CEN/TC 250/SC 3 N 2637

CEN/TC 250/SC 3Eurocode 3 - Design of steel structuresE-mail of Secretary: susan.kempa@din.deSecretariat: DIN

EN 1993-1-6 First Draft

Date of document 2018-05-02

Expected action CommentDue Date 2018-06-29

Background

Dear Member,

Please find attached the First Draft of the revision of EN 1993-1-6 for commenting. Please ensure that you write any comments into the CEN comments template and forward them to your NSB for submission via the CIB ballot. The CIB ballot will close on 29th June 2018. We kindly ask for giving comments only to the obviously changed parts and not to the original document text.

Kind regards

Susan KempaSecretary CEN/TC 250/SC 3

prEN 1993-1-6:2017

CEN/TC 250 Date: 2018-04 prEN 1993-1-6 :2018 Secretariat: BSI

Eurocode 3 — Design of steel structures — Part 1-6 : Strength and Stability of Shell Structures

Eurocode 3 — Calcul des structures en acier — Partie 1.6 : Resistance et Stabilité des Coques

Eurocode 3 — Bemessung und Konstruktion von Stahlbauten — Teil 1.6 : Aus Schalen

ICS:

Version V13 April 2018

CCMC will prepare and attach the official title page.

TC WI :2016 (E)

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NOTES ON PROGRESS This standard has been transferred onto the new template (courtesy of DIN) Many minor editorial amendments have been made throughout to improve the ease of use and clarity of expression. Many EDITORIAL NOTES, highlighted in green, and other items have been introduced to alert the reader to issues that will be discussed further by the Project Team and Working Group, and where helpful comments are welcome. Other similar items under discussion in the PT and WG are shown with bold red or blue highlights All items in the Systematic Review classed as 1 or 2 have been addressed in the revised text. New material in relation to Mandate Sub-Tasks ST1, ST2, ST3, ST4 and ST6 has been added New material in relation to stainless steel shells has been added in D.1.2.3 to remedy this previous omission An effort has been made to introduce distinctions between the different tolerance requirements according to the buckling limit state being addressed. These tolerances are excessively strict for shells that are not susceptible to buckling, whilst only some categories of tolerance are relevant to shells that are clearly only susceptible to a specific buckling mode (associated with the geometry and load case). Significant changes have been introduced into sub-clause 8.4. All references to “expression” have been changed to “Formula” All reference to the term “section” have been removed (as per CEN rules) All instances of “shall” have been checked for correct usage, and “must” has been eliminated. The correct format for all definitions has been adopted. Equations that were not in Mathtype have been re-drafted in Mathtype where found Figures that had become reduced to “photo” version have been traced to the original and restored as Word Picture editable format Section 8.5, which describes the simplest hand calculation process for shell buckling, has been carefully re-structured to make it easier to follow by making the sequence of clauses correspond to the steps in the process, and by clarifying where each step must be conducted repeatedly to deal with the three membrane stress components. It is expected that some material in the early chapters may be eliminated when it has been satisfactorily treated in EN 1993-1-14. However, some matched statements may be needed in both this standard and EN 1993-1-14 to ensure that the meaning in this standard is always clear. It is expected that the early clauses (sections) of this standard will be revised to eliminate areas where there is something close to repetition. This will be undertaken for the Second Draft.

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Contents Foreword...........................................................................................................................................................................5

1 General ................................................................................................................................................................6 1.1 Scope............................................................................................................................................................................6 1.2 Normative references...........................................................................................................................................7 1.3 Terms and definitions ..........................................................................................................................................8 1.4 Symbols ................................................................................................................................................................... 15 1.5 Sign conventions.................................................................................................................................................. 19

2 Basis of design and modelling..................................................................................................................20 2.1 General..................................................................................................................................................................... 20 2.2 Types of analysis ................................................................................................................................................. 20 2.3 Shell boundary conditions............................................................................................................................... 23

3 Materials and geometry .............................................................................................................................23 3.1 Material properties............................................................................................................................................. 23 3.2 Design values of geometrical data................................................................................................................ 24 3.3 Geometrical tolerances and geometrical imperfections ..................................................................... 24

4 Ultimate limit states in steel shells ........................................................................................................25 4.1 Ultimate limit states to be considered........................................................................................................ 25 4.2 Design concepts for the limit states design of shells ............................................................................ 27

5 Stress resultants and stresses in shells ................................................................................................30 5.1 Stress resultants in the shell........................................................................................................................... 30 5.2 Modelling of the shell for analysis................................................................................................................ 30 5.3 Types of analysis ................................................................................................................................................. 33

6 Plastic failure limit state (LS1) ................................................................................................................34 6.1 Design values of actions ................................................................................................................................... 34 6.2 Stress design.......................................................................................................................................................... 34 6.3 Design by global numerical MNA or GMNA analysis ............................................................................ 36 6.4 Design using standard formulae ................................................................................................................... 37

7 Cyclic plasticity limit state (LS2).............................................................................................................37 7.1 Design values of actions ................................................................................................................................... 37 7.2 Stress design.......................................................................................................................................................... 38 7.3 Design by global numerical MNA or GMNA analysis ............................................................................ 38 7.4 Design using standard formulae ................................................................................................................... 39

8 Buckling limit state (LS3) ..........................................................................................................................39 8.1 Design values of actions ................................................................................................................................... 39 8.2 Special definitions and symbols.................................................................................................................... 39 8.3 Buckling-relevant boundary conditions .................................................................................................... 40 8.4 Buckling-relevant geometrical tolerances................................................................................................ 41 8.5 Stress design.......................................................................................................................................................... 50 8.6 Design using reference resistances.............................................................................................................. 54 8.7 Design by global numerical analysis using MNA and LBA analyses............................................... 56 8.8 Design by global numerical analysis using GMNIA analysis.............................................................. 60

9 Fatigue limit state (LS4) .............................................................................................................................66 9.1 Design values of actions ................................................................................................................................... 66 9.2 Stress design.......................................................................................................................................................... 66 9.3 Design by global numerical LA or GNA analysis..................................................................................... 68

Annex A (normative) Membrane theory stresses in shells ..........................................................................69

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A.1 General .....................................................................................................................................................................69 A.2 Unstiffened cylindrical shells..........................................................................................................................70 A.3 Unstiffened conical shells.................................................................................................................................72 A.4 Unstiffened spherical shells ............................................................................................................................73

Annex B (normative) Formulae for plastic reference resistances .............................................................74 B.1 General .....................................................................................................................................................................74 B.2 Unstiffened cylindrical shells..........................................................................................................................75 B.3 Ring stiffened cylindrical shells .....................................................................................................................77 B.4 Junctions between shells ..................................................................................................................................80 B.5 Circular plates with axisymmetric boundary conditions....................................................................82

Annex C (normative) Formulae for linear elastic membrane and bending stresses...........................84 C.1 General .....................................................................................................................................................................84 C.2 Clamped base unstiffened cylindrical shells ............................................................................................85 C.3 Pinned base unstiffened cylindrical shells ................................................................................................87 C.4 Internal conditions in unstiffened cylindrical shells.............................................................................90 C.5 Ring stiffener on cylindrical shell .................................................................................................................92 C.6 Circular plates with axisymmetric boundary conditions....................................................................94

Annex D (normative) Formulae for use in buckling stress design.............................................................96 D.1 Unstiffened cylindrical shells of constant wall thickness ...................................................................96 D.2 Unstiffened cylindrical shells of stepwise variable wall thickness .............................................. 110 D.3 Unstiffened lap jointed cylindrical shells................................................................................................ 115 D.4 Unstiffened complete and truncated conical shells............................................................................ 117

Annex E (normative) Formulae for use in reference resistance design................................................ 122 E.1 Cylindrical shells under global bending .................................................................................................. 122 E.2 Complete and partial spherical shells ...................................................................................................... 127

Bibliography .............................................................................................................................................................. 133

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Foreword This European Standard EN 1993-1-6, Eurocode 3: Design of steel structures: Part 1-6: Strength and stability of shell structures, has been prepared by Technical Committee CEN/TC 250 “Structural Eurocodes », the Secretariat of which is held by BSI. CEN/TC 250 is responsible for all structural Eurocodes. This European Standard shall be given the status of a National Standard, either by publication of an identical text or by endorsement, at the latest by xxxxx, and conflicting National Standards shall be withdrawn at latest by March 20xx. This Eurocode supersedes EN 1993-1-6 (2007). According to the CEN-CENELEC Internal Regulations, the National Standard Organizations of the following countries are bound to implement this European Standard: Austria, Belgium, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Romania, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland and United Kingdom. National annex for EN 1993-1-6 This standard gives alternative procedures, values and recommendations with notes indicating where national choices may have to be made. Therefore the National Standard implementing EN 1993-1-6 should have a National Annex containing all Nationally Determined Parameters to be used for the design of steel structures to be constructed in the relevant country. National choice is allowed in EN 1993-1-6 through: − 6.3 (7) − 7.3.1 (1) − 9.2.1 (4)

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1 General

1.1 Scope (1) EN 1993-1-6 gives basic design rules for plated steel structures that have the form of a shell of revolution. (2) This standard is intended for use in conjunction with EN 1993-1-1, EN 1993-1-2, EN 1993-1-3, EN 1993-1-4, EN 1993-1-9 and the relevant application parts of EN 1993, which include: • Part 3.1 for towers and masts; • Part 3.2 for chimneys; • Part 4.1 for silos; • Part 4.2 for tanks; • Part 4.3 for pipelines. (3) This standard defines the characteristic and design values of the resistance of the structure. (4) This standard is concerned with the requirements for design against the ultimate limit states of: • plastic failure; • cyclic plasticity; • buckling; • fatigue. (5) Overall equilibrium of the structure (sliding, uplifting, overturning) is not included in this standard, but is treated in EN 1993-1-1. Special considerations for specific applications are included in the relevant application parts of EN 1993. (6) The provisions in this standard apply to axisymmetric shells and associated circular or annular plates and to beam section rings and stringer stiffeners where they form part of the complete structure. General procedures for computer calculations of all shell forms are covered. Detailed formulae for the hand calculation of unstiffened cylinders, cones and spherical domes are given in the Annexes. (7) Whilst the hand calculations of this standard apply only to unstiffened shells (formed from isotropic plates), the global analysis procedures may be used to design stiffened shells. (8) Cylindrical and conical panels are not explicitly covered by this standard. However, the provisions of 8 can be applicable if the appropriate boundary conditions are duly taken into account. (9) This standard is intended for application to steel shell structures. Where no standard exists for shell structures made of other metals, the provisions of this standards may be applied provided that the appropriate material properties are duly taken into account. (10) The provisions of this standard are intended to be applied within the temperature range defined in the relevant EN 1993 application parts. The maximum temperature is restricted so that the influence of creep can be neglected if high temperature creep effects are not covered by the relevant application part. (11) The provisions in this standard apply to materials that satisfy the brittle fracture provisions given in EN 1993-1-10.

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(12) The provisions of this standard apply to structural design under actions that can be treated as quasi-static in nature. (13) In this standard, it is assumed that both wind loading and bulk solids flow can, in general, be treated as quasi-static actions. (14) Dynamic effects should be taken into account according to the relevant application part of EN 1993, including the consequences for fatigue. However, the stress resultants arising from dynamic behaviour are treated in this part as quasi-static. (15) The provisions in this standard apply to structures that are constructed in accordance with EN 1090-2, though the buckling-related tolerance requirements of this standard may differ from those of EN 1090-2. (16) This standard does not cover the aspects of leakage. (17) This standard is intended for application to structures within the following limits: • design metal temperatures within the range −50°C to +150°C; • radius to thickness ratios within the range 50 to 2000. NOTE It should be recognised that the stress design rules of this standard may be rather conservative if applied to some geometries and loading conditions for relatively thick-walled shells. NOTE Thinner shells may be designed using these provisions but the provisions have not been verified for such thin shells EDITORIAL NOTE: THE LOWER LIMIT OF r/t = 20 SEEMS TOO LOW, AND INVADES THE TERRITORY OF EN 1993-1-1. IT SHOULD PROBABLY BE RAISED TO PERHAPS 200 OR 300

EDITORIAL NOTE: THE UPPER LIMIT SHOULD HAVE A NOTE TO SAY THAT THINNER SHELLS MAY BE DESIGNED USING THESE PROVISIONS BUT THE PROVISIONS HAVE NOT BEEN VERIFIED FOR SUCH THIN SHELLS

1.2 Normative references (1) This European Standard incorporates, by dated or undated reference, provisions from other publications. These normative references are cited at the appropriate places in the text and the publications are listed hereafter. For dated references, subsequent amendments to or revisions of any of these publications apply to this European Standard only when incorporated in it by amendment or revision. For undated references the latest edition of the publication referred to applies. EN 1090-2 Execution of steel structures and aluminium structures – Part 2: Technical requirements for steel structures; EN 1990 Basis of structural design; EN 1991 Eurocode 1: Actions on structures; Part 1-4: Wind actions Part 4: Actions on silos and tanks; EN 1993 Eurocode 3: Design of steel structures: Part 1.1: General rules and rules for buildings;

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Part 1.2: Structural fire design; Part 1.3: Cold formed thin gauged members and sheeting; Part 1.4: Stainless steels; Part 1.5: Plated structural elements; Part 1.9: Fatigue strength of steel structures; Part 1.10: Selection of steel for fracture toughness and through-thickness properties; Part 1.12: Additional rules for the extension of EN 1993 up to steel grades S 700 Part 2: Steel bridges; Part 3.1: Towers and masts; Part 3.2: Chimneys; Part 4.1: Silos; Part 4.2: Tanks; Part 5: Piling. 1.3 Terms and definitions The terms that are defined in EN 1990 for common use in the Structural Eurocodes apply to this standard. Unless otherwise stated, the definitions given in ISO 8930 also apply in this standard. Supplementary to EN 1993-1-1, for the purposes of this standard, the following definitions apply: 1.3.1 Structural forms and geometry

1.3.1.1 shell a structure or a structural component formed from a curved thin plate 1.3.1.2 shell of revolution a shell whose geometric form is defined by a middle surface that is formed by rotating a meridional generator line around a single axis through 2π radians 1.3.1.3 complete axisymmetric shell a shell composed of a number of parts, each of which is a shell of revolution 1.3.1.4 shell segment a shell of revolution in the form of a defined shell geometry with a constant wall thickness: a cylinder, conical frustum, spherical frustum, annular plate, toroidal knuckle or other form

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1.3.1.5 shell panel an incomplete shell of revolution: the shell form is defined by a rotation of the generator about the axis through less than 2π radians 1.3.1.6 middle surface the surface that lies midway between the inside and outside surfaces of the shell at every point, which is the reference surface for analysis, and can be discontinuous at changes of thickness or at shell junctions, leading to eccentricities that may be important to the shell structural behaviour: in a shell stiffened on either one or both surfaces, the reference middle surface is still taken as the middle surface of the curved shell plate 1.3.1.7 junction the line at which two or more shell segments meet: it can include a stiffener, which may be treated as a junction at the circumferential line of attachment of a ring stiffener to the shell 1.3.1.8 stringer stiffener a local stiffening member that follows the meridian of the shell, representing a generator of the shell of revolution, provided to increase the stability, or to assist with the introduction of local loads, but not intended to provide a primary resistance to bending effects caused by transverse loads 1.3.1.9 rib a local member that provides a primary load carrying path for bending down the meridian of the shell, representing a generator of the shell of revolution, used to transfer or distribute transverse loads by bending 1.3.1.10 ring stiffener a local stiffening member that passes around the circumference of the shell of revolution at a given point on the meridian, normally assumed to have no stiffness for deformations out of its own plane (meridional displacements of the shell) but to be stiff for deformations in the plane of the ring, and provided to increase the stability or to introduce local loads acting in the plane of the ring 1.3.1.11 base ring a structural member that passes around the circumference of the shell of revolution at the base and provides a means of attachment of the shell to a foundation or other structural member, needed to ensure that the assumed boundary conditions are achieved in practice 1.3.1.12 ring beam or ring girder a circumferential stiffener that has bending stiffness and strength both in the plane of the shell circular section and normal to that plane, acting as a primary load carrying structural member and provided for the distribution of local loads into the shell

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1.3.2 Limit states

1.3.2.1 plastic failure limit state the ultimate limit state where the structure develops zones of yielding in a pattern such that its ability to resist increased loading is deemed to be exhausted It is closely related to a small deflection theory plastic limit load or plastic collapse mechanism. 1.3.2.2 tensile rupture the ultimate limit state where the shell plate experiences gross section failure due to tension 1.3.2.3 cyclic plasticity the ultimate limit state where repeated yielding is caused by cycles of loading and unloading, leading to a low cycle fatigue failure where the energy absorption capacity of the material is exhausted 1.3.2.4 buckling the ultimate limit state where the structure suddenly loses its stability under membrane compression and/or shear, leading either to large displacements or to the structure being unable to carry the applied loads 1.3.2.5 fatigue the ultimate limit state where many cycles of loading cause cracks to develop in the shell plate that by further load cycles may lead to rupture 1.3.3 Actions

1.3.3.1 axial load externally applied loading acting in the axial direction 1.3.3.2 radial load externally applied loading acting normal to the surface of a cylindrical shell 1.3.3.3 internal pressure component of the surface loading acting normal to the shell in the outward direction, whose magnitude can vary in both the meridional and circumferential directions (e.g. under solids loading in a silo, see EN 1991-4) 1.3.3.4 external pressure component of the surface loading acting normal to the shell in the inward direction whose magnitude can vary in both the meridional and circumferential directions (e.g. under wind, see EN 1991-1-4)

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1.3.3.5 hydrostatic pressure pressure varying linearly with the axial coordinate of the shell of revolution, which is deemed to have its axis vertical 1.3.3.6 wall friction load meridional component of the surface loading acting on the shell wall due to friction connected with internal pressure (e.g. when solids are contained within the shell, see EN 1991-4) 1.3.3.7 local load point applied force or distributed load acting on a limited part of the circumference of the shell and over a limited height 1.3.3.8 patch load local distributed load acting normal to the shell 1.3.3.9 suction uniform net external pressure due to the reduced internal pressure in a shell with openings or vents under wind action (see EN 1991-1-4) 1.3.3.10 partial vacuum uniform net external pressure due to the removal of stored liquids or solids from within a container that is inadequately vented (see EN 1991-4) 1.3.3.11 thermal action temperature variation either down the shell meridian, or around the shell circumference or through the shell thickness 1.3.4 Stress resultants and stresses in a shell

1.3.4.1 membrane stress resultants the membrane stress resultants are the forces per unit width of shell that arise as the integral of the distribution of direct and shear stresses acting parallel to the shell middle surface through the thickness of the shell, such that under elastic conditions, each of these stress resultants induces a stress state that is uniform through the shell thickness, resulting in three membrane stress resultants at any point (see Figure 1.1(e))

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1.3.4.2 bending stress resultants the bending stress resultants are the bending and twisting moments per unit width of shell that arise as the integral of the first moment of the distribution of direct and shear stresses acting parallel to the shell middle surface through the thickness of the shell, such that under elastic conditions, each of these stress resultants induces a stress state that varies linearly through the shell thickness, with value zero and the middle surface, resulting in two bending moments and one twisting moment at any point 1.3.4.3 transverse shear stress resultants the transverse stress resultants are the forces per unit width of shell that arise as the integral of the distribution of shear stresses acting normal to the shell middle surface through the thickness of the shell, such that under elastic conditions, each of these stress resultants induces a stress state that varies parabolically through the shell thickness, resulting in two transverse shear stress resultants at any point (see Figure 1.1(f)) 1.3.4.4 membrane stress the membrane stress is defined as the membrane stress resultant divided by the shell thickness (see Figure 1.1(e)) 1.3.4.5 bending stress the bending stress is defined as the bending stress resultant multiplied by 6 and divided by the square of the shell thickness (only meaningful for conditions in which the shell is elastic) 1.3.5 Types of analysis

1.3.5.1 global analysis an analysis that includes the complete structure, rather than individual structural parts treated separately 1.3.5.2 membrane theory analysis an analysis that predicts the behaviour of a thin-walled shell structure under distributed loads by assuming that only membrane forces satisfy equilibrium with the external loads 1.3.5.3 semi-membrane theory analysis an analysis that predicts the behaviour of an unsymmetrically loaded or supported thin-walled cylindrical shell structure by assuming that only membrane forces and circumferential bending moments satisfy equilibrium with the external loads 1.3.5.4 linear elastic shell analysis (LA) an analysis that predicts the behaviour of a thin-walled shell structure on the basis of the small deflection linear elastic shell bending theory, related to the perfect geometry of the middle surface of the shell

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1.3.5.5 linear elastic bifurcation (eigenvalue) analysis (LBA) an analysis that evaluates the linear bifurcation eigenvalue for a thin-walled shell structure on the basis of the small deflection linear elastic shell bending theory, related to the perfect geometry of the middle surface of the shell (note that the term eigenvalue in this standard does not relate to a vibration mode) 1.3.5.6 geometrically nonlinear elastic analysis (GNA) an analysis based on the principles of shell bending theory applied to the perfect structure, using a linear elastic material law but including nonlinear large deflection theory for the displacements that accounts full for any change in geometry due to the actions on the shell, including a bifurcation eigenvalue check at each load level 1.3.5.7 materially nonlinear analysis (MNA) an analysis based on shell bending theory applied to the perfect structure, using the assumption of small deflections, as in 1.3.5.4, but adopting an ideal elastic plastic material law (idealised perfectly plastic response after yield) 1.3.5.8 geometrically and materially nonlinear analysis (GMNA) an analysis based on shell bending theory applied to the perfect structure, using the assumptions of nonlinear large deflection theory for the displacements and a fully nonlinear elastic-plastic-hardening material law, where appropriate, and in which a bifurcation eigenvalue check is included at each load level 1.3.5.9 geometrically nonlinear elastic analysis with imperfections included (GNIA) an analysis with imperfections explicitly included, similar to a GNA analysis as defined in 1.3.5.6, but adopting a model for the geometry of the structure that includes the imperfect shape (i.e. the geometry of the middle surface includes unintended deviations from the ideal shape), as well as imperfections in the boundary conditions and residual stresses as potential imperfections where appropriate and including a bifurcation eigenvalue check at each load level 1.3.5.10 geometrically and materially nonlinear analysis with imperfections included (GMNIA) an analysis with imperfections explicitly included, based on the principles of shell bending theory applied to the imperfect structure (i.e. the geometry of the middle surface includes unintended deviations from the ideal shape), including nonlinear large deflection theory for the displacements that accounts fully for any change in geometry due to the actions on the shell and a fully nonlinear elastic-plastic-hardening material law, where appropriate, as well as imperfections in the boundary conditions and residual stresses as potential imperfections where appropriate and including a bifurcation eigenvalue check at each load level

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1.3.6 Stress categories used in stress design

1.3.6.1 Primary stresses the stress system required for equilibrium with the imposed loading, consisting primarily of membrane stresses, but under some situations bending stresses may also be required to achieve equilibrium 1.3.6.2 Secondary stresses stresses induced by internal compatibility or by compatibility with the boundary conditions, associated with imposed loading or imposed displacements (temperature, prestressing, settlement, shrinkage), and not not required to achieve equilibrium between an internal stress state and the external loading 1.3.7 Special definitions for buckling calculations

1.3.7.1 capacity curve the algebraic description of the resistances of all systems from elastic imperfect slender systems through elastic-plastic to fully plastic and hardening systems, characterised through the capacity parameters αG, αI, β, η0, ηp, λo and χh (Formulae 8.30 - 8.33) 1.3.7.2 critical buckling resistance the smallest bifurcation load determined assuming the idealised conditions of elastic material behaviour, small deflection theory (no change of geometry), perfect geometry, perfect load application, perfect support, material isotropy and absence of residual stresses (LBA analysis) 1.3.7.3 critical buckling stress the membrane stress associated with the critical buckling resistance 1.3.7.4 plastic reference resistance the plastic limit load, determined assuming the idealised conditions of rigid-plastic material behaviour, small deflection theory (no change of geometry), perfect geometry, perfect load application, perfect support and material isotropy (modelled using MNA analysis) 1.3.7.5 characteristic buckling resistance the load associated with buckling in the presence of the geometrical and structural imperfections that are inevitable in practical construction, inelastic material behaviour where appropriate, and follower load effects if relevant (defined in terms of the characteristic values of the modulus and yield stress of the material) 1.3.7.6 characteristic buckling stress the membrane stress associated with the characteristic buckling resistance

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1.3.7.7 design buckling resistance the design value of the buckling load, obtained by dividing the characteristic buckling resistance by the partial factor for resistance 1.3.7.8 design buckling stress the membrane stress associated with the design buckling resistance 1.3.7.9 fabrication tolerance quality class the category of fabrication tolerance requirements that is assumed in design (see 8.4) 1.3.7.10 key value of the stress the value of stress in a non-uniform stress field that is used to characterise the complete pattern of varying stresses in a buckling limit state assessment 1.4 Symbols (1) In addition to those given in EN 1990 and EN 1993-1-1, the following symbols are used: (2) Coordinate system, see Figure 1.1: r radial coordinate, normal to the axis of revolution; x meridional coordinate; z axial coordinate; θ circumferential coordinate; φ meridional slope: angle between axis of revolution and normal to the meridian of the shell. (3) Pressures: pn normal to the shell; px meridional surface loading parallel to the shell; pθ circumferential surface loading parallel to the shell, (4) Line forces: Pn load per unit circumference normal to the shell; Px load per unit circumference acting in the meridional direction; Pθ load per unit circumference acting circumferentially on the shell,

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(5) Membrane stress resultants: nx meridional membrane stress resultant; nθ circumferential membrane stress resultant; nxθ membrane shear stress resultant, (6) Bending stress resultants: mx meridional bending moment per unit width; mθ circumferential bending moment per unit width; mxθ twisting shear moment per unit width; qxn transverse shear force associated with meridional bending; qθn transverse shear force associated with circumferential bending, (7) Stresses: σx meridional stress; σθ circumferential stress; σeq von Mises equivalent stress (can also take negative values during cyclic loading); τ, τxθ in-plane shear stress; τxn, τθn meridional, circumferential transverse shear stresses associated with bending. (8) Displacements: u meridional displacement; v circumferential displacement; w displacement normal to the shell surface; βφ meridional rotation, see 5.2.2. (9) Shell dimensions: d internal diameter of shell; L total length of the shell; ℓ length of shell segment; ℓg gauge length for measurement of imperfections; ℓgθ gauge length in circumferential direction for measurement of imperfections; ℓgw gauge length across welds for measurement of imperfections; ℓgx gauge length in meridional direction for measurement of imperfections;

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ℓR limited length of shell for buckling strength assessment; r radius of the middle surface, normal to the axis of revolution; t thickness of shell wall; tmax maximum thickness of shell wall at a joint; tmin minimum thickness of shell wall at a joint; tave average thickness of shell wall at a joint; β apex half angle of cone.

Normal

Circumferential

Meridional

Directions

w

v

u

Displacements

n

θ

x

Coordinates

Membrane stresses

σx

σθ

τxθ

σx

σθz

pθ

pn

px

θ

φ

Surface pressuresTransverse shear

stresses

τθn

τxn

Figure 1.1: Symbols in shells of revolution

(10) Tolerances, see 8.4: e eccentricity between the middle surfaces of joined plates; Ue unintended eccentricity tolerance parameter; Ur out-of-roundness tolerance parameter; Un initial dimple imperfection amplitude parameter for numerical calculations; U0 initial dimple tolerance parameter; Δw0 tolerance normal to the shell surface. (11) Properties of materials: E Young’s modulus of elasticity;

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Esh tangent strain hardening modulus; feq von Mises equivalent strength; fy yield strength; fu ultimate strength; ν Poisson’s ratio. (12) Parameters in strength assessment: C coefficient in buckling strength assessment; D cumulative damage in fatigue assessment; F generalised action; FEd action set on a complete structure corresponding to a design situation (design values); FRd calculated values of the action set at the maximum resistance condition of the structure (design values); Rcr critical buckling resistance ratio (defined as a load factor on design loads using LBA analysis); Rk characteristic reference resistance ratio (used with subscripts to identify the basis): defined as a load factor on design loads using the ratio (FRk / FEd); Rpl plastic reference resistance ratio (defined as a load factor on design loads using MNA analysis); Rplf plastic failure resistance ratio (defined as a load factor on design loads using GMNA analysis); RGMNA buckling resistance ratio determined in a GMNA analysis; k calibration factor for nonlinear analyses; k power of interactions in buckling strength interaction formulae (D.1.6); n number of cycles of loading; α elastic buckling reduction factor in buckling strength assessment; αG geometric reduction factor; αI imperfection reduction factor; β plastic range factor in buckling interaction; γ partial factor; Δ range of parameter when alternating or cyclic actions are involved; εp plastic strain; η interaction exponent for buckling; η0 value of interaction exponent at λ = 0λ ; ηp value of interaction exponent at λ = pλ ; λ relative slenderness of shell; λs complete shell relative slenderness for the complete shell (multiple segments);

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0λ squash limit relative slenderness (value of λ− above which resistance reductions due to instability or change of geometry occur); pλ plastic limit relative slenderness (value of λ− below which plasticity affects the stability);

ω first relative length parameter for a shell; Ω second relative length parameter for a shell; χ buckling reduction factor including elastic-plastic effects in buckling strength assessment; χs complete shell buckling reduction factor including elastic-plastic effects in a complete shell. (13) Subscripts: E value of stress or displacement (arising from design actions); F actions; M material; R resistance; cr critical buckling value (see 1.3.7.1); d design value; int internal; k characteristic value; max maximum value; min minimum value; nom nominal value; pl plastic value; s for a complete shell, potentially with multiple segments; u ultimate; y yield. (14) Further symbols are defined where they first occur.

1.5 Sign conventions (1) Outward direction positive: internal pressure positive, outward displacement positive, except as noted in (4). (2) Tensile stresses positive, except as noted in (4). NOTE Compression is treated as positive in EN 1993-1-1. (3) Shear stresses positive as shown in Figures 1.1 and D.1.

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NOTE Although the directions of direct stresses differ between Figures 1.1 and D.1, the direction of shear is retained unchanged. (4) For simplicity, in 8 and Annex D, compressive stresses are treated as positive. For these cases, both external pressures and internal pressures are treated as positive where they occur. 2 Basis of design and modelling

2.1 General (1)P The basis of design shall be in accordance with EN 1990, as supplemented by the following. (2) In particular, the shell should be designed in such a way that it will sustain all actions and satisfy the following requirements: • overall equilibrium; • equilibrium between actions and internal forces and moments, see 6 and 8; • limitation of cracks due to cyclic plastification, see 7; • limitation of cracks due to fatigue, see 9. (3) The design of the shell should satisfy the serviceability requirements set out in the appropriate application standard (EN 1993 Parts 3.1, 3.2, 4.1 and 4.2). (4) The shell may be proportioned using design assisted by testing. Where appropriate, the requirements are set out in the appropriate application standard (EN 1993 Parts 3.1, 3.2, 4.1 and 4.2). (5) All actions should be introduced using their design values according to EN 1991 and EN 1993 Parts 3.1, 3.2, 4.1 and 4.2 as appropriate. 2.2 Types of analysis

2.2.1 General (1) One or more of the following types of analysis should be used as detailed in 4, depending on the limit state and other considerations: • Global analysis, see 2.2.2; • Membrane theory analysis, see 2.2.3; • Semi-membrane theory analysis, see 2.2.4; • Linear elastic shell analysis, see 2.2.5; • Linear elastic bifurcation analysis, see 2.2.6; • Geometrically nonlinear elastic analysis, see 2.2.7; • Materially nonlinear analysis, see 2.2.8; • Geometrically and materially nonlinear analysis, see 2.2.9;

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• Geometrically nonlinear elastic analysis with imperfections explicitly included, see 2.2.10; • Geometrically and materially nonlinear analysis with imperfections explicitly included, see 2.2.11. 2.2.2 Global analysis (1) In a global analysis simplified treatments may be used for certain parts of the structure. 2.2.3 Membrane theory analysis (1) A membrane theory analysis may be used provided that the following conditions are met: • the boundary conditions are appropriate for transfer of the stresses in the shell into support reactions without causing unacceptable bending effects; • the shell geometry varies smoothly in shape (without discontinuities); • the loads have a smooth distribution (without locally concentrated or point loads). (2) A membrane theory analysis does not meet the requirements of compatibility of deformations at boundaries or between shell segments of different shape or between shell segments subjected to different loading. However, the resulting field of membrane forces satisfies the requirements of equilibrium of the primary stresses (useful for LS1). 2.2.4 Semi-membrane theory (1) A semi-membrane theory analysis may be used when a long cylindrical shell is subject to a circumferentially varying load with a variation more rapid than harmonic 1 and subject to axial displacements at a boundary (e.g. wind loading or discretely supported shells). 2.2.5 Linear elastic shell analysis (LA) (1) The linearity of the theory results from the assumptions of a linear elastic material law and small deformation theory. Small deformation theory implies that the assumed geometry remains that of the undeformed structure. (2) An LA analysis satisfies compatibility in the deformations as well as equilibrium. The resulting field of membrane and bending stresses satisfy the requirements of primary plus secondary stresses (useful for LS1, LS2, LS3 and LS4). 2.2.6 Linear elastic bifurcation analysis (LBA) (1) The conditions of 2.2.4 concerning the material and geometric assumptions are met. However, this linear bifurcation analysis obtains the lowest eigenvalue at which the shell may buckle into a different deformation mode, assuming no change of geometry, no change in the direction of action of the loads, and no material degradation. Imperfections of all kinds are ignored. This analysis provides the elastic critical buckling resistance Rcr see 8.6, 8.7 and 8.8 (useful for LS3), which can be interpreted as a load amplification factor Rcr on the design value of the loads FEd. (2) This perfect shell elastic critical load should always be determined when the limit state LS3 is verified using GMNIA analysis, see 8.8.

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(3) Where this analysis is used as the basis for an LBA-MNA design procedure, the requirement to examine multiple eigenvalues and the corresponding eigenmodes (not only the lowest eigenvalue) should be undertaken (see 6.3 and 8.7) 2.2.7 Geometrically nonlinear elastic analysis (GNA) (1) A GNA analysis satisfies both equilibrium and compatibility of the deformations under conditions in which the change in the geometry of the structure caused by loading is included. The resulting field of stresses matches the definition of primary plus secondary stresses (useful for LS2). (2) Where compression or shear stresses are predominant in some part of the shell, a GNA analysis delivers the elastic buckling load of the perfect structure, including changes in geometry, that may be of assistance towards a check of the limit state LS3, see 8.8. (3)P Where this analysis is used for a buckling load evaluation, the eigenvalues of the system shall be checked throughout the loading path to ensure that the numerical process does not fail to detect a bifurcation in the load path. 2.2.8 Materially nonlinear analysis (MNA) (1) The result of an MNA analysis leads to the plastic reference limit load, which can be interpreted as a load amplification factor Rpl on the design value of the loads FEd. This analysis provides the plastic reference resistance ratio Rpl used in 8.6, 8.7 and 8.8. (2) An MNA analysis may be used to verify limit state LS1. (3) An MNA analysis may be used to give the plastic strain increment Δε during one cycle of cyclic loading that may be used to verify limit state LS2. (4) This perfect shell plastic limit load should always be determined when the limit state LS3 is verified using GMNIA analysis, see 8.8. 2.2.9 Geometrically and materially nonlinear analysis (GMNA) (1) The result of a GMNA analysis, analogously to 2.2.7, gives the geometrically nonlinear plastic failure load of the perfect structure and the plastic strain increment, that may be used for checking the limit states LS1 and LS2. (2) Where compression or shear stresses are predominant in some part of the shell, a GMNA analysis gives the elastic-plastic buckling load of the perfect structure. This perfect shell buckling load should always be determined when the limit state LS3 is verified using GMNIA analysis, see 8.8. (3) Where this analysis is used for a buckling load evaluation, the eigenvalues of the system should be checked to ensure that the numerical process does not fail to detect a bifurcation in the load path. 2.2.10 Geometrically nonlinear elastic analysis with imperfections explicitly included (GNIA) (1) A GNIA analysis is used in cases where compression or shear stresses dominate in the shell. It delivers elastic buckling loads of the imperfect structure, that may be of assistance in checking the limit state LS3, see 8.7. (2) Where this analysis is used for a buckling load evaluation (LS3), the eigenvalues of the system should be checked to ensure that the numerical process does not fail to detect a bifurcation in the load path. Care should be taken to ensure that the local stresses do not exceed values at which material nonlinearity may affect the behaviour. NOTE GNIA analysis is often useful for very thin shells where plasticity plays no role in the ultimate limit state.

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2.2.11 Geometrically and materially nonlinear analysis with imperfections explicitly included (GMNIA) (1) A GMNIA analysis is used in cases where compression or shear stresses are dominant in the shell. It delivers elasto-plastic buckling loads for the "real" imperfect structure, that may be used for checking the limit state LS3, see 8.7. (2) Where this analysis is used for a buckling load evaluation, the eigenvalues of the system should be checked to ensure that the numerical process does not fail to detect a bifurcation in the load path. (3) Where this analysis is used for a buckling load evaluation, additional LBA, MNA and GMNA analyses of the perfect shell should always be conducted to ensure that the slenderness is properly recognised and that the degree of imperfection sensitivity of the structural system is identified.

2.3 Shell boundary conditions (1) The boundary conditions assumed in the design calculation should be chosen in such a way as to ensure that they achieve a realistic or conservative model of the real construction. Special attention should be given not only to the constraint of displacements normal to the shell wall (deflections), but also to whether the displacements in the plane of the shell wall (meridional and circumferential) are adequately constrained because of the significant effect these displacements can have on the shell strength and buckling resistance. NOTE The buckling resistance of a shell is often sensitive to any minor flexibility in the boundary conditions, making the modelling of realistic boundary conditions more critical than for a simple load-deformation analysis. (2) In shell buckling (eigenvalue) calculations (limit state LS3), the definition of the boundary conditions should refer to the incremental displacements during the buckling process, and not to total displacements induced by the applied actions before buckling. (3) The boundary conditions at a continuously supported lower edge of a shell should take into account whether local uplifting of the shell is fully prevented or not. (4) The shell edge rotation βφ should be particularly considered in short shells and in the calculation of secondary stresses in longer shells (according to the limit states LS2 and LS4). (5) The boundary conditions set out in 5.2.2 should be used in computer analyses and in selecting formulae from Annexes A to D. (6) The structural connections between shell segments at a junction should be such as to ensure that the boundary condition assumptions used in the design of the individual shell segments are satisfied. 3 Materials and geometry

3.1 Material properties (1) The material properties of steels should be obtained from the relevant application standard. (2) Where a material that has a nonlinear stress-strain curve is involved and a buckling analysis is carried out under stress design (see 8.5), the initial tangent value of Young´s modulus E should be replaced by a reduced value. If no better method is available, the secant modulus at the 0,2% proof stress should be used when assessing the elastic critical load or elastic critical stress.

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EDITORIAL NOTE: IT IS UNCLEAR WHETHER THIS PROVISION OF THE 0,2% PROOF STRESS IS ALWAYS CONSERVATIVE, AND THE LOCATION IN WHICH THE STRESS IS TO BE EVALUATED IS NOT DEFINED, NOR IS THE STRESS COMPONENT (von Mises?).

EDITORIAL NOTE: IT WOULD BE GOOD TO REPLACE THIS PROVISION WITH A REQUIREMENT THAT THE PEAK von Mises STRESS STATE IS USED TO DETERMINE THE LOCAL TANGENT MODULUS, AND THAT THIS VALUE IS THEN USED TO ASSESS THE RESISTANCE. (3) In a global numerical analysis using material nonlinearity, the 0,2% proof stress should be used to represent the yield stress fy in all relevant formulae. The stress-strain curve should be obtained from EN 1993-1-5 Annex C for carbon steels and EN 1993-1-4 Annex C for stainless steels. (4) The material properties apply to temperatures not exceeding 150°C. EDITORIAL NOTE: THIS TEMPERATURE IS NOW MATCHED TO THAT DEFINED IN 1.1 (16)

EDITORIAL NOTE: SHOULD THERE BE A NOTE THAT REFERS INSTEAD TO EN 1993-1-2 ? (5) For the mechanical properties of the structural carbon steels S 235, S 275, S 355, S 420 and S 460 and also for weathering steel grades S 235, S 275 and S 355, see EN 1993-1-1. For the properties of these steels at higher temperatures see EN 13084-7. EDITORIAL NOTE: IS THIS LIST SUFFICIENTLY COMPLETE? WHAT SHOULD BE ADDED ? (6) For steel grades other than those given in EN 13084-7, the mechanical properties used at temperature of steel higher than 150°C should be based on reliable information. (7) The variation of properties of steels at temperatures above 150°C are given for carbon steels in EN 1993-1-2: 3.2, and for stainless steels in EN 1993-1-2: 3.3. (8) For stainless steels, the rules of EN 10088-4 or EN 1993-1-4 should be applied. 3.2 Design values of geometrical data (1) The thickness t of the shell should be taken as defined in the relevant application standard. If no application standard is relevant, the nominal thickness of the wall, reduced by the prescribed value of the corrosion loss and ignoring any coatings, should be used. (2) The thickness ranges within which the rules of this standard may be applied are defined in the relevant EN 1993 application parts. (3) The middle surface of the shell should be taken as the reference surface for loads. (4) The simple radius r of the shell should be taken as the nominal radius of the middle surface of the shell, measured normal to the axis of revolution. NOTE This radius varies with position on the axis in all shells that are not simply cylindrical. (5) The buckling design rules of this standard should not be applied outside the ranges of the r/t ratio set out in 1.16, or where stricter restrictions apply, as defined in 8 or Annex D or in the relevant EN 1993 application parts. 3.3 Geometrical tolerances and geometrical imperfections (1) Tolerance values for the deviations of the geometry of the shell surface from the nominal values are defined in the execution standard EN 1090-2. Relevant categories for the design of shells for the ultimate limit state of buckling (LS3) are: • local dimples (local normal deviations from the nominal middle surface);

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• out-of-roundness (deviation from circularity); • eccentricities (deviations from a continuous middle surface in the direction normal to the shell across the junctions between plates); • deviations of the base of a shell from full contact with the support. NOTE The requirements for execution are set out in EN 1090, but a fuller description of these tolerances is given here because of the critical relationship between the form of the tolerance measure, its amplitude and the evaluated buckling resistance of the shell structure. (2) If the limit state of buckling (LS3, as described in 4.1.3) is one of the ultimate limit states to be considered, the buckling-relevant geometrical tolerances should be carefully observed in order to keep the geometrical imperfections within specified limits. These buckling-relevant geometrical tolerances and the conditions to which they are relevant are identified and quantified in 8 or in the relevant EN 1993 application parts. (3) Calculation values for the deviations of the shell surface geometry from the nominal geometry, as required for geometrical imperfection assumptions (complete shell imperfections or local imperfections) for buckling design by global GMNIA analysis (see 8.8), should be derived from the specified geometrical tolerances. Relevant rules are given in 8.8 or in relevant EN 1993 application parts. 4 Ultimate limit states in steel shells 4.1 Ultimate limit states to be considered

4.1.1 LS1: Plastic failure limit state (1) The limit state of the plastic failure should be taken as the condition in which the capacity of the structure to resist the actions on it is exhausted by plasticity in the material. NOTE The plastic failure resistance differs from the plastic reference resistance. The plastic reference resistance is found as the plastic collapse load obtained from a mechanism based on small displacement theory using an ideal elastic-plastic material law. (2) The limit state of tensile rupture should be taken as the condition in which the shell wall experiences gross section tensile failure, leading to separation of the two parts of the shell. (3) In the absence of fastener holes, verification at the limit state of tensile rupture may be assumed to be covered by the check for the plastic failure limit state. However, where holes for fasteners occur, a supplementary check in accordance with 6.2 of EN 1993-1-1 or EN 1993-1-8 should be carried out. (4) In verifying the plastic failure limit state, plastic or partially plastic behaviour of the structure may be assumed (i.e. elastic compatibility considerations may be neglected). NOTE Since the plastic failure limit state includes change of geometry, it may be noted that this limit state may also capture snap-through buckling, which may occur in the elastic state. The plastic reference resistance does not include change of geometry, so this apparent anomaly does not occur. NOTE The plastic failure limit state does not include considerations of bifurcation, so no checks for bifurcation are required when a GMNA analysis is used to assess the plastic failure limit state LS1. (5) All relevant load combinations should be accounted for when checking LS1.

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(6) One or more of the following methods of analysis (see 2.2) should be used for the calculation of the design stresses and stress resultants when checking LS1: − membrane theory; − formulae in Annexes A and B; − linear elastic analysis (LA); − materially nonlinear analysis (MNA); − geometrically and materially nonlinear analysis (GMNA).

4.1.2 LS2: Cyclic plasticity (1) The limit state of cyclic plasticity should be taken as the condition in which repeated cycles of loading and unloading produce repeated yielding in tension and in compression at the same point, thus causing plastic work to be repeatedly done on the structure, eventually leading to local cracking by exhaustion of the energy absorption capacity of the material. NOTE The stresses that are associated with this limit state develop under a combination of all actions and the compatibility conditions for the structure. (2) All variable actions (such as imposed loads and temperature variations) that can lead to yielding, and which might be applied with more than three cycles in the life of the structure, should be accounted for when checking LS2. (3) In the verification of this limit state, compatibility of the deformations under elastic or elastic-plastic conditions should be considered. (4) One or more of the following methods of analysis (see 2.2) should be used for the calculation of the design stresses and stress resultants when checking LS2: − formulae in Annex C; − elastic analysis (LA or GNA); − MNA or GMNA to determine the plastic strain range. (5) Low cycle fatigue failure may be assumed to be prevented if the procedures set out in this standard are adopted.

4.1.3 LS3: Buckling (1) The limit state of buckling should be taken as the condition in which all or part of the structure suddenly develops large displacements normal to the shell surface, caused by loss of stability under compressive membrane or shear membrane stresses in the shell wall, leading to inability to sustain any increase in the stress resultants, and possibly cause total collapse of the structure. A MORE CAREFUL DESCRIPTION OF THIS PARAGRAPH WAS NEEDED (2) One or more of the following methods of analysis (see 2.2) and buckling resistance assessment should be used for the calculation of the design stresses and stress resultants when checking LS3:

− membrane theory for axisymmetric conditions only (for exceptions, see relevant application parts of EN 1993); − formulae in Annexes A and D;

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− reference resistance design, where the formulae in Annex E refer to the specific geometry, loading and boundary conditions of the structure; − linear elastic analysis (LA), which is a minimum requirement for stress analysis under general loading conditions with formulae in Annex D (except where the stress analysis of the load case is given in Annex A, or where the buckling condition is treated as a special case in Annex D); − linear elastic bifurcation analysis (LBA), which is required for shells under general loading conditions if the critical buckling resistance is to be used in an LBA-MNA assessment, or a GMNIA assessment; − materially nonlinear analysis (MNA), which is required for shells under general loading conditions if the true reference plastic resistance (rather than a lower bound estimate taken from an LA analysis) is to be used in an LBA-MNA assessment; − GMNIA, together with supporting MNA, LBA and GMNA analyses, and using appropriate imperfections and calculated calibration factors. (3) All relevant load combinations causing compressive membrane or shear membrane stresses in the shell should be accounted for when checking LS3. (4) Because the strength under limit state LS3 depends strongly on the quality of construction, the strength assessment should take account of the associated requirements for execution tolerances. For this purpose, three classes of geometrical tolerances, termed “fabrication quality classes” are given in 8.

4.1.4 LS4: Fatigue (1) The limit state of fatigue should be taken as the condition in which repeated cycles of increasing and decreasing stress lead to the development of a fatigue crack without yielding. (2) The following methods of analysis (see 2.2) should be used for the calculation of the design stresses and stress resultants when checking LS4: − formulae in Annex C, using stress concentration factors; − elastic analysis (LA or GNA), using stress concentration factors. (3) All variable actions that will be applied with more than Nf cycles in the design life time of the structure according to the relevant action spectrum in EN 1991 in accordance with the appropriate application part of EN 1993-3 or EN 1993-4, should be accounted for when checking LS4. The value of Nf should be taken as Nf = 10 000.

4.2 Design concepts for the limit states design of shells

4.2.1 General (1) The limit state verification should be carried out using one of the following: − stress design; − design using standard formulae; − design by global numerical analysis. (2) Account should be taken of the fact that elastic-plastic material responses induced by different stress components in the shell have different effects on the failure modes and the ultimate limit states. The stress components should therefore be placed in stress categories with different limits. Stresses that develop to

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meet equilibrium requirements should be treated as more significant than stresses that are induced by the compatibility of deformations normal to the shell. Local stresses caused by notch effects in construction details may be assumed to have a negligibly small influence on the resistance to static loading. (3) The categories distinguished in stress design should be primary, secondary and local stresses. Primary and secondary stress states may be replaced by stress resultants where appropriate. (4) In a global analysis, the primary and secondary stress states should be replaced by the limit load and the strain range for cyclic loading. (5) In general, it may be assumed that primary stress states control LS1, LS3 depends strongly on primary stress states but may be affected by yielding caused by secondary stress states, LS2 depends on the combination of primary and secondary stress states, and local surface stresses govern LS4. 4.2.2 Stress design

4.2.2.1 General (1) Where the stress design approach is used, the limit states should be assessed in terms of three categories of stress: primary, secondary and local. The categorisation is performed, in general, on the von Mises equivalent stress at a point, but buckling stresses cannot be assessed using this value. 4.2.2.2 Primary stresses (1) The primary stresses should be taken as the stress system required for equilibrium with the imposed loading. They may be calculated from any realistic statically admissible determinate system. The plastic failure limit state (LS1) should be deemed to be reached when the primary stress reaches the yield strength throughout the full thickness of the wall at a sufficient number of points, such that only the strain hardening reserve or a change of geometry would lead to an increase in the resistance of the structure. (2) The calculation of primary stresses should be based on any system of stress resultants, consistent with the requirements of equilibrium of the structure. It may also take into account the benefits of plasticity theory. Alternatively, since linear elastic analysis satisfies equilibrium requirements, its predictions may also be used as a safe representation of the plastic failure limit state (LS1). Any of the analysis methods given in 5.3 may be applied. (3) Because limit state design for LS1 allows for full plastification of the cross-section, the primary stresses due to bending moments may be calculated on the basis of the plastic section modulus, see 6.2.1. Where there is interaction between stress resultants in the cross-section, interaction rules based on the Ilyushin yield criterion may be applied. (4) The primary stresses should be limited to the design value of the yield strength, see 6 (LS1). 4.2.2.3 Secondary stresses (1) In statically indeterminate structures, account should be taken of the secondary stresses, induced by internal compatibility and compatibility with the boundary conditions, that are caused by imposed loading or imposed displacements (temperature, prestressing, settlement, shrinkage). NOTE As the von Mises yield condition is approached, the local strains in the structure increase without further increase in the stress state. (2) Where cyclic loading causes plasticity, and several loading cycles occur, consideration should be given to the possible reduction of resistance caused by the secondary stresses. Where the cyclic loading is of such a magnitude that yielding occurs both at the maximum load and again on unloading, account should be taken of a possible failure by cyclic plasticity associated with the secondary stresses.

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(3) If the stress calculation is carried out using a linear elastic analysis that allows for all relevant compatibility conditions (effects at boundaries, junctions, variations in wall thickness, misalignment of the middle surface etc.), the stresses that vary linearly through the thickness may be taken as the sum of the primary and secondary stresses and used in an assessment involving the von Mises yield criterion, see 6.2. NOTE The secondary stresses are never needed in a evaluation without inclusion of the primary stresses. (4) The secondary stresses should be limited as follows: − The sum of the cyclic change in the primary and secondary stresses (including bending stresses) should be limited to 2fyd for the condition of cyclic plasticity (LS2: see 7); − The membrane component of the sum of the primary and secondary stresses should be limited by the design buckling resistance (LS3: see 8). − The sum of the primary and secondary stresses (including bending stresses) should be limited to the fatigue resistance (LS4: see 9).

4.2.2.4 Local stresses (1) The highly localised stresses associated with stress raisers in the shell wall due to notch effects (holes, welds, stepped walls, attachments, and joints) should be taken into account in a fatigue assessment (LS4). (2) For construction details given in EN 1993-1-9, the fatigue design may be based on the nominal linear elastic stresses (sum of the primary and secondary stresses) at the relevant point. For all other details, the local stresses may be calculated by applying stress concentration factors (notch factors) to the stresses calculated using a linear elastic stress analysis. (3) The local stresses should be limited according to the requirements for fatigue (LS4) set out in 9. 4.2.3 Design using standard formulae (1) Where this design concept is used, the limit states may be represented by standard formulae that have been derived from either membrane theory, plastic mechanism theory, linear elastic analysis and geometrically and materially nonlinear analysis with explicit imperfections. (2) The membrane theory formulae given in Annex A may be used to determine the primary stresses needed for assessing LS1 and LS3. (3) The formulae for plastic design given in Annex B may be used to determine the plastic reference resistances needed for assessing LS1. (4) The formulae for linear elastic analysis given in Annex C may be used to determine stresses of the primary plus secondary stress type needed for assessing LS2 and LS4. An LS3 assessment may be based on the membrane part of these formulae. (5) The formulae for reference resistance design given in Annex E may be used to give direct assessment of the design buckling resistance for assessing LS3.

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4.2.4 Design by global numerical analysis (1) Where a global numerical analysis is used, the assessment of the limit states should be carried out using one of the alternative types of analysis specified in 2.2 (but not membrane theory analysis) applied to the complete structure. (2) Linear elastic analysis (LA) may be used to determine stresses or stress resultants, for use in assessing LS2 and LS4. The membrane parts of the stresses found by LA may be used in assessing LS3. LS1 may be assessed using LA, but LA only gives an approximate and safe estimate and its results should be interpreted as set out in 6. (3) Linear elastic bifurcation analysis (LBA) may be used to determine the critical buckling resistance of the structure, for use in assessing LS3. (4) A materially nonlinear analysis (MNA) may be used to determine the plastic reference resistance, and this may be used for assessing LS1. Under a cyclic loading history, an MNA analysis may be used to determine plastic strain incremental changes, for use in assessing LS2. The plastic reference resistance is also required as part of the assessment of LS3, and this may be found from an MNA analysis. (5) Geometrically nonlinear elastic analyses (GNA and GNIA) include consideration of the deformations of the structure, but none of the design methodologies of 8 permit these to be used without a GMNIA analysis. A GNA analysis may be used to determine the elastic buckling load of the perfect structure. A GNIA analysis may be used to determine the elastic buckling load of the imperfect structure. (6) Geometrically and materially nonlinear analysis (GMNA and GMNIA) may be used to determine collapse loads for the perfect (GMNA) and the imperfect structure (GMNIA). The GMNA analysis may be used in assessing LS1, as detailed in 6.3. The GMNIA collapse load may be used, with additional consideration of the results of LBA, MNA and GMNA analyses to assess LS3 as detailed in 8.8. Under a cyclic loading history, the plastic strain incremental changes taken from a GMNA analysis may be used for assessing LS2. 5 Stress resultants and stresses in shells

5.1 Stress resultants in the shell (1) In principle, the eight stress resultants in the shell wall at any point should be calculated and the assessment of the shell with respect to each limit state should take all of them into account. However, the shear stresses τxn, τθn due to the transverse shear forces qxn, qθn are insignificant compared with the other components of stress in almost all practical cases, so they may usually be neglected in design. (2) Accordingly, for most design purposes, the evaluation of the limit states may be made using only the six stress resultants in the shell wall nx, nθ, nxθ, mx, mθ, mxθ. Where the structure is axisymmetric and subject only to axisymmetric loading and support, only nx, nθ, mx and mθ need be used. (3) If any uncertainty arises concerning the stress to be used in any of the limit state verifications, the von Mises equivalent stress on the shell surface should be used. 5.2 Modelling of the shell for analysis

5.2.1 Geometry (1) The shell should be represented by its middle surface. (2) The radius of curvature should be taken as the nominal radius of curvature. Imperfections should be neglected, except as set out in 8 (LS3 buckling limit state).

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(3) An assembly of shell segments should not be subdivided into separate segments for analysis unless the boundary conditions for each segment are chosen in such a way as to represent interactions between them in a conservative manner. (4) A base ring intended to transfer local support forces into the shell should not be separated from the shell it supports in an assessment of limit state LS3. (5) Eccentricities and steps in the shell middle surface should be included in the analysis model if they induce significant bending effects as a result of the membrane stress resultants following an eccentric path. (6) At junctions between shell segments, any eccentricity between the middle surfaces of the shell segments should be considered in the modelling. (7) A ring stiffener should be treated as a separate structural component of the shell, except where the spacing of the rings is closer than 1,5 rt . (8) A shell that has discrete stringer stiffeners attached to it may be treated as an orthotropic uniform shell, provided that the stringer stiffeners are no further apart than 5 rt . (9) A shell that is corrugated (vertically or horizontally) may be treated as an orthotropic uniform shell provided that the corrugation wavelength is less than 0,5 rt . (10) A hole in the shell may be neglected in the modelling provided its largest dimension is smaller than 0,6 rt (11) The overall stability of the complete structure should be verified as detailed in EN 1993 Parts 3.1, 3.2, 4.1, 4.2 or 4.3 as appropriate. 5.2.2 Boundary conditions (1) The appropriate boundary conditions should be used in analyses for the assessment of limit states according to the conditions shown in Table 5.1. For the special conditions needed for buckling calculations, reference should be made to 8.3. (2) Rotational restraints at shell boundaries may be neglected in modelling for limit state LS1, but should be included in modelling for limit states LS2 and LS4. For short shells (see Annex D), the rotational restraint should be included for limit state LS3. (3) Support boundary conditions should be checked to ensure that they do not cause excessive non-uniformity of transmitted forces or introduced forces that are eccentric to the shell middle surface. Reference should be made to the relevant EN 1993 application parts for the detailed application of this rule to silos and tanks. (4) When a global numerical analysis is used, the boundary condition for the normal displacement w should also be used for the circumferential displacement v, except where special circumstances make this inappropriate.

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Table 5.1: Boundary conditions for shells

Boundary condition code

Simple term

Description Displacements normal to the shell surface

Meridional displacements

Meridional rotation

BC1r Clamped radially restrained meridionally restrained rotation restrained w = 0

u = 0 βφ = 0

BC1f radially restrained meridionally restrained rotation free w = 0

u = 0 βφ ≠ 0

BC2r radially restrained meridionally free rotation restrained w = 0

u ≠ 0 βφ = 0

BC2f Pinned radially restrained meridionally free rotation free w = 0

u ≠ 0 βφ ≠ 0

BC3r radially free meridionally free rotation restrained w ≠ 0

u ≠ 0 βφ = 0

BC3f Free edge radially free meridionally free rotation free w ≠ 0

u ≠ 0 βφ ≠ 0

NOTE The circumferential displacement v is closely linked to the displacement w normal to the surface, so separate boundary conditions are not identified for these two parameters (see (4)) but the values in Column 4 should be adopted for displacement v. 5.2.3 Actions and environmental influences (1) Actions should all be assumed to act at the shell middle surface. Eccentricities of load should be represented by static equivalent forces and moments at the shell middle surface. (2) Local actions and local patches of action should not be represented by equivalent uniform loads except as detailed in 8 for buckling (LS3). (3) The modelling should account for whichever of the following are relevant:

− local settlement under shell walls; − local settlement under discrete supports; − uniformity / non-uniformity of support of structure; − thermal differentials from one side of the structure to the other;

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− thermal differentials from inside to outside the structure; − wind effects on openings and penetrations; − interaction of wind effects on groups of structures; − connections to other structures; − conditions during erection.

5.2.4 Stress resultants and stresses (1) Provided that the radius to thickness ratio is greater than (r/t)min = 25, the curvature of the shell may be ignored when calculating the stress resultants from the stresses in the shell wall. 5.3 Types of analysis (1) The design should be based on one or more of the types of analysis given in Table 5.2. Reference should be made to 2.2 for the conditions governing the use of each type of analysis.

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Table 5.2: Types of shell analysis

Type of analysis Shell theory Material law Shell geometry Membrane theory of shells membrane equilibrium not applicable perfect

Semi-membrane theory of shells linear circumferential bending, membrane shear and axial stretching linear perfect Linear elastic shell analysis (LA) linear bending and stretching linear perfect Linear elastic bifurcation analysis (LBA) linear bending and stretching linear perfect Geometrically non-linear elastic analysis (GNA) non-linear linear perfect Materially non-linear analysis (MNA) linear ideal elastic-plastic (Esh< 10-4E) perfect Geometrically and materially non-linear analysis (GMNA) non-linear fully non-linear perfect Geometrically non-linear elastic analysis with imperfections (GNIA) non-linear linear imperfect Geometrically and materially non-linear analysis with imperfections (GMNIA) non-linear fully non-linear imperfect

6 Plastic failure limit state (LS1)

6.1 Design values of actions (1)P The design values of the actions shall be based on the most adverse relevant load combination (including the relevant γF and ψ factors). (2) Only those actions that represent loads affecting the equilibrium of the structure need be included. 6.2 Stress design

6.2.1 Design values of stresses (1) Although stress design is based on an elastic analysis and therefore cannot accurately predict the plastic failure limit state, it may be used, on the basis of the lower bound theorem, to provide a conservative assessment of the plastic collapse resistance which is used to represent the plastic failure limit state, see 4.1.1.

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(2) The Ilyushin yield criterion may be used, as detailed in (6), that comes closer to the true plastic collapse state than a simple elastic surface stress evaluation. (3) At each point in the structure the design value of the stress σeq,Ed should be taken as the highest primary stress determined in a structural analysis that considers the laws of equilibrium between imposed design load and internal forces and moments. (4) The primary stress may be taken as the maximum value of the stresses required for equilibrium with the applied loads at a point or along an axisymmetric line in the shell structure. (5) Where a membrane theory analysis is used, or where a linear bending theory analysis (LA) is used subject to the conditions defined in (6), the resulting two-dimensional field of stress resultants nx,Ed, nθ,Ed and nxθ,Ed may be represented by the equivalent design stress σeq,Ed obtained from: 2 2 2

, , , , , ,1

3eq Ed x Ed Ed x Ed Ed x Edn n n n nt θ θ θσ = + − ⋅ +

... (6.1) EDITORIAL NOTE: THE PARAGRAPH BELOW HAS BEEN ARRANGED TO TRY TO ENSURE THAT THE ANALYST USING LA OR GNA IS NOT DISADVANTAGED RELATIVE TO THOSE USING MEMBRANE THEORY ALONE. THE RULE HERE TRIES TO IDENTIFY SITUATIONS WHERE BENDING IS REALLY IMPORTANT, SO THAT THE ILYUSHIN RULE SHOULD DEFINE PLASTICITY, AND TO DISTINGUISH THESE SITUATIONS FROM THOSE WHERE BENDING IS NOT CRITICAL. IT MAY ALSO BE NOTED THAT IN AN LA TREATMENT LOCAL BENDING PRODUCES HIGHER LOCAL MEMBRANE STRESSES THAN MEMBRANE THEORY WOULD IDENTIFY, SO THIS TREATMENT IS STILL NOT QUITE AS ECONOMIC AS THE MEMBRANE THEORY METHODOLOGY WHICH IS WIDELY USED FOR SMALLER SHELLS. (6) Where an LA or GNA analysis is used, and the magnitude of the largest von Mises surface stress found using Formulae (6.2) to (6.4) exceeds n times the von Mises membrane stress found using Formula (6.1) at the same location, the equivalent stress should be taken as the value determined using Formulae (6.2) to (6.4). The recommended value of n is n = 3.

2 2 2eq,Ed x,Ed θ,Ed x,Ed θ,Ed xθ,Ed3σ σ σ σ σ τ= + − ⋅ + ... (6.2) in which:

2( / 4)

n m

t tσ = ±x,Ed x,Ed

x,Ed , , ,, 2( / 4)

Ed EdEd

n m

t t

θ θθσ = ± ... (6.3)

, ,, 2( / 4)

x Ed x Edx Ed

n m

t t

θ θθτ = ± , ... (6.4)

NOTE Formulae (6.2) to (6.4) provide a simplified conservative equivalent stress for design purposes. EDITORIAL NOTE: THE PUBLISHED VERSION OF THE STANDARD INCLUDED THE TRANSVERSE SHEAR STRAINS IN THIS LIST. THEY HAVE BEEN REMOVED AS GENERALLY IRRELEVANT IN THIN SHELLS, AND POTENTIALLY CONFUSING TO THE USER.

6.2.2 Design values of resistance (1) The von Mises design strength should be taken from:

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feq,Rd = fyd = fyk / γM0 ... (6.5) (2) The partial factor for resistance γM0 should be taken from the relevant application standard. EDITORIAL NOTE: SHOULD THE VALUE OF γM0 BE CHOSEN OR STATED AS DIFFERENT FOR TENSILE RUPTURE AND BENDING, SINCE THE CONSEQUENCES OF RUPTURE IN A SHELL ARE GREAT BUT THOSE FOR BENDING YIELDING ARE QUITE LIMITED. SHOULD WE CHANGE RUPTURE TO γM2 AND CHANGE TO UTS ? (3) Where no application standard exists for the form of construction involved, or the application standard does not define the relevant values of γM0, the value of γM0 should be taken from EN 1993-1-1. (4) Where the material has a nonlinear stress strain curve, the value of the characteristic yield strength fyk should be taken as the 0,2% proof stress. (5) The effect of fastener holes should be taken into account in accordance with 6.2.3 of EN 1993-1-1 for tension and 6.2.4 of EN 1993-1-1 for compression. For the tension check, the resistance should be based on the design value of the ultimate strength fud. 6.2.3 Stress limitation (1)P In every verification of this limit state, the design stresses shall satisfy the condition:

σeq,Ed ≤ feq,Rd ... (6.6) 6.3 Design by global numerical MNA or GMNA analysis (1)P The design plastic failure resistance shall be determined as a load factor Rpl,d applied to the design values FEd of the combination of actions for the relevant load case. (2)P The plastic reference resistance shall be determined using an MNA analysis as a load factor Rpl applied to the design values FEd of the combination of actions for the relevant load case. (3) The design values of the actions FEd should be determined as detailed in 6.1. The relevant load cases should be formed according to the required load combinations. (4) In an MNA or GMNA analysis based on the design yield strength fyd, the shell should be subject to the design values of the load cases detailed in (2), progressively increased by the load ratio R until the plastic failure limit condition is reached at Rplf. (5) Where an MNA analysis is used, the load ratio RMNA may be taken as the largest value attained in the analysis, with no effect of strain hardening. This load ratio is identified as the plastic reference resistance ratio Rpl in 8.8. (6) A GMNA analysis may not be used to establish the plastic reference resistance Rpl, which is required in 8.6 as part of the LBA-MNA design method. (7) Where a GMNA analysis is used, if the analysis predicts a maximum load followed by a descending path, the maximum value should be used to determine the load ratio RGMNA. Where a GMNA analysis does not predict a maximum load, but produces a progressively rising action-displacement relationship, the load ratio RGMNA should be taken as no larger than the value at which the maximum von Mises equivalent plastic true strain in the structure attains the value εmps = nmps (fyd / E). EDITORIAL NOTE: A CLEARER DEFINITION OF THIS STRAIN LIMIT SHOULD BE FOUND, PERHAPS RELATING IT TO SOME PROPORTION OF THE FRACTURE STRAIN

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NOTE The National Annex may choose the value of nmps. The value nmps = ( )66 /15ydf− is recommended where fyd is in MPa. THE ABOVE EXPRESSION, TAKEN FROM ECCS EDR5, REPLACED THE 2007 PUBLISHED EN 1993-1-6 NOTE (8) The characteristic plastic failure resistance Rplf,k should be taken as either RMNA or RGMNA according to the analysis that has been used. (9)P The design plastic failure resistance FR,plf,d shall be obtained from:

R,plf,k plf,k EdR,plf,d plf,d Ed

M0 M0γ γ⋅

= = = ⋅F R F

F R F … (6.7)

where: M0γ is the partial factor for resistance to plasticity according to 6.2.2. (10)P It shall be verified that:

, , , , or 1≤ = ⋅ ≥Ed R plf d plf d Ed plf dF F R F R ... (6.8) 6.4 Design using standard formulae (1) For each shell segment in the structure represented by a basic loading case as given by Annex A, the highest von Mises membrane stress σeq,Ed determined under the design values of the actions FEd should be limited to the stress resistance according to 6.2.2. (2) For each shell or plate segment in the structure represented by a basic load case as given in Annex B, the design value of the actions FEd should not exceed the resistance FRd based on the design yield strength fyd. (3) Where net section failure at a bolted joint is a design criterion, the design value of the actions FEd should be determined for each joint. Where the stress can be represented by a basic load case as given in Annex A, and where the resulting stress state involves only membrane stresses, FEd should not exceed the resistance FRd based on the design ultimate strength fud, see 6.2.2(5). 7 Cyclic plasticity limit state (LS2)

7.1 Design values of actions (1) Unless an improved definition is used, the values of the actions for each load case should be chosen as the characteristic values of those parts of the total actions that are expected to be applied and removed more than three times in the design life of the structure. (2) Where an elastic analysis or the formulae from Annex C are used, only the varying part of the actions between the extreme upper and lower values should be taken into account. (3) Where a materially nonlinear computer analysis is used, the varying part of the actions between the extreme upper and lower values should be considered to act in the presence of the design values of the coexistent permanent parts of the load.

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7.2 Stress design

7.2.1 Design values of stress range (1) The shell should be analysed using an LA or GNA analysis of the structure subject to the two extreme design values of the actions FEd. For each extreme load condition in the cyclic process, the stress components should be evaluated. From adjacent extremes in the cyclic process, the design values of the change in each stress component Δσx,Ed,i, Δσθ,Ed,i, Δτxθ,Ed,i on each shell surface (represented as i=1,2 for the inner and outer surfaces of the shell) and at any point in the structure should be determined. From these changes in stress, the design value of the von Mises equivalent stress range on the inner and outer surfaces should be found from: 2

iEd,θ,x 2

Edθ,iEd,θ,iEd,x,2

iEd,x,iEd,eq, 3 τΔσΔσΔσΔσΔσΔ ++⋅−= ... (7.1) (2) The design value of the stress range Δσeq,Ed should be taken as the largest change in the von Mises equivalent stress changes Δσeq,Ed,i, considering each shell surface in turn (i=1 and i=2 considered separately). (3) At a junction between shell segments, where the analysis models the intersection of the middle surfaces and ignores the finite size of the junction, the stress range may be taken at the first physical point in the shell segment (as opposed to the value calculated at the intersection of the two middle surfaces). NOTE This allowance is relevant where the stress changes very rapidly close to the junction. 7.2.2 Design values of resistance (1) The von Mises equivalent stress range resistance Δfeq,Rd should be determined from:

Δfeq,Rd = 2 fyd ... (7.2) in which fyd = fyk / γM4 ... (7.3) (2) The partial factor for resistance γM4 should be taken from the relevant application standard.

7.2.3 Stress range limitation (1)P In every verification of this limit state, the design stress range shall satisfy: Δσeq,Ed ≤ Δfeq,Rd ... (7.4)

7.3 Design by global numerical MNA or GMNA analysis

7.3.1 Design values of total accumulated plastic strain (1) Where a materially nonlinear global numerical analysis is used, the shell should be subject to the design values of the varying and permanent actions detailed in 7.1. An MNA analysis should be used for this purpose. EDITORIAL NOTE: THE PUBLISHED VERSION OF EN 1993-1-6 CONTAINED THE TWO NOTES: WE NEED TO CONFIRM THE DELETION OR AMENDMENT OF THESE NOTES, IF IT IS APPROPRIATE

WE SHOULD TRY TO ELIMINATE NATIONAL ANNEX VARIATIONS WHERE POSSIBLE NOTE The National Annex may give recommendations for a more refined analysis.

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(2) The total accumulated von Mises equivalent plastic strain εp,eq.Ed at the end of the design life of the structure should be assessed. (3) The total accumulated von Mises equivalent plastic strain may be determined using an analysis that models all cycles of loading during the design life. (4) Unless a more refined analysis is carried out, the total accumulated von Mises equivalent plastic strain εp,eq,Ed may be determined from: εp,eq,Ed = n Δεp,eq,Ed ... (7.4) where:

n is the number of cycles of loading in the design life of the structure; Δεp,eq,Ed is the largest increment in the von Mises equivalent plastic strain during one complete load cycle at any point in the structure, occurring after the third cycle. (5) It may be assumed that “at any point in the structure” means at any point not closer to a notch or local discontinuity than the thickest adjacent plate thickness.

7.3.2 Total accumulated plastic strain limitation (1) Unless a more sophisticated low cycle fatigue assessment is undertaken, the design value of the total accumulated von Mises equivalent plastic strain εp,eq,Ed should satisfy the condition: εp,eq,Ed ≤ np,eq (fyd / E) ... (7.5) where fyd is the design value of the yield stress according to 7.2.2. The value of np,eq should be taken as np,eq = 25.

7.4 Design using standard formulae (1) For each shell segment in the structure, represented by a basic loading case as given by Annex C, the highest von Mises equivalent stress range Δσeq,Ed considering both shell surfaces under the design values of the actions FEd should be determined using the relevant formulae given in Annex C. The further assessment procedure should be as detailed in 7.2. 8 Buckling limit state (LS3)

8.1 Design values of actions (1)P All relevant combinations of actions causing compressive membrane stresses or shear membrane stresses in the shell wall shall be taken into account. 8.2 Special definitions and symbols (1) Reference should be made to the special definitions of terms concerning buckling in 1.3.7. (2) In addition to the symbols defined in 1.4, additional symbols should be used in this 8 as set out in (3) and (4). (3) The stress resultant and stress quantities should be taken as follows:

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nx,Ed, σx,Ed are the design values of the acting buckling-relevant meridional membrane stress resultant and stress (positive when compression); nθ,Ed, σθ,Ed are the design values of the acting buckling-relevant circumferential membrane (hoop) stress resultant and stress (positive when compression); nxθ,Ed, τxθ,Ed are the design values of the acting buckling-relevant shear membrane stress resultant and stress. (4) Buckling resistance parameters for use in stress design: σx,Rcr is the meridional elastic critical buckling stress; σθ,Rcr is the circumferential elastic critical buckling stress; τxθ,Rcris the shear elastic critical buckling stress; σx,Rk is the meridional characteristic buckling stress; σθ,Rk is the circumferential characteristic buckling stress; τxθ,Rk is the shear characteristic buckling stress; σx,Rd is the meridional design buckling stress; σθ,Rd is the circumferential design buckling stress; τxθ,Rd is the shear design buckling stress. NOTE This is a special convention for shell design that differs from that detailed in EN1993-1-1. (5) The sign convention for use with LS3 should be taken as compression positive for meridional and circumferential stresses and stress resultants. 8.3 Buckling-relevant boundary conditions (1) For the buckling limit state, special attention should be paid to the boundary conditions which are relevant to the incremental displacements of buckling (as opposed to pre-buckling displacements). (2) Examples of situations in which the different boundary conditions of Table 5.1 arise are illustrated in Figure 8.1. These conditions apply only to the restraint of buckling displacements. NOTE Displacements induced during the principal loading (pre-buckling displacements) place a weak demand on strict adherence to the precise boundary condition. Thus, full fixity is rarely achieved and is not critical to the pre-buckling condition where the chief design consideration is the magnitude of induced stresses. But the ultimate limit state of buckling in shells places a much greater burden on strict attainment of the assumed boundary conditions. This consideration is critical to a safe design to LS3.

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roof

BC2f

BC2f

bottomplate

no anchoring

BC2f

BC2f

no anchoring

BC2f

BC1f

closely spacedanchors

a) tank without anchors b) silo without anchors c) tank with anchors open

no stiffening ring

BC3

BC1f

closely spacedanchors

BC1r

BC1r

welded from both sides

end plates with highbending stiffness

BC2f

BC2f d) open tank with anchors e) laboratory experiment f) section of long ring-stiffened cylinder Figure 8.1: Schematic examples of boundary conditions for limit state LS3

EDITORIAL NOTE: IT IS EXPECTED THAT SOME ADDITIONAL EXAMPLE SITUATIONS WILL BE ADDED TO FIGURE 8.1 IN THE SECOND DRAFT TO PROVIDE MORE CLARITY AND EASE OF USE FOR DESIGNERS 8.4 Buckling-relevant geometrical tolerances

8.4.1 General (1) Unless specific buckling-relevant geometrical tolerances are given in the relevant EN 1993 application parts, the following tolerance limits should be considered if design for LS3 is a requirement for the structure. NOTE 1 The characteristic buckling stresses determined hereafter include imperfections that are based on the amplitudes and forms of geometric tolerances that are expected to be met during execution. NOTE 2 The geometric tolerances given here are those that are known to have a large impact on the safety of the structure.

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(2) The tolerance requirements for LS3 are divided into four categories, as set out in 8.4.2 to 8.4.5. Not all of these tolerance requirements are appropriate to all shells, as the needs depend on the stress states that will develop within the shell. (3) Each tolerance is therefore identified below for its relevance to a particular structure. The tolerances that are required to meet the resistance requirements should be clearly communicated to the fabricator and the relevant authority as part of the design documentation. (4) The fabrication tolerance quality class should be chosen as Class A, Class B or Class C according to the tolerance definitions in 8.4.2, 8.4.3, 8.4.4 and 8.4.5. The description of each class relates only to the resistance evaluation and not to other considerations (e.g. aesthetics or functional performance). NOTE1 The tolerances defined here currently match those specified in the execution standard EN 1090, but not all are needed for every structure. These tolerance requirements relate to specific failure modes, so distinctions are needed between the categories and amplitudes according to the susceptibility of the structure to each potential failure mode in relation to its geometry, loading condition and boundary conditions. The requirements are consequently more carefully defined in this standard than in EN 1090. NOTE2 The relationship between the each imperfection amplitude and the evaluated resistance is vital to the resistance evaluation, so lower fabrication qualities may be acceptable for some categories of imperfection, whilst maintaining a resistance evaluation that relates to a higher fabrication quality class. NOTE3 The tolerances defined here are also required for the application of defined imperfection assumptions in GMNIA analyses given in 8.8. For application in 8.8, the adopted imperfection amplitude must be related to the manner in which the tolerance is specified in this sub-clause 8.4. DELETE THIS OLD CLAUSE (3) Each imperfection category should be classified separately: the lowest fabrication tolerance quality class obtained corresponding to a high tolerance, should then govern the entire design. EDITORIAL NOTE: THIS RULE IN THE PUBLISHED EN 1993-1-6 (2007) IS VERY EXPENSIVE FOR DESIGNS THAT INVOLVE SIMPLE LOADING CONDITIONS. UNDER THE MANDATE, WE MUST FIND A BETTER WAY TO TREAT THE PROBLEM. A FIRST ATTEMPT AT AN IMPROVED SPECIFICATION IS OFFERED HERE. (5) Each imperfection category should be classified separately: the quality class of the complete structure should relate to its attainment with respect to the tolerance category that is most relevant to the buckling mode being assessed. (6) The different tolerance categories may each be treated independently, and no interactions need normally be considered. (7) It should be established by representative sample checks on the completed structure that the measurements of the geometrical imperfections are within the geometrical tolerances stipulated in 8.4.2 to 8.4.5. (8) Sample imperfection measurements should be undertaken on the unloaded structure (except for self weight) and, where possible, with the operational boundary conditions. NOTE Where the measurement systems indicated in Figures 8.2, 8.3 and 8.4 are used, the strict interpretations given in sub-clauses 8.4.2, 8.4.3, 8.4.4 and 8.4.5 are relevant. Where a laser scan of the structure has been undertaken, the results can be interpreted using an notional measuring stick of the same lengths, but only 95% of the measurements are required to fall within the tolerance limits. EDITORIAL NOTE: THE ABOVE MIGHT SEEM A VERY LAX REQUIREMENT IN THE CONTEXT OF THE DANGER OF SHELL BUCKLING, BUT IT MUST ALSO BE RECOGNISED THAT THE ONLY TOLERANCES THAT REALLY MATTER ARE THOSE LOCATED IN THE PART OF THE STRUCTURE THAT IS MOST CRITICALLY LOADED WITH RESPECT TO BUCKLING, AND THAT, IN GENERAL, IT IS VERY DIFFICULT

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TO DESIGN A SHELL TO BE ONLY JUST STRONG ENOUGH TO SATISFY THE STRICT LIMIT STATE REQUIREMENTS SINCE THESE SHELLS ARE CONSTRUCTED FROM PLATES OF PRE-DETERMINED THICKNESS AND SIZE. MOREOVER, A SINGLE DIMPLE, EVEN AT THE CRITICAL LOCATION, IS NOT SUFFICIENT TO REDUCE THE BUCKLING RESISTANCE DOWN TO THE VALUE DEFINED AS THE CHARACTERISTIC VALUE IN THE RULES OF THIS STANDARD. (9) If the measurements of geometrical imperfections do not satisfy the geometrical tolerances stated in 8.4.2 to 8.4.4, any correction steps, such as straightening, should be investigated and decided individually. NOTE Before a decision is made in favour of straightening to reduce geometric imperfections, it should be noted that this can cause additional residual stresses. The degree to which the design buckling resistance is essential to the design resistance verification should also be considered. 8.4.2 Out-of-roundness tolerance (1) The out-of-roundness tolerance is important where the identified most critical mode in the shell arises under circumferential compression. This tolerance may be reduced by one fabrication tolerance class where other stress states dominate the critical mode. (2) The out-of-roundness should be assessed in terms of the parameter Ur (see Figure 8.2) given by:

max min

nomr

d dU

d

−= ... (8.1) where:

dmax is the maximum measured internal diameter, dmin is the minimum measured internal diameter, dnom is the nominal internal diameter. (3) The measured internal diameter from a given point should be taken as the largest distance across the shell from the point to any other internal point at the same axial coordinate. An appropriate number of diameters should be measured to identify the maximum and minimum values.

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dmax

dmindnom

dnom

dmin

dmax

a) flattening b) unsymmetrical Figure 8.2: Measurement of diameters for assessment of out-of-roundness (4) The out-of-roundness parameter Ur should satisfy the condition:

Ur ≤ Ur,max ... (8.2) where: Ur,max is the out-of-roundness tolerance parameter for the relevant fabrication tolerance quality class. The values of Ur,max are given in Table 8.1.

Table 8.1: Recommended values for out-of-roundness tolerance parameter Ur,max

Diameter range d [m] ≤ 0,50m 0,50m < d [m] < 1,25m 1,25m ≤ d [m]

Fabrication tolerance quality class

Description Recommended value of Ur,max

Class A Excellent 0,014 0,007 + 0,0093(1,25−d) 0,007 Class B High 0,020 0,010 + 0,0133(1,25−d) 0,010 Class C Normal 0,030 0,015 + 0,0200(1,25−d) 0,015

8.4.3 Unintended eccentricity tolerance (1) The unintended eccentricity tolerance is important in circumferential joints where the identified most critical mode in the shell arises under meridional compression. This tolerance may be reduced by one fabrication tolerance class where other stress states dominate the critical mode. (2) The unintended eccentricity tolerance in circumferential joints may be increased to designer-defined quantities where formal provision for joint eccentricity is made according to the provisions of Annex D.2.1.2 and D.3. (3) At joints in shell walls perpendicular to membrane compressive forces, the unintended eccentricity should be evaluated from the measurable total eccentricity etot and the intended offset eint from: ea = etot − eint ... (8.3)

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where: etot is the eccentricity between the middle surfaces of the joined plates, see Figure 8.3c; eint is the intended offset between the middle surfaces of the joined plates, see Figure 8.3b; ea is the unintended eccentricity between the middle surfaces of the joined plates. (4) The unintended eccentricity ea should be less than the maximum permitted unintended eccentricity

ea,max for the relevant fabrication tolerance quality class. The values of ea,max are given in Table 8.2. Table 8.2: Recommended values for maximum permitted unintended eccentricities

Fabrication tolerance quality class

Description Recommended values for maximum permitted unintended eccentricity ea,max Class A Excellent 2 mm Class B High 3 mm Class C Normal 4 mm

EDITORIAL NOTE: THESE VALUES INVOLVE VERY STRICT TOLERANCES ON LARGE STRUCTURES COMPOSED OF THICK PLATES (e.g. 25mm plate on a 25m diameter structure). THEY SHOULD BE RECONSIDERED UNDER THE MANDATE (5) The unintended eccentricity ea should also be assessed in terms of the unintended eccentricity parameter Ue given by:

or a ae e

av

e eU U

t t= = ... (8.4)

where: tav is the mean thickness of the thinner and thicker plates at the joint.

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t

t

ea

perfect jointgeometry

eint

tmax

tmin

perfect jointgeometry

etot

tmax

t min

perfect jointgeometry

ea = etot – eint

a) unintended eccentricity when there is no change of plate thickness b) intended offset at a change of plate thickness without unintended eccentricity

c) total eccentricity (unintended plus intended) at change of plate thickness Figure 8.3: Unintended eccentricity and intended offset at a joint (6) The unintended eccentricity parameter Ue should satisfy the condition:

Ue ≤ Ue,max ... (8.5) where: Ue,max is the unintended eccentricity tolerance parameter for the relevant fabrication tolerance quality class. The values of Ue,max are given in Table 8.3.

Table 8.3: Recommended values for unintended eccentricity tolerances

Fabrication tolerance quality class Description Recommended value of Ue,max Class A Excellent 0,14 Class B High 0,20 Class C Normal 0,30 NOTE Intended offsets are treated within D.2.1.2 and lapped joints are treated within D.3. These two cases are not treated as imperfections within this standard.

EDITORIAL NOTE: THESE TOLERANCES COULD BE REPLACED OR AMENDED IN THE LIGHT OF EVIDENCE ON THE RESISTANCE REDUCTION ASSOCIATED WITH MISALIGNMENT OR UNINTENDED ECCENTRICITIES. SOME RELAXATION ON TOLERANCE SHOULD BE INTRODUCED WHERE THE DESIGN INCLUDES A RESISTANCE REDUCTION FOR SPECIFIC MAGNITUDES OF UNINTENDED ECCENTRICITY

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8.4.4 Dimple tolerances (1) The dimple tolerance is important for compressive stresses in all directions, but the tolerance measurement relevant to each direction of stress is different. The tolerance requirement may be reduced by one fabrication tolerance class when the limitations defined in (3) show that the specific measurement relates to a stress state that does not dominate the critical mode. (2) A dimple measurement gauge should be used in every position (see Figure 8.4) in both the meridional and circumferential directions. The meridional gauge should be straight, but the gauge for measurements in the circumferential direction should have a curvature equal to the intended radius of curvature r of the middle surface of the shell. (3) The depth Δw0 of initial dimples in the shell wall should be measured using gauges of length g which should be taken as follows: a) Wherever meridional compressive stresses are present, including across welds, measurements should be made in both the meridional and circumferential directions, using the gauge of length gx given by:

gx = 4 rt ... (8.6) b) Where circumferential compressive stresses or shear stresses occur, circumferential direction measurements should be made using the gauge of length gθ given by: gθ = 2,3 (2 rt)0,25, but gθ ≤ r ... (8.7) where: is the meridional length of the shell segment between boundaries that are either BC1 or BC2. c) Additionally, where the shell radius to thickness ratio is less than r/t = 400, measurements should be made across welds in both the circumferential and meridional directions, using the gauge length gw: gw = 25 t or gw = 25 tmin , but with gw ≤ 500mm ... (8.8) where:

tmin is the thickness of the thinnest plate at the weld. EDITORIAL NOTE: THIS GAUGE LENGTH, AS DEFINED IN THE PUBLISHED EN 1993-1-6 (2007) PRESENTS DIFFICULTIES OF APPLICATION IN RELATING THE IMPERFECTION AMPLITUDE TO THE RESISTANCE, AS THE LITERATURE ON IMPERFECTION SENSITIVITY UNDER EXTERNAL PRESSURE DOES NOT APPEAR TO ADDRESS IMPERFECTION AMPLITUDES DEFINED IN THIS WAY. IT ALSO PRESENTS DIFFICULTIES IN THE APPLICATION OF GMNIA ANALYSES DEFINED IN 8.8. THE GAUGE LENGTH AND APPROPRIATE LITERATURE SHOULD BE RECONSIDERED BY THE PROJECT TEAM AND WORKING GROUP.

EDITORIAL NOTE: THIS SHORT GAUGE ACROSS A WELD HAS A SIGNIFICANT IMPACT ONLY ON THICKER SHELLS. A VERY CONSERVATIVE ESTIMATE OF THE RADIUS TO THICKNESS RATIO AT WHICH IT COULD BE APPLICABLE IS r/t = 400, SO THE EXCLUSION HERE IS ADDED FOR THINNER

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SHELLS BOTH TO REDUCE THE MEASURING EFFORT AND TO PREVENT VERY LOCAL DEEP IMPERFECTIONS FROM BEING DEEMED TO BE DAMAGING TO THE SHELL BUCKLING RESISTANCE.

(4) The depth of initial dimples should be assessed in terms of the dimple parameters U0x, U0θ, U0w given by: 0 0x x gxU w= Δ 0 0 gU wθ θ θ= Δ 0 0w w gwU w= Δ ... (8.9) (5) The value of the dimple parameters U0x, U0θ, U0w should satisfy the conditions: 0 0,maxxU U≤

0 0,maxU Uθ ≤ 0 0,maxwU U≤ ... (8.10) where:

U0,max is the dimple tolerance parameter for the relevant fabrication tolerance quality class. The values of U0,max are given in Table 8.4. Table 8.4: Recommended values for dimple tolerance parameter U0,max

Fabrication tolerance quality class Description Recommended value of U0.max Class A Excellent 0,006 Class B High 0,010 Class C Normal 0,016

Δwox inward

gx

r

t

Δwox

gx

t

a) Measurement on a meridian (see 8.4.4(2)a) b) First measurement on a circumferential circle (see 8.4.4(2)a)

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Δwox inward

gx

r

t

weld

Δwoθ

gθ

t

c) First measurement on a meridian across a weld (see 8.4.4(2)a) d) Second measurement on circumferential circle (see 8.4.4(2)b)

gw

weld

Δwow

t

Δwox or Δwoθ or Δwow

gx or gθ or gw

t weld

e) Second measurement across a weld with special gauge (see 8.4.4(2)c) f) Measurements on circumferential circle across weld (see 8.4.4(2)c)

Figure 8.4: Measurement of depths Δw0 of initial dimples

8.4.5 Interface flatness tolerance (1) The interface flatness tolerance is only important for meridional compressive stresses at the base of a shell. The tolerance requirement may be ignored when meridional compression does not play a significant role in the assessed critical mode of buckling. (2) Where another structure continuously supports a shell (such as a foundation), its deviation from flatness at the interface should not include a local slope in the circumferential direction greater than βθmax. The value of βθmax should be taken as βθmax = 0,1 % = 0,001 radians. EDITORIAL NOTE: THE FOLLOWING IS THE AUSTRIAN RECOMMENDATION, WHICH IS PROPOSED TO BE ADOPTED AS A SIGNIFICANT IMPROVEMENT ON THE CURRENT RULE

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(3) The measured value of βθ should be found as βθ = [0.8 / √(rt)] u0 in which u0 is the separation of the shell from the foundation measured using a measuring gauge length of ℓg in the circumferential direction of the support circle. The length ℓg should be ℓg = 4√(rt): (4) The value of βθmax should be obtained from u0max as u0max = 0.02 √(rt) but ≤ 15 mm where r and t are the radius and wall thickness of the lowest course of the cylinder. (5) This tolerance measurement is only relevant to limit the deviations of the foundation of a shell structure that is susceptible to buckling under axial compression. 8.5 Stress design

8.5.1 Design values of stresses (1) The rules given here apply to a cylindrical shell segment that may be part of a larger structure. The shell length is taken as the length of the segment alone. (2) The design values of stresses for the shell segment σx,Ed, σθ,Ed and τxθ,Ed should be taken as the key values of these three basic compressive and shear membrane stresses obtained from a linear shell analysis (LA). Under purely axisymmetric conditions of loading and support, and in other simple load cases, membrane theory may generally be used. (3) The key values of membrane stresses should be taken as the maximum value of each stress component at a single axial coordinate in the shell segment, unless specific provisions are given in Annex D or in the relevant application part of EN 1993. NOTE In some cases (e.g. stepped walls under circumferential compression, see Annex D.2.3), the key values of membrane stresses are fictitious and larger than the real maximum values. (4) For basic loading cases the three membrane stress components in the shell segment may be taken from Annex A or Annex C. 8.5.2 Design resistance (buckling strength) (1) The elastic critical buckling stress for each component of membrane stress in the shell segment should be obtained as σx,Rcr, σθ,Rcr and τxθ,Rcr from the relevant formulae in Annex D. NOTE Where the membrane stress state effectively only involves one of these three basic components, it is sufficient to address that alone, without conducting the following triple calculation for possible interactions between the different components. (2) Where no appropriate formulae are given in Annex D, the elastic critical buckling condition may be extracted from a numerical LBA analysis of the shell segment under the buckling-relevant combinations of actions defined in 8.1. For the conditions that this analysis should satisfy, see 8.7.2 (5) and (6). NOTE Where an LBA analysis is used, and no single stress component is clearly dominant in the critical state, it is more satisfactory to follow the LBA-MNA design process of 8.7.2. (3) The relative slenderness of the shell segment λ for each stress component should be separately determined from:

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Rcrx,ykx /σλ f= , Rcr,yk / θθ σλ f= , ( ) Rcr,xyk /3/ θτ τλ f= ... (8.12) (4) The elastic-plastic buckling reduction factor for each component χx, χθ and χτ should be separately determined as a function of the corresponding relative slenderness of the shell segment λ = λx , θλ , or τλ from: ( ) ( )1/ h0h −−= χλλχχ when 0λλ ≤ ... (8.13)

χ = 1 − β η

λλλλ

−−

0p

0 when p0 λλλ << ... (8.14) χ =

2λα when λλ ≤p ... (8.15)

where the relevant capacity parameters for each separate component of the membrane stress are: α the elastic imperfection reduction factor for that component; β the plastic range factor for that component; η the interaction exponent for that component;

0λ the squash limit relative slenderness for that component. χh the hardening limit for that component. (5) The value of the plastic limit relative slenderness pλ for each component λpx , θλp and τλp , should be determined, using the relevant values of α and β from:

βαλ−

=1p ... (8.16) (6) The value of η is sometimes defined as a single value, but is otherwise defined by two limiting values

η0 and ηp, with η determined as ( )λ η η λ η λ η

ηλ λ

− + −=

−

p o p o o p

p o

... (8.17) where:

η0 is the value of η at 0λ λ= for that component;; ηp is the value of η at λ λ= p for that component. NOTE 1 The values of these capacity parameters for each component should be taken from Annex D.

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NOTE 2 Formula (8.15) describes the elastic buckling stress, accounting for geometric imperfections. In this case, where the behaviour is entirely elastic, the characteristic buckling stresses may alternatively be determined directly from σx,Rk = αx σx,Rcr, σθ,Rk = αθ σθ,Rcr, and τxθ,Rk = ατ τxθ,Rcr. (7) The characteristic buckling stress for each component should be obtained by multiplying the characteristic yield strength by the elastic-plastic buckling reduction factors χ: σx,Rk = χx fyk , σθ,Rk = χθ fyk , τxθ,Rk = χτ fyk / 3 ... (8.12) (8) The buckling resistance should be derived from the values for the three basic membrane stress components as defined in 1.3.7. The design buckling stress for each component should be obtained from: σx,Rd = σx,Rk/γM1, σθ,Rd = σθ,Rk/γM1, τxθ,Rd = τxθ,Rk/γM1 ... (8.11) (9) The partial factor for resistance to buckling γM1 should be taken from the relevant application standard. (10) Where no application standard exists for the form of construction involved, or the application standard does not define the relevant values of γM1, the value of γM1 should not be taken as smaller than γM1 = 1,2. NOTE The value γM1 =1,2 should be used. The partial factor γM1 has been raised from 1,1 to 1,2 because shell buckling is a phenomenon that may occur without warning, leading to a serious sudden loss of load-bearing capacity.

EDITORIAL NOTE: THE FOLLOWING WAS A PROPOSAL FROM CROATIA. THE PROJECT TEAM CHOSE TO ADOPT 1,2 FOR ITS SIMPLICITY AND EASE OF USE The value γM1 =1,2 should be used. The partial factor γM1 should be increased by factor 1,1 if there is a risk of losing load-bearing capacity without any warning. EDITORIAL NOTE: THE FOLLOWING IS A PROPOSAL FROM DENMARK. THE PROJECT TEAM FELT THAT IT WAS BEYOND THE SCOPE OF THE STANDARD AND IN CONFLICT WITH THE NEED FOR EASE OF USE (RELIABILITY CLASSES ARE NOT USED ELSEWHERE IN THIS STANDARD). The following formulae for γMi are used, including the factor (γ0) for the partial factors for resistances (cf. Danish National Annex to EN 1990, Table A1.2(B+C)): γM1 = 1,2·γ0·γ3 γM2 = 1,35·γ0·γ3 Where there is a risk that no warning of failure is given, the partial factor should be multiplied by a factor 1,1. The factor γ0 takes account of the combination of actions, cf. National Annex to EN 1990, Table A1.2(B+C). Limit state STR/GEO STR Combination of actions 1 2 3 4 5 γ0 1,0 1,0 KFI KFI 1,2·KFI The factor γ3 takes account of the level of checking of the product. The reduced level of checking is not used. Extended level of checking: γ3 = 0,95 Normal level of checking: γ3 = 1,00 The partial factors are determined in accordance with the National Annex to EN 1990, Annex F, where γM = γ1 γ2 γ3 γ4.

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γ1 takes into account the type of failure γ2 takes into account the uncertainty related to the design model γ3 takes into account the scope of checking γ4 takes into account the variation of the strength parameter for resistance. When determining γ1, the following types of failure have been assumed: γM1: Warning of failure without residual resistance γM2: No warning of failure For accidental and seismic design situations the following values are used: γM1 = 1,0 γM2 = 1,0 8.5.3 Stress limitation (buckling strength verification) (1) Although buckling is not a purely stress-initiated failure phenomenon, the buckling limit state, within this sub-clause, should be represented by limiting the design values of membrane stresses. The influence of bending effects on the buckling strength may be neglected provided they arise as a result of meeting boundary compatibility requirements. In the case of bending stresses from local loads or from thermal gradients, special consideration should be given. (2) Depending on the loading and stressing situation, one or more of the following checks for the key values of single membrane stress components should be carried out:

σx,Ed ≤ σx,Rd, σθ,Ed ≤ σθ,Rd, τxθ,Ed ≤ τxθ,Rd ... (8.19) (3) If more than one of the three buckling-relevant membrane stress components are present under the actions under consideration, the following interaction check for the combined membrane stress state should be carried out: , , , , ,

, , , , ,1

xk k kx Ed x Ed Ed Ed x Ed

ix Rd x Rd Rd Rd x Rd

kθ τ

θ θ θ

θ θ θ

σ σ σ σ τσ σ σ σ τ

− + + ≤

... (8.20) where σx,Ed, σθ,Ed and τxθ,Ed are the interaction-relevant groups of the significant values of compressive and shear membrane stresses at a single location in the shell segment and the values of the buckling interaction parameters kx, kθ , kτ and ki are given in Annex D.1.6. (4) Where σx,Ed or σθ,Ed is tensile, its value should be taken as zero in Formula (8.20). NOTE For axially compressed cylinders with internal pressure (leading to circumferential tension) special provisions are made in Annex D. The resulting value of σx,Rd accounts for both the strengthening effect of internal pressure on the elastic buckling resistance and the weakening effect of the elastic-plastic elephant’s foot phenomenon (Formula D.43). If the tensile σθ,Ed is then taken as zero in Formula (8.20), the buckling strength is accurately represented. (5) The locations and values of each of the buckling-relevant membrane stresses to be used together in combination in Formula (8.20) are defined in Annex D.1.6.

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(6) Where the shell buckling condition is not included in Annex D, the buckling interaction parameters may be conservatively estimated using: kx = 1,0 + χx2 …(8.21) kθ = 1,0 + χθ

2 …(8.22) kτ = 1,5 + 0,5 χτ

2 …(8.23) ki = (χx χθ)2 …(8.24) NOTE These rules may sometimes be very conservative, but they include the two limiting cases which are well established as safe for a wide range of cases: a) in very thin shells, the interaction between σx and σθ is approximately linear; and b) in very thick shells, the interaction becomes that of von Mises.

8.6 Design using reference resistances

8.6.1 Principle (1) Because buckling is not controlled by a single membrane stress at a single location, but depends on the stress state throughout a zone large enough for a buckle to form and which can also include significant plasticity, the buckling limit state, within this sub-clause, is represented by the design value of the actions, augmented to the point of buckling and applied to the specific defined conditions. (2) The influence of membrane and bending effects, and of plasticity and geometric imperfections, are all included by applying the defined values of the capacity parameters to the two reference resistances. NOTE The background to the method of Reference Resistance Design may be found in Bibliography 4. 8.6.2 Design value of actions (1) The design values of actions should be taken as in 8.1(1). 8.6.3 Design value of resistance (1) The design buckling resistance should be determined from the reference elastic critical resistance Rcr and the reference plastic resistance Rpl for the geometry and load case, together with the capacity parameters αs, βs, ηs, λos and χhs as defined in Annex E. (2) The plastic reference resistance Rpl is defined in Annexes B and E for specific geometries, load cases, and boundary conditions and may only be used for these specific cases. (3) The value of Rpl for a given load case, involving as appropriate the loading PnEd, PxEd, pnEd, FEd, etc. should be obtained as follows. Where there is more than one loading component, the ratios between different loading components should be retained in fixed proportions, with one nominated as the leading load FEd. The plastic collapse load should then be determined for the magnitude of the leading load as FRpl. The plastic reference resistance should then be found as the ratio

Rplpl EdFR F= (8.25)

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(4) The elastic critical buckling load FRcr is defined in Annex E for specific geometries, load cases, and boundary conditions and may only be used for these specific cases. (5) The elastic critical reference resistance Rcr for the same given load case should be obtained from the elastic critical buckling load for the magnitude of the leading load FRcr. The elastic critical reference resistance should then be found as the ratio Rcrcr EdFR F= (8.26)

(6) The relative slenderness of the shell should be found as λ = pl

scr

R

R (8.27)

(7) The value of the plastic limit relative slenderness ,s pλ should be determined from: ,

1

αλβ

=−

ss p

s (8.28)

, ,α α α=s s I s G ... (8.29) where: αs,G is the geometric reduction factor for the complete shell; αs,I is the imperfection reduction factor for the complete shell; βs is the plastic range factor for the complete shell; (8) The elastic-plastic buckling reduction factor χs should be determined as a function of the relative slenderness of the shell sλ from:

( ) ( ), , , 1χ χ λ λ χ= − −s s h s s o s h when ,λ λ≤s s o (8.30)

,

, ,

1

ηλ λ

χ βλ λ

−= − −

s

s s os s

s p s o when , ,λ λ λ< <s o s s p (8.31)

2χ α λ=s s s when ,λ λ≤s p s (8.32) with: ( ), , . . , ,

, ,

λ η η λ η λ ηη

λ λ

− + −=

−

s s p s o s p s o s o s ps

s p s o

... (8.33)

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where: ,λs o is the squash limit relative slenderness for the complete shell;

χs,h is the hardening limit for the complete shell. ηs,o is the value of ηs at ,λ λ=s s o for the complete shell; ηs,p is the value of ηs at ,λ λ=s s p for the complete shell. NOTE 1 The values of these capacity parameters should be taken from Annex E. NOTE 2 Formula (8.32) describes the elastic buckling condition, accounting for geometric nonlinearity and geometric imperfections. In this case, where the behaviour is entirely elastic, the characteristic buckling resistance may alternatively be determined directly from Rk = αs Rcr. (9) The characteristic resistance of the shell should be determined from: k s plR Rχ= (8.34) (10) The design resistance of the shell should then be determined from:

1/d k MR R γ= (8.35) where: γM1 is the partial factor for resistance to buckling according to 8.5.2 (9) and (10).

8.6.4 Buckling strength verification (1) The following verification of the resistance of the shell structure to the defined loading should be undertaken: 1dR ≥ (8.36) 8.7 Design by global numerical analysis using MNA and LBA analyses

8.7.1 Design value of actions (1) The design values of actions should be taken as in 8.1 (1). 8.7.2 Design value of resistance (1) The design buckling resistance should be determined from the amplification factor Rd applied to the design values FEd of the combination of actions for the relevant load case. (2) The design buckling resistance Rd d EdF R F= ⋅ should be obtained from the plastic reference resistance , = ⋅R pl pl EdF R F and the elastic critical buckling resistance , = ⋅R cr cr EdF R F , combining these to find the characteristic buckling resistance = ⋅Rk k EdF R F . The partial factor 1Mγ should then be used to obtain the design resistance.

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(3) The plastic reference resistance Rpl (see Figure 8.5) should be obtained using a materially non-linear analysis (MNA) as the plastic limit load under the applied combination of actions. This load ratio Rpl may be taken as the largest value attained in the analysis, using an ideal elastic-plastic material law. NOTE Improved methods for accurate evaluation of the plastic limit load are available, which permit progressively better estimates of the true value of Rpl to be obtained without the analysis having to approach the plateau closely (see Bibliography 3). (4) When conducting the MNA calculation, the complete shell should be considered as an assembly of individual segments, with each segment represented by a simple shell geometry (e.g. cylinder, cone, sphere, toroid etc.). The plastic collapse mechanism should be examined to determine the location and form of the collapse mode. The segment that exhibits the largest plastic displacements should be taken as the critical segment for plastic collapse. EDITORIAL NOTE IT IS UNCLEAR WHETHER THE ABOVE IS SUFFICIENTLY GENERAL, AND WHETHER IT IS ONLY A GUIDE RATHER THAN A RULE. MORE DISCUSSION OF THIS PROVISION IS NEEDED. NOTE It is possible that the location of the plastic collapse mode found in this way may not be the same as the location of the most imperfect buckling mode. However, if the plastic reference load found in the above manner is adopted in the complete shell buckling assessment, the conservatism of the outcome is secure when this result is combined with a buckling mode in a different part of the shell (see Bibliography 2). NOTE Where the shell has only one segment, many of the issues identified in the following paragraphs do not apply and the complete process can be undertaken without the checks described in (4), (8), (9), (110 and (12).

Rpl small displacement theory plastic limit load

Rcr from linear elastic bifurcation

LA

Load factor on design

actions R

Deformation

MNA

LBA

Rpl estimated from LA

Figure 8.5: Definition of plastic reference resistance ratio Rpl and critical buckling resistance ratio

Rcr derived from global MNA and LBA analyses (5) Where it is not possible to undertake a materially non-linear analysis (MNA), the plastic reference resistance ratio Rpl may be conservatively estimated from linear shell analysis (LA) conducted using the design values of the applied combination of actions using the following procedure. The evaluated membrane stress resultants nx,Ed, nθ,Ed and nxθ,Ed at any point in the shell should be used to estimate the plastic reference resistance from: 2 2 2, , , , ,3

ykpl

x Ed x Ed Ed Ed x Ed

t fR

n n n n nθ θ θ

⋅=

− ⋅ + + ... (8.37)

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The lowest value of plastic resistance ratio so calculated should be taken as the estimate of the plastic reference resistance ratio Rpl. NOTE A safe estimate of Rpl can usually be obtained from an LA analysis as follows. The three points in the shell are identified where each of the three buckling-relevant membrane stress resultants attains its highest value. Formula (8.37) is then applied separately to the stress state at each of these three points to obtain three separate estimates of Rpl. The relevant estimated value of Rpl can then be found as the lowest of these three estimates. (6) The elastic critical buckling resistance ratio Rcr should be determined from an eigenvalue analysis (LBA) applied to the linear elastic calculated stress state in the geometrically perfect shell (LA) under the design values of the load combination. The lowest eigenvalue (bifurcation load factor) should be taken as the elastic critical buckling resistance ratio Rcr (see Figure 8.5), but subject to the fuller process given in (8) and (9) when the shell consists of multiple segments. (7) It should be verified that the eigenvalue algorithm that is used is reliable at finding the eigenmode that leads to the lowest eigenvalue. In cases of doubt, neighbouring eigenvalues and their eigenmodes should be calculated to obtain a fuller insight into the bifurcation behaviour of the shell. The analysis should be carried out using software that has been authenticated against benchmark cases with physically similar buckling characteristics. (8) When conducting an LBA calculation, the complete shell should be considered as an assembly of individual segments, with each segment represented by a simple shell geometry. (9) A sufficient number of eigenvalues Rcr should be calculated (progressively rising from the lowest eigenvalue) and their corresponding eigenmodes examined to ensure that a buckling mode in each segment in which compressive stresses or shear are present has been found at least once. However, when the highest of these eigenvalues exceeds 10 times the smallest eigenvalue, it is not necessary to pursue the calculation further. (10) For each segment, the local value for the elastic imperfection reduction factor αs should be determined using relevant formulae given in Annexes D and E. The corresponding assessed elastic imperfect buckling resistance αsRcr should be found for each segment. The segment that has the lowest value of αsRcr should be deemed to be the critical segment for elastic buckling. This elastic critical buckling resistance Rcr and elastic imperfection reduction factor αs should be taken as the values for the complete shell. NOTE A multi-strake wind turbine support tower may be an assembly of individual cylindrical and truncated conical shell segments, while a pressure vessel may be an assembly of cylindrical and spherical shell segments. In some cases, the procedure defined in (8) and (9) may require very many eigenmodes to be calculated, depending on the complexity of the structure, since a shell segment that is not the most critical may still exhibit a large number of modes with similar eigenvalues. . (11) The segment that is deemed critical for elastic buckling may be chosen as the only one on which to base the design, since this leads to a conservative outcome for the whole system. (12) Alternatively, a better assessment may be made by using two separate sets of capacity parameters λs,0, αs, βs, ηs and χs,h for the segment identified as critical for plastic collapse in (3) and the segment identified as critical for elastic buckling in (5), using a consistent treatment for each individually. These two sets of factors may then be used to provide two separate estimates for χs. The lower estimate for χs should be retained in the design evaluation. If the same segment is critical for both cases, only one estimate is obtained and is sufficient. (13) The relative slenderness λ s for the complete shell should be determined from:

R,pl R,cr pl cr/ /λ = =s F F R R ... (8.38)

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EDITORIAL NOTE: THE NOTATION USING A SUBSCRIPT ov SEEMED RATHER CLUMSY IN THE PUBLISHED STANDARD AND HAVE BEEN REPLACED BY THE SIMPLER NOTATION s FOR EITHER “SYSTEM” OR “SHELL” . (8) The complete shell elastic-plastic buckling reduction factor χs should be determined as

( )s , s s,0 s s s, , , , ,χ χ λ λ α β η= s hf using 8.6.3 (7) and (8), to obtain the lowest value for the complete structural system. Here αs is the complete shell elastic imperfection reduction factor, βs is the complete shell plastic range factor, ηs is the complete shell interaction exponent, s,0λ is the complete shell squash limit relative slenderness and χs,h is the complete shell hardening limit. (9) The evaluation of the factors s,0λ , s,0λ , αs, βs, ηs and χs,h should take account of the imperfection sensitivity, geometric nonlinearity, material hardening and other aspects of the particular shell buckling case. Conservative values for these capacity parameters should be determined by comparison with known shell buckling cases (see Annex D) that have similar buckling modes, similar imperfection sensitivity, similar geometric nonlinearity, similar yielding sensitivity and similar post-buckling behaviour. The value of αs should also take account of the appropriate fabrication tolerance quality class. NOTE Care should be taken in choosing an appropriate value of αs when this approach is used on shell geometries and loading cases where snap-through buckling may occur. Such cases include conical and spherical caps and domes under external pressure or on supports that can displace radially. The appropriate value of αs should also be chosen with care when the shell geometry and load case produce conditions that are highly sensitive to changes of geometry, such as at unstiffened junctions between cylindrical and conical shell segments under meridional compressive loads (e.g. in chimneys). The commonly reported elastic shell buckling loads for these special cases are normally based on geometrically nonlinear analysis applied to a perfect or imperfect geometry, which predicts the snap-through buckling load. By contrast, the methodology used here adopts the linear bifurcation load as the reference elastic critical buckling resistance, and this is often much higher than the snap-through load. The design calculation must account for these two sources of reduced resistance by an appropriate choice of the complete shell elastic imperfection reduction factor αs. This choice must include the effect of both the geometric nonlinearity (that can lead to snap-through) and the additional strength reduction caused by geometric imperfections. (10) If the provisions of (9) cannot be achieved beyond reasonable doubt, appropriate tests should be carried out, see EN 1990, Annex D. (11) If specific values for the capacity parameters αs, βs, ηs, s,0λ and χs,h are not obtainable according to (9) or (10), the values for an axially compressed unstiffened cylinder may be adopted, see D.1.2.2. Where snap-through is known to be a possibility, appropriate further reductions in αs should be considered. (12) The characteristic buckling resistance ratio Rk should be obtained from: Rk = χs Rpl ... (8.39) where:

Rpl is the plastic reference resistance ratio.

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(13) The design buckling resistance ratio Rd should be obtained from: Rd = Rk/γM1 ... (8.40) where:

γM1 is the partial factor for resistance to buckling according to 8.5.2 (9) and (10). 8.7.3 Buckling strength verification (1) It should be verified that:

or 1Ed Rd d Ed dF F R F R≤ = ⋅ ≥ ... (8.41) 8.8 Design by global numerical analysis using GMNIA analysis

8.8.1 Design values of actions (1) The design values of actions should be taken as in 8.1 (1). 8.8.2 Design value of resistance (1) The design buckling resistance should be determined as a load factor R applied to the design values FEd of the combination of actions for the relevant load case. (2) The characteristic buckling resistance ratio Rk should be found from the imperfect elastic-plastic buckling resistance ratio RGMNIA, adjusted by the calibration factor kGMNIA where necessary. The design buckling resistance ratio Rd should then be found using the partial factor γM1. (3) To determine the imperfect elastic-plastic buckling resistance ratio RGMNIA, a GMNIA analysis of the geometrically imperfect shell under the applied combination of actions should be carried out, accompanied by an eigenvalue analysis to detect possible bifurcations in the load path. NOTE Where plasticity has a significant effect on the buckling resistance, care should be taken to ensure that the adopted imperfection mode induces some pre-buckling shear strains, because the shear modulus is very sensitive to small plastic shear strains. In certain shell buckling problems (e.g. shear buckling of annular plates), if this effect is omitted, the eigenvalue analysis may give a considerable overestimate of the elastic-plastic buckling resistance. (4) An LBA analysis should first be performed on the perfect structure to determine the elastic critical buckling resistance ratio Rcr of the perfect shell. (5) An MNA analysis, adopting a perfect elastic-plastic material representation, should next be performed on the perfect structure to determine the perfect plastic reference resistance ratio Rpl. (6) The LBA and MNA resistance ratios should then be used to establish the complete shell relative slenderness λ s according to Formula 8.38. (7) A GMNA analysis should then be performed on the perfect structure to determine the perfect elastic-plastic buckling resistance ratio RGMNA. This resistance ratio should be used later to verify that the effect of the chosen geometric imperfections has a sufficiently deleterious effect to give confidence that the lowest resistance has been obtained. The GMNA analysis should be carried out under the applied combination of actions, accompanied by an eigenvalue analysis to detect possible bifurcations in the load path. (8) The imperfect elastic-plastic buckling resistance ratio RGMNIA should be found as the lowest load factor R obtained from the three following criteria C1, C2 and C3, see Figure 8.6:

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• Criterion C1: The maximum load factor on the load-deformation-curve (limit load); • Criterion C2: The bifurcation load factor, where this occurs during the loading path before reaching a limit point on the load-deformation-curve; • Criterion C3: The largest tolerable deformation, where this occurs during the loading path before reaching a bifurcation load or a limit load. (9) The largest tolerable deformation should be assessed relative to the conditions of the individual structure. If no other value is available, the largest tolerable deformation may be deemed to have been reached when the greatest local rotation of the shell surface (slope of the surface relative to its original geometry) attains the value β. The value of β should be taken as β = 0,1 radians. EDITORIAL NOTE: THIS METHOD OF ASSESSING A LARGEST TOLERABLE DEFORMATION SHOULD BE RECONSIDERED BY THE PROJECT TEAM AND WORKING GROUP.

Deformation

Load factor on design

actions R

RGMNIA is the lowest of these alternative measures

largest tolerable deformation

C1

C2

C2

C3

C4 First yield safe estimate

Figure 8.6: Definition of buckling resistance from global GMNIA analysis (10) Using the Criterion C4, a conservative assessment of the imperfect elastic-plastic buckling resistance ratio RGMNIA may be obtained using an elastic GNIA analysis of the geometrically imperfect shell under the applied combination of actions. The lowest load factor RGNIA.y should be obtained according to Criterion C4. Criterion C4: The load factor RGNIA.y at which the equivalent stress at the most highly stressed point on the shell surface reaches the design value of the yield stress fyd = fyk/γM0, see Figure 8.6. NOTE It should be noted that GMNA, GMNIA and GNIA analyses must always be undertaken with eigenvalue checks throughout the load path to ensure that any possible bifurcation is detected. (11) In formulating the GMNIA (or GNIA) analysis, appropriate allowances should be considered for incorporation into the model to cover the effects of imperfections that cannot be avoided in practice, including: a) geometric imperfections, such as:

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− deviations from the nominal geometric shape of the middle surface (pre-deformations, out-of-roundness); − irregularities at and near welds (minor eccentricities, shrinkage depressions, rolling curvature errors); − deviations from nominal thickness; − lack of evenness of supports. b) material imperfections, such as: − residual stresses caused by rolling, pressing, welding, straightening etc.; − inhomogeneities and anisotropies. NOTE Further possible negative influences on the imperfect elastic-plastic buckling resistance ratio RGMNIA, such as ground settlements or flexibilities of connections or supports, are not classed as imperfections in the sense of these provisions. (12) Imperfections should be allowed for in the GMNIA analysis by including appropriate additional quantities in the analytical model for the numerical computation. (13) The imperfections should generally be introduced by means of equivalent geometric imperfections in the form of initial shape deviations perpendicular to the middle surface of the perfect shell, unless a better technique is used. The middle surface of the geometrically imperfect shell should be obtained by superposition of the equivalent geometric imperfections on the perfect shell geometry. (14) The pattern of the equivalent geometric imperfections should be chosen in such a form that, under the defined loading condition, it has the most unfavourable effect on the imperfect elastic-plastic buckling resistance ratio RGMNIA of the shell. (15) The information given in 8.4 on the relevance of different forms of imperfection to stress states in the shell should be used to inform the choice of imperfection form. (16) If the most unfavourable pattern cannot be readily identified beyond reasonable doubt, the analysis should be carried out for a sufficient number of different imperfection patterns, and the worst case (lowest value of RGMNIA) should be identified. (17) Equivalent geometrical imperfections that are parallel to the shell middle surface (introducing membrane forces) should also be considered (e.g. imperfections of the bottom face of a vertical cylindrical shell) (18) The eigenmode-affine pattern may be used unless a different unfavourable pattern can be justified.

EDITORIAL NOTE THE FOLLOWING IS A PROPOSAL FROM GERMANY. THE PROJECT TEAM CHOSE TO ADOPT IT AS NOTE2. “When choosing suitable imperfection patterns normal to the shell mid-surface as in (13), due consideration should be given to “collapse affine” patterns as well as “eigenmode affine” patterns (as in 15). Further, “post-buckling affine” patterns should be considered.” NOTE1 The eigenmode affine pattern is the critical buckling mode associated with the elastic critical buckling resistance ratio Rcr based on an LBA analysis of the perfect shell under the defined loading condition. NOTE2 It is possible that “collapse affine” patterns and “post-buckling affine” patterns derived from GNA calculations may lead to lower resistance evaluations. NOTE3 Equivalent geometrical imperfections should also be considered that consist of a boundary unevenness (e.g. imperfections of the bottom face of a vertical cylindrical shell)

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(19) The pattern of the equivalent geometric imperfections should, if practicable, reflect the constructional detailing and the boundary conditions in an unfavourable manner. EDITORIAL NOTE THE FOLLOWING IS A PROPOSED ADDITIONAL PARAGRAPH FOLLOWING THE PROJECT TEAM MEETING IN NOVEMBER 2017. (20) Imperfection patterns that have been demonstrated to be severe for shell buckling in relation to the shell geometry and loading conditions may be taken to be sufficient. NOTE EN 1090-2 does not specify imperfection patterns, and only defines measuring stick lengths and amplitudes which relate directly to Formulae 8.42 and 8.43 only for conditions of uniform axial compression. EDITORIAL NOTE THE FOLLOWING PROPOSALS FROM FRANCE SHOULD BE CONSIDERED BY THE PROJECT TEAM AND WORKING GROUP FOR ADOPTION, THOUGH THIS STATEMENT IS PROBABLY MORE APPROPRIATE IN EN 1993-1-14, WHICH HAD NOT EVEN BEEN PROPOSED WHEN THIS FRENCH PROPOSAL WAS MADE.

For possible transfer to EN 1993-1-14 The finite element mesh and the choice of element type should permit the accurate modelling of the geometry of the imperfections, and also the potentially local character of the buckling mode. EDITORIAL NOTE THE FOLLOWING PROPOSAL FROM IRELAND SHOULD BE CONSIDERED BY THE PROJECT TEAM AND WORKING GROUP FOR ADOPTION. NOTE Further appropriate patterns of imperfections should be considered by the designer based on available authoritative research when the design is being verified. (21) Notwithstanding (14) to (19), patterns may be excluded from the investigation if they can be eliminated as unrealistic because of the method of fabrication, manufacture or erection. NOTE For example, eigenmode imperfections relating to shear buckling modes are not commonly found in fabricated shells, so modes of this kind can be adopted at a lower amplitude or set aside as improbable. (22) Modification of the adopted mode of geometric imperfections to include realistic structural details (such as axisymmetric weld depressions) should be explored. NOTE Where axial compression dominates in considerations for design against buckling, the very local nature of the weld depression, coupled with its extremely severe effect on buckling resistance even when not extending around a large part of the shell, indicate that it should be a serious choice for the imperfection form in welded structures. (23) The sign of the equivalent geometric imperfections should be chosen in such a manner that the maximum initial shape deviations are unfavourably oriented towards the centre of the shell curvature. (24) The amplitude of the adopted equivalent geometric imperfection form should be interpreted in a manner consistent with the tolerance measurements defined in 8.4.3. To achieve this when using a calculated eigenmode, post-buckling mode or collapse-affine mode, an appropriate calibration must be undertaken using a notional measuring stick to deduce the relationship between the peak deformation of the mode (1,0) and the magnitude that would be measured by the tolerance measurement (usually > 1,0). (25) The amplitude of the adopted equivalent geometric imperfection form should be taken as dependent on the fabrication tolerance quality class. The deviation of the geometry of the equivalent imperfection from the perfect shape Δw0,eq, interpreted as in (24), should be the larger of Δw0,eq,1 and Δw0,eq,2, as defined by:

Δw0,eq,1 = g Un1 ... (8.42) Δw0,eq,2 = ni t Un2 ... (8.43)

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EDITORIAL NOTE FOR AXIAL COMPRESSION, THE VALUE FROM FORMULA 8.43 HAS NO VALUE EXCEPT FOR EXTREMELY THICK SHELLS THAT LIE OUTSIDE OUR SCOPE (r/t < 39). WE SHOULD RECONSIDER THE VALUES PROPOSED. HOW SHOULD THESE VALUES BE DEVISED FOR OTHER SHELL BUCKLING SITUATIONS? MOST CASES WORTHY OF GMNIA ARE NOT UNIFORM STRESS STATES. where:

g is all relevant gauge lengths according to 8.4.4 (2); t is the local shell wall thickness; ni is a multiplier to achieve an appropriate tolerance level; Un1, Un2

are the dimple imperfection amplitude parameters for the relevant fabrication tolerance quality class. The value of ni should be taken as ni = 25. The values of the dimple tolerance parameters Un1 and Un2 are given in Table 8.5. EDITORIAL NOTE THE VALUE OF ni MAY BE APPROPRIATE, BUT THE OUTCOME ΔW0,EQ,2 GIVEN BY FORMULA (8.43) IS ONLY USEFUL WHEN THE MODELLED IMPERFECTION IS LOCATED AT A WELDED JOINT.

Table 8.5: Recommended values for dimple imperfection amplitude parameters Un1 and Un2

Fabrication tolerance quality class Description Recommended value of Un1

Recommended value of Un2 Class A Excellent 0,010 0,010 Class B High 0,016 0,016 Class C Normal 0,025 0,025

EDITORIAL NOTE THE ABOVE VALUES ARE LINKED TO THE TOLERANCE MEASUREMENTS FOR AXIAL COMPRESSION AND THE HAND CALCULATION RULE IN ANNEX D ON AXIAL COMPRESSION.

IS THIS A SUFFICIENT OR APPROPRIATE JUSTIFICATION FOR UNIVERSAL ADOPTION FOR ALL LOAD CASES AND STRESS STATES?

THE EXTERNAL PRESSURE IMPERFECTION MEASUREMENTS LEAD TO QUITE DIFFERENT OUTCOMES. (26) The amplitude of the geometric imperfection in the adopted pattern of the equivalent geometric imperfection should be interpreted in a manner which is consistent with the gauge length method, set out in 8.4.4 (2), by which it is defined. (27) Additionally, it should be verified that an analysis that adopts an imperfection whose amplitude is 10% smaller than the value Δw0,eq found in (21) does not yield a lower value for the ratio RGMNIA. If a lower value is obtained, the procedure should be repeated to find the lowest value of the ratio RGMNIA as the amplitude is varied. (28) If follower load effects are possible, either they should be incorporated in the analysis, or it should be verified that their influence is negligible. (29) For each calculated value of the imperfect elastic-plastic buckling resistance RGMNIA, the ratio of the imperfect to perfect resistance (RGMNIA/RGMNA) should be determined and compared with values of α found

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using the procedures of 8.5 and Annex D, to verify that the chosen geometric imperfection has a deleterious effect that is comparable with that obtained from a lower bound to test results. NOTE Where the resistance is dominated by plasticity effects, the ratio (RGMNIA/RGMNA) will be much larger than the elastic buckling reduction factor α, and no close comparison can be expected. However, where the resistance is controlled by buckling phenomena that are substantially elastic, the ratio (RGMNIA/RGMNA) should be only slightly higher than the value determined by hand calculation, and the features that have led to a substantially higher value should be examined carefully. (30) The reliability of the numerically determined imperfect elastic-plastic buckling resistance ratio RGMNIA should be checked by one of the following alternative methods: a) by using the same program to calculate values RGMNIA,check for other shell buckling cases for which characteristic buckling resistance ratio values Rk,known are known. The check cases should use comparable imperfection assumptions and be similar in their buckling controlling parameters (such as relative shell slenderness, post-buckling behaviour, imperfection-sensitivity, geometric nonlinearity and material behaviour); b) by comparison of calculated values (RGMNIA,check) against test results (Rtest,known). The check cases should satisfy the same similarity conditions given in (a). NOTE 1 Other shell buckling cases for which the characteristic buckling resistance ratio values Rk,known are known may be found from the scientific literature on shell buckling. It should be noted that the hand calculations of 8.5 and Annex D are derived as general lower bounds on test results, and these sometimes lead to such low assessed values for the characteristic buckling resistance that they cannot be easily obtained numerically. NOTE 2 Where test results are used, it should be established that the geometric imperfections present in the test may be expected to be representative of those that will occur in practical construction. (31) Depending on the results of the reliability checks, the calibration factor kGMNIA should be evaluated, as appropriate, from:

k,known test,knownGMNIA GMNIA

GMNIA,check GMNIA,check or = =

R Rk k

R R ... (8.44)

where: Rk,known is the known characteristic value; Rtest,known is the known test result; RGMNIA,check is the calculation outcome for the known buckling case or the test buckling case, as appropriate. (32) Where test results are used to determine kGMNIA, and the calculated value of kGMNIA exceeds 1,0, the adopted value should be kGMNIA = 1,0. (33) Where a known characteristic value based on existing established theory is used to determine kGMNIA, and the calculated value of kGMNIA lies outside the range 0,8 < kGMNIA < 1,2, this procedure should not be used. The GMNIA result should be deemed invalid, and further calculations undertaken to establish the causes of the discrepancy. (34) The characteristic buckling resistance ratio should be obtained from:

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Rk = kGMNIA RGMNIA ... (8.45) where: RGMNIA is the calculated imperfect elastic-plastic buckling resistance ratio; kGMNIA is the calibration factor.

8.8.3 Buckling strength verification (1) The design buckling resistance ratio Rd should be obtained from: Rd = Rk/γM1 ... (8.46) where:

γM1 is the partial factor for resistance to buckling according to 8.5.2 (9) and (10). 2) It should be verified that: or 1≤ = ⋅ ≥Ed Rd d Ed dF F R F R ... (8.47)

9 Fatigue limit state (LS4)

9.1 Design values of actions (1) The design values of the actions for each load case should be taken as the varying parts of the total action representing the anticipated action spectrum throughout the design life of the structure. (2) The relevant action spectra should be obtained from EN 1991 in accordance with the definitions given in the appropriate application parts of EN 1993. 9.2 Stress design

9.2.1 General (1) The fatigue assessments defined in EN 1993-1-9 should be used, except as provided here. (2) The partial factor for resistance to fatigue γMf should be taken from the relevant application standard. EDITORIAL NOTE THE PARAGRAPH IN EN 1993-1-6 (2007) IS NOW DIVIDED INTO TWO TO SEPARATE OUT THE DIFFERENT REQUIREMENTS, FOLLOWING THE PROJECT TEAM MEETING IN NOVEMBER 2017 (3) Where no application standard exists for the form of construction involved, or the application standard does not define the relevant values of γMf, the value of γMf should be taken from EN 1993-1-9. (4) It is recommended that the value of γMf should not be taken as smaller than γMf = 1,1, except where the National Annex specifies a lesser value. EDITORIAL NOTE SHOULD THIS PERMISSION FOR THE NATIONAL ANNEX TO DEFINE THE PARTIAL FACTOR BE RETAINED OR REMOVED?

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9.2.2 Design values of stress range (1) Stresses should be determined by a linear elastic analysis of the structure subject to the design values of the fatigue actions. (2) In each verification of the limit state, the design value of the fatigue stress should be taken as the larger stress range Δσ of the values on the two surfaces of the shell, and based on the sum of the primary and the secondary stresses. (3) Depending upon the fatigue assessment carried out according to EN 1993-1-9, either nominal stress ranges or geometric stress ranges should be evaluated. (4) Nominal stress ranges may be used if 9.2.3 (2) is adopted. (5) Geometric stress ranges should be used for construction details that differ from those of 9.2.3 (2). (6) The geometric stress range takes into account only the complete shell geometry of the joint, excluding local stresses due to the weld geometry and internal weld effects. It may be determined by use of geometrical stress concentration factors given by formulae. (7) Stresses used for the fatigue design of construction details with linear geometric orientation should be resolved into components transverse to and parallel to the axis of the detail. 9.2.3 Design values of resistance (fatigue strength) (1) The design values of resistance obtained from the following may be applied to structural steels in the temperature range up to 150° C. EDITORIAL NOTE: THIS UPPER LIMIT HAS NOW BEEN ALSO PLACED IN 1.1(16) FOR CONSISTENCY, WHICH IS A LOWER TEMPERATURE AND IS USED AS A SAFE LIMITATION (2) The fatigue resistance of construction details commonly found in shell structures should be obtained from EN 1993-3-2 in classes and evaluated in terms of the stress range ΔσE, appropriate to the number of cycles, in which the values are additionally classified according to the quality of the welds. (3) The fatigue resistance of the detail classes should be obtained from EN 1993-1-9. 9.2.4 Stress range limitation (1) In every verification of this limit state, the design stress range ΔσE should satisfy the condition:

γFf ΔσE ≤ ΔσR / γMf ... (9.1) where: γFf is the partial factor for the fatigue loading (normally γFf =1,0) (see EN 1993-1-9); γMf is the partial factor for the fatigue resistance (see EN 1993-1-9); ΔσE is the equivalent constant amplitude stress range of the design stress spectrum; ΔσR is the fatigue strength stress range for the relevant detail category and the number of cycles of the stress spectrum.

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(2) As an alternative to (1), a cumulative damage assessment may be made for a set of m different stress ranges Δσi (i = 1,m) using the Palmgren-Miner rule: Dd ≤ 1 ... (9.2) in which:

1/

m

d i ii

D n N=

= ... (9.3) where:

ni is the number of cycles of the stress range Δσi; Ni is the number of cycles of the stress range γFf γMf Δσi to cause failure for the relevant detail category. (3) In the case of combination of normal and shear stress ranges the combined effects should be considered in accordance with EN 1993-1-9.

9.3 Design by global numerical LA or GNA analysis (1) The fatigue design on the basis of an elastic analysis (LA or GNA analysis) should follow the provisions given in 9.2 for stress design. However, the stress ranges due to the fatigue loading should be determined by means of a shell bending analysis, including the geometric discontinuities of joints in constructional details. (2) If a three dimensional finite element analysis is used, the notch effects due to the local weld geometry should be eliminated.

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Annex A (normative)

Membrane theory stresses in shells

A.1 General

A.1.1 Action affects and resistances The action effects or resistances calculated using the formulae in this annex may be assumed to provide characteristic values of the action effect or resistance when characteristic values of the actions, geometric parameters and material properties are adopted. A.1.2 Notation The notation used in this annex for the geometrical dimensions, stresses and loads follows 1.4. In addition, the following notation is used. Roman upper case letters

Fx axial load applied to the cylinder Fz axial load applied to a cone M global bending moment applied to the complete cylinder (not to be confused with the moment per unit width in the shell wall m) Mt global torque applied to the complete cylinder V global transverse shear applied to the complete cylinder Roman lower case letters g unit weight of the material of the shell pn distributed normal pressure px distributed axial traction on cylinder wall t shell wall thickness

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Greek lower case letters φ meridional slope angle σx axial or meridional membrane stress (= nx/t) σθ circumferential membrane stress (= nθ/t) τ membrane shear stress (= nxθ/t)

A.1.3 Boundary conditions (1) The boundary condition notations should be taken as detailed in 2.3 and 5.2.2. (2) For these formulae to be strictly valid, the boundary conditions for cylinders should be taken as radially free at both ends, axially supported at one end, and rotationally free at both ends. (3) For these formulae to be strictly valid for cones, the applied loading should match a membrane stress state in the shell and the boundary conditions should be taken as free to displace normal to the shell at both ends and meridionally supported at one end. (4) For truncated cones, the boundary conditions should be understood to include components of loading transverse to the shell wall, so that the combined stress resultant introduced into the shell is solely in the direction of the shell meridian. A.1.4 Sign convention (1) The sign convention for stresses σ should be taken everywhere as tension positive, though some of the Figures illustrate cases in which the external load is applied in the opposite sense. A.2 Unstiffened cylindrical shells

A.2.1 Uniform axial load A.2.2 Axial load from global bending

A.2.3 Friction load

FFxx == 22ππrr PPxx

FFxx == 22ππrr PPxx

M = πr2 Px,max

M = πr2 Px,max

Px,max

Px,max

px(x)

Px Px

l

2x

xF

rtσ

π= − 2x

M

r tσ

π= ± 0

1 l

x xp dxt

σ = − ⋅

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A.2.4 Uniform internal pressure A.2.5 Variable internal pressure

pn

pn(x)

nr

ptθσ = ⋅ ( ) ( )n

rx p x

tθσ = ⋅ A.2.6 Uniform shear from torsion A.2.7 Sinusoidal shear from

transverse force

Pθ

Mt = 2πr2 Pθ

Pθ(θ)

V = πr Pθ,max

22tM

r tτ

π= max

V

rtτ

π= ±

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A.3 Unstiffened conical shells

A.3.1 Uniform axial load A.3.2 Axial load from global bending

A.3.3 Friction load

Pz,2

Pz,1

β

Fz = 2πr2 Pz,2

Fz = 2πr1 Pz,1

Mt = 2πr 22 Pθ,2

Mt = 2πr 21 Pθ,1

Pθ,1

Pθ,2

px(x)

Px

x1 = r1

sinβ x2 = r2

sinβ

xlocal

x

Px x1

x2

r2

r1

2 cosz

xF

rtσ

π β= −

⋅

σθ = 0 ,max 2 cosx

M

r tσ

π β= ±

⋅

σθ = 0 2

1

11

1 x

x xx

p x dxx t

σ = − ⋅ σθ = 0

A.3.4 Uniform internal pressure A.3.5 Linearly varying internal pressure

r2

β

Px

pn

r

r2S

PPxx

pn γ

r2S is the radius at the fluid surface 2

2

2 cosx nrr

pt r

σβ = − ⋅ 2

2 2 3sin 6 3

s sx

r rr r

t r

γσβ

= − − + ⋅ cosn

rp

tθσβ

=⋅ 2( )

sin sr

r rtθ

γσβ

= + −⋅

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A.3.6 Uniform shear from torsion A.3.7 Sinusoidal shear from transverse force

Mt = 2r22 Pθ,2

Mt = 2πr12 Pθ,1

Pθ,1

Pθ,2

V = πr2 Pθ,2,max

V = πr1 Pθ,1,max

Pθ,1(θ)

Pθ,2(θ)

22tM

r tτ

π= max

V

rtτ

π= ±

A.4 Unstiffened spherical shells

A.4.1 Uniform internal pressure A.4.2 Uniform self-weight load

pn

φ φο

nφ

γ

2np R

tφσ = 1

1 cos

R

tφγσ

φ = − +

2np R

tθσ = 1cos

1 cos

R

tθγσ φ

φ = − − + where

R is the radius of the sphere; γ is the unit weight of the material of construction; φ is the local meridional slope of the shell; σφ is the meridional membrane stress.

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Annex B (normative)

Formulae for plastic reference resistances

B.1 General

B.1.1 Resistances The resistances calculated using the formulae in this annex may be assumed to provide characteristic values of the plastic reference resistance when characteristic values of the geometric parameters and material properties are adopted. B.1.2 Notation The notation used in this annex for the geometrical dimensions, stresses and loads follows 1.4. In addition, the following notation is used. Roman upper case letters

Ar cross-sectional area of a ring PR characteristic value of small deflection theory plastic mechanism resistance Roman lower case letters b thickness of a ring effective length of shell which acts with a ring r radius of the cylinder se dimensionless von Mises equivalent stress parameter sm dimensionless combined stress parameter sx dimensionless axial stress parameter sθ dimensionless circumferential stress parameter Subscripts r relating to a ring

B.1.3 Boundary conditions (1) The boundary condition notations should be taken as detailed in 5.2.2. (2) The term “clamped” should be taken to refer to BC1r and the term “pinned” to refer to BC2f.

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B.2 Unstiffened cylindrical shells

B.2.1 Cylinder: Radial line load

o

o

PnR

r

t

Reference quantities: o = 0,975 rt The plastic resistance PnR (force per unit circumference) is given by:

02nR

yP t

fr

=

B.2.2 Cylinder: Radial line load and axial load

m

m

PnR

r

t

PxR

PxR

Reference quantities: x

xy

Ps

f t= o = 0,975 rt

Range of applicability: −1 ≤ sx ≤ +1

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Dependent parameters: If Pn > 0 (outward) then: A = + sx − 1,50 If Pn < 0 (inward) then: A = − sx − 1,50 sm = A + ( )2 21 xA A s+ −

If sx ≠ 0 then: m = sm o The plastic resistance PnR (force per unit circumference) is given by: 2

nRy

m

P tf

r=

B.2.3 Cylinder: Radial line load, constant internal pressure and axial load

m

m

r

t

pnR PnR

PxR

PxR Reference quantities: x

xy

Ps

f t= n

y

p rs

f tθ = ⋅ o = 0,975 rt se = 2 2

x xs s s sθ θ+ − Range of applicability: −1 ≤ sx ≤ +1 −1 ≤ sθ ≤ +1

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Dependent parameters: Outward directed ring load Pn > 0 Inward directed ring load Pn < 0

Condition Formulae Condition Formulae

se < 1,00 and sθ ≤ 0,975

A = + sx − 2sθ − 1,50 sm = A + ( )2 21 eA A s+ −

0 1m

ms

sθ

= −

se < 1,00 and sθ ≥ −0,975

A = − sx + 2sθ − 1,50 sm = A + ( )2 21 eA A s+ −

0 1m

ms

sθ

= +

se = 1,00 or sθ > 0,975

m = 0,0 se = 1,00 or

sθ < −0,975 m = 0,0

The plastic resistance is given by (Pn and pn always positive outwards): 2

nRn y

m

P tp f

r+ =

B.3 Ring stiffened cylindrical shells

B.3.1 Ring stiffened Cylinder: Radial line load

m b

m

r

tPnR

Ar The plastic resistance PnR (force per unit circumference) is given by: ( 2 )r m

nR yA b t

P fr

+ + =

m = o = 0,975 rt

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B.3.2 Ring stiffened Cylinder: Radial line load and axial load

m

b

m

r

t PnR

PxR

PxR

Ar

Reference quantities: x

xy

Ps

f t= o = 0,975 rt

Range of applicability: −1 ≤ sx ≤ +1 Dependent parameters: If Pn > 0 then: A = + sx − 1,50 If Pn < 0 then: A = − sx − 1,50 sm = A + ( )2 21 xA A s+ −

If sx ≠ 0 then: m = sm o The plastic resistance PnR (force per unit circumference) is given by: ( 2 )r m

nR yA b t

P fr

+ + =

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B.3.3 Ring stiffened cylinder: Radial line load, constant internal pressure and axial load

m

bm

r

t

pn

PnR

PxR

PxR

Ar

Reference quantities: x

xy

Ps

f t= n

y

p rs

f tθ = ⋅ o = 0,975 rt se = 2 2

x xs s s sθ θ+ − Range of applicability: −1 ≤ sx ≤ +1 −1 ≤ sθ ≤ +1 Dependent parameters:

Outward directed ring load Pn > 0 Inward directed ring load Pn < 0

Condition Formulae Condition Formulae

se < 1,00 and sθ ≤ 0,975

A = + sx − 2sθ − 1,50 sm = A + ( )2 21 eA A s+ −

0 1m

m

s

sθ

= −

se < 1,00 and sθ ≥ −0,975

A = − sx + 2sθ − 1,50 sm = A + ( )2 21 eA A s+ −

0 1m

m

s

sθ

= +

se = 1,00 or sθ > 0,975

m = 0,0 se = 1,00 or

sθ < −0,975 m = 0,0

The plastic resistance is given by (PnR and pn always positive outwards): PnR + pn (b + 2m) = fy ( 2 )r mA b t

r

+ +

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B.4 Junctions between shells

B.4.1 Junction under meridional loading only (simplified)

Px,c

Ar

β

tc

thts

Px,h

PPxx,,ss

r

Range of applicability: 2 2 2c s ht t t≤ + |Px,s| << tsfy, |Px,h| << thfy and |Px,c| << tcfy` Dependent parameters:

2

2 2c

s h

t

t tη =

+ ψs = ψh = 0,7 + 0,6η2 − 0,3η3

For the cylinder oc = 0,975 crt For the skirt os = 0,975 ψs srt For the conical segment oh = 0,975 ψh

coshrt

β The plastic resistance is given by:

PxhR r sinβ = fy (Ar + octc + osts + ohth)

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B.4.2 Junction under internal pressure and axial loading Px,c

pn,c

pn,h

Ar

β

tc

thts

Px,h

PPxx,,ss

r

Reference quantities: ,= x c

xcy c

Ps

f t ,= x s

xsy s

Ps

f t ,= x h

xhy h

Ps

f t

,θ = ⋅n c

cy c

p rs

f t sθs = 0 ,

cosθ β= ⋅

⋅n h

hy h

p rs

f t

for i = c, s, h in turn sei = 2 2i xi xi is s s sθ θ+ − in which the subscripts c, s and h refer to the cylinder, skirt and hopper respectively. Range of applicability:

−1 ≤ sxi ≤ +1 −1 ≤ sθi ≤ +1 Equivalent thickness evaluation: Lower plate group thicker 2 2 2

c s ht t t≤ + Upper plate group thicker 2 2 2c s ht t t> +

2

2 2c

s h

t

t tη =

+

ψc = 1,0 ψs = ψh = 0,7 + 0,6η2 − 0,3η3

2 2

2s h

c

t t

tη += ψc = 0,7 + 0,6η2 − 0,3η3 ψs = ψh = 1,0

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Dependent parameters: For the cylindrical segments oi = 0,975 ψi irt For the conical segment oh = 0,975 ψh

cosirt

β

For each shell segment i separately

Condition Formulae

sei < 1,00 and sθi ≥ −0,975

Ai = − sxi + 2sθi − 1,50 smi = Ai + ( )2 21i eiA A s+ − mi = oi ( )

1mi

i

s

sθ+

sei = 1,00 mi = 0,0 sθi < −0,975 mi = 0,0 Plastic resistance is given by:

PxhR r sinβ = fy (Ar + mctc + msts + mhth) + r (pncmc + pnhmh cosβ) B.5 Circular plates with axisymmetric boundary conditions

B.5.1 Uniform load, simply supported boundary

2, 1,625( )n R y

tp f

r=

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B.5.2 Local distributed load, simply supported boundary

rx

bt

F

Uniform pressure pn on circular patch of radius b F = pn π b2

2

2R yF K t fπ= with 4 1

1,0 1,10 1,15( ) or K=3

b b bK

r r t= + + ⋅ whichever is the lesser

B.5.3 Uniform load, clamped boundary

pn,R = 3,125 2( )

t

rfy

B.5.4 Local distributed load, clamped boundary

rrxx

bbtt

FF

Uniform pressure pn on circular patch of radius b F = pn π b2 2

2R yF K t fπ= with

4 11,40 2,85 2,0 or K=

3

b b bK

r r t = + + ⋅

whichever is the lesser

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Annex C (normative)

Formulae for linear elastic membrane and bending stresses

C.1 General

C.1.1 Action effects The action effects calculated using the formulae in this annex may be assumed to provide characteristic values of the action effect when characteristic values of the actions, geometric parameters and material properties are adopted. C.1.2 Notation The notation used in this annex for the geometrical dimensions, stresses and loads follows 1.4. In addition, the following notation is used. Roman characters

b radius at which local load on plate terminates r outside radius of circular plate x axial coordinate on cylinder or radial coordinate on circular plate Greek symbols σeq,m von Mises equivalent stress associated with only membrane stress components σeq,s von Mises equivalent stress derived from surface stresses σMT reference stress derived from membrane theory σbx meridional bending stress σbθ circumferential bending stress σsx meridional surface stress σsθ circumferential surface stress τxn transverse shear stress associated with meridional bending Subscripts n normal r relating to a ring y first yield value

C.1.3 Boundary conditions (1) The boundary condition notations should be taken as detailed in 5.2.2.

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(2) The term “clamped” should be taken to refer to BC1r and the term “pinned” to refer to BC2f. C.2 Clamped base unstiffened cylindrical shells

C.2.1 Cylinder, clamped: uniform internal pressure

pn

r

t

x MT n

rp

tθσ = BC1r Maximum σsx Maximum σsθ Maximum τxn Maximum σeq,s Maximum σeq,m

±1,816 σMTθ +1,080 σMTθ 1,169 t r σMTθ 1,614 σMTθ 1,043 σMTθ C.2.2 Cylinder, clamped: axial loading

Px

r

t

x

xMTx

P

tσ = BC1r

Maximum σsx Maximum σsθ Maximum τxn Maximum σeq,s Maximum σeq,m1,545 σMTx +0,455 σMTx 0,351 t r σMTx 1,373 σMTx 1,000 σMTx C.2.3 Cylinder, clamped: uniform internal pressure with axial loading

Px

r

t

x

pn

MT nr

ptθσ =

xMTx

P

tσ = BC1r

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Maximum σeq,m = σMTθ 2

1 MTx MTx

MT MTθ θ

σ σσ σ

− +

Maximum σeq,m = k σMTθ MTx

MTθ

σσ

−2,0 0 0 2,0 Outer surface controls Inner surface controls

k 4,360 1,614 1,614 2,423 Linear interpolation may be used between values where the same surface controls C.2.4 Cylinder, clamped: hydrostatic internal pressure

rr

tt

xx ppnn00 pp

0MT n

rp

tθσ = BC1r Maximum σsx Maximum σsθ Maximum τxn Maximum σeq,s Maximum σeq,m

kx σMTθ kθ σMTθ kτ t r σMTθ keq,s σMTθ keq,m σMTθ p

rt kx kθ kxθ keq,s keq,m 0 1,816 1,080 1,169 1,614 1,043 0,2 1,533 0,733 1,076 1,363 0,647

C.2.5 Cylinder, clamped: radial outward displacement

rt

x

w

MTwE

rθσ = BC1r

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Maximum σsx Maximum σsθ Maximum τxn Maximum σeq,s Maximum σeq,m 1,816 σMTθ 1,545 σMTθ 1,169 t r σMTθ 2,081 σMTθ 1,000 σMTθ C.2.6 Cylinder, clamped: uniform temperature rise

r

t

x σMTθ = α E T BC1r

Maximum σsx Maximum σsθ Maximum τxn Maximum σeq,s Maximum σeq,m 1,816 σMTθ 1,545 σMTθ 1,169 t r σMTθ 2,081 σMTθ 1,000 σMTθ C.3 Pinned base unstiffened cylindrical shells

C.3.1 Cylinder, pinned: uniform internal pressure

r

t

x

pn

MT n

rp

tθσ = BC1f Maximum σsx Maximum σsθ Maximum τxn Maximum σeq,s Maximum σeq,m ±0,585 σMTθ +1,125 σMTθ 0,583 t r σMTθ 1,126 σMTθ 1,067 σMTθ

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C.3.2 Cylinder, pinned: axial loading

Px

r

t

x

xMTx

P

tσ = BC1f

Maximum σsx Maximum σsθ Maximum τxn Maximum σeq,s Maximum σeq,m +1,176 σMTx +0,300 σMTx 0,175 t r σMTx 1,118 σMTx 1,010 σMTx C.3.3 Cylinder, pinned: uniform internal pressure with axial loading

Px

r

t

x

pn

MT nr

ptθσ =

xMTx

P

tσ = BC1f

Maximum 2

, 1 MTx MTxeq m MT

MT MTθ

θ θ

σ σσ σσ σ

= − +

Maximum σeq,s = k σMTθ MTx

MTθ

σσ

−2,0

−1,0 −0,5 0,0

0,25 0,50 1,00 2,0 k 3,146 3,075 1,568 1,126 0,971 0,991 1,240 1,943

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C.3.4 Cylinder, pinned: hydrostatic internal pressure

rt

x pn,0

p 0MT n

rp

tθσ = BC1f Maximum σsx Maximum σsθ Maximum τxn Maximum σeq,s Maximum σeq,m kx σMTθ kθ σMTθ kτ t r σMTθ keq,s σMTθ keq,m σMTθ

p

rt

kx

kθ kτ

keq,s keq,m 0 0,585 1,125 0,583 1,126 1,067 0,2 0,585 0,873 0,583 0,919 0,759 Linear interpolation in

p

rt

may be used for different values of p. C.3.5 Cylinder, pinned: radial outward displacement

rt

x

w

MTwE

rθσ = BC1f Maximum σsx Maximum σsθ Maximum τxn Maximum σeq,s Maximum σeq,m

±0,585 σMTθ 1,000 σMTθ 0,583 t r σMTθ 1,000 σMTθ 1,000 σMTθ

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C.3.6 Cylinder, pinned: uniform temperature rise

r

t

x

w

σMTθ = α E T w = α r T BC1f Maximum σsx Maximum σsθ Maximum τxn Maximum σeq,s Maximum σeq,m

±0,585 σMTθ 1,000 σMTθ 0,583 t r σMTθ 1,000 σMTθ 1,000 σMTθ C.3.7 Cylinder, pinned: rotation of boundary

r

t

x

βφ

θ φσ β= ⋅MT

tE

r BC1f Maximum σsx Maximum σsθ Maximum τxn Maximum σeq,s Maximum σeq,m

±1,413 σMTθ 0,470 σMTθ 0,454 t r σMTθ 1,255 σMTθ 0,251 σMTθ C.4 Internal conditions in unstiffened cylindrical shells

C.4.1 Cylinder: step change of internal pressure

rt-x

pn

x

MT nr

ptθσ =

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Maximum σsx Maximum σsθ Maximum τxn Maximum σeq,s Maximum σeq,m ±0,293 σMTθ 1,062 σMTθ 0,467 t r σMTθ 1,056 σMTθ 1,033 σMTθ

C.4.2 Cylinder: hydrostatic internal pressure termination

rt-x

pn1

xrt

1MT nr

ptθσ =

pn1 is the pressure at a depth of rt below the surface Maximum σsx Maximum σsθ Maximum τxn Maximum σeq,s Maximum σeq,m kx σMTθ kθ σMTθ kτ t r σMTθ keq,s σMTθ keq,m σMTθ

kx kθ kτ keq,s keq,m −1,060 0,510 0,160 1,005 0,275

C.4.3 Cylinder: step change of thickness

r

t1 --x

pn x

t2

1MT n

rp

tθσ =

Maximum σsx Maximum σsθ Maximum τxn Maximum σeq,s Maximum σeq,m

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kx σMTθ kθ σMTθ kτ t r σMTθ keq,s σMTθ keq,m σMTθ 1

2

t

t

kx kθ kτ keq,s keq,m 1,0 0,0 1,0 0,0 1,0 1,0 0,8 0,0256 1,010 0,179 1,009 0,895 0,667 0,0862 1,019 0,349 1,015 0,815 0,571 0,168 1,023 0,514 1,019 0,750 0,5 0,260 1,027 0,673 1,023 0,694 C.5 Ring stiffener on cylindrical shell

C.5.1 Ring stiffened cylinder: radial force on ring The stresses in the shell should be determined using the calculated value of w from this clause introduced into the formulae given in C.2.5. Where there is a change in the shell thickness at the ring, the method set out in 8.2.2 of EN 1993-4-1 should be used.

bm

bm

b

deformations

P

r

t

Ar

wr

w = wr bm = 0,778 rt ( 2 )r m

wE P r

r A b b t

⋅ = + +

( 2 )rr m

P r

A b b tθσ ⋅=+ +

C.5.2 Ring stiffened cylinder: axial loading The stresses in the shell should be determined using the calculated value of w from this clause introduced into the formulae given in C.2.5 and C.2.2.

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bm

bm

b

deformations

nx

nx

r

t

Ar

wr

wo

xMTx

n

tσ =

w = wr − wo 0 MTx

rw

Eνσ= −

bm = 0,778 rt

0( 2 )

( 2 )m

rr m

b b tw w

A b b t

+=+ +

0 ( 2 )r

r m

Aw w

A b b t= −

+ + r

rw

Erθσ =

C.5.3 Ring stiffened cylinder: uniform internal pressure The stresses in the shell should be determined using the calculated value of w from this clause introduced into the formulae given in C.2.5 and C.2.1.

bm

bm

b

deformations

r

t

Ar wr

wo

pnR

t

rpnMT =θσ

w = wr − wo 0 MT

rw

Eθσ= bm = 0,778 rt

wr = wo (1−k) w = −wo k

( 2 )r

r m

A

A b b tκ =

+ +

rr

wE

rθσ = Maximum σsx Maximum σsθ Maximum τxn Maximum σeq,s Maximum σeq,m

kx σMTθ kθ σMTθ kτ t r σMTθ keq,s σMTθ keq,m σMTθ κ kx kθ kτ keq,s keq,m 1,0 1,816 1,080 1,169 1,614 1,043 0,75 1,312 1,060 0,877 1,290 1,032 0,50 0,908 1,040 0,585 1,014 1,021 0,0 0,0 1,000 0,0 1,000 1,000

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C.6 Circular plates with axisymmetric boundary conditions

C.6.1 Plate with simply supported boundary: uniform load

t

pn

rx

w

deflected shape

4

30,696 np r

wEt

= max. 2

1,238σ =

bx nr

pt

max. 2

1,238θσ =

b nr

pt

2

, 0,808 =

n y yt

p fr

C.6.2 Plate with local distributed load: simply supported boundary

rrxx

bbtt

FF

ww

ddeefflleecctteedd sshhaappee

Uniform pressure pn on circular patch of radius b F = pn π b2 b < 0,2 r

2

30,606

Frw

Et=

max. σbx = max. σbθ = 0,621 2(ln 0,769)

F b

rt+

21,611

(ln 0,769)y y

tF f

br

=+

C.6.3 Plate with fixed boundary: uniform load

tt

ppnn

rrxx

ww

deflected shape

4

30,171 np r

wEt

= 2

0 ( )nr

pt

σ = 2

, 1,50( )n y yt

p fr

= (at edge) Maximum σbx at centre Maximum σbθ at centre Maximum σeq at centre Maximum σbx at edge Maximum σbθ at edge Maximum σeq at edge 0,488 σo 0,488 σo 0,488 σo 0,75 σo 0,225 σo 0,667 σo

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C.6.4 Plate with fixed boundary: local distributed load

rrxx

bbtt

FF

ww

deflected shape

Uniform pressure pn on circular patch of radius b F = pn π b2 b < 0,2 r

2

30,217

Frw

Et= 0 2

F

tσ = 2

1,611(ln )

y yt

F fbr

= at centre Maximum σbx at centre Maximum σbθ at centre Maximum σeq at centre Maximum

σbx at edge Maximum σbθ at edge Maximum

σeq at edge 00,621(ln )

b

rσ 00,621(ln )

b

rσ 00,621(ln )

b

rσ 0,477 σo 0,143 σo 0,424 σo

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Annex D (normative)

Formulae for use in buckling stress design

D.1 Unstiffened cylindrical shells of constant wall thickness

D.1.1 Notation and boundary conditions (1) Geometrical quantities cylinder length between defined boundaries r radius of cylinder middle surface t thickness of shell Δwk characteristic imperfection amplitude

nxθ=τt

nx=σxt

nθ=σθt

nθx=τt

t

r

t

n,w

x,u

θ,v

w

x,u

θ,v Figure D.1: Cylinder geometry, membrane stresses and stress resultants (2) The relevant boundary conditions are set out in 2.3, 5.2.2 and 8.3.

D.1.2 Meridional (axial) compression

D.1.2.1 Critical meridional buckling stresses (1) The following formulae may only be used for shells with boundary conditions BC 1 or BC 2 at both edges. (2) The length of the shell segment is characterised in terms of the dimensionless length parameter ω: r

r t rtω = = ... (D.1)

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(3) The elastic critical meridional buckling stress, using a value of Cx from (4), (5) or (6), should be obtained from: , 0,605x Rcr x

tEC

rσ = ... (D.2)

(4) For medium-length cylinders, which for meridional compression are defined by: 1,7 ≤ ω ≤ 0,5 r

t ... (D.3)

the factor Cx should be taken as: Cx = 1,0 ... (D.4) (5) Short cylinders under meridional compression are defined by: ω < 1,7 ... (D.5) The factor Cx should be taken as: Cx = 1,36 −

2

1,83 2,07

ω ω+ ... (D.6)

EDITORIAL NOTE: LONG CYLINDERS SHOULD PROBABLY BE RE-DEFINED AS THOSE SUBJECT TO EULER BUCKLING, SINCE RECENT EVIDENCE SHOWS THAT IMPERFECT CYLINDERS ARE UNAFFECTED BY THE DROP IN CRITICAL BUCKLING STRESS DEFINED HERE. THAT WOULD CHANGE THE DEFINITION OF “LONG” TO ω > 2.86 (r/t) AND WOULD ELIMINATE FORMULAE D.8, D.9 AND D.10 AS WELL AS TABLE D.1.

THE LONG CYLINDER STRENGTH RULE COULD THEN BE REMOVED AND REPLACED BY A STATEMENT THAT REFERENCE SHOULD BE MADE TO EN 1993-1-1 FOR LONGER CYLINDERS. (6) Long cylinders under meridional compression are defined by:

ω > 0,5 r

t ... (D.7)

The factor Cx should be found as: Cx = Cx,N ... (D.8) in which Cx,N is the greater of: Cx,N = 1 + 0,2

1 2xb

t

C rω −

... (D.9) and

Cx,N = 0,60 ... (D.10) where Cbx is a parameter depending on the boundary conditions and being taken from Table D.1.

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Table D.1: Parameter Cbx for the effect of boundary conditions on the elastic critical meridional buckling stress in long cylinders

Case Cylinder end

Boundary condition

Cbx 1 end 1 end 2 BC 1 BC 1 6 2 end 1 end 2 BC 1 BC 2 3 3 end 1 end 2 BC 2 BC 2 1

(7) Cylinders under a combination of uniform axial compression and uniform bending may be treated using the provisions of 8.6 and Annex E.1. The following simpler conservative treatment for stress design may be used as an alternative. (8) For long cylinders as defined in (6), under a combination of uniform axial compression and uniform bending the factor Cx may be obtained from: , ,

,xE N xE M

x x NxE xE

C Cσ σσ σ

= +

... (D.11)

where: σxE is the design value of the total acting meridional stress σx,Ed ; σxE,N is the component of σx,Ed that derives from axial compression (circumferentially uniform component); σxE,M is the component of σx,Ed that derives from tubular global bending (peak value of the circumferentially varying component). (9) The following simpler and more conservative Formula may also be used in place of Formula (D.11):

,0,60 0,40 xE Mx

xEC

σσ

= +

... (D.12)

D.1.2.2 Meridional buckling capacity parameters (1) The meridional squash limit slenderness 0xλ should be taken as: 0xλ = 0,10 ... (D.13)

EDITORIAL NOTE THIS VALUE IS ADOPTED AS A SAFE ONE RELATIVE TO TEST DATA AND TO FINITE ELEMENT CALCULATIONS USING LOW STRAIN HARDENING. BUT IT IS MUCH LOWER THAN THE VALUE IN EN 1993-1-1. SHOULD IT BE INCREASED TO AGREE WITH EN 1993-1-1 ALTHOUGH WE CANNOT JUSTIFY IT BY CALCULATION USING THE PERMITTED STRESS-STRAIN CURVE?

EDITORIAL NOTE THE FOLLOWING ARE AN UPGRADED INTERPRETATION BASED ON MORE RECENT FINITE ELEMENT CALCULATIONS. AS THE INTERACTION OF AXIAL COMPRESSION AND BENDING IS

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FURTHER DEVELOPED IN ANNEX E, THE FORMULATION MAY BE FURTHER IMPROVED, ESPECIALLY TO ALLOW FOR A VARYING VALUE OF ETA WITH SLENDERNESS, WHICH LEADS TO A MUCH BETTER MATCH TO THE DATA. PLEASE NOTE THAT THE IMPERFECTION SENSITIVITY IS NOW DEEMED TO REACH AN ASYMPTOTIC MINIMUM AT ALPHAI OF 0,06. (2) The meridional elastic imperfection reduction factor αx = αxGαxI should be obtained from:

0,83xGα = ... (D.14) ( )0,85

0,930,06

1 2,7α = +

+ ΔI

kw t ... (D.15)

in which Δwk is the characteristic imperfection amplitude given by: 1kw rt Q tΔ= ... (D.16)

where: Q is the meridional compression fabrication quality parameter. (3) The fabrication quality parameter Q should be taken from Table D.2 for the specified fabrication tolerance quality class.

Table D.2: Values of fabrication quality parameter Q

Fabrication tolerance quality class

Description Q Class A Excellent 40 Class B High 25 Class C Normal 16 (4) The plastic range factor βx should be obtained from:

0,951

1 1,12( / )β = −

+ Δx

kw t ... (D.17)

(5) The interaction exponent ηx should be obtained from: ( )0,87

5,8

1 4,6 /η =

+ Δ kw t ... (D.18)

(6) The hardening limit χ xh should be taken as: 1,15χ =xh ... (D.19)

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EDITORIAL NOTE THE LIMIT IN (D.20) MAY BE REDEFINED RELATIVE AFTER FURTHER STUDY OF EXISTING FINITE ELEMENT CALCULATIONS. WHILST THIS SLENDERNESS RANGE MAY APPEAR TO BE THE DOMAIN OF EN 1993-1-1, CONDITIONS WHERE BUCKLING OCCURS UNDER HIGH TENSILE STRESSES IN ONE DIRECTION AND SMALL COMPRESSION IN ANOTHER BUCKLE WITH EXTENSIVE PLASTICITY WHILST STILL VERY THIN. (7) For long cylinders that satisfy the special conditions of D.1.2.1 (8), the meridional squash limit slenderness 0xλ may be obtained from:

⎯ 0xλ = 0,10 + 0,10 ,xE M

xE

σσ

... (D.20) where:

σxE is the design value of the meridional stress σx,Ed ; σxE,M is the component of σx,Ed that derives from tubular global bending (peak value of the circumferentially varying component).

EDITORIAL NOTE THIS RULE (7) MUST BE REVISED TO ACCORD WITH WHATEVER CHANGE IS MADE ABOVE ON AXIAL COMPRESSION, AND TO MATCH ANNEX E.1 FOR PURE BENDING (8) Cylinders need not be checked against meridional shell buckling if they satisfy:

0,242 xyk

r EC

t f≤ ... (D.21)

EDITORIAL NOTE WE SHOULD INCLUDE HERE THE EULER LIMITATION AND ALSO DIRECT THE USER TO EN 1993-1-1 D.1.2.3 Stainless steel cylinders under meridional compression (1) With the exceptions defined here, the above formulae may be applied to shells under meridional compression constructed from austenitic and austenitic-ferritic stainless steels with all boundary conditions. (2) For austenitic stainless steels at ambient temperatures, the capacity parameters β and η (Formulae D.17 and D.18) should be taken instead as

( )λ η η λ η λ ηη

λ λ

− + −=

−

p o p o o p

p o

... (D.22) in which ηo = 1,1 and the value of β and ηp should be taken from Table D.3. The value of λp should still be obtained from Formula 8.16 but using the values of β, ηo and ηp defined here.

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Table D.3: Values of capacity parameters β and ηp for austenitic stainless steels

Fabrication tolerance quality class

Description β ηp Class A Excellent 0,614 -0,002 Class B High 0,720 0.056 Class C Normal 0,799 -0.095 (3) For austenitic-ferritic and ferritic stainless steels in the slenderness range 0,4 ≤ λ ≤ 1,1, the value of

χx determined for carbon steels using the procedures of 8.5 should be reduced by 5%. NOTE When assessing the meridional compression buckling resistance of shells with nonlinear stress-strain curves, the use of either a reduced value of elastic modulus Ered or the secant modulus at the 0.2% proof stress, as indicated in 3.1(2), may produce very conservative buckling resistances for stainless steel shells at both low and high slendernesses. D.1.3 Circumferential (hoop) compression

D.1.3.1 Critical circumferential buckling stresses (1) The following formulae may be applied to shells with all boundary conditions. (2) The length of the shell segment should be characterised in terms of the dimensionless length parameter ω: l r l

r t rtω = = ... (D.23)

(3) Medium-length cylinders under circumferential compression are defined by 1,63 θω < r

Ct

... (D.24) except where they are short according to (5). (4) The elastic critical circumferential buckling stress should be obtained from:

, 0,92RcrC t

Er

θθσ

ω =

... (D.25)

(5) The lower limit of the medium length domain under circumferential compression depends on the boundary conditions and is given in Table D.4. The factor Cθ should be taken from Table D.4, see 5.2.2 and 8.3. NOTE For most boundary conditions in short cylinders, Table D.4 gives a safe estimate of the value of Cθ. However, for cylinders with boundary conditions BC1 at both ends, this is not a safe assumption and the short cylinder treatment using Cθs should always be adopted.

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Table D.4: External pressure buckling factors for medium-length cylinders Cθ

Case Cylinder end Boundary condition Value of Cθ Lower limit of medium length ω 1 end 1 end 2 BC 1 BC 1 1,5 80

2 end 1 end 2 BC 1 BC 2 1,25 70 3 end 1 end 2 BC 2 BC 2 1,0 130 4 end 1 end 2 BC 1 BC 3 0,6 15 5 end 1 end 2 BC2 BC3 See Table D.5 0 6 end 1 end 2 BC 3 BC 3 0 0

(6) For short cylinders, the elastic critical circumferential buckling stress should be obtained instead from: , 0,92 sRcr

C tE

rθ

θσω

=

... (D.26) (7) The factor Cθs should be taken from Table D.5, with a value that depends on the boundary conditions, see 5.2.2 and 8.3.

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Table D.5: External pressure buckling factors for short cylinders Cθs

* Use of Table D.4 is unsafe for values of ω below the value given here

Case Boundary condition at each end

Formula for Cθs Lower validity limit of Table D.4 values of ω

End 1 End 2 BC1r BC1r 2 3

1,6 19 201,50

ω ω ω− + − 80 *

BC1r BC1f 2 3

1,4 10 2,31,50

ω ω ω− + − 80 *

BC1f BC1f 2 3

1,45 5,8 2,91,50

ω ω ω− + + 90 *

BC1r BC2r 2 3

0,86 5,9 8,91,25

ω ω ω+ + + 70

BC1f BC2r 2 3

0,96 0,51 16,31,25

ω ω ω+ − + 70

BC1r BC2f 2 3

8,3 4,71,25

ω ω+ − 25

BC1f BC2f 2 3

3,35 1,951,25

ω ω+ + 15

BC2r BC2r 2 3

2,60 1,13 13,21

ω ω ω+ + + 130

BC2r BC2f 2 3

1,96 1,00 6,611

ω ω ω+ + + 130

BC2f BC2f 2 3

1,3 1,65 0,591

ω ω ω+ + + 60

BC1r BC3 2,08

1,130,60

ω+ 15

BC1f BC3 0,60 Both short and medium lengths

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BC2r BC3 2 3

2,18 6,19 11,20,050

ω ω ω+ − + Both short and medium lengths BC2f BC3

0,29 1,5 ω

+

t t

r r Both short and medium lengths

where l

rtω =

(8) Long cylinders under circumferential compression are defined by: 1,63 θω > r

Ct

... (D.27) (9) The elastic critical circumferential buckling stress for long cylinders should be obtained from:

42

, 0,275 2,03RcrCt r

Er t

θθσ

ω = + ⋅

... (D.28)

D.1.3.2 Circumferential buckling capacity parameters (1) The circumferential elastic buckling reduction factor αθ should be taken from Table D.6 for the specified fabrication tolerance quality class. Table D.6 : Values of αθ based on fabrication quality

Fabrication tolerance quality class

Description αθ

Class A Excellent 0,75 Class B High 0,65 Class C Normal 0,50 EDITORIAL NOTE: IT IS EXPECTED THAT A RELATIONSHIP BETWEEN IMPERFECTION AMPLITUDE AND THE ELASTIC BUCKLING REDUCTION FACTOR WILL BE DEVISED FOR THE SECOND DRAFT. THE CHIEF CURRENT DIFFICULTY LIES IN THE MISMATCH OF ANALYTICAL PREDICTIONS FOR IMPERFECTION SENSITIVITY WITH THE RELEVANT TOLERANCE MEASUREMENT IN 8.4 (2) The circumferential squash limit slenderness 0θλ , the plastic range factor βθ, the interaction exponent ηθ and the hardening limit θχ h should be taken as:

0 0,40θλ = 0,60θβ = 1,00θη = 1,10θχ =h ... (D.29)

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(3) Cylinders need not be checked against circumferential shell buckling if they satisfy: 0,21

yk

r E

t f≤ ... (D.30)

qw,max

qw(θ)

qqeeqq

a) wind pressure distribution around shell circumference b) equivalent axisymmetric pressure distribution

Figure D.2: Transformation of typical wind external pressure load distribution

EDITORIAL NOTE THIS SUB-CLAUSE ON THE TREATMENT OF WIND SHOULD BE TRANSFERRED TO EITHER THE SILOS OR TANKS STANDARD.

SINCE TANKS ARE SLIGHTLY MORE SUSCEPTIBLE TO WIND THAN SILOS, IT SEEMS BEST TO PUT IT THERE, AND TO INCLUDE STEPPED WALL CONSTRUCTION AT THE SAME TIME (4) The non-uniform distribution of pressure qw resulting from external wind loading on cylinders (see Figure D.2) may, for the purpose of shell buckling design, be substituted by an equivalent uniform external pressure:

qeq = kw qw,max ... (D.31) where qw,max is the maximum wind pressure, and kw should be found as follows: 0,46 1 0,1w

C rk

tθ

ω

= + ⋅

... (D.32) with the value of kw not outside the range 0,65 ≤ kw ≤ 1, and with Cθ taken from Table D.4 according to the boundary conditions. (5) The circumferential design stress to be introduced into 8.5 follows from:

, ( )Ed eq sr

q qtθσ = +

... (D.33)

where:

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qs is the internal suction caused by venting, internal partial vacuum or other phenomena. D.1.4 Shear

D.1.4.1 Critical shear buckling stresses (1) The following formulae should be applied only to shells with boundary conditions BC1 or BC2 at both edges. (2) The length of the shell segment should be characterised in terms of the dimensionless length parameter ω: l r l

r t rtω = = ... (D.34)

(3) The elastic critical shear buckling stress should be obtained from: ,

10,75x Rcr

tEC

rθ ττω =

... (D.35) (4) Medium-length cylinders under membrane shear are defined by:

10 8,7r

tω≤ ≤ ... (D.36)

(5) The factor Cτ for medium-length cylinders should be found as: Cτ = 1,0 ... (D.37) (6) Short cylinders under membrane shear are defined by: ω < 10 ... (D.38) (7) The factor Cτ for short cylinders should be found as:

3

421Cτ ω

= + ... (D.39) (8) Long cylinders under membrane shear are defined by:

8,7r

tω > ... (D.40)

(8) The factor Cτ for short cylinders should be found as: 1

3

tC

rτ ω= ... (D.41) D.1.4.2 Shear buckling capacity parameters (1) The shear elastic buckling reduction factor ατ should be taken from Table D.7 for the specified fabrication tolerance quality class.

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Table D.7: Values of ατ based on fabrication quality

Fabrication tolerance quality class

Description ατ

Class A Excellent 0,75 Class B High 0,65 Class C Normal 0,50 (2) The shear squash limit slenderness 0τλ , the plastic range factor βτ, the interaction exponent ητ and the hardening limit χτh should be taken as: 0τλ = 0,40 βτ = 0,60 ητ = 1,0 χτh = 1,0 ... (D.42) (3) Cylinders need not be checked against shear buckling if they satisfy:

0.67

0,17yk

r E

t f

≤

... (D.43)

D.1.5 Meridional (axial) compression with coexistent internal pressure

D.1.5.1 Pressurised critical meridional buckling stress (1) The elastic critical meridional buckling stress σx,Rcr may be assumed to be unaffected by the presence of internal pressure and may be obtained as specified in D.1.2.1. D.1.5.2 Pressurised meridional buckling capacity parameters (1) The pressurised meridional buckling stress should be verified analogously to the unpressurised meridional buckling stress as specified in 8.5 and D.1.2.2. However, the unpressurised elastic buckling reduction factor αx should be replaced by the pressurised elastic buckling reduction factor αxp. (2) The pressurised elastic buckling reduction factor αxp should be taken as the smaller of the two following values:

αxpe is a factor covering pressure-induced elastic stabilisation; αxpp is a factor covering pressure-induced plastic destabilisation. (3) The factor αxpe should be obtained from:

0,5(1 )

0,3/s

xpe x xs x

p

pα α α

α

= + −

+ ... (D.44)

,

ss

x Rcr

p rp

tσ =

... (D.45)

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where: ps is the smallest design value of local internal pressure at the location of the point being assessed, guaranteed to coexist with the meridional compression; αx is the unpressurised meridional elastic buckling reduction factor according to D.1.2.2; σx,Rcr is the elastic critical meridional buckling stress according to D.1.2.1 (3).

EDITORIAL NOTE THE ABOVE FORMULAE WILL BE REVISED FOR THE SECOND DRAFT USING THE OUTCOME OF EXTENSIVE RECENT CALCULATIONS (4) The factor αxpe should not be applied to cylinders that are long according to D.1.2.1 (6). In addition, it should not be applied unless one of the following two conditions are met: • the cylinder is medium-length according to D.1.2.1 (4); • the cylinder is short according to D.1.2.1 (5) and Cx = 1 has been adopted in D.1.2.1 (3). (5) The factor αxpp should be obtained from:

( )

++

+−

−=

1

21,1

12,1

111

2x

2

2/3

2

2x

xpp ss

s

s

pg λλ

α ... (D.46) ,

gg

x Rcr

p rp

tσ =

... (D.47) 1

400

rs

t= ⋅ ... (D.48)

where: pg is the largest design value of local internal pressure at the location of the point being assessed that can coexist with the meridional compression;

xλ is the dimensionless shell slenderness parameter according to 8.5.2 (3); σx,Rcr is the elastic critical meridional buckling stress according to D.1.2.1 (3).

EDITORIAL NOTE THE ABOVE FORMULAE WILL BE REVISED FOR THE SECOND DRAFT USING THE OUTCOME OF EXTENSIVE RECENT CALCULATIONS D.1.6 Combinations of meridional (axial) compression, circumferential (hoop) compression

and shear (1) The buckling interaction parameters to be used in 8.5.3 (3) may be obtained from: kx = 1,25 + 0,75 χx ... (D.49) kθ = 1,25 + 0,75 χθ ... (D.50)

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kτ = 1,75 + 0,25 χτ ... (D.51) ki = (χx χθ)2 ... (D.52) where:

χx, χθ, χτ are the elastic-plastic buckling reduction factors defined in 8.5.2, using the buckling parameters given in D.1.2 to D.1.4. (2) The three membrane stress components should be deemed to interact in combination at any point in the shell, except those adjacent to the boundaries. The buckling interaction check may be omitted for all points that lie within the boundary zone length R adjacent to either end of the cylindrical segment. The value of R is the smaller of: R = 0,1L ... (D.53) and R ≤ 0,16 r rt ... (D.54) (3) Where checks of the buckling interaction at all points is found to be onerous, the following provisions of (4) and (5) permit a simpler conservative assessment. If the maximum value of any of the buckling-relevant membrane stresses in a cylindrical shell occurs in a boundary zone of length R adjacent to either end of the cylinder, the interaction check of 8.5.3 (3) may be undertaken using the values defined in (4). (4) Where the conditions of (3) are met, the maximum value of each of the buckling-relevant membrane stresses occurring within the free length f (that is, outside the boundary zones, see Figure D.3a) may be used in the interaction check of 8.5.3 (3), with the free length defined as: f = L − 2R ... (D.55)

(5) For long cylinders as defined in D.1.2.1 (6), the interaction-relevant groups introduced into the interaction check may be restricted further than the provisions of paragraphs (3) and (4). The stresses deemed to be in interaction-relevant groups may then be restricted to any region of length int falling within the free remaining

length f for the interaction check (see Figure D.3b), with the region length defined as:

int = 1,3 r r t ... (D.56)

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R

R

L f

σθ

σx

τ

L

f

σθ

σx

τ

R

int

R a) in a short cylinder b) in a long cylinder Figure D.3: Examples of interaction-relevant groups of membrane stress components (6) If (3)-(5) above do not provide specific provisions for defining the relative locations or separations of interaction-relevant groups of membrane stress components, and a simple conservative treatment is still required, the maximum value of each membrane stress, irrespective of location in the shell, may be adopted into Formula (8.20).

D.2 Unstiffened cylindrical shells of stepwise variable wall thickness

D.2.1 General

D.2.1.1 Notation and boundary conditions (1) In this clause the following notation is used: L overall cylinder length r radius of cylinder middle surface j an integer index denoting the individual cylinder segments with constant wall thickness (from j = 1 to j = n) tj the constant wall thickness of segment j of the cylinder j the length of segment j of the cylinder (2) The following formulae may only be used for shells with boundary conditions BC 1 or BC 2 at both edges (see 5.2.2 and 8.3), with no distinction made between BC 1 and BC 2.

D.2.1.2 Geometry and joint offsets (1) Provided that the wall thickness of the cylinder increases progressively stepwise from top to bottom (see Figure D.5a), the procedures given in this sub-clause D.2 may be used.

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(2) Intended offsets e0 between plates of adjacent segments (see Figure D.4) may be treated as covered by the following formulae provided that the intended value e0 is less than the permissible value e0,p which should be taken as the smaller of: e0,p = 0,5 (tmax – tmin) ... (D.57)

and

e0,p = 0,5 tmin ... (D.58) where: tmax is the thickness of the thicker plate at the joint; tmin is the thickness of the thinner plate at the joint. NOTE These restrictions correspond to a) a smooth surface on one side (D.57), and b) a limitation that the thicker plate should not be thicker than twice the thickness of the thinner plate (D.58).

EDITORIAL NOTE THIS SUB-CLAUSE SHOULD BE REVIEWED IN THE LIGHT OF MORE RECENT CALCULATIONS (3) For cylinders with permissible intended offsets between plates of adjacent segments according to (2), the radius r may be taken as the mean value of all segments . (4) For cylinders with overlapping joints (lap joints), the provisions for lap-jointed construction given in D.3 should be used.

ttmmiinn

ttmmaaxx

ee00

Figure D.4: Intended offset e0 in a butt-jointed shell

D.2.2 Meridional (axial) compression (1) Each cylinder segment j of length j should be treated as an equivalent cylinder of overall length L = j and of uniform wall thickness t = tj according to D.1.2. (2) For long equivalent cylinders, as governed by D.1.2.1 (6), the parameter Cxb should be conservatively taken as Cxb = 1, unless a better value is justified by more rigorous analysis. EDITORIAL NOTE THIS SUB-CLAUSE WILL BE IMPROVED FOR THE SECOND DRAFT BY EXPLOITING MORE RECENT RESEARCH

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D.2.3 Meridional (axial) compression with co-existent internal pressure (1) Each cylinder segment j of length j should be treated as an equivalent cylinder of overall length L = j and of uniform wall thickness t = tj according to D.1.5. D.2.4 Circumferential (hoop) compression

D.2.4.1 Critical circumferential buckling stresses

EDITORIAL NOTE THE FOLLOWING TREATMENT IS COMPLICATED AND RELIES ON DATA READ FROM CHARTS (FIGURE D.6). IT SHOULD BE REPLACED BY THE WEIGHTED SMEARED METHOD, AND POSSIBLY MOVED TO THE TANKS STANDARD EN 1993-4-2. (1) If the cylinder consists of three segments with different wall thickness, the procedure of (4) to (7) should be applied to the real segments a, b and c, see Figure D.5b. (2) If the cylinder consists of only one segment (i.e. constant wall thickness), D.1 should be applied. (3) If the cylinder consists of two segments of different wall thickness, the procedure of (4) to (7) should be applied, treating two of the three fictitious segments, a and b, as being of the same thickness. (4) If the cylinder consists of more than three segments with different wall thicknesses (see Figure D.5a), it should first be replaced by an equivalent cylinder comprising three segments a, b and c (see Figure D.5b). The length of its upper segment, a, should extend to the upper edge of the first segment that has a wall thickness greater than 1,5 times the smallest wall thickness t1, but should not comprise more than half the total length L of the cylinder. The length of the two other segments b and c should be obtained as follows:

b = a and c = L − 2a, if a ≤ L/3 ... (D.59) b = c = 0,5 (L − a), if L/3 < a ≤ L/2 ... (D.60)

1 2 3

j

t1

tn

t2

tt33

tj L

t4

c

b

a

tc

tb

ta

eff

ta

(a) Cylinder of stepwise variable wall thickness (b) Equivalent cylinder comprising three segments (c) Equivalent single cylinder with uniform wall thickness Figure D.5: Transformation of stepped cylinder into equivalent cylinder

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(5) The fictitious wall thicknesses ta, tb and tc of the three segments should be determined as the weighted average of the wall thickness over each of the three fictitious segments: 1

a j jaa

t t=

... (D.61) 1

b j jbb

t t=

... (D.62) 1

c j jcc

t t=

... (D.63) (6) The three-segment-cylinder (i.e. the equivalent one or the real one respectively) should be replaced by an equivalent single cylinder of effective length eff and of uniform wall thickness t = ta, see Figure D.5c. The effective length should be determined from:

eff = a / κ ... (D.64) in which κ is a dimensionless factor obtained from Figure D.6. (7) For cylinder segments of moderate or short length, the elastic critical circumferential buckling stress of each cylinder segment j of the original cylinder of stepwise variable wall thickness should be determined from: , , , ,

aRcr j Rcr eff

j

t

tθ θσ σ

=

... (D.65) where σθ,Rcr,eff is the elastic critical circumferential buckling stress derived from D.1.3.1 (3), D.1.3.1 (5) or D.1.3.1 (7), as appropriate, of the equivalent single cylinder of length eff according to paragraph (6). The factor Cθ in these formulae should be given the value Cθ = 1,0. NOTE Formula D.65 may seem strange in that the resistance appears to be higher in thinner plates. The reason is that the whole cylinder bifurcates at a single critical external pressure, and Formula D.65 gives the membrane stress in each course at that instant. Since the external pressure is axially uniform, these stress values are smaller in the thicker courses. It should be noted that the design membrane circumferential stress, with which the resistance stresses will be compared in a design check, is also smaller in the thicker courses (see Figure D.7). If a stepped cylinder is elastic and under uniform external pressure, the ratio of the design membrane circumferential stress to the design resistance stress is constant throughout all courses. (8) The length of the shell segment is characterised in terms of the dimensionless length parameter ωj:

j jj

j j

r

r t rtω = =

... (D.66) (9) Where the cylinder segment j is long, a second additional assessment of the buckling stress should be made. The smaller of the two values derived from (7) and (10) should be used for the buckling design check of the cylinder segment j.

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11..2255 11..2255

11..0000

00..7755

00..5500

11..0000

00..1100 00..1155 00..2200

11..0000

00..7755 00..7755

00..550000..5500

00..2255

00

00..4400

00..2200

00..1100 00..1155

00..3300 00..3333

00..2255

11..2255aa

LL ==

00..4400 00..5500

00..2255 00..3300 00..3333

22..00

11..5500 11..7755

11..2255

11..00 ttbb

ttaa ==

00 00

11..2255

11..5500 11..5500 11..2255

11..00

ttbb

ttaa ==

ttcc

ttaa

ttcc

ttaa

ttcc

ttaa

22..2255 22..00

22..5500

22..00

22..5500

22..225522..5500

b = c

22..2255

11..7755

00..2255

11..7755

ttbb

ttaa == 11..00

a = b a = b

aa

LL ==aa

LL ==

Figure D.6: Factor κ for determination of the effective length eff (10) The cylinder segment j should be treated as long if:

1,63jj

r

tω > ... (D.67)

in which case the elastic critical circumferential buckling stress should be determined from: 42

, ,1

0,275 2,03jRcr j

j j

t rE

r tθσω

= + ⋅

... (D.68) D.2.4.2 Buckling strength verification for circumferential compression in a stepped wall (1) For each cylinder segment j, the conditions of 8.5 should be met, and the following check should be carried out:

σθ,Ed,j ≤ σθ,Rd,j ... (D.69) where: σθ,Ed,j is the key value of the circumferential compressive membrane stress, as detailed in the following clauses; σθ,Rd,j is the design circumferential buckling stress, as derived from the elastic critical circumferential buckling stress according to D.1.3.2. (2) Provided that the design value of the circumferential stress resultant nθ,Ed is constant throughout the length L, the key value of the circumferential compressive membrane stress in the segment j, should be taken as the simple value:

σθ,Ed,j = nθ,Ed / tj ... (D.70) (3) If the design value of the circumferential stress resultant nθ,Ed varies within the length L, the key value of the circumferential compressive membrane stress should be taken as a fictitious value σθ,Ed,j,mod determined from the maximum value of the circumferential stress resultant nθ,Ed anywhere within the length L divided by the local thickness tj (see Figure D.7), determined as:

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σθ,Ed,j,mod = max (nθ,Ed) / tj ... (D.71)

LL ttjj nnθθ,,EEdd

nnθθ,,EEdd,,mmoodd

σσθθ,,EEdd,,jj

σσθθ,,EEddjj,,mmoodd

Figure D.7: Key values of the circumferential compressive membrane stress in cases where nθ,Ed

varies within the length L

D.2.5 Shear

D.2.5.1 Critical shear buckling stresses (1) If no specific rule for evaluating an equivalent single cylinder of uniform wall thickness is available, the formulae of D.2.4.1 (1) to (6) may be applied. (2) The further determination of the elastic critical shear buckling stresses may on principle be performed as in D.2.4.1 (7) to (10), but replacing the circumferential compression formulae from D.1.3.1 by the relevant shear formulae from D.1.4.1. D.2.5.2 Buckling strength verification for shear (1) The rules of D.2.4.2 may be applied, but replacing the circumferential compression formulae by the relevant shear formulae. D.3 Unstiffened lap jointed cylindrical shells

D.3.1 General

D.3.1.1 Definitions

D.3.1.1.1 Circumferential lap joint A lap joint that runs in the circumferential direction around the shell axis. D.3.1.1.2 meridional lap joint A lap joint that runs parallel to the shell axis (meridional direction). D.3.1.2 Geometry and stress resultants (1) Where a cylindrical shell is constructed using lap joints (see Figure D.8), the following provisions may be used in place of those set out in D.2. (2) The following provisions apply both to lap joints that increase, and to those that decrease the radius of the middle surface of the shell.

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(3) Where the lap joint runs in a circumferential direction around the shell axis (circumferential lap joint), the provisions of D.3.2 should be used for meridional compression. (4) Where many lap joints run in a circumferential direction around the shell axis (circumferential lap joints) with changes of plate thickness down the shell, the provisions of D.3.3 should be used for circumferential compression. (5) Where a continuous lap joint runs parallel to the shell axis (unstaggered meridional lap joint), the provisions of D.3.3 should be used for circumferential compression. (6) In other cases, no special considerations are needed for the influence of lap joints on the buckling resistance.

ttmmiinn

tmax

Figure D.8: Lap jointed shell

D.3.2 Meridional (axial) compression (1) Where a lap jointed cylinder is subject to meridional compression, with circumferential lap joints, the buckling resistance may be evaluated as for a uniform or stepped-wall cylinder, as appropriate, but with the design resistance reduced by the factor 0,70. (2) Where a change of plate thickness occurs at the lap joint, the design buckling resistance may be taken as the same value as that of the thinner plate as determined in (1). EDITORIAL NOTE THIS SUB-CLAUSE WILL BE ENHANCED IN THE SECOND DRAFT BY EXPLOITING MORE RECENT CALCULATIONS D.3.3 Circumferential (hoop) compression (1) Where a lap jointed cylinder is subject to circumferential compression across continuous meridional lap joints, the design buckling resistance may be evaluated as for a uniform or stepped-wall cylinder, as appropriate, but with a reduction factor of 0,90. (2) Where a lap jointed cylinder is subject to circumferential compression, with many circumferential lap joints and a changing plate thickness down the shell, the procedure of D.2 should be used without the geometric restrictions on joint eccentricity, and with the design buckling resistance reduced by the factor 0,90. (3) Where lap joints are used in both directions, with staggered placement of the meridional lap joints in alternate strakes or courses (as in brickwork construction), the design buckling resistance should be evaluated as in (2), but no further resistance reduction need be applied.

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EDITORIAL NOTE CAN WE MAKE ANY BETTER ESTIMATES THAN WERE MADE WHEN THIS SUB-CLAUSE WAS DRAFTED IN ABOUT 1997? D.3.4 Shear (1) Where a lap jointed cylinder is subject to membrane shear, the buckling resistance may be evaluated as for a uniform or stepped-wall cylinder, as appropriate, without any special allowance for the lap joints. D.4 Unstiffened complete and truncated conical shells

D.4.1 General

D.4.1.1 Notation In this clause the following notation is used: h is the axial length (height) of the truncated cone; L is the meridional length of the truncated cone (=h/cosβ); r is the radius of the cone middle surface, perpendicular to axis of rotation, that varies linearly down the length; r1 is the radius at the small end of the cone; r2 is the radius at the large end of the cone; β is the apex half angle of cone.

β β

L

tr1

r2

r h

n,w

x,u

θ,v

nxθ=τtnx=σxt

nθx=τt

nθ=σθt

Figure D.9: Cone geometry, membrane stresses and stress resultants

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D.4.1.2 Boundary conditions (1) The following formulae should be used only for shells with boundary conditions BC 1 or BC 2 at both edges (see 5.2.2 and 8.3), with no distinction made between them. They should not be used for a shell in which any boundary condition is BC 3. (2) The rules in this sub-clause D.4 should be used only for the following two radial displacement restraint boundary conditions, at either end of the cone: “cylinder condition” w = 0 ... (D.72) “ring condition” u sin β + w cos β = 0 ... (D.73) D.4.1.3 Geometry (1) Only truncated cones of uniform wall thickness and with apex half angle β ≤ 65° (see Figure D.9) are covered by the following rules. D.4.2 Design buckling stresses

D.4.2.1 Equivalent cylinder (1) The design buckling stresses that are needed for the buckling strength verification according to 8.5 may all be found by treating the conical shell as an equivalent cylinder of length e and of radius re in which e and re depend on the type of membrane stress distribution in the conical shell. D.4.2.2 Meridional compression (1) For cones under meridional compression, the equivalent cylinder length e should be taken as:

e = L ... (D.74) (2) The equivalent cylinder radius at any buckling relevant location re should be taken as: cose

rr

β= ... (D.75)

(3) The characteristic imperfection amplitude Δwk, which may be needed for tolerance controls, should be taken as: t

Qw ⋅

= λΔ 22

k ... (D.76) in which Q is the meridional compression fabrication quality parameter, t is the local thickness and λ is the shell slenderness. The values of λ and Q should be taken as those for the equivalent cylinder, with the value of Q taken from Table D.2. D.4.2.3 Circumferential (hoop) compression (1) For cones under circumferential compression, the equivalent cylinder length e should be taken as:

e = L ... (D.77)

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(2) The equivalent cylinder radius re should be taken as: 1 2( )

2coser r

rβ

+= ... (D.78) D.4.2.4 Uniform external pressure (1) For cones under uniform external pressure q, that have either the boundary conditions BC1 at both ends or the boundary conditions BC2 at both ends, the following procedure may be used to produce a more economic design. (2) The equivalent cylinder length e should be taken as the lesser of:

e = L ... (D.79) and 2 (0,53 0,125 )

siner β

β = +

... (D.80) where the cone apex half angle β is measured in radians. (3) For shorter cones, where the equivalent length e is given by Formula (D.77), the equivalent cylinder radius re should be taken as:

1 20,55 0,45

coser r

rβ

+ =

... (D.81) (4) For longer cones, where the equivalent length e is given by Formula (D.79), the equivalent cylinder radius re should be taken as:

21 0,1

0,71 coser r

ββ

−=

... (D.82) where β is measured in radians (5) The buckling strength verification should be based on the notional circumferential membrane stress:

,e

Edr

qtθσ =

... (D.83)

in which q is the external pressure, and no account is taken of the meridional membrane stress induced by the external pressure. D.4.2.5 Shear (1) For cones under membrane shear stress, the equivalent cylinder length e should be taken as:

e = h ... (D.84) (2) The equivalent cylinder radius re should be taken as:

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11

1 cose gg

r rρ βρ

= + − ⋅

... (D.85) in which:

1 2

12gr r

rρ += ... (D.86)

D.4.2.6 Uniform torsion (1) For cones under membrane shear stress, where this is produced by uniform torsion (inducing a shear that varies quadratically down the meridian), the following procedure may be used to produce a more economic design, provided ρu ≤ 0,8 and the boundary conditions are BC2 at both ends. (2) The equivalent cylinder length e should be taken as: e = L ... (D.87) (3) The equivalent cylinder radius re should be taken as:

( ) 0,42,51 2 12cose ur r

r ρβ

+ = −

... (D.88) in which:

2

sinu

L

r

βρ = ... (D.89) D.4.3 Buckling strength verification

D.4.3.1 Meridional compression (1) The buckling design check should be carried out at that point of the cone where the combination of design meridional membrane stress σx,Ed and design meridional buckling stress σx,Rd according to D.4.2.2 is most critical. (2) In the case of meridional compression caused by a constant axial force on a truncated cone, both the small radius r1 and the large radius r2 should be considered as possible locations for the most critical position. (3) In the case of meridional compression caused by a constant global bending moment on the cone, the small radius r1 should be taken as the most critical. (4) The design meridional buckling stress σx,Rd should be determined for the equivalent cylinder according to D.1.2. D.4.3.2 Circumferential (hoop) compression and uniform external pressure (1) Where the circumferential compression is caused by uniform external pressure, the buckling design check should be carried out using the design circumferential membrane stress σθ,Ed determined using Formula (D.83) and the design circumferential buckling stress σθ,Rd according to D.4.2.1 and D.4.2.3 or D.4.2.4.

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(2) Where the circumferential compression is caused by actions other than uniform external pressure, the calculated stress distribution σθ,Ed(r) should be replaced by a fictitious enveloping stress distribution σθ,Ed,env(r) that everywhere exceeds the calculated value, but which would arise from a fictitious uniform external pressure. The buckling design check should then be carried out as in paragraph (1), but using σθ,Ed,env as it varies with recosβ, instead of σθ,Ed. (3) The design buckling stress σθ,Rd should be determined for the equivalent cylinder according to D.1.3. D.4.3.3 Shear and uniform torsion (1) In the case of shear caused by a constant global torque on the cone, the buckling design check should be carried out using the design membrane shear stress τxθ,Ed at the point with r = re cosβ and the design buckling shear stress τxθ,Rd according to D.4.2.1 and D.4.2.5 or D.4.2.6. (2) Where the shear is caused by actions other than a constant global torque (such as a global shear force on the cone), the calculated stress distribution τxθ,Ed(r) should be replaced by a fictitious enveloping stress distribution τxθ,Ed,env(r) that everywhere exceeds the calculated value, but which would arise from a fictitious global torque. The buckling design check should then be carried out as in (1), but using τxθ,Ed,env as it varies with recosβ, instead of τxθ,Ed. (3) The design shear buckling stress τxθ,Rd should be determined for the equivalent cylinder according to D.1.4.

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Annex E (normative) Formulae for use in reference resistance design

E.1 Cylindrical shells under global bending

E.1.1 General

E.1.1.1 Scope (1) The following rules apply to uniform unstiffened cylindrical shells subjected to global bending. (2) The rules given here apply only to the range r

t≤ ≤25 3000 (E.1)

E.1.1.2 Notation In this subclause the following notation is used (see Figure E.1): r is the radius of the cylinder middle surface; t is the uniform thickness of the cylinder; L is the length of the cylinder; M is the bending moment acting on the cylinder.

Figure E.1 — Cylinder under global bending

E.1.1.3 Boundary conditions (1) The rules given here are strictly applicable to cylinders with fixed end boundary conditions BC1r. Cylinders with BC1f may also be treated if the dimensionless length ω is greater than 5. E.1.1.4 Loading conditions (1) The following rules apply to global bending characterised by the maximum moment M (see Figure E.1).

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E.1.2 Buckling resistance under uniform global bending

E.1.2.1 Plastic reference resistance (1) The plastic reference moment should be obtained from: 2

R,pl y,k4 M r t f= (E.2) E.1.2.2 Elastic critical buckling resistance (1) The elastic critical buckling moment crR,M is given by:

1

EM C rt C Ert

ν= ≈

−2 2

R,cr m m21,813 1,90 (E.3)

where the factor Cm accounts for the difference between the linear bifurcation bending moment and the classical elastic critical bending moment. (2) The value of mC may be taken conservatively as: 1C

ω= +m 2

4 for BC1r boundaries (E.4) 1

1ω

= +Cm 7,5 for BC1f boundaries (E.5) where the first dimensionless length is given by:

L

rtω = (E.6)

E.1.2.3 Buckling capacity parameters (1) The geometrical reduction factor Gα depends on the second dimensionless length of the cylinder Ω, which should be determined as: ωΩ = =L t t

r r r (E.7)

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(2) The value of Gα should be determined as Gα = 0,9 when Ω < 0,5 (E.8) Gα = ( ) ( )

( )( ) ( )

( )

0,59

2,08

1,364

0,98

3,5 1 0,405 0,5..

3,6 1 0,416 0,5

2 1 1 6,87 10 7

25,2 1 0,145 7

−

− Ω − Ω −+

+ Ω − Ω − + × − Ω

− − Ω

when 0,5 ≤ Ω ≤ 7,0 (E.9) Gα = 0,516 when 0,70< Ω (E.10) (3) The imperfection reduction factor Iα should be obtained from:

( )0,72

1

1 /α =

+ ΔI

wka t (E.11)

in which 2.5

2.5

4,550,66

1.56

+ Ω=

+ Ω

a (E.12) where kwΔ is the characteristic imperfection amplitude:

rtwQ1

k =Δ (E.13) where: Q is the fabrication quality parameter given in Table E.1.

Table E.1 — Values of fabrication quality parameter Q

Quality Class Description Q Class A excellent 40 Class B high 25 Class C normal 16 NOTE For manufactured tubes, tests may indicate that the relevant value of Q is different from the above. The National Annex may define an appropriate value of Q. (4) The elastic buckling reduction factor α should be found as: Iααα G= (E.14) (5) The plastic range factor β should be found as:

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( ) 8,0k /2,11

60,01twΔ

β+

−= (E.15) (6) The interaction exponent η and the hardening limit χh should be found as:

( )η = + Δ0 65 0 2, , kw t and h 1 2χ = , (E.16) (7) The squash limit relative slenderness 0λ should be taken as:

30,00 =λ (E.17) EDITORIAL NOTE THE VALUES OF η, β, λo, and χh ARE CURRENTLY UNDER REVISION AND MAY BE REVISED FOR THE SECOND DRAFT .

E.1.3 Buckling resistance under global bending with axial compression

EDITORIAL NOTE FORMULAE HERE SHOULD, IF POSSIBLE, BE MADE COMPATIBLE WITH THOSE ADOPTED FOR EN 1993-1-1, THOUGH THE TREATMENT HERE MAY LEAD TO LESS CONSERVATIVE OUTCOMES WHERE APPROPRIATE. COMPATIBILITY WITH THE PROVISIONS OF THIS STANDARD FOR PURE AXIAL COMPRESSION WILL ALSO BE ACHIEVED IF POSSIBLE.

E.1.3.1 Plastic reference resistance (1) The plastic reference resistance should be taken using the plastic resistances under uniform compression and uniform bending as : E.1.3.2 Elastic critical buckling resistance (1) The elastic critical buckling moment should be taken from Formula E.3. (2) The elastic critical axial force should be taken from 8.2.1 as : EDITORIAL NOTE THE PARAMETERS FOR GLOBAL BENDING WITH AXIAL COMPRESSION ARE CURRENTLY IN PREPARATION FOR THE SECOND DRAFT.

E.1.3.3 Buckling capacity parameters (1) The geometrical reduction factor Gα should be determined as:

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E.1.4 Buckling resistance under global bending with moment gradient

EDITORIAL NOTE THE PARAMETERS FOR GLOBAL BENDING WITH SHEAR ARE CURRENTLY IN PREPARATION FOR THE SECOND DRAFT.

E.1.4.1 Plastic reference resistance (1) The plastic bending reference resistance should be used and taken as defined in E.1.2.1. (2) The plastic shear reference resistance should be taken as ... E.1.4.2 Elastic critical buckling resistance (1) The elastic critical buckling moment should be taken as defined in E.1.2.2. (2) The elastic critical shear should be taken as E.1.4.3 Buckling capacity parameters (1) The geometrical reduction factor Gα should be determined as: EDITORIAL NOTE ALL FORMULAE FROM HERE ON WILL NEED TO BE RE-NUMBERED AND CAREFUL CHECKS MADE THAT APPROPRIATE REFERENCES TO THEM ARE CORRECT E.1.5 Buckling resistance evaluation

E.1.5.1 Characteristic buckling resistance (1) The characteristic buckling resistance should be determined according to 8.6.3, with the leading load EdF taken as the applied bending moment EdM , the reference plastic resistance FR,pl taken as MR,pl (Formula (E.2)) and the reference elastic critical resistance FR,cr taken as MR,cr (Formula (E.3)). (2) The reference resistances are then found as:

Ed

plR,pl M

MR = and

Ed

crR,cr M

MR = (E.18)

(3) The relative slenderness λ is given by: crR,

plR,

cr

plM

M

R

R==λ (E.19)

(4) The characteristic buckling resistance or the buckling moment is given by: plk RR χ= or plR,kR, MM χ= (E.20) where:

χ is the elastic-plastic buckling reduction factor according to 8.6.3 (5).

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E.1.5.2 Buckling strength verification (1) The buckling verification is then:

1M1

kd ≥=

γR

R (E.21) where the safety factor γM1 is taken as defined in 8.5.2 (9) and (10). E.2 Complete and partial spherical shells

E.2.1 General

E.2.1.1 Scope (1) The following rules apply to spherical shells and spherical caps under internal vacuum or uniform external pressure with different boundary conditions. The wall thickness of the spherical shell should not vary significantly. The shell is unstiffened. EDITORIAL NOTE IN THE CONTEXT OF THE USE OF THIS SUB-CLAUSE ON TANKS AND SIMILAR STRUCTURES, IT IS DESIRABLE THAT THE BOUNDARY SUPPORT CONDITIONS ARE AMENDED TO ADDRESS THE PRESENCE OF AN EAVES RING, AND THE VERTICAL SUPPORT FROM A CYLINDRICAL SHELL

THIS SHOULD BE POSSIBLE DURING THE PERIOD BEFORE THE SECOND DRAFT

(2) The rules are limited to the ranges given by: 0003100 s ≤≤

t

r (E.22) φ ≤ 135° (spherical caps) with the special addition of φ = 180° (complete sphere) (E.23) No lower limit on the range of φ is given, but very flat spherical caps should be checked by means of plate bending analysis. The test of Formula (E.22) defines the corresponding limit of application. (3) The shell segments should be connected by welded butt-joints or by bolted symmetrical double-lap-joints or the shell should consist of a single spherical element without any interior joints.

E.2.1.2 Notation In this sub-clause the following notation is used (see Figure E.2): rs is the radius of the sphere (shell middle surface); r is the simple radius of the shell middle surface = r(x), perpendicular to the axis of rotation;

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r0 is the radius of the base circle of the spherical cap; t is the thickness of the shell; φ is the semi-angle of the spherical cap.

Key 1 spherical cap 2 circumference 3 base circle 4 complete sphere

Figure E.2 — Spherical shell geometry

E.2.1.3 Support and boundary conditions (1) The rules given here are applicable only to shells that are supported as indicated in Figure E.3 with the following boundary conditions: SC 1: complete sphere without support or complete sphere with meridional support around a complete circumference; SC 2: spherical cap with clamped edges; SC 3: spherical cap with edges with displacement restraint in both the meridional direction and normal to the shell middle surface, and flexurally pinned; SC 4: spherical cap with edges with displacement restraint in the meridional direction, but free normal to the shell middle surface, and flexurally pinned; SC 5: spherical cap with edges free to displace in the plane of the base circle.

Figure E.3 — Illustrations of the different support conditions

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E.2.1.4 Loading conditions (1) The following apply only to uniform internal vacuum or external pressure loading p perpendicular to the shell wall (see Figure E.4). The design value of the pressure difference between the inside and outside surfaces pEd should be taken as the key value.

a) Complete sphere subjected to uniform internal vacuum or external pressure b) Spherical cap subjected to uniform external pressure Figure E.4 — Loading on spherical shells and caps (2) For the loading cases of self-weight or snow, the procedures here may be used to obtain a conservative estimate of resistance if the value of the pressure load p is taken as the maximum surface load normal to the middle surface of the shell.

E.2.2 Tolerances for spherical shells (1) The geometrical tolerances are classified into three Fabrication Tolerance Quality Classes A to C. (2) For the buckling relevant tolerances, the provisions of Sub-clause 8.4 apply by taking the radius rs of the spherical shell in place of the cylinder radius r and the diameter 2rs instead of the diameter d of the cylinder. The measurement of dimples (8.4.4) should be performed in both the meridional and circumferential directions using the stick lengths gx given by Formula (8.6) and gw given by Formula (8.8). It is not necessary to use the stick length θg given by Formula (8.7) at all. (3) The tolerance limits for each fabrication tolerance quality class given in subclause 8.4 should be used.

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E.2.3 Buckling design

E.2.3.1 Limitation on buckling calculations (1) It is not necessary to check the resistance to buckling in shells that satisfy either of the following conditions: Spherical shells that satisfy the condition: c

ky,

s20

Cf

E

t

r ⋅≤ (E.24) Very flat spherical caps that satisfy the condition:

trr

r

/1,1

ss

0 ≤ (E.25) E.2.3.2 Elastic critical buckling resistance (1) The elastic critical buckling pressure p

R,cr is given by:

( )2

sc

2crR,

13

2

⋅⋅⋅

−=

r

tEC

v

p (E.26) where the factor Cc depends on the support conditions and should be taken from Table E.4. EDITORIAL NOTE THIS EXPRESSION, WITH HIGH SENSITIVITY TO THE BOUNDARY CONDITIONS, SHOULD BE REPLACED BY ONE THAT WILL MATCH LBA CALCULATIONS DONE BY FE ANALYSTS.

THE NET OUTCOME IN GNA MAY NEED TO BE THE SAME, BUT GN PHENOMENA SHOULD NOT BE USED IN DEFINING THE ELASTIC CRITICAL STATE, SINCE THE USERS OF THE MNA/LBA METHOD MIGHT WELL ASSUME THAT THE ONLY GN REDUCTION IS GIVEN BY FORMULA E.28

IT IS EXPECTED THAT THE SECOND DRAFT WILL INCLUDE THE SUPPORT CONDITION OF A SPHERICAL SHELL SUPPORTED ON A CYLINDER WITH A STIFFENING RING AT THE JUNCTION. FOR THIS CASE, THE LBA AND GNA CALCULATIONS WILL BE CLEARLY SEPARATED.

Table E.4 — Values of Cc for different support conditions

Support condition SC

SC 1

SC 2

SC 3

SC 4

SC 5

Cc 1,0 0,8 0,7 0,4 0,1 Applicable for complete sphere 135≤φ E.2.3.3 Plastic reference resistance (1) The plastic reference resistance should be obtained from:

splky,plR,

2r

tCfp ⋅⋅= (E.27)

in which the factor Cpl is a function of the support conditions and should be taken from Table E.5.

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Table E.5 — Values of plC for different support conditions

Support condition SC

SC 1

SC 2

SC 3

SC 4

SC 5

Cpl 1,0 0,9 0,9 0,8 0,2 Applicable for complete sphere 135≤φ E.2.3.4 Buckling capacity parameters for uniform external pressure (1) The geometric reduction factor Gα is given as:

70,0G =α (E.28) (2) The imperfection reduction factor Iα should be obtained from: ( ) 75,0

k /90,111

twΔα

+=I (E.29)

in which kwΔ is the characteristic imperfection amplitude defined by: tr

Qw sk

1=Δ (E.30) where:

Q is the fabrication quality parameter given in (3). (3) The fabrication quality parameter Q should be taken from Table E.6 for the specified fabrication tolerance quality. Table E.6 — Values of fabrication quality parameter Q

Quality class Description Q Class A excellent 40 Class B high 25 Class C normal 16 (4) The elastic buckling reduction factor α should be found as: Iααα G= (E.31) (5) The squash limit relative slenderness 0λ , the plastic range factor β, the interaction exponent η and the hardening limit χh should be taken as:

0 0,20λ = 0,70β = 0,1=η ,hχ = 1 0 (E.32)

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E.2.3.5 Characteristic buckling resistance (1) The characteristic buckling resistance should be determined according to 8.6.3, with the leading load FEd taken as the applied external pressure pEd, the reference plastic resistance FR,pl taken as pR,pl (Formula (E.27)) and the reference elastic critical resistance FR,cr taken as pR,cr (Formula (E.26)). (2) The reference resistances are given by:

Ed

plR,pl p

pR = and

Ed

crR,cr p

pR = (E.33)

(3) The relative slenderness λ is given by: cr

pl

crR,

plR,R

R

p

p==λ (E.34)

(4) The characteristic buckling resistance and the buckling pressure are given by: plk RR χ= or plR,kR, pp χ= (E.35) where:

χ is the elastic-plastic buckling reduction factor according to 8.6.3(5). E.2.4 Buckling strength verification (1) The buckling verification is then:

1M1

kd ≥=

γR

R or R,kR,d E,dM1pp pγ

= ≥ (E.36) where the safety factor γM1 is taken as defined in 8.5.2 (9) and (10).

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Bibliography

(1) ECCS EDR5 (2013) European Recommendations for Steel Construction: Buckling of Shells, 5th edition revised second impression, Edited by J.M. Rotter and H. Schmidt, European Convention for Constructional Steelwork, Brussels, 388 pp. (2) Rotter, J.M. (2011) “Shell buckling design and assessment and the LBA-MNA methodology”, Stahlbau, Vol. 80, Heft 11, Nov., pp 791-803. (3) Doerich, C. and Rotter, J.M. (2011) “Accurate determination of plastic collapse loads from finite element analyses”, Journal of Pressure Vessel Technology, Vol. 133, Issue 1, February, pp. 011202-1 to 10. (4) Rotter, J.M. (2016) “The new method of Reference Resistance Design for shell structures”, Proc. SDSS 2016, International Colloquium on Stability and Ductility of Steel Structures, 30 May - 1 June, Timisoara, Romania, pp 623-630.