On the stress problem of large elliptical cutouts and cracks in circular cylindrical shells

Numerical solutions are presented for stresses around an elliptical hole in a long, thin, circular cylindrical shell subjected to axial tension for both the symmetric orientations of the hole with respect to the shell. The method of analysis involves obtaining a series solution to the governing shell equations in terms of Mathieu functions by the method of separation of variables and satisfying the boundary conditions numerically term by term in a Fourier series formulation. Results are presented in the form of charts from which stress concentration factors can be directly read over a wide range of the two parameters, namely, axis ratio of the ellipse and a curvature parameter defining the hole size with respect to dimensions of the shell. An interesting feature of the investigation is the analysis of limiting cases of circumferential and axial cracks for axial tension and internal pressure loadings respectively. The method developed involves determining the solution completely in elliptic coordinates and then determining the singular stresses by carrying out a transformation to polar coordinates with crack tip as the origin through a Taylor series expansion. Membrane and bending stress intensity factors are computed and plotted over a sufficiently wide range of the curvature parameter extending from small to large sized cracks. As an outcome of the analysis, a “hybrid” technique has been developed by which singularity conditions at the crack tip can be handled effectively in dealing with boundary conditions in crack problems.