The Shallow Shell Simplification of the Timoshenko-Reissner Model for Circular Cylindrical Shells and Its Application to the Vibrations and Buckling Problems

Abstract eng:
A circular cylindrical shell made of a transversely isotropic linearly elastic material is studied with the help of the Timoshenko-Reissner model, which takes into account transversal shear. The model is described by system of differential equations of the10th order. Functions with two different indexes of variation are among the solutions of the system. With assumptions made for shallow shells the system splits into two separate sub-systems. The solutions of each of them have only one index of variation. The error of the simplified system for shallow shells compared to the general Timoshenko-Reissner system is estimated. Then we study a thin cylindrical panel in assumption that one of its rectilinear edges is free or weakly supported and the curvilinear edges are simply supported. Two similar problems are solved: 1) free vibration and 2) buckling of cylindrical panel under axial compression. Specifically, we are interested in the case when the eigenfunction is localized near the weakly supported rectilinear edge. Earlier these problems were solved for isotropic shell by means of the Kirchhof-Love theory, when six cases of weakly supported edges, for which the localized solution exists, were found [1]. Here we study same six variants of the boundary conditions for transversely isotropic shells with shear modulus that is comparatively small in transverse direction. We compare the results obtained with the TimoshenkoReissner and the Kirchhof-Love models. The special attention is paid to the effect of the 5th boundary condition, which is missed in the Kirchhof-Love model. References [1] P.E. Tovstik, A.L. Smirnov, Asymptotic Methods in Buckling Theory of Elastic Shells, World Scientific Publishing Co Ltd., 2001.

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