On a strain gradient elastic Timoshenko beam model

Considering the influence of the microstructure, the Timoshenko beam model is revisited, invoking Mindlin's strain gradient strain energy density function. The equations of motion are derived and the bending equilibrium equations are discussed. The adopted strain energy density function includes new terms. Those terms introduce the strong effect of the beam cross‐section area. The influence of those terms is more evident in thin beams where the cross‐section area is far bigger than its moment of inertia. Applications have been worked out exhibiting the difference of the present theory not only from the classical Timoshenko beam, but also from the existing variations including couple stresses. The solution of the static problem, for a simply supported beam loaded by a force at the middle of the beam, is defined and the first (least) eigen‐frequency is found. The present model is proved to be stiffer. Considering the influence of the microstructure, the Timoshenko beam model is revisited, invoking Mindlin's strain gradient strain energy density function. The equations of motion are derived and the bending equilibrium equations are discussed. The adopted strain energy density function includes new terms. Those terms introduce the strong effect of the beam cross‐section area. The influence of those terms is more evident in thin beams where the cross‐section area is far bigger than its moment of inertia. Applications have been worked out exhibiting the difference of the present theory not only from the classical Timoshenko beam, but also from the existing variations including couple stresses.