Design by Analysis Manual_EUR19030EN.pdf

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Introduction

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1 Introduction 1.1 Background Pressure vessel design has been historically based on Design By Formula. Standard vessel configurations are sized using a series of simple formulae and charts. In addition to the Design by Formula route, many national codes and standards for pressure vessel and boiler design do provide for a Design By Analysis (DBA) route, where the admissibility of a design is checked, or proven, via a detailed investigation of the structure's behaviour under the external loads (or ‘actions’) to be considered. Nevertheless Design By Formula remains the dominant approach. In an increasingly technically sophisticated society, it may be asked why this should be the case? All these DBA routes in the major codes and standards in the pressure equipment field are based on the rules first proposed in the ASME Pressure Vessel and Boiler Code, which was formulated in the late 1950’s before being released, originally for nuclear applications, in 1964. All these routes lead to the same well-known problems, especially the stress categorisation problem[1-6], and all are outof-step with the continuing development of computer hardware and software. Further, all are focused on pressure, and possibly, and to a limited extent, temperature, treating other actions in an inflexible manner, giving them marginal attention only. The DBA route in the proposal of CEN's unfired pressure vessel standard prEN 13445-3 tries to avoid these problems: 1. 2. 3. 4.

by addressing the failure modes directly by allowing for non-linear constitutive models by applying a multiple safety factor format for the incorporation of actions other than pressure by specifying mainly the principal technical goals of the standard together with some application rules as possible methods for the fulfilment of these goals.

In the new proposal of a European Standard, two documents are included concerning design by analysis: • •

Document prEN 13445-3, Annex B, Direct Route for Design by Analysis Document prEN 13445-3, Annex C, Stress Categorisation Route for Design by Analysis

For various reasons SG-DC of the Working Group (WGC) of the CEN Technical Committee TC54 decided to use in the new European Standard an approach similar to the one used in Eurocodes (for steel structures), using the notions of principles and application rules as well as the notion of partial safety factors. One reason is that the DBA-approach is flexible and simplifies the incorporation of constructional requirements (wind, snow, earthquake, etc.), if required, in a consistent manner. Another reason is, that there has been considerable criticism of the ASME stress classification (or categorisation) method, which is used in principle in almost all countries: One solution to the annoying problem of stress classification is to apply limit analysis, as proposed in the rules for DBA. Limit analysis does not require categorisation into primary and secondary

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stresses and it gives a unique result (which stress categorisation in general does not). The calculations can be made using existing software, but no doubt special software could be readily developed if there were sufficient demand. Nevertheless, part of the usual stress categorisation approach is included in the Standard, as an application rule. The DBA route is included in the new European Standard • • • • •

as a complement to Design by Formula (DBF) for cases not covered there as a complement for cases requiring superposition of environmental actions - wind, snow, earthquake, etc. as a complement for fitness-for-purpose cases where (quality related) manufacturing tolerances are exceeded as a complement for cases where local authorities require detailed investigations, e. g. in major hazard situations, for environmental protection reasons as an alternative to the Design by Formula route.

For the time being, this route is restricted to sufficiently ductile steels and steel castings with calculation temperatures below the creep range. The main concepts are dealt with here in detail, because • • • • •

it is a real alternative to DBF, as stated above, with many advantages many concepts are new in pressure equipment design it may be used as a yard-stick for DBF solutions, to show possible improvements some concepts have already influenced the DBF-section, their discussion will shed light onto some DBF-details it may lead to an improved design philosophy by indicating more clearly the critical failure modes, especially of importance for in-service inspections.

1.2 Aims From the point of view of an analyst or designer, the rules in the new European Standard are quite general, and in fact as mentioned above this is intentional. In broad terms, in the context of the Direct Route either an admissibility check, or a check on maximum allowable load, has to prepared on the basis on either detailed elastic-plastic finite element analysis or some method of estimating plastic failure loads for gross and progressive plastic deformation. In principle this seems straightforward, but in practice can be difficult. The aim of this study has been to provide guidelines on the application of elastic-plastic analysis (in its broadest sense) to the Standard and in doing so to highlight possible problem areas and suggest methods of resolving these. This has been achieved using a new collection of ten benchmark problems. These example problems have been chosen to be typical of cases where design by formula cannot be used. A substantial part of this document provides detailed, step-by-step, studies of each of the example problems. This study has been undertaken by experts either in the research and development of design by analysis itself or in its practical use. This expertise is apparent in the review of the current state-ofthe-art of pressure vessel design by analysis in Section 2, which highlights unexpected, but now

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well-known, problem areas, and in the detailed description of the analysis procedures and their application to the Standard in Sections 3 and 7. The solutions reported in Section 7 were carried out independently, although unusual results were re-checked. It is intended that this document should be read in conjunction with the Standard, and can be looked on as a supplement. Particular emphasis is placed on the expected readership, and the expertise and knowledge required of them. While the Standard itself is fairly simple and transparent, the writers have also been aware that the current state-of-the-art in finite element analysis technology, and the expected continuing increase in sophistication, renders elastic-plastic analysis ever more routine. However the issue here is whether or not the analyst/ designer understands the underlying mechanics. Many users of elastic-plastic finite element analysis are unaware of the assumptions and approximations of the Classical Theory of Plasticity, which are embodied in most finite element software. They generally do not recognise the implications of the neglect of the Bauschinger effect and hysteresis, the assumptions concerning yield in compression in general, or that the basic mathematical models of initial and subsequent yield are approximations which are valid in some situations but not in others. At a more basic level, very few analysts are even aware of the fundamental assumptions of the engineering yield stress itself, for example it is measured from a tensile test and arbitrarily used as a reference to develop multiaxial yield criteria. Further, plasticity in metals is a shear mechanism, yet we use yield measured in tension rather than torsion and the measured value can be difficult to identify and is usually subjective. An overview of the contents of this study is given in the next sub-section, followed by some additional comments on the expected readership and recommendations of how the document should be used.

1.3 Overview This document is divided into nine Sections, including this Introduction: Section 2 provides an overview of the current state of design by analysis, as typified by the ASME Pressure Vessel and Boiler Code. The ASME Code offers two routes to design by analysis, the socalled elastic route and an inelastic route which requires the calculation of limit and shakedown loads – these are briefly summarised, together with definitions of basic terminology. Following this a short discussion of the most common method of analysis, using finite element techniques, is provided. This is not intended as an introduction to the finite element method applied to pressure vessels, rather several issues related to choice of element type are raised since they have implications for code interpretation – specifically the two main problem areas of the elastic route: linearisation and categorisation. These problem areas are then discussed in some detail, to give the reader an insight into the nature of major difficulties in application of what seems a fairly simple and straightforward set of design by analysis rules. Following this discussion, application problems with the inelastic route are then examined, in particular the difficulty of extracting meaningful plastic design loads from elastic-plastic finite element analysis. This Section then concludes with an introduction to the novel features of the new European standard in relation to design by analysis. In Section 3 a description of the various procedures used in this document to satisfy the analysis requirements is given. Some detail is provided on using the results of elastic-plastic analysis in the Direct Route for the checks on both gross plastic deformation and progressive deformation. In the

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case of the latter, problems with estimating shakedown loads when shell elements are used, or when there are stress singularities are discussed in some detail. The use of deviatoric maps to assist the shakedown analysis is also described. As an alternative to elastic-plastic analysis a new technique for directly estimating limit and shakedown loads from elastic finite element analysis alone is also used. This technique – the elastic compensation method – is briefly described in the context of the requirements of the Standard. Also, the treatment of shell elements is discussed. This Section also reviews various other issues related to the practical use of the Standard – in particular wind action, the stress categorisation route and checks against fatigue and instability. In Section 4 a simple example – a circular plate – is used to describe and discuss each step in the application of the Standard before proceeding to the main examples examined in this document. Sections 5, 6 and 7 contain the main body of this study – the detailed application of the European Standard to ten benchmark problems. Section 5 gives a specification of each example, followed by a summary of the results of the analysis and application of the Standard in Section 6. Section 7 provides the detailed results for each benchmark problem using the analysis procedures described in Section 3. Finally, in Section 8, recommendations and concluding remarks are given. This covers comments on the appropriateness and difficulties with the methodology, software requirements, expertise and knowledge expected by the analyst and various warnings. For example, it is apparent that the fatigue rules – which are used for both design by formula and design by analysis – need special care. Appended to the report are various Annexes, specifically a bibliography, analysis input files (where appropriate) and excerpts of the Standard. 1.4 How to read this document This document is not aimed at the complete novice, but two broad types of reader are envisaged. It is presumed that anyone starting to read this has a basic familiarity with the concepts of plasticity theory and the behaviour of structures under plastic strain. In addition, familiarity with the practice of elastic finite element analysis for pressurised components, preferably with basic experience of elastic-plastic analysis is suggested. Also it is recommended that the reader should read the European Standard in some detail beforehand, if necessary. It is then envisioned that the reader will either already be broadly familiar with pressure vessel design by analysis and elastic-plastic finite element analysis and is comfortable with the Standard (whom we will call the Expert), or has read the Standard and has some basic experience of elastic design by analysis (whom we will call the Novice). In the case of the Expert, it is anticipated that this reader will begin with Section 5, the specification of the examples, followed by Section 6, the analysis summary and then initially carry out his own analysis and code check. It is possible that some reference will have to be made to Section 3 on procedures if substantial variation from the results reported here are obtained, or if details on application of the Standard need to be clarified. In the case of the Novice, it is expected that more or less the whole document will be carefully read, from Section 2 through to 7 before carrying out his own analyses. (Of course only a few of the

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benchmarks may be read in detail so that the Novice may test his understanding of the basic principles and procedures on the remainder). Finally it is also expected that engineering managers may wish to review Section 8, which deals with recommendations – in particular the discussion of assumed expertise on the part of the analyst/designer.

1.5 Literature [1] J. L. Hechmer & G. L. Hollinger, "Considerations in the calculations of the primary plus secondary stress intensity range for Code stress classification," "Codes & Standards and Applications for Design and Analysis of Pressure Vessel and Piping Components" Ed. R. Seshardi, ASME PVP Vol. 136,1988. [2] A. Kalnins & D. P. Updike, "Role of plastic limit and elastic plastic analyses in design", ASME PVP-Vol. 210-2 Codes and Standards and Applications for Design and Analysis of Pressure Vessel & Piping Components, Ed. R. Seshardi & J. T. Boyle, 1991. [3] A. Kalnins & D. P. Updike, "Primary stress limits ion the basis of plasticity", ASME PVP-Vol. 230, Stress Classification, Robust Methods and Elevated Temperature Design, Ed. R. Seshardi & D. L. Marriott, 1992. [4] A. Kalnins, D. P. Updike & J. L. Hechmer, "On Primary Stress in Reducers", ASME PVP-Vol. 210-2, pp. 117-124 [5] D. Mackenzie & J. T. Boyle, "Stress Classification: A Way Forward", IMechE Presentation 5.5.92 [6] T.P. Pastor & J.L. Hechmer: “ASME task group report on primary stress” Proc. ASME PVP Conf., 1994, Minneapolis, 277, 67-78.

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2. Design by Analysis The aim of this section is to summarise issues related to the current use of design by analysis in order to put the new European rules in context. The concept of design by analysis was first formulated in the US ASME Pressure Vessel and Boiler Code in the early 1960’s; with almost forty years of use various critical problem areas have arisen, most of which have been addressed in the new European rules. These problem areas are discussed in the following since they highlight implicit difficulties with an apparently simple and straightforward set of design rules. In the following the approach devised by ASME is briefly summarised, followed by a description of the usual methods by which the rules are implemented and a discussion of the problem areas which arise. After this the differences in implementation of design by analysis rules in the European Standard are described. 2.1 Design by analysis: the current Stress Categarisation route The design by analysis procedure is intended to guard against eight possible pressure vessel failure modes by performing a detailed stress analysis of the vessel. The failure modes considered are: 1. Excessive elastic deformation including elastic instability. 2. Excessive plastic deformation. 3. Brittle fracture. 4. Stress rupture/creep deformation (inelastic). 5. Plastic instability - incremental collapse. 6. High strain - low cycle fatigue. 7. Stress corrosion. 8. Corrosion fatigue. Most of the design by analysis guidelines given in the codes relates to design based on elastic analysis – this is the so-called elastic route. Essentially it was recognised when the rules were being developed that only elastic stress analysis was feasible. In the 1960s, most designers were restricted to linear elastic stress analysis, and in the case of pressure vessel design most analysis was defined in terms of elastic shell discontinuity theory (also known as the influence function method). The nature of elastic shell analysis impinges significantly upon the way the above failure modes are treated in the Code. Thus, rules were developed to help the designer guard against the various failure mechanisms using elastic analysis alone. These guidelines guard against three specific failure modes - gross plastic deformation, incremental plastic collapse (ratchetting) and fatigue. These failure modes are precluded by failure criteria based on limit theory, shakedown theory and fatigue theory respectively. It is essential to appreciate at the beginning, the excessive plastic deformation and incremental plastic collapse cannot be dealt with simply in an elastic analysis, as the failure mechanism is inelastic. In addition, the type of loading causing the stress can significantly affect the level of permissible stress. Ideally, these inelastic failure modes should be assessed by an appropriate analysis which adequately models the mechanism of failure. In this approach the designer is required to classify the calculated stress into primary, secondary and peak categories and apply specified allowable stress limits. The magnitude of the allowable values assigned to the various stress categories reflect the nature of their associated failure mechanisms, therefore it is essential that the categorisation procedure is performed correctly.

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Stress categorisation (sometimes, classification) is probably the most difficult aspect of the design by analysis procedure and, paradoxically, the problem has become more difficult as stress analysis techniques have improved. When the design by analysis procedure was introduced, the dominant analysis technique in pressure vessel design was thin shell discontinuity analysis or the influence function method. This is reflected in the definitions of stress categories given in the Codes, which are based on the assumption of shell theory stress distributions; membrane and bending stress. It is therefore difficult to equate the calculated stresses and the code categories unless the design is based on shell analysis. The various stress categories are described first in the following: 2.1.1 Stress Categories The object of the elastic analysis is to ensure that the vessel has adequate margins of safety against three failure modes: gross plastic deformation, ratchetting and fatigue. This is done by defining three classes or categories of stress, which have different significance when the failure modes are considered. These three stress categories are assigned different maximum allowable stress values in the code: the designer is required to decompose the elastic stress field into these three categories and apply the appropriate stress limits. The total elastic stress which occurs in the vessel shell is considered to be composed of three different types of stress primary, secondary and peak. In addition, primary stress has three specific sub-categories. The ASME stress categories and the symbols used to denote them in the code are given below; (1) Primary Stress General Primary Membrane Stress, Pm Local Primary Membrane Stress, PL Primary Bending Stress, Pb (2) Secondary Stress, Q (3) Peak Stress, F and depend on location, origin and type. Before we can give a proper definition of these stresses, we must first give some terminology: Gross Structural Discontinuity: A gross structural discontinuity is a source of stress or strain intensification that affects a relatively large portion of a structure and has a significant effect on the overall stress or strain pattern or on the structure as a whole. Examples of gross structural discontinuities are: ∗ end to shell junctions, ∗ junctions between shells of different diameters or thickness, ∗ nozzles. Local Structural Discontinuity: A local structural discontinuity is a source of strain intensification that affects a relatively small volume of material and does not have a significant effect on the overall stress or strain pattern or on the structure as a whole. Examples of local structural discontinuities are:

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small fillet radii, small attachments, partial penetration welds.

Normal Stress: The normal stress is the component of stress normal to the plane of reference; this is also referred to as direct stress. Usually the distribution of normal stress is not uniform through the thickness of a part, so this stress is considered to be made up in turn of two components one of which is uniformly distributed and equal to the average value of stress across the thickness of the section under consideration, and the other of which varies with the location across the thickness. Shear Stress: The shear stress is the component of stress acting in the plane of reference. Membrane Stress: The membrane stress is the component of stress that is uniformly distributed and equal to the average value of stress across the thickness of the section under consideration. Bending Stress: The bending stress is the component of stress that varies linearly across the thickness of section under consideration. With this terminology as background, we now can define primary, secondary and peak stresses properly. Primary Stresses: A primary stress is a stress produced by mechanical loading only and is so distributed in the structure that no redistribution of load occurs as a result of yielding. It is a normal stress or a shear stress developed by the imposed loading, that is necessary to satisfy the simple laws of equilibrium of external and internal forces and moments. The basic characteristic of this stress is that it is not self-limiting. Primary stresses that considerably exceed the yield strength will result in failure, or at least in gross distortion. A thermal stress is not classified as a primary stress. Primary stresses are divided into ‘general’ and ‘local’ categories. The local primary stress is defined hereafter. Typical examples of general primary stress are: ∗ The average stress in a cylindrical or spherical shell due to internal pressure or to distributed live loads, ∗ The bending stress of a flat cover without supporting moment at the periphery due to internal pressure. Primary Local Membrane Stress: Cases arise in which a membrane stress produced by pressure or other mechanical loading and associated with a primary together with a discontinuity effect produces excessive distortion in the transfer of load to other portions of the structure. Conservatism requires that such a stress be classified as a primary local membrane stress even though it has some characteristics of a secondary stress. A stressed region may be considered as local if the distance over which the stress intensity exceeds 110% of the allowable general primary membrane stress does not extend in the meridional direction more than 0.5 times (according to BS5500 - 1 time according to ASME and CODAP) the square root of R times e and if it is not

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closer in the meridional direction than 2.5 times the square root of R times e to another region where the limits of general primary membrane stress are exceeded. R and e are respectively the radius and thickness of the component. An example of a primary local stress is the membrane stress in a shell produced by external load and moment at a permanent support or at a nozzle connection. Secondary Stresses: Secondary stresses are stresses developed by constraints due to geometric discontinuities, by the use of materials of different elastic moduli under external loads, or by constraints due to differential thermal expansion. The basic characteristic of secondary stress is that it is self-limiting. Local yielding and minor distortions can satisfy the conditions that cause the stress to occur and failure from one application of the stress is not to be expected. Examples of secondary stresses are the bending stresses at dished end to shell junctions, general thermal stresses. Peak stresses: Peak stress is that increment of stress which is additive to the primary-plussecondary stresses by reason of local discontinuities or local thermal stress including the effects (if any) of stress concentration. The basic characteristic of peak stresses is that they do not cause any noticeable distortion and are only important to fatigue and brittle fracture in conjunction with primary and secondary stresses. A typical example is the stress at the weld toe. 2.1.2 Stress intensity Pressure vessels are subject to multiaxial stress states, such that yield is not governed by the individual components of stress but by some combination of all stress components. Most Design by Formula rules make use of the Tresca criterion but in the DBA approach a more accurate representation of multiaxial yield is required. The theories most commonly used to relate multiaxial stress to uniaxial yield data are the Mises criterion and the Tresca criterion. ASME chose the Tresca criterion for use in the design rules since it is a little more conservative than Mises and sometimes easier to apply. For simplicity we will consider a general three-dimensional stress field described by its principal stress components, which we will denote σ1, σ2 and σ3, and define the principal shear stresses: 1 τ 1 = (σ 2 − σ 3 ) 2

1 τ 2 = (σ 3 − σ 1 ) 2

1 τ 3 = (σ 1 − σ 2 ) 2

According to the Tresca criterion yielding occurs when 1 τ = max(τ 1, τ 2 , τ 3 ) = σ Y 2 where σY is the uniaxial yield stress obtained from tensile tests.

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In order to avoid the unfamiliar (and unnecessary) operation of dividing both calculated and yield stress by two, a new term called "equivalent intensity of combined stress" or simply Stress Intensity was defined: Stress differences, S12, S23 and S31 are equated to twice the principal shear stress given above, such that: S12 = (σ 1 − σ 2 ) = τ 3

S23 = (σ 2 − σ 3 ) = τ 1

S31 = (σ 3 − σ 1 ) = τ 2

The Stress Intensity, S is then defined as the maximum absolute value of the stress differences, that is S = max (|S12|, |S23|, |S31|), so that the Tresca criterion reduces to: S = σY Once an analysis has been performed, the Stress Intensity for each stress category is evaluated and used in the design stress limits. 2.1.3 Stress limits The primary stress limits are provided to prevent excessive plastic deformation and provide a factor of safety on the ductile burst pressure (ductile rupture) or plastic instability (collapse). The primaryplus-secondary stress limits are provided to prevent progressive plastic deformation leading to collapse, and to validate the application of elastic analysis when performing the fatigue analysis. The allowable stresses in the Codes are expressed in terms of design stress Sm. The tabulated values of Sm given in the Code are based on consideration of both the yield stress and ultimate tensile strength of the material. Sm is notionally two-thirds of the "design" yield strength σY. Code allowable stresses for primary and secondary stress combinations are shown in the following table in terms of both Sm and σY.

ALLOWABLE STRESS

STRESS INTENSITY 2/3 k

σY

General primary membrane, Pm

k Sm

Local primary membrane, PL

1.5 k Sm

k σY

Primary membrane plus bending

1.5 k Sm

k σY

3 Sm

2 σY

(Pm + PB) or (PL + Pb) Primary plus secondary (Pm + PB + Q) or (PL + Pb + Q)

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In addition to these allowables, when fatigue is considered relevant the total sum of (PL+Pb+Q+F) should be less than an allowable fatigue stress intensity range, Sa. The value of the k factor depends on the load combinations experienced by the vessel. For load combinations including design pressure, the dead load of the vessel, the contents of the vessel, the imposed load of the mechanical equipment and external attachment loads the k factor has a value of 1. When earthquake, wind load or wave load are added to the above, a k value of 1.2 is used. Special limits are also stipulated for hydraulic testing. Under design load conditions k = 1 and the maximum value of the primary stress combinations is yield the yield stress of the material. Primary stress is yield limited to ensure gross plastic deformation does not occur. The primary plus secondary stress combinations have a much higher allowable stress: twice the yield stress of the material. Primary plus secondary stress is limited to ensure shakedown of the vessel. Because of the different allowable values for primary and primary plus secondary stress, it is essential that the calculated elastic stress is correctly decomposed into the various categories. This is one of the most difficult problems encountered in DBA and has potentially critical effect on the final design. If primary stresses are classified as secondary the design may be unsafe, whilst if secondary stresses are classified as primary the design will be over-conservative. The code provides explicit classification guidance for certain typical vessel geometries and load through Table 4.120.1 Classification of stresses for some typical cases. In situations other than these cases the designer must rely on the basic code definitions of primary, secondary and peak stress and his own judgement to properly classify the elastic stress. In fact some of the stress classifications recommended in Table 4.120.1 have been in doubt for some time, and must be used with care.

2.2 Design by Analysis: the ASME inelastic route The ASME VIII Division 2 rules for inelastic analysis are given in Appendix 4-136 Applications of Plastic Analysis. These rules “provide guidance in the application of plastic analysis and some relaxation of the basic stress limits which are allowed if plastic analysis is used.” The rules for inelastic analysis considered here pertain to calculation of permissible loads for gross plastic deformation only. Rules are given in the Code for shakedown analysis but in practice shakedown analysis is difficult and it is simpler to apply the 3Sm limit to an elastic analysis. Two types of analysis may be used to calculate allowable loads for gross plastic deformation: limit analysis and plastic analysis. Limit analysis is used to calculate the limit load of a vessel. By definition, the analysis is based on small deformation theory and an elastic-perfectly plastic (or rigid-perfectly plastic) material model. Plastic analysis is used to determine the plastic collapse load of a vessel. The analysis is based on a model of the actual material stress-strain relationship and may assume small or large deformation theory as required. Material models can vary in complexity (or degree of approximation) from simple bilinear kinematic hardening models to more complex curves defining the actual stress-strain curve in a piecewise continuous manner. Including strain hardening in the analysis may give a higher plastic collapse load than the limit load but in the design by analysis procedure the allowable load is dependent on the criterion of plastic

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collapse used. Including large deformation effects in the analysis may increase or decrease the calculated allowable load depending on the geometry of the vessel. Some structural configurations exhibit geometrical strengthening when non-linear geometry is considered whilst others exhibit geometric weakening. The expression ‘plastic collapse load’ is to some extent a misnomer, as a real vessel may not physically collapse at this load level, hence the ‘plastic collapse load’ is often referred to simply as the ‘plastic load’. 2.2.1 Limit analysis ASME VIII Division 2 Appendix 4-136.3 Limit Analysis states: “The limits on general membrane stress intensity ...local membrane stress intensity ... and primary membrane plus primary bending stress intensity ... need not be satisfied at a specific location if it can be shown by limit analysis that the specified loadings do not exceed two-thirds of the lower bound collapse load. The yield strength to be used in these calculations is 1.5Sm.” Thus allowable load Pa is Pa =

2 PLim where PLim is the limit load of the vessel. 3

Clearly, if the limit load can be calculated this procedure is much simpler to apply than the elastic analysis stress categorisation procedure. However, there are two additional requirements that must be satisfied when applying this approach. Firstly, the effects of plastic strain concentrations in localised areas of the structure such as points where plastic hinges form must be assessed in light of possible fatigue, ratchetting and buckling failure. Secondly, the design must satisfy the minimum wall thickness requirements given in the design by rule section of the Code. In effect, the design by rule formulae for wall thickness have priority over design by analysis calculations. 2.2.2 Plastic analysis ASME VIII Division 2 Appendix 4-136.5 Plastic Analysis states: “The limits of general membrane stress intensity ...local membrane stress intensity ... and primary membrane plus primary bending stress intensity ... need not be satisfied at a specific location if it can be shown by limit analysis that the specified loadings do not exceed two-thirds of the plastic analysis collapse load determined by application of 6-153, Criterion of Collapse Load (Appendix 6) [Mandatory Experimental Stress Analysis], to a load deflection or load strain relationship obtained by plastic analysis.” Thus allowable load Pa is Pa =

2 PP , where PP is the plastic load of the vessel. 3

Calculating plastic loads is more problematic than calculating limit loads as no rigorous definition of what constitutes a plastic load is given. Instead, the twice elastic slope criterion as used in experimental analysis is prescribed.

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2.3 Analysis methods for the ASME approach Design by analysis procedures do not specify particular implementation tools: it has been left to the analysts to choose the technique they feel most appropriate. Shell discontinuity analysis was the primary tool in the early days of design by analysis, where stresses could easily be categorised in terms of shell-type membrane and bending stress. By now analysis techniques have developed significantly and although shell discontinuity analysis is still used very often in structural analysis, it is replaced more and more by computer based numerical methods. The most widely used technique in contemporary pressure vessel design is the finite element method, a powerful technique allowing the detailed modelling of complex vessels. Shell discontinuity analysis and the finite element method are discussed in relation to pressure vessel design by analysis in the following sections. 2.3.1 Shell discontinuity analysis Shell discontinuity analysis was the primary means of stress analysis in the early days of design by analysis procedures. Although largely replaced by finite element analysis, shell discontinuity analysis remains a useful tool for simple geometries, and indeed many engineering software companies still supply programs for discontinuity analysis. V

Forces

H

u q Semi-infinite cylinder

Cone

Displacements

v

Hemisphere

Flat end

Figure 2.1.: Shell discontinuity forces and moments.

Shell discontinuity analysis is primarily used to evaluate shell membrane and bending stresses for axisymmetric vessels under internal pressure. It makes use of the fact that typical vessel configurations are composed of regular parts - spheres, cylinders, cones and flat ends in particular. For pressure loading, simple regular shapes exhibit mainly membrane stress. However, at junctions local bending (and additional membrane) stresses are generated. These stresses are called discontinuity stresses for obvious reasons. Shell discontinuity analysis allows these junction

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stresses, and their effect in the vessel, to be readily calculated using a simple engineering force method. This force method uses analytical solutions for the local bending and shear stress close to junctions which allow so-called edge forces and moments to be related to edge displacements and rotations, Figure 2.1. These edge relations are evaluated for each part of the vessel and then assembled at junctions. Continuity of displacement and rotation between parts then allows the edge forces and moments at the junction to be derived and finally the stresses in the various parts can be calculated.

2.3.2 Finite Elements for Pressure Vessel Analysis In creating a model, element selection and mesh definition are crucial aspects of finite element analysis. The type of element used in a finite element analysis for pressure vessel design can greatly influence the design procedure, so a brief overview is given here. Most commercial programs include large finite element libraries, however, in pressure vessel design the most common element types are 3-D solid, axisymmetric and shell elements. 2.3.2.1 3-D Solid Elements Solid (or continuum) elements are based on the mathematical theory of elasticity, which describes the behaviour of a deformable component under load assuming small deformation and strain. The most general theory is three dimensional, but under specific circumstances certain two dimensional reductions are possible. 3-D solid elements are used to model real three-dimensional structures such as the part model of a nozzle-vessel intersection shown in Figure 2.2.

Figure 2.2: 3-D solid model of a nozzle-vessel intersection.

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Elastic 3-D solid elements are based on 3-D elasticity theory. A general system of forces acting on a three dimensional elastic body sets up internal forces within the body, which vary with position throughout the body. The state of stress at a point in the body is fully defined by six components: Direct stresses: σx, σy, σz Shear stresses: τxy, τyz, τzx σY

as illustrated in Figure 2.3. Three degrees of freedom are defined at each node of a 3-D solid element: orthogonal displacements ux, uy and uz. Displacement throughout the domain of the element is defined in terms of these nodal displacements by the interpolation functions used in the element formulation. Most commercial finite element packages offer solid elements based on two different orders of interpolation: •

τ XY τYZ

τ XY σX

τYZ σZ Y

Z

τ ZX

τZX

X

Figure 2.3: Stresses acting on a differential cube of 3-D elastic continua.

8 node linear element: Figure 2.4.

Each element has 24 (8 node x 3) associated degrees of freedom.

uy

8 NODE LINEAR

ux uz

ISOPARAMETRIC 3-D SOLID BRICK ELEMENT

Deformed geometry

Y X Z

Original geometry

ELEMENT DEFORMATION

Figure 2.4: Linear 3-D solid brick element

Edges/sides must be straight/ plane before deformation, as the geometry is defined by linear equations. When the element is loaded it deforms such that the sides remain straight, as the displacements of the element are also defined by linear equations.

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2.11

20 node quadratic element: Figure 2.5.

ORIGINAL GEOMETRY STRAIGHT SIDES

Deformed geometry

20 NODE QUADRATIC ISOPARAMETRIC 3-D SOLID BRICK ELEMENT

Original geometry

ELEMENT DEFORMATION

ORIGINAL GEOMETRY QUADRATICALLY CURVED SIDES

Figure 2.5: Quadratic 3-D solid brick element.

Each element has 60 (20 nodes x 3) associated degrees of freedom. Edges/sides may be defined as quadratic curves/surfaces, as the geometry of the element is defined by quadratic interpolation. This means that the 20 node brick can more closely model the true shape of a curved body than the simpler 8 node element. When the 20 node element is loaded the sides may deform quadratically, as the displacements of the element are also defined by quadratic interpolation. Both 8 node and 20 node brick elements may be degenerated to give elements of wedge and tetrahedron shape by defining two or more nodes at the same position, as shown for the 8 node element in Figure 2.6. In general, degenerate elements do not perform as well as the brick elements, however they can be used to model areas of a structure which cannot be meshed using brickshaped elements. Solid elements based on tetrahedral geometry (as opposed to being degenerate bricks) are also available in many finite element programs.

4 NODES

BRICK

2 NODES 2 NODES

TETRAHEDRON

WEDGE OR PRISM

Figure 2.6: Degenerating brick geometry to give wedge and tetrahedron geometry

3-D solid elements can, hypothetically, be used to model any type of structure but well-designed 3-D solid models are usually large in terms of computing requirements. Hardware limitations etc. tend to restrict their use to situations where simplified models (as discussed below) are not viable; for example, thick non-axisymmetric vessels, thick axisymmetric vessels under non-axisymmetric boundary conditions (loading and support) or perhaps thinner vessels with unusual or significant geometric details. Modern finite element programs make it relatively simple (although perhaps time-consuming) to create complex 3-D models of pressure vessels. The most significant problem in practical design by analysis using solid models is not in model creation but interpretation of the results of the analysis in the light of code requirements.

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As stated above, solid elements are based on elasticity theory, in which stress at a point is defined in terms of six stress components: σx, σy, σz, τxy, τyz, τzx. These stresses vary continuously throughout the body and in thick pressure components the through-wall distribution is non-linear. This form of stress distribution is significantly different from that envisaged when design by analysis was first implemented, which implicitly assumed a linear through thickness, shell-type stress distribution that could be decomposed into membrane and bending constituents. This incompatibility in format between the stresses calculated in the solid model and those required for design by analysis procedures often makes it extremely difficult for the designer to classify the calculated stresses as primary, secondary and peak and apply the appropriate category stress limits. 2.3.2.2 Axisymmetric Elements Whilst, ideally, all three dimensional structures can be modelled using 3-D solid elements to a greater or lesser degree of accuracy, it is not always necessary to perform a complete three dimensional analysis. In certain classes of structure, advantage can be taken of the structure geometry and loading to reduce the problem to two dimensions. Such structures can be analysed using much simpler and smaller finite element models.

Y

r

The most useful class of three-dimensional vessel which can be analysed using two-dimensional elements is the body of revolution. Here geometric and material properties are symmetric about the symmetry axis subject to loading symmetric about the symmetry axis, as illustrated for an example structure in Figure 2.7.

X

Z

Figure 2.7: Axisymmetric structure.

This type of structure is called an axisymmetric structure. In finite element practice the component geometry is defined so that the Y axis is the axis of rotational symmetry. For convenience, the geometry, loading, stresses and strains of the component are defined in polar co-ordinates: distance r in the radial direction R from the origin, circumferential or hoop position θ in the circumferential direction Θ and meridional or height position y in the axial direction Y. When a rotationally symmetric body is loaded symmetrically about the symmetry axis - that is, loads are applied radially or vertically and uniformly with respect to circumferential position, as illustrated in Figure 2.8, points in the body can move radially in or out and vertically up or down. The material does not move sideways, in the Θ (hoop) direction, nor rotate as there are no loads causing the body to deform in this manner.

ELEVATION PLAN

Figure 2.8: Axisymmetric loading

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As the points on any R-Y plane can move only in that plane, they have two degrees of freedom: radial displacement ur and vertical displacement uy. As the behaviour will be the same at all R-Y planes in the body, the behaviour of the entire component is fully defined if the behaviour of any such plane is defined. Consequently, in an axisymmetric analysis only a single 2-D section of body need be analysed. In finite element practice, axisymmetric models are created in the global X-Y plane, as shown in Figure 2.9.

Y uy ux

X Figure 2.9: Axisymmetric degrees of freedom

The state of strain at a point in an axisymmetric body may be defined by considering the deformation defined above. As deformation is symmetric about the Y axis, no strains are present which would give rise to nonsymmetrical deformation. Thus, no shear strains can arise perpendicular to the X-Y plane. Under this condition, the number of stresses at a point reduces from six (3-D) to four: σx, σy, σθ, τry, as illustrated in Figure 2.10.

Clearly, the form of stress distribution calculated in an axisymmetric solid analysis is similar to that calculated in 3-D analysis. Consequently, the same incompatibility exists between the form of calculated stress results and the form required by design by analysis rules as discussed for 3-D analysis above. Y u2 u1

σθ

σy τ ry

X

σr σθ

σr σy

Figure 2.10: Axisymmetric element stresses.

A range of axisymmetric solid elements are available in most commercial finite element programs, as illustrated in Figure 2.11.

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The linear quadrilateral element has four nodes, each with two degrees of freedom. Thus, each linear element has 8 associated degrees of freedom, compared with 24 for a linear 3-D solid. The quadratic element has eight nodes with two degrees of freedom. Thus, each quadratic element has 16 associated degrees of freedom, compared with 60 for a linear 3-D solid. Clearly, the use of Y

LINEAR TRIANGLE (CONSTANT STRAIN TRIANGLE)

r X Z QUADRATIC TRIANGLE

LINEAR QUAD

QUADRATIC QUAD

Figure 2.11: Typical elements

for axisymmetric

analysis. axisymmetric elements leads to smaller models in terms of degrees or freedom or, if preferred, permits a finer mesh for the same model size. An axisymmetric model of the 3-D nozzle intersection shown in Figure 2.2 is shown in Figure 2.12. The use of axisymmetric elements allows the analyst to produce a finer mesh through the thickness of the vessel wall without creating an excessively large model. Care must be taken when defining loads for axisymmetric models. Forces may be defined as a total applied force or on a per radian basis. The program user’s manual should be consulted to check the situation for the software in use.

Y

It is worth noting that many pressure vessel problems relate to axisymmetric structures Figure 2.12: Axisymmetric model of nozzle intersection. under non-axisymmetric loading. For linear elastic analysis, it is possible to treat this as an axisymmetric problem, and model the loading using Fourier series around the circumference. Some commercial finite element software offers this capability through modifications to the basic axisymmetric element, known as an harmonic element. X

Solid elements are extremely versatile, powerful, and suitable for a wide range of applications. However, there are two significant factors which limit the use of solid elements in practical finite element analysis, particularly for pressure vessel problems. These are:

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•

The aspect ratio of solid elements should ideally be 1 - that is, the element is a cube - but in practice limited to 2 in the case of linear elements and 5 in the case of quadratic elements.

•

Solid elements do not respond well to bending loads, and at least three linear or two quadratic elements must be used through thickness when bending is present.

Taken together, it becomes clear that these factors often make it impracticable to model thin shell structures such as the longitudinally supported vessel shown in Figure 2.13 using solid elements.

Figure 2.13: Shell model of vessel.

In general, shell structures are thin in one direction and carry both membrane (in-plane) and bending (out-of-plane) loads. The load-carrying capacity of a shell mainly is derived from its membrane strength, but it is impossible to construct real shell structures, such as pressure vessels, without inherent bending during loading - for example, the junction between a cylinder and a spherical end cap gives a discontinuity in curvature which induces a bending stress (called a discontinuity stress). Bending stresses can also result from mechanical and thermal loading - for example, piping forces on a nozzle. Several solid elements are required through the shell thickness to adequately represent bending behaviour but these elements cannot themselves be thin or they will violate aspect ratio requirements of the formulation. Consequently, a large number of solid elements are required in order to model even simple shell structures. These shell analysis problems are avoided by using special shell elements, which incorporate assumptions about the nature of the bending in the formulation.

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2.3.2.3 Thin Shell Elements The traditional method of analysing shell structures is to simplify the behaviour of the structure by assuming an appropriate thin shell theory in which the behaviour of the three dimensional structure is described in terms of the deformation of a doubly curved reference surface. This reduces the number of degrees of freedom required to model the real behaviour of the structure as only one element is used through thickness. In addition, the definition of the finite element model is also considerably simplified, as only the mid-surface has to be defined and meshed with surface or area elements. However, the reduction of a real three dimensional shell structure to a reference surface model is considerably more complicated than reducing three dimensional elasticity to axisymmetry or plane strain, as discussed earlier. In general, shell structures are doubly curved and the radii of curvature are not constant throughout the body. Shell structures support both membrane forces, which act in the plane of the shell, and bending forces or moments which arise due to out-of-plane loading. These are shown diagrammatically in Figure 2.14.

MEMBRANE MODES

BENDING MODES

The membrane and bending forces are Figure 2.14: Thin shell bending & membrane modes. coupled throughout the shell, but their relative magnitude differs with position in the structure. In certain locations membrane action may predominate, whilst in other locations, most notably at structural supports and discontinuities, bending becomes more significant. In flat plates and in the theory of shallow shells, membrane and bending actions can be sensibly de-coupled - this simplification allows a simpler element formulation, and for this reason flat plate elements are commonly used in the analysis of general shell structures. It is possible to simplify the real three-dimensional problem if certain assumptions are made about how the thin plate or shell deforms, particularly during bending. The most significant assumption of thin plate or shell theory is that straight lines initially perpendicular to the mid-surface remain straight during deformation, and follow either the Kirchhoff hypothesis or the Mindlin hypothesis. In the Kirchhoff hypothesis, when the shell deforms, straight lines normal to the mid-surface rotate, but so that they remain straight and normal to the deformed mid-surface, as illustrated in Figure 2.15. In the less restrictive (but more complex) Mindlin Hypothesis these lines are not required to remain normal during deformation, such that the influence of shear strain can be better represented.

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PLANE SECTIONS REMAIN PLANE AND NORMAL TO THE NEUTRAL AXIS

KIRCHHOFF HYPOTHESIS

PLANE SECTIONS REMAIN PLANE BUT NOT NECESSARILY NORMAL TO THE NEUTRAL AXIS

MINDLIN HYPOTHESIS

Figure 2.15: Kirchhoff and Mindlin Hypotheses

The Kirchhoff hypothesis is common in shell theory, and most published results for thin pressure vessels are based on this. However, some modern finite element formulations for thin shells use the Mindlin hypothesis, (essentially since numerical analysis is used and there is no need to be so restrictive). In addition, the Mindlin hypothesis represents shell behaviour more accurately in the vicinity of discontinuities and restraints, where transverse shear effects are more significant. The effect of using one of these simplifying hypotheses in the finite element method is that the deformation at any point can be defined if the displacement and rotation of the mid-surface are known. The rotation at any point on the mid-surface is defined by interpolating between rotational degrees of freedom defined at the nodes. Therefore beam, plate and shell elements have translational and rotational degrees of freedom. Only these bending elements have rotational degrees of freedom: solid elements do not need rotational degrees of freedom to define the element deformation (although it is possible to formulate solid elements which include rotational degrees of freedom to enhance performance but at the cost of increasing the number of degrees of freedom per element). A great deal of work has been undertaken on formulating shell elements, and it is an indication of the complexity of the problem that no single type of formulation has been universally accepted as being the best. Classical shell theory produces equations which are difficult to solve and which are remarkably sensitive to slight variations in shape (which are common in the approximate finite element method). A large number of different approaches have been developed over the years but basically only three types of shell element are used in practice: •

Facet (flat) shell elements, formed by combining membrane and plate bending elements

•

Curved shell elements, based on classical shell theory

•

Reduced (or degenerate) solid (continuum) iso-parametric elements which directly take account of thinness and the Mindlin hypothesis in their formulation

The most popular are flat elements and reduced solid elements, both of which appear in commercial software.

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Flat (Plate) Elements The geometry of a doubly curved shell surface can be approximated by a faceted surface formed by connecting flat triangular elements together at their vertices. A flat three-noded triangular shell element would have six degrees of freedom per node: in an element co-ordinate system (x,y,z), as shown in Figure 2.16, there would be three translational degrees of freedom, (ux, uy, uz), and three rotational degrees of freedom, (φx, φy, φz), giving a total of eighteen degrees of freedom per element. This element can be used to represent a shell by including both membrane, (ux, uy, φz), and bending, (uz, φx, φy), degrees of freedom. Membrane stiffness is derived from simple plane stress conditions, with the added drilling degree of freedom, φz. The most significant aspect of the derived shell element is that the membrane and bending stiffness are uncoupled, although there is a degree of coupling when the elements are assembled. uz Y

Z

uy ux

K

Z Y

X

I X

J Figure 2.16: Triangular flat plate element

Used on their own, it has been found that triangular shell elements based on plate bending elements do not perform very well, having an artificially high bending stiffness and spurious torsion modes. Other flat shell elements have been formulated, among the most common of which is the BatozRazzaque element. This is a quadrilateral element formed from four flat shell elements such that the diagonals are continuous, Figure 2.17. This formulation works well and can be found in many commercial programs.

Figure 2.17: Batoz Razzaque quadrilateral flat shell element

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Reduced Continuum Elements The reduced continuum shell elements are similar to the 3-D solid elements discussed above but with the Mindlin hypothesis (plane sections remain plane or linear interpolation) applied through the thickness. Under this condition, if the deformation of the element mid-surface is known, the deformation at any other point can be defined through the element shape functions. In this way, the 3-D element is reduced to a shell-type area element. Such elements are usually called reduced continuum, or degenerate solid, or sometimes thick shell, elements. In practice, the element’s curved mid-surface is approximated from the given nodal co-ordinates and this can affect the performance and accuracy of this type of element if it has a poor shape. 2.3.2.4 Discussion At first sight the most appropriate choice of finite element may seem obvious for a given vessel under consideration. However when the choice is examined in the light of the pressure vessel design by analysis elastic route, where limits are placed on membrane and bending stress and stresses, or indeed parts of stresses, must be categorised, various well-known difficulties arise. These problem areas lie at the heart of criticism of the ASME design by analysis rules and consequently are discussed in more detail in the following: 2.4 Implementation Problems of the Stress Categorisation Route 2.4.1 Overview of Problems Once the linear elastic analysis of a part is complete and the immediate results for stresses and strains obtained, there is the need to satisfy the design by analysis rules. As mentioned previously this is not necessarily as straightforward as it may at first seem. Specifically, there is a requirement to obtain membrane and bending components of primary stress and the calculated stresses must be categorised. This does not present a problem in cases where the analysis utilises thin shells. However, for analysis (in particular finite element analysis) utilising solid models (2 or 3 dimensional) where the calculated stress can not be easily identified as membrane, bending or peak the problems of linearisation and categorisation become apparent. Difficulties implementing this area of the design by analysis rules have become increasingly evident to both designers and analysts[1]. This section examines the practical problems associated with the implementation of these design by analysis rules. 2.4.2 Linearisation The design by analysis criteria, as formulated nearly thirty years ago, is based on the behaviour of thin shells and includes the notion of membrane and bending stress. Inherent in this understanding is the assumption that membrane and bending stress act on a plane under the Kirchhoff hypothesis that plane sections remain plane during bending. The shell type membrane and bending stresses cause gross distortions under primary loads and strain enhancement under secondary loads. Most of our understanding of basic pressure vessel geometry and components come from our knowledge of their behaviour as thin shells. A consequence of this understanding is the possibility that portions of total stress, identified as membrane or bending (or peak) can be categorised as primary or secondary.

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For example, in the case of a nozzle in a spherical shell (Figure 2.18) with area compensated reinforcement, only the membrane stress is primary, despite significant bending in the shell close to the nozzle. The bending stress is secondary (since only the membrane stress and the hoop stress in the nozzle are required to satisfy equilibrium with the internal pressure). In this case, it is essential to consider membrane and bending stress for the correct categorisation.

2.20

Reinforcement

M R BENDING

t

p Pm = pr t MEMBRANE

r

Pm is PRIMARY M R is SECONDARY

Figure 2.18: Stress categories for a nozzle.

l

z

o

p

If the analysis is based on thin shell finite elements, then there is no difficulty in identifying membrane and bending stress, as they are part of the underlying theory, Figure 2.19.

TOP

y x

k

m

n

i TOP TOP MID

BOT MID

BOT

x

y

Difficulties arise when thin shell analysis is not used and the finite element analysis is based on axisymmetric or three-dimensional solid elements. Z X In general, unless the section is indeed thin, the Figure 2.19: Shell membrane and bending stress stresses on a through thickness line are not linear, and further plane sections do not remain plane during bending. Over the years it has become common practice to linearise the calculated through thickness stresses in order to separate membrane and bending components. j

Y

BOTTOM

A technique for linearising stress was first suggested by Kroenke[2,3], and has been adopted in several finite element postprocessors. A stress classification line (or plane) or supporting line segment is chosen and the stresses are linearised along this line. The supporting line segment (SLS) or classification line is the smallest segment joining the two sides of the wall where the stress is to be linearised. Outside of gross structural discontinuity regions, the SLS is normal to the wall mean surface, i.e. its length is equal to the thickness of the wall in the analysis. There are difficulties with this procedure – which seems straightforward, but again is a fundamental difficulty - this will be discussed in more detail in the following. Pressure vessel design codes are not particularly helpful on the problem of linearisation. ASME III & VIII admit a non-linear bending stress, but also contains some ambiguities: bending stress is described as a normal stress - and it is bending stress that may need linearisation. In Paragraph NB-3215 a note is provided to the effect that “.. membrane stress intensity is derived from the stress components averaged across the thickness of the section. The averaging shall be performed at the component level ...”. This implies that only stress components may be linearised (by definition this could include shear stress), and not derived principal values. However, through omission from the

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code, it may be argued that shear stress should not be linearised. Inclusion of shear stress linearisation will mostly affect the surface stress: in practice linearisation of the normal stress only is adopted to modify the surface stress in application of the design criteria. The basic procedure for stress linearisation is the selection of a stress classification plane or supporting line segment on which shell type stress will be evaluated. A stress classification line (or plane of referencesee page 31) is identified through a section and the non-linear stress distribution along this line is linearised in order to extract membrane and bending stress, as shown in Figure 2.20. The stress classification line (CL) lies along a local axis X3; the origin is located at the mid-point of the CL (i.e. at radius Rc); the abscissa of a point on the supporting line segment is designated x3.

R

X

c

3

Stress classification line or supporting line segment of length h.

Figure 2.20: Stress classification line

In practice, the linearisation procedure is performed automatically by special postprocessors. For simplicity, some basic postprocessors (in particular self written ones) may require the finite element mesh to be created so that a line of nodes lie along the chosen classification line, making it relatively simple to extract stress results. In postprocessors that are more complex, the classification line need not pass through a line STRESS CLASSIFICATION of nodes. LINE The classification line in Figure 2.21 is defined from node Ni at the inner surface to No at the No Ni outer. The path of the classification line does not pass through a line of nodes: it cuts through the elements. Advanced linearisation postprocessors use the location of the surface nodes to define the path through the elements and then apply interpolation functions to the appropriate nodal stresses to calculate the stress along the path. Two possible procedures Figure 2.21: Interpolation of a classification line for linearisation have been suggested, Kroenke which has been discussed by numerous analysts[4] and a more refined version by Gordon[5].

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2.4.2.1 Kroenke’s Procedure Kroenke’s procedure makes reference to familiar beam bending stress - a uniaxial stress - and attempts to define an equivalent linear stress distribution on the classification line (CL). Consider a typical stress distribution along a classification line as in Figure 2.22. If x3 measures local distance along the classification line then the equivalent linearised stress is,

(σ )

ij L

= ax3 + b .

The membrane stress component is given by the formula nonlinear stress

σ

distribution

σp

linearised stress

b

σb

a

σm classification line

X3 x3

e/2

e/2

Figure 2.22: Typical Stress Distribution

(σ ) ij

m= b =

1 e

e 2 e − 2

∫

σ ij dx3 .

The membrane force per unit length of the membrane stress component is equal to that from the calculated FE stress component. The bending stress component is given by

(σ )

ij b

= a ⋅ x3 =

12 x3 e3

e 2 e − 2

∫

σ ij x3 dx3 .

The maximum and minimum bending stresses can then be evaluated (for x3 = ± e / 2 )

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(σ )

ij b , s

6 =± 2 e

e 2 e − 2

∫

Page 2.23

σ ij x3 dx3

The bending moment per unit length of the calculated FE stress component is equal to

(σ )

ij b , s

⋅

e e2 = ∫ 2e σ ij x3 dx3 − 6 2

The linearised stress (σ ij )L is found by adding membrane and bending stresses

(σ ) = (σ ) + (σ ) ij L

ij m

ij b

The bending stress of this equivalent linear stress distribution vanishes at x3 = 0 . The peak value of stress at a point is the difference between the total stress and the sum of the membrane and bending stresses

(σ ) = (σ ) − (σ ) = (σ ) − [(σ ) + (σ ) ] . ij

p

ij

ij L

ij

ij m

ij b

2.4.2.2 Gordon’s Procedure– for axisymmetric problems In Kroenke’s procedure for axisymmetric problems, the shell wall is assumed (locally) straight in the meridional direction. In some circumstances the meridional curvature is finite. Gordon suggested a modification to Kroenke’s procedure to allow for this. Gordon’s procedure for an axisymmetric case is the same, in principle, as the case above, except for the fact that there is more material at a greater radius than at a smaller radius. The neutral axis is shifted radially outward to accommodate for this. Consider the axisymmetric section of a vessel wall as shown in Figure 2.23. ρ is defined as the radius of curvature of the mid-surface of the shell. In the case of an axisymmetric straight section such as a cylinder or cone, ρ = ∞. Adopting the following notation: ρ − radius of meridional curvature R1 – radius of circumferential curvature z – axial co-ordinate r – radial co-ordinate θ – angle in hoop direction φ – angle in meridional direction X3 – local co-ordinate containing the classification line X2 – local co-ordinate normal to classification line x3 – co-ordinate along classification line e – shell thickness

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R – radial co-ordinate of a point of the classification line Rc – radial co-ordinate of mid-surface point z R Rc X3 Axis of symm.

φ

X2

ρ e z

R1

r

centreline

θ

r

Figure 2.23: Geometry for finite curvature

From an axisymmetric analysis the following stresses would be obtained in the local classification line co-ordinates: σX2 - (local) meridional stress σX3 - (local) radial stress σθ - hoop stress σX2X3 - (local) shear stress The other shear stresses would be zero in an torsion-free axisymmetric analysis. The aim is to obtain membrane and bending components of these stresses, denoted by subscripts m and b respectively, evaluated from the average stress across the section and the beam type bending stress. The membrane component of the (local) meridional stress on the classification line is given by

(σ ) X2

m

=

FX 2 AX 2

=

e 2 e − 2

∫

σ X 2 ⋅ (R1 + x 3 ) ⋅ ∆θ ⋅ dx3 R1 ⋅ ∆θ ⋅ e

=

e 2 e − 2

∫

σ X 2 R ⋅ ∆θ ⋅ dx3 Rc ⋅ e ⋅ ∆θ

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where the area AX 2 of a small segment extending over the angle ∆θ in hoop direction is given by A X 2 = Rc ⋅ e ⋅ ∆θ . In this linearisation the bending stress component of the (local) meridional stress on the classification line vanishes at x3 = x f , where x f is the x3 - co-ordinate of the resultant of a constant stress distribution σ X 2 and of the centroid of the considered area. x f is given by xf =

e2 e 2 cosφ , = 12 R1 12 Rc

thus the bending stress component is given by

(σ )

X2 b

=

M X 2 ( x3 − x f ) Im

,

where M X 2 = ∫ 2e (x 3 − x f )σ X 2 R ⋅ ∆θ ⋅ dx3 e

−

2

and  e2  I m = Rc e ⋅ ∆θ ⋅  − x 2f  ,  12  which leads to

(σ )

X2 b

=

x3 − x f  e2 2 Rc ⋅ e − x f   12 

e 2

∫ (x

3

− x f )σ X 2 R ⋅ dx3 .

e − 2

The hoop stress is evaluated in a similar manner to the above; however in this case the meridional curvature, ρ must be taken into account: The membrane component of the hoop stress is given by

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(σ )

θ m

=

Fθ = Aθ

Design by Analysis e 2 e − 2

∫

σ θ ( ρ + x3 ) ⋅ ∆φ ⋅ dx3 =

ρ ⋅ e ⋅ ∆φ

e 2

1 σθ e ∫e −

Page 2.26

 x  ⋅ 1 + 3  dx3 ρ 

2

where the area Aθ of a small segment extending over die angle ∆φ in meridional direction is given by Aθ = ρ ⋅ e ⋅ ∆φ . In this linearisation the bending stress component of hoop stress on the classification line vanishes at x3 = xh , where x h is the x3 - co-ordinate of the resultant of a constant stress distribution σ θ and of the centroid of the considered area, where x h is given by xh =

e2 . 12 ⋅ ρ

Thus the bending stress component is given by

(σ θ )b

M θ ( x3 − x h ) , Ih

=

where e

M θ = ∫ 2e ( x3 − x h ) ⋅ σ θ ⋅ ( ρ + x 3 ) ⋅ ∆φ dx3 , −

2

and  e2  I h = ρ e ⋅ ∆ϕ ⋅  − x h2  ,  12  which leads to

(σ θ )b

=

x3 − x h e 2 e − x h   12  2

e 2

∫ (x −

e 2

3

 x  − x h )σ θ ⋅ 1 + 3  ⋅ dx3 . ρ 

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(Local) radial stress on the classification line is treated in a special way: in most situations the radial stress will equal the applied pressure at the internal surface and be free (zero) at the outer surface. Therefore, membrane stress may be evaluated,

(σ )

1 e2 X3 e σ X dx 3 e ∫− 2 3 but it is questionable whether a bending stress should be evaluated. Either this should be taken as zero = m

(σ )

= 0

X3 b

or as the simple difference between actual and averaged value

(σ ) =σ X3

b

X3

( )

− σ X3

m

which may not be linear. Similarly, an average membrane shear stress can be determined along the classification line

(σ

X 2X3

)

= m

1 2e e σ X X Rdx3 Rc e ∫− 2 2 3

Since the shear stress would be expected to be nearly parabolic (from basic elasticity theory), and zero at the surface, the bending stress should be taken as zero

(σ

X2X3

) =0 b

The surface values of shear therefore only contribute to peak stress. The development of Gordon’s procedure given here is in terms of stress components in a local coordinate system (X3, θ, X2). In practice these would be transformed into the global (r, θ, z) coordinate system, to give global linearised stress components. Once the global stress components have been linearised, the principle stresses and stress intensity can then be evaluated, as total values and as averaged membrane and surface bending stresses. 2.4.2.3 Discussion As mentioned previously, several finite element programs contain post processing options to directly calculate the equivalent linearised stresses on any prescribed classification line. Usually all stress components are linearised. A short consideration of the linearisation procedure immediately brings several possible problem areas to mind. .

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• Selecting the stress classification line. This should be a line through the vessel wall, which would be expected to yield shell type deformations, namely straight lines remaining straight. However close to discontinuities some warping and shear would be expected (and indeed observed in the finite element calculations) and the concept of averaged membrane and linearised bending stress is tenuous. Ideally, inner, outer surfaces, and, thus, the mid-surface should be Normal to LINE 1 Normal to Inner Surface parallel, with the stress classification line Outer Surface perpendicular to these surfaces. Of course LINE 2 in some situations ambiguities can arise, as illustrated in Figure 2.24. • Selecting which stress components should be linearised. The global stress Figure 2.24: Ambiguous classification lines components, including shear, may be linearised - but what about the principal stresses? With principal stresses, there is the obvious problem that principal directions can alter from point to point through the thickness unless the classification line is far from a discontinuity. As mentioned previously, the ASME Code itself implies that the linearisation should be performed on the global component stresses: “... membrane stress intensity is derived from the stress components averaged across the thickness of the section. The averaging shall be performed at the component level ...”. There is the related question as to whether linearising the component stresses then calculating the principal stresses from the linearised components is consistent with linearising the principal stresses - plane sections remaining plane should provide this consistency. BS5500 implies that all stress components should be linearised. Selection of the classification line has received little attention in the literature. However, selection of which stress components to linearise has been examined, on behalf of the ASME Code committee, by G L Hollinger and J L Hechmer[6]. Hechmer & Hollinger analysed a representative axisymmetric vessel problem and examined several different methods of stress linearisation. Two methods appear which were identified as being both conservative and consistent: either linearise the two normal stress components on a line (in the hoop and meridional directions) and use the total normal radial and total shear stress at the surface, or linearise the meridional principal stress and use the total stresses for the other principal direction (the exact technique is not wholly clear from the paper). Neither of these would appear to be common practice. 2.4.2.4 Three Dimensional Problems Three-dimensional solid finite element analysis poses a significant problem for stress linearisation. In 3-D analysis it is necessary to find a consistent stress classification plane, which again could cause problems near the very features the designer is concerned with (fillets and gross structural discontinuities). There is the added problem of defining exactly what should be meant by plane sections remaining plane in this case. Three possibilities arise: firstly the stress components at a point are directly used to evaluate the stress differences and stress intensity; this is easy, but the

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subsequent categorisation of these stress intensities is not. Secondly, stresses are linearised along radial lines to obtain beam type membrane and bending stresses; this suffers from the same problems mentioned above. Thirdly, selected planes are specified and two sets of stresses on distinct lines (on the plane) are used to evaluate shell type direct and bending stress on a plane. To the writers knowledge, no commercial post processors offer three dimensional stress linearisation capability over a plane, only along a line, only one by a vessel manufacturer[7]. In the case of the three dimensional problem, Hechmer & Hollinger[8] analysed a complex nozzle shell assembly using brick elements and examined the consequences of the three different assessment methods described above to calculate the stress intensity. As expected their study demonstrated a wide variation in the calculated results for the various methods (mainly because there are many possibilities open to the analyst) - a variation in stress intensity of over 35% was noted in this example. The results are indeed inconclusive: stress at a point calculation is easiest to apply but the results are not always conservative while stress along a line calculation is more advantageous with respect to Code rules. 2.4.2.5 Linearisation Guidelines It has been apparent for some time that there are deficiencies in the rules for design by analysis when the finite element method is used. In particular this has highlighted problems with the design criteria and the underlying philosophy of assessment. Over the past few years, the US Pressure Vessel Research Council (PVRC) has funded a project to consider recommendations for updating the ASME Code. It is worthwhile reviewing some of these recommendations; a summary has been given by Hechmer & Hollinger[9]. The short term recommendations consisted of six sections. The second, fourth, fifth and sixth recommendations are related to linearisation problems for primary stress and three-dimensional problems. The first and third recommendations are of a more fundamental implication since they relate to the use of finite element methods for design by analysis using the existing ASME - Code criteria. The project members have been very careful with the wording of the recommendations, and some interpretation is required. These recommendations consider essential pressure vessel components, which are basic structural elements: •

•

•

Shells of revolution and circular plates with either constant or variable thickness (transition elements) normally connecting one structural element to another. Smooth junctions - where the model represents the actual geometry for example connecting fillet or blend radius. Sharp junctions - where the model does not represent actual geometry, such as sharp corners or notches, as shown in Figure 2.25.

Basic Structural Element

Smooth Junctions Sharp Junction Fillet Transition Element

Sharp Junction

Basic Structural Element

Figure 2.25: Pressure vessel ‘elements’

Blend

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The first and third recommendations are summarised below: First recommendation: This relates to the use of finite element analysis (FEA) in pressure vessel design by analysis. It is recommended that for the majority of pressure vessel components, which are basic structural elements, FEA is inappropriate. Pm stresses should be calculated using general equilibrium considerations, with Pm+Pb evaluated by hand calculations for conditions where Pm is small (for example in flat plates). FEA is appropriate for calculating PL+Pb stresses near discontinuities (see third recommendation below) and for the calculation of P+Q stresses in general. Notably it is only in complex components where basic structural analysis does not exist that FEA is recommended as appropriate for Pm and Pm+Pb stress evaluation. “... the thrust is that the designer should be applying his ingenuity to calculating equilibrium stresses, not to extracting stresses from a general finite element model ...”. Third recommendation: This relates to the locations in a pressure vessel where stress evaluations for Code compliance should be considered. It is recommended that it is appropriate to perform Pm+Pb (PL+Pb) and P+Q evaluations in basic structural elements, but inappropriate in discontinuity type transition regions. If there is a smooth junction then the stresses should be evaluated in the row of elements adjacent to the junction (or the line of nodes at the junction). When there is a sharp junction, the evaluation must be far enough from the junction so that the stresses are not affected by the notch behaviour. This recommendation should eliminate the need to linearise erratic stress distributions; “... the thrust ... is that plastic collapse and gross strain concentration will not occur in stiff transition regions; they will occur in the more flexible shell elements ... the purpose of the P+Q limits is to validate the fatigue analysis by precluding strain concentration and ratchet. It is highly unlikely that ratchet could occur in a transition element ...” The first recommendation is rather subtle. In the light of the ASME Code (as it stands), finite element analysis is only appropriate in certain special cases in primary stress calculation - in general, equilibrium and shell discontinuity analysis are to be preferred. However, FEA is appropriate for secondary (and peak) stress evaluation. In the context of the discussion given this may be interpreted further as follows: finite element analysis may be used to evaluate the overall stress distribution for shakedown and fatigue assessment but the analyst should use simple calculations and strength of materials arguments to extract the primary stress components. In other words, elastic finite element analysis should not be used as the basis for categorisation or evaluation of primary stress. The third recommendation also needs careful interpretation and is the most intriguing of all those provided by the PVRC project. The implication to the writers is clear - ignore the calculated stresses in sharp transition regions, since they will not affect the post yield failure mechanisms. The mid term recommendations aim to provide additional tools and procedures to assist the designer in making better use of the existing ASME Code rules, specifically to address the problems of categorisation and linearisation directly through finite element analysis. Finally the long-term recommendations aim for a more fundamental assessment of the ASME Code philosophy and criteria and require extensive new research. It is felt that new rules should be based on specific quantities required to prevent a failure mechanism, perhaps moving away from simple elastic analysis and stress evaluation. For example, the limits based on shell type membrane and

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bending stress are difficult to understand and often misinterpreted, while the secondary limits are probably oversimplified and over-conservative, particularly in the presence of combined load. Considerable research on shakedown and ratchetting over the past twenty five years has confirmed this. 2.4.3 Problems with categorisation The process of stress categorisation (or classification) is difficult, as stress may be composed of both primary and secondary parts as seen in Figure 2.18 for the nozzle reinforcing pad. It is not sufficient just to categorise a particular stress corresponding to a given load condition, but also to categorise segments of the stress. This prospect is not inviting, and indeed rarely done in practice unless specified in Code rules (as in the case of the nozzle). We have reached a familiar problem - how should finite element (or otherwise) calculated stress be categorised? This is usually left to experience or strength of materials type arguments if this is possible. It is usually possible with simple strength of materials analyses or shell discontinuity analysis to separate primary and secondary stress with the understanding of the fundamental failure mechanisms that the Code addresses, since the equilibrium calculations were done manually. This is not obvious with finite element results, and in particular with the results of using continuum elements. The question is what can be done to ease this problem. An obvious solution would be to provide additional Code rules. While this is likely to be the case in the long term, it does not help the designer who must carry out pressure vessel design with the current rules. Briefly, the evaluation rules in this route can be summarised. Membrane and other primary membrane stresses are not allowed to approach yield since beyond yield there is the possibility of a catastrophic plastic collapse – for example bursting under internal pressure. The total (membrane plus bending) stress can increase fifty percent above the membrane limit since there is some safety margin here, but is still yield limited. Discontinuity and thermal stresses (or strain controlled stresses) must be limited to ensure shakedown under cyclic load; thus the range of secondary stress is limited to twice yield (or some smaller proportion for particular components). The peak stress must be limited to ensure a sufficient fatigue life, and certain other failure criteria may need to be addressed depending on the operating temperature - for example creep rupture at high temperature, fast fracture at low temperature. At this level categorisation is straightforward: any sustained stress that, subject to overload, would lead to plastic collapse is primary. The remaining stress (or indeed proportion of stress) can be classified as secondary and is subject only to the shakedown criterion (and fatigue limit). The problem arises because this design by analysis route relies upon elastic analysis. Elastic analysis on its own cannot characterise the nature of the stress since it is not clear what failure mechanisms can arise; it is left to the designer to do this. In addition, this approach does not make use of the ductility of pressure vessel steels, resulting in a wholly inconsistent (conservative) margin of safety[10]. In the absence of any meaningful information the designer is led to classify all stresses as primary and base redesign on this.

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It is useful to view the categorisation problem as part of the basic requirement to avoid failure by the various failure mechanisms: the problem of categorisation should then have a different interpretation. The stress system in the component should be such that shakedown is achieved and the fatigue limits satisfied for all stresses. In fact, these are the basic design requirements. The categorisation problem can then be interpreted as the need to isolate those stress systems that could cause gross plastic collapse - that is the primary stresses. The distinction here is subtle - there is no real need to identify a calculated stress as being primary or secondary; it is only necessary to identify the primary stresses. One solution to this difficulty is to calculate the limit load of the vessel by inelastic analysis. Limit load assessment and calculation of principal stress has been discussed by several authors using a variety of methods, notably Marriott[11], Kalnins & Updike[12], Mackenzie & Boyle[13], Seshadri[14] Ponter & Carter[15] and Zeman et. al.[16].

2.5 Implementation problems of the ASME inelastic route Inelastic finite element analysis[17,18] is more difficult than linear analysis and requires considerably greater computing resources. Essentially, the non-linear problem is solved in a piecewise manner using incremental solution techniques. The procedure usually requires the analyst to define an appropriate number of load steps, equilibrium iterations within load steps and convergence criteria defining the required accuracy of the solution. Poor choice for any of these parameters can lead to lack of convergence or indeed “convergence” to the wrong answer. In addition, it is difficult to make a priori engineering estimates of the inelastic response and to verify results of the analysis through simple calculations. There is also a shortage of non-linear benchmarks, which the analyst can use to assess the accuracy of the analysis procedures. There are two types of inelastic analysis methods, which may be used to guard against gross plastic deformation: limit analysis and plastic analysis. Limit analysis is based on an elastic-perfectly plastic material model and small deformation theory. The assumption of perfect plasticity sometimes causes convergence problems in non-linear analysis and in practice a bilinear hardening material model with a low value of plastic modulus Ep (1/10000 of E) is often used. This analysis determines the limit load PL of the vessel. The allowable load Pa then is defined as a specified fraction of the limit load. Plastic analysis is based on the ‘actual’ non-linear stress-strain relationship of the vessel material, including non-linear geometry effects if desired. This analysis determines the plastic collapse load Pφ. However, determination of the ‘plastic collapse load’ is not straightforward – to understand this, some basic concepts are required.

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2.5.1 Plastic design loads The aim with the ASME inelastic route is to estimate limit and shakedown loads directly, which can then be used to characterise an allowable load (Sec. 2.2). To begin with these are defined: The First Yield Load Py The first yield load Py is defined as the load for which the material of the pressure vessel first yields (from the virgin stress-free state) at the most highly stressed point. Because only one point of the material is at yield, the surrounding elastic material restrains the vessel from plastic deformation as a whole. The Limit Load P0 The classical definition of a limit load P0 according to limit analysis is an idealized one, a mathematical one. This “theoretical limit load” is the maximum load solution to an analytical model of the structure which embodies the following conditions: the strain-displacement relations are those of small displacement theory (first order); the material response is rigid plastic or elastic-perfectly-plastic (Fig.2.26), the internal stresses and applied forces are related by the usual linearised equations of equilibrium which ignore changes in geometry due to deformations.

stress (force)

rigid-perfectly plastic

strain (extension)

elastic-perfectly plastic

stress (force)

• • •

strain (extension)

Fig 2.26 : Elastic-perfectly-plastic and rigid-plastic deformation curves

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A (sufficiently) small region in an elastic-perfectly plastic material behaves either elastically (if stressed below yield), or plastically (for stresses at yield). At loads above the first yield load, P > Py, but less than the limit load, P < P0, a region of material may have stresses at yield, but this region is still restrained by the remaining rigid portions of material in the vessel. When the load is increased to the limit value P0, the plastic region has grown to an extent such that the rigid region has either disappeared or has become insufficient to restrain the plastic region from motion. The load for which overall plastic deformation of the vessel occurs is called the limit load. According to limit analysis theory, it is impossible to have loads greater than the limit load for a perfectly plastic material. The Plastic Collapse Load Pc The plastic collapse load Pc is applied to the actual structure or vessel consisting of an actual strain hardening material. It includes the effects of geometry change due to large deformations. At this load, significant plastic deformation occurs for the structure or vessel as a whole (un-contained plastic flow). The cause is the plastic region in the vessel, who now has grown to a sufficient extent such that the surrounding elastic regions no longer prevent overall plastic deformation from occurring. When this occurs, it may constitute a real failure, in the sense that the structure then can no longer fulfil its intended function. The plastic collapse load can be used as a realistic basis for design; an efficiently designed structure will be proportioned so that the external (operational) actions would have to be increased by a specified factor (safety factor) in order to produce failure. The limit load for an idealised structure then can be an approximation for the plastic collapse load for the actual vessel, when it is largely plastic at small deflections. The Ultimate Load Pu At the plastic collapse load, the vessel does not necessarily collapse. Therefore, the adjective, “collapse”, is unfortunate. The terminology of plastic deformation load or just plastic load would be more meaningful. The load at which the vessel actually collapses is the ultimate load Pu. An example of an ultimate load is the burst pressure for a cylindrical vessel of sufficient ductility. The Plastic Instability Load Ppi Plastic instability loads can be of two types: • •

of the material instability type, and of the structural instability type.

Plastic material instability corresponds for example to necking of a tensile specimen at the ultimate load. The plastic structural instability load, depends upon the yield strength of the material, and is accompanied by significant changes in shape of the structure or vessel. The plastic instability load is important because its value is often less than the limit load. The Shakedown Load PS All the above load definitions are for monotonic increasing loads. The shakedown load refers to cyclic loading and is considered briefly because it is important to know the relative margin of safety on shakedown of a design.

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If upon loading the structure beyond yield into the plastic range to a load value P > Py, and upon unloading, a residual stress distribution is produced in the structure such that further cycles of load to value P produce only elastic changes in stress, the structure is said to shakedown. The highest value of P for which shakedown occurs is called the shakedown load Ps. Failure to shakedown, i.e. P > Ps, leads to either progressive plastic flow called ratchetting, or to low cycle fatigue failure. 2.5.2 Limit Analysis Theory as Applied to Pressure Vessels Consider a typical pressure vessel loaded by internal pressure, a perfectly-plastic material, small deflections and increasing the pressure p. At small values of p, the vessel material will be elastic and deformation of the vessel will increase in proportion to p. However, as the pressure is continually increased, a region of the vessel becomes plastic and the rate of deformation begins to increase, but deformation of the vessel as a whole is usually still restrained by the surrounding elastic material. Finally, upon further increase in pressure, a limit pressure or (in this case) a plastic collapse pressure is reached, where the plastic zone has grown sufficiently large so that the deformation has suddenly begun to increase with little or no additional increase in pressure. The problem then is this: What will be the magnitude of the limit pressure of a particular pressure vessel? This is an important question in designing a vessel with a sufficient margin of safety. As described above, the problem from the beginning of loading involves initially elastic, then elastic-plastic, and finally largely plastic behaviour. This is an involved and complicated loading process. The theory of limit analysis, an idealised theory, enables the limit pressure to be found by considering: • •

a rigid ideal-plastic, or a linear elastic ideal-plastic material, characterised by a sharply defined yield limit (no strain hardening material), the small displacement theory (any effect of geometry change of the shell due to deformation is neglected).

These limitations must be kept in mind when applying limit analysis theory to certain problems where the effects of strain hardening and geometry change may be important. 2.5.3 Elastic-Plastic Theory as Applied to Pressure Vessels If effects of strain hardening and geometry change are important, an elastic-plastic analysis is to be applied. Their influence on the load-deflection curvature is discussed. Geometry Effects Fig 2.27 shows a comparison between: a) b) c) d)

a small-deflection rigid-plastic limit load analysis, a small-deflection elastic-plastic analysis, a large-deflection elastic-plastic analysis, with geometrical weakening, a large-deflection elastic-plastic analysis, with geometrical strengthening,

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e) a large-deflection elastic analysis, with geometrical weakening, f) a large-deflection elastic analysis, with geometrical strengthening. 2

f

1.8

d

e 1.6

a b

1.4

a b

load (bar)

1.2

c

c

1

d e

0.8

f

0.6 0.4 0.2 0 0

0.5

1

1.5

2

2.5

deflection (mm)

Fig 2.27 : Influence of geometrical effects

The small deflection elastic-plastic solution b) approaches the small-deflection rigid-plastic limit load solution a) as expected. The large-deflection elastic-plastic solution, with geometrical strengthening, gives a value higher than the limit load. The large-deflection elastic-plastic solution, with geometrical weakening, gives a value lower than the limit load. Effect of Strain Hardening The effect of strain hardening is to increase the pressure capability above the limit load predicted by the perfectly-plastic analysis, including the large deflection effect. Fig 2.28 shows that a higher slope of the plastic part of the load-deflection curvature corresponds to a higher strain hardening effect.

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1.8

6% 4% 2% 0%

1.6 1.4 1.2 load (bar)

0% 1

2% 4%

0.8

6%

0.6 0.4 0.2 0 0

0.5

1

1.5

2

2.5

deflection (mm)

Fig 2.28: Influence of strain hardening effect

2.5.4 Estimation of plastic loads Using an elastic-plastic analysis including strain hardening and large deflections or equivalently considering experimental analysis of an actual vessel, one is confronted with the problem of defining a realistic measure of plastic loads. A number of estimations have been used. These are reviewed next. The discussion refers to pressure loading, but the same definitions can be applied to other types of loadings. The Limit Pressure p0 Characteristic for the limit pressure definition according to the rigid perfectly-plastic theory is (with p = pressure and δ = deflection) dp/dδ = ∞ or dδ/dp = 0 for p < p0, and dp/dδ = 0 or dδ/dp = ∞ for p = p0. Characteristic for the limit pressure definition according to the elastic perfectly-plastic theory is dp/dδ > 0 for p < p0,

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and dp/dδ = 0 or dδ/dp = ∞ for p = p0. These definitions only apply for small-deflection analyses. The Tangent-Intersection Pressure pti (Fig 2.29) 1.6 1.4 Pti = 1.35 bar 1.2

load (bar)

1 0.8 0.6 0.4 0.2 0 0

0.5

1

1.5

2

2.5

deflection (mm)

Fig 2.29 : Tangent-Intersection method

The tangent-intersection pressure is the pressure at the intersection of the two tangents, drawn to the elastic and plastic parts of the pressure-deflection curves. The value of the pressure obtained by this method is sensitive to the localisation of the tangent-point in the plastic range The 1% Plastic Strain Pressure p1 The plastic pressure is defined as the pressure with an equivalent plastic strain of 1%. Methods based upon an absolute maximum strain not only will depend on the material assumed, but more significantly on the geometry: •

•

Material: e.g. a 1% plastic strain is ten times the yield point strain if the yield point stress is 150 MPa, but five times the yield point strain if the yield point stress is 300 MPa. Consequently, the relative size of the elastic and plastic zones will differ and the shape of the pressure-deflection response curves will differ. Geometry: Ellipsoidal heads have been found to deform less than torispherical or toriconical heads. Whereas a torispherical vessel may reach a 1% strain at a certain pressure, the ellipsoidal vessel may reach the same pressure at a lower strain.

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At a yield hinge location, strains will be larger than at other locations. Consequently, the selection of a strain gauge location on an experimental vessel presents a variable when yield hinge locations are not known precisely or a priori. Thus, in summary, a strain basis for defining a plastic pressure may be subject to error in locating the exact location of maximum strain. Also, strain is a local phenomenon that is not indicative of plastic work. The Twice-Elastic-Deformation Pressure p2y A plastic pressure is defined to be the pressure at which the deflection or strain reaches twice the value of the elastic deflection or elastic strain at the first yield pressure py. Thus, p2y depends upon py. Exact determination of py using a computer analysis should not be a problem. In experiments however, determining the elastic limit on the load deflection curve may be subject to error. The Twice-Elastic-Slope Pressure pϕ A plastic pressure is defined to be the value at the intercept of a line drawn from the origin of a pressure-deformation curve at a slope of twice the slope of the elastic portion of the curve (see Fig 2.30). 1.6 Pφ = 1.39 bar

1.4 1.2

load (bar)

1 y = 2.2x 0.8

y = 1.1x

0.6 0.4 0.2 0 0

0.5

1

1.5 deflection (mm)

Fig 2.30 : Twice-Elastic-Slope method

2

2.5

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The 0.2% Offset Strain Pressure p.2 The 0.2% offset strain pressure is a test pressure that causes a permanent strain of 0.2%. The Proportional Limit Definition ppl Ppl is a test pressure defined as the pressure causing the displacement versus pressure curve to deviate from linearity. The displacement of the vessel is to be measured at the weakest point, the most highly stressed point, giving the lowest value of ppl. Analytical calculations can determine this pressure correctly. It will not necessarily be equal to the first yield pressure py. Experimental measures are subject to error in determining the point of deviation from linearity. Values of ppl up to 30% greater than py can be estimated from an experimental curve. This method of determining a plastic pressure will generally give a lower bound to the plastic pressure found by most other methods. The Plastic-Instability Pressure ppi This is an actual plastic collapse pressure and not just an estimate of a plastic pressure. It may be identical to the limit pressure if large deflection effects are small, e.g. when the vessel is relatively thick. However, the plastic-instability pressure may be less than the small-deflection limit pressure as in the case of a large-deflection elastic-plastic solution, with geometrical weakening (see Fig 2.27, curve c). The plastic instability is defined by a zero slope on the pressure-deflection curve. A large-deflection elastic-plastic analysis is required to detect ppi. It will also be detected in experiments on actual vessels and it is possible to have plastic instability pressures less than lowerbounds to the limit pressure where the latter are based on small-deflection analyses. It may occur that some of the above estimations of the limit pressures will be non-conservative estimates of the real plastic collapse pressure, an instability pressure, if the estimates were applied to smalldeflection theoretical results. 2.5.5 Inelastic Progressive Plastic Deformation - Shakedown Within the DBA approach the determination of the limit load, for a given constitutive law, is one step. Proving that Progressive Plastic Deformation (PD) will not occur, or – more stringent – that neither PD nor Accumulating Plasticity (AP) will occur, in other words, proving that the structure under consideration will shake down to pure elastic behaviour under cyclic varying actions, is another step. Considering this proof, which is to be obtained through numerical simulation, the following information may be useful: - In this proof the constitutive law may be, but needs not to be the same law as used in the determination of the limit load. Normally the structure shakes down under cyclic actions which are to be specified as functions of a single parameter. This parameter determines the sequence of the actions and quite often this parameter is time. The proof of shakedown is easier to perform than the proof against PD. Being conservative this approach yields the proof that neither PD nor AP occurs.

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- In the inelastic simulation the proof of shakedown can be performed by applying the action cycle repeatedly. - The proof may also be performed using Melan's shakedown principle: •

Using an equivalent linear elastic structure, a given cyclic action results in a corresponding cyclic stress field. Additionally a time-independent self-equilibrating stress field should be found, such that, using superposition of both stress fields, the stress intensity does not exceed the yield limit at any time in the cycle.

•

This approach is especially attractive in those particular cases where an appropriate selfequilibrating stress field is already known. A thermal stress field may serve as an example, as well as the difference between a purely elastically determined stress field and the corresponding field using plastic constitutive laws.

•

In many cases the proof may be performed using the check of primary + secondary stresses used in a linear elastic DBA route, against the so-called 3f-criterion. The fulfilment of this criterion is a necessary condition for shakedown. It is considered accurate enough for most cases especially in combination with some other checks. However care should be taken whenever the cyclic action contains a non-negligible time-invariant part e.g. a large contribution of selfweight.

2.5.6 Discussion Again it can be seen that apparently simple requirements of the inelastic route can be problematic. Limit or shakedown analysis could be used to directly estimate the limit and shakedown loads, but until recently this was difficult if not impossible for complex structures. If elasto-plastic finite element analysis is used there remains the problem of defining the plastic load – there are various estimations as described above. The twice-elastic-slope method recommended by ASME has been shown to give inconsistent results. The European standard aims to remove some of these problem areas. In the following an overview of the new rules is given:

2.6 Design by Analysis in the European Standard 2.6.1 General The European Standard has introduced the possibility of satisfying the requirement to avoid various failure mechanisms directly through the detailed rules embodied in the new Direct Route, while retaining the ‘conventional’ elastic route which uses stress categorisation. In addition it also introduces several new concepts to help overcome the known difficulties with the current design by analysis approach and to assist the formulation of the Direct Route. In particular the notion of an ‘action’ rather than a force, and the inclusion of ‘partial safety factors’ is a novel and welcome addition to the area of pressure vessel design by analysis. In the following some background to these new concepts is provided, followed by a summary of the required design checks, with some explanation if required.

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2.6.2 New concepts 2.6.2.1 Principles and application rules Like in the Eurocode (for steel structures) distinction is made between principles and application rules. Principles comprise general statements, definitions and requirements for which there is no alternative, and requirements and analytical models for which no alternative is permitted (unless specifically stated). Application Rules are generally recognised rules which follow the principles and satisfy their requirements; alternatives are allowed provided it is shown that they accord with the relevant principle. Typical examples are the primary and the primary & secondary stress criteria of the stress categorisation approach, which are stated here, in slightly modified forms, as application rules. 2.6.2.2 Actions This term, which replaces the old term loadings, denotes all thermo-mechanical quantities imposed on the structure causing stress or strain, like forces (including pressure), temperature changes and imposed displacements. Actions are classified by their variation in time: • • • •

permanent actions (G) variable actions (Q) exceptional actions (E) operating pressures and temperatures ( p, T ) - although these are variable actions, they are considered separately to reflect their special characteristics (variation in time, random properties, etc.).

The notion variable actions encompasses actions of quite different characteristics – from those actions which are deterministically related to pressure and/or temperature, via actions not correlated with pressure or temperature but with well defined (bounded) extreme values, to actions which can be described only as stochastic processes not correlated with pressure or temperature, like wind loads. Actions with a deterministic relationship with pressure and/or temperature shall be combined in the pressure/temperature action and the relationship, exact or approximate, shall be used. The characteristic values of actions describe the regime of actions which envelops all the actions that can occur under reasonably foreseeable conditions. The characteristic values are used in determining the design values of the actions, and they depend on the actions' (statistical) properties. The characteristic values of permanent actions are usually their mean values (or credible extreme values). The characteristic values of variable actions are defined as mean values, or p% percentiles, of extreme values, and values specified in relevant codes for wind, snow, earthquake may be used; usually they are adapted to Eurocode concepts anyway. The upper characteristic value of pressure shall not be smaller than the lesser of the set pressure of the protecting device or the highest credible pressure that can occur under normal and upset conditions (reasonably foreseeable), and the upper characteristic value of the temperature not smaller than the highest credible temperature (under the same conditions). Therefore, the (limited) pressure excursion

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(overpressure) that occurs if a safety valve opens need not be included in the (maximum) characteristic value of pressure; it is taken care of in the partial safety factors. 2.6.2.3 Partial safety factors To allow for an easy, straightforward combination of pressure action with environmental ones, and, at the same time, to give the flexibility, expected from a modern code, to adjust safety margins to differences in action variation, likelihood of action combinations, consequences of failure, differences of structural behaviour and consequences in different failure modes, uncertainties in analyses, a multiple safety factor format was introduced, using different partial safety factors for different actions, different combinations of actions, different failure modes and corresponding resistances of the structure. Examples of partial safety factors are given in the following Table. The corresponding combination rules for e.g. Design Check GPD-OC Global Plastic Deformation – Operating Conditions are: • • • •

all permanent actions shall be included in each load case each pressure action shall be combined with the most unfavourable variable action each pressure action shall be combined with the corresponding sum of variable actions; stochastic actions may be multiplied by the combination factor. favourable actions shall not be considered.

The partial safety factors of pressure and resistances are calibrated with respect to the DBF results; no attempt has been made to justify the partial safety factors by probabilistic investigations or decision theory under uncertainty; if pressure is the only action the approach can be transformed to a nominal design stress one.

Actions Permanent γ G Unfavourable Favourable Pressure γ P Variable γ Q

Partial safety factors Design check GPD-OC GPD-HT

Combination factor ψ (stochastic actions) Resistance γ R (Temperature γ T ) 1

1.35 1.0 1.2 (1.0) 1.5 (1.0)

1.35 1.0 1.0 -

0.91

1.01

1.25 (1.0)

1.05 (1.0)

If not specified differently in the relevant code of environmental actions.

2.6.2.4 Design checks – effects of actions Design checks are investigations of the structure's safety under the influence of specified combinations of actions - the design load cases - with respect to specified limit states (representing

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one or more failure modes). Characteristic values of the actions are multiplied by the corresponding partial safety factors to obtain their design values and their combined design effect (on the structure) is evaluated: E d ( γ G G , γ p p , γ Q Q ,... , a d ,... )

In the design checks these design effects are compared with the corresponding design resistances, obtained by dividing the resistance of the structure, corresponding to the action's combination, by the relevant partial safety factor of the resistance: E d ≤ R d = R ( G , p , Q ,... , a d , ) / γ R

This comparison can, in general, be performed in actions, in stress resultants (generalized stresses) or in stresses. The resistances are related to the limit states - states beyond which the part no longer satisfies the design performance requirements. 2.6.2.5 Design checks – resistances Design checks are designated by the failure modes they deal with. The following ones are incorporated in the first issue of the standard: • • • • •

gross plastic deformation (GPD), with corresponding failure modes ductile rupture and, for "normal" designs, also excessive local strains progressive plastic deformation (PD) instability (I) fatigue (F) static equilibrium (SE).

Checks against gross plastic deformation The design resistances are given by the lower-bound limit loads for • proportional increase of all actions • a linear-elastic ideal-plastic material (or a rigid ideal-plastic one) • first-order theory • Tresca's yield criterion and associated flow rule • specified design strength parameters. Design strength parameters R M and partial safety factors of the resistances γ R are chosen such that for the simplest structures and pressure action only DBA and DBF results agree. The only exception are steels, other than austenitic ones with A 5 ≥ 3 0 % , where the design strength parameter R M is given by R eH , T or R p 0 . 2 , T and γ R = 1.25 for R eH / R m ≤ 0.8 and γ R = 1.5625 R eH / R m otherwise.

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If the procedure used to determine the limit action does not give an (absolute) maximum in the region with maximum absolute values of principal strains less than 5%, the boundary maximum, for which the maximum absolute value of the principal strains equals 5%, shall be used. As an application rule the "usual" primary stress criterion is given, formulated in stresses and - for structures where the concept of stress resultants is applicable - in stress resultants and local (technical) limit loads. These checks (against GPD) are considered also to encompass Excessive Yielding, provided "usual" design details (with not too severe strain concentrations) exist. Checks against progressive plastic deformation On repeated application of specified action cycles PD shall not occur for • a linear-elastic ideal-plastic material • first-order theory • Mises' yield condition and associated flow rule • specified design strength parameters RM. A slight modification of the "usual" 3 f criterion is given as application rule; it is noted that this application rule, which is derived from shakedown considerations, is only a necessary condition for the fulfilment of the principle, but is considered, together with all the other checks, to be sufficient to achieve the principle's goal - avoidance of ratchetting in the structure. Check against fatigue failure Reference is made to the Fatigue Assessment section of the Standard. Instability Static equilibrium The usual checks against overturning and (rigid body) displacement are stated explicitly, using the partial safety factors given in the other checks. 2.6.3 Application remarks Whether the Direct Route or the Stress Categorisation Route is followed, it is imperative that all stated checks are considered: •

•

in the Direct Route: At least the five given checks – sometimes it may be necessary to include additional ones, like excessive deformation (leakage). Not all of the checks will require calculations, but all must be considered – e. g. it may be obvious that instability can be excluded. in the Stress Categorisation Route:

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Check of Primary Stresses Check of Primary + Secondary Stresses Check of Total Stresses (Primary + secondary + peak stresses) - Fatigue Usually it is required to perform each of these checks for different load cases – for different combinations of coincident actions, as well as for different characteristic values of actions, e. g. different pressure – temperature pairs. The Design by Analysis route may be chosen to prove conformity of a design also for a part of a component, suitably selected and limited; and with appropriate boundary conditions.

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2.7 REFERENCES [1] W. C. Kroenke, J. L. Hechmer, G. L. Hollinger & A. J. Pedani, “Component evaluation using the finite element method. In “Pressure Vessel & Piping Design: A Decade of progress, 19701980”, Chap. 2.11, ASME, 1981. [2] W. C. Kroenke, “Classification of finite element stresses according to ASME Section III stress categories,” Proc 94th ASME Winter Annual Meeting, 1973. [3] W. C. Kroenke et al, “Interpretation of finite element stresses according to ASME III,” ASME Tech. Paper 75-PVP-63, 1975. [4] N.V.L.S. Sarma, G. L. Narasaiah & G. Subhash, “A computational approach for the classification of FEM axisymmetric stresses as per ASME Code,” Proc ASME Pressure Vessel & Piping Conf, Pittsburgh, 1988. [5] J. L. Gordon, “OUTCUR: An automated evaluation of two dimensional finite element stresses according to ASME,” ASME Paper 76-WA/PVP-16, 1976. [6] J. L. Hechmer & G. L. Hollinger, “Considerations in the calculations of the primary plus secondary stress intensity range for Code stress classification,” “Codes & Standards and Applications for Design and Analysis of Pressure Vessel and Piping Components” Ed R. Seshadri, ASME PVP Vol.136, 1988. [7] B. W. Leib, “An automatic surface element generator for calculating membrane and bending stresses from three dimensional finite element results,” Proc 4th Int Conf on “Structural Mechanics in Reactor Technology”, San Francisco, 1977. K. H. Hsu & D A McKinley “SOAP - a computer program for classification of three dimensional finite element stresses on a plane,” Proc ASME Pressure Vessel & Piping Conference, Nashville, 1990. [8] J. L. Hechmer & G. L. Hollinger, “Three dimensional stress criteria - a weak link in vessel design and analysis,” ASME Special Publ. PVP 109 “A Symposium on ASME Codes and Recent Advances in Pressure Vessel and Valve Technology” Ed J. T. Fong, 1986. J. L. Hechmer & G. L. Hollinger, “Three dimensional stress criteria -application of Code rules,” ASME Special Publ. PVP 120 “Design and Analysis of Piping, Pressure Vessels and Components” Ed W. E. Short, 1987. J. L. Hechmer & G. L. Hollinger, “Code evaluation 3D stresses on a plane, “ Codes & Standards and Applications for Design & Analysis of Pressure Vessels & Piping, ASME PVP-Vol.161, 1989. [9] J.L. Hechmer & G.L. Hollinger, “Three dimensional stress criteria,” ASME PVP-Vol.210-2 Codes and Standards and Applications for Design and Analysis of Pressure Vessel & Piping Components, Ed R. Seshadri & J.T. Boyle, 1991. G. Hollinger, “Summary of three dimensional stress classification,” Proc Int Conf on Pressure Vessel Technology, Dusseldorf, 1992.

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[10] R.L. Roche, “Practical procedures for stress classification,” Int Journ Press Vess & Piping, Vol.37, 27-44, 1989. [11] D.L. Marriott, “Evaluation of deformation or load control of stresses under inelastic conditions using elastic finite element analysis,” Proc ASME Pressure Vessel & Piping Conf, Vol.136, Pittsburgh, 1988. [12] A. Kalnins & D.P. Updike, “Role of plastic limit and elastic plastic analyses in design,” ASME PVP-Vol.210-2 Codes and Standards and Applications for Design and Analysis of Pressure Vessel & Piping Components, Ed R. Seshadri & J.T. Boyle, 1991. A. Kalnins & D.P. Updike, “Primary stress limits on the basis of plasticity,” ASME PVPVol.230, Stress Classification, Robust Methods and Elevated Temperature Design, Ed R. Seshadri & D.L. Marriott, 1992. [13] D. Mackenzie & J. T. Boyle, “A computational procedure for calculating primary stress for the ASME B&PV code,” Trans ASME, Jrn Pressure Vessel Tech, Vol. 116, No. 4, 1994. D. Mackenzie, J. Shi, R. Hamilton & J. T. Boyle, "Simplified lower bound limit analysis of pressurised cylinder-cylinder intersection Shells using a generalised yield criteria", Int Jrn of Pressure Vessels & Piping, 67, pp. 219-226, 1996. J. T. Boyle, R. Hamilton, J. Shi, & D. Mackenzie, "A simple method of calculating Limit Loads for thin axisymmetric shells", Trans. American Society of Mechanical Engineers (ASME), Jrn Pressure Vessel Technology, Vol. 119, No.2, pp. 236-242, 1997. [14] R. Seshadri & C.P.D. Fernando “Limit loads of mechanical components and structures using the GLOSS r-node method”, Proceedings of ASME PVP, Vol. 210-2, pp. 125-134, 1991. [15] A.R.S. Ponter, K.F. Carter, “Limit state solutions, based upon linear elastic solutions with a spatially varying elastic modulus”, Jrn of Computer Methods in Applied Mechanics and Engineering, Vol.140, No.3-4, pp.237-258, 1997. A.R.S. Ponter, K.F. Carter, “Shakedown state simulation techniques based on linear elastic solutions”, Jrn of Computer Methods in Applied Mechanics and Engineering, Vol.140, No.3-4, pp.259-279, 1997. [16] T. Seibert, J. L. Zeman, „Analytischer Zulässigkeitsnachweis von Druckgeräten“, Techn. Überwachung, Bd. 35 (1994) Nr. 5, 222-228. W. Poth, J. L. Zeman, „Grenztragfähigkeit der Innendruckeinwirkung“, Konstruktion 48 (1996), 219-223.

Zylinder-Kegel-Verbindung

unter

J. L. Zeman, „Ratcheting limit of flat end cylindrical shell conections under internal pressure“, Int J Pres Ves & Piping 68 (1996), 293-298. R. Preiss, F. Rauscher, D. Vazda, J. L. Zeman, „The flat end to cylindrical shell connection – limit load and creep design“, Int J Pres Ves & Piping, 75 (1998),715-726 [17] D. Mackenzie, J. T. Boyle & R. Hamilton, “Application of Inelastic Finite Element Analysis to Pressure Vessel Design”, International Conference on Pressure Vessel Technology, Volume 2, ASME 1996.

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[18] J. C. Gerdeen, “A critical Evaluation of Plastic Behaviour Data and a United Definition of Plastic Loads for Pressure Components”, WRC Bulletin 254, November 1979, ISSN 0043-2326. [19] A. Kalnins, D. Updike & J.L. Hechmer, “On Primary Stress in Reducers”, ASME PVP-Vol. 210-2, pp. 117-124 [20] D. Mackenzie, J.T. Boyle, J. Spence, "Some Recent Developments in pressure vessel Design by Analysis" Proc IMechE, Part E, Journal of Process Mech Eng, 1994, 208, 23-30.

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3 Procedures 3.1 General Procedures, used in section 7 in the various checks, which are of a more general nature and which would have to be repeated in the design checks frequently, are collected here in this section. Also included are general remarks on the various routes and possibilities within one route. 3.2 The Two Routes There are two approaches for carrying out a DBA that cover both the direct route method and stress categorisation method. If in a design the values of the actions are defined, a check on admissibility of the actions, i.e. an adequacy check, can be made. Alternatively, if only geometry is defined but not the magnitude of the actions, DBA may be used to calculate maximum allowable values of the actions.

the the the the

3.2.1 Admissibility check for defined actions For a given design where the actions are defined, DBA may be used to check if the defined actions are admissible. It is not necessary in this case to calculate an upper limit of the actions, although by doing so one will automatically check the admissibility of the defined actions. In the admissibility check it is only required to show that the defined actions do not exceed the limits defined in the applicable section of the code. For example in the direct route check against GPD, for admissibility it is only necessary to show that (for the defined actions with appropriate safety factors) the resulting elasto-plastic stress field is an equilibrium stress field where the absolute maximum total principal strain does not exceed 5%. (For the GPD-check using elastic compensation admissibility of the actions is shown if the maximum stress in the redistributed equilibrium stress field does not exceed the design resistance). In the case of a stress categorisation route, admissibility is shown if the linearised stress categories do not exceed the limits defined in the code rules, prEN 13445-3 Annex B. 3.2.2 Determination of maximum allowable actions For a given design where only the type of action is specified, DBA may be used to calculate the maximum allowable actions. The maximum allowable actions are given by the actions that place the structure at the defined limits as specified in the applicable section of the code. In the direct route the maximum allowable action according to GPD is usually calculated using limit analysis. In elasto-plastic analysis the actions are increased until the limit is reached, where loss of equilibrium occurs or the absolute maximum total principal strain exceeds 5%. The value of the action at this limit with the appropriate partial safety factors applied is the maximum allowable action according to the GPD-check. (In elastic compensation, the redistributed equilibrium stress field is scaled, along with the applied action to the design resistance; the value of the scaled action with the appropriate safety factors applied is the maximum allowable action). In the check against PD, a lower bound of the maximum allowable actions can be calculated by finding the maximum actions for which the structure will shake down (see the procedure for Melan’s theorem below).

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For the stress categorisation approach the maximum allowable action can be found by scaling the applied action (by evoking the linear proportionality of the elastic solution) to a point where the linearised stress categories are at the limits defined in the code rules, prEN 13445-3 Annex B.

3.3 Direct Route (using elasto-plastic calculations) 3.3.1 Check against Gross Plastic Deformation (GPD) Generally, two different ways of performing calculations according to the GPD-check are possible: First, if the actions for a given structure are specified, a real check can be performed, to show that the actions are admissible (or not) under application of prEN 13445-3 Annex B. This procedure was applied within the GPD-check of the examples 2, 5 and 6. The second possibility is to calculate the limit actions for a given structure, and using these limit values then to determine the maximum admissible actions according to prEN 13445-3 Annex B. The latter procedure will be useful, if the structure is to be used with extreme actions. This procedure was applied within the GPD-check of the examples 1.1, 1.2, 1.3, 1.4, 3.1, 3.2 and 4. As stated in the application rule in prEN 13445-3 Annex B.9.2.2, which fulfils the principle in the check against GPD - prEN 13445-3 Annex B.9.2.1 – the limit load has to be determined using Tresca’s yield condition with associated flow rule and first order theory. Since there is no standardised subroutine for Tresca’s yield condition in the ANSYS® software, a made-to-order subroutine of an ANSYS® distributor was used. Unfortunately, for some structures the subroutine showed bad convergence and/or long computation times. The cause may be the edges in Tresca’s yield surface. If no subroutine for Tresca’s yield condition is available or if it shows too bad convergence, Mises’ yield condition can be used instead. Since the maximum ratio of the Mises equivalent stress to the Tresca equivalent stress for the same load is 2 3 , a multiplication of the design resistance with 3 2 will always lead to conservative results. Furthermore, if the result of the check against PD (where Mises’ yield condition is allowed) is used in the check against GPD, instead of a separate calculation, and if there is only one partial safety factor γ R for the considered structure, multiplication of the limit pressure from the check against PD with 3 2 leads to the same result as the multiplication of the material strength parameter. Of course, since no partial safety factors are used in the check against PD they have to be taken into consideration by scaling down the PD-check results. Of course, if the check against GPD is performed first, the results can be used in the check against PD, using the relevant factors. As stated in the application rule in prEN 13445-3 Annex B.9.2, the maximum absolute value of the principle strains must not exceed 5%. To fulfil this demand in cases where the results from the PDcheck are used, such a value of the pressure from the check against PD shall be used that the maximum absolute value of the principle strains calculated with this pressure (and the material strength parameter used for the check against PD) does not exceed 5 %. If this procedure leads to too conservative results (because of the restraint by the 5 % - limit) an additional FE-calculation using the scaled down material strength parameter is required to determine the exact 5% - limit pressure. Because of this strain limitation in the GPD-check, there is now in general a difference between

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•

the design resistance, defined as ratio of the resistance (of the specified model) and the appropriate partial safety factor, where the resistance is determined with (specified) material strength parameters

•

the resistance of a corresponding model but with material strength parameters replaced by the corresponding design material strength parameters, obtained by dividing the material strength parameters by the partial safety factors of the resistance.

Usually the difference is very small, and it is recommended that the resistance in the second route may be used as design resistance; this recommendation has already been used in the examples. Performing a real GPD-check only, e.g. no limit actions are determined, the admissibility of specified actions acting on a specified structure is shown, if the maximum absolute value of the principal strains does not exceed 5%, under usage of a linear-elastic ideal-plastic material law in the FE-calculations. If the FE-model consists of shell elements, usually the mid-surface of the structure is modelled. Therefore, the practical relevance of the results in points (nodes) on the intersection curve of two shells depends on the kind of geometry of the structure. For example, for (cylindrical) main shell – (cylindrical) nozzle intersections the results in nodes of the intersection curve should not be used, since they do not correspond to points of the real structure with the real geometry under consideration of the reinforcement due to the weld - see Figure 3.1. Therefore, if usage of solid elements or of submodelling is not possible, the results in the so called “evaluation” cross-sections should be used for the determination of the 5% principal strain limit. Figure 3.1: Shell intersection

As stated in the principle in prEN 13445-3 Annex B.9.2.1, the design resistance (limit action) should be obtained from calculations with proportional increase of all design actions. The limit action is independent of the action history, but with strain limitation, i.e. if the strain limitation governs, the limit will depend on the action’s history. In the case of constant moment load and varying internal pressure load (examples 3.1 and 3.2), where the strain limitation does govern, the deviation from the standard’s procedure – proportional loading – is the only sound one, the moment being constant during all action cycles. 3.3.2 Check against Progressive Plastic Deformation (PD) 3.3.2.1 General Again, two different ways of performing calculations to check PD are possible: First, if the action cycles for a given structure are specified, a real check can be performed, showing that the actions are admissible (or not) under application of prEN 13445-3 Annex B. This procedure was applied within the PD-check of the examples 2, 5 and 6. The second possibility is to calculate the limit

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actions (in the sense of PD) for a given structure, and afterwards using these limit actions to determine the maximum admissible actions according to prEN 13445-3 Annex B. The latter procedure will be useful, if the structure is to be used with extreme actions. This procedure was applied within the PD-check of the examples 1.1, 1.2, 1.3, 1.4, 3.1, 3.2 and 4. Principally, the application rule in prEN 13445-3 Annex B.9.3.2 corresponds to the well-known criterion for the sum of the primary and secondary stresses in stress categorisation – the (often) socalled “3f–criterion” (where f stands for the allowable stress). This criterion is an upper bound criterion for shakedown, and, therefore, the requirement given in this application is only a necessary and not a sufficient condition for the fulfilment of the principle – prEN 13445-3 Annex B.9.3.1 [1], [2] . Usage of this application rule could be the easiest way of applying the check against PD if only one action is considered, but if more than one action and/or additional thermal stresses have to be considered, its usage could be difficult and uncertain. Therefore, and for guideline purposes, usually another possibility of fulfilling the principle – Melan’s shakedown theorem – is employed. Melan’s shakedown theorem states [1], [2]: The structure will shake down for a given cyclic action, if a time-invariant self-equilibrating stress field can be found such that the sum of this stress field and the cyclically varying stress field determined with the (unbounded) linear-elastic constitutive law for the given cyclic action is compatible with the yield condition – the equivalent stress nowhere and at no time exceeds the (yield) material parameter. Using this theorem the principle is fulfilled, since Progressive Plastic Deformation (PD) and Alternating Plasticity (AP) are the two possible inadaption modes if a structure does not shake down under a cyclic load set. One advantage of using Melan’s theorem, in comparison with the application rule, is, that the admissibility is shown for all points of the structure, if the determined self-equilibrating stress field is superposed onto the linear-elastic stress fields of the different states using the postprocessor of the FE-software. A problem arises if the maximum allowable action given by the shakedown limit is lower than the one resulting from the check against GPD. In this case, detailed examination of the structure’s behaviour under a cyclic loading with the maximum allowable action according to the check against GPD, i.e. determination of the structures inadaption mode – progressive plastic deformation and/or alternating plasticity - would be necessary. If it can be shown that the inadaption mode is given by alternating plasticity only, the action would be admissible. Unfortunately, the possibility of such examinations is restricted, on one hand because of the hardware and time limits for performing cyclic calculations and on the other hand because of a lack of generally applicable theorems. Determining generally applicable theorems in this field is a present topic of research, and, therefore, they should be available in the future. 3.3.2.2 Problems in performing the shakedown check using shell elements in the FE-model [3] Usage of stress resultants of technical theories of structures, i.e. generalised stresses, to verify that progressive plastic deformation (PD) does not occur is often not appropriate, because •

the validity of the corresponding theorem[4] seems to be restricted to passive (unloading) processes [2], [5], [6];

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(local) interaction surfaces are available for rotational-symmetric shells under rotationalsymmetric loading only.

Therefore, a shakedown check using stresses has to be carried out quite often. However, if shell elements are used, the only stress results available in the postprocessor are surface stresses (top / bottom of the shell, i.e. outer / inner surface). Usage of Melan’s shakedown theorem and corresponding self-equilibrating stress states to ensure that a structure shakes down under a given cyclic load path could then be non-conservative, because the fulfilment of the theorem can only be proven for surface stresses but not in the interior of the shell model. In fact, when the absolute maximum of the self-equilibrating stress distribution in a cross-section is located in the interior, the above stated non-conservatism is possible. In the elasto-plastic simulation of a structure's behaviour during the loading or unloading half cycle, three phenomena can occur: (1) if the surface equivalent stress in at least one point of the structure remains, or becomes equal to the material strength parameter, yielding occurs during the half cycle, on the surface and/or in the interior; (2) if the surface equivalent stress in all points of the structure is less than the material strength parameter, no yielding occurs on the surfaces during the half cycle, but yielding inside of crosssections cannot be excluded; (3) only if the stress state on the surface after the half cycle is identical to the one resulting from purely elastic loading or unloading of the structure, no stress redistribution due to plastic deformation occurs, neither at the beginning nor at the end of the half cycle. Therefore, it is assured that the half cycle is purely elastic and the structure has shaken down. There follows that to verify in a strict manner that a structure modelled by shell elements shakes down, one has to show that condition (3) is fulfilled. If computation of the half cycle shows that condition (1) is fulfilled, further cycles are necessary until condition (2) is fulfilled - if actually possible. Afterwards, to verify that the structure shakes down, one has to show that condition (3) is fulfilled. However, using this procedure can be difficult if considerable numerical errors, for example due to extrapolations and averaging, are present in the stress plots. Alternatively, if the FE-software confirms that no further plastic strains occur during a half cycle (after some initial load cycles) - requiring a suitable parameter in the computation output -, it is proven that the structure has shaken down, independently of possible numerical errors in the postprocessor. Additionally, the practical relevance of the results in points (nodes) on the intersection curve of two shells depends on the kind of geometry of the structure, and, therefore, care must be taken when using these results. Using “evaluation” cross-sections in a similar manner as in the check against GPD is not a satisfying solution, since the stationarity of stress cycles cannot be proven if the whole structure is not considered. A possibility of showing that Melan’s theorem is fulfilled by performing the shakedown check at the shell model, i.e. only at the surface but not in the interior of the structure, is to use submodeling. Submodeling is a finite element technique to obtain more accurate results in a region; by performing an analysis of a coarse model (shell model), interpolating the results to the (cut-)

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boundaries of the fine model (solid element model), which only represents the critical part of the structure, and by computing the stress distribution in the fine model afterwards. If this stress distribution confirms that the maximum stresses at the critical parts of the structure are on the surface of the model, the shakedown check using the shell model fulfils the theorem. 3.3.2.3 Problems concerning stress singularities Performing linear-elastic calculations – as necessary for a shakedown check –, stress singularities [8] can arise specific points of a weld modelled without fillets. Since it is stated in the standard – see prEN 13445-3 Annex B.3.9.3.2 – that the check against PD can be performed for a stress-concentration-free structure, different possibilities of avoiding the stress singularities are possible: •

modelling the welds with fillets corresponding to the weld influence zones - see Figure 3.2. This procedure is suitable, if 2-D FE-models are used.

•

modelling the welds with fillets, which are completely inside of the weld – see Figure 3.3. This procedure is suitable if 3-D FE-models are used, and if modelling of fillets corresponding to the weld influence zones would be too difficult and time consuming.

Figure 3.3: Fillet inside the weld

Figure 3.2: Fillets-weld influence zones

3.3.2.4 Use of the deviatoric map (for constant principal stress axes) [7] The deviatoric projection is a simple tool in plasticity theory for visualising stress states vis-a-vis to yield conditions. In principle, it is the projection of the stress point in the (three – dimensional,

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Cartesian) space of principal stresses – with co-ordinates in the directions of the unit vectors e1 , e 2 , e3 equal to the three principal stresses – onto the deviatoric plane, also often called π -plane, i.e. the plane which is normal to the hydrostatic axis, given by equal principal stresses. The coordinates of this projection point in the co-ordinate system e1 , e 2 , e3 equal the principal values of the stress deviator. Using this deviatoric projection as a tool, this projection point can be obtained quite simply by vector addition of σ 1 e1d , σ 2 e2d , and σ 3 e3d , with arbitrary scale, see Figure 3.4; quite conveniently σ i = σ i / RM , i = 1, 2, 3 , can be used instead of σ i , where RM is the appropriate relevant strength parameter. The vector d from the origin of the deviatoric map to a specific stress point in a Cartesian coordinate system, with e y = e2d and e x to the right, has the components (σ 1 − σ 3 ) 3 / 2 in the direction of e x [− (σ 1 + σ 3 ) / 2 + σ 2 ] in the direction of e y if the standardised principal stresses are used. If the considered vector d corresponds to the stress in a specific point of the structure due to a specific action, the location of its end point vis-a-vis a limit curve, e. g. the unit circle with centre in the origin which corresponds to Mises' yield condition, visualises clearly the instantaneous behaviour of the structure under the considered action: If the end point is not outside of the limit curve, the stress state is compatible with the corresponding yield condition; if it is on the limit curve plastic deformation may occur "at the structural point". If the action changes - and the orientation of the principal axes does not change -, the stress state may change, and the vector d may change.

e2d= e y

d

e3

d

e2d

ex d

e3

e1d

e1 d

Figure 3.4: Deviatoric projection

If a cyclic action results in a cyclic stress state, the vector will describe a closed curve in this deviatoric map. A curve completely inside of the limit curve corresponds to only elastic stress states, portions on the limit curve correspond to plastic flow, portions outside are possible only in case of hardening, or for fictitious stress states, e. g. for a fictitious unlimited linear elastic constitutive law.

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If two actions A1 and A2 act simultaneously and both result in stress states (at a specific point of the structure) with the same principal axes, the vector property of the deviatoric mapping can be used: • If d1 corresponds to A1 , and d 2 to A2 (determined with a fictitious unlimited linear elastic constitutive law), then d = α1 d1 + α 2 d 2 corresponds to the stress state due to α1 A1 + α 2 A2 , where α1 and α 2 may be functions of time. • If d 3 corresponds to A for a specific constitutive law, e. g. a linear elastic – ideal plastic one or one used in an elastic compensation procedure, and d 4 corresponds to A for a different constitutive law, e. g. a linear elastic one, then d = d 3 - d 4 corresponds to a self equilibrating stress state Melan's shakedown theorem can be reformulated in the vectors of the deviatoric mapping: If a value β can be found such that the end point of the vector β ( d 3 - d 4 ) + α1 (t ) d 1 + α 2 (t ) d 2 for the prescribed cyclic action α1 (t ) A1 + α 2 (t ) A2 is never outside the limit curve, then the structure will, at the considered structural point P, shake down to purely elastic action. Even the necessary condition for shakedown, resulting immediately from Melan's shakedown theorem and usually designated as 3f-criterion, can be easily visualised: If the largest diameter of the path of the vector's end point – the stress path -, α 1 (t ) d 1 + α 2 (t ) d 2 , corresponding to a cyclic action α (t ) 1 + α 2 (t ) A in a specific 2RM , then the structure, with the specified RM, cannot shake down under this cyclic action. The extension of this procedure to more than two actions, or to more than one self-equilibrating stress, including, of course, those due to thermal stresses, is obvious and straightforward. 3.3.2.5 Shakedown Analysis for a single varying action (internal pressure) In the following procedure, it is assumed that the only action acting on the structure under consideration is internal pressure, and that it varies between zero and the maximum admissible pressure for shakedown PSmax SD. The problem in using Melan’s theorem is to find an optimal self-equilibrating stress field. Often, the optimal, or a near optimal, stress field can be found from the stress fields at the limit load: The difference of the linear-elastic stress field at (or near) the limit pressure of the structure (σ ij ) le ,l and the elasto-plastic stress field at (or near) the limit pressure (σ ij ) ep,l is a self-equilibrating stress field (σ ij ) res : (σ ij ) res = (σ ij ) ep,l − (σ ij ) le,l

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Since one endpoint of the considered linear-elastic load cycle is the point PS = 0, the selfequilibrating stress field used in Melan’s theorem must not violate the yield condition itself. Therefore, the self-equilibrating stress field (σ ij ) res has to be scaled with a factor β such that it does not violate the yield condition: (σ ij ) res ,co = β ⋅ (σ ij ) res . If the uncorrected self-equilibrating stress field according to the limit load (σ ij ) res does not violate the yield condition, the shakedown load is equal to the limit load. After this correction of the self-equilibrating stress field, the linear-elastic stress field with the possible greatest value of internal pressure has to be determined, such that the superposition with the corrected self-equilibrating stress field does not violate the yield condition. Because of the linearity, this can be done exactly for a fixed point of the structure. The stress field at a lower bound shakedown limit (σ ij ) SD is found as (σ ij ) SD = (σ ij ) res ,co + α ⋅ (σ ij ) le ,l . Note: The scaling factors α and β can be determined easily using the equivalent stress plots (or listings). Note: Another possibility of finding a self-equilibrating stress state is given by elasto-plastic unloading of the structure within an FE-calculation. 3.3.2.6 Shakedown analysis for one constant action (nozzle moment) and a single varying action (internal pressure) In the following procedure, the structures under consideration are cylinder-cylinder intersections, where a constant moment load M is acting at the nozzle and the cyclic load is given by the internal pressure, which varies between zero and the maximum admissible pressure for shakedown PSmax SD. Again, the problem in using Melan’s theorem is the determination of an optimal self-equilibrating stress field. The difference of the elasto-plastic stress field (σ ij ) ep ,( M + p ) , which corresponds to a loading state with the constant moment M and an internal pressure near the limit state of the structure, and the linear-elastic stress field at this state (σ ij ) le ,( M + p ) , the stress field (σ ij ) res ,( M + p ) = (σ ij ) ep ,( M + p ) − (σ ij ) le,( M + p ) is a self-equilibrating stress field. One endpoint of the considered linear-elastic load cycle is the point ( M , p = PS max SD ), the other endpoint is given by ( M , p = 0 ), where p = PS max SD is the maximum admissible pressure for

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shakedown of the structure. Therefore, to fulfil Melan’s theorem, the following conditions have to be met: φ [ β ⋅ (σ ij ) res ,( M + p ) + (σ ij ) M ,le + α ⋅ (σ ij ) p ,le ] ≤ 0 , φ [ β ⋅ (σ ij ) res , ( M + p ) + (σ ij ) M ,le ] ≤ 0 , where φ is the yield condition, (σ ij ) M ,le is the linear-elastic stress field corresponding to the moment M , (σ ij ) p ,le is the linear-elastic stress field corresponding to an arbitrary value of internal pressure p , and β and α are the factors which have to be determined such that the conditions are fulfilled. Thus, the maximum admissible internal pressure according to shakedown is given by PS max SD = α ⋅ p . Defining and using load cases in ANSYS®, the superposition of stress fields can be done easily. Unfortunately, in the examples quite often no combination of stress states, i.e. no factors β and α , could be found, such that the two conditions above were fulfilled. Often, the Mises’ equivalent stresses of the combined load cases were too high either at the outer surface of the weld-fillet or at the inner edge of the nozzle shell intersection. In this case, the conclusion is, that the chosen equilibrating stress field was not an appropriate one. Therefore, a linear combination of self-equilibrating stress fields – the one according to the limit state (σ ij ) res ,( M + p ) and the one according to moment load only (σ ij ) res , M – is used to fulfil Melan’s theorem. This procedure is permissible because of the following attributes of self-equilibrating stress fields: The sum of two self-equilibrating stress fields is a self-equilibrating stress field. The multiple of a self-equilibrating stress field is a self-equilibrating stress field. Using this procedure, the necessary conditions are given by φ [ β 1 ⋅ (σ ij ) res ,( M + p ) + β 2 ⋅ (σ ij ) res , M + (σ ij ) M ,le + α ⋅ (σ ij ) p ,le ] ≤ 0 , φ [ β 1 ⋅ (σ ij ) res , ( M + p ) + β 2 ⋅ (σ ij ) res , M + (σ ij ) M ,le ] ≤ 0 , where the maximum admissible internal shakedown pressure is now given by PS max SD = α ⋅ p . The self equilibrating stress field according to the moment only, (σ ij ) res , M , is given by the difference of the elasto-plastic stress field at this state (σ ij ) ep, M and the linear-elastic stress field at this state (σ ij ) le, M : (σ ij ) res , M = (σ ij ) ep, M − (σ ij ) le , M . For the determination of the factors β1 , β 2 and α the deviatoric maps of the stress states, i.e. the co-ordinates of a stress point given by its principal stresses, at the critical locations of the structure

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are used, since due to the increased number of factors and the different critical locations load case operations using the FE-software directly are not feasible. Nevertheless, the validity of the determined self-equilibrating stress field should be checked by superposition with the linear-elastic stress fields in the postprocessor of the FE-software. 3.3.2.7 Shakedown Analysis for problems including thermal stresses Since thermal stress fields are self-equilibrating stress fields they can be used in Melan’s theorem directly – e.g. multiplied with a suitable factor as part of a self-equilibrating stress, which is given by the sum of different self-equilibrating stress fields. A suitable factor for a thermally induced selfequilibrating stress field can be –0.5, since states with and without thermal stresses (e.g. the zero stress state) have to be considered within the cycle. If one part of the structure is highly influenced by the thermal stresses and another part by nonthermal action induced stresses, use of the sum of two self-equilibrating stress fields can be suitable – one being a thermally induced self-equilibrating stress field and the other one being induced by the non-thermal action. 3.3.2.8 Literature [1]

Zeman, J.L.: Repititorium Apparatebau – Grundlagen der Festigkeitsberechnung; Oldenbourg, 1992.

[2]

Gokhfeld, D.A., Cherniavsky, O.F.: Sijthoff & Noordhoff, 1980.

[3]

Preiss, R.: On the Shakedown Analysis of Nozzles Using Elasto-Plastic FEA; Intl. J. Pres. Ves. & Piping 76 (1999), 421-434.

[4]

König, J.A.: Shakedown of Elastic-Plastic Structures; Elsevier, 1987.

[5]

Burgreen, D.: Design Methods for Power Plant Structures; C.P. Press, 1975.

[6]

Preiss, R.: Das Einspielverhalten des Balkens mit Rechteckquerschnitt bei Biege- und Normalkraftbelastung; Bericht Nr. 13, Institut für Apparate- und Anlagenbau (A&AB), TU Wien, 1998.

[7]

Zeman, J.L., Preiss, R.: The Deviatoric Map – A Simple Tool in DBA. Intl. J.Pres. Ves. & Piping 76 (1999) 339-344.

[8]

Rammerstorfer, F.G.: On the Modeling of Cracks by Finite Elements; Scand. J. of Metallurgy, 12 (1983) p. 293 – 298.

Limit Analysis of Structures at Thermal Cycling;

3.4 Wind actions If a wind load is specified as an action on a structure, its effects have to be considered in the checks against GPD and PD. Since wind loading is three-dimensional, or at least not rotational-symmetric, the corresponding FE-model has to be three-dimensional too. A possibility to avoid a computation time intensive 3-D model in the elasto-plastic calculation is to use a 3-D model only for the

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calculation of the linear-elastic stresses caused by the wind. Furthermore, the GPD check and the PD (or SD) check can be performed using an axial-symmetric model without consideration of the wind effects (which would not be possible within this model), with decreased design resistance (or material strength parameters) given by the difference of the original design resistance (or material strength parameters) minus the maximum linear-elastic equivalent stresses in the structure due to the wind load. This procedure is admissible due to the positive definiteness of the equivalent stress, i.e. the sum of the maximum equivalent stresses of two stress tensors is greater or equal to the maximum equivalent stress of the sum of the two stress tensors. Of course, the usefulness of this approach is limited by the margin of the stresses caused by the wind action. In general, the local wind load per unit area is given by the product of the stagnation pressure times the local drag coefficient. Figure 3.5 shows a typical distribution of the (standardised) pressure distribution for a cylindrical structure with smooth surface as a function of the angle α from the stagnation point. The corresponding global drag coefficient is given by π c= ⋅ c p ,e ⋅ cos α dα 180 ∫0 180

The wind load component normal to the wind direction calculated with the global drag coefficient equals the one calculated with the local drag coefficients, but local effects due to the real pressure distribution are neglected in this approach.

1 0.5

cp,e

0 -0.5 -1 -1.5 0

25

50

75

100

125

150

175

Figure 3.5: Wind pressure distribution

To apply this pressure distribution, or a similar non rotational-symmetric one, performing an FE analysis is time consuming. It is, therefore, customary to use the following approximate procedure: With respect to the centre of an interesting cross-section, the resultant moment and (shear) force of the wind action above are calculated. At the upper end of a sufficiently long shell above the section of interest, the resultant moment and shear force are applied (often the shear force can be

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neglected). Of course, at cross-sections in some distance from the reference one, the results are inaccurate. This approach was followed in example 2 throughout the whole of the checks. To check the deviation from the non rotational-symmetric approach, the latter one was used also for the stress calculation due to the wind in example 2, but only for comparison – not in the DBA design checks. The difference in the wind effects is a remarkable 29% - the more accurate one giving the larger values.

3.5 Direct Route (using Elastic Compensation) 3.5.1 Check against Global Plastic Deformation GPD Elastic compensation[1-6] calculates bounds of the limit load and shakedown load for a - structure for a given load set by using iterative elastic FE-analysis. This method is a generalisation of the technique proposed by Marriot [5] for estimating lower bound limit loads on pressure vessel applications. The procedure involves calculating a series of elastic equilibrium stress fields where the stress is redistributed by altering the elastic modulus of each element based upon the maximum unaveraged nodal stress from the previous iteration, thus σ nom σ e max where E is the elastic modulus, i the iteration number, σnom is some nominal value, and σemax the maximum unaveraged nodal stress in that element from the previous solution. The resulting redistributed stress fields are equilibrium stress fields. By definition, if the equivalent stress anywhere in the equilibrium stress field does not exceed the yield stress of the material then that stress field relates to a lower bound on the limit load. Therefore, scaling the applied loads by the amount given by maximum stress in the redistributed stress field to the yield stress of the material will give the limit load, i.e. σy AL = Aap ⋅ σ max where AL is the limit load for the action(s), Aap is the applied load to the FE-model, σy is the yield strength of the material and σmax the maximum unaveraged nodal stress in the model. Due to the simplicity of this method, it lends itself to application in design checks against GPD according to the direct route method in prEN 13445-3 Annex B. Ei = Ei −1

As this method of determining limit loads is wholly elastic it is very simple to apply different yield criteria to the analysis without the convergence difficulties associated with elasto-plastic analysis. In the rules for the check against GPD, the analysis is required to be based on Tresca’s yield condition and associated flow rule, first order theory and an elastic-perfect plastic material model. This can be performed directly using the elastic compensation procedure for solid models. For analyses utilising shell elements, elastic compensation cannot be applied directly in the same method as described above. As a shell has only one element through thickness, it is not possible to modify the elastic modulus through the thickness. To allow the application of elastic compensation to shell elements a generalised yield model is adopted in the analysis. Ilyushin's[6] generalised yield model for a doubly curved shell is used which is based upon Mises' condition and associated flow

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rule. A brief overview of Ilyushin’s generalised yield model is given at the end of this sub-section. As the code specifies the application of Tresca’s, some modification of the results is required when the analysis is based upon Mises' condition. As stated in 3.3.1 the application of a factor of √3/2 to the design stress will result in a conservative result. This method can also be used as a check on the results for solid models utilising Tresca’s condition. As with the elasto-plastic method above, elastic compensation may be applied in two ways to the check against GPD. First, for specified actions a check on the admissibility of the load set can be made by checking that the equilibrium stress fields satisfy the lower bound limit load theorem. If the maximum equivalent stress anywhere in the equilibrium stress field remains below the design strength of the material then the specified loading is admissible. Second, the limit on the applied action(s) may be found using the above procedure and the maximum admissible action(s) can be determined according to prEN 13445-3 Annex B. Where there are multiple actions applied, the second case, where the actual limit is calculated, becomes more complex. For example in problems 3.1 and 3.2 there is a constant moment action and an internal pressure action. The limit on the pressure has to be found. In elastic compensation, the applied load set is scaled to give the limit load set. Therefore, in multiple action conditions one analysis is not sufficient to define the limit load, as the ratio of the loads at the limit is not already known. In this situation multiple analyses are made for different ratios of applied load and a limit locus is constructed that describes the limit state for all combinations of load. In the case of problems 3.1 and 3.2, with the constant moment known, the limit pressure can be found directly from the limit locus. As the code rules in prEN 13445-3 Annex B address elasto-plastic analysis and not any simplified method, some problems arise in applying elastic compensation. The application rule for GPD in prEN 13445-3 Annex B.9.2 states that the maximum absolute value of principal strain should not exceed 5%. As elastic compensation is not a displacement-based approach the values of strain are not accurate in the equilibrium stress fields and cannot be used. Therefore, it is possible for structures that are ‘stiff’ well into the plastic range (where the limit according to the code is defined by the 5% maximum principal strain limit) that elastic compensation non is conservative. In this situation the elastic compensation result would be close to the limit defined by loss of equilibrium in the elasto-plastic analysis (for an elastic-perfect plastic material). This can be noted in problem 3.1 where the elastic compensation result is very much higher than the elasto-plastic result defined by the limit on the principal strain. However, if the elasto-plastic result were defined by the tangent intersection method with no limit on the principal strain, the results would be similar. 3.5.2 Check against Progressive Plastic Deformation (PD) The principle in the check against PD according to prEN 13445-3 Annex B.9.3 is fulfilled if the structure can be shown to shake down, as progressive plastic deformation and alternating plasticity are the two possible failure modes if the structure fails to shakedown. It is possible using the elastic compensation procedure to calculate lower bounds on the shakedown load using elastic compensation. As used in the elasto-plastic check described in 3.3.2 above Melan’s shakedown theorem is also used in the elastic compensation procedure. The Mises limit stress field for the applied action(s) is calculated using elastic compensation as described above. As the zeroth iteration in elastic compensation is the true elastic stress field (i.e. no modulus modification), the residual stress field can be found by subtracting the elastic stress field from the redistributed limit

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stress field. When the limit stress field and applied loads are scaled to give the limit, the maximum stress in the limit field equals the yield stress of the material. Therefore, if the maximum stress in the residual stress field is less than the maximum stress in the limit stress field then the shakedown load is the same as the limit load. If, however, the maximum stress in the residual stress field is greater than the maximum stress in the limit stress field, then the shakedown load will be lower than the limit load. The value of the shakedown load in this case can be calculated by invoking the linear proportionality of the FE-solution, thus σy Ash = Aap ⋅ σ res max where Ash is the shakedown limit on the action, Aap is the applied value of the action, σy the yield strength of the material, and σres max is the maximum residual stress in the model. As with the check against GPD there are two possible ways of performing calculations in the PDcheck. First, by checking for admissibility of the action(s) by checking if the maximum stress in the residual stress field is lower than the design resistance for the PD-check. Second, by calculating the maximum shakedown load as described above, and then determining from that the maximum allowable shakedown load according to the code rules for the check against PD. As with the GPD calculations for elastic compensation with multiple actions described above, loci have to be constructed to describe the shakedown limits for all ratios of the actions. 3.5.3 Ilyushin’s generalised yield function In the usual classical theory of thin shells, stretching and bending stress resultants, Figure 3.6, are used, usually defined as 1 T /2 σ i dz T ∫− T / 2 1 T /2 Mi = 2 ∫ σ i z dz T −T / 2

Ni =

i = x, y, xy i = x, y, xy

where T is the shell thickness and σ x , σ y , σ xy , are in-plane stress components. z

M x ds y

y

x

N x ds y

M y ds x N y ds x

Figure 3.6: Thin shell variables M x ds y T

N x ds y

ds x ds y

N xyds y

N yx ds x

M y ds x

N y ds x

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The Mises yield function σ

2 x

− σ xσ

y

+σ

2 y

+ 3σ

2 xy

=σ

2 Y

where σ Y is the yield stress, can be approximated by a function of the thin-shell stress resultants, and, in this manner, a generalized yield surface obtained. This function – the generalized yield function – is conveniently expressed by means of the functions QN , QM and QNM : QN = Nx2 + Ny2 − Nx Ny + 3Nxy2 QM = Mx2 + My2 − Mx My + 3Mxy2 1 1 QNM = Nx Mx − Nx My − Ny Mx + Ny My + 3Nxy Mxy 2 2 Initially Ilyushin[6] presented an exact generalized yield model, but this was too complex for practical use (at that time) so a linear approximation was proposed (usually referred to as Ilyushin's generalised yield model):

∑ IL = QN + QM + 2

QNM 2 =σ Y 3

3.5.4 Literature [1]

Mackenzie D., Boyle J. T. et al: A simple method of estimating limit loads by iterative elastic analysis I, II & III, Int. J. Pres. Ves. & Piping 53 (1993) 77-142.

[2]

Mackenzie D., Nadarajah C. Shi J. & Boyle J. T.: Simple bounds on limit loads by elastic finite element analysis, Trans. ASME J. Pres. Ves. Tech 115 1993 27-31.

[3]

Mackenzie D., & Boyle J. T.: A simple method of estimating shakedown loads for complex structures, Proc. ASME PVP, Denver 1993.

[4]

Hamilton R., Mackenzie D., Shi J & Boyle J. T.: Simplified lower bound limit analysis of pressurised cylinder/cylinder intersections using generalised yield criteria, Int. J. Pres. Ves. & Piping 67 (1996) 219-226.

[5]

Marriot D. L.: Evaluation of deformation or load control of stresses under inelastic conditions using elastic finite element analysis, Proc. ASME PVP Conf., Vol 136, Pittsburgh 1988.

[6]

Ilyushin A. A.: Plasticity (Russian), Gostekhizda, Moscow, 1948 and Plasticite (French), Eyrolles, Paris 1956.

3.6 Stress Categorisation Route 3.6.1 General DBA based on stress categories is today well established, even if the method is not used as much as it should be, in order to optimise the design of pressure equipment. During the years the method has

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been developed, but still some work can be done, for example to improve and increase the table on typical cases. The method has been criticised for difficulties with the stress classification, but for thin-walled structures the classification is usually not a problem. The problems will occur in thick-walled structures, with complicated geometry, for example big valves and pumps. 3.6.2 Evaluation procedure The analysis is performed in three steps: Step 1 – Primary stresses Assure that the primary stresses caused by mechanical design loads will fulfil the requirements. The allowable design stress f is based on the design temperature, but no thermal effects or discontinuities are included. Step 2 –Secondary stresses Assure that the 3f (shakedown) criterion is fulfilled. Here f is based on t*, where t* is a kind of balanced value taking into account maximum and minimum temperatures during the considered operating cycle or cycles. The 3f-criterion shall be satisfied by any equivalent stress range resulting from variation of primary + secondary stresses between two normal operating conditions. A secondary stress is a stress which is self-equilibrating. It occurs at large discontinuities, but does not include stress concentrations. A secondary stress can be caused both by mechanical and thermal loads. It can be stated that if the secondary stress range, caused by for example thermal effects, is large, this will limit the pressure bearing capacity of the structure. Step 3 –Fatigue assessment The fatigue assessment can be directly based on the results from step 2. This is of the utmost importance as fatigue is the most important failure mode in pressure vessel technology. The fact that the requirements from step 2 are fulfilled will for most weld classes guarantee a fatigue life of more than 500 load cycles. For a more exact fatigue assessment, if required, the results from step 2 shall be used in a detailed assessment of fatigue life. 3.6.3 Notations In the plots, collected in the discussion of examples 1.1, 1.2, 1.3, 1.4, 2, the following notations are used: Gen.Dir. Inside (I), Outside (O) Circum Dir I, O Membrane (Tresca) Shear Tresca I, O

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3.7 Check against Instability Of the various approaches to perform the instability check three have been used here: 3.7.1 Usage of DBF or well-proven handbook formulae There are cases where the formulae of the DBF section or well-proven formulae in handbooks can be used, directly or indirectly, exactly or as an approximation. Wherever possible, this approach should be used, and if it is only used as a complement to one of the other approaches, or to obtain a reasonably good starting point for a classical (eigenvalue) procedure. It shall always be ascertained that the formulae used for imperfection-sensitive structures are not simply theoretical results but do contain appropriate reduction (knockdown) factors for imperfection-induced and for possible plasticity-induced effects. The partial safety factor for the resistance in the I-check, as specified in the draft standard, does not incorporate imperfection effects; these have to be taken into account in the determination of the resistances directly. In some cases the formulae, of the DBF section or of the handbooks, can be used directly – see, for instance, the buckling of the smaller cylinder of Example 2, or the case radial external pressure on inner shell of Example 6. In other cases it may be necessary to use results of FEM calculations, e. g. reactions between components, thermal stresses, as actions in the DBF or handbook models – see, for instance, the ring buckling check in Example 2, and the axial loading case of the inner shell of Example 6; in the latter example the axial force (per unit length) can be obtained by simple hand calculations directly as well. 3.7.2 Classical (eigenvalue) approach Modern software allows for the determination of classical (birfurcation) buckling loads. Appropriate reduction factors for both imperfection-induced and plasticity-induced effects have to be applied – these may be taken from the same sources as before – if necessary as approximations only. It is especially important to ascertain that an eigenvalue problem does exist for the structure and the load case considered. It is equally important to ascertain that the model used does allow for the appropriate buckling deformations - a symmetric model will not allow for non-symmetric buckling modes. Care is necessary to ascertain that the software used can reach the relevant buckling modes, and that it does so. For instance in Example 6, the jacket is subject to internal pressure. If pressure is applied on all walls of the jacket, the internal pressure action on the outside wall rendered non-convergence or non-usable results – the buckling modes of the internal wall, which is under outside pressure and for which buckling modes do exist, are not obtained. 3.7.3 Fully nonlinear approach In this approach a geometrically nonlinear analysis with nonlinear constitutive law is used, the initial imperfections of the structure are applied as initial geometry.

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In case of the investigation of a real structure these initial imperfections may be the actual imperfections (in detail) or the relevant buckling shapes obtained via one of the two former results but scaled-up by some measures of the actual imperfections, like out-of-roundness, flatness, peaking. In case of a virtual structure, specified by the drawings and relevant codes and standards, these initial imperfections may be the relevant buckling shapes obtained via one of the two former methods but here scaled-up using the allowed deviation (measures). Unfortunately the constitutive law to be used is not yet specified in prEN 13445-3; it was agreed that the very same model as in the PD-check should be used. The maximum action, corresponding to collapse of the structure, is the characteristic value of the resistance of the structure – to be divided by the relevant partial safety factor to obtain the design resistance. No additional reduction is required. There are various approaches for this proof: If the sufficient resistance against instability of a structure for a set of specified actions A1, . . ., An shall be proven, the design values of these actions shall be determined first, and then it shall be shown that the resistance of the structure (the limit carrying capacity) obtained by proportional increase of all design actions, is large enough, such that the limit values of the actions divided by the partial safety factor of the resistance are larger than the design values. With the (limit) multiplication factor MF obtained by means of the fully nonlinear model for the limit carrying capacity, this requirement can be written symbolically as  A1d  γ A1 A1  γ A1 A1 ⋅ MF    ...   ....   ..... /γ . ≤  =  R  ...   ....   .....        And  γ An An  γ An An ⋅ MF  If just a specific load case shall be proven to be admissible, it is not necessary to obtain the limit carrying capacity – it is only required to show that MF is not smaller than γ R , i. e. that the set of actions γ A1 γ R A1 , . . . ., γ An γ R An can be carried. 3.7.4 Literature [1] Samuelson, L. A., Eggwertz, S.: Shell Stability Handbook. Elsevier, London 1992. [2]

Bushnell, D.: Computerized buckling analysis of shells. Martinus Nijhoff Publishers, Dordrecht 1985.

[3]

Baker, E. H., Kovalevsky, L., Rish, F. L.: Structural Analysis of Shells. Krieger Publishing Comp., Malabar, USA 1986.

[4]

Ross, C. T. F.: Pressure Vessels under External Pressure. Statics and Dynamics. Elsevier, London 1990.

[5]

Como, M., Grimaldi, A.: Theory of Stability of Continuous Elastic Structures, CRS Press, Boca Raton 1995.

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[6]

Schmidt, H., Krysik, R.: Towards Recommendations for Shell Stability Design by Means of Numerically Determined Buckling Loads. Int. Coll. Buckling. Elsevier, London 1991.

[7]

Budiansky, B.: Theory of Buckling and Post-Buckling Behaviour of Elastic Structures. In: Advances in Applied Mechanics, Vol. 14, Academic Press, New York 1974.

[8]

Hutchinson, J. W.: Plastic Buckling. In: Advances in Applied Mechanics, Vol. 14, Academic Press, New York 1974

3.8 Check against Fatigue In the F-check of the DBA section of prEN 13445-3 reference is made to the fatigue calculations of the DBF route. Unfortunately this section, reprinted in Annex A2, is not easily readable. Some misprints are still in this version, but they are not always obvious as such, and they are often quite misleading. The flow-sheets on the following pages should help, misprints we discovered are corrected, nonunique approaches are deleted. The main changes are: •

•

Structural stresses to be determined by quadratic extrapolation in all cases, into the hot spot or the point of maximum equivalent stress, respectively. The extrapolation shall be performed as stated in Fig. 18-3 of prEN 13445-3, i. e. with pivot point distances from the critical point of 0.4e, 0.9e, 1.4e, e being the thickness at the critical point. The "thickness" to be used in the thickness correction formulae is the shortest distance from the crack initiation site (critical point) to the other wall to which the crack is likely to grow. In case of more than one potential crack direction, the largest of these shortest distances shall be used. K eff = K t if the equation for K eff gives a value greater than K t

•

Simplified procedures for using Class 100 design data for unwelded regions is deleted.

•

The flow-sheets are for repeated action in form of one and the same cycle, superposition with other cycles is not directly included, nor is the contribution to the overall damage of cycles with more than 2.106 repetitions in the case of unwelded regions. The results given for the various examples are for neutral environment – environmental assisted corrosion, including the cracking types, is excluded. For further reading, the booklet by Niemi, E.: Stress Determination for Fatigue Analysis of Welded Components. Abington Publ. , Cambridge 1995, and the literature cited there, can be recommended. The "definition" of structural stress given there differs from the one used here, but the numerical differences in the results are usually small.

21

DBA Design by Analysis

Data tmax = ..… °C tmin = ….. °C t* = 0,75 tmax + 0,25 tmin =….. °C Rm = ….. MPa Rp0,2/t* = ….. MPa

Page 3.21

Procedures

Rz = ….. µm (table 18-8) en = ….. mm ∆σD = ….. MPa (table 18-10 for N ≥ 2.106 cycles) N = …..…(for the first iteration) ∆σR = ….. MPa (allowable stress range for N< 2.106 cycles)

Stresses ∆σeq,t (total or notch equivalent stress range) = ….. MPa ∆σstruc (structural equivalent stress range) = ….. MPa (obtained by quadratic extrapolation)

ó

eq

σeqmax = ….. MPa (maximum notch equivalent stress)

= ..... MPa (mean notch equivalent stress )

Theoretical elastic stress concentration factor Kt Kt =∆σeq,t / ∆σstruc = …..

Effective stress concentration factorKeff K

eff

= 1 +

1,5 (K

t

− 1)

 Äó struc 1 + 0,5 K  t  Äó D

  

= .....

but not largen than Kt 18.8 Plasticity correction factor ke mechanical loading If ∆σstruc > 2 Rp0,2/t* k

Thermal loading

 Äó  struc = 1+ A0  − 1 e  2 R p0,2/t *  

with A0 =



0,5 for 800 MPa ≤ Rm ≤ 1000 MPa 0,4 for Rm ≤ 500 MPa  R − 500  0,4 +  m for 500 MPa ≤ R m ≤ 800 MPa  3000 

A0 = ….. ke = ….. ∆σtotal = ke.∆σeq,t = ….. MPa Else ∆σtotal = ∆σeq,t = ….. MPa

If ∆σstruc > 2 Rp0,2/t* 0,7 k = υ 0,4 0,5 + Äó  eq, l     R p0,2/t * 

kυ = ….. ∆σtotal = kυ.∆σeq,t = ….. MPa Else ∆σtotal = ∆σeq,t = ….. MPa

If both mechanical and thermal loadings are to be considered, the correction has to be made on each component of the stress tensors. ke and kí are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated. ∆σeq,struc = Äó

total

K

t

(for usage 18-11-3)

18-10-6-2 Temperature correction factor ft*

18-11-1-1 Surface finish correction factor fs

For t* > 100 °C ft* = 1,03 – 1,5.10-4 t* -1,5.10-6 t*2 = ….. Else ft* = 1

fs = Fs[0,1ln(N)-0,465] if N < 2.106 fs = Fs if N ≥ 2.106 with Fs = 1- 0,056 [ln (Rz)]0,64[ln(Rm)] +0,289 [ln (Rz)]0,53 = ….. fs = …..

22

Unwelded material – Ferritic steel Fatigue calculation

DBA Design by Analysis

Page 3.22

18-11-1-2 Thickness correction factor fe en ≤ 25 mm

25 mm ≤ en ≤ 150 mm fe = Fe

[0,1ln(N)-0,465]

en ≥ 150 mm fe = 0,7217[0,1ln(N)-0,465] if N< 2.106

N < 2.106

if

fe = Fe if N ≥ 2.106 with Fe = (25/en)

fe = 1

fe = 0,7217 if N ≥ 2.106

0,182

= .....

fe = …..

fe = …..

18-11-1-3 Mean stress correction factor fm If ∆σstruc < 2 Rp0,2/t* and σeqmax < Rp0,2/t*

If ó If ó

eq eq

and ó

> 0 then ó < 0 then ó

eq

= ó

For N ≥ 2.106 cycles See figure 18-14

eq, r

eq, r eq, r

= Rp0,2/t* =

∆σ eq, t 2

∆σ eq, t 2 - Rp0,2/t*

= ….. MPa

For N ≤ 2.106 cycles M = 0,00035 Rm – 0,1 = ….. if –Rp0,2/t* ≤ ó

fm = …..

If ∆σstruc >2 Rp0,2/t*

If ∆σstruc < 2 Rp0,2/t* and σeqmax > Rp0,2/t*

eq

≤

Äó

Äó

R then 2(1 + M )

 M(2 + M )  2ó   eq  fm = 1 1 + M  ÄóR     

0,5

if

R ≤ ó ≤R p0,2/t* then eq 2(1 + M )

2ó 1 + M 3 M  eq  f = − = …. = ….. m 1 + M 3  ÄóR 





fm = 1

18-11-2-1 Overall correction factor fu f .f .f .f fu = s e m t * = …..

Äó

K eff

eq, struc

/f

u

=

18-11-3 Allowable number of cycles N 2

  4, 6⋅10 4  if N ≤ 2.106 cycles N =  ∆σ eq , struc − 0 , 63 R + 11 , 5 m  f u 

N = ∞ if Äó

eq, struc

/f

u

≤ ∆σ

D

N = …..

N is obtained by iterations. If the value of N oscillates, a difference equal to 1 % between two iterations is acceptable. If the values decrease monotonously, the difference must be less than 0,001 % between two iterations.

v

Welded material – Ferritic steel Fatigue calculation

DBA Design by Analysis

Data tmax = ..… °C tmin = ….. °C t* = 0,75 tmax + 0,25 tmin =….. °C Rm = ….. MPa Rp0,2/t* = ….. MPa

Page 3.23

23

en = ….. mm ∆σD (5.106cycles) = ….. MPa (class …..) equivalent stresses or principal stresses m = 3 C = ….. m = 3 C⊥ = ….. C// = ….. m = 5 C = ….. m = 5 C⊥ = ….. C// = …..

Stresses ∆σstruc = ….. MPa (structural equivalent stress range,determined by extrapolation) [or ∆σstruc ⊥ = ….. MPa and ∆σstruc// = ….. MPa]

18.8 Plasticity correction factor ke

Thermal loading

mechanical loading

If ∆σstruc > 2 Rp0,2/t*

If ∆σstruc > 2 Rp0,2/t* k

 Äó  struc = 1+ A0  − 1 e  2 R p0,2/t *  

with A0 =

k



0,5 for 800 MPa ≤ Rm ≤ 1000 MPa 0,4 for Rm ≤ 500 MPa  R − 500  0,4 +  m for 500 MPa ≤ R m ≤ 800 MPa  3000 

A0 = ….. ke = ….. ∆σ = ke ∆σstruc = ….. MPa Else ∆σ = ∆σstruc = ….. MPa

υ

= 0,5 +

0,7 0,4

 Äó   struc   R p0,2/t * 

kυ = ….. ∆σ = kυ ∆σstruc = ….. MPa Else ∆σ = ∆σstruc = ….. MPa

If both mechanical and thermal loadings are to be considered, the correction has to be made on each component of the stress tensors . ke and kí are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated. 18-10-6-1 Thickness correction factor few en ≤ 25 mm

25 mm ≤ en ≤ 150 mm few = (25/en)

few = 1

0,25

few = …..

en ≥ 150 mm

= ..... few = 0,639

DBA Design by Analysis

24

Unwelded material – Austenitic steel Fatigue calculation

Page 3.24

18-10-6-2 Temperature correction factor ft* For t* > 100 °C Else ft* = 1

ft* = 1,03 – 1,5.10-4 t* -1,5.10-6 t*2 = …..

18-10-6-3 Overall correction factor fw fw = few.ft* = ….. Data Rz = ….. µm (table 18-8) 18-10-7 Allowable number of cycles N tmax = ..… °C e n = ….. mm tmin = ….. °C . 6 ∆σ 8 Äó t* = =0,75 + 0,25 tminN=….. …..tmax MPa = ∞ °C if < ∆σ10 , else D = ….. MPa (table 18-10 for N ≥ 2 10 cycles) fw N = …..…(for the first iteration) Rm = ….. MPa ∆σR = ….. MPa (allowable stress range for N < 2.106 cycles) Rp1,0/t* = ….. MPa 6 6 Äó . . If Äó > ∆σ then If < ∆σ 5 10 cycles 5 10 cycles and other If Äó < ∆σ5.106 cycles and all other fw fw f w

Stresses m = 3 and C (C⊥ or C//) = ….. cycles with Äó > ∆σ5.106 cycles cycles with Äó < ∆σ5.106 cycles fw fw ∆σeq,t (total or notch equivalent stress range) = ….. MPa then then ∆σstruc (structural equivalent stress range) = ….. MPa (obtained by extrapolation) m = 5 and C (C⊥ or C//) = ….. C (maximum σeqmax = ….. MPa N ó = .....CMPa (mean notch equivalent stressN ) = = ∞ notch equivalent stress) m = ….. N= eq m = …..  ∆σ   f   w  Theoretical

 ∆σ   fw

  

elastic stress concentration factor Kt

Kt =∆σeq,t / ∆σstruc = …..

Effective stress concentration factorKeff K

eff

1,5 (K

= 1 +

with C from Table 18-7 of prEN 13445-3, 1 + 0,5

t

− 1)

 Äó struc K  t  Äó  D

  

= .....

in dependence of the (weld) class, given by prEN 13445-3, Tables 18.4 and 18.5, respectively. but not larger than Kt 18.8 Plasticity correction factor ke mechanical loading If ∆σeq,l > 2 Rp1,0/t*

  Äó eq, l k = 1 + 0, 4  − 1 e  2 R p1,0/t *  ke = ….. ∆σtotal = ke.∆σeq,t = ….. MPa Else ∆σtotal = ∆σeq,t = ….. MPa

Thermal loading If ∆σeq,l > 2 Rp1,0/t* 0,7 0,4 0,5 + Äó  eq, l     R p1,0/t *  kυ = ….. ∆σtotal = kυ.∆σeq,t = ….. MPa Else ∆σtotal = ∆σeq,t = ….. MPa k = υ

If both mechanical and thermal loadings are to be considered, the correction has to be made on each component of the stress tensors. ke and k í are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated.

DBA Design by Analysis

∆σeq,struc = Äó

total

K

t

Unwelded material – Austenitic steel Fatigue calculation

25 Page 3.25

(for usage in 18-11-3)

18-10-6-2 Temperature correction factor ft*

18-11-1-1 Surface finish correction factor fs

For t* > 100 °C ft* = 1,043 – 4,3.10-4 t* = ….. Else ft* = 1

fs = Fs[0,1ln(N)-0,465] if N < 2.106 fs = Fs if N ≥ 2.106 with Fs = 1- 0,056 [ln (Rz)]0,64[ln(Rm)] +0,289 [ln (Rz)]0,53 = ….. fs = …..

Welded material – Austenitic steel Fatigue calculation

DBA Design by Analysis

18-11-1-2 Thickness correction factor fe en ≤ 25 mm 25 mm ≤ en ≤ 150 mm

26 Page 3.26

en ≥ 150 mm

[0,1ln(N)-0,465]

fe = 0,7217[0,1ln(N)-0,465] if N < 2.106 fe = Fe if N ≥ 2.106

fe = Fe if N < 2.106 6 . fe = Fe if N ≥ 2 10 with Fe = (25/en)0,182 = .....

fe = 1

fe = …..

fe = …..

18-11-1-3 Mean stress correction factor fm If ∆σstruc < 2 Rp1,0/t* If ∆σstruc >2 Rp1,0/t* If ∆σstruc < 2 Rp1,0/t* Data and σeqmax < Rp1,0/t* and σ > R en =eqmax ….. mmp1,0/t* tmax = ..… °C ∆σD (5.106cycles) = ….. MPa (class …..) tmin = ….. °C equivalent stresses or principal stresses t* = 0,75 tmax + 0,25 tmin =….. °C ∆σeq, t Rm = ….. MPa m = 3 C = ….. m = 3 C⊥ = ….. If ó > 0 then ó = Rp1,0/t* - 2 eq eq, r Rp1,0/t* = ….. MPa C// = ….. ∆ ó eq, t m = 5 C = ….. m = 5 C⊥ = ….. If ó < 0 then ó = 2 - Rp1,0/t* eq eq, r C// = ….. Stresses and ó = ó = ….. MPa eq range,determined eq, r ∆σstruc = ….. MPa (structural equivalent stress by extrapolation) [or ∆σstruc⊥ = ….. MPa and ∆σstruc// = ….. MPa] For N ≥ 2.106 cycles See figure 18-14 18.8 Plasticity correction factor ke

fm = ….. Äó

e

= 1 + 0, 4 

Äó Thermal loading R ≤ ó ≤R if p1,0/t* then eq If2∆σ (1 +strucM>) 2 Rp1,0/t*

mechanical loading Äó R then if –Rp1,0/t* ≤ ó ≤ eq 2(1 + M )

If ∆σstruc > 2 Rp1,0/t*

k

For N ≤ 2.106 cycles M = 0,00035 Rm – 0,1 = …..

struc

 M(2 + M )  2ó   eq  fm = 1  1 + M  ÄóR    − 1

 2 R p1,0/t *

0,5



18-11-2-1 Overall correction factor fu f .f .f .f fu = s e m t * = ….. A0 = …..K eff ke = ….. ∆σ = ke ∆σ MPa of cycles N struc = …..number 18-11-3 Allowable Else ∆σ = ∆σstruc = ….. MPa





Äó

 2ó  1 + M 3 M  eq  f = − = …. = ….. m 1 + M 0,7 3  ÄóR  k =   υ 0,4 0,5 +  Äó   struc   R p1,0/t *  eq, struc

fm = 1

/ f = ….. u kυ = ….. ∆σ = kυ ∆σstruc = ….. MPa Else ∆σ = ∆σstruc = ….. MPa

2

4, 6⋅10 4 If both mechanical are to be considered, the correction has to be made on each component of the  loadings N = ∆σ eq , struc and thermal if N ≤ 2.106 cycles N = ∞Äó ifstruc Äóis the full/ mechanical f ≤ ∆σ and thermal stress tensors . k and k are to be calculated with the above formulas where eq, struc u D  f u −e 0 , 63 Rm +í 11,5  equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated.

18-10-6-1 N = ….. Thickness correction factor few N is obtained by iterations. If the value of N oscillates, a difference equal to 1 % between two iterations is en ≤If25 25 mm ≤ en ≤ 150 en ≥ 150 acceptable. themm values decrease monotonously, the mm difference must be less thanmm 0,001 % between two 0,25 iterations. few = (25/en) = ..... few = 1

few = …..

few = 0,639

DBA Design by Analysis

Welded material – Austenitic steel Fatigue calculation

27 Page 3.27

28

DBA Design by Analysis

Page

Procedures

3.28

18-10-6-2 Temperature correction factor ft* For t* > 100 °C Else ft* = 1

ft* = 1,043 – 4,3.10-4 t* = …..

18-10-6-3 Overall correction factor fw fw = few.ft* = ….. 18-10-7 Allowable number of cycles N Äó fw

If

= ….. MPa Äó fw

N = ∞ if

> ∆σ5.106 cycles then

m = 3 and C (C⊥ or C//) = …..

N=

C  ∆σ   f   w 

m

= …..

Äó fw

< ∆σ108, else: If

Äó fw

< ∆σ5.106 cycles and other

cycles with

Äó fw

> ∆σ5.106 cycles

then m = 5 and C (C⊥ or C//) = ….. N = C m = …..  ∆σ   fw

  

If

Äó fw

< ∆σ5.106 cycles and all other

cycles with

Äó fw

< ∆σ5.106 cycles

then N=∞

with C from Table 18-7 of prEN 13445-3, in dependence of the (weld) class, given by prEN 13445-3, Tables 18.4 and 18.5, respectively.

DBA Design by Analysis

1

Illustrative Example

Page 4.1

4 Illustrative Example 4.1 Introduction A simple illustrative example is described in this Section - a circular plate under the action of a uniform pressure action which varies between zero and a maximum value. This Section should be read in conjunction with prEN13445-3, Annex B. The checks which have to be considered according to this are the check against global plastic deformation (GPD), the check against progressive plastic deformation (PD), the fatigue (F) check, the check against instability (I), and the static equilibrium check (SE). In the following the maximum admissible pressure according to the GPD-check and the maximum pressure for shakedown of the structure, with a load variation between zero and this maximum, are determined. If the structure shakes down under a given action cycle, the admissibility of this action cycle against PD is proven – see subsection 3.3.2.1 of section 3 – Procedures. For the calculation of the allowable number of action cycles, in the F-check, an upper value for the pressure equal to 90% of the maximum allowable pressure (given by the other checks) is used: see also subsection 5.1 of section 5 – Case Specification. For this structure and the specified action cycle, the I-check and the SE-check are not required.

4.2 Problem Specification The dimensions of the plate are: diameter 500 mm, thickness 25 mm. The design temperature is specified as 20°C. The plate is assumed to be clamped at the edge. The material of the plate is the ferritic steel P265GH according to EN 10028-2, the surface is assumed to be machined with RZ = 50 µm . The material strength parameter is RM = 255 MPa , it corresponds to the upper yield strength of P265GH at room temperature for a thickness range of 16 mm to 40 mm according to EN 10028-2. The modulus of elasticity is given by E = 212 GPa (see subsection 5.4 of section 5 – Case Specification).

4.3 Finite element model and boundary conditions Since the radius to thickness ratio of the plate is relatively small, and since the application of the checks can be shown more clearly, 2-D axisymmetric solid elements are used instead of plate elements for the finite element model: see Figure 4.1. The mesh is fairly dense, and would be expected to yield accurate solutions for elastic and inelastic analysis without further refinement. The commercial finite element analysis system ANSYS© is used here.

DBA Design by Analysis

2

Illustrative Example

Page 4.2

Figure 4.1: Finite element model

The boundary conditions require some thought. In the technical theory of structures (beams, plates and shells) the notion of a “clamped edge” of a plate has a specific meaning – displacement in thickness direction and tangent rotation of the mid-plane at the plate’s edge are zero. In a solid model a “clamped edge” has to be modelled appropriately and suitable boundary conditions chosen. In the present example, which of course is chosen just to illustrate DBA and not modelling, the vertical and horizontal displacements in the nodes at the plate’s edge were constrained to zero. These boundary conditions seem to be fairly reasonable, but they create a localised stress concentration near the clamped edge. 4.4 Determination of the maximum admissible pressure according to the GPD-Check In general terms, the principles specified in DBA, as given in prEN 13445-3, Annex B, require that the design effect of actions does not exceed the corresponding design resistance. The design effect of actions is the relevant response of the model, with a specified material model, to the relevant design actions. What the relevant effect actually is depends on the design check under consideration. The design actions are products of specified characteristic values of actions and corresponding partial safety factors of actions. Which design actions are relevant depends on the load cases that have to be considered: In the GPD-check it is the carrying capacity of the model that matters, that is, the carrying capacity in terms of actions. Therefore, the design effects are the design actions themselves. In the simple example considered here there is only one action – pressure - and only one load case to be considered – maximum allowable pressure. In this specific case the design principle can here be rephrased to read: The design action, given by the product of the maximum allowable pressure and the corresponding partial safety factor, shall not be larger than the design resistance which is given by the ratio of the maximum pressure the model can carry to the partial safety factor of the resistance. Following the proposal for change in this check, as stated at the beginning of Annex 2 of this manual, there is a side-condition to be taken into account when determining this maximum pressure: the side-condition is that the maximum absolute value of the principal strains for this maximum pressure must not exceed 5%.

DBA Design by Analysis

3

Illustrative Example

Page 4.3

4.4.1 Approach using Mises yield criterion In the GPD–check, Tresca's yield criterion is specified. Depending on the analysis software being used, if no subroutine for this yield criterion is available, or if it is available but shows bad convergence, Mises’ yield criterion can be used instead: Since the maximum ratio of the Mises equivalent stress to the Tresca equivalent stress for the same load is 2 3 , a GPD-check with Mises' yield criterion, with the design material strength parameter multiplied by

3 2 , will always lead to conservative results.

Furthermore, if the result of the check against PD (where Mises’ yield criterion is allowed) is used in the check against GPD (instead of a separate calculation) and if, like in this problem, there is only one partial safety factor of the resistance γ R , multiplication of the limit pressure, from the check against PD, with 3 2 leads to the same result as the multiplication of the material strength parameter. Of course, since no partial safety factors are used in the check against PD, they have to be taken into account by scaling down the PD-check results. As stated in the application rule in prEN 13445-3 Annex B.9.2, the maximum absolute value of the principal strains must not exceed 5%. To fulfil this requirement in cases where the results from the PD-check are used, a value of the pressure from the check against PD shall be used such that the maximum absolute value of the principal strains calculated with this pressure (and the material strength parameter used for the check against PD) does not exceed 5 %. Since in the final loadstep the maximum principal strain of the elasto-plastic calculation for the PDcheck, which corresponds to a pressure of 8.48 MPa, was about 11%, a lower pressure value had to be used in the GPD-check. Figure 4.2 shows the principal strain distribution for a pressure of 8.43 MPa - the maximum value is approximately equal to the allowed 5%. Figure 4.2: Distribution of principal strain

The partial safety factor for the resistance γ R according to prEN-13445-3 Annex B, Table B.9-3 is 1.25 , and the partial safety factor for pressure action without a natural limit is given by γ P = 1.2 , according to prEN-13445-3 Annex B, Table B.9-2.

DBA Design by Analysis

4

Illustrative Example

Page 4.4

Thus, the maximum pressure according to the GPD-check is, in this approach, given by

PS max GPD =

8.43 3 8.43 3 ⋅ = ⋅ = 4.87 MPa . γ P ⋅ γ R 2 1.2 ⋅ 1.25 2

4.4.2 Approach using Tresca’s yield criterion In prEN 13445-3 Annex B, Tresca’s yield criterion is prescribed. Unfortunately most commercial software does not include this criterion in elasto-plastic calculations (although this could be available in future releases). The approach given in Sec.4.4.1 is a simple work-around, but one which leads to conservative results. To show the possibilities given by the Standard, an analysis was performed with Tresca’s yield criterion – the special routine was provided by an ANSYS© distributor: The design material strength parameter is given by 255 / γ R = 255 / 1.25 = 204 MPa . In this analysis, a first order theory and a linear-elastic ideal-plastic material with design material strength parameter of 204 MPa were used. The pressure was increased until either an absolute maximum was obtained or the maximum absolute value of the principal strains reached 5%. In this example the second condition governed – at a pressure of 6.07 MPa the maximum absolute value of the principal strains reached 5%. Therefore, the maximum allowable pressure according to this approach for the GPD-check is given by PS max GPD , T = 6.07 / γ p = 6.07 / 1.2 = 5.06 MPa. This value is about 4% larger than the one obtained in Sec.4.4.1. Finally, it should be noted that since most commercial software do not offer this approach directly, the result obtained in Sec.4.4.1 is used in the F-check. 4.5 Check against PD The PD-check was performed by way of a shakedown check using Melan’s shakedown theorem – see subsections 3.3.2.1 and 3.3.2.5 of Section 3 (Procedures) for further details. The elasto-plastic finite element analysis was carried out as required in prEN 13445-3 Annex B, Sec. B.9.3.1, using Mises’ yield condition and associated flow rule, a linear-elastic ideal-plastic constitutive law with a design material strength parameter of 255 MPa, and first order theory. By defining and using load cases in ANSYS®, the superposition of stress fields can be done easily. Therefore, the first load step of the analysis was defined to be at a very low load level (0.1 MPa), so that there was a linear-elastic response of the structure. All other linear-elastic stress fields can then be determined easily by multiplication with a suitable scale-up factor.

5

DBA Design by Analysis

Illustrative Example

Page 4.5

The analysis was carried out using the arc-length method, using a maximum vertical displacement of 25 mm in the middle of the plate as termination criterion. The termination criterion was fulfilled for a pressure of 8.48 MPa , and this pressure was used as limit pressure. Figure 4.3 shows the elasto-plastic Mises equivalent stress distribution for the limit pressure of 8.48 MPa .

Figure 4.3: Elasto-plastic Mises‘ equivalent stress distribution

Figure 4.4 shows the linearelastic Mises’ equivalent stress distribution for the limit pressure – the stress maximum is located near the edge of the plate.

Figure 4.4: Linear-elastic Mises’ equivalent stress distribution for the limit pressure

DBA Design by Analysis

6

Illustrative Example

Figure 4.5 shows the Mises’ equivalent stress distribution of the corrected residual stress field, the scaling factor β is given by 0.719 (see subsection 3.3.2.5 of section 3 – Procedures).

Figure 4.5: Mises‘ equivalent stress distribution of the corrected residual stress field

Figure 4.6 shows the Mises’ equivalent stress distribution at the lower bound shakedown limit. The scaling factor α is given by 0.854 (see subsection 3.3.2.5 of section 3 – Procedures).

Figure 4.6: Mises’ equivalent stress distribution at the lower bound shakedown limit

Page 4.6

7

DBA Design by Analysis

Page 4.7

Illustrative Example

Thus, the shakedown limit pressure is given by PS max SD = 8.48 ⋅ 0.854 = 7.24 MPa . This value is already well above the value of the maximum allowable pressure of 4.87 MPa, given in Sec.4.4.1, or 5.06 MPa in 5.2. Thus, a further (complicated) investigation of the PD behaviour is not required. A simulation with pressure cycling between 0 and 8.0 MPa has shown that the model shakes down under this cyclic action to steady-state behaviour after four action cycles, within the numerical accuracy that can be expected. Of course, since the maximum pressure, 8.0 MPa, is larger than 7.24 MPa, the value obtained for elastic shakedown, the model does not shake down to elastic behaviour, but to a purely cyclic behaviour, where at the end of each cycle the stress distribution is equal to the one at the beginning, and where in two distinct and nonconnected regions alternating plasticity occurs such that the strain increment over one cycle is zero in every point – within the numerical accuracy. Figure 4.7 shows this steady-state Mises equivalent stress distribution for maximum pressure, and Figure 4.8 the deviatoric mappings of the stress state in the node of maximum accumulated plastic strain, for maximum and minimum pressure. The connection line already passes close to the origin – an indication that steady-state behaviour with alternating plasticity is almost reached.

Figure 4.7: Steady state equivalent stress distribution σ hoop

minimum pressure

maximum pressure

Figure 4.8: Deviatoric map

σ normal

σ radial

8

DBA Design by Analysis

Illustrative Example

Page 4.8

Figure 4.9 finally shows the evolution, over four cycles, of the radial strain, for maximum pressure and for minimum pressure, respectively.

Figure 4.9: Total radial strain versus load history

Note: Within the framework of the technical theory of plates the model does sustain theoretically pressure cycles from zero to the limit pressure (according to the GPD-check) without progressive plastic deformation.

4.6 Fatigue (F) – Check The number of allowable action cycles has been determined for a lower value of the pressure equal to Pop, inf = 0 MPa and an upper value of Pop, sup = 0.9 ⋅ PS max GPD = 0.9 ⋅ 4.87 = 4.38 MPa . The location where the linear-elastic stress maximum occurs is near the edge of the model. The structural equivalent stress range at this point is obtained by quadratic extrapolation. Figure 4.10 shows the corresponding pivot points: node 65, node 67 and node 69, the stress shown in this plot is Mises’ equivalent stress for a pressure of 0.1 MPa . The details of the fatigue calculation are given on the pages 4.9 and 4.10 – the fatigue calculation sheets.

9

DBA Design by Analysis

Data tmax = 20°C tmin = 20°C t* = 0,75 tmax + 0,25 tmin = 20°C Rm = 410 MPa Rp0,2/t* = 255 MPa

Page 4.9

Illustrative Example

Rz = 50 µm (table 18-8) en = 25 mm ∆σD = 279.3 MPa (table 18-10 for N ≥ 2.106 cycles) N = 100 000 (for the first iteration) ∆σR = 407.8 MPa (allowable stress range for N< 2.106 cycles)

Stresses ∆σeq,t (total or notch equivalent stress range) = 344.5 MPa ∆σstruc (structural equivalent stress range) = 312.3 MPa (obtained by quadratic extrapolation)

ó

eq

σeqmax = 344.5 MPa (maximum notch equivalent stress)

= 172.75 MPa (mean notch equivalent stress

Theoretical elastic stress concentration factor Kt Kt =∆σeq,t / ∆σstruc = 1.1031

Effective stress concentration factorKeff K

eff

1,5 (K

= 1 +

1 + 0,5 K

t

− 1)

 Äó struc  t Äó  D

  

= 1 . 0957

but not largen than Kt 18.8 Plasticity correction factor ke mechanical loading If ∆σstruc > 2 Rp0,2/t*

Thermal loading

 Äó  struc k = 1+ A0  − 1 e  2 R p0,2/t *  with A0 =

0,5 for 800 MPa ≤ Rm ≤ 1000 MPa 0,4 for Rm ≤ 500 MPa  R − 500  0,4 +  m for 500 MPa ≤ R m ≤ 800 MPa  3000 

A0 = ….. ke = ….. ∆σtotal = ke.∆σeq,t = ….. MPa Else ∆σtotal = ∆σeq,t = 344.5 MPa

If ∆σstruc > 2 Rp0,2/t* 0,7 k = υ 0,4 0,5 +  Äóeq, l     R p0,2/t * 

kυ = ….. ∆σtotal = kυ.∆σeq,t = ….. MPa Else ∆σtotal = ∆σeq,t = ….. MPa

If both mechanical and thermal loadings are to be considered, the correction has to be made on each component of the stress tensors. Ke and kí are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated. ∆σeq,struc = Äó

total

K

t

(for usage in 18-11-3 and 18-11-2-1) = 312.3

18-10-6-2 Temperature correction factor ft*

18-11-1-1 Surface finish correction factor fs

For t* > 100 °C ft* = 1,03 – 1,5 10-4 t* -1,5 10-6 t*2 = ….. Else ft* = 1

fs = Fs[0,1ln(N)-0,465] if N < 2.106 fs = Fs if N ≥ 2.106 with Fs = 1- 0,056 [ln (Rz)]0,64[ln(Rm)] +0,289 [ln (Rz)]0,53 = 0.7889 fs = 0.8539

DBA Design by Analysis

Page 4.10

Illustrative Example

18-11-1-2 Thickness correction factor fe en ≤ 25 mm

25 mm ≤ en ≤ 150 mm fe = Fe[0,1ln(N)-0,465] if N < 2.106 fe > Fe if N ≥ 2.106

en ≥ 150 mm fe = 0,7217[0,1ln(N)-0,465] if N< 2.106 fe = 0,7217 if N ≥ 2.106

with Fe = (25/en)0,182 = .....

fe = 1

fe = …..

fe = …..

18-11-1-3 Mean stress correction factor fm If ∆σstruc < 2 Rp0,2/t* and σeqmax < Rp0,2/t*

If ∆σstruc >2 Rp0,2/t* If ∆σFig.4.10: struc < 2 Rp0,2/t* Pivot points for quadratic interpolation and σeqmax > Rp0,2/t*

If ó If ó

eq eq

and ó

> 0 then ó < 0 then ó = ó

eq

For N ≥ 2 106 cycles See figure 18-14

eq, r

∆σ eq, t 2 = 82.75

∆σ eq, t 2 - Rp0,2/t*

=

= 82.75 MPa

For N ≤ 2 106 cycles M = 0,00035 Rm – 0,1 = …..

if –Rp0,2/t* ≤ ó

fm = …..

eq, r

eq, r

= Rp0,2/t* -

eq

≤

Äó

Äó

R then 2(1 + M )

 M(2 + M )  2ó   eq  fm = 1 1 + M  ÄóR     

0,5

R ≤ ó ≤R p0,2/t* then eq 2(1 + M )

if

 2ó  1 + M 3 M  eq  f = − = …. m = 0.9826 1+ M 3  ÄóR  



fm = 1

18-11-2-1 Overall correction factor fu f .f .f .f fu = s e m t * = 0.7658

Äó

K eff

eq, struc

/f

u

= 407.8

18-11-3 Allowable number of cycles N 2

  4, 6⋅10 4  if N ≤ 2 106 cycles N =  ∆σ eq , struc − 0 , 63 ⋅ R + 11 , 5 m  f u 

N = ∞ if Äó

eq, struc

/f

u

≤ ∆σ

D

N = 81600

N is obtained by iterations. If the value of N oscillates, a difference equal to 1 % between two iterations is acceptable. If the values decrease monotonously, the difference must be less than 0,001 % between two iterations.

DBA Design by Analysis

5

Specification of examples

5.1

General

Specification of Examples

Page 5.1

In this section the specifications are given of ten examples, dealt with by the group. The various design checks had been assigned to the members a priori, differences discussed, and, where necessary, supplemented by additional investigations and corrected. The results are summarized in the next section, the details in the section thereafter. The specifications are complete, but following the examples in details, or using them as benchmarks, it is necessary to consult standards, like material standards, drafts of other parts of prEN13455, etc. As a help, physical properties of materials used in the examples are collected in an annex to this section. The geometries specified in the drawings are already those to be used in the analyses, i. e. the thicknesses given are already analysis thicknesses, allowances – for tolerances and, if relevant, corrosion – have already been deducted. Should the corrosion allowance be required for specifying the weld regions: A value of 1.00 mm was used for ferritic steels, and 0 for austenitic ones. The proposal for Detailed Fatigue Analysis states that in the (fatigue) analysis extreme operating values of actions rather than design values should be used. For the calculation of the allowable number of action cycles an upper value for the pressure Pop, sup equal to 90% of the maximum allowable pressure PS is specified in the examples here. In cases where the maximum allowable pressure PS can be determined by the Design by Formulae (DBF) section of the CEN TC 54 proposal of an Unfired Pressure Vessel standard, PS has not been specified here. The maximum allowable pressure according to this DBF proposal - PS max DBF - shall be used as characteristic value in the design checks for Gross Plastic Deformation (GPD), Progressive Plastic Deformation (PD) or Shakedown (SD): Pc = PS max DBF In the other cases values for PS are specified. Note: Unfortunately, because of the combination of ideas and designations from Euronorm 3 on one hand, and those from the Pressure Equipment Directive (PED) and EN 764 on the other hand, one has to distinguish between the design values of (the action) pressure – obtained by multiplying the characteristic values of pressure by the relevant partial safety factors – and the design pressure Pd - the maximum pressure at the top of the equipment specified by the manufacturer and used for the determination of the calculation pressures, mainly within the framework of DBF.

DBA Design by Analysis

Specification of Examples

Page 5.2

To minimize the possibility of confusing the two, the notion design pressure – relevant to DBF – is never used here. The specified values for PS or those determined by DBF shall be used as upper characteristic values; the design value for (the action) pressure shall not be called design pressure and especially not be denoted by Pd ! (It could be denoted by APd ). In all cases the admissibility of the design shall be checked and proven first by the simplest means possible for the example – to show that quite often DBA can be quite short and simple, if only the admissibility is to be be proven. In all examples a normal hydraulic test is presupposed, i. e. the checks against GPD for testing conditions do not require separate calculations. Specified pressures and temperatures are to be considered as pairs. If other actions are specified, they are considered to form, with pressures and temperatures, triplets, etc. Should the alternative of using any primary stress field be used in the check against GPD, the principal strains corresponding to this primary stress field shall be limited by ± 5% - as specified in the tangent intersection procedure. In general, this requires primary stress fields obtained by (inelastic) FEM. 5.2

General notations

Where possible, the notations of the CEN proposal for the Unfired Pressure Vessel Standard, prEN 13445, are used, and shall be used: •

Analysis thickness: Nominal thickness minus allowances – manufacturing, δ e , and corrosion, erosion, c : ea = en − δ e − c

•

Maximum allowable pressure: The maximum pressure (on top of the vessel) specified (by the manufacturer) for design, for normal operating conditions: PS . This pressure PS constitutes an upper limit for the set-pressure of the safety valve – if there is only one -, or for the maximum pressure (at the top of the compartment) that can occur under reasonably foreseeable conditions – if no safety valve is required. It shall be used in the design checks against GPD and PD, or SD.

•

Maximum operating pressure: Pop, sup This value – specified directly, or as being equal to 90% of PS max DBF - shall be used as upper value (of full pressure cycles) for cases with cyclic pressure.

•

Maximum allowable pressure according to prEN 13445-3, Annex 5.B: PS max DBF

•

Maximum allowable pressure according to DBA: PS max DBA For GPD and PD/SD checks, and, if relevant, for checks against instabilty (I), only.

•

Characteristic value of moment: Maximum reasonably foreseeable value of external moment; (in general equal to the "usual" design moment): M c

•

Allowable number of cycles: Number of cycles (for specified actions) allowed by prEN 13445-3, Section Detailed Fatigue Analysis.

DBA Design by Analysis

Specification of Examples

•

Modulus of elasticity: E E at 100°C, say: E100

•

Mean coefficient of (linear) thermal expansion: α α between 20°C and 100°C, say: α 20, 100

Page 5.3

Weld symbols according to EN 22553. 5.3

Designations

DBA. DBF GPD PD SD I F NLG PS PSmaxDBF PSmaxGPD

Pop, sup

Design by Analysis Design by Formulae Gross Plastic Deformation Progressive Plastic Deformation Shakedown Instability Fatigue Non-Linear Geometry Maximum alloxable pressure Maximum allowable pressure according to prEN 13445-3 Section DBF Maximum allowable pressure according to Gross Plastic Deformation using DBA Maximum allowable pressure according to Progressive Plastic Deformation using DBA Maximum operating pressure.

APd

Design value for the pressure action

Pap Mc Tc E α λ h a σ

Applied pressure to elastic compensation analysis Characteristic value of moment Calculation temperature Modulus of elasticity Mean coefficient of (linear) thermal expansion Heat conduction coefficient Heat transfer coefficient Thermal diffusivity (temperature conductivity) Stress..... σ ij , σ i , σ e , σ e, max , σ nom

ε

Strain..... ε ij , ε e

PSmaxSD:

DBA Design by Analysis

5.4

Specification of Examples

Page 5.4

Specifications of examples

Example No. 1.1: Thick unwelded flat end 1. Material:

P280GH according to EN 10222-2 The relevant heat treatment dimension is specified as 101.6 mm (in deviation from EN 10222-1).

2. Actions:

Pressure PS = PS max DBF = 17 MPa x) Temperature Tc = 20°C

3. Operational cycles:

T = const , p varying from 0 to Pop ,sup = 0.9 ⋅ PS

4. Geometry:

See Fig. 5.1

Fig. 5.1

x)

A not very reasonable result: The end thickness is large and the ratio of admissible pressure to nominal design stress is outside the graphs and the scope of DBF. Extrapolation was necessary.

DBA Design by Analysis

Specification of Examples

Example No. 1.2: Thin unwelded flat end 1. Material:

P280GH according to EN 10222-2 The relevant heat treatment thickness is specified as 101.6 mm (in deviation from EN 10222-2).

2. Actions:

Pressure PS = PS max DBF = 4.2 MPa Temperature Tc = 20°C

3. Operational cycles:

T = const , p varying from 0 to Pop ,sup = 0.9 ⋅ PS

4. Geometry:

See Fig. 5.2

Fig. 5.2

Page 5.5

DBA Design by Analysis

Specification of Examples

Example No. 1.3: Welded-in flat end without nozzle 1. Material:

P265GH according to EN 10028-2

2. Actions:

Pressure PS = PS max DBF = 12.7 MPa Temperature Tc = 20°C

3. Operational cycles:

T = const , p varying from 0 to Pop ,sup = 0.9 ⋅ PS

4. Geometry:

See Fig. 5.3 and 5.4

Fig. 5.3

Page 5.6

DBA Design by Analysis

Specification of Examples

Fig. 5.4

Example No. 1.4: Welded-in flat end with nozzle 1. Material:

Plate and Shell: P265GH according to EN 10028-2 Nozzle: P265 according to prEN 10216-2

2. Actions:

Pressure PS = PS max DBF = 7.9 MPa Temperature Tc = 20°C

3. Operational cycles:

T = const , p varying from 0 to Pop ,sup = 0.9 ⋅ PS

4. Geometry:

See Fig.5.5, 5.6 and 5.7

Page 5.7

DBA Design by Analysis

Specification of Examples

Fig. 5.5

Fig.5.6

Fig. 5.7

Page 5.8

DBA Design by Analysis

Specification of Examples

Page 5.9

Example No. 2: Storage tank (cone-cylinder junctions) 1. Material:

Shell: X6CrNiTi 18-10 (1.4541) according to prEN 10028-7 Reinforcing ring, foot ring: P235GH according to EN 10028-2; Note: the different thermal expansion coefficients shall be considered.

Hydrostatic pressure pH , medium density ρ M = 1000 kg m3 minimum medium level hMIN and maximum medium level hMAX see Fig. 5.8. Note: no longitudinal stress in the main cylindrical shell caused by hydrostatic pressure. Temperature in service Tc = 60°C ; Temperature before complete filling of the vessel 20°C. Internal pressure during draining (see also Fig. 5.9) PS = 0.06 MPa ; Note: longitudinal stress in the main cylindrical shell caused by internal pressure acting on the upper end of the vessel.

2. Actions:

Dead load (self weight and insulation): Insulation: qd = 220 N m 2 (weight force / surface of the vessel), insulation thickness 200 mm; dead weight of roof including insulation and reinforcing ring 26,15 kN. Wind load (limit value): stagnation pressure qW depending on qW = 0.81 kN / m 2 height h: 0 m ≤ h ≤ 6 m:

Wind force:

6 m < h ≤ 10 m :

qW = 0.88 kN / m 2

10 m < h ≤ 15 m :

qW = 0.94 kN / m 2

15 m < h ≤ 25 m :

qW = 1.02 kN / m 2

Wi = c ⋅ qW ,i ⋅ Ai

where c = 0.44 and Ai = projection of the surface of the vessel in wind direction. 3. Detail to be investigated: wide and narrow ends of cone 4. Operational cycles:

See Figure 5.9. Note: It is ascertained that internal pressure can be increased only if the medium height is below hMIN. The internal pressure increases slowly; for safety reasons both extremes shall be considered, the very slow (dotted line) and very fast (full line) pressure increases.

5. Geometry:

See Fig.5.8.

DBA Design by Analysis

Specification of Examples

Fig. 5.8

Page 5.10

DBA Design by Analysis

Specification of Examples

Fig. 5.9

Example No. 3.1: Thin-walled cylinder-cylinder intersection 1. Material:

P295GH according to EN 10028-2

2. Actions:

Pressure PS = PS max DBF = 0.28 MPa Nozzle longitudinal moment M c = 15644.4 Nm (moment vector normal to plane through both cylinder axes). Temperature Tc = 50°C

3. Operational cycles:

A) T = const , p varying from 0 to Pop, sup = 0.9 ⋅ PS , M c = const and B) T = const , M varying from 0 to 26400 Nm, p = const = 1.28MPa (for comparison with experimental results). Crotch corner surface machined: R z = 50 µm

Page 5.11

DBA Design by Analysis 4. Geometry:

Specification of Examples

See Fig.5.10.

Note: Checks against GPD and PD, or SD, to be performed for constant longitudinal moment only.

Fig. 5.10

Page 5.12

DBA Design by Analysis

Specification of Examples

Page 5.13

Example No. 3.2: Thick-walled cylinder-cylinder intersection 1. Material:

Shell: P265GH according to EN 10028-2 Nozzle: 11CrMo9-10 according to prEN 10216-2

2. Actions:

Pressure PS = PS max DBF = 14.09 MPa Nozzle longitudinal moment M c = 711.1 Nm (moment vector normal to plane through both cylinder axes). Temperature Tc = 50°C

3. Operational cycles:

A) T = const , p varying from 0 to Pop, sup = 0.9 ⋅ PS , M c = const and B) T = const , M varying from 0 to1200 Nm, p = const = 24MPa (for comparison with experimental results). Crotch corner surface machined: R z = 50 µm

4. Geometry:

See Fig 5.11.

Note: See note in Example No. 3.1. Fig. 5.11

Fig. 5.11

DBA Design by Analysis

Specification of Examples

Page 5.14

Example No. 4: Dished end with nozzle in knuckle region 1. Material:

X6CrNiMoTi 17-12-2 (1.4571) according to prEN 10028-7

2. Actions:

Pressure PS = PS max DBF = 0.583 MPa Temperature Tc = 180°C

3. Operational cycles:

T = const , p varying from 0 to Pop,sup = 0.9 ⋅ PS

4. Geometry:

See Fig. 5.12 and 5.13.

Fig. 5.12

Fig. 5.13

DBA Design by Analysis

Specification of Examples

Page 5.15

Example No. 5: Nozzle in spherical end with cold medium injection 1. Material:

Shell: 11CrMo9-10 according to EN 10028-2 Nozzle reinforcement: 11CrMo9-10+QT according to prEN 10216-2 Nozzle: P265 according to prEN 10216-2

2. Actions:

Pressure PS = 0.9 ⋅ PS max DBF = 0.9 ⋅ 13.01 = 11.71 MPa Temperature of medium inside the vessel TS = 325°C (constant in operation). Temperature of injected cold medium TN = 80°C . Location of different heat transfer coefficient for cold medium injection see Fig. 5.15. The outer surface of the vessel is insulated ideally. Heat transfer coefficients: -) medium to vessel wall, and to nozzle if there is no injection: hS = 1.16 kW m 2 K -) cold (injection) medium to nozzle wall during injection: hN = 10.8 kW m 2 K .

3. Operational cycles:

See Fig. E 5.15. Cold medium injection takes place for 10 minutes. The time between the injection cycles is long enough such that temperature reaches , stationarity. After 500 injection cycles one shutdown (and startup) should be considered. At shutdown and startup, pressure and temperature are decreased or increased in phase, respectively. Temperature changes during shutdown and startup are slow, and therefore thermal stresses can be neglected. 4. Geometry:

See Fig. 5.14

DBA Design by Analysis

Specification of Examples

Fig. 5.14

Fig. 5.15

Page 5.16

DBA Design by Analysis

Specification of Examples

Page 5.17

Example No. 6: Jacketed vessel with jacket on cylindrical shell only, and flat annular end plates 1. Material:

X6CrNiTi 18-10 (1.4541) according to prEN 10028-7

2. Actions:

Inner space: Outer space:

3. Operational cycles:

See Fig. 5.18. Inner space: Outer space:

Pressure PS = −0.1 / + 1.3 MPa Temperature Tc = TS = 160°C Pressure PS = 0 / + 0.5 MPa Temperature Tc = TS = 160°C

Pop ,sup = 1.1 MPa

Pop ,inf = 0 MPa

Top,sup = TS

Top ,inf = 20°C .

Pop ,sup = 0.45 MPa

Pop ,inf = 0 MPa

Top,sup = TS

Top,inf = 10°C

A pressure in the inner space below atmospheric can occur independently and repeatedly in operation, and an underpressure will occur concurrently with an outer space temperature of 10°C (whereby an inner space temperature value of 160°C shall be used). This case shall be included as a normal operating condition in the check against GPD, I, PD, or SD, but not in the fatigue check. A pressure in the outer space below atmospheric cannot occur, but a minimum pressure of 0 bar cannot be excluded. This case shall also be included as a normal operating condition in the check against GPD, I, PD, or SD (with temperatures in the inner and outer space of 160°C). Note: Top ,sup and Top ,inf are medium temperatures. The wall temperatures shall be determined using heat transfer coefficients of h i = 1.16 kW m 2 K on inside of inner vessel wall and

h o = 14.4 kW m 2 K on all surfaces of the inside of the jacket. Jacket and main vessel outside of jacket are insulated ideally. Note: checks against GPD, I, PD or SD shall be performed usingthe PS values. Note: only steady state thermal stresses shall be considered. The maximum allowable out-of-roundness of the inner cylindrical shell is specified in prEN13445-3 as (D+1250) / 200 = (2780 + 1250) / 200 = 20,15 mm, where D is the mean shell diameter. 4. Details to be investigated:

Jacket and jacketed part of inner vessel

5. Geometry:

See Fig. 5.16 and 5.17.

DBA Design by Analysis

Specification of Examples

Fig. 5.16

Fig. 5.17

Page 5.18

DBA Design by Analysis

Specification of Examples

160°C

160°C

Fig. 5.18

Page 5.19

DBA Design by Analysis

Specification of Examples

5.5 Appendix: Physical properties of some materials P 235 GH

P 265 GH

Page 5.20

DBA Design by Analysis

P 295 GH

11CrMo9-10

Specification of Examples

Page 5.21

DBA Design by Analysis

1.4541

1.4571

Specification of Examples

Page 5.22

1

DBA Design by Analysis

Page

Analysis Summary

6.1

The following tables summarise the checks carried out for the ten examples, specified in Section 5. All fatigue results of welded regions are obtained for testing group 1 and, in general, based on the principal stress range approach; in cases where the equivalent stress range approach is used this is stated in a footnote. Results are designated in terms of the routes used as follows: DBF DBS DRC SC NLG

Design by formulae Design by analysis using the direct route elasto-plastic approach Design by analysis using the direct route elastic compensation approach Design by analysis using the stress categorisation approach For information only: Result for non-linear geometry in the elasto-plastic approach

and in terms of the checks performed: GPD PD I F SE

Check against gross plastic deformation Check against progressive plastic deformation Check against instability Calculation of allowable number of cycles according to the F-check Check for static equilibrium

The result summaries are tabulated as follows: Table 6.1 Table 6.2 Table 6.3 Table 6.4 Table 6.5 Table 6.6 Table 6.7 Table 6.8 Table 6.9 Table 6.10

Example 1.1 Example 1.2 Example 1.3 Example 1.4 Example 2 Example 3.1 Example 3.2 Example 4 Example 5 Example 6

“Thick unwelded flat end” “Thin unwelded flat end” “Welded-in flat end without nozzle” “Welded-in flat end with nozzle” “Storage tank (cylinder-cone junction)” “Thin walled cylinder-cylinder intersection” “Thick walled cylinder-cylinder intersection” “Dished end with nozzle in knuckle region” “Nozzle in spherical shell with cold medium injection” “Jacketed vessel: jacket on cylindrical shell with flat annular end plates”

Table 6.1: Analysis Summary for Example 1.1: “Thick unwelded flat end” Project Member and Analysis Type Design Check GPD PS(MPa)

St. A

A&AB

WTCM

DBF

DRS

DRS

DRC

DRS

17[1]

60.87

58.02

60.1

62.8

101.45

-

102.1

PD PS(MPa) I F (cycles) SE

Strathclyde

TKS

RWTUV

SC

NLG

CETIM

58

-

-

69.7 57.5[3]

-

-

-

infinity

-

infinity

[2]

Not Required infinity

infinity

-

Not Required

[1] Conservative result as the geometry is at the limit allowed by DBF. [2] Thick cylinder and head modelled using shell elements with pressure applied at inner radius (primary stresses are limiting). [3] Thick cylinder and head modelled using shell elements with pressure applied at mean radius (primary stresses are limiting).

2

DBA Design by Analysis

Page

Analysis Summary

6.2

Table 6.2: Analysis Summary for Example 1.2: “Thin unwelded flat end” Project Member and Analysis Type Design Check GPD PS(MPa)

St. A

A&AB

WTCM

DBF

DRS

DRS

DRC

5.7

6.18

7.9

-

4.2

PD PS(MPa) I F (cycles)

Strathclyde

TKS

RWTUV

CETIM

DRS

SC

NLG

4.5

5.7

5.6[1]

5.6

-

7.25

-

-

-

-

375200

CETIM

Not Required 360000

-

-

SE

-

-

309600

Not Required

[1] Primary stresses are limiting

Table 6.3: Analysis Summary for Example 1.3: “Welded-in flat end without nozzle” Project Member and Analysis Type Design Check GPD PS(MPa)

St. A

A&AB

WTCM

TKS

RWTUV

DBF

DRS

DRS

DRC

DRS

SC

NLG

12.6

12.41

10.8

12.48

12.3[1]

12.4

-

13.3

-

13.3

-

-

-

-

14673[2]

TKS

RWTUV

CETIM

SC

NLG

12.7

PD PS(MPa) I F (cycles)

Strathclyde

Not Required 13581

[2]

6910

-

SE

-

-

4665

Not Required

[1] Shakedown is limiting [2] Equivalent stress range approach used.

Table 6.4: Analysis Summary for Example 1.4: “Welded-in flat end with nozzle” Project Member and Analysis Type Design Check GPD PS(MPa)

St. A

A&AB

WTCM

DBF

DRS

DRS

DRC

DRS

9.9

10.0

8.42

9.92

12.2

-

12.8

-

21010

-

7.8

PD PS(MPa) I F (cycles) SE

Strathclyde

10.0

[1]

9.9[2]

-

-

-

-

26947[3]

Not Required 28600[3]

Not Required

[1] Primary stresses are limiting [2] 5% maximum absolute value of principal strain exceeded at this load. [3] Equivalent stress range approach used.

-

14570

3

DBA Design by Analysis

Page

Analysis Summary

6.3

Table 6.5: Analysis Summary for Example 2: “Storage tank (cylinder-cone junction)” Design Check

Project Member and Analysis Type

Load Case 1 2 3

St. A

A&AB

Strathclyde

TKS

CETIM

DBF

DRS

DRC

SC

GPD PS(MPa)

Admissible

Admissible

-

PD PS(MPa)

Admissible

-

-

GPD PS(MPa)

Admissible

Admissible

-

PD PS(MPa)

Admissible

-

-

GPD PS(MPa)

Admissible

Admissible

-

PD PS(MPa)

Admissible

-

-

I

Admissible

Admissible

-

-

-

-

F (cycles)

-

-

-

830

1984[2]

-

SE

-

Admissible[1]

-

-

-

[1] Bolting required for wind load only. [2] Equivalent stress range approach used. Load Cases: (1) Hydrostatic pressure at maximum medium level, dead weight, wind load, and for PD check only, thermal stresses. (2) Hydrostatic pressure at minimum medium level, draining pressure, dead weight and wind load, and for PD check only, thermal stresses. (3) Draining pressure, dead weight and wind load, and, for PD check only, thermal stresses.

Table 6.6: Analysis Summary for Example 3.1: “Thin walled cylinder-cylinder intersection” Project Member and Analysis Type Design Check GPD PS(MPa)

St. A

A&AB

DBF

DRS

0.28

PD PS(MPa)

0.39 0.5

Strathclyde DRC 0.467 1.34

I F (cycles)

[1]

[2]

WTCM

DRS

SC

0.383

0.45

-

CETIM -

Not Required load A

96500[3]

-

-

-

-

122770[3]

load B

1710[3]

-

-

-

-

1042[3]

SE

Not Required

[1] The 5% maximum absolute principal strain rule can currently not be used in elastic compensation, resulting in a much higher load corresponding to loss of equilibrium between the external applied loads and internal stresses and strains. See Chapter 3 for more details. [2] This value exceeds the 2RM limit placed on shakedown at the discontinuity, for more information see the analysis details. [3] Equivalent stress range approach used. load (A): constant moment, varying pressure load (B): constant pressure, varying moment

4

DBA Design by Analysis

Page

Analysis Summary

6.4

Table 6.7: Analysis Summary for Example 3.2: “Thick walled cylinder-cylinder intersection” Project Member and Analysis Type Design Check GPD PS(MPa)

St. A

A&AB

DBF

DRS

DRC

DRS

SC

14.79

11.5

15.1

14.25

17.65

13.7

-

14.09

PD PS(MPa)

Strathclyde

I F (cycles)

Strathclyde

WTCM

CETIM

-

-

-

-

Not Required [1]

load A

-

70865

load B

-

1160

-

-

-

59494[1]

55474[1]

-

-

-

-

790[1]

SE

Not Required

[1] Equivalent stress range approach used load (A): constant moment, varying pressure load (B): constant pressure, varying moment

Table 6.8: Analysis Summary for Example 4: “Dished end with nozzle in knuckle region” Project Member and Analysis Type Design Check GPD PS(MPa)

St. A DBF 0.583

PD PS(MPa)

A&AB

Strathclyde

WTCM

DRS

DRC

DRS

0.375

0.41[1]

0.42

[5]

0.289 0.375[6]

0.271

I

[3]

RWTUV

CETIM

SC

NLG

0.263

0.42[2]

-

-

-

-

880[4]

Not Required

F (cycles)

2000

[4]

4072

-

SE

-

-

Not Required

[1] The 5% maximum absolute principal strain rule can currently not be used in elastic compensation, resulting in a much higher load corresponding to loss of equilibrium between the external applied loads and internal stresses and strains. See Chapter 3.3 for more details. [2] 5% maximum absolute value of principal strain exceeded at this load. [3] Modelled with shell elements – thus simplified geometry and discontinuities. [4] Equivalent stress range approach used. [5] Shakedown limit pressure [6] Allowable number of cycles: 1440. See Chapter 7.9 for more details.

Table 6.9: Analysis Summary for Example 5: “Nozzle in spherical shell with cold injection” Project Member and Analysis Type Design Check PS

[1] max

(MPa)

GPD PD

St. A

A&AB

JRC

DBF

DRS

DRS

13.013

13.04

13.125

-

Admis.

Admis.

-

[5]

N-A

[5]

N-A

I F (cycles) SE

Strathclyde DRC 11.1

[2]

DRS 11.43

WTCM

TKS

SC

SC

[2]

N-A

N-A

-

-

CETIM -

[3]

N-A

[4]

N-A

-

Not Required -

-

-

-

-

-

13

9.8[6]

Not Required

[1] Maximum allowable pressure according to the GPD-check. [2] Defined internal pressure, PS is 0.9xPSmaxDBF = 11.71MPa, therefore, load is non-admissible. [3] No pressure allowable due to the magnitude of the thermal stresses according to the conditions of the SC Route. [4] The allowable pressure for the primary stress criterion only is 12.8 MPa using ANSYS and 11.7 MPa using BOSOR. [5] The thermal stress prevents the structure from shaking down at the applied load and is therefore non-admissible according to the PD-check. [6] 9.8 full cycles (start up – shut down + 500 cold media injection).

5

DBA Design by Analysis

Page

Analysis Summary

6.5

Table 6.10: Analysis Summary for Example 6: “Jacketed vessel: jacket on cylindrical shell with flat annular end plates” Project Member and Analysis Type Load Case PS 1

Design Check

[1] maxGPD (MPa)

GPD

St. A

A&AB

DBF

DRS

DRC

1.35 Admissible

1.3 [3]

PD 2

GPD

[3]

PD 3

GPD

Admissible

PD 4 5

WTCM

RWTUV

NLG

DRS

NLG

1.19

2.0

1.26

1.23

-

NonAdmissible Admissible

Admissible

NonAdmissible -

NonAdmissible -

-

Admissible -

NonAdmissible -

NonAdmissible -

-

Admissible

NonAdmissible Admissible

Admissible

Admissible

Admissible

Admissible

Admissible

-

Admissible

-

-

-

-

-

Admissible Admissible

Strathclyde

-

CETIM

-

-

PD

-

Admissible

-

-

-

-

-

GPD

Admissible

Admissible

Admissible

Admissible

Admissible

Admissible

-

Admissible

-

-

-

-

-

-

PD I

-

Admissible

Admissible

-

-

-

-

-

F (cycles)

-

20506

-

-

20695

-

67953[2]

-

SE

Not

Required

[1] Maximum calculated allowable pressure in the dished end (pressure limiting component) [2] Equivalent stress range approach used [3] To meet DBF requirements the thickness of upper stiffener should be increased from 13.5 to 28 mm and the dimensions of the lower stiffener from 20x110 mm2 to 41x125 mm2.

Load Cases: (1) Inner chamber internal pressure Outer chamber internal pressure temperature in both chambers (2) Inner chamber internal pressure Outer chamber internal pressure temperature in both chambers (3) Inner chamber internal pressure Outer chamber internal pressure temperature in inner chamber temperature in outer chamber

PSi = 1.3 MPa PSo = 0.5 MPa TSi = 160oC PSi = 1.3 MPa PSo = 0.0 MPa TSi = 160oC PSi = -0.1 MPa PSo = 0.5 MPa TSi = 160oC TSo = 10oC

(4) Inner chamber internal pressure Outer chamber internal pressure temperature in inner chamber temperature in outer chamber (5) Inner chamber internal pressure Outer chamber internal pressure temperature in inner chamber temperature in outer chamber

PSi = 1.3 MPa PSo = 0.5 MPa TSi = 160oC TSo = 10oC PSi = 0.0 MPa PSo = 0.5 MPa TSi = 20oC TSo = 10oC

DBA Design by Analysis

Analysis Details

Page 7.1

7 Analysis Details 7.1 General Details of some of the various checks by the members of the group are compiled in this section. The page numbering is consecutively throughout the whole section. Since each contribution is a more or less self-contained part, each is preceeded by a summary page, which shows in the page head the example number, in the Analysis Type box the checks dealt with, and in the Member box an acronym for the responsible group member. To ease browsing, a letter is added in brackets to the page number to indicate the responsible group member, and the paragraph numbering is new for each new contribution. The usual order is •

GPD-Check and PD-Check

•

Stress Categorisation Route

•

F-Check

•

I-Check, where applicable

•

SE-Check, where applicable

For the GPD-check and the PD-check there are usually two approaches given •

Direct route using elastic compensation

•

Direct route (using non-linear calculations)

The stress categorisation route follows after the PD-check. The fatigue results are all based on linear-elastic FE-calculations. If the F-check given is from a group member different from the one whose PD-check is given, these linear-elastic calculations are separate ones, just for the F-check. The stress components used are stated, but they cannot be deduced always from other plots exactly. To avoid unnecessary problems, we state here again that the formulae in the F–checks differ from those given in section 18 of prEN 13445-3, which is reproduced in Annex 2. The formulae used here agree with those in the flow-sheets given in subsection 3.8 of this manual, and the flow-sheets have been used directly. The main changes in the formulae are stated in subsection 3.8 of section 3 – Procedures, and they are repeated at the beginning of Annex 2. Since some details on partial safety factors for actions and on characteristic values are missing in the subsection on the I-check in prEN13445-3 Annex B, some additional material had been agreed upon, used, and described in subsection 3.7 of section 3 – Procedures. .

DBA Design by Analysis

Analysis Type:

Analysis Details

Page 7.2(S)

Example 1.1

GPD-Check and PD-Check Direct Route using Elastic Compensation

Member: Strathclyde

FE-Software:

ANSYS 5.4

Element Types:

8 – node, 2-D structural axisymmetric solid.

Boundary Conditions:

No vertical displacement in the undisturbed end of the shell remote from the flat end. Axisymmetry from elements

Model and Mesh:

Number of elements – 339 Height of model – 500 mm

Results: Maximum internal pressure according to the GPD-check: PSmax GPD = 60.1 MPa Check against PD: Shakedown limit pressure PSmax GPD = 102.1 MPa

DBA Design by Analysis

Analysis Details Example 1.1

Page 7.3(S)

1. Finite Element Mesh Element divisions for the finite element mesh were defined parametrically to allow modification of the element density. Analysis was carried out for a coarse mesh density and a much finer mesh density to allow any alterations on the results to be noted. Finite element models were created using linear 4-node 2-D axisymmetric solids, the analysis was repeated with higher order 8-node structural 2-D axisymmetric solids, any alterations this may have on the results could also be noted. The allowable pressure according to GPD and the shakedown pressure according to PD were calculated from the model giving the greatest lower bounds. Here, the fine density mesh with 8node higher order elements gave the highest lower bounds. Boundary conditions applied to the model reflect axisymmetry, applied via a key option when defining the element type in the FE-software (axisymmetry around the vertical axis Y). The nodes at the undisturbed end of the cylindrical shell have their vertical degree of freedom constrained to zero to ensure that plane sections remain plane. 2. Material properties Material strength parameter RM = 255 MPa , modulus of elasticity E = 212 GPa . 3. Determination of the maximum admissible pressure according to the GPD-Check Using the application rule in prEN 13445-3 Annex B.9.2.2 to check against GPD, the principle is fulfilled when for any load case the combination of the design actions do not exceed the design resistance. This may be shown by calculating the limit load. The limit load has to be determined using Tresca‘s yield condition and associated flow rule. As elastic compensation is based on a series of elastic equilibrium stress fields, it is a relatively simple procedure to derive a lower bound limit load directly from the Tresca yield model. From prEN 13445-3 Annex B, Table B.9-3 the partial safety factor, γR on the resistance is 1.25. Therefore, the design material strength parameter is given by RM/γR = 204 MPa. The analysis was carried out using the elastic compensation method conforming to the direct route rules for GPD in Annex B: linear elastic - ideal plastic material law, Tresca’s yield condition and associated flow rule and first order theory. In each elastic compensation iteration an equilibrium stress field is produced where the elastic modulus of each subsequent iteration is defined by the previous elastic solution. In this way some regions in the FE-model may be systematically stiffened or weakened depending upon the stress magnitudes in the previous solution. The result is, that every equilibrium stress field is a lower bound of the

Figure 7.2.1-1: Limit Stress Field (Tresca)

Analysis Details

DBA Design by Analysis

Example 1.1

Page 7.4(S)

limit load, the stress field giving the greatest limit load is taken as the closest to the actual limit state and defines the limit load in the analysis. Convergence of the equilibrium stress field will occur typically between 8 and 15 iterations, i.e. no further benefit in the limit load will be noted with subsequent iterations. The total computing time to run the analysis on a 300 MHz Pentium two processor with 128 Mb RAM Windows NT workstation was 70 seconds. The stress field was shown to converge after eight iterations giving a lower bound on the pressure limit of 72.1 MPa. Figure7.2.1-1 shows the limit stress (intensity) field according to the Tresca criterion. The limit pressure is given by scaling the limit stress field so that the stress anywhere in the model does not exceed the design materialstrength, 204 MPa, i.e. the applied pressure is scaled by the factor (204/28.304) 7.207. According to prEN 13445-3 Annex B, Table B.9-2 for pressure loads (without natural limit) the partial safety factor γp is 1.2. Thus, the internal pressure limit according to failure by GPD is PS max GPD =

72.1 = 60.1 MPa 1.2

It is also possible to determine a limit pressure from the check against PD. In this case, the elastic compensation is based on Mises‘ yield condition. The partial safety factor on the resistance γR is not applied for the PD-check. However, as the analysis is wholly elastic it is possible to scale the stress fields at any time (similarly as was done above). The maximum ratio of Mises' equivalent stress to Tresca's equivalent stress for the same load is 2/√3. Therefore, applying a factor of √3/2 to the yield stress in the Mises analysis (or to the limit load, as the analysis is elastic) will always lead to a conservative result. From the Mises analysis the limit load was found to be 102.1 MPa, and with the partial safety factors γR = 1.25 and γp = 1.2, the internal pressure limit according to GPD can be found as PS max GPD =

102.1 3 ⋅ = 58.95MPa γ p ⋅γ R 2

4. Check against PD In this check the principle in prEN13445-3 B.9.3.1 is fulfilled if the structure can be shown to shake down. When a structure has been shown to shake down, the failure modes of progressive plastic deformation and alternating plasticity can not occur. In elastic compensation the load at which the structure will shake down is simple to calculate. Based on Melan’s shakedown theorem, the self-equilibrating residual stress field that would result after a loading cycle can be calculated by subtracting the linear-elastic stress field at the limit pressure from the limit stress field. The residual stress

Figure 7.2.1-2: Residual Stress Field (Mises)

DBA Design by Analysis

Analysis Details Example 1.1

Page 7.5(S)

field is in effect the resulting stress from an elastic unloading from the limit state back to zero pressure. If no stress in the residual field violates the yield condition, i.e. if there is no equivalent stress above the material (yield) parameter, then the shakedown load is equal to the limit load. Where the residual stress field of a structure does exceed the yield condition, the shakedown limit can be calculated easily from the stress plots because of the linearity. The residual stress field of the compensation analysis using Mises’ criterion is shown in Figure 7.2.1-2. Because the applied load is arbitrary and the resulting stress fields are scaled to the yield condition, the maximum residual stress is then compared to the maximum stress in the limit stress field, Figure 7.2.1-3. The maximum residual equivalent stress of 21.246 MPa is smaller than the 24.971 MPa for the limit stress field. The shakedown limit is therefore the same as the calculated limit load from the Mises condition, given by scaling up the load by a factor of material yield parameter to maximum stress in the limit field (255/24.971) = 10.21. With an applied load of 10 MPa the shakedown limit is 102.1 MPa.

Figure 7.2.1-3: Limit Stress Field (Mises)

5. Check against GPD Using Non-linear Analysis A check against GPD was also performed for the same FE - model using conventional non-linear analysis. In this way, a direct comparison may be made between the two limit approaches. The FE geometry, mesh and boundary conditions are the same as those used in the elastic compensation analysis. Material non-linearities were applied corresponding to the material strength parameter, 204 MPa and perfect plasticity. A ramped load is applied and the analysis runs until the applied load is such that convergence can no longer occur due to unrestrained displacement – Gross Plastic Deformation. It is assumed that the last converged solution is the limit

Figure 7.2.1-4: Limit Stress Field (non-linear analysis)

Analysis Details

DBA Design by Analysis

Example 1.1

Page 7.6(S)

load. Here, the last converged solution was at a load of 87 MPa, using the same method as above the allowable pressure according to the GPD-Check using Mises' criterion is 87 3 ⋅ = 62.8MPa γp 2 The result offers a small benefit to the allowable pressure calculated using elastic compensation, however the analysis is more difficult. Figure 7.2.1-4 shows the Mises equivalent stress at the limit load. Analysis time to calculate limit load using non-linear FE - analysis was 290 seconds. PS max GPD =

6. Additional Comments Additional analysis was completed to ascertain any effect on the results for different mesh density and for lower order elements (4-node). Limit loads and shakedown were calculated using Mises' condition, the results are summarised in Table 7.2.1-1. Number of Elements 119 339 119 339

Element Type 4 node 4 node 8 node 8 node

Number of Iterations 8 8 8 8

Lower Bound Limit Load 56.9 58.8 57 58.95

Shakedown Load 98.5 101.9 98.75 102.1

Processor Time 40.7 107 70.2 190.3

Table 7.2.1-1. Limit and shakedown analysis summary

Both the lower bound limit load and the shakedown load were calculated using the same method as described above. As can be seen from Table 7.2.1-1, essentially no difference is noted between the results for the different element types. For this geometry, higher order elements offer no benefit over the lower order elements. The geometry is simple and at the smaller mesh density the 4-node elements fit the curvature well, therefore little change in the results would be expected. A difference in the results can be noted between the two mesh densities. A slightly larger shakedown and limit load result is obtained from the analysis using the higher mesh density, although small, approximately 3%. In general, the results for both the limit load and shakedown calculations show very little sensitivity to the element type and element density for this geometry. As the geometry in this problem is outside the scope of DBF, the DBA calculations are a quick and simple alternative for this simple problem. The two elastic compensation methods used to calculate the lower bound limit load (direct from Tresca's criterion or via a correction of Mises') show good correlation; the Mises corrected value is slightly conservative as would be expected. However, carrying out the Mises elastic compensation will give both the limit load and shakedown load in one analysis. Good agreement was shown between the elastic compensation results and the non-linear analysis results. Processing time is longer for the non-linear analysis however, due to the simplicity of the model this time was also short.

DBA Design by Analysis Analysis Type:

Analysis Details

Page 7.7(S)

Example 1.1

GPD - Check and PD - Check

Member:

Direct Route using elasto-plastic FE calculations

A&AB

FE-Software:

ANSYS® 5.4

Element Types: 4 – node, 2 – D axisymmetric solid PLANE42

Boundary Conditions:

Zero vertical displacement in the nodes at the undisturbed end of the shell. Symmetry boundary conditions in the nodes in the centre of the flat end.

Model and Mesh:

Whole height of model: 751.2 mm Number of elements: 1294

Results: Maximun allowable pressure according to the GRD-Check: PSmaxGPD =60.87 Mpa Shakesown limit pressure:

PSmaxGPD =101.45 Mpa

DBA Design by Analysis

Analysis Details Example 1.1

Page 7.8(S)

1. Elements, mesh fineness, boundary conditions The model of the structure is shown on the preceeding page, a total number of 1294 4-node axisymmetric solid elements, PLANE42 in ANSYS® 5.4, was used. The linear shape function of these elements is sufficient, since • the number of elements in the (linear-elastic) high stressed region is large, • the computation time in an analysis using nonlinear material properties is much larger for elements with midside nodes and quadratic shape functions, although, close to the limit load, the results are almost identical compared to those using elements with a linear shape function, • there is no need to compute linear-elastic peak stresses very exactly, because the check against PD can be carried out using the stress-concentration-free structure (according to prEN 13445-3 Annex B.9.3.2) and the structure’s geometry is modelled exactly in the example considered. The boundary conditions applied to the model are symmetry ones in the nodes in the centre of the plate (where the horizontal direction is perpendicular to the plane of symmetry) and constraining the vertical degree of freedom in the nodes at the undisturbed end of the cylindrical shell to zero.

2. Determination of the maximum admissible pressure according to the GPD-Check The partial safety factor γ R according to prEN-13445-3 Annex B, Table B.9-3 is 1.25 . Therefore, the analysis using Tresca’s yield condition (delivered by an ANSYS® distributor) was carried out with a linear-elastic ideal-plastic material law, a design material strength parameter of 204 MPa (corresponding to a material strength parameter of RM = 255 MPa according to EN 10222-2) for the shell and the plate, associated flow rule, and first order theory. The elastic modulus used in all calculations is E = 212 GPa . The analysis was carried out using the arc-length method, which showed faster convergence for the considered structure than the Newton - Raphson method; since at the limit load the structure is fully plastified in the shell and in the adjacent part of the plate, a maximum horizontal displacement of 10 mm at the upper face of the shell was used as termination criterion. To restrict the computation time in a reasonable manner, the analysis was terminated after 17 hours on a Compaq® Professional Workstation 5000 with two Pentium Pro® processors and 256 MB RAM. The last convergent solution showed an internal pressure of 73.05 MPa – this pressure was used as limit pressure. Figure 7.2.2-1 shows the horizontal displacement in the upper end of the shell versus the internal pressure.

Analysis Details

DBA Design by Analysis

Example 1.1

Page 7.9(S)

Figure 7.2.2-1

Figure 7.2.2-2 shows the distribution of Tresca's equivalent stress at this limit pressure. Because of the almost full plastification in the shell there is a small region where, due to numerical effects, the equivalent stress exceeds 204 MPa , but this has no effect on further analyses. The maximum absolute value of the principal strains in the structure at this limit pressure is 1.4 %, smaller than 5%, as required in the standard. According prEN 13445-3 Annex B, Table B.9-2, the partial safety factor for pressure (without natural limit), γ P , is 1.2. Figure 7.2.2-2

Therefore, the (internal) allowable pressure according to GPD is given by PS max GPD =

73.05 = 60.87 MPa. 1.2

A less time-consuming method to determine a limit pressure according to GPD is given by usage of the limit pressure result from the check against PD (see chapter 2 of section 3 - Procedures). With the partial safety factors γ R = 1.25 and γ P = 1.2 , the internal limit pressure according to GPD is, in this approach, PS max GPD =

102.89 3 102.89 3 ⋅ = ⋅ = 59.4 MPa . γ P ⋅ γ R 2 1.2 ⋅ 1.25 2

DBA Design by Analysis

Analysis Details Example 1.1

Page 7.10(S)

The maximum absolute value of the principal strains in the structure, at the limit pressure used here, and for the design material strength parameter of the check against PD-check, is 4 %, smaller than 5%, as required in the standard. 3. Check against PD The elasto-plastic FE analysis was carried out as stated in prEN 13445-3 Annex B, Sec. B.9.3.1, using Mises’ yield condition and associated flow rule, a linear-elastic ideal-plastic constitutive law with a material strength parameter of 255 MPa for shell and plate, and first order theory. By defining and using load cases in ANSYS®, the superposition of stress fields can be done easily. Therefore the first load step of the analysis was defined at a very low load level (5 MPa), so that there was linear-elastic response of the structure. All other linear-elastic stress fields can then be determined easily by multiplication with a suitable scale-up factor. Again, the analysis was carried out using the arc-length method; since at the limit load the structure is fully plastified in the shell and in the adjacent part of the plate, a maximum horizontal displacement of 10 mm at the upper face of the shell was used as termination criterion.

Figure 7.2.2-3

The computation time of the limit load was 1 hour and 15 minutes on the Compaq® Professional Workstation 5000 with two Pentium Pro® processors and 256 MB RAM. The termination criterion was fulfilled for an internal pressure of 102.89 MPa – this pressure was used as limit pressure. Figure 7.2.2-3 shows the horizontal displacement in the undisturbed shell versus the internal pressure; according to this figure the limit state is reached. Figure 7.2.2-4 shows the elasto-plastic Mises equivalent stress distribution at the limit pressure of 102.89 MPa .

Figure 7.2.2-4

DBA Design by Analysis

Analysis Details Example 1.1

Figure 7.2.2-5 shows the linearelastic Mises equivalent stress distribution for the limit pressure – the stress maximum is located in the fillet.

Figure 7.2.2-5

Figure 7.2.2-6 shows the Mises equivalent stress distribution of the corrected residual stress field, the scaling factor β is given by 0.96 (see subsection 3.3.2.5 of section 3 – Procedures). The site of the stress maximum is now located at the outer surface of the cylinder.

Figure 7.2.2-6

Page 7.11(S)

Analysis Details

DBA Design by Analysis

Example 1.1

Figure 7.2.2-7 shows the Mises equivalent stress distribution at the lower bound shakedown limit. The scaling factor α is given by 0.986 (see subsection 3.3.2.5 of section 3 – Procedures).

Figure 7.2.2-7

Thus, the shakedown limit pressure is given by PS max SD = 0.986 ⋅ 102.89 = 101.45 MPa .

Page 7.12(S)

DBA Design by Analysis

Analysis Details

Page 7.13(S)

Example 1.1

Analysis Type:

Member: Stress Categorization Route

FE-Software:

BOSOR

Element Types:

Axisymmetric shell elements.

TKS

Model and Mesh:

As shown in diagram – Shell numbers in calculation model

Remark: In this example the structure is very thick walled. As BOSOR operates with thin-walled elements the pressure acting on element 3 and 4 has been reduced with the factor Ri / Rm = 250.4 / 301.2

Results: Maximum admissible action according to the Stress Categorization Route: With reduction factor (Calc NO 11E) - Internal pressure PSmax SC = 69.7 MPa

Analysis Details

DBA Design by Analysis

Page 7.14(S)

Example 1.1

-5.22E+01 1.25E+03 MIN: MAX:

JOB NO11E 99-08-25 10.54.26

0 4.26E+02 MIN: MAX:

WINDOW (X,Y):

UNDEFORM.

DEFORM.

ALL SHELLS

*GEOMETRY PLOT

POSTBOSOR 1.04

The following figure shows the deformed model, the figures thereafter the distribution of stresses – membrane and membrane and membrane plus bending - in the surfaces of the various parts of the model. With the designation list in subsection 3.6 the various plots are self-explaining. The plots are for a pressure of 50 MPa. The limiting part is the cylindrical shell, the general membrane stress criterion is the governing one – see the membrane stress distribution in shell 4 on the last page of this contribution.

2

.20

.40

.40

.60

.60

.80

.80

1.00

1.00

1.20

1.20

1.40

1.40

10

10

2

2

JOB NO11E 99- 08-18 14.09.44

-8.03E+01 1.32E+02

10

.40

.50

.60

.70

.80

.90

1.00

1.10

1.20

1.30

-.75

-.50

2

.20

.20

.40

.40

.60

.60

.80

.80

1.00

1.00

1.20

1.20

1.40

1.40

10

10

2

2

-8.03E +01 1.32E+02

3.54E+01 1.32E+02 JOB NO11E 99-08-18 14.11.10

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 1

TRE SCA

*STRESSES*

POSTBOSOR 1.04

JOB NO1 1E 99-08-18 14.09.05

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 1

GEN.DIR

*STRESSES*

POSTBOSOR 1.04

Example 1.1

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 1

CIRCUMF

*STRESSES*

POSTBOSOR 1.04

JOB NO11E 99-08-18 14.08.15

2.61E+01 2.61E+01

2

Analysis Details

-.75

-.50

-.25

.00

.25

.50

.75

1.00

1.25

10

.20

MIN: MAX:

-.25

.00

.25

.50

.75

1.00

1.25

10

Shell No. 1

FUNC.VALUES:

"MEMBR"

SHELL 1

COMP.-ST

*STRESSES*

POST BOSOR 1.04

DBA Design by Analysis Page 7.15(S)

Stresses

1.50

1

1.50

1.75

1.75

2.00

2.00

2.25

2.25

2.50

2.50

2.75

2.75

3.00

3 .00

3.25

3.25

3.50

3.50

10

10

2

2

1.12E+01 2.61E+01 1.50

1.75

2.00

2.25

2.50

2.75

3.00

3.25

3.50

10

2

JOB NO11E 99-08-18 14.14.30

-3.08E+01 8.29E+01

.20

.40

.60

.80

1.00

1.20

1.50

1.75

2.00

2.25

2.50

2.75

3.00

3.25

3.50

10

2

"INSIDE"

2.05E+00 1.75E+02 JOB NO11E 99-08-25 10.56.02

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

Example 1.1

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 2

*STRESSES*

POSTBOSOR 1.04

SHELL 2 1.40

2

-1.23E+02 1.75E+02 JOB NO11E 99-08-18 14.13.40

MIN: MAX:

FUNC.VALUES:

TRESCA

1.60

10

-1.00

CIRCUMF

*STRESSES*

POST BOSOR 1.04

JOB NO11E 99-08-18 14.13.02

MIN: MAX:

FUNC.VALUES:

-.50

.00

.50

"INSIDE" "OUTSIDE"

Analysis Details

- 2.00

.00

2.00

4.00

6.00

8.00

10

12.00

14.00

16.00

"MEMBR"

SHELL 2 1.00

SHELL 2

*STRESSES*

POST BOSOR 1.04

GEN.DIR

1.50

2

COMP.-ST

*STRESSES*

10

Shell No. 2

18.00

20.00

22.00

24.00

26.00

POSTBOSOR 1.04

DBA Design by Analysis Page 7.16(S)

Stresses

4.60

4.60

10

10

2

2

0

4.00

4.20

4.20

4.40

4.40

4.60

4.60

10

10

2

2

*STRESSES*

POSTBOSOR

"INSIDE" 60.00

JOB NO11E 99-08-18 14.17.57

1.98E+01 8.18E+01 20.00

"INSIDE"

1.98E+01 8.18E+01 JOB NO11E 99-08-18 14.18 .32

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

Example 1.1

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

1.04

JOB NO11E 99-08-18 14.17.05

SHELL 3

3.8

4.00

SHELL 3

3.60

3.80

-1.36E+01 6.94E+01

Analysis Details

20.00

3.60

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 3

GEN.DIR

*STRESSES*

POST BOSOR 1.04

TRESCA 70.00

80.00

1

CIR CUMF

*STRESSES*

POSTBOSOR 1.04

JOB NO11E 99-08-18 14.15.21

30.00

4.40

4.40

30.00

4.20

4.20

40.00

4.00

4.00

-1.00

40.00

3.80

3.80

3.72E+01 7.06E+01

.00

1.00

2.00

3.00

4.00

5.00

50.00

3.60

3.60

MIN: MAX:

FU NC.VALUES:

"MEMBR"

SHELL 3

6.00

50.00

60.00

70.00

80.00

40.00

45.00

50.00

*STRESSES* COMP.-ST

10

Shell No. 3

55.00

60.00

65.00

70.00

POSTBOSOR 1.04

DBA Design by Analysis Page 7.17(S)

Stresses

2

.60

.60

.80

.80

1.00

1.00

1.20

1.20

1.40

1.40

(σ eq )Pm ≤ f

Mem. 122MPa

10

3

10

3

6.99E+01 1.27E+02

JOB NO11E 99-08-18 14.21.11

6.66E+01 1.29E+ 02

.70

.80

.90

1.00

1.10

1.20

10

10.00

20.00

30.00

40.00

50.00

60.00

2

.60

.60

.80

.80

1.00

1.00

1.20

1.20

1.40

1.40

10

10

3

3

9.08E+00 6.52E+01

6.66E+01 1.29E+ 02 JOB NO11E 99-08-18 14.21.41

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 4

TRESCA

*STRESSES*

POSTBOSOR 1.04

JOB NO11E 99-08-18 14.20.30

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 4

GEN .DIR

*STRESSES*

POSTBOSOR 1.04

Example 1.1

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 4

CIRCUMF

*STRESSES*

POSTBOSOR 1.04

JOB NO11E 99-08-18 14.19.47

MIN: MAX:

FUNC.VALUES:

"MEMBR"

SHELL 4

COMP.-ST

*STRESSES*

POSTBOSOR 1.04

Analysis Details

.70

.80

.90

1.00

1.10

1.20

10

.70

.80

.90

2

Shell No. 4

1.00

1.10

1.20

10

DBA Design by Analysis Page 7.18(S)

Stresses

DBA Design by Analysis

Analysis Summary Example 1.1 / F - Check

Analysis Type:

Member: F - Check

FE –Software:

Page 7.19 (A)

A&AB

ANSYS 5.4

Element Types: 4 – node, 2 – D axisymmetric solid PLANE42 Boundary Conditions:

Zero vertical displacement in the nodes at the undisturbed end of the shell. Symmetry boundary conditions in the nodes in the centre of the flat end.

Model and Mesh:

Height of the model : 1000 mm

Whole height of model: 751.2 mm Number of elements: 1294

Results: Fatigue life = infinity

1

DBA Design by Analysis

Analysis Details Details Analysis Example 1.1 1.2 / F -/ Check Example Rev.0

Page 7.20 (A)

, Data tmax = 20 °C tmin = 20°C t* = 0,75 tmax + 0,25 tmin = 20 °C Rm = 460 MPa Rp0,2/t* = 255 MPa

Rz = 50 µm (table 18-8) en = 101,6 mm ∆σD = 310.8 MPa (table 18-10 for N ≥ 2.106 cycles) N = 2.106 (for the first iteration)

Stresses Critical point: Point of maximum equivalent stress (Tresca) ∆σeq,t (total or notch equivalent stress range) = 75.55 MPa for ∆p=15.3 MPa ∆σstruc (structural equivalent stress range obtained by quadratic extrapolation from the shell side into the critical point) = 116.28 MPa

ó

eq

= 37.78 MPa (mean notch equivalent stress )

∆σstruc = ∆σeq, t if ∆σstruc > ∆σeq, t,

σeqmax = 75.55 MPa (maximum notch equivalent stress)

∆σstruc = 75.55 MPa

Theoretical elastic stress concentration factor Kt

Effective stress concentration factorKeff

Kt =∆σeq,t / ∆σstruc = 1.0 K

eff

18.8 Plasticity correction factor ke mechanical loading If ∆σstruc > 2 Rp0,2/t*

1,5 (K

t

− 1)

 Äó struc  1 + 0,5 K   t Äó D  

= 1 .0

Thermal loading

 Äó  struc k = 1+ A0  − 1 e  2 R p0,2/t *  with A0 =

= 1+

0,5 for 800 MPa ≤ Rm ≤ 1000 MPa 0,4 for Rm ≤ 500 MPa  R − 500  0,4 +  m for 500 MPa ≤ R m ≤ 800 MPa  3000 

If ∆σstruc > 2 Rp0,2/t* 0,7 k = υ 0,4 0,5 +  Äóeq, l     R p0,2/t * 

A0 = ….. ke = ….. ∆σtotal = ke.∆σeq,t = ….. MPa

kυ = ….. ∆σtotal = kυ.∆σeq,t = ….. MPa

Else ∆σtotal = ∆σeq,t = 75.55 MPa

Else ∆σtotal = ∆σeq,t = ….. MPa

If both mechanical and thermal loadings are to be considered, the correction has to be made on each component of the stress tensors. Ke and kí are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then tensors are added and the new stress range is calculated. ∆σeq,struc = Äó

total

K = 75.55 MPa (for usage in 18-11-3) t

18-10-6-2 Temperature correction factor ft* For t* > 100 °C ft* = 1,03 – 1,5 10-4 t* -1,5 10-6 t*2 = ….. Else ft* = 1

18-11-1-1 Surface finish correction factor fs fs = Fs[0,1ln(N)-0,465] if N < 2.106, fs = Fs if N ≥ 2.106, with Fs = 1- 0,056 [ln (Rz)]0,64[ln(Rm)] +0,289 [ln (Rz)]0,53 Fs = 0,7735 fs = 0,7735

DBA Design by Analysis

2

Analysis Details

Page 7.21 (A)

Example 1.1 / F - Check

18-11-2 Thickness correction factor fe en ≤ 25 mm 25 mm < en ≤ 150 mm Fe = 1 Fe = (25/en)0.182 = 0.7748 For N ≥ 2.106 : fe = Fe For N < 2.106: fe = 1 fe = Fe[0,1ln(N)-0,465] fe = 0.7748

en > 150 mm Fe = 0.7217

fe = 0,7217[0,1ln(N)-0,465] fe = …..

18-11-1-3 Mean stress correction factor fm If ∆σstruc < 2 Rp0,2/t* and σeqmax < Rp0,2/t*

If ó If ó

eq eq

and ó

> 0 then ó < 0 then ó

eq

= ó

For N ≥ 2 106 cycles See figure 18-14

eq, r

eq, r eq, r

= Rp0,2/t* =

∆ó eq, t 2

∆ó eq, t 2 - Rp0,2/t*

= …..

For N ≤ 2 106 cycles M = 0,00035 Rm – 0,1= 0,061 if –Rp0,2/t* ≤ ó

fm = 1

If ∆σstruc >2 Rp0,2/t*

If ∆σstruc < 2.Rp0,2/t* and σeqmax > Rp0,2/t*

eq

≤

Äó

 M(2 + M )  2ó   eq  fm = 1 1 + M  ÄóR     

18-11-2-1 Overall correction factor fu f .f .f .f fu = s e m t * = 0.5993

K eff

Äó

R then 2(1 + M )

if

R ≤ ó ≤R p0,2/t* then eq 2(1 + M )

0,5 = …..

fm =

2ó 1 + M 3 M  eq  − = …. 1+ M 3  ÄóR 

∆σequ, struc / fu = 126.1 MPa

18-11-3 Allowable number of cycles N 2

    4   4.6 ⋅ 10 N=   Äó  eq, struc  − 0,63R + 11,5   m f  u  N=∞

if N ≤ 2.106, N = ∞ if ∆σeq, struc / fu ≤ ∆σD





fm = 1

Analysis Details

DBA Design by Analysis Analysis Type

Page 7.22 (S)

Example 1.2

GPD-Check and PD-Check

Member:

Direct Route using Elastic Compensation

FE-Software:

ANSYS 5.4

Element Types:

8 – node, 2-D structural axisymmetric solid.

Boundary Conditions:

Strathclyde

No vertical displacement in the undisturbed end of the shell remote from the flat end. Axisymmetry from elements

Model and Mesh: Number of elements – 384 Height of model – 500 mm

Results: Maximum internal pressure according to the GPD-Check: Check against PD:

PSmax GPD = 4.5 MPa

Shakedown limit pressure PSmax SD = 7.25 MPa

DBA Design by Analysis

Analysis Details Example 1.2

Page 7.23 (S)

1 Finite Element Mesh The geometry model was constructed according to the problem specification for example 1.2. Element divisions for the finite element mesh were defined parametrically to allow modification of the element density. Analysis was carried out for various mesh densities to note any alterations this may have on the results. Finite element models were created using linear 4-node 2-D axisymmetric solids, the analysis was repeated with higher order 8-node structural 2-D axisymmetric solids, any alterations this may have on the results could also be noted. The allowable pressure according to GPD and the shakedown pressure according to PD were calculated from the model giving the greatest lower bounds. Here, the fine density mesh with 4-node higher order elements gave the highest lower bounds. Boundary conditions applied to the model are axisymmetry, applied via a key option when defining the element type in the FE-software (axisymmetry around the vertical axis Y). The nodes at the undisturbed end of the cylindrical shell have their vertical degree of freedom constrained to ensure that plane sections remain plane. 2 Material properties Material strength parameter RM = 255 MPa, modulus of elasticity E = 212 GPa. 3 Determination of the maximum admissible pressure according to the GPD-Check Using the application rule in prEN-13445-3 Annex B.9.2.2, to check against GPD the principle is fulfilled when for any load case the combination of the design actions do not exceed the design resistance. This may be shown by calculating the limit load. The limit load has to be determined using Tresca‘s yield condition and associated flow rule. As elastic compensation is based upon a series of elastic equilibrium stress fields, it is a relatively simple procedure to derive a lower bound limit load direct from the Tresca yield model. From prEN-13445-3 Annex B, Table B.9-3 the partial safety factor γR on the resistance is 1.25. Therefore, the design or material strength parameter is given by RM/γR = 204 MPa. The analysis was carried out using the elastic compensation method conforming to the direct route rules for GPD in Annex B: - linear elastic ideal plastic material law, Tresca’s yield condition and associated flow rule and first order theory. In each elastic compensation iteration an equilibrium stress field is produced where the elastic modulus of each subsequent iteration is defined by the previous elastic solution. In this way regions of the FE-model may be systematically stiffened or weakened depending upon the stress magnitudes in the previous solution. The result is that every equilibrium stress field is a lower bound of the limit load, the stress

Figure 7.3.1-1 Limit Stress Field (Tresca)

Analysis Details

DBA Design by Analysis

Example 1.2

Page 7.24 (S)

field giving the greatest limit load is taken as the closest to the actual limit state and defines the limit load in the analysis. Convergence of the equilibrium stress field will occur typically between 8 and 15 iterations, i.e. no further benefit in the limit load will be noted with subsequent iterations. The total computing time to run the analysis on a 300 MHz Pentium two processor with 128 Mb RAM Windows NT workstation was 106 seconds. The stress field was shown to converge after twelve iterations giving a lower bound on the limit pressure of 5.4 MPa. Figure 7.3.1-1 shows the limit stress (intensity) field according to the Tresca condition. The limit pressure is given by scaling the limit stress field so that the stress anywhere in the model does not exceed the design material strength, 204 MPa, i.e. the applied pressure (10 MPa) is scaled by the factor (204/378.7) 0.54. According to prEN-13445-3 Annex B, Table B.9-2 for pressure loads (without natural limit) the partial safety factor γp is 1.2. Thus, the internal pressure limit according to failure by GPD is 5.4 = 4.5MPa 1.2 It is also possible to determine a limit pressure from the check against PD. In this case, the elastic compensation is based on Von Mises‘ yield criterion. The partial safety factor on the resistance γR is not applied for the PD-check. However, as the analysis is wholly elastic it is possible to scale the stress fields at any time (similarly as was done above). The maximum ratio of Mises' equivalent stress to Tresca's equivalent stress for the same load is 2/√3. Therefore applying a factor of √3/2 to the yield stress in the Mises analysis (or to the limit load, as the analysis is elastic) will always lead to a conservative result. From the Mises analysis the limit load was found to be 7.5 MPa and with the partial safety factors γR = 1.25 and γp = 1.2, the internal pressure limit according to GPD can be found as PS max GPD =

PS max GPD =

7.5 3 ⋅ = 4.33MPa γ p ⋅γ R 2

4 Check against PD In the check for progressive plastic deformation, the principle in prEN13445-3, B.9.3.1 is fulfilled if the structure can be shown to shake down. When a structure has been shown to shake down, the failure modes of progressive plastic deformation and alternating plasticity can not occur. In elastic compensation the load at which the structure will shakedown can be calculated simply. Based on Melan’s shakedown theorem, the self-equilibrating residual stress field that would result after a loading cycle can be calculated by subtracting the linear-elastic stress field at the limit pressure from the limit stress field. The residual stress field is in effect

Figure 7.3.1-2 Residual Stress Field (Mises)

DBA Design by Analysis

Analysis Details Example 1.2

Page 7.25 (S)

the resulting stress from an elastic unloading from the limit state back to zero pressure. If no stress in the residual field violates the yield condition, i.e. if there is no stress above the design material (yield) strength, then the shakedown load is equal to the limit load. Where the residual stress field of a structure does exceed the yield condition, the shakedown limit can be calculated easily from the stress plots because of the linearity. The residual stress field from the elastic compensation analysis using Mises' criterion is shown in Figure 7.3.1-2. Because the applied load is arbitrary and the resulting stress fields are scaled to the yield condition, the maximum residual stress is compared with the maximum stress in the limit stress field, Figure 7.3.1-3. The maximum residual stress of 249.5 MPa is lower than that of 339.5 MPa for the limit stress field. The shakedown limit is therefore the same as the calculated limit load from the Mises condition, given by scaling up the load by a factor of material yield parameter to maximum stress in the limit field (255/339.5) 0.75. With an applied load of 10 MPa the shakedown limit is 7.5 MPa.

Figure 7.3.1-3 Limit Stress Field (Mises)

5 Check on GPD Using Non-linear Analysis A check against GPD was also performed for the same FE-model using conventional non-linear analysis. In this way, a direct comparison may be made between the two limit approaches. The FEgeometry, mesh and boundary conditions are the same as those used in the elastic compensation analysis. Material non-linearities were applied corresponding to the design material strength parameter, 204 MPa, and perfect plasticity. A ramped load is applied and the analysis runs until the applied load is such that convergence can no longer occur due to unrestrained displacement – Gross Plastic Deformation. It is assumed that the last converged solution is the limit load. Here, the last converged

Figure 7.3.1-4 Limit Stress Field (non-linear analysis)

Analysis Details

DBA Design by Analysis

Example 1.2

Page 7.26 (S)

solution was at a load of 7.9 MPa, using the same method as above the allowable pressure according to GPD using Mises' criterion is 7.9 3 ⋅ = 5.7 MPa γp 2 The result offers a considerable benefit to the allowable pressure calculated using elastic compensation. Figure 7.3.1-4 shows the Mises equivalent stress at the limit. Analysis time to calculate limit load using non-linear FE-analysis was 360 seconds. PS max GPD =

6

Additional Comments

Additional analysis was completed to ascertain any effect on the results for different mesh density and for higher order elements (8-node). Limit loads and shakedown were calculated using Mises condition, the results are summarised in Table 7.3.1-1. Number of Elements 151 151 267 267 383 383

Element Type 4 node 8 node 4 node 8 node 4 node 8 node

Number of Iterations 8 8 8 8 12 12

Lower Bound Limit Load 4.7 4.4 5.0 4.8 5.2 5.1

Shakedown Load 6.8 6.4 7.25 6.9 7.5 7.4

Processor Time 50 87.3 85.5 152.1 184 204

Table 7.3.1-1. Limit and shakedown analysis summary

Both the lower bound limit load and the shakedown load were calculated using the same method as described above. As can be seen from Table 7.3.1-1, the choice of element type and mesh density can make a considerable difference to the results (maximum difference of 14%). The greatest difference is noted over the various mesh densities, with the highest density mesh giving the greatest lower bound on the limit pressure and shakedown pressure. It can be noted that the higher order elements give a lower limit pressure than the lower order elements, although the difference is less significant than that due to mesh density. As the element density increases the difference in limit pressure and shakedown pressure for the different element type is reduced. The utilisation of elastic compensation in this DBA calculation has given a small increase in the allowable pressure over that given by the DBF calculation. The two methods used to calculate the lower bound limit load (direct from Tresca's criterion or correction of Mises') show good correlation; the Mises corrected value is slightly conservative as would be expected. However, carrying out the Mises elastic compensation will give both the limit load and shakedown load in one analysis. Non-linear calculations offer a considerable benefit in terms of higher allowable pressure than the pressures calculated by DBF and elastic compensation in this example.

DBA Design by Analysis

Analysis Details

Page 7.27 (A)

Example 1.2

Analysis Type:

Member:

Direct Route using elasto-plastic FE calculations

FE-Software:

A&AB

ANSYS® 5.4

Element Types: 4 – node, 2 – D axisymmetric solid PLANE42 Boundary Conditions:

Zero vertical displacement in the nodes at the unsdisturbed end of the shell; Symmetry boundary conditions in the nodes in the centre of the flat end.

Model and Mesh:

Whole height of model: 476,2 mm Number of elements 2640

Results: Maximum allowable pressure according to GPD: PS max GPD = 5.7 MPa Shakedown limit pressure:

PS max SD = 7.9 MPa

DBA Design by Analysis

Analysis Details Example 1.2

Page 7.28 (A)

1. Elements, mesh fineness, boundary conditions The model of the structure is shown on page 1, a total number of 2640 4-node axisymmetric solid elements – PLANE42 in ANSYS® 5.4 - was used. The linear shape function of these elements is sufficient, since • the number of elements in the (linear-elastic) high stressed region is large, • the computation time in an analysis using nonlinear material properties is much higher for elements with midside nodes and quadratic shape functions, although, close to the limit load, the results are almost identical compared to those using elements with a linear shape function, • there is no need to compute linear-elastic peak stresses very exactly, because the check against PD can be carried out using the stress-concentration-free structure (according to pr EN 13445-3 Annex B.9.3.2) and the structure’s geometry is modelled exactly in the example considered. The boundary conditions applied in the model are symmetry ones in the nodes in the centre of the plate (where the horizontal direction is perpendicular to the plane of symmetry) and constraining the vertical degree of freedom in the nodes at the undisturbed end of the cylindrical shell to zero. 2. Determination of the maximum admissible pressure according to GPD Since the subroutine using Tresca’s yield condition showed bad convergence, the results from the check against PD (see subsection 3.2 of section 3 – Procedures) have been used. The partial safety factor γ R according to prEN 13445-3 Annex B, Table B.9-3 is 1.25 and the partial safety factor for pressure (without natural limit) according to prEN 13445-3 Annex B, Table B.9-2 γ P is 1.2 . Therefore, the (internal) allowable pressure according to the GPD-check is given by PS max GPD =

9.904 3 9.904 3 ⋅ = ⋅ = 5.72 MPa. γ P ⋅γ R 2 1.2 ⋅ 1.25 2

The maximum principal strain, at the limit pressure used here and for the design material strength parameter of the check against PD, is about 3 %, the condition in prEN 13445-3 Annex B.9.2.2 is fulfilled. 3. Check against PD The elasto-plastic FE analysis was carried out as stated in prEN 13445-3 Annex B, Sec. B.9.3.1, using Mises’ yield condition and associated flow rule, an linear-elastic ideal-plastic law with a design material strength parameter of 255 MPa for shell and plate (according to EN 10222-2 and for the structure turned from one forged part), and first order theory. The elastic modulus used in the calculations is E = 212 GPa. By defining and using load cases in ANSYS®, the superposition of stress fields can be done easily. Therefore an early load step of the analysis was defined at a very low load level (2 MPa), so that there was linear-elastic response of the structure. All other linear-elastic stress fields can then be determined easily by multiplication with a suitable scale-up factor.

DBA Design by Analysis

Analysis Details Example 1.2

Page 7.29 (A)

The analysis was carried out using the arc-length method; since at the limit load the structure is fully plastified in the plate, a maximum vertical displacement at the middle of the plate of 10 mm was used as termination criterion. The computation time of the limit load was 1 hour and 15 minutes on the Compaq® Professional Workstation 5000 with two Pentium Pro® processors and 256 MB RAM. The termination criterion was fulfilled at an internal pressure of 9.904 MPa – this pressure was used as limit pressure. Figure 7.3.2-1 shows the vertical displacement at the middle of the plate versus the internal pressure; according to this figure the limit state is reached.

Figure 7.3.2-1

Figure 7.3.2-2 shows the elastoplastic Mises equivalent stress distribution at the limit pressure of 9.9 MPa . Here, because of the almost full plastification in the plate there are small regions near the mid-plane of the plate, where, due to numerical effects, the equivalent stress exceeds 255 MPa, but this has no effect on further analyses.

Figure 7.3.2-2

DBA Design by Analysis

Analysis Details Example 1.2

Figure 7.3.2-3 shows the linearelastic Mises equivalent stress distribution for the limit pressure – the stress maximum is located in the fillet.

F Figure 7.3.2-3

Figure 7.3.2-4 shows the Mises equivalent stress distribution of the corrected residual stress field. The used scaling factor β is 0.66 (see subsection 3.3.2.5 of section 3 Procedures), the maximum stress is located again in the fillet.

Figure 7.3.2-4

Page 7.30 (A)

Analysis Details

DBA Design by Analysis

Example 1.2

Figure 7.3.2-5 shows the Mises equivalent stress distribution at the lower bound shakedown limit. The scaling factor α is given by 0.802 (see subsection 3.3.2.5 of section 3 Procedures). The shakedown limit pressure is given by PS max SD = 0.802 ⋅ 9.904 = 7.943 MPa

F Figure 7.3.2-5

Page 7.31 (A)

Analysis Details

DBA Design by Analysis

Example 1.2

Analysis Type: Stress Categorization Route

Page 7.32 (T) Member: TKS

FE-Software: BOSOR

Element Types: Axisymmetric shell elements

Model and Mesh:

As shown in diagram – Shell numbers in Calculation Model

Results: Maximum admissible action according to the Stress Categorization Route: Internal pressure PSmax SC = 5.6 MPa

Analysis Details

DBA Design by Analysis

Page 7.33 (T)

Example 1.2

-4.73E+01 6.00E+02 MIN: MAX:

JOB NO12 99-08-25 10.59.05

-1.16E-19 3.02E+02 MIN: MAX:

): WINDOW (X,Y

UNDEFORM.

DEFORM.

ALL SHELLS

*GEOMETRY PLOT

POSTBOSOR 1.04

The following figure shows the deformed model. The figure following the next show the distributions of stresses – membrane and membrane plus bending – in the surfaces of the various parts of the model. With the designation list in subsection 3.6 the plots are self-explaining. The calculation pressure used is 4.2 MPa. The limiting part is the flat end, and the critical point is the center, where the primary membrane plus bending stress is governing – see the 4th figure after the next. The allowable design stress used is 170 MPa. The membrane plus bending stress at the plate to cylinder junction has been classified as secondary – see the classification table of Annex C of prEN 13445-3.

2

.50

.50

1.00

1.00

1.50

1.50

2.00

2.00

2.50 10

2.50 10

2

2

7.60E+00 1.48E+01

JOB NO12 99-08-20 14.35.45

-1.62E+02 1.91E+02

10

.80

1.00

1.20

1.40

1.60

1.80

2.00

2.20

2.40

-2.00

-1.50

2

.50

1.00

1.50

2.00

2.50 10

2

.50

1 .00

191MPa (σ eq )P ≤ 1.5 f

1.50

2.00

2.50 10

2

"INSIDE"

-2.17E+02 2.47E+02

6.37E+01 2.47E+ 02 JOB NO12 99-08-20 14.36.51

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 1

TRESCA

*STRESSES*

POSTBOSOR 1.04

JOB NO12 99-0 8-20 14.34.52

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

Example 1.2

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 1

CIRCUMF

*STRESSES*

POST BOSOR 1.04

JOB NO12 99-08-20 14.34.18

MIN: MAX:

FUNC.VALUES:

- 1.00

-.50

.00

.50

1.00

Analysis Details

-1.50

- 1.00

-.50

.00

.50

1.00

1.50

10

8.00

9.00

10.00

"MEMBR"

SHELL 1

1.50

SHELL 1

*STRESSES*

POSTBOSOR 1.04

GEN.DIR

2.00

2

COMP.-ST

*STRESSES*

10

Shell No. 1

11.00

12.00

13.00

14.00

POSTBOSOR 1.04

DBA Design by Analysis Page 7.34 (T)

Stresses

1

2.60

2.60

2.70

2.70

2.80

2.80

2.90

2.90

3.00

3.00

10

10

2

2

4.36E+00 7.07E+00 2.6

0

2.70

2.80

2.90

3.00

10

2

JOB NO12 99-08-20 14.39.24

-2.35E+00 1.57E+01

10.00

15.00

20.00

25.00

2.70

2.80

2.90

3.00

10

2

5.93E+00 3.38E+01 JOB NO12 99-08-20 14.39.53

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 2

Example 1.2

MIN: MAX:

FU NC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 2

TRESCA

*STRESSES*

CIRCUMF

*STRESSES*

2.60

-2.52E+01 3.38E+01 JOB NO12 99-08-20 14.38.30

MIN: MAX:

FUNC.VALUES:

POSTBOSOR 1.04

30.00

-2.00

POSTBOSOR 1.04

JOB NO12 99-08-20 14.37.56

MIN: MAX:

FUNC.VALUES:

-1.00

.00

1.00

"INSIDE" "OUTSIDE"

S*

Analysis Details

-.20

.00

.20

.40

.60

.80

1.00

1.20

1.40

10

4.50

5.00

"MEMBR"

SHELL 2 2.00

SHELL 2

*STRESSE

POSTBOSOR 1.04

GEN.DIR

3.00

1

COMP.-ST

*STRESSES*

10

Shell No. 2

5.50

6.00

6.50

7.00

POSTBOSOR 1.04

DBA Design by Analysis Page 7.35 (T)

Stresses

1

4.00

4.00

5.00

5.00

6.00

6.00

7.00

7.00

8.00

8.00

10

10

2

2

4.30E+00 1.09E+01

JOB NO12 99-08-20 14.44.02

-2.16E+00 1.09E+01

5.00

7.50

10.00

12.50

15.00

17.50

20.00

22.50

25.00

- 1.50

-1.00

-.50

.00

.50

1.00

1.50

4.00

4.00

5.00

5.00

6.00

6.00

7.00

7.00

8.00

8.00

10

10

2

2

-1.73E+01 2.59E+01

S*

3.14E+00 2.59E+ 01 JOB NO12 99-08-20 14.44.29

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 3

TRESCA

*STRESSES*

POSTBOSOR 1.04

JOB NO12 99-08-20 14.41.11

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 3

GEN.DIR

*STRESSE

POSTBOSOR 1.04

Example 1.2

MIN: MAX:

FU NC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 3

CIRCUMF

*STRESSES*

POSTBOSOR 1.04

JOB NO12 99-08-20 14.40.29

MIN: MAX:

FUNC.VALUES:

"MEMBR"

SHELL 3

2.00

1

Analysis Details

-.20

.00

.20

.40

.60

.80

1.00

10

5.00

6.00

*STRESSES* COMP.-ST

2.50

10

Shell No. 3

7.00

8.00

9.00

10.00

POSTBOSOR 1.04

DBA Design by Analysis Page 7.36 (T)

Stresses

DBA Design by Analysis

Analysis Details

Page

Example 1.2 / F-Check

7.37 (C) Member:

Analysis Type: F-Check

CETIM

FE –Software: ABAQUS/Standard version 5.8.1 Element Types: Quadratic axisymmetric 8 nodes elements (CAX8). 1161 nodes and 344 elements Boundary Conditions: The nodes at the high of the cylindrical part are locked in the vertical direction.

Model and Mesh:

Height of the model : 1000 mm

Results: Fatigue life = 375200 cycles

DBA Design by Analysis

Data tmax = 20 °C tmin = 20°C t* = 0,75 tmax + 0,25 tmin = 20 °C Rm = 460 MPa Rp0,2/t* = 255 MPa

Analysis Details Example /F-Check Example1.21.2 / Rev.0

2

Page 7.38 (C)

Rz = 50 µm (table 18-8) en = 25,8 mm ∆σD = 310.8 MPa (table 18-10 for N ≥ 2 106 cycles) N = 2.106 (for the first iteration) ∆σR = 353.4 MPa (allowable stress range for N cycles at the 10th iteration)

Stresses ∆σeq,t (total or notch equivalent stress range) = 271.6 MPa ∆σstruc (structural equivalent stress range) = ∆σeq,l (linearised equivalent stress range) = 230.2 MPa

ó

eq

= 135.8 MPa (mean notch equivalent stress )

Theoretical elastic stress concentration factor Kt

σeqmax = 271.6 MPa (maximum notch equivalent stress) Effective stress concentration factorKeff

Kt =∆σeq,t / ∆σstruc = 1.1798 K

eff

18.8 Plasticity correction factor ke mechanical loading If ∆σeq,l > 2 Rp0,2/t*

1,5 (K

t

− 1)

 Äó struc  1 + 0,5 K   t Äó D  

= 1 . 1877

Thermal loading

 Äó eq, l  k = 1+ A0  − 1 e  2 R p0,2/t *    with A0 =

= 1+

0,5 for 800 MPa ≤ Rm ≤ 1000 MPa 0,4 for Rm ≤ 500 MPa  R m − 500  0,4 +  3000  for 500 MPa ≤ R m ≤ 800 MPa

A0 = ….. ke = ….. ∆σtotal = ke.∆σeq,t = ….. MPa Else ∆σtotal = ∆σeq,t = 271.66 MPa

If ∆σeq,l > 2 Rp0,2/t* 0,7 k = υ 0,4 0,5 +  Äóeq, l     R p0,2/t *  kυ = ….. ∆σtotal = kυ.∆σeq,t = ….. MPa Else ∆σtotal = ∆σeq,t = ….. MPa

If both mechanical and thermal loadings are to be considered, the correction has to be made on each component of the stress tensors. Ke and kí are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated. ∆σeq,struc = Äó

total

K

t

= 228.7 MPa (for using in 18-11-3)

18-10-6-2 Temperature correction factor ft* For t* > 100 °C ft* = 1,03 – 1,5 10-4 t* -1,5 10-6 t*2 = ….. Else ft* = 1

18-11-1-1 Surface finish correction factor fs fs = Fs[0,1ln(N)-0,465] with Fs = 1- 0,056 [ln (Rz)]0,64[ln(Rm)] +0,289 [ln (Rz)]0,53 Fs = 0.7735 fs = 0.8104

DBA Design by Analysis

3

Analysis Details

Page 7.39(C)

Example 1.2 / F-Check

18-11-1-2 Thickness correction factor fe en ≤ 25 mm 25 mm ≤ en ≤ 150 mm fe = Fe[0,1ln(N)-0,465]

en ≥ 150 mm fe = 0,7217[0,1ln(N)-0,465]

with Fe = (25/en)0.182 = 0.9943

fe = 1

fe = 0.9953

fe = …..

18-11-1-3 Mean stress correction factor fm If ∆σstruc < 2 Rp0,2/t* and σeqmax < Rp0,2/t*

If ó If ó

eq eq

and ó

> 0 then ó < 0 then ó

eq

= ó

For N ≥ 2.106 cycles See figure 18-14

eq, r

eq, r eq, r

= Rp0,2/t* =

Äóeq, t 2

∆ó eq, t 2 - Rp0,2/t*

= 119.2 MPa

For N ≤ 2.106 cycles M = 0,00035 Rm – 0,1= 0,061 if –Rp0,2/t* ≤ ó

fm = …..

If ∆σstruc >2 Rp0,2/t*

If ∆σstruc < 2 Rp0,2/t* and σeqmax > Rp0,2/t*

eq

≤

Äó

R then 2(1 + M )

 M(2 + M )  2ó   eq  fm = 1 1 + M  ÄóR     

0,5

Äó

if

R ≤ ó ≤R p0,2/t* then eq 2(1 + M )

2ó 1 + M 3 M  eq  f = − = …. = 0.9592 m 1+ M 3  ÄóR 





fm = 1

18-11-2-1 Overall correction factor fu f .f .f .f fu = s e m t * = 0.6514

K eff

18-11-3 Allowable number of cycles N 2

    4   4.6 ⋅ 10 N=   Äó  eq, struc  − 0,63R + 11,5   m f   u

10

     2,7 R m + 92  N=   if N ≤ 2.106 cycles Äó  eq, struc   f   u 

if 2.106 ≤ N ≤ 108cycles

N = 375200 N is obtained by iterations. If the value of N oscillates, a difference equal to 1 % between two iterations is acceptable. If the values decrease monotonously, the difference must be less than 0,001 % between two iterations.

Analysis Details

DBA Design by Analysis

Page 7.40 (S)

Example 1.3

Analysis Type: GPD- and PD-Check

Member: Strathclyde

Direct Route using Elastic Compensation

FE-Software:

ANSYS 5.4

Element Types:

8 – node, 2-D structural axisymmetric solid.

Boundary Conditions:

No vertical displacement in the undisturbed end of the shell remote from the flat end. Axisymmetry from elements

Model and Mesh:

Number of elements – 1039 Total height – 500 mm

Results: Maximum internal pressure according to the GPD-Check: Check against PD:

PSmax GPD = 10.8 MPa

Shakedown limit pressure PSmax SD = 13.3 Mpa

DBA Design by Analysis

Analysis Details Example 1.3

Page 7.41 (S)

1. Finite Element Mesh Finite element models were created using 1040 linear 4-node 2-D axisymmetric solids. The allowable pressure according to GPD and the shakedown pressure according to PD were calculated using elastic compensation procedures and the rules prEN-13445-3 Annex B. Boundary conditions applied in the model are axisymmetry, applied via a key option when defining the element type in the FE-software (axisymmetry around the vertical axis Y). The nodes at the undisturbed end of the cylindrical shell had their vertical degree of freedom constrained to zero to ensure that plane sections remain plane. 2. Material properties Material strength parameter RM = 255 MPa (cylindrical shell), RM = 245 MPa (flat end), modulus of elasticity E = 212 GPa (shell and flat end). 3. Determination of the maximum admissible pressure according to the GPD-Check Using the application rule in prEN-13445-3 Annex B.9.2.2, to check against GPD the principle is fulfilled when for any load case the combination of the design actions does not exceed the design resistance. This may be shown by calculating the limit load. The limit load has to be determined using Tresca‘s yield condition and associated flow rule. As elastic compensation is based upon a series of elastic equilibrium stress fields, it is a relatively simple procedure to derive a lower bound limit load direct from the Tresca yield model. From prEN-13445-3 Annex B, Table B.9-3 the partial safety factor, γR on the resistance is 1.25. Therefore, the design material strength parameters given by RM/γR are 204 MPa for the shell and 196 MPa for the plate. The analysis was carried out using the elastic compensation method conforming to the direct route rules for GPD in Annex B: linear elastic ideal plastic material law, Tresca’s yield condition and associated flow rule and first order theory. In each elastic compensation iteration an equilibrium stress field is produced where the elastic modulus of each subsequent iteration is defined by the previous elastic solution. In this way regions of the FE-model may be systematically stiffened or weakened depending upon the stress magnitudes in the previous Figure 7.4.1-1: Limit Stress Field (Tresca) solution. The result is that every equilibrium stress field is a lower bound of the limit load, the stress field giving the greatest limit load is taken as the closest to the actual limit state and defines the limit load in the analysis. In problems such as this, where there are materials with different properties the modulus modification has a modified procedure that takes account of the different material properties. This modified method calculates the limit pressure for

Analysis Details

DBA Design by Analysis

Example 1.3

Page 7.42 (S)

each component with a different material, allowing the component giving the lowest limit load to define the limit for the whole model. Convergence of the equilibrium stress field will occur typically between 8 and 15 iterations, i.e. no further benefit in the limit load will be noted with subsequent iterations. The total computing time to run the analysis on a 300 MHz Pentium two processor with 128 Mb RAM Windows NT workstation was 180 seconds. Figure 7.4.1-1 shows the limit stress (intensity) field according to the Tresca condition. The maximum stresses in the plate and shell are 152.0 MPa and 152.3 MPa respectively, and the limit pressure for each component is given by using the linear proportionality: Rd σ max Pap is the applied load, Rd the design material strength, and σmax the maximum redistributed stress for the component with design resistance Rd. The limit loads for the plate and shell are as follows: PL = Pap ⋅

plate

shell

PL = Pap ⋅

Rd 196 = 10 ⋅ = 13.0 MPa σ max 151.1

PL = Pap ⋅

Rd 204 = 10 ⋅ = 13.4 MPa σ max 152.3

Therefore, for this stress field the limiting component is the plate and the limit load for the structure is 13.0 MPa. According to prEN-13445-3 Annex B, Table B.9-2 for pressure loads (without natural limit) the partial safety factor γp is 1.2. Thus, the internal pressure limit according to the check against GPD is It is also possible to determine a limit pressure from the check against PD. In this case, the elastic compensation is based on Mises‘ yield condition. The partial safety factor on the resistance γR is not applied for the PD-check. From the Mises analysis in the PD-check, the limit load for the structure was found as 17.55 MPa using the same method as above. The maximum ratio of Mises' 13.0 13.0 PS max GPD = = = 10.8 MPa 1.2 γP equivalent stress to Tresca's equivalent stress for the same load is 2/√3. Therefore, applying a factor of √3/2 to the yield stress in the Mises analysis (or to the limit load, as the analysis is elastic) will always lead to a conservative result. Applying partial safety factors γR = 1.25 and γp = 1.2, the internal pressure limit according to the GPD-check can be found to be 4. Check against PD PS max GPD =

17.55 3 17.55 3 ⋅ = ⋅ = 10.13 MPa γ p ⋅γ R 2 1.2 ⋅ 1.25 2

In the check for progressive plastic deformation, the principle in prEN-13445-3 B.9.3.1 is fulfilled if the structure can be show to shake down. When a structure has been shown to shake down, the failure modes of progressive plastic deformation and alternating plasticity cannot occur.

DBA Design by Analysis

Analysis Details Example 1.3

Page 7.43 (S)

In elastic compensation, the limit on the load at which the structure will undergo shakedown can be calculated simply. Based on Melan’s shakedown theorem, the self-equilibrating residual stress field that would result after a loading cycle can be calculated by subtracting the linear-elastic stress field at the limit pressure from the limit stress field. The residual stress field is in effect the resulting stress from an elastic unloading from the limit state back to zero pressure. If no stress in the residual field violates the yield condition, i.e. if there is no stress above the design material strength parameter, then the shakedown load is equal to the limit load. Where the residual stress field of a structure does exceed the yield condition, the shakedown limit can be calculated easily from the stress plots because of the linearity. The residual stress field from the elastic compensation analysis using Mises'condition is shown in Figure7.4.1-2. In the check against PD there are no partial safety factors applied to the material strength or actions. Therefore, the design material strength is 245 MPa for the plate and 255 MPa for the shell respectively. The maximum stress in the residual field is greater than the stress defining the limit load in the PD analysis. Figure 7.4.1-2: Equivalent Residual Stress Field The shakedown limit will be less than the limit load given by the PD-check. By invoking the elastic proportionality the shakedown load is given as Figure 7.4.1-3 shows the limit stress field for the PD-check. This can be used to calculate the limit load for the structure based on Mises' condition criterion and associated flow rule. The maximum stress in the plate is 139.64 MPa and by invoking the proportionality of the elastic solution, the limit load is 17.55 MPa. This can be Rd 245 = 10 ⋅ = 13.3MPa σ max 184.7 used in a check against GPD as shown above. PS max SD = Pap ⋅

5. Check against GPD Using Non-linear Analysis

Figure 7.4.1-3: Equivalent Limit Stress Field.

DBA Design by Analysis

Analysis Details Example 1.3

Page 7.44 (S)

A check against GPD was also performed for the same FE-model using conventional non-linear analysis. In this way, a direct comparison may be made between the two limit approaches. The FEgeometry, mesh and boundary conditions are the same as those used in the elastic compensation analysis. Material non-linearities were applied corresponding to the design material strength parameters, 204 MPa for the shell, 196 MPa for the plate and perfect plasticity. A ramped load is applied and the analysis runs until the applied load is such that convergence can no longer occur due to unrestrained displacement – Gross Plastic Deformation. It is assumed that the last converged solution is the limit load. Here, the last converged solution was at a load of 17.29 MPa. Using the same method as above the allowable pressure according to the GPD-check using Mises' condition is

Figure 7.4.1-4 Limit Stress Field (non-linear analysis)

The result offers a considerable benefit to the allowable pressure calculated using elastic compensation. Figure 7.4.1-4 shows the equivalent stress distribution at the limitpressure. Considerable plasticity can be noted in the plate with some additional plasticity spreading into the stronger material in the shell. Analysis time to calculate limit load using non-linear FE-analysis 17.29 3 ⋅ = 12.48MPa γp 2 was 360 seconds. PS max GPD =

DBA Design by Analysis

Analysis Details

Member:

Analysis Type:

A&AB

Direct Route using elasto-plastic FE calculations

FE-Software:

Page 7.45 (S)

Example 1.3

ANSYS® 5.4

Element Types: 4 – node, 2 – D axisymmetric solid PLANE42 Boundary Conditions:

Zero vertical displacement in the nodes of the undisturbed end of the shell; Symmetry boundary conditions in the nodes in the centre of the flat end.

Model and Mesh:

Whole height of model: 443 mm Number of elements: 2690

Results:

Maximum allowable pressure according to GPD:

PS max GPD = 12.6 MPa

Shakedown limit pressure:

PS max SD = 13.3 MPa

DBA Design by Analysis

Analysis Details Example 1.3

Page 7.46 (S)

1. Elements, mesh fineness, boundary conditions The model of the structure is shown on page 1, a total number of 2690 4-node axisymmetric solid elements – PLANE42 in ANSYS® 5.4 - was used. The linear shape function of these elements is sufficient, since • the number of elements in the (linear-elastic) high stressed region is large, • the computation time in an analysis using nonlinear material properties is much higher for elements with midside nodes and quadratic shape functions, although, close to the limit load, the results are almost identical compared to those using elements with a linear shape function, • there is no need to compute linear-elastic peak stresses very exactly, because the check against PD can be carried out using the stress-concentration-free structure (according to prEN 13445-3 Annex B.9.3.2) and the structure’s geometry is modelled exactly in the example considered. The boundary conditions applied in the model are symmetry ones in the nodes in the centre of the plate (where the horizontal direction is perpendicular to the plane of symmetry) and constraining the vertical degree of freedom in the nodes at the undisturbed end of the cylindrical shell to zero.

2. Determination of the maximum admissible pressure according to GPD According to prEN 13445-3 Annex B, Table B.9-3, the partial safety factor γ R is 1.25 . Therefore, the analysis using Tresca’s yield condition (delivered by an ANSYS® distributor) was carried out with a linear-elastic ideal-plastic material law, design material strength parameters of 204 MPa for the shell and 196 MPa for the plate (corresponding to material strength parameters according to EN 10028-2 of 255 MPa for the shell and 245 MPa for the plate), associated flow rule, and first order theory. For simplification, the boundary between the two materials was assumed to be in the plane of the upper surface of the plate. The elastic modulus of the cylindrical shell and the flat end used in all calculations was E=212 GPa. The analysis was carried out using the arc-length method, which showed faster convergence for the considered structure than the Newton - Raphson method. Since at the limit load the structure is fully plastified in the plate and in the shell adjacent to the plate, a maximum vertical displacement in the middle of the plate of 10 mm was used as termination criterion. To restrict computation time in an appropriate manner, the analysis was terminated after 15 hours on a Compaq® Professional Workstation 5000 with two Pentium Pro® processors and 256 MB RAM.

DBA Design by Analysis

Analysis Details Example 1.3

At termination time the last convergent solution showed an internal pressure of 15.12 MPa , and this pressure was used as limit pressure. Figure 7.4.2-1 shows the vertical displacement in the middle of the plate versus the internal pressure.

Figure 7.4.2-1

Figure 7.4.2-2 shows Tresca's equivalent stress for this limit pressure. Because of the almost full plastification in the plate, there are small regions near the mid-plane of the plate, where, due to numerical effects, the equivalent stress slightly exceeds 196 MPa , but this has no effect on further analyses.

F Figure 7.4.2-2

Page 7.47 (S)

Analysis Details

DBA Design by Analysis

Example 1.3

Page 7.48 (S)

As shown in Figure 7.4.2-3 the maximum absolute value of the principal strains in the structure is less than 5 % - as required in the standard. According to prEN 13445-3 Annex B, Table B.9-2, the partial safety factor for pressure (without a natural limit) γ P is given by 1.2. Therefore, the allowable (internal) pressure according to the GPDcheck is

PS max GPD =

15.12 = 12.6 MPa. 1.2

Figure 7.4.2-3

A less time-consuming method to determine a limit action according to the GPD-check is given by usage of the limit pressure result form the check against PD (see subsection 3.2 of section 3 Procedures). With the partial safety factors γ R = 1.25 and γ P = 1.2 , the allowable (internal) pressure according to the GPD-check is, in this approach, PS max GPD =

21.37 3 21.4 3 ⋅ = ⋅ = 12.34 MPa , γ P ⋅ γ R 2 1.2 ⋅ 1.25 2

where 21.37 MPa is the pressure calculated in the check against PD for which the maximum absolute value of the principal strains does not exceed 5%. 3. Check against PD The elasto-plastic FE analysis was carried out as stated in prEN 13445-3 Annex B, Sec. B.9.3.1 using Mises’ yield condition and associated flow rule, a linear-elastic ideal-plastic constitutive law with design material strength parameters of 255 MPa for the shell and 245 MPa for the plate (for simplification the boundary between the materials was assumed again to be in the plane of the upper surface of the plate), and first order theory. By defining and using load cases in ANSYS®, the superposition of stress fields can be done easily. Therefore the first load step of the analysis was defined at a very low load level, so that there was

DBA Design by Analysis

Analysis Details Example 1.3

Page 7.49 (S)

linear-elastic response of the structure. All other linear-elastic stress fields can then be determined easily by multiplication with a suitable scale-up factor. The analysis was carried out using the arc-length method. Since at the limit load the structure is fully plastified in the plate and in the shell adjacent to the plate, a maximum vertical displacement in the middle of the plate of 10 mm was used as termination criterion. The computation time of the limit load was 1 hour and 10 minutes on the Compaq® Professional Workstation 5000 with two Pentium Pro® processors and 256 MB RAM. The termination criterion was fulfilled at an internal pressure of 21.43 MPa – this pressure was used as limit pressure. Figure 7.4.2-4 shows the vertical displacement in the middle of the plate versus the internal pressure; according to this figure the limit state is reached. Figure 7.4.2-4

Figure 7.4.2-5 shows the elastoplastic Mises equivalent stress distribution at the limit pressure of 21.43 MPa . Because of the almost full plastification in the plate, there are small regions near the mid-plane of the plate, where, due to numerical effects, the equivalent stress exceeds 245 MPa , but this has no effect on further analyses.

Figure 7.4.2-5

DBA Design by Analysis

Analysis Details Example 1.3

Figure 7.4.2-6 shows the distribution of the linearelastic Mises’ equivalent stress for the limit pressure – the stress maximum is located in the groove.

Figure 7.4.2-6

Figure 7.4.2-7 shows the Mises equivalent stress of the corrected residual stress field, with a used scaling factor β given by 0.444 (see subsection 3.3.2.5 of section 3 - Procedures), the stress maximum is again located in the groove.

1 . Figure 7.4.2-7

Page 7.50 (S)

Analysis Details

DBA Design by Analysis

Page 7.51 (S)

Example 1.3

Figure 7.4.2-8 shows the Mises equivalent stress distribution for the lower bound shakedown limit. The scaling factor α is given by 0.621 (see subsection 3.3.2.5 of section 3 - Procedures), and the shakedown limit pressure by PS max SD = 0.621 ⋅ 21.43 = 13.3 MPa

The greatest equivalent stress is again located in the groove, the scaling factor α is limited by the yield stress of the plate.

Figure 7.4.2-8

Figure 7.4.2-9 shows a plot of the stress states in the groove’s surface ( ϕ = 0° is adjacent to the shell, ϕ = 180° is adjacent to the plate) in the deviatoric map, i.e. given in isometric coordinates by the principal stresses σ 1 (perpendicular to the groove's surface), σ 2 (tangential to the groove's surface), and hoop stress σ2 σ3. It can easily be seen, that the sum (thick green curve) of the linear-elastic stress (thin green curve, which corresponds to an internal pressure of 133 MPa ) and of the corrected selfequilibrating stress (yellow curve) does not violate the yield condition, which is given by the circle.

ϕ = 180°

σ3

ϕ = 0° F i

F Figure 7.4.2-9

σ1

DBA Design by Analysis

Analysis Details Example 1.3

Page 7.52 (S)

Note: Use of the application rule for the check against PD for constant principal stress directions, see prEN 13445-3 Annex B.9.3.2, leads to a limit pressure of PS =

2 ⋅ RM 2 ⋅ 245 ⋅ 21.43 = 13.4 MPa , ⋅ 21.43 = 783.8 783.8

where 783.8 MPa is the maximum Mises equivalent stress of the linear-elastic solution for an internal pressure of 21.43 MPa . Since the requirement given in the application rule is only a necessary condition for the fulfilment of the principle, i.e. it is a necessary but not a sufficient condition for shakedown, the result using the application rule usually differs from that using Melan’s theorem, but in this example the difference of 0.1 MPa is negligible.

4. Comments As shown by this DBA calculation, the DBF result is in this case optimal, since the maximum allowable internal pressure according to DBF is practically the same as the one according to DBA.

DBA Design by Analysis

Analysis Details Example 1.3

Analysis Type: Stress Categorisation Route

FE-Software: BOSOR

Element Types:

Axisymmetric shell elements

Model and Mesh:

As shown in diagram – Shell Numbers in Calculation Model

Results: Maximum admissible action according to the Stress Categorization Route: Internal pressure PSmax SC = 12.3 MPa

Page 7.53 (S)

Member: TKS

Analysis Details

DBA Design by Analysis

Page 7.54 (S)

Example 1.3

The radius R 18 at the periphery of the flat end has been simulated by two strait lines, see the figure of the model. This is acceptable, as BOSOR calculates the structural stresses, and this is a simplification, which has negligible influence of the structural stresses. The following figure shows the model's displacements. The plots thereafter the distributions of stresses – membrane and membrane plus bending – in the various parts of the model. The plots are for a pressure of 15 MPa. The critical points is the connection between the cylinder and the flat end. At this point the secondary stresses in the cylinder are limited by the shakedown criteria.

-3.00E+01 3.00E+02 MIN: MAX:

JOB NO132 99-08-25 11.02.14

-2.20E-19 2.23E+02 MIN: MAX:

WINDOW (X,Y):

UNDEFORM.

DEFORM.

ALL SHELLS

*GEOMETRY PLOT

POSTBOSOR 1.04

The used allowable design stress is f = 255 MPa.

2

.25

.25

.50

.50

.75

.75

1.00

1.00

1.25

1.25

1.50

1.50

1 .75

1.75

2.00

2.00

10

10

2

2

1.94E+01 6.53E+01

JOB NO132 99-08-18 16.20.12

-2.52E+02 2.91E+02

.50

1.00

1.50

2.00

2.50

3.00

10

-2.00

2

.25

.50

.75

1.00

1.25

1.50

1.75

2.00

10

2

.25

.50

.75

1.00

1.25

1.50

1.75

2.00

10

2

"INSIDE"

-2.53E+02 3.23E+02

2.02E+01 3.23E+02 JOB NO132 99-08-18 16.20.43

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 1

TRESCA

*STRESSES*

POSTBOSOR 1.04

JOB NO132 99- 08-18 16.19.21

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

Example 1.3

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 1

CIRCUMF

*STRESSES*

POST BOSOR 1.04

JOB NO132 99-08-18 16.17.23

MIN: MAX:

FUNC.VALUES:

-1.00

.00

1.00

Analysis Details

-2.00

-1.00

.00

1.00

2.00

10

20.00

25.00

30.00

3 5.00

"MEMBR"

SHELL 1

SHELL 1

*STRESSES*

POSTBOSOR 1.04

GEN.DIR

2.00

3.00

2

COMP.-ST

*STRESSES*

10

Shell No. 1

40.00

45.00

50.00

55.00

60.00

65.00

POSTBOSOR 1.04

DBA Design by Analysis Page 7.55 (S)

Stresses

2

2.22

2.22

2.24

2.24

2.26

2.26

2.28

2.28

2.30

2.30

2.32

2.32

2.34

2.34

2.36

2.36

2.38

2.38

10

10

2

2

8.55E+01 9.01E+01

JOB NO132 99- 08-18 16.23.18

-1.59E+02 1.66E+02

10

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

5.50

6.00

-4.00

2.22

2.24

4

2.26

2.28

2.30

2.32

2.34

2.36

2.38

10

2

2

2.22

2.2

2.26

2.28

2.30

624 MPa ∆(σ eq )P + Q ≤ 3f

2.32

2.34

2.36

2.38

10

2

"INSIDE"

-4.52E+02 6.24E+02

JOB NO132 99-08-18 16.24 .03

1.39E+02 6.24E+02

FUNC.VALUES: MIN: MAX:

1.04

"OUTSIDE"

"INSIDE"

SHELL 2

TRESCA

*STRESSES*

POSTBOSOR

JOB NO132 99-08-18 16.22.37

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

Example 1.3

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 2

CIRCUMF

*STRESSES*

POSTBOSOR 1.04

JOB NO132 99-08-18 16.21.27

MIN: MAX:

FUNC.VALUES:

-2.00

.00

2.00

S*

Analysis Details

- 1.50

-1.00

-.50

.00

.50

1.00

1.50

10

86.00

8 6.50

87.00

87.50

"MEMBR"

SHELL 2

4.00

SHELL 2

*STRESSE

POSTBOSOR 1.04

GEN.DIR

2

COMP.-ST

*STRESSES*

6.00

10

Shell No. 2

88.00

88.50

89.00

89.50

90.00

POSTBOSOR 1.04

DBA Design by Analysis Page 7.56 (S)

Stresses

2

2.50

2.50

3.00

3.00

3.50

3.50

4.00

4.00

4.50

4.50

5.00 10

5.00 10

2

2

8.05E+01 1.54E+02

JOB NO132 99- 08-18 16.26.09

-7.24E+01 1.60E+02

.50

1.00

1.50

2.00

2.50

10

-1.00

-.50

.00

.50

1.00

1.50

2.00

2.50

10

2

2

2.50

2.50

3.00

3.00

3.50

3.50

4.00

4.00

4.50

4.50

2

5.00 10 2

5.00 10

-1.37E+02 2.98E+02

9.31E+00 2.98E+02 JOB NO132 99-08-18 16.26.38

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 3

TRESCA

*STRESSES*

POSTBOSOR 1.04

JOB NO132 99- 08-18 16.25.29

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 3

GEN.DIR

*STRESSES*

POSTBOSOR 1.04

Example 1.3

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 3

CIRCUMF

*STRESSES*

POSTBOSOR 1.04

JOB NO132 99-08-1 8 16.24.40

MIN: MAX:

FUNC.VALUES:

"MEMBR"

SHELL 3

COMP.-ST

*STRESSES*

POSTBOSOR 1.04

Analysis Details

-.50

-.25

.00

.25

.50

.75

1.00

1.25

1.50

10

.90

1.00

1.10

2

Shell No. 3

1.20

1.30

1.40

1.50

10

DBA Design by Analysis Page 7.57 (S)

Stresses

DBA Design by Analysis

Analysis Details Example 1.3 / F-Check

Analysis Type: F-Check

FE –Software:

Page 7.58(A) Member: A&AB

ANSYS 5.4

Element Types: 4 – node, 2 – D axisymmetric solid PLANE42 Boundary Conditions: Zero vertical displacement in the nodes of the undisturbed end of the shell; Symmetry boundary conditions in the nodes in the centre of the flat end.

Model and Mesh:

Results:

Fatigue life N = 6373 cycles (welded part)

DBA Design by Analysis

Analysis Details Example 1.3 / F-Check

Page 7.59 (A)

DataCritical point: Weld end to shell, inside; Principal stress range approach en = 21,5 mm ∆σD (5.106 cycles) = 46 MPa (class 63) equivalent stresses or m = 3 C = ……..

tmax = 20 °C tmin = 20 °C t* = 0,75 tmax + 0,25 tmin = 20 °C Rm = 410 MPa Rp0,2/t* = 255 MPa

principal stresses m = 3 C⊥ = 5.1011 C// = ….. m = 5 C⊥ = ….. C// = …..

m = 5 C = …….. Stresses ∆σstruc = 416.7 MPa (structural equivalent stress range, determined by extrapolation) 18.8 Plasticity correction factor ke

Thermal loading

mechanical loading

If ∆σeq,l > 2 Rp0,2/t*

If ∆σstruc > 2 Rp0,2/t* k

 Äó  struc = 1+ A0  − 1 e  2 R p0,2/t *  

with A0 =

k



0,5 for 800 MPa ≤ Rm ≤ 1000 MPa 0,4 for Rm ≤ 500 MPa  R − 500  0,4 +  m for 500 MPa ≤ R m ≤ 800 MPa  3000 

A0 = ….. ke = ….. ∆σ = ke ∆σeq,l = ….. MPa Else ∆σ = ∆ σstruc = 416.7 MPa

υ

= 0,5 +

0,7 0,4

 Äó   struc   R p0,2/t * 

kυ = ….. ∆σ = kυ ∆σeq,l = ….. MPa Else ∆σ = ∆σstruc = ….. MPa

If both mechanical and thermal loadings are to be considered, the correction has to be made on each component of the stress tensors . Ke and kí are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated. 18-10-6-1 Thickness correction factor few en ≤ 25 mm few = 1

25 mm ≤ en ≤ 150 mm few = (25/en)0,25 = few = …..

en ≥ 150 mm few = 0,639

DBA Design by Analysis

Analysis Details Example 1.3 / F-Check

Page 7.60 (A)

18-10-6-2 Temperature correction factor ft* For t* > 100 °C Else ft* = 1

ft* = 1,03 – 1,5 10-4 t* -1,5 10-6 t*2 = …..

18-10-6-3 Overall correction factor fw fw = few.ft* = 1

18-10-7 Allowable number of cycles N Äó fw

If

= 416.7 MPa Äó fw

> ∆σ5.106 cycles then

M = 3 and C (C⊥ or C//) = 5.1011

N=

C  ∆σ   f   w 

m

= 6910 cycles

If

< ∆σ5.106 cycles and other

Äó fw

cycles with

Äó fw

> ∆σ5.106 cycles

then M = 5 and C (C⊥ or C//) = ….. N = C m = …..  ∆σ   fw

If

Äó fw

< ∆σ5.106 cycles and all other

cycles with

Äó fw

< ∆σ5.106 cycles then

N=∞

  

Note: All unwelded regions are less critical Note: In this example both approaches – the equivalent stress range approach and the principal stress range approach – have been used. The maximum principal stress is the tangential stress component and it is positive, the minimum one – pressure – is negative and small. Therefore, the difference in the results is small.

DBA Design by Analysis

Analysis Details Example 1.3 / F-Check

Page 7.61(A)

Data Critical point: Weld end to shell, inside; Equivalent stress range approach tmax = 20 °C tmin = 20 °C t* = 0,75 tmax + 0,25 tmin = 20 °C Rm = 410 MPa Rp0,2/t* = 255 MPa

en = 21,5 mm ∆σD (5 106cycles) = 46 MPa (class 63) equivalent stresses or 11 . m = 3 C = 5 10 m = 5 C = 1.08.1015

principal stresses m = 3 C⊥ = C// = ….. m = 5 C⊥ = ….. C// = …..

Stresses ∆σstruc = 428.1 MPa (structural equivalent stress range,determined by extrapolation)

18.8 Plasticity correction factor ke mechanical loading If ∆σstruc > 2 Rp0,2/t*

If ∆σstruc > 2 Rp0,2/t*

 Äó  struc k = 1+ A0  − 1 e  2 R p0,2/t *  with A0 =

Thermal loading

k

0,5 for 800 MPa ≤ Rm ≤ 1000 MPa 0,4 for Rm ≤ 500 MPa  R − 500  0,4 +  m for 500 MPa ≤ R m ≤ 800 MPa  3000 

A0 = ….. ke = ….. ∆σ = ke ∆σeq,l = ….. MPa Else ∆σ = ∆ σstruc = 428.1 MPa

υ

= 0,5 +

0,7 0,4

 Äó   struc   R p0,2/t * 

kυ = ….. ∆σ = kυ ∆σeq,l = ….. MPa Else ∆σ = ∆σstruc = ….. MPa

If both mechanical and thermal loadings are to be considered, the correction has to be made on each component of the stress tensors . Ke and kí are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated. 18-10-6-1 Thickness correction factor few en ≤ 25 mm few = 1

25 mm ≤ en ≤ 150 mm few = (25/en)0,25 = few = …..

en ≥ 150 mm few = 0,639

DBA Design by Analysis

Analysis Details Example 1.3 / F-Check

Page 7.62(A)

18-10-6-2 Temperature correction factor ft* For t* > 100 °C Else ft* = 1

ft* = 1,03 – 1,5 10-4 t* -1,5 10-6 t*2 = …..

18-11-3 18-10-6-3 Allowable Overallnumber correction of cycles factorNfw fw = few.ft* = 1

18-10-7 Allowable number of cycles N Äó fw

= 416.7 MPa

If

Äó fw

> ∆σ5.106 cycles then

M = 3 and C (C⊥ or C//) = 5.1011

N=

C  ∆σ   fw

  

m

= 6373 cycles

If

< ∆σ5.106 cycles and other

Äó fw

cycles with

Äó fw

> ∆σ5.106 cycles

then M = 5 and C (C⊥ or C//) = ….. N = C m = …..

If

Äó fw

< ∆σ5.106 cycles and all other

cycles with

Äó fw

< ∆σ5.106 cycles then

N=∞

 ∆σ   f   w 

Note: All unwelded regions are less critical Note: In this example both approaches – the equivalent stress range approach and the principal stress range approach – have been used. The maximum principal stress is the tangential stress component and it is positive, the minimum one – pressure – is negative and small. Therefore, the difference in the results is small.

Analysis Details

DBA Design by Analysis

Page 7. 63 (S)

Example 1.4

Member: Strathclyde

Analysis Type: Direct Route using Elastic Compensation

FE-Software:

ANSYS 5.4

Element Types:

8 – node, 2-D structural axisymmetric solid.

Boundary Conditions:

No vertical displacement in the undisturbed end of the shell remote from the flat end. Axisymmetry from elements

Model and Mesh:

Results: Maximum internal pressure according to the GPD-Check Check against PD:

PSmax GPD = 8.42 MPa

Shakedown limit pressure PSmax SD = 12.8 MPa

DBA Design by Analysis

Analysis Details Example 1.4

Page 7. 64 (S)

1. Finite Element Mesh Element divisions for the finite element mesh were defined parametrically to allow modification of the element density. The analysis was carried out on two models; one with fillets at the discontinuities at the nozzle the other without. Mesh densities for the two models remained similar: 655 nodes for the fillet model 627 nodes for the model with no fillets at the nozzle. Finite element models were created using linear 4-node 2-D axisymmetric solids, the analysis was repeated with higher order 8-node structural 2-D axisymmetric solids, any alterations this may have on the results could also be noted. The allowable pressure according to the GPD-Check and the shakedown pressure according to the PD-check were calculated from the model giving the greatest lower bounds. The analysis discussed is for the model containing fillets at the nozzle with 4-node elements. Boundary conditions applied to the model are axisymmetry, applied via a key option when defining the element type in the FE-software (axisymmetry around the vertical axis Y). The nodes at the undisturbed end of the cylindrical shell have their vertical degree of freedom constrained to ensure that plane sections remain plane. 2. Material properties Material strength parameter RM = 255 MPa (cylindrical shell), RM = 245 MPa (flat end), RM = 265 MPa (nozzle), modulus of elasticity E = 212 GPa (all parts of the structure). 3. Determination of the maximum admissible pressure according to the GPD-Check Using the application rule in prEN-13445-3 Annex B.9.2.2 to check against GPD the principle is fulfilled when for any load case the combination of the design actions do not exceed the design resistance. This may be shown by calculating the limit load. The limit load has to be determined using Tresca‘s yield condition and associated flow rule. As elastic compensation is based upon a series of elastic equilibrium stress fields, it is a relatively simple procedure to derive a lower bound limit load direct from the Tresca yield model. From prEN-13445-3 Annex B, Table B.9-3 the partial safety factor, γR on the resistance is 1.25. Therefore, the design material strength parameters given by RM/γR were: 204 MPa for the shell, 196 MPa for the plate and 112MPa for the nozzle. The analysis was carried out using the elastic compensation method conforming to the direct route rules for GPD in Annex B: linear elastic ideal plastic material law, Tresca’s yield condition and associated flow rule and first order theory. In each elastic compensation iteration an equilibrium stress field is produced where the elastic modulus of each subsequent iteration is defined by the previous elastic solution. In this way regions of the FE- model may be Figure 7.5.1-1: Limit Stress Field (Tresca)

Analysis Details

DBA Design by Analysis

Example 1.4

Page 7. 65 (S)

systematically stiffened or weakened depending upon the stress magnitudes in the previous solution. The result is that every equilibrium stress field is a lower bound of the limit load, the stress field giving the greatest limit load is taken as the closest to the actual limit state and defines the limit load in the analysis. In problems such as this, where there are materials with different properties the modulus modification has a modified procedure that takes account of the different material properties. This modified method calculates the limit pressure for each component with a different material, allowing the component giving the lowest limit load to define the limit for the whole model. Convergence of the equilibrium stress field will occur typically between 8 and 15 iterations, i.e. no further benefit in the limit load will be noted with subsequent iterations. The total computing time to run the analysis on a 300 MHz Pentium two processor with 128 Mb RAM Windows NT workstation was 180 seconds. The stress field was shown to converge after eight iterations giving a lower bound on the limit pressure of 10.1 MPa. Figure 7.5.1-1 shows the limit stress (intensity) field according to the Tresca condition. The limit pressure is given by scaling the limit stress field so that the stresses anywhere in the model do not exceed the material resistance for that component. The analysis showed that the plate was the first component to fail. The limit load can be calculated by scaling the applied pressure (10 MPa) by the factor 1.01 (design material strength/max stress: 196/195). According to prEN-13445-3 Annex B, Table B.9-2 for pressure loads (without natural limit) the partial safety factor γp is 1.2. Thus, the internal pressure limit according to the GPD-check is

PS max GPD =

10.1 = 8.42 MPa 1.2

It is also possible to determine a limit pressure from the check against PD. In this case, the elastic compensation is based on Mises‘ yield condition. The partial safety factor on the resistance γR is not applied for the PD-check. However, as the analysis is wholly elastic it is possible to scale the stress fields at any time (similarly as was done above). The maximum ratio of Mises' equivalent stress to Tresca's equivalent stress for the same load is 2/√3. Therefore, applying a factor of √3/2 to the yield stress in the Mises analysis (or to the limit load, as the analysis is elastic) will always lead to a conservative result. From the Mises analysis the limit load was found to be 13.9 MPa and with the partial safety factors γR = 1.25 and γp = 1.2, the internal pressure limit according to the GPDcheck can be found as PS max GPD =

13.9 3 ⋅ = 8.03 MPa γ p ⋅γ R 2

4. Check against PD In the check for progressive plastic deformation, the principle in prEN-13445-3 B.9.3.1 is fulfilled if the structure can be show to shake down. When a structure has been shown to shake down, the failure modes of progressive plastic deformation and alternating plasticity can not occur. In elastic compensation the load at which the structure will shakedown can be calculated simply. Based on Melan’s shakedown theorem, the self-equilibrating residual stress field that would result after a loading cycle can be calculated by subtracting the linear-elastic stress field at the limit pressure from the limit stress field. The residual stress field is in effect the resulting stress from an

DBA Design by Analysis

Analysis Details Example 1.4

Page 7. 66 (S)

elastic unloading from the limit state back to zero pressure. If no stress in the residual field violates the yield condition, i.e. if there is no stress above the design material strength, then the shakedown load is equal to the limit load. Where the residual stress field of a structure does exceed the yield condition, the shakedown limit can be calculated easily from the stress plots because of the linearity. The residual stress field from the elastic compensation Mises analysis is shown in Figure 7.5.1-2. Because the applied load is arbitrary and the resulting stress fields are scaled to the yield condition, the maximum residual stress is compared to the maximum stress in the limit stress field, Figure 7.5.1-3. As stated above, the presence of different materials requires here some modification. A shakedown pressure is calculated for each component with a different material and the component giving the lowest shakedown limit defines the limit for the structure. The analysis showed that the plate may be the first component to suffer from progressive plastic deformation. In many of the iterations the maximum residual stress was greater than the maximum from the limit stress field, thus the shakedown limit is less than the limit load. As all the iterations are lower bounds, the stress field yielding the greatest value of shakedown load is used to define the limit. With an applied load of 10 MPa the shakedown limit is 12.8 MPa.

Figure 7.5.1-2: Residual Stress Field (Mises)

Figure 7.5.1-3: Limit Stress Field (Mises)

5. Check against GPD Using Non-linear Analysis A check against GPD was also performed for the same FE-model using conventional non-linear analysis. In this way, a direct comparison may be made between the two limit approaches. The FEgeometry, mesh and boundary conditions are the same as those used in the elastic compensation analysis. Material non-linearities were applied corresponding to the design material parameters,

DBA Design by Analysis 204 MPa for the shell, 196 MPa for the plate, 112 MPa for the nozzle and perfect plasticity. A ramped load is applied and the analysis runs until the applied load is such that convergence can no longer occur due to unrestrained displacement – Gross Plastic Deformation. It is assumed that the last converged solution is the limit load. Figure 7.5.1-4 shows the Mises equivalent stress distribution at the limit load. Here, the last converged solution was at a load of 13.75 MPa, using the same method as above the allowable pressure according to the GPD-check using Mises condition is At this load, there is a considerable amount of plasticity and a check on the maximum total principal strain is required to be below +/- 5%. Figure 7.5.1-5 shows that the maximum absolute value of the total principal strain is 4.7% and is within the limit: Therefore, PS max GPD =

Analysis Details Example 1.4

Page 7. 67 (S)

Figure 7.5.1-4: Limit Stress Field (non-linear analysis)

13.75 3 ⋅ = 9.92 MPa γp 2

The result offers a considerable benefit to the allowable pressure calculated using elastic compensation. Analysis time to calculate limit load using non-linear FE-analysis was 360 seconds. Figure 7.5.1-5: Maximum total principal strain

Analysis Details

DBA Design by Analysis

Example 1.4

Page 7. 68 (S)

6. Additional Comments

Additional analysis was completed to ascertain any effect on the results for higher order elements (8-node) and omission of the fillets at the nozzle discontinuities . Limit loads and shakedown values were calculated using Mises' condition, the results are summarised in the following table. Use of Fillets

Element Type

No No Yes Yes

4 node 8 node 4 node 8 node

Number of Iterations 8 8 8 8

Lower Bound Limit Load 8.08 7.96 8.42 7.9

Shakedown Load 12.8 12.3 12.8 12.3

Processor Time 273 284 278 289.7

Table 7.5.1-1: Limit and shakedown analysis summary

Both the lower bound limit load and shakedown load are calculated using the same method as described above. As can be seen from the table above, the choice of element type and application of fillets is of negligible difference to the results. The utilisation of elastic compensation in this DBA calculation has given an increase in the allowable pressure over that given by the DBF calculation. The two methods used to calculate the lower bound limit load (direct from Tresca or correction of Mises) show good correlation; the Mises corrected value is slightly conservative as would be expected. However, carrying out the Mises elastic compensation will give both the limit load and shakedown load in one analysis. Non-linear DBA calculations offer a further benefit than elastic compensation in terms of higher allowable pressure than those pressures calculated by DBF in this example.

DBA Design by Analysis

Analysis Details

Page 7. 69 (S)

Example 1.4

Member:

Analysis Type: Direct Route using elasto-plastic FE calculations

FE-Software:

A&AB

ANSYS® 5.4

Element Types: 4 – node, 2 – D axisymmetric solid PLANE42 Boundary Conditions:

Zero vertical displacement in the nodes of the unsdisturbed end of the shell; Longitudinal stress corresponding to pressure force on the nozzle's closed end.

Model and Mesh:

Length of shell: 400 mm Length of nozzle (weld incl.):150 mm Total length of model: 593 mm Number of elements: 3693

Results:

Maximum allowable pressure according to GPD:

PS max GPD = 9.9 MPa

Shakedown limit pressure:

PS max SD = 12.2 MPa

DBA Design by Analysis

Analysis Details Example 1.4

Page 7. 70 (S)

1. Elements, mesh fineness, boundary conditions The model of the structure is shown on page 1, a total number of 3693 4-node axisymmetric solid elements – PLANE42 in ANSYS® 5.4 - was used. To avoid stress singularities in the linear-elastic calculation, the boundary between the weld and the nozzle and the weld and the plate, respectively, is modelled with fillets corresponding to a thickness of the weld influence zone of 2 mm. Therefore, the corresponding fillet radii are given by R1 = 7 mm and R2 = 10.5 mm - see Figure 7.5.2-1. Linear-elastic calculations with different mesh finenesses in the high stress regions showed, that the used mesh fineness is sufficient (doubling the number of elements in this regions leads to a difference in the stress results of approximately 1 %). Furthermore, the used linear shape function of the elements is here sufficient, since the computation time in an analysis using non-linear material properties is much higher for elements with mid-side nodes and quadratic shape functions, and, close to the limit load, the results are almost identical. The boundary condition applied in the model is constraining the vertical degree of freedom in the nodes at the undisturbed end of the cylindrical shell to zero. At the end of the nozzle a longitudinal stress corresponding to a closed end is applied. Figure 7.5.2-1

2. Determination of the maximum allowable pressure according to GPD Since the subroutine using Tresca’s yield condition showed bad convergence, the results from the check against PD, see chapter 2 of subsection 3 -Procedures, have been used. The partial safety factor γ R according to prEN 13445-3 Annex B, Table B.9-3 is 1.25 , and the partial safety factor for pressure (without a natural limit) according to prEN 13445-3 Annex B, Table B.9-2 γ P is 1.2 . Therefore, the (internal) allowable pressure according to the GPD-check is by PS max GPD =

17.18 3 17.18 3 ⋅ = ⋅ = 9.92 MPa. γ P ⋅ γ R 2 1.2 ⋅ 1.25 2

The maximum principal strain for this state is about 5 %, see Figure 7.5.2-2, the condition in prEN 13445-3 Annex B.9.2.2 is fulfilled.

DBA Design by Analysis

Analysis Details Example 1.4

Page 7. 71 (S)

Figure 7.5.2-2

3. Check against PD The elasto-plastic FE analysis was carried out as stated in prEN 13445-3 Annex B, Sec. B.9.3.1, using Mises’ yield condition and associated flow rule, a linear-elastic ideal-plastic constitutive law with design material strength parameters of 255 MPa for the shell (according to EN 10028-2), 245 MPa for the plate (according to EN 10028-2), and 265 MPa for the nozzle (according to prEN 10216-2), and first order theory. For simplicity, the boundaries between the materials were assumed to be in the planes of the lower and upper surface of the plate. The elastic modulus used in the calculations was E=212 GPa for all parts of the structure. Defining and using load cases in ANSYS®, the superposition of stress fields can be done easily; therefore an early load step of the analysis was defined at a very low load level (1 MPa), so that there was linear-elastic response of the structure. All other linearelastic stress fields can then be determined by multiplication with a suitable scale-up factor. The analysis was carried out using the arclength method. Since at the limit load the structure is fully plastified in the plate and in the nozzle adjacent to the plate, a Figur 7.5.2-3

DBA Design by Analysis

Analysis Details Example 1.4

Page 7. 72 (S)

maximum vertical displacement at the inner edge of the plate of 5 mm was used as termination criterion. The computation time of the limit load was 54 minutes on the Compaq® Professional Workstation 5000 with two Pentium Pro® processors and 256 MB RAM. The termination criterion was fulfilled at an internal pressure of 17.18 MPa – this pressure was used as limit pressure. Figure 7.5.2-3 shows the vertical displacement at the inner edge of the plate versus the internal pressure. According to this figure the limit state is reached. Figure 7.5.2-4 shows the elastoplastic Mises' equivalent stress distribution at the limit pressure of 17.18 MPa . Figure 7.5.2-4

Figure 7.5.2-5 shows the linearelastic Mises equivalent stress distribution at the limit pressure – the stress maximum is located at the junction of nozzle and weld.

Figure 7.5.2-5

DBA Design by Analysis

Analysis Details Example 1.4

Figure 7.5.2-6 shows the Mises equivalent stress distribution of the corrected residual stress field, the scaling factor β is given by 0.538 (see subsection 3.3.2.5 of section 3 Procedures). The maximum (allowable) stress, which is used for the determination of β , is now located in the stress relief groove of the plate.

Figure 7.5.2-6

At the lower bound shakedown limit the scaling factor α is now determined by the maximum stress at the junction between the nozzle and the weld – see Figures 7.2.5-7 and 7.2.5-8 – and given by 0.710 (see subsection 3.3.2.5 of section 3 - Procedures).

Figure 7.5.2-7

Page 7. 73 (S)

Analysis Details

DBA Design by Analysis

Example 1.4

Figure 7.5.2-8

Thus, the shakedown limit pressure is given by PS max SD = 0.710 ⋅ 17.18 = 12.2 MPa .

Page 7. 74 (S)

Analysis Details

DBA Design by Analysis

Example 1.4

Page 7. 75 (S)

Member:

Analysis Type: Stress Categorization Route

FE-Software: BOSOR

Element Types:

Axisymmetric shell elements.

Model and Mesh:

As shown in diagram – Shell Numbers in Calculation Model

Results: Maximum admissible action according to Stress Categories: Internal pressure PSmax SC = 10.0 MPa

TKS

Analysis Details

DBA Design by Analysis

Page 7. 76 (S)

Example 1.4

-4.33E+01 4.50E+02 MIN: MAX:

JOB NO14 99-08-25 11.05.54

1.00E+02 2.23E+02 MIN: MAX:

WINDOW (X,Y):

UNDEFORM .

DEFORM.

ALL SHELLS

*GEOMETRY PLOT

POSTBOSOR 1.04

The following figure shows the deformed model. The figures thereafter show the distribution of stresses – membrane and membrane plus bending – in the surfaces of the various parts of the model. With the designation list in subsection 3.6 the various plots are self-explaining. The plots are for a pressure of 7.9 MPa. The limiting part is the nozzle close to the flat end. There the limiting stress category is the membrane stresses at the (spatial) local stress – see the membrane stress plot on the page after this one. The allowable design stress used is f = 170 MPa , the allowable value for the local membrane stress 187 MPa.

2

148 MPa

.20

.20

.40

.40

1.0f

1.1f = 188MPa

.60

.60

.80

.80

1.00

1.00

1.20

1.20

Limit for localmembran

10

10

2

2

7.52E+01 1.84E+02

1.04

JOB NO14 99-08-19 12.56.58

7.17E+01 2.62E+02

1.00

1.50

2.00

2.50

3.00

10

-2.00

-1.00

.00

1.00

2.00

3.00

10

2

2

.20

.20

.40

.40

.60

.60

.80

.80

1.00

1.00

1.20

1.20

10

10

2

2

-2.26E+02 3.06E+02

7.17E+01 3.28E+02 JOB NO14 99-08-19 12.57.38

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 1

TRESCA

*STRESSES*

POSTBOSOR 1.04

JOB NO14 99-0 8-19 12.55.14

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 1

GEN.DIR

*STRESSES*

POSTBOSOR 1.04

Example 1.4

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 1

CIRCUMF

*STRESSES*

POSTBOSOR 1.04

JOB NO14 99-08-19 12.54. 31

MIN: MAX:

FUNC.VALUES:

"MEMBR"

SHELL 1

COMP.-ST

*STRESSES*

POSTBOSOR

Analysis Details

.80

1.00

1.20

1.40

1.60

1.80

2.00

2.20

2.40

2.60

10

.80

1.00

2

Shell No. 1

1.20

1.40

1.60

1.80

10

DBA Design by Analysis Page 7. 77 (S)

Stresses

0

2

1.

1.40

40

1.60

1 .60

1.80

1.80

2.00

2.00

2.20

2.20

2.40

2.40

10

10

2

2

6.43E+00 3.81E+01

1.40

1.60

1.80

1.80

2.00

2.00

2.20

2.20

2.40

2.40

10

10

2

2

"INSIDE"

*STRESSES*

POSTBOSOR 1.04

JOB NO14 99-08-19 13.07.05

-1.17E+02 1.36E+02

.20

.40

.60

.80

1.00

1.20

1.40

"INSIDE"

1.39E+01 1.99E+02 JOB NO14 99-08-19 13.07.48

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

Example 1.4

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

-1.59E+02 1.99E+02 JOB NO14 99-08-19 13.03.09

SHELL 2 1.60

1.60

SHELL 2

2

1.40

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 2

TRESCA

1.80

10

-1.50

- 1.00

-.50

.00

.50

1.00

GEN.DIR

*STRESSES*

POST BOSOR 1.04

CIRCUMF

*STRESSES*

POSTBOSOR 1.04

JOB NO14 99-08-19 13.01.51

MIN: MAX:

FUNC.VALUES:

"MEMBR"

SHELL 2

1.50

2

Analysis Details

-1.00

-.50

.0

.50

1.00

10

10.00

15.00

*STRESSES* COMP.-ST

10

Shell No. 2

20.00

25.00

30.00

35.00

POSTBOSOR 1.04

DBA Design by Analysis Page 7. 78 (S)

Stresses

2.50

1

2.50

3.00

3.00

3.50

3.50

4.00

4.00

4.50

4.50

5.00

5.00

2

10 2

10

4.34E+01 8.13E+01

JOB NO14 99-08-19 13.14.53

-9.46E+01 9.30E +01

.50

1.00

1.50

2.00

2.50

3.00

3.50

10

- 2.00

-1.00

.00

1.00

2.00

2.50

2

2.50

3.00

3.00

3.50

3.50

4.00

4.00

4.50

4.50

5.00

5.00

10

10

2

2

-2.65E+02 3.61E+02

9.48E+00 3.61E+02 JOB NO14 99-08-19 13.15.27

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 3

TRESCA

*STRESSES*

POSTBOSOR 1.04

JOB NO14 99-08-19 13.12.56

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 3

GEN.DIR

*STRESSES*

POST BOSOR 1.04

Example 1.4

MIN: MAX:

FUNC.VALUES:

"OUTSIDE"

"INSIDE"

SHELL 3

CIRCUMF

*STRESSES*

POSTBOSOR 1.04

JOB NO14 99-08-19 13.11.30

MIN: MAX:

FUNC.VALUES:

"MEMBR"

SHELL 3

3.00

2

Analysis Details

-8.00

-6.00

-4.00

-2.00

.00

2.00

4.00

6.00

8.00

10

45.00

50.00

55.00

*STRESSES* COMP.-ST

10

Shell No. 3

60.00

65.00

70.00

75.00

80.00

POSTBOSOR 1.04

DBA Design by Analysis Page 7. 79 (S)

Stresses

DBA Design by Analysis

Analysis Summary

Page 7.80 (A)

Example 1.4 / F-Check

Analysis Type:

Member: F-Check

FE –Software:

A&AB

ANSYS 5.4

Element Types: 4 – node, 2 – D axisymmetric solid PLANE42 Boundary Conditions: Zero vertical displacement in the nodes of the undisturbed end of the shell;

Model and Mesh:

Length of shell: 400 mm Length of nozzle (weld incl.):150 mm Total length of model: 593 mm Number of elements: 3693

Results: Fatigue life N = 21010 cycles (welded part – shell-plate junction)

DBA Design by Analysis

Analysis Details Example 1.4 / F-Check

Page 7.81 (A)

Three locations, where local stress maxima occur, are of interest with regard to the check against fatigue: •

the weld toe in the nozzle (inside) at the nozzle-plate junction (node 3052 – see Figure 7.5.4-1)

•

the weld toe in the nozzle (outside) at the nozzle-plate junction (node 3038 – see Figure 7.5.4-2)

•

the weld toe at the inside of the shell at the shell-plate junction (node 1438 – see Figure 7.5.4-3).

At these points the structural (equivalent and principal) stresses were determined by quadratic extrapolation. The figures 7.5.4-1, 7.5.4-2 and 7.5.4-3 show the corresponding nodes, which are used as pivot points for the extrapolation. The distances between these points are approximately the ones recommended in Figure 18-3 of prEN 13445-3.

Figure 7.5.4-1

DBA Design by Analysis

Figure 7.5.4-2

Figure 7.5.4-3

Analysis Details Example 1.4 / F-Check

Page 7.82 (A)

DBA Design by Analysis

Analysis Details Example 1.4 / F-Check

Page 7. 83 (A)

Data Welded region / Principal stress range approach / Critical point: Node 3052 en = 13,7 mm tmax = 20 °C tmin = 20 °C ∆σD (5 106cycles) = 52 MPa (class 71) t* = 0,75 tmax + 0,25 tmin = 20 °C equivalent stresses or principal stresses Rm = 410 MPa m=3 C= m = 3 C⊥ = 7.16.1011 Rp0,2/t* = 265/245 MPa C// = ….. Used: Rp0,2/t* = 245 MPa m=5 C= m = 5 C⊥ = ….. C// = ….. Stresses Critical point: Weld toe in nozzle (inside): Node 3052 ∆σstruc = 225,6 MPa (structural equivalent stress range,determined by extrapolation) (obtained by quadratic extrapolation on inside of nozzle)

18.8 Plasticity correction factor ke Thermal loading

mechanical loading

If ∆σstruc > 2 Rp0,2/t*

If ∆σstruc > 2 Rp0,2/t* k

 Äó  struc = 1+ A0  − 1 e  2 R p0,2/t *  

with A0 =

k



0,5 for 800 MPa ≤ Rm ≤ 1000 MPa 0,4 for Rm ≤ 500 MPa  R m − 500  0,4 +  3000  for 500 MPa ≤ R m ≤ 800 MPa

A0 = ….. ke = ….. ∆σ = ke ∆σstruc = ….. MPa Else ∆σ = ∆ σstruc = 225.6 MPa

υ

= 0,5 +

0,7 0,4

 Äó   struc   R p0,2/t * 

kυ = ….. ∆σ = kυ ∆σstruc = ….. MPa Else ∆σ = ∆σstruc = ….. MPa

If both mechanical and thermal loadings are to be considered, the correction has to be made on each component of the stress tensors . ke and k í are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated. 18-10-6-1 Thickness correction factor few en ≤ 25 mm few = 1

25 mm ≤ en ≤ 150 mm few = (25/en)0,25 = few = …..

en ≥ 150 mm few = 0.639

DBA Design by Analysis

Analysis Details

Page 7.84 (A)

Example 1.4 / F-Check

18-10-6-2 Temperature correction factor ft* For t* > 100 °C Else ft* = 1

ft* = 1,03 – 1,5 10-4 t* -1,5 10-6 t*2 = …..

18-10-6-3 Overall correction factor fw fw = few.ft* = 1 18-10-7 Allowable number of cycles N Äó fw

If

= 225.6 MPa Äó fw

> ∆σ5.106 cycles then

m = 3 and C (C⊥ or C//) = 7,16.1011

N=

C  ∆σ   fw

  

m

= 62380 cycles

If

Äó fw

< ∆σ5.106 cycles and other

cycles with

Äó fw

> ∆σ5.106 cycles

then m = 5 and C (C⊥ or C//) = ….. N = C m = …..

If

Äó fw

< ∆σ5.106 cycles and all other

cycles with

Äó fw

< ∆σ5.106 cycles

then N=∞

 ∆σ   f   w 

Since large differences between the principal stress range approach and the structural equivalent stress one are expected, both approaches are used here. The details of the structural equivalent stress range approach are given on the next two pages.

DBA Design by Analysis

Analysis Details Example 1.4 / F-Check

Page 7. 85 (A)

Welded region/ Structural equivalent stress range approach Critical point: Node 3052 Data en = 13,7 mm tmax = 20 °C tmin = 20 °C ∆σD (5.106cycles) = 52 MPa (class 71) t* = 0,75 tmax + 0,25 tmin = 20 °C equivalent stresses or principal stresses 11 Rm = 410 MPa . m = 3 C = 7.16 10 m = 3 C⊥ = Rp0,2/t* = 265/245 MPa C// = ….. Used: Rp0,2/t* = 245 MPa 15 . m = 5 C = 1.96 10 m = 5 C⊥ = ….. C// = ….. Critical point: Weld toe in nozzle (inside): Node 3052 Stresses ∆σstruc = 350,7 MPa (structural equivalent stress range) (total value used, since quadratic extrapolation on inside of nozzle gives larger value)

18.8 Plasticity correction factor ke Thermal loading

mechanical loading

If ∆σstruc > 2 Rp0,2/t*

If ∆σstruc > 2 Rp0,2/t* k

 Äó  struc = 1+ A0  − 1 e  2 R p0,2/t *  

with A0 =

k



0,5 for 800 MPa ≤ Rm ≤ 1000 MPa 0,4 for Rm ≤ 500 MPa  R − 500  0,4 +  m for 500 MPa ≤ R m ≤ 800 MPa  3000 

A0 = ….. ke = ….. ∆σ = ke ∆σstruc = ….. MPa Else ∆σ = ∆ σstruc = 350.7 MPa

υ

= 0,5 +

0,7 0,4

 Äó   struc   R p0,2/t * 

kυ = ….. ∆σ = kυ ∆σstruc = ….. MPa Else ∆σ = ∆σstruc = ….. MPa

If both mechanical and thermal loadings are to be considered, the correction has to be made on each component of the stress tensors . ke and k í are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated. 18-10-6-1 Thickness correction factor few en ≤ 25 mm few = 1

25 mm ≤ en ≤ 150 mm few = (25/en)0,25 = few = …..

en ≥ 150 mm few = 0.639

DBA Design by Analysis

Analysis Details

Page 7.86 (A)

Example 1.4 / F-Check

18-10-6-2 Temperature correction factor ft* For t* > 100 °C Else ft* = 1

ft* = 1,03 – 1,5 10-4 t* -1,5 10-6 t*2 = …..

18-10-6-3 Overall correction factor fw fw = few.ft* = 1 18-10-7 Allowable number of cycles N Äó fw

= 350.7 MPa

For N ≥ 2 1066 cycles If Äó >figure ∆σ5.10 18-14 cycles then fSee w

If

m = 3 and C (C⊥ or C//) = 7.16.1011

cycles with

N=

C  ∆σ   fw

  

m

fm = 1

= 16600 cycles

Äó fw

< ∆σ5.106 cycles and other Äó fw

> ∆σ5.106 cycles

If

Äó fw

< ∆σ5.106 cycles and all other

cycles with

Äó fw

< ∆σ5.106 cycles then

Äó then R ≤ ó ≤R if p0,2/t* then m = 5 and C (C⊥ or C//) = ….. N = eq ∞ 2(1 + M ) C N= m = …..  ∆σ  2ó   1 + M 3 M  eq   fw  fm = − = …. 1+ M 3  ÄóR 





fm = 1

DBA Design by Analysis

Analysis Details Example 1.4 / F-Check

Page 7.87 (A)

Data Welded region / Principal stress range approach Critical point: Node 3038 en = 13,7 mm tmax = 20 °C tmin = 20 °C ∆σD (5 106cycles) = 52 MPa (class 71) t* = 0,75 tmax + 0,25 tmin = 20 °C equivalent stresses or principal stresses Rm = 410 MPa m=3 C= m = 3 C⊥ = 7.16.1011 Rp0,2/t* = 265/245 MPa C// = ….. Used: Rp0,2/t* = 245 MPa m=5 C= m = 5 C⊥ = ….. C// = ….. Stresses Critical point: Weld toe in nozzle (outside): Node 3038 ∆σstruc = 318,0 MPa (structural equivalent stress range) (maximum value used, since quadratic extrapolation on outside of nozzle gives larger value) 18.8 Plasticity correction factor ke Thermal loading

mechanical loading

If ∆σstruc > 2 Rp0,2/t*

If ∆σstruc > 2 Rp0,2/t* k

 Äó  struc = 1+ A0  − 1 e  2 R p0,2/t *  

with A0 =

k



0,5 for 800 MPa ≤ Rm ≤ 1000 MPa 0,4 for Rm ≤ 500 MPa  R − 500  0,4 +  m for 500 MPa ≤ R m ≤ 800 MPa  3000 

A0 = ….. ke = ….. ∆σ = ke ∆σstruc = ….. MPa Else ∆σ = ∆ σstruc = 318.0 MPa

υ

= 0,5 +

0,7 0,4

 Äó   struc   R p0,2/t * 

kυ = ….. ∆σ = kυ ∆σstruc = ….. MPa Else ∆σ = ∆σstruc = ….. MPa

If both mechanical and thermal loadings are to be considered, the correction has to be made on each component of the stress tensors . ke and k í are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated. 18-10-6-1 Thickness correction factor few en ≤ 25 mm few = 1

25 mm ≤ en ≤ 150 mm few = (25/en)0,25 = few = …..

en ≥ 150 mm few = 0.639

DBA Design by Analysis

Analysis Details

Page 7.88 (A)

Example 1.4 / F-Check

18-10-6-2 Temperature correction factor ft* For t* > 100 °C Else ft* = 1

ft* = 1,03 – 1,5 10-4 t* -1,5 10-6 t*2 = …..

18-10-6-3 Overall correction factor fw fw = few.ft* = 1 18-10-7 Allowable number of cycles N Äó fw

= 318,0 MPa

If

Äó fw

> ∆σ5.106 cycles then

m = 3 and C (C⊥ or C//) = 7.16.1011

N=

C  ∆σ   fw

  

m

= 22270 cycles

If

Äó fw

< ∆σ5.106 cycles and other

cycles with

Äó fw

> ∆σ5.106 cycles

then m = 5 and C (C⊥ or C//) = ….. N = C m = …..  ∆σ   fw

If

Äó fw

< ∆σ5.106 cycles and all other

cycles with

Äó fw

< ∆σ5.106 cycles then

N=∞

  

Since large differences between the principal stress range approach and the structural equivalent stress one are expected, both approaches are used here. The details of the structural equivalent stress range approach are given on the next two pages

DBA Design by Analysis

Analysis Details Example 1.4 / F-Check

Page 7.89 (A)

Data Welded region / Structural equivalent stress range approach / critical point: node 3038 en = 10,25 mm tmax = 20 °C tmin = 20 °C ∆σD (5 106cycles) = 52 MPa (class 71) t* = 0,75 tmax + 0,25 tmin = 20 °C equivalent stresses or principal stresses Rm = 410 MPa m = 3 C = 7.16.1011 m = 3 C⊥ = ….. Rp0,2/t* = 265/245 MPa C// = ….. Used: Rp0,2/t* = 245 MPa m = 5 C = 1,96 1015 m = 5 C⊥ = ….. C// = ….. Stresses Critical point: Weld toe in nozzle (outside): Node 3038 ∆σstruc = 225,6 MPa (structural equivalent stress range) (obtained by quadratic extrapolation on outside of nozzle)

18.8 Plasticity correction factor ke Thermal loading

mechanical loading

If ∆σstruc > 2 Rp0,2/t*

If ∆σstruc > 2 Rp0,2/t* k

 Äó  struc = 1+ A0  − 1 e  2 R p0,2/t *  

with A0 =

k



0,5 for 800 MPa ≤ Rm ≤ 1000 MPa 0,4 for Rm ≤ 500 MPa  R − 500  0,4 +  m for 500 MPa ≤ R m ≤ 800 MPa  3000 

A0 = ….. ke = ….. ∆σ = ke ∆σstruc = ….. MPa Else ∆σ = ∆ σstruc = 289.1 MPa

υ

= 0,5 +

0,7 0,4

 Äó   struc   R p0,2/t * 

kυ = ….. ∆σ = kυ ∆σstruc = ….. MPa Else ∆σ = ∆σstruc = ….. MPa

If both mechanical and thermal loadings are to be considered, the correction has to be made on each component of the stress tensors . ke and k í are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated. 18-10-6-1 Thickness correction factor few en ≤ 25 mm few = 1

25 mm ≤ en ≤ 150 mm few = (25/en)0,25 = few = …..

en ≥ 150 mm few = 0.639

DBA Design by Analysis

Analysis Details

Page 7.90 (A)

Example 1.4 / F-Check

18-10-6-2 Temperature correction factor ft* For t* > 100 °C Else ft* = 1

ft* = 1,03 – 1,5 10-4 t* -1,5 10-6 t*2 = …..

18-10-6-3 Overall correction factor fw fw = few.ft* = 1 18-10-7 Allowable number of cycles N Äó fw

= 289.1 MPa

If

Äó fw

> ∆σ5 106 cycles then

m = 3 and C (C⊥ or C//) = 7.16.1011

N=

C  ∆σ   fw

  

m

= 29630 cycles

If

Äó fw

< ∆σ5.106 cycles and other

cycles with

Äó fw

> ∆σ5.106 cycles

then m = 5 and C (C⊥ or C//) = ….. N = C m = …..

If

Äó fw

< ∆σ5.106 cycles and all other

cycles with

Äó fw

< ∆σ5.106 cycles then

N=∞

 ∆σ   f   w 

Due to extrapolation differences this value is larger than the one via the principal stress range approach.

DBA Design by Analysis

Analysis Details Example 1.4 / F-Check

Page 7.91 (A)

Data Welded region / Principal stress range approach / critical point: node 1438 en = 21,5mm tmax = 20 °C tmin = 20 °C ∆σD (5 106cycles) = 52 MPa (class 71) t* = 0,75 tmax + 0,25 tmin = 20 °C equivalent stresses or principal stresses Rm = 410 MPa m=3 C= m = 3 C⊥ = 7.16.1011 Rp0,2/t* = 265/245 MPa C// = ….. Used: Rp0,2/t* = 245 MPa m=5 C= m = 5 C⊥ = ….. C// = ….. Stresses Critical point: Weld toe at inside of shell: Node 1438 ∆σstruc = 234,2 MPa (structural equivalent stress range) (obtained by quadratic extrapolation on inside of shell)

18.8 Plasticity correction factor ke Thermal loading

mechanical loading

If ∆σstruc > 2 Rp0,2/t*

If ∆σstruc > 2 Rp0,2/t* k

 Äó  struc = 1+ A0  − 1 e  2 R p0,2/t *  

with A0 =

k



0,5 for 800 MPa ≤ Rm ≤ 1000 MPa 0,4 for Rm ≤ 500 MPa  R − 500  0,4 +  m for 500 MPa ≤ R m ≤ 800 MPa  3000 

A0 = ….. ke = ….. ∆σ = ke ∆σstruc = ….. MPa Else ∆σ = ∆ σstruc = 324.2 MPa

υ

= 0,5 +

0,7 0,4

 Äó   struc   R p0,2/t * 

kυ = ….. ∆σ = kυ ∆σstruc = ….. MPa Else ∆σ = ∆σstruc = ….. MPa

If both mechanical and thermal loadings are to be considered, the correction has to be made on each component of the stress tensors . ke and k í are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated. 18-10-6-1 Thickness correction factor few en ≤ 25 mm few = 1

25 mm ≤ en ≤ 150 mm few = (25/en)0,25 = few = …..

en ≥ 150 mm few = 0.639

DBA Design by Analysis

Analysis Details

Page 7.92 (A)

Example 1.4 / F-Check

18-10-6-2 Temperature correction factor ft* For t* > 100 °C Else ft* = 1

ft* = 1,03 – 1,5 10-4 t* -1,5 10-6 t*2 = …..

18-10-6-3 Overall correction factor fw fw = few.ft* = 1 18-10-7 Allowable number of cycles N Äó fw

= 324.2 MPa

If

Äó fw

> ∆σ5.106 cycles then

m = 3 and C (C⊥ or C//) = 7.16.1011

N=

C  ∆σ   fw

  

m

= 21010 cycles

If

Äó fw

< ∆σ5.106 cycles and other

cycles with

Äó fw

> ∆σ5.106 cycles

then m = 5 and C (C⊥ or C//) = ….. N = C m = …..  ∆σ   f   w 

If

Äó fw

< ∆σ5.106 cycles and all other

cycles with N=∞

Äó fw

< ∆σ5.106 cycles then

DBA Design by Analysis

Analysis Details Example 2

Analysis Type: Direct Route using Elastic Compensation

FE-Software:

ANSYS 5.4

Element Types:

4 – node, 3-D isoparametric shells.

Page 7.93 (S)

Member: Strathclyde

Boundary Conditions: Symmetry boundary conditions along the vertical cutting plane of the model. All degrees of freedom at open end of small cylinder have displacements constrained to 0. Model and Mesh: Number of elements - 3570

Results: All load conditions admissible according to checks against GPD and PD.

DBA Design by Analysis

Analysis Details Example 2

Page 7.94 (S)

1. Finite Element Mesh The geometry model was constructed according to the problem specification. Half symmetry was used with finite element models created using 3570 low order 4-node 3-D shell elements. Calculations to check the admissibility of the defined load cases were carried out according to the GPD- and PD-check rules in the code. As half symmetry was utilised, symmetry boundary conditions were applied to the model along the vertical cutting plane. The nodes at the open end of the narrow cylinder (bottom of storage tank) have all their degrees of freedom constrained to zero displacement (fully fixed). When internal pressure is applied, a corresponding thrust is applied to the top open edge of the elements of the thick cylinder to model the closed end condition. The mass of the top cover and insulation of the top cover are also applied as a pressure over this edge. Hydrostatic pressure is modelled by applying a pressure gradient over the internal surface of the cylinder corresponding to ρgh. The wind loading is calculated as a total wind force for each section of the tank and distributed evenly over the nodes in that section of the tank (in the direction of the wind). Figure 7.6.1-1 shows the resulting Mises equivalent stress distribution resulting from the wind load. Dead weight is applied by defining a density that includes the insulation and density of the Figure 7.6.1-1: Equivalent Stress for Wind Loading. material. Acceleration equal to gravity is then applied. 2. Material properties: •

Shell (X6Cr Ni Ti 18-10): material strength parameter RM = 224 MPa , modus of elasticity E = 193GPa , coefficient of linear thermal expansion α = 16.4 ⋅ 10 −6 1 / K .

•

ring (P235 GH): material strength parameter RM = 202 MPa, modulus of elasticity E = 209 GPa, coefficient of linear thermal expansion α = 12.2 ⋅ 10 −6 1 / K .

3. Admissiblity check against to GPD. Using the application rule in prEN-13445-3 Annex B.9.2.2 to check against GPD, the principle is fulfilled when for any load case the combination of the design actions do not exceed the design resistance. The analysis was carried out using the elastic compensation method conforming to the direct route rules for GPD in Annex B: linear elastic ideal - plastic material law, Tresca’s yield condition and associated flow rule and first order theory. For shell elements it is not currently

DBA Design by Analysis

Analysis Details Example 2

Page 7.95 (S)

possible to calculate limit loads based directly on Tresca’s condition from elastic compensation. Models utilising shells have only one element through thickness. Instead of carrying out the analysis using a Tresca or Mises model directly, a generalised yield model is used which considers the element's thickness. In elastic compensation, the Ilyushin generalised yield model is used in the calculation of limit stress fields. Ilyushin's model is based Mises' condition, the limit load will require correction to meet the code rules on the use of the Tresca condition. The maximum ratio of Mises' equivalent stress to Tresca's equivalent stress for the same load is 2/√3. Therefore, applying a factor of √3/2 to the yield stress in the Mises analysis will always lead to a conservative result for the Tresca condition. Admissibility checks are required for the following three load cases: (1) Hydrostatic pressure at maximum medium level, dead weight, wind load. (2) Hydrostatic pressure at minimum medium level, draining pressure, dead weight and wind load. (3) Draining pressure, dead weight and wind load The full model is used in the check against GPD for the above three load cases. Materials defined for the analysis have proof strengths of 224 MPa and 202 MPa for the shell and ring respectively. From prEN-13445-3 Annex B, Table B.9-3, the partial safety factor γR on the resistance is 1.18 and 1.25 for the shell and ring respectively. The design material strength parameter used in the calculations are given by applying the partial safety factors and Mises' correction, i.e. the design material strength parameter for the shell is 164.4 MPa and for the ring it is 140 MPa. For the loading, according to prEN-13445-3 Table 5.B.9.2 the partial safety factors are Hydrostatic pressure (pressure with natural limit) Draining pressure (pressure without natural limit) Dead weight (action with unfavourable effect) Wind load (variable action) In the specification, the weight of the insulation is 220 N/m2 (weight per unit surface area) or 110 kg/m2. To apply this as a density to the model, the value has to be divided by the thickness of the shell. The resulting densities for the insulation are then added to those of the steel, 7930 kg/m3 for the shell and 7850 kg/m3 for the ring. A pressure equivalent to the weight of the roof is applied over the top edge of the shell. The partial safety factor for the dead weight is applied to the densities and roof weight for use in the model. The equivalent stress distribution for the dead weight only is shown in Figure 7.6.1-2.

γp = 1.0 γp = 1.2 γG = 1.35 γG = 1.0

Figure 7.6.1-2: Equivalent stress for dead weight

DBA Design by Analysis

Analysis Details Example 2

Page 7.96 (S)

Load case 1 For load case 1, hydrostatic pressure at maximum medium level of 19.68 m, dead weight and wind load corresponding to that defined in the specification are checked against GPD. In checking for admissibility of the applied loads using elastic compensation a lower bound limit stress field must have an Ilyushin function less than 1 where the Ilyushian function f(IL) is Where σe is the element stress and Rd the design material strength parameter. σ f ( IL) = e Rd Figure 7.6.1-3 shows the limit Ilyushin stress field (where the scale is given by the square of the Ilyushin function) for the applied loading. The maximum square of the Ilyushin function of 0.33 is less than 1, and therefore, the loading is below the limit load and the load case is admissible according to the GPDcheck.

Figure 7.6.1-3: Limit Ilyushin field (Load case 1)

Load case 2 For load case 2 the actions are: hydrostatic pressure at minimum medium level, draining pressure, dead weight and wind load. Draining pressure is 0.06 MPa with a partial safety factor of 1.2, giving a design value for the pressure action of Apd = 0.072. Figure 7.6.1-4 shows the limit Ilyushin stress field for load case 2. The maximum square of the Ilyushin function of 0.73 is less than the 1 therefore the action is below the limit. Admissibility for load case 2 is shown. Figure 7.6.1-4: Limit Ilyushin field (Load case 2)

DBA Design by Analysis

Analysis Details Example 2

Page 7.97 (S)

Load case 3 For load case 3 the actions are: draining pressure, dead weight and wind load. Figure 7.6.1-5 shows the limit Ilyushin stress field for load case 3. The maximum square of the Ilyushin function of 0.73 is less than the 1, and therefore, the action is below the limit. The loading is very similar to that in load case 2, as can be seen from the resulting stress plot. Thus, the admissibility for load case 3 is shown. Figure 7.6.1-5: Limit Ilyushin field (Load case 3)

4. Admissibility check against PD In the check for progressive plastic deformation, the principle in prEN-13445-3 B.9.3.1 is fulfilled if the structure can be shown to shake down. In the check against progressive plastic deformation the thermal stresses have to be considered. Thermal stresses arise in this problem due to the different thermal expansion coefficients of the shell and support ring. Figure 7.6.1-6 shows the Mises equivalent (thermal) stress distribution resulting from the different thermal expansion at a temperature of 60oC. For the PD-check there are no partial safety factors on the material strength, giving design strengths of 224 MPa and 202 MPa for the shell and reinforcing ring respectively. Checks against PD are for the load cases defined above in the GPD-check. Since there are no Figure 7.6.1-6: Equivalent (thermal) stress distribution. partial safety factors on the actions in the check against PD, the results for each of the load cases is found by superposition of the thermal result on the load cases defined above.

DBA Design by Analysis

Analysis Details Example 2

Page 7.98 (S)

Figures 7.6.1-7 through 7.6.1-9 show the elastic equivalent stress distribution for the three load cases as follows. Figure 7.6.1-7 – Load case 1. The maximum equivalent stress in the shell is 225 MPa, approximately the upper limit of the elastic range for the material. As the model does not pass into the plastic range, admissibility against PD of load case 1 is shown.

Figure 7.6.1-7: Elastic Equivalent Stress for Load Case 1

Figure 6.7.1-8 – Load case 2. The maximum equivalent stress in the model is 196.3 MPa. The maximum stress is within the elastic range, proving the admissibility of load case 2 against PD.

Figure 7.6.1-8: Elastic Equivalent Stress for Load Case 2

DBA Design by Analysis

Analysis Details Example 2

Page 7.99 (S)

Figure 7.6.1-9 – Load case 3. As with the check against GPD, the resulting stress distribution is similar to that of load case 2. The maximum stress in the model is 196.3 MPa, within the elastic range of the materials. Therefore, the admissibility in the check against PD for load case 3 is shown.

Figure 7.6.1-9: Elastic Equivalent Stress for Load Case 3

DBA Design by Analysis

Analysis Details

Page 7.100 (S)

Example 2

Analysis Type:

Member:

Direct Route using elasto-plastic FE calculations

A&AB

FE-Software:

ANSYS® 5.4

Element Types:

4-node, 2-D axisymmetric solid PLANE42 for axisymmetric model and 8-node, 3-D shell elements for 3-D model used for the wind load calculation.

Boundary Conditions:

♣Axisymmetric model: No vertical displacement in the nodes at the lower end of the model; ♣3-D model:• Symmetry boundary conditions in the symmetry plane of the structure; • no vertical displacement in the nodes at the lower end of the model; • rigid region concerning vertical displacements at the upper end of the model to apply the moment caused by the wind action; • vertical displacement set to 0 for dummy end of beam.

Model and Mesh:

Axialsymmetric model: Height of upper and lower cylinder 500 mm Total number of elements: 3965; 3-D model: Height of upper cylinder 1500 mm, height of lower cylinder 500 mm Total number of elements: 5361.

Results:

Actions and action cycles, as given in the specification, are admissible according to the GPD- and SD-checks.

DBA Design by Analysis

Analysis Details Example 2

Page 7.101 (S)

1. Elements, mesh fineness, boundary conditions Since the wind action, a variable action according to EN 13445-3 Annex B.7.1, had to be taken into account when performing the checks against GPD and PD, a 3-D model representing the interesting part of the structure, i.e. the cone-cylinder intersections, has been used. Under usage of this model, the linear-elastic stress corresponding to the moment caused by the wind action was determined and further used in all other calculations, which were carried out under usage of an axisymmetrical model. In elasto-plastic calculations, the yield strengths of the linear-elastic ideal-plastic material law used in the calculations were the (design) material strength parameters decreased by the value of the maximum equivalent linear-elastic stress caused by the wind action. Since this latter value is small, this conservative approach could be used. The 3D-model consisted of a total number of 3560 8-node shell elements SHELL93 and one elastic beam element BEAM4 (which is necessary to create a rigid region). The two cone-cylinder intersections with the stiffener ring at the narrow end of the cone and the adjacent lower cylindrical parts with a length of 500 mm and the adjacent upper cylindrical part with a length of 1500 mm have been modelled. Because of the symmetry of the structure and the action, only one half of the structure was considered. To investigate the influence of the length of the cylindrical part at the upper end on the stress results, calculations with different lengths were performed (always using the same moment value). The (stress) results were the same for shell lengths of 1500 and 2500 mm, and about 10 % smaller for a length of 500 mm. Therefore, the model with an upper cylindrical shell length of 1500 mm was used. The vertical displacements in the nodes in the lower end of the model were constrained to zero (corresponding to an undisturbed membrane stress state), and a symmetry boundary condition was applied to all nodes in the symmetry plane of the structure. To apply the moment, a beam element (3D elastic beam BEAM4 in ANSYS® 5.4) was attached to the structure at the centre of the structure’s upper end. To transfer the moment from the beam to the structure’s model , the nodes in the upper cross-section of the structure (slave nodes) have been rigidly linked to the lower node of the beam as master node (under usage of the “rigid region” command in ANSYS®, where the vertical displacement was associated to the corresponding constraint equations). To obtain a stable model, the vertical displacement of the upper (dummy) end of the beam element was constrained to zero. For the axisymmetric model of the structure a total number of 3965 4-node axisymmetric elements, PLANE42 in ANSYS® 5.4, were used. As boundary condition the vertical displacement in the nodes at the lower end of the model were constrained to zero. According to the specification, different material properties were used for the stiffener ring (and the corresponding welds) and the shell. The welds between stiffener ring and shell were modelled with fillets according to weld influence zone of about 2 mm (a corresponding fillet radius of 5 mm was used). To avoid stress singularities, the shell welds at the cone cylinder junctions were modelled with appropriate fillet radii.

Analysis Details

DBA Design by Analysis

Example 2

Page 7.102 (S)

2. Admissibility check against GPD The admissibility checks have to be carried out for the following 3 load cases: LC1: Wind, dead weight and hydrostatic pressure according to maximum medium level LC2: Wind, dead weight, hydrostatic pressure according to minimum medium level and draining pressure LC3: Wind, dead weight and draining pressure Note: The admissibility of the load case wind, dead weight and hydrostatic pressure corresponding to minimum medium level is shown by the admissibility of LC1. To avoid the usage of a computation time intensive 3-D model in the elasto-plastic calculations, the maximum linear-elastic equivalent stresses in the shell and the stiffener ring due to the wind action were calculated using the 3-D shell model. Furthermore, the GPD checks were performed with the axisymmetrical model without application of the wind action (which would not be possible within this model) and with decreased design material strength parameters given by the difference of the original design material strength parameters minus the maximum linear-elastic equivalent stresses in the shell and the stiffener ring due to the wind action. This procedure is admissible due to the positive definiteness of the equivalent stress: The sum of the maximum equivalent stresses of two stress tensors is greater or equal to the maximum equivalent stress of the sum of the two stress tensors. The elastic moduli used in the models are given by 193 GPa for the shell and 209.5 GPa for the stiffener ring, respectively. The moment, which has to be applied to the upper cross-section of the 3-D model and which corresponds to the wind action on the vessel above the wide end of the cone, was calculated with a partial safety factor of γ Q = 1.0 , since the wind action is specified in the specification is as a limit value (see the specification of the example): M W = ∑ Wi ⋅ ai = c ⋅ ∑ qW ,i ⋅ Ai ⋅ a i = c ⋅ d ges ⋅ ∑ qW ,i ⋅ hi ⋅ a i , where qW , i

is the wind stagnation pressure depending on height h (specified) 0 m ≤ h ≤ 6 m:

qW = 0.81 kN / m 2

6 m < h ≤ 10 m :

qW = 0.88 kN / m 2

10 m < h ≤ 15 m :

qW = 0.94 kN / m 2

Wi

15 m < h ≤ 25 m : qW = 1.02 kN / m 2 the wind force on section i: Wi = c ⋅ qW ,i ⋅ Ai

c

the drag coefficient

Ai

the projection of the surface of the vessel in wind direction: Ai = d ges ⋅ hi

ai

the vertical distance of the resultant wind force of Wi from the wide end of the cone,

d ges

the vessel diameter including insulation, d ges = 6900 mm .

c = 0.44 (specified)

and

Analysis Details

DBA Design by Analysis

Example 2

Page 7.103 (S)

Therefore, the moment corresponding to the wind and with respect to the centre point of the cross section at the cone’s wide end is given by M W = 356.4 MNmm . Since only half of the structure was modelled, half of this moment was applied in the FE-model. Figure 7.6.2-1 shows the Mises’ equivalent stress distribution in the shell model according to the wind action. The maxima are given by 26.9 MPa in the shell at the wide end of the cone and by about 9 MPa in the ring.

Figure 7.6.2-1

Since the subroutine using Trecas’s yield condition showed bad convergence, all elasto-plastic calculations were carried out using Mises’ yield condition and associated flow rule, a linear-elastic ideal-plastic material law, first order theory; under usage of the Newton-Raphson method, and scaled down material strength parameters with the factor 3 / 2 (see subsection 3.2 of section 3 Procedure). The partial safety factors according prEN 13445-3 Annex B, Table B.9.3, are given by γ R = 1.18 for the shell material and γ R = 1.25 for the ring material. Therefore, the design material strength parameters used in the calculations are given by Rd =

224 3 ⋅ − 26.9 = 137.4 MPa 1.18 2

Rd =

202.8 3 ⋅ − 9 = 131.5 MPa 1.25 2

for the shell, and for the ring.

where RM = 224 MPa is the material strength parameter of the shell according to prEN 10028-7 and RM = 202.8 MPa the material strength parameter of the ring according to EN 10028-2, both determined for the calculation temperature of the vessel.

DBA Design by Analysis

Analysis Details Example 2

Page 7.104 (S)

According to prEN 13445-3 Annex B, Table B.9.2, the partial safety factor for the hydrostatic pressure is given by γ P , hydro = 1.0 , the partial safety factor for the draining pressure by γ P , drain = 1.2 , and the partial safety factor for the dead weight by γ G = 1.35 . For the application of the dead weight, densities of 7930 kg / m 3 for the shell material, 7850 kg / m 3 for the ring material, and 110 kg / m 3 for the insulation were used (the densities used in the input file were the sum of the densities of the corresponding material and the insulation multiplied by the factor γ G = 1.35 ). The dead weight of the part of the structure above the modelled part was calculated to be 239 MN ; a corresponding pressure of 1.96 MPa multiplied by the factor γ G = 1.35 was applied in the FE- model. The internal (draining) pressure applied in the FE-model is given by the product 0.06 ⋅ 1.2 = 0.072 MPa ; a corresponding longitudinal stress was applied at the upper end of the model. Figure 7.6.2-2 shows the Mises’ equivalent stress distribution for LC1. The corresponding maximum absolute value of the principal strains is 0.09 %, and, therefore, the admissibility of this load case according to the GPD-check is shown. Figure 7.6.2-2

Figure 7.6.2-3 shows the Mises’ equivalent stress distribution for LC2. The corresponding maximum absolute value of the principal strains is 0.32 %, and, therefore, the admissibility of this load case according to the GPD-check is shown.

Figure 7.6.2-3

DBA Design by Analysis

Analysis Details Example 2

Page 7.105 (S)

Figure 7.6.2-4 shows the Mises’ equivalent stress distribution for load case LC3. The corresponding maximum of the absolute values of the principal strains is 0.32 %, and, therefore, the admissibility of this load case according to the GPD-check is shown. The computation time of the elasto-plastic solutions were about 7 minutes on the Compaq® Professional Workstation 5000 with two Pentium Pro® processors and 256 MB RAM for every state. Figure 7.6.2-4

3. Admissibility check against PD In the following, it is shown that Melan’s shakedown theorem is fulfilled for all load cases under consideration, i.e. the structure shakes down, and, therefore, progressive plastic deformation cannot occur. The following load cases have to be considered, including the thermal stress according to the different thermal expansion coefficients of the shell and the ring material. LC1: Wind, dead weight and hydrostatic pressure according to maximum medium level LC2: Wind, dead weight, hydrostatic pressure according to minimum medium level and draining pressure LC3: Wind, dead weight and draining pressure. Again, to avoid the usage of a computation time intensive 3-D model in the calculations, the maximum linear-elastic equivalent stresses in the shell and the stiffener ring due to the windaction, calculated using the 3-D shell model, were used. Therefore, the calculations for the shakedown check were carried out without the wind action, using the axisymmetrical model, and with decreased material strength parameters, given by the difference of the original material strength parameters and the value of the maximum linear-elastic equivalent stresses in the shell and the stiffener ring due to the wind action. Thus, the reduced (design) material strength parameters used in the shakedown check are given by

DBA Design by Analysis

Analysis Details Example 2

RM = 224 − 26.9 = 197.1 MPa

for the shell, and

RM = 202.8 − 9 = 193.8 MPa

for the ring.

Page 7.106 (S)

The thermal expansion coefficients to be used in the calculation, are α = 16.4 ⋅ 10 −6 1 K for the shell material, and α = 12.2 ⋅ 10 −6 1 K for the ring material. Figure 7.6.2-5 shows the linear-elastic (thermal) Mises equivalent stress distribution according to the different thermal expansion coefficients at the temperature 60°C.

Figure 7.6.2-5

The following figures show the linear-elastic Mises’ equivalent stress distributions of the considered load cases: Figure 7.6.2-6 shows results for LC1, and Figure 7.6.2-7 for LC3. The results of LC2 are not shown since they are practically the same as for LC3. The results for all these load cases were calculated by superposition of the load cases for the different loads.

DBA Design by Analysis

Figure 7.6.2-6

Figure 7.6.2-7

Analysis Details Example 2

Page 7.107 (S)

DBA Design by Analysis

Analysis Details Example 2

Page 7.108 (S)

As can be seen from these figures, the behaviour of the structure is completely elastic for LC1, but in LC2 and in LC3 the maximum Mises’ equivalent stress is larger than the used material strength parameter, and, therefore, some plastification will occur. From the behaviour of the structure it was concluded, that the stresses due to the draining pressure were the reason for the plastification. Therefore, a self-equilibrating stress field based on internal pressure solutions was used to show that Melan’ s shakedown theorem is fulfilled. An elasto-plastic FE calculation was carried out for a value of the internal pressure of 0.1 MPa . And the self-equilibrating stress field, shown in Figure 7.6.2-8 was found by the difference of the elasto-plastic stress field minus the corresponding linear-elastic stress field. The computation time of the elasto-plastic solution was 10 minutes on the Compaq® Professional Workstation 5000 with two Pentium Pro® processors and 256 MB RAM. Figure 7.6.2-8

Using this self-equilibrating stress field, Melan’s theorem could not be fulfilled for LC1, and, therefore, the self-equilibrating stress field was multiplied with a suitable factor, which was found to be 0.4. Using this corrected self-equilibrating stress field, Melan’s theorem was fulfilled for all load cases under consideration. Figure 7.6.2-9 shows the sum of the corrected self-equilibrating stress field and the linear-elastic stress field of LC1, and Figure 7.6.2-10 shows the sum corresponding LC2. The sum of the corrected self-equilibrating stress field and the linear-elastic stress field of LC3 is not shown since it is practically the same as the sum corresponding to LC2.

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Figure 7.6.2-9

Figure 7.6.2-10

Analysis Details Example 2

Page 7.109 (S)

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Analysis Details

Analysis Type:

Member: F-Check

FE –Software:

Page 7.110 (C)

Example 2 / F-Check

CETIM

ABAQUS / Standard version 5.8.1

Element Types: Quadratic axisymmetric 8-node elements . 4743 nodes and 1126 elements Boundary Conditions: The base ring of the vessel is fixed in the vertical direction.

Model and Mesh:

Results: Fatigue life N = 1984 cycles (junction shell/stiffener)

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Analysis Details Example 2 / F-Check

Page 7.111 (C)

1. Introduction: The calculations performed were linear-elastic ones. In the welded zones, the mesh takes into account the nodes where the results are extrapolated. A quadratic extrapolation was used, with the first node at a distance equal to 0,4 en (en = thickness of the shell) from the weld, the second at 0,9 en of the weld and the third node at 1,4 en of the weld. 2. Material parameters: X6CrNiTi 18-10 EN 10028-7 for the shell, Rm = 520 MPa, R p1.0 / t = 224 MPa . P235 GH EN 10028-2 for reinforcing, Rm = 360 MPa, Rp 0.2 / t = 202.8 MPa. 3. Operating Cycles: The load cases used in determining the cycles are. § § § § § § §

LC 1 : media level = hmax ; internal pressure = 0 ; temperature = 60 °C LC 2 : media level = hmin ; internal pressure = 0 ; temperature = 60 °C LC 3 : media level = hmax ; internal pressure = 0 ; temperature = 60 °C LC 4 : media level = hmin ; internal pressure = 0 ; temperature = 60 °C LC 5 : media level = 0 ; internal pressure = 0,6 bar ; temperature = 60 °C LC 6 : media level = 0 ; internal pressure = 0 ; temperature = 20 °C LC 7 : media level = hmax ; internal pressure = 0 ; temperature = 60 °C

Stress range : The are five (welded) zones of possible interest.

weld 5 weld 3 weld 4

weld 2 weld 1

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Analysis Details Example 2 / F-Check

Page 7.112 (C)

The selection of welds to be investigated in detail was based on the principal stress range in the steps and the class of the weld. Since only the wide end and the narrow and of the cone are specified as to be investigated, there remains: § §

Weld 3 : Äó = 224 MPa – class 63 Weld 5 : Äó = 253,1 MPa – class 80

With the list of load cases given above the full cycle, relevant at the two welded regions, can be written as §

Weld 3, weld 5: 1 x (LC 6 - LC 3), 99 x (LC 3 – LC 2) and 1 x (LC 6 - LC 5)

The structural stress components and the equivalent structural stresses for the load cases and the load case differences at the two welds were obtained via quadratic extrapolation, and are Weld 3 (structural stresses obtained with quadratic extrapolation) steps 2 3 5 6 6-3 3-2 6-5

ó11 (MPa) -5,2 -37 -4,3 0 37 -31,8 4,3

ó22 (MPa) -35,8 -203,3 -31,6 0 203,3 -167,5 31,6

ó33 (MPa) -34,6 -117,8 -33,5 0 117,8 -83,2 33,5

óTresca (MPa) 166,3 157,7 29,2

Weld 5 (structural stresses obtained with quadratic extrapolation) steps 2 3 4 4-3 3-2

ó11 (MPa) 0 -0,2 -20,1 -19,9 -0,2

ó22 (MPa) 0 -0,4 276,9 277,3 -0,4

ó33 (MPa) 0 83,3 -1,1 -84,4 83,3

óTresca (MPa) 361,7 83,5

Since thermal stresses are negligeable in the weld region 5, the load cases used for the two regions are different. It is still not obvious which of the two welded regions is the more critical one, both need detailed investigation, see the following-sheets.

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Data

Analysis Details Example 2 / F-Check

Page 7.113 (C)

Weld 3 / Equivalent stress range approach en = 8 mm ∆σD (5 106cycles) = 46 MPa (class 63) equivalent stresses or m = 3 C = 5.1011

tmax = 60 °C tmin = 20 °C t* = 0,75 tmax + 0,25 tmin = 50 °C Rm = 520 MPa Rp1,0/t* = 224 MPa

principal stresses m = 3 C⊥ = ….. C// = ….. m = 5 C⊥ = ….. C// = …..

m = 5 C = 1,08.1015

Stresses ∆σstruc = 116,3 / 135,7 / 29,2 MPa (structural equivalent stress range, determined by extrapolation) 18.8 Plasticity correction factor ke Thermal loading

mechanical loading

If ∆σstruc > 2 Rp1,0/t*

If ∆σstruc > 2 Rp1,0/t*

 Äó

k

e

= 1 + 0, 4 

struc

 2 R p1,0/t * 



− 1

k



ke = ….. ∆σ = ke ∆σstruc = ….. MPa Else ∆σ = ∆ σstruc = 166,3 / 135,7 / 29,2 MPa

υ

= 0,5 +

0,7 0,4

 Äó   struc   R p1,0/t * 

kυ = ….. ∆σ = kυ ∆σstruc = ….. MPa Else ∆σ = ∆σstruc = ….. MPa

If both mechanical and thermal loadings are to be considered, the correction has to be made on each component of the stress tensors . ke and k í are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated. 18-10-6-1 Thickness correction factor few en ≤ 25 mm few = 1

25 mm ≤ en ≤ 150 mm few=(25/en)0,25 = few = …..

en ≥ 150 mm few = 0,639

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Analysis Details

Page 7.114 (C)

Example 2 / F-Check

18-10-6-2 Temperature correction factor ft* For t* > 100 °C Else ft* = 1

ft* = 1,043 – 4,3 10-4 t* = …..

18-10-6-3 Overall correction factor fw fw = few.ft* = 1

18-10-7 Allowable number of cycles N Äó fw

If

= 166,3 / 135,7 / 29,2 MPa

Äó fw

> ∆σ5.106 cycles then

M = 3 and C (C⊥ or C//) = 5.1011

N=

C  ∆σ   fw

  

m

= 108720 / 200090 cycles

Äó fw

=29.2 ⇒ N = ∞

If

Äó fw

< ∆σ5.106 cycles and other

cycles with

Äó fw

> ∆σ5.106 cycles

then M = 5 and C (C⊥ or C//) = N= C m =  ∆σ   fw

If

Äó fw

< ∆σ5.106 cycles and all other

cycles with

Äó fw

< ∆σ5.106 cycles

then N=∞

  

N = ∞ since 29.2 < ∆σ108 cycles

The complete operating cycle corresponds to 1 cycle for the stress ranges 166,3 MPa and 29,2 MPa and 99 cycles for the stress range 135,7 MPa. The global allowable number of cycles is equal to N with : N 99 N N + + =1 108720 200090 ∞ ⇒ N = 1984 cycles (each cycle is equal to 100 variations between hmax and hmin and 1 complete draining and filling)

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Analysis Details Example 2 / F-Check

Weld 5 / Equivalent stress range approach Data en = 6 mm tmax =60 °C ∆σD (5 106cycles) = 59 MPa (class 80) tmin = 20 °C equivalent stresses or t* = 0,75 tmax + 0,25 tmin = 50 °C 12 . Rm = 520 MPa m = 3 C = 1,02 10 Rp1,0/t* = 224 MPa m = 5 C = 3,56.1015

Page 7.115 (C)

principal stresses m = 3 C⊥ = ….. C// = ….. m = 5 C⊥ = ….. C// = …..

Stresses ∆σstruc = 361,7 / 83,5 MPa (structural equivalent stress range, determined by extrapolation) 18.8 Plasticity correction factor ke Thermal loading

mechanical loading

If ∆σ struc > 2 Rp1,0/t*

If ∆σstruc > 2 Rp1,0/t*

 Äó

k

e

= 1 + 0, 4 

struc

 2 R p1,0/t *



− 1

k



ke = ….. ∆σ = ke ∆σstruc = ….. MPa Else ∆σ = ∆σstruc = 361,7 / 83,5 MPa

υ

= 0,5 +

0,7 0,4

 Äó   struc   R p1,0/t * 

kυ = ….. ∆σ = kυ ∆σstruc = ….. MPa Else ∆σ = ∆σstruc = ….. MPa

If both mechanical and thermal loadings are to be considered, the correction has to be made in each component of the stress tensors . ke and k í are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated. 18-10-6-1 Thickness correction factor few en ≤ 25 mm few = 1

25 mm ≤ en ≤ 150 mm few=(25/en)0,25 = few = …..

en ≥ 150 mm few = 0,639

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Analysis Details

Page 7.116 (C)

Example 2 / F-Check

18-10-6-2 Temperature correction factor ft* For t* > 100 °C Else ft* = 1

ft* = 1,043 – 4,3 10-4 t* = …..

18-10-6-3 Overall correction factor fw fw = few.ft* = 1 18-10-7 Allowable number of cycles N Äó fw

If

= 361,7 / 83,5 MPa Äó fw

> ∆σ5.106 cycles then

M = 3 and C (C⊥ or C//) = 1,02.1012

N=

C  ∆σ   fw

  

m

= 21555 / 1752000 cycles

If

Äó fw

< ∆σ5.106 cycles and other

cycles with

Äó fw

> ∆σ5.106 cycles

then M = 5 and C (C⊥ or C//) = ….. N = C m = ….. cycles  ∆σ   fw

  

If

Äó fw

< ∆σ5.106 cycles and all other

cycles with

Äó fw

< ∆σ5.106 cycles

then N=∞

The complete operating cycle corresponds to 1 cycle for the stress ranges 361,9 MPa and 99 cycles for the stress range 83,7 MPa. The global allowable number of cycles is equal to N with : N 99 N + =1 21555 175200 ⇒ N = 9718 cycles (each cycle is equal to 100 variations between hmax and hmin and 1 complete draining and filling)

Analysis Details

DBA Design by Analysis

Example 2

Page 7.117(A)

• Stability check of ring at small end of cone: Load case: Self weight of vessel, vessel fully filled, wind: Maximum radial force resultant at inner ring surface: (diameter 4000 mm), determined from FE-analyses (Direct Route using elasto-plastic FE, corresponding linear-elastic calculation): qr = −99 N / mm This value corresponds to the characteristic value of the actions. The partial safety factors for the actions are γ G = 1.35 for self weight γ p , hydro = 1.0 for the hydrostatic pressure due to filling γ a = 1.0 for the wind action, see pages 7.104. Since the force (per unit length) due to the filling weight is much larger than the forces due to wind moment and self weight, see page 7.120, the characteristic value of the effective radial force given above is used also as design value, i. e. γ a = 1.0 is used for all actions in the following. If a more accurate result is required, the contribution of self weight, given by nsw , could be multiplied by γ G = 1.35 . This results in a design value of the effective radial force of q re , d =

n f + n w + γ G ⋅ n sw n f + n w + n sw

⋅ q re = 1.023 q re = 97.6 N / mm

with values for n f , n w , n sw of page 7.120. Effective radial force at center of gravity of ring (diameter 4150) q re = q r 4000 / 4150 = 95.4 N / mm Allowable radial force (per unit length) acc. to Section 8 of prEN 13445-3, where qr corresponds to pH LsH : Eq. (8.4.3-48): pH LSH = 3 E I eH / R3 . For simplicity the ring only is considered: R = 2000 mm, I s = 8.4375.106 mm 4 :

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Analysis Details Example 2

Page 7.118(A)

p H LSH = 610.66 N / mm Safety factor according to 8.2.3 : k = 1.5 Coefficient for hot bent stiffener according to 8.4.3.5a : k f = 1.2 Allowable radial force: qra = pH LsH /(k k f ) = 339.3 N / mm : Conclusion: q ra > q re , d = 95 → check o.k. • Stability check of small cylindrical shell: Maximum axial force resultant (per unit length): Load case: Vessel fully filled, self weight of vessel and wind: nmax = − n w + n sw + n f = 328.0 N / mm, see SE-check, page 7.120. Allowable axial stress resultant (per unit length): Roark’s Formulas for Stress and Strain, 6th edition, Table 35 Case 15: Theoretical value: nath = ( Ee 2 / r ) / (3 (1 − v 2 )) 0.5 = 3745 N / mm Practical value: n pr = 1498 N / mm with reduction (knock-down) factor 0.4 (see Roark) With a safety factor k = 1.5 , assuming that the pressure test cannot be performed with the required test pressure, the allowable axial stress resultant na = 998.8 N / mm results. There follows: Allowable value > Maximum value: 998.8 > 328.0 → check o. k. The civil engineering standard DIN 18800 gives, for the boundary conditions •

shell clamped at lower end

•

shell radial displacement restrained (to zero) at upper end,

a theoretical value of

DBA Design by Analysis

Analysis Details Example 2

Page 7.119(A)

nath = 3763 N / mm and a reduction factor of 0.544, and, thus, a reduced (practical) value of na , pr = 2048 N / mm With a partial safety factor of γ I = 1.5 , assuming that the pressure test cannot be performed with the required test pressure, an allowable axial force per unit length of na , dr = 1366 N / mm

results; thus, the conclusion is, that the allowable (design resistance) value is larger than the (design value) of the maximum axial force (per unit length), na , dr = 1366 > 328 = nmax → check o.k .

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Analysis Details Example 2

Page 7.120(A)

Moment due to wind with respect to centre of base: M W = 582.95 MNmm Corresponding maximum axial force resultant (per unit length, in shell d m = 3992 mm ): nw = 4 M w /(d m2 π ) = 46.58 N / mm Total weight of vessel (exclusive base plate): Fsw = 269.44 kN Corresponding axial force resultant: nsw = Fsw /(d mπ ) = −21.29 N / mm Weight of filling above conical shell: F f = 3.263 MN Corresponding axial force resultant: n f = −260.17 N / mm • Check against overturning: Load case: Vessel empty, self weight of vessel, and wind: nw + nsw = 24.73 N / mm > 0 → Bolting required Bolt force for 8 bolts: FB = (nw + nsw ) d m π / 8 = 38.8 kN Check of bolt load: Load case: Vessel empty, self weight of vessel, and wind: FB = 38.8 N Load case: Vessel empty, self weight of vessel, internal pressure and wind: Axial force resultant for internal pressure: n p = p d i / 4 = 59.76 N / mm Maximum bolt force for 8 bolts: FB = (n w + n sw + n p ) d mπ / 8 = 133.3 kN

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Analysis Details Example 2

Page 7.121(A)

Check of maximal pressure under base ring: Load case: Self weight of vessel, vessel fully filled and wind: Maximal pressure under base ring (minimum pretensioning of bolts, stiff foundation, triangular pressure distribution): − p b = 4 (−2nw + n sw + n f ) / (4120 − 3864) = −5.86 MPa • Check against displacement: Load case: Vessel empty, self weight of vessel and wind Horizontal wind force (at base): Fw = 52.68 kN Friction force: FF = µ Fsw = 80.8 kN with friction coefficient µ = 0.3 FF > Fw → check o.k. ; bolting required only because of wind moment.

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Analysis Details

Page 7.122 (S)

Example 3.1 / GPD- & PD-Check

Analysis Type: Direct Route using Elastic Compensation

Member: Strathclyde

FE-Software:

ANSYS 5.4

Element Types:

8 – node, 3-D isoparametric shells.

Boundary Conditions:

Symmetry boundary conditions in the plane along the longitudinal direction of the shell. Hoop displacements in nodes at both ends of the shell constrained to zero. Longitudinal displacements in the nodes at one end of the shell constrained to zero, longitudinal displacements coupled in the nodes at the other end of the shell (plane sections remain plane). A rigid region is set up on the top surface nodes of the nozzle to apply the bending moment.

Model and Mesh:

Number of elements - 576

Results: Maximum admissible action according to the GPD-Check: Internal pressure PSmax GPD = 0.467 MPa (for the constant moment of 15644.4 Nm) Check against PD: Shakedown limit pressure PSmax SD = 1.34 MPa

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Analysis Details Example 3.1 / GPD- & PD-Check

Page 7.123 (S)

1. Finite Element Mesh Element divisions for the finite element mesh were defined parametrically to allow modification of the element density. Half symmetry was used with finite element models created using high order 8-node 3-D shell elements. The allowable pressure according to GPD and the shakedown pressure according to PD were calculated for this model. Boundary conditions applied to the model are symmetry about the xz plane therefore the applied moment is half the specified value. The nodes at both ends of the main cylinder are constrained in the hoop direction but are allowed to move radially. The nodes at one end of the main cylinder were constrained fully in the longitudinal direction with the nodes at the other end coupled in the longitudinal direction, thus maintaining plane sections remaining plane. Pressure was applied over the whole internal structure with the equivalent thrust that would be produced from closed ends applied at the longitudinally coupled degree of freedom end of the cylinder. The moment applied to the intersecting nozzle was modelled by creating a rigid region over the top edge of the nozzle via constraint equations. A mass element was created in the centre at the top of the nozzle (to act as the master node to which the moment is applied) with the nodes on the top edge of the nozzle (slave nodes) connected to it by constraint equations. The moment applied to the master node would then be transmitted by the rigid constraint region evenly to the slave nodes on the nozzle. This results in the application of a pure moment to the top surface of the nozzle with no warping, the cross-section is forced to remain plane. 2. Material properties Material strength parameter RM = 272 MPa , modulus of elasticity E = 210.125 GPa 3. Determination of the maximum admissible pressure according to the GPD-Check Using the application rule in prEN 13445-3, Annex B.9.2.2 to check against GPD the principle is fulfilled when for any load case the combination of the design actions does not exceed the design resistance. This may be shown by calculating the limit load. The limit load has to be determined using Tresca‘s yield condition and associated flow rule. However, for shell elements the limit load from elastic compensation is calculated using a generalised yield criterion based on Mises'. From prEN-13445-3 Annex B, Table B.9-3 the partial safety factor γR on the resistance is 1.25. Therefore, the design material strength parameter is given by RM/γR = 217.6 MPa. The analysis was carried out using the elastic compensation method conforming to the direct route rules for GPD in Annex B: linear elastic - ideal plastic material law, Tresca’s yield condition and associated flow rule and first order theory. For shell elements it is currently not possible to calculate limit loads based directly on Tresca’s condition from elastic compensation. Models utilising shells have only one element through thickness. Therefore, it is not possible to carry out an elastic compensation analysis in the same way as in solid elements. Instead of carrying out the analysis using a Tresca or Mises model directly, a generalised yield model was used which considers the elements thickness. In elastic compensation, the Ilyushin generalised yield model is used in the limit load calculation. Ilyushin's model is based upon Mises' condition, the limit load will require correction to meet the code rules on the use of the Tresca condition. The maximum ratio of Mises' equivalent stress to Tresca's equivalent stress for the same load is 2/√3. Therefore, applying a factor of √3/2 to the yield stress in the Mises analysis (or to the limit load, as the analysis is elastic) will always lead to a conservative result on the Tresca condition. In effect the Mises yield locus is reduced to fit within the Tresca yield locus.

Analysis Details

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Example 3.1 / GPD- & PD-Check

Page 7.124 (S)

In each elastic compensation iteration an equilibrium stress field is produced where the elastic modulus of each subsequent iteration is defined by the previous elastic solution based upon the Ilyushin generalised yield model. In this way regions of the FE-model may be systematically stiffened or weakened depending upon the stress magnitudes in the previous solution. The result is that every equilibrium stress field is a lower bound of the limit load, the stress field giving the greatest limit load is taken as the closest to the actual limit state and defines the limit load in the analysis. Convergence of the equilibrium stress field will occur typically between 8 and 15 iterations, i.e. no further benefit in the limit load will be noted with subsequent iterations. In the check against GPD, the loading considered is a constant moment and increasing pressure, i.e. the limit pressure is to be calculated. As elastic compensation is based on linear-elastic analysis, it is not possible to calculate the limit load directly where there are two separate actions with one constant. The elastic compensation procedure would scale the combined load stress field and would therefore scale both sets of loads. However, by carrying out a series of elastic compensation analyses where the ratio of the load sets are altered for each analysis a limit locus can be constructed. The limit locus will then describe the limit loads for any ratio of the load sets. Correction of the analysis based on Mises' condition was applied to the design material strength parameter to give a corrected value of (217.6) ( 3 / 2 ) = 188.5 MPa.

Pressure (MPa)

The total computing time to run each analysis on a 300 MHz Pentium two processor with 128 Mb RAM Windows NT workstation was 750 seconds. The stress fields were shown to converge after fourteen iterations, eight separate analyses were carried out to describe the limit locus. Figure 7.7.1-1 shows the pressure moment limit locus from which the design pressure is found. According to prEN-13445-3 Annex B, Pressure-Moment Limit Locus Table B.9-2 for permanent actions with an unfavourable 1.4 effect, e. g. the moment, the 1.2 partial safety factor γG is 1.35. Therefore, the design 1 value for the moment action, 0.8 AMd, is the allowable APd =0.56 MPa 0.6 moment (defined in the specification) multiplied by 0.4 γG giving AMd=21120 Nm. 0.2 AMd=21120 Nm From Figure 7.7.1-1, for a 0 design moment action of 0 5000 10000 15000 20000 25000 30000 21120 Nm a design value for Moment (Nm) the pressure action, APd, of 0.56 MPa results. For a Figure 7.7.1-1: Pressure Moment Limit Locus pressure without natural limit the partial safety factor γp, given by prEN 13445-3, Table B.9-2, is 1.2. Thus the allowable pressure is given as 0.56 0.56 = = 0.467 MPa γp 1.2 By carrying out a final elastic compensation analysis using the calculated value of design pressure, a check of the result can be made. Figure 7.7.1-2 shows the limit stress field based on the Ilyushin function at the design actions. The contour units are dimensionless and are termed Ilyushin PS max GPD =

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Analysis Details Example 3.1 / GPD- & PD-Check

Page 7.125 (S)

function. The Ilyusin function represents the ratio of actual equivalent stress to the yield stress of the material, i.e. σe σY Where f(IL) is the Ilyushin function, σe is the element equivalent stress, and σY the yield stress of the design material (or in this analysis the design material strength parameter). Therefore, for the applied loading to be a lower bound on the limit load the Ilyushin function anywhere in the redistributed limit field cannot exceed 1. In Figure Figure 7.7.1-2: Ilyushin Function Limit Field 7.7.1-2 it can be seen, that the maximum square of the Ilyushin function is 1.001. Therefore, for the given moment, the allowable pressure loading of 0.467 MPa fulfils the check against GPD. The plot also shows the extent of plasticity on and around the nozzle at the limit. f ( IL) =

4. Check against PD In the check against progressive plastic deformation, the principle in prEN-13445-3 B.9.3.1 is fulfilled if the structure can be shown to shake down. In elastic compensation the load at which the structure will shake down is simple to calculate. Based on Melan’s shakedown theorem, the self-equilibrating residual stress field that would result after a loading cycle can be calculated by subtracting the linear-elastic stress field at the limit pressure from the limit stress field. The residual stress field is in effect the resulting stress from an elastic unloading from the limit state back to zero pressure. If no stress in the

Figure 7.7.1-3: Residual Ilyushin Field

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Analysis Details Example 3.1 / GPD- & PD-Check

Page 7.126 (S)

residual field violates the yield condition, i.e. if there is no stress above the design material strength parameter, then the shakedown load is equal to the limit load. Where the residual stress field of a structure does exceed the yield condition, the shakedown limit can be calculated easily from the stress plots because of the linearity.

Pressure (MPa)

The residual stress field Pressure-Moment Shakedown Locus from the elastic 1.8 compensation analysis using the Ilyushin 1.6 function is shown in 1.4 Figure 7.7.1-3. It can be P SH=1.34 MPa 1.2 seen that the maximum 1 residual Ilyushin function 0.8 is 0.388 which is less than 0.6 the yield condition of 1. 0.4 Therefore, there is no equivalent stress greater 0.2 M=15644.4 Nm than the design material 0 strength parameter and the 0 10000 20000 30000 40000 50000 shakedown load is equal Moment (Nm) to the limit load. As with Figure 7.7.1-4: Pressure-Moment Shakedown Locus the check against GPD, a shakedown locus can be created covering all of the pressure-moment combinations. Due to the linearity, the limit locus in Figure 7.7.1-1 can be scaled up to meet the PD rules: Mises' condition and no partial safety factor on the material strength. Figure 7.7.1-4 shows the shakedown pressuremoment locus. As there are no partial safety factors applied to the actions for the PD-check, the applied moment is 15644.4 Nm, giving a maximum shakedown pressure of PSmaxSD = 1.34 MPa. This load gives a maximum stress at the discontinuity of 1124 MPa, which is greater than the 2 RM placed on the shakedown load as defined in the code. The equivalent stress near to (but not on) the discontinuity is 628 MPa, which is also greater than the 2 RM-limit. The value of the shakedown load calculated by elastic compensation using the generalised yield criterion is non-conservative and may not be used to give a lower bound shakedown load. Therefore, the use of elastic compensation in this shakedown check is inconclusive.

5. Check on GPD Using Non-linear Analysis A check against GPD was also performed for the same FE-model using conventional non-linear analysis. In this way, a direct comparison may be made between the two limit approaches. The FEgeometry, mesh and boundary conditions are the same as those used in the elastic compensation analysis. Material non-linearities were applied corresponding to the material strength parameter, 188.45 MPa and pseudo-perfect plasticity. Limit analysis using shell elements can have problems with convergence, so a small value of plastic modulus is applied. The pressure is ramped and the analysis continues to converge to a pressure of 0.55 MPa. The maximum absolute value of the principal strain may not exceed 5%. The maximum absolute value of the principal strain occurs at

DBA Design by Analysis the intersection of the two cylinders. Due to the simplification in modelling with shells, the intersection of the two cylinders is not used for the strain evaluation. An evaluation cross-section is taken at the location of the weld toe for assessment of the strain (see subsection 3.3.1). At this position the load of 0.55 MPa exceeds the 5% limit. At a load of 0.455 MPa the value of maximum principal strain is within the 5% limit, Figure 7.7.1-5. With a constant moment of 21120 Nm, the maximum allowable load according to the GPD-check using non-linear analysis is given as 0.455 PS max GPD = = 0.38MPa γp Figure 7.7.1-6 shows the equivalent stress at the limit defined by the maximum allowable principal strain. The extent of the plasticity region covers the area immediately round the intersection of the two cylinders. The allowable load calculated using the non-linear analysis is considerably lower than that calculated using elastic compensation. The structure can still support further increase in load above the maximum principal strain limit of +/- 5%. Elastic compensation has no such limit applied to the analysis and gives a limit load based upon the point at which deformations become unconstrained.

Analysis Details Example 3.1 / GPD- & PD-Check

Page 7.127 (S)

Figure 7.7.1-5: Maximum Principal Strain at 5% limit

Figure 7.7.1-6: Equivalent Stress At Limit

Running on the same equipment as the elastic compensation analysis, the non-linear analysis required a CPU time of 4874 seconds.

6. Additional Comments As the geometry in this problem is outside the scope of DBF, the DBA calculations are a quick and simple alternative for this simple problem.

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Analysis Details Example 3.1 / GPD- & PD-Check

Page 7.128 (S)

Pressure (MPa)

The load given by the non-linear analysis is considerably lower than that given by elastic compensation. The reason is the limit on maximum principal strain. Elastic compensation has no limit on the maximum 0.6 principal strain and the limit load is not restricted 0.5 by this. If the tangent intersection method is 0.4 considered again and the limit load calculated with 0.3 no restriction on the maximum principal strain 0.2 (Figure 7.7.1-7) a limit pressure of 0.51 MPa is 0.1 found. Giving a maximum allowable pressure 0 according to the GPD0 10 20 30 40 50 Check of 0.425 MPa, a Displacement (mm) value closer to that calculated by elastic Figure 7.7.1-7: Pressure-Displacement Graph (no principal strain limit) compensation. Usage of the generalised yield function in elastic compensation for shells works successfully in performing a conventional limit analysis. When the code rules are considered and the restriction on plastic strain is applied, the limit is considerably reduced. The structure is still relatively stiff. Elastic compensation does not take account of this and will give a limit load based upon the true limit state of the structure when deformations become unbounded. This may be seen by considering the limit defining stress fields of each method. The elastic compensation limit field (Figure 7.7.1-2) shows a large plastic region covering most of the nozzle and extending far into the main cylinder. The non-linear limit field shows a considerably smaller extent of plasticity, an area restricted to the region immediate to the cylinder intersection. If the stress field for the non-linear analysis is Figure 7.7.1-8: Non-linear equivalent stress at P=0.51 MPa examined at the limit load given by elastic compensation, Figure 7.7.1-8, it can be seen that the extent of the plasticity region is similar, for the same load, to that for elastic compensation, Figure 7.7.1-2.

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Analysis Details

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Example 3.1 / GPD- & PD-Check

Analysis Type:

Member:

Direct Route using elasto-plastic FE calculations

FE-Software:

ANSYS® 5.4

Element Types:

Model: 8 – node structural shell SHELL93

A&AB

Submodell: 8 – node, 3 - D structural solid SOLID45 Boundary Conditions:

♣Model: • Symmetry b.c. in the symmetry plane in longitudinal direction of the shell; • vertical displacement in the middle-nodes of the flat ends of the shell constrained to 0; • longitudinal displacement in the middle node of the left flat end of the shell constrained to 0; • coupling of the rotational DOF about x-axis (see Figure) at the end of the nozzle. ♣Submodel: • Symmetry b.c. in the symmetry plane in longitudinal direction of the shell; • b.c. at the cut-boundaries according to submodelling.

Model and Mesh:

Model: Number of shell elements: 1324 Geometry: as in example specification Submodel: Number of solid elements: 13600 (4 in thickness direction) Geometry: see Analysis Details

Results: Max. allowable pressure acc. to GPD, for constant moment, PS max GPD = 0.39 MPa Shakedown pressure, for constant moment, PS max SD = 0.51 MPa

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Analysis Details Example 3.1 / GPD- & PD-Check

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1. Elements, mesh fineness, boundary conditions Principally, the model size and the boundary conditions used for modelling a shell-nozzle intersection strongly influence the calculation results; the influence increases with increasing ratio of nozzle diameter to shell diameter and with decreasing wall thickness. The example under consideration was used to investigate these influences. Some results are given briefly in the following. •

For a shell-nozzle intersection with an inner shell diameter of 496 mm, an inner nozzle diameter of 367 mm, and wall thicknesses of 4 mm, the linear-elastic behaviour of FE models with different shell lengths and with boundary conditions as applied in Example 3.2 of this project, was studied, with internal pressure action only. The maximum of Mises’ equivalent stress showed a strong dependence of the length of the shell – the difference was about 12 % for models with 1500 mm and 2000 mm total length of shell, respectively. Therefore, a restriction of the model size to minimise computation time is not admissible. The location of the maximum of Mises’ equivalent stress was the lowest point of the intersection curve (in the symmetry plane normal to the main shell axis).

•

Since the example was specified initially to compare results with fatigue tests, performed at WTCM-CRIF (a member of the project group), a model with a geometry exactly the same as used in the test was studied too. This model had a greater shell thickness near the ends, and flat ends at the shell and the nozzle (see specification of examples). Again, the linear-elastic behaviour of this model was studied under internal pressure action only: The maximum Mises equivalent stress was about half of the one of the model stated above (with the same total length of the shell). Moreover, the site of the maximum of the Mises equivalent stress was completely different – it was the saddle point of the intersection curve.

•

The limit pressure of this latter model, corresponding to the fatigue tests, was almost twice the one of the other model (the limit pressures were determined with the very same material strength parameter, Mises’ yield condition and constant nozzle moment).

As a straightforward conclusion: The knowledge of the real “boundary conditions” of a real shellnozzle intersection with such extreme diameter ratios and small wall thicknesses is a basic requirement for FE-calculations of practical significance. Therefore, the FE-model used for Example 3.1 had exactly the same geometric features as the one tested (which are the ones stated in the specification of the examples). Due to the large ratio of diameter to wall thickness of the structure, 8-node shell elements SHELL93 have been used in the check against GPD and for the “coarse model” of the shakedown check. To model the whole structure with solid elements using an appropriate number of elements in wall thickness direction would result in an FE-model with too many elements and nodes and, therefore, in an unacceptable long computation time. To evaluate the 5% principal strain limit in the check against GPD in the so called “evaluation” cross – section (see chapter 1 of the Procedure Guide) exactly, 8 rows of smaller elements corresponding to the weld geometry were located at the intersection curve – see Figure 7.7.2-1.

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Analysis Details Example 3.1 / GPD- & PD-Check

Page 7.131 (S)

Because of the symmetry of the structure and the actions, only the half structure was modelled. Therefore, the moment applied to the model was half of the specified one. In the plane of the symmetry a symmetry boundary condition was applied to all nodes. To apply the moment on the top of the nozzle properly, the rotational degrees of freedom about the axis normal to the plane of symmetry of all nodes of the flat end of the nozzle were coupled. Furthermore, the middle node of the left flat end of the shell was supported in vertical and horizontal direction and the middle node of the right flat end of the shell was supported in vertical direction. Figure 7.7.2-1

The shakedown check was performed by submodel technique, and, for this purpose, the coarse model was identical with the shell model described above. The submodel consisted of 13600 8-node brick solid elements SOLID45. To obtain proper stress results, 4 elements were arranged over the wall thickness. The submodel had a total length of 440 mm, the highest cross-section of the nozzle had a distance of 280 mm to the centreline of the shell, and the angle between the lower cutboundary of the shell and the x-axis was 30°. To avoid stress concentrations, the weld surface was modelled with a fillet of 4 mm radius and the inner edge of the intersection was modelled with a fillet of 2mm radius. In the plane of symmetry a symmetry boundary condition was applied to all nodes, the boundary conditions of the cut-boundaries were interpolated by the software from the corresponding displacements of the coarse model. 2. Determination of the maximum allowable pressure according to GPD Since the subroutine for Tresca's yield condition showed bad convergence, the check against GPD was carried out using Mises' yield condition (see subsection 3.2 of section 3 - Procedure) only. Because of the different partial safety factors for the constant moment in the GPD- and the PDchecks, the results from the PD-check cannot be used, and a separate calculation had to be carried out for that particular purpose. Because of the fact, that the moment is specified as being constant, the limit pressure was determined for constant moment and increase of pressure only, and not for proportional increase of all actions as required in prEN 13445-3 Annex B.9.2.1. The partial safety factor γ G for permanent actions (with an unfavourable effect) is 1.35. Therefore, the moment to be applied in the FE-model is given by 15644.4 = 10560 Nm. 2 The analysis was carried out with a linear-elastic ideal-plastic material law, Mises’ yield condition, a design material strength parameter of 188.45 MPa for the shell and the nozzle (corresponding to a material strength parameter of 272 MPa according to EN 10028-2) , associated flow rule, and first M = 1.35 ⋅

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Analysis Details Example 3.1 / GPD- & PD-Check

Page 7.132 (S)

order theory. The analysis was performed in two parts: In the first, the moment load was applied; in the second, internal pressure was applied additionally, and increased until a state near the limit state was reached. The elastic modulus used in the calculations was E = 210.125 GPa . To restrict computation time to reasonable values, the analysis was terminated after 7 hours on the Compaq® Professional Workstation 5000 with two Pentium Pro® processors and 256 MB RAM. At termination time the last convergent solution showed an internal pressure of 0.645 MPa . Since the maximum absolute value of the principal strain – located in the “evaluation” crosssection – exceeded 5% at the last load level – the absolute principal strain value was 17% - a lower load value with appropriate strains had to be used as limit value. For a value of the internal pressure of 0.472 MPa the maximum absolute value of the principal strains was less than 5%. Figure 7.7.2-2 shows the corresponding distribution of the Mises equivalent stress, and Figure 7.7.2-3 the corresponding distribution of the maximum absolute principal strain.

Figure 7.7.2-2

According to prEN 13445-3 Annex B, Table B.9-2, the partial safety factor for pressure action (without natural limit) γ P is 1.2. Thus, the maximum allowable pressure according to the GPDcheck is given by PS max GPD =

0.472 = 0.393 MPa. 1.2 Figure 7.7.2-3

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Analysis Details Example 3.1 / GPD- & PD-Check

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3. Check against PD The following procedure corresponds to the one given in subsection 3.3.2.6 of section 3 – Procedures. The load case with a constant moment M = 15644.4 / 2 = 7822.2 Nm , as in the specification, and an internal pressure of p = 1.2 MPa was found to be relatively close to the limit state of the structure, and, therefore, will be called the “limit state” in the following. Since there is no possibility within the submodel technique to calculate different load steps during one analysis, separate calculations for the coarse model and the submodel were necessary to obtain the required stress states: Internal pressure and linear-elastic calculation; moment only and linearelastic calculation; moment only and elasto-plastic calculation; limit state and elasto-plastic calculation. The computation times were • 2 minutes for the coarse model and 5 minutes for the submodel for the load case internal pressure and linear-elastic calculation, • 2 minutes for the coarse model and 5 minutes for the submodel for the load case moment only and linear-elastic calculation, • 20 minutes for the coarse model and 55 minutes for the submodel for the load case moment only and elastic-plastic calculation, • 1hour and 3 minutes for the coarse model and 3hours and 12 minutes for the submodel for the load case limit state and elastic-plastic calculation, on the Compaq® Professional Workstation 5000 with two Pentium Pro® processors and 256 MB RAM. The elasto-plastic FE-analyses were carried out as stated in prEN 13445-3 Annex B, Sec. B.9.3.1 - using Mises’ yield condition and associated flow rule, a linear-elastic ideal-plastic law with a design material strength parameter of 272 MPa for shell and nozzle, and first order theory. A linear combination of self equilibrating stress fields – the one according to the limit state (σ ij ) res ,( M + p ) and the one according to moment action only (σ ij ) res , M – was used to fulfil Melan’s theorem. For the determination of the factors β 1 , β 2 and α , the deviatoric mappings of the stress states, i.e. the coordinates of a stress point given by its principal stresses, at the critical locations of the structure were used, since, due to the increased number of factors and the different critical locations, load case operations using the FE-software are not feasible. The two critical locations of the structure are: The inner surface of the shell-nozzle junction (path I in Figure 7.7.2-4, the fillet is excluded), where the maximum linear-elastic calculated stress arises in the case of internal pressure only (location on the symmetry plane of the structure -see Figure 7.7.2-4), and the intersection curve of the shell and the nozzle on the outer surface of the structure (path A in Figure 7.7.2-5), where the maximum linear-elastic calculated stress arises in the case of moment load only (see Figure 7.7.2-5). As can be seen from the following deviatoric mappings, it is necessary to consider the whole intersection curve from point X to point Y, a consideration only of the point where the maximum stress arises would result in non-conservative results. The self-equilibrating stress field according to the moment (σ ij ) res , M is equal to 0 at the inner surface (paths I). Therefore, the necessary conditions (see also subsection 3.3.2.6 of section 3 – Procedures) are given by

DBA Design by Analysis

Analysis Details Example 3.1 / GPD- & PD-Check

φ [ β 1 ⋅ (σ ij ) res , ( M + p ) + (σ ij ) M ,le + α ⋅ (σ ij ) p ,le ] ≤ 0 φ [ β 1 ⋅ (σ ij ) res ,( M + p ) + (σ ij ) M ,le ] ≤ 0 .

The optimal scaling factors β 1 and α can be determined from these two conditions – for the equality sign: β1 = 0 , α = 0.858 ; The first result β 1 = 0 shows that the self-equilibrating stress field according to the limit state (σ ij ) res ,( M + p ) cannot be used, because of the distribution of the linear-elastic calculated stresses due to the moment action only at path I. Therefore the maximum admissible internal pressure (at path I) is given by the condition that the sum of the two linear-elastic stress distributions are completely inside of the limit circle in the Figure 7.7.2-4 deviatoric map, which renders the value of α for path I. This behaviour can be visualised directly in Figure 7.7.2-6, which shows the stress distribution in the deviatoric map for the part of path I located at the shell, where the maximum stresses arise: •

linear-elastic stress according to moment and pressure action (σ ij ) le ,( M + p ) – thin green line,

•

elasto-plastic stress according to moment and pressure action (σ ij ) ep ,( M + p ) – thick green line,

•

linear-elastic stress according to moment action only (σ ij ) le, M – orange line,

•

self-equilibrating stress field according to moment and pressure action (σ ij ) res ,( M + p ) – violet line.

Figure 7.7.2-5

Page 7.134 (S)

DBA Design by Analysis

Analysis Details Example 3.1 / GPD- & PD-Check

Page 7.135 (S)

Path I, shell

Figure 7.7.2-6

For the points at the outer surface (path A) the necessary conditions are now given by φ [ β 2 ⋅ (σ ij ) res , M + (σ ij ) M ,le + α ⋅ (σ ij ) p,le ] ≤ 0 , φ [ β 2 ⋅ (σ ij ) res , M + (σ ij ) M ,le ] ≤ 0 . Figure 7.7.2-7 shows the stress distribution in the deviatoric map for path A: •

linear-elastic stress according to pressure load 0.858 ⋅ (σ ij ) le, p – thin green line,

•

linear-elastic stress according to moment load only (σ ij ) le, M – thin orange line,

•

elasto-plastic stress according to moment load (σ ij ) ep, M – thick orange line,

•

self-equilibrating stress according to moment load only (σ ij ) res , M – thin violet line.

It can easily be seen, that the critical point at the outer surface is point X, because of its large value of linear-elastic stress due to the pressure load, and not the point with the maximum linear-elastic stress due to the moment load only. Furthermore, since in X the self-equilibrating stress according to moment load only is 0, the admissible pressure to fulfil Melan’s theorem is, for the chosen stress distribution, very low – i.e., the admissible value of α for path A is (independently from β 2 ) close to 0.

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Analysis Details

Page 7.136 (S)

Example 3.1 / GPD- & PD-Check

The conclusion is, that the used self-equilibrating stress field, according to moment only, is not an optimal one, and, therefore, another self-equilibrating stress field, according to a moment only, was used – the one which corresponds to a moment of M = 14632 Nm . This moment is scaled up from the moment in the specification with the factors γ R , γ G and 2 / 3 , and is , therefore, smaller than the limit moment of the structure. The reason for the usage of this moment is the plastification in point X in this loading case, and, therefore, a resulting non-zero self-equilibrating stress according to this moment in point X.

X

X Y

Y X=Y

Figure 7.7.2-7

Since elasto-plastic results cannot be scaled up, an additional FE-calculation with this larger moment had to be carried out. The corresponding computation times were 1 hour 12 minutes for the coarse model and 4 hours and 7 minutes for the submodel. The new self-equilibrating stress field according to this larger moment M is again practically 0 at path I, and, therefore, does not change the admissibility conditions at this path. At path A now, a value of β 2 = 1 for the self-equilibrating stress field is used, and a new corresponding value of α for the linear-elastic stress due to pressure (which must be smaller than 0.858) is determined. This value was found to be 0.425 and, therefore, a lower bound shakedown pressure is given by PS max SD = α ⋅ p = 0.425 ⋅ 1.2 MPa = 0.51 MPa .

Analysis Details

DBA Design by Analysis

Page 7.137 (S)

Example 3.1 / GPD- & PD-Check

Figure 7.7.2-8 shows some corresponding stress distributions in the deviatoric map for path A: •

linear-elastic stress according to the large moment (σ ij ) le , M – thin orange line,

•

elasto-plastic stress according to large moment (σ ij ) ep , M – thick orange line,

•

self-equilibrating stress according to the large moment (σ ij ) res , M – thin violet line,

•

sum of the self-equilibrating stress according to the large moment (σ ij ) res , M and the linear-elastic stress according to the specified moment (σ ij ) le, M – thin yellow line,

•

sum of the self-equilibrating stress according to the large moment (σ ij ) res , M and the linear-elastic stress according to the specified moment (σ ij ) le, M and the linear-elastic stress field according to the pressure 0.51 ⋅ (σ ij ) le, p - thin red line.

X

X Y

Y

Y X

Figure 7.7.2-8

Note: It is recommended to check always the validity of used ( or obtained) self-equilibrating stress fields, determined by the procedure described above, in the postprocessor of the FE-software - by superposition with the linear-elastic stress fields according to the load cases under consideration.

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Analysis Details

Page 7.138 (S)

Example 3.1 / GPD- & PD-Check

Analysis type: Stress Categorization Route

Member:

WTCM Materials and Properties: Material strength parameter Rp0.2/t = 272 MPa Modulus of elasticity E = 210125 MPa. FE- Software:

ALGOR.

Element types:

19116 8-node brick elements.

Boundary conditions:

yz-plane (longitudinal plane of symmetry) fixed in x (Tx), the shell ends fixed in z, one end in y (Ty and Tz), rotations free.

Model and mesh:

Maximum admissible internal pressure according to the Stress Categorization Route: Internal pressure = 0.45 MPa for a constant moment Mc = 15644.4 Nm.

DBA Design by Analysis

Analysis Details Example 3.1 / GPD- & PD-Check

Page 7.139 (S)

Three classification lines (CL) are to be considered: - CL 1: through point 1 with coordinates (200.218 , 20.7107 , 146.301) and point 2 with coordinates (202.649 , 22.5705 , 149.674). The maximum Tresca stress occurs in point 1 and is 309.87 MPa for a combination of longitudinal moment (15644.4 Nm) and internal pressure (0.28 MPa). - CL 2: through point 3 with coordinates (198.258 , 0 , 148.788) and point 4 with coordinates (200.694 , 0 , 152.153). The maximum Tresca stress occurs in point 3 and is 270.81 MPa for a longitudinal moment (15644.4 Nm) only. - CL 3: through point 5 with coordinates (16.7046 , 191.002 , 247.433) and point 6 with coordinates (15.8173 , 190.6 , 251.379). The maximum Tresca stress occurs in point 5 and is 603.52 MPa for an internal pressure of (1.28 MPa) only.

In the figure above the locations of the classification lines are shown. The CL’s are situated at the nozzle-cylinder intersection. Three CL’s are considered : CL2 at the location of highest stress for a longitudinal moment loading only, which is situated in the transversal plane of symmetry. CL3 at the location of highest stress for internal pressure loading only, situated near the longitudinal plane of symmetry (2 or 3 degrees from this longitudinal plane) and CL1, at the location of the highest stress, for a combination of longitudinal moment and internal pressure

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Analysis Details Example 3.1 / GPD- & PD-Check

Page 7.140 (S)

For the stress classification, the following procedure was followed: - for each load acting on the vessel, calculate the elementary stresses Φij (i,j = 1,2,3) in the different points on the different CL’s. - for each load acting on the vessel and along each CL, calculate the membrane stress components Φij,m and the bending stress components Φij,b. - Classify the membrane stress components Φij,m in (Φij)Pm, (Φij)PL or (Φij)Qm and the bending stress components Φij,b in (Φij)Pb or (Φij)Qb. - Calculate the sum of the stresses classified in this way for the set of loads acting simultaneously on the vessel. The stresses resulting from this summation are designated (Γij)Pm, (Γij)PL, (Γij)Pb, (Γij)Qm, (Γij)Qb. - From this deduce: (Γij)Pm, (Γij)PL, (Γij)P, (Γij)P+Q. - Calculate the following equivalent stresses: (Φeq)Pm or (Φeq)PL, (Φeq)P, (Φeq)P+Q. According to table 19C-2, the following classification must be used: for internal pressure: PL and Qb, for a local load: PL and Pb. - Verify the admissibility of the equivalent stresses. The results of this procedure are given on the next pages. Two loads are considered, which act simultaneously: an internal pressure (initial value = 1.28 MPa) and a longitudinal moment of 15644.4 Nm. The stresses are calculated for those two individual actions, and a stress classification along the CL’s 1, 2 and 3 is applied. The results of the calculations are shown in the next tables.

Analysis Details

DBA Design by Analysis

Example 3.1 / GPD- & PD-Check

Page 7.141 (S)

1. Pressure (1.28 MPa) a. Along CL3

pressure = 0.45 MPa pressure = 1.28 MPa pressure = 0.45 MPa 6

5

6

(σij)m

382,1

477,4

134,3

167,8

-203,9

227,7

-71,7

80,1

27,2

59,3

9,6

S12

-53,8

-24

S23

-7,1

S31

-25,5

5 S11 S22 S33

(σij)b

(σij)m

429,75

-47,65

151,1

-16,8

11,9

-215,8

4,2

-75,9

20,8

43,25

-16,05

15,2

-5,6

-18,9

-8,4

-38,9

-14,9

-13,7

-5,2

-45,1

-2,5

-15,9

-26,1

19

-9,2

6,7

-29,3

-9,0

-10,3

-27,4

1,9

-9,6

0,7

b.Along CL2

S11

(σij)b

pressure = 0.45 MPa pressure = 1.28 MPa pressure = 0.45 MPa 3

4

5

6

(σij)m

(σij)b

(σij)m

(σij)b

-23,2

20,7

-8,2

7,3

-1,25

-21,95

-0,4

-7,7

S22

-139

12,5

-48,9

4,4

-63,25

-75,75

-22,2

-26,6

S33

-48,3

22,4

-17,0

7,9

-12,95

-35,35

-4,6

-12,4

S12

-3,1

-3,4

-1,1

-1,2

-3,25

0,15

-1,1

0,1

S23

4,3

3,8

1,5

1,3

4,05

0,25

1,4

0,1

S31

33,8

-22,1

11,9

-7,8

5,85

27,95

2,1

9,8

a. Along CL1

pressure = 0.45 MPa pressure = 1.28 MPa pressure = 0.45 MPa 1

2

5

6

(σij)m

(σij)b

(σij)m

(σij)b

S11

-23,5

35,1

-8,3

12,3

5,8

-29,3

2,0

-10,3

S22

-140,1

53,6

-49,3

18,8

-43,25

-96,85

-15,2

-34,0 -19,1

S33

-44,1

64,5

-15,5

22,7

10,2

-54,3

3,6

S12

-28,6

-31,3

-10,1

-11,0

-29,95

1,35

-10,5

0,5

S23

43,7

32,7

15,4

11,5

38,2

5,5

13,4

1,9

S31

25

-39,6

8,8

-13,9

-7,3

32,3

-2,6

11,4

The FE-calculations for internal pressure only are performed with a pressure of 1.28 MPa. Since this is a linear elastic analysis, one can calculate the stresses for all other pressures (e.g. 0.45 MPa) from this FE-calculation.

Analysis Details

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Example 3.1 / GPD- & PD-Check

Page 7.142 (S)

2. Longitudinal moment (15644.4 Nm) a. Along CL3 5

6

(σij)m

(σij)b

S11

61.4

142.4

101.9

-40.5

S22

-127.5

111.2

-8.15

-119.35

S33

11.8

29

20.4

-8.6

S12

-18.6

-1.4

-10

-8.6

S23

-1

-19.6

-10.3

9.3

S31

-6

-9

-7.5

1.5

3

4

(σij)m

(σij)b

S11

0

0

0

0

S22

0

0

0

0

S33

0

0

0

0

S12

-85.8

-23.1

-54.45

-31.35

S23

104.7

31.1

67.9

36.8

S31

0

0

0

0

b. Along CL2

a. Along CL1 1

2

(σij)m

(σij)b

S11

25.5

-12.3

6.6

18.9

S22

23.1

-27.2

-2.05

25.15

S33

63.2

-4.1

29.55

33.65

S12

-79.2

-22.5

-50.85

-28.35

S23

96.7

26.6

61.65

35.05

S31

-30.6

15.8

-7.4

-23.2

The FE-calculations for longitudinal moment only are performed for the given moment of 15644.4 Nm.

Analysis Details

DBA Design by Analysis

Example 3.1 / GPD- & PD-Check

Page 7.143 (S)

3. Pressure (0.45 MPa) + longitudinal moment (15644.4 Nm) a. Along CL3 5

6

(Σij)PL

(Σij)Qb

(Σij)P+Q

(Σij,PL)1

S11

195.7

310.2

253.0

-57.3

310.2

(Σij,PL)2

S22

-199.2

191.3

-4.0

-195.2

-199.2

(Σij,PL)3

256.23 (Σij,P+Q)1 314.006 54 43.382 (Σij,P+Q)2 -15.011 (Σij,P+Q)3 -207.23

S33

21.4

49.8

35.6

-14.2

49.8

S12

-37.5

-9.8

-23.7

-13.8

-37.5

(σeq)Pl

271.241 MPa

S23

-3.5

-35.5

-19.5

16.0

-35.5

S31

-15.0

-19.3

-17.1

2.2

-19.3

3

4

(Σij)PL

(Σij)Qb

(Σij)P+Q

(Σij,PL)1

S11

-8.2

7.3

-0.4

-7.7

-8.2

(Σij,PL)2

S22

-48.9

4.4

-22.2

-26.6

-48.9

(Σij,PL)3

75.678 (Σij,P+Q)1 100.739 (Σij,P+Q)2 -0.075 0 (Σ -102.88 ij,P+Q)3 -174.76

S33

-17.0

7.9

-4.6

-12.4

-17.0

S12

-86.9

-24.3

-55.6

-31.3

-86.9

(σeq)Pl

178.554 MPa

S23

106.2

32.4

69.3

36.9

106.2

S31

11.9

-7.8

2.1

9.8

11.9

1

2

(Σij)PL

(Σij)Qb

(Σij)P+Q

S11

17.2

0.0

8.6

8.6

17.2

S22

-26.2

-8.4

-17.3

-8.9

-26.2

(σeq)P+Q 521.236 MPa

b.Along CL2

(σeq)P+Q 275.503 MPa

a. Along CL1

S33

47.7

18.6

33.1

14.6

47.7

S12

-89.3

-33.5

-61.4

-27.9

-89.3

S23

112.1

38.1

75.1

37.0

112.1

S31

-21.8

1.9

-10.0

-11.8

-21.8

(Σij,PL)1 108.744 (Σij,P+Q)1 165.111 (Σij,PL)2 8.359 (Σij,P+Q)2 7.641 (Σij,PL)3 -92.702 (Σij,P+Q)3 -134.05 (σeq)Pl

201.446 MPa

(σeq)P+Q 299.163 MPa

The calculated equivalent stresses must meet the assessment criteria: (σeq)Pl

<

1.5f

(σeq)Pl+Qb

<

3f

The highest equivalent stresses are reached on CL3. These stresses are used to meet the assessment criteria. With f = Rp0.2/t / 1.5 = 272 / 1.5 MPa = 181.3 MPa, the assessment criteria are met for an internal pressure of 0.45 MPa and a longitudinal moment of 15664.4 Nm.

DBA Design by Analysis

Analysis Details Example 3.1 / F-Check

Analysis Type:

Page 7.144 (C) Member:

F-Check (Equivalent structural stress approach)

CETIM

FE –Software: ABAQUS / Standard version 5.8.1 Element Types: Quadratic shell rectangular 8-node and triangular 6-node elements . 13247 nodes and 3362 elements Boundary Conditions: The model takes into account a plane of symmetry in the axial direction. In this plane, the perpendicular displacement to this plane and the rotations in this plane are locked. One end of the horizontal shell is locked in the vertical and the axial direction. The other end is locked in the vertical direction. Model and Mesh:

Results: Fatigue life N = 122770 cycles (for cyclic pressure) 1042 cycles (for cyclic moment)

DBA Design by Analysis

Data Cyclic pressure tmax = 50 °C tmin = 50 °C t* = 0,75 tmax + 0,25 tmin = 50 °C Rm = 460 MPa Rp0,2/t* = 272 MPa

Analysis Details Example 3.1 / F-Check

en = 4 mm ∆σD (5.106cycles) = 52 MPa (class 71) equivalent stresses or m = 3 C = 7,16.1011

Page 7.145 (C)

principal stresses m = 3 C⊥ = ….. C// = ….. m = 5 C⊥ = ….. C// = …..

m = 5 C = 1,96.1015

Stresses ∆σstruc = 180 MPa (structural equivalent stress range, determined by extrapolation) Note: As the maximum stresses are obtained in the weld, it is not necessary to combine the constant loading with the varying loading.

18.8 Plasticity correction factor ke Thermal loading

mechanical loading

If ∆σstruc > 2 Rp0,2/t*

If ∆σstruc > 2 Rp0,2/t* k

 Äó  struc = 1+ A0  − 1 e  2 R p0,2/t *  

with A0 =

k



0,5 for 800 MPa ≤ Rm ≤ 1000 MPa 0,4 for Rm ≤ 500 MPa  R − 500  0,4 +  m for 500 MPa ≤ R m ≤ 800 MPa  3000 

A0 = ….. ke = ….. ∆σ = ke ∆σ struc = ….. MPa Else ∆σ = ∆ σstruc = 180 MPa

υ

= 0,5 +

0,7 0,4

 Äó   struc   R p0,2/t * 

kυ = ….. ∆σ = kυ ∆σ struc = ….. MPa Else ∆σ = ∆σstruc = ….. MPa

If both mechanical and thermal loadings are to be considered, the correction has to be made on each component of the stress tensors . ke and k í are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated. 18-10-6-1 Thickness correction factor few en ≤ 25 mm few = 1

25 mm ≤ en ≤ 150 mm few = (25/en)0,25= few = …..

en ≥ 150 mm few = 0,639

DBA Design by Analysis

Analysis Details

Page 7.146 (C)

Example 3.1 / F-Check

18-10-6-2 Temperature correction factor ft* For t* > 100 °C Else ft* = 1

ft* = 1,03 – 1,5 10-4 t* -1,5 10-6 t*2 = …..

18-10-6-3 Overall correction factor fw fw = few.ft* = 1 18-10-7 Allowable number of cycles N Äó fw

If

= 180 MPa

Äó fw

N = ∞ if

> ∆σ5.106 cycles then

m = 3 and C (C⊥ or C//) = 7,16 1011

N=

C  ∆σ   fw

  

m

= 122770 cycles

Äó fw

< ∆σ108, else:

If

Äó fw

< ∆σ5.106 cycles and other

cycles with

Äó fw

> ∆σ5.106 cycles

then m = 5 and C (C⊥ or C//) = ….. N = C m = …..  ∆σ   f   w 

If

Äó fw

< ∆σ5.106 cycles and all other

cycles with then N=∞

Äó fw

< ∆σ5.106 cycles

DBA Design by Analysis

Data Cyclic moment tmax = 50 °C tmin = 50 °C t* = 0,75 tmax + 0,25 tmin = 50 °C Rm = 460 MPa Rp0,2/t* = 272 MPa

Analysis Details Example 3.1 / F-Check

en = 4 mm ∆σD (5.106cycles) = 52 MPa (class 71) equivalent stresses or 11 . m = 3 C = 7,16 10

Page 7.147 (C)

principal stresses m = 3 C⊥ = ….. C// = ….. m = 5 C⊥ = ….. C// = …..

m = 5 C = 1,96.1015 Stresses ∆σstruc = 761 MPa (structural equivalent stress range, determined by extrapolation)

Note: As the maximum stresses are obtained in the weld, it is not necessary to combine the constant loading with the varying loading.

18.8 Plasticity correction factor ke Thermal loading

mechanical loading

If ∆σstruc > 2 Rp0,2/t*

If ∆σstruc > 2 Rp0,2/t* k

 Äó  struc = 1+ A0  − 1 e  2 R p0,2/t *  

with A0 =

k



0,5 for 800 MPa ≤ Rm ≤ 1000 MPa 0,4 for Rm ≤ 500 MPa  R − 500  0,4 +  m for 500 MPa ≤ R m ≤ 800 MPa  3000 

A0 = 0,4 ke = 1,1596 ∆σ = ke ∆σstruc = 882,4 MPa Else ∆σ = ∆σstruc = …..

υ

= 0,5 +

0,7 0,4

 Äó   struc   R p0,2/t * 

kυ = 1,0887 ∆σ = kυ ∆σstruc = Else ∆σ = ∆σstruc = ….. MPa

If both mechanical and thermal loadings are to be considered, the correction has to be made on each component of the stress tensors . ke and k í are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated. 18-10-6-1 Thickness correction factor few en ≤ 25 mm few = 1

25 mm ≤ en ≤ 150 mm few = (25/en)0,25= few = …..

en ≥ 150 mm few = 0,639

DBA Design by Analysis

Analysis Details

Page 7.148 (C)

Example 3.1 / F- Check

18-10-6-2 Temperature correction factor ft* For t* > 100 °C Else ft* = 1

ft* = = 1,03 – 1,5 10-4 t* -1,5 10-6 t*2 = …..

18-10-6-3 Overall correction factor fw fw = few.ft* = 1 18-10-7 Allowable number of cycles N Äó fw

If

= 882,4 MPa Äó fw

N = ∞ if

> ∆σ5.106 cycles then

M = 3 and C (C⊥ or C//) = 7,16.1011

N=

C  ∆σ   f   w 

m

= 1042 cycles

Äó fw

< ∆σ108, else:

If

Äó fw

< ∆σ5.106 cycles with other

cycles where

Äó fw

> ∆σ5.106 cycles

then M = 5 and C (C⊥ or C//) = ….. N = C m = ….. cycles  ∆σ   f   w 

If

Äó fw

< ∆σ5.106 cycles with all other

cycles where then N=∞

Äó fw

< ∆σ5.106 cycles

DBA Design by Analysis

Analysis Details

Page 7.149 (S)

Example 3.2

Member:

Analysis Type: Direct Route using Elastic Compensation

Strathclyde

FE-Software:

ANSYS 5.4

Element Types:

8 – node, 3-D isoparametric solids.

Boundary Conditions:

Symmetry boundary conditions in the plane along the longitudinal direction of the shell. Hoop displacements in nodes at both ends of the shell constrained to zero. Longitudinal displacements in the nodes at one end of the shell constrained to zero, longitudinal displacements coupled in the nodes at the other end of the shell (plane sections remain plane). A rigid region is set up on the top surface nodes of the nozzle to apply the bending moment.

Model and Mesh:

Length of shell – 500 mm Length of nozzle – 80mm Number of elements - 1307

Results: Maximum admissible action according to the GPD-Check: Internal pressure PSmax GPD = 11.5 MPa (for the constant moment of M c = 711.1 Nm ) Check against PD: Shakedown limit pressure PSmax SD = 13.7 MPa

DBA Design by Analysis

Analysis Details Example 3.2

Page 7.150 (S)

1. Finite Element Mesh The geometry model was constructed according to the problem specification for example 3.2. Element divisions for the finite element mesh were defined parametrically to allow modification of the element density. Half symmetry was used with finite element models created using low order 8node 3-D isoparametric solid elements. The maximum allowable pressure according to GPD and the shakedown pressure according to PD were calculated for this model. Boundary conditions applied to the model are symmetry about the xz plane, therefore the applied moment is half the specified value. The nodes at both ends of the main cylinder are constrained in the hoop direction but are allowed to move radially. The nodes at one end of the main cylinder were constrained fully in the longitudinal direction with the nodes at the other end coupled in the longitudinal direction, thus maintaining plane sections remaining plane. Pressure was applied over the whole internal structure with the equivalent thrust that would be produced from closed ends applied at the longitudinally coupled degree of freedom end of the cylinder. The moment applied to the intersecting nozzle was modelled by creating a rigid region over the top edge of the nozzle via constraint equations. A mass element was created in the centre at the top of the nozzle (to act as the master node to which the moment is applied) with the nodes on the top edge of the nozzle (slave nodes) connected to it by constraint equations. The moment applied to the master node would then be transmitted by the rigid constraint region evenly to the slave nodes on the nozzle. Resulting in the application of a pure moment to the top surface of the nozzle with no warping, the cross-section remains plane. 2. Material properties: -

-

Shell (P265GH): material strength parameter RM = 234 MPa , modulus of elasticity E = 210,125 GPa. Nozzle (11CrMo9-10): material strength parameter RM = 343MPa, modulus of elasticity E = 210,125 GPa.

3. Determination of the maximum admissible pressure according to the GPD-Check Using the application rule in prEN-13445-3 Annex B.9.2.2 to check against GPD, the principle is fulfilled when for any load case the combination of the design actions do not exceed the design resistance. This may be shown by calculating the limit load. The limit load has to be determined using Tresca‘s yield condition and associated flow rule. As elastic compensation is based upon a series of elastic equilibrium stress fields, it is a relatively simple procedure to derive a lower bound limit load direct from the Tresca yield model. From prEN 13445-3 Annex B, Table B.9-3 the partial safety factor, γR on the resistance is 1.25. Therefore, the design material strength parameter given by RM/γR for the shell and nozzle is 187.2 MPa and 274.4 MPa, respectively. The analysis was carried out using the elastic compensation method conforming to the direct route rules for GPD in Annex B: linear elastic - ideal plastic material law, Tresca’s yield condition and associated flow rule and first order theory. In each elastic compensation iteration an equilibrium stress field is produced where the elastic modulus of each subsequent iteration is defined by the previous elastic solution. In this way regions of the FE-model may be systematically stiffened or weakened depending upon the stress magnitudes in the previous solution. The result is that every equilibrium stress field is a lower bound of the limit load, the stress field giving the greatest limit load is taken as the closest to the actual limit state and defines

Analysis Details

DBA Design by Analysis

Example 3.2

Page 7.151 (S)

the limit load in the analysis. In problems such as this, where there are materials with different properties, the modulus modification has a modified procedure that takes account of the different material properties. This modified method calculates the limit pressure for each component with a different material, allowing the component giving the lowest limit load to define the limit for the whole model. Convergence of the equilibrium stress field will occur typically between 8 and 15 iterations, i.e. no further Pressure-Moment Limit Locus benefit in the limit load will be noted with subsequent 16 iterations. 14

Pressure (Mpa)

In the check against GPD, 12 APd = 12 MPa the loading considered is 10 constant moment and 8 increasing pressure, i.e. the 6 limit pressure is to be calculated. As elastic 4 compensation is based upon 2 AMd= 960 Nm linear-elastic analysis it is 0 not possible to calculate the 0 200 400 600 800 1000 1200 1400 limit load directly, where the Moment (Nm) load set contains two Figure 7.8.1-1: Pressure Moment Limit Locus separate loads with one constant. The elastic compensation procedure will scale the combined load stress field therefore both sets of loads are scaled. However, by carrying out a series of elastic compensation analyses, where the ratio of the load sets are altered for each analysis, a limit locus can be constructed. The limit locus will then describe the limit load for any ratio of the loads within the load set. The total computing time to run each analysis on a 300 MHz Pentium two processor with 128 Mb RAM Windows NT workstation was approximately 458 seconds. The stress fields were shown to converge after eight iterations, fourteen separate analyses were carried out to describe the limit locus. Figure 7.8.1-1 shows the pressure moment limit locus from which the design pressure is found. The limit locus follows an unusual pan, this may be a result of the mixed material model. When the analysis results are studied, it can be shown that the failing component moves from the shell when the pressure is predominant to the nozzle when the moment is predominant. Resulting in a transition in the limit defining Figure 7.8.1-2: Tresca Limit Stress Field locus from that defining the limit of the shell to the locus defining the limit of the nozzle.

Analysis Details

DBA Design by Analysis

Example 3.2

Page 7.152 (S)

According to prEN-13445-3 Annex B, Table B.9-2 for permanent actions with an unfavourable effect (this covers the moment) the partial safety factor γG is 1.35. Therefore, the design value for the moment action, AMd, is the allowable moment (defined in the specification) multiplied by γG, giving AMd=960 Nm. From Figure 7.8.1-1, for a design moment action of 960 Nm the limit locus gives a design value for the pressure action, APd, of 12 MPa. For a pressure without natural limit the partial safety factor, γp, given by Table B.9-2 is 1.2. The allowable pressure is given as By carrying out a final elastic compensation analysis using the calculated value of design pressure, a check on the result can be made. Figure 7.8.1-2 shows the limit stress intensity field based on 12 PS max GPD = = 10 MPa γp

Pressure (MPa)

Tresca's criterion at the design loads. The maximum stress intensity in the shell is 188 MPa, which is approximately equal to the material strength parameter for the shell. The maximum stress in the nozzle is 228 MPa, which is lower than the design strength parameter for the nozzle. Clearly, the shell is the limit defining component with the stress in the nozzle still some way below its material strength parameter. Therefore, Mises Limit Locus for the given moment, the allowable pressure loading of 10 25 MPa fulfils the check against 20 GPD. P = 19.9 MPa L

The maximum pressure according 15 to GPD can also be calculated from the limit load results of the 10 PD check using Mises' condition, 5 where a lower bound on the A = 960 Nm Tresca limit load is given by 0 applying a factor of √3/2 to the 0 200 400 600 800 1000 1200 1400 1600 1800 yield strength. Elastic Moment (Nm) compensation based on Mises' Figure 7.8.1-3: Mises Limit Locus condition was carried out as described in the PD-check, the resulting limit locus is as shown in Figure7.8.1-3. The calculated limit pressure for a constant design moment action of 960 Nm is PL = 19.9 MPa. The maximum allowable pressure according to the GPD-check is obtained by applying the partial safety factors for the material and pressure (as before) along with the Mises correction: Md

PS max GPD =

19.9 3 19.9 3 ⋅ = ⋅ = 11.5 MPa γ p ⋅γ R 2 1.2 ⋅ 1.25 2

The value for maximum allowable pressure will be taken from the Mises analysis, analysis based on Mises' and Tresca's condition both give lower bounds on the limit load. Therefore, the maximum will be used in this analysis.

Analysis Details

DBA Design by Analysis

Example 3.2

Page 7.153 (S)

Similarly as with the Tresca locus the Mises limit locus shows the transition from failure occurring in the shell to failure occurring in the nozzle at large moments, hence the unusual shape of the Mises limit locus. Mises Shakedown Locus

4. Check against PD 18 16 14 Pressure (MPa)

In the check for progressive plastic deformation, the principle in prEN13445-3 Annex B.9.3.1 is fulfilled if the structure can be shown to shakedown.

12 10

Psh= 13.7 MPa

8

6 In elastic compensation the load at 4 which the structure will shakedown 2 M = 711.1 Nm is simple to calculate. Based on 0 Melan’s shakedown theorem, the 0 200 400 600 800 1000 1200 1400 1600 1800 self-equilibrating residual stress Moment (Nm) field that would result after a Figure 7.8.1-4: Mises Shakedown Locus loading cycle, can be calculated by subtracting the linear-elastic stress field at the limit pressure from the limit stress field. The residual stress field is in effect the resulting stress from an elastic unloading from the limit state back to zero pressure. If no stress in the residual field violates the yield condition, i.e. if there is no equivalent stress above the material (yield) parameter, then the shakedown load is equal to the limit load. Where the residual stress field of a structure does exceed the yield condition, the shakedown limit can be calculated easily from the stress plots because of the linearity of elastic compensation.

As with the check against GPD, a shakedown locus can be created covering all pressure-moment combinations. A second series of elastic compensation calculations are made based on Mises' condition. Design material strength parameters of 234 MPa and 343 MPa were used for shell and nozzle respectively. The resulting pressure-moment shakedown locus is shown in Figure 7.8.1-4. As there are no partial safety factors applied to the actions for the PD-check then the applied moment is 711.1 Nm, giving a maximum shakedown pressure PSmaxSD = 13.7 MPa. As with the limit loci the shakedown locus follows an unusual path. Here the shakedown limit was always defined by the shell, however the graph shows the effect the different material yield parameters have on the result. It may also be noted that when we consider the moment only the limit load and the shakedown load are similar, and as the pressure becomes a higher percentage of the combined loading the shakedown load becomes considerably less than the limit load.

5. Additional Comments Application of the DBA rules to this problem gives little benefit over the result obtained from DBF, in-fact the results given by elastic compensation are less than that obtained from DBF.

DBA Design by Analysis

Analysis Details Example 3.2

Page 7.154 (S)

Elastic compensation in this problem shows some unusual results. Maximum allowable pressure is greater when calculated from the corrected Mises PD limit load than that given by the Tresca calculation. By definition, this should not be possible, as the application of the correction should make the Mises calculated value a lower bound on the Tresca value. However, both calculations are lower bounds on the limit load and the pressure given from the Mises model defines the limit. An interesting observation in the limit loci of elastic compensation results is the transition in failure from one component to another. The non-linear analysis calculates a limit load slightly higher than that given by DBF and considerably higher than those given by elastic compensation. Elastic compensation in this problem is too conservative.

DBA Design by Analysis

Analysis Details

Page 7.155 (S)

Example 3.2

Member:

Analysis Type: Direct Route using elasto-plastic FE calculations

A&AB

FE-Software:

ANSYS® 5.4

Element Types:

8 – node (brick element), 3 – D structural solid SOLID45; 3 – D elastic beam BEAM4 (to apply the moment load).

Boundary Conditions:

• Symmetry boundary conditions in nodes in the symmetry plane in longitudinal direction of the shell; • hoop displacements in nodes at both ends of the shell constrained to 0; • longitudinal displacement in one node of one end of the shell constrained to 0; • rigid region concerning vertical displacements at the end of the nozzle to apply the moment; • vertical displacement set to 0 for dummy end of beam element.

Model and Mesh: Length of shell: 200 mm Length of nozzle: 100 mm Number of elements: 3281 (1 beam element)

Results: Max. allowable pressure acc. to GPD, for constant moment, PS max GPD = 14.79 MPa Shakedown pressure, for constant moment, PS max SD = 17.65 MPa

DBA Design by Analysis

Analysis Details Example 3.2

Page 7.156 (S)

1. Elements, mesh fineness, boundary conditions To model the structure, 3-D 8-node brick elements – SOLID45 in ANSYS® 5.4 – have been used. For comparison, a higher order version of this elements - the 20-node brick element SOLID95 with midside nodes – was used additionally. To show the differences for these two types of elements, linear-elastic and elasto-plastic calculations have been carried out, leading to the following results: • the linear-elastic peak stresses differ by about 5 to 8 % (for pressure as well as for moment action), the larger values were obtained by the model with the 20-node elements; • the elasto-plastic results near the limit action of the structure are almost identical, i.e. nearly independent of the kind of elements used; • the time for computing the results, using the Compaq® Professional Workstation 5000 with two Pentium Pro® processors and 256 MB RAM, applying an internal pressure of 26 MPa at constant moment (which is very close to the limit state of the structure) was 1 hour 15 minutes using the 8-node elements, and 63 hours 15 minutes using the 20-node elements. Because of the major difference in computation time and the fact that there is no need to compute linear-elastic peak stresses very exactly, because the check against PD can be carried out using the stress-concentration-free structure (according to prEN 13445-3 Annex B.9.3.2), the 8-node elements were used for all calculations. Because of the possibility of using a stress-concentration-free structure in the check against PD, the weld was modelled with a fillet (radius 6.5 mm) located entirely inside the weld. At the inner edge of the cylinder-cylinder intersection a fillet with 1mm radius was used. Note: Modelling details of a structure to avoid stress singularities or peak stresses influence linearelastic calculations, but do not affect results near the limit load of the structure. The nozzle of the considered structure is set-on. Therefore, using the material characteristics of the shell also for the welding zone leads to conservative results - the material of the nozzle is stronger than the material of the shell. Calculations with the moment action only showed that the region of the shell which is influenced by the moment is rather small – see Figure 7.8.2-1, which shows the Mises equivalent stress of the elasto-plastic calculation with moment load only. Therefore, FE-models with different lengths of the shell render almost identical results in the intersection region. Since the opening in the shell is small and the wall thicknesses are considerable, the results obtained for internal pressure are independent of the kind of boundary condition at the end of the nozzle (rigid region or free) and also of the length of the shell and the nozzle. Therefore, to save computation time, the total length of the model was restricted to 200 mm. The length of the set-on nozzle in the model was 100 mm (including the weld).

DBA Design by Analysis

Analysis Details Example 3.2

Page 7.157 (S)

To apply the moment load, the 3-D elastic beam BEAM4 of ANSYS® 5.4 was attached to the structure at the centre of the nozzle’s upper end. To transfer the moment from the beam to the nozzle, the nodes in the upper cross-section of the nozzle (slave nodes) have been linked to the lower node of the beam (master node) rigidly. The constraining equations of this “rigid region” couple the vertical displacements of the nodes in the upper cross section of the nozzle with the rotation in the node of the beam caused by the moment load. As a consequence of this boundary condition the upper cross-section of the nozzle remains plane, the cross-section can ovalize, but it cannot warp. Figure 7.8.2-1

Because of the symmetry of the structure and the actions, only half of the structure was modelled, and, therefore, the applied moment was half of the specified value. In the plane of symmetry a symmetry boundary condition was applied to all nodes. As boundary condition in the end cross-sections of the shell, the hoop displacements were set to zero. This allows for warping and ovalization, although ovalization is constrained to some extent, see also Example 4. In this example this boundary condition is considered to be appropriate, the main shell being considered to be a part of a piping. Setting the hoop displacements to zero and creating additionally rigid regions in the end cross-sections of the shell leads to plane end crosssections and avoids warping. Since linear-elastic calculations showed that the maximum stresses in the model with the boundary condition for rigid ends are lower than the ones for boundary condition without, the one without rigid ends was used, for reasons of conservativity. To obtain a stable model, the horizontal displacement in one node in one end cross-section of the main shell has been constrained to zero. The vertical displacement in the node of the upper (dummy) end of the elastic beam has been constrained to zero too.

2. Determination of the maximum allowable pressure according to GPD The partial safety factors γ R according to EN-UFPV Annex B, Table B.9-3 are 1.25 . Therefore, the analysis was carried out with a linear-elastic ideal-plastic material law, Tresca’s yield condition (delivered as a subroutine by an ANSYS® distributor) with design material strength parameters of 187.2 MPa for the shell and 274.4 MPa for the nozzle (corresponding to material strength parameters of 187.2 MPa for the shell, according to EN 10028-2 and 343 MPa according to prEN 10216-2 for the nozzle), associated flow rule, and first order theory. For simplicity, the boundary

Analysis Details

DBA Design by Analysis

Example 3.2

Page 7.158 (S)

between the two materials was assumed to be at the upper end of the weld. The elastic modulus used for the shell and the nozzle was E = 210,125 GPa. Because of the fact, that the moment is constant, the limit pressure was determined for constant moment and increase of pressure only, and not for proportional increase of all actions as required in prEN 13445-3 Annex B.9.2.1. The partial safety factor γ G for permanent actions (with an unfavourable effect) is 1.35. Therefore, the moment to be applied to the FE-model is given by M = 1.35 ⋅

711.1 = 480 Nm. 2

The analysis was carried out with the Newton-Raphson method using the initial stiffness matrix in every equilibrium iteration, which showed the best convergence using Tresca’s yield condition for the structure considered. The analysis was carried out in two parts: in the first, the moment load was applied, and in the second, internal pressure and corresponding longitudinal stresses at the ends of the shell and at the end of the nozzle were applied additionally, and increased until a state near the limit state was reached. To restrict computation time to reasonable values, the analysis was terminated after 6 hours on the Compaq® Professional Workstation 5000 with two Pentium Pro® processors and 256 MB RAM. At termination time the last convergent solution showed an internal pressure of 17.66 MPa – this pressure was used as limit pressure. This value is close to the theoretical limit pressure of the undisturbed cylindrical shell, which is given by 18.34 MPa for Tresca’s yield condition. The Figure 7.8.2-2 shows the distribution of Tresca's equivalent stress for a pressure of 17.66 MPa . Figure 7.8.2-2

As shown in Figure 7.8.2-3, the maximum absolute value of the principal strains in the structure is less than 5 %, as required in the standard. According to EN-UFPV Annex B, Table B.9-2, the partial safety factor for pressure actions (without natural limit) γ P is 1,2. Therefore, the maximum allowable pressure according to GPD is PS max GPD =

17.66 = 14.72 MPa. 1.2

DBA Design by Analysis

Analysis Details Example 3.2

Page 7.159 (S)

If a subroutine for Tresca’s yield condition is not available, Mises’ yield condition has to be used for the check against GPD (see subsection 3.2 of section 3 - Procedures). Since the partial safety factor for the constant moment is different from 1, the results from the check against PD cannot be used, and an additional calculation has to be carried out for that particular purpose. In this case, the analysis was carried out with a linear-elastic ideal-plastic material law, Mises’ yield condition with design material strength parameters of 162.1 MPa for the shell and 237.6 MPa for the nozzle (again, the boundary between the two materials was assumed to be at the upper end of the weld), associated flow rule, and first order theory. Figure 7.8.2-3

The analysis was carried out in the same way as the one using Tresca’s yield condition, except that the kind of stiffness matrix used in the Newton-Raphson method was program-chosen. After two hours computation time, the analysis on the Compaq® Professional Workstation 5000 with two Pentium Pro® processors and 256 MB RAM was terminated, because at this time the last convergent solution showed an internal pressure of 17.91 MPa , which is close to the theoretical limit pressure of the undisturbed cylindrical shell, which is given by 18.73 MPa for Mises' yield condition. Since the maximum absolute value of the principal strains exceeds 5 % at the last load level – the maximum absolute principal strain value is 14 % – a lower load level with appropriate strains had to be used as limit value. For an internal pressure of 17.75 MPa the maximum absolute value of the principal strain in the structure is less than 5 % - as required in the standard – see Figure 7.8.2-4. Figure 7.8.2-4

Analysis Details

DBA Design by Analysis

Example 3.2

Page 7.160 (S)

Figure 7.8.2-5 shows the distribution of Mises equivalent stress at this pressure of 17.75 MPa . According to prEN 13445-3 Annex B, Table B.9-2 the partial safety factor for pressure (without natural limit) γ P is given by 1.2, and, therefore, the maximum allowable pressure according to GPD by PS max GPD =

17.75 = 14.79 MPa 1.2

Figure 7.8.2-5

3. Check against PD The following procedure corresponds to the one given in subsection 3.6 of section 3 Procedures. The used loading state with constant moment M = 711.1 / 2 = 355.5 Nm as in the specification and internal pressure of p = 26 MPa is close to the limit state of the structure, since the theoretical limit pressure of the undisturbed shell is 27.04 MPa for Mises’ yield condition and a material strength parameter of 234 MPa . To obtain the required stress states and the elasto-plastic stress state corresponding to the moment only during one analysis – which is advantageous to realise load case operations in ANSYS®5.4 the following load path was used in the FE analysis: state 1

p = 0 MPa, M = 35.5 Nm

(linear-elastic path)

state 2

p = 2.6 MPa, M = 35.5 Nm

(linear-elastic path)

state 3

p = 0 MPa, M = 0 Nm

(linear-elastic path)

state 4

p = 0 MPa, M = 355.5 Nm

(elasto-plastic path)

state 5

p = 26 MPa, M = 355.5 Nm

(elasto-plastic path).

The required linear-elastic stress field corresponding to the limit state is obtained by scaling the stress field of state 2. The one corresponding to an internal pressure of p = 26 MPa is obtained by scaling the stress field of state 1, and the one corresponding to the moment of M = 35.5 Nm by scaling the stress field of the difference of the stress fields of the states 2 and 1.

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Analysis Details Example 3.2

Page 7.161 (S)

The computation time was 3 hours and 10 minutes on the Compaq® Professional Workstation 5000 with two Pentium Pro® processors and 256 MB RAM. The elasto-plastic FE analysis was carried out as stated in prEN 13445-3 Annex B, Sec. B.9.3.1 - using Mises’ yield condition and associated flow rule, a linear-elastic ideal-plastic law with (design) material strength parameters of 234 MPa for the shell and 343 MPa for the nozzle (again, the boundary between the two materials was assumed to be at the upper end of the weld), and first order theory. In the case under consideration, no factors α and β (see subsection 3.3.2.6 of section 3 – Procedures) were found, so that Melan’s theorem was fulfilled. Therefore, a linear combination of self equilibrating stress fields – the one according to the limit state (σ ij ) res ,( M + p ) and the one according to moment load only (σ ij ) res , M – was used to fulfil Melan’s theorem. For the determination of the factors β 1 , β 2 and α β (see subsection 3.3.2.6 of section 3 – Procedures) the deviatoric maps of the stress states, i.e. the coordinates of a stress point given by its principal stresses, at the critical locations of the structure are used, since due to the increased number of factors and the different critical locations load case operations using the FE software are not feasible. The two critical locations are: The outer surface of the weld fillet (path A in Figure 7.8.2-6) with the adjacent part of the nozzle (path N in Figure 7.8.2-6), and the inner surface of the shell of the junction and the nozzle (path I in Figure 7.8.2-6) with the adjacent part of the nozzle (path Z in Figure 7.8.2-6). To ensure, that the critical part of the structure is not in a part of the undisturbed shell, a corresponding point at the inner surface was also taken into account.

Figure 7.8.2-6

Since the self-equilibrating stress field according to the moment load (σ ij ) res , M is equal to 0 in the critical point at the inner surface (paths I and Z), the point of the linear-elastic stress distribution for moment and pressure load with greatest distance to the centre in deviatoric map, see Figure7.8.2-7, the necessary conditions in this point are now given by

Analysis Details

DBA Design by Analysis

Page 7.162 (S)

Example 3.2

φ [ β 1 ⋅ (σ ij ) res , ( M + p ) + (σ ij ) M ,le + α ⋅ (σ ij ) p ,le ] ≤ 0 φ [ β 1 ⋅ (σ ij ) res ,( M + p ) + (σ ij ) M ,le ] ≤ 0 . Thus, the optimal scaling factors β 1 and α can be determined from these two conditions β 1 = 0.567 , α = 0.679 . Figure 7.8.2-7 shows the stress distribution in the deviatoric map for the inner surface: •

linear-elastic stress according to moment and pressure (σ ij ) le ,( M + p ) – thin green line,

•

elasto-plastic stress according to moment and pressure (σ ij ) ep ,( M + p ) – thick green line,

•

linear-elastic stress according to moment only (σ ij ) le, M – yellow line,

•

self-equilibrating stress according to moment and pressure (σ ij ) res ,( M + p ) – black line.

nozzle

connection line (no physical meaning)

fillet

Figure 7.8.2-7

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Analysis Details Example 3.2

Page 7.163 (S)

For the critical point on the outer surface (paths A and N), the point of the linear-elastic stress distribution for moment and pressure with greatest distance to the centre in deviatoric map – see Figure 7.8.2.-8 -, the necessary conditions are given by φ [0.569 ⋅ (σ ij ) res ,( M + p ) + β 2 ⋅ (σ ij ) res , M + (σ ij ) M ,le + 0.679 ⋅ (σ ij ) p ,le ] ≤ 0 , φ [0.569 ⋅ (σ ij ) res ,( M + p ) + β 2 ⋅ (σ ij ) res , M + (σ ij ) M ,le ] ≤ 0 . It can be shown, that the first condition is fulfilled for all values of β 2 ≥ 0 , and, therefore, the second condition renders – for the equality sign: β 2 = 2.4 . Figure 7.8.2-8 shows the stress distribution in the deviatoric map for the outer surface: •

linear-elastic stress according to moment and pressure (σ ij ) le ,( M + p ) – thin green line,

•

elasto-plastic stress according to moment and pressure (σ ij ) ep ,( M + p ) – thick green line,

•

linear-elastic stress according to moment only (σ ij ) le, M – thin yellow line,

•

self-equilibrating stress according to moment and pressure (σ ij ) res ,( M + p ) – thin black line

•

self-equilibrating stress according to moment only (σ ij ) res , M – thin red line.

nozzle

connection line (no physical meaning)

fillet Figure 7.8.2-8

DBA Design by Analysis

Analysis Details Example 3.2

Page 7.164 (S)

Thus, the limit pressure according to the shakedown condition is not smaller than PS max SD = α ⋅ p = 0.679 ⋅ 26 MPa = 17.65 MPa .

4. Comments If the mechanical properties as specified in EN 10028-2 at calculation temperature (50°C) are used instead of the ones at ambient temperature, there is a remarkable difference in the corresponding DBF results. Generally, the influence of the total model length (length of the shell) of shell-nozzle intersections on the calculation results increases with the ratio of the nozzle to shell diameter. In the example considered the influence is still negligible.

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Analysis Details

Page 7.165 (S)

Example 3.2

Member:

Analysis Type: Stress Categorization Route

Strathclyde

FE-Software:

ANSYS 5.4

Element Types:

8 – node, 3-D isoparametric solids.

Boundary Conditions:

Symmetry boundary conditions in the plane along the longitudinal direction of the shell. Hoop displacements in nodes at both ends of the shell constrained to zero. Longitudinal displacements in the nodes at one end of the shell constrained to zero, longitudinal displacements coupled in the nodes at the other end of the shell (plane sections remain plane). A rigid region is set up on the top surface nodes of the nozzle to apply the bending moment.

Model and Mesh:

Length of shell – 500 mm Length of nozzle – 80mm Number of elements - 2560

Results: Maximum admissible action according to the Stress Categorisation Route Internal pressure PSmax SC = 14.25 MPa

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Analysis Details Example 3.2

Page 7.166 (S)

1. Finite Element Mesh Boundary conditions applied to the model are symmetry about the xz plane. Therefore the applied moment is half the specified value. The nodes at both ends of the main cylinder are constrained in the hoop direction but are allowed to move radially. The nodes at one end of the main cylinder were constrained fully in the longitudinal direction with the nodes at the other end coupled in the longitudinal direction, thus maintaining plane sections remaining plane. Pressure was applied over the whole internal structure with the equivalent thrust that would be produced from closed ends applied at the longitudinally coupled degree of freedom end of the cylinder. The moment applied to the intersecting nozzle was modelled by creating a rigid region over the top edge of the nozzle via constraint equations. A mass element was created in the centre at the top of the nozzle (to act as the master node to which the moment is applied) with the nodes on the top edge of the nozzle (slave nodes) connected to it by constraint equations. The moment applied to the master node is then transmitted by the rigid constraint region evenly to the slave nodes on the nozzle, resulting in the application of a pure moment to the top surface of the nozzle with no warping, the cross-section remains plane. 2. Material properties Shell: Material strength parameter RM = 234 MPa , modulus of elasticity E = 210.125 GPa Nozzle: Material strength parameter RM = 343 MPa , modulus of elasticity E = 210.125 GPa 3. Determination of the Maximum Allowable Pressure Annex C of prEN 13445-3 defines the rules and methods for the interpretation of stresses calculated on an elastic basis and the verification of their admissibility by means of appropriate assessment criteria. Elastic stresses are linearised along stress classification lines into membrane and bending components. The membrane and bending components are then categorised depending upon the location of the stress classification line. Figure 7.8.3-1 shows the selected classification lines. Classification lines 1-3 are within the region of local primary membrane stress occurring as a result of the discontinuity, prEN 13445-3 Annex C. Classification line 4 is at a considerable distance from the discontinuity where no local effects occur. Two separate elastic analyses are carried out one for each action; a bending moment applied to the nozzle of 711.1 Nm and for an applied pressure of 10 MPa. Superposition may be used to combine the moment and pressure load case. As the analysis is elastic, the pressure load case may be scaled to the limit of admissibility to find the maximum allowable pressure according to the stress category approach. Figure 7.8.3-2 shows the stress intensity in the structure for moment load of 711.1 Nm only.

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Analysis Details

Page 7.167 (S)

Example 3.2

CL 2 CL 3 CL 1

CL 4

Figure 7.8.3-1: Position of Classification Lines

Figure 7.8.3-2: Stress Intensity for Moment Load (711.1Nm)

DBA Design by Analysis

Analysis Details Example 3.2

Page 7.168 (S)

Figure 7.8.3-3: Stress Intensity for Internal Pressure (10MPa)

Figure 7.8.3-3 shows the stress intensity of the structure for an applied pressure of 10 MPa. Figure 7.8.3-4 Shows the stress intensity in the structure when the above load cases are superimposed with a scale factor of 1.425 applied to the pressure load case i.e. the internal pressure is 14.25 MPa.

Figure 7.8.3-4: Stress intensity for Combined Load (Moment 711.1 Nm, Pressure 14.25 MPa)

Analysis Details

DBA Design by Analysis

Example 3.2

Page 7.169 (S)

The assessment criteria of Annex C states that the following limits be placed on the stress categories: (σ eq ) Pm ≤ f (σ eq ) PL ≤ 1.5 f (σ eq ) P b ≤ 1.5 f (∆σ eq ) P +Q ≤ 3 f σeq denotes Tresca's equivalent stress intensity, Pm denotes general primary membrane stress, PL is, local primary membrane stress, Pb primary bending stress, P+Q primary plus secondary stress and f is the value of the nominal design stress. Clause 6 of prEN 13445-3 gives details on the value to be taken as the nominal design stress f; 6.2.1 states that the value of f is taken as 0.2% proof strength of the material at calculation temperature with a safety factor of 1.5. Therefore, for the shell f = 234/1.5 = 156 MPa and for the nozzle f = 343/1.5 = 228.7 MPa. For the classification lines in Figure 7.8.3-1, the linearised stress components fall into the stress categories as shown in the following Table (given from Table C-2 in prEN 13445-3 Annex C) Stress Categories for Linearised Stresses

Classification Type of Stress line 1 σij,m σij,b 2 σij,m σij,b 3 σij,m σij,b 4 σij,m σij,b

Pressure Load

Moment Load

PL Qb PL Qb PL Qb Pm Qb

PL Qb PL Qb PL Qb Pm Pb

The following Table summarises the stress categorisation results for a moment load of 711.1 Nm and pressure of 14.25 MPa. Stress linearisation along the chosen classification lines was carried using the post processing linearisation routines within the ANSYS program.

Analysis Details

DBA Design by Analysis

Example 3.2

Page 7.170 (S)

Results for Stress Categories on Linearised Stresses

Classification line 1

2

3

4

Type of stress

Stress category

membrane bending mem+bend membrane bending mem+bend membrane bending mem+bend membrane bending mem+bend

(σeq)PL (σeq)Qb (∆σeq)P+Q (σeq)PL (σeq)Qb (∆σeq)P+Q (σeq)PL (σeq)Qb (∆σeq)P+Q (σeq)Pm (σeq)Qb (∆σeq)P+Q

Linearised Stress Intensity (MPa) 211.3 24.17 234.7 91.67 33.46 120.3 227.9 26.25 253.5 156.0 3.82 158.8

Limit on stress categories (MPa) 1.5f = 234 3f = 468 1.5f = 343 3f = 686.1 1.5f = 234 3f = 468 f = 156 3f = 468

This Table shows that the limiting stress category is the primary membrane stress in the shell resulting from the internal pressure ((σeq)Pm on classification line 4). General primary membrane stress is limited to f (156 MPa in the shell) and at an internal pressure of 14.25 MPa this limit is reached, therefore PS max SC = 14.25 MPa

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Analysis Summary

Page 7.171 (A)

Example 3.2 / F-Check

Analysis Type:

Member: Check against Fatigue

A&AB

FE –Software:

ANSYS 5.4

Element Types:

8 – node (brick element), 3 – D structural solid SOLID45; 3 – D elastic beam BEAM4 (to apply the moment load).

Boundary Conditions: • Symmetry boundary conditions in nodes in the symmetry plane in longitudinal direction of the shell; • hoop displacements in nodes at both ends of the shell constrained to 0; • longitudinal displacement in one node of one end of the shell constrained to 0; • rigid region concerning vertical displacements at the end of the nozzle to apply the moment; • vertical displacement set to 0 for dummy end of beam element. Model and Mesh:

Length of shell: 200 mm Length of nozzle: 100 mm Number of elements: 3281 (1 beam element)

Results: Fatigue life N = 70865 cycles (for cycling pressure) 1160 cycles (for cycling moment)

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1

Analysis Details

Page 7.172 (A)

Example 3.2 / F-Check

Data

Cyclic pressure tmax = 50 °C tmin = 50 °C t* = 0,75 tmax + 0,25 tmin =50 °C Rm = 410 MPa Rp0,2/t* = 234 MPa

Rz = 50 µm (table 18-8) en = 21.2 mm ∆σD = 279,3 MPa (table 18-10 for N ≥ 2.106 cycles) N = 200000 (for the first iteration) ∆σR = 293.4 MPa (allowable stress range for N cycles) at the last iteration

Stresses

The maximum total stress range occurs in the crotch corner – an unwelded region ∆σeq,t (total or notch equivalent stress range) = 361.7 MPa ∆σstruc (structural equivalent stress range) = 293.4 MPa (obtained by quadratic extrapolation along the main shell inside surface)

ó

eq

σeqmax = 442.2 MPa (maximum notch equivalent stress)

= 402.0 MPa (mean notch equivalent stress )

Theoretical elastic stress concentration factor Kt Kt =∆σeq,t / ∆σstruc = 1.2329

Effective stress concentration factorKeff K

eff

= 1 +

1,5 (K

t

− 1)

 Äó struc 1 + 0,5 K  t Äó D 

  

= 1 . 2120

18.8 Plasticity correction factor ke Thermal loading

mechanical loading If ∆σeq,l > 2 Rp0,2/t*

 Äó  struc k = 1+ A0  − 1 e  2 R p0,2/t *  with A0 =

0,5 for 800 MPa ≤ Rm ≤ 1000 MPa 0,4 for Rm ≤ 500 MPa  R m − 500  0,4 +  3000  for 500 MPa ≤ R m ≤ 800 MPa

A0 = ….. ke = ….. ∆σtotal = ke.∆σstruc = ….. MPa Else ∆σtotal = ∆σstruc = 361.7 MPa

If ∆σeq,l > 2 Rp0,2/t* 0,7 k = υ 0,4 0,5 +  Äó   struc   R p0,2/t * 

kυ = ….. ∆σtotal = kυ.∆σsruc = ….. MPa Else ∆σtotal = ∆σstruc = ….. MPa

If both mechanical and thermal loadings are to be considered, the correction has to be made on each component of the stress tensors. ke and k í are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated. 18-10-6-2 Temperature correction factor ft*

18-11-1-1 Surface finish correction factor fs

For t* > 100 °C ft* = 1,03 – 1,5 10-4 t* -1,5 10-6 t*2 = ….. Else ft* = 1

fs = Fs[0,1ln(N)-0,465] with Fs=1- 0,056 [ln (Rz)]0.64[ln(Rm)] +0,289 [ln (Rz)]0,53 = 0.7889 fs = 0.8568

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Analysis Details

Page 7.173 (A)

Example 3.2 / F-Check

18-11-1-2 Thickness correction factor fe en ≤ 25 mm 25 mm ≤ en ≤ 150 mm fe = Fe[0,1ln(N)-0,465]

en ≥ 150 mm fe = 0,7217[0,1ln(N)-0,465]

with Fe = (25/en)0.182 =

fe = 1

fe = …..

fe = …..

18-11-1-3 Mean stress correction factor fm If ∆σstruc < 2 Rp0,2/t* and σeqmax < Rp0,2/t*

If ó If ó

> 0 then ó

eq

< 0 then ó

eq

ó

= ó

eq

eq, r

For N ≥ 2.106 cycles See figure 18-14

eq, r eq, r

= Rp0,2/t* =

∆ó eq, t 2

∆σ eq, t 2 - Rp0,2/t*

= 53.14 MPa

For N ≤ 2.106 cycles M = 0.00035 Rm – 0,1 = 0.0435 if –Rp0,2/t* ≤ ó

fm = …..

If ∆σstruc >2 Rp0,2/t*

If ∆σstruc < 2 Rp0,2/t* and σeqmax > Rp0,2/t*

eq

≤

Äó

R then 2(1 + M )

 M(2 + M )  2ó   eq  fm = 1 1 + M  ÄóR     

0,5

Äó

if

R ≤ ó ≤R p0,2/t* then eq 2(1 + M )

2ó 1 + M 3 M  eq  f = − = …. = 0.9892 m 1+ M 3  ÄóR 





fm = 1

18-11-2-1 Overall correction factor fu f .f .f .f fu = s e m t * = 0.6993

K eff

18-11-3 Allowable number of cycles N 2

    4   4.6 ⋅ 10 N=   Äó  eq, struc  − 0.63R + 11,5   m f   u

     2.7 R m + 92  N=   if N ≤ 2.106 cycles Äó  eq, struc    f  u 

10

if 2.106 ≤ N ≤ 108cycles

N = 70865 cycles N is obtained by iterations. If the value of N oscillates, a difference equal to 1 % between two iterations is acceptable. If the values decrease monotonously, the difference must be less than 0,001 % between two iterations.

DBA Design by Analysis

Analysis Details Example 3.2 / F-Check

Cyclic moment / Principal stress range approach Data en = 7,5 mm tmax = 50 °C ∆σD (5.106cycles) = 52 MPa (class 71) tmin = 50 °C equivalent stresses or t* = 0,75 tmax + 0,25 tmin = 50 °C Rm = 540 MPa m=3 C= Rp0,2/t* = 343 MPa m=5 C= Stresses

Page 7.174 (A)

principal stresses m = 3 C⊥ = 7.16.1011 C// = ….. m = 5 C⊥ = ….. C// = …..

Critical point: Weld nozzle to shell, weld toe in branch longitudinal symmetry plane

∆σstruc = 797.8 MPa (structural equivalent stress range) (obtained by quadratic extrapolation on outside of nozzle) 18.8 Plasticity correction factor ke Thermal loading

mechanical loading

If ∆σstruc > 2 Rp0,2/t*

If ∆σstruc > 2 Rp0,2/t* k

 Äó  struc = 1+ A0  − 1 e  2 R p0,2/t *  

with A0 =

k



0,5 for 800 MPa ≤ Rm ≤ 1000 MPa 0,4 for Rm ≤ 500 MPa  R m − 500  0,4 +  3000  for 500 MPa ≤ R m ≤ 800 MPa

A0 = 0.4133 ke = 1.0674 ∆σ = ke ∆σstruc = 851.5 MPa Else ∆σ = ∆σstruc = …..

υ

= 0,5 +

0,7 0,4

 Äó   struc   R p0,2/t * 

kυ = ∆σ = kυ ∆σstruc = Else ∆σ = ∆σstruc = ….. MPa

If both mechanical and thermal loadings are to be considered, the correction has to be made on each component of the stress tensors . ke and k í are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated. 18-10-6-1 Thickness correction factor few en ≤ 25 mm few = 1

25 mm ≤ en ≤ 150 mm fe = (25/en)0.25 = few = …..

en ≥ 150 mm few = 0.639

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Analysis Details

Page 7.175 (A)

Example 3.2 / F-Check

18-10-6-2 Temperature correction factor ft* For t* > 100 °C Else ft* = 1

ft* = = 1.03 – 1.5.10-4 t* -1.5.10-6 t*2 = …..

18-10-6-3 Overall correction factor fw fw = few.ft* = 1 18-10-7 Allowable number of cycles N Äó fw

If

= 851.5 MPa Äó fw

> ∆σ5.106 cycles then

m = 3 and C (C⊥ or C//) = 7.16.1011

N=

C  ∆σ   f   w 

m

= 1160 cycles

If

Äó fw

< ∆σ5.106 cycles and other

cycles with

Äó fw

> ∆σ5.106 cycles

then m = 5 and C (C⊥ or C//) = ….. N = C m = ….. cycles  ∆σ   f   w 

If

Äó fw

< ∆σ5.106 cycles and all other

cycles with then N=∞

Äó fw

< ∆σ5.106 cycles

DBA Design by Analysis

Analysis Details Example 4

Analysis Type: Direct Route using Elastic Compensation

Page 7.176 (S)

Member: Strathclyde

FE-Software:

ANSYS 5.4

Element Types:

8 – node, 3-D isoparametric shell elements

Boundary Conditions:

Symmetry boundary conditions along the vertical cutting plane of the model Hoop and longitudinal displacements in nodes at the open end of the cylindrical shell constrained to zero Hoop displacement in nodes at open end of nozzle constrained to zero Longitudinal displacements coupled at nodes on open end of nozzle (plane sections remain plane)

Model and Mesh:

Number of elements - 1002 Results: Maximum admissible action according to the GPD-Check:Internal pressure PSmax GPD = 0.41 MPa Check against PD:

Shakedown limit pressure PSmax SD = 0.27 MPa

DBA Design by Analysis

Analysis Details Example 4

Page 7.177 (S)

1. Finite Element Mesh Element divisions for the finite element mesh were defined parametrically to allow modification of the element density. Half symmetry was used with finite element models created using high order 8-node 3-D shell elements. The allowable pressure according to GPD and the shakedown pressure according to PD were calculated for this model. Boundary conditions applied to the model are symmetry about the vertical plane as both geometry and loading are symmetrical about this plane. The nodes at the open end of the cylinder are constrained in the hoop direction and the longitudinal direction. The nodes on the top surface of the nozzle are constrained in the hoop direction and are coupled in the longitudinal direction (plane sections remain plane). Internal pressure is applied over the entire inside surface of the model. A thrust is applied to the top edge of the elements on the nozzle to model the thrust on the closed end resulting from the internal pressure. It is important to model a large shell length as the tangential nozzle has a large effect down the length of the cylinder. 2. Material properties Material strength parameter RM = 202 MPa ; modulus of elasticity E = 183.6 GPa 3. Determination of the maximum admissible pressure according to the GPD-Check Using the application rule in prEN-13445-3 Annex B.9.2.2 to check against GPD the principle is fulfilled when for any load case the combination of the design actions do not exceed the design resistance. This may be shown by calculating the limit load. The limit load has to be determined using Tresca‘s yield condition and associated flow rule. However, for shell elements the limit load from elastic compensation is calculated using a generalised yield condition based on Mises'. From prEN 13445-3 Annex B, Table B.9-3 the partial safety factor, γR for the resistance is 1.25. With a proof strength of 202 MPa the design material parameter is 161.6 MPa. The analysis was carried out using the elastic compensation method conforming to the direct route rules for GPD in Annex B: linear elastic - ideal plastic material law, Tresca’s yield condition and associated flow rule and first order theory. For shell elements it is not currently possible to calculate limit loads based directly on Tresca’s condition from elastic compensation. Models utilising shells have only one element through thickness. Therefore, it is not possible to carry out an elastic compensation analysis in the same way as in solid elements. Instead of carrying out the analysis using a Tresca or Mises model directly a generalised yield model is used which considers the elements thickness. In elastic compensation, the Ilyushin generalised yield model is used in the limit load calculation. Ilyushin's model is based upon Mises' condition the limit load will require correction to meet the code rules on the use of the Tresca condition. The maximum ratio of Mises' equivalent stress to Tresca's equivalent stress for the same load is 2/√3. Therefore, applying a factor of √3/2 to the yield stress in the Mises analysis (or to the limit load, as the analysis is elastic) will always lead to a conservative result on the Tresca condition. In effect, the Mises yield locus is being reduced to fit within the Tresca yield locus. The applied design material parameter, considering Mises' correction is 140 MPa. In each elastic compensation iteration an equilibrium stress field is produced where the elastic modulus of each subsequent iteration is defined by the previous elastic solution based upon the Ilyushin generalised yield model. In this way regions of the FE-model may be systematically stiffened or weakened depending upon the stress magnitudes in the previous solution. The result is that every equilibrium stress field is a lower bound of the limit load, the stress field giving the

Analysis Details

DBA Design by Analysis

Example 4

Page 7.178 (S)

greatest limit load is taken as the closest to the actual limit state and defines the limit load in the analysis. Convergence of the equilibrium stress field will occur typically between 8 and 15 iterations, i.e. no further benefit in the limit load will be noted with subsequent iterations. In the check against GPD, the loading considered is increasing pressure, i.e. the limit pressure is to be calculated. The total computing time to run each analysis on a 300 MHz Pentium two processor with 128 Mb RAM Windows NT workstation was 820 seconds. The stress fields were shown to converge after ten iterations. Iteration number five gives the highest lower bound on the limit load and the square of the squared Ilyusin function distribution is shown in Figure 7.9.1-1. The contour units are dimensionless and are termed the square of the Ilyushin function, where the Ilyushin function represents the ratio of actual stress to the yield stress of the material, i.e. σe σY Where f(IL) is the Ilyushin function, σe is the element stress and σY the yield stress of the material (or in this analysis the design material strength parameter). Therefore, for the applied loading to be a lower bound on the limit load the Ilyushian function anywhere in the redistributed limit field cannot exceed 1. Therefore, the limit multiplier on the applied load is given as f ( IL) =

Figure 7.9.1-1: Limit Ilyushian Stress Field

1 ILmax Where PL is the limit load, Pap the applied load and ILmax the maximum of the Ilyushin function in the limit field. For an applied load of 1 MPa the maximum squared Ilyushin function in the limit PL = Pap ⋅

field in Figure 7.9.1-1 is 4.142 giving a limit pressure of PL = 1 ⋅

1 4.142

= 0.491 MPa

Application of the partial safety factor given in prEN-13445-3 Annex B for a pressure without natural limit to the value of the design action is γp =1.2. The allowable load according to GPD is thus given by PS max GPD =

0.491 0.491 = = 0.41 MPa γp 1.2

DBA Design by Analysis

Analysis Details Example 4

Page 7.179 (S)

4. Check against PD In the check against progressive plastic deformation, the principle in prEN-13445-3 Annex B.9.3.1 is fulfilled if the structure can be shown to shake down. In elastic compensation the load at which the structure will shake down is simple to calculate. Using Melan’s lower bound shakedown theorem, as self-equilibrating stress field the residual stress field that would result after a loading cycle can be used. It can usually be calculated by subtracting the linear-elastic stress field at the limit pressure from the limit stress field. The residual stress field is in effect the resulting stress from an elastic unloading from the limit state back to zero pressure. If no stress in the residual field violates the yield condition, i.e. if there is no equivalent stress above the material parameter, then the shakedown load is equal to the limit load. Where the residual stress field of a structure does Figure 7.9.1-2: Residual Stress Field (Mises) exceed the yield condition, the shakedown limit can be calculated by evoking the proportionality of the linear elastic solution. It is not practical to calculate the shakedown load using a generalised yield model as used for calculating the limit pressure. In calculating the shakedown load the limit equilibrium stress fields are created based on the generalised yield model, with the residual stress fields calculated using Mises' equivalent stress. For the material strength parameter of 202 MPa and an applied load of 0.345 MPa, the residual stress field is shown in Figure 7.9.1-2. The maximum equivalent residual stress is 256.8 MPa and the equilibrium limit stress field for this iteration has a maximum of 237.8 MPa. Therefore, the residual stress field violates the yield condition. Invoking the proportionality of the elastic solution gives an allowable load according to PD of 202 = 0.271 MPA 256.8 For the same applied load of 0.345 MPa the maximum elastic Mises equivalent stress is 474.8 MPa, giving a load to first yield of 0.147 MPa. The upper limit on the pressure according to PD given in the application rule in Annex B.9.3.2 is the pressure where the elastic stress is limited to twice RM. For case 4 the upper limit on the pressure according to this application rule is 0.294 MPa. PS max SD = 0.345 ⋅

5. Check against GPD Using Non-linear Analysis A check against GPD was also performed for the same FE-model using conventional non-linear analysis. In this way, a direct comparison may be made between the two limit approaches. The FE geometry, mesh and boundary conditions are the same as those used in the elastic compensation analysis. Material non-linearities were applied corresponding to the design material parameter,

DBA Design by Analysis

Analysis Details Example 4

Page 7.180 (S)

140 MPa and again perfect plasticity. The pressure is ramped until the analysis terminates due to nonconvergence, i.e. equilibrium can no longer be maintained between the external applied forces and the internal forces. The limit load can be defined as the last converged solution in the analysis. Figure 7.9.1-3 shows the equivalent stress, at the top surface of the shell, for the last converged solution (prior to loss of equilibrium). The corresponding load is 0.572 MPa. Figure 7.9.1-3: Equivalent Stress Field Prior to Loss of Equilibrium (shell top)

According to the rules for the GPD-check in prEN13445-3 Annex B, the maximum principal strain must not exceed +/- 5%. Figure 7.9.1-4 shows the absolute maximum total principal strain on the top shell surface, as can be seen the maximum principal strain is 13.85%. According to the code, the load at the loss of equilibrium is too high. At a load of 0.502 MPa the maximum absolute value of principal strain is within the 5% limit. Figure 7.9.1-5 shows the Mises equivalent stress field at the Figure 7.9.1-4: Principal Strain Field Prior to Loss of Equilibrium (shell top) design value of the pressure 0.502 MPa.

DBA Design by Analysis

Analysis Details Example 4

Page 7.181 (S)

Figure 7.9.1-5: Equivalent Stress (shell top) at Design Value of Pressure (0.502MPa)

Applying the rules for the GPD-check in prEN-13445-3 Annex B, the allowable load is found by applying the partial safety factors on the design value of the action. The limit analysis was carried out using Mises' condition and correction was applied to the material strength parameter as with the elastic compensation solution above. The allowable load according to GPD using non-linear analysis is given as PS max GPD =

0.502 0.502 = = 0.418 MPa γp 1.2

Running on the same equipment as the elastic compensation analysis, the non-linear analysis required a CPU time of 3810 seconds.

DBA Design by Analysis

Analysis Details

Page 7.182 (S)

Example 4

Analysis Type:

Member:

Direct Route using elasto-plastic FE calculations

A&AB

FE-Software:

ANSYS® 5.4

Element Types:

Model: 8-node structural shell SHELL93 Submodel:10-node, 3-D tetrahedral structural solid SOLID87

Boundary Conditions:

♣Model: • Symmetry b.c. in the symmetry plane of the structure and in the horizontal plane at the lower end of the cylinder; • horizontal displacement in one node at the lower end of the cylinder constrained to 0. ♣Submodel: • Symmetry b.c. in the symmetry plane of the structure; • b.c. at the cut boundaries according to submodelling.

Model and Mesh: Model:

Number of shell elements: 1002

Submodel: Number of solid elements: 19526 Geometry: see Analysis Details

Results:

Max. allowable pressure, according to GPD, PS max GPD = 0.375 MPa Shakedown pressure PS max SD = 0.289 MPa , Number of allowable action cycles with a maximum pressure of PS max GPD : N PD = 1440

DBA Design by Analysis

Analysis Details Example 4

Page 7.183 (S)

1. Model geometry, boundary conditions, elements The structure is modelled with the nozzle closed by a flat end, as specified. Without the flat end relatively large warping and ovalization of the nozzle’s upper end occur. Due to the large diameter-to-wall thickness ratio of the structure 8-node shell elements SHELL93 have been used for the GPD check and for the shakedown check. Additionally, a submodel was used to ensure that the maxima of the self-equilibrating stress state used in the shakedown check were located in the surface of the structure (see subsection 3.3.3.2 of section 3 - Procedures). To model the whole structure with solid elements and using an appropriate number of elements in wall thickness direction would result in an FE-model with too many elements and nodes, and, therefore, in an unacceptable long computation time. Because of the non-rotational symmetry of the structure, model details and boundary conditions can influence the results of the calculations strongly. Generally, whenever a cylindrical shell structure is prone to ovalization and/or warping model details and boundary conditions require special attention, and even investigation. The influence of three different kinds of boundary conditions and four different lengths of the cylindrical shell adjacent to the dished end on linear-elastic stress results and on the internal limit pressure was investigated, the results are given briefly in the following. Because of the problem’s symmetry, for all calculations a model comprising half of the structure was used. Following prEN 13445-3 Annex B.9.2.1 a linear-elastic ideal-plastic constitutive law with a design material strength parameter of 138.5 MPa (see the determination of the maximum allowable pressure according to GPD) was used. The modulus of elasticity used in the calculations was 183.6 GPa. In all cases the 5 % strain limit criterion - see prEN 13445-3 Annex B.9.2.2 – was governing the calculation of the (limit) pressure. The different shell lengths used were: 500mm, 1000mm, 2000mm and 3000mm. The boundary conditions considered were: (a) Symmetry boundary condition in the nodes in the symmetry plane of the structure; vertical (longitudinal) and hoop displacements in the nodes in the end of the cylindrical shell constrained to zero, i.e. no warping and practically no ovalization of the cylinder is possible. (b) Symmetry boundary condition in the nodes in the symmetry plane of the structure; vertical (longitudinal) displacements in the nodes in the end of the cylindrical shell constrained to zero, i.e. no warping but ovalization of the cylinder is possible. (c) Symmetry boundary condition in the nodes in the symmetry plane of the structure; symmetry boundary condition (vertical displacement, longitudinal and hoop rotations constrained to zero) in the nodes in the end of the cylindrical shell constrained to zero, i.e. no warping but ovalization of the cylinder is possible. Table 7.9.2-1 shows the calculation results corresponding to the different shell lengths and boundary conditions, i.e. the (limit) pressure where the maximum absolute principal strain does not exceed 5 %, and the maximum Mises linear-elastic equivalent stress according to an internal pressure of 0.1 MPa.

DBA Design by Analysis

Shell length

boundary condition (a) (limit) pressure

lin.-el. 0.1MPa

Analysis Details

Page 7.184 (S)

Example 4

boundary condition (b) (limit) pressure

lin.-el. 0.1MPa

boundary condition (c) (limit) pressure

lin.-el. 0.1MPa

[mm]

[MPa]

[MPa]

[MPa]

[MPa]

[MPa]

[MPa]

500

5.8

129.1

4.5

137.4

4.5

137.4

1000

5.7

132.5

4.5

138.4

4.5

138.4

2000

5.0

137.7

4.5

138.6

4.5

138.6

3000

4.5

138.6

4.5

139.0

4.5

139.0

Table 7.9.2-1

As can be seen from this table, the results are for shell lengths greater 500 mm independent from the length of the cylindrical shell in the case of boundary condition (b) or (c), and the results are practically the same for both boundary conditions. If boundary condition (a) is used, the results depend on the length of the cylindrical shell, the structure’s stiffness increases with decreasing cylinder length, and the results are different from those obtained by using boundary condition (b) or (c), with the exception of the longest model. Therefore, boundary condition (c) and a cylindrical shell length of 1000 mm was used for all further calculations. Additional remark: Generally, if a 90° - model is used instead of a 180° - model, the structure becomes slightly stiffer, the (limit) pressure of the 90° - model is about 3 % to 10% larger than the corresponding one of the 180° - model. The submodel used for the check of the self-equilibrating stress field in the shakedown check consisted of 19526 elements SOLID87. To obtain proper stress results, elements with midsidenodes and a maximum global element size of 5 mm were used in the (free) meshing of the structure. The submodel is bounded by a cylinder with radius 186,5 mm and by a horizontal plane, located 85 mm below the flat end. In the plane of symmetry a symmetry boundary condition was applied to all nodes, the boundary conditions of the cut-boundaries were interpolated by the software from the corresponding displacements of the coarse (shell) model. 2. Determination of the maximum allowable pressure according to the GPD-check Since the subroutine for Tresca’s yield condition showed bad convergence, the check against GPD was carried out using Mises’ yield condition only. To evaluate the 5% limit of the absolute principal strain more accurately than with the results obtained from the shakedown check, a separate calculation was carried out (see subsection 3.3.1 of section 3 - Procedures). The analysis was carried out with a linear-elastic ideal-plastic material law, Mises' yield condition with a design material strength parameter of 138.5 MPa for the shell, the nozzle and the flat end, associated flow rule and first order theory. The value for the design material strength parameter is equal to the 1.0 % proof strength of X6CrNiMoTi 17-12-2 according to prEN 10028-7 at 180°C, 200 MPa , divided by the partial safety factor for the material γ R = 1.25 and multiplied with the factor Mises’ yield condition was used instead of Tresca’s).

3 2 (since

DBA Design by Analysis

The last convergent solution showed an internal pressure of 0.457 MPa after a computation time of about 3 hours on the Compaq® Professional Work-station 5000 with two Pentium Pro® processors and 256 MB RAM. Since the maximum absolute value of the principal strain exceeded 5% at the last load level – the absolute principal strain value was 5.8 % (but only in a very small part of the structure) - a lower load value with appropriate strains had to be used as limit value. Figure 7.9.2-1

At an internal pressure of 0.451 MPa the maximum absolute value of the principal strain was about 5%. Figure 7.9.2-1 shows the distribution of the corresponding Mises’ equivalent stress, and Figure 7.9.2-2 shows the distribution of the corresponding maximum absolute principal strain – the maximum value is located in the symmetry plane of the structure, on the inside of the nozzle, slightly above the intersection curve.

Figure 7.9.2-2

Analysis Details Example 4

Page 7.185 (S)

Analysis Details

DBA Design by Analysis

Example 4

Page 7.186 (S)

According to prEN 13445-3 Annex B, Table B.9-2, the partial safety factor for pressure loads (without natural limit) γ P is 1.2. Therefore, the maximum allowable pressure according to GPD is PS max GPD =

0.451 = 0.375 MPa. 1.2

3. Check against PD The elasto-plastic FE-analysis was carried out as stated in prEN 13445-3 Annex B.9.3.1 - using Mises’ yield condition and associated flow rule, a linear-elastic ideal-plastic constitutive law with design material strength parameters of 200 MPa for the shell , the flat end and the nozzle, and first order theory. To ensure that the maxima of the used self-equilibrating stress field at the critical parts are located in the surface of the structure (see subsection 3.3.3.2 of section 3 – Procedures), a submodel was used. According to this proof, the shell model can be used for the shakedown check. By defining and using load cases in ANSYS®, the superposition of stress fields can be done easily. Therefore, the first load step of the analysis was defined at a very low load level ( 0.1 MPa ), so that there was linear-elastic response of the structure. All other linear-elastic stress fields due to pressure can then be found easily by scaling. The analysis was carried out using the NewtonRaphson method. A pressure close to the (unknown) limit pressure was found to be 0.651 MPa after a computation time of 3 hours on the Compaq® Professional Workstation 5000 with two Pentium Pro® processors and 256 MB RAM. Figure 7.9.2-3 shows the elasto-plastic Mises’ equivalent stress field for the pressure of 0.651 MPa .

Figure 7.9.2-3

Analysis Details

DBA Design by Analysis

Example 4

Page 7.187 (S)

Figure 7.9.2-4 shows the linear-elastic Mises’ equivalent stress field for the pressure of 0.651 MPa – the stress maximum is located in the symmetry plane of the structure, on the inside of the nozzle, slightly above the intersection curve. The second region with large linear-elastic stresses is a part of the intersection curve.

Figure 7.9.2-4

As a self-equilibrating stress field of the structure, the difference of the elasto-plastic and the linearelastic stress fields corresponding to the pressure of 0.651 MPa was used. A check with the submodel verified, that the corresponding stress maxima are located in the surface of the structure. The self-equilibrating stress field of the submodel was generated by performing two calculations: An elastic-plastic one and a corresponding linear-elastic one. Creating two load cases, copying the linear-elastic load case into the working directory of the elastic-plastic load case and superposition of the load cases renders the stress field. The corresponding computation times were 7 minutes for the linear-elastic and 4 hours and 5 minutes for the elastic-plastic calculation on the Compaq® Professional Workstation 5000 with two Pentium Pro® processors and 256 MB RAM. Since the stress maxima are located in the surface of the structure, the self-equilibrating stress field obtained by the shell model was scaled down – the scaling-down factor, for which the yield condition is not violated by the stress field, is given by β = 0.281 (see subsection 3.3.2.5 of section 3 - Procedures).

DBA Design by Analysis

Analysis Details Example 4

Figure 7.9.2-5 shows the Mises‘ equivalent stress distribution of this scaled self-equilibrating stress field.

Figure 7.9.2-5

Figure 7.9.2-6 shows the Mises’ equivalent stress distribution at the lower bound shakedown limit, the scaling factor α is given by 0.443 (see subsection 3.3.2.5 of section 3 Procedures).

Thus, the shakedown limit pressure is given by PS max SD = 0.443 ⋅ 0.651 = 0.289 MPa

Figure 7.9.2-6

Page 7.188 (S)

Analysis Details

DBA Design by Analysis

Page 7.189 (S)

Example 4

The determination of the allowable pressure according to the PD - check using the application rule given in prEN 13445-3, Annex B.9.3.2, leads to a limit pressure of 0.289 MPa – the very same value as obtained by performing the SD - check. Since the limit pressure according to the GPD check is larger than the one resulting from the SD-check and the application rule in Annex B.9.3.2, further investigations of the cyclic behaviour of the structure were performed. FE - calculations with an internal pressure cycling between a maximum value of 0.375 MPa and a minimum value of 0 MPa were performed. The behaviour of the structure was considered for 4 full action cycles (the numbers 0, 2, 4, and 8 in the load history correspond to an internal pressure of 0 MPa, and the numbers 1, 3, 5 and 7 to an internal pressure of 0.375 MPa; these numbers also correspond to the calculated half-cycles). Plastification occured only in the final part of each half cycle, close to the maximum and minimum pressures, as can be seen from the MxPl – parameter from the solution history information file (“jobname.mntr”) written by the FE software ANSYS®. To investigate whether alternating plasticity and/or progressive plastic deformation occurs, the cyclic stress and strain behaviour of the structure has to be considered. A necessary (but not sufficient) condition for alternating plasticity is that the stress states for the upper (or lower) extreme values of the actions remain constant, a necessary and sufficient condition for alternating plasticity is that the sum of the plastic strains (for every component) during a full cycle is zero. To identify the critical location of the structure, i.e. the locations where detailed investigations have to be performed, it is useful to create a “difference - load case”, i.e. a fictitious load case where the results are given by the differences of the results of two real load cases for the same action. By consideration of the stresses and the strains of this “difference - load case” the critical location can usually be identified, but quite often only the stresses can be used for this purpose, since the differences in the (plastic) strains are too small. Figure 7.9.2-7 shows the Mises equivalent stress of the “difference - load case” created with the load states 5 and 7 (maxiumum pressure). As can be seen from this figure, there are two locations where the stress differences are noticeable: around node 2503, which is located in the symmetry plane of the structure, and around node 1161, which is located in the intersection seam of the shell and the nozzle.

1

2

MX

MX

Figure 7.9.2-7

ANSYS 5.5.3 OCT 4 1999 09:09:50 NODAL SOLUTION STEP=3 SUB =1 SEQV (AVG) PowerGraphics EFACET=1 AVRES=Mat DMX =.226E-03 SMN =.506E-06 SMX =5.205 .506E-06 .578302 1.157 1.735 2.313 2.892 3.47 4.048 4.626 5.205 .506E-06 .578302 1.157 1.735 4.048 4.626 5.205

Analysis Details

DBA Design by Analysis

Example 4

Page 7.190 (S)

For plotting and listing the plastic strains which occur in these locations during the cycles, the time history post processor of the FE - software was used. To obtain accurate results without averaging, the element solution values, i.e. the strains calculated in the integration points and copied to the adjacent node of the element, are used. Since one node belongs to more than one element the corresponding maximum values shall be used in this procedure. Figure 7.9.2-8 shows the principal plastic strains on the inner and outer surfaces of the shell model in the critical nodes 2503 and 1161 versus the load history. 1

1

EPPL1

EPPL1

EPPL2 EPPL2

EPPL3

1

EPPL3

1

EPPL1 EPPL1

EPPL2

EPPL2 EPPL3

EPPL3

Figure 7.9.2-8 Figure 7.9.2-8 shows clearly, that, in the considered history “interval”, alternating plasticity and progressive plastic deformation occur on the inner surface in node 2503 and on the outer surface in node 1161, and that progressive plastic deformation only occurs on the outer surface in node 2503, and that the plastic strains remains constant on the inner surface in node 1161.

The strain differences of the full cycles decrease with increasing cycle number, and, therefore, it is possible that the structure shakes down to constant cyclic behaviour, i.e. alternating plasticity, after a certain number of action cycles, but this behaviour can only be verified by a large number of simulation cycles, and, therefore, is unsuitable for a practical procedure. Thus, the maximum strain difference can be used to calculate an allowable number of cycles, using the following proposal

Analysis Details

DBA Design by Analysis

Example 4

Page 7.191 (S)

concerning the PD - check in prEN 13445-3, Annex B: „If it can be shown via cyclic simulations, that the maximum/minimum principle strain after the specified number of cycles is less than +-5% PD does “technically” not occur.“ Thus, the allowable number of cycles N PD i for a specific location of the structure and for the ith principal strain (i = 1, 2, 3) can be calculated according to the following formulae: N PD i = where

n (ε i el ) c (ε i pl ) n (∆ε i pl ) ( n− 2) n

5 − (ε i el )c − (ε i pl ) n (∆ε i pl )( n − 2 ) n

,

number of half-cycles in the simulation, maximum linear-elastic ith principal strain occuring during the full cycle in [%], plastic ith principal strain strain after the nth half-cycle in [%], difference of the ith plastic principal strain for the last full cycle in [%].

According to the number of calculated cycles for the structure under consideration, the above formulae is now given by N PD i =

5 − (ε i el ) c − (ε i pl ) 8 (∆ε i pl ) 6 8

.

Table 7.9.2-2 lists the strain values and the corresponding number of allowable action cycles for the two critical nodes and the inner and outer surface of the structure, respectively. The most critical location is given by node 1161 on the outer surface, where the difference of the 1st plastic principal strain for the last action cycle is calculated to be 2.92.10-3 %. Thus, the number of allowable action cycles for an maximum internal pressure of 0.375 MPa is given by N PD = 1440 . Note: If this number of action cycles is too small, further cyclic simulations – with probably decreasing plastic strain differences – would be necessary.

Analysis Details

DBA Design by Analysis

Example 4

Page 7.192 (S)

node 1161, element 945 outer surface

node 1161, element 945 inner surface

node2503, element 959 outer surface

node2503, element 959 inner surface

(ε1 el)c

0,11302

not relevant

0,11273

0,11311

(∆ε1 pl)24

5,71 . 10-3

0

0,34 . 10-3

7,92 . 10-3

(∆ε1 pl)46

3,93 . 10-3

0

0,32 . 10-3

4,14 . 10-3

(∆ε1 pl)68

2,92 . 10-3

0

0,23 . 10-3

2,25 . 10-2

(ε1 pl)8

0,68170

not relevant

0,04036

0,34479

NPD1

1440

∞

21059

2018

(ε2 el)c

-0,02212

not relevant

-0,0184

-0,0273

(∆ε2 pl)24

-4,34 . 10-3

0

-0,12 . 10-3

-7,41 . 10-3

(∆ε2 pl)46

-3,03 . 10-3

0

-0,11 . 10-3

-3,93 . 10-3

(∆ε2 pl)68

-2,24 . 10-3

0

-0,08 . 10-3

-2,14 . 10-3

(ε2 pl)8

0,02476

not relevant

-0,01405

0,01051

NPD2

2211

∞

62100

2319

(ε3 el)c

-0,11612

not relevant

-0,0421

-0,11242

(∆ε3 pl)24

-1,42. 10-3

0

-0,22 . 10-3

-0,52 . 10-3

(∆ε3 pl)46

-0,93. 10-3

0

-0,21 . 10-3

-0,16 . 10-3

(∆ε3 pl)68

-0,72. 10-3

0

-0,16 . 10-3

-0,10 . 10-3

(ε3 pl)8

-0,70646

not relevant

-0,02631

-0,33427

NPD3

5802

∞

30825

45583

Table 7.9.2-2

DBA Design by Analysis

Analysis Details

Page 7.193 (S)

Example 4

Analysis Type:

Member:

Stress Categorisation Route

WTCM

Materials and Properties: Material strength parameter Rp1.0/t = 200 MPa, modulus of elasticity E = 183600 MPa. FE- Software:

ALGOR

Element types:

7088 shell elements

Boundary conditions: yz-plane fixed in x (Tx), the shell ends fixed in z (Tz), xz-plane fixed in y (Ty) and fixed rotations (Rxyz) Model and mesh:

Maximum admissible internal pressure according to the Stress Categorisation Route: Internal pressure = 0.263 MPa.

DBA Design by Analysis

Analysis Details Example 4

Page 7.194 (S)

Six classification lines (CL) are considered : -

CL 1: through point 1 with coordinates (995;0;40). CL 2: through point 2 with coordinates (983.053;60.062;68.961). CL 3: through point 3 with coordinates (984.062;77.4474;56.3427). CL 4: through point 4 with coordinates (909.304;139.843;166.708). CL 5: through point 5 with coordinates (985.185;54.281;62.2765). CL 6: through point 6 with coordinates (886.55;149.268;180.84).

In the figure above the locations of the six classification lines are shown. The CL’s are situated at the nozzle-cylinder intersection. They are drawn through one point, perpendicular to the shell surface. The stresses are calculated at the inner and outer diameter. These values are used to calculate the membrane and bending components of the stresses.

Analysis Details

DBA Design by Analysis

Example 4

Page 7.195 (S)

For the stress classification, the following procedure was followed : -

-

-

for each load acting on the vessel, calculate the elementary stresses Φij (i,j = 1,2,3) in the different points on the different CL’s. for each load acting on the vessel and each CL, calculate the membrane stress components Φij,m and the bending stress components Φij,b. Classify the membrane stress components Φij,m in (Φij)Pm, (Φij)PL or (Φij)Qm and the bending stress components Φij,b in (Φij)Pb or (Φij)Qb. Calculate the sum of the stresses classified in this way for the set of loads acting simultaneously on the vessel. The stresses resulting from this summation are designated (Γij)Pm, (Γij)PL, (Γij)Pb, (Γij)Qm, (Γij)Qb. From this deduce: (Γij)Pm, (Γij)PL, (Γij)P, (Γij)P+Q. Calculate the following equivalent stresses: (Φeq)Pm or (Φeq)PL, (Φeq)P, (Φeq)P+Q. According to table C-2, the following classification must be used for internal pressure: PL and Qb. Verify the admissibility of the equivalent stresses.

In the case under consideration only one load is considered: internal pressure with an initial value = 0.583 MPa. The stresses were calculated for this load and a stress classification along the CL’s 1 to 6 is applied. The results of the calculations are shown in the following tables. CL 1 through point 1 (995;0;40) - Pressure = 0.583 MPa. 1

1'

(Σij)PL

(Σij)Qb

(Σij)P+Q

S11

148,4

-456

-153,8

302,2

-456

S22

180

-538,5

-179,25

359,25

-538,5

S33

0

0

0

0

0

S12

-161,2

289,9

64,35

-225,55

289,9

S23

0

0

0

0

0

S31

0

0

0

0

0

(Σij,PL)1

0

(Σij,P+Q)1

0

(Σij,PL)2

-100,929

(Σij,P+Q)2

-204,43

(Σij,PL)3

-232,121

(Σij,P+Q)3

-790,07

(σeq)Pl

232,121 MPa

(σeq)P+Q

790,07

MPa

Analysis Details

DBA Design by Analysis

Example 4

CL 2 through point 2 (983.053;60.062;68.961) Pressure = 0.583 MPa. 2

2'

(Σij)PL

(Σij)Qb

(Σij)P+Q

S11

-345,5

280,6

-32,45

-313,05

-345,5

S22

-444,1

304,2

-69,95

-374,15

-444,1

S33

0

0

0

0

0

S12

3,8

28,4

16,1

-12,3

28,4

S23

0

0

0

0

0

S31

0

0

0

0

0

(Σij,PL)1

0

(Σij,P+Q)1

0

(Σij,PL)2

-26,486

(Σij,P+Q)2

-337,905

(Σij,PL)3

-75,914

(Σij,P+Q)3

-451,695

(σeq)Pl (σeq)P+Q

75,914

MPa

451,695 MPa

CL 3 trough point 3 (984.062;77.4474;56.3427) - Pressure = 0.583 MPa. 3

3'

(Σij)PL

(Σij)Qb

(Σij)P+Q

S11

-485,2

47,1

-219,05

-266,15

-485,2

S22

-216,6

378,5

80,95

-297,55

378,5

S33

0

0

0

0

0

S12

98,5

16,3

57,4

41,1

98,5

S23

0

0

0

0

0

S31

0

0

0

0

0

(Σij,PL)1

83,615

(Σij,P+Q)1

389,591

(Σij,PL)2

0

(Σij,P+Q)2

0

(Σij,PL)3

-221,715

(Σij,P+Q)3

-496,291

(σeq)Pl

305,33

(σeq)P+Q

MPa

885,882 MPa

Page 7.196 (S)

Analysis Details

DBA Design by Analysis

Example 4

CL 4 through point 4 (909.304;139.843;166.708) - Pressure = 0.583 MPa. 3

3'

(Σij)PL

(Σij)Qb

(Σij)P+Q

S11

324,8

293,9

309,35

15,45

324,8

S22

215

-28,3

93,35

121,65

215

S33

0

0

0

0

0

S12

3,2

-70,8

-33,8

37

-70,8

S23

0

0

0

0

0

S31

0

0

0

0

0

(Σij,PL)1

347,522

(Σij,P+Q)1

359,492

(Σij,PL)2

55,178

(Σij,P+Q)2

180,308

(Σij,PL)3

0

(Σij,P+Q)3

0

(σeq)Pl

347,522 MPa

(σeq)P+Q

359,492 MPa

CL 5 through point 5 (985.185;54.281;62.2765) - Pressure = 0.583 MPa. 3

3'

S11

-290,4

S22

-524,2

S33

(Σij)PL

(Σij)Qb

(Σij)P+Q

205,2

-42,6

-247,8

-290,4

222,6

-150,8

-373,4

-524,2

0

0

0

0

0

S12

8,7

83,3

46

-37,3

83,3

S23

0

0

0

0

0

S31

0

0

0

0

0

(Σij,PL)1

0

(Σij,P+Q)1

0

(Σij,PL)2

-7,596

(Σij,P+Q)2

-281,675

(Σij,PL)3

-185,804

(Σij,P+Q)3

-532,925

(σeq)Pl

185,804 MPa

(σeq)P+Q

532,925 MPa

Page 7.197 (S)

Analysis Details

DBA Design by Analysis

Example 4

Page 7.198 (S)

CL 6 through point 6 (886.55;149.268;180.84) - Pressure = 0.583 MPa. 3

3'

(Σij)PL

(Σij)Qb

(Σij)P+Q

S11

298,8

233,2

266

32,8

298,8

S22

272,2

-51,9

110,15

162,05

272,2

S33

0

0

0

0

0

S12

-4,9

-96

-50,45

45,55

-96

S23

0

0

0

0

0

S31

0

0

0

0

0

(Σij,PL)1

302,142

(Σij,P+Q)1

337,674

(Σij,PL)2

74,008

(Σij,P+Q)2

233,326

(Σij,PL)3

0

(Σij,P+Q)3

0

(σeq)Pl

302,142 MPa

(σeq)P+Q

337,674 MPa

The calculated equivalent stresses must meet the assessment criteria: (σeq)Pl

<

1.5f

(σeq)Pl+Qb

<

3f

With f = Rp1.0/t / 1.5 = 200 / 1.5 MPa = 133.3 MPa, the assessment criteria are met if the internal pressure is reduced to 0.263 MPa (the most severe CL is CL3).

DBA Design by Analysis

Analysis Details

Page 7.199 (A)

Example 4 / F-Check

Analysis Type:

Member: Check against Fatigue A&AB

FE –Software:

ANSYS® 5.4

Element Types: Model: 8-node structural shell SHELL93 Boundary Conditions: • Symmetry b.c. in the symmetry plane of the structure and in the horizontal plane at the lower end of the cylinder; • horizontal displacement in one node at the lower end of the cylinder constrained to 0.

Model and Mesh:

Model:

Number of shell elements: 1002

Results: Fatigue life N = 4072 cycles

DBA Design by Analysis

Data

Analysis Details Example 4 / F-Check

Page 7.200 (A)

Welded region / Structural equivalent stress range approach en = 8,5 mm ∆σD (5.106cycles) = 52 MPa (class 71) equivalent stresses or 11 . m = 3 C = 7.16 10

tmax = 180 °C tmin = 180 °C t* = 0,75 tmax + 0,25 tmin = 180 °C Rm = 540 MPa Rp1,0/t* = 202 MPa

principal stresses m = 3 C⊥ = ….. C// = ….. m = 5 C⊥ = ….. C// = …..

m = 5 C = 1.96.1015

Stresses Critical point: Weld toe in nozzle, maximum stress range point at outside nozzle ∆σstruc = 491.1 MPa (structural equivalent stress range) (obtained by quadratic extrapolation along the outside nozzle generatrix) Note: In this example the extrapolation pivot distances, as specifies in the draft standard, are too small to be used compared to the used, mesh-size; the three closest nodal points have been used for extrapolation.

18.8 Plasticity correction factor ke Thermal loading

mechanical loading

If ∆σstruc > 2 Rp1,0/t*

If ∆σstruc > 2 Rp1,0/t*

 Äó  struc k = 1 + 0, 4  − 1 e  2 R p1,0/t * 

k

ke = 1,0862 ∆σ = ke ∆σstruc = 533.5 MPa Else ∆σ = ∆σstruc = …..

υ

= 0,5 +

0,7 0,4

 Äó   struc   R p1,0/t * 

kυ = ….. ∆σ = kυ ∆σstruc = ….. Else ∆σ = ∆σstruc = ….. MPa

If both mechanical and thermal loadings are to be considered, the correction has to be made on each component of the stress tensors . ke and k í are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated.

18-10-6-1 Thickness correction factor few en ≤ 25 mm few = 1

25 mm ≤ en ≤ 150 mm few = (25/en)0.25 = few = …..

en ≥ 150 mm few = 0.639

DBA Design by Analysis

Analysis Details

Page 7.201 (A)

Example 4 / F-Check

18-10-6-2 Temperature correction factor ft* For t* > 100 °C Else ft* = 1

ft* = 1,043 – 4,3 10-4 t* = 0.9656

18-10-6-3 Overall correction factor fw fw = few.ft* = 0.9656 18-10-7 Allowable number of cycles N Äó fw

If

= 552.5 MPa Äó fw

> ∆σ5 10 6 cycles then

M = 3 and C (C⊥ or C//) = 7.16.1011

N=

C  ∆σ   fw

  

m

= 4246 cycles

If

Äó fw

< ∆σ5.10 6 cycles and other

cycles with

Äó fw

> ∆σ5.10 6 cycles

then M = 5 and C (C⊥ or C//) = ….. N = C m = ….. cycles  ∆σ   fw

If

Äó fw

< ∆σ5.10 6 cycles and all other

cycles with

Äó fw

< ∆σ5.10 6 cycles

then N=∞

  

In this example a large difference between the structural equivalent stress range approach and the principal stress range approach is to be expected. Therefore both approaches have been applied. Details on the principal stress range approach are given on the next two pages. The minimum total stress range is in the unwelded region, at the outside of the nozzle, in the symmetric plane of the model, and slightly above the weld. The details of the corresponding fatigue calculation is given on the two pages after the next two.

DBA Design by Analysis

Analysis Details Example 4 / F-Check

Page 7.202 (A)

Data

Welded region / Principal stress range approach en = 8.5 mm tmax = 180 °C tmin = 180 °C ∆σD (5.106cycles) = 52 MPa (class 71) t* = 0,75 tmax + 0,25 tmin = 180 °C principal stresses or Rm = 540 MPa m = 3 C⊥ = 7.16 1011 Rp1,0/t* = 202 MPa C// = m = 5 C⊥ = C// = …..

Stresses

equivalent stresses m = 3 C = 7,16 1011 ….. m = 5 C = 1,96 1015 …..

Critical point: Weld toe in nozzle, maximum stress range point at outside nozzle

∆σstruc = 432.9 MPa (structural equivalent stress range) (obtained by quadratic extrapolation along the outside nozzle generatrix) Note: In this example the extrapolation pivot distances, as specifies in the draft standard, are too small to be used, compared to the used mesh-size; the three closest nodal points have been used for extrapolation.

18.8 Plasticity correction factor ke Thermal loading

mechanical loading

If ∆σstruc > 2 Rp1,0/t*

If ∆σstruc > 2 Rp1,0/t*

 Äó  struc k = 1 + 0, 4  − 1 e  2 R p1,0/t * 

k

ke = 1,0286 ∆σ = ke ∆σstruc = 445.3 MPa Else ∆σ = ∆σstruc = …..

υ

= 0,5 +

0,7 0,4

 Äó   struc   R p1,0/t * 

kυ = ….. ∆σ = kυ ∆σstruc = ….. Else ∆σ = ∆σstruc = ….. MPa

If both mechanical and thermal loadings are to be considered, the correction has to be made on each component of the stress tensors . ke and k í are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated.

18-10-6-1 Thickness correction factor few en ≤ 25 mm few = 1

25 mm ≤ en ≤ 150 mm few = (25/en)0.25 = few = …..

en ≥ 150 mm few = 0.639

DBA Design by Analysis

Analysis Details

Page 7.203 (A)

Example 4 / F-Check

18-10-6-2 Temperature correction factor ft* For t* > 100 °C Else ft* = 1

ft* = 1,043 – 4,3 10-4 t* = 0.9656

18-10-6-3 Overall correction factor fw fw = few.ft* = 0.9656

18-10-7 Allowable number of cycles N Äó fw

If

= 461.1 MPa Äó fw

> ∆σ5.10 6 cycles then

M = 3 and C (C⊥ or C//) = 7.16.1011

N=

C  ∆σ   fw

  

m

= 7303 cycles

If

Äó fw

< ∆σ5.10 6 cycles and other

cycles with

Äó fw

> ∆σ5.10 6 cycles

then M = 5 and C (C⊥ or C//) = ….. N = C m = ….. cycles  ∆σ   fw

If

Äó fw

< ∆σ5.10 6 cycles and all other

cycles with

Äó fw

< ∆σ5.10 6 cycles

then N=∞

  

This value is noticeably larger than the result for the structural equivalent stress range approach. The details of the calculation for the maximum total stress range in the unwelded region is given on the next two pages.

DBA Design by Analysis

Analysis Details

Page 7.204 (A)

Example 4 / F-Check

Data tmax = 180 °C tmin = 180 °C t* = 0,75 tmax + 0,25 tmin =180 °C Rm = 540 MPa Rp0,2/t* = 202 MPa

Rz = 200 µm (table 18-8) en = 8.5 mm ∆σD = 361.2 MPa (table 18-10 for N ≥ 2 106 cycles) N = 7000 (for the first iteration) ∆σR = 1049.6 MPa (allowable stress range for N cycles) at the last iteration

Stresses ∆σeq,t (total or notch equivalent stress range) = 808.0 MPa ∆σstruc (structural equivalent stress range) = 808.0 MPa (the maximum occurs in a region which is stress-concentration-free) ó

eq

σeqmax = 808.0 MPa (maximum notch equivalent stress)

= 404.0 MPa (mean notch equivalent stress )

Theoretical elastic stress concentration factor Kt Kt =∆σeq,t / ∆σstruc = 1

Effective stress concentration factorKeff K

eff

= 1 +

1,5 (K

t

− 1)

 Äó struc 1 + 0,5 K  t Äó D 

  

= 1

18.8 Plasticity correction factor ke Thermal loading

mechanical loading If ∆σstruc > 2 Rp0,2/t*

 Äó  struc k = 1+ A0  − 1 e  2 R p0,2/t *  with A0 =

0,5 for 800 MPa ≤ Rm ≤ 1000 MPa 0,4 for Rm ≤ 500 MPa  R m − 500  0,4 +  3000  for 500 MPa ≤ R m ≤ 800 MPa

A0 = ….. ke = 1.0862 ∆σtotal = ke.∆σstruc = 877.6 MPa Else ∆σtotal = .....

If ∆σstruc > 2 Rp0,2/t* 0,7 k = υ 0,4 0,5 +  Äó   struc   R p0,2/t * 

kυ = ….. ∆σtotal = kυ.∆σsruc = ….. MPa Else ∆σtotal = ∆σstruc = ….. MPa

If both mechanical and thermal loadings are to be considered, the correction has to be made on each component of the stress tensors. ke and k í are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated. ∆σeq.struc= ∆σtotal / Kt = 877.6 MPa (for age in 18-11-3)

18-10-6-2 Temperature correction factor ft*

18-11-1-1 Surface finish correction factor fs

For t* > 100 °C ft* = 1,03 – 1,5 10-4 t* -1,5 10-6 t*2 = 0.9656 Else ft* = 1

fs = Fs[0,1ln(N)-0,465] with Fs=1- 0,056 [ln (Rz)]0,64[ln(Rm)] +0,289 [ln (Rz)]0,53 Fs = 0.6751 fs = 0.8568

DBA Design by Analysis

Analysis Details

Page 7.205 (A)

Example 4 / F-Check

18-11-1-2 Thickness correction factor fe en ≤ 25 mm 25 mm ≤ en ≤ 150 mm fe = Fe[0,1ln(N)-0,465]

en ≥ 150 mm fe = 0,7217[0,1ln(N)-0,465]

with Fe = (25/en)0.182 =

fe = 1

fe = …..

fe = …..

18-11-1-3 Mean stress correction factor fm If ∆σstruc < 2 Rp0,2/t* and σeqmax > Rp0,2/t*

If ∆σstruc < 2 Rp0,2/t* and σeqmax < Rp0,2/t*

If ó If ó

eq eq

ó

> 0 then ó < 0 then ó

eq

= ó

For N ≥ 2 106 cycles See figure 18-14

eq, r

= Rp0,2/t* -

∆ó eq, t 2

∆σ eq, t 2 - Rp0,2/t*

=

=

For N ≤ 2.106 cycles M = 0.00035 Rm – 0.1 = 0.0435 if –Rp0,2/t* ≤ ó

fm = 1

eq, r

eq, r

If ∆σstruc >2 Rp0,2/t*

eq

≤

Äó

R then 2(1 + M )

 M(2 + M )  2ó   eq  fm = 1 1 + M  ÄóR     

Äó

if

R ≤ ó ≤R p0,2/t* then eq 2(1 + M )

0,5 =

fm =

2ó 1 + M 3 M  eq  − = …. 1+ M 3  ÄóR 





fm = 1

18-11-2-1 Overall correction factor fu f .f .f .f ∆σeq, struc / fu = 1049.5 MPa fu = s e m t * = 0.8362

K eff

18-11-3 Allowable number of cycles N 2

    4   4.6 ⋅ 10 N=   Äó  eq, struc  − 0,63R + 11.5   m f   u

if N ≤ 2.106 cycles N = ∞ if ∆σ eq, struc < ∆σ2.106

N = 4072 cycles N is obtained by iterations. If the value of N oscillates, a difference equal to 1 % between two iterations is acceptable. If the values decrease monotonously, the difference must be less than 0,001 % between two iterations.

DBA Design by Analysis

Analysis Details Example 4 / GPF- & PD-Check

Analysis Type: Direct Route using Elastic Compensation

Page 7.176 (S)

Member: Strathclyde

FE-Software:

ANSYS 5.4

Element Types:

8 – node, 3-D isoparametric shell elements

Boundary Conditions:

Symmetry boundary conditions along the vertical cutting plane of the model Hoop and longitudinal displacements in nodes at the open end of the cylindrical shell constrained to zero Hoop displacement in nodes at open end of nozzle constrained to zero Longitudinal displacements coupled at nodes on open end of nozzle (plane sections remain plane)

Model and Mesh:

Number of elements - 1002 Results: Maximum admissible action according to the GPD-Check:Internal pressure PSmax GPD = 0.41 MPa Check against PD:

Shakedown limit pressure PSmax SD = 0.27 MPa

DBA Design by Analysis

Analysis Details Example 4 / GPF- & PD-Check

Page 7.176 (S)

1. Finite Element Mesh Element divisions for the finite element mesh were defined parametrically to allow modification of the element density. Half symmetry was used with finite element models created using high order 8-node 3-D shell elements. The allowable pressure according to GPD and the shakedown pressure according to PD were calculated for this model. Boundary conditions applied to the model are symmetry about the vertical plane as both geometry and loading are symmetrical about this plane. The nodes at the open end of the cylinder are constrained in the hoop direction and the longitudinal direction. The nodes on the top surface of the nozzle are constrained in the hoop direction and are coupled in the longitudinal direction (plane sections remain plane). Internal pressure is applied over the entire inside surface of the model. A thrust is applied to the top edge of the elements on the nozzle to model the thrust on the closed end resulting from the internal pressure. It is important to model a large shell length as the tangential nozzle has a large effect down the length of the cylinder. 2. Material properties Material strength parameter RM = 202 MPa ; modulus of elasticity E = 183.6 GPa 3. Determination of the maximum admissible pressure according to the GPD-Check Using the application rule in prEN-13445-3 Annex B.9.2.2 to check against GPD the principle is fulfilled when for any load case the combination of the design actions do not exceed the design resistance. This may be shown by calculating the limit load. The limit load has to be determined using Tresca‘s yield condition and associated flow rule. However, for shell elements the limit load from elastic compensation is calculated using a generalised yield condition based on Mises'. From prEN 13445-3 Annex B, Table B.9-3 the partial safety factor, γR for the resistance is 1.25. With a proof strength of 202 MPa the design material parameter is 161.6 MPa. The analysis was carried out using the elastic compensation method conforming to the direct route rules for GPD in Annex B: linear elastic - ideal plastic material law, Tresca’s yield condition and associated flow rule and first order theory. For shell elements it is not currently possible to calculate limit loads based directly on Tresca’s condition from elastic compensation. Models utilising shells have only one element through thickness. Therefore, it is not possible to carry out an elastic compensation analysis in the same way as in solid elements. Instead of carrying out the analysis using a Tresca or Mises model directly a generalised yield model is used which considers the elements thickness. In elastic compensation, the Ilyushin generalised yield model is used in the limit load calculation. Ilyushin's model is based upon Mises' condition the limit load will require correction to meet the code rules on the use of the Tresca condition. The maximum ratio of Mises' equivalent stress to Tresca's equivalent stress for the same load is 2/√3. Therefore, applying a factor of √3/2 to the yield stress in the Mises analysis (or to the limit load, as the analysis is elastic) will always lead to a conservative result on the Tresca condition. In effect, the Mises yield locus is being reduced to fit within the Tresca yield locus. The applied design material parameter, considering Mises' correction is 140 MPa. In each elastic compensation iteration an equilibrium stress field is produced where the elastic modulus of each subsequent iteration is defined by the previous elastic solution based upon the Ilyushin generalised yield model. In this way regions of the FE-model may be systematically stiffened or weakened depending upon the stress magnitudes in the previous solution. The result is that every equilibrium stress field is a lower bound of the limit load, the stress field giving the

Analysis Details

DBA Design by Analysis

Example 4 / GPF- & PD-Check

Page 7.176 (S)

greatest limit load is taken as the closest to the actual limit state and defines the limit load in the analysis. Convergence of the equilibrium stress field will occur typically between 8 and 15 iterations, i.e. no further benefit in the limit load will be noted with subsequent iterations. In the check against GPD, the loading considered is increasing pressure, i.e. the limit pressure is to be calculated. The total computing time to run each analysis on a 300 MHz Pentium two processor with 128 Mb RAM Windows NT workstation was 820 seconds. The stress fields were shown to converge after ten iterations. Iteration number five gives the highest lower bound on the limit load and the square of the squared Ilyusin function distribution is shown in Figure 7.9.1-1. The contour units are dimensionless and are termed the square of the Ilyushin function, where the Ilyushin function represents the ratio of actual stress to the yield stress of the material, i.e. σe σY Where f(IL) is the Ilyushin function, σe is the element stress and σY the yield stress of the material (or in this analysis the design material strength parameter). Therefore, for the applied loading to be a lower bound on the limit load the Ilyushian function anywhere in the redistributed limit field f ( IL) =

Figure 7.9.1-1: Limit Ilyushian Stress Field

1 ILmax cannot exceed 1. Therefore, the limit multiplier on the applied load is given as PL = Pap ⋅

Where PL is the limit load, Pap the applied load and ILmax the maximum of the Ilyushin function in the limit field. For an applied load of 1 MPa the maximum squared Ilyushin function in the limit PL = 1 ⋅

1

= 0.491 MPa 4.142 field in Figure 7.9.1-1 is 4.142 giving a limit pressure of Application of the partial safety factor given in prEN-13445-3 Annex B for a pressure without natural limit to the value of the design action is γp =1.2. The allowable load according to GPD is PS max GPD = thus given by

0.491 0.491 = = 0.41 MPa γp 1.2

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Analysis Details Example 4 / GPF- & PD-Check

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4. Check against PD In the check against progressive plastic deformation, the principle in prEN-13445-3 Annex B.9.3.1 is fulfilled if the structure can be shown to shake down. In elastic compensation the load at which the structure will shake down is simple to calculate. Using Melan’s lower bound shakedown theorem, as self-equilibrating stress field the residual stress field that would result after a loading cycle can be used. It can usually be calculated by subtracting the linear-elastic stress field at the limit pressure from the limit stress field. The residual stress field is in effect the resulting stress from an elastic unloading from the limit state back to zero pressure. If no stress in the residual field violates the yield condition, i.e. if there is no equivalent stress above the material parameter, then the shakedown load is equal to the limit load. Where the residual stress field of a structure does Figure 7.9.1-2: Residual Stress Field (Mises) exceed the yield condition, the shakedown limit can be calculated by evoking the proportionality of the linear elastic solution. It is not practical to calculate the shakedown load using a generalised yield model as used for calculating the limit pressure. In calculating the shakedown load the limit equilibrium stress fields are created based on the generalised yield model, with the residual stress fields calculated using Mises' equivalent stress. For the material strength parameter of 202 MPa and an applied load of 0.345 MPa, the residual stress field is shown in Figure 7.9.1-2. The maximum equivalent residual stress is 256.8 MPa and the equilibrium limit stress field for this iteration has a maximum of 237.8 MPa. Therefore, the residual stress field violates the yield condition. Invoking the proportionality of the elastic solution gives an allowable load according to PD of 202 = 0.271 MPA 256.8 For the same applied load of 0.345 MPa the maximum elastic Mises equivalent stress is 474.8 MPa, giving a load to first yield of 0.147 MPa. The upper limit on the pressure according to PD given in the application rule in Annex B.9.3.2 is the pressure where the elastic stress is limited to twice RM. For case 4 the upper limit on the pressure according to this application rule is 0.294 MPa. PS max SD = 0.345 ⋅

5. Check against GPD Using Non-linear Analysis A check against GPD was also performed for the same FE-model using conventional non-linear analysis. In this way, a direct comparison may be made between the two limit approaches. The FE geometry, mesh and boundary conditions are the same as those used in the elastic compensation analysis. Material non-linearities were applied corresponding to the design material parameter,

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Analysis Details Example 4 / GPF- & PD-Check

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140 MPa and again perfect plasticity. The pressure is ramped until the analysis terminates due to nonconvergence, i.e. equilibrium can no longer be maintained between the external applied forces and the internal forces. The limit load can be defined as the last converged solution in the analysis. Figure 7.9.1-3 shows the equivalent stress, at the top surface of the shell, for the last converged solution (prior to loss of equilibrium). The corresponding load is 0.572 MPa. Figure 7.9.1-3: Equivalent Stress Field Prior to Loss of Equilibrium (shell top)

According to the rules for the GPD-check in prEN13445-3 Annex B, the maximum principal strain must not exceed +/- 5%. Figure 7.9.1-4 shows the absolute maximum total principal strain on the top shell surface, as can be seen the maximum principal strain is 13.85%. According to the code, the load at the loss of equilibrium is too high. At a load of 0.502 MPa the maximum absolute value of principal strain is within the 5% limit. Figure 7.9.1-5 shows the Mises equivalent stress field at the Figure 7.9.1-4: Principal Strain Field Prior to Loss of Equilibrium (shell top) design value of the pressure 0.502 MPa.

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Analysis Details Example 4 / GPF- & PD-Check

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Figure 7.9.1-5: Equivalent Stress (shell top) at Design Value of Pressure (0.502MPa)

Applying the rules for the GPD-check in prEN-13445-3 Annex B, the allowable load is found by applying the partial safety factors on the design value of the action. The limit analysis was carried out using Mises' condition and correction was applied to the material strength parameter as with the elastic compensation solution above. The allowable load according to GPD using non-linear analysis is given as PS max GPD =

0.502 0.502 = = 0.418 MPa γp 1.2

Running on the same equipment as the elastic compensation analysis, the non-linear analysis required a CPU time of 3810 seconds.

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Analysis Details Example 4 / GPF- & PD-Check

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Analysis Details

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Example 5

Analysis Type: Direct Route using Elastic Compensation

7.206 (W) Member: Strathclyde

FE-Software:

ANSYS 5.4

Element Types:

8 – node, axisymmetric 2-D isoparametric solids.

Boundary Conditions:

Constraint on the hoop displacement at the cut plane. Axisymmetry

Model and Mesh:

Number of elements - 611

Results: Maximum admissible action according to the GPD-Check: Internal pressure PSmax GPD = 11.1 MPa Check against PD:

Outside the scope of elastic compensation

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Analysis Details Example 5

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1. Finite Element Mesh Finite element models were created using 8-node 2-D axisymmetric solids. The model used in the analysis contained 620 elements. The allowable pressure according to the GPD-check and admissibility of the DBF pressure and thermal stresses according to the PD-check required assessment. Boundary conditions applied to the model are axisymmetry, applied via a key option when defining the element type in the FE- software (axisymmetry around the vertical axis, Y). The nodes at the undisturbed end of the spherical shell have their hoop degree of freedom constrained to ensure that plane sections remain plane. 2. Material properties Shell: Material strength parameter RM = 230 MPa ; modulus of elasticity E = 200 GPa Nozzle (reinforced part): Material strength parameter RM = 284 MPa ; modulus of elasticity E = 200 GPa Nozzle (unreinforced part): Material strength parameter RM = 147.5 MPa ; modulus of elasticity E = 200 GPa 3. Determination of the maximum admissible pressure according to GPD Using the application rule in prEN-13445-3 Annex B.9.2.2 to check against GPD, the principle is fulfilled when for any load case the combination of the design actions does not exceed the design resistance. This may be shown by calculating the limit load. The limit load has to be determined using Tresca‘s yield condition and associated flow rule. As elastic compensation is based upon a series of elastic equilibrium stress fields, it is a relatively simple procedure to derive a lower bound limit load direct from the Tresca yield model. In the check against GPD only the pressure load has to be considered in this problem. The thermal stresses that arise due to the cold medium injection are self-equilibrating and, therefore, do not affect the pressure limit of the vessel. From prEN-13445-3 Annex B, Table B.9-3 the partial safety factor, γR on the material resistance is 1.25. At a temperature of 180 oC the proof strengths of the materials for the shell, nozzle reinforcement and nozzle are 230 MPa, 284 MPa and 147.5 MPa, respectively. The analysis was carried out using the elastic compensation method conforming to the direct route rules in the GPDCheck in Annex B: linear elastic - ideal plastic material law, Tresca’s yield condition and associated flow rule and first order theory. In each elastic compensation iteration an equilibrium stress field is produced where the elastic modulus of each subsequent iteration is defined by the previous elastic solution. In this way regions of the FE-model may be systematically stiffened or weakened depending upon the stress magnitudes in the previous solution. The result is that every equilibrium stress field is a lower bound of the limit load, the stress field giving the greatest limit load is taken as the closest to the actual limit state and defines the limit load in the analysis. In problems such as this, where there are materials with different properties, the modulus modification requires a modified procedure that takes account of the different material properties. This modified method calculates the limit pressure for each component with a different material, allowing the component giving the lowest limit load to define the limit for the whole model. Convergence of the equilibrium stress field will occur typically between 8 and 15 iterations, i.e. no further benefit in the limit load will be noted

Analysis Details

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Example 5

7.208 (W) with subsequent iterations. The total computing time to run the analysis on a 300 MHz Pentium II processor with 128 Mb RAM Windows NT workstation was 197 seconds. The stress field was shown to converge after ten iterations. The highest limit pressure supplied from iteration number 3. Figure 7.10.1-1 shows the limit stress (intensity) field according to the Tresca condition. The limit pressure is given by scaling the limit stress field so that the equivalent stress anywhere in the model does not exceed the material strength parameter for that component: RM , σ max Where PL is the limit load, Pap the applied load in the elastic compensation analysis, RM is the yield strength of the component and σmax is the maximum stress intensity in that component. In this problem the shell gives the lowest limit pressure and defines the limit for the whole structure. For a pressure of 10 MPa the maximum stress intensity in the shell from the redistributed stress field is 138.3 MPa giving a limit pressure of PL = Pap ⋅

Application of the partial safety factors given in prEN 13445-3

Figure 7.10.1-1: Limit Stress Field (Tresca)

RM 230 = 10 ⋅ = 16.63 MPa σ max 138.3 Annex B to the limit pressure is necessary to calculate the allowable pressure according to GPD. The partial safety factor on the material yield strength is γR = 1.25, for a pressure without natural limit the partial safety factor on the action is γp =1.2. The allowable pressure according to the GPDcheck is thus given by PL = Pap ⋅

16.63 0.734 = = 11.1 MPa γ p ⋅ γ R 1.2 ⋅ 1.25 It is also possible to determine a limit pressure using Mises' condition and associated flow rule. However, due to the fact that the Tresca yield envelope lies within the Mises yield envelope, correction is required. The maximum ratio of Mises' equivalent stress to Tresca's equivalent stress for the same load is 2/√3. Therefore, applying a factor of √3/2 to the yield stress in the Mises analysis (or to the limit load, as the analysis is elastic) will always lead to a conservative result. From the Mises analysis the limit load was found to be 17.57 MPa and with the partial safety factors γR = 1.25 and γp = 1.2, the internal pressure limit according to the GPD-check can be found PS max GPD =

PS max GPD = as,

17.57 3 17.57 3 ⋅ = ⋅ = 10.14 MPa γ p ⋅ γ R 2 1.2 ⋅ 1.25 2

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Analysis Details

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Example 5

7.209 (W) As would be expected, using a corrected Mises analysis gives a lower bound on the Tresca limit pressure . The value calculated using the Tresca based elastic compensation is used for this problem. 4. Check against PD In the check against progressive plastic deformation, thermal stresses created by the cold medium injection will affect the residual stress and shakedown. Elastic compensation deals solely with structural loads and cannot be used in any assessment involving thermal transients. Therefore, this check is outside the scope of the direct route of DBA using elastic compensation. 5. Check on GPD Using Non-linear Analysis A check against GPD was also performed for the same FE-model using conventional non-linear analysis. In this way, a direct comparison may be made between the two limit approaches. The FE-geometry, mesh and boundary conditions are the same as those used in the elastic compensation analysis. Material nonlinearities were applied corresponding to the material yield strengths, 230 MPa (shell), 284 MPa (nozzle reinforcement), 147.5 MPa (nozzle) and perfect plasticity. Mises' yield condition and associated flow rule was used in the nonlinear analysis. Therefore, correction of Mises' yield surface is required, as the check against GPD requires the use of Figure 7.10.1-2: Limit Equivalent Stress Field (shell top) Tresca’s yield condition. As stated above the maximum ratio of Mises' equivalent stress to Tresca's equivalent stress for the same load is 2/√3. Therefore, applying a factor of √3/2 to the yield stress in the Mises analysis (or to the limit load) will always lead to a conservative result. Pressure is ramped until the analysis terminates due to non-convergence, i.e. equilibrium can no longer be maintained between the external applied forces and the internal forces. The limit load can be defined as the last converged solution in the analysis. Figure 7.10.1-2 shows the equivalent stress at the top surface for the last converged solution. The corresponding pressure is the limit one,

Figure 7.10.1-3: Limit Equivalent Strain Field (shell top)

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Analysis Details

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Example 5

7.210 (W) PL = 19.8 MPa. According to the rules for the GPD check in prEN-13445-3 Annex B, the maximum principal strain must not exceed +/- 5%. Figure 7.10.1-3 shows the maximum total principal strain on the top shell surface, as can be seen the maximum principal strain is 4.66%. The limit pressure for the structure is therefore 0.948 MPa. Applying the rules for the GPD-check in prEN 13445-3 Annex B, the allowable pressure is found by applying the partial safety factors on the action and material strength parameter as with the elastic compensation solution above. The limit analysis was carried out using Mises' condition and correction is required to bring the result within the bounds of the Tresca condition as stated above. The allowable load according to the GPD-Check using non-linear analysis is given as Running on the same equipment as the elastic compensation analysis, the non-linear analysis 19.8 3 19.8 3 ⋅ = ⋅ = 11.43 MPa γ p ⋅ γ R 2 1.2 ⋅ 1.25 2 required a CPU time of 155 seconds. PS max GPD =

6. Additional Comments The utilisation of elastic compensation in this DBA-calculation gives a lower allowable pressure according to the GPD-check than that given by the DBF-calculation. Close correlation was shown between the various methods used to calculate the maximum allowable pressure according to GPD. The lowest value was calculated by elastic compensation based on Mises' yield condition, and the highest by the non-linear analysis. Elastic compensation could not be used in the admissibility check against progressive plastic deformation. The loading cycles involved thermal transients that could not be dealt with within the elastic compensation routines.

Analysis Type: Direct Route using elasto-plastic FE calculations

Member: A&AB

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Analysis Details Example 5

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FE-Software:

ANSYS® 5.4

Element Types:

4 – node, 2 – D axisymmetric solid PLANE42 (structural analysis) and PLANE55 (thermal analysis), respectively.

Boundary Conditions (structural analysis): Zero meridional displacement in the nodes at the undisturbed end of the hemispherical shell; Longitudinal stress in the end of the nozzle according to a closed end. Model and Mesh:

Total number of elements 1889

Results:

1.

Action cycles as given in the specification not admissible according to the checks against PD.

Elements, mesh fineness, boundary conditions

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Analysis Details

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Example 5

7.212 (W) To model the structure a total number of 1889 4-node axisymmetric elements – PLANE42 in ANSYS® 5.4 for the structural analysis and PLANE55 for the steady-state thermal analysis – was used. The hemispherical part of the structure was modelled up to an angle of 30°, measured from the vertical axis of the structure. The boundary condition applied in the model for the structural analysis is given by restraining the meridional displacement in the nodes at the undisturbed end of the shell to 0. Furthermore, a longitudinal stress according to a closed end was applied at the end of the nozzle. The boundary conditions used for the transient thermal analysis are given in the next chapter. The welds between the shell and the nozzle reinforcement, and the nozzle reinforcement and the nozzle, respectively, were modelled with fillets corresponding to weld influence zone thicknesses of 3 mm between shell and nozzle reinforcement, and 2 mm between nozzle reinforcement and nozzle.

2.

Transient thermal analysis

The transient thermal analysis according to the cold media injection (bulk temperature 80°C) in the vessel with an initially uniform temperature of 325° was carried out considering the specified 10 minutes of injection and the following 10 minutes. For the injection phase the heat transfer coefficient of the medium to the nozzle wall was specified as 10.8 kW/m2K. The heat transfer coefficient of the medium to the vessel wall, and to the nozzle wall if there is no injection, was specified as 1.16 kW/m2K. The outer surface of the vessel was assumed to be insulated perfectly. The density of the materials was specified to be 7.85 kg/dm3. For the input of the temperature dependent specific heat, the following interpolation knots were used for the linear interpolation in the FE software: 20°C: 461 J/(kg.K), 100°C: 479 J/(kg.K), 200°C: 499 J/(kg.K), 300°C: 517 J/(kg.K), 400°C: 531 J/(kg.K). Furthermore, temperature dependent thermal conductivities were used, and the following interpolation knots were used for the material P265GH: 20°C: 51 W/(m.K), 100°C: 50.8 W/(m.K), 200°C: 48.7 W/(m.K), 300°C: 45.8 W/(m.K), 400°C: 42.5 W/(m.K), and for the material 11CrMo9-10: 20°C 34.9 W/(m.K), 100°C: 37.3 W/(m.K), 200°C: 38.2 W/(m.K), 300°C: 37.8 W/(m.K), 400°C: 36.6 W/(m.K). Note: Since the structural and thermal calculations were carried out with the consistent mm-t-s unit system (1 t corresponds to 1000 kg) – as usual in structural calculations if the stress output shall be in MPa -, the input unit is given for the thermal conductivity by mW/(mm.K), for the heat transfer coefficient by mW/(mm2.K), for the specific heat by mJ/(t.K), and for the density by t/mm3. In the non-linear transient analysis the transient load is step-changed, i. e. the load is step-changed at the first substep of the corresponding load steps (at the beginning and the end of the injection) to the value of this load step. To obtain proper results, a minimum of 10 substeps was used each time for the following time intervals, given in [s], where 0 s corresponds to the beginning of the injection and 600 s to the end of the injection: [0, 0.1]; [0.1, 1]; [1, 10]; [10, 100]; [100, 600]; [600, 600.1]; [600.1, 601]; [601, 610]; [610, 700]; [700, 1200].

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Figure 7.10.2-1 shows the temperature field at the time 0.1 s, Figure 7.10.2-2 the temperature field at 1 s, and Figure 7.10.2-3 the one at 10 s. Figure 7.10.2-4 shows the temperature field at 100 s, and Figure 7.10.25 the one at 600 s, and Figure 7.10.2-6 the one at 1200 s.

Figure 7.10.2-1

Figure 7.10.2-2

Analysis Details Example 5

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DBA Design by Analysis Figure 7.10.2-3 Figure 7.10.2-4

Analysis Details Example 5

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Analysis Details Example 5

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Figure 7.10.2-5 Figure 7.10.2-6

3.

Admissibility check against GPD

Since the subroutine for Tresca's yield condition showed bad convergence, the check against GPD was carried out using Mises' yield condition (see subsection 3.3.1 of section 3 – Procedures) only. Thus, the elastic-plastic calculation was carried out using Mises' yield condition and associated flow rule, a linear-elastic ideal-plastic constitutive law, first order theory, and under usage of the Newton-Raphson method. According to prEN 13445-3 Annex B, Table B.9.3 the partial safety factor for the materials is given by γ R = 1.25 , and, therefore, the design material strength parameters (corresponding to the relevant material strength parameters for a temperature of 325°C – the maximum calculation temperature of the structure) are given by RM 3 230 3 ⋅ = ⋅ = 159.35 MPa for the shell, γR 2 1.25 2 RM 3 284 3 ⋅ = ⋅ = 197.11 MPa for the nozzle reinforcement, γR 2 1.25 2 RM 3 147.5 3 ⋅ = ⋅ = 102.19 MPa for the nozzle. γR 2 1.25 2

For the welds the lower value of the welded parts was used. The modulus of elasticity used for all parts of the structure was E = 192 GPa.

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Analysis Details

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Example 5

7.216 (W) The last convergent solution showed an internal pressure value of 15.65 MPa, which is close to the theoretical limit pressure of the corresponding undisturbed hemispherical shell of 15.95 MPa. Thus, 15.65 MPa was used as limit pressure in the check against GPD, since the maximum absolute value of the principal strains was 3.3%, and, therefore, the requirement of the standard was fulfilled. The computation time of this (internal pressure) limit load was 10 minutes on the Compaq® Professional workstation 5000 with two Pentium Pro® processors and 256 MB RAM. Figure 7.10.2-7 shows the distribution of the Mises' equivalent stress at the limit pressure. According to prEN 13445-3 Annex B, Table B.9.2 the partial safety factor for the pressure is γ P = 1.2 , and, therefore, the admissible (internal) pressure according to the check against GPD is given by 15.65 = 13.04 MPa PS max GPD = 1.2 Figure 7.10.2-7

4.

Admissibility check against PD

The linear-elastic stress states due to an internal pressure 0.9 ⋅ PS max DBF = 0.9 ⋅ 13.01 = 11.71 MPa and due to the thermal stresses according to the temperature distributions of the transient thermal analysis have been calculated, using temperature dependent elastic moduli and coefficients of thermal expansion. As input the following interpolation knots were used for the material P265: Elastic modulus: 20°C: 212 GPa, 100°C: 207 GPa, 200°C: 199 GPa, 300°C: 192 GPa, 400°C: 184 GPa; Coefficient of thermal expansion: 20°C: 11.9e-6 1/K, 100°C: 12.5e-6 1/K, 200°C: 13e-6 1/K, 300°C: 13.6e-6 1/K, 400°C: 14.1e-6 1/K.

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Analysis Details Example 5 Figure 7.10.2-8

As input the following interpolation knots were used for the other material: 11CrMo9-10: Elastic modulus: same values as for P265; coefficient of thermal expansion: 20°C: 11.5e-6 1/K, 100°C: 12.1e-6 1/K, 200°C: 12.7e-6 1/K, 300°C: 13.2e-6 1/K, 400°C: 13.6e-6 1/K.

Figure 7.10.2-9

Figure 7.10.2-8 shows the Mises' equivalent stress distribution for time = 0 s; figure 7.10.2-9 the Mises' equivalent stress distribution for 0.1 s; figure 7.10.2-10 the Mises' equivalent stress distribution for 1 s; figure 7.10.2-11 the Mises' equivalent stress distribution for 10 s; figure 7.10.2-12 the Mises' equivalent stress distribution for 100 s; figure 7.10.2-13 the Mises' equivalent stress distribution for 600 s; and figure 7.10.2-14 the Mises' equivalent stress distribution for 1200 s.

Figure 7.10.2-10

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Analysis Details Example 5

Figure 7.10.2-11

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Figure 7.10.2-12

Figure 7.10.2-13

Analysis Details Example 5

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Analysis Details Example 5

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Figure 7.10.2-14

As can be seen easily from these figures, the stresses are very large, and the variation of the stresses (the difference between a stress state under consideration and the zero-stress-state, which corresponds to shutdown) larger than twice the relevant design material strength parameters, which are given according to prEN 13445-3 Annex B.9.3.4 by the arithmetic mean of the yield or 1% proof strength for the highest and lowest calculation temperatures at the position under consideration during the whole action cycle, in more than one point of the structure for more than one instant of time. The corresponding design material strength parameters are given by 270 MPa for the shell, 319.75 MPa for the nozzle reinforcement, and 206.25 MPa for the nozzle. At the locations where the large stresses arise – on the inner surface of the nozzle reinforcement – no local structural discontinuity, exists. Thus, the application rule in prEN 13445-3 Annex B.9.3.2 cannot be fulfilled, and, therefore, the cycle cannot be shown by this procedure to be admissible according to the check against PD. Since the stress range is larger than twice the design material strength parameter, there follows that the structure cannot shake down under the given action cycle, and, therefore, the behaviour of the structure was investigated by performing cyclic elastic-plastic FE-calculations as follows.

Analysis Details

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Example 5

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For this calculation, an elasto-plastic constitutive law with the following (guaranteed) yield strength values according to the relevant material standards was used: P265 – nozzle – according to prEN 10216-2: 20°C: 265 MPa, 100°C: 226 MPa, 200°C: 192 MPa, 300°C: 154 MPa, 325°C: 147.5 Mpa. 11CrMo9-10 – shell - according to EN 10028-2: 20°C: 310 MPa, 250°C: 255 MPa, 300°C: 235 MPa, 325°C: 230 MPa. 11CrMo9-10 – reinforcement - according to prEN 10216-2: 20°C: 355 MPa, 100°C: 323 MPa, 200°C: 304 MPa, 300°C: 289 MPa, 325°C: 284 MPa. Figure 7.10.2-15 shows the simulated cycles. At first a start-up from 0 MPa and 20°C to 11.71 MPa and 325°C, one thermal cycle (injection cycle) and one shutdown cycle (unloading and reloading) were applied. Afterwards three thermal cycles and one shutdown cycle were appended and then these cycles were repeated once again. The calculation of the whole cycling needed about 2 hours on a Pentium Pro with 200 MHz and 128 MB memory.

Figure 7.10.2-15

The accumulated plastic strain after this load history is shown in Figure 7.10.2-16. The maximum strain after the complete action history is already 4%.

Figure 7.10.2-16

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Analysis Details Example 5

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To check if there is progressive plastic deformation in the next step, only the last 3 thermal cycles and the shutdown cycle afterwards were considered. Therefore, the results of point 1 were subtracted from the results of point 4 (Figure 7.10.2-15). The difference in the absolute plastic strain (Figure 7.10.2-17) is about 0.004% (Mises' equivalent strain). But, by considering the absolute values of the total strain (Figure 7.10.2-16), it can be seen that there may be a numerical problem. The difference is too small in comparison to the absolute value to achieve good numerical results.

Figure 7.10.2-17 (values to be divided by 1.5)

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Analysis Details Example 5

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To obtain better insight into this problem, the difference in the displacements for these three thermal and one shutdown cycle was plotted (Figure 7.10.2-18). If the large displacement at the corner is neglected, and only the largest mean displacement over the wall thickness considered, the value is about 0.0002 mm. This value is mainly due to the growing of the pipe diameter near the reinforcement. But there may be also numerical problems as well.

Figure 7.10.2-18

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Analysis Details Example 5

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For progressive plastic deformation to occur, there must be a failure mechanism, and, therefore, at least one cross section must fail (plastify as a whole). To check this behaviour, the accumulated plastic strains (Mises) for these three thermal and one shutdown cycles were plotted (Figure 7.10.2-19). It can be seen that there is one cross section (connection of pipe to the reinforcement piece) with plastification during this cycling of almost the whole cross section. In this cross section progressive plastic deformation is likely. In the region above, where the large deformation were seen (Figure 7.10.2-18), the radial growing of the displacements may stop after further cycling.

Figure 7.10.2-19

To obtain information whether the structure develops to stable cycling, the time history behaviour of a point at the inside of the pipe, slightly above the corner with the maximal plastic strain increment, was investigated. The plot of the plastic equivalent strain (Mises) vs. time (Figure 7.10.2-20) shows that there are relevant differences between the plastic strains before and after a thermal cycle at the three thermal cycles after the first shutdown. For the considered three thermal cycles after the second shutdown, this amount of plastic strain is very small in comparison with the plastic strain accumulated at the cycle.

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Analysis Details Example 5

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Figure 7.10.2-20

The history of accumulated equivalent plastic strain (Figure 7.10.2-21) shows also that the amount of plastic strain due to the closed cycles is much larger than the amounts of strain due to progressive plastic deformation (if there were progressive plastic deformation).

Figure 7.10.2-21

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Analysis Details Example 5

Page

7.226 (W) To determine whether the problem with progressive plastic deformation is more relevant at the thermal cycle or at the shutdown cycle, the accumulated plastic strains at the last thermal cycle (Figure 7.10.2-22) and at the last shutdown cycle (Figure 7.10.2-23) were plotted. It can be seen that the problem results from the thermal cycle with the same picture as above. At the shutdown cycle only a little plastification in a small part of the cross section can be discovered, i. e. t, the problem is due to the thermal cycling.

Figure 7.10.2-22

Figure 7.10.2-23

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Example 5

7.227 (W)

Result: Due to numerical problems, it is difficult to judge whether progressive plastic deformation occurs. A large number of cycles may be needed to reach an (almost) steady-state cycle, and because of the large number of necessary life cycles the very small strain differences (small in comparison to the absolute strain) of one cycle may be important. Therefore, in this example an exact answer cannot be given. But it can be seen that the strain accumulated during the thermal cycle is much larger than the strain due to possible progressive plastic deformation. Therefore, the fatigue calculation, which must consider the plastic strain accumulated within the cycle, should give appropriate numbers of cycles, e.g. Fatigue and not PD is likely to be the relevant failure mechanism. On the other hand, it does not seem to be appropriate to approve of a design for which in an operating cycle only a very small part of a cross-section remains elastic. A fatigue calculation starting from results of a linear elastic analysis may in such a case give results which are not relevant for the fatigue life. To avoid this type of numerical difficulties, appropriate application rules are required.

Analysis type:

Stress Categorization Route

Member: WTCM

DBA Design by Analysis

Analysis Details Example 5

Page 7.228 (W)

Materials and Properties: For shell and nozzle reinforcement: Rp0.2/t = 215 MPa and E = 190000 MPa at 325 °C. For nozzle: Rp0.2/t = 147.5 MPa and E = 190000 MPa at 325 °C. FE- Software:

ALGOR.

Element types:

2-D axisymmetric elements

Boundary conditions: No x- and z-displacement (Tx and Tz). No rotations (Rx, Ry and Rz ). Model and mesh:

Maximum admissible actions according to the Stress Categorisation Route: Superposition of internal pressure and cold injection not admissible

DBA Design by Analysis

Analysis Details Example 5

Page

7.229 (W) The following two figures show the temperature distribution after timestep 1 (5 sec) and 59 time steps (295 sec), respectively.

Fig. 7.10.3-1: Temperature distribution after 5 sec of cold injection

Fig. 7.10.3-2: Temperature distribution after 295 sec of cold injection

DBA Design by Analysis

Analysis Details Example 5

Page 7.230 (W)

Initially, four classification lines were chosen as shown in the figure below. The stress distribution shown corresponds to an internal pressure of 11.712 MPa.

Fig. 7.10.3-3: Tresca's equivalent stress for internal pressure

For the stress classification, the following procedure was followed:

-

for each load acting on the vessel, calculate the elementary stresses Φij (i,j = 1,2,3) in the different points on the different CL’s. for each load acting on the vessel and along each CL, calculate the membrane stress components Φij,m and the bending stress components Φij,b. Classify the membrane stress components Φij,m in (Φij)Pm, (Φij)PL or (Φij)Qm and the bending stress components Φij,b in (Φij)Pb or (Φij)Qb. Calculate the sum of the stresses classified in this way for the set of loads acting simultaneously on the vessel. The stresses resulting from this summation are designated (Γij)Pm, (Γij)PL, (Γij)Pb, (Γij)Qm, (Γij)Qb. From this deduce: (Γij)Pm, (Γij)PL, (Γij)P, (Γij)P+Q. Calculate the following equivalent stresses:

-

(Φeq)Pm or (Φeq)PL, (Φeq)P, (Φeq)P+Q.

-

According to Annex C of prEN13445-3, the following classification shall be used: - for internal pressure: PL and Qb, - for a thermal load: Qm and Qb. - Verify the admissibility of the equivalent stresses.

DBA Design by Analysis

Analysis Details

Page

Example 5

7.231 (W) Two loads are considered, which act simultaneously: an internal pressure (initial value = 11.712 MPa) and a thermal load. This thermal load is divided in timesteps to simulate the cooling effect of the injected cold medium inside the vessel. A total of 100 timesteps is considered and each timestep is 5 seconds long. Only timestep 1 (after 5 seconds), timestep 2 (after 10 seconds) and timestep 59 (after 295 seconds) are taken into account because at those timesteps, the Tresca equivalent stresses are the largest. The stresses were calculated for those two individual loads and a stress classification along the CL’s 1, 2, 3 and 4 was applied. On scrutinising the results of these calculations, it was observed that even without internal pressure, the assessment criteria can not be met. The elastic Mises stress equivalent distribution for timestep 1 is shown in the figure below.

Fig. 7.10.3-4: Mises' equivalent thermal stress after 5 sec of cold injection

The classification is most severe along CL 1. The results are shown in the next tables:

Analysis Details

DBA Design by Analysis

Point on CL Stress comp. S11 S22 S33 S12 S23 S31

1 (MPa) 487,8 1,9 349,4 0,3 0 0

2 (MPa) 138,5 101,2 131,4 3,6 0 0

Page

Example 5

3 (MPa) 9,1 49,3 -22,3 5,7 0 0

4 (MPa) -5,1 -74,9 -121,6 10,4 0 0

Membrane Timestep 1 stress comp. (MPa) σ11,m 58,668 σ22,m -37,319

Bending Timestep 1 stress comp. (MPa) σ11,b -244,25 σ22,b -182,693

σ33,m

-32,526

σ33,b

-283,088

σ12,m

13,161

σ12,b

21,658

σ13,m

0

σ13,b

0

σ23,m

0

σ23,b

0

i,j

(Σij)Qm

(Σij)Qb

(Σij)P+Q

11

58,668

-244,25

302,918

22

-37,319

-182,693

-220,012

33

-32,526

-283,088

-315,614

12

13,161

21,658

34,819

13

0

0

0

23

0

0

0

σ1,P+Q

305,226

MPa

σ2,P+Q

-222,32

MPa

σ3,P+Q

-315,614

MPa

σeq,P+Q=

620,84

MPa

5 (MPa) -17,6 -178,0 -192,4 24,6 0 0

7.232 (W)

6 (MPa) -99,6 -192,5 -229,1 47,4 0 0

With f = Rp0.2/t / 1.5 = 147.5 / 1.5 MPa = 98.3 MPa (in the nozzle Rp0.2/t = 147.5 MPa), the assessment criteria (σeq,P+Q < 3f = 295 MPa) are not met. The cold medium injection is too severe for any kind of internal pressure, according to the Stress Categorisation Route.

Analysis Details

DBA Design by Analysis

Page 7.206 (S)

Example 5 / GPD- & PD-Check

In the shell, CL 3 is the most severe. The results for thermal stresses only (after 295 sec of cold injection) are shown in the next tables:

S11 S22 S33 S12 S23 S31

1 74,9 47,0 498,6 -21,4 0 0

2 12,6 -7,7 361,3 5,9 0 0

3 42,2 -2,1 295,4 -28,2 0 0

4 58,1 0,5 253,0 -32,2 0 0

5 69,7 0,6 214,1 -35,8 0 0

6 76,8 -2,6 177,1 -37,3 0 0

7 80,6 -8,8 141,7 -36,6 0 0

8 81,8 -17,5 108,1 -33,7 0 0

S11 S22 S33 S12 S23 S31

9 79,5 -20,8 103,7 -34,6 0 0

10 77,7 -33,2 70,1 -30,1 0 0

11 73,1 -46,4 38,6 -24,0 0 0

12 64,2 -58,3 8,9 -15,9 0 0

13 44,9 -61,3 -18,8 5,1 0 0

14 -1,9 -26,5 -34,4 1,3 0 0

15 -12,9 -25,9 -41,8 -18,7 0 0

16 -18,3 -29,5 -52,8 -47,6 0 0

σ11,m

Timestep 59 (MPa) 57,668

σ11,b

Timestep 59 (MPa) -6,84

σ22,m σ33,m

150,407

σ22,b σ33,b

-231,551

σ12,m

-25,469

σ12,b

4,608

σ13,m

0 0

σ13,b

0 0

-20,525

σ23,m

-37,305

σ23,b

i,j

(Σij)Qm

(Σij)Qb

(Σij)P+Q

11

57,668

-6,84

64,508

-20,525

-37,305

-57,83

12

150,407 -25,469

-231,551 4,608

381,958 -30,077

13

0

0

0

23

0

0

0

σ1,P+Q

381,958

MPa

σ2,P+Q

71,503

MPa

σ3,P+Q

-64,825

MPa

σeq,P+Q=

446,783

MPa

22 33

With f = Rp0.2/t / 1.5 = 215 / 1.5 MPa = 143.3 MPa (in the shell Rp0.2/t = 215 MPa), the assessment criteria (σeq,P+Q < 3f = 430 MPa) are not met. The cold medium injection is too severe for any value of internal pressure.

DBA Design by Analysis

Analysis Details Example 5 / F-Check

Analysis Type:

Page 7.234 (C) Member:

Check against Fatigue

CETIM

FE –Software: ABAQUS / Standard version 5.8.1 Element Types: Quadratic axisymmetric 8-node elements . 1241 nodes and 352 elements Boundary Conditions: The mid-plane of the spherical part is locked in the vertical direction.

Model and Mesh:

Results: Fatigue life N = 9,8 full cycles (start up – shut down + 500 cold media injection)

DBA Design by Analysis

Analysis Details Example 5 / F-Check

Page 7.235 (C)

One (full) operating cycle consists of 1 start up – shut down cycle and 500 cold media injections. Two points show themselves as possibly critical: One at the inside of the nozzle reinforcement, slightly below the crotch corner – see the following figures. The following drawing shows the Tresca equivalent stress for the thermal loading only (cold media injection).

Notch stress = 681MPa

This figure shows the position and the value of the maximum total equivalent stress for cold media injection, i. e. thermal stresses only. The corresponding value for cold media injection plus (maximum) pressure action is 881.2 MPa. This point is in a region where the theoretical stress concentration factor K t = 1 . The fatigue results for this point are shown on the following four pages. The other possibly critical point, and certainly a point of interest, is at the weld toe of the weld seam nozzle reinforcement to nozzle. The calculation details for the structural equivalent stress range approach are given on the two pages after the next four.

DBA Design by Analysis

Analysis Details Example 5 / F-Check

1

Page 7.236 (C)

Data Critical point: Unwelded region / Inside nozzle reinforcement Rz = 200 µm (table 18-8) tmax = 325 °C en = 57.5 mm (distance from critical point to outside weld surface) tmin = 20 °C t* = 0,75 tmax + 0,25 tmin =248,75 °C ∆σD = 361,2 MPa (table 18-10 for N ≥ 2.106 cycles) Rm = 540 MPa N = 2300 (for the first iteration) Rp0,2/t* = 296,2 MPa ∆σR = 1303,0 MPa (allowable stress range for N cycles) at the 2nd iteration Stresses

Start up – shut down + 1 cold media injection ∆σeq,t (total or notch equivalent stress range) = 881,2 MPa ∆σstruc (structural equivalent stress range) = 881,2 MPa

ó

eq

σeqmax = 881,2 MPa (maximum notch equivalent stress)

= 440,6 MPa (mean notch equivalent stress )

Theoretical elastic stress concentration factor Kt Kt =∆σeq,t / ∆σstruc = 1

Effective stress concentration factorKeff K

eff

= 1 +

1,5 (K

t

− 1)

 Äó struc  1 + 0,5 K   t  Äó D  

= 1

18.8 Plasticity correction factor ke Thermal loading

mechanical loading If ∆σ struc > 2 Rp0,2/t*

 Äó  struc − 1 k = 1 + A0  e  2 R p0,2/t *  with A0 =

0,5 for 800 MPa ≤ Rm ≤ 1000 MPa 0,4 for Rm ≤ 500 MPa  R − 500  0,4 +  m for 500 MPa ≤ R m ≤ 800 MPa  3000 

A0 = 0,4133 ke = 1,2015 ∆σtotal = ke.∆σeq,t = ….. MPa Else ∆σtotal = ∆σeq,t = …..

If ∆σ struc > 2 Rp0,2/t* 0,7 k = υ 0,4 0,5 +  Äó   struc   R p0,2/t * 

kυ = 1,1033 ∆σtotal = kυ.∆σeq,t = ….. MPa Else ∆σtotal = ∆σeq,t = ….. MPa

If both mechanical and thermal loadings are to be considered, the correction has to be made on each component of the stress tensors. Ke and kí are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated. For this loading, we have both mechanical and thermal loadings. The combination of the two correction factors and the tensors gives the following result : ∆σtotal = 992,6 MPa ∆σ struc = Äó K = 992,6 MPa (for using in 18-11-3) total t 18-10-6-2 Temperature correction factor ft*

18-11-1-1 Surface finish correction factor fs

For t* > 100 °C ft* = 1,03 – 1,5 10-4 t* -1,5 10-6 t*2 = 0,8999 Else ft* = 1

fs = Fs[0,1ln(N)-0,465] with Fs = 1- 0,056 [ln (Rz)]0,64[ln(Rm)] +0,289 [ln (Rz)]0,53 =0,675 fs = 0,8868

DBA Design by Analysis

Analysis Details Example 5 / F-Check

18-11-1-2 Thickness correction factor fe en ≤ 25 mm 25 mm ≤ en ≤ 150 mm

en ≥ 150 mm

fe = Fe[0,1ln(N)-0,465] few = (25/en)0.25 = 0.8593 fe = 0,9547

fe = 1

1 Page 7.237 (C)

fe = 0,7217[0,1ln(N)-0,465] fe = …..

18-11-1-3 Mean stress correction factor fm If ∆σstruc < 2 Rp0,2/t* and σeqmax > Rp0,2/t*

If ∆σstruc < 2 Rp0,2/t* and σeqmax < Rp0,2/t*

If ó If ó

eq eq

and ó

> 0 then ó < 0 then ó

eq

= ó

For N ≥ 2 106 cycles See figure 18-14

eq, r

= Rp0,2/t* =

∆σ eq, t 2

Äóeq, t 2 - Rp0,2/t*

= ….. MPa

For N ≤ 2.106 cycles M = 0,00035 Rm – 0,1 = …..

if –Rp0,2/t* ≤ ó

fm = …..

eq, r

eq, r

If ∆σstruc >2 Rp0,2/t*

eq

≤

Äó

R then 2(1 + M )

 M(2 + M )  2ó   eq  fm = 1  ÄóR  1 M +    

Äó

if

0,5 = …..

R ≤ ó ≤R p0,2/t* then eq 2(1 + M )

2ó 1 + M 3 M  eq  fm = − = …. 1+ M 3  ÄóR 





fm = 1

18-11-2-1 Overall correction factor fu f .f .f .f ∆σstruc / fu = 1303 MPa fu = s e m t * = 0,7619

K eff

18-11-3 Allowable number of cycles N 2

    4.6 ⋅ 10 4   N=   Äó  struc − 0,63R + 11,5  m  f   u 

if N ≤ 2.106 cycles

N = ∞ if ∆σstruc / fu ∆σ 2.106

N = 2229 cycles (second iteration with an error of 1%) N is obtained by iterations. If the value of N oscillates, a difference equal to 1 % between two iterations is acceptable. If the values decrease monotonously, the difference must be less than 0,001 % between two iterations.

DBA Design by Analysis

Analysis Details

Page 7.238 (C)

Example 5 / F-Check

Data Critical point: Unwelded region / Inside nozzle reinforcement Rz = 200 µm (table 18-8) tmax = 325 °C en = 57.5 mm (distance from critical point to outside weld surface) tmin = 80 °C t* = 0,75 tmax + 0,25 tmin =263,75 °C ∆σD = 361,2 MPa (table 18-10 for N ≥ 2.106 cycles) Rm = 540 MPa N = 200000 (for the first iteration) Rp0,2/t* = 294,1 MPa ∆σR = 986,0 MPa (allowable stress range for N cycles) at the 7th iteration Stresses

Cold media injection ∆σeq,t (total or notch equivalent stress range) = 681 MPa ∆σstruc (structural equivalent stress range) = 681 MPa

ó

eq

σeqmax = 881.2 MPa (maximum notch equivalent stress)

= 540.7 MPa (mean notch equivalent stress )

Theoretical elastic stress concentration factor Kt Kt =∆σeq,t / ∆σstruc = 1

Effective stress concentration factorKeff K

eff

= 1 +

1,5 (K

t

− 1)

 Äó struc  1 + 0,5 K   t  Äó D  

= 1

18.8 Plasticity correction factor ke mechanical loading

Thermal loading

If ∆σstruc > 2 Rp0,2/t*

 Äó  struc − 1 k = 1 + A0  e  2 R p0,2/t *  with A0 =

0,5 for 800 MPa ≤ Rm ≤ 1000 MPa 0,4 for Rm ≤ 500 MPa  R − 500  0,4 +  m for 500 MPa ≤ R m ≤ 800 MPa  3000 

A0 = ….. ke = ….. ∆σtotal = ke.∆σeq,t = ….. MPa Else ∆σtotal = ∆σeq,t = …..

If ∆σstruc > 2 Rp0,2/t* 0,7 k = υ 0,4 0,5 +  Äó   struc   R p0,2/t * 

kυ = 1,0405 ∆σtotal = kυ.∆σeq,t = 708,6 MPa Else ∆σtotal = ∆σeq,t = ….. MPa

If both mechanical and thermal loadings are to be considered, the correction has to be made on each component of the stress tensors. ke and k í are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated. ∆σ struc = Äó K = 708,6 MPa (for using in 18-11-3) total t 18-10-6-2 Temperature correction factor ft*

18-11-1-1 Surface finish correction factor fs

For t* > 100 °C ft* = 1,03 – 1,5 10-4 t* -1,5 10-6 t*2 = 0,8861 Else ft* = 1

fs = Fs[0,1ln(N)-0,465] with Fs = 1- 0,056 [ln (Rz)]0,64[ln(Rm)] +0,289 [ln (Rz)]0,53 = 0,675 fs = 0,8597

DBA Design by Analysis

Analysis Details

Page 7.239 (C)

Example 5 / F-Check

18-11-1-2 Thickness correction factor fe en ≤ 25 mm 25 mm ≤ en ≤ 150 mm fe = Fe[0,1ln(N)-0,465] few = (25/en)0.25 = 0.8593 fe = 1 fe = 0,9434

en ≥ 150 mm fe = 0,7217[0,1ln(N)-0,465] fe = …..

18-11-1-3 Mean stress correction factor fm

If ó If ó

eq eq

and ó

> 0 then ó < 0 then ó

eq

= ó

For N ≥ 2 106 cycles See figure 18-14

eq, r

eq, r eq, r

= Rp0,2/t* =

∆σ struc 2

∆σ struc - Rp0,2/t* 2

= ..… MPa

For N ≤ 2.106 cycles M = 0,00035 Rm – 0,1 = ..… if –Rp0,2/t* ≤ ó

fm = …..

If ∆σstruc >2 Rp0,2/t*

If ∆σstruc < 2 Rp0,2/t* and σeqmax > Rp0,2/t*

If ∆σstruc < 2 Rp0,2/t* and σeqmax < Rp0,2/t*

eq

≤

Äó

R then 2(1 + M )

 M(2 + M )  2ó   eq  fm = 1 1 + M  ÄóR     

Äó

if

0,5 = …..

R ≤ ó ≤R p0,2/t* then eq 2(1 + M )

fm =

2ó 1 + M 3 M  eq  − = …. 1+ M 3  ÄóR 





fm = 1

18-11-2-1 Overall correction factor fu f .f .f .f fu = s e m t * = 0,7187 ∆σstruc / fu = 986.0 MPa

K eff

18-11-3 Allowable number of cycles N 2

    4,6 104   N=  Äó   struc − 0,63R + 11,5  m  f u 

if N ≤ 2.106 cycles

N = ∞ if ∆σstruc / fu ∆σ 2.106

N = 4898 cycles N is obtained by iterations. If the value of N oscillates, a difference equal to 1 % between two iterations is acceptable. If the values decrease monotonously, the difference must be less than 0,001 % between two iterations. The complete operating cycle corresponds to 1 cycle with start up and shut down and 499 cycles with the cold media injection. The global allowable number of cycles is equal to N with : N 499 N + =1 2229 4898 ⇒ N = 9.8 full cycles

DBA Design by Analysis

Analysis Details Example 5 / F-Check

Data Critical point: Weld seam nozzle to reinforced nozzle en = 14,3mm tmax = 325 °C tmin = 20 °C ∆σD (5.106cycles) = 46 MPa (class 631) t* = 0,75 tmax + 0,25 tmin = 248,75 °C equivalent stresses or Rm = 410 MPa m = 3 C = 5.1011 Rp0,2/t* = 171.5 MPa m = 5 C = 1.08.1015

Stresses

Page 7.240 (C)

principal stresses m = 3 C⊥ = ….. C// = ….. m = 5 C⊥ = ….. C// = …..

Start up – shut down + 1 cold media injection

∆σstruc =422.9 MPa (structural equivalent stress range, determined by extrapolation)

18.8 Plasticity correction factor ke Thermal loading

mechanical loading

If ∆σstruc > 2 Rp0,2/t*

If ∆σstruc > 2 Rp0,2/t* k

 Äó  struc = 1+ A0  − 1 e  2 R p0,2/t *  

with A0 =

k



0,5 for 800 MPa ≤ Rm ≤ 1000 MPa 0,4 for Rm ≤ 500 MPa  R − 500  0,4 +  m for 500 MPa ≤ R m ≤ 800 MPa  3000 

A0 = 0,4 ke = 1,0932 ∆σ = ke ∆σstruc = .......MPa Else ∆σ = ∆σstruc = …..MPa

υ

= 0,5 +

0,7 0,4

 Äó   struc   R p0,2/t * 

kυ =1,0571 ∆σ = kυ ∆σstruc = .....

If both mechanical and thermal loadings are to be considered, the correction has to be made on each component of the stress tensors . ke and k í are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated. For this loading, we have both mechanical and thermal loadings. The combination of the two correction factors and the tensors given the following result: ∆σ struc = 449.3 MPa

18-10-6-1 Thickness correction factor few en ≤ 25 mm few = 1

25 mm ≤ en ≤ 150 mm fe = (25/en)0.25 = few = …..

en ≥ 150 mm few = 0,639

DBA Design by Analysis

Analysis Details

Page 7. 241 (C)

Example 5 / F-Check

18-10-6-2 Temperature correction factor ft* For t* > 100 °C Else ft* = 1

ft* = 1,03 – 1,5 10-4 t* -1,5 10-6 t*2 = 0,8999

18-10-6-3 Overall correction factor fw fw = few.ft* = 0,8999 18-10-7 Allowable number of cycles N Äó fw

If

= 499.3 MPa

Äó fw

> ∆σ5.106 cycles then

m = 3 and C (C⊥ or C//) = 5.1011

N=

C  ∆σ   fw

  

m

= 4018 cycles

If

Äó fw

< ∆σ5.106 cycles and other

cycles with

Äó fw

> ∆σ5.106 cycles

then m = 5 and C (C⊥ or C//) = ….. N = C m = …..  ∆σ   f   w 

If

Äó fw

< ∆σ5.106 cycles and all other

cycles with then N=∞

Äó fw

< ∆σ5.106 cycles

DBA Design by Analysis

Analysis Details Example 5 / FE-Check

Data Critical point: Weld seam nozzle to reinforced nozzle en = 14,3 mm tmax = 325 °C tmin = 80 °C ∆σD (5.106cycles) = 46 MPa (class 63) t* = 0,75 tmax + 0,25 tmin = 263,75 °C equivalent stresses or 11 Rm = 410 MPa . m = 3 C = 5 10 Rp0,2/t* = 166,3 MPa

principal stresses m = 3 C⊥ = ….. C// = ….. m = 5 C⊥ = ….. C// = …..

m = 5 C = 1,08.1015 Stresses

Page 7. 242 (C)

Cold media injection

∆σstruc = 360,7 MPa (structural equivalent stress range, determined by extrapolation) 18.8 Plasticity correction factor ke Thermal loading

mechanical loading

If ∆σstruc > 2 Rp0,2/t*

If ∆σstruc > 2 Rp0,2/t* k

 Äó  struc = 1+ A0  − 1 e  2 R p0,2/t *  

with A0 =

k



0,5 for 800 MPa ≤ Rm ≤ 1000 MPa 0,4 for Rm ≤ 500 MPa  R − 500  0,4 +  m for 500 MPa ≤ R m ≤ 800 MPa  3000 

A0 = ….. ke = ….. ∆σ = ke ∆σ struc = ….. MPa Else ∆σ = ∆σstruc = .....

υ

= 0,5 +

0,7 0,4

 Äó   struc   R p0,2/t * 

kυ = ….. ∆σ = kυ ∆σ struc = ….. MPa Else ∆σ = ∆σstruc = ….. MPa

If both mechanical and thermal loadings are to be considered, the correction has to be made on each component of the stress tensors . ke and k í are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated. 18-10-6-1 Thickness correction factor few en ≤ 25 mm few = 1

25 mm ≤ en ≤ 150 mm few = (25/en)0,25= few = …..

en ≥ 150 mm few = 0,639

DBA Design by Analysis

Analysis Details

Page 7. 243 (C)

Example 5 / F-Check

18-10-6-2 Temperature correction factor ft* For t* > 100 °C Else ft* = 1

ft* = 1,03 – 1,5 10-4 t* -1,5 10-6 t*2 = 0,8861

18-10-6-3 Overall correction factor fw fw = few.ft* = 0,8861 18-10-7 Allowable number of cycles N Äó fw

If

= 416.3 MPa

Äó fw

> ∆σ5.106 cycles then

m = 3 and C (C⊥ or C//) = 5.1011

N=

C  ∆σ   fw

  

m

= 6929 cycles

If

Äó fw

< ∆σ5.106 cycles and other

cycles with

Äó fw

> ∆σ5.106 cycles

then m = 5 and C (C⊥ or C//) = ….. N = C m = …..  ∆σ   fw

  

If

Äó fw

< ∆σ5.106 cycles and all other

cycles with

Äó fw

< ∆σ5.106 cycles

then N=∞

The complete operating cycle corresponds to 1 cycle with start up and shut down and 499 cycles with the cold media injection. The global allowable number of cycles is equal to N with: N 499 N + =1 4018 6929 ⇒ N = 13.8 full cycles This value is larger than the one for the unwelded region – the unwelded region governs the fatigue life.

DBA Design by Analysis

Analysis Details Example 6

Analysis Type: Direct Route using Elastic Compensation

Page 7.244 (S)

Member: Strathclyde

FE-Software:

ANSYS 5.4

Element Types:

4 – node, axisymmetric 2-D isoparametric solids for model 1 8 – node 3-D structural shell selements for model 2

Boundary Conditions: Model 1: Axisymmetry about Y axis. Bottom nodes on shell and jacket constrained longitudinally (symmetry). Nodes at centre of head constrained (axisymmetry) Model and Mesh: Axisymetric solid mesh (model 1)

Number of elements - 660

Results: Check against GPD:

Actions not admissible.

Analysis Details

DBA Design by Analysis

Example 6

Page 7.245 (S)

Number of elements - 435

Boundary Conditions:

Model 2: Symmetry conditions on both longitudinal cut edges. Nodes on open end of cylinder constrained longitudinally and in the hoop direction (displacement allowed in the radial direction) Shell element mesh

Results: Check against GPD:

Actions not admissible.

DBA Design by Analysis

Analysis Details Example 6

Page 7.246 (S)

1. Finite Element Mesh Two different finite element models of the geometry were created. The first one a 2-D solid axisymmetric model, the second a partial 3-D shell model utilising a quarter rotational symmetry and higher order 8-node elements. The 2-D model used in the analysis was created using 660 2-D 4-node solid axisymmetric elements. Boundary conditions for this model involved symmetry on the free end of the cylinder and jacket, axisymmetry and constraint on the radial displacement of the nodes at the centre of the dished end. The 3-D model was created using 435 3-D 8-node isoparametric shell elements. The boundary conditions on this model involved symmetry along the three cutting planes. The pressure combinations described in the problem specification were applied in accordance with prEN 13445-3 Annex B and checks are performed against GPD and PD. 2. Admissibility checks against GPD Using the application rule in EN-UFPV Annex B.9.2.2, to check against GPD, the principle is fulfilled when for any load case the combination of the design actions do not exceed the design resistance. Analysis is performed using Tresca‘s yield condition and associated flow rule. Elastic compensation is generally used to calculate lower bounds on the shakedown and limit loads. In the case of multiple loads, the process involves constructing yield loci to describe the structural limits for every load combination. However, if the applied loads are known, it is a simple procedure to perform an admissibility check against GPD. In the check against GPD the pressure action only has to be considered in this problem. The thermal stresses that arise due to the differential temperatures in the two chambers are selfequilibrating, and do not affect the pressure limit of the vessel. From prEN 13445-3 Annex B, Table B.9-3 the partial safety factor, γR for the material resistance is 1.2. At a temperature of 160 oC the proof strength of the material is 194 MPa. The analysis was carried out using the elastic compensation method conforming to the direct route rules for the GPDcheck in Annex B: linear elastic - ideal plastic material law, Tresca’s yield condition and associated flow rule and first order theory. In each elastic compensation iteration an equilibrium stress field is produced where the elastic modulus of each subsequent iteration is defined by the previous elastic solution. In this way regions of the FE-model may be systematically stiffened or weakened depending upon the stress magnitudes in the previous solution. The result is, that every equilibrium stress field is a lower bound of the limit load, the stress field giving the greatest limit load is taken as the closest to the actual limit state and defines the limit load in the analysis. For this problem the load cases are defined and the admissibility of the load cases are checked against GPD. To check against GPD using elastic compensation, the loads are applied according to the code rules for GPD and the resulting redistributed equilibrium stress fields found. For the load cases to be admissible, the stress anywhere in the model should have a stress no greater than the design material strength parameter. Convergence of the equilibrium stress field will occur typically between 8 and 15 iterations, i.e. no further benefit in the equilibrium stress field will be noted with subsequent iterations. The total computing time to run an analysis on the 2-D solid model on a 300 MHz Pentium two processor with 128 Mb RAM Windows NT workstation was 240 seconds. The equilibrium stress field was shown to converge after fourteen iterations. According to the action combinations laid down in the problem specification, three checks against GPD have to be performedfor the following load cases:

DBA Design by Analysis

Analysis Details Example 6

LC1:

Inner chamber internal pressure Outer chamber internal pressure temperature in both chambers

PSi = 1.3 MPa PSo = 0.5 MPa TSi = 160oC

LC2:

Inner chamber internal pressure

PSi = 1.3 MPa

Outer chamber internal pressure temperature in both chambers

PSo = 0.0 MPa TSi = 160oC

Inner chamber internal pressure

PSi = -0.1 MPa

Outer chamber internal pressure temperature in inner chamber temperature in outer chamber

PSo = 0.5 MPa TSi = 160oC TSo = 10oC

LC3:

Page 7.247 (S)

Check for LC 1 According to the rules for checks against GPD, the partial safety factors applied to this analysis are as follows. As stated above the partial safety factor used to give the design resistance is γR = 1.2, giving a design material strength parameter of 161.67 MPa. According to Table B.9-2 the partial safety factor for pressure loads without natural limit is γP = 1.2. Therefore, the pressures applied to the model for the analysis are γP. PSi = 1.56 MPa and γP. PSo = 0.6 MPa. In the elastic compensation analysis using the 2-D model, the 14th iteration gave the lowest equilibrium stress field, shown in Figure 7.11.1-1. The plot shows that the main area of plasticity occurs in the knuckle region of the dished end. The maximum equivalent stress in the model is 176.6 MPa, which is greater than the design material strength parameter. Therefore, the defined loads are inadmissible according to the check against GPD using elastic compensation. It should be noted that the maximum stress is very near the design strength and this shows that the load case is very near the maximum limit. Lower bounds calculated using elastic compensation can be conservative and it may be that under different analysis types the loads may be found admissible. As a check, the same analysis was carried out on a shell element model as described above. In elastic compensation applied to shell elements, a different yield function is used, as there is only one element modelling the thickness. A generalised yield function is used in the analysis; the Ilyushin function is based on Mises' condition and correction is required to Figure 7.11.1-1: Equilibrium stress (intensity) field at design make it a lower bound action.

on the Tresca condition. Therefore, the design material strength parameter is scaled down by a factor of √3/2 to make the analysis a lower bound of the Tresca yield condition. The design material strength parameter applied to the shell analysis is therefore 140 MPa. Running on the same equipment as the solid model above, the equilibrium stress field converged after fourteen iterations with iteration fourteen giving the lowest

DBA Design by Analysis

Analysis Details Example 6

Page 7.248 (S)

maximum for the Ilyushin function. Figure 7.11.1-2 shows the Ilyushin equilibrium stress field, the contours relate to the squared Ilyushin function, where f(IL)2 is the square of the Ilyushin function from the contour plot, σ e the maximum equivalent stress in the equilibrium stress field and Rd the design material strength parameter. For the load case to be admissible according to the check 2

σ  f ( IL) =  e  Figure 7.11.1-2: Ilyushin equilibrium stress field  Rd  against GPD, the maximum Ilyushin function can be no greater than 1. From Figure 7.11.1-2 the maximum Ilyushin function is 1.08 (√1.167) - the load case is found inadmissible according to the GPD-check. As with the 2-D solid model the shell model shows that the load case is very near the pressure limit for the structure. 2

Check for LC 2 As with the check for load case 1, the partial safety factors are applied according to the check against GPD laid down in the rules. As the operating temperature is TSi = 160 oC the design material strength parameter remains 161.67 MPa. Applying the safety factor of γP = 1.2 on the actions gives pressures, used in the analysis , of γP. PSi = 1.56 MPa and γP. PSo = 0. Elastic compensation carried out on the 2-D model using the Tresca condition gave the lowest equilibrium stress field in iteration fourteen, shown in Figure 7.11.1-3. As would be expected, with the inner chamber pressure the same as in load 1, there is considerable plasticity shown in the knuckle region as in LC1. The maximum equivalent stress in the equilibrium stress field is 176.79 MPa, occurring in the knuckle region, which is approximately the same as in LC 1. The maximum stress is above the design material strength parameter and the elastic compensation analysis deems the load case inadmissible. Invoking the proportionality of the linear result from elastic compensation, the maximum PS max GPD = P ⋅

Figure 7.11.1-3 Equilibrium stress field at design load

Rd 161.67 = 1.3 ⋅ = 1.19 MPa σ max 176.79

DBA Design by Analysis

Analysis Details Example 6

Page 7.249 (S)

allowable load in the dished end can be found. The load multiplier can be given by the ratio to the design material strength parameter to the maximum equivalent stress found in the model. This renders Check for LC 3 In this load case, the temperature of the material will vary as the inner chamber is at 120oC and the outer chamber at 10oC. The material strength parameters will, therefore, vary throughout the structure. However, a conservative approach is to take the lowest material strength parameter and apply it over the whole structure. The lowest material strength parameter is 194 MPa, given at the temperature of 160oC. As before, the resulting design material strength parameter is 161.67 MPa. Applying the safety factor to the actions results in design pressures of γP. PSi = -0.12 MPa and γP. PSo = 0.6 MPa. Elastic compensation carried out on the 2-D solid model resulted in the lowest equilibrium stress field being obtained at iteration fourteen, shown in Figure 7.11.1-4. The maximum equivalent stress found is 115.2 MPa occurring in the relief groove in the outer chamber. As the maximum equivalent stress is considerably lower than the design material strength parameter, the load case is deemed admissible according to GPD. Figure 7.11.1-5: Equivalent stress at design actions

3. Check against GPD Using Non-linear Analysis Although the check against GPD via elastic compensation deemed the first two load steps inadmissible, the maximum equivalent stress was very close to the design strength parameter in both cases. As a check, a conventional non-linear analysis based on Mises' condition was performed for the first two load cases, to assess if the elastic compensation results are too conservative. Figure 7.11.1-4: Equilibrium stress (intensity) field at designaction.

Check for LC 1 Partial safety factors, design material strength parameters and action as in the elastic compensation analysis above. As the analysis is based on Mises' condition, a factor of √3/2 was applied to the design strength to make it a lower bound of the Tresca condition. The design material strength parameter is thus 140 MPa. An elastic-perfect plastic material model was used in accordance with the code. Non-linear analysis was carried out on the 2-D solid model with the pressures ramped simultaneously. Figures 7.11.1-5 shows the equivalent stress at the applied design actions.

Figure 7.11.1-6: Equivalent Stress at Limit Load

DBA Design by Analysis

Analysis Details Example 6

Page 7.250 (S)

The solution continued to converge to the design load indicating that the structure was still within the limit. As can be seen from Figure 7.11.1-5 the plasticity at the knuckle region of the dished end is extensive as was shown in the elastic compensation result. For the loading to be admissible, the maximum absolute value of total principal strain must not be greater than 5%. The maximum value of principal total strain is 0.35%. Therefore, from the non-linear analysis the load case is admissible according to the GPD-check. Figure 7.11.1-6 shows the equivalent stress at the limit pressure of 1.654 MPa. The resulting maximum allowable pressure for the cylinder head is 1.654/γP = (1.654/1.2), i.e. PSmaxGPD=1.378 MPa. The maximum total principal strain at this load is 1.1%. The same procedure was carried out for load case 2, the non-linear result converged up to the design loads and the maximum total principal strain was 0.38%. This would be expected as the pressure on the ellipsoidal head is limiting in both load cases 1 and 2. Therefore, the admissibility of load case 2 according to GPD using non-linear analysis is proved. 4. Admissibility check against PD. Presently it is not possible to apply the elastic compensation shakedown procedure to load sets involving thermal stresses, as a result of differential thermal expansions. Elastic compensation at present can not deal with thermally induced stress.

DBA Design by Analysis

Analysis Details Example 6

Analysis Type:

Page 7.251 (S)

Member:

Direct Route using elasto-plastic FE calculations

A&AB

FE-Software:

ANSYS® 5.4

Element Types:

4 – node, 2 – D axisymmetric solid PLANE42 (structural analysis) and PLANE55 (thermal analysis), respectively.

Boundary Conditions (structural analysis): ♣Whole model: Symmetry boundary conditions in the nodes in the centre of the dished end and in the nodes in the horizontal symmetry plane of the structure. ♣Partial model – dished end: Symmetry boundary conditions in the nodes in the centre of the dished end, and vertical displacement in the nodes in the edge of the cylindrical part constrained to 0. Model and Mesh:

Total number of elements 2833

DBA Design by Analysis

Results:

Analysis Details Example 6

Page 7.252 (S)

Actions and action cycles, as given in the specification, are admissible according to the GPD- and SD-checks.

1. Elements, mesh fineness, boundary conditions To model the structure (taking account of the symmetries), a total number of 2833 4-node axisymmetric elements – PLANE42 in ANSYS® 5.4 for the structural analysis and PLANE55 for the steady-state thermal analysis - was used (see figure on previous page). The boundary conditions applied to the model are symmetry ones in the nodes in the centre of the dished end (with the horizontal direction perpendicular to the plane of symmetry), and symmetry ones in the nodes in the horizontal symmetry plane of the structure. The welds of the stiffening rings and the weld between the inner vessel and the flat end of the jacket were modelled with fillets completely inside of the welds. In cases where thermal stresses need not to be included, e.g. in the GPD-check, the adjacent vessel does not influence the dished end’s stresses and strains significantly. Therefore, partial models of the dished end and the jacket with the jacketed part of the inner vessel could be used to determine the admissible pressure according to GPD conservatively. This was done additionally for guideline purposes for the dished end (see chapter 4), using a model with a total number of 1776 4-node axisymmetric PLANE42 elements. A thermal analysis was not necessary, since the model (i.e. this part of the whole structure) had a uniform temperature of 160°C during all operating conditions – as can bee seen from the results of the thermal analysis of the whole model. The model of the dished end is shown in Figure 7.11.2-1. To use partial models for the shakedown check could lead to non-conservative results, since the linear-elastic stress distribution including the thermal stresses of the whole structure is different from the ones using partial models. Figure 7.11.2-1

2. Steady-state thermal analysis The steady-state thermal analysis was carried out for medium (bulk) temperature of 10°C inside the jacket and 160°C inside the inner vessel. The corresponding heat transfer coefficients were specified by hi = 1.16 kW/(m2.K) on the inside of the vessel wall, ho = 14.4 kW/(m2.K) on the inside of the jacket on all surfaces, and the outer surface of the jacket and the vessel outside of the jacket

DBA Design by Analysis

Analysis Details Example 6

Page 7.253 (S)

were perfectly insulated. A temperature dependent thermal conductivity was used. For the input the following interpolation knots were used for the linear interpolation in the FE-software: 10°C: 14.11 W/(m.K); 20°C: 14.3 W/(m.K); 100°C: 15.8 W/(m.K); 160°C: 16.82 W/(m.K). Note: Since the structural and thermal calculations were carried out using the consistent mm-t-s unit system (1 t corresponds to 1000 kg) – as usual in structural calculations if the stress output is given in terms of MPa, the input unit for the thermal conductivity is given by mW/(mm.K), and for the heat transfer coefficient by mW/(mm2.K). Figure 7.11.2-2 shows the temperature distribution in the structure, the dished end shows a uniform temperature of 160°C after some distance above the jacket, and the jacket shows a uniform temperature of 20°C slightly away from the inner vessel wall. Figure 7.11.2-2

3. Admissibility check against GPD of the whole structure The admissibility checks have to be carried out for the following 4 states: LC1:

Internal pressure in the inner chamber internal pressure in the outer chamber temperature in both chambers

PS i = 1.3 MPa , PS o = 0.5 MPa , TS i = TS o = 160°C .

LC2:

Internal pressure in the inner chamber internal pressure in the outer chamber temperature in both chambers

PS i = 1.3 MPa , PS o = 0 MPa , TS i = TS o = 160°C .

LC3:

Internal pressure in the inner chamber internal pressure in the outer chamber temperature in the inner chamber temperature in the outer chamber

PS i = − 0.1 MPa , PS o = 0.5 MPa , TS i = 160°C , TS o = 10°C .

LC4:

Internal pressure in the inner chamber internal pressure in the outer chamber temperature in the inner chamber temperature in the outer chamber

PS i = 0 MPa , PS o = 0.5 MPa , TS i = 20°C . TS i = 10°C .

DBA Design by Analysis

Analysis Details Example 6

Page 7.254 (S)

Since the subroutine for Tresca’s yield condition showed bad convergence, the check against GPD was carried out using Mises’ yield condition (see subsection 3.2 of section 3 - Procedures) only. Thus, the elastic-plastic calculation was carried out using Mises’ yield condition and associated flow rule, a linear-elastic ideal-plastic material law, first order theory, and under usage of the Newton-Raphson method In the admissibility check of state LC1 the partial safety factor according to prEN 13445-3 Annex B, Table B.9-3, is given by γ R = 1.2 , and, therefore, a design material strength parameter given by RM 3 194 3 ⋅ = ⋅ = 140 MPa γR 2 1.2 2 was used (corresponding to a material strength parameter of 194 MPa for a temperature of 160°C according to prEN 10028-7). According to prEN 13445-3 Annex B, Table B.9-2, the partial safety factor for pressure without natural limit) is given by γ P = 1.2 . Thus, the check was carried out with internal pressures of γ P ⋅ PS i = 1.56 MPa for the inner, and γ P ⋅ PS o = 0.6 MPa for the outer chamber, respectively.

Figure 7.11.2-3

The corresponding distribution of the Mises equivalent stress is shown in Figure 7.11.2-3. There is plastification in the knuckle region of the dished end, and a very small zone of plastification in the stress relief groove of the flat end of the jacket. As can be seen in Figure 7.11.2-4, the maximum value of the absolute principal strains in the structure is 0.38 %, and, therefore, the admissibility of this state in the GPD check is shown.

Figure 7.11.2-4

Analysis Details

DBA Design by Analysis

Example 6

Page 7.255 (S)

Since the temperatures in LC 2 are equal to those of state LC1, and the pressures acting on the structure are a part of those applied in state LC 1, the admissibility of state LC2 is shown by the admissibility of LC 1. In the admissibility check of state LC3 a partial safety factor γ R = 1.2 for the material, and temperature dependent design material strength parameters RM 3 , ⋅ γR 2 were used, where RM corresponds to the 1% proof strength values according to prEN 10028-7 at the considered temperatures. Thus, as input the following interpolation knots were used for the linear interpolation in the FE software: 20°C: 173.2 MPa; 100°C: 150.1 MPa; 150°C: 141.45 MPa; 160°C: 140 MPa. The modulus of elasticity was also specified in temperature dependent form. To use all of those temperature dependent values in the calculation, the results from the thermal analysis were used as input for the structural analysis, but since no thermal stresses were considered in the GPD-check, the coefficient of thermal expansion was set to zero. The partial safety factor of pressure without a natural limit is given by γ P = 1.2 , and with a natural limit by 1.0 . Thus, the check was carried out with pressures given by 1.0 ⋅ PS i = − 0.1 MPa and γ P ⋅ PS o = 0.6 MPa for inner and outer chamber, respectively. The resulting distribution of the Mises equivalent stress is shown in Figure 7.11.2-5. Only a very small zone of plastification in the stress relief groove and on the outer surface of the end plate of the jacket can be observed. The corresponding maximum of the absolute values of the principal strains in the jacket and the jacketed part of the inner vessel is 0.116 %, and, therefore, the admissibility of this state in the GPD-check is shown. Figure 7.11.2-5

Since the temperatures in LC 4 are smaller compared with those of state LC3, thermal stresses have no influence on the GPD-check, and the pressures acting on the structure are a part of those applied in state LC 3, the admissibility of state LC4 is shown by the admissibility of LC 3.

DBA Design by Analysis

Analysis Details Example 6

Page 7.256 (S)

4. Maximum admissible pressure of the dished end according to GPD Since the subroutine using Tresca’s yield condition showed bad convergence, Mises’ yield condition and associated flow rule, a linear-elastic ideal-plastic law with a design material strength parameter of 194 MPa, first order theory, and the Newton-Raphson method was used, and, therefore, the result was scaled down with the factor 3 / 2 (see subsection 3.2 of section 3 Procedures). The material’s partial safety factor is given by γ R = 1.2 , and the partial safety factor for pressure (without a natural limit) by γ P = 1.2 . The computation time of the limit load was 12 minutes on the Compaq® Professional Workstation 5000 with two Pentium Pro® processors and 256 MB RAM, and the internal limit pressure of the structure was found to be 2.25 MPa. Figure 7.11.2-6 shows the corresponding Mises’ equivalent stress distribution. The corresponding maximum absolute principal strain was 1.4%, and, thus, the condition in prEN 13445-3 Annex B.9.2.2 is fulfilled . Figure 7.11.2-6

Therefore, the allowable (internal) pressure according to the GPD-check is given by

PS max GPD =

2.25 3 17.18 3 ⋅ = ⋅ = 1.35 MPa . γ P ⋅ γ R 2 1.2 ⋅ 1.2 2

DBA Design by Analysis

Analysis Details Example 6

Page 7.257 (S)

5. Admissibility check against PD In the following, it is shown that Melan’s shakedown theorem is fulfilled for a cycle through all states under consideration, i.e. the structure shakes down, and, therefore, progressive plastic deformation cannot occur. The following states have to be considered, including possible thermal stresses : LC1:

Internal pressure in the inner chamber internal pressure in the outer chamber temperature in both chambers

PS i = 1.3 MPa , PS o = 0.5 MPa , TS i = TS o = 160°C .

LC2:

Internal pressure in the inner chamber internal pressure in the outer chamber temperature in both chambers

PS i = 1.3 MPa , PS o = 0 MPa , TS i = TS o = 160°C .

LC3:

Internal pressure in the inner chamber internal pressure in the outer chamber temperature in the inner chamber temperature in the outer chamber

PS i = − 0.1 MPa , PS o = 0.5 MPa , TS i = 160°C , TS o = 10°C .

LC4:

Internal pressure in the inner chamber internal pressure in the outer chamber temperature in the inner chamber temperature in the outer chamber

PS i = 1.3 MPa , PS o = 0.5 MPa , TSi = 160°C , TS o = 10°C .

LC5:

Internal pressure in the inner chamber internal pressure in the outer chamber temperature in the inner chamber temperature in the outer chamber

PSi = 0 MPa , PS o = 0.5 MPa , TSi = 20°C , TS o = 10°C .

A linear-elastic calculation was carried out for all these states. Corresponding to the states, either a uniform temperature or the temperature distribution according to the thermal analysis was used. The modulus of elasticity and the coefficient of thermal expansion were specified temperature dependent, with the following interpolation knots used for the linear interpolation in the software: Modulus of elasticity: 10°C: 197 GPa; 20°C: 196 GPa; 100°C: 190 GPa; 160°C: 185.2 GPa. Coefficient of thermal expansion: 10°C: 16.05e-6 1/K; 20°C: 16.1e-6 1/K; 100°C: 16.7e-6 1/K; 160°C: 17e-6 1/K. The following figures show the linear-elastic Mises equivalent stress distributions for these states: Figure 7.11.2-7 – state LC1, Figure 7.11.2-8 - state LC2, Figure 7.11.2-9 – state LC3, Figure 7.11.2-10 – state LC4, and Figure 7.11.2-11 – state LC5. The relevant design material strength parameter for the PD-check is given by RM = 217 MPa , which corresponds to the arithmetic mean of the 1% proof strength values R p1, 0 / t for the highest

DBA Design by Analysis

Analysis Details Example 6

Page 7.258 (S)

(160°C) and the lowest (10°C and 20°C, respectively) calculation temperatures during the whole action cycle. Taking this design material strength parameter into account, it can be seen from the Figures 7.11.27 to 7.11.2-11, that the behaviour of the jacket and the jacketed part of the inner vessel is completely elastic in the states LC1, LC2 and LC5, but in the states LC3 and LC4 the maximum Mises equivalent stress is larger than the corresponding design material strength parameter, and, therefore, plastification will occur. The dished end is completely elastic in the state LC3 and LC5, but in the states LC1, LC2 and LC4 the maximum Mises equivalent stress is larger than the design material strength parameter, and, therefore, plastification Figure 7.11.2-7 will occur.

Figure 7.11.2-8

DBA Design by Analysis

Figure 7.11.2-9

Figure 7.11.2-10

Analysis Details Example 6

Page 7.259 (S)

DBA Design by Analysis

Analysis Details Example 6

Page 7.260 (S)

Figure 7.11.2-11

Figure 7.11.2-12 shows the Mises equivalent stress distribution of the linear-elastic calculated thermal stresses only.

Figure 7.11.2-12

DBA Design by Analysis

Analysis Details Example 6

Page 7.261 (S)

From the behaviour of the structure, it was concluded that the thermal stresses are the reason for the plastification in the jacket and in the jacketed part of the inner vessel, whereas the internal pressure is the reason for the plastification in the dished end. Therefore, a pressure induced self-equilibrating stress field alone seems to be not sufficient to fulfil Melan’s theorem with the 5 linear-elastic stress fields given above, and the same applies for a thermal induced stress field (which is a selfequilibrating stress field) alone. To obtain a better understanding of the situation, the stresses in the critical points of the structure were drawn in a deviatoric map – see Figure 7.11.2-14. These critical points are given by the locations where the maximum (equivalent) stresses of the different load cases occured: • • • •

LC1 and LC2: node 93 at the inner side of the knuckle region of the dished end (designation “A” in Figure 7.11.2-14), LC3: node 2694 at the inside of the inner vessel wall (designation “D” in Figure 7.11.2-14), LC4: node 441 at the outer surface of the end plate of the jacket (designation “B” in Figure 7.11.2-14), LC5: node 751 in the stress relief groove of the annular end plate of the jacket (designation “C” in Figure 7.11.2-14).

The locations of these critical points are shown in Figure 7.11.2-13.

Figure 7.11.2-13

Analysis Details

DBA Design by Analysis

Page 7.262 (S)

Example 6

B

LC4

LC3

Thermal B C

C

B

C

Res(O) Res(I) A

LC2

C C

D

LC1 D

B

B D

C A

C

LC5 D C

0.32 Res(O)

B

C

D

-0.34 Thermal

D

A D

Figure 7.11.2-14

To obtain a pressure induced self-equilibrating stress field for the knuckle region of the dished end, a linear-elastic and an elasto-plastic FE-calculation, with an internal pressure of 2 MPa applied to the main shell of the structure, was carried out. The difference of a elasto-plastic stress field and linear-elastic one is a self-equilibrating stress field; Figure 7.11.2-15 shows the corresponding equivalent stress distribution. Point A is the only one of the critical locations where this selfequilibrating stress field is not zero, the corresponding deviatoric point is designated in Figure 7.11.2-14 by [Res(I)]. The sums of this self-equilibrating stress field and all stress states of point A are inside the deviatoric limit circle.

DBA Design by Analysis

Analysis Details Example 6

Page 7.263 (S)

Figure 7.11.2-15

An appropriate self-equilibrating stress field for the application of Melan’s theorem for the points B and D is given by the thermal stress field multiplied by –0.34. The corresponding deviatoric mapping of this stress field for the critical locations is shown in Figure 7.11.2-14. Unfortunately, the sum of this self-equilibrating stress field and the stress state of point C for LC5 is outside the limit circle. Therefore, a third self-equilibrating stress field was used: to obtain a self-equilibrating stress field, which influences only the behavior of point C, a linear-elastic and an elasto-plastic FE calculation, with an internal pressure of 1.2 MPa applied only in the jacket of the structure, was carried out. The difference of the corresponding elasto-plastic and linear-elastic stress fields is a self-equilibrating stress field; Figure 7.11.2-16 shows the corresponding equivalent stress distribution. Point C is the only one of the critical locations, in which this self-equilibrating stress field is not zero. The corresponding deviatoric point is designated in Figure 7.11.214 by [Res(O)]. The sums of this self-equilibrating stress field multiplied by 0.32 - see Figure 7.11.2-14 -, the thermal stress field multiplied by –0.34 and all stress states of point C are inside the deviatoric limit circle. Figure 7.11.2-16

DBA Design by Analysis

Analysis Details Example 6

Page 7.264 (S)

The conclusion is, that a selfequilibrating stress field, which is given by the sum of the selfequilibrating stress fields [Res(I)], 0.32 [Res(O)] and –0.34 [Thermal] can be used to fulfil Melan’s theorem for all critical locations, and, therefore, the structure shakes down in these points for the action cycle under consideration. Figure 7.11.2-17 shows the equivalent stress distribution of this selfequilibrating stress field.

Figure 7.11.2-17

To show that Melan’s theorem is fulfilled for all points of the structure, this self-equilibrating stress field is superposed to the linear-elastic stress fields of the load cases. The corresponding stress distributions are shown in the Figures 7.11.2-18, 7.11.2-19, 7.11.2-20, 7.11.2-21 and 7.11.2-22. As can be seen, the maximum equivalent stress nowhere exceeds 212 MPa, and, therefore, it is shown that the structure shakes down in all points for the action cycle under consideration.

Figure 7.11.2-18

DBA Design by Analysis

Figure 7.11.2-19

Figure 7.11.2-20

Analysis Details Example 6

Page 7.265 (S)

DBA Design by Analysis

Figure 7.11.2-21

Figure 7.11.2-22

Analysis Details Example 6

Page 7.266 (S)

DBA Design by Analysis

Analysis Details Example 6

Page 7.267 (S)

DBA Design by Analysis Analysis Type:

Analysis Details Example 6 / F-Check

Page 7.268 (A) Member:

Check against fatigue

A&AB

FE –Software:

ANSYS 5.4

Element Types:

4 – node, 2 – D axisymmetric solid PLANE42 (structural analysis) and PLANE55 (thermal analysis), respectively.

Boundary Conditions (structural analysis): Symmetry boundary conditions in the nodes in the centre of the dished end and in the nodes in the horizontal symmetry plane of the structure. .

Model and Mesh:

Total number of elements: 2833

Results: Fatigue life N = 20506 cycles

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Analysis Details

Page 7.269(A)

Example 6 / F-Check

In each operating cycle four different action states are taken on: AS1: P0i = 0; P0o = 0;

T0i = 20° C; T0o = 20° C;

AS2: P0i = 0; T0i = 20° C; P0o = 4.5 bar; T0i = 20° C; AS3: P0i = 11 bar; T0i = 160° C; P0o = 4.5 bar; T0o = 160° C; AS4: P0i = 11 bar; T0i = 160° C; P0o = 4.5 bar; T0o = 10° C; Where P0i and T0i denote operating pressure and temperature in the inner chamber, and P0o and T0o operating pressure and temperature in the jacket. A full operating cycle corresponds to the series AS1 – AS2 – AS3 – AS4 – AS3 – AS2 - AS1. The maximum equivalent stress range occurs for AS1 – AS4 in the knuckle region of the dished end. A rough, conservative check shows that the fatigue life in this point is larger than the one for the critical welded regions – the weld seams of the outer jacket shell to the annular end plates and the weld seams of the annular end plates to the main shell. The fatigue class of the former is 40 – the inside can not be visually inspected – and the fatigue class of the latter is 71 – welding from both sides. Again, a rough check shows that the weld seam outer jacket shell to end plates govern the fatigue life, and the critical point is the weld toe (on inside). Therefore, details for this point are given in the following. The prinicipal stresses normal and parallel to the weld seam for the four action states are given in the following table (in the order of their occurence in the operating cycle)

σl σc

AS1 0

AS2 138.3

AS3 114.2

AS4 -12.2

AS3 114.2

AS2 138.3

AS1 0

0

-39.0

66.7

228.0

66.7

-39.0

0

There are two stress ranges for the principal stress normal to the weld

∆σ struc = 150.5 MPa ∆σ struc = 138.3 MPa

(AS3 – AS4) (AS1 – AS2)

For the principal stress parallel to the weld only one stress range results. Since for this weld fatigue class 80 seems to be justified – with proper testing for full penetration and freedom from significant flaws – this principal stress does not govern the fatigue life (N = 86060 cycles).

DBA Design by Analysis

Analysis Details Example 6 / F-Check

Page 7.270 (A)

Data Welded region / Weld toe of jacket to annular plate: Principal stress range approach en = 8 mm tmax = 160 °C ∆σD (5.106cycles) = 46 MPa (class 63) tmin = 10 °C principal stresses or equivalent stresses t* = 0,75 tmax + 0,25 tmin = 122.5°C 11 . Rm = 520 MPa m = 3 C⊥ = 1.28 10 m=3 C= Rp1,0/t* = 202.6 MPa C// = ….. m = 5 C⊥ = ….. m=5 C= C// = ….. Stresses

Cycle: AS3 – AS4 – AS3

∆σstruc = 150.5 MPa (structural equivalent range) (obtained by quadratic extrapolation on the jacket shell inside) 18.8 Plasticity correction factor ke Thermal loading

mechanical loading

If ∆σstruc > 2 Rp1,0/t*

If ∆σstruc > 2 Rp1,0/t*

 Äó  struc − 1 k = 1 + 0, 4  e  2 R p1,0/t * 

k = υ

ke = ….. ∆σ = ke ∆σstruc = ….. MPa Else ∆σ = ∆σstruc = 150.5 MPa

0,5 +

0,7 0,4

 Äó   struc   R p1,0/t * 

kυ = ….. ∆σ = kυ ∆σstruc = ….. MPa Else ∆σ = ∆σstruc = ….. MPa

If both mechanical and thermal loadings are to be considered, the correction has to be made on each component of the stress tensors . ke and k í are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated. 18-10-6-1 Thickness correction factor few en ≤ 25 mm few = 1

25 mm ≤ en ≤ 150 mm few = (25/en)0.25 = few = …..

en ≥ 150 mm few = 0,639

DBA Design by Analysis

Analysis Details

Page 7.271 (A)

Example 6 / F-Check

18-10-6-2 Temperature correction factor ft* For t* > 100 °C Else ft* = 1

ft* = 1,043 – 4,3 10-4 t* = 0.9903

18-10-6-3 Overall correction factor fw fw = few.ft* = 1 18-10-7 Allowable number of cycles N Äó fw

If

= 152.0 MPa

Äó fw

> ∆σ5.106 cycles then

m = 3 and C (C⊥ or C//) = 1.28.1011

N=

C  ∆σ   fw

  

m

= 36470 cycles

If

Äó fw

< ∆σ5.106 cycles and other

cycles with

Äó fw

> ∆σ5.106 cycles

then m = 5 and C (C⊥ or C//) = ….. N = C m = …..  ∆σ   f   w 

If

Äó fw

< ∆σ5.106 cycles and all other

cycles with then N=∞

Äó fw

< ∆σ5.106 cycles

DBA Design by Analysis

Analysis Details Example 6 / F-Check

Page 7.272(A)

Data Welded region / Weld toe of jacket to annular plate: Principal stress range approach en = 8 mm tmax = 160 °C tmin = 20 °C ∆σD (5 106cycles) = 46 MPa (class 63) t* = 0,75 tmax + 0,25 tmin = 125 °C principal stresses or equivalent stresses Rm = 500 MPa m = 3 C⊥ = 1.28.1011 m=3 C= Rp1,0/t* = 202 MPa C// = ….. m = 5 C⊥ = ….. m=5 C= C// = ….. Stresses

Cycle: AS1 – AS1

∆σstruc = 138,3MPa (structural equivalent stress range) (obtained by quadratic extrapolation on the jacket shell inside) 18.8 Plasticity correction factor ke Thermal loading

mechanical loading

If ∆σstruc > 2 Rp1,0/t*

If ∆σstruc > 2 Rp1,0/t*

 Äó  struc − 1 k = 1 + 0, 4  e  2 R p1,0/t * 

k = υ

ke = ….. ∆σ = ke ∆σstruc = ….. MPa Else ∆σ = ∆σstruc = 138.3 MPa

0,5 +

0,7 0,4

 Äó   struc   R p1,0/t * 

kυ = ….. ∆σ = kυ ∆σstruc = ….. MPa Else ∆σ = ∆σstruc = ….. MPa

If both mechanical and thermal loadings are to be considered, the correction has to be made on each component of the stress tensors . ke and k í are to be calculated with the above formulas where Äóeq,l is the full mechanical and thermal equivalent stress range. The factor ke is applied to the mechanical stress tensors and the factor kυ is applied to the thermal stress tensors. Then both tensors are added and the new stress range is calculated. 18-10-6-1 Thickness correction factor few en ≤ 25 mm few = 1

25 mm ≤ en ≤ 150 mm few = (25/en)0.25 = few = …..

en ≥ 150 mm few = 0.639

DBA Design by Analysis

Analysis Details

Page 7.273 (A)

Example 6 / F-Check

18-10-6-2 Temperature correction factor ft* For t* > 100 °C Else ft* = 1

ft* = 1,043 – 4,3 10-4 t* = 0.9892

18-10-6-3 Overall correction factor fw fw = few .ft* = 0.9892 18-10-7 Allowable number of cycles N Äó fw

If

= 139.8 MPa Äó fw

> ∆σ5.106 cycles then

m = 3 and C (C⊥ or C//) = 1.28.1011

N=

C  ∆σ   f   w 

m

= 46845cycles

If

Äó fw

< ∆σ5.106 cycles and other

cycles with

Äó fw

> ∆σ5.106 cycles

then m = 5 and C (C⊥ or C//) = ….. N = C m = ….. cycles

If

Äó fw

< ∆σ5.106 cycles and all other

cycles with

Äó fw

< ∆σ5.106 cycles

then N=∞

 ∆σ   f   w 

Combing the two subcycles to the full operating cycle (with allowed number of cycles N) gives, with

n1 n N N + 2 = + = 1, N1 N 2 N1 N 2 finally N = 20506 cycles

DBA Design by Analysis

Analysis Details Example 6 / I-Check

Analysis Type:

Page 7.274 (A)

Member:

I-Check

A&AB

FE-Software:

ANSYS® 5.4

Element Types:

8 – node structural shell elements SHELL93

Boundary Conditions:

Symmetry b.c. in the nodes in the vertical symmetry plane of the structure; the displacements in one node at the junction of the inner shell and its stiffener ring in this symmetry plane was constrained to zero in horizontal and vertical direction, and in the other one constrained to zero in vertical direction only.

Model and Mesh:

Total number of elements 4608

Results:

Actions as given in the specification are admissible according to the I-Check

DBA Design by Analysis

Analysis Details Example 6 / I-Check

Page 7.275 (A)

1. General The following approach was used here to show the admissibility of the specified actions against instability: A input to the fully nonlinear analysis of the structure with initial imperfections, an eigenvalue buckling calculation of the perfect structure (with linear-elastic constitutive law) was performed, and the so obtained buckling shape of the first buckling mode is scaled with regard to the specified maximum (admissible) out-of-roundness of the structure and the result used for the imperfect geometry. The latter analysis was performed using the nonlinear geometry and a linearelastic ideal-plastic constitutive law. 2. FE-model To include the critical non-rotational symmetric buckling mode forms, and the fact that the buckling forms are not necessarily symmetric to the horizontal symmetry plane of the structure, an FE-model representing half of the structure was used. Because of the large diameter to wall thickness ratio of the structure shell elements, i.e. the 8–node SHELL93 elements in ANSYS® 5.4, were used. In the nodes in the vertical symmetry plane of the structure symmetry boundary conditions were applied. Additionally, to avoid rigid body motions, the displacement in one node at the junction of the inner shell and its stiffener ring in the symmetry plane was constrained to zero in horizontal and vertical direction, and the other one constrained to zero in vertical direction only. The FE-model used for the eigenvalue buckling approach was created as usual by the preprocessor of the FE-software. The model used for the fully nonlinear analysis, which included the shape imperfections corresponding to the first eigenvalue buckling mode scaled by the allowable out-of-roundness, was created by use of the macro IMPER (see Annex 5: Model and Solution Input), which determined the geometry i.e. the coordinates of the nodes - of the imperfect structure using the perfect geometry and the results of the eigenvalue buckling calculation. In this procedure the FE-mesh is detached from the geometric lines and areas, which had been used to create the model of the perfect structure. Therefore, the different loads which have to be applied on different parts of the structure in the fully nonlinear analysis, have to be applied directly on the corresponding elements. To make the selection of these elements easy, the real constant number notation in ANSYS®5.4 was used to perform this selection - see picture on preceeding page, where the different colours correspond to different real constant numbers. 3. Eigenvalue buckling approach An eigenvalue buckling analysis predicts the theoretical buckling strength (the bifurcation point) of an (geometrically) ideal and linear-elastic structure. This approach corresponds to the textbook approach for elastic buckling analysis. In the case under consideration the reason for performing the eigenvalue buckling analysis is two-fold: to have a comparison value for other approaches, and to obtain a relevant buckling mode shape, to be used as an initial imperfection in the model of the fully nonlinear analysis. The first step of an eigenvalue buckling analysis is the calculation of the static solution for “unit load(s)”, i.e. small values of the loads for which the buckling mode shall be calculated. In the next step the program calculates multiplication factors for these unit loads corresponding to the relevant buckling modes and the buckling mode shapes.

DBA Design by Analysis

Analysis Details Example 6 / I-Check

Page 7.276 (A)

The relevant load case, for which the instability check has to be performed is LC 3: Internal pressure in the inner chamber PS i = − 0.1 MPa , internal pressure in the outer chamber PS o = 0.5 MPa , temperature in the inner chamber TS i = 160°C , temperature in the outer chamber TS o = 10°C . The modulus of elasticity used in the calculations was E = 191 GPa , a value which corresponds to the mean temperature of the wall of the buckling-critical inner cylinder. Since the program failed to calculate eigenvalue buckling modes for the case of internal pressure applied to the outside wall of the jacket, and since the results were to be used as imperfection of the geometry for the fully nonlinear approach only, the action used for the eigenvalue calculation was external pressure acting on the inside wall of the jacket. The value of this “unit external pressure” was 0.7 MPa , the calcualted multiplication factor was 7.18, and, therefore, the lowest eigenvalue buckling pressure corresponding to the model and load used was 5.02 MPa . Figure 7.11.4-1 shows the deformed shape of this first eigenvalue buckling mode, the deformation is scaled up in order to show the (bifurcation) buckling mode shape clearly.

Figure 7.11.4-1

DBA Design by Analysis

Analysis Details Example 6 / I-Check

Page 7.277 (A)

4. Fully Nonlinear Analysis A nonlinear buckling analysis is simply a nonlinear static analysis, which can be extended to a point where the structure reaches its limit load or maximum load carrying capacity, or where simply the admissibility of specified actions is shown. The latter simpler approach was followed here. In the present analysis an initial imperfect geometry, geometric nonlinearities (2nd order theory) and a linear elastic – ideal plastic constitutive law was used. With the exception of temperature, the partial safety factors for actions in the instability check are equal to those of the GPD-check, i.e. the partial safety factor for internal pressure is γ P = 1.2 and for external pressure (vacuum) the partial safety factor is γ P = 1.0 ; for thermal stresses the partial safety factor is γ T = 1.0 . Since the hydraulic pressure test can be performed as required, the partial safety factor for the resistance of the structure against instability is γ R = 1.25 . Since only the admissibility of the specified actions has to be shown , all characteristic values of the actions were multiplied with the corresponding partial safety factors for the actions and the partial safety factor for the resistance (see subsection 3.7.3 of section 3 - Procedures). Thus, the following actions were applied: • internal pressure of 0.5 ⋅ 1.2 ⋅ 1.25 = 0.75 MPa in the jacket, • external pressure of − 0.1 ⋅ 1.0 ⋅ 1.25 = −0.125 MPa on the inside of the main shell, • temperature distribution: as an approximation of the temperature distribution calculated in the steady state thermal analysis, the following temperature distribution have been used for the shell model: main shell outside of the jacket T = 160°C , end rings of the jacket and jacket’s outside wall T = 10°C , main shell outside surface in jacket region T = 20°C , main shell inside surface in jacket region T = 102.6°C . This approximate temperature distribution, multiplied by the partial safety factor for the resistance γ R = 1.25 was applied to FE-model. To obtain the (initial) imperfect geometry of the structure, the results and the ANSYS® 5.4 database of the eigenvalue buckling calculation were used. The macro IMPER generated an (initial) imperfect geometry according to the first buckling mode shape with a maximum deviation of 20.15 mm from the perfect geometry; this value corresponds to the specified maximum out-of-roundness of the main vessel cylindrical wall. The fully nonlinear calculation was performed with temperature dependent elastic modulus and thermal expansion coefficients, where the following interpolation knots were used in the input: Elastic modulus: 0°C: 198 GPa; 20°C: 196 GPa; 100°C: 190 GPa; 200°C: 182 GPa. Thermal expansion coefficient: 0°C: 16.0e-6 1/K; 20°C: 16.1e-6 1/K; 100°C: 16.7e-6 1/K; 200°C: 17.2e-6 1/K. For the linear elastic – ideal plastic constitutive law the following design material strength parameter values, corresponding to the 1%-yield strength of X6CrNiTi18-10 according to prEN 10028-7, were used as interpolation knots: 0°C: 240 MPa; 20°C: 240 MPa; 100°C: 208 MPa; 150°C: 196 MPa; 200°C: 186 MPa. The analysis was performed using the Newton-Raphson method. The computation time until the final load level was reached was 6 hours and 31 minutes on the Compaq® Professional Workstation 5000 with two Pentium Pro® processors and 256 MB RAM.

DBA Design by Analysis

Figure 7.11.4-2 shows the distribution of the Mises equivalent stress of the whole structure, and figure 7.11.4-3 of the jacketed part of the main shell, according to the final load level. Figure 7.11.4-4 shows the corresponding maxi-mum principal plastic strain distribution of the whole structure. Since the final load level was reached via a convergent solution of the fully nonlinear FE analysis, admissibility of the actions under consideration against instability is shown. F

Figure 7.11.4-3 igure 7.11.4-2

Analysis Details Example 6 / I-Check

Page 7.278 (A)

DBA Design by Analysis

Figure 7.11.4-4

Analysis Details Example 6 / I-Check

Page 7.279 (A)

DBA Design by Analysis

Analysis Details Example 6 / I-Check

Page 7.280 (A)

5. Handbook formulae approach Load Case 3, Inner shell: • Axial stress resultant: FE-calculation: Axial stress resultant (per unit length) na = −22.66 N / mm This value results from the actions for load case 3, i. e. for characteristic values. The partial safety factors for the internal pressure is 1.0, for the external 1.2. The partial safety factor for thermal stresses is not specified, the value 1.0 is considered appropriate. To avoid new calculations a conservative design axial stress of na , d = − (1.2) ( 22.66) = −27.2 N / mm is used. Critical value: Roark’s Formulas for Stress and Strain, 6th edition, Table 35, Case 15: ncrth = E (e 2 r ) /(3(1 − ν 2 )) 0.5 E = 191 GPa, e = 20mm, r = 1390mm , ν = 0.3 According to the civil engineering code for steel structures DIN 18800 a reduction factor of 1.0 can be used in this case of a small relative slenderness ratio: A practical value of na , pr = na , th = 33266 N / mm results; there follows: Stability check for axial stress: na , d / n a , pr = 0.0008 • External pressure (difference): Characteristic values of pressures: internal pressure: pi , c = −0.1 MPa external pressure: p e, c = 0.5 MPa Design value of pressure difference: p d , d = (1.0 ) (−0.1) − (1.2) (0.5) = −0.7 MPa

DBA Design by Analysis

Analysis Details Example 6 / I-Check

Page 7.281 (A)

Critical value: Roark’s Formulas for Stress and Strain, 6th edition, Table 35, Case 19b: p crth = 0.807 E (e 2 / lr ) (e / r ) 0.5 ⋅ (1 − v 2 ) −0.75 l = 1492 p crth = 3.83 MPa pcr = 3.06 MPa with a knock-down factor 0.8 According to the civil engineering code for steel structures DIN 18800 the theoretical value is given by 4.7 MPa, the reduction factor by 0.745. Thus p d , pr = 3.5 MPa , and p d , d / p d , pr = 0.2 • Stability check: na , d / n a, pr + p d , d / p d , pr = 0.2 < 1 / γ I = 0.8 , with γ I = 1.25 , since the required pressure test can be performed for the relevant action, the pressure difference. Note: The influence of the temperature moment on the critical inner shell – due to the temperature gradient over the wall – is not included in this result, but the safety margin seems to be sufficient. Note: The boundary conditions used in both buckling cases are: • Radial displacement at both ends of (inner) shell zero • No other constraints – end sections may warp. In the actual shell the radial displacement of the end section is restrained by the annular end rings of the jacket (and the outer shell), but not totally. But, nevertheless, the first boundary condition is considered to be a good one. The axial displacement is restrained by the ends of the vessel and, to a minor extent, by the annular end rings of the jacket and the outer shell. The second boundary condition is considered to be a (poor) conservative one.

DBA Design by Analysis

Analysis Details Example 6 / I-Check

Page 7.282 (A)

1

DBA Design by Analysis

Recommendations

Page 8.1

8 Recommendations 8.1 Overview In this final Section, the experience the writers have gained from this study, and general and specific recommendations for the aspiring analyst/designer or engineering manager are summarised. To begin with, various recommendations concerning the methodology of design by analysis – specifically elastic-plastic finite element analysis – are proposed. This is followed by a brief discussion of software requirements (although it is noted that most commercial finite element systems in common use in the industry have sufficient capability). Then a discussion on the requirements – expertise and engineering or technical knowledge – is provided. It is argued that the analyst plays a critical role in the design by analysis process, and that careful attention should be paid to their appropriate training, ideally through Continuing Professional Development. Finally, a few concluding remarks are made.

8.2 Methodology Commercial finite element analysis systems which are capable of carrying out the analyses required for design by analysis using the Direct Route are common (this is discussed further in sub-section 8.3). Current desktop PC-based computing hardware, running Windows 98 or NT, with twin Pentium III processors and analysis software optimised to use them, and large fast multi-gigabyte hard discs, are more than adequate for the majority of vessels and actions which need to undergo design by analysis. The weak areas of the design by analysis procedure relates more to the expertise of the analyst and the suitability (or correctness) of the analysis. The former will be discussed in more detail in sub-section 8.4, while the latter is examined here. To begin with, one crucial but misunderstood aspect of modern finite element analysis needs to be emphasised: Our practical knowledge of the behaviour of modern finite element formulations, and what constitutes a ‘suitable’ finite element model, is almost exclusively concerned with linear elastic behaviour with small deformations. This (limited) core expertise is also the case with the majority of practising finite element analysts. Put simply, a finite element model which may be suitable for linear elastic, small deformation behaviour under simple actions may be wholly unsuitable for elastic-plastic analysis, as well as large deformations and multiple actions. Even an adapted finite element mesh (constructed during the kind of adaptive finite element procedures available in many commercial systems) is only really valid for linear elastic, small deformation behaviour. For example, in thin-walled vessels it is common for relatively small weak areas (related to the formation of plastic hinges) to develop. When this occurs, finite element theory would recommend re-meshing these areas if detailed plastic strain levels are required. Such a re-mesh is not common. Similarly, experience with the effect of poorly shaped elements is almost exclusively based on linear elastic analysis with limited actions. Most commercial systems provide various checks and corrections for element shape, according to common industry guidelines. The validity of these checks for elastic-plastic behaviour is largely unknown. It remains good advice to try to make the element shape in regions of extensive plasticity close to its natural ‘square’ shape, but this requires analysts skill, and arguably could suggest a re-mesh once the plastic region is identified. Also,

2

DBA Design by Analysis

Recommendations

Page 8.2

unless the analyst is familiar with the approximations and assumptions of the Classical Theory of Plasticity, which is contained in the majority of commercial systems, he may be led to believe that the strain levels being predicted for multiple actions are realistic. In regions which are essentially compressive or highly constrained, this can be far from the truth! As a final example, it is often not fully appreciated, or conveniently forgotten, that the majority of common commercial finite element analysis systems are based on the displacement formulation, where displacement is interpolated within an element in terms of the nodal values. As a consequence stresses are discontinuous from element to element and in general equilibrium is only satisfied at nodes. When this is placed in the context of a highly nonlinear material, this implies that the stresses are even less reliable in nonlinear analysis than in linear, and that it is very easy for errors to propagate as load increases. Very few commercial systems include mixed formulations, where both stress and displacement are interpolated and equilibrium is satisfied, although these are common in the theoretical literature. It is fairly easy to construct problems where the displacement formulation in commercial software fails to converge to the correct solution for nonlinear analysis[1] . The ten benchmark examples studied in Section 7, can for the most part, be modelled with well structured finite element meshes. However, in the following some basic recommendations, or warnings, are summarised. After some general comments, the discussion is split into two parts: the first looks at more general modelling issues for nonlinear finite element analysis, while the second deals with issues specific to this study: 8.2.1 General comments The novice may only have a general idea of the role of advanced analysis and simulation in engineering design. A common view is that the aim is to simulate, and thereby quantify, the actual behaviour of the component or structure. This is far from the truth. In the design of pressure vessels it has to be remembered at all times that the aim is to check the code requirements. As a consequence the code rules tell us that small details, such as welds, may be omitted. Further, as discussed in detail in Section 2, the results of stress categorisation have to be post-processed in a particular manner – the code stress limits apply to membrane and bending stress, and except for fatigue ignore the peak. If the stress categorisation route is used, then the allowable loads are based on shakedown and limit analysis, which assume perfect plasticity to introduce a failure mechanism in the component. Perfect plasticity is a hypothetical material behaviour. Indeed the Classical Theory of Plasticity is an approximation. In general any finite element model is an approximation, and this is even more true for nonlinear behaviour. Since any model is an approximation, engineering judgement, both in the modelling and assessment phase, is essential. One approach, or modelling and assessment framework, for pressure vessel design by analysis which the analyst could refer to is the following[2]: • • • •

Ensure there is a good understanding of how the finite element model behaves Interpret the model behaviour into real structural behaviour Ensure there is a good understanding of how the real structure behaves Judge the fitness of purpose of the real structure

3

DBA Design by Analysis

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This needs some more explanation: To begin with, a clear distinction should be made throughout between the real structure on the one hand and two different models of it on the other: The first model is the physical model, quite often also called analysis model, and is deduced from the real structure by an abstraction, or idealisation, process with regard to geometry, boundaries and boundary conditions, constitutive laws, etc. This idealisation quite often also requires assumptions on material properties or even constitutive laws, which are unknown and even not determinable for the real structure - the real constitutive law of base metal or ‘zones’ of weldments, the real deviations from the ideal geometry and so on. The second model is the mathematical model - a mathematical description of the physical model using the basic principles of mechanics. In case of finite element analysis this mathematical model is obtained by means of the FEA software. Also one of two analysis situations can arise: Firstly, if the starting point is a real existing structure, the actual geometry is known, especially the actual thicknesses are known or can be determined. Nevertheless, it may be necessary to use a different geometry in the physical model - not only because of the necessary idealisation, but also because of the requirement in some checks to deal with safety/behaviour of the real structure for the whole design life. Therefore, it may be necessary to use in some checks actual thicknesses minus allowances for corrosion, erosion, i. e. for allowances specified in the design specification. In the context of the European Standard obvious checks where the full specified thickness allowances shall be taken into account are the checks against Global Plastic Deformation (GPD), Instability (I); for reasons of simplicity the same physical model may be used in the checks against Fatigue (F), and Progressive Plastic Deformation (PD) or in the Shakedown check (SD). Secondly in a design situation the real structure is not known: the starting point is a ‘virtual structure’, specified by the design drawings, material lists, directly or indirectly referred to standards (for materials, tolerances of pre-products, shape deviations, allowed manufacturing deviations or defects, etc.). If the starting point is the ‘virtual’ model not only the allowances for corrosion, erosion shall be deducted, but also those for material tolerances, manufacturing tolerances, e. g. for dished ends, cones with knuckles, where the minimum specified thickness for the knuckle should be used, a value which should have been specified in the drawing anyway. In both cases it may also be necessary to take into account allowed shape deviations. In the Standard, checks where this may be necessary are the Instability Check, and possibly the Fatigue Check. Further, additional actions, not accounted for in the physical model, especially non-physical ones, may play an important role in the behaviour of the real structure, for example some types of local corrosion, including the various cracking types, embrittlement, etc. Thus, while it should always be kept in mind that the behaviour being investigated is the behaviour of the physical model of the real structure, the analysis or simulation tells us something about the behaviour of the real structure, but should not be confused with that. If the analyst always keeps in mind the distinction between the real structure, or the virtual one, and the models, an understanding

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of the behaviour of these models will help to better understand the behaviour of the real structure, even in cases not taken into account in the model. If we look at something, our brain uses our experience to transform the sense-impression into a picture of something meaningful - at least it does that in normal situations. Similarly the analyst does need (a lot of) experience to transform finite element results – that is, results of the mathematical model, into something meaningful with regard to the actual response of the real structure under imposed actions and constraints. Transition from the physical model to the mathematical one is made easy by software and requires just some vocational training in using the software. The creation and assessment of the physical model requires much expertise and much theoretical knowledge; it also requires knowledge of the software's capabilities, assumptions and requirements. This can be clearly seen in the models used in the Instability Checks in Section 7. Once the analysis has been carried out, the results from the physical model must be assessed in the context of the actual vessel and the rules given in the standard. However it is at this stage that the analyst very often makes a major blunder – a situation which is in fact made worse by the fact that a specific design code is being used! “… at first sight it might appear that the ideal result … is a single answer: ‘Yes, the maximum stress is below the allowable’ … the post-processors of most FE programs are capable of sorting results so that the maximum and minimum of any calculated values can be extracted easily. However by looking at these values alone the analyst would learn little about how the structure works. Even worse, he would be unable to form an opinion about how well his idealisation has simulated the real structure …” “… it is not possible to determine if an analysis is right or wrong, only that the results are fit for their purpose …” [2] . Thus, the code checks are done and, hopefully, satisfied – very often that is the main test of the model! It is often not easy to have a good understanding of how the real structure behaves and to relate this to the models. This is especially true in the case of nonlinearity, and thermal actions in particular. In any case, the first task facing the analyst is always to justify the assumptions made during modelling, particularly those assumptions regarding constraints. These assumptions may also have a physical basis: has an initial strain, or residual stress from welding, been justifiably ignored? Is the sequence of loading important: the nonlinear behaviour of a pressure vessel which is heated then pressurised, is much different if the load sequence is reversed. Does a change to the modelling assumption have no effect, or only a trivial one which does not effect code compliance? If there is an effect then further investigation, re-modelling and justification must be made. Then the influence of the chosen mathematical – choice of element type, mesh design, load modelling and so on should be examined. While a professional and competent analyst can build up a personal body of understanding, this is most often for linear elastic behaviour, and this can only provide an indication that the finite element models are suitable to begin with. In the following some general modelling issues for nonlinear analysis are discussed: 8.2.2 General modelling issues There are surprisingly few sources to which the designer/analyst can refer to give general help with nonlinear finite element analysis, and elastic-plastic behaviour in particular. Probably the best discussions have been given by Adams & Askenazi[3] , Cook[4] and NAFEMS[5] – these are summarised briefly here, but the reader is strongly advised to consult these sources directly. Written

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by experts, the advice given in theses sources are in basic agreement on the modelling issues which arise for nonlinear finite element analysis. A good summary of good modelling practice has been given by Adams & Askenazi[3]. Part of Chapter 15: “Nonlinear Analysis” is reproduced here: A few guidelines for meshing and model building are appropriate or more appropriate for nonlinear problems. Building the model correctly in the first place can greatly enhance the speed of the solution and the likelihood that convergence will be obtained. Most of these guidelines centre around the concept of keeping the nonlinear model as simple as possible. Due to the length of a nonlinear solution compared to a linear one, every attempt should be made to reduce the model size without compromising solution accuracy. This is where your linear preliminary studies may be useful. Expect that you will run a nonlinear model many times, both to achieve convergence and to adjust the final behaviour. Smart modelling can save hours of run time. Keep the following points in mind while building the model for a nonlinear solution. Use symmetry wherever possible. This is a good idea in a linear model and a great idea in a nonlinear model. The guidelines for using symmetry as described earlier in this book should be used, such as avoiding symmetry in a nonlinear buckling problem or a when a dynamic solution is required. Use beam, shell, or planar idealisations whenever possible. If a problem is marginal from the standpoint of using an idealisation, it may still be worth-while to consider the simplification as a test model and learn as much as you can from it. Region your model to use the nonlinear material model only where required. A perfectly valid technique is to restrict the use of nonlinear elements to regions where plasticity is expected. This may require you to mesh the model with multiple properties so that, in reality, two separate materials are used. Chapters 5 and 7 discuss this technique in more detail. By limiting the use of nonlinear elements to the areas that require it, you can speed up the model by forcing the solver to iterate on only a subset of the mesh. Refine and smooth the mesh in areas of high strain. Nonlinear solutions are sensitive to element distortion, and the discontinuity these elements cause can force the solver to unnecessarily iterate a higher strain, plastic solution when none should have been required. If an explicit nonlinear transient solution is required, distorted elements may skew the time stepping algorithm unnecessarily as well. The mesh at any contact region should be refined to capture the contact stresses that will be developed. As the contact area gets smaller, the need for more refinement increases. You may be able to "un-refine" the mesh where stress is low to reduce the model size as long as the overall model stiffness is not affected. If large deformations are expected to distort elements such that their accuracy maybe called into question, it may be worthwhile to manually distort the elements in the opposite direction somewhat before starting the solution. In this way, the final shape will be closer to the ideal shape. Use your judgement to determine if this is necessary for accuracy or even to prevent the solution from failing due to the presence of highly distorted elements. Always check your software documentation for element types allowed in a nonlinear solution. Many codes restrict the use of elements in a nonlinear solution to a subset of those available in a linear solution. The use of higher order elements is discussed below. Certain line, rigid, and speciality elements may be restricted for use in linear or dynamic solutions only. Take the time to read your documentation to understand which elements can and cannot be used in a nonlinear problem. Convergence problems can sometimes be resolved simply by changing element types to those better suited for a nonlinear problem. If you must use a solid model for a nonlinear solution, keep the following additional points in mind while constructing the mesh. Take another look at small or insignificant features in the model. Due to the speed of most linear solutions, you may have developed the habit of leaving in some fillets or features simply because it took more time to remove them than they added to the run time. While this is acceptable in a linear model, it

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can cause trouble in a nonlinear run. Consider that in addition to the increased length of a nonlinear solution, many iterations may be required to achieve convergence. While this is not an issue in linear analysis, any run time penalty due to an excessive mesh from unnecessary features will be paid for in each iteration.

Even though current analysis software makes nonlinear analysis easy, much more attention must be given to suitability of the model, its representation of the real component and interpretation of this behaviour. It is harder to predict the structural response and a good understanding of the response may develop only after carrying out several trial analyses. “… much more than in linear analysis, the nature of a problem may become clear only after solving it. At the outset the types and extent of the nonlinearity may not be apparent. Even if they are, the appropriate elements, mesh layout, solution algorithms, and load steps may not be. Accordingly, an attempt to solve a nonlinear finite element problem in ‘one go’ is likely to fail ……”[4] , or at worst be very misleading. A converged solution, with appealing stress and strain contours may not be physically, or even mathematically, correct! It is always advisable to make generous use of test cases and pilot studies before the main analysis, then examine the effect of adjustments to all modelling assumptions. And above all, to ask for help! All of this to get a ‘feel’ for the behaviour of the model, and to test if it represents the real component. The selection of suitable test cases and pilot studies is not easy for nonlinear analysis, so justification of the analysis is even more difficult.

8.3 Software requirements From the preceding discussion and Sections 3 & 7, it is clear that, as a basic requirement the analysts should have access to finite element analysis software which includes: 1. 2. 3. 4. 5. 6.

Three dimensional and axisymmetric solid and shell elements Thermal stress analysis Small and large deflection elastic and elastic-plastic analysis Instability and buckling capability The ability to handle combined actions and complex load histories A variety of conventional models of classical plasticity, including Tresca and von Mises initial yield and some subsequent yield models (for example isotropic and kinematic hardening)

For some of the specialist procedures described in Section 3, the software should preferably also include: 7. A macro language for enhanced post-processing, or to control procedures such as elastic compensation 8. Built-in procedures for stress linearisation (which could of course be developed using a macro language if necessary) In general most modern commercial finite element software should be capable of the basic requirements. A brief summary is given below: •

ABAQUS (elastic, plastic and creep calculations with beam, shell or solid elements),

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•

ALGOR (elastic and plastic calculations with beam, shell or solid elements, it can compute limit loads using inelastic calculations),

•

CASTEM 2000 (elastic, plastic and creep calculations with beam, shell or solid elements, it can compute limit loads using elastic calculations),

•

SYSTUS (elastic, plastic and creep calculations with beam, shell or solid elements),

•

ANSYS (elastic, plastic and creep calculations with beam, shell or solid elements),

•

NASTRAN (elastic, plastic and creep calculations with beam, shell or solid elements),

•

CASTOR (elastic calculations with beam, shell or solid elements),

•

COSMOS (elastic, plastic and creep calculations with beam, shell or solid elements),

•

MAGICS (elastic calculations with beam, shell or solid elements),

•

CADSAP (elastic calculations with beam, shell or solid elements),

•

SAMCEF (elastic, plastic and creep calculations with beam, shell or solid elements),

•

FE-PIPE (elastic calculations with beam, shell or solid elements, especially for pressure vessel components),

•

BOSOR4 (program for stress, buckling and vibration of complex shells of revolution).

A fuller comparison can be found in the text by Adams & Askenazi [3] .

8.4 Analyst requirements As mentioned in the Introduction, Section 1, the expertise of the analyst/designer plays a critical role in the whole design by analysis process. While the Standard itself, and to some extent most commercial finite element software, undergo rigorous quality assessment, this is not the case with the analyst. Indeed it is wholly possible for a relatively new and inexperienced engineer or analyst to be allocated the task of design by analysis. Further, it is equally typical to team an experienced pressure vessel designer with an experienced finite element analyst who has no specific familiarity with the requirements of pressure vessel design, with neither having any real experience of design by analysis, or inelastic analysis. This is just as much a recipe for disaster as the novice. Typical professional training of engineering graduates across Europe does not examine inelastic material behaviour and constitutive modelling, or inelastic stress analysis in any great detail (if at all), and only a few cover the detailed behaviour of elastic-plastic components and structures, let alone pressure vessel design. Training in practical finite element analysis is common, but tends to emphasise modelling and the interface to CAD. Training in finite element theory beyond linear elasticity is rare. As a consequence, analysts must gain the necessary experience for pressure vessel

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design by analysis on the job, hopefully apprenticed to an expert, and through Continuing Professional Development Courses. That this additional training is rarely professionally accredited has long been a concern of finite element educators [6] . The experienced pressure vessel analyst and designer is uncommon in the industry. One of the failings of modern engineering in general is that the gap between technical knowledge of engineering principles and analysis software capabilities is growing! This has been expressed well by Adams & Askenazi [3] (italics added by the writers for emphasis): “… this literal explosion of speed and capabilities has been a mixed blessing for design engineers. From the near-seamless, near-painless integration with CAD, a growing number of design analyst non-specialists with a part-time interest in FEA has emerged. The growth of this segment of the industry is accelerating rapidly as technology providers scramble to fill the need for easy-to-use FEA software tools. Some of these tools are so easy to use, little thought is required to develop stress contours on parts with complex geometry. With little thought comes little chance of accuracy. The early analysts sweated every node and element and paid dearly for lengthy run times. Good practices were developed to ensure boundary conditions and properties were well thought out before a model was submitted. Hand calculations and results correlation increased users' awareness of the capabilities and limitations of their tools. Today, new users tend to believe that any result that looks right probably is right. Material properties are assigned carelessly and boundary conditions are applied more out of convenience than based on actual environmental interactions. Is the role of the analysis specialist doomed? Hardly! Most design analysts would agree that their usage of the technology is, by the nature of their responsibilities, only surface level. Many do not want nor do they have time to pursue more complex analysis types or assembly simulation with ambiguous interactions. Engineers have come to expect the instant gratification provided by photo-realistic CAD models and rapid prototyping systems. Consequently, most design analysts who learn to appreciate the complexity of FEA hesitate to jump head first into a problem that requires lengthy setup, and run times which often yield less than spectacular data …” “… an editorial in the July 9, 1998 issue of MACHINE DESIGN magazine addressed the subject of designers and FEA, This editorial's premise was essentially, 'Why not let designers do FEA without understanding engineering theory?’ The discussion was supported by an example of a failed aerospace project conducted by expert analysts that relied a little too heavily on analytical results. The second example described an aerospace project completed by engineers who were admittedly weak on theory and chose to use "make-and-break" methods instead of analysis to great success. Based on these two data points, the author concluded that because expert analysts cannot always get it right, designers without grounding in fundamentals can do no worse. This somewhat irresponsible position is contradictory, assuming that the reader is objective in his/her views on the subject. Taken to its extreme, one might conclude that because successful products have been developed by individuals with no engineering background whatsoever and products developed by highly trained and competent engineers have experienced failures, a basic engineering education should not be a requirement for aircraft development! …” “ … ‘Good users will readily admit they need to know more. Those who feel satisfied are probably in trouble.’ Surprisingly, few design analysts model, analyse, and interpret results efficiently or even accurately. However, most do not realise it unless a major mistake costs their company money …”

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The engineering manager thinking of embarking on design by analysis should ponder the above quotes very carefully indeed! So what are the educational requirements of the analyst? The following are considered to be essential elements of a competent analyst’s basic training for pressure vessel design by analysis: 1. An accredited undergraduate degree in mechanical/civil/plant engineering (preferably to the highest SARTOR professional level) 2. A knowledge of the classical theory of plasticity, including multiaxial behaviour, subsequent yield and standard elastic-plastic constitutive models 3. An understanding of the behaviour of standard components in the plastic range including the basic concepts of limit analysis, failure mechanisms, plastic hinges and shakedown 4. A basic knowledge of finite element theory and common element formulations 5. A few years experience of linear elastic finite element analysis of complex structures and components, working with a knowledgeable analyst 6. An understanding of the procedures of elastic-plastic finite element analysis 7. A basic knowledge of the behaviour and stress systems in common thin-walled pressurised components 8. A basic knowledge of buckling in simple plates and shells 9. An understanding of the aims and requirements of the various national codes and standards, including the European Standard The following would be considered desirable elements: 10. A basic understanding of the behaviour of thin curved shells and the behaviour of more complex pressurised components under mechanical and thermal loading 11. A basic understanding of modern constitutive models for elastic-plastic material behaviour, and the limitations of the classical theory of plasticity 12. A basic understanding of finite element theory for nonlinear analysis, in particular both material and geometrical nonlinearities 13. An understanding of the procedures for, but also the limitations of, finite element instability and buckling analysis 14. A basic understanding of plasticity in the presence of large deformations, and the limitations of associated constitutive and numerical models 15. Experience with a range of analysis tools and commercial finite element software 16. Basic, but first hand, experience of pressure vessel manufacture, construction, operation and maintenance These recommendations may seem to be rather rigorous, but it must be remembered that many pressure vessels which need to be considered for design by analysis through the Direct Route may be high integrity and consequently a high level of technical competence may be required. Of course it may be argued that a team of engineers with a pooled competence may be sufficient, but in some of the writers’ experience this may not be satisfactory. It is well understood in engineering education that a higher level of technical knowledge and competence will be required by tomorrow’s analyst and designer. This requirement is prompted not only by the availability, and relative ease of use, of modern analysis tools, but also by the growing need for industrial

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competitiveness at the same time as increased public demand for safety. It is equally well understood that only a basic introduction to a range of technical skills can be given in accredited degree level engineering education, for example as detailed by SARTOR. The only way tomorrow’s engineer (or indeed anyone in the workplace) can acquire these skills would be through lifelong learning and Continuing Professional Development, both of which are seen as crucial. The question of accreditation of engineering competence in the use of national codes and standards must be faced at some time, as the level required increases.

8.5 Concluding remarks The new approach to DBA, as laid down in CEN's prEN 13445-3, seems to be a major step forward in design by analysis. Problems still contained in the present proposal have been identified in this European research project, initiated by EPERC. This project has shown that the approach is sound, that the approach gives much insight into the behaviour of the vessel, into the safety margins against failure modes, and, therefore, that this approach can lead to improved designs and improved in-service inspection procedures. This project has also shown that the time effort required is, even with presently available hard and software, affordable. It is clear to the writers of this document, that if the new European Standard recommendations for Design by Analysis are to succeed and become more widely used, with the accompanying increased safety and reliability, there is a real need for Continuing Professional Development Courses. 8.6 Literature [1] J.N.C. Guerreiro, A.F.D. Loula & J.T. Boyle: “Finite element methods in stress analysis for creep” General Lecture, Creep in Structures IV, Proceedings of IUTAM Symposium, Cracow, 1990. Ed M Zyczkowski, Springer, 1991 [2] NAFEMS: “How to Interpret Finite Element Results” National Agency for Finite Element Methods & Standards, 1990 [3] V. Adams & A. Askenazi: “Building Better Products with Finite Element Analysis” Onward Press, 1999 [4] R.D. Cook, D.S. Malkus, M.E. Plesha: “Concepts and Applications of Finite Element Analysis”, John Wiley, 3rd Ed, 1994 [5] E. Hinton et al, Eds: “NAFEMS Introduction to Nonlinear Stress Analysis” National Agency for Finite Element Methods & Standards, 1992 [6] J.T. Boyle et al, Eds: “Finite Element Analysis; Education & Training” Elsevier Applied Science, 1991

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Annex 1: Bibliography

Page A1.1

In this Annex, three tables are presented as a literature review on design by analysis: •

Table A1.1: Books concerning DBA, limit and shakedown analysis.

•

Table A1.2: DBA using the finite element method (FEM), numerical and/or analytical procedures.

•

Table A1.3: Theory and/or numerical methods for limit and shakedown analysis.

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Annex 1: Bibliography

A1.2

Table A1.1: Books concerned with DBA, limit and shakedown analysis AUTHOR Bolt, S.E.; Bryson, J.W.; Corum, J.M.; Gwaltney, R.C.

TITLE

SOURCE

THEORETICAL AND EXPERIMENTAL STRESS ANALYSIS OF ORNL THIN-SHELL CYLINDER-TO-CYLINDER MODEL NO. 3

Oak Ridge National Laboratory - 5020

Bolt, S.E.; Bryson, J.W.; Gwaltney, THEORETICAL AND EXPERIMENTAL STRESS ANALYSIS OF ORNL THIN-SHELL R.C. CYLINDER-TO-CYLINDER MODEL NO. 4

Oak Ridge National Laboratory - 5019

Bolt, S.E.; Bryson, J.W.; Gwaltney, THEORETICAL AND EXPERIMENTAL STRESS ANALYSIS OF ORNL THIN-SHELL R.C. CYLINDER-TO-CYLINDER MODEL NO. 2

Oak Ridge National Laboratory - 5021

Burgreen,D.

DESIGN METHODS FOR POWER PLANT

C.P. Press, 1975

Burgreen,D.

PRESSURE VESSEL ANALYSIS

C.P. Press, 1979

Chen,W.F.; Han,D.J.

PLASTICITY FOR STRUCTURAL ENGINEERS

Springer, 1988

Corum, M.J.; Bolt, S.E.; THEORETICAL AND EXPERIMENTAL STRESS ANALYSIS OF ORNL THIN-SHELL Greenstreet, W.E.; Gwaltney, R.C. CYLINDER-TO-CYLINDER MODEL NO. 1

Oak Ridge National Laboratory - 4553

Gokhfeld, D.A.; Cherniavsky, O.F. LIMIT ANALYSIS OF STRUCTURES AT THERMAL CYCLING

Sijthoff & Noordhoff, 1980

Kamenjarzh, J.

LIMIT ANALYSIS OF SOLIDS AND STRUCTURES

CRC-Press, Florida, 1996

König, J.A.

SHAKEDOWN OF ELASTIC PLASTIC STRUCTURES

Elsevier, 1987

Save, M.A.; Massonnet, C.E.;

PLASTIC ANALYSIS AND DESIGN OF PLATES, SHELLS AND DISKS

North-Holland, 1972

Sawczuk, A.; Jaeger, T.

GRENZTRAGFÄHIGKEITSTHEORIE DER PLATTEN

Springer, 1963

Zeman, J.L;

REPETITORIUM APPARATEBAU/GRUNDLAGEN DER FERSTIGKEITSBERECHNUNG

Oldenbourg, 1992

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Table A1.2: DBA using FEM, analytical and/or numerical procedures AUTHOR

TITLE

SOURCE

Hamilton, J.; Boyle, J.T., Shi, J., Mackenzie, D.

SHAKEDOWN LOAD BOUNDS BY ELASTIC FINITE ELEMENT ANALYSIS

PVP - Vol. 343, p. 205 f.f.

Zeman, Josef L.

SOME ASPECTS OF THE WORK OF THE EUROPEAN WORKING GROUPS RELATING TO BASIC PRESSURE VESSEL DESIGN

IJPV&P, 70, p.3-10

Mackenzie, D.; Boyle, J.T.;

A SIMPLE METHOD OF ESTIMATING SHAKEDOWN LOADS FOR COMPLEX STRUCTURES

PVP-Vol. 265, p. 89-94

Hollinger, G.L.

SUMMARY OF THREE DIMENSIONAL STRESS CLASSIFICATION

7.th ICPVT, Vol. 1, p. 454 f.f.

SIMPLIFIED LOWER BOUND LIMIT ANALYSIS OF PRESSURISED Hamilton, R.; Mackenzie, CYLINDER/CYLINDER INTERSECTIONS USING GENERALISED IJPV&P, 67, p. 219-226 D.; Shi, J.; Boyle, J. T.; YIELD CRITERIA Mackenzie, D.; Nadarajah, SIMPLE BOUNDS ON LIMIT LOADS BY ELASTIC FINITE C.; Shi, J.; Boyle, J. T.; ELEMENT ANALYSIS

Shi, J.; Mackenzie, D.; Boyle, J. T.;

A METHOD OF ESTIMATING LIMIT LOADS BY ITERATIVE ELASTIC ANALYSIS. III-TORISPHERICAL HEADS UNDER INTERNAL PRESSURE

IJPVT, Vol. 115, p. 27-31

IPV&P, 53, p. 121-142

ROUND ROBIN CALCULATIONS OF COLLAPSE LOADS - A Yamamoto, Y.; Asada, S.; TORISPHERICAL PRESSURE VESSEL HEAD WITH A CONICAL Okamoto, A. TRANSITION

IJPVT, Vol. 119, p. 503 f.f.

Hollinger, G.L..; Hechmer, CODE STRESS CLASSIFICATION FOR EVALUATION BY 3D J.L. METHODS

2-Part Workshop; February 4,1998 PVRC Meeting; San Diego, CA

Mackenzie, D.; Boyle, J.T.

ASSESSMENT OF CLASSIFICATION PROCEDURES FOR FINITE ELEMENT STRESSES

7th ICPVT, Vol.1, p. 346-358

Seibert, T.; Zeman, J.L.

ANALYTISCHER ZUVERLÄSSIGKEITSNACHWEIS VON DRUCKGERÄTEN

Technische Überwachung, Bd. 35 (1994), Nr. 5, p. 222 f.f.

NUMERICAL ANALYSIS OF THE ELASTIC-PLASTIC BEHAVIOUR Yeom, D.J.; Robinson, M.; OF PRESSURE VESSELS WITH ELLIPSOIDAL AND TORISPHERICAL HEADS

IJPV&P, 65, p. 147-156

Porowski, J.S.; Kasraie, B.; PRIMARY STRESS EVALUATIONS FOR REDUNDANT Bielawski, G.; O'Donnell, STRUCTURES W.J.; Badlani, M.L.

PVP-Vol. 265, p. 185 f.f.

Tanaka, M.; Yamamoto, Y.

to be presented at ASME PVP 1998

2D AND 3D COLLAPSE EVALUATION OF SMALL RADIAL AND OBLIQUE NOZZLE TO SPHERICAL SHELL INTERSECTIONS

Boyle, J.T.; Hamilton, R.; A SIMPLE METHOD OF CALCULATING LOWER-BOUND LIMIT Mackenzie, D.; LOADS FOR AXISYMMETRIC THIN SHELLS

IJPVT, Vol. 119, p. 236-242

Page A1.3

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Table A1.2: DBA using FEM, analytical and/or numerical procedures (continued) AUTHOR

TITLE

SOURCE

Mackenzie, D.; Boyle, J.T.; Hamilton, R.;

APPLICATION OF INELASTIC FINITE ELEMENT ANALYSIS TO PRESSURE VESSEL DESIGN

8th IC PVT, Vol. 2, p. 109-115

Preiss, R.; Rauscher, F.; Vazda, D.; Zeman, J.L.

THE FLAT END TO CYLINDRICAL SHELL CONNECTION - LIMIT LOAD AND CREEP DESIGN

IJPV&P 75 (1998), p 715-726

COMPARISON BETWEEN LINEAR-ELASTIC AND LIMIT Dixon, R.D.; Perez, E. H. ANALYSIS METHODS FOR THE DESIGN OF HIGH PRESSURE VESSELS

PVP-Vol. 344, p.43 f.f.

Mackenzie, D.; Boyle, J.T.

COMPUTIONAL PROCEDURES FOR CALCULATING PRIMARY STRESS FOR THE ASME B&PV CODE

PVP-Vol. 265, p. 177 f.f.

Okamoto, A.

PRIMARY STRESS EVALUATION PROCEDURE FOR NOZZLETOCYLINDER JUNCTURE UNDER PRESSURE LOADING

to be presented at ASME PVP 1998

Nadarajah, C.; Mackenzie, D.; Boyle, J.T.;

LIMIT AND SHAKEDOWN ANALYSIS OF NOZZLE/CYLINDER INTERSECTIONS UNDER INTERNAL PRESSURE AND IN-PLANE MOMENT LOADING

IJPV&P, 68, p. 261-272

Lapsley, C.; Mackenzie, D.; Tooth, A. S.;

LOCAL LOADING OF CYLINDERS: LIMIT ANALYSIS

8th ICPVT, Vol. 2, p. 55-59

Kalnins, A.; Updike, P.D.; PLASTIC STRAINING AT NOZZLES IN PRESSURE VESSELS Park, I.

PVP-Vol. 353, p. 143 f.f.

Kalnins, A.; Updike, P.D.

EFFECT OF HYDROSTATIC PROOF TEST ON SHAKEDOWN OF TORISPHERICAL HEADS UNDER CYLIC PRESSURE

PVP-Vol.353, p.217 f.f.

Mackenzie, D.; Boyle, J. T.;

A METHOD OF ESTIMATING LIMIT LOADS BY ITERATIVE ELATSIC ANALYSIS: I-SIMPLE EXAMPLES

IJPV&P, 53, p. 77-95

Nadarajah, C.; Mackenzie, D.; Boyle, J.T.;

A METHOD OF ESTIMATING LIMIT LOADS BY ITERATIVE ELASTIC ANALYSIS. II-NOZZLE SPHERE INTERSECTIONS WITH INTERNAL PRESSURE AND RADIAL LOAD

IJPV&P, 53, p. 97-119

Scavuzzo, R. J.; Lam, P. C.; Gau, J.S.;

RATCHETING OF PRESSURIZED PIPING SUBJECTED TO SEISMIC PVP-Vol. 197, p. 77-84 LOADING

Lietzmann, A.; Rudolph, J.; Weiß, E.;

FAILURE MODES OF PRESSURE VESSEL COMPONENTS AND THEIR CONSIDERATION IN ANALYSES

Chemical Engineering and Processing, 35, p. 287-293

Jin-Guang-Teng

PLASTIC COLLAPSE AT LAP JOINTS IN PRESSURIZED CYLINDERS UNDER AXIAL LOAD

Journal of Structural Engineering, Vol. 120, No. 1, p. 23-44

Porowski, J.S.; O'Donnell, WEIGHT-SAVING PLASTIC DESIGN OF PRESSURE VESSELS W.J.; Reid, R.H.;

IJPVT, Vol. 119, p. 161-166

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DBA Design by Analysis

Annex 1: Bibliography

Page A1.5

Table A1.2: DBA using FEM, analytical and/or numerical procedures (continued) AUTHOR

TITLE

SOURCE

Miklus S.; Kosel, F.;

PLASTIC COLLAPSE OF PIPE BIFURCATION

IJPV&P, 48, p. 79-92

Naderan-Tahan, K.; Robinson, M.;

PLASTIC LIMIT PRESSURES FOR NEIGHBOURING RADIAL NOZZLES IN A SPHERICAL PRESURE VESSEL

J. of Process Mechanical Engineering, Vol. 210, p. 75-78

A COMPARISON OF THE STRESS RESULTS FROM SEVERAL Porter, A.M.; Martens, D. COMMERCIAL FINITE ELEMENT CODES WITH ASME SECTION VIII, DIVISION 2 REQUIREMENTS

http://www.dynamicanalysis.co m/pvp96a.htm

Updike, D.P.; Kalnins, A.

ULTIMATE LOAD ANALYSIS FOR DESIGN OF PRESSURE VESSELS

PVP-Vol.353, p. 289 f.f.

Teng, J.-G.

CONE-CYLINDER INTERSECTION UNDER PRESSURE: AXISYMMETRIC FAILURE

Journal of Engineering Mechanics, Vol. 120, No. 9, p. 1896 f.f

Tabone, C. J.; Mallett, R. H.;

PRESSURE PLUS MOMENT LIMIT LOAD ANALYSIS FOR A CYLINDRICAL SHELL NOZZLE

PVP - Vol. 120, p. 131-136

Shalaby, M.A.; Younan, M.Y. A.

LIMIT LOADS FOR PIPE ELBOWS WITH INTERNAL PRESSURE UNDER IN-PLANE CLOSING BENDING MOMENT

IJPVT, Vol. 120 , p. 35-42

Broyles, R.K.

FINDING DESIGN ACCETABILITY USING FINITE ELEMENT ANALYSIS

PVP-Vol. 353, p. 151 f.f.

Babu, S.; Iyer, B.K.

INELASTIC ANALYSIS OF COMPONENTS USING A MODULUS ADJUSTMENT SCHEME

IJPVT, Vol 120, p. 1-5

Seshadri, R.

ROBUST STRESS-CLASSIFICATION OF PRESSURE COMPONENTS PVP-Vol. 265, p. 155 f.f. USING THE GLOSS METHODS

Teng, J.-G.

Journal of Engineering CONE-CYLINDER INTERSECTION UNDER INTERNAL PRESSURE: Mechanics, Vol. 121, No. 12, p. NONSYMMETRIC BUCKLING 1298 f.f.

Seshadri, R.; Fernando, C. LIMIT LOADS OF MECHANICAL COMPONENTS AND P. D.; STRUCTURES USING THE GLOSS R-NODE METHOD

IJPVT, Vol. 114, p. 201-208

Rogalska, E.; Kakol, W.; Guerlement, G.;Lamblin, D.;

LIMIT LOAD ANALYSIS OF PERFORATED DISKS WITH SQUARE PENETRATION PATTERN

IJPVT, Vol. 119, p. 122-126

Tong, R.; Wang, X.;

SIMPLIFIED METHOD BASED ON THE DEFORMATION THEORY FOR STRUCTURAL LIMIT ANALYSIS - I. THEORY AND FORMULATION

IJPV&P, 70, p. 43-49

Tong, R.; Wang, X.;

SIMPLIFIED METHOD BASED ON THE DEFORMATION THEORY FOR STRUCTURAL LIMIT ANALYSIS - II. NUMERICAL APPLICATION AND INVESTIGATION ON MESH DENSITY

IJPV&P, 70, p. 51-58

6

DBA Design by Analysis

Annex 1: Bibliography

Page A1.6

Table A1.2: DBA using FEM, analytical and/or numerical procedures (continued) AUTHOR

TITLE

SOURCE

Teng, J.G.; Rotter, J.M.,

PLASTIC COLLAPSE OF RESTRAINED STEEL SILO HOPPERS

J. Construct. Steel Research , 14, p. 139-158

Teng, J.-G.; Rotter M.

COLLAPSE BEHAVIOR AND STRENGTH OF STEEL SILO TRANSITION JUNCTIONS. PART 2: PARAMETRIC STUDY

Journal of Structural Engineering, Vol. 117, No.12, p. 3605 f.f.

Mello, R.M.; Griffin, D.S.;

PLASTIC COLLAPSE LOADS FOR PIPE ELBOWS USING INELASTIC IJPVT, Aug. 1974, p.177-183 ANALYSIS

Teng, J.-G.; Rotter, J. M.

COLLAPSE BEHAVIOR AND STRENGTH OF STEEL SILO TRANSITION JUNCTIONS. PART 1: COLLAPSE MECHANICS

Journal of Structural Engineering, Vol.117, No.12, p. 3587 f.f.

Adinarayana, N.; Alwar, R.S.;

ELASTOPLATIC STRESS AND STRAIN PREDICTIONS FOR AXISYMMETRIC PRESSURE VESSEL BODIES UNDER TRANSIENT THERMAL LOADS

Journal of Strain Analysis, Vol. 31, No. 2, p. 81-89

Weiß, E; Joost, H.

BESTIMMUNG DER TRAGLAST VON BEHÄLTERN UNTER ZUSÄTZLICHEN LASTEN MITTELS FEM

14. CAD-FEM User's Meeting, 1996, Bad Aibling, Germany, Art. 1-9

Weiß, E; Postberg, B.

MODELLIERUNG VON RATCHETTING-EFFEKTEN BEZOGEN AUF 15. CAD-FEM User`s Meeting, KOMPONENTEN DER CHEMIE- UND KRAFTWERKSTECHNIK 1997, Fulda, Germany, Art. 2-11

Galletly, G.D.; Aylward, R.W.;

PLASTIC COLLAPSE AND THE CONTROLLING FAILURE PRESSURES OF THIN 2:1 ELLIPSOIDAL SHELLS SUBJECTED TO INTERNAL PRESSURE

IJPVT, Vol. 101, 64-72

Radhamohan, S.K.; Galletly, G.D.;

PLASTIC COLLAPSE OF THIN INTERNALLY PRESSURIZED TORISPHERICAL SHELLS

IJPVT, Vol. 101, p. 311-319

Seshadri, R.; Mangalaramanan, S., P.;

LOWER BOUND LIMIT LOADS USING VERIATIONAL CONCEPTS: IJPV&P, 71, p. 93-106 THE M -METHOD

Weiß, E.; Lietzmann, A.; Rudolph, J.

ELASTISCHE UND ELASTISCH PLASTISCHE FESTIGKEITSANALYSEN GEWÖLBTER BÖDEN MIT UND OHNE STUTZEN IM KREMOENBEREICH

VGB Kraftwerkstechnik 75 (1995), Heft 6, p.. 549 f.f.

Zarrabi. K.;

PLASTIC COLLAPSE PRESSURES FOR DEFECTED CYLINDRICAL VESSELS

IJPV&P; 60, p. 65-69

Hayakawa, T.; Yoshida, T.; COLLAPSE PRESSURE FOR THE SMALL END OF A CONE3rd ICPVT, Vol. 1, p. 149 f.f. Mii, T. CYLINDER JUNCTION BASED ON ELASTISC-PLASTIC ANALYSIS

Weiß, E.; Lietzmann, A.; Rudolph, J.;

FESTIGKEITSANALYSE UND BEANSPRUCHUNGSBEWERTUNG FÜR KOMPONENTEN DES DRUCKBEHÄLTERBAUS

ELASTIC-PLASTIC BEHAVIOUR OF A SIMPLY SUPPORTED Webster, J.J.; Sahari, B.B.; CIRCULAR PLATE SUBJECTED TO STEADY TRANSVERSE Hyde, T.H.; PRESSURE AND CYCLIC LINEAR RADIAL TEMPERATURE VARIATION

Technische Überwachung, Bd. 36, Nr. 11/12, p. 424-430

I.J.Mech.Sci., Vol. 29, No. 8, p. 553-544

7

DBA Design by Analysis

Annex 1: Bibliography

Page A1.7

Table A1.2: DBA using FEM, analytical and/or numerical procedures (continued) Gerlach, H.D.; Ludeke, M.

COMPARISON OF CONCEPTS FOR STRESS ANALYSIS OF COMPONENTS IN PLANT TECHNOLOGY

10th MPA seminar, 11/12 October 1984, Stuttgart, Germany

Gerlach, H.D.; Ludeke, M.

INTERNATIONAL COMPARISON OF STRESS ANALYSIS CONCEPTS FOR CHEMICAL PLANT COMPONENTS

6th ICPVT, Vol. 1 (Design and Analysis), Beijing, 1988, 407421

Ringelstein, K.H.; Baues

COMPARISON AND APPLICATION OF INTERNATIONAL TECHNICAL CODES, SPECIFICALLY PRESENTED WITH A VIEW TO ASME BOILER AND PRESSURE VESSEL CODE AND THE APPLICABLE GERMAN RULES, PART 3: DESIGN MINISTERY OF RESEARCH AND TECHNOLOGY

Project no. RS 150 345, June 1980

Roche, R.

FACTEURS MECANIQUES ET METALLURIQUES DE LA RUPTURE, NOTAMMENT DANS L'INDUSRIE NUCLEAIRE

17ème colloque de métallurgie (Saclay, juin 1974)

Roche; R.

CRITERES MODERNES D’APPRECIATION DA LA SECURITE DES Conférence du 7 février 1974, APPAREILS A PRESSION, NOTAMMENT DANS LE DOMAINE salle Chaleil, 11 avenue Hoche, NUCLEAIRE Paris 8

Cousseran, P.; Lebey, J.; Moulin, D.;Roche, R.;Clement, G.

NOUVELLES EXPERIENCES SUR l’EFFET DE ROCHET - UNE REGLE PRATIQUE DE PREVENTION

IIIème Congrès national sur la technologie des appareils à pression, AFIAP 1-2-3 octobre 1980

Roche, R.

SPECIALISATION MECANIQUE DES STRUCTURES APPLIQUEES AU GENIE NUCLEAIRE

Note technique EMT/SMTS/79/08, 26 janvier 1979

document FRAMATOME

FONDEMENTS DES REGLES LES DE PREVENTION DE L’ENDOMMAGEMENT DES MATERIELS MECANIQUES INTRODUCTION AUX REGLES DE LA CONCEPTION ET D’ANALYSE DU RCC-M

octobre 1991

Debaene, J.B.

METHODES D’ANALYSE DES APPAREILS A PRESSION

Techniques de l’ Ingénieur, Chapitre A846

Kroenke, W.C.; Addicott, INTERPREATION OF FINITE ELEMENT STRESSES ACCORDING G.W.; Hinton, B.M. TO ASME SECTION III

ASME paper 75-PVP-63 (1975)

Hollinger, G.L.; Hechmer, THREE DIMENSIONAL STRESS CRITERIA J.M.

PVRC 1991 Phase 1 Report

Hollinger, G.L.; Hechmer, 3-D STRESS CRITERIA: GUIDELINES FOR APPLICATION J.M.

PVRC 1995 Phase 2 Report

Pastor, T.P.; Hechmer, J.L.

PVP-vol. 277, ASME (1994)

ASME TASK GROUP REPORT ON PRIMARY STRESSES

Porowski, J.S.; Kasraie, B.; Bielawski, G.; PRIMARY STRESS EVALUATIONS FOR REDUNDANT O'Donnell, W.J.; Badlani, STRUCTURES M.L.

PVP-vol. 265, ASME (1993)

8

DBA Design by Analysis

Annex 1: Bibliography

Page A1.8

Table A1.2: DBA using FEM, analytical and/or numerical procedures (continued) Hollinger, G.L.; Hechmer, THREE DIMENSIONAL STRESS CRITERIA J.M.

PVRC 1991 Phase 1 Report

Hollinger, G.L.; Hechmer, 3-D STRESS CRITERIA: GUIDELINES FOR APPLICATION J.M.

PVRC 1995 Phase 2 Report

Pastor, T.P.; Hechmer, J.L.

PVP-vol. 277, ASME (1994)

ASME TASK GROUP REPORT ON PRIMARY STRESSES

Porowski, J.S.; Kasraie, PRIMARY STRESS EVALUATIONS FOR REDUNDANT B.; Bielawski, G.; O'Donnell, W.J.; Badlani, STRUCTURES M.L.

PVP-vol. 265, ASME (1993)

Zeman, J.L., Preiss; R.

THE DEVIATORIC MAP - A SIMPLE TOOL IN DESIGN BYANALYSIS

Preiss, R.

ON THE SHAKEDOWN ANALYSIS OF NOZZLES USING ELASTOIJPV&P, 76, p. 421-434 PLASTIC FEA

Save, M.

LIMIT ANALYSIS OF PLATES AND SELLS: RESEARCH OVER TWO J. Struct. Mech., 13 (3&4), p. DECADES 343-370

Bree, J.

PLASTIC DEFORMATION OF A CLOSED TUBE DUE TO I.J.Mech.Sci., Vol. 31, No. INTERACTION OF PRESSURE STRESSES AND CYCLIC THERMAL 11/12, p. 865-892 STRESSES

Kalnins, A.; Updike, D.P.

SHAKEDOWN AND STRESS RANGE OF TORISPHERICAL HEADS PVP-Vol. 2, p. 25-31 UNDER CYCLIC INTERNAL PRESSURE

Hollinger, G.L.; Hechmer, J.M.

ASSESSMENT OF THE ASME CODE STRESS LIMITS FOR 3D, SOLID ELEMENT, FINITE ELEMENT ANALYSIS: SUMMARY OF THE PVRC PROJECT

PVP-Vol. 360, ASME (1998)

Kalnins, A.; Updike, P.D.

PLASTICITY AND CHANGING GEOMETRY IN PRESSURE VESSEL DESIGN

PVP-360, ASME (1998)

Blachut, J.; Ramachandra, L. S.; Krishnan, P. A.

EXPERIMENTAL AND NUMERICAL INVESTIGATION OF PLASTIC LOADS FOR INTERNALLY PRESSUREISED VESSEL HEADS

PVP-360, ASME (1998)

Teng, J.G.

PLASTIC COLLAPSE BEHAVIOUR AND STRENGTH OF CONECYLINDER AND CONE-CONE INTERSECTIONS

PVP-360, ASME (1998)

IJPV&P, 76, p. 339-344

Mohamed, A.I.; APPLICATION OF ITERATIVE ELASTIC TECHNIQUES FOR Megahed, M.M.; Bayoumi, L.S.; Younan, ELASTIC-PLASTIC ANALYSIS OF PRESSURE VESSELS M.Y.A..

IJPVT, Vol 121 (1999), p. 2429

Kalnins, A.; Updike, P.D.

SHAKEDOWN OF TORISPHERICAL HEADS USING PLASTIC ANALYSIS

IJPVT, Vol 120 (1998), p 431437

Sanal, Z.

GEOMETRISCH UND PHYSIKALISCH NICHTLINEARE ANALYSE Stahlbau 67 (1998), Heft 6 VON DRUCKBEHÄLTERN

9

DBA Design by Analysis

Annex 1: Bibliography

Page A1.9

Table A1.2: DBA using FEM, analytical and/or numerical procedures (continued) AUTHOR

TITLE

SOURCE

Zeman, J.L.

DIE VERBINDUNG MANTEL-EBENER BODEN UNTER DRUCKEINWIRKUNG

Technische Überwachung, Bd. 35 (1994), Nr. 11/12, p. 450 f.f.

Zeman, J.L

DIE VERBINDUNG MANTEL-EBENER BODEN MIT UNVERSTÄRKTEM KONZENTRISCHEN AUSSCHNITT UNTER DRUCKEINWIRKUNG

Technische Überwachung, Bd. 36 (1995), Nr. 4, p. 153 f.f.

Zeman, J.L.

RATCHETING LIMIT OF FLAT END TO CYLINDRICAL SHELL CONNECTIONS UNDER INTERNAL PRESSURE

IJPV&P, 68 (1996), p. 293-298

Poth, W.; Zeman, J.L.

GRENZTRAGFÄHIGKEIT DER ZYLINDER-KEGEL-VERBINDUNG UNTER INNENDRUCKEINWIRKUNG

Konstruktion, 48, p. 219-223

Myler, P.; Robinson, M.

LIMIT ANALYSIS OF INTERSECTING CONICAL PRESSURE VESSELS

IJPV&P, 18, p. 209-240

Updike, D.P.; Kalnins, A.; Hechmer, J.

ON PRIMARY STRESSES IN CONICAL REDUCERS

PVP-Vol. 210-2, p. 117 f.f.

Kalnins, A.; Updike, P.D. PRIMARY STRESSES FROM SIMPLE LAWS OF EQUILIRIUM

PVP-Vol. 353, p. 259 f.f.

Kalnins, A.; Updike, D.P. ON PRIMARY STRESS CALCULATIONS

PVP-Vol. 265, p. 167-176

Kalnins, A.; Updike, D.P.

ROLE OF PLASTIC LIMIT AND ELASTIC-PLASTIC ANALYSIS IN DESIGN

PVP-Vol.210-2, p. 135 f.f.

Moreton, D.N.

THE RATCHETTING OF A CYLINDER SUBJECTED TO INTERNAL PRESSURE AND ALTERNATING AXIAL DEFORMATION

Journal of Strain Aanalysis, Vol. 28, No. 4, p. 277-282

Cinquini, C.; Zanon, P.

LIMIT ANALYSIS OF CIRCULAR AND ANNULAR PLATES

Ingenieur-Archiv , 55, p. 157-175

Save, M.

LIMIT ANALYSIS AND DESIGN OF CONTAINMENT VESSELS

Nuclear Engineering and Design, 79, p. 343-361

Dinno, K. S.; Al-Zabin, S. A.

LIMIT ANALYSIS OF CIRCULAR STEEL SILOS

Bull. Int. Ass. Shells&Spatial Structures, V 38, 3. Nov. 87, p 37-45

Leckie, F. A.; Penny, R.K.

SHAKEDOWN LOADS FOR RADIAL NOZZLES IN SPHERICAL PRESSURE VESSELS

I. J.Solids & Structures, Vol.3, p. 743-755

Guerlement, G.; Lamblin, LIMIT ANALYSIS OF CYLINDRICAL SHELL WITH REINFORCING D.O.; Save, M. A:; RINGS

Engineering. Stuct., Vol. 9, p. 146156

10

DBA Design by Analysis

Annex 1: Bibliography

Page A1.10

Table A1.2 DBA using FEM, analytical and/or numerical procedures (continued) AUTHOR

TITLE

SOURCE

Ghorashi, M.

LIMIT ANALYSIS OF CIRCULAR PLATES SUBJECTED TO ARBITRARY ROTATIONAL SYMMETRIC LOADINGS

I. J. Mech. Sci., Vol. 36, No. 2, p. 8794

Srinivasan, M. A.;

LIMIT STRENGTH OF LOCALLY LOADED SPHERICAL DOMES: RADIAL LOAD AT A CIRCULAR CUT-OUT

I. J. Solids & Stuctures, Vol. 24, No. 7, p. 723-734

Updike, D.P.; Kalnins, A.

LOCAL PLASTIC COLLAPSE IN STAYED TUBESHEETS WITH REGULAR SUPPORT ARRAYS

PVP-Vol. 235, p. 195-202

Biron, A.; Veillon, J.

INFLUENCE OF HEAD THICKNESS ON YIELD PRESSURE FOR CYLINDRICAL PRESSURE VESSELS

IJPVT, May 1974, p. 113-120

Foo, S. S. B.

ON THE LIMIT ANALYSIS OF CYLINDRICAL SHELLS WITH A SINGLE CUTOUT

IJPV&P, Vol.49, p. 1-16

Cocks, A.C.F.

LOWER-BOUND SHAKEDWON ANALYSIS OF A SIMPLY SUPPORTED PLATE CARRYING A UNIFORMLY DISTRIBUTED LOAD AND SUBJECTED TO CYCLIC THERMAL LOADING

I.J.Mech.Sci, Vol.26, No.9, p.471475

Robinson, M.

LOWER-BOUND LIMIT PRESSURES FOR THE CYLINDERCYLINDER INTERSECTION: A PARAMETRIC SURVEY

IJPVT, Vol. 100, p. 65-73

McDonald, C.K.; McDonald, R.E.

NONLINEAR EFFECTS ON THE DISCONTINUITY STRESSES IN A CYLINDRICAL SHELL WITH A FLAT HEAD CLOSURE

IJPVT, February 1974, p. 44-46

Save, M. A.

LIMIT ANALYSIS AND DESIGN: AN UP-TO-DATE SUBJECT OF ENGINEERING PLASTICITY

Conf. Title: Plasticity Today, June 1983, Udine, Italy; publ. by Elsevier, p. 767-785

Karadeniz, S.; Ponter, A.R.S.; Carter, K.F.;

THE PLASTIC RATCHETING OF THIN CYLINDRICAL SHELLS SUBJECTED TO AXISYMMETRIC THERMAL AND MECHANICAL LOADING

IJPVT, Vol. 109, p. 387-393

Jiang, W.

THE ELASTIC PLASTIC ANALYSIS OF TUBES-I: GENERAL THEORY

IJPVT, Vol. 114, p. 213-221

Jiang, W.

THE ELASTIC-PLASTIC ANALYSIS OF TUBES-II: VARIABLE LOADING

IJPVT, Vol. 114, p. 222-228

Jiang, W.

THE ELASTIC-PLASTIC ANALYSIS OF TUBES-III: SHAKEDOWN ANALYSIS

IJPVT, Vol. 114, p. 229-235

Jiang, W.

THE ELASTIC PLASTIC ANALYSIS OF TUBES-IV: THERMAL RATCHETTING

IJPVT, Vol. 114, p. 236-245

SHELL-STIFFENER INTERACTION. APPLICATION TO SIMPLY Cinquini, C.; Guerlement, G.; Lamblin, SUPPORTED CYLINDRICAL SHELLS UNDER UNIFORM D.; PRESSURE

I. J. Solids & Structures, Vol. 21, Nr. 5, p. 447-465

11

DBA Design by Analysis

Annex 1: Bibliography

Page A1.11

Table A1.3: Theory and/or numerical methods for limit and shakedown analysis WRITER

Rust, W.

Franco, J.R.Q; Ponter, A.R.S.

Franco, J.R.Q.; Ponter, A.R.S.

TITLE MATERIALMODELLE FÜR ANSYS: PULVERMETALLURGIE, ZYKLISCHE PLASTIZITÄT, TRESCA A GENERAL APPROXIMATE TECHNIQUE FOR THE FINITE ELEMENT SHAKEDOWN AND LIMIT ANALYSIS OF AXISYMMETRIC SHELLS. PART 2: NUMERICAL APPLICATIONS A GENERAL APPROXIMATE TECHNIQUE FOR THE FINITE ELEMENT SHAKEDOWN AND LIMIT ANALYSIS OF AXISYMMETRIC SHELLS: PART 1: THEORY AND FUNDAMENTAL RELATIONS

SOURCE 15.CAD-FEM Users' Meeting, Oktober 1997, Fulda, Germany, Art. 1-26

International Journal For Numerical Methods in Engineering, Vol. 40, p. 3515-3536, 1997

International Journal for Numerical Methods in Engineering, Vol. 40, p. 3495-3513, 1997

Morelle,P.; Fonder, SHAKEDOWN AND LIMIT ANALYSIS OF SHELLS - A G. VARIATIONAL AND NUMERICAL APPROACH

Lecture Notes in Engineering 26. Publ. by Springer-Verlag, p. 381-405, (Conf: Shell and Spatial Structures, Louvain, Belg., 1986)

Stumpf, H.; Le Khanh Chau

ON SHAKEDOWN OF ELASTOPLASTIC SHELLS

Quarterly of Applied Mathematics, Vol. XLIX, No. 4, p. 781-793

Berak, E. G.; Gerdeen, J. C.

A FINITE ELEMENT TECHNIQUE FOR LIMIT ANALYSIS OF PVP - Vol. 124, p. 1-12 STRUCTURES

Borges, L. A.; A NONLINEAR OPTIMIZATION PROCEDURE FOR LIMIT Zouain, N.; Huespe, ANALYSIS A. E.;

Eur. J. Mech., A/Solids, 15, No. 3, p. 487-512

Esslinger, M.; Geier, B.; Wendt, U.

BERECHNUNG DER TRAGLAST VON Stahlbau , 3, p. 76-80 ROTATIONSSCHALEN IM ELASTOPLASTISCHEN BEREICH

Tin-Loi, F.; Pulmano, V. A.

LIMIT LOADS OF CYLINDRICAL SHELLS UNDER HYDROSTATIC PRESSURE

J. of Structural Engineering, Vol. 117, No. 3, p. 643-656

Biron, A.

REVIEW OF LOWER-BOUND LIMIT ANALYSIS FOR PRESSURE VESSEL INTERSECTIONS

IJPVT, August 1977, p. 413-418

Staat, M.; Heitzer, M.

LIMIT AND SHAKEDOWN ANALYSIS USING A GENERAL PURPOSE FINITE ELEMENT CODE

Proc. of NAEFEMS Wordl Congress '97, Vol.1, p. 522-533

Save, M.

ATLAS OF LIMIT LOADS OF METAL PLATES, SHELSS AND Elsevier DISKS

12

DBA Design by Analysis

Annex 1: Bibliography

Page A1.12

Table A1.3: Theory and/or numerical methods for limit and shakedown analysis (continued) WRITER

TITLE

SOURCE

O'Donnell, W.J.; YIELD SURFACES FOR PERFORATED MATERIALS Porowski, J.

J. of Applied Mechanics, March 1973, p. 263 f.f.

Stein, E.; Zhang, SHAKEDOWN ANALYSIS FOR PERFECTLY PLASTIC G.; Mahnken, R. AND KINEMATIC HARDENING MATERIALS

E. Stein: "Progress in Computational Analysis of Inelastic Structures", CISM 1993, p.175-244

Oschatz, A.

MISES-FLIEßBEDINGUNGEN FÜR ROTATIONSSCHALEN

König, J. A.

ON EXACTNESS OF THE KINEMATICAL APPROACH IN THE STRUCTURAL SHAKEDOWN AND LIMIT Ingenieur Archiv 52 (1982), p. 421-428 ANALYSIS

Koiter, W. T.

Progress in Solid Mechanics, Vol. 1, p. 165GENERAL THEOREMS FOR ELASTIC-PLASTIC SOLIDS 221; Edited by I.N. Sneddon & R. Hill, North Holland, 1960

König; J.A.

SHAKEDOWN THEORY OF PLATES

Archiwum Mechaniki Stossowanej 5, 21 (1969), p. 623-637

Hübel, H.

ELASTIC-PLASTIC CYLINDRICAL SHELL UNDER AXISYMMETRIC LOADING-ANALYTICAL SOLUTION

IJPV&P, 29, p. 67-81

Polizzotto, C.

A STUDY ON PLASTIC SHAKEDOWN OF STRUCTURES: PART II - THEOREMS

J. of Applied Mechanics, Vol. 60, p. 324-330

Polizzotto, C.

A STUDY ON PLASTIC SHAKEDOWN OF STRUCTURES: PART I - BASIC PROPERTIES

J. of Applied Mechanics, Vol. 60, p. 318-323

Ploizzotto C.

ON THE CONDITIONS TO PREVENT PLASTIC SHAKEDOWN OF STRUCTURES: PART II - THE PLASTIC SHAKEDOWN LIMIT LOAD

J. of Applied Mechanics, Vol. 60, p. 20-25

Polizzotto C.

ON THE CONDITIONS TO PREVENT PLASTIC SHAKEDOWN OF STRUCTURES: PART I - THEORY

J. of Applied Mechanics, Vol. 60, p. 15-19

Seshadri M.

LOWER BOUND LIMIT LOADS USING VARIATIONAL CONCEPTS: THE M-alfa METHOD

Int. J. Ves. & Piping 71, 1997

Ponter, A.R.S.; Carter, K.F.

LIMIT STATE SOLUTIONS, BASED UPON LINEAR ELASTIC SOLUTIONS WITH A SPATIALLY VARYING ELASTIC MODULUS

Computer methods in applied mechanics and engineering

Ponter, A.R.S.; Carter, K.F.

SHAKEDOWN STATE SIMULATION TECHNIQUES BASED ON LINEAR ELASTIC SOLUTIONS

Computer methods in applied mechanics and engineering

Plancq, D.

ETUDE ELASTIQUE ET ANALYSE LIMITE DES PIQUAGES ET DES TES

Thesis (1997), Cetim

Technische Mechanik 5 (1984), Heft 1, p. 56 f.f.

DBA Design by Analysis

Annex 2: Draft CEN prEN 13445-3

Page A 2.1

We are grateful to CEN for their permission to include the Clause 18, Annexes B and C of the draft unfired pressure vessel standard prEN 13445-3, which constitute an essential reference for the anlayses within this manual.

DBA Design by Analysis

Annex 2: Clause 18 Draft CEN prEN 13445-3

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18 Detailed assessment of fatigue life 18.1 Purpose 18.1.1 This clause presents rules for the detailed fatigue assessment of pressure vessels and their components which are subjected to repeated fluctuations of stress. 18.1.2 The assessment procedure assumes that the vessel has been designed in accordance with all other requirements of this standard. 18.1.3 These rules are only applicable to the ferritic and austenitic steels specified in Part 2 of EN 13445. 18.1.4 These rules are not applicable to testing group 4 pressure vessels. For testing group 3 welded joints, see the special provisions in 18.10.2.1. 18.1.5 This method is not intended for design involving elastic follow-up (See reference [1]). 18.2 Specific definitions The following definitions are in addition to those in clause 3: 18.2.1 fatigue design curves: Curves given in this clause of ∆σR against N for welded and unwelded material, and of ∆σR/Rm against N for bolts. 18.2.2 discontinuity: A shape or material change which affects the stress distribution. 18.2.3 gross structural discontinuity: A structural discontinuity which affects the stress or strain distribution across the entire wall thickness. 18.2.4 local structural discontinuity: A discontinuity which affects the stress or strain distribution locally, across a fraction of the wall thickness. 18.2.5 nominal stress: The stress which would exist in the absence of a discontinuity. NOTE 1: Nominal stress is a reference stress which is calculated using elementary theory of structures. It excludes the effect of structural discontinuities (e.g. welds, openings and thickness changes). See figure 18-1. NOTE 2: The use of nominal stress is permitted for some specific weld details for which determination of the structural stress would be unnecessarily complex. It is also applied to bolts. NOTE 3: The nominal stress is the stress commonly used to express the results of fatigue tests performed on laboratory specimens under simple unidirectional axial or bending loading. Hence, fatigue curves derived from such data include the effect of any notches or other structural discontinuities (e.g. welds) in the test specimen. 18.2.6 notch stress: The total stress located at the root of a notch, including the non-linear part of the stress distribution. NOTE 1: See figure 18-1 for the case where the component is welded, but notch stresses may similarly be found at local discontinuities in unwelded components. NOTE 2: Notch stresses are usually calculated using numerical analysis. Alternatively, the nominal or structural stress is used in conjunction with the effective stress concentration factor, Keff.

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18.2.7 equivalent stress: The uniaxial stress which produces the same fatigue damage as the applied multi-axial stresses. NOTE: It is calculated using a failure criterion; the Tresca criterion is applied in this clause. 18.2.8 stress on the weld throat: The average stress on the throat thickness in a fillet or partial penetration weld. NOTE 1: In the general case of a non-uniformly loaded weld, it is calculated as the maximum load per unit length of weld divided by the weld throat thickness and it is assumed that none of the load is carried by bearing between the components joined. NOTE 2: The stress on the weld throat is used exclusively for assessment of fatigue failure by cracking through weld metal in fillet or partial penetration welds. 18.2.9 stress range (∆ ∆σR): The value from maximum to minimum in the cycle (see figure 18-2[withdrawn]) of nominal, principal or equivalent stress, depending on the component and as defined in this clause.

1 Nominal stress; 2 Structural stress; 3 Notch stress; 4 Extrapolation to give structural stress at potential crack initiation site. Figure 18-1: Distribution of nominal, structural and notch stress at a structural discontinuity

1 One cycle; ∆σ Stress range

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18.2.10 structural stress: The linearly distributed stress across the section thickness which arises from applied loads (forces, moments, pressure, etc.) and the corresponding reaction of the particular structural part. NOTE 1: Structural stress includes the effects of gross structural discontinuities (e.g. branch connections, cone/cylinder intersections, vessel/end junctions, thickness change, deviations from design shape, presence of an attachment). However, it excludes the notch effects of local structural discontinuities (e.g. weld toe) which give rise to non-linear stress distributions across the section thickness. See figure 18-1. NOTE 2: For the purpose of a fatigue assessment, the structural stress shall be evaluated at the potential crack initiation site. NOTE 3: Structural stresses may be determined by one of the following methods: numerical analysis (e.g. finite element analysis (FEA)), strain measurement or the application of stress concentration factors to nominal stresses obtained analytically. Guidance on the use of numerical analysis is given in reference [2]. NOTE 4: Under high thermal stresses, the peak stress rather than the linearly distributed stress should be considered. 18.2.11 weld throat thickness: The minimum thickness in the weld cross-section.

18.3 Specific symbols and abbreviations The following are in addition to those in clause 4. C

is the constant in equation of fatigue design curves for welded components;

D

is the cumulative fatigue damage index;

E

is the modulus of elasticity at maximum operating temperature;

Fe, Fs

are coefficients;

fb

is the overall correction factor applied to bolts;

fc

is the compressive stress correction factor;

fe

is the thickness correction factor in unwelded components;

few

is the thickness correction factor in welded components and bolts;

fm

is the mean stress correction factor;

fs

is the surface finish correction factor;

ft *

is the temperature correction factor;

fu

is the overall correction factor applied to unwelded components;

fw

is the overall correction factor applied to welded components;

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Annex 2: Clause 18 Draft CEN prEN 13445-3

g

is the depth of groove produced by weld toe grinding;

Keff

is the effective stress concentration factor given by equation 18.7-1

Km

is the stress magnification factor due to deviations from design shape;

Kt

is the theoretical elastic stress concentration factor;

ke

is the plasticity correction factor for stress due to mechanical loading;

kν

is the plasticity correction factor for stress due to thermal loading;

M

is the mean stress sensitivity factor;

m

is the exponent in equations of fatigue design curves for welded components;

N

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is the allowable number of cycles obtained from the fatigue design curves (suffix i refers to life under ith stress range);

n

is the number of applied stress cycles (suffix i refers to number due to ith stress range);

R

is the mean radius of vessel at point considered;

Rmin

is the minimum inside radius of cylindrical vessel, including corrosion allowance;

Rmax

is the maximum inside radius of cylindrical vessel, including corrosion allowance;

Rz

is the peak to valley height;

r

is the radius of groove produced by weld toe grinding;

Sij

is the difference between either principal stresses (σi and σj) or structural principal stresses (σstruc,i and σstruc,j) as appropriate;

tmax

is the maximum operating temperature;

tmin

is the minimum operating temperature;

t*

is the assumed mean cycle temperature;

∆εT

is the total strain range;

∆σ

is the maximum principal stress range (suffix i refers to ith stress range);

∆σeq

is the equivalent stress range (suffix i refers to ith stress range);

∆σR

is the stress range obtained from fatigue design curve;

∆σstruc

is the structural stress range;

∆σeq,l

is the equivalent stress range corresponding to variation of equivalent linear distribution;

∆σeq,t

is the notch (or total) equivalent stress range;

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∆σeq,nl

is the stress range corresponding to variation of non-linear part of the stress distribution;

δ

is the total deviation from mean circle of shell at seam weld;

δ1

is the offset of centre-lines of abutting plates;

θ

is the angle between tangents to abutting plates at a seam;

σ

is the direct stress or stress range as indicated (suffix w applies to weld);

(σ eq ) op

is the equivalent stress due to operating pressure (for specific use in 18.4.6)

σeqmax

is the maximum equivalent stress;

σeqmin

is the minimum equivalent stress;

σ

eq

is the mean equivalent stress;

σ

eq, r

is the reduced mean equivalent stress for elastic-plastic conditions;

σstruc1

is a structural principal stress (1, 2 , 3 apply to the axes) at a given instant;

σ1

is a principal stress (suffices 1, 2 , 3 apply to the axes) at a given instant;

σV1, σV2

are stress ranges obtained in the example of reservoir cycle counting in 18.9.3;

τ

is the shear stress or stress range as indicated (suffix w applies to weld);

18.4 Limitations 18.4.1 Where a vessel is designed for fatigue, the method of manufacture of all components,

including temporary fixtures and repairs, shall be specified by the manufacturer 18.4.2 There are no restrictions on the use of the fatigue design curves for vessels which operate at subzero temperatures, provided that the material through which a fatigue crack might propagate is shown to be sufficiently tough to ensure that fracture will not initiate from a fatigue crack. 18.4.3 These rules are only applicable to vessels which operate at temperatures below the creep range of the material. Thus, the fatigue design curves are applicable up to 380 °C for ferritic steels and 500 °C for austenitic stainless steels. 18.4.4 It is a condition of the use of these rules that all regions which are fatigue-critical are accessible for inspection and non-destructive testing, and that in-service inspection shall be performed at not later than 20 % of the allowable fatigue life. 18.4.5 Corrosive conditions are detrimental to the fatigue lives of steels. Environmentally-assisted fatigue cracks can occur at lower levels of fluctuating stress than in air and the rate at which they propagate can be higher. The fatigue strengths specified do not include any allowances for corrosive conditions. Therefore, where corrosion fatigue is anticipated and effective protection from the corrosive medium cannot be guaranteed, a factor should be chosen, on the basis of experience or testing, by which the fatigue strengths given in these rules should be reduced to compensate for the corrosion. If, because of lack of experience, it

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is not certain that the chosen fatigue strengths are low enough, the frequency of inspection should be increased until there is sufficient experience to justify the factor used. 18.4.6 For water conducting parts made from non-austenitic steels, operating at temperatures exceeding 200 °C, conservation of the magnetite protective layer shall be ensured. This will be obtained if the stress range on the surface in contact with water always fulfils the following conditions:

( )

σ eqmax ≤ σ eq

( )

σ eqmin ≥ σ eq

op

op

(

)

…(18.4-1)

(

)

…(18.4-2)

+ 200 N / mm 2 − 600 N / mm 2

NOTE: It is assumed that under the operating conditions at which the magnetite layer forms, there is no stress in that layer.

18.4.7 Where vibration (e.g. due to machinery, pressure pulsing or wind) cannot be removed by suitable strengthening, support or dampening, it shall be assessed using the method in this clause. 18.5 General 18.5.1 A fatigue assessment shall be made at all locations where there is a risk of fatigue crack

initiation. NOTE: It is recommended that the fatigue assessment is performed using operating rather than design loads. 18.5.2 In fatigue, welds behave differently from plain (unwelded) material. Therefore the

assessment procedures for welded and unwelded material are different. 18.5.3 Plain material might contain flush ground weld repairs. The presence of such repairs can

lead to a reduction in the fatigue life of the material. Hence, only material which is certain to be free from welding shall be assessed as unwelded. 18.5.4 A typical sequence in the design of a vessel for fatigue is shown in table 18-1. 18.5.5 The fatigue life obtained from the appropriate fatigue design curves (for welded

components, unwelded components and bolts) for constant amplitude loading is the allowable number of cycles. 18.5.6 For calculation of cumulative damage under variable amplitude loading, D is given by: D=

n1 n 2 + +...... = N1 N1

∑ Nii n

...(18.5-1)

The following condition shall be met: D ≤1

...(18.5-2)

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Table 18-1: Summary of fatigue assessment process Task 1. Design vessel for static loads 2. Define fatigue loading

3.

Identify locations of vessel to be assessed

4.

At each location, establish stress range

5.

At each location, establish design stress range spectrum Identify fatigue strength data

6.

7.

Note relevant implications and inform relevant manufacturing and inspection personnel

8.

Extract allowable fatigue lives from fatigue design and perform assessment

9.

Further action if location fails assessment

Comment Gives layout, details, sizes

Relevant clause(s) Main Standard

Based on operating specification, secondary effects identified by manufacturer, etc. Structural discontinuities, openings, joints (welded, bolted), corners, repairs, etc. a) Calculate stress ranges b) Apply plasticity correction factors where relevant c) Apply overall correction factor

18.5

Perform cycle counting operation

a) Welded material b) Unwelded material c) Bolted material a) Inspection requirements for welds b) Control of or assumptions about misalignment c) Acceptance levels for weld flaws a) Welded material b) Unwelded material c) Bolts d) Assessment method a) Re-assess using more refined stress analysis b) Reduce stresses by increasing thickness c) Change detail d) Apply weld toe dressing (if appropriate)

18.5

Welded: 18.6, 18.8 and 18.10.6; Unwelded: 18.7, 18.8 and 18.11.2; Bolts: 18.7.2,18.12.2.2. 18.9

18.10, tables 18-4 & 18-5 18.11 18.12 Tables 18-4 or 18-5 18.10.4 18.10.5 18.10, table 18-7 18.11, table 18-10 18.12 18.5.5, 18.5.6 18.6 (welded), 18.7 (unwelded)

Table 18-4 or 18-5 18.10.2.2

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18.6 Welded material 18.6.1 Stresses For the assessment of simple attachments and aligned seam welds, principal stresses may be calculated elastically from the nominal stresses. In the case of weld metal in directly loaded fillet or partial penetration joints, but not butt joints (e.g. seams), use is made of the nominal stress range on the throat as shown in 18.6.3. For all other welded components, structural principal stresses shall be determined. They shall be: - either calculated using elastic theory from the structural stresses at the potential crack initiation site, taking account of all membrane, bending and shearing stresses; - or deduced from strains measured on the vessel and converted to linear-elastic conditions. Where the structural principal stress is obtained by detailed stress analysis (e.g. FEA) or by measurement, it is to be determined from the principal stress which acts closest to the normal to the weld by extrapolation using the procedures detailed in figure 18-3. NOTE1: In arriving at the structural principal stress, it is necessary to take full account of the structural discontinuities (e.g. nozzles) and all sources of stress. The latter may result from: global shape discontinuities such as cylinder to end junctions, changes in thickness and welded-on rings; deviations from intended shape such as ovality, temperature gradients, peaking and misaligned welds (note some misalignment is already included in some of the fatigue design curves). Methods in this clause and in the published literature (see references [3] - [7]) provide estimates of such stresses for many geometries, or at least enable a conservative assessment to be made. NOTE 2: For nozzles being assessed using principal stresses directly, three possible stress concentrations due to structural discontinuities should be considered and estimates of the stresses made as follows: a) At the crotch corner, a stress concentration factor may be applied to the nominal hoop stress in the shell to determine the maximum structural stress which is circumferential with respect to the nozzle. b) At the weld toe in the shell, the stresses in the shell acting in all radial directions with respect to the nozzle should be considered in order to determine the maximum structural stress in the shell. Stresses in the shell as a result of mechanical loading on the nozzle as well as pressure should be considered. c) At the weld toe in the branch, the maximum structural stress range in the branch should be calculated. Again, the possibility of mechanical as well as pressure loading should be considered. NOTE 3: Since the maximum range of stress on the weld throat can be expressed as a vector sum, ∆σ is the scalar value of the greatest vector difference between different stress conditions during the cycle.

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Locations of stresses for determination of structural stress by extrapolation to point of stress concentration (weld toe in this case): a) low bending stress component, gauge length ≤ 0,2e, linear extrapolation; b) high bending stress component, stiff elastic foundation, gauge length ≤ 0,2e, quadratic extrapolation; c) gauge length > 0,2e, linear extrapolation where "gauge length" refers to size of strain gauge or FE mesh. Figure 18-3: Extrapolation to obtain structural stress from FEA or strain gauge results 18.6.2 Stress range in parent material and butt welds 18.6.2.1 Options For the assessment of simple attachments and aligned seam welds, the nominal equivalent stress range (see table 18-4a) and 18-4e)) or the nominal principal stress range (see table 18-5a) and 18-5e)) can be used. This shall be calculated in the same way as structural stress ranges (see equations 18.6-4, 18.6-5 and 18.6-6) using nominal principal stresses instead of structural principal stresses. For all other welded components, depending on the calculation method: - either the principal stress range shall be determined from the range of the structural principal stresses and used with table 18-5; - or the equivalent stress range shall be calculated from the range of the equivalent stresses determined from the structural principal stresses and used with table 18-4. Tension stresses are considered positive and compression stresses negative. In both cases, an important aspect is whether, under multiple load actions, the directions of the structural principal stresses remain constant or not. Where applicable, the elastically calculated principal or equivalent stress range shall be modified by the plasticity correction factors given in 18.8. NOTE: For welded components, the full stress range is used regardless of applied or effective mean stress. The fatigue design curves incorporate the effect of tensile residual stresses; post-weld heat treatment is ignored in the fatigue analysis. 18.6.2.2 Equivalent stress range ∆σeq

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18.6.2.2.1 Structural principal stress directions constant When the structural principal stress directions are constant, ∆σeq shall be calculated as follows. The variation with time of the three structural principal stresses shall be established. The variation with time of the three principal stress differences shall be calculated as follows: S12 = σ struc1 − σ struc2

...(18.6-1)

S23 = σ struc2 − σ struc3

...(18.6-2)

S31 = σ struc3 − σ struc1

...(18.6-3)

Applying Tresca's criterion, ∆σeq is:

(

∆σ eq = max S12 max − S12 min ; S23 max − S23 min ; S31max − S31min

)

...(18.6-4)

NOTE: A typical example is shown in figures 18-4(a) and (b). ∆σeq is twice the greatest shear stress range and occurs on one of the three planes of maximum shear.

(a): Typical variation with time of the structural principal stresses

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(b): Variation with time of the principal stress differences and the resulting ∆ σ eq

(c): For unwelded components, the variation with time of the difference between the structural principal stresses which determine ∆σeq (i.e. σ struc1 and σ struc3 in this case) and the resulting mean σ

eq

Figure 18-4: Typical example of stress variation when the principal stress directions remain constant 18.6.2.2.2 Structural principal stress directions change When the structural principal stress directions change, ∆σeq shall be calculated as follows.

Determine the variation with time of the six stress components (three direct and three shear) with reference to some convenient fixed axes. For each stress component, calculate the maximum variation. The structural principal stress ranges, S12, S23 and S31, are then calculated from the resulting stress variations. ∆σeq is determined as before from equation 18.6-4.

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18.6.2.3 Principal stress range ∆σ 18.6.2.3.1 Application If the potential fatigue crack initiation site is at the weld toe or on the surface of the weld, the structural stress range in the material adjacent to the weld is required for the fatigue assessment. Since σstruc3 = 0, use is made of the two structural principal stresses σstruc1 and σstruc2 acting essentially (i.e. within 45°) parallel and normal to the direction of the weld respectively, on each material surface. 18.6.2.3.2 Structural principal stress directions constant Where the directions of the structural principal stresses remain fixed, ∆σ is determined as follows. ∆σ struc1 = σ struc1max - σ struc1min

...(18.6-5)

∆σ struc2 = σ struc2max - σ struc2min

...(18.6-6)

NOTE: Both principal stress ranges may need to be considered, depending on their directions. 18.6.2.3.3 Structural principal stress directions change When the structural principal stress directions change during cycling between two load conditions, ∆σ shall be calculated as follows. Determine the three stress components (two direct and one shear) at each load condition with reference to some convenient fixed axes. For each stress component, calculate the maximum difference between the stresses. Calculate the principal stresses from the resulting stress differences. NOTE: Both principal stress ranges may need to be considered, depending on their directions. Where cycling is of such a complex nature that it is not clear which two load conditions will result in the greatest value of ∆σ, they shall be established by carrying out the above procedure for all pairs of load conditions. Alternatively, it is conservative to assume that ∆σ is the difference between the algebraically greatest and smallest principal stresses occurring during the whole loading cycle regardless of their directions, and to assume the lowest classification for the detail (see Table 18.5). 18.6.3 Stress range in directly loaded fillet or partial penetration welds ∆σ is the maximum range of stress on the weld throat. Where stress cycling is due to the application and removal of a single load,

(

∆σ = σ w 2 + τ w 2

)

1/ 2

...(18.6-7)

where σw is the normal stress range on the weld throat and τw is the shear stress range on the weld throat. Where stress cycling is due to more than one load source, but the directions of the stresses remain fixed, ∆σ is determined from the maximum range of the load per unit length of the weld. Where the direction of the stress vector on the weld throat changes during the cycle between two extreme load conditions, ∆σ is the magnitude of the vector difference between the two stress vectors.

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Where cycling is of such a complex nature that it is not clear which two load conditions will result in the greatest value of ∆σ, then the vector difference should be found for all pairs of extreme load conditions. Alternatively, it is conservative to assume: ∆σ = [( σ max - σ min )2 + ( τ 1max - τ 1min )2 + ( τ 2 max - τ 2 min )2 ]1 / 2

...(18.6-8)

where τ1 and τ2 are the two components of shear stress on the weld throat. Directly loaded fillet and partial penetration welds do not need to be assessed if the effective weld throat thickness is such that the stress range in the weld does not exceed 0,8 times the stress range in the plate. Welds not subject to direct loading (e.g. attachments) do not need to be assessed if the effective weld throat is at least 0,7 times the thickness of the thinner part joined by the weld.

18.7 Unwelded components and bolts 18.7.1 Unwelded components 18.7.1.1 Stresses For unwelded components, equivalent effective notch stresses only shall be determined. They shall be calculated using structural principal stresses which incorporate the full effect of gross and local structural discontinuities. The equivalent effective notch stress can be obtained by the use of Keff given by: K eff = 1 +

15 , (K t − 1) ∆σ struc 1 + 0,5 K t ⋅ ∆σ D

(18-7.1)

where ∆σD = ∆σR for N ≥ 2.106 cycles for unwelded material. The effective stress concentration factor Keff is used in subclause 18.11.2 for the calculation of the overall correction factor. If the notch stresses are calculated directly by analysis (e.g. FEA) or determined experimentally (e.g strain gauges), the structural and peak stresses should be separated (as described in Annex C) to give the total stress as follows: σ total = σ struc + σ peak

(18.7-2)

The theoretical stress concentration factor shall be calculated as follows: Kt =

σ peak σ total =1+ σ struc σ struc

(18.7-3)

If the total stress is calculated directly by analysis (e.g. FEA) the model shall include the notch in sufficiently fine details. If the equivalent notch stresses are determined directly by analysis (e.g. FEA) the model must include the notch in sufficiently fine detail. If they are determined experimentally (eg. strain gauges), measurements must be made within the notch, or sufficiently close to enable the notch stress to be established by extrapolation (see reference [2]). Strains shall be converted to stresses assuming linear elastic conditions and, for this case, no plasticity correction is required.

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The equivalent stress range ∆σeq and equivalent mean stress σ eq shall be determined. Two methods are given for this depending on whether, under multiple load actions, the directions of the structural principal stresses remain constant or not. Tension stresses are considered positive and compression stresses negative.A 18.7.1.2 Principal stress directions constant When the principal stress directions remain constant, ∆σeq shall be determined per 18.6.2.2.1 and equation (18.6-4). The corresponding mean equivalent stress σ eq Bis the maximum value of the average of the two structural principal stresses, σstruc,i and σstruc,j, which produced ∆σeq. Thus: σ eq =

1 2

[(σ

struc,i

)

(

)min]

+ σ struc, j + σ struc,i + σ struc,j max

...(18.7-4)

NOTE: A typical example is shown in figure18-4(c). σ eq is twice the mean value of the direct stress, averaged over time, normal to the plane of maximum shear stress range. 18.7.1.3 Principal stress directions change When the principal stress directions change, ∆σeq shall be calculated as detailed in 18.6.2.2.2. ∆σeq and

σ

eq

Care determined from equations (18.6-4) and (18.7-4) respectively.

18.7.2 Bolts For bolts, ∆σ is the maximum nominal stress range arising from direct tensile and bending loads on the core cross-sectional area, determined on the basis of the minor diameter. For pre-loaded bolts, account may be taken of the level of pre-load, with ∆σ based on the full applied load rather than the fluctuating portion of that load. For a bolt pre-tensioned to its minimum proof load, ∆σ may be based on 20 % of the maximum applied load. NOTE: The fatigue design curve for bolts takes account, for any form of thread, of the stress concentrations at the thread root.

18.8 Plasticity correction factors 18.8.1 Elastic-plastic conditions For any component, if the calculated pseudo-elastic structural stress range for both welded joints and unwelded parts exceeds twice the yield strength of the material under consideration, i.e. if ∆σ eq,l > 2Rp0,2/ t* , see note, it shall be multiplied by a plasticity correction factor. The correction factor for mechanical loading is ke and for thermal loading it is kν. NOTE: This applies to ferritic steels; for austenitic steels, use Rp1,0/t* . 18.8.1.1 Mechanical loading For mechanical loading, the corrected stress range is ke ∆σ, where:

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 ∆σ  eq,l   k e = 1 + A0  − 1  2Rp0,2/ t*   

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...(18.8-1)

where: A0 = 0,5 for ferritic steels with 800 ≤ Rm ≤ 1000(N / mm 2 ) ; = 0,5 for ferritic steels with Rm ≤ 500(N / mm 2 ) and for all austenitic steels (see .........note in 18.8.1); = 0,4 +

(Rm − 500) 3000

for ferritic steels with 500 ≤ Rm ≤ 800(N / mm 2 ) .

The procedure for determining the mean equivalent stress to allow for elastic-plastic conditions is shown in figure 18-5 and applied in 18.11.

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(*) For unwelded parts, σ or ∆σ values are notch stresses or stress ranges (**)This applies to ferritic steels; for austenitic steels, use Rp1,0/t* . Figure 18-5: Modifications to mean equivalent stress to allow for elastic-plastic conditions due to mechanical loadings

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18.8.1.2 Thermal loading In the case of a thermal stress distribution which is non-linear through the material thickness, both the nonlinear and the equivalent linear stress distributions shall be determined for each stress component. Using the linearised stress range ∆σeq,l, kν shall be calculated by: kν =

0 ,7 0,4 0,5 +  ∆σ  eq,l    R   p0,2/ t* 

...(18.8-3)

The corrected stress range shall be either kν . ∆σeq,l for welded joints or kν . ∆σeq,t for unwelded zones. 18.8.1.3 Elastic-plastic analysis If the total strain range ∆εT (elastic plus plastic) due to any source of loading is known from theoretical or experimental stress analysis, correction for plasticity is not required and

∆σ = E . ∆ ε T

...(18.8-4)

18.9 Fatigue action 18.9.1 Loading 18.9.1.1 All sources of fluctuating load acting on the vessel or part shall be identified. NOTE: Such loads are: fluctuations of pressure; variations in contents; temperature transients; restrictions of expansion or contraction during temperature variations; forced vibrations; and variations in external loads. Account shall be taken of all operational and environmental effects defined in the purchase specification. 18.9.2 Simplified cycle counting method 18.9.2.1 Loads shall be grouped into specific loading events. Loading events shall be independent of each other and shall be considered separately. 18.9.2.2 A loading specification shall be prepared stating for each loading event the stress range (calculated from 18.5, 18.6, 18.7 and 18.8 as appropriate for the component and load) and number of cycles for each load. As shown in figure 18-6 and table 18-3, the stress ranges shall be plotted or tabulated against number of cycles. The loading with the lowest number of cycles shall be plotted or tabulated at the top and the cycles summed as shown.

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∆σ ∆σ4

∆σ3

∆σ2 ∆σ1

0 ∆σ n

n4

c4

n3

c3

n2

c2

c1

n1

n

combined stress range number of applied cycles

c4 cycles of ∆σ4 + ∆σ3 + ∆σ2 + ∆σ1 c3 cycles of ∆σ3 + ∆σ2 + ∆σ1 c2 cycles of ∆σ2 + ∆σ1 c1 cycles of ∆σ1 Figure 18--6 Simplified counting method NOTE: An example is shown in table 18-3

Table 18-3: Example of determination of stress cycles using simplified cycle counting method Loading 4

Individual loadings Stress range No of cycles Example n4 Full pressure σ4 range

Number A

3

σ3

n3

Temperature difference

B

2

σ2

n2

C

1

σ1

n1

Pressure fluctuation Mechanical loading

D

Loading events Stress range No of cycles c4 = n4 ∆σ4 + ∆σ3+ ∆σ2+ ∆σ1 c3 = n3 - n4 ∆σ3+ ∆σ2+ ∆σ1 c2 = n2 -n3 ∆σ2 + ∆σ1 n4 c1 = n1 -n2 -n3 ∆σ1 -n4

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Annex 2: Draft CEN prEN 13445-3

Page A 2.20

18.9.3 Reservoir cycle counting method 18.9.3.1 As an alternative to the simplified counting method given in 18.9.2, the more accurate reservoir cycle counting procedure may used. 18.9.3.2 For each loading event, derive the variation with time of either the structural principal stress or the equivalent stress as specified in 18.6 or 18.7 for the component. 18.9.3.3 Plot the peak and trough values for two occurrences of the event as shown in figure 18-7. 18.9.3.4 Mark the highest peak stress in each cycle and join the two peaks together with a straight line. If there are two or more equal highest peaks in a cycle, mark only the first such peak in the occurrence. 18.9.3.5 Join the two marked points and consider only that part of the plot which falls below this line, like the section of a full reservoir. 18.9.3.6 Drain the reservoir from the lowest point leaving the water that cannot escape. If there are two or more equal lowest points, drainage may be from any one of them. 18.9.3.7 List one cycle having a stress range, σV1, equal to the vertical height of water drained. 18.9.3.8 Repeat the step in 18.9.3.7 successively with each remaining body of water until the reservoir is emptied, listing one cycle at each draining operation. 18.9.3.9 List all the individual stress ranges in descending order of magnitude, σV1, σV2, σV3, σV4 etc. Where two or more cycles of equal stress range occur, record them separately. This provides the design stress range spectrum.

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Annex 2: Draft CEN prEN 13445-3

Page A 2.21

Figure18-7: Reservoir cycle counting method

18.10 Fatigue strength of welded components 18.10.1 Classification of weld details 18.10.1.1 Use of the tables Welds shall be classified to tables 18-4 and 18-5 according to whether the stress range is calculated from equivalent or principal stresses. The sketches in tables 18-4 and 18-5 indicate the potential mode of cracking corresponding to the position and direction of the fluctuating stress shown. All deviations from the ideal shape (misalignment, peaking, ovality etc.) shall be included in the determination of the stresses. NOTE1: A detail may appear several times in the tables because of the different modes in which it might fail. In general, fatigue strength depends on: the direction of the fluctuating stress relative to the weld detail; the locations of possible fatigue crack initiation at the detail; the geometrical arrangement and proportions of the detail; and the methods of manufacture and inspection. NOTE2: The fatigue life of a vessel or part of a vessel may be governed by one particular detail. Therefore, the classes of other details which experience the same fatigue loading need be no higher. For example, the potentially high class attainable from perfectly-aligned seams may not be required if overall fatigue life is governed by fillet welds. 18.10.1.2 Classification of weld details to be assessed using equivalent stress range Weld details and their corresponding classes for use in assessments based on equivalent stress range are given in table 18-4. The classification refers either to fatigue cracking in the parent metal from the weld toe or end, which shall be assessed using ∆σeq in the parent metal adjacent to the potential crack initiation site,

Annex 2: Draft CEN prEN 13445-3

DBA Design by Analysis

Page A 2.22

or to fatigue cracking in the weld itself from the root or surface, which shall be assessed using ∆σeq in the weld. Since ∆σeq has no direction, the class indicated in table 18-4 refers to the least favourable stressing direction for the particular weld detail and mode of fatigue cracking shown. Table 18.4. Class of weld details for use with structural equivalent stress range

Table 18.4(a) Seam welds Detail No.

1.1

Joint type

Sketch of detail

Class

Full penetration butt weld flush ground, including weld repairs

Comments

Testing group 1 or 2

Testing group 3

90

71

Weld proved free from surfacebreaking flaws and significant sub-surface flaws (see annex 18xx) by non-destructive testing. few=1

80

63

Weld proved free from significant flaws (see annex 18xx) by non-destructive testing and, for welds made from one side, full penetration.

Fatigue cracks usually initiate at weld flaws 1.2

Full penetration butt weld made from both sides or from one side on to consumable insert or temporary non-fusible backing

In case of misalignment, see clause 18.10.4.

1:3 1.3

Weld proved free from significant flaws by nondestructive testing (see annex 18xx).

e 80

63

Effect of centre-line offset included in calculated stress

63

40

Class includes effect of centreline offset of e/10 and therefore its effect is neglected when calculating the structural stress.

e

1:3

For other cases of misalignment, see detail 1.2. 1.4

Weld proved free from significant flaws (see annex 18xx) by non-destructive testing

α

80 71

63 56

α ≤ 30° α > 30°

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Table 18.4(a) Seam welds cont'd... Detail No.

Joint type

Sketch of detail

Comments

Class Testing

Testing

or 2 1.5

Full penetration butt

63

If full penetration can be assured.

40

If inside cannot be visually inspected and full penetration cannot be assured.

side without backing

In all cases 40

1.6

In case of misalignment, see clause 18.10.4. Circumferential seams only (see 5.7) Minimum throat = shell thickness.

Full penetration butt welds made from one side onto permanent backing. 56

Weld root pass inspected to ensure full fusion to backing.

40

Single pass weld 40

1.7

Joggle joint

56

Weld root pass inspected to ensure full fusion to backing.

40

Single pass weld 40

Cont'd...

In all cases Circumferential seams only (see 5.7) Minimum throat = shell thickness.

In all cases

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Annex 2: Draft CEN prEN 13445-3

Page A 2.24

Table 18.4(b) Shell to head or tubesheet Detail No.

Joint type

Sketch of detail

Testing group 1 or 2 2.1

Comments

Class Testing group 3

Welded-on head

Head plate must have adequate through-thickness properties to resist lamellar tearing. Full penetration welds made from both sides: 71 80

63 63

- as-welded - weld toes dressed (see 18.10.2.2) Partial penetration welds made from both sides:

32

32

- based on stress range on weld throat

63

63

- weld throat ≥ 0,8 x shell thickness Full penetration welds made from one side without back-up weld:

63

63

40

40

2.2

Welded-on head with relief groove 80

63

- if the inside weld can be visually inspected and is proved to be free from overlap or root concavity. - if the inside cannot be visually inspected and full penetration cannot be assured. - in all cases Weld proved free from significant flaws (see annex 18XX) by NDT. Full penetration welds: made from both sides, or from one side with the root pass ground flush. Made from one side:

63

- if the inside weld can be visually inspected and is proved to be free from weld overlap and root concavity.

40

- if the inside cannot be visually inspected. 40

- in all cases.

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Annex 2: Draft CEN prEN 13445-3

Page A 2.25

18.4(b) Shell to head or tubesheet cont'd ... Detail No.

Joint type

Sketch of detail

Testing group 1 or 2 2.3

Set-in head

Comments

Class Testing group 3

(a)

Full or partial penetration welds made from both sides. Refers to fatigue cracking from weld toe in shell: 71 80

63 63

- as-welded; - weld toes dressed (see 18.10.2.2). Partial penetration welds made from both sides:

(b) 32

32

- refers to fatigue cracking in weld, based on weld throat stress range.

63

63

- weld throat ≥0,8 x head thickness.

(c) Full penetration weld made from one side without back-up weld: 63

- if the inside weld can be visually inspected and is proved to be free from overlap or root concavity.

40

- if the inside cannot be visually inspected. 40

- in all cases.

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DBA Design by Analysis

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Table 18.4(c) Branch connections Detail No.

3.1

Joint type

Sketch of detail

Crotch corner

Comments

Class Testing group 1 or 2

Testing group 3

100

100

Assessment by the method for unwelded parts is the normal approach. However, simplified assessment using class 100 according to clause 18.11.2.2 is allowed. few = 1.

Crack radiates from corner into piece, sketches show plane of crack 3.2

Weld toe in shell

Full penetration welds: 71 80 63

- as welded - weld toes dressed (see 18.10.2.2) - in all cases Partial penetration welds:

63

63

71

63 32

- weld throat ≥ 0,8 x thinner thickness of connecting walls, as welded - weld toes dressed (see 18.10.2.2) Fillet and partial penetration welds

3.3

Stressed weld metal

32

3.4

Weld toe in branch

71

As-welded

80

Weld toes dressed (see 18.10.2.2). 63

In all cases en = branch thickness in equation 18.10-6

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Table 18.4(d) Jackets Detail No.

Joint type

Sketch of detail

Testing group 1 or 2 4.1

Comments

Class Testing group 3

Jacket connection weld with shaped sealer ring

Full penetration weld to be proved free from significant flaws (see annex 18xx) by non-destructive testing Welded from one side: 63

- multi-pass weld with root pass inspected to ensure full fusion;

40

- single pass weld.

71

40

- in all cases.

56

Welded from both sides or from one side with back-up weld.

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Annex 2: Draft CEN prEN 13445-3

Page A 2.28

Table 18.4(e) Attachments Detail No.

5.1

5.2

5.3

Joint type

Sketch of detail

Class for use with: Structural equivalent stress

Nominal equivalent stress

Testing group 1, 2, 3

Testing group 1, 2, 3

Attachment of any shape with an edge fillet or bevel buttwelded to the surface of a stressed member, with welds continuous around the ends or not

Comments

For details with welds continuous around ends, one class increase if weld toes dressed (see 18.10.2.2) 71

56

L ≤ 160mm, t ≤ 55mm

71

50

L > 160mm

For details with welds continuous around ends, one class increase if weld toes dressed (see 18.10.2.2)

Attachments of any shape with surface in contact with stressed member, with welds continuous around ends or not 71

56

L ≤ 160mm, W ≤ 55mm

71

50

L > 160mm, W ≤ 55mm

71

45

L > 160mm, W ≤ 55mm

Continuous stiffener

For full penetration welds, one class increase if weld toes dressed (see 18.10.2.2). 71

56

t ≤ 55mm

71

50

t > 55mm

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Table 18.4(f) Supports Detail No.

6.1

6.2

6.3

6.4

6.5

Joint type

Sketch of detail

Comments

Class Testing group 1 or 2

Testing group 3

Support on either horizontal or vertical vessel

71

71

As-welded.

80

80

Weld toe in shell dressed (see 18.10.2.2)

Trunnion support

71

71

As-welded.

80

80

Weld toe in shell dressed (see 18.10.2.2)

71

71

As-welded.

80

80

Weld toe in shell dressed (see 18.10.2.2)

Saddle support

Skirt support

Leg support (with or without reinforcing pad) with fillet weld to vessel continuous all around.

Welded from both sides:

71

71

as-welded;

80

80

weld toe in shell dressed (see 18.10.2.2).

56

56

Welded from one side

71

71

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Table 18.4(g) Flanges and pads Detail No.

Joint type

Sketch of detail

Testing group 1 or 2 7.1

Comments

Class Testing group 3

Full penetration butt welded neck flange or compensation flange with welding lug.

Weld proved free from surface-breaking and significant sub-surface flaws (see annex 18xx) by nondestructive testing. 80

63

Weld made from both sides or from one side with back-up weld or onto consumable insert or temporary backing. Weld made from one side:

63 40 40 7.2

- if full penetration can be assured - if the inside cannot be visually inspected. - in all cases. Full penetration welds:

Welded flange 71 80

63 63

- as-welded; - weld toe dressed (see 18.10.2.2) Partial penetration welds:

7.3

63

63

32

32

Set-in flange or pad

- weld throat ≥ 0,8 x shell thickness; - weld throat < 0,8 x shell thickness.

Full penetration weld: 71 80

63 63

- as-welded; - weld toe dressed (see 18.10.2.2). Fillet welded on both sides:

63

63

32

32

- weld throat ≥ 0,8 x shell thickness. - weld throat < 0,8 x shell thickness.

DBA Design by Analysis

Annex 2: Draft CEN prEN 13445-3

Page A 2.31

Table 18.4(g) Flanges and pads concluded

Detail No.

7.4

Joint type

Set-in flange or pad, welded from both sides

Sketch of detail

Comments

Class Testing group 1 or 2 63

Testing group 3

32

32

63

weld throat ≥ 0,8 x shell thickness. weld throat < 0,8 x shell thickness.

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DBA Design by Analysis

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Table 18.5. Class of weld details for use with principal stress range Table 18.5(a) Seam welds Detail No.

1.1

Joint type

Sketch of detail

Comments

Class

Full penetration butt weld flush ground, including weld repairs

Testing group 1 or 2

Testing group 3

90

71

90

71

Weld proved free from surface-breaking flaws and significant sub-surface flaws (see annex 18xx) by nondestructive testing.

few = 1. Fatigue cracks usually initiate at weld flaws 1.2

Full penetration butt weld made from both sides or from one side on to consumable insert or temporary non-fusible backing

1.3

80

63

80

63

80

71

63

40

80

63

80

63

80

63

71

56

80

71

e

1.4

Weld proved free from significant flaws (see annex 18xx) by non-destructive testing. In case of misalignment, see clause 18.10.4.1.

Class includes effect of centre-line offset of e/10, due to thickness change. Effect of misalignment to be included in calculated stress. For other cases of misalignment, see detail 1.2.

Weld proved free from significant flaws (see annex 18xx) by non-destructive testing.

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Table 18.5(a) Seam welds cont'd.. Detail No.

Joint type

Sketch of detail

Testing group 1 or 2 1.5

Full penetration butt welds made from one side without backing

Comments

Class

80

Testing group 3

71

63

1.6

Full penetration butt welds made from one side onto permanent backing

40

40

63

63

56

40

(a)

Weld proved to be full penetration and free from significant flaws (see annex 18xx) by non-destructive testing. If full penetration can be assured. If inside cannot be visually inspected. In case of misalignment, see clause 18.10.4. Circumferential seams only (see clause 5.7). Backing strip to be continuous and, if attached by welding, tack welds to be ground out or buried in main butt weld, or continuous fillet welds are permitted. Minimum throat = shell thickness. Weld root pass shall be inspected to ensure full fusion to backing. Single pass weld.

40

40

(b)

Circumferential seams only (see clause 5.7). 63

63

Backing strip attached with discontinuous fillet weld.

few = 1

1.7

Joggle joint

Circumferential seams only (see clause 5.7). 63

63

Minimum throat = shell thickness.

few = 1 56

40

40

40

Weld root pass shall be inspected to ensure full fusion. Single pass weld.

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Page A 2.34

Table 18.5(b) Shell to head or tubesheet Detail No

Joint type

For principal stresses acting essentially normal to the weld Sketch of detail

2.1

Class Testing group 1 or 2

Comments Testing group 3

Welded-on head

Head plate must have adequate through-thickness properties to resist lamellar tearing.

71 80

63 63

Full penetration welds made from both sides: - as-welded; - weld toes dressed (see 18.10.2.2). Partial penetration welds made from both sides:

63

63

32

32

- refers to fatigue cracking in shell from weld toe - refers to fatigue cracking in weld, based on stress range on weld throat Full penetration welds made from one side without backup weld:

2.2

63

40

- if the inside weld can be visually inspected and is proved free from weld overlap and root concavity.

40

40

- if the inside cannot be visually inspected. Full penetration welds

80

63

Made from one side with the inside weld ground flush

Welded-on head with relief groove

Made from one side: 63

40

- if the inside weld can be visually inspected and is proved free from weld overlap and root concavity. 40

- if the inside cannot be visually inspected.

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Page A 2.35

Table 18.5(b) Shell to head or tubesheet cont'd..

Detail No

Joint type

For principal stresses acting essentially normal to the weld Sketch of detail

2.3

Set-in head

Class Testing group 1 or 2

Comments Testing group 3

(a)

Full penetration weld made from both sides: refers to fatigue cracking from weld toe in shell: 71

63

- as-welded;

80

63

- weld toes dressed (see 18.10.2.2).

(b) Partial penetration welds made from both sides:

(c)

32

32

- refers to fatigue cracking in weld, based on weld throat stress range

71

71

- refers to fatigue failure in shell.

63

63

- refers to fatigue failure in head.

56

40

Full penetration weld made from one side

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Table 18.5(c) Branch connections Detail No.

3.1

Joint type

Sketch of detail

Crotch corner

Comments

Class Testing group 1 or 2

Testing group 3

100

100

Assessment by the method for unwelded parts based on equivalent stress is the normal approach. However, simplified assessment, using class 100, according to clause 18.11.2.2, still based on equivalent stress, is allowed few = 1.

Crack radiates from corner. Sketches show plane of crack. 3.2

3.3

Weld toe in shell

Stressed weld metal

Full penetration welds: 71

63

- as-welded;

80

71

- weld toes dressed (see 18.10.2.2)

63

63

Partial penetration welds

Continuous weld stressed along its length

Based on stress range parallel to weld on weld cross-section few = 1. 71

71

Full penetration weld

71

71

Partial penetration weld

32

32

Based on stress range on weld throat. few = 1.

Weld metal stressed normal to its length

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Table 18.5(c) Branch connections cont'd.. Detail No.

3.4

Joint type

Sketch of detail

Weld toe in branch

Comments

Class Testing group 1 or 2

Testing group 3

71

63

As-welded;

80

71

Weld toes dressed (see 18.11.2.2) en = branch thickness in Eq. 18.10-6

Table 18.5(d) Jackets Detail No

Joint type

For principal stresses acting essentially normal to the weld

Sketch of detail

4.1

Class Testing group 1 or 2

Comments Testing group 3 Full penetration weld to be proved free from significant flaws (see annex 18xx) by nondestructive testing

Jacket connection weld with shaped sealer ring

Welded from one side: 63

- multi-pass weld with root pass inspected to ensure full fusion;

40

- single pass weld.

71

40

- in all cases

56

Welded from both sides or from one side with back-up weld.

Annex 2: Draft CEN prEN 13445-3

DBA Design by Analysis

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Table 18.5(e) Attachments

Detail No.

5.1

Joint type

Attachment of any shape with an edge fillet or bevel - butt welded to the surface of a stressed member, with welds continuous around the ends or not

Sketch of detail

Class for use with: Structural equivalent stress range

Nominal equivalen t stress range

Testing group 1, 2, 3

Testing group 1, 2, 3

Comments

For details with welds continuous around ends, one class increase if weld toes dressed (see 18.10.2.2)

Stresses acting essentially parallel to weld:

L ≤ 160mm 71

56

71

50

L > 160mm few = 1.

One class increase if weld toes dressed (see 18.10.2.2)

Stresses acting essentially normal to weld: 71

56

t ≤ 55mm

71

50

t > 55mm few = 1.

5.2

For details with welds continuous around ends, one class increase if weld toes dressed (see 18.10.2.2)

Attachments of any shape with surface in contact with stressed member, with welds continuous around ends or not

L ≤ 160mm, W ≤ 55mm

71

56

71

50

71

45

L > 160mm, W ≤ 55mm L > 160mm, W > 55mm

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Page A 2.39

Table 18.5(e) Attachments cont'd..

Detail No.

5.3

Joint type

Continuous stiffener

Sketch of detail

Class for use with:

Comments

Structural equivalent stress range

Nominal equivalent stress range

Testing group 1 or 2

Testing group 1 or 2

Testing group 3

Testing group 3 Based on stress range parallel to weld in stiffener. few = 1.

Stresses acting essentially parallel to weld:

80

71

80

71

Full penetration weld.

71

71

71

71

Partial penetration weld.

Stresses acting essentially normal to weld

For full penetration welds, one class increase if weld toes dressed (see 18.10.2.2). t ≤ 55mm 71

71

56

56

71

71

50

50

t> 55mm

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DBA Design by Analysis

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Table 18.5(f) Supports Detail No

Joint type

For principal stresses acting essentially normal to the weld Sketch of detail

6.1

6.2

6.3

6.4

6.5

Class Testing group 1 or 2 71

Support on either horizontal or vertical vessel

Trunnion support

Saddle support

Comments Testing group 3 71

As-welded;

80

80

Weld toe in shell dressed (see 18.10.2.2)

71

71

As-welded;

80

80

Weld toe in shell dressed (see 18.10.2.2)

71

71

As-welded;

80

80

Weld toe in shell dressed (see 18.10.2.2)

Skirt support

Welded from both sides:

Leg support (with or without reinforcing pad) with fillet weld to vessel continuous all around.

a b

71

71

As-welded;

80

80

Weld toe in shell dressed (see 18.10.2.2).

56

56

Welded from one side.

a) 80

a) 80

Refers to fatigue cracking in the shell.

b) 71

b) 71

Refers to fatigue cracking in the leg.

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DBA Design by Analysis

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Table 18.5(g) Flanges and pads Detail No

Joint type

For principal stresses acting essentially normal to the weld

Sketch of detail

7.1

Class Testing group 1 or 2

Comments Testing group 3

Full penetration butt welded neck flange or compensation flange with welding lug.

Weld proved free from surfacebreaking and significant subsurface flaws (see annex 18xx) by non-destructive testing. 80

63

Weld made from both sides or from one side with back-up weld or onto consumable insert or temporary backing. Weld made from one side:

63 40

7.2

Welded flange

40

- if full penetration can be assured; - if the inside cannot be visually inspected.

Full penetration welds:

a

a) 71

a) 63

as-welded;

a) 80

a) 63

weld toes dressed (see 18.10.2.2. Partial penetration welds:

a

b

a

a) 63

a) 63

b) 32

b) 32

- refers to fatigue cracking from weld toe; - refers to fatigue cracking in weld, based on stress range on weld throat.

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Table 18.5(g) Flanges and pads cont’d Detail No

7.3

Joint type

For principal stresses acting essentially normal to the weld Sketch of detail

Class Testing group 1 or 2

Comments Testing group 3

Set in flange or pad

Full penetration weld: 71

63

- as-welded

80

63

- weld toes dressed (see 18.10.2.2). Fillet welded from both sides:

7.4

Set-in flange or pad, welded from both sides

63

63

- refers to fatigue cracking from weld toe

32

32

- refers to fatigue cracking in weld, based on stress range on weld throat .

a) 63

a) 63

Refers to fatigue cracking from weld toe

a) 32

a) 32

Refers to fatigue cracking in weld, based on stress range on weld throat .

b) 71

b) 71

Based on hoop stress in shell at weld root. few = 1.

a

b

18.10.1.3 Classification of weld details to be assessed using principal stress range Weld details and their corresponding classes for use in assessments based on principal stress range are given in table 18-5. The fatigue strengths of weld details for which the relevant potential failure mode is by fatigue cracking from the weld toe or weld surface are expressed in terms of the principal stress range on the parent metal surface adjacent to the crack initiation site (see 18.6.2.3.1). Short or discontinuous welds, where the relevant potential failure is by fatigue cracking from the weld end or weld toe into the parent metal, are assessed on the basis of the maximum principal stress range, ∆σ, and classified on the basis that the weld is orientated in the least favourable direction with respect to ∆σ. Continuous welds (e.g. seams, ring stiffener welds) may be treated differently if the maximum principal stress range acts in the direction which is within 45° of the direction of the weld. Then, the weld can be classified as being parallel to the direction of loading with respect to the maximum principal stress range and normal to the direction of loading with respect to the minimum principal stress range.

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18.10.1.4 Exclusions The classification tables do not include any bolts which are welded. The assessment method in this clause is not applicable to such bolts. 18.10.2 Change of classification 18.10.2.1 Welds in testing group 3 Welds in testing group 3 shall be assessed according to tables 18-4 and 18-5. 18.10.2.2 Weld toe dressing Fatigue cracks readily initiate at weld toes on stressed members partly because of the stress concentration resulting from the weld shape but chiefly because of the presence of inherent flaws. The fatigue lives of welds which might fail from the toe can be increased by locally machining and/or grinding the toe to reduce the stress concentration and remove the inherent flaws. The classification of fillet welds (including full penetration welds with reinforcing fillets) may, where indicated in tables 18-4 and 18-5, be raised when dressing of the toe is carried out according to the following procedure. Tables 18-4 and 18-5 include the revised class.

Figure 18-7: Weld toe dressing

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The weld toe is machined using a rotating conical tungsten-carbide machining burr. In order to ensure that weld toe flaws are removed, the required depth of machining is 0,5 mm below any undercut (see figure 187). The area should be inspected using dye penetrant or magnetic particle. Such inspection is facilitated if the machined toe is ground using emery bands, a measure which also improves fatigue life. The resulting profile should produce a smooth transition from the plate surface to the weld, as shown in figure 18-8, with all machining marks lying transverse to the weld toe. Toe dressing only affects the fatigue strength of a welded joint as regards failure from the weld toe. The possibility of fatigue crack initiation from other features of the weld (e.g. weld root in fillet welds) should not be overlooked. Weld toe dressing cannot be assumed to be effective in the presence of any corrosive environment which can cause pitting in the dressed region. 18.10.2.3 Dressing of seam welds Dressing or flush grinding of the seam welds justifies an upgrade from Class 80 to Class 90. A fatigue strength higher than Class 90 cannot be justified because of the possible presence of weld flaws which are too small for reliable detection by non-destructive inspection methods but are of sufficient size to reduce the fatigue strength of the joint. The detrimental effect of misalignment can, to some extent, be alleviated by weld toe dressing (see 18.10.2.2). Previously buried flaws revealed by dressing, which could reduce the fatigue strength of the joint, should be assessed (see 18.10.5). 18.10.3 Unclassified Details Details not fully covered in tables 18-4 and 18-5 shall be treated as Class 32 unless superior resistance to fatigue is proved by special tests or reference to relevant fatigue test results. To justify a particular design ∆σR-N curve, tests must be performed on specimens which are representative of the design, manufacture and quality of the relevant detail in the actual vessel. Test stress levels must be chosen to result in lives no more than 2x106 cycles and the geometric mean fatigue life obtained from tests performed at a particular stress range must not be less than the life from the ∆σR-N curve at that stress multiplied by the factor F from table 18-6. Table 18-6: Fatigue test factor F related to 97,7% probability of survival Number of test results 1

F 12,5

2

10,5

3

9,8

4

9,4

10

8,8

18.10.4 Deviations from design shape Discontinuities and departures from the intended shape of a vessel (i.e. "misalignments") will cause local increases in pressure-induced stresses in shells, as a result of secondary bending, and hence reduce fatigue life. This is true even if the allowable assembly tolerances given in Part 4 of this standard are met.

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Departures from intended shape include misalignment of abutting plates, an angle between abutting plates, roof-topping where there is a flat at the end of each plate, weld peaking and ovality (see figure 18-9). In most cases these features cause local increases in the hoop stress in the shell but deviations from design shape associated with circumferential seams cause increases in the longitudinal stress.

Figure 18-9: Deviations from design shape at seam welds

NOTE : When stresses greater than yield arise as a result of deviation from design shape, the pressure test will lead to an improvement in the shape of the vessel due to plastic deformation. However, vessels made from materials with yield strengths considerably higher than the specified minimum are less likely to benefit in this way. The beneficial effect of the pressure test on the shape of the vessel cannot be predicted and therefore if some benefit is required in order to satisfy the fatigue analysis, it is necessary to measure the actual shape after pressure test. Similarly, strain measurements to determine the actual stress concentration factor should be made after pressure test. The influence of misalignment must be considered at the design stage using one of the following approaches. In each case, the aim is to deduce assembly tolerances which are consistent with the required fatigue life. a) Assume values for misalignment, calculate the resulting secondary bending stresses, and include them in the calculation of structural stress for the detail under consideration. Adopt the class from table 18-4 or 18-5 and check the fatigue life. If unacceptable, tighten some or all of the tolerances to meet the required life. b) For a detail of nominal class C1, determine the class actually needed to meet the required fatigue life, C2. Then, the allowable increase in stress due to misalignments is Km = C1/C2. Assembly tolerances which result in Km ≤ C1/C2 can then be deduced.

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A conservative estimate of Km is: Km = 1 + A1 + A2 + A3 + A4

...(18.10-1)

where A1 caters for misalignment and is given by:  6 δ   e nx1   A1 =  1    en1   enx1 + enx2 

...(18.10-2)

in which δ1 is the offset of the centre lines of abutting plates; en1 ≤ en2 where en1 and en2 are the nominal thicknesses of the two abutting plates; x is 1,5 for a sphere or circumferential seam in a cylinder and 0,6 for a longitudinal seam in a cylinder. A2 caters for ovality in cylinders and is given by: A2 =

3 (Rmax − Rmin )   P 1− ν 2  e 1 + 3   2⋅R   2E   en  

(

)

      

...(18.10-3)

where R is the mean radius A3 caters for poor angular alignment of plates in spheres and is given by: R θ   en  A3 = 49

0,5

...(18.10-4)

where θ is the angle between tangents to the plates, at the seam (in degrees); A4 caters for local peaking and is given by: A4 =

6δ en

...(18.10-5)

where δ is the deviation from true form, other than above, and other terms are defined in figure 18-8. NOTE: This estimate of A4 ignores the beneficial reduction of the peaking due to pressure and is therefore conservative. Corrections due to non-linear effects, which reduce A4 , are permissible [11]. In the case of seam welds, the incorporation of a transition taper at a thickness change does not affect the value of A1.

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Equation 18.10-1 will overestimate Km if local bending is restricted, for example: at short shape imperfections, when there will be a stress redistribution around the imperfection; at imperfections in short cylindrical vessels, which can get support from the ends; adjacent to attachments which stiffen the shell. However, special analysis must be performed to justify lower Km values. The effect of departures from design shape for which Km ≤ 2 can be ignored if the weld toe is burr machined using the procedure given in 18.10.2.1. 18.10.5 Welding flaws Fatigue cracks can propagate from welding flaws and, therefore, depending on the required fatigue life, the flaws tolerated in Parts 4 and 5 of this standard may or may not be acceptable. Thus, in fatigue-loaded vessels the following apply: a) Planar flaws are unacceptable. b) Acceptance levels for embedded non-planar flaws and geometric imperfections are given in annex in preparation. NOTE: All other flaws can be assessed using an established fitness-for-purpose flaw assessment method, such as that in reference [8]. The fatigue strengths of welds containing flaws can be expressed in terms of the classification system in 18.10.1. Thus, they can be readily compared with those of other weld details. 18.10.6 Correction factors 18.10.6.1 To take account of material thickness en > 25 mm, few shall be calculated as follows:

f ew

 25  =    en 

0,25

...(18.10-6)

For en > 150 mm, the value of few for en = 150 mm applies. NOTE: In all cases, fatigue cracking from the toe of the weld in the stressed member is being considered. Thus, the correction is not required (i.e. few = 1) for some details, see tables 18-4 and 185. 18.10.6.2 For operating temperatures above 100 °C, ft* is given by: - for ferritic materials: f t * = 103 , − 15 , ⋅ 10 −4 t * −15 , ⋅ 10 −6 t * 2

...(18.10-7)

- and for austenitic materials: f t* = 1043 , − 4,3 ⋅ 10 −4 t *

...(18.10-8)

where t * = 0,75t max + 0,25t min NOTE: temperatures in equation (18.10-9) are all in degree Celsius. ft* is illustrated in figure 18-10.

...(18.10-9)

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18.10.6.3 The overall correction factor for welded components, fw, shall be calculated as follows: fw = few ⋅ ft*

...(18.10-10)

ft*

Mean cycle temperature t*, °C Figure 18-10: Correction factor ft*

18.10.7 Fatigue design curves Fatigue strength is expressed in terms of a series of ∆σR-N curves in figure 18-11, each applying to particular construction details. The curves are identified by the fatigue strength value ∆σR (N/mm2) at fatigue life N = 2x106 cycles. The design curves have the form as shown in figure 18-12 and conform to the equation: N =

C ∆ σ Rm

...(18.10-11)

where m and C are constants whose values are given in table 18-7. Different values apply for fatigue lives up to 5x106 cycles and for lives above 5x106 cycles. For constant amplitude loading, the endurance limit (i.e. stress range below which the fatigue life can be assumed to be infinite) corresponds to the stress range at 5x106 cycles. The corresponding stress range for variable amplitude loading is that at 108 cycles. NOTE: Alternative curves and constant amplitude endurance limits are permissible if they can be justified. For lives above 2x106 cycles the curves, which are consistent with reference [9], are conservative.

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To obtain the permissible number of load cycles, N, at a specified stress range, ∆σeq or ∆σ, the following shall be calculated.

N =

C  ∆σ eq     fw 

...(18.10-12)

m

or

N =

C  ∆σ     fw 

...(18.10-13)

m

Alternatively, for use as a design curve to obtain the allowable stress range for a specified number of applied load cycles, n,  C ∆ σ eq or ∆σ ≤ ∆ σ R ⋅ f w =    n

1/ m

⋅ fw

...(18.10-14)

NOTE1: The curves have been derived from fatigue test data obtained from appropriate laboratory specimens, tested under load control or, for applied strains exceeding yield (low cycle fatigue), under strain control. Continuity from the low to high cycle regime is achieved by expressing the low cycle fatigue data in terms of the pseudo-elastic stress range (i.e. strain range multiplied by elastic modulus, if necessary corrected for plasticity (see 18.8)). Such data are compatible with results obtained from pressure cycling tests on actual vessels. NOTE2: The fatigue strength design curves are approximately three standard deviations of log N below the mean curve, fitted to the original test data by regression analysis. Thus, they represent probability of failure of approximately 0,5 % with 99 % confidence. 10000 E=2,09x105 N/mm2

(N/mm²)

1000

32

4 0 4 5 5 0 5 6 6 3 7 1 8 0 9 0 10 0

2

100

1

10 1,0E+02

1,0E+03

1,0E+04

1,0E+05

1,0E+06

1,0E+07

1,0E+08

N

Figure 18-11: Fatigue design curves for welded components: (1) curves for assessing variable 6 amplitude loading; (2) For constant amplitude loading, endurance limit = ∆ σ R at 5x10 cycles. N.B.: For N>2x10 cylces, alternative curves and ∆σR values are permissible, see clause 18.10-6. 6

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Figure 18-12: The form of the fatigue design curves for welded components

Table 18-7: Coefficients of the fatigue design curves for welded components Class

Constants of ∆σR - N curve*

100

For 102 < N < 5x106 m C 3,0 2,00 x 1012

90

3,0

1,46 x 1012

5,0

80

3,0

1,02 x 1012

5,0

3,56 x 10

71

3,0

7,16 x 1011

5,0

1,96 x 10

63

3,0

5,00 x 1011

5,0

56

3,0

3,51 x 1011

50

3,0

45 40 32 *

Stress range at N cycles, N/mm2 For 5x106 < N < 108 m C 5,0 1,09 x 1016

5 x 106 74

108 40

6,41 x 1015

66

36

15

59

32

15

52

29

1,08 x 10

15

46

26

5,0

5,98 x 1014

41

23

2,50 x 1011

5,0

3,39 x 10

14

37

20

3,0

1,82 x 1011

5,0

2,00 x 1014

33

18

3,0

1,28 x 10

11

5,0

1,11 x 1014

29,5

16

6,55 x 1010 3,0 5 For E = 2,09x10 N/mm2

5,0

3,64 x 1013

24

13

18.11 Fatigue strength of unwelded components 18.11.1 Correction factors

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18.11.1.1 Surface finish correction factor To take account of surface finish, fs shall be calculated as follows: f s= Fs( 0,1⋅ ln N − 0,465 )

...(18.11-1)

where Fs = 1 − 0,056 (ln R z )

0,64

⋅ ln Rm + 0,289 (ln R z )

0,53

...(18.11-2)

and Rz is the peak-to-valley height (µm). If not specified, the manufacturing-related peak-to-valley heights in table 18-8 shall be used in equation 18.11-2. For polished surfaces with a peak-to-valley height Rz < 6 µm, assume fs = 1. Values of fs for as-rolled plate are given in figure 18-13. Table 18-8 Base values for peak-to-valley heights Surface condition Rolled or extruded Machined Ground, free of notches

RZ, µm 200 50 10

Figure 18-13: Correction factor fs for as-rolled plates 18.11.1.2 Thickness correction factor For wall thicknesses 25 mm < en ≤ 150 mm, fe is: f e= Fe(0,1⋅ lnN −0,465)

...(18.11-3)

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where  25   F e =    en 

0,182

...(18.11-4)

For en > 150 mm, the value of fe for e = 150 mm applies. 18.11.1.3 Correction factor to take account of the influence of mean stress 18.11.1.3.1 Elastic range 6 For ∆σ eq,l < 2Rp0,2/t * and σeq max < Rp0,2/t* , the mean stress correction factor fm for N ≤ 2x10 cycles is to

be determined for rolled and forged steel as a function of the mean stress sensitivity M from:  M (2 + M )  2σ R    fm = 1 − 1 + M  ∆σ R   when R p 0,2/t * ≤ σ R ≤

0,5

...(18.11-5)

∆σR 2( 1 + M)

or fm =

1+ M / 3 M − 1+ M 3

when

 2σ R   ∆σ R

  

...(18.11-6)

∆σR ≤ σ R ≤ R p 0,2/t * 2(1+ M)

where for rolled and forged steel: M = 0,00035 Rm - 0,1

...(18.11-7)

For N ≥ 2x106 cycles, fm shall be taken from figure 18-14. NOTE: fm is independent of stress range. 18.11.1.3.2 Partly plastic range For ∆σ eq,l < 2Rp0,2 / t * and σeq max > Rp 0,2/t* , equation (18.11-5) or (18.11-6) shall also be used to determine fm, although the reduced mean equivalent stress, as calculated from equations (18.11-8) or (18.11-9) shall be used instead of σ eq . See figure 18-5 . If σ eq > 0 , σ eq,r = R p 0,2/t * − If σ eq < 0 , σ eq, r =

∆ σ eq 2

∆ σ eq 2

− R p 0,2/t *

...(18.11-8)

...(18.11-9)

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Figure 18-14: Correction factor fm to take account of mean stress in unwelded material for N > 2x106 cycles 18.11.2 Overall correction factor for unwelded components 18.11.2.1 Main procedure The overall correction factor for unwelded components, f u, shall be calculated as follows: fu =

f s ⋅ f e ⋅ f m ⋅ f t* K eff

...(18.11-9)

in which fs, fe, and fm are given in 18.11.1.1 - 3 respectively; and ft* is given in 18.10.5.2. 18.11.2.2 Simplified Procedure A simplified procedure for the fatigue assessment of unwelded steel is permissible using the class 100 design data for welded components, independently of material static strength or surface finish. The data are used in conjunction with equation 18.10-12, with fw replaced by fu. If the applied stress is partly compressive, it is permissible to assume that the relevant value of ∆σeq is the sum of the tensile component and 60 % of the compressive component. Thus, for mean stress σ eq the correction factor fu becomes f e ⋅ f t * ⋅ f c / Keff in which:  σ eq , -  f c = 125 2 ∆σR 

   

fe is given in 18.11.1.2 and ft* in 18.10.5.2.

...(18.11-10)

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18.11.3 Design data The fatigue strengths of unwelded components are expressed in terms of a series of ∆σR-N curves, each applying to a particular tensile strength of steel, as given in figure 18-15. The fatigue design curves in figure 18-15 have the form: 2

  4,6. 104 N =   ,   ∆ σ R - 0,63 Rm + 115

...(18.11-11)

for lives up to 2x106 cycles. For N ≥ 2x106 cycles, values of ∆σR are given in table 18-10. For cumulative damage calculations using equation 18.5-1, the curves are linear for N = 2x106 to 108 cycles, and have the form: 10

 2,7. R m . + 92  N=  ∆ σ eq  

...(18.11-12)

Values of ∆σR at (and beyond) 108 cycles for selected tensile strengths are included in table 18-10. To obtain the allowable number of load cycles, N, at a specified stress range ∆σeq,     4 4,6 ⋅ 10   N =   ∆ σ eq  - 0,63 Rm + 115 ,   fu 

2

...(18.11-13)

Alternatively, for use as a design curve to obtain the allowable stress range for a specified number of load cycles, n,  4,6 ⋅ 104 ∆ σ eq ≤ ∆ σ R ⋅ f u =   n

 + 0,63 Rm - 115 ,  ⋅ fu 

...(18.11-14)

NOTE1: The curves have been derived from fatigue test data obtained from unnotched polished ferritic and austenitic rolled and forged steel specimens at room temperature, under alternating (mean load = 0) load control or, for applied strains exceeding yield (low-cycle fatigue), strain control. NOTE2: Compared with the mean curve fitted to the original data, the curves incorporate safety factors of 10 on fatigue life and 1.5 on stress range.

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Table 18-10: Stress range ∆ σR for N ≥ 2x106 cycles for unnotched test bars of ferritic and austenitic rolled and forged steels at room temperature and zero mean stress

∆σR = constant, N/mm2

Tensile strength Rm, N/mm2

N ≥ 2x106 275 400 525 650

400 600 800 1000

N ≥ 108 (for cumulative damage calculations) 162 236 310 385

Stress range, ∆σ R ,N/mm2

10 000

Rm (N/mm2) 1000 800 600 400

1 000

100 1,0E+02

1,0E+03

1,0E+04

1,0E+05

1,0E+06

1,0E+07

1,0E+08

Fatigue life, N, cycles Figure 18-15: Fatigue design curves for unwelded ferritic and austenitic forged and rolled steels (mean stress = 0)

18.12 Fatigue strength of steel bolts 18.12.1 Scope These rules apply only to axially-loaded steel bolts. They do not apply to other threaded components such as flanges, ends or valves. 18.12.2 Correction factors 18.12.2.1 For bolt diameters > 25 mm, the correction factor fe shall be calculated using equation 18.11-3, with en put equal to the bolt diameter. For bolt diameters ≤ 25 mm, fe = 1. 18.12.2.2 Overall correction factor for bolts fb shall be calculated as follows:

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f b = f e ⋅ f t*

Page A 2.56

...(18.12-1)

in which fe is given in 18.12.2.1 and ft* is given in 18.10.5.2. 18.12.3 Design data The fatigue strength of axially loaded bolts is expressed in terms of the ratio: ∆σ maximum nominal stress range = nominal ultimate tensile strength of bolt material Rm The single design curve 3

 ∆σ R    ⋅ N = 285  Rm 

...(18.12-2)

with an endurance limit at 2x106 cycles, shown in figure 18-15, is used for any thread form (machined, ground or rolled) and core diameters up to 25 mm. However, regardless of the actual tensile strength of the bolt material, Rm should never be assumed to exceed 785 N/mm2.

Figure 18-16: Fatigue design curve for bolts NOTE: The design curve has been derived from fatigue test data obtained from axially-loaded threaded connections. The design curve is three standard deviations of log N below the mean curve, fitted to the original test data by regression analysis. Thus, the curve represents a failure probability of approximately 0,5 % with 99 % confidence. To obtain the allowable number of load cycles, N, at a specified stress range, ∆σ:

 R . f b N ≤ 285  m   ∆σ 

3

...(18.12-3)

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Alternatively, for use of the design curve to obtain the allowable stress range, ∆σ, for a specified number of load cycles, n, 1/ 3

 285  ∆σ ≤ ∆σ R. f b = Rm    n 

⋅ fb

...(18.12-4)

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Annex B (informative)

Direct route for design by analysis (This annex refers to clause 5). B.1 Purpose Design-by-analysis (DBA) provides rules for the design of any component under any action. It may be used: - as an alternative to design-by-formula (see 5.4.1), - as a complement to design-by-formula for: - cases not covered by that route; - cases involving superposition of environmental actions; - where DBA is required, e.g. by local authorities where a potential major hazard is involved or for environmental reasons; - exceptional cases where the manufacturing tolerances given in clause 5 of Part 4 are exceeded. . NOTE1: In the last item, any deviations beyond tolerance limits shall be clearly documented. NOTE2: The method given in this clause is used in structural design, e.g. Eurocode No.3 Design of Steel Structures. Some aspects are applied in the Danish code for pressure vessels, DS 458.

It is pre-supposed that this annex will be used with conformity assessment modules G, B1+D, or B1+F. B.2 Specific definitions

The following definitions are in addition to those in clause 3. B.2.1 action: Physical influence which causes stress and/or strain in a structure. B.2.2 application rule: Procedure to determine whether a principle is satisfied. B.2.3 characteristic value: Representative value which takes account of the variation of an action. B.2.4 coefficient of variation: Measure of statistical dispersion (standard deviation divided by mean); B.2.5 design check: Assessment of a component for a load case by means of an application rule. B.2.6 effect: Response (e.g. stress, strain, displacement, resultant force or moment, equivalent stress resultant) of a component to a specific action. B.2.7 load case: A combination of coincident actions. B.2.8 loading type: Classification of loading based on frequency and duration. B.2.9 limit state: Structural condition beyond which the design performance requirements of a component are not satisfied.

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B.2.10 partial safety factor: Factor which depends on the limit state and is applied to either an effect or a resistance to obtain a design value. B.2.11 principle: General statement, definition or requirement for a given failure mode for which there is no alternative, unless specifically stated otherwise. B.2.12 resistance: the limiting value for a given limit state of an action or effect. B.2.13 structure: All load carrying parts relevant to the component, e.g. the whole vessel, its load carrying attachments, supports and foundations.

B.3 Specific symbols and abbreviations

The following symbols and abbreviations are in addition to those in clause 4. B.3.1 Subscripts d

is design

I

is ith value

inf is lowest (infimum) j

is jth value

k

is kth value

E is effect G is permanent action P is pressure action Q is variable action R

is resistance

sup is highest (supremum) X

is exceptional action

B.3.2 Symbols

A

is action (general term)

A5 is minimum rupture elongation

a

is any structural dimension

DBA Design by Analysis D

is fatigue damage

E

is effect (general term)

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Ep is plastic modulus R

is resistance (general term)

RM is design strength parameter σ

is stress

ε

is strain

γ

is partial safety factor

ψ

is a combination factor for pressure and variable actions.

B.4 Method

The method comprises the following stages: a) specify the relevant failure mode and limit state, taking account of the loading type, see B.5; NOTE: There may be more than one failure mode.

b) specify the principle, see B.6; c) select an appropriate application rule, see B.6; d) using the detailed information in B.9, carry out the design check as follows: - define the load case and specify the actions, see B.7.1; - determine the characteristic value and calculate the design value of each action, see B.7.2 and B.7.3; - calculate the effect of the actions, see B.7.4; - calculate the resistance of the component, see B.8; - indicate whether or not the principle is satisfied. e) if the principle is not satisfied, repeat the design check using amended loading, geometry or material.

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B.5 Failure modes and limit states

The main failure modes are listed in table B.5-1 with the relevant limit states. The latter are classified according to whether the loading is short term, long term or cyclic. Individual failure modes only are given in table B.5-1. Combinations of failure modes, e.g. e.g. fatigue - plastic rupture, creep - plastic rupture, creep - fatigue, shall be considered separately.

A limit state is classified as either an ultimate or a serviceability limit state. An ultimate limit state is a structural condition (of the component or vessel) beyond which the safety of personnel could be endangered. NOTE1: Ultimate limit states include: failure by gross plastic deformation; rupture caused by fatigue; collapse caused by instability of the vessel or part of it; loss of equilibrium of the vessel or any part of it, considered as a rigid body, by overturning or displacement; and leakage which affects safety. NOTE2: In the case of collapse, some states prior to collapse are considered as collapse and also classified as ultimate limit states. A serviceability limit state is a structural condition (of the component or vessel) beyond which the service criteria specified for the component are no longer met.

NOTE1: Serviceability limit states include: - deformation or deflection which adversely affects the use of the vessel (including the proper functioning of machines or services), or causes damage to structural or non-structural elements; - leakage which affects efficient use of the vessel but does not compromise safety or cause an unacceptable environmental hazard. NOTE2: Depending upon the hazard, leakage may create either an ultimate or a serviceability limit state.

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Table B.5-1: Classification of failure modes and limit states

Loading type

Failure mode Brittle fracture Ductile rupture3) Excessive deformation 14) Excessive deformation 25) Excessive deformation 36) Excessive local strains7) Instability8) Progressive plastic def.9) Alternating plasticity10) Creep rupture Creep-Excessive def. 111) Creep-Excessive def. 212) Creep-Excessive def. 313) Creep instability Erosion, corrosion Environmentally assisted cracking14) Creep Creep-Excessive def. 111) Creep-Excessive def. 212) Creep-Excessive def. 313) Creep instability Erosion, corrosion Environmentally assisted cracking14) Fatigue Environmentally assisted fatigue

Short term Single Multiple application application U U S, U 1) U S U U, S 2) U U

Long term Single Multiple application application

Cyclic

U S, U 1) U S U, S 2) S U U S, U 1) U S U, S 2) S U U U

U indicates ultimate limit state. S indicates service limit state. 1) In case of risk due to leakage of content (toxic, inflammable, steam etc.) 2) In case of sufficient post-instability load carrying capacity 3) Unstable gross plastic yielding or unstable crack growth 4) Excessive deformations at mechanical joints. 5) Excessive deformations resulting in unacceptable transfer of load. 6) Excessive deformations related to service restraints. 7) Resulting in crack formation or ductile tearing, by exhaustion of material ductility 8) Elastic, plastic, or elastic-plastic 9) Progressive plastic deformations (or ratcheting) 10) Alternative plasticity (see also clause 6) 11) Creep-Excessive deformation at mechanical joints 12) Creep-Excessive deformation resulting in unacceptable transfer of load 13) Creep-Excessive deformation related to service restraints 14) Stress corrosion cracking (SCC), Hydrogen induced cracking (HIC), Stress orientated hydrogen induced cracking (SOHIC).

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B.6 Principles, application rules and design checks

For each failure mode, a single principle is stated to ensure that the limit state is not exceeded. The principle requires that for any load case either the: - combination of the design actions does not exceed the design resistance or - the design effect of the design actions does not exceed the design resistance. Coincident actions are combined in a load case. All relevant load cases shall be considered.

For each principle, one or more application rules are given to indicate different means by which an assessment can be made. The most relevant application rule or rules shall be selected. It is permissible to use other application rules provided they accord with the relevant principle and are at least equivalent with regard to the resistance, serviceability and durability of the vessel. The assessment of a component against a load case by means of an application rule comprises a design check. The principles, application rules and design checks are specified in detail in B.9. NOTE: Proof testing or non-destructive testing , additional to that specified in Part 5, should be specified which is appropriate to the stress level in the component and the failure mode.

B.7 Actions B.7.1 Classification Actions are classified into the following four types: - permanent; - variable (other than temperature, pressure and actions related to them deterministically, i.e. not involving probability); - exceptional (see 5.3.5 and 6.1.1); - temperature, pressure and actions related to them deterministically. NOTE1: Mechanical, physical, chemical or biological actions may have an influence on the safety of a vessel. However, in DBA, only those which cause stress or strain are considered. Examples are: volume forces (e.g. self-weight), surface forces (pressures, surface loadings, etc.), singular forces (resultants representing e.g. imposed surface forces), line forces, point forces, temperature changes, displacements imposed on the vessel at connections, foundations, due to e.g. temperature changes, settlement. NOTE2: Examples of permanent actions are: self-weight of a structure and associated fittings, ancillaries and fixed equipment. NOTE3: Examples of variable actions are: imposed loads, wind or snow loads NOTE4: Examples of exceptional actions are: actions on secondary containment due to failure of primary containment or exceptional earthquake actions. NOTE5: Actions which may be either permanent or variable are: temperature changes, imposed loads or displacements. Temperature changes have a dual role in that they may cause stress in the structure and also change its resistance.

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NOTE6: Environmental attack (whether internal or external) may reduce the safety or serviceability of a vessel. This should be taken into account in the selection of materials, provision of additional wall thickness (see 5.2.2), or specification of appropriate material parameters in the determination of resistance (see B.8.2) . Although operating pressures and temperatures are variable actions, they have special characteristics with regard to their variation in time, random properties, etc. Because there is usually a strong correlation between operating pressure and temperature, they shall be considered to act simultaneously, and the pressure - temperature dependence shall be defined appropriately. NOTE: Pressure-temperature dependence may be stated either in the form of coincident pairs or in the form of a functional relationship between fluid pressure and temperature.

With actions which consist of permanent and variable parts, the parts shall be considered individually. Variable actions may include actions of quite different characteristics, e.g.: - actions which are related to pressure and/or temperature in a deterministic way. These shall be combined in the pressure/temperature action and the relationship, exact or approximate, shall be used. -actions which are not correlated with pressure or temperature but have well defined (bounded) extreme values; - actions, like wind loads, which can be described only as stochastic (i.e. random) processes and are not correlated with pressure or temperature. B.7.2 Characteristic values

The requirements for determining the characteristic values of different types of action are given in table B.7-1 and in the following. The characteristic values of pressure and temperature describe the pressure-temperature regime that envelops those pressures and temperatures which can occur under reasonably forseeable conditions, see figure B.7-1. The following characteristic values shall always be specified: - the upper characteristic value of the pressure (Psup) - the lower characteristic value of the pressure (Pinf) - the upper characteristic value of the temperature (Tsup) - the lower characteristic value of the temperature (Tinf).

The self-weight of the structure and of non-structural parts may be calculated on the basis of nominal dimensions and mean unit masses. For wind and snow loadings, the values specified in relevant codes may be used. In load cases where thermal stresses (constant or transient) have an influence on the safety of the structure, the characteristic values of coincident pressure / temperature shall be the extreme values of operating pressure and temperature that can reasonably be expected to occur under normal operating conditions over the life of the vessel.

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Table B.7-1: Characteristic values for different types of action

Action Permanent

Coefficient of variation ≤0,1 (note 1) >0,1 (note 3)

Variable Exceptional

Characteristic value

Gk

Mean of extreme values

(note 2) G k, sup

Permanent

Variable

Symbol

≤0,1 (note1) >0,1 -

G k, inf

(note 2)

Qk

(note 2) Qk

(note 2) Psup

Pressures and temperatures

Tsup

-

Upper limit with 95% probability of not being exceeded (see note 4); Lower limit with 95% probability of being exceeded (see note 4). Mean of extreme values

Pinf Tinf

97% percentile of extreme value in given period (see note 5) Shall be individually specified Reasonably forseeable highest pressure (see note 7) Reasonably forseeable highest temperature Reasonably forseeable lowest pressure (see note 6) Reasonably forseeable lowest temperature

NOTE1: The mean of the extreme values may also be used when the difference between the reasonably foreseeable highest value and the lowest one is not greater than 20% of their arithmetic mean value. NOTE2: The k subscript in table B.7-1 indicates that there are usually several actions in a load case and they are individually numbered. NOTE3: Also applies where the actions are likely to vary during the life of the vessel (e.g. some superimposed permanent loads) NOTE4: If a statistical approach is not possible, the highest and lowest credible values may be used. NOTE5: For variable actions which are bounded, the limit values may be used as characteristic values. NOTE6: This value is usually either zero or -1,0 (for vacuum conditions). NOTE7: This may be, for example, the set pressure of the relief valve.

For temperature values which are not environmentally imposed and in cases where a combination of Psup and Tsup is uneconomic, it may be necessary to specify characteristic pressure - temperature pairs, e.g. (Psup,i, Tsup,i), (Pinf,i, Tinf,i), which determine an envelope of the (P, T) - regime of the reasonably foreseeable extreme values, see figure B.7-1.

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P Psup1 1 Tsup 1 Tinf 5 Tinf 1

Psup 2 2

Psup 5 5 Pinf 3

3

Tsup 2

Tinf 4 Tsup 3 4

Pinf 4 T

Figure B.7-1: Typical plot of coincident temperatures and pressures NOTE: Characteristic values should be stated clearly in the technical documentation. B.7.3

Design value

The design value Ad of an action is expressed in general terms as: Ad = γ A ⋅ A

... (B.7-1)

where A is the characteristic value of the action (permanent, variable, exceptional or pressure) and γA is the relevant partial safety factor as given in B.9 for the considered limit state. NOTE: γ A takes account of the following: - the possibility of non-conservative deviation of the actions from their characteristic values; - the uncertainty of the models which describe the physical phenomena for the action and effect; - uncertainty in any stochastic models of the action; - whether the action has a favourable or an unfavourable effect. (For example, in one load case the action due to the weight of a component might be opposing the pressure force and therefore has a favourable effect on reducing stress. In another, the weight might be acting with the pressure and so has an unfavourable effect. In the two load cases, γ A would have a different value). B.7.4 Design effect For each load case, the effect of all the design actions is combined to give the design effect. This is a function of the design actions ( Ad ) and the dimensions ( ad ). It is expressed in general terms as: E d = E( Ad , ad ,....) = E(γ P ⋅ P, γ G ⋅ G, γ Q ⋅ Q, γ X ⋅ X , ad......)

... (B.7-2)

The calculation requires both geometric data and material properties. For geometric data, nominal values for individual dimensions and properties, rather than minimum values, may be used. For strength related data, ReH, Rp, Rm etc., the minimum guaranteed values specified in the material codes or material data sheets shall be used in the calculations. For the other properties, e.g. modulus of elasticity or coefficient of linear thermal expansion, nominal or mean values may be used.

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For components clad on the inside and subjected to internal pressure, the inside surface for pressure action shall be taken as the nominal inner face of the cladding. For components clad on the outside and subjected to external pressure, the outside surface for pressure action shall be taken as the outer face of the base metal.

B.8 Design resistance B.8.1 Resistance For the calculation of resistance: - the nominal values of the geometric data may be used with the exception of thicknesses for which the nominal values minus the allowances shall be used; - the minimum guaranteed values shall be used for strength data, i.e.: ReH , Rp0,2/ t , Rp1,0 / t , Rm / t ; - for other properties, e.g. modulus of elasticity, coefficient of linear thermal expansion, nominal or mean values may be used.

The resistance shall be determined by increasing in the same proportion all the design actions in the load case. NOTE: When all the design actions in the load case are increased in the same proportion, the path in the action space is a straight line passing through the origin and through the point the co-ordinates of which represent the design actions of the load case. B.8.2 Design resistance The design resistance of a component is expressed in general terms as Rd =

R γR

...(B.8-1)

where γ R is the partial safety factor for the resistance. NOTE: γ R takes account of the following: - the possibility of a non-conservative deviation in the material properties or geometrical data; - the possibility of inaccuracies in the model for the calculation of resistance; - the possibility of deterioration not explicitly accounted for; - the failure mode, or failure modes under consideration, and the use in some cases of a strength parameter which is only approximate for the failure mode; - the level of hazard which is considered acceptable. In some design checks, the design resistance is obtained directly as a function of the values of material strength parameters divided by γ R , e.g.  ReH  Rd = R , ad ,....  γR 

...(B.8-2)

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B.9 Design checks B.9.1 General A design check shall be carried out against a limit state for one of the following circumstances: - normal situations where normal conditions apply; - special situations where conditions for construction, erection, repair or testing apply; - exceptional situations, see 5.3.5. All relevant requirements of all the following design checks shall be fulfilled. Subclause B.9.2 applies to failure by gross plastic deformation (GPD) in either operation or test. Design details which would cause severe strain concentration or elastic follow-up shall not be present. NOTE: Avoidance of severe stress concentrations is specified because only the effects of excessive local yielding are included in subclause B.9.2. The other subclauses apply as follows. For failure by progressive plastic deformation (PD), see B.9.3; by instability, see B.9.4; by fatigue, see B.9.5, and by overturning and global displacement (rigid body motions), see B.9.6. NOTE1: The design checks in the following are not exhaustive. NOTE2: In some cases, it may be necessary to investigate additional limit states. For example, with austenitic stainless steel, failure by GPD should be checked (as an ultimate limit state) but leakage may also need to be checked (as either an ultimate or a serviceability limit state), see table 5.B.9-3. B.9.2 Gross Plastic Deformation (GPD) B.9.2.1 Principle The principle is as follows:

For any load case, either the combination of the design actions shall not exceed the design resistance: Ad ≤ Rd

...(B.9-1)

or the design effect of the actions shall not exceed the design resistance: E d ≤ Rd In either case, the design resistance shall be obtained from calculations assuming: - proportional increase of all design actions, see note in B.8.1; - first order theory; - a linear-elastic ideal-plastic material or a rigid ideal-plastic material; - Tresca's yield criterion (maximum shear stress criterion) and associated flow rule; - design strength parameter RM as specified in B.9.2.5 or B.9.2.6; - partial safety factor γ R as specified in B.9.2.5 or B.9.2.6.

...(B.9-2)

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B.9.2.2 Application rule 1 - lower-bound limit load NOTE: Application rule 1 comprises the whole of this subclause. The design action shall be less than the lower-bound limit load divided by γ R . To avoid possible computational difficulties when using a computer, a linear-elastic linear-hardening material with a plastic modulus Ep equal to E/10 000 (or a similar small value) may be used, instead of the linear ideal plastic one, see figure B.9-1. The lower bound limit is given by the tangent-intersection, see figures B.9-2 and B.9-3. If there is no maximum in the region of principal strains less than + 5 %, the greatest tangent intersection value shall be used with one tangent through the origin, the other through a point where the maximum principal strain does not exceed + 5 %.

σ

ε Figure B.9-1 Simplified stress-strain model for computation of the lower bound limit load

action

action

deflexion Figure B.9-2: Limit load determination: plot

deflexion Figure B.9-3: Limit load determination: plot

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of action versus deflection initially non-linear

B.9.2.3 Application rule 2 - maximum primary stress intensity NOTE: Application rule 2 comprises the whole of this subclause. The maximum primary stress intensity in the structure shall not be greater than

RM . γR

NOTE: Stress intensity is the uniaxial stress equivalent to the multi-axial stress state and is defined as twice the maximum shear stress. It is therefore is the difference between the algebraically largest principal stress and the algebraically smallest at a point. A primary stress field is any stress field which just satisfies the equilibrium equations (at any point throughout the structure). The main characteristic of a primary stress field is that it is not self-limiting. The main problem of determining a primary stress field corresponding to a given imposed load is that it has no unique solution. In cases of structures, where the concept of stress resultants is applicable, e.g. beams, plates, shells, this requirement may be verified in terms of stress resultants (generalised stresses) and local (technical) limit loads. Examples of local (technical) limit load sets and the equations for the allowable resultants are given in table B.9-1.

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Table B.9-1. Equations for allowable stress resultants (Tresca yield criterion) 1. Beam, rectangular cross-section, bending moment MB2 (direction of axis of symmetry), normal force FN1 (direction of beam axis) Limit bending moment MB2pl = RM a2a12 / (4γR ) ... (B.9-3) Limit normal force FN1pl = RM a2a1 / γR ... (B.9-4) Allowable stress resultants MB2/ MB2pl + (Fnl / FN1pl )² ≤ 1 ... (B.9-5) (See figure B.9-4 for a plot of this equation). 2. Plate, thickness e, bending moments (per unit length) m1, m2, m12 Limit bending moment Allowable stress resultants

mpl = RM e2 / (4γR ) mpl ( m1 + m2) - m1m2 + m12² ≤ mpl ²

... (B.9-6) ... (B.9-7)

- mpl ( m1 + m2) - m1m2 + m12² ≤ mpl ²

... (B.9-8)

( m1 - m2)² + 4 m12² ≤ mpl ²

... (B.9-9)

(See figure B.9-5 for a plot of the surface formed by these equations).

3. Plate, thickness e, rotational symmetry, bending moments (per unit length) mr, mφ Limit bending moment Allowable stress resultants

mpl = RM e2 / (4γR ) mr≤ mpl

... (B.9-10) ... (B.9-11)

mϕ≤ mpl

... (B.9-12)

mr- mϕ≤ mpl (See figure B.9-6 for a plot of these equations).

... (B.9-13)

4. Shell, thickness e, rotational symmetry, bending moments (per unit length) mS, mθ (=µ µ mS), membrane forces (per unit length) nS, nθ Limit bending moment Limit normal force Allowable stress resultants

mpl = RM e2 / (4γR) npl = RM e / γR ns/npl ≤ 1

... (B.9-14) ... (B.9-15) ... (B.9-16)

nθ/npl ≤ 1

... (B.9-17)

ns - nθ /npl ≤ 1

... (B.9-18)

ms/ mpl +(ns/npl )² ≤ 1

... (B.9-19)

2ms/ mpl + 2(nθ/npl -1)² + 2(nθ/npl - 2ns/npl -1)² ≤ 2 ... (B.9-20) (See figure B.9-7 for a plot of the surface formed by these equations). 5. Shell, special case of 4, without moment Limit normal force Allowable stress resultants

npl = RM e / γR

... (5.B.9-21)

ns ≤ npl

... (B.5-22)

nθ ≤ npl

... (B.5-23)

ns - nθ ≤ npl (See figure B.9-8 for a plot of these equations).

... (B.5-24)

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Figure B.9-4

Figure B.9-5

Figure B.9-6

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Figure B.9-7

Figure B.9-8

B.9.2.4 Particular requirements

In design checks against GPD, structural strength may only be attributed to cladding in the case of integrally-bonded cladding, see 5.5.2. B.9.2.5 Design check against failure by GPD in operation a) Partial safety factors against actions shall be as given in table B.9-2. Table B.9-2: Partial safety factors against actions for GPD for load cases in operation Action Permanent

Condition For actions with an unfavourable effect

Partial safety factor γ G = 135 ,

Permanent

For actions with an favourable effect

γ G = 10 ,

Variable

For unbounded variables

γ Q = 15 ,

Variable

For bounded variable actions and limit values

γ Q = 10 ,

Pressure

For actions without a natural limit

γ P = 12 ,

Pressure

For actions with a natural limit, e.g. vacuum

γ P = 10 ,

If only part of the pressure is subject to a natural limit, e.g. static head, this part may be multiplied by γp = 1,0

and the remainder by γp = 1,2.

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b) Combination rules shall be as follows. All permanent design actions shall be included in each load case. Each pressure design action shall be combined with the most unfavourable variable design action. Each pressure design action shall be combined with the corresponding sum of the variable design actions; the design values of stochastic actions, see B.7-1 and table B.7-1, may be multiplied by the combination factor Ψ = 0,9. NOTE: Since it is most unlikely that all the variable actions would be at their maximum together, they may each be multiplied by Ψ = 0,9 when their total is combined with pressure. Favourable variable actions shall not be considered. c) Design material strength parameter (RM) and partial safety factor ( γ R ) shall be as given in table B.9-3. Table B.9-3: RM and γ R for GPD for load cases in operation γR

Material

RM

Ferritic1 steel

ReH or Rp0,2 / t

1,25 for

Rp0,2/ t Rm / 20

≤ 0,8

 Rp0,2 / t   otherwise 1,5625   Rm / 20  Austenitic steel (30%