A NONCLASSICAL REDDY-LEVINSON BEAM MODEL BASED ON A MODIFIED COUPLE STRESS THEORY

A microstructure-dependent, nonclassical Reddy-Levinson (R-L) beam model is developed using a variational formulation based on Hamilton's principle. A modified couple stress elasticity theory is used, which contains a material length scale parameter and enables the new beam model to capture the size effect. Also, the Poisson effect is incorporated in the current beam model, which differs from other R-L beam models. The newly developed nonclassical R-L model reduces to the existing classical elasticity-based R-L model when the material length scale parameter and Poisson's ratio are both taken to be zero. In addition, the current R-L beam model recovers the nonclassical Bernoulli-Euler beam model based on the same modified couple stress theory when the normality assumption is reinstated. To illustrate the new R-L beam model, the static bending and free vibration problems of a simply supported beam under a concentrated load are analytically solved by directly applying the general formulas derived. The numerical results for the static bending problem reveal that both the deflection and rotation predicted by the current nonclassical R-L model are smaller than those predicted by the classical R-L model. Also, the differences in both the deflection and rotation predicted by the two models are very large when the beam thickness is small, but they diminish with increasing beam thickness. For the free vibration problem, the numerical results show that the natural frequency predicted by the current R-L model is higher than that by the classical R-L model, and the difference is significant only for very thin beams. These predicted trends of the size effect at the micron scale agree with those observed in experiments.