Analytical and numerical solution for a elastic pipe bend at in-plane bending with consideration for the end effect

The authors proposed an analytical method for the analysis of the end effect in a pipe bend loaded by a bending moment with consideration for the action of internal pressure. The method is based on the use of simplifying hypotheses and is reduced to the solution of a system of fourth-order differential equations along the axial coordinate with respect to unknown coefficients in the expansion for tangential displacements. An approximate analytical solution, which has a trapezoidal structure and is written in terms of Krylov’s functions, has been obtained. Boundary conditions are formulated in terms of the tangential and longitudinal displacements and axial and shearing stress resultant. For the flexibility factor, analytical solutions are presented in the case where a bend is approximated by a rigid restraint on both ends. To verify the analytical solution and its applicability limits, two numerical procedures were developed, which are based on the finite difference method and the reduction to the Kochi problem by the expansion of the unknowns in the Fourier series over the circumferential coordinate. The authors compare the results obtained with data from the literature, discuss the advantages and disadvantages of the methods, and present recommendations for their practical application