Simplified Finite Element Assessment of Beams With and Without Shear Deformation

This paper presents a simplified finite element (FE) methodology for solving accurately beam models with (Timoshenko) and without (Bernoulli-Euler) shear deformation. Special emphasis is made on showing how it is possible to obtain the exact solution on the nodes and a good accuracy inside the element. The proposed simplifying concept, denominated as the equivalent distributed load (EDL) of any order, is based on the use of Legendre orthogonal polynomials to approximate the original or acting load for computing the results between the nodes. The 1-span beam examples show that this is a promising procedure that allows the aim of using either one FE and an EDL of slightly higher order or by using an slightly larger number of FEs leaving the EDL in the lowest possible order assumed by definition to be equal to 4 independently of how irregular the beam is loaded.