Closed-form solution of Euler–Bernoulli frames in the frequency domain.

This paper presents the frequency domain formulation of the Green's Functions Stiffness Method (GFSM) for plane Euler–Bernoulli frames subjected to arbitrary external loads (forces and bending moments). The GFSM is a generalization of the Stiffness Method (SM) or the Finite Element Method (FEM) that computes the closed-form analytical structural response by decomposing it into a homogeneous and a fixed or particular responses. The homogeneous response is obtained from the displacements at the element ends in the absence of external loads, while the fixed response is calculated using the Green's functions of fixed elements through a superposition integral that includes the external loads. The GFSM can be categorized as a mesh reduction method that generalizes the Spectral Element Method (SEM), in which the structural response is first obtained in the frequency domain and then converted to the time domain using the Fast Fourier Transform algorithm. Compared to the SEM, the significant advantage of the GFSM is that the frequency domain structural response is exact and avoids the need for dense meshes when external loads have complex patterns. Three examples are presented to demonstrate the effectiveness of the GFSM, each of which shows excellent agreement when compared to the FEM solution even using far fewer elements. • The Green's Functions Stiffness Method is a numerical method employed to compute the analytical closed-form response of structures. • The Green's Functions Stiffness Method reduces the number of elements required in a model compared to the Finite Element Method. • The Green's Functions Stiffness Method is a particular version of the Spectral Element Method that uses Green's Functions to obtain the dynamic response of structures. • Reduction in computing time is obtained using the Green's Functions Stiffness Method compared with traditional numerical methods. [ABSTRACT FROM AUTHOR]