## 1 Introduction  
Pile-supported wharf (PSW) is one of the main port facilities for transferring shiploads and ship berthing. PSWs installed in seismically active regions are prone to damage and loss of functionality due to earthquakes. The 1964 Niigata earthquake caused damage to PSWs due to lateral spreading [1] and the 1989 Loma Prieta earthquake caused the significant destruction of PSWs in the Port of Oakland, including pile head failure, settlement, and lateral spreading [2]. The 2010 Haiti earthquake affected the Port de Port-au-Prince when a PSW shifted toward the sea, sinking the gantry crane [3]. Such losses of PSW serviceability immediately affect the functions of entire ports, resulting in considerable economic losses. To mitigate the threat of seismic damage to a PSW, its seismic performance must be evaluated. During earthquakes, the piles of PSWs are subjected not only to the inertial force transmitted from the wharf's deck but also to the kinematic force caused by the ground displacement of the soil slope. The latter is usually more destructive because the lateral resistance of the pile is reduced during massive earthquakes [4]. In the case of piles embedded in a soil slope, a significant bending moment is generated by kinematic force [5]. Comparing the natural periods of PSWs with and without soil slopes, the former is shorter than the latter. Moreover, because the amplitude of the spectral acceleration (SA) is generally significant in the short period range, neglecting the soil slope in seismic design may result in dangerous outcomes, even if the kinematic force is negligible. The seismic performance of PSWs can be precisely analyzed using soil-structure finite element analysis (FEA). Previous studies, such as by [PERSON] and [PERSON] [6], [PERSON] and [PERSON] [7], and [PERSON] et al. [8], have discussed the seismic responses of PSWs using this method. In FEA, the structure and soil layer are modeled, and the complex problem is solved, such as the interaction between the structure and soil in the case of lateral spreading occurring by earthquakes. In design practice, however, FEA is not usually applied because it is time-consuming and requires advanced simulation programs. On the contrary, frame analysis (FA) is a simpler method for analyzing the seismic response of the frame structure, such as PSW; therefore, SA is the standard method applied in design practice. However, SA has shortcomings including, e.g., the ground deformation due to earthquakes cannot be considered. The inertial force estimated from SA is applied to the PSW deck as a static load. The magnitude of SA is dependent on the damping coefficient and the value commonly employed by PSW engineering design is 0.03-0.05, with a typical PSW width of 20-60 m [9; 10]. As for PSWs that exceed 100 m in width, the commonly estimated damping coefficient may not be appropriate. Consequently, establishing a method for predicting the damping coefficient of wide PSWs is necessary. Several studies, such as by [PERSON] [11] and [PERSON] et al. [12], have identified potential factors that may affect the damping of a structure, including material damping, structural connections, foundations, building height, and the hysteresis of yielding components. [PERSON] and [PERSON] [13] found that the larger the foundation depth, the greater the damping coefficient. [PERSON] [11] proposed methods to predict the damping coefficient as in Equations (1) and (2). \[\zeta=\frac{0.75}{H}+380\frac{x}{H}-0.0014:\text{Reinforced concrete buildings} \tag{1}\]  
\[\zeta=\frac{0.52}{H}+380\frac{x}{H}+0.0023:\text{Steel buildings} \tag{2}\]  
where \(H\) is the building height (m) and \(x\) is the deformation at the peak of the building (m). However, these equations were developed specifically for buildings. A simplified method for evaluating the optimum damping coefficient for the design practice of PSWs has not yet been proposed. Several studies have addressed the impact of kinematic forces on pile design.