# A Novel Reliability Analysis Approach under Multiple Failure Modes Using an Adaptive MGRP Model
## 4 Case Studies
### Motor Hanger Reliability Analysis under Multiple Failure Models  
As a key component of the railway vehicle, the function of the motor hanger is to suspend the motor and bear various excitations during train operation. Any form of failure will directly affect the service safety of the train. Therefore, a typical motor hanger is studied to verify the engineering application of the proposed method. Due to the physical model of the motor hanger being difficult to obtain, its failure models can only be simulated by the finite element (FE) simulation method. Then, the FE model of the motor hanger is built according to the characteristics and load cases, as shown in Figure 6. This FE model consists of beam188, shell181, solid185, rigid and rhe3 elements. It is meshed to have 50,879 elements, including 21,766 triangular and quadrilateral shell elements, and 28,973 tetrahedral and hexahedral solid elements. The ANSYS software is employed to perform the structure strength analysis; the stress and displacement nephogram are plotted in Figure 7.  
As can be seen from Figure 7, the maximum stress and displacement of the motor hanger are 316.85 MPa and 3.52 mm, respectively, which meet the standard requirements of the maximum allowable stress of 345 MPa and the maximum displacement of 4 mm. However, there are many uncertain factors in the manufacture and operation of the motor hanger, which is easy to make its stress and displacement exceed the allowable range and failure occurs. Therefore, the reliability analysis of the motor hanger is carried out with the  
Figure 6: Structure characteristic and FE model of the motor hanger. (**a**) Geometric model of the motor hanger; (**b**) FE model of the motor hanger. Figure 7: Static strength analysis of motor hanger. (**a**) Stress nephogram; (**b**) Displacement nephogram. help of Matlab and ANSYS collaborative computing platforms, considering the stress and displacement failure modes. Their failure probability limit state equation can be provided as follows:  
\[P_{f}=\Pr\biggl{(}\min_{x=1,2}G_{\chi}(x)\leq 0\biggr{)}=\Pr\{\min\left( \begin{array}{c}345-y_{SEQV}(x)\leq 0\\ 4-y_{USIM}(x)\leq 0\end{array} \right)\biggr{\}} \tag{32}\]  
where \(x\) is the random variables, \(y_{SEQV}(\bullet)\) is the maximum Mises stress, and \(y_{USIM}(\bullet)\) is the maximum displacement. First, two plate thicknesses and a stochastic load, which have a greater impact on the two failure modes, are selected as random variables. Second, the initial samples are obtained by the RMQGS method, and the ANSYS software is repeatedly called to determine the true response value of the performance function. Finally, the MRGP model is continuously updated iteratively by the proposed method until the reliability accuracy meets the requirements. The reliability analysis results are shown in Figure 8 and Table 3.  
Figure 8 depicts the CDF curves of the motor hanger with limit state function and updated samples. The reliability of the motor hanger under the two failure models of stress and displacement is 0.9969, i.e., the failure probability is 0.0031. This shows that the motor hanger can maintain high structural performance under the fluctuation of uncertainty factors. It can be seen from Figure 8b that the updated sample sizes of the four methods are 9, 21, 13, and 10, respectively, and the reliability accuracy is the same, indicating that the proposed method can achieve high-precision reliability analysis with the least number of iterations. Table 3 provides the comparative quantification results of the four methods. The accuracy of the proposed method is the same as that of the MRGP + U method, but the efficiency is improved by 18.18%. Moreover, the MRGP + EFF method is slightly lower than the proposed method in both accuracy and efficiency, while the MRGP + H method has a larger difference. In a word, the proposed method has good engineering applicability in dealing with the reliability of the motor hanger with multiple failure modes.