Refined zigzag theory for homogeneous, laminated composite, and sandwich beams derived from Reissner’s mixed variational principle

A highly accurate and computationally attractive shear-deformation theory for homogeneous, laminated composite, and sandwich laminates is developed for the linearly elastic analysis of planar beams. The theory is derived using the kinematic assumptions of Refined Zigzag Theory (RZT) and a two-step procedure that implements Reissner’s Mixed Variational Theorem (RMVT). The basic expression for the transverse-shear stress that satisfies a priori the equlibrium conditions along the layer interfaces is obtained from Cauchy’s equilibrium equations. The resulting transverse-shear stress consists of second-order derivatives of the two rotation variables of the theory, which subsequently are restated as the unknown stress functions. As the first step in fulfilling RMVT, the Lagrange-multiplier functional is minimized with respect to the unknown stress functions, resulting in the stress functions consisting of first-order derivatives of the kinematic variables. Subsequently, the second term of RMVT is minimized, producing four beam equilibrium equations and consistent boundary conditions. For any number of material layers the new theory maintains only four kinematic variables. The theory is labeled RZT(m), where the superscript (m) stands for mixed formulation. The RZT(m) can accurately model the axial stretch, bending, and transverse-shear deformations, without shear-correction factors. Analytic solutions are derived for simply supported beams subjected to transverse-normal and transverse-shear tractions on the top and bottom surfaces. It is demonstrated that RZT(m) has a wide range of applicability which includes sandwich construction and the laminates with embedded thin compliant layers that can potentially model progression of delaminations. The main advantage of RZT(m) over RZT is in the superior predictions of transverse-shear stresses that are obtained directly from the low-order transverse-shear strain measures of the theory without resorting to a post-processing integration procedure. Importantly, the methodology can be readily extended to plate theory, and it can be applied effectively for developing simple and efficient C0-continuous finite elements.