Influence of circular hole on the shell stress state for arbitrary Gaussian curvature
The work is devoted to determining isotropic shell of stress-strain state for arbitrary Gaussian curvature with a circular hole, located in the center of the structure. An axial tension or an internal pressure is applied to the surface of the shell. The isotropic shallow shell theory equations were used, which coincide with the isotropic shell theory equations with a large measure of variability. The integral Fourier transformation and the theory of generalized functions were applied. As a result the problem was reduced to solving the system of boundary integral equations. One benefit of using the method of boundary integral equations for the study of shell stress-strain state weakened by a hole is the ability to define the unknown quantities ​​directly on the contour of the hole, not evaluating them on the whole surface of the shell. To obtain the kernels of the singular integral equations, integral representations of displacements and shallow isotropic shells static equations fundamental solutions were used. As the unknown functions, a combination of displacements, rotation angles and their derivatives were used. Analytical calculations are considerably simplified if it is assumed to take into account not four unknown functions on the contour, as it is customary, but five. In this paper it is chosen to use the differential equation which relates the unknown functions as the fifth equation of the boundary integral equations system. In order to obtain numerical solution of the problem, the method of mechanical quadratures for systems of integral equations and finite difference method for the fifth differential equation were used to reduce a problem to a system of linear algebraic equations. The stress concentration factors values ​​depending on the isotropic shell curvature are given. Also the results were compared with other researchers.