Seattle, Washington
June 14, 2015
June 14, 2015
June 17, 2015
978-0-692-50180-1
2153-5965
Mechanics
13
26.1712.1 - 26.1712.13
10.18260/p.25048
https://peer.asee.org/25048
2250
Warping Deformation Caused by Twisting Non-circular Shafts ABSTRACTThis project is a combined analytical and experimental activity to study warping deformationthat occurs in shafts of non-circular cross section subjected to torsion. This is a supplementalactivity for the junior level Mechanics of Materials course. The students see the warping insquare sections and their pronounced effects in ductile materials such as aluminum when thestresses exceed the material yield. The students are made aware of the fact that warping has asignificant role in structural design.As a part of the project the students are see that the basic theory of torsion as presented in theMechanics of Materials course that states that cross-sections distort in plane during twisting issomething that holds only for circular cross sections where every point in the cross-sectionrotates (in plane) about the center of twist. For non-circular cross sections one needs to addresswarping which is assumed to vary with the rate of twist and is a function only of the position onthe cross section and not on the lengthwise coordinate. The students are then exposed to thedifferent ways the warping behavior is modeled analytically through partial differentialequations. The warping function model leads to Laplace’s equation which is hard to solve evenfor a rectangular boundaries. Next they are introduced to the St. Venant stress functionformulation which leads to Poisson’s equation which does have an analytical solution forrectangular cross section in terms of an infinite series involving products of hyperbolic andsinusoidal functions. The expressions for the stress function and its derivatives (the in-planeshear stress components) are then numerically evaluated and plotted for the cross section. Theplots show that the shear stresses vanish at the corners, a result that is counterintuitive from thestudy of torsion of a member of circular cross section. It is inferred that these constrainingeffects 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 pronouncedwarping as the shear stresses due to the torsional loads exceed the torsional yield strength of thematerial.
Chattopadhyay, S. (2015, June), Warping Deformation Caused by Twisting Non-circular Shafts Paper presented at 2015 ASEE Annual Conference & Exposition, Seattle, Washington. 10.18260/p.25048
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