FORM Reliability Example

A worked example of FORM analysis on a simple limit state — transforming to standard-normal space, finding the design point and computing the reliability index.

Probabilistic & Reliability6 min read

worked exampleFORMreliability indexdesign pointstandard normal

Problem

A limit state g = R - S, where R ~ N(350, 25^2) and S ~ N(200, 20^2), both normal and independent. Compute the reliability index and failure probability using FORM.

Given

R ~ N(350, 25^2)
S ~ N(200, 20^2)
Independent
g = R - S

Solution

For a linear limit state with normal variables,
FORM is exact:

Mean of g:  mu_g = 350 - 200 = 150
Std of g:   sigma_g = sqrt(25^2 + 20^2) = sqrt(625 + 400) = sqrt(1025) = 32.0

Reliability index:
  beta = mu_g / sigma_g = 150 / 32.0 = 4.69

Failure probability:
  Pf = Phi(-beta) = Phi(-4.69) = 1.36e-6

Design point (in original space):
  R* = 350 - 25 * (25/32.0) * 4.69 = 350 - 91.6 = 258.4 MPa
  S* = 200 + 20 * (20/32.0) * 4.69 = 200 + 58.6 = 258.6 MPa

The design point is the most probable failure point:
R* ≈ S* ≈ 258 MPa (capacity meets demand at failure).

Interpretation

The reliability index of 4.69 corresponds to a failure probability of approximately 1.4 x 10^-6. The design point shows that failure is most likely when both R and S are approximately 258 MPa — the resistance is low and the demand is high. This is an illustrative example.