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A MODEL FOR ROCK MASS BULKING AROUND UNDERGROUND EXCAVATIONS
by
JAIRO GOMEZ-HER-DEZ
Thesis submitted in partial fulfillment
of the requirement for the degree of
Master of Applied Science (M.A.Sc.)
School of Graduate Studies
Laurentian University
Sudbury, Ontario, Canada
@ JAIRO GOMEZ-HERN~DEZ, 200 1
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The process of stress-induced fracturing around underground excavations is associated
with dilation of the failed rock. Different attempts to account analytically for this dilation
have been made in the framework of the plasticity theory. This theory has been developed
mostly for metals and strain-hardening rnaterials, but brittle failure of rock-like materials
is characterized by strain-softening behavior, which may violate stability principles of
plasticity.
The most comrnon concept used in continuum mechanics has been the dilation angIe,
which sets a ratio between plastic volumetric and deviatoric strain rates or increments. In
spite of the validity of this concept, there are other components of dilation, e.g. an-
isotropic dilation, which cannot yet be taken into account in the context of continuum
mechanics. As noted by Kaiser et al. (2000), the volume increase of stress-fractured rock
near an excavation results from three sources: (1) dilation due to new fracture growth, (2)
shear along existing fractures or joints, and most importantly, (3) dilation due to
geometric incompatibilities when blocks of broken rock move relative to each other as
they are forced into the excavation. This dilation process is called Rock Mass Bulking
(Kaiser et ul. 1996), and is quantified by a Bulking Factor (Kaiser et al. 1996), defined as
the percentage increase in radial deformation due to fracturing inside the faiIure zone
extending to a depth of failure (dl).
iii
To develop a model for the calculation of the Bulking Factor ( B e , it is necessary to
consider two options: (1) develop a non-continuum theory for rock mass behavior, or (2)
adapt continuum mechanics principles to the problem and introduce an empirical
component to calibrate the model. In this thesis, the second approach is adopted. A semi-
empirical Rock Mass Bulking Mode1 (RMBM) was developed, using as starting concepts
dilation angle, plastic strain rates, effective defonnation modulus, effective Poisson's
ratio, Griffith locus, and the definition of BF introduced by Kaiser et al. (1996). The
model was calibrated in order to obtain BFs in accordance with experimental data, and
case studies were used to account for bulking around underground excavations.
ACKNOWLEDGEMENTS
1 wish to express my appreciation to the following people:
My wife Tatiana and Our children Elizabeta and Daniel, and my family in my home
country, for their love and emotional support.
Dr. Peter Kaiser, my supervisor, for giving me the opportunity to accomplish this work,
for his advict:, his scientific papers, and his financial support.
Dr. Dougal McCreath, for his support as graduate program coordinator, and his
contributions through scientific papers; Dr. Derek Martin, for his course in Rock
Mechanics and his scientific papers; and Dr. S.K. Sharan, for his course in Numencal
Modeling and his motivating discussions and suggestions.
I would like to acknowledge and thank MIRARCO staff, who have assisted me to make
Sudbury and Canada my second home. In this manner, everyone, directly or indirectly,
has contributed to the development of my thesis.
Finally, 1 would like to thank Laurentian University and the School of Graduate Studies
for their financial support, and for providing me the opportunity to obtain a Master degree
of Applied Science in Minera1 Resources Engineering.
CONTENTS
... ABSTRACT ...................................................................................................................... 111
ACKNOWLEDGEMENTS ................................................................................................ v 1 . INTRODUCTION ..................................................................................................... 10 2 . LITERATURE REVIEW ........................................................................................ 19
2.1 Principles of plasticity theory ..................................................................... 19 ..................................................................................................... 2.2 Rock dilation 21 ..................................................................................................... 2.3 Griffith locus 22
....................................................... 2.4 Dilation and displacements around tunnels 23 2.5 Depth of failure around tunnels ....................................................................... 24
...................................................................................... 2.6 BrittIe and ductile flow 24 ................................................................................................... 2.7 Bulking factor 25
3 . MODEL DEVELOPMENT ................................................................................ 26 ................................................................................. 3.1 MATERIAL BEHAVIOR 26
.................................................. 3.1.1 Rock mass types and in-situ stress levels 26 3.1.2 Griffith locus ............................................................................................. 26 3.1.3 Rock dilation ....................... ... ................................................................ 34 3.1.4 Yield function ............................................................................................ 41
................................................................................................... 3.1.5 Flow rule 45 3.2 TUNNEL BEHAVIOR ................................................................................. 48
3.2.1 Depth of failure ......................................................................................... 48 3.2.2 Bulking factor ...................................................................................... 50
4 . PARAMETRIC ANALYSIS ................................................................................... 54 ........................................................................................ 5 . MODEL VERIFICATION 60
5.1 Rock type I ........................................................................................................ 61 ....................................................................................................... 5.2 Rock type I1 64
5.3 Rock type 111 ...................................................................................................... 68 5.4 RocktypeIV ................................................................................................... 71
............................................................................................. 6 . TUNNEL MODELING 74 ............................................................................................................... CONCLUSIONS 84
FUTURE WORK .............................................................................................................. 86 ............................................................................................................. BIBLIOGRAPHY 87
................................................................................................................. 1 . References 87 .................................................................................................. 2 . Related publications 94
APPENDIX A GRJFFITH LOCUS ............................................................................... 107 APPENDIX B DILATION ANGLE ............................................................................ 112 APPENDIX C FRICTION COEFFICIENT .................................................................. 117
........................................................................................ APPENDIX D FLOW RULE 121 APPENDIX E DEPTH OF FAILURE .......................................................................... 126
...................................... APPENDIX F FLOW CHART TO DEVELOP THE RMBM 130 APPENDIX G ROCK MASS BULKING MODEL ...................................................... 131 APPENDIX H SPREADSHEET FOR NOMOGRAMS ............................................... 137
LIST OF FIGURES
Figure 1 . 1 Figure 1.2
Figure 1.3 Figure 3.1
Figure 3.2
Figure 3.3
Figure 3.4 Figure 3.5
Figure 3.6
Figure 3.7
Figure 3.8 Figure 3.9 Figure 3.10 Figure 3.1 1 Figure 3.1 2 Figure 3.1 3 Figure 3.14 Figure 3.1 5 Figure 3.1 6
Figure 4.1 Figure 4.2 Figure 4.3 Figure 4.4
Figure 5.1
Figure 5.2 Figure 5.3 Figure 5.4
Figure 5.5 Figure 5.6 Figure 5.7
............... Illustration of starting concepts for Bulking Factor calculation Illustration of rock mass bulking concept around underground excavations ...........................................................................
................... Examples of tunnel convergence due to rock mass bulking Examples of tunnel instability and brittle failure as a function of RMR and the ratio of the maximum far-field stress 01 to the unconfined compressive
. ......................................*...... strength oc (after Martin et al 1999) A stress-strain diagram obtained frorn a single uniaxial compression test of
....................................... Lac du Bonnet granite (after Martin 1993) Young's modulus and Poisson's ratio as a function of damage for an
............................................... unconfined test (after Martin 1993) Mechanisms for damage initiation (After Kaiser et al . 2000) .................
............................... Griffith locus for three confinement stress levels
Comparison of the measured crack damage locus and the predicted Griffith locus at various confining stresses (after Martin 1993) ............... Effective deformation modulus. effective Poisson's ratio and plastic strains
.............................................................................. for rock II Plastic strain rates for rock II ...................................................... DiIation angle and dilation factor for rock II .................................... Plastic strain rate vector plot in 01-03 space for rock types I to IV .......... Plastic strain rate vector plot in q-a3 space for rock type 1 .................. Linear Mohr-Coulomb yield criterion (Modified from Ogawa et al . 1987) . Friction strength component concept ............................................. Yield function and plastic potential ............................................. Depth of failure for four rock types and a hydrostatic stress field ............ Stress condition and geometry of the analyzed problern (after Ogawa et al . 1987) ............................................................................... Influence of support pressure @) on bulking factor (B F) ...................... Influence of excavation radius (ri) on bulking factor (BF) ..................... Influence of rock uniaxial strength (a, ) on bulking factor (BF) ............... Influence of in-situ stress (p, ) on excavation wall displacement (u, ) and bulking factor (BF) .................................................................. Equivalent tunnel radius (a) and baggage (A) concepts (after Kaiser et al . 1996) ................................................................................... Severe sidewall bulking (after Spearing et al . 1994) ........................... Test tunneI under high vertical stress (after Spearing et al . 1994) ............ Radial displacements and bulking factor for Witwatersrand Gold Mine openings in Rock Type 1 (South Africa) .......................................... Fracturing around a tunnel at Kloof Gold mine ..................................
................... Location of instruments at Silver Shaft (after Barton 1983) Radial displacements and bulking factor for a Kloof Gold Mine opening (South Africa) and the Silver shaft in Rock Type II (USA) ...................
vii
Figure 5.8 Figure 5.9
Figure 5.10
Figure 5.1 1
Figure 5.12
Figure 5.13
Figure 6.1
Figure 6.2
Figure 6.3
Figure 6.4
Figure 6.5
Figure 6.6
Figure 6.7
.......................... Instrumentation at UK coalmine (after Singh 1983).. Instrumentation at Nottinghamshire coalmine site 3 (after Whittaker et al. 1983). ................................................................................. Radial displacements and bulking factor for some coal mine openings in
................................................................ Rock Type III (UK).. Measured radial displacement profile at chainage 2263 (after Pelli et al. 1991).. .................................................................................. Instrumentation at Nottinghamshire coalmine site I (after Whittaker et al. 1983). ................................................................................... Radial displacements and bulking factor for Donkin Morien Tunnel
.................... (Canada) and a coalmine opening in Rock Type IV (UK).. Combination of stress concentration factor, depth of failure chart, bulking and convergence chart for support selection to control failing ground in
........................... over-stressed brittle rock (after Kaiser et al. 2000). Bulking factor as a function of distributed support capacity (after Kaiser et al. 1996). .............................................................................. Depth of failure (dl), excavation wall displacement (unV) and bulking factor (BF) for very good rock (I).. ........................................................ Depth of failure (dJ), excavation wall displacement (u,) and bulking factor (BF) for good rock (a[).. ............................................................. Depth of failure (df), excavation wall displacement (u,) and bulking factor (BF) for fair rock (III) ................................................................ Depth of failure (df), excavation wall displacement (unV) and bulking factor (BF) for poor rock (N) .............................................................. Depth of failure (df), excavation wall displacement (u,) and bulking factor (BF) for rock I, IT, ID, and IV.. .............................................
viii
LIST OF TABLES
............................................................................................. Table 3.1 Rock mass types 27 ......................................................................... Table 3.2 Rock mass constants 1 and q 44
......................................... Table 3.3 Values of GSI and mi used to obtain Figure 3.13. 44 .................................................................................. Table 3.4 Coefficients Ar and Br. 4 9
The purpose of this work is to develop a semi-empirical mode1 for Rock Mass Bulking
around underground excavations in brittle rock. It is well known that the process of
stress-induced fracturing around the excavations is associated with volume increase
(dilation) of the failed rock, which depends on such factors as rock mass quality, in-situ
stress, support pressure and excavation radius. In rock mechanics, most attempts to
account for this dilation have been made in the framework of the plasticity theory, using
the flow rule concept (Gerongiannopoulos and Brown 1978, Elliot and Brown 1986,
Michelis and Brown 1985, Maier and Hueckel 1979).
The parameter most widely used to measure dilation is the dilation angle, which sets the
ratio between plastic volumetric strain rates and the associated plastic deviatoric strain
rates (Vermeer and de Borst 1984). The way strain rates (and consequently dilation) are
associated with the yield stresses is called the flow rule. In plasticity theory, a yield
function F ( { O ) ) , where o are the stress components, is defined such that F=O at yield.
Following yield, strains are not uniquely defined by the current stress state, but depend on
the stress history. However, it is possible to relate plastic strain increments (rates) to the
current state of stress through a plastic potential function Q ( ( O } ) . If Q=F, the flow rule is
said to be us~ociated and the vector of plastic strain rates is orthogonal to the yield
surface at yield (Elliot and Brown 1986). Using this flow mle implies that the dilation
angle is equal to the friction angle of the material. These concepts were developed
initially for metals and materials that behave as perfectly plastic or as strain hardening
materials. However, rocks and rock masses often show strain-softening behavior, which
may violâie stabiiity principles of plasticity (Drucker 1966).
Different approaches have been adopted to develop constitutive models for rock-like
materials. Some of them consider that these materials may exhibit strain softening
characteristics, and the important fact that the plastic strain rate vector is not normal to the
yield function, meaning that the plastic potential and the yield function are not the same
(Maier and Hueckel 1979). In this case, the fiow rule is non-associated. However, even
using these concepts, the theory is still based on the continuum mechanics principles, and
there are rock mass behaviors that cannot be described in the framework of this theory.
The problem is that, unless the rock fails in a ductile manner, brittle failure is in essence a
non-continuum process. As noted by Kaiser et al. (2000), the volume increase of stress-
induced fracturing near the excavation results from three sources: (1) dilation due to new
fracture growth, (2) shear along existing fractures or joints, and rnost importantly, (3)
dilation due to geometric incompatibilities when blocks of broken rock move relative to
each other as they are forced into the excavation.
Following Kaiser et al. (1996), we cal1 this dilation process Rock Mass Bulking, and it
can be quantified by the Bulking Factor (BF), defined as the percentage increase in radial
deforrnation due to fracturing inside the failure zone extending to the depth of failure (df).
The following equation is used as the starting point for this work and can be found in the
above-mentioned paper:
(Eqn 1.1)
where u, - Radial displacements at excavation wall; udf - Radial displacements at elasto-
plastic boundary defined by the depth of failure df (Figure 1.1).
Figure 1.1 Illustration of starting concepts for Bulking Factor calculation.
This equation allows calculating the BF at any point inside of the failed zone, if UW is
replaced by ur, which represents the radial displacement at any point inside the failed
zone. The mode1 developed in this thesis includes elastic displacements, which are
discounted from the total displacements for rock mass bulking calculation, as they are
assumed not to lead to dilation. The final equation, developed to calculate the Bulking
Factor, is the following:
(Eqn 1.2)
where u: and u i - Radial total and elastic displacements in the plastic zone
E respectively; uz and ure - Radial total and elastic displacements at elasto-plastic
boundary respectively; Te and ri - Radii of the plastic zone and of the excavation.
The relationship between the concept of Bulking Factor and the concept of dilation angle
lies in the fact that dilation angle is one of the parameters that characterize the
constitutive behavior of rock material, while BF is a parameter that characterizes the
constitutive behavior of the rock mass under different engineering conditions. Of course,
it is expected to have greater values of BF for greater values of dilation angle. The
concept of BF is necessary for tunnel design because excavation wall displacements are
of paramount importance in the engineering behavior of underground excavations and
ground support.
Figure 1.2 is an illustration of what bulking looks like for different rock types. Having a
look at this picture, we can understand the importance of BF for tunnel design. Note that
there are two sources for displacements: (1) the continuum displacement of the rock
toward the excavation, related to its deformation modulus, and (2) the plastic
displacement due to the volume increase of the failed rock (BF). Note that, under the
same in-situ stress conditions, most of the displacement in good rock is due to its volume
increase, while in poor rock most of the displacernent is related to its low deformation
modulus.
Figure 1.3 shows some examples of the effect of volume increase of the failed rock
around underground excavations. If bulking is not controlled in time and with good
engineering rneans, the excavation may collapse or the support system may be destroyed
by large defomations.
To develop a model for the calculation of the Bulking Factor (BF), we have to consider
two options: (1) develop a non-continuum theory for rock mass behavior, or (2) adapt
continuum mechanics principles to the problem and introduce an empirical component to
calibrate the model. In this thesis, the second approach is adopted.
In Chapter 2, a literature review of the problem is presented. Not al1 the papers included
in the Bibliography are explained here, but only those from which some basic concepts,
used in this work, are adopted.
calculation in the failed zone as a function of rock mass quality, in-situ stress, support
pressure, and excavation radius.
In Chapter 4 the model is subjected to parametric analyses, which allows calibrating the
BF as a function of the above-mentioned parameters.
In Chapter 5, the model is verified through comparing its results with case studies. Seven
cases were analyzed from mines in different countries, covering a wide range of rock
m a s types, from hard rock metal mines (Rock I) to soft rock coalmines (Rock IV). It is
important to note that more cases could be included, but the idea was to use the cases
where extensometer data are available. NevertheIess, the model is open for further
calibration as more convergence and extensometer measurements from underground
excavations are exarnined.
In Chapter 6, nomograms are presented to facilitate the use of this work for engineering
purposes. The concepts developed in the work of Kaiser et al. (2000) were fundamental to
establish these nomograms. They correlate in-situ stress, elastic stress concentration at
excavation wall, depth of fadure, wall displacements, support pressure, and Bulking
Factor. Four nomograms for four types of rock mass (1, TI, ID, IV) were developed.
(Courtesy J. Henning) (Courtesy J. Henning)
(Courtesy J. Henning) (Courtesy T. Villeneuve)
Figure 1.3 Examples of tunnel convergence due to rock mass bulking.
Fiinally, this thesis yields a series of conclusions and suggestions for future work.
Continuum mechanics continues to be a very useful means for rock mechanics studies,
though many empirical considerations still have to be integrated to arrive at meaningful
results. Eventually, a non-continuum rock mechanics theory seems to be a necessity to
account for the process of transition from continuum to discontinuum.
2.1 PRlNCPLES OF PLASTlCfTY THEORY
The developed in this thesis model for rock mass bulking borrows some concepts from
the theory of plasticity, but it is not an analytical solution. As mentioned before, it is a
semi-empirical model based on some analytical tools, with a further empirical calibration
with case studies. Some important concepts of plasticity theory deserve special attention.
In most of the works, dilation is quantified using the concepts of plastic strain rates or
plastic strain increments. Gerongiannopoulos and Brown (1978) explain that plastic
strains are irrecoverable and once yield has taken pIace, the relationships between stresses
and strains in the material may not be unique. Therefore, the plasticity theory most
commonly used in engineering is one in which stresses are related to plastic strain
increments (rates). Other important concepts used in this work are Flow Rule, Yield
Function and Plastic Potential. The Flow Rule is defined as the stress-plastic strain
increment (rate) relationship (see Section 3.1.5). The Yield Function is defined as the
States of stress under which plastic flow can occur (see Section 3.1.4). The concept of
Plastic Potential combines the yield function and the flow rule, allowing the plastic strain
rates to be calculated from differentiation of the plastic potential function (See Appendix
D)
Elliot and Brown (1986) analyze the problem of strain hardening and strain softening.
They note that rocks and rock masses often show strain-softening characteristics and that
modeling this behavior presents a number of difficulties because plasticity is a continuum
theory, but strain softening in a continuum may cause instability. According to these
researchers, at low confining pressures, brittle behavior accompanied by volumetric
dilation predominates, while at high confining pressures ductile work hardening behavior
accompanied by sample contraction prevails.
Similar concepts can be found in the work of Michelis and Brown (1985). They use the
concept of plasticity, which is developed for engineering materials that behave in either a
work softening or a work hardening manner. The yield equation they developed takes into
account the characteristics of rock-like materials and volumetric strains due to shear
strains, showing that dilation is a fundamental factor to be considered in the constitutive
models for rock.
Softening behavior, which has been proven to be a very important feature of brittle rock
behavior, is analyzed arnong others by Maier and Hueckel (1979). They show that the
flow rule for rock-like materials is predominantly non-associated, rneaning that the plastic
strain rate vector, in the strain space superimposed to the stress space, is not directed as
the outward normal to the yield surface (see Figure 3.14). Moreover, the direction of the
plastic strain rate vector is related to the plastic dilation. It changes, depending upon the
rock hardening behavior (expansion of the yield surface), the softening behavior
(shrinking of the yield surface) or the perfect plasticity, when the yield surface remains
unaltered. This work shows how important it is to consider dilation in the constitutive
mode1 of the material.
Other important work on strain softening is by Prevost and Hoeg (1975). They explain the
concept of isotropic hardening or softening, which means that the initial yield surface
expands or contracts uniformly. Moreover, they suggest that strain softening, which is
predominant in brittle rock behavior, can be accounted for by a gradua1 loss in shearing
resistance after the peak strength has been reached.
2.2 ROCK DILA TION
The idea of analytically considering volume increase as rock fails around underground
excavations has been considered by many authors through the associated and non-
associated flow rules (Brown et al. 1983). The basic concept to account for volume
increase has been the dilation angle.
Brace et al. (1966), based on laboratory experiments on granite, aplite, and marble,
showed how vohmetric changes took place as rock was loaded. Dilation is the increase of
volume relative to elastic changes caused by deformation.
An important work on rock dilation, after Vermeer and de Borst (1984), defines dilation
as the change in volume that is associated with shear distortion of an element in the
material. They point out that a suitable parameter for characterizing a dilatant material is
the dilation angle y. According to this work, the dilation angle concept was introduced by
Hansen (1 958) and represents the ratio of plastic volume change over plastic shear strain.
Their research shows that for soils, rock, and concrete, the dilation angle is generally
lower than the friction angle. An equation to calculate dilation angle is suggested, based
on a linear Mohr-Coulomb yield equation, which, after substitution of friction angle by
dilation angle, transforms into the plastic potential. Based on laboratory data, dilation 3 .
angle shouid fa11 between 0" and 20" whether we are dealing with soils, concrete or rock.
Hoek and Brown (1997) confirm the fact that dilation angle is lower than the friction
angle. Based on their wide experience, they suggest some typical values for very good,
average, and poor quality rock. For very good rock, they suggest that the dilation angle is
about 114 of the friction angle; for the average quality rock, the value suggested is 1/8 and
poor rock seems to have a negligible dilation angle.
2.3 GRIFFITH LOCUS
The Griffith locus equation, applied by Berry (1960) for tensile conditions, by Cook
(1965) for uniaxial compression, and by Martin (1993) for triaxial compression, was used
to model the stress-strain behavior of rock. According to Cook (1965), non-elastic
behavior can be described by Griffith locus in the strain-stress plane, and when the slope
of this locus, adao, is greater than zero, the material is intrinsically brittle and liable to
spontaneous fracture. The Griffith locus 'concept is also consistent with the Mohr-
Coulomb yield criterion. Cook's (1965) paper shows that this criterion is based on energy
conservation principles. The locus, based on a crack sliding model, defines the strain-
stress path along which a material fails due to the concurrent extension of a number of
identical cracks.
Kerneny and Cook (1986) developed the concept of Griffith locus and combined it with
non-linear deformation and effective modulus concepts. They show how important the
locus is to express the behavior of a solid containing many cracks under any boundary
conditions. Al1 stress-strain behavior of the rock under load, including initial elastic
behavior, peak strength, and post failure, can be modeled using the Griffith locus and the
effective modulus concept. These analytical tools are in good agreement with laboratory
tests on rock (Martin 1993, Wawersik 1 968).
2.4 DILA TlON AND DISPLACEMENTS A ROUND TUNNELS
Different approaches have been undertaken to introduce the concept of dilation in plastic
solutions for tunnels (Brown et al. 1983). Most of them refer to associated and non-
associated flow d e s . Ladanyi (1974) introduced the concept of dilation to analyze the
plastic volumetric strains that take place around underground circular excavations under
hydrostatic in-situ stress. He used the associated flow rule combined with the Mohr-
Coulomb straight-Iine yield criterion.
Brown et al. (1983) developed closed form solutions for stresses and displacements
around circular excavations under hydrostatic in-situ stresses. They used the associated
flow rule for an elastic-brittle-plastic rnodel, combined with Hoek-Brown yield criterion.
It is interesting to note that they attempted to get a solution for an elastic-softening-plastic
model. Wowever, to solve that problem, because of algebraic complexity, it was not
possible to get a closed solution for the complete stress and strain distribution around a
circuIar excavation. Instead, they developed a stepwise solution.
Two works that are used for the present thesis are those of Ogawa (1986) and Ogawa and
Lo (1987). In these works elasto-plastic solutions for stresses and displacements around
circular openings are developed applying the non-associated flow rule, combined with
Hoek-Brown or Mohr-Coulomb yield criteria. A constant Dilation Factor (Ny) is
included in the solutions to account for the volume increase as rock fails.
2.5 DEPTH OF FAILURE AROUND TUNNELS
Kaiser et al. (1996) and Martin et al. (1999) conducted a research to establish the main
parameters that influence the depth of failure in brittle rock. Based on case studies and on
the concepts of zero friction strength component and cohesion loss, they show that the use
of a rock mass failure criterion with frictional parameters (m>O, according to Hoek-
Brown) significantly under-predicts the depth of brittle failure. They proposed to use the
so-called brittle rock mass parameters (m=O and s=0.11) to mode1 the depth of failure,
and the results were shown to have a good agreement with field observations. The
empirical equation developed by thern for the depth of stress-induced failure around
underground excavations is considered in this thesis.
2.6 BRITTLE AND DUCTILE FLOW
Martin et al. (1999) and Diederichs (2000) note that, unlike ductile materials, in which
shear slip surfaces can form while continuity of material is maintained, brittle failure
deals with materials for which continuity must first be disrupted before kinematically
feasible mechanisms can form.
Following Mogi (1966), brittle behavior is characterized by a sudden and appreciable
drop of the stress-strain curve after the peak load point, while ductile behavior is
characterized by a stress-strain curve with a rnonotonic changing, positive slope, after the
yield point. Mogi shows how confinement influences the transition from brittle to ductile
behavior.
According to Cook (1965), the Griffith locus allows modeling a material behavior for a
rock that fails as a brittle material, and whose brittIeness reduces as it fails or as a
consequence of an increase in the confinement stress.
The concept of Bulking Factor (BF) was introduced in the Canadian Rockburst Research
Program 1990-1995, developed by CAMIRO Mining Division (Kaiser et al. 1996). The
Bulking Factor provides the percentage increase in radial deformation due to fracturing
inside the failure zone. They note that bulking of rocks around underground excavations
results from three sources: (1) dilation due to new fracture growth, (2) shear along
existing fractures or joints, and most importantly, (3) dilation due to geometric
incompatibilities when blocks of broken rock rnove relative to each other as they are
forced into excavation. They show that support plays a fundamental role in controlling
bulking around tunnels. For example, an effective reinforcement system, providing a
distributed support capacity of >200 kPa, can reduce bulking, in hard brittle rock, from
as much as 30-60%, for unsupported rock, to less than 3%.
3.1 MATERIAL BEHA VIOR
3.1 -1 ROCK MASS TYPES AND IN-SITU STRESS LEVELS
Based on rock mass rating (Bieniawski 1976) and the GSI classification developed by
Hoek et al. (1998) four types of rock masses are considered in this thesis. The basic
characteristics of the rock masses are presented in Table 3.1.
Another factor that is important to caIibrate the model for Rock Mass Bulking is the in-
situ stress level cornpared to the rock strength, as the depth of failure, which influences
bulking, depends on the stress level and induced stress around the excavation. The mode1
developed in this work was calibrated in such a manner that it is applicable to the
intermediate to high in-situ stress ranges (Figure 3.1).
3.1 -2 GRIFFITH LOCUS
To develop a model for Rock Mass Bulking, it is necessary to model, first-of-all, the rock
stress-strain behavior. As pointed out by Cook (1965), and Kemeny and Cook (1986),
stress-strain behavior may be modeled using Griffith locus. Many researchers have
validated this assumption. Wawersik (1968) experimentally measured the stress-strain
Table 3.1 Rock mass types.
--
ROCK MASS 1 Il ÏÏI N
TYPE (VERYGOOD) (GOOD) (FAIR) (POOR)
GSI
o c
mi
Note: GSI - Geological Strength Index; Oc - Uniaxial compression strength of the rock
(MPa); mi - Intact rock constant; m, s - Hoek-Brown rock mass strength constants; # -
Rock mass friction angle (O); k = (l+sin@)/(l-sin@); C - Cohesion (MPa); 0' - Rock
mass uniaxial strength (MPa); E - Young's Modulus (GPa); v - Poisson's ratio.
Massive (RMR > 75)
Moderately Fractured (50 > RMR < 75)
Linear elastlc tesponse. Faillng or diding of blocks and wedges.
Highly Fractured
Unravelllng of blacks from the excavation surface.
Squeezlng and sweliing rocks. Elastlc/plasllc continuum.
Figure 3.1 Examples of tunnel instability and brittle faiture as a function of RMR and the ratio of the
maximum far-field stress 01 to the unconfined compressive strength a, (after Martin et al. 1999).
locus for different rocks and Martin (1993) did the same for the Lac du Bonnet granite
and other rock types. These works have demonstrated the applicability of the Griffith
locus to mode1 rock mass stress-strain behavior.
Martin (1 993) established general relationships between crack growth observations and
measured strains during loading of laboratory samples in compression (Figure 3.2).
Axial Stress (MPa)
I 1 -0.2 4.16 - 1 -0.06 -0.04 0 0.1 : 0.2 ; 0.3 ] 0.4
tateral Strain (%) 0.2 C
I
- Axial Straln Gaugs
-- Latetal Strain Gaoga
Figure 3.2 A stress-strain diagram obtained from a single uniaxial compression test of Lac du Bonnet
granite (after Martin 1993).
He notes that cracks initiate at about 0.40, (region Il), and then grow in a stable manner
until about 0.8% (region m), where sample diIatancy begins. In region IV unstable
cracking leads to a stage characterized by macroscopic crack sliding as a shear plane
develops in the sample. As rock .fails, both the Young's modulus and Poisson's ratio
change (Figure 3.3). The former reduces as the rock fails and the latter increases. These
concepts are important in rock behavior modeIing and are taken into account in this work.
Lac du Bonnet Granite UFtL 420 Level 1
Peak(~) Sample MB1-21.065 u3 = O MPa
Poisson's ratio
P d 0.6 $-
Y 2
Figure 3.3 Young's modulus and Poisson's ratio as a function of damage for an unconfined test (after
Martin 1993).
The works after Cook (1965), Kemeny and Cook (1986), Wawersik (1968), and Martin
(1993) have demonstrated theoretically and experimentally that a crack sliding mode1 can
be used for the analysis of rock dilation. It has been shown (Martin 1993) that dilation is
related to macroscopic crack sliding, and that the Griffith locus can be used to mode1
this process (Figure 3.4).
.( O'
t Sliding Crack Mode1 T e d e Crack Mode1
Figure 3.4 Mechanism for damage initiation (After Kaiser et al. 2000).
Figure 3.5 presents the Griffith loci for three confinement stress levels. As a material
with a given density of cracks is loaded, it follows a stress-strain path given by the
Young's modulus of the rock, until it intersects the Griffith locus, and fracture initiation
occurs. As rock fails, its crack density increases, which can be captured through the
increase of crack length (c) and a resulting decrease in the Young's modulus (dashed
lines).
Another important issue to be considered is the influence of confinement on rock m a s
behavior. As shown in Figure 3.6, confinement influences the strength and deformation
properties of the rock. As pointed out before, for a given value of crack length, the post-
0.5 1.5 2 2.5 3 3.5 4 Normalized Axial strain
Figure 3.5 Griffith locus for three confinement stress levels (after Martin 1997).
peak slope of the locus, given by addo, is reduced by the confining stress. In other
words, confined rock fails in a less brittle manner (Cook 1965).
Griffith Locus
G= 20 GPa a= 1 ~ / r n ~
Adal Strain (%)
Figure 3.6 Cornparison of the measured crack damage locus and the predicted Griffith locus at various confining stresses (after Martin 1993).
Based on energy principles, Cook (1965) developed the basic equation for the Griffith
locus in uniaxial compression:
(Eqn 3.1)
where v - Poisson's ratio; E - Young's modulus; $ - Friction angle; n - crack density; G =
En(l+ V ) - Shear modulus; p = tan$ - Coefficient of friction; 8 = (1/2)atan(l/,u) - Critical
crack angle.
Martin (1993) expanded this equation for axi-symmetric triaxial compression (e = Q):
where c is the half crack length.
This equation is expressed in tems of normal, shear, and principal stresses and was
reduced by the author of this thesis to the following form in tems of only principal
stresses:
o, - ~VO, P lof + J O , a, + LO: Eir = +
2G(l +v) a, - O, (Mo, - NO,)' (Eqn 3.3)
where I, JJ L, MJ N, and P depend on material properties of the rock (Appendix A, Eqn
A. 14).
The Griffith locus is used in this thesis and represents the starting point for the
development of a mode1 of the stress-strain behavior of a brittle rock mass. A detailed
derivation of the equations can be followed in the Appendix A.
3.1.3 ROCK DILATION
The Griffith locus represents a convenient tool to mode1 axial stress-strain rock behavior.
Nevertheless, as a sample is loaded in the axial direction, it deforms not only in this
direction, but also in the lateral direction. Dilation is a phenornenon that can be captured
only if it is known what is happening in the axial and Iateral directions as the rock is
loaded. As was shown above, the Young's modulus decreases while Poisson's ratio
increases when the rock fails, and their resulting values are called effective modulus and
effective Poisson's ratio, respectively. The effective modulus for a sample in axi-
symmetric triaxial compression (a2 = 03) can be calculated from Hooke's law, assuming
that axial strain follows the Griffith locus:
1 = - [(a, - 2 ~ 0 , )]
E 11
(Eqn 3.4)
where E,, is given in Eqns 3.1 or 3.3.
The influence of cracks on the Poisson's ratio are analyzed in this thesis following Walsh
J.B. (1965), and the basic equation that relates the Poisson's ratio to the effective modulus
of deformation in compression is used in this work:
(Eqn 3.5)
The combination of the Griffith locus (Eqn 3.2), the effective modulus of deformation
(Eqn 3.4), and the effective Poisson's ratio (Eqn 3 3 , allows developing an equation for
dilation angle under axi-symmetric triaxial compression stress conditions (Q = c3). After
rearranging for plane strain conditions (see Appendixes B and D), the equation for
dilation angle, used to develop the rock mass bulking model, is obtained:
(Eqn 3.6)
where Evpr and E,, are the volumetric and deviatoric plastic strain rates respectively. A
dilation factor Nyl, which is mentioned in the following sections, can now be calculated:
(Eqn 3.7)
The dilation angle is based on the ratio between volumetric and deviatoric plastic strain
rates. Thus, it is indirectly a function of the rock properties and the confinement. Eqn 3.6
can be calibrated in such a manner that the peak values of the dilation angles match
experimental results. The mode1 was adjusted to produce peak dilation angle values
ranging between 0" and 20°, as suggested by many researchers like Vermeer and de Borst
(1984) and Hoek and Brown (1997). In reality these values may be greater when large
geometric incornpatibilities (e.g. during spalling of rock) are encountered. A fuIl
derivation of the equation for the dilation angle can be followed in the Appendix B.
Figure 3.7 shows a graphical representation of the equations developed in Appendix B.
Note that as the rock deforrns, the effective deformation modulus decreases and the
Poisson's ratio increases, two important aspects that have been observed experimentally
(Martin 1993) and nurnerically (Diederichs 2000). This figure also shows the
development of plastic axial and lateral strains as the crack length increases.
Figure 3.7 Effective deformation modulus, effective Poisson's ratio and plastic strains for rock II.
Figure 3.8 shows the behavior of plastic strain rates as the sample deforms. The axial
strain rates are positive and the lateral strain rates are negative, which is consistent with
the adopted compression positive rule. From the axial and lateral strain rates the
volumetric and deviatoric strain rates can be calculated.
Figure 3.9 shows the behavior of the dilation angle and the dilation factor. Note that as
failure starts, the dilation angle and the dilation factor have the highest values, and as the
rock fails, they decrease. Another important feature to see is how the confinement stress
influences dilation. Note how the increase of the confinement leads to the decrease in
dilation.
Figure 3.8 Plastic strain rates for rock II.
In Figure 3.10, using the mathematical mode1 for dilation, developed in this thesis, it is
illustrated how plastic strain rate vector is influenced by confinement. Plastic volumetric
strain rates E~~~ (horizontal component) and plastic deviatoric strain rates E,, (vertical
component) were calculated for stresses given by the Mohr-Coulomb yield function (see
section 3.1.4), for the four types of rock rnentioned before. Note how the plastic strain
rate vectors (short lines on the yield curve) change in direction as confinement increases.
At low confinement, volumetric strain rates are negative (dilation) and, as confinement
increases, they turn positive (contraction). This means that as confinement increases the
rock behavior changes from bulking (dilating) to contracting.
Figure 3.9 Dilation angle and dilation factor for rock II.
For comparative purposes, the spalling limits after Diederichs (2000) and Kaiser et al.
(2000) and the strain weakening/ductile function after Mogi (1966) are superimposed on
Figure 3.10. It is useful to see that at iow confinement it is expected to have greater
volume increase and that the change from brittle to ductile behavior is accompanied by a
change in dilation. These aspects should be treated more deeply in more advanced
research studies (see for exarnple Diederichs 2000).
Figure 3.10 Plastic strain rate vector plot in oi-a3 space for rock types 1 to IV (Table 3.1).
Substituting in the mode1 for dilation angle, developed in this thesis, the yield function by
the constant ratios q/030f 20, 10, and 3.4, it was noted that, for a selected type of rock,
lines of constant stress ratio correspond to lines of constant bulking or dilation, and that
the Mogits (1966) ratio corresponds to a transition from contraction to dilation. This can
be seen in Figure 3.1 1, where the plastic strain rates are superimposed to the lines of
constant principal stress ratio.
Figure 3.11 Plastic strain rate vector plot in ai-03 space for rock type 1 (Table 3.1)
The states of stress under which failure can occur are defined by a yield function. Recent
works have emphasized the fact that brittle failure is a process of cohesion loss and
friction mobilization (Martin 1993, Kaiser et al. 2000). The Griffith locus, which is used
as one of the components for the development of this work, is in accordance with this
principle. Based on energy principles, Cook (1965) shows that the condition for failure of
a crack-sIiding mode1 can be expressed by the following equation (Appendix A):
(Eqn 3.8)
where s, on - Shear and normal stress; v - Poisson's ratio; G - Shear modulus; p -
Coefficient of friction; a - Fracture surface energy; c - Crack half length.
Eqn 3.8 can' be reduced to:
(Eqn 3.9)
where SI represents the material's intrinsic strength or cohesion (right-hand side part in
Eqn 3.8). Note that Eqn 3.9 is the weIl-known linear Coulomb equation, with cohesive
and friction strength components, and that as the crack Iength increases, the cohesion
decreases, simulating the process of cohesion loss. Regarding friction rnobilization,
however, the Griffith locus concept does not include explicitly this aspect of failure. This
is the reason why in this work friction is treated as a constant as rock fails, and faiIure is
treated as a cohesion loss process (Figure 3.12).
From back analysis of tunnel displacement observations (see Chapter 5), it has been
found that the better the quality of the rock is, the less the mobilized friction strength
component should be. This can be modeled introducing an empirical friction coefficient
(k), which is a function of rock mass properties.
TAU
PEAK STRENGTH
POSTPEAK STBEWGTH
SIGMA 3 SIGMA 1 aBsIDVAL SIGMA I P M SIGMA
Figure 3.12 Linezir Mohr-Coulomb yield criterion (modified from Ogawa et al. 1987).
To mode1 the friction strength component in Eqn 3.9, the rock mass friction angle @ is
multiplied by the friction coefficient k (p = tan(k#)). An explanation of the derivation of
Eqn 3.10 can be found in Appendix C, and the coefficient k can be calculated from the
following equation:
(Eqn 3.10)
where A, q - Rock mass constants; GSI - Geological Strength Index (Hoek et al. 1998);
mi - Intact rock constant (Hoek et al. 1998); oc - Uniaxial compression strength Mf a.
The constants L and q were established for the above-mentioned four types of rock mass,
and are presented in Table 3.2. A graphical representation of Eqn 3.10 with the
coefficients of Table 3.2 is presented in Figure 3.13. Note that a constant GSI and mi was
taken for each rock mass type, as listed in Table 3.3.
Table 3.2 Rock mass constants il and q.
Rock Type A rl
I Very Good
II Good
DI Fair
IV Poor
Table 3.3 Values of GSI and mi used to obtain Figure 3.13.
Rock Type GSI mi
1 Very Good
II Good
IiI Fair
IV Poor
3.1.5 FLOW RULE
The state of stress under which plastic flow can occur is defined by a yield function, and
the stress-plastic strain increment relations are known as flow rule. Gerogiannopoulos and
Brown (1978) point out that in the early stages of the development of plasticity theory,
the yield function and flow rule were treated independently, but later they were combined
using the concepts of plastic potential and normality. According to that approach, the
plastic strain rates (incrernents) are obtained by partial differentiation of the yietd
function, which serves as a plastic potential. The identification of the yield function as
the plastic potential does have some theoretical justification in that it permits certain
Figure 3.13 Friction strength component concept.
uniqueness theorems of the plasticity theory to be proved. The geornetrical interpretation
of the concept of plastic potential is that if the stresses and plastic strain rates
(increments) are CO-axial, then the strain rate vector will be perpendicular to the yield
surface (a representation of the yield function on principal stress axes) since it is
proportional to the gradient of the yield function (Gerogiannopoulos and Brown 1978).
This result is known as the principle of normality (Drucker 1952, 1964), and the flow rule
is called associated flow rule. If a linear Mohr-Coulomb yield function is assumed
(Figure 3.14) and identified as a plastic potential, the angle iy that the plastic strain rate
vector forms with respect to the vertical axis, is equivalent to the friction angle used in the
Mohr-Coulomb equation, formed between the horizontal axis and the yield function.
Vermeer and de Borst (1 984) point out that for soils, rock and concrete the principle of
normality, as formulated by Drucker (1952, 1964) is disproved, and that the angle iy is
lower than the friction angle of the material. This means that the yield function cannot be
used as a plastic potential, and that the plastic strain rates (increments) cannot be
calculated from the derivation of the yield function. A different equation for the plastic
potential is needed, In this case, the flow nile is called non-associated (Figure 3.14).
PUSTIC STRAIH RATE VECiOR
PLASTIC POTEMTIAL
PLASTIC VOLUMETRlC STRAIN RATE AND HYDROSTATIC STRESS
Figure 3.14 Yield function and plastic potentiaI (modified from Ogawa et al. 1987).
Following Ogawa et al. (1987), and based on Eqn 3.6, a non-associated flow rule is used
in this thesis. It is useful to note that an equation for a plastic potential is not needed in
this case. The plastic strain rates are calculated based on the material deformation
behavior that follows the Griffith locus and the assumptions of effective deformation
modulus and Poisson's ratio. An explanation of the flow rule, as it is used in this thesis,
can be found in Appendix D.
3.2 TUIVNELBEHAVDR
3.2.1 DEPTH OF FAILURE
The depth of failure is a very important parameter to be considered for bulking factor
analysis. Eqn 1.1 shows that the bulking factor is related to the depth of failure, i.e. the
difference between plastic radius and excavation radius. The depth of failure indicates the
amount of rock that fails, and the bulking factor is meant to calculate how this rock
increases in volume. Practical experience indicates that in hard brittle rock, for typical
and economically achievable support pressures (less than 2.0 MPa), the depth of failure is
essentially independent of the support pressure (Kaiser et al. 2000).
Following Ogawa (1986) and Martin et al. (1999), and using case study data (see Chapter
5), for unsupported excavations the radius of the plastic (damaged) zone can be calculated
using the following equation (Appendix E):
-= Ar-+Br 5 [ o . ) (Eqn 3.11)
where A, Br - Rock constants; p, - In-situ hydrostatic stress; a, - Uniaxial compression
strength; r, - Radius of elastic-plastic boundary; n - Excavation radius.
This equation is similar in structure to the one developed by Martin et al. (1999) for
brittle rock, which for hydrostatic in-situ stress is expressed as:
(Eqn 3.12)
Eqn 3.12 is a particular case of Eqn 3.1 1, with A r 2 . 5 and Br0.49. Hence, Eqn 3.1 1 is
adopted as a general equation for the depth of failure in this work. The coefficients have
to be established for different kinds of rock. Based on case studies presented later in this
thesis, the coefficients for four types of rock. were established and are listed in Table 3.4.
The graphical representation of Eqn 3.1 1, with the coefficients in Table 3.4, together with
Eqn 3.12 suggested by Martin et al. (1 999), are shown in Figure 3.15.
Table 3.4 Coefficients Ar and Br.
Rock type Ar Br
1 Very Good 2.3 0.48
II Good 3.2 0.50
III Fair 3.8 0.70
IV Poor 4.6 0.72
Figure 3.15 Depth of faiiure for four rock types and a hydrostatic stress field.
3.2.2 BULKING FACTOR
Eqn 3.6 for rock dilation is used to develop a Rock Mass Bulking Mode1 (RMBM). This
equation is transformed in order to be included in an analytical solution for dispIacements
around underground circular excavations. To be consistent with the material behavior
adopted in this thesis, the problem should be solved taking into account the stress-strain
material behavior given by the Griffith locus. Unfortunately, that solution is extremely
compiex from the mathematical point of view. It is useful to remember that Brown et al.
(1983) attempted to get a solution for an elastic-softening-plastic mode1 but because of
algebraic complexity had to adopt a stepwise approach. To simplify the problem, an
elastic-brittle-plastic stress-strain mode1 for the rock, with peak and residual strength
values and with cohesion and friction, can be adopted. The solution of Ogawa (1986) for
displacements was used for this purpose.
The geometry of the problem is presented in Figure 3.16. A circular excavation under
hydrostatic in-situ. stress is analyzed. At the wall of the excavation a support pressure can
be appljed. Under some bowndary conditions, e.g. at sufficiently high in-situ stress, a
plastic zone may develop around the excavation. From the analytical point of view, the
depth of failure depends on rock strength, the stress level, and the magnitude of the
support pressure.
The following equation represents the solution for radial dispIacements in the plastic zone
(Ogawa 1986):
(Eqn 3.13)
where N, - Dilation factor; B, - Integration constant;&: - Radial elastic strains;~: -
Tangential elastic strains; iy - Dilation angle in plane strain conditions.
STRESS CONDITION 1 GEOMETRY
Figure 3.16 Stress condition and geometry of the analyzed problem (after Ogawa et al. 1987).
As shown in Appendix D, the term (1-siny) in Eqn 3.13 can be substituted by the term
(1- tan^;). After rearranging, the equation used to calculate the total radial displacements
in the plastic zone around the circular excavation takes the forrn:
(Eqn 3.14)
The basic difference between Eqn 3.13 and Eqn 3.14 is that the dilation angle used in Eqn
3.13 is a constant value, while in Eqn 3.14 the dilation angle is a function of the rock
properties and confinement and is calculated from Eqn 3.6. In Ogawa's (1986) solution
y must be prescribed as an input parameter, whereas it is derived from rock properties in
the solution presented in this thesis.
The solution of Eqn 3.14, taking into account Eqn 1 .l, produces the equation for Bulking
Factor:
where UT and uf - Radial total and elastic displacements in plastic zone; u: and un
- Radial total and elastic displacements at elasto-plastic boundary; re and ri - Radii of the
plastic zone and of the excavation.
Note the sirnilarity between Eqn 3.15 and Eqn 1.1. The same general expression as
suggested by Kaiser et al. (1996) is used to calculate the Bulking Factor. The difference
between these equations is that in the latter consideration is made for elastic
displacements and the bulking factor can be calculated at any point inside of the failed
zone. Eqn 3.15 is used in the following sections, and a detailed derivation of the mode1
cm be found in Appendix F.
In Chapter 3, a model for rock mass bulking was developed. The Bulking Factor (BF), a
pararneter that rneasures rock mass bulking, can be calculated from Eqn 3.15. In spite of
the simplicity of the final equation, the rnodel is relatively cornplex. More than ten
variables are involved in this solution. For some of them correIations were found, but
other variables had to be fixed as average assumed values. Appendix F shows the
compIete model, which was called RMBM (Rock Mass Bulking Mode]). The following is
an example of the input parameters needed for the model:
INPUT PARAMETRS
RT=1 Rock Type (1, II[, m, IV) a,= 200 Uniaxial compressive strength, MPa p,=80 In-situ hydrostatic stress, MPa p=O Radial support pressure, MPa ri=2. 1 Tunnel radius, m
It is useful to analyze how the RMBM behaves as a function of the input parameters. A
pararnetric analysis was undertaken for this purpose. Note in Figure 4.1 that the Bulking
Factor (BF) is greater for good quality rock than for poor quality rock. As pointed out by
Kaiser et al. (2000), values of BF in the order or 30-60% can be expected in unsupported
drifts in hard brittle rock. Poor rock is expected to have very Iow volume increase during
failure. Note how BF is sensitive to support pressure (p) , as practical experience confirms,
and how this sensitivity is higher for hard brittle rock.
The influence of the excavation radius (r i ) on BF is shown in Figure 4.2. In Very Good (I)
and Good (II) rock, as the excavation radius increases, the depth of failure also increases,
and more highly dilatant rock fails around the excavation, leading to greater values of BF.
In Fair (III) and Poor (IV) rock, in spite of the fact that the depth of failure increases as
the excavation radius increases, it is expected to have lower values of BF due to the low
dilation angle that characterizes this rock.
Figure 4.3 shows the influence of the rock uniaxial strength (a,) on BF. Note that the
better the quality of the rock, the greater the BF is. This result is in agreement with
practical experience, and can be better understood considering that higher strength rock
has higher dilation angles and leads to lower depths of failure (Hoek and Brown 1997).
An important issue of the developed RMBM is the relation between radial displacements
(u,,) and Bulking Factor (BF) under the same in-situ stress conditions. Note in Figure 4.4
that poor quality rock mass is expected to have bigger wall displacements, but lower BF.
This is explained by the fact that, as was shown before, poor quality rock mass has lower
dilation angles than good quality rock, but the depth of failure is larger. Note as well that
the increase of in-situ stress (p,) leads to an increase in BF. This can be explained by the
fact that as the depth of failure increases, the rock in the failed zone is subjected to further
deformations, and it dilates more. The increase of the rate of dilation is greater than the
rate of increase of the depth of failure.
Figure 4.1 Influence of support pressure @) on bulking factor (BF).
Figure 4.2 Influence of excavation radius (ri) on bulking factor (BF).
Figure 4.3 Influence of rock uniaxial strength (0') on bulking factor (BF).
Figure 4.4 Influence of in-situ stress @,) on excavation wall displacement (u,) and bulking factor (BF).
Part of the process of calibrating the mode1 included a detailed analysis of several case
studies. For each of the rock types, information was found about displacements around
excavations, in-situ stress, opening radius, and rock mass properties. Some assumptions
had to be made because not al1 the information required was always available. In
particular, the assumption of hydrostatic in-situ stresses is seldom satisfied. Nevertheless,
the developed in this thesis mode1 can be applied to non-hydrostatic in-situ stresses,
considering that the depth of failure, a parameter involved in bulking factor calculations,
is insensitive for a range of stress ratios (K,) from about 1 to 5 (Detournay and St. John
1988, Martin et al. 1999, Kaiser et al. 2000). In addition, for engineering applications of
this inodel, aIlowance must be made for the fact that under anisotropic in-situ stress
conditions the damaged zone is localized.
Another question that rnay arise concerns is the fact that excavations usually are not
circular. Kaiser et al. (1996, 2000) have demonstrated that, for the purpose of depth of
failure calculation, mining and civil excavations can be approximated as circular openings
using the concepts of equivalent tunnel radius and baggage (Figure 5.1). These principles
are adopted in this work,
bag-gage
Figure 5.1 Equivalent tunnel radius (a) and baggage (A) concepts (after Kaiser et al. 1996).
5.1 ROCK TYPE I
Some information about South African hard rock mines at depths in excess of 4000 m
was analyzed. Unfortunately, extensometer data for these mines was not available, but the
papers of some researchers (Spearing et al. 1994, 1995, Speers et al. 1996, Stacey et al.
1998) allowed establishing the basic input parameters needed to calculate the Bulking
Factor using the Rock Mass Bulking Model (RMBM). The cases reported in the
mentioned papers refer to a quartzite with unconfined compressive strength (oc) ranging
from 100 to 250 MPa. For Rock Type 1, a quartzite with a,= 200 MPa was selected.
Figure 5.2 Severe sidewall bulking (rifter Spearing et al. 1994).
At depths greater than 4000 m, the in-situ stress rises to values of the order of 150 MPa
and sidewall displacements of 300 to 500 mm are typically encountered. Note the high
volume increase of the failed rock due to geometric incompatibilities (Figure 5.2), as
noted by Kaiser et al. (2000).
Figure 5.3 shows the final profile of an initially square tunnel under high in-situ stress.
Note that the failed zone is localized (horizontal notches or breakouts).
Figure 5.3 Test tunnel under high vertical stress (after Spearing et al. 1994).
In Figure 5.4, displacements and Bulking Factor (BF), as they are modeled in the RMBM,
are shown. As noted above, sidewall displacements of 300 to 500 mm are typically
encountered in deep South African mines, and BF values are in general agreement with
the observations of Kaiser et al. (2000) regarding bulking factors greater than 30% in
hard rock mines. In the right-hand side picture three curves are shown. The middle one
shows average values of BF and the others show empirically assumed upper and Iower
boundaries.
Figure 5.4 Radial displacements and bulking factor for Witwatersrand Gold Mine openings in Rock
Type 1 (South Africa).
5.2 ROCK TYPE //
Two case studies were analyzed. The first corresponds to a tunnel in quartzite with oc
ranging between 108 to 220 MPa at the Kloof Gold Mine in South Africa, at a depth of
approximately 2300 m (Sevume 1999). Extensometeres were installed after the tunnel
was created, and the data thus only reflects mining-induced defomations.
Vertical st~èss 90 MPa Vertical stress 94 MPa
Vertical st~ess 95 MPa
Figure 5.5 Fracturing around a tunnel at Kloof Gold mine (after Sevome 1999).
Using RMBM, the displacements that could take place when the tunnel was built were
calculated, and the data presented in Sevume's (1999) paper was reanalyzed to account for
the total displacements, due to initial and mining-induced stresses. Note in Figure 5.5 how
mining induced stresses lead to an increase in the depth of failure, and that the failed zone
is localized. The fracture patterns shown were obtained from borehole petroscope
observations and compared with the measured displacements.
The second case study relates to data from a shaft in argillite quartzite at a depth of
approximately 1500 m (Barton et al. 1983). Three fifty foot long MPB extensometers
Figure 5.6 Location of instruments at Silver Shaft (after Barton 1983).
were instalIed close to the face. The shaft sinking continued while continuously
monitoring the extensometers, which had grouted anchors at radial depths of 50, 30, 15,
10, 5 and 3 feet. Figure 5.6 shows the location of instruments around the shaft. Figure 5.7
shows the calculated displacements and BF for both Kloof Gold Mine and Silver Shaft.
Note in this figure that at the Kloof Mine the in-situ stress is greater and the support
pressure lower than in the Silver Shaft. Consequently, the radia1 displacements and the
bulking factor are greater at the Kloof Mine.
* Extensometer data.
Figure 5.7 Radial displacements and bulking factor for a Moof Gold Mine opening (South Africa)
and the SiIver shaft in Rock Type II (USA).
For this rock type, information from coalmines in the UK was analyzed. According to
Brady (1993), back analysis based on observed excavation performance suggests that the
stress field in the UK coalfields is approximately hydrostatic. One site considered here
was situated at a depth of 500 m and the excavation was supported by steel arch sets.
lole\\ . Horizontal
Coa
Coal
3-D Probe Surveys 1
Mudstone and
lrregular Ironstone
Figure 5.8 Instrumentation at UK coaImihe (after Singh 1983).
A section of roadway was instmmented in a length of about lOOm using borehole
extensometers (Figure 5.8). The data presented in this figure was taken after the
excavation was built and thus corresponds to displacements due to mining induced
stresses. Taking into consideration the findings of Wilson (1977) about mining induced
stresses in long-wall excavations, the input data for RMBM was obtained. Calculated
displacements and BF can be seen in Figure 5.10.
Another case study was reported by Whittaker et al. (1983). It relates an investigation in
the UK, which was conducted to monitor the development of yield zones associated with
three major access drivages in Nottinghamshire Coalfield undertaken by the National
Coal Board. The tunnels were monitored using extensometers and were also supported
with steel arches.
Sandy Si1 ts tom
Siltstone with Sands tone lagers
Figure 5.9 Instrumentation at Nottinghamshire coalmine site 3 (after Whittaker et al. 1983).
Figure 5.9 shows some features of the site. Figure 5.10 shows displacements and BF
around the excavation in Rock Type III. No extensometer data is available for this second
case but the vaIues of maximum wall displacement and depth of failure reported in the
paper were used to calibrate the model.
* Extensorneter data.
Figure 5.10 Radial displacements and bulking factor for some coal mine openings in Rock Type ni
(UW*
5.4 ROCK TYPE IV
The case reported by Pelli et al. (1991) refers to a mine access tunnel for the Donkin-
Morien coalmine in Cape Breton Island, Nova Scotia, Canada. The tunnel was driven to a
maximum depth of 200 m below the seabed in layered sedimentary rock of Carboniferous
age. Some extensometer data at chainage 2263 in interbedded siltstone-mudstone rocks
were analyzed. Figure 5.1 1 shows a profile of radiai displacements at the tunnel crown.
The calibrated displacements and the BF after the RMBM are shown in Figure 5.13.
Interbedded sandstone- E siltstone
Mudstone ----
km In terbedded 1 -. si1 tstone-mudstone O 9'&
/ I 1
7' Displacernent ( Downword 1
4" (mm) 1
Figure 5.11 Measured radial displacement profile at chainage 2263 (after Pelli et al. 1991).
Another case study, reported by Whittaker et al. (1983) refers to a main access tunneI
in Nottinghamshire Coalfield in the UK. The coalfield is the same as reported in Section
5.3, the difference being that the rock mass is poorer in this case (Figure 5.12).
Tunnel. Nidth = 5 m
Rock Type -- -
Mudstone
Rider Coal
Seatearth
Mdin Coal
Seatearth
Figure 5.12 Instrumentation at Nottinghamshire coalmine site 1 (after Whittaker et al. 1983).
Data presented in the paper was rearranged because part of the displacements was due to
bed separation in the roof, a process that has littIe to do with rock mass bulking.
Displacements and BF representative for this case are shown in Figure 5.13.
* Extensometer data.
Figure 5.13 Radial displacements and bulking factor for Donkin Morien Tunnel (Canada) and a coalmine opening in Rock Type IV (UK).
The purpose of this thesis, as was noted at the beginning, is to develop a closed form
solution that allows calculating the Bulking Factor (BF) around underground excavations,
and to establish how rock mass quality, in-situ stress conditions, support pressure, and
excavation radius influence it. The parametric anaIysis presented in Chapter 4 allows the
understanding of these relationships. In this chapter, the goal is to illustrate how in-situ
stress conditions, depth of failure, support pressure, maximum displacements, and BF are
related. The nornograms presented here provide a practical means to predict the wall
deformations and the BF for different types of rock.
The starting point to develop these nomograms is the conceptual work by Kaiser et al.
(2000). Note how the in-situ stress, the stress concentration at the excavation wall, the
depth of failure and the wall displacements are related (Figure 6.1). Any increase of in-
situ stress (for example mining induced stress) leads to an increase of the depth of failure
and of the wall displacements around the excavation. Note also that the increase of
support pressure leads to a decrease of the BF (Figure 6.2). These principles, which
characterize the mode1 developed in this work, are shown to be valid in the foIlowing
nomograms. It is important to note here that the application of the nomograms is limited
to a depth of failure not greater than about one excavation radius.
b Martin, 1989 a ompp a ~ a y , 4984 x PeIII et al, 1991 4 stacey L de Jongh. 1977
Depth of failure dfh
Figure 6.1 Combination of stress concentration factor, depth of failure chart, bulking and convergence chart for support selection to control failing ground in over-stressed briffle rock (after
Kaiser et al. 2000).
The reason is that at greater depths, due to confinement increase, friction mobilization is
expected to play a role different frorn that considered in the assumptions made in this
thesis.
Distributed support capacity &Pa)
Figure 6.2 Bulking factor as a function of distributed support capacity (after Kaiser et al. 1996).
Figures 6.3 to 6.6 show rock mass behavior for Rock types 1 to IV, in a format similar to
that of Figure 6.1. In the nomogram for rock 1 (Figure 6.3), the in-situ stress @,) is
assumed to be in the order of 0.50, (remember that for Rock 1, oc ranges from 200 to 300
MPa). Because the mode1 was developed for a circular excavation in a hydrostatic in-situ
stress field, and the depth of failure (dj) is related to the maximum elastic stress
concentration around the excavation ( q n e 2 p , ) , the depth of failure should be about 0.63
times the excavation radius. If the excavation is unsupported (support pressure pl = O
MPa), the wall displacement is expected to be about 20% of the excavation radius. This
displacement results in a BF of about 31%. If the excavation was supported with light
support, say mechanical bolts with mesh (p2=0.05 MPa), the wall displacement would be
about 10% of the excavation radius and the BF would be in the order of 16%. If the
excavation was supported with yielding support, Say frictional bolts with mesh but
without grouted rebar @3=0.2 MPa), the wall displacements would be about 3% of
excavation radius with a BF of about 4%. If the excavation was supported with heavy
strong support with rock mass reinforcement (p4=0.5 MPa), then displacements would be
about 0.8% of excavation radius and the BF would be reduced to about 1%. Similar
explanations are valid for the rest of the rock types and can be seen in the correspondent
nomograms (Figures 6.4 to 6.6).
A cornparison of the four nomograms leads to some interesting conclusions about rock
mass bulking and deformation around underground excavations. Note that the range of
the relation p& decreases from Rock 1 to Rock IV. This is because, in poor quality rock,
a lower in-situ stress is necessary to reach the same depth of failure than in good rock.
Note that if the rock yields, the displacements are greater for Rock 1 than for Rock IV for
the same ratio of p&&. This difference is due to the greater volume increase, expressed
through BF, which is expected to occur in good brittle failing rock, for which the dilation
angle is greater. Note also that the sensitivities of displacements and BF to support
pressure are greater in Rock 1 than in Rock IV. This can be explained by the fact that, in
good hard rock, volumetric expansion is mostly related to the development of geometric
incompatibilities, when blocks of broken rock move relative to each other as they are
forced into excavation (Kaiser et al. 1996). This process can be controlled from its very
beginning by an adequate support system. The interested reader rnay reproduce the
nomograms presented in Figures 6.3 to 6.6. A MatlabB spreadsheet based on the Rock
Mass Bulking Model can be found in Appendix G.
Figure 6.3 Depth of failure (dl), excavation wall displacement (u,) and bulking factor (BF) for very
good rock (1).
Figure 6.4 Depth of failure (dl), excavation walI displacement (u,) and bulking factor (BF) for good
rock (II).
Figure 6.5 Depth of failure (df), excavation wall displacement (u,) and bulking factor (BF) for fair
rock (III).
Figure 6.6 Depth of failure (dl), excavation wall displacement (u,) and bulking factor (BF) for poor
rock (IV).
Figure 6.7 presents a combined nomogram illustrating the behavior of the four rock types
for an unsupported excavation (pl = O MPa). In order for the four types of rock to be
included in the same nomogram, it is necessary to consider two different scenarios. In
case A, a circular excavation in Rock 1 and II is subjected to an in-situ stress of 80 MPa.
In case B, a tunnel in Rock III and N is stressed only to 10 MPa. In this example, an in
situ stress of 80 MPa corresponds to a ratio p d a , of 0.4 in Rock 1, while it equals about
0.47 for Rock II, because o, is Iower in the Iatter case. As would be expected, the depth of
failure is greater for Rock II than for Rock 1, and the wall displacements are greater in
Rock II than in Rock 1. Regarding BF, note that it is geiter for Rock 1 than for Rock II,
due to the greater dilation angle for the former. A similar reasoning can be undertaken for
Rock III and Rock IV as presented in the same nomogram.
Figure 6.7 Depth of failure (df), excavation wall displacement (uw) and bulking factor (BF) for rock
1, II, III, and IV.
CONCLUSIONS
The purpose of this thesis was to develop a closed form solution to calculate the Bulking
Factor (BR of rock failing around underground excavations. The BF is defined as the
percentage increase in radial deformation due to fracturing inside a failure zone extending
to a stress-induced depth of failure (Kaiser et al. 1996).
Regarding material behavior, some assumptions were adopted. The stress-strain rock
behavior was modeled assuming that under load, the axial strain follows the Griffith
locus. For lateral strain calculations, the concepts of effective deformation modulus and
effective Poisson's ratio were adopted. The former was based on the observation that
failing rock becomes more deformable. The latter was based on a crack mode1 after
Walsh (1965) leading to an increase in Poisson's ratio, reaching a maximum value of 0.5
at large axial deformations.
For the modeling of the tunnel behavior, a plane strain elasto-plastic solution for circular
excavations under hydrostatic in-situ stress was adopted (Ogawa 1986). The constant
dilation angle used in Ogawa's (1986) solution was replaced in this thesis by a dilation
angle that depends on rock properties and confinement stress. This approach in modeling
rock mass bulking removes the need to estimate a dilation angle.
Using as a starting point the work by Kaiser et al. (2000), several nomograms were
developed to provide a practical means to predict the bulking factor and the related wall
deformations for different types of rock. In these nomograms, the in-situ stress, the stress
concentration at the excavation wall, the depth of failure, the wall displacements, the
support pressure, and the bulking factor are related.
Because the model was developed using concepts based on rock brittle behavior, it is
expected to work better for depths of failure of less than about one excavation radius. In
addition, it is expected to have better results in very good (0, good (II), and fair (III) rock
types. The ductile nature of rock IV makes it less qualified to be analyzed with the
RMBM, but the model still may be useful for depths of failure much less that one
excavation radius.
The Rock Mass Bulking Mode1 (RMBM) is flexible, in the sense that it can be adjusted to
case study data, and calibrated in such a manner that the expected Bulking Factor around
underground excavations can be obtained as a function of in-situ stress, rock m a s
quality, excavation radius and support pressure.
The M M can find application for the development of deformation based support
selection methods in civil and mining tunneling.
The mode1 developed in this thesis evolves from continuum mechanics principles and is
calibrated in an empirical way to match field data. However, in reaIity, rock mass bulking
is a non-continuum process that introduces anisotropy in rock behavior. Hence, a method
consistent with a non-continuum theory should be developed. Such a theory must lead to
constitutive laws that covers ductile and brittle behavior, and includes dilation as a
pervasive rock mass property of failed rock. Future work should focus on the
development of numerical methods that take this principle into account and introduce the
anisotropic effect of bulking in a consistent manner.
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GRIFFITH LOCUS
A detailed derivation of the equations for the Griffith locus can be found in the Appendix
B of Martin's (1993) Ph.D. thesis. In a simplify rnanner some basic equations are
explained here.
Cook (1965) developed the basic equation for the locus, based on energy principles. He
shows that strain-stress linear relationships for rocks with a given crack density and crack
iength have the following form:
0 + n(1 -v)a[(sin2 28)/4+ psin2 @(sin 26)/2 -2p2 sin4 8]c2n & = (Eqn A.1)
2(1+ v)G 2G
where v - Poisson's ratio; E - Young's modulus; c - crack half length; n - crack density;
G= E/2(1+ v ) - Shear modulus; ,u = tan# - Coefficient of friction; @ - Friction angle; 0 =
(l/S)atan(l/,u) - Critical crack angle.
If the rock is subjected to load, it deforms and the work done in the crack closure process
is given by:
where t - Shear stress; a, - Normal stress.
According to Cook (1965), the criterion for crack extension, or failure, is
where a - Fracture surface energy.
From Eqn A.2 and Eqn A.3 it follows that
and the critical crack length for failure to occur is:
(Eqn A.2)
(Eqn A.3)
(Eqn A.4)
(Eqn AS)
which can be rearranged to obtain:
(Eqn A.6)
In Eqn A.6 the right-hand term represents the cohesion strength component. It decreases
as the crack length (c) increases, and can be called So:
(Eqn A.7)
From Eqn A.6 and Eqn A.7 it follows that
7 = su + Pa,, (Eqn A.8)
which represents a linear Mohr-Coulomb yield criterion. This equation can be expressed
in terms of principal stresses:
O,= 0 3 + (0, + a3)sin$ + 2S,,cos# (Eqn A.9)
Substituting Eqn A.5 into Eqn A.l, Cook (1965) gets the Griffith locus in uniaxial
compression:
(Eqn A.10)
Martin (1993) developed Eqn A. 10 for axi-symmetric triaxial compression (n=o3):
- 6, - 2~0, - W 2 (Eqn A.11) €1, - G G
This equation is expressed in tems of normal, shear, and principal stresses. Considering
that
and
(Eqn ~ . 1 2 )
(Eqn A.13)
Eqn A. 1 1, combined with Eqn AS, can be reduced to the following form in terms of only
principal stresses:
where
(Eqn A.14)
M = F - - ' N = F + - p=- 32a2~n A=-- sin2 28 p sin 28 cos 28 2 2 n(l - v ) 4 4
p sin 28 B = -
C = p2(l -cos28)' = p2(1 + ~ 0 ~ 2 8 ) ~ 4 2 2
F = sin 28 + p cos 28
2
Using deformation data from case studies (see Chapter 51, the values for the crack density
(11) and the fracture surface energy in compression (a) were back analyzed. Taking into
consideration the relationship between fracture surface energy in compression (a) and in
tension (1)3, established by Martin (1993), the mode 1 (KI,) and mode II (KIICI fracture
toughness were calculated from the following equations:
The values used in th are pre ,sented in the fol1
(Eqn A.16)
owing table:
(Eqn A.15)
Table A.l Rock mass parameters a, n, KI" and KIIF
1 ROCK TYPE 1 n (crackslmA3) 1 a (MJImA2) 1 & . ( ~ ~ a d m ) 1 KIlc ( ~ ~ a d m ) 1
DILATION ANGLE
An important principle to develop the rock mass bulkjng model was to avoid the need for
introducing a fixed value of dilation angle as a mode1 input parameter. The problem is
that the dilation angle concept is well known in plasticity theory, but there are no standard
methods for its calculation in rock mechanics. It is not a constant parameter (dilation
angle changes as rock fails), and most importantly, it is a function of confinement stress.
Moreover, the scarcity of data on this parameter shows that its calculation is relatively
complicated and requires special laboratory procedures.
If a stress-strain model for rock behavior is assumed, it can be calibrated to model the
dilation angle. This approach may be valid if the model gives values of dilation angle
that are similar to those-observed in reality, and if they can be confinned with case study
data from underground excavations. In this thesis, a stress-strain model was assumed and
an equation for the dilation angle was developed. The mode1 was calibrated with case
studies and the concepts developed by Vermeer and de Borst (1984) and Hoek and Brown
(1997), regarding non-associated plasticity, were taken into account. The equation for
dilation angle developed in this work allows calculating this parameter as a function of
rock properties and the confinement. It was then transformed and introduced in an
analytical solution for displacements around underground excavations (Ogawa 1986) to
calibrate the Rock Mass BuIking Mode1 with field data.
As was mentioned in Chapter 3, as brittle rock fails its crack density increases, the
Young's rnodulus decreases and Poisson's ratio increases, and their resulting values are
called effective modulus and effective Poisson's ratio, respectively. For axi-symmetric
triaxial compression (a2 = 03), assurning that the axial strain follows the Griffith locus
given by Eqn A. 1 1, the effective modulus can be calculated as follows:
(Eqn B.1)
where v - Poisson's ratio; 0, and aj - maximum and minimum principal stresses; 81, -
axial strain according to the Griffith locus.
The influence of cracks on the Poisson's ratio is analyzed in this thesis after Walsh J.B.
(1965), and the basic equation that relates Poisson's ratio to effective modulus of
deformation in compression is used in this work:
where E = Young's modulus.
(Eqn B.2) .
By combining the Griffith locus, the effective modulus of deformation and the effective
Poisson's ratio equations, the lateral strains can be obtained:
(Eqn B.3)
Because dilation is studied from the plasticity point of view, Le., dilation is a non-elastic
phenornenon, elastic strains are calculated and discounted from the total strains to obtain
the plastic strains. The elastic strains can be calculated from the equations:
The plastic strains are obtained as:
(Eqn B.4)
(Eqn B.5)
(Eqn B.6)
(Eqn B.7)
The plasticity theory most commonly used in engineering relates stresses to plastic strain
increments or plastic strain rates. The dilation angle is then calculated from plastic strain
rates, as can be seen in Vermeer and de Borst (1984), Schanz and Vermeer (1996) and
others. As pointed out by Maier and Hueckel (1979), the plastic strain rates are
understood as derivatives with respect to any monotonously increasing function, a
parameter that controls the sequence of the loading process. Following Cook (1965), the
crack length c can be used as a parameter that monotonously increases as failure develops
in the mode1 (Appendix A). Plastic axial and lateral strain rates may be obtained from:
E,,, =as,, lac (Eqn B.S)
Having axial and lateral plastic strain rates, the volumetric and deviatoric plastic strain
rates are calculated as:
(Eqn B.10)
(Eqn B.ll)
The dilation angle for axi-symmetric triaxial compression stress conditions is obtained as
the ratio between volumetric (Eqn B.10) and deviatoric (Eqn B. l l ) plastic strain rates,
and can be caIculated from the following equation:
(Eqn B.12)
Under plane strain conditions, the volumetric and deviatoric plastic strain rates are
calculated from the following equations (Ogawa 1987, Vermeer and de Borst 1984):
- &qpr - ' ~ p r - &3pr
and the dilation angle is calculated as follows:
(Eqn B.13)
(Eqn B.14)
(Eqn B.15)
It is evident that a difference exists between the dilation angle y' calculated for axi-
symmetric triaxial compression stress conditions (01-03) according to Eqn B.12, and the
dilation angle calculated for plane strain conditions according to Eqn B.15. Cornparhg
these equations it can be found that s ( 2 1 3 ) ~ ' .
FRICTION COEFFICIENT
As mentioned in Chapter 3, the introduction of the equation for dilation angle into
Ogawa's (1986) solution for displacements makes the mode1 complicated from the
mathematical point of view. More than ten variables are involved in this solution. For
some of them correlations were found, others were back analyzed, and others had to be
fixed as assumed average values. It was noted that the Bulking Factor is a function of the
rock mass quality and that it is highly sensitive to the value of the rock mass friction
angle.
Considering that an increase in Geological Strength Index (GSI), uniaxial compressive
strength of the rock (O,) and intact rock parameter mi al1 lead to an increase of rock
quality (Hoek and Brown 1997), a Quality Parameter QP = GSI x a, x mi can be
introduced. Rock type IV is characterized by a low value of QP, while rock type 1 is
characterized by a high value. The sensitivity of the bulking factor to the rock mass
friction angle can be easily observed in Figure C. 1. Giving values of GSZ, a,, and mi to a
rock rnass, and plotting the relationship between the friction angle (the independent
variable) and the bulking factor, it is seen that for a given value of QP (it means for each
curve in the figure) an appropriate value of the friction angle must be introduced in order
to have realistic values of bulking factor. Figure C. 1 shows that as the quality of the rock
increases, lower values of the friction angle have to be used.
Figure C.l Relationship between rock mass quality, friction angle and the Bulking Factor (the friction angle is given as a % of the original rock mass friction angle).
In this figure, the zone limited by the upper and lower straight lines is assumed to
correspond to the values of Bulking Factor that can be expected around underground
excavations in a range of rocks from poor (rock IV) to very good quality (rock 1). The
middle straight line corresponds to the average values of BF. For example, if the Bulking
Factor for rock type II (see Table 3.1) is calculated, the friction angle to be used in the
mode1 should be between 0.03 and 0.3 % of the rock mass friction angle. If greater
values of the friction angle are used, the calculated Bulking Factor is much lower than
observed in the field. Following Hoek and Brown (1997), Vermeer and de Borst (1984),
Kaiser et al. (2000), and case studies in this work, poor quality rock is expected to have a
Bulking Factor near zero, while good quality rock may have values of BF greater than 10-
20 %. Having the BF as the criteria to be met, a relationship was established between the
rock mass quality parameter QP and a Friction Coefficient (k), which was introduced in
order to mode1 the friction component in the yield equation. As noted in Chapter 3, the
friction coefficient (k) is multiplied by the original rock mass friction angle in order for
the mode1 to match the displacement data of the case studies. Different kinds of
relationships between QP and k were analyzed. The general equation found has the
following forrn:
Considering that QP = GSI x o, x mi, and calibrating the equation with the deformation
data from case studies, the final equation that correlates the Friction Coefficient (k) with
GSI, 0,. and mi is the following:
k = a
(GSI X O c X III)" (Eqn C.2)
The values of the constants rZ and q were obtained from case study back analysis, using
deformation extensometer data (see Chapter 3, and are presented in the following table:
Table C.l Rock mass constants I . and q.
--
Rock Type A 4'
1 Very Good
II Good
iII Fair
IV Poor
FLOW RULE
As noted in Chapter 3, the state of stress under which plastic flow can occur is defined by
a yield function, and the stress-plastic strain rates relation is known as flow rule. This
concept is important because it is related to the deformations of a rock mass under load
conditions. The plane strain analytical solution of Ogawa (1986) for displacements
around underground circular excavations was used in this work. The solution presented
by Ogawa (1986) was transformed and empirically calibrated to match case study
deformation data (see Chapter 5). As a part of such calibration, the dilation factor N, used
in Ogawa's (1986) solution (see Eqn D.8 and Eqn D.9), based on a constant dilation angle
and a non-associated flow rule, had to be replaced by a dilation factor based on a variabIe
dilation angle (see Appendix B), maintaining the non-associated flow rule as a vaIid
assumption for rocks (Vermeer and de Borst 1984).
OGAWA'S (1986) APPROACH.
In Ogawa's (1986) plane strain solution a linear Mohr-Coulomb yield function is assumed
as a plastic potential and its derivatives with respect to the principal stresses allow
calculating the plastic strain rates. Following Ogawa (1986) and Vermeer and de Borst
(1984), the yield equation f is the following:
where C and @ are the cohesion and friction angle respectivefy .
The plastic potential can be expressed as:
g = (0, - G) - (cq + b3)siny + const = O
(Eqn D. 1)
(Eqn D.2)
where y is the dilation angle (see Eqn A.9 for cornparison).
Differentiating Eqn D.2 with respect to the principal stresses, the plastic strain rates can
be calcuIated as follows:
ag E ~ , , , = - = -(1+ sin yl) 2%
(Eqn D.3)
(Eqn D.4)
Under plane strain conditions, the volumetric and deviatoric plastic strain rates are
calculated from the following equations (Ogawa 1987, Vermeer and de Borst 1984):
(Eqn D.6)
According to Ogawa (1986) and Vermeer and de Borst (1984), the equation for the
dilation angle, based on the plastic potential given in Eqn D.2, is the following:
(Eqn D.7)
~ n d the dilation factor N, used in Ogawa's (1986) solution is calculated as follows:
(Eqn D.8)
Taking into account Eqn D.8, Ogawa (1986) derives the following equation for radial
displacements in the plastic zone:
(Eqn D.9)
In Eqn D.9 the term (1-siny) cornes frorn Eqn D.8, which is based on the plastic potential
given in Eqn D.2. The dilation angle is given as a constant input value, and therefore, the
dilation factor N, is aIso a constant value.
PRESENT APPROACH.
In the present thesis, the solution of Eqn D.9 requires the term (1-siny) to be substituted
for a properly determined one, considering that the dilation factor Nv in this work is a
function of a dilation angle that is not a constant value, does not corne from a plastic
potential, and is a function of rock properties and confinement. In this case, under plane
strain conditions, the dilation angle can be calculated from Eqn B.15.
Taking into account that:
(Eqn D.lO)
and substituting Eqn D.10 into Eqn B.15, the equation for the dilation factor, used in this
thesis, is obtained:
(Eqn D.ll)
If the Eqn D.11 is used, instead of the Eqn D.8 for the dilation factor N,, Eqn D.9
transforrns into:
(Eqn D.12)
The Iimits of integration in Eqn D.12 are the excavation radius (ri) and the radius of the
plastic zone (r,). The former is an input parameter in the Rock Mass Bulking Mode1
(RMBM), and the latter is calculated from Eqn E.4.
The basic difference between Eqn D.9 and Eqn D.12 is that the dilation angle and the
dilation factor are constants in the first equation, while in the second equation they are a
function of the rock properties and confinement. In Ogawa's (1986) solution, y must be
prescribed as an input parameter, whereas it is derived from rock parameters in the
solution presented in this thesis.
DEPTH OF FAILURE
The bulking factor is related to the depth of failure, the difference between plastic radius
and excavation radius. The depth of failure indicates the amount of rock that fails, and the
Bulking Factor is meant to calculate how this rock increases in volume. Practical
experience indicates that in hard brittle rock, for the currently used levels of support
pressures (less than 2.0 MPa), the depth of failure is independent of the support pressure,
while in poor quality rock masses the depth of failure decreases as the support pressure
increases. The approach assumed in this thesis is based on the fact that in the rock mass
bulking mode1 presented above, the calculated depth of failure has to match the one
obtained from case studies.
In the solution suggested by Ogawa (1986), the radius of the plastic zone is calculated
according to the following equation:
(Eqn E.l)
w here,
2 sin tPr X , =
1 - sin Gr (Eqn E.2)
and
(Eqn E.3)
In these equations, pi - support pressure; p, - in-situ hydrostatic stress; C and Cr - Peak
and residual cohesion respectively; @ and #r - Peak and residual friction angle
respective1 y.
For simplicity and practical purposes, it is reasonable to calculate the maximum depth of
failure that may be expected around underground excavations. It can be calculated
assuming that there is no support pressure applied on the excavation wall.
The parameters C, Cr, $, and & can be considered as a function of the uniaxial
compression strength oc. Regarding the friction angle, as noted in Chapter 3, in this thesis
it is assumed that it does not change as rock fails, and the peak and residual values equal,
$ = @r. This assumption is based on the Griffith locus, for which failure is a cohesion loss
process, while friction is treated as a constant as rock fails. It is possible then to express
the Eqn E. 1 in a more general and simplified form:
(Eqn E.4)
where Ar and Br - Rock constants; p, - In-situ hydrostatic stress; oc - Uniaxial
compression strength; r, - Radios of the failed zone; ri - Excavation radius.
The coefficients Ar and Br are then calculated in such a manner as to match the depth of
failure obtained from extensometer data in case studies (see Chapter 5). A trial and error
approach was assumed and the coefficients are presented in the following Table:
Table E.1 Coefficients A, and Br.
Rock type Ar Br
1 Very Good 2.3 0.48
II Good 3.2 0.50
ïU Fair 3.8 0.70
IV Poor 4.6 0.72
Eqn E.4 is of the same form as the empirical solution for the depth of failure developed
by Martin et al. (1999):
(Eqn E.5)
where a,,,= 301-a3.
For hydrostatic in-situ stress, where o,=o3=p, and onU=2p,, Eqn ES transforms to:
(Eqn E.6)
Hence, Eqn E.6 is a particular case of Eqn E.4, with coefficient Ap2.5 and Br0.49.
FLOW CHART TO DEVELOP THE RMBM
1 Griffith locus 1 Eqn A.11 4
Effective deformation Effective Poisson's
Axial elastic strains Eqn B.4 Eqn B.3
Lateral elastic strains Eqn B.5
strains Eqn B.6lB.7
1 Plastic axiavlateral 1 strain rates
VolumetricIDeviatoric strain rates Ean B. lO/B. 11
volumetric rates Dilation angle triaxial stress Eqn B. 12
Plane strain plastic deviatoric rates Eqn B. 14
displacements Ean D.9 ?-
Dilation angle plane strain conditions strain
Ogawa's solution depth of failure Ecin E. 1
Yield equation Eqn 3.8
Case study data "7
RMBM Eqn 3.15
ROCK MASS BULKING MODEL (RMBM)
The Rock Mass Bulking Mode1 is presented in this Appendix. In spite of its length, the
mode1 is pretty simple. It was written in the MATLAB" code (The Math Works, Inc.,
1999) and can be reproduced in any programming language. It takes seconds to calculate
the Bulking Factor in the failed zone around an underground circular excavation.
To make calculations with this spreadsheet, the needed input information is the Rock
Type (1 to IV), the uniaxiaI compressive strength of the rock in MPa, the in-situ
hydrostatic stress in MPa, the radial support pressure on the excavation in MPa, and the
tunnel radius in meters.
The output of the mode1 is two curves: the total displacements and the Bulking Factor as a
function of the plastic zone radius.
%SELECT ROCK MASS TYPE (RT)
RT=l %Rock Q p e Gc=200 %Uniaxial compressive strenqth, MPa po=80%In-situ hydrostatic strees,MPa p=O%Radial support pressure, MPa ri=2.1%Tunnel radious,m
%DETERMINE RADIOUS OP THE PLASTIC ZONE (ml
re=Ar:ri.* I(Brbpo./Gc)+Cr) ."Dr; iE (re./ri)<al;
'NO PLASTIC ZONE' break;
end;
% CALCULATE PARAMETERS FUNCTION OP MATERIAL PROPERTIES AND BOUNDARY CONDITIONS
% CALCULATE RADIAL DISPLACEMENT FOR DIFFERENT VALOES OP rpps
rpps=ri:(O.Ol*re):ra 6Radious of plastic zone A-(rpps./ri)."N~; K6=A.*K5-KI; K7=A.*M3+M6; KB=A.*Ml+M5 Sl=Pn.*A.+2+M8.*A+M9; S2=(M10.*A."2+Mll~*A+M12)./~~A.'M13+M141.*4)./((A.*M1+MS)."2)
%CALCULATE RADIAL ELASTIC DISPLACEMENT AT te
0 CALCULATE INTEGRATION CONSTANT 80. (FOR rpps-re)
$ CALCULATE DISPLACEMENT AT rppS
%CALCULATE RADIAt ELAÇTIC DISPLRCEMENTS AT rpps
%CRLCULATE BVLKING FACTOR AT rppS
8PM=(((urpps-urppse)-~urppsre-urppsere~l.I~re-ril~.*100; O Value of Bulking Pactor at r p p e BPMAXn((ri-urppsrel ./(re-ri)).*lOO;\BP which cl0Ses the excavation bf1=5.*k.a-0,2423; %Equation for upper boundary of 'ideal" bulking factor bf2=4.9979.gk.4-0.1861;%Equation for gidealg bulking factor bÉ3=5.0017.*k.n-0.1240;%Equation for lower boundary of 'ideal- bulking factor BQ~=BFM.*(~+I (bEl-bf2).lbf2l);%Equation for upper boundary of calculated bulking factor BP2nBPM.*(l+((bf3-bf2).IbE2)):%EQUation for lower boundary of calculated bulking factor BFaIBF2; BPM; BPI] %Equation for Bulking Pactor Interval %$%%%%%%%%%%%%%%%%%%%%%%%%%%%%a%%%%%%%%%%%%0%%%%%%%%%O%%%%%%%%%%%~ if BFN(l,l)>=BFMAX
'WCAVATION CLOSED BY RûCK MAÇS BULKXNG' 'RADIAL WALL DISPLACPIPFP EQUALS EXCAVATION RADIOUS' break
end %%%%a%âaaaa%ea~%%%%%%a%%%%%%%%%%%a%~%%%%%%%%%%%%%~%%%~%%%%%~%%%%%%
SPREADSHEET FOR NOMOGRAMS
The following set of equations allows producing the nomogram presented in Figure 6.3
for Rock Type 1. For the other types of rock the format is the same, the only difference is
that under the subtitle INPUT PARAMETRS the values change for each type of rock. These
values are the following:
For rock 1:
RT=1 %Rock Type
Gci=200 %Uniaxial compressive strength, MPa
po=0:1:150 %In-situ hydrostatic stress,MPa
For rock JI:
RT=2 %Rock Type
Gci=170 %Uniaxial compressive strength, MPa
po=0 : 1: 85 %1n-situ hydrostatic stress ,MPa
For rock III:
RT=3 %Rock Type
Gci=60 %Uniaxial compressive strength, MPa
po=0:0.5:30 %In-situ hydrostatic stress,MPa
For rock IV:
RT=4 %Rock Type
Gci=36 %Uniaxial compressive strength, MPa
po=0:0.1:18 %In-situ hydrostatic stress,MPa
%ANALYZE CASE SlUDY: ROCK 1.
%CREATE A NOMWRAM FOR PRACTXCAL APPLICATIONS
%,*,*,~,*,*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*:.*.*.*.*.*.*
%SELECT ROCK MASS TYPE (RT)
%INPUT PARAMETERS
tXNPüT PARAHETERS
RT-1 %Rock Type
Gci=200 OUniaxial compreesive strength, HPa
po=O:l:150%In-ritu hydrostatic strsss,MPa
p=O%Radial support pressure. MPa
ric2.0%?unnel radious,m
plot(dt,Uw,'b','LineWidth',l.5i
xlabel(*df/rio)
ylabelt'~\fontsize(l5)u)[\fontsize~lO)rw)/[\fontsizs(l5~r)(\fontsize(lO)i) {\fontaize(lS)%)')
hold on
0%%0%%00%%0%%%%%%0%%%a%%%%%%%%%%%%%0%%00%%%0%%%%%%%%%%%%%%%%%%%0%#
%INPUT PARAMETERS
NINPUT PARAMETERS
RT=l \Rock Type
Gci=ZOO %Uniaxial coupressive strcngth, MPa
po=O:l:150%fn-situ hydrostatic stress,MPa
p=0.2%Radial support pressure, MPa
ri=2.00hinnal radious,m
0%%0%00%%%00%00%%%%%%%%%00%%%%%%%%%%0%%%%%%00%%%%0%%%%%%0%%%0%00%%
INTRODUCE HODEL (SEE BEMW 1
%%%$00%%%%0$%0%%%%9%0%%%0%%%%%%%%%%00%%0%%000%%%%%%%0%%%%%%0%%t00N
plotidf,Urw, Ir'. lLineWidth'.l.5)
hold on
%%%0%%%%%00%%%~%%%%~%%0%%a%%%%%%%%%0%0%%%%a0%%%0%0%~0%t%%%%%%%at%0
RT-1 %Rock Type
Gci=200 %Uniaxial compressive etrength, MPa
po=0:1:1500fn-situ hydrostatic stress,HPa
p=O.S%Radial support pressure, MPa
ri=2.0%?'unntl radious,m
%%%%%%%%%t%%%%%%%%%%%%%%%%%%%t%%%$%%%%%t%%%%%%%%%%%%%b%%%%%%%%%%%%
INTRODUCE MODEL (SEE B W W
%0%%%%%%%%%%%%%%%%$%%%%%%%%%%%%%%k8%%%%%t%%%%%%%%%%0%0%%%%%%%%%%%%
plot (df , U w , ' g ' , ' LineWidth' ,las)
..................................................................
%INPüT PARAMETERS
RT=1 %Rock Typa
Gci.200 %Uniaxial comprassive strenqth, MPa
po~0:1:150%In-situ hydrostatic streas,MPa
p=O%Radial support pressure, MPa
rima. O â T u ~ a l radious, rn
%INPUT PARAMETERS
RT=1 %Rock Type
Gci=POO %Uniaxial compressive strength, MPa
po=O:l:lSO%In-situ hydrostatic srress.MPa
p*0.05%Radial support pressure, MPa
ri.2. OtTunnel radiaus, m
%INPUT PARAMETERS
RTnl %Rock Type
Gcia200 %Uniaxial compressive etrength, MPa
po=O:l:150%In-situ hydrostatic stres6,MPa
pr0.20Radial support pressure, MPa
ri-2.00'hinnel radious,rn
%INPUT PARAMETERS
RT=l %Rock Type
Gci=ZOO %Uniaxial compressive strangth, MPa
po=0:1:150aIn-situ hydrostatic stress,MPa
p=0.5%Radial support pressure, MPa
rinZ.O%ninnel radious, m
~ ~ a % a ~ ~ ~ a ~ % ~ a a % % ~ ~ a ~ ~ a % a % % a ~ % % % % % % ~ s u % ~ % t % e e % % % % % a % % % % t % t % ~ h % % % a % t
INTRODUCE MODEL (SEE BELûW 1
%%%%0%%%%%%~%%%%%%%%%%%%%0%~%%%%~%%~%%%0%66%%%~%%%%%00%8%%~%%%%%%%
plot iBFM,Urw, 'g' , 'LineWidth0 .l.Si
%%%a~ta%%%o~a%%â%%%~%~e%%ta~a%%%%%%%%~a%e%a~ataa%%%~%~%%%%%a%a%%%%
Gmax=2*po/ Gci;
GmaxiGmax(l.dfi~l1
a%%à%%a%%%t$aa%%â%%e%%%%%%%%aaa~%e%%~%%%%~e%%%%a%a%%%e%aa%%%%a%%t%
aubplot(2,2.21
plot (Cmax, Gmax, 'b' , @LineWidth4 ,l. SI
xlabel(o(\fontsizc(lSlp}~\fonteizeil0}o)/(\tontsize~l8l\si~~(\~ont~ize~lO~c~'~
ylabel(~(\fontsize(l8)\si~)(\fontsize(lO)mkx)/(\fontsize(l8)\si~)~\fontsize~lO)c)')
MODEL
PART A)
end
tDETERMINE RADIOUS OF THE PLASTIC ZONE (ml
i f [re./rilc=l;
'NO PLASTIC ZONE'
break;
end;
a CALCULATE PARAHETERS FUNCTION OP MATERIAL PROPERTIES AND BOVNDARY CONDITIONS
Q16S7-tM23. .A+M24). / (A. *Ml+M5)
$-----------------------------------------------------------------------------------
Q116b-2. *K3 . * (A . +M3+M6). / (A.*Ml+M5) ; Q116C=K4 .* (A. *K5-Kl) . / (A.*Ml+M5) Q116D=(Q16SlO,*Q16Sll)./(MS5.*A+H26); Q116E-N1.*Q16S8.*Q16Sll./~M25.*A+M26)."2
Q116F- (M25. *A+M26). *Q16S10. /hi. /Q16SBB12 ; Q116Q=Q116F- lK3. IQ16SB)
Q116H~Q16S8.*(num+Ql6S9).*(l+num+Q16S9)./tM25.*A+M26); Q1161=(2.*num+2.*Ql6S9).*(A.*K5-K1+A:M3+M6)+A.*M3+M6
Q116JmN3. *Q16S8.*Qll6G. *Q116I./fM25. *A+M26)
Q116K=Q116B+Q116C+Q16S6-Q16S4+4.*Q16S3+Ql6S2+K3-Qll6DtQll6EtQll6~+Qll6H-l./2.*K4
Q116L~Q116D-Ql16E-Q116J-Q116H+l./2."KO-Ql6S7+Ql6S6-Ql6S4+4.*Ql6S3+Ql6S2+K3
Q116=Q116K. /Qll6L
%-------------------------------------------------------------------------------
Hl=l./(l+Q116); HZiNP.*Q116./(l+Q116); H3n(l-Q116)./(1+Q116); H4=Hl.*NP.*Q116
HS=(re.AH1).^2; H6=iri."H1).*2; H7=2.*H6.*K13.*ri.'(Hl,*NP),*riinH4
HBm2:H5.*K13.*ra.̂ (Hl.tNPI.*rs.'H4; H9=H5:KlO.*NP; HlO=H6.*KlO.+NP
L1=l./2.~K14.*rpps.nNP+1./2.9K15; L2=~-K13.*rpps."(H3+NP)-KlO.*rpps.*H3).*(2+NP+NP.*Q116)
Al=-H9 + H10; A2=2.*H10-2.*H9-HB-2.'HS.*KlO+H7+2.*H6.*KlO
A3~-H8-2.*H5.*KlO-H9+H7+2.*H6.*K10+HlO; XlaLl./LZ; X2xAl.*Q116.^2: X3aAZ.*Q116+A3
X=X1 . (X2+X3 $-----------------------------------------------------------------------------------------
% Y
Q21=MlO.*[A)."Z+MIl.*A+M12; Q22=(M1O.*(A).n2+Ml1.*A+M12)./(A..M13+M14).n4./(A.*M1+M5.n2
Q23r(M15.+(A).n2+M16.*A+M17)./(A.*M13+M14.n5./~A.*M1+M5; Q24=(M18.*A+M19)./(A.*M13+Ml4)."4./(A.*Ml+M5)
Q25=M2O.*IA).^2+MSl.*A+M22; Q26=(M20,*(A).A2+~l.*A+M22)./Pn./(A.*Ml+M5).A2
Q27=(M23.*A+MZ4)./(A.*Ml+MS); Q28=(MlO.*(A).*2+Mll.*A+M12)./(A.*M13+Ml4).~4./(A.*Ml+M5)
Q29n(M20.*(A)."2+M2l.*A+M22)./PnPn/(A.'M1+M5)
$--------------------"--""----------------"----*--------""------"----"---"-*---------------------
Q210Ba(M20.*A.*2+M21.*A+M22)Q21o8ao.1(Em..(A.'M/(Em.*(A.*M~+M~))
Q2lOC+(MlO.*A."2+M~1.*A+Ml2)./(~~A.~M13+M14).n4~.*(A.'Ml+M5))
QZlO=N3.*(1-(M25.*A+M26) ./(Q2lOB+Q2lOCII/hi)
$-----------------------------------------------------------------------------------------------
Q2llB= (MZ3. *A+M24). / (A. *Ml+MS
Q211C=(M20.+A.n2+M21.*A+M22)./(Em.*(A.*M1+M5).A2)
Q211Di(M18.*A+M19)./(I(A.*M13+M14).n4).*(A.bM1+M5))
Q211E-(4.*M15.*A.n2+4.*M16.*A+4.*M17)./(((A.*M13+M14).A5).*(A.*M1+M5))
Q211F-(MlO.*A.n2+Ml1.*A+Ml2~./((IA.*M13+H14).A4).+~A.'M1+M5).^2)
% CALCULATE INTEûRATION CONSTANT BO. (FOR rgpsnre)
rppsnre %Radious of p l a s t i c zone
$0$%%%0%%%%%%08
INTRODUCE PART B)
0%0000%00%91%%0
0 CALCLLATE RROIAL DISPLACEMENT AT rppB
%CALCULATE RADIAL ELASTIC DISPUCEMPFPS AT rppS
% CALCULATE BULKXNG FACTOR AT rppe
BPM=~((urpps-urppsel-~urppsre-urppserell./~re-ril).+100; % Value of Bulking Factor at rppe
BFHAX=((ri-urppsre)./ire-rilI.*lOO;&BP which closes the excavation
bfl=S.*k.^-0.2423; %Equation for upper boundary of 'ideal- bulking factor
bf2-4.9979. *k.n-0.1861;%Epuation for *idealg bulking factor
bf3=5.0017.gk.~-0.1240;%Equation for lower boundary of 'ideal* bulking factor
~Fl=~m.*ll+llbfl-bf2)./bf2ll;%Epuation for Umar boundary of calculated bulkinp factor
BP2=~m.*(l+[(bf3-bf2l./bf2)I;%Equatfon for lower boundary of calculated bulkinp factor
BPn[BP2; BFM: BFl];%Equation for Bulking Factor Interval
%%%%%%%%â%%%%aa%aa%a~%%%a%%%~a%%%~aaao%%~aa%%aa%%%%~%%%%~aaaz%%a%%
if BPM(l.l)>=BPMAX
'EXCAVATION CMSED BY ROCK KFSS BOLKING'
' RADIAL WALL DISPLACMe3T EQUALS EXCAVATION RADIOUS '
break
end
%%â%%%%%%%%%%a%$%a%zs%%%aaaIr%~%%%a%%%%%%%%%%%c%%%a%a%%~%%a%%%%%%%%
dfm (ru-ri1 /ri;
dfndf (l,df<=l);
Gmx=2 .po/Gci;
Gmax-Gmaxi 1, dt-11