Elastic buckling and free vibration of nonlocal strain gradient Euler-Bernoulli beams using Laplace transform

Previous studies have shown that Lim's nonlocal strain gradient differential model would lead to some inconsistencies for the static bending of Bernoulli-Euler and Timoshenko beams, though it is widely applied to address both softening and toughening size-effect. In this work, nonlocal strain gradient integral model is applied to study the elastic buckling and free vibration of Bernoulli-Euler beam under different boundary conditions. The differential governing equation and corresponding boundary conditions are derived via the Hamilton's principle, and the relation between strain and nonlocal stress is expressed in integral form. The Laplace transform technique is applied to solve the integro-differential equations directly, and the explicit expression for bending deflections and moments is obtained with six unknown constants. The nonlinear characteristic equations to determine the characteristic buckling load and vibration frequency are derived explicitly in consideration of corresponding boundary conditions. The buckling load and vibration frequency from present model are validated against to the existed results in literature. Consistent softening and toughening responses respect to length-scale parameters can be obtained for both elastic buckling and free vibration.