## 3 The Robustness Quantification Framework for Precast Concrete Frames under Seismic Loading
### Robustness Quantification  
When the failure probability of a precast concrete structural system can be calculated, the failure probabilities of the considered intact and damaged structures can be obtained following the same procedures, as shown in Figure 2. Specifically, the main procedures are as follows:  
1. Collect the necessary basic inputs for robustness quantification, including geometric and material properties, boundary conditions, building class, site class, seismic design category and so on;
2. With the basic inputs at hand, perform structural analysis and determine the resistances (\(R_{i}\)), load effect coefficients (\(a_{ij}\), \(b_{ij}\)) of structural element \(i\) subjected to the distributed and pointed loads (\(q_{j}\), \(P_{j}\));
3. Set up the limit state function of element \(i\);
4. By considering the main sources of uncertainties, calculate the failure probability \(P(F_{i})\) of element \(i\) using reliability analysis methods (e.g., the first-order reliability method) or simulation techniques;
5. Calculate the failure probability \(P_{f}\) of the structure using the beta-unzipping method at level 1;
6. Repeat procedures (2)-(5) to obtain the failure probabilities \(P_{f(\text{intact})}\) and \(P_{f(\text{damaged})}\) of the intact and damaged structures;
7. Quantify the structural robustness by adopting the redundancy and vulnerability-based robustness indices based on Equations (1)-(3) accordingly.  
In the development of the proposed robustness quantification framework, two main assumptions are used: (1) the failure of a structure is directly related to the sequential  
Figure 2: Calculation procedures within the robustness quantification framework.  
failures of structural elements; and (2) the structural internal forces are approximately believed to be linearly related to the external loads (i.e., the limit state function shown in Equation (4)). Based on these assumptions, the beta-unzipping method is used to facilitate the calculation of the failure probability of the structure of interest. The first assumption shows that the proposed framework can be directly used to quantify the structural robustness of frame structures since sequential failure mechanisms among frame beams and columns are theoretically and experimentally found in frame structures. The framework is also applicable for other various structures if such structures can be explicitly simplified to be structures composed of discrete components (e.g., frame-wall structures, truss structures and timber structures). The other assumption describes the scope of the structural behaviours that can be considered in the process of robustness quantification. Obviously, elastic behaviours for all structures can be considered. In case the nonlinear effects are under consideration, the framework can also be adopted because approximately the analysis of a specific nonlinear state of interest can also be viewed as an elastic analysis. In this case, the load effect coefficients in Equation (4) are calculated separately. With respect to the failure modes of structural components, all kinds of failure modes including the flexural, shear, compression (or tensile) failure modes can be considered in the framework.  
The limitation of the proposed framework lies mainly in two ways: on the one hand, the robustness is quantified in an approximate way by which the nonlinear effects in concrete structures cannot be fully considered; on the other hand, it is difficult to explicitly consider the coupling effect of axial force, bending moment and shear force because the consideration of such effect would highly increase the complexity of the calculation of the correlation coefficients between structural components. Therefore, the robustness quantification framework proposed in this work is recommended for those designers who need to evaluate the structural safety of the designed structure from the viewpoint of robustness because the proposed framework herein is theoretically rigorous, easy to use and generally applicable in most situations.