# Calculation Method of Rotational Constraint Stiffness for a New Tower-Pier Connected System
## 2 The Plane Rotation Model of UHS  
UHS is mainly composed of a bottom basin and top plate. The top plate is connected to the bridge tower by bolts and shear tenons. The connection between the bottom basin and the concrete pier is performed by high-strength bolts, and the concrete is then poured to seal the anchor. The geometric model of the UHS is shown in Figure 1.  
Most research on bridge bearings mainly focuses on the vertical plane rotation of the bearings [37; 38; 39; 40], while only few studies have been performed to explore the properties of the horizontal-plane rotation, as Figure 2 shows. The top plate and the bottom basin are in frictional contact under a vertical force \(F\). When the torque \(M\) transmitted by the bridge tower to the UHS increases to the critical torque \(M_{R}\), the friction behavior between the top plate and the bottom basin has converted from static friction to dynamic friction. The top plate rotates around the UHS \(z\)-axis due to the rigid connections, and is adopted between the bottom basin and the concrete pier. During the process of rotation, the critical torque will vary with the variation in the vertical force \(F\). In addition, the critical torque also depends on the friction coefficient and the size of the UHS. Figure 1: Geometric model of UHS. Figure 2: The rotation model of the UHS. The stress decomposition of UHS is shown in Figure 3, where \(q_{0}\) is the uniform distributed force at the top plate (that is, the equivalent uniform load of F) and \(q\) is the uniform force transmitted from the top plate to the bottom basin. \(R_{0}\) is the spherical radius of UHS, \(R_{1}\) is the radius of the vertical projection surface of the arc-contact surface, and \(R_{2}\) is the radius of the arc-contact surface, \(R_{2}=R_{0}\times\theta\times\frac{\pi}{180^{7}}\). The differential idea is used to solve the critical torque of the UHS. The contact surface between the top plate and the bottom basin is unfolded, as shown in Figure 4, where \(ds\) is the area of the micro-ring with radius \(r\) and \(ds=2\pi rdr\). Make the following assumptions in the calculation:  
1. The pressure on the contact surface is evenly distributed along the surface. 2. Both the top plate and the bottom basin are rigid bodies. According to Figures 3 and 4, the vertical uniform distribution force of the peak point of the arc surface is \(q_{v}=q\) and the vertical uniform distribution force of the lowest point of the arc surface is \(q_{v}=\frac{q}{cosh}\). For the convenience of calculation, the vertical uniform distribution force of any point on the arc surface is approximated as \(q_{v}=\left(q+\frac{q}{cosh}\right)/2\), then the uniform distribution force on the surface is as follows:  
\[q=2q_{v}/\left(1+\frac{1}{\cos\theta}\right) \tag{1}\]  
The pressure of the micro-ring in Figure 4 can be obtained according to Equation (1):  
\[\mathrm{d}F=q\mathrm{d}s=2\pi qr\mathrm{d}r \tag{2}\]  
Figure 4: Unfolding plane of the contact surface between the top plate and bottom basin of UHS. Figure 3: Force transmission diagram: (**a**) the force transmitted from the tower to the top plate; (**b**) the force transferred from the top plate to the bottom basin. The friction force \(df\) on the micro-ring can be obtained according to [PERSON]'s friction formula (\(f=\mu\times F\)) and Equation (2):  
\[\mathrm{d}f=\mu dF=2\pi\mu qr\mathrm{d}r \tag{3}\]  
Then, the torque of the friction force on the micro-ring to the center of the arc-contact surface is as follows:  
\[\mathrm{d}M=df\times r=2\pi\mu qr^{2}\mathrm{d}r \tag{4}\]  
The torque generated by the total friction force on the arc-contact surface to the center of the circle can be obtained by integrating Equation (4):  
\[M_{R}=\int_{0}^{R_{2}}\mathrm{d}M=\int_{0}^{R_{2}}2\pi\mu qr^{2}\mathrm{d}r= \frac{2}{3}\pi\mu q{R_{2}}^{3} \tag{5}\]  
Substituting Equation \(q_{v}=F/\left(\pi{R_{2}}^{2}\right)\) into Equation (1):  
Figure 1:0  
Substituting Equation (6) into Equation (5), the critical torque can be obtained:  
Figure 1:1  
**3.