Hencky bar-chain model for buckling and vibration analyses of non-uniform beams on variable elastic foundation

This paper is concerned with the development of the Hencky bar-chain model (HBM) for the buckling and vibration analyses of non-uniform beams resting on partial variable elastic foundation. The HBM allows analysts to obtain beam buckling and vibration solutions by solving a set of algebraic equations instead of a differential equation. To get the HBM, we resort to the first order central finite difference model (FDM) because both discrete models have phenomenological similarities. Based on bending moment, shear force and deflection equivalence between the two discrete beam models, the expressions for the internal spring stiffness, the end spring stiffness and the elastic foundation stiffness for the HBM are obtained. Some example problems are considered to demonstrate the simplicity of the HBM for buckling and vibration analyses of non-uniform beams on partial variable elastic foundation by taking the number of segments of the HBM to infinity. The effects of the foundation length and stiffness on the buckling load and natural frequencies are also discussed. Naturally, the HBM can be used to obtain solutions for articulated beams or beam-like structures with repetitive cells as well.