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