FG Baustatik, Stahlbau, FEM
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- Ratcheting (5)
- Dehnschwingbreite (4)
- Vereinfachte Fließzonentheorie (4)
- Zarka-Methode (3)
- cyclic loading (3)
- pipe bend (3)
- Ermüdung (2)
- Simplified Theory of Plastic Zones (2)
- Zarka's method (2)
- post-shakedown quantities (2)
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Die Vereinfachte Fließzonentheorie beruht auf der Zarka-Methode und gestattet die näherungsweise Ermittlung der elastisch-plastischen Verzerrungen, Spannungen und Verformungen bei monotoner oder zyklischer Belastung. Bei zyklischer Belastung wird sofort der Einspielzustand berechnet, ohne das Belastungshistogramm Zyklus für Zyklus inkrementell durchrechnen zu müssen. Als Berechnungsaufwand fallen lediglich einige modifizierte linear elastische Analysen sowie „lokale“ Berechnungen an, so daß gegenüber den herkömmlichen inkrementellen Berechnungen nach der exakten Fließzonentheorie ein erheblicher Gewinn an Rechenzeit möglich ist. Bei einigen Beispielrechnungen wurde nur etwa 1/10.000 der Rechenzeit benötigt, um sowohl die Dehnungsschwingbreite (zur Ermittlung der Ermüdungsausnutzung) als auch die akkumulierten Verzerrungen (für einen Ratcheting-Nachweis) in guter Näherung abschätzen zu können. Die Vereinfachte Fließzonentheorie wurde mittels einer user-subroutine und einigen Makros in ANSYS implementiert. Es werden die Grundlagen der Vereinfachten Fließzonentheorie dargestellt, ihre Implementierung in ANSYS und einige Beispielrechnungen.
Es wird die Vereinfachte Fließzonentheorie als Weiterentwicklung der Zarka-Methode vorgestellt sowie ihre Implementierung in das FE-Programm ANSYS. Anwendungsbeispiele belegen die gute Ergebnisqualität bei gleichzeitig geringem Berechnungsaufwand für die Ermittlung der plastischen Dehnschwingbreite und der akkumulierten plastischen Verzerrungen bei zyklischer Belastung.
Die Vereinfachte Fließzonentheorie gestattet bei zyklischer Belastung die näherungsweise Ermittlung der elastisch-plastischen Dehnungsschwingbreite und der durch einen Ratcheting-Mechanismus akkumulierten Verzerrungen sowie aller daraus ableitbaren Größen wie etwa Verformungen im elastischen und plastischen Einspielzustand. Sie beruht auf der Zarka-Methode. Im Gegensatz zu den in den technischen Regelwerken der Anlagentechnik zugelassenen vereinfachten Berechnungsmethoden (wie etwa die Anwendung des Faktors Ke) kann sie neben der Werkstoffverfestigung auch den Einfluß der individuellen Konfiguration von Bauteilgeometrie und Belastungsart auf das plastische Verhalten der Struktur erfassen. Sie ist gleichermaßen geeignet, globale Struktureffekte, lokale Kerbeffekte und Einflüsse aus der unterschiedlichen Querdehnungszahl im Elastischen und im Plastischen zu berücksichtigen. Als Berechnungsaufwand fallen lediglich einige modifizierte linear elastische Analysen sowie „lokale“ Berechnungen an. Eine Reihe von Beispielen zeigt, daß sowohl die Dehnungsschwingbreite als auch die akkumulierten Verzerrungen mit geringem Berechnungsaufwand in guter Näherung abgeschätzt werden können.
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.
Die Zarka-Methode zur vereinfachten Berechnung elastisch-plastisch beanspruchter Tragwerke wird vorgestellt. Ihre historische Entwicklung und ihr theoretisches Fundament werden dargelegt. An einigen Beispielen (Lochscheibe, dickwandiger Zylinder unter Innendruck, Griffith-crack, Bree-Modell) wird die Notwendigkeit ihrer Weiterentwicklung belegt, aber auch ihr Potential zur Abschätzung sog. post-shakedown quantities mittels weniger linear elastischer Analysen.
In case of cyclic loading, strain may accumulate due to a ratcheting mechanism until the state of shakedown is possibly achieved. Design Codes frequently require strain limits to be satisfied at the end of the specified lifetime of the structure. However, this requirement is sometimes tied to misleading prerequisites, and little guidance is provided on how the strains accumulated in the state of shakedown can be calculated. Incremental elastic-plastic analyses which require to go step-by-step through many cycles of a given load histogram are rather costly in terms of engineering time and numerical effort. As an alternative, the Simplified Theory of Plastic Zones (STPZ) is used in the present paper. Being a direct method, effects from load history are disregarded. The theory is described shortly and exemplarily applied to a simplification of a pipe bend and a straight pipe, both subjected to combinations of several loads which vary independently from each other so that a multidimensional load domain is represented. It is shown that the Simplified Theory of Plastic Zones is well suited to provide reasonable estimates of strains accumulated in the state of elastic shakedown at the cost of few linear elastic analyses.
As elastic-plastic fatigue analyses are still time consuming the simplified elastic-plastic analysis (e.g. ASME Section III, NB 3228.5, the French RCC-M code, paragraphs B 3234.3, B
3234.5 and B3234.6 and the German KTA rule 3201.2, paragraph 7.8.4) is often applied. Besides linearly elastic analyses
and factorial plasticity correction (Ke-factors) direct methods are an option. In fact, calculation effort and accuracy of results are growing in the following graded scheme: a) linearly elastic analysis along with Ke correction, b) direct methods for the determination of stabilized elastic-plastic strain ranges and c) incremental elastic-plastic methods for the determination of stabilized elastic-plastic strain ranges. The paper concentrates on option b) by substantiating the practical applicability of the simplified theory of plastic zones STPZ (based on Zarka’s method). Application relevant aspects are particularly addressed. Furthermore, the applicability of the STPZ for arbitrary load time histories in connection with an appropriate cycle counting method is discussed. Note, that the STPZ is applicable both for the determination
of (fatigue relevant) elastic-plastic strain ranges and (ratcheting relevant) locally accumulated strains. This paper concentrates on the performance of the method in terms of the determination of elastic-plastic strain ranges and fatigue usage factors. The additional performance in terms of locally accumulated strains and ratcheting will be discussed in a future publication.
As elastic-plastic fatigue analyses are still time consuming the simplified elastic-plastic analysis (e.g. ASME Section III, NB 3228.5, the French RCC-M code, paragraphs B 3234.3, B 3234.5 and B3234.6 and the German KTA rule 3201.2, paragraph 7.8.4) is often applied. Besides linearly elastic analyses and factorial plasticity correction (Ke-factors) direct methods are an option. In fact, calculation effort and accuracy of results are growing in the following graded scheme: a) linearly elastic analysis along with Ke correction, b) direct methods for the determination of stabilized elastic-plastic strain ranges and c) incremental elastic-plastic methods for the determination of stabilized elastic-plastic strain ranges.
The paper concentrates on option b) by substantiating the practical applicability of the simplified theory of plastic zones STPZ (based on Zarka’s method) and – for comparison – the established Twice Yield method. Application relevant aspects are particularly addressed. Furthermore, the applicability of the STPZ for arbitrary load time histories in connection with an appropriate cycle counting method is discussed.
Note, that the STPZ is applicable both for the determination of (fatigue relevant) elastic-plastic strain ranges and (ratcheting relevant) locally accumulated strains. This paper concentrates on the performance of the method in terms of the determination of elastic-plastic strain ranges and fatigue usage factors. The additional performance in terms of locally accumulated strains and ratcheting will be discussed in a future publication.
Life assessment of a structure subject to cyclic loading rests on quantifying strain accumulated prior to shakedown and the strain range experienced after plastic shakedown has been achieved. Few methods exist to predict these quantities. Zarka's method is one of these methods. It is evaluated by analyzing several examples of structures and comparing the quality of the results obtained and the numerical effort required with evolutive analyses by using a commercial Finite Element program.