Tensegrity structures have a high degree of indeterminacy. The occurrence of initial partial failure at one point of these structures can lead to the propagation of failure throughout the structure. One of the influential factors in the propagation of initial failure is the initial starting point of failure. The first failure can occur in a member that is considered "critical member" and causes further damage to the structure, or it can occur in a member that has caused minor damage to the structure and maintained the overall stability of the structure. Identification of critical members in tensegrity barrel-vaults, due to the application of these structures in the roof covering of meeting halls, passenger terminals, industrial halls, aircraft hangars, etc., can lead to important and effective results in preventing the occurrence of progressive collapse and possible damages. Therefore, in this paper, the critical members of a two-layer tensegrity barrel-vaults consisting of square simplexes are identified by performing nonlinear dynamic analysis using Abaqus software. Abaqus software uses the Model Change settings to simulate the initial failure. The stability of the structure under the dynamic effects of failure of different members is investigated, and finally, the critical and non-critical members are introduced. It was found that the failure of members of the modules, which are closer to the center of the structure in the longitudinal direction of the barrel vault, shows more critical behavior than the members of other modules. Also, as observed in the results of analysis, members with maximum stress are not among the most critical members.