Second Order Reliability Method (SORM)

The following figure presents a graphical representation of SORM.

Compared to the First Order Reliability Method (FORM), SORM results in a better estimation of the reliability index when the failure function is nonlinear at the MPP. SORM improves FORM by including the curvature of the failure surface in the estimation using the approximation given in Breitung (1984):

P_{f} \approx Φ (- β) Π_{i = 1}^{n} \sqrt{(1 + β k_{i})}

In this equation $Φ$ is the standard normal distribution function, $β$ is the reliability-index obtained using FORM, and $k_{i}$ k_i are the principle curvatures of the failure surface in the standard normal space at the MPP. The principle curvature is obtained from the second-order derivatives of the failure function with respect to the random variables.