A new Timoshenko beam model based on modified gradient elasticity: Shearing effect and size effect of micro-beam

Abstract To investigate the shearing effect and size effect of micro-beams, a new Timoshenko beam model based on the modified gradient elasticity (MGE) is developed by using the variational principle. The new model can be simplified to the MGE Bernoulli-Euler beam model when the shearing effect is neglected, also can be simplified to the classical Timoshenko beam model when two internal length scales vanish. To solve this new model, a central finite difference method with virtual nodes is developed. Three numerical examples of the homogeneous beams and axially functionally graded (AFG) beams are considered in this paper. The size effect can be captured by the new model, i.e. the deflection, rotation angle and shearing angle decrease with the internal length scales increasing. The shearing effect, which plays an important role in deflection for a micro-beam, cannot be neglected even for the slender beam.