## 2 The Methodologies for Robustness Quantification
### Robustness Index Selection  
An insight into the available robustness indices shows that most of the structural property-based and performance-based robustness indices are deterministic ones. For example, the stiffness matrix based robustness index proposed by [PERSON] and [PERSON] [28], some of the redundancy-based robustness indices proposed by [PERSON] and [PERSON] [29], and the energy-based robustness index proposed by [PERSON] and [PERSON] [30] are all deterministic because only the structural configurations (including geometries, material properties etc.), as shown in Figure 1. When robustness quantification is performed with respect to precast concrete structures under seismic loading, however, the stochastic characteristics of the earthquake motions should not be ignored. Therefore, in this work, the deterministic robustness indices are not adopted.  
When quantifying structural robustness within a risk-based methodology, it is necessary to quantify the direct and indirect consequences before and after the local damage (e.g., the central column removal scenario). It is worth emphasizing that the quantification of consequences is rather subjective and for the same situation, the consequence quantification may vary from person to person. This is because in the estimation of consequences, even if the structure and its context are given, the consequence estimation is still uncertain as it is associated with quantifiable and unquantifiable constitutes [31; 32]. Therefore, to minimize the complexities the risk-based robustness index is also ignored in this paper and only the probability- or reliability-based robustness indices are provided herein. The inclusion of all robustness indices is not the scope of this work and the readers are suggested to refer to, e.g., [PERSON] et al. [2] when of interest.  
In order to account for the random nature, [PERSON] and [PERSON] [27] and [PERSON] [33] proposed some probabilistic measures related to structural redundancy, \(RI\), defined by  
\[RI=\frac{P_{f(\text{damaged})}-P_{f(\text{intact})}}{P_{f(\text{intact})}} \tag{1}\]  
in which \(P_{f(\text{damaged})}\) is the probability of failure for a damaged structural system and \(P_{f(\text{intact})}\) is the probability of failure of an intact structural system. This redundancy index provides a measure of the robustness of a structural system. The index takes values between 0 and infinity, with smaller values indicating larger robustness.  
They also considered the following redundancy related factor \(\beta_{R}\) defined by  
\[\beta_{R}=\frac{\beta_{\text{intact}}}{\beta_{\text{intact}}-\beta_{\text{ damaged}}} \tag{2}\]in which \(\beta_{\text{intact}}\) and \(\beta_{\text{damaged}}\) are, respectively, the reliability indices of the intact and damaged system. The system is very robust if the probabilistic redundant index is close to infinity and the system is not robust if the probabilistic redundancy index is close to 1.  
[PERSON] [34] proposed several quantitative measures of system vulnerability and damage tolerance as robustness indicators. According to the author, vulnerability and damage tolerance are considered reciprocals. Take the vulnerability \(V\) of a system for example, which is defined as  
\[V=\frac{P(r_{d},S)}{P(r_{0},S)} \tag{3}\]  
where \(r_{0}\), \(r_{d}\) are the resistances of the undamaged system and the damaged system, respectively; \(P(r,S)\) denotes the probability of failure of the system in a state \(r\) with respect to the loading \(S\). The vulnerability of a system can be used to quantify the robustness of the system. The value of \(V\) varies from 1 when the damage has a null impact on the system (robust) and to infinity when it has a huge impact on the system (non-robust).