Engineering and Construction

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Road and railway truss bridges can either be underslung (deck at top chord level), through (deck at bottom chord level) or semi-through.

Limits on headroom, navigation height and construction depth will determine whether an underslung or through truss will be the most appropriate. For large-span bridges the through type is often adopted as ample headroom will be available to permit direct lateral restraint to the top compression chord using cross bracing.
For short-span bridges, however, the underslung type is most appropriate provided the navigation height or construction depth limits are satisfied. Underslung trusses are usually more economical than either through or semi-through trusses as the deck structure performs the dual function of directly supporting the traffic loads and providing lateral restraint to the compression chords. In the case of short-span through trusses the span-to-depth ratio may be uneconomically low if the top chord is restrained by cross-beams, as sufficient traffic headroom must be provided. In such a situation it is more economical to brace the top compression chords by U-frame action.

A span-to-depth ratio of between 6 and 8 for railway bridges and between 10 and 12 for road bridges offers the most economical design. In general terms the pro- portions should be such that the chords and web members have approximately an equal weight.

The bay widths should be proportioned so that the diagonal members are inclined at approximately 50° or slightly steeper. For large-span trusses  subdivision of the bays is necessary to avoid having excessively long web members.

The Pratt, Howe and Warren trusses, Fig. 19.3(a)–(c), have an economic span range of between 40m and 100m for railway bridges and up to 150m for road bridges. For the shorter spans of the range the Warren truss requires less material than either the Pratt or the Howe trusses. For medium spans of the range the Pratt or Howe trusses are both more favourable and by far the most common types. For large spans the modified Warren truss and subdivided Parker (inclined chord) truss are the most economical.

Fig. 19.3 Common types of bridge trusses: (a) Pratt, (b) Howe, (c) Warren, (d) modified Warren, (e) Parker, (f) K, (g) diamond, (h) Petit

For spans of between 100m and 250m the depth of the truss may be up to four times the economic bay width, and in such a case the K-, diamond or Petit (subdivided Pratt or Howe) trusses are more appropriate. For the shorter spans of the range the diamond or Petit trusses, by their nature, are subject  to very high sec- ondary stresses. In such a case the K-truss, with primary truss members at all nodes, is more appropriate as joint deflections are uniform, greatly reducing the secondary stresses.

For spans greater than 150m variable depth trusses are normally adopted for economy.

The spacing of bridge trusses depends on the width of the carriageway for road bridges and the required number of tracks for railway bridges, in addition to con- siderations regarding lateral strength and rigidity. However, in general the spacing should be limited to between  1/18 and 1/20 of the span, with a minimum of 4m to 5mfor through trusses and approximately 1/15 of the span, with a minimum of 3m to 4m, for underslung trusses.

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For pitched-roof trusses such as the Pratt, Howe or Fink, Fig. 19.1(a), (b) and (c), the most economical span-to-depth ratio (at apex) is between 4 and 5, with a span range of 6m to 12m, the Fink truss being the most economical at the higher end of the span range. Spans of up to 15m are possible but the unusable roof space becomes excessive and increases the running costs of the building. In such circum- stances the span-to-depth ratio may be increased to about 6 to 7, the additional steel weight (increase in initial capital expenditure) being offset by the long-term savings in the running costs. For spans of between 15m and 30m, the mansard truss, Fig. 19.1(d), reduces the unusable roof space but retains the pitched appearance and
offers an economic structure at span-to-depth ratios of about 7 to 8.

Fig. 19.1 Common types of roof trusses: (a) Pratt – pitched, (b) Howe, (c) Fink, (d) mansard, (e) Pratt – flat, (f) Warren, (g) modified Warren, (h) saw-tooth

The parallel (or near parallel) chord trusses (also known as lattice girders) such as the Pratt or Warren, Fig. 19.1(e) and (f), have an economic span range of between 6m and 50m, with a span-to-depth ratio of between 15 and 25 depending on the intensity of the applied loads. For the top end of the span range the bay width should be such that the web members are inclined at approximately 50° or slightly steeper.

For long, deep trusses the bay widths become too large and are often subdivided with secondary web members.

For roof trusses the web member intersection points with the chords should ideally coincide with the secondary transverse roof members (purlins). In practice this is not often the case for economic truss member arrangements, thus resulting in the supporting chord being subject to local bending stresses.

The most economical spacing for roof trusses is a function of the span and load intensity and to a lesser extent the span and spacing of the purlins, but as a general rule the spacing should be between 1/4 and 1/5 of the span, which results in a spacing of between 4m and 10m for the economic range of truss spans.

For short-span roof trusses between 6m and 15m the minimum spacing should be limited to 3–4m.

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Trusses are now infrequently used for road bridges in the UK because of high fabrication and maintenance costs. However, the recent award-winning Brinnington railway bridge (Fig. 19.2) demonstrates that they can still be used to create efficient and attractive railway structures. In many parts of the world, particularly in developing countries where labour costs are low and material costs are high, trusses are often adopted for their economy in steel. Their structural form also lends itself to transportation in small components and piece-small erection,which may be suitable for remote locations.

Fig. 19.2 Brinnington Railway Bridge

Some of the most commonly used trusses suitable for both road and rail bridges are illustrated in Fig. 19.3. Pratt, Howe and Warren trusses, Fig. 19.3(a), (b) and (c), which are discussed in section 19.2.1, are more suitable for short to medium spans.

The economic span range of the Pratt and Howe trusses may be extended by subdividing the diagonals and the deck support chord as shown for the Petit truss, Fig. 19.3(h), although this often gives rise to high secondary stresses for short to medium spans. In the case of the Warren truss the unsupported length of the chords may be too great for the economic span and depth range; in such a case the modified Warren truss, Fig. 19.3(d), is more appropriate.

The variable depth type truss such as the Parker, Fig. 19.3(e), offers an aesthetically pleasing structure.With this type of truss its structural function is emphasized and the material is economically distributed at the cost of having expensive fabrication due to the variable length and variable inclination of the web and top chord members.

For economy the truss depth is ideally set at a fixed proportion of the span.

As the span increases the truss depth and bay width increase accordingly. The bay width is usually fixed by providing truss nodes on the centrelines of the deck crossbeams, thus avoiding high local bending stress in the deck chord. For large-span bridges with an economical spacing for the deck cross-beams, the height of the truss may be as much as four times the bay width. In such a case a subdivided form of truss will be required to avoid very long uneconomical compression web members, or tensile members subjected to load reversal due to moving live loads. The diamond, Petit and K-trusses, Fig. 19.3(g), (h) and (f), are just three types which can be used.

The diamond and Petit trusses have the advantage of having shorter diagonals than the K-truss. The main disadvantage of trusses such as the diamond or Petit, which have intermediate bracing members connected to the chords away from the main joints, is that they give rise to high secondary stresses for short to medium spans due to differential joint deformation caused by moving live loads.The K-truss is far superior in this respect.

Fig. 19.3 Common types of bridge trusses: (a) Pratt, (b) Howe, (c) Warren, (d) modified Warren, (e) Parker, (f) K, (g) diamond, (h) Petit

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The most common use of trusses in buildings is to provide support to roofs, floors and such internal loading as services and suspended ceilings. There are many types and forms of trusses; some of the most widely used are shown in Fig. 19.1.The type of truss adopted in design is governed by architectural and client requirements, varied in detail by dimensional and economic factors.

The Pratt truss, Fig. 19.1(a) and (e), has diagonals in tension under normal verti- cal loading so that the shorter vertical web members are in compression and the longer diagonal web members are in tension. This advantage is partially offset by the fact that the compression chord is more heavily loaded than the tension chord at mid-span under normal vertical loading. It should be noted, however, that for a light-pitched Pratt roof truss wind loads may cause a reversal of load thus putting the longer web members into compression.

The converse of the Pratt truss is the Howe truss (or English truss), Fig. 19.1(b).

The Howe truss can be advantageous for very lightly loaded roofs in which reversal of load due to wind will occur. In addition the tension chord is more heavily loaded than the compression chord at mid-span under normal vertical loading.The Fink truss, Fig. 19.1(c), offers greater economy in terms of steel weight for long-span high-pitched roofs as the members are subdivided into shorter elements. There are many ways of arranging and subdividing the chords and web members under the control of the designer.

The mansard truss, Fig. 19.1(d), is a variation of the Fink truss which has the advantage of reducing unusable roof space and so reducing the running costs of the building. The main disadvantage of the mansard truss is that the forces in the top and bottom chords are increased due to the smaller span-to-depth ratio.

However, it must not escape the designer’s mind that any savings achieved in steel weight by introducing a greater number of smaller members may, as is often the case, substantially increase fabrication and maintenance costs.

The Warren truss, Fig. 19.1(f), has equal length compression and tension web members, resulting in a net saving in steel weight for smaller spans. The added advantage of the Warren truss is that it avoids the use of web members of differing length and thus reduces fabrication costs.For larger spans the modified Warren truss, Fig. 19.1(g), may be adopted where additional restraint to the chords is required (this also reduces secondary stresses). The modified Warren truss requires more material than the parallel-chord Pratt truss, but this is offset by its symmetry and pleasing appearance. The saw-tooth or butterfly truss, Fig. 19.1(h), is just one of many examples of trusses used in multi-bay buildings, although the other types described above are equally suitable.

Fig. 19.1 Common types of roof trusses: (a) Pratt – pitched, (b) Howe, (c) Fink, (d) mansard, (e) Pratt – flat, (f) Warren, (g) modified Warren, (h) saw-tooth

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The development of highly automated workshops in recent years has reduced the fabrication costs of plate girders very considerably; box girders and trusses still have to be fabricated manually, with consequently high fabrication costs. Optimum use of material is made, compared with rolled sections, as the girder is fabricated from plates and the designer has greater freedom to vary the section to correspond with changes in the applied forces. Thus variable depth plate girders have been increasingly designed in recent years. Plate girders are aesthetically more pleasing than trusses and are easier to transport and erect than box girders.

There are only a very few limitations in the use of plate girders. Compared with trusses they are heavier,more difficult to transport and have larger wind resistance.

The provision of openings for services is also more difficult. The low torsional stiffness of plate girders makes them difficult to use in bridges having small plan radius.

Plate girders can sometimes pose problems during erection because of concern for the stability of compression flanges.

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Introduction
Plate girders are employed to support heavy vertical loads over long spans for which the resulting bending moments are larger than the moment resistance of available rolled sections. In its simplest form the plate girder is a built-up beam consisting of two flange plates, fillet welded to a web plate to form an I-section (see Fig. 17.1).

The primary function of the top and bottom flange plates is to resist the axial com- pressive and tensile forces caused by the applied bending moments; the main function of the web is to resist the shear. Indeed this partition of structural action is used as the basis for design in some codes of practice.

For a given bending moment the required flange areas can be reduced by increasing the distance between them.Thus for an economical design it is advantageous to increase the distance between flanges. To keep the self-weight of the girder to a minimum the web thickness should be reduced as the depth increases, but this leads to web buckling considerations being more significant in plate girders than in rolled beams.

Plate girders are sometimes used in buildings and are often used in small to medium span bridges. They are designed in accordance with the provisions contained in BS 5950: Part 1: 20001 and BS 5400: Part 32
respectively. This chapter explains current practice in designing plate girders for buildings and bridges; refer- ences to the relevant clauses in the codes are made

Fig. 17.1 Elevation and cross-section of a typical plate girder

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One solution to the problem of accommodating services within a  restricted floor  depth is to run the services through openings in the floor beams. Since the size of hole necessary in the beam web will then typically represent a significant proportion of the clear web depth, it may be expected that it will have an effect on structural performance.The easiest way of visualizing this is to draw an analogy between a beam with large rectangular web cut-outs and a Vierendeel girder. Figure 16.16 shows how the presence of the web hole enables the beam to deform locally in a similar manner to the shear type deformation of a Vierendeel panel.

Fig. 16.16 Vierendeel-type action in beam with web openings: (a) overall view, (b) detail of deformed region

These deformations, superimposed on the overall bending effects, lead to increased deflection and additional web stresses.

A particular type of web hole is the castellation formed when a UB is cut, turned and rewelded as illustrated in Fig. 16.17. For the normal UK module geometry this leads to a 50% increase in section depth with a regular series of hexagonal holes.

Fig. 16.17 Castellated beam: (a) basic concept, (b) details of normal UK module geometry

Other geometries are possible, including a further increase in depth through the use of plates welded between the two halves of the original beam. Some aspects of the design of castellated beams are covered by the provisions of BS 5950, while more detailed guidance is available in a Constrado publication.

Based on research conducted in the USA, a comparatively simple elastic method for the design of beams with web holes, including a fully worked example, is available.

This uses the concept of an analysis for girder stresses and deflections that neglects the effects of the holes, coupled with checking against suitably modified limiting values. The full list of design checks considered in Reference 14 is:

(1) web shear due to overall bending acting on the reduced web area
(2) web shear due to local Vierendeel bending at the hole
(3) primary bending stresses (little effect since overall bending is resisted princi-
pally by the girder flanges)
(4) local bending due to Vierendeel action
(5) local buckling of the tee formed by the compression flange and the web adjoining the web hole
(6) local buckling of the stem of the compression tee due to secondary bending
(7) web crippling under concentrated loads or reactions near a web hole; as a simple guide,Reference 14 suggests that for loads which act at least (d/2) from the edge of a hole this effect may be neglected
(8) shear buckling of the web between holes; as a simple guide, Reference 14 sug- gests that for a clear distance between holes that exceeds the hole length this effect may be neglected
(9) vertical deflections; as a rough guide, secondary effects in castellated beams may be expected to add about 30% to the deflections calculated for a plain web beam of the same depth (1.5D). Beams with circular holes of diameter (D/2) may be expected to behave similarly, while beams with comparable rectangular holes may be expected to deflect rather more.

As an alternative to the use of elastic methods, significant progress has been made in recent years in devising limit state approaches based on ultimate strength  conditions.A CIRIA/SCI design guide dealing with the topic principally from the point of composite beams is now available. If some of the steps in the 24-point design check of Reference 15 are omitted, the method may be applied to non-composite beams, including composite beams under construction.

The governing condition for a stocky web in the vicinity of a hole is taken as excessive plastic deformation near the opening corners and in the web above and below the opening as illustrated in Fig. 16.18. A conservative estimate of web strength may then be obtained from a moment–shear interaction diagram of the type shown as Fig. 16.19.Values of M0 and V1 in terms of the plastic moment capacity and plastic shear capacity of the unperforated web are given previouly for both plain and reinforced holes; M1 may also be determined in this way. Solution of these equations is tedious, but some rearrangement and simplification are possible so that an explicit solution for the required area of reinforcement may be obtained.

However, the whole approach is best programmed for a microcomputer, and a program based on the full method of Reference 15 is available from the SCI.

Fig. 16.18 Hole-induced failure

Fig. 16.19 Moment–shear interaction for a stocky web in the vicinity of a hole

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In situations where a relatively lightly loaded beam is required such as a purlin or sheeting rail spanning between main frames supporting the cladding in a portal frame, it is common practice to use a cold-formed section produced cold from flat steel sheet, typically between about 1mm and 6mm in thickness, in a wide range of shapes of the type shown in Fig. 16.12. A particular feature is that normally each section is formed from a single flat bent into the required shape; thus most available sections are not doubly symmetric but channels, zeds and other singly sym- metric shapes. The forming process does, however, readily permit the use of quite complex cross-sections, incorporating longitudinal stiffening ribs and lips at the edges of flanges. Since the original coils are usually galvanized, the members do not normally require further protective treatment.

Fig. 16.12 Typical cold-formed section beam shapes

The structural design of cold-formed sections is covered by BS 5950: Part 5,which permits three approaches:

(1) design by calculation using the procedures of the code, section 5, for members in bending
(2) design on the basis of testing using the procedures of section 10 to control the testing and section 10.3 for members in bending
(3) for three commonly used types of member (zed purlins, sheeting rails and lattice joists), design using the simplified set of rules given previously.

In practice option (2) is the most frequently used, with all the major suppliers providing design literature, the basis of which is usually extensive testing of their product range, design being often reduced to the selection of a suitable section for a given span, loading and support arrangement using the tables provided.

Most cold-formed section types are the result of considerable development work by their producers. The profiles are therefore highly engineered so as to produce a near optimum performance, a typical example being the ranges of purlins produced by the leading UK suppliers. Because of the combination of the thin material and the comparative freedom provided by the forming process, this means that most sections will contain plate elements having high width-to-thickness ratios. Local buckling effects, due either to overall bending because the profile is non-compact, or to the introduction of localized loads, are of greater importance than is usually the case for design using hot-rolled sections. BS 5950: Part 5 therefore gives rather more attention to the treatment of slender cross-sections than does BS 5950: Part 1. In addition, manufacturers’ design data normally exploit the post-buckling strength observed in their development tests.

The approach used to deal with sections containing slender elements in BS 5950 is the well accepted effective width technique. This is based on the observation that plates, unlike struts, are able to withstand loads significantly in excess of their initial elastic buckling load, provided some measure of support is available to at least one of their longitudinal edges. Buckling then leads to a redistribution of stress, with the regions adjacent to the supported longitudinal  edges attracting higher stresses and the other parts of the plate becoming progressively less effective, as shown in Fig. 16.13. A simple design representation of the condition of  Fig. 16.13 consists of replacing the actual post-buckling stress distribution with the approximation shown in Fig. 16.14. The structural properties of the member (strength and stiffness) are then calculated for this effective cross-section as illus- trated in Fig. 16.15. Tabulated information in BS 5950 for steel of yield strength 280N/mm2 makes the application of this approach simpler in the sense that effective widths may readily be determined, although cross-sectional properties have still to be calculated.The use of manufacturer’s literature removes this requirement. For beams, Part 5 also covers the design of reinforcing lips on the usual basis of ensuring that the free edge of a flange supported by a single web behaves as if
both edges were supported; web crushing under local loads, lateral–torsional buckling and the approximate determination of deflections take into account any loss of plating effectiveness.

Forzed purlins or sheeting rails section 9 of BS 5950: Part 5 provides a set of simple empirically based design rules.Although easy to use, these are likely to lead to heavier members for a given loading, span and support arrangement than either of the other permitted procedures.A particular difference of this material is its use of unfactored loads, with the design conditions being expressed directly in member property requirements.

Fig. 16.13 Loss of plating effectiveness at progressively higher compressive stress

Fig. 16.14 Effective width design approximation

Fig. 16.15 Effective cross-section

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Frequently beam design will be constrained by a need to keep the beam depth to a minimum. This restriction is easy to understand in the context of floor beams in a multi-storey building for which savings in overall floor depth will be multiplied several times over, thereby permitting the inclusion of extra floors within the same overall building height or effecting savings on expensive cladding materials by reducing building height for the same number of floors. Within the floor zone of buildings with large volumes of cabling, ducting and other heavy services, only a fraction of the depth is available for structural purposes.
Such restrictions lead to a number of possible solutions which appear to run contrary to the basic principles of beam design. However, structural designers should remember that the main framing of a typical multi-storey commercial building  typically represents less than 10% of the building cost and that factors such as the efficient incorporation of the services and enabling site work to proceed rapidly  and easily are likely to be of greater overall economic significance than trimming steel weight.

An obvious solution is the use of universal columns as beams.While not as structurally efficient for carrying loads in simple vertical bending as UB sections, as illustrated by the example of Table 16.8, their design is straightforward. Problems of web bearing and buckling at supports are less likely due to the reduced web d/t ratios.




Lateral–torsional buckling considerations are less likely to control the design of laterally unbraced lengths because the wider flanges will provide greater lateral  stiffness (L/ry values are likely to be low).Wider flanges are also advantageous for supporting floor units, particularly the metal decking used frequently as part  of a composite floor system.

Difficulties can occur, because of the reduced depth, with deflections, although dead load deflections may be taken out by precambering the beams. This will not assist in limiting deflections in service due to imposed loading, although composite action will provide a much stiffer composite section.Excessive deflection of the floor beams under the weight of wet concrete can significantly increase slab depths at mid-span, leading to a substantially higher dead load. None of these problems need cause undue difficulty provided they are recognized and the proper checks made at a sufficiently early stage in the design.

Another possible source of difficulty arises in making connections between shallow beams and columns or between primary and secondary beams.The reduced web depth can lead to problems in physically accommodating sufficient bolts to carry the necessary end shears.Welding cleats to beams removes some of the dimensional tolerances that assist with erection on site as well as  interfering with the smooth flow of work in a fabricator’s shop that is equipped with a dedicated saw and drill line for beams. Extending the connection beyond the beam depth by using seating cleats is one solution, although a requirement to contain the connection within the beam depth may prevent their use.

Beam depths may also be reduced by using moment-resisting beam-to-column connections which provide end fixity to the beams; a fixed end beam carrying a central point load will develop 50% of the peak moment and only 20% of the central deflection of a similar simply-supported beam. Full end fixity is unlikely to be a realistic proposition in normal frames but the replacement of the notionally pinned beam-to-column connection provided by an arrangement such as web cleats, with a substantial end plate that functions more or less as a rigid connection, permits the development of some degree of continuity between beams and columns. These arrangements will need more careful treatment when analysing the pattern of internal moments and forces in the frame since the principles of simple construction will no longer apply.

An effect similar to the use of UC sections may be achieved if  the flanges of a UB of a size that is incapable of carrying the required moment are reinforced by welding plates over part of its length.Additional moment capacity can be provided where it is needed as illustrated in Fig. 16.11; the resulting non-uniform section is stiffer and deflects less. Plating of the flanges will not improve the beam’s shear capacity since this is essentially provided by the web and the possibility of shear  or indeed local web capacity governing the design must be considered. A further development of this idea is the use of tapered sections fabricated from plate. To be economic, tapered sections are likely to contain plate elements that lie outside the limits for compact sections.

Because of the interest in developing longer spans for floors and the need to improve the performance of floor beams, a number of ingenious arrangements have developed in recent years.

Fig. 16.11 Selective increase of moment capacity by use of a plated UB

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One (or more) of a number of distinct limiting conditions may, in theory, control the design of a particular beam as indicated in Table 16.3, but in any particular practical case only a few of them are likely to require full checks. It is therefore convenient to consider the various possibilities in turn, noting the conditions under which each is likely to be important. For convenience the various phenomena are first considered principally within the context of using standard hot-rolled sections, i.e. UBs, UCs, RSJs and channels.




1 Moment capacity, Mc
The most basic design requirement for a beam is the provision of adequate in-plane bending strength. This is provided by ensuring that Mc for the selected section exceeds the maximum moment produced by the factored loading.Determination of Mc, which is linked to section classification.

For a statically determinate structural arrangement, simple considerations of statics provide the moment levels produced by the applied loads against which Mc must be checked. For indeterminate arrangements, a suitable method of elastic analysis such as moment distribution or slope deflection is required. The justification for using an elastically obtained distribution of moments with, in the case of compact or plastic cross-sections, a plastic cross-sectional resistance has been fully discussed by Johnson and Buckby.

2 Effect of shear
Only in cases of high coincident shear and moment, found for example at the internal supports of continuous beams, is the effect of shear likely to have a significant influence on the design of beams.

Shear capacity Pv is normally calculated as the product of a shear strength pv, often taken for convenience as 0.6py which is close to the yield stress of steel in shear of 1/÷3 of the uniaxial tensile yield stress, and an appropriate shear area Av.

The process approximates the actual distribution of shear stress in a beam web as well as assuming some degree of plasticity.While suitable for rolled sections, it may not therefore be applicable to plate girders.An alternative design approach, more suited to webs containing large holes or having variations in thickness, is to work from first principles and to limit the maximum shear stress to a suitable value; BS 5950: Part 1 uses 0.7py. In cases where d/t > 63, shear buckling limits the effectiveness of the web and reference to Chapter 17 should be made for methods of determining the reduced capacity.

In principle the presence of shear in a section reduces its moment capacity. In practice the reduction may be regarded as negligibly small up to quite large fractions of the shear capacity Pv. For example, BS 5950: Part 1 requires a reduction in Mc for plastic or compact sections only when the applied shear exceeds 0.6Pv, and permits the full value of Mc to be used for all cases of semi-compact or slender  sections.

Figure 16.2 illustrates the application of the BS 5950: Part 1 rule for plastic or compact sections to a typical UB and UC having approximately equal values of plastic section modulus Sx. Evaluation of the formula in cases where Fv/Pv > 0.6 first requires that the value of Sv, the plastic modulus of the shear area Av (equal to tD in this case), be determined. This is readily obtained from the tabulated values of S for rectangles given in Reference 2 corresponding to a linear reduction from Sx to (Sx - Sv).

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Beams are possibly the most fundamental type of member present  in a civil engineering structure. Their principal function is the transmission  of vertical load by means of flexural (bending) action into, for example, the columns  in a rectangular building frame or the abutments in a bridge which support them.

Table 16.1 provides some idea of the different structural forms suitable for use  as beams in a steel structure; several of these are illustrated in Fig. 16.1. For modest spans, including the majority of those found in buildings, the use of standard  hot-rolled sections (normally UBs but possibly UCs if minimizing floor depth is a prime consideration or channels if only light loads need to be  supported) will  be sufficient. Lightly loaded members such as the purlins supporting the roof  of a portal-frame building are frequently selected from the range of proprietary cold-formed sections produced from steel sheet only a few millimetres thick, nor- mally already protected against corrosion by galvanizing, in a variety of highly efficient shapes, advantage being taken of the roll-forming process to produce sections with properties carefully selected for the task they are required to perform. For spans in excess of those that can be achieved sensibly using ready-made sections some form of built-up member is required. Castellated beams, formed by profile cutting of the web and welding to produce a deeper section, typically 50% deeper using the standard UK geometry, are visually attractive but cannot withstand high shear loads unless certain of the castellations are filled in with plate. The range of spans for which UBs may be used can be extended if cover plates are welded to both flanges.

Fig. 16.1 Types of beam cross-section

Alternatively a beam fabricated entirely by welding plates together may be employed allowing variations in properties by changes in depth, for example, flange thickness, or, in certain cases where the use of very thin webs is required, stiffening to prevent premature buckling failure is necessary.A full treatment of the specialist aspects of plate-girder design is provided later. If spans are so large that a single member cannot economically be provided, then a truss may be a suitable alternative. In addition to the deep truss fabricated from open hot-rolled sections, SHS or both, used to provide long clear spans in sports halls and supermarkets, smaller prefabricated arrangements using RHS or CHS provide an attractive alter- native to the use of standard sections for more modest spans.


Since the principal requirement of a beam is adequate resistance to vertical bending, a very useful indication of the size of section likely to prove suitable may be obtained through the concept of the span to depth ratio.This is simply the value of the clear span divided by the overall depth. An average figure for a properly designed steel beam is between 15 and 20, perhaps more if a particularly slender form of construction is employed or possibly less if very heavy loadings are present.

When designing beams, attention must be given to a series of issues, in addition to simple vertical bending, that may have some bearing upon the problem.Torsional loading may often be eliminated by careful detailing or its effects reasonably regarded as of negligible importance by a correct appreciation of how the structure actually behaves; in certain instances it should, however, be considered. Section classification (allowance for possible local buckling effects) is more involved than is the case for struts since different elements in the cross-section are subject to different patterns of stress; the flanges in an I-beam in the elastic range will be in approximately uniform tension or compression while the web will contain a stress gradient.

The possibility of members being designed for an elastic or a plastic state, including the use of a full plastic design for the complete structure, also affects section classification.Various forms of instability of the beam as a whole or of parts subject to locally high stresses, such as the web over a support, also require attention. Finally certain forms of construction may lend themselves to the appearance of unacceptable vibrations; although this is likely to be affected by the choice of beams, its coverage later on floors.

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Strut design is a relatively straightforward design task involving choice of cross-sectional type, assessment of end restraint and thus effective length, calculation of slenderness, determination of compressive strength and hence checking that the trial section can withstand the design load. Certain subsidiary checks may also be required part way through this process to ensure that the chosen cross section is not slender (or make suitable allowances if it is) or to guard against local failure in compound members.Thus only limited opportunities occur for the designer to use judgement and to make choices on the grounds of economy.

Essentially  these are restricted to control of the effective length, by introducing intermediate restraints where appropriate, and the original choice of cross section.

However, certain other points relating to columns may well have a bearing on the overall economy of the steel frame or truss. Of particular concern is the need to be able to make connections simply. In a multi-storey frame, the use of heavier UC  sections thus may be advantageous, permitting beam-to-column connections to be made without the need for stiffening the  flanges or web. Similarly in order to accommodate beams framing into the column web an increase in the size of UC may eliminate the need for special detailing.

While compound angle members were a common feature of early trusses, maintenance costs due both to the surface area requiring painting and to the incidence of corrosion caused by the inherent dirt and moisture traps have caused a change to the much greater use of tubular members. If site joints are kept to the minimum tubular trusses can be transported and handled on site in long lengths and a more economic as well as a visually more pleasing structure is likely to result.

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Individual members may be combined in a variety of ways to produce a more efficient compound section. Figure 15.15 illustrates the most common arrangements. In each case the concept is one of providing a compound member whose overall slenderness will be such that its load-carrying capacity will significantly exceed the sum of the axial resistances of the component members, i.e. for the case of Fig. 15.15(b) the laced strut will be stronger than the four corner angles treated separately.

The detailed rules contained in these clauses are essentially of two types:

(1) Covering construction details such as the arrangements for interconnection in a general ‘good practice’ manner
(2) Quantitative rules for the determination of the overall slenderness, limits necessary for component slenderness, forces for which the interconnections should be designed, etc.

BS 5400: Part 3 also contains specific rules for the design of:

(1) Batten struts
(2) Laced struts
(3) Struts connected by perforated cover plates
(4) Struts consisting of back-to-back components.

These are somewhat more detailed than those of BS 5950: Part 1, particularly in the matter of determining suitable design forces for the interconnections, i.e. battens, lacings, etc.

Fig. 15.15 Typical arrangements for compound struts: (a) closely spaced, (b) laced or battened

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The design of two types of strut requires that certain additional points be considered:

(1) built-up sections or compound struts (Fig. 15.13), for which the behaviour of the individual components must be taken into account

(2) angles, channels and tees (Fig. 15.14), for which the eccentricity of loading produced by normal forms of end connection must be acknowledged.

In both cases, however, it will often be possible to design this more complex type of member as an equivalent single axially-loaded strut.

Fig. 15.13 Built-up struts

Fig. 15.14 Geometrical properties of an angle section

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The most important property of a strut as far as the determination of its load- carrying capacity is concerned is its slenderness, l, defined as the ratio of its effective length, LE, divided by the appropriate radius of gyration, r. Codes of practice such as BS 5950 used to place upper limits on l, as indicated in Table 15.1, so as to avoid the use of flimsy construction, i.e. to ensure that a member which will ordinarily be subject only to axial load does have some limited resistance to an acci- dental lateral load, does not rattle, etc. Although not now explicitly stated in the 2000 version of the code, it is still good practice to aim for robust construction.


Similarly, for BS 5400: Part 3, no actual limits are specified but the user is provided with a general advisory note to the effect that construction should be suitably robust. By contrast, BS 5400 does place upper bounds on local plate slenderness to avoid consideration of local buckling (see Table 15.2).

Strut design will normally require that, once a trial member has been selected and its loading and support conditions determined, attention be given to whichever of the following checks are relevant for the particular application:

(1) Overall flexural buckling – largely controlled by the slenderness ratio, l, which is a function of member length, cross-sectional shape and the support condi- tions provided; also influenced by the type of member
(2) Local buckling – controlled by the width-to-thickness ratios of the component plate elements (see Chapter 13); with some care in the original  choice of member this need not involve any actual calculation
(3) Buckling of component parts – only relevant for built-up sections such as laced and battened columns; the strength of individual parts must be checked, often by simply limiting distances between points of interconnection
(4) Torsional or torsional-flexural buckling – for cold-formed sections and in extreme cases of unusually shaped heavier open sections, the inherent low  torsional stiffness of the member may make this form of buckling more critical than simple flexural buckling.

In principle, local buckling and overall buckling (flexural or torsional) should always be checked. In practice, provided cross sections that at least meet the semi-compact limits for pure compression are used, then no local buckling check is necessary since the cross section will be fully effective.

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Various types of steel section may be used as struts to resist  compressive loads; Fig. 15.1 illustrates a number of them. Practical considerations such as the methods to be employed for making connections often influence the choice, especially for light members. Although closed sections such as tubes are theoretically the most efficient, it is normally much easier to make simple site connections, using the minimum of skilled labour or special equipment, to open sections. Typical arrangements include:

(1) light trusses and bracing – angles (including compound angles back to back) and tees
(2) larger trusses – circular hollow sections, rectangular hollow sections, compound sections and universal columns
(3) frames – universal columns, fabricated sections e.g. reinforced UCs
(4) bridges – box columns
(5) power stations – stiffened box columns.

Fig. 15.1 Typical column cross sections

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Introduction
Members subject to compression, referred to as either ‘columns’ or ‘struts’, form one of the basic types of load-carrying component.They may be found, for example, as vertical columns in building frames, in the compression chords of a bridge truss or in any position in a space frame.

In many practical situations struts are not subject solely to compression but, depending upon the exact nature of the load path through the structure, are also required to resist some degree of bending. For example, a corner column in a building is normally bent about both axes by the action of the beam  loads, a strut in a space frame is not necessarily loaded concentrically, the compression chord of a roof truss may also be required to carry some lateral loads. Thus many compression members are actually designed for combined loading as beam-columns. Notwith-standing this, the ability to determine the compressive resistance of members is of fundamental importance in design, both for the struts loaded only in compression and as one component in the interaction type of approach normally used for beam- column design.

The most significant factor that must be considered in the design of struts is buckling. Depending on the type of member and the particular application under consideration, this may take several forms. One of these, local buckling of individual plate elements in compression, has already been considered in Chapter 13.Much of this chapter is devoted to the consideration of the way in which buckling is handled in strut design.

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Determination of member sizes
Although the engineer may use his experience or other methods to determine preliminary member sizes, the graphs in Figs 11.5, 11.6 and 11.7 will be used for this example. In order to use these graphs, four parameters are required:

Fig. 11.8 Portal frame design example

It is suggested that the next stage is to check the overall stability of the frame. The main reason for this is that the only way to correct insufficient stability is to change the main member sizes. If any other checks are not satisfied, additional bracing can frequently be used to rectify the situation, without altering the member sizes.

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The design process explained here assumes that the reader is familiar with the basic concepts of limit state steel design, and the basic analysis and design of continuous beams. These notes take the form of a worked example.A method of determining preliminary member sizes has been included.

1 Methods of analysis BS 5950: Part 1allows two main methods of analysis of a structure:

(1) Linear elastic. The frame is analysed either by hand or by computer assuming linear elastic behaviour. Once the forces, moments and shears have been derived by elastic analysis the ultimate capacity of each section is checked using the rules given in Section 4 of the Code.

(2) Simple plastic theory. The frame is analysed using the basic principles of simple plastic theory.Once the forces,moments and shears have been derived by analysis the member capacities are checked. Those containing plastic  hinges are checked in accordance with Section 4.

Elastic analysis
Using elastic analysis it is to be expected that the structure will be heavier than that designed by plastic methods, but less stability bracing will be needed. It may well be that the final details will also be more simple.

It will remain the engineer’s responsibility to ensure that stability is provided both locally and in the overall condition. BS 5950: Part 12 provides no specific rules regarding the stability of the frame as a whole; this means that the engineer must ensure that the stability is checked using the general rules for all frames. He must also check that the movement of the frame under all loading cases is not sufficient to cause damage to adjacent construction, i.e. brick walls or cladding, the serviceability limit state of deflection.

Plastic analysis
The method of calculating the ultimate load of a portal frame is described in many publications.The main essence of the method is to assume that plastic ‘hinges’ occur at points in the frame where the value of M/Mp is at its highest value, the load being considered as increasing proportionally until the failure or ultimate state is reached.

Because of the straining at the hinge points it is essential that the local buckling and lateral distortion do not occur before failure. Failure is deemed to have taken place when sufficient hinges have formed to create a mechanism.

The member capacities are calculated using the rules given in Section 4 of the Code but with the additional restrictions applied to hinge positions. In addition, positive requirements are put on checking frame stability for both single-bay and multi-bay frames. Plastic designed frames are lighter than elastic designed frames, providing deflection is not a governing point; however, additional bracing may well be required.

2 Stability
With the use of lighter frames, various aspects of stability take a more prominent part in the design procedures.As far as portal frames are concerned the following areas are important:

(1) overall frame stability, in that the strength of the frame should not be affected by changes in geometry during loading (PD effect)
(2) snap-through stability, in multi-bay frames (three or more), where the effects of continuity can result in slender rafters
(3) plastic hinge stability, where the member must be prevented from moving out of plane or rotating at plastic hinges
(4) rafter stability, ensuring that the rafter is stable in bending as an unrestrained beam
(5) leg stability, where the leg below the plastic hinge must be stable
(6) haunch stability, where the tapered member is checked to ensure that the inner (compression flange) is stable.

3 Selecting suitable members for a trial design
The design of a portal frame structure is in reality a process of selecting suitable members and then proving their ability to perform in a satisfactory manner. Inex- perienced engineers can be given some guidance to estimate initial member sizes.

In order to speed the initial selection of members, three graphs have been produced to enable simple pin-based frames to be sized quickly.

These graphs have been prepared making the following assumptions:

(1) plastic hinges are formed at the bottom of the haunch in the leg and near the apex in the rafter, the exact position being determined by the frame geometry
(2) the depth of the rafter is approximately span/55 and the depth of the haunch below the eaves intersection is 1.5 times rafter depth
(3) the haunch length is 10% of the span of the frame, a limit generally regarded as providing a balance between economy and stability
(4) the moment in the rafter at the top of the haunch is 0.87Mp, i.e. it is assumed that the haunch area remains elastic
(5) the calculations assume that the calculated values of Mp are provided exactly by the sections and that there are no stability problems. Clearly these conditions will not be met, and it is the engineer’s responsibility to ensure that the chosen sections are fully checked for all aspects of behaviour.

The graphs cover the range of span/eaves height between 2 and 5 and rise/span of 0 to 0.2 (where 0 is a flat roof). Interpolation is permissible but extrapolation is not.

The three graphs give:

Figure 11.5: the horizontal force at the feet of the frame as a proportion of the totalfactored load wL, where w is the load/unit length of rafter and L is the span of the frame
Figure 11.6: the value of the moment capacity required in the rafters as a proportion of the load times span wL^2
Figure 11.7: the value of the moment capacity required in the legs as a proportion of the load times span wL^2.

Fig. 11.5 Rise/span against horizontal force at base for various span/eaves heights

Fig. 11.6 Rise/span against required Mp of rafter for various span/eaves heights

Fig. 11.7 Rise/span against required Mp of leg for various span/eaves heights

The graphs are non-dimensional and may be used with any consistent set of units.

In the worked example kilonewtons and metres are used.

Method of use of the graphs

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General
The formulae given in this section are based on Professor Kleinlogel’s Rahmen-formeln  and Mehrstielige Rahmen.

The formulae are applicable to frames which are symmetrical about a central vertical axis, and in which each member has constant second moment of area.

Formulae are given for the following types of frame:

Frame I Hingeless rectangular portal frame.
Frame II Two-hinged rectangular portal frame.
Frame III Hingeless gable frame with vertical legs.
Frame IV Two-hinged gable frame with vertical legs.

The loadings are so arranged that dead, snow and wind loads may be reproduced on all the frames. For example, wind suction acting normal to the sloping rafters of a building may be divided into horizontal and vertical components, for which appropriate formulae are given, although all the signs must be reversed because the loadings shown in the tables act inwards, not outwards as in the case of suction.

It should be noted that, with few exceptions, the loads between node or panel points are uniformly distributed over the whole member. It is appreciated that it is normal practice to impose loads on frames through purlins, siderails or beams. By using the coefficients in Fig. 11.1, however, allowance can be made for many other symmetrically placed loads on the cross-beams of frames I and II shown, where the difference in effect is sufficient to warrant the corrections being made. The inde- terminate BMs in the whole frame are calculated as though the loads were uniformly distributed over the beam being considered, and then all  are adjusted by multiplying by the appropriate coefficient in Fig. 11.1. It may be of interest to state why these adjustments are made. In any statically indeterminate structure the inde- terminate moments vary directly with the value of the following quantity:


Where the loaded member is of constant cross section, EI may be ignored.

Fig. 11.1 Conversion coefficients for symmetrical loads

Consider, as an example, the case of an encastré beam of constant cross section and of length L carrying a UDL of W. Then the area of the free BM diagram is


If, however, W is a central point load, the area of the free BM diagram is

The fixed end moments (FEM) due to the two types of loadings are WL/12 and WL/8 respectively, thus demonstrating that the indeterminate moments vary with the area of the free BM diagram and proving that the indeterminate moments are in the proportion of 1 : 1.5.

No rules can be laid down for the effect on the reactions of a change in the mode of application of the load, although sometimes they will vary with the indeterminate moments. Consider a simple rectangular portal with hinged feet. If a UDL placed over the whole of the beam is replaced by a central point load of the same magnitude, then the knee moments will increase by 50% with a corresponding increase in the horizontal thrusts H, while the vertical reactions V will remain the same.

Although the foregoing remarks relating to the indeterminate moments resulting from symmetrical loads apply to all rectangular portals, the rule applies for asym- metrical loads imposed upon the cross-beam of a rectangular portal frame with hinged feet. If a vertical UDL on the cross-beam is replaced by any vertical load of the same magnitude, then the indeterminate moments vary with the areas of the respective free BM diagram.

No doubt readers who use the tables frequently will learn short cuts, but it is not inappropriate to mention some. For example, if a UDL of W over the whole of a single-bay symmetrical frame is replaced by a UDL of the same magnitude of W over either the left-hand or right-hand half of the frame, the horizontal thrust at the feet is unaltered. If the frame has a pitched roof then the ridge moment will also be unaltered.

The charts in the Appendix have been prepared to assist in the design of rectangular frames or frames with a roof pitch of 1 in 5.