## 3 Load-Bearing Analysis of Columns under Impact Load
### Cross-Sectional Bending Moment M
Under the impact of 4 different initial kinetic energies (0.2, 0.5, 1 and 2 kJ), as \(m\) increases from 0.2 t to 40 t, \(t_{\rm d}\) increases from 0.05 s, 0.03 s, 0.02 s, and 0.02 s to 0.66 s, 0.56 s, 0.49 s, and 0.45 s, respectively. Figure 17b shows that at impact energy \(E_{\rm k4}\), the plastic hinge is still not formed at the bottom of the specimen, the overall damage is very slight, and the bending moment values for different combinations of \(m\) and \(v_{0}\) are quite different from the ultimate bending moment of the cross-section. Under \(E_{\rm k2}\) and \(E_{\rm k3}\), the bending moment values are still small when \(m\) is 0.2 t, and as \(m\) subsequently changes, the bending moment increases to a certain level and then remains constant. At a large impact kinetic energy \(E_{\rm k1}\), the bending moment  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(E_{\rm k}\) (kJ) & \begin{tabular}{c} **Specimen** \\ **Number** \\ \end{tabular}  & \(m\) **(t)** & \(v_{0}\) **(m/s)** & \(t_{\rm d}\) **(s)** & \(u_{\rm max}\) **(mm)** & \(M_{\rm max}\) **(kN-m)** \\ \hline \multirow{8}{*}{2} & \multirow{8}{*}{2} & DZ150-\(E_{\rm k1}\)-0.2 & 0.2 & 4.47 & 0.05 & 60.1 & 15.4 \\  & DZ150-\(E_{\rm k1}\)-0.5 & 0.5 & 2.83 & 0.07 & 65.6 & 15.6 \\  & DZ150-\(E_{\rm k1}\)-1 & 1 & 2.00 & 0.1 & 72.7 & 15.9 \\  & DZ150-\(E_{\rm k1}\)-4 & 4 & 1.00 & 0.2 & 72.9 & 16.23 \\  & DZ150-\(E_{\rm k1}\)-8 & 8 & 0.71 & 0.28 & 72.8 & 15.93 \\  & DZ150-\(E_{\rm k1}\)-12 & 12 & 0.58 & 0.35 & 74.4 & 15.99 \\  & DZ150-\(E_{\rm k1}\)-16 & 16 & 0.50 & 0.4 & 74.2 & 16.05 \\  & DZ150-\(E_{\rm k1}\)-20 & 20 & 0.45 & 0.45 & 74.4 & 16.04 \\  & DZ150-\(E_{\rm k1}\)-30 & 30 & 0.36 & 0.57 & 74.4 & 16.1 \\  & DZ150-\(E_{\rm k1}\)-40 & 40 & 0.32 & 0.66 & 74.6 & 16.08 \\ \hline \multirow{8}{*}{1} & \multirow{8}{*}{1} & DZ150-\(E_{\rm k2}\)-0.2 & 0.2 & 3.16 & 0.03 & 34.8 & 14.5 \\  & DZ150-\(E_{\rm k2}\)-0.5 & 0.5 & 2.00 & 0.04 & 38.9 & 16.1 \\  & DZ150-\(E_{\rm k2}\)-1 & 1 & 1.41 & 0.07 & 41.4 & 15.7 \\  & DZ150-\(E_{\rm k2}\)-4 & 4 & 0.71 &completely reaches the same level as that under static loading (16.41 kN\(\cdot\)m, obtained by the FE calculation). Similarly, as seen in Figure 17c,d, \(u_{\text{max}}\) increases with an increase in \(m\) when it is small (i.e., 0.2 to 1 t), while the maximum displacement reached by the specimen is similar to the subsequent changes in \(m\) (1 to 40 t). For the responses of the specimen with the same \(m\) values and different \(E_{\text{k}}\) values, \(u_{\text{max}}\) increases linearly with an increasing \(E_{\text{k}}\), and the change slopes of the \(E_{\text{k}}\)-\(u_{\text{max}}\) curves with different impact masses are similar, except for slightly small change slopes with small \(m\) values. At the same \(E_{\text{k}}\), the overall maximum displacement of the column is slightly small when \(m\) is small and \(v_{0}\) is large. The reason can be derived from analyzing the concrete damage in Figure 18. Under the impact with a small \(m\) and large \(v_{0}\), the specimen is severely damaged locally at the impact point. When \(m\) increases to 1 t, the minimum concrete damage occurs at the impact point, and the external impact energy is mostly dissipated by the overall deformation. Figure 17: Influence law of the impact energy on each response index. Figure 19 shows the variation curve of the bottom bending moment \(M\) with \(u_{1}\). A comparison of the \(M\)-\(u_{1}\) curves with different \(E_{\mathrm{k}}\) values indicates that the specimen does not enter the plastic hinge deformation stage under the minimum impact kinetic energy \(E_{\mathrm{k}4}\) and is still in the small damage stage, and the \(M\)-\(u_{1}\) curve rebounds before reaching the peak.