A New Bernoulli-Euler Beam Model Based on a Modified Couple Stress Theory

A new model for the bending of a Bernoulli-Euler beam is developed using a modified couple stress theory. A variational formulation based on the total minimum potential energy principle is employed. The new model contains an internal material length parameter and can capture the size effect, unlike the classical Bernoulli-Euler beam model. The former reduces to the latter when the material length parameter is set to zero. As a direct application of the new model, a cantilever beam problem is solved. It is found that the rigidity of the cantilever beam predicted by the new model is larger than that predicted by the classical beam model. The difference between the deflections predicted by the two models is very significant when the beam thickness is small (below 10 μm), but is diminishing with the increase of the beam thickness. This demonstrates that the new model can indeed predict the size effect at the micron scale observed in bending tests.