Formulation of the Green’s functions stiffness method for Euler–Bernoulli beams on elastic Winkler foundation with semi-rigid connections

The Green's Functions Stiffness Method (GFSM) is a method to compute the analytic closed-form response (reactions, displacements, and internal forces fields) of structures. It merges the strengths of the stiffness method (SM) (an exact relation between forces and displacements at the ends of elements) with those of Green's functions (computation of closed-form analytic structural response due to any arbitrary load). Its formulation is based on the decomposition of the structural response into homogeneous and fixed parts. The former depends on the degrees of freedom (joints displacements and rotations) and yields the stiffness matrix, while the latter depends on the external loads and is related to the fixed end forces vector. The element response is computed directly from the nodal displacements. First, the displacement fields are computed, and from their derivatives, the internal forces fields are obtained. This paper presents the formulation of the GFSM for Euler–Bernoulli beams on elastic Winkler foundation with semi-rigid connections and, as a particular case, for Euler–Bernoulli beams with semi-rigid connections. Additionally, two examples, and conclusions are presented.