Limit analysis and design of containment vessels

In the introduction, the theory of plastic analysis of shells is briefly recalled. Minimum-volume design for assigned load factor at plastic collapse is then considered and optimality criteria are derived for plates and shells of continuously varying or piecewise-constant thickness.In the first part, containers made of metal are examined. Analytical and numerical limit analysis solutions and corresponding experimental results are considered for various types of vessels, including intersecting shells. Attention is given to experimental post-yield behavior. Some tests up to fracture are discussed. New theoretical and experimental results of limit analysis of stiffened cylindrical vessels are presented, in which reinforcing rings are treated as discrete structural element (no smearing out) and due account is taken of their strong curvature. Cases of collapse by instability under internal pressure are pointed out. Minimum-volume design of circular plates and cylindrical shells is then formulated and various examples are presented of sandwich and solid metal structures. Containers of piecewise-constant thickness are given particular attention. Available experimental evidence on minimum-volume design of plates and shells is reviewed and commented upon.The second part deals with reinforced concrete vessels. Cylindrical containers are studied, from both points of view of limit analysis and of limit design with minimum volume of reinforcement. The practical use of the latter solutions is discussed.A third part reviews other loading cases (including cyclic and impact loads) and gives indications on corresponding theories, formulations and solution methods.The last part is devoted to a discussion of the limitations of the methods presented, within the frame of the “limit states” design philosophy, which is first briefly recalled. Considerations on further research in the field conclude the paper.     

Buckling sensitivity analysis of cracked thin plates under membrane tension or compression loading

Thin-walled structural components such as plates and shells are commonly used in practical applications such as aerospace, naval, nuclear power plant, pressure vessels, mechanical and civil engineering structures and so on, and the safety assessment of such structures must carefully consider all the phenomena which can decrease the bearing capacity of such elements. Among them, the presence of cracks in thin-walled structures can heavily affects their safety factor with respect to the more common modes of failure such as buckling or fracture. For very thin plate, buckling collapse under compression or even under tension, apart fracture or plastic failure in this last case, can easily take place: the presence of flaws such as through-the-thickness cracks can sensibly modify such ultimate loads. In the paper the effects of cracks’ length and orientation on the buckling loads of rectangular elastic thin-plates - characterised by different boundary conditions and by various Poisson's ratio - under tension and compression, is considered. For tensioned flawed plates a fracture-buckling and a plastic-buckling collapse maps are obtained. After a short explanation of the buckling phenomena in plates, several FE numerical parametric analyses results are presented in terms of critical load multiplier in compression or in tension in cracked plates. The obtained results are discussed and some interesting and useful conclusions regarding the sensitivity to cracks’ presence of buckling loads of thin plates under compression or tension (or fracture in this last case) are explained. The interesting case of tensioned cracked plates is considered by studying the easiest collapse between fracture, plastic flow and buckling: in such cases some failure-type maps are finally determined.     

Influence of flow stress choice on the plastic collapse estimation of axially cracked steam generator tubes

The influence of the choice of flow stress on the plastic collapse estimation of axially cracked steam generator (SG) tubes is considered. The plastic limit and collapse loads of thick-walled tubes with external axial semi-elliptical surface cracks are investigated by three-dimensional non-linear finite element (FE) analyses. The limit pressure solution as a function of the crack depth, length and tube geometry has been developed on the basis of extensive FE limit load analyses employing the elastic–perfectly plastic material behaviour and small strain theory. Unlike the existing solutions, the newly developed analytical approximation of the plastic limit pressure for thick-walled tubes is applicable to a wide range of crack dimensions. Further, the plastic collapse analysis with a real strain-hardening material model and a large deformation theory is performed and an analytical approximation for the estimation of the flow stress is proposed. Numerical results show that the flow stress, defined by some failure assessment diagram (FAD) methods, depends not only on the tube material, but also on the crack geometry. It is shown that the plastic collapse pressure results, in the case of deeper cracks obtained by using the flow stress as the average of the yield stress and the ultimate tensile strength, can become unsafe.     

Post-collapse cross-sectional flattening of thick pipes in plastic bending

Post-collapse cross-sectional flattening of a thick pipe at the centre of a plastic hinge formed during excessive bending due to unrestrained pipe whip is analysed with an aim to ascertain the extent of flow choking and consequent reduction of blowdown force. Based on the experimentally observed similarity between the plastic collapse mode of the critical pipe section during bending and plastic deformation of a ring under lateral compression, the effect of excessive bending on pipe section flattening is simulated using some well known analytical models for section collapse. A theoretical relationship between post-collapse bend curvature and section flattening is proposed for relatively thick pipes made of strain hardening material. The calculations made using the above relationship are found to compare well with those observed experimentally.     

Measurement of the yield and tensile strengths of neutron-irradiated and post-irradiation recovered vessel steels with notched specimens

Tensile circumferentially notched bars are examined as test specimens for measuring the yield and tensile strengths of nuclear pressure vessel steels under several conditions of irradiation and temperature that a vessel can experience during its service life, including recovery post-irradiation treatment. For all the vessel steels, notch geometries and conditions explored, it has been found that notched specimens fail by plastic collapse, and simple formulae have been derived that allow the yield and tensile strengths to be determined from the yielding and plastic collapse load of a notched specimen. Values measured in this way show good agreement with those measured by the standard tensile test method.     

Effect of local wall thinning on the collapse behavior of pipe elbows subjected to a combined internal pressure and in-plane bending load

The objective of this study was to investigate the effect of local wall thinning on the collapse behavior of pipe elbows subjected to a combined internal pressure and in-plane bending load. This study evaluated the global deformation behavior and collapse moment of the elbows, which contained various types of local wall-thinning defects at their intrados or extrados, using three-dimensional elastic–plastic finite element analysis. The analysis results showed that the global deformation behavior of locally wall-thinned elbows was largely governed by the mode of the bending and the elbow geometry rather than the wall-thinning parameters, except for elbows with considerably large and deep wall thinning that showed plastic instabilities induced by local buckling and plastic collapsing in the thinned area. The reduction in the collapse moment with wall-thinning depth was considerable when local buckling occurred in the thinned areas, whereas the effect of the thinning depth was small when ovalization occurred. The effects of the circumferential thinning angle and thinning length on the collapse moment of elbows were not major for shallow wall-thinning cases. For deeper wall-thinning cases, however, their effects were significant and the dependence of collapse moment on the axial thinning length was governed by the stress type applied to the wall-thinned area. Typically, the reduction in the collapse moment due to local wall thinning was clearer when the thinning defect was located at the intrados rather than the extrados, and it was apparent for elbows with larger bend radius.     

The application of a kinematic hardening model for assessing ratchetting in pressurized pipes

Concerns within the nuclear power generation industry regarding the possibility of incremental collapse or ratchetting incurred in pressure vessels and large pressurized piping runs during seismic disturbance has led to a programme of experimental work to simulate component and material behaviour under such conditions.As part of this programme, the plastic deformation of thin-walled cylinders has been experimentally examined for the loading conditions of ±1% cyclic axial strain with hoop stresses of approximately of the initial unixial yield stress.Two materials similar to those used in the pipework of pressurized-water reactor nuclear plant in the UK have been tested, namely type 304S11 stainless steel and En6 low carbon steel. Under the loading conditions, both materials incurred plastic hoop ratchet strains to varying degrees. These ratchet strains were compared with the limiting ratchet strains predicted by the Prager-Ziegler model of kinematic work hardening.It was concluded that this model could not be satisfactorily used for design purposed as it did not consistently either overestimate or underestimate the measured ratchet strains. Furthermore, the manner in which the model reaches a limit is not observed in the experimental results.     

Buckling assessment procedure for large diameter vessel with local thin area subjected to combined pressure and external moment

The newly developed p-M diagram provides a means for readily evaluating the collapse and/or buckling load of pressure equipment with external flaws simultaneously subjected to internal pressure, p and external bending moment, M due to earthquake, etc. In this paper, some FEAs for large diameter vessels with an external flaw were conducted under (1) pure external bending moment, and (2) subjected simultaneously to both internal pressure and external bending moment, in order to determine the plastic collapse load by applying the twice-elastic slope (TES) as recommended by the ASME and to determine the buckling load. The p-M line adopted in the Ibaraki FFS rule based on the measured yield stress indicates that the safety margin for the TES loads at the LTA is about 1.2-1.8, and 1.5-2.0 for the buckling loads. The Ibaraki FFS rule that prevents buckling by applying Donnell's equation of FS = 2 can assure adequate levels of integrity and safety.     

Plastic collapse and LBB behavior of statically indeterminate piping system subjected to a static load

The plastic collapse and LBB behavior of statically indeterminate piping system were investigated in this study, compared with those of the statically determinate piping system. Special attention was paid to evaluate the crack opening displacement after a crack penetrated wall thickness. The main results obtained were as follows:

1. The reduction of ultimate strength caused by a crack was relatively small in the statically indeterminate piping system. The main reason is thought to be that a sufficient redistribution of the bending moment occurs in this system.

2. A method to evaluate the crack opening displacement after crack penetration in a pipe with a non-penetrating crack was proposed. From this method, it was known that the crack opening displacement could be evaluated by using the incremental plastic rotation angle.

3. The acceptable defect size considering the deformation of a pipe was estimated by comparing the plastic moment at the defective part and the gross yielding moment at the non-defective part.

The effect of internal pressure on in-plane collapse moment of elbows

Elastic–plastic finite element analysis has been carried out to evaluate collapse moments of six elbows with elbow factors varying from 0.24 to 0.6. Collapse moment is obtained by twice elastic slope method from the moment vs. end-rotation curve. The effect of internal pressure on in-plane closing/opening collapse moment is studied. For various elbow factors and level of internal pressure, total 60 and 54 cases are analysed for closing and opening mode of bending moment, respectively. Based on these results, two closed-form equations are proposed to evaluate the collapse moments of elbows under combined internal pressure and in-plane closing and opening bending moment.     

Fracture behavior of straight pipe and elbow with local wall thinning

Fracture behavior of pipes with local wall thinning is very important for the integrity of nuclear power plant. Then we studied the fracture behavior of straight pipe and elbow with local wall thinning. For the straight pipe, failure mode, limit load and allowable wall thinning limit based on plastic deformation ability have been studied systematically. Twenty two straight pipe specimens were tested. The failure mode was divided into four types; cracking, local buckling, ovalization and plastic collapse (ovalization+buckling). Maximum load was successfully evaluated using plastic section modulus and modified flow stress, in dependent to failure mode. For the elbow, plastic collapse and low cycle fatigue fracture by reversed loading have been tested using ten specimens. Observed failure modes were ovalization and local buckling under monotonic loading, and were local buckling and cracking under cyclic loading, especially local buckling promoted crack initiation. Test results were compared with ASME design curve and allowable limit of local wall thinning will be discussed.     

Characterization of plastic collapse load determination in circumferentially through-wall cracked elbows

The solutions of cracked elbow are shown to be excessively conservative and on occasion, non-applicable to the cases for which they are intended. The objective of the work described in this paper is to use the 3D non-linear finite element method (FEM) backed up with experimental results to determine the collapse limit load. Non-linear finite element analyses were performed considering both material and geometrical non-linearity using the advanced fracture analysis code WARP3D. Various alternative methods are used to determine plastic collapse loads based on the FEM calculated load–displacement curves. The predicted collapse loads are compared to collapse loads determined by available solutions and finally these are compared to experimental results. The work can be considered as the source of the benchmark data that helped to shape the engineering treatment of piping elbows in design codes.     

Leak-before-break and plastic collapse behaviour of statically indeterminate pipe system with circumferential crack

Much research has been carried out on Leak-Before-Break (LBB) behavior of pipes with cracks. However, most studies have been made on statically determinate pipe systems. Few studies have been made on LBB behavior of statically indeterminate pipe systems. Most pipe systems in nuclear power plants have supports and restraints, thus they can be considered as statically indeterminate pipe systems. From above points of view, LBB and plastic collapse behaviors of statically indeterminate pipe with circumferential crack and compliance were studied in this paper. A new method is proposed to analyze and evaluate the LBB and plastic collapse behavior of a statically indeterminate structure. The pipe system of which one end is clamped and the other is supported with compliance was analyzed. The main results obtained are as follows: (1) By combining the limit analysis theory and elastic–plastic fracture mechanics, the effects of crack size, compliance and fracture toughness on load deflection behaviors to failure and structural integrity of statically indeterminate pipe system have been analyzed quantitatively and easily. (2) When a crack grows in a statically indeterminate pipe before plastic collapse, load drop conditions can be derived quantitatively, as a function of J_IC, dJ/da, flow stress, crack size, pipe span length, compliance and flexural rigidity of the pipe. (3) The analytic method developed in this research is useful and convenient to evaluate the LBB and tearing instability behavior of a statically indeterminate pipe system. (4) LBB resolves easily for statically indeterminate pipes with a crack, even when it does not resolve for statically determinate pipes with the same crack. That results from the fact that bending moment redistribution during the fracture process occurs easily for statically indeterminate pipe systems, and its redistribution restrains plastic deformation of the cracked weak section.     

Flow-induced plastic collapse of stacked fuel plates

Flow-induced plastic collapse of stacked fuel plate assemblies was first noted in experimental nuclear reactors such as the Oak Ridge National Laboratory High Flux Reactor Assembly and the Engineering Test Reactor (ETR). The ETR assembly is a stack of 19 thin flat rectangular fuel plates separated by narrow channels through which a coolant flows to remove the heat generated by the nuclear fission of the fuel within the plates. The uranium alloyed plates have been noted to buckle laterally and plastically collapse at the system design coolant flow rate of 10.7 m/s, thus restricting the coolant flow through adjacent channels. In this paper a methodology and criterion are developed for predicting the plastic collapse of ETR fuel plates. The criterion is compared to some experimental results and the Miller critical velocity theory.     

An elasto-plastic fracture mechanics based model for assessment of hydride embrittlement in zircaloy cladding tubes

This paper describes a finite element based fracture mechanics model to assess how hydrides affect the integrity of zircaloy cladding tubes. The hydrides are assumed to fracture at a low load whereas the propagation of the fractured hydrides in the matrix material and failure of the tube is controlled by non-linear fracture mechanics and plastic collapse of the ligaments between the hydrides. The paper quantifies the relative importance of hydride geometrical parameters such as size, orientation and location of individual hydrides and interaction between adjacent hydrides. The paper also presents analyses for some different and representative multi-hydride configurations. The model is adaptable to general and complex crack configurations and can therefore be used to assess realistic hydride configurations. The mechanism of cladding failure is by plastic collapse of ligaments between interacting fractured hydrides. The results show that the integrity can be drastically reduced when several radial hydrides form continuous patterns.     

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.     

Displacement estimates of pipe elbows prior to plastic collapse loads

The non-differentiability of the plastic dissipation calculated through the Von-Mises yield function leads to convergence difficulties when using mathematical programming to solve Köiter's kinematical formulation of the classical limit analysis. This problem is avoided replacing the plastic dissipation by the strain energy of a fictitious viscoelastoplastic material with a nearly infinite Young modulus. Classical limit analysis can only give information about the limit load multiplier and the plastic collapse mechanism. Based on Zarka's method and using the finite element method and mathematical programming, it is possible to obtain not only the limit load but also an estimate of the elastoplastic displacements. This is very useful because construction codes usually impose limits on the electroplastic displacements. Some pipeline systems are examined using a 1-dimensional shell type finite element to illustrate the procedure. The results obtained are compared with simplified analytical solutions and with alternative numerical results using 2-dimensional shell elements and realistic materials.     

Probabilistic methods in plastic structural analysis

The redistribution of stresses in ductile structures, although beneficial from the safety viewpoint, introduces another source of uncertainty, which requires specific methods when the probabilistic approach to reliability evaluation is followed.Practicable procedures have been developed for structures that satisfy the classical assumptions of plastic limit analysis. In particular, two theorems that allow to find rigorous upper and lower bouds on the probability of full plastic collapse under given loads, are presented. Other methods for probabilistic limit analysis are also indicated, including in particular a specifically developed parametric simulation procedure.The last part of the paper is devoted to the reliability analysis of plastic structures subject to loads varying (slowly) in time.It is recalled first that probabilistic limit analysis can be easily extended to the shakedown—incremental collapse problem, provided the loads vary within a finite domain: however, the significance of such an approach for stochastically varying loads is questioned. In fact, as time increases, the probability also increases that the loads cross any given threshold. Therefore, it is more appropriate to speak of “plastic adaption” rather than “shakedown”, and to focus the attention on the probability of reaching, in any given time interval, a certain permanent deformation. Again, only approximate solutions (in the form of upper and lower bounds) can be found to this question, but this appears to be a more rational and promising approach to the problem.     

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 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.