## 7 Impact of Climate Change on Structural Reliability  
The dependency of structural reliability on the non-stationary nature of climatic loads and its evolution over time is a modern research topic inside the structural engineering community [45; 46]. A methodology to estimate time-dependent reliability of aging structures accounting for non-stationarity of structural loads and for the variability of structural deterioration phenomena is, for example, proposed in [24]. This method assumes the deterioration of structural resistance as a function of time by means of the following expression:  
\[R(t)=R_{0}\;D(t) \tag{9}\]  
where \(R_{0}\) is the initial resistance and \(D(t)\) is a suitable degradation function. The non-stationarity of the structural loads is taken into account assuming both the mean load intensity \(\overline{q}(t)\) and the mean occurrence rate \(\lambda(t)\) as functions of time. The hazard function or conditional failure rate at time \(t\), \(h(t)\), i.e., the probability of structural failure during interval \(t+dt\) given that the structure survives over \(t\), is defined as in [24]:  
\[h(t)=\lambda\left(\overline{t}\right)\left[1-F_{s}\left(R_{0}\;D\left( \overline{t}\right),\overline{t}\right)\right] \tag{10}\]  
where \(F_{s}\) is the cumulative distribution function (CDF) of the load intensity, often described by an extreme value type I distribution. The hazard function can be also related to the time-dependent structural reliability \(L(t)\) as reported in [47]. Indeed, it can be written as the ratio between the probability density function of the time to failure, i.e., the derivative of \(1-L(t)\), and the structural reliability function \(L(t)\), arriving to the following expression:  
\[h(t)=\frac{d(1-L(t))}{dt}\frac{1}{L(t)}=-\frac{dL(t)}{dt}\frac{1}{L(t)}=-\frac {d(\ln(L(t))}{dt} \tag{11}\]  
Finally, combining Equations (10) and (11), the time-dependent structural reliability becomes:  
\[L(t)=\exp\left(-\int_{0}^{t}\lambda\left(\overline{t}\right)\left[1-F_{s}\left( R_{0}\;D\left(\overline{t}\right),\overline{t}\right)\right]d\overline{t}\right) \tag{12}\]  
and the probability of failure trivially results:  
\[P_{f}(t)=1-L(t) \tag{13}\]  
In this study, the time-dependent structural reliability is evaluated introducing in Equation (12) the variation over time of \(F_{s}\) obtained following the above-described approaches. Among the outcomes of the analysis, particularly relevant are the factors of changes pertaining to the \(n\)th time window, \(FC_{\overline{q}}(n)\) and \(FC_{\alpha_{q}}(n)\), affecting the mean load intensity, \(\overline{q}\), and the standard deviation, \(\alpha_{q}\), of the annual maxima of the climatic action under examination. Assuming an extreme value type I distribution, its location and its scale parameter, \(\mu(t)\) and \(\sigma(t)\), are given by:  
\[\mu(t)=FC_{\overline{q}}(n)\;\overline{q}-0.5772\frac{\sqrt{6}}{\pi}FC_{ \alpha_{q}}(n)\;\sigma_{q}; \tag{14}\]  
\[\sigma(t)=\frac{\sqrt{6}}{\pi}FC_{\alpha_{q}}(n)\;\sigma_{q}; \tag{15}\]  
where \(t\) is the starting time of the considered time window:  
\[t=(n-1)\;\Delta t  
The results are shown in Figure 12, where the red line, the blue line, and the green line illustrate the evolution over time of the factors of change for mean value, \(\overline{q}\), standard deviation, \(\sigma_{q}\), and characteristic value, \(q_{k}\), respectively, referring to the RCP8.5 scenario.  
Adopting the same probabilistic models of resistance and load intensity used in [24] and reported in Table 2, the variation of the probability of failure over time has been then evaluated, considering the resistance a deterministic linear degradation function:  
\[D(t)=1-a\ t \tag{19}\]  
where \(a\) is the degradation rate parameter.