## Mechanical Properties of Ring-Friction Pendulum Isolation Bearings
### Vertical Seismic Isolation Performance of the Ring Spring-Friction Pendulum Bearing
When the vertical deformation of the loop spring is s, the axial force on the ring spring can be expressed as follows:  
\[F=\frac{2\pi EK_{\mathrm{c}}\tan\beta}{n_{0}(\frac{D\prime_{01}}{A_{1}}+\frac{D \prime_{02}}{A_{2}})}s \tag{2}\]  
The stiffness of the ring spring is calculated as follows [25]:  
\[K_{\mathrm{v}}=\frac{2\pi EK_{\mathrm{c}}\tan\beta}{n_{0}(\frac{D\prime_{01}}{ A_{1}}+\frac{D\prime_{02}}{A_{2}})} \tag{3}\]  
where \(\beta\) is the cone angle of the circle; \(K_{\mathrm{c}}\) is a calculation factor \((K_{\mathrm{c}}=\tan(\beta\pm\rho))\), \(\rho\) is the angle of friction, take \"+\" when the spring is under pressure and \"-\" when the spring is restored; \(D^{\prime}{}_{01}\) and \(D^{\prime}{}_{02}\) are the diameters of the central rings of the inner and outer rings, respectively; \(A_{1}\) and \(A_{2}\) are the cross-sectional areas of the inner and outer rings, respectively; and \(n_{0}\) is the number of groups of circles. In order to further verify the mechanical properties of the ring springs, we conducted numerical simulation studies of the ring springs in Abaqus software. We calculated the total self-weight of the structure and the overlying soil of each section of the Dakai station as 860,548 kg, and each ring spring needed to bear 5.6 \(\times\) 10\({}^{6}\) N according to the design of the rare earthquake with a seismic fortification intensity of 7 degrees. The sizes of the ring springs are shown in Figure 4a. The total height of the ring spring was 2340 mm, and the maximum diameter of the ring was 1500 mm. The maximum vertical compression of the ring spring was 480 mm in the normal working condition, and the interval between each ring was 2 mm. The vertical compression in the ultimate bearing capacity condition was 500 mm. The material of the ring spring was ISO6 Si8 spring steel, which is widely used in the mechanical engineering. Its density is 7850 kg/m\({}^{3}\), modulus of elasticity is 206 GPa, Poisson's ratio is 0.33 The elastic-plastic model of Abaqus was used for the ring spring, with yield stress 1375 MPa and the mesh cell of type C3D8I. The normal contact between the rings was hard contact, and separation was allowed after contact. According to the experimental results of Professor [PERSON] of Tongji University, \"sandblasting without lubricant treatment\" should be performed between the inner and outer rings. The toroidal spring is more beneficial to structural building seismic isolation, so a penalty function was chosen tangentially between the contact surfaces of the toroidal spring. The friction coefficient was entered as 0.1 [26]. A fixed restraint was set at the bottom of the ring spring, a displacement load was applied to the upper surface, and the rotation of all circles around the central axis was restricted; the three-dimensional entity diagram is shown in Figure 4b. In order to accurately observe the variation of stress in the ring along the thickness direction, the mesh of the ring needs to be detailed. We divided the mesh into four columns in the thickness direction of the ring, as shown in Figure 5. After the model was built, we loaded it with vertical displacements, starting from 0 to 480 mm and 500 mm and then unloaded it. Finally, we output the load-displacement curves for both conditions. As seen in Figure 6, there was little discrepancy between the formula calculation and finite element simulation in the normal working condition (480 mm), which can confirm the correctness of the formula. The loading stiffness of the ring spring in the normal working condition was 13.2 kN/mm. In the ultimate limit condition (500 mm), the load and displacement were correlated linearly during the initial loading, and the finite element simulation results and the formula calculation results were basically consistent. After loading to 480 mm, the spring stiffness rose slowly and then increased sharply when the load approached 500 mm. The final bearing capacity was 8013.7 kN.