.. _use-case-cantilever-beam: The cantilever beam model ========================= We are interested in the the vertical deviation of a diving board created by a child diver. We consider a child whose weight generates a force approximately equal to 300N (i.e. almost 30 kg). Because of the uncertainties in the weight of the person, we consider that the force is a random variable. The length of the diving board is between 2.5 m and 2.6 m. The Young modulus is uncertain and between 65 and 75 GPa, which corresponds to the fiberglass material, a material often used for diving boards. Uncertainties in the production of the material are taken into account in the Young modulus and the section modulus of the board. We consider a cantilever beam defined by its Youngâ€™s modulus :math:E, its length :math:L and its section modulus :math:I. One end of the cantilever beam is built in a wall and we apply a concentrated bending load :math:F at the other end of the beam, resulting in a deviation :math:Y. .. figure:: ../_static/beam.png :align: center :alt: beam geometry :width: 25% The beam geometry **Inputs** * :math:E : Young modulus (Pa), Beta(:math:\alpha = 0.9, :math:\beta = 3.5, a = :math:65.0 \times 10^9, :math:b = 75.0 \times 10^9) * :math:F : Loading (N), Lognormal(:math:\mu_F=300.0, :math:\sigma_F=30.0, shift=0.0) * :math:L : Length of beam (m), Uniform(min=2.5, max= 2.6) * :math:I : Moment of inertia (:math:m^4), Beta(:math:\alpha = 2.5, :math:\beta = 4.0, :math:a = 1.3 \times 10^{-7}, :math:b = 1.7 \times 10^{-7}). In the previous table :math:\mu_F=E(F) and :math:\sigma_F=\sqrt{V(F)} are the mean and the standard deviation of :math:F. We assume that the random variables E, F, L and I are dependent and associated with a gaussian copula which correlation matrix is : .. math:: R = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -0.2 \\ 0 & 0 & -0.2 & 1 \end{pmatrix} In other words, we consider that the variables L and I are negatively correlated : when the length L increases, the moment of intertia I decreases. **Output** The vertical displacement at free end of the cantilever beam is: .. math:: Y = \dfrac{F\, L^3}{3 \, E \, I} A typical event of interest is when the beam deviation is too large which is a failure : .. math:: Y \ge 0.30 (m) Load the use case ----------------- We can load this classical model from the use cases module as follows : .. code-block:: python >>> from openturns.usecases import cantilever_beam >>> # Load the cantilever beam example >>> cb = cantilever_beam.CantileverBeam() API documentation ----------------- See :class:~openturns.usecases.cantilever_beam.CantileverBeam. Examples based on this use case ------------------------------- .. raw:: html
.. only:: html .. figure:: /auto_reliability_sensitivity/central_dispersion/images/thumb/sphx_glr_plot_central_tendency_thumb.png :alt: :ref:sphx_glr_auto_reliability_sensitivity_central_dispersion_plot_central_tendency.py .. raw:: html
.. toctree:: :hidden: /auto_reliability_sensitivity/central_dispersion/plot_central_tendency .. raw:: html
.. only:: html .. figure:: /auto_meta_modeling/polynomial_chaos_metamodel/images/thumb/sphx_glr_plot_chaos_cantilever_beam_integration_thumb.png :alt: :ref:sphx_glr_auto_meta_modeling_polynomial_chaos_metamodel_plot_chaos_cantilever_beam_integration.py .. raw:: html
.. toctree:: :hidden: /auto_meta_modeling/polynomial_chaos_metamodel/plot_chaos_cantilever_beam_integration .. raw:: html
.. only:: html .. figure:: /auto_reliability_sensitivity/reliability/images/thumb/sphx_glr_plot_estimate_probability_directional_sampling_thumb.png :alt: :ref:sphx_glr_auto_reliability_sensitivity_reliability_plot_estimate_probability_directional_sampling.py .. raw:: html
.. toctree:: :hidden: /auto_reliability_sensitivity/reliability/plot_estimate_probability_directional_sampling .. raw:: html
.. only:: html .. figure:: /auto_reliability_sensitivity/reliability/images/thumb/sphx_glr_plot_estimate_probability_form_thumb.png :alt: :ref:sphx_glr_auto_reliability_sensitivity_reliability_plot_estimate_probability_form.py .. raw:: html
.. toctree:: :hidden: /auto_reliability_sensitivity/reliability/plot_estimate_probability_form .. raw:: html
.. only:: html .. figure:: /auto_reliability_sensitivity/reliability/images/thumb/sphx_glr_plot_estimate_probability_importance_sampling_thumb.png :alt: :ref:sphx_glr_auto_reliability_sensitivity_reliability_plot_estimate_probability_importance_sampling.py .. raw:: html
.. toctree:: :hidden: /auto_reliability_sensitivity/reliability/plot_estimate_probability_importance_sampling .. raw:: html
.. only:: html .. figure:: /auto_reliability_sensitivity/reliability/images/thumb/sphx_glr_plot_estimate_probability_randomized_qmc_thumb.png :alt: :ref:sphx_glr_auto_reliability_sensitivity_reliability_plot_estimate_probability_randomized_qmc.py .. raw:: html
.. toctree:: :hidden: /auto_reliability_sensitivity/reliability/plot_estimate_probability_randomized_qmc .. raw:: html
.. only:: html .. figure:: /auto_meta_modeling/kriging_metamodel/images/thumb/sphx_glr_plot_kriging_beam_arbitrary_trend_thumb.png :alt: :ref:sphx_glr_auto_meta_modeling_kriging_metamodel_plot_kriging_beam_arbitrary_trend.py .. raw:: html
.. toctree:: :hidden: /auto_meta_modeling/kriging_metamodel/plot_kriging_beam_arbitrary_trend .. raw:: html
.. only:: html .. figure:: /auto_meta_modeling/kriging_metamodel/images/thumb/sphx_glr_plot_kriging_beam_trend_thumb.png :alt: :ref:sphx_glr_auto_meta_modeling_kriging_metamodel_plot_kriging_beam_trend.py .. raw:: html
.. toctree:: :hidden: /auto_meta_modeling/kriging_metamodel/plot_kriging_beam_trend .. raw:: html
.. only:: html .. figure:: /auto_meta_modeling/kriging_metamodel/images/thumb/sphx_glr_plot_kriging_cantilever_beam_hmat_thumb.png :alt: Create a copula :ref:sphx_glr_auto_meta_modeling_kriging_metamodel_plot_kriging_cantilever_beam_hmat.py .. raw:: html
.. toctree:: :hidden: /auto_meta_modeling/kriging_metamodel/plot_kriging_cantilever_beam_hmat .. raw:: html
.. only:: html .. figure:: /auto_meta_modeling/kriging_metamodel/images/thumb/sphx_glr_plot_kriging_cantilever_beam_thumb.png :alt: :ref:sphx_glr_auto_meta_modeling_kriging_metamodel_plot_kriging_cantilever_beam.py .. raw:: html
.. toctree:: :hidden: /auto_meta_modeling/kriging_metamodel/plot_kriging_cantilever_beam