## 3 Model Uncertainty Parameters
### The Uncertainty Factor of the Vibration Isolation System  
Regarding the yield strength of the fixed supports, after controlling for failure modes, the source of uncertainty is directly related to the material strength of the steel fuse protection component. Following [PERSON] [7], it is assumed that the yield strength values follow a log-normal distribution with a coefficient of variation of 0.08. In FEM, this is achieved by adjusting the limit values of the MinMax material properties. For the friction coefficient of the supports, according to [PERSON] [18], the recommended values for the design friction coefficient of polytetrafluoroethylene (PTFE) materials, lubricated with silicone grease under different compressive stresses and temperatures, range from 0.005 to 0.027. Given the lack of extensive sample data, but with a clear understanding of the upper and lower bounds, it can be assumed that the sample follows a uniform distribution within the limits of 0.005 and 0.027. This is achieved in FEM by altering the yield force of the bilinear model. Concerning the yield displacement of the supports, identified as the breaking \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Classification** & **No.** & **Name** & **Sign** \\ \hline \multirow{8}{*}{Seismic isolation system} & 1 & Limiting force & \(F_{limit}\) \\  & 2 & Bearing friction coefficient & \(\mu_{bearing}\) \\  & 3 & Yielding displacement of bearing & \(d_{baring}\) \\  & 4 & Bearing hardening coefficient & \(\alpha_{baring}\) \\  & 5 & Yielding force of steel damping & \(F_{dampper}\) \\  & 6 & Yielding displacement of & \(d_{dampper}\) \\  & & steel damping & \\  & 7 & Hardening coefficient of & \(\rho_{dampper}\) \\  & & steel damping & \(F_{dampper}\) \\  & 8 & Collision stiffness of left unite & \(K_{d\#1}\) \\  & 9 & Collision stiffness of right unite & \(K_{d\#2}\) \\  & 9 & Slot gap of the end of beam & \(\Delta\) \\ \hline \multirow{8}{*}{Non-isolation system} & 10 & Damping ratio & \(\zeta\) \\  & 11 & Volumetric weight coefficient & \(\lambda_{up}\) \\  & 12 & Pier diameter & \(D\) \\ \cline{1-1}  & 13 & Sand resistance & \(P_{ult\_sun}\) \\ \cline{1-1}  & 14 & Clay resistance & \(P_{ult\_clay}\) \\ \cline{1-1}  & 15 & Clay deformation & \(y_{50\_day}\) \\ \cline{1-1}  & 16 & Elastic modulus & \(E\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Uncertain parameters of bridge structure model. displacement of the fuse pin, previous experiments conducted by the research team [19] indicated an equivalent yield displacement of 0.4 mm, reflecting the displacement when the steel fuse protection component experiences shear failure. Considering that assembly gaps of 1 to 2 mm are generally present during the installation of support components, it is concluded that the equivalent yield displacement of the supports, taking assembly uncertainty into account, follows a uniform distribution between 1.4 mm and 2.4 mm. In FEM, this is implemented by changing the yield displacement parameters of the bilinear model. Regarding the ratio of post-yield stiffness to elastic stiffness, referred to as the hardening coefficient \(\alpha\), theoretically, the dynamic frictional force generated by sliding behavior on a planar friction pair should remain approximately constant, resulting in a hardening coefficient close to zero. This behavior is likely influenced by the dynamic friction coefficient, which was previously observed to be 0.0015. The experiments utilized sinusoidal wave loading, which at lower amplitudes resulted in greater relative sliding velocities, facilitating the formation of a lubricant film [20]. During larger amplitude loading, the relative sliding velocities diminished, thereby increasing the dynamic friction coefficient, leading to a certain degree of post-yield stiffness in the experimental results. It is assumed that the hardening coefficient of the supports follows a uniform distribution within the range of 0 to 0.0015. In FEM, this is accomplished by modifying the hardening parameters of the bilinear material model.