Mixed finite element formulation for bending of laminated beams using the refined zigzag theory

Abstract

This study presents a mixed finite element formulation for the stress analysis of laminated composite beams based on the refined zigzag theory. The Hellinger–Reissner variational principle is employed to obtain the first variation of the functional that is expressed in terms of displacements and stress resultants. Due to C0 continuity requirements of the formulation, linear shape functions are adopted to discretize the straight beam domain with two-noded finite elements. The proposed formulation is shear locking free from nature since it introduces displacement and stress resultant terms as independent field variables. A monolithic solution of the global finite element equations is preferred, hence the stress resultants are directly obtained from the solution of these equations. The in-plane strain measures of the beam are obtained directly at the nodes over the compliance matrix and stress resultants by avoiding error-prone spatial derivatives. Following, transverse shear stresses are calculated from the equilibrium equations at the post-processing level. This simple but effective finite element formulation is first verified and tested for convergence behavior. The robustness of the approach is shown through some examples and its accuracy in predicting the displacement and stress components is revealed.

Mixed Finite Elements for Higher-order Laminated Cylindrical and Spherical Shells

“…Ozutok and Madenci [34] combined the mixed finite element method (MFEM) with HSDT for static analysis of laminated composite beams. Kutlu et al [35][36][37] and Aribas et al [38][39][40] have illustrated the benefits of MFEM-derived formulations while precisely predicting stress resultants and components in diverse structural configurations. Kutlu and Omurtag [41] applied a MFE formulation based on the Hellinger-Reissner principle for the static behavior of elliptic plates under large deflection conditions.…”

Stress Analysis of Laminated HSDT Beams Considering Bending Extension Coupling

“…Ozutok and Madenci [25] combined mixed finite element method (MFEM) and HSDT to perform static analysis of laminated composite beams. Kutlu et al [26][27][28] and Aribas et al [29][30][31] revealed some advantages of MFEM based formulations on the prediction of accurate stress resultants and stress components of various beam, helix and plate type structures. Shao et al [32] proposed a new HSDT for the free vibration analysis of laminated composite beams.…”

A C0 continuous mixed FE formulation for bending of laminated composite plates based on unified HSDT

“…In this study, a mixed finite element formulation [50][51][52][53] is employed to investigate the bending behavior of laminated composite plates using the Hellinger-Reissner variational principle based on HSDT. The formulation follows a unified form of HSDTs; thus, various shear functions (trigonometric, exponential, or polynomial) can be easily considered.…”

“…Obtaining stress resultants directly from the FE solution paves the way for calculating the strain measures from compliance relations instead of error-prone numerical derivatives of kinematic variables. In recent studies [43,50], it is shown that this approach performs superior on the prediction of stress components in laminated structures modeled by RZT [39,46,56]. The details of the proposed numerical procedure, namely the Mixed Higher-Order Plate Theory (MHPT), are introduced in Section 2.…”