A novel piecewise approximate analytical Lagrangian approach for non-uniform functionally graded and discontinuous composite beams with internal springs

The primary strategy for optimizing beam design relies on varying cross-sectional geometry and mechanical properties. Furthermore, to achieve more realistic modeling, internal springs are integrated into beam elements to simulate embedded semi-rigid connections, hinges, and concentrated damage (cracks). Although rigorous approaches have been formulated to model these types of members, their intricate mathematical derivations and methods present significant implementation challenges. Therefore, to provide a simplified alternative, a piecewise approximate analytical approach is formulated in this study to predict the linear-elastic static behavior of in-plane Euler-Bernoulli beams featuring generalized mechanical and geometric variations, including multi-stepped variations, variable centerlines, and internal springs. The model is derived from a Lagrangian variational principle, where the actual energy is approximated using a base beam and a piecewise complementary beam. This yields a piecewise equivalent beam that represents the actual beam, thereby simplifying and shortening the mathematical derivations. Artificial Discontinuities (Ds) are strategically introduced to represent variable geometric and mechanical characteristics associated with the beam subdivision; besides, internal springs are modeled using physical Ds, capturing the behavior of semi-rigid connections, hinges, and concentrated damage. Governing differential equations are derived from the variational formulation via the calculus of variations, from which piecewise approximate analytical expressions are subsequently obtained. Solutions in fixed-order polynomial forms offer a distinct and straightforward alternative that simplifies practical implementation. The efficiency of the developed approximate method is further assessed and validated through numerical examples, where accurate results are provided using sufficient strategically located Ds.