Addition laws of failure probability and their applications in reliability analysis of structural system with multiple failure modes

Compared with the methods solving failure probability of the structural system with multiple failure modes, those solving failure probability of a single failure mode are simpler and more well-developed, thus in order to employ the latter to establish the former, the addition laws of the failure probability are derived mathematically by use of the basic principles of probability theory. In the derived addition laws, the failure probability of the structural system with n failure modes is expressed as a combination of the failure probabilities of 2 ⁿ −1 single failure modes. Therefore, the failure probability of the structural system with multiple failure modes can be solved by the well-developed methods for the failure probability of a single failure mode. After reviewing the boundary theories, such as the second-order boundary, the third-order boundary, and the linear programming based boundary for analyzing the failure probability of the structural system with multiple failure modes, the derived addition laws are applied to evaluate several former order joint failure probability involved in those boundary theories. Additionally, a new small-scale linear programming based boundary theory which can sufficiently reduce the scale of the linear programming model involved is proposed. Two numerical examples, including a series and a parallel structural system, are employed to demonstrate the accuracy and efficiency of proposed techniques.