Coupled approximate long and short solutions versus exact Navier and Galerkin ones for cylindrical shell under radial load: Fid-Bau Portal

Coupled approximate long and short solutions versus exact Navier and Galerkin ones for cylindrical shell under radial load

Orynyak, Igor / Bai, Yulia

Abstract

Abstract Cylindrical shell under concentrated force is a fundamental problem, which provides the understanding of general patterns of deformation and allow to consider many other types of loading and geometries. The common idea of existing approaches is expansion of the looking for solutions into Fourier series with respect to $n φ$ , where $φ$ is a circumferential coordinate, and $n$ is the circumferential wave number. This reduces the problem to the 8th order even differential equation as to axial coordinate. Yet the finding of relevant 8 eigenfunctions and exact relations of 8 constants of integration with boundary conditions are still beyond the possibilities of analytical treatment. Most analytical solutions are based on Vlasov-like solution of simplified 4th order equations with unknown ranges of their accuracies. The goal of paper is threefold. First, to give the exact solutions for infinite shell, which are based on: (a) the well-known Navier approach, and (b) the Galerkin methods with application of decaying exponential functions. Second, to modernize the hypotheses of Vlasov’s theory (long-wave axial solution) and supplement it with the short solution, and commonly use both in formulation of boundary conditions, i.e., to take into account that their interaction occurs not due to shear force only but other force and displacement parameters may interact too. Third, to perform the tedious comparison of exact Galerkin and Navier solutions with approximate ones for the whole range of circumferential waves $n$ , assess their accuracy and establish the accuracy of plate-like solution for the larger number of $n$ .

Graphical abstract Display Omitted

Highlights • Exact Navier solution is obtained for concentrated radial force in infinite shell. • Application of decaying exponential functions in Galerkin method is proved to be very effective. • Long solution is supplemented by short solution and high accuracy of their coupled application is established. • Simple Plate-like solution is very accurate for bigger number of circumferential waves.