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
The input to the problem are the following (these are the known or given):
Hence the torsional rigidity is known since is given (material) and is from above (geometry).
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
where the linear twist
Hence (2) becomes
Where is given by (1)
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.
Hence only and are non-zero.
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
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