validation report of hoek-brown model implemented in plaxis
TRANSCRIPT
June 2009 / CGG_IR011_2009
Client:
Computational Geotechnics Group
Institute for Soil Mechanics and Foundation Engineering
Graz University of Technology
Plaxis
P.O. Box 572
2600 AN Delft
The Netherlands
Ao. Univ.-Prof. Helmut F. Schweiger
M.Sc. Ali Nasekhian
Validation Report of Hoek-Brown
Model Implemented in Plaxis
COMPUTATIONALGEOTECHNICSGROUP 1
Project-Nr.: CGG_IR011_2009
Validation Report of Hoek-Brown Model
Implemented In Plaxis
Client:
Plaxis P.O. Box 572 2600 AN Delft The Netherland
Ao. Univ.-Prof. Helmut F. Schweiger M.Sc. Ali Nasekhian
Computational Geotechnics Group Institute for Soil Mechanics and Foundation Engineering Graz University of Technology
Graz, am 18.June 2009 Helmut F. Schweiger
COMPUTATIONALGEOTECHNICSGROUP 2
CONTENTS
1 SCOPE OF THE REPORT ………………………………………………………….……. 3
2 VALIDATION SCHEME ……………………………………………………………........... 3
3 HOEK-BROWN MODEL (HB-MODEL) …………………………………………………. 4
4 TRIAXIAL TEST …………………………………………………………………………… . 6 4.1 Stress path and yield surface check ……………………………..………... . 6 4.2 Comparison with lab data……………………………………………….…….. 8
4.3 HB-model in compression and extension mode…....…………………….. 10 4.4 HB-Model and safety factor…………………………………………………… 13
5 EVALUATION HB-MODEL IN BOUNDARY VALUE PROBLEMS……….……..……. 14
5.1 Circular opening under hydrostatic pressure …………………..………… 14 5.2 Slope stability ………………………….……………………………..…….…… 19
6 REFERENCES …………………………………….…………………………..……….….... 23
7 APENDIX A: PLAXIS FILES …………………………………………………………….… 24
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1 SCOPE OF THE REPORT
The objective of this report is to validate the Hoek-Brown model implemented in Plaxis using an
MMTFILE to assign input parameters instead of the normal Plaxis interface. To do this, first a
validation scheme was provided as given in section (2) to evaluate different aspects and features of
the Hoek-Brown model based on reliable references. This scheme incorporates both, element tests
and boundary value problems. In element tests several properties of an elastic perfectly plastic model
such as elastic part, limit strength of material, stress path, drained and undrained conditions have
been assessed. Afterwards, an analytical solution of a circular tunnel under hydrostatical stress has
been compared to the results of the numerical model using the Plaxis HB-model. Accordingly, both the
stress and displacement field of a boundary value problem have been checked.
2 VALIDATION SCHEME
The validation scheme is divided into 3 parts. First, a triaxial test is modelled numerically and
according to HB properties of the intact or jointed rock (mentioned in the references) a compression or
extension test is performed and the results are compared with theoretical Hoek-Brown curves or with
other user-defined HB models such as FLAC, as well as experimental data obtained from lab tests. In
the second and third part of this scheme two boundary value problems including a simple slope and a
circular deep tunnel under hydrostatic pressure have been considered. The validation scheme is
briefly explained in the following.
a. Triaxial Test
The following items have been taken into account:
Whether HB-model complies with the theoretical (σ1- σ3) curves or not?
Comparison with lab data
(Madhavi, 2004)
Comparison with other user-defined model implemented in FLAC (using FISH).
(Madhavi, 2004)
Modelling triaxial compression and extension test to check whether shear strength
reduction scheme works or not (c-φ reduction). (Benz et al., 2008)
Hard rock mass Fair quality Poor quality
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b. Boundary Value Problem – Simple Slope
Comparison between Bishop, MC and HB with two different slope angles (35.5° and 75°) under
drained and undrained conditions in terms of F.O.S. (Benz et al. 2008)
The following items will be checked:
Arclength control Ignore undrained behaviour c-φ reduction
c. Boundary Value Problem – Circular opening under hydrostatic pressure (2D)
Plastic radius around the opening stress and displacement field (ur,σθ,σr)
(Carranza 2004, Carranza et al. 1999 & Sharan 2008)
3 HOEK-BROWN MODEL (HB-MODEL)
The Hoek-Brown model is an elastic perfectly plastic model with non-associated flow rule. Deformation
prior to yielding is assumed to be linear elastic governed by the elastic parameters E and n. The yield
function f for the Hoek-Brown model is given by:
a
cibciHB smfwithff )(~)(~ 3
331 +=−−=σσ
σσσσ
which is derived from the generalized Hoek-Brown failure criterion.
Figure 1 Hoek-Brown failure criterion in principal stress space (left) and in the deviatoric plane (right)
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The Hoek-Brown failure criterion was introduced in the early eighties to describe the shear strength of
intact rock as measured in triaxial tests (Hoek & Brown 1980). The failure criterion for intact rock
defines the combination of major and minor principal stresses (σ1 and σ3) at failure to be:
ciici mσσ
σσσ 331 1++=
(1)
In the equation above, σci is the unconfined compressive strength of the rock and the coefficient mi is a
parameter that depends on the type of rock (normally 5 ≤ mi ≤ 40). Both parameters, σci and mi, can be
determined from regression analysis of triaxial test results). The Hoek-Brown failure criterion was later
extended to define the shear strength of jointed rock masses. This form of the failure criterion, that is
normally referred to as the generalized Hoek-Brown failure criterion, is
a
ciici sm )( 3
31 ++=σσ
σσσ (2)
The coefficients mb, s and a in equation (2) are semi empirical parameters that characterize the rock
mass. In practice, these parameters are computed based on an empirical index called the Geological
Strength Index or GSI. This index lies in range 0 to 100 and can be quantified from charts based on
the quality of the rock structure and the condition of the rock surfaces (Marinos & Hoek 2000). In the
latest update of the Hoek-Brown failure criterion, the relationship between the coefficients mb, s and a
in equation (2) and the GSI is as follows (Hoek, Carranza-Torres, & Corkum 2002)
⎟⎠⎞
⎜⎝⎛
−−
=D
GSImm ib 1428100exp
(3)
⎟⎠⎞
⎜⎝⎛
−−
=D
GSIs39100exp
(4)
( )3/2015/
61
21 −− −+= eea GSI
(5)
In equations (3) and (4) D is a factor that depends on the degree of disturbance to which the rock has
been subjected due to blast damage and stress relaxation. This factor varies between 0 and 1.
The model parameters are listed in Table (1).
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Table 1 Parameters for the HB-Model ____________________________________________________________________ Nr. Name Unit Description ____________________________________________________________________ 1 Gref [kN/m2] Elastic Shear Modulus 2 ν - Poisson`s Ratio 3 σci [kN/m2] Unconfined Compressive Strength 4 mi - Hoek-Brown Parameter 5 GSI - Geological Strength Index 6 m - Power Law Exponent 7 Pref - Reference Stress 8 D - Disturbance Factor ____________________________________________________________________
4 TRIAXIAL TEST
4.1 Stress path and yield surface check
Modelling a triaxial test numerically is a simple way to check whether the implemented material model
is able to model the strength of a rock sample according to the Hoek-Brown criterion and its input
parameters. To do so, properties of an average quality rock mass were chosen which are given below.
σci=80 MPa
mi=12
GSI=50
Gref=3600000 kN/m2
n=0.25 ; D=0
Figure 2 Triaxial test modelled in Plaxis with prescribed displacement
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Hoek-Brown Element Test Results
0
10
20
30
40
50
60
0 10 20 30 40 50 60
p' (MPa)
q' (M
Pa)
Drained-Plaxis ResultsFailure EnvelopeUndrained-Plaxis results
3
1
σ 3=5
.2M
Pa
σ 3=1
.7 M
Pa
σ 3=1
.1 M
Pa
Undrained
Figure 3 Hoek-Brown failure criterion and HB-model results in (p´-q) space
Comparison between FEM model and Hoek_Brown failure envelope
0
10
20
30
40
50
60
70
80
90
-5 0 5 10 15 20 25
Minor principal stress (MPa)
Maj
or p
rinci
pal s
tres
s (M
Pa)
HB Failure Envelope Plaxis Results
σ 3=0
.050
MPa
σ 3=1
.1
σ 3=5
.2 σ 3=1
0.1
σ 3=1
7.1
Figure 4 Hoek-Brown failure criterion and HB-model results in terms of principal stress
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A prescribed displacement method was applied to simulate the vertical loading in the triaxial test.
Incremental multipliers and additional steps of the loading phase are chosen such that in the sample
roughly 10% strain occurs. This procedure was repeated for five different confining stresses to ensure
that in different stress levels correct failure is predicted. The results of the simulation have been
illustrated in Figures (2) to (4). The stress path of the sample under different confining stress in (p’-q)
space is 3 vertical to 1 horizontal while in terms of principal stress it goes up straight to reach to yield
surface.
One run was carried out under undrained condition at σ3=10.1 MPa. As depicted in Figure (3) the
respective stress path moves up vertically and finally touches the failure surface.
4.2 Comparison with lab data In the next phase, the HB-model is employed to simulate real laboratory triaxial test results to evaluate
the model for a real intact and jointed rock specimen.
Intact Kota sandstone with linear stress-strain response and Gypsum Plaster (block jointed sample
with two sets of joints inclined at 30°/60° with a joint frequency of 20 per metre depth) exhibiting highly
nonlinear stress-strain response were selected and their stress-strain responses at different confining
pressures were calculated from numerical analysis. The lab data were adopted from papers presented
by Madhavi(2004) and Brown(1970). First, according to the lab results of the ultimate strength of
jointed Plaster samples an attempt has been made to fit the best failure envelope over lab results in
principal stress space. Figure (5) shows these fitted curves and the respective Hoek-Brown properties
as well as rock properties presented by Madhavi(2004). HB properties corresponding to the two-point
fitted curve (which is the best fitted-curve for the range 0<σ3<4 MPa) were adopted for further
analyses. The stress-strain curve is depicted in Figure (6) which indicates reasonable agreement with
experimental data.
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HB Properties of jointed Sample 30°/60° Plaster Gypsum
0
10
20
30
40
50
60
70
0 2 4 6 8 10 12 14Minor principal stress (MPa)
Maj
or p
rinci
pal s
tres
s (M
Pa)
Balanced fitting
Lab Data
Madhavi Fitting
Two-Point Fitting
BH_Model
BH_Model
Balanced Fitting:σci=20 MpaGSI=50 mi=15s=0.0039a=0.505
Madhavi Fitting:σci=21 MpaGSI=20 mi=0.402s=0.0001a=0.544
Two-Point Fitting:σci=20 MpaGSI=35 mi=15s=0.0007a=0.516
σ3=1379 kPa
σ3=3447 kPa
Figure 5 Hoek-Brown failure envelopes and HB-model results in terms of principal stress for jointed
sample Plaster
HB-Model vs Lab DataSample 30°/60° Plaster Gypsum
0
2000
4000
6000
8000
10000
12000
0,0% 0,5% 1,0% 1,5% 2,0% 2,5% 3,0% 3,5% 4,0% 4,5% 5,0%
Axial Strain
Prin
cipa
l Str
ess
Diff
eren
ce k
Pa
Lab DataLab DataHB_ModelHB_Model
|σ1-σ3
|
σ3=1379kPa
σ3=3447kPa
Figure 6 Numerical and experimental stress-strain results for jointed sample Plaster
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Hoek-Brown Properties of jointed Plaster sample:
σci=20 MPa
GSI=35; mi=15
s=0.0007; a=0.516
Hoek-Brown Properties of intact Kota sandstone:
σci=70 MPa
GSI=100; mi=22
Besides jointed Plaster sample, an intact rock namely Kota sandstone was selected. This type of rock
sample behaves roughly linearly elastic during the test. As depicted in Figure (7) triaxial test has been
carried out with two different confining stresses σ3=1 & σ3=5 MPa with the elastic modulus equal to
2.34 and 2.81 GPa respectively. Therefore this is just a check on the linear range of the model.
HB-Model vs Lab Dataintact Kota sandstone
0
20000
40000
60000
80000
100000
120000
0.0% 0.2% 0.4% 0.6% 0.8% 1.0% 1.2% 1.4% 1.6% 1.8% 2.0%
Axial Strain
Prin
cipa
l Str
ess
Diff
eren
ce k
Pa
Lab DataLab DataHB_ModelHB_Model
|σ1-σ
3|
σ3=5MPa & E=2.81 GPa
σ3=1MPa & E=2.34 GPa
HB Properties:σci=70 MpaGSI=100 mi=22
Figure 7 Numerical and experimental stress-strain results for intact Kota sandstone
4.3 HB-Model in compression and extension mode
In general, stress paths can be classified according to the type of loading and its direction. Two main
types of stress path are axial compression and axial extension. In this section, the performance of the
HB-model is checked for compression and extension paths in a triaxial test. Two different confining
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stresses σ3=71 & σ3=188 kPa have been used. The following properties for the rock element are
considered.
σci=30 MPa
mi=2
GSI=5
Eref= 5000 MPa
n=0.3 ; D=0
The objective of this test is to follow a simple stress path of a rock element from initial state to the yield
surface. In compression test, upon applying hydrostatic pressure to develop the initial stress, a
prescribed displacement is applied until the failure is reached. In the extension test, after producing
the initial stress the vertical pressure is decreased until failure. The developed stresses in the
specimen by this method coincides with yield surface obtained from Hoek-Brown criterion.
The respective stress paths of both compression and extension tests have been illustrated in Figures
8 and 9 in principal stress and (p’-q) space respectively.
The stress path with respect to the extension and compression test are distinguished by acronyms
“TXE” and “TXC”, respectively.
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Compression and Extension Triaxial Test
0.000
100.000
200.000
300.000
400.000
500.000
-100 0 100 200 300 400 500 600
Minor principal stress kPa
Maj
or p
rinci
pal s
tress
kP
a
Hoek-Brown CriterionHB-Model in Plaxis
E (Mpa)=5000ν=0,3σci(Mpa)=30GSI=5mi=2D=0
TXE
TXC
TXE
TXC
Figure 8 Compression and Extension stress path in a triaxial test _ in terms of principal stress
Compression and Extension Triaxial Test
0
50
100
150
200
250
300
-50 0 50 100 150 200 250 300
p' kPa
q' k
Pa
Compression Extension HB_Model
Yield Surface for Compression Test
Yield Surface for extension Test
TXC: Triaxial Compression TXE: Triaxial Extension
TXE
TXE
TXC
TXC
E (Mpa)=5000ν=0,3σci(Mpa)=30GSI=5mi=2D=0
Figure 9 Compression and Extension stress path in a triaxial test _ in (p´-q) space
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4.4 HB-Model and safety factor
The safety factor can be defined as the ratio of available strength to mobilised strength in terms of the
deviatoric stress q. In order to reduce the shear strength of HB material the Hoek-Brown criteria in the
(p´-q) space can be considered by dividing q by the required safety factor.
To investigate the validity of the phi-c reduction scheme for the HB-model the following procedure was
considered.
A triaxial compression test is modelled using HB material properties as in the previous section. The
yield surface of the material along with its reduced one, by factor of 1.31, is depicted in Figure (10). On
the dashed line three points were selected which their respective principal stresses are given in the
following table:
Table 2 Specifications of the selected points_ units in kPa ____________________________________________________________________ Point p’ q σ1 σ3 FOS(Plaxis) ____________________________________________________________________ 1 142.8 125.6 227 101 1.31
2 203.4 158.9 309 150 1.30
3 247 180.9 368 187 1.29 ____________________________________________________________________
Compression Triaxial Test
0
50
100
150
200
250
300
-50.000 0.000 50.000 100.000 150.000 200.000 250.000 300.000
p' kPa
q' k
Pa
Compression yield surfaceHB_ModelFS=1.31
E (Mpa)=5000ν=0,3σci(Mpa)=30GSI=5mi=2D=0
1
2
3
Yield Surface for Compression Test
Figure 10 Shear strength reduced by factor of safety and stress path of three different stress states
Three different triaxial tests are simulated separately and a phi-c reduction phase is followed after
each test. The safety factors given by Plaxis for each test are presented in Table (2) which indicate
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very reasonable agreement with the assumed safety factor equal to 1.31. Also, Figure (10) shows that
at the end of each test the stress path touches the reduced yield surface (green dashed line).
5 EVALUATING HB-MODEL IN BOUNDARY VALUE PROBLEMS
5.1 Circular opening under hydrostatic pressure
In this section, the HB-model is validated against an analytical solution of a boundary value problem.
Exact closed form solution for the elastio plastic behaviour of rock mass with the generalized form of
the Hoek-Brown failure criterion for a circular opening subjected to a hydrostatic in situ stress has
been presented by Carranza-Torres (2004). In order to simulate the aforementioned boundary value
problem a 4-meter circular deep tunnel under 15 MPa hydrostatic pressure was considered. The
model consists of 850 15-noded triangular elements with a total dimension of 50×25 m and
homogeneous HB material with following material parameters.
Intact rock parameters:
HB constant, mi [-] 10
Uniaxial compression strength, σci [MPa] 30
Geological strength index, GSI [-] 50
Hydrostatic pressure, p0 [MPa] 15
Young's modulus, E [MPa] 5700
Poisson's ratio,ν [-] 0.3
Rock mass parameters:
HB constant mb 1.6767
Parameter s 0.0038
Parameter a 0.5057
Parameter D 0
First, initial stresses are generated in the mass and then the tunnel is excavated in the next stage. The
stress field and the measure of plastic radius developed around the tunnel are investigated assuming
two different support pressures inside the opening, namely 0 and 2.5 MPa.
Plastic points and relative shear stresses for both support pressures (0 and 2.5 MPa) are illustrated in
Figures (11) and (12) respectively. It follows that the extent of the plastic points in both cases are in
reasonable agreement with the analytical plastic radius which are 2.58 m and 3.79 m in the case of
Pi=2.5 and Pi=0 respectively.
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Figure (13) and (14) compare radial and tangential stresses obtained from FE-analysis with the
analytical solution along a horizontal cross section for both support pressures. It can be seen that the
agreement is almost perfect.
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Rpl=3.79 m
Rpl=3.79 m
Figure 11 Plot of plastic points and relative shear stress for support pressure equal to 0
a) Relative shear stress b) Plastic points
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Rpl= 2.58 m
25 m
50 m
Figure 12 Plot of plastic points for support pressure equal to 2.5 MPa
C O M P UT A TI O N A LGEOTECHNICSG R O UP 18
Elasto-Plastic Stress Distribution (after Carranza-Torres)
0.0
5.0
10.0
15.0
20.0
25.0
0.0 5.0 10.0 15.0 20.0 25.0 30.0
Distance from Tunnel Center [m]
Stre
ss [M
Pa]
Radial Stress (Exact)
Tangential Stress (Exact)
Plaxis HB-Model Radial Stress
Plaxis HB-Model Tangential Stress
Figure 13 Radial and tangential stresses in numerical model and analytical solution
Pi=0 MPa
Elasto-Plastic Stress Distribution (after Carranza-Torres)
0.0
5.0
10.0
15.0
20.0
25.0
30.0
0.0 5.0 10.0 15.0 20.0 25.0 30.0
Distance from Tunnel Center [m]
Stre
ss [M
Pa]
Radial Stress (Exact)
Tangential Stress (Exact)
Plaxis HB-Model Radial Stress
Plaxis HB-Model Tangential Stress
Figure 14 Radial and tangential stresses in numerical model and analytical solution
Pi=2.5 MPa
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5.2 Slope stability
A simple slope with homogeneous rock mass material has been considered to evaluate correctness of
safety factor obtained from phi-c reduction procedure in Plaxis. To evaluate the HB-model, the safety
factor obtained from the same boundary value problem using a Mohr-coulomb material was adopted
as a comparative reference value. Corresponding MC strength parameters c, φ are derived according
to Hoek et al. (2002) suggestions which is based on general stress level of the problem. To consider a
wide range of stress levels different kind of slopes were selected including different inclinations and
heights.
The physical properties of the FE-models as well as strength material parameters of the MC and HB
model have been tabulated in the following and the safety factors of the corresponding analyses are
presented in table (4).
In Figure(15) safety factor versus number of steps for several analyses have been shown.
Table 3 Physical and mechanical properties of the considered slope models
HB properties MC equivalent
properties Model properties
σci mi GSI D C φ Tension cut-off
H* α** Element type
Num. of elements
Additional Steps
Ana
lysi
s N
o.
[MPa] [-] [-] [-] [kPa] [-] [kPa] [m] [°] [-] [-] [-]
1 30 2 5 0.0 - - - 10 35.5 15-node 2573 200
2 - - - - 20 21 0.0 10 35.5 15-node 2573 150
3 40 10 45 0.9 - - - 32 75 6-node 3238 150
4 - - - - 180 38 0.0 32 75 6-node 3238 150
5 40 10 45 0.9 - - - 32 75 15-node 1024 300
6 Same as analysis No. 5 but the width of the model is wider 15-node 922 220
7 - - - - 180 40 Full Tension 32 75 15-node 1024 200
8 40 10 45 0.9 - - - 12 75 15-node 918 300
9 - - - - 180 38 0.0 12 75 15-node 918 200 * Height of slope
** Angle of slope
Table 4 Table of results
Analysis No. 1 2 3 4 5 6 7 8 9
HB-model 1.34 - 2.5 - 2.48 2.62 - 5.32 - Factor of
safety MC-model - 1.33 - 1.56 - - 1.75 - 5.16
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From this example the following can be concluded:
1) In the case of α=35.5°, H=10 m both HB and MC model give the same safety factor.
2) In the case of α=75°, H=32 m there is a large discrepancy between HB and MC model.
Analyses No. 3 & 4 give safety factor values for the HB and MC model of 2.50 and 1.56
respectively. To describe this significant difference some comparative results in both models
such as plastic points failure mode, yield surface of respective models as well as stress level
of 3 points chosen on the failure line of the slope (E,F,G) have been depicted in Figures (16) &
(17).
Figure (17) implies that in the MC model tension failure mode affects the stability and the
angle of failure line is lower than the corresponding HB model. It should be noted that these
plots are associated with the end of phi-c reduction phase.
3) To ensure that modelling attributes such as element type and number of additional steps don’t
play a significant role in aforementioned difference analyses No. 5 & 6 were carried out with
15-noded element and larger additional steps. It is shown that these setting parameters do not
make a considerable difference.
4) In analysis No.7, MC strength parameters were selected in a way that MC-failure criterion lies
above the HB failure line at any stress range as illustrated in Figure (16). Even with this set of
MC parameters (φ=40°, c=180 kPa) the safety factor (1.75) is much less than the HB one
which is not expected. In addition, Arc-length option was switched off, nevertheless minor
changes were observed.
5) In a couple of analyses (No. 8 & 9) keeping the angle of slope 75° the height of the slope was
reduced to 12m checking that whether the high stress level of the toe has caused this
difference. Similar to the case (α=35.5°, H=10 m) there is no considerable difference between
the MC and HB safety factor which are 5.16 and 5.32 respectively. Figure (18) shows the plot
of plastic points and incremental strain after the phi-c reduction phase. In contrast to analyses
No. 3 & 4, in both HB and MC models shear failure mode causes instability.
6) Figure (15) shows that regarding safety factor there is some fluctuation in analysis No. 5 which
indicates a large number of steps are required to gain a stable safety factor.
In summary, the strength reduction scheme (phi-c reduction procedure) within the HB model were
examined with element tests and two reference boundary value problem and in most cases the results
of the obtained factor of safety are satisfactory. However, for the special case (α=75°, H=32 m) further
investigations are required.
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Factor of safety vs. calculation steps
0.0
1.0
2.0
3.0
4.0
5.0
6.0
0 50 100 150 200 250 300 350
Step
Msf
Slope 35.5 °_HB model
Slope 35.5 °_MC model
Slope75 °_HB model_H=32m
Slope75 °_MC model_H=32m
Slope75 °_HB model_H=12m
Slope75 °_MC model_H=12m
Figure 15 Safety factor versus number of steps
0
500
1000
1500
2000
2500
-300 -200 -100 0 100 200 300 400 500 600 700 800 900 1000 1100 1200
p´(kPa)
q (k
Pa)
HB-Model
MC-Model, c=180& =38
MC-Model, c=180& =40
HB-reduced byS.F.=2.5
MC-reduced byS.F.=2.5
MC-reduced byS.F.=1.57
HB-reduced byS.F.=1.57
Point E
Point F
Point G
ϕ
ϕ
E
F
G
Figure 16 Failure line of both MC and HB model in (p´-q) space
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Contour of incremental strains Plastic PointsSlope 75°_MC model_H=32m Slope 75°_MC model_H=32m
Contour of incremental strains Plastic PointsSlope 75°_HB model Slope 75°_HB model
Point E
Point F
Point G
Figure 17 Plot of plastic points and incremental strain contours, H=32 m
Slope 75°_MC model_H=12m Slope 75°_MC model_H=12m
Slope 75°_HBmodel_H=12m Slope 75°_HBmodel_H=12m
Figure 18 Plot of plastic points and incremental strain contours, H=12 m
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6 REFERENCES
Benz T., Schwab R., Kauther RA., Vermeer PA., 2008. A Hoek–Brown criterion with intrinsic
material strength factorization. Int J Rock Mech Min Sci; 45(2), 210–22.
Brown E.T., Trollope D.H., 1970. Strength of model of jointed rock. J. Soil Mech. Found. Div., ASCE;
96 (2), 685-704.
Carranza-Torres C., 2004. Elasto-plastic solution of tunnel problem using the generalized form of the
Hoek–Brown failure criterion. Int J Rock Mech Min Sci; 41(3), 480–1.
Hoek E., Carranza-Torres C., Corkum B., 2002. Hoek–Brown failure criterion—2002 edition. In:
Proceedings of the North American rock mechanics Symposium, Toronto.
Madhavi Latha G., Sitharam T.G., 2004. Comparison of failure criteria for jointed rock masses, In:
Proceedings of SINOROCK 2004 Symposium, Int. J. Rock Mech. Min. Sci; 41( 3).
Sharan S.K., 2008. Analytical solutions for stresses and displacements around a circular opening in a
generalized Hoek-Brown rock, Int. J. Rock Mech. Mining Sciences, Vol. 45, 78-85.
Carranza-Torres C., Fairhurst C., 1999. The elasto-plastic response of underground excavations in
rock masses that satisfy the Hoek–Brown failure criterion. Int J Rock Mech Min Sci; 36, 777–809.
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7 APPENDIX A: PLAXIS FILES
The Plaxis project files of all mentioned analyses have been enclosed to the present report and the file
names are presented in this appendix corresponding to report sections.
Section 4.1
Project name Description
ElementTest-Triaxial(mmt).PLX Drained analysis
Triaxial(mmt)undrained.PLX Undrained analysis
Section 4.2
Project name Description
TCT(mmt)Plaster-Sigma-1379 jointed Plaster, σ3=1379
TCT(mmt)Plaster-sigma-3447 jointed Plaster, σ3=3447
TCT(mmt)Kota-sigma-1000 Kota Sandstone, σ3=1000
TCT(mmt)Kota-sigma-5000 Kota Sandstone, σ3=5000
Section 4.3 All calculations have been carried out in one Plaxis file namely “TET(mmt)Extension.PLX”
Section 4.4 All calculations have been carried out in one Plaxis file namely “TET(mmt)Extension_FOS.PLX”
Section 5.1
Project name Description
Opening-Pi_0.PLX Circular deep tunnel with no internal support
Opening-Pi_2500.PLX Circular deep tunnel with 2500 kPa internal pressure
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Section 5.2
Project name Description
Slope35.5.PLX Analysis No. 1
Slope35.5-MC.PLX Analysis No. 2
Slope75.PLX Analysis No. 3
Slope75-MC.PLX Analysis No. 4
Slope75(15-noded).PLX Analysis No. 5
Slope75(15-noded)wide.PLX Analysis No. 6
Slope75-MC(15-nod)T23phi40.PLX Analysis No. 7
Sloope75(15n-12h).PLX Analysis No. 8
Slope75-MC(15n-12h).PLX Analysis No. 9