Vertical deflection of a tube

Description

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

simply supported beam

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

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 D and d are:

I = \frac{\pi (D^4-d^4)}{32}.

The vertical deflection at point x=a is:

g_1(X) = - F \frac{a^2 (L-a)^2}{3 E L I},

where X=(F,L,a,D,d,E). The angle of the tube at the left end is:

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:

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

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 \vect{X} = (F, L, a, D, d, E) and Y = (y, \theta_L, \theta_R).

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)

inputDistributionJointDistribution

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