Vibration of multilayered beams using sinus finite elements with transverse normal stress

A family of sinus models is presented for the analysis of laminated beams in the framework of free vibration. A three-noded finite element is developed with a sinus distribution with layer refinement. The transverse shear strain is obtained by using a cosine function avoiding the use of shear correction factors. This kinematic accounts for the interlaminar continuity conditions on the interfaces between the layers, and the boundary conditions on the upper and lower surfaces of the beam. A conforming FE approach is carried out using Lagrange and Hermite interpolations. It is important to notice that the number of unknowns is independent of the number of layers. Vibration mechanical tests for thin and thick laminated and sandwich beams are presented in order to evaluate the capability of these new finite elements to give accurate results with respect to elasticity or finite element reference solutions. Both convergence velocity and accuracy are discussed and these new finite elements yield very accurate results at a low computational cost for various boundary conditions. In particular, the two models including the transverse normal effect have the capability to take into account the thickness mode shape.