Green’s-function-based spectral super-elements for dynamic finite-element analysis of framed structures

This paper develops a finite-element-compatible frequency-domain procedure for constructing spectral super-elements for the dynamic analysis of framed structures composed of linear reciprocal members subjected to arbitrary external loading. The formulation starts from the frequency-dependent analytical shape functions obtained from the homogeneous governing equations, from which the corresponding dynamic stiffness-matrix coefficients are derived. In standard Dynamic Stiffness Method (DSM) and Spectral Element Method (SEM) formulations, distributed and non-nodal concentrated loads can be incorporated through equivalent nodal or fixed-end load vectors; however, reconstruction based exclusively on the homogeneous shape functions does not generally recover the analytical strong-form intra-element displacement and internal-force fields of the loaded member. The proposed approach provides an analytical completion of the conventional spectral element by decomposing the response into homogeneous and fixed, or particular, components. The homogeneous response is described by the analytical shape functions, whereas the fixed response is recovered through Green’s Functions (GFs) constructed directly from the same homogeneous solution and its associated boundary stiffness relations. Consequently, neither an independent derivation of the GFs nor a load-specific particular solution is required for each new element formulation. The resulting spectral super-elements preserve the standard finite-element degrees of freedom, connectivity, and assembly structure while providing analytical frequency-domain displacement and internal-force fields under arbitrary intra-element loading. Numerical examples involving a non-uniform rod, a uniform Euler–Bernoulli beam, a Timoshenko frame, and a uniform Euler–Bernoulli beam on a Winkler foundation demonstrate excellent agreement with independent analytical, SEM, and FEM reference solutions, while substantially reducing the number of elements required to represent the structural members.