# A New SORM Method for Structural Reliability with Hybrid Uncertain Variables
## 4 Numerical Examples
### A Practical Application Example--Space Docking Mechanism  
In the space docking mechanism, the docking lock is a key component which realizes the rigid connection and separation of the spacecraft, as shown in Figure 3. With the interaction of the docking lock system on the two spacecrafts the connection force is generated/released between the two spacecrafts and the rigid connection or separation is realized. Each set of locks is composed of an active lock hook and a passive lock hook. The active lock hook is driven by a motor and matches with the passive lock hook corresponding to the target spacecraft. The passive lock hook is installed on the disc spring to match with the active lock hook corresponding to the target spacecraft [22]. The load \(F\) is defined as an interval variable that ranges \([30 KN,40 KN]\) the structural locking latch. The elastic modulus of the material E, Poisson's ratio \(\mu\), and the density \(\rho\) are random variables shown in Table 2. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{2}{c}{**Failure Probability Interval**} & \multicolumn{2}{c}{**Relative Error**} \\ \cline{2-5}  & **Lower Boundary** & **Upper Boundary** & **Lower Boundary** & **Upper Boundary** \\ \hline MC(10\({}^{7}\)) & 0.0591 & 0.4421 & & \\ Uncertainty FORM in [22] & 0.0537 & 0.4672 & 9.13\% & 5.68\% \\ UPSORM & 0.0634 & 0.4192 & 7.28\% & 5.18\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison result. Figure 3: Space docking mechanism. The reliability problem is formulated using the state function as:  
\[g=b-y(F,E,\mu,\rho)\]  
where \(b\) indicates the yield strength (1300 Mpa), and the contact stress \(y(F,E,\mu,\rho)\) on the connective surface of the two latches. Note that the four independent random variables are assumed and the max value of the contact stress \(y(F,E,\mu,\rho)\) is a function of the four variables \(F,E,\mu,\rho\), which cannot be an explicit expression. A simplified model for a finite element analysis is established to estimate the max contact stress \(y(F,E,\mu,\rho)\), as shown in Figure 4, which is the data of the red area in the left figure. In order to ensure the accuracy of the collected data, the test is used, and it is suggested that the damage position of the structure (the right picture) is the distribution area of the maximum contact stress (the left picture), which verifies the correctness of the collected data in this case  
In this case, the new method is employed to compute the failure probability of the latch, comparing the result to the computation made by the MC method with \(10^{7}\) samples to evaluate the accuracy of the calculation results. The results are shown in Table 3.  
Table 3 summarizes the estimation results for the structural latch computed by Monte Carlo (MC) simulation, the uncertainty FORM, and the proposed UPSORM. As shown in Table 3, the lower and upper bounds are computed for the failure probability: \([4.2203\times 10^{-6},4.4190\times 10^{-5}]\), which is measured by the proposed UPSORM after 154 iterations. It is matched to the results obtained by the MC simulation, showing that all the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Random Variables** & **Distribution Type** & **Mean** & **Standard Deviation** \\ \hline elastic modulus E (Pa) & normal & \(117\times 10^{9}\) & \(117\times 10^{7}\) \\ [PERSON]’s ratio \(\mu\) & normal & 0.3 & 0.003 \\ density \(\rho\) (kg/m\({}^{3}\)) & normal & \(4.81\times 10^{3}\) & \(0.481\times 10^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The list of random variables. Figure 4: The stress nephogram and test result for the structural latch. failure samples were correctly identified. Yet, the relative errors of 7.88% and 7.55% mean that the adoption of the second-order Taylor expansion improves the efficiency with a tradeoff in estimation accuracy. The lower and upper boundaries between the MC and UPSORM are different, as shown in Table 3, induced by the nonlinear limit-state function.