A simple parabolic formula that describes the opening profile of a sufficiently long circumferential crack in a pipe subjected to bending and tension was found after a proper normalization of the second-order asymptotic expansion of the solutions of Sanders and Alabi [5-9]. Here the crack is sufficiently long in the sense that approximately the crack angle is larger than a small parameter which has the same order of magnitude as the ratio of the wall thickness to the pipe radius, and the pipe is sufficiently long enough not to consider the end effects. These linear elastic solutions can be superposed for combined bending and tension loading. The smile parabolic formula is shown to be a good approximation of the crack opening profile and can be used to evaluate the leak rate for crack length up to one-half of the circumference.
Approximate Crack Opening Displacement Solutions for Long Circumferential Cracks in Pipes Subjected to Bending and Tension
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Yoo, S. H., and Pan, J. (May 1, 1992). "Approximate Crack Opening Displacement Solutions for Long Circumferential Cracks in Pipes Subjected to Bending and Tension." ASME. J. Pressure Vessel Technol. May 1992; 114(2): 178–180. https://doi.org/10.1115/1.2929026
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