## 3 Model Uncertainty Parameters
### The Uncertainty Factor of the Non-Isolation Device System  
In this study, the Boulanger model is employed to simulate the soil confinement mechanism, where the ultimate resistance and ultimate deformation of the soil are two critical parameters. The distribution characteristics of the uncertainty in the ultimate resistance of sandy soil are referenced from [PERSON] et al. [8], assuming that the ultimate resistance of sandy soil follows a normal distribution with a coefficient of variation of 0.1. The ultimate deformation of sandy soil is expected to exhibit a corresponding distribution characteristic based on the resistance distribution. In contrast, the situation with clay is different: the ultimate deformation of clay is typically not explicitly related to its ultimate resistance, and its parameter values are highly empirical. Building upon the assumption in [PERSON] et al. [8] that the ultimate resistance of clay follows a normal distribution with a coefficient of variation of 0.3, this study similarly assumes that its ultimate deformation adheres to the same statistical distribution. This is implemented in FEM by altering the maximum force parameters of the soil spring.  
Regarding the geometric dimensions of the piers and piles, various studies present differing descriptions. [PERSON] et al. [22] suggest that the diameter parameter follows a normal distribution with a coefficient of variation of 10%. Similarly, [PERSON] et al. [23,24] argue for a normal distribution of the diameter parameter but with a coefficient of variation of 5%. Conversely, [PERSON] [25] claims that this coefficient of variation is only 7%. As construction techniques improve and mature, the quality inspection and assessment requirements for bridge pier components are becoming increasingly stringent. Referring to the latest standards published by the Ministry of Transport of the People's Republic of China, specifically the \"Highway Engineering Quality Inspection and Assessment Standards\" [17], for cast-in-situ construction, the cross-sectional dimension deviations should be controlled within \(\pm\)20 mm; for prefabricated and assembled cases, the requirements are even more stringent, with dimensional deviations limited to \(\pm\)15 mm.  
For example, if a bridge pier with a diameter of 3.2 m must meet the manufacturing precision requirement of \(\pm\)20 mm under the assumption of normal distribution, its coefficient of variation must be controlled to within 4%. Given that the regulatory standards for the control of geometric dimensions are absolute and do not vary with the target dimensions of components, this study assumes that the diameter of the pile--following the maximum allowable deviation as specified in the inspection and assessment standards--follows a uniform distribution within this range. The distribution characteristics of the parameters such as the elastic modulus of concrete, unit weight, and damping ratio are referenced from previous studies [2,7], with the details provided in Table 2.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Classification** & **No.** & **Random Variable** & **Parameter** & **Lower Bound (5\%)** & **Median (50\%)** & **Upper Bound (95\%)** \\ \hline \multirow{8}{*}{Seismic isolation system} & 1 & \(F_{\text{mixed}}\) (\(\Delta N\)) & \(LN\) (\(\mu=1250\), \(\text{cov}=8\%\)) & 1093 & 1246 & 1421 \\  & 2 & \(F_{\text{Dining}}\) & \(U\) (0.004, 0.0