1-g Shaking Table Test Study on the Influence of Soil–Caisson Dynamic Interaction (SCDI) on the Caisson Foundation Motion

Abstract

Caisson foundations are commonly used as the tower foundations in many long-span bridges. However, the seismic performance analysis of bridge structures using caisson foundations typically assumes that the tower is fixed at the base, applying the free-field ground acceleration to the base. Consequently, the impact of soil–caisson dynamic interaction (SCDI) on the caisson foundation motion is not considered. To investigate the SCDI effects on the motion of the caisson foundation, two different systems of 1 g shaking table model tests were carried out: a free-field system model test and a soil–caisson system model test. The test results show that an increase in the peak acceleration of the table input seismic wave is associated with a greater influence of SCDI on the motion of the caisson foundation. Compared with the free-field ground motion, the SCDI effects reduce the amplitude of the horizontal acceleration of the caisson foundation motion but introduce a significant rotational component. Additionally, both effects are frequency-dependent and become more significant with increasing frequency. The shaking table test study presented in this paper reveals several crucial features of SCDI that influence the motion of the caisson foundation, enhancing the comprehension of the mechanism of SCDI and providing essential data support for subsequent theoretical and numerical simulation studies.

1. Introduction

Foundations of long-span bridges are required to bear heavy static and dynamic loads; therefore, deep foundations are usually adopted to ensure the safety of these bridges. As a typical type of deep foundations, caisson foundations, characterized by their strong integrity, high stiffness, good stability, and reliable seismic performance, have been used in many long-span bridges, such as the 1915 Çanakkale Bridge in Turkey, the Akashi Strait Bridge in Japan, the Taizhou Yangtze River Highway Bridge, and the Shanghai-Suzhou-Nantong Yangtze River Bridge in China.

The seismic design of long-span bridges with caisson foundations must account for the influence of soil–caisson dynamic interaction (SCDI) because the SCDI effects can significantly alter the dynamic responses of bridge structures through the following ways: (1) kinematic interaction, modifying the motion of caisson foundations; (2) hysteretic and radiative damping, affecting the energy dissipation; and (3) elongation of the structural fundamental period [1,2,3,4]. However, the traditional seismic design method for long-span bridges with caisson foundations does not consider the SCDI effects. Instead, the base of the tower is assumed to be fixed at the top surface of the caisson, and the ground motion acceleration response spectrum or time history of the free field, which is not affected by the structural vibration or the scattering of waves around the foundation, is directly applied to the base of the tower to analyze the seismic response of the bridge. Researchers found that the dynamic interaction between soil and caisson foundations may lead to a significant reduction in the horizontal acceleration amplitude [1,2,3,5,6,7,8,9] of the foundation motion compared to that of the free-field ground motion, and also induce the rotation component of the foundation [3,10].

Furthermore, based on the data from a separately conducted shaking table test, the contribution of the tower top displacement due to foundation rotation to the maximum tower top displacement can be more than 70%, and neglecting the foundation rotation may underestimate the displacement seismic demands of bridge structures, despite the reduction in horizontal acceleration of the free-field ground motion [11,12]. Therefore, the influence of SCDI on the foundation motion needs to be considered when performing the seismic response analysis of bridge structures.

Many researchers have uesd theoretical and numerical methods to examine the influence of dynamic interaction between soil and foundations on the motion of rigidly embedded foundations subjected to dynamic loading. For example, Thau et al. [13] derived the dynamic equations of motion of a rigid embedded rectangular foundation in an elastic half-space under plane strain conditions using the Laplace and Kantorovich–Lebedev transformations. Similarly, Elsabee et al. [14] parametrically investigated the motion of a rigid, massless, cylindrical embedded foundation under the action of a vertical shear wave by using the three-dimensional finite element method and proposed the simplified formulas for foundation translation and rotation; Mita et al. [15] combined the Green’s function for the half-space continuum with a finite element to study the motion of a square-embedded foundation in a homogeneous viscoelastic half-space, and presented numerical results of the foundation motion in the form of a table; Karabalis et al. [16] employed the boundary element method to examine the foundation motion of a three-dimensional rigid embedded foundation of arbitrary shape situated on a linearly elastic, homogeneous, isotropic half-space; Conti et al. [3,10] investigated the effect of dynamic interaction between soil and embedded foundations on rectangular foundation motions using theoretical analysis and 2D plane strain numerical simulations, and proposed frequency-dependent kinematic interaction functions, which relate the embedded foundation motions to the free-field surface motions.

However, the above studies are subject to one or more of the following issues: (1) the soil is usually assumed to be linearly elastic, homogeneous, and isotropic, and the influence of confining pressure and soil nonlinearity on the soil parameters is not considered; (2) the contact between the soil and the structure is usually assumed to be perfect, and the effects of separation and slip between the soil and the foundation are not considered; (3) the above studies are usually conducted in the plane strain conditions or using two-dimensional numerical models, and it has been demonstrated that a three-dimensional model is more accurate and real for examining the dynamic interaction between soil and foundations; and (4) the embedment of the foundation in the aforementioned study is relatively shallow, with the maximum embedment depth of 12 m. However, the embedment depth of the caisson foundation employed for long-span bridges is typically greater than 40 m, and the influence of the embedment depth on the motion of caisson foundations is more significant. Therefore, it is necessary to verify the applicability of the above research results on the effect of dynamic interaction between soil and foundations on the motion of shallow embedded foundations to the deep embedded caisson foundations of long-span bridges in real situations. Adopting the experimental method to study the influence of SCDI effects on the motion of caisson foundations can address the issues mentioned above in a more satisfactory manner.

The experimental research on the SCDI effects has not yet been fully conducted. Gaudio et al. [17,18,19] employed the centrifuge model test, which can simulate stress levels equal or similar to those of the prototype, and effectively reproduce the physical properties of the prototype, to analyze the seismic performance of the bridge pier with the caisson foundation. However, they paid more attention to the SCDI effects on the seismic performance indexes such as maximum acceleration, maximum displacement at the top of the pier, and maximum bending moment at the bottom of the pier, etc., without focusing on the SCDI effects on the motion of the caisson foundation. In addition, the prototype embedment depth of the caisson foundation in their test was only 11 m, which is much smaller than the embedment depth of the caisson foundation for the long-span bridge. Therefore, it is difficult to use the centrifuge model test to study the SCDI effects on the motion of the large-size caisson foundation, due to the limitation of the capacity and size of the centrifuge.

The 1 g shaking table model tests, with larger geometry and higher bearing capacity than centrifuge model tests, allow precise loading, control, and observation [20,21], and are more suitable for the study of large-size caisson foundations. Additionally, the 1 g shaking table model test method can realistically reproduce the whole process of seismic loading action and serves as the most direct approach for studying the characteristics of the SCDI and its impact on the caisson foundation motion. Therefore, this paper adopts the method of 1 g shaking table model test to study the influence of SCDI, and the novelty is shown: By carefully designing the model dimensions, and configuring the model soil to meet the specific similarity relationship according to the similarity theory, the 1 g three-dimensional shaking table model test for the deeply embedded caisson foundation of the long-span bridge can simultaneously consider the influence of the nonlinear soil, the complex three-dimensional contact relationship between the soil and the caisson, the ultra-large embedment of the long-span bridge foundation, and other factors. Thus, the influence of SCDI on the motion of the caisson can be evaluated more truly and accurately.

A series of 1 g shaking table model tests were conducted in this study, including a free-field system model test and a soil–caisson system model test, in which the free-field system model test consisted of soil only and no structure included. First, the test setup, the design of the model soil and the model caisson, the arrangement of measuring points, and the loading cases of the tests are introduced. Next, the variation in characteristic parameters of the motion of the caisson foundation under seismic loading is presented. Then, the distributions of peak accelerations along the vertical direction in different system model tests are compared. Finally, the impacts of kinematic interaction on the horizontal acceleration and rotational component of the caisson foundation motion are investigated.

2. Test Setup

2.1. Shaking Table

The tests were conducted at the State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, using a shaking table with a vertical load capacity of 30 t within the multi-point shaking table test system, featuring a table size of 6 m × 4 m. The shaking table has three degrees of freedom (two translational and one rotational) with a displacement of +/−500 mm, a maximum acceleration of 1.5 g, and an operating frequency range of 0.1 to 50 Hz. The shaking table system can simulate the occurrence of real ground motions, adjusting the table input according to the acceleration signal of the ground target seismic wave in an iterative process until the target ground acceleration response is achieved.

2.2. Model Soil Container

To minimize the boundary effect of the model soil container in the test, the test used a laminar shear soil container, which is composed of a steel base plate and a 16-layer steel frame with a height of 10 cm and a gap of 1.2 cm between the layers. The steel frames are constructed from rectangular hollow steel pipes welded together to reduce inertia while providing sufficient constraint for the $K_0$ condition. The cross-section width of the frame is 10 cm, while the wall thickness is 0.8 cm. The physical model of the soil container is shown in Figure 1.

Short steel reinforcements are welded to the bottom of the soil container to increase the roughness of the container bottom, reducing the relative slip between the model soil and the bottom of the container. To prevent soil leakage, the inner wall of the soil container is covered with a rubber membrane. Additionally, the inner side of the rubber membrane is covered with a 5 cm thick foam layer to further reduce the boundary effect of the model soil container. In order to reduce the influence of the boundary effect when studying the SCDI through shaking table tests, it is necessary to ensure that the relationship between the size of the soil volume and the size of the caisson meets certain requirements, including [21,22]: The ratio of the plane area of the soil volume to the base area of the caisson is greater than 5, the ratio of the vertical size of the soil volume to the embedded depth of the caisson is greater than 3, and the ratio of the size of the soil volume to the structure size is usually between 3 and 10 in the vibration direction of the seismic wave. To meet the aforementioned requirements, a laboratory investigation was conducted to ascertain the suitability of existing soil containers stored in the laboratory, taking into account the dimensions and embedment of the caisson, and finally determined that the plane size of the soil container was 2.8 m × 2.3 m, with a vertical size of 1.68 m.

2.3. Sensors and Data Acquisition System

The objective of this study is to examine the influence of SCDI on the motion of the caisson foundation. To this end, an instrumentation system comprising accelerometers and tensile-wire displacement meters was employed to measure the acceleration response of the soil and the caisson, and the vertical displacement of the top of the caisson, respectively. The range of the tensile-wire displacement meters is 0~1100 mm, with an accuracy of 0.03 mm. All sensors were connected to the multi-channel data acquisition system of the shaking table to ensure signal synchronization, with a sampling frequency of 256 Hz.

3. Test Design

The shaking table tests aim to investigate the influence of SCDI on the caisson foundation motion. The contents of the experimental study can be summarized as follows: (1) Studying the influence of SCDI on the characteristic parameters of horizontal acceleration response of the caisson foundation motion; (2) Examining the distribution pattern of horizontal acceleration along the vertical direction in different system model tests; and (3) Investigating the influence of SCDI on the horizontal acceleration and rotational component of the caisson foundation motion.

3.1. Design of Scale Factors

The prototype caisson foundation in the model test is from the middle tower of the Taizhou Yangtze River Highway Bridge. The dimensions of the prototype caisson foundation are 56.2 m in length, 44.1 m in width, and 76 m in height, with an embedment depth of 55 m. According to similarity theory [20,23], the simulation of SCDI effects should mainly consider three aspects: geometric similarity, physical similarity, and mechanical similarity. Considering the large geometric dimensions of the caisson foundation, the design of the length scale factor incorporates considerations such as the size and bearing capacity of the shaking table, the dimensions of the model soil container, the boundary effect, the model manufacturing process, and other factors, ultimately leading to the determination of the length scale factor for the model test, which is set at $S_l=1/80$.

The results of the dynamic triaxial and resonant column tests of model soil conducted by Yan Xiao et al. [24,25] showed that adding sawdust to sand soil reduced both the density and dynamic shear modulus. This finding is consistent with the requirement of similarity. Thus, a mixture of sawdust and dry sand was chosen as the model soil, with a mass ratio of 1:2.5. This ratio was then used to calculate the scale factors for both the modulus of elasticity ($S_E=1/32$) and the density ($S_{\rho }=1/2$). Once the scale factors of length, modulus of elasticity, and density are determined, the remaining physical quantities can be deduced according to the Buckingham π law. The scale factors of the model soil are listed in

Table 1. Scale factors of the model soil and the model structure.

Types	Physical Quantities	Symbols	Scale Factors of the Model Soil	Scale Factors of the Model Structure
Material properties	Strain	Sε	1	1
	Stress	Sσ	1/32	1/10
	Elastic modulus	SE	1/32	1/10
	Poisson’s ratio	Sμ	1	1
	Density	Sρ	½	½
Geometry properties	Length	Sl	1/80	1/80
	Area	SA	1/6400	1/6400
	Linear displacement	Sl	1/80	1/80
Loading	Force	SF	1/204,800	1/64,000
	Moment	SM	1/16,384,000	1/5,120,000
Dynamic properties	Mass	Sm	1/1,024,000	1/1,024,000
	Time	St	1/20	1/20
	Frequency	Sω	20	20
	Damping	Sd	1/640,000	1/640,000
	Velocity	Sv	¼	¼
	Acceleration	Sa	5	5

.

The use of organic glass in the fabrication of model structures is common due to its favorable characteristics, such as low density, homogeneous composition, high strength, low modulus of elasticity, and ease of processing and fabrication. In the design of the model structure, the following basic physical quantities are chosen: $S_l=1/80$, $S_{\rho }=1/2$, and $S_a=5$. According to the material properties of organic glass, the elastic modulus scale factor $S_E$ and the density scale factor $S_{\rho }$ can be determined as 1/10 and 1/2, respectively. However, the elastic modulus scale factor $S_E$ does not fully satisfy the similarity theory. Preliminary finite element analysis indicates that the elastic modulus scale factor $S_E$ is approximately 1/10 or 1/32, which has minimal impact on the dynamic response of displacement and acceleration of the caisson foundation and the soil, primarily due to the significant stiffness discrepancy between the caisson foundation and the soil. The stiffness difference between the caisson foundation and the soil means that the caisson foundation can be considered a rigid body in the soil, regardless of whether the elastic modulus scale factor $S_E$ of the model structural material is 1/10 or 1/32. Therefore, the elastic modulus scale factor of the model structure, $S_E$, is set to 1/10 for this experiment. Once the length, elastic modulus, and density scale factors are determined, the remaining scale factors can be calculated using the Buckingham π law. The scale factors of the model structure are also presented in Table 1.

3.2. Model Materials and Design

3.2.1. Model Structure

The model design of the caisson foundation follows the scale factors outlined in Table 1. The model material is organic glass with a modulus of elasticity of 3 Gpa and a density of 1190 kg/m³. To ensure that the roughness of the contact surface between the prototype and the model is the same, a cement mortar layer with a density of 2000 kg/m³ is sprayed on the lateral and bottom surfaces of the organic glass model. According to the density scale factor, the equivalent density of the caisson foundation model is determined to be 1300 kg/m³. This calculation determines the size of the organic glass model to be 697.5 mm × 528 mm × 936 mm. The thickness of the cement mortar layer is approximately 14 mm, resulting in a total size of the caisson foundation model of 725 mm × 555 mm × 950 mm, meeting the length scale factor of 1/80. Figure 2 illustrates the dimensions of the caisson foundation model (without the cement mortar layer), as well as the physical view of the caisson foundation model.

3.2.2. Model Soil

The soil profile at the prototype caisson is multilayered, with the shear wave velocity ($v_s$) increasing with depth ($z$). Although simulating a detailed layered soil profile is not the focus of this paper, it is important to ensure the distribution of soil stiffness along the depth. To model the $v_s(z)$ distribution, the prototype soil profile is idealized as a single-layer model soil, where $v_s$ increases parabolically with depth $z$. The tests were conducted using a model soil consisting of a mixture of dry sand and sawdust, where the mass ratio of sand to sawdust was 2.5:1. The grading curves of dry sand, mixed model soil, and sawdust are shown in Figure 3. The results of the resonance column tests on synthetic model soils [25] indicated a nonlinear increase in the shear modulus $G_0$ with confining pressure ${{\sigma }_0}^{\prime }$, following the relationship:

$$G_0=1251.63\times {({{\sigma }_0}^{\prime })}^{0.684}\left(\mathrm{kPa}\right)$$

(1)

The key mechanical parameters of the model soil are determined through the direct shear test. The effective internal friction angle $\phi$ is 30.7°, and the effective cohesive strength $c$ is 0.54 kPa.

In the free-field system mode test, a physical model with a depth of 1.68 m was constructed layer by layer, with each layer measuring 11.2 cm in thickness. The density $\rho$ of the model soil was controlled at 910 kg/m³, which meant that the mass of the model soil needed for each layer was 556.5 kg. Manual tamping and mechanical vibration were used to compact each soil layer until the target thickness was achieved.

In the soil–caisson system model test, the model soil was built up to the caisson base elevation, and then the caisson foundation was placed in the pre-designed position. Subsequently, each layer of model soil above the caisson base was constructed until the target thickness was reached.

3.3. Instrumentation Layout

A dense 3D instrumentation layout for the free-field system and soil–caisson system model tests is depicted in Figure 4. In the free-field system model test, 42 acceleration sensors were placed in the soil to measure the acceleration response of the soil, as illustrated in Figure 4a.

For the soil–caisson system model test, 14 accelerometers and 6 displacement meters were installed on the caisson to measure the acceleration response of the caisson and the vertical displacement of the top of the caisson, respectively. Additionally, 33 accelerometers were installed in the soil to measure the acceleration response of the soil, as illustrated in Figure 4b.

3.4. Loading Method and Cases

Shaking table model tests were conducted to obtain the dynamic response of the free-field and soil–caisson systems, respectively. The load excitations in the tests included white noise, as well as Chichi (CHY012 H1*, Taiwan, 1999) and Kobe (Abeno station H1*, Hanshin, Japan, 1995) ground motions selected from the Pacific Earthquake Engineering Research Center database (https://ngawest2.berkeley.edu, accessed on 1 March 2022). The latter two waves were used as target waves at the surface, and scaled in time and acceleration according to the scale factors in Table 1, where the time scale factor S_t and acceleration scale factor S_a are 1/20 and 5, respectively. The control signals of the shaking table were adjusted iteratively to ensure that the acceleration response of a steel rod at the ground surface matched the target waves. The final control signals of the shaking table were then recorded. In the case where the peak acceleration of the surface target wave was 0.5 g, the time histories and Fourier spectra of the shaking table input acceleration are depicted in Figure 5.

Subsequently, the model test of the soil–caisson system was conducted. In the soil–caisson system model test, the position of the steel rod was shifted from the center of the soil to the location away from both the caisson and soil container boundaries. For 0.1 g~0.5 g cases, the time histories of the shaking table input acceleration recorded in the free-field system were directly used as the input seismic waves. Additionally, for 0.75 g~1.5 g cases, the table inputs of the free-field system at 0.5 g were amplified until the peak acceleration response of the steel rod reached 0.75 g, 1.0 g, and 1.5 g, respectively.

The loading cases for the free-field and soil–caisson system model tests are presented in

Table 2. Loading cases of the free-field and soil–caisson system model tests.

Test System	Ground Motion	Peak Acceleration (g)
Free-Field	White noise	0.1
	Chichi Kobe	0.1, 0.25, 0.5
Soil–Caisson	White noise	0.1
	Chichi Kobe	0.1, 0.25, 0.5, 0.75, 1.0, 1.5

. The white noise sweep test was performed on the system before and after each set of seismic wave cases to evaluate changes in the dynamic properties of the system. A total of three white noise cases, numbered WN1 to WN3, were performed in each system test.

4. Test Results and Discussion

To investigate the influence of SCDI on the motion of the caisson foundation, the characteristic parameters of the motion of the caisson foundation in the soil–caisson system model test under seismic loading are examined. Additionally, the section also examines the distribution pattern of peak acceleration in the vertical direction and the transfer functions of horizontal acceleration and rotational component of the caisson foundation motion.

4.1. Characteristic Parameters of the Caisson Motion

For structural systems subjected to seismic loading, the frequency response function $H_{fre}(i\omega )$ is employed to identify the characteristic frequency $f_c$, defined as the frequency corresponding to the peak of the amplitude of $H_{fre}(i\omega )$ of the caisson motion, while the characteristic damping ratio ${\xi }_c$, defined as the damping ratio at characteristic frequency $f_c$, is calculated using the half-power method [21,26]. Based on power $(S_{xx})$ and the cross-power $(S_{xy})$ spectral density functions of the input and output acceleration signals within the structural system, calculating $H_{fre}(i\omega )$:

$$H_{fre}(i\omega )={\displaystyle \frac{S_{xy}}{S_{xx}}}$$

(2)

where $x$ and $y$ denote the acceleration signals recorded at the table surface and the top measurement point (A-C-10) of the caisson foundation, respectively.

In this section, the influence of the peak acceleration of the shaking table input acceleration on the characteristic parameters of the motion of the caisson foundation is investigated by calculating the $H_{fre}(i\omega )$ under different loading cases.

The $H_{fre}(i\omega )$ is calculated to obtain the characteristic parameters such as characteristic frequency $f_c$ and characteristic damping ratio ${\xi }_c$, of the motion of the caisson. These results, along with the displacement amplitude at the top of the caisson, are listed in

Table 3. Characteristic parameters of the motion of the caisson under different loading cases.

Loading Case	Characteristic Frequency (Hz)	Characteristic Damping Ratio (%)	Displacement Amplitude (mm)
WN1	6.33	2.04	2.03
Chichi 0.1 g	7.30	7.69	0.93
Chichi 0.25 g	5.95	10.00	1.99
Chichi 0.5 g	5.32	10.00	4.45
Chichi 0.75 g	2.86	20.00	8.28
Chichi 1.0 g	2.75	20.00	16.86
Chichi 1.5 g	1.68	33.33	22.69
WN2	6.45	2.0	2.25
Kobe 0.1 g	6.86	7.69	0.19
Kobe 0.25 g	6.85	7.69	0.43
Kobe 0.5 g	6.66	8.33	0.68
Kobe 0.75 g	5.74	11.11	1.36
Kobe 1.0 g	5.17	18.75	4.33
WN3	6.43	2.0	1.96

. From the results shown in Table 3, it can be seen that: (1) As the peak acceleration of the input wave increases, the $f_c$ of the motion of the caisson foundation decreases, while the displacement amplitude and ${\xi }_c$ exhibit an increasing trend. (2) The characteristic parameters of the motion of the caisson foundation under the Chichi wave differ significantly from those under the Kobe wave. Under the Chichi wave, the $f_c$ of the motion of the caisson foundation is smaller than that under the Kobe wave, but the displacement amplitude and ${\xi }_c$ are larger than that under the Kobe wave.

The observations mentioned above are due to the fact that for the Chichi wave, the dominant frequency of the table input acceleration is smaller, approximately 12 Hz. Under the Chichi wave, the soil experiences more low-frequency cycles, resulting in larger displacements, leading to increased shear stress and greater plasticity within the soil, causing a notable decrease in the shear modulus and an increase in the damping ratio of the soil. As a consequence, the $f_c$ of the motion of the caisson foundation reduces, while the ${\xi }_c$ increases. In contrast, for the Kobe wave, the dominant frequency of the table input acceleration is higher, approximately 23.1 Hz, resulting in smaller soil displacements. The soil within the system remains mostly elastic for Kobe 0.1 g~Kobe 0.5 g cases, with displacement amplitudes less than 0.68 mm and characteristic frequencies even higher than the fundamental frequency in the white noise case. Therefore, the $f_c$ and ${\xi }_c$ of the motion of the caisson foundation do not change significantly with increasing peak acceleration of the table input acceleration. However, for Kobe 0.75 g~Kobe 1.0 g cases, the soil undergoes a certain extent of plasticity, as indicated by displacement amplitudes of 1.36 mm and 4.33 mm, respectively. Moreover, the $f_c$ of the motion of the caisson foundation is at least 10% lower than the fundamental frequency of the white noise case, leading to significant changes in the $f_c$ and ${\xi }_c$ of the caisson foundation’s motion response. Notably, the extent of plasticity observed in the soil for Kobe 0.75 g~Kobe 1.0 g cases is lower than that for Chichi 0.75 g~Chichi 1.0 g cases, where the displacement amplitudes at the top of the caisson exceed 8 mm, and the characteristic frequencies are below 3.0 Hz.

In this section, the characteristic parameters of the motion of the caisson foundation under different seismic loading conditions in the soil–caisson system are calculated based on the frequency response function $H_{fre}(i\omega )$ and the half-power method. The results show that the effects of SCDI have a significant impact on the characteristic parameters of the caisson motion, specifically related to the peak value of the shaking table input acceleration signals.

4.2. Distribution of Acceleration Magnification Factors

In both the free-field system and the soil–caisson system model tests, acceleration responses of the soil and the caisson foundation at different depths were measured. Acceleration magnification factors (AMFs) along the vertical direction are calculated by comparing peak accelerations measured at the different depths with the peak acceleration of the table excitations. The AMF distribution results of the free-field (referred to as “FF”) system and soil–caisson (referred to as “SC”) system model tests are illustrated by the dashed and solid lines in Figure 6, respectively.

4.2.1. Distribution of AMFs of the Free-Field System Model Test

Results of the free-field system tests, shown in Figure 6, indicate that the AMFs of the soil exhibit a “C”-shaped distribution along the vertical direction. This distribution pattern, which is characterized by a peak acceleration decrease in the middle soil layer, followed by an increase near the surface, is consistent with the test and numerical simulation results of Wu et al. [20,27,28]. To better understand this distribution pattern, frequency response functions $H_{fre}(i\omega )$ of different embedment measurement points A-2-1 to A-15-1, as shown in Figure 4a, in the free-field system test are presented in Figure 7.

The results in Figure 7 illustrate how seismic waves propagate in the free-field of sandy soil. As seismic waves travel from the bottom to the surface, high-frequency components are attenuated, while low-frequency components are amplified. This amplification is particularly noticeable closer to the surface. This propagation law can provide insights into the distribution of AMFs along the vertical direction. In the middle soil layer, the filtered high-frequency components carry more energy than amplified low-frequency components, resulting in a reduction in the peak acceleration. Near the surface, low-frequency components are further amplified due to reflection and scattering at the soil–air interface, significantly amplifying the peak acceleration of the surface soil layer.

Moreover, in Figure 7b, it is evident that the amplitude of $H_{fre}(i\omega )$ at the surface measurement point is higher around 20 Hz under the Kobe wave. This is attributed to the close proximity between the dominant frequency of the Kobe wave, approximately 23.1 Hz, and the second-order frequency of the soil layer, approximately 18.54 Hz. Consequently, the acceleration response near the second-order frequency of the soil layer is amplified.

As the table input acceleration increases, trends of AMFs at the surface measurement points vary with different seismic waves. Characteristic parameters of the dynamic response at the surface measurement point A-15-1, as shown in Figure 4a, of the free-field system can be found in

Table 4. Characteristic parameters of the free-field ground motion under different loading cases.

Loading Case	Characteristic Frequency (Hz)	Characteristic Damping Ratio (%)	Displacement Amplitude (mm)
WN1	6.33	3.06	3.85
Chichi 0.1 g	6.52	8.33	0.87
Chichi 0.25 g	6.29	9.1	2.32
Chichi 0.5 g	5.50	11.1	5.2
WN2	6.35	2.04	3.33
Kobe 0.1 g	6.58	8.33	0.43
Kobe 0.25 g	6.74	8.33	0.58
Kobe 0.5 g	6.50	9.09	0.63
WN3	6.37	2.04	3.82

. For the Chichi wave, AMFs at the ground surface decrease as the table input acceleration increases, which is attributed to the lower dominant frequency of the table input acceleration of the Chichi wave, about 12.5 Hz, resulting in greater soil displacement, as shown in Table 4. Consequently, the shear strain of the soil increases, and the soil experiences evident plastic behavior, which can be seen from the fact that the characteristic frequency of the surface response is clearly smaller than the fundamental frequency in the white noise case, leading to a decrease in the dynamic shear modulus and an increase in the damping ratio of the soil, resulting in greater soil energy dissipation and a corresponding decrease in the AMFs at the surface.

As for the Kobe wave, it is evident that AMFs at the ground surface demonstrate a positive correlation with the increase in the table input acceleration. This is due to the higher dominant frequency of the table input acceleration for the Kobe wave, about 23.1 Hz, leading to small soil displacement, as shown in Table 4. As a result, the soil behaves elastically, with the characteristic frequency of the surface response exceeding the fundamental frequency in the white noise case. Therefore, there is minimal change in the energy dissipation of the soil, but the input energy of the system increases, leading to an increase in AMFs at the surface.

4.2.2. Distribution of AMFs of the Soil–Caisson System Model Test

Upon examining the test results of the soil–caisson system presented in Figure 6, it is evident that the AMFs still show a “C”-shaped distribution pattern under Chichi and Kobe waves. However, the AMFs at the surface measurement point within the soil–caisson system are generally smaller than those observed in the free-field system, indicating that the presence of the caisson foundation has an attenuating effect on the peak acceleration at the surface of the soil. Furthermore, the results in Figure 6 illustrate that the AMFs of the soil–caisson system subjected to the Chichi wave are typically greater than those under the Kobe wave. This is because the Chichi wave is richer in low-frequency content, as illustrated in Figure 5, and the soft soil, such as sand, amplifies low-frequency waves more effectively.

In addition, it has been noted that the soil–caisson system subjected to the Chichi wave exhibits larger AMFs at the ground surface with smaller table input acceleration, indicating a more significant amplification effect of the system on the input excitation in this scenario. The mechanism behind this phenomenon is consistent with that observed in the free-field system tests, where an increase in table input acceleration leads to a decrease in AMFs at the ground surface. However, under the Kobe wave, the amplification effect of the system to the input excitations initially increases and then subsequently decreases with the increase in the table input acceleration. This behavior occurs because the soil remains essentially elastic in the Kobe 0.1 g~Kobe 0.5 g cases, as reflected in the displacement amplitude and characteristic frequency data in Table 3, where the energy dissipation of the soil undergoes minimal changes, resulting in an increase in the AMFs at the surface with the increase in the input energy to the system. In contrast, under the Kobe 0.75 g~Kobe 1.0 g cases, the soil–caisson system exhibits significant plastic behavior, as evidenced by the characteristic frequencies of the motion response of the caisson foundation and substantial changes in displacement amplitudes, illustrated in Table 3. Consequently, as the table input acceleration increases, the dynamic shear modulus of the soil decreases, the damping ratio increases, and the energy dissipated by the soil increases, leading to a reduction in AMFs at the surface.

In this section, a comparative analysis of the distributions of AMFs in the free-field system and the soil–caisson system is presented. The findings demonstrate that the SCDI effects exert a considerable influence on the peak acceleration within the caisson embedment depth and significantly attenuate the peak value of the surface acceleration response.

4.3. The Effect of Soil–Caisson Kinematic Interaction

In the framework of the substructure approach, soil–caisson dynamic interaction can be decoupled into kinematic interaction and inertial interaction. The kinematic interaction arises from the substantial stiffness of the caisson foundation, leading to incompatible deformation between the caisson foundation and the surrounding soil. On the other hand, the inertial interaction is a result of the inertial force of the foundation itself and those transmitted to the foundation by the superstructure.

The SCDI effects, which modify the incident seismic wave field applied to the caisson foundation, result in a deviation of the caisson foundation motion from the free-field ground motion. This deviation is primarily attributed to the kinematic interaction [3,4,10,29,30], as the effect of the inertial interaction on the foundation motion is mainly concentrated around the natural frequencies of the system [29,31].

Compared to free-field ground motion, the impact of the kinematic interaction on the motion of the caisson foundation can be observed in two distinct aspects: it creates a difference in horizontal acceleration between the caisson foundation and the free-field ground motion, and it introduces a rotational component to the foundation’s motion. The effect of the kinematic interaction on the motion of caisson foundations is further explored from two specific aspects mentioned above.

4.3.1. Effect on the Horizontal Acceleration Motion

Firstly, the influence of the kinematic interaction on the horizontal acceleration motion of the caisson foundation is investigated by comparing the acceleration response of the free-field surface measurement point with that of the corresponding elevation measurement point on the caisson. The results are presented in Figure 8.

As illustrated in Figure 8a,b, in the case of Chichi 0.1 g, the horizontal acceleration of the caisson foundation is virtually identical to that of the free-field ground motion in terms of waveform and amplitude, indicating that the influence of the kinematic interaction is negligible. However, as the acceleration peak increases, although the horizontal acceleration waveform remains similar between the caisson foundation motion and free-field ground motion, there is a noteworthy reduction in the amplitude of the horizontal acceleration of the caisson foundation compared to that of the free-field ground motion. Specifically, the horizontal acceleration amplitude of the caisson foundation is reduced by 2.38%, 34.57%, and 26.24% for the Chichi 0.1 g, Chichi 0.25 g, and Chichi 0.5 g cases, respectively, compared to the amplitude of the free-field ground motion in the corresponding cases.

To better understand the impact of the kinematic interaction on the horizontal acceleration of the caisson foundation motion, a transfer function (TF) between the acceleration response of the free-field ground motion (FFM) and the caisson foundation motion (CFM) at the same elevation is computed. In this analysis, the FFM serves as the input signal, while the CFM serves as the output signal. In practice, the transmissibility function [31,32] Equation (3) and the coherence function [9] Equation (4) are employed to estimate the TF and evaluate the influence of noise on the signals, respectively. The coherence function close to unity indicates minimal noise effect and a strong correlation between the two signals. Reliable transfer function estimates are considered at frequencies of ${\gamma }^2\ge 0.8$ [9,32].

$$\left|H_{gf}(\omega )\right|=\sqrt{{\displaystyle \frac{S_{gg}}{S_{ff}}}}$$

(3)

$${\gamma }^2(\omega )={\displaystyle \frac{{\left|S_{gf}(\omega )\right|}^2}{S_{gg}(\omega )S_{ff}(\omega )}}$$

(4)

where $S_{gg}$ and $S_{ff}$ are the power spectral density functions of the acceleration response of the CFM ($A_{gg}$) and the acceleration response of the FFM ($A_{ff}$) at the ground surface, respectively, and $S_{gf}$ is the cross-power spectral density function.

Results of the transfer function $\left|H_{gf}(\omega )\right|$ between $A_{gg}$ and $A_{ff}$ under Chichi and Kobe waves are presented in Figure 8c,d, where the results of $\left|H_{gf}(\omega )\right|$ with a high coherency are also demonstrated with dots at the same time. From the results in Figure 8, it can be seen that: (1) The influence of noise on the signals in the tests is negligible; (2) When the frequency is less than 15 Hz, the transfer function approaches unity. As frequency increases, the transfer function shows a decreasing trend, indicating that the caisson foundation filters high-frequency components of the incident seismic waves; (3) The first three orders of natural frequencies of the soil–caisson system are 6.4 Hz, 19 Hz, and 29.5 Hz, respectively, and close to the natural frequency of the system, the transfer function demonstrates distinct fluctuations, attributed to inertial interaction [29,31,32]. (4) The transfer function exhibits clear frequency dependence, with the dominant frequency and peak acceleration of the table input wave significantly influencing the deviation of $A_{gg}$ and $A_{ff}$.

The observations mentioned above are due to the fact that the dominant frequency and peak acceleration of the seismic wave affect the extent of plasticity of the soil around the caisson foundation, which corresponds to the different shear modulus, affecting the contact stiffness between the caisson foundation and the surrounding soil, resulting in frequency-dependent radiation damping and energy dissipation as the waves emanating from the vibrating caisson foundation propagate into the soil medium.

4.3.2. Effect on the Rotation Component

Under seismic loading, the horizontal displacement response of the soil at various embedment depths in the free field exhibits differences in amplitudes and phases, referred to as the “embedment depth effect” [31]. These disparities lead to an asynchronization of the soil motion at different embedment depths, i.e., “pseudo-rotation”. The stiffness difference between the caisson foundation and the surrounding soil results in the kinematic interaction, modifying the rotation of the caisson foundation distinct from the free-field “pseudo-rotation”.

This section explores the impact of the kinematic interaction on the rotation component of the caisson foundation. It presents the rotation component of the caisson foundation and its comparison with the free-field “pseudo-rotation”, as depicted in Figure 9. The rotation histories of the caisson foundation are calculated from the responses of the vertical displacement meters U1-U6 (as shown in Figure 4b) positioned at the top of the caisson, as well as the horizontal distances between the displacement meters.

Figure 9a illustrates the comparison between the rotation amplitude of the caisson foundation and the amplitude of the free-field “pseudo-rotation” under different Chichi cases. The results clearly show that the caisson foundation exhibits a distinct rotational response under seismic loading, especially when the peak acceleration of the table input acceleration is larger than 0.8 g. However, for the same table input acceleration, the rotation amplitude of the caisson foundation is smaller than that of the free-field “pseudo-rotation”, which indicates that the kinematic interaction suppresses the rotation component of the caisson foundation compared to the free-field “pseudo-rotation”.

For a deeper understanding of how the kinematic interaction affects the rotational component of the caisson foundation, the relationship between the rotational response of the caisson foundation and the “pseudo-rotation” response of the free-field is investigated in the frequency domain by using the transfer function of Equation (5) and the coherence function of Equation (6), and the results of the transfer function $\left|H_{cp}(\omega )\right|$ under Chichi and Kobe waves are presented in Figure 9b,c, where the results of $\left|H_{cp}(\omega )\right|$ with a high coherency are also demonstrated with dots at the same time.

$$\left|H_{cp}(\omega )\right|=\sqrt{{\displaystyle \frac{S_{cr}}{S_{pr}}}}$$

(5)

$${\gamma }^2(\omega )={\displaystyle \frac{{\left|S_{cp}(\omega )\right|}^2}{S_{cr}(\omega )S_{pr}(\omega )}}$$

(6)

where $S_{cr}$ and $S_{pr}$ are the power spectral density functions of the caisson foundation rotation response (${\theta }_{cr}$) and the free-field “pseudo-rotation” response (${\theta }_{pr}$), respectively, and $S_{cp}$ is the cross-power spectral density function.

From the results shown in Figure 9b,c, it can be seen that: (1) In the Chichi 0.1 g and Kobe 0.1 g cases, the rotation angle of the caisson is less than 0.1°, and the effect of noise on the signal is more pronounced. As the peak acceleration of the table input wave increases gradually, the angle of rotation of the caisson increases, and the impact of noise on the signal decreases gradually. (2) The rotational transfer function starts at zero and gradually increases with increasing frequency. (3) Similar to the transfer function $\left|H_{gf}(\omega )\right|$, the transfer function $\left|H_{cp}(\omega )\right|$ also displays frequency-dependent behavior, consistently staying below unity at all frequencies. These observations show that the rotational response of the caisson foundation is lower than the free-field “pseudo-rotation” response, indicating the influence of the kinematic interaction. This finding aligns with the conclusions obtained from Figure 8a.

Moreover, the amplitude of the transfer function $\left|H_{cp}(\omega )\right|$ under the Chichi wave is smaller than that under the Kobe wave, which indicates that the transfer function is influenced by the dominant frequency of the table input wave. Additionally, as the peak acceleration of the table input wave increases, the value of the transfer function decreases, implying a greater discrepancy between the caisson rotation ${\theta }_{cr}$ and the free-field “pseudo-rotation” ${\theta }_{pr}$. The impact of the dominant frequency and peak acceleration on the transfer function is also due to their influence on the contact stiffness between the caisson foundation and the surrounding soil, as described in Section 4.3.1 affecting the propagation of the outgoing waves emanating from the oscillating caisson foundation.

In this section, the relationship between the caisson foundation motion and the free-field ground motion is quantified by means of the horizontal transfer function $\left|H_{gf}(\omega )\right|$ and the rotation transfer function $\left|H_{cp}(\omega )\right|$, thus illustrating the influence of the SCDI effects on the motion of the caisson foundation, in which both transfer functions show frequency dependence, with amplitudes less than one, and the horizontal transfer function is close to unity at low frequencies, and the amplitude of the horizontal transfer function shows a tendency to decrease with increasing frequency, whereas the rotation transfer function amplitude starts from zero and increases with increasing frequency.

5. Conclusions

In this paper, the influence of SCDI on the motion of the caisson foundation is studied. To this end, a series of 1 g shaking table model tests were conducted on free-field and soil–caisson systems. The test results demonstrate the significant influence of SCDI on the characteristic parameters of the motion of the caisson foundation, the distribution of AMFs along the vertical direction in different test systems, the horizontal acceleration, and the rotational component of the caisson foundation. The main conclusions are summarized as follows:

(1) The larger the peak acceleration of the shaking table input waves, the greater the influence of SCDI on the motion of the caisson foundation, resulting in a significant decrease in the characteristic frequency and an apparent increase in the characteristic damping ratio of the motion of the caisson foundation.

(2) The results of the model tests conducted on free-field and soil–caisson systems indicate that the AMFs along the vertical direction exhibit a “C”-shaped distribution. Additionally, the SCDI effects exert a considerable influence on the peak acceleration within the caisson embedment depth, significantly attenuating the acceleration response at the ground surface, especially in high peak acceleration cases.

(3) The horizontal and rotational transfer functions are used to quantify the relationship between the motion of the caisson foundation and the free-field ground motion, while considering the SCDI effects. At lower frequencies, the horizontal transfer function is nearly equal to unity, and although the inertial interaction causes significant fluctuations around the natural frequencies of the system, the horizontal transfer function shows a decreasing trend with increasing frequency, because the effective size of the caisson foundation relative to the seismic wavelength increases as frequency increases, leading to a stronger filtering effect from the caisson foundation. Additionally, the rotational transfer function starts from zero and becomes amplified as frequency increases. In general, the SCDI effects reduce the amplitude of horizontal acceleration of caisson foundation motion, but also introduce a rotational component, which may reduce the maximum absolute acceleration of the tower top and increase the relative displacement between the tower top and tower base for bridge structures, which has been confirmed in the author’s subsequent research.

The results and conclusions of this paper provide preliminary evidence of the influence of SCDI on caisson foundation motions. To further develop analytical expressions for the transfer functions relating the caisson foundation motion to the free-field ground motion, an extensive parametric study based on numerical methods is necessary. The current shaking table tests provide valuable data support for validating analytical expressions and numerical methods. Furthermore, the implementation of this study paves the way for discussing how to modify the free-field ground motion and introduce the rotation component when assessing the seismic performance of long-span bridges with caisson foundations.