Vertical deflection of a tube

Description

We consider the deflection of a tube under a vertical stress.

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:

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 and are:

The vertical deflection at point is:

where . The angle of the tube at the left end is:

and the angle of the tube at the right end is:

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.

API documentation

class DeflectionTube

Data class for the deflection of a tube model.

Examples

>>> from openturns.usecases import deflection_tube
>>> # Load the deflection tube model
>>> dt = deflection_tube.DeflectionTube()
>>> print("Inputs:", dt.model.getInputDescription())
Inputs: [F,L,a,De,di,E]
>>> print("Outputs:", dt.model.getOutputDescription())
[Deflection,Left angle,Right angle]

Attributes:

dimConstant, the dimension of the problem.

dim=6

modelSymbolicFunction

Model of the deflection. The model has input dimension 6 and output dimension 3. More precisely, we have and .

XFNormal distribution

ot.Normal(1, 0.1)

XENormal distribution

ot.Normal(200000, 2000)

XLDirac distribution

ot.Dirac(1.5)

XaDirac distribution

ot.Dirac(1.0)

XDDirac distribution

ot.Dirac(0.8)

XdDirac distribution

ot.Dirac(0.1)

inputDistributionComposedDistribution

The joint distribution of the input parameters.

Examples based on this use case

Calibration of the deflection of a tube

Calibration of the deflection of a tube