A Size-dependent Bernoulli-Euler Beam Formulation based on a New Model of Couple Stress Theory

In this paper, a size-dependent formulation for the Bernoulli-Euler beam is developed based on a new model of couple stress theory presented by Hadjesfandiari and Dargush. The constitutive equation obtained in this new model, consists of only one length scale parameter that is capable of capturing the micro-structural size effect in predicting the mechanical behavior of the structure. Having one length scale parameter is claimed to be an advantage of the model in comparison with the classical couple stress theory. The governing equations and boundary conditions of the Bernoulli-Euler beam are developed using the variational formulation and the Hamilton principle. The static bending and free vibration problems of a Bernoulli-Euler beam with various boundary conditions are solved. Numerical results demonstrate that the value of deflection predicted by the new model is lower than that of the classical theory. It is also found that natural frequencies obtained by the present couple stress model are higher than those predicted by the classical theory. The differences between results obtained by the present model and the classical theory become significant as the thickness of the beam gets close to the length scale parameter of the beam material.