Hencky Bar-Chain Model for Buckling and Vibration of Beams with Elastic End Restraints

This paper presents the Hencky bar-chain model (HBM) for buckling and vibration analyses of Euler-Bernoulli beams with elastic end restraints. The Hencky bar-chain comprises rigid beam segments (of length a = L/n where L is the total length of beam and n the number of beam segments) connected by frictionless hinges with elastic rotational springs of stiffness EI/a where EI is the flexural rigidity of the beam. The elasticity and the mass of the beam are concentrated at the hinges with rotational springs. The key contribution of this paper lies in the modeling of the elastic end restraints of the Hencky bar-chain that will simulate the same buckling and vibration results as that furnished by the first-order central finite difference beam model (FDM) which was earlier shown to be analogous to the HBM. The establishment of such a physical discrete beam model allows one to obtain solutions for beam-like structure with repetitive cells (or elements) as well as to calibrate the Eringen's coefficient e₀ in the nonlocal beam theory that captures the small length scale effect.