1.. _use-case-deflection-tube: 2 3Vertical deflection of a tube 4============================= 5 6 7Description 8----------- 9 10We consider the deflection of a tube under a vertical stress. 11 12.. figure:: ../_static/simply_supported_beam.png 13 :align: center 14 :alt: simply supported beam 15 :width: 50% 16 17 A simply supported beam 18 19The parameters of the model are: 20 21* F : the strength, 22* L : the length of the tube, 23* a : position of the force, 24* D : external diameter of the tube, 25* d : internal diameter of the tube, 26* E : Young modulus. 27 28The following figure presents the internal and external diameter of the tube: 29 30.. figure:: ../_static/tube-diameters.png 31 :align: center 32 :alt: tube diameters 33 :width: 50% 34 35 Tube diameters 36 37The 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: 38 39.. math:: 40 41 I = \frac{\pi (D^4-d^4)}{32}. 42 43 44The vertical deflection at point :math:`x=a` is: 45 46.. math:: 47 g_1(X) = - F \frac{a^2 (L-a)^2}{3 E L I}, 48 49 50where :math:`X=(F,L,a,D,d,E)`. 51The angle of the tube at the left end is: 52 53.. math:: 54 g_2(X) = - F \frac{b (L^2-b^2)}{6 E L I}, 55 56 57and the angle of the tube at the right end is: 58 59.. math:: 60 g_3(X) = F \frac{a (L^2-a^2)}{6 E L I}. 61 62The following table presents the distributions of the random variables. These variables are assumed to be independent. 63 64======== ========================== 65Variable Distribution 66======== ========================== 67F Normal(1,0.1) 68L Normal(1.5,0.01) 69a Uniform(0.7,1.2) 70D Triangular(0.75,0.8,0.85) 71d Triangular(0.09,0.1,0.11) 72E Normal(200000,2000) 73======== ========================== 74 75 76References 77---------- 78 79 * Deflection of beams by Russ Elliott. http://www.clag.org.uk/beam.html 80 * https://upload.wikimedia.org/wikipedia/commons/f/ff/Simple_beam_with_offset_load.svg 81 * https://en.wikipedia.org/wiki/Deflection_(engineering) 82 * https://mechanicalc.com/reference/beam-deflection-tables 83 * https://en.wikipedia.org/wiki/Second_moment_of_area 84 * Shigley's Mechanical Engineering Design (9th Edition), Richard G. Budynas, J. Keith Nisbettn, McGraw Hill (2011) 85 * Mechanics of Materials (7th Edition), James M. Gere, Barry J. Goodno, Cengage Learning (2009) 86 * 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. 87 88Load the use case 89----------------- 90 91We can load this classical model from the use cases module as follows : 92 93.. code-block:: python 94 95 >>> from openturns.usecases import deflection_tube as deflection_tube 96 >>> # Load the tube deflection model 97 >>> dt = deflection_tube.DeflectionTube() 98 99API documentation 100----------------- 101 102See :class:`~openturns.usecases.deflection_tube.DeflectionTube`. 103 104Examples based on this use case 105------------------------------- 106 107 108.. raw:: html 109 110 <div class="sphx-glr-thumbcontainer" tooltip=""> 111 112.. only:: html 113 114 .. figure:: /auto_calibration/least_squares_and_gaussian_calibration/images/thumb/sphx_glr_plot_calibration_deflection_tube_thumb.png 115 :alt: 116 117 :ref:`sphx_glr_auto_calibration_least_squares_and_gaussian_calibration_plot_calibration_deflection_tube.py` 118 119.. raw:: html 120 121 </div> 122 123.. toctree:: 124 :hidden: 125 126 /auto_calibration/least_squares_and_gaussian_calibration/plot_calibration_deflection_tube 127 128