Use of Numerical Simulation in the Development of Empirical Predictions of Liquefaction Behavior

Steven L. Kramer and David A. Baska

University of Washington

ABSTRACT Soil liquefaction has been an interesting and important topic in geotechnical earthquake engineering for more than 35 years. Ever since its consequences were so vividly observed in the 1964 Niigata and Alaska earthquakes, geotechnical engineers have sought to develop accurate procedures for predicting the behavior of liquefiable soils. Research has advanced on several fronts – experimental laboratory element testing, physical model testing, field performance investigation, and numerical simulation. Each of these approaches to the liquefaction problem has advanced the profession’s understanding of liquefaction behavior and contributed, directly and indirectly, to current procedures for evaluation of liquefaction hazards. The complementary nature of these approaches has not always been fully recognized. This paper describes some aspects of liquefiable soil behavior in which empirical procedures could be improved by considering the results of numerical simulations. The manner in which numerical simulation can be used to guide the development of empirical predictions of liquefaction behavior is illustrated through the problem of lateral spreading displacement. Widely used empirical relationships have been developed using multiple linear regression on databases of observed lateral spreading case histories. These relationships have identified a set of statistically significant parameters and used them to develop relatively simple, easy-to-use predictive equations. The form of some of these parameters, however, lead to displacement predictions that can be inconsistent with the known behavior of liquefiable soils. It is possible to use numerical simulations that capture the mechanics of the lateral spreading problem to identify an improved form of a predictive equation for lateral spreading displacement. The equation can then be calibrated against empirical case history data. The result is a predictive equation that is consistent with observed lateral spreading behavior, but also consistent with the mechanics of liquefiable soils.

Use of Numerical Simulation in the Development of Empirical Predictions of Liquefaction Behavior

Steven L. Kramer and David A. Baska

University of Washington Introduction Soil liquefaction has been an interesting and important topic in geotechnical earthquake engineering for more than 35 years. Ever since its consequences were so vividly observed in the 1964 Niigata and Alaska earthquakes, geotechnical engineers have sought to develop accurate procedures for predicting the behavior of liquefiable soils. Research has advanced on several fronts – experimental laboratory element testing, physical model testing, field performance investigation, and numerical simulation. Each of these approaches to the liquefaction problem has advanced the profession’s understanding of liquefaction behavior and contributed, directly and indirectly, to current procedures for evaluation of liquefaction hazards. The complementary nature of these approaches has not always been fully recognized. This paper describes some aspects of liquefiable soil behavior in which empirical procedures could be improved by considering the results of numerical simulations. Liquefaction and Performance The performance of many important structures can be influenced by soil liquefaction. Because infrastructure and commercial structures and facilities are often constructed adjacent to rivers, lakes, bays, etc. where liquefiable soils are commonly encountered, evaluation of liquefaction hazards is an important part of their design.

To date, liquefaction hazard evaluation has focused on evaluation of the susceptibility of soils to liquefaction and on the conditions required to trigger liquefaction. With respect to liquefaction, design has generally been oriented toward elimination of the possibility of liquefaction. Relatively less attention has been paid to the problem of predicting the effects of liquefaction. As earthquake-resistant design moves increasingly toward a performance-based approach, however, prediction of the influence of liquefaction on the performance of structures becomes increasingly important. Performance is related to effects; hence, improved prediction of the effects of liquefaction is of great contemporary importance. Previous Approaches Liquefaction Potential Early approaches to the evaluation of liquefaction hazards were based on programs of laboratory testing, principally cyclic triaxial testing. Cyclic triaxial tests on isotropically consolidated, reconstituted specimens were performed to evaluate the relationship between loading characteristics and pore pressure generation. Even the earliest of these tests (Figure 1) showed a clear relationship between factors such as soil density, shear stress, and effective confining pressure on the rate of pore pressure generation. Defining liquefaction as the attainment of 100% pore pressure ratio (or some limiting cyclic strain amplitude in the case of

very dense test specimens), liquefaction resistance curves (Figure 2) could be plotted and used to characterize liquefaction resistance.

Figure 1. Results of early cyclic triaxial test (Seed and Lee, 1966).

Early procedures for evaluation of liquefaction potential were based on laboratory tests. The results of such tests, performed using cyclic triaxial and cyclic simple shear devices, were typically expressed in terms of cyclic resistance curves which showed the level of cyclic shear stress required to produce liquefaction in a given number of harmonic stress cycles. Because earthquakes do not produce harmonic loading, procedures were developed (reference) to express an equivalent number of stress cycles as a function of earthquake magnitude (Figure 2). Later, increased recognition of the difficulties of correlating laboratory test results to field behavior led to the development of liquefaction evaluation procedures based on field evidence. These procedures typically express the cyclic resistance ratio corresponding to liquefaction in a M7.5 earthquake as a function of an insitu test parameter such as (N1)60 (Figure 3) or qc1. Evaluation of liquefaction potential for other magnitudes is accomplished using magnitude scaling factors.

Earthquake Magnitude (m)

Figure 2. Variation of number of equivalent cycles with earthquake magnitude (after Liu et al., 2001).

Figure 3. Variation of CRR with (N1)60 for M7.5 earthquakes (after Liu et al., 2001).

Lateral Spreading Deformations At sloping sites and sites adjacent to free slopes, liquefaction phenomena frequently lead to horizontal ground movement commonly referred to as lateral spreading. Lateral spreading has caused significant damage to buildings, bridges, levees, buried pipelines, and other constructed facilities in past earthquakes. Prediction of the performance of existing structures, and performance-based design of new structures, requires reliable prediction of the permanent deformations associated with lateral spreading.

Lateral spreading displacements are most commonly predicted using the empirical relationships developed by Bartlett and Youd (1992) and recently updated by Youd et al. (1999). These relationships were assembled by (1) compiling a database of lateral spreading case histories, (2) developing an extensive list of parameters that could potentially influence lateral spreading displacement, (3) developing a suite of potential predictive relationships involving various combinations of the parameters (natural and transformed), (4) performing regression analyses on the case history database, and (5) identifying the predictive relationship that was most consistent with the case history data. This process resulted in two predictive relationships – one for slopes with constant inclination (ground slope model) and one for slopes with free faces (free face model). The predictive relationship for the ground slope model is

log DH = -16.213 + 1.532M – 1.406 log R* - 0.012R + 0.388 log S (1) + 0.540 log T15 + 3.413 log (100-F15) – 0.795 log (D5015 + 0.1 mm)

where M = moment magnitude, R* = , R = , S = ground slope in percent, T15 is the cumulative thickness of all sublayers with (N1)60 ≤ 15, F15 is the average fines content of the materials contributing to T15, and D5015 is the average D50 value for the materials contributing to T15. Note that the T15 parameter, which proved to be most effective in predicting the observed lateral spreading displacements, implies behavior that is inconsistent with the observed behavior of potentially liquefiable soils. For example, it implies a sharp discontinuity in behavior of materials with (N1)60 values above and below 15, and implies no sensitivity of lateral spreading displacement to (N1)60 values for values below 15, or for values above 15. Laboratory tests indicate that a relatively smooth variation of performance with soil density should be anticipated. UWsand Soil Model The UWsand soil model was developed to provide a simple constitutive model that could represent the actual behavior of liquefiable soils, including phase transformation behavior, and be calibrated using information that is commonly available to practicing geotechnical engineers. For liquefiable soils, this information is typically limited to measures of soil density and penetration resistance. UWsand is a one-dimensional model, i.e. it assumes that particle motion occurs in a single plane. This assumption is valid for most cases of site response analysis and is commonly made in geotechnical earthquake engineering practice. Model Characteristics The UWsand model assumes that yield occurs at a constant stress ratio, η = q/p’. Hardening behavior is explicitly controlled to match the rate of shear modulus degradation implied by a standard modulus reduction curve. By fitting a rational function to the Seed-Idriss upper bound modulus reduction curve, an algebraic expression for plastic shear modulus is obtained. The UWsand flow rule is written in such a way that the plastic volumetric strain component is zero when the effective stress path is on the PTL, i.e. when η = ηcv. With the UWsand flow rule, the plastic volumetric strain is negative (contraction) when η < ηcv, and positive (dilation) when η > ηcv (Figure 4). Under cyclic loading conditions, unloading-reloading behavior is modeled by applying the Cundall-Pyke hypothesis to the plastic shear modulus. Under undrained conditions, total volumetric strain increments are assumed to be zero.

For each stress increment, an incremental change in porewater pressure is computed from the incremental volumetric strain, the bulk modulus of water, and the soil porosity. Drained analyses can be performed by assigning a very low value to the bulk modulus of the pore fluid.

Figure 4. Schematic illustration of flow rule for UWsand model.

Figures 5 and 6 illustrate the response predicted by the UWsand model for elements of

soil under symmetric, (τstatic = 0, τcyc = 10 kPa) and asymmetric (τstatic = 5, τcyc = 10 kPa) loading. Note that the cyclic stress amplitude is the same for both cases, but the presence of a static stress causes plastic shear strains to accumulate in one direction.

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Figure 5. Response of UWsand model to symmetric loading.

Figure 6. Response of UWsand model to asymmetric loading.

Model Calibration

The UWsand model has been developed and calibrated to be consistent with the observed field behavior of liquefiable sands. Like all constitutive models, the UWsand model can be used to simulate the results of a laboratory test on an element of soil. Such laboratory data is available

for specific sands upon which tests have been performed, but not for typical sands. It is possible, however, to use field performance data to develop equivalent liquefaction resistance curves, i.e. to put the field-based liquefaction criteria in a laboratory-based framework. Because of the widespread acceptance of using penetration resistance to assess liquefaction resistance (Youd et al., 2001), (N1)60 was selected as the material parameter in the constitutive model. A large database of case histories exists that relates the (N1)60 of the soil to observations of liquefaction during previous earthquakes. The case history data may be used to approximate the number of cycles to initial liquefaction as a function of (N1)60. This was accomplished in the following steps:

1. Assume a particular "class" of typical sand, say clean sand (FC < 5 percent). 2. Assume a particular value of (N1)60 and determine CRRM for different earthquake

magnitudes using Figure 2 and the magnitude scaling factor (Figure 3).

3. Compute the number of equivalent cycles for each earthquake magnitude.

4. Plot CRR vs. number of equivalent cycles to initial liquefaction.

The final result of this process is a set of field-based liquefaction resistance curves that can be used for model calibration. Calibration of UWsand against this data will ensure that the model predicts initial liquefaction in the right number of equivalent stress cycles for soils of different (N1)60 and fines content. Calibration of the UWsand model involves correlating several other constants and expressions within UWsand to (N1)60.

Number of Cycles to Liquefaction Matching the number of cycles to initial liquefaction as a function of (N1)60 and CRR was one of the main goals of the model calibration process. The accuracy of the match provided by UWsand is shown in Figure 7; the circles are data points computed using UWsand. The excellent agreement between the discrete points predicted by UWsand and the curves shows that the UWsand model is capable of modeling the development of initial liquefaction reasonably.

Figure 7. Comparison of cyclic resistance ratio predicted by UWsand

with values backcalculated from field data.

Rate of Pore Pressure Generation

DeAlba et al. (1975) performed cyclic loading tests in the laboratory that showed that the pore pressure ratio in soil elements subjected to uniform harmonic loading increased rapidly in the first few cycles of loading, increased at a slower rate during intermediate cycles of loading, and then increased quickly again as initial liquefaction was approached. Figure 8 demonstrates the ability of UWsand to match the rate of pore pressure generation measured in laboratory tests. Figure 8(a) shows that model simulations for a range of (N1)60 values generally falls within the mean and upper bound limits observed by DeAlba et al. for the case of Nl = 5. Figure 8(b) shows that for the case of Nl = 20, the model simulations generally fall within the mean and lower bound limits of the experimental data. Of primary importance in this validation step is the ability of the model to compute the characteristic rapid increase in excess pore pressure observed during the early and later loading cycles of the laboratory tests.

Figure 8. Rates of excess pore pressure

generation (a) NL = 5 cycles and various (N1)60 values, and (b) NL = 20 cycles and various (N1)60

values. Model Behavior The ability of the UWsand model to represent the behavior of typical liquefiable sands can be illustrated through a series of parametric analyses. The model should be capable of representing the increased resistance to liquefaction of dense soils relative to loose soils, as characterized by (N1)60. It should also be capable of representing the cumulative permanent strain that develops when an element of soil is subjected to cyclic loading in the presence of a static shear stress.

The effects of (N1)60 on rates of pore pressure generation, stress path, and stress-strain behavior are illustrated in Figures 9 and 10. Figure 9 shows the response of an element of soil with (N1)60 = 5 to a cyclic shear stress that produces initial liquefaction in 5 cycles. Figure 10 shows that a cyclic stress more than six times higher is required to produce initial liquefaction in 5 cycles in a (N1)60 = 20 soil.

Figure 9. Pore pressure generation, effective stress

path, and stress-strain plots for (N1)60 = 5 and 5 cycles to initial liquefaction.

Figure 10. Pore pressure generation, effective stress path, and stress-strain plots for (N1)60 = 20 and 5 cycles to initial

liquefaction. The static shear stress also plays an important role in the response of liquefiable soils.

The presence of a static shear stresses causes a symmetric cyclic shear stress to induce more shear strain in one direction than the other, thereby resulting in the accumulation of permanent strains. The performance of UWsand for various cases involving static shear stresses is shown in Figures 11 and 12. Figure 11 shows that permanent strains decrease in a smooth, continuous manner with increasing (N1)60. Figure 12 shows that permanent strains increase with increasing static shear stress level.

Figure 11. Validation of UWsand permanent shear strain results with a cyclic shear stress of 10 kPa and a

Figure 12. Validation of UWsand permanent shear strain results with N = 5 and a cyclic shear stress of 12

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static shear stress of 5 kPa. N increases from 2 (uppermost plot) to 25 (lowest plot).

kPa. Static shear stress increases in 2 kPa increments from 0 kPa (uppermost plot) to 8 kPa (lowest plot).

Permanent shear strains should also increase with increased static shear stress for

transient loading such as that produced by earthquake shaking. Using the shear stress time history from an equivalent linear ground response analysis as input to UWsand superimposed upon different levels of static shear stress produces the response shown in Figures 13 and 14. The stress-strain plots generally show an increase in permanent strain with increasing static shear stress.

Figure 13. Stress-strain diagrams for transient loading with ground slopes increasing from 0 degrees at the top to 5 degrees at the bottom.

Figure 14. Effective stress paths for transient loading with ground slopes increasing from 0 degrees (uppermost plot) to 5 degrees (lowest plot).

The results of these analyses indicate that the UWsand model is capable of capturing the

most important characteristics of liquefiable soil behavior, and to do so in a manner that is consistent with the performance of typical liquefiable soils as observed in actual earthquakes. When implemented into a nonlinear, effective stress-based site response analysis, it offers the potential for improved understanding of how the mechanics of liquefiable soils influence performance in the field.

Simulation of Lateral Spreading

The UWsand model has been incorporated into a one-dimensional finite difference site response program called WAVE (Horne, 1996). WAVE allows specification of an initial (static) shear stress profile, which can be used to simulate sloping ground conditions. By using WAVE with a suitably chosen static shear stress profile, the phenomenon of lateral spreading can be numerically simulated.

To develop a better understanding of how the mechanics of the liquefaction process affect lateral spreading, a series of parametric analyses were performed. These analyses assumed infinite slope conditions and a wide variety of subsurface conditions. A generic four-layer

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profile (Figure 15) was used in the parametric analyses. A surficial non-liquefiable layer of thickness, d, was underlain by a liquefiable layer of thickness, h. The liquefiable layer was underlain by a 6-m-thick layer of dense ((N1)60 = 50) sand, which was underlain by bedrock. To investigate the influence of the thickness and depth of the liquefiable layer on lateral displacement, analyses were performed with a matrix of d-h pairs. To investigate the influence of soil density on displacement, a variety of (N1)60 profiles were used. Effect of Soil Density Soil density is known to play a strong role in determining the liquefaction potential of soils, and can be expected to have a significant influence on the permanent displacement produced by lateral spreading. Analyses of multiple profiles with constant (N1)60 values in the liquefiable layer illustrated the effect of soil density on lateral spreading displacement. The analyses showed (Figure 16) that displacements varied relatively smoothly with (N1)60, and that there was no discontinuity in displacement at (N1)60 = 15 or any other value.

Figure 16. Variation of computed ground surface

displacement with (N1)60 for d = 3 m, h = 6 m. Effect of Liquefiable Layer Thickness and Depth Permanent displacements result from integration of the permanent strain that develops at different depths in liquefiable soil layers. Therefore, permanent displacements can intuitively be expected to increase with increasing liquefiable layer thickness. Less intuitive is the effect of liquefiable layer depth on permanent displacement. WAVE analyses performed holding d constant and varying h (Figure 17a) confirmed the expectation that permanent displacement increases with increasing liquefiable layer thickness. Analyses with constant h and variable d (Figure 17b) showed a modest trend of decreasing permanent displacement with increasing liquefiable layer depth.

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An Equivalent Thickness Parameter

The observations that lateral spreading displacements decrease with increasing (N1)60, and that they do so in a continuous manner, along with the observation that deep liquefiable layers produce less displacement than otherwise identical shallow layers suggests that an improved method for characterization of soil profiles can be developed. The concept of characterizing a profile by means of an equivalent thickness, such as the T15 parameter introduced by Bartlett and Youd (1992), has proven useful and popular. The results of the WAVE analyses suggest, however, that an equivalent thickness parameter should be a continuous function of (N1)60 and depth.

An approach to liquefiable profile characterization based on the summation of contributions from different sublayers was developed. With this approach, the equivalent thickness is defined as

( )hzNfT im

iii∑=

=1,* (2)

where Ni is the (N1)60-cs value of the ith sublayer, zi is the depth to the center of the ith sublayer, hi is the thickness of the ith sublayer, and m is the total number of sublayers in the profile. Using the WAVE-generated database of computed lateral spreading displacements, the suitability of a number of potential expressions for f(Ni, zi) were investigated. Through this process it was determined that displacement predictions based on an expression of the form

∑==

m

izikN ik eeT

121* (3)

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produced the best agreement with the computed displacements. This form implies that hi = 1.0 m, i.e. that the profile is divided into 1 m thick sublayers.

Using the coefficients obtained by regression on the numerical database, the variation of lateral spreading with (N1)60 would be as shown in Figure 18. The figure also shows how the commonly used T15 parameter would vary. Clearly, the continuous variation of T* is more consistent with the mechanics of liquefiable soil than the step function produced by T15. Of course, the variation of T15 is unlikely to be so abrupt in an actual soil deposit in which (N1)60 would vary with depth.

Predicted Displacement (m)

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Figure 18. Relationship between computed (WAVE) and predicted lateral spreading displacements using T* expression of Equation (3): (a) scatter plot, and (b) plot of residuals.

Analyses are currently underway to develop an improved predictive equation for lateral

spreading displacement. The equation will include an expression for T* of the form shown in Equation (3), but the coefficients will be determined based on regression on a database of actual lateral spreading case histories.

The intent of this approach is to develop an empirical predictive equation that is consistent with the known mechanics of liquefiable soil, but is calibrated against actual case histories. The accuracy of the approach will depend on the degree to which the mechanisms that produced the lateral spreads in the case history database are consistent with those represented by the WAVE analyses. The extent to which lateral spreading displacements in the case history database were influenced by pore pressure redistribution is not known, but may be considerable for a number of case histories. Because the WAVE analyses did not consider the effects of pore pressure redistribution, its existence can be expected to increase the dispersion in lateral spreading displacement predictions. Consideration of pore pressure redistribution effects is complicated, however, by the lack of information on permeability and other factors that influence pore pressure redistribution in the case history database.

Summary and Conclusions Numerical simulation offers the potential for reliable a priori predictions of the

performance of geotechnical systems. Full realization of this potential, however, will require further advances in constitutive modeling, numerical analysis, and in tools and techniques for site characterization. Given the natural randomness through available site characterization techniques, the use of empirical methods of performance prediction will continue to be an important part of geotechnical practice. Numerical simulation techniques can be used to investigate the mechanics of geotechnical performance and to determine (?) the form of an empirical predictive equation that is consistent with those mechanics. Calibration of such equations by field data can be expected to produce more accurate predictions of performance in future earthquakes than equations that are not consistent with mechanics. References Bartlett, S.F. and Youd, T.L. (1992). "Empirical analysis of horizontal ground displacement

generated by liquefaction-induced lateral spread," Technical Report NCEER-92-0021, National Center for Earthquake Engineering Research, Buffalo, New York.

DeAlba, P., Chan, C.K., and Seed, H.B. (1975). "Deternimation of soil liquefaction

characteristics by large-scale laboratory tests," Report No. EERC 75-14, Earthquake Engineering Research Center, University of California, Berkeley.

Horne, J. C. (1996). “Effects of Liquefaction-Induced Lateral Spreading on Pile Foundations,”

Ph.D. Dissertation, University of Washington. Liu, A.H., Stewart, J.P., Abrahamson, N.A., Moriwaki, Y. (2001). “Equivalent number of

uniform stress cycles for soil liquefaction analysis,” Journal of Geotechnical & Geoenvironmental Engineering, ASCE, 127 (12), in press.

Seed, H.B. and Lee, K.L. (1966). “Liquefaction of Saturated Sands during Cyclic Loading,”

Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 92, No. SM6, pp. 105-134.

Youd, T. L., Hansen, C., and Bartlett, S (1999). "Revised MLR Equations for Predicting Lateral

Spread Displacement," 7th U.S. - Japan Workshop on Earthquake Resistant Design of Lifeline Facilities and Countermeasures Against Soil Liquefaction, August, Seattle, WA, T.D. O'Rourke, J.P. Bardet, M. Hamada, (Eds.), Technical Report MCEER-99-0019, Multidiciplinary Center for Earthquake Engineering Research, 99-114.

Youd, T.L., Idriss, I.M., Andrus, R.D., Arango, I., Castro, G., Christian, J.T., Dobry, R., Finn,

W.D.L., Harder, L.F., Hynes, M.E., Ishihara, K., Koester, J.P., Liao, S.S.C., Marcuson, W.F. III, Martin, G.R., Mitchell, J.K., Moriwaki, Y., Power, M.S., Robertson, P.K., Seed, R.B., and Stokow, K.H. II (2001). “Liquefaction resistance of soils: Summary report from the 1996 NCEER and 1998 NCEER/NSF workshops on evaluation of liquefaction resistance of soils,” Journal of Geotechnical and Geoenvironmental Engineering, Vol. 127, No. 10, pp. 817-833.

IntroductionLiquefaction and PerformancePrevious ApproachesLiquefaction PotentialNumber of Cycles to Liquefaction

Simulation of Lateral SpreadingEffect of Soil DensityEffect of Liquefiable Layer Thickness and DepthAn Equivalent Thickness Parameter

Summary and ConclusionsReferences