A new Bernoulli–Euler beam model based on a reformulated strain gradient elasticity theory

Abstract

A new non-classical Bernoulli–Euler beam model is developed using a reformulated strain gradient elasticity theory that incorporates both couple stress and strain gradient effects. This reformulated theory is first derived from Form I of Mindlin’s general strain gradient elasticity theory. It is then applied to develop the model for Bernoulli–Euler beams through a variational formulation based on Hamilton’s principle, which leads to the simultaneous determination of the equation of motion and the complete boundary conditions and provides a unified treatment of the strain gradient, couple stress and velocity gradient effects. The new beam model contains one material constant to account for the strain gradient effect, one material length scale parameter to describe the couple stress effect and one coefficient to represent the velocity gradient effect. The current non-classical beam model reduces to its classical elasticity-based counterpart when the strain gradient, couple stress and velocity gradient effects are all suppressed. In addition, the newly developed beam model includes the models considering the strain gradient effect only or the couple stress effect alone as special cases. To illustrate the new model, the static bending and free vibration problems of a simply supported beam are analytically solved by directly applying the general formulas derived. The numerical results reveal that the beam deflection predicted by the current model is always smaller than that by the classical model, with the difference being large for very thin beams but diminishing with the increase of the beam thickness. Also, the natural frequency based on the new beam model is found to be always higher than that based on the classical model.