Chapter 3 Models and Finite Elements for Thin-walled Structures M. Bischoff 1 , W. A. Wall 2 , K.-U. Bletzinger 1 and E. Ramm 3 1 Lehrstuhl f¨ ur Statik, TU M¨ unchen, M¨ unchen, Germany 2 Lehrstuhl f¨ ur Numerische Mechanik, TU M¨ unchen, Garching, Germany 3 Institut f¨ ur Baustatik, Universit¨ at Stuttgart, Stuttgart, Germany 1 Introduction 1 2 Mathematical and Mechanical Foundations 5 3 Plates and Shells 10 4 Dimensional Reduction and Structural Models 12 5 Finite Element Formulation 50 6 Concluding Remarks 70 7 Related Chapters 71 Acknowledgments 71 References 71 Further Reading 75 Appendix 76 1 INTRODUCTION 1.1 Historical remarks Thin-walled structures like plates and shells are the most common construction elements in nature and technology. This is independent of the specific scale; they might be small like cell membranes and tiny machine parts or very large like fuselages and cooling towers. This preference to Encyclopedia of Computational Mechanics, Edited by Erwin Stein, Ren´ e de Borst and Thomas J.R. Hughes. Volume 2: Solids, Structures and Coupled Problems.  2004 John Wiley & Sons, Ltd. ISBN: 0-470-84699-2. apply walls as thin as possible is a natural optimization strategy to reduce dead load and to minimize construction material. In addition to the slenderness, the advantageous effect of curvature is utilized in shell structures allowing to carry transverse loading in an optimal way, a design prin- ciple already known to the ancient master builders. Their considerable heuristic knowledge allowed them to create remarkable buildings, like the Roman Pantheon (115–126) and the Haghia Sophia (532–537) in Constantinople, still existing today. It was not before the Renaissance that scientists began to mathematically idealize the structural response, a process that we denote nowadays as modeling and simulation. Already, Leonardo da Vinci (1452–1519) stated (Codex Madrid I) a principle that later on emerged to a beam model. The subsequent process, associated with names like Galileo (1564–1642), Mariotte (1620–1684), Leibniz (1646–1716), Jakob I Bernoulli (1654–1705), Euler (1707–1783), Coulomb (1736–1806), and Navier (1785–1836), led to what we call today Euler–Bernoulli beam theory (Timoshenko, 1953; Szabo, 1979). This development was driven by the ingenious ideas to condense the complex three-dimensional situation to the essential ingredients of structural response like stretching, bending, torsion, and so on, and to cast this into a manageable mathematical format. The inclusion of transverse shear deformation is attributed (1859) to Bresse (1822–1883) and extended (1921) to dynamics by Timoshenko (1878–1972), whose name has established itself as a common denomination for this model. Extensions to further effects like uniform and warping torsion, stability problems,