The regular prismatic bar under torsion: A new approach based on caustics

Abstract

A new approach to the general torsion problem of the regular prismatic bar was introduced, based on extensive and meticulous experimental evidence, indicating that the twisted prismatic bars, except the cylindrical ones, present, a longitudinal warping, following a sinusoidal type of curves for the warping displacements with their extrema displaced towards the corners. As a guide for the establishment of the method the solutions of the elliptical and the equiangular triangular bars were used. It was found that the regions inside the inscribed circles tangent to the sides of their undeformed cross-sections constituted regions whose warped surfaces constitute undulated forms following sinusoidal surfaces. This property was proved exact for the exact solutions for the elliptic and triangular cross-sections and it is assumed by induction valid for any regular cross-section. Comparing the warping displacements along the inscribed circle to a square or regular polygon cross section with the classical Timoshenko solution for the square bar it was found an exact coincidence in these displacements. It was further shown that these circles and all concentric full circles constitute the initial curves of the caustics formed from reflections of the warped cross-sections of the twisted bars. The differences between the stresses and displacements between the existing approximate solutions and the new approach appear in the partly curved triangles lying outside the inscribed circles. For the square cross-section it was shown that thew-displacements between Timoshenko's solution and the present one were less than 5 percent. Experimental evidence with the Moire method and comparison with the exact solution for the equiangular triangular cross section proved that the present approach is closer to reality and therefore presents higher accuracy than any approximate method. Furthermore, it presents the advantage that the full circles inscribed inside the cross-section constitute the initial curves forming caustics in space when a parallel light beam is reflected on the warped cross-section of the twisted bars. interesting properties of caustics, proved in this paper, help down-rightly for a thorough study of the form and properties of the twisted bars.