Transverse shear warping functions for anisotropic multilayered plates

TL;DR

This paper develops and validates analytical transverse shear warping functions for anisotropic multilayered plates using a variational approach, improving accuracy over existing formulas especially for thick and complex laminates.

Contribution

It introduces a new variational method to derive warping functions for multilayered anisotropic plates, with analytical solutions for single layers and a semi-analytical approach for multilayers, validated by finite element simulations.

Findings

Analytical warping functions involve hyperbolic functions and differ from Reddy's formula.

Finite element simulations confirm the accuracy of the derived warping functions.

The method provides relevant warping shapes for angle-ply laminates despite some assumptions.

Abstract

In this work, transverse shear warping functions for an equivalent single layer plate model are formulated from a variational approach. The part of the strain energy which involves the shear phenomenon is expressed in function of the warping functions and their derivatives. The variational calculus leads to a differential system of equations which warping functions must verify. Solving this system requires the choice of values for the (global) shear strains and their derivatives. A particular choice, which is justified for cross-ply laminates, leads to excellent results. For single layer isotropic and orthotropic plates, an analytical expression of the warping functions is given. They involve hyperbolic trigonometric functions. They differ from the z - 4/3z3 Reddy's formula which has been found to be a limit of present warping functions for isotropic and moderately thick plates. When the h/L and/or the G13/E1 ratios significantly differ from those of isotropic and moderately thick plates, a difference between present warping functions and Reddy's formula can be observed, even for the isotropic single layer plate. Finite element simulations agree perfectly with the present warping functions in these cases. For multilayer cross-ply configurations, the warping functions are determined using a semi-analytical procedure. They have been compared to results of 3D finite element simulations. They are in excellent agreement. For angle-ply laminates, the above process gives warping functions that seem to have relevant shapes, even if the choice for global shear values cannot be justified in this case. No finite element comparison has been presented at this time because it is difficult to propose boundary conditions and prescribed load that permit to isolate the shear phenomenon.