# An Adaptive Kriging-Based Fourth-Moment Reliability Analysis Method for Engineering Structures
## 3 Results
### Reliability Analysis of Implicit State Function for Rolling Bearing  
Rolling bearing is a complex assembly composed of rollers, raceways, and cages whose working conditions and dimensional parameters affect the contact load between rollers and raceways, and thus, the performance of rolling bearings [40]. Existing mechanical modeling methods offer improved accuracy in calculating the contact load between the roller and raceway compared to previous methods, but they still fall short of the precision achieved by the finite element method. This study proposes establishing a finite element model based on input working conditions and size parameters to accurately calculate the contact load between the roller and raceway. The determination of the bearing qualification is based on whether the contact load exceeds a critical value, presenting a typical complex implicit equation of state problem. This paper takes the deep groove ball bearing 6016 as an example, and its working conditions are as follows: outer diameter, \(D=125\) mm; inner diameter, \(d=80\) mm; number of balls, \(Z=20\); rotating speed, \(n=1000\) r/min; radial force, \(F_{r}=500\) N; material GCr15; elastic modulus, \(E=208\) GPa; Poisson's ratio, \(v=0.3\); and hardness, \(HB=200\). We selected the design parameters and constructed the input parameter vector \(\mathbf{X}=[D_{m},D_{w},R_{i},R_{o}]^{T}\). Ten 6016 bearings with the same precision produced by the same manufacturer were selected and cut into 10 groups of specimens by EDM wire cutting, and the dimensional parameter data of the specimens were obtained by measuring with the CMM, as shown in Figure 4, respectively. According to the calculation results of these 10 groups of data, the moments of each size parameter in \(\mathbf{X}\) were obtained. The remaining parameters were obtained according to the method described in reference [41] and are shown in Table 2.  
Based on the data presented in Table 2, a total of 100 sets of input parameter vector samples were randomly generated. From these, 20 sets were selected as the training sample set and analyzed using a finite element model for stress simulation. The interactions between the inner ring and the roller, as well as between the roller and the outer ring, were considered in this model, with all contact surfaces assumed to be smooth. All components were made of GCr15 material. The outer ring was constrained and completely fixed, while a radial concentrated force was applied to the inner surface of the inner ring coupled to the center point. The maximum contact stress \(\sigma_{max}(\mathbf{X})\) under the current operating conditions was then determined, as shown in Figure 5. An initial Kriging model was established between the training sample set and \(\sigma_{max}(\mathbf{X})\), which was subsequently updated and optimized using the proposed method. Figure 3: Reliability for different U function thresholds. Assuming the maximum threshold value of contact stress \(\hat{\sigma}\) based on material properties, the contact state function of rolling bearing 6016 can be defined as follows:  
\[g_{6016}=\hat{\sigma}-\sigma_{max}(\mathbf{X}) \tag{24}\]  
Obviously, when \(g_{6016}\geq 0\), it is in a safe state, and when \(g_{6016}<0\), it is in a failed state. For the above 6016 bearing contact state function, the reliability results calculated using the method proposed in this paper are compared with the results obtained using the AK-MCS method, as shown in Table 3. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Bearing Parameters** & **Mean** & **Variance** \\ \hline Ball diameter \(D_{w}\) (mm) & 13.4940 & 4.0 \(\times\) 10\({}^{-6}\) \\ Centre circle diameter \(D_{m}\) (mm) & 102.4874 & 2.5 \(\times\) 10\({}^{-5}\) \\ Inner ring groove radius \(R_{i}\) (mm) & 7.1010 & 1.0 \(\times\) 10\({}^{-6}\) \\ Outer ring groove radius \(R_{o}\) (mm) & 6.9960 & 1.0 \(\times\) 10\({}^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Random parameters of 6016 bearing input. Figure 4: CMM measurement of 6016 bearing specimen.