## 3 The Robustness Quantification Framework for Precast Concrete Frames under Seismic Loading
### Failure Criterion and the Beta-Unzipping Method  
Structural collapse can be chosen as a basis for defining the failure of a structural system. Then it should be determined which phenomenon, i.e., failure criterion is appropriate as an indicator of structural collapse. Difficulties in quantitatively determining a failure criterion arise from the fact that the meaning of the collapse of an entire structure is unclear. It is well known that the failure of a member can be defined as its incapability to sustain the external loads to which it is subjected. On the one hand, it is well accepted that a column fails when it cannot resist the sustaining loads. For a frame structure consisting of beams, columns and slabs, it is also clear a column failure might lead to failure of its adjacent beams and slabs and the loads sustained by the failed column will be resisted by other members. If the other components cannot sustain the loads, then the structure collapses. On the other hand, a beam failure could be seen as a local failure and does not necessarily trigger the failure of the structure. Therefore, for frame structures, it is of significance to define the system failure and clarify the relationships between the system failure and the failures of structural components. Approximately, a linear relationship between the internal forces and the external loads can be assumed for structural members when static analysis is performed with respect to the structure. At this point, the failure probabilities of the structural members can be evaluated from the applied loads and resistance statistics using various reliability analysis methods (e.g., the widely used first-order reliability method). Almost all real structures are statically indeterminate and therefore, failure of a structural component results in a redistribution of the internal forces and the structure might survive due to the existence of several possible load paths. The (complete) failure or collapse of a structural system normally occurs after the failure of several structural components. These several membersfail progressively, and their failures form a failure sequence. Such a failure sequence among structural members can be considered a failure mode of the structure. In this situation, the entire structure can be viewed as a series system in which each system component can be viewed as a parallel subsystem. Each of the system components in series represents a failure mode (i.e., a failure sequence) of the entire structure, while the components in the parallel subsystem represent those failure events of the structural elements that constitute the specific failure sequence. After evaluating the failure probability of the first member in the sequence, the member is removed (in case of brittle failure) or replaced by forces and/or moments (in case of ductile failure) equal to that at the failure of the member. The damaged structure can be analysed again and the corresponding failure probability of each member can be evaluated. Then the joint probability of the failure of two members can also be evaluated. This progressive procedure can be continued until a mechanism level is reached. The failure probability of the structure is the probability of the union of the failure events of each failure mode. In principle, the procedure mentioned above can be used to estimate the system reliability. However, the number of failure modes for even a simplified real structure (e.g., a two-bay three-storey plane frame) is extremely large, thus making the analysis time-consuming. Alternatively, to obtain a reasonable estimate of the system reliability methods such as the branch-and-bound method and the beta-unzipping method can be adopted [35]. These methods are based on consideration of only the most critical paths and are briefly described in the following. The main characteristic of the branch-and-bound method is the procedure for the identification of the failure sequences which are the most probable to occur and the evaluation of upper and lower bounds for each failure sequence.