# Application of Sensitivity Analysis to Progressive Collapse Resistance of Planar Truss Structures
## 5 Application of Sensitivity Analysis with Lower Chords as the Statistical Objects  
### Discussion of Analysis Results  
Figure 14 shows the sensitivity analysis results for the two types of trusses with the lower chords as the statistical objects. In both types of trusses, the midspan upper chord has the largest importance coefficient, followed by the web members, and its importance coefficient decreases from support to midspan. The above results are consistent with the analysis of the progressive collapse resistance mechanism. After the failure of the upper chord in the midspan, the structure undergoes significant vertical deformation, and the tension developed in the lower chords multiplies. It is also consistent with the engineering experience that the internal force of the rod at the support is generally larger, and the section of the web member is usually smaller in the structural design. Therefore, the lower chords as the statistical objects of sensitivity analysis can effectively evaluate the important elements of the planar truss structures and ensure a well-targeted analysis. Furthermore, for the bar structure with many elements, compared with calculating the importance coefficients by counting the mean value of sensitive indices of all remaining elements, only counting the stress changes of the lower chords will reduce the computational cost, and better meet the requirement of determining important elements easily and quickly in the Alternate Path method.  
Figure 15 shows the comparison of the evaluation results of the two trusses using the energy-based judgment method in the literature [24] and the sensitivity analysis method with the lower chords as the statistical objects. The calculated results have been normalized.  
Figure 14: Importance coefficient (statistical objects: Lower chords): (**a**) test truss, (**b**) steel roof truss.  
The two methods are basically consistent in the important elements evaluation results. Both evaluation results show that the upper chords are the most important in the two trusses, followed by the web members, and the lower chords are the least important. It proves the rationality of using the lower chords as the statistical objects for sensitivity analysis. Further correlation analysis was performed on the calculated results of the two methods for the two types of trusses, and the Pearson correlation coefficient \(\rho\) was calculated for both. The results are shown in Table 2.  
It can be seen from the table that the \(\rho\) of the calculated results of the two methods for the two types of trusses are 0.987 and 0.92, respectively, both greater than 0.90, indicating that the results obtained by the two methods have the same trend, and there is a significant positive correlation, which is also illustrated by the original data in Figure 15. Meanwhile, the scatter diagram (Figure 16) also intuitively shows a significant linear relationship between the calculation results.  
The above comparison with the calculation results in the literature [24] and correlation analysis fully illustrate the accuracy of selecting the lower chords as the statistical objects for sensitivity analysis, which can judge important elements of the planar truss structures.  
\begin{table}
\begin{tabular}{c c} \hline
**Truss Type** & \(\rho\) \\ \hline Test truss & 0.987 \\ Steel roof truss & 0.92 \\ \hline \end{tabular}
\end{table}
Table 2: Pearson correlation coefficient.  
Figure 16: Relationship between energy method and sensitivity analysis method: (**a**) Test truss, (**b**) steel roof truss.  
Figure 15: Relative importance coefficient: (**a**) Test truss, (**b**) steel roof truss.