Flexural analysis of laminated beams using zigzag theory and a mixed inverse differential quadrature method

Zigzag theories are robust equivalent single layer models for stress analysis of laminated structures. They provide highly accurate stress predictions, especially in combination with through-thickness higher-order shear deformation theories, with significant reduction of computational time in comparison with finite element modelling or layer-wise theories [1]. However, the interlaminar continuity of transverse stresses is usually not guaranteed via the constitutive relations. Indefinite integration from Cauchy’s equilibrium equations can address this issue but the inconsistency between transverse stresses from this equilibrium integration and those from constitutive relations remain. By employing a compact form of the Hellinger-Reissner mixed variational principle, the Cauchy’s equilibrium conditions can be included in the variational statement to provide a variationally consistent solution. This approach was implemented successfully for analysing stresses in highly heterogeneous laminated beams [1]. It is worth mentioning that the governing equations resulting from this variational statement include fourth-order derivatives of the functional unknowns, which necessitate multiple successive differentiations in numerical methods. This current study presents an inverse DQM (iDQM) recently proposed by the authors [2,3] to investigate the flexural behaviour of constant and variable angle tow (VAT) laminated beams. In the iDQM approach, derivatives of primary quantities are approximated, thereby reducing the order of differentiation that needs to be performed. Moreover, the necessity of using higher-order axial displacements for better predictions of stresses near the boundaries is examined. Together with refined zigzag theory (RZT) and Murakami zigzag theory (MZZF), third- and fifth-order global displacement fields are employed in this example for stress prediction of moderately thick sandwich and VAT beams. Numerical results from the first-order mixed iDQM, i.e. first derivatives of functional unknowns are approximated using Lagrange polynomials, are verified with those obtained from conventional DQM and a high-fidelity 3D finite element solution (Abaqus commercial software). With the same third-order zigzag displacement fields (HR3-RZT/HR3-MZZF), improvement is observed in using iDQM for solving the governing equations. Moreover, the results also show that a fifth-order zigzag model (HR5-MZZF) is needed for predicting stresses in the vicinity of boundaries in VAT beams.