Critical behavior of flat and stiffened shell structures through different kinematical models: A comparative investigation Eugenio Ruocco a,n , Massimiliano Fraldi b a Department of Civil Engineering, Second University of Naples, Italy b Department of Structural Engineering, University of Naples Federico II, Italy article info Article history: Received 4 November 2011 Received in revised form 27 June 2012 Accepted 27 June 2012 Available online 20 August 2012 Keywords: Buckling Stiffened shell structures Koiter–Sanders model Von Karman model Kantorovich method Finite strip method abstract In the present work buckling stresses of prismatic flat and stiffened shell structures are derived within the framework of the Kantorovich approach, making reference to both Von Karman and Koiter–Sanders theories, the latter exploiting the Green–Lagrange strain tensor which is needed if the expected buckling modes involve comparable in-plane and out-of-plane displacements. Additionally, in order to highlight the contribution of each nonlinear term of the Koiter–Sanders model, two further inter- mediate choices are also considered, namely an enhanced Von Karman model and a spurious model which collects selected terms from different theoretical approaches, generally adopted in literature with the aim of simplifying the numerical analyses. The obtained results show how in buckling problems where the weight of in-plane displacements cannot be neglected the Von Karman model tends to overestimate the critical load, while the three considered alternative models result substantially equivalent, at least from the practical standpoint. & 2012 Elsevier Ltd. All rights reserved. 1. Introduction Plate and shell structures are frequently used as structural components in a large number of engineering applications. If compressive forces are exerted on such structures alone or in combination with other boundary conditions, the design process requires the evaluation of the critical stress level associated to the buckling load. In the analysis of geometrically complex structures, as well as in the case of combined loads, numerical strategies are necessa- rily required and, because of its relative simplicity and versatility in modeling, the Finite Element Method (FEM) is, among others, the most employed computational routine for solving both research problems and practical applications in mechanics and structural analysis [1]. However, FEM-based approaches can sometimes result in very time-consuming procedures, especially at the initial stages of a design process [2], the alternative numerical methods turning potentially more attractive. When the structures are prismatic, constituted by flat or curved plate components rigidly connected along their longitudinal edges to form arbitrary cross-section profiles, the finite strip method (FSM) can represent for example a very competitive alternative to the FEM in terms of accuracy, computational times and ease of data preparation [3]. The FSM exists in literature in a number of variants: the main two can be distinguished by the nature of the displacement field, and are referred to Semi-analytical Finite Strip Method (SaFSM) and Polynomial Finite Strip Method (PFSM) [4]. In both the models longitudinal displacements are described through trigonometric functions. In transversal direc- tion SaFSM uses analytical solutions of the equilibrium equations in the form of displacements, whereas PFSM adopts suitable polynomial functions. SaFSM and PFSM can be seen as comple- mentary procedures, each one exhibiting specific advantages and disadvantages. For example, numerical codes which exploit PFSM starting from approximate energy-based or work-based FSM approaches are extremely versatile, but the solution of the problem, as it happens in FEM analyses, strongly depends on the adopted discretization and can be thus affected by numerical instabilities [5]. On the contrary, numerical codes based on SaFSM approaches are generally more stable regardless of the discretiza- tion adopted, offering reliable results with the minimum number of elements required for a proper geometrical description of the structure, but they can be applied only when some closed form solutions of the equilibrium equations are available. Many efforts have been spent in finding such solutions, starting from the pioneer work by Williams and Wittrick [6], who derived an exact solution for isotropic, flat, Kirchhoff plate subjected to uniaxial uniform in-plane load, by explicitly solving the governing equili- brium equations. Various other solutions have been successively Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/tws Thin-Walled Structures 0263-8231/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tws.2012.06.016 n Corresponding author. E-mail addresses: eugenio.ruocco@unina2.it (E. Ruocco), fraldi@unina.it (M. Fraldi). Thin-Walled Structures 60 (2012) 205–215