Solution for Second-Moment Reliability Index

This paper presents a general method for determination of the magnitude of the second-moment reliability index in a probabilistic design procedure. The minimum distance from the origin of a transformed parameter space to a failure surface (defined by a particular failure criterion) is deduced from consideration of the geometry of the failure surface and that of a sphere centered on the origin. The result is expressed in the original (untransformed) design parameters and is expressed in terms of the gradient of the failure surface and the means and covariances of the design parameters. The solution is shown to coincide with previous analysis for linear failure conditions and a general method for numerical solution in the case of many parameters and nonlinear failure criteria is also considered.