FG Baustatik, Stahlbau, FEM
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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.
The Simplified Theory of Plastic Zones (STPZ) is a direct method based on Zarka's method, primarily developed to estimate post-shakedown quantities of structures under cyclic loading, avoiding incremental analyses through a load histogram. In a different paper the STPZ has previously been shown to provide excellent estimates of the elastic–plastic strain ranges in the state of plastic shakedown as required for fatigue analyses. In the present paper, it is described how the STPZ can be used to predict the strains accumulated through a number of loading cycles due to a ratcheting mechanism, until either elastic or plastic shakedown is achieved, so that strain limits can be satisfied. Thus, a consistent means of estimating both, strain ranges and accumulated strains is provided for structural integrity assessment as required by pressure vessel codes. The computational costs involved typically consist of few linear elastic analyses and some local calculations. Multilinear kinematic hardening and temperature dependent yield stresses are accounted for. The quality of the results and the computational burden involved are demonstrated through four examples.
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. In addition, the strain range is required for performing fatigue analyses in case of plastic shakedown. However, little guidance is usually provided by Design Codes on how the accumulated strains and strain ranges are to be calculated, and some of the guidelines implemented in Design Codes are not well founded and may therefore be misleading. This is, for example, true for the ASME B&PV Code, Section III. Of course, strains and strain ranges can be determined by means of incremental elastic-plastic analyses, which require to go step-by-step through many cycles of a given load histogram until the state of shakedown is reached. This is rather costly in terms of engineering time and numerical effort. As an alternative, simplified methods can be adopted, e.g. the Simplified Theory of Plastic Zones (STPZ) as used in the present paper. Being a direct method, effects from load history are disregarded. The theory is described shortly and illustrated by some examples. 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 and plastic shakedown at the cost of few linear elastic analyses.
Progressive deformation (ratcheting) can occur as a response to variable loads as soon as the elastic limit is exceeded. If this is the case, strains and displacements accumulate in the event of cyclic loading in each load cycle. Widely known as triggers for ratcheting and already being considered in some design codes are configurations, in which a structure is subjected to at least two different types of load, namely a constant load (the primary load) and a superimposed cyclic load. In this paper, another mechanism that generates ratcheting is introduced. It can be attributed solely to the effect of a single load. In the simplest case, this can be explained by the successive activation of (an infinite number of) plastic hinges if a load of constant magnitude is moved in space. The increments of strains and displacements can decrease or increase from cycle to cycle, when the material is hardening, or if elastic foundation is present, or if the equilibrium condition is formulated for the deformed system (second-order theory) or if “large” rotations are taken into account (third-order theory).
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.