Abstract
Solution for cylindrical shell under concentrated force is a fundamental problem which allow to consider many other cases of loading and geometries. Existing solutions were based on simplified assumptions, and the ranges of accuracy of them still remains unknown. The common idea is the expansion of them into Fourier series with respect to circumferential coordinate. This reduces the problem to 8th order even differential equation as to axial coordinate. Yet the finding of relevant 8 eigenfunctions and exact relation of 8 constant of integrations with boundary conditions are still beyond the possibilities of analytical treatment. In this paper we apply the decaying exponential functions in Galerkin-like version of weighted residual method to above-mentioned 8th order equation. So, we construct the sets of basic functions each to satisfy boundary conditions as well as axial and circumferential equilibrium equations. The latter gives interdependencies between the coefficients of circumferential and axial displacements with the radial ones. As to radial equilibrium, it is satisfied only approximately by minimizations of residuals. In similar way we developed technique for application of Navier like version of WRM. The results and peculiarities of WRM application are discussed in details for cos2j concentrated loading, which methodologically is the most complicated case, because it embraces the longest distance over the cylinder. The solution for it clearly exhibits two types of behaviors – long-wave and short-wave ones, the analytical technique of treatment of them was developed by first author elsewhere, and here was successfully compared. This example demonstrates the superior accuracy of two semi analytical WRM methods. It was shown that Navier method while being simpler in realization still requires much more (at least by two orders) terms than exponential functions.
Highlights
The distribution of deformations in elastic infinite cylindrical shell under concentrated radial force is a classical problem which potentially establish the basis for solving shells under different boundary conditions, loadings, and material properties
The convergence to the correct result at x = 0 go slowly with increase of number of terms, and perfect accuracy, see Table 1
This paper is devoted to the interesting classical problem of cylinder under concentrated radial force and in first time establish correct values of stresses and displacements for particular loading (11d)
Summary
The distribution of deformations in elastic infinite cylindrical shell under concentrated radial force is a classical problem which potentially establish the basis for solving shells under different boundary conditions, loadings, and material properties. More correct and simpler approach was suggested in work [5] where the expansion in Fourier series was applied in axial direction as well, and this unnecessitated the simplification of governing equations This method was applied by author and other investigators [6] to cylinders of finite length at specific boundary conditions. Theory of Vlasov became a very popular one, and several its modifications were applied to the problem of interest [8, 9] These papers were aimed only at derivation of the maximal deflection of the point of force application and their relative accuracies can be explained by overwhelming contribution of terms at smaller number of in expansion as to circumferential coordinate. The experience gained at preparation of work [15] will be used thoughtfully this paper
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