Closed-form solution of Timoshenko frames using the Green’s Function Stiffness Method

Abstract

This paper presents the formulation of the Green’s Function Stiffness Method (GFSM) for the static analysis of Timoshenko frames subjected to arbitrary external loads and bending moments. The GFSM is a novel analytic method that provides closed-form solutions of structures by combining the strengths of the Stiffness Method (SM) (exact relation between forces and displacements at the element ends) and Green’s functions (GF) (analytical closed-form structural response to any arbitrary external load). The GFSM is based on the classical idea of decomposing the solution of differential equations into homogeneous and particular (fixed) solutions, with the latter computed through superposition integrals using Green’s functions of fixed elements as kernels. Its principal aim is the computation of the displacements fields, while the internal forces are calculated directly from their derivatives. The GFSM is directly related to the SM and the Finite Element Method (FEM), making its numerical implementation easily performed from either of them. Additionally, the basic ideas behind the GFSM can be easily extended to other physical fields. An example is presented to show the advantages of the proposed method.