## 3 The Robustness Quantification Framework for Precast Concrete Frames under Seismic Loading
### Failure Element and Limit State Function  
In a concrete frame structure, the basic structural elements are beams and columns. As mentioned above, each structural element is possible to fail in the structural analysis. The term failure element refers to any failed beam or column of the considered frame. When a precast concrete frame structure is subjected to permanent loads, imposed live loads and seismic loads, each section of a structural element is possibly subjected to a bending moment, a shear force and an axial force.  
For columns, each column is usually subjected to compression or the interaction of bending moment and axial force, and the failure of a column is determined by the state of its section. Hence the limit state for a column section is governed by the interaction curve of the column: the column does not fail if the couple of the moment and axial force is located within the interaction curve of the column; otherwise, the column fails. The failure of columns due to shear in the case of vertical loadings is generally neglected since the shear force for each column is relatively small compared to the corresponding shear resistance. However, the shear force for each column under seismic loadings is significant and might result in large horizontal displacements and thereafter the collapse of the upper story sitting on the corresponding lower floors. In this case, the failure event of the considered column due to shear can be linked to that due to the combination of the moment and axial force, which forms a new failure event, i.e., a series system with two failure events.  
For beams, each beam is commonly subjected to bending moments and shear forces at its ends and to a negligible axial force. Theoretically, bending failure or shear failure might occur in each beam section. In most cases, the shear failure is not allowed to occur before the flexural failure in normal design cases. Therefore, only the bending failures are considered for beams while shear failures are ignored by assuming adequate shear design. Note that in case shear failures and failures due to the interaction of bending moments and axial forces (e.g., when prestressing is applied to precast beam members) should be considered, all the possible failure events should be linked together to form a series failure event for the considered beam.  
Mathematically, the limit state of a structural element (a beam or a column) can be defined as follows [35]:  
\[G_{i}=R_{i}-\sum_{j=1}^{m}a_{ij}q_{j}-\sum_{j=1}^{m}b_{ij}P_{j} \tag{4}\]  
where \(G_{i}\) is the safety margin of the \(i\)th (\(i=1\), \(2\), \(3\), \(\ldots\)\(n\)) structural element; \(R_{i}\) is the resistance of the \(i\)th structural element; \(a_{ij}\) (or \(b_{ij}\)) is the load effect coefficient for the \(i\)th element due to the \(j\)th (\(j=1\), \(2\), \(3\), \(\ldots\)\(m\)) external distributional load \(q_{j}\) (or pointed load \(P_{j}\)); \(i\) and \(j\) are reference numbers of the structural component and the distributed and pointed loads, respectively; and \(m\) and \(n\) denote the total numbers of the distributed and pointed loads and the structural components, respectively.