Influence of site conditions on seismic design parameters for foundations as determined via nonlinear site response analysis

Abstract

Site conditions, including geotechnical properties and the geological setting, influence the near-surface response of strata subjected to seismic excitation. The geotechnical parameters required for the design of foundations include mass density (ρ), damping ratio (β_s), shear wave velocity (V_s), and soil shear modulus (G_s). The values of the last three parameters are sensitive to the level of nonlinear strain induced in the strata due to seismic ground motion. In this study, the effect of variations in soil properties, such as plasticity index (PI), effective stress (σ′), over consolidation ratio (OCR), impedance contrast ratio (ICR) between the bedrock and the overlying strata, and depth of soil strata over bedrock (H), on seismic design parameters (β_s, V_s, and G_s) was investigated for National Earthquake Hazards Reduction Program (NEHRP) site classes C and D, through 1D nonlinear seismic site response analysis. The Morris one-at-a-time (OAT) sensitivity analysis indicated that β_s, V_s, and G_s were significantly influenced by variations in PI, while ICR affected β_s more than it affected V_s and G_s. However, the influence of H on these parameters was less significant. It was also found that variations in soil properties influenced seismic design parameters in soil type D more significantly than in soil type C. Predictive relationships for β_s, V_s, and G_s were derived based on the 1D seismic site response analysis and sensitivity analysis results. The β_s, V_s, and G_s values obtained from the analysis were compared with the corresponding values in NEHRP to determine the similarities and differences between the two sets of values. The need to incorporate PI and ICR in the metrics for determining β_s, V_s, and G_s for the seismic design of foundations was highlighted.

Keywords

site effects / 1D seismic site response analysis / sensitivity analysis / foundations / shear wave velocity / soil shear modulus

Introduction

The design of foundations under seismic loading requires the use of effective geotechnical properties that are compatible with the intensity of seismic ground motion [¹]. The seismic design of foundations is dependent not only on the intensity of seismic ground motion but also on the geological setting and geotechnical properties of the soil strata. The influence of these two factors is usually referred to as the “site effect” [²–⁴]. According to Romo et al. [²], parameters related to the geological setting include rock type, soil deposit thickness (H), and the stratigraphic characteristics of soil deposits. The geotechnical parameters of interest include shear wave velocity (V_s), the nonlinear stress–strain (τ–γ) behavior of the soil deposits, the impedance contrast ratio (ICR) between the bedrock and overlaying soil material, over consolidation ratio (OCR), and the effective stress (σ€′) of the soil strata. Seismic design codes in the USA (National Earthquake Hazards Reduction Program (NEHRP) [⁵]) and Europe (EC-8 [⁶]) recognize this fact by classifying the soil deposits into various site classes based solely on the following three parameters computed in the top 30 m of the strata: (i) V_s30, (ii) average standard penetration test (SPT) N values, and (iii) average undrained shear strength {\overline{s}}_{\mathrm{u}}. In contrast, the Japanese (JRA [⁷]) and Australian/New Zealand (AS/NZ 1170 [⁸]) codes incorporate the fundamental period of soil deposits overlying the bedrock as well in site classification. The NEHRP [⁵] seismic design provisions are adopted in several building and bridge design codes in the USA (e.g., ASCE 7, IBC, and AASHTO). Based on the travel time-averaged shear wave velocity in the top 30 m of the strata (V_s30), site conditions are classified into six categories (A to F) according to NEHRP [⁵]: A (hard rock; V_s30>1500 m/s), B (rock; 760<V_s30<1500 m/s), C (soft rock or very dense soil; 360<V_s30<760 m/s), D (stiff soil; 180<V_s30<360 m/s), E (soft clay soil; V_s30<180 m/s), and F (highly organic or highly plastic soft soils that require site-specific investigations).

Soil parameters required for seismic foundation design include hysteretic soil damping ratio (β_s), shear wave velocity (V_s), soil shear modulus (G_s), and the mass density of the soil strata (ρ) [⁹]. The first three parameters are dependent on the level of soil strain, and their corresponding values can be obtained via 1D seismic site response analysis or from Tables 19.3-1 to 19.3-3 in the NEHRP provisions [⁵]. The effective β_s, V_s, and G_s values in the NEHRP provisions are tabulated for the six site classes, which not only have a significant variation in V_s30, as previously mentioned, but also ignore the geological setting parameters (i.e., V_rock, H, and ICR) and geotechnical index properties, such as plasticity index (PI), over-consolidation ratio (OCR), and effective stress (σ′). Studies [¹⁰–¹⁷] have shown that the aforementioned parameters can have a significant impact on the site amplification factors used for constructing seismic design spectra. Boaga et al. [¹⁸] conducted a parametric study to determine the influence of soil damping ratio on seismic ground response, while Hamidpour and Soltani [¹⁹] conducted a probabilistic assessment of seismic ground motion considering uncertainty in soil properties. Tsang et al. [²⁰] demonstrated that the period-shift due to soil-structure interaction and damping of the soil strata is most sensitive to the level of shaking and PI. They also noted that the sensitivity of these parameters to PI is a relatively less-known fact. Fatahi and Tabatabaiefar [²¹] noted that the performance of rigid concrete frames found in soft soils is significantly dependent on soil plasticity.

This study was conducted to investigate the dependence of variations in geological setting parameters and the geotechnical properties of the strata on the effective soil parameters used for the seismic design of foundations (i.e., β_s, V_s, and G_s). Predictive relationships for the effective values of β_s, V_s, and G_s to be used in foundation design were derived based on the results of 1D nonlinear site response analysis and sensitivity analysis. In the final part of this study, the computed and predicted values of β_s, as well as the reductions in V_s and G_s with respect to small strain values, were compared with the tabulated values in the NEHRP provisions [⁵]. Moreover, potential limitations of the values stipulated in NEHRP [5] were identified.

Methodology

As previously indicated, site conditions are classified into six categories (A–F) in NEHRP [⁵], based on V_s30. Site Classes A and B are rock sites with a V_s30 of more than 760 m/s. Shallow spread footing-type foundations are commonly used at these site classes. Site class C represents significantly dense soil or soft rock with a V_s30 between 360 and 760 m/s. Shallow spread foundations are suitable for this site class for high values of V_s30, while deep foundations (piles or caissons) can be used for the low values of V_s30. Site class D has a V_s30 ranging from 180 to 360 m/s; the use of pile-group foundations is common in this site class. Whereas, site classes E and F require specialized design of foundations. This study focused on determining the effective values of β_s, V_s, and G_s for foundation design in site classes C and D, which are characterized by a wide variation in V_s30, PI, OCR, and σ′.

Procedure

The sensitivity of the effective values of β_s, V_s, and G_s to variations in V_s30, PI, OCR, σ′, H, and ICR under varying levels of seismic intensity was analyzed in this study. The procedure involved 1D nonlinear site response analysis for various combinations of the input parameters, followed by a global sensitivity analysis to determine the importance of each parameter for the effective design values of β_s, V_s, and G_s. The steps adopted in this study are explained in the following subsections.

Bedrock parameters

NEHRP [⁵] classifies rock profiles as classes A and B, very dense soil/soft rock as class C, and normal soil as class D. Based on the classification provided by the South African Council on Scientific and Industrial Research (CSIR) [²²], the rock profiles of NEHRP site classes A, B, and C were subdivided into categories, as listed in Table 1. CSIR categorizes rocks into classes I–V, with class I indicating exceptionally strong rock (V_s>3353 m/s) and class V indicating weak rock (600<V_s<780 m/s). The CSIR rock classes were matched to the corresponding NEHRP site class in Table 1 along with the salient mechanical properties of each rock class. The average representative values of the mass density (ρ) and Poisson’s ratio (n) listed in Table 1 for various rock classes were obtained from previous studies [²³–²⁵]. Serafim and Pereira [²⁶] proposed the following relationship for computing the modulus of elasticity, E (GPa), of jointed rock masses:

(1)E={\mathrm{10}}^{(RMR-\mathrm{10})/\mathrm{40}},

where RMR is the rock mass rating as per the CSIR classification and is listed in Table 1. The following relationships were used to compute G and V_s:

(2)G={\displaystyle \frac{E}{\mathrm{2}(\mathrm{1}+v)}},

(3)V_{\mathrm{S}}=\sqrt{{\displaystyle \frac{G}{\rho }}}.

The characteristics of the soil-rock interface influence wave propagation in the soil strata from the bedrock, and the effect of these characteristics is represented by the impedance contract ratio (ICR), which is defined as follows:

(4)ICR={\displaystyle \frac{{\rho }_{\mathrm{rock}}\times V_{\mathrm{rock}}}{{\rho }_{\mathrm{soil}}\times V_{\mathrm{soil}}}},

where V_soil is the shear wave velocity of the soil layer in contact with the bedrock. The range of ICR values for the soil profiles used in this study are presented in Section 2.1.2.

Selection of soil profiles

Selection of the shear wave velocity profile of the soil strata for performing 1D seismic site response analysis is an important task because the composition of the strata has a direct influence on wave propagation. Seismic site response analysis requires detailed geotechnical characterization of the strata. Several studies have shown that soil profiles belonging to the same soil category, and thus the same V_s30 range, can exhibit variable seismic responses [²⁷,²⁸]. Therefore, proper characterization of representative soil strata is important for 1D seismic site response analysis. Santamarina et al. [²⁹] proposed the following relationship for the shear wave velocity profile along depth:

(5)V_{\mathrm{S}}=V_{\mathrm{S}\mathrm{0}}{\mathit{\sigma }}_{\mathrm{v}\mathrm{0}}^{\mathit{\eta }},

where V_S0 is the shear wave velocity at the free surface, and η is a parameter of the model. The total vertical stress (σ_v0) caters for the depth-dependency of shear wave velocity with the assumption of a constant value for soil mass density (ρ) (1900 kg/m³). Corigliano et al. [³⁰] used the V_s model developed by Santamarina et al. [²⁹] and employed Monte Carlo simulations to propose a parabolic V_s profile based on a target V_s30 for shallow (≤50 m deep) soil profiles; the hyperbolic V_s profile based on the work of Gibson [³¹] and Awojobi [³²] was also used.

However, in this study, V_s profiles for various site classes were adopted from the works of Douglas et al. [³³] and Toro [³⁴] and are depicted in Fig. 1. Douglas et al. [³³] proposed V_s profiles for various site classes based on statistical analysis of 858 real soil profiles from western USA, France, and Japan. Toro [³⁴], on the other hand, adopted the maximum likelihood procedure to generate V_s models for the Geomatrix and V_s30 classifications, using data from 557 soil profiles in the USA. A noteworthy characteristic of these profiles is the gradual increase in V_s with depth, without abrupt changes in the V_s of adjacent layers. It was further assumed that all layers in a single soil profile had the same PI values. These assumptions were made to limit the number of seismic site response simulations and to enable a clear comparison of the effect of basic variables (PI, ICR, and H) on the response parameters (β_s, V_s, and G_s).

NEHRP [⁵] site classes C and D cover a broad range of V_s30 values (180–760 m/s). These site classes were further subdivided into five categories to clearly reflect the effect of V_s30 on β_s, V_s, and G_s, which are listed in Table 2 along with the variations in the other geological and geotechnical parameters considered in this study. The median as well as the 10% and 90% confidence limits of the Douglas and Toro profiles were used to generate additional soil profiles for the proposed V_s30 ranges, as listed in Table 2. Specifically, the 90% and median profiles for EC-8 [6] Class B were adopted for soil profiles C_high and C_avg, respectively, for NEHRP [⁵] site class C. Similarly, the 90%, median, and 10% profiles of EC-8 [⁶] Class C soil were adopted to represent soil profiles D_high, D_avg, and D_low, respectively, for NEHRP site class D. The shear wave velocity of a few layers in these profiles was adjusted to match the target travel time-based average V_s30 for each site class [³⁵]. The travel-time averaged V_s30 is the shear wave velocity based on the time for the shear wave to travel from a depth of 30 m to the ground surface. V_s30 is calculated as follows:

(6)V_{\mathrm{s}\mathrm{3}\mathrm{0}}={\displaystyle \frac{\mathrm{30}}{\mathrm{\sum }\left(d_i/V_{\mathrm{s}i}\right)}}.

Note that V_s30 is not the same as the weighted average of the shear wave velocity of individual layers for the top 30 m depth, which is computed as follows:

(7)V_{\mathrm{s30�weighted}}={\displaystyle \frac{{\displaystyle \sum \left(V_{\mathrm{s}i}{d}_i\right)}}{\mathrm{30}}}.

The mechanical properties of the considered soil profiles are listed in Table 3. Two depths of soil strata, 40 m and 110 m, were considered in this study, as depicted in Fig. 1. The ICR values for type C soil in this study were 1–6.75 and 1.35–4 for the 40-m- and 110-m-deep strata, respectively. The ICR values for type D soil were 1–16 and 1.21–18 for the 40-m- and 110-m-deep strata, respectively.

Variation in geotechnical parameters

The geotechnical parameters whose variation was considered in the study included PI, OCR, and σ′, as listed in Table 2. The selected values of PI (0, 15, and 60) correspond to the generally recognized limits for non-cohesive soils (i.e., sand) and soils with low/medium and high plasticity, respectively. The variations in these geotechnical parameters necessitated the selection of appropriate soil modulus reduction and damping (MRD) curves for use in the site response analysis. Guerreiro et al. [³⁶] compared the MRD curves obtained by Vucetic and Dobry [³⁷], Darendeli [³⁸], and Ishibashi and Zhang [³⁹] and concluded that the first two sets of MRD curves were better-suited for practical applications. The effect of σ′ on MRD curves was accounted for in the Ishibashi & Zhang and Darendeli curves, while the effect of OCR was considered only in the Darendeli curves. On the contrary, the Vucetic and Dobry curves covered a wider range of PI values but did not consider σ′ and OCR. Darendeli [³⁸] noted that OCR had a rather insignificant effect on MRD curves for non-plastic soils and only a minor impact on MRD curves for plastic soils. Therefore, a constant value of OCR ( = 1) was adopted in this study for all soil profiles. The following two values of σ′ were considered in the Darendeli curves used in this study: 2 atm for non-cohesive (sandy) soils (PI = 0) and 4 atm for cohesive soils (PI = 15 and 60). The chosen σ′ values resulted in MRD curves that closely matched those obtained by Vucetic and Dobry [³⁷] and by Seed and Idriss [⁴⁰] for cohesive and non-cohesive soils, respectively. Constant σ′-based MRD curves were used for all soil layers in a stratum because the effect of σ′ on the variability in the MRD curves has minimal impact on the seismic site response analysis results, when compared with the influence of uncertainties associated with shear wave velocity, soil properties, and seismic input motions [⁴¹,⁴²].

Selection of seismic ground motions

Twenty-one actual far-field ground motions with PGA values of 0.036g–0.47g were selected based on PGA, V_s, and fault distance (>10 km per FEMA [⁴³] for far-field motions) for performing the 1D nonlinear seismic site response analysis. The acceleration response spectra of the strong motions are depicted in Fig. 2, while their salient features are listed in Table A-1 in Appendix A. The seismic ground motions were grouped into design basis earthquakes (DBE), functional evaluation earthquakes (FEE), and maximum credible earthquakes (MCE) based on median PGA values of 0.17g, 0.32g, and 0.42g, respectively. A few ground motions were scaled to match the target median PGA values, as shown in Table A-1. The average V_s values of the selected seismic records in the DBE, FEE, and MCE groups were 724, 703, and 792 m/s respectively, which were very close to the V_s values of 760 and 800 m/s for engineering bedrock, as per NEHRP [⁵] and EC-8 [⁶], respectively. The values of V_rock adopted in this study were between 600 and 3353 m/s, and ideally, the V_s value of the selected seismic ground motions should be in this range as well. However, recorded seismic ground motions on engineering bedrock are scarce [⁴⁴,⁴⁵]. Therefore, seismic ground motions with V_s values sufficiently close to that of the engineering bedrock definition were used in the study such that the selected V_s values also matched the target PGA values and satisfied the far-field fault distance criteria. The seismic ground motions used in this study were obtained from the PEER NGA-West2 website [⁴⁶].

1D seismic site response analysis

The previously mentioned soil profiles and bedrock parameters were used for conducting 1D site response analysis using the DEEPSOIL [⁴⁷] code, for the seismic ground motions listed in Table A-1. More than 2500 cases were analyzed considering all combinations of the five soil profiles, three PI values, two stratum depths, and 4–5 ICR values, for 21 seismic ground motions [5 × 3 × 2 × (4–5) × 21>2500]. DEEPSOIL was used to perform a 1D nonlinear seismic response analysis of the soil columns in the time domain. It solved the resulting equations of motion through step-by-step time-integration using the Newmark-β method [⁴⁸] with an average acceleration assumption, because integration is unconditionally stable for this method [⁴⁷]. The backbone or skeletal curve defining the soil constitutive stress strain (τ–γ) model in DEEPSOIL is the modified Kondner–Zelasko (MKZ) model. The MKZ model, a modification of the hyperbolic model developed by Kondner and Zelasko [⁴⁹], was introduced by Matasović and Vucetic [⁵⁰] and is expressed as follows:

(8)\tau \left(\mathit{\gamma }\right)={\displaystyle \frac{G_{\mathrm{0}}\mathit{\gamma }}{\mathrm{1}+\alpha {\left({\displaystyle \frac{\mathit{\gamma }}{{\mathit{\gamma }}_{\mathrm{ref}}}}\right)}^s}},

where τ is shear stress, γ is shear strain, G₀ is maximum (small strain) shear modulus, γ_ref is reference strain (γ_ref€=€τ₀/G₀, τ₀ is the soil shear strength), and α and s are the curve-fitting parameters that adjust the shape of the backbone curve. The hysteretic behavior of soil in DEEPSOIL is defined in accordance with the Masing [⁵¹] or extended Masing rules [⁵²,⁵³], and the unloading, reloading, and cyclic degradation relationship is expressed as follows:

(9)\tau ={\displaystyle \frac{\mathrm{2}{G}_{\mathrm{0}}\left({\displaystyle \frac{\mathit{\gamma }-{\mathit{\gamma }}_{\mathrm{rev}}}{\mathrm{2}}}\right)}{\mathrm{1}+\alpha {\left({\displaystyle \frac{\mathit{\gamma }-{\mathit{\gamma }}_{\mathrm{rev}}}{\mathrm{2}{\mathit{\gamma }}_{\mathrm{ref}}}}\right)}^s}}+{\tau }_{\mathrm{rev}},

where τ_rev and γ_rev are the reversal point stress and strain, respectively. Details regarding the implementation of these rules are provided elsewhere [⁵⁴]. The hysteretic soil damping (β_s) for nonlinear analysis in DEEPSOIL is calculated using the MKZ soil constitutive model in conjunction with the Masing rules, as follows [⁵⁵]:

(10){\mathit{\beta }}_{\mathrm{s}}={\displaystyle \frac{\mathrm{4}}{\mathrm{\pi }}}{\displaystyle \frac{{\displaystyle {\int }_{\mathrm{0}}^{{\gamma }_{\mathrm{c}\mathrm{0}}}f}\left(\mathit{\gamma }\right)\mathrm{d}\mathit{\gamma }}{f\left({\mathit{\gamma }}_{\mathrm{c}\mathrm{0}}\right){\mathit{\gamma }}_{\mathrm{c}\mathrm{0}}}}-{\displaystyle \frac{\mathrm{2}}{\mathrm{\pi }}},

where f(γ) = τ, which is the initial backbone curve, and γ_c0 is the strain amplitude for which β_s is computed. Damping in soil at small strains (<0.01%) cannot be adequately captured from the hysteretic behavior, because hysteresis loops are almost non-existent. Therefore, small-strain viscous damping is added to DEEPSOIL based on the Rayleigh damping formulation [⁵⁶]; thus, the damping matrix C becomes C = a₀M + a₁K, where M and K are the mass and stiffness matrices, respectively, a_{\mathrm{0}}={\displaystyle \frac{\mathrm{4}\mathrm{\pi }\xi }{T_{\mathrm{1}}+T_{\mathrm{2}}}} and a_{\mathrm{1}}={\displaystyle \frac{T_{\mathrm{1}}{T}_{\mathrm{2}}\xi }{\mathrm{\pi }\left(T_{\mathrm{1}}+T_{\mathrm{2}}\right)}}. T₁ is the fundamental period of the strata (T₁ = 4H/V_s, where H is the stratum depth, and V_s is the travel-time-averaged shear wave velocity in the strata), T₂ is the period of another target mode, which is commonly taken as 0.2T₁ [⁴⁷], and ξ is the viscous soil damping ratio. In this study, the values of T₁ were 0.24, 0.31, 0.39, 0.52, 0.87 s and 0.45, 0.61, 0.69, 0.86, 1.41 s for the 40-m- and 110-m-deep strata, respectively, corresponding to site classes C_high, C_avg, D_high, D_avg and D_low.

Skeletal MRD curves obtained by Darendeli [³⁸] were used for the soil profiles included in this study. Parameters α and s in Eq. (8), which correlate the skeletal MRD curves and the constitutive MKZ relationship, were obtained using the built-in curve-fitting feature of DEEPSOIL [⁴⁷]. The option that simultaneously matches both modulus reduction and damping curves with an acceptable error was utilized because the damping values are unreasonably over-estimated when only the modulus reduction curve is matched for the MKZ modeling. Figure 3 presents the skeletal curves and fitted MKZ curves along with the values of the curve parameters. A damping ratio of 1% was assigned to the bedrock in the 1D seismic site response analysis.

Computation of average seismic soil design parameters and sensitivity analysis

In a nonlinear time-history analysis, all response parameters (γ, G_s, β_s, etc.) change during the passage of a seismic wave. However, only one effective value of these parameters needs to be adopted for the foundation design. For the equivalent-linear site response analysis procedure, it is common to choose an effective soil strain (γ_eff) value of 0.65γ_max [⁵⁷]. Therefore, in this study, this value of γ_eff was used to obtain the values of G/G₀ and β_s from the skeletal backbone curves for each layer in the soil column after completion of the 1D nonlinear site response analysis. Thereafter, a weighted average was used to determine the seismic design values of β_s and G_s in various layers of the top 30 m strata, as indicated in Section 3. However, the value of V_s was computed considering the travel-time-averaged shear wave velocity in the top 30 m strata.

The aforementioned average values were then used to conduct sensitivity studies to determine the impact of variations in PI, ICR, and H on β_s, V_s, and G_s, as discussed in Section 4. Based on the 1D seismic site response analysis and sensitivity analysis results, predictive relationships for β_s, V_s, and G_s were derived using multi-variable regression analysis in Section 5.

Comparison of seismic soil design parameter values with NEHRP

In the final part of the study, the values of β_s, V_s, and G_s determined from the 1D site response analysis results were compared with the corresponding values recommended by NEHRP [⁵]; conclusions regarding the applicability of the NEHRP recommended values were drawn and are presented in Section 6.

Results and discussion

1D seismic site response analysis results

Variations in β_s, V_s, and G_s along the stratum depth for various analyses, as outlined in Table 2, were determined by conducting a 1D seismic site response analysis for the chosen ground motions, as indicated in Section 2. The weighted damping ratio in the top 30 m depth was computed using the following relationship for a particular seismic ground motion analysis:

(11){\mathit{\beta }}_{\mathrm{s}}={\displaystyle \frac{{\displaystyle \sum d_i}{\mathit{\beta }}_i}{\mathrm{30}}}.

The weighted soil shear modulus in the top 30 m strata was also computed using an expression similar to Eq. (11). The time-averaged shear wave velocity in the top 30 m strata (V_s) was computed using Eq. (6). The top 30 m depth was selected for computing these parameters because soil properties at this depth are used by NEHRP [⁵] to characterize site classes.

After computing the weighted damping ratio, soil shear modulus, and time-averaged V_s in the top 30 m strata for each seismic ground motion, the average values of seven ground motions in the respective seismic intensity categories (DBE, FEE, and MCE were determined for each combination of PI, V_rock, site classification, and stratum depth. Figure 4 presents these results in a 3D pictorial form for β_s, V_s, and G_s. A side-by-side comparison of the results for the 40-m- and 110-m-deep strata is presented in Fig. 4. The following observations were made.

a)‚ β_s increased, while V_s and G_s decreased with an increase in PGA.

b)‚ β_s decreased with an increase in PI. This indicates that non-plastic (sandy) soils exhibited more damping than plastic (clayey) soils. In contrast, V_s and G_s increased with increasing PI. This indicates that non-plastic (sandy) soils in the same site class had smaller V_s values than plastic (clayey) soils.

c)‚ β_s increased with a decrease in V_s30, that is, softer soils exhibited more damping; the reverse was true for V_s and G, which decreased as the soil became softer.

d)‚ An increase in ICR or V_rock within a soil type and for the same PI value resulted in an increase in β_s. However, an increase in ICR within a soil type and for the same PI value decreased V_s and G_s.

e) ‚The impact of ICR on the variation in β_s, V_s, and G_s was greater in soil type D than in soil type C.

f)‚There was no significant difference in the values of β_s, V_s, or G_s between the 40-m and 110-m strata profiles.

Discussion of 1D seismic site response analysis results

Damping ratio, β_s

Figure 5 presents the disaggregated results for the variation in β_s from the 3D graphs presented in Fig. 4. Separate plots for each earthquake intensity were obtained for the five site classes, PI values, and V_rock for both the 40-m and 110-m strata. The numerical values along the x-axis in these plots correspond to V_rock. The NEHRP-recommended values for β_s for each earthquake intensity and site class are also shown on these plots for comparison. The β_s values exhibited a wide variation due to the variation in PI for each earthquake intensity for all site classes. However, the difference was particularly pronounced for site class D. Additionally, β_s exhibited a sharp increase with an increase in V_rock, and increasing earthquake intensity also increased β_s values. Differences in stratum depth (H), however, seemed to have a relatively minor influence on the β_s values. The NEHRP values agreed sufficiently with the β_s values only for site class C_high, whereas for the Class D soils, the difference between the NEHRP values and the analysis results was significant, particularly for soil profiles with a PI of 0 and 15.

Shear wave velocity, V_s

The disaggregated values of V_s from Fig. 4 are presented in Fig. 6 to illustrate the variations in V_s for the three seismic intensities with respect to PI, V_rock, and site class. The NEHRP-recommended design values of V_s for site classes C and D are also plotted for comparison. It was observed that V_s generally decreased with a reduction in V_s30 and PI. A consistent difference was observed in the V_s values due to the variation in PI for all site classes and earthquake intensities. The variation in V_s due to the variation in V_rock was milder than the variation in β_s values. According to a comparison between the V_s results for the 40-m and 110-m strata, the stratum depth has a rather minor impact on V_s. The NEHRP-recommended design values of V_s did not generally cover the observed variation in the V_s values, especially the difference increased significantly in the soil profiles of site class D.

Soil shear modulus, G_s

Figure 7 presents the disaggregated results for G_s from Fig. 4, along with the design values recommended by NEHRP [5]. The observations regarding the variation in G_s with respect to variations in PI, V_rock, earthquake intensity, and stratum depth were similar to those in the case of V_s; this was due to the relationship between the two parameters, which is expressed in Eq. (3).

Effect of earthquake intensity on design parameters

The effect of earthquake intensity on variation in the design values of β_s, V_s, and G_s is recognized by NEHRP [5]. The effect of earthquake intensity on variation of β_s, V_s, and G_s as observed from Fig. 4 is presented in Figs. 8, 9, and 10, respectively, for various V_rock values. The variation in V_rock significantly affected the β_s value for all values of PI in all site classes, except in some cases of C_high and D_low, as depicted in Fig. 8. However, the variation in V_rock had a moderate impact on the V_s and G_s values for non-plastic soils (PI = 0) and a minor impact for medium and highly plastic soils, as shown in Figs. 9 and 10, respectively.

A strong positive dependence of β_s on earthquake intensity was noted for all three PI values, as depicted in Fig. 8, except for soil class C_high with medium and high plasticity (PI = 15 and 60). In contrast, V_s and G_s exhibited a sharp decrease with an increasing seismic intensity for non-plastic and medium plastic soils, as depicted in Figs. 9 and 10, respectively. However, the decrease in V_s and G_s with increasing seismic intensity was less prominent for highly plastic soils.

Sensitivity analysis for seismic soil design parameters

This section presents the results of the sensitivity analysis for the seismic soil design parameters (β_s, V_s, and G_s) with respect to PI, V_rock, and stratum depth (H). It is noteworthy that the influence of these parameters is not considered for the design values of β_s, V_s, and G_s in the NEHRP [⁵] provisions.

The sensitivity analyses for β_s, V_s, and G_s with variations in PI and V_rock values were conducted for both values of H based on Figs. 5, 6, and 7, respectively. These figures plot the average values of a parameter (β_s, V_s, or G_s) for the seven seismic ground motions of a given seismic intensity (DBE, FEE, or MCE for a particular value of V_rock and PI and H. The Morris method [⁵⁸], which is a non-statistical global sensitivity analysis technique, was used in this study due to its lower computational cost, ease of result interpretation, and unambiguous determination of sensitivity [⁵⁹–⁶¹]. This method is particularly suitable for models whose behavior is influenced by a small number of input parameters. The Morris method is a one-at-a-time (OAT) sensitivity analysis technique in which one parameter is varied at a time, and its effect on the output is examined through the computation of the first derivative with respect to each input parameter. Morris termed these derivatives (\partial y/\partial x_i) the elementary effects (EE), which are denoted as d_i herein. The Morris sensitivity parameters (SP) are the mean (µ_i) and standard deviation (σ_i) of the EEs. A large µ_i indicates a significant overall influence on the output, while a large value of σ_i is indicative of nonlinear effects or interaction with other parameters. In a model with r EEs, the µ_i and σ_i values for the ith parameter are estimated as follows:

(12){\mu }_i={\displaystyle \sum _{j=\mathrm{1}}^r\frac{d_i^j}{r}},

(13){\sigma }_i=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{j=\mathrm{1}}^r{\left(d_i^j-{\mu }_i\right)}^{\mathrm{2}}}}{r-\mathrm{1}}}}.

The Morris plot presents the relationship between µ_i and σ_i for each input parameter, and the sensitivity of each input parameter is visually discerned by placing the most influential parameter(s), which have a higher µ_i, on the right side of the plot. Mohanty and Codell [⁶¹] suggested normalizing the EE with respect to the mean value of the output (i.e., d_i/\overline{y}) for a direct comparison of input parameter sensitivities that have different units.

The various aspects of the sensitivity study are presented in the following sections. The normalized EEs are determined in Section 4.1, and the Morris SPs (µ_i and σ_i) are computed in Section 4.2. A summary of the sensitivity analysis is presented in Section 4.3.

Computation of elementary effects (EE)

The following three EEs were considered in this study: PI, ICR, and H, and the influence of each EE on β_s, V_s, and G_s was evaluated.

Effect of PI

Soil index properties, especially PI, considerably influence the seismic behavior of a stratum, as soils with different PI values exhibit markedly different stress–strain and damping behaviors with increasing ground acceleration [³⁷–³⁹]. However, the design values of β_s, V_s, and G_s in NEHRP [⁵] are dependent only on V_s30 and the level of seismic excitation; the soil index properties are ignored. The EE of PI is presented in Fig. 11 as the percentage difference in the values of β_s, V_s, and G_s due to the variation in the PI values for i set of conditions. This EE is computed using Eq. (14), with V_s as an example parameter:

(14){\left(\mathrm{d}{V}_{\mathrm{s},\mathrm{P}\mathrm{I}}^i\right)}_{\mathrm{SC}}={\displaystyle \frac{\max {\left|V_{\mathrm{s},P{I}_{j+\mathrm{1}}}^i-V_{\mathrm{s},P{I}_j}^i\right|}_{j=\mathrm{1},...,k}}{{\displaystyle \frac{\mathrm{1}}{k}}{\displaystyle \sum _{j=\mathrm{1}}^k{V}_{\mathrm{s},P{I}_j}^i}}}

where SC= site class, i = same conditions in terms of ICR and seismic intensity for a particular site class and stratum depth, and j is the index for the PI values that has k values for which V_s is computed. The numerical values on the x-axis of the plots in Fig. 11 correspond to V_rock.

The following observations were made:

i) The differences in the values of β_s, V_s, and G_s due to the variation in PI values were generally greater for a stronger seismic excitation for both stratum depths; except for the β_s values for class D soil profiles where the difference was greater for FEE and DBE motions.

ii) β_s was affected the most for class C soils, while G_s was affected the most for class D soils, due to the variation in PI. In both cases, the maximum difference was greater than 100%.

iii) The average differences in the β_s values due to the PI variation for class C and D soil sites were 75% and 66%, respectively, when averaged for the two stratum depths. The average differences were 12% and 35% for V_s and 24% and 70% for G_s for class C and D soil sites, respectively. This indicates that the PI variation had the greatest effect on the β_s values in both site classes.

iv) The differences in the values of β_s, V_s, and G_s due to PI variation increased with increasing V_rock for all site classes. The only exception was for β_s for site class D, which presented a decreasing trend with increasing V_rock.

Effect of ICR

The impedance contrast between rock and soil strata causes dispersal of propagating seismic waves and impacts the seismic amplification factors and seismic soil parameters (β_s, V_s, and G_s) used in foundation design [⁶²]. The EEs of ICR on β_s, V_s, and G_s are presented in Fig. 12 for l set of conditions. The differences in the values of β_s, V_s, and G_s due to the EE of V_rock are computed using Eq. (15), with G_s as an example parameter.

(15){\left(\mathrm{d}{G}_{\mathrm{s},\mathrm{I}\mathrm{C}\mathrm{R}}^l\right)}_{\mathrm{SC}}={\displaystyle \frac{\max {\left|G_{\mathrm{s},IC{R}_{m+\mathrm{1}}}^l-G_{\mathrm{s},IC{R}_m}^l\right|}_{m=\mathrm{1},...,n}}{{\displaystyle \frac{\mathrm{1}}{n}}{\displaystyle \sum _{m=\mathrm{1}}^n{G}_{\mathrm{s},IC{R}_m}^l}}},

where l are the same conditions of PI and seismic intensity for a SC and H, and m is the index for ICR that has n values for which G_s is determined.

As shown in Fig. 12, the ICR affected the β_s values the most, with a maximum difference of approximately 60%. The V_s values were affected the least by variation in ICR, with a maximum difference of slightly more than 25%. The maximum difference for G_s was approximately 55%. The average differences in the design parameter values due to the ICR variation for site class C were 35%, 4.5%, and 8% for β_s, V_s, and G_s, respectively. The average differences for site class D was 33%, 11%, and 22% for β_s, V_s, and G_s, respectively. The difference was observed to decrease with increasing PI values, except for β_s in site class D, for which the difference increased with increasing PI.

To determine a correlation between ICR and β_s, V_s, and G_s, each parameter was plotted against ICR considering the average value of three seismic intensities (DBE, FEE, and MCE) for the three values of PI in Fig. 13, with H = 40 m. The coefficient of determination (R²) was observed to vary between 0.32 and 0.41 for the regression lines fitted for β_s. The corresponding coefficient of correlation (R) was between 0.57 and 0.64, respectively. R was between 0.5 and 0.8, indicating that a moderate correlation existed between ICR and β_s [63]. On the contrary, R² varied between 0.19 and 0.22 for the regression lines plotted for V_s and G_s. The corresponding R was between 0.43 and 0.47, which is less than 0.5. Therefore, V_s and G_s were weakly correlated with ICR [⁶³]. This conclusion was confirmed through the sensitivity analysis presented in Section 4.2.

Effect of stratum depth

The EE of stratum depth on β_s, V_s, and G_s was computed using Eq. (16), with β_s as an example parameter:

(16){\left(\mathrm{d}{\mathit{\beta }}_{\mathrm{s},\mathrm{H}}^p\right)}_{\mathrm{SC}}={\displaystyle \frac{\left|{\mathit{\beta }}_{\mathrm{40}}^p-{\mathit{\beta }}_{\mathrm{110}}^p\right|}{\mathrm{avg}\left({\mathit{\beta }}_{\mathrm{40}}^p·{\mathit{\beta }}_{\mathrm{110}}^p\right)}}

where p are the same conditions of PI, ICR, and seismic intensity for a SC, and H is the stratum depth. Figure 14 presents the results of Eq. (16) for compatible values of V_rock. In some cases, the V_rock values did not match exactly in the two stratum depths, but were sufficiently close. For example, in site class D_avg, the differences in the parameter values for V_rock = 967 m/s in 110-m strata and V_rock = 760 m/s in the 40-m strata is reported for V_rock value of 967 m/s in Fig. 14. The average differences in the values of β_s, V_s, and G_s due to differences in the stratum depth for the soil profiles in site class C were determined to be 8%, 1%, and 2%, respectively. Similarly, the average difference for the profiles in site class D were 9.7%, 5.6%, and 1.7% for β_s, V_s, and G_s, respectively. These relatively small values indicate that β_s, V_s, and G_s are less sensitive to variation in values of H, as verified through the sensitivity analysis presented in Section 4.2.

Sensitivity analysis based on Morris OAT method

The Morris SPs, µ_i and σ_i, for the EEs determined in Section 4.1 were computed using Eqs. (12) and (13); the Morris plots for these SPs are presented in Fig. 15(a), 15(b), and 15(c) for β_s, V_s, and G_s, respectively. The values of the SPs are also listed in Table 4, in which the SP values for PI and ICR are the average values for the two stratum depths. Each pair of SPs in Fig. 15 refers to a particular site class.

As shown in Fig. 15(a), the effects of PI, ICR, and stratum depth on β_s are not only suitably ranked but also clearly segregated, with PI and H being the most and least influential parameters, respectively. A higher σ value for PI may be indicative of nonlinear effects.

The Morris plot for V_s is presented in Fig. 15(b) and it was noted that the sensitivity rank of parameters was the same as for β_s, with PI and H being the most and least influential parameters, respectively. The values of µ for all parameters were approximately 50% of those for β_s. However, the variation in σ was similar to that in β_s, which indicated that V_s was not as sensitive as β_s to the variations in the EEs.

Figure 15(c) presents the Morris plot for G_s; the µ values for PI were noted to be as high as those for β_s, indicating a high influence of G_s on the design values. However, the σ values for PI were higher than those for β_s, indicating more nonlinear effects or interactions with other parameters. The SPs had the lowest values of H, which indicated that stratum depth had the least significant influence on G_s.

Summary of sensitivity analysis

Table 5 lists the sensitivity indicators (SIs) for β_s, V_s, and G_s for the EEs of PI, ICR, and H, based on the SPs listed in Table 4. The SIs were based on the µ values in Table 4 and were defined as follows: negligible (<12%); low (12%–25%); moderate (25%–50%); high (50%–80%) and very high (>80%).

Based on the defined SIs, the variation in PI had a high to very high effect on β_s for both site classes C and D. The PI variation had moderate and negligible effects on V_s in site classes D and C, respectively. The variation in PI had high and low-to-moderate effects on G_s in site classes D and C, respectively.

Variations in ICR had a moderate effect on β_s for all site classes except D_low, for which the effect was negligible. The effect of ICR variation on V_s was negligible for all site classes except site class D_avg, for which the effect was low. For G_s, the effect of ICR was negligible for site class C and low to moderate for site class D.

The impact of H on β_s, V_s, and G_s was found to be the least, and as the m values for this parameter were was less than 12%, the effect of H on the values of the design parameters was considered to be negligible for both site classes.

Predictive relationship for values of design parameters

Based on the results of the sensitivity analysis presented in Section 4, the variation in PI values had the greatest impact on the values of β_s, V_s, and G_s. The ICR had moderate impact on β_s values, while its effects on V_s and G_s were largely negligible. Finally, stratum depth did not have a significant impact on the values of β_s, V_s, and G_s. Predictive relationships for the values of β_s, V_s, and G_s were derived through regression of the 1D nonlinear site response analysis results while considering the sensitivity analysis results.

Predictive relationship for β_s

Predictive relationships for β_s were determined by considering the impact of PI and ICR based on the sensitivity analysis results. Stratum depth (H) was excluded from the predictive relationships for β_s, based on the SA results. Because β_s was found to be highly sensitive to PI (refer to Table 5), three separate relationships were derived for each of the three PI values considered in this study. The general form of the relationship for β_s is given by Eq. (17):

(17){\mathit{\beta }}_s=\left(c_{\mathrm{1}}{V}_{s\mathrm{3}\mathrm{0}}+c_{\mathrm{2}}\mathit{\alpha }+c_{\mathrm{3}}\right)A^{\left(c_{\mathrm{4}}{V}_{s\mathrm{3}\mathrm{0}}+c_{\mathrm{5}}\mathit{\alpha }+c_{\mathrm{6}}\right)},

where β_s is soil damping ratio (%), α is ICR [1≤α≤18], V_s30 is small strain shear wave velocity that defines a site class [180≤V_s30≤600 m/s], A is peak ground acceleration [0.04 g≤A≤0.46 g], and c₁–c₆ are the regression coefficients listed in Table 6. Linear interpolation can be used to determine c₁–c₆ for other PI values.

Figure 16 presents the values of β_s from the 1D site response analysis and the values estimated using Eq. (17). The ICR term was not included in the estimated values in Fig. 16(a) but was included in the values in Fig. 16(b). Ignoring ICR had a detrimental effect on β_s for values between 7%–15%. More than 90% of the estimated values were plotted with±2% of the analysis results, considering the effects of both PI and ICR. The measure of closeness, λ, between the estimated values and the analysis results was computed using the concept of weighted least square errors, which is shown in Eq. (18):

(18)\mathit{\lambda }=\mathrm{1}-{\displaystyle \frac{\mathrm{1}}{n}}{\displaystyle \sum _{i=\mathrm{1}}^n\sqrt{{\left[\mathrm{1}-{\left({\displaystyle \frac{{\mathit{\beta }}_{\mathrm{estimated}}}{{\mathit{\beta }}_{\mathrm{analysis}}}}\right)}_i\right]}^{\mathrm{2}}}},

where λ was found to be 0.79 and 0.84 for Figs. 16(a) and 16(b), respectively. The estimated values of β_s improved by 5% when ICR was included in the predictive equation.

Predictive relationships for V_s and G_s

The results of the sensitivity analysis indicated that PI variations had moderate and high effects on V_s and G_s, respectively. Contrarily, the effects of ICR on V_s and G_s, were negligible and low respectively; therefore ICR was not included in the predictive relationships for V_s and G_s. Similarly, stratum depth (H) had a negligible effect on both these design parameters and was not included in the predictive relationships either. Additionally, V_s and G_s are related through Eq. (3). Therefore, one predictive relationship of the form given by Eq. (19) was used for both parameters. This relationship considers the influence of PI in addition to that of two other parameters (i.e., V_s30 and PGA) that are considered in NEHRP [⁵] for prescribing the design values of V_s and G_s:

(19)V_{\mathrm{s}}\mathrm{or}{G}_{\mathrm{s}}=\left\{\left(c_{\mathrm{1}}P+c_{\mathrm{2}}\right)V_{\mathrm{s}\mathrm{3}\mathrm{0}}^{\mathrm{2}}+\left(c_{\mathrm{3}}P+c_{\mathrm{4}}\right)V_{\mathrm{s}\mathrm{3}\mathrm{0}}+c_{\mathrm{5}}P+c_{\mathrm{6}}\right\}A+\left(c_{\mathrm{7}}P+c_{\mathrm{8}}\right)V_{\mathrm{s}\mathrm{3}\mathrm{0}}+\left(c_{\mathrm{9}}P+c_{\mathrm{10}}\right),

where V_s is the design value of the shear wave velocity (m/s), G_s is the design value of the shear modulus (MPa), P = PI, V_s30 = small strain shear wave velocity (m/s), A= peak ground acceleration, and c₁–c₁₀ are the regression coefficients listed in Table 7. The applicable ranges of V_s30 and A are the same as those for β_s.

Figures 17 and 18 present the 1D seismic site response analysis and the estimated values of V_s and G_s, respectively. The ±10% bounds with respect to the analysis values are also plotted in these figures. Almost all data points fall within these bounds when PI is included in the estimation of these design parameters. The closeness, λ, between the analysis and the estimated values, which is calculated using an equation similar to Eq. (18), was determined for V_s and G_s with and without the consideration of PI. The values of λ determined from Fig. 17 for V_s were 0.94 and 0.88 for the cases with and without PI, respectively. Similarly, the values of λ determined from Fig. 18 for G_s were 0.89 and 0.77 for the cases with and without PI, respectively. The higher values of λ for the cases that included PI highlighted the importance of PI in determining the design values of these parameters for a given site class and level of seismic intensity. Ignoring the effect of PI resulted in overall errors of 6% and 12% in V_s and G_s, respectively.

Comparison with NEHRP values

NEHRP prescribes the design values of β_s, V_s, and G_s based on the site class (as determined from V_s30) and seismic intensity listed in Tables 19.3-1 to 19.3-3 [⁵]. Figures 19, 20, and 21 compare the average values of β_s, V_s, and G_s, respectively, obtained from the 1D site response analysis with the corresponding NEHRP-recommended values and the proposed values, which are based on Eqs. (17) and (19), respectively. The design parameter values were plotted separately for site classes C and D for the 40-m- and 110-m-deep strata. A comparison between the NEHRP, analysis and the predicted values of β_s, V_s, and G_s is presented in the following section.

Comparison of β_s values

Figures 19(a) and 19(b) show the average values of β_s for site classes C and D respectively. It was noted that β_s values were generally underestimated by NEHRP for non-plastic and medium plastic soils; however, the β_s values recommended by NEHRP [⁵] were closer to the analysis results and estimated values for highly plastic soils (PI = 60). The difference between the analysis-derived and NEHRP values of the damping ratio was greater for site class D than for site class C. The values of β_s estimated using Eq. (17) were in close agreement with the NEHRP values for a PI of 60 for both site classes. However, there was a considerable discrepancy between the two sets of values for non-plastic and medium plastic soils.

Comparison of V_s/V_s30

Figure 20 presents a comparison of the V_s/V_s30 ratio for values obtained from the 1D site response analysis, the NEHRP-recommended values, and the ones computed using the predictive relationship in Eq. (19), for both site classes. The NEHRP values were generally higher for non-plastic soils for both site classes C and D. The discrepancy was significant (more than 90%) for class D soils under the MCE seismic intensity. However, for highly plastic soils, the NEHRP values were close to the analysis-derived values or in some cases, less than the latter. The discrepancy between the analysis-derived and NEHRP values generally increased with increasing seismic intensity. The predicted values agreed sufficiently with the NEHRP values for site class C. However, the predicted values were slightly higher than the NEHRP values for highly plastic soils and significantly lower than the NEHRP values for non-plastic and medium plastic soils for site class D.

Comparison of G_s/G₀

A comparison of the NEHRP shear modulus reduction (G_s/G₀) values with the analysis-derived and predicted values is presented in Fig. 21. The NEHRP values of G_s/G₀ matched closely with the analysis-derived and predicted values, which were representative of medium plastic soils for site class C. The G_s/G₀ values, in NEHRP, for non-plastic and highly plastic soils were over- and under-estimated, respectively, for this site class. The NEHRP values of G_s/G₀ were generally over-estimated for non-plastic and medium plastic soils for site class D. The discrepancy between the NEHRP and analysis-derived values was significant (more than 150% in some cases) for non-plastic and medium plastic soils for this site class. However, for highly plastic soils, the NEHRP values were generally conservative compared with the analysis-derived and predicted values for site class D.

Discussion

From Figs. 19, 20, and 21, it is evident that the NEHRP-recommended design values for β_s, V_s, and G_s, respectively, were not representative of certain soil types. It was found that the NEHRP values of β_s were generally low when compared with the analysis-derived and predicted values. The discrepancy was substantial for non-plastic and medium plastic soils in site class D. Because lower values of the damping ratio are recommended by NEHRP, the seismic design of foundations based the NEHRP guidelines may be conservative, considering that a lower damping ratio may necessitate a larger-than-required foundation. However, the NEHRP-recommended design values for V_s and G_s were generally higher than the analysis-derived and predicted values for most soil sub-classes investigated in this study. This discrepancy was significant for non-plastic and medium plastic soils in both site classes C and D. Because the NEHRP-recommended design values were higher than the analysis-derived values, the seismic design of foundations based on the NEHRP guidelines could be unconservative (and may be unsafe) because foundation stiffness is directly related to V_s and G_s, and a smaller-than-required foundation would be designed for higher values of V_s and G_s.

The NEHRP values for the three design parameters were observed to be relatively close to the analysis-derived and predicted values for highly plastic soils (PI = 60). The exception was the damping ratio in site class D, which was underestimated. The results of the sensitivity analysis presented in Section 4 indicate that β_s, V_s, and G_s were most sensitive to the variation in PI. Therefore, PI should also be included along with the current parameters of V_s30 and seismic intensity in the metric for selecting the values of foundation design parameters, viz., β_s, V_s, and G_s.

Conclusions

In this study, 1D nonlinear seismic site response analysis was conducted for five soil profiles included in NEHRP site classes C and D, for various combinations of PI and ICR and for two stratum depths, under 21 seismic ground motions. A sensitivity analysis was conducted to identify the factors that affect the seismic soil design parameters (β_s, V_s, and G_s). The following conclusions were drawn from this study.

1)‚The influence of PGA, soil type, ICR, stratum depth, and PI on the values of the seismic soil design parameters (β_s, V_s, and G_s) was investigated. The sensitivity analysis indicated that variations in PI influenced the effective design values of β_s, V_s, and G_s the most. The effects of variations in ICR on β_s and on V_s and G_s were moderate and low to moderate, respectively. In contrast, variations in stratum depth had an insignificant-to-low effect on β_s, V_s, and G_s.

2)‚Predictive relationships for the seismic design values of β_s, V_s, and G_s were proposed based on the results of the 1D seismic site response analysis as well as the sensitivity analysis. The predicted parameter values matched sufficiently with the analysis-derived values, as evidenced by the closeness index (λ) values of 0.84, 0.94, and 0.89 for β_s, V_s, and G_s, respectively.

3)‚The NEHRP-recommended values of β_s were generally lower than the analysis-derived and predicted values. The discrepancy increased with decreasing soil plasticity and increasing seismic intensity. Based on this discrepancy, the lower β_s values prescribed in NEHRP may result in a higher seismic response and hence, a more conservative foundation design.

4)‚The NEHRP-recommended design values for V_s and G_s were higher than the analysis-derived as well as the predicted values. Similar to the case of the β_s values, the discrepancy increased with decreasing soil plasticity and increasing seismic intensity. As the soil shear modulus values recommended in NEHRP were greater than the analysis-derived values, the seismic response may be under-predicted, which would lead to an unconservative foundation design.

5)‚The NEHRP-recommended values nearly matched the analysis-derived values for highly plastic soils (PI = 60). The plasticity index was found to be the most influential parameter affecting the effective seismic design values of β_s, V_s, and G_s. Therefore, PI should be included in the metric for determining the effective design values of β_s, V_s, and G_s, along with V_s30 and seismic intensity, which are currently present in the NEHRP guidelines.

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