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June 20, 2014

1 Introduction

1.1 Problem setup

1.1.1 What are the assumptions?

1.1.2 What is the input and what is the output?

1.1.3 The output from the problem (the things we need to calculate)

2 Analytical solution using Prandtl stress function

2.1 Stress components

2.2 Strain components

2.3 Determining the twist angle

2.4 Displacement calculations

3 References

1.1 Problem setup

1.1.1 What are the assumptions?

1.1.2 What is the input and what is the output?

1.1.3 The output from the problem (the things we need to calculate)

2 Analytical solution using Prandtl stress function

2.1 Stress components

2.2 Strain components

2.3 Determining the twist angle

2.4 Displacement calculations

3 References

For MAE 207, Computational methods. UCI. Fall 2006

We consider bar made of isotropic martial with rectangular cross section subjected to twisting torque . The following diagram illustrate the basic geometry.

Experiments show that rectangular cross sections do wrap and that cross sections do not remain plane as shown in this diagram (in the case of a circular cross section, cross section do NOT wrap).

This is another diagram showing a bar under torsion

- 1.
- The twist rate (called in this problem) and deﬁned as where is the twist angle is assumed to be constant.
- 2.
- Cross section can wrap also in the direction (i.e. the cross section does not have to remain in the plane) but if this happens, all cross sections will wrap in the section by the same amount.
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- Material is isotropic

The input to the problem are the following (these are the known or given):

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- The width and height of the cross section.
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- Material Modulus of rigidity or sheer modulus which is the ratio of the shearing stress to the shearing strain
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- The applied torque
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- the torsion constant for the a rectangular cross section. For a rectangular section of
dimensions it is given by

(1)

Hence the torsional rigidity is known since is given (material) and is from above (geometry).

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- The stress distribution in the cross section (stress tensor ﬁeld). Once this is found then using the material constitutive relation we can the strain tensor ﬁeld.
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- The angle of twist as a function of (the length of the beam).

First we solve for the Prandtl stress function by solving the Poisson equation

Where is the sheer modulus and is the twist rate (which was assumed to be constant).

The boundary conditions ( at any point on the edge of the cross section and at the ends of the beam) is an arbitrary constant. We take this constant to be zero. Hence at the cross section boundary we have

The analytical solution to the above equation is from book Theory of elasticity by S. P. Timoshenko and J. N. Goodier

| (2) |

where the linear twist

Hence (2) becomes

Hence

and

Timoshenko gives the maximum sheer stress, which is as

Given that is Young's modulus for the material, is Poisson's ratio for the material, and we can now obtain the strain components from the constitutive equations (stress-strain equations) since we have determined the stress components from the above solution.

If we look at a cross section of the bar at some distance from the end of the bar, the angle that this speciﬁc cross section has twisted due to the torque is .

This angle is given by the solution to the equation

But is the linear twist and is given by hence the above equation becomes

Hence

Where is the constant of integration. Assuming at we obtain that

and using the expression given in equation (1) above we can determine for each

we see that

Hence

- 1.
- Mathematica Structural Mechanics help page
- 2.
- MIT course 16.20 lecture notes. MIT open course website.
- 3.
- Theory of elasticity by S. P. Timoshenko and J. N. Goodier. chapter 10