1. PURE COMPRESSION WITHOUT BUCKLING
1.1
Stub Columns
Stub (or stocky) columns are characterized by very low slenderness, are not affected by buckling and can be
designed to the yield stress fy.
If local buckling does not affect the compression resistance (as can be assumed for Class 1, 2 and 3 crosssections), the mode of failure of such members corresponds to perfect plastic behaviour of the whole crosssection, which theoretically occurs when each fibre of the cross-section reaches fy. It is to be noted that
residual stresses and geometric imperfections are practically without influence on the ultimate strength of
this kind of column and that most experimental stub columns fail above the yield stress because of strainhardening.
The maximum compression resistance Nmax is, therefore, equal to the plastic resistance of the cross-section:
Nmax = Npl = Aeff·fy
Aeff is the effective area of the cross-section.
Eurocode 3 considers that columns are stocky when their reference slenderness  is such that   0.2 .
They are represented by a plateau on the ECCS curves.
1.2
Effective Area
The effective area of the cross-section used for design of compression members with Class 1, 2 or 3 crosssections, is calculated on the basis of the gross cross-section using the specified dimensions. Holes, if they
are used with fasteners in connections, need not be deducted.
2. STABILITY OF SLENDER STEEL COLUMNS
Depending on their slenderness, columns exhibit two different types of behaviour: those with high
slenderness present a quasi-elastic buckling behaviour whereas those of medium slenderness are very
sensitive to the effects of imperfections.
2.1
Euler Critical Stress
If lcr is the critical length, the Euler critical load Ncr is equal to:
N cr 
 2 EJ
lcr 2
and it is possible to define the Euler critical stress cr as:
 cr 
N cr  2 EJ

A
A  lcr 2
By introducing the radius of gyration, i 
J
A
l
, and the slenderness,   cr
i
, for the relevant buckling
mode, the previous equation becomes:
 2E
 cr  2
cr
Plotting the curve cr as a function of  on a graph, with the line representing perfect plasticity,  = fy, shown,
it is interesting to note the idealized zones representing failure by buckling, failure by yielding and safety.
Figure 2-1
The intersection point P, of the two curves represents the maximum theoretical value of slenderness of a
column compressed to the yield strength. This maximum slenderness (sometimes called Euler slenderness),
called 1 in Eurocode 3, is equal to:
1  
E
 93.9
fy
where:  
235
.
fy
A non-dimensional representation of this diagram is obtained by plotting /fy as a function of /1 (Figure
2-2); this is the form used for the ECCS curves. The coordinates of the point P are, therefore (1;1).
Figure 2-2
2.2
Buckling of Real Columns
The real behaviour of steel columns is rather different from that described in the previous section and
columns generally fail by inelastic buckling before reaching the Euler buckling load. The difference in real and
theoretical behaviour is due to various imperfections in the "real" element: initial out-of-straightness,
residual stresses, eccentricity of axial applied loads and strain-hardening. The imperfections all affect
buckling and will, therefore, all influence the ultimate strength of the column. Experimental studies of real
columns give results as shown in Figure 2-3. Compared to the theoretical curves, the real behaviour shows
greater differences in the range of medium slenderness than in the range of large slenderness. In the zone
of the medium values of  (representing most practical columns), the effect of structural imperfections is
significant and must be carefully considered. The greatest reduction in the theoretical value is in the region
of Euler slenderness 1. The lower bound curve is obtained from a statistical analysis and represents the safe
limit for loading.
Figure 2-3
Slender columns
A column can be considered slender if its slenderness is larger than that corresponding to the point of
inflexion of the lower bound curve, shown in Figure 2-3.
The ultimate failure load for slender columns is close to the Euler critical load Ncr. As this is independent of
the yield stress, slender columns are often designed using the ratio:  2 
Alcr 2
J
, a geometrical
characteristic independent of the mechanical strength.
Individual or star-battened angles or small flat plates have generally a poor second moment of area about
the minor axis, relative to their cross-section area. This can result in large slenderness with high sensitivity to
buckling and explains why classical cross-bracing systems, using these shapes, are only designed in tension.
Columns of medium slenderness
Columns of medium slenderness are those whose behaviour deviates most from Euler's theory. When
buckling occurs, some fibres have already reached the yield strength and the ultimate load is not simply a
function of slenderness; the more numerous the imperfections, the larger the difference between the actual
and theoretical behaviour. Out-of-straightness and residual stresses are the imperfections which have the
most significant effect on the behaviour of this kind of column.
Residual stresses can be distributed in various ways across the section. They are produced by welding, hotrolling, flame-cutting or cold-forming; Figure 2-4(a) shows some of the stress patterns that can occur.
Figure 2-4
Residual stresses combined with axial stresses are shown in Figure 2-4(b). If the maximum stress n reaches
the yield stress fy, yielding begins to occur in the cross-section. The effective area able to resist the axial load
is, therefore, reduced.
Alternatively, an initial out-of-straightness eo, produces a bending moment giving a maximum bending stress
B (see Ошибка! Источник ссылки не найден.(a)), which when added to the residual stress, R gives the
stress distribution shown in Ошибка! Источник ссылки не найден.(b). If max is greater than the yield stress
the final distribution will be part plastic and part of the member will have yielded in compression, as shown
in Ошибка! Источник ссылки не найден.(c).
Figure 2-5
3. THE EUROPEAN BUCKLING CURVES
3.1
Reference Slenderness
The reference slenderness  is defined as the following non-dimensional parameter for Class 1, 2 or 3 crosssections:


1
where  and 1 are defined in Section 2.1.
 can also be written in the following form:
2 fy
f
2
  2  y
 E  cr
or  
3.2
fy
 cr
Basis of the ECCS Buckling Curves
From 1960 onwards, an international experimental programme was carried out by the ECCS to study the
behaviour of standard columns. More than 1000 buckling tests, on various types of members (I, H, T, U,
circular and square hollow sections), with different values of slenderness (between 55 and 160) were studied.
A probabilistic approach, using the experimental strength, associated with a theoretical analysis, showed that
it was possible to draw some curves describing column strength as a function of the reference slenderness.
The imperfections which have been taken into account are: a half sine-wave geometric imperfection of
magnitude equal to 1/1000 of the length of the column and the effect of residual stresses relative to each
kind of cross-section.
The European buckling curves (a, b, c or d) are shown in Figure 3-1. These give the value for the reduction
factor  of the resistance of the column as a function of the reference slenderness for different kinds of crosssections (referred to different values of the imperfection factor ).
Figure 3-1
The mathematical expression for  is:

1
  2  
2
1


where:   0.5  1     0.2   

2

The imperfection factor  depends on the shape of the column cross-section considered, the direction in
which buckling can occur (y axis or z axis) and the fabrication process used on the compression member (hotrolled, welded or cold-formed); values for , which increase with the imperfections, are given in Table 1.
Table 1
 Curve a represents quasi perfect shapes: hot-rolled I-sections (h/b > 1.2) with thin flanges (tf  40 mm) if
buckling is perpendicular to the major axis; it also represents hot-rolled hollow sections.
 Curve b represents shapes with medium imperfections: it defines the behaviour of most welded boxsections; of hot-rolled I-sections buckling about the minor axis; of welded I-sections with thin flanges (tf 
40 mm) and of the rolled I-sections with medium flanges (40 < tf  100 mm) if buckling is about the major
axis; it also concerns cold-formed hollow sections where the average strength of the member after
forming is used.
 Curve c represents shapes with a lot of imperfections: U, L, and T shaped sections are in this category as
are thick welded box-sections; cold-formed hollow sections designed to the yield strength of the original
sheet; hot-rolled H-sections (h/b  1.2 and tf  100 mm) buckling about the minor axis; and some welded
I-sections (tf  40 mm buckling about the minor axis and tf > 40 mm buckling about the major axis).
 Curve d represents shapes with maximum imperfections: it is to be used for hot-rolled I-sections with very
thick flanges (tf > 100 mm) and thick welded I-sections (tf > 40 mm), if buckling occurs in the minor axis.
Table 2 and Table 3 help the selection of the appropriate buckling curve as a function of the type of crosssection, of its dimensional limits and of the axis about which buckling can occur. For cold-formed hollow
sections, fyb is the tensile yield strength and fya is the average yield strength. If the cross-section in question
is not one of those described, it must be classified analogously.
It is important to note that the buckling curves are established for a pin-ended, end loaded member.
Table 2 – From EN1993-1-1:2005
Table 3 – From EN1993-1-3:2007
3.3
Equivalent Initial Bow Imperfection
To study a column using second order theory, it is necessary to choose geometrical imperfections (initial outof-straightness and eccentricities of loading) and mechanical imperfections (residual stresses and variations
of the yield stress). Eurocode 3 proposes values for a bow imperfection, eo, whose effect is equivalent to a
combination of the two previous kinds of imperfections.
If the column is designed using elastic analysis, eo is as follows:

 A
for plastic design of cross-sections

 WA
for elastic design of cross-sections.
eo     0.2 
W pl
or,
eo     0.2 
el
4. ANALYTICAL FORMULATION OF THE EUROPEAN BUCKLING CURVES
4.1
Introduction
The behaviour of real steel structures is always different from that predicted theoretically; the main reasons
for this discrepancy are:
̶ geometrical imperfections, due to defects causing lack of straightness (crookedness or bow imperfection
after the fabrication process), unparallel flanges, asymmetry of cross-section etc;
̶ material imperfections, due to residual stresses (caused by the rolling or fabrication process) or material
inelasticity;
̶ deviation of applied load from idealised position due to imperfect connections, erection tolerances or lack
of verticality of the member (initial out-of-plumb or system imperfection).
Of the above, some are important in the buckling of slender columns (geometrical imperfections), others in
the compression of stub columns (material inelasticity) and others in the buckling of columns of medium
slenderness (geometric imperfections and residual stresses). The behaviour of these three types of columns
has been described previously.
In reality, all the imperfections act together simultaneously and their effect depends on their individual
intensity and on the slenderness of the column. An experimental study of many columns with various
characteristics gives the results shown in Figure 4-1. The results of the tests should be below the Euler
buckling curve because initial out-of-straightness, eccentricity of applied loads and residual stresses all
decrease the allowable buckling load; for small slenderness (stub columns), however, it is possible to find
some results above the yield stress line because of possible strain-hardening. A safety curve obtained through
a statistical analysis is always situated under the minimum experimental values and has the form shown in
Figure 4-1; the plateau is necessary to limit the allowable stress to the yield value. This is the general form of
the European buckling curves.
Figure 4-1
4.2
Initial Deflection
Assuming that the initial deflection of a pin-ended column of length l, has a half sine-wave form with
magnitude eo (Figure 4-2), the initial deformation along the column can be written as:
x
yo  eo sin 

 L 
Figure 4-2
The differential equation for the deformation of such a pin-ended column loaded by an axial force N is:
d 2 y N  y  yo 

0
d 2x
EJ
Combining this with the expression for yo, and taking into account the boundary conditions, the solution of
this equation is:
y
x 
sin 

 N cr

 L 

1


 N

eo
The maximum total deflection, e, of the column is then:
e  eo 
eo
 N cr

 1

 N


eo

N 
1 

N cr 

and the ratio 1/(1 - N/Ncr) is generally called the "amplification factor".
Taking into account the maximum bending moment, Ne, due to buckling, the equilibrium of the column
requires that:
N Ne

 fy
A W
where fy is the yield stress.
If N is the maximum axial load acting in the member, limited by buckling, and b the corresponding normal
stress (b = N/A), this becomes:
N N  e A 
 e A 
 
  b b 
  fy
A A W 
 W 
or, introducing cr, the Euler critical stress (cr = 2E/2) and including the value of e:
b b 
eo
A
 fy
 b  W
1 

  cr 

which can be written as:
 cr   b    f y   b    b   cr  o
e A
W
This equation is the basic form of the Ayrton-Perry formula.
4.3
Eccentricity of the Applied Load
If the axial compression load is applied with an eccentricity ec on an initially straight pin-ended column (Figure
4-3), a bending moment (N·ec) is introduced which increases the buckling effect. This effect obviously
increases along with axial load.
Figure 4-3
It is possible to show that the total maximum deflection e of the column is equal to:
e  ec 
ec
J
N 
cos  

 2 EJ 
1
and the "amplification factor" to:

N 
cos  

 2 N cr 
Now, if the combined effect of the initial deflection and of the eccentricity of loading is considered, the stress
is approximately equal to:
b b 
eo  ec  0.23ec
b
 cr A

 b 
1 

  cr 
W
 fy
This relationship is correct within a few percent for all values of b from 0 to cr.
4.4
Ayrton-Perry formula
The classical form of the Ayrton-Perry formula is:
 cr   b    f y   b      b   cr
e A
This is the form of equation at the end of chapter 4.2, where   o
.
W
The coefficient  represents the initial out-of-straightness imperfection of the column but it can also include
other defects such as residual stresses in which case it is called the "generalized imperfection factor".
It is possible to write the Ayrton-Perry formula under another form:
  cr


 N   1  N    N  cr


fy
 fy


where: N 
2
If  
fy
 cr

b
fy
then, dividing by
1  N    1  N     N
 cr
, gives:
fy
2
2
2

2

or:  N  N     1  1  0
This form leads to the European formulation.
4.5
Generalized Imperfection Factor
The generalized imperfection factor takes into account all the relevant defects in a real column when
considering buckling: geometric imperfections, eccentricity of applied loads and residual stresses; inelastic
properties are not considered because they only influence stub columns. The generalized imperfection factor
can be expressed through the coefficient  representing the effect of deflections:
L A
 W

where   L
eo
, represents the equivalent geometrical imperfection (which is the ratio of the length over
the equivalent initial curvature of the column).
Then using L  lcr    i , W  J and i 2  J ,  can be written as:
v
A



 v
 i
where (i/v) is the relative diameter of the inertia ellipse in the axis where buckling occurs.
E
, introducing the plateau N  1 when    o , the previous relationship can be written as:
fy
As    


90.15    o
 v

 i
because all the European buckling curves were established with fy = 255 MPa (the real value of the yield stress
having a very small influence).
4.6
European Formulation


2
2

2

Using  expressed as       o , the smallest solution of  N  N     1  1  0 is:
N




2
2
2
2
1      o    1      o     4


2
2
Multiplying by the conjugated term and choosing  o  0.2 , this relationship gives the European formulation:


1
   
2
2

 1
where:


  0.5 1     0.2   

2

and  is the reduction factor considered in Eurocode 3.
The different shapes of cross-sections used to design steel columns have the coefficient  varying from 0.21
to 0.76 and it is possible to represent the real behaviour of all classical columns using the four curves (a, b, c
and d) shown in Figure 4-4,  increasing with the imperfections.
Figure 4-4
 takes into account two kinds of imperfections (geometrical and mechanical). It can be written as  = 1 +
2, where 1 represents the mechanical and 2 the geometrical imperfections. Considering only the
geometrical imperfections, the European buckling curves were established with an initial curvature equal to
L/1000; this gives
2 
90.15
1000  i
 v


Considering now the equivalent initial deflection: eo = L/, linked to the generalized imperfection factor by
e A
means of the relation   o
and using equation       o , gives:
W
W
eo     0.2
A




which represents the equivalent initial bow imperfection of a pin-ended column including the initial
crookedness and the effect of residual stresses; this has to be taken into account in a second order analysis.
5. ESEMPIO
Si analizza una colonna in HEA140 (S355), alta 10 m e con schema di semplice appoggio; il piano di interesse
per l’inflessione è quello attorno all’asse forte.
I dati principali sono:
A  31.42 cm 2
Wel , y  155.36 cm 3
 y  2.282
a y  0.34 ,
da cui si ottiene un’imperfezione eo, se il progetto della sezione trasversale viene condotto al limite plastico,
di:

 A  39.09 mm
eo     0.2 
W pl
Partendo dunque da una deformata iniziale della colonna sinusoidale, con massimo in mezzeria pari a e o ed
eseguendo un’analisi non lineare geometrica con carico assiale Nc di compressione via via crescente, il valore
di tale carico per cui vale la seguente relazione:
Nc M  Nc 

 f y  355 MPa
A
W pl , y
risulta di 184 kN con momento massimo in mezzeria M, funzione di Nc, di 51.587 kNm.
Tale momento è anche ottenibile come amplificazione della parte al primo ordine, N c eo , per il fattore
amplificativo
1
:

N 
 1  c 
N cr , y 

eo
39.09
M  Nc
 184 
 51.1 kNm
184 


Nc 
1 

 1 

214.13 

N
cr , y 

In conclusione, il valore del carico per instabilità è di 184 kN che, rispetto al valore di resistenza per
schiacciamento f y A  1115.41 kN , ne rappresenta il 16.5%, cioè il fattore .