Elasticity/Torsion of triangular cylinder

Example: Equilateral triangle edit

Torsion of a cylinder with a triangular cross section

The equations of the three sides are


Let the Prandtl stress function be


Clearly,   at the boundary of the cross-section (which is what we need for solid cross sections).

Since, the traction-free boundary conditions are satisfied by  , all we have to do is satisfy the compatibility condition to get the value of  . If we can get a closed for solution for  , then the stresses derived from   will satisfy equilibrium.

Expanding   out,


Plugging into the compatibility condition




and the Prandtl stress function can be written as


The torque is given by


Therefore, the torsion constant is


The non-zero components of stress are


The projected shear stress


is plotted below

Stresses in a cylinder with a triangular cross section under torsion

The maximum value occurs at the middle of the sides. For example, at  ,


The out-of-plane displacements can be obtained by solving for the warping function  . For the equilateral triangle, after some algebra, we get


The displacement field is plotted below

Displacements   in a cylinder with a triangular cross section.