIGURES

Figure 1.1. Roof squeezing during hydroelectric project in Nathpa Jhakri, India (After Hoek, 2000) 2

Figure 1.2. Squeezing deformation in tunnel, Taiwan (After Hoek, 2000) 2

Figure 1.3. Collapse of Mucha tunnel in Taiwan (After Hoek, 2000) 2

Figure 1.4. Squeezing rock reduces the tunnel cross section in the Saint Martin La Porte access adit (Lyon-Turin Base Tunnel). The photographs show re-mining works needed in order to cope with the difficult conditions encountered 3

Figure 1.5. Idealized creep curve showing axial strain versus time including the three major creep stages: I) primary creep, II) secondary creep, and III) tertiary creep 5

Figure 2.1. Simulation of rock creep behavior: (a) spring and dashpot model (Burger substance) simulating creep behavior for rock loaded in uniaxial compression; and (b) typical creep curve for Burger substance 9

Figure 2.2. Basic Rheological Models: (a) Spring, (b) Dash Pot (After Dusseault & Fordham, 1993) 11

Figure 2.3. Visco-Elastic Creep Models: (a) Maxwell Model, (b) Kelvin Model, (c) Generalized Maxwell Model, (d) Generalized Kelvin Model, (e) Burgers Model (after Goodman, 1989) 12

Figure 2.4. Visco-Elasto-Plastic Creep Model (a) Model Constitutions, (b) Strain vs Time 13

Figure 2.5. Characteristics of Rock Creep: (a) Phenomenological Behaviour of Rock under Uniaxial Tension, (b) Typical Creep Behaviour under Confining Stresses. (after Dusseault & Fordham, 1993) 15

Figure 3.1. Basic types of elements used in rheological models 21

Figure 3.2. Rheology curves of the Hooke’s and Newton’s elements 22

Figure 3.3. Maxwell rheological model with demonstration of the stress and strain behaviour 24

Figure 3.4. Kelvin and Voight rheological model with demonstration of the stress and strain behaviour 25

Figure 3.5. Bingham rheological model with demonstration of the flow curve 26

Figure 3.6. Examples of the stress-strain behaviour under different conditions together with illustration of the connection with creep curves 28

Figure 3.7. Illustration of the primary, secondary and tertiary phases of the creep 29

Figure 3.8. Solution of the strained-based differential equation in the form of relation between degree of primary consolidation and time factor 30

xi

Figure 3.9. Variation of e with log t under a given load increment, and definition of coefficient of secondary compression 32

Figure 4.1. Intact rock creep movements extrapolated from results of uniaxial creep laboratory tests of short duration, assuming creep obeys the creep law ( ). For the upper curve

( ) the axial stress corresponded to 40% of the compressive strength (Pusch and Hökmark 1992) 34

Figure 5.1. Hypothetical stress-deformation path of a discontinuity at a constant normal stress (After Amadei and Curran 1982) 36

Figure 5.2. Shear stress versus normal stress for a schematic direct shear test 38

Figure 5.3. The increase in friction angle for a joint in granite with wear of the surface (After Lajtai 1991) 42

Figure 5.4. The increase in the friction angle following stationary contact in a direct shear test (After Lajtai 1991) 43

Figure 5.5. Axial strain versus time for a joint in water-saturated granite under constant principal stress difference (s1–s3 = 60.5 MPa) and confining pressure (s3 = 20.7 MPa). Note that the creep rate after 30 hours is very small. The slope of the curve gives an approximate value of 1·10–11 s–1. (After Wawersik 1974) 44

Figure 5.6. Shear creep behaviour of an excavation-induced tensile fracture in porphyry lava with a stepwise increase in shear stress. The normal stress was 1 MPa. The sample was tested at the ambient humidity of 50%. (Malan et al. 1998) 45

Figure 5.7. Creep curves for intact and jointed samples at 0.4 sc stress level. parameter q represents the angle between the direction of the applied axial stress and the direction of the normal to the joint plane (Schwartz and Kolluru 1981) 46

Figure 5.8. Creep displacements of an artificial discontinuity in shale. The uppermost curve in the top graph is the lower most curve in the bottom graph (After, Bowden and Curran 1984) 47

Figure 5.9. Shear creep behaviour of a discontinuity with bedding plane gouge -collected from Hartebeestfontein mine. The normal stress was 1 MPa. The gouge thickness was 2 mm and the humidity 50% (After Malan et al. 1998) 48

Figure 5.10. The effect of shear stress magnitude on the creep behaviour. The percentages indicate the magnitude of the shear stress relative to the shear strength (0.28 MPa). The normal stress was 0.5 MPa and the gouge thickness 2 mm (After Malanet al. 1998) 49

Figure 5.11. The effect of shear stress magnitude on the creep behaviour of a gouge filled fracture. The percentages indicate the magnitude of shear stress relative to the shear strength (0.28 MPa). The normal stress was 0.5 MPa and the gouge thickness 2 mm (After Malan et al. 1998) 49

Figure 5.12. Initial creep velocity (v after 1 min) as a function of filling thickness. The filling material was Kaolin with a clay content of 0.75 and a consistency index of 0.75. The normal stress was 1 MPa and the ratio (R = t/tR) was varied between 0.7 and 0.9. (After Höwing and Kutter 1985) 50

xii

Figure 5.13. Steady-state creep rate as a function of filling thickness. The filling material was a medium to fine sand. The (t/tp) ratio was 0.86 for the 0.5 mm thickness, while 0.87 for the 1 mm and 2 mm thickness. The samples were tested at 50% humidity. (After Malan et al. 1998) 51

Figure 5.14. Initial creep velocity (v after 1 min) as a function of particle size distribution of the filling material. The filling thickness was 2 mm and the consistency index 0.75. The normal stress was 1 MPa. The diagram on the left shows a shear stress ratio R = 0.7 and the diagram on the right a ratio R = 0.9. (After Höwing and Kutter 1985) 52

Figure 6.1. Section equipped with four pins to measure the convergences in a horseshoe-shaped tunnel 54

Figure 6.2. Graphs for the analysis of distance versus time, convergence versus time and convergence versus distance 56

Figure 6.3. Convergences versus time at three adjacent stations of the Fréjus Tunnel 57

Figure 6.4. Instability of blocks at the periphery of a tunnel (unstable key blocks are hatched) 58

Figure 6.5. Radial displacement versus the distance to the face of excavation in the elastic axisymmetric case 60

Figure 6.6. Variation of ( ) versus x ( ) in the elastic axisymmetric case 60

Figure 6.7. The equivalent plane strain problem 61

Figure 6.8. Distributions of radial stress and tangential stress according to the value in the elastic axisymmetric case 62

Figure 6.9. The stress and strain paths at the wall of the tunnel in the elastic axisymmetric case 62

Figure 6.10. The ground characteristic curve (1) and the support characteristic curve, (2) in the elastic axisymmetric case 63

Figure 6.11. Buckling failures in laminated rocks 67

Figure 6.12. The shape of the plastic zone according to the value 69

Figure 6.13. Stress and strain paths at the wall of a tunnel in the elasto perfectly plastic axisymmetric case 72

Figure 6.14. The intersection of the ground and the support characteristic curves in the elastoplastic case 73

Figure 6.15. The curve giving the radial displacement in the elastoplastic case may be obtained by a simple homothety of the curve in the elastic case, according to Corbetta et al 73

Figure 6.16. Determination of the deformation at distance d from the excavation face, according to Corbetta et al 74

Figure 6.17. A trilinear stress-strain law for strain-softening material 75

xiii

Figure 6.18. The elastic zone (I), strain-softening plastic zone (II) and residual zone (III) around a tunnel 76

Figure 6.19. Stress and strain paths at the wall of the tunnel in the elasto strain-softening plastic axisymmetric case without residual plastic zone 77

Figure 6.20. Stress and strain paths at the wall of the tunnel in the elasto strain-softening plastic axisymmetric case with a residual plastic zone 77

Figure 6.21. Stress and strain paths at the wall of the tunnel in the elasto brittle axisymmetric case 79

Figure 6.22. Allowance for the dead weight of the yielded rock 81

Figure 6.23. Various time functions ( ) 83

Figure 6.24. Some unidimensional rheological models used for analyzing time-dependent underground deformations 84

Figure 6.25. Ground characteristic curves for some rheological models 85

Figure 6.26. Kelvin model in the axisymmetric case: (a) radial displacement for an unsupported

tunnel, (b) pressure acting on a support installed at 86

Figure 6.27. Determination of the supporting pressure with a viscoelastic model by the use of the ground characteristic curves 87

Figure 7.1. Two-dimensional distribution of stresses around an opening in an elastic body 90

Figure 7.2. Fictitious support pressure coefficient 90

Figure 7.3. The distributions of radial stress and tangential stress according to the value in the elastic axisymmetric case for two values for and 92

Figure 7.4. Variation in tangential boundary stress in the sidewalls and also roof and floor of a

tunnel with variation in the ratio of applied stresses 93

Figure 7.5. Variation in redial boundary stress in the sidewalls and also roof and floor of a tunnel with variation in the ratio of applied stresses 94

Figure 7.6. " a " constant values for variation of Geological Strength Index (GSI) 95

Figure 7.7. Cross-section showing the broken and the elastic zones 97

Figure 8.1. The location of the Seymareh Dam site, Iran 100

Figure 8.2. Summarized geology of the Symareh 101

Figure 8.3. Location of Power systems 104

Figure 8.4. Water supply and power generation system 105

xiv

Figure 8.5. Seymareh power tunnel during construction 106

Figure 8.6. Monitoring system of the Seymareh power tunnel, a) Tape Extensometer, b) Rod Extensometer c) Load Cell 107

Figure 8.7. Reference points grouted into drill holes in rock 107

Figure 8.8 the location of the four monitoring mean station under study 108

Figure 8.9 the three reference points used for measuring of convergence in the tunnel 108

Figure 8.10 the results of convergence monitoring of the tunnel walls (in a period of 100 days) 109

Figure 8.11. Meshing of the numerical modelling (7932 triangular elements) 110

Figure 8.12. Horizontal displacements (X component) in tunnel walls for four sections in case of time independent (mm)

111

Figure 8.13 Distribution of the plastic zones in four sections 112

Figure 8.14 Horizontal displacements (X component) of side walls during 100 days, in case of time dependent

113

Figure 8.15. Horizontal displacements (X component) of side walls during 100 days, Analytical solution

115

Figure 8.16. Comparison between the experimental data (monitoring), numerical analysis (FEM) and analytical solution

116

Figure A.1. Screenshot of COMSOL Multiphysics Software 4.2a topography modelling of surfaces, Mount St. Helens, US 124

Figure A.2. available add-on products for COMSOL Multiphysics software 125

Figure A.3. Effective plastic strain in the soil sample 127

Figure B.1. Relationships between major and minor principal stresses for Hoek-Brown and equivalent Mohr-Coulomb criteria 134

Figure B.2. Relationship for the calculation of σ′3max for equivalent Mohr-Coulomb and Hoek-Brown parameters for tunnels 136

xv

ACKNOWLEDGEMENTS

I would like to extend my sincerest thanks to my thesis supervisor, Professor Adolfo Foriero, for his

guidance and support throughout my work in this research.

The data of Seymareh power tunnel is provided by the Iran Water & Power Resources Development

Company (IWPCO). Also some of the geotechnical and instrumentation data is provided by Mahab Ghodss

Consulting Engineering Company. I also would like to thank them, especially their technical offices at the dam

site for their assistance and providing valuable data.

Special thanks are due to my dear wife Elham and my dear children Arvin and Shervin. The study would

not have been possible without their encouragement and full supports.

1

1 INTRODUCTION

Time-dependent deformations in a hard rock mass may be manifested as creep of the intact material

and /or creep along the discontinuities. In intact crystalline rock at normal temperatures creep is primarily the

product of time-dependent micro fracturing of the rock; which will produce both shear and volumetric strains if

the rock is dilatant (Schwartz and Kolluru 1981, Rummel 1982). The principal factors controlling creep rates in

intact rock are applied deviator stress, effective confining pressure and moisture content (Kirby and

McCormick 1984). Creep in rock joints occurs as normal compression and shear movements along the

discontinuities. Factors that govern the creep rates in discontinuities are the character of the joints and the

geometry of the joint system with respect to the excavation (Ladanyi 1993). The total time-dependent response

of a rock mass depends on the relative contribution from each source.

It is usually considered that creep deformations in fractures overshadow intact rock creep in hard rock

masses. The reason is that the shear strength of a fracture generally is less than that of intact rock, in

particular along those that are coated or filled with material (Bowden and Curran 1984, Höwing and Kutter

1985). The movements will arise along the joint system with most unfavourable conditions and direction with

respect to the excavation. In rock masses with high deviatoric in situ stresses and a sparse, interlocked

fracture system, a possibility exists that intact rock creep could dominate.

Time-dependent creep deformation of a rock mass around a tunnel has been encountered frequently. It is

particularly common in tunnels excavated in soft rock, heavily sheared weak rock masses or rock masses

subjected to high in-situ stresses (Verman, et al., 1988; Bai et al., 1991; Phien-wej, 1991; Barla, 1995;

Matsushita & Mizoguchi,1995; Bhasin & Grimstad,1996). Numerous case histories of severe creep movement

have been observed in the world in recent years.

Creep deformation in fault and shear fractured zones are one of the frequently encountered difficulties in

long tunnel construction, which tend to cause failure of supporting systems due to excessive deformation and

cavern. Excessive deformation would necessitate re-mining of the tunnel cross section, thus imposing impacts

such as extra cost, extended time schedule and safety risk on the project. Furthermore, as the ground stability

is in critical condition during re-mining, the slightest negligence would lead to major cavern. Creep deformation

is common to extremely poor rock mass under high overburden in a tunnel alignment. However this

phenomenon is not limited to tunnels with high overburden. Figures 1.1, 1.2 and 1.3 show the tunnel collapses

due to creep deformation in some of tunnels in the world.

2

Figure 1.1 Roof squeezing during hydroelectric project in Nathpa Jhakri, India (After Hoek, 2000)

Figure 5.2 Squeezing deformation in tunnel, Taiwan (After Hoek, 2000)

Figure 1.3 Collapse of Mucha tunnel in Taiwan (After Hoek, 2000)

3

These case histories demonstrate that tunnels excavated in soft rock or heavily sheared rock masses may

experience severe squeezing. The excessive tunnel convergence due to the long term creep movement of the

surrounding rock mass can cause severe damages to the support works which endangers the stability or

service function of the tunnel. In many cases, the tunnel convergence is so large that it requires the

replacement of the damaged supports or even the re-mining of the intruding rock mass. The remedial

measures are expensive and difficult to carry out. It is one of the major causes of delay in construction

programmes and cost over-runs in tunnelling projects.

In the tunnel that develops severe squeezing due to the creep of the surrounding rock mass, it is often very

difficult to control the tunnel movement to a very small rate prior to the construction of the secondary or

permanent concrete lining. Therefore, the secondary lining may be subject to increasing rock pressure until

final equilibrium is reached or damage to the structure occurs. However, the current practices of primary rock

support and secondary lining design are highly empirical. They can not consider the effects of a creeping rock

mass on the stability of a tunnel and the safety of the lining structure.

The term “squeezing” originates from the pioneering days of tunnelling through the Alps. It refers to the

reduction of the tunnel cross section that occurs as the tunnel is being excavated (Figure 1.4). Based on the

work of a Commission of the International Society for Rock Mechanics (ISRM), which has described squeezing

and the main features of this mechanism, it is agreed that “squeezing of rock” stands for large time dependent

convergence during tunnel excavation. This happens when a particular combination of material properties and

induced stresses causes yielding in some zones around the tunnel, exceeding the limiting shear stress at

which creep starts. Deformation may terminate during construction or continue over a long period of time.

Figure 1.4. Squeezing rock reduces the tunnel cross section in the Saint Martin La Porte access adit (Lyon-Turin Base Tunnel). The photographs show re-mining works needed in order to cope with the difficult conditions encountered

4

The magnitude of tunnel convergence, the rate of deformation, and the extent of the yielding zone around

the tunnel depend on the geological and geotechnical conditions, the in situ state of stress relative to rock

mass strength, the groundwater flow and pore water pressure, and the rock mass properties. Squeezing is

therefore synonymous with yielding and time dependence, and often is largely dependent on the excavation

and support techniques being used. If the support installation is delayed, the rock mass moves into the tunnel

and a stress redistribution takes place around it. On the contrary, if deformation is restrained, squeezing will

lead to long term load build-up of the support system.

Deformation analysis of tunnels during excavation and after can be done using computational or

experimental methods. Practical difficulties exist for tunnel engineers having to consider complex geology and

unpredictability in material behaviors, leading frequently to mismatch between numerical prediction and field

measurement results. Time required for computational analysis, especially when nonlinear material behavior

has to be considered, is another obstacle even considering advance in computation speed in these days. On

the other hand, in a modern observational method for tunneling, geology and geo-mechanical properties of the

rocks are monitored, tunnel face observation for rock mass condition recorded, as well as the presence of

underground water checked. Recently, numerical methods based on the finite elements technique have been

used widely for the analysis of stress and deformation of tunnels including plasticity, fracture mechanics,

viscoelasticity and many others. A number of works have been reported in geotechnical papers. The most

signification program in the process of developing practical analysis tools for underground excavations and

tunnels has occurred in the area of two-dimensional stress and deformation analysis. But the development of

the three-dimensional stress and deformation analysis codes is difficult, because of the mathematical

complexity of their formulations and the programming skills required for their efficient implementation.

A number of computer programs, based on the boundary elements method, finite elements method and

finite difference are now available commercially. While some of the programs model the rock mass as a linear

elastic, isotropic and homogeneous body, others are capable of handling anisotropy, arbitrary non-

homogeneity and geometric as well as material non-linearity.

1.1 Concepts

Creep is defined as the time-dependent deformation of rock under a load that is less than the short term

strength of the rock. Creep strain can seldom be recovered fully when loads are removed, thus it is largely

plastic deformation.

Creep stages: Rock creep is usually expressed as a creep function relating strain to time. In its most

general form, this creep function can be expressed as (Jaeger and Cook 1976):

5

( ) ( ) ( )

where, is the total creep strain; ( ) is the primary or transient creep, is the secondary or steady-

state creep, and ( ) is the tertiary creep or accelerating creep which occurs just prior to failure of the

sample. The general form of these three stages is illustrated in Figure 1.5. In the primary stage, the creep

velocity decreases steadily with time, in the secondary stage the creep velocity is constant with time, and

finally in the tertiary stage the creep velocity increases with time.

Figure 1.5. Idealized creep curve showing axial strain versus time including the three major creep stages: I) primary creep, II) secondary creep, and III) tertiary creep.

Any of the three stages described above may dominate the deformation process, depending on the

material and the testing conditions. The schematic curve in Figure 1.5 illustrates a case where the applied load

is :

• Sufficiently high to take the rock sample through all creep stages,

• Constant over time.

Here, “sufficiently high” means that the load constitutes a sufficiently high fraction of the short-term

strength. If that fraction is low, then it may not even suffice to take the rock sample into the secondary creep

stage, in which case the creep movement will slow and eventually die out.

Creep movements around openings in the repository do not take place under constant load conditions. The

effect of creep movements is to reduce the stress anisotropy such that rock volumes, or parts of individual

fractures that have entered stage II, are likely to fall back into stage I.

6

Generaly, most formulations of creep in rock suggested in the literature can be separated into two main

categories, empirical creep functions, based upon curve fitting of experimental data, and rheological creep

functions, based upon creep behaviour models composed of assemblages of elastic springs, viscous

dashpots, plastic slider and brittle yield elements.Common forms of these two types of creep functions are

described by Dusseault and Fordham (1993) and Ladanyi (1993). These will be more discussed in the next

chapters.

1.2 Objectives

The pressure on the lining of an underground space or a tunnel increases with time. This is due to the

time-dependent mechanical behaviour of the surrounding rocks or soil. If, however, the tunnel support is

installed at the tunnel face immediately after its excavation, the pressure increases with progress of the tunnel

face. The increase of displacements and pressures with time are brought about, on one hand, by the progress

of the tunnel face, and on the other hand, by the time-dependent mechanical properties of the rock masses.

The models take into account both, or separately these two factors. The rheological behaviours of rock are

viscoelastic, visco-plastic or include the swelling characteristics for some type of rocks. The effects of these

two different factors must be clearly separated in the time-dependent analysis of tunnel support.

In the case of a circular tunnel driven through a homogeneous isotropic linear visco-elastic medium, Panet

proposed an analytical solution for the convergence of the tunnel walls as a function of two parameters, the

distance to the face and the time. For a similar problem, a closed form solution for the pressure acting on

tunnel support structures was given by Sakurai. Sulem et al. presented an analytical solution for time-

dependent displacements in a circular tunnel, in case of homogeneous and isotropic. For the sake of

simplicity, they have considered a linear law (Mohr-Coulomb failure criterion).

In this study, after describing a summary of the time-dependent behaviour in tunnels, especially the tunnels

excavated in fractured rocks and introducing theories of creep generally in intact rocks and fractured rocks, a

new relationship for generalizing of the Sulem Fictitious support pressure coefficient in case of non-

homogeneous and anisotropic will be proposed, and consequently a closed form solution for the determination

of wall displacements and ground pressure acting on the tunnel support for modifying the Sulem equations

using non-linear Generalized Hoek-Brown failure criterion in case of a time-dependent stress-strain and failure

behaviour will be presented. This method is based on the analysis of the interaction between the ground which

tends to deform toward the excavation and the support which limits the closure by exerting a confining

pressure.

7

As a case study, "Seymareh power tunnel" will be introduced and by means of numerical modelling of

Seymareh power tunnel using of COMSOL Multiphysics software, validity of the proposed equations will be

evaluated. The achieved results of numerical modelling, analytical solutions and experimental data

(instrumentation) will be compared and discussed.

At the end, after summary of achieved results and conclusions, some recommendations for future studies

are proposed.

Generally, the purpose of this study is to develop a practical method to evaluate time-dependent behaviour

of tunnels in fractured rocks and to estimate the long term displacements around the tunnels and the stresses

induced due to the creep phenomenon.

1.3 Scope of work

Generally, the rock mass consists of intact rock blocks and discontinuities around them. Both of them

contribute to the creep movement of the rock mass, but the effect of the discontinuities and joints is much

more effective rather than the rock blocks, unless the rock blocks are soft. On the base of practical

experiences, the tunnels which are constructed in the rock of good quality seldom develop time-dependent

creep movement. Most tunnels that experienced creep movements are excavated in soft rock formations or

heavily fractured rock masses.

In this study, it will be focused in the creep phenomenon on the tunnels which are constructed in fractured

rock masses. Since the results of the study is based on previous studies, such as generalized Hoek-Brown

failure criterion; validity of the results depends on estimation accuracy of the initial parameters and engineering

judgement in determination of rock mass properties.

1.4 Methodology

The second chapter is about literature review. An introduction to the time-dependent behaviour models is

presented in this chapter. Theoretical background and also previous researches are discussed in chapter two.

Chapter three is dedicated to the creep theories and models, involving the overview of basic rheological,

phenomenological and numerical creep models.

Study of creep phenomenon in intact rocks is discussed in chapter four. Furthermore, Influence of time in

rock behaviour, such as shear strength and displacements is discussed in this chapter.

8

Also, study of creep phenomenon in fractured rocks is presented in chapter five. Some effective

parameters in creep movements, like influence of deviator stress, influence of confining pressure, influence of

water content, influence of loading rate and influence of temperature on creep parameters, is studied and

discussed in chapter five.

Chapter six describes time-dependent behavior of tunnels and presents some behavioural models in tunnel

engineering.

An analytical solution for time-dependent displacements in tunnels is proposed in chapter seven. In fact,

analytical solution of Sulem et al for time-dependent displacements in circular tunnels is modified for general

condition of heterogeneity and anisotropy. Kirsch equations for estimation of induced stresses around a tunnel,

modification of Sulem fictitious support pressure coefficient (), generalized Hoek-Brown failure criterion and

finally Sulem equations modification using non-linear Generalized Hoek-Brown failure criterion are presented

in this chapter.

As a case study, Seymareh power tunnel is introduced in chapter eight. At the start of chapter eight there is

a basic description of the Seymareh power tunnel from the viewpoint of geotechnical & geomechanical

features of the Seymareh region. Deformation analysis of Seymareh power tunnel by using numerical

modelling (COMSOL Multiphysics software) and consequently comparison of numerical results with

experimental data and analytical results is discussed.

The final chapter, Chapter nine, summarizes the main conclusions and comments based on chapters

seven and eight. The new improved findings which can be helpful for the other creep studies are presented.

Finally some recommendations are given for further studies.

Seymareh power tunnel data (as a case study) is provided by the Iran Water & Power Resources

Development Company1 (IWPCO). Also some of the geotechnical and instrumentation data is provided by

Mahab Ghodss consulting engineering company2.

1. Iran Water and Power Resources Development Company (IWPCO) is the major employer of big dam and hydro electric power

plants in Iran (www.iwpco.ir)

2. Mahab Ghodss is a consulting engineering company which provides engineering consultancy services particularly in water

industry (www.mahabghodss.com)

9

2 LITERATURE REVIEW

2.1 Theoretical Background, Time-dependent behavior Models

The creep behavior of rock can be simulated by models comprising springs and dashpots, with the spring

representing elastic strain and the dashpot representing viscous strain. The best simulation of rock creep is

provided by a combination of a spring and a dashpot in series and another spring and dashpot in parallel

known as a Burger model (Figure 2.1).

Figure 2.1 Simulation of rock creep behavior: (a) spring and dashpot model (Burger model) simulating creep behavior for rock loaded in uniaxial compression; and (b) typical creep curve for Burger model.

The axial strain with time, (t), in a Burger substance subjected to a constant axial stress is (Goodman,

1980):

( ) {

[ ( )]

} ( )

10

where [ ( )] is the bulk modulus (assumed to be independent of time), determines the

rate of delayed viscosity, determines the rate of viscous flow, determines the amount of delayed

elasticity, and is the elastic shear modulus.

Values for the viscoelastic constants can be obtained by conducting creep tests either in the laboratory, or

in situ by means of radial jacking tests or plate jacking tests. The general procedure is to measure both the

elastic strain, and the strain with time from which the strain rate and the intercepts and (Figure 2-1) are

determined. The viscoelastic constants are calculated from these measured results using equations 2.2 and

2.3 (Goodman, 1980):

{

} ( )

{

} ( )

The constants and are determined from equation 2.4 where q is the positive distance between the

creep curve and the line asymptotic to the secondary creep curve at any time t.

(

)

( )

A semilog plot of versus has intercept ( ) and slope (- ). Generaly, in order to

describe rock creep, many mathematical models have been developed. These models can be grouped into

three categories of visco-elastic model, visco-elasto-plastic model, and finally empirical model.

2.1.1 Visco-elastic Model

Visco-elastic models are simple rheological models which generally comprise basic mechanical models,

such as a spring and dash pot, as shown in Fig. 2.2, to simulate a range of time-dependent behaviour. These

models include:

elastic model

viscous model

Maxwell model

Kelvin model

Generalized Maxwell model

Generalized Kelvin model

Burger model, and others

11

These are described and illustrated in Table 2.1 and Fig. 2.3. More complex models can be derived from

the combination of basic mechanical models of springs and dash pots, where = stress, = strain,

= Young‘s modulus, = bulk modulus, = shear modulus, = viscosity, = time, subscript denotes

Kelvin model, subscript denotes Maxwell model. However, they do introduce more parameters and the

physical meaning can become unclear and hard to use in practical applications. Goodman (1980, 1989)

concluded that for many practical purposes for rocks, the Burger creep model is preferable and will suffice for

the description of most rock creep behaviour if proper parameters were selected.

Figure 2.2 Basic Rheological Models: (a) Spring, (b) Dash Pot (after Dusseault & Fordham, 1993)

Table 2.1- Visco-Elastic Creep Models

Type Property Stress-Strain-Time Relation in One Dimension

Spring Elastic

{

}

Newtonian Viscous ( )

Maxwell Visco-elastic ( )

Kelvin Visco-elastic ( )

[ (

)]

Generalized Maxwell Visco-elastic ( )

[ (

)]

Generalized Kelvin Visco-elastic ( )

[ (

)]

Burger Visco-elastic ( )

[ (

)]

12

Figure 2.3 Visco-Elastic Creep Models: (a) Maxwell Model, (b) Kelvin Model, (c) Generalized Maxwell Model, (d) Generalized Kelvin Model, (e) Burger Model (after Goodman, 1989)

2.1.2 Visco-elasto-plastic Model

The visco-elasto-plastic model extends the linear visco-elastic model to cover the non-linear viscosity when

the stress is high or the rock approaches yield (Zienkiewicz & Cormeau, 1974; Bodner & Partom, 1975; Kanchi

et al.,1978; Gioda,1981; Gioda & Cividini, 1981; Cividini, et al., 1991). The typical model consists of a Kelvin

model in series with a Bingham model, as shown in Figure 2.4 (Bai,et al., 1991). If the applied stress does not

cross the yield limit, it will exhibit the same creep behaviour as those described in a Kelvin model. But if the

stress is over the threshold, the non-linear, visco-plastic behavior will be mobilized. The stress-strain-time

relationship of the visco-elasto-plastic model (see Figure 2.4) in one dimensional form is as follows:

when

( )

[ (

)] ( )

13

when

( )

[ (

)] ( )

( )

where denotes the yielding strength or failure strength. The tertiary creep is introduced by considering

the loss of shear strength and apparent viscosity of the rock with increasing shear strength.

(a)

(b)

Figure 2.4 Visco-Elasto-Plastic Creep Model (a) Model Constitutions, (b) Strain vs Time

2.1.3 Empirical Model

Empirical creep models were generally derived from the observed relationship between stress-strain-time

in the creep test, and the parameters involved can be determined by curve fitting of the test data obtained in

the laboratory or field. Sometimes they can possess very complicated mathematical forms with a large number

of parameters. Also, the models are normally developed for certain types of rock only and may not be

applicable to other rock types or testing conditions. Table 2.2 lists some of the empirical creep laws reviewed

by Farmer (1983), where A, B, C, D are constant parameters. Senseny (1983) reviewed eight empirical creep

14

laws for rock salt and concluded that although all of them were capable of adequately reproducing laboratory

creep strain measurements at constant stress and temperature, none of them had firm physical bases.

Table 2.2 - Empirical Models (After Farmer ,1983)

type Empirical Models

1

2

3

4

5

6

7

8 ( )

9 ( )

10 ( )

11 ( )

12 ( )

13 ( )

14 ( )

15 [ ( )]

16 [ ( )] [ ( )]

17

18

19

20

21 [ ( )]

22 [ ( )]

23 [ ( )]

24 [ ( )]

15

2.2 Previous Researches

Creep is the time-dependent deformation behaviour of a solid material under sustained load. It can be

noticed almost in all construction materials and natural media such as metal, concrete, bitumen, soil and rock.

The research of creep can be dated back to the nineteenth century as described by Dusseault & Fordham

(1993). They mainly focused on the progressive rupture of metal under tensile stress. The study of creep

behaviour of rock started early this century. The main approaches in the early days of rock creep research

were to observe the creep behaviour of rock

Specimen subjected to compression, tension, bending and torsion in a laboratory (Clark, 1965; Jaeger &

cook, 1969 &1979; Vyalov, 1970; Cogan, 1976; Lama & Vutukuri, 1978; Langer, 1979; Goodman, 1980 &

1989; Farmer,1983; Cruden, et al., 1987; Dusseault & Fordham, 1993; Cristescu, 1993; Wilson, 1993). The

general rock creep phenomenon under a sustained load on a rock sample can be divided into three stages as

shown in Figure 2.5. The sample exhibits instantaneous elastic strain when a constant load is applied. If the

load is sustained longer or the stress level increases, the primary (transient) creep or the attenuating creep

occurs. At high stress, secondary creep or steady state creep becomes apparent. If the applied stress

approaches or passes the yield limit or the material strength, the strain will increase rapidly and a tertiary creep

or accelerating creep will appear and lead to eventual failure of the specimen.

Figure 2.5 Characteristics of Rock Creep : (a) Phenomenological Behaviour of Rock under Uniaxial Tension, (b) Typical Creep Behaviour under Confining Stresses. (after Dusseault & Fordham, 1993)

Early research work on rock creep was concentrated on the time-dependent post-failure strength of

originally intact rock (Bieniawski 1970) and jointed samples (Kaiser and Morgenstern 1979). This topic is not

considered to be relevant under the stress and strain conditions encountered in a repository. A number of

studies of time-dependent shear strength in discontinuities have been related to the investigation of

earthquake mechanisms, focusing on time-dependent shear strength in deep faults (Dieterich 1972). Other

16

applications for studies of the time-dependent shear strength in discontinuities have been the stability of rock

slopes and long-term strength of hard rock masses (Lajtai and Gadi, 1989, Lajtai 1991). The investigations

show that under certain conditions the frictional resistance of a discontinuity may increase and impede the

development of steady-state creep. This fact is of some importance for the current review and a description of

the studies is given below. (Dieterich 1972) performed direct shear tests on greywacke, granite, quartzite and

porous sandstone. His investigations focused on the relationships between the duration of stationary contact

and the static friction coefficient of unfilled and gouge filled rock joints. Samples were prepared by sawing

blocks with subsequent lapping to obtain flat and parallel slip surfaces of a required roughness. The duration of

stick was varied at a constant normal stress by holding the shear stress slightly below the peak strength. The

static friction at the end of a desired interval was then measured by rapidly increasing the shear stress until the

block moved. The time interval was between 1 sec and 24 hours and the normal stress was varied between 2

and 85 MPa. The results show that the coefficient of static friction of joints is time independent for a clean

rough joint surface while a joint with gouge exhibits a highly time-dependent behaviour. Static friction increases

with the time that adjacent blocks remain in stationary contact. Dieterich suggests that compaction of the

gouge separating the slip surfaces is time controlled and determines frictional strength. One implication of the

work is that, in gouge filled joints mainly primary creep should be expected, since the joint friction coefficient

increases with time.

Lajtai and Gadi (1989) studied the time-dependence on friction in fractures planes. They performed direct

shear tests on smooth, planar rock blocks of Lac du Bonnet granite. The test specimens were cut by diamond

saw and ground to the required size with subsequent polishing of the shear surface. The sample size

amounted to 40 mm by 90 mm. The maximum displacement amounted to 500 mm achieved by repeated

shearing of the same surface in steps of 25 mm. The normal load ranged from 0.2 to 8 MPa. No steady state

displacement was observed in any of the tests. The frictional resistance in the tests increased with both

displacement and time. The increase of the frictional resistance of an initially smooth and polished surface

during continuing shearing displacement is due to wear. The results confirm the findings by Dieterich (1972)

that most likely creep is transient in rock joints under shear conditions that accumulate gouge fill.

Creep movements in unfilled rock joints have been studied by a number of workers with reference to rock

engineering and design of underground structures. Uniaxial tests and direct shear tests have been performed

to investigate creep in fractures close to the ground surface (Amadei and Curran 1982, Bowen and Curran

1984, Schwartz and Kolluru 1981, 1982). Triaxial tests have been performed to investigate creep in deep

seated fractures and faults under high confining pressure (Wawersik 1974, Solberg et al. 1978). The work of

the above mentioned researchers is described below (Amadei and Curran 1982) conducted triaxial and direct

shear tests on unfilled and clean rock joints under a variety of stress states and surface conditions. The rock

types tested were oven dried (120°C) sandstone, limestone, marble and granite. Triaxial creep tests were

17

performed on artificially jointed cylindrical samples of approximately 54 mm diameter (NX size), with a

diamond saw cut at a 30-degree angle with respect to the axis of the sample. A few intact rock samples were

also tested in order to separate creep of intact rock from creep along the joint. The direct shear tests

comprised samples 127 mm by 127 mm in size, including an artificially prepared horizontal saw cut. Stresses

used in the experiments ranged up to 9.5 MPa normal stress and 5.7 MPa shear stress in the triaxial tests and

0.5 MPa normal stress and 0.3 MPa shear stress in the direct shear tests. Creep displacements were recorded

over periods ranging from 5 to 300 hours. In general, the creep behaviour of unfilled discontinuities was found

to be similar to that of an intact rock. Of special importance for the current study is that the results show the

occurrence of a threshold value, related to the applied shear stress – shear strength ratio. Beyond a ratio of

approximately 0.5 the creep movement increases greatly for a joint in a marble sample.

(Bowden and Curran 1984) investigated the creep behaviour by direct shear tests on artificially prepared

fractures in clastic shale. The mineralogical composition of the shale consisted primarily of clay minerals (illite

and chlorite, 75–90%) with minor amounts of crystalline silicates (5–20%) and lesser amounts of calcite,

dolomite, and heavy minerals. The shale blocks were cut parallel to the bedding and lapped until planar and

uniform. The sample size was 200 mm by 200 mm. Factors studied were the relative responses of sheared

and unsheared joints to normal loading and stress relief, and the influence of time-dependent displacements of

constant normal and sub-critical shear stresses. The test duration was typically 4–5 days, with a normal stress

of 1 or 2 MPa and shear stresses maintained constant throughout the tests. The most important result for the

current study is the observation of a threshold value, related to the applied shear stress/shear strength ratio.

The determined curves display a creep rate, which is relatively insensitive to changes in stress/strength ratios

below 0.7, and which decreases rapidly with time. The value of the ratio is higher than the value presented by

(Amadei and Curran 1982). The divergence probably depends on the distinction in discontinuity roughness

between a sample in marble and a sample in shale. The conditions of a discontinuity in marble are most likely

closer to that of hard rock. Schwartz and Kolluru (1981, 1982) investigated the influence of shear stress levels

on the creep of artificial cohesive and non-cohesive joints. Uniaxial creep tests were conducted on intact

samples of gypsum plaster and samples with discontinuities inclined at angles between 0 and 60 degrees to

the axial stress. The samples were rectangular, 32 mm by 32 mm by 121 mm in dimension, with a single joint

through its midpoint. Samples with a joint with small cohesive strength were produced by casting specimens in

a mould with a PVC insert with an inclined face. The liquid plaster was poured into one half, vibrated, and

allowed to set. The insert was then removed and replaced with more liquid plaster cast directly against the first

half. Samples with a non-cohesive joint were produced by casting the two halves of the specimens as

individual blocks and then stacking one on top of the other. The tests with jointed specimens were conducted

at a stress level up to 0.4 over a period of approximately 60 hours. Although a synthetic rock material was

used, the findings from the tests are of interest, since they bring about a general understanding of the time-

18

dependent mechanism in fractures. Schwartz and Kolluru (1982) present a simple theoretical model that at

least in part can explain the observed joint creep behaviour. As the first team of researchers they concluded

that creep appears to depend not only on the applied shear stress to shear strength ratio but also on the

absolute shear and normal stress levels across the discontinuity. Malan et al. (1998) have later confirmed this

statement. Wawersik (1974) studied time-dependent deformation on air-dry and water-saturated samples of

Westerly granite and Nugget sandstone. Triaxial tests were performed on cylindrical specimens, 25.5 mm in

diameter and approximately 60 mm in length. Most tests were on intact specimens, but a few were performed

on samples including one artificially induced joint (roughness ±0.5 mm) oriented at a 30° angle with respect to

the sample axes. The joints were formed by inducing tensile fractures in a block of rock, gluing this fracture

together, coring a cylindrical test sample at a proper orientation, and then dissolving the epoxy glue with

solvent prior to testing. The tests were carried out under 20.7 MPa confining pressure with a deviator stress of

60.5 MPa and 77.3 MPa, respectively. Axial and radial strains were measured over periods up to 30 hours.

The results show that the creep behaviour of jointed rock has the same character as that of the intact rock, but

that the magnitude of strains is much larger. They also demonstrate that the instantaneous joint deformation

exceeds the time-dependent response by several orders of magnitude. However, the test period is small when

compared to the periods of interest for a repository.

Solberg et al. (1978) simulated creep along a fault by testing a cylindrical sample of Westerly granite, 63.5

mm in length and 25.4 mm in diameter, containing a saw cut joint inclined at 30°, filled with gouge of 1 mm

thickness from crushed Westerly granite. The stresses applied on the samples were very high, 400 MPa

confining pressure and up to 1,080 MPa deviator stress, in an attempt to simulate conditions along a real fault.

The results showed that primary creep was followed by constant rate secondary creep at a stress ratio of 0.85

of the joint shear strength. Furthermore, the results show that at a confining pressure of several 100 MPa the

crushed gouge filling material becomes densely compact and its shear strength approaches that of intact rock

at the same confining pressure. The applicability of the findings are limited with regard to the current study,

since the stress level in the laboratory test far exceeds the expected stress condition around a repository. In

addition, our concern is mainly with threshold values related to primary creep rather than with secondary creep

as in the experiment.

Studies of creep movements in filled rock joints have been reported by two teams of researchers. The main

purpose of the investigations has been progressive movements of large natural rock slides (Höwing and Kutter

1985) and creep deformations around deep level mines in hard rock (Malan et al. 1998). The work of these

studies is described below. Höwing and Kutter (1985) carried out direct shear tests with smooth saw-cut rock

surfaces, and a few samples of tensile fractures. The rock type used in all cases was medium grained

sandstone. The aim of the study was to determine those factors that predominantly influence the creep

behaviour of filled fractures. The factors varied in the tests were: type of filling material, width of the filling

19

layer, normal and shear stress, and sample size. The filler material selected for the creep tests included clay,

silt and sand. The filler width ranged from 1 to 10 mm and the sample size was 100 mm by 100 mm. The

period of a test taks between 2 hours and 7 days with a normal stress generally between 1 and 2 MPa. The

results, which are considered to be of great interest for the current study, show that the creep rate of a filled

rock discontinuity is primarily determined by the particle size distribution. The percentage of clay in the filler is

much more critical for the creep velocity than the filler width. Malan et al. (1998) performed laboratory

investigations of the creep in fractures in hard rock. Unfilled mining-induced tensile fractures in lava, along with

gouge-filled saw cut discontinuities in quartzite, were examined in direct shear tests. The gouge material used

in the tests was natural as well as artificially obtained by crushing quartzite. The particle size distribution of the

gouge filling used corresponds to medium to fine grained sand. Factors that were studied were the effect of the

shear stress to shear strength ratio (t/ts), the effect of the magnitudes of the shear and normal stresses, and

the effect of gouge thickness. The tests were conducted using a normal stress of 0.5, 1.0 and 1.5 MPa.

The shear stress was increased in a stepwise fashion with a period of 24 or 48 hours between load

increases. The test duration was up to 5 days. Although Malan et al. have concentrated the investigation on

steady-state creep and our concern is mainly with primary creep the findings are considered valuable. The

results clearly show that the primary and secondary creep rate increases with increasing shear stress to shear

strength ratio. This creep rate is also affected by the gouge thickness and the absolute values of shear and

normal stress. The results further display, which in comparison with gouge filled discontinuities, the creep

behaviour of unfilled fractures in hard rock is negligible.

20

3 THEORIES OF CREEP

The fundamental problem of creep’s theories is the determination of the stress-strain-time relationship.

Because of the complexity of the creep phenomenon and the large number of factors influencing it, several

creep theories have been developed to describe creep behaviour in a wide range of real materials. These

theories differ in one way in the equations of state correlating stress, strain and time:

( ) ( ) ( )

And in another in the way about how to obtain these equations or how to describe creep phenomena

(which is more or less connected). From this point of view it is possible to divide creep theories into different

groups. Maslov (1968) has identified two approaches, the phenomenological and the physico-mechanical. The

physico-mechanical approach is described as “the method based on investigation into the nature of the

observed phenomena and the related laws, and utilisation of certain physical properties of soil and typical

parameters of the process”. A typical quantity introduced in the physico-mechanical approach is viscosity

according to Maslov. This decomposition seems to be logical. However, after analysing several creep theories

from these two groups one can find the incorrectness of this conclusion. The problem is that after

mathematical modification and parameters’ substitution of some creep theories from the phenomenological

group, the same creep theory can be physico-mechanical.

Feda (1992) has generally divided all rheological problems into macro and micro-rheology, where

macro-rheology is subdivided into the method of rheological models and method of integral representation. It is

obvious from the names of groups that the mathematical modelling of the real material’s structure is defined as

micro-rheology issue whereas in macro-rheology the structure of material is not represented. A similar

structure as introduced by Feda will be used for the description of the basic creep theories. The present author

has divided creep theories into the four groups - rheological models, models based on phenomenological

behaviour (for simplicity the term “phenomenological models” will be used here, although there are not

phenomenological from the Maslov’s point of view), numerical models and micro-rheology.

21

3.1 Rheological models

All materials possess properties of elasticity, plasticity and viscosity. These properties can act in different

combinations depending on the current material and situation. The behaviour of the real material is extremely

complex and a description of the deformation often becomes a questionable task. Hence deformation of the

real materials is described here by their simplified models known as rheological models (also models of ideal

bodies). Qualitatively there is similarity between rheological properties of various materials. The rheological

study of one material can be extremely useful for carrying out experimental and theoretical studies of another.

Theoretically one can say that rheological models for the viscous soil behaviour are historically based on the

viscous behaviour of fluids, gases, metals and lately mainly concrete. The foundation of the science of

rheology was laid by Newton in 1687, in his laws of motion of an ideal viscous fluid, today known as Newtonian

fluid. Newton found a linear relation that existed between the flow rate and flow resistance of an ideal viscous

fluid. The viscosity of the Newtonian ideal viscous fluid can be determinated by a relation between stress and

the rate of flow, shown in Equation (3.2):

( )

where is the viscous strain, time, the coefficient of viscosity (dynamic or effective viscosity) and

the rate of viscous strain. In rheological models the Newtonian fluid is simulated as a viscous element,

consisting of a cylinder with a perforated piston filled with a viscous fluid. Illustration of this element can be

found in Figure 3–1-c:

a) Hooke’s elastic element b) Plastic Saint Venant element c) Newton’s viscous element

Figure 3.1 Basic types of elements used in rheological models

Generally the relation between stress and rate of flow is considered to be linear or non-linear, see Figure 3.2.

22

Figure 3.2 Rheology curves of the Hooke’s and Newton’s elements

In rheological models the properties of elasticity are described through the ideal elastic Hooke medium

simulated as an elastic spring, as shown in Figure 3–1-a. The characteristic feature of the ideal elastic Hooke

medium is a linear relation between stress and strain:

( )

where is Young’s modulus or modulus of elasticity and is the elastic strain. However, in rheological

models this relation can be non-linear as well, as shown in Figure 3–2. Equation (3.3) represents the simplest

equation of state, which can be written both in differential and integral form.

The ideal rigid plastic Saint Venant body is used for the description of plastic behaviour. The body is

simulated by the dry Coulomb friction that appears on a rough surface when a load is moved on it under the

action of a horizontal force producing a stress equal to the plastic flow limit . When the

deformation of rigid plastic body is zero, while at it experiences unlimited plastic strain. Illustration of

the plastic Saint Venant element can be found in Figure 2–1-b.

The rheological behaviour of an ideal body which is determined by the rheological equation of state

includes stress and strain and their derivatives with respect to time:

(

) ( )

This equation combines Newton’s viscous, Hooke’s elastic and Saint Venant’s plastic elements, depending

on the given rheological model. As mentioned above, the rheological models are constructed by combining the

constitutive behaviour of these three basic elements. Mathematical relations of rheological models can then be

depicted in graphical forms, using the symbols of the elementary materials described above. Such a

visualisation serves well as an introduction into the problems and defines the problems in a clear manner. A lot

1 – nonlinear Hooke’s element;

2 - linear Hooke’s element;

3 - non-linear Newton’s element;

4 - linear Newton’s element

23

of different rheological models have been introduced for the mathematical description of the stress-strain-time

behaviour of soils. Nevertheless, the basic behaviour of the soils can be described according to three

elementary rheological models; Maxwell, Kelvin and Voight, and Bingham. All these models will be introduced

in the following sections.

3.1.1 Maxwell rheological model

The Maxwell visco-elastic rheological model (Maxwell body) was first presented in 1868 by James C.

Maxwell. The model consists of Hooke’s elastic element connected in series with Newton’s viscous element,

see Figure 2–3. After the application of a certain stress, the elastic spring will produce immediate strain in

contradiction to the viscous element where the immediate strain will be zero. As the stress continues to act,

viscous flow due to the movement of the performed piston in the viscous fluid will occur and so-called delayed

strain will be produced. The total strain will be the sum of immediate and delayed strains (primary and

secondary strains respectively), and the stress will be equal on both the basic elements:

( )

( )

The same relation as shown in Equation (3.5) can be also applied to the strain rates. Hence the total strain

rate is the sum of the strain rates of Newton’s viscous and Hooke’s elastic elements:

( )

which leads, after combination with Equations (2.2) and (2.3), to the following differential equation

according to Maxwell:

( )

or, after rearrangement:

( )

The parameter is the time constant called relaxation time, which determined the time-dependent

behaviour of the serially connected components of the Maxwell model. After integration of this differential

equation for the initial condition for time one can obtained the time-dependent stress relaxation

function, formulated as:

( ) (

) ( )

24

At the time , the following relation can be found:

( ) ( )

It means that in the time corresponding to the relaxation time of the Maxwell model the value has

decreased to 36.8% of the initial stress , i.e. the stress has already decreased to 63.2% of the initial value.

Based on the previous equation one can see that the parameter can characterise speed of the relaxation

of given material. Example of very small molecules with extremely short relaxation time can be, for example,

water with .

Deformation of the Maxwell body for a constant stress , i.e. for the creep behaviour, can be determined

from the relation based on Equation (3.8):

( )

( )

where ( ) is strain of the body at time t and is a strain at time . The creep behaviour of the

Maxwell rheological model during loading and subsequent unloading, with linear viscosity, is shown in Figure

3–3.

Figure 3.3 Maxwell rheological model with demonstration of the stress and strain behaviour

3.1.2 Kelvin and Voight rheological model

Kelvin and Voight rheological model is a visco-elastic model which consists of Hooke’s elastic element

connected in parallel with Newton’s viscous element, see Figure 3–4. Some authors describe this model as the

Kelvin model, and others as the Voight model or visco-elastic solid model. In this model, elastic and viscous

elements can only be deformed together and to the same extent. Hence instantaneous deformation of this

rheological model is zero. Stress will be distributed between both model components:

( )

25

( )

After the combination of equations, the differential equation according to Kelvin and Voight will be:

( )

Solving this equation for the constant stress one arrives at the relation, also called the creep function:

( )

[ (

)]

[ (

)] ( )

where is a constant called the time of retardation which determines the timedependent deformation

behaviour of the parallel connected components of the Kelvin and Voight model. Solving this equation at the

time point leads to the result:

( )

[

]

( )

This means that in the creep phase at the time corresponding to the retardation time the value of the creep

strain has increased to 63.2% of the maximum strain , which will be finally reached at the end of the load

interval. Similarly like relaxation time for relaxation tests, the retardation time is relevant in creep tests

and is unique for all materials.

The creep behaviour in the Kelvin and Voight rheological model for the linear viscosity is illustrated in

Figure 3–4.

Figure 3.4 Kelvin and Voight rheological model with demonstration of the stress and strain behaviour

3.1.3 Bingham rheological model

The Bingham visco-plastic rheological model consists of a Saint Venant rigid plastic element connected in

parallel with Newton’s viscous element, see Figure 3–5. This model introduces a yield point, often called a

26

“Bingham yield point”, which describes the transition from the state of rest to the state of flow. The state

equation of this model, the Bingham model equation is then:

( )

where is the yield stress and is so called “Bingham viscosity”. It is important to note, that the

“Bingham viscosity” is not a physical viscosity value of the investigated sample, but in this case it is no more

than a calculated coefficient used for the curve approximation (sometimes also called “Bingham flow

coefficient”).

Figure 3.5 Bingham rheological model with demonstration of the flow curve

After computers became widely used, this rheological model is not used so often mainly because of

problems with the physical meaning of the introduced parameters. Besides the “Bingham viscosity” also

“Bingham yield point” describes the transition from the state of rest to the state of flow relatively inaccurately. It

means that this model should only be used for very simple quality control tests. A flow curve according to the

Bingham model for the linear “Bingham viscosity” is shown in Figure 2–5.

There are a lot of different rheological models which more or less combine or use the basic ideas of

Maxwell, Kelvin/Voight and Bingham bodies. Some rheological models proposed for the characterisation of the

stress-strain-time soil behaviour are Murama and Shibata model (1958), Schiffman model (1959), Christensen

and Wu model (1964), Abdel-Hady and Herrin model (1966) or modified Komamura-Huang model (1974).

The advantages and disadvantages of the different rheological models are neatly recapitulated in.

According to Feda (1992), the method of rheological models is flexible in modelling different time effects. In

such a way, even complex mechanical behaviour may be fitted by a complicated rheological model. One is,

however, forced to use different models for loading and unloading, for volumetric and deviatoric creep and

their cross-effects, etc. The dependence of the mechanical behaviour of geomaterials on the stress and strain

paths has the consequence that for different paths different models must be used. There are also other

27

inconsistencies, such as the effect of the time being treated as a stress effect. Rheological models are,

therefore, not a visualisation of the structural changes to which the material is subjected in the deformation

process, but rather they serve only as a formal description of its phenomenological behaviour. The main

advantage of the rheological models is that they illustrate different constitutive relations in a graphical,

accessible form and allow their transformation by changing the position of various rheological elements in the

total scheme.

3.2 Phenomenological models

Observation of experimental creep curves obtained by testing a number of identical specimens under

creep or relaxation conditions has led to many theories of creep, which can be described as phenomenological

models. Generally speaking, these models are based on the study of phenomenological stress-strain-time soil

behaviour (stress vs. “instantaneous” strain, strain vs. time and stress vs. time curves) in various laboratory

conditions. This behaviour is finally described by an aforementioned equation of state, see Equation (3.1),

which can be purely empirical, physical or combined.

The simplest phenomenological models were the first models that tried to explain the creep behaviour of

the soil material. These models were purely empirical, trying to characterise the stress-strain or strain-time

curves by regression. Application of these models were obviously possible only for given conditions. Modern

models based on phenomenological soil behaviour are using physical quantities or trying to introduce physical

meaning to the used parameters. This makes them more usable in various cases.

3.2.1 Application to stress-strain behaviour

Stress-strain history plays a significant role in creep behaviour. The type of the creep behaviour is fully

connected with the stress-strain situation and boundary conditions, e.g. in the case of the oedometric stress-

strain conditions the volumetric creep, without secondary and tertiary creep phases will be expected. From this

point of view it is necessary to understand and evaluate the stress-strain situation for estimating and

understanding the following creep behaviour. Typical stress-strain curves plotted for a specific instant of time

in a case of the consolidation and deviatoric behaviour together with the connection between stress-strain and

strain-time (creep) curves at instant time for a case of the deviatoric creep tests are shown in Figure 3–6.

28

Figure 3.6 Examples of the stress-strain behaviour under different conditions together with illustration of the connection with creep curves

One can find a lot of relations between stress and strain, based on approximations with power,

exponential, logarithmic, hyperbolic or other empirical functions. One of the first relation between stress and

strain was proposed by Bernoulli in 1694 as a simple power function:

( )

where the parameter n is non-dimensional without any physical meaning, while parameter represents the

strain of the body at stress . A more general power function was derived by Dupen in 1811, as:

( )

where and are constants. This function describes the deformation of specimens under uniaxial

compression and tension, but is also applicable for the determination of the shear deformation. Different

phenomenological models for stress-strain behaviour have been proposed by many authors. Beside power

functions, exponential functions were used by Riccati or later by Meschyan, logarithmic functions were used by

Lesslie, hyperbolic functions by Timoshenko, etc. A lot of different stress-strain relations based on

approximation of the stress-strain curves are described in (Meschyan, 1995).

3.2.2 Application to strain-time behaviour

According to the acting stress it is possible to divide creep behaviour into volumetric and deviatoric (or

shear) creep. Here, volumetric creep is caused by the constant volumetric stress and deviatoric creep is

a) Typical stress-strain diagram in the case of:

1 - Deviatoric behaviour,

2 - Consolidation

b) Set of creep curves connected with curve at

instant time for a case no. 1

(deviatoric creep tests)

29

caused by the constant deviatoric stress. There is also a different way of dividing the creep behaviour, shown

in Figure 3–7, which is based on the type of the strain-time behaviour. According to the shape of the strain-

time curve, one can divide creep into the primary, secondary and tertiary phases. The primary phase, in some

literature also called transient or fading, can be defined as a creep deformation during which the strain rate

decreases continuously with time. Deformation at a constant rate (material flow) is denoted as the secondary

phase, and sometimes also called non-fading. In the case of the tertiary or the accelerated phase the strain

rate is continuously increasing and this leads to the creep rupture. Generally, volumetric creep consists only of

the primary phase of the creep deformation, i.e. it tends to stabilise. Deviatoric creep may or may not consist

all three phases, depending on the shear mobilisation. If the deviatoric stress is low, then only primary creep

phase will appear, but after crossing some level of the shear mobilisation primary phase will be followed by the

secondary phase which can lead to the tertiary phase and creep rupture.

Figure 3.7 Illustration of the primary, secondary and tertiary phases of the creep

According to the classical theory, consolidation of saturated soils is divided into the primary and secondary

phase, where the primary phase deals with the pore pressure dissipation with time after a load application,

whereas secondary phase is mainly connected to the rearrangement and better packing of the soil mineral

particles and aggregates. Based on the classical theory of consolidation, one can express the primary phase

with the strain-based differential equation proposed by Janbu (1965):

(

) ( )

where is depth and is coefficient of strain diffusivity. The strain based degree of primary consolidation

is then defined as:

∫ ( )

∫

( )

30

where is strain in time .

Equation (3.11) was numerically solved in 1965 for three different primary strain distributions with depth.

Based on the solution, one can write a general equation for the strain-based degree of primary

consolidation :

( ) ∑

(

) ( )

( )

where is time factor, and is an integer connected with . Three different primary strain distributions

are defined by value as:

The solution in graphical form of the relation between degree of primary consolidation and time factor

is shown in the Figure 3.8:

Figure 3.8 Solution of the strained-based differential equation in the form of relation between degree of primary consolidation and time factor

It is possible to write for the time dependent primary strain in the given point:

( )

( )

After simplification of the relation between and one can write two approximations for different degree

of primary consolidation (Janbu, 1965):

31

( )

√

√

( )

and

( )

(

) ( )

where is oedometric modulus, is coefficient of consolidation and is the height of the soil. The

simplest models of the secondary phase of consolidation (volumetric creep) or generally the primary creep

phase (according to the creep strain-time behaviour) are based on the approximation of strain-time curves. In

this case the strain-time curves can be approximated by exponential or logarithmic functions:

( ) [ ( )] ( )

( ) ( ) ( )

where , , , and are experimentally determined parameters. Equation (2.27) was proposed by

F. Kohlrausch in 1863 for the description of the glass-reinforced fibre creep. Later it was widely used in the

theory of the concrete and clayey soils creep (Maslov, 1940). Logarithmic equation (Equation 2.28) was

suggested by Buisman. For the evaluation of the secondary consolidation, the coefficient of secondary

compression has been very often used. The coefficient can be defined from the variation of the void

ratio e with time t for a given load increment, see Figure 2.9:

(

) ( )

where is change of void ratio and and is time. The magnitude of the secondary deformation can

be calculated as:

(

) ( )

32

where:

( )

is void ratio at the end of primary consolidation and is thickness of clay layer.

Figure 3.9 Variation of e with log t under a given load increment, and definition of coefficient of secondary compression.

33

4 STUDY OF CREEP PHENOMENON IN INTACT ROCKS

4.1 Introduction

The effect of time on the mechanical behaviour of intact rocks has two main origins:

• Dislocation of defects in the crystalline structure,

• Stable micro fracturing.

Which one of the two processes being the most important for the observed macroscopic rock response

depends mainly on factors such as the stress level relative to the rock strength, the temperature relative to the

melting temperature and confining pressure relative to the brittle- ductile transition pressure (Landanyi, 1993).

The subdivision of rock behaviour into brittle, semi-brittle and ductile regimes, based on the stress-strain

curves from compressive tests on rock samples, is also generally applied to time-dependent deformation.

Under long-term loading, crystalline rocks are considered to be in a brittle state at temperatures below

100–150°C (Pusch and Hökmark, 1992, Eloranta et al., 1992). In this regime, the time effects are considered

to be controlled by the rate of the slow growth of micro cracks. That creep strains are due to micro cracking, is

clear from direct measurements of cracks in rocks that have undergone creep and from a large number of

indirect measurements, such as acoustic emissions, the velocity of elastic waves and permeability (Kirby and

McCormick, 1984). Crystalline rock with low porosity does not display significant steady-state creep under

brittle conditions (Carter et al., 1981). The dominant mechanism of micro-crack generation and propagation

along grain boundaries, either stabilises or accelerates (Dusseault, 1993). Cracks extend from the stress

concentrators- cracks, pores, and grain boundaries; and the rate of growth decreases as the number of

concentrators decreases by cracking. Thus creep rates decrease with time in a hardening stage. However, if

the crack density exceeds some critical value such that cracks are close enough to interact on a large scale,

crack coalescence occurs to form a macroscopic fracture on which failure occurs (Kirby and McCormick 1984).

The often reported decrease in creep rate is illustrated schematically in Figure 4.1, which shows log time

creep behaviour for granite, extrapolated from laboratory test results, obtained from uniaxial load tests with

axial loads between 1/3 and 2/3 of the uniaxial compression strength (Pusch and Hökmark, 1992). The two

curves correspond to two different experiments performed on different rock types and with different loads,

giving different values of the parameter A in the log-time creep law. The figure should be interpreted with

34

caution, since extrapolating laboratory test data to long time periods is an uncertain procedure. Here it is

sufficient to conclude that the creep strain after 100,000 years seems to be too small to have any potential of

influencing the convergence of repository openings.

Figure 4.1. Intact rock creep movements extrapolated from results of uniaxial creep laboratory tests of short duration, assuming creep obeys the creep law ( ). For the upper curve ( ) the axial stress corresponded to

40% of the compressive strength (Pusch and Hökmark 1992)

The main factors controlling brittle creep rates, i.e. applied deviator stresses, effective confining pressure

and moisture content, are discussed below.

4.2 Influence of deviator stress

Increasing the applied deviatoric stress increases the creep rate. Different rocks require different levels of

deviatoric stress to creep at the same rate. It has also been observed that several rocks will not show

significant creep unless the deviatoric stress is above some threshold value (Dusseault and Fordham, 1993).

According to (Damjanac and Fairhurst, 2010) the threshold deviatoric stress for brittle crystalline rock is about

40–60% of its compressive strength.

4.3 Influence of confining pressure

The effect of an increased confining stress is to decrease the creep rate, because confinement acts to

suppress the tension stresses associated with crack growth (Damjanac and Fairhurst 2010). It is also assumed

35

that the confining pressure increases the energy barrier to be overcome for continuous crack propagation. In

addition, high confining pressures result in the closure of cracks, which reduces the transport of fluids and

gases through the rock. This has a limiting effect on stress corrosion, and hence on the creep rate, because

corrosive agents cannot easily reach new crack surfaces that are successively created during creep

(Krantz, 1980).

4.4 Effects of water content

The presence of pore water in rocks is found to affect their behaviour in two ways:

• It enhances the rate of crack formation under stress.

• It produces a temporarily internal confinement.

4.5 Effect of the rate of crack formation

The effect of the rate of crack formation is considered to be partly due to water favouring stress corrosion

at the crack tips and partly due to the internal stresses induced by changing the rock humidity. The result is an

increased short- and long-term deformability and a reduced long-term strength (Lajtai et al., 1987).

4.6 Effect of dilatancy hardening

The effect of temporary internal confinement increases the rock strength. This effect, known as ‘dilatancy

hardening’, is related to the possibility of pore water drainage at a given applied strain rate (Brace and Martin,

1968).

36

5 STUDY OF CREEP PHENOMENON IN FRACTURED ROCKS

5.1 Introduction

The creep displacement in a discontinuity is a function of both the normal stress ( ) and the shear stress

( ) of the fracture. Under a constant normal stress, creep deformation is expected to increase as the shear

stress is increased and, for a constant shear stress, to decrease if the normal stress is increased (Amadei and

Curran, 1982).

The long-term shear behaviour of a discontinuity at a constant normal stress is shown schematically in

Figure 5.1. The shear model suggests that creep displacements leading to failure may occur for any shear

stress level between the peak strength and the residual strength. For a constant applied shear stress, the

distance AB represents the creep deformation necessary for failure. If the peak and residual shear strengths

and the stiffnesses and are constant, then the joint creep deformation necessary to cause failure will be

independent of time. However, observed time-dependency of the static coefficient implies that peak shear

strength is a function of time (Dieterich 1972, Lajtai and Gadi 1989, Lajtai 1991).

Figure 5.1. Hypothetical stress-deformation path of a discontinuity at a constant normal stress (After Amadei and Curran 1982).

37

The creep movement in a discontinuity seems to be of a different nature depending on whether the fracture

is planar or rough, and on whether it is clean or filled. Creep in planar joints is presumed to be controlled

mainly by an adhesion-frictional mechanism (Bowden and Curran 1984).

In an unfilled rough discontinuity, the process of time-dependent shearing of individual asperities may

explain creep. The time element in this process is due to stress concentrations at asperities along the joint

surface that cause slip as the asperities yield progressively and the shear stresses are redistributed to other

intact asperities (Schwartz and Kolluru 1982, Ladanyi 1993).

In a filled discontinuity, the creep is influenced both by the filler shear strength and the shear strength of

the filler-rock interface. The creep rate is dependent on the ratio between the filling thickness and the

amplitude of undulation. If the particle size is fine and the filling thickness large enough, the roughness of the

discontinuity becomes ineffective. This may produce conditions similar to those of a smooth discontinuity,

where creep is controlled mainly by the characteristics of the filling material (Höwing and Kutter 1985, Malan et

al. 1998).

5.2 Fracture shear strength

Shearing along a natural unfilled discontinuity is resisted by friction caused by over-riding or fracturing of

microscopic asperities. Larger scales may also involve undulation and rock bridges. Smooth and rough joint

walls will usually result in widely different shear strength and deformation characteristics.

The original theory of friction assumes that the material on either side of the contact surface is rigid. The

gouge that accumulates on a sliding surface, however, is evidence of intense deformation and the initial

surface roughness changes rapidly as the surface wears (Lajtai and Gadi, 1989). Under the influence of a

steadily increased shearing force and a constant normal load, the mobilized friction resistance depends on:

1. the initial condition of the surface,

2. wear of the surface through displacements,

3. time and velocity related effects.

The interpretation of the shear strength usually involves some variation of Mohr-Coulomb’s linear equation,

in which the peak shear strength is given by the following expression:

( )

where is the joint cohesive strength, is the effective normal stress and the peak friction angle. The

expression can also be used for the residual shear strength by taking and substituting the peak friction

angle by the residual friction angle , see Figure 5.2. The peak shear strength envelopes for rock joints are

38

in reality not linear but strongly curved. Both and are stress dependent variables and also scale dependent

(Barton and Choubey 1977).

Figure 5.2. Shear stress versus normal stress for a schematic direct shear test

5.2.1 Unfilled discontinuities

Shear strength of planar joints

The shear strength of two flat rock samples in contact is a result of frictional forces between contact points

of mineral grains in two rock surfaces. Different geological minerals have different coefficients of friction. The

measured friction between two rock samples is an average of the friction forces of the grains composing the

rock sample.

The shear strength of two smooth planar surfaces in fresh rock is usually described by the following

equation (Barton 1974):

( )

where is basic friction angle. The basic friction angle represents the minimum shear resistance of a

smooth planar surface in fresh rock, and is considered to be a material constant dependent on the mineralogy

and the moisture of the rock. The parameter is usually approximately equal to the residual friction angle ( ).

However, the residual friction angle can be lower than the basic friction angle due to weathering or alteration

effects.

The basic friction angle for some typical rock types listed by (Barton and Choubey 1977) is presented in

Table 5.1. The values indicate that a basic friction angle in the interval 25°–35° can be used as a reference

39

value when estimating the shear strength of unfilled joints in common unweathered rocks. The basic friction

angle for samples of granite from Äspö Hard Rock Laboratory has been evaluated to 28.7° ± 2.7° (Olsson,

1998).

Table 5.1. Basic friction angles of various unweathered rocks obtained from flat surfaces (Modified after Barton and Choubey 1977)

Rock type Moisture

Conditions Basic friction angle

(deg)

Basalt

dry 35–38

wet 31–36

Fine-grained granite

dry 31–35

wet 29–31

Coarse-grained granite

dry 31–35

wet 31–33

Gneiss

dry 26–29

wet 23–26

Shear strength of rough joints

A natural discontinuity surface in hard rock usually includes undulation and asperities. The joint surface

roughness causes a displacement to occur in the normal direction, i.e. perpendicular to the shear direction.

The behaviour can be represented by a dilatation angle ( ). The degree to which a rock joint dilates when

sheared has a significant influence on the joint shear strength. At low normal stresses, where rough joints

suffer insignificant or relatively little damage during shear, the following equation may be used as a first

approximation to the peak shear strength (Patton 1966):

( )

where is effective normal stress, the basic friction angle and the peak dilation angle. Equation 5.3

is not valid at high normal stresses, when the strength of the asperities will be exceeded and the teeth will tend

40

to break off, resulting in a shear strength behaviour closely related to the intact material strength (Hoek et al.,

1994).

According to (Barton, 1971) the peak dilation angle at low normal stress is related to the applied normal

stress ( ) and the uniaxial strength ( ) as:

( ) ( )

Using Equation 5.4 for a discontinuity in granite, under an applied normal stress in the interval 2 – 4 MPa

with a uniaxial stress of = 150 MPa, gives a peak dilation angle estimate in the interval of 16 – 19°. The

mean peak dilation angle from test samples of joints in granite from Äspö Hard Rock Laboratory, under a

similar normal stress, has been evaluated to 13.8° ± 3.2° (Olsson, 1998).

5.2.2 Filled discontinuities

Rock joints may naturally be filled with frictional or cohesive material. The filling material is rarely of a better

quality than that of the rock blocks. In general, filling materials are of poor quality: sand, silt or clay. Filling

materials can be classified with respect to the way of formation:

1. Sedentary filling (stationary) – originating through local alteration or weathering along the

discontinuity. Stationary, formed at the location.

2. Sedimentary filling (transported) – originating through soils washed in by circulating water.

3. Gouge filling- originating through soils generated by shear movement.

The shear strength can be reduced drastically when part or the entire surface is covered by soft filling

material. The most obvious effect of a filling material is to separate the joint surfaces and thereby reduce the

contact between opposing asperities. However, the shear strength will also be influenced by the surface

texture, the nature of the filling material and the characteristics of the joint-fill interface (Papaliangas et al.

1990).

The shear strength decreases with increasing filling thickness (Papaliangas et al. 1990, Pereira 1990). For

planar surfaces a thin clay coating will result in a significant shear strength reduction. For a rough or undulating

joint, the filling thickness has to be approximately twice the amplitude of the undulation before the shear

strength is reduced to that of the filling material (Phien - wej et al. 1990).

Barton (1974) has presented a comprehensive review of the shear strength of filled discontinuities. Friction

angles of typical filling minerals can also be found in the reference work Landolt-Börnstein (Rummel 1982). A

summary of the shear strength of typical discontinuity fillings is presented in Table 5.2 and Table 5.3. These

values can be used for estimating reference values of shear strength for a coated or filled joint.

41

Table 5.2. Shear strength of filled discontinuities and filling material (Modified after Barton1974)

Rock type Description Cohesion

(MPa)

Friction angle

(deg)

Basalt Clayey basaltic breccia 0.24 42

Diorite and porphyry Clay gouge (2% clay, PI=17%) 0 26.5

Granite Clay filled faults 0–0.1 24–45

Granite Sandy loam fault filling 0.05 40

Schists and quartzites Stratification with thin clay 0.61–0.74 41

Schists and quartzites Stratification with thick clay 0.38 31

Table 5.3. Friction angle of mineral as a function of dry and wet surface

(Modified after Rummel1982)

Rock type Surface condition Friction angle

(deg)

Biotite , Canada

cleavage, dry 17

cleavage, wet 7.4

Chlorite, Vermont, USA

polished, dry 28

polished, wet 12

Muscovite

cleavage, dry 23

cleavage, wet 13

5.2.3 Time-dependent shear strength

The friction angle of a joint surface is not a constant, but a variable that depends on time-related

parameters such as displacement rate, accumulated displacement and the duration of stationary contact. Any

process that increases the ‘true area of contact’ in the discontinuity will increase the frictional resistance of the

joint (Lajtai and Gadi 1989).

Displacement-strengthening involves change of the sliding surface into one that is in equilibrium with the

imposed velocity. Once such a surface is created, the angle of friction will no longer change with further

42

displacement. The displacement required to reach equilibrium conditions can, however, be quite large. Shear

tests performed by Lajtai and Gadi (1989) on granite blocks with initially smooth and polished surfaces, gave

friction angle increases of approximately 14 degrees for an accumulated displacement of 500 mm, see

Figure 5.3.

Figure 5.3. The increase in friction angle for a joint in granite with wear of the surface (After Lajtai 1991)

For clean, rough joint surfaces, the friction coefficient is independent of the duration of stationary contact,

while for joints in which some gouge has accumulated it increases with the contact duration (Dieterich 1972).

One explanation put forward for the observed increase in frictional resistance, during stationary contact, is that

it is attributed to time-dependent compaction and subsequent strength increase of the gouge material

accumulated in the discontinuity. Experiments by Dieterich (1972) on rough as well as smooth joint surfaces

showed that the time-dependence of the static coefficient of friction could be expressed as:

( ) ( ) ( )

where

is the initial coefficient of friction, is a constant and is the duration of contact in seconds.

Values for

and for granite and quartzite are given in Table 5.4. The increase of with time appears to be

independent of normal stress. Using the equation for a discontinuity in granite under 2 days stationary contact

will result in an increase of 5–6 degrees in the friction angle. This is in general agreement with results by Lajtai

and Gadi (1998) that show an increase of 3–4 degrees during a shear loading delay of 2–4 days (Figure 5.4).

43

Table 5.4. Values for coefficients

and for granite and quartzite at a normal stress of approximately 10 MPa

(After Dieterich 1972)

Rock type Coefficient of static friction,

Constant,

Sandstone 0.73 0.015

Granite 0.79 0.022

Quartzite 0.84 0.020

Greywacke 0.83 0.012

The test results presented above indicate that, in rock joints under shear conditions that accumulate gouge

fill, creep is most likely transient. The reason is that the time-dependence of the static coefficient of friction

implies that the peak shear strength will increase with time. This will give a reduction in the ratio of the applied

shear stress to the peak shear strength, which is found to control the magnitude of creep displacements

(Amadei and Curran 1982). The described state will, however, not arise if the characteristics of a joint are a

product of earlier shearing under a similar loading condition since wear and compaction of the accumulated

filling will not occur in these circumstances.

Figure 5.4. The increase in the friction angle following stationary contact in a direct shear test. (After Lajtai 1991)

44

It should be noted that the time strengthening achieved through a delay in loading is temporary. Once the

friction resistance is overcome and displacement resumes its pre-delay rate, the friction resistance drops back

to the level established before the loading delay (Lajtai and Gadi 1989).

5.3 Creep tests

5.3.1 Unfilled discontinuities

The typical creep behaviour of an unfilled discontinuity is presented in Figure 5.5. The test was conducted

on a sample of water-saturated Westerly granite containing one joint inclined at 30° relative to the axial

compression (Wawersik 1974).

Figure 5.5. Axial strain versus time for a joint in water-saturated granite under constant principal stress difference (s1–s3 = 60.5 MPa) and confining pressure (s3 = 20.7 MPa). Note that the creep rate after 30 hours is very small.

The slope of the curve gives an approximate value of 1·10–11 s–1. (After Wawersik 1974)

The results show an instantaneous elastic strain followed by a transient and steady-state creep. The

instantaneous joint deformations exceed the time-dependent response by several orders of magnitude. The

steady-state creep is in the actual experiment just a few percents of the total strain. Another example of creep

tests performed on an unfilled joint is shown in Figure 5.6. The examined fracture is an excavation-induced

rough discontinuity in porphyry lava with very high uniaxial compressive strength (435 MPa). The fracture was

closely matched and not subjected to any previous shear movements. The normal load was 1.0 MPa (Malan et

al., 1989).

45

Figure 5.6. Shear creep behaviour of an excavation-induced tensile fracture in porphyry lava with a stepwise increase in shear stress. The normal stress was 1 MPa. The sample was tested at the ambient humidity of 50%. (Malan et al. 1998)

The sample failed at a shear stress of 1.7 MPa giving an apparent friction angle of 59.5°. The total

displacement before failure was only about 10 μm. Unexpected trends of creep in the steady-state creep

phase of the first two loading stages are explained by Malan et al. (1998) by an effect of slight changes in

temperature (±1°C) that have an influence on the instrumentation.

Several researchers report that the creep behaviour of unfilled discontinuities has the same general

character as that of intact rock, but with a strain magnitude that is larger (Amadei and Curran 1982, Schwartz

and Kolluru 1981, 1982, Wawersik 1974). An example of these types of results is presented in Figure 5.7. The

creep curves in the figure are from four different configurations: intact rock, a cohesive joint oriented at =30°,

a cohesive joint oriented at =60°, and a cohesion less joint oriented at =30°. All four tests were conducted

at a stress magnitude of 0.4 . The parameter is the angle between the normal to the joint plane and the

direction of the applied stress. All curves exhibit a characteristic non-linear primary creep followed by

secondary steady-state creep. In all samples including a joint, the creep strains were higher but not very much

higher, than for the intact sample. Increasing from 30° to 60° or decreasing the cohesion to zero also lead to

higher creep (Schwartz and Kolluru 1981).

A possible explanation for the agreement in creep behaviour between intact rock and unfilled

discontinuities could be a similarity in the creep process. The observed state with decreasing time-dependent

displacement can in both cases be explained by stress concentrators that are exhausted by micro fracturing of

the rock material.

46

Figure 5.7. Creep curves for intact and jointed samples at 0.4 stress level. parameter represents the angle between the direction of the applied axial stress and the direction of the normal to the joint plane (Schwartz and Kolluru 1981)

Stress dependence

Several studies indicate that significant creep in fractures will not occur unless the shear stress magnitude

is above some threshold value (Amadei and Curran 1982, Bowden and Curran 1984, Schwartz and Kolluru

1982).

The results from a test of marble, performed by (Amadei and Curran 1982) show a limit of the shear

stress/strength ratio ( ) of approximately 0.5, beyond which there is a plain increase in the creep

movement along the discontinuity. For low values of the shear/strength ratio, creep displacement of the intact

rock dominates.

Figure 5.8 shows the results of several shear tests performed at shear stress strength ratios ranging

between 0.6 and approximately 1.0 on artificial discontinuities in shale (Bowden and Curran 1984). The test

results display creep rates which are small and decrease rapidly with time for stress/strength ratios less than

0.7, while for ratios larger than 0.9 the creep displacements become significant. Amadei (1979) hypothesized

that the ratio of the applied shear stress to the peak shear strength ( ) was the critical factor governing

discontinuity creep. However, later laboratory investigations have shown that the creep rate is not only a

function of shear stress to shear strength ratio, but also of the absolute values of the shear and normal

stresses.

47

Figure 5.8. Creep displacements of an artificial discontinuity in shale. The uppermost curve in the top graph is the lower most curve in the bottom graph (After Bowden and Curran 1984)

Influence of joint surface condition

Conditions of the joint surface such as roughness, wall strength, moisture content and weathering, have an

influence on the peak shear strength of the discontinuity. By a change in the degree of interlocking or a

change of the strength of individual joint asperities, the ratio of the applied shear stress to shear strength of the

joint is changed, which in turn influences the creep rate. Results from creep tests that in any real sense

demonstrate the influence of the surface condition of unfilled joints have not been found. This is probably due

to difficulties in reproducing identical surface conditions from one test to another.

5.3.2 Filled discontinuities

The creep behaviour of a discontinuity with gouge filling in a shear test performed by Malan et al. (1998) is

shown in Figure 5.9. The shear stress was increased stepwise and both primary and secondary creep can be

observed. The sample failed at a shear stress of about 0.5 MPa giving a friction angle of approximately 26.5

degrees. The total shear displacement before failure amounts to about 1.1 mm. The normal displacement

recorded in the test indicated compaction of the gouge layer with increasing shear stress, despite the constant

normal stress.

Opinion is divided about the number of creep stages that can be identified in the case of filled

discontinuities. According to test results by Höwing and Kutter (1985), no secondary creep stage is found for

filled rock discontinuities, i.e. no transition phase with constant creep velocity between primary and tertiary

creep was observed. Malan et al. (1998), on the other hand, focus on steady-state creep in their presentation

of results. The reason for the divergence is probably differences in the test conditions, i.e. initial conditions, the

48

stress and displacement magnitude, the duration of the tests and the grain size distribution of the filling

material. The factors of importance for the creep rate of filled discontinuities seem to be:

1. shear stress/shear strength ratio,

2. absolute stress magnitudes,

3. gouge thickness,

4. grain size distribution.

The influence of these factors on the creep behaviour is discussed in figure 5.9.

Figure 5.9. Shear creep behaviour of a discontinuity with bedding plane gouge -collected from Hartebeestfontein mine. The normal stress was 1 MPa. The gouge thickness was 2 mm and the humidity 50% (After Malan et al. 1998)

Stress dependence

Malan et al. (1998) investigated the effect of the magnitude of the shear stress and normal stress on a

gouge filled fracture. The tests were conducted by increasing the shear load, in a stepwise fashion, under a

normal stress at three different magnitudes: 0.5, 1.0 and 1.5 MPa. In Figure 5.10 incremental shear

displacements for different shear load increments under a normal stress of 0.5 MPa are illustrated. The

magnitude of instantaneous response was very prominent in the test and increased with the shear stress to

shear strength ratio ( ).

49

Figure 5.10. The effect of shear stress magnitude on the creep behaviour. The percentages indicate the magnitude of the shear stress relative to the shear strength (0.28 MPa). The normal stress was 0.5 MPa and the gouge thickness 2 mm

(After Malan et al., 1998)

The test also showed a steady-state creep rate that increased with the shear stress magnitude, see Figure

5.11. The behaviour in tests with normal loads of 1.0 MPa and 1.5 MPa was similar. It was, however, observed

that the magnitudes of the primary and steady-state creep rate are not only a function of the shear stress to

shear strength ration ( ), but also of the absolute values of the shear and normal stresses. For a 1.5 MPa

normal load, the steady-state rate at a ( ) ratio of 0.93 was more than 3 times larger than that for tests with

a 0.5 MPa normal load.

Figure 5.11. The effect of shear stress magnitude on the creep behaviour of a gouge filled fracture. The percentages indicate the magnitude of shear stress relative to the shear strength (0.28 MPa). The normal stress was 0.5 MPa and the

gouge thickness 2 mm (After Malan et al., 1998)

50

Malan et al. (1998) suggest the following relation between the steady state creep rate and the stress/

strength ratio:

(

)

( )

where and are parameters that depend on the thickness and type of filling, in addition to the absolute

stress level.

Influence of filling thickness

The effect of the filling thickness of the shear creep in a fracture has been investigated by Höwing and

Kutter (1985) and Malan et al. (1998). Figure 5.12 presents the results by Höwing and Kutter (1985) with kaolin

filling and Figure 5.13 the results by Malan et al. (1998) with medium to fine grained sand filling.

Figure 5.12. Initial creep velocity (v after 1 min) as a function of filling thickness. The filling material was Kaolin with a clay

content of 0.75 and a consistency index of 0.75. The normal stress was 1 MPa and the ratio (R = ) was varied between 0.7 and 0.9. (After Höwing and Kutter, 1985)

51

Figure 5.13. Steady-state creep rate as a function of filling thickness. The filling material was a medium to fine sand. The

( ) ratio was 0.86 for the 0.5 mm thickness, while 0.87 for the 1 mm and 2 mm thickness. The samples were tested

at 50% humidity. (After Malan et al., 1998)

The test results clearly illustrate that creep is dependent of the filling thickness. Both initial creep velocity

and steady-state creep rate increase with the filling thickness.

The results are most likely dependent upon the asperity amplitude of the fracture surface in relation to the

grain size of the infilling. The reason for increased creep velocity with thicker infilling, is probably that the

interlocking effects of opposing asperities are reduced. This means that the creep behaviour of the fracture

approaches the creep behaviour of the filling material (Phien- wej, 1990, Pereira, 1990 Papliangas et al.,

1990).

Influence of particle size distribution

According to results by Höwing and Kutter (1985) the creep rate in the primary and tertiary phase is directly

dependent on the clay content of the infilling. Figure 5.14 presents the initial creep velocity in a triangular

concentration chart for grain-size fractions. Lines of equal creep velocity run parallel to lines of equal clay

content in the diagrams. The creep rate of soils with coarse grains, i.e. with a clay content between 0.1–0.3, is

rather small. The /Höwing and Kutter 1985/ explanation for the observed results is that with increasing clay

content the coarse grains are more and more enclosed by clay particles. The friction between the coarse

particles is therefore increasingly reduced.

The increase of the creep velocity in the primary phase ( ) with the clay content can be expressed by an

exponential function (Höwing and Kutter 1985):

52

( )

where, and are coefficients that are determined empirically and is the clay content.

Figure 5.14. Initial creep velocity (v after 1 min) as a function of particle size distribution of the filling material. The filling thickness was 2 mm and the consistency index 0.75. The normal stress was 1 MPa. The diagram on the left shows a

shear stress ratio R = 0.7 and the diagram on the right a ratio R = 0.9. (After Höwing and Kutter 1985)

53

6 DEFORMATION ANALYSIS IN TUNNELS

6.1 INTRODUCTION

Some thirty years ago, most tunnel supports and linings were designed using empirical rules defining

ground loads acting on the supporting structures. This approach is still popular and interest in it has been

renewed by the use of rock mass classifications systems. Correlations between rock mass conditions and the

type of support used are based on case histories. This approach perpetuates existing practice, even if

overconservative or unsatisfactory.

Most recent tunnel design relies on more elaborate analysis of the complex rock-support

interaction. This analysis must take into account not only the properties of the rock mass, but

also the structural behavior of the supporting structure and the construction procedures. Stresses

and displacements in the rock mass surrounding tunnels depend on the rock mass properties, the in

situ initial stress field, the stiffness of the lining or support and the timing of its installation.

A complete analysis requires a large amount of data and information which may be taken into

account by numerical modelling. However, in most cases it is difficult to obtain all the necessary data

before construction. Many of the parameters that influence the analysis are ill-defined and, up to

now, no constitutive model has been shown to be successful at simulating all the aspects of rock

behavior important to tunneling. Some rock properties may only be evaluated by back-analyzing

field data.

A good understanding of the deformations caused by an underground excavation requires

simultaneously knowledge of the rock-support interaction and interpretation of field data. Formerly, the main

purpose of the monitoring carried out during tunnel construction was to measure the

ground pressures acting on the tunnel lining. Modern tunneling practice emphasizes the monitoring

of the displacements occurring during and after the construction. The measurement of the walls’

Since the Chapter 7 is based on the deformation equations of Panet on tunnels, Chapter 6 is extracted from the "Comprehensive Rock Engineering" book, Volume 1, chapter 27, Understanding deformation in tunnels; Marc Panet, SIMECSOL, Le Plessis Robinson, France.

54

displacements, commonly called "convergence", is the one most used on a tunnel site. Monitoring and

interpretation of convergence tend to become an integral part of tunnel engineering.

6.2 CONVERGENCE MEASUREMENTS

In many instances, the convergence of the tunnel walls is the easiest and most reliable parameter

to measure at a tunnel working site. The convergence is the variation of the distance between two

opposite pins in the tunnel wall (Figure 6.1):

( ) ( ) ( )

where ( ) is the convergence at time t between any two opposite points A and B (for example, 1 and 3 or

1 and 4, etc.), is the initial measurement of the distance between the two points A and B and ( ) is the

measurement at time of the distance between the two points A and B. The convergence is said to be positive

if the distance between the two points has decreased and negative if it has increased.

Figure 6.1. Section equipped with four pins to measure the convergences in a horseshoe-shaped tunnel

Convergence pins are installed at a number of points on the periphery of the tunnel cross section

and the displacements between two pins are measured with an invar wire or an invar tape. The most

common convergence meters are equipped with a dynamometer that enables a constant value of the wire

tension to be maintained and a dial gauge extensometer. More advanced techniques under

study make use of laser waves.

55

The absolute and precise knowledge of the distance between the opposite points is not necessary and

usually is not measured. The variations of the distance may be measured over a long period with an accuracy

of 10-5, i.e. 0.1 mm for a distance of 10 m. If greater accuracy is necessary, the measurements require special

care. The convergences between several anchored points on one section of the tunnel may be measured. This

would give an idea of the deformation of the section, but the data are generally insufficient for the

determination of a unique solution. The measurements of the vertical and the horizontal convergences are

often required by tunneling engineers; the horizontal convergence is easy to obtain, but the vertical

convergence is difficult to measure because the anchored point, installed on the floor of the tunnel, is made

unstable or even destroyed by the traffic of the construction equipment.

The convergence of a tunnel is due to: (i) the effect of the face advance; and (ii) the time-dependent

behavior of the rock mass. To carry out a complete analysis of the convergence in the simple case of a full-

face excavation, it is useful to plot the data on three different graphs (Figure 6.2), giving: (i) the distance x

between the convergence station and the face of excavation versus time t; (ii) the convergence C versus t; and

(iii) C versus x.

The effect of the face advance decreases rapidly as x increases; the slope of the convergence curve is

very steep close to the face. Therefore, it is necessary to carry out the first measurements of convergence as

close as possible to the excavation face. However, in practice, with some construction methods this condition

may prove difficult or even impossible to fulfill; this is the case when the tunnel is excavated with a shield

tunneling machine and a segment erector is used. When the excavation face is far ahead of the convergence

station, the influence of the face vanishes and the convergence is controlled by the rheological behavior of the

rock mass and the stiffness of the support.

The distance of influence of the face may be evaluated by a careful examination of the shape of the curve

of the convergence versus time after an interruption of the excavation. While the progress of the face is

stopped, the convergence goes on because of the time-dependent properties of the rock mass, as shown in

Figure 2 for the two periods - and - . When the advance is resumed, a discontinuity in the slope of the

curve C versus appears if the face is close enough to the convergence station; such is the case at time .

However, when the face is far ahead of the convergence station, this discontinuity does not appear (time ).

For example, during the excavation of the Fréjus Tunnel, the work was interrupted for 22 days

at point PM 5117.5. An examination of the convergences of three stations at distances of 14m, 37.5m and

75.5m from the face respectively shows clearly that the change of the slope of the convergence curve versus

time is noticeable for the two closest stations and is not for the more distant one (Figure 6.3). The periods of

interruption of the excavation may also be used to analyze the time-dependent properties of the rock mass.

56

Figure 66.2. Graphs for the analysis of distance versus time, convergence versus time and convergence versus distance

When the tunnel is not excavated by a full-face method, but by a multi-drift type of excavation, the analysis

is more complex because it is necessary to consider the effect of the different sequences of excavation.

Careful examination of the convergence curves enables the site engineer to make a sound judgment about the

sequence of excavation and the efficiency of the various techniques of supports, e.g. steel arches, rock bolting

and shotcrete. At the Alberg Tunnel, the convergence rate was used to determine whether additional bolting

was required and what should be the optimal length of the rock bolts. Kaiser and Morgenstern (1983) showed

it is possible to distinguish three modes of tunnel behavior from an analysis of convergence: pre-failure mode,

stable yield zone propagation mode and unstable mode. Analysis of the variation of the rate of convergence

versus time proves to be very useful.

57

Figure 6.3. Convergences versus time at three adjacent stations of the Fréjus Tunnel

6.3 MODELS USED IN ROCK TUNNEL ENGINEERING

The issues concerning the methods for geotechnical design of tunnels include stability, loads on

supporting systems, water inflows and ground movements. Many site engineers think that empirical

analysis of convergence is sufficient to manage the excavation of a tunnel and do not care much

about more sophisticated analysis. However, models may prove very useful in obtaining a better

understanding of the interaction between the rock mass and the support, in order to optimize the

design.

In rock engineering, modelling always implies simplification; it is necessary to select from all geological

data those which are really meaningful for the understanding of the rock mass behavior in relation to the

design of the structure. For this purpose, it may be useful to carry out preliminary studies on some simple

models to test the true significance of some parameters of the rock mass. The type of model selected should

be able to answer precise questions.

In rock tunneling, two basic types of instabilities can be distinguished; (i) the failures of blocks or wedges

under gravity are controlled by the pre-existing structure of the rock mass. Discontinuities divide the rock mass

into discrete blocks which may translate or rotate. At the wall of an excavation, these blocks may move in

response to unbalanced forces; under the effect of gravity, some may fall from the crown or rotate towards the

58

excavation in the side (Figure 6.4). These movements are analyzed by a simple graphic method using

stereographic projection or by more sophisticated two-dimensional or three-dimensional distinct element

models. These structurally controlled failures are the only ones which occur in many underground excavations

at low depth or in hard rocks. (ii) On the contrary, at great depth failures are induced by high initial stresses

and may affect the rock mass by spalling or popping with possible rock bursts; the effect of discontinuities in

the rock mass is secondary.

In many situations, however, it is necessary to model the behavior of a jointed rock mass in order to design

the support or the lining to resist stress-induced deformations. The question of the significance of an isotropic

or anisotropic equivalent continuum model in representing the behavior of a jointed rock mass was raised early

in the study of rock mechanics. The art of modelling jointed rock masses relies on the ability of the rock

engineer to anticipate the most important mechanisms of deformation and modes of failures, in order to

identify the main features of a rock mass for a specific problem.

Figure 6.4. Instability of blocks at the periphery of a tunnel (unstable key blocks are hatched)

When designing a tunnel it is essential to understand the deformations which occur close to the face of

excavation, this situation is typically three-dimensional. As the power of computers increases, three-

dimensional finite element analyses become possible at an acceptable cost for the design of important

underground structures. However, other factors must also be considered: the determination of a stress-

dependent rock mass constitutive model for a true triaxial situation, modelling of the construction procedures

and the overwhelming task of plotting and analyzing the results. In a three-dimensional analysis, it is

necessary to make so many assumptions and simplifications that it is doubtful if it can actually improve on a

two-dimensional analysis, which partially accounts for the three-dimensional effect. However, with the

continuous progress made in geotechnical numerical modelling, three-dimensional analysis may become a

practical tool in the future for tunnel design.

59

The present finite element programs offer a range of constitutive laws which may be used to simulate the

behavior of a rock mass; the simpler ones are linear elastic but some others have the capability of kinematic

hardening and allow for strain softening. Numerical modelling is more and more extensively used in tunnel

design. Finite element analyses are useful inasmuch as they can account for the initial stress gradient due to

depth as well as the irregularities introduced by the geometry, the inhomogeneities due to the geology and the

different stages of the construction procedure. Closed-form solutions may appear to be old-fashioned

techniques only useful for university teaching. However, they give very valuable guidelines for design and also

allow one to check the validity of numerical models.

6.4 DEFORMATIONS CAUSED BY A TUNNEL DRIVEN IN AN ELASTIC MEDIUM

6.4.1 The Axisymmetric Case

The rock-support interaction may easily be treated by solving a simple axisymmetric problem. A circular

tunnel of radius is excavated in an homogeneous isotropic medium, where the initial stresses are isotropic

and equal to . The distribution of stresses and strains close to the excavation face is three-dimensional. The

effect of the proximity of the face has been approximated by different authors in different ways.

The field of displacements caused by the excavation may be obtained by numerical modelling; then it is

possible to analyze the distribution of the radial displacement versus the distance to the excavation face

(Figure 6.5). If no support is installed, the radial displacement far behind the face is given by:

( )

( )

where is the shear modulus of elasticity of the ground. At some distance from the excavation face, we

may write:

( ) ( ) ( ) ( )

The variation of between 0 and 1 is given in Figure 6.5. It may be remarked that, for , .

The measured convergence is then equal to:

( ) ( ) ( )

[( ) ( )] ( )

if the initial measure is carried out at a distance from the face. If , the following expression is a

good approximation for ( ) (see Figure 6.12).

( ) ( ) [ (

)

] ( )

60

where . The slope of the tangent of the curve for is:

(

)

( )

( )

Then the tangent at the curve for goes through the point with coordinates:

( ) ( )

Figure 6.5. Radial displacement versus the distance to the face of excavation in the elastic axisymmetric case

A simple geometric construction may help to determine the asymptotic value C (Figure 6.6).

Figure 6.6. Variation of ( ) versus x ( ) in the elastic axisymmetric case

61

Then:

( ) [ (

)

] ( )

The three-dimensional problem may be approximated by an equivalent plane strain problem. The effect of

the proximity of the face, which limits the displacements, is simulated by a fictive radial stress which decreases

from its initial value for to zero as (Figure 6.7).

The state of deformation at a distance from the face is evaluated by applying at the wall of the cylindrical

cavity:

( ) ( )

Figure 6.7. The equivalent plane strain problem

In polar coordinates, the radial stress and the tangential stress (Figure 6.8) are given by:

(

) (

) ( )

62

Figure 6.8. Distributions of radial stress and tangential stress according to the value in the elastic axisymmetric case

And the displacement by:

( )

At the tunnel wall, for :

( ) ( )

( )

The stress and strain paths at the wall of the tunnel are represented in Figure 6.9.

Figure 6.9. The stress and strain paths at the wall of the tunnel in the elastic axisymmetric case

63

If we plot versus for each value of , we obtain the convergence curve or the ground characteristic

curve. In the simple elastic case, it is a straight line (Figure 6.10).

Figure 6.10. The ground characteristic curve (1) and the support characteristic curve, (2) in the elastic axisymmetric case

In this approach, the ground-support interaction may be easily analyzed. A support installed at a distance

from the face ( is called the unsupported span) limits by its stiffness the radial displacement for . The

stiffness of the supporting structure may be characterized by a stiffness modulus given by :

( )

being the external pressure acting on the support and:

( ) ( ) ( )

Then, at equilibrium, the ground pressure acting on the support may be obtained by solving the two

equations (Figure 6.10):

[ ( )] ( )

Then:

( )

( )

In these formulae it appears that the final equilibrium depends on the deformation characteristics of the

ground, the stiffness of the support, the unsupported span and the initial state of stress. The larger the ratio

64

is, the greater the supporting pressure and the smaller the final convergence. If is larger than the

distance of influence of the excavation face, the supporting pressure is equal to zero and the convergence is

maximum, as in the unsupported case.

6.4.2 Anisotropy of the Initial State of Stress (Einstein and Schwartz, 1979)

When the initial state of stress is not isotropic, the problem is no longer axisymmetric. Let us consider the

case where the tunnel axis is a principal direction for the initial state of stress and, in the plane orthogonal to

the tunnel axis, the initial principal stresses are and

:

( )

The displacements due to the advancing face may be approximated by an equivalent plane strain problem

in which the radial stress and the shear stress acting on the tunnel wall are decreased from their initial

value to zero. Then the deformations due to the tunnel excavation are computed by applying on the tunnel wall

a normal stress and a shear stress given by:

( )[( ) ( ) ]

( )( )

( )

where ( ) is a parameter which is increased from 0 to 1 as the face advances. The components and

of the displacement of a point whose cylindrical coordinates are are given by:

( ) [( )

( ) (

( )

) ]

( )[ ] [

( )

]

( )

At the tunnel wall for :

( )[( ) ( )( ) ]

( )( )( )

( )

65

If a support is installed at a distance from the face corresponding to the interaction between the

ground and the support depends on the rigidity of the support. If we assume that the stress acting on the outer

face of the support is given by:

( )

These stresses bring about displacements of the outer face of the support given by:

( )

( )

Between and there are the relationships:

( )

( )

( )

where is the compression modulus of rigidity of the support and is the bending modulus of rigidity of

the support. For an elastic annulus of thickness , and are given by:

( )

( )

( )

If we assume a perfect adherence between the ground and the lining, the values of are given by:

( )

( )[( ) ( )]( )

( ) [ ( ) ]

( )[( ) ( )]( )

( )

[ ( ) ] [ ( ) ] ( )[( ) ( )]( )

If we assume a no friction condition between the ground and the lining ( ), are given by:

( )

( )[( ) ( )]( )

66

[ ( ) ] ( )[( ) ( )]( ) ( )

The total displacements at a distance from the face are:

( ) ()

( ) () ( )

( ) and ( ) may be calculated from equation (6.20) and (), () from equations (6.22),

(6.23),(6.25) or (6.26).

6.4.3 Anisotropic Elastic Rock Mass

Anisotropy of deformability and strength is a very common characteristic of intact rocks and rock masses.

In many rocks, it stems from the preferred orientation of minerals. Foliation and schistosity make schists,

slates and many other metamorphic rocks highly anisotropic. Rock masses, even if they are made of isotropic

rocks, may have an anisotropic behavior because of the presence of bedding planes or orientated sets of

joints.

In many situations, e.g. for bedded or foliated rock masses, an orthotropic anisotropy with an axis of

symmetry (transverse isotropy) is assumed. The corresponding elastic stress-strain relationships require the

knowledge of five independent constants.

Let be an axis parallel to the axis of symmetry and and two orthogonal axes in the plane of

isotropy normal to ; then in the linear tensor relationship, , may be written as:

[

( )

]

( )

Among the five elastic characteristics, the shear modulus is one of the most important parameters. For

a low value of this parameter, the analytical solution derived by Lekhnitskii (1983) shows high stress

67

concentrations in the zones where the wall of excavation is tangential to the plane . For tunnels driven

in laminated rocks, failure by buckling occurs often in these zones (Figure 6.11). These types of failures are

much more dependent on the relative orientation of the tunnel and of the schistosity. The size of the opening is

also an important factor; under identical conditions, a small gallery may be stable but buckling failures would

be encountered in a larger excavation.

Figure 6.11. Buckling failures in laminated rocks

The convergence is usually larger in the direction . In the elastic case, if the initial stress is isotropic,

the radial displacement for a tunnel whose axis coincides with is given, with a good approximation, by the

expression:

( )

where and are:

[

√

]

[

√

] ( )

with:

√

68

√

( )

The combination of the anisotropy of the rock mass and the anisotropy of the initial stresses may create

the most adverse conditions for the stability of a tunnel excavated in schistose rock masses; they bring about

very large convergences in the direction perpendicular to the schistosity due to buckling failures.

6.5 DEFORMATIONS CAUSED BY A TUNNEL DRIVEN IN A YIELDING ROCK MASS

6.5.1 The Development of a Plastic Zone

The elastic theory of the stress distribution around underground openings demonstrates that the deviatoric

stresses are maximum at the periphery of the excavation and decrease with the distance from the tunnel.

Therefore, the rock may yield in the overstressed zone surrounding the excavation. Since Fenner, many

authors have developed an elasto-plastic theory to study the zone of yielded ground. Many solutions giving the

stress distribution and the displacements in the axisymmetric case are available.

The most elaborate closed-form solutions incorporate more complex and more realistic rock mass

behavior, taking into account the elastic parameters, the criterion for the peak strength of the rock mass and

the rock mass behavior in the yielded zone with a strain softening law, a postpeak dilatancy and a residual

strength criterion.

The most commonly used criterion for the rock mass is the Mohr-Coulomb criterion. It may be conveniently

written:

( )

where is the major principal stress, is the minor principal stress, is the uniaxial compressive

strength, ( ) ( ) and is the internal friction angle.

Hoek and Brown (1988) proposed a nonlinear empirical criterion, which is more and more used in rock

mechanics practice:

( )

( )

In a general way, the yield criterion of the rock mass may be written:

( ) ( )

In the simple axisymmetric case, the plastic zone appears for a value A, of the parameter A given by:

69

(( ) ( )

) ( )

with the Mohr-Coulomb criterion:

[

] ( )

and the Hoek and Brown criterion

[( )

] ( )

where .

For the unsupported tunnel, there is no plastic zone if for and , . This condition

gives .

When there is a plastic zone, it is important in practice to distinguish three cases (Figure 6.12). (i) The

excavation face is completely included in the plastic zone ( ) with the stability of the face becoming

critical; this situation is mainly met in soft ground tunneling, but is very rare in rock tunneling. (ii) The plastic

zone appears behind the excavation face ( ). If a sufficiently stiff support is installed close enough to

the face, the plastic zone does not exist; however, such a support is rarely necessary to limit the convergence

and is usually not economic. (iii) The intermediate case, where . For , the plastic zone

has a radius .

Figure 6.12. The shape of the plastic zone according to the value

70

For the analysis of the deformations in an elasto-plastic problem, it is generally assumed that the increment

of the total strain is the sum of the increment of the elastic strain and of the plastic strain:

( )

For the sake of simplicity of derivation, two different assumptions are often made. The first one neglects

the elastic strain in the plastic zone (rigid plastic behavior in the plastic zone):

( )

In the second assumption, only the elastic strain developed before the yielding of the rock is considered.

Neglecting the elastic strain in the plastic zone and assuming that the principal strains and satisfy the

law:

( )

then if , the volumetric strain is zero, but if , there is a dilatancy in the plastic zone due to the

loosening of the rock mass.

The determination of results from the integration of equation (6.8)

( )

taking into account the boundary condition for :

( )

Then, for :

( )

And for :

(

)

( )

The radial displacement is a function of the radius of the plastic zone which depends on and

therefore on .

For the convergence ( ), the same empirical law as in the elastic case may be used, where

. By curve fitting, it is possible to obtain an estimate of . The larger is the longer is the

distance of influence of the excavation face.

71

6.5.2 The Ground-Support Interaction for an Elasto Perfectly Plastic Medium

The stress distribution in the plastic zone may be derived from the integration of the equation of equilibrium

in the axisymmetric case:

( )

For a Mohr-Coulomb criterion of plasticity:

{[( ) ( )] (

)

[( ) ( )]}

( )

and:

[

[( ) ( )]

]

( )

Introducing a plastic potential with a dilatancy angle , the plastic strain increments may be written:

( )

where ( ) ( ). The dilatancy angle is smaller than or equal to the internal

angle of friction . The value corresponds to no volumetric plastic strain.

The axisymmetric condition gives the differential equation:

[(

) (

) ] ( )

where and

This equation may be integrated with the boundary condition for :

( )

It gives:

{(

)

( ) [(

)

]}

72

( )( )

[

] [(

)

(

)

] (

)

( )

The stress and strain paths at the wall of the tunnel are shown in Figure 6.13.

Figure 6.13. Stress and strain paths at the wall of a tunnel in the elasto perfectly plastic axisymmetric case

The parametric equations of the ground characteristic curve are:

( )

{(

)

( ) [(

)

]}

( )( )

[

] [(

)

] ( )

for

where is given by equation (6.47).

The intersection of the rock mass characteristic curve and of the support characteristic curve gives, as in

the elastic case, the supporting pressure at equilibrium. The rock mass characteristic curve is no longer

straight (Figure 6.14).

73

Figure 6.14. The intersection of the ground and the support characteristic curves in the elasto-plastic case

It is important to note that, when plastic deformations occur, the parameter is no longer equal to ( )

( ) In order to use the convergence-confinement method, it is necessary to know the radial displacement

of the wall at the unsupported distance from the face. Corbetta, Bernaud and Nguyen Minh (1991) have

proposed recently a useful method to determine this, on the basis of the results of numerical models of a

circular tunnel driven in an elasto perfectly plastic ground with a Poisson’s ratio equal to 0.5. They remarked

that the curve ( ) may be derived, with a good approximation, by a simple homothety of center from

that obtained in the elastic case (Figure 6.15). The ratio of the homothety is equal to:

( )

( )

Figure 6.15. The curve giving the radial displacement in the elastoplastic case may be obtained by a simple homothety of the curve in the elastic case, according to Corbetta et al. (1991)

74

With a simple geometric construction it is possible to place the support characteristic curve (Figure 6.16).

Figure 6.16. Determination of the deformation at distance d from the excavation face, according to Corbetta et al. (1991)

6.5.3 The Ground-Support Interaction for a Tunnel Driven in an Elasto-plastic Rock Mass

with a Strain-softening or Brittle Behavior

When a rock mass is strained beyond the peak strength, it exhibits a strain softening down to the residual

strength. This behavior has been modeled by a trilinear stress-strain law (Figure 6.17):

(i) a linear elastic behavior up to the peak strength; the peak strength criterion is:

( ) ( )

75

Figure 6.17. A trilinear stress-strain law for strain-softening material

(ii) a linear strain softening down to the residual strength; the residual strength criterion is:

( ) ( )

The strain softening is characterized by a dilation angle .

(iii) a perfectly plastic behavior beyond the residual strength with a dilation angle

.

The rock mass around the tunnel is divided into two or three regions. When the rock mass has been highly

strained close to the wall of the excavation, it is divided into three zones: the elastic zone, the strain-softening

zone and the zone where the rock mass has reached its residual strength (Figure 6.18a). Only the elastic zone

and the strain-softening zone exist when the deformations around the tunnel are not large enough for the

residual strength of the rock mass to be reached (Figure 6.18b). By installing an adequate support close

enough to the face of excavation, this second situation can be achieved by the tunneling engineer.

Assuming Mohr-Coulomb criteria both for peak strength

( )

and for residual strength:

( )

The stress and strain paths at the wall of the tunnel are given in Figures 6.19 and 6.20 for both cases. If we

take the same angle of friction for the peak and the residual strength ( ) and no cohesion in the

residual state, the yield criterion in the softening zone may be written:

(

) ( )

76

where

. The residual state is reached for .

The strain increments in the strain-softening zone are the sum of the elastic and the plastic ones If the

dilation angle is constant, the strain increments may be written in finite terms:

{

[( )

( )]

[( )

( )]

( )

then:

[(

) ( ) (

) ( )] ( )

and :

{

[( )

( )]} ( )

Figure 6.18. The elastic zone (I), strain-softening plastic zone (II) and residual zone (III) around a tunnel

77

Figure 6.19. Stress and strain paths at the wall of the tunnel in the elasto strain-softening plastic axisymmetric case without residual plastic zone

Figure 6.20. Stress and strain paths at the wall of the tunnel in the elasto strain-softening plastic axisymmetric case with a residual plastic zone

78

The boundary conditions for are:

( )

( ) (

)

( )

( )

There is no closed-form solution in the more general case; only stepwise numerical solutions are available.

However, a closed-form solution may be given in the simple brittle model where it is assumed that the strength

drops discontinuously from the peak strength to the residual strength.

The radial stress and the tangential stress in the plastic zone are given by:

( ) (

)

( ) (

)

( )

and the radius of the plastic zone is determined by the boundary condition for ,

( ) .

then:

(

)

( )

It must be remarked that there is no possibility of stability without support; when the parameter tends to

1, the radius of the broken zone increases indefinitely.

Integration of equation (6.52) gives an expression for :

[(

)

( )] [(

)

(

)]

( )( )

( ) [(

)

(

)

] (

)

( )

in which ( ) (

). Figure 6.21 gives the stress and strain paths at the wall of

the tunnel in this particular case. Then the parametric equations of the ground characteristic curve are:

( )

79

[ ( ) (

)

( )]

( )( )

( ) [(

)

] ( )

for

Figure 6.21. Stress and strain paths at the wall of the tunnel in the elasto brittle axisymmetric case

If a Hoek and Brown peak strength criterion is assumed, the associated residual strength criterion may be

written:

( )

( )

Then the expression of the radius of the broken rock is given by:

{[

( ) ]

[

( ) ]

} ( )

The radial displacement up is calculated, assuming that the principal strains in the broken zone are:

80

( )

and:

( )

and being the postpeak strain increments with .

Then is the solution of the differential equation:

( )

( )

The solution of this equation, with the boundary condition ( ) for , is:

{

[(

)

]} ( )

Then the ground characteristic curve is defined by the parametric equations:

( )

{

[(

)

]} ( )

where is given by equation (6.68).

In all the derivations given above, it has been assumed that the principal stress parallel to the axis of the

tunnel remains the intermediate principal stress; it must be kept in mind that this is not always the case.

As the plastic criterion is independent of the intermediate principal stress, two cases can be considered

when computing the plastic strain rate:

If then:

( )

If then:

81

( )

( )

It can be shown that this situation may occur if:

( )

( )

6.5.4 Allowance for the Weight of the Yielded Rock

In all the equations of the ground characteristic curve given above, the supporting pressure is decreasing

monotically with the convergence; this conclusion is not in agreement with field evidence. In fact, in the

derivations, the gravity forces are considered to be included in the initial state of stress it is necessary to allow

for the dead weight of the loosened rock above the tunnel (Figure 6.22). Pacher (1964) suggested including

the weight of the rock in the yielded zone as an additional component of the ground characteristic curve:

( ) ( )

where is the unit weight of the loosened rock.

Figure 6.22. Allowance for the dead weight of the yielded rock

This correction is a gross simplification; it may be somewhat improved by using the static Caquot solution

in the yielded zone, assuming that the rock in the yielded zone can be characterized by its residual strength.

82

The relative importance of this additional load on the support depends directly on the yielded zone radius

and therefore on the convergence of the opening; if the convergence is too large, the supporting pressure

given by the intersection of the characteristic curves is small but the additional load due to the weight of the

yielded zone is large and, , a stiff support installed close to the face limits the convergence too

much. Then the supporting pressure is no longer decreasing monotically with the convergence. Another

consequence of allowing for the weight of the yielded zone is that the required supporting pressure increases

with the tunnel size.

6.6 TIME-DEPENDENT DEFORMATIONS

6.6.1 Introduction

In the preceding sections, the ground deformations around tunnels have been analyzed without taking into

consideration the time-dependent behavior of the rock mass. However, for many rock masses, the

deformations which bring about an increase of the loads on the support are due not only to the progress of the

excavation but also to the rheology of the rock mass.

Rock rheology is a complex process which depends on the type of rock, the temperature and the mean

and deviatoric stresses. A great variety of creep formulations have been proposed to describe the creep tests

data. If we consider uniaxial creep tests, the equation of the total strain at a given time due to an increase in

the axial stress from time may be written as:

( ) ( ) ( )

where is the instantaneous strain due to the increase of the axial strain and is the delayed strain.

For many types of rock, such as limestones, shales, rock salt or potash, the creep law is often written:

( ) (

)

( ) ( )

where is the creep exponent of stress with its value usually in the range 2-5 and ( ) is the time

function. When the strain increase with time is not limited, the following expression of ( ) is considered

(Figure 6.23a):

( ) ( )

where is the creep exponent of time and is less than or equal to 1.

83

Figure 6.23. Various time functions ( )

If the strain tends towards an asymptotic value as time increases, the time function may be taken as equal

to:

( ) (

) ( )

(Figure 23b) or:

( ) (

)

( )

(Figure 23c).

Equation (6.81) with corresponds to the linear Kelvin-Voigt model; is the relaxation time.

Equation (6.82) has proved to be the most adequate for the interpretation of convergence data measured on

several sites.

In theoretical analyses, the models used most often to represent time-dependent behavior are

unidimensional and consist of the association of simple elastic, viscous or plastic elements, as shown in Figure

6.24. Among these rheological models, those where the delayed deformations are viscoelastic and those

where the delayed deformations are viscoplastic can be distinguished.

84

Figure 6.24. Some unidimensional rheological models used for analyzing time-dependent underground deformations

Numerous studies have been devoted to the application of such rheological models to the analysis of time-

dependent deformations of spherical or cylindrical cavities. However, it must be acknowledged that their

application to actual underground works meets serious difficulties; in situ measurements very rarely agree over

a long period of time with the rheological parameters deduced from the data obtained in the laboratory tests.

In the analysis of delayed deformations of underground works it is necessary to compare the rate of

excavation with the strain rate of the rheological model, which may be characterized by the time . If is the

rate of excavation and the distance of influence of the face, a time excavation parameter:

( )

may be introduced. To analyze the ground-support interaction, it is necessary to consider the relative

values of , and ; is the time of installation of the support or more precisely the time when the support

comes into contact with the ground. In many long term studies is much smaller than ; then it is assumed

that the process of excavation is instantaneous.

When delayed deformations occur, the ground characteristic curve corresponding to the unsupported

tunnel is not unique; for each rate of excavation, there exists a curve which lies between two boundary curves

corresponding respectively to an instantaneous excavation ( ) and to an infinitely low rate of excavation

( ). Figure 6.25 gives the ground characteristic curves for various rheological models.

85

Knowing the radial displacement at the time , the conditions of the long term equilibrium for a lined tunnel

are often determined by the intersection of the support characteristic curve with the ground characteristic curve

corresponding to ; as will be indicated below, this procedure is not always acceptable and may lead to

too small a value for the support pressure.

Figure 6.25. Ground characteristic curves for some rheological models

6.6.2 The Viscoelastic Models

With the Maxwell model (Figure 6.25a) the increase of convergence is linear with time as long as the

tunnel is unsupported. If a stiff support is installed, a relaxation of the deviatoric stresses in the ground occurs

and the support pressure acting on the lining increases asymptotically up to the initial stress . Maxwell and

related models are mainly used for designing underground works in salt or potash rocks. With the Kelvin-Voigt

model (Figure 6.25b) delayed strains are limited. In the case of no support, the convergence increases with

time up to an asymptotic value (Figure 6.26a).

86

Figure 6.26. Kelvin model in the axisymmetric case: (a) radial displacement for an unsupported tunnel,

(b) pressure acting on a support installed at

If a support with a stiffness compression modulus is installed at time , after the excavation, which is

supposed to be instantaneous, the increase of the support pressure with time is given by:

(

) (

) [ (

)] ( )

Such a model was used for the design of the lining of the Channel Tunnel in the Chalk Marl.

With the viscoelastic models, the intersection of the support characteristic curve and the ground

characteristic curve for an infinitely low rate of excavation gives a correct solution of the final equilibrium of the

ground-support interaction (Figure 6.27).

87

Figure 6.27. Determination of the supporting pressure with a viscoelastic model by the use of the ground characteristic curves

6.6.3 The Viscoplastic Models

The viscoplastic models associate an instantaneous elasticity and a viscous plasticity. The one most often

used is the Bingham model (Figure 6.24d).

In a general way the strain rate is given by:

( )

where

( )⟨ ( )⟩

( )

( )

is a scalar equal to the viscoplastic strain, ( ) is the viscoplastic criterion; the variation of with

represents the strain softening behavior of the rock mass, ( ) is the plastic potential which may take into

account a viscoplastic volumic dilation, and ( ) is the viscosity which may be constant or may vary with to

represent the tertiary creep.

Berest and Nguyen Minh (1983) analyzed the axisymmetric case in a Bingham visco-plastic medium with

a Tresca criterion. For a lined tunnel it is necessary to distinguish two cases according to the stiffness of the

lining.

If ( ) ( ) the lining is soft and after the installation of the lining, the viscoplastic zone is

increasing. The final convergence, plastic radius and supporting pressure may be obtained by the nonviscous

88

elastoplastic solution given by the application of the convergence-confinement method, introducing the radial

displacement at the time of installation of the lining. However, if ( ) ( ) the lining is stiff

and after the installation of the lining, the radius of the viscoplastic zone decreases. The supporting pressure is

larger than that given by the nonviscous elastoplastic solution.

If the time of excavation is of the same order as the time parameter of the rheological deformations

and if the installation of the support is within the distance of influence of the face of excavation ( ), then

a complete step-by-step numerical analysis is necessary when a visco- plastic model is used. Such an

analysis was used to interpret the data obtained at different scales in an experimental gallery for nuclear waste

disposal at Mo] (Belgium), and it proved to be very effective (Rousset, 1988).

For the analysis of the convergences measured on sites, it has to be assumed that the convergence

depends both on the distance to the face of excavation and on the time t; the following expression was

proposed:

( ) ( )[ ( )] ( )

where and are constant, ( ) is a function depending only on , and ( ) is a function depending

only on .

In various situations, the functions:

( ) (

)

( )

and:

( ) (

)

( )

allow for fitting the field measurements with a great accuracy. The final convergence is then given by:

( ) ( )

A great variety of models are available to analyze the deformations around underground excavations,

taking into account both the influence of the proximity of the face of excavation and the time-dependent

properties of the rock mass. As a general rule, the best choice is always the simplest one which can simulate

the key elements of the anticipated deformations. The selection of the appropriate constitutive law is a matter

of sound engineering judgment. It is quite unnecessary to use the more sophisticated finite element programs

and the more elaborate constitutive laws to carry out a good tunnel design. When the outcome of a numerical

model does not agree with the judgment of an experienced tunnel engineer, it is best discounted; however,

before doing this, be sure the tunnel engineer is really experienced.

89

7 AN ANALITICAL SOLUTION FOR DEFORMATIONS OF TUNNELS

Panet and Sulem (1987) for determining of deformations in tunnels have assumed that "The tunnel has a

circular cross section and around the tunnel, the rock is homogeneous and isotropic and also the tunnel is

deep enough to consider that the stress distribution is homogenous". But in almost real cases, the stresses

distribution is not homogeneous and isotropic. On the other hand, for the sake of simplicity, they have

considered a linear law. They have assumed the rock fails according to the Mohr--Coulomb failure criterion,

while the failure criterion in almost rocks is not linear, especially in fractured rocks. In this chapter, we will try to

modify the Panet and Sulem equations for generalizing of the problem (nonhomogeneous and anisotropic

situation) and will propose some new equations based on the Generalized Hoek-Brown failure criterion (2002)

which is more appropriate for fractured rocks.

7.1 Kirsch equations for estimation of induced stresses around a tunnel

In order to calculate the stresses, strains and displacements induced around excavations in elastic

materials, it is necessary to turn to the mathematical theory of elasticity. This requires that a set of equilibrium

and displacement compatibility equations be solved for given boundary conditions and constitutive equations

for the material. The process involved in obtaining the required solutions can become quite complex and

tedious. One of the earliest solutions for the two-dimensional distribution of stresses around an opening in an

elastic body was published by Kirsch (1898) for the simplest cross sectional shape, the circular hole. A full

discussion on the derivation of the Kirsch equations, as they are now known, is given by Jaeger and Cook

(1969) and no attempt will be made to reproduce this discussion here. The final equations are presented in

figure 7.1, using a system of polar coordinates in which the stresses are defined in terms of the tractions acting

on the faces of an element located by a radius and a polar angle . Stress components at point ( ):

Radial stress

[( )( ) ( )( ) ] ( )

Tangential stress

[( )( ) ( )( ) ] ( )

90

Figure 7.1. Two-dimensional distribution of stresses around an opening in an elastic body

7.2 Modification of Panet fictitious support pressure coefficient ()

Inside the excavated tunnel that (Figure 7.2, advance parameter) is equal 1, is zero. But in the case

≠1 in above equations assuming a=0.

Figure 7.2. Fictitious support pressure coefficient

91

Therefore, according to the Kirsch equations, in the plane-strain problem, a radial stress and a

tangential stress are applied on the tunnel wall in the case of no support. The radial stress and a

tangential stress simulate this face effect and this fictitious temporary support is given by:

Radial stress

[( )( (

)) ( ) ( (

) (

)) ] ( )

Tangential stress

[( )( (

)) ( )( (

)) ] ( )

where the parameter is increased from 0 to 1.

As the distance from the tunnel increases, the influence of the opening upon the stresses in the rock

decreases. Plots of the ratio of against the distance along the horizontal axis of the stressed model

for two values of and are given in figure 7.3. These plots show that the stress concentrating

effect of the tunnel dies away fairly rapidly and that. at , the ratio of induced to applied stress is very

close to unity. This means that, at this distance from the tunnel boundary, the stresses in the rock do not see

the influence of the opening. This fact has been utilised by those concerned with model studies of stresses

around underground excavations. The general rule is that the minimum size of the model should be 3 to 4

times the maximum dimension of the tunnel in the model.

In particular case K=1 (isotropic):

[ (

)] ( )

[ (

)] ( )

and around the tunnel boundary :

[ ] ( )

[ ] ( )

which are Panet equations in isotropic and homogeneous conditions.

92

Figure 7.3. The distributions of radial stress and tangential stress according to the value in the elastic axisymmetric case for two values for and

As the equations of 7.3 and 7.4 show, the radial and tangential stresses are function of (ratio of

horizontal applied stress to vertical applied stress). Only in isotropic condition ( ), the radial and

tangential stresses are independent of . in other cases, radial and tangential stresses around the tunnel

boundary are given by:

[( )( ) ( )( ) ] ( )

93

[( )( ) ( )( ) ] ( )

In the roof and floor of tunnel, and respectively, equations 7.9 and 7.10 reduce to:

( ) ( )

[ ( )] ( )

In the sidewalls of the tunnel, and , equations 7.9 and 7.10 become:

( ) ( )

[ ( )] ( )

Equations 7.12 and 7.14 (tangential stresses) are plotted in figure 7.4 which shows variation in tangential

boundary stress in the sidewalls and also roof and floor of a tunnel with variation in the ratio of applied

stresses.

Figure 7.4. Variation in tangential boundary stress in the sidewalls and also roof and floor of a tunnel

with variation in the ratio of applied stresses

94

Also equations 7.11 and 7.13 (radial stresses) are plotted in figure 7.5 which shows variation in redial

boundary stress in the sidewalls and also roof and floor of a tunnel with variation in the ratio of applied

stresses. In case of , the value of radial stress around boundary of tunnel is zero.

Figure 7.5. Variation in redial boundary stress in the sidewalls and also roof and floor of a tunnel with variation in the ratio of applied stresses

7.3 Generalized Hoek-Brown failure criterion

The Generalized Hoek-Brown failure criterion for jointed rock masses is defined by (Hoek and

Brown, 2002):

(

)

( )

where and

are the maximum and minimum effective principal stresses at failure, is a reduced

value of the material constant and is given by:

[

] ( )

and are constants for the rock mass given by the following relationships:

[

] ( )

[ ] ( )

95

is a factor which depends upon the degree of disturbance to which the rock mass has been subjected by

blast damage and stress relaxation. It varies from 0 for undisturbed in situ rock masses to 1 for very disturbed

rock masses. And finally is the uniaxial compressive strength of the intact rock pieces.Since theoretically

GSI (Geological Strength Index) varies from 0 for laminated/sheared rocks to 100 for intact or massive rocks,

so constant will vary from 0.5 to 0.67 approximately (figure 7.6).

In order to use the Hoek-Brown criterion for estimating the strength and deformability of jointed rock

masses, three properties of the rock mass have to be estimated. These are:

Uniaxial compressive strength of the intact rock pieces,

Value of the Hoek-Brown constant for these intact rock pieces, and

Value of the Geological Strength Index GSI for the rock mass.

Figure 7.6. " a " constant values for variation of Geological Strength Index (GSI)

7.4 Panet equations modification using non-linear Generalized Hoek-Brown failure

criterion

As long as is smaller than a certain value , the yield criterion is not reached in any point of the rock

mass. Using the generalized equations in the roof and floor of tunnel, and respectively

(equations 7.11 and 7.12), and also the generalized Hoek-Brown failure criterion (equation 1.7), When =

the yield criterion is reached at , the elastic stresses at the wall must then also satisfy the yield criterion:

( ) ( )

[ ( ) ] ( )

96

(

)

( )

[ ( ) ] ( )

( ( )

)

Since parameter in generalized Hoek-Brown failure criterion for most of rocks is close to 0.5, therefore in

last equation it is assumed that , so:

[( ) ]

(

( )

)

( )

[ ( )

] [( )

]

therefore will be:

[ ( )

] √[ ( )

]

[( )

]

(7.22)

And as well as in the sidewalls of the tunnel, and , the equations 1.13 and 1.14

become:

( ) ( )

[ ( ) ] ( )

then:

[ ( ) ] ( )

( ( )

)

( )

[ ( )

] [( )

]

97

therefore will be:

[ ( )

] √[ ( )

]

[( )

]

(7.25)

For a broken zone of radius will develop around the tunnel (Figure 7.7).

Figure 7.7. Cross-section showing the broken and the elastic zones

The equation of equilibrium is:

( )

where and are cylindrical co-ordinates, is radial stress and is tangential stress.

Within the broken zone, the equation of equilibrium (Equation 7.26) is integrated by substituting the non-

linear Generalized Hoek-Brown failure criterion (Equation 7.15) with respect to the boundary conditions:

( )

( )

in the roof and floor of tunnel:

98

Boundary conditions: { ( ) [ ( )]

( ( ) ) [ ]

so:

∫

∫

( ( ) )

( )

∫

( )

( ( ) )

( )

(

) [

( )

]

[ ( )]

then:

[ (

)]

(

) {

[ ( )] } [ ( )] ( )

And also in the sidewalls of the tunnel:

Boundary conditions: { ( ) [ ( )]

( ( ) ) [ ]

so:

∫

∫

( ( ) )

( )

∫

( )

( ( ) )

( )

(

) [

( )

]

[ ( )]

then:

[ (

)]

(

) {

[ ( )] } [ ( )] ( )

It is clear that for two cases will be:

(

)

99

The radius of the broken zone is obtained by writing the continuity of the radial stress field across the

elastic-plastic boundary:

In the roof and floor:

( )

{ [ ( )]

} {{ [ ( )]

} {[ ( )] }}

(7.30)

In the sidewalls:

( )

{ [ ( )]

} {{ [ ( )]

} { [ ( )] }}

(7.31)

where is radial stress in the elastic-plastic boundary:

[ ]

( )

The radial displacement at is :

( )

[

( )] ( )

where

(

)

depends only on the face advance effect and [

( )] depends only on the

creep effect.

=

( ) : Shear modulus

: Creep modulus

( ) : Creep function

100

8 CASE STUDY SEYMAREH POWER TUNNEL

8.1 Introduction

Seymareh dam & hydroelectric power plant project is an arch dam under construction on the Seymareh

River in Darreh-shahr County, Ilam Province, Iran (Figure 8.1). The primary purpose of the dam is

hydroelectric power generation. Studies for the dam were carried out in the mid to late 1970s and construction

began on the diversion works in 1997. In 2006, concrete placement began and on 19 May 2011, the dam

began to impound the river. The dam's first generator is expected to be operational in 2013. The power plant

will hold three 160 MW Francis turbine-generators with an installed capacity of 480 MW. The annual income

from electricity generated by the project will be 42.5 million dollars and income due to the regulation of 3,215

million cubic metres of water will be 550 million dollars, making this one of the most important economic

infrastructure projects in the area.

Figure 8.1. The location of the Seymareh Dam site, Iran

101

The dam will be a 180 m tall variable-radius arch type with a crest length of 202 m. The dam's crest width

will be 6 m and the base width 28 m while the volume of concrete in the dam structure will be 550,000 m3. The

dam sits at the head of a 27,886 km2 catchment area and will create a reservoir with a 3,200,000,000 cubic

meter capacity. The reservoir's surface area will be 69.5 km2 and its length 40 km. The dam will have a main

and auxiliary spillway. The main spillway will be controlled by two radial gates and have a maximum discharge

capacity of 5,763 m3/s. The auxiliary spillway will be uncontrolled and have a discharge capacity of 2,467 m3/s.

8.2 Geotechnical & Geological Features of the Region

The dam site is located on the north slope of the Ravandi anticline normally extending from northwest to

southeast, but in the dam site, it locally tends toward east and west because of the tectonic forces. The

formations existing in the area are namely, Bakhtiari,

Gachsaran, and Asmari according to their age from new to

old. The foundation rocks are Asmari Formation limestones

which have a – dip at the entrance of the gorge

(dam axis) gradually decreasing to – downstream,

near the anticline axis. The geology of the project area is

summarised in Figure 8.2. Of particular note is the Asmari

Formation which constitutes the main foundation rock at the

Seymareh Dam and comprises 572 m of alternating

massive too thinly bedded grey to light brown fossiliferous

limestone, microcrystalline limestone, dolomitic limestone,

marly limestone and marlstone. It is unconformably overlain

by Gachsaran evaporites. The Asmari Formation is divided

into three rock units according to the engineering

characteristics of the rock mass (Figure 8.2).

Upper Asmari (As.3): 150 m, medium to thinly bedded,

crystalline, bioclastic limestone and marly limestone.

Middle Asmari (As.2): 238 m, massive to thickly bedded

and karstified microcrystalline, dolomitic limestone and

marly limestone. Figure 8.2. Summarized geology of the Symareh

(Mahab Ghodss, 2002)

102

Lower Asmari (As.1): 188 m, medium to thickly bedded fossiliferous marly limestone and

microcrystalline limestone.

Most rock instabilities are related to the middle and upper parts of the Asmari Formation.

The river valley is U shaped with relatively steep slopes which overhang in some places. The slopes are

nearly vertical up to 760 m, above which the angle decreases to where weak marlstone interbeds are

present. In the site area, the river is between 35 and 40 m wide. The main lithology at the dam foundation is

the Asmari Formation (limestones). The limestones are typically strong, supporting the high near-vertical

slopes.

The Asmari is mainly formed from thick lime containing fairly thick to thick layers with a significant number

of void dolomitic limestone and some marly layers in between. According to Petro-logical differentials and

thickness of layers, the Asmari is divided into upper, middle, and lower parts through the dam site. The rock

existing in the original form (bedrock) is of less permeability, but several faults and many joints have

increasingly caused secondary permeability.

Angle of slope of the north limb (location of dam site) gradually increases from 20-25 degree at a 300 m

distance from the axis of the anticline to 30-35 degree in the area of the dam axis, and finally reaches 45-50

degree northward where the Asmari is covered with the alluvial deposit or Gachsaran deformation. The south

limb of the anticline is quite orderly, and the angle of the layers varies from 10 to 25 degree. An overview of the

field surveys and geological maps show that the right side of the Ravandi anticline (toward the river) is of a

steep slope in the north limb and of a gentle slope in south limb. But the left side has a gentle slope in the

north limb and of a steep slope in south limb while being far from the river.

Two major joint sets (J1 & J2) and a minor joint set (J3) plus bedding planes form the discontinuities of the

rock mass. The geometrical features of the joint sets and bedding planes are as follows (Table 8.1). Moreover,

Table 8.1. Characteristics of rock mass discontinuities in the Seymareh dam site (Mahab Ghodss, 2002)

Joint set Dip Direction

(Deg.) Dip

(Deg.) Spacing

(m) Opening

(m) Filling

Major joint set - J1 (parallel to anticline axis)

170-175 65-75 0.55 2-20 Clay, calcite

Major joint set - J2 (vertical to anticline axis)

270-275 80-90 0.65 2-20 Clay, calcite

Minor joint set - J3 120-130 70-80 1.4 2-20 Clay, calcite

Bedding planes 010-020 25-35 0.35-3 2 Clay

103

The geomechanical parameters of rock mass in the Seymareh dam site are as follows (Table 8.2).

Table 8.2. Geomechanical parameters of rock mass in the Seymareh dam site (Mahab Ghodss, 2002)

Rock class

Cohesion (MPa)

Angle of internal friction (D)

Poisson’s ratio

Modulus of Deformation (GPa)

I 1.5 45 0.25 9.0

II 0.4 39 0.28 5.5

III 0.15 29 0.32 2.5

According to the laboratory and in-situ mesurements such as dilatometry, plate load test, in situ direct

shear test, direct shear test in direction of joints and discountinous etc., some of the geotechnical features of

the Asmari rock formation in direction of the power tunnel are summarized in table 3.

Table 8.3. Some of the geotechnical features of the Asmari rock formation in direction of the Seymareh power tunnel (Mahab Ghodss, 2002)

Features Unit Quantity

Density (Original) tonnes/cubic meter 2.59

Density (Saturated) tonnes/cubic meter 2.63

Porosity percentage 4

Poisson’s ratio unitless 0.3

Modulus of Deformation GPa 6

Modulus of Elasticity (Young) GPa 9

Uniaxial Compressive Strength MPa 50

Cohesion MPa 0.5

Angle of internal friction degree 35

RMR unitless 53

Q unitless 7.42

8.3 Specifications of Powerhouse Waterways

The power plant is located 2.5 km downstream of the dam axis at the left bank and on the south side of the

Ravandi anticline (Figure 8.3). Water is supplied to the power plant through a 1546 m long power tunnel with

an internal diameter of 11 m and three 150 m long penstock tunnels, each with an internal diameter of 6.4 and

4.9 m at the beginning and contact with the power plant respectively. There is also a 71 m high vertical shaft

with an internal diameter of 8 m and a surge tank with a diameter varying from 77.75 to 40 m at a 1350 m

distance from the inlet of power tunnel (Figure 8.4).

104

Figure 87.3. Location of Power systems

The power tunnel is in a direct line for 20.55 m first part and then curves in plan within a radius of 150 m at

a 22.80 degree central angle. The curve is 59.69 m in length. And thence, the tunnel extends for a 1193.48 m

long direct route. After that, the tunnel is divided into two tunnels in a 33.39 m length, one is directly connected

to the power plant and the other is divided into two branches after a 25.7 m long extension.

Figure 8.4. Water supply and power generation system

105

Each of these three branch tunnels (penstocks) has a concrete section 6.4 m in diameter. The concrete

section is 70.1, 53.1, and 67.8 m in the first, second, and third penstock tunnel respectively. The diameter of

each penstock is changed from 6.4 to 5.7 m in a 10 m long section and thence extends 150 m long in the new

diameter. At the end, a 10 m long transition changes the diameter from 5.7 to 4.9 m in a linear direction.

8.4 Excavation Method

In attention to the geological structure of the surrounding rock mass and the cost & safety factors,

controlled blasting method has performed for excavation of Seymareh power tunnel (Figure 8.5). In tunneling,

road and railroad cuts, it is almost importance that remaining rock is of high quality in order to avoid rockfall,

rockslides and excessive maintenance work.

Before the advent of tunnel boring machines (TBM), drilling and blasting was the only economical way of

excavating long tunnels through hard rock, where digging is not possible. Even today, the method is still used

in the construction of tunnels, such as in the construction of the Seymareh power tunnel. The decision whether

to construct a tunnel using a TBM or using a drill and blast method includes a number of factors such as:

Tunnel length

Managing the risks of variations in ground quality

Required speed of construction

The required shape of the tunnel

Figure 8.5. Seymareh power tunnel during construction

Tunnel length is a key issue that needs to be addressed because large TBMs for a rock tunnel have a high

capital cost, but because they are usually quicker than a drill and blast tunnel the price per metre of tunnel is

106

lower. This means that shorter tunnels tend to be less economical to construct with a TBM and are therefore

usually constructed by drill and blast. Managing ground conditions can also have a significant effect on the

choice with different methods suited to different hazards in the ground.

8.5 Geotechnical instrumentation & monitoring

The designer of geotechnical construction works with naturally occurring materials, and does not know

their exact engineering properties. He may carry out tests in the laboratory on the samples picked up from the

field, and sometimes change the naturally occurring materials to make them more suitable for his needs. But

his structural design will essentially be based on the values of engineering properties of the materials tested by

him. Therefore, as construction progresses and exact geotechnical conditions observed or behaviour

monitored by means of instrumentation, the design judgments are evaluated and changes made, if necessary.

Hence observations by means of monitoring instruments during geotechnical construction are an integral part

of the design process. Instrumentation is a tool to assist with these observations. They are our eyes and ears

inside the rock. Instrumentation is used to measure the response (deformation, stress etc.) of soil or rock to

changes in loading or support arrangements, and from the measurements taken, the need for modifications to

the loading or support arrangements is determined.

Generally, three type of instrumentation include Tape Extensometer, Rod Extensometer and Load Cell is

used for monitoring of the Seymareh power tunnel (Figure 8.6).

Figure 8.6. Monitoring system of the Seymareh power tunnel, a) Tape Extensometer (Slope Indicator Co.), b) Rod Extensometer (Soil Instruments Ltd.), c) Load Cell (Slope Indicator Co.)

107

The tape extensometer (Figure 8.6a) is used to detect and monitor changes in the distance between two

reference points. Typical applications include:

Monitoring convergence of tunnel walls.

Monitoring deformations in underground openings.

Monitoring displacement of retaining structures, bridge supports, and other structures.

Reference points are stainless steel eyebolts that are threaded into groutable or expansion anchors.

Reference points may also be bolted to the structure. Reference points are positioned to reveal the magnitude

and direction of movements. The figure 8.7 shows typical locations for reference points. It is important to

protect the points once they have been installed, since any change in the position or the condition of the points

will affect the repeatability of the system.

Figure 8.7. Reference points grouted into drill holes in rock

The rod extensometer (figure 8.6b) monitors changes in the distance between one or more downhole

anchors and a reference head at the borehole collar. Typical applications include:

Monitoring settlement in foundations.

Monitoring subsidence above tunnels and mines.

Monitoring heave in excavations.

Monitoring the stability of tunnels and other underground openings.

Monitoring deformation in abutments and walls.

The load cells (figure 8.6c) are designed to measure loads in tiebacks, rock bolts, and cables. Typical

applications include:

Proof testing and performance monitoring of tiebacks, rock bolts, and other anchor systems.

Monitoring loading of vertical supports in underground openings.

108

8.6 Data obtained by Monitoring

For monitoring of the Seymareh power tunnel, several mean stations were established. In this study, it is

just used the data collected form four monitoring mean station which are located the chainages of 0+225.50

km (section A-A), 0+350.50 km (section B-B), 0+820.80 km (section C-C) and 1+041.50 km (section D-D)

(before the surge tank structure in chainage of 1+193.1 km). Figure 8.8 demonstrates the location of the

sections (mean monitoring stations) under study.

Figure 8.8 the location of the four monitoring mean station under study

The overburden thickness in the sections of monitoring station A-A, B-B, C-C and D-D are 142.88, 184.92,

219.15 and 178.27 meter respectively. For measuring the convergence of power tunnel, it is used the three

reference points as shown in figure 8.9.

Figure 8.9 the three reference points used for measuring of convergence in the tunnel

109

The results of monitoring of the tunnel walls (in a period of 100 days) for four sections show that the

convergence of tunnel walls has stabilized after 30 days approximately (figure 8.10).

Figure 8.10 Results of convergence monitoring of the tunnel walls in a period of 100 days (Seymareh monthly reports, 2003)

Although the curves are increasing (positive trend), but the rate of the displacements values are very low.

The maximum values of the convergence for four sections of the tunnel in different lines are demonstrated in

table 8.4.

Table 8.4. Maximum values of the convergence in four sections of the tunnel, mm (Seymareh monthly reports, 2003)

Section Chainage (km) Overburden

thickness (m) Displacement

Line 1-2 Displacement

Line 2-3 Displacement

Line 3-1

A-A 0+225.50 142.88 3 20 4

B-B 0+350.50 184.92 8 41 5

C-C 0+820.80 219.15 10 42 6

D-D 1+041.50 178.27 6 28 3

The maximum displacement is caused in line 2-3 of section C-C (42 mm). Generally, the maximum

displacements are caused in line 2-3 of all the sections. The maximum displacements of the other lines (lines

110

1-2 and 3-1) in comparison of the line 2-3 are not considerable. This means the horizontal stress is bigger that

the vertical stress in all sections. In attention to the displacements values of table 8.4, it is obvious that the

ratio of the horizontal stresses and vertical stresses ( ) is more than one. Furthermore, the displacements

values of line 1-2 (except of section A-A) are more than displacements values of line 3-1. This means the

stresses around the left side of the tunnel is more than the right side.

8.7 Deformation Analysis using Numerical Modelling

For comparison between the analytical solution, experimental data and numerical solution, several

numerical models were built using COMSOL Multiphysics finite elements software (Apendix-A). The analyses

were done in two cases; time independent and time dependent. The radius of excavated tunnel is 6 meter.

Since the problem is symmetric in two directions, only a quarter of tunnel is modeled. The meshing of the

numerical modelling is shown in figure 8.11. The Tunnel is modelled with 7932 triangular elements.

Figure 8.11. Meshing of the numerical modelling (7932 triangular elements)

Some of the material properties which are used in the model is summarised in table 8.5.

Table 8.5. Some of the material properties in the sections under study

Property Unit Quantity

Modulus of Elasticity ( E ) GPa 9

Poisson’s ratio ( ) Unit-less 0.3

Density ( ) Kg/m3 2600

Cohesion ( c ) MPa 0.5

Angle of internal friction ( ) degree 35

111

Time independent numerical analysis (instantaneous)

Time independent numerical analyses (instantaneous) are done for four sections in stationary condition.

Figure 8.12 shows the horizontal displacements (X component) in tunnel walls for four sections under study in

case of time independent. Maximum displacements caused in tunnel walls are -10.441 mm in section A-A, -

20.654 mm in section B-B, -31.901 mm in section C-C and -18.78 mm in section D-D. It is clear that the

horizontal displacements (in X direction) in the roof and floor of the tunnel are absolutely zero.

Figure 8.12. Horizontal displacements (X component) in tunnel walls for four sections in case of time independent (mm)

Also, the results of the numerical analysis show the ratios of the horizontal stresses and vertical stresses

are more than one. In the next section, it will be presented a comparison between the monitoring data and

numerical analysis results.

The distribution of the plastic zones is shown in figure 8.13. The approximate radius of plastic zone in four

sections are 7.3, 7.9, 8.2 and 7.8 meter, respectively for sections A-A, B-B, C-C and D-D.

112

Figure 8.13 Distribution of the plastic zones in four sections

Time dependent numerical analysis

In the design for structural integrity one is often most interested in the primary and secondary stages of

creep. A commonly used equation of state representation of these two creep stages is provided by the

Norton-Bailey law, sometimes called power law creep law. This is given by:

where is creep rate coefficient, is stress exponent and is hardening exponent. Time dependent

numerical analyses are done for four sections by means of COMSOL multiphysics software. In these analyses,

general conditions of the problem are the same, and only the analyses are done in a period of 100 days. For

the numerical analysis, as creep parameters (Material model: Norton-Bailey), we set (Table 8.6):

Table 8.6. the Norton-Bailey creep parameters

[ ]

10 5 0.7

113

Figure 8.14 shows the results obtained in case of time dependent (horizontal displacements, 100 days).

Figure 8.14 Horizontal displacements (X component) of side walls during 100 days, in case of time dependent

According to the results obtained of the time dependent analyses (figure 8.13), the rates of the horizontal

displacements in the sections A-A and C-C decrease after 20 days, while in the sections B-B and D-D they

decrease after 10 days (approximately). The maximum displacements for four sections A-A, B-B, C-C and D-D

after 100 days are 11.25mm, 21.60mm, 32.51mm and 19.20mm respectively.

8.8 Analytical solution

Since the tunnel is driven in fractured rocks, the generalized Hoek-Brown failure criterion (non-linear) is

used in analytical solution. As it is mentioned in previous sections, the controlled blasting method was

performed for excavation of the tunnel. Therefore, the disturbance factor D (in generalized Hoek-Brown failure

criterion) is assumed equal to 0.7 (good blasting, Appendix-B). According to the rock mechanics reports of the

Seymareh tunnel (second phase study), the Hoek-Brown parameters for intact rocks are extracted (Table 8.7):

Table 8.7. the Hoek-Brown parameters for Seymareh intact rocks

48.3 55 13.5

114

Therefore:

[

] [

]

[

] [

]

[ ]

[ ]

As it is mentioned, the power tunnel passes through the Ravandi anticline and therefore the stresses

distribution are not only a function of the overburden. In this situation, other geotectonics and geomechanical

forces can effect on the stresses distribution in region. According to rock mechanics report of the Seymareh

dam project (second phase study and complementary studies), the ratio of horizontal applied stress ( ) to

vertical applied stress ( ) are shown in table 8.8.

Table 8.8. the ratio of horizontal applied stress to vertical applied stress ( )

Section A-A

Section B-B

Section C-C

Section D-D

1.2 1.2 1 1.1

So the value of in the roof and floor of tunnel ( and respectively) using equation

7.22, and also in the sidewalls of the tunnel ( and ) using equation 7.25 for four sections is

summarized in table 8.9.

Table 8.9. Value of in the roof, floor and sidewalls of the tunnel in four sections

Section A-A

Section B-B

Section C-C

Section D-D

roof and floor 0.7842 0.8213 0.7637 0.7940

sidewalls 0.7725 0.8102 0.7554 0.7887

Equation 7.30 is used for estimating of the radius of the broken zone (plastic area) in the roof and floor of

tunnel and equation 7.30 is used in the sidewalls of the tunnel as well as. The results are shown in table 8.10.

Table 8.10. Value of (radius of the plastic zone) in the roof and floor and in the sidewalls of tunnel in four sections (m)

Section A-A

Section B-B

Section C-C

Section D-D

roof and floor 6.42 6.29 6.87 6.54

sidewalls 7.69 8.19 8.23 7.84

As it is mentioned in previous chapter (equation 7.33), the radial displacement ( ) at is :

115

( )

[

( )]

By having shear modulus and creep modulus and assuming: ( ) (

)

With T=0.2 day and ; the curves of the radial displacements during 100 days are plotted

(Figure 8.15).

Figure 8.15. Horizontal displacements (X component) of side walls during 100 days, Analytical solution

Since the radial displacements in the roof and floor of the tunnel are very low, so we have neglected them

and we have just presented the curves of the radial displacements for sidewalls of the tunnel (Figure 8.14).

8.9 Comparison of the Results obtained

The results obtained from comparison between experimental data (monitoring), numerical analyses (FEM)

and analytical solution (proposed equations) show although the field data, which are collected through the

monitoring, are very close to the analytical solution results (approximately), but there is a significant difference

between both of them and numerical modelling results (Figure 8.16). Such differences were predictable;

116

because the influence of the other activities such as excavation of shaft (vertical tunnel and surge tank) or

excavation of the ground of power plant in the numerical analysis is not considered.

Figure 8.16. Comparison between the experimental data (monitoring), numerical analysis (FEM) and analytical solution

It is obvious that other activities such as excavation of shaft and surge tank and also excavation of mean

tunnel from other direction, which were under operation at the same time, can have an effect on the

deformation of tunnel and consequently on the results of monitoring. For instance, during excavation of the

power tunnel, excavation of the ground of power plant (for construction of a 95m 47.5m 51.5m power plant)

was taking place. Therefore such huge unloading could affect the results of monitoring. Beside, blasting

operation for unloading could affect the deformation of tunnel too.

On the other hand, the initial data which are used in numerical analysis and analytical solution such as

Hoek-Brown failure criterion parameters ( , GSI, ) or the ratio of the horizontal stresses to vertical

stresses ( ) for each section are not quite accurate, because they are extracted as a representative

of the rock mass of region, not for a particular section of the tunnel. However, the accuracy of the initial data

may effect on the final results in numerical analysis and analytical solution as well.

117

SUMMARY, CONCLUSIONS & RECOMMENDATIONS

9.1 Summary and Conclusions

Time-dependent deformations in a rock mass may be manifested as creep of the intact material or creep

along the discontinuities. In intact crystalline rock at normal temperatures creep is primarily the product of time-

dependent micro fracturing of the rock; which will produce both shear and volumetric. The principal factors

controlling creep rates in intact rock are applied deviator stress, effective confining pressure and moisture

content. Creep in rock joints occurs as normal compression and shear movements along the discontinuities.

Factors that govern the creep rates in discontinuities are the character of the joints and the geometry of the

joint system with respect to the excavation.

It is known that ground pressure on the lining of a tunnel increases with time. This is due to the time-

dependent mechanical properties of the surrounding medium. If, however, the tunnel support is installed at the

tunnel face immediately after its excavation, the pressure increases with progress of the tunnel face.

In the case of a circular tunnel driven through a homogeneous isotropic linear visco-elastic medium, Panet

and Sulem proposed an analytical solution for the convergence of the tunnel walls as a function of two

parameters, the distance to the face and the time. For a similar problem, a closed-form solution for the

pressure acting on tunnel support structures was given by Sakurai.

Panet and Sulem, for determining of deformations in tunnels, have assumed that "The tunnel has a circular

cross section and around the tunnel, the rock is homogeneous and isotropic and also the tunnel is deep

enough to consider that the stress distribution is homogenous". But in all real cases, the stresses distribution

around the tunnel is not homogeneous and isotropic. In this study, some equations are proposed for

modification of the Panet and Sulem equations in case of nonhomogeneous and anisotropic (generalizing of

the problem).

Seymareh power tunnel which is considered as a case study is a part of the powerhouse waterways

system of the Seymareh dam and hydroelectric power plant project. The project is located in west of Iran. The

monitoring data of power tunnel which are collected during excavation of tunnel compared with the results of

numerical modelling and analytical solution results as well as.

The results obtained from comparison show although the field data, which are collected through the

monitoring, are very close to the analytical solution results (approximately), but there is a significant difference

between both of them and numerical modelling results. It was predictable; because the influence of the other

118

activities such as excavation of shaft and surge tank in the numerical analysis and also analytical solution is

not considered. It is obvious that other activities such as excavation of shaft and surge tank and also

excavation of mean tunnel from other direction, which were under operation at the same time, can effect on

the results of monitoring. On the other hand, the initial data which are used in numerical analysis and analytical

solution such as Hoek-Brown parameters and the ratio of the horizontal stresses to vertical stresses for each

section are not quite accurate; because they are extracted as a representative of the rock mass of region, not

for a particular section of the tunnel.

Generally, based on the research carried out in this study, the following conclusions were obtained:

Since the Panet and Sulem equations are valid only in homogeneous and isotropic situations,

therefore in this study, a closed-form solution is proposed for modification of the Panet and Sulem

equations in nonhomogeneous and anisotropic situations.

Panet and Sulem have assumed the rock fails according to the Mohr--Coulomb failure criterion,

while the failure criterion in almost rocks, especially in fractured rocks, is not linear. In this study,

some new equations have been proposed based on the non-linear Generalized Hoek-Brown

failure criterion which is more appropriate for fractured rocks.

A comparison between field data, numerical analysis and analytical solution in power tunnel of the

Seymareh project shows the analytical results are very close to the field data, but there is a

difference between both of them and numerical results.

9.2 Recommendations for future research

The following recommendations for further research can be established based on this study:

New studies using other non-linear failure criteria which are more appropriate for jointed and

fractured rocks, could be done. In the new studies must be attention paid to the geotechnical and

geomechanical properties of the rock mass surrounding the tunnel.

In this study, role of the water in the proposed equations is not considered. In future studies, more

research on deformation of tunnels in saturated/unsaturated environments can be done.

When a tunnel is designed, it is essential to understand the deformations which occur close to the

face of excavation and the rock-support interaction as well. This situation is typically three-

dimensional; therefore a special attention should be paid to the study of deformation in tunnels in

three-dimensional situation.

However the goal of this study is development of analytical solution of deformation in tunnels on

general conditions and pursuit of the study could lead to more development in this field.

119

BIBLIOGRAPHY

Anagnostou G., 2007, " Practical Consequences of the Time-dependency of Ground Behavior for Tunneling", Rapid Excavation & Tunneling Conference Proceedings (RETC).

Aristorenas, G. V., 1992, " Time-dependent behavior of tunnels excavated in shale", Doctor of Philosophy thesis, Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, U. S.

Barla G., Bonini M., Debernardi D., 2010, "Time Dependent Deformations in Squeezing Tunnels", International Journal of Geoengineering Case Histories, Vol.2, Issue 1, p.40-65.

Bernaud D., Brosset G., 1996, "The new implicit method for tunnel analysis", International Journal for Numerical and Analytical Methods in Geomechanics, Vol.20, No.9, pp. 673–690.

Bernaud D., Brosset G., 1992, "La nouvelle méthode implicite pour l'étude du dimensionnement des tunnels", Revue Française de Géotechnique, Vol.60, No.2, pp. 5–26.

Brown E. T., Bray J. W., Ladanyi B. et al., 1983, "Ground response curves for rock tunnels", Journal of Soil Mechanics and Geotechnical Engineering, ASCE, Vol.109, No.1, pp. 15–39.

Carranza C., Fairhurst C., 2000, "Application of the convergence-confinement method of tunnel design to rock masses that satisfy the Hoek-Brown failure criterion", Tunnelling and Underground Space Technology, Vol.15, No.2. pp. 187–213.

Carranza C., Fairhurst C., 1999, "The elastoplastic response of underground excavation in rock masses that satisfy the Hoek-Brown failure criterion", International Journal of Rock Mechanics and Mining Sciences, Vol.36, No.6, pp. 777–809.

Corbette F., Bernaud D., Nguyen Minh D., 1991, "Contribution à la methode convergence confinement par le principe de la similitude", Revue Française de Geotechnique, Vol.54, No.4, pp. 5–12.

Cristescu N.D., 2009, "Time Effects in Rock Mechanics", Proceedings of the SEM Annual Conference, New Mexico, USA.

Cristescu N. D., Hunsche U., 1998, "Time Effects in Rock Mechanics", Wiley Series in Materials, Modelling and Computation, John Wiley & Sons.

Deffayet M., Robert A., 1989, "Dimensionnment et réalisation d'un tunnel à partir de mesures de

convergence", Proceedings of the Colloque Organisé à Paris，Tunnels et Micro-tunnels en Terrain

Meuble-du Chantier à La Théorie, Paris, pp. 316–324.

Eisenstein Z., Branco P., 1991, "Convergence-confinement method in shallow tunnels", Tunnelling and Underground Space Technology, Vol.6, No.3, pp. 343–346.

Fahimifar A., Monshizadeh Tehrani F., Hedayat A., Vakilzadeh A., 2010, "Analytical solution for the excavation of circular tunnels in a visco-elastic Burger’s material under hydrostatic stress field", Tunnelling and Underground Space Technology Journal, Vol. 25, pp. 297–304.

Gaudin B., Polacci J. P., Panet M. et al., 1981, "Souténement d'une galerie dans les marnes du Cenomanien", Proceedings of the 10th International Conference on Soil Mechanics and Foundation Engineering, Rotterdam, A. A. Balkema, pp. 293–296.

Giraud A., Picard J.M., Rousset G., 1993, "Time dependent behaviour of tunnels excavatedin porous rock mass", International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, Vol. 30, No. 7, pp. 1453–1459.

120

Glamheden R., Hökmark H., 2010, "Creep in jointed rock masses", State of knowledge report, Swedish Nuclear Fuel and Waste Management Company.

Goodman R. E., 1980, "Introduction to Rock Mechanics", John Wiley & Sons, New York, U.S.

Guenot A., Panet M., SULEM J., 1985, "New aspects in tunnel closure interpretation" Proceedings of the 26th US Symposium on Rock Mechanics, Research and Engineering Applications in Rock Masses, pp. 455–460.

Havel F., 2004, "Creep in soft soils", Doctor of Philosophy thesis, Department of Civil and Transport Engineering, The Faculty of Engineering Science and Technology, Norwegian University of Science and Technology, Norway.

Hoek E., 2010, "Rock-Support Interaction analysis for tunnels in weak rock masses", Hoek's Corner, Tunnels in Weak Rock, http://www.rocscience.com/education/hoeks_corner.

Hoek E., 2000, " Big Tunnels in Bad Rock ", ASCE 2000 Civil Engineering Conference and Exposition, Seattle, U.S.

Katz, O. and Reches, Z., 2002, "Pre-failure damage, timedependent creep and strength variations of a brittle granite", Proceedings, 5th International Conference on Analysis of Discontinuous Deformation, Ben-Gurion University, Balkema, Rotterdam.

Kitagawa T., Kumeta T., Ichizyo T. et al. , 1991, "Application of convergence-confinement analysis to the study

of preceding displacement of a squeezing rock tunnel", Rock Mechanics and Rock Engineering，Vol.24,

No.1, pp. 31–51.

Kolymbas D., 2005, "Tunnelling and tunnel mechanics, a rational approach to tunnelling", Heidelberg, Springer-Verlag.

Konietzky H., 2004, "Numerical Modelling of Discrete Materials in Geotechnical Engineering, Civil Engineering, and Earth Sciences ", A. A. Balkema Publishers.

Ladanyi B., Gill D. E., 1988, "Design of tunnel linings in a creeping rock", International Journal of Mining and Geological Engineering, No.6, pp. 113-126.

Lee Y. L., 1994, "Prise en compte des non-linéarité de comportement des sols et roches dans la modélisation du creusement d'un tunnel", Doctor of Philosophy thesis, Département de Génie Civil, Ecole Nationale des Ponts et Chaussées, Paris.

Mahab Ghodss Consulting Engineering Company, 2002, “Rock mechanics reports of the Seymareh tunnel (second phase study)”, Technical report.

Maranha J. R., Vieira A., 2005, "Modelling the time behaviour of a tunnel using a viscoplastic bounding surface model", 8th international Conference on Computational Plasticity, Barcelona.

Nadimi S., Shahriar K., Sharifzadeh M., Moarefvand P., 2011, "Triaxial creep tests and back analysis of time-dependent behavior of Siah Bisheh cavern by 3-Dimensional Distinct Element Method", Tunnelling and Underground Space Technology Journal, Vol. 26, pp. 155–162.

Nguyen MINH D., Corbette F., 1991, "New calculation methods for lined tunnels including the effect of the front face", Proceedings of the 7th International Congress on Rock Mechanics, Aachen, Vol.1, pp. 335-338.

Nguyen MINH D., Guo C., 1993, "A ground support interaction principle for constant rate advancing tunnels", Proceedings of the Eurock 93, Balkema Rotterdam, pp. 171–177.

Oreste P. P., 2005, "Back-analysis techniques for the improvement of the understanding of rock in underground constructions", Tunnelling and Underground Space Technology, Vol.20, No.1, pp. 7-21.

121

Oreste P. P., 2003, "Analysis of structural interaction in tunnels using convergence-confinement approach", Tunnelling and Underground Space Technology, Vol.18, No.4, pp. 347–363.

Oreste P. P., Peila D., 1997, "Modelling progressive hardening of shotcrete in convergence-confinement approach to tunnel design", Tunnelling and Underground Space Technology, Vol.12, No.3, pp. 425–431.

Oreste P. P., Peila D., 1996, "Radial passive rockbolting in tunneling design with a new convergence-confinement model", International Journal of Rock Mechanics Mining Sciences and Geomechanics Abstracts, Vol.33, No.5, pp. 443–454.

Pan, Y.W., Dong, J.J., 1991, "Time-dependent tunnel convergence - II. Advance rate and tunnel-support interaction", International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, Vol. 28, No. 6, pp. 477–488.

Panet M., 2001, "Recommendation on the convergence-confinement method", Association Française des Tunnels et de l'Espace Souterrain (AFTES), Paris.

Panet M., 1995, "Calcul des tunnels par la méthode de convergenceconfinement", Press de l'école Nationale des Ponts et Chaussées, Paris.

Panet M., 1993, "Understanding deformations in tunnels",Comprehensive Rock Engineering, Vol.1, pp. 663-690.

Panet M., 1979, "Time-dependent deformations in underground works", Proceeding of the 4th Congress of ISRM, pp. 279–289.

Sakurai S., 1997, "Lessons learned from filed measurements in tunnelling", Tunnelling and Underground Space Technology, Vol.12, No.4, pp. 453-460.

Sakurai S., 1978, "Approximate time-dependent analysis of tunnel support structure considering progress of tunnel face", International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 2, pp. 159-175.

Sakurai S., Akutagawa S., Takeuchi K. et al., 2003, "Back analysis for tunnel engineering as a modern observational method", Tunnelling and Underground Space Technology, Vol. 18, No.2, pp. 185-196.

Schubert W., Button E. A., Sellner P. J., Solak T., 2003, " Analysis of Time Dependent Displacements of Tunnels ", Vol. 21, No. 5.

Singh D. P., 1975, "A Study of Creep of Rocks", International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, Vol. 12, pp. 271-276

Starfield A. M., McClain W. C., 1973, "Project Salt Vault: A Case Study in Rock Mechanics", International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, Vol. 10, pp. 641-657.

Sulem J., Panet M., Guenot A., 1987, "An Analytical Solution for Time-dependent Displacements in a Circular Tunnel", International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, Vol. 24, No. 3, pp. 155—164.

Sulem J., Panet M., Guenot A., 1987, " Closure Analysis in Deep Tunnels", International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, Vol. 24, No. 3, pp. 145—154.

Yu C., 1998, "Creep Characteristics of Soft Rock and Modelling of Creep in Tunnel", Doctor of Philosophy thesis, Department of Civil and Environmental Engineering, University of Bradford, U. K.

Zhao T., Ma Y., Tan Y., Chen Y., 2009, "Creep characteristic simulation of deep soft rock roadway and long-term mechanical analysis of lining support", Journal of Coal Science & Engineering, Vol.15, No.2, pp. 193–196.

122

APPENDICES

Appendix-A

COMSOL Multiphysics Software

Appendix-B

Hoek-Brown failure criterion-2002 edition

123

Appendix-A

COMSOL Multiphysics Software

COMSOL Multiphysics software is a finite element analysis, solver and Simulation software / FEA

Software package for various physics and engineering applications, especially coupled phenomena,

or multiphysics. COMSOL Multiphysics also offers an extensive interface to MATLAB and its toolboxes for a

large variety of programming, preprocessing and post processing possibilities. The packages are cross-

platform (Windows, Mac,Linux). In addition to conventional physics-based user interfaces, COMSOL

Multiphysics also allows for entering coupled systems of partial differential equations (PDEs). The PDEs can

be entered directly or using the so-called weak form (see finite element method for a description of weak

formulation). An early version (before 2005) of COMSOL Multiphysics was called FEMLAB.

The COMSOL was started by graduate students to Germund Dahlquist based on codes developed for a

graduate course at the Royal Institute of Technology (KTH)[1] in Stockholm, Sweden.

The COMSOL Multiphysics is an integrated environment for solving systems of time-dependent or

stationary second order in space partial differential equations in one, two, and three dimensions. Moreover,

such equations may be coupled in an almost arbitrary way. Comsol Multiphysics provides sophisticated (and

convenient) tools for geometric modelling. Therefore, for many standard problems, there exist predefined so-

called application modes which act like templates in order to hide much of the complex details of modelling by

equations. The application modes make use of the language used in the respective engineering discipline.

The COMSOL Multiphysics simulation environment facilitates all the steps in the modelling process –

defining your geometry, meshing, specifying your physics, solving, and then visualizing your results. It also

serves as a platform for the application specific modules.

Model set-up is quick, thanks to a number of predefined physics interfaces for applications ranging from

fluid flow and heat transfer to structural mechanics and electrostatics. Material properties, source terms, and

boundary conditions can all be spatially varying, time-dependent, or functions of the dependent variables. You

can freely mix physics interfaces into new multiphysics combinations as well as couple with any application

specific module.

124

Figure A.1. Screenshot of COMSOL Multiphysics Software 4.2a topography modelling of surfaces, Mount St. Helens, US

As an alternative to writing your own simulation code, the COMSOL Multiphysics user interface gives you

the option to specify your own partial or ordinary differential equations (PDEs or ODEs) and link them with

other physics interfaces. When combined with the CAD Import Module or one of the LiveLink products, this

enables you to run custom simulations on CAD models from industry-standard formats.

Collaboration within design teams is made easy with the CAD interoperability tools for COMSOL. The CAD

Import Module brings in all major CAD formats directly into the COMSOL Desktop where you can simulate

your design accurately, using a real-world multiphysics model.

The CAD Import Module includes the Parasolid® geometry kernel which enables powerful repair and

defeaturing tools. The interactive repair feature assures that imported geometries are mathematically correct

for simulation. And, in order to cut down on unnecessary details in your CAD geometries, defeaturing tools that

remove fillets, small faces, sliver faces, as well as spikes or short edges are included.

Several add-on products are available for COMSOL Multiphysics which are demonstrated in figure A.2.

The COMSOL Multiphysics is a powerful interactive environment for modelling and solving all kinds of

scientific and engineering problems based on partial differential equations (PDEs). With this product you can

easily extend conventional models for one type of physics into multiphysics models that solve coupled physics

phenomena—and do so simultaneously. Accessing this power does not require an in-depth knowledge of

mathematics or numerical analysis. It is possible to build models by defining the relevant physical quantities—

such as material properties, loads, constraints, sources, and fluxes—rather than by defining the underlying

equations. COMSOL Multiphysics then internally compiles a set of PDEs representing the entire model. You

125

access the power of COMSOL Multiphysics as a standalone product through a flexible graphical user

interface, or by script programming in the MATLAB language.

Figure A.2. available add-on products for COMSOL Multiphysics software

As noted, the underlying mathematical structure in COMSOL Multiphysics is a system of partial differential

equations. In addition to the physics mode and the modules, it is provided three ways of describing PDEs

through the following PDE modes:

Coefficient form, suitable for linear or nearly linear models

General form, suitable for nonlinear models

Weak form, for models with PDEs on boundaries, edges, or points, or for models using terms with

mixed space and time derivatives.

Using the application modes in COMSOL Multiphysics, it can perform various types of analysis including:

Stationary and time-dependent analysis

Linear and nonlinear analysis

Eigenfrequency and modal analysis

To solve the PDEs, COMSOL Multiphysics uses the proven finite element method (FEM). The software

runs the finite element analysis together with adaptive meshing and error control using a variety of numerical

solvers. A more detailed description of this mathematical and numerical foundation appears in the COMSOL

Multiphysics User’s Guide and in the COMSOL Multiphysics Modelling Guide.

126

PDEs form the basis for the laws of science and provide the foundation for modelling a wide range of

scientific and engineering phenomena. Therefore it can be used COMSOL Multiphysics in many application

areas, just a few examples being:

Acoustics

Bioscience

Chemical reactions

Diffusion

Electromagnetics

Fluid dynamics

Fuel cells and electrochemistry

Geophysics

Heat transfer

Microelectromechanical systems (MEMS)

Microwave engineering

Optics

Photonics

Porous media flow

Quantum mechanics

Radio-frequency components

Semiconductor devices

Structural mechanics

Transport phenomena

Wave propagation

Many real-world applications involve simultaneous couplings in a system of PDEs —multiphysics. For

instance, the electric resistance of a conductor often varies with temperature, and a model of a conductor

carrying current should include resistive-heating effects.

one of the unique features in COMSOL Multiphysics is something it is refered to as extended multiphysics:

the use of coupling variables to connect PDE models in different geometries. This represents a step toward

system-level modelling. Another unique feature is the ability of COMSOL Multiphysics to mix domains of

different space dimensions in the same problem. This flexibility not only simplifies modelling, it also can

decrease execution time.

Geomechanics Module

Nonlinear material add-on to the Structural Mechanics Module. Includes material models for concrete, rock,

and soil such as Cam Clay, Drucker-Prager, Mohr-Coulomb, Matsuoka-Nakai, and Lade-Duncan. In addition,

127

large strain plastic deformation and creep is included. The Geomechanics Module is a specialized add-on to

the Structural Mechanics Module for simulation of geotechnical applications such as tunnels, excavations,

slope stability, and retaining structures. The module features tailored user interfaces to study deformation,

creep, plasticity, and failure of soils and rocks, as well as their interaction with concrete and human-made

structures. A variety of material models for soils are provided: Cam-Clay, Drucker-Prager, Mohr-Coulomb,

Matsuoka-Nakai, and Lade-Duncan. In addition to the built-in plasticity models, user-defined yield functions

can be created by the versatile equation-based user interface provided by the COMSOL

Multiphysics environment. Dependencies of a computed temperature field as well as other field quantities can

be blended into these material definitions.

The Geomechanics Module also makes available very powerful tools for modelling concrete and rock

materials: the Willam-Warnke, Bresler-Pister, Ottosen, and Hoek-Brown models are available as built-in

options and can also be adapted and extended to a more general class of brittle materials. The Geomechanics

Module can easily be combined with analysis from other Modules such as the porous media flow,

poroelasticity, and solute transport functionality of the Subsurface Flow Module. The triaxial test is one of the

most common tests used in laboratory soil testing. The soil sample is normally placed inside a rubber

membrane and then compressed maintaining a radial pressure. In this model, a vertical displacement and a

confinement pressure are applied on the sample and the static response and the collapse load for various

confinement pressures are studied. The material is modeled with the soil plasticity feature, and the Drucker-

Prager criterion. The analysis can be simplified by considering the intrinsic axial symmetry of the model. Figure

A.3 shows the Effective plastic strain in the soil sample. The red zone shows the volume of soil that under

plastic deformation.

Figure A.3. Effective plastic strain in the soil sample.

128

CAD Import Module

Facilitates the reading of most industry-standard CAD formats. Includes add-in packages that support the

file formats for specific CAD programs' geometry kernels. Also switches COMSOL's native geometry engine

into the Parasolid engine. For bidirectional CAD links see LiveLink products for CAD. For CATIA V5, the File

Import for CATIA V5 is needed.

LiveLink Products for CAD

Adds bidirectional links to major CAD software packages. Allows for geometric (and other) parameters to

be swept for from within COMSOL, encompassing CAD package. Includes the CAD Import Module.

LiveLink for MATLAB

Integrates COMSOL Multiphysics with MATLAB thereby extending COMSOL's simulation tools with

scripting capabilities. Allows for saving COMSOL models as M-files. Such M-files still require COMSOL

Multiphysics to be installed since these are calling high-performance library functions proprietary to COMSOL.

COMSOL generated M-files can be combined with any MATLAB M-files for advanced postprocessing,

statistical and probabilistic analysis, for-loops, optimization, and combination with other MATLAB add-on

products.

LiveLink for Excel

Adds a COMSOL Multiphysics tab and specialized toolbar to the Excel ribbon for controlling the

parameters, variables and mesh, or running a simulation. Includes the capability of import/export of Excel files

for parameter and variable lists in the COMSOL GUI.

Material Library

Internal material property database with more than 2500 materials and 20,000 properties. The database

contains temperature dependence of electrical, thermal, and structural properties of solid materials. The

material library can also accept files generated by the MatWeb material property database. This is a

searchable database of over 59,050 material data sheets, including property information on thermoplastic and

thermoset polymers, metals, and other engineering materials. MatWeb is a division of Automation Creations,

Inc. (ACI) of Blacksburg, Virginia

Nonlinear Structural Materials Module

Nonlinear material add-on to Structural Mechanics Module. Includes elastoplasticity, hyperelastic

material, viscoplasticity, and creep. Also allows for large-strain plastic deformation.

129

Structural Mechanics Module

Performs classical stress-strain analysis with full multiphysics capabilities. Comprises large deformation

(geometrically nonlinear deformations], shell, beam and contact abilities; all able to be freely coupled to other

physics such as in the case of fluid-structure interaction - which comes as a built-in tool. Static, time-domain,

frequency-domain, eigenmode, nonlinear analysis included. Nonlinear material models included when

combined with Nonlinear Structural Materials Module or Geomechanics Module.

130

Appendix-B

Hoek-Brown failure criterion-2002 edition

Evert Hoek

Consulting Engineer, Vancouver, Canada

Carlos Carranza-Torres

Itasca Consulting Group Inc., Minneapolis, USA

Brent Corkum

Rocscience Inc., Toronto, Canada

ABSTRACT: The Hoek-Brown failure criterion for rock masses is widely accepted and has been applied in

a large number of projects around the world. While, in general, it has been found to be satisfactory, there are

some uncertainties and inaccuracies that have made the criterion inconvenient to apply and to incorporate into

numerical models and limit equilibrium programs. In particular, the difficulty of finding an acceptable equivalent

friction angle and cohesive strength for a given rock mass has been a problem since the publication of the

criterion in 1980. This paper resolves all these issues and sets out a recommended sequence of calculations

for applying the criterion. An associated Windows program called “RocLab” has been developed to provide a

convenient means of solving and plotting the equations presented in this paper.

1. INTRODUCTION

Hoek and Brown (1980) introduced their failure criterion in an attempt to provide input data for the analyses

required for the design of underground excavations in hard rock. The criterion was derived from the results of

research into the brittle failure of intact rock by Hoek (1968) and on model studies of jointed rock mass

behaviour by Brown (1970). The criterion started from the properties of intact rock and then introduced factors

to reduce these properties on the basis of the characteristics of joints in a rock mass. The authors sought to

link the empirical criterion to geological observations by means of one of the available rock mass classification

schemes and, for this purpose; they chose the Rock Mass Rating proposed by Bieniawski (1976).

Because of the lack of suitable alternatives, the criterion was soon adopted by the rock mechanics

community and its use quickly spread beyond the original limits used in deriving the strength reduction

relationships. Consequently, it became necessary to re-examine these relationships and to introduce new

elements from time to time to account for the wide range of practical problems to which the criterion was being

131

applied. Typical of these enhancements were the introduction of the idea of “undisturbed” and “disturbed” rock

masses Hoek and Brown (1988), and the introduction of a modified criterion to force the rock mass tensile

strength to zero for very poor quality rock masses (Hoek, Wood and Shah, 1992).

One of the early difficulties arose because many geotechnical problems, particularly slope stability issues,

are more conveniently dealt with in terms of shear and normal stresses rather than the principal stress

relationships of the original Hoek-Brown riterion, defined by the equation:

(

)

( )

where and

are the major and minor effective principal stresses at failure, is the uniaxial

compressive strength of the intact rock material and and are material constants, where for intact

rock.

An exact relationship between equation 1 and the normal and shear stresses at failure was derived by J.

W. Bray (reported by Hoek, 1983) and later by Ucar (1986) and Londe1 (1999).

Hoek (1990) discussed the derivation of equivalent friction angles and cohesive strengths for various

practical situations. These derivations were based upon tangents to the Mohr envelope derived by Bray. Hoek

(1994) suggested that the cohesive strength determined by fitting a tangent to the curvilinear Mohr envelope is

an upper bound value and may give optimistic results in stability calculations. Consequently, an average value,

determined by fitting a linear Mohr-Coulomb relationship by least squares methods, may be more appropriate.

In this paper Hoek also introduced the concept of the Generalized Hoek-Brown criterion in which the shape of

the principal stress plot or the Mohr envelope could be adjusted by means of a variable coefficient a in place of

the square root term in equation 1.

Hoek and Brown (1997) attempted to consolidate all the previous enhancements into a comprehensive

presentation of the failure criterion and they gave a number of worked examples to illustrate its practical

application.

In addition to the changes in the equations, it was also recognised that the Rock Mass Rating of Bieniawski

was no longer adequate as a vehicle for relating the failure criterion to geological observations in the field,

particularly for very weak rock masses. This resulted in the introduction of the Geological Strength Index (GSI)

by Hoek, Wood and Shah (1992), Hoek (1994) and Hoek, Kaiser and Bawden (1995). This index was

subsequently extended for weak rock masses in a series of papers by Hoek, Marinos and Benissi (1998),

Hoek and Marinos (2000) and Marinos and Hoek (2001).

1 Londe’s equations were later found to contain errors although the concepts introduced by Londe were extremely important in the

application of the Hoek-Brown criterion to tunnelling problems (Carranza-Torres and Fairhurst, 1999)

132

The Geological Strength Index will not be discussed in the following text, which will concentrate on the

sequence of calculations now proposed for the application of the Generalized Hoek Brown criterion to jointed

rock masses.

2. GENERALIZED HOEK-BROWN CRITERION

This is expressed as:

(

)

( )

where is a reduced value of the material constant and is given by:

[

] ( )

and are constants for the rock mass given by the following relationships:

[

] ( )

[ ] ( )

is a factor which depends upon the degree of disturbance to which the rock mass has been subjected by

blast damage and stress relaxation. It varies from 0 for undisturbed in situ rock masses to 1 for very disturbed

rock masses. Guidelines for the selection of are discussed in a later section.

The uniaxial compressive strength is obtained by setting in equation 2, giving:

( )

and, the tensile strength is:

( )

Equation 7 is obtained by setting

in equation 2. This represents a condition of biaxial

tension. Hoek (1983) showed that, for brittle materials, the uniaxial tensile strength is equal to the biaxial

tensile strength.

Note that the “switch” at for the coefficients and (Hoek and Brown, 1997) has been

eliminated in equations 4 and 5 which give smooth continuous transitions for the entire range of values.

133

The numerical values of and , given by these equations, are very close to those given by the previous

equations and it is not necessary for readers to revisit and make corrections to old calculations.

Normal and shear stresses are related to principal stresses by the equations published by Balmer (1952).

( )

(

) √

( )

where:

( ) ( )

3. MODULUS OF DEFORMATION

The rock mass modulus of deformation is given by:

( ) (

)√

(( ) ) ( )

Equation 11a applies for . For use equation 11b:

( ) (

) (( ) ) ( )

Note that the original equation proposed by Hoek and Brown (1997) has been modified, by the inclusion of

the factor , to allow for the effects of blast damage and stress relaxation.

4. MOHR-COULOMB CRITERION

Since most geotechnical software is still written in terms of the Mohr-Coulomb failure criterion, it is

necessary to determine equivalent angles of friction and cohesive strengths for each rock mass and stress

range. This is done by fitting an average linear relationship to the curve generated by solving equation 2 for a

range of minor principal stress values defined by , as illustrated in Figure B.1. The fitting

process involves balancing the areas above and below the Mohr-Coulomb plot. This results in the following

equations for the angle of friction and cohesive strength :

[ (

)

( )( ) ( )

] ( )

134

[( ) ( )

]( )

( )( )√ ( ( ) ) ( )( )

( )

Where:

Figure B.1. Relationships between major and minor principal stresses for Hoek-Brown and equivalent Mohr-Coulomb criteria.

Note that the value of , the upper limit of confining stress over which the relationship between the

Hoek-Brown and the Mohr-Coulomb criteria is considered, has to be determined for each individual case.

Guidelines for selecting these values for slopes as well as shallow and deep tunnels are presented later.

The Mohr-Coulomb shear strength , for a given normal stress , is found by substitution of these values

of and in to the equation:

( )

The equivalent plot, in terms of the major and minor principal stresses, is defined by:

( )

135

5. ROCK MASS STRENGTH

The uniaxial compressive strength of the rock mass is given by equation 6. Failure initiates at the

boundary of an excavation when is exceeded by the stress induced on that boundary. The failure

propagates from this initiation point into a biaxial stress field and it eventually stabilizes when the local

strength, defined by equation 2, is higher than the induced stresses and

. Most numerical models can

follow this process of fracture propagation and this level of detailed analysis is very important when

considering the stability of excavations in rock and when designing support systems.

However, there are times when it is useful to consider the overall behaviour of a rock mass rather than the

detailed failure propagation process described above. For example, when considering the strength of a pillar, it

is useful to have an estimate of the overall strength of the pillar rather than a detailed knowledge of the extent

of fracture propagation in the pillar. This leads to the concept of a global “rock mass strength” and Hoek and

Brown (1997) proposed that this could be estimated from the Mohr-Coulomb relationship:

( )

With and determined for the stress range giving:

[ ( )]( )

( )( ) ( )

6. DETERMINATION OF

The issue of determining the appropriate value of for use in equations 12 and 13 depends upon the

specific application. Two cases will be investigated:

1. Tunnels − where the value of is that which gives equivalent characteristic curves for the two

failure criteria for deep tunnels or equivalent subsidence profiles for shallow tunnels.

2. Slopes – here the calculated factor of safety and the shape and location of the failure surface have to be

equivalent.

For the case of deep tunnels, closed form solutions for both the Generalized Hoek-Brown and the Mohr-

Coulomb criteria have been used to generate hundreds of solutions and to find the value of that gives

equivalent characteristic curves.

For shallow tunnels, where the depth below surface is less than 3 tunnel diameters, comparative numerical

studies of the extent of failure and the magnitude of surface subsidence gave an identical relationship to that

obtained for deep tunnels, provided that caving to surface is avoided.

136

The results of the studies for deep tunnels are plotted in Figure B.2 and the fitted equation for both cases is:

[

]

( )

where is the rock mass strength, defined by equation 17, is the unit weight of the rock mass and

is the depth of the tunnel below surface. In cases where the horizontal stress is higher than the vertical stress,

the horizontal stress value should be used in place of .

Equation 18 applies to all underground excavations, which are surrounded by a zone of failure that does

not extend to surface. For studies of problems such as block caving in mines it is recommended that no

attempt should be made to relate the Hoek-Brown and Mohr-Coulomb parameters and that the determination

of material properties and subsequent analysis should be based on only one of these criteria.

Similar studies for slopes, using Bishop’s circular failure analysis for a wide range of slope geometries and

rock mass properties, gave:

[

]

( )

where is the height of the slope.

Figure B.2. Relationship for the calculation of σ′3max for equivalent Mohr-Coulomb and Hoek-Brown parameters for tunnels.

137

7. ESTIMATION OF DISTURBANCE FACTOR D

Experience in the design of slopes in very large open pit mines has shown that the Hoek-Brown criterion

for undisturbed in situ rock masses ( ) results in rock mass properties that are too optimistic (Sjöberg,

Sharp and Malorey, 2001, and Pierce, Brandshaug and Ward, 2001). The effects of heavy blast damage as

well as stress relief due to removal of the overburden result in disturbance of the rock mass. It is considered

that the “disturbed” rock mass properties (Hoek and Brown, 1988), in equations 3 and 4, are more

appropriate for these rock masses.

Lorig and Varona (2001) showed that factors such as the lateral confinement produced by different radii of

curvature of slopes (in plan) as compared with their height also have an influence on the degree of

disturbance. Sonmez and Ulusay (1999) back-analysed five slope failures in open pit coal mines in Turkey and

attempted to assign disturbance factors to each rock mass based upon their assessment of the rock mass

properties predicted by the Hoek-Brown criterion. Unfortunately, one of the slope failures appears to be

structurally controlled while another consists of a transported waste pile. The authors consider that the Hoek-

Brown criterion is not applicable to these two cases.

Cheng and Liu (1990) report the results of very careful back analysis of deformation measurements, from

extensometers placed before the commencement of excavation, in the Mingtan power cavern in Taiwan. It was

found that a zone of blast damage extended for a distance of approximately 2 m around all large excavations.

The back-calculated strength and deformation properties of the damaged rock mass give an equivalent

disturbance factor .

From these references it is clear that a large number of factors can influence the degree of disturbance in

the rock mass surrounding an excavation and that it may never be possible to quantify these factors precisely.

However, based on their experience and on an analysis of all the details contained in these papers, the

authors have attempted to draw up a set of guidelines for estimating the factor D and these are summarised in

Table B.1. The influence of this disturbance factor can be large. This is illustrated by a typical example in

which , and . For an undisturbed in situ rock mass surrounding a tunnel

at a depth of 100 m, with a disturbance factor , the equivalent friction angle is while the

cohesive strength is . A rock mass with the same basic parameters but in highly disturbed

slope of 100 m height, with a disturbance factor of , has an equivalent friction angle of

and a cohesive strength of . Note that these are guidelines only and the reader would be

well advised to apply the values given with caution. However, they can be used to provide a realistic starting

point for any design and, if the observed or measured performance of the excavation turns out to be better

than predicted, the disturbance factors can be adjusted downwards.

138

Table B.1 Guidelines for estimating disturbance factor D

139

8. CONCLUSION

A number of uncertainties and practical problems in using the Hoek-Brown failure criterion have been

addressed in this paper. Wherever possible, an attempt has been made to provide a rigorous and

unambiguous method for calculating or estimating the input parameters required for the analysis. These

methods have all been implemented in a Windows program called “RocLab” that can be downloaded (free)

from www.rocscience.com. This program includes tables and charts for estimating the uniaxial compressive

strength of the intact rock elements ( ), the material constant and the Geological Strength Index ( ).