materials Article Stress-Based FEM in the Problem of Bending of Euler–Bernoulli and Timoshenko Beams Resting on Elastic Foundation Zdzisław Wi˛eckowski* and Paulina Swi´ ˛atkiewicz Department of Mechanics of Materials, Łód´zUniversity of Technology, 90-924 Łód´z,Poland;
[email protected] * Correspondence:
[email protected] Abstract: The stress-based finite element method is proposed to solve the static bending problem for the Euler–Bernoulli and Timoshenko models of an elastic beam. Two types of elements—with five and six degrees of freedom—are proposed. The elaborated elements reproduce the exact solution in the case of the piece-wise constant distributed loading. The proposed elements do not exhibit the shear locking phenomenon for the Timoshenko model. The influence of an elastic foundation of the Winkler type is also taken into consideration. The foundation response is approximated by the piece-wise constant and piece-wise linear functions in the cases of the five-degrees-of-freedom and six-degrees-of-freedom elements, respectively. An a posteriori estimation of the approximate solution error is found using the hypercircle method with the addition of the standard displacement-based finite element solution. Keywords: finite element method; stress-based approach; Euler–Bernoulli beam; Timoshenko beam; shear locking 1. Introduction Citation: Wi˛eckowski,Z.; Beams are commonly used in engineering practice; thus, they often are a matter of Swi´ ˛atkiewicz,P. Stress-Based FEM in interest of advanced research and numerous papers. The Euler–Bernoulli beam theory is the Problem of Bending of the most basic theory applied to slender beams that is based on assumption that straight Euler–Bernoulli and Timoshenko lines normal to the mid-plane before deformation remain straight and normal to it after Beams Resting on Elastic Foundation.