SIST EN 1993-1-6:2007

SLOVENSKI STANDARD
01-julij-2007
1DGRPHãþD
SIST ENV 1993-1-6:2001
Evrokod 3: Projektiranje jeklenih konstrukcij - 1-6.del: Trdnost in stabilnost
lupinastih konstrukcij
Eurocode 3 - Design of steel structures - Part 1-6: Strength and Stability of Shell
Structures
Eurocode 3 - Bemessung und Konstruktion von Stahlbauten - Teil 1-6: Festigkeit und
Stabilität von Schalen
Eurocode 3 - Calcul des structures en acier - Partie 1-6: Résistance et stabilité des
structures en coque
Ta slovenski standard je istoveten z: EN 1993-1-6:2007
ICS:
91.010.30 7HKQLþQLYLGLNL Technical aspects
91.080.10 Kovinske konstrukcije Metal structures
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

EUROPEAN STANDARD
EN 1993-1-6
NORME EUROPÉENNE
EUROPÄISCHE NORM
February 2007
ICS 91.010.30; 91.080.10 Supersedes ENV 1993-1-6:1999
English Version
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: Eurocode 3 - Bemessung und Konstruktion von
Résistance et stabilité des structures en coque Stahlbauten - Teil 1-6: Festigkeit und Stabilität von Schalen
This European Standard was approved by CEN on 12 June 2006.
CEN members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this European
Standard the status of a national standard without any alteration. Up-to-date lists and bibliographical references concerning such national
standards may be obtained on application to the CEN Management Centre or to any CEN member.
This European Standard exists in three official versions (English, French, German). A version in any other language made by translation
under the responsibility of a CEN member into its own language and notified to the CEN Management Centre has the same status as the
official versions.
CEN members are the national standards bodies of Austria, Belgium, Bulgaria, Cyprus, Czech Republic, Denmark, Estonia, Finland,
France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal,
Romania, Slovakia, Slovenia, Spain, Sweden, Switzerland and United Kingdom.
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Management Centre: rue de Stassart, 36  B-1050 Brussels
© 2007 CEN All rights of exploitation in any form and by any means reserved Ref. No. EN 1993-1-6:2007: E
worldwide for CEN national Members.

EN 1993-1-6: 2007 (E)
Contents Page
1. General 4
1.1 Scope 4
1.2 Normative references 5
1.3 Terms and definitions 6
1.4 Symbols 11
1.5 Sign conventions 15
2 Basis of design and modelling 15
2.1 General 15
2.2 Types of analysis 15
2.3 Shell boundary conditions 17
3 Materials and geometry 18
3.1 Material properties 18
3.2 Design values of geometrical data 18
3.3 Geometrical tolerances and geometrical imperfections 18
4 Ultimate limit states in steel shells 19
4.1 Ultimate limit states to be considered 19
4.2 Design concepts for the limit states design of shells 20
5 Stress resultants and stresses in shells 23
5.1 Stress resultants in the shell 23
5.2 Modelling of the shell for analysis 23
5.3 Types of analysis 26
6 Plastic limit state (LS1) 26
6.1 Design values of actions 26
6.2 Stress design 26
6.3 Design by global numerical MNA or GMNA analysis 27
6.4 Direct design 28
7 Cyclic plasticity limit state (LS2) 28
7.1 Design values of actions 28
7.2 Stress design 29
7.3 Design by global numerical MNA or GMNA analysis 29
7.4 Direct design 30
8 Buckling limit state (LS3) 30
8.1 Design values of actions 30
8.2 Special definitions and symbols 30
8.3 Buckling-relevant boundary conditions 31
8.4 Buckling-relevant geometrical tolerances 31
8.5 Stress design 38
8.6 Design by global numerical analysis using MNA and LBA analyses 40
8.7 Design by global numerical analysis using GMNIA analysis 43
9 Fatigue limit state (LS4) 48
9.1 Design values of actions 48
9.2 Stress design 48
EN 1993-1-6: 2007 (E)
9.3 Design by global numerical LA or GNA analysis 49
ANNEX A (normative) 50
Membrane theory stresses in shells 50
A.1 General 50
A.2 Unstiffened cylindrical shells 51
A.3 Unstiffened conical shells 52
A.4 Unstiffened spherical shells 53
ANNEX B (normative) 54
Additional expressions for plastic collapse resistances 54
B.1 General 54
B.2 Unstiffened cylindrical shells 55
B.3 Ring stiffened cylindrical shells 57
B.4 Junctions between shells 59
B.5 Circular plates with axisymmetric boundary conditions 62
ANNEX C (normative) 63
Expressions for linear elastic membrane and bending stresses 63
C.1 General 63
C.2 Clamped base unstiffened cylindrical shells 64
C.3 Pinned base unstiffened cylindrical shells 66
C.4 Internal conditions in unstiffened cylindrical shells 68
C.5 Ring stiffener on cylindrical shell 69
C.6 Circular plates with axisymmetric boundary conditions 71
ANNEX D (normative) 73
Expressions for buckling stress design 73
D.1 Unstiffened cylindrical shells of constant wall thickness 73
D.2 Unstiffened cylindrical shells of stepwise variable wall thickness 83
D.3 Unstiffened lap jointed cylindrical shells 88
D.4 Unstiffened complete and truncated conical shells 90

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/TC250 « Structural
Eurocodes », the Secretariat of which is held by BSI. CEN/TC250 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 August 2007, and conflicting National Standards shall
be withdrawn at latest by March 2010.
This Eurocode supersedes ENV 1993-1-6.
According to the CEN-CENELEC Internal Regulations, the National Standard Organizations of the
following countries are bound to implement this European Standard: Austria, Belgium, Bulgaria Cyprus,
Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy,
EN 1993-1-6: 2007 (E)
Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Romania, 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:
− 3.1.(4)
− 4.1.4 (3)
− 5.2.4 (1)
− 6.3 (5)
− 7.3.1 (1)
− 7.3.2 (1)
− 8.4.2 (3)
− 8.4.3 (2)
− 8.4.3 (4)
− 8.4.4 (4)
− 8.4.5 (1)
− 8.5.2 (2)
− 8.5.2 (4)
− 8.7.2 (7)
− 8.7.2 (16)
− 8.7.2 (18) (2 times)
− 9.2.1 (2)P
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 1993-1-3, 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.
EN 1993-1-6: 2007 (E)
(4) This Standard is concerned with the requirements for design against the ultimate limit states of:
plastic limit;
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 expressions for
the hand calculation of unstiffened cylinders and cones are given in the Annexes.
(7) Cylindrical and conical panels are not explicitly covered by this Standard. However, the
provisions can be applicable if the appropriate boundary conditions are duly taken into account.
(8) 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.
(9) 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.
(10) The provisions in this Standard apply to structures that satisfy the brittle fracture provisions
given in EN 1993-1-10.
(11) The provisions of this Standard apply to structural design under actions that can be treated as
quasi-static in nature.
(12) In this Standard, it is assumed that both wind loading and bulk solids flow can, in general, be
treated as quasi-static actions.
(13) 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.
(14) The provisions in this Standard apply to structures that are constructed in accordance with
EN 1090-2.
(15) This Standard does not cover the aspects of leakage.
(16) This Standard is intended for application to structures within the following limits:
design metal temperatures within the range −50°C to +300°C;
radius to thickness ratios within the range 20 to 5000.

NOTE: It should be noted 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.
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
EN 1993-1-6: 2007 (E)
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 ;
EN 1993 Eurocode 3: Design of steel structures:
Part 1.1: General rules and rules for buildings;
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 4.3: Pipelines;
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. The shell can be of any length.
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.
EN 1993-1-6: 2007 (E)
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.
Where the shell is stiffened on either one or both surfaces, the reference middle surface is still taken
as the middle surface of the curved shell plate. The middle surface 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.
1.3.1.7 junction
The line at which two or more shell segments meet: it can include a stiffener. The circumferential line
of attachment of a ring stiffener to the shell may be treated as a junction.
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. It is provided to increase the stability, or to assist with the introduction of local loads. It
is 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. It is 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. It is normally assumed to have no stiffness for deformations out of its own
plane (meridional displacements of the shell) but is stiff for deformations in the plane of the ring. It is
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. It is 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. It is a primary load carrying structural member, provided for
the distribution of local loads into the shell.
1.3.2 Limit states
1.3.2.1 plastic limit
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.
EN 1993-1-6: 2007 (E)
1.3.2.4 buckling
The ultimate limit state where the structure suddenly loses its stability under membrane compression
and/or shear. It leads 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. Its magnitude
can vary in both the meridional and circumferential directions (e.g. under solids loading in a silo).
1.3.3.4 external pressure
Component of the surface loading acting normal to the shell in the inward direction. Its magnitude can
vary in both the meridional and circumferential directions (e.g. under wind).
1.3.3.5 hydrostatic pressure
Pressure varying linearly with the axial coordinate of the shell of revolution.
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).
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.
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.
1.3.3.11 thermal action
Temperature variation either down the shell meridian, or around the shell circumference or through
the shell thickness.
EN 1993-1-6: 2007 (E)
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. Under elastic conditions, each of these stress resultants induces a stress state
that is uniform through the shell thickness. There are three membrane stress resultants at any point
(see figure 1.1(e)).
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. 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. There are 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. Under elastic conditions, each of these stress resultants induces a stress state that varies
parabolically through the shell thickness. There are 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. It is 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 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.
1.3.5.4 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. It should be noted that, where an eigenvalue is mentioned, this does not
relate to vibration modes.
1.3.5.5 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
EN 1993-1-6: 2007 (E)
accounts full for any change in geometry due to the actions on the shell. A bifurcation eigenvalue
check is included at each load level.
1.3.5.6 materially nonlinear analysis (MNA)
An analysis based on shell bending theory applied to the perfect structure, using the assumption of
small deflections, as in 1.3.4.3, but adopting a nonlinear elasto-plastic material law.
1.3.5.7 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 nonlinear elasto-plastic material law. A
bifurcation eigenvalue check is included at each load level.
1.3.5.8 geometrically nonlinear elastic analysis with imperfections included (GNIA)
An analysis with imperfections explicitly included, similar to a GNA analysis as defined in 1.3.4.5,
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). The
imperfection may also cover the effects of deviations in boundary conditions and / or the effects of
residual stresses. A bifurcation eigenvalue check is included at each load level.
1.3.5.9 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 full for any change in geometry due to the actions on the shell and a nonlinear elasto-
plastic material law. The imperfections may also include imperfections in boundary conditions and
residual stresses. A bifurcation eigenvalue check is included at each load level.
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. This consists primarily of
membrane stresses, but in some conditions, 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). These stresses are 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 critical buckling resistance
The smallest bifurcation or limit load determined assuming the idealised conditions of elastic material
behaviour, perfect geometry, perfect load application, perfect support, material isotropy and absence
of residual stresses (LBA analysis).
1.3.7.2critical buckling stress
The membrane stress associated with the critical buckling resistance.
1.3.7.3 plastic reference resistance
The plastic limit load, determined assuming the idealised conditions of rigid-plastic material
behaviour, perfect geometry, perfect load application, perfect support and material isotropy (modelled
using MNA analysis).
EN 1993-1-6: 2007 (E)
1.3.7.4 characteristic buckling resistance
The load associated with buckling in the presence of inelastic material behaviour, the geometrical and
structural imperfections that are inevitable in practical construction, and follower load effects.
1.3.7.5 characteristic buckling stress
The membrane stress associated with the characteristic buckling resistance.
1.3.7.6 design buckling resistance
The design value of the buckling load, obtained by dividing the characteristic buckling resistance by
the partial factor for resistance.
1.3.7.7 design buckling stress
The membrane stress associated with the design buckling resistance.
1.3.7.8 key value of the stress
The value of stress in a non-uniform stress field that is used to characterise the stress magnitudes in a
buckling limit state assessment.
1.3.7.9 fabrication tolerance quality class
The category of fabrication tolerance requirements that is assumed in design, see 8.4.
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:
p normal to the shell;
n
p meridional surface loading parallel to the shell;
x
p circumferential surface loading parallel to the shell;
θ
(4) Line forces:
P load per unit circumference normal to the shell;
n
P load per unit circumference acting in the meridional direction;
x
P load per unit circumference acting circumferentially on the shell;
θ
(5) Membrane stress resultants:
n meridional membrane stress resultant;
x
n circumferential membrane stress resultant;
θ
n membrane shear stress resultant;
xθ
(6) Bending stress resultants:
m meridional bending moment per unit width;
x
EN 1993-1-6: 2007 (E)
m circumferential bending moment per unit width;
θ
m twisting shear moment per unit width;
xθ
q transverse shear force associated with meridional bending;
xn
q transverse shear force associated with circumferential bending;
θn
(7) Stresses:
σ meridional stress;
x
σ circumferential stress;
θ
σ von Mises equivalent stress (can also take negative values during cyclic loading);
eq
τ, τ in-plane shear stress;
xθ
τ , τ meridional, circumferential transverse shear stresses associated with bending;
xn θn
(8) Displacements:
u meridional displacement;
v circumferential displacement;
w displacement normal to the shell surface;
β meridional rotation, see 5.2.2;
φ
(9) Shell dimensions:
d internal diameter of shell;
L total length of the shell;
ℓ length of shell segment;
ℓ gauge length for measurement of imperfections;
g
ℓ gauge length in circumferential direction for measurement of imperfections;
gθ
ℓ gauge length across welds for measurement of imperfections;
gw
ℓ gauge length in meridional direction for measurement of imperfections;
gx
ℓ limited length of shell for buckling strength assessment;
R
r radius of the middle surface, normal to the axis of revolution;
t thickness of shell wall;
t maximum thickness of shell wall at a joint;
max
t minimum thickness of shell wall at a joint;
min
t average thickness of shell wall at a joint;
ave
β apex half angle of cone;
EN 1993-1-6: 2007 (E)
θ
v
Circumferential
n
w
Normal
x
Meridional
u
Directions Displacements
Coordinates
z
σ
θ
p
θ σ
x
τ
xn
θ
τ
xθ
p
n
τ
θn
φ
σ
p x
x
σ
θ
Transverse shear
Surface pressures Membrane stresses
stresses
Figure 1.1: Symbols in shells of revolution

(10) Tolerances, see 8.4:
e eccentricity between the middle surfaces of joined plates;
U accidental eccentricity tolerance parameter;
e
U out-of-roundness tolerance parameter;
r
U initial dimple imperfection amplitude parameter for numerical calculations;
n
U initial dimple tolerance parameter;
Δw tolerance normal to the shell surface;
(11) Properties of materials:
E Young’s modulus of elasticity;
f von Mises equivalent strength;
eq
f yield strength;
y
f ultimate strength;
u
ν Poisson’s ratio;
(12) Parameters in strength assessment:
C coefficient in buckling strength assessment;
D cumulative damage in fatigue assessment;
F generalised action;
F action set on a complete structure corresponding to a design situation (design
Ed
values);
F calculated values of the action set at the maximum resistance condition of the
Rd
structure
(design values);
r characteristic reference resistance ratio (used with subscripts to identify the basis):
Rk
defined as
EN 1993-1-6: 2007 (E)
the ratio (F / F );
Rk Ed
r plastic reference resistance ratio (defined as a load factor on design loads using
Rpl
MNA
analysis);
r critical buckling resistance ratio (defined as a load factor on design loads using LBA
Rcr
analysis);
NOTE: For consistency of symbols throughout the EN1993 the symbol for the reference resistance
ratio r is used instead of the symbol R . However, in order to avoid misunderstanding, it needs to be
Ri Ri
noted here that the symbol R is widely used in the expert field of shell structure design.
Ri
k calibration factor for nonlinear analyses;
k power of interaction expressions in buckling strength interaction expressions;
n number of cycles of loading;
α elastic imperfection reduction factor in buckling strength assessment;
β plastic range factor in buckling interaction;
γ partial factor;
Δ range of parameter when alternating or cyclic actions are involved;
ε plastic strain;
p
η interaction exponent for buckling;
−
λ relative slenderness of shell;
−
λ overall relative slenderness for the complete shell (multiple segments);
ov
− −
λ squash limit relative slenderness (value of λ above which resistance reductions due
to instability or change of geometry occur);
− −
λ plastic limit relative slenderness (value of λ below which plasticity affects the
p
stability);
ω relative length parameter for shell;
χ buckling reduction factor for elastic-plastic effects in buckling strength assessment;
χ overall buckling resistance reduction factor for complete shell;
ov
(13) Subscripts:
E value of stress or displacement (arising from design actions);
F actions;
M material;
R resistance;
cr critical buckling value;
d design value;
int internal;
k characteristic value;
max maximum value;
min minimum value;
nom nominal value;
pl plastic value;
EN 1993-1-6: 2007 (E)
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.
(4) For simplicity, in section 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 sections 6 and 8;
limitation of cracks due to cyclic plastification, see section 7;
limitation of cracks due to fatigue, see section 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, 4.2, 4.3).
(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, 4.2, 4.3).
(5) All actions should be introduced using their design values according to EN 1991 and EN 1993
Parts 3.1, 3.2, 4.1, 4.2, 4.3 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 section 4,
depending on the limit state and other considerations:
Global analysis, see 2.2.2;
Membrane theory analysis, see 2.2.3;
Linear elastic shell analysis, see 2.2.4;
Linear elastic bifurcation analysis, see 2.2.5;
Geometrically nonlinear elastic analysis, see 2.2.6;
Materially nonlinear analysis, see 2.2.7;
Geometrically and materially nonlinear analysis, see 2.2.8;
Geometrically nonlinear elastic analysis with imperfections included, see 2.2.9;
Geometrically and materially nonlinear analysis with imperfections included, see 2.2.10.

EN 1993-1-6: 2007 (E)
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 should only 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 significant 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 necessarily fulfil the 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
primary stresses (LS1).
2.2.4 Linear elastic shell analysis (LA)
(1) The linearity of the theory results from the assumptions of a linear elastic material law and the
linear small deflection theory. Small deflection 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 (LS2 and LS4).
2.2.5 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 r , see 8.6 and 8.7 (LS3).
R
cr
2.2.6 Geometrically nonlinear elastic analysis (GNA)
(1) A GNA analysis satisfies both equilibrium and compatibility of the deflections 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 (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 in checking the limit state LS3, see 8.7.
(3) Where this analysis is used for a buckling load evaluation, the eigenvalues of the system must
be checked to ensure that the numerical process does not fail to detect a bifurcation in the load path.
2.2.7 Materially nonlinear analysis (MNA)
(1) The result of an MNA analysis gives the plastic limit load, which can be interpreted as a load
amplification factor r on the design value of the loads F . This analysis provides the plastic
Rpl Ed
reference resistance ratio r used in 8.6 and 8.7.
Rpl
(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.
EN 1993-1-6: 2007 (E)
2.2.8 Geometrically and materially nonlinear analysis (GMNA)
(1) The result of a GMNA analysis, analogously to 2.2.7, gives the geometrically nonlinear plastic
limit 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 elasto-plastic buckling load of the perfect structure, that may be of assistance in
checking the limit state LS3, see 8.7.
(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.
2.2.9 Geometrically nonlinear elastic analysis with imperfections 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 must be taken to ensure that the local stresses do not exceed values at which material
nonlinearity may affect the behaviour.
2.2.10 Geometrically and materially nonlinear analysis with imperfections 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, an additional GMNA analysis of the
perfect shell should always be conducted to ensure 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 the constraint of the displacements in the plane of the shell wall (meridional and
circumferential) because of the significant effect these have on shell strength and buckling resistance.
(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 prevented or not.
(4) The shell edge rotation β should be particularly considered in short shells and in the
φ
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
expressions from Annexes A to D.
EN 1993-1-6: 2007 (E)
(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 materials with nonlinear stress-strain curves are 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.
(3) In a global numerical analysis using material nonlinearity, the 0,2% proof stress should be used
to represent the yield stress f in all relevant expressions. The stress-strain curve should be obtained
y
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.
NOTE: The national annex may give information about material properties at temperatures exceeding
150°C.
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, 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 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.
(5) The buckling design rules of this Standard should not be applied outside the ranges of the r/t
ratio set out in section 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 standards due to the requirements of serviceability. Relevant items are:
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),
local dimples (local normal deviations from the nominal middle surface).

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 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, additional buckling-relevant geometrical tolerances have to be observed in order to keep
the geometrical imperfections within specified limits. These buckling-relevant geometrical tolerances
are quantified in section 8 or in the relevant EN 1993 application parts.
EN 1993-1-6: 2007 (E)
(3) Calculation values for the deviations of the shell surface geometry from the nominal geometry,
as required for geometrical imperfection assumptions (overall imperfect
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