Exact solution of the Michell’s problem for the pipelines

To improve accuracy and reliability in strength computation of the most loaded pipeline sections carried out using the theory of elastic rods, methods of the mathematical theory of elasticity are used. Their corresponding boundary value problems are solved numerically or analytically. Well-known examples of the analytical solutions include solution of the Lame problem for a pipe exposed to internal and external pressure and solution of the Saint-Venant problem on the bending of a prismatic rod with the annular cross-section sealed at one end and loaded with a transverse force at the other. The Mitchell’s problem is a natural continuation of the Saint-Venant problem, as it considers the stress state in a prismatic rod uniformly loaded along the lateral surface. The scientific and technical literature provides solutions to the Michell’s problem for a general case of the prismatic rod with the arbitrary cross-section. Therefore, practical application of the available computation formulas for the pipeline annular cross-section is problematic. The paper considers and solves the Michell’s problem in a formulation satisfying the pipeline transportation requirements, when the outer and inner side surfaces are exposed to the uniformly distributed tangential stresses arising from the axial displacement and pipeline twisting in the ground, as well as in transportation of the highly viscous products. Exact solution to the Michell’s problem is obtained in stresses. An approximate solution to the problem was preliminarily found by the methods of strength of materials. The paper shows that the theory of elastic rods provides reliable results for stresses, as well as fairly accurate results for the displacements, despite the cross sections warping. The results obtained could also be useful in developing the logically consistent mathematical models of elastic rods, work on which is still underway.
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