Introduction to Elasticity/Torsion of triangular cylinder

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Example: Equilateral triangle[edit | edit source]

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.