# Single Layer Modelling and Effective Stiffness Estimations of Laminated Plates

• Holm Altenbach
• Johannes Meenen
Part of the International Centre for Mechanical Sciences book series (CISM, volume 448)

## Abstract

Two-dimensional laminated plate theories can be derived by reducing the three-dimensional equations of continuum mechanics or by postulating the existence of a two-dimensional Cosserat surface (the so-called direct approach). Theories derived by both approaches can be further classified into equivalent single-layer theories and layer-wise theories which possess individual unknowns for each layer. Layer-wise theories generally lead to better approximations than equivalent single-layer theories, but the large number of independent unknowns and governing equations increases the amount of computational power which is necessary to find a numerical solution, and the possibilities to obtain simple analytical solutions are limited.

Introducing assumptions into the principle of virtual displacements, classical and refined plate theories such as the classical laminated plate theory based on extended Kirchhoff’s assumptions, the first order shear deformation theories considering the transverse shear deformations of the plate, the refined plate theories working with additional degrees of freedom and the power series approach can be derived. After a brief overview of different approaches the modelling with respect to the scale size is discussed. Then, refined plate theories derived from the principle of virtual displacements are presented in detail. In contrast to this approach, by exploiting a direct formulation based on the theory of deformable surfaces and the introduction of five independent degrees of freedom for each material point of this surface, a theory similar to the first order shear deformation plate theory is developed. Finally, the extension of classical and refined plate theories to non-mechanical loadings is explained for the example of the laminated plate under thermal loadings.

Although many different plate theories result in identical stiffness properties for bending, tension/compression, in-plane shear and torsion, differences are present in the formulation of the transverse stiffness. From the analysis of three-dimensional problems, it is known that the accuracy of results obtained from single-layer theories is strongly dependent on the correct determination of the effective stiffness. It is shown that the classical stiffness tensors are identical for several different formulations of laminated plate theories. For the calculation of these properties closed solutions can be obtained. For the transverse shear stiffness closed solutions can be obtained too, but the comparison of numerical solutions with experimental data or calculations based on the three-dimensional theory sometimes does not lead to a satisfying agreement. For the first order shear deformation theory, a method for the determination of the transverse shear stiffness tensors is proposed, which leads to a Sturm-Liouville problem. For special cases the stiffness expressions are estimated and compared with results obtained from other authors. The approach is applicable to laminated plates as well as to sandwich plates.

The paper is limited to the calculation of laminated and sandwich plates. The effective properties of the individual layers are assumed to be known with respect to the global coordinate system. The estimation of these effective properties from the properties of the different constituents of each laminae is briefly presented and the transformation of the effective properties from a local coordinate system to a global coordinate system is discussed.

## Keywords

Transverse Shear Plate Theory Laminate Plate Sandwich Plate Shear Deformation Theory
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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