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1
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Recent developments in steel structures:
Introduction to Eurocode 3 & Elliptical hollow sections
Dr Leroy GardnerImperial College London
South Africa, 2009
2
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Introduction to Eurocode 3
2
3
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
• Introduction
• Load combinations
• Frame stability
• Key design rules
Outline:
Overview
4
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
• EN 1990 – Basis of structural design
• EN 1991 – Actions of structures
Scope of Eurocodes
The first 2 codes are material independent:
• A total of 10 codes (comprising 58 documents)
Scope of structural Eurocodes:
3
5
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
• EN 1992 – Design of concrete structures
• EN 1993 – Design of steel structures
• EN 1994 – Design of composite structures
• EN 1995 – Design of timber structures
• EN 1996 – Design of masonry structures
• EN 1997 – Geotechnical design
• EN 1998 – Design of structures for earthquakes
• EN 1999 – Design of aluminium structures
Scope of Eurocodes
Remaining 8 codes focus on materials:
6
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
• Codes are published by CEN in 3 languages:
• English
• French
• German
• All codes originally developed in English, and then ‘exactly’ translated
• Other participating counties will translate through National standards bodies
Eurocodes
4
7
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
National Annexes
National Annexes:
• Every Eurocode will contain a National Annex
• National choice
• Non-Contradictory Complementary Information
• Majority of UK NAs now published
8
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
• All codes are published – 58 documents
• 48 have a UK National Annex
• Conflicting steel design standards (e.g. BS 5950) to be withdrawn in 2010
Present situation
Present situation of Eurocodes:
5
9
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Efficiency of EC3 vs BS 5950
• The common design rules are similar and thus lead to similar results
• Where vertical loading governs, it has been estimated that structures designed to EC3 will be 6-8% lighter.
Comparing the results of design to Eurocode 3 and BS 5950:
10
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Computer aided analysis and design
Computer aided analysis and design:
• Greater codification of structural analysis• FE may be implemented through e.g. ‘the general method’• Semi-rigid joints in structural analysis• Beam-column formulations need to be automated
6
11
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Buildings and bridges
• Philosophy in structural Eurocodes not to repeat material
• BS 5950 vs BS 5400
• Owing to commonality of rules, it will be easier to make the transition between the two disciplines using Eurocode.
12
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Loading and load combinations
7
13
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
• EN 1990 – Basis of design
• Includes load combinations
• EN 1991 – Actions on structures
• Dead loads, imposed loads, wind loads etc
EN 1990 and EN 1991
EN 1990 and EN 1991:
14
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Load combinations
Equation 6.10:
∑∑1>i
i,ki,0i,Q1,k1,QPj,k1≥j
j,G Q"+"Q"+"P"+"G ψγγγγ
1.35 x Permanent actions
‘to be combined with’ Actions due to
prestressing
1.5 x Leading variable action
1.5 x combination factor x Other variable actions
8
15
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Load combinations for a typical structure from Equation 6.10:
Load combinations – Eq. 6.10
EHF
Dead + Wind (uplift)
D + I + W(wind leading)
D + I + W(imposed leading)
Dead + Imposed
Combination WindImposedDead
1.35 1.5 1.0
1.0 1.5 1.0
1.35 1.5 0.75 1.0
1.5 1.01.35 1.05
For Eq. 6.10b, Dead load factor reduces to 1.25.
16
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Remarks on loading and load combinations:
Remarks on load combinations
• Presentation of load combinations unfamiliar
• Idea of leading variable actions and combination factors etc is new
• Other than format and notation, loading codes are similar to existing BS
• Four basic load combinations arise
9
17
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Overview of Eurocode 3
18
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
EN 1993: Eurocode 3
Eurocode 3 contains six parts:
• EN 1993-1 Generic rules
• EN 1993-2 Bridges
• EN 1993-3 Towers, masts & chimneys
• EN 1993-4 Silos, tanks & pipelines
• EN 1993-5 Piling
• EN 1993-6 Crane supporting structures
10
19
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
EN 1993-1
Eurocode 3: Part 1 has 12 sub-parts:
• EN 1993-1-1 General rules
• EN 1993-1-2 Fire
• EN 1993-1-3 Cold-formed thin gauge
• EN 1993-1-4 Stainless steel
• EN 1993-1-5 Plated elements
• EN 1993-1-6 Shells
20
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
EN 1993-1
• EN 1993-1-7 Plates transversely loaded
• EN 1993-1-8 Joints
• EN 1993-1-9 Fatigue
• EN 1993-1-10 Fracture toughness
• EN 1993-1-11 Cables
• EN 1993-1-12 High strength steels
11
21
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Axes convention
Different axes convention:
ZYMinor axis
YXMajor axis
XAlong the member
Eurocode 3BS 5950
22
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Labelling convention
Labelling convention:
b
h d
tw
tf
r
y y
z
z
t
b
r
h y y
z
z
12
23
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Subscripts
Extensive use of sub-scripts – generally helpful:
• ‘Ed’ means design effect (i.e. factored member force or moment)
• ‘Rd’ means design resistance
So,
• NEd is an axial force
• NRd is the resistance to axial force
Sometimes tedious e.g. Ac,eff,loc
24
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Structural analysis
13
25
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Analysis types:
• First order elastic• Second order elastic• First order plastic• Second order plastic
Analysis types
26
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Second order effects
• if they increase the action effects significantly
• or modify significantly the structural behaviour
EN 1993-1-1 Clause 5.2.1(2) states that deformed geometry (second order effects) shall be considered:
14
27
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
For elastic analysis:
where
αcr is the factor by which the design loading would have to be increased to cause elastic instability in a global mode (λcr in BS 5950-1)
FEd is the design loading on the structure
Fcr is the elastic critical buckling load for global instability based on initial elastic stiffness.
10FF
Ed
crcr ≥=α
Limits for ignoring deformed geometry
28
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
• Portals with shallow roof slopes
• Beam and column frames (each storey)
where
HEd horizontal reaction at bottom of the storey
VEd total vertical load at bottom of the storey
δH,Ed storey sway when loaded with horizontal loads (eg wind, equivalent horizontal forces)
h storey height
⎟⎟⎠
⎞⎜⎜⎝
⎛
δ⎟⎟⎠
⎞⎜⎜⎝
⎛=α
Ed,HEd
Edcr
hVH
Simple estimate for αcr
Simple estimate for αcr may be applied to:
15
29
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Limits for treatment of second order effects depend on αcr: Ed
crcr F
F=α
Second order effects more accurately
Second order effects by approximate means
First order only
Achievement
Second order analysisαcr<3
First order analysis plus amplification or effective length method
10>αcr>3
First order analysisαcr>10
ActionLimits on αcr
Frame stability
30
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Global initial sway imperfections:
Global imperfections for frames
mh0 ααφ=φ
factorsreductionareand
200/1valuebasictheiswhere
mh
0
αα
=φ
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31
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Global imperfections and checks
Global imperfections:
•Much easier to apply as equivalent horizontal forces φNEd, where NEd is the design compressive force in the column
Checks:
• Check cross-sections, members and joints based on maximum forces and moments from analyses
32
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Cross-section classification
17
33
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Cross-section classification
• Cross-sectional resistance and rotation capacity are limited by the effects of local buckling.
• Eurocode 3 (and BS 5950) account for the effects of local buckling through cross-section classification.
• The classifications from BS 5950 of plastic, compact, semi-compact and slender are replaced in Eurocode 3 with Class 1, Class 2, Class 3 and Class 4, respectively.
34
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Cross-section classification
Deformation
Moment
Mel
Mpl
Class 1
Class 2
Class 4
Class 3
Eurocode 3 defines four classes of cross-section:
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35
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Definition of compressed widths – flat widths:
Compressed widths c
c
(a) Outstand flanges (b) Internal compression parts
c
c
Rolled
Welded c
Rolled
Welded
yf/235=ε
Limits on slenderness e.g. c/t ≤ 9ε
36
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Class 4 cross-sections
• Contain slender plate elements
• Local buckling in elastic range
• Effective width concept
Class 4 cross-sections:
Channel in pure compression
19
37
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Compression members
38
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Elastic buckling theory
Two bounds for perfect columns: Yielding and buckling
Afy
Non-dimensional slenderness
Material yielding (squashing)
Euler (critical) buckling Ncr
NEd
NEd
Lcr
Load
20
39
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Imperfections
Geometric imperfections
Residual stresses
40
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Residual stresses
Welding
Hot-rolling
21
41
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Buckling curves
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 0.5 1 1.5 2 2.5
Curve a0 Curve a Curve b Curve c Curve d
Curve a0
Red
uctio
n fa
ctor
χ
Non-dimensional slenderness λ
42
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Cross-section resistance
sections4ClassforfA
N
ectionss3or21,ClassforAf
N
0M
yeffRd,c
0M
yRd,c
γ=
γ=
• Cross-section resistance in compression Nc,Rd depends on cross-section classification:
γM0 is specified as 1.0 in EN 1993
This value will also be adopted in the UK
22
43
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Compression buckling resistance Nb,Rd:
1M
yRd,b
fAN
γ
χ=
1M
yeffRd,b
fAN
γ
χ=
Column buckling
for Class 1, 2 and 3
for (symmetric) Class 4
Reduction factor
44
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Member buckling
Calculate non-dimensional slenderness λ
3and2,1ClassforN
fA
cr
y=λ
4ClassforN
fA
cr
yeff=λ
Ncr is the elastic critical buckling load for the relevant buckling mode based on the gross properties of the cross-section
23
45
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Non-dimensional slenderness
The non-dimensional slenderness used in EC3 is derived from the geometric slenderness (l/r), normalised by λ1.
y
1
fEr/l
π=
λλ
=λ
λ1 is the slenderness at which the elastic buckling stress is equal to the yield stress
46
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Buckling reduction factor -Graphical approach
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 0.5 1 1.5 2 2.5
Curve a0 Curve a Curve b Curve c Curve d
Curve a0
Red
uctio
n fa
ctor
χ
Non-dimensional slenderness λ
0.63
≈0.88
24
47
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
• Calculate reduction factor, χ
1)(
15.022 ≤
λ−ϕ+ϕ=χ
))2.0(1(5.0 2λ+−λα+=ϕ
Column buckling
Imperfection factor α
Plateau length
48
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Imperfection factor α
0.760.490.340.210.13Imperfection factor α
dcbaa0Buckling curve
Imperfection factors α for 5 buckling curves:
Buckling curves are chosen from a selection table depending on type of section and axis of buckling
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Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Column buckling design procedure
Design procedure for column buckling:
1. Determine design axial load NEd
2. Select section and determine geometry
3. Classify cross-section (if Class 1-3, no account need be made for local buckling)
4. Determine effective (buckling) length Lcr
5. Calculate Ncr and Afy
50
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Column buckling design procedure
6. Non-dimensional slenderness
7. Determine imperfection factor α
8. Calculate buckling reduction factor χ
9. Design buckling resistance
10. Check
cr
y
NfA
=λ
1M
yRd,b
fAN
γ
χ=
0.1NN
Rd,b
Ed ≤
26
51
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Beams
52
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Non-dimensional slenderness
Beam behaviour analogous to yielding/buckling of columns.
M
Wyfy
Material yielding (in-plane bending)
Elastic member buckling Mcr
Lcr
MEd MEd
Non-dimensional slenderness LTλFully laterally restrained beams fail by in-plane bending
27
53
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
In-plane bending
• Class 1 & 2 cross-sections:
0M
yplplRd,c
fWMM
γ==
0M
yelelRd,c
fWMM
γ==
• Class 3 cross-sections:
54
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
In-plane bending
• Class 4 cross-sections:
0M
yeffRd,c
fWM
γ=
28
55
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Shear resistance Vpl,Rd
The usual approach is to use the plastic shear resistance Vpl,Rd
The plastic shear resistance is essentially defined as the yield strength in shear multiplied by a shear area Av:
0M
yvRd,pl
)3/f(AV
γ=
≈ 0.6 fy
56
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Shear buckling
The resistance of the web to shear buckling should also be checked, though this is unlikely to affect cross-sections of standard hot-rolled proportions.
Shear buckling need not be considered provided:
0.1;f
235where
websdunstiffene for 72th
y
w
w
=η=ε
ηε
≤
29
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Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Lateral torsional buckling
Lateral torsional buckling
Lateral torsional buckling is the member buckling mode associated with slender beams loaded about their major axis, without continuous lateral restraint.
If continuous lateral restraint is provided to the beam, then lateral torsional buckling will be prevented and failure will occur in another mode, generally in-plane bending (and/or shear).
58
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Lateral torsional buckling
Can be discounted when:
• Minor axis bending
• CHS, SHS, circular or square bar
• Fully laterally restrained beams
• < 0.2 (or 0.4 in some cases)LTλ
30
59
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Lateral torsional buckling
Eurocode 3 design approach for lateral torsional buckling is analogous to the column buckling treatment.
The design buckling resistance Mb,Rd of a laterally unrestrained beam (or segment of beam) should be taken as:
1M
yyLTRd,b
fWM
γχ=
Reduction factor for LTB
60
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
LTB curves
4 buckling curves for LTB (a, b, c and d)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 0.5 1 1.5 2 2.5
Curve a
Curve b
Curve c
Curve d
Red
uctio
n fa
ctor
χLT
Non-dimensional slenderness LTλ0.2
31
61
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
cr
yyLT M
fW=λ
Non-dimensional slenderness
• Buckling curves as for compression (except curve a0)
• Wy depends on section classification• Mcr is the elastic critical LTB moment
• Calculate lateral torsional buckling slenderness:
62
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Mcr under uniform moment
G is the shear modulusIT is the torsion constantIw is the warping constantIz is the minor axis second moment of areaLcr is the buckling length of the beam
5.0
z2
T2
cr
z
w2
cr
z2
0,cr EIGIL
II
LEIM ⎥
⎦
⎤⎢⎣
⎡
π+
π=
Under uniform moment the elastic critical lateral torsional buckling moment Mcr is:
32
63
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Mcr under non-uniform moment
Numerical solutions have been calculated for a number of other loading conditions. For uniform doubly-symmetric cross-sections, loaded through the shear centre at the level of the centroidal axis, and with the standard conditions of restraint described, Mcr may be calculated by:
5.0
z2
T2
cr
z
w2
cr
z2
1cr EIGIL
II
LEICM ⎥
⎦
⎤⎢⎣
⎡π
+π
=
64
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
For hot-rolled doubly symmetric I and H sections without destabilising loads, may be conservatively simplified to:
LTλ
LTofassessmentSimplified λ
1
z
1z
1LT 9.0
C19.0
C1
λλ
=λ=λ
y1zz f
E;i/L π=λ=λ
As a further simplification, C1 may also be conservatively taken = 1.0.
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Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Design procedure for LTB
Design procedure for LTB:
1. Determine BMD and SFD from design loads
2. Select section and determine geometry
3. Classify cross-section (Class 1, 2, 3 or 4)
4. Determine effective (buckling) length Lcr –depends on boundary conditions and load level
5. Calculate Mcr and Wyfy
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Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Design procedure for LTB
6. Non-dimensional slenderness
7. Determine imperfection factor αLT
8. Calculate buckling reduction factor χLT
9. Design buckling resistance
10. Check for each unrestrained portion
cr
yyLT M
fW=λ
1M
yyLTRd,b
fWM
γχ=
0.1MM
Rd,b
Ed ≤
34
67
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Sources of further information
68
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Sources of further information
• http://www.eurocodes.co.uk/
• http://www.steel-sci.org/publications/
• http://www.access-steel.com/
• http://www.steel-ncci.co.uk/
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Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Observations on UK implementation:
Implementation in UK
• Existing British Standards withdrawn in 2010
• Public projects must be designed to Eurocodes from 2010
• ICE training courses running since 2005
• Demand for EC courses growing; demand for BS courses reducing
• Many participants now engaged in EC designs
• Design guidance and software
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Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Conclusions:
Conclusions
• Advanced design codes
• Greater in scope
• Biggest change since limit states
• Unfamiliar format
• Guidance material and training emerging
• Basis for other National design codes
36
71
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Elliptical hollow sections (EHS)
72
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Overview
• Introduction/ strategy
• Structural investigations
• Design rules
• Conclusions
Outline contents:
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73
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Introduction
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Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
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L. Gardner
• Full-scale structural testing
• Non-linear numerical modelling
• Theoretical investigations
• Development of structural design rules
• Eurocode, BS, AISI
Stages of developments:
Strategy
38
75
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
• Local buckling and section classification• Cross-section compression resistance
• Cross-section bending resistance
• Shear resistance
• Combined bending and shear
• Flexural buckling of columns
• Concrete-filled EHS
Structural scenarios addressed:
Areas examined
76
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
SHS CHSAxial loading
EHSRHSBending
Axial + bending
Elliptical hollow sections
39
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Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Elliptical hollow sections
a
a
b b
z
y
Elliptical hollow sections (EHS):
• Combine merits of RHS and CHS
• Elegant appearance (CHS)
• Differing rigidities about principal axes (RHS)
78
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. GardnerHeathrow Airport, UK Jarrold store, UK
Applications
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Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. GardnerMadrid Airport, Spain Society Bridge,
Scotland
Applications
80
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Cross-section classification
41
81
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Compression (and minor axis bending):
Elastic critical buckling
b
b
a a
z
y)1(3r
tE2
maxcr
ν−=σ
1az
by 22
=⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛
)1(3ba
tE
22cr
ν−⎟⎟⎠
⎞⎜⎜⎝
⎛=σ
rmax
rmax is the maximum local radius of curvature
82
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
a
a
b b
Major axis bending:
Elastic critical buckling
rmax
Maximum compression
Compression
Tension
z
y
Buckling initiates
For a/b < 1.15, buckling initiates at extreme fibre
42
83
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Material testing and geometric measurements:
Testing
Tensile coupons Geometric measurements
84
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Compression tests:
Testing
43
85
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
• Nonlinear FE package ABAQUS
• Stage 1: Validation against tests– Measured material properties– Measured geometry– Measured geometric imperfection
FE modelling
86
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
• Stage 1: Validation against tests– Full loading history and failure models
well predicted
• Stage 2: Parametric studies (>800 FE results in total)– Cross-section slenderness– Member slenderness– Aspect ratios (for all tests, a/b = 2)
FE modelling
44
87
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
0
600
1200
1800
0 6 12 18 24
End shortening δ (mm)
Load
N (k
N)
FE
Test
FE validation
88
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Compression
De/tε2
F u/F
y
De = 2rmax = 2a2/b, ε = (235/fy)0.5
0.0
0.5
1.0
1.5
2.0
0 30 60 90 120 150 180 210 240 270
2a
2b
EHSCHSFE
Class 1-3 Class 4
45
89
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Minor axis bending tests:
Testing
90
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Minor axis bending – M/Mel
De/tε2
Mu/M
el
De = 2rmax = 2a2/b, ε = (235/fy)0.5
0.0
0.5
1.0
1.5
2.0
2.5
0 20 40 60 80 100 120 140 160 180 200 220 240 260
EHSCHSFE
2a
2b
Class 4Class 1-3
46
91
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Minor axis bending – M/Mpl
De/tε2
Mu/M
pl
De = 2rmax = 2a2/b, ε = (235/fy)0.5
0.00.20.40.60.81.01.21.41.61.82.0
0 20 40 60 80 100 120 140 160 180 200 220 240 260
2a
2bClass 1-2 Class 3-4
EHSCHSFE
92
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Minor axis bending – Rotation capacity
De/tε2
Rot
atio
n ca
paci
ty
De = 2rmax = 2a2/b, ε = (235/fy)0.5
0369
1215182124273033
0 20 40 60 80 100 120 140 160 180 200 220 240 260
2a
2b
EHSCHSFE
Class 1 Class 2-4
47
93
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Radii of curvature:
Major axis bending
2b
2a
0.125a2/b0.22a
0.4a2/b 0.42a
0.65a2/b
a2/b
94
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Summary of proposed measures of slenderness:
Slenderness parameters
Corresponding point on cross-sectionEquivalent diameterLoading
2a
2bAxial compression De = 2a2/b
2a
2bMinor axis bending (z-z) De = 2a2/b
2b
2aMajor axis bending (y-y)
De = 0.8a2/b a/b > 1.36
2b
2a
De = 2b2/a a/b ≤ 1.36
48
95
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Summary of proposed slenderness limits:
Slenderness limits
90ε2Not applicableDe/tAxial compression
De/tMajor axis bending (y-y)
140ε270ε250ε2
De/tMinor axis bending (z-y)
Class 3Class 2Class 1
Proposed slenderness limits
Diameter ratio
Type of compression
loading
96
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Shear resistance
49
97
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Shear tests
• A total of 24 three-point bending tests performed. Constant aspect ratio = 2.0.
• 12 minor axis tests, 12 major axis tests
• Three variations in cross-section slenderness (thickness)
• Four variations in span length (to investigate moment-shear interaction)
98
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Shear tests
• A total of 24 three-point bending tests performed
L/2 L/2
F
Moment gradient
Uniform shear
Uniform shear
50
99
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Plastic shear resistance
• Design plastic shear resistance Vpl,Rd
0M
yvRd,pl
3/fAV
γ=
where Av= Shear areafy = Material yield strengthγM0= 1.0
100
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Plastic shear resistance
z
y
θ
r
π=
τ=∴
πτ
=
τ×××=
θ×τ××θ××= ∫π
A2VA
A2V
tr4V
)sin()tdr(2V
0v
0
0
00
Shear area for circular hollow sections: Av=2A/π
51
101
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Plastic shear resistance
b b
a
a
z
y
For shear along z-z
a a
b
b
z
y
For shear along y-y
Av = (4b-2t)t Av = (4a-2t)t
102
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Design guidance
0.0
0.5
1.0
1.5
0.00 0.25 0.50 0.75 1.00 1.25
Vu/Vpl,Rd
Mu/M
pl,R
dor
Mu/M
el,R
d
Shear along y-yShear along z-z
Proposed shear-moment interaction
52
103
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Design guidance
• For VEd < 50% Vpl,Rd
– Full bending resistance (Mel,Rd or Mpl,Rd)
• For VEd > 50% Vpl,Rd
– Reduced bending resistance
fyr = (1-ρ)fy
where ρ = (2VEd/Vpl,Rd – 1)1/2
104
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Column buckling
53
105
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Column Tests
• A total of 24 column tests performed, constant aspect ratio = 2.0
• 12 minor axis tests, 12 major axis tests
• Three variations in cross-section slenderness (thickness)
• Four variations in column length
106
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Knife edgeLoad cell
LVDT
Strain gauge
CL
Hydraulic jack
Testing
54
107
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Load-deflection response
0
200
400
600
800
1000
0 20 40 60 80 100 120
2nd order rigid plastic
2nd order elastic
Lateral deflection at mid-height ω (mm)
Load
N (k
N)
Elastic buckling
Plastic yielding
108
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
0
250
500
750
0 15 30 45 60
Lateral deflection at mid-height ω (mm)
Load
N (k
N)
Test
FE
FE validation
55
109
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
0.0
0.5
1.0
1.5
0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8
Member slenderness λ
Buckling about z-z
Buckling about y-y
Nu/
Ny
or N
u/N
eff
z
y
EC3 – curve ‘a’
Column buckling curve
110
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Conclusions:
Conclusions
• New and rare addition to hot-rolled range
• Significant testing and FE modelling programmes
• Design rules developed for primary structural configurations
• Development of comprehensive design rules ongoing
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111
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
Recent developments in steel structures:
Introduction to Eurocode 3 & Elliptical hollow sections
Dr Leroy GardnerImperial College London
South Africa, 2009
112
Eurocode 3
Background
Design rules
Elliptical tubes
Introduction
Cross-section
Member
L. Gardner
ReferencesReferences:
Gardner, L. & Nethercot, D. A. (2005). Designers’Guide to EN 1993-1-1: Eurocode 3: Design of Steel Structures. Thomas Telford Publishing.
Trahair, N. S., Bradford, M. A., Nethercot, D. A. & Gardner, L. (2008). The behaviour and design of steel structures to EC3. Fourth Edition. Taylor & Francis.
Gardner, L. and Chan, T. M. (2007). Cross-section classification of elliptical hollow sections. Steel and Composite Structures. 7(3), 185-200.