Size-dependent nonlinear bending of tapered cantilever microbeam based on modified couple stress theory

The Euler-Bernoulli beam theory is adopted in conjunction with modified couple stress theory (MCST) in this paper to formulate a beam element for size-dependent nonlinear bending analysis of a tapered cantilever microbeam subjected to end force/moment. The microbeam is considered to be linearly tapered in both the width and height directions. The element is derived in the context of the co-rotational approach in which the internal force vector and the tangent stiffness matrix are firstly derived in an element attached coordinate system and then transferred to the global one by the transformation matrices. An incremental/iterative procedure is adopted in conjunction with the arc-length method to compute the response of the microbeam. The obtained results show that the formulated element is efficient, and it is capable to model accurately the size-dependent nonlinear response of the microbeam by using just several elements. It is shown that the size effect plays an important role on the large deflection response, and the displacements of the microcantilever are overestimated by ignoring the influence of the micro-scale size effect. The effects of the material length scale parameter and the tapered ratio on the nonlinear behavior of the microbeam are studied in detail and highlighted.