## 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 Canakkale 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 used 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, [PERSON] 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, [PERSON] 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; [PERSON] 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; [PERSON] 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; [PERSON] 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.