Three-dimensional exact solution for the vibration of functionally graded rectangular plates

Bending, buckling and free vibration of laminated composite and sandwich beams: A critical review of literature

A review of displacement and stress based refined theories for isotropic and anisotropic laminated plates is presented. Various equivalent single layer and layerwise theories for laminated plates are discussed together with their merits and demerits. Exact elasticity solutions for the plate problems are cited, wherever available. Various critical issues related to plate theories are presented, based on the literature reviewed.

A Review of Refined Shear Deformation Theories for Isotropic and Anisotropic Laminated Beams

A three-dimensional mixed variational principle is used toderive a Kth-order two-dimensional lineartheory for an anisotropichomogeneouspiezoelectric(PZT)plate.Themechanicaldisplacements, theelectricpotential,theinplane components of the stress tensor, and the in-plane components of the electric displacement are expressed as a e niteseries of orderK inthethickness coordinate bytakingLegendrepolynomialsas thebasisfunctions. However, the transverse shear stress, the transverse normal stress, and the transverse electric displacement are expressed as a e nite series of order (K +2) in the thickness coordinate. The formulation accounts for the double forces without moments that may change the thickness of the plate. Results obtained by using the plate theory are given for the bending of a cantilever thick plate loaded on the top and the bottom surfaces by uniformly distributed 1) normal tractions and 2) tangential tractions. Results are also computed for the bending of a cantilever thick PZT beam loaded by 1) a uniformly distributed charge density on the top and the bottom surfaces and 2) equal and opposite normal tractions distributed uniformly only on a part of the beam. The seventh-order plate theory captures well the boundary-layer effects near the clamped and the free edges and adjacent to the top and the bottom surfaces of a thick orthotropic cantilever beam with the span to the thickness ratio of two. Also, through-the-thickness variation of the transverse shear and the transverse normal stresses agree well with those computed from the analytical solution of the three-dimensional elasticity equations. The governing partial differential equations are second order, so that Lagrange basis functions can be used to solve the problem by the e nite element method.

Higher-Order Piezoelectric Plate Theory Derived from a Three-Dimensional Variational Principle

This study investigates and reviews approaches to modelling laminated composite plates. It explores theories that have been proposed and developed and assesses their suitability and functionality. The particular focus in this study has been on normal stresses and the through-thickness distributions of transverse shear. These are important for composite plates as stress-induced failures can occur in three different ways. Therefore, it is essential to understand and calculate transverse shear and normal stress through the thickness of the plate accurately. In this study, previous laminated composite plate theories are categorised and reviewed in a general sense, i.e. not problem specific, and the advantages and disadvantages of each model are discussed. This research mainly focuses on how accurate and efficient the models can predict the transverse shear. It starts with displacement-based theories from very basic models such as Classical laminate plate theory to more complicated and higher-order shear deformation theory. Models are furthermore categorised by how the models consider the overall laminate. In this article, the theories are divided into two parts: Single layer theory, where the whole plate is considered as one layer; and Layerwise theory, where each layer is treated separately. The models based on zig-zag and Discrete Theories are then reviewed, and finally the mixed (hybrid) plate theories are studied.

The development of laminated composite plate theories: a review

A general theory for the accurate stress analysis of homogeneous and laminated composite beams

This paper proposes a method for improving the performance of two-dimensional, higher order theories of homogeneous and laminated composite plates. The method is based on the appropriate specification of through-thickness shape functions which are suitable for the accurate stress analysis of plates, on the basis of a general six-degrees-of-freedom plate theory (G6DOFPT). G6DOFPT takes both transverse shear and transverse normal deformation effects into consideration using three shape functions, each of which is associated with one of the three unknown displacement components. Specification of these three shape functions is achieved by solving the corresponding equations of three-dimensional elasticity for a simply supported plate. Thus, the shape functions correspond to the exact elasticity solution presented in previous work. Hence, no results are shown for simply supported plates because, in that case, the present method will yield the results of the exact elasticity analysis. Instead, new results are p...

Accurate Stress Analysis of Laminated Plates Combining a Two-Dimensional Theory with the Exact Three-Dimensional Solution for Simply Supported Edges:

Exact three-dimensional elasticity solutions for bending of moderately thick inhomogeneous and laminated strips under normal pressure

This paper proposes a method for substantially improving the performance of the higher-order plate and shell theories in connection with the accurate stress analysis of homogeneous and laminated composite structural elements. The presentation of the method is based on the equations of the “general five-degrees-of-freedom” shear deformable plate theory. Since the method is entirely new, it is initially applied to the solution of the problem of simply supported plates deformed by cylindrical bending, for which there exists an exact elasticity solution [12]. Hence, its reliability is substantially validated by means of appropriate comparisons between numerical results based on the present plate theory and this exact elasticity solution. Moreover, the one-dimensional version of the present plate theory, employed for the cylindrical bending of plates, is considered as a general three-degrees-of-freedom shear deformable beam theory. This advanced beam theory is used for an accurate stress analysis of two-layered composite beams having one of their edges rigidly clamped and the other either rigidly clamped, free of tractions or simply supported. This final set of applications can be thought of alternatively as a stress analysis of two-layered plates deformed in cylindrical bending and subjected to several, different sets of edge boundary conditions.

A method for improving the stress analysis performance of one- and two-dimensional theories for laminated composites

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A general theory for the accurate stress analysis of homogeneous and laminated composite beams

Accurate Stress Analysis of Laminated Plates Combining a Two-Dimensional Theory with the Exact Three-Dimensional Solution for Simply Supported Edges:

Exact three-dimensional elasticity solutions for bending of moderately thick inhomogeneous and laminated strips under normal pressure

A method for improving the stress analysis performance of one- and two-dimensional theories for laminated composites