## 2 Theoretical Background
### Classical Reliability Assessment  
The classical term of \"reliability\" deals with the complement of the probability of failure (= 1 - \(P_{f}\) ) for the given uncertainties of the system [11], but more properly, it is the probability of safety of the structure over a given period of time [12]. The limit-state function \(G(X)\) (sometimes also referred to as performance function) can be expressed by two domains such as the safe portion (\(D_{s}\)) and failure portion \((D_{f})\)[13]:  
\[X\in D_{S}\Leftrightarrow G(X)>0 \tag{1}\]  
\[X\in D_{f}\Leftrightarrow G(X)\leq 0 \tag{2}\]  
The time-independent (Figure 2a) failure probability is described by a combined probability density function (PDF) counting the random vector of the safe variables as:  
\[P_{f}=\mathbb{P}[R(X)-S(X)\leq 0]=\int_{-\infty}^{\infty}F_{R}(\delta)f_{S}( \delta)d\delta=1-\int_{-\infty}^{\infty}F_{S}(\delta)f_{R}(\delta)d\delta \tag{3}\]where \(F_{R}\) and \(F_{S}\) represents the Cumulative Density Functions (CDFs), and the randomness of R and S is expressed by \(f_{R}\) and \(f_{S}\)[12].  
With the consideration of the random function of time, the probabilistic analysis of structural capacity will change as shown in Figure 2b. The structural capacity decreases with time due to the material degradation, while the reliability index for the safety check will also be decreased. A time-varying structural load can be idealized as an order of randomly occurring pulses like \(S_{1}\), \(S_{2}\),... \(S_{n}\) at times \(t_{1}\), \(t_{2}\)...\(t_{n}\) with \(\tau\) (small duration when the degradation due to load is zero) as shown in Figure 2b. [PERSON] [14] explained the  
Figure 1: Flowchart of the safety index determination framework.  
Figure 2: Illustration of the reliability analysis for basic random variables; (**a**) time-invariant model; (**b**) time-variant model.  
several methods of the time-integrated and discrete approaches for the reliability analysis, where the time-variant reliability analysis is defined as the following equation:  
\[P_{f}(t)=\mathbb{P}[R(X(t))<S(X(t))]=\int_{0}^{\infty}F_{R,t}(\delta)f_{S,t}( \delta)d\delta \tag{4}\]  
where \(F_{R,t}\) and \(f_{S,t}\) are the impromptu CDF of R and the impromptu PDF of S at time t respectively explained by [PERSON] et al. [15].  
To determine the failure probability \(P_{f}\), several methods have been discussed previously [16; 17; 18], where three strategies are available to evaluate it such as approximation, simulation and adaptive surrogate-modeling-based methods. The FORM, i.e., first-order reliability method and SORM, i.e., second-order reliability method which is the type of approximation method and the Monte Carlo simulation (MCS), Importance Sampling (IS) [19], Latin Hyperbolic Sampling (LHS) [20] and Subset Simulation (SS) [21] methods are the type of simulation method. Alongside, Adaptive Kriging Monte Carlo Sampling (AK-MCS) can be discussed under adaptive surrogate-modeling-based methods that are based on building a Kriging (aka Gaussian process regression) surrogate model from a small initial sampling of the input vector X available in [PERSON], Schobi and Sudret [13].