Two Constitutive Models for Simulation of Liquefaction in Sandy Soils

Prepared by Jack Montgomery

March 19, 2010

ECI 284 Term Project

Instructor: Boris Jeremić

Introduction: Cyclic loading of loose, granular materials tends to cause the material skeleton to contract. For saturated samples in which water is not allowed to drain, the total volume of the sample will remain relatively unchanged. As the skeleton tries to contract, forces will be transferred onto the pore fluid generating positive pore water pressure. As pore pressure increases, the effective stress of the sample will be reduced, as will the frictional strength. Different researchers have used different terms to describe this type of behavior, but the term liquefaction will be used in this paper. Liquefaction, and its associated strength loss, can be detrimental to geotechnical structures founded on loose, granular soils. The recently published Delta Risk Management Strategy found the greatest risk to levees in the California Delta Region is from earthquakes (DWR 2009) and a component of this risk was due to liquefaction. Multiple dams across the United States are being studied or rehabilitated due to seismic loading deficiencies (e.g. Success Dam, Perlea et al. 2008). One major consideration in many of these projects is the possibility of foundation liquefaction leading to the slumping or cracking of the earthen embankments. Although the mechanics behind liquefaction have been studied extensively over the last 50 years, there is still a great deal of uncertainty associated with its prediction and evaluation. Some of the rather large uncertainty in the analysis of liquefaction comes from the variability of earthquake loading and the heterogeneity of natural soil deposits. Because of these difficulties, current practice often uses empirical correlations based on case histories where liquefaction has been observed. These correlations often relate in-situ penetration resistance to the soils resistance to the triggering of liquefaction, termed the cyclic resistance ratio (e.g. Youd et al. 2001) and residual strength of the soil (e.g. Seed and Harder 1990). After the possibility of triggering has been evaluated, the consequences of liquefaction must be considered. Often displacements are estimated using the Newmark Sliding Block method or empirical correlations based on limit equilibrium slope stability analyses. An alternative to these simplified methods is the use of a constitutive model to predict the soil’s response to seismic loading. These models are often implemented in some type of numerical analysis software, such as a finite element program, which can capture many aspects of a problem such as pore pressure generation, complex deformation patterns and the development of shear strains. Different models have been developed for different applications and models currently used in practice range from relatively simple pore-pressure generation models, which count stress-reversal cycles, to the more advanced, fully-coupled, effective stress plasticity models. Each model has its benefits and drawbacks and it is important to examine these before use.

Constitutive Models: Two constitutive models for the evaluation of liquefaction will be evaluated in this paper.

This work is the first step of many to determine the effectiveness of each model at predicting liquefaction-induced deformations of small earthen embankments. This paper will briefly describe both models, focusing on the constitutive laws and the selection of the respective input

parameters. The four basic components of a constitutive model are the elastic response, the yield function, plastic flow and hardening laws. Both models will be used to simulate a cyclic, undrained direct simple shear test on a medium-loose sample of sand. Details about can be found in the respective references mentioned below.

UBCSAND: The first model examined for this paper is UBCSAND. This model was developed at the

University of British Columbia and is outlined in multiple references (e.g. Puebla et al. 1998, Byrne et al. 2004, Beaty 2009). UBCSAND is an incremental elastic-plastic model which is controlled by changes in the effective stress ratio. The model was developed based on plasticity theory and observations from laboratory testing on sands. The model is currently in use on multiple large scale projects, such as the evaluation of liquefaction potential at Success Dam (Perlea et al. 2004). The version of the model used here is 904aR as described by Beaty (2009). One major change with the new version is that reloading cycles which do not follow a stress reversal behave plastically as opposed to remaining elastic as predicted by previous versions. This allows plastic strains to be accumulated without having full stress reversals. This model is formulated for plane strain conditions and depends only on the shear and normal effective stress. Elasticity: The elastic strains for this model are a function of changes in either the shear or normal effective stress. The relationship between stresses and strain is controlled by the shear and bulk moduli. Both moduli are isotropic and non-linear, meaning they are a function of the current mean stress. Yield Surface: The yield surface for a model controls the boundary between elastic and plastic behavior. For UBCSAND, the yield surface can be described as radial lines which extend outward from the origin at a constant stress ratio. At the onset of loading the current stress ratio is very small so each increment of loading produces an elastic-plastic response. The yield surface is illustrated schematically in Figure 1. Plastic Flow: As the stress ratio of the soil moves outside the yield surface plastic strains will develop. This model utilizes a non-associated flow rule, meaning that the direction of plastic strains is not dependent on the current slope of the yield surface. Plastic shear strains are computed from the current stress ratio through a hyperbolic relationship as shown in Figure 2. The magnitude of the plastic volumetric strains is coupled to the shear strain through the angle of dilation. Volumetric strains can either be contractive or dilative and this is determined by the relation of the soil to the critical state line as defined by the constant volume friction angle. Soils at a stress ratio below the constant volume friction angle will undergo contractive behavior while soils above will tend to dilate. Soils at the critical state line will not experience volumetric strains. This is consistent with critical state theory and is illustrated in Figure 3.

Hardening Laws: The yield surface begins as a small radial line extending from the origin, but as the soil is plastically sheared the yield surface “opens” as shown in Figure 1. The maximum and minimum stress ratios seen by the soil are tracked separately, so the hardening law is tracked independently for positive and negative stress ratios. If the soil is unloaded it will behave elastically until the sign of the shear stress reverses. At this point the soil will be “reloading” until it reaches its previous maximum or minimum stress ratio. Reloading does generate plastic strains, but with a shear modulus that is significantly stiffer than for first time or virgin loading. Input Parameters: For this study only the basic input parameters were modified to calibrate the model to the soil of interest. Values for these parameters are given later in the paper, but each will be briefly described here.

Clean Sand SPT blow count – The corrected SPT blow count after considering the effect of fines on the measured penetration resistance. The correction proposed by Idriss and Boulanger (2008) was used in this paper. UBCSAND is internally calibrated to the Youd et al. (2001) relationship for calculating the CRR for a given soil from the clean sand blow count, so this blow count directly affects the number of cycles to cause liquefaction.

Constant Volume Friction Angle – The constant volume or critical state friction angle controls the direction of the plastic flow for UBCSAND as shown in Figure 3.

Elastic Shear Modulus Number – The elastic shear modulus number represents the small-strain shear stiffness, Gmax, for a given soil and is used in the non-linear relationship between the shear modulus and mean confining stress. This is an optional input parameter, but to maintain consistency between the two models it was set. Gmax is based on the equivalent shear wave velocity as correlated from the blow counts and the density of the material.

Elastic Bulk Modulus Number – The elastic bulk modulus number was set for similar reasons to the elastic shear modulus number. This number is related to Gmax through Poisson’s ratio. Assumed values for these parameters are listed in later sections.

PM4SAND: The second constitutive model used in this paper is a bounding surface plasticity model

for sand developed at UC Davis by Professor Ross Boulanger (Boulanger 2010). PM4SAND was modified from a model proposed by Dafalias and Manzari (2004), to improve its ability to predict the soil behavior important to geotechnical earthquake engineering, as seen in empirical and case history based correlations. This model was modified at the equation level to better predict the response of sandy soils to laboratory testing and to fit published design relationships and correlations. This model is formulated for plane strain problems and is controlled by the

current stress ratio (q/p’). The soil response is compatible with critical state soil mechanics through the use of the relative state parameter and the critical state line formulate by Bolton (1986). This model is in the final stages of development, so its applications for practice have not yet been evaluated. Elasticity: The elastic strains for this model are a function of changes in either the deviatoric or volumetric stresses. The relationship between stresses and strain is controlled by the shear and bulk moduli. Both moduli are isotropic and non-linear, meaning they are a function of the current mean stress. Yield Surface: The yield surface for PM4SAND is a tiny cone in stress-space whose center is defined by a term called the back-stress ratio. The yield surface along with bounding and dilatancy surfaces, described below, are illustrated schematically in Figure 4. Plastic Flow: The flow rule for PM4SAND is non-associated and is separated into deviatoric and volumetric components. Deviatoric strains are computed based on the plastic modulus which is besed on the distance from the current stress state to the bounding surface. The location of the bounding surface is related to the relative state of the soil, so as the soil is sheared the bounding surface will also move towards the critical state line. Volumetric strains are related to the deviatoric strains through the dilation term. The sign of the dilation term is controlled by the distance to the dilatancy, or phase transformation, line. If the soil lies above the line it will become dilative, below the line the soils will be contractive. Accumulation of strains is affected in this model by a fabric dilatancy tensor. This tensor is modified due to deviatoric strains during dilation and gives the soil a “memory” of previous loading. The fabric tensor modifies the plastic modulus and the dilation response of the soil to better predict the accumulation of shear strains with additional cyclic loading and the effects of sustained static shear stresses. Details about this fabric tensor can be found in Dafalias and Manzari (2004) and Boulanger (2010). Hardening Laws: This model accounts for hardening and softening by kinematic rotation of the yield surface in the stress space. This rotation is accomplished by updating the back-stress ratio which defines the center of the yield surface. The rate of hardening is controlled by the distance to the bounding surface and the plastic modulus, so it will also be affected by the fabric tensor defined above. Input Parameters: For this study only the basic input parameters were modified to calibrate the model to the soil of interest. Values for these parameters are given later in the paper, but each will be briefly described here.

Relative Density – The relative density of the material is the primary input variable for PM4SAND and determines the dilatancy and stress-strain response of

the soil. For this paper the relative density was related to the corrected SPT resistance through the relationship proposed by Idriss and Boulanger (2008).

Constant Volume Friction Angle – The constant volume or critical state friction angle controls the slope of the critical state line in PM4SAND. This is an optional parameter, but was set to maintain consistency with UBCSAND in which it is a required parameter.

Elastic Shear Modulus Number – The elastic shear modulus number represents Gmax for a given soil and is used in the non-linear relationship between stiffness and mean confining stress. Gmax is based on the equivalent shear wave velocity as correlated by the blow counts and the density of the material.

Poisson’s Ratio – This number is used to calculate the small-strain bulk modulus based on the shear modulus described above.

Numerical Implementation: Each of the models has been implemented into the commercial finite difference program

FLAC, Fast Lagrangian Analysis of Continua (Itasca 2004) by their respective authors. In FLAC the finite difference method is used, instead of the more commonly known finite element method, to solve the differential equations associated with each problem. Each differential term is replaced with an algebraic equation and this set of equations is solved in FLAC using an explicit integration technique. 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. 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. An assumption is made here 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. Details on FLAC and the computational methods can be found in the user manual.

By implementing each of the above models in FLAC, the response of the soil skeleton is fully coupled with the pore fluid through the bulk modulus of water. Changes in stress computed by the constitutive model are be translated into strains and displacements by FLAC, which will strain the pore fluid generating positive or negative pressures in an undrained test. This type of coupled approach allows one model to capture many complex aspects of soil behavior as opposed to evaluating the pore fluid and soil skeleton separately, as is done with some other models (Dawson et al. 2001). The details of one simulation performed for this paper will be given in the next section.

Simulation: Each of the above models was used to simulate a direct simple shear (DSS) test on a

sample of saturated silty sand. The parameters for the model were based on blow counts and gradations obtained from the foundation of the Torishima Dike on the Yodo River in Japan, believed to have liquefied during the 1995 Kobe earthquake (Matsuo 1996). Little data was available for this material and the parameters needed for the models were based on the SPT blow counts and fines content. Generic parameters for silty sands were selected based on data from the EPRI manual 6800. The selection of input parameters for each model was guided by Beaty (2009) and Boulanger (2010), respectively. An effort was made to use similar input parameters for each model, to limit the differences in the response which might be caused by the input parameters. Table 1 summarizes the input parameters used for each model and the source from which they were obtained.

In order to determine the liquefaction response of the soil, an undrained cyclic direct simple shear test was simulated. The soil sample was modeled as a single element in FLAC. The test was conducted by first isotropically consolidating the sample under a confining stress of 1 atmosphere. After reaching equilibrium under these consolidation stresses, a constant rate of strain, in this case .5% per second when FLAC is run in dynamic mode, was applied to the top edge of the model. The bottom boundary of the element was fixed in place. This strain rate was kept constant until the sample reaches the desired cyclic stress ratio (CSR, defined as shear stress divided by effective vertical consolidation stress) and then the direction of the strain was reversed. This process is repeated for many cycles and the changes in normal stress, shear stress, pore pressure, and strain are monitored. Plots of the response of the sample confined at 1 atmosphere and loaded with a CSR of 0.17 are shown in Figures 6-8. The response of each model in shown separately and will be discussed below.

During the writing of this report, it was noted that some of the simulations had a tendency to undergo very small dilation upon first loading. This response was examined carefully by the author and Dr. Boulanger and was considered to be a numerical artifact, which had no real effect on the solution. This dilation occurred for only the first one to two steps out of hundreds of thousands, but it did generate very small negative pore water pressures. In FLAC, the default value for the tensile limit of water is zero meaning if any negative pore water pressure develops, cavitation is assumed to occur and the pore water pressure is set to zero and the saturation of the element is reduced. When FLAC automatically reduces the saturation of the element, the test is no longer completely undrained and significant contractive volumetric strains occurred before the sample volume was reduced to the point where it became saturated again. This problem was resolved by setting the tensile limit of water to a value less than zero or by fixing the saturation of the element. This explanation is included here to remind the reader when dealing with numerical modeling it is important to carefully examine the predicted response to determine if it the predicted response is being adversely affected by the selection of input parameter or boundary conditions.

Table 1. Summary of input parameters used in creating the simulation.

Property Description Value Units Source N(1,60) Corrected Blow Count 12 Matuso, 1996

FC Fines Content 20% Matuso, 1996 Pa Atmospheric Pressure 101.3 kPa

Vs1 Shear Wave Velocity 165 m/s Andrus and Stokoe, 2000 e Void Ratio 0.6 EPRI Manual 6800 ν Poisson’s Ratio 0.3 EPRI Maual 6800

Both Models ρ(dry) Dry Density 1.75 t/m^3 EPRI Manual 6800

Go Maximum Shear Modulus

5.07E+04 kPa Formula from Boulanger, 2010

Go/Pa Normalized Shear Modulus 500

K Maximum Bulk Modulus

1.10E+05 kPa Formula from Boulanger, 2010

K/Pa Normalized Bulk Modulus 1083.3 n Porosity 0.625

Φ'cv Constant Volume Friction

Angle 30 degrees EPRI Manual 6800

CRR(7.5)

Cyclic Resistance Ratio for M = 7.5 or 3% strain in 15

cycles 0.17 Idriss and Boulanger, 2008 UBCSAND

N(1,60)cs Clean Sand Blow Count

16.5 Idriss and Boulanger, 2008 PM4SAND

Dr Relative Density 51% Idriss and Boulanger, 2008

hp Used to calibrate model to

CRR 0.8 Boulanger, 2010

Results: Figures 6 through 8 shown the computed response of the soil when subjected to the

simulated simple shear test described above. Each model accurately captured the general trend observed in cyclic, undrained simple shear tests on medium-loose sands (eg. Byrne et al. 2004, Idriss and Boulanger 2008). The soil first responds rather stiffly to the cyclic loading, almost in an elastic manner, but subsequent cycles begin to generate pore water pressure. As the effective stress on the soil is reduced, the soil begins to soften and with relatively few cycles large plastic strains can develop. At this point the pore pressure ratio is near 1.0 and the largest shear strains are observed as the soil is unloaded and attempts to contract, generating more pore water pressure. At The high pore pressures reduce the effective stress to near zero (the minimum stress is limited in PM4SAND) and the soil sample will be near the origin on the stress paths shown in Figure 6. As the shear stress begins to reverse, the sample will first generate large plastic shear

strains in the new direction of loading due to the very low stiffness. The stress ratio will then cross the phase transformation line, where the soil begins to dilate, which reduces the pore water pressure and increases the effective stress, stiffening the soil response. At the target CSR, the sample would sit at the tip of the characteristic “banana curves” shown in Figure 7. As the sample is unloaded, it will again begin to contract and generate large pore pressures, bringing the stress ratio back to the origin in Figure 6, but now the stress path will be approaching from the opposite side. This general stress path repeats with each subsequent cycle of loading generating additional shear strains each time.

The description of soil behavior above describes the general trends noted in laboratory tests, but the formulation of each model leads to different ways of approximating these trends. UBCSAND accurately predicts the stiff initial response of the soil and then the subsequent softening as the pore pressures increase. At this point however, the loading will be almost entirely unloading and reloading as the maximum and minimum stress ratios have already been reached. Because of this UBCSAND “locks” into a repeating stress-strain loop after reaching an ru=1 as shown by Figures 7 and 8. This response is due to the fact that the model has no way to accumulate damage caused by subsequent cycles of loading. The stress path shown in Figure 6 is repeated for each cycle and the model can no longer differentiate between the cycles, so the same response is computed each time. For the case shown in Figures 6 through 8 this resulted in a maximum shear strain of 1.5% regardless of how many cycles of loading were performed. Some of this response could likely be adjusted by modifying some of the optional input parameters, but without accurate lab data to guide that process, it was not attempted here. Figure 9 shows the results of an actual test plotted against the results of a UBCSAND simulation published in Beaty (2009). This shows that the actual soil continues to accumulate larger and larger shear strains as subsequent loads are applied, while UBCSAND predicts a repeating loop.

PM4SAND produces simulated results which look very similar to published test results. The soil transitions more smoothly from contractive to dilative responses and the stress-strain response shows the model continues to accumulate strains with subsequent cycles of loading. This accumulation of strain is controlled by the fabric tensor discussed in previous sections, which allows for cumulative strains to be developed. This is done by tracking “damage” to the soils fabric caused by the loading cycles. The default values for the fabric terms were used in this paper, but the rate at which plastic strains accumulate can be controlled by the user if desired. As stated before, without accurate lab data to calibrate to this was not attempted here. A characteristic “bump” has been noticed by some looking at the stress-strain curves in Figure 7. The bump can be seen immediately after the sample crosses the x-axis and is present in both models. This corresponds to the point where the sample transitions from a contractive to a dilative response and the location is determined by the constant volume friction angle in UBCSAND and the dilatancy line in PM4SAND.

Conclusion: Two constitutive models have been reviewed for the simulation of liquefaction in sandy

soils. The formulation and basic input parameters of each model have been explored and the computed response of each has been presented and briefly compared to previously published test results. Both UBCSAND and PM4SAND replicate the general response of soils subjected to cyclic, undrained simple shear tests, but each model attempts to replicate this response in different ways. These two models represent two different levels of complexity in the world of constitutive models and these results seem to suggest that the more complex model, PM4SAND better replicates the expected response. It is not possible to draw absolute conclusions about the application of each model based on one simulation of a lab test. A goal of future research will be to evaluate whether the additional complexity inherent in a bounding surface plasticity model such as PM4SAND produces more accurate results in a field-scale model.

Figures:

Figure 1. Yield surfaces for UBCSAND (Beaty 2009)

Figure 2. Relationship between stress ratio and plastic shear strains for UBCSAND (Beaty 2009)

Figure 3. Direction of plastic shear strains for UBCSAND (Beaty 2009)

Figure 4. Critical state line and relative state parameter used in PM4SAND (Boulanger 2010)

Figure 5. Various surfaces used in PM4SAND (Boulanger 2010) PM4SAND UBCSAND

Figure 6. Stress paths predicted by each model. Y-axis is shear stress in kPa and x-axis is vertical effective stress in kPa

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Figure 7. Shear stress-strain relationships predicted by each model. Y-axis is shear stress in kPa and x-axis is shear strain in decimal form.

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Figure 8. Pore pressure ratio generated with cyclic loading cycles. Y-axis is ru as defined by the change in pore pressure divided by the vertical consolidation stress and the x-axis is the number of cycles.

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Figure 9. Predicted response from UBCSAND plotted against actual test results (from Beaty 2009)

References: Beaty, M. (2009). Summary of UBCSand constitutive model: Versions 904a and 904aR. Draft Document.

Beaty Engineering LLC.

Bolton, M.D. (1986). “The strength and dilatancy of sands,” Geotechnique, Vol. 36(1): 65-78.

Boulanger, R.W. (2010). A sand plasticity model for earthquake engineering applications. Draft Document. Report UCD/CGM-10-01, UC Davis, Davis, California.

Byrne, P.M., Park, S.S., Beaty, M., Sharp, M.K., Gonzalez, L., and Abdoun, T. (2004). “Numerical modeling of liquefaction and comparison with centrifuge tests,” Canadian Geotechnical Journal, Vol. 41(2):193‐211.

Dafalias, Y.F. and Manzari, M.T. (2004). “Simple plasticity sand model accounting for fabric change effects,” Journal of Engineering Mechanics, ASCE, 130(6), 622-634.

Dawson, E.M., W.H. Roth, S. Nesarajah, G. Bureau, and C.A. Davis. (2001). “A practice-oriented pore pressure generation model,” Proceedings, 2nd FLAC Symposium on Numerical Modeling in Geomechanics. Oct. 29-31, 2001, Lyon, France.

DWR. (2009). Delta Risk Management Study. California Department of Water Resources. February, 2009. Available online at: http://www.drms.water.ca.gov 

EPRI. (1990). Manual on Estimating Soil Properties for Foundation Design. Electric Power Research Institute. August, 1990.

Idriss, I.M. and Boulanger, R.W. (2008). Soil Liquefaction During Earthquakes. Earthquake Engineering Research Institute. 2008.

Perlea, V.G., D.C. Serafini, E.M. Dawson, and M.H. Dawson. (2008). “Deformation analysis for seismic retrofit of an embankment dam,” Proceedings, Geotechnical Earthquake Engineering and Soil Dynamics IV. May 18-22, 2008, Sacramento, CA.

Puebla, H., Byrne, P.M. and Phillips, R. (1997). “Analysis of CANLEX liquefaction embankments: prototype and centrifuge models,” Canadian Geotechnical Journal, Vol. 34: 641-657.