A simplified technique for shakedown limit load determination of a large square plate with a small central hole under cyclic biaxial loading

A simplified technique for determining the shakedown limit load of a structure was previously developed and successfully applied to benchmark shakedown problems involving uniaxial states of stress ( [Abdalla et al., 2007a], [Abdalla et al., 2007b] and [Abdalla et al., 2007c]). In this paper, the simplified technique is further developed to handle cyclic biaxial loading resulting in multi-axial states of stress within the large square plate with a small central hole problem. Two material models are adopted namely: an elastic-linear strain hardening material model obeying Ziegler's linear kinematic hardening (KH) rule and an elastic-perfectly-plastic (EPP) material model. The simplified technique utilizes the finite element (FE) method and employs small displacement formulation to determine the shakedown limit load without performing lengthy time consuming full elastic-plastic cyclic loading FE simulations or conventional iterative elastic techniques. The simplified technique is utilized to generate the shakedown domain for the plate problem subjected to cyclic biaxial tension along its edges. The outcomes of the simplified technique showed very good correlation with the results of analytical solutions as well as full elastic-plastic cyclic loading FE simulations. Material hardening showed no effect on the shakedown domain of the plate in comparison to employing EPP-material.     

Interaction buckling-progressive deformation

The interaction between buckling and ratchetting under cyclic additional loading is discussed for elastic-plastic structures. It is shown that progressive plastic deformation can lead to the buckling of a structure as a consequence of the resulting additional displacement. This interaction is particularly strong in perfect plasticity. In the case of kinematic hardening materials, an estimate of the primary loading is given in order to prevent the risk of ratchetting. It is shown that elastic and plastic shake-down theorems still hold when the primary load is smaller than the smallest critical load of tangent modulus.     

Application of material models and assessment of strains in a surgeline under cyclic thermal loading

Inelastic constitutive models in commercial finite element (FE) programs are examined with respect to their capability of describing cyclic thermal loading. Neither isotropic nor linear kinematic hardening alone gives correct answers. Therefore, a model with combined isotropic and kinematic hardening based on Chaboche is implemented in an FE program and validated by appropriate experiments. This model is applied to assess the safety of a piping loop in a nuclear power plant subjected to cyclic loading. The numerical simulation shows a purely elastic behaviour of the surgeline after two cycles (elastic shakedown) as indicated by previous strain gauge measurements.     

Shakedown analysis of elastic-plastic structures

Behavior of elastic-plastic structures under repetitive and fluctuating loads is considered in this paper. Plastic deformation either stabilizes after a finite number of cycles or continues during the cycling. In the first case the structure is said to have shaken down to the boundary at the loading program. If plastic deformation does not stabilize an elastic-plastic structure becomes unserviceable due to either alternating plasticity when yielding occurs repeatedly in the opposite senses or accumulation of plastic strains and progressive increase of permanent displacements. This paper attempts to survey the shakedown theory, including the accommodation of a structure to the prescribed loading range as well as inadaptation and unserviceability.The notion of shakedown, incremental collapse, ratchetting and alternating plastic deformation are first illustrated with examples. The fundamental theorems on shakedown and inadaptation are presented next, attention being directed to generalizations of the classical Melan and Koiter theorems. Applications of the theorems are given. Available methods for determining the shakedown range on generating self-equilibrated stress fields are discussed. Special techniques applicable to problems involving both dead and fluctuating loads are invoked. Recent studies accounting for inertia forces, geometrical changes, material hardening, variation of material properties with temperature, displacement assessment, etc. are referred to.     

Elastic-plastic deformation of cylindrical pressure vessels under cyclic loading

A numerical procedure based on the finite element method and incremental solution approach is presented for analyzing cylindrical pressure vessels deformed in the state of generalized plane strain. The structures considered are subjected to internal pressure, thermal gradients and axial forces, which are cyclic in nature. The materials are assumed to obey the von Mises yield criterion with kinematic strain hardening which, during the process of plastic deformation, the yield surface is permitted to translate in the stress space and may change its size as a function of temperature. Based on Drucker's postulates for stable materials in the incremental theory of plasticity, a noniso-thermal flow rule is deduced in the cylindrical coordinates. By use of the virtual work principle, the finite element equations with displacement formulations are derived. The numerical solution is then carried out by a step-by-step approach for small loading increments and an iteration scheme is employed for each loading step. Furthermore, the analysis method is extended to the determination of limit load and shakedown load of a vessel. Several problem cases are presented.     

Upper bound methods for use in the design and assessment of axisymmetric thin shells subjected to cyclic thermal loading

Upper bound methods are used to calculate interaction diagrams showing the structural behaviour of axisymmetric thin shell components under cyclic thermal loading conditions and constant mechanical load. After a review of these upper bound methods and their implementation, a number of typical examples of thermal loading situations are examined in detail, including stationary thermal cycling, Bree type thermal loading and the effects of moving temperature fronts, as well as the effects of changes in shell thickness. It is demonstrated how interaction diagrams calculated by this method may be used in the design and assessment of components under these loading conditions.     

Elasto-plastic behavior of pipe subjected to steady axial load and cyclic bending

The elasto-plastic behavior of a pipe subjected to a steady axial force and a cyclic bending moment is studied. By using two parameters c and d, which describe the elasto-plastic interfaces of beam cross-section, the boundary curve equations between various types of elasto-plastic behavior, such as shakedown, plastic fatigue, ratcheting, and plastic collapse, are derived. The results are applicable for beams of any cross-section with two orthogonal axes of symmetry. As a result, the load regime diagram for a pipe is obtained, which gives an intuitive picture of the elasto-plastic behavior of the pipe under a given combination of constant axial load and cyclic bending moment.     

Mathematical programming methods for displacement bounds in elasto-plastic dynamics

In elastic-plastic structures subjected to dynamic external actions, if unbounded plastic deformations are developed, either local failure due to plastic fatigue (alternating plasticity) or gradual divergence of the deformed configuration (incremental collapse) will occur. Therefore, the boundedness in time of total plastic strains, and hence of total plastic work (usually referred to as adaptation or shakedown) is necessary for structural safety, in the sense that it rules out the occurrence of the above critical phenomena. Necessary and sufficient conditions for shakedown have been established by several authors. However, in many instances adaptation is not sufficient to ensure safety. In fact, even if plastic deformations can be proved to be finite, they can exceed some critical limit or exhaust the material ductility. In particular, for dynamic loading histories that cease after a certain time, a structure will certainly shakedown under any load amplitude, so that a safety criterion based on this event is clearly meaningless. Typical histories of this kind are earthquake or blast loadings.When the loading history is known, it is possible, in principle, to assess safety by following the actual plasto-dynamic evolution of the system, but this is often a laborious task and in several cases it provides far more information than is actually needed. On this remark rests the interest of methods capable to provide some essential information, such as upper bounds on maximum deflections or strains, through a moderate computational effort. In recent years several alternative techniques have been developed to bound from the above various quantities of interest. With the exception of very simple situations, the best bounds that can be obtained through any one of these methods involve the solution of constrained optimization problems.In this paper a study of several deformation bounding techniques is performed. The problem is formulated and the main previous results are outlined first with reference to general continua made of hardening materials. Then a class of discrete structural models (such as some finite element discretizations) is considered and, on this basis, two categories of deformation bounding techniques are described from the previous main results. All these techniques, some of which are new, permit the optimization of the upper bound by solving one or more mathematical programming problems of special forms. Some of the bounding procedures are shown to have merely theoretical interest, since they lead to cumbersome numerical procedures or to very coarse bounds. The formulations that appear to have practical application are compared from various standpoints (type of loading history, different hardening rules, influence of second order geometric effects, quantities to be bounded) and first assessment of their practical usefulness is attempted. Generalizations to second-order geometric and thermal effects and to situations in which the time history is not completely known are envisaged.     

Low cycle fatigue and ductile fracture for Japanese carbon steel piping under dynamic loadings

Dynamic fracture behavior of circumferentially cracked pipe is important to evaluate the structural integrity of nuclear piping from the viewpoint of the LBB concept under seismic conditions. Fracture tests have been conducted for Japanese carbon steel (STS410) circumferentially through-wall cracked pipes that are subjected to monotonic or cyclic bending loads at room temperature. In the monotonic-loading tests, the maximum load to failure increases slightly with increasing loading rate. The failure cycles can be expressed simply by ratio of the load amplitude to the plastic collapse load. Fracture analysis has been also conducted to model the pipe tests. A new equation for calculating ΔJ for a circumferentially through-wall cracked pipe subjected to bending has been proposed. The failure cycles under cyclic loads are satisfactorily evaluated using an elastic-plastic fracture mechanics parameter ΔJ.     

Deformation bounds for elastic-plastic solids within and out of the creep range

A bounding principle for elastic-perfectly plastic creeping and noncreeping structures subjected to mechanical and/or thermal loads varying below or above the shakedown limit is presented. This principle contains some free “perturbation functions” which, suitably chosen, enable it to specialize, so generating bounds on a variety of deformation measures (such as inelastic work dissipated within any portion of the body, inelastic strains and displacements), some of which are new results, others recover or generalize known results. The resulting bounding technique possesses a quite unified character which is useful for computational purposes. The concept of “pseudo-plastic” strain is shown to be crucial for the derivation of bounds applicable above the shakedown limit.     

On a simplified inelastic analysis of structures

In this paper two main problems are considered: the derivation of cyclic constitutive relations during inelastic regime where hardening, softening and creep can occur, and the development of the eventual periodical state in the structure during cyclic thermodynamical loadings.We give a very simple and practical framework to solve these problems in one unique manner.Its essential feature consists in the introduction of a family of internal parameters which characterize local inelastic mechanisms and the family of transformed internal parameters which are linearly linked to the previous ones through a symmetrical non-negative matrix and are indeed the opposite of the associated residual stresses. Thanks to that, the treatment of the local plastic or viscoplastic yield conditions can be easily made from only the classical simple purely elastic (or viscoelastic) analysis.This property allows important results during cyclic loadings: conditions for elastic shakedown, plastic shakedown, ratcheting and bounds for the limiting state.Several examples are given in the text.     

A bounding technique for dynamic plastic deformations of damped structures

A bounding principle for elastic-perfectly plastic creeping and noncreeping structures subjected to mechanical and/or thermal loads varying below or above the shakedown limit is presented. This principle contains some free “perturbation functions” which, suitably chosen, enable it to specialize, so generating bounds on a variety of deformation measures (such as inelastic work dissipated within any portion of the body, inelastic strains and displacements), some of which are new results, others recover or generalize known results. The resulting bounding technique possesses a quite unified character which is useful for computational purposes. The concept of “pseudo-plastic” strain is shown to be crucial for the derivation of bounds applicable above the shakedown limit.     

FEM-computation of load carrying capacity of highly loaded passive components by direct methods

Limit and shakedown theorems are exact theories of classical plasticity for the direct computation of safety factors or of the load carrying capacity under constant and varying loads. Simple versions of limit and shakedown analysis are the basis of all design codes for pressure vessels and pipings. Using finite element methods (FEM), more realistic modeling can be used for a more rational design. The methods can be extended towards optimum plastic design. In this paper, we present a first implementation of limit and shakedown analyses for perfectly plastic material into a general purpose FEM program. Limit and shakedown loads are obtained for a square plate with a hole and for a thin tube. Interaction diagrams are calculated and the results are compared with known analytic solutions.     

Shakedown and ratchetting under tension–torsion loadings: analysis and experiments

Structural design analyses are conducted with the aim of verifying the exclusion of ratchetting. To this end it is important to make a clear distinction between the shakedown range and the ratchetting range. The performed experiment comprised a hollow tension specimen which was subjected to alternating axial forces, superimposed with constant moments. First, a series of uniaxial tests has been carried out in order to calibrate a bounded kinematic hardening rule. The load parameters have been selected on the basis of previous shakedown analyses with the PERMAS code using a kinematic hardening material model. It is shown that this shakedown analysis gives reasonable agreement between the experimental and the numerical results. A linear and a nonlinear kinematic hardening model of two-surface plasticity are compared in material shakedown analysis.     

Asymptotic techniques in elastic-plastic analysis of structures

Elastic-plastic structures can nowadays be analyzed with the powerful numerical procedures of the finite element method. Nevertheless, in many engineering applications, analytical expressions capable of predicting with sufficient accuracy the stress distributions, the extent of the plastic zones and the load displacement behavior could be of great practical value. For simple structures and loading stages not too far from the elastic limit, such analytical expressions may be obtained by using perturbation methods and asymptotic expansions. A small dimensionless parameter is defined as the ratio of a length characterizing the extent of the narrow plastic zone, to a conveniently chosen typical dimension of the structure. Stresses and displacements are formally expanded as asymptotic series in terms of powers of . For each order of magnitude, the exact basic relations lead to a separate set of simplified differential equations which can be integrated analytically or numerically by using standard procedures. The method is very general and can be applied to several classes of plastic behaviour and of structural problems.Three examples of very simple structures are chosen in particular to illustrate the applicability of the perturbation method to engineering problems: (1) For a simply supported beam under a concentrated load, is chosen as the ratio of the largest axial extension of the plastic zone to twice the beam length. (2) The case of a long thin cylindrical shell subject to a gradually increasing radial ring of load is algebraically more involved but can be treated similarly. (3) For a simply supported circular plate under a ring of load, is defined as the thickness ratio of the central plastic zone which is assumed to be sufficiently thin in the first stage of loading beyond the elastic limit. Both Tresca's and von Mises' yield criteria with the associated flow rule (rate theory) can be used.The method can also be applied to more involved geometries (in conjunction with linear FE-procedures) and more complex material behaviour.     

Investigation of the ratcheting phenomenon for dominating bending loads

For components subjected to strong mechanical and cyclically thermal loads, proof of a limitation of permanent strain is of great significance due to the danger of deformation progressing from cycle to cycle (ratcheting). Within the framework of investigations concerning the load situation of the first wall of a fusion reactor, experimental and theoretical studies on ratcheting were carried out. Progressive deflections were identified under suitable conditions on internally cooled beam-shaped components subjected to cyclic heating and a bending load. A simple beam model can be used for systematic theoretical analysis in this case. Its treatment leads to the development of so-called Bree diagrams with which the structural behaviour can be characterized as a function of the applied primary and secondary stresses. The peculiarities arising under predominant bending loads will be discussed.     

The influence of transient thermal loading on the Bree plate; A simplified method of analysis

The paper describes a simplified technique, based upon an extension of the upper bound shakedown theorem, for the evaluation of ratchet boundaries for a plate subject to a through-thickness temperature transient and in-plane loading. The study of a range of cases indicates that transient effects can have a significant effect upon the deformation properties and that strain growth can occur for zero applied loads at quite moderate levels of thermal loading. As a result the identification of transient thermal stresses as F stresses in ASME design codes does not seem acceptable.     

Practical remarks about cyclic loadings on an elastic-plastic structure

A complete study of the classical symmetrical three-bar system for perfect plastic and kinematical hardening materials is presented. Based on this elementary example, a simple practical approach for the analysis of cyclic thermomechanical loading problems for elastic-plastic structures is discussed.     

Experimental investigation of the failure behavior of notched wall-thinned pipes

The present study performed full-scale pipe tests using 100A Schedule 80 pipe specimens with simulated notched and circular wall thinning to investigate the failure behavior of notched wall-thinned pipes. The tests were conducted under both monotonic and cyclic bending moments at a constant internal pressure of 10 MPa at room temperature. The failure pattern, load carrying capacity, deformation ability, and fatigue strength of the notched wall-thinned pipes were evaluated by comparing results to those of circular wall-thinned pipes. The investigation showed that the effect of the type of thinning on the failure behavior was more sensitive under cyclic loading conditions than under monotonic loading conditions. The load carrying capacity of pipes with notched wall thinning was approximately the same or slightly less than that of pipes with circular wall thinning when the thinning area was subjected to tensile stress. However, when the thinning area was subjected to compressive stress, the load carrying capacity of pipes with notched wall thinning was greater than that of pipes containing circular wall thinning. The deformation ability and fatigue strength increased proportionally with the axial length of the thinning defect, and thus these properties were significantly reduced in notched wall-thinned pipes.