A new Bernoulli–Euler beam model based on modified gradient elasticity

A new Bernoulli–Euler beam model is developed based on modified gradient elasticity theory. The governing equation and boundary conditions, which contain two internal length scales (i.e., \(l_{x}\) and \(l_{z})\), are derived by the variational principle. The new model can be simplified to the classical beam theory when the two internal length scales vanish. The numerical examples of cantilever beams subjected to two typical loadings are presented. Results show that the size effect can be captured by the new model, and the deflection decreases with the internal length scales increasing. The influence of \(l_{z}\) (the internal length scale along the beam thickness direction) on deflection is much greater than that of \(l_{x}\) (the internal length scale along the beam length direction), and the increment of stiffness is mainly controlled by \(l_{z}\). The new beam model is convenient for engineering applications and designs.