B04 Overcoming Hurdles that Limit the Application of ...

Overcoming Hurdles that Limit the Application of Nonlinear Seismic Ground Response Analysis in Engineering Practice Jonathan P. Stewart, On-Lei Annie Kwok, Youssef M.A. Hashash, Neven Matasovic, Robert Pyke, Zhiliang Wang and Zhaohui Yang ABSTRACT One-dimensional seismic ground response analyses are often performed using equivalent-linear procedures, which require few, generally well-known parameters (shear wave velocity, modulus reduction and damping versus shear strain, and soil density). Nonlinear analyses provide a more robust characterization of the true nonlinear soil behavior, but their implementation in practice has been limited, which is principally a result of poorly documented and unclear parameter selection and code usage protocols. Moreover, the benefits and potentials of nonlinear analysis relative to equivalent linear are not well defined. In this paper, we present preliminary results of a “benchmarking study” of nonlinear ground response analysis procedures. Key issues that are discussed include: (1) the use of a simple curve-fitting parameter to describe the shape of the backbone curve, avoiding the need to determine dynamic shear strength (which is often unavailable); (2) strategies for managing excessive large-strain material damping that occurs when Masing’s rule is applied to the backbone curve to evaluate hysteretic damping; (3) specification of input motion as “outcropping” (i.e., equivalent free-surface motions) versus “within” (i.e., motion recorded at depth in a vertical array); and (4) specification of viscous damping, specifically the target value of the viscous damping ratio and the frequencies for which the viscous damping produced by the model matches the target.

_____________ Jonathan P. Stewart, Associate Professor, Dept. of Civil & Environmental Engr., Univ. of Calif., Los Angeles On-Lei Annie Kwok, Graduate Student Researcher, Dept. of Civil & Environmental Engr., Univ. of Calif., Los Angeles Youssef M.A. Hashash, Associate Professor, Dept. of Civil & Environmental Engr., Univ. of Illinois, Urbana-Champaign Neven Matasovic, Associate, GeoSyntec Consultants, Huntington Beach, California Robert Pyke, Consulting Engineer, Lafayette, California Zhiliang Wang, Senior Engineer, Geomatrix Consultants, Oakland, California Zhaohui Yang, Engineer, URS Corporation, Oakland, California

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Fifth National Seismic Conference on Bridges & Highways, San Francisco, CA, September 18-20, 2006

INTRODUCTION Nonlinear ground response analysis is seldom used in practice by non-expert users because parameter selection and code usage protocols are often elaborate, the effect of parametric variability on the analysis results is generally unknown, and the benefits of nonlinear analysis relative to the widely-used equivalent-linear analysis are generally unquantified and unclear. In this paper we report results of a benchmarking project for nonlinear ground response analysis codes organized through the Pacific Earthquake Engineering Research (PEER) center Lifelines program. The objective of the project is to “de-mystify” nonlinear ground response analysis routines for Geotechnical Earthquake Engineers by providing clear and well documented code usage protocols, verifying the codes at different strain conditions, and investigating the benefits of nonlinear analysis relative to equivalent-linear analysis. The project is utilizing five leading nonlinear codes: DEEPSOIL (Hashash and Park, 2001, 2002, Park and Hashash 2004), D-MOD_2 (Matasovic, 2006), OpenSees (Ragheb, 1994; Parra, 1996; Yang, 2000; McKenna and Fenves, 2001), SUMDES (Li et al., 1992) and TESS (Pyke, 2000). This paper is concerned with the first objective noted above – the technical underpinnings of clear code usage protocols. We begin with the issue of parameterizing the nonlinear backbone curve, which is closely linked to the problem of excessive material damping at large strain that occurs when some unloading and reloading rules (e.g. Masing’s rules) are applied to the backbone curve. Second, we describe the method by which input motions should be specified in nonlinear analysis (outcropping versus within) – a significant source of confusion even among many experts users. Lastly, we provide some preliminary guidelines on the specification of viscous damping, a parameter that can significantly affect analysis results in some cases yet which lacks a clear link to fundamental soil properties. KEY ISSUES IN NONLINEAR GROUND RESPONSE ANALYSIS Parameters Describing Nonlinear Material Behavior Backbone Curve The classical definition of reference strain is the ratio of shear strength to initial small strain secant shear modulus (Hardin and Drnevich, 1972). The parametric description of the nonlinear backbone curve in the past has generally required the specification of this reference strain along with a number of curve fitting parameters. A practical problem with this approach is that the shear strength at rapid strain rate, needed to define reference strain, is often not available. At least for low to moderate strains levels, a practical alternative is to use a hyperbolic fit to the backbone curve in which a “pseudo-reference strain” (γr) is used as a fitting parameter. We use the term pseudo-reference strain to avoid confusion with reference strain as defined by Hardin and Drnevich (1972). Pseudo reference strain is defined as the shear strain at which G/Gmax = 0.5. This definition arises from hyperbolic fits of G/Gmax curves according to 1 (1) G / Gmax = a 1 + (γ γ r )

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where a is a fitting parameter generally taken as 0.92. The advantages of using pseudo reference strain are that (1) γr can be readily evaluated from material-specific modulus reduction curves evaluated from laboratory testing and (2) lacking material-specific testing, empirical relationships exist to predict γr as a function of basic parameters such as PI, overburden stress, and overconsolidation ratio (Darendeli, 2001). Because pseudo reference strains are determined from modulus reduction curves that are typically defined for strains less than 1%, a backbone curve described by a hyperbolic curve fit using γr would not necessarily be expected to accurately represent soil behavior at large strain, including the shear strength. We investigate this problem by examining the degree to which the shear strength implied by the use of Eq. 1 (approximately Gmax × γr) is realistic. This is done using ratios of Gmax to shear strength, for which empirical relationships are available from Weiler (1988). The ratios from Weiler are for soils with OCR of 1 to 5, confining pressures from 1 to 5 tsf and PI from 15 to 45. Weiler’s undrained shear strengths (Su) are based on direct simple shear testing. Weiler’s Gmax / Su ratio is compared to the inverse of Darendeli’s (2001) estimate of γr (which is approximately the ratio of Gmax to the large-strain asymptote of the hyperbolic curve, taken as shear strength). As observed from Figure 1, the Gmax/effective-strength ratios implied by pseudo reference strain γr are significantly higher than those from Weiler for an overburden stress of σ =1 tsf. This bias implies that the shear strength implied by γr is underestimated by Darendeli’s relationships at σ = 1 tsf. This bias disappears at larger overburden pressures (σ = 5 tsf). Accordingly, at relatively shallow depths, the use of backbone curves derived from the pseudo reference strain parameter may overestimate the soil nonlinearity at large strains, which could significantly affect the results of ground response analyses.

Figure 1. Comparison of Gmax / Su ratio from Weiler (1988) to inverse of pseudo reference strain (1/γr) from Darendeli (2001). Quantity 1/γr is approximately the ratio of Gmax to the shear strength implied by the use of pseudo reference strain for fitting nonlinear backbone curves.

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Our general recommendation is to use Eq. 1 with the true reference strain (Gmax/Su using measured shear strengths) when shear strength data is available. When this data is not available, Darendeli’s estimate of pseudo reference strain can be used in Eq. 1, but the results should only be trusted for shear strains less than about 1%. Damping Curve Masing’s rules (Masing, 1926) and extended Masing rules (Vucetic, 1990; Pyke, 1979) are employed in nonlinear analysis in conjunction with the backbone curve to describe unloading, reloading and cyclic degradation behavior of soil. Material damping is directly proportional to the shape of the loop, and hence is sensitive to the shape of the backbone curve and unload/reload rules. The damping at large strain that results from the use of Masing or extended Masing rules tends to be over-estimated relative to laboratory measurements. There are three schools of thought on managing the over-estimation of damping. One approach is to select model parameters for the backbone curve (and hence modulus reduction curves) that optimally fit the target data and accept the resulting overestimation of damping using Masing’s rules. Another approach is to again use Masing’s rules and select model parameters that optimize the fitting of modulus reduction and damping curves simultaneously (across the strain range of interest). Figure 2 shows the difference in the fitted modulus reduction and damping curves (relative to target data) when different fitting approaches are employed. 1

G / Gmax

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0 60

Damping

50 40 30 20 10 0 0.0001

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Figure 2. Different approaches in fitting modulus reduction and damping curves in nonlinear analysis The third approach is to introduce an additional parameter that changes the shape of the unload/reload curves so that both modulus reduction and damping curves can be fit simultaneously. There are several alternative unloading/reloading schemes which allow fitting of damping curves. Lo Presti et al. (2006) allows unloading and reloading curves to have the same shape as the backbone curve, but with a scaling factor of n (for the original Masing criteria, n = 2). Lo Presti et al. provide

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recommendations for estimating n as a function of soil type, strain level and number of cycles for the motion. Wang et al. (1980) suggested another approach in which a damping correction factor is applied to the damping calculated by Masing’s rule. These unload/reload rules are not yet implemented in the nonlinear codes considered in this study, which are the codes most often used in engineering practice. The relative merits of the three alternative approaches are being investigated as part of the validation phase of this project (results not yet available). Specification of Input Motion Input motions are specified at the bottom of the 1D site profile in nonlinear analyses. There has been confusion regarding whether the motions specified at the base of the profile should represent an outcropping condition (i.e., equivalent free-surface motions that are twice the amplitude of the incident wave due to full reflection) or a within condition (i.e., the sum of the incident waves and downward propagating waves reflected from overlying layer interfaces). A closely related question is whether the base condition (representing the material below the site column) should be elastic or rigid. For some of the codes used in this research, past practice had been to calculate within motions at the profile base using equivalent-linear analyses, and specify those motions as the input for nonlinear analysis along with an appropriate elastic base. For other codes, full outcropping motions were used with an elastic base. To clarify this issue, we exercised nonlinear codes for cases with known elastic solutions. An illustrative example of those calculations is a single soil layer (thickness = 30 m; Vs = 300 m/s) overlying an elastic half-space with Vs = 600 m/s. All soil properties were taken as elastic, and viscous damping was set to zero. A sinusoidal acceleration history matching the site frequency of 2.5 Hz was specified at the top of the halfspace for time domain analyses (i.e., the nonlinear codes, but used with linear backbone curves). The same motion was specified as outcropping for linear frequency domain analyses (SHAKE91, Idriss and Sun, 1992). Figure. 3 (bottom frame) shows the input acceleration history, while the middle frame shows the resulting within motion at the layer interface from frequency-domain analyses. Since the 2.5 Hz motion has a node (i.e., zero amplitude at all times) at the interface depth, the within motion decays to null upon achieving a steady state condition. As shown in Figure. 3 (top frame), surface acceleration histories obtained from both frequency and time domain analyses match well. These and other similar results suggest that input motions should be specified for an outcropping condition in time domain analyses, and used with an elastic base. Other work (not presented for brevity) has similarly shown that within motions can be used with a rigid base, which would be appropriate practice for applying recorded downhole motions in time domain analyses.

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Surface Acceleration (g)

SHAKE91 All Nonlinear Codes

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Figure 3. Acceleration histories for the one-layer problem Specification of Viscous Damping

Viscous Damping Ratio (%)

In most nonlinear codes, some form of viscous damping is used to provide for damping in the analysis at very small strains where the hysteretic damping from the non-linear soil models is nearly zero (an exception is TESS, which does not require viscous damping). There are a number of options for modeling viscous damping, which vary by code (Table 1). As illustrated in Figure 4, there are three principal issues: (1) the form of the damping formulation (simplified versus full or extended Rayleigh damping; Park and Hashash, 2002); (2) the target viscous damping ratio (labeled ζtar in Figure 3) that is matched at specified target frequencies; and (3) the matching frequencies (one, two, and four for the cases of simplified, full, and extended Rayleigh damping, respectively). Full Rayleigh Damping

Target Damping Ratio, ξtar

Extended Rayleigh Damping

Simplified Rayleigh Damping (Stiffness Proportional)

f1

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Figure 4. Schematic illustration of viscous damping models (after Park and Hashash, 2004) Paper No. B04


TABLE I. AVAILABLE VISCOUS DAMPING FORMULATION FOR NONLINEAR CODES AND SUMMARY OF ANALYSES DISCUSSED IN TEXT Nonlinear Code Viscous Damping Option Viscous Damping Best Match to Formulation Considered Frequency Domain in Current Analyses Solution for all Three Sites D-MOD_2 Simplified & Full Simplified (fs*; fm*; fp*) Full (5×fs) at ζtar=5% Full (fs + 3×fs; fs + 5×fs) DEEPSOIL Simplified, Full & Extended Simplified (fs; fm; fp) Full (5×fs) at ζtar=5% Full (fs + 3×fs; fs + 5×fs) OpenSees Simplified & Full Simplified (fs; fm; fp) Full (5×fs) at ζtar=5% Full (fs + 3×fs; fs + 5×fs) SUMDES Simpified (Target frequency Simplified ζtar=1% fixed at 1 Hz) TESS No viscous damping *fs, fm and fp represent site frequency, mean frequency and predominant frequency of motion respectively.

Few protocols are available for guiding users in the selection of the model-type and parameters described above. One set of guidelines has been presented by Hashash and Park (2004) in which the model parameters are selected through an iterative process in which frequency and time domain elastic solutions are matched over a frequency range of interest (because frequency domain analyses using equivalent viscous damping provide a “correct” response against which the viscous damping formulation for the time domain can be calibrated). The procedure is implemented through a convenient user interface. We have developed preliminary procedures that supplement the Hashash and Park guidelines. Such procedures are needed in the absence of a readily available interface, or to provide a good starting point for iterative analyses using the Hashash and Park (2004) procedure. To address this issue, results of linear time domain analyses for actual site profiles are compared to solutions from linear frequency domain analyses with a specified amount of equivalent viscous damping (fixed at 5%). The three selected sites represent a broad range of site conditions: shallow stiff soil over rock, soft clay overlying stiffer sediments and rock, and very deep stiff soils typical of the LA basin. A broadband synthetic is used as an outcropping input motion for all analyses. For all codes, target damping ratios of 0.5% and 5% were used – the motivation being to evaluate whether the target viscous damping ratio should match the small strain material damping or a much smaller value. Both simplified and full Rayleigh damping formulations were used for codes with both options (see Table 1 for details). The target frequencies considered were selected in accordance with guidelines that had been used previously by code developers (details in Table 1). Shallow Stiff Site: Simi Valley Knolls School The upper 14 m of Simi Valley Knolls School is composed of silty sand which has shear wave velocities of about 300 m/s and is underlying by sandstone (site frequency fs = 6.4 Hz). Figure. 5 compares 5% damped acceleration response spectra of surface motions from the frequency domain solution (developed using SHAKE91; Idriss and Sun, 1992) with time-domain results from D-MOD_2. The trends in the D-MOD_2 results are similar to those of DEEPSOIL and OPENSEES. Comparing the left and right frames, the results are relatively insensitive to the target damping ratio for this shallow site. Both simplified and full Rayleigh damping formulations are reasonably

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effective, although simplified with the target frequency set to the mean frequency of the input motion overdamps at short periods. Additional analyses were performed for SUMDES and TESS (the results are not shown for brevity). For SUMDES, simplified Rayleigh damping is used with the target frequency fixed at 1 Hz. As shown in Table 1, the best fit was generally obtained for a target viscous damping of 1%, which bears no clear relationship to the equivalent viscous damping ratio from the frequency domain analyses. In the case of TESS, strain rate parameters (denoted VG and VT) are used that produce a damping effect. Values of VG and VT between the 1.0 and 2.0 times the equivalent viscous damping ratio generally provide the best fit. Soft Clay Medium Deep Site: Treasure Island The Treasure Island site has a 16 m layer of San Francisco Bay Mud that overlies stiffer sands and clays. The site frequency is dominated by the soft clay layer, and is 1.06 Hz. As shown in Figure.6, the analysis results in this case indicate a much better match for ζtar = 5% than 0.5%. The greater sensitivity to ζtar (relative to the Simi Valley site) results from the thicker site profile, and the results indicate that the ζtar should be selected to match the equivalent viscous damping from the frequency domain analysis. Simplified Rayleigh damping generally overdamps at low periods, although the results are reasonable when the target frequency is set at the site frequency. Full Rayleigh damping would appear to be preferred, with the results being fairly insensitive to the second target frequency (3fs or 5fs). Deep Stiff Site: La Cienega The La Cienega site consists of bedded sands, silts, and clays that gradually increase in stiffness with depth. We model the upper 305 m of the profile, which has a site frequency of 0.45 Hz, although the true first mode site frequency is much lower because crystalline bedrock occurs at great depth. As shown in Figure 7, we again see a high sensitivity to ζtar (with 5% providing the better match). Simplified Rayleigh damping is most effective when the target frequency is set to the mean frequency of the input motion (fm), and overdamps otherwise. Full Rayleigh damping generally provides an improved fit, with a slight preference towards the second frequency being 5fs.

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Pseudo Spectral Acceleration (g)

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Target viscous damping ratio used in DMOD_2 is 0.5%

Target viscous damping ratio used in DMOD_2 is 5% Input Outcropping Within (SHAKE) Surface (SHAKE) Surface (DMOD2, Simplified, fs)

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Figure 5. Comparison of response spectra for Simi Valley Knolls School. Results are shown for D-MOD_2, but similar results were obtained for DEEPSOIL and OPENSEES. Pseudo Spectral Acceleration (g)

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Figure 6. Comparison of response spectra for Treasure Island. Results are shown for D-MOD_2, but similar results were obtained for DEEPSOIL and OPENSEES.

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Pseudo Spectral Acceleration (g)

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Figure 7. Comparison of response spectra for La Cienega. Results are shown for D-MOD_2, but similar results were obtained for DEEPSOIL and OPENSEES. CONCLUSION This paper represents a progress report on the development of parameter selection and code usage protocols for time-domain nonlinear ground response analysis procedures. The participants in the project represent a broad range of code developers, academics, and practitioners. Work to date that is summarized in this paper has led to the following conclusions: 1. Up to a shear strain level of approximately 1%, the pseudo reference strain parameter (γr) as estimated from empirical relationships can provide an efficient method of characterizing nonlinear backbone curves. To replicate the stress-strain behavior at larger strains, it is important to estimate the shear strength and define a true reference strain (Gmax/Su) for use with Eq. 1. 2. Input motions for nonlinear codes should be specified as outcropping with an elastic base or within with a rigid base. Use the former for recordings made at the ground surface, and the latter for downhole records. 3. A good starting point for the estimation of viscous damping is to use a full Rayleigh damping formulation with the target damping ratio set to the small strain material damping and the two target frequencies to the site frequency and 5 times the site frequency. The use of simplified Rayleigh damping is discouraged. These recommendations may be subject to revision when the validation phase of the project is completed. ACKNOWLEDGEMENTS Financial support for the benchmarking of nonlinear ground response analysis procedures was provided by PEER Lifelines project 2G02, which is sponsored by the Pacific Earthquake Engineering Research Center’s Program of Applied Earthquake Engineering Research of Lifeline Systems. The PEER Lifelines program, in turn, is supposed by the State Energy Resources Paper No. B04


Conservation and Development Commission and the Pacific Gas and Electric Company. This work made use of Earthquake Engineering Research Centers Shared Facilities supported by the National Science Foundation under Award #EEC-9701568. In addition, the support of the California Department of Transportation’s PEARL program is acknowledged. This project has benefited from the helpful suggestions of an advisory panel consisting of Drs. Susan Chang, I.M. Idriss, Steven Kramer, Faiz Makdisi, Geoff Martin, Lelio Mejia, Walter Silva, and Joseph Sun. DEEPSOIL development was supported in part by the Earthquake Engineering Research Centers Program of the National Science Foundation under Award Number EEC-9701785; the Mid-America Earthquake Center. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. REFERENCES Darendeli (2001). “Development of a new family of normalized modulus reduction and material damping curves,” Ph.D. Dissertation, Univ. of Texas. Hardin, B.O. and Drnevich, V.P. (1972). “Shear modulus and damping in soils: design equations and curves,” J. of Soil Mechanics and Foundation Engrg. Div., ASCE, 98 (SM7) No. SM7, 667-692 Hashash, Y.M.A. and Park, D. (2001). "Non-linear one-dimensional seismic ground motion propagation in the Mississippi embayment." Engrg. Geology, 62(1-3), 185-206. Hashash, Y.M.A. and Park, D. (2002). "Viscous damping formulation and high frequency motion propagation in nonlinear site response analysis." Soil Dynamics and Earthquake Engrg., 22(7), 611-624. Idriss, I.M. and Sun, J.I. (1992). SHAKE91: A computer program for conduction equivalent linear seismic response analyses of horizontally layered soil deposits, Center for Geotechnical Modeling, Univ. of California, Davis. Li, X.S., Wang, Z.L. and Shen, C.K. (1992). SUMDES: A nonlinear procedure for response analysis of horizontally-layered sites subjected to multi-directional earthquake loading. Dept. of Civil Engin. Univ. of Calif., Davis Lo. Presti, D.C.F., Lai, C.G. and Puci I. (2006). “ONDA: Computer code for nonlinear seismic response analyses of soil deposits.” Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 132(2), 223-236 Masing, G. (1926). “Eigenspannungen and verfertigung beim messing,” Proc. 2nd Int. Congress on Applied Mechanics, Zurich. Matasovic, N. (2006).“D-MOD_2 – A Computer Program for Seismic Response Analysis of Horizontally Layered Soil Deposits, Earthfill Dams, and Solid Waste Landfills,” User’s Manual, GeoMotions, LLC, Lacey, Washington McKenna, F. and Fenves, G.L (2001). “The OpenSees command language manual, version 1.2.” Pacific Earthquake Engrg. Research Center, Univ. of Calif., Berkeley. (http://opensees.berkeley.edu). Park, D. and Y. M. A. Hashash (2004). "Soil damping formulation in nonlinear time domain site response analysis." Journal of Earthquake Engineering 8(2): 249-274. Parra, E. (1996). "Numerical modeling of liquefaction and lateral ground deformation including cyclic mobility and dilation response in soil systems." PhD Dissertation, Dept. of Civil Engrg., Rensselaer Polytechnic Institute, Troy, NY. Pyke, R.M. (1979). "Nonlinear soil models for irregular cyclic loadings" J. of the Geotechnical Engineering Division, ASCE, Vol. 105, No. GT6, pp715-726 Pyke, R.M. (2000). “TESS Users' Manual,” TAGA Engineering Software Services, Lafayette, CA. Ragheb, A. M. (1994)."Numerical analysis of seismically induced deformations in saturated granular soil strata." PhD Dissertation, Dept. of Civil Engrg., Rensselaer Polytechnic Institute, Troy, NY. Wang, Z.L., Han, Q.Y. and Zhou G.S. (1980). “Wave propagation method of site seismic response by visco-elastoplastic model,” Proceedings: Seventh World Conference on Earthquake Engineering, V2, 379-386 Weiler, W.A. (1988). “Small strain shear modulus of clay,” Proceedings, ASCE Conference on Earthquake Engineering and Soil Dynamics II: Recent Advances in Ground Motion Evaluation, Geotechnical Special Publication 20, ASCE, New York, 331-345. Yang, Z. (2000). "Numerical modeling of earthquake site response including dilation and liquefaction." PhD Dissertation, Dept. of Civil Engrg. and Engrg. Mech., Columbia University, NY, New York. Youngs, R.R., 2001. Software validation report for SHAKE04, Geomatrix Consultants

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B04 Overcoming Hurdles that Limit the Application of ...

B04 Overcoming Hurdles that Limit the Application of ...

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