ABSTRACT
This paper uses thin plate theory to address a problem in warping for which the solution by Timoshenko anticipates that the suppression of warping in a thin rectangular section will produce a stress system which is the solution of a third order differential equation although he has not set up the appropriate differential equation. In the light of thin walled member theory, inconsistencies are perceived in the form of these equations because they do not allow the Saint-Venant stresses to decay in proportion to the reduction in the rate of the angle of twist. The paper shows that if we use the same assumptions as are used for flexural torsion in thin-walled members, the fundamental parameters of the differential equation can be established from first principles, leading to a solution of Timoshenko’s problem which is consistent with full thin walled member theory as well as with Timoshenko’s seminal solution in the field, that of an I-section built in at one end, free to warp at the other and subjected to a constant torsion moment along its length. The solution is ten percent different from Timoshenko’s energy solution based on his field but coincides with a different way of using the information in his field. The approximations implicit in the solution are discussed demonstrating that it is not an exact solution but is explicit and is consistent with thin walled member theory.
