First-Order Reliability Method

FORM estimates a failure probability by finding the most probable failure point and linearizing the limit state there.

The reliability problem

Reliability analysis estimates the probability that a performance function g(X) falls below zero (failure) given uncertain inputs X. The first-order reliability method (FORM) does this efficiently by transforming to standard normal space, locating the failure point closest to the origin, and approximating the failure surface by its tangent plane there.

The design point

Kronos motion — materials first

In standard normal space the joint density is highest at the origin and decays radially, so the point on the limit-state surface nearest the origin is the most probable failure point (the design point). Its distance from the origin, the reliability index beta, sets the failure probability estimate P_f approximately equal to Phi(-beta), where Phi is the standard normal CDF.

python

from scipy.stats import norm
# beta = distance from origin to design point (found by optimization)
Pf = norm.cdf(-beta)

Finding beta

Transform inputs to independent standard normals (Rosenblatt or Nataf transform)
Solve a constrained optimization: minimize the norm subject to g = 0
The optimum's norm is beta; its direction gives sensitivity to each input

Sensitivity output

A useful byproduct is the set of importance factors, the squared components of the design-point unit vector, which show how much each input contributes to the failure probability. This directs where to tighten tolerances or reduce uncertainty.

Limits

FORM's linearization is accurate only when the limit state is nearly linear near the design point and the probability is dominated by a single failure point. Strong curvature calls for the second-order method (see SORM), and multiple comparable failure points require multi-point or sampling approaches. FORM can also converge to a local, non-global design point, so restarts from different initial guesses are prudent.