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8 3.1 General One-dimensional effective-stress SRA is usually preceded by at least one of two types of total stress analysis (TSA): (i) equivalent-linear analysis, or (ii) (total-stress) nonlinear analysis. The equivalent-linear analysis is used because (i) it is the simplest and most validated type of SRA available, so the potential for errors associated with model assembly is the lowest, and (ii) it is based on a correct solution for viscous damping and therefore may be used to fine-tune6 the parameters of viscous damping models that are inherent to nonlinear programs. Once the non- linear SRA model is assembled and checked,7 it becomes a reference case for the evaluation of the effects of PWP generation and dissipation on the site response. Most soil deposits consist of alternating layers of sand, clay, and silt. Sands are rarely clean, and groundwater is rarely at the ground surface. Therefore, in ESA, only fully saturated layers of sand, silty sand, and sometimes silt need to be modeled with effective-stress constitutive models. Other layers can be modeled using simple, nonlinear total-stress constitutive models with param- eters derived from available modulus reduction and damping curves, or even as an elastoplastic material with moduli developed from geophysical measurements and strength evaluated from the results of some type of in-situ test, including a standard penetration test (SPT) or CPT. The theoretical background required for understanding total-stress equivalent-linear and nonlinear analysis is outlined in Ishihara (1996) and Towhata (2008) and is presented in greater detail in Matasovic and Hashash (2011; 2012). The following sections build on Matasovic and Hashash (2011) and expand the content as appropriate for basic understanding and application of 1D nonlinear analysis with PWP generation and dissipation. Due to the relative complexity of nonlinear analysis, the explanation is provided separately for the dynamic response model of the site and for the constitutive models that simulate the cyclic response of soils in the profile. 3.2 Equivalent-Linear Analysis The equivalent-linear model, referred to herein as the S-I model, was first introduced by Seed and Idriss (1970), while the equivalent-linear approach to total-stress site response analysis was first introduced by Schnabel et al. (1972), and it has remained substantially the same since. The soil properties needed for equivalent-linear site response analysis are shear wave velocity Vs, mass density Ï, modulus reduction and damping (curves), and, for computational purposes, an initial estimate of damping. The analysis is based on a closed-form solution of the wave propagation equation. To generate suitable input for a closed-form solution, the input ground motion needs to be converted into a series of sinusoidal motions of different frequencies within the program using a fast Fourier transformation (FFT) algorithm. The calculations are performed for each frequency, and hence these analyses are commonly referred to as occurring in the frequency domain.8 C H A P T E R 3 Site Response Analysis â Theoretical Background
Site Response Analysis â Theoretical Background 9  Soil nonlinearity is accounted for by modulus reduction and damping curves. Modulus reduc- tion and damping curves are developed from the results of drained testing, including cyclic direct simple shear (CyDSS), cyclic triaxial (CyTX), resonant column (RC)/torsional shear, and other testing methods, or by in-situ testing (modulus reduction only; see Section 6.2.4 and Figure 6-3). Ideally, the results of in-situ measurement of shear wave velocity and unit weight, interpreted in terms of low-strain shear modulus (Gmax), should match results of laboratory testing in the low strain range (especially of the RC testing). Mismatch is often an indication of sample disturbance. There are generic modulus reduction and damping curves that can be selected from technical literature based on known soil index properties, over-consolidation ratios (OCRs), or confining stress. Examples of generic modulus reduction and damping curves that are a function of the soil plasticity index (PI) are shown in Figure 3-1 (modified from Vucetic and Dobry, 1991). Nowadays, there are modulus reduction and damping curves available for almost any type of idealized natural soil or artificial soil-like material, such as mine tailings or municipal solid waste, and for confining stresses of up to 1,600 kPa (approximately up to 85 m of overburden). Examples of stress-dependent curves can be found in EPRI (1993), Darendeli (2001), and Menq (2003). The main source of uncertainty related to modulus reduction and damping curves is a lack of information in the very low (i.e., up to 0.001%; damping only) and the high (i.e., beyond, say, 0.5% to 1%) strain ranges. Most damping curves are not plotted below 0.001%, and both modulus reduction and damping curves are not presented beyond 0.8% to 1.0% shear strain, including the Vucetic and Dobry (1991) curves shown in Figure 3-1. The broad availability of modulus reduction and damping data is one of the main reasons that equivalent-linear analysis remains the most popular type of site response analysis. The simplicity of use of available software, the stability (or apparent stability) of numerical modeling solutions, and user familiarity with the input parameters are also important considerations. Modulus reduction and damping curves may be directly entered into a program as a series of discrete values or selected from a database of such curves accompanying most equivalent-linear programs. Because there is usually no constitutive law involved (modulus reduction and damp- ing curves are, in most cases,9 developed by nonlinear regression of data from multiple sources or by engineering judgment; see, e.g., Figure 6-3 showing WLA modulus reduction and damp- ing curves for silty sand), they do not qualify to be called âconstitutive models.â However, for practical reasons, especially when parameters of nonlinear constitutive models are developed Figure 3-1. Generic modulus reduction and damping curves (modified from Vucetic and Dobry, 1991).
10 Seismic Site Response Analysis with Pore Water Pressure Generation: Guidelines by fitting of these curves, they are referred to as the âSeedâIdrissâ constitutive model and are abbreviated in the balance of this study as the S-I constitutive model. Equivalent-linear analysis is used in this study as the reference analysis for validation of non- linear models (see Section 7.2.3.2) and, when required, for fine-tuning of the viscous damping model parameters. These parameters should be representative of a broad range of frequencies, as required by the nonlinear software used for this study and discussed in the next section and further in Section 7.2.3). Suggestions have also been made on how to use this type of analysis to evaluate the results of more complicated nonlinear and nonlinear with PWP generation analyses. Both suggested uses of equivalent-linear analysis are subject to certain limitations (e.g., a maxi- mum shear strain that can be handled in equivalent-linear analysis of approximately 1.0%) that are further explained later in this report. 3.3 Nonlinear Analysis â Dynamic Response Model Nonlinear site response analysis originated in structural dynamics. The first nonlinear models were modified structural models of a kind used to calculate the response of moment-resisting frames or of multi-degree-of-freedom oscillators. As such, these programs were written to solve a dynamic equation of motion (see, e.g., Newmark and Rosenblueth, 1971) in the time domain. The dynamic equation of motion is commonly written as: u uM C K u f t+ + = -p c ` j8 8 8B B B$ $ $. . . (Equation 1) where [M], [C], and [K] are the mass matrix, viscous damping matrix, and (nonlinear) stiffness matrix, respectively; {u}, {ů}, and {ü} are the displacements, velocities, and accelerations of the mass [M] relative to the base; and f(t) = {üg} is a forcing function (vector) that is a proxy for the acceleration of the base. In order to develop elements of matrices [M], [C], and [K] (diagonal only in 1D analyses) and further solve the dynamic equation of motion for displacement, velocity, and acceleration (histories), it is necessary to discretize the domain of interest, which is herein a soil column. There are several options for discretization, including lumped-mass discretization, finite element method (FEM) discretization (mass can either be distributed over layer thickness or lumped), and finite difference method (FDM) grid10 discretization. The thickness of the layers in both lumped-mass and FEM/FDM discretization has an impact on calculated site response. The layer thickness determines the maximum and minimum fre- quency that can be propagated through a soil column. If the layer is too thick, higher frequency components of the input ground motion may be filtered, and thus the calculated ground response may be underestimated. The maximum frequency ( fmax) is the highest frequency that the layer can propagate. For a layer i, it may be calculated as: f T h1 4max mini i s ii= = V` j (Equation 2) where Timin is the minimum period of a given layer i. Assuming fmax = 25 Hz (a common assump- tion for the Western United States, WEUS), the maximum layer thickness in the WEUS is cal- culated as: h 100WEUS maxi s i = V` ` `j j j (Equation 3)
Site Response Analysis â Theoretical Background 11  For the Central and Eastern United States (CEUS), an assumption of fmax = 50 Hz is commonly used, and the maximum layer thickness is calculated as: h 002EUSC maxi s i = V` ` `j j j (Equation 4) There are no requirements for a minimum layer thickness. However, in practice, most engi- neers use the minimum layer thickness equal to the standard resolution of geophysical sounding [0.5 m for OYO Corporation suspension logging and 1.5 m for seismic CPT (sCPT); see, e.g., Appendix B-1]. Figure 3-2 schematically shows a soil deposit represented in 1D. The deposit is discretized in n layers (layers i = 1 through i = n) of thickness hi in accordance with the principles explained previously. In this example of soil profile discretization, with the exception of the top and bottom layers, layer mass m is lumped at the layer interfaces. Soil stiffness is represented by a hypothetical spring placed in the middle of each soil layer. In 1D analysis, an assembly of these springs repre- sents a diagonal of the stiffness matrix [K]. Soil springs can be representative of a linear-elastic material, but also of nonlinear material with PWP generation, as explained in Section 3.4 (Nonlinear Analysis â Constitutive Models). A separate model is required for calculation of PWP dissipation. This separate model typically requires generation of a finer mesh (i.e., with a mesh that typically has more layers than its counter part used to account for soil layering and stiffness). In principle, damping in the soil can be captured through the hysteretic loops of the consti- tutive model used to model cyclic soil behavior (i.e., by the stiffness term [K] of Equation 1). However, viscous damping (term [C] in Equation 1) is also required. This is because (i) viscous damping is a necessary input into a linear-elastic analysis that is an integral part of all time- domain nonlinear site response analysis programs, (ii) viscous damping increases the numerical stability of several numerical schemes used to solve Equation 1 across the full duration of shaking, and (iii) constitutive models tend to underpredict soil damping at low strains (i.e., strains less than 0.01%). Viscous damping can be entered directly into most nonlinear site response analysis pro- grams as a coefficient c evaluated based on engineering judgment. However, viscous damping Figure 3-2. Damped lumped-mass stiffness system with transmitting boundary (Sample 1D Dynamic Response Model) (Matasovic, 1993).
12 Seismic Site Response Analysis with Pore Water Pressure Generation: Guidelines is frequency dependent (i.e., it changes during shaking, especially when soil is modeled by non- linear soil models with PWP generation and dissipation where soil period is reduced as shaking progresses), and therefore, a better way to evaluate it is by means of an appropriate model. The most commonly used model for assigning viscous damping into site response analysis programs is the Rayleigh damping model (Rayleigh and Lindsay, 1945). The Rayleigh damping model assumes that viscous damping is both mass and stiffness proportional and, therefore, can be expressed as: c m kR Ra b= + (Equation 5) where c is the Rayleigh damping, αR and βR are the Rayleigh damping coefficients, and m and k are diagonal elements of the mass and stiffness matrices, respectively. The Rayleigh damping coefficients can be evaluated as: T n n4 1R star :a p r= +a `k j R T SS V X WW (Equation 6) T n1R star :b p r= +` `j j R T SS V X WW (Equation 7) where Ts is the predominant (i.e., fundamental) period of soil deposit, n = an odd integer (1, 3, 5, . . . 11), and ξtar is the target damping ratio (typically between 0.5 and 5%). If both αR and βR are used (i.e., required as input in software), the damping formulation is both mass and stiffness proportional and is referred to as the âfullâ Rayleigh damping. If one of those coefficients is set to zero, the damping formulation is referred to as the âsimplifiedâ Rayleigh damping. Figure 3-3 illustrates how, under various assumptions, Rayleigh damping changes with fre- quency f. When the full formulation is used, ξtar is matched at two prespecified frequencies (typically at fs = 1/Ts and 5 à fs) and remains close to the target frequency in between. If the simplified formulation is used, the system may be overdamped (OD) or underdamped (UD) over a relatively broad range of frequencies. Based on back analysis of several total-stress case histories, Kwok et al. (2007) developed guid- ance on what target damping ratio should be selected (answer: small strain material damping) and what the matching frequencies should be (answer: fs and, for the first approximation, 5fs). Figure 3-3. Schematic illustration of viscous damping ratio change with frequency.
Site Response Analysis â Theoretical Background 13  Once information on soil deposit layering, groundwater elevation, mass density, stiffness, and viscous damping has been evaluated, matrices [M], [C], and [K] can be assembled, and Equation 1 can be solved. Equation 1 (i.e., the dynamic equation of motion) can be solved in several ways, including directly by numerical integration and numerically by means of the FEM or FDM. In any case, the solution of Equation 1 calls for temporal discretization (i.e., the system of coupled equations is discretized temporally as a series of discrete timesteps) and solved on a step-by-step basis. Because calculations are performed for each timestep, this type of calculation is referred to as a âsolution in time domain.â Many programs offer a choice of up to 25 numerical integration schemes (e.g., the program OpenSees), but provide little guidance on which one to choose for a given application. 3.4 Nonlinear Analysis â Constitutive Models 3.4.1 Total-Stress Constitutive Models The total-stress CMs, here referred to as the CM sub-models (or sub-CMs), include (i) simple stressâstrain models intended for a stand-alone use [examples of such models are the linear- elastic MohrâCoulomb (LE-MC) model and the Darendeli (2001) model (updated/expanded in 2022 as the Wang and Stokoe, 2022 model)]; (ii) simple stressâstrain models that have been com- bined with semi-empirical PWP generation models such as the Pyke (1979) model and the MKZ model (Matasovic, 1993; Matasovic and Vucetic, 1993); and (iii) simple stressâstrain models that can be invoked by running advanced constitutive models in a total-stress mode. Advanced CMs that are based on the theory of plasticity are explained later. In its basic form, simple stressâstrain models and more advanced semi-empirical models describe the behavior of the initial loading curve, as indicated in Figure 3-4 by a solid blue line. By reversing signs along the abscissa and ordinate, this curve can be extended to the opposite side of the chart. Such an extended curve, indicated in Figure 3-4 by a dashed green line, is referred to as a backbone curve. max Figure 3-4. Initial loading and backbone curve.
14 Seismic Site Response Analysis with Pore Water Pressure Generation: Guidelines Cyclic loading and reloading can be accounted for by applying rules for loading and reload- ing that are commonly referred to as the Masing rules (Masing, 1926). The Masing rules were originally developed for metals (e.g., for brass). They were modified to more accurately simulate cyclic response of sandy and clayey soils under a variety of loading and unloading conditions. Examples of such modifications can be found in Pyke (1979), Vucetic (1990), and Phillips and Hashash (2009). The latter modification of Masing rules compensates for calculated overestima- tion of hysteretic damping at large shear strains. The cyclic total-stress stressâstrain relationship (i.e., a simple total-stress constitutive model) can be coded in a spreadsheet. The coded equations can be manipulated such that data are plotted in the form of modulus reduction curves. The corresponding damping curves can be approxi- mated [i.e., calculated by the Jacobsen 1929 equation (see Ishihara, 1996), which approximates actual cyclic loops with an ellipse]. Relevant equations and example calculations are presented in Matasovic and Vucetic (1993), Darendeli (2001), Khosravifar et al. (2018), and Wang and Stokoe (2022). Simple total-stress constitutive models are, in some form, coded in all nonlinear programs. 3.4.2 Effective-Stress Constitutive Models 3.4.2.1 General Cyclic loading of saturated soils is, beyond the volumetric threshold cyclic shear strain γtvp, accompanied by PWP change. The typical values of γtvp for saturated sand, as evaluated and reported by Vucetic (1994), is approximately 0.01%. If, in a given soil layer, the excess PWPs are sufficiently large, the soil stiffness and strength can be significantly reduced. Large reductions in soil stiffness and strength accompany soil liquefaction. In a complete nonlinear effective-stress site response analysis, the response of the soil to cyclic loading includes calculation of excess PWP during cyclic shearing of the soil as well as calculation of dissipation of this excess PWP during and after the cessation of shearing. In some software, excess PWP generation is a feature of a comprehensive constitutive model, while PWP dissipa- tion or redistribution (e.g., redistribution between sand and clay layers) is a separate feature. In some software, a separate PWP dissipation model, based on Terzaghiâs theory of consolidation, is incorporated. There are two general classes of effective-stress constitutive models: (i) semi-empirical models that are coupled with total-stress stressâstrain models with a set of rules, and (ii) advanced effective-stress models whereby the model formulation is in terms of effective-stress, and the PWP change is computed as the change between total stresses (or loads) and effective stresses via the soil constitutive model. In other words, these models couple volumetric strain of the soil skeleton (i.e., dilation/contraction) with shear strains, resulting in the organic development of excess PWP. The terminology used to distinguish between PWP generation and dissipation model types and input types is, unfortunately, software dependent. Terms used in software employed in this study include âexternalâ versus âinternal, âexogenousâ versus âendogenous,â âinorganicâ versus âorganic,â and âdecoupledâ versus âcoupled.â Explanations are provided in respective software manuals or white papers that accompany the software. 3.4.2.2 Semi-Empirical Models In this class of constitutive models, PWP generation and associated effects, such as degrada- tion of soil stiffness and strength due to excess PWP buildup, are computed independently of the stressâstrain model. At the beginning of shaking (i.e., at time t = 0), stressâstrain relationships of the soil are identical to that of the total-stress models since PWP is zero. As shaking progresses, excess PWP is generated. As excess PWP is generated, soil strength and stiffness are degraded. This degradation is accounted for by a set of rules.
Site Response Analysis â Theoretical Background 15  Several rules for soil strength and stiffness degradation due to PWP buildup are available. These rules include the degradation index δ, which is an empirical parameter proposed by Idriss et al. (1978). The intensity of this parameter can be evaluated by laboratory testing and can be incorporated into stressâstrain models to (indirectly and in a very simplified manner) account for PWP buildup. Nowadays, δ is used to degrade the strength and stiffness of soft clays. An extended version of the degradation index parameter, with δ as a function of PWP, can be used to couple semi-empirical PWP models with stressâstrain models. Different formulations can be developed for degradation of strength and stiffness. An extended version of the degrada- tion index parameter is presented for sand and silty sand in Matasovic (1993) and Matasovic and Vucetic (1993). A version of this parameter for soft clays is presented in Matasovic and Vucetic (1992) and Matasovic and Vucetic (1995a). Several semi-empirical models for PWP generation have been proposed, starting with the pioneering work by Martin et al. (1975), and later by Martin and Seed (1979) and Dobry et al. (1985). The last two models were developed based on strain-controlled CyDSS and CyTX testing. The Dobry et al. (1985) model was later modified by Vucetic (1986) to allow for quasi- two-dimensional shaking effects and γtvp. Matasovic (1993) modified and incorporated the Vucetic (1986) version of the Dobryâs model into the more comprehensive semi-empirical MKZ model (Matasovic, 1993; Matasovic and Vucetic, 1993) that is one of the constitutive models used in this study. The effect of cyclic degradation on soil stiffness and strength is illustrated in Figure 3-5 using the MKZ constitutive model (Matasovic, 1993; Matasovic and Vucetic, 1993). The initial back- bone curve is the basis for development of the initial hysteresis loop. As cyclic shear strain Figure 3-5. Schematic representation of cyclic stressâstrain response with reduction in strength and stiffness due to PWP buildup (Matasovic, 1993).
16 Seismic Site Response Analysis with Pore Water Pressure Generation: Guidelines increases beyond γtvp, generation of excess PWP is initiated. Excess PWP degrades both soil strength and stiffness, as schematically illustrated in Figure 3-5. The dynamic equation of motion (Equation 1) is solved on a step-by-step basis, with the corresponding parameter used in calcu- lations. This includes both tangent and secant shear moduli at time t (Gmt and Gst, respectively) and (degraded) cyclic shear stress at time t (Ïct). Ultimately, Ïct is reduced to the residual shear strength of liquefied soil. The class of PWP generation models outlined here requires evaluation and use of an equivalent number of cycles to represent earthquake shaking. The equivalent number of cycles is evaluated internally by the program using yet another set of rules and, in some cases, additional parameters (e.g., threshold shear strain is used in the MKZ model; correction factors for initial static stresses in the Roth/Dames & Moore model; Dawson and Mejia, 2012). To eliminate the potential draw- back of cycle counting, Polito et al. (2008) proposed an energy-based model (known as the GMP model) for the generation of PWP based on a large number of advanced laboratory tests. There are three advantages of this class of semi-empirical effective-stress CMs: (i) with respect to their counterparts developed based on the theory of plasticity, they are easier to understand and use, (ii) generic sets of parameters are available (which can be selected based on grain size distribution and soil index properties), and (iii) parameters of PWP models can be developed and checked independently of stressâstrain model parameters (or overall advanced CM parameters), which facilitates their use and quality control of input data. Semi-empirical models, however, cannot simulate within-cycle dilation,11 which has been repeatedly observed to produce spikes in ground motions. These spikes often control high-frequency ground motions in liquefiable soils, and ignoring their development introduces potentially significant underprediction bias. 3.4.2.3 Advanced Models Advanced CMs are developed based on the theory of plasticity and are, therefore, commonly referred to as plasticity-based models. This class of CMs was originally proposed for clays by Roscoe et al. (1963) and was further developed by Dafalias and Popov (1975; 1976) and Krieg (1975) to model the effects of cyclic loading on metal specimens. It was later adapted for model- ing soil behavior by Mróz et al. (1979), and for cyclic behavior of sand by Aboim and Roth (1982), Herrmann et al. (1982), Dafalias (1986), and Pestana (1994). An in-depth review of these constitutive models is provided in Witthoeft (2009). Modern theory of plasticity-based formulations of the CMs is developed in 2D or 3D effective- stress space. Excess PWPs are computed as the difference between effective stresses and total stresses in the domain of interest. Basic principles are outlined and explained in Potts and ZdravkoviÄ (1999) and further, with an explanation of the bounding surface models and their application in site response analysis, in Borja et al. (2002). An example of a modern plasticity- based bounding surface CM can be found in Elgamal et al. (2001). This and other modern CMs are capable of simulating complex soil behavior under a variety of loading conditions. Key ele- ments of these models include yield surfaces, flow rules, and hardening (or softening) rules. They can also simulate dilation. Three theories of plasticity-based CMs are used in this study. All three fall within the class of multi-yield surface models, and two also fall within the class of bounding surface models. Models with two yield surfaces work well when subjected to sinusoidal excitations, but it is not clear how well they work when subjected to irregular excitations such as accelerogram-induced stress or strain histories. A common feature of the bounding surfaceâtype model is a dilatancy surface, which provides for both dilation and contraction. The most important feature is con- traction (i.e., the feature responsible for the generation of excess PWP). Dilation is generally a secondary effect.
Site Response Analysis â Theoretical Background 17  There are three advantages of this class of effective-stress CMs: (i) physical interpretation of the real-world phenomenon is, arguably, more accurate, (ii) the models can simulate dila- tion, and (iii) once set for 1D applications and properly calibrated (using element tests; see Sec- tion 7.2.2), they can be used for 2D or 3D applications. Disadvantages include that (i) they are relatively difficult to understand and use, (ii) they can be difficult to validate (an element test is required for validation), and (iii) for soils other than clean sands (e.g., Ottawa sand), the full set of generic material parameters is often not available or is indirectly evaluated by converting normalized/standardized SPT blow counts [(N1)60] to their clean sand equivalent [(N1)60âCS] and then using correlations between the necessary properties and (N1)60âCS. The latter is implemented in one of the CMs used herein (PM4SAND; Boulanger and Ziotopoulou, 2013a; 2013b). For many other models, evaluation of material parameters requires significant specialty expertise and interpretation of the results of advanced laboratory testing. 3.5 Nonlinear Analysis â Other Models Generation of seismically induced PWP (i.e., excess PWP) in different saturated soils12 occurs at different rates and intensities within a soil profile. Therefore, soon after the onset of shaking, there is a difference in hydraulic gradients within the profile. To reach an equilibrium, redistribution of PWP occurs. The rate of this redistribution is a function of material hydraulic properties. Redistribution of excess PWP occurs simultaneously with their generation. In 1D SRA analysis, PWP redistribution is often treated as a diffusion problem and is, in many cases, accounted for by solving Terzaghiâs differential equation of consolidation by means of the finite difference method. An early model for PWP dissipation and redistribution between saturated sand layers was introduced by Martin and Seed (1979). Input parameters include (saturated) hydraulic conduc- tivity and fitting parameters of effective stressâdependent (constrained) rebound modulus of the soil. The rebound modulus of soil is, like saturated hydraulic conductivity, evaluated by specialty (advanced) laboratory testing. For most practical applications, parameters of the effective stressâ dependent rebound modulus of soil model can be evaluated from tabulated values provided by the authors for a range of relative densities (Dr) of clean sand. The Martin and Seed (1979) model has limited application in soil profiles with alternating layers of highly permeable soils (e.g., sand or silty sand) and soils of low hydraulic conductiv- ity (e.g., clay or silt). (Such profiles are referred to as composite soil profiles.) This is because the Martin and Seed (1979) model is effective stressâdependent and hence is not applicable to saturated soils of low hydraulic conductivity, where use of a constant value of rebound modulus is appropriate. A numerical solution for accounting for a PWP redistribution within composite soil deposit is presented in detail in Matasovic (1993) and in an abbreviated form in Matasovic and Vucetic (1995b). Terzaghiâs differential equation of consolidation is a 1D model. Therefore, in 2D software, the effects of simultaneous PWP generation and distribution are addressed through a relatively complex mechanism of coupled fluid deformation that is based on the Biot theory of consolida- tion.13 However, details on how excess PWP is generated during local shear response are vague or are not provided at all in most 2D and 3D software. For example, Laera and Brinkgreve (2015) describe the methodology used for the calculation of coupled flow, but when this method- ology is used with the generalized hardening soil model (GHS, a variant of HSsmall constitutive model; see Section 4.3.4) or with UBCSAND (Beaty and Byrne, 1998; 2011), it is not clear that the solution simultaneously involves dynamic flow, coupled flow, and these advanced constitu- tive models.
18 Seismic Site Response Analysis with Pore Water Pressure Generation: Guidelines 3.6 Development of Model Parameters Documentation for most CMs includes tables of generic material parameters. For advanced CMs, two generic parameter tables are usually provided, one for stress-controlled testing and another one for strain-controlled testing. The models are set up to establish the modulus reduc- tion and damping (MRD) behavior first, including a provision to adjust the parameters to be compatible with standard MRD relations from the literature. A spreadsheet-type tool is used to compare the standard (or developed from measurements) MRD curves to those generated by the CM. Spreadsheets and other tools are available to further compare measured and calcu- lated histories, including stressâstrain and PWP response. Such tools are available for download and include those provided by Matasovic (2018), Khosravifar et al. (2018), and Boulanger et al. (2022) (Python script). Tabulated values of CM parameters are available for sands of various grain size distribution curves and relative densities (MKZ); loose sand, medium-dense sand, dense sand, and very dense sand (UCSDSAND3); and clean sands of various relative densities (PM4SAND). Relative density is often not available from the results of field investigations. Therefore, it is indirectly evaluated by converting normalized/standardized SPT blow counts (N1)60 to their clean sand equivalent (N1)60âCS and then using correlations between the necessary properties and (N1)60âCS. For some CMs (not used herein), evaluation of material parameters requires significant specialty expertise and results of advanced laboratory testing.