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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).
The Simplified Theory of Plastic Zones (STPZ) may be used to determine post-shakedown quantities such as strain ranges and accumulated strains. The principles of the method are summarized succinctly and the practical applicability is shown by the example of a pipe bend subjected to internal pressure and cyclic in-plane bending.
Cyclic loading may cause elastic-plastic strains to accumulate if a ratcheting mechanism is present. After a number of cycles, strain accumulation may cease so that a state of either elastic or plastic shakedown is reached. Determination of strains and other quantities in the state of shakedown (post-shakedown quantities) by means of incremental analyses is costly. Direct methods such as the Simplified Theory of Plastic Zones (STPZ) aim at providing estimates of the post-shakedown quantities, bypassing cycle-by-cycle analyses. If geometric effects such as stress stiffening play a role, determination of accumulated strains even becomes more complicated. The STPZ has been further developed in order to account for the combination of plasticity and stress stiffening with respect to elastic shakedown. The theory is described and illustrated using examples such as a pipe bend subjected to cyclic in-plane bending. The implications of cyclic as opposed to constant stress stiffness are discussed. The effect of stress stiffening on ratcheting interaction diagrams (RID), separating regions of elastic and plastic shakedown in the space of loading parameters, is discussed.
The behavior of elastic-plastic structures under cyclic loading can be determined by incremental elastic-plastic analyses where a given load histogram is analysed cycle-by-cycle until shakedown is achieved. Many cycles may be required for this, if a ratchet mechanism is causing plastic strains to accumulate in each cycle until either elastic or plastic shakedown is achieved. The complexity of the elastic-plastic response of a structure is further increased if geometric effects are present. It may be very costly to get the accumulated strains and strain ranges in the state of shakedown by incremental analyses so that simplified or direct methods have been developed as an alternative. The Simplified Theory of Plastic Zones (STPZ) has proven itself for estimating the required quantities in the state of elastic and plastic shakedown within the framework of the 1st order theory, i.e., if the equilibrium conditions are satisfied for the undeformed structure. It is shown in this paper, how the STPZ can be expanded to capture 2nd order effects introduced by the equilibrium of the deformed structure. Some examples are used to demonstrate its applicability and the quality of the results, but also its limitations with respect to determining the accumulated strains and elastic-plastic strain ranges.
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.
The Simplified Theory of Plastic Zones (STPZ) may be used to determine post-shakedown quantities such as strain ranges and accumulated strains at plastic or elastic shakedown. The principles of the method are summarized. Its practical applicability is shown by the example of a pipe bend subjected to constant internal pressure along with cyclic inplane bending or/and cyclic radial temperature gradient. The results are compared with incremental analyses performed step-by-step throughout the entire load history until the state of plastic shakedown is achieved.
Das Phänomen Ratcheting - Auswirkung plastischen Materialverhaltens bei ortsveränderlicher Belastung
(2017)
Bei Belastungsänderungen kann eine progressive Deformation (Ratcheting) auftreten, sobald plastische Beanspruchungen im Tragwerk existieren. Dann akkumulieren sich Dehnungen und Verformungen im Falle zyklischer Belastung in jedem Belastungszyklus. Dieser Vorgang begrenzt die Lebensdauer eines Tragwerks, ist aber unabhängig von einer eventuell ebenfalls auftretenden Ermüdungsschädigung als eigenständige mögliche Schadensursache zu betrachten. Bekannt als Auslöser von Ratcheting und in manchen Regelwerken bereits berücksichtigt sind Konfigurationen, bei denen ein Tragwerk mindestens zwei unterschiedlichen Belastungsarten unterworfen ist, nämlich einer konstanten Belastung (der Primärlast) und einer überlagerten zyklischen Belastung. Selbst wenn letztere klein ist und für sich alleine keine plastischen Deformationen hervorruft, kann sie durch Zusammenwirkung mit der Primärlast dennoch eine progressive Deformation in Gang setzen.
In der vorliegenden Arbeit wird ein weiterer, Ratcheting erzeugender Mechanismus vorgestellt, der allein auf ortsveränderliche Wirkung einer einzelnen Lastgröße zurück zu führen ist. Im einfachsten Fall lässt sich dieser erklären durch die sukzessive Aktivierung von (gegebenenfalls unendlich vielen existierenden) Fließgelenken. Die Inkremente der Dehnungen und Verformungen können von Zyklus zu Zyklus ab- oder zunehmen, wenn die Verfestigung des Werkstoffs berücksichtigt wird, elastische Bettung vorliegt, die Formulierung des Gleichgewichts am verformten System erfolgt (Theorie II. Ordnung) oder die wahre Verformungsgeometrie (Theorie III. Ordnung) berücksichtigt wird.
Many pressure vessel and piping components have to withstand high internal pressures and are therefore thick-walled so that geometric effects such as stress stiffening need not be accounted for. However, thin-walled or moderately thick structures may be sensitive to these effects. Design Codes such as the ASME Boiler and Pressure Vessel Code usually provide little guidance on when they are to be accounted for. In the opinion of the authors, this effect deserves more attention. Therefore, the purpose of this paper is to illuminate the effect of stress stiffening by investigating some examples, with particular attention to elastic-plastic strain ranges and the plastic strain range enhancement factor Ke used for fatigue analyses.
Many pressure vessel and piping components have to withstand high internal pressures and are therefore thick-walled so that geometric effects such as stress stiffening need not be accounted for. However, thin-walled, or moderately thick structures may be sensitive to these effects. Design Codes such as the ASME Boiler and Pressure Vessel Code usually provide little guidance on when they are to be accounted for. In general, the effects of stress stiffening are difficult to estimate even for experienced engineers and can only be estimated by detailed finite element analyses. In the opinion of the authors, this effect deserves more attention. This is particularly true for simplified elastic-plastic methods for fatigue and ratcheting assessment of structures subjected to cyclic loading.
Cyclic loading may cause elastic-plastic strains to accumulate if a ratcheting mechanism is present. After a number of cycles, strain accumulation may cease so that a state of either elastic or plastic shakedown is reached. Determination of accumulated strains, strain ranges and other quantities in the state of shakedown (post-shakedown quantities) by means of incremental analyses is costly, in particular if geometric effects such as stress stiffening play a role. Direct methods aim at providing estimates of the post-shakedown quantities, bypassing cycle-by-cycle analyses. These methods claim to deliver the post-shakedown quantities with high accuracy and low computational effort.
The Simplified Theory of Plastic Zones can account for the combination of plasticity and stress stiffening. The theory is described and illustrated by examples. The Simplified Theory of Plastic Zones (STPZ) has proven itself for estimating the post-shakedown quantities in the state of elastic and plastic shakedown within the framework of the 1st order theory, i.e. if the equilibrium conditions are satisfied for the undeformed structure.
In this paper, the results of elbows subjected to various loading parameters are compared, considering and neglecting stress stiffening. Thus, the results show the influence 2nd order effects can generate. It is further shown that the STPZ can capture 2nd order effects introduced by the equilibrium of the deformed structure. Some examples are used to demonstrate its applicability and the quality of the results, e.g. for a pipe bend subjected to cyclic in-plane bending.
Cyclic and over-elastic loading can lead to an accumulation of plastic strains. If there is a cyclic load, which is driven by a single parameter, the lifecycle design can be very costly in terms of computational effort. If more than one cyclic load parameter is to be taken into account, which is then a multi-parameter loading, this task can become even more complex and costly. To solve this problem efficiently, different techniques are proposed. One of these techniques is based on step-by-step calculations of the strain ranges for a reduced set of loadings. Once these strain ranges are known, the accumulated state for each individual load case can be estimated using the Simplified Theory of Plastic Zones (STPZ), which requires just a few linear elastic analyses. It is shown that cyclic loads, which occur in intervals, can be replaced by interval-free calculations, which reduce the computational effort enormously. All these techniques lead to a procedure, which delivers good estimations in terms of post-shakedown quantities with very low computational effort compared to incremental step-by-step calculations. The results of the STPZ are presented by an example. A thick-walled cylinder is loaded with a constant axial force and subjected to cyclic shear and cyclic internal pressure. In general, for structures exhibiting ratcheting, hundreds or more load cycles must be analysed via step-by-step calculations until the shakedown state is reached. Using the STPZ, post-shakedown quantities, including strain ranges and accumulated strains can be estimated efficiently and the structure can be designed according to the rules of the ASME Codes. The computational effort and the quality of the results of the STPZ are compared with a step-by-step calculation.