Warping Function and Torsion of Non-Circular Cylinders edit

Warping functions are quite useful in the solution of problems involving the torsion of cylinders with non-circular cross sections.

For such problems, the displacements are given by

 

where   is the twist per unit length, and   is the warping function.

The stresses are given by

 

where   is the shear modulus.

The projected shear traction is

 

Equilibrium is satisfied if

 

Traction-free lateral BCs are satisfied if

 

or,

 

The twist per unit length is given by

 

where the torsion constant