A strain-displacement mixed formulation based on the modified couple stress theory for the flexural behaviour of laminated beams.

Luan Trinh, Rainer Groh, Giovanni Zucco, Paul M Weaver

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A novel strain-displacement variational formulation for the flexural behaviour of laminated composite beams is presented, which accurately predicts three-dimensional stresses, yet is computationally more efficient than 3D finite element models. A global third-order and layer-wise zigzag profile is assumed for the axial deformation field to account for the effect of both stress-channelling and stress localisation. The axial and couple stresses are evaluated from the displacement field, while the transverse shear and transverse normal stresses are computed by the interlaminarcontinuous equilibrium conditions within the framework of the modified couple stress theory. Then, axial and transverse force equilibrium conditions are imposed via two Lagrange multipliers, which correspond to the axial and transverse displacements. Using this mixed variational approach, both displacements and strains are treated as unknown quantities, resulting in more functional freedom to minimise the total strain energy. The differential quadrature method is used to solve the resulting governing and boundary equations for simply-supported and clamped laminated beams. For the simply-supported case, numerical results from this variational formulation agree well with those from a Hellinger-Reissner stress-displacement mixed model found in the literature and the 3D elasticity solution given by Pagano. For the clamped laminate, the additional curvature associated with the couple stress is important to accurately predict localised stresses near clamped ends, which is confirmed by a high-fidelity 3D finite element model.
Original languageEnglish
Article number107740
Number of pages22
JournalComposites Part B: Engineering
Issue number15 March 2020
Early online date7 Jan 2020
Publication statusE-pub ahead of print - 7 Jan 2020


  • Strain-displacement mixed formulation
  • modified couple stress
  • laminated beam
  • stress analysis
  • zigzag theory


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