# An Adaptive Kriging-Based Fourth-Moment Reliability Analysis Method for Engineering Structures
## 3 Results
### Reliability Analysis under Smooth Time-Varying State Function for Rolling Bearing  
The efficiency advantage of the proposed AK-FM method over the AK-MCS method will be more pronounced when extended to the time-varying reliability problem, considering the cumulative cost of reliability computation at successive single time points.  
On the basis of the rolling bearing contact reliability calculation example, by considering the rolling bearing smooth time-varying fatigue accumulation damage [41], we constructed the 6016 bearing fatigue state function \(g_{f6016}\) as follows:  
\[g_{f6016}=1-\frac{\Delta D_{rb}(\mathbf{X})}{T}\times t \tag{25}\]  
By using the method proposed in this paper to calculate the reliability and setting the time interval at \(\Delta t=10\) h, the curve of reliability over time can be obtained as shown in Figure 6.  
We selected the time period of 1000-1100 h, where the reliability curve changes more drastically, and the reliability curve over time calculated using the method proposed in this paper is compared with the reliability curve calculated using the AK-MCS method, as shown in Figure 7.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{\begin{tabular}{c} **Initial Sample** \\ **Size** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Increased** \\ **Sample Size** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **U Function** \\ **Value** \\ \end{tabular} } & \multicolumn{2}{c}{\
\begin{tabular}{c} **Number of** \\ **State Function** \\ **Calls** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Number of State** \\ **Function Predictions** \\ \end{tabular} } & \multirow{2}{*}{**Reliability**} \\ \hline AK-FM & 20 & 4 & 4.4659 & 24 & 1000 & 0.7087 \\ AK-MCS & 20 & 4 & 4.4659 & 24 & \(10^{6}\) & 0.7086 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of contact reliability results for 6016 bearings.  
Figure 6: Smooth time-varying fatigue reliability curve of 6016 bearing.  
It can be seen that in the interval of 1000-1100 h, the results of the two methods are basically the same. The time points t = 1040 h, t = 1050 h, and t = 1060 h are further selected to compare the reliability results of the proposed method with the state function prediction of 1000 and 10,000 times and the AK-MCS reliability calculation with the state function prediction of \(10^{6}\) times, as shown in Table 4.  
From the results in Table 4, it can be seen that, on the one hand, the computational results of the method proposed in this paper have less error with the AK-MCS method at different time points, while the number of times the state function is predicted is smaller. On the other hand, it is obvious that the results of the proposed method for 10,000 state function predictions are more accurate relative to the proposed method for 1000 state function predictions, indicating that the accuracy of the proposed method increases with the increase in the number of state function predictions. It is demonstrated that the proposed AK-FM method can be used for reliability computation under a smooth time-varying state function, and that it can accurately and efficiently perform the reliability computation at a single time point with the appropriate number of state function prediction sets.