## 3 First-Passage Reliability Problem
### Mathematical Basis
\begin{split}\lim_{\text{d}x\to 0}\
u^{+}(x)\text{d}x& =\Pr\left\{\Omega(x)>Y(x)\bigcap\Omega(x+\text{d}x)<Y(x+\text{d}x )\right\}\\ &=\Pr\left\{\Omega(x+\text{d}x)-\dot{Y}(x)\text{d}x<\dot{Y}(x)< \Omega(x)\right\}\\ &=\int_{\dot{\Omega}(x)}^{\infty}\left[\dot{Y}(x)-\dot{\Omega}(x )\right]f_{Y\dot{Y}}\left[\Omega(x),\dot{Y}(x)\right]\text{d}\dot{Y}(x)\text{d }x\end{split} , \tag{6}\]  
where \(\dot{Y}(x)\) and \(\dot{\Omega}(x)\) are the differentials of \(Y(x)\) and \(\Omega(x)\) respectively. Rearranging Equation (6) yields  
\[\
u^{+}(x)=\int_{\Omega(x)}^{\infty}\left(\dot{Y}-\dot{\Omega}\right)f_{Y\dot{ Y}}\left(\Omega,\dot{Y}\right)\text{d}\dot{Y}. \tag{7}\]Since \(Y(x)\) is a stationary Gaussian process, \(Y(x)\) and \(\dot{Y}(x)\) are mutually independent. Thus, it follows,  
\[f_{Y\dot{Y}}(x,\dot{x})=\frac{1}{2\pi\sigma_{Y}\sigma_{\dot{Y}}}\exp\left\{- \frac{1}{2}\left(\frac{x^{2}}{\sigma_{Y}^{2}}+\frac{\dot{x}^{2}}{\sigma_{\dot{Y }}^{2}}\right)\right\}, \tag{8}\]  
where \(\sigma_{Y}\) and \(\sigma_{\dot{Y}}\) are the standard deviations of \(Y(x)\) and \(\dot{Y}(x)\), respectively. Combining Equations (7) and (8) gives  
\[\
u^{+}(x)=\frac{1}{2\pi\sigma_{Y}}\exp\left[-\frac{\Omega^{2}(x)}{2\sigma_{Y }^{2}}\right]\cdot\left\{\sigma_{\dot{Y}}\exp\left(-\frac{\dot{\Omega}^{2}(x) }{2\sigma_{\dot{Y}}^{2}}\right)-\sqrt{2\pi}\Omega(x)\left[1-\Phi\left(\frac{ \dot{\Omega}(x)}{\sigma_{\dot{Y}}}\right)\right]\right\}, \tag{9}\]  
where \(\Phi(\,)\) is the cumulative density function (CDF) of a standard normal distribution. Assume that the upcrossings of \(Y(x)\) to \(\Omega(x)\) are temporally independent and are rare, with which the Poisson process can be used to describe the occurrence of the upcrossings. Let \(N_{X}\) be the number of upcrossings within the length \([0,X]\), and it follows,  
\[\Pr(N_{X}=i)=\frac{1}{i!}\left\{\int_{0}^{X}\
u^{+}(x)\mathrm{d}x\right\}^{i} \exp\left\{-\int_{0}^{X}\
u^{+}(x)\mathrm{d}x\right\},\quad i=0,1,2,\ldots. \tag{10}\]  
By noting the fact that the structural deformation reliability within \([0,X]\) equals the probability of \(N_{X}=0\), one has  
\[\mathbb{L}(X)=[1-\mathbb{P}(0)]\exp\left\{-\int_{0}^{X}\
u^{+}(x)\mathrm{d}x \right\}, \tag{11}\]  
where \(\mathbb{P}(0)\) is the failure probability (the probability of deformation exceeding the limit) at \(x=0\). Specifically, as \(\mathbb{P}(0)\) is typically small enough, it follows that ([31; 32; 33])  
\[\mathbb{L}(X)=\exp\left\{-\int_{0}^{X}\
u^{+}(x)\mathrm{d}x\right\}. \tag{12}\]  
A similar form of Equation (12) (with the terms \(X\) and \(x\) denoting time) has been extensively used for the estimate of structural time-dependent reliability [31; 32; 44; 45; 46]. In this paper, Equation (12) is adopted with modifications to assess the deformation reliability of retaining structures. Based on Equation (12), combining the concept of \"point control\", the deformation reliability, \(\mathbb{L}_{\text{nme}}(X)\), is formulated as follows,  
\[\mathbb{L}_{\text{nme}}(X)=\exp\left\{-\int_{0}^{X}\
u^{+}(x)\mathrm{d}x \right\}\cdot\prod_{i=1}^{n}\mathbb{I}(y_{i}\geq y_{\text{lim}}). \tag{13}\]  
By comparing Equations (3) and (13), it can be seen that the proposed method in Equation (13) has moved the deformation limit state from \"point control\" to \"surface control\". Note that the subscript \"nme\" in Equation (13) means that \(\mathbb{L}_{\text{nme}}(X)\) has not (n) considered the measurement (m) error (e). The impact of ME on the deformation reliability is addressed later, in Section 4.3.