reliability-based-design

name	reliability-based-design
description	Reliability-based design optimization (RBDO) — FORM/SORM reliability index β, FOSM, probability of failure P_f, Monte Carlo simulation, design point, sensitivity factors, ASCE 7 load combinations.
metadata	{"priority":7,"promptSignals":{"phrases":["reliability based design","RBDO","reliability index","FORM reliability","probability of failure","FOSM method","structural reliability"],"minScore":3}}

Reliability-Based Design — Complete Skill

Fundamental Concepts

Limit State Function

Performance function (limit state): g(X) = R(X) - S(X) [R = resistance; S = load/demand; X = vector of random variables]

g > 0: safe domain; g < 0: failure domain; g = 0: limit state surface

Probability of failure: P_f = P(g(X) ≤ 0) = ∫_{g≤0} f_X(x) dx

Reliability: P_s = 1 - P_f

First-Order Second-Moment Method (FOSM)

Assume linear g and normal distributions: μ_g = g(μ_X) [mean of g evaluated at mean values] σ_g = √(Σ (∂g/∂X_i)² × σ_Xi²) [standard deviation of g]

Reliability index β: β = μ_g / σ_g

Probability of failure (for linear, normal): P_f = Φ(-β) [Φ = standard normal CDF]

β	P_f
1.0	0.159
2.0	0.023
3.0	0.00135
3.5	2.3 × 10⁻⁴
4.0	3.2 × 10⁻⁵
4.75	10⁻⁶

Target β values: Structural failure (building, ultimate): β = 3.5–4.0 (P_f ≈ 10⁻⁴ to 10⁻⁵) Nuclear safety: β = 5–6 (P_f ≈ 10⁻⁷) Serviceability limit state: β = 1.5–2.5

First-Order Reliability Method (FORM)

Non-linear g, non-normal variables:

1. Transform variables to standard normal space U: u_i = (X_i - μ_Xi)/σ_Xi
2. Transform non-normal distributions to equivalent normal (Rackwitz-Fiessler)
3. Find design point u* = argmin ||u|| s.t. g(u) = 0
4. β = ||u*|| (distance from origin to limit state in U-space)
5. P_f ≈ Φ(-β)

Design point (most probable failure point, MPFP): Point on g = 0 with maximum probability density → closest to origin in standard space Found by constrained optimization (HL-RF algorithm or gradient methods)

Sensitivity factors α_i = -∂u/∂β:* α_i² = fraction of variance contributed by variable X_i |α_i| near 1: dominant variable; near 0: negligible

Second-Order Reliability Method (SORM)

Improves on FORM when limit state is highly nonlinear Fits second-order surface (paraboloid) to g at design point Breitung formula: P_f ≈ Φ(-β) × Π (1 - β κ_i)^(-0.5)

κ_i = principal curvatures of limit state at design point More accurate for P_f < 10⁻⁶ or when g is highly nonlinear

Monte Carlo Simulation (MCS)

Direct simulation: Sample N realizations of X from f_X(x) Count failures: N_f = #{i : g(x_i) ≤ 0} P_f ≈ N_f / N; σ_P_f ≈ √(P_f(1-P_f)/N)

Required samples for coefficient of variation δ: N ≥ 1/(P_f × δ²) [δ = 0.1 for 10% CV; δ = 0.05 for 5% CV]

Example: P_f = 10⁻⁴, δ = 0.1 → N = 10^6 → expensive Variance reduction: Importance Sampling (IS); Latin Hypercube; Subset Simulation

Importance Sampling (IS)

Sample near design point u*; apply importance weights Reduces N by 10²–10⁴ for small P_f h_sampling(u) = φ(u - u*) [shift to center sampling at design point]

Subset Simulation (Au & Beck)

Adaptively estimate P_f as product of conditional probabilities P_f = p₁ × p₂ × ... × p_m where each p_i ≈ 0.1 Efficient for P_f < 10⁻⁶; N ≈ 10,000–50,000 total samples

Reliability-Based Design Optimization (RBDO)

Problem formulation: Minimize: cost C(d, X) Subject to: P_f,i = P(g_i(d,X) ≤ 0) ≤ P_f,target,i for each constraint i d_lower ≤ d ≤ d_upper

d = deterministic design variables; X = random variables

RIA (Reliability Index Approach)

Constraints: β_i(d) ≥ β_target,i Compute β by FORM for each constraint evaluation Nested loop: outer optimization over d; inner FORM for each β

PMA (Performance Measure Approach)

Invert problem: find g* such that β = β_target Constraint: g*(d) ≥ 0 (much easier to evaluate than β) More stable convergence; preferred in practice

Sensitivity of Reliability

Importance factor for variable X_i (FORM): γ_i = ∂β/∂μ_Xi [sensitivity of β to mean shift] = -α_i / σ_Xi [per unit of mean; positive = increasing mean increases β]

Uncertainty importance: The contribution of each random variable to P_f variability High |α_i| → focus design changes and testing on that variable

Load and Resistance Factor Design (LRFD)

Calibrated to target reliability β: Φ R_n ≥ Σ γ_i Q_i

Φ = resistance factor (< 1); accounts for resistance uncertainty γ_i = load factors (> 1); accounts for load uncertainty

AISC steel: Φ = 0.9 (tension yielding); 0.75 (fracture); 0.85 (compression) ACI concrete: Φ = 0.9 (flexure); 0.75 (shear); 0.65 (compression)

ASCE 7-22 Load Combinations:

1. 1.4D
2. 1.2D + 1.6L + 0.5(L_r or S or R)
3. 1.2D + 1.6(L_r or S or R) + (L or 0.5W)
4. 1.2D + 1.0W + L + 0.5(L_r or S or R)
5. 1.2D + 1.0E + L + 0.2S
6. 0.9D + 1.0W (overturning check)
7. 0.9D + 1.0E (seismic uplift check)

D = dead; L = live; L_r = roof live; S = snow; W = wind; E = seismic; R = rain

Load Combination Calibration

β = 3.5 for structural failure; ASCE 7 factors calibrated to achieve this For materials/conditions not covered by AISC/ACI: use FORM/MCS to verify β ≥ target

Time-Variant Reliability (Degradation)

Corrosion: thickness t(T) = t₀ - C_r × T [C_r = corrosion rate; varies/year] → P_f increases with time as resistance decreases

Fatigue crack growth: da/dN = C(ΔK)^m; P_f increases as a approaches a_c

Time-variant P_f: P_f(T) = P(g(X, T) ≤ 0) = ∫₀^T λ_f(t) dt [for Poisson failure process] λ_f(t) = time-varying failure rate from degradation

First passage problem: P_f = P(R(t) < S at any t ∈ [0,T]) Requires outcrossing rate ν⁺ calculation

Output

Provide: reliability index β, probability of failure P_f = Φ(-β), sensitivity factors α_i² (% contribution of each variable to P_f), design point values x* (most probable failure combination), LRFD equivalent load and resistance factors if applicable, required sample size N for Monte Carlo at target P_f precision, dominant failure mode sensitivity (which variable change reduces P_f most per unit investment), and comparison to target reliability per applicable code (ASCE 7, ISO 2394, EN 1990).