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

Akif Kutlu*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)1712-1722
Number of pages11
JournalProceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications
Issue number7
Publication statusPublished - Jul 2021

Bibliographical note

Publisher Copyright:
© IMechE 2021.


  • laminated beam
  • mixed finite element
  • Refined zigzag theory
  • stress analysis


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