| Two formulations have been developed for the determination of the reliability of general redundant structural systems, including systems with brittle components. One is the failure mode approach, which is based on ways in which a structure can fail. The other is the stable configuration approach, which is based on ways in which a structure can carry an applied load. For a class of structures, in which the load effects on the surviving elements never decrease with the failures of other elements, simplified formulations are introduced. Such simplifications for this type of structure result from the fact that the sequence in which the elements may fail is irrelevant. Results of the simplified formulation are generally on the conservative side.;The failure mode approach results in the probability of failure being determined as the probability of the union of the individual failure modes (paths). Each failure mode is composed of the intersection of events; all the branches in the path must fail.;The stable configuration approach leads to the probability of the intersection of the failures of the cuts; if any cut survives there is no possible path from the initial node to the terminal node of the failure graph. The failure of a cut is the union of the branches comprising the cut.;To aid in the development of the formulations, the failure graph concept is introduced, a failure graph being a directed graph of all possible sequences of component failures that lead to the prescribed limit state. Each path from the initial node to the terminal node of the graph represents a failure mode. A branch represents a component failure and each node of the graph (except the terminal node) represents a stable configuration of a structure. A cut is defined as a set of branches containing one branch from every path. |
