New finite-difference-based discrete models for Euler and Timoshenko beam problems

Abstract

Although the finite difference method (FDM) has been widely employed in structural analysis for its rigorous mathematical formulation, its underlying physical interpretation remains a topic of continued investigation. Historically, the Hencky bar-chain model (HBM) was proposed as a discrete physical analogue of the central first-order FDM, representing a system of rigid segments interconnected by frictionless hinges and elastic rotational springs. However, the end rotational springs in the HBM do not correspond to actual physical springs, further clarification is thus required. In this study, we develop two novel finite-difference-based discrete models (FDDMs) tailored for Euler−Bernoulli and Timoshenko beam problems. By applying appropriate finite difference schemes to discretize the spatial derivatives in the potential energy function, these FDDMs emerge as well-posed and fully independent discrete beam models incorporating actual end springs. Consequently, they serve as true physical counterparts to their respective FDMs. This work not only introduces innovative discrete beam models aligned with finite difference formulations, but also offers a deeper understanding of the method.