# Vulnerability Analysis of Column-Supported Reinforced Concrete Silo Structures
## 3 Determination of Silo Damage Index and Seismic Demand Parameters
### Definition of Damage States  
The degree of structural damage and the deformation have a one-to-one relationship and can be quantified by a specific function. In this section, structural damage under earthquake action was determined, and the damage grade was classified through a multiple IDA on the silo structure. Based on the selected parameters related to functional requirements, the limit state of the silo was quantified using the peak top displacement (\(\delta\)) of the supporting column in each IDA process. Further, the median value of each IDA curve and its derivative function were obtained, as shown in Figure 7b. This derivative function truly reflects the deterioration in the lateral stiffness of the silo structure with the peak top displacement [34]. By observing the change in the lateral stiffness, five damage levels of the column-supported reinforced concrete silo could be defined: intact, minor damage, moderate damage, severe damage, and collapse; the corresponding limit states are LS\({}_{1}\)-LS\({}_{4}\), respectively, as listed in Table 3. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Performance Level** & **Qualitative Description** & \begin{tabular}{c} **Displacement Limit at the** \\ **Top of the Column** \\ \end{tabular} & **Element Drift Ratio** \\ \hline No damage & No damage, and the silo structure remained & 0–1.0 cm & 0–1/710 \\ Minor damage & \begin{tabular}{c} \\ \end{tabular} & \begin{tabular}{c} Bending or shear-type cracks appeared at the top \\ of the supporting column and at the location \\ connecting the ring beam \\ \end{tabular} & 1.0–4.0 cm & 1/710–1/180 \\ Moderate damage & \begin{tabular}{c} \\ \end{tabular} & \begin{tabular}{c} The concrete of the supporting column spalled \\ early and needed to be repaired. \\ \end{tabular} & 4.0–6.5 cm & 1/180–1/110 \\ Severe damage & \begin{tabular}{c} \\ \end{tabular} & \
\begin{tabular}{c} The concrete fell off over a large area, and the \\ structure exceeded its yield capacity. \\ \end{tabular} & 6.5–14.0 cm & 1/110–1/50 \\ Collapse &
\begin{tabular}{c} The structure had collapsed or was in the danger \\ of collapsing \\ \end{tabular} & 14.0 cm & 1/50 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Qualitative and quantitative descriptions of the damage states of the silo structure. Figure 7: (a) IDA curves of the silo model. ; (b) Median IDA curve (black), its derivative (red), and the defined damage limit states (blue). From the curve shown in Figure 7b and the IDA damage parameter results of the reinforced concrete silos, the following analysis can be performed to further confirm the rationality of the limit state delineated in Table 3.  
No damage. The first significant reduction (65%) in the lateral stiffness occurred at \(\delta\) = 1.0 cm, before which the IDA curve was in the initial linear elastic range and the silo structure remained intact. Subsequently, the structure began showing tensile cracks and changed to a slightly damaged state in terms of deformation. Further, the tensile damage (\(D_{1}\)) of the structure can also be observed in the IDA results, and the displacement at the top of the column corresponding to different seismic waves fluctuated at approximately 1.0 cm (Figure 8). Minor damage: When \(\delta\) = 4.0 cm, the derivative value of the IDA curve in Figure 7b decreases significantly, with the lateral stiffness of the silo decreasing by approximately 22.5%. As a result, using the top displacement value at this point as the limit value can help distinguish the minor and moderate damage states. In the case of minor damage, the supporting columns, ring beams, and silo walls typically bear a high tensile force. Figure 9 shows the extension of tensile damage, with the most significant damage occurring from the bottom of the support column to a height of 2/3, and the corresponding top displacement fluctuates at approximately 4.0 cm.