A new Bernoulli-Euler beam model based on a simplified strain gradient elasticity theory and its applications

A new Bernoulli-Euler beam model based on a simplified strain gradient elasticity theory is established in the current investigation. The generalized Euler-Lagrange equations and corresponding boundary conditions are naturally derived from the Hamilton's principle. Then axial wave propagation of small scale bars, static bending of cantilever beams, buckling and free vibration of simply supported beams are analytically solved by using the simplified strain gradient beam theory. The influences of the Poisson's effect as well as the weak non-local strain gradient elastic effect are discussed. The Poisson's effect is found to increase with the increase of the beam thickness in the buckling analysis, while the higher-order bending moment induced by stretch strain gradient has an insignificant influence on the critical buckling load in our numerical analysis. However, the effect of the higher-order bending moment is very significant on axial wave propagation and static bending of micro-scale beams. The current work is very helpful in understanding the microstructure-related size dependent phenomenon.