## 2 The Methodologies for Robustness Quantification
### Philosophy of Robustness Quantification  
To illustrate how the structural robustness quantification can be performed, it is first considered how a structure can be analysed or assessed. When a structure is to be designed, the characteristics of the structure such as geometries, selected materials and boundaries are determined. Based on these characteristics the (internal) structural properties regarding the structure itself such as the slenderness and the axial/bending stiffness can be obtained. If the (external) environments such as attributes of loading (e.g., magnitude, direction or distribution) are considered, then the structural performances (or responses) such as load-bearing capacity, the displacement and the stored energy can be calculated. Furthermore, probabilistic properties such as the failure probability or reliability of the structure will be known if the uncertainties of the basic variables are taken into account. Moreover, in case damage occurs due to accidental events posed on the structure, the structural properties, structural performances and probabilistic properties can be assessed again for the damaged structure; structural risk analysis with respect to the original structure can also be conducted when the consequences due to the accidental events are considered. As mentioned, it is generally accepted that the process of robustness quantification comprises the identification and modelling of exposures, the formulation and calculation of damages and the assessment of the disproportionality. Therefore, the event tree used, e.g., in [PERSON] et al. [26] can be adopted as a tool to analyse the framework of robustness quantification. Based on this tool, the classification of the methodologies for robustness quantification is realized by exploring the complexity of the analysis method, as shown in Figure 1. Specifically, the methodologies can be classified into four categories by performing different analyses or assessments for a structure:  
* Structural-property-based: given an accidental exposure, the considered structure might be damaged. The possibility of damage is only related to the (internal) structural properties (the simplest case is related to the load-bearing capacity). If the structure is damaged, the loss directly due to the exposure initiation (e.g., a loss of a column) is seen as the direct consequence. In this case, the robustness quantification is structural-property-based. * Structural-performance-based: consider that in practice the structure is subjected to (external) loadings such as self-weight and wind load, then damage in the structure is possible to trigger a follow-up failure or collapse of the whole structure. Note that the loss from the follow-up failure or collapse (e.g., the monetary value of the structure, fatalities due to collapse, etc.) is seen as the indirect consequence. In this case, the probability of collapse is calculated conditionally on the damage initiation as well as the sustained loadings on the structure before the damage occurs, and the afterwards robustness quantification is seen as structural-performance-based. * Probability- or reliability-based: if the uncertainties corresponding to the internal and external characteristics are taken into account, the probability of damage or collapse can be obtained and then either a property-based or a performance-based methodology is updated to be probability- or reliability-based. * Risk-based: if a methodology is to compare the changes of direct and indirect risks based on consequences, then this methodology is seen as a risk-based approach. Except for the classification mentioned above, the methodologies can also be classified into deterministic and probabilistic ones by considering whether or not the effect of uncertainties is involved, as shown in Figure 1. Note that some researchers also proposed generalized/generic forms into which the robustness measures based on, e.g., structural properties and/or structural performances can be fitted.