## 4 Robustness Quantification of a Precast Concrete Frame: An Illustration Example
### Reliability Index and Robustness Index  
With the stochastic properties from the resistance side and from the load side (Table 1), the failure probability of each element can be obtained according to Equation (4). Specifically, the resistance of each beam or column element is calculated by sectional analysis and the load effect coefficients are obtained by static analysis [42], followed by the failure probability of each element based on the first-order reliability method. In the sectional analysis, each section is subdivided into discrete layers to consider the differences in the constitutional model between the precast concrete and the cast-in-place concrete. Note that  
Figure 3: Skeleton of the considered precast concrete frame: geometry, applied loads and numbering of structural elements. the nonlinear effect of concrete is implicitly considered in the sectional analysis. A beam or column element is deemed to fail when its ultimate strength reaches the applied load effects and the strain hardening effects of concrete members are ignored for simplicity purposes. With the failure probabilities of all beams and columns available, the system reliability of the frame can be calculated following the beta-unzipping method at level 1. All the analyses and calculations are performed using software Python. Before the calculation of the reliability indices needed for robustness quantification, the system failure probability of the frame under the design loading conditions is first calculated. This failure probability is \(4.24\times 10^{-4}\) (\(\beta=3.34\)) showing that the precast frame is well designed. Then the failure probability of the intact and damaged frames under the severe earthquake condition is examined. The failure probability of each failure element is given in Table 2. As mentioned above, a notional central column removal scenario is assumed to define the damaged state of the considered frame. Specifically, it is assumed column No. 2 is removed from the frame in this illustration example. For the intact precast concrete frame, the contribution of those elements with failure probabilities smaller than \(1\times 10^{-3}\) to the system failure probability can be neglected and other elements can be seen as critical elements ((\(\Delta\beta=1.33\)). Note that theoretically all the failure elements should be considered for accuracy purposes. However, the inclusion of elements with relatively low failure probabilities will sharply increase the computational costs. The neglection of these failure elements normally results in a good balance between simplicity and accuracy. In this example, these elements are shown in Figure 4, in which the failure elements are ascending ordered with respect to reliability index. Figure 4 shows that among all the elements, elements No. 4, No. 6, No. 7, No. 9, No. 11, No. 2 and No. 12 are more vulnerable elements with their reliability indices smaller than 2.0. Based on Equations (8) and (9), the correlation coefficient matrix with respect to the failure elements for the intact structure is obtained, as shown in Table 3. It can be seen from Table 3 that the beam elements (especially adjacent beam elements) are usually highly correlated. For example, the correlation coefficient between Beams No. 4 and No. 9 in the same bay is 0.895 and the correlation coefficient for the adjacent Beams No. 4 and No. 6 on the first floor is as high as 0.933.