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4th IASPEI / IAEE International Symposium:

Effects of Surface Geology on Seismic Motion August 2326, 2011 University of California Santa Barbara

NUMERICAL MODELING OF LIQUEFACTION EFFECTS

Ross W. Boulanger Ronnie Kamai Katerina Ziotopoulou University of California University of California University of California Davis, CA 95616 Davis, CA 95616 Davis, CA 95616 USA USA USA ABSTRACT Development and initial applications of a sand plasticity model for nonlinear seismic deformation analyses involving liquefaction are described. The model follows the basic framework of the stress-ratio controlled, critical state compatible, bounding surface plasticity model for sand presented by Dafalias and Manzari (2004). Modifications were implemented to improve its ability to approximate stress strain responses important to geotechnical earthquake engineering applications. Initial applications of the model have included one-dimensional site response analyses of down-hole arrays where liquefaction was triggered and two-dimensional analyses of dynamic centrifuge model tests involving liquefaction and lateral spreading. An overview of the sand model is presented, followed by example results from these initial validation studies.

INTRODUCTION Numerical modeling of liquefaction effects is performed in practice and research using a wide range of constitutive models and numerical procedures. The constitutive models vary in complexity, depending on which aspects and details of liquefaction behavior they are intended to approximate. For example, there are uncoupled total-stress models that track cyclic stresses for determining when to trigger liquefaction and impose a residual strength; these types of phenomenological models may be useful for evaluating how liquefaction-induced strength loss affects stability or deformations of a structure, but are not capable of simulating realistic stress-strain behaviors and thus cannot generate responses which arise from those behaviors. Coupled, effective-stress models can provide more realistic simulations of stress-strain behaviors, but they often require considerably more engineering effort for calibration and use. Simulation results can also be significantly affected by the details of the numerical procedures, such as the specification of boundary conditions, input motions, numerical damping, pore water flow, and mesh geometry updating. Consequently, there is a need in practice to further develop constitutive models that provide realistic approximations of soil behavior while being easy to use, and to develop more standardized protocols for the calibration and validation of constitutive models and numerical modeling procedures. In this paper, the development and initial applications of a sand plasticity model for numerical modeling of liquefaction effects are described. The sand plasticity model, PM4-Sand, is a stress-ratio controlled, critical state compatible, bounding surface plasticity model that follows the basic framework by Dafalias and Manzari (2004) with modifications that improve its ability to approximate stress strain responses important to geotechnical earthquake engineering applications. The motivation behind the model, its basic formulation, and examples of its constitutive responses are presented. Initial applications of the PM4-Sand model have included one-dimensional site response analyses of down-hole arrays where liquefaction was triggered and two-dimensional analyses of dynamic centrifuge model tests involving liquefaction and lateral spreading. Example results are presented from the site response analyses of the Wildlife Liquefaction Array during the 1987 Superstition Hills Earthquake and from two-dimensional analysis of a dynamic centrifuge model test by Kamai et al. (2008). PM4-SAND PLASTICITY MODEL Development of the PM4-Sand model was guided by the need in practice for models that can reasonably approximate, and be quickly

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calibrated to, the engineering design relationships that are commonly used for estimating the stress-strain behaviors of liquefiable soils. Stress-strain behaviors that can be important for applications include the following items.

The cyclic resistance ratio (CRR) against triggering of liquefaction, which is commonly estimated based on SPT and CPT penetration resistances with case-history-based liquefaction correlations. The CRR is the cyclic stress ratio (e.g., CSR =

cyc/ 'vc, with cyc = horizontal cyclic shear stress, 'vc = vertical consolidation stress) that is required to trigger liquefaction in a specified number of equivalent uniform loading cycles.

The response under the irregular cyclic loading histories produced by earthquakes, which is approximately represented by the relationship between CRR and number of equivalent uniform loading cycles. This aspect of behavior also directly relates to the magnitude scaling factors (MSF) that are used with liquefaction correlations in practice.

The dependence of CRR on effective confining stresses and sustained static shear stresses. These aspects of behavior are represented by the K and K correction factors, respectively, that are used with liquefaction correlations in practice.

The accumulation of shear strains after triggering of liquefaction. Evaluations of reasonable behavior are often based on comparisons to laboratory tests results for similar soils in the literature.

The strength loss as a consequence of liquefaction, which may involve explicitly modeling phenomena such as void redistribution or empirically accounting for it through case history-based residual strength correlations.

The small-strain shear modulus which can be obtained through in-situ shear wave velocity measurements. The shear modulus reduction and equivalent damping ratio relationships prior to triggering of liquefaction. These aspects of

behavior are commonly estimated using empirical correlations derived from laboratory test results for similar soils. The drained and undrained monotonic shear strengths, which may be estimated using correlations to SPT and CPT

penetration resistances. The volumetric strains during drained cyclic loading or due to reconsolidation following triggering of liquefaction, both of

which may be estimated using empirical correlations derived from laboratory test results for similar soils. The utility of a nonlinear soil model in practice is dependent on: (1) its ability to approximate the above behaviors over a broad range of conditions and, (2) the level of engineering effort required for calibrating the model. For example, a single geotechnical structure can have strata or zones of sand ranging from very loose to dense under a wide range of confining stresses, drainage conditions, and loading conditions, such that the engineering effort is greatly reduced if the constitutive model can reasonably approximate stress-strain behaviors under all these conditions. If the model cannot approximate the trends across all these conditions, then extra engineering effort is required by the need to decide what behaviors should be prioritized in the calibration process, and sometimes by the need to repeat the calibrations for the effects of different initial conditions within the same geotechnical structure. Model Formulation The PM4-Sand model follows the basic framework of the stress-ratio controlled, critical state compatible, bounding-surface plasticity model for sand presented by Dafalias and Manzari (2004). On a q-p space, the model has a narrow stress-ratio based elastic cone and three other key surfaces the bounding, dilation and critical-state surfaces (Fig. 1). The locations of the dilation and bounding surfaces are dependent on the state of the soil (Fig. 2), such that they move towards the critical state surface in a scissor-like movement when the soil is sheared towards critical state, with all three surfaces coinciding when the soil reaches critical state. A series of modifications and additions to the model were incorporated to improve its ability to approximate the trends observed in empirical correlations commonly used in practice. These constitutive modifications included:

revising the fabric formation/destruction to depend on plastic shear rather than plastic volumetric strains; adding fabric history and cumulative fabric formation terms; modifying the plastic modulus relationship and making it dependent on fabric; modifying the dilatancy relationships to provide more distinct control of volumetric contraction versus expansion behavior; adding sedimentation effects for improved estimation of reconsolidation strains following liquefaction; providing a constraint on the dilatancy during volumetric expansion so that it is consistent with Boltons (1986) dilatancy

relationship; modifying the elastic modulus relationship to include dependence on stress ratio and fabric history; modifying the logic for tracking previous initial back-stress ratios (i.e., loading history effect); recasting the critical state framework to be in terms of a relative state parameter index (i.e., R in Fig. 2) rather than a state

parameter; simplifying the formulation by restraining it to plane strain without Lode angle dependency for the bounding and dilation

surfaces; and providing default values for all but three primary input parameters.

The constitutive equations and model behaviors are provided in Boulanger (2010).

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There are three primary parameters that are most important for model calibration, and a secondary set of 17 parameters that may be modified from their default values in special circumstances. The three primary model input parameters are: the shear modulus coefficient, Go, which should