## 4 Seismic Vulnerability and Sensitivity Analysis
### Effect on Member Response  
The prototype structure of this study is an old school building reflecting the inadequate construction practices of South Korea in the 1980s. The seismic vulnerability of this structure may increase due to the attainment of member level limit states. The shear demand and capacity of a RC column were monitored to assess the seismic vulnerability at the local level. Shear demand was used as an EDP, while the shear capacity was selected as a limit state function. Regarding the shear capacity of the RC column, the shear strength model by ACI 318-19 [25] was utilized.  
Figure 9 compares the fragility curves at the global and local levels of the selected structure. The exceedance probability of the serviceability limit state at the global level was always higher than that of the shear failure state at the local level, which means the shear failure of the RC column does not occur until the longitudinal reinforcement yields. On the contrary, the exceedance probability of the shear failure state was higher than that of the damage control limit state until \(S_{a}\) reaches 0.64 g. Note that the design spectral acceleration in the Korean Building Code [26], corresponding to the fundamental period of the reference structure, was 0.64 g as shown in Figure 9. For the spectral acceleration of 0.64 g, the exceedance probability of the serviceability state was 83.5%, while those of the damage control and shear failure limit states were 30.7% and 30.9%, respectively. In addition, Figure 9 clearly indicates that the likelihood of shear failure of RC members was much higher than reaching the collapse prevention limit at the global level. Hence, local failure may occur before the structural instability on the system level occurs, and thus the effect of construction quality defects on the shear vulnerability of RC members should be considered for the selected structure.  
For each uncertain variable shown in Table 1, Figure 10 illustrates the fragility curve for the local failure state and its sensitivity swing. Compared with other variables, it was observed that the effect of the variability in the concrete strength (\(f_{c}^{\prime}\)) on the vulnerability curve was insignificant at the local level as opposed to the global level. However, variabilities in the yield strength (\(f_{y}\)) and longitudinal reinforcement ratio (\(\rho_{l}\)) notably affect the local failure mode as shown in Figure 10b,c. These variables are related to the moment capacity of the RC column. The RC column with low yield strength and longitudinal reinforcement ratio attains the ultimate flexural strength before reaching the shear capacity, resulting in a low probability of shear failure. Figure 10d clearly shows that the most significant effect on the shear vulnerability is due to the volumetric ratio of transverse reinforcement (\(\rho_{w}\)), which is one of the governing parameters for the ductility and shear capacity of RC columns. For example, in the fragility curve for \(\rho_{w}\), the exceedance probability at \(S_{a}\) of 1.0 g varied from 0.29 to 0.66,  
Figure 9: Seismic vulnerability curves at the global and local levels.  
resulting in a swing of 0.37, and the corresponding swing in the fragility curve for \(f_{c}^{\prime}\), \(f_{y}\), and \(\rho_{l}\) were 0.03, 0.25, and 0.25.  
This observation can be confirmed in Figure 11, which shows the relative sensitivity on the shear vulnerability curve for the selected uncertain variables. For instance, at an \(S_{a}\) of 1.0 g, the relative sensitivities of rebar yield strength, longitudinal rebar ratio, and volumetric ratio of stirrup were 0.28, 0.28, and 0.41. Thus, these three parameters affect the shear vulnerability almost equally. Therefore, it could be inferred that quality control on these three parameters is essential to avoid the shear failure of RC columns.  
Figure 10: Sensitivity swing for shear vulnerability curves: (**a**) concrete strength; (**b**) yield strength; (**c**) longitudinal reinforcement ratio; (**d**) volumetric ratio of transverse reinforcement.