## 1 Introduction  
In order to prevent damage to bridge structures by seismic excitation, various types of countermeasures to bridge collapses are adopted in earthquake-prone regions. Focusing on Japan as a representative country, the Great Hanshin earthquake of 1995, which was a strong local earthquake, caused the unseating of many elevated bridges in urban areas. Due to the extensive damage to bridges caused by this earthquake, the guidelines for countermeasures to bridge unseating were significantly modified [1]. At present, the bridge unseating prevention system (Figure 1) is well defined, specifies the girder seating length and requires bridge restrainers and lateral displacement constraint structures to be installed on bridges, according to the Japanese seismic design specifications for highway bridges [2]. The function of this system is to prevent superstructures from being displaced from substructures because of massive earthquakes beyond the assumptions of aseismic design specifications. Bridge restrainers, which are the focus of this paper, are expected to ensure that the relative displacement between the superstructure and substructure does not exceed the girder seating length. Recently, many bridge restrainers have been installed in new bridges and seismically retrofitted bridges, as shown in Figure 2. In the existing design for bridge restrainers, the design load is considered to be 1.5 times the dead load reaction force. However, there is no specific evidence for this regulation. Furthermore, it seems that similar measures for using bridge restrainers as equipment to prevent reseating have been adopted in other earthquake-prone regions in the U.S. [3]. Considering the working situation of a bridge restrainer, the actual maximum acting load on the bridge restrainer is defined by the shared mass of the superstructure, cable stiffness, acting speed, cushioning type, and so on because the superstructure has various collision speeds. However, these influences are not reflected in the existing design method. Instead, a stereotypical and straightforward approach has been used for a long time. Therefore, there is concern that the acting load may temporarily exceed the design load, depending on the situation. To provide an overview of the existing research, [PERSON] et al. investigated road bridges that collapsed due to large earthquakes in the past and reported that simply supported girders with plural spans were a big part of numerous collapsed road bridges. They also reported that the collapse of piers accounted for 86% of bridge unseatings [4]. Through a model experiment, [PERSON] and [PERSON] reported that girder bounding had the potential of causing a bridge to collapse when bridge restrainers were not installed [5]. Extensive research on the acting speeds and loads of bridge restrainers, and the cushioning effect of the rubber, was conducted by [PERSON] and [PERSON] [6]. Some possibilities beyond the design load are indicated by considering dynamic acting, and there is a need for changing the specifications of the cable and rubber depending on the acting speed. Various numerical analyses have been conducted to examine the effect of restrainers under seismic loads [7; 8], showing that cable restrainers are useful for reducing bearing deformations or for preventing  
Figure 1: Bridge unseating prevention device (**a**) for the bridge’s axis direction and (**b**) for the bridge’s lateral axis direction. Figure 2: Installation status of cable-type bridge restrainers (**a**) for connecting the girder and abutment and (**b**) for connecting the girders to each other. girder unseating problems. [PERSON] et al. proposed a simplified and idealized mechanical model for cable bridge restrainers, considering the nonlinear behavior and fracture of the cable, pointing out that the cable length, clearance length, and cable section area should be appropriately selected, depending on the level of seismic intensity [9]. [PERSON] and [PERSON] proposed a design method for cable hinge restrainers.