Elasticity/Warping of elliptical cylinder

Example 2 : Elliptical cylinder

edit

Choose warping function

 

where   is a constant.


Equilibrium ( ) is satisfied.


The traction free BC is

 

Integrating,

 

where   is a constant.


This is the equation for an ellipse with major and minor axes   and  , where

 

The warping function is

 

The torsion constant is

 

where

 

If you compare   and   for the ellipse, you will find that  . This implies that the torsional rigidity is less than that predicted with the assumption that plane sections remain plane.


The twist per unit length is

 

The non-zero stresses are

 

The projected shear traction is

 
 
Shear stresses in the cross section of an elliptical cylinder under torsion

For any torsion problem where   is convex, the maximum projected shear traction occurs at the point on   that is nearest the centroid of  .

The displacement   is

 
 
Displacements ( ) in the cross section of an elliptical cylinder under torsion