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Reliability Index
-----------------
The generalized reliability index :math:`\beta_{gen}` is used under the
following context: let :math:`\inputRV` be a probabilistic input
vector with joint density probability :math:`\pdf`, let :math:`\model(\inputRV)` be the limit state function of
the model and let :math:`\cD_f = \{\vect{x} \in \Rset^d \,
/ \, \model(\vect{x}) \le 0\}` be an event whose probability
:math:`P_f` is defined as:
.. math::
:label: PfX6
P_f = \int_{\model(\vect{x}) \le 0} \pdf\, d\vect{x}.
The generalized reliability index is defined as:
.. math::
\beta_{gen} = \Phi^{-1}(1-P_f) = -\Phi^{-1}(P_f),
where :math:`\Phi` is the CDF of the normal distribution with zero mean and unit variance.
As :math:`\beta_{gen}` increases, :math:`P_f` decreases rapidly.
According to the method used to evaluate :math:`P_f`, the generalized reliability index differs:
- when :math:`P_f` has been obtained from the :ref:`form_approximation` approximation, then :math:`\beta_{gen}` is
equal to the Hasofer-Lindt reliability index :math:`\beta`, which is the distance of the design point from the
origin of the standard space,
- when :math:`P_f` has been obtained from a :ref:`sorm_approximation` approximation, then :math:`\beta_{gen}` is
equal to :math:`\beta_{Breitung}`, :math:`\beta_{Tvedt}` or :math:`\beta_{Hohenbichler}`,
- when :math:`P_f` has been obtained from another technique (Monte Carlo simulations, importance samplings,...), we
get the generalized index :math:`\beta_{gen}`.
.. topic:: API:
- See :class:`~openturns.FORMResult`
- See :class:`~openturns.SORMResult`
.. topic:: Examples:
- See :doc:`/auto_reliability_sensitivity/reliability/plot_estimate_probability_form`
.. topic:: References:
- Cornell, "A probability-based structural code," Journal of the American Concrete
Institute, 1969, 66(12), 974-985.
- O. Ditlevsen, 1979, "Generalized Second moment reliability index,"
Journal of Structural Mechanics, ASCE, Vol.7, pp. 453-472.
- O. 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.
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