Revisiting the elasticity solution for a simply supported beam under sinusoidal load

This article presents an alternate final solution to the deflection profile of a simply supported beam under sinusoidal load based on the theory of elasticity. It begins with a review of the same problem found in typical graduate textbooks, which ends with an elasticity solution that is valid only for moderately thick beams, and thereafter provides an alternative ending for providing a more accurate deflection profile that is valid for very thick beams. Plotted results show evidence on the deficiency of the textbook solution for very thick beams, thereby limiting its use as a verifier for the Mechanics of Materials solution. Unlike the existing simplified elasticity model, the exact model does not reduce to the Mechanics of Materials model when the Poisson’s ratio of the beam material is −1. In addition to being a better verifier to the Mechanics of Materials solution, the proposed exact elasticity solution can be easily reduced to the simplified elasticity solution that is currently adopted in some graduate textbooks.