FG Baustatik, Stahlbau, FEM
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For a life prediction of structures subjected to variable loads, frequently encountered in mechanical and civil engineering, the cyclically accumulated deformation and the elastic-plastic strain ranges are required. The Simplified Theory of Plastic Zones (STPZ) is a direct method which provides the estimates of these and all other mechanical quantities in the state of elastic and plastic shakedown. The STPZ is described in detail, with emphasis to the fact that not only scientists but engineers working in practice and advanced students are able to get an idea of the possibilities and limitations of the STPZ. Numerous illustrations and examples are provided to support your understanding.
Für eine Lebensdauervorhersage veränderlich belasteter Tragwerke, etwa des Anlagen- und Maschinenbaus sowie des Bauingenieurwesens, werden die zyklisch akkumulierten Verzerrungen und ggf. auch die elastisch-plastischen Dehnschwingbreiten benötigt. Die Vereinfachte Fließzonentheorie (VFZT) ist eine direkte Methode, die Abschätzungen dieser und aller anderen mechanischen Größen im elastischen und im plastischen Einspielzustand liefert. Das vorliegende Buch stellt die VFZT ausführlich dar und legt Wert darauf, dass sich nicht nur Wissenschaftler, sondern auch in der Praxis tätige Ingenieure sowie Studierende höherer Semester ein Bild von den Möglichkeiten und Grenzen machen können. Zahlreiche Abbildungen und Anwendungsbeispiele unterstützen das Verständnis.
The European Fast Reactor (EFR) collaboration with the EFR Associates Design and Construction Rules Committee, and the R&D Agreement (AGT9B) produced significant developments in design-by-analysis procedures for high temperature plant. Many of these developments are judged to be relevant to the non-creep conditions of LWR plant, and for this reason, the CEC DGXI Working Group Codes and Standards supported this study to review and make recommendations on their potential application for improving LWR design code procedures. The topics considered are judged to be those where the most significant and relevant developments have been made and the list, although not exhaustive, is as follows:
- Negligible creep criteria
- Design-by-analysis procedures for weldments
- Shakedown design rules
- Design-by-analysis methods for tubeplates
- Buckling rules
- Interaction diagrams for assessing ratcheting
- Rules for the prevention of elastic follow-up in piping
- Strain range enhancement
- Constitutive equations for inelastic analysis
- Margins on Level D criteria
- Zarka's method
Design codes require to demonstrate that certain limits of inelastic deformation are not exceeded during the service life of a structure. If the loading is cyclic, inelastic strains may accumulate cycle by cycle and may exceed specified allowables after a number of cycles before a stationary state (elastic or plastic shakedown) is achieved, or the inelastic strains may grow unboundedly due to a ratchetting mechanism.
In principle, inelastic deformations can be calculated by performing evolutive (step-by-step) inelastic analyses. These require specific information, which is, however, not always available, such as detailed constitutive modelling and loading history. Furthermore, evolutive inelastic analyses are very costly. Therefore, simplified inelastic analyses are desirable to provide at least partial information about structural behaviour: more specifically, upper bounds on, or estimates of, elastic-plastic-creep deformations.
Some simplified methods are envisaged by design codes (as pointed out in Chapter 2). However, they are based on specific configurations of geometry and loading or they adopt heuristic assumptions, the reasonability of which is not always evident for general applicability within the scope of these codes. Accordingly, design codes seem to require improvements.
Several simplified methods are reviewed in the present Report and might serve as alternatives to those suggested by design codes. Simplified methods can be grouped in two classes: (a ) procedures intended to determine a safety factor against a critical event of the global structure (such as collapse); (b ) techniques apt to provide information on local quantities (such as inelastic strain) associated to inelastic structural responses. Procedures of class (a ) and some of their recent extensions are briefly surveyed in Chapter 3. Subclasses of category (b ) are discussed in the subsequent Chapters, with emphasis on their operative peculiarities and on their practical usefulness or potentialities.
Upper bounds in plasticity (Chapter 4 and Appendix A) can be computed by various approaches, basically by satisfying a set of equations and inequalities and by carrying out some optimisation procedure. To within the consequences of modelling errors, residual post shakedown quantities are guaranteed to be bounded from above: this circumstance is referred to by the adjective "rigorous". A number of bounding inequalities can be proved. Usually, the better (lower) the bound, the more expensive is its computation. However, computational advantages over other simplified approaches can hardly be ascertained in general.
Upper bounds in creep (Chapter 5) are based on general rigorous mechanical foundations, but the applications available mostly concern particular cases employing "ad hoc" imaginative, sometimes heuristic assumptions which are not easy to transfer to other cases. Mostly, elastic-perfectly plastic material behaviour and the Bailey-Orowan creep model are assumed.
Some simplified methods, such as the British shakedown method (Chapter 6), intend to estimate residual stress fields after elastic shakedown (the British method also, in certain circumstances, after plastic shakedown). They adopt empirically corroborated conjectures, rather than rigorous arguments. Material hardening is neglected.
Zarka's method (Chapter 7 and 8 and Appendix B) provides estimates of the mean strain in case of elastic shakedown and, in addition, a lower and an upper estimate of strain range in case of plastic shakedown, by adopting some heuristic assumptions. The validity of these assumptions is difficult to assess in practical applications. Material hardening is required.
The method developed by Ladevèze and coworkers (Chapter 9 and Appendix C) is not, strictly speaking, a simplified method, in the sense that it provides the same kind of information as rigorous evolutive analyses. The simplification lies in the solution process, which can be stopped after a few iterations since each one of these concern the whole
time interval of interest. Thus estimates are achieved of the structural inelastic response over a large time interval (much larger than the time step in an evolutive analysis). General material models are admitted.
None of the simplified methods reviewed in the present Report can directly be recommended for general practical use in nuclear design situations. However, some of them seem to have at least no less potentialities than the methods mentioned so far by design codes. Further work is necessary to clarify the conditions under which they are advantageous.
Engineering structures exposed to high temperature environment exhibit time dependent behaviour due to time dependent material behaviour. In order to avoid full inelastic analyses, simplified inelastic analysis methods are desirable.
The concept of elastic follow-up was introduced to allow determination of quantities serving as measure for life assessment of structures subjected to creep conditions without performing full time dependent structural analyses. Various simplified methods are described in the literature to estimate enhancement of creep strain and of creep damage arising from the fact that so-called secondary stresses caused by displacement-controlled loading do not necessarily relax with time in the same way as stresses do that are caused by strain-controlled loading, but rather show some characteristics of stress-controlled loading.
After providing a definition of elastic follow-up, the fundamental principles of elastic follow-up are compiled. A parameter "q" is introduced as a measure of elastic follow-up and is derived for some examples. Effects of non-uniform temperature distribution are discussed as well as effects resulting from multiaxial stress states. A number of methods to quantify elastic follow-up by simplified methods adopted by design codes or proposed in the literature are reviewed for general structures (Part I) and for piping (Part II).
Engineering structures subjected to high cyclic straining necessitating fatigue analysis play an important role in many industries. If the proportionality limit of the material is exceeded, nonlinearity of the material behaviour is to be taken into account. The structural response can then be calculated using either rigorous inelastic analysis methods, where stresses and strains are calculated on a step-by-step basis throughout a given load history, or by employing simplified methods of analysis. In the latter case a plastic strain range enhancement factor Ke is often used to obtain an elastic-plastic strain range based on fictitious elastic stress analyses.
Plastic behaviour of simple academic and practical structures is investigated to identify the basic features which determine the factor Ke:
(a) geometry of the structure
(b) kind of loading
(c) load level
(d) material behaviour.
The factor Ke is quantified for different geometries and kinds of loading. Parameter studies are performed to quantify the effects of load level, material models and hardening characteristics.
The background of some factors Ke established in Nuclear Design Codes or proposed in the literature is reviewed.
The behaviour of plastic structures can be attributed to global, localised and multiaxiality effects. Global structural effects comprise uniform reduction and redistribution of section forces and moments due to plasticity, including redistribution of stress across a section. Localised effects are concentrated to a very small volume of the material not affecting section forces and moments (e.g. notch effects). Multiaxiality effects arise from different Poisson's ratio associated with elastic and plastic behaviour.
Emphasis is laid on global structural effects rather than localised effects due to notches and fillets. Therefore structures exhibiting geometric stiffness discontinuities are considered to be "smooth" in the sense that the specific geometry of the transition between parts of different stiffnesses is disregarded.