Seismic response of deep soft soil sites with varying shear wave velocities

Abstract

The variations in seismic response between deep soft soil sites with different shear wave velocities were not fully understood. This study focuses on the seismic response of deep soft soil sites in the lower reaches of the Yangtze River, China. A nonlinear dynamic finite element model was developed for two representative deep soft soil sites with borehole profiles and the shear wave velocity tested by the single borehole method. Two nonlinear cyclic constitutive models are used and thus compared through the site seismic response. To accurately calibrate the nonlinear cyclic model parameters, resonant column tests were conducted on 21 soil samples collected from the two boreholes. The results show that the peak ground acceleration (PGA) under low-frequency (Liuan) input motion was higher for soft soil sites compared to that under medium- and high-frequency (Kobe and Nahanni) input motions. The PGA amplification factor for deep soft soil sites under different input motions can be approximated by an exponential function. The peak ground acceleration tends to be lower as the equivalent shear wave velocity (V_se) decreases. The shapes of the spectral acceleration were similar for the two sites, despite a substantial difference in the V_se between them. Additionally, a crossover point was observed in the spectral acceleration for the two sites. The period corresponding to this crossover point increased with increasing intensity of input motions, indicating that the sites became softer with higher intensity and thus generally exhibited a longer characteristic period of the spectral acceleration. This paper also highlights the significance of selecting nonlinear constitutive models and the precise calibration of model parameters in the seismic response analysis of deep soft soil sites, providing a scientific basis for future similar site analyses.

1 Introduction

Soil is a stratified geological body formed through long and complex geological processes. Due to differences in formation environments, the dynamic properties of soil layers within a site exhibit significant heterogeneity (Shiuly, 2019). Deep soft soil deposits are widely distributed in the downstream plains of rivers, which are densely populated, economically developed, and home to significant infrastructure. In the event of a strong earthquake, this area could suffer substantial economic losses and casualties, making seismic safety for deep soft soil sites a critical concern. Existing studies indicate that variations in local soil conditions can lead to significant differences in site seismic responses. Particularly for deep soft soil sites, there are still differing perspectives on seismic response (Zahoor et al., 2024). Thus, studying the impact of soft soil sites on seismic response is of great theoretical and practical importance for seismic microzonation and disaster mitigation in the area of earthquake engineering (Shiuly et al., 2014; Shiuly and Narayan, 2012).

Currently, scholars have conducted several numerical analyses on the seismic response of soft soil sites. (Pires, 1996). emphasized that the constitutive model in seismic response analysis of soft soil sites needs to account for the damping characteristics of soils under large strain conditions. (Villalobos and Romanel, 2019). found that soft soil sites typically reduce short-period spectral accelerations while increasing long-period accelerations. However, (Sun et al., 2019), found that soft soil interlayers do not always reduce peak acceleration within soil layers, as this is highly influenced by the frequency of the input seismic motion. (Cavalieri et al., 2021). analyzed the impact of soil-structure interaction effects on the seismic response of soft soil sites. (Silahtar, 2023). discovered that the spectral accelerations for soft soil sites exceeded the requirements of the Turkish Building Earthquake Code. Xiao et al. (2022) observed that soft soil sites exhibit a more significant weakening effect on peak ground acceleration compared to bedrock sites. Qiao et al. (2023) explored the influence of soil type on site seismic response but assumed the soil profile to be a uniform and single ideal layer. These prior studies have primarily focused on shallow soft soil layers (less than 100 m), leaving limited research on the seismic response of deep soft soil layers exceeding 100 m. While Zhang et al. (2023) examined the seismic response of deep soft soil layers in the Shanghai area, they did not clarify how the parameters for the nonlinear constitutive model were calibrated. Chen et al. (2013) pointed out the need to fully consider the effects of large, distant earthquakes on deep soft soil sites, and further research is needed on the differences in seismic response among deep soft soil sites with varying equivalent shear wave velocities.

Seismic response analysis of soil layers is essential for the seismic safety evaluation of major projects, as the scientific accuracy of these results is critical for earthquake-resistant design. In recent developments, deep learning techniques have been applied to compute site seismic responses (Choi et al., 2024). However, in engineering practice, the most widely used method for seismic response calculation of soil layers at present is the one-dimensional seismic response analysis (Ansal et al., 2024; Shiuly et al., 2017). The one-dimensional seismic response analysis method is divided into frequency-domain equivalent linear methods and time-domain nonlinear methods. Studies have shown that the differences between these two methods can become significant under large strain of soils (Chen et al., 2022; Yee et al., 2013; Kaklamanos et al., 2015; Kim et al., 2016). While many studies have applied the frequency-domain equivalent linear method to site seismic response analysis, there has been less application of time-domain nonlinear methods to deep soft soil sites. The existing time-domain nonlinear methods need improvement when calculating the seismic response of soft soil sites under high-intensity seismic inputs (Griffiths et al., 2016), particularly regarding the damping characteristics of soil under low and high strain conditions (Phillips and Hashash, 2009; Yniesta et al., 2017), which involves the crucial issue of selecting an appropriate nonlinear model. (Groholski et al., 2016). stressed that inaccuracies in the soil maximum shear stress described by the nonlinear constitutive model could lead to underestimation or overestimation of seismic response. Therefore, nonlinear constitutive models require careful calibration of model parameters in seismic response analysis.

This study focuses on the seismic response of deep soft soil sites in the lower reaches of the Yangtze River, China. A nonlinear dynamic finite element model was developed for two representative deep soft soil sites with borehole profiles and shear wave velocity tested using the single borehole method. According to the “GB 50011-2010: Code for Seismic Design of Buildings,” these two deep soft soil sites are classified as Type IV, but their shear wave velocities differ significantly. Two nonlinear cyclic constitutive models were used and compared through site seismic response analysis. To accurately calibrate the nonlinear cyclic model parameters, resonant column tests were conducted on 21 soil samples collected from the two boreholes. This paper discusses in detail the effects of model selection, input seismic intensity and frequency, and the average equivalent shear wave velocity on seismic response of deep soft soil sites. The main objective of this study is to investigate the seismic response of deep soft soil sites with varying equivalent shear wave velocities (V_se), using a nonlinear dynamic finite element model and calibrated constitutive models. By comparing the seismic responses under different input motions, the study aims to provide a scientific basis for seismic site analysis and improve the understanding of how V_se affects peak ground acceleration (PGA) and spectral characteristics of deep soft soil sites.

2 Finite element modeling of two soft soil sites

A nonlinear dynamic finite element model was developed using borehole profiles and resonant column tests to calibrate the parameters of two constitutive models (Wakai and Ugai, 2004; Hardin and Drnevich, 1972). The following sections present the site configuration, governing equation, nonlinear cyclic constitutive models, and input seismic motions.

2.1 Site configurations

The borehole data were obtained, from the lower reaches of the Yangtze River, near Haimen, China. The profile of the two selected typical boreholes, ZK1 and ZK2, is shown in Figure 1. The boreholes revealed that the upper soil layers with a thickness of several tens of meters mainly consist of mud, silty sand, silty soil with interlayers of muddy silty clay, muddy clay with interlayers of silty soil, and silt with fine sand layers. The lower layers are mostly fine sand and gravelly medium to coarse sand, with some areas containing thick silty clay. The depth of the ZK1 and ZK2 boreholes reached 150 m. Table 1 shows that the overburden thickness of the ZK1 and ZK2 boreholes is 111.5 m and 95 m, respectively. The calculation domain and the shear wave velocity tested by the single borehole method for two soft soil sites are provided in Figure 1. The equivalent shear wave velocity for these two boreholes was calculated to be 147.6 m/s and 123.2 m/s. According to the “Code for Seismic Design of Buildings” (GB50011-2010), both the ZK1 and ZK2 boreholes are classified as soft soil sites (Class IV).

FIGURE 1

2.2 Governing equation for the dynamic response

A finite element model was established in the fully coupled dynamic effective stress finite element analysis software called UWLC (Cai et al., 2002; Forum 8 Co. Ltd, 2005; Xu et al., 2021; Xu et al., 2023a; Xu et al., 2024; Xue et al., 2023), which enables both effective stress and total stress analyses. In this study, the total stress analysis method was used for the site seismic response analysis, and the governing equation for the seismic response is given by Equation 1 (Biot, 1956; Pastor et al., 1990):where M is the mass matrix, C is Rayleigh damping matrix, K is the stiffness matrix, u is the displacement vector, and f^u is the external load vector. The mass coefficient and stiffness coefficient in Rayleigh damping matrix were calculated using a damping ratio of 0.03 at two frequencies (f₁ = 0.5 Hz and f₂ = 5 Hz) (Wakai and Ugai, 2004; Xu et al., 2023a).

In the finite element (FE) analysis, a static analysis should be performed to provide the initial stress for the site’s seismic response. In the dynamic analysis, Multi-Point Constraints (MPC) boundaries were applied to the lateral sides of the FE model, which is fixed at the bottom.

2.3 Nonlinear cyclic constitutive models

This study employed two nonlinear cyclic constitutive models, the Ugai and Wakai model (Wakai and Ugai, 2004) and the Hardin and Drnevich model (Hardin and Drnevich, 1972), to simulate the nonlinear and hysteretic behavior of the soil. The two models and the differences between these two models are discussed in detail as follows.

2.3.1 Hardin and Drnevich (1972) model

Nonlinear cyclic constitutive models generally consist of backbone and hysteretic curves. The backbone curve of Hardin and Drnevich (1972) model, describing the relationship between the shear stress τ and the shear strain γ, was described by Equation 2:where G₀ are τ_f are the initial shear modulus and shear strength of soils, which are given by Equations 3, 4, respectively:where G_0,rm, c, φ, and R_f are four model constants, P_a is the standard atmospheric pressure and was taken as 100 kPa in this study, Θ is Lode angle. G₀ is determined by , where ρ is natural soil density and V_s is the shear wave velocity.

The hysteretic curve of Hardin and Drnevich (1972) model is also described by Equation 5:

2.3.2 Ugai and Wakai (2004) model

The backbone curve of the Wakai and Ugai, 2004 (UW) model is the same as that of the Hardin and Drnevich (1972) (HD) model, but the hysteresis curve of the UW model differs from that of the HD model and is given by Equation 6:where b and n are two model constants that can be used to accurately control the damping ratio of soils (Xu et al., 2023b). When , the hysteresis curve of the UW model can degrade into that of the HD model, where (Wakai and Ugai, 2004).

The effectiveness of the HD model in dynamic analysis has been validated by simulating the acceleration data measured at the Wildlife site. The results show that the simulated peak ground acceleration and the peak values of the surface acceleration response spectrum match well with the observed data. A detailed description of the model can be found in the literature (Xu et al., 2013; Xu et al., 2014). Similarly, the UW model has also been proven effective in site seismic response analysis and seismic slope failure analysis (Wakai et al., 2010; Iino et al., 2024).

The Hardin and Drnevich (1972) model is a classic dynamic constitutive model, but it tends to overestimate soil damping under large strains. In contrast, the UW model introduces two parameters, b and n, which address this issue and allow for precise control of soil damping. In this study, a comparative analysis of the differences between the UW model and HD model results was conducted based on accurately calibrated model parameters.

2.3.3 Model calibration

To accurately calibrate the nonlinear model parameters, resonant column tests were conducted on 21 soil samples collected from the two (ZK1 and ZK2) boreholes. The testing instrument used was the GZZ-50 resonant column apparatus, which operates based on the principle of torsional free vibration. The test process and data acquisition are controlled by a computer, ensuring high testing accuracy. The test procedure follows (ASTM D4015-15, 2015). Figures 2, 3 shows the experimental normalized shear modulus (G/G_max)- shear strain (γ) and the damping ration (λ) – shear strain (γ) curves of various soils at the ZK1 and ZK2 boreholes, respectively. The UW model parameters calibrated based on the test data are shown in Table 2. By adjusting the model parameters, the simulated modulus and damping curves were fitted to the experimental values with an R-squared value greater than 0.94. The simulation results using the model parameters are plotted in Figures 2, 3, where it can be seen that the model simulations agree well with the experimental results.

FIGURE 2

FIGURE 3

2.4 Input seismic motions

Considering the impact of the spectral characteristics of input motions on site seismic response, this study selected three input motions with distinctly different frequency characteristics: the Liuan input motion, which is rich in low-frequency components; the Kobe input motion, which has a uniform frequency distribution; and the Nahanni input motion, which is rich in high-frequency components. Figure 4 shows the acceleration time history curves and Fourier amplitude for the Liuan, Kobe, and Nahanni input motions with peak horizontal acceleration of 0.1 g. It can be seen that the Liuan, Kobe, and Nahanni input motions represent low-frequency, medium-frequency, and high-frequency waves, respectively.

3 Results and discussions

This study focuses on the impact of input motion frequency, horizontal input motion amplitude, and equivalent shear wave velocity on the seismic response of soft soil sites (see Table 3), including peak ground acceleration (PGA), peak ground acceleration (PGA) amplification factor, acceleration response spectrum, and Fourier spectrum. The PGA amplification factor is defined as the ratio of PGA to PHA, where PHA is the peak horizontal acceleration of input motions.

The effect of input motion frequency is investigated by inputting seismic motions with three distinctly different frequency components, as shown in Figure 4. The PHA of input motions was adjusted to 0.02 g, 0.05 g, 0.1 g, 0.2 g, 0.3 g, 0.4 g, 0.5 g, 0.6 g, and 0.8 g in the finite element analyses.

FIGURE 4

Both soft soil sites are classified to Class IV according to GB50011–2010, but they differ significantly in the equivalent shear wave velocity shown in Table 1. Therefore, the effect of equivalent shear wave velocity is studied by comparing the seismic responses of these two soft soil sites. Although both the Wakai and Ugai (2004) (UW) model and the Hardin and Drnevich (1972) (HD) model can simulate the behavior of soil under cyclic loading and unloading conditions, there is a significant difference in the damping characteristics of the soils simulated by the two models. This study investigated the influence of nonlinear model selection on the seismic response of soft soil sites.

3.1 Comparison between the results using two constitutive models

Figure 5 shows the comparison between the peak ground acceleration (PGA) and the PGA amplification factor at the ground surface using two constitutive models under different input motions. The configuration of the ground is from the ZK1 borehole. As the peak horizontal acceleration (PHA) of input motions increased, the PGA calculated by both models showed an increasing trend. However, when the PHA exceeded approximately 0.1 g, the rate of increase in PGA became less pronounced, and after PHA reached 0.5 g, a decreasing trend was observed. At PHA = 0.02 g, the PGA calculated using the UW model was higher than that from the HD model. But as PHA increased further, the PGA from the UW model became significantly lower than that from the HD model. Additionally, the difference in PGA between the two models grew as PHA increased. This indicates that the choice of constitutive models can introduce systematic biases in the seismic response of sites. To avoid this issue, it is recommended that some resonant column tests be conducted in seismic safety assessments to rigorously calibrate the parameters of constitutive models.

FIGURE 5

Figure 5 also shows that the PGA amplification factor shows a gradually decreasing trend as the PHA increased. However, the difference in the PGA amplification factors calculated by the two constitutive models under low-frequency and high-frequency waves was much greater than that under medium-frequency waves. Additionally, the rate at which the PGA amplification factor decreased with increasing PHA varied depending on the input motion. Under high-frequency input motion, the PGA amplification factor was smaller, but its rate of decrease was the slowest. In contrast, the rate of decrease in the PGA amplification factor was fastest under medium-frequency input motion. Overall, when PHA exceeded 0.3 g, the PGA amplification factor for soft soil sites tended to stabilize, with all amplification factors falling below 1. The minimum PGA amplification factors calculated by the HD model and the UW model were approximately 0.45 and 0.25 respectively.

Figure 6 shows the comparison between the acceleration response spectra at the ground surface using two nonlinear constitutive models at various PHAs. In this study, a uniform damping ratio of 5% was used for the calculation of the acceleration response spectra. It can be observed that the dominant periods of the response spectra obtained from the HD model and the UW model were nearly identical, both around 3 s. However, as PHA increased, the difference in spectral acceleration (S_a) at these dominant periods between the two models became more pronounced. In the long-period range of the response spectrum, such as beyond 5 s, the spectral accelerations calculated by the UW model were higher than those from the HD model, despite the HD model generally yielding higher PGA values. Due to the larger soil damping simulated by the HD model compared to the UW model, the shear strain in the soil calculated by the HD model is smaller, leading to an overestimation of the soil’s shear modulus. This explains that the HD model produced a higher short-period response in the acceleration response spectrum. Therefore, the HD model tends to be conservative for short-period structures, but for long-period structures, it may still underestimate the seismic effects.

FIGURE 6

Figure 7 further shows that Fourier amplitude of the ground acceleration calculated by the HD model was significantly higher than those calculated by the UW model at 0.2 Hz, while for frequencies below 0.2 Hz, the trend of Fourier amplitude is reversed. This also accounts for the higher PGA amplification factor obtained with the HD model and the elevated long-period spectral acceleration observed with the UW model.

FIGURE 7

3.2 Effect of input motion frequency

Given that a deep soft soil site might result in a stronger response to long-period seismic motions, the UW model was selected for subsequent calculations and analyses. Figure 8 shows the effect of input motion frequency on the PGA and acceleration amplification factor (PGA/PHA). Overall, the PGA under low-frequency (Liuan) input motion was higher for soft soil sites compared to that under medium- and high-frequency (Kobe and Nahanni) input motions. When PHA was below 0.05 g, the PGA amplification factor under high-frequency seismic motion was significantly lower than under low-and medium-frequency input motions. However, when PHA exceeded 0.05 g, the differences in the PGA amplification factors for different input motions were not significant. The PGA amplification factor for the soft soil site under different input motions can be approximated by an exponential function, as shown by Equation 7:where a₁, a₂, and a₃ are fitting parameters and were taken as 3.90, −10.48, and 0.38, respectively. Moreover, the PGA amplification factors from the two references were given in Figure 8B. The range of PGA amplification factors calculated in this study was quite similar to the results obtained for deep soft soil sites by Zhang et al. (2023). In contrast, Tang et al. (2024) observed higher PGA amplification factors for soft soil sites in their shaking table tests, likely due to the shallower soft soil layer in their model (equivalent to a prototype depth of 75 m). Therefore, the effect of soft soil thickness on site seismic response requires careful consideration.

FIGURE 8

Figure 9 shows the effect of input motion frequency on the amplification factor (S_a/PGA) of the spectral acceleration at various PHAs. A noticeable amplification zone forms around 1 s in the acceleration response spectra under different input motions. As the PHA increased, the amplification zone expanded and shifted towards the long-period region. Overall, the amplification factor in the period range above 1 s gradually increased with increasing PHA. This is because the nonlinearity of the soil layer intensifies as the input intensity increases.

FIGURE 9

Additionally, it is particularly noteworthy that the shapes of the spectral acceleration amplification factor (S_a/PGA) under high- and medium-frequency input motions were similar, while their shapes differ significantly from those dominated by low-frequency input motions. The amplification factor under low-frequency (Liuan) seismic motion was generally above 2 for periods longer than 1 s, and there was a noticeable peak around 3 s with a value of approximately 5. In contrast, the amplification factors for medium- and high-frequency seismic motions were generally below 2 at around 2 s. This indicates that Liuan input motion, with abundant low-frequency components, may be close to the natural frequency of soft soil sites, leading to a resonance effect. This resonance effect was particularly evident when the PHA was relatively high.

Figure 10 shows that under low-frequency (Liuan) seismic motion, the low-frequency components of the ground acceleration Fourier amplitude had higher values, while under high-frequency (Nahanni) seismic motion, the high-frequency components exhibited higher values. Moreover, under mid-frequency (Kobe) seismic motion, the Fourier amplitude of ground acceleration below 0.2 Hz was the lowest, due to the minimal spectral value in this frequency range for the Kobe input motion (see Figure 4). Additionally, as PHA increased, the Fourier amplitude of ground acceleration below 1 Hz under high-frequency (Nahanni) seismic motion gradually rise, approaching the results seen under low-frequency (Liuan) seismic motion. This is because a higher PHA enhanced the nonlinearity of soils, resulting in increased spectral accelerations in the long-period range.

FIGURE 10

3.3 Effect of shear wave velocity

To study the impact of equivalent shear wave velocity (V_se) on the seismic response of the same site classification, this study selected the ZK2 borehole with V_se = 147.6 m/s for comparison. The V_se of the ZK2 borehole was much higher than the V_se of the ZK1 borehole (123.2 m/s), as shown in Table 1.

Figure 11 illustrates that the variation pattern in PGA for the two sites with increasing PHA was quite similar. However, in general, as PHA increased, the PGA and the amplification factor for the site with a lower V_se were relatively smaller. Figure 12 reveals that the shapes of the spectral acceleration were similar for the two sites, despite a substantial difference in the V_se between them. Additionally, a crossover point was observed in the spectral acceleration for the two sites. When T was less than the period corresponding to this crossover point, the spectral accelerations were lower for the site with a lower V_se. In contrast, when T exceeded the period of the crossover point, the spectral accelerations become relatively higher for the site with a lower V_se. Moreover, the spectral acceleration crossover point signifies the transition in seismic response characteristics between the two sites. This point shifts to longer periods as input motion intensity increases, indicating that the sites became softer with higher PHA and thus generally exhibited a longer characteristic period in the acceleration response spectrum.

FIGURE 11

FIGURE 12

Figure 13 provides a comparison of Fourier amplitude of ground acceleration for the two soft soil sites under the same input motion. The variation in Fourier amplitudes with frequency was fundamentally consistent with the variation in spectral acceleration with period. Specifically, the soft soil site with a higher V_se showed higher Fourier amplitudes in the high-frequency range, while the Fourier amplitudes were lower in the relatively low-frequency range.

FIGURE 13

4 Conclusion

The scope of this study is limited to two specific deep soft soil sites, which may not represent all deep soft soil conditions, further research using field data or full-scale testing is needed to ensure broader applicability of the findings. The findings of this study suggest that the selection of appropriate nonlinear constitutive models and the accurate calibration of model parameters are essential for reliable seismic response analysis of deep soft soil sites. The results provide a scientific basis for improving seismic hazard assessments and site-specific analyses, particularly in regions with deep soft soil conditions similar to those in the lower reaches of the Yangtze River, China.

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