Elasticity/Torsion of triangular cylinder

Example: Equilateral triangle

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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

 

Therefore,

 

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.