## 3 The Time-Varying Global Reliability Analysis Based on the HM-IMEM
### The High-Order Moments-Based Improved Maximum Entropy Method
#### 3.2.4 The Structural Reliability Analysis  
Using Equation (7) and combing the first \(s\)-order raw moments \(M_{z,k}\) obtained from Equation (14) with the integral limits \(\varsigma^{\prime}\) and \(\varsigma\) calculated by Equation (16a), the objective function of the unconstrained optimization problem can be expressed as follows:  
\[L=-H_{s}+\lambda_{0}\left(\int_{\varsigma}^{\varsigma^{\prime}}f_{Z}(z)\mathrm{ d}z-1\right)+\sum_{k=1}^{n}\lambda_{k}\left[\int_{\varsigma}^{\varsigma^{\prime}}z^{k }f_{Z}(z)\mathrm{d}z-M_{Z,k}\right]. \tag{18}\]  
Similarly, the maximum entropy PDF\(f_{Z}(z)\) can be estimated by the Newton method as  
\[f_{Z}(z)=\exp\left(\lambda_{0}-\sum_{k=1}^{s}\lambda_{k}z^{k}\right). \tag{19}\]Then, based on the maximum entropy PDF of Equation (19) and the lower integral bound \(\varsigma\), the structural failure probability and reliability index can be calculated, namely  
\[\left\{\begin{array}{l}P_{f}=P_{f}(Z\leq 0)=\int_{\varsigma}^{0}\exp\left( \lambda_{0}-\sum\limits_{q=1}^{s}\lambda_{q}z^{q}\right)\mathrm{d}z\\ \beta{=}-\Phi^{-1}\left(P_{f}\right)\end{array}\right. \tag{20}\]  
in which \(\Phi^{-1}(\cdot)\) denotes the inverse function of the cumulative distribution function of standard normal variable \(\Phi\) (\(\cdot\)).  
Numerical Implementation of the HM-IMEM for Time-Varying Global Reliability of In-Service Transmission Towers  
By combining the established time-varying equivalent performance function of in-service transmission towers with the proposed HM-IMEM, the time-varying global reliability analysis of in-service transmission towers under action of random wind loads can be carried out in the following implementation steps (see also the flowchart in Figure 1):  
\[\begin{array}{l}\begin{array}{l}\begin{array}{l}\begin{array}{l}\begin{array}{l} \begin{array}{  
**Step1:** Discretization of the design reference period.  
The design reference period \(T=[t_{0},t_{m}]\) is equally divided into \(m+1\) intervals, each interval is \(\Delta t=(t_{m}-t_{0})/\text{m}\) in length, and the design reference period \(T\) can be discretized into \(\{t_{0},t_{1}=t_{0}+\Delta t,\,\ldots,\,t_{m}=t_{0}+m\cdot\Delta t\}\) intervals.  
**Step2:** The equivalent description of the time-varying performance function.  
According to Equation (5a), the equivalent performance function \(Z(t_{r})\) of the in-service transmission tower at \(\text{t}_{r}=\text{t}_{0}\) can be obtained.  
**Step3:** The statistical moments estimation for \(Z(t_{r})\).  
Based on Equation (14) and taking \(s=2\), the first s-order moments of the \(Z(t_{r})\) are calculated at \(t_{r}=t_{0}\).  
**Step4:** Calculation of integral limits.  
Based on Equations (15) and (16c), the integral limits \(\varsigma\) and \(\varsigma^{\prime}\) of the HM-IMEM can be obtained.  
**Step5:** Calculation of the maximum entropy PDF.  
With the first s-order moments, the maximum entropy PDF can be estimated by Equation (19).  
**Step6:** Convergence mechanism.  
If the Equation (17) is satisfied, the next step will be implemented; otherwise, go back to step (3) and make \(s=s+2\).  
**Step7:** Estimation of the failure probability and reliability index for the in-service transmission tower at \(t_{r}\).  
The failure probability and reliability index of the in-service transmission tower at \(t_{r}\) can be calculated according to Equation (20), but with a time parameter, namely  
\[\left\{\begin{array}{l}P_{f}(t_{r})=\int_{\zeta(t_{r})}^{0}\exp\left(\lambda _{0}(t_{r})-\sum\limits_{q=1}^{s(t_{r})}\lambda_{q}(t_{r})z^{q}(t_{r})\right) \text{d}z\\ \\ \beta(t_{r})=\ -\Phi^{-1}\Big{(}P_{f}(t_{r})\Big{)}\end{array}\right.. \tag{21}\]  
**Step8:** Calculation of the time-varying failure probability and reliability index of the in-service transmission tower.  
Repeating steps (2)-(7), and calculating the failure probability, \(\{P_{f}(t_{1}),P_{f}(t_{2}),\,\ldots\,\,P_{f}(t_{m})\}\), and the reliability index, \(\{\beta(t_{1}),\,\beta(t_{2}),\,\ldots\,\,\beta(t_{m})\}\) at \(r=1,\,2,\ldots,\,\text{m}\), respectively.