Unified finite-difference-based discrete model for vibration and buckling of beams with geometric discontinuities

A unified finite-difference-based discrete model (FDDM) is developed to solve the vibration and buckling problems of beams with geometric discontinuities, including cracks and stepped cross-sections. Using this unified model, discontinuous beams are represented as assemblies of rigid segments connected by frictionless hinges, equivalent rotational springs and lumped masses. Based on the discrete model, the discrete potential and kinetic energy functions are established, and Lagrange’s equation is then employed to formulate a set of discrete algebraic equations for determining the vibration frequencies and buckling loads of beams. This unified FDDM is thoroughly demonstrated to be a well-posed and fully independent discrete model that is exactly equivalent to the conventional finite difference method (FDM). The effectiveness and accuracy of the proposed discrete model are examined through several illustrative examples. This is the first unified discrete model that can achieve exact equivalence with FDM for the analysis of imperfect beams with geometric discontinuities.