Reliability Index

The generalized reliability index \beta is used under the following context: \vect{X} is a probabilistic input vector, \pdf its joint density probability, \vect{d} a deterministic vector, g(\vect{X}\,,\,\vect{d}) the limit state function of the model, \cD_f = \{\vect{X} \in \Rset^n \, / \, g(\vect{X}\,,\,\vect{d}) \le 0\} the event considered here and g(,) = 0 its boundary.
The probability content of the event \cD_f is P_f:

(1)P_f = \int_{g(\vect{X}\,,\,\vect{d}) \le 0} \pdf\, d\vect{x}.

The generalized reliability index is defined as:

\beta_g = \Phi^{-1}(1-P_f) = -\Phi^{-1}(P_f).

As \beta_g increases, P_f decreases rapidly.

These indices are available:

  • \beta_{FORM} the FORM reliability index, where P_f is obtained with a FORM approximation (refer to ~): in this case, the generalized reliability index is equal to the Hasofer-Lindt reliability index \beta_{HL}, which is the distance of the design point from the origin of the standard space,

  • \beta_{SORM} the SORM reliability index, where P_f is obtained with a SORM approximation : Breitung, Hohenbichler or Tvedt (refer to ),

  • \beta_g the generalized reliability index, where P_f is obtained with another technique : Monte Carlo simulations, importance samplings,…

API:

Examples:

References:

  • Cornell, “A probability-based structural code,” Journal of the American Concrete Institute, 1969, 66(12), 974-985.

    1. Ditlevsen, 1979, “Generalized Second moment reliability index,” Journal of Structural Mechanics, ASCE, Vol.7, pp. 453-472.

    1. Ditlevsen and H.O. Madsen, 2004, “Structural reliability methods,” Department of mechanical engineering technical university of Denmark - Maritime engineering, internet publication.

  • Hasofer and Lind, 1974, “Exact and invariant second moment code format,” Journal of Engineering Mechanics Division, ASCE, Vol. 100, pp. 111-121.