LIMIT AND SHAKEDOWN ANALYSIS OF PLATES AND SHELLS INCLUDING UNCERTAINTIES Von der Fakultät für Maschinenbau der Technischen Universität Chemnitz genehmigte Dissertation zur Erlangung des akademischen Grades Doktoringenieur (Dr.-Ing.) vorgelegt von MSc. Thanh Ngọc Trần geboren am 03. Februar 1975 in Nam Dinh, Vietnam eingereicht am 12. Dezember 2007 Gutachter: Prof. Dr.-Ing. Reiner Kreißig Prof. Dr.-Ing. Manfred Staat Prof. Dr.-Ing. Christos Bisbos Tag der Verteidigung: 12. März 2008 Trần, Thanh Ngọc Limit and shakedown analysis of plates and shells including uncertainties Dissertation an der Fakultät für Maschinenbau der Technischen Universität Chemnitz, Institut für Mechanik und Thermodynamik, Chemnitz 2008 149 + vii Seiten 55 Abbildungen 28 Tabellen 162 Literaturzitate Referat The reliability analysis of plates and shells with respect to plastic collapse or to inadaptation is formulated on the basis of limit and shakedown theorems. The loading, the material strength and the shell thickness are considered as random variables. Based on a direct definition of the limit state function, the nonlinear problems may be efficiently solved by using the First and Second Order Reliability Methods (FORM/SORM). The sensitivity analyses in FORM/SORM can be based on the sensitivities of the deterministic shakedown problem. The problem of the reliability of structural systems is also handled by the application of a special barrier technique which permits to find all the design points corresponding to all the failure modes. The direct plasticity approach reduces considerably the necessary knowledge of uncertain input data, computing costs and the numerical error. Die Zuverlässigkeitsanalyse von Platten und Schalen in Bezug auf plastischen Kollaps oder Nicht-Anpassung wird mit den Traglast- und Einspielsätzen formuliert. Die Lasten, die Werkstofffestigkeit und die Schalendicke werden als Zufallsvariablen betrachtet. Auf der Grundlage einer direkten Definition der Grenzzustandsfunktion kann die Berechnung der Versagenswahrscheinlichkeit effektiv mit den Zuverlässigkeitsmethoden erster und zweiter Ordnung (FROM/SORM) gelöst werden. Die Sensitivitätsanalysen in FORM/SORM lassen sich auf der Basis der Sensitivitäten des deterministischen Einspielproblems berechnen. Die Schwierigkeiten bei der Ermittlung der Zuverlässigkeit von strukturellen Systemen werden durch Anwendung einer speziellen Barrieremethode behoben, die es erlaubt, alle Auslegungspunkte zu allen Versagensmoden zu finden. Die Anwendung direkter Plastizitätsmethoden führt zu einer beträchtlichen Verringerung der notwendigen Kenntnis der unsicheren Eingangsdaten, des Berechnungsaufwandes und der numerischen Fehler. Schlagworte: Limit analysis, shakedown analysis, exact Ilyushin yield surface, nonlinear programming, first order reliability method, second order reliability method, design point Archivierungsort: http://archiv.tu-chemnitz.de/pub/2008/0025 ACKNOWLEDGEMENTS This work has been carried out at the Biomechanics Laboratory, Aachen University of Applied Sciences, Campus Jülich. The author gratefully acknowledges the Deutscher Akademischer Austausch Dienst (DAAD) for a research fellowship award under the grant reference A/04/20207. The author is indebted to Prof. Dr.-Ing. M. Staat who has been the constant source of caring and inspiration for his helpful guidance and encouragement. His commitment and assistance were limitless and this is greatly appreciated. The author would like to express his deep gratitude to Prof. Dr.-Ing. R. Kreißig for giving him the permission to complete Doctorate of Engineering at the Chemnitz University of Technology and for kindly assistance and supervision. The author would like to thank Prof. Dr.-Ing. C. Bisbos, Aristotle University of Thessalonoki, Greece for having kindly accepted to review this thesis. The author is thankful to Dr.-Ing. Vũ Đức Khôi for help and advice, to Ms Wierskowski and Ms Dronia for their programming as part of their diploma theses in some parts of FEM source code. The author’s thanks are also extended to Prof. Dr. rer. nat. Dr.-Ing. S. Sponagel and to the other colleagues at the Biomechanics Laboratory for their helpful assistance. The author is immensely indebted to his father Trần Thanh Xuân and his mother Nguyễn Thị Hòa who have been the source of love and discipline for their inspiration and encouragement throughout the course of his education including this Doctorate. Last but not least, the author is extremely grateful to his wife Mrs. Nguyễn Thị Thu Hà who has been the source of love, companionship and encouragement, to his daughters My and Ly who have been the source of joy and love. iii iv TABLE OF CONTENTS INTRODUCTION ........................................................................................................ 1 1. FUNDAMENTALS.................................................................................................. 3 1.1 Basic concepts of plasticity................................................................................. 3 1.1.1 Elastic and rigid perfectly plastic materials................................................. 3 1.1.2 Fundamental principles in plasticity............................................................ 4 1.1.3 Drucker’s postulate...................................................................................... 6 1.1.4 Yield criteria ................................................................................................ 7 1.1.5 Plastic dissipation function in local variables.............................................. 8 1.2 Normalized shell quantities ................................................................................ 9 1.2.1 Reference quantities..................................................................................... 9 1.2.2 Stress quantities ........................................................................................... 9 1.2.3 Strain quantities ......................................................................................... 10 1.2.4 Stress-Strain relation.................................................................................. 11 1.3 Exact Ilyushin yield surface.............................................................................. 12 1.3.1 Derivation .................................................................................................. 12 1.3.2 Description of the exact Ilyushin yield surface ......................................... 14 1.3.3 Reparameterization .................................................................................... 16 1.3.4 Plastic dissipation function ........................................................................ 18 1.3.5 Reformulation ............................................................................................ 19 2. MATHEMATICAL FORMULATIONS OF LIMIT AND SHAKEDOWN ANALYSIS IN GENERALIZED VARIABLES ....................................................... 21 2.1 Theory of limit analysis .................................................................................... 22 2.1.1 Introduction................................................................................................ 22 2.1.2 General theorems of limit analysis ............................................................ 23 2.2 Theory of shakedown analysis.......................................................................... 24 2.2.1 Introduction................................................................................................ 24 2.2.2 Definition of load domain.......................................................................... 25 2.2.3 Fundamental of shakedown theorems........................................................ 27 2.2.4 Separated shakedown limit ........................................................................ 30 2.2.5 Unified shakedown limit............................................................................ 33 v Table of Contents 3. DETERMINISTIC LIMIT AND SHAKEDOWN PROGRAMMING.................. 38 3.1 Finite element discretization............................................................................. 39 3.2 Kinematic algorithm ......................................................................................... 41 4. PROBABILISTIC LIMIT AND SHAKEDOWN PROGRAMMING................... 49 4.1 Basic concepts of probability theory ................................................................ 50 4.1.1 Sample space.............................................................................................. 50 4.1.2 Random variables ...................................................................................... 50 4.1.3 Moments .................................................................................................... 51 4.2 Reliability analysis............................................................................................ 53 4.2.1 Failure function and probability .............................................................. 53 4.2.2 First- and Second-Order Reliability Method ............................................. 55 4.3 Calculation of design point............................................................................... 58 4.4 Sensitivity of the limit state function................................................................ 61 4.4.1 Mathematical sensitivity............................................................................ 62 4.4.2 Definition of the limit state function.......................................................... 63 4.4.3