Eringen’s small length scale coefficient for vibration of axially loaded nonlocal Euler beams with elastic end restraints

Abstract This paper presents the calibration of Eringen’s small length scale coefficient e0 for elastically restrained beams (in the context of buckling and axially loaded vibration) on the basis of an interesting connection between a discrete beam model (DBM) and the Eringen’s nonlocal beam model (ENBM). The DBM is formulated from the use of the central finite difference method which has been shown to be equivalent to the Hencky bar-chain model. By solving the discrete beam formulation using the theory of linear difference equations and matching the buckling loads and natural frequencies of the DBM with those of the ENBM, the small length scale coefficient e0 may be calibrated for buckling and vibration beam problems. It is found that by applying the traditional nonlocal continuous boundary conditions, e0 varies with respect to the boundary conditions. However, e0=1/6${e_0} = 1/\sqrt 6$ for purely free vibration problems while e0=1/12${e_0} = 1/\sqrt {12}$ for buckling problems, irrespective of the boundary conditions, when the continualized discrete boundary conditions are applied. For both traditional continuous and continualized discrete boundary conditions, the value of e0 is found to be dependent on the axial load.