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
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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.
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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.
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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
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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
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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
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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
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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.
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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
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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
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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 :
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( )
[
( )]
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.
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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.
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APPENDICES
Appendix-A
COMSOL Multiphysics Software
Appendix-B
Hoek-Brown failure criterion-2002 edition
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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.
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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 :
[ (
)
( )( ) ( )
] ( )
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[( ) ( )
]( )
( )( )√ ( ( ) ) ( )( )
( )
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.
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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 ( ).