Structural reliability
Deterministic design asks whether the margin is positive. Reliability asks what the probability is that it is negative — the question a code safety factor is really a proxy for. FORM transforms every variable to standard normal, finds the closest point on the failure surface, and reports that distance as β.
| Quantity | Value | Note |
|---|---|---|
| Mean resistance | 100.0 | σ 15.0 |
| Mean dead load | 30.0 | σ 8.0 |
| Mean live load | 20.0 | σ 6.0 |
| Mean margin | 50.00 | R − D − L |
| Standard deviation | 18.028 | √(σR² + σD² + σL²) |
| Reliability index β | 2.7735 | margin / σ |
| Probability of failure | 0.00277289 | Φ(−β) |
| Safety factor | 2.000 | on the means |
| Direction cosine, R | 0.8321 | share of uncertainty |
| Direction cosine, D | -0.4438 | |
| Direction cosine, L | -0.3328 | |
| Design point uR | -2.3077 | standard normal coordinates |
| Design point uD | 1.2308 | |
| Design point uL | 0.9231 | |
| Monte Carlo Pf | 0.002300 | 40,000 samples |
| Resistance for β = 3.5 | 113.1 | the target index |
The direction cosines are the useful output: they say which variable is driving the risk. If the resistance term dominates, buy stronger material; if the load term does, the answer is a different structural form.
g = R − D − L · Failure when g < 0.
β = (μR − μD − μL)/√(σR²+σD²+σL²) · Exact for linear, normal variables.
Pf = Φ(−β) · β = 3.5 is about 2.3×10⁻⁴.
min ||u|| subject to g(u) = 0 · Hasofer-Lind.
A clear path from inputs to results.
Deterministic design asks whether the margin is positive. Reliability asks what the probability is that it is negative — the question a code safety factor is really a proxy for. FORM transforms every variable to standard normal, finds the closest point on the failure surface, and reports that distance as β.
How to use
- Enter the mean and standard deviation of the resistance.
- Enter the dead and live load means and standard deviations.
- Read the reliability index, the failure probability and the design point.
- Use the direction cosines to see which variable is driving the risk.
The formula
Reference: Hasofer & Lind (1974); Ditlevsen & Madsen, Structural Reliability Methods; EN 1990 for the β = 3.5 calibration.
Worked example
A member with a mean safety factor of 2.0
- Resistance 100 ± 15, dead load 30 ± 8, live load 20 ± 6.
- All normal and independent.
- 1
The mean margin is 100 − 30 − 20 = 50.
- 2
The standard deviation of the margin is √(225 + 64 + 36) = 18.03.
- 3
The reliability index is 50 / 18.03 = 2.774.
- 4
The failure probability is Φ(−2.774) = 2.77×10⁻³, which 40,000 Monte Carlo samples confirm at 2.30×10⁻³.
- 5
That is short of the β = 3.5 target, which would need a mean resistance of 113.1 for the same variability.
β 2.774 · Pf 2.8×10⁻³ · resistance 113.1 needed for β = 3.5.
Interpreting the result
A safety factor of 2.0 sounds comfortable and produces a failure probability of 0.3% — about one in 360. The index is the honest number, and the direction cosines are the useful one: they say whether to buy stronger material or change the structural form.
Assumptions & limitations
- A linear limit state g = R − D − L.
- All three variables normal and independent.
- The means and standard deviations are known, not estimated.
- Monte Carlo uses 40,000 seeded samples.
Frequently asked questions
Why is β = 3.5 the target?
It corresponds to a failure probability of about 2.3×10⁻⁴ over the design life, which is the calibration behind the partial factors in the Eurocodes and most national codes.
What are direction cosines?
The unit vector from the origin to the design point. Each component squared is that variable’s share of the total uncertainty, so they tell you where to spend money.
What if the variables are not normal?
FORM transforms them — lognormal, Gumbel, Weibull — into equivalent normals at the design point. The linear case here is the exact one; the transformation is where the approximation enters.
Why does Monte Carlo differ slightly?
Sampling error. At 40,000 samples the standard error is about 2.6×10⁻⁴, so the two agree within the noise — which is the point of running both.
Methodology last reviewed 2026-10-22. Results are informational and not financial or engineering advice — see our disclaimer.