## 3 The Robustness Quantification Framework for Precast Concrete Frames under Seismic Loading
### Uncertainty Characterization and Failure Probability
\begin{table}
\begin{tabular}{c c c c c} \hline \(X\) & **Description** & **Distribution Type** & **Mean \(\mu_{X}\)** & **std.**\(\sigma_{X}\) \\ \hline \(G\) & Permanent load & Normal & specified & 0.05\(\mu_{X}\) \\ \(Q\) & Variable load (50 year) & Gumbel & 0.6 \(Q_{k}\) & 0.35\(\mu_{X}\) \\ \(F\) & Equivalent static shear force & Gumbel & 1.0 \(F_{k}\) & 0.3 \(\mu_{X}\) \\ \(a_{\mathrm{s}}\) & Concrete cover & Beta * & specified & 10 mm \\ \(f_{\mathrm{ck}}\) & Concrete strength & Lognormal & \(f_{\mathrm{ck}}+2\,\sigma\) & 0.15\(\mu_{X}\) \\ \(E_{\mathrm{c}}\) & Concrete modulus of elasticity & Lognormal & \(22[(f_{\mathrm{ck}}+8)/10]^{0.3}\) & 0.15\(\mu_{X}\) \\ \(\epsilon_{\mathrm{cu}}\) & Concrete ultimate compressive strain & Lognormal & specified & 0.15\(\mu_{X}\) \\ \(f_{\mathrm{y}}\) & Steel yield strength & Lognormal & \(f_{\mathrm{yk}}+2\,\sigma\) & 30 MPa \\ \hline \end{tabular}  
* Lower bound \(a=0\); upper bound, \(b=3\mu_{X}\).  
\end{table} 
Table 1: Uncertainty characterization of random variables. reliability index \(\beta\in[\beta_{\min},\beta_{\min}+\Delta\beta]\) are selected to constitute a series system. \(\Delta\beta\) is a prescribed positive number determined by the user. As suggested in [35], a value of 3.00 can be an option for \(\Delta\beta\). When the Monte Carlo simulation technique is used for the calculation of the failure probabilities, it is often the case that the computation is extremely time-consuming for some elements with high safety margins. Since the failure probabilities of these elements do not greatly contribute to the system failure probability, it is not necessary to calculate them accurately and the estimation with an upper bound is acceptable. In the selection of critical elements, these elements with low failure probability contributions are ignored. Therefore, the value for \(\Delta\beta\) is not fixed in this work but changes from case to case. According to the beta-unzipping method and level 1, the selected critical elements form a series system. For a series system including \(n\) elements, the failure event and the failure probability of the \(i\)th (\(i=1\), 2, 3..., \(n\)) element are, respectively, denoted as \(F_{i}\) and \(P(F_{i})\). Then the failure probability of this system is expressed as:  
\[P_{f}=P(F_{1}\cup F_{2}\cup\cdots\cup F_{n}) \tag{5}\]  
To obtain a good estimate of the system failure probability, the Ditlevsen bounds [35] can be used:  
\[ \begin{split} P_{f}\leq\sum\limits_{i=1}^{n}P(F_{i})-\sum \limits_{i=2}^{n}\max\limits_{r<i}\{P(F_{i}\cap F_{r})\}\\ P_{f}\geq P(F_{1})+\sum\limits_{i=2}^{n}\max\limits\{P(F_{i})- \sum\limits_{r=1}^{i-1}P(F_{i}\cap F_{r}),0\}\end{split}  \tag{6}\]  
The calculation of the Ditlevsen bounds requires an estimate of the probability of the intersection of \(F_{i}\) and \(F_{r}\), i.e., \(P(F_{i}\cap F_{r})\). In case all the variables in the limit state function, i.e., Equation (4) of the \(i\)th element are normally distributed (if not, for example, the Rackwitz-Fiessler algorithm [40] can be applied), the probability of the intersection is given by:  
\[P(F_{i}\cap F_{r})=\int_{-\infty}^{-\beta_{i}}\int_{-\infty}^{-\beta_{r}}\frac {1}{2\pi\sqrt{1-\rho_{i,r}^{2}}}e^{-\frac{1}{2(1-\rho_{i,r}^{2})}(\beta_{1}^{ 2}+\beta_{r}^{2}-2\rho_{i}\beta_{r})} \tag{7}\]  
with  
\[\rho_{i,r}=\frac{\sum\limits_{j=1}^{m}a_{ij}a_{ij}\text{Var}[q_{j}]}{\sigma_{ G_{i}}\sigma_{G_{r}}} \tag{8}\]  
where \(\sigma_{G_{i}}\) and \(\sigma_{G_{r}}\) are the standard deviation of the safety margin of the \(i\)th and \(r\)th element, respectively. Since the random variables in the limit state functions are (transformed to be) normally distributed, the standard deviation of the safety margin of the \(i\)th element is:  
\[\sigma_{G_{i}}=\sqrt{\sum\limits_{j=1}^{m}a_{ij}^{2}\text{Var}[q_{j}]} \tag{9}\]  
According to Equations (4) and (8), the calculation of the coefficient of correlation for beam members can be performed.