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Numerical and Analytical Investigation Into the Plastic Buckling Paradox for Metal Cylinders Rabee Shamass

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Numerical and Analytical Investigation Into the Plastic Buckling Paradox for Metal Cylinders Rabee Shamass Numerical and Analytical Investigation into the Plastic Buckling Paradox for Metal Cylinders A thesis submitted for the degree of Doctor of Philosophy By Rabee Shamass School of Engineering, Design and Physical Sciences Brunel University London March 2016 Declaration This thesis entitled “Numerical and Analytical Investigation into the Plastic Buckling Paradox of Metal Cylinder”, is submitted to the Department of Mechanical, Aerospace and Civil Engineering, School of Engineering, Design and Physical Sciences, Brunel University of London, Kingston Lane, Uxbridge, UB8 3PH, UK, for the degree of Doctor of Philosophy. The research has not formed the basis of a submission for any other degree. Publications based on this thesis: Conference papers Shamass, R., Alfano, G. and Guarracino, F. (2013) ‘A numerical investigation into the plastic buckling of circular cylinders’. Shell Structures: Theory and Application, 3, pp.441-444. Journal articles Shamass, R., Alfano, G. and Guarracino, F. (2014) ‘A numerical investigation into the plastic buckling paradox for circular cylindrical shells under axial compression’ Engineering Structures, 75, 429-447. Shamass, R., Alfano, G. and Guarracino, F. (2015) ‘An analytical insight into the buckling paradox for circular cylindrical shells under axial and lateral loading’, Mathematical problems in engineering, ID 514267:1-11. Shamass, R., Alfano, G. and Guarracino, F. (2015) ‘An investigation into the plastic buckling paradox for circular cylindrical shells under non-proportional loading’, Thin-Walled Structures, 95, 347-362. Shamass, R. Alfano, G. and Guarracino, F. (2016) ‘On the Elastoplastic Buckling Analysis of Cylinders under Non-proportional Loading via the Differential Quadrature Method’, International journal of Structural Stability and Dynamics, (online ready). Book Chapter Shamass, R., Alfano, G. and Guarracino, F. (2016). Investigation on the plastic buckling paradox for metal cylinders. In: B.G. Falzon et al., ed., Buckling and Postbuckling Structures II: Experimental, Analytical and Numerical Studies. Imperial College Press/World Scientific, Accepted. Rabee Shamass Date Abstract It is widely accepted that, for many buckling problems of plates and shells in the plastic range, the flow theory of plasticity either fails to predict buckling or significantly overestimates buckling stresses and strains, while the deformation theory, which fails to capture important aspects of the underlying physics of plastic deformation, provides results that are more in line with experimental findings and is therefore generally recommended for use in practical applications. This thesis aims to contribute further understanding of the reasons behind the seeming differences between the predictions provided by these two theories, and therefore provide some explanation of this so-called ‘plastic buckling paradox’. The study focuses on circular cylindrical shells subjected to either axial compression or non-proportional loading, the latter consisting of combined axial tensile stress and increasing external pressure. In these two cases, geometrically nonlinear finite-element (FE) analyses for perfect and imperfect cylinders are conducted using both the flow and the deformation theories of plasticity, and the numerical results are compared with data from widely cited physical tests and with analytical results. The plastic buckling pressures for cylinders subjected to non-proportional loading, with various combinations of boundary conditions, tensile stresses, material properties and cylinder’s geometries, are also obtained with the help of the differential quadrature method (DQM). These results are compared with those obtained using the code BOSOR5 and with nonlinear FE results obtained using both the flow and deformation theories of plasticity. It is found that, contrary to common belief, by using a geometrically nonlinear FE formulation with carefully determined and validated constitutive laws, very good agreement between numerical and test results can be obtained in the case of the physically more sound flow theory of plasticity. The reason for the ‘plastic buckling paradox’ appears to be the over-constrained kinematics assumed in many analytical and numerical treatments, such as BOSOR5 and NAPAS, whereby a harmonic buckling shape in the circumferential direction is prescribed. Acknowledgment I am very grateful to Dr. Giulio Alfano, my supervisor from Brunel University London, and to Dr. Federico Guarracino, my superviosor from University of Naples in Italy, for their continued encouragement and their helpful guidance in doing this research. Their patience, understanding, and immense knowledge helped me in all the time of research. Without their timely advices, my thesis work would not be finished. I owe Dr. Giulio Alfano a debt of gratitude for their careful reading of this thesis and the corrections and suggestions he has made. I would like to express my warm appreciation for Dr. Mustafa Batikha, my teacher from Damascus University in Syria, who is always encouraging me through the difficult times. Finally, truly unbounded thanks are due to Imane, my dear wife, who with her love, understanding and patience supported me in finishing this research. Table of Contents Chapter 1 ....................................................................................................... 1 Introduction .................................................................................................... 1 1.1 What is buckling? ................................................................................. 1 1.2 Plastic buckling and its importance in engineering applications ........... 2 1.3 Plastic-buckling paradox ...................................................................... 5 1.4 Aims and objectives ............................................................................. 7 1.5 Summary of methodology .................................................................... 7 1.6 Outline of the thesis ............................................................................. 8 Chapter 2 ..................................................................................................... 10 Literature Review ......................................................................................... 10 2.1 The equilibrium differential equations ................................................. 10 2.2 Nonlinear equilibrium paths ................................................................ 12 2.2.1 The adjacent equilibrium criterion ................................................ 14 2.2.2 The Trefftz criterion ..................................................................... 15 2.3 Elastic versus plastic buckling ............................................................ 16 2.3.1 Elastic Buckling ........................................................................... 17 2.3.1.1. Stable symmetric buckling ........................................................ 17 2.3.1.2. Unstable symmetric buckling .................................................... 18 2.3.1.3. Asymmetric buckling ................................................................ 19 2.3.2 Plastic Buckling ........................................................................... 20 2.4 Key milestones in the development of buckling theories .................... 21 2.4.1 Elastic buckling ............................................................................ 21 2.4.2 Post-buckling behaviour .............................................................. 23 2.4.3 Inelastic buckling ......................................................................... 24 2.5 Early development in Plasticity .......................................................... 27 2.6 Computational Plasticity ..................................................................... 30 2.7 Inelastic material behaviour –Elasto-plasticity based on the flow theory 31 2.7.1 Plastic stress-strain relations ....................................................... 31 2.7.2 Classical yield surfaces ............................................................... 33 2.7.3 The Prandtl-Reuss equations for plane stress assumption ......... 38 2.8 Inelastic material behaviour –Elasto-plasticity based on the deformation theory ....................................................................................... 39 2.8.1 The Hencky equations for plane stress assumption .................... 41 2.9 Plastic buckling paradox .................................................................... 44 2.10 A brief overview of the underlying methods of BOSOR5 ................ 52 Chapter 3 ..................................................................................................... 55 A Detailed Numerical Investigation into the Plastic Buckling Paradox for Circular Cylindrical Shells under Axial Compression ................................... 55 3.1. Introduction ........................................................................................ 55 3.2. Test samples and finite-element modelling ........................................ 56 3.2.1 Geometry and elements .............................................................. 56 3.2.2 Constitutive relationship and material constants .......................... 59 3.2.3 Large displacement formulation ................................................... 65 3.2.4 Solution strategy .......................................................................... 65 3.2.5 Imperfection sensitivity analysis .................................................
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