The Green’s functions differential transform element method: A new semi-analytical approach for the analysis of framed structures

Abstract

This paper presents the Green’s Functions Differential Transform Element Method (GFDTEM), a novel semi-analytical technique for analyzing linearly elastic framed structures subjected to arbitrary loads. As a powerful mesh reduction strategy, the GFDTEM uniquely integrates fundamental principles from the Finite Element Method (FEM), the Differential Transform Method (DTM), and the Green’s Functions Stiffness Method (GFSM). This synergistic combination yields a highly efficient and accurate modeling approach for complex structural systems. The method discretizes the structure into elements connected at nodes, following the FEM scheme, and characterizes each element by means of stiffness matrices, shape functions, and fixed-end forces. Within each element, the local response is systematically approximated by high-order polynomials derived via the DTM, while the challenging treatment of arbitrary loads is elegantly accomplished analytically using Green’s functions, a hallmark of its semi-analytical power. A defining feature of the GFDTEM is that its formulation relies solely on the solution of the homogeneous form of the governing differential equations -namely, the shape functions and stiffness matrix coefficients- leading to a compact, general and streamlined representation. Its versatility, efficiency and FEM-based formulation allows the GFDTEM to be seamlessly embedded within incremental-iterative linearisation schemes, granting considerable potential for extension to geometrically and materially nonlinear problems. The effectiveness and accuracy afforded by the GFDTEM are validated with illustrative examples for axially non-uniform rods, Euler-Bernoulli beams and frames.