# An Efficient Time-Variant Reliability Analysis Method with Mixed Uncertainties
## 4 A New Analysis Method of Structural Time-Variant Reliability with Random and Interval Variables
### Model Equivalence Proof  
According to the studies of [PERSON] [39] on the asymptotic approximation of polynomial integrals, [PERSON] et al. [40] on first-order and second-order reliability analysis of series structures, and [PERSON] [41] on structural safety methods, it is seen that MPP has the highest probability density at all points on the limit state function. Therefore, (\(\mathbf{X}^{\prime}\), \(\mathbf{Y}^{\prime}\)) is also the optimal solution to the following problem:  
\[\left\{ \begin{array}{l}\underset{\mathbf{X},\mathbf{Y}}{\text{max}}\ f_{ \mathbf{X},\mathbf{Y}}(\mathbf{X},\mathbf{Y})\\ \text{s.t.}\ g(\mathbf{X},\mathbf{Y},t_{i})=0\end{array} \right. \tag{30}\]  
where \(f_{\mathbf{X},\mathbf{Y}}\) denotes the probability density function (PDF) of random variables \(\mathbf{X}\) and \(\mathbf{Y}\). Since all variables in the limit state function are independent of each other, Equation (30) can be modified as the following equation, based on the processing approach of the series system:\[\left\{ \begin{array}{l}\max\limits_{\mathbf{X}\mathbf{Y}}f_{\mathbf{X}}(\mathbf{X })f_{\mathbf{Y}}(\mathbf{Y})\\ \text{s.t.}\ g(\mathbf{X},\mathbf{Y},t_{i})=0\end{array} \right. \tag{31}\]  
Since the random variable \(\mathbf{Y}\) is uniformly distributed, the probability density function \(f_{\mathbf{Y}}(\mathbf{Y})\) of the above-mentioned equation is a normal number. Therefore, Equation (31) can be rewritten into the following equation:  
\[\left\{ \begin{array}{l}\max\limits_{\mathbf{X}\mathbf{Y}}f_{\mathbf{X}}( \mathbf{X})\\ \text{s.t.}\ g(\mathbf{X},\mathbf{Y},t_{i})=0\\ Y_{j}^{L}\leq Y_{j}\leq Y_{j}^{R},j=1,2,\ldots,n\end{array} \right. \tag{32}\]  
For the constrained optimization problem of Equation (32), [PERSON] and [PERSON] [42] proposed a numerical optimization approach according to the Karush-Kuhn-Tucker necessary conditions. Therefore, the optimal (\(\mathbf{X}^{{}^{\prime}}\), \(\mathbf{Y}^{{}^{\prime}}\)) must meet the following equation:  
\[\left\{ \begin{array}{l}-\
abla_{\mathbf{X}_{k}}f_{\mathbf{X}}(\mathbf{X})+ \lambda_{1}\
abla_{\mathbf{X}_{k}}g(\mathbf{X},\mathbf{Y},t_{i})=0,k=1,2, \ldots,n\\ \lambda_{1}\
abla_{\mathbf{Y}_{i}}g(\mathbf{X},\mathbf{Y},t_{i})+\lambda_{3j }-\lambda_{2j}=0,j=1,2,\ldots,n\\ \lambda_{2j}\left(Y_{j}^{L}-Y_{j}\right)=0,j=1,2,\ldots,n\\ \lambda_{3j}\left(Y_{j}-Y_{j}^{R}\right)=0,j=1,2,\ldots,n\\ g(\mathbf{X},\mathbf{Y},t_{i})=0\\ Y_{j}^{L}\leq Y_{j}\leq Y_{j}^{R},j=1,2,\ldots,n\end{array} \right. \tag{33}\]  
where, \(\lambda_{1}\), \(\lambda_{2i}\), \(\lambda_{3j}\) represent Lagrange multipliers. Secondly, the reliability of the original mixed time-invariant model is investigated. As described above, in order to compute the maximum failure probability through the first-order second-moment approach, the MPP \(\textbf{U}\) and the corresponding optimal value \(\mathbf{X}^{{}^{\prime}}\) should be obtained. Moreover, the following optimization issues should be resolved:  
\[\left\{ \begin{array}{l}\beta=\min\limits_{\mathbf{U}}\lVert\mathbf{U}\rVert \\ \text{s.t.}\ \underset{\mathbf{Y}\in\mathbf{Y}^{{}^{\prime}}}{\text{min}}g( \mathbf{U},\mathbf{Y},t_{i})=0\end{array} \right. \tag{34}\]  
Therefore, according to the properties of MPP, \(\mathbf{X}^{{}^{\prime}}\) is the optimal solution of the following problem:  
\[\left\{ \begin{array}{l}\max\limits_{\mathbf{X}\mathbf{Y}}f_{\mathbf{X}}( \mathbf{X})\\ \text{s.t.}\ \underset{\mathbf{Y}\in\mathbf{Y}^{{}^{\prime}}}{\text{min}}g( \mathbf{X},\mathbf{Y},t_{i})=0\end{array} \right. \tag{35}\]  
where \(\mathbf{Y}\) is the interval variable.