FG Baustatik, Stahlbau, FEM
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- Ratcheting (15)
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Im Zuge einer Ermüdungsanalyse plastisch beanspruchter Komponenten von Kernkraftwerken werden Dehnungserhöhungsfaktoren Ke zur Ermittlung der plastischen Dehnschwingbreite aus elastisch berechneten Beanspruchungen verwendet. Ausgehend von einer Kritik an der derzeit üblichen Vorgehensweise nach ASME Code wird eine Modifikation vorgeschlagen.
Der Vorschlag basiert wie der Ke-Faktor des ASME Code (der auch von diversen KTA-Regeln übernommen worden ist) auf dem Konzept eines einfachen Faktors. Er beseitigt die potentielle Unkonservativität des ASME Code bei Kerben (Rundungsradien) und reduziert gleichzeitig dessen Überkonservativität in vielen anderen Anwendungsbereichen, indem drei unterschiedliche Effekte berücksichtigt und individuelles Werkstoffverhalten erfasst werden können. Zudem beruht er im Gegensatz zu Ke-ASME auf möglichst realitätsnahen Werkstoffdaten, die eher den Charakter von Mittel- als von Mindestwerten besitzen. Ferner ist die Anwendung dieses Vorschlages insofern einfacher als die des ASME Code, als die Suche nach der ungünstigsten Orientierung eines Schnittes durch die Wand, die i. allg. nicht einfach zu identifizieren ist, entfällt.
Eine Reihe detaillierter elastisch-plastischer Vergleichsrechnungen für hinsichtlich Bauteilgeometrie und Belastung typische Problemstellungen bei schnellen und Leichtwasserreaktoren bestätigen die Konservativität des Vorschlages.
Es wird ein Potential zur weiteren Reduzierung der Konservativität bei Anwendung auf Bauteile von Leichtwasserreaktoren im Rahmen eventueller zukünftiger Entwicklungsarbeiten aufgezeigt.
Es werden Ke,r-Faktoren als Dehnungserhöhungsfaktoren für vereinfachte elastisch-plastische Ermüdungsanalysen vorgeschlagen, die realistischer sind als in Regelwerken vorhandene Ke-Faktoren. Letztere dienen einer meistens konservativen Erfassung der auftretenden elasto-plastischen Dehnungen, sie sind aber weder physikalisch nachvollziehbar, noch werkstoffgerecht formuliert. Die bisher exemplarisch durchgeführten Verifikationsrechnungen belegen, dass die vorgeschlagenen realistischen Ke,r-Faktoren leicht bestimmbar sind und zudem ausreichend konservative Ergebnisse liefern. Somit liegt mit dieser neuen Methode ein beachtenswertes Potential vor, das jedoch noch durch weitere Verifikationsrechnungen abgesichert werden muss, bevor es in das KTA-Regelwerk eingebunden werden kann.
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.
This paper is intended to provide an overview of different aspects of ratcheting under cyclic loading below the creep range. It distinguishes between material ratcheting and structural ratcheting, each being characterized by several different phenomena which appear in different configurations of materials, states of stress, structural geometries and loadings. The systematic compilation of these phenomena presented in the paper may help to improve understanding between material researches, developers of inelastic methods of analysis, structural analysts and design code committees. Above all, a certain degree of knowledge about the different mechanisms of ratchetting is important for a structural analyst to be able to choose an appropriate analytical method for assessing the ratcheting phenomena involved in a specific design problem.
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).
According to the relevant KTA-Rules, e.g. KTA 3201.2, strain correction factors — Ke-factors — have to be used in the fatigue analysis of pressurised components if the strain intensity ranges are determined by elastic analyses, and if in this case the range of primary plus secondary stress intensity exceeds a certain limit. This limit is three times the design stress intensity value, Sm, and thus approximately corresponds to twice the value of the 0.2% strain limit. The relations given in the above-mentioned rules to determine the Ke-factors for considering plastification have proved to be very conservative in many cases compared with the strain intensity ranges that were determined by complete elastic–plastic analyses. In order to improve the validity of the fatigue analysis, the topic of `Performance of fundamental work to prepare concrete proposals for realistic Ke,r-factors (strain correction factors) to consider plastification at large strain amplitudes' was one of the subjects of the BMU project SR 2063. In summary, the result was that the proposed realistic Ke,r-factors present a real alternative to the Ke-factors of the regulations; the latter serve a mostly conservative registration of the observed elastic–plastic strain but cannot be explained in terms of physics and are not formulated in a manner adequately specific of any material. The exemplary verification calculations that have been performed so far show, furthermore, that the proposed realistic Ke,r-factors can be easily determined and also deliver sufficiently conservative results. This new method therefore has great potential which, however, still has to continue to be verified by further calculations before it can be included in the KTA-Rules.
Es werden drei vereinfachte Methoden zur Berechnung elastisch-plastischer Dehnschwingbreiten infolge zyklischer Belastung von Komponenten der Kerntechnik vorgestellt. Es handelt sich um die Methoden von Roche, Seshadri und Zarka. Diese werden mit dem Faktor Ke des KTA-Regelwerks sowie dem im Vorhaben SR 2063 unterbreiteten Vorschlag zu seiner Modifikation verglichen. Die ihnen zugrunde liegenden Annahmen werden kritisch durchleuchtet. Ihre Anforderungen an das Werkstoffgesetz im plastischen Bereich sowie der erforderliche Berechnungsaufwand werden herausgestellt. Einfache Beispielrechnungen zeigen, durch Gegenüberstellung mit den exakten Werten, die erzielbare Genauigkeit.
Es wird eine vereinfachte Fließzonentheorie vorgestellt, mit der das plastische Verhalten eines Tragwerks berechnet werden kann. Sie lässt sich nicht nur auf Stabwerke, sondern auch auf Flächentragwerke unter beliebiger Belastung anwenden. Das zugrunde gelegte Werkstoffgesetz ist bilinear, wodurch Verfestigung erfasst werden kann. Die Theorie beruht auf dem Konzept transformierter interner Variabler nach Zarka, mit dem das plastische Problem in ein geeignet formuliertes elastisches Problem überführt wird. Damit fällt oft nur eine weitere elastizitätstheoretische Berechnung an mit modifizierten elastischen Werkstoffparametern und mit (in Form von Anfangsdehnungen) modifizierter Belastung. Das Ergebnis kann gegebenenfalls iterativ verbessert werden, bis das "exakte" Ergebnis erreicht ist. Mehrere Beispiele erläutern die Methode.
If a mechanical structure is to be designed for operation under cyclic loading, primarily two kinds of failure must be guarded against: (1) low cycle fatigue which may occur due to strains cycling between two states (controlled by the strain range exceeding twice the yield limit);
(2) ductility exhaustion which may occur due to accumulating strain from one load cycle to another.
These two kinds of failure are local failure modes so that strains need to be calculated and then assessed by comparison with code allowables such as the 1%, 2% and 5% strain limits set by the ASME nuclear codes. Elastic-plastic strains can be calculated by incremental (or step-by-step or evolutive) analyses. Unfortunately, this can be extremely costly if thousands of cycles are required to achieve shakedown. Therefore, simplified elastic-plastic analysis methods are desired allowing to obtain specific information at reduced effort, nevertheless accounting for the main features controlling strain such as kinematic hardening. Zarka’s method, early versions of which are available since twenty years, appears promising to provide both strain ranges and accumulated strains in the saturated cycle, i.e. after shakedown has been achieved. However, several attempts to use this method in the nuclear industry failed to qualify the method as a reliable analysis tool. This was due to several reasons:
(1) the publications describing the method were written in a highly scientific language the design engineers in industry were not familiar with;
(2) in some cases Zarka’s method provided excellent results (compared with incremental analyses), but bad ones in others.
Nevertheless, there remained some interest to uncover the potential of this method. For that purpose some calculations are performed for simple configurations of structure and loading (so that the structural response can be interpreted relatively easily). More insight into the performance of the method may thus be gained in terms of computational steps to be followed, the numerical effort required, the quality of the results obtained, and the sensibility with respect to material data and load level.
The basic idea of Zarka's method is to redefine the elastic-plastic problem by an equivalent elastic problem with suitably defined modified elastic material parameters and initial strains. This requires estimating (and iteratively improving) the geometry of the plastic zone and of transformed internal variables. A particular class of material models is admitted, the simplest of which is the linear kinematic hardening model.