Closed-form solution for non-uniform Euler–Bernoulli beams and frames

Abstract

Non-uniform elements, such as non-prismatic and those made from composite materials, are commonly used in structures. This is because the variation in their internal forces makes uniform elements less effective to withstand the external loads. This paper presents the formulation of the Green’s Functions Stiffness Method (GFSM) for the static analysis of non-uniform Euler–Bernoulli frames subjected to arbitrary external loads and bending moments. The GFSM is a mesh reduction method that combines the strengths of the Stiffness Method (SM) and the Finite Element Method (FEM) with those of Green’s functions, resulting in an analytic method for obtaining closed-form solutions of reticular structures. As a particular case, the GFSM is closely related to the Transcendental Finite Element Method (TFEM), which allows to obtain closed-form solutions from an implementation of the latter by only making changes at the post-processing stage. Two examples are provided with the analysis of a beam and a plane frame. Additionally, two appendices are included with the particularization of the GFSM for some specific non-uniform elements, and general stepped elements.