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10 ses and comparison with other analysis models (e.g., Seed NONLINEAR TOTAL STRESS SITE RESPONSE et al. 1988; Idriss 1990; Dickenson et al. 1991; Idriss and ANALYSIS Hudson 1993; Kavazanjian and Matasovic 1995; Darragh and Idriss 1997; Rathje and Bray 2001; Baturay and Stewart In nonlinear site response analysis, the nonlinear behavior 2003). Based on the findings of these studies and authors' of the soil during cyclic loading can be represented, which experience, the following critique applies to the equivalent- makes it possible to move away from the inherent linear linear analysis: approximation of the equivalent-linear analysis approach. Cyclic hysteretic soil behavior during unloading and reload- Equivalent-linear analysis is a total-stress analysis; ing is also represented in the nonlinear site response anal- hence, it does not account for pore pressure generation ysis. Furthermore, nonlinear analysis makes it possible to and its effect on material properties during shaking; explicitly include soil strength and the effects of seismic This method is not recommended when the levels of pore water pressure generation on soil strength and stiff- shaking-induced shear strains are "high." There is no ness. These options have significant effects on site response consensus on the limiting ("high") shear strain level. in areas of very high seismicity (e.g., PGA 0.4 g) and/or Studies by the authors have shown that results of equiv- when soft and/or potentially liquefiable soils are present in alent-linear and nonlinear analyses start to diverge at local soil deposits. strains as a low as 0.1%0.2%. At strains greater than 0.5%1% at any depth (layer) within the soil profile, In total stress site response analysis, the explicit inter- the equivalent-linear analysis results are not neces- action of pore fluid with the soil matrix is neglected. This sarily reliable. Recently, Assimaki et al. (2008) pro- is an acceptable simplification under many conditions and posed the use of a frequency index that measures the is numerically efficient. Kwok et al. (2007) conducted a frequency content of incident ground motion relative detailed study of the total stress nonlinear site response anal- to the resonant frequencies of the soil profile. This ysis software. The study provides an excellent review of the index is then used in conjunction with the rock-outcrop various issues associated with this type of analysis. Kwok et peak ground acceleration (PGA) to identify conditions al. (2007) noted the following nonlinear software that were where incremental nonlinear analyses, including the evaluated as a part of the PEER 2G02 Project (20052007; equivalent-linear approach, should be used instead of J. Stewart Project Director): DEEPSOIL (Hashash and Park, approximate methodologies. 2001, 2002; Hashash et al. 2011); D-MOD_2 (Matasovic 2006; later upgraded to D-MOD2000); OpenSees (Ragheb Despite its apparent shortcomings, the total-stress equiv- 1994, Parra 1996, Yang 2000); SUMDES (Wang 1990, Li alent-linear analysis is likely to remain a tool of choice for et al. 1992); and TESS (Pyke 2000). The study was able many practicing engineers and may have a slightly differ- to identify key controlling parameters that are common to ent and expanded role. In particular, this approach is now all software. The study found that when input is properly used not only as the "first approximation" of site response, controlled, most of the software provided similar results. but also for calibration of more advanced models, including Hashash et al. (2010) provide a description of recent develop- nonlinear and effective-stress analyses. ments in nonlinear site response analysis and highlight key steps and issues required for conducting such analyses. Many of the modulus reduction and damping curves were based on small strain data (testing shear strain typically In nonlinear site response analysis the dynamic equation reaching 0.5% to 1.0%). These curves are then extrapolated of motion is solved in the time domain. The equation is com- at strain levels exceeding 0.5%1%. However, the results monly written as: of site response analyses increasingly show calculated strains increasing to 1.0%, especially in soft soils. CalTrans [M ] {} + [C ] {} + [K ] {u} = - [M ] {g}(1) (Jackura 1992) recognized that the implied strength associ- ated with the extended curves might either underestimate where [M ], [C ], and [K ] are the mass matrix, viscous or overestimate the actual strength of soils, so the agency damping matrix, and nonlinear stiffness matrix, respec- developed an-in house simplified procedure for "extension" tively; {u}, {}, and {} are, respectively, the displacements, of modulus reduction and damping curves. Recently, Chiu velocities, and accelerations of the mass [M ] relative to the et al. (2008) and Hashash et al. (2010) proposed advanced base, and {g} is the acceleration of the base. procedures to remedy this arbitrary extrapolation of sub- ject curves in both equivalent-linear and nonlinear analysis. The stiffness matrix [K ] is derived from the nonlinear soil Most recently, Stokoe (K.H. Stokoe, personal communica- constitutive model selected to represent cyclic soil response. tion 2011) pointed out this problem and urged that a remedy In principle, all damping in the soil can be captured through approach be developed. the hysteretic loops in the soil constitutive model. However,

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11 as a practical matter, most available soil constitutive mod- gration to solve the same dynamic equation. Program PSNL, els cannot properly represent measured soil damping at low currently under development (S.L. Kramer personal com- strains and significantly underestimate damping at these low munication, 2011; see description in Anderson et al. 2011) strains. Therefore, it is necessary to add damping through models soil profile as a continuum and can simulate dilation. the use of velocity proportional viscous damping [C ]. Most nonlinear codes are formulated to calculate site The dynamic equation of motion can be solved by response in one horizontal direction of shaking, although numerical integration. The numerical integration calls for some such as SUMDES (Wang 1990) and OpenSees temporal discretization (i.e., system of coupled equations (Ragheb 1994; Parra 1996; Yang 2000) allow for multidi- is discretized temporally) and solution by one of the avail- rectional shaking. able time-stepping schemes. Examples of time-stepping schemes include Wilson's algorithm (Clough and Penzien A sample 2-D finite element model (OpenSees) is shown 1993) and numerous variations of Newmark's algorithms in Figure 5. Such a model allows for simultaneous applica- (Newmark 1959). tion of excitation in both horizontal and vertical directions. To solve the equation of motion, it is necessary to dis- cretize the domain of interest, which in this case is the soil column. Two different approaches for discretization of the soil domain are available: (1) lumped mass discretization and (2) finite element discretization. Figure 4 shows a lumped mass model that depicts a hori- FIGURE 5 Mesh representation of 2-D Nonlinear Site zontally layered soil deposit (i.e., a soil deposit that can be Response Analysis of Embankment (Nikolaou 2011). represented by a 1-D model). Soil mass is lumped at the layer interfaces, and soil stiffness is represented by (nonlinear) Regardless of the discretization method used in nonlinear springs. The figure shows the hysteretic damping inherent site response analysis, the thickness of sublayers in a model with nonlinear springs and viscous damping, which is also has important consequences. The layer thickness determines part of the model. This model is used in nonlinear analy- the maximum frequency that can be propagated through a sis software such as DESRA-2/DESRA-2C, DESRAMOD, soil column. If the layer is too thick, the discretized domain DESRAMUSC, SUMDES, TESS, D MOD_2/D-MOD2000, may filter important components of the ground motion and and DEEPSOIL, and in many Japanese programs that are not thus underestimate the ground response. If layer thickness is reviewed here. too small, the computational cost can be too high. Therefore, as a practical matter, 1-D nonlinear site response models will usually have greater (i.e., finer) discretization than their 2-D and 3-D model counterparts, and thus will propagate higher frequencies and filter less of the input ground motion. The survey results in this study indicate that users of 2-D and 3-D software are not always aware of this important limita- tion. The graphical user interfaces can help alert the user to the maximum frequency that can be propagated. Several 1-D software (e.g., DEEPSOIL and D MOD2000) have such alerts incorporated in graphical users interfaces. Nonlinear Constitutive Models with Hysteretic Damping FIGURE 4 Lumped mass discretization for 1-D Nonlinear Hysteretic Site Response Model (Matasovic 1993). Nonlinear total stress site response analysis is generally done with relatively simplified soil constitutive models. A number of other programs discretize the soil domain These models evolved from the early stress-strain rela- by means of finite elements; the details of this approach are tionships of Ramberg and Osgood (1943) and Kondner not discussed here. For dynamic problems, the equations of and Zelasko (1963). The hyperbolic model introduced by motions are solved using an explicit time marching integra- Duncan and Chang (1970) for axial soil behavior, which tion algorithm. For example, TESS (Pyke 2000) and FLAC was based on the above-cited shear stress and strain behav- (Itasca 2005) use an explicit finite difference to solve the ior models, was accompanied by sets of generic material wave propagation problem. Programs such as OpenSees properties and hence allowed for an elegant and simple (Ragheb 1994; Parra 1996; Yang 2000), ABAQUS, and way to capture soil nonlinearity at small axial strains. All PLAXIS use a finite element method with explicit time inte- three models provided the basis for constitutive models

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12 that are presently in use. These models (Pyke 1979: Mata- sovic 1993; Matasovic and Vucetic 1993; Darendeli 2001) provide for better simulation of nonlinear stress-strain behavior and also allow for simulation of cyclic loading and reloading in accordance with certain rules. The stress- strain relationship in these models is generally established by: an initial loading curve; a series of rules that describe the backbone curve (see Figure 6 for definition of backbone curve); and unloading-reloading behavior rules required to establish cyclic loops. The most widely used rules are the Masing rules (Masing 1926) and extended Masing rules (Pyke 1979; Wang et al. 1980; Vucetic 1990). The extended Masing rules, including unloading-reloading rules, are used in several 1-D site response analysis soft- ware (DESRA 2C, TESS, D-MOD_2, DESRAMOD, D MOD2000, and DEEPSOIL). FIGURE 7 Evaluation of proposed damping reduction factor (a) modulus reduction and (b) damping curve using Darendeli's curves for cohesionless soils as target. FIGURE 6 Backbone curve as stress-shear strain relationship Note :MR = modulus re-matching only with extended Masing for monotonic loading. rules, MRD = approximate match of both modulus and damping with extended Masing rules, MRDF = modulus reduction and damping matching with non-Masing rules (after It has been long noted that the use of Masing rules, and Hashash et al. 2010). to some extent extended Masing rules, leads to an overes- timation of soil damping at large strains. This, in turn, may result in an underestimation of calculated ground motion Assimaki and co-workers (e.g., Assimaki et al. 2009) have intensity. To compensate for this phenomenon, Darendeli also made important contributions to a number of the above- (2001) proposed the introduction of reduction factors in cited issues related to site response. Their work includes the the development of his family of standard curves. Phillips incorporation of uncertainty in site response analysis and and Hashash (2009) used a similar approach to introduce addresses the issues related to unloading-reloading rules and a modification to the Masing rules (MRDF) and employed damping at larger strains. that in the MRDF model used in DEEPSOIL. Figure 7 from Hashash et al. (2010) illustrates the limitations of the Mas- Borja et al. (1999, 2002) developed a software called ing rules (MR) and extended Masing rules (MRD) in terms SPECTRA, a 1-D nonlinear total stress site response analy- of overestimation of damping at large strain levels (MR at sis program that uses a bounding surface plasticity model strains > 0.1%; MRD at strains > 1%) and improve- to simulate stress-strain behavior. SIREN (Oasys 2006) and ment in matching of both damping and modulus reduction LS Dyna (LSTC 1988) can be used to perform total stress curves with MRD and MRDF. Matasovic (1993) showed site response analysis. that using MR rather than MRD may result in a higher computed surface acceleration response and/or a shift in Viscous Damping Models the response spectrum when relatively high shear strains are induced in the profile. Philips and Hashash (2009) Most available constitutive models show very small hysteretic further showed that MRD, as compared with MRDF, may damping at small strains, which is inconsistent with measured result in a higher computed surface acceleration response soil behavior. Viscous damping is introduced to compensate and/or a shift in the response spectrum at strain levels for this deficiency. The amount of viscous damping is typi- exceeding approximately 1%. cally selected such that the sum of hysteretic and viscous