# Optimisation of Cable Dome Structure Design for Progressive Collapse Resistance
## 2 Dynamic Response and Model Analysis of Progressive Collapse
### Dynamic Response and Collapse Model Analysis for the Entire Process of Structural Collapse  
Because of space limitations, analyses were performed only for the displacement response, internal force response, energy response, and collapse model of the structure after the OHC was removed. 1. Displacement response: Figure 2(a),b illustrate the vertical displacement response when the equivalent force began to be unloaded (t = 44 s) and when the structure reached its final state with damping (t = 200 s), respectively. The results indicated the following. First, the node displacement was consistent with the initial structure once \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline
**Member** & **DC1** & **DC2** & **DC3** & **RC1** & **RC2** & **RC3** & **OS** & **MS** & **IS** & **OHC** & **IHC** & **IUS** & **ILS** \\ \hline Area (mm\({}^{2}\)) & 2490 & 853 & 605 & 1840 & 1360 & 853 & 7800 & 4670 & 4670 & 7470 & 3320 & 3320 & 3320 \\ Prestress (kN) & 466.6 & 208 & 105.9 & 682.2 & 473.1 & 370 & \(-158\) & \(-70.4\) & \(-36.2\) & 1403.2 & 625.7 & 1190.1 & 305.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters and initial pre-stress of initial model elements. Figure 1: Investigated cable dome structure. Figure 2: Whole progress of the equivalent force. the structure reached a balance after the cable was replaced with an equivalent force. Second, Nodes 2 and 9, which were connected to the OHC, rapidly moved towards the sides after the OHC was removed (equivalent force). Their movement resulted in varying degrees of horizontal displacement in other OHC elements and all members connected to the OHC. Third, the failure of the OHC and the resultant reduction in the OS support led to a large vertical displacement in the upper and lower nodes of all OSs. Additionally, Nodes 3 and 2 in the OS, which were directly connected to the failed member, were vertically displaced by 5.03 m and 2.49 m, respectively. The displacement of other OS nodes in the cable-truss frames negatively correlated with their distance from the failed member. 2. Energy response: Figure 4, which depicts the change in internal kinetic energy over a period of 200 s, reveals the following results: First, the structure exhibited a certain kinetic energy response and remained at an imbalanced state after the failed member was replaced by an equivalent force, and the kinetic energy gradually decreased with damping and reached 0 at 44 s. Second, when equivalent force unloading began at t = 44 s, the structure transitioned again from a balanced state to an imbalanced state, with the redistribution of the internal force that contributed to the generation of a relatively large kinetic energy, which gradually decreased with damping. At approximately 75 s, the structure approached a balanced state, with a kinetic energy of 0. The text following an equation need not be a new paragraph. Please punctuate equations as regular text. 3. Internal force response: Figure 5, which depicts the time-dependent internal force change in members in the cable-truss frame adjoining the left-hand side of the failed OHC, reveals the following results. First, the internal forces of the RC1, OHC, and OS, all of which were directly connected to Node 2, reached 0 after the OHC was removed. These members continued to exert an effect on the structure after the redistribution of the structure's internal forces. The OHC experienced the largest loss of prestress, with a reduction of 90%. Second, the internal forces of other members exhibited varying degrees of reduction, with the largest reduction (90%) observed in the DC1. 4. Collapse model: After the OHC was removed, the structure produced a substantial dynamic response. Almost all components exhibited a displacement greater than 1/50 of the span. When the structure reached its final balanced state, all prestress was lost, and the structure completely collapsed. The largest displacement was observed at Node 17 (a node on the OS), which exhibited a vertical displacement of 5.58 m.  
Figure 3: Displacement response.