New Orleans, Louisiana
June 26, 2016
June 26, 2016
August 28, 2016
This project is primarily an experimental study on the membrane analogy and the sand-heap analogy associated respectively with warping and plasticity when a non-circular bar is twisted. This is a supplemental lab activity for the junior level Mechanics of Materials course. When a prismatic bar of arbitrary cross-section is twisted, it gets deformed into helical curves as the cross sections rotate during the torsion. The analogy between the elastic torsion of a bar and the small deflection of a laterally loaded membrane is what is known as the membrane analogy. The membrane is stretched by a uniform tension and is attached to a die whose edge plane is the same as the cross section of the twisted bar. A uniform lateral pressure is applied to the membrane to produce a deflection. The resultant shear stress at any point is G (modulus of rigidity) times the greatest slope at the corresponding point of the deflected membrane, and the torque is 2G times the volume bounded by the deflected membrane and the cross-sectional plane. Except for a circular cylindrical bar, the cross-sections become warped and the amount of warping depends on the angle of twist per unit length. The students are made aware of the fact that warping in non-circular sections plays a significant role in structural design. The shear stresses vanish at the corners, when a square bar is twisted: a result that is counterintuitive from the study of torsion of a bar 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. As the torque is increased, the material passes from the elastic to plastic state. There is a limit to the ability of the cross section to withstand increasing torque. This feature is not typically addressed in the junior level Mechanics of Materials course, but we felt the need for it because this deformation can be readily shown in a torsion experiment with a ductile material such as aluminum. When the torque is increased to a critical value, the resultant shear stress attains the yield value ‘k’ at one or more points, somewhere along the boundary of the cross section. The extension of the membrane analogy to elastic plastic torsion leads to sand-heap analogy. We can erect a roof of constant slope k/G and its base identical to the boundary of the cross section. The base is filled with a stretched membrane loaded by uniform surface pressure as before. When the intensity of the pressure becomes sufficiently high, the membrane begins to touch the roof. This corresponds to the state of initial yielding. At the limit the membrane fills the entire volume, when the fully plastic state has been attained. Experimental estimates for the elastic and inelastic torques in twisting an aluminum square shaft (½ in x ½ in) are in reasonable agreement with the values predicted by the membrane analogy and the sand heap analogy.
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