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13 damping is equal to the total damping measured for the given leigh damping frequencies, so this frequency-independent soil type. Historically, the important role of viscous damp- approach eliminates a potentially confusing step in input ing in site response analysis was not well understood; it was development. thought that this was mostly needed for numerical stability. This assumption led to significant confusion in the way it was employed and, in some cases, led to unrealistic results in nonlinear site response analysis resulting from either over or under damping. Viscous damping represents soil damping at a very small strain, so its value is generally small, typically in the range of 0.5% to 5%. It can be directly obtained from the intercept of the damping curve with the vertical axis in the damping versus shear strain curve. The most commonly used formulation for evaluation of viscous damping is Rayleigh damping. The Rayleigh damp- ing is frequency dependent and can be evaluated as: c = R m + R k(2) FIGURE 8 Schematic illustration of viscous damping change with frequency (after Park and Hashash 2004). where R and R are the Rayleigh damping coefficients (Rayleigh and Lindsay 1945) and m and k are elements of the mass and stiffness matrices, respectively. NONLINEAR SITE RESPONSE ANALYSIS WITH PORE WATER PRESSURE CHANGE Figure 8 illustrates how Rayleigh damping, expressed through c, changes with frequency. The viscous damp- The cyclic loading of saturated soils is accompanied by pore ing ratio can be brought closer to a constant value of the water pressure (pwp) generation and dissipation. If the gen- target damping ratio (tar) by specifying c at only one erated pore water pressures are sufficiently large, the soil frequency (e.g., at f 2 in Figure 8), which is termed the stiffness and strength are significantly reduced and ulti- simplified Rayleigh damping formulation; at two frequen- mately, in some soils, liquefaction can occur. In nonlinear cies (at f 1 and f 2), which is termed the full Rayleigh damp- site response analysis with pwp generation, the response ing formulation (Hudson 1994); and at four frequencies of the soil to cyclic loading accounts for the generation of (at f 1 through f4), which is termed the extended Rayleigh excess pwp during cyclic shearing of the soil as well as dis- damping formulation (Clough and Penzein 1993; Park and sipation of these excess pore water pressures during and Hashash 2004). Park and Hashash (2004) have shown that after the cyclic loading. The representation of dissipation/ the use of simplified Rayleigh damping results in signifi- redistribution of pwp influences soil stiffness (modulus) cant errors and the extended Rayleigh is computationally and strength (shear stress) during shaking, which results in expensive; hence, they suggest the use of full Rayleigh a more realistic simulation of site response. The pwp dis- damping formulation. Kwok et al. (2007) recommended sipation/redistribution is discussed in a later section. This use of the full Rayleigh damping formulation in nonlinear section discusses pwp generation. (total stress) site response analysis whereby the first fre- quency is equal to the fundamental frequency of the soil The influence of pwp changes during cyclic loading is column, and the second frequency is equal to 5 times the incorporated in soil constitutive modeling in two ways: (1) fundamental frequency. Full Rayleigh damping is avail- semi-empirical pwp generation models used in combination able in a number of software, including ABAQUS, Cyber- with total stress soil models; and (2) effective-stress models Quake (Modaressi and Foerster 2000), D-MOD2000, whereby the pwp change is computed as the change between DEEPSOIL, FLAC, OpenSees, SIREN (Oasys 2006), LS- total stresses (or loads) and effective stresses, computed DYNA (LSTC 1988). through the soil constitutive model. Philips and Hashash (2009) introduced a new viscous Semi-Empirical Pore Water Pressure Generation Models damping formulation that is independent for frequencies, which is more consistent with the current understanding of In this class of models, pwp generation is calculated using soil response within the seismic frequency range of inter- semi-empirical models. At the beginning of shaking (i.e., at est (Park and Hashash 2008). This formulation, used in time t = 0), stress-strain relationships of the soil are identi- DEEPSOIL, does not require the user to select frequencies. cal to that of the total stress models because pwp is zero. As Many users of site response analysis are not fully aware shaking progresses, pwp is generated and cyclic degradation of the implications associated with the selection of Ray- (of clay microstructure) starts. Subsequently, the effects of
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14 pwp generation and, in some models, of cyclic degradation clay is of much lower intensity than in sand; and that in over- are included by degradation of soil strength and stiffness. consolidated clay, both positive and negative (suction) pwp Some models use different factors for degradation of soil may develop (e.g., Matasovic and Vucetic 1992). Generic strength and stiffness. For example, Matasovic (1993) and sets of material parameters for this model are provided in Matasovic and Vucetic (1995b) proposed degradation index the D-MOD2000 package. functions that degrade strength and stiffness of sands at dif- ferent rates, whereas the concept of the degradation index (Idriss et al. 1978) is used to degrade strength and stiffness of soft clays. A number of pwp generation models have been devel- oped, starting with Martin and Seed (1978) and Martin et al. (1975). The Martin and Seed (1978) model was imple- mented in early iterations of FLAC by Dr. Wolfgang Roth (Roth and Inel 1993; W. Roth, personal communication, 2011). The Martin et al. (1975) pwp generation model was used in DESRA-2 (Lee and Finn 1978). A more recent example of the semi-empirical pwp model for saturated sand is the Dobry et al. (1985) model. This model was based on straincontrolled cyclic direct simple shear and cyclic triaxial testing. The model was later modified by Vucetic FIGURE 9 Stress-strain behavior modeling illustrating (1986) to allow for quasi-2-D shaking and further by Mata- stiffness degradation with the MKZ Constitutive Model sovic (1993) to more accurately model pwp-induced deg- (Matasovic 1993; Matasovic and Vucetic 1993). radation of shear modulus and shear stress. The Vucetic (1986) modification of this pwp model has been success- Advanced Effective-Stress-Based Models fully incorporated in DESRAMOD (Vucetic 1986), and the Matasovic (1993) modification has been incorporated in Another class of soil constitutive models used in site D-MOD (Matasovic 1993; Matasovic and Vucetic 1995b), response analysis is effective-stress models. In these D MOD_2 (Matasovic 2006) and D-MOD2000 (Matasovic models, the formulation of the constitutive law is devel- and Ordonez 2007), and DEEPSOIL (Hashash et al. 2011). oped in effective-stress space, and pwp is computed as the The pwp generation models described require the use of an difference between effective-stresses and total stresses equivalent number of cycles to represent earthquake shak- in the domain of interest. Examples of plasticity-based ing. Polito et al. (2008) introduced an energy-based model constitutive laws include Roscoe and Schofield (1963), (GMP model) for the generation of pwp based on a large Mroz (1967), Roscoe and Burland (1968), Prevost (1977), number of laboratory tests, which does not require the devel- Dafalias and Popov (1979), Pestana (1994), Whittle and opment of an equivalent number of cycles. This model has Kavvadas (1994), Byrne et al. (1995), Manzari and Nour been implemented in DEEPSOIL (Hashash et al. 2011) com- (1997), Beaty and Byrne (1998), and Elgamal et al. (2001). bined with the degradation index framework introduced by These advanced constitutive models are capable of simu- Matasovic (1993). With the exception of the modified Dobry lating complex soil behavior under a variety of loading et al. (1985) model, as implemented in D-MOD2000, there is conditions. Key elements of these models include yield limited information to guide the user in selecting the appro- surfaces, flow rules, and hardening (or softening) laws. priate pwp model parameters. A review of advanced constitutive models with appli- cation in site response analysis is provided in Potts and The effect of cyclic degradation on soil stiffness and Zdravkovi (1999). strength is illustrated in Figure 9 for the MKZ constitutive model (Matasovic 1993; Matasovic and Vucetic 1995b). The Generic material parameters for advanced constitu- initial hysteretic loop shown in the figure refers to the first tive models are often not available. Evaluation of material cycle of cyclic loading (i.e., at time t = 0). The subsequent parameters for these models requires significant expertise degraded hysteretic loop refers to any subsequent cycle (i.e., and detailed site-specific soil properties. Examples of site at time t) for which enough pwp has built up to degrade both response programs that incorporate advanced constitutive initial shear modulus Gmo and initial shear stress co at cor- models are DYNA1D (Prevost 1989), SUMDES (Li et al. responding shear strain co. 1992), SPECTRA (Borja and Wu 1994), AMPLE (Pestana and Nadim 2000), CYCLIC 1-D (Elgamal et al. 2004), An example of a pwp model for clay is the Matasovic and CyberQuake (Modaressi and Foerster 2000; Foerster and Vucetic (1995a) model. This model was based on the results Modaressi 2007; Lopez-Caballero et al. 2007) and the of cyclic simple shear testing. It can be noted that pwp in ground response module in the OpenSees simulation plat-