Elastic buckling and free vibration analysis of functionally graded Timoshenko beam with nonlocal strain gradient integral model

Abstract

In this work, nonlocal strain gradient integral model is applied to study the elastic buckling and free vibration response of functionally graded (FG) Timoshenko beam which is made of two constituents varying along thickness direction. The differential governing equations and corresponding boundary conditions are derived via the Hamilton's principle, and the relations between nonlocal stresses and strains are expressed as integral equations. The Laplace transform technique is applied to solve directly the integro-differential equations, and the explicit expression for bending deflections and moments, as well as cross-sectional rotation and shear force is expressed with eight unknown constants. The nonlinear characteristic equations to determine the characteristic buckling load and vibration frequency are derived explicitly in consideration of corresponding boundary conditions and constitutive constraints. The buckling load and vibration frequency from present model are validated against to the existed results in literature. Consistent softening and toughening responses with respect to length-scale parameters can be obtained.