.. _use-case-deflection-tube: Vertical deflection of a tube ============================= Description ----------- We consider the deflection of a tube under a vertical stress. .. figure:: ../_static/simply_supported_beam.png :align: center :alt: simply supported beam :width: 50% A simply supported beam The parameters of the model are: * F : the strength, * L : the length of the tube, * a : position of the force, * D : external diameter of the tube, * d : internal diameter of the tube, * E : Young modulus. The following figure presents the internal and external diameter of the tube: .. figure:: ../_static/tube-diameters.png :align: center :alt: tube diameters :width: 50% Tube diameters The area moment of inertia of the cross section about the neutral axis of a round tube (i.e. perpendicular to the section) with external and internal diameters :math:D and :math:d are: .. math:: I = \frac{\pi (D^4-d^4)}{32}. The vertical deflection at point :math:x=a is: .. math:: g_1(X) = - F \frac{a^2 (L-a)^2}{3 E L I}, where :math:X=(F,L,a,D,d,E). The angle of the tube at the left end is: .. math:: g_2(X) = - F \frac{b (L^2-b^2)}{6 E L I}, and the angle of the tube at the right end is: .. math:: g_3(X) = F \frac{a (L^2-a^2)}{6 E L I}. The following table presents the distributions of the random variables. These variables are assumed to be independent. ======== ========================== Variable Distribution ======== ========================== F Normal(1,0.1) L Normal(1.5,0.01) a Uniform(0.7,1.2) D Triangular(0.75,0.8,0.85) d Triangular(0.09,0.1,0.11) E Normal(200000,2000) ======== ========================== References ---------- * Deflection of beams by Russ Elliott. http://www.clag.org.uk/beam.html * https://upload.wikimedia.org/wikipedia/commons/f/ff/Simple_beam_with_offset_load.svg * https://en.wikipedia.org/wiki/Deflection_(engineering) * https://mechanicalc.com/reference/beam-deflection-tables * https://en.wikipedia.org/wiki/Second_moment_of_area * Shigley's Mechanical Engineering Design (9th Edition), Richard G. Budynas, J. Keith Nisbettn, McGraw Hill (2011) * Mechanics of Materials (7th Edition), James M. Gere, Barry J. Goodno, Cengage Learning (2009) * Statics and Mechanics of Materials (5th Edition), Ferdinand Beer, E. Russell Johnston, Jr., John DeWolf, David Mazurek. Mc Graw Hill (2011) Chapter 15: deflection of beams. Load the use case ----------------- We can load this classical model from the use cases module as follows : .. code-block:: python >>> from openturns.usecases import deflection_tube as deflection_tube >>> # Load the tube deflection model >>> dt = deflection_tube.DeflectionTube() API documentation ----------------- See :class:~openturns.usecases.deflection_tube.DeflectionTube. Examples based on this use case ------------------------------- .. raw:: html
.. only:: html .. figure:: /auto_calibration/least_squares_and_gaussian_calibration/images/thumb/sphx_glr_plot_calibration_deflection_tube_thumb.png :alt: :ref:sphx_glr_auto_calibration_least_squares_and_gaussian_calibration_plot_calibration_deflection_tube.py .. raw:: html
.. toctree:: :hidden: /auto_calibration/least_squares_and_gaussian_calibration/plot_calibration_deflection_tube