# Calculation Method of Rotational Constraint Stiffness for a New Tower-Pier Connected System
## 2 The Plane Rotation Model of UHS
### RCS Calculation Model of UHS  
In order to establish the calculation model of the RCS under the joint influence of multi-parameters, according to the influence law of each parameter (\(R_{2}\), \(\mu\), \(F\)) on the RCS from Section 3.2, that is, RCS has a linear relationship with the radius of the contact surface, the friction coefficient of the contact surface, and the vertical force, respectively, the calculation model of the RCS with multi-parameters were obtained by coupling the three influence parameters into the initial model (Equations (8)-(10)), as shown in Equations (11)-(13):  
\[K_{M}=\frac{123.609}{\mu\times R_{2}}\mu\times F\times R_{2}=1340\mu\times F \times R_{2}=\xi_{1}\times\mu\times F\times\theta\times R_{0}\times\frac{\pi}{ 180^{\circ}} \tag{11}\]  
\[K_{M}=\frac{2021.6}{\mu\times F}\mu\times F\times 1000R_{2}=1347.7\mu\times F \times R_{2}==\xi_{2}\times\mu\times F\times\theta\times R_{0}\times\frac{\pi} {180^{\circ}} \tag{12}\]  
\[K_{M}=\frac{8246074}{F\times R_{2}}\mu\times F\times R_{2}=1340.8\mu\times F \times R_{2}=\xi_{3}\times\mu\times F\times\theta\times R_{0}\times\frac{\pi} {180^{\circ}} \tag{13}\]  
It can be seen that the results of the three Equations (11)-(13) are very close to each other, which verifies that the modeling method of coupling multi-factors after univariate analysis is advisable. In order to unify the calculation models, take the average value calculated by Equations (11)-(13) as the calculation model of RCS:  
\[K_{M}=\xi\times\mu\times F\times\theta\times R_{0}\times\frac{\pi}{180^{\circ }}=\frac{\xi_{1}+\xi_{2}+\xi_{3}}{3}\times\mu\times F\times\theta\times R_{0} \times\frac{\pi}{180^{\circ}}=1342.8\mu\times F\times\theta\times R_{0}\times \frac{\pi}{180^{\circ}} \tag{14}\]  
where the unit of RCS \(K_{M}\) is kN\(\cdot\)m/rad, and the unit of vertical force \(F\) is kN. Among them, \(\xi\) is a non-dimensional coefficient. It should be noted that the radius of the arc-contact surface \(R_{2}\) directly affects the RCS. Both the central angle \(\theta\) and curvature radius \(R_{0}\) jointly affect the arc-contact surface radius \(R_{2}\), which indirectly affects the RCS. That is, when \(R_{2}\) is constant, changing \(R_{0}\) cannot affect the RCS. Therefore, Equation (14) can be finally written as:  
\[K_{M}=1342.8\mu\times F\times\theta\times R_{2} \tag{15}\]  
where \(R_{2}=R_{0}\times\theta\times\frac{\pi}{180^{\circ}}\). The friction coefficient \(\mu\) in the model could be any value, which depends on the lubrication effect between the top plate and the bottom basin of the UHS. \(F\) could be the vertical force generated by different bridge structures or induced at the different construction. \(R_{2}\) should be determined according to the geometric parameters of the contact surface between the top plate and the bottom basin of the UHS used in different structures. Therefore, this model is applicable to the calculation of the horizontal rotational stiffness of any structures using UHS. In practical applications, the geometric parameters \(R_{0}\) and \(\theta\) can be easily obtained according to the design data, the friction coefficient \(\mu\) is provided by the manufacturer, and the vertical force \(F\) can also be easily obtained at each stage according to the structural calculation. Therefore, the RCS can be easily obtained by Equation (15) in this study. Engineers input the calculated RCS into the mechanical model as a boundary parameter in order to perform the mechanical calculation of the structure construction stage, and then the mechanical response of the whole structure can be obtained accurately and fast. This method avoids complicated modeling and calculation work, thus greatly reducing the calculation cost, contributing to the project by saving time and shortening the construction period and thus running smoothly.