A Simple Unified Treatment of the Torsion Problem in Bars of Rectangular Cross-Section and Levy's Solution in Classical Plate Theory

Abstract

A survey of the literature reveals that the problems stated in the title are solved independently: in the case of the torsion problem the trigonometric terms are of the type (cos (nπx)/2a, n = 1,3,5,…) while when implementing Levy's solution one uses terms of the type (sin (nπx)/a, n = 1,3,5,…). In the first case the coordinate system coincides with the axis of symmetry of the rectangle while, in the second, the y-axis coincides with the left, vertical side of the rectangle and the x-axis is the horizontal axis of symmetry of the configuration. It is shown in the present study that a basic, unified approach is possible for solving both problems using the coordinate system mentioned in the second place and expressing the dependent variable in terms of the infinite series

$\sum _i^{\infty }sin\frac{n\pi x}{a}{Y}_n\left(y\right)$