## Limit States of Materials and Structures: Direct MethodsDieter Weichert, Alan Ponter To predict loading limits for structures and structural elements is one of the oldest and most important tasks of engineers. Among the theoretical and numericalmethodsavailableforthispurpose,so-called“DirectMethods”,- bracing Limit- and Shakedown Analysis, play an eminent role due to the fact that they allow rapid access to the requested information in mathematically constructive manners. The collection of papers in this book is the outcome of a workshop held at Aachen University of Technology in November 2007. The individual c- tributions stem in particular from the areas of new numerical developments renderingthemethodsmoreattractive forindustrialdesign,extensionsofthe general methodology to new horizons of application, probabilistic approaches and concrete technological applications. The papers are arranged according to the order of the presentations in the workshop and give an excellent insight into state-of-the-art developments in this broad and growing ?eld of research. The editors warmly thank all the scientists, who have contributed by their outstanding papers to the quality of this edition. Special thanks go to Jaan Simon for his great help in putting together the manuscript to its ?nal shape. |

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### Contents

1 | |

A Decomposition Approach | 23 |

Gurson Model for Porous Pressure Sensitive Materials | 44 |

A Direct Method for the Determination of Effective Strength Domains for Periodic ElasticPlastic Media | 67 |

Stochastic Limit Load Analysis of ElastoPlastic Plane Frames | 87 |

Limit Load Analysis of Plane Frames Under Stochastic Uncertainty | 112 |

Reliability Analysis of Inelastic Shell Structures Under Variable Loads | 135 |

Static Shakedown Theorem for Solids with TemperatureDependent Elastic Modulus | 157 |

On Shakedown of Structures Under Variable Loads with a Kinematic Nonlinear and Nonassociated Hardening Rule | 179 |

Limit Analysis of Orthotropic Laminates by Linear Matching Method | 197 |

Mechanical Surface Treatments and Life Improvement | 221 |

Force Method Based Procedures in the Limit Equilibrium Analysis of Framed Structures | 233 |

Shakedown Analysis of Composite Materials Based on Nonlinear Mathematical Programming | 253 |

A Nonlinear Programming Approach to Shakedown Analysis for a General Yield Condition | 271 |

to LargeScale Problems with Selective Algorithm | 288 |

### Common terms and phrases

algorithm applied approach behaviour coeﬃcient composite materials composites Comput constraints convergence convex cost factor cycle basis cyclic loading deﬁned denotes diﬀerent displacement dissipated power e-mail eﬀective elastic equilibrium ﬁctitious ﬁnite element ﬁrst ﬁxed ﬂow formulation friction angle Gaussian integral homogeneous incremental inequation Inﬂuence Inglebert iteration kinematic shakedown analysis limit analysis limit and shakedown Linear Matching Method linear programming load factor load multiplier lower bound material mathematical programming matrix Mécanique Mech mesh Mises modulus non-linear programming numerical obtained optimization problem orthotropic materials penalty costs plastic strain plate Porous quadratic residual stress satisﬁed selective algorithm shakedown analysis shakedown limit shakedown load shakedown theorem shear shear stress shot peening solution Springer Science+Business Media static stochastic strain rate stress ﬁeld structure tensor Tsai-Wu type unit cell upper bound values variables vector velocity Weichert yield condition yield criterion yield stress yield surface