# Application of Sensitivity Analysis to Progressive Collapse Resistance of Planar Truss Structures
## 1 Introduction  
Most current studies on progressive collapse focus on frame structures [1, 2, 3], and relatively few studies focus on large-span space structures. However, with the wide application of space structures, progressive collapse accidents occur frequently [4, 5]. Such structures are generally used in densely populated public buildings, so the progressive collapse study of large-span space constructions is necessary. The Alternate Path (AP) method is the most extensively utilized method in progressive collapse analysis. The primary issue of this method is to identify the critical element, which is the initial failure element. There is a clear definition of initial failure elements of frame structures in the existing guidelines for progressive collapse design [6, 7, 8]. However, there are no such codified recommendations for non-frame structures such as the large-span space structure. Therefore, for non-frame structures with numerous elements, an effective method to identify important elements is significant. There are many studies on the methods of element importance evaluation. [PERSON] et al. [9] identified the key element as those presenting the greatest intersection between the vulnerability to progressive collapse and element importance. The calculation formulas of the vulnerability coefficient and importance coefficients were given by the failure probability of an element and the probability of occurrence of the failure path. The stiffness-based judgment method determines the importance of a single element by studying its contribution to the overall structural stiffness (e.g., tangential stiffness). The tangential stiffness matrix contains structurally specific information, and its singularity can reflect structural safety [10]. However, the topology judgment that purely depends on the structural stiffness is divorced from the load effect, and the complex algorithm involved has a high threshold in engineering applications. The method of element importance judgmentconsidering the conventional load and connecting the \"AP method\" concept is more in line with the actual situation of progressive collapse and has engineering practicability. For example, [PERSON] [11] defined sensitivity as the structural response changes caused by initial local failure, which was inversely proportional to the redundancy of the structure. The quantitative formula of structural redundancy was given theoretically. [PERSON] [12] defined the node displacement response change after the initial failure as the object of sensitivity analysis to calculate the importance coefficient of the element. According to the sensitivity analysis, the structure robustness is evaluated by the change of structural strain energy after element failure, and the element importance is inversely proportional to the robustness coefficient, performed by [PERSON] et al. [13]. [PERSON] et al. [14] obtained the maximum load coefficient from the load-deflection curve. The element or node sensitivity was calculated by the structural ultimate bearing capacity coefficient difference before and after the failure of the element or node. A suitable identification method for determining the critical member by calculating a criticality index based on node buckling was established by [PERSON] et al. [15]. In the above method, when judging the element's importance by sensitivity analysis, all remaining elements except the damaged ones are consistently taken as statistical objects without considering the differences between elements. The mean value of the sensitive indices of statistical objects reflects the importance of the damaged element. Nevertheless, the average state of all remaining elements does not accurately reflect the significance of the damaged elements in the structure. As shown later in this study, when the statistical objects are different in the same structure, using the above method will obtain entirely different evaluation results, and the method accuracy is poor.