SORM

SORM improves on FORM by fitting a quadratic surface at the design point, correcting for curvature of the failure boundary.

Adding curvature

The second-order reliability method (SORM) starts from the FORM design point but approximates the limit-state surface with a paraboloid rather than a plane. Capturing the curvature of the failure boundary corrects the FORM probability, which can be significantly biased when the boundary bends toward or away from the origin.

The curvature correction

Kronos motion — design envelope

Breitung's asymptotic formula multiplies the FORM estimate by a factor involving the principal curvatures kappa_i of the limit state at the design point: P_f approximately equals Phi(-beta) times the product over i of (1 + beta*kappa_i)^(-1/2). Curvature that wraps the surface around the origin increases the failure probability; curvature away from it decreases the probability.

Getting the curvatures

Compute the Hessian of the performance function at the design point
Rotate to align one axis with the design-point direction
Extract the principal curvatures of the remaining subspace

When SORM matters

For small failure probabilities with a strongly curved limit state, FORM can err by a large relative factor while SORM stays accurate. The correction is most valuable at high reliability index, where the tangent-plane approximation drifts furthest from the true curved boundary.

Cautions

SORM requires second derivatives, which are costly and noisy for black-box models, and it still assumes a single dominant failure point. When the Hessian is unavailable or multiple failure modes exist, sampling-based methods such as subset simulation or surrogate-based reliability are more robust. SORM is best seen as a cheap curvature correction to FORM, not a general-purpose estimator.