An Examination of the FLAC Software:

Undrained Triaxial Test on Cam-clay

Prepared by Jack Montgomery

June 8, 2010

ECI 280A Term Project

Instructor: Boris Jeremi

Introduction

Many problems in geotechnical engineering are too complex to be properly evaluated

using analytical solutions or physical models. Complexity may be due to non-linear material

properties, complex loading patterns or non-standard geometries. For these problems

numerical simulations are often used to examine the forces and displacements from various

types of loading. Many software packages are available for this type of simulation, but one that

is commonly used in practice is FLAC. FLAC, Fast Lagrangian Analysis of Continua, is distributed

by Itasca Consulting Group and has the ability to solve a wide range of geotechnical problems

involving dynamic loading, multiple material models, structural elements and pore fluid. In

order to better understand the inner workings of the software, the formulation will be

examined, some of the methods will be discussed and finally an undrained triaxial test on a

sample of Cam-clay will be simulated.

FLAC: Fast Lagrangian Analysis of Continua

FLAC is a commercially available, two-dimensional finite difference software program. In

FLAC, the finite difference method is used, instead of the more commonly understood finite

element method, to solve the differential equations associated with each problem. Each

differential term is replaced with an algebraic equation called a finite difference approximation

and this set of equations is solved in FLAC using an explicit integration scheme. The applied

forces are then divided into a series of incremental forces, referred to as time steps. The explicit

integration solves each equation of motion at each element for each time step with no

iterations (FLAC 2001).

After solving the equations for each element, the calculated velocities and

displacements are sent to the constitutive model to calculate stresses and strains. These

stresses and strains are then used to create new equations of motions. Because there is no

iteration this method assumes that the changes in each element do not affect the neighboring

elements within a time step. This assumption allows each element to be evaluated

independently. To ensure this assumption is valid, a small enough time step must be used so

that information would not passed between the elements within a time step. This method is

in contrast to the implicit method which solves the equations of motion for all elements at

once. This requires iteration to find the solution and may take more computational effort for

each time step, but significantly larger time steps may be used compared with explicit

integration scheme. Two of the most important features are the finite difference method and

the explicit integration scheme. These will be explored further in the following sections.

Finite Difference Method

One important difference between FLAC and some of the more commonly understood

finite element programs is FLAC uses the finite difference solution for solving differential

equations. The Finite Difference method is a numerical solution scheme for solving the

governing equations of a continuum body (Bathe 1996). This technique was pioneered in the

1920s as a method of solving nonlinear hydrodynamic equations (for more on development

see Thom and Apelt, 1961). Differential terms in the equations are replaced by algebraic

equations called finite difference approximations. These approximations are defined as the

difference between field variables at two discrete points in space. The finite difference method

has no shape functions, as finite elements do, so a linear change is assumed between the two

points (Bathe 1996). This would be equivalent to a finite element with a linear shape function.

This simplification can cause numerical errors in areas of high gradients since the

approximation can average out the changing variables. In these areas many elements may be

required to properly capture the response. One example of this can be seen in liquefaction

modeling where using only one row of elements can cause excess pore pressures to be

averaged out with the layers above and below. In an extreme case this could mask the

liquefaction phenomenon completely. This can be avoided by using two or more rows of

elements for any material.

An example will be used to illustrate how the finite difference technique can be applied

to a uniaxial bar (see Figure 1). This example is worked out by Bathe (1996) for both finite

difference and finite elements, but only the finite difference solution is presented here. In this

case the governing equation of the bar is the equation of motion. This equation requires finding

the second derivative of displacement. Consider three nodes on the bar spaced equally at a

distance h and two sub-nodes located midway between each of the main nodes. The second

derivative of the center node is approximated by considering the change in displacements over

three nodes and two points half-way between these nodes. First, the first derivative of

displacements at the two sub-nodes is found by considering the difference in displacements

between the left node and the center node. This approximates the derivative as the change in

displacements divided by the distance between the nodes. The second derivative at the central

node is found by taking the derivative between the first derivatives at the sub-nodes divided by

the distance between them. The first derivatives of the sub-nodes were found as described

above in terms of displacements at all three nodes. When the equations are combined the

result can be seen in Figure 1. One issue that can be immediately seen is the need for a virtual

node outside the bar to properly impose boundary conditions on the bar.

The finite difference method has been used for many years and has some distinct

advantages and disadvantages for solving differential equations. One of the most important

advantages is the simplicity of the formulations. This method can be easily coded into programs

and requires little memory to perform calculations. This simplicity makes it easy to recalculate

the equations of motions at each step, so more steps can be used without great penalty in the

form of calculation time. This makes it an ideal method for combination with the explicit

integration scheme which requires small step sizes. There are also no shape functions in the

finite difference method as there are in the finite element method. This means that variables

are undefined within the elements, but values can be assumed to vary linearly between nodes.

In FLAC, quadrilateral elements are specified by the user, but are subdivided into two triangular

elements internally. Given the lack of shape functions, these are analogous to constant strain,

triangular elements in finite elements. One distinct disadvantage of this approach is that many

elements may be required to properly capture areas where variables are changing rapidly. This

is especially important in problems like liquefaction. If only one row of elements is liquefiable,

the excess pore pressures will be averaged out by the nodes above and below the liquefied

elements. More details on the use of the finite difference method in FLAC can be found in the

users manual, Theory and Background (FLAC 2001).

Explicit Integration

There are two main types of numerical integration used in finite element or finite

difference solutions. These are the explicit and implicit methods, respectively. In numerical

solutions integration of the equations is performed at both the global and constitutive levels.

The global integration is concerned with the response of the entire system, while the

constitutive integration is concerned with the material response. The main difference between

the two methods is that the implicit method uses iterations to ensure equilibrium at each step.

On the global level, iterations ensure that neighboring elements are all in equilibrium with each

other and the applied loads. On the constitutive level, iterations are performed to ensure that

when the material is yielding it finishes on the yield surface. These iterations are not concerned

with the accuracy of the solution, but do ensure equilibrium at each step which may increase

the accuracy of the solution when compared with the explicit method in which the solution may

not be on the yield surface at all. The explicit method relies on small time steps to ensure any

errors in equilibrium are small and can be neglected.

On the global level, FLAC uses the explicit integration scheme which uses no iterations

to find equilibrium among the elements. This means that changes in stresses within a single

element will not affect the calculated displacements, and therefore the neighboring elements,

until the next step. This assumption is valid as long as the time step is small enough to ensure

that the calculation wave moves faster than the physical wave of the loading (FLAC 2002). This

means if the system is being loaded by an earthquake, the time step must be small enough so

the wave could not propagate through an element within the time step. Over multiple steps the

loading would propagate upward just as it would physically. This time step is calculated

automatically by the program to ensure stability and accuracy of the solution.

For all material models included with the FLAC software explicit integration is used at

the constitutive level. If a user was to define their own model they could include some sort of

implicit integration. Doing this would ensure equilibrium at the expense of computational time

and memory requirements. Explicit integration ensures accuracy by using a small time step

which is just as important at the constitutive level as it is at the global level. At the constitutive

level, the KarushKuhnTucker conditions for constitutive models (Karush 1939, Kuhn-Tucker

1951) require that the stress state of a material be on or within the yield surface at all times.

When a material attempts to cross the yield surface, plastic strains will develop and the stress

state will change. In constitutive modeling, this is handled by using an elastic predictor to allow

the stress state to cross the yield surface and then uses a plastic corrector to bring the material

back to the yield surface (Figure 2). The implicit method would use iterations to ensure that the

stress state ends on the yield surface. Explicit integration will draw a tangent line at the location

where the material crossed the yield surface and will come back to that line. If the time step is

small the error between the tangent plan and the yield surface will likely be small (Ortiz and

Popov 1985). Errors will be most significant with irregular yield surfaces and at bifurcation

points. This error is easy to examine at the element level, but can be lost in a large problem

with many elements. This is one of the many reasons single element tests are important to

check the accuracy and stability of the numerical solution.

Numerical Simulation

A single element test will be conducted on a sample of Cam-clay. The simulation will

serve two purposes. First, the results will be compared to analytical solutions published by

David Muir Wood (1990) to gauge the accuracy of the FLAC solution compared with what the

model should be predicting. This process is called verification of the software. Verification is

used to ensure that the model and software are functioning correctly numerically (Muir Wood

2004). A complimentary process to verification is validation of the model. Validation compares

the model response to some actual test result or case history to see if the model is capturing

the desired behavior. A simple way to think of it is verification is checking whether the model is

working right, validation checks whether it is the right model. Although validation is a crucial

step in any numerical simulation, it is a test of the appropriateness of the model, not of the

accuracy of the software program. For this reason it will not be explored further here. More

about verification and validation can be found in Oberkampf et al. (2002).

The second purpose of these analyses is to gauge the effect of different size time steps.

As was discussed earlier the size of the time step directly affects the accuracy and stability of an

explicit integration solution. In FLAC the minimum time step size is calculated automatically to

ensure numerical stability. The calculated size is based on the relative stiffness of the materials,

but an easy way to adjust the equivalent time step is to adjust the rate of loading so that more

load is applied during a given time step. For this report an undrained triaxial test will be

simulated using material parameters from Muir Wood (1990) and the Cam-clay material model

implemented in FLAC. Each component of the simulation will be described in the following

sections.

Cam-clay Constitutive Model

The modified Cam-clay model was developed by Roscoe and Burland (1968) and is a

modification of the original Cam-clay model developed by Roscoe et al. 1958. Because the

original model is not considered in this report, the modified Cam-clay model will simply be

referred to as Cam-clay. This model is an elasto-plastic constitutive model with a nonlinear

hardening and softening law which depends on the pre-consolidation pressure of the soil. The

model determines the response of the soil based on the specific volume or void ratio, a

deviator stress and a mean effective stress. Cam-clay is an associated plastic flow model in

which the yield surface is defined as an ellipsoid in q-p space with no strength at the origin and

pre-consolidation pressure (i.e. isotropic consolidation). Within the yield surface the material is

elastic and as the stress state crosses the yield surface both plastic volumetric and deviatoric

strains will develop. The material is either incrementally contractive or dilative depending on

whether it is dense or loose of critical state. The material will harden or soften depending on

the volumetric strains. At critical state the sample will undergo only deviatoric strain and

therefore will not harden or soften. This model has a relatively simple formulation and is

compatible with critical state soil mechanics and certain idealized clays. The formulation is

shown graphically in Figure 3. Further details about Cam-clay can be found in Muir Wood

(1990).

Simulated Triaxial Test

An isotropically consolidated, undrained triaxial test was simulated on a sample of Cam-

clay. In FLAC this was simulated with a single element using the axisymmetric option within the

software. The boundary conditions selected are shown in Figure 4. The horizontal axis of the

element is considered to be the radial direction and the vertical axis is the axial direction. The

nodes are numbered clockwise from the lower right corner as node 1, 2, 3 and 4. Each node has

two degrees of freedom in the axial and radial direction. Node 1 and 2 are fixed in the radial

direction to simulate the axisymmetric element. Node 3 is free to move in both directions and

nodes 1 and 3 are fixed in the axial direction to simulate the end platen. To summarize, Node 1

is fixed, Node 2 is fixed radially, Node 3 is free and Node 4 is fixed axially. A constant confining

pressure is applied between nodes 3 and 4 to simulate the cell pressure and a constant rate of

axial strain was applied to the top of the element. As a consequence of the constant strain

assumption the total axial load is not controlled directly. The effects of this will be examined in

the results section of this report.

The internal stresses in the element are initialized to the confining pressure to simulate

an isotropically consolidated sample. The sample is initially saturated and the bulk modulus of

water is defined to simulate the compressibility of water. Groundwater flow is turned off to

simulate an undrained test. As plastic volumetric strains try to accumulate, the sample is unable

to change volume due to saturation and the pore pressure will increase or decrease depending

on whether the sample is trying to contract or dilate. Two samples were simulated to capture

this behavior. One was a lightly over-consolidated sample (OCR =1.5) which is loose of critical

and a heavily over-consolidated sample (OCR = 20) which is dense of critical. The material

properties were taken from published test results in Muir Wood (1990). These properties are

summarized in Table 1.

Analytical Solution

To validate the results of the numerical solutions an analytical solution was used from

Muir Wood (1990). This solution solves for the stresses in an undrained triaxial test at yielding,

and critical state. A formula is also presented to calculate the stress path of the material

between the initial yield point and critical state. The results of the analytical solutions for both

the highly and lightly over-consolidated samples are presented in Figure 5. The predicted

responses are consistent with general critical state theory in that the loose sample contracts

and generates positive pore pressure, while the over-consolidated sample dilates and

generates negative pore pressure. The calculations needed for the analytical solution were

performed using the Mathcad software package and are shown in Appendix A.

Numerical Results

The numerical simulation was conducted using three different loading rates. The loading

was applied as a constant rate of strain at the top of the model. The slowest loading rate was

0.01% strain per step which was increased to 0.5% strain per step and then to 10% strain per

step. This increase in loading was meant to raise the equivalent step size of the simulation and

gauge the effects on the results. Figure 6 through Figure 8 show the response of the lightly

over-consolidated sample to each of the three loading rates and Figure 9 through Figure 11

show the response of the heavily over-consolidated sample. The figures show that the effective

stresses predicted by the numerical models are in excellent agreement regardless of the step

size. This is not surprising as Cam-clay has a very simple yield surface in which tangent lines will

closely approximate the actual surface. This may not be the case if the loading crossed near the

apex of the yield surface. Figure 12 shows the pore pressure response for both the lightly and

heavily over-consolidated samples. Pore pressures are only shown for the smallest step size

because the other paths were very erratic as suggested by the total stress paths shown in the

other figures. As expected the lightly over-consolidated sample undergoes contraction and

positive pore pressure generation. The heavily over-consolidated sample experiences

contraction at first and then dilation. Selected points from the analytical and numerical

solutions are shown in Tables 2 and 3 for comparison.

Although the effective stress path is in good agreement regardless of the step size the

total stress paths show how large steps can adversely affect the results. This triaxial test was

simulated by applying and axial strain while holding the radial confining stress constant. This is

defined as a conventional triaxial test and the total stress path will rise at a slope of 3q to 1p

(Powrie 2004). Only the step size of 0.01% produces a reasonable total stress path while the

larger step sizes produce total stress paths which oscillate around the correct line. The reasons

for this response will be explored next. In an undrained test pore pressures represent the

sample attempting to change volume, but because the test is undrained this contraction or

dilation occurs in the form of pore pressure. The pore pressure can be thought of as

representative of the volumetric strains which would occur in a drained test. When the sample

is loaded very quickly the pore pressure will suddenly spike or drop in response to the imposed

strains. This would normally lower the effective stress of the sample, but in FLAC the

constitutive model is only given displacements and velocities from which it produces stress and

strains. Because the rate of strain is controlled by the loading, FLAC adjusts the total stress to

maintain the proper strain rate. This adjustment occurs as the sample tries to maintain

equilibrium under the large strains. In reality, the samples would likely form tension cracks

when the effective stress dropped to zero, but the numerical solution prevents this and the

error manifests itself in the total stress.

Conclusions

The formulation of the FLAC software package has been explored and the accuracy of

the Cam-clay material model was examined by simulating an undrained triaxial test on a lightly

and heavily over-consolidated sample. The results of these simulations show that FLAC does an

excellent job in calculating the effective stress paths predicted by the analytical solution. It does

this regardless of the step size chosen, but it can be seen from the total stress paths that the

predicted strains are not accurate for the larger step sizes. These numerical errors are due to

the large strains generated during each step.

The dependence on step size shown in these simulations is not a surprising result. The

accuracy of any explicit integration scheme is dependent on the size of the step chosen. This is

a limitation of the integration method and does not speak to the accuracy of the software

package. The implementation of the Cam-clay constitutive model appears to be functioning

properly and the numerical methods in FLAC match well with the analytical solutions. As with

all explicit integration schemes the step size must be small to ensure numerical stability and

accuracy.

Table 1. Selected material parameters for Cam-clay

Material Parameter Value

Lambda 0.088

Kappa 0.031

Gamma 2.058

N 2.097

p0' 150

M 0.882

phi' 22.6

G 3000

Table 2. Results for lightly over-consolidated Cam-clay.

Analysis Method

OCR = 1.5

qyield pyield qcs p'cs ucs

Analytical 62.2 100.0 73.2 83.0 41.4

Small T-Step 62.3 100.0 73.0 83.3 41.0

Medium T-Step 62.2 100.0 73.1 83.1 36 to 46

Large T-Step 62.3 100.1 73.1 82.9 ?

Table 3. Results for heavily over-consolidated Cam-clay.

Analysis Method

OCR = 20

qyield pyield qcs p'cs ucs

Analytical 28.8 7.5 29.4 33.3 -16.0

Small T-Step 28.8 7.5 29.4 33.1 -15.7

Medium T-Step 28.8 7.5 29.3 33.3

-12.5 to

-19.5

Large T-Step 28.6 7.5 29.3 33.3 ?

Finite Difference Example

+

= 0

|1/2 = 1

|+1/2 =+1

| =|+1/2 |1/2

2

| =1

2(+1 2 + 1)

=

(+1 + 2 1)

Figure 1. Finite difference example for a bar in uniaxial tension (after Bathe, 1996)

Figure 2. Schematic showing one large explicit step.

Elastic Predictor

Plastic Corrector

m

q

P

Figure 3. Schematic of Cam-clay constitutive model.

Figure 4. Boundary conditions for the axi-symmetric triaxial test

0

10

20

30

40

50

60

70

80

90

100

0 50 100 150

q (

kP

a)

p' (kPa)

Cam-clay Constitutive Model

CSL

Yield Surface

Dilative Contractive

P0 = Pre-consolidation pressure = 150 kPa

Figure 5. Analytical solution to Cam-Clay loading.

Figure 6. Numerical solutions of effective stress path for low OCR Cam-clay triaxial test.

0

10

20

30

40

50

60

70

80

90

100

0 50 100 150

q (

kP

a)

p' (kPa)

Analytical Response

Low OCR

High OCR

CSL

Yield Surface

0

10

20

30

40

50

60

70

80

90

100

0 50 100 150

q (

kP

a)

p' (kPa)

Low OCR - Effective Stress

Low OCR

High Velocity - Effective

Medium Velocity - Effective

Low Velocity - Effective

CSL

Yield Surface

Figure 7. Total and effective stress paths for low OCR Cam-clay

Figure 8. Combined plot of all stress paths for low OCR Cam-clay.

0

10

20

30

40

50

60

70

80

-50 0 50 100 150 200 250

q (

kP

a)

p' (kPa)

Low OCR - Numerical Stress Paths

High Velocity - Effective

High Velocity - Total

Medium Velocity - Effective

Medium Velocity - Total

Low Velocity - Effective

Low Velocity - Total

0

10

20

30

40

50

60

70

80

90

100

0 50 100 150

q (

kP

a)

p' (kPa)

Combined Plot

Low OCR

High Velocity - Effective

High Velocity - Total

Medium Velocity - Effective

Medium Velocity - Total

Low Velocity - Effective

Low Velocity - Total

CSL

Yield Surface

Figure 9. Numerical solutions of effective stress path for low OCR Cam-clay triaxial test.

Figure 10. Total and effective stress paths for high OCR Cam-clay

0

10

20

30

40

50

60

70

80

90

100

0 50 100 150

q (

kP

a)

p' (kPa)

High OCR - Effective Stress

High OCR

High Velocity - Effective

Medium Velocity - Effective

Low Velocity - Effective

CSL

Yield Surface

0

5

10

15

20

25

30

35

-150 -100 -50 0 50 100 150

q (

kP

a)

p' (kPa)

High OCR - Numerical Stress Paths

High Velocity - Effective

High Velocity - Total

Medium Velocity - Effective

Medium Velocity - Total

Low Velocity - Effective

Low Velocity - Total

Figure 11. Combined plot of all stress paths for high OCR Cam-clay.

Figure 12. Pore pressure response of both the high and low OCR samples

0

10

20

30

40

50

60

70

80

90

100

0 50 100 150

q (

kP

a)

p' (kPa)

High OCR - Combined Plot

High OCR

High Velocity - Effective

High Velocity - Total

Medium Velocity - Effective

Medium Velocity - Total

Low Velocity - Effective

Low Velocity - Total

CSL

Yield Surface

-20

-10

0

10

20

30

40

50

0 500000 1000000 1500000 2000000

u (

kP

a)

Steps

Pore Pressure Development

High OCR

Low OCR

References:

Bathe, K.J. (1996). Finite Element Procedures. New Jersey. Prentice-Hall, 1996.

FLAC User Manual. (2001). Itasca Consulting Group. Minnesota. 2001.

Kuhn H. W. and A.W Tucker. (1951). "Nonlinear programming". Proceedings of 2nd Berkeley

Symposium. Berkeley: University of California Press. pp. 481492.

Karush W. (1939). Minima of Functions of Several Variables with Inequalities as Side

Constraints. M.Sc. Dissertation. Dept. of Mathematics, Univ. of Chicago, Chicago, Illinois.

Muir Wood, D. (1990). Soil Behaviour and Critical State Soil Mechanics. Cambridge University

Press. 1990.

Oberkampf W.L., Trucano T.G. and Hirsch C. (2002). Verification, Validation and Predictive

Capability in Computational Engineering and Physics. Invited Paper: Foundations for

Verification and Validation in the 21st

Century Workshop. Laurel, Maryland: Johns Hopkins

Univeristy. October 22-23, 2002.

Powrie, W. (2004). Soil Mechanics: Concepts and Applications. Spon Press. 2004.

Roscoe, K. H.; Schofield, A. N.; Wroth, C. P. (1958), "On the Yielding of Soils", Geotechnique 8:

2253

Roscoe, K.H. and Burland, J.B. (1968). On the generalized stress-strain behaviour of wet clay.

Engineering Plasticity. Cambridge: Cambridge University Press. pp. 535-609.

Thom, A. and C. J. Apelt, Field Computations in Engineering and Physics. London. Van Nostrand,

1961

Appendix A: Mathcad Calculations