## 4 Non-Probabilistic Time-Dependent Reliability Assessment  
For the structural safety, the comparison between the actual performance \(u_{i}(t)\) and the criteria value \(u_{i}^{cr}(t)\) over the analyzed time \([0,T]\) are usually of more concern. Nevertheless, once the uncertainty factors originating from insufficient samples are taken into account, traditional deterministic analysis or the time-invariant probabilistic reliability theory may not be feasible. In view of this, we further define the non-probabilistic time-dependent reliability index as  
\[R_{s}(T)=1-P_{f}(T)=Pos\bigg{\{}\forall t\in[0,T]:\underset{i\leq r}{min}[g_{ i}(t,\mathbf{u}(t))=u_{i}^{cr}(t)-u_{i}(t)]>0\bigg{\}} \tag{46}\]  
where \(R_{s}(T)\) and \(P_{f}(T)\) are the measurements of safety and failure, respectively, \(Pos\{\cdot\}\) means the possibility of a specific event, \(\underset{i\leq r}{min}[\cdot]\) represents the minimum, and \(g_{i}(t,\mathbf{u}(t))\) stands for the time-varying limit state.  
Figure 1: The novel optimization-based strategy for determination of the smallest rotary rectangular domain.  
It is extremely hard to obtain the analytical solution of Equation (46) in general. To solve this problem, some approximation methods should be used. Based on the classical first-passage theory in random process, Equation (46) can be translated by  
\[R_{s}(T)=1-P_{f}(T)=1-\underset{i\leq r}{max}\big{[}Pos\big{\{}(g_{i}(0,\mathbf{u}(0 ))<0)\cup\big{(}N_{i}^{+}(0,T)>0\big{)}\big{\}}\big{]} \tag{47}\]  
where \(\underset{i\leq r}{max}[\cdot]\) represents the maximum, \(g_{i}(0,\mathbf{u}(0))\) means the initial state, and \(N_{i}^{+}(0,T)>0\) denotes the number of upcrossings of zero-value by the processes \(g_{i}(t,\mathbf{u}(t))\) from the safe domain to the failure domain during \([0,T]\).  
The basic simplification in the first-passage theory is that the crossing from the non-failure into the failure domain at each moment is regarded as independent from others. Under such circumstances, \(R_{s}(T)\) arrives at  
\[R_{s}(T)=1-P_{f}(T)=1-\underset{i\leq r}{max}\left[Pos_{i}(0)-\sum_{j=1}^{N} PI(E_{i,j})\right],N=\frac{T}{\Delta t} \tag{48}\]  
where \(Pos_{i}(0)\) means the possibility of failure at the initial moment \(t=0\) and \(PI(E_{i,j})\) is the possibility index of the event \(E_{i,j}\), which can be expressed as \(g_{i}((j-1)\Delta t,\mathbf{u}((j-1)\Delta t))>0\)\(\cap g_{i}(j\Delta t,\mathbf{u}(j\Delta t))\leq 0\).  
It is obvious that the possibility index \(PI(E_{i,j})\) in Equation (48) depends on the interference of the limit-state functions (\(g_{i}((j-1)\Delta t)>0\) and \(g_{i}(j\Delta t)\leq 0\)) and the feasible rectangular domain generated from \(g_{i}((j-1)\Delta t)\) and \(g_{i}(j\Delta t)\). The possibility index of \(E_{i,j}\) is defined as the ratio of the interference area to the feasible rectangular area after normalization, i.e.,  
\[PI(E_{i,j})=Pos\big{\{}g_{i}((j-1)\Delta t,\mathbf{u}((j-1)\Delta t))>0\cap g_{i} (j\Delta t,\mathbf{u}(j\Delta t))\leq 0\big{\}}=\left.\frac{A_{\text{ interference}}}{A_{total}}\right|_{i,j} \tag{49}\]  
Figure 2 embodies the novel solution method, in which  
\[g_{i}^{\zeta}(t,\mathbf{u}(t))=u_{i}^{cr}(t)-u_{i}^{\zeta}(t),g_{i}^{\prime}(t, \mathbf{u}(t))=u_{i}^{r}(t),\rho_{g_{i}}((j-1)\Delta t,j\Delta t)=\rho_{u_{i}}((j -1)\Delta t,j\Delta t) \tag{50}\]  
Substitution of Equation (49) into Equation (48) yields  
\[R_{s}(T)=1-\underset{i\leq r}{max}\left[Pos_{i}(0)-\sum_{j=1}^{N}\frac{A_{ \text{interference}}}{A_{total}}\right|_{i,j}\right] \tag{51}\]  
As mentioned above, to form the limited samples, the entire theoretical contents from quantization and propagation of static and dynamic uncertainties to structural safety estimation based on non-probabilistic time-dependent reliability have been expounded (as shown in Figure 3). In the next two sections, the effectiveness of the proposed method will be proven by numerical and experimental examples.