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

Based on recent researches (e.g. [1–3]) 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 pres- ent simple analytical approach while preserving the accuracy of the results within the frame of the exist- ing 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 func- tions (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 sub- jected 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. The above-mentioned analyses left, however, open questions; indeed, Messina [3] noticed certain boundary conditions as mainly respon- sible for a low convergence of eigenvalues to the expected exact three-dimensional counterparts. After investigating the mentioned influence of boundary conditions [3], this writer became aware of static analyses of possible relevant interest. In particular, this writer noticed with extreme interest the paper by Vel and Batra [5] to which the attention of this note is essentially addressed. Vel and Batra [5] analysed the generalized plane strain deformations from a static point of view and by using the Eshelby– Stroh formalism. In spite of the different nature of the problem (static in [5], dynamic in [1,3]) we must accept the fact that a possible slack convergence at exact values in a static analysis is a particular aspect of its dynamic counterpart and vice versa. Therefore, this author discusses the performance of the Eshelby–Stroh formalism, illustrated by Vel and Batra [5], in comparison to the capability of an analytical model to deal with multilayered plates as if they were made up of a single layer architecture [1,3] within the frame of static deformations. Interestingly the comparison will show how sometimes the explicit introduction of the required natural boundary conditions can reduce the accuracy of the results. Moreover, it is shown how the GPSFs bring remarkable benefits when used to model multilayered plates; the benefits can even increase when the GPSFs are used in multiple dimensions rather than only through the thickness of the plate. In passing, and finally, the present comparison shows a slight disagreement with Vel and Batra [5] for few transver- sal stress component distributions; this slight disagreement, however, is definitely of extremely minor importance when compared to the clean and helpful frame offered by the analysis of [5].