Application of exponential functions in weighted residuals method in structural mechanics. Part 3: infinite cylindrical shell under concentrated forces

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.

Applications of randomly selected sets of exact Voight's solutions for vibration of thin plates

“…Make some explanations as to the choice of Voigt functions in SES method. For 2 N = the number of functions should be equal to 16 As it follows from the Table 1 both Galerkin methods demonstrate a good convergence even at very small number of unknowns. This is because the trial functions exactly satisfy the boundary conditions for this task.…”

“…Their application is described in details in works [15][16][17]. Introduce the following designations for exponential functions: ( ) ( )…”

“…From other hand, the principally new family of trial functions were suggested by present authors in works [15][16][17] where the specially constructed exponential functions, EF, were successfully applied in Galerkin method for various structural mechanics problems. Remind that Galerkin method is a special kind of WRM, where the weight functions coincide with trial ones [3].…”