# Application of Sensitivity Analysis to Progressive Collapse Resistance of Planar Truss Structures
## 5 Application of Sensitivity Analysis with Lower Chords as the Statistical Objects
### Progressive Collapse Analysis  
The nonlinear dynamic Alternate Path method was used for the progressive collapse analysis of the two types of trusses. The structural dynamic response after removing elements is simulated by implicit dynamic analysis. Figure 17 shows the remaining structure deformation of the test truss under the two working conditions of the failure of web members W2 and W5. Under the two working conditions, the remaining structure regained stable equilibrium without progressive collapse. However, under the failure condition of the web member W2, the vertical deflection of the remaining structure at the node directly connected to the initial failure is larger, indicating that the remaining structure bears a greater unbalanced load. A more extensive catenary bearing capacity should be provided to complete the redistribution of the unbalanced load to regain the stable equilibrium state. According to the importance coefficient Figure 10, the results of the lower chords as the statistical objects show that the importance coefficient of web member W2 is 6.53, which is larger than that of W5 (5.15), in good agreement with the progressive collapse analysis. However, when the upper chords are the statistical objects, the importance coefficient of web member W5 is 4.13, much larger than that of W2 (1.82). According to Figure 18, it can be seen that, due to the instability of UC2 under pressure and the failure of the arch mechanism, an excessive stress change of the upper chords occurs. Meanwhile, progressive collapse did not occur under this working condition, indicating that the catenary action fully developed in LC3 is critical to completing the redistribution of the remaining structural internal force.  
The deformation diagram Figure 19 also demonstrated that the evaluation results with the lower chords as the statistical objects are consistent with the progressive collapse analysis. When the lower chords are the statistical objects, after the failure of TC4, it is the element with the largest importance coefficient, the vertical deformation of the remaining structure is more significant, and the safety reserve is smaller than WM2. Once again, the rationality and accuracy of selecting the lower chords as the statistical objects in the sensitivity analysis of the planar truss structures are verified.  
Figure 17: Structural deformation diagram under two working conditions: (a) W2 failure condition, (b) W5 failure condition.  
Figure 18: LC3, UC2 strain time history.  
Furthermore, it can be seen from the sensitivity analysis after the failure of the important element TC4 (Figure 20). The change in internal force in the lower chords is the most significant, followed by its adjacent web members WM9 and WM10, and the change in other rods is relatively tiny. It indicates that after the failure of important elements, the remaining structure dissipates the unbalanced load at the local failure through the internal force redistribution. Therefore, the progressive collapse can be avoided by determining the redistribution direction of the remaining structural internal force after the failure of important elements and taking effective measures to stop the spread of initial local failure in the redistribution direction of the internal force.