Beams with Quadratic Function’s Variation of Height

Abstract

This paper extends the utilisation of the standard beam finite element model with two nodes by presenting derivations of a closed-form stiffness matrix and a load vector for slender beams where the height variation is given as a general quadratic function. All the required stiffness matrix coefficients were obtained from two cantilever substructures: clamped at both the right and left-ends, respectively. Each substructure was subjected to a vertical upward force and an anticlockwise bending moment at the free end and transverse displacement and rotation at the free-end due to both acting loads were evaluated. From the expressions obtained, the nodal force and moment were afterwards vice-versly expressed as functions of nodal displacement and rotation. Further, both supports’ reactions due to the applied loads were evaluated from the basic equilibrium, thus completing the coefficients of two rows of the stiffness matrix. The derived matrix and vector thus define an ‘exact’ finite element for this kind of height variation and all the obtained terms are written entirely in closed-symbolic forms. Numerical examples demonstrated that the elaborated solutions may be effectively implemented for structural analyses, as the presented expressions produced excellent results that were confirmed independently by more thorough 2D models.