## 4 Robustness Quantification of a Precast Concrete Frame: An Illustration Example
### Reliability Index and Robustness Index
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Element**} & \multicolumn{2}{c}{**Failure Probability**} & \multicolumn{2}{c}{**Reliability Index**} \\ \cline{2-5}  & **Intact** & **Damaged** & **Intact** & **Damaged** \\ \hline Beam No. 4 & \(8.55\times 10^{-2}\) & \(6.25\times 10^{-1}\) & \(1.37\) & \(-0.32\) \\ Beam No. 6 & \(7.51\times 10^{-2}\) & \(1.61\times 10^{-2}\) & \(1.44\) & \(2.14\) \\ Beam No. 9 & \(5.14\times 10^{-2}\) & \(5.63\times 10^{-1}\) & \(1.63\) & \(-0.16\) \\ Beam No. 11 & \(4.57\times 10^{-2}\) & \(2.37\times 10^{-2}\) & \(1.69\) & \(1.98\) \\ Beam No. 14 & \(1.25\times 10^{-4}\) & \(7.44\times 10^{-2}\) & \(3.66\) & \(1.44\) \\ Beam No. 15 & \(4.17\times 10^{-5}\) & \(1.29\times 10^{-3}\) & \(3.93\) & \(3.01\) \\ Column No. 1 & \(3.48\times 10^{-3}\) & \(2.53\times 10^{-1}\) & \(2.70\) & \(0.67\) \\ Column No. 2 & \(4.28\times 10^{-2}\) & N/A & \(1.72\) & N/A \\ Column No. 3 & \(7.38\times 10^{-3}\) & \(9.52\times 10^{-2}\) & \(2.44\) & \(1.31\) \\ Column No. 5 & \(2.14\times 10^{-2}\) & \(3.74\times 10^{-2}\) & \(2.02\) & \(1.78\) \\ Column No. 7 & \(6.19\times 10^{-2}\) & \(1.75\times 10^{-1}\) & \(1.54\) & \(0.93\) \\ Column No. 8 & \(1.55\times 10^{-2}\) & \(1.66\times 10^{-2}\) & \(2.16\) & \(2.13\) \\ Column No. 10 & \(8.72\times 10^{-3}\) & \(1.01\times 10^{-1}\) & \(2.38\) & \(1.27\) \\ Column No. 12 & \(2.89\times 10^{-2}\) & \(4.42\times 10^{-2}\) & \(1.90\) & \(1.70\) \\ Column No. 13 & \(3.82\times 10^{-3}\) & \(1.39\times 10^{-2}\) & \(2.67\) & \(2.20\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Failure probability of structural elements. Figure 4: The series system representing the intact precast concrete frame. Further, the failure probability, as well as the reliability of the damaged system, is calculated as:  
\[P_{f(\text{damaged})}=7.42\times 10^{-1}\]  
\[\beta_{\text{damaged}}=-\Phi^{-1}\Big{(}P_{f(\text{damaged})}\Big{)}=-0.65\]  
When the failure probabilities of the intact and damaged precast concrete frame are obtained, the quantification of the robustness indices for the precast concrete frame can be proceeded according to Equations (1)-(3):  
\[RI=\frac{P_{f(\text{damaged})}-P_{f(\text{intact})}}{P_{f(\text{intact})}}=2.88\]  
\[\beta_{R}=\frac{\beta_{\text{intact}}}{\beta_{\text{intact}}-\beta_{\text{damaged }}}=0.57\]  
\[V=\frac{P(r_{d},S)}{P(r_{0},S)}=3.88\]  
According to the definition of the redundancy-based robustness index \(RI\) in Equation (1), a value close to zero indicates absolute robust and a value approaching infinity indicates absolute non-robust. In the case of the prescribed precast concrete frame herein, the value of \(RI\) is 2.88 which is relatively small when compared with infinity or compared with the \(RI\) values found in other studies using the same calculation philosophy [42]. Similar results are found when the reliability-based robustness index \(\beta_{R}\) and the vulnerability-based robustness index \(V\) are analysed. Therefore, the results of indices \(RI\), \(\beta_{R}\) and \(V\) indicate that the prescribed precast concrete frame is relatively robust. It is worth noting that, for these three robustness indices, there are still no standard values with respect to robustness indices to clearly identify whether a structure is robust or non-robust. This is because the probabilistic robustness indices are very sensitive to the sources of structural and non-structural variations including the structure type, the loading combinations, the used materials as well as methods used for structural analysis and mathematical calculations. More efforts are still needed for the scientific communities to reach a consensus on how to determine such standard values by further research.