SORM: Second-Order Reliability Method (SORM)

In TesiproV: Reliability Analysis Methods for Structural Engineering

Second-Order Reliability Method (SORM)

Description

Second-Order Reliability Method (SORM) for the approximation of failure probabilities in structural reliability analysis.

The SORM extends the First-Order Reliability Method (FORM) by incorporating second-order information of the limit-state function at the design point. By accounting for local curvature effects, SORM generally provides more accurate failure probability estimates for nonlinear limit-state functions.

Failure is defined as g(x) < 0 (see FORM). The curvature correction is applied to the FORM result.

The method evaluates the principal curvatures of the limit-state surface in standard normal space and applies an asymptotic correction to the FORM probability of failure.

The curvatures are determined numerically from second derivatives of the limit-state function in standard normal space. For distributions that are not smooth, e.g. empirical distributions ("emp"), finite differences are dominated by sampling noise. If the second derivatives depend strongly on the step size, a local quadratic regression around the design point is used instead. The curvatures, and thus the SORM result, are then less precise than for smooth distributions and should be checked against FORM or a simulation method.

Usage

SORM(lsf, lDistr, debug.level = 0)

Arguments

lsf	objective function with limit state function in form of function(x) { x[1] + x[2] + ... }.
lDistr	list of distributions (distribution objects of TesiproV)
debug.level	If 0 no additional info if 2 high output during calculation

Value

SORM returns an object containing the following elements:

• beta: Hasofer-Lind reliability index.

• pf: Estimated probability of failure.

• u_points: Design point(s) in the standard normal space (u-space).

• dy: Gradient(s) of the limit-state function at the design point.

Author(s)

(C) 2021-2026 K. Nille-Hauf, T. Feiri, M. Ricker, T. Lux – Hochschule Biberach (until 2022), TU Dortmund University - Chair of Structural Concrete (since 2023)

References

Breitung, K. (1989). Asymptotic approximations for probability integrals. Probabilistic Engineering Mechanics, 4(4), 187-190.

Cai, G. Q., & Elishakoff, I. (1994). Refined second-order reliability analysis. Structural Safety, 14(4), 267-276.

Hohenbichler, M., Gollwitzer, S., Kruse, W., & Rackwitz, R. (1987). New light on first- and second-order reliability methods. Structural Safety, 4, 267-284.

Tvedt, L. (1990). Distribution of quadratic forms in normal space - Applications to structural reliability. Journal of Engineering Mechanics, 116(6), 1183-1197.

Marelli, S., & Sudret, B. (2014). UQLab: A framework for uncertainty quantification in Matlab. In Proceedings of the 2nd International Conference on Vulnerability, Risk Analysis and Management (ICVRAM2014), Liverpool, United Kingdom, 2554-2563.

Lacaze, S., & Missoum, S. (2015). CODES: A toolbox for computational design (Version 1.0). Retrieved from https://sites.arizona.edu/codes/

Wu, X. Z. (2017). Implementing statistical fitting and reliability analysis for geotechnical engineering problems in R. Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 11(2), 173-188.

TesiproV documentation built on Oct. 28, 2026, 5:13 p.m.