Warping Deformation Caused by Twisting Non-circular Shafts

This project is a combined analytical and experimental activity to study warping deformation in shafts of non-circular cross section subjected to torsion. This is a supplemental activity for the junior level Mechanics of Materials course. The students see the warping in square sections and their pronounced effects in ductile materials such as aluminum when the stresses exceed the material yield. The students are made aware of the fact that warping plays a significant role in structural design. As a part of the project, the students are exposed to the basic theory of torsion as presented in the Mechanics of Materials course. For non-circular cross sections one needs to address warping which is assumed to vary with the rate of twist and is a function only of the position on the cross section and not on the lengthwise coordinate. The students are then exposed to the different ways the warping behavior is modeled analytically through partial differential equations. The warping function model leads to Laplace's equation which is hard to solve even for a rectangular boundaries. Next they are introduced to the St. Venant stress function formulation which leads to the Poisson's equation. This equation has an analytical solution for rectangular cross section in terms of an infinite series involving products of hyperbolic and sinusoidal functions. The expressions for the stress function and its derivatives (the inplane shear stress components) are then numerically evaluated and plotted for the cross section. The plots show that the shear stresses vanish at the corners, a result that is counterintuitive from the study of torsion of a member of circular cross section. It is inferred that these constraining effects cause the warping deformations to appear in torsion of shafts of non-circular cross-section. A further extension of the work in the experimental area demonstrates pronounced warping as the shear stresses due to the torsional loads exceed the torsional yield strength of the material.