Validation and Parametric Study of a Soil–Structure Interaction Nonlinear Simplified Model

Abstract

This paper established a numerical model for the structure system in a soil–structure interaction using MATLAB 23.2 and ANSYS 24.1 software and validated the effectiveness and accuracy of a simplified nonlinear calculation model for soil–structure interaction through the numerical simulation results. Meanwhile, a list of key parameters was identified according to their influence on the simplified model, including the free field area, site soil parameters, equivalent soil area, superstructure size and number of stories, and input seismic waves. On this basis, a series of parametric investigations were carried out with the validated simplified model, by varying the selected key parameters. The research findings demonstrated that the simplified calculation model exhibited excellent accuracy and efficiency, and was capable of reflecting the nonlinear response of the structural system. It is applicable to the field of dynamic response analysis of the superstructure, considering soil–structure interaction.

1. Introduction

The majority of national standards (e.g., [1,2,3]) currently assume a rigid foundation for the analysis of seismic performance in the superstructure, with the soil–structure interaction (SSI) being largely overlooked. However, due to the flexible nature of foundation soil, the dynamic response and failure modes of structures under seismic effects differ from those assumed with rigid foundations. Numerous studies (e.g., [4,5,6,7,8,9]) have demonstrated that the damage to structures constructed on soft soil foundations is significantly exacerbated during earthquakes due to the influence of SSI. Currently, research on SSI is predominantly conducted through experimental studies and theoretical analyses. Experimental studies include prototype tests (e.g., [10,11]), centrifuge methods (e.g., [12,13,14]), forced vibration tests (e.g., [15,16,17]), and shaking table tests (e.g., [18,19,20,21]). Theoretical analysis methods (e.g., [22,23,24,25]) primarily employ analytical formulations and numerical analysis techniques. Due to the complexity of SSI research and the uncertainty of foundation soil parameters, the consideration of SSI effects increases the complexity of structural numerical models and computational redundancy. Consequently, in engineering applications, a simplified model calculation method that is both convenient and simple, yet accurately reflects structure responses, is required.

In recent years, numerous scholars have proposed various analytical models and corresponding research methods for the study of the soil–structure interaction. Lysmer and Richart [26] proposed the SR model (Swing-Rocking Model), which simplifies the superstructure into a centralized mass, and set horizontal and rotating springs at the base position to simulate the horizontal movement and rotation of the foundation. The free field system adopted the shear particle system, with the acceleration of the top mass of the free field system serving as the input seismic motion for the foundation. This model is relatively simple and straightforward to employ, yet it is unable to simulate the nonlinear soil–foundation interaction. McClelland and Focht [27] proposed the Winkler model, which treats piles as beams in the soil. The site soil is divided into two regions: the near field and the far field. The near field region employs nonlinear springs and dampers to simulate the nonlinear soil–structure interaction. The far-field region employs linear springs and dampers to simulate the free field. This model considers the nonlinear soil–structure interaction, enabling the analysis of the vertical vibration effects of pile groups. However, as a single-parameter model, the Winkler model is unable to simulate the continuous behavior of the ground and is therefore only capable of predicting vertical displacement at specific points. In addition, the determination of spring parameters is difficult, prompting numerous scholars to improve the Winkler model. Nogami et al. [28] used an approximate frequency-independent parameter model as the far-field unit to simulate the linear elastic free field and solve the non-dynamic soil–pile response. Novak [29] used frequency-dependent springs to simulate the foundation soil impedance. Li et al. [30] developed a simplified model to calculate the lateral nonlinear dynamic soil–pile interaction. All the above models effectively simulate the soil–structure interaction mechanism.

Penzien and Scheffy [31] proposed the Penzien model, which consists of three parts: the superstructure, the pile and the equivalent soil. The superstructure is simplified to a lumped-mass model and the site soil is divided into equivalent soil layers according to certain principles. The pile group is combined into a single pile, which is divided into different units according to the soil layers, and the equivalent soil around the pile is simplified into a centralized mass. The soil–structure interaction is represented by springs and dampers. This model can better reflect the stiffness, damping and nonlinear properties of the soil, but the determination of the model parameters is relatively complex. Sun et al. [32] transformed the single pile of the Penzien model into a multi-pile model and added a free field system outside the structural system to improve the applicability of the model. Yongmei et al. [33] improved the superstructure from a multi-mass system to a beam element system, allowing for the quantitative analysis of the superstructure with the improved Penzien model. Ge et al. [34] proposed an improved Penzien model that takes into account the dynamic nonlinearity between the soil and the superstructure, and used an ANSYS numerical simulation to verify the soil nonlinearity in pseudo-dynamic substructure tests. However, further validation is required before this model can be widely applied to a wider range of conditions.

In this paper, the improved Penzien model was validated using MATLAB and ANSYS software to construct structural models that take into account soil–structure interaction, using a real building as an illustrative example. On this basis, the effects of the parameters of the components in the simplified model were investigated. This parametric study included the free field system, site soil properties, equivalent soil area, superstructure dimensions and number of stories, and seismic waves.

2. Improved Penzien Model

2.1. Model Details

The improved Penzien model consists of two parts: a structural system, including the superstructure, pile, and equivalent soil, and the free field system. The free field system is divided into multiple soil layers based on soil properties and actual conditions, with half of the mass taken from each of the adjacent two soil layers and concentrated at their interface. The superstructure, the pile, and the soil–pile interaction are the same as in the Penzien model. The improved Penzien model is based on four assumptions: (1) only the horizontal degrees of freedom between the foundation soil and the free field are considered, (2) the vertical and torsional effects of the structure are neglected, (3) the group piles corresponding to the raft foundation move synchronously and can be combined into a single pile, and (4) interactions between piles are ignored. The dynamic equation of the improved Penzien model under seismic effects is represented by Equation (1):

$$M\ddot{U}+C\dot{U}+KU=-M{\ddot{U}}_g+C_h{\dot{U}}^G+K_h{U}^G$$

(1)

where $U,\dot{U},\ddot{U}$ denote the displacement, velocity, and acceleration of the structural system, respectively. $U^G$ and ${\dot{U}}^G$ represent the displacement and velocity of the free field system. These are solved using the ANSYS finite element model. M, C, and K are the mass, damping, and stiffness matrices of the structural system. $C_h$ and $K_h$ are the damping and stiffness matrices of the pile–soil interaction. ${\ddot{U}}_g$ denotes the input acceleration due to seismic forces.

In the simplified model based on the improved Penzien model, both the superstructure and the soil can exhibit nonlinear behavior. The superstructure is simplified into a lumped-mass model with layer shear, and its nonlinear behavior is represented through a nonlinear layer shear force model. The nonlinearity of the soil is modeled through a nonlinear soil dynamics simulation within the free field system. The calculation method of specific parameter values of the simplified model under seismic effects are referenced from the paper written by Ge et al. [34].

2.2. Calculation Process of Improved Penzien Model

To develop a computational program for a simplified model using MATLAB, the process involves the following steps: (1) Utilizing ANSYS software to construct the free field model for seismic analysis, extracting the parameters and responses of each soil layer at every moment. (2) The shear modulus G and damping ratio D, obtained in the previous step, are employed to calculate the stiffness and damping of the equivalent soil and the pile–soil interaction stiffness and damping in the simplified model. (3) The mass and stiffness matrices of the structural system are formed by combining the mass and stiffness of the superstructure, pile, and equivalent soil. The damping of each subpart is then computed to create the damping matrix of the structural system. (4) Implement the dynamic time–history method within MATLAB to establish a nonlinear simplified computational model for soil–structure interaction. (5) Inputting selected seismic accelerations, solving motion equations, and conducting a seismic analysis of the simplified soil–structure interaction model. The calculation flowchart is shown in Figure 1.

3. Improved Penzien Model Validation

3.1. Validation of the Nonlinear Model of Superstructure

Before validating the nonlinear simplified calculation model for soil–structure interaction, it is necessary to verify the soil mechanics’ nonlinearity, the nonlinear model of the superstructure, and the accuracy of the time–history dynamic integration method. The soil was based on the Davidenkov equivalent linear model [35] to demonstrate its dynamic nonlinearity, which had been validated in shaking table tests [36,37,38]. The correctness of the nonlinear model of the superstructure, the time–history dynamic integration method, and the ANSYS layered shear model were validated using the 3-degrees-of-freedom (3-DOF) example employed by Yan [39]. The time–history dynamic integration method included both the central difference method and the Newmark method. The COMBIN40 element was used in the ANSYS layered shear model. The COMBIN40 element consists of two springs and a damper in parallel, effectively simulating the bilinear restoring force model.

3.1.1. Model Overview

The 3-DOF system adopted a bilinear restoring force model with a damping ratio of ${\xi }_1=0.05$ and ${\xi }_2=0.07$. The damping was calculated as $c=aM+bK$, with a = 0.05 and b = 0.07. In the model, the elastic–plastic stage corresponded to the second stiffness, and the stiffness coefficient of this stage was $\alpha =0.2$. The yield displacements for the first to third stories were 0.03, 0.02, and 0.01, respectively. The input seismic wave was the Kobe wave, and the first 20 s of the wave were selected. The time step was set to 0.02 s. The mass matrix and the initial stiffness matrix were given by Equations (2) and (3).

$$\mathrm{M}=\left[\begin{array}{ccc}{m}_1& 0& 0\\ 0& m_2& 0\\ 0& 0& m_3\end{array}\right]=\left[\begin{array}{ccc}8\times {10}^5& 0& 0\\ 0& 6\times {10}^5& 0\\ 0& 0& 6\times {10}^5\end{array}\right]$$

(2)

$$K=\left[\begin{array}{ccc}{k}_1+k_2& -k_2& 0\\ -k_2& k_2+k_3& -k_3\\ 0& -k_3& k_3\end{array}\right]=\left[\begin{array}{ccc}10\times 10^7& -4\times 10^7& 0\\ -4\times 10^7& 8\times 10^7& -4\times 10^7\\ 0& -4\times 10^7& 4\times 10^7\end{array}\right]$$

(3)

3.1.2. Comparison of Results

The 3-DOF bilinear model was analyzed for seismic effect using the COMBIN40 element in ANSYS software. In the MATLAB software, the seismic effect analysis was conducted using both the central difference method and the Newmark method for nonlinear systems, and compared with data from the reference literature [39] for hysteresis curves, displacements, velocities, and acceleration time–history responses. Due to space limitations, only the hysteresis curve, displacement, velocity, and acceleration time–history curves of the first story were shown, respectively, as shown in Figure 2. The comparisons of the peak responses for each story in MATLAB, ANSYS and the reference data were presented in

Table 1. Comparison of peak responses with the literature data (based on [39]).

Peak Responses	Story Number	Literature Data	MATLAB Newmark Method	Relative Difference (%)	MATLAB Central Difference Method	Relative Difference (%)	ANSYS Model	Relative Difference (%)
Peak displacements (m)	1	0.158	0.161	−2.14	0.164	−3.75	0.166	−4.76
	2	0.200	0.195	2.60	0.200	−0.18	0.210	−5.24
	3	0.338	0.374	−9.55	0.345	−2.12	0.345	−1.95
Peak velocities (m/s)	1	1.090	1.073	1.59	1.075	1.36	1.051	3.56
	2	1.180	1.159	1.77	1.164	1.35	1.113	5.65
	3	0.980	0.976	0.39	0.970	1.04	0.923	5.80
Peak accelerations (m/s2)	1	7.450	7.451	−0.02	7.475	−0.33	7.075	5.04
	2	9.870	9.696	1.76	9.698	1.75	9.618	2.55
	3	8.890	8.448	5.23	8.887	0.03	8.795	1.06

.

From Table 1, it can be observed that the maximum relative differences in peak displacements, velocities, and accelerations are 9.55%, 5.8%, and 5.23%, respectively. There were two main sources of these errors: Firstly, these different integration methods introduced errors after the structure entered the nonlinear phase. Secondly, the literature comparison data from the literature were obtained by extracting data points from figures using GETDATE (Transact-SQL) software, which could introduce some inaccuracies. The relative differences in peak responses for each story were within 5%. As shown in Figure 2, the hysteresis curves and the time–history curves of the first story obtained by different methods closely match, indicating the effective representation of the dynamic response characteristics of the structure by the bilinear restoring force model used for the superstructure in the simplified model. Moreover, the simplified model of the superstructure was both effective and accurate. Additionally, both the central difference method and the Newmark method for nonlinear systems were reliable and met usage requirements. The results obtained using the COMBIN40 element in ANSYS software to simulate the superstructure model were also dependable. The results obtained using the Newmark method and the central difference method in MATLAB software generally aligned. However, the Newmark method is more commonly employed in engineering calculations and is more reliable for MDOF structural systems. Therefore, in subsequent seismic response analyses, MATLAB software will exclusively use the Newmark method.

3.2. Validation of the Soil–Structure Interaction Simplified Model

3.2.1. Model Overview

The superstructure was a single-span steel-framed structure with six stories, each with a uniform height of 2 m, with both the span and depth measuring 2 m. The construction material selected was Q235 steel. The pile consisted of circular concrete piles that penetrated the soil to reach the bedrock. The diameter of the piles was 0.45 m, and the concrete used for the piles was C30, with a platform area of 5 m². The site soil was classified as type II, comprising three layers of varying materials, starting from the surface and extending towards the bedrock: 4 m of clay, 4 m of silty clay, and 8 m of silty soil. The parameters of various soils were calculated after fitting, according to the test data of Yuan et al. [40]. The detailed material parameters for each soil layer are provided in

Table 2. Soil material parameters (based on [40]).

Soil Type	Shear Wave Velocity (m/s)	Density (kg/m3)	Poisson’s Ratio (-)	Damping Ratio (-)	Shear Modulus (MPa)	Elasticity Modulus (MPa)
Dense sandy	350	2100	0.30	0.01	257	669
Medium dense sandy	340	2010	0.30	0.01	232	604
Silty soil	338	2010	0.30	0.01	230	598
Silty clay	210	1930	0.32	0.01	85	224
Clay	135	1820	0.35	0.01	33	89

.

3.2.2. Simplified Model Validation

In the numerical simulation conducted using ANSYS software, the superstructure was represented by COMBIN40 elements, while the equivalent single pile was modeled using BEAM188 elements with a diameter of 0.636 m. The soil was represented by SOLID185 elements with a vertical thickness of 16 m. The horizontal extent, informed by the study of Lu et al. [41], was selected as 100 m by 100 m. The boundary condition was set as fixed at the base and free along the perimeter.

After the six-story framework structure model was simplified into a shear-cut model, the superstructure did not reach the nonlinear stage under the seismic effect of the EL-central wave with a peak acceleration of 0.4 g. The seismic responses of the third, fourth, and sixth stories, which were representative of the structure, were compared and analyzed.

Table 3. Peak structure responses.

Story Number	Peak Displacements (m)			Peak Velocities (m/s)			Peak Accelerations (m/s2)
	ANSYS	MATLAB	Relative Difference (%)	ANSYS	MATLAB	Relative Difference (%)	ANSYS	MATLAB	Relative Difference (%)
3	0.082	0.070	13.97	0.806	0.736	8.74	12.421	12.287	1.08
4	0.097	0.087	10.87	1.018	0.967	5.01	13.780	13.888	−0.78
6	0.114	0.106	7.25	1.269	1.248	1.61	15.157	14.939	1.44

presents a comparison of peak displacements, velocities, and accelerations. Table 3 indicates that the greatest difference in peak displacements occurs in the third story, at 13.97%. Similarly, the greatest difference in peak velocity occurs in the third story, at 8.74%, while the greatest difference in peak accelerations occurs in the sixth story, at 1.44%. The results from the two models were highly comparable, with the relative differences in peak responses of each story not exceeding 14%. Figure 3 illustrates that the displacement, velocity, and acceleration time–history curves of the MATLAB model and the ANSYS model were largely congruent, with only minor discrepancies in amplitude.

A comparison of the response spectra in Figure 4 reveals that the overall response spectra of the two models were consistent. In the short-period range, the displacement, velocity, and acceleration response spectra values of each story in the MATLAB model were found to be largely consistent with those of the ANSYS model. In the peak regions, the response spectra values for displacement, velocity, and acceleration in the MATLAB model were slightly higher than those in the ANSYS model. In the long-period range, the velocity and acceleration response spectra obtained from the MATLAB simplified model were essentially identical to those from the ANSYS model, while the displacement response spectra obtained from the MATLAB simplified model were slightly lower than those from the ANSYS model. This finding was consistent with the conclusions drawn by Chen et al. [42] in the comparison of simplified models and FE models.

The discrepancies between the MATLAB and ANSYS models can be attributed to three main factors. Firstly, the modeling methods employed for the soil differed between the two models. In the MATLAB model, the soil was divided into the free field system and the equivalent soil according to the difference between the far field and the near field. The seismic analysis of the free field system enabled the extraction of parameters such as shear modulus, damping ratio, velocity, and displacement for each soil layer, which were subsequently employed in the dynamic equation calculations of the structural system. Furthermore, the stiffness and damping of the equivalent soil were incorporated into the structural system stiffness and damping matrices, thereby allowing for the consideration of the influence of soil properties on the structural system. In the ANSYS model, the soil was represented using an actual finite element model. Once the overall structural system model had been established, the soil was meshed to create numerous soil elements. Subsequently, the entire model was subjected to a seismic analysis. Secondly, the modeling methods employed for the pile differ between the two models. In the MATLAB model, the pile was simplified into a lumped-mass model based on soil stratification. In the ANSYS model, the pile was simulated using BEAM elements. Thirdly, the seismic wave input methods differed. In the MATLAB model, the seismic analysis of the structural system involved the calculation of seismic forces for all mass points, with consideration given to the soil–structure interaction within the dynamic equations. In contrast, the ANSYS model input seismic acceleration from all nodes at the bottom bedrock surface, which may introduce filtering effects.

Although discrepancies existed between the two models due to differences in soil modeling, pile modeling and seismic input methods, the nonlinear soil–structure interaction simplified calculation model was capable of rapidly and effectively reflecting the structure response under seismic loads, while considering soil–structure interaction. The simplified model has been sufficiently validated for both soil and superstructure nonlinearity. Consequently, it can be employed as a computational model for nonlinear systems in the field of soil–structure interaction research, offering effective guidance for design and optimization in practical engineering applications with the objective of ensuring structure safety and stability under seismic effects.

4. Parametric Study of the Simplified Model

In order to study the influence of each component in the soil–structure interaction simplified model, five variables, such as the free field system, site soil parameters, equivalent soil area, superstructure size and number of stories, and input seismic waves were altered, respectively, to explore their variation patterns in terms of structure response and frequency. When changing a single parameter, the remaining parameters were consistent with those used in Section 3.2.

4.1. Free Field System

Theoretically, the larger the range of the free field system selected, the more it can reflect the response under real conditions. However, once a certain range is reached, the results obtained from the seismic analysis of the free field system will not vary significantly. To investigate the impact of the free field system range on the results, four free field systems of different areas were selected as working conditions, with specific information shown in

Table 4. Conditions of the free field area.

Condition	Condition 1	Condition 2	Condition 3	Condition 4
Free Field Area (m × m)	20 × 20	50 × 50	100 × 100	200 × 200

. Seismic analysis was conducted under EL-central waves with peak accelerations of 0.1 g, 0.2 g, 0.3 g, and 0.4 g, respectively. The peak velocities and displacements of each soil layer were extracted and are presented in

Table 5. Peak displacements of each layer in the free field system (for input 0.1 g).

Soil Layer	Peak Displacements in (m) for Condition				Relative Difference between Adjacent Conditions (%)
	1	2	3	4	1–2	2–3	3–4
1	0.004293	0.002807	0.002389	0.002340	52.94	17.49	2.09
2	0.003683	0.002473	0.002114	0.002071	48.95	16.96	2.07
3	0.003017	0.002073	0.001773	0.001736	45.56	16.89	2.13
4	0.002400	0.001698	0.001455	0.001424	41.34	16.71	2.15
5	0.001818	0.001337	0.001154	0.001130	35.96	15.86	2.09
6	0.001299	0.001016	0.000895	0.000879	27.86	13.51	1.84
7	0.000763	0.000643	0.000580	0.000571	18.59	10.99	1.58
8	0.000366	0.000333	0.000307	0.000303	9.73	8.52	1.39

,

Table 6. Peak velocities of each layer in the free field system (for input 0.1 g).

Soil Layer	Peak Velocities in (m/s) for Condition				Relative Difference between Adjacent Conditions (%)
	1	2	3	4	1–2	2–3	3–4
1	0.048120	0.039859	0.040681	0.041306	20.73	−2.02	1.51
2	0.041122	0.035265	0.035782	0.036245	16.61	−1.44	1.28
3	0.033539	0.027727	0.029867	0.030177	20.96	−7.16	1.03
4	0.026604	0.025408	0.024263	0.024452	4.71	4.72	0.77
5	0.020128	0.019599	0.018996	0.019108	2.70	3.17	0.58
6	0.015398	0.015442	0.014382	0.014470	−0.28	7.37	0.61
7	0.008485	0.008828	0.009044	0.009100	−3.89	−2.38	0.62
8	0.004141	0.004430	0.004674	0.004702	−6.53	−5.23	0.58

,

Table 7. Peak displacements of each layer in the free field system (for input 0.2 g).

Soil Layer	Peak Displacements in (m) for Condition				Relative Difference between Adjacent Conditions (%)
	1	2	3	4	1–2	2–3	3–4
1	0.008613	0.005641	0.004810	0.004715	52.70	17.27	2.01
2	0.007393	0.004967	0.004260	0.004177	48.83	16.60	1.99
3	0.006062	0.004169	0.003580	0.003508	45.40	16.46	2.05
4	0.004828	0.003421	0.002944	0.002885	41.12	16.21	2.07
5	0.003661	0.002699	0.002340	0.002294	35.67	15.33	2.02
6	0.002620	0.002054	0.001817	0.001785	27.54	13.06	1.79
7	0.001540	0.001302	0.001177	0.001159	18.27	10.65	1.55
8	0.000738	0.000676	0.000624	0.000616	9.23	8.29	1.37

,

Table 8. Peak velocities of each layer in the free field system (for input 0.2 g).

Soil Layer	Peak Velocities in (m/s) for Condition				Relative Difference between Adjacent Conditions (%)
	1	2	3	4	1–2	2–3	3–4
1	0.095767	0.080551	0.079388	0.078713	18.89	1.47	−0.86
2	0.081936	0.070831	0.069967	0.069460	15.68	1.24	−0.73
3	0.066947	0.059182	0.058586	0.058629	13.12	1.02	0.07
4	0.053203	0.048217	0.047762	0.048140	10.34	0.95	0.78
5	0.040326	0.037993	0.037521	0.037755	6.14	1.26	0.62
6	0.028869	0.028602	0.028397	0.028581	0.93	0.72	0.64
7	0.017027	0.017974	0.017867	0.017902	−5.27	0.60	0.19
8	0.008271	0.009192	0.009249	0.009300	−10.02	−0.62	0.55

,

Table 9. Peak displacements of each layer in the free field system (for input 0.3 g).

Soil Layer	Peak Displacements in (m) for Condition				Relative Difference between Adjacent Conditions (%)
	1	2	3	4	1–2	2–3	3–4
1	0.012988	0.008544	0.007284	0.007148	52.01	17.31	1.89
2	0.011159	0.007511	0.006461	0.006342	48.58	16.24	1.88
3	0.009165	0.006308	0.005443	0.005340	45.28	15.89	1.94
4	0.007315	0.005189	0.004491	0.004405	40.95	15.55	1.96
5	0.005559	0.004104	0.003580	0.003513	35.44	14.63	1.92
6	0.003983	0.003129	0.002782	0.002735	27.29	12.47	1.72
7	0.002345	0.001987	0.001803	0.001777	18.00	10.19	1.51
8	0.001126	0.001033	0.000957	0.000944	9.01	7.94	1.35

,

Table 10. Peak velocities of each layer in the free field system (for input 0.3 g).

Soil Layer	Peak Velocities in (m/s) for Condition				Relative Difference between Adjacent Conditions (%)
	1	2	3	4	1–2	2–3	3–4
1	0.142888	0.116661	0.115396	0.116920	22.48	1.10	1.30
2	0.122457	0.103125	0.101950	0.102169	18.75	1.15	0.21
3	0.100316	0.086415	0.085703	0.086548	16.09	0.83	0.98
4	0.079948	0.071252	0.070221	0.070757	12.21	1.47	0.76
5	0.060770	0.056297	0.055373	0.055733	7.95	1.67	0.65
6	0.043550	0.042440	0.041911	0.042179	2.62	1.26	0.63
7	0.025707	0.026608	0.026439	0.026585	−3.39	0.64	0.55
8	0.012435	0.013686	0.013722	0.013792	−9.14	−0.26	0.51

,

Table 11. Peak displacements of each layer in the free field system (for input 0.4 g).

Soil Layer	Peak Displacements in (m) for Condition				Relative Difference between Adjacent Conditions (%)
	1	2	3	4	1–2	2–3	3–4
1	0.017440	0.011529	0.009821	0.009650	51.27	17.39	1.77
2	0.015005	0.010154	0.008728	0.008577	47.78	16.33	1.77
3	0.012348	0.008517	0.007373	0.007241	44.97	15.52	1.82
4	0.009882	0.007014	0.006105	0.005995	40.90	14.87	1.85
5	0.007532	0.005564	0.004884	0.004796	35.37	13.93	1.82
6	0.005406	0.004250	0.003799	0.003738	27.20	11.88	1.64
7	0.003187	0.002704	0.002465	0.002430	17.88	9.70	1.46
8	0.001534	0.001408	0.001310	0.001293	8.92	7.55	1.31

and

Table 12. Peak velocities of each layer in the free field system (for input 0.4 g).

Soil Layer	Peak Velocities in (m/s) for Condition				Relative Difference between Adjacent Conditions (%)
	1	2	3	4	1–2	2–3	3–4
1	0.189622	0.153720	0.149134	0.150906	23.36	3.08	1.17
2	0.162823	0.134586	0.132064	0.133476	20.98	1.91	1.06
3	0.133791	0.113253	0.111477	0.112493	18.13	1.59	0.90
4	0.106995	0.093698	0.091829	0.092491	14.19	2.04	0.72
5	0.081611	0.074298	0.072705	0.073152	9.84	2.19	0.61
6	0.058564	0.056117	0.055160	0.055459	4.36	1.74	0.54
7	0.034603	0.035260	0.034927	0.035095	−1.86	0.95	0.48
8	0.016761	0.018174	0.018183	0.018267	−7.78	−0.05	0.46

.

Based on Table 5, Table 6, Table 7, Table 8, Table 9, Table 10, Table 11 and Table 12, for peak displacements under seismic effects at various peak accelerations, the relative difference between Condition 1 and Condition 2 exceeded 20%, except for the bottom layer, with a maximum of 52.94%. The relative difference between Condition 2 and Condition 3 ranged from 8% to 20%, with a maximum of 17.49%. The relative difference between Condition 3 and Condition 4 was within 2.20%, with a maximum of only 2.15%. The maximum relative difference between Condition 1 and Condition 2 for peak velocities under seismic effects at various peak accelerations was 23.36%. The maximum relative difference between Condition 2 and Condition 3 was 4.72%. The maximum relative difference between Condition 3 and Condition 4 was 1.51%.

In summary, the relative difference between Condition 1 and Condition 2 was substantial, indicating that when the side length of the free field system increased from 20 m to 50 m, the resulting differences were significant. The maximum relative differences in displacement and velocity between Condition 2 and Condition 3 were 17.49% and 4.72%, respectively, indicating that there was still a noticeable difference when the soil body side length was 50 m compared to 100 m. The maximum relative differences in displacement and velocity between Condition 3 and Condition 4 were 2.15% and 1.51%, respectively, showing that the results were very similar when the soil body side length was 100 m compared to 200 m. Therefore, it can be concluded that the results obtained with a 100 m side length free field system were essentially consistent with those obtained with a 200 m side length system, and a free field side length of 100 m meets the requirements without needing to increase the size further. The superstructure’s span and depth were both 2 m, and the 100 m × 100 m free field system’s side length was 50 times the layout size of the superstructure. The side length of the free field system was 6.25 times the thickness of the free field.

4.2. Site Soil Parameters

The properties of the site soil had a significant impact on the soil–structure interaction simplified model, with the mass of the site soil being factored into the calculation of the equivalent soil mass. Following the completion of a seismic analysis in a free field system, the obtained shear modulus and damping ratio were used to determine the stiffness and damping of the equivalent soil, as well as the stiffness and damping of the soil–pile interaction. The free field system consists of various types of soil layers. Following the seismic action analysis, the shear modulus, damping ratio, velocity, and displacement results for each layer of soil were used in the structural system calculations, thereby influencing the response of the structural system within the simplified model. For different combinations of soil types, a total of eight different types of site soils were selected as eight different conditions, with the properties of the site soil gradually becoming softer from Condition 1 to Condition 8. The types of site soil compositions for these eight conditions are shown in

Table 13. Types of site soil composition.

Thickness (m)	4	4	8
Condition 1	Dense sandy	Dense sandy	Dense sandy
Condition 2	Medium dense sandy	Medium dense sandy	Dense sandy
Condition 3	Silty soil	Medium dense sandy	Dense sandy
Condition 4	Silty soil	Silty soil	Dense sandy
Condition 5	Clay	Silty soil	Dense sandy
Condition 6	Clay	Silty clay	Medium dense sandy
Condition 7	Clay	Clay	Silty soil
Condition 8	Clay	Clay	Silty clay

, and the parameters for the various soils are depicted in Table 2. Seismic action analyses were conducted under EL-central waves with peak accelerations of 0.1 g, 0.2 g, 0.3 g, and 0.4 g. Due to the limited space of the article, only the peak responses of each soil layer for peak seismic accelerations of 0.1 g and 0.4 g are shown in Figure 5.

Figure 5 shows that under the influence of seismic waves with different peak accelerations, the peak displacements for Conditions 1 to 4 were relatively close to each other, while the curves for Conditions 5 to 8 were significantly higher than those for the first four Conditions. For peak velocities and accelerations, the peaks for each layer in Conditions 2 to 4 were quite similar, with the curve for Condition 1 being slightly higher than those for Conditions 2 to 4. The curves for Conditions 5 to 8 were significantly higher than those for Conditions 2 to 4, and there was a clear distinction between adjacent curves in Conditions 5 to 8. The maximum peak responses under the different conditions all occurred in the first layer of the free field system; therefore, only the relative differences in the first layer conditions were analyzed, and the results are presented in

Table 14. The free field 1st layer peak displacements in (cm) for condition.

Seismic Acceleration	1	2	3	4	5	6	7	8
0.1 g	0.13199	0.12471	0.12491	0.12704	0.15983	0.22143	0.33524	0.46969
0.2 g	0.26562	0.25044	0.25088	0.25551	0.32101	0.44454	0.67540	0.93991
0.3 g	0.40189	0.37788	0.37862	0.38640	0.48447	0.67065	1.02352	1.41073
0.4 g	0.54116	0.50733	0.50846	0.52014	0.65048	0.90076	1.38067	1.88178

,

Table 15. Relative difference between adjacent conditions (%) of the free field 1st layer peak displacements.

Seismic Acceleration	1–2	2–3	3–4	4–5	5–6	6–7	7–8
0.1 g	5.83	−0.16	−1.67	−20.52	−27.82	−33.95	−28.63
0.2 g	6.06	−0.17	−1.81	−20.40	−27.79	−34.18	−28.14
0.3 g	6.36	−0.19	−2.02	−20.24	−27.76	−34.48	−27.45
0.4 g	6.67	−0.22	−2.25	−20.04	−27.79	−34.76	−26.63

,

Table 16. The free field 1st layer peak velocities in (cm/s) for condition.

Seismic Acceleration	1	2	3	4	5	6	7	8
0.1 g	2.94696	2.34589	2.36825	2.50786	3.13581	3.59393	4.69157	5.32391
0.2 g	5.79372	4.61552	4.66073	4.94772	6.14699	7.04792	9.19737	10.41477
0.3 g	8.49859	6.77604	6.84652	7.29171	8.97619	10.30576	13.44678	15.26319
0.4 g	11.07289	8.83540	8.93559	9.54552	11.65110	13.39956	17.47332	19.84260

,

Table 17. Relative difference between adjacent conditions (%) of the free field 1st layer peak velocities.

Seismic Acceleration	1–2	2–3	3–4	4–5	5–6	6–7	7–8
0.1 g	25.62	−0.94	−5.57	−20.03	−12.75	−23.40	−11.88
0.2 g	25.53	−0.97	−5.80	−19.51	−12.78	−23.37	−11.69
0.3 g	25.42	−1.03	−6.11	−18.77	−12.90	−23.36	−11.90
0.4 g	25.32	−1.12	−6.39	−18.07	−13.05	−23.31	−11.94

,

Table 18. The free field 1st layer peak accelerations in (m/s²) for condition.

Seismic Acceleration	1	2	3	4	5	6	7	8
0.1 g	1.16347	1.04627	1.04937	1.06718	1.12363	1.20535	1.34852	1.52690
0.2 g	2.32815	2.13087	2.13319	2.15903	2.28586	2.41394	2.66726	3.02455
0.3 g	3.49249	3.18218	3.20088	3.23067	3.40469	3.58465	3.95295	4.48361
0.4 g	4.65324	4.24889	4.25415	4.26863	4.55841	4.85031	5.21881	5.90990

and

Table 19. Relative difference between adjacent conditions (%) of the free field 1st layer peak accelerations.

Seismic Acceleration	1–2	2–3	3–4	4–5	5–6	6–7	7–8
0.1 g	11.20	−0.30	−1.67	−5.02	−6.78	−10.62	−11.68
0.2 g	9.26	−0.11	−1.20	−5.55	−5.31	−9.50	−11.81
0.3 g	9.75	−0.58	−0.92	−5.11	−5.02	−9.32	−11.84
0.4 g	9.52	−0.12	−0.34	−6.36	−6.02	−7.06	−11.69

. A final calculation of the first two orders of frequency of the structural system was conducted based on the parameters of soil density, shear modulus, and other parameters to form the mass and stiffness of the equivalent soil. The first-order frequency was 11.369 Hz, and the second-order frequency of the structure is shown in

Table 20. The second-order frequencies of structural systems under different peak seismic accelerations.

The Second-Order Frequencies in (Hz) for Input	0.1 g	0.2 g	0.3 g	0.4 g
1	31.520	31.255	30.900	30.511
2	31.514	31.255	30.903	30.515
3	31.513	31.254	30.901	30.514
4	31.502	31.240	30.884	30.491
5	31.686	31.448	31.119	30.755
6	26.590	26.300	25.883	25.387
7	23.701	23.419	23.035	22.607
8	19.874	19.717	19.505	19.270

. A comparison of the second-order frequency of adjacent conditions is shown in

Table 21. Relative difference between adjacent conditions (%) of the second-order frequencies.

Seismic Acceleration	1–2	2–3	3–4	4–5	5–6	6–7	7–8
0.1 g	0.018	0.004	0.035	−0.584	16.082	10.865	16.147
0.2 g	0.001	0.004	0.044	−0.666	16.368	10.954	15.809
0.3 g	−0.009	0.003	0.058	−0.761	16.824	11.003	15.325
0.4 g	−0.013	0.003	0.077	−0.866	17.453	10.949	14.764

.

Table 14 and Table 15 demonstrate that, in the free field, the peak displacements of the first layer in Condition 1 were between those of Conditions 4 and 5. Apart from Condition 1, the peak displacements in Conditions 2 to 8 increased continuously, with the relative differences between adjacent conditions in Conditions 4 to 8 all exceeding 20, with a maximum of 34.76. Table 16 and Table 17 demonstrate that, for the peak velocities of the first layer in the free field, the peak velocities in Condition 1 were between those of Conditions 4 and 5. The relative differences between adjacent conditions in Conditions 4 to 8 all exceeded 10, with a maximum of 23.40. According to Table 18 and Table 19, the peak accelerations of the first layer in the free field in Condition 1 were between those of Conditions 5 and 6. The relative differences between adjacent conditions from Conditions 4 to 8 all exceeded 5, with a maximum of 11.84.

As shown in Table 20 and Table 21, changes in the ground conditions influenced the second natural frequency of the structural system. The second natural frequencies for Conditions 1 to 5 were found to be quite similar under the influence of the same peak seismic wave, with the relative differences all within 0.1. However, the frequencies for Conditions 5 to 8 gradually decreased and differed significantly, with the greatest difference occurring between Conditions 5 and 8, with a relative difference of approximately 60. The relative differences between Conditions 5 and 8 all exceeded 10, with the maximum being 17.453. This indicated that as the site soil properties changed from Condition 5 onwards, the softening of the soil had an increasing effect on the frequency reduction in the structural system. Under the influence of seismic waves with different peak accelerations, the peak response in Condition 1 mainly fell between those of Conditions 4 and 5. This indicated that the response of soils with a certain stiffness to seismic action can still be significant. The results for Conditions 2 to 4 showed minimal differences, indicating that the response of these three soil types to seismic action was similar. Conversely, the results for Conditions 5 to 8 differed significantly from those for Conditions 2 to 4, and there were also significant differences between adjacent conditions within Conditions 5 to 8. This indicated that once the soil properties reached those characterized in Condition 5, the softer the soil, the greater the response under seismic effects. In addition, the influence of the soil on the frequency of the structural system increased. Therefore, when the soil softness reaches the level of Condition 5, it was essential to consider the soil–structure interaction. In Condition 5, the soil composition from top to bottom was 4 m of clay, 4 m of silt and 8 m of dense sand.

In summary, within a certain range, the softer the soil layers, the greater the response of the free field system under seismic effects. This led to a greater reduction in the frequency of the structural system and increased the effect of soil–structure interaction. Therefore, it was important to consider the soil–structure interaction on soft foundations where the soil softness was lower than in Condition 5.

4.3. Equivalent Soil Area

The mass, stiffness, and damping of the equivalent soil contributed to forming the overall mass, stiffness, and damping matrices of the simplified model. Consequently, selecting different ranges of equivalent soil areas had a certain impact on the overall structural system of the simplified model. In the majority of cases, the footing area of the superstructure was selected as the equivalent soil area. The equivalent soil, treated with group piles and forming an equivalent single pile area, was selected as the reference. The equivalent single pile radius was 0.318 m. Five different equivalent soil areas were selected as five conditions. The specific condition information was shown in

Table 22. Conditions of equivalent soil area.

Condition	Condition 1	Condition 2	Condition 3	Condition 4	Condition 5
Equivalent soil area	No consideration of equivalent soil	The pile area	10 times the pile area	The footing area	20 times the pile area

. Under the action of EL-central seismic waves with peak accelerations of 0.1 g, 0.2 g, 0.3 g, and 0.4 g, the frequency changes in the structural system were analyzed. Since the first-order frequencies of the simplified model structural system were all 11.369 Hz under different conditions and different peak seismic waves, only the second-order frequencies of the simplified model structural system were extracted and presented in

Table 23. The second-order frequencies of structural systems with different equivalent soil areas.

Condition	0.1 g		0.2 g		0.3 g		0.4 g
	Frequency (Hz)	Relative Difference (%)	Frequency (Hz)	Relative Difference (%)	Frequency (Hz)	Relative Difference (%)	Frequency (Hz)	Relative Difference (%)
1	21.084	-	21.084	-	21.084	-	21.084	-
		−24.69		−24.69		−23.83		−21.68
2	27.997		27.996		27.682		26.921
		1.80		2.80		3.62		3.04
3	27.503		27.233		26.715		26.127
		3.19		3.62		3.71		3.81
4	26.653		26.281		25.760		25.168
		1.37		1.79		1.84		1.83
5	26.294		25.819		25.295		24.715

.

From Table 23, it can be observed that when the equivalent soil was not considered, the frequency under different peak acceleration seismic waves were identical. This was because when not considering the equivalent soil, the soil parameters did not contribute to forming the frequency of the structural system. Under different peak acceleration seismic waves, the relative difference between Condition 1 and Condition 2 was significant, with a maximum relative difference of up to 24.69. The difference between Condition 2 and Condition 4 was relatively small, while the difference between Condition 4 and Condition 5 was the smallest, with relative differences all below 1.90.

The results of the calculations conducted under the assumption that the equivalent soil was not considered were found to be significantly different from those obtained when the equivalent soil was taken into account. This indicated that the equivalent soil should be included in the simplified model. Upon consideration of the equivalent soil, it became evident that this had a significant impact on the second-order frequency of the structural system. The results with equivalent soil areas of the pile area, 10 times the pile area, 20 times the pile area, and 5 m² differed to a relatively small degree. The results were closest when the equivalent soil area was 20 times the pile area and 5 m², with relative differences all below 1.90. This indicated that when considering the equivalent soil, the area of 20 times the single pile area can be selected based on the area of the equivalent single pile, or an empirical value slightly larger than the superstructure area could be chosen.

4.4. Superstructure Size and Number of Stories

The mass, stiffness and damping of the superstructure not only contributed to the overall mass, stiffness and damping matrix of the simplified model, but also determined the number of lumped-mass points in the structural system based on the number of stories in the superstructure. The dimensions of the superstructure dictated parameters such as its mass and stiffness. Therefore, structural systems formed by superstructures with different numbers of stories could be completely different. Two structural shapes were selected for the superstructure, with each floor having a span, depth and height of either 2 m or 3 m. The superstructure was then configured with 1, 3 and 6 stories to produce simplified models of the soil–structure interaction systems, with the corresponding conditions shown in

Table 24. Structural system frequencies.

Condition	Dimensions per Story	Stories Number	Frequency (Hz)
			First-Order	Second-Order
Condition 1	2 m × 2 m × 2 m	1	25.168	47.161
Condition 2	2 m × 2 m × 2 m	3	20.989	25.168
Condition 3	2 m × 2 m × 2 m	6	11.369	25.168
Condition 4	3 m × 3 m × 3 m	1	22.819	26.762
Condition 5	3 m × 3 m × 3 m	3	11.910	22.819
Condition 6	3 m × 3 m × 3 m	6	6.452	18.980

.

The frequencies of the structural system are presented in Table 24. When the dimensions of each story of the superstructure were 2 m × 2 m × 2 m, the first-order frequencies of the structural system for one story, three stories, and six stories were 25.168 Hz, 20.989 Hz, and 11.369 Hz, respectively. Similarly, when the dimensions of each layer of the superstructure were 3 m × 3 m × 3 m, the first-order frequencies of the structural system for one layer, three stories, and six stories were 22.819 Hz, 11.910 Hz, and 6.452 Hz, respectively. When the structural form of each layer remained unchanged, the relative differences between Condition 1 and Condition 3 were 121.37, between Condition 2 and Condition 3 were 84.61, between Condition 4 and Condition 6 were 253.68, and between Condition 5 and Condition 6 were 84.61. When the number of stories of the superstructure was six, the relative difference between Condition 3 and Condition 6 was 76.22. It was observed that both changes in the number of stories and the dimensions of the superstructure had a significant impact on the entire structural system, resulting in considerable differences among simplified models. This highlighted the importance of superstructure design in the seismic analysis of soil–structure interaction simplified models.

4.5. Input Seismic Waves

When performing a free field system analysis, seismic acceleration had to be input to obtain the response of each layer of the free field system and the soil parameters. In addition, when solving the dynamic equations of the structural system, seismic acceleration was also input to solve these equations. Consequently, the input seismic waves resulted in variations in the dynamic equations. In the simplified model, seismic waves were input for both the free field system analysis and the structural system analysis under seismic effects. In this paper, four seismic waves were selected for analysis: the EL-Central wave, the Kobe wave, the Taft wave, and the TS wave. The initial 10 s of each wave were utilized, with peak accelerations of 0.1 g and 0.4 g applied, resulting in a total of eight conditions for the seismic analysis of the simplified model.

The peak displacements, velocities, and accelerations of the superstructure obtained from the time–history analysis under the action of different seismic waves are presented in

Table 25. Peak displacements of each story of the superstructure (for input 0.1 g).

Story Number	Peak Displacements in (cm) under Seismic Waves				Relative Difference with EL Wave (%)
	EL Wave	Kobe Wave	Taft Wave	TS Wave	Kobe Wave	Taft Wave	TS Wave
6	2.500	1.995	1.451	1.401	20.17	41.95	43.94
5	2.363	1.870	1.351	1.311	23.65	42.81	44.49
4	2.091	1.632	1.163	1.147	23.17	44.38	45.16
3	1.704	1.300	0.908	0.935	21.42	46.70	45.12
2	1.209	0.901	0.612	0.675	18.67	49.37	44.19
1	0.630	0.459	0.303	0.360	15.80	51.94	42.88

,

Table 26. Peak velocities of each story of the superstructure (for input 0.1 g).

Story Number	Peak Velocities in (cm/s) under Seismic Waves				Relative Difference with EL Wave (%)
	EL Wave	Kobe Wave	Taft Wave	TS Wave	Kobe Wave	Taft Wave	TS Wave
6	0.311	0.216	0.169	0.178	30.50	45.72	42.83
5	0.284	0.205	0.156	0.165	14.37	45.20	42.06
4	0.239	0.183	0.133	0.140	18.11	44.51	41.57
3	0.182	0.149	0.107	0.108	22.73	41.22	40.76
2	0.121	0.106	0.077	0.074	26.27	36.36	39.03
1	0.061	0.055	0.040	0.038	28.21	33.80	37.54

,

Table 27. Peak accelerations of each story of the superstructure (for input 0.1 g).

Story Number	Peak Accelerations in (m/s2) under Seismic Waves				Relative Difference with EL Wave (%)
	EL Wave	Kobe Wave	Taft Wave	TS Wave	Kobe Wave	Taft Wave	TS Wave
6	3.723	2.870	2.580	2.148	22.92	30.71	42.30
5	3.658	2.605	2.353	1.984	16.98	35.68	45.78
4	3.466	2.145	1.902	1.838	8.84	45.14	46.97
3	3.056	1.718	1.571	1.590	4.19	48.59	47.98
2	2.320	1.219	1.184	1.177	1.80	48.97	49.24
1	1.432	0.638	0.624	0.632	0.47	56.44	55.88

,

Table 28. Peak displacements of each story of the superstructure (for input 0.4 g).

Story Number	Peak Displacements in (cm) under Seismic Waves				Relative Difference with EL Wave (%)
	EL Wave	Kobe Wave	Taft Wave	TS Wave	Kobe Wave	Taft Wave	TS Wave
6	10.595	7.982	5.804	5.605	24.67	45.22	47.09
5	9.451	7.481	5.404	5.246	23.65	42.82	44.49
4	8.665	6.526	4.653	4.587	22.37	46.31	47.06
3	7.015	5.200	3.633	3.740	20.81	48.21	46.68
2	4.838	3.603	2.450	2.700	18.67	49.35	44.18
1	2.521	1.838	1.212	1.440	15.80	51.92	42.88

,

Table 29. Peak velocities of each story of the superstructure (for input 0.4 g).

Story Number	Peak Velocities in (cm/s) under Seismic Waves				Relative Difference with EL Wave (%)
	EL Wave	Kobe Wave	Taft Wave	TS Wave	Kobe Wave	Taft Wave	TS Wave
6	1.248	0.866	0.676	0.712	30.60	45.80	42.92
5	1.136	0.822	0.623	0.658	14.37	45.20	42.06
4	0.967	0.732	0.531	0.559	17.89	45.14	42.23
3	0.736	0.596	0.427	0.431	22.45	41.96	41.50
2	0.485	0.423	0.308	0.296	26.27	36.41	39.05
1	0.244	0.221	0.161	0.152	28.22	33.83	37.53

and

Table 30. Peak accelerations of each story of the superstructure (for input 0.4 g).

Story Number	Peak Accelerations in (m/s2) under Seismic Waves				Relative Difference with EL Wave (%)
	EL Wave	Kobe Wave	Taft Wave	TS Wave	Kobe Wave	Taft Wave	TS Wave
6	14.939	11.478	10.314	8.591	23.17	30.96	42.49
5	14.635	10.418	9.408	7.934	16.98	35.72	45.79
4	13.888	8.579	7.606	7.352	8.83	45.23	47.06
3	12.278	6.868	6.281	6.358	4.15	48.85	48.22
2	9.272	4.874	4.731	4.708	1.78	48.97	49.22
1	5.722	2.550	2.491	2.525	0.43	56.47	55.87

. From the data in Table 25, Table 26, Table 27, Table 28, Table 29 and Table 30, it can be observed that there were certain differences in the peak responses of the structure when subjected to different seismic waves with the same peak acceleration. When the peak acceleration was 0.1 g, the relative differences in peak displacements for the Kobe, Taft, and TS waves were, in comparison to the EL-central wave, 23.65, 51.49, and 45.16, respectively. For velocity, the relative differences were 30.50, 45.72, and 42.83, respectively. For acceleration, the relative differences were 22.92, 56.44, and 55.88, respectively. When the peak acceleration was 0.4 g, the relative differences in peak displacements for the Kobe, Taft, and TS waves, in comparison to the EL-central wave, were 23.65, 51.92, and 45.16, respectively. For velocity, the relative differences were 30.49, 45.71, and 42.82, respectively. For acceleration, the relative differences were 22.97, 56.47, and 55.87, respectively.

The comparison of the Fourier amplitude spectra of the EL wave and the TS wave, which exhibited significantly different peaks, revealed that the EL wave had a higher distribution in the high-frequency range, whereas the TS wave was mainly distributed in the low-frequency range. Given that the first-order frequency of the structural system was 11.369 Hz, the EL wave aligned more closely with the structural system frequency distribution. Similarly, a comparison of the Fourier amplitude spectra of the Taft wave and the TS wave showed that both were predominantly distributed in the low-frequency range, with fewer distributions in the high-frequency range, resulting in similar response outcomes.

In conclusion, the response of the structural system to the same peak acceleration varied significantly when subjected to different seismic waves, particularly when the frequency distribution of the seismic waves was closer to that of the structural system. This could lead to larger structure responses.

5. Conclusions

This paper presented an improved Penzien model for calculating soil–structure interaction. The effectiveness and accuracy of the computational model were validated through the establishment of simplified models of a 6-story single-span steel frame in MATLAB and ANSYS software. The models were subjected to dynamic nonlinear analysis. In the improved Penzien model, the superstructure was represented by a bilinear restoring force model, while the soil was represented by the Davidenkov equivalent linear model that accounted for its dynamic nonlinearity. The soil–structure interaction was represented by springs and damping. Based on model verification, the parameters of the model components were studied, leading to the following conclusions:

(a) A comparison of the results of the seismic analysis of the 3-DOF superstructure from the literature, MATLAB software using the central difference method and Newmark method, and ANSYS layer shear model indicated that the relative differences in the peak responses of each layer among the four results were mostly within 5. This confirmed that the adoption of a bilinear restoring force model for the superstructure effectively reflected its nonlinear response and mechanical behavior. Furthermore, the utilization of two time–history dynamic integration methods in MATLAB served to demonstrate the effectiveness and accuracy of the approach, while the feasibility of simulating the superstructure using the COMBIN40 element in ANSYS was validated.

(b) A simplified model of a six-story steel frame structure was established using MATLAB and ANSYS software, and a seismic analysis was carried out. The time–history curves and wave forms under the two models were essentially identical, with only a difference in amplitude. The maximum relative difference in peak displacements, velocities, and accelerations was 13.97, 8.74, and 1.44, respectively. This indicated that the nonlinear simplified calculation model of soil–structure interaction was more effective in reflecting the structure response under seismic effects.

(c) When the length of the free field system was 50 times the size of the superstructure, the results obtained met the accuracy requirements of the calculation. It was observed that, within a certain range, the presence of softer soil layers in the site soil resulted in a greater structure response under seismic effects and a more significant frequency reduction. This indicated that the soil–structure interaction became more pronounced. It was of paramount importance to consider equivalent soil, with the range being selectable as 20 times the area of the equivalent single pile or the area of the superstructure’s footing. The superstructure significantly impacted the simplified model, with changes in the number of superstructure layers and dimensions leading to significant variations in the structural system frequency. Therefore, when studying soil–structure interaction, it was imperative to develop and simulate the superstructure comprehensively. Moreover, the greater the degree of overlap between the high-frequency range of the input seismic waves and the structural system frequency, the more pronounced the response of the structure under seismic effects.

(d) For the improved Penzien model, it is then applied to the substructure pseudo-dynamic test method, covering both theoretical research and experimental research, to further improve the effectiveness and reliability of the model in practical engineering applications.