# FINITE ELEMENT METHOD APPLIED TO HIGHER-ORDER ZIG-ZAG THEORY FOR LAMINATED BEAMS

## Palavras-chave:

Laminated beams, ESL theories, Finite Element Method, Higher-Order Zig-Zag Theory## Resumo

Due to the continuing importance of laminated materials in civil, naval, mechanical

and aerospace engineering, the development of structural analysis theories of laminated beams

has been an active area of research. The well-known classical theories of Euler-Bernoulli and

Timoshenko have limitations because they do not present a field of shear deformation or by the

incorrect consideration of such deformations, without respecting the nullity of the shear stress

in the edges of the beam. Thus, high-order theories have emerged to remedy the limitations of

classical theories, especially the Equivalent Single Layer (ESL) theories. In ESL theory, the

displacement function in the thickness coordinate are assumed to Class C1

. This feature

provides a discontinuous shear stress field at the interfaces of adjacent layers with different

materials. In the 1980s, DiSciuva introduced a new class of laminated theories, where a zig-zag

function is added to the ESL theories to describe the displacement in the thickness coordinate.

This new theory allows the continuity of interlaminar tensions and a number of variables

independent of the number of layers of the beam. In this way, the present work seeks to present

the complete development of a finite element model of several ESL-Zig-Zag theories. Thus, a

unified displacement field will be used that allows the simultaneous development and

comparison of several refined theories found in the literature. After obtaining the governing

equations, the finite element model is constructed using the Lagrange and Hermite polynomial

functions. Finally, to show the good efficiency of the finite element model, numerical results

are shown and compared with the exact solution of the elasticity of Pagano (1970).