Stochastic Analysis & Reliability
When loads and materials are uncertain — what's the probability of failure?
1. What is it?
Stochastic analysis quantifies how uncertainty in inputs (loads, material properties, geometry) affects the output (stress, displacement, life):
- FORM (First-Order Reliability Method): computes the reliability index β and probability of failure Pf.
- Monte Carlo simulation: runs thousands of random samples to build a failure distribution.
- Sensitivity analysis: identifies which uncertain variables matter most.
"This bracket has a 5% probability of failure under the load spectrum — is that acceptable?"
2. When do I use it?
| Use when | Don't use for |
|---|---|
| Loads are random or variable | Deterministic loads (use Static Structural) |
| Material properties vary | Single value known (use Static Structural) |
| Safety factor is insufficient | Simple pass/fail check (use Static Structural) |
| Risk-based design | Code-based design (use Static Structural) |
3. The flow — every action described
Step 1: Design (geometry)
What you do: Draw your part or import a CAD file.
How:
- Click Design tab
- Pick a primitive or import geometry
Why: The solver needs to know the shape.
Step 2: Boundary (material + supports + loads + uncertainty)
What you do: Define deterministic and random variables.
How:
- Click Boundary tab
- Material: Define mean and standard deviation for E, ν, σY
- Supports (BCs): Fix faces/edges
- Loads: Define mean and standard deviation for forces/pressures
- Limit state: Define the failure criterion (e.g., σ_vm < σY)
Why: The solver needs to know which variables are uncertain and how.
Step 3: Mesh
What you do: Cut the part into small cells.
How:
- Click Mesh tab
- Adjust mesh size
- Click Generate
Why: The solver works on tiny triangles/quads.
Step 4: Simulate → Stochastic
What you do: Run the reliability analysis.
How:
- Click Simulate tab
- Select Stochastic
- Choose FORM or Monte Carlo
- Set number of samples (Monte Carlo) or convergence tolerance (FORM)
- Click Run
Why: The solver computes the probability of failure.
Step 5: Results
What you do: Read the reliability index and sensitivity.
How:
- Click Results tab
- View reliability index β — higher is safer
- View probability of failure Pf — lower is safer
- View sensitivity — which variables matter most?
Why: The reliability index tells you how safe the part is.
4. Reading your results
Reliability index β
| Value | Meaning |
|---|---|
| β < 2.0 | High risk — Pf > 2.3% |
| β = 2.0–3.5 | Moderate risk — Pf = 0.23%–2.3% |
| β = 3.5–5.0 | Low risk — Pf = 0.02%–0.23% |
| β > 5.0 | Very low risk — Pf < 0.02% |
Good: β > required (typically 3.5 for structural, 4.5 for aerospace). Bad: β < 2.0 — redesign needed.
Probability of failure Pf
| Value | Meaning |
|---|---|
| Pf < 0.001 | Acceptable for most applications |
| Pf > 0.01 | Unacceptable — redesign |
Sensitivity
Shows which uncertain variables contribute most to failure probability.
5. Boundaries of truth
What's validated
| Case | Reference | Error |
|---|---|---|
| FORM on linear-normal limit state | Hasofer-Lind exact | < 0.01% |
| Determinism across repeated calls | Bit-identical | 0 |
| Honest-failure path (insensitive limit state) | Returns null | Correct |
What's NOT covered
- Non-Gaussian distributions — only normal and lognormal
- System reliability — multiple failure modes
- Time-dependent reliability — corrosion, fatigue
- Spatial variability — random fields
Common mistakes
| Mistake | Fix |
|---|---|
| Too few Monte Carlo samples | Increase samples for better accuracy |
| Wrong distribution type | Choose normal, lognormal, or uniform |
| Limit state insensitive to variables | Check that variables appear in limit state |
| No convergence tolerance | Set tolerance for FORM iteration |
| Ignoring correlation | Define correlation between variables |
See also
- [Static Structural](static-structural.md) — for deterministic analysis
- [Fatigue](vibration-fatigue.md) — for time-dependent reliability
- [Fracture](fracture.md) — for crack reliability
Keep exploring
Open the interactive workspace — mesh, solve, validate and export in the browser.
Start a guided solve