Shell analysis of thin-walled pipes. Part I – Field equations and solution

In this first of two companion papers, the governing equilibrium conditions for a thin-walled pipe subjected to general loading are developed. The internal strain energy is expressed in terms of general displacement fields under the common kinematic assumptions of thin shell theory. The formulation is quite general and involves a complete description of membrane and shell bending deformations. The unknown displacement fields are expressed as infinite Fourier series. Thus, they fully capture warping deformations, the variation of tangential and longitudinal displacements in the longitudinal and transverse directions, and the radial expansion induced by effects such as internal pressure. The equilibrium conditions and boundary conditions are then derived by evoking the stationarity conditions of the total potential energy functional. The resulting equilibrium conditions are observed to couple the displacement contributions (radial, tangential, and longitudinal displacements) within each Fourier mode, while uncoupling the contributions of a given Fourier mode i   from those of other Fourier modes j≠ij≠i. An exact solution of the field equations is developed. The solution is applicable for general loading and boundary conditions. The applicability of the solution is demonstrated by solving two practical problems. A comparison with other finite element solutions demonstrates the validity and accuracy of the solution.