## 1 Introduction  
Recently, a large number of bridges with tall reinforced concrete (RC) piers have been constructed and used in bridges that commonly cross ocean bays and deep canyons with a pier height surpassing 50 m. These bridges have hollow sections that significantly reduce the volume and weight of concrete piers. The diminution of a structure's self-weight provides several advantages. For example, excessive hydration heat produced by cement hydration can be reduced to mitigate the thermal cracking of massive concrete piers and thus may increase the durability of concrete [1,2]. In countries like China, most bridges with hollow sections are located in seismic areas. For example, more than 40% of bridges in southwest China have high RC hollow piers with a height of more than 40 m [3,4]. The damage or failure of hollow RC bridge piers has been reported in several earthquakes, such as the devastating magnitude 7.9 (Wenchuan) earthquake in 2008 [5]. However, current seismic design codes provide guidelines for short to medium-height piers instead of hollow tall bridge piers [4,6]. The high slenderness ratio and axial load ratio of hollow, large mass RC pierscould produce different seismic performance compared with normal concrete piers. For example, considerable seismic inertial forces could be produced given the mass of piers is larger than that of its superstructure [7]. Many experimental studies were thus conducted to understand the seismic properties of hollow RC tall piers in the world. For example, seismic studies for hollow RC bridge piers began in the 2000s in Korea [8]. Since then, the research interest in the study of the seismic performance of hollow RC bridge piers has only increased. In the past decades, many quasi-static tests in studies, such as [PERSON] et al. [9], [PERSON] et al. [10], [PERSON] et al. [11], and [PERSON] et al. [12], have been performed to understand the seismic behaviors (e.g., failure, ductility capacity) of bridge piers subjected to the combined unilateral and axial loads. The results have shown that the seismic performance of RC box piers varies according to different aspects such as the aspect ratio, amount of transverse reinforcement, and the shape of cross-ties. In practical applications, the actual seismic waves consist of three directional components in orthogonal directions. This fact indicates that the seismic performance of a structure, including the ultimate load-bearing capacity, the stiffness degeneration, and the strength degradation could be influenced by more than one directional seismic excitation. Several studies have shown that the failure characteristics of structures under multi-direction loadings are more complicated due to the multi-axial coupling effect [13; 14; 15; 16; 17; 18]. The crack resistance, energy absorption capacity, damage propagation, and ductility of the RC bridge columns under biaxial horizontal cyclic loading usually differs markedly from that under uniaxial loading [14; 15; 16; 17; 19]. Note that most of these studies focus on the bridge piers with solid sections. To understand the seismic performance of hollow RC box piers, few studies have been performed on the multi-dimensional seismic performance. A number of related topics, such as the seismic performance degradation induced by coupling effect from multi-dimensional loads, have been investigated [5; 20]. The reinforced concrete box pier was certified to have good bending properties in multi-dimensional seismic loads. However, there have been no detailed studies on the multi-dimensional seismic damage index or the corresponding seismic design methods. Consequently, there is no effective method for the multi-dimensional seismic design of bridges with reinforced concrete box piers. The displacement at the top of a bridge pier is currently used as the seismic performance evaluation index in the Chinese Seismic Design Code for Highway Bridge JTG/B 02-01 (JTG/T B02-01-2008).