A first reliability analysis#

Build a resistance-minus-load model, run FORM and check its result against an exact normal-distribution calculation.

Before you start: basic Python and the v2 installation. This example uses only PySTRA’s core dependencies.

Define the physical event#

Let resistance \(R\sim N(10,1^2)\) and load \(S\sim N(5,1^2)\) be independent, in the same force unit. Failure occurs when \(g(R,S)=R-S\leq0\). The constructor arguments below are the mean and standard deviation, not variance.

[1]:

import numpy as np
from scipy.stats import norm
import pystra as ra

model = ra.StochasticModel()
model.add_variable(ra.Normal("R", 10.0, 1.0))
model.add_variable(ra.Normal("S", 5.0, 1.0))
response = ra.LimitState(lambda R, S: R - S)

Run and inspect the analysis#

The workflow is model → analysis → result. Check convergence before using the probability or design point.

[2]:

analysis = ra.FORM(model=model, limit_state=response)
result = analysis.run()
assert result.converged, result.message
print(f"Reliability index: {result.beta:.4f}")
print(f"Failure probability: {result.failure_probability:.6g}")
print("Physical design point:", dict(zip(result.variable_names, result.design_point_x)))

Reliability index: 3.5355
Failure probability: 0.000203476
Physical design point: {'R': np.float64(7.500000000000835), 'S': np.float64(7.500000000001386)}

Check the known answer#

Here \(R-S\) is normal with mean 5 and variance \(1^2+1^2=2\). Its exact reliability index is \(5/\sqrt{2}\) and its failure probability is \(\Phi(-5/\sqrt{2})\). FORM is exact for this plane in independent normal coordinates.

[3]:

exact_beta = (10.0 - 5.0) / np.sqrt(1.0**2 + 1.0**2)
exact_probability = norm.sf(exact_beta)
np.testing.assert_allclose(result.beta, exact_beta, rtol=1e-6)
np.testing.assert_allclose(result.failure_probability, exact_probability, rtol=1e-6)
print(f"Exact failure probability: {exact_probability:.6g}")

Exact failure probability: 0.000203476

Interpretation#

The failure probability is about 0.0002035 for the event described by these input distributions. It is not automatically an annual probability: that interpretation requires an annual event model. The physical design point is approximately \(R=S=7.5\).

For nonlinear models, numerical convergence does not establish approximation accuracy. Continue with comparing FORM, SORM and simulation, interpreting results, the reliability API and design-point theory.

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