PLAXIS Hoek and Brown Validation.pdf

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


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


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


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)


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).


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


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


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.


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


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


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.


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


Figure 9 Compression and Extension stress path in a triaxial test _ in (p´-q) space


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


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


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.


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.


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


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

C O M P UT A TI O N A L G E OT E C H N I C SG R O UP 19

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


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.


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


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


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.


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


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


Opening-Pi_0.PLX Circular deep tunnel with no internal support

Opening-Pi_2500.PLX Circular deep tunnel with 2500 kPa internal pressure


Section 5.2


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

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Transcript of PLAXIS Hoek and Brown Validation.pdf