Comparison of Seismic Site Factor Models Based on Equivalent Linear and Nonlinear Analyses and Correction Factors for Updating Equivalent Linear Results for Charleston, South Carolina

Abstract

In practice, site-specific one-dimensional (1D) seismic site response analyses are conducted to compute surface acceleration time histories considering shear wave velocity profile, modulus reduction, damping, and site-specific ground motions. The computed surface responses depend not only on the geologic and seismic characteristics but also on the type of 1D analysis (i.e., equivalent linear or nonlinear) and the software. Equivalent linear analysis (EQLA) is preferred by practicing engineers because the analysis procedure is well defined, but the accuracy of the results is questionable for certain geologic and input motion characteristics. On the other hand, nonlinear analysis (NNLA) is accurate for any geologic and input motion characteristics, but it is complicated because certain steps in the analysis procedure are complicated and not well defined. The objective of this study is to compare the responses computed from EQLA and NNLA procedures and make recommendations on when to use EQLA and NNLA, considering Charleston, South Carolina; geology; and seismicity. About 18,000 NNLAs (DMOD2000 and DEEPSOIL) and EQLAs (SHAKE2000) were performed, considering variations in shear wave velocity profiles, shear modulus reduction curves, damping curves, and ground motions. Based on the results from each software, three seismic site factor models were developed and compared with the published models. Results show that the EQLAs produced conservative estimates compared to the NNLAs. It is also observed that the site factor model based on EQLA diverges from the models based on NNLA even at the lowest amplitude shaking considered in the study (0.05 g), particularly for profiles with low shear wave velocity. This indicates that soils behave nonlinearly even at low amplitude shaking. Although a similar shear stress/shear strain model is used in DMOD2000 and DEEPSOIL, the site factor models show significant differences. Finally, an easy-to-use chart was developed to select suitable software and analysis types for accurately computing the surface responses based on the peak ground acceleration (PGA) of the input motion at the reference rock outcrop and average shear wave velocity in the top 30 m.

1. Introduction

Charleston, South Carolina (SC), is one of the most seismically active areas in the eastern United States. Seismic site coefficients that are used in the design of transportation and other structures are typically computed by conducting site response analysis considering site-specific geologic and seismic conditions. Several researchers have focused on developing inputs for site response analysis, such as shear wave velocity, modulus reduction, and damping models [1,2,3,4], and also synthetic and real ground motions [5,6]. Such models are being developed not only at local scales but also at state and regional scales. The estimate of the surface response depends upon the characteristics of the ground motions, properties of the soil profile, and the theory implemented in the software used for the analysis. Typically, equivalent linear (EQL) or nonlinear (NNL) site response analysis is conducted to estimate surface acceleration time history given the seismic motion at the bedrock or reference outcrop and the properties of the soil profile. However, no well-defined guidelines exist for choosing the analysis procedure and software. The EQLA procedure requires a few well-defined input parameters and accurately predicts the surface response when the soil profile behaves within the linear range, which depends not only on the soil profile properties but also on the characteristics of the ground motion. On the other hand, the NNLA procedure is more accurate for a wide range of soil properties and input motion characteristics, but the procedure is not well defined and is the least preferred [7,8].

The reference outcrop spectral acceleration (S_Outcrop) and the seismic site factors (F_T) are used to develop an acceleration design response spectrum (ADRS), an input for the seismic design of transportation and other structures. The F_T at a period (T) is the ratio between the computed surface spectral acceleration (S_Site) and the S_Outcrop in the same period. The site factors computed using EQLA and NNLA codes can vary significantly, especially when the soil column experiences large shear strains (i.e., behaves in the nonlinear range). Thus, it has been a discussion for years under which condition the complicated NNLA should be performed over the simple EQLA to predict the responses accurately.

Based on the survey conducted among the practitioners, Kramer and Paulsen [9] suggested that the EQLA can be used for a shear strain of 1–2% and PGA_Outcrop of 0.3–0.4 g. Based on their recommendation, Aboye et al. [10,11] developed seismic site factor models specific to the Charleston, SC, area from thousands of 1D EQLA (SHAKE2000 for PGA_Outcrop ≤ 0.3 g) and NNLA (DMOD2000 for PGA_Outcrop > 0.3 g) site response analysis results. However, several others have reported that EQLA and NNLA results differ at shear strain and earthquake amplitude less than 2% and 0.3 g, respectively. Tokimatsu and Sugimoto [12] reported a strong nonlinearity from their study using the Holocene dune with a V_S of 310–350 m/s. It showed only about 0.3% shear strain from the down-hole array data collected during the 2007 Niigata-ken Cheutsu-oki earthquake. The NNLA and EQLA results diverge due to nonlinear behavior for strong ground motions (i.e., for high amplitude shaking) and/or for soft (low V_S) soil sites. Stewart et al. [8] reported that the EQLA results diverge from NNLA results at around PGA_Outcrop values of 0.1 g to 0.2 g. Hartzell et al. [13], Ardoino et al. [14], and Hashash et al. [15] recommended that NNLA must be performed for soft soil sites, which will behave nonlinearly even at low PGA_Outcrop levels. Hartzell et al. [13] suggested to use NNLA for soil profiles with site classes D and E. Kaklamanos et al. [16] observed that EQLA produced acceptable estimates up to 0.4% shear strain levels and/or around a 0.1 g PGA_Outcrop level, while in Kaklamanos et al.’s study [17], it was observed that NNLA and EQLA results differed even at 0.05% strain. These observations led Matasovic and Hashash [7] to conduct a comprehensive survey to collect data on the use of different site response analysis software and procedures used by researchers and practitioners. They collected information on the current practice for conducting site-specific response analysis. It was found that NNLA and EQLA results begin to diverge at around 0.2% strain, and that the EQLA results are unreliable beyond 0.5% strain. Their survey revealed a trend among practitioners using NNLA procedures for site classes E and F, while some use EQLA procedures for up to 1% strain. Based on the survey outcomes, Matasovic and Hashash [7] strongly differed from that opinion by claiming that soils would be too close to failure with a shear strain of 1%, and a high level of nonlinearity is expected. In summary, the survey results indicated that well-defined site-specific guidelines must be developed to select the appropriate type of site response program and procedure.

The abovementioned guidelines are site-specific because of the unique geologic and seismic conditions for each project site. This study estimates the differences in surface accelerations and corresponding seismic site factors computed from EQLA and NNLA, considering the geology and seismicity of Charleston, SC, and makes recommendations for selecting a suitable analysis procedure and computer program. First, three separate seismic site factor models were developed based on SHAKE2000, DEEPSOIL, and DMOD2000 results. Then, a threshold chart was developed to select the most suitable site-specific response analysis procedure. The uniqueness of this chart is that it uses V_S30 (the soil parameter) and PGA_Outcrop (the ground motion parameter), commonly available data for site response analysis, to choose the analysis procedure and the difference in computed results between EQLA and NNLA procedures.

2. Charleston, SC, Geology and Seismology

2.1. Geology and Dynamic Material Properties

Charleston, SC, situated in the Atlantic Coastal Plain, has subsurface stratigraphy characterized by layers of oceanward thickening sediments that span 700 to 1000 m thick [18]. The unconsolidated quaternary sediment near the surface lies on a weakly lithified tertiary layer. Below the tertiary is the hard Mesozoic/Paleozoic basement rock that extends to a great depth.

Among many material properties, shear wave velocity profile, damping, shear modulus reduction, density, and plasticity index are critical for site response analysis. Accurate compilation of these properties for such deep profiles is quite challenging; the uncertainties in the dynamic properties must be considered to produce reasonable results for an area. A 137 m deep soil profile in Charleston, SC, was selected. This profile is situated over a soft-rock half-space with an assumed shear wave velocity (V_S) of 700 m/s, based on the statistical analysis by Aboye et al. [11]. The profiles from Aboye et al. [11] are derived from in situ V_S measurements compiled by Andrus et al. (2006). Most of these Vs measurements were obtained using the seismic cone penetration test method, while others were taken using the seismic downhole, spectral analysis of surface waves, suspension logger, and seismic refraction methods. The mean shear wave velocity profile was then randomized to consider uncertainties in shear wave velocity, modulus reduction, damping, and the thickness of the quaternary layers. The twenty-eight shear wave velocity (V_S) profiles and other data, such as layer number, thickness, total unit weight (γ_t), and plasticity index (PI), generated for this study are shown in Figure 1. The thick line in the figure shows the reference V_S profile, and the thin lines show the other profiles generated considering ±1, −2, and −3 standard deviations (σ) of the V_S (natural logarithm). The standard deviation (σ) of ln(V_S) is used because Vs data typically follow lognormal distributions. The average values of σ of ln(V_S) are 0.32 for the Wando formation and range from 0.14 to 0.31 for the tertiary-age sediments. The figure also shows the variation in the thickness of the quaternary layer from 0, 10, 20, and 30 m considered in this study. Additionally, for this investigation, the mean normalized shear modulus (G/G_max) and damping (D) variation with shear strain (γ) for each layer were obtained using the relationships from Zhang et al.’s study [19] and are shown in Figure 2a and 2b, respectively. Zhang et al.’s [19] relationships are selected because they are developed based on resonant column and torsional shear test results from 122 specimens comprising the quaternary, tertiary, and older residual/saprolite soils from South Carolina. For the half-space with a V_S of 700 m/s, G/G_max(1.0) and D (0.5%) were assumed to vary linearly with shear strain [20].

2.2. Seismology and Ground Motions

In SC, a local dominant seismic source clustered by paleoliquefaction events is designated as a repeated-large-magnitude-earthquake (RLME) zone of Charleston [21]. In 1886, Charleston, SC, was hit by a devastating earthquake considered the largest earthquake in the Central and Eastern United States (CEUS). Gheibi and Gassman [22] studied the geotechnical data from various paleoliquefaction sites in the Charleston area and estimated a magnitude of M 5.5 to 7.2 for prehistoric earthquakes in the Charleston seismic source zone.

Figure 3 shows the 7.5 min quadrangle and the Woodstock fault zone, which may have contributed to the catastrophic event. Moreover, in the past 6000 years, this area experienced several other liquefaction-inducing earthquakes, and the recurrence rate of an 1886-like earthquake is about 500 years [23]. According to the South Carolina Department of Natural Resources (SCDNR), several minor earthquakes have recently occurred around the Columbia and Charleston areas [24].

Since there are no strong-motion records available in the Charleston, SC, area, researchers and practitioners use synthetic ground motions to evaluate the seismic hazard and to develop region-specific site coefficient/amplification models [25,26,27,28]. In this study, site-specific acceleration time histories and response spectra for Charleston, SC, are generated by the computer program SCENARIO_PC [5,20], which was developed for the South Carolina Department of Transportation (SCDOT) to carry on site response analyses. SCENARIO_PC uses a point-source stochastic model [29] to generate a synthetic ground motion incorporating site-specific geophysical and crustal parameters such as the rock model, earthquake moment magnitude, source-to-site distance, and return period as the inputs. The hazard model to generate a uniform hazard spectrum/target response spectrum in the SCNEARIO_PC is based on the Probabilistic Seismic Hazard Analysis (PSHA) procedure proposed by Cornell [30]. A thick outcropping layer of soft rock with a V_S of 760 m/s (termed as the B-C boundary) is considered the geologic realistic rock model type [18]. In the Charleston region, considering a return period of 2% in 50 years, which corresponds to a Safety Evaluation Earthquake (SEE), the deaggregation data indicate that the seismic hazard across all spectral periods is primarily influenced by events with a moment magnitude (M_w) ranging from 7.2 to 7.4 and source-to-site distance (R) between 3.8 and 22.4 miles.

The centers of 12 quadrangles were chosen to generate the representative time histories, as shown in Figure 3. A total of 144 synthetic acceleration time histories on the top of the soft rock outcrop were generated for these locations. Then, the ground motions were scaled to PGA_Outcrop values of 0.1, 0.2, 0.3, 0.4, and 0.5 g to be consistent with the amplitude range recommended by the National Earthquake Hazard Reduction Program (NEHRP). Since the seismic hazard in the Charleston area is dominated by only the CEUS Seismic Source Characterization narrow seismic source [21], Park et al. [31] proposed that arbitrary scaling could be employed when the seismic hazard is primarily influenced by a single earthquake source zone. A sample synthetic acceleration time history and corresponding response spectra for a quadrangle are shown in Figure 4. Li et al. [32] determined the seismic amplification factors at various soil and rock sites using recorded and synthetic ground motions. They reported no remarkable bias in amplification factors resulting from site-specific synthetic ground motions compared to real ground motions with similar geophysical properties. NUREG-0800-SRP 3.7.1 [33] and ASCE 7–16 [34] allow the application of synthetic ground motions as an alternative in seismic hazard analysis in the absence of real strong motion records for the study area. One may consider different suites of ground motions to develop more accurate results for their location of interest.

3. Site Response Analysis Software and Model Parameter Calibration

The EQLA was conducted using the widely used computer program known as SHAKE2000 [35], whereas NNLA was conducted using the nonlinear site response analysis programs DMOD2000 [36] and DEEPSOIL [37]. Although both software use similar equations and models, the authors have observed differences in the results obtained from DMOD2000 and DEEPSOIL. Since these software are extensively used by practicing engineers and researchers, we used both software to conduct NNLA and compared the results to quantify the differences.

3.1. Equivalent Linear Site Response Analysis Using SHAKE2000

The cyclic stress/strain response is estimated in SHAKE2000 using an equivalent linear shear modulus (G) and damping ratio (ξ). An iterative procedure is utilized to determine the values of G and ξ during the simulation for a given strain. The process begins with an initial estimate of G and ξ, followed by a linear trial analysis that produces a shear strain time history for each layer. Next, the maximum observed shear strain is used to compute the effective shear strain. Finally, the G and ξ corresponding to the effective shear strain values are estimated based on the input G/Gmax-γ and D-γ curves and are used for the next trial run. This process repeats until convergence is achieved, and the final G and ξ values are used for the final run of the entire earthquake excitation. It should be noted that this method cannot accurately account for soil stiffness changes or predict the failure response.

3.2. Nonlinear Site Response Analysis Using DMOD2000 and DEEPSOIL

Selecting suitable constitutive models and corresponding model parameters is a crucial aspect of NNLA. Since both DMOD2000 and DEEPSOIL have the same constitutive models implemented, comparing the results will reveal the differences between them due to the other underlying techniques and algorithms. The equation of motion is solved using Newmark’s beta (β) method. However, DEEPSOIL offers an added advantage with its advanced algorithm that accurately estimates small strain (viscous) and hysteretic damping, giving it a theoretical edge over DMOD2000.

3.2.1. Constitutive Model and Shear Modulus Reduction

The initial backbone curve shown mathematically in Equation (1) and schematically in Figure 5 is represented by the Modified KZ (MKZ) model [36], which is implemented in both software.

$${\tau }^{\ast }={\displaystyle \frac{{G^{\ast }}_{max}}{1+\alpha {\left(\frac{\gamma }{{\gamma }_r}\right)}^S}}$$

(1)

where τ^* = τ/σ′_vc, G^*_max = G_max/σ′_vc, γ_r = τ_max/G_max, σ′_vc is the initial vertical effective stress, G_max is the initial shear modulus of the soil, s and α are model parameters, τ is the shear stress, τ_max is the maximum shear stress, and γ is the shear strain. The material degradation with the repeated cycle of loading is incorporated into this model using degradation index functions, as shown in Figure 5 [38].

The reference shear strain model proposed by Hashash and Park [39], shown in Equation (2), is incorporated in DEEPSOIL.

$${\gamma }_r=a{\left({\displaystyle \frac{{\sigma }_{vc}^{\prime }}{{\sigma }_{ref}}}\right)}^b$$

(2)

where ${\sigma }_{vc}^{\prime }$ is the effective vertical stress, ${\sigma }_{ref}$ is the reference confining pressure, and a and b are the model parameters.

3.2.2. NNL Governing Equation and Viscus Damping Formulation

A fully coupled governing equation for the dynamic behavior of soil will have the damping matrix naturally formed during the derivation of the partial differential equations using balance equations. However, the theories implemented in DMOD2000 and DEEPSOIL are loosely coupled, i.e., the damping matrix does not appear in the governing equation. Therefore, an external damping matrix is incorporated externally (C_R) to improve the mathematical representation of the real problem, as shown in Equation (3).

$$\mathit{M}\ddot{\mathit{u}}+{\mathit{C}}_{\mathit{R}}\dot{\mathit{u}}+\mathit{K}\mathit{u}=\mathit{f}$$

(3)

where M is the mass matrix, C_R is the external viscous damping matrix, K is the nonlinear stiffness matrix, f is the excitation at the base of the layer, and $u,\dot{u},\mathrm{and}\hspace{0.17em}\ddot{u}$ are the relative displacement, velocity, and acceleration vectors, respectively. The external damping matrix is calculated using the full Rayleigh damping formulation, which is a mass and stiffness proportional damping matrix, as shown in Equation (4).

$$C_R={\alpha }_R\mathit{M}+{\beta }_R\mathit{K}$$

(4)

where ${\alpha }_R$ and ${\beta }_R$ are mass and stiffness coefficients calculated using Equations (5) and (6), respectively [40].

$${\alpha }_R={\xi }_{tar}\left({\displaystyle \frac{4\pi }{T}}\right)\left({\displaystyle \frac{n}{(n+1)}}\right)$$

(5)

$${\beta }_R=\left({\xi }_{tar}T\right)\left({\displaystyle \frac{1}{\pi (n+1)}}\right)$$

(6)

where ${\xi }_{tar}$ is the target damping ratio obtained through calibration and the range is 0.1 to 5%, $T$ is the fundamental period of the soil profile given by $T=4H/V_{S,avg}$, H is the thickness of the soil profile, $V_{S,avg}$ is the weighted average shear wave velocity, and n is an odd integer (1, 3, 5, 7, …) that is related to the mode of excitation of the soil column. Since DMOD2000 accounts only for hysteretic damping, viscous damping was introduced using a Rayleigh damping formulation. To implement this, DMOD2000 requires n values corresponding to the first (i.e., n = 1) and higher modes, along with a single target damping ratio (ξ_tar) applied uniformly across all layers. The suitable pair of n and ξ_tar values were computed by comparing the results of DMOD2000 and SHAKE2000 by applying low PGA_Outcrop motions. The idea behind this calibration procedure is that the NNLA and EQLA responses will be similar at low amplitude shaking because the soil column will behave within the linear elastic range at low amplitude shaking.

The spectral accelerations computed by SHAKE2000 and DMOD2000 with acceleration time history with PGA_Outcrop values of 0.01 g and 0.1 g and V_S30 values of 100, 200, and 295 m/s are presented in Figure 6. Figure 6a shows that for all n and ξ_tar combinations (5 and 0.5, 7 and 0.5, and 7 and 0.75), the DMOD2000 results matched well with SHAKE2000. However, differences were observed for low-velocity profiles, even with applied acceleration time history with very low PGA_Outcrop values of 0.05 g and 0.1 g. Therefore, the two profiles with low V_S30 values (100 and 200 m/s) and an acceleration time history with PGA_Outcrop of 0.001 g were selected for further calibration of n and ξ_tar. Figure 6b,c show the comparison of computed spectral accelerations for V_S30 = 100 and 200 m/s profile and acceleration time history with PGA_Outcrop = 0.001 g. Additionally, it can also be noted from these figures that changing n shifts the central frequencies at which the target damping ratio is applied. When n corresponds to a lower mode (n = 5), more damping is applied to higher frequencies, leading to lower spectral accelerations at short periods (higher frequencies) compared to n = 7 with the same target damping ratio, while increasing the damping ratio from 0.5% to 0.75% at the same n results in more damping at higher frequencies. Since n determines the frequency range affected, increasing the ξ from 0.5% to 0.75% further suppresses the response in the range influenced by the Rayleigh curve (<0.2 s). This explains why the n = 5, ξ = 0.75% curve damps out more high-frequency content than n = 7, ξ = 0.5%.

The figure shows that, for the same sets of n and ξ_tar combinations (5 and 0.5, 7 and 0.5, and 7 and 0.75), SHAKE2000 and DMOD2000 produced very similar results. It is worth mentioning that in some simulations with motion with high PGA_Outcrop and/or soil profile with low V_S30 (soft), DMOD2000 experienced numerical instability. Such an issue was avoided by altering the n and ξ_tar values, which is the reason for considering multiple sets of calibrated n and ξ_tar combinations during the DMOD2000 simulations.

Phillips and Hashash [41] and Hashash et al. [15] reported that Rayleigh damping tends to over-dampen the system in deep profiles, and thus, computed responses might be under-predicted. DEEPSOIL has a frequency-independent small-strain damping estimation scheme, shown in Equation (7) [41], which addresses the over-damping issue and eliminates the need for calibrating the Rayleigh damping parameters.

$$C_R=\mathit{M}\sum _{b=0}^{N-1}{a}_b{\left({\mathit{M}}^{-1}\mathit{K}\right)}^b$$

(7)

where N is the number of modes/frequencies and a_b is a scalar value associated with a constant damping ratio (ξ_n) throughout the soil profile, as shown in Equation (8).

$${\xi }_n={\displaystyle \frac{1}{4\pi f_n}}\sum _{b=0}^{N-1}{a}_b{\left(2\pi f_n\right)}^{2b}$$

(8)

For the scalar index b = ½, the ξ_n in Equation (8) reduces to ξ_n = a_b/2, which is frequency-independent; thus, a frequency-independent damping matrix is obtained [41].

3.2.3. Hysteretic Damping Formulation

In DEEPSOIL, the simultaneous modulus reduction and damping curve fitting (MRD-DS) feature was used to calibrate MKZ constitutive model parameters α and s. This feature employs a reduction factor (R_F), as shown in Equation (9), to modify the extended Masing [42] loading/unloading stress/strain relationship [41].

$$R_F=b_1{\left({\displaystyle \frac{G_{\mathrm{secant}}}{G_{\max }}}\right)}^{0.1}$$

(9)

where b₁ is a variable related to soil and input ground motion characteristics and G_secant is the secant shear modulus corresponding to the maximum shear strain. R_F is multiplied by the damping computed using Masing’s criteria to obtain the modified ξ_Masing value for MRD-DS in DEEPSOIL, which increases the flexibility of the model to fit both the modulus reduction and damping curves simultaneously with better accuracy than the modulus reduction (MR-DM) or modulus reduction and damping (MRD-DM) feature available in DMOD2000.

The modulus reduction and damping curves fitted using MR-DM, MRD-DM and MRD-DS with Zhang et al.’s research [19] for a soil layer at 4–10 m depth of the reference profile (V_S30 = 295 m/s) are shown in Figure 7. The MR-DM approach in DMOD2000 fitted Zhang et al.’s [19] G/G_max-γ curve well, as seen in Figure 7a, but showed a significant difference in fitting the D-γ curve, as seen in Figure 7b. It is also observed that the difference is substantial at high shear strain. The MRD-DM matching scheme in DMOD2000 showed a significant difference in fitting G/G_max-γ and D-γ. On the other hand, the MRD-DS scheme in DEEPSOIL fitted the G/G_max-γ and D-γ curves well. This parametric study indicates that the DEEPSOIL theory and algorithm are flexible enough to simultaneously fit the reference G/G_max-γ and D-γ over the entire strain range.

4. Analyses, Results, and Discussion

4.1. Comparison SHAKE2000, DMOD2000, and DEEPSOIL Results

A set of sample response spectra computed using SHAKE2000, DEEPSOIL, and DMOD2000 by applying motion with a PGA_Outcrop of 0.05 g and mean G/G_max-γ and D-γ for site classes E (V_S30 = 134 m/s), D (V_S30 = 295 m/s), and C (V_S30 = 406 m/s) are shown in Figure 8a, b, and c, respectively. Similar results for a PGA_Outcrop of 0.5 g are shown in Figure 8d–f. Several observations were made from these figures: (1) the EQLA and NNLA results diverge at higher amplitude shaking (PGA_Outcrop of 0.5 g) and (2) the low shear wave velocity profiles (softer profiles) show significant differences between EQLA and NNLA even at low amplitude shaking (PGA_Outcrop of 0.05 g). This indicates that the amplitude of shaking and the shear wave velocity (stiffness) must be considered together when selecting the analysis procedure for accurately computing the surface responses.

The difference between the EQLA and NNLA results decreases with increasing shear wave velocity (i.e., increasing stiffness). (3) The EQLA results show higher peak values but lower values at low periods as seen on response spectra plots, and (4) the DMOD2000 and DEEPSOIL results diverge at lower periods when analyzed with higher amplitude motion (PGA_Outcrop = 0.5 g).

4.2. Development of Site Factor Models

All of the profiles presented in Figure 1 were analyzed using SHAKE2000, DMOD2000, and DEEPSOIL. Based on the results from each software, three seismic site factor models were developed for six spectral period ranges: (a) ≤0.01 s for F_PGA, (b) 0.01–0.4 s for F_0.2 or F_a, (c) 0.41–0.8 s for F_0.6, (d) 0.81–1.2 s for F₁ or F_v, (e) 1.21–2.0 s for F_1.6, and (f) 2.01–4.0 s for F_3.0. The variation of F_PGA, F_0.2, and F_1.0 with V_S30 computed from the SHAKE2000 results are presented in Figure 9, Figure 10 and Figure 11, respectively. Similar curves were produced from the DEEPSOIL and DMOD2000 results but are not presented here. Three distinct features were observed from these plots: (a) F_T increases with V_S30 for low V_S30 profiles, (b) there is a peak F_T within a range of V_S30, and (c) F_T decreases with V_S30 beyond the peak. The data from each software were fitted with Equation (10a) for V_S30 < V_S30P, (10b) for T ≤ 0.2 s and V_S30 ≥ V_S30P, and (10c) for T > 0.2 s and V_S30 ≥ V_S30P. It should be noted that these equations have different regression coefficients for the SHAKE2000, DMOD2000, and DEEPSOIL results.

$$\begin{array}{cc}{F}_T=\left({\displaystyle \frac{F_p}{V_{S30P}}}\right)V_{S30}& for V_{S30}<V_{S30P}\end{array}$$

(10a)

$$\begin{array}{cc}{F}_T={\displaystyle \frac{\left(F_p-1\right)\left(760-V_{S30}\right)}{760-V_{S30P}}}+1& for T \le 0.2 \mathit{sec}and V_{S30}\ge V_{S30P}\end{array}$$

(10b)

$$\begin{array}{cc}{F}_T=a+b({\mathit{exp}}^{c{V}_{\mathrm{S}30}})& for T>0.2 \mathit{sec}and V_{S30}\ge V_{S30P}\end{array}$$

(10c)

where F_T is the median site factor value, F_p is the peak F_T value, T is the spectral period, V_S30P is the V_S30 corresponding to F_p, and a, b, and c are the fitting parameters. The values of F_p, V_S30P, b, and c are calculated using Equations (11a), (11b), (11c), and (11d), respectively.

$$F_p=x_1{S}_{Outcrop}+x_2$$

(11a)

$$V_{S30P}=x_3{S}_{Outcrop}+x_4$$

(11b)

$$b={\displaystyle \frac{1-a}{{\mathit{exp}}^{760c}}}$$

(11c)

$$c={\displaystyle \frac{\mathit{ln}\left(\frac{1-a}{F_p-a}\right)}{760-V_{S30P}}}$$

(11d)

where x₁, x₂, x_3, and x₄ are another set of regression coefficients that are listed in

Table 1. Regression coefficients for site factor models based on SHAKE2000, DMOD2000, and DEEPSOIL results.

		Regression Coefficients *
Spectral Period, T (s)	SOutcrop	x1 (g−1)	x2	x3 (g−1·m/s)	x4 (m/s)	a
0.0	PGAOutcrop	−1.91, −1.39, −1.40	1.95, 1.62, 1.27	200, 270, 270	170, 174, 174	-
0.2	Ss	−0.79, −0.76, −0.61	2.00, 1.97, 1.48	129, 84, 84	195, 207, 207	0.65
0.6	S0.6	−2.26, −2.52, −2.92	2.86, 2.68, 2.71	207, 139, 156	156, 183, 182	0.85
1.0	S1	−2.39, −2.50, −3.40	3.43, 2.89, 3.41	129, 124, 97	153, 147, 156	0.90
1.6	S1.6	−4.46, −4.92, −4.92	3.49, 3.22, 3.21	198, 323, 324	121, 113, 133	0.97
3.0	S3.0	−8.2, −4.389, −0.97	2.80, 2.10, 2.21	394, 346, 482	80, 85, 131	0.99

* Comma-separated values are for models based on SHAKE2000, DMOD2000, and DEEPSOIL, respectively.

.

4.3. Comparison of the Seismic Site Factor Models

The seismic site factor models developed based on the SHAKE2000, DMOD2000, and DEEPSOIL results for F_PGA, F_a (i.e., F_0.2), and F_v (i.e., F_1.0) are shown in Figure 12, Figure 13 and Figure 14, respectively. Each figure contains six sub-figures corresponding to six S_Outcrop (or PGA_Outcrop in the case of Figure 12) ranges. Similar figures for F_0.6, F_1.6, and F_3.0 were also generated but are not presented here. Several observations were made based on the comparison of the site factor models. Firstly, SHAKE2000 models generally produce more conservative estimates than the NNLA counterparts DMOD2000 and DEEPSOIL, even at low amplitude shaking, except for the F_0.2, as seen in Figure 13 and softer profiles where the DMOD2000 model predicts higher values than the SHAKE2000 model. Secondly, there are significant differences between the NNLA models generated from the DMOD2000 and DEEPSOIL software. The DEEPSOIL model shows lower values than DMOD2000 in low periods (T = 0.0 and 0.2 s). On the other hand, the DEEPSOIL model shows higher values than DMOD2000 for T ≥ 1.0 s. Finally, the difference between the three models decreases when the profiles behave within the linear range where the responses computed from the EQLA and NNLA programs are expected to be similar. These observations are consistent with the computed surface responses shown in Figure 8.

Aboye et al.’s [11] site factor models are plotted in Figure 12, Figure 13 and Figure 14 for comparison purposes. It should be noted that the Aboye et al.’s [11] site factor models adopted by the SCDOT were developed by combining the EQLA and NNLA results. A comparison of Aboye et al.’s [11] site factor models with the models generated in this study shows that both NNLA models generally fall below Aboye et al.’s [11] model. This indicates that Aboye et al.’s [11] site factor models generally predict conservative estimates relative to the models developed in this study.

Generally, the models based on EQLA results are higher than those of NNLA. This implies that the site factors generated entirely based on EQLA results may overestimate the responses. It was also observed that the site factor models developed based on DMOD2000 and DEEPSOIL are significantly different. This indicates that users must know the differences before choosing the computer program for site response analysis. A recent study [43] came to a similar conclusion about varying results from several NNLA programs. This indicates that the widely used computer programs for site response analysis must be validated against experimental results and/or recordings for various soil and shaking conditions before using them. Based on the observations, it is suggested that the site factors be computed based on both EQLA and NNLA and that engineering judgments be made to decide the final site factors for the subsequent seismic analysis of the structure [7].

4.4. Possible Reasons for the Difference Between Seismic Site Factor Models

To further understand the differences between the EQLA and NNLA results presented in Figure 12, Figure 13 and Figure 14, first, the maximum shear strains (${\gamma }_{Max\_Profile}$) from the considered profiles for all six PGA_Outcrop levels (i.e., 0.05, 0.1, 0.2, 0.3, 0.4, and 0.5 g in six sub-plots) from each software were plotted against the respective V_S30 values, as shown in Figure 15. ${\gamma }_{Max\_Profile}$ is the maximum shear strain observed in the soil profile during the entire excitation.

As seen in Figure 15, higher ${\gamma }_{Max\_Profile}$ values were observed in the softer profiles, especially for V_S30 ≤ 200 m/s from the three computer programs (SHAKE2000, DMOD2000, and DEEPSOIL), while the ${\gamma }_{Max\_Profile}$ values are insignificant for the stiffer profiles. For profiles with V_S30 ≤ 200 m/s, ${\gamma }_{Max\_Prrofile}$ values were greater than 0.1% even for a PGA_Outcrop of 0.05 g, a threshold suggested by many researchers [7,16]. This explains why the site factor models (Figure 12, Figure 13 and Figure 14) based on the EQLA and NNLA results are starting to diverge even at low amplitude shaking for the softer profile cases. Shear strain estimates from SHAKE2000 are lower than those computed from both NNLA programs. This indicates that SHAKE2000 uses a ‘higher equivalent linear modulus and/or lower damping.’ Consequently, the EQLA program predicts higher site factors. However, results from low-amplitude shakings yield low spectral acceleration values, leading to typically higher site factors for low-amplitude cases than high-amplitude ones. This contributes to the contrasts observed between different site factor models at low amplitude shaking, as seen in Figure 12, Figure 13 and Figure 14. Nonetheless, higher amplitude shaking (e.g., 0.3, 0.4, and 0.5 g) show strains much higher than 0.5%, challenging the ability of the EQLA procedure to accurately represent the stress/strain behavior, as suggested by Matasovic and Hashash [7]. The high shear strains result in high hysteretic damping in the system; thus, a smaller amplification of the ground motion at the surface was observed from the NNLA compared to the EQLA for the softer profiles. Even for the stiffer profiles, a slight deviation between the EQLA and NNLA models is observed.

4.5. Possible Reasons for the Difference Between DMOD2000 and DEEPSOIL Site Factor Models

Although the theoretical frameworks are similar in both computer programs, differences exist in how the input modulus reduction and damping are fitted by the analytical models and the computation of small-strain damping. To investigate the effect of these differences, several analyses were conducted using the reference profile (V_S30 = 295 m/s) with the ground motion amplitude scaled to PGA_Outcrop of 0.5 g.

Two cases were considered for the DEEPSOIL analyses. For Case I, the MRD-DS fitting option for the MKZ model parameter calibration and frequency-independent small-strain damping were considered. For case II, the MR-DM fitting option for MKZ model parameter calibration and the full Rayleigh damping for computing small-strain damping were considered. Thus, the damping estimation method for Case II in DEEPSOIL is the closest to the method implemented in DMOD2000. Figure 16a compares the computed surface spectral accelerations for DMOD200 and DEEPSOIL: Cases I and II. The DEEPSOIL: Case I and DMOD2000 responses are very close, while DEEPSOIL: Case I shows a deviation in lower and higher periods. Figure 16b compares the computed shear strains for a layer at 10 m depth, where the largest shear strain concentration is observed for each case. Here also, the DEEPSOIL: Case II and DMOD2000 results are very close, while DEEPSOIL: Case I shows deviation, as expected. These observations reveal the following: (a) both DMOD2000 and DEEPSOIL produce very similar responses when the damping estimation techniques are similar, and (b) the incorporation of advanced damping estimation techniques (both for small- and large-strain ranges) are the sources of the differences between the DEEPSOIL and DMOD2000 results. The above observation supports that both DMOD2000 and DEEPSOIL can produce similar results, given that the damping formulations used in these software are close to each other.

However, DEEPSOIL predicting higher shear strain values than DMOD2000 implies the relative difference observed in their surface responses. Again, the layer in the reference profile (V_S30 = 295 m/s) at 10 m depth is selected to compare DMOD2000 and DEEPSOIL (here, the MRD-DS fitting and frequency-dependent techniques are used in DEEPSOIL) responses. The shear strain time history presented in Figure 17a shows that DEEPSOIL generally produces a higher strain than DMOD2000.

The corresponding damping versus strain curves from DMOD2000 (MR-DM-fitted) and DEEPSOIL (MRD-DS-fitted) are plotted in Figure 17b. The higher strain predicted by DEEPSOIL may be attributed, in part, to the fact that higher shear strain in DEEPSOIL minimizes the observed damping differences between the two programs. For example, after 7.7 s of excitation, the shear strain computed by DEEPSOIL is 0.37%, as seen in Figure 16a. According to Figure 17b, the corresponding damping value for 0.37% strain from the DEEPSOIL results is 15.5%, while DMOD2000 shows 28.8%. This shows a 13.3% difference in damping. On the other hand, the shear strain computed by DMOD2000 is 0.27% at 7.7 s, and the corresponding damping is 25.5%. Thus, the difference in the estimated damping between DEEPSOIL and DMOD2000 is 10%, and it could be higher (~13.5%) if both programs produce closer shear strain estimates.

5. Site Response Analysis Tool Selection Criterion—Threshold Chart

Based on a rigorous research survey, the need for a site-specific comprehensive guideline to select an appropriate site response analysis procedure and method is at stake. Although a few approaches have emerged [7,16,17,44], more work is necessary to develop a site-specific comprehensive guideline that can be easily used by practitioners. In this study, an easy-to-use chart was developed by comparing EQLA and NNLA results.

5.1. Key Steps

The following is a step-by-step procedure for developing the threshold chart:

(a)

The area under the surface response spectral acceleration plots between the period of 0.01 and 4.0 s, the shaded area shown in Figure 18, was calculated for each simulation result from each software.

(b)

For each of the V_S30 cases, the arithmetic means of the area from all cases of ground motions were calculated from each simulation result for all six PGA_Outcrop levels.

(c)

For each V_S30, the corresponding averaged area ratios of SHAKE2000 to DMOD2000 and SHAKE2000 to DEEPSOIL were computed for all six PGA_Outcrop levels.

(d)

The threshold chart, shown in Figure 19, was developed by considering V_S30 and PGA_Outcrop as the x- and y-axes, respectively. Then, the area ratios from SHAKE2000 to DMOD2000 and SHAKE2000 to DEEPSOIL were plotted using different markers for three distinct ranges identified: (a) less than 1.1 (10%), (b) between 1.1 to 1.2 (10 and 20%), and (c) greater than 1.2 (20%). Higher V_S30 cases fall in the ’less than 10%’ range for PGA_Outcrop values of up to 0.5 g.

The profiles with low V_S30 values fall in the ‘between 10 to 20%’ and/or ‘greater than 20%’ ranges even at a low PGA_Outcrop of 0.1 g shaking. The results with PGA_Outcrop ≥ 0.4 g cases show more than a 20% difference for the profiles with VS30 ≤ 300 m/s; thus, they are labeled as ’greater than 20%’. For the analyses with V_S30 < 200 m/s and PGA_Outcrop ≥ 0.1 g, at least a 10% difference between the EQLA and NNLA programs was observed, and for PGA_Outcrop ≥ 0.3 g, the difference is greater than 20%. In this study, the regions with ≤10%, >10%, <20%, and ≥20% differences were separated with three significant color patches. These regions provide information on whether to switch to NNLA programs such as DEEPSOIL or DMOD2000 or to use EQLA programs such as SHAKE2000 for a given set of site-specific V_S30 and PGA_Outcrop values. Similarly, two other threshold charts were developed for the mean + 1σ and mean−1σ G/G_max-γ and D-γ cases based on the same SHAKE2000, DMOD2000, and DEEPSOIL results and are presented in Figure 19b,c.

It should be noted here that the threshold charts account for the spectral acceleration difference over the 0.01 to 4.0 s period range rather than specifying period-dependent distinctions such as the ones suggested by Kaklamanos et al. [16] and Kim et al. [44]. As is seen from Figure 19a–c, the analyses with the mean and mean ± 1σ G/G_max-γ and D-γ produce significantly different results. On the other hand, Figure 19b shows that the NNLA and EQLA programs produce similar estimates for most V_S30 and PGA_Outcrop conditions for mean + 1σ G/G_max-γ and D-γ. The observations are completely opposite for the mean−1σ G/G_max-γ and D-γ analyses, as seen in Figure 19c. These mean ± 1σ cases represent two extremes of dynamic soil properties (G/G_max-γ and D-γ) and are less likely to be found in real situations. However, the threshold chart for the mean case shown in Figure 19a is recommended for general application, especially when the engineers do not have accurate G/G_max-γ and D-γ curves for the project site. Similar threshold charts were generated based on the spectral acceleration area ratios computed using the entire 0.01 to 10 s period range and were found to be similar to the cases shown in Figure 19a–c.

Finally, regression equations for the recommended threshold chart were developed to easily estimate the differences between the EQLA and NNLA results. The area ratios of SHAKE2000 to DMOD2000 and SHAKE2000 to DEEPSOIL were plotted against V_S30 and grouped into six PGA_Outcrop levels, as shown in Figure 20. A clear deviation from ‘1.0’ at around 200 m/s or below was observed from the plots, especially for PGA_Outcrop values of 0.05, 0.1, 0.2, and 0.3 g. For PGA_Outcrop values of 0.4 and 0.5 g, the deviation starts at higher V_S30 values. This supports the observations of this study that the softer profiles (i.e., low-V_S30 profiles) begin to behave nonlinearly at small shaking amplitudes while the stiffer profiles (i.e., high-V_S30 profiles) behave within the linear range even at significantly higher shaking amplitudes. Based on the results, Equations (12a)–(12c) were developed using regression analysis.

$$\mathrm{Area} \mathrm{ratio} \left(\mathrm{EQL}/\mathrm{NNL}\right)=a{\left(V_{S30}\right)}^{(-0.8)}+b$$

(12a)

$$\mathrm{where} a=\left(30.68\right)PG{A}_{Outcrop}+4.353$$

(12b)

$$b=\left\{\begin{array}{c}0.935, when PG{A}_{Outcrop}\le 0.2g\\ 0.905, when PG{A}_{Outcrop}>0.2g\end{array}\right.$$

(12c)

The coefficient of determination (R²) values shown in Figure 20 for each PGA_Outcrop value supports the quality of these fits. Equation (12) can be used to estimate the difference between the SHAKE2000 and DMOD2000 or SHAKE2000 and DEEPSOIL responses. Therefore, Equation (12) and the threshold chart shown in Figure 19a provide a guideline for selecting a more appropriate site-specific response analysis tool for the project. The chart requires the PGA_Outcrop of the ground motion and V_S30 of the shear wave velocity profile as the inputs. These two parameters are readily available to the engineer before performing a site-specific response analysis. This is an advantage over the other commonly available protocols [7,9,16,17], which use the shear strain to select the appropriate site response program.

5.2. Validations and Limitations

The results of this study generally agree with those of Kaklamanos et al. [16,17], although the site’s geologic and seismic conditions are different. From Figure 15, shear strains ≥ 0.4% were observed for simulations with V_S30 ≤ 200 m/s and PGA_Outcrop ≥ 0.1 g. Kaklamanos et al. [16,17] also observed similar differences between NNLA and EQLA at this strain level.

Interestingly, these cases (V_S30 ≤ 200 m/s and PGA_Outcrop ≥ 0.1 g) fall within the >10% and <20% and ≥20% regions of the proposed threshold chart (Figure 19), which encourages the use of NNLA programs for such conditions. Afacan et al. [43] and Brandenberg et al. [44] studied the seismic site response of soft clays over a wide strain range using centrifuge tests. They also evaluated the performance of several EQLA and NNLA programs. Figure 21a,b present two cases where the EQLA and the NNLA responses computed using DEEPSOIL were compared with the measured responses from the centrifuge model for PGA_Outcrop values of 0.28 g and 0.55 g [44]. In the case of a PGA_Outcrop of 0.28 g, the EQLA and NNLA results reasonably matched the measured response spectra, as seen in Figure 21a. On the other hand, only the NNLA predictions matched the measured response spectra well for the PGA_Outcrop of 0.55 g, as seen in Figure 21b.

In this study, the area ratios between the EQLA and NNLA responses for PGA_Outcrop values of 0.28 g and 0.55 g were calculated, and the threshold chart was used to check their applicability and accuracy. As seen in Figure 21a, for a PGA_Outcrop of 0.28 g, the calculated area ratio (~6%) fell within the 10% region, suggesting that both the EQLA and NNLA programs will produce similar results. On the other hand, for a PGA_Outcrop of 0.55 g, the area ratio (~13%) falls above the 10% region, as seen in Figure 21b, which suggests that the NNLA programs will produce better results. This validates the use of the area ratio as a valuable indicator for selecting the appropriate analysis procedure.

Although the proposed chart is a valuable tool for selecting site response analysis programs for accurate prediction, several limitations exist. The primary limitations are as follows: (a) only the conditions specific to the Charleston, SC, area are considered, although a similar tool can also be developed for other parts of the world by following and/or adjusting the procedure presented here, and (b) the differences seen between the NNLA programs suggests that the use of other available NNLA programs (i.e., those other than DMOD2000 and DEEPSOIL) may produce different results.

6. Conclusions

Three new seismic site factor models were developed and compared based on 18,000 site response analyses using SHAKE2000, DMOD2000, and DEEPSOIL. The NNLA site factor models predicted lower values than the EQLA model, even at the smallest shaking amplitude considered in this study. This indicates the presence of nonlinearity even at a small level of shaking for softer profiles (low V_S30) and emphasizes the need for the selection of an appropriate site response code for a site-specific analysis. The programs based on nonlinear theories generated a much higher shear strain in the profile, implying more significant hysteretic damping primarily for the softer profile cases. The difference observed in the surface amplifications between the NNLA programs may be related to the difference in the damping formulation implemented in these codes. However, the site factor models developed based on the NNLA programs fall below those proposed by Aboye et al. [11]. Finally, a unique threshold chart was proposed based on the comparisons between the EQLA and NNLA results and verified. Practicing engineers can use the proposed chart to select an appropriate site response analysis procedure for their projects in the Charleston, SC, area.