.. _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 = \text{Prob}(G(R,F) \leq 0). The exact :math:P_f is .. math:: P_f = 0.02920. Load the use case ----------------- We can load this classical model from the use cases module as follows : .. code-block:: python >>> from openturns.usecases import stressed_beam as stressed_beam >>> # Load the use case axial stressed beam >>> sb = stressed_beam.AxialStressedBeam() API documentation ----------------- See :class:~openturns.usecases.stressed_beam.AxialStressedBeam. Examples based on this use case ------------------------------- .. raw:: html
.. only:: html .. figure:: /auto_reliability_sensitivity/reliability/images/thumb/sphx_glr_plot_axial_stressed_beam_quickstart_thumb.png :alt: :ref:sphx_glr_auto_reliability_sensitivity_reliability_plot_axial_stressed_beam_quickstart.py .. raw:: html
.. toctree:: :hidden: /auto_reliability_sensitivity/reliability/plot_axial_stressed_beam_quickstart .. raw:: html
.. only:: html .. figure:: /auto_reliability_sensitivity/reliability/images/thumb/sphx_glr_plot_axial_stressed_beam_thumb.png :alt: :ref:sphx_glr_auto_reliability_sensitivity_reliability_plot_axial_stressed_beam.py .. raw:: html
.. toctree:: :hidden: /auto_reliability_sensitivity/reliability/plot_axial_stressed_beam .. raw:: html
.. only:: html .. figure:: /auto_reliability_sensitivity/reliability/images/thumb/sphx_glr_plot_estimate_probability_lhs_thumb.png :alt: :ref:sphx_glr_auto_reliability_sensitivity_reliability_plot_estimate_probability_lhs.py .. raw:: html
.. toctree:: :hidden: /auto_reliability_sensitivity/reliability/plot_estimate_probability_lhs .. raw:: html
.. only:: html .. figure:: /auto_reliability_sensitivity/reliability/images/thumb/sphx_glr_plot_estimate_probability_monte_carlo_thumb.png :alt: :ref:sphx_glr_auto_reliability_sensitivity_reliability_plot_estimate_probability_monte_carlo.py .. raw:: html
.. toctree:: :hidden: /auto_reliability_sensitivity/reliability/plot_estimate_probability_monte_carlo