A Green’s function driven mesh reduction technique for obtaining closed-form solutions of uniform Euler–Bernoulli beams on two-parameter elastic foundations

This paper presents the formulation of the Green’s Function Stiffness Method (GFSM) for the static analysis of linearly elastic uniform Euler–Bernoulli beams on two-parameter elastic foundations subjected to arbitrary external loads. The GFSM is a mesh-reduction method closely related to the Finite Element Method (FEM) family, offering a means to compute closed-form solutions for framed structures. It is based on a strong-form formulation and decomposes the element-level response into homogeneous and fixed (particular) components, the latter obtained analytically using Green’s functions of fixed-end elements. The method retains essential FEM features — including shape functions, stiffness matrices, and fixed-end force vectors — while extending the capabilities of the Transcendental Finite Element Method (TFEM), a FEM variant that employs exact shape functions. In this context, the GFSM serves as a post-processing enhancement that transforms the approximate TFEM solution into an exact closed-form. A defining characteristic of the GFSM is that its formulation relies solely on the solution of the homogeneous form of the governing differential equations — specifically, the shape functions and stiffness matrix coefficients that constitute the core of the TFEM. The effectiveness of the GFSM is demonstrated through two examples, where its results are compared against those obtained from TFEM with varying levels of mesh refinement.