On size-dependent Timoshenko beam element based on modified couple stress theory

Due to complexity of geometry of small-scale structures as well as size effect phenomenon in such scale, size-dependent finite element method can be employed as a powerful numerical technique to investigate the mechanical behavior of such structures. Recently, two distinct size-dependent Timoshenko beam elements have been proposed based on modified couple stress theory. The first beam element is a two-node element which has 3-DOF (degrees of freedom) at each node. The second beam element is also a two-node element, but it has 2-DOF at each node. Since enough verification and convergence studies have not been performed on the proposed beam elements, the present study aims to examine the accuracy, reliability and stability of aforementioned beam elements in the static bending. To that end, the cantilevered, simply supported and doubly clamped Timoshenko beams are chosen as the case studies. It is observed that the results obtained via the 6-DOF beam element are in excellent agreement with those obtained via the other solutions for the three case studies. In addition, it is found that the rate of convergence increases by ascending the influence of size-dependency. Although the 4-DOF beam element presents stable solutions, the acquired results reveal that the 4-DOF beam element is incapable of verifying the results obtained based on the other solutions for the three case studies. Moreover, it is found that the 4-DOF beam element error has an ascending trend with respect to the size-dependent shear deformation. Finally the reason for inaccuracy of the 4-DOF beam element is discussed.