A (3,2) reduced degree-of-freedom unified zigzag laminated beam theory

Abstract

A (3,2) unified zigzag beam theory is developed with a reduced number of degree-of-freedom. Comparing to previous methods in the field of zigzag beam theory, the main novelty in this paper's method is that a more general non-vanishing top/bottom surface's shear stress boundary conditions are satisfied automatically in strong form. The bottom surface shear stress condition and the interface shear stress continuity conditions are used to uniquely determine the coefficients of zigzag functions. For the top surface shear stress condition, it is used to eliminate one degree-of-freedom, changing the 7°-of-freedom (3,2) zigzag beam to a 6°-of-freedom (3,2) zigzag beam. The zigzag coefficients are derived with an explicit formulation. Since the proposed method's formula is based on the unified beam theory, the formulation can be applied to any specific beam theory. The corresponding zigzag coefficients are also dependent on the specific beam theory's thickness basis function.In the numerical test section, several benchmark problems are solved to verify the accuracy. It is observed that the proposed beam has accurate solution for both thick and thin beams. The shear stress accuracy is also good for both vanishing and non-vanishing shear stress boundary conditions on top/bottom surfaces.