# A Novel Reliability Analysis Approach under Multiple Failure Modes Using an Adaptive MGRP Model
## 3 Reliability Analysis Approach under Multiple Failure Modes
### MRGP-SS Based Structure Reliability Analysis  
For engineering structures with multiple failure modes, the relationship of each failure mode is mainly in the forms of series, parallel, and series-parallel, etc. Then, their failure probability expression can be provided as follows:  
\[P_{f}=\Pr(G_{\chi}(\mathbf{x}),\chi=1,2,\cdots,m)=\left\{\begin{array}{ll}\Pr \biggl{(}\min_{\chi=1,2,\cdots,m}G_{\chi}(x)\leq 0\biggr{)},&\text{series}\\ \Pr\biggl{(}\max_{\chi=1,2,\cdots,m}G_{\chi}(x)\leq 0\biggr{)},&\text{ parallel}\\ \Pr\biggl{(}\min_{\chi=1,2,\cdots,m}G_{\chi}(x)\leq 0\biggr{)},&\text{series-parallel} \end{array}\right. \tag{18}\]  
where \(G_{\chi}(\bullet)\) is the \(\chi\)th limit state equation, which is built using adaptive MRGP, and m is the number of the limit state equation.  
The traditional failure probability is mostly based on MC simulation to ensure the accuracy of the results, which is a time-consuming computer-aided program. However, the reliability solution under multiple failure modes belongs to the problem of small failure probability; its efficiency and accuracy have a great impact on the applicability of engineering applications. For these purposes, a novel reliability approach combined with the adaptive MRGP model and SS method is proposed. The principle of SS is to convert the problem of small failure probability into the product of larger conditional probability, if the intermediate events of each sequence satisfy \(f_{1}\supset f_{2}\supset\cdots\supset f_{n}=f\), the target failure probability can be expressed as follows:  
\[P_{F}=P(F_{1})\prod_{a=2}^{m}P(F_{a}|F_{a-1}) \tag{19}\]  
where \(F=\left\{\,G_{\chi}(x)\leq b\right\}\,\) is the target failure event, b is the structural response threshold, \(F_{1}\) is the first failure event, \(F_{a}\) is the middle failure event, and \(\,P(F_{a}|F_{a-1})\,\) is conduction probability.  
In Equation (19), the initial value of conditional probability is usually selected from an interval of \([0.1,0.3]\). Meanwhile, the value \(P(F_{1})\) is determined as follows:  
\[P(F_{1})\approx\widetilde{P}_{1}=\frac{1}{N}\left[\sum_{b=1}^{N}I_{F_{1}}(\mathbf{ x}_{b})\right] \tag{20}\]  
\[I_{F_{1}}=\left\{\begin{array}{ll}1,&\mathbf{x}_{b}\in F_{1}\\ 0,&\mathbf{x}_{b}\
otin F\end{array}\right. \tag{21}\]  
where \(\left\{x_{b}|b=1,2,\ldots,N\right\}\) is a set of sample points of independent and identically distributed, and \(I_{F_{1}}\) is the indicative function.  
Moreover, the Markov chain Monte Carlo simulation is conducted based on an improved Metropolis-Hastings algorithm to produce the condition samples, and the estimated value of conditional probability in a level can be expressed as follows:  
\[P(F_{a}|F_{a-1})\approx\widetilde{p}_{a}=\frac{1}{N}\left[\sum_{b=1}^{N}I_{F_{ b+1}}(\mathbf{x}_{b})\right] \tag{22}\]  
where \(\widetilde{p}_{a}\) is the predicted failure probability in a level.  
The estimated failure probability of the target failure event is given as follows:  
\[P_{F}\approx p_{0}(a-1)\frac{N_{F}(a)}{N} \tag{23}\]  
where \(N_{F}(a)\) is the number of sample points, which fall into the failure region in a level.