# An Efficient Time-Variant Reliability Analysis Method with Mixed Uncertainties
## 3 Analysis Method of Structural Time-Variant Reliability Model with Random and Interval Variables
### Formulation of Stochastic Process Discretization with Mixed Variables  
According to the definition of time-variant reliability [13; 32; 38], the design reference period \(T\) can be divided into \(m\) uniformly equal periods. Consequently, each period can be \(\tau=T/m\). The resistance stochastic process \(R(t)\) and dynamic load \(Q(t)\) can be discretized to obtain \(m\) random variables. It should be noted that the median value of the resistance in the \(i\)-\(th\) period equals the \(R_{i}\) value. Figure 1 illustrates distributions of resistance \(R(t)\) and variable load \(Q(t)\) over time. The instantaneous value of the dynamic load \(Q_{i}\) can be obtained statistically in the period. Generally, \(Q_{i}\) can be regarded as an independent and identically distributed function. Based on the reliability theory of the series system, Equation (11) can be rewritten in the form below:  
\[P_{f}(T)=1-P\left\{\underset{i=1}{\overset{m}{\cap}}[g(R_{i}(t,\mathbf{Y}),Q _{i}(t,\mathbf{Y}),\mathbf{A})\geq 0]\right\} \tag{12}\]  
From the engineering point of view, \(Q^{\prime}\) is \(Q_{i}\)'s s the maximum value. It should be indicated that \(Q_{i}\) is independently and evenly distributed. Based on the principle of mechanism statistics, the probability distribution function \(F_{Q_{T}}(x)\)of the maximum load effect \(Q_{T}\) in the reference period \(T\) can be formulated as:  
\[F_{Q_{T}}(x)=\left[F_{Q_{T}}(x)\right]^{m} \tag{13}\]  
In practical problems, the loading effect \(Q_{i}\) is usually approximated to the extreme value type I distribution with parameters \(\alpha_{\tau}\) and \(u_{\tau}\), subjected to the extreme value type I distribution with parameters \(\alpha_{T}\) and \(u_{T}\). This can be mathematically expressed in the form below:  
Figure 1: Distribution of resistance \(R(t)\) and variable load \(Q(t)\) over time. where \(\phi\) is the standard normal distribution function and \(\mathbf{U}\) is the standard normal space variable. By substituting Equation (20) into Equation (19), the limit state function in standard normal space can be obtained:  
\[G(\mathbf{U},\mathbf{Y})=g^{\prime}\left(F_{\mathbf{X}}^{-1}(\phi(\mathbf{U}), \mathbf{Y},t_{i})\right) \tag{21}\]  
As the coexistence of random and interval variables, the limit state function (21) turns into a band in the parameter space as shown in Figure 2. The lower and upper boundaries are as follows:  
\[ \begin{split} S_{L}:\min_{\mathbf{Y}}g(\mathbf{U},\mathbf{Y},t_ {i})\\ S_{U}:\max_{\mathbf{Y}}g(\mathbf{U},\mathbf{Y},t_{i})\end{split}  \tag{22}\]  
where \(S_{L}\) and \(S_{U}\) represent the lower and upper boundaries, respectively. The limit state region in Figure 2 shows failure probability \(P_{f}\) and reliability index \(\beta\) obtained from different \(\mathbf{Y}\)-values of the limit state function on \(S_{L}\) and \(S_{U}\) planes. It should be indicated that both of these parameters are range values, in the form below:  
\[\beta\in\left[\beta^{L},\beta^{U}\right] \tag{23}\]  
\[P_{f}\in\left[P^{L},P_{f}^{U}\right]=\left[\phi\left(-\beta^{U}\right),\phi \left(-\beta^{L}\right)\right] \tag{24}\]  
where \(\beta^{U}\), \(\beta^{L}\), \(P_{f}^{U}\) and \(P_{f}^{L}\) represent the upper and lower interfaces of the reliability index and the upper and lower boundaries of failure probability, respectively. Generally, the following two optimization problems are set to calculate the values of \(\beta^{L}\) and -\(\beta^{U}\):  
\[ \begin{split}\beta^{L}=\min_{\mathbf{U}}\lVert\mathbf{U}\rVert\\ \text{s.t.}\min_{\mathbf{Y}}G(\mathbf{U},\mathbf{Y},t_{i})=0\\ \beta^{U}=\min_{\mathbf{U}}\lVert\mathbf{U}\rVert\\ \text{s.t.}\max_{\mathbf{Y}}G(\mathbf{U},\mathbf{Y},t_{i})=0 \end{split}  \tag{25}\]  
Figure 2: Limit state region. It should be reminded that time-variant failure probability falls in an interval. In most practical engineering problems, only the maximum failure probability is concerned. Consequently, only Equation (25) needs to be solved and the governing equations are reduced to the following expression:  
\[