# Calculation Method of Rotational Constraint Stiffness for a New Tower-Pier Connected System
## 1 Introduction  
A cable-stayed bridge is formed by three main structural elements: deck, towers, and cable-stays. According to the difference in the connection patterns between the tower, main beam, and pier, the cable-stayed bridge can be divided into four systems: a floating system, semi-floating system, tower-beam consolidation system, and rigid frame system [1; 2; 3]. As bridge engineers pay more and more attention to bridge aesthetics, many special-shaped cable-stayed bridges have been designed and built, such as Australia's Batman Bridge, Spain's Alamillo Bridge, China's Hunan Hongshanmiao Bridge, and so on [4; 5; 6; 7; 8; 9; 10; 11]. These are mostly irrational in mechanics due to their strange appearance, while they are generally regarded as landmarks of the city because of their special appearance. For these bridges, the pier and tower become whole through rigid connection. Huge internal forces will be generated at the root of the tower under loads [12], which need to be resisted by increasing the size of the cross-section of the tower. However, the increased sizes will lead to an increased cost for the project. Engineers have tried to solve the above problem through structural optimization [13]. For the Tianjin Tuanbo New Bridge [14], built in China in 2011, and the Shanghai Wenzaobang Bridge [15; 16], built in China in 2014, the structural designs were optimized by changing the structural system, and the one-way hinge support was used to connect thebridge tower and pier in the longitudinal direction of the bridge. So, the bridge tower is allowed to rotate along the longitudinal direction of the bridge, which releases the huge bending moment at the root of the bridge tower, makes the force in the bridge more reasonable, and the tower becomes lighter and more shapely. These two cases provide new ideas to solve the problem of unreasonable force in the STCSB. However, for a special-shaped cable-stayed bridge with asymmetry in both the transverse and longitudinal directions, the root of the tower will suffer a large bending moment in both the transverse and longitudinal directions under loads [12]. The one-way hinge at the tower bottom cannot solve the problem of the bending moments generated in the transverse direction. Therefore, the Sanya Landscape Bridge innovatively adopts universal hinge support (UHS) to replace the traditional connection mode between the bridge tower and pier [17; 18]. UHS is composed of a top plate and bottom basin. The huge bending moment at the root of the tower can be released through the rotation of the top plate, and this rotation behavior is similar to a gyrostat [19; 20]. The mechanical properties of the bridge structure using UHS are improved, and the structural safeties of the steel bridge tower and concrete pier are greatly upgraded. In addition, the bridge tower size can have more design space, which also creates more possibilities in the structural form and aesthetics. This design is an innovation in the connection between the tower and the pier of the cable-stayed bridge, and it is also a new connection pattern between different materials, that is, a steel tower and concrete pier. In order to simplify the calculation, the boundary conditions are usually idealized when the bridge structure is designed and calculated. However, in practical applications, considering the great difficulty when replacing the UHS at the tower bottom during service, in order to synchronize the service life of the UHS with that of the bridge, rubber or ultra-high molecular weight polyethylene slide plates are not used between the top plate and the bottom basin, which eliminates the impact of material aging on the durability of the UHS [21; 22; 23; 24] and makes it meet the requirement of not being replaced within the design service life. Unfortunately, not using slide plates will lead to greater friction between the top plate and the bottom basin, which is a non-ignorant factor for in structural safety calculations.