ESDEP WG 7
ELEMENTS
To describe the different kinds of steel columns and to explain the procedures involved in the design of compression members.
Lecture 6.1: Concepts of Stable and Unstable Elastic Equilibrium
Lecture 7.2: CrossSection Classification
Lecture 7.3: Local Buckling
Lecture 7.6: Builtup Columns
Lectures 7.10: Beam Columns
Lecture 7.12: Trusses and Lattice Girders
Worked Example 7.5: Column Design
Different kinds of compression members (uniform and nonuniform crosssections and builtup columns) are described. The differences between stocky and slender columns are explained and the basis for design, using the European buckling curves, is outlined.
Columns are vertical members used to carry axial compression loads. Such structural elements are found, for example, in buildings supporting floors, roofs or cranes. If they are subjected to significant bending moments in addition to the axial loads, they are called beamcolumns.
The term 'compression' member is generally used to describe structural components subjected only to axial compression loads; this can describe columns (under special loading conditions) but generally refers to compressed pinended struts found in trusses, lattice girders or bracing members.
This lecture deals with compression members and, therefore, concerns very few real columns because eccentricities of the axial loads and, above all, transverse forces, are generally not negligible. Nevertheless, compression members represent an elementary case which leads to the understanding of the term of compression in the study of beamcolumns (Lectures 7.10.1 and 7.10.2), frames (Lecture 7.11), and trusses and lattice girders (Lecture 7.12).
Because most steel compression members are rather slender, buckling can occur; this adds an extra bending moment to the axial load and must be carefully checked.
The lecture briefly describes the different kinds of compression members and explains the behaviour of both stocky and slender columns; the buckling curves used to design slender columns are also given.
For optimum performance compression members need to have a high radius of gyration, i, in the direction where buckling can occur; circular hollow sections should, therefore, be most suitable in this respect as they maximise this parameter in all directions. The connections to these sections are, however, expensive and difficult to design.
It is also possible to use square or rectangular hollow sections whose geometrical properties are good (the square hollow sections being the better); the connections are easier to design than those of the previous shape, but again rather expensive.
Hotrolled sections are, in fact, the most common crosssections used for compression members. Most of them have large flanges designed to be suitable for compression loads. Their general square shape gives a relatively high transverse radius of gyration i_{z} and the thickness of their flanges avoids the effect of local buckling (care has to be taken in the case of light Hsections and high strength steel). The open shape, produced by traditional rolling techniques, facilitates beamtocolumn and other connections.
Welded box or welded Hsections are suitable if care is taken to avoid local flange buckling. They can be designed for the required load and are easy to connect to other members; it is also possible to reinforce these shapes with welded cover plates.
Figure 1 illustrates all the shapes mentioned above.
It should be noted that:
Members with changes of crosssection within their length, are called nonuniform members; tapered and stepped members are considered in this category.
In tapered members (Figure 2) the crosssection geometry changes continuously along the length; these can be either open or box shapes formed by welding together elements, including tapered webs, or flanges, or both. A classical example is that of a hotrolled H or Isection whose web is cut along its diagonal; a tapered member is obtained by reversing and welding the two halves together.
Stepped columns (Figures 3) vary the crosssection in steps. A typical example of their use is in industrial buildings with overhead travelling cranes; the reduced cross section is adequate to support the roof structure but must be increased at crane level to cater for the additional loads. Stepped columns can also be used in multistorey buildings to resist the loads in the columns at the lower levels.
Builtup columns are fabricated from various different elements; they consist of two or more main components, connected together at intervals to form a single compound member (Figure 4). Channel sections and angles are often used as the main components but it is also possible to use I or Hsections; they are laced or battened together with simple elements (bars or angles or smaller channel sections) and it is possible to find columns where both methods are combined (Figure 5). Columns with perforated plates (Figure 6) are also considered in this category.
The advantage of this system is that it gives relatively light members which, because most of the steel is placed quite far from the centre of gravity of the crosssection have a relatively high radius of gyration. The most important disadvantage comes from the high fabrication costs involved. These members are generally designed for large structures where the compression members are long and subjected to heavy loads. It is to be noted that buckling of each individual element must be checked very carefully.
Figure 7 shows some closely spaced builtup members and gives details of starbattened angle members. These are not so efficient as the previous ones because of their smaller radius of gyration; however, the ease with which these can be connected to other members may make their use desirable. They behave in compression in a similar way to those described above, hence their inclusion in this section.
Builtup columns may have uniform or nonuniform crosssections; it is possible, for example, to find stepped or tapered builtup columns (Figure 8).
The design of builtup columns will be examined in Lecture 7.6.
Stub (or stocky) columns are characterised by very low slenderness, are not effected by buckling and can be designed to the yield stress f_{y}.
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 crosssection reaches f_{y}. 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 N_{max} is, therefore, equal to the plastic resistance of the crosssection:
N_{max} = N_{pl} = A_{eff} f_{y} (1)
A_{eff} is the effective area of the crosssection, see Section 3.2.
Eurocode 3 [1] considers that columns are stocky when their reference slenderness is such that £ 0,2 ( is defined in Section 5.1). They are represented by a plateau on the ECCS curves [2], see Section 5.2.
The effective area of the crosssection used for design of compression members with Class 1, 2 or 3 crosssections, is calculated on the basis of the gross crosssection using the specified dimensions. Holes, if they are used with fasteners in connections, need not be deducted.
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.
If l_{cr} is the critical length, the Euler critical load N_{cr} is equal to:
N_{cr} = (2)
and it is possible to define the Euler critical stress s_{cr} as:
s_{cr} = (3)
By introducing the radius of gyration, i = , and the slenderness, l = l_{cr}/i, for the relevant buckling mode, Equation (3) becomes:
s_{cr} = (4)
Plotting the curve s_{cr} as a function of l on a graph (Figure 9), with the line representing perfect plasticity, s = f_{y}, shown, it is interesting to note the idealised zones representing failure by buckling, failure by yielding and safety.
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 l_{1} in Eurocode 3, is equal to:
l_{1} = P [E/f_{y}]^{1/2} = 93,9e (5)
where: e = [235/f_{y}]^{1/2} (6)
l_{1} is equal to 93,9 for steel grade F_{e} 235 and to 76,4 for steel grade F_{e} 355.
A nondimensional representation of this diagram is obtained by plotting s/f_{y} as a function of l/l _{1} (Figure 10); this is the form used for the ECCS curves (see Section5.2). The coordinates of the point P are, therefore, (1,1).
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 outofstraightness, residual stresses, eccentricity of axial applied loads and strainhardening. 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 11. 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 l (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 l_{1}. The lower bound curve is obtained from a statistical analysis and represents the safe limit for loading.
a. 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 11.
The ultimate failure load for slender columns is close to the Euler critical load N_{cr}. As this is independent of the yield stress, slender columns are often designed using the ratio: l^{2} = (A l_{cr}^{2})/I, a geometrical characteristic independent of the mechanical strength.
Individual or starbattened angles or small flat plates have generally a poor second moment of area about the minor axis, relative to their crosssection area. This can result in large slenderness with high sensitivity to buckling and explains why classical crossbracing systems, using these shapes, are only designed in tension.
b. 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. Outofstraightness 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, flamecutting or coldforming; Figure 12a shows some of the stress patterns that can occur.
Residual stresses combined with axial stresses are shown in Figure 12b. If the maximum stress s_{n} reaches the yield stress f_{y}, yielding begins to occur in the crosssection. The effective area able to resist the axial load is, therefore, reduced.
Alternatively, an initial outofstraightness e_{o}, produces a bending moment giving a maximum bending stress s_{B} (see Figure 13a), which when added to the residual stress, s_{R} gives the stress distribution shown in Figure 13b. If s_{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 Figure 13c.
The reference slenderness is defined as the following nondimensional parameter for Class 1, 2 or 3 crosssections:
= l /l_{1} (7)
where l and l_{1} are defined in Section 4.1.
can also be written in the following form:
^{2} = l^{2}f_{y}/p^{2}E = f_{y}/s_{cr } (8)
or = (9)
From 1960 onwards, an international experimental programme was carried out by the ECCS to study the behaviour of standard columns [2]. 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 sinewave 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 crosssection.
The European buckling curves (a, b, c or d) are shown in Figure 14. These give the value for the reduction factor c 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 a).
The mathematical expression for c is:
c = 1/ {f + [f^{2}  ^{2}]^{1/2}} ^{ }£ 1 (10)
where: f = 0,5 [1 + a (  0,2) + ^{2}] (11)
Table 1 gives values of the reduction factor c as a function of the reference slenderness .
The imperfection factor a depends on the shape of the column crosssection 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 coldformed); values for a, which increase with the imperfections, are given in Table 2.
Curve a represents quasi perfect shapes: hotrolled Isections (h/b > 1,2) with thin flanges (t_{f} £ 40mm) if buckling is perpendicular to the major axis; it also represents hotrolled hollow sections.
Curve b represents shapes with medium imperfections: it defines the behaviour of most welded boxsections; of hotrolled Isections buckling about the minor axis; of welded Isections with thin flanges (t_{f} £ 40mm) and of the rolled Isections with medium flanges (40 < t_{f} £ 100mm) if buckling is about the major axis; it also concerns coldformed 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 boxsections; coldformed hollow sections designed to the yield strength of the original sheet; hotrolled Hsections (h/b £ 1,2 and t_{f} £ 100mm) buckling about the minor axis; and some welded Isections (t_{f} £ 40mm buckling about the minor axis and t_{f} > 40mm buckling about the major axis).
Curve d represents shapes with maximum imperfections: it is to be used for hotrolled Isections with very thick flanges (t_{f} > 100mm) and thick welded Isections (t_{f} > 40mm), if buckling occurs in the minor axis.
Table 4 helps 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 coldformed hollow sections, f_{yb} is the tensile yield strength and f_{ya} is the average yield strength. If the crosssection 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 pinended, end loaded member; it is necessary carefully to evaluate the buckling lengths if the boundary conditions are different, see Lecture 7.7.
To study a column using second order theory, it is necessary to choose geometrical imperfections (initial outofstraightness and eccentricities of loading) and mechanical imperfections (residual stresses and variations of the yield stress). Eurocode 3 proposes values for a bow imperfection, e_{o}, whose effect is equivalent to a combination of the two previous kinds of imperfections [1].
If the column is designed using elastic analysis, e_{o} is as follows:
e_{o} = a (^{ } 0,2) W_{pl}/A for plastic design of crosssections
or,
e_{o} = a (  0,2) W_{el}/A for elastic design of crosssections (12)
If it is designed with an elasticplastic analysis (elastoplastic or elasticperfectly plastic), the values of e_{o} are functions of the buckling length L, and are given in Table 3.
To design a simple compression member it is first necessary to evaluate its two effective lengths, in relation to the two principal axes, bearing in mind the expected connections at its ends. Secondly, the required second moment of area to resist the Euler critical loads should be calculated to give an idea of the minimum crosssection necessary. The verification procedure should then proceed as follows:
Buckling resistance of a compression member is then taken as:
N_{b.Rd} = c A f_{y}/g_{M1} (13)
the plastic resistance as:
N_{pl.Rd} = A f_{y}/g_{M0} (14)
and the local buckling resistance as:
N_{o.Rd} = A_{eff} f_{y}/g_{M1} (15)
If these are higher than the design axial load, the column is acceptable; if not, another larger crosssection must be chosen and checked.
In addition, torsional or flexuraltorsional buckling of the column must be prevented.
[1] Eurocode 3: " Design of Steel Structures": ENV 199311: Part 1.1: General rules and rules for buildings, CEN, 1992.
Table 1 Reduction factors
Reduction factor c 

Curve a 
Curve b 
Curve c 
Curve d 

0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9 2,0 2,1 2,2 2,3 2,4 2,5 2,6 2,7 2,8 2,9 3,0 
1,0000 0,9775 0,9528 0,9243 0,8900 0,8477 0,7957 0,7339 0,6656 0,5960 0,5300 0,4703 0,4179 0,3724 0,3332 0,2994 0,2702 0,2449 0,2229 0,2036 0,1867 0,1717 0,1585 0,1467 0,1362 0,1267 0,1182 0,1105 0,1036 
1,0000 0,9641 0,9261 0,8842 0,8371 0,7837 0,7245 0,6612 0,5970 0,5352 0,4781 0,4269 0,3817 0,3422 0,3079 0,2781 0,2521 0,2294 0,2095 0,1920 0,1765 0,1628 0,1506 0,1397 0,1299 0,1211 0,1132 0,1060 0,0994 
1,0000 0,9491 0,8973 0,8430 0,7854 0,7247 0,6622 0,5998 0,5399 0,4842 0,4338 0,3888 0,3492 0,3145 0,2842 0,2577 0,2345 0,2141 0,1962 0,1803 0,1662 0,1537 0,1425 0,1325 0,1234 0,1153 0,1079 0,1012 0,0951 
1,0000 0,9235 0,8504 0,7793 0,7100 0,6431 0,5797 0,5208 0,4671 0,4189 0,3762 0,3385 0,3055 0,2766 0,2512 0,2289 0,2093 0,1920 0,1766 0,1630 0,1508 0,1399 0,1302 0,1214 0,1134 0,1062 0,0997 0,0937 0,0882 
Table 2 Imperfection factors
Buckling curve 
a 
b 
c 
d 
Imperfection factor a 
0,21 
0,34 
0,49 
0,76 
Table 3 Equivalent initial bow imperfections
Buckling curve 
Elastoplastic 
Elasticperfectly plastic 
a b c d 
L/600 L/380 L/270 L/180 
L/400 L/250 L/200 L/150 
Table 4: Selection of buckling curve for a crosssection