Compression Members
Local Buckling and Section Classification
Summary:
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Structural sections may be considered as an assembly of individual plate elements.
Plate elements may be internal (e.g. the webs of open beams or the flanges of boxes) and others
are outstand (e.g. the flanges of open sections and the legs of angles).
Loaded in compression these plates may buckle locally.
Local buckling may limit the section capacity by preventing the attainment of yield strength.
Premature failure (by local buckling) may be avoided by limiting the width to thickness ratio (or
slenderness) of individual elements within the cross section.
This is the basis of the section classification approach.
EC3 defines four classes of cross-section.
The class into which a particular cross-section falls depends on the slenderness of each element
and the compressive stress distribution.
Objectives:
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Sections may fail by compressive buckling of plates within the section.
Distinguish between internal and outstand elements.
Demonstrate that plate slenderness and edge restraints control the buckling behaviour.
Sketch the relationship between normalised ultimate compressive stress and normalised plate
slenderness
Explain the meaning of different section classifications.
Derive a result from EC3 Tables for hot rolled sections.
Use the section classification method to choose appropriate sections.
Describe the effective width approach for Class 4 sections.
References:
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Eurocode 3: Design of steel structures Part 1.1 General rules and rules for buildings
The Behaviour and Design of Steel Structures, Chapter 4- Local buckling of thin plate elements,
N S Trahair and M A Bradford, E & FN Spon, Revised Second Edition 1994
Contents:
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Introduction
Classification
Behaviour of plate elements in compression
Effective width approach to design of Class 4 sections
Concluding summary
1. Introduction
Structural sections, rolled or welded, may be considered as an assembly of individual plate elements.
Most of these elements (figure 1), if in compression, can be separated into two categories:
Internal or stiffened elements: these elements are considered to be simply supported along two edges
parallel to the direction of compressive stress.
Outstand or unstiffened elements; these elements are considered to be simply supported along one edge
and free on the other edge parallel to the direction of compressive stress.
Outstand
Internal
Internal
Outstand
Internal
Web
Web
Web
Flange
(a) Rolled I-section
Flange
Flange
(b) Hollow section
Internal
(c) Welded box section
Figure 1 - Internal or outstand elements
As the plate elements in structural sections are relatively thin compared with their width, when loaded
in compression (as a result of axial loads and/or from bending) they may buckle locally.
The disposition of any plate element within the cross section to buckle may limit the axial load carrying
capacity, or the bending resistance of the section, by preventing the attainment of yield.
Avoidance of premature failure arising from the effects of local buckling may be achieved by limiting
the width-to-thickness ratio for individual elements within the cross section.
2. Classification
EC3 defines four classes of cross section.
The cross section class depends upon the slenderness of each element (defined by a Eurocode 3
width-to-thickness ratio) and the compressive stress distribution i.e. uniform or linear. 5. 3.2 (1)
or 5.5.2
The classes are defined as performance requirements for bending moment resistance:
• Class 1 - cross-sections that can form a plastic hinge with the required rotational capacity for plastic
analysis.
• Class 2 - cross-sections that, although able to develop a plastic moment, have limited rotational
capacity and are therefore unsuitable for plastic design.
• Class 3 - cross-sections that the calculated stress in the extreme compression fibre can reach yield but
local buckling prevents the development of the plastic moment resistance.
• Class 4 - cross-sections that in which local buckling limits the moment resistance (or compression
resistance for axially loaded members). Explicit allowance for the effects of local buckling
is necessary.
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Table 1 summarises the classes in terms of behaviour, moment capacity and rotational capacity.
Moment
Resistance
Model of
Behaviour
Moment
Plastic moment
on gross section
M pl
Local
Buckling
fy
M
Mpl
Sufficient
1
1
Plastic moment
on gross section
fy
Mpl
Local
Buckling
M
Mpl
1
Mpl
fy
Mel
2
Local
Buckling
M
Mpl
fy
Mel
3
φ
φpl
1
Plastic moment on
effective section
Mpl
None
1
φ
Moment
φ
φpl
1
Elastic moment
on gross section
φ
φpl
Limited
φ
Moment
1
φrot
φpl
φ
Moment
Class
Rotation Capacity
M
Mpl
None
4
1
Local
Buckling
φ
1
φ
φpl
Mel elastic moment resistance of cross-section
Mpl plastic moment resistance of cross-section
M applied moment
φ rotation (curvature) of section
φpl rotation (curvature) of section required to generate fully plastic stress distribution
across section
Table 1 - Cross-section classifications in terms of moment resistance and rotation capacity.
The moment resistances for the four classes defined above are:
for Classes 1 and 2: the plastic moment (Mpl = Wpl . fy)
for Class 3: the elastic moment (Mel = Wel . fy)
for Class 4: the local buckling moment (Mo < Mel).
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3. Behaviour of plate elements in compression
A thin flat rectangular plate subjected to compressive forces along its short edges has an elastic critical
buckling stress (σcr ) given by:
σ cr
kσ π 2 E ⎛ t ⎞
=
⎜ ⎟
12(1 − ν 2 )⎝ b ⎠
2
(1)
Where
kσ is the plate buckling parameter which accounts for edge support conditions,
stress distribution and aspect ratio of the plate - see figure 2a.
ν= Poisson’s coefficient, E = Young’s modulus
3.2.5 (1)
3.2.5 (1)
L
t
(b)
(a)
b
Simply supported on
all four edges
Buckling coefficient k
5
b
4
Simply supported
edge
b
L
Free
Exact
3
k = 0.425 + (b/L)2
2
L
(c)
1
0.425
Free
edge
0
(d)
1
2
3
4
5
Plate aspect ratio L / b
Figure 2 - Behaviour of plate elements in compression. (Trahair and Bradford)
The elastic critical buckling stress (σcr ) is thus inversely proportional to (b/t)2 and analogous to the
slenderness ratio (L/i) for column buckling.
Open structural sections comprise a number of plates that are free along one longitudinal edge (figure
2b) and tend to be very long compared with their width. These plates buckled shape is seen in figure 2c.
The relationship between aspect ratio and buckling parameter for a long thin outstand element of this
type is shown in figure 2d.
The buckling parameter tends towards a limiting value of 0.425 as the plate aspect ratio increases.
For a section to be classified as class 3 or better the elastic critical buckling stress (σcr ) must exceed the
yield stress fy . From equation (1) (substituting ν = 0.3 and rearranging) this will be so if
b/t < 0,92 (k σ E/f y )
0,5
(2)
This expression is general as the effect of stress gradient, boundary conditions and aspect ratio are all
encompassed within the buckling parameter kσ.
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Table 2 gives values for kσ for internal and outstand elements under various elastic stress distributions.
Buckling factor kσ
Support conditions at long edges
Clamped + clamped
6,97
Clamped + simply supported
5,40
Simply supported + simply supported
4,00
Clamped + free
1,28
Simply supported + free
0,43
Free + free
(b/a)2
a
Various support
conditions
b
a/b >> 1
σ
1
σ
σ
2
I
σ
2
σ
1
1
σ
2
III
II
σ = is maximum stress, compression
ψ =σ2 / σ1
Case I
Internal element
Case II
Outstand element
Case III
Outstand element
+1
1>ψ>0
0
0 > ψ>
-1
4,0
8,02
1,05 + ψ
7,81
7,81+6,29ψ+9,78ψ2
23,9
0,43
0,57-0,21ψ+0,07ψ2
0,57
0,57-0,21ψ+0,07ψ2
0,85
0,43
0,578
ψ+0,34
1,70
1,7-5ψ+17,1ψ2
23,8
Table 2 - Buckling factors and stress distribution.
The elastic-plastic behaviour of a perfect plate element subject to uniform compression may be
represented by a normalised load-slenderness diagram where normalised ultimate load, Np , and
normalised plate slenderness, λ p , are given by:
= σult / fy
0,
λ p = (f y / σ cr )
(3)
Np
5
(4)
Substituting equation (1) for σcr into (4), and replacing fy with 235/ε2 (so that the expression may be used
for any grade of material) the normalised plate slenderness, λp, may be expressed as
⎛
⎞
b/t
⎟
=⎜
⎜ 28.4ε k ⎟
σ ⎠
⎝
(5)
where b is the appropriate width for the type of element and cross-section type.
⎛ fy
λ p = ⎜⎜
⎝ σ cr
⎞
⎟
⎟
⎠
0 .5
5
Figure 3 shows the relationship between Np and λ p .
Np
=
σu
fy
1
Class 3
Class 2
Class 1
Euler Buckling Stress
0,5 0,6
0,9
1,0
λ
p
Figure 3 - Dimensionless representation of the elastic-plastic buckling stress.
For normalized plate slenderness less than one, the normalised ultimate load can reach its squash load.
For greater values of λ p , Np decreases as the plate slenderness increases, the ultimate stress sustained
being limited to the elastic critical buckling stress, σcr.
The actual behaviour is somewhat different from the ideal elastic-plastic behaviour due to:
i. initial geometrical and material imperfections,
ii. strain-hardening of the material,
iii. the postbuckling behaviour.
These factors require λ p values to be reduced. This is made to delay the onset of local buckling until the
requisite strain distribution through the section (yield at the extreme fibre or fully plastic distribution)
has been attained.
EC3 uses the following normalised plate slenderness’ as limits for classifications:
5.2.1.4 (7)
Class 1 λ p < 0,5
Class 2 λ p < 0,6
Class 3 λ p < 0,9 for elements under a stress gradient; this is further reduced to 0,74 for
elements in compression throughout.
By substituting the appropriate values of kσ into equation (5) and noting the λ p to be used for each class,
limiting b/t ratios can be calculated.
Tables 4-7 are EC3 extracts giving the limiting proportions for compression elements from class 1 to 3.
When any of the compression elements within a section fail to satisfy the limit for class 3 the whole
section is classified as class 4 (commonly referred to as slender), and local buckling should be taken
into account in the design using an effective cross section.
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a. Webs: (internal elements perpendicular to axis of bending)
tf
tw
Axis of
Bending
tw
d
tw
d
tw
d
h
d = h-3t (t = tf = t w)
Web subject to
bending
Class
Web subject to
compression
+ fy
Stress
distribution in
element
(compression
positive)
+ fy
d
fy
Web subject to bending
and compression
αd
d
h
fy
-
+ fy
d h
h
fy when α > 0,5:
d/t w <_ 396ε/(13α − 1)
when α < 0,5:
d/t w _< 36ε/α
-
1
d/t w <_ 72ε
d/t w <_ 33 ε
2
d/t w <
_ 83 ε
d/t w <_ 38 ε
when α > 0,5:
when α < 0,5:
d/t w <_ 41,5ε/α
+fy
+ fy
+ fy
Stress
distribution in
element
(compression
positive)
d/t w _
< 456ε/(13α − 1)
d/2
d/2
d
h
fy -
ψ fy -
+
when ψ > −1:
d/t w _< 42ε/(0,67 + 0,33ψ)
_ 42 ε
d/t w <
d/t w <_ 124 ε
3
h
d
h
_ −1:
when ψ <
d/t w _< 62ε/(1 − ψ)
ε = 235 / f
y
(−ψ )
fy
235
275
355
ε
1
0,92
0,81
Table 4 - Maximum width-to-thickness ratios for compression elements.
b. Internal flange elements:
(internal elements parallel to axis of bending)
b
b
tf
axis of
bending
Class
b
tf
tf
b
Section in bending
Type
Stress distribution
in element and
across section
(compression
positive)
Section in compression
fy
+
-
tf
fy
+
-
- +
1
Rolled hollow section
Other
2
Rolled hollow section
Other
Stress distribution
in element and
across section
(compression
positive)
- +
_
<33ε
_<33ε
_<38ε
_<38ε
(b - 3t f )/ t f
b / tf
(b - 3t f )/ t f
b/tf
fy
+
-
+
-
Rolled hollow section
Other
ε = 235/ f y
fy
fy
- +
3
_<42ε *
_
<42ε
*
_
<42ε
_
<42ε
(b - 3t f)/ t f
b / tf
(b - 3t f )/ tf
b / tf
fy
ε
(b - 3t f )/ t f
b / tf
- +
*
_
(b - 3t f)/ t f <42ε
_<42ε
b / tf
_<42ε
_<42ε
235
275
335
1
0,92
0,81
* For a cross section in compression with no bending the classification 1,2,3 are irrelevant
and hence the limit is the same in each case.
Table 5 - Maximum width-to-thickness ratios for compression elements.
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c. Outstand flanges:
c
c
c
c
tf tf
tf
Welded sections
Rolled sections
Type of section
Class
2
ε = 235/ f y
+
-
c
Rolled
c/t f _
< 10ε
_ 10ε
c/t f <
α
Welded
c/t f _< 9ε
c/t f <_ 9e
α
Rolled
_ 11ε
c/t f <
_ 11ε
c/t f <
α
Welded
_ 10ε
c/t f <
c/t f <_ 10ε
α
Stress distribution
in element
(compression positive)
3
Flange subject to
compression and bending
Tip in
Tip in
compression
tension
αc
αc
+
c
c
Flange subject
to compression
Stress distribution
in element
(compression positive)
1
tf
10 ε
α α
_ 9ε
c/t f <
α α
_
c/t f <
11ε
α α
c/t f <
_ 10ε
α α
_
c/t f <
+
-
c
+
-
c
Rolled
_ 15ε
c/t f <
_ 23ε k σ
c/t f <
Welded
_ 14ε
c/t f <
c/t f <
_ 23 ε k σ
+
-
c
+
-
For k σ see figure 2d
and table 8
fy
235
275
355
ε
1
0,92
0,81
Table 6 - Maximum width-to-thickness ratios for compression elements.
h
d. Angles:
Refer also to c.
'Outstand flanges'
(Table 6)
(Does not apply to
angles in continuous
contact with other
components).
t
b
t
Section in compression
Class
fy
+
-
fy
Stress distribution
across section
(compression positive)
+
t
h
b+h
≤ 15 ε :
≤ 115
, ε
t
2t
3
e. Tubular sections:
d
t
Section in bending and/or compression
d / t ≤ 50 ε 2
Class
1
d / t ≤ 70 ε 2
d / t ≤ 90 ε 2
2
3
ε = 235/ f y
fy
ε
235
275
355
1
0,92
0,81
ε2
1
0,85
0,66
Table 7 - Maximum width-to-thickness ratios for compression elements.
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4. Effective width approach to design of Class 4 sections
For members with Class 4 sections the effect of local buckling on global behaviour at the ultimate limit
state is such that the elastic resistance, calculated on the assumption of yielding of the extreme fibres of
the gross section (criteria for Class 3 sections), cannot be achieved.
Figure 4 gives the moment deflection curve for a point loaded beam (Class 4).
Figure 4 - Moment versus deflection curve of a pointed loaded beam.
The reason for the reduction in strength is that local buckling occurs at an early stage in parts of the
compression elements of the member; the stiffness of these parts in compression is thereby reduced and
the stresses are distributed to the stiffer edges, see Figure 5.
Figure 5 – Strain/stress distribution of a member with deck plate local buckling in compression.
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To allow for the reduction in strength the actual non linear distribution of stress is taken into account by
a linear distribution of stress acting on a reduced "effective plate width" leaving an "effective hole"
where the buckle occurs, Figure 5.
By applying this model an "effective cross-section" is defined for which the resistance is then calculated
as for Class 3 sections (by limiting the stresses in the extreme fibres to the yield strength).
The effective widths beff are calculated on the basis of the Winter formula:
beff = ρ .b
Reduction coefficient ρ depends on the plate slenderness
p
defined by plate bucking theory, Figure 6.
Figure 6 – Reduction coefficient ρ for the effective width.
Cross-sections with class 4 elements may be replaced by an effective cross-section, taken as the gross
section minus holes where the buckles may occur, and then designed in a similar manner to class 3
sections using elastic cross-sectional resistance limited by yielding in the extreme fibres.
Effective widths of compression elements may be calculated by use of a reduction factor ρ which is
dependent on the normalised plate slenderness λ p (which is in turn dependent on the stress distribution
and element boundaries through application of the buckling parameter kσ) as follows:
( )
( )
⎛ λ − 0,22 ⎞
p
⎟
ρ = ⎜⎜
2
⎟
λp
⎝
⎠
(6)
The reduction factor ρ may then be applied to outstand or internal element as shown in Tables 8 and 9.
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Stress distribution
(compression positive)
beff
Effective width b eff
1 > ψ ≥ 0:
σ1
σ2
beff = ρ c
c
bt
bc
ψ < 0:
σ1
beff = ρbc = ρc / (1 − ψ )
σ2
beff
ψ = σ 2 /σ1
1
0
-1
1 ≥ ψ ≥ −1
Buckling factor k σ
0,43
0,57
0,85
0,57 − 0,21ψ + 0,07ψ 2
beff
1 > ψ ≥ 0:
σ1
σ2
beff = ρ c
c
beff
ψ < 0:
σ1
beff = ρbc = ρc / (1 − ψ )
σ2
bc
bt
ψ = σ 2 /σ1
1
1>ψ > 0
0
Buckling factor k σ
0,43
0,578
ψ + 0,34
1,70
0 > ψ > −1
1,7 − 5ψ + 171
, ψ2
-1
23,8
Table 8 - Effective widths of outstand compression elements.
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Stress distribution
(compression positive)
σ1
Effective width b eff
ψ = 1:
σ2
b = b - 3t
beff = ρ b
be1 = 0,5 b eff
be2 = 0,5 b eff
be2
be1
b
1 > ψ >_ 0 :
σ1
σ2
b e1
b = b - 3t
beff = ρ b
2b
b e1 = eff
5- ψ
b e2 = beff - be1
be2
b
bc
bt
ψ < 0:
σ1
b = b - 3t
beff = ρ bc = ρ b / (1 - ψ )
σ2
b e1
be1 = 0,4b eff
be2 = 0,6b eff
be2
b
ψ = σ2 /σ1
1
1>ψ > 0
0
Buckling
factor k σ
4,0
8,2
1,05 + ψ
7,81
Alternatively, for
_ ψ >_ - 1:
1>
0 >ψ > - 1
-1
7,81- 6,92ψ + 9,78ψ 2 23,9
kσ =
- 1>ψ > - 2
5,98 (1 -ψ )2
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[(1 + ψ )2 + 0,112(1 - ψ )2 ]0,5 + (1 + ψ )
Illustrated as rhs.
For other sections b = d for webs
b = b for internal flange elements (except rhs)
Table 9 - Effective widths of compression elements
Figure 7 shows examples of effective cross-sections for members in compression or bending.
Notice that the effective cross-section centroidal axis may shift relative to the gross cross-section. For
bending members this will be considered when calculating the effective section properties.
For axial force members the shift of the centroidal axis will give rise to a moment that should be
accounted for in member design.
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Centroidal axis of
gross cross-section
Centroidal axis of
gross cross-section
Centroidal axis of
effective cross-section
eN
Non-effective zones
Gross cross-section
(a) Class 4 cross-sections - axial force
eM
Centroidal axis
Non-effective zone
Centroidal axis of
effective section
Non-effective zone
eM
Centroidal axis
Centroidal axis of
effective section
Gross cross-section
(b) Class 4 cross-sections - bending moment
Figure 7 - Effective cross-sections for class 4 in compression and bending
6. Concluding summary
•
Structural sections may be considered as an assembly of individual plate elements.
•
When loaded in compression these plates may buckle locally.
•
Local buckling within the cross-section may limit the load carrying capacity of the section by
preventing the attainment of yield strength.
•
Premature failure (from local buckling) may be avoided by limiting the width to thickness ratio or slenderness - of individual elements within the cross section.
•
This is the basis of the section classification approach. EC3 defines 4 classes of cross-section.
•
The class into which a particular cross-section falls depends upon the slenderness of each
element and the compressive stress distribution.
Additional reading
[1] Salmon, C.G., Johnson, J.E., "Steel Structures. Design and Behaviour", Harper et Row, New York.
[2] Dubas, P., Gehri, E., "Behaviour and Design of Steel Plated Structures", Pub. 44, ECCS, TC8, 1986.
[3] Bulson, P.S., "The Stability of Flat Plates" Chatto and Windus, London.
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