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.. _use-case-stressed-beam:
A simple stressed beam
======================
We consider a simple beam stressed by a traction load F at both sides.
.. figure:: ../_static/axial-stressed-beam.png
:align: center
:alt: use case geometry
:width: 50%
Beam geometry
The geometry is supposed to be deterministic; the diameter D is equal to:
.. math:: D=0.02 \textrm{ (m)}
By definition, the yield stress is the load divided by the surface. Since the surface is :math:`\pi D^2/4`, the stress is:
.. math:: S=\frac{F}{ \pi D^2/4}
Failure occurs when the beam plastifies, i.e. when the axial stress gets larger than the yield stress:
.. math:: R - \frac{F}{ \pi D^2/4} \leq 0
where :math:`R` is the strength.
Therefore, the limit state function :math:`G` is:
.. math:: G(R,F) = R - \frac{F}{\pi D^2/4},
for any :math:`R,F \in \mathbb{R}`.
The value of the parameter :math:`D` is such that:
.. math:: D^2/4 = 10^{-4},
which leads to the equation:
.. math:: G(R,F) = R - \frac{F}{10^{-4} \pi}.
We consider the following distribution functions.
======== ================================================================================
Variable Distribution
======== ================================================================================
R LogNormal( :math:`\mu_R= 3 \times 10^6`, :math:`\sigma_R=3 \times 10^5` ) [Pa]
F Normal( :math:`\mu_F=750` , :math:`\sigma_F=50`) [N]
======== ================================================================================
where :math:`\mu_R=E(R)` and :math:`\sigma_R^2=V(R)` are the mean and the variance of :math:`R`.
The failure probability is:
.. math:: P_f = \Prob{G(R,F) \leq 0}.
The exact :math:`P_f` is
.. math:: P_f = 0.02920.
API documentation
-----------------
.. currentmodule:: openturns.usecases.stressed_beam
.. autoclass:: AxialStressedBeam
:noindex:
Examples based on this use case
-------------------------------
.. minigallery:: openturns.usecases.stressed_beam.AxialStressedBeam
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