Analytical solution for the axisymmetric buckling of cylindrical shells Miguel Lagos • Raj Das Received: 1 April 2014 / Accepted: 10 May 2014 Ó Springer Science+Business Media Dordrecht 2014 Abstract Closed-form analytical solutions for thin shell buckling problem are useful in a wide range of analysis and design problems. In this paper, the profile of a cylindrical shell in the post-buckling regime of axisymmetric deformation is analysed, and the solu- tion is shown to be a Jacobi elliptic sine function, for any load and axial deformation. The exact solution of the non-linear differential equation for the thin elastic shell profile holds for any deformation, up to the limit in which the shell is almost flattened by the applied load. Closed-form expressions are derived also for the load dependent axial deflection and stored energy. The analytical solution of the buckling loads and deformed profile are found to agree well with an equivalent numerical solution. Results show that an axially compressed cylindrical shell exhibits ideal behaviour for a safety shock energy absorber. Keywords Buckling  Cylindrical shell  Elliptic functions  Energy absorption 1 Introduction Buckling of cylindrical shells is an important problem in mechanics and is commonly encountered in several applications, such as pressure vessels, aerospace and civil structures. Although this system has been a common subject in the specialized literature, only numerical solutions have been reported (Kim and Kim 2002; Hunt et al. 2003; Paschero and Hyer 2009; Pinna and Ronalds 2003; Simitses 1986; Wullschleger and Meyer-Piening 2002). It is envisaged that an analytical solution of this buckling problem will have several advantages in designing thin shell curved structural members. To address this, a closed-form analytical solution for the axisymmetric buckling of an elastic cylindrical shell under an axial load, applied uni- formly on its edges, is developed here. Instead of applying directly the methods of the standard theory of elasticity, the present approach resorts to the analysis of the equilibrium of forces at an elementary sector of the shell. Although both procedures are conceptually equivalent, dealing with forces instead of stresses allows for a much simpler derivation of the equation for the shell profile, because forces convey an implicit first integration. In turn, the applied load, shell profile and axial deflection are actually mean values, and hence integrals over the strain fields. The non-linear equation for the buckled axisymmetric thin shell profile can be solved by analytical means and yields a Jacobi elliptic sine function. Closed-form expressions are also derived for the buckling critical load and M. Lagos  R. Das Faculty of Engineering, University of Talca, Curico ´, Chile e-mail: mlagos@utalca.cl R. Das (&) Department of Mechanical Engineering, University of Auckland, Auckland, New Zealand e-mail: r.das@auckland.ac.nz 123 Int J Mech Mater Des DOI 10.1007/s10999-014-9259-9