uji triaxial

GEO-SLOPE International Ltd, Calgary, Alberta, Canada www.geo-slope.com

SIGMA/W Example File: MCC - triaxial tests.doc (pdf) (gsz) of 13

Modified Cam-clay triaxial test simulations

1 Introduction

This example simulates a series of triaxial tests which can be used to verify that Modified Cam-Clay

constitutive model is functioning properly. The simulations include:

Consolidating the sample to an initial isotropic stress state

A drained strain-controlled test

An undrained test with pore-pressure measurements

A load-unload-reload test

Consolidation with a back-pressure

The verification includes comparisons with hand-calculated values and discussions relative to the

Cam-Clay theoretical framework.

2 Feature highlights

GeoStudio feature highlights include:

Using the axisymmetric option to simulate a triaxial test

Displacement-type boundary conditions to simulate a strain-controlled test

Using the MCC model with a Load/Deformation analysis with no pore-pressure changes due

to the loading

A fully coupled analysis with specified initial pore-pressures

A fully coupled analysis with zero-flow boundary conditions to simulate an undrained test

3 Included files

Full details of this example and the GeoStudio files are included as:

MCC-triaxial tests.gsz MCC-triaxial tests.pdf

The specifics of each analysis are available in the GeoStudio data file.

4 General Methodology

The simulated shearing phases are preceded by the simulation of the consolidation phase of a triaxial test.

Consolidation is isotropic with the confining pressure equal to 100 kPa or 150 kPa. The isotropic stress

state is simulated by applying a normal stress on the top and on the right side of the sample equal to 100

kPa or 150 kPa. The consolidation stage is set as the “Parent”; that is, the initial condition, for the

subsequent simulations involving shearing.



metres (x 0.001)

-10 0 10 20 30 40

metr

es (

x 0

.001)

-10

0

10

20

30

40

50

60

Figure 1 Triaxial test configuration for establishing initial stress state

The shearing phase of the analysis is simulated as a strain-rate controlled test. The definition of the

strain-rate involves defining the number of „time‟ steps and the displacement that occurs over each step.

Although the „time‟ steps are being defined, it is more appropriate to think of the time steps as load steps.

Absolute time has no meaning in the context of these analyses. The number of load steps defined in the

shear stage simulations is generally 50 or 25 and the incremental y-displacement of the top of the

specimen is defined as -0.0002 m (per load step), where the negative sign indicates downward

displacement. Consequently, 50 and 25 load steps multiplied by a y-displacement of -0.0002 m per load

step results in a total vertical displacement of 0.01 m and 0.005 m, respectively.

Symmetry is assumed about the vertical and horizontal centre-lines; consequently, only ¼ of the specimen

is simulated. The dimensions of the simulation portion of the specimen are 0.025 m by 0.05 m, which is

half of the width and height of a conventionally sized triaxial specimen. Total vertical y-displacements of

0.01 m and 0.005 m produce axial strains of 0.2 (or 20%) and 0.1 (10%), respectively.

Notice that Linear-Elastic parameters are used when setting up confining stresses; non-linear models are not

required for this and the value of E is not relevant

5 Initial yield surface with OCR 1.25

The Cam-Clay properties are in the data file.

The material has a Ø of 25.84 degrees and an OCR of 1.25. A Ø of 25.84 is equivalent to an Μ value of

1.0.



The first step in using the Cam-clay models is to establish the yield surface created sometime in the past

under some particular stress state condition. In the field this would be some past insitu condition. This is

referred to as the past or initial yield surface.

SIGMA/W uses the initial vertical stress specified together with a specified Ø value and a specified OCR

(over-consolidation ratio) value to establish the past or starting yield surface.

The initial confining stress is 100 kPa. The past vertical effective stress then is,

σ΄y = 100 x OCR = 100 x 1.25 = 125 kPa

Ko = 1- sin Ø = 1 – 0.44 = 0.56 (formula in SIGMA/W code)

σ΄x = σ΄z = 125 x 0.56 = 70.52

σ΄mean = (125 + 70.52 + 70.52) /3 = 88.68

The shear stress q at the past mean stress is,

2 2 21( ) ( ) ( )

2y x z y x zq = 54.48 kPa

This is one point on the yield surface created by the past stresses. Since the sample is slightly

over-consolidated, the past stress state was higher than the current stress state. Currently, the sample is

isotropically consolidated to 100 kPa (σ΄x = σ΄y = σ΄z).

The past maximum-mean stress (pre-consolidation pressure) is,

2 2 2 2 2 2

2 2

1 154.48 1 88.68 122.15

1 88.68cp q p

p

p΄x is at the center of the yield surface where q is at its maximum value on the initial yield surface; p΄x =

122.15 / 2 = 61.08 kPa.

Now that p΄c is known, we can draw the initial yield surface for an assumed range of p΄ values between

0.0 and p΄c using the equation,

2 2 2

cq p p p

Figure 2 shows the initial or past yield surface. The three means stresses computed above are marked on

the diagram. The maximum q value occurs where the yield surface passes through the CSL (critical state

line), as it properly should.

The qpast and p΄past values form the starting point for establishing the initial yield surface.



0

10

20

30

40

50

60

70

80

0 10 20 30 40 50 60 70 80 90 100 110 120 130

Mean stress - p'

Sh

ea

r s

tre

ss

- q

CSL

pc = 122.15

ppast

= 88.68

px = 61.08

1

1

qpast

= 54.48

qmax

= 61.08

Figure 2 Initial yield surface for the past stresses at OCR = 1.25 (produced in EXCEL)

6 Analysis 1 – Drained-load deformation

A drained test can be done with a Load/Deformation analysis, which does not involve any changes in

pore-pressures due to the loading (straining).

During drained loading, the yield surface continues to increase in size as Figure 3. The total stress path

(which is equal to the effective stress path in this case) on a q-p‟ plot for a triaxial test will have a slope of

1h:3v. This being the case, the stress path intersects the CSL at 150 kPa.



0

20

40

60

80

100

120

140

160

0 20 40 60 80 100 120 140 160 180 200

Mean stress - p'

Sh

ear

str

ess -

q

CSL

Effective stress path &

Total stress path

3

1

Figure 3 Total stress path and final yield surface under drained loading

The final deviatoric stress (q) will be 150 kPa. The final vertical (y) stress will be the confining stress

plus the deviatoric stress; that is, 100 + 150 = 250 kPa.

Moreover, this being a drained test, the sample will undergo some volumetric strain.

The next three figures from SIGMA/W confirm this behavior.

Stress path

De

via

tori

c S

tre

ss (

q)

(kP

a)

Mean Effective Stress (p') (kPa)

0

50

100

150

100 110 120 130 140 150

Figure 4 Stress path under drained loading



Y stress - strain

Y-T

ota

l S

tre

ss (

kP

a)

Y-Strain

100

150

200

250

Figure 5 Vertical stress versus vertical strain

volumetric:axial strain

Volu

metr

ic S

train

Y-Strain

0

0.02

0.04

0.06

0.08

0.1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Figure 6 Volumetric strain versus load step number

7 Analysis 2 – Drained with fixed pwp

This analysis is a repeat of the previous drained test simulation, but now using the fully coupled

formulation with pore-pressure specified as a constant equal to zero pore-pressure. The specified pore-

pressure becomes a specified hydraulic boundary condition. Notice in Figure 7 how the hydraulic

boundary condition is specified at all nodes and outside edges.



metres (x 0.001)

-10 0 10 20 30 40

metr

es (

x 0

.001)

-10

0

10

20

30

40

50

60

Figure 7 Hydraulic boundary conditions for coupled drained simulation

q versus strain

De

via

tori

c S

tre

ss (

q)

(kP

a)

Y-Strain

0

50

100

150

Figure 8 Deviatoric stress versus y-strain for coupled drained test

The results from the Coupled analysis are identical to the previous Load/Deformation analysis. This

verifies that two different formulations give matching results.

This example also illustrates that drained conditions in a Coupled analysis can be simulated by fixing the

pore-pressures with a specified boundary condition.

8 Analysis 3 – Undrained OCR 1.25

The previous coupled analysis is now repeated, but the hydraulic boundary conditions are specified as no

flow. Actually, leaving the hydraulic boundary conditions undefined has the effect of zero flow across the



perimeter. By not allowing flow to exit from the sample, this analysis simulates a consolidated-undrained

test with pore-pressure measurements.

Based on theoretical consideration for the MCC model, the effective stress path should be vertical until it

meets the past maximum yield surface as in Figure 9. Once the effective stress path meets the past yield

surface, the path bends to the left and continues to rise slightly until it hits the CSL. The total stress path

again is straight line rising at a 1h:3v slope. The different between the total and the effective stress paths

is the excess pore-pressure.

The final total mean stress is 122 kPa. The effective mean stress where the effective stress path hits the

CSL is 66 kPa. The final pore-pressure therefore is 122 – 66 = 56 kPa.

0

10

20

30

40

50

60

70

80

90

0 10 20 30 40 50 60 70 80 90 100 110 120 130

Mean stress - p'

Sh

ea

r s

tre

ss

- q

u = 56 kPa

3

1Effective stress path

Total stress path

Figure 9 Stress paths for undrained loading

Two SIGMA/W output graphs below confirm these values. The SIGMA/W q-p΄ plot ends at p΄ = q = 66

kPa (Figure 10).

The SIGMA/W pore-pressure versus y-strain plot indicates the maximum pore-pressure is 56 kPa (Figure

11).



Stress pathD

evia

tori

c S

tre

ss (

q)

(kP

a)


0

10

20

30

40

50

60

70

65 70 75 80 85 90 95 100 105

Figure 10 Effective stress under undrained loading

pore-pressure:axial strain

Po

re-W

ate

r P

ressu

re (

kP

a)

Y-Strain

0

10

20

30

40

50

60

0 0.02 0.04 0.06 0.08 0.1 0.12

Figure 11 Excess pore pressure in undrained test

The volumetric strain for this test is zero, as it properly should be.

The sample, however, undergoes some plastic strain, which results in some strain-hardening and the yield

surface consequently expands such that it passes thorough the point where the effective stress path meets

the CSL.

9 Analysis 4 – Load-unload-reload

The MCC model treats the soil as elastic when the stress state is under the past maximum yield surface.

In an undrained test, the effective p‟:q stress path is vertical inside the yield locus whether the loading

path is one of unloading or loading. Figure 12 reveals that this is indeed the case: the stress path resumes

its non-linear behavior once the yield locus is crosses upon re-loading.



p':q stress pathD

evi

ato

ric

Str

ess

(q

) (k

Pa

)


0

10

20

30

40

50

60

70

65 70 75 80 85 90 95 100 105

Figure 12 Effective loading – unloading stress path

10 Initial yield surface with OCR 5.0

In this analysis, the sample is subjected to a cell pressure of 150 kPa and the back pressure is set to 50

kPa. The cell pressure is simulated with normal stress boundary conditions equal to 150 kPa. The back

pressure is specified as a material activation PWP equal to 50 kPa. This makes the effective confining

consolidation stress equal to 100 kPa.

The next step is to simulate a consolidated undrained test with pore-pressure measurements for an over-

consolidated soil.

In this test, we will set the confining stress to 150 kPa with a back pressure of 50 kPa. The effective

confining stress is again 100 kPa. This is discussed further in the next section.

With and initial effective confining stress at 100 kPa, the past vertical effective stress is,

σ΄y = 100 x OCR = 100 x 5.0 = 500 kPa

Ko = 1- sin Ø = 1 – 0.44 = 0.56 (formula in SIGMA/W code)

σ΄x = σ΄z = 500 x 0.56 = 282.07

σ΄mean = (500 + 282.07 + 282.07) /3 = 354.71

The shear stress q at the past mean stress is,

2 2 21( ) ( ) ( )

2y x z y x zq = 217.93

This is one point on the yield surface created by the past stresses. Since the sample is slightly

over-consolidated, the past stress state was higher than the current stress state. Currently, the sample is

isotropically consolidated to 100 kPa (σ΄x = σ΄y = σ΄z).

The past maximum-mean stress (pre-consolidation pressure) is,

2 2 2 2 2 2

2 2

1 1217.9 1 354.7 488.6

1 354.7cp q p

p



p΄x is at the center of the yield surface where q is at its maximum value on the initial yield surface; p΄x =

488.6 / 2 = 244.30 kPa.

Now that p΄c is known, we can draw the initial yield surface for an assumed range of p΄ values between

0.0 and p΄c using the equation,

2 2 2

cq p p p

Figure 13 shows the initial or past yield surface. The three p‟ stresses computed above are marked on the

diagram. The maximum q value occurs where the yield surface passes through the CSL (critical state line)

as it properly should.

The qpast and p΄past values form the starting point for establishing the initial yield surface.

0

50

100

150

200

250

300

350

0 50 100 150 200 250 300 350 400 450 500

p' - kPa

q -

kP

a

px = 244.30

ppast

= 354.71

pc = 488.61

CSL

qpast

= 217.93

Figure 13 Initial yield surface for OCR = 5

Now the effective stress path starts at 100 kPa and rises vertically until it hits the initial yield surface.

The soil behaves in a elastically up to this point, as illustrated in Figure 14.

After meeting the yield surface, the effective stress path bends to the right and rises slightly until it

intersects the total stress path. This is the point at which the excess pore-pressure is zero. After this point

the excess pore-pressure diminishes until the effective stress path intersects the CSL. Beyond this point

the soil behaves in a plastic manner with no further change in load or pore-pressure.



0

50

100

150

200

250

300

350

0 50 100 150 200 250 300 350 400 450 500

p' - kPa

q -

kP

a

CSL

Effective stress path

Total stress path

u = 0.0

u = -39

Figure 14 Effective stress for over-consolidated case (produced in EXCEL)

Of significance in this case is the fact that the effective stress path remains below the initial yield surface.

This is in response to dilation that occurs once the stress path meets the initial yield surface.

11 Analysis Undrained OCR 5.0

The following graphs from SIGMA/W confirm this behavior.

The effective stress path is vertical until meets the initial yield surface. Then it bends over to the right and

continues to the right until the path meets the CSL.

Stress path

De

via

tori

c S

tre

ss (

q)

(kP

a)


0

50

100

150

200

250

80 100 120 140 160 180 200 220

Figure 15 Effective stress path with OCR = 5



pore-pressure:axial strainP

ore

-Wa

ter

Pre

ssu

re (

kP

a)

Y-Strain

0

20

40

60

80

100

120

0 0.05 0.1 0.15 0.2 0.25

Figure 16 Pore-pressure variations with OCR = 5

The pore-pressure starts at 50 kPa, which is the starting back pressure. The pore-pressure then rises until

the effectives stress path meets the initial yield surface. After that, the pore pressure diminishes due to the

tendency for dilation until it approaches the CSL. The ending pore-pressure is around 10 KPa. Without

the initial back pressure, the ending pore-pressure would be around -40 kPa.

In SIGMA/W the MCC model is actually formulated only for saturated conditions and in the SIGMA/W

formulation, this means the pore-pressure must be positive. This is the reason for the back pressure. The

pore-pressure in this case can fall below the initial value and yet remain positive, so the SIGMA/W results

can be compared with the hand-calculated values in Figure 14.

uji triaxial

Documents