Modal analysis and superposition for dynamic response of structures with discontinuities using HybriDFEM

The dynamic characterization of structures using discrete models, as well as the application of modal superposition to compute their dynamic response, has been rarely explored in the literature. This is at odds with the international relevance of discrete models in structural assessment, and the multiple fields of application of modal analysis and superposition, from structural health monitoring to seismic engineering. This paper introduces a 2D discrete formulation, developed within a finite element framework, to address this gap. Initially conceived for nonlinear static analyses as HybriDFEM (Hybrid Discrete-Finite Element Method), it is now augmented with a procedure to compute the mass matrix, natural frequencies, mode shapes, and response-related quantities such as modal and dynamic contribution factors or effective modal mass. Moreover, using the structural tangent stiffness matrix in the eigenvalue problem allows tracking the evolution of natural frequencies and modes in structures loaded into their nonlinear material and geometric range. The formulation is validated through several examples, where it compares well with results from engineering beam theories, refined finite element models, and numerical time-integration methods. In an application example studying the evolution of modal properties of a progressively damaged frame, HybriDFEM is coupled with finite elements, highlighting its novel approach to integrating discrete and finite elements for enhanced structural modal analysis and superposition.