Universal Journal of Mechanical Engineering 1(4): 128-132, 2013 http://www.hrpub.org DOI: 10.13189/ujme.2013.010404 The Deformation of Cylindrical Shells Subjected to Radial Loads Using Mixed Formulation and Analytic Solutions Luisa R. Madureira 1,* , Elza M. M. Fonseca 2 , Francisco Q. Melo 3 1 University of Porto, Faculty of Engineering, Department of Mechanical Engineering, Portugal 2 Polytechnic Institute of Bragança , Department of Applied Mechanics, Portugal 3 University of Aveiro, Department of Mechanical Engineering, Portugal *Corresponding Author: luisa.madureira@fe.up.pt Copyright © 2013 Horizon Research Publishing All rights reserved. Abstract The objective of this work is to contribute with a simple and reliable numerical tool for the stress analysis of cylindrical vessels subjected to generalized forces using a mixed formulation. Variational techniques coupled with functional analysis are used to obtain an optimized solution for the shell displacement and further stress field evaluation using a combination of unknown analytic functions with Fourier expansions. A large cylindrical shell subjected to pinching loads is considered. These elements are intended to provide accurate modelling of the initially circular pipes response. Because of this behaviour, the bend’s cross-section abandons its original roundness, turning into an oval or noncircular configuration. In addition, the initially plane cross-section, tends to deform out of its own plane. These two deformation patterns are termed ovalization and warping, respectively. In this work the results for the radial displacement and section ovalization are analysed where the solution has six terms for an acceptable accuracy. The transverse displacement presents important dependence on the shell thickness vs radius, where in the case of thin shells the ovalization is restricted to a local area and this is the case analysed. The proposed method leads to accurate results with low complex input data. The conclusions of this work are that the definition of the load system and boundary conditions are easily processed as the method has pre-defined possibilities for each load case or edge boundary conditions. An analytic solution is obtained and a low number of terms in the Fourier series show good accuracy as can be seen by a comparison with finite element methods. Keywords Piping Engineering; Fourier Series; System Of Differential Equations; Boundary Conditions 1. Introduction The solution here proposed deals with the combination of unknown analytic functions with Fourier expansions, where the former depend on the axial shell coordinate and the trigonometric terms are dependent upon the cylinder circumferential polar angle [1]. With this formulation and the evaluation of the total energy a system of ordinary differential equations is obtained and solved where the solution is analytic after evaluation of eigenvalues and eigenvectors. The boundary conditions are then used to achieve the final solution. The cylindrical shell deformations include axial, circumferential and shear membrane terms ε xx , ε θθ , and γ xθ . In this formulation the shell is assumed to be circumferentially inextensible, so ε θθ =0. Due to the shell elasticity the definition of the displacements in cylindrical thin pipes leads to a deformation field originating internal forces, which for the geometry considered, include the membrane axial force N xx , the circumferential force N θθ and the shear force N xθ . and also the bending moments N xx , N θθ and the twist moment N xθ .[1].They are related to the deformations by the elasticity matrix as shown by Eq.(1): 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 2 xx xx xθ xθ θ θ xx xx xθ x A N ε S N γ D M k D M k M k D θ θ θ ν                         =                   −             (1) The constants A, S, and D in Eq.(1) are given by: ( ) ( ) 2 3 2 1 21 12 1 Eh Eh Eh A S D = = = − + − ν ν ν (2) where E and ν are, respectively, the material Young´s modulus and Poisson´s ratio while h is the pipe thickness (assumed uniform). The terms presented in Eq.(1) together with internal forces complete the solution unknowns. The displacement field consists of the axial, the circumferential and the transverse shell displacements u, v and w respectively. These displacements result from the superposition of the so-called beam type with the ones assigned to the transverse section distortion. The first group results from the bending of a pipe,