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## Understanding Non-linear Finite Element Analysis Through Illustrative Benchmarks

The use of Finite Element software has become very widespread in engineering and its range of applicable has greatly increased. Indeed, it is nowadays difficult to find many engineering applications for which the Finite Element method has not been used or at least attempted. With this rapid increase in the use of the Finite Element  method, it’s increasingly automated procedures, and its user-friendly graphical interfaces, the expectations of FE users have risen, sometimes unrealistically. Inexperienced Finite Element users may be under the impression that they can easily tackle non-linear problems without having to gain a good understanding of the background theory.

NAFEMS has published many benchmarks on non-linear problems, usually devised to address a particular type of non-linearity, such as plasticity, geometric non-linearity, etc. Non-linear FE theory has been included in many FE textbooks.

This book is written to fill the gap between the benchmark reports and the textbooks of Finite Element theory, by presenting a carefully selected set of non-linear benchmarks and providing a brief coverage of the background non-linear theory covered in the benchmarks.

### Contents

 CHAPTER 1 - INTRODUCTION 1 1.1 SCOPE AND LAYOUT OF THE BOOK 1 1.2 CLASSIFICATION OF NON-LINEARITIES 2 1.3 NON-LINEAR FE PROCEDURES 7 CHAPTER 2 - PLASTICITY (TIME-INDEPENDENT MATERIAL NON-LINEARITY) 13 2.1 INTRODUCTION 13 2.2 STRESS-STRAIN RELATIONSHIPS 14 2.3 REVIEW OF ELASTO-PLASTICITY THEORY 21 2.4 FUNDAMENTAL 2D PLASTICITY BENCHMARK 31 2.5 3D PLASTICITY BENCHMARK 39 2.6 KINEMATIC HARDENING BENCHMARK 45 2.7 PRESSURISED CYLINDER PLASTICITY BENCHMARK 51 2.8 CYCLIC PLASTICITY BENCHMARK 57 2.9 RIGID PUNCH PLASTICITY BENCHMARK 66 CHAPTER 3 - CREEP (TIME-DEPENDENT MATERIAL NON-LINEARITY) 71 3.1 INTRODUCTION 71 3.2 REVIEW OF CREEP THEORY 72 3.3 CONSTANT-LOAD CREEP BENCHMARKS 83 3.4 STRESS RELAXATION CREEP BENCHMARKS 87 3.5 VARIABLE-LOAD CREEP BENCHMARKS 92 3.6 PRESSURISED CYLINDER CREEP BENCHMARK 96 3.7 TORSIONAL CREEP BENCHMARK 98 3.8 THERMALLY-INDUCED CREEP BENCHMARK 102 CHAPTER 4 - GEOMETRIC NON-LINEARITY 105 4.1 INTRODUCTION 105 4.2 REVIEW OF GEOMETRIC NON-LINEARITY THEORY 106 4.3 SNAP-THROUGH GNL BENCHMARK 116 4.4 BIFURCATION BENCHMARK 124 4.5 SNAP-BACK BENCHMARK 134 4.6 RIGID-BODY ROTATION BENCHMARK 138 4.7 STRAIGHT CANTILEVER BENCHMARK 140 4.8 CURVED CANTILEVER BENCHMARK 146 4.9 Z-SHAPED 3D CANTILEVER BENCHMARK 149 4.10 TORSIONAL BUCKLING 3D BENCHMARK 151 4.11 PIPE COLLAPSE BENCHMARK 154 4.12 STIFFENED CYLINDRICAL PANEL BENCHMARK 158 CHAPTER 5 - PRACTICAL GUIDELINES 161 5.1 QUESTIONS TO BE ADDRESSED BY THE USER 161 5.2 DIFFICULTIES ASSOCIATED WITH NON-LINEAR PROBLEMS 162 5.3 NON-LINEAR BENCHMARKS 163 REFERENCES 165 GLOSSARY 167

### Document Details

Reference R0080 Becker. A English Analyst Student Publication 1st January 2001 Global