Three-Dimensional Free Vibration Analysis of Cross-Ply Laminated Rectangular Plates with Free Edges Through a Displacement-Based Approach

Abstract

This paper presents a model belonging to a theory that recently appeared in the literature dealing with accurate solutions of freely vibrating laminated plates. The present model, which is derived from a displacement-based variational approach, investigates both free edge boundary conditions and the possibility of obtaining accurate results without explicitly incorporating stress interlaminar continuity conditions where they are in principle requested. These investigations are carried out within the frame of multi-layered plates where the literature is lacking in relevant three-dimensional results.In spite of the multi-layer nature of the plate, the model is developed as if the same plate were virtually made of a single layer because a suitable set of piecewise-smooth functions is used. Once the model is theoretically introduced, natural frequencies are evaluated for the case of plates with simply supported and free edge boundaries. The evaluations are compared with those few existing values of exact or approximate three-dimensional models, new results are provided for future comparisons and, finally, the performance of the model is tested with respect to extreme cases which could also deal with interesting engineering applications without reverting to non-immediate numerical schemes.

On the performance of advanced three-dimensional models for cylindrically bent plates subjected to arbitrary boundary conditions

“…The cylindrical bending quantities of the infinite plate shown in Figure1 are considered regardless of the y coordinate, namely the ones in which all the plate cross-sections deform in an identical pattern and for which there is not the dependence on the y coordinate. Therefore, the displacement field is established through the dependence (2):…”

“…There exist several ways to build suitable admissible bases once the essential edge boundary conditions (9) and (10) are established for each displacement component; here the bases are assumed orthonormal and are built by using a recursive procedure (e.g. [2]). Equations (11) provide explicit expressions in the non dimensional domain j 2 [-1/2, 1/2] for three essential-type boundary conditions:…”

“…[1][2][3]) aimed at approaching the three-dimensional dynamical values (natural frequencies and mode shapes) of multilayered plates subjected to classical although arbitrary boundary conditions, the interest of the present communication is based on the attempt to test the present simple analytical approach while preserving the accuracy of the results within the frame of the existing modern calculators. Such an attempt has been previously pursued by the present author through the classical theorem of virtual displacements along with the adoption of the global piecewise smooth functions (GPSFs) [4]; as is shown by Messina and Rollo [2] and Messina [3], such a formalization has allowed one to deal with a multilayered architecture as if the plates were made up of a single layer subjected to classical arbitrary boundary conditions with, these latter, even dealt with through only three distinct functional bases. The fact that any explicit continuity conditions of the relevant interfacial stress components are not explicitly introduced could be worth mentioning.…”

Three-dimensional free vibration of multi-layered piezoelectric plates through approximate and exact analyses

“…In particular, the functional bases ( X, Y ) should fulfil the essential boundary conditions at ( x, y ) = − L x,y /2, L x,y /2 and are assumed to be classical polynomials (smooth functions); in this case, any inner condition is not present due to the absence of in-plane material/geometric discontinuities. Instead, the functional z -bases should fulfil the essential boundary conditions at ( z ) = − h /2, h /2 and are assumed smooth, such as classical polynomials, only when single-layer plates are taken into account; when multilayers are stacked, such z -functional bases must correspond to GPSFs (Messina, 2002; Messina and Rollo, 2010) in order to potentially allow continuous or discontinuous stresses.…”

“…GPSFs stand for global piecewise-smooth functions and recall the suitable functional bases that have the aim of modelling multi-layered plates as if they were virtually made up of a single layer; the idea, which was first introduced by Messina (2002), is based on the idea of considering cusps as additional features in an expansion series; in this regard, an algorithmic graph (Messina, 2002) joins piecewise-smooth functions in order to obtain an expansion series that contains discontinuous derivatives; this expansion is able to model non-smooth functions (displacement components and electric quantities) without suffering from the so-called Gibbs phenomenon. For the sake of brevity, the GPSFs are not shown here; the readers can refer to the works of Messina (2002, 2005, 2011) and Messina and Rollo (2010) for further details.…”