Design of Columns and Struts in Structural Steel

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    Columns and struts carry load primarily in compression along their length, and are found in most building structures. Columns, sometimes referred to as stanchions, and struts are structural elements which support compressive loads primarily along their longitudinal axes. Such members are present in the structure of almost all buildings from the temples of ancient civilisations to present day frame structures.

At the ancient monument of Stonehenge all stone posts are compression members of huge proportions. Compression members (struts) are also an integral part of trusses and space frames.

n suspension structures, cable-stayed roof structure and tent structures compression members play an essential role. They generally take the form of a mast or tower providing support to the cables or membrane acting in tension of the mast. Columns are an essential part of modern framed buildings. n some instances, these columns may additionally be required to carry lateral wind load and bending due to eccentricity of the end reaction of floor beams. However, axial compression is normally the predominant effect. The combination of bending and compression complicates the behaviour of the column and is not considered here.

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    jor a given axial load, steel grade, column length and end conditions, the procedure for sizing a column section is based on trial and error, and can be described as a sequence of steps. The method of calculating suitable column sizes to satisfy the requirements of BS 5950 is based on a trial and error routine. This is because both strength and actual stress depend on the crosssection size. nitially the design load and effective length must be determined. The design load is generally derived from the beam reactions, under factored load conditions. The effective length is based on height between restraints and the end conditions. The method of member selection can then be summarised as a number of steps. Step 1: Choose a trial section. The trial section should be estimated to be of adequate size to carry the applied load. This may appear to be quite arbitrary, but with a little experience the initial guess will become quite accurate. The only penalty for an µincorrect' guess at this stage is the need to repeat the calculation procedure with a more informed guess. At this stage the class of cross-section can be checked to eliminate the risk of local buckling. To do this it is necessary to calculate b/T and d/t values for the section and check that these are within certain limits. Rolled steel sections will normally be classified as plastic, compact or semi-compact and need no further consideration. Step 2: Determine slenderness ratios jind x and y as follows: lx = LE/rx and ly = LE/ry Ensure that lx and ly are within maximum permissible values. jor a member resisting normal dead and imposed load in a building, this value would be 180. f necessary reselect a bigger section to comply with this condition. Step 3: Determine the compressive strength from the appropriate strut table. Select the relevant strut table with reference to Table 25 and the type of section (, H, hot rolled, welded section etc.) used. The compressive strength pc can then be read directly from the appropriate table 27 (a), (b), (c) or (d) for the calculated value of slenderness ratio and the specified material strength.

Step 4: Compare axial load capacity with the applied design axial load, and adjust the trial section size if necessary, repeating the checks from step 2. jind the compressive resistance of the section. Pc = Ag . pc and compare with the design axial load. f Pc is either smaller than the required value (or too large), a larger section (or smaller) than the previous section must be adopted and steps 2 to 5 are repeated until a satisfactory result is obtained. Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y

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        The radius of gyration is a convenient parameter, providing a measure of the resistance of a cross-section to lateral buckling. The radius of gyration is not a physical entity in itself. t is a relationship derived to make prediction of column behaviour easy. t is simply the square root of the second moment of area of the section divided by the cross-sectional area of the section. Thus, r = Ö (/A) where r = the radius of gyration  = the moment of inertia or the second moment of area of the section A = the cross sectional area of the section The radius of gyration is related to the size and shape of the cross-section. The shape and size of the cross-section of the compression member determines the least radius of gyration of the section. f all the material of the section is concentrated to produce a solid section of small overall dimensions the section will have a smaller radius of gyration than a hollow member with the material distributed further away from the centre of the cross section. This can be illustrated by comparing solid cross-sections (a) with cross-sectional shapes in which the overall dimensions are much larger (b), but the cross-sectional area is the same. n each case cross-section (b) has a larger radius of gyration than (a) and the buckling strength is therefore increased.

Columns will buckle in the direction of least cross-sectional stiffness. t should also be noted that when a column of rectangular cross-section is loaded it will buckle in the direction of the smaller dimension in cross-section. A column of square cross-section will be equally prone to buckling in x and y directions. This is because the cross section will offer equal resistance to buckling in the direction x and y. (n practice the presence of imperfections will cause the column to buckle preferentially in one direction.) jor a rectangular cross-section, the tendency to buckle will be in the direction of the smaller dimension, that is perpendicular to the line yy.

A special range of sections (UCs) is manufactured principally for use as columns; these sections have similar resistance to lateral buckling about both axes of the cross-section. Hot-rolled steel sections, such as Universal Beams (UBs), channels and joists, are deeper in one direction than the other. These sections are ideal for use as beams, where the main strength of the section is required in the direction of bending. f these sections are used as columns, their strength will be determined by their ability to resist buckling in the weaker direction. This

suggests that rolled steel sections which are efficient in bending are less good at resisting lateral buckling as columns. jor this reason, a different set of sections are produced with more comparable resistance to lateral buckling in xx and yy directions. These are called Universal Columns (UC) and are normally used for columns carrying primarily axial load. However, in practice, columns may have to resist lateral bending as well as axial load and, if this is significant, a Universal Beam section may prove to be more economical. Other cross-section shapes are often used for struts, and hollow sections are very efficient in compression. jor small scale columns, joists or channel sections may be used, whilst lighter sections such as angles are more common for use as struts in trusses. The use of hollow sections is also becoming increasingly popular. Although connections involving such sections are more difficult, they are efficient in compression and are often preferred for aesthetic reasons. Apart from standard rolled sections, it is possible to fabricate sections to suit any specific requirement although there is a cost penalty in doing so.

Although generally not justifiable on the grounds of cost, tapered columns can be efficient in resisting buckling. n a compression member the resistance to buckling can be increased by shaping the member in such a way as to strengthen the point which is most affected by buckling. jor example, in a column with both ends pinned, the point most susceptible to buckling deformation would be at the mid height. This reduces gradually towards the end of the column. deally the column could be shaped by making the mid point of the largest cross-section and gradually reducing the crosssectional size towards the ends. Similarly, for a column fixed at both ends, the susceptible part is the central 70% of its column length. t is possible to strengthen this part to make a more stable column overall by making a tapered section or by using tension wires. However, the cost of such

extra fabrication is likely to be very high and only justified on the basis of architectural expression. Y Y Y

   
     
 

 Steel columns are often slender because of the high strength of the material. Columns constructed in traditional materials such as stone and brick tend to be of large crosssectional size relative to their length. A column in structural steel may also be stocky but the much greater strength of steel compared with stone enables the safe design of much more slender columns. The load capacity of a stocky column is related to its material strength and area of cross-section When loaded, a stocky column will shorten elastically until a point is reached when any further increase in load causes a disproportionately high reduction in length. At this stage the column is described as having reached the yield point. On further loading the column will reach its collapse load. jor the column to behave in this way it must be very short in relation to the size of its cross-section. Under this condition, the failure of the column is due to the failure of steel as a material in compression and the axial load capacity of the column to carry load is determined as follows: BY

Axial load capacity of the column = yield stress of steel in compression x cross-sectional area of column

However, in practice such columns are rarely found since the high strength of steel requires a relatively small cross-sectional area for the column. A typical steel column is therefore likely to be slender. This contrasts with columns constructed in relatively weak materials which require a large cross-section and are therefore generally stocky.

n practice, the failure load of steel columns is associated with buckling; this is related to the column slenderness. n columns of practical proportions failure occurs well before the crushing strength of steel is reached. As the compressive stress is gradually increased, a value is reached at which the column, instead of just axially shortening in length, buckles and deforms perpendicular to its axis. This value of load is called the buckling load. When a column has reached its buckling load it has effectively failed as a structural element and is incapable of sustaining the load. The aim of column design, therefore, is to predict the load at which the column may collapse and to ensure that there is always an adequate factor of safety compared with the applied load. Y Y Y

+          The restraint at the ends of a column has a significant influence on the effective length and therefore the buckling strength. Because of the significance of the effective length, the way a column buckles under load will depend on how the column is constructed. f one end of a slender column is fixed at its base and the other end is totally free of any restraint, the column will be extremely prone to lateral buckling under load. n contrast, a column which is fixed at both ends will be much more stable. End conditions can be idealised as free, pinned or fixed. dealised end conditions for a column are described as free, pinned or fixed. Each of these conditions is described below. Pinned ends allow free rotation. When a column end is connected in such a way that the joint allows the column to rotate freely but restrains it from translation (horizontal movement), the connection could be described as being pinned. Such a joint could be produced in practice in several ways. jor example the figure below shows a typical arrangement where the column is free to rotate in the xx direction, but it is secured in position at point A.

jixed ends do not allow any rotation. A fixed end does not allow the column end to move in any manner, ie. to translate or to rotate. Such a joint could be bolted or welded.

jree ends allow both lateral movement and rotation. Such a condition can apply to only one end of a column and the other end must be fixed to ensure stability of the column. A free end is totally unrestrained and both translation and rotation may take place.

The end conditions influence the shape of the buckled form; the effective length corresponds to that part of the column which deforms as a single curve in the shape of a pin ended buckle. The type of end condition will dictate how much restraint to lateral buckling is offered to the column. The pin ended column can buckle more freely than a fixed ended column and is therefore less strong. The effective length describes the part of the column which is free to buckle laterally. jor pinned ends, the effective length LE and the actual length L are the same. jor partial or nominally full rotational end restraint, the effective length LE = 0.85 L and 0.7 L respectively. These figures are established by laboratory experiment. jor a cantilever column the effective length can be deduced by comparing with a column which has both the ends pinned. When the deflection configuration is completed, it becomes clear that the effective length LE is 2L.Table 24 of BS 5950 lists a number of end conditions for columns and the related factors for the effective lengths. Y Y

        

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 àulti-storey columns are generally restrained at every floor level, and the effective length is therefore based on storey height. n multi-storey frame buildings the columns are restrained by the floor beams framing in at each storey. t is therefore storey height and not overall building height which is significant in determining effective lengths. Thus the column length between floors should be considered and restraints in both the xx and yy directions are to be considered in arriving at the appropriate value of the effective length LE. A practical column in a building has many restraints offered by various building components working in unison. Their overall effect cannot be accurately measured. A guess based on experience is the only possibility, and in that, the BS 5950 recommendations provide safe and sensible values for calculation of the strength of columns.

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 Special consideration may need to be given to the effective length of columns in single storey buildings to account for practical construction details. jor single storey shed type buildings, columns are often fixed at the base, ie. restrained against rotation and translation, and normally carry a roof structure, such as a truss, at the top, providing partial restraint against rotation only. f the lateral stability of the building as a whole is dependent on the fixity at the base, the top of the column is not restrained against translation. n the horizontal plane of the building there may be elements which effectively restrain the column from lateral buckling at certain intervals along the height. Normally, these restraints are provided at the positions where the sheeting rails are connected, because the sheeting rails themselves can then become part of the restraining elements. The effective length for such a column will need to be carefully considered.

àajor axis buckling is related to the full height of the column. jor buckling about the xx axis, the whole column is capable of distortion. Hence the length of the column which is capable of distortion in one unit is the whole length L rather than a single part of it. The base of the column is fixed and the top of the column is not fully free to distort. The top can sway laterally taking the truss with it, but cannot rotate due to the restraint from the roof truss. Hence, the effective length of the column axis xx is 1.5 L àinor axis buckling is related to the distance between longitudinal restraints, such as may be provided by sheeting rails. jor buckling about the yy axis the column has different degrees of restraint over different sections. The bottom length L1, can be described as effectively held in position at both ends and

restrained in direction at the base only. Therefore, the effective length of the column axis yy for the length L1 is equal to 0.85 L1.jor the lengths L2 and L3, the column can be described as being effectively held in position at both ends, but not restrained in direction. Therefore, the effective length of the column axis yy for the lengths L2 and L3 are 1.0 L2 and 1.0 L3 respectively.The final LE to be adopted for calculation would be the largest of the three values 0.85 L1, 1.0 L2 and 1.0 L3. Y Y Y Y

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 Columns are often subject to some bending in addition to compression. Two factors contribute towards the bending in a practical column; these are eccentricity of real beam connection details, and the effect of wind loading. Bending can develop in columns due to eccentricity of beam connections; this is accounted for by assuming that the beam reaction is applied at a prescribed distance from the face of the column. The way the floor beams are connected to a column causes inevitable eccentric application of loads on the column. Beam end reactions are generally applied on the supporting brackets or cleats and consequently they are at a distance e from the centre of the column. This eccentric load would cause the column to bend towards the applied load resulting in tension on one face and compression on the other along with a uniform compression all along the section due to the direct effect of the compressive load alone. Whether both faces will be in compression, albeit unequal amounts of compression, or the outside face will be in tension would depend on the value of the beam reaction and the value of eccentricity.

The effect of connection eccentricity is negligible for internal columns supporting an approximately symmetrical arrangement of beams. n the figure below, the column would be subjected to bending in directions opposed to each other. f W1 and W2 were equal and they were an equal distance away from the column centre, ie. e1 and e2 were equal, then the bending produced by the two beams would be equal and in the opposite directions and would cancel each other out. f the column is an internal column with beams running in xx and yy axes the column would experience a combined effect of some or all of the following: Direct compression due to vertical loads imposed by the end reactions of beams 1, 2, 3 and 4. BY Compression and tension due to bending about the xx axis. BY Compression and tension due to bending about the yy axis. BY

n principle a simplified interaction relationship is used to assess the strength of a column in such cases, but this is beyond the scope of the material covered here. n buildings which depend on rigid frame action for lateral stability, bending will develop in the columns due to wind loads. All building structures need to resist horizontal loads due to wind. f the structure cannot readily transfer the wind loads to braced parts of the building such as stair towers, lift wells and shear walls, it is generally necessary to rely on rigid frame action. The effect of this is to cause bending in both the beams and columns. n such cases the bending moment distribution must be determined by analysis, and the columns designed for combined axial load and bending. Y

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 Simplified procedures can be used to estimate the required size of column section. t may not be necessary to perform detailed calculations to determine column sizes, and simpler methods can be adopted. These include rules of thumb which enable a simple estimate of approximate section sizes, and safe load tables which provide a more rigorous approach to estimating column sizes. Rules of thumb provide an estimate of the required cross-section in relation to function and position within the building. A typical structural frame for a multi-storey building may require column sizes which vary depending on the number of storeys supported. Thus the columns on the top storey supporting the roof only are likely to be much smaller than those on the ground floor which support all of the intermediate floors. Safe load tables provide more rigorous guidance on required sizes. The compressive strength of a column is related primarily to its cross-sectional area and slenderness ratio, and the material strength. Because of the importance of slenderness ratio in relation to buckling, it is not possible to calculate the maximum compressive strength for any cross-section, unless this is related to the effective length of the column. However, by assuming a range of different values, the compressive strength can be published as a function of effective length. Such tables enable the section size required for a given axial load and effective length to be read directly. Y Y Y Y Y Y Y Y Y

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   Column buckling was first investigated by Euler who established that the buckling strength is inversely proportional to the square of the slenderness ratio. n 1757 Leonard Euler propounded a theory for calculating the strength of an axially loaded column pinned at both ends. The relationship he established was as follows: Pe = pie2E/L2 (1) where, Pe = Euler collapse load. E = àodulus of elasticity of the material of construction of the column.  = àoment of inertia of the column section. L = Length of the pin-ended column. The corresponding buckling stress is obtained simply by dividing the collapse load by the crosssectional area of the column, A. Thus: p = Pe/A = pie2E/AL2 Since the moment of inertia () and the cross-sectional area (A) are both dependent solely on the geometry of the cross-section they can be combined into a single variable. This is defined as the radius of gyration r of the column section and is related to  and A as follows:  = A.r2 (2) Substituting equation (2) in (1) pe = pie2E./(L/r)2 (3) This relationship can be represented graphically as a graph of buckling stress against (L/r). Clearly the value of (L/r) is of considerable importance in determining the ability of a column to carry load without buckling.

The relationship between buckling strength and slenderness ratio depends on the support conditions at the column ends. The above expressions describing the theoretical buckling behaviour of columns are valid for pinned ends only. Similar expressions can be obtained for other end conditions. jor instance the buckling stress of a fixed ended column is given by: Pe = 4pie2E / (L/r)2 which can be written as Pe = pie2E /(0.5 L/r)2 The influence of different end conditions can most conveniently be accounted for by using the concept of an effective length. t can be seen that the buckling strength of a fixed ended column is the same as for a pin ended column of half the length. This introduces the concept of effective length which can be defined as the length of an equivalent pin ended column with the same buckling strength. Thus to extend the relationship (3) to include columns with end conditions other than pinned ends the value for length L should be substituted by effective length LE. Thus: Pe = pie2E.A/ (LE/r)2 (4) The ratio LE/r is called the slenderness ratio of the section and the larger the value of this slenderness ratio, the smaller is the value of the collapse load Pe of the column.

t is also worth noting that, because the slenderness ratio is squared, a relatively small increase in its value can cause a large reduction in Pe. Thus, in theory, the buckling load of a column with a slenderness ratio of 160, is just one quarter that of a column with a slenderness ratio of 80. Because the slenderness ratio is dependent on effective length and radius of gyration, these two items both influence the Euler critical load. A column of 5m height with both ends pinned, will carry four times as much load as a similar sized column with fixed base and free top. jor stocky columns the buckling stress can exceed the material strength and the dominant failure mode is therefore 'squashing'. n spite of the elegance of Euler's formula in explaining the parameters contributing to the strength of a compression member, it is only effective in predicting the collapse load of columns with large values of slenderness ratio. jor stocky columns, Euler's collapse load Pe exceeds the yield stress of steel, and the column fails by crushing or 'squashing'. jor example, for a value of LE/r=20, and taking the modulus of elasticity for steel E as 205000N/mm2, the compressive stress obtained from Euler's formula would be as follows: pe = Pe/A = pie2E/(LE/r)2 pe = pie2 x 20500 / 400 = 5120 N/mm2 This value of Euler collapse stress is much larger than the yield stress of steel (210 - 450N/mm2) specified in BS 5950. The failure of stocky columns is therefore determined by material yield stress, whilst for slender columns failure corresponds to the Euler stress. jor columns with intermediate slenderness ratios both buckling and yielding contribute to collapse. Clearly both these conditions must be accounted for in any design rule for columns.

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Columns are commonly found in many types of building. Columns carry load principally by axial compression. The strength of stocky columns is related to material strength. The strength of slender columns is limited by buckling. n practice steel columns have to allow for both buckling and material failure, and for interaction between the two. The resistance of a cross-section to buckling is represented by its radius of gyration. End conditions influence buckling behaviour and are accounted for by using an effective length. n practice columns are subject to a combination of compression and bending. Because buckling resistance and actual stress are both related to the size of the crosssection, iterative design procedures must be used.