Finite element formulation for analysis of pipes based on thin shell theory

A general solution for the stress-deformation analysis of pipes subjected to general loading conditions is developed. The solution is based on the assumptions of thin-walled shell theory and is limited to straight pipes of prismatic cross-section made of linearly elastic isotropic material subjected to general loading. The principle of stationary potential energy is used in conjunction with general Fourier series expansion for the displacement fields to formulate the equilibrium conditions and boundary conditions. The equilibrium equations for each Fourier mode are observed to be uncoupled from other modes, a feature that is exploited in formulating a general closed form solution for the displacement fields. The analytical solution developed is then successfully adopted to solve two pipe problems. Comparisons with established finite element solutions demonstrate the ability of the model to accurately capture complex shell behaviour with a remarkably small number of degrees of freedoms. However, the algebraic manipulations in the analytical solution are found quite tedious for hand calculations, even for simple problems. In order to remedy this limitation and to make the solution scheme amenable to more complex problems, a finite element solution is formulated based on the analytical solution developed. The expressions for the displacement fields obtained are used to formulate a series of exact shape functions which relate the intermediate displacements within a finite element to the nodal displacements. Using the exact shape functions, the principle of stationary potential energy is adopted to formulate the stiffness matrices and associated energy equivalent load vectors for pipe finite elements. The new finite element is tested for a variety of practical loading conditions including point loads, gravity loads, internal and external pressure, twisting deformation, axial deformation, transverse deformations, and combinations thereof. Results are found in excellent agreement with those based on shell finite elements in ABAQUS. The pipe element developed is shown to be free of discretization errors and is thus computationally efficient. The element can very accurately predict the displacements and stresses of a pipe with only a very few elements and a few Fourier modes