# Application of Sensitivity Analysis to Progressive Collapse Resistance of Planar Truss Structures
## 3 Judgment of Critical Elements Based on Sensitivity Analysis
### Critical Elements  
The removal of the failed elements is defined as the damage parameter for sensitivity analysis. The failure element is introduced through the Model Change command in ABAQUS, and the implicit analysis module is used to conduct the structural nonlinear static analysis. The Mises equivalent stress before and after the structural damage is extracted. The element-sensitive indices are obtained by calculating the stress difference before and after the structural damage by Equation (1). The importance coefficient of the failed element under corresponding statistical objects is obtained by calculating the mean value of the sensitive indices under different statistical objects by Equation (2). Figure 10  
Figure 8: LC1 strain time history.  
Figure 7: LJ2 displacement time history.  
Figure 9: UCI strain time history.  
shows the calculation results of the importance coefficients of the two types of trusses under different statistical objects.  
As shown in Figure 10, the sensitivity analysis results of different statistical objects are significantly different for the same truss.  
Under different statistical objects, the element importance coefficient changes in different trends. As in Figure 10a, when the statistical objects are the lower chords or all elements, the upper chords are the most crucial element in the structure, followed by the web members. The lower chords are the least important. However, when the statistical objects are the upper chords, the importance coefficient of web member W5 is 4.13, which is the most important element in the structure. In Figure 10b, when the statistical objects are the lower chords, the elements' importance is in the order of the upper chord at the midspan, the web member at the support, and the lower chords. When the upper chords are the statistical objects, the web member at the support is the most important element in the structure, followed by the upper chords.  
Under different statistical objects, the elements' importance is different. As in Figure 10a, when the statistical objects are the lower chords, the importance coefficient of web member W2 is 6.53, higher than that of web member W5 (5.15). When the statistical objects are the upper chords, \(\alpha_{\text{W2, UC}}=1.82\), less than \(\alpha_{\text{W5, UC}}=4.13\). Similarly, in Figure 10b, when the statistical objects are the upper chords, the importance coefficient of the web member WM2 at the support is the largest, reaching 32.59. However, the importance coefficient of the midspan upper chord is the greatest under other statistical objects.  
Under different statistical objects, the positions of important elements are quite different. When the statistical objects are the lower chords, the maximum importance coefficient of the web member appears at the support. When the statistical objects are the upper chords, the maximum importance coefficient of the web member appears at the midspan, as shown in Figure 10a. In Figure 10b, when the web members are used as the statistical objects, the web member with the largest importance coefficient appears at the midspan, \(\alpha_{\text{WMS, WM}}=1.475\). However, under the other three statistical objects, the importance coefficient of the web member at the support is the largest.  
The above analysis shows that when using the existing sensitivity analysis method to determine the important elements in the same structure, the magnitude, the variation trend of the element importance coefficients, and the distribution location of the important elements are quite different under different statistical objects. The choice of statistical objects will significantly impact the sensitivity analysis results.