## 3 Load-Bearing Analysis of Columns under Impact Load
### Cross-Sectional Bending Moment M
Under \(E_{\mathrm{k}3}\), the specimen has just passed through the elastic stage and enters the plastic deformation stage characterized by the yielding of longitudinal rebars, and a plastic hinge has been formed at the bottom cross-section of the column, indicating that the damage to the specimen is more severe than that under \(E_{\mathrm{k}4}\). There is no significant difference between the maximum bending moments reached under the four different \(E_{\mathrm{k}}\) levels. The main difference is that under the small impact kinetic energy \(E_{\mathrm{k}4}\), the specimens have little subsequent deformation after reaching the maximum bending moment and then rebound immediately, while under the large \(E_{\mathrm{k}1}\), the bottom of the column experiences a long period of stable development after entering the plastic hinge deformation stage. It is worth noting that under the impact of \(E_{\mathrm{k}1}\), the \(M\)-\(u_{1}\) curves with different \(m\) values differ at the end part. The curves show a secondary decrease caused by the non-rebound process when they develop stably to a large displacement, because the concrete in the compression zone of the specimen has been damaged very severely and the flexural strength of the plastic hinge cannot be maintained. As shown in Figure 16b, for the slightly damaged specimens, the maximum bending moment has not reached the ultimate flexural capacity of the bottom cross-section, and hence, the plastic hinge does not occur in the specimens. For specimens with moderate or severe damage, the bending moment reached the maximum flexural capacity. Therefore, to evaluate the moderate or severe damage to specimens subjected to horizontal impact, use of the maximum bending moment alone can only determine whether a plastic hinge is formed; it cannot reflect the degree of damage in the subsequent deformation. In this case, the two types of damage states need to be judged by examining whether there is a secondary descending stage in the \(M\)-\(u_{1}\) curve. Figure 18: Damage of the column under \(E_{\mathrm{k}1}\) (peeq contour). In summary, the development trend of the \(M\)-\(u_{1}\) curve is similar under the same \(E_{\rm k}\) and different combinations of \(m\) and \(v_{0}\). On the other hand, as is known from the internal force analysis in the previous subsection, the \(M\)-\(u_{1}\) curve under static and dynamic loads shows a consistent development path. Therefore, the performance level of the column in the impact process can effectively be evaluated by the development path of the \(M\)-\(u_{1}\) curve. As shown in Figure 20, when the bottom cross-sectional bending moment of the column has not yet reached the ultimate value, that is, before the characteristic point 1 is reached, the specimen is in the elastic state (stage 1), and the damage is relatively slight; when the ultimate bending moment is reached, the plastic hinge appears in the specimen, which enters the plastic damage state (stage 2); finally, when the deformation of the specimen is too large and exceeds characteristic point 2, the bending moment shows a second drop, which indicates that the specimen has been damaged very severely (stage 3). Figure 20: Schematic diagram of the \(M\)-\(u_{1}\) curve under an impact load. Figure 19: \(M\)-\(u_{1}\) curve under different kinetic energies \(E_{\rm k}\).