pystra.sorm.Sorm#

class Sorm(stochastic_model=None, limit_state=None, analysis_options=None, form=None)[source]#

Bases: AnalysisObject

Second Order Reliability Method (SORM).

Approximates the failure surface in standard normal space using a quadratic surface, improving on the linear FORM approximation. Two approaches are available:

Curve-fitting (fit_type='cf', default): computes the Hessian of the limit state function at the design point, extracts principal curvatures as eigenvalues, and applies the Breitung formula.

Point-fitting (fit_type='pf'): locates fitting points directly on the failure surface on both sides of each principal axis using Newton iteration, then computes curvatures from their positions. This yields asymmetric curvatures (different on the positive and negative sides of each axis).

Both methods report the Breitung [Breitung1984] and the Hohenbichler-Rackwitz [Hohenbichler1988] failure probabilities and generalised reliability indices.

Parameters:

stochastic_model (StochasticModel, optional) – The stochastic model with random variables and correlations.
limit_state (LimitState, optional) – The limit state function.
analysis_options (AnalysisOptions, optional) – Options controlling the analysis.
form (Form, optional) – A pre-computed FORM result. If None, FORM is run automatically.

betaHL#

Hasofer-Lind reliability index (from FORM).

Type:: float or ndarray

kappa#

Principal curvatures. For curve-fitting, a 1-D array of length nrv - 1. For point-fitting, the sorted average of the positive and negative curvatures.

Type:: ndarray

kappa_pf#

Asymmetric curvatures from point-fitting, shape (2, nrv - 1) with rows [kappa_minus, kappa_plus]. None for curve-fitting.

Type:: ndarray or None

fit_type#

The fitting method used: 'cf', 'pf', or None if not yet run.

Type:: str or None

pf2_breitung#

Failure probability from the Breitung approximation.

Type:: float or ndarray

betag_breitung#

Generalised reliability index from the Breitung approximation.

Type:: float or ndarray

pf2_breitung_m#

Failure probability from the modified Breitung (Hohenbichler-Rackwitz).

Type:: float or ndarray

betag_breitung_m#

Generalised reliability index from the modified Breitung.

Type:: float or ndarray

Class constructor

Methods

`computeHessian`	Computes the matrix of second derivatives using forward finite difference, using the evaluation of the gradient already done for FORM, at the design point
`evaluateLSF`	For use in computing the Hessian without altering the FORM object.
`gram_schmidt`	Creates an orthonormal matrix using the modified Gram-Schmidt process.
`init_run`	Initialise the Nataf transformation before the analysis loop.
`orthonormal_matrix`	Computes the rotation matrix of the standard normal coordinate space where the design point is located at Beta along the last axis.
`pf_breitung`	Calculates the probability of failure and generalized reliability index using [Breitung1984] formula.
`pf_breitung_m`	Calculates the probability of failure and generalized reliability index using Brietung's formula ([Breitung1984]) as modified by Hohenbichler and Rackwitz [Hohenbichler1988].
`run`	Run SORM analysis.
`run_curvefit`	Run SORM analysis using curve fitting
`run_pointfit`	Run SORM analysis using point-fitting.
`showDetailedOutput`	Print detailed FORM/SORM comparison to the console.
`showResults`	Print a compact summary of the SORM results.

run(fit_type='cf')[source]#

Run SORM analysis.

Parameters:

fit_type ({'cf', 'pf'}, optional) –

Fitting method to use (default 'cf'):

'cf' — Curve-fitting via Hessian eigenvalues.
'pf' — Point-fitting via Newton iteration on the failure surface.

Raises:

ValueError – If fit_type is not 'cf' or 'pf'.

run_curvefit()[source]#: Run SORM analysis using curve fitting

run_pointfit()[source]#

Run SORM analysis using point-fitting.

Finds fitting points on the limit state surface on both the positive and negative sides of each principal axis in the rotated standard normal space. Curvatures are computed from the positions of these points, producing asymmetric curvatures that are stored in kappa_pf.

The generalized Breitung formula for asymmetric curvatures is:

\[p_{f2} = \Phi(-\beta) \prod_{i=1}^{n-1} \frac{1}{2} \left[ (1 + \beta \kappa_i^+)^{-1/2} + (1 + \beta \kappa_i^-)^{-1/2} \right]\]

Notes

Based on the point-fitting implementation contributed by Henry Nguyen (Monash University, PR #65).