Comparative Experiment and Analysis of a Base-Isolated Structure with Small Aspect Ratio on Multi-Layered Soft Soil Foundation and Rigid Foundation

Abstract

Through conducting a comparative experimental study of small-aspect-ratio isolated structure models on multi-layered soft soil foundations and rigid foundations, this paper investigates the influence of soil–structure interaction (SSI) effects on the seismic response of small-aspect-ratio isolated structures on multi-layered soft soil foundations. An energy balance equation for isolated structure systems considering SSI effects is proposed, and the impact of SSI effects on the energy dissipation response of small-aspect-ratio isolated structures on multi-layered soft soil foundations is analyzed in depth. The analysis results reveal that SSI effects on multi-layered soft soil foundations reduce the first-order natural frequency of the isolated structure system and significantly increase the damping ratio of the system. Furthermore, the rotational effect of the isolated structure foundation is significant on multi-layered soft soil foundations, and the isolation layer has a certain amplification effect on the rotational effect of the foundation. The study shows that SSI effects on multi-layered soft soil foundations may either increase or decrease the seismic response of isolated structures. Moreover, due to the influence of SSI effects, the ratios of kinetic energy, damping energy dissipation, and hysteresis deformation energy dissipation of the isolated structure on multi-layered soft soil foundations are significantly different from those on rigid foundations. The research concludes that the influence of SSI effects is more significant during large earthquakes, where the ratios of kinetic energy and damping energy dissipation of the isolated structure increase, the hysteresis deformation energy dissipation ratio of the isolation layer decreases, and the magnitude of the decrease is related to the characteristics of the input seismic motion. This research has significant implications for improving the seismic design theory of small-aspect-ratio isolated structures on multi-layered soft soil foundations.

1. Introduction

An effective seismic mitigation technique, isolated structures have been widely adopted in building engineering. By utilizing isolation bearings and energy dissipation devices between the superstructure and foundation, isolated structures exhibit excellent energy dissipation capabilities, reducing seismic energy input to the superstructure and decreasing its seismic response. However, previous studies have indicated that the self-vibration period and damping parameters of isolated structure systems on soft soil foundations may change due to soil–structure interaction (SSI) effects, leading to altered seismic response characteristics [1,2,3,4,5,6,7]. Consequently, the energy dissipation characteristics of isolated structures on soft soil foundations may not match those assumed for rigid foundations. To address this, this study conducted shaking table model tests of isolated structure systems on soft soil and rigid foundations, and provides a comparative analysis of the vibration characteristics of isolated structures on both types of foundations. The mechanism and laws of the SSI effect on energy dissipation of isolated structures on soft soil foundations are also extracted, providing significant implications for improving the seismic design theory of isolated structures on soft soil foundations.

Numerous domestic and foreign researchers have devoted significant attention to investigating the dynamic characteristics of isolated structures on soft soil foundations, resulting in a vast body of research and fruitful outcomes. The current research methods primarily focus on theoretical analysis and numerical simulation. Regarding theoretical analysis, researchers such as Constantinou et al. [8] and Novak et al. [9] employed simplified analytical models to investigate the impact of soil–structure interaction effects on the dynamic characteristics of isolated structures, highlighting the significance of considering these effects in the isolated structure system. Recently, Kokusho [10], Maravas et al. [11], and Spyrakos et al. [12] developed relevant theoretical analysis models by simulating soil–structure interaction through springs and dampers and provided specific simplified calculation methods, which offer more reliable scientific techniques for the theoretical analysis of this scientific problem. In terms of numerical analysis, Sayed et al. [13] investigated the significant influence of foundation stiffness on isolated structures using numerical calculation methods. Kyung et al. [14] verified the feasibility of establishing a time-domain numerical calculation method for the dynamic interaction between soil and isolated structures using coupled finite element and boundary element methods. Hokmabadi et al. [15] constructed a three-dimensional finite element analysis model of soil–pile–structure dynamic interaction based on model tests, and using FLAC3D, studied the impact of soil–structure interaction effects on the dynamic twisting and swinging of the structure. Bhagat et al. [16] conducted three-dimensional finite element analysis considering material and geometric nonlinearity and investigated the seismic performance of steel-reinforced concrete foundation-isolated structures under bidirectional near-fault and far-fault motion. Chinese scholars such as Zhang et al. [17], Wang et al. [18], Zhu et al. [19], and Liu et al. [20] comprehensively studied the impact of soil–structure interaction effects on the isolation effect of foundation-isolated structures using finite element analysis methods. However, the aforementioned theoretical analysis and numerical simulation methods made certain assumptions and simplifications, and the research outcomes lack verification from measured data and model tests.

Model tests that consider the interaction between foundation soil and isolated structure have been recognized as an effective means to verify numerical calculation models and theoretical analysis methods. In recent years, several scholars have conducted model tests on the dynamic response of isolated structures on different types of foundations. The authors of this paper, along with Zhuang et al. [21,22,23], conducted vibration table model tests on the seismic response of isolated structures on rigid and general foundations. The results indicated that the isolation effect of isolated structures was lower when considering the SSI effect than on a rigid foundation. Li et al. [24] conducted comparative vibration table tests on high-rise isolated structures on rigid and soft soil foundations, and studied the seismic response characteristics and isolation effects of high-rise isolated structures on soft soil foundations. He et al. [25] analyzed the test results under different peak seismic actions through vibration table tests on high-aspect-ratio isolated structures and compared them with numerical analysis results and limit-theory results to provide the aspect ratio limit of high-rise isolated structures. Xu et al. [26] conducted a vibration table model test on eccentrically isolated foundation structures on soft soil foundations considering the SSI effect under far-field long-period seismic motion. The test results showed that compared with the rigid foundation structure system, the period extension of isolated structures on soft soil foundations was relatively small, leading to reduced isolation effect. However, currently conducted model tests lack specialized research on the seismic performance of small-aspect-ratio isolated structures on flexible foundations, especially on the energy dissipation characteristics of such structures, which cannot comprehensively reflect the seismic mechanism of small-aspect-ratio isolated structures on flexible foundations.

Based on a thorough review of existing shaking table tests and theoretical studies on the seismic response mechanism of isolated structures, this paper presents a shaking table test plan for small-aspect-ratio isolated structures on a multi-layered soft soil foundation. The design takes into account the requirement of the Code for Seismic Design of Buildings that the aspect ratio of isolated structures should be less than 4. The results of the shaking table tests are compared and analyzed with the results of previous tests conducted by the authors on a rigid foundation. The seismic response law of the small-aspect-ratio isolated structure model on the multi-layered soft soil foundation is investigated. To achieve a comprehensive understanding of the isolation mechanism and performance of isolated structures on flexible foundations, this paper proposes an energy analysis method to establish an energy response balance equation of the isolated structure system, considering the dynamic interaction between soil and structure (SSI effect). By analyzing the energy dissipation response of the shaking table model tests of isolated structure systems on both multi-layered soft soil foundation and rigid foundation, the influence of SSI effect on the energy dissipation response of small-aspect-ratio isolated structure on multi-layered soft soil foundation is studied. The results are expected to contribute to the improvement of seismic design theory of isolated structures on flexible foundations.

2. Shaking Table Test Design

2.1. Similitude Ratio of SSI System

In shaking table tests that involve soil–structure dynamic interaction, it is often difficult to achieve a fully similar relationship between the test parameters and the prototype parameters of the model when multiple materials are involved in the experiment. Therefore, it is necessary to develop a similarity determination method for multi-medium coupling systems in soil–structure dynamic interaction shaking table tests. To ensure similarity of the main parameters of the soil–structure interaction system and considering the test objectives, the geometric length, elastic modulus, and acceleration are chosen as the basic physical quantities for the model structure, while the shear modulus, density, and acceleration are chosen as the basic physical quantities for the model foundation. By using the Buckingham theorem, the similarity ratios of other physical quantities are derived. The similarity relationship of the model system is presented in

Table 1. Similitude ratios of the test system.

Types	Physical Quantity	Similitude Relationship	Similitude Ratio
			Model Structure	Model Foundation
Geometric characteristics	Length, l	$S_l$	1/20	1/20
	Displacement, r	$S_x=S_l$	1/20	1/20
Material properties	Elastic modulus, E	$S_E$	1	1/4
	Equivalent density, ρ	$S_{\rho }$	20	1
	Mass, m	$S_m=S_{\rho }{S}_l^3$	1/400	1/8000
	Stress, σ	$S_{\sigma }=S_E{S}_{\epsilon }^{}$	1	1/4
	Shear modulus, G	$S_G$	1	1/4
Dynamic characteristics	Time, t	$S_t=\sqrt{S_l/S_a}$	1/4.47	1/4.47
	Frequency, ω	$S_f=1/S_t$	4.47	4.47
	Acceleration, a	$S_a$	1	1

.

2.2. Model Structure and Isolation Bearing

The upper structure of the isolated structure model is comprised of a 4-story steel frame system, which utilizes square steel tubes for columns and H-shaped steel for beams. The dimensions of the steel frame model are depicted in Figure 1, with a longitudinal dimension of 0.8 m, a transverse dimension of 0.6 m, and a height of 2.1 m. The bottom floor has a height of 0.6 m, while the remaining floors have a height of 0.5 m each. Steel plates are installed on each floor to simulate floor slabs. The model is excited in the longitudinal direction, and the longitudinal aspect ratio of the structure model is 2.625, while the transverse aspect ratio is 3.5, both of which satisfy the requirements of a small-aspect-ratio isolated structure. The weight of the upper steel frame model is 0.32 t. In order to effectively consider the effects of gravity on the seismic response of the model structure, the model is equipped with a weight of 0.736 t on each floor, resulting in a total weight of 3.68 t.

Based on the stress similarity ratio of the rubber isolation bearings and the total weight of the model structure, lead-core rubber isolation bearings with a diameter of 100 mm were selected. The design details of the isolation bearings can be found in [22]. Four lead-core rubber isolation bearings were installed in the isolation layer, and their geometric dimensions and appearance are depicted in Figure 2a,b. The physical parameters of the lead-core rubber bearings are listed in

Table 2. Physical properties of the model lead–rubber bearings.

Physical Quantity	Value	Physical Quantity	Value
Shear modulus of rubber G (N/mm)	0.6	First form factor S1	19.2
Bulk modulus of rubber Eb (N/mm2)	1960	Second form factor S2	3.48
Vertical elastic modulus of rubber E0 (N/mm2)	1.8	Diameter of pencil lead (mm)	8
Rubber hardness correction factor K	0.77	Diameter of bearing (mm)	100

. Prior to the test, basic mechanical performance tests were conducted on the lead-core rubber bearings to determine the average horizontal stiffness and vertical stiffness of the four isolation bearings, which were found to be 0.278 kN/mm and 197.9 kN/mm, respectively.

2.3. Model Pile Foundation and Model Soil

The model foundation in this paper is composed of six concrete square piles, each with a length of 0.8 m and a cross-sectional area of 0.035 m × 0.035 m. The pile cap has a plan dimension of 1.2 m × 1.0 m × 0.1 m. Reinforcement details for both the pile cap and pile foundation are depicted in Figure 3. The pile foundation is arranged in a specific pattern and the model is presented in Figure 4.

The experimental model soil used in this study is composed of three stratified layers with a total thickness of 130 cm. The top layer, which is 30 cm thick, is dry sand with a density of 1760 kg/m³ and a water content ranging from 8.2% to 9.0%. The middle layer, which is 40 cm thick, is clay with a density of 1933 kg/m³ and a water content ranging from 27.2% to 30.0%. The bottom layer, which is 60 cm thick, is saturated dense sand with a density of 1920 kg/m³ and a water content ranging from 26.2% to 27.0%. This stratified soil constitutes a multi-layered soft soil foundation. The physical and mechanical parameters of the soil layers in the model are shown in

Table 3. Physical and mechanical parameters of the model soil layers.

Soil Layer	Thickness (m)	Density, ρ (kg/m3)	Shear Modulus, G (MPa)	Friction Angle (°)
Top sand layer	0.3	1760	11.3	27
Soft clay	0.4	1933	3.91	18
Bottom sand layer	0.6	1920	27.6	28

. The preparation of the model soil involves controlling the water content and compactness, with the water sinking method and manual layer filling employed. Prior to the loading tests, the average shear wave velocity of the soil was measured using an SDMT wave velocity tester, and was about 35–40 m/s, indicating that the model soil meets the requirements for simulating flexible foundation tests. The model soil box has a net size of 3.5 m (vibration direction) × 2 m (lateral direction) × 1.7 m (height), with a layered shear deformation soil box configuration developed by the Institute of Geotechnical Engineering of Nanjing Tech University [27]. The model box is constructed with 15 layers of rectangular planar steel frames, with grooves placed between each layer of steel frames to accommodate steel balls that form support points capable of free sliding. This design allows for the generation of horizontal relative deformation between the layers of the model soil box, thus allowing for almost unconstrained shear deformation of the soil. As a result, the reflection of waves from the boundary is greatly reduced, leading to better simulation of soil boundary conditions.

2.4. Measurement Points and Loading Method

During the experimental testing, various parameters were measured to evaluate the dynamic behavior and performance of the model structure. These measurements included the acceleration and horizontal displacement of the model structure, the pressure and horizontal force acting on the isolation bearings, the acceleration of the model foundation soil, the vertical and horizontal acceleration components of the model foundation base, the contact pressure of the pile–soil interface, and the strain of the pile. The layout of the sensors used for measurement is depicted in Figure 5, which includes horizontal and vertical accelerometers labeled A1 to A17 and V1 to V4, respectively, and displacement transducers labeled S1 to S5. The shear deformation of the model structure and soil box was also measured using the target method [28], and the soil pressure was measured using sensors labeled T1 to T6. In addition, the strain gauges labeled E1 to E8 were used for measuring the strain in the pile.

The El-Centro earthquake wave (El wave), Kobe earthquake wave (KB wave), and Nanjing artificial earthquake wave (NJ wave) were utilized as input motions for the shaking table test. The EL wave was originally recorded during the 1940 Imperial Valley earthquake in the United States, with an original peak acceleration of 0.349 g and a duration of strong shaking of about 26 s. The KB wave was recorded by the Marine Meteorological Station during the Kobe earthquake of 1995 in Japan. The north–south horizontal acceleration record was used as the input wave for the shaking table in this experiment, and the original peak acceleration of the Kobe wave was 0.85 g, with a duration of strong shaking of about 10 s. The NJ wave was artificially synthesized by the Jiangsu Earthquake Engineering Research Institute under typical site conditions for the Nanjing Metro. The acceleration time history and Fourier spectra of the three waves are shown in Figure 6 and Figure 7, where the KB wave has the smallest bandwidth, the NJ wave has the widest bandwidth, and the bandwidth of the EL wave is in the middle. The time step for inputting the earthquake motion was adjusted according to a model time similarity ratio of 1:4.47, with an adjusted time step of 0.0045 s (the original time step was 0.02 s). In order to determine the dynamic characteristics of the model system on the software layer and the changes in the seismic isolated structure, white-noise scanning was conducted before and after the experiment to obtain the natural frequency and damping ratio of the model system. During the experiment, loading was carried out in a step-by-step manner, and the specific loading scheme is shown in

Table 4. Loading conditions of shaking table tests.

Test Sample No.	Loading No.	Seismic Wave	Peak Bedrock Acceleration of the Input Motion (G)
1	JTWN1	White noise	0.05
2	JTEL1	El-Centro	0.05
3	JTNJ1	Nanjing	0.05
4	JTKB1	Kobe	0.05
5	JTEL2	El-Centro	0.15
6	JTNJ2	Nanjing	0.15
7	JTKB2	Kobe	0.15
8	JTEL3	El-Centro	0.3
9	JTNJ3	Nanjing	0.3
10	JTKB3	Kobe	0.3
11	JTEL4	El-Centro	0.5
12	JTKB4	Kobe	0.5
13	JTWN2	White noise	0.05

.

The authors previously conducted a shaking table test on a small-aspect-ratio isolated structure model on a rigid foundation, which is described in [22]. The similarity ratio of the model structure used in this paper, as presented in Table 1, is the same as that of the model structure on a rigid foundation. Moreover, the structural dimensions, floor mass, basic parameters of the isolation bearings, and layout of the upper-structure measurement points of the isolated structure model in this paper are consistent with those of the small-aspect-ratio isolated structure model on a rigid foundation.

3. Analysis of Test Results

3.1. Dynamic Characteristics of the Isolated Structure

The first-order natural frequency and damping ratio of the base-isolated structure model system were determined on a multi-layered soft soil foundation by white-noise scanning with an amplitude of 0.05 g before and after testing. Using the results obtained from the base-isolated structure model system on a rigid foundation, which were previously completed by the authors [22],

Table 5. First-order natural frequency and damping ratio of model systems.

Test Condition	Type of Foundation
	Multi-Layered Soft Soil Foundation		Rigid Foundation
	Frequency (Hz)	Damping Ratio (%)	Frequency (Hz)	Damping Ratio (%)
Before test	2.4	14.8	2.65	8.3
After test	2.27	18.4	2.62	8.8

presents a comparison of the first-order natural frequency and damping ratio of the base-isolated structure model system on both the multi-layered soft soil foundation and the rigid foundation. Table 5 indicates that the SSI effect has a significant influence on the dynamic characteristics of the base-isolated structure on the multi-layered soft soil foundation and no influence on the base-isolated structure on a rigid foundation. The following observations were made. (1) The SSI effect results in a decrease of 9.5% in the first-order natural frequency of the isolated structure compared to the rigid foundation, but it significantly increases the damping ratio by 78.3%. (2) The dynamic characteristic parameters of the model structure system before and after the test are different. The parameters of the base-isolated structure on a rigid foundation have only slightly changed, with the first-order natural frequency remaining essentially the same and the damping ratio slightly increasing. However, the parameters of the base-isolated structure on the multi-layered soft soil foundation have changed significantly, with a significant decrease in the first-order natural frequency and a significant increase in the damping ratio of the model system.

Further comparison of the experimental results of the model’s dynamic characteristics with those reported in reference [24] reveals that the effect of soil–structure interaction (SSI) on the dynamic characteristics of high-aspect-ratio isolated structures on soft soil foundation differs from that on small-aspect-ratio isolated structures in this paper, due to the difference in stiffness of the soft soil foundation and the aspect ratio of the isolated structure. Specifically, the damping ratio of high-aspect-ratio isolated structures on soft soil foundation increased by 36% compared with that on a rigid foundation, and there was no significant change in the model’s dynamic characteristics before and after the experiment. Comparison with [24] and analysis shows that the effect of SSI on the dynamic characteristics of isolated structures on soft soil foundation is closely related to the stiffness of the foundation soil and the aspect ratio of the isolated structure.

3.2. Seismic Acceleration Response of Isolated Structures

The acceleration peak amplification factor (AMF) is a dimensionless parameter that quantifies the degree of amplification or attenuation of the peak floor acceleration response of the upper structure relative to the input acceleration peak of the isolated structure base. This parameter is an important indicator of the seismic performance of isolated structures, which feature an isolation layer that dissipates a significant portion of the seismic energy during an earthquake, leading to a reduction in energy transmitted to the upper structure. As a result, the seismic response of the upper structure is mainly elastic [29]. In this study, the AMF of the isolated structure is compared on two different types of foundation: a rigid foundation and a multi-layered soft soil foundation. To achieve this, an interpolation method based on experimental measurements is used to calculate the AMF of the isolated structure on a multi-layered soft soil foundation. Specifically, the corresponding model base acceleration peak (PGA) values are set to 0.1 g, 0.2 g, and 0.3 g, respectively, to ensure that the isolated structure has equivalent base acceleration peaks on both types of foundations [30]. Figure 8 presents a comparison of the AMFs of the isolated structure on a multi-layered soft soil foundation and a rigid foundation, with layer number 0 in the figure denoting the model base.

Through comparative analysis of the AMFs of an isolated structure on a multi-layered soft soil foundation and a rigid foundation (as illustrated in Figure 8), due to the effect of SSI, the AMFs of the isolated structure on a multi-layered soft soil foundation and a rigid foundation exhibits a significant difference. Specifically, the El-Centro motion excitation results in a noticeable increase in the AMF on a multi-layered soft soil foundation relative to that on a rigid foundation. Under Kobe motion excitation, the AMF increases on a multi-layered soft soil foundation in large earthquakes (PGA = 0.3 g), but is smaller than that on a rigid foundation in small earthquakes (PGA = 0.1 g). With Nanjing motion excitation, the AMF slightly increases on a multi-layered soft soil foundation compared to that on a rigid foundation. The above analysis indicates that SSI effects on a multi-layered soft soil foundation may either increase or decrease the acceleration response of an isolated structure.

Further analysis of Figure 8 reveals that the AMF of the isolated structure on a rigid foundation decreases as PGA increases, indicating a better isolation effect with larger-input seismic motion peaks. However, the isolation effect of the isolated structure on a multi-layered soft soil foundation differs significantly from that on a rigid foundation. Specifically, under Nanjing motion excitation, the AMF of the isolated structure on a multi-layered soft soil foundation decreases as PGA increases and shows a similar isolation effect to that on a rigid foundation. In contrast, under El-Centro and Kobe motion excitations, the AMF of the isolated structure on a multi-layered soft soil foundation increases as PGA increases, indicating a worse isolation effect with larger-input seismic motion peaks and a reduction in the isolation effect of the isolated structure on a multi-layered soft soil foundation. The above analysis shows that the type and peak of input seismic motion have a significant impact on the isolation effect of the isolated structure on a multi-layered soft soil foundation. These findings differ from those reported in [21] on the isolation effect of high-rise isolated structures on soft soil foundations.

3.3. Interstory Displacement Response of Isolated Structures

Figure 9 illustrates the interstory displacement responses of isolated structures on multi-layered soft soil and rigid foundations, where the isolation layer is represented by a floor location of 0. The distribution characteristics of the maximum interstory displacement of the isolated structures on the rigid and multi-layered soft soil foundations are similar, as observed in Figure 9, with larger interstory displacement at the bottom isolation layer and smaller displacement in other upper layers. However, the maximum interstory displacement responses of the isolated structures on the two foundations differ due to the influence of SSI and the characteristics of the input seismic motion. Specifically, under the El-Centro and Kobe motions with a large earthquake intensity (PGA = 0.3 g), the maximum interstory displacement response of the isolated structures on the multi-layered soft soil foundation significantly increases compared to that on the rigid foundation, with the largest displacement increase observed in the isolation layer. Meanwhile, under the Nanjing motion with a large earthquake intensity, the maximum interstory displacement responses of the isolated structures on the two foundations are similar. Under El-Centro motion and Nanjing motion with a small earthquake intensity (PGA = 0.1 g), the maximum interstory displacement response of the isolated structures on the multi-layered soft soil foundation slightly increases compared to that on the rigid foundation. However, under the Kobe motion with a small earthquake intensity, the maximum interstory displacement response of the isolated structures on the multi-layered soft soil foundation decreases compared to that on the rigid foundation.

3.4. Hysteresis Curve of Seismic Isolated Structure Lead Rubber Bearing

Figure 10 depicts the hysteresis behavior of lead–rubber bearings (LRBs) in isolated structures on both multi-layered soft soil and rigid foundations under varying levels of large earthquake intensities (PGA = 0.3 g). The figure reveals that the hysteresis loop area of LRBs on the multi-layered soft soil foundation decreases by different degrees compared to that on the rigid foundation, implying a lower energy dissipation capacity of LRBs on the former under large earthquakes. The extent of this reduction might be attributed to the features of the input seismic motion.

3.5. Rotational Effects of Foundation and Isolation Layer

Previous research has shown that soil–structure dynamic interaction results in translational and rotational movement of building foundations relative to the subgrade, leading to significant changes in the dynamic response characteristics of the upper structure. Preliminary findings from shaking table model tests on an isolated structure system with general soil foundation carried out by the authors of this paper suggest that when taking into account the effect of soil–structure interaction (SSI), the foundation and isolation layer of an isolated structure experience rotational movement, whereas the relative translational movement of the foundation with respect to the subgrade is negligible [18]. To investigate the impact of the rotational movement of the foundation and isolation layer on the seismic response of the upper structure on a soft soil foundation, vertical accelerometers (V1, V2, V3, and V4) were placed at the top of the isolation foundation and isolation layer, respectively, in accordance with the method of reference [18]. The rotational acceleration of the pile cap and isolation layer were calculated using Equations (1) and (2), and the results are presented in

Table 6. Rotational acceleration amplitude of pile cap and isolation layer with multi-layered soft soil foundation.

Input Motion	PGA (g)	${\mathit{\theta }}_{1,\mathbf{m}\mathbf{a}\mathbf{x}}^{″}$ (rad/s−2)	${\mathit{\theta }}_{2,\mathbf{m}\mathbf{a}\mathbf{x}}^{″}$ (rad/s−2)	${\mathit{\theta }}_{2,\mathbf{m}\mathbf{a}\mathbf{x}}^{″}/{\mathit{\theta }}_{1,\mathbf{m}\mathbf{a}\mathbf{x}}^{″}$
El-Centro	0.1	0.347	0.418	1.200
Nanjing		0.356	0.435	1.220
Kobe		0. 414	0. 432	1.04
El-Centro	0.2	0.459	0.716	1.56
Nanjing		0.556	0.659	1.190
Kobe		0.810	0.980	1.210
El-Centro	0.3	1.113	1.762	1.580
Nanjing		0.940	0.920	0.980
Kobe		1.129	1.502	1.330

. The table shows the ratio of the peak rotational acceleration of the isolation layer to that of the foundation.

$$\underset{}{\overset{••}{{\displaystyle {\theta }_1}}}=\frac{\underset{}{\overset{••}{{\displaystyle V_1}}}+\underset{}{\overset{••}{{\displaystyle V_2}}}}{\underset{}{\overset{}{{\displaystyle L_1}}}}$$

(1)

$$\underset{}{\overset{••}{{\displaystyle {\theta }_2}}}=\frac{\underset{}{\overset{••}{{\displaystyle V_3}}}+\underset{}{\overset{••}{{\displaystyle V_4}}}}{\underset{}{\overset{}{{\displaystyle L_2}}}}$$

(2)

where L₁ is the horizontal distance between accelerometers V1 and V2, L₂ is the horizontal distance between accelerometers V3 and V4. ${\ddot{V}}_1~{\ddot{V}}_4$ are the vertical peak accelerations recorded by accelerometers V1, V2, V3, and V4.

Based on Table 6, it can be observed that the peak rotational acceleration (PRA) of the pile cap with multi-layered soft soil foundation ranges from 0.347 to 1.129 rad/s² in this test. According to the literature [21], the PRA of t the pile cap with sandy soil foundation ranges from 0.062 to 0.355 rad/s², while the PRA of the cap with rigid foundation is very small and can be neglected [22]. This indicates that the softer the foundation soil, the more significant the rotational effect on the isolated structure system’s pile cap.

Based on Table 6, it is observed that the isolation layer has a certain amplification effect on the rotational acceleration response of pile cap. The ratio of the PRA of the isolation layer to the pile cap increases with increasing peak ground acceleration (PGA) for El-Centro motion and Kobe motion inputs, indicating that the rotational effect of the isolation layer becomes more significant with increasing PGA, particularly for El-Centro motion input. Conversely, for the Nanjing motion input, the ratio of the PRA of the isolation layer to the pile cap decreases with increasing PGA, indicating that the rotational effect of the isolation layer weakens with increasing PGA and even shows a “vibration reduction” phenomenon when the input acceleration peak is 0.3 g. These phenomena demonstrate that the amplification effect of the isolation layer varies under different seismic motions and is related to the characteristics of the input earthquake motion.

Due to the limitations of the model test, the above findings still need to be further verified through numerical simulation and theoretical analysis.

3.6. Energy-Based Structural Response Analysis

3.6.1. Energy Response Equation of Isolated Structures Considering SSI Effect

When accounting for soil–structure interaction (SSI) in non-isolated structures, the inertial forces of the upper structure are transmitted to the foundation through the base, inducing local deformations in the foundation that lead to translation and rotation of the base relative to the foundation. Long-term analysis of earthquake observation data by the American scholar Sivanovic revealed that soil–structure interaction during earthquakes is significant, characterized mainly by foundation rocking [31]. To address this phenomenon, Wu et al. developed a mechanical model and a multi-point simplified analysis model [32] (as shown in Figure 11a,b) for non-isolated structures when considering SSI effects. Test results in this paper and in the literature [21] show that there exists a rotational effect of the isolated structure foundation and isolation layer on soil foundations, and the rotational response of the isolation layer and isolated structure foundation are not the same. The literature [21] indicates that acceleration response of the isolated structure on soil foundation is coupled with the rocking component of the isolation layer and the elastoplastic deformation component of the isolated structure. Therefore, a mechanical model considering SSI effects on soil foundations for isolated structures can be established (as shown in Figure 12a), and the corresponding multi-point simplified analysis model is shown in Figure 12b. In the figures, $h_i$ represents the distance from the centroid of each upper structure layer to the isolation layer, $h_0$ is the height of the isolation layer, $u_i$ is the horizontal displacement of the upper structure relative to the isolation layer, $m_i$, $k_i$, and $c_i$ are the mass, stiffness, and damping of each upper structure layer, $m_0,$ $k_0$, and $c_0$ are the mass, horizontal stiffness, and damping of the isolation layer, $u_0$ is the horizontal displacement of the isolation layer relative to the foundation, ${\mu }_f$ and ${\theta }_1$ are the horizontal displacement and rotation angle of the foundation center of gravity relative to the foundation, $\theta$ is the rotational angle of the isolation layer, and ${\mu }_g$ is the horizontal ground displacement.

Based on the simplified analysis model shown in Figure 12b, the motion differential equations for the seismic response of an isolated structure considering SSI effects can be established under horizontal seismic motions:

$$[M](\left\{\stackrel{··}{u}\right\}+\left\{\stackrel{··}{u_0}\right\}+\left\{h\right\}\stackrel{··}{\theta }+\left\{{\stackrel{··}{u}}_f\right\})+[C](\left\{\stackrel{·}{u}\right\}+\left\{\stackrel{·}{u_0}\right\})+\left\{f_s\left(u,\stackrel{·}{u}\right)\right\}=-[M]\left\{{\stackrel{··}{u}}_g\right\}$$

(3)

where [M] is the mass matrix of the upper structure of the isolated structure (including the mass $m_0$ of the isolation layer), [C] is the viscous damping matrix of the isolated structure (including the damping coefficient $c_0$ of the isolation layer); ${\mu }_f$ is the displacement vector of the foundation relative to the ground, $\theta$ is the rotation angle of the isolation layer, $\left\{\ddot{u}\right\}$ and $\left\{\dot{u}\right\}$ are the acceleration and velocity vectors of the upper structure particles relative to the isolation layer (excluding the displacement caused by the rotation of the isolation layer), ${\mu }_0$ is the horizontal displacement vector of the isolation layer relative to the foundation, $\left\{f_s\left(u,\dot{u}\right)\right\}$ is the vector of viscoelastic restoring forces of the upper structure (including the viscoelastic restoring force $f_d$ of the isolation layer), ${\ddot{u}}_g$ is the ground acceleration, and $\left\{h\right\}$ is the distance vector from the centroid of each upper structure layer to the isolation layer.

The relative energy response equation for an isolated structure system considering SSI effects can be obtained by integrating the relative displacement of each particle with respect to the ground at both ends of Equation (3) over the duration of the earthquake excitation [0, t]:

$${\displaystyle \sum _{j=0}^N{\displaystyle {\int }_0^t{m}_j(\stackrel{··}{u_j}+\stackrel{··}{u_0}+h_j\stackrel{··}{\theta }+{\stackrel{··}{u}}_f)\stackrel{·}{x_j}}dt}+{\displaystyle \sum _{j=1}^N{\displaystyle {\int }_0^t{c}_j\stackrel{·}{(u_j}+\stackrel{·}{u_0})\stackrel{·}{x_j}dt}}+{\displaystyle {\int }_0^t{c}_0\stackrel{·}{u_0}\stackrel{·}{x_0}dt}+{\displaystyle \sum _{j=1}^N{\displaystyle {\int }_0^t{f}_{sj}\left(u,\stackrel{·}{u}\right)}{\stackrel{·}{x}}_{j}dt+}{\displaystyle {\int }_0^t{f}_d}{\stackrel{·}{x}}_{0}dt=-{\displaystyle \sum _{j=0}^N{\displaystyle {\int }_0^t{m}_j{\stackrel{··}{u}}_g}\stackrel{·}{x_j}dt}$$

(4)

where j = 0 denotes the isolation layer, and the right-hand term $E_i^{SSO}$ is the total input energy of the earthquake motion:

$$E_i^{sso}=-{\displaystyle \sum _{j=0}^N{\displaystyle {\int }_0^t{m}_j{\stackrel{··}{u}}_g}\stackrel{·}{x_j}dt}$$

(5)

The first four terms on the left-hand side of the equation are:

Kinetic energy of the isolated structure system considering SSI effects $E_k^{SSO}$:

$$E_k^{sso}={\displaystyle \sum _{j=0}^N{\displaystyle {\int }_0^t{m}_j(\stackrel{··}{u_j}+\stackrel{··}{u_0}+h_j\stackrel{··}{\theta }+{\stackrel{··}{u}}_f)\stackrel{·}{x_j}}dt}={\displaystyle \sum _{j=0}^N{\displaystyle {\int }_0^t{m}_j\stackrel{··}{x_j}\stackrel{·}{x_j}}dt}$$

(6)

Viscous damping energy of an isolated structure system considering SSI effects $E_c^{SSO}$:

$$E_c^{sso}={\displaystyle \sum _{j=1}^N{\displaystyle {\int }_0^t{c}_j(\stackrel{·}{u_j}+\stackrel{·}{u_0})\stackrel{·}{x_j}dt}}+{\displaystyle {\int }_0^t{c}_0\stackrel{·}{u_0}\stackrel{·}{x_0}dt}$$

(7)

Total deformation energy of the isolated structure system considering SSI effects $E_s^{SSO}$:

$$E_s^{sso}={\displaystyle \sum _{j=1}^N{\displaystyle {\int }_0^t{f}_{sj}}{\stackrel{·}{x}}_{j}dt}=E_v^{sso}+E_h^{sso}$$

(8)

where $E_h^{SSO}$ is the viscoelastic hysteresis energy dissipation of the system, and $E_v^{SSO}$ is the elastic strain energy of the system.

Hysteresis energy of isolation layer $E_d^{SSO}$:

$$E_d^{sso}: E_d^{sso}={\displaystyle {\int }_0^t{f}_d}{\stackrel{·}{x}}_{0}dt$$

(9)

At any given time t, the energy balance for each part of an isolated structure system considering SSI effects should be maintained, and this can be mathematically represented as:

$$E_i^{sso}=E_k^{sso}+E_c^{sso}+E_s^{sso}+E_d^{sso}$$

(10)

Currently, the design of isolated structures assumes rigid foundations and neglects the effects of SSI. The differential equation [33] for the motion of an isolated structure on a rigid foundation under horizontal earthquake motion can be formulated as follows:

$$[M]\left\{\stackrel{··}{x}\left(t\right)\right\}+[C]\left\{\stackrel{·}{x}\left(t\right)\right\}+\left\{f_s\left(x,\stackrel{·}{x}\right)\right\}=-[M]\left\{{\stackrel{··}{x}}_g\left(t\right)\right\}$$

(11)

where [M] is the mass matrix of the isolated structural system, which includes the mass $m_0$ of the isolation layer. [C] is the viscous damping matrix of the isolated structural system, including the damping coefficient of the isolation layer, $\ddot{x}\left(t\right)$,$\dot{x}\left(t\right)$, and $x\left(t\right)$ are the acceleration, velocity, and displacement of each mass particle with respect to the ground. The displacement of the isolation layer with respect to the ground is $x_0\left(t\right)$, $f_s(x,\dot{x})$ is the vector of hysteretic and restoring forces of the isolated structure, which includes the hysteretic and restoring forces $f_d$ of the isolation layer, $\ddot{x_g\left(t\right)}$ is the ground motion acceleration.

By integrating both sides of Equation (6) over the earthquake motion duration [0, t] with respect to the relative displacement of the mass particle, the relative energy response equation of the isolated structure system on a rigid foundation can be obtained:

$${\displaystyle \sum _{j=0}^N{\displaystyle {\int }_0^t{m}_j\stackrel{··}{x_j}}{\stackrel{·}{x}}_{j}dt}+{\displaystyle \sum _{j=0}^N{\displaystyle {\int }_0^{t}c{\stackrel{·}{x}}_j{\stackrel{·}{x}}_j}dt}+{\displaystyle \sum _{j=1}^N{\displaystyle {\int }_0^t{f}_{sj}}{\stackrel{·}{x}}_{j}dt}+{{\displaystyle {\int }_0^t{f}_d\stackrel{·}{x}}}_{0}dt=-{\displaystyle \sum _{j=0}^N{\displaystyle {\int }_0^{t}m{\stackrel{··}{{}_{j}x}}_g}{\stackrel{·}{x}}_{j}dt}$$

(12)

where j = 0 denotes the isolation layer.

The left-hand side of the equation consists of four terms, namely, the kinetic energy of the isolated structural system $E_k^0$:

$$E_k^o={\displaystyle \sum _{j=0}^N{\displaystyle {\int }_0^t{m}_j\stackrel{··}{x_j}}{\stackrel{·}{x}}_{j}dt}={\displaystyle \sum _{j=0}^N\frac{1}{2}{m}_j{({\stackrel{·}{x}}_j)}^2}$$

(13)

The viscous damping energy $E_c^0$:

$$E_c^o={\displaystyle \sum _{j=0}^N{\displaystyle {\int }_0^{t}c{\stackrel{·}{x}}_j{\stackrel{·}{x}}_j}dt}$$

(14)

The total deformation energy of the isolated structural system $E_s^0$:

$$E_s^o={\displaystyle \sum _{j=1}^N{\displaystyle {\int }_0^t{f}_{sj}}{\stackrel{·}{x}}_{j}dt}=E_v^o+E_h^o$$

(15)

where $E_h^0$ is the hysteresis deformation energy of the structural system and $E_v^0$ is the elastic strain energy of the structural system.

The hysteresis deformation energy of the isolation layer $E_d^0$:

$$E_d^o={{\displaystyle {\int }_0^t{f}_d\stackrel{·}{x}}}_{0}dt$$

(16)

The right-hand side of the equation is the total input energy of the earthquake motion $E_i^0$:

$$E_i^o=-{\displaystyle \sum _{j=0}^N{\displaystyle {\int }_0^{t}m{\stackrel{··}{{}_{j}x}}_g}{\stackrel{·}{x}}_{j}dt}$$

(17)

At any given time t, the energy balance for each component of the isolated structure system on a rigid foundation must be maintained, and can be mathematically expressed as:

$$E_i^o=E_k^o+E_c^o+E_s^o+E_d^o$$

(18)

Comparing the energy compositions of Equations (5) and (8), two main changes can be observed in the energy response equation of the isolated structure system considering SSI effects: (1) compared with the case of a rigid foundation, the kinetic energy composition of the isolated structure system considering SSI effects includes additional components of base translation and isolation layer rotation; and (2) existing studies have shown that the damping ratio of the isolated structure system on a soil foundation differs significantly from that on a rigid foundation due to SSI effects [21,22,23]. Consequently, the energy dissipation due to damping of the isolated structure system considering SSI effects is different from that on a rigid foundation. Therefore, under a certain total energy input, the hysteresis energy dissipation of the isolated structure system isolation layer may undergo significant changes.

3.6.2. Parameters for Calculating Model Energy Dissipation

Based on the experimental results presented in this paper, it can be inferred that the parameters of the isolation layer and upper structure in the model system are equivalent for both a rigid foundation and a multi-layered soft soil foundation, as evidenced in

Table 7. Parameters of the isolation layer in the model system.

Horizontal Equivalent Stiffness of Isolation Layer (N/mm)	Equivalent Viscous Damping Ratio of Isolation Layer (%)	Vertical Stiffness of Isolation Layer (N/mm)
1111	8.3	791,600

and

Table 8. Mass and stiffness of each story of the upper structure in the model system.

Floor Location	Density (kg)	Stiffness (N/mm)	Story Height (m)
4	800	23,040	0.5
3	800	23,040	0.5
2	800	23,040	0.5
1	800	16,000	0.6

, respectively. The first-order natural frequency and damping ratio of the isolated structure on a rigid foundation were determined to be $f_1=2.65 \mathrm{H}\mathrm{z}$ and ${\xi }_1$ = 0.083, respectively, whereas those on a multi-layered soft soil foundation were found to be $f_1=2.4 \mathrm{H}\mathrm{z}$ and ${\xi }_1$ = 0.148. The model system employed a Rayleigh damping ratio, and its corresponding damping matrix is expressed as follows:

$$[C]=\alpha [M]+\beta [K]$$

(19)

where $\alpha =2{\omega }_1{\omega }_2\left({\xi }_1{\omega }_2-{\xi }_2{\omega }_1\right)/({\omega }_2^2-{\omega }_1^2)$, $\beta =2\left({\xi }_2{\omega }_2-{\xi }_1{\omega }_1\right)/({\omega }_2^2-{\omega }_1^2)$, ${\omega }_1$ and ${\omega }_2$ are the first and second natural circular frequencies of the model system, ${\xi }_1$ and ${\xi }_2$ are the corresponding damping ratios at the first and second circular frequencies of the model system, and the damping ratio is taken as ${\xi }_1$ = ${\xi }_2$ according to [22]. Ground acceleration ${\ddot{u}}_g$: The acceleration response at measurement point A12 is taken. Upper structure acceleration ${\ddot{x}}_j$ relative to the ground surface: The difference in acceleration response between measurement points (A1–A5) on the upper structure and the surface measurement point A12 of the foundation soil layer is taken, and the corresponding value is denoted by $x_j={\int }_0^t{\ddot{x}}_{j}dt$. Foundation acceleration ${\ddot{u}}_f$ relative to the ground surface: The difference in acceleration response between measurement points A7 and A12 is taken. Isolation layer acceleration ${\ddot{u}}_0$ relative to the foundation: The difference in acceleration response between measurement points A1 and A7 is taken, and the corresponding value is denoted by ${\dot{u}}_0={\int }_0^t{\ddot{u}}_{0}dt$. Rotational acceleration response $\ddot{\theta }$ of the isolation layer: Calculated using Equation (2), where ${\ddot{V}}_3$ and ${\ddot{V}}_4$ are the measured acceleration responses at measurement points V3 and V4, respectively, and L is the horizontal distance between measurement points V3 and V4. Acceleration response ${\ddot{u}}_j$ of the upper structure mass relative to the isolation layer: Calculated using the equation ${\ddot{u}}_j={\ddot{x}}_j-{\ddot{u}}_0-{\ddot{u}}_f-h_j\ddot{\theta }$, and the corresponding value is denoted by ${\dot{u}}_j={\int }_0^t{\ddot{u}}_{j}dt$. The hysteresis recovery force $f_d$ of the isolation layer is obtained by measuring the three-way force sensor above the isolation bearing.

3.6.3. Energy Dissipation Analysis of Isolated Structures on Rigid Foundations and Multi-Layered Soft Soil Foundations

Based on the seismic response of the isolated structure model on multi-layered soft soil foundations presented in this study, as well as the response of the isolated structure model on rigid foundations described in literature [22], an energy analysis of the seismic response time history was conducted for both isolated structure models on different types of foundations using the calculation method outlined in Section 3.6.1.

Table 9. Analysis of energy dissipation composition of isolated structures on multi-layered soft soil foundation and rigid foundation when El-Centro motion is applied.

Type of Foundation	Actual PGA (g)	Total Input Energy Ei (N·m)	Kinetic Energy Ek (N·m)	Deformation Energy Es (N·m)	Viscous Damping Energy Ec (N·m)	Hysteresis Energy of Isolation Layer Ed (N·m)
Rigid foundation	0.131	15.9	1.5	0.2	1.0	13.3
	0.235	55.2	4.6	0.6	2.9	47.1
	0.344	123.6	9.0	1.0	5.6	108.0
Multi-layered soft soil foundation	0.112	13.4	1.6	0.2	1.4	10.3
	0.203	46.0	6.4	0.9	6.4	32.4
	0.327	127.3	25.5	2.8	19.6	79.4

,

Table 10. Analysis of energy dissipation composition of isolated structures on multi-layered soft soil foundation and rigid foundation when Kobe motion is applied.

Type of Foundation	Actual PGA (g)	Total Input Energy Ei (N·m)	Kinetic Energy Ek (N·m)	Deformation Energy Es (N·m)	Viscous Damping Energy Ec (N·m)	Hysteresis Energy of Isolation Layer Ed (N·m)
Rigid foundation	0.094	9.3	2.2	0.1	0.9	6.1
	0.187	34.4	5.4	0.6	2.7	25.8
	0.274	86.1	9.0	1.1	5.8	70.2
Multi-layered soft soil foundation	0.118	16.2	1.4	0.2	1.2	13.5
	0.220	60.9	6.7	1.3	6.8	46.1
	0.390	193.3	38.9	5.0	25.3	124.1

and

Table 11. Analysis of energy dissipation composition of isolated structures on multi-layered soft soil foundation and rigid foundation when Nanjing motion is applied.

Type of Foundation	Actual PGA (g)	Total Input Energy Ei (N·m)	Kinetic Energy Ek (N·m)	Deformation Energy Es (N·m)	Viscous Damping Energy Ec (N·m)	Hysteresis Energy of Isolation Layer Ed (N·m)
Rigid foundation	0.113	14.9	1.8	0.3	1.1	11.7
	0.237	79.8	7.3	1.8	4.9	65.8
	0.321	117.7	9.8	1.6	6.7	99.6
Multi-layered soft soil foundation	0.072	5.6	0.8	0.1	0.6	4.1
	0.132	20.6	2.7	0.3	2.0	15.6
	0.260	84.6	8.3	0.8	7.8	67.8

provide calculations for the total input energy and energy dissipation of each component for isolated models on rigid and multi-layered soft soil foundations subjected to different seismic actions. The measured peak ground acceleration (PGA) on rigid foundations is the measured peak acceleration response value of the vibration table, whereas the measured PGA on multi-layered soft soil foundations is the measured peak acceleration response value of the A12 point on the ground surface. Based on the results obtained in Table 9, Table 10 and Table 11, a comparison of the energy dissipation of the isolated structure models on rigid and multi-layered soft soil foundations is presented in Figure 13, Figure 14 and Figure 15. In these figures, R_k (R_k = E_K/E_i), R_s (R_s = E_s/E_i), R_c (R_c = E_c/E_i), and R_d (R_d = E_d/E_i) represent the ratios of kinetic energy, structural deformation energy, damping energy, and energy dissipated by hysteretic behavior of the isolation layer, respectively. The portion of the total energy input to the model system that is converted into kinetic energy and elastic strain energy is not dissipated. The energy dissipation of the model system mainly occurs in the form of damping and hysteretic energy dissipation. A comprehensive analysis of Figure 13, Figure 14 and Figure 15 reveals that the energy dissipation of the isolated structure models on rigid and multi-layered soft soil foundations is not the same and follows certain rules.

• The seismic input energy of isolated structures on rigid foundations is primarily absorbed by the hysteresis energy dissipation (E_d) of the isolation layer, and during the strong seismic motion, the hysteresis energy dissipation of the isolation layer exceeds 0.8. The kinetic energy of the isolated structure is relatively small compared to the energy dissipation of the isolation layer’s hysteresis (R_k) and damping (R_c). The deformation energy dissipation (R_s) of the upper structure of the isolation system is minimal and can be ignored. The energy dissipation distribution of the isolated structure system on a multi-layered soft soil foundation is different from that on a rigid foundation. The hysteresis energy dissipation of the isolation layer (R_d) still accounts for a large proportion, but there is a significant change in the kinetic energy of the isolated structure compared to the energy dissipation of the isolation layer’s hysteresis (R_k) and damping (R_c). The deformation energy dissipation (R_s) of the upper structure of the isolation system is relatively small.
• The dynamic kinetic energy ratio R_k of an isolated structure on a rigid foundation decreases with increasing PGA of the input seismic motion, while the variation pattern of R_k for an isolated structure model system on the multi-layered soft soil foundation is significantly different from that on a rigid foundation, mainly manifested as follows: the dynamic kinetic energy ratio R_k of the isolated structure system on the multi-layered soft soil foundation increases with increasing PGA of the El-Centro motion and the Kobe motion, and decreases with increasing PGA of the Nanjing motion. This phenomenon is consistent with the experimental results in Table 5, which show that the ratio of the PRA of the isolation layer to the pile cap increases with increasing PGA of the El-Centro motion and the Kobe motion, while the ratio of the PRA of the isolation layer to the pile cap decreases with increasing PGA of the Nanjing motion. The above analysis indicates that the dynamic kinetic energy ratio R_k of the isolated structure model system on the multi-layered soft soil foundation is related to the strength of the isolation layer rotation effect. When the rotation effect of the isolation layer is enhanced, the dynamic kinetic energy ratio R_k of the isolated model system increases, while it decreases when the rotation effect of the isolation layer is weakened.
• The damping energy ratio R_c of the isolated structure on the multi-layered soft soil foundation is not equivalent to that on the rigid foundation, and it is dependent on the damping ratio of the isolated structure. The damping energy ratio R_c of the isolated structure under different input motion ranges from 0.045 to 0.076 on the rigid foundation, whereas it ranges from 0.073 to 0.154 on the multi-layered soft soil foundation. This indicates that the damping energy ratio R_c of the isolated structure on the multi-layered soft soil foundation is significantly greater compared to that on the rigid foundation. The main reason for this discrepancy is that the damping ratio of the isolated structure on the multi-layered soft soil foundation increases significantly due to the influence of soil–structure interaction (SSI) effects, resulting in an increase in the damping energy ratio R_c of the isolated structure.
• The hysteretic deformation energy dissipation ratio R_d of the isolation layer on the multi-layered soft soil foundation is lower than that on the rigid foundation during the strong seismic motion, with values ranging from 0.624 to 0.801 and 0.835 to 0.874, respectively, under different seismic motions. Two factors contribute to this phenomenon. Firstly, the energy response equation of the isolated structure on the multi-layered soft soil foundation differs significantly from that on the rigid foundation, with kinetic energy components that include the translation of the foundation and the rotation of the isolation layer. Experimental results from Section 3.5 demonstrate that the rotational acceleration response of the isolation layer on the multi-layered soft soil foundation is significant (as shown in Table 5), and the kinetic energy of the isolated structure is closely related to the strength of the rotational effect of the isolation layer. Secondly, the SSI effect greatly influences the dynamic characteristics of the isolated structure on the multi-layered soft soil foundation. The damping ratio of the isolation system on the multi-layered soft soil foundation is significantly higher than that on the rigid foundation, resulting in an increase in the damping energy dissipation of the isolated structure. This indirectly reduces the proportion of hysteretic deformation energy dissipation of the isolation layer, which is particularly evident during the strong seismic motion. Therefore, the hysteretic deformation energy dissipation ratio R_d of the isolation layer on the multi-layered soft soil foundation is lower than that on the rigid foundation during the strong seismic motion, under a certain total input energy of seismic motion.
• The energy dissipation allocation pattern of the isolated structure on the multi-layered soft soil foundation may be the same or opposite to that on the rigid foundation, which depends on the characteristics and peak value of the seismic motion applied. When the El-Centro motion and the Kobe motion are applied on the multi-layered soft soil foundation, the ratio of hysteretic deformation energy dissipation R_d of the isolation layer decreases with the increasing PGA of the input seismic motion, while the corresponding damping energy dissipation ratio R_c and kinetic energy ratio R_k increase. That is, the greater the peak value of the input seismic motion, the worse the energy dissipation performance of the isolation layer, which is completely opposite to the variation law of R_d on the rigid foundation. On the other hand, when the Nanjing motion is applied on the multi-layered soft soil foundation, the R_d ratio of hysteretic deformation energy dissipation of the isolation layer increases with the increasing PGA of the seismic motion applied, while the corresponding damping energy dissipation ratio R_c and kinetic energy ratio R_k decrease. That is, the greater the PGA of the input seismic motion applied, the better the energy dissipation performance of the isolation layer, which is the same as the variation law of R_d on the rigid foundation.

The phenomenon described above can be explained as follows: isolated structures use a flexible isolation layer between the building’s foundation and upper structure to extend the fundamental period of the upper structure. This approach avoids the frequency band range of ground motion, reduces resonance effects, blocks the transmission of seismic energy to the upper structure, and minimizes the seismic response of the structure. However, for isolated structures built on soft soil foundations, the spectral characteristics of ground motion may decrease in frequency, and the natural frequency of the isolated structure may change due to soil–structure interaction (SSI) effects, which can make it challenging to avoid the main frequency range of the ground motion. As a result, resonance effects may impact the energy dissipation of the isolated structure. In the model test of isolated structures on the multi-layered soft soil foundation, the Fourier spectra of three input seismic motions measured at the A6 point on the shaking table (with an acceleration peak value of 0.05 g) are presented in Figure 16. The Fourier spectrum of the Kobe motion is primarily composed of low-frequency components, the Fourier spectrum of the El-Centro motion is primarily composed of mid-low-frequency components, and the Fourier spectrum of the Nanjing motion has the widest frequency band, with mid-high-frequency components as the primary component. After filtering through the multi-layered soft soil foundation soil in the isolated structure model test, the low-frequency components in the frequency spectrum of surface point A12 of soil layer are significantly enhanced. The seismic Fourier spectrum is shown in Figure 17. It can be observed that the main frequency range of surface point A12 is 3.2 Hz to 7.4 Hz when El-Centro motion is applied, 2.8 Hz to 6.2 Hz when Kobe motion is applied, and 12.3 Hz to 15.0 Hz and 17.6 Hz to 22.3 Hz when Nanjing motion is applied. The first-order natural frequency of the isolated structure on the multi-layered soft soil foundation measured in the test is 2.4 Hz, which is closer to the main frequency range of Kobe motion and El-Centro motion, but relatively far from the main frequency range of the Nanjing motion. The test results of the isolated structure on the multi-layered soft soil foundation show that with increasing PGA of the seismic motion applied, the phenomenon of the frequency spectrum of surface point A12 shifting towards low frequency becomes more apparent. This indicates that as the PGA of the seismic motion applied increases, the first-order natural frequency of the isolated structure will continuously approach the main frequency range of the seismic motion when Kobe motion and El-Centro motion are applied, and the effect of resonance will continuously increase, causing the seismic response of the isolated structure to continuously increase. The corresponding kinetic energy and damping energy of the isolated structure will increase. Under the condition of the total energy remaining constant, the hysteresis deformation energy consumption of the isolation layer will decrease. When Nanjing motion is applied, the first-order natural frequency of the seismic isolated structure effectively avoids the main frequency range of the seismic motion, reducing the resonance effect, and the seismic response of the isolated structure is reduced. The corresponding kinetic energy and damping energy of the isolated structure will continuously decrease. Under the condition of the total energy remaining constant, the hysteresis deformation energy consumption of the isolation layer will increase. Therefore, the influence of the input seismic motion’s spectral characteristics on the energy dissipation allocation of the isolated structure can be observed as follows: when the mid-to-low-frequency components dominate the spectral characteristics of the input seismic motion, the SSI effect has a greater effect on the energy dissipation of the isolated structure, resulting in a continuous reduction in the hysteresis deformation energy consumption of the isolation layer, while the damping energy consumption ratio and kinetic energy ratio increase, leading to a significant decrease in the isolation efficiency. Conversely, when the mid- to high-frequency components dominate the spectral characteristics of the input seismic motion (such as Nanjing motion), the SSI effect has a smaller effect on the energy dissipation of the isolated structure, resulting in a continuous increase in the hysteresis deformation energy consumption of the isolation layer, while the damping energy consumption ratio and kinetic energy ratio decrease.

4. Conclusions

Based on shaking table tests of small-aspect-ratio isolated structures on the multi-layered soft soil foundation and the rigid foundations, this paper compares and analyzes the seismic response of small-aspect-ratio isolated structures on a multi-layered soft soil foundation and a rigid foundation. It proposes an energy response balance equation for the isolated structure considering the SSI effect and thoroughly analyzes the impact of the SSI effect on the energy dissipation response of small-aspect-ratio isolated structures on multi-layered soft soil foundation. The main conclusions obtained are:

• Due to the SSI effect, the first-order natural frequency of the isolated structure on the multi-layered soft soil foundation is reduced compared to that on the rigid foundation, while the damping ratio is significantly increased compared to that on the rigid foundation. The magnitude of this effect is closely related to the stiffness of the foundation soil and the aspect ratio of the isolated structure.
• On the site of multi-layered soft soil foundation, the SSI effect can either increase or decrease the acceleration response of the isolated structure, depending on the characteristics and peak value of the input earthquake motion.
• The isolated structure system’s pile cap on multi-layered soft soil foundations has significant rotational acceleration response, and the isolation layer has a certain amplification effect on the rotational acceleration response of pile cap.
• The equation has a clear concept, and the energy response analysis of the model test system shows that it effectively reflects the energy distribution law of each part of soil-isolated structure dynamic interaction system.
• Isolated structures on the rigid foundation primarily dissipate energy by the hysteresis deformation energy of the isolation layer, and as the input seismic motion increases, the hysteresis energy ratio of the isolation layer also increases. This indicates that the stronger the seismic motion, the better the isolation efficiency. However, for the isolated structures on the multi-layered soft soil foundation, although the energy dissipation is still mainly by the hysteresis deformation energy of the isolation layer, the hysteresis deformation energy of the isolation layer on the multi-layered soft soil foundation is reduced during strong seismic motion compared to that on the rigid foundation. This indicates that the isolation efficiency of the isolation layer on a soft soil foundation is reduced.
• The effect of SSI on the energy dissipation of isolated structures on the multi-layered soft soil foundation is related to the characteristics and peak values of the seismic motion applied. Under the action of seismic motion with mainly low-frequency components, the SSI effect has a significant impact on the energy dissipation of the isolation layer, leading to a continuous decrease in the hysteresis energy ratio of the isolation layer, an increase in the damping energy ratio and kinetic energy ratio of the isolated structure, and a significant decrease in the seismic isolation efficiency of the isolation layer. On the other hand, under the action of seismic motion with mainly high-frequency components, the SSI effect has a smaller impact on the energy dissipation of the isolated structure system, resulting in a continuous increase in the hysteresis energy ratio of the isolation layer and a decrease in the damping energy ratio and kinetic energy ratio.