# Vulnerability Analysis of Column-Supported Reinforced Concrete Silo Structures
## 4 Vulnerability Analysis of the Column-Supported Reinforced Concrete Silos
### IDA Based on PGA  
Numerical simulations of the column-supported reinforced concrete silo subjected to x-direction ground motion in the empty-, half-, and full-silo states produced the dynamic response of the silo under different acceleration excitations. Next, the peak top displacement of the supporting column was extracted under different PGAs for each seismic wave and used as the seismic demand response of the silo. Correspondingly, the relationship curves between the PGA and the peak top displacement (\(\delta\)) in the empty-, half-, and full-silo states were drawn, as shown in Figure 14.  
As shown in Figure 12, it is evident that:  
1. The IDA curves of the silo structure under the three conditions have certain discreteness, which gradually increases with the increase in the amount of stored material. 2. The slope of the IDA curve clearly reflects the change in the structural stiffness. In fact, the slope decreases steadily with the increase in the ground motion under different conditions; the trend is first linear and then nonlinear. Only a few IDA curves can be characterized by a zigzag rise with a nearly constant slope, which may be related to the randomness and spectral characteristics of the corresponding seismic waves. 3. The stiffness degradation of the silo is most gentle in the empty state. However, in the full state, it enters the plastic deformation stage the earliest, and the structural stiffness also decreases most rapidly. The damage effect is also the most evident. Further, the IDA curve for the half-silo state contains two characteristic forms of the IDA curves for the empty- and full-silo states. The existence and volume of the stored grain are directly related to the damage to the silo. The greater the amount of stored grain, the earlier the silo enters the plastic state and the more evident the damage effect. Figure 14: Cluster of IDA curves. Considering the uncertainty in the ground motion, the displacement of the top column under the same intensity and distinct ground motion has a certain discreteness. Clearly, the reliability of the results can be verified through a statistical analysis of the degree of discreteness and the mean value of the IDA curve cluster. In this section, the median \(\upeta\)d and the standard deviation \(\upbeta\)d of the displacement at the top of different columns under the same ground motion intensity in the IDA curve cluster were calculated using the IM criterion. Figure 13 shows the 16%, 50%, and 84% quantile curves, with the peak acceleration of the ground motion as the Y-axis and the median \(\upeta\)e as the X-axis. Figure 15 shows the 16%, 50%, and 84% quantile curves, with the peak acceleration of the ground motion as the Y-axis and the median \(\upeta\)e as the X-axis. Evidently, the 50% quantile curve reflects the mean level of the dynamic response of the silo under different seismic waves. In comparison, the 16% and 84% quantile curves can measure the degree of dispersion in the cluster of IDA curves. In view of the quantile curves for the three working states, selecting the IDA curves obtained from the PGA as the ground motion parameter has little global discreteness. In the empty-silo state (Figure 14a), the silo was in the elastic stage as the PGA was less than 0.60 g, and the peak acceleration of the ground motion was linearly correlated with the top displacement of the supporting column. When the PGA was greater than 0.60 g, the mean value of the displacement changed to 3.93 cm, approaching moderate damage, and the rigidity and bearing capacity of the silo decreased evidently; however, no evident yielding phenomenon could be observed. In the half state (Figure 14b), when the PGA was less than 0.30 g, the peak acceleration of the ground motion in the elastic stage had a linear relationship with the top displacement of the supporting column. When it was greater than 0.30 g, the average displacement changed to 3.83 cm, and the silo came close to exhibiting moderate damage.