Finite Element Modeling for Vibration of Initially Stressed Non-Local Euler-Bernoulli Beams

Abstract

This paper is concerned with a precise finite element model for vibration analysis of initially stressed micro/nano beams based on nonlocal Euler-Bernoulli and Eringen's nonlocal elasticity theory. For this purpose analytical solutions for the exact dynamic shape functions has been derived by also use of Hamiltonian's principle for the governing equations. The solution is applicable to various initial stresses such as tensile and compressive and scaling effect. The exact dynamic shape functions have been constructed to obtain analytic expressions for the exact dynamic element stiffness matrix components. Numerical results are displayed to indicate the effects of initial stresses and scaling effect parameters on the vibration characteristics of initially stressed clamped nonlocal Euler-Bernoulli beams. For the first time in literature, this study presents such an element formulation that it provides adequate and accurate representation of the vibration behavior of initially stressed micro/nano beams based on nonlocal Euler-Bernoulli beam theory.