Structural Systems Building

Lecture 14.1.1: Single- Storey Buildings: Introduction and Primary Structure OBJECTIVE/SCOPE • • • • To describe the anatomy of single-storey buildings, cladding, secondary elements and main frame structures. To clarify how the loads are supported. To present the basic principles of design and analysis of a single storey building. To give a design example. PREREQUISITES Lecture 1B.1: Process of Design Lecture 1B.3: Background to Loadings Lecture 1B.5.1: Introduction to Design of Industrial Buildings Lecture 9.1: Thin-walled Members and Sheeting RELATED LECTURES Lecture 14.1.2: Single-Storey Buildings: Envelope and Secondary Structure Lecture 14.2: Analysis of Portal Frames: Introduction and Elastic Analysis Lecture 14.3: Analysis of Portal Frames: Plastic Analysis Lecture 14.4: Crane Runway Girders Lecture 14.5: Space Structure Systems Lecture 14.6: Special Single-Storey Structures SUMMARY Single-storey steel buildings are used to accommodate many functions such as factories, leisure facilities, and supermarkets. The structure consists of several elements. The lecture examines the function of each element, gives general indications of the cost as a proportion of the total cost of the building. The lecture also gives information on loading, on the different factors affecting the concept of the main frame, and on considerations relating to the construction process of the structure. 1. INTRODUCTION Steel construction is used for the majority of non-domestic single-storey buildings, the proportion is as high as 90% in the UK and 70% in France. This large proportion is due to the ability to design relatively light, long span, durable structures in steel which are easy to erect safely and quickly. The developments in steel cladding and light-gauge purlin and rail systems in recent years have enabled architects and engineers to create economical, attractive designs for a wide range of applications and budgets. The rate of change of any activity is very rapid as technology develops. Clients expect therefore their buildings to have a lifetime several times longer than the initial layout. A primary requirement is therefore flexibility of planning which results in a demand for as few columns as possible. The ability to provide spans up to 60m, but more commonly around 30m, using steel has proved very popular for commercial and leisure buildings. The lightness and flexibility of this kind of steel structure reduces the sizes and the costs of foundations and makes them less sensitive to the geotechnical characteristics of the soil. The brief for the design of the majority of single storey industrial buildings is essentially to provide a structure which is without, or has a limited number of, internal columns. In principle the requirement is for the construction of four walls and a roof for a single or multi-bay structure. The walls can be formed of steel columns with cladding which may be of profiled or plain sheet, precast concrete, or masonry. The designer considers a system of beams or frameworks (latticed or traditional) in structural steel to support the cladding for the roof. Use is made of hot rolled hollow sections (circular, rectangular) and traditional sections (I sections, angles, etc.) and also cold formed sections. Light latticed frameworks for the roof of an industrial building provide a neat, efficient, structure which is simple to design, economic to execute and frequently satisfies architectural requirements. Whilst the structural envelope and the design are 'basically' simple, it is essential to ascertain correctly the loads applied to the structure and to predict the load paths from the sheeting to the purlins and side rails, through the roof girder to the column and finally to the foundation and supporting soil. 2. ANATOMY AND CONCEPTION OF THE STRUCTURE The skeleton of a typical single-storey building is shown in the diagram (Figure 1). It consists of three major elements: cladding for both roof and walls; secondary steel to support the cladding and form framing for doors, windows and the like; and the main frame of the structure, including all necessary bracing. In addition, the building requires foundations which have to be designed and built to transmit all the loads to the soil. 2.1 Cladding The primary functions of the cladding are to provide shade, shelter, and an attractive appearance. The cladding is therefore arguably the most important element of the structure. It is likely to be some 50% of the total cost even if a fairly basic specification is used. The frame members, both primary and secondary are there to support the cladding and services. The requirements for walls and roof are somewhat different although both need to be weather tight, durable and provide insulation. The weathertightness of the roof is clearly paramount, particularly for lower pitches. In the case of walls the appearance will be the highest priority in making the selection. Cladding formed from metal sheets has emerged as the most popular choice since its introduction in the 1970's. Steel is the most usual substrate with aluminium as a more expensive second choice. The higher coefficient of expansion of aluminium can lead to difficulties in some circumstances. 2.2 Secondary Elements In the normal single-storey building the cladding is supported on secondary members, which transmit the loads back to main structural steel frames. An economic solution is provided by the use of cold-formed light gauge sections [1]. These sections are of proprietary shape and are produced to order on "computer numerically controlled" rolling machines. These processes are extremely efficient since the components are delivered to site pre-engineered to the exact requirements. As a result fabrication and erection times are minimised and material wastage is eliminated. With high volume rolling the material content of purlins and rails is a very significant part of the cost and manufacturers have developed shapes which are extremely material efficient. The most common are zeds, modified Zeds and sigma shapes as shown in Figure 2. Spans are commonly 5-8m but longer spans up to 12m can be achieved. It is also traditional to use hot-rolled sections as I or C shapes, not so sensitive to the effect of local instability. The centre to centre distance of the portal frames is then reduced, in general, to 5 or 6m. Generally, the cost breakdown of various elements will be cladding 50%; purlins and rails 10%; mainframe 30%; and foundations 10%. These figures are indicative, as clearly there is considerable variation in specification and cost of the cladding which is the largest single element. Cladding can vary from 10 ecu/m2 to over 150 ecu/m2. The remainder of this lecture discusses the main frame of the structure. Cladding and secondary elements are considered in Lecture 14.1.2. 2.3 The Main Frame of the Structure The loads are transferred from the sheeting onto the purlins and rails, which in turn are supported by a primary (or main) structure. These loads can be obtained from the relevant codes. They will include: • • • • • • the self weight of the cladding, of the secondary elements (purlins and rails) and of the frame itself, the applied loads from services, etc, the vertical and horizontal loads from cranes, the snow loads, the wind loads, the earthquake effects, in some areas. When the length of the structure is not too great, say 50 to 100m depending on temperature range, it is normal practice not to consider the effects produced by changes of temperature. It is usual practice, also, not to consider differential settlement of foundations if they are below 2cm. Wind can of course cause pressure, suction and drag loads on the cladding. Figure 3 illustrates the various components of load to consider in the design. The primary structural frames will normally be spaced from 5 to 8m centres. Although larger spacings are growing in popularity, an analysis of a purlin manufacturers' sales shows that about 6m remains the most popular spacing. The most straight forward single-storey structural form consists of a pair of vertical columns supporting a spanning beam. To be practical, there is a need for a fall in the roof finish to provide adequate drainage. For the small spans for which this arrangement is suitable, the fall can be achieved in the finishes or by a nominal slope in the beam. This form of construction, is shown in Figure 4. Stability against wind loading can be provided by the cladding which is fixed from frame to frame. This simple form of structure is used for small workshops. It is only suitable for spans up to approximately 12m. For longer spans this simple solution becomes, in general, uneconomic. Then portal frames and lattice trusses are more competitive solutions. Figure 5 shows some options commonly used for portal frames. The more popular solutions are pinned bases if there is no crane to be supported and the fully rigid version when it is necessary to support crane loads and to obtain smaller horizontal displacements. Less weight of steel results, in any case, from fixed based frames, but the additional cost of the foundations of the supports can be higher than the saving of steel. Figure 6 shows an example of solid column with a typical lattice girder. Whichever solution is chosen, all loads have to be transferred to the ground in a coherent fashion in even the simplest of buildings. The provision of bracing is covered in Lecture 14.1.2. Restraints to the main structure against out-of-plane buckling can be provided by the purlins connected to the top flange of the rafter and by the rails connected to the outer column flange. Figure 7 shows three typical ways of providing adequate restraint. Plastic design of portal frames brings limitations on the spacing of restraints of about 1,8 to 2m. At plastic hinge locations, it is necessary to provide stays from the rail to the inner flange in order to avoid the lateral buckling of the compressed flange. The cladding spans between the purlins and commercially available profiles are economic in this range of spacing to meet the requirements for walkability, strength (corrugation depths), and drainage. Where lattice structures are used, secondary bending in the top boom is avoided if the purlins are supported at the node points. Spacings of about 1,8m are often found convenient and economic. It is also possible to use the cladding as a stressed skin in order to transmit the horizontal forces generated in the roof by the wind and the tendency of the frame to out-of-plane buckling. 2.3.1 Simplest Frames The cross-section shown in Figure 4 is undoubtedly the simplest framing solution which can be used to provide structural integrity to single-storey buildings. Used predominately in spans of up to 10m, where flat roof construction is acceptable, the frame comprises standard hot-rolled sections having simple or moment-resisting joints. Flat roofs are notoriously difficult to weatherproof, since deflections of the horizontal cross-beams induce ponding of rainwater on the roof which tends to penetrate the laps of the traditional cladding profiles and, indeed, any weakness of the exterior roofing fabric. To counteract this, either the cross-member is cambered to provide the required fall across the roof, or the cladding itself is laid to a predetermined fall, again facilitating drainage of surface water off the roof. Due to the need to control excessive deflections, the sections tend to be somewhat heavier than those required for strength purposes alone, particularly if the cross-beam is designed as simply supported. In its simplest form, the cross-beam is designed as spanning between columns. For gravity loadings the latter are in direct compression apart from a small bending moment at the top of the column due to the eccentricity of the beam connection. The crossbeam acts in bending due to the applied gravity loads, the compression flange being restrained either by purlins, which support the roof sheet, or by a proprietary roof deck which may span between the main frames and which must be adequately fastened. Resistance to lateral loads is achieved by the use of a longitudinal wind girder, usually situated within the depth of the cross-beam. This transmits load from the top of the columns to bracing in the vertical plane, and thence to the foundation. The bracing is generally designed as a pin-jointed frame, in keeping with the simple joints used in the main frame. Buildings which employ the use of beam-and-column construction often have brickwork cladding in the vertical plane. With careful detailing, the brickwork can be designed to provide the vertical sway bracing, acting in a similar manner to the shear walls of a multi-storey building. Resistance to lateral loading can also be achieved either by the use of rigid connections at the column/beam joint or by designing the columns as fixed-base cantilevers. Rigid connections and rigid column/foundation joints reduce also the deflection of the beam and need less weight of structural steel for the frame. 2.3.2 Portal frames As explained above, the two most popular arrangements are the portal frame with pinned bases, if there is no crane to be supported, and the fully rigid portal frame, which is often used if there is a crane. These forms are both functional and economic. In- plane stability is derived from the provision of moment-resisting connections at the top and at the beam-to-column connections for the first situation and also at the base in the second one [2]. The falls required to the roof are naturally provided by the cladding carried on purlins which, in turn, are supported by the main frame members. Architectural pressures lead to the use of the flattest slopes compatible with weathertightness. The most common slope is about 6°, but slopes as low as 1° have been used. The frames are constructed from I-section rafters and columns with haunches at the connections at the eaves as illustrated in Figure 1. The haunch length is approximately 10% of the span and can be formed from welded plate or more commonly a cutting from a rolled section. The depth at the column face is typically slightly deeper than the rafter section. The design of these frames is dealt with in Lecture 14.2 and Lecture 14.3. Portal frames can be also built with tapered rather than haunched sections. Frames of this type are common in the USA and are being used more frequently in Europe. The sections are fabricated from plate on automated welding machines. The ability to vary web thickness, flange dimensions and section depth results in high material efficiency. Deep slender sections are used to maximise economy. Suitable design methods are described in Lecture 14.3. In addition to material economies there are benefits in reduced deflections resulting from the high in-plane stiffness of the deep sections. Portal frames are particularly economic up to 40m spans and, where the internal planning permits, multi-bay configurations of 20-30m spans are effective. They have been used for frames up to 75m span. 2.3.3 Lattice Trusses Figure 8 shows a typical frame built with lattice trusses. Lattice structures are lighter than the portal alternatives for spans greater than 25m but the additional workmanship increases fabrication costs [3]. Figure 9 gives an indication of the relative material weights. It is not possible to be definitive but based on structural requirements, lattice systems are likely to be cost effective for spans over 50 m. Because lattice girders have a much larger second moment of area and section modulus than a corresponding I section of a similar weight, they have greater stiffness and resistance to load. These enhanced properties are however accompanied by higher fabrication and erection costs due to the required effort for the connections. Spans up to 80m have been realised in that way. When deciding the size of different elements of the lattice girder, the engineer should be aware that stress reversals are likely due to wind effects. Frames are normally positioned at 6,0m to 8m centres. These spacings generally provide economic solutions for the cold-formed purlin and side rail arrangements. Generally a decision is taken early during the process of conceptual design on the type(s) of member(s) to be used for the latticed frame. There are many alternatives: a. Hollow sections - circular or rectangular. b. Traditional sections - angles, tees, channels, I sections. c. Combination of (a) and (b). The selected truss should reflect not only the design aim to produce the lightest frame but also fabrication and erection requirements. Examples of composite form are shown in Figure 10 where the booms are I sections and the internal members are RHS. The I sections enable easy connection of services to the truss and easy connection to columns. Bracing in the plane of the roof can be provided using simple in-plane members and simple connections, or by using the relative stiffness of an I or H section. The specific advantages of hollow sections (and tubes) when compared with traditional sections (I sections, channels, angles, etc.) are the high strength to weight ratio, maximum efficiency in tension, efficiency as struts, good torsional properties, appearance and maintenance. In considering use of CHS or RHS the designer should remember that some fabricators are not fully equipped to fabricate circular hollow sections. (Connection costs are considerably reduced if RHS beams are selected, with CHS or RHS web members). The main disadvantages of CHS and RHS are the higher cost of connections at some nodes and the relative difficulties of making on-site connections for services (electrical, etc.). In addition basic costs are higher than traditional sections on a tonnage basis (overall however frames of lighter weight are produced). The relative slopes of the internal members should be considered in relation to the detailing and fabrication process. If they are parallel to each other then the angle of cut at each end is identical for all members. The final decision on the type(s) of member(s) to be used may be influenced by aesthetics and not cost. Early industrial buildings were built with saw tooth truss layouts illustrated in Figure 11. The vertical (or nearly vertical) elements were glazed and north facing to allow the maximum daylight with minimum direct sun light. In modern times the need for natural lighting has diminished with modern lighting systems and while 10% of area of roof lighting through translucent panes is common, many take advantage of the more reliable and controllable conditions with no natural lighting at all. Generally, the columns to the frame are I or H sections (Figure 12a). The latter have a greater transverse stiffness than the former and are preferred in cases of biaxial bending. When the building incorporates an overhead travelling crane of high capacity, a built-up load or battened column may be used (Figure 12b). In some cases the built-up columns are continued by I sections over the level of the crane supporting structure in order to reduce the costs (Figure 12c). 3. LOADING 3.1 External Gravity Loads The dominant gravity load is from snow. The general case is the application of a basic uniform load, but with sloping roofs having multiple spans and parapets, the action of drifting snow has to be considered. The basic loading is variable according to location. Design information is currently given in national load codes and will be addressed in Eurocode 1, in due course. The main frame design for portals can be carried out using the uniform load case but the variable loads caused by drifting are to be applied to cladding and purlins. The effects of drifting are idealized into triangular loadings with formulae given for the various effects of valleys, parapets, upstands, etc. Early tests carried out in the UK established that equivalent uniformly distributed loads can be used for the purlins design. In the areas of high local load, consideration has to be given as to whether to reduce purlin spacing or to increase the gauge. Where practicable the reduction of spacing is preferable as it prevents the dangers and disruption involved with identification and production of different thicknesses of purlin supplied to one job. For portal frames the frame strength will usually be determined by the snow load case, unless the eaves height is large in relation to span. 3.2 Wind Loads With lightweight cladding and purlins and rails, wind loads are important. Cladding and its fasteners are designed for the local pressure coefficient, for example as given in BS6399: Part 3 and other national codes. Purlins and main frames are designed using the relevant statistical factors, but not additional local coefficients. Care must be taken to include the total effect of both internal and external pressure coefficients. 3.3 Internal Gravity Loads Service loads for lighting, etc., are reasonably assumed to be globally 0,6kN/m2. As service requirements have increased, it has become necessary to consider carefully the provision to be made. Most purlin manufacturers can provide proprietary clips for hanging limited point loads to give flexibility of layout. Where services and sprinklers are required, it is normal to design the purlins for a global service load of 0,1 0,2kN/m2 with a reduced value for the main frames to take account of likely spread. Particular items of plant must be treated individually. The specifying engineer should make a realistic assessment of the need as the elements are sensitive and, while the loads may seem low, they represent a significant percentage of the total and affect design economy accordingly. 3.4 Cranes Where moving loads such as cranes and conveyors are present (Figure 13), in addition to the gravity loads, the effects of acceleration and deceleration have to be taken into account in the design. A quasi-static approach is generally used in which the moving loads are enhanced and treated as static loads in the design. The enhancement factors to be used depend on the particular plant and its acceleration and braking capacity. Manufacturers must be consulted where heavy, high speed or multiple cranes are being used. To take into account dynamic effects due to cranes, the maximum vertical loads and the horizontal forces are increased by specific factors which can be found in national guidance. The repeated movement of a crane gives rise to fatigue conditions. Fatigue effects are restricted to the local areas of support, the crane beam itself, support bracket and the connection to main columns. It is not normal to design the whole frame for fatigue as the stress levels due to crane travel are relatively low. 3.5 Other Actions In certain areas, the effects of earthquakes should be considered. In those countries affected, there are maps which identify the seismic level of each zone and standards to evaluate structural behaviour. Eurocode 8 [5] deals specifically with this problem. In common practice, it is not necessary to take into account differential settlement of less than 2,5cm. If differential settlement exceeds 2,5cm, its effects must be examined, both from the structural and functional points of view. In less ductile structures, such as those constructed with sections not in Class 1 or 2, it is always important to evaluate the sensitivity of the structure in relation to possible differential settlement. It is also general practice not to take into account the effects of temperature when the maximum dimension of the building is less than 40 to 50m, or when expansion joints have been used which separate the structure into zones which do not exceed this dimension. Elsewhere, it is important to evaluate the effects of variations in temperature. It is also necessary to ensure that the characteristics of the finished structure, both the systems of fastening and the seals in the envelope, are compatible with the inevitable deformations due to change in climate. 4. FABRICATION Important factors which must be considered at the conceptual and detailing stages are the questions of workshop facilities and space and of transport between workshop and site. Whilst large girders and/or large sections may appear to be desirable in order to reduce the number of site connections, the use of large members can often reduce the number of fabricators who can tender for a given project. The lengths of members available from stockists or direct from the mill can vary. When long lengths are deemed desirable it is necessary to check their availability. Generally standard sections can be obtained in a reasonable time but there are likely to be delays and additional cost if non-standard sections are required. The use of slightly heavier sections and standardisation of section size may lead to a cheaper fabricated structure. When large prefabricated trusses are used it may be necessary to provide lifting points (eyes) which are located to minimise stresses induced during lifting. All parts which are shop painted need to be handled carefully to avoid damaging the coating. Latticed girders are made up of long, basically slender members, and may therefore be subject to severe distortion due to welding unless care is taken during the fabrication process. It is essential that the design engineer notes that: i. bulk orders of minimum size variation are cheaper than small orders of many different sizes. ii. the number of pieces to be handled should be kept to a minimum. iii. weld distortion and tolerances should be allowed for. iv. automated fabrication is generally cheaper. v. careful design can minimise transportation costs. vi. specifications need to be realistic to reduce costs. vii. good quality control is essential. 5. TRANSPORTATION In each country there are specific lengths and widths of structures that may be transported without any problems, e.g. widths up to approximately 3 m, lengths up to 15m. For larger dimensions a police notification or special permission is required. It should be noted that the various police authorities specify different periods when abnormal loads are allowed to move through their districts. If neighbouring "times" are significantly out of phase and general traffic hold-ups cause disruption to the movement of abnormal loads, it is possible for loads to be delayed by up to 24 hours. If one or more cranes and associated erection staff are unable to work because of these enforced delays, the additional costs can be very significant. Some towns and cities place length restrictions on materials which can be moved by road. Girders can be fabricated and despatched lying flat. The overall height allowed for the load is dependent upon the route travelled and the clear height of any bridges likely to be encountered. Rail transport can accommodate long pieces, but width and height are more restricted. To limit the length of units being transported, trusses may be divided into welded parts (2, 3 or more parts) which may be bolted together on site. The complete rafter can be then craned into position. For export where shipment is involved, pieces up to the same dimensions as for road transport may be accommodated. It should be appreciated, however, that shipping charges are often based on volume rather than weight. There are often relatively severe restrictions on the length of a piece that can be carried in the hold of a ship. The ship's engineer may refuse to carry steelwork as deck cargo. It may be found more economical to despatch the steel in pieces for subsequent assembly on site. 6. ERECTION In considering the erection of a framework, the designer seeks an economic but safe process. The cost of erection can be a significant proportion of the cost of a steelwork contract. It is often useful for a designer with little or no experience of steel erection to discuss possible solutions with a contractor. The latter will know how to build and avoid the possibility of collapse and how to satisfy the insurer. Potentially steelwork erection is hazardous and good control is required to ensure safety of the erectors. Latticed girders can conveniently be assembled on the ground and lifted into place. However, since cranes are likely to be used, the effect of dynamic lifting loads or stress reversals should not be ignored. All steel structures are likely to require temporary bracing, which may be part of the permanent system. The temporary systems need to be carefully designed. The following points summarise the process: a. Structural steelwork erection is an operation requiring meticulous planning from conception to completion. b. The operation involves many disciplines and requires co-operation and communication between all those involved. c. It is an operation which is dependent upon the personal competencies of all those involved to ensure the contract is completed without accident, on time and within the budget. 7. CONCLUDING SUMMARY • • • • • • At the most basic level single-storey structures have to provide shade and shelter for designated activity in the building. Steel provides the means to obtain economical buildings with large column-free spaces. The structural systems are discussed. The structural system is clad to resist the weather. The cladding is supported on secondary cold-formed sections which, in turn are supported on the main frame. The span range is from 6m to 100m. To cover this range, structural systems are available ranging from simple beams and columns through portal frames to lattice trusses. Lateral stability can be achieved either by bracing systems or moment resisting joints in the frames. The proportions of costs for a simple shed are approximately: 50% cladding, 10% rails, 30% main frame 10% foundations. 8. REFERENCES [1] Davis, J. M. and Raven, G. K., 'Design of Cold Formed Steel Purlins', IABSE Thin Walled Metal Structures in Buildings Colloquium, Stockholm 1986. [2] Dowling, P. J. et al, 'A Development in the Automated Design and Fabrication of Portal Framed Industrial Buildings', Institution of Structural Engineers, London, October 1982. [3] Horridge, J. F. and Morris, L. J., 'Comparative Costs of Single Storey Steel-Framed Structures', Institution of Structural Engineers, London, July 1986. 9. WIDER READING [1] Ballio, G. and Mazzolani, F. M., 'Theory and Design of Steel Structures', Chapman and Hall, 1983. [2] Eurocode 3: "Design of Steel Structures": ENV 1993-1-1: Part 1, General rules and rules for buildings, CEN 1992. [3] Dowling, P. J., Knowles, P. R., and Owen, S. G. W., "Structural Steel Design", Butterworths, 1988. [4] Steel Designers' Manual, fifth edition. The Steel Construction Institute, 1992. [5] Eurocode 8: "Structures in Seismic Regions - Design", CEN (in preparation). Lecture 14.1.2: Single Storey Buildings: Envelope and Secondary Structure OBJECTIVE/SCOPE To describe further the functions and characteristics of the cladding and the elements of the secondary structure in single-storey buildings and to give guidance of purlin/wind designs. PREREQUISITES None. RELATED LECTURES Lecture 1B.5.1: Introduction to Design of Simple Industrial Buildings Lectures 9: Thin-Walled Construction Lecture 14.2: Analysis of Portal Frames: Introduction and Elastic Analysis Lecture 14.3: Analysis of Portal Frames: Plastic Analysis Lecture 14.4: Crane Runway Girders SUMMARY The functions and characteristics of the elements of the envelope and secondary structure of single storey buildings are described. The derivation of the load-carrying resistance of cladding using manufacturers' tables is presented and the design of purlins using manufacturers information is outlined. Finally the requirements for main frame bracing are discussed. 1. INTRODUCTION Lecture 14.1.1 provided a brief outline of the functions of the various elements of single-storey buildings. Design methods for frames of the various configurations are presented in other lectures and are also described in conventional text books. The derivation of design and selection of commercially available cladding, purlin and rail systems are not generally covered elsewhere. The lecture reviews the available systems and selection criteria. It concludes with a description of bracing systems used in single-storey buildings. 2. CLADDING SYSTEMS Steel-based cladding sheets are formed from a substrate with layers of galvanising, primer and colour coating as indicated in Figure 1. For the weather face the coating will normally be polyvinyl chloride (PVC), polyvinyl fluoride (PVF2) or, where price is a constraint, the less durable acrylic coatings. PVC and PVF2 can be expected to have a life to first maintenance of 10 to 25 years depending on environmental conditions. Light colours should be used on roofs to minimise heat absorption and thermal movement. PVC should not be used in latitudes less than 49° [1]. There are four main categories of cladding system: • • • • single skin trapezoidal (Figure 2a) double skin trapezoidal (Figure 3) standing seam/secret fix (Figure 4) composite panels (Figure 2b). Where double skin systems are used the liner sheet, normally as thin as 0,4mm, has a similar coating build-up but with a light coloured polyester paint to the inner face. In addition to providing weather protection the cladding provides insulation to the building. The thickness varies according to the insulation required with typical values of 0,45 w/m2° C being obtained from 80mm of fibreglass or mineral wool. The equivalent in composite panels would normally be obtained from 50mm thickness. Manufacturer's literature should be considered for particular cases. 2.1 Roof Cladding The characteristics of the various roof systems are: Single Skin Trapezoidal Roofing This is the most basic form of cladding construction. The main problem is to prevent condensation since it is difficult to exclude damp air from the underside of the sheeting. The internal appearance is utilitarian since the insulation usually takes the form of unrolled thermal quilting supported on wire mesh or rigid boards supported on T Bars. Double Trapezoidal Shell Roof Construction This is the most common form of cladding both for walls and roofs, with pitches down to approximately 5°. It consists of a steel liner sheet fastened to the purlins or rails with an outer weather sheet held apart by Zed spacers. The gap created contains insulation normally of fibreglass or mineral fibre. Sealing of the liner skin or addition of a vapour barrier and breather membrane can be employed to improve performance in situations where there is a higher condensation risk. Manufacturers normally supply a co-ordinated range of accessories and rooflights to match their particular profiles. A typical construction is shown in Figure 3. Standing Seam and Concealed Fix Roofs As steel has replaced asbestos cement in sheeting, lower roof slopes have become possible and the appearance of buildings has greatly improved, both in form and colour treatment. It is feasible to disguise the sloping roofs and provide the rectangular appearance recently favoured by planners, engineers and architects who have become increasingly involved in this type of building. The problems associated with drifted snow and internal gutters, together with the wasted heated in the space in a sloping roof encouraged the development of systems which remain weathertight at slopes down to 1°. These developments have been achieved by minimising laps and through fasteners. The current range of standing seam and concealed fastener systems are provided in long lengths, up to 32m with special transport arrangements, and use of clips to hold the sheeting down to the purlins. Typical construction is shown in Figure 4. The outer sheeting is used to provide the weather skin in both single and double construction, as previously described. Composite or Sandwich Panels This is the most recently developed form of cladding and consists of panels where the insulating foam is integral with the two metal skins [2]. The foam, which during manufacture is pumped into the space between the skins, expands to totally fill the void and adheres to both liner and outer sheet (Figure 2b). This composite action gives robust stiff panels, with the further benefit of rapid erection since the whole skin is fastened in one operation. Composite panels are available with traditional type trapezoidal outer skins and through fasteners, while more recent developments include standing seam variations allowing slopes down to 1°. 2.2 Wall Cladding The systems used for walls are similar in construction to those for roofing. Since the sheets will normally be vertical the weatherproofing requirements are less rigorous. Another important difference compared to roofs is the lower angle of incidence of the sun allowing the full range of colours to be used without excessive surface temperatures, so giving the architect the widest possible scope for creative use of colour. Given appropriate attention to joint detailing, wall cladding can be fixed horizontally, diagonally or in combinations. The range of construction includes, as for roofs, single skin, double skin and composite systems, with both conventional and secret fastenings. Appearance is a major consideration in the selection of the particular type to be used. All reputable manufacturers supply brochures giving details of the systems offered. General advice can be obtained from literature. 3. RESISTANCE OF CLADDING TO LOADS Non-composite Systems Since the sheets are rolled formed, the designer has to select the profiles from the various manufacturers predetermined shapes. The sheets must be strong enough to resist the various loads which will be applied in construction and during service without damage or excessive deflection. For simple profiles, calculation methods are available in some national codes and European recommendations. Most manufacturers carry out full scale tests to determine the strength of their various profiles. The test regimes for steel sheeting are designed to take account of the various limiting criteria. The criteria relate to shear of the webs combined with bending at internal supports for shorter spans and mid-span bending for longer, simple spans. Sheets are therefore tested with jacking loads and spreader beams, air bags or vacuum boxes with spans representative of the shortest and longest support spans anticipated in practice. The maximum allowable shears and moments are derived together with deflection behaviour and load tables for service conditions drawn up for pressure and suction conditions. One important facet of a roof sheet is its ability to withstand local point loads both to sustain personnel walking on it during construction and also in service for maintenance purposes. It is this aspect which governs the minimum thickness acceptable for a roof sheet. This thickness is between 0,6 and 0,7mm. Current approval regimes include a point load test. Sheeting which passes this test will be suitably robust for construction loads but care needs to be taken with walking on PVF2 and painted finishes. Wall cladding is often 0,5mm thick. To allow selection of the appropriate profile and thickness the majority of profilers (manufacturers) publish catalogues containing suggested fixing and sealing details together with allowable loads for the range of span conditions for both pressure and suction actions. In addition to resisting externally applied loads, conventional cladding with screw fasteners can provide lateral restraint to the purlins and side rails supporting it. In this application care should be taken in the specification of the fasteners concerning prevention of corrosion around the connections. Standing seam or concealed fastener systems have clips which hold the sheets down but allow expansion and contraction of the sheets and so provide a lower restraint value to the rails. Where these systems are used it is necessary to provide supplementary restraints to the purlins. At present there is no generally accepted guidance on the amount of restraint provided by these newer forms of cladding. Advice on the necessary inclusion of additional restraint in the form of sag bars is available from rail manufacturers relating to their particular products. Where a metal liner tray is used in conjunction with a cladding system fixed by clips, adequate rail restraint is normally provided if the tray is fastened in the conventional manner. Composite Systems The use of foam filled panels has become more common over the past few years. They are more complicated to evaluate in design since the properties of the foam and its adhesion to the two metal skins are fundamental to the performance of the panels. In addition to the normal applied loads, the inherent insulating qualities of the panels leads to significant temperature gradients which have to be taken into account in the panel design. Calculation methods for analysis are available but the shapes of practical panels are outside of their current scope. It is normal therefore for extensive testing to be carried out by manufacturers in order to obtain type approval from the appropriate national authorities. The tests, in addition to normal load tests, include thermal cycling on full scale panels and foam evaluation to ensure the essential adhesion to the skins is obtained. Quality assurance regimes are an essential part of the manufacturing process. The results of the tests are collated and simplified to produce load carrying tables. 4. SHAPES OF PURLINS AND RAILS Although some long-span cladding systems are available which will span up to 6m, most portal frame structures are designed with purlins and rails supporting the cladding and spanning between the main frames. The spans between frames are between 4,5 and 10m with 6 to 7,5m being most popular. Roof decks of larger span are used primarily in flat roof systems and are supported directly on the trusses. The purlins can be designed as continuous beams supported by three or more portal frames. Because of the difference in the value of the support reactions, it is necessary to stagger the joints to obtain equal loads on the main frames. 4.1 Cold-Formed Shapes The purlins and rails are commonly supplied by specialist manufacturers who deliver pre-cut and punched coldrolled items together with the necessary sag bars, cleats or rafter stays. Material economy is vitally important to the manufacturers whose output is often in excess of 80km/week. They have therefore developed and refined by testing a variety of shapes. The more common ones are Zeds, modified Zeds and Sigma shapes illustrated in Figure 7. Depending on the slope, anti-sag bars may be necessary to provide one or two intermediate supports in the direction of the weak axis of the beam (Figure 8). Generally, the anti-sag bars are connected using diagonal ties to the portal frames to avoid an increase of the load in the purlins at the top. Zeds The Zed section was the first shape to be introduced. It is material efficient but its major disadvantage is that the principal axes are inclined to the web and so out-of-plane forces are generated. When asbestos covered roofs with slopes of the order of 10° were normal these forces were advantageous and opposed to the down slope forces. Modified Zeds As lower roof pitches have been introduced, modified Zeds have been developed with the inclination of the principal axis considerably reduced, so enhancing overall performance. Stiffening has also been introduced, improving material efficiency. These more complicated shapes have to be produced by rolling rather than by press braking. Sigma Shape The first shapes to be employed were the simple Zed and Channel since they could be formed by press brakes. As described above the Zed shape has been modified to overcome its major disadvantage. The Channel has not achieved much use, since, although its principal axes are parallel to the major elements, the shear centre lies well outside of the section. Undue twisting under load results which can be reduced by shaping the web and creating a sigma shape such that the shear centre is approximately coincident with the load application line. One manufacturer now produces an economical second generation product of this configuration using rolling techniques. 4.2 Hot-Rolled Shapes Classical I-beams such as IPE I CDN or HE can also be used as purlins. I-beams and channel shapes can be used as rails. These shapes are less sensitive than the previous ones to the effect of local instabilities and there is preferences to use them in some countries. However, cold-formed sections are generally the most economical solution. 5. RESISTANCE OF PURLINS AND RAILS TO LOADS Cold-formed sections manufactured from thin gauge material are particularly prone to twisting and buckling due to factors which are directly related to the section's shape. The torsional constant of all thin gauge sections is low due to it being a function of the cube of the thickness; in the case of lipped channels the shear centre is eccentric to the point of application of load thus inducing a twist on the section; in the case of zeds the principal axes are inclined to the plane of the web thus inducing bi-axial bending effects. Clearly the above secondary effects, induced by the primary load application, affect the load-carrying resistance of the section. When used in service the support system is subject to downward loading due to dead and live loads arising from the weight of the cladding, snow, services, etc., and uplift if the design wind pressure is greater than the dead load of the system. Therefore, for a typical double span system as shown in Figure 9, most of the compression flange is directly restrained laterally and against rotation by the cladding for downward loading, but it is not so restrained in the case of load reversal. In supporting the external fabric of the building, the purlins and side rails gain some degree of restraint against twisting and rotation from the cladding and the method of its fastening to the supporting members. In addition, the connection of the support members to the main frame also has a significant effect on the load-carrying resistance of the section. Economical design, therefore, must take account of the above effects. Several approaches are used for the design of purlin and rail systems. a. Design by calculation based on an elastic analysis as detailed in the relevant codes of practice. This approach neglects any beneficial effect of cladding restraint for the uplift case and in the compression zone adjacent to the central support in double span arrangements. It is normally confined to "one-off" situations where material savings do not justify more rigorous solutions. b. Design by calculation based on a rational analysis which accounts for the stabilising influence of the cladding, plasticity in the purlin as the ultimate load is approached and the behaviour of the cleat at the internal support. The effects, however, are difficult to quantify. Although understanding of the restraint of purlins by cladding is improving and methods for including the cladding/purlin interaction are available, the methods are necessarily conservative. They are included in Eurocode 3: Part 1.3 [4] but are likely to be used in the short term only by small to medium volume users where potential material savings do not justify a major test programme. In some countries, despite the almost traditional nature of these forms of construction, it is not permitted to take account of cladding restraint to the purlins in the section design if the cladding is to be supplied and fixed by differing suppliers. This restriction is due more to the apportionment of responsibility should there be failures than due to lack of technical knowledge. This situation is unusual since a great deal of construction depends on the interaction of elements supplied and fixed by different contractors. c. Design on the basis of full-scale testing Manufacturers differ in the design methods they use. The methods are based on the observed behaviour of the system under test. For volume production, design by testing is the approach which is normally used. Although this approach is expensive, maximum economy of material can be achieved and the cost of the testing can be spread over several years of production. Design by testing involves the "fine-tuning" of theoretical expressions for the collapse load of the system. For example, although the sections involved are inevitably slender, the collapse mechanism which occurs with a well developed two-span system is essentially as represented in Figure 10. The published paper [3] on one manufacturer's system shows that the collapse load will be: Wc = f (M1, M2, x, L) with x = f (M1, M2, L) and θp = f (Wc, M1, L) The performance of a two-span system is considerably enhanced if some redistribution of bending moment from the internal support is taken into account. The moment-rotation characteristic at the support depends very much on the cleat detail and the section shape. The characteristics of the central support can be found by testing a simply supported beam subject to a central point load applied through a cleat so as to simulate the behaviour of the central support of a double span system. From this test, the load-deflection characteristics can be plotted well beyond the deflection at which first yield occurs. A lower bound empirical expression can then be found for the support moment, M1, based on an upper limit of rotational capacity. A similar expression can be found for M2, the internal span moment, again on the basis of a test on a simply supported beam subject to a uniformly distributed load, applied by the use of a vacuum rig, or perhaps sand bags. The design expressions can then be confirmed by the execution of numerous full-scale tests. The results of the full-scale tests are then condensed into easy to use load span tables which are detailed in the purlin manufacturers' design and detail literature. The tabular format is typical of that contained in purlin manufacturers' technical literature. The table is generally prefaced by explanatory notes regarding fixing condition and lateral restraint requirements, the latter being particularly relevant where load reversal occurs. Conditions which arise in practice, and which are not covered in the technical literature are best dealt with by the manufacturer's Technical Service Department, which should be consulted for all non-standard cases. Anti-sag bars are used on longer spans to assist erection and improve performance under wind suction. Side rail design is essentially identical to that for purlins, and load resistances are again arrived at via test procedures. The practical construction problem of levelling side rails in the field, to remove self weight deflection about the weak axis of the section, is overcome by the use of a tensioned wire system incorporating the use of tube struts, typically at mid-span for spans of 6 - 7m, and third-points of the span for spans of 7 - 8m and above. Typical methods are shown in Figure 12. 6. MAIN FRAME BRACING Returning to the requirements for the main frame which were discussed in Lecture 14.1.1, most frame configurations for larger spans have moment resisting joints at the column/roof member connections. In addition to assisting with deflection control and reducing member sizes, this arrangement provides inherent resistance to lateral in-plane loads such as those from side winds and crane movements. Bracing, both horizontal and vertical, must be provided to transfer to the foundations the horizontal loads due to wind possibly earthquake and out-of-plane loads. It is possible to use the cladding and wall panels instead of braces for this purpose. In that case stressed skin design should be used (Lecture 9.5). When masonry is used as all or part of the vertical cladding, it is feasible to use that element as part of the bracing system. The bracing can be single diagonals or cross members. If the former system is adopted, the members are designed to support compressive and tensile loads. When cross members are used only the members in tension are assumed to be effective, those in compression being designed to satisfy the slenderness criteria. Bracing may be located either at mid-length of the building (Figures 13a, 14a, and 14c) or at its ends (Figures 13b, and 14b). Bracing at the ends helps for erection purposes, since it provides a stable structure at one end of the building when the erection starts. Its disadvantage is that it prohibits the free movement of the building due to temperature so that stresses in the members, sometimes of considerable amount, may result. By putting the braces at the mid-length this disadvantage disappears since the building is able to expand freely. However additional temporary braces should be provided during the erection stage to stabilise the first part of the building. The horizontal forces have to "travel" from their point of application in the gable to the brace thus causing compression in the purlins. 7. CONCLUDING SUMMARY • • • • A variety of cladding systems is available manufactured from pre-coated steel. These systems are normally supported by light gauge purlins and rails but hot-rolled profiles are also used. The theoretical design procedures for these systems are improving although large volume manufacturers still carry out full-scale testing as a basis for design in order to achieve material economy. Many single-storey frame arrangements have inherent in-plane resistance to side loads but bracing systems have to be provided to transfer longitudinal loads to the foundations. 8. REFERENCES [1] Colourcoat in Building, British Steel Strip Products, Newport, Wales. [2] ECCS Recommendations for Sandwich Panels: Part 1 Design, Part 2 Good Practice. [3] Davies, J. M. and Raven G. K., 'Design of Cold Formed Steel Purlins', IABSE "Thin Walled Metal Structures in Buildings" Colloquium, Stockholm 1986. [4] Eurocode 3: "Design of Steel Structures" Part 1.3, Cold Formed Thin Gauge Members and Sheeting, CEN (in preparation). PANELS Wind loads in kN/m2 WIND Deflection limit Span condition Thickness of Span 'L' (in metres) profile (mm) 1,6 L/90 0,5 0,55 0,6 0,7 0,8 0,5 0,55 0,6 0,7 0,8 0,5 0,55 0,6 0,7 0,8 3,00 3,47 3,96 4,98 6,02 2,70 2,94 3,61 4,55 5,14 3,28 3,54 4,41 5,56 6,24 1,8 2,30 2,60 2,91 3,61 4,21 2,19 2,41 2,93 3,68 4,20 2,67 2,92 3,58 4,52 5,11 2,0 1,66 1,88 2,11 2,61 3,05 1,81 2,01 2,43 3,05 3,49 2,22 2,45 2,97 3,74 4,26 2,2 1,24 1,40 1,57 1,95 2,27 1,52 1,71 2,04 2,56 2,95 1,87 2,09 2,51 3,15 3,61 2,4 0,94 1,07 1,20 1,49 1,73 1,30 1,46 1,74 2,17 2,51 1,60 1,80 2,14 2,68 3,08 2,6 0,73 0,83 0,93 1,16 1,35 1,12 1,27 1,50 1,87 2,17 1,39 1,56 1,81 2,25 2,62 2,8 0,58 0,66 0,73 0,91 1,06 0,97 1,11 1,30 1,62 1,90 1,14 1,29 1,44 1,79 2,08 3,0 0,46 0,52 0,59 0,73 0,85 0,85 0,97 1,14 1,42 1,66 0,91 1,04 1,16 1,44 1,68 L/150 0,5 0,55 0,6 0,7 0,8 0,5 0,55 0,6 0,7 0,8 0,5 0,55 0,6 0,7 0,8 1,96 2,22 2,48 3,07 3,58 2,70 2,94 3,61 4,55 5,14 3,28 3,54 4,41 5,56 6,24 1,36 1,54 1,73 2,14 2,49 2,18 2,41 2,94 3,68 4,20 2,62 2,92 3,32 4,11 4,79 0,98 1,11 1,24 1,54 1,80 1,81 2,01 2,42 3,05 3,49 1,90 2,15 2,40 2,98 3,47 0,73 0,82 0,92 1,14 1,33 1,52 1,71 2,04 2,56 2,95 1,42 1,60 1,79 2,22 2,59 0,55 0,62 0,70 0,87 1,01 1,30 1,46 1,74 2,18 2,51 1,08 1,22 1,37 1,70 1,98 0,42 0,48 0,54 0,67 0,78 1,08 1,22 1,37 1,70 1,98 0,84 0,95 1,06 1,32 1,54 0,33 0,37 0,42 0,52 0,61 0,86 0,97 1,08 1,35 1,57 0,66 0,75 0,84 1,05 1,22 0,26 0,29 0,33 0,41 0,48 0,69 0,78 0,87 1,08 1,26 0,53 0,60 0,67 0,84 0,98 Figures shown in bolder type are limited by deflection. All other loads are limited by stress. ROOF PANELS Maximum loads in kN/m2 deflection limit L/100 PANEL SPAN THICKNESS CONDITION (mm) FACING DETAILS MATERIAL THICKNESS (mm) Outer Inner 1,6 1,8 2,0 2,5 3,0 SPAN 'L'(m) 37 STEEL 0,5 0,4 + 2,54 - 3,02 2,16 2,59 2,39 2,85 1,66 2,26 2,16 2,56 2,39 2,85 1,93 2,25 2,16 2,59 2,39 2,85 1,93 2,26 3,40 3,97 3,48 4,05 3,23 3,75 2,68 2,86 3,10 3,30 1,85 2,25 2,06 2,48 1,21 1,74 1,85 2,21 2,06 2,48 1,62 1,91 1,85 2,25 2,06 2,48 1,62 1,91 2,98 3,52 3,07 3,58 2,51 3,20 2,29 2,49 2,65 2,86 1,28 1,63 1,42 1,82 1,30 1,63 1,46 1,82 1,08 1,32 1,30 1,63 1,46 1,82 1,08 1,32 2,17 2,63 2,38 2,79 1,38 2,00 1,65 1,86 1,90 2,13 0,76 1,22 0,82 1,36 0,94 1,22 1,06 1,36 0,76 0,96 0,94 1,22 1,06 1,36 0,76 0,96 1,59 1,99 1,69 2,23 0,78 1,20 1,24 1,47 1,44 1,68 0,6 0,4 + 2,80 - 3,30 ALUMINIUM 0,7 0,5 + 2,32 - 2,70 STEEL 0,5 0,4 + 2,54 -3,02 0,6 0,4 + 2,80 - 3,30 ALUMINIUM 0,7 0,5 + 2,34 - 2,64 STEEL 0,5 0,4 + 2,54 - 3,02 0,6 0,4 + 2,80 - 3,30 ALUMINIUM 0,7 0,5 + 2,34 - 2,70 60 STEEL 0,5 0,4 + 3,91 - 4,55 0,6 0,4 + 4,01 - 4,65 ALUMINIUM 0,7 0,5 + 3,85 - 4,33 STEEL 0,5 0,4 + 3,20 - 3,37 0,6 0,4 + 3,70 - 3,88 ALUMINIUM 0,7 0,5 + 2,91 - 3,00 2,44 2,56 3,07 3,35 3,48 3,84 2,89 3,17 2,08 2,23 2,64 2,93 3,04 3,35 2,49 2,78 1,48 1,68 1,93 2,23 2,22 2,54 1,81 2,11 1,11 1,34 1,47 1,78 1,70 2,02 1,36 1,68 STEEL 0,5 0,4 + 3,62 - 3,90 0,6 0,4 + 4,01 - 4,47 ALUMINIUM 0,7 0,5 + 3,42 - 3,68 WALL PANELS Maximum loads in kN/m2 deflection limit L/100 L/100 PANEL SPAN THICKNESS CONDITION (mm) 37 FACING DETAILS MATERIAL THICKNESS (mm) Outer STEEL 0,5 Inner 0,4 1,5 + 2,78 - 2,78 STEEL 0,5 0,4 + 2,78 - 2,78 STEEL 0,5 0,4 + 2,78 - 2,78 60 STEEL 0,5 0,4 + 3,40 -3,40 2,0 1,69 1,97 2,09 2,09 2,09 2,09 2,55 2,55 2,5 0,96 1,06 1,67 1,67 1,66 1,67 1,81 2,04 3,0 1,26 1,39 1,11 1,38 1,18 1,39 3,5 0,90 1,08 0,78 0,94 0,79 0,91 SPAN 'L'(m) STEEL 0,5 0,4 + 3,40 - 3,40 2,55 2,55 2,55 2,55 2,04 2,04 2,04 2,04 1,70 1,70 1,70 1,70 1,33 1,33 1,33 1,33 STEEL 0,5 0,4 + 3,40 - 3,40 Notes to Load Tables 1. 2. 3. 4. 5. 6. 7. + reload indicate pressure or snow loading - ve load indicates suction loading. Indicates values too low for normal applications. The permissible loads take account of dead and imposed loading, including their long term effects, and differential thermal loading. Values for loads at intermediate spans can be obtained by linear interpolation. The concealed fixing system can be used for all + ve loading conditions. Under suction loading the clip capacity is limited to 4kN wall and 6kN roof. Where the clip capacity is exceeded, additional through fasteners or reduced purlin rail spacings may be required. If in doubt contact WBC Technical Department. For concealed fix aluminium panels contact WBC in every case. The roof loadings are based on the standard colour range. For variations to this range consult the Technical Department. When designing roofs with pitches between 1 ° and 4° a deflection limit of span/200 should be used to prevent ponding. 8. CW 1200 values are for all four wall panels. ULTIMATE LOAD TABLES INTRODUCTION The following load tables have been prepared to assist the designer in specifying cold formed sections. The tabulated values are only valid for use with the fixing details and recommendations proposed by the manufacturer. This information has been prepared as a supplement to the main handbook for ease of reference in the design office. USE OF TABLES 1. 2. 3. 4. 5. 6. The load tables show the ultimate load for double span sections in terms of a UDL per span. Section self weight has not been subtracted in the loads shown. Loadings have also been tabulated that will produce the noted deflection ratio. Loads shown assume lateral restraint to the top flange of the section and that the beams are fixed in accordance with manufacturers instructions. Ultimate reversal loads may be obtained by multiplying the loads shown by a factor of 0,8. Interpolation of the ultimate loads shown is permissible on a linear basis. Note: These values are typical and should not be used in design. LOAD FACTORS The following load tables have been prepared to assist the designer in specifying cold formed sections. The tabulated values are only valid for use with the fixing details and recommendations proposed by the manufacturer. This information has been prepared as a supplement to the main handbook for ease of reference in the design office. USE OF TABLES 1. The load tables show the ultimate load for double span sections in terms of a UDL per span. 2. 3. 4. 5. 6. Section self weight has not been subtracted in the loads shown. Loadings have also been tabulated that will produce the noted deflection ratio. Loads shown assume lateral restraint to the top flange of the section and that the beams are fixed in accordance with manufacturers instructions. Ultimate reversal loads may be obtained by multiplying the loads shown by a factor of 0,8. Interpolation of the ultimate loads shown is permissible on a linear basis. Note: These values are typical and should not be used in design. LOAD FACTORS Loading Dead load Dead load restraining uplift or overturning Dead load acting with wind and imposed loads combined Imposed load Imposed load acting with wind load Wind load Wind load acting with imposed load Forces due to temperature Factor 1,4 1,0 1,2 1,6 1,2 1,4 1,2 1,2 PURLINS Span (m) 4,5 B120/150 A140/155 A140/165 *A140/180 A170/160 A170/170 *A170/180 11,89 15,35 17,16 19,85 20,40 22,70 24,97 Section UDL Deflection L/200 7,77 13,67 14,56 15,89 20,40 22,70 24,97 5,0 B120/150 A140/155 A140/165 *A140/180 A170/160 A170/170 *A170/180 10,79 13,99 15,62 18,04 18,63 20,71 22,76 6,29 11,07 11,79 12,87 18,02 19,16 20,30 5,5 A140/155 A140/165 *A140/180 A170/160 A170/170 *A170/180 A200/160 A200/180 *A200/200 12,84 14,33 16,54 17,14 19,03 20,90 21,27 23,88 28,10 9,15 9,75 10,64 14,89 15,84 16,78 21,27 23,88 27,43 6,0 A140/155 A140/165 *A140/180 A170/160 A170/170 *A170/180 A200/160 A200/180 *A200/200 A230/180 A230/200 *A230/240 11,87 13,23 15,26 15,87 17,60 19,32 19,81 22,11 25,98 26,44 31,11 40,14 7,69 8,19 8,94 12,51 13,30 14,10 18,38 20,72 23,05 28,92 32,18 38,68 6,5 A170/160 A170/170 *A170/180 A200/160 A200/180 *A200/200 14,77 16,37 17,95 18,43 20,58 24,15 10,66 11,34 12,01 15,66 17,65 19,64 Lecture 14.2: Analysis of Portal Frames: Introduction and Elastic Analysis OBJECTIVE/SCOPE To present the basic principles of design of portal frames using elastic analysis; the scope includes tapered portal frames. The lecture is illustrated by a design example. PREREQUISITES Lectures 2.3: Engineering Properties of Steels Lecture 2.4: Steel Grades and Qualities Lectures 3.1: General Fabrication of Steel Structures Lectures 6: Applied Stability Lecture 7.2: Cross-Section Classification Lecture 7.3: Local Buckling Lectures 7.9: Unrestrained Beams Lectures 10: Composite Construction Lecture 11.1.1: Connections in Buildings Lectures 11.4: Analysis of Connections RELATED LECTURES Lecture 11.6: Moment Connections for Continuous Framing Lectures 14.1: Single-Storey Buildings SUMMARY Introduction; the economic advantages of using tapered, I-section portal frames with slender webs and Class 2 or 3 flanges; summary of fabrication methods with cross-references to fabrication lectures. Methods of analysis for uniform and tapered portal frames; the need to consider bending moments envelopes and non-symmetric uplift conditions. Methods of resistance assessment for cross-sections. Slender webs under combined bending, shear and compression. Connection design; requirements for secondary structure including bracing. 1. INTRODUCTION Portal frames are single storey, single or multi-bay frames with pitched or flat roof (Figure 1). This lecture presents the elastic analysis and design of portal frames, considering mainly the case of single bay and pitch roof, which is the most common in practice. The next lecture presents the design of portal frames using plastic analysis. Where the frame comprises Class 1 rolled sections and the design is governed by strength, plastic analysis will lead to the greatest economy. As shown in Figure 2a, plastic redistribution is utilised to make maximum use of the frame resistance. However, there are situations where plastic analysis cannot be used: i. The frame comprises Class 1 sections but stiffness (deflections) governs design. ii. The frame comprises Class 2 or higher sections. Where prismatic sections are used, e.g. rolled sections, the elastic distribution of moments is such that the structure is relatively inefficient, as shown in Figure 2b. The beam section has to be chosen to satisfy the governing moment at the eaves; away from there the stresses in the beam are low. Economy of material is best achieved in an elastically designed frame by the use of tapered, welded sections. As shown in Figure 2c, the strength envelope can then be made a close fit to the bending moment envelope and stresses are high throughout the frame. Figure 3 shows a general arrangement of such as structural system [1]. Once the concept of using fabricated, non-prismatic members is accepted, the designer has the freedom to choose the following independent geometric parameters: • • • • • member depth member shape, i.e. variation in depth around frame top flange width and thickness (possibly varying around frame) bottom flange width and thickness (possibly varying around frame) web thickness (possibly varying around frame). Some form of optimisation has to be used to determine these parameters, taking account of practical restraints. In practice it is found that greatest economy is generally obtained by using: Class 2 or 3 flanges Class 3 or 4 webs, with a maximum depth (d) to thickness (t) ratio up to 200, and without any stiffening. Within these restrictions considerable weight savings, compared to plastic design, can be achieved, as illustrated in Figure 4. Of course minimising weight does not necessarily minimise cost. In this instance, overall economy of construction will only be achieved if automated fabrication is adopted. Recent developments in this context are summarized later in this lecture. The behaviour of sections with slender elements is inherently more complex than that of stocky Class 1 sections, due to local buckling and cross-section distortion. Special design procedures are required and more attention has to be devoted to stabilising the frame by appropriate bracing from the secondary structure. These topics are addressed later in the lecture. The complexity of these strength checks and the greater complexity of analysis make it essential that computer aided design is adopted. Thus, before tapered portal frames can be contemplated, a considerable investment is required in both automatic fabrication and computer aided design. Given this investment, there are considerable advantages in the use of welded tapered portal frames, compared to rolled section portal frames: • • • They can achieve significant weight and cost economies. They are inherently stiffer because the optimised welded sections are considerably deeper than the rolled sections having the same resistance. Automated design and fabrication can readily be incorporated into a fully-computerised manufacturing system, with very rapid and accurate cost estimation, computerised stock control and waste minimisation, computerised drawings and computer controlled equipment. 2. ELASTIC ANALYSIS OF PORTAL FRAMES The elastic analysis of prismatic portal frames may readily be carried out by computer. Manual methods and charts [2,3] are also used. The analysis of tapered portal frames is somewhat more complex. The variation of stiffness around the hyper-static frame influences the distribution of moments and associated shear and axial forces. Procedures for manual analysis do exist [4] but in practice analysis is usually carried out by computer. Figure 5 shows a typical analytical model; beams and columns are divided into shear elements; each element is considered as a prismatic member and is assigned the average properties over its length. The analysis provides a full distribution of moments, shear and axial forces of each element. With tapered sections, economy dictates that section depth, and therefore resistance, are locally reduced where possible. For example, location 11 in Figure 5 is at a point of contraflexure under symmetrical vertical loading. These zones of reduced resistance do not exist in prismatic frames and designers are therefore not accustomed to thinking about the full envelope of bending moments on such frames. Figure 6 illustrates the bending moment envelopes that exist on a typical tapered frame. It is clear that, near to regions of minimum depth it may be pattern loading and other non-symmetrical cases that govern the design. 2.1 Serviceability Limit States Under the combinations of actions to be considered for serviceability limit states, Eurocode 3 [5] recommends that the deflections of portal frames do not exceed the following limiting values: • • Horizontal deflections at the top of the columns for portal frames without gantry frames : h/150 for other single storey buildings : h/300 where h is the height of the column. Total vertical deflections in beams : h/200 where L is the span of the beam Usually, for portal frames with a pitched roof, the deflection criteria for beams is not a critical one. All roofs with a slope of less than 5% should be checked to ensure that rain water cannot collect in pools. In this check, due allowance should be made for possible construction inaccuracies and settlements of foundations, deflections of roofing materials, deflections of structural members and the effects of precamber, if any. Where the roof slope is less than 3%, additional calculations should be made to check that collapse cannot occur due to the weight of water collected in pools which may be formed due to the deflection of structural members and roofing material or retained by snow. 2.2 Imperfections For ultimate limit states calculations, the effects of frame imperfections should be allowed for in global frame analysis by means of an equivalent geometric imperfection in the form of an initial sway (Figure 7a) determined from: Φ = {0,5 + 11nc}0,5 1200 = 1/200 where nc is the number of columns for single bay portal frames In practice, it is more convenient and accepted in Eurocode 3 [5], to replace the initial sway imperfection by a system of equivalent horizontal forces, as shown in Figure 7b. Member imperfections are taken into account when carrying out the global frame analysis by incorporating the appropriate bow in the columns, as shown in Figure 8. The value eo,d of the bow depends on the type and dimension of the column section, and is given in Eurocode 3 [5]. The member imperfections may not be considered in the global frame analysis when < 0,5 {Afy / Nsd}0,5 where Nsd is the design value of the compressive force is the in-plane non-dimensional slenderness calculated using a buckling length equal to the system length. 2.3 Second Order Global Analysis Second order global analysis, including frame imperfections (sway or horizontal forces) and member imperfections (bow) is the most exact and convenient method for portal frames with tapered members. Some existing computer software will automatically: • • • Calculate the section properties at mid-length of each beam and column element, knowing the extreme values of web and flange dimensions and the numbers of equal length elements in beams and in columns to be taken. Calculate the position of each node when the sway at the top of the columns and the bow at mid-length of the columns are given. Perform second order elastic analysis of the plane frame. After having obtained the results of the computer calculation, there is no need for further hand calculation of effective length and in-plane buckling of members. The only checks needed are: • • • The values of the stresses in the members, which should not exceed the yield stress fy; The out-of-plane buckling of columns, unless there are walls, rails or cladding preventing it; The lateral-torsional buckling of members. If the frame comprises Class 4 sections, the reduced effective cross-section must be determined. This does not apply to Class 1, 2 and 3 sections. As the second order analysis is non-linear, the composite calculations must be carried out for all the load combinations. There is no possibility of superposing the effects of elementary loads. 2.4 First Order Global Analysis The so-called 'simplified methods' of analysis can also be adopted, but they present several disadvantages: • • • They require more hand calculations They are less accurate: approximations are needed They are on the safe side compared to second-order methods so they are not economical, as long as the steel weight is concerned. In these 'simplified methods' a first order global elastic analysis is performed by computer calculations (or by hand, but only for prismatic members, single bay portal frames) taking into account frame imperfections (usually horizontal forces). As single bay portal frames are not very stiff, they must usually be classified as sway frames. For most load cases this situation occurs when: Vcr < 10 Vsd where Vsd is the design value of the total vertical load Vcr is its elastic critical value for failure in a sway mode. After the first order global analysis has been carried out, two possibilities for complementary hand calculations are found in Eurocode 3: First Possibility - "Amplified sway moments" method 1. 2. 3. Increase the moments due to horizontal forces by multiplying them by the ratio 1/(1 - Vsd / Vcr). Calculate the in-plane buckling lengths of members, considering the non-sway mode. Check the in-plane buckling resistance of members, using the appropriate buckling formulae given in Eurocode 3. Second Possibility - "Sway mode buckling lengths" method 1. 2. 3. Increase the moments in the beams and the beam-to-column connections, due to horizontal forces, by multiplying them by 1,2. Calculate the in-plane buckling lengths of members, considering the sway mode. Check the in-plane buckling resistance of members, using the appropriate buckling formulae given in Eurocode 3 [5]. The formulae that have to be applied for points 2 and 3 (buckling lengths and buckling resistance) are given in Eurocode 3 for prismatic members, but they do not apply to tapered members. Approximate conservative solutions may be used for tapered members. Finally the complementary checks presented for the second order global analysis must also be done by hand calculation, i.e. stresses, out-of-plane buckling and lateral-torsional buckling. If the frame comprises Class 4 sections, the reduced effective cross-section must be determined. 3. SPECIAL FEATURES OF BEHAVIOUR FOR TAPERED PORTAL FRAMES AND ASSOCIATED DESIGN RULES • Local Buckling of Web The web slenderness (d/t) may be as high as 200. The web is therefore subject to local buckling under bending, shear and compression of the type shown in Figure 9. This form of instability does not often occur in other forms of building structure. • • Local Flange Buckling Normally flanges are restricted to Class 2 or 3 and local flange buckling need not be considered. Distortional Buckling of the Frame between Points of Restraint As shown in Figure 10 buckling of a rafter or column between stays is a complex interaction of local web buckling, lateral-torsional buckling and cross-section distortion. Recent studies [6] have considered these effects directly and resulted in advanced tentative design approaches. However, a much simpler model that provides good agreement with experimental results is that based on the reduced effective cross-section shown in Figure 11. The strength check is then as follows: i. Determine the stresses on the reduced effective cross-section from the bending and axial compression. ii. Ensure the compressive resistance of the Tee section is greater than the applied compression, assuming that it has an effective length equal to the stay spacing. iii. Check the tension flange to ensure that it is not yielding. Yielding will only govern if the section is highly nonsymmetric. iv. Check the resistance of the web under combined compression, bending and shear, using stress resultants for the first two actions that are based on i. above. • Eaves Connection The behaviour of the eaves connection is best understood by following the development of the connection utilised by a leading UK fabricator [7]. Originally the diagonal connection plane shown in Figure 12a was considered. Inplane behaviour was satisfactory but there was a major stability problem at the inside corner. Lack of continuity between the flanges along the joint, exacerbated by any lack-of-fit from end-plate distortion, led to premature outof-plane failure at this point. This problem could be controlled by direct staying to this corner but such a solution is costly in the absence of any convenient purlin or siderail for that stay. Stability of the inside corner for this elastic design can be achieved by continuing one or other of the inside (compression) flanges through to the relevant tensile load path, as shown in Figure 12b. Generally some stiffening will be required at point X in Figure 12b. In the absence of a diagonal stiffener, because the corner panel is slender, the high corner shears are resisted by the tension field action shown in Figure 12c. A local reduction in effective lever arm results which will cause a sharp increase in compression in the immediate vicinity of the inside corner, causing premature failure. It was decided to provide a substantial diagonal stiffer, as shown in Figure 12d, which could resist the combined compression of the two flanges by direct triangulation of forces. This stiffener carries out the multiple functions of maintaining the lever arm around the corner, stiffening the flange so that it may resist the horizontal compression from the incoming flange and stabilising the slender web. • Stay Forces Traditional design criteria for stay forces are empirical. Typical design values are 21/2% of the maximum resulting force in the compressed area of the braced sections, distributed among the restraints along the length of the element. This approach has been demonstrated to be satisfactory by experience. However, the deep, slender sections that are a feature of tapered, fabricated frames, with a high ratio of Ixx to Iyy, are more prone to buckling. Previous work [7, 8] has suggested that a figure of 2% at each brace might be more appropriate. 4. PRACTICAL DESIGN AND FABRICATION OF TAPERED PORTAL FRAMES The potential economic advantage of tapered portal frames can only be realised if the structural sophistication is matched by efficient office procedures and fabrication. Design, drafting, estimating and stock control need to be computerised. The design procedures need to encompass analysis, resistance and serviceability checks and optimisation. Drafting should produce full drawings based on detailing guidelines specified by the workshop. As a by-product data is generated for all numerically controlled workshop machines. Accurate estimating follows readily from the foregoing data. At tender stage it can be used for the preparation of precise cost and time information without the preparation of full drawings and numerically controlled data. Stock control can be utilised to minimise waste; this is particularly important for the efficient cutting of tapered webs. Figure 13 shows a typical cutting pattern for a web plate. Use of stock materials can be introduced as a constraint on the design optimisation. The heart of the semi-automatic fabrication is the efficient welding of the flanges to the web. Welding is usually based on the single-sided submerged-arc process shown in Figure 14. The essential features of the process are shown in Figure 15. Further details are given in Lecture 3.4 on Fabrication. However, to achieve overall efficiency this manufacturing process has to be supported by: • • • efficient materials handling for the web and flange components. numerically controlled cutting for the web - and also for the flanges if they are stripped from plate. semi-automatic butt welding for flange and web joints. 5. CONCLUDING SUMMARY • • • Tapered portal frames fabricated by automatic welding can be utilised to create aesthetic and economical industrial buildings. Greatest economy is likely to be achieved with Class 2 or 3 flanges and Class 3a webs, with a shape that gives a bending strength distribution that is a close fit to the bending moment envelope. Such frames must be analysed elastically. • • The behaviour of fabricated sections with slender webs is more complex than that of rolled sections; the resistance checks must take account of local buckling, cross-section distortion and the interaction between the primary and secondary structure through the stays. This form of construction will be economic if there is a substantial investment in a 'production engineering' approach to management, design, estimating and fabrication. 6. REFERENCES [1] Dowling, PJ, Mears, T.F, Owens, G.W, and Raven, G.K. "A development in the automated design and fabrication of portal framed industrial buildings". The Structural Engineer, London, Vol. 60A. No. 10, October 1982. [2] Kleinlogel, Mehrstielige Rahmen, Band I and II Berling, Verlag von Withelm, Ernst & Sohn. [3] Owens, G.W and Knowles, P.R (Ed) "Steel Designers Manual" Blackwells Scientific Press, Oxford 1991. [4] "Metal building systems manual", Cleveland, Ohio, Metal Building Manufacturers Association 1981. [5] Eurocode 3: "Design of Steel Structures": ENV 1993-1-1, Part 1.1, General Principles and Rules for Buildings, CEN 1992. [6] Chung, K.F and Owens, G.W., "Distortional Instability of very Slender Web Beams". Proc. Forth Rail Bridge Centenary Conference Developments in Structural Engineering Edited by B.H.V Topping, Chapman and Hall London Volume II, Pg 747-757. [7] "Reference 1 discussion". The Structural Engineer, London, Volume 61A, Number 10, December 1983. [8] Owens, G.W and Dowling, P.J., "Full scale testing of tapered portal frames". IStructE/BRE Seminar on Structural Assessment, BRE, Watford, England 1987. Lecture 14.3: Analysis of Portal Frames: Plastic Analysis OBJECTIVE/SCOPE To present the basic principles of portal frame design using the rigid-plastic method of analysis which are then demonstrated in a design example. PREREQUISITES Lecture 2.3: Engineering Properties of Metals Lecture 2.4: Steel Grades and Qualities Lecture 7.2: Cross-Section Classification Lectures 7.9: Unrestrained Beams Lectures 7.10: Beam Columns RELATED LECTURES Lecture 11.6: Rigid Moment Connections for Buildings Lectures 14.1: Anatomy & Analysis of Single Storey Buildings Lecture 14.13: Rigid Jointed Frame Design SUMMARY The basic principles of rigid-plastic analysis are presented with reference to plastic hinges, effects of combination of bending, ± axial forces and ± shear forces, "free" and "reactant" bending moment diagrams, hinge history and collapse mechanisms, and settlement of supports. The principles are developed for a semi-continuous beam and extended to a flat portal and to a pitched portal. Design rules for a pitched portal frame are discussed followed by a design example. 1. THE MODERN STEEL PORTAL FRAME Figures 1 and 2 illustrate typical modern steel portal frame buildings. They may consist of: • • • • • Thermally insulated colour-coated steel cladding system. Hot or cold-rolled steel purlins. Hot-rolled beam steel sections. Steel grade S275. Usually a 1 in 10 roof slope for architectural/planning requirements. In the UK such structures are frequently designed using the simple rigid-plastic method of analysis. This lecture describes the design of portal frames fabricated from beams and designed using the simple rigid-plastic method of analysis. References to clauses in Eurocode 3 [1] are given in the text usually in brackets, e.g. (clause 5.3.1). 2. REQUIREMENTS FOR PLASTIC ANALYSIS The use of the plastic method of analysis in the design of steel structures is possible due to the ability of structural steel to sustain considerable deformation without fracture. A typical stress/strain curve is shown in Figure 3. Beams subjected to bending moments must be symmetrical about the axis in the plane of loading, (Clause 5.3.3.(1)), and comply with certain dimensional properties for a plastic hinge to be developed and maintained, (Clauses 5.3.2 and 5.3.3). A summary of these criteria is shown in Figure 4. In Figures 5a, 5b and 5c the development of a plastic hinge in an I-section is shown. Any attempt to apply more bending moment to the section, once the complete section is fully plastic, causes the member to act as if a hinge had occurred at that point. This hinge action is called a plastic hinge. At a plastic hinge the steel section maintains the plastic bending moment and also can undergo considerable joint rotation thus causing any additional bending moments to be transferred to other parts of the member or structure. Figure 5 is for the situation where only a bending moment exists and the bending moment resistance at the plastic hinge condition is called the Plastic Moment of Resistance (PMR) of the section. In Clause 5.4.5.1: Mpl.Rd = Wpl fy /γM0 = the design plastic resistance moment of the gross section = PMR. Fastener holes in the tension flange may reduce the PMR - see Clause5.4.5.3. The PMR of a section is reduced when a bending moment is co-existent with an axial force or a large shear force. In Figure 6 the stress distribution for bending and co-existent axial force is shown. It should be noted that an axial force usually causes only a small reduction to the gross section PMR. Typically an axial force of PMR by only 2%. reduces the Formulae for the reduced PMR (MN.Rd) are given in Clause 5.4.8. MN.Rd = Mpl.Rd [1 - (NSd /Npl.Rd )2] For the design condition MSd ≤ MN.Rd The above equation can be re-arranged to give the following interaction equation: Modified formulae for bending and coexistent shear are given in Clause 5.4.7. It should be noted that where the shear force < 50% of Vpl.Rd then the gross PMR is not reduced. The predominant action effects in portal frame members are bending moments. Axial and shear forces usually have a negligible effect on the moment of resistance. However large shear forces do occur in the column head as a result of the eaves haunch connection. It is very common practice NOT to check the column head for the combined effect of bending moment and shear force even when the column has a bending moment plastic hinge in the column immediately below the column head. However, it is common practice to stiffen the column head if the column web shear stress is > fy /√3, i.e. > 0,6 fy. The verification of this practice requires further clarification because it has been reported that, although post-yield strength can justify it, excessive deformation may occur due to second order effects on the portal frame [2]. The test reported in [2] does have a very high shear stress in the column web which is NOT appropriate to a typical portal frame column head. It is necessary to have certain restrictions on fabrication to ensure that hardened material does not occur within the locality of a plastic hinge. Clause 7.3 lists restrictions relating to: • • • • • Flame cut or sheared edges*) Punched holes Hard marking Temporary welded attachments Surface repair by welding. (This requirement affects the supply conditions for the material.) It is also specified that "All locations where restrictions on hardening are required should be clearly indicated on the drawings". Plastic Design is for the ultimate limit state condition and includes structure and member stability checks. Serviceability limit state conditions also require checking. 3. APPLICATION OF PLASTIC ANALYSIS TO A BEAM In Figure 7 the beam has an elastic bending moment diagram with the maximum bending moment at Position 2. The first plastic hinge forms at Position 2 at a load of . At this stage of loading only one plastic hinge has been formed and the beam has been reduced to two simply supported beams for any additional loading. The effect of adding more loading is shown in Figure 8(c). This additional loading has the effect of causing a plastic hinge to form near the midspan of each beam. The exact position of the sag hinge can be determined and the value of the collapse load , i.e. . If compared with an elastic analysis in which the section remains elastic, the additional resistance of adopting a plastic design for this particular example is: i.e. 67% where is the shape factor (approx. 1,15 for an I section). However, if compared to an elastic analysis in which the full plastic resistance of the section is assumed (which is normally the case), the increase in strength by adopting a plastic analysis for this example reduces to: i.e. 45% The shear force at Position 2 might reduce the PMR and therefore Fp. In Figure 7(d) the plastic hinge history is shown. It can be seen that, if plastic hinges are not acceptable at working loads then, in this example the working load must not be more than In Figure 8(d) the collapse mechanism is shown from which it should be noted that: • • • times the collapse load. Pin joints ≈ hinges. Adjacent hinges are open/close/open ... The number of plastic hinges required to collapse one span is 2, i.e. r+1, where r is the number of redundancies. The effect of foundation settlement on plastically designed structures is: • • Not to change the collapse load. To change the load at which the first hinge forms. However the serviceability loading conditions should be checked for deflections and to determine whether or not a plastic hinge has formed. In Figure 8(a) the effect of settlement of the Support 2 is shown from which it can be seen that the hogging moment at Support 2 is reduced thus: • • • Increasing the sagging deflections of the beam. Increasing the load at which the first plastic hinge forms. Settlement influences the moment distribution for an elastic calculation. The load at which the first plastic hinge forms at Position 2 is: F1 = (PMR + PL/2) where P is the equivalent vertical load due to frame settlement δ . Theoretically there is a possibility that the first hinge to form could be the sag hinge if there is a large enough settlement at Position 2. Shear lag effects are covered in Clause 5.4.2.3 and apply to elastic and plastic analyses. If the length between points of zero moment is less than 10 times the width of the I beam flange then not all of the flange width is effective. The above example illustrated in Figure 7 is for a beam that is in one continuous length from Support 1 to Support 3. If connections are introduced at Support 2, then reference should be made to Eurocode 3, Clause 6.9: "Beam-toColumn Connections". Beam-to-column connections are classified by their moment resistance and rotational stiffness characteristics: Moment resistance (Clause 6.9.6.3) Nominally pinned Full strength Partial strength Rotational stiffness (Clause 6.9.6.2) Nominally pinned Rigid Semi-rigid Full strength and partial strength can each be rigid or semi-rigid. The example in Figure 7 can be classified thus: Supports 1 & 3 Support 2 Nominally pinned Full strength (plastic hinge) Nominally pinned Rigid If at Support 2 the connection was "semi-rigid" then the moment of resistance of the connection might not be equal to the design plastic moment resistance of the connected beam and would therefore be "partial strength". It can be seen that if the Mφ characteristics of the connection are too flexible, then the connection itself becomes a plastic hinge because the collapse mechanism moment has not reached Fp L/11,66. In such circumstances the moments at the other hinge positions are larger and a larger steel beam is required. The Mφ characteristics also determine the hinge history and the deflections at serviceability limit state. This discussion of the behaviour of a continuous beam includes some of the features of plastic design that have to be considered in the Plastic Design of Portal Frames. 4. APPLICATION OF PLASTIC ANALYSIS TO A FLAT TOP PORTAL FRAME As an example Figure 9(a) gives the general details a flat top portal frame. It can be seen that vertical and horizontal reactions are required at the pinned bases for the frame to perform effectively if analysed elastically or plastically. The value of H determines the position of the "reactant" bending moment on the bending moment diagram. For elastic analysis the value of H is determined from the relative stiffnesses of the steel members. In contrast, for plastic analysis it is determined from the relative bending resistances. The analysis of the development of plastic hinges (the history) has to be checked for its effect on the serviceability performance of the frame. In Figure 9(b) the beam and column are of the same steel section, whereas in Figure 9(c) the column has been arbitrarily chosen to have a bending resistance twice that of the beam. On Figure 9(c) it can be seen that the beam has to be strengthened at its connection to the column for a length of at least 0,09175L. This strengthening can be achieved by fabricating a haunch as shown in Figure 9(d). For stability reasons, the haunch is usually designed to remain elastic along its length when the frame is at the required ULS, see Annex A(c) iv. It is not necessary or desirable to have both hinges PMR1 and PMR2 adjacent to the haunch. The effective length factor in the haunch stability check is a function of the values of the bending moments at the haunch ends [3]. The design in Figure 9(c) shows that for a haunch length ≈ 10% of the span then the beam resistance need only be 50% of the column resistance. Depending on the frame span/height ratio this design can provide cost savings compared with the constant section solution from Figure 9(b). The moment at the beam-to-column connection can be reduced by providing a 'weaker', or partial strength connection, (see Lectures 14.10 and 14.11). This reduction has the effect of increasing the size of beam required, but reduces the forces which are transferred through the connection. Where the connection has relatively low moment resistance, it may be possible to alleviate the need for a column web stiffener. Since the fabrication of stiffening elements can be a costly and labour intensive operation, overall economies may result. 5. THE PRINCIPLE OF VIRTUAL WORK In Section 4 the required PMRs of the sections are derived by the manipulation of the Bending Moment diagrams. The Principle of Virtual Work is an alternative method and requires the collapse mechanism to be assumed from which the steel sections PMRs can be calculated from the knowledge that: Σ Mθ = Σ Wδ Internal work done = External work done. where each Mθ = Plastic hinge PMR x hinge rotation. and each Wδ = Applied load x distance travelled. The application to the first flat portal shown in Figures 9a and 9b is thus: Assume mechanism based on Figure 9b and that the hinge rotation of the column head is θ. Hence: Σ Mθ eaves = Mθ midspan + 2Mθ eaves + Mθ = 4Mθ Σ Wδ = 4Mθ = M= = required PMR 6. PLASTIC ANALYSIS METHODS So far only simple structures have been used as examples for the application of plastic analysis. The analysis method used is called the (simple) rigid-plastic method of analysis. In Clause 5.2.1.4 three methods of plastic analysis are given. Their differences are summarised in Table 1. Calibration of the elasto-plastic method has demonstrated that it is not necessary to use this more exact method for many (but not all) practical structural frames, including portal frames. Simple empirical rules have been developed that can be used with the rigid-plastic method and produce satisfactory economical designs. 7. APPLICATION OF THE SIMPLE RIGID-PLASTIC METHOD OF ANALYSIS TO THE DESIGN OF A PITCHED PORTAL FRAME From the previous Sections it can be seen that the simple rigid-plastic method of analysis is purely the manipulation of the bending moment resistances of the steel members by superimposing the "Reactant" bending moment on top of the "Free" bending moment. For portal frames this manipulation can be achieved by graphical means. This procedure was the sole means of design prior to electronic calculators and computers becoming available. This graphical method can be applied to virtually any loading combination, including hurricane winds. As a consequence of the simplicity of the method, other criteria have to be checked as listed below: a. Plastic hinges should preferably not form at the serviceability limit state because they would have to be taken into account when checking frame deflections. Frame deflections can be near to maximum acceptable limits and therefore the occurrence of plastic hinges is not desirable [4]. The occurrence of the first plastic hinge can be determined from an elastic analysis of the frame. A Serviceability Limit State (SLS) for portal frames is specified in 4.2.2.(4) but consideration should also be given to the effect of the SLS deflections on side wall cladding or masonry and also the racking of the roof cladding in bays adjacent to stiff gable frames. There is no need to check the deflections of portal frames at loadings between SLS and ULS conditions. b. For some frames and loading combinations it is possible for a plastic hinge to "form" and "unform" and not take part in any collapse mechanism. This phenomena should show up in computer programs based on the classical stiffness method where loading is incrementally "applied" to a frame and the plastic hinge history determined. Some computer programs have been reported to get this aspect wrong [4], and also to give incorrect results because of insufficient accuracy from the computer processor. An important point about "unformed" hinges is that they should be accounted for in the member stability checks. c. The suitability of the simple rigid-plastic method of analysis to the actual portal frame being designed has to be determined because second order effects caused by the frame deflections might reduce the actual ULS resistance of the frame by too much. Clause 5.2.6.3 permits simple rigid-plastic analysis to be used with an indirect allowance for second-order effects providing: Elastic critical load ratio, VSd/Vcr ≤ 0,20 where VSd = design value of the total vertical load Vcr = elastic critical value for failure in a sway mode In the case of portal frames, this only applies to frames in which either: • no plastic hinges occur in the columns, or • the columns satisfy limitations on the in-plane slenderness given in 5.2.7. The indirect allowance for second-order effects involves amplifying all the internal forces and moments by the factor in 5.2.6.2(3): Application factor = d. If VSd/Vcr ≤ 0,10, the frame can be classified as non-sway and provided it has adequate resistance to failure in a sway mode, no further checking for frame stability is required. e. Most portal frames have elastic critical load ratios between 0,10 and 0,20. They are, therefore, sway frames and can be analysed by the simple rigid-plastic method provided the internal forces and moments are amplified. Amplification factors tend to be around 1,1. f. Member stability has to be checked. The check is usually in the two areas of rafter haunch plus adjacent rafter and the columns, particularly where the compression flanges are unrestrained. g. The loads applied to portal frames including wind loads, are usually classed as static loads. There is usually no need to check for alternating plasticity on building structures, see Clause 5.2.1.4.(11). Annex A gives some design rules for simple portal frames. 8. CONCLUDING SUMMARY 1. 2. 3. Class 1 steel sections allow the use of the rigid-plastic method of analysis giving further increase in the efficient use of steel. The rigid-plastic method of analysis is a simple yet powerful well - proven method for designing the portal frame to satisfy ULS requirements. SLS requirements can be checked by an elastic analysis. Modern design codes, such as Eurocode 3 [1], use lower ratios for ULS/Working loads than earlier codes. Thus second order effects need to be proved to be insignificant. This requirement has made it necessary to ensure that the elastic critical load ratio is satisfactory. There is a need for simple accurate design aids for deriving the elastic critical load ratios. Portal frames fabricated from hot rolled I beams and with tapered eaves haunches can provide economical attractive structures without the need for substantial investment in production machinery. The tapered eaves haunch provides the means for varying the sizes of the rafter and column sections to suit the dimensional proportions of the structure. 4. 5. 9. REFERENCES [1] Eurocode 3: "Design of Steel Structures": European Prestandard ENV1993-1-1: Part 1.1 General rules and rules for buildings, CEN, 1992. [2] Morris, L. J. and Newsome, C. P., "Bolted Corner Connection subject to an out-of-balance moment - The behaviour of the column web panel". International Conference, Teesside Polytechnic, Middlesborough, Cleveland 6-9th April 1981. Additional Papers Volume. [3] Draft Revision Amendment No. 2 to BS 5950: Part 1: 1990. [4] Davies, J. M., "False Mechanisms in Elastic-Plastic Analysis". The Structural Engineer, page 268, August 1988. [5] Morris, L. J. and Nakane, K., "Member Stability in Portal Frames", pages 305-336 of "Steel Framed Structures", Narayanan, R. Elsevier Applied Science Publishers. 10. ADDITIONAL READING 1. 2. Baker, J., Horne, M. R. and Heyman, J., "The Steel Skeleton. Vol II. Plastic Behaviour and Design". Cambridge University Press, 1956 reprinted 1965. Morris, L. J. and Randall, A. L., "Plastic Design". Constrado 1975. The Steel Construction Institute, Ref. SCI-P-026 (plus SCI-P-027). ANNEX A SOME DESIGN RULES FOR SIMPLE PORTAL FRAMES Figure 10 shows the plastic bending moment diagram and collapse mechanism for a simple portal frame from which the following points are made and expanded into design rules: (a) The loading combination shown, of Dead + Snow + Equivalent Horizontal Forces (for frame imperfections), is usually the governing criterion but it does depend upon the intensities of Dead, Snow and Wind loads and the h/L ratio. (b) Two plastic hinges are required in the actual portal frame to form a collapse mechanism since two other hinges are already available at the pinned bases. The 'sag' hinge occurs in the rafter near the apex (MR). The 'hog' hinge can occur in the rafter at the haunch toe (M1) or in the column (MS). The designer can usually choose which one by appropriate selection of column and haunch properties. (c) The haunch is an important member and a number of points need to be considered (see also Figure 11): i. A haunch length of L/10 is a good first guess at what it should be. A shorter haunch increases the rafter section closer to the column section size, whilst a longer haunch reduces the rafter size but may cause problems in fabrication or in not being able to achieve the resistance ME. ii. The haunch is usually fabricated from a cutting from another hot rolled section and welding it to the rafter and the end plate. The cutting provides a "3 flange haunch" and is preferred to a "2 flange haunch" achieved from a plate insert because of its superior stability characteristics [5]. The toe angle should be kept above 7° to minimise distortion due to the effect of residual stresses that are "released" during fabrication. iii. A cutting from the rafter section creates a symmetric section. Hence compressive stresses are high and may give rise to a need for closely spaced restraints to the cutting compression flange. iv. The shape of the haunch and the bending moment diagram along its length causes it to be fairly constantly stressed and so a plastic hinge would form along the complete haunch length and may cause instability problems. One way of overcoming this is for the haunches to remain elastic along their complete lengths when the frame is at its required ULS. If this approach is adopted, the haunch cutting compression flange is at lower stresses. Hence stability is easier to ensure without a lot of restraints. In addition, the frame stiffness is increased and SLS deflections are reduced. v. The flange and end plates of the haunch connection are usually designed using yield line analyses. Other criteria, e.g. bolt prying loads, and local tensile stresses in the rafter web and welds, and the need for local stiffeners (Lecture 11.6), also need to be considered. Grade 8.8 high tensile bolts are commonly used. The top group provide the tensile force which has a lever arm down to the point of rotation at the cutting compression flange. Other limiting factors on the size of bolts are dimensions for access for tightening, edge distances and the thickness of column flange and haunch end plate. vi. All welds in a haunch connection should be fillet welds. Butt welds are more expensive and can create problems due to shrinkage, e.g. the point of rotation might move up from the required position. vii. The haunch is designed as a rigid connection. Eurocode 3 [1] requirements for connections are given in Table 5.2.1. A portal frame being "continuous framing" and "rigid-plastic" global analysis requires a "Full Strength" connection in accordance with Clause6.4.3.2. 6.4.3.2 Subclause (1) requires the strength of a full-strength connection should be "at least equal to that of the member connected". If the rotation capacity of a full-strength connection is limited, the design resistance of the connection must be at least 1,2 times the design plastic resistance of the member (6.4.3.2(2)) to allow for the possibility of the members being overstrength. However, the rotation capacity of a connection adjacent to a haunch need not be checked provided that the connection is capable of resisting the maximum moments and forces that would result if one or more of the plastic hinges located in the members were overstrength, due to the relevant members having an actual yield strength 1,2 times the specified value. For example, if the eaves hinge forms in the column, the moment resistance of the connection must be at least 1,2 times the plastic moment resistance of the column section. Member stability based on the new moment distribution would not have to be checked. (d) If the 'hog' hinge is in the column (MS) then the stability check will require more restraints than if the column remained elastic. This can be important if the client requires full bay width doors in the side of a building. (e) The modelling of the haunch in the frame analysis is shown by the dotted lines 1-2-3-4 for simplicity. An alternative is 1-5-4. Criteria Rigid-Plastic COMPUTER ELASTIC-PLASTIC ElasticPerfectly Plastic BASED Elasto-Plastic 1. First Order Effects i) Bending moments. ii) Effect of axial forces on member bending capacity. iii) Effect of shear forces on member bending capacity. * Optional * Optional * * * 2. Plastic Hinge Environment i) Linear elastic members up to sudden formation of hinges at fyWpl. ii) Concentrated at hinge position. iii) Plasticity spreads across the section and partially along the member as the bending moment increases through fyWel to fyWpl. iv) Hinge history available. * * - * * - * 3. Second Order Effects in ULS Analysis i) Deflections at main nodes due to first order bending moment effects. ii) No change in member EI values. iii) Loss of member stiffness due to combined effect of bending, axial (& shear?) forces AND member displaced shape. iv) Strain hardening at plastic hinge locations. * * * - * - included in iii * - - * Table 1: Plastic analysis methods ` Lecture 14.4: Crane Runway Girders OBJECTIVE/SCOPE To present the structural functions of the crane runway girder and to give design guidance on the girder and on its various components. PREREQUISITES Lectures 1B.5: Introduction to Design of Industrial Buildings Lectures 6.6: Buckling of Real Structural Elements Lectures 7.9: Unrestrained Beams Lectures 8.4: Plate Girder Behaviour and Design Lectures 11: Connection Design: Static Loading RELATED LECTURES Lectures 12: Fatigue Lecture 14.1.1: Single Storey Buildings: Introduction and Primary Structure Lecture 14.1.2: Single Storey Buildings: Envelope and Secondary Structure Lecture 14.3: Analysis of Portal Frames: Plastic Analysis SUMMARY Crane runway girders are usually regarded as a part of the building structure and are designed accordingly. A more realistic approach is to regard the crane runway girders as a part of the mechanical transport system in which the dominant component is the crane itself. There is a very strong interaction between the moving and the stationary parts of the crane system. There can be no successful design of either the crane itself or the crane runway girders if they are treated as separated structures. The forces imposed on the girders by the crane are in part caused by the behaviour of the crane itself, especially in regard to the vertical and lateral stiffness of the girder. The transfer of the crane wheel reactions to the crane runway girder induces a complex pattern of stresses in the upper part of the girder and leads to early service failures if not taken into consideration in the design. 1. INTRODUCTION In designing cranes, rails, runway girders and the supporting structure, the most important parameters are the maximum and most frequently occurring weights to be lifted, the speed and acceleration and the free height below the crane. The maximum wheel loads are determined by the net capacity of the crane together with the dead weight of the crane and dynamic effects. Handling facilities in simple portal frame buildings are often provided by light overhead travelling cranes carried on crane runway girders supported on brackets secured to the columns, see Figure 1a. The maximum capacity of cranes supported in this manner is about 100kN. Above this capacity, it is better to provide a separate leg or to increase the depth of the column below the crane runway girder to give adequate support. When an overhead travelling crane is introduced into a building, special care must be taken to ensure that the building is adequately braced in both directions. It is also worth mentioning that, where heavy cranes are involved, the crane runway girders may be subjected to severe fatigue conditions. 1.1 The Crane Runway Girder and the Structure The support method of the crane runway girder depends on the magnitude of the reactions being transmitted, in relation to the strength of the structural framing of the building. Some typical arrangements ranging from the lightest to the heaviest are shown in Figure 1. A separate crane column, as shown in Figures 1b and 1d is attractive for heavy cranes because it permits the effect of the crane to be considered isolated. However therein lies a danger, since the displacement of the building column could induce overstress in the connection between the two columns. A correct and more realistic approach is to analyse the columns as one. Careful consideration should be given to the transfer of the horizontal forces from the top flange of the girder to the column. This connection should: • • • safely resist the horizontal reactions allow free rotation at the support of the crane runway girder allow lateral adjustment of the crane runway girder after completion of the building. A very important aspect is the need for adjustment. It is impossible to erect building frames to the tolerance required by the crane manufacturer and it is therefore essential that the whole crane runway girder can be adjusted up to 10mm with respect to the building columns. Therefore, slotted holes and shims are required, as shown in Figure 2. Free rotation at the supports of crane runway girders is important in order to prevent bending and torsional moments in the columns. Rotation at the supports of a continuous girder can be realised by appropriate, flexible detailing as shown in Figure 3. Rotation at the end of a simply supported girder results in a longitudinal movement of the top flange in relation to the centre line. The member which connects the top flange to the building column must therefore be capable of allowing free longitudinal movement without becoming overstressed. A simple flexible plate may be satisfactory when the movements are less than 1mm, but a connection with slotted holes is a safer solution in most cases (see detail B, Figure 2). Another vital aspect is that the distance between the two columns of a portal frame at the height of the rail changes with the loading. The change in distance between two load cases can easily amount to 1/180 of the column-height. The wheel flange clearances must therefore be much larger than immediately expected (often 50 mm or more are recommended). Longitudinal bracing of the building and crane runway girders can be arranged in several different ways: • • • vertical bracing used as building and as crane runway girder bracing. vertical bracing bays with direct connection to the brackets and positioned in the plane of the crane runway girder (for heavy cranes). vertical bracing in the planes of both crane runway girder and building columns (for very heavy cranes only). If the last method is used, there must be an effective restraint to the crane brackets to prevent torsion in the column. This restraint is normally obtained by a horizontal truss, as shown in Figure 4. The ideal place for the braced bay is half-way between the expansion joints in the crane runway girder, or in the middle of the building, see Figure 5. This arrangement prevents the build up of axial compressive forces due to temperature rise, which could cause buckling of the crane runway girders. Furthermore, it forces the expansion in two directions, and thereby minimises the total movement. Only the columns below the crane runway girder are deformed. It is the magnitude of the secondary stresses associated with this deformation which limits the distance between the expansion joints. The maximum allowable distance between the expansion joints depends on the horizontal longitudinal displacement capacity of the columns bearing the crane runway girder - see Figure 5. A method of transferring the axial forces in a simply-supported girder directly across the joint at the support is shown in Figure 2.3. The detail also shows an effective method of supporting the girders by using load bearing stiffeners. Attention has to be paid to the local eccentricity of the bearing stiffener with regard to the web of the bracket. 2. TYPE OF CRANES The most common types of cranes running on elevated runway girders are: • • Top running bridge cranes consisting of a single or a double girder spanning between the end carriages (Figure 6a). Underslung bridge crane with special end carriages where the wheels are running on the bottom flange of the runway girders (Figure 6b). 2.1 Classification of Cranes Loads from crane wheels have a static and a dynamic component. Both components are functions of time and vary with crane position and the magnitude of the load. The loads handled by the crane consist of a spectrum of light, medium and heavy loads. The dynamic forces due to acceleration and braking, hoisting and unevenness of the rails also vary from installation to installation. To ensure economical design of cranes, they are normally divided into several classes depending on the frequency of their use, the average ratio of the loads lifted to the safe load, and the dynamic effects experienced in service. In this way it is possible to assess the fatigue risk to the crane and its runway girder during its design life. Classification is based on two factors: • • Frequency of use. State of loading (ratio of magnitude of actual or assumed load to the safe working load). Selection of values for frequency of use and state of loading determines the final classification of a crane. 3. CRANE RAILS The crane rail and its interaction with the top flange of the girder has a very strong influence on the performance of the crane. It is, therefore, important to know what type of crane is going to be applied when designing the crane rail and runway girder. Loading characteristics should be adopted which are in accordance with the crane which will probably be installed. These characteristics can be obtained from manufacturers manuals. In practice it is sometimes impossible to prepare the design of the crane and the crane runway girder at the same time because the crane is ordered much later than the building structure. The result may be a poor design leading to problems such as excessive wear of the crane rail and crane wheel flanges or fatigue cracking in the upper web of the girder. The crane rail must meet the requirements for protecting the top flange from wear and for distributing the wheel loads evenly over the greatest possible length of contact. The crane rail must therefore have: • • an adequate wear resistance. a high flexural rigidity. Two types of crane rail are shown in Figure 2: • • block rail. specially rolled rail section. 3.1 Rail Splices There are two types of splice: • • Splices which join individual lengths. Expansion splices. Longer rail lengths can be obtained rather by welding than by bolting. Welded splices are normally superior to bolted splices because the welded joint avoids a gap and gives a step-free running surface. Special care is required in the welding operation if there are high carbon and manganese contents in the steel. Expansion joints in rails must be provided on long runways when rails are fixed to the girders. They should coincide with joints in the main girder. A gradual transfer of wheel load from one rail to another is ensured if the ends of the rail are bevelled as shown in Figure 7. 3.2 Rail Fastenings Various types of rail fastenings are shown in Figure 8. The traditional approach is to provide a fastening which restrains the rail in all directions. The fastening of block rails is always by shop welding. The fastening of specially rolled rail sections is normally obtained by a fully rigid clamp or by welding the rail to the flange of the crane runway girder. Welding has the advantage that the rail can be accurately located on the girder centreline due to the fact that lateral adjustment is possible. However the use of welding gives problems in some cases. For example: • • Renewal may be difficult. In simply-supported joints crane runway girders occur at each support if shop welded. • • • Site welding is necessary if continuous crane runway girders are used. This problem is solved if site welding is located at positions where the bending moments are minimal, in which case the stress situation in the welds is less critical. The welds can induce fatigue cracks. When higher strength steel has been specified, the welding operation is more difficult. Modern practice tends towards a fastening which gives partial restraint, as shown in Figure 8c. The rail is restrained in the vertical and lateral direction, but the clamps allow the rail to move in the longitudinal direction. Figure 9 shows a very economical method, for heavy duty applications, of obtaining lateral restraint by site welding 'steering' plates between the clamps instead of using high strength bolts in the clamps to eliminate the possibility of movement. This type of fixing has to be checked for its influence on the fatigue of the crane runway girder. 4. LOADS ON THE CRANE RUNWAY GIRDER The static wheel loads are exceeded during operation of the crane as a result of impact, inertial effects and other dynamic effects. These effects can also result in lateral forces at the top of the crane rail. The main factors to be considered are: • • • • • acceleration and deceleration of the crane bridge and the crab. degree of control over the hoisting speed. off-vertical lifting at the start of hoisting, see Figure 10. tendency of the crane to travel obliquely, see Figure 11. condition of the rail surface and the width of rail joints. These dynamic effects can be approximated by multiplying the static wheel loads with an appropriate factor which may range from 1,0 to 2,0. Oblique travelling of the crane can also induce lateral loads, as shown in Figure 11. The forces on the rail are acting in opposite directions on each wheel of the end carriage and depend on the ratio of crane span to wheel base. The longitudinal forces due to crane acceleration and braking should be verified by calculations, when data on masses of the moving parts and their accelerations are known. The end stops placed on the crane runway girder must be designed to take the crane buffer force. The buffer force is calculated from the kinetic energy of the mass of the crane, but without the lifted load due to the fact that it is suspended from the ropes. Another approach is to use electronic devices to stop the cranes at the ends, yielding a more beneficial loading situation for the structure supporting the crane runway girder. Other loads that need to be considered are: • • Catwalks and ladders attached to the girder. Power supply cabling and cable trays. For more quantitative information on loads to be taken into account in designing a crane runway girder, national codes or crane manufacturer's documentation should be referred to. 4.1 Transfer of Loads to the Top Flange The loads transmitted to the rail produce a triaxial stress state in the flange and the upper part of the web. The stress components are: • • • • • Compressive stress in the longitudinal direction of the flange. Compressive stress in the web in the vertical direction. Local bending stress in the flange in the longitudinal direction. Local bending stress in the web in the transverse direction. Shear stresses in the web. To make a realistic assessment of the stresses, the following design hints could be given: • • Wheel load should be distributed over a length equal to twice the rail depth. The stresses in the web should be calculated with an assumption for the eccentricity of the wheel with respect to the centre of the web, which might occur at the supports or when the crane and/or the rail have • • • seriously suffered wear. Eccentricity of the rail to the runway girder usually has to be prevented by connecting them together with very small tolerances (preferably shop welding). Welds connecting the flange to the web should be checked for a combination of vertical stresses and bending stresses due to eccentricity (of the wheel load) in addition to shear. To avoid the necessity to move the rail from its location above the web, alignment of the whole crane runway girder should be possible. Therefore, slotted holes and shims are applied, see Figure 2. If welded crane runway girders are used, a full penetration butt weld should be used for the top flange to web joint to give resistance to fatigue. 5. SELECTION OF THE CRANE RUNWAY GIRDER During the conceptual stage of the design of the crane runway girder the fundamental questions are: • • • • Should a simply-supported or a continuous girder be used? Should a solid web girder or a latticed girder be used? Should a single or double web construction be used? Should high strength steel be used? In some countries, simply-supported girders are preferred; in others continuous girders. When continuous girders are used, special attention should be paid to: • • differential settlement between adjacent footings. This should be limited to L/600. erection, especially when site welding is adopted. Figure 12 shows some cross-sections used for crane runway girders. For small spans and light-to-medium crane loads, it is normally possible to use rolled-beam sections. In some cases reinforcement may be necessary to give resistance to lateral forces (Figure 12a-c). Single web plate girders are suitable for the majority of heavier cranes. Their insufficient resistance to lateral forces is normally solved by introducing horizontal bracing, as shown in Figure 12d. Plate box girders are popular for the crane itself but are seldom used for the crane girder. The rail must be situated directly over the inner web of the box girder, so that transverse flexural stresses in the top flange plate are avoided, as shown in Figure 12e. High strength steel is seldom used in crane runway girders because fatigue considerations limit the permissible stresses quite severely and thus reduce the economical advantages (the fatigue strengths of mild and high strength steel for welded structures are the same). Additionally, deflection and lateral-torsional buckling considerations also prevent the designer from gaining advantage from using high strength steel. 5.1 Optimum Girder Proportions A general set of rules to assist the choice of optimum depth of crane runway girders cannot be given due to the variety of load cases and the differences in the cross-sections normally used. As a rough guideline, the usual range of girder depth-to-span ratios is between 8 and 14. The deflection limitation may dictate a larger depth, especially where spans are long. 6. DESIGN OF THE CRANE RUNWAY GIRDER The design of crane runway girders has some special aspects which are not often present in the design consideration of other types of girder: • • • • combination of concentrated loads and bending moments. combination of lateral loads and lateral-torsional buckling. combination of web buckling and plate bending stresses due to torsion induced by the rail eccentricity and lateral forces. design is required against early fatigue failure. The degree of refinement required in considering these special effects during design, depends very much on the class of the crane. One of the most important decisions in connection with the design is to determine how far to go in minimising the mass of steel. Good design must take into consideration all costs during the design life of the crane installation. A very light design may promise a low first cost, but could give rise to large maintenance costs resulting from a need for frequent repairs. 6.1 Crane Runway Girder-to-Column Details The predominant loading is vertical. The crane runway girder is normally directly supported by its seated connection on the column or by means of a bracket. The best way to secure a direct flow of stresses from the crane runway girder to the column or bracket below, with a minimum of eccentricity, is by means of welded brackets, as shown in Figure 2. The next principal loading is transverse. Figure 13a shows a dangerous detail frequently used on lighter crane girders to resist lateral forces. Figure 13b illustrates the reversible strain to which the girder web is subjected - an action leading to the result shown in Figure 13c. The failure could easily be prevented by simply connecting the top flange directly to the column, as shown in Figure 14. The top flange acts as a horizontal beam delivering its reaction to the column. Another effect caused by this bad detail is shown in Figure 15. The vertical deflection of the crane girder rotates its ends on the column seat. If the connection is not designed for that purpose the result is high shear on the upper fasteners, and local tension in the web, which could lead to failure in that area of the web. A continuous girder offers a possible solution to the rotation problem when a flexible detail as in Figure 3 is chosen. 6.2 Rigidity Requirements The following maximum values for the deflection of the crane girder must normally not be exceeded in order to avoid undesirable dynamic effects and to secure the function of the crane: • • Vertical deflection at midspan, due to maximum wheel reactions without duty features L/700 Horizontal deflection at midspan due to maximum wheel reactions multiplied by the duty factor L/600 In the absence of more detailed calculations it is acceptable to assume that the top flange resists the whole horizontal force. The rigidity requirement for horizontal deflection is essential to prevent oblique travelling of the crane. The vertical deflection is normally limited to a value not greater than 25 mm to prevent excessive vibrations caused by the crane operation and crane travel. 6.3 Web Stiffeners It becomes uneconomical to use unstiffened webs when girder depths increase, because a relatively large proportion of the girder material is in the web. Web stiffeners serve the purpose of: • • preventing buckling in the web. adding rotation capacity to the top flange. Twisting of the top flange caused by lateral forces has to be resisted by the web alone, if no web stiffeners are present. When the girder is relatively deep and the lateral forces are high, it will not be possible to omit web stiffeners. The distance between the stiffeners must not be so large that twisting of the top flange becomes too large at the mid-point. The method of attaching the stiffeners to the web and the flanges must be detailed carefully to prevent fatigue failure. Fatigue in the tensile flange can be averted by providing a gap of 4t between the end of the stiffener and the bottom flange, as shown in Figure 16. However there will still be a possibility of fatigue in the web at the termination of the stiffener. However, the detail shown in Figure 17 is normally considered to be the best solution. The stiffener should be welded to the compression flange so that relative movement of the flange in relation to the web due to lateral forces is totally prevented. The stiffener should be coped a maximum of 200 mm. 6.4 Lateral Forces and Lateral-Torsional Buckling The simultaneous effects of torsion induced by lateral forces and lateral-torsional buckling can be considered in several ways. It is often difficult to decide how rigorously the structural calculations should be done. Lateral forces due to off-vertical lifting, inertial effects and oblique travelling can only be estimated approximately. Values obtained from relevant codes together with the use of duty factors given in the Codes is the only means at the designer's disposal. Torsion in the section is caused by: • • lateral force acting at the rail head level. eccentricity of the vertical force due to tolerances dependent on the fabrication of the rail to the girder (see Section 4.1). The geometry of the top flange should be chosen from those alternatives that offer the best torsional resistance and the best lateral stiffness. 6.5 Fatigue Considerations Crane runway girders are subjected to repetitive stressing and unstressing. The number of stress cycles that certain parts of the crane runway girder is subjected to may be two to four times the number of crane passages because each passage of the wheels causes stress fluctuations. This effect is one of the reasons why special care must be paid to the detailing of the top part of the crane runway girder. The number of the crane passages is not easy to estimate. For design purposes it is assumed that the number of stress fluctuations corresponds to the class of the crane as specified in the Codes. The critical details in fatigue design are the stiffener-to-flange, the stiffener-to-web, and the flange-to-web connections where severe concentrations of stresses exist. The following recommendations are made: • • • welds attaching the stiffeners to the girder web should be terminated at a distance from the flanges to reduce the stress concentration (see Figure 17). welds connecting the web to the top flange should be full penetration butt welds, although fillet welds are sometimes used for light, primarily static cranes. flange reinforcement using cover plates leads to poorer fatigue life. 7. CONCLUDING SUMMARY Crane runway girders require a special care in design and detailing. They should be regarded as a mechanical item. The uncertainties, especially regarding the transverse loads and the transfer of forces to the girders, have to be clearly recognised. In the following some guidance in obtaining the proper design is given: • • • • • Simplified calculations are adequate for light load cranes, but more rigorous analyses are required for heavy load cranes. The depth of structural investigations can be decided from the class of the crane. Although minimum weight design may provide an economical solution to many design problems, this is not the case in the design of crane runway girders where the overall costs must include the maintenance costs. Attention must be made to detailing which may reduce the fatigue life of the crane runway girder. This consideration applies especially to the top region of the girder. Welded fabrication should be given a more rigorous inspection than the rest of the building structure. No further welding attachments should be allowed during the lifetime of an intensively used crane girder. 8. ADDITIONAL READING 1. 2. 3. 4. Petersen, C., Stahlbau, Friedr. Vieweg & Sohn, 1988. Dubas, P. and Gehri, E., Stahlhochbau, Springs-Verlag, 1988. Gorene, Crane Runway Girders, Steel Construction, Vol. 10, No 4. Mueller, J. E., Lessons from Crane Runways, Steel Construction, Vol.10, No 4. Lecture 14.5: Space Structure Systems OBJECTIVE/SCOPE To describe different types of spatial truss systems, and the design parameters to be considered. To give guidance on initial sizing and on analysis methods. To describe fabrication and erection procedures. PREREQUISITES Lecture 1B.3: Background to Loadings Lecture 6.3: Elastic Instability Modes Lecture 7.12: Trusses and Lattice Girders RELATED LECTURES Lectures 13: Tubular Structures Lecture 14.6: Special Single Storey Structures SUMMARY The lecture provides an historical background and an overview of different types of spatial truss systems: doublelayer grids, barrel vaults and domes. Design parameters are introduced and some rules for initial sizing are described. The principles of different methods of analysis are given. The lecture concludes by describing aspects of fabrication and erection particular to these structures. 1. INTRODUCTION 1.1 Definitions For this lecture, trusses are defined as structural systems in which the members are interlinked so that they are only subject to axial compressive or tensile forces. This definition assumes that no action is applied directly onto the members. All loads are applied to the joints which are known as 'nodes'. In case it is impossible to guarantee the coincidence of member axis, the bending effect resulting from this must be evaluated. It is particularly important to ensure that the axes of the members coincide (Figure 1). Only perfect pins could completely ensure compliance with this loading condition. The technological construction of assemblies deviates to some extent from this theoretical situation and, in effect, is one of the main difficulties associated with these structural systems. The lecture is concerned mainly with truss systems for roofs, which span in two directions (termed 'space structures'). Other arrangements are possible, such as continuous systems, based on the Vierendeel girder (Figure 2) in which diagonal bracing members are unnecessary because the bending behaviour is predominant, rather than the axial one; the resulting voids can be used to accommodate mechanical and electrical services. 1.2 Historical Background Until the 1960s, almost all truss systems were two-dimensional. They had developed from timber roofs, which themselves had evolved from a basic triangular arrangement to more complex shapes (Figure 3). The need to lighten long tie beams and reduce bending stresses (Figure 3a) had led to the introduction of a suspender (Figure 3b). A similar concern to reduce bending of the rafters led to the introduction of diagonal members (Figure 3c). By dividing the suspended member in two, the familiar arrangement of Figure 3d was obtained. The use of metal became dominant in the 19th century for all types of structures except domestic buildings. Articulated systems (Figure 4) commonly used for roofs of railway stations were perfect examples of the twodimensional triangulated system. The second half of the 19th century was characterised by some remarkable achievements, for example the Garabit viaduct in France (Figure 5) and the Maria Pia bridge in Oporto, both designed by Eiffel. Although spatial systems were proposed early in the 20th century, their use in practice has arisen from the more recent development of computer methods for analysis, the functional need for spaces free of columns and from demands of architectural appearance. 1.3 Different Types of System 1.3.1 Introduction Different types of spatial truss systems are normally classified according to their general shape. The following may therefore be distinguished [1]: • • • two dimensional grids cylindrical vaults domes. In each case it is advisable to distinguish between single and double or even triple layer grids. The number of layers depends on the span. A third characteristic lies in the geometry chosen for the system of members in the layers and possibly in the composition of the bracing of the layers. 1.3.2 Two dimensional grids 1.3.2.1 Single layer grids These grids are mentioned only as a reminder that these systems are beam grillages which work in bending and torsion, rather than under axial compression and tension. Depending on the directions assigned to the members, grids may be identified in two, or three directions (Figure 6). Grids in two diagonal directions are more rigid (beams follow the direction of the principal stresses of the equivalent plates) and are widely used. Utilisation is restricted to about 10m of span. 1.3.2.2 Double layer grids These grids comprise two systems of members on two parallel levels (upper and bottom layer). Both these systems are interlinked by bracing members (web members) (Figure 7). Two types of double layer grids may be distinguished (Figure 8): • • lattice grids where there are always top and bottom chords in the same vertical plane (Figure 8a). spatial grids, made up of triangular based pyramids, square or hexagonal (Figure 8b). Two kinds may be identified: one where the layer geometries are identical though displaced (offset grids), and the other where the layer geometries are different (differential grids). These systems are suitable for spans up to 100m. For greater spans, it is necessary to incorporate triple layer grids, to avoid long members otherwise necessary with the increased depth. The size of the constituent modules depends on several factors, principally: span, load, cladding system, type of node, transportation and erection facilities. For spans of between 30 and 40m, member lengths of about 1,5m to 3m are acceptable. The advantages of double layer grids are numerous: • • • • • they are three dimensional structures which can withstand loads from any direction. they are hyperstatic, and buckling of some compression members does not cause the whole to collapse as has been demonstrated by mathematical models and experiments. their rigidity minimises deflections. they have a very good fire resistance. their composition allows factory pre-fabrication in modular elements, which are easily transported. Fabrication precision ensures ease of assembly and erection. • • • • they allow a wide choice of support positions owing to modular construction. the space between the two layers may be used to install electricity, electrical and thermal piping, etc. installation is carried out by bolting and may be done whatever the atmospheric conditions. they provide indisputable aesthetic qualities. From an economic point of view, it is important to have a minimum number of nodes. It is therefore necessary to compromise between this criteria and those determined by the choice of module sizes. 1.3.3 Cylindrical vaults In the history of construction, cylindrical vaults appear as an evolution of arches. The use of metal has enabled construction to be carried out with factory prefabricated elements which may be assembled on site. The first example to recall is the Crystal Palace which was erected by Joseph Paxton for the Great Exhibition in 1851. This shape has proved to be suitable for roofs of halls, railway stations and sports facilities, e.g. in-door tennis courts. Maximum efficiency may be attained for shapes with rectangular surfaces and a length/width ratio of between 1 and 2. The optimum shape (rise/span ratio) is in the region of 0,15 to 0,20 (Figure 9). Several layer geometries are possible (Figure 10). In practice, three directional systems offer the most advantages. They may usually be analysed by assuming pin-jointed behaviour for the nodes. This assumption does not hold, however, for some systems where bending rigidity must be taken into account, e.g. for a vault composed of prefabricated elements rigidly joined by high strength bolts. The sensitivity of these systems to asymmetrical actions, in particular to wind, should not be underestimated. Such actions can even bring about force reversal in the members, which is an additional reason for choosing a three directional geometry in which the length of all elements is identical. In the same way, choice of support conditions along the boundaries influences force distribution. Economical spans for single layer vaults are in the region of 20m. Spans may be increased by inserting diagonal elements. They reach 60m for double layer systems, in some cases even more. Appropriate weights for double layer systems vary between 0,13 and 0,25kN/m2 depending on the intended shape, support conditions and on the geometry of the sheets (for a uniform load of between 0,75 and 1,50kN/m2). 1.3.4 Domes Domes constitute one of the most ancient forms of construction. However big or small their size, the outline of the two-dimensional support is normally circular on plan. Skeletal dome structures can be classified into several categories depending on the orientation and position of principal members. The four more popular types are: ribbed domes, Schwedler domes, three-way grid domes and parallel lamella domes (Figure 11). Domes are of special interest to architects and engineers because they provide the maximum enclosed volume for minimum surface area. In the last 25 years construction with steel sections has largely replaced reinforced concrete. The first two examples concern single and double layer systems. These systems permit spans of about 40m and more than 100m respectively. Some double layer solutions have encouraged 'record' spans of more than 200m. The accurate analysis of domes has only been possible with the introduction of computers which have allowed an accurate study of elastic behaviour. It is important to note that large span single layer domes subjected to asymmetrical actions can experience global instability effects. In addition to consideration of local buckling of compression elements, specific attention should be drawn to possible global 'snap-through' buckling (Figure 12). The action of wind is not very well known; application of factored actions to a non-critical load case does not necessarily cover the most unfavourable situation and recognition of an appropriate excess weight does not necessarily bring about the most unfavourable situation. From the physical point of view, it is important to stress the difference in behaviour of domes with respect to cylindrical vaults. The sensitivity to asymmetrical actions and the resistance to global buckling phenomena is strictly connected to the rigidity of the geometrical shape. For cylindrical vaults, their surface has a single curvature, so it can be developed on plan. In contrast, domes, having a double curvature, resist any actions by virtue of the shape itself. Definition of the geometric arrangement of elements, whether for a single or a double layer construction, is a difficult problem to resolve. Research is aimed at using fewer different lengths for the members. Moreover, it is important to check that the polygons defined are as similar as possible, in order to facilitate cladding. Fuller designed one of the first big developments: the dome of the American Pavilion in Montreal (1967) which is composed of two interconnected layers and constructed with welded tubular sections; the resulting structure is 5/8 of a total sphere, with a diameter of 75m. 2. DESIGN OF SPATIAL TRUSS SYSTEMS 2.1 Conceptual design At the conceptual design stage it is important to define the geometry of the structure, given that determining support positions is an important factor in the strength and rigidity of the system. Geometric design parameters are: • the overall shape; flat shape or assembly of small flat shapes (so called "polyhedric surfaces"), curved surfaces (generally positive double curved surfaces; one of the curvatures may be equal to zero, i.e. the case of cylindrical vaults). For curved surfaces, a rise/span ratio should be fixed to satisfy both mechanical and architectural criteria, e.g. in order to avoid domes which are too shallow. geometry of the cladding supports. the number of layers; structural depth/span ratio influences the weight, strength and cost. frequency of mesh, i.e. number of geometric elements for a given length. • • • The choices made directly influence the number of members converging on each node and on the connecting angles between these members; these two parameters determine the feasibility of the nodes. Too many elements meeting at different angles and lack of repetition are hindrances to construction efficiency. The choice of spacing of grids should be related to geometric connections, particularly the connection of the two-dimensional sides of a polyhedric surface. Choice of support conditions does not pose any specific problems, but it does affect behaviour. It should be noted that, because of the low weight of these structures, the weight of the equipment supported by the structure should not be negligible. Similarly, actions resulting from the method of construction should be carefully examined. The effects of concentrated actions and of partially distributed loads, which are greater when the total loading is not symmetrically distributed, should be examined. Except in specific cases, dynamic actions can be replaced by enhanced static actions. 2.2 Design Method Every project is undoubtedly an individual case. Nevertheless, it is possible to establish sequential steps in the development of the design. Most importantly it should be noted that, for a shape to be covered in any pre-determined way, two distinct methods can be used to define the general surface: • • either the overall surface geometry is defined a priori: a geometric division must then be made, e.g. a geodesic division for domes. or the generating module is laid as before and multiplication of this module provides the final geometry. Once the overall geometry has been established, the designer must decide on the number of layers. This design depends basically on the free span, and also on the geometric distribution of the members in and between the layers. Choice of frequency of mesh is important for reasons of resistance and cost, as well as for aesthetic reasons. Choice in the geometry of the network of members directly influences behaviour of the systems. For example, in the case of double layer grids, an examination of different geometric arrangements has confirmed the importance of adopting an arrangement where directions of the members in the two layers are set at 45°. It is therefore possible to examine the structural behaviour under appropriate combinations of actions. Choice of support conditions has a big influence on the distribution of internal forces and size of the deflections. The possible use of multi-point supports is an important advantage of spatial trusses. Definition of the areas of the cross-sections of the elements may lead to a process of optimisation suitable for the design model. 2.3 Initial Sizing It is possible to provide further information in the case of double layer two dimensional grids. The curves in Figure 13 illustrate the variation in weight suitable for different structural depths with the slenderness ratio for seven different geometries (Figures 14-20). Calculations were made for the case of a square outline with peripheral supports under each node, assuming a uniformly distributed load. In addition, the following data have been assumed: • • take a depth/span ratio of 1:15 in relation to the free span where there is a working load of 1,50kN/m2;. consider a self weight of about 0,15 to 0,20kg/m2 for spans up to 30m. It is also important to consider the relation between the size of the mesh element and depth of the grid. 2.4 Choice of the Structural System As soon as the overall shape of the structure is defined, it is necessary to choose the structural system. A wide variety of structural systems is available, which can fulfil the geometrical requirements of the design. However, they cannot all provide the load carrying capacity which is required to resist the most unfavourable design loading conditions. The sizing of the cross-section of members is a task of the designer, whereas the producer of the structural system must guarantee that the node-member combination should be able to give a joint of full strength type. Only by means of appropriate qualification procedures can this guarantee be realised. 2.5 Qualification Procedure The qualification procedure should demonstrate that the system for the space structure is based on a basic node member joint which is a full strength type. The demonstration can be done by calculation and by tests. On the one hand, it is possible to model the node-member joint by means of finite element techniques. However, it is not prudent to rely solely on these numerical results. On the other hand, only experimental evidence can fully demonstrate the actual behaviour of the system. For these reasons, the qualification procedure should be based mainly on laboratory tests, the results validating the calculations. A suggested procedure could be based on the following phases: • • • • monoaxial tests on the node; monoaxial tests on the member-to-node connection; bending and shear tests on full-scale structural units; bending tests, monotonic and cyclic, in the elastic range (including temperature effects) up to collapse on full-scale prototypes. 3 ANALYSIS OF SPACE TRUSS SYSTEMS 3.1 Different Analysis Methods The objective of an analysis of truss systems is to determine the values of the variables necessary for sizing purposes and for those required to size their supports. The variables generally required may be: • • • compressive and tensile forces in the members in the system. node displacements. values of support reactions. The study has to be made for several cases of actions and combinations of actions. The most unfavourable cases are used as the basis for design. Depending on the type of problem being examined, it is not necessary to determine all unknown quantities. The methods of analysis available can be classified as follows: i. Method of joints: applicable to two-dimensional systems which are internally isostatic (Figure 21), for which the reactions have previously been determined. It is possible to determine the forces in all members. They can be determined by analytical or graphical methods (see Figures 22 and 23). ii. Method of sections: in certain cases, this method can give direct access to internal forces in a limited number of members selected by the designer. It is used, for example, for preliminary design work in order to assess the maximum forces in a triangulated system (Figure 24). iii. Displacement method: This is the most general method. It is applicable to two-dimensional and spatial systems, isostatic or hyperstatic. It gives values for all unknown quantities: internal forces, displacements, reactions. 3.2 Design Assumptions In addition to use of principles which are common to all structures in respect of actions and their combinations, Eurocode 3 [2] allows a perfectly pin-jointed model for global analysis. Furthermore, it is considered that all actions are applied at the systems nodes. It is sufficient to assume linear-elastic response. Calculations in the elasto-plastic range can be done by using simplified models of the members such as is shown in Figure 25. The results show that a reasonably stable response of compression elements (after the ultimate load of a member is reached) can be achieved and the forces transferred to the adjacent members from that which has reached its ultimate resistance. The methods described in the following paragraphs are all based on the application of equilibrium to the forces applied by the members to the nodes. The pin-jointed analysis used for triangulated systems simplifies the formulae for equilibrium: in general, the three equations relating to translational equilibrium are necessary. 3.3 Limit of Validity of the Methods Described Attention is drawn to the fact that the methods described do not take account of: • • • • any non-coincidence of the axes of the members forming a node. flexure induced by the application of external actions but along the length of members, rather than at nodes. secondary bending moments due to the effective rigidity of the joints which no longer corresponds with the assumption of pin-jointed behaviour. non-linearity due to geometry and/or to material. It is therefore necessary to evaluate carefully the significance of factors associated with failure to ensure the appropriateness of the assumed models, either by making additional calculations, or if applicable, by using more detailed calculation models, e.g. the generalised displacements model which takes account of stiffness in flexure and in torsion, when the given joint is far from the pin-jointed assumption. 3.4 Displacement Method This is the most general method applicable in all cases of space structures. The behaviour of the materials is assumed to be elastic and linear. The principle of the method lies in resolving a system of linear equilibrium equations as follows: [K] {D} = {F} where [K] is the structure stiffness matrix. {D} is the unknown displacement vector. {F} is the vector of known actions. The components of {D} corresponding to the fixed supports are zero. The corresponding equations, the second components of which are the reactions, do not therefore appear in the corresponding system of linear equations. Determination of the displacements enables the internal forces to be calculated. This method is normally implemented by computer software. 4. FABRICATION OF SPACE TRUSSES 4.1 Introduction The very nature of spatial trusses encourages research into maximum standardisation, linked to individual component manufacture, and requires particular attention to problems of precision. It is advantageous to design spatial trusses with a minimum number of different members; the same criteria should be used for the nodes. It is common to use members with the same section sizes, independent of their different stress state due to their location in the structure. For tubular sections, it seems reasonable, however, to keep the same external diameter and to vary the wall thickness. Installation methods may cause greater deformations and forces during erection than after completion. The designer should consider the erection phases when sizing the elements. 4.2 The Structural System The structural system is characterized by the combination of three main components: • • • member node connection Members Hollow sections are essential for a number of reasons. In particular, tubular sections are generally used because of their large and uniform radius of gyration. Nodes The dream of a 'universal' node has not yet been realised. Several parameters govern the design of nodes. Nodes can be connected mainly by welding, bolting or by special fabrication. Except where pretensioned bolts are used, bolted connections reduce the resistance of net-sections. Some authorities prefer welding for large spans, even if it is difficult to guarantee the quality of welds in site. One of the determining factors in the choice of nodes is the number of members to be assembled. Apart from structural influences on the node itself, this problem is linked to the way the members are connected to the node and to considerations of space and ease of installation. The regularity of geometry resulting from the node determines the entire geometry of the structure. Significant progress has been made in this area through computerisation linking design and fabrication. As a result, it is possible to fabricate nodes by varying the angles of incidence of the members. Five 'kinds' of nodes may be identified, Figure 26: • • • • • plate nodes (Figure 26a) folded nodes (Figure 26b) cast nodes (Figure 26c) nodes of extruded aluminium section (Figure 26d) special connections with spherical nodes (Figure 26e) Plated and folded nodes are usually connected to the member ends by means of bolted connections, but also welding can be done. The node is a critical element when evaluating the cost of spatial structures: one node for 2,5-3,0m2 would seem to be an economical solution. Connections The node-to-member joining system determines how the ends of the members must be treated. Five processes may be described, for example, (Figure 27): • • • • • Straight cutting (Figure 27b) Profiled cutting (Figure 27c) Squashed and drilled (Figure 27d) Addition of a connection plate (Figure 27e) Special fixing: threading, welding, or bolt crushing (Figure 27a) It is clear that not all these systems give full strength joints. 4.3 Methods of Fabrication and Erection Methods used may be listed in three categories: a. Erection of separate members, each one lifted into position and connected to the work already assembled. b. Erection of sub-assemblies: this is an intermediate stage whereby the members are connected in sub-assemblies, either in the factory or on site. The sub-assemblies are lifted into final position and connected to the work already assembled. c. Lifting of the whole space structure, which is assembled on the ground on site. Various methods may be considered ranging from the use of vertical construction parts as lifting masts to cranes. The choice of one of these three methods depends on: • • • the nature of the project in terms of type of structure and size. operational conditions: actual layout of the site, available means of lifting, transport costs, experience, etc. safety. In a) and b) it is essential to predict the need for any temporary supports that may be necessary where the structure achieves stability only when it is complete. The many phases in erection should be carefully examined so as to avoid intermediate structural behaviour which is less favourable than that for the final state of the structure. Lifting the whole space frame has the following advantages: • • • the greater part of the work is carried out on the ground, thus aiding control of the operation, especially the making of welded joints. the use of heavy hoisting machinery is required for a shorter period, which may reduce final costs. in some cases, the structure with other equipment attached may be lifted together. Hoisting is a critical stage in erection. Lifting points should be carefully examined. Lifting should be carried out in the best meteorological conditions and certainly not in wind. Once in position, the structure should be connected to the work already erected. Precise regulating devices should be planned in advance in order to facilitate connection and fixing. The lifting stage can be a determining factor in the design of the structure. A new approach has been used in Barcelona for the erection of the dome of the Olympic Palace. It involves the fabrication of the dome on the ground in five portions which are temporarily pinned to each other (Figure 28). The central portion is then lifted and the remaining segments of the dome locked in the final position. Different methods of execution may be considered depending on the type of structure and place of installation. For example, it is possible to carry out assembly using a launching method borrowed from bridge construction, etc. There is no limit to the list of solutions. The erection method chosen depends on the imagination and know-how of the designer within a particular context. 5. CONCLUDING SUMMARY • • The lecture has been concerned with space roof structures in which members are subject to internal axial forces. The structures may take the form of: ⋅ two-dimensional grids. ⋅ cylindrical vaults. ⋅ domes. • • Unsymmetrical patterns of loading must be considered, including combinations of actions that may arise during erection. The displacement method of analysis, implemented by computer software, is the most convenient approach to determine internal forces, displacements of nodes and support reactions. • Repetition of nodes and members permits the use of standard components and reduces the costs associated with design, detailing and fabrication. 6. REFERENCES [1] Makowski, Z.S.: "Structures Spatiales en Acier", Centre Belge-Luxembourgeois d'Information de l'Acier, 1964. [2] Eurocode 3: "Design of Steel Structures": ENV 1993-1-1: Part 1.1: General rules and rules for buildings, CEN, 1992. 7. ADDITIONAL READING 1. 2. 3. 4. 5. 6. 7. Mainstone, R. "Developments in Structural Form", MIT Press 1975. Makowski, Z.S. "Space Frames and Trusses" from Constructional Steel Design, Elsevier Applied Science, 1992. Makowski, Z.S. "Analysis, Design and Construction of Braced Domes". Granada, 1984. Fuller, R.B., Marks, R. "The Dymaxion World of R.B Fuller". Anchor Books 1973. Motro, R. "Optimisation de Structures Spatiales et Application à des Grilles à Double Nappe". Revue du Centre Technique Industriel de la Construction Métallique. No. 2, Juin 1976, pp. 24-36. Livesley, R.K. "Matrix methods of structural analysis". Pergamon Press, 1964. Tsuboi, Y. "Analysis, Design and Realisation of Space Frames, a state-of-art report". Bulletin de l'IASS n 84/85 Avril, Août 1984, Volume XXV-1/2. Lecture 14.6: Special Single Storey Structures OBJECTIVE/SCOPE To describe less known forms of single storey steel structures, including cable and tension structures. PREREQUISITES None RELATED LECTURES Lecture 14.5: Space Structure Systems Lecture 15C.4: Guyed Masts SUMMARY The lecture examines some less-common structural systems, not described earlier. Variants on standard forms used for industrial buildings are briefly described. Cable-stayed and tension structures are described in more detail. 1. INTRODUCTION 1.1 General Most single storey structures are portal or trussed frames, Figure 1. These structural forms have been shown to provide economic and effective solutions for routine construction. However, alongside these commonplace forms a wide range of single storey structures have been developed for special applications. Although the number of special structures is relatively small, they encompass a wide range of structural forms. Many of these structures result from the need to cover large areas, typically for sports, exhibition, industrial or commercial purposes. As such, they are characterised by large uninterrupted spans and relatively light imposed loading. They are frequently lightweight, reflecting their designers' intention to maximise structural efficiency. Structural considerations are, therefore, to the fore in determining their architectural form. The lightness of these structures requires close attention to be paid to aspects of structural action and response that are not significant for more conventional structures. Attention has to be given to dynamic and fatigue effects, to load reversal and to uplift. In this latter regard, elements primarily designed for tension may need to be braced to resist compressive forces, and elements may require pretensioning to ensure that stiffness is maintained under reversible or dynamic loading. Members and connections may require detailing for fatigue. Temperature effects and second order effects relating to cable extension or relaxation may also need to be considered. It is convenient to categorise these forms in terms of their structural systems. These include direct-force systems and mixed systems combining flexural elements with major direct-force components, in addition to systems which exploit the potential of curved or folded forms. In some cases, the foundations play a more complex role than usual in equilibrating the force system and in limiting deformations. It is proposed in this lecture to outline some of the more common systems encountered. Additional material on some of these topics will be found in Lecture 14.5 on Space Frames, and Lecture 15C.4 on Guyed Masts and Towers. The slides on special single-storey structures provide a wide range of examples of the application of the principles outlined in this Lecture. The Lecture commenced with a brief review of some non standard approaches which have been used for industrial buildings in the past - the saw tooth roof, the umbrella and butterfly roof and the arch roof. The dome is also reviewed. 1.2 Safety With large-span structures, the consequences of failure are more likely to be catastrophic than is the case for normal structures and safety therefore merits particularly careful consideration at all stages of the life of the structure. The more severe consequences of failure can be attributed partly to the large scale and partly to the lack of structural redundancy encountered in many of the systems adopted for large-span structures. Increased redundancy offers the prospect of greater stiffness and strength, and the alternative load paths associated with redundancy offer a reduced likelihood of total collapse. Against this must be balanced the capacity of statically determinate structures to accommodate secondary effects (shrinkage, creep, relaxation, temperature changes, settlement, lack of fit, etc.) without distress. Special structures require critical assessment of the applicability of standard loading codes, particularly in regard to wind. Construction techniques, materials and detailing may be non-conventional and introduce additional uncertainties. Non-linear effects - relating to material or geometrical factors - may have a bearing on the behaviour and detailing of the structure. Inspection during construction, and periodic inspection and maintenance of the structure in service, assume added importance. 2. AN OUTLINE OF OLDER TYPES OF SPECIAL SINGLE STOREY STRUCTURES Some of the more interesting types of older, single-storey steel structures are briefly described below. These types have been used extensively in the past, and although their use today is rather limited because of the development of other systems, they still have a significant interest for the designer. In certain cases they can still provide satisfactory solutions from the viewpoints of functionality, economy and aesthetics. 2.1 The Saw-tooth Roof This structural system (Figure 2) was frequently used in the past, mainly for industrial buildings. Its use is now rather limited because its main advantage of uniform daylighting into the building, is achieved by new roofing products that can provide daylighting efficiently through flat roofs, or by artificial lighting. The saw-tooth roof is costly to construct, wasteful of heat, and requires many internal gutters. 2.2 The Umbrella and Butterfly Roofs These two structural systems (Figure 3) are conceptually very similar to the saw-tooth roof, and they are used to cover large floor areas with a minimum of internal columns. It is still common to use this construction in portal frames with, for example, alternate valley columns omitted (Figure 4). 2.3 Arched Roofs Arches have long been recognized as being highly efficient and economic structural systems for covering large spans in buildings (Figure 5). The main problem of an arch is the thrust developed at its supports. The thrust can be of considerable magnitude for long spans. It increases as the ratio of rise to span becomes smaller and can be resisted only by appropriate foundations, or by an arch tie. Portal frames benefit from arching action, which has led to the use of tied frames (Figure 6). Care has to be taken to design the rafter against the considerable compressive force that arises. Also the tie member must not buckle if wind uplift causes reversal of force in this member. 2.4 Prestressed Frames Prestressing provides an option, if uncommon, for creating a more favourable regime of bending moments in large span frames, as well as for deflection control (Figure 7). The additional compressive forces must obviously be taken into consideration in determining the bracing requirements of members. 2.5 Domes Domes are three-dimensional structures used to cover buildings with a circular floor plan. Their cross-section may be spherical, elliptical, parabolic, etc. Steel framed domes have been extensively used for public assembly buildings such as auditoria, arenas, gymnasia, exhibition halls, etc. There are many types of dome (Figure 8). A common type is the ribbed spherical dome which consists of main meridional ribs and horizontal (parallel) rings. The rings decrease in diameter going from the base to the top of the dome, concluding with the compression ring at the top. This arrangement, whereby the ribs are stopped at the compression ring instead of being continuous over the pole, is used practically in all dome designs. It has three advantages. Firstly, it makes possible the connection of all ribs converging to the same point. Secondly, a large opening is formed which can be used for lighting and ventilation. Thirdly, the structure becomes statically determinate. Apart from the compression ring at the top, the primary system of ribs and rings, may be subjected to further modifications. For example, auxiliary members, such as purlins and diagonal bracings may be added in the plane of the roof surface. The bracings are usually placed in alternate bays. They are an important aid during erection, which is usually done by using only one scaffold tower at the centre of the dome. Another modification concerns the connections between the ribs and rings. Theoretically they should be pinned but in practice they are moment resisting. Finally, the geometry of the dome is usually modified, so that only the panel points lie on the true spherical surface. All members between these points are made straight, resulting in considerable fabrication savings. This type of dome is essentially a direct force structure, its individual members being subjected primarily to axial tension or compression (membrane action). For symmetrical loading, the global analysis can be made by simple statics (Figure 9). However this method is invalid for analysis under unsymmetrical loading. The most important unsymmetrical loading is wind pressure. Economy depends primarily upon the rise-to-span ratio of the dome, and the number of ribs and rings to be used. A rise-to-span ratio around 0,13, i.e. when the spherical radius is equal to the dome diameter, provides the maximum structural economy according to some authors. 3. CABLE AND TENSION STRUCTURES These structural systems play an interesting role in modern construction and are examined in detail below. 3.1 General High strength steel cables have been used extensively over the past twenty five years for space roof structures. There are two different possibilities when using steel cables in roof structures. The first possibility, consists of using the cables only for suspension of the main roof structure, which can be either conventional, e.g. beams, cantilevers, etc., or a space frame. In this case, the main roof structure, instead of being supported, is actually suspended from steel cables above the roof, which transmit the tensile forces to appropriate anchorages (Figure 10). They are cable-stayed roofs. There are many examples of this type of construction used as industrial buildings where the roof structure, either as a single or as a double cantilever, is suspended from cables, which in turn are anchored on robust pylons above the roof level. In this type of construction, the cables behave as simple suspension elements, while the roof structure itself behaves like a normal load resisting unit, subject to moments, shears, and other kinds of action effect. It is expected that the suspending elements remain in tension, even under wind uplift, due to the dead weight of the roof. The second possibility is represented by those roof structures where the steel cables are effective members of the roof structure itself, and not just conveyors of forces from the structure to the anchorages. In this type of construction (tension structures), the cables themselves resist the various external loads. Their particular behaviour has deeply influenced the structural forms used and has imposed new methods of execution. Tension structures may be categorised as: (a) Single-layer cable systems (Figure 11a) (b) Double-layer prestressed cable truss systems (Figure 11b) (c) Prestressed tensile membrane systems (Figure 12) Tension structures are used to cover stadia, arenas, swimming pools, recreation halls and other buildings where a large area for public assembly and exceptional aesthetic effect are required simultaneously. There are some particular problems associated with these cable-stayed and tension roof structures. A first problem derives from the fact that the cable is flexible. It assumes a shape compatible with the applied loads whilst architectural and building requirements demand that the structure has a definite form. Any deviations from that form due to the action of the applied loads, must be kept to a minimum. To meet this requirement, a pretension must be introduced into the structure, which must be compatible with the desired shape, and when combined with the applied loads, must maintain the deformation between specified limits. Design may therefore involve use of mathematical 'form-finding' procedures, implemented by appropriate software. Another feature of these structures is their geometrically non-linear behaviour. Deformations play an essential role in the analysis and the principle of superposition of effects is not valid. Finally, an important problem associated with these structures is their sensitivity to aerodynamic instability, e.g. flutter. This sensitivity imposes special requirements on the design and the constructional details of these systems, particularly those which use membranes made of lightweight fabric as cladding. The requirements of stiffness under transverse loading and anchorage are major form determinants for cable structures, and these are examined in the following sections. 3.2 Stiffness Under Transverse Loading Single cable structures are characterised by their flexibility, Figure 13. They require stiffening to prevent a change of shape with each variation in load and to make them capable of resisting uplift due to wind, Figure 14. Gusty winds can produce oscillations, unless damping is provided to the structure. The principal methods of providing stability are the following: (i) Additional permanent load supported on, or suspended from, the roof, sufficient to neutralise the effects of asymmetrical variable actions or uplift Figure 14a). This arrangement has the drawback that it eliminates the lightweight nature of the structure, adding significant cost to the entire structure. (ii) Rigid members acting as beams, where permanent load may not be adequate to counteract uplift forces completely, but where there is sufficient flexural rigidity to deal with the nett uplift forces, whilst availing of cables to help resist effects of gravity loading (Figure 14b). (iii) Rigid surfaces behaving as inverted shells or vaults, where uplift forces are countered by the in-plane compressive rigidity of the structure (Figure 14c). (iv) Secondary cables prestressing the main cables so that these remain in tension under all conditions of load. Such prestressing can take a variety of forms: ⋅ a stayed (guyed) arrangement, wherein the main cable is stayed to other elements or to the ground, as in the case of guyed trusses (Figure 14d). ⋅ A planar arrangement of suspension and stabilising cables, with opposite curvatures cables, Figure 14e. This structure reacts elastically to all changes of shape provoked by the externally applied loads. This principle can be extended to permit creation of space trusses, or structures of revolution. ⋅ An orthogonal or diagonal arrangement of suspension and stabilising cables, with opposite curvatures, forming an anticlastic (saddle-shaped) surface, Figure 14f and 15. Figures 14 and 15 show the application of these general principles to cable and cable-stayed systems, whilst Figure 16 details the structural actions of prestressed cable truss systems. Accurately defined, a cable truss system has a triangulated structural form which increases stiffness, particularly under non-symmetric loading. However, the term is also frequently applied to the cables with opposite curvature shown in Figure 14e. The orthogonal or diagonal arrangement of anticlastic cables shown in Figure 15 can also be extended to the conical form shown in Figure 17. The increasing use of horizontal ring cables, from Figure 17a to 17c enhances stiffness against asymmetric loading. Because of the difficulty of anchoring a large number of cables at a point, the top is usually flattened as shown in Figure 17d. The use of anticlastic cable nets is further enhanced by the use of internal arches, Figure 18. The use of conical forms can be extended to create exciting doubly curved surfaces by the use of multiple high points and/or interior anchorages, Figure 19. The Pavilion of the Federal Republic of Germany, Expo 1967, Montreal by Frei Alto and Rolf Gutbred was an outstanding example of the former. 3.3 Anchorage Cable stayed structures generate a requirement for the anchoring of tension forces. Some of the commoner solutions are: (i) Vertical and horizontal reactions provided by axially loaded elements - stayed columns used with ground anchors (Figure 20a). (ii) Vertical and horizontal reactions provided by flexural elements i.e. cantilever columns (Figure 20b) or legged columns (Figure 20c). (iii) Vertical columns acting with horizontally loaded edge beams which transfer horizontal reactions to rigid diaphragms (Figure 20d). (iv) Inclined walls, or vertical cylindrically curved walls (Figure 21a). (v) Form-related boundary shapes, creating, in some cases, a closed self-equilibrating system of tension and compressive forces and requiring no tension ground anchors (Figure 21b). The magnitude of forces in stayed columns and in diagonal stay restraining cables is reduced by inclining the columns. In some symmetrical structures lateral thrust is balanced by means of struts at foundation level. Some tension anchorage possibilities are illustrated in Figure 22. 4. ADDITIONAL SPECIAL STRUCTURE CATEGORIES 4.1 Hangars Cantilever construction is used extensively for large hangars, the column free interior and facade permitting the required ease of access and flexibility of use. Sliding doors around the perimeter are vertically supported on rollers at ground level and laterally supported by the roof structure. The deflection of the cantilever roof structure must be allowed for on the design of the doors. Single or double cantilever systems may be used. In the single cantilever system, substantial foundations must be provided to counteract the overturning moment. With symmetrical double cantilever systems, the permanent loads on either side of the central block balance each other. The central block which frequently contains offices and circulation areas, may be used to counteract the effects of unsymmetrical loading. In regard to the composition of the cantilever, roof structures may be categorised as follows: (a) Pure cantilever systems, formed of varying-depth trusses, girders, folded plates and shells (Figure 23a). (b) Cable supported structures, supporting any of the above structures types (Figure 23b). In addition to gravity loading and temperature effects, the roof structure is exposed to wind loading on its upper and lower surfaces. In the case of cable-stayed structures, particular attention must be paid to uplift. 5. CONCLUDING SUMMARY • • • Standard industrial forms can be varied to provide structures capable of spanning considerable distances. Curved forms, arches or domes, provide further possibilities. Cable staying extends the spanning possibilities of conventional trusses or truss frames. • • Tensile structures open up a large repertoire of dynamic structural possibilities for medium to large-span structures. Tensile structures may be planar or anticlastic, membrane or cable net structures. 6. ADDITIONAL READING 1. Duncan, I., "Other Structural Applications of Steel", Chapter 5 - Steel Designers Manual, 5th ed., Blackwell Scientific Publications, Oxford, 1992. 2. Schlaich, J., "Cable and Membrane Structures for Buildings", Paper presented at Conference on Tension Structures, IStructE, London, 1988. 3. Bergermann, R., "Cable Membrane Roof for the Arena in Zaragoza, Spain", Structural Engineering International, Vol 2 No 4, pp 238-241, IABSE, Zurich, 1992. 4. Krijgsman, A., "Design of Economical Large Spans": The Heerenveen Skating Rink and the Arnhem Burgers Bush, Delft University of Technology, Netherlands. 5. Bascialla, E., "New Roof for the G Meazze Football Stadium, Italy, Construzioni Metalliche, February 1991, pp 8-21. 6. Morley, S., "A Stadium for the Nineties, Steel Construction Today, Vol 5, No. 4, July 1991. 7. Lau, J. M., "Design and Construction of a Cable-Stayed Steel Roof Structure for Yishin Indoor Stadium and Sports Complex", Paper presented at International Conference on Steel and Aluminium Structures, ICSAS, Singapore, May 1991. 8. Finzi, L., "Football Stadiums in Italy", IABSE Structures, Zurich, 1990. 9. "Olympics, Barcelona 1992: Barcelona - A City Regenerated", The Architectural Review Feature, August 1992, London. 10. Lan, T. T., "Space Structures for Sports Buildings", Proceedings of the International Colloquium, Beijing, China, October 1987, London, Elsevier Applied Science. Lecture 14.7: Anatomy of Multi-Storey Buildings OBJECTIVE/SCOPE To describe the different functions accommodated by multi-storey buildings and to present the various building elements. PRE-REQUISITES Lectures 1B.7: Introduction to Design of Multi-Storey Buildings Lecture 3.5: Fabrication/Erection of Buildings Lecture 4A.3: Practical Corrosion Protection for Buildings Lecture 4B.4: Practical Ways of Achieving Fire Resistance of Steel Structures RELATED LECTURES Lecture 14.9: Methods of Analysis for Multi-Storey Frames Lecture 14.10: Simple Braced Non-Sway Multi-Storey Buildings Lecture 14.11: Influence of Connections on Behaviour of Frames Lecture 14.12: Simplified Method for Low-Rise Frames Lecture 14.13: Design of Multi-Storey Frames with Partial Strength and Semi-Rigid Connections Lectures 14.15: Tall Building Design SUMMARY This lecture describes the anatomy of multi-storey buildings. The range of building types in this category is described. The anatomy of a typical building is described by considering the individual elements of structure, finishes and services. 1. INTRODUCTION Multi-storey steel-framed buildings can accommodate a wide range of functions and architectural treatment. The term multi-storey refers to structures with more than one storey and covers building used for many different purposes including: • • • Apartments Office developments Shopping centres • • • Car parks Schools and universities Hospitals Although the basic anatomy of each building is similar, they may have different requirements for column grid, services, and internal/external finishes. For example, a car park may be designed with floors of moderate spans and will have minimal requirements for cladding, finishes and services, whereas a prestige office development may need large column free areas with air conditioning and under-floor cabling for computers. The structure will generally be more economic if large-spans are avoided, hence providing a shorter path between the point of application of loads and the ground. The speed and economy of construction can also be increased by the large degree of vertical and/or horizontal repetition common in the structural systems of multi-storey buildings. The individual contributions of major components to the overall building cost can vary significantly with building function, size and architectural treatment. However they are generally within the indicative ranges given below: Foundations Steel Skeleton Floor Structure Cladding/Finishes Services 2. PRIMARY STRUCTURE The structural frame is provided to transmit vertical and horizontal loads from their point of application to the foundations by the most efficient path with the minimum impact on the economy and function of the other elements of the building. 2.1 Vertical Load-Bearing Elements Figure 1 illustrates the principal structural elements of a typical multi-storey building. 5% to 10% 10% to 20% 5% to 10% 15% to 40% 15% to 40% 2.1.1 Floors The floor slab usually spans one way and is, either simply supported or continuous. It is supported by 'secondary' steel beams, typically at 2,5m to 3,5m centres. Several different types of slab can be used, most of which can be designed to act compositely with the supporting beams if adequate shear connection is provided. The system illustrated in Figure 2 is commonly used, where a concrete topping (lightweight or dense) is cast in-situ on profiled steel decking acting as permanent formwork and reinforcement to the concrete. Steel bars are included in the slab to prevent cracking and to provide reinforcement in the event of degradation of the decking in a fire. This form of slab construction is particularly popular for multi-storey buildings where rapid construction is required. The demands on cranage are low as many sheets of steel decking may be lifted at a time and the concrete topping may be placed by pump. For spans up to about 3,5m temporary propping may not be required. The steel decking is available in a number of different profiles, many of which include systems for supporting services below. The overall weight of this system is low leading to possible economies in the supporting frame and foundations, particularly if lightweight concrete topping is used. It is however relatively expensive and not suitable for situations where a ceiling is not required. Other types of floor slab construction can offer advantages in certain circumstances where, for example, speed of construction and cranage are not problems, or a larger span is required. In-situ concrete, cast on temporary formwork, either reinforced or pre-tensioned is suitable for larger two-way spans and where a smooth soffit finish is needed. Precast concrete units also provide a smooth soffit and some of the different types available are illustrated in Figure 3. These systems require more cranage and on-site storage space than profiled steel sheet, but they can be used for larger spans. Timber floors are not common in steel buildings. A new floor system, so-called 'Slim Floor', has been developed during the 1980's and early 1990's. It is used for long span slabs, allowing for the elimination of secondary beams (Figure 4). The primary beam has a typical built-up cross-section, which is designed in order to directly support the floor slab on the bottom flange (see Figure 5a). According to the different systems produced in Europe, distribution can be made between open and closed sections (Figure 5b); in particular types 1, 2, 3, 4 are open, single-symmetric I beams, types 5, 6, 7 are closed top-hat beams, type 8 is open top-hat beams. Steel beams are integrated with in-situ concrete, providing composite action. Typical dimensions for 'Slim Floor' are the following: Floor (Beam) Depth Beam Span 200 mm 260 mm 5m 7m Slab Span 7m 9m 2.1.2 Structural frame The floor slab is normally supported by 'secondary' steel beams, spanning between 'primary' beams which are in turn supported by the columns. Bays of the frame are normally rectangular with the secondary beams spanning the greater dimension. For long-span slab systems the secondary beams are sometimes omitted, as for example in the 'Slim Floor' system illustrated in Figure 4. Shear studs are welded to the top flange of both primary and secondary beams to provide composite action between the beams and the concrete slab. The slab then acts as the compression flange to the beams for loading applied after the slab has matured. When profiled steel decking is used, the studs are often welded through the decking after it has been placed. Column spacing depends on building function, but is usually between 5m and 10m. Closer centred columns may be used in an external 'tube' stability system for a tall building as described later (see Section 2.2.3). The simplest and hence normally most economic system, is for both primary and secondary beams to be rolled Isections in the same horizontal plane, designed as simply-supported, and with simple bolted connections between them and to the columns. The steel skeleton must be protected against fire. Typical solutions of protection are shown in Figure 6. Columns filled with cast concrete can be designed for composite action (Figure 6a). Beams can be protected in different ways (Figure 6b): by sprayed vermiculite, by concrete encasement, by filled concrete or by box-shaped cladding. In most buildings, the need to accommodate services has a major effect on the design of the floor system. In an office building, the office floor areas may have air-conditioning ducting, pipes for fire sprinklers, and cabling for electrics, telephones and computers distributed horizontally above and/or below the floor from risers in service cores. The major services are normally under-floor in zones up to 500mm deep, with electric and communication services above within raised floor zones up to 200mm deep. The cores may also house toilets, lifts and stairs, with their requirements for water, sewerage and ventilation. In most cases it is possible to provide separate service zones below and/or above the floor structure. Buildings with large spans and/or restrictions on storey height will require a different design approach. Holes may have to be cut in webs of the beams to accommodate services, a costly operation, or one of the systems illustrated in Figure 7 may be adopted: • • • • A dual-plane grillage system, where the secondary beams run over the primaries, allows two-way service distribution with the structural zone. This system provides continuous construction in both directions if twin primaries are supported by brackets on each side of the columns. Composite action can normally only be between the secondary beams and the slab, but if 'stub-girders' are welded to the top flange of the primaries, it is provided in both directions. Services may be accommodated near the supports whilst maintaining mid-span bending resistance for simply-supported primary or secondary beams if they are notched at their ends, or tapered in depth. As tapered beams are fabricated from plate, they can have smaller top flanges for composite action, and can also be tapered in plan. Haunched beams can provide a service zone at mid-span and may be appropriate in situations where sway frames are used. Trusses and castellated beams naturally allow the passage of services, larger services being accommodated in trusses if the bracing is omitted at mid-span. It may not be possible to maintain uniform column spacing at all levels of the building. For example, column-free space may be required at ground floor for a conference area. 'Transfer' beams or trusses can be provided to transmit the loads from above to adjacent columns. They may alternatively be at a higher level, with floors below suspended from hangers (Figure 8). The columns are supported at their base by foundations. Several different foundation types can be used to support multi-storey buildings, the selection for a particular case being dependent on column load, soil resistance and settlement limitations. The most common types of foundations, illustrated in Figure 9, are: • • • 'Pad Footings', where an individual base of mass or reinforced concrete is provided under each column is the simplest option, where the supporting ground is good. For greater loads, or poor ground, the individual pads may be connected to form a continuous 'Raft'. This system may also provide improved resistance to water. Alternatively, where ground conditions are poor the load carrying capacity of the individual pads or raft may be increased by installing piles to create individual 'Pile Caps' or a 'Piled Raft'. 2.2 Horizontal Load-Bearing Elements Although the primary function of the structure is to support vertical loads, it must incorporate a lateral stability system to resist horizontal forces, which commonly are wind loads and, in some countries, earthquakes. Some typical systems are illustrated in Figures 10 and 11, where steel bracings and reinforced concrete cores are used as stabilizing elements, respectively. 2.2.1 Braced systems Staircases and lifts (provided for access and escape), toilets, plant rooms and risers for heating, air-conditioning, electrical and public health services, each require penetrations through, and support from, the floor structure of the building. They are usually located together to form one or more 'service cores'. The resulting walls, required for compartmentation and fire separation, may be used to as the lateral stability elements of the structure. Possible forms of construction include: • • • • Conventional in-situ reinforced concrete shear-walls or cores Slip-formed or jump-formed reinforced concrete shear walls or cores Braced steel frame Stiffened steel plate walls. Concrete walls can lead to longer construction periods, unless systems such as slip-forming are adopted which allow the core construction to proceed in advance of the erection of the steel frame. Steel to concrete connections can lead to problems if cast-in fixings are out of tolerance. Braced steel frames which are erected at the same time as the other elements of the frame, are most commonly used. Bracing geometry is normally concentric (see Figure 12), allowing the use of simple connections. Eccentric bracing requiring bending resistance in the sections and joints may sometimes be used, however, to allow for increased openings. They can also provide major ductility, when required for seismic resistant structures. All stability systems use the floor plate as a diaphragm to transfer lateral loads from their point of application to the bracing elements, as illustrated in Figure 13. The designer should ensure that the floor is capable of performing this function. The bracing system must ensure lateral stability in two main directions and also torsional stability. The correct location of such elements is a fundamental pre-requisite in the bracing system design (see Figure 14). For long floors without expansion joints, the designer should avoid longitudinal walls at both ends of the building, as they will restrict thermal movements, and hence attract large forces. Bracing systems provide the most economical solution for medium rise multi-storey buildings (see Section 2.2.2). 2.2.2 Frame systems Alternatively, stability can be provided by moment resisting frames formed by some or all of the columns and intersecting floor beams, as illustrated in Figure 15. These systems, which may be designed as either 'Rigid' or 'Semi-rigid' frames, which can sometimes offer advantages in terms of increased flexibility of internal layout, are described in detail in Lectures 14.13 and 14.14. 2.2.3 Tall buildings Some examples of stability systems used for tall buildings are illustrated in Figure 16. A combination of core and frame systems is common. Advantage can be taken of closely-spaced external mullions framed by relatively deep spandrels between windows acting together to form a 'Perforated Tube'. 'Outrigger Frames', where the central core is connected at one or more levels to the external columns with deep trusses may also be used. The diagram of Figure 16 illustrates how the bending moment in the core is reduced when this system is adopted. Further discussion on tall building design is included in Lecture 14.15. 3. SECONDARY ELEMENTS AND FINISHES Figure 17 shows a section through a typical floor deck. The surface treatment applied to the concrete slab will depend on the flooring system required. Power floating provides a smooth finish ready for carpets or tiles. A trowelled finish is suitable if a raised floor is to be installed to accommodate electrical and computer services. The surface is tamped to provide a key if a concrete screed is to be applied. A false ceiling can conceal air-conditioning duct-work and other services suspended from the floor. Staircase flights and half landings are normally prefabricated in steel or precast concrete for economy and ease of construction, but they may also be constructed of reinforced concrete. Toilet modules may also be prefabricated and supplied to site in an enclosure complete with all fixtures, fittings, services and finishes. A variety of external finishes may be used. Common facade treatments which may be supported by the steel frame include: • • • • Masonry panels. Precast concrete panels. Curtain walling. Prefabricated steel/aluminium cladding panels. The chosen cladding system must perform a number of different functions for each building type. For a car park it must be robust and provide natural ventilation, whereas, in an office building, it must be water-tight and provide adequate insulation and natural light without excessive solar gain. For tall buildings it must be able to be installed and maintained without the need for scaffolding. In all cases, it must be capable of accommodating the movements of the building frame and of satisfying the aesthetic requirements of the architecture. Roofs may be flat or sloping, and clad with a variety of materials to provide insulation and waterproofing. 4. PERFORMANCE REQUIREMENTS All elements of the structure should be designed by considering the ultimate and serviceability limit states at which they would become unfit for their intended use. Limit states are considered in Lecture 14.9. 5. CONCLUDING SUMMARY • This lecture has described the anatomy of typical multi-storey steel-framed buildings. The most suitable system for a particular building is that which satisfies the performance requirements with the minimum adverse effects on: ⋅ Cost of construction ⋅ Speed of construction ⋅ Interference with services ⋅ Maintenance costs ⋅ Architecture. 6. REFERENCES [1] Hart, F., Henn, W. and Sontag, H.: "Multi-Storey Building in Steel" (second edition), Collins, London, 1982. 7. ADDITIONAL READING 1. 2. Ballio, G. and Mazzolani, F.M.: "Theory and Design of Steel Structures", Chapman & Hall, London, 1983. Iyengar, S. H., Baker, W. F. and Sinn, R.: "Multi-Storey Buildings", from Constructional Steel Design, Elsevier, London, 1992. Lecture 14.8: Classification of Multi-Storey Frames OBJECTIVE/SCOPE: To provide definitions that make it possible to point out the essential characteristics of a framed structure. Attention is given to joint behaviour and to the problem of the choice of structural model depending on the loads acting on the structure. PREREQUISITES Lecture 7.11: Frames Lecture 11.6: Moment Connections for Continuous Framing Lecture 11.7: Partial Strength Connections for Semi-Continuous Framing RELATED LECTURES Lecture 14.9: Methods of Analysis for Multi-Storey Frames Lecture 14.10: Simple Braced Non-Sway Multi-Storey Buildings Lecture 14.13: Design of Multi-Storey Frames with Partial Strength and Semi-Rigid Connections Lecture 14.14: Methods of Analysis of Rigid Jointed Frames SUMMARY The following subjects are discussed: • • • • Bracing systems Framed systems Braced and unbraced frames Sway and non-sway frames. In particular the differences between braced and unbraced frames are analysed as well as the differences between sway and non-sway frames. The different behaviour of connections of structural members (girders, columns and bracings) is considered, in order to define the behaviour of rigid and semi-rigid frames. 1. INTRODUCTION Before discussing rigid jointed frames, some definitions are introduced since the same meaning is not always associated with the same words in different countries. It is necessary sometimes also to define the structure in a particular manner in order to use conventional simplified analytical models in the analysis and design of the structure. The evolution of computer methods and hardware in fact allows any type of analytical evaluation, e.g. elastic and inelastic analysis including any type of inelastic model and of imperfection, to be undertaken. It might be said therefore that it is not necessary to go through definition of systems and to simplify models of analysis. For a braced frame, for example, it is not necessary to separate frame and bracing behaviour since both can be analysed with one computer model. On the other hand, simple models are useful for preliminary design and for checking computer results in the design office. The definitions below describe what is meant by a bracing system, what a framed system represents and when a framed system can be considered to be braced by another system. Sway and non-sway frames are defined. An explanation of why a braced frame is often considered equivalent to a non-sway frame is given. In all these cases the following steps are followed: • • • first the common practice and what is commonly intended with that definition is described. secondly an engineering definition which attempts to provide a quantitative measure for that definition is given finally the definition provided by Eurocode 3 [1] is explained. 2. BRACING SYSTEMS 2.1 Introduction In common design practice and in design guides and manuals, bracing systems are very often identified with triangulated trusses or with concrete cores or shear walls which are present in buildings to accommodate shafts and staircases. It is very common to find bracing systems represented as shown in Figure 1. This assumption is based on engineering common sense which tends to represent reality in an average sense without referring to more general and mathematical definitions which would include all the possible cases. The simplification of representing a bracing system by a triangulated truss also arises because in steel structures, in contrast to concrete structures where all the joints are naturally continuous, the most immediate way of making connections between members is to hinge one member to the other. As a result structures are created which need bracing systems in order to prevent failure mechanisms forming. Based on this simplifying consideration, all the joints of Figure 1 can be assumed to be hinged. Therefore bracing can only be obtained by use of triangulated trusses or concrete cores or, exceptionally, by a very strong frame. There are several reasons for having hinged connections in the steel system: a. from the point of view of ease of fabrication and erection, it is more convenient simply to join the webs of the members without connecting the flanges. b. it is simpler to use bolted connections which do not need the deformations in the connections to be minimised. c. from the design point of view there is a dramatic simplification in the calculations if the resisting systems can be separated into systems resisting vertical actions and systems resisting horizontal actions. In addition, if all the girders are hinged to the columns, the sizing of the simply supported girders and the columns is a simple task. d. from the economic point of view, it is more convenient to reduce the horizontal drift by means of bracing systems added to the hinged construction than to use framed systems with rigid jointed connections. This consideration is even more important for steel structures with reinforced concrete cores where the core itself can act as the bracing system. In conclusion, a topological definition of a bracing system equates a bracing system to a triangulated truss or to a shear wall. This definition covers the majority of actual cases but is not sufficient to clarify the function of a bracing system. For this purpose a definition based on the requirements of a bracing system is provided below. 2.2 Engineering Definition A bracing system can be defined as a structural system capable of resisting horizontal actions and limiting horizontal deformations. On the basis of this definition, all the systems shown in Figure 2 can be considered bracing systems. Within one building more than one of these systems can be present. In that case some systems are more effective than others in resisting horizontal loads, the others are neglected. The definition allows a simple frame, and even a column, to be considered as a bracing system. The column or the frame may not have enough strength or stiffness to resist the horizontal actions with reasonable sizing of its members (columns and girders) and then to satisfy the strength and serviceability checks which require limited interstorey and global drifts. In this case it is necessary to add other bracing systems to the frame itself. 2.3 Eurocode Definition When the words "bracing system" are used in the Eurocode, they refer to a system for preventing lateral instability of beams or compression members and a triangulated truss is shown as an example, see Fig. 5.2.5 of Eurocode 3 [1]. According to 5.2.5.1 bracing systems may be triangulated frames, rigid-jointed frames or shear walls/cores, see Figure 2. 3. FRAMED SYSTEMS 3.1 Introduction In common practice, (continuous) framed systems are considered to be assemblies of beams and columns in which all the joints are completely continuous. In contrast truss or simple framed systems are those in which all the joints can be considered hinged. A purely rigid jointed frame system is characterised by joints between frame members and by no additional bracing system. The frame itself has to resist all the actions, vertical as well as horizontal. At the same time, it has to provide the required stiffness to the structure in order to limit deformations within the allowable values. Even though the detailing of all the connections results in a less economic structure, framed systems have some benefits: a. the connections are more ductile and therefore the structure performs better in earthquakes. b. from the architectural and functional points of view, it can be advantageous not to have any triangulated truss in the structure. Actual structures do not always fall into the two categories defined above. Connections themselves actually behave semi-rigidly and therefore the hinged and framed conditions are only idealisations. An engineering definition is needed to define when a semi-rigid connection can be assumed to be hinged and when it can be assumed to be rigid. The practice in different countries in the design of connections and framed systems varies. The approaches used are different in different parts of the world. In some countries, e.g. USA, the concept of semi-rigid connections dates back to the 1930's when the first studies on semi-rigid riveted connections were carried out by Johnston. In these countries also the code allows the use of semi-rigid connections (type 3 connections) and introduces the concept of wind design (type 2 connections). In wind design the connection is assumed to be capable of transmitting only part of the bending moments (those due to the wind and not those due to vertical loads). The approach of semi-rigid connections, in use for several years, is also adopted, for example, in UK, Australia, Canada, and Netherlands. In other countries, e.g. Italy, France, Spain, Greece, these concepts have not been introduced and therefore semi-rigid connections are not widely adopted although they are included in Eurocode 3 [1]. 3.2 Engineering Definition To determine whether a system can be considered continuously framed or not, the effects of the connection on the frame behaviour has to be considered, taking into account that the ideal condition of hinged or rigid connection does not correspond to reality. These effects are not for immediate evaluation since the behaviour of the frame can be different depending on what is being considered: • • • • vertical load-carrying resistance. horizontal load-carrying resistance. stability. performance in seismic situations. In some cases the strength of the connection is the important consideration whilst in other cases the flexibility of the connection plays a major role. Sometimes the elastic behaviour of the connection is sufficient to determine its effects on the behaviour of the frame whilst in other cases the complete inelastic behaviour of the connection is needed for evaluating frame effects. These cases cannot be fully covered in this lecture. Study of the other Lectures 14 will give a more complete view of the different aspects. In recent years this subject has been given much attention from research devoted either to the analytical and to the experimental aspects. In the remainder of this lecture the effects on frame behaviour are described and some indicative values for connection characteristics are suggested as a basis for assuming a system as a rigidly jointed frame. A connection between two or more different members has to transmit all the internal actions through the members, i.e. axial load, shear load and bending moment in the case of a plane frame. However, the term "semi-rigid connection" is only used to refer to the bending flexibility in the discussion below. In Figure 3, taken from [4], connection behaviour is shown qualitatively by definition of the regions in which the connection can be assumed as hinged flexible or semi-rigid. This qualitative representation suggests that fully welded connections, extended end plate, and top and bottom flange splices could be considered as rigid restraints. The results of computer analyses for portal and multi-storey frames with semi-rigid connections, with respect to the elastic critical load of the frame, are given in [5]. For partial strength semi-rigid connections, some results [5] indicate that, for rigid jointed frames, within the ranges indicated before, some degree of flexibility can be allowed, but no partial strength is advised. Partial strength connections are considered to be within the semi-rigid or flexible range. 3.3 Eurocode Definition In Eurocode 3 [1] it is first stated, within the Design Assumptions (5.2.2), that: "the assumptions made in the global analysis of the structure shall be consistent with the anticipated type of behaviour of the connections". Then, in 5.2.2.2, simple framing is defined for frames where the connections between the members may be assumed not to develop moments. In the global analysis, members may be assumed to be effectively pin connected. Further, in 5.2.2.3, when continuous framing is defined, it is stated that: "Elastic analysis should be based on the assumption of full continuity, with rigid connections which satisfy the requirements given in 6.4.2.2". The same statement is given for the other methods suggested for the analysis, i.e rigid plastic and elasto-plastic. These methods are discussed further in Lecture 14.14. In 6.4.2.2, when rigid connections are defined, the following principle is given: "A rigid connection shall be so designed that its deformation has no significant influence on the distribution of internal forces and moments in the structure, nor on its overall deformation". The application rules provided state: "The deformation of rigid connections should be such that they do not reduce the resistance of the structure by more than 5%". When beam-to-column connections are classified, in 6.9.6.2 first it is stated that: "A beam-to-column connection may be classified as rigid or nominally pinned on the basis of particular or general experimental evidence, or significant experience of previous satisfactory performance in similar cases or by calculation based on test evidence". Then, the following is suggested as an application rule: "A beam-to-column connection in a braced frame, or in an unbraced frame which satisfies the condition specified in (5), may be considered to be rigid compared to the connected beam, if the rising portion of its moment characteristics lies above the solid line on the appropriate diagram of Fig. 6.9.8. of Eurocode 3" (Figure 4). This rule suggests that, in order to have the moment-rotation curve of the connection lying above the solid line, it has to result, for values of < 2/3 (which means for values of the moment M smaller than 67% of the plastic beam moment MplRd), K has to be greater than 25. In fact (see definitions in Figure 4): K= > 25 This value of 25 guarantees the stiffness characteristic of the connection. Some other requirements are also given in order to ensure sufficient strength. The fact that the entire moment-rotation curve has to be above the solid line indicates that the entire nonlinear curve of the connection is to be checked. The further requirement for unbraced frames, given in Clause (5) of 6.9.6.2 of Eurocode 3, discussed below, is somewhat cumbersome. The meaning of this requirement is that, for unbraced frames, the beams must have an adequate stiffness otherwise the frame will be too flexible, even with rigid connections. This consideration is obvious for an engineer who knows which typology to choose for each case. However, since all the serviceability and ultimate limit states have to be satisfied, this requirement might be considered automatically fulfilled. Clause (5) states: "The line given in Figure 6.9.8(a) (Figure 4 of this lecture) for unbraced frames may be used only for frames in which every storey satisfies: Kb / Kc > 0,1 where Kb is the mean value of Ib/Lb for all the beams at the top of that storey Kc is the mean value of Ic/Lc for all the columns in that storey Ib is the second moment of area of a beam Ic is the second moment of area of a column Lb is the span of beam Lc is the storey height of a column. 4. BRACED AND UNBRACED FRAMES 4.1 Introduction In Section 2 the requirements of a bracing system are described and a common definition is given. Sometimes the term "bracing system" is inappropriately identified with the term "braced frame". It is clear that the definitions of the two terms are different. The word "braced" in the second case is used as an adjective to the word "frame" and therefore at least two structures have to be identified: a bracing and a frame. A braced frame is commonly intended as a frame to which a triangulated truss is attached. The fact that in reality there is no clear cut distinction between hinged structures with bracing systems and purely rigid jointed framed structures calls for a more exact definition which allows distinctions to be made between: • • purely hinged bracing systems. rigid jointed framed systems. • • semi-rigid frames. braced frames. For the first three types reference should be made to Sections 2 and 3 above. The definition of braced frames is discussed below. 4.2 Engineering Definition The main function of a bracing system is: to resist horizontal actions, and is derived from the separation of the resisting systems: vertical and horizontal. In some cases the vertical system also has some capability to resist horizontal actions. It is necessary therefore, from an engineering point of view, to identify the two sources of resistance and to compare their behaviour with respect to the horizontal actions. Sometimes this identification is not obvious since the bracing is integral within the frame and therefore there is only one structure. However, even in this case, it is possible to make some assumptions in order to define the two structures to be compared. The examples given below clarify these concepts. Figures 5 and 6 represent structures in which it is easy to define, within one system, two sub-assemblies which identify the bracing system and the system to be braced. In particular, a structure is shown in Figure 5 where there is a clear separation of functions: the horizontal loads are carried by the first hinged sub-assembly (A) and the vertical loads are carried out by the second one (B). In Figure 6, in contrast, since the second sub-assembly (B) is able to resist horizontal actions as well as vertical actions, it is necessary to assume that practically all the horizontal actions are carried by the first sub-assembly (A) in order to define this system as braced. In this case the first sub-assembly is defined as a bracing system if its lateral stiffness expressed by the spring constant Ka is considerably higher than the one of the second sub-assembly Kb (in this case a braced frame or system): Ka » Kb (1) This relation can be easily applied to the system of Figure 5 since the constant Kb is equal to zero and therefore the relation is certainly satisfied. For the system in Figure 6, the stiffnesses of both sub-assemblies have to be calculated and compared. 4.3 Eurocode Definition The following definition is provided in 5.2.5.3 of Eurocode 3 [1]: "the frame can be classified as braced if the bracing system reduces its horizontal displacement by at least 80%" which means that the stiffness of the two systems have to be compared and the following relationship satisfied: Ka > 0,8 (Ka + Kb) or Ka > 4 Kb 5. SWAY AND NON-SWAY FRAMES 5.1 Introduction Before defining sway frames and non-sway frames, it is useful to note the common design practice for evaluating safety of structures against stability. It is often convenient to isolate the columns from the frame and treat the stability of columns and the stability of frames as independent problems. For this purpose it is assumed the columns are restricted at their ends from horizontal displacements and therefore are only subjected to end moments and axial loads as transferred from the frame. It is then assumed that the frame, possibly by means of a bracing system, satisfies global stability checks and that the global stability of the frame does not affect the column behaviour. This gives the commonly assumed non-sway frame. This approach has led to years of research spent in the field of behaviour of columns and beam-columns. Design books, guidance documents, and even codes and recommendations, when speaking of stability of columns or stability of frames, commonly use the terms: "sway frames", "non-sway frames", "sway restricted columns" and "sway columns". To explain the concept of sway, as opposed to non-sway, figures such as Figures 7 and 8 are used. The frame of Figure 7 is considered to be the non-sway type and the one of Figure 8 is considered to be the sway type. This form of representation, which is based on common practice and common engineering sense, leads to the erroneous assumption that non-sway frames and braced frames are perfectly equivalent and therefore that one definition can be used instead of the other without causing any misunderstanding. 5.2 Engineering Definition The equivalence between "braced" and "non-sway" frames cannot be established in general since the two terms refer to different aspects of the behaviour of the structure. The fact that the definitions of "sway" and "non-sway" appears when the problem of stability of columns and frames is evaluated suggests that these definitions are part of a simpler treatment of this problem. The concept of braced and unbraced frames can be defined in engineering terms by means of comparison of the stiffness of the systems, as in previous sections, and has no straightforward implications for stability. The concept of sway frames is not intrinsic to the structure: it is based only on its mechanical properties. In fact the seismic meaning of the term "non-sway frame" has no real significance. It is only valid in an "engineering" sense. There is no structure, braced or unbraced, in which there are no sway displacements. The displacements can only be small enough, for particular design purposes, to be considered equal to zero in an engineering sense. Another reason for defining "sway" and "non-sway frames" is the need to adopt conventional analysis in which all the internal actions are computed on the basis of the undeformed shape of the structure. To make this assumption it is necessary that second order effects are negligible, i.e. no significant moments arise due to the action of vertical loads on the deformed shape of the structure. This definition can be shown to be equivalent to the previous one since the vertical design loads cause no significant moments if their value is not close to the elastic critical load of the structure. When there is interaction between global and column behaviour, it is not possible to isolate the column. The column or the frame then has to be assumed to be the "sway" type. Unfortunately, research has been limited in this field and therefore extrapolation of the same procedures already used for non-sway frames to sway frames has been used. As a result inaccuracies occur also due to the fact that the actual behaviour is inelastic and is therefore affected by all types of imperfections, i.e. cross-section, column and frame imperfections. In addition the inelasticity in the columns prevents the use of the familiar concept of "effective length". The design of sway frames has to consider the structure as a whole. On the basis of those considerations, the following definitions can be established for sway and non-sway frames: A non-sway frame is a structure which, from the points of view of stability and the definition of the internal action, can be considered to have small interstorey displacements. Therefore column buckling is independent by frame buckling, i.e. the problems can be uncoupled. This definition will be true if the safety factor against overall buckling is sufficiently large that global buckling can be neglected when carrying out the check against column buckling. On the basis of this definition, it is clearly that to be a non-sway frame is not a characteristic intrinsic of the frame since the safety factor against critical load depends on the magnitude of the design vertical loads acting on the structure. Whilst it is possible to define whether a frame is braced or not by evaluating the stiffness of its members, in order to evaluate whether a frame is the non-sway type, i.e. second order effects can be neglected, the design vertical loads have to be known. This is understandable since even a very flexible structure has no second order effects if the vertical loads are practically equal to zero. 5.3 Eurocode Definition The definition provided by 5.2.5.3 of Eurocode 3 [1] is: "A frame can be classified as non-sway if its response to in-plane horizontal forces is sufficiently stiff for it to be acceptably accurate to neglect any additional internal forces or moments arising from horizontal displacements of its nodes". Examination of this definition does not immediately reveal the relation between sway and instability. However, the Eurocode also provides the following application rule: "A frame may be classified as non-sway for a given load case if the elastic critical load ratio Vsd/Vcr for that load case satisfies the criterion: where Vsd is the design value of the total vertical load and Vcr is its elastic critical value for failure in a sway mode." This application rule confirms that the definition of a frame as non-sway depends on the vertical loads. Furthermore it establishes that a safety factor against overall buckling equal to 10 is enough for considering the problem uncoupled from column buckling. 6. CONCLUDING SUMMARY • • Some definitions have been provided in order to clarify the meaning of words which sometimes are inappropriately used or referred to different structures. The importance of introducing sway and non-sway frames and the implications on design and analysis have been given. 7. REFERENCES [1] Eurocode No. 3: "Design of Steel Structures": ENV 1993-1-1: Part 1.1: General Rule and Rules for Buildings, CEN, 1992. [2] Astaneh, A., Demand and supply of ductility in steel shear connections, Journal of Constructional Steel Research, vol. 14, 1989. [3] Cosenza, E., DeLuca, A., Faella, C., Nonlinear behaviour of framed structures with semi-rigid joints, Costruzioni Metalliche, 199-211. 1984. [4] Cosenza, E., DeLuca, A., Faella, C., Inelastic buckling of semi-rigid sway frames, Structural connections: stability and strength, London, Elsevier Applied Science, 1989. [5] Cosenza, E., DeLuca, A., Faella, C., Elastic buckling of semi-rigid sway frames, Structural connections: stability and strength, London, Elsevier Applied Science, 1989. 8. ADDITIONAL READING 1. 2. 3. 4. Ballio, G. and Mazzolani, F.M. Theory and Design of Steel Structures, Chapman & Hall, London, 1983. Davison, J.B. and Nethercot, D.A. Overview of connection behaviour, Structural connections: stability and strength, London, Elsevier Applied Science, 1989. Dowling, P.J., Knowles, P.R., Owens, G.W., Structural Steel Design, Butterworths, London, 1988. Galambos, T.V. Guide to Stability Design Criteria for Metal Structures, 4th Edition, John Wiley & Sons, New York, 1988. Lecture 14.9: Methods of Analysis for Multi-Storey Frames OBJECTIVE/SCOPE To present the factors which are taken into account in the analysis of multi-storey buildings, and the methods which can be adopted. PRE-REQUISITES Lecture 1B.3: Background to Loadings Lectures 1B.7: Introduction to Design of Multi-Storey Buildings Lectures 5: Computer Aided Design and Manufacture RELATED LECTURES Lecture 14.7: Anatomy of Multi-Storey Buildings Lecture 14.10: Simple Braced Non-Sway Multi-Storey Buildings Lecture 14.13: Design of Multi-Storey Frames with Partial Strength & Semi-Rigid Connections Lecture 14.15: Tall Building Design SUMMARY Methods of analysing the elements of structure described in Lecture 14.7 are presented. The following aspects are considered: • • • • • • • Purpose of Analysis Performance Requirements Loading Effects Analysis Assumptions Hand Methods Computer Methods Future Systems 1. INTRODUCTION Lecture 14.7 described the assembly of the structural elements forming the anatomy of multi-storey buildings. This lecture examines the procedures which should be adopted to conduct an analysis of the structure. It should be noted that specific methods of analysis for certain types of frames (e.g. rigid jointed, wind-moment, braced and partial strength) are, for clarity, presented in the appropriate subsequent lectures. The objective of this particular lecture is therefore to introduce the general philosophy of frame analysis and to identify those considerations which are common to all frame types. 2. OBJECTIVES AND PRELIMINARY CONSIDERATIONS The objectives of analysis are: • • • To gain a better understanding of the structural behaviour under the action of the applied loading. To obtain sets of applied actions which are in equilibrium with the reactions and which can be used for the design of individual structural elements. To predict the static and dynamic movements of the structure and their effect on the building finishes and occupant comfort. In order determine whether the structure is satisfactory, the structural response is compared with the appropriate codified limits, e.g. member stresses, local and overall deflection response, stability criteria, etc. Clearly it is important that the model assumed in the analysis is indeed representative of the way in which the proposed structure would behave. As a result, it is important that prior to conducting an analysis, consideration is given to the following: • • • • • • • An appraisal of the building construction. For example, the type of floor construction and its ability to distribute lateral forces, or the nature of the cladding and how it will distribute the applied wind loading. The overall building geometry and proposed foundation construction. For large structures, expansion joints may be required thereby disrupting the overall integrity of the structure. For structures with top storeys at different levels, and hence significantly different load intensities over the 'footprint', the potential for ground settlement due to soil-structure interaction or the eventual need for structural articulation should be considered in the analysis (Figure 1). It is particularly important that the load path through the structure is identified, i.e. how the permanent and variable actions are transferred to the primary structural frame and how these actions are subsequently transferred by the frame to the foundations (Figure 2). The nature and magnitude of the permanent and variable actions and the co-existent combinations which are likely to have the most detrimental effects on the structure. An understanding of building economy including fabrication and erection procedures. Specifically, the ideal arrangement of the primary structural frame and whether the nature of the frame, e.g. simple or rigid beam to column connections, is appropriate for the structure being considered. Overall frame stability and ability to resist lateral forces. Specifically, whether bracings are present within the structure or whether lateral forces have to be resisted by frame action. 3. ACTIONS The following categories of loading are considered for the analysis of multi-storey buildings: 3.1 Permanent Actions These actions typically comprise the following: • • • • • • • Self-weight of structural elements (slabs, beams, columns, bracing). Floor finishes. False ceiling. Services. Fixed partitions. Fire protection. Cladding. Information on the characteristic load for proprietary products can be obtained from manufacturer's literature. 3.2 Variable Actions - Imposed Load Minimum imposed loading is based on the usage of the building being considered, and specified in local regulations. Buildings are often designed for imposed loads specified in excess of these minima, to increase the flexibility of future usage. Items to consider are: • • • • • • People. Furniture. Moveable partitions. Plant. Storage. Snow/roof loading. Usually only static analysis is required for imposed load, but dynamic analysis may be necessary to assess the effects of: • • • • Seismic actions. Vehicle impact. Transmission of noise and vibrations from plant within the building or from adjacent roads or railways. Floor vibrations due to walking, usually assessed by considering the effect of a standard 'heel drop'. Figure 3 presents generally accepted limits (annoyance thresholds) for vibrations induced by walking. 3.3 Variable actions - Wind Loads The magnitude of the design wind loading to be applied to a building depends on the location and local topography of the site. The following terms are used to describe the different types of wind loading: • • Mean: Static effect due to steady wind flow around the building. Usually based on windspeed averaged over a period of one hour. The magnitude decreases with increase in ground roughness upwind of the site. Gust: Dynamic effect of gusts on the building. The magnitude of this component reduces as building size increases since an individual gust is not large enough to envelop the whole of a large building. Magnitude increases with ground roughness. • Resonant: Dynamic effect due to the flexibility of the building interacting with the applied gusts. Magnitude increases with building height, and reduces with mass and damping. Cross-wind effects may also be present: • • • List: Static and dynamic 'aerofoil' effect due to building shape. Buffeting: Dynamic effect due to fluctuations in wind direction. Vortexes: Dynamic effect of vortices being shed on alternate sides downwind of the building. For low and medium rise multi-storey buildings, complex wind analysis is usually unnecessary, and equivalent static loads defined in local regulations or Eurocode 1 (when available) may be used. Cross-wind effects are not normally significant for ultimate limit state analysis, but can have an important influence on comfort. Figure 4 presents some acceptance levels for wind induced vibrations. 3.4 Seismic Actions Seismic action need normally not be considered in northern Europe, but is the dominating action in southern Europe. It is a function of: • • • • Regional seismicity. Local ground conditions. Building period (a function of height, mass and lateral stiffness). Structural form and materials. Buildings subjected to seismic forces should be designed not only for strength but also for ductility. Details for the seismic resistant design are given in Lectures 17. 3.5 Temperature Depending on the size and layout of the structure, thermal strains due to change in temperature can be significant. For example, if a long building has stiff bracing frames at each end, differential thermal strains will be generated between the superstructure and the foundations and large internal forces will be generated in beams and bracings (Figure 5a). To avoid this inconvenience, the location of bracings in the middle of the structure allows for free expansion of members without any restraint (Figure 5b), so no additional internal forces arise. Thermal effects can be taken into account by adopting an appropriate temperature range (depending on the building location and usage) and a thermal coefficient of expansion for steel. 4. LIMIT STATES The structure should be analysed to ensure that the probability of reaching any of the limit states at which it would become unfit for its intended use within the design life of the building is acceptably low. 4.1 Ultimate Limit State The ultimate limit state considers the strength and stability requirements of the structure - essentially a collapse criterion. Individual types of characteristic loadings are multiplied by the relevant factors to derive the design loads, and applied in the most unfavourable realistic combination for the element or overall structural response being considered. A simplified expression when considering only the most unfavourable variable action is of the following form: Fd = where Fd is the design action Gk is the characteristic permanent load Qk is the characteristic variable load γG is the permanent load factor γQ is the variable load factor The load factors proposed in Eurocode 3 [1] when designing for the ultimate limit state are given in Table 1. It should be noted that permanent actions are effectively the self weight of the elements of the structure and of nonstructural elements, whilst variable actions are the applied loads, e.g. wind load, imposed floor load, settlement load, etc. Where more than one variable actions are considered to act co-existently, the load factors are reduced to reflect the reduced probability of this combination of loading actually occurring on the real structure, i.e.: Fd = Figure 6 shows how these load factors may be applied to determine the most critical design conditions for a range of different frame types. The allowance for imperfections is explained in Section 6. 4.2 Serviceability Limit State The serviceability limit state for steelwork addresses the following: • • • deformations or deflections which adversely affect the appearance or the effective use of the structure (including the proper functioning of machines or services). vibration, oscillation or sway which causes discomfort to the occupants of a building or damage to its contents. deformations, deflections, vibrations, oscillations or sway which causes damage to finishes or nonstructural elements. Eurocode 3 [1] presents guidelines to ensure that these limits are not exceeded. When assessing the serviceability performance of the structure subject to a single variable load in addition to the permanent (self weight) load, the load factors are unity. The structure is effectively analysed using the characteristic actions. However, where the serviceability response of the structure is being assessed with more than one variable action, or where the dynamic response is being considered, Eurocode 3 [1] introduces three types of combinations of loading: • Rare combination • Frequent combination • Quasi-permanent combination The factors in the above formula (ψo, ψ1 and ψ2 ) vary and are detailed in the individual National Application Documents for Eurocode 3 of the various Member States. The rare combination, as the name suggests, considers a larger proportion of the variable action compared to that assumed in the frequent combination. It is a requirement in Eurocode 3 that the rare combination of loading is considered when conducting an appraisal of the member or overall frame deflections, whereas the frequent combination of load is considered when the dynamic response of the structure is being assessed. 5. FRAME CLASSIFICATION When analysing a principle structural frame, it is important to classify it as being either sway or non-sway, braced or unbraced. The classification of the frame determines the method of global analysis which may be employed and the influence of secondary effects on frame actions (see Lecture 14.8). 5.1 Classification as Braced or Unbraced The frame is classified as braced or unbraced depending on the relative stiffness of the bracing system providing the resistance to lateral forces. The criterion for a braced frame is that the bracing system is at least five times stiffer than the lateral stiffness of the frame. It should be noted that this requirement will automatically be satisfied for simple frames incorporating bracing systems. In the absence of a bracing system a simple frame has zero lateral stiffness. For a frame which is deemed to be braced, the bracing system must be designed to resist all lateral loads applied to the structure including those arising due to frame imperfections, see Section 6. A frame which is classified as braced is automatically designed as non-sway. A frame which is deemed to be unbraced must be further classified as either sway or non-sway. 5.2 Classification as Sway or Non-Sway A frame is classified as non-sway for a given load case if the following criterion is satisfied: where Vsd is the design value of the total vertical load Vcr is the elastic critical load for failure of the frame in a sway mode. For multi-storey beam-and-column plane frames, with beams connecting each column at each storey level, classification as non-sway can be made for a given load combination, simply by determining whether the following condition is satisfied: where δ is the deflection over a storey height (interstorey drift) h is the storey height V is the vertical reaction at the bottom of a storey H is the horizontal reaction at the bottom of a storey. Frames classified as sway must subsequently be checked for overall sway stability using the procedures given in Eurocode 3 [1]. 6. ALLOWANCE FOR IMPERFECTIONS The effects of frame imperfections must be included in the global analysis of any frame. Effectively, the imperfections are treated as a load case to be used in conjunction with all the critical load combinations acting on the frame. In Eurocode 3 [1], imperfection effects are quantified in terms of an initial sway rotation at the base of the column (see Figure 7), but may then be converted to an equivalent horizontal force (see Figure 8). The initial sway imperfections φ may be determined directly from Table 3. The number of columns nc includes only the main load bearing columns (i.e. only those which carry over 50% of the average load taken by all the columns in the frame being considered) which extend through all the storeys considered when evaluating ns. Similarly, the number of storeys ns only includes those floor and roof levels which are connected to all the columns included in evaluating nc. A single value of φ is adopted for the whole frame. If the column arrangement is such that it is possible to calculate more than one value, it is permissible to choose the value which has the most beneficial effect. Alternatively, any other choice will be conservative. The values of φ range from an upper bound of 1/200 to a lower limit of about 1/630. The equivalent horizontal force φF at each roof and floor level is calculated by multiplying the proportion of the vertical load, F, applied at that level by the initial sway imperfection, φ (see Figure 8). The equivalent horizontal forces may be applied in any horizontal direction, but only in one direction at a time. At the supports, the equivalent horizontal forces obtained by multiplying the vertical reactions by φ are applied so that the equivalent horizontal forces on the whole frame form a closed system, which results in a net horizontal reaction of zero in the absence of actual horizontal loads. It should be emphasised that the resulting equivalent horizontal forces should be applied in addition to any other horizontal forces which may be acting. 7. ANALYSIS MODEL AND METHOD The modelling of structural systems will be covered in Annex H in a future addendum to Eurocode 3 [2]. In the meantime the following general principles apply. It is conventional to assume that the members in a principle structural frame have centre lines intersecting at a point. The manner in which the design actions are distributed around the frame is dependant on the nature of the connections and the measures employed to resist lateral loading. Table 3, extracted from Eurocode 3, illustrates the different methods of global analysis which are permitted depending on the type of framing and connection. 7.1 Simple Framing The method for analysis and design of simple framing is dealt with at length in Lecture 14.10. The frame incorporates a separate bracing system which is designed to resist lateral loading and provide lateral stability to the part of the structure resisting gravity load (see Figure 9). The model assumes the following: • • • The members intersecting at a joint are pin connected. The necessary flexibility in connections may result in some non-elastic deformation of the connecting parts (other than the fasteners). The supporting columns are not subjected to any direct moment transferred through the connection. As a result, the structure is statically determinate. The internal forces and moments are therefore determined from a consideration of statics. Simple frames are invariably designed as non-sway (see Section 5) and, as a result, the effect of frame deformations on the distribution of internal forces and moments can be ignored. 7.2 Continuous Frames Such structures are statically indeterminate. A detailed appraisal of the analysis and design of partial-strength (or partially continuous) frames is presented in Lecture 14.13, whilst those where full continuity is assumed at the connection are addressed in Lecture 14.14. For continuous frames, the internal forces can be determined using either: a. elastic global analysis. b. plastic global analysis. Whilst elastic methods of global analysis can be used throughout, plastic global analysis is only applicable where the frame members are of the appropriate classification to enable the development of plastic hinges. 7.2.1 Elastic first order analysis In a first-order analysis, the additional secondary actions due to the deformation of the structure (see Figure 10) are ignored. This assumption is only valid in the following cases: a. Where the frame is classified as braced (see Section 5.1). b. Where the frame is classified as non-sway (see Section 5.2). c. Where a design method is used in which an indirect allowance (amplification) is made for the occurrence of second-order effects. In this latter case, the sway moments obtained from a first order linear elastic analysis are multiplied by the ratio: (Formula 1) where Vsd is the design value of the total vertical load. Vcr is the elastic critical value in a sway mode. For conventional framed structures, the ratio Vsd/Vcr can be determined as follows: 1/[1 - Vsd/Vcr], for Vsd/Vcr ≤ 0,25 The first order elastic method of analysis is a convenient approach. Most design offices possess computer software capable of performing this method of analysis on large highly indeterminate structures. Figures 11 - 13 show typical graphical output from first order analysis made using computer software. As an alternative, hand calculations can be performed on appropriate subframes within the structure (see Figure 14) comprising a significantly reduced number of members. These simple calculations are also recommended to make some physical check of the out-put of the computer. When conducting the analysis of an isolated subframe it is important that: a. The subframe is indeed representative of the structure as a whole. b. The selected boundary conditions are appropriate. c. Account is taken of the possible adverse effects between adjacent subframes. 7.2.2 Plastic global analysis Plastic global analysis may be made by means of either: • • Rigid-plastic, methods (i.e. simple plastic) or Elastic-plastic methods. Elastic-plastic methods require quite sophisticated computer programs, which enable second order effects to be taken into account. As a design tool, Eurocode 3 [1] does present a method of rigid plastic analysis which takes into account secondary effects by amplifying the design internal forces and moments by means of the ratio given in Formula (1). The method is only applicable where frames have fixed bases and the only hinge to form in the column is at the base (see Figure 15) and the VSd/Vcr ratio does not exceed 0,2. By means of this method the design is based on an incomplete mechanism in which the columns are designed to remain elastic at the calculated plastic hinge moment. For building structures in which the required rotations are not calculated, all members containing plastic hinges shall have Class 1 cross-sections at the plastic hinge location. 8. CONCLUDING SUMMARY • • • The main objectives of the analysis of multi-storey buildings have been identified. The various loadings have been given and the limit states for verification have been discussed. Analytical models and methods relevant to the frame characteristics have been presented. 9. REFERENCES [1] Eurocode 3: "Design of Steel Structures": ENV1993-1-1: Part 1.1: General rules and rules for buildings, CEN, 1992. [2] Eurocode 3: "Design of Steel Structures": Annex H: Modelling of Building Structures for Analysis (in preparation). Permanent actions (γG) Favourable effect γF,inf Unfavourable effect γF,sup 1,0*) 1,35*) Variable actions (γQ) Leading variable action 0,0 1,5 Accompanying actions 0,0 1,5 variable Table 1: Load factors in Eurocode 3 Type of framing Simple Method of global analysis Pin joints Types of connections Nominally pinned Nominally pinned Continuous Elastic Rigid Nominally pinned (6.4.3.1) Rigid-Plastic Full-strength Nominally pinned Elastic-Plastic Full-strength Rigid Nominally pinned Semi-continuous Elastic Semi-rigid Rigid Nominally pinned Rigid-Plastic Partial-strength Full-strength Nominally pinned Elastic-Plastic Partial-strength Semi-rigid Partial-strength Rigid Full-strength Semi-rigid Full-strength Rigid Nominally pinned Table 2: Methods of global analysis given in Eurocode 3 for different types of frame and connection nc ns 1 2 3 4 5 6 2 3 4 5 6 7 8 1/200 1/240 1/275 1/300 1/315 1/325 1/220 1/260 1/300 1/325 1/350 1/360 1/230 1/275 1/315 1/345 1/365 1/375 1/240 1/285 1/325 1/355 1/375 1/390 1/245 1/290 1/335 1/365 1/385 1/400 1/250 1/295 1/345 1/375 1/400 1/400 1/255 1/300 1/375 1/400 1/400 1/400 ns is the number of storeys nc is the number of columns per plane φ = k c k s φ0 with φ0 = 1/200 kc = √{0,5 + 1/nc} ≤ 1 ks = √{0,2 + 1/nc} ≤ 1 Table 3: Sway imperfection φ Lecture 14.10: Simple Braced Non-Sway Multi-Storey Buildings OBJECTIVE/SCOPE To identify design, construction, cost and fabrication considerations which are specific to non-sway multi-storey steel frames. PREREQUISITES Lecture 1A.2: Introduction to Steel's Role in Construction in Europe Lecture 1A.3: Introduction to Structural Steel Costs Lecture 1B.1: Process of Design Lectures 1B.7: Introduction to Design of Multi-Storey Buildings Lecture 2.4: Steel Grades and Qualities Lectures 3.1: General Fabrication of Steel Structures Lecture 3.5: Fabrication/Erection of Buildings Lecture 4A.1: General Corrosion Lectures 4B: Protection: Fire Lectures 5: Computer Aided Design and Manufacture Lectures 7: Elements Lectures 11.4: Analysis of Connections RELATED LECTURES Lectures 3.2: Erection Lecture 6.3: Elastic Instability Modes Lectures 10: Composite Construction Lectures 11: Connection Design: Static Loading Lecture 14.7: Anatomy of Multi-Storey Buildings Lecture 14.9: Methods of Analysis for Multi-Storey Frames SUMMARY The preliminary considerations and the sequence of design of simple frame construction are presented. Bracing systems (type/location/construction), preferred structural grids, connections (appropriate types for simple frames), the availability of diaphragm action and frame erection, are described. In discussing the sequence of design, the treatment of loading (permanent, variable, imperfections), beam design (design actions, ultimate and serviceability criteria), column design (design actions, effective length), bracing systems (stiffness requirements, construction details), and connections (classification; category), are covered. 1. INTRODUCTION Simple braced non-sway frames probably offer at present the most cost effective structural solution for multi-storey steel buildings. These frames are composed of one or more bracing systems and a simple framing attached to it. The beam-to-column joints are nominally pinned so that the frame is considered to be 'simple'. As a simple frame is not able to resist any horizontal loads, the lateral stability of the entire structure is provided by the bracing system(s), while vertical loads are resisted by both the frame and the bracing system. In most cases, the bracing system's response to in-plane horizontal forces is sufficiently stiff such that the effects of horizontal displacements on equilibrium (second order effects) may be neglected. The structure, therefore, may be classified as non-sway. Figure 1 shows the principal components - simple frame and bracing system - of such a structure. The fabrication of beam-to-column connections in braced, pin-jointed, multi-storey frames is relatively straightforward. The connections are fabricated using simple elements without the need for labour intensive welded stiffeners such as are required moment resisting connections. The current economic trend is such that the ratio of labour cost to that of the material is increasing at an increasing rate (see Figure 2). In addition, direct and indirect labour costs for a typical steel frame constitute as much as 5070% of the cost of the erected steelwork. As a result, the cost of the increased total weight of a simple steel frame, when compared to a moment resisting equivalent, is often offset by the reduced cost of the steelwork fabrication. In many cases, therefore, simple braced steel frames offer the most cost effective structural solution. This particular lecture examines the design and construction considerations which are specific to simple braced nonsway frames and demonstrates the design approach which should be adopted in accordance with Eurocode 3 [1]. 2. ELEMENTS OF THE STRUCTURE The multi-storey structure under consideration is composed of two separate sub-systems, usually a bracing system and a simple frame. 2.1 Bracing Systems The main purpose of a bracing system is to provide the lateral stability of the entire structure. It has to resist, therefore, all lateral loading due to external forces, e.g. wind, imposed deformation, e.g. temperature, earthquake and the effects of imperfections on the simple bracing. For a non-sway frame, the bracing system must, in addition, be stiff enough so that second order effects need not be taken into account in the analysis. In terms of Eurocode 3 [1], this requirement means that either criterion (1): Vsd/Vcr ≤ 0,1 (1) where Vsd is the design value of the total vertical load on the structure and Vcr is the elastic critical value of the bracing system for failure in a sway mode or, for each storey, criterion (2) (2) where δ is the interstorey drift h is the storey height H is the total horizontal reaction at the bottom of the storey V is the total vertical reaction at the bottom of the storey (for both the bracing system and the simple frame) should be satisfied for all load cases under consideration. Possible configurations of bracing systems as shown in Figure 3 are: • • • Braced-bay frames Reinforced concrete cores/walls Rigid jointed frames The systems most appropriate for a particular structure depend very much on the structural layout, the availability of service cores, and the effect on the construction programme. (a) Braced-bay frames (Figure 3a) Braced-bays are located such that they have minimum impact on the structural layout, but taking into account the manner in which the frame is to be erected, the distribution of horizontal forces and the location of any movement joints in the structure. Braced-bay systems comprise diagonal, cross, 'K' and eccentric bracing arrangements. The advantage of triangulated systems is that the bracing elements are subjected only to tension ('cross') or tension and compression ('K' and 'diagonal') in the absence of bending moments. Consequently, the members are relatively light providing a very stiff overall structural response. In the case of eccentric bracing, the system relies, in part, on flexure of the horizontal beam elements. This particular arrangement provides a more flexible overall response which is most effective under seismic loading conditions. Where a single diagonal (as opposed to 'cross') brace is used, it must be capable of resisting both tensile and compressive axial forces to allow for the alternating direction of wind load. Under these conditions, it is recommended that the bracing member has a minimum slenderness ratio of 250 to prevent the self-weight deflection of the brace limiting its compressive resistance. Although many different section shapes can be used as compression braces, a circular hollow section is the most efficient structurally. It should be noted that, in addition, hollow sections offer a greater resistance to corrosion and can be more aesthetically pleasing than open sections. In a cross-braced system, the brace members are required to resist tension only. Consequently, very light, solid tiebars or flats can be used. Figure 4 illustrates fixing details appropriate to cross braced systems whilst Figure 5 shows construction details appropriate for K-braced systems. (b) Reinforced concrete shear walls and cores (Figure 3b) Shear walls are constructed from reinforced in-situ concrete and are positively connected to the steel structure to resist horizontal forces. Such walls often serve a second important function of compartmenting the structure to limit the spread of fire. Concrete cores are provided in multi-storey structures to house lifts, stairwells and service ducts. Both concrete shear walls and cores may be provided with or without openings depending on the functioning requirements of the building. (c) Rigid jointed frames In cases where braces or concrete walls would disturb the functioning of the building, rigid jointed moment resisting frames may provide the lateral stability of the building. The design of such frames is described in Lecture 14.14. The location of the bracing systems in plan within the structure will influence the efficiency with which the lateral forces can be resisted. The most appropriate position for the bracing systems is in the periphery of the building (Figure 6a) since this arrangement provides the largest overall torsional resistance. The minimum number of bracing planes required for the provision of translational and torsional stability is 3. These planes should not meet at one point since this could be a pole of rotation. When stiff cores are used, they should preferably be located in the centre of the building (Figure 6b), since it can expand freely in both directions. The torsional stability is then provided by the torsional rigidity of the core. In some cases eccentrically placed service cores in conjunction with supplementary bracing systems are present (Figure 6c). In that case the possibly different flexibilities of the bracing systems and the core must be taken into account in the analysis, since the stiffer element will attract a larger share of the applied horizontal load. 2.2 Simple Frames The simple frame consists of a series of beams and columns, forming a structural grid. The elements of the simple frame are assumed to resist gravity loads only. (a) The structural grid Usually the layout of the structural grid is dictated by the intended use of the structure and architectural requirements. Notwithstanding this, there are a number of factors to be borne in mind when choosing column spacings in a simple frame. In all structural framing, the greatest cost effectiveness can be achieved through a high repetition of similarly fabricated components. Essentially, a regular column grid is significantly less expensive than a non-regular grid for a given floor area, see Figure 7. Not surprisingly, orthogonal arrangements of beams and column grids, as opposed to skewed, result in the most cost effective layout. In addition, the greatest economies can be achieved when the column grids are rectangular - not square. In this configuration, the secondary beams should span in the longer direction and the primary beams in the shorter. This arrangement has the effect of reducing to a minimum, the number of beam-to-beam connections and the number of individual members per unit area of supported floor. By definition, the beams in a simple frame are assumed to be simply supported between columns. The most cost effective maximum span will be dependent on the applied load, the type of beam system employed and the restrictions on structural floor depth. This latter consideration will be influenced by restrictions on overall building height and the requirements for services either above or below the structural floor zone. Lecture 14.7 introduced the different types of floor and beam construction which are commonly in use. Figure 8 shows the practical span ranges for these different systems when used in office construction. Not surprisingly, systems in which the structural steel members act compositely with an in-situ concrete slab achieve the longest spans. Columns are conventionally located at spacings in multiples of 2,5m or 3,0m (e.g. typical bay sizes 9m x 6m, 7,5m x 15m etc). This latter dimension conforms with the approximate maximum span of conventional steel permanent formwork (as used in composite construction) when unpropped. Figure 9 shows typical arrangements of primary and secondary beams supporting a composite floor using permanent decking. (b) Connections As discussed above, the beam-to-column and beam-to-beam connections are assumed to function as 'pins' and should therefore be detailed accordingly. It should be noted that a 450mm deep steel 'I' section spanning 6m experiences a 10mm shift between the upper and lower flange at the connection when subjected to the ultimate uniformly distributed load (Figure 10). Consequently, connection details should be adopted which can accommodate this magnitude of rotation without transferring significant moment. Figure 11 shows three connection types which satisfy this criterion - namely web cleat, fin plate and partial-depth end-plate connections. The design moments in columns of simple frames tend to be rather small when compared to the axial compression. Consequently, bearing and end-plate splice connections and nominal 'pin' base connections (see Lecture 11.5) are used extensively. Access restrictions permitting, column splices should be provided at 12-16m intervals along the column, with the splice located at approximately 500mm above the nearest structural floor level. Columns splices are particularly expensive to fabricate. There is invariably greater economic advantage in retaining a heavier column section through a number of levels rather than introducing a splice to reduce the weight of the upper column segments. Simple beam-to-concrete walls/cores connections should be able to carry loads as soon as the connection is made. Some connection types that may be used for such cases are shown in Figure 12 as follows: • Projecting cleats are embedded in the wall (Figure 12a). • Steel components are embedded in the wall and connector elements are welded to them on site (Figure 12b). • Pockets are made in the core wall in which the ends of the beams engage (Figure 12c). 2.3 Floors Floors should be able to transmit the vertical loads to the supporting framing beams. However, in the multi-storey simple, braced frames discussed here, lateral loads applied to the external envelope of the building should be transferred to the vertical bracing systems. This transfer occurs via diaphragm action of the floors. Consequently, the floors in these frames have, in addition, to transmit horizontal loads. The adequacy of the floor to act as a diaphragm depends very much on the type of floor system employed. Pre-cast concrete floor planks with a non-structural screed have limited resistance to the racking effects of diaphragm action. In such cases, supplementary bracing systems in plan are required to distribute lateral forces adequately. Supplementary bracing systems can result in a significant increase in fabrication costs and, as such, should be avoided. Where precast concrete floor units are employed, sufficient diaphragm action can be achieved in most cases by using a lightly reinforced structural concrete topping. Composite concrete floors, incorporating permanent metal decking, provide excellent diaphragm action. In addition, it should be noted that a correctly fixed decking, with the appropriate side lap stitching, provides an adequate floor diaphragm during the construction stage. 3. DESIGN OF THE STRUCTURE 3.1 Loads and Their Combination The loads to be considered for the analysis of multi-storey buildings are: • Permanent loads (G) including the self weight of: - structural elements - secondary non-structural elements - services • • • • Imposed floor loads (Q) Wind loads (W) Seismic loads (E) Snow loads (S) for the design of the roof. The effects of imperfections should be allowed for in the analysis of the frame in the form of an initial sway imperfection φ determined from: φ = kc ks φo (3) where φ = 1/200 kc, ks are factors depending on the number of storeys and columns per plane. These imperfections are usually accounted for in the form of equivalent horizontal forces FH at floor levels according to: FH = φ Fs (4) where Fv is the design vertical load at the floor level under consideration. The design includes verifications in two different limit states, the serviceability limit state and the ultimate limit state. The most usual combinations of actions for the type of buildings under discussion are: • Serviceability limit state Beam deflections Frd = G + Q (5) Frd = Q (6) It should be noted that the deflection limits are different for the combinations (5) and (6) • Vibrations Frd = G + ψ1 Q (7) • Inter-storey drift VHd = W (+ FH from Equation (4)) (8) • Ultimate limit state For the simple frame FHd = 1,35 G + 1,5 Q (9) For the bracing system Vr,H,d = 1,35 G + 1,5 Q + ψo 1,5 W (10) = 1,35 G + 1,5 ψo Q + 1,5 W (11) = G + Q + E (12) Detailed information on the combination rules is provided by the relevant rules of Eurocodes 1 [2], 3 [1] and 8 [3]. 3.2 Beam Design Beams are designed as simple spans, neglecting any continuity at the supports. When the beam moments and shears are known from the analysis, the beam dimensions may be determined according to the provisions of Eurocode 3 [1] for steel beams or Eurocode 4 [4] for composite beams. It should be noted that most conventional types of floor slab construction will provide adequate positional restraint to the top (compression flange) of the beam. Consequently, the beams can be designed without taking into account reductions in moment resistance due to lateral-torsional buckling effects. Under the full characteristic loading (load factors equal unity), the total central deflection of the beam, δmax (Equation (5)), and the deflection of the beam due solely to imposed load (Equation (6)), δ2, should satisfy the limits in Table 4.1 in Eurocode 3 (reproduced here as Table 1). It should be noted that the deflection check is performed using the 'rare combination' of loading (Equation (6)). Depending on the use of the structure, it is necessary to check the dynamic sensitivity of the floor beams. Clause 4.3.2 of Eurocode 3 states that, where the total deflection of the beam is less than 28mm, the dynamic sensitivity is satisfactory for foot traffic (walking), and when less than 10mm, the sensitivity is adequate for rhythmic loading (dance floor). It should be noted that these limits are based on a 'frequent' combination of loading (Equation (7)). When assessing the deflection and dynamic sensitivity of secondary beams, it is important to include that component due to the deflection of the supporting beams. Whether it is the strength, deflection or dynamic sensitivity which controls the design will depend on the span-todepth ratio of the beam. Figure 13 gives typical span ranges for beams in office buildings for which these design criteria (strength, deflection, vibration) may be dominant. 3.3 Column Design The ultimate axial load in columns is derived from the cumulative total of the ultimate support reactions from those beams which frame into the column. Although there is assumed to be no direct transfer of bending moment from the beam to column, it is a requirement in most European countries that a nominal moment is transferred. This moment is assumed to be equal to the vertical beam reaction multiplied by the offset to the interface between the beam and the column. For a beam connection to the major axis of the column, this offset is equal to half the depth of the column. In certain countries, an increased eccentricity of beam reaction is considered, e.g. in the U.K. the eccentricity of a major axis connection is assumed equal to Dc/2 + 100mm. In all cases, the applied nominal moment is divided between the upper and lower column segments in proportion to their flexural stiffnesses. The precise manner in which simple frames should be modelled and, in particular, the actions to be considered in the column design, will be addressed in the forthcoming Annex H of Eurocode 3 [5]. As the column node points are positionally restrained, the maximum effective column length (Le) for buckling considerations is 1,0L, where 'L' is the length of the column segment. In cases where the adjacent column segments are under-utilised in terms of load resistance, the residual flexural stiffness of these members may result in an effective length of < 1,0L for the segment under consideration. Such a situation arises where the column section is continuous through a restraint point and the column segments on each side of the restraint point are of different length. Table 2 summaries the reductions in effective length which may be considered for different a/L ratios. However, the location of column splices and the degree of stiffness continuity should be borne in mind when taking this enhancing effect into account. The effects of patterned imposed loading are not usually considered in the design of the frames. Consideration of patterned loading will depend on the specific requirements in individual countries. However, in cases where nominal moments are applied to the column, it is a requirement that a value of βM.LT = 1,1 (Eurocode 3: clause 5.5.4) is adopted when considering the lateral-torsional buckling of the column. This is a conservative measure which guards against particular patterns of load being applied in the real structure which would cause the column to be subjected to a single curvature moment distribution which is a particularly onerous arrangement in lateral-torsional buckling considerations. 3.4 Bracing System As discussed in Lecture 14.8, the bracing system must satisfy certain criteria in order that the frame may be correctly classified as braced and non-sway. In Eurocode 3 (clause 5.2.5.3(2)), a frame is 'braced' if the bracing system reduces the lateral deflection of the frame by 80%. Clearly, this criterion will always be satisfied in the case of simple frames which behave as mechanisms in the absence of a bracing system. Additionally, by satisfying criterion (2) the bracing system is classified as non-sway so that second order effects can be neglected in the analysis. The stiffness of the bracing system is not only governed by Equation (2) but also, and more often, in the serviceability verification which requires that both the interstorey shifts and the lateral deflections of the structure as a whole must be limited, the limits depending on the sensitivity of the structural elements to shear deformations. The limits recommended by Eurocode 3 [1] are: h/300 for the interstorey drifts ho/500 for the structure as a whole where h is the storey height ho is the overall height of the building When considering the ultimate limit state, the bracing system must be capable of safely transferring the factored lateral loads safely down to the foundations. An all steel braced-bay system will, more often than not, comprise a trussed lattice. The design of the internal bracing members is therefore similar to the procedure in Lecture 7.12 which is devoted to the design of lattice girders. Quite often, a horizontal member in a latticed bracing system serves also as a floor beam. This particular member will be subjected to primary bending (due to gravity loads) and axial compression (due to wind and imperfection load). The resistance of the element should therefore be checked as a beam-column (in accordance with clause 5.5.4 of Eurocode 3) using the load factors appropriate for gravity plus lateral load, as discussed in Section 3.1 above. Particular care should be taken in the modelling of concrete cores. Some points to take into account are: (a) The use of the effective width rather than the full width for the flanges if they are very wide (Figure 14). The effective width to be considered for the determination of the bending properties is a function of the b/t ratio of the flange, the building height and the form of the bending moment diagram along the height. Relevant provisions are given in Eurocode 3 [1]. (b) The inclusion of the torsional properties of the cores. The torsional resistance of hollow sections (Figure 14b) is provided mainly rotational torsion, while that of channel or I sections (Figure 14a) mainly by non-uniform, warping torsion, i.e. through opposite bending of the flanges. Depending on the software used, the modelling of the core as a single vertical element should be examined to determine if it is adequate. It may be necessary to introduce more elements to represent the properties of core. (c) The inclusion in the analysis of the bending resistance which is provided by the staircases between the floors. A staircase provides some resistance to relative floor displacement. Its inclusion leads to a reduction of interstorey drifts but leads to additional reinforcement for the staircase. Models for determining the equivalent static properties of a staircase may be taken from the literature. 3.5 Connections The connections should be designed and detailed to prevent excessive transfer of moment between the beams and columns. Such connections should comply with the classification for a 'nominally pinned connection' in terms of both strength and rigidity, see clauses 6.4.3 and 6.4.2 respectively of Eurocode 3 [1]. The beam-to-column and beam-to-beam connections are designed principally to resist the shear due to the vertical beam reaction. Depending on the connection detail adopted, it may also be necessary to consider an additional bending moment resulting from the eccentricity of the bolt line from the supporting face. Invariably, these connections will conform with the Category A: Bearing type designation in clause 6.5.3.1 of Eurocode 3. This category is applicable to connections comprising non-preloaded bolts for which the principal design criteria are the shear and bearing resistance, see Table 6.5.2 in Eurocode 3 [1]. In almost all cases, the connections in simple frames are made on-site using conventional bolts in clearance holes. These bolts slip into bearing when subjected to load. Such connections are used also in those parts of the structure which are subjected to reversible wind loads, e.g. braced-bay frames - see clause 6.3(3). The design of bolted steelwork connections is presented in full in Lectures 11. 4. ERECTION The connections between the various elements in a simple frame are relatively simple to fabricate and can be easily made on site. Consequently, simple frames can be erected very quickly and efficiently. However, due to the lack of moment transfer between the beams and columns, maintaining the stability of the structure during erection can present a number of problems. Often, it is necessary to incorporate temporary bracing into the structure to provide the necessary lateral resistance until such time as the permanent bracing system is installed. Clearly the layout of the structure and in particular, the location of the permanent bracing systems, will influence the sequence of fabrication, delivery of materials to site and erection. The implications on the construction programme should, therefore, be considered when developing the conceptual design of the main structural frame. It should be noted that one of the main benefits of using slip formed concrete cores constructed in advance of the steelwork, is that they can totally eliminate the need for temporary bracing. One of the main issues affecting the efficiency and speed of erection is the number of individual elements. A large number of elements will require extensive use of cranes, a major cost item in the erection process. A structural grid and frame arrangement should, therefore, be selected which reduces to a minimum the number of elements to be erected. The erection of steel frames is discussed in detail in Lectures 3.2. 5. CONCLUDING SUMMARY • • • • • Simple, braced, non-sway steel frames often offer the most cost effective structural solutions for multistorey buildings. The main elements of the structure are the simple frame and the bracing system. The simple frame is composed of simply supported beams and columns which resist vertical loads only. The bracing system may be a braced frame, a concrete core, wall or a moment resisting frame. It provides the lateral stability of the structure. All elements of the structure should be determined such that they resist the applied actions in both serviceability and in ultimate limit state conditions. 6. REFERENCES [1] Eurocode 3: "Design of Steel Structures": ENV1993-1-1: Part 1.1: General rules and rules for buildings, CEN 1992. [2] Eurocode 1: "Basis of Design and Actions on Structures"; ENV 1991-1, basis of design (in preparation). [3] Eurocode 8: "Structures in Seismic Regions - Design" (in preparation). [4] Eurocode 4: "Design of Composite Steel and Concrete Structures": ENV 1994-1-1: Part 1, General rules and rules for buildings, CEN, 1992. [5] Eurocode 3: "Design of Steel Structures": Annex H: Modelling of building structures for analysis (in preparation). 7. ADDITIONAL READING 1. 2. 3. Hart, F., Henn, W. and Sontag, H.: "Multi-Storey Buildings in Steel, Colins, London, 1982. Dowling, P. J., Knowles, P. R. and Owens, G. W.: "Structural Steel Design", Butterworths, London, 1988. Petersen, Ghr: Stahlbauten, Vieweg, 1990. Recommended limiting values for vertical deflections Condition Limits δmax Roofs generally Roofs frequently carrying personnel other than for maintenance Floors generally L/200 L/250 L/250 δ2 L/250 L/300 L/300 L/350 Floors and roofs supporting plaster or other brittle finish or non- L/250 flexible partitions Floors supporting columns (unless the deflection has been included in L/400 the global analysis for the ultimate limit state) Where δmax can impair the appearance of the building L/250 L/500 - Table 1: Deflection limits for beams in Eurocode 3 Column Frame 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 a/L Lt/L 0,70 0,73 0,76 0,79 0,82 0,85 0,88 0,91 0,94 0,97 1,0 0,50 0,53 0,57 0,61 0,65 0,70 0,75 0,81 0,87 0,93 1,0 Lt/L 0,70 0,72 0,74 0,77 0,79 0,81 0,84 0,87 0,91 0,95 1,0 Lt/L Table 2: Effective length of continuous braced columns Lecture 14.11: Influence of Connections on Behaviour of Frames OBJECTIVE/SCOPE To present and illustrate procedures for the design of multi-storey buildings relating to the behaviour of beams, columns and connections. Special attention is paid to the influence of the design of partial-strength connections on the frame behaviour. PREREQUISITES Lecture 14.8: Classification of Multi-Storey Frames RELATED LECTURES Lecture 11.6: Moment Connections for Continuous Framing Lecture 11.7: Partial Strength Connections for Semi-Continuous Framing Lecture 14.10: Simple Braced Non-Sway Multi-Storey Buildings SUMMARY The influence of partial strength and semi-rigid connections on the design of building frames of structural steel is considered. Plastically designed connections in elastically designed frames and elastically designed connections in plastically designed frames are discussed. 1. INTRODUCTION The main structural elements of steel framed multi-storey structures are the columns, the beams and their connections. Conventionally the beam-to-column connections are considered to be either pinned or rigid. In the case of pinned or 'simple' connections, the frames have to be stabilised by appropriate bracing systems. Such frames are named braced frames by Eurocode 3. Their design is treated in detail in lecture 14.10. The term 'rigid' in this context implies that the connection is capable of resisting moments with a high stiffness, i.e., the connection flexibility has a negligible influence on the distribution of movements in the frame connections. When the connections are rigid, the overall stability may be provided by the frame itself without the inclusion of specific bracing systems. These rigid-jointed or moment resisting frames are treated in lecture 14.14. Although the idealisation of connection stiffness as pinned or rigid has been applied exclusively in the past it is generally recognised that the real behaviour of the connections is never as ideal as assumed in the analysis (Figure1). The two cases, pinned and fully rigid, actually represent extremes of connection behaviour. In reality, the connections behave somewhere between those limits, that is they behave as semi-rigid [1, 2]. A further classification of moment resisting connections relates to their strength. A 'full-strength' connection is a connection that can at least develop the bending strength of the elements it connects. A 'partial-strength' connection has a lower design strength than that of the elements it connects. The rotation capacity of a moment-resisting connection can also be important. For example a beam with partialstrength end connections can be designed plastically if the connection rotation capacity is sufficient to ensure the development of an effective hinge at midspan. Figure 2 shows the moment/rotation diagram of a beam to column connection. For design purposes the real connection behaviour can be represented by a bi-linear or tri-linear diagram in which the following properties can be distinguished. • The design resistance of the connection • The stiffness of the connection when subjected to small moments • The stiffness of the connection when subject to ultimate moments • The rotation capacity The influence of connections on frame behaviour is treated separately for partial-strength and semi-rigid connections. Braced frames are in general designed based on strength conditions and unbraced frames are generally designed based on stability and deformation conditions. Therefore partial strength connections are mainly applicable for braced frames and semi-rigid connects for unbraced frames. 2. CLASSIFICATION OF CONNECTIONS 2.1 Influence of Connection Flexibility on Elastic Frame Stability The end rotation of a simply supported beam under a uniform vertical load and external negative moments M (Figure 3) is given by φ= (1) The (M-φ) response represented by equation 1 is a straight line which is called the beam line as show in Figure 2b). In real frames the end moment restraint is provided by the rigidity of the connection S = M/φ as shown in Figure 2c and therefore the actual end moment and the end rotation of the beam is given by the intersection of the beam line with the connection characteristic as shown in Figure 2b. For practical design situations the actual non-linear connection behaviour has to be approximated. Various approximate connection characteristics are in Figure 4. The connection behaviour is characterised by its moment resistance MRd, its rotational capacity φcd and its rigidity s = M/φ. In order to determine if the connection flexibility S-1 needs to be included in the overall frame analysis it is important to examine its influence on the behaviour of the frame [3]. This is studied subsequently for single storey braced and unbraced frames as shown in Figure 5a. Figure 5b presents the relationship between the relative connection-to-beam rigidity and the relative beam-to column rigidity ρ in order that the flexibility of the connection reduces the Euler buckling load of the rigid frame by 5%. An example of the evaluation of the curve for an unbraced frame is given by considering point x, where ρ = 1.4 and = 25. Assume kb = 10 If ρ = 1.4; kc = 10/1,4= 7.14 If = 25; s = kb = 250 =∞ First examine case with fully rigid connection k1 = kb = 10 From EC3 Annex E η1 = 1.0; η2 = = = 0.417 = 2.391 Examine case with semi-rigid connection For symmetrical double curvature in beam, end rotational stiffness = , therefore effective stiffness of connection in calculation of k1 is Hence, based on ratios of to . k1 = . kb = From EC3 Annex E η1 = 1.0 η2 = Hence reduction of elastic critical load in the presence of such connection flexibility is given by: = = of 25. i.e. for a frame with ρ = 1.4, a 5% reduction in elastic critical load factor is caused by 2.2 Influence of Connection Flexibility on Frame Strength In real frames the most important quantity is the ultimate limit load Fu of the frame rather than the Euler buckling load Fe. This can be found by the Merchant-Rankine formula (lecture 7.7) for the unbraced frame. If Fpl is the plastic failure load of the frame then according to the above formula the ultimate load is given by: (2) From equation 2 it is obvious that a 5% drop of the Euler buckling load Fe due to connection flexibility leads to a drop in load carrying capacity of the frame Fu by not more than 5%. The previous discussion leads to the observation that for a given frame configuration, depending on the parameter ρ , the minimum required connection stiffeness may be determined from Figure 5b that does not lead to a decrease of the frame ultimate capacity by more than 5%. If the actual stiffeness of the connection is smaller than that one determined from Figure 5b, the connection flexibility shall be taken into account in the frame analysis. In Eurocode 3, a further simplification is made, see Figure 6. Constant boundary factors and unbraced frames rather than values dependent on ρ. are chosen for the braced For braced frames the boundary value of is equal to 8 as shown in Figure 6b. That means that if s ≥ 8kb a connection may be treated as rigid, otherwise as semi-rigid. This is illustrated in Figure 6a. For unbraced frames the relevant value of is 25, which means that if s ≥ 25kb a connection may be treated as rigid, otherwise as semi-rigid. Figure 6b illustrates this case. According to Figure 5b the boundary value = 25 for unbraced frames covers only the cases when ρ ≥ 1,4. For ρ < 1,4 this value is unsafe. This situation is investigated below. Frames for which ρ < 0.1 are not realistic, so the value ρ = 0.1 can be used as a boundary. Figure 7 shows the relationship between ρ and (FE( on = 25 is 85% of the Euler buckling load for )/FE( = ∞). =∞. 100%. When ρ =0.1, the Euler buckling load based The carrying resistance of the frame based on the Merchant-Rankine formula has reduced as follows: 1/Fu( = ∞) = 1/Fpl + 1/FE( = ∞) = ∞) where Fu( 1/Fu( where: X = FE( 1/Fu( Fu( = ∞) is the carrying resistance of the frame with rigid connections = 1/Fpl + 1/XFpl = ∞) / Fpl = ∞) = (1+ 1/X) / Fpl = ∞) = {X/(X + 1)}/Fpl = 25 The following relationship holds for connections where 1/Fu( Fu( = 25) = 25) = (1+ 1/0,85X)Fpl = {0,85X/(1 + 0,85X)}Fpl The reduction of the carrying resistance is: ∆ = {Fu( =∞) - Fu( = 25)}/Fu( = ∞ ) . 100% Even for a very slender frame, namely FE( 5.6%. = ∞) = Fpl, so X = 1, the reduction ∆ = 8% and for X = 2 the reduction ∆ = It can be concluded that = 25 is a sufficiently safe boundary value for practical frames for the rotation stiffness of beam-to-column connection in unbraced frames in order to consider them as rigid. In Figure 6 the boundaries for the distinction between rigid and semi-rigid behaviour of the connection follows a trilinear rather than a bi-linear characteristic. The reasons are: a) experimental evidence shows that a beam-to column end-plate connection behaves elastically for up to at least 2/3 of its moment resistance b) The beam characteristic is also not linear up to the plastic moment. Plastification in the beam theoretically starts at about Wel/Wpl = 0,9 for I sections. Due to residual stresses a more practical figure is 70% of the beam plastic moment. It is therefore considered reasonable to alleviate the bi-linear characteristic of the connection by a third branch. 2.3 Influence of Connection Strength on Frame Behaviour As far as the moment resistance of beam-to-column connections is concerned, classification is simpler. If the moment resistance of the beam-to-column connection is equal to the plastic resistance of the connected beam, the connection is considered as full-strength. If not, the connection is considered as partial-strength. On the basis of the boundary for the rotation stiffness and the boundary for the moment resistance a bi-linear moment-rotation characteristic is achieved. If the moment-rotation characteristic of a beam-to-column connection lies on the left hand side and above the boundary lines, this connection can be classified as rigid and full-strength. 3. MODELLING OF THE CONNECTION Connections may be designed either by elastic or by plastic theory. Since they are modelled mostly by rotational springs as shown in Figure 4a, it is important to determine the spring characteristic representing the behaviour of the connection. From the lay-out of the connections, its moment-rotation characteristic may be determined on the basis of Annex JJ of Eurocode 3. However, the initial estimate of the characteristic has to be modified with the type of global analysis (elastic or plastic) and the method of connection verification (elastic or plastic) as shown in Figure 8 reproduced from Annex JJ. If the global analysis is elastic the spring behaviour is considered to be linear since the same applies also to the behaviour of the other structural elements (beams and columns). For this type of analysis the joint may be verified either on the basis of elastic or on the basis of plastic theory. In the former case, only the linear part of the connections characteristic is taken into account ie., the design moment of the connection is only 2/3 of its ultimate moment and its stiffness is equal to the initial value (Figure 8a). In the latter case the design moment of the connection is equal to its plastic moment but its stiffness is reduced by the factor χ as shown in Figure 8b to take into account the non-linearities. In the case of plastic global analysis, with partial strength joints, the spring behaviour of the connection is considered as bi-linear, since the same applies also to the behaviour of the beams and the columns of the frame. In that case the non-linear part of the connections is taken into account as shown in Figure 8c. Obviously such connections should possess sufficient rotation capacity in order to be able to undergo the resulting plastic relations. A last case is when plastic global analysis is applied with full strength joints. In this case no plastic hinge is formed in the joint and the joint model is linear according to Figure 8a or 8b. In order to be sure that no plastic hinge will form in the joint and bearing in mind the possible overstrength of the beam, the following conditions shall apply: For Figure 7(a): 2/3 MRd ≥ 1,2 Mpl,b For Figure 7(b): MRd ≥ 1,2 Mpl,b The greatest problem in the practical design of frames with semi-rigid connection lies in the determination of the connection flexibility since this is only possible once the connection is designed. This means that there must be an iterative process between frame analysis and joint dimensions until a connection is found with the characteristics assumed in the analysis ie., one that is able to transmit the forces and moment and undergo the required rotation determined by the analysis. Specifically in the case of unbraced frames an additional parameter shall be taken into account in the modelling of connections. As illustrated in Figure 9, the deformation characteristic of the joint is different for the various loading conditions (vertical loading, horizontal loading) due to the large flexibility of the column web when submitted to shear. Since this case appears only for horizontal loading the joint behaviour is much softer in that case than in the case of vertical loading. In the normal case of both vertical and horizontal loading, the actual spring stiffness lies between the two values. This makes the practical design of unbraced frames with semi-rigid connections almost impossible. 4. CONCLUDING SUMMARY • • • • • Connections are classified in Eurocode 3 according to their stiffness as pinned, semi-rigid or rigid and according to their strength as full or partial strength. In conventional design of frames connections are considered as either pinned or rigid. If semi-rigid connections are introduced into the design, their flexibility should be taken into account in the global analysis. Thus connections need to be designed prior to final verification of the frame. For braced frames the introduction of semi-rigid connections may lead to more economical results due to the optimisation of the steel material for the beams. For unbraced frames the use of semi-rigid connections is associated with problems in the global analysis and leads to larger horizontal displacements which may cause problems in the overall stability. Therefore these connections are not recommended for such frames. 5. REFERENCES [1] Chen, W F: Joint flexibility on steel frames, Elsevier Applied Science Publications Ltd, 1987 [2] Bjorhorde, R., Brozzetti, J., Colson, A.: Connections in Steel Structures, Elsevier Applied Science Publishers Ltd, 1988 [3] Background Document 6.09 to Eurocode 3, Beam to Column Connections, Commission of the European Communities, 1989 Lecture 14.12: Simplified Method of Design for Low-Rise Frames OBJECTIVE/SCOPE To describe the assumptions of the wind-moment method for the design of unbraced frames. To summarise the justifications of the method. To show the method of analysis and appropriate design rules. PREREQUISITES Lecture 6.3: Elastic Instability Modes Lecture 7.2: Cross-section Classification Lectures 7.8: Restrained Beams Lectures 7.10: Beam Columns Lecture 14.8: Classification of Multi-Storey Frames RELATED LECTURES Lecture 1B.2.1: Design Philosophies Lectures 1B.7: Introduction to Design of Multi-Storey Buildings Lecture 11.7: Partial Strength Connections for Semi-Continuous Framing Lecture 14.7: Anatomy of Multi-storey Buildings SUMMARY The wind-connection method for unbraced frames renders the structure statically determinate and thereby avoids interaction between global analysis and member design. It assumes that the connections act as pins under gravity load, whilst under horizontal load the connections behave as rigid joints. Studies on frames designed by the method have given a suitable range of application. The typical behaviour of windmoment frames is described and those aspects that require particular attention to achieve a satisfactory design are identified. The method of global analysis is explained and appropriate rules for member design are summarised. 1. INTRODUCTION In some countries, e.g. UK, Australia and USA, a simplified method for the design of "low-rise frames" has been in use for many years. This is known as the "wind-moment" or "wind-connection" method. It is recognised within relevant national recommendations and the experience achieved in its application throughout these countries has been satisfactory. As a result, it is widely used. The method is not universally applied though. In the UK its use is now limited to simple frames, regular geometry, with a maximum of 8 floors and 4 spans, maximum heights of 5m between floors, maximum span distances of 12m and maximum relative lengths of 0,5 and 2,0 between spans. In many other countries, this simplified method is not widely known and is not included in their standards, hence its application is not possible. It is not specifically included in the current version of Eurocode 3 [1]. Despite this, it seemed useful to dedicate one lecture to this method and to develop a practical application example. The method continues to be widely used, at least in those countries mentioned previously, and the lecture is therefore an interesting piece of background information for many lecturers. Perhaps the inclusion of this Lecture may also widen the long-standing and inconclusive debate regarding the type of simplification and level of rigour that should be adhered to in design methods laid down in structural codes. 2. THE METHOD Where a steel frame is unbraced, an established design technique is to rely on the rotational stiffness of the connections to provide resistance to wind, even though such restraint is ignored under the action of gravity loads. This approach is termed the 'wind-moment' or 'wind-connection' method. In its usual form the method assumes: • • under gravity load the connections act as pins (Figure 1a) under wind load the connections behave as rigid joints, with points of contraflexure at the mid-height of columns and mid-length of beams (Figure 1b) Members and connections are proportioned initially to withstand gravity load. The internal forces and moments due to gravity load and wind (Figures 2a and 2b) are then combined in appropriate load cases. The design for strength is completed by amending the initial section sizes and other details for the members and connections, to withstand the combined effects. No calculation is made for second-order moments due to the 'P-∆' effect. It is assumed that these moments can be accounted for by using effective column lengths greater than the true lengths, for axes about which sway can occur. For serviceability, sway deflections are calculated assuming connections are rigid. The advantage of the method is its simplicity. As the frame is rendered statically determinate, internal moments and forces are not dependent on the relative stiffnesses of the members. The need to repeat the analysis to correspond to changed section sizes is thereby avoided. Consequently, the method has been used extensively [2, 3], although it has not been verified as a generally applicable approach. The justification of the method has been partly in the fact that buildings designed on this basis have proved satisfactory in use. In recent years, the method has been regarded as a form of semi-rigid design and analytical justification has been carried out on this basis [4-7]. The conclusions are as follows: • Beams These members tend to be overdesigned for the following reason. Beam design is usually governed by the sagging internal moment due to gravity load. The semi-rigid nature of the connections causes hogging support moments to arise (Figure 3). As the usual form of the method assumes zero support moments (Figure 2a), no advantage is taken of the reduction in sagging moment. • Columns These members tend to be underdesigned, due to the detrimental effect on such members of the hogging moments developed in the beams. These moments particularly affect external columns and other members subject to unbalanced loading. However, as columns are also designed to support axial load, the underdesign of the columns is not as significant as the overdesign of the beams. • Connections The beam-to-column connections will generally be underdesigned. This is because the internal moment for at least one end of a beam will be greater than that predicted by the method, due to the hogging moments due to gravity loads (see above). As beams are usually governed by mid-span moment, whilst connections are sized only for end moment, the connections will generally be only 'partial-strength' with respect to the beams. • Sway deflections These deflections are larger than those predicted assuming rigid joints. This is because of the semi-rigid and partialstrength nature of the connections. • Frame stability The onset of frame instability will be above the design load level in low and medium-rise frames. For repeated variations of loading expected during the lifetime of the structure, such as reversals of wind load, the frame will 'shake down' with connections then behaving elastically. Some justification for the method is also given by rigid-plastic theory [8]. According to this theory, the collapse condition has the following characteristics: ⋅ a mechanism of plastic hinges has formed ⋅ the internal moments and forces are in equilibrium with the applied loads ⋅ nowhere does the internal moment exceed the plastic moment of resistance Provided that the second and third conditions are satisfied, the Lower-Bound Theorem [8] states that the applied loads are either less than or equal to the loads which collapse the frame. These conditions are met by the windmoment method, which will therefore provide safe designs, provided that the frame also satisfies the assumptions of rigid-plastic theory: ⋅ the effect of deflections on equilibrium can be neglected ⋅ collapse does not occur as a result of any form of buckling These assumptions therefore indicate those aspects of design that require particular attention if the wind-moment method is to provide frames of adequate strength. An analytical study has been made of frames designed by the method using a limit states approach [7]. This lecture describes how the method can be used in a manner consistent with Eurocode 3 [1]. The recommendations cover global analysis and member design. From the results of the studies, these recommendations are expected to result in designs for low- and medium-rise frames which possess adequate resistance. 3. SCOPE Frame Layout The method applies to steelwork which can be idealised as a series of unbraced plane frames. The range of application is restricted to multi-storey plane frames in which: • • • • • • • the frame consists principally of horizontal beams and vertical columns (Figure 4) the frame does not exceed eight storeys the number of bays does not exceed four the width of each bay is constant over the height of the frame the frames are effectively braced against out-of-plane sway at roof level and at each floor level the beam grids comprise only primary beams (Figure 5), or arrangements of primary and secondary beams as shown in Figures 6a and 6b the flooring and roofing span in the directions shown in Figures 5 and 6 The limitations concerning the number of storeys and bays have been chosen on basis of References [6, 7]. The main reasons for these limitations are: • • • the comparative rarity of unbraced construction in taller structures lack of experimental evidence on the behaviour of joints connecting sections of large size unwillingness to accept an approximate method for taller structures The arrangement of beams in Figure 6b reduces the gravity load carried by the beams forming part of the plane frame. This leads to smaller beam sections in the plane frame, compared to the grids shown in Figures 5 and 6a. All three arrangements have been considered in the study [7] which forms the basis of the recommendations. Grids which do not conform to one of these arrangements are outside the scope. This limitation is required because their possible effect on the stiffness and strength of beams relative to columns has not been studied. Frame dimensions Based on the dimensions of the frames studied, the range of application is limited according to the following: • • • The maximum column height is 6,0m for bottom storeys and 5,0m in the others The maximum span is 12,0m and the minimum 4,50m The ratio between the greatest and the smallest bay width of the frame is not more than 2 The actual height of a column should be limited because sway stiffness is inversely proportional to the square of the length of the member. Other limitations are given in Table 1. The limits are not unduly restrictive in practice. Structural sections Sections may be in S235, S275 or S355 steel, or in steel having similar structural properties. The same grade of steel should be used for all sections in a frame. In addition: • • • • hot-rolled I or H sections should be used for horizontal members HE, universal column or similar sections should be used for vertical members sections should be orientated such that loads in the plane of the frame tend to cause bending about the major axis All cross-sections, both for columns and for beams must be Class 1 The reasons for these limitations are now given. The method does not provide an exact calculation of column end moments. HE, UC or similar sections with substantial buckling resistance moment should therefore be used for these members. Column sections should be orientated as recommended because the study [7] did not examine structures in which the beams frame into the column web. There is not yet an accepted method for predicting the behaviour of such connections. The recommended orientation of the beam sections is usual practice. It is necessary to adhere to this to provide stiffness in the plane of the frame. Beam-to-column connections Extended end-plate (Figure 7a) or flush end-plate (Figure 7b) connections should be used. Connections in windmoment frames form part of the subject matter of Lecture 11.5. It is important to note that since the method results in partial-strength connections which are required to deform plastically, each connection should be designed to have sufficient rotation capacity. Column bases Columns should be rigidly connected to foundations by bases designed in accordance with usual practice for this type of construction. Frames with pinned bases are excluded from the recommendations given here. This is because columns with pinned bases require large effective lengths if they are to be safely designed. Such members also cause large sway deflection in the bottom storey of the structure. Loading The range of application is restricted to the following values of loading: • • • the total unfactored dead load plus unfactored imposed load should not exceed 12,5 kN/m2 wind loads should be based on a wind speed corresponding to a 3-second gust speed of at least 37m/s, measured at 10m above ground in an open situation for a return period of 50 years the wind load should not be such that it controls the design of any beam The tendency to underdesign columns and connections, because of neglect of end moments due to gravity load (Figure 3), is increased if the wind load is low. The restrictions on maximum gravity load and minimum wind speed restrict this tendency. If the wind load is so high that it begins to govern the design of the beams, the frame is best designed as rigidjointed. This is because the serviceability limit on sway is likely to control the design. 4. GLOBAL ANALYSIS FOR ULTIMATE LIMIT STATES Load combinations The following load combinations should be used in design: • • • 1,35 (Dead load) + 1,50 (Imposed load) 1,35 (Dead load + Imposed load + Wind load) 1,35 (Dead load) + 1,50 (Wind load). Frame imperfections should be taken into account by applying equivalent horizontal forces according to the Eurocode 3 rules [1]. Unbalanced ("chequer-board") gravity loading may be critical in the design of internal columns, and should therefore be considered in multi-bay frames. The load combination 1,35 (Dead load) + 1.50 (Wind load) will usually govern the design of the connections as moment-resisting components. Internal moments and forces due to gravity load Under gravity load, allowance should be made for the partial-fixity of the connections between a beam and a column by an end restraint moment equal to 10% of the maximum sagging moment in the beam, assuming the beam to be simply supported. In addition, the shear force at the end of the beam should be assumed to act on the column at a distance of 100 mm from the face of the column. Internal moments in beams though should be calculated for a span equal to the distance between centre-lines of columns. Each column has to be designed to resist the algebraic sum of the restraint moments from the beams at the same level on each side of the column, in addition to moments due to eccentricity of connections. The net moment applied at any one level should be divided between the column lengths above and below the level in proportion to the stiffness of each length. The moments applied to the column due to partial-fixity and eccentricity should be assumed to have no effect at the levels above and below the level at which they are applied. The assumption above of an end moment equal to 10% of the free moment combined with moments due to eccentricity of connections offsets to some extent the tendency of the method to underdesign columns and connections. Internal moments and forces due to horizontal load. Horizontal design loads arise due to: • • practical imperfections such as lack of verticality which are represented by notional horizontal forces wind load The analysis for internal moments and forces due to horizontal load should be by the "portal method" [2]. The following assumptions are made: • • • horizontal loads are applied at floor levels there is a point of contraflexure at the mid-height of each column there is a point of contraflexure at the mid-length of the each beam • each bay acts as a simple portal and the total horizontal load is divided between the bays in proportion to their spans The assumed points of contraflexure for a single bay frame are shown in Figure 1b. Analysis by "portal method" Consider the upper part of a single bay frame shown in Figure 8a. The upper part of the diagram in Figure 8b shows the forces on the portion of the frame above the points of contraflexure A and D, whilst the lower part shows the forces on the portion ABCDEF of the frame. In the derivation which follows, compressive axial force is denoted by N. The compressive axial forces at A and D are obtained by taking moments about either D or A and by considering vertical equilibrium. Taking moments about A: NDL = W1 H1 /2 (1) Hence: ND = W1 H1 /(2L) (2) Since ND + NA = 0 it follows that: NA = -W1 H1 /(2L) (3) As these forces are now known, the axial forces at C and F can be calculated by taking moments about either F or C for the region ABCDEF and by vertical equilibrium. Taking moments about C: (NF - ND) L = W1 (H1 + H2)/2 + W2 H2 /2 (4) Substituting for ND from Equation (2) and re-arranging: NF = W1 H1 /L + (W1 + W2 ) H2 /(2L) (5) By vertical equilibrium, NC + NF = 0 and NC can therefore be found. The moments in the columns at roof level are clearly given by W1 H1 /4. For equilibrium, these moments are also the moments at each end of the roof beams. The moment in the upper column at B due to the force at A is also W1 H1 /4, whilst that in the lower column at B due to the force at C is (W1 + W2) H2 /4. The same values apply in the leeward side of the frame. It follows that the moment at each end of the beam BE, which resists the sum of column end moments at B or E, is given by: MBE = MEB = (W1 H1 + (W1 + W2) H2)/4 (6) A similar calculation procedure can be followed for other storeys in the frame, or for other bays of a multi-bay structure. The resulting bending moment diagram for a complete single bay frame is shown in Figure 2b. 5. DESIGN OF BEAMS FOR ULTIMATE LIMIT STATES As the wind-moment method can be justified in part as a method of plastic design, cross-sections must be able to form plastic hinges and participate in collapse mechanisms. To prevent premature failure by local buckling, sections must therefore be Class 1, Plastic. The moment resistance though should be restricted to 90% of the plastic moment, to provide restraint to the columns. 6. DESIGN OF COLUMNS FOR ULTIMATE LIMIT STATES Effective lengths Effective lengths for compression resistance, Pc: For in-plane behaviour (bending about major axis): LE = 1,5 L (7) For out-of-plane behaviour (bending about minor axis): LE = 1,0 L (8) These effective lengths are nominal values which, in conjunction with the other recommendations, were found in the studies [7] to result in adequate column sections. The value for out-of-plane behaviour is based in the frame being effectively held against out-of-plane sway. Equivalent slenderness for buckling resistance moment, Mb: The slenderness λLT should be calculated assuming that the effective length factors k and kw given in Eurocode 3 [1] are both unity, i.e. no fixity. In addition, the factor C1 needed for the calculation of λLT for a rolled H-section should be taken as unity. These factors are consistent with simple design. As the end moments are not calculated by exact analysis, the factor C1 should be taken as the most pessimistic value, which corresponds to single curvature bending. Design moments Under each combination, the column end moment should be taken as the sum of: • • • net (i.e. out-of-balance) moment due to gravity loads and eccentricity of connections net (i.e. out-of-balance) moment due to restraint moments from the beams arising under gravity loads moment due to horizontal load As the horizontal load may reverse, the total moment should be calculated by addition of the numerical magnitudes of the component moments. Class of section Sections should be Class 1, Plastic, for the same reason as for the design of beams. Overall buckling check The proposed sections should satisfy the checks for combined bending and axial compression given in Eurocode 3, including that in which lateral-torsional buckling is the potential failure mode. The formulae include allowance for the beneficial effects of moment gradient by means of equivalent uniform moment factor, β. As the end moments are not calculated by exact analysis, the factor β should have the most pessimistic value, which corresponds to single curvature bending. 7. DESIGN FOR SERVICEABILITY LIMIT STATE General The significance of sway deflections in the design of unbraced frames is influenced by the ratio of gravity load to wind load. Even though the design of some frames will be governed by limitation of sway, for others a design made for the ultimate limit state will be adequately stiff. Design codes give recommended limits on deflections, but these limits are not performance criteria; rather, the limits are intended for comparison with the results of calculations, usually on bare frames. The justification for these limits rests on the satisfactory performance of structures in practice. Analyses accounting for connection flexibility shows that sway deflections in wind-moment frames are significantly larger than those predicted assuming rigid joints, and yet, as far as is known, structures incorporating such frames do not exhibit distress in practice. The increase in deflection given by such analyses is dependent on the moment-rotation (M - ψ) relationship of the connections. The calculated increase in deflection is large if the M - ψ relationship is represented by an elasticplastic approximation (Figure 9). A lesser increase is calculated if M - ψ is represented by a non-linear curve without a plateau (Figure 10). In shape the latter conforms more closely to the experimental behaviour of connections (Figure 11). Even with an elastic-plastic approximation, studies [7] show that wind-moment frames subject to only light wind load will not deflect more than the Eurocode 3 [1] limit of 1/500th of the total height. For frames with high wind load, in which deflections are critical, the non-linear representation of connection behaviour gives an increase in deflection of up to 60% [7]. Recommendations In accordance with usual practice, the design made for the ultimate limit state should be analysed as an elastic rigidjointed frame to determine sway deflections. If the deflections are unacceptable, then the design should be revised to provide additional stiffness: • • connection details should be changed to provide greater stiffness so that they may be taken as rigid member sections should be increased If the rigid-frame deflections are acceptable, the values should be increased by 60%. If the increased values are unacceptable, then the frame should be redesigned, following the recommendations in the previous paragraph. Otherwise, the design of the members is complete. 8. CONCLUDING SUMMARY • • • • • The assumptions of wind-moment method render the frame statically determinate. The need for repeated analysis to correspond to changed section sizes is thereby avoided. The method tends to overdesign beams and underdesign columns and beam-to-column connections. This has to some extent been accounted for by introducing a minimum end restraint moment under gravity load. Analytical studies have shown that low and medium-rise frames designed by the method have adequate overall strength. Design recommendations have been presented which are consistent with Eurocode 3. These recommendations aim to achieve a more uniform safety level in all parts of the frame. The connections should be designed to have sufficient rotation capacity to deform plastically as part of a plastic-hinge mechanism. 9. REFERENCES [1] Eurocode 3: "Design of Steel Structures": ENV 1993-1-1: Part 1.1, General rule and rules for buildings, CEN, 1992. [2] Construction Steel Research and Development Organisation, Steel Designers' Manual, Crosby Lockwood Staples, 1972. [3] American Institute of Steel Construction, Manual of Steel Construction, AISC Chicago, 1980. [4] Nethercot, D. A., "Joint Action and the Design of Steel Frames", The Structural Engineer Vol. 63A, No. 12, December 1985, pp. 371-379. [5] Gerstle, K. H., "Flexibly Connected Steel Frames, Steel Framed Structures", Stability and Strength (ed R. Narayanan), Elsevier, 1985, pp205-239. [6] Ackroyd, M., "Design of Flexibly Connected Unbraced Steel Building Frames", Journal of Constructional Steel Research Vol 8, 1987, pp 281-286. [7] Anderson, D., Reading, S. J., Najafi, A. and Kavianpour, K., "Wind-Moment Design of Unbraced Frames", Steel Construction Today Vol 6, No. 4, July 1992, pp. 159-164. [8] Neal, B. G., "The Plastic Methods of Structural Analysis", Chapman and Hall, 1977. Table 1 Relative dimensions Bay width: storey height (bottom storey) Bay width: storey height (above bottom storey) 0,90 2,50 Minimum 0,75 Maximum 2,00 Greatest bay width: smallest bay width 1,00 2,00 Table 2: Maximum column height Bottom storeys Other storeys 6,0 m 5,0 m Lecture 14.13: Design of Multi-Storey Frames with Partial Strength and Semi-Rigid Connections OBJECTIVE/SCOPE To present and illustrate procedures for the design of multi-storey buildings relating to the behaviour of beams, columns and connections. Special attention is paid to the influence of the design of partial-strength connections on the frame behaviour. PREREQUISITES Lectures of Group 6: Applied Stability Lectures of Group 11: Connections: Static Loading RELATED LECTURES Lecture 11.7: Partial Strength Connections for Semi-Continuous Framing SUMMARY The influence of partial strength and semi-rigid connections in the design of building frames of structural steel is considered. Plastically designed connections in elastically designed frames and elastically designed connections in plastically designed frames are discussed. The background to relevant parts of EC3 is also presented. 1. INTRODUCTION Steel structures for multi-storey buildings consist mainly of columns and beams. A multi-storey structure is shown in Figure 1. Within the structure two directions can be distinguished: a direction in which beams and columns are arranged into frames, and a direction in which the frames are connected to floors or beams in order to form a threedimensional structure. It is common practice to ensure the stability of the frames out-of-the-plane by triangulation or by shear walls, cores and the like. The frames are in general connected to the bracing structure with pinned connections. In the plane, the frame can be stabilized by a bracing system, or can be stable in itself (unbraced frame). The mechanical behaviour of a connection is represented in Line I of Figure 2. The figure shows a moment rotation diagram of a T-connection in a frame (beam-column connection). The real connection behaviour can be represented by a bi- or tri-linear diagram in which a number of properties can be distinguished. • • • • The (design) resistance of a connection. The stiffness of the connection when subjected to small moments, Line III of Figure 2. The stiffness of the connection when subjected to ultimate moments, Line II of Figure 2. The rotation capacity. A connection can have a lower design strength than the beam or column which it connects. In that case, a connection is partial-strength. If the stiffness of the connection has a significant influence on the stability and deformations of a frame, the connection should be regarded as semi-rigid or even nominally pinned. The influence of the connections on the frame behaviour is treated separately for partial-strength connections and semi-rigid connections. Braced frames are in general designed based on strength conditions and unbraced frames are in general designed based on stability and deformation conditions. Therefore, partial strength connections are mainly used in braced frames, and semi-rigid connections in unbraced frames. 2. CLASSIFICATION OF CONNECTIONS In Eurocode 3, criteria are given for nominally pinned, semi-rigid and rigid beam-to-column connections, when the distribution of forces and moments in the structure is determined using elastic or plastic theory. The structural properties of beam-to-column connections, such as stiffness, resistance and rotation capacity, should be in accordance with the assumptions made in the design of the structure. The structural properties of a beam-to-column connection were indicated qualitatively in Section 1 above. The classification of beam-to-column connections provides the designer with a quick answer to the question how a certain beam-to-column connection, given its the properties in a moment rotation diagram, will behave in the structure. This behaviour can be rigid or semi-rigid with respect to stiffness, full-strength or partial-strength with respect to moment resistance and ductile or brittle with respect to rotation capacity. These properties are illustrated in Figure 3. In this figure Mplb.Rd represents the design plastic moment of the beam. The boundaries between the areas of behaviour of beam-to-column connections are discussed below. The distribution of forces and moments in a structure is influenced by the flexibility of the connections in that structure. Stability, deformations, and displacements are also similarly influenced. The flexibility of beam-to-column connections may be neglected in some cases, i.e. the connection may be assumed to be rigid in some cases and a hinge in others. The assumption which can be made depends on the stiffness ratio between the beam-to-column connection and the connected beams and columns. The structural behaviour will be analyzed by showing the relationship between the parameters and ρ. The parameter is the relative rotation stiffness = , in which s is the rotation stiffness of the beam-to-column connection and is the flexural stiffness of the beam. The parameter ρ is the ratio between the flexural stiffness , in which is the flexural stiffness of the beam and is the of the beam and the column, i.e. flexural stiffness of the column. For one-bay single-storey frames, both braced and unbraced, the relationships between shown in Figure 4. and ρ can be determined as The relationship can be determined at a constant ratio between the Euler buckling load of the frame with semi-rigid connections and the Euler buckling load of the same frame but now with rigid connections. For this ratio, and according to Eurocode 3, the value 0.95 is chosen. In Figure 4 the relationship between and ρ is shown for the unbraced as well as for the braced frame. The mathematical background of these relationships is given in the background report to this subject of Eurocode 3 (9). With the aid of the Merchant-Rankine formula (Lecture 7.7) for the unbraced frame, the influence of reduction of 5% in the Euler buckling load on the carrying resistance of the frame can be shown. The reduction is a consequence of the semi-rigidity of the beam-to-column connections. Suppose Fpl is the load at plastic failure of the frame and FE( = ∞) is the Euler buckling load of the frame with perfectly rigid connections. Then FE( ) = 0.95 FE ( = ∞) is the Euler bucking load of the frame with semi-rigid connections which have flexibility such that the Euler buckling load is lower by 5% compared to the situation with perfectly rigid connections. Suppose that FE ( = ∞) = k . Fpl. Using connections with a semi-rigidity such that the Euler buckling load reduces by 5%, the following holds: FE ( ) = 0.95 . k . Fpl. On the basis of the Merchant-Rankine formula for the determination of the carrying resistance of the unbraced frame, , it can be shown that the carrying resistance will drop by not more than 5%. Figure 4 gives the relationship between the geometry of the frame and the ratio of flexural stiffnesses between the connection and the beam for those rotation stiffnesses of the connection which can be assumed to be perfectly rigid, because the flexibility of the connection causes a drop of the carrying resistance of the frame of not more than 5%. To verify the design of the frame against the requirements, all the geometrical data are required. These data are the sections used as columns and beams together with the lay-out of the beam-to-column connections. From the lay-out of the connections the moment-rotation relationship can be determined on the basis of Annex J of Eurocode 3. With that data the parameters and ρ can be determined and, via Figure 4, the influence the connection stiffness has on the distribution of forces and moments, and on the stability of the frame can be shown. The design may have to be made when not all the data are known and it is difficult to estimate before hand what influence the connections will have on the behaviour of the frame. In Eurocode a further simplification is given for this case to enable the connection influence to be estimated. By choosing a constant boundary value for the factor frames the boundary value is The lines of = 8 and , it becomes independent of the parameter ρ . For braced = 25. = 8 covers the ρ - = 8, and for unbraced frames the boundary value is = 25 are drawn in Figure 4. It can be seen that the boundary value relationship for braced frames completely. The boundary value relationship only if ρ ≥ 1.4. For ρ < 1.4 the boundary value investigated. = 25 for unbraced frames covers the ρ - = 25 is, in principle, unsafe. This situation will now be Frames for which ρ < 0.1 are not realistic, so the value ρ = 0.1 can be used as a boundary. Figure 5 shows the relationship between ρ and (FE ( based on = 25)/FE ( = ∞)) . 100%. When ρ = 0.1 the Euler bucking load = ∞. = 25 would be not more than 85% of the Euler buckling load if the value for The carrying resistance of the frame based on the Merchant-Rankine formula has reduced as follows: where Fcr ( = ∞) is the carrying resistance of the frame: (2) The following relationship holds: (3) The reduction of the carrying resistance is: (4) Even for a relatively slender frame, namely FE ( reduction ∆ = 5.6%. = 25) = Fpl, so X = 1 the reduction ∆ = 8% and for X = 2 the It can be concluded that = 25 is a sufficiently safe boundary value for the rotation stiffness of beam-to-column connections in unbraced frames in order to consider them as rigid, ensuring that FE/Fpl ≥ 1. As far as the moment resistance of beam-to-column connections is concerned, classification is simpler. If the moment resistance of the beam-to-column connection is equal to the plastic resistance of the connected beam, the connection is considered as full-strength. If not, the connection is considered as partial-strength. On the basis of the boundary for the rotation stiffness and the boundary for the moment resistance a bi-linear moment-rotation characteristic is achieved, as shown by the dotted lines in Figure 6. If the moment-rotation characteristic of a beam-to-column connection lies on the left hand side and above the boundary lines, this connection can be classified as rigid and full-strength. The bi-linear boundary is rather severe for classifying beam-to-column connections, when it is compared with the moment-rotation characteristic of a beam section. If the moment acting on a beam section exceeds the elastic moment resistance, Me = 0.85 Mpl for an I-section, then plastification will develop and the stiffness will decrease. If residual stresses are taken into account plastification will commence at M = 0.7 Mpl. So it is reasonable to cut off the bi-linear characteristic with a third branch. Tests show that beam-to-column connections with end-plates have an elastic behaviour up to at least 2/3 of the moment resistance of the connection. On this basis, full-strength connections are required to behave elastically up to at least 2/3 Mpl,beam. Theoretically the plastic resistance of a beam section is reached at an infinitively large rotation of the plastic hinge. In practice a large percentage of the plastic moment resistance is reached at a relative small rotation. In Figure 6 the boundaries are shown for braced and unbraced frames as given in Eurocode 3. In Figure 6 both axes of the moment-rotation characteristics are normalised by dividing the moment by the plastic moment resistance of the beam, so = M/Mpl; beam and by dividing the rotation by a reference rotation, so: (5) If the portion of its moment rotation characteristic which is used lies below the appropriate line in Figure 6, a beam to column connection should be classified as semi-rigid, according to Eurocode 3 (Clause 6.9.6.2), unless it also satisfies the requirements for a nominally pinned connection. The requirements for rotational capacity are different for unbraced frames and braced frames. For unbraced frames, rotation capacity has to be calculated with help of global frame analyses. The calculation is as follows: Assume a beam has a pinned connection at both ends and is loaded by an equivalent distributed load transverse to the axis as shown in Figure 7. The ends of the beam are loaded with external moments M. The rotation at the end support due to the load will be: (6) In Figure 8 the M-φ rotation characteristics of three connections with strengths MRd (I,II and III) are shown. Line II represents a connection with a stiffness which fits the elastic model as shown in Figure 7. No plastification occurs in the beam when reaching the equilibrium situation. The stiffness of this connection is: (7) Provided that: (8) where Mpl.b is the moment resistance of the beam at mid span, this stiffness can be used in three situations. For S < Sb (Line III): Plastification will occur in the mid span of the beam and, the required rotational capacity . As a matter of fact, this will always occur in plastically designed connections. It is important to check, equals whether the beam at mid span has sufficient rotational capacity. In other words: Is the section at mid span an Eurocode 3 Class 1 section (see Lecture 7.8.1(i). For S = Sb (Line II), no plastic hinges will occur in the structure. For S > Sb (Line I) plastic hinges will occur in the connections. In that case, the rotational capacity of the connections has to be equal or greater than: (9) This is the rotation at the intersection between the horizontal branch of Line I and Line (a) in Figure 8. If the moment resistance of the connection is determined conservatively, a positive effect is produced in the required rotational capacity of the connection. For very flexible connections (S < Sb), the same effects arises if stiffness is underestimated. A classification with respect to rotation capacity is not yet possible for unbraced frames. The rotation capacity need not be checked only in cases where the moment resistance of the beam-to-column connection is larger than 1.2 Mpl;beam. The plastic hinge will always form on the beam section adjacent to the connection. In other cases the rotation capacity should be checked if redistribution of moments is taken into account. For unbraced frames the required rotation capacity has to be calculated and checked against the rotation capacity which is available in the connection. For braced frames the required rotation capacity can be determined by looking at the beam mechanisms which will form. It appears that the classification described above can be used safely also for multi-storey frames and multi-bay frames (9). 3. RELATION BETWEEN FRAME AND CONNECTION BEHAVIOUR Table 1 gives a summary of the possible relationships which may be adopted in designing frames and connections. Frames and connections can be designed either by elastic theory or by means of plastic theory. It is even possible to perform the calculations for frames on the basis of elastic theory and for connections on the basis of plastic theory, or the other way around. There is, however, one limitation. A connection which remains elastic up to failure can only be used in frames designed by plastic theory provided that the calculated moment resistance is higher than that of the connected beam, and the beam sections have to be of the Class 1 type (see Lecture 7.8.1(i)). Therefore, a full-strength connection is required in order to achieve sufficient plastic deformation adjacent to the connection. The summary given in Table 1 only holds for partial-strength connections. When the force distribution in a connection is based on elastic theory, the connection will in general be stiffer than a connection designed in accordance with plastic theory. Clearly, this statement only holds when the connection in reality fulfils the assumptions used in the elastic or plastic theory calculations. It can be illustrated as follows. In using elastic theory, the hypothesis of Bernoulli holds (cross-sections remain plane). Such a hypothesis is employed to calculate the force distribution in the bolts connecting the end plate to the column flange. In reality, the end plate and the column flange have to remain plane. If this is realised, the connection is stiff because the only deformation is caused due to the elongation of the bolts. Using plastic theory, it is necessary for the components in the connection to deform sufficiently in order to obtain a redistribution of forces and the formation of a failure mechanism inside the connection. The consequence is that such a connection is in general less stiff than a connection designed according to elastic theory. Table 1 shows which relationships between the design methods for frames and connections need further explanation. The behaviour of elastically designed frames with elastically designed fully-rigid connections is well known. Connections calculated on the basis of elastic theory which remain elastic up to failure are not allowed in frames designed by means of plastic theory, unless the moment resistance of the connections is higher than that of the connected beams. On the other hand, connections designed in accordance to plastic theory can always be used in elastically designed frames. In Table 1, references are given relating to the design of braced frames according to the elastic theory, taking second-order effects into account. These references discuss the use of the stiffness of the connection together with the bending stiffness of the beam in order to reduce the (elastic) effective length of the column. In (10) it is shown that this approach leads to a strength requirement for both the connection and the beam. 4. PLASTICALLY DESIGNED CONNECTIONS IN ELASTICALLY DESIGNED FRAMES In calculating the force distribution in a frame using elastic theory, the ultimate carrying resistance is not calculated. Instead the forces and moments due to the design load are computed. If the ratio n between the Euler buckling load and the design load is small, say smaller than 10, (see Lecture 7.7) second-order effects have to be taken into account. This can be done by multiplying the components of the moments due to side-sway (in the case of unbraced frames) by the amplification factor n/(n-1). If the moments, calculated in this way, are smaller than the ultimate moment resistances of the various connections, then the connections meet the design requirements. In the connections, designed in accordance with plastic theory, plastic deformations will occur when the ultimate moment resistance is reached. This effect is taken into account in calculating the Euler buckling load using the bilinear approximation of the connection behaviour, which leads to a safe result (8). In the case of braced frames, the moments in the connections due to loading will perhaps be underestimated if a low rotational stiffness is used. Therefore, one should take an upper bound for the connection stiffness when calculating safe values for the moments in the connections. This approach is in contradiction to the advice given earlier, namely the use of the secant stiffness (bi-linear approximation). It is shown in the literature that using the bi-linear approximation for the rotational stiffness of the connection in calculating the moments in the connection is still safe provided that the connection possesses sufficient deformation capacity. In the case of unbraced frames the use of a lower bound for the rotational stiffness of the connection will lead to higher values for the moments in the connections due to the increased second-order effects. However, the first-order elastic moment in the connection decreases when a lower rotational stiffness is used. A lower bound for the rotational stiffness of connections in an elastic design of an unbraced frame does not necessarily lead to a safe elastic calculation of the moment in the connection. It is therefore necessary to use connections with sufficient rotational capacity even in an elastically designed frame. 5. ELASTICALLY DESIGNED CONNECTIONS IN PLASTICALLY DESIGNED FRAMES In frames where more than one plastic hinge is necessary in order to reach the plastic failure mechanism, the first, second and subsequent plastic hinges have to rotate until the last plastic hinge is formed. This requirement holds for braced as well as for unbraced frames. Partial-strength connections which remain elastic up to failure are not to be used, because they possess insufficient deformation capacity. A method is given above for the calculation of the required rotational capacity of connections in braced frames. The required rotational capacity in unbraced frames is larger, and should in fact be calculated for the actual geometry of the unbraced frame. In most cases, however, an upper bound for the required rotational capacity is about 0.04 radians. 6. CONCLUDING SUMMARY • Real connection behaviour is characterised by: ⋅ design resistance ⋅ stiffness ⋅ rotation capacity • • • • • • Connections that are partial strength and/or semi-rigid influence significantly frame behaviour. Partial strength connections may be used in braced frames. Semi-rigid connections may be used in unbraced frames. It is possible to define tri-linear boundaries between rigid and semi-rigid behaviour. Rotation capacity is an important criterion for the safe use of partial strength connections. It is possible to apply elastically designed connections in plastically designed frames and vice versa. The consequences of this application are treated in the last paragraphs of the lecture. 7. REFERENCES 1. Zoetemeijer, P., "A design method for the tension side of statically loaded bolted beam-to-column connections", Monogr. Heron 20 (No. 1), Stevin Lab Techn. Univ. Delft / TNO Building and Construction Research 1974. Zoetemeijer, P., "Bolted connections with flush end-plates and haunched beams", Test and Limit State Design Methods, Rept 6-81-15, Dept. Civ. Eng. Tech. Univ. Delft, 1981. Zoetemeijer, P., "Bolted beam to column connections with flush end-plates and haunched beams", Test and Limit State Design Methods, Rept 6-81-23, Dept. Civ. Eng. Tech. Univ. Delft, 1981. Eurocode 3, " Design of steel structures" -Edited draft issue 3, November 1990. Tautchnig, A., Entwicklung eines neuen makromechanischen Knotenmodells and Erstellung eines darauf aufbauenden EDV-Programmes zur berechnung van Stahlskeletellragwerken unter Berücksischtigung nichtlinearer Nachgiebigkeiten der Verbindungselemente ins besondere bei steifenloser Bauweise. PhD Thesis, University of Innsbruck. Sugimoto, H., Chen, W. F., "Small end restraint effects on strength of H-columns", J. Struct Div. ASCE 108 (1982) 661-81. Jones, S. W., Kirby, P. A. and Nethercot, D. A., "Columns with semi-rigid joints", J. Struct Div. ASCE 108 (1982) 361-72. Bijlaard, F. S. K., Zoetemeijer, P., "Joint characteristics and structural response of frames, Steel structures Recent research advances and their application to design", Elsevier (1986) 109-133. Meijer, H. S., "Influence of the rotational stiffness of column-beam connections on the behaviour of braced and unbraced frames" (in Dutch), Technical University Eindhoven, Netherlands, 1990. 2. 3. 4. 5. 6. 7. 8. 9. 10. Snijder, H. H., Bijlaard, F. S. K. and Stark, J. W. B, "Use of the elastic effective length theory for stability checks of columns and consequences for checks on beams in braced frames, instability and plastic collapse of steel structures", edited by L J Morris, London, Granada, 1983, 152-63. Table 1 Summary of Possible Design Relationships between Frames and Connections Partial-strength connection Elastic Design Rigid Frame Elastic theory First-order Braced Common practice Semi-rigid Plastic Design Full-strength Partial-strength MomentNo particular No particular rotation has to problems to problems to be be known be expected expected See Ref 4 Moment-rotation characteristic has to be known No particular problems to be expected See Ref 5 Unbraced Second-order Braced effects included Unbraced See Ref 4 See Refs 6, 7 See Ref 4 Momentrotation characteristic has to be known Not allowed See Ref 4 Plastic theory First-order Braced No particular problems to be expected Moment-rotation characteristic has to be known Unbraced Second-order Braced effects Unbraced included See Ref 4 See Refs 1-3 Moment-rotation characteristic has to be known Lecture 14.14: Methods of Analysis of Rigid Jointed Frames OBJECTIVE/SCOPE: To give the designer a deeper understanding of how to analyse a rigid jointed frame structure and to describe the checks to be performed according to Eurocode 3 [1]. PREREQUISITES Lecture 7.11: Frames Lecture 14.8: Classification of Multi-Storey Frames RELATED LECTURES Lecture 14.2: Analysis of Portal Frames: Introduction and Elastic Analysis Lecture 14.3: Analysis of Portal Frames: Plastic Analysis Lecture 14.7: Anatomy of Multi-Storey Buildings Lecture 14.10: Simple Braced, Non-Sway Multi-Storey Buildings SUMMARY In this lecture the following subjects are treated: • • • • • • • First order elastic global analysis Second order elastic global analysis Rigid plastic analysis Elastic-perfectly plastic analysis Elastic-plastic analysis Possibilities and limitations for the choice of method Calculation of internal forces and moments. The discussion is closely related to the Eurocode 3 [1] approach to the analysis and design of framed structures. 1. INTRODUCTION In this lecture the methods of structural analysis are discussed. Structural analysis provides the internal forces to be used in safety checks. In the choice of the structural analysis method, different levels characterized by different degrees of accuracy can be obtained. A very refined degree of analysis, and hence very accurate methods of analysis, are useless in most of the actual cases in which the common simplified assumptions still hold. It is for this reason that most codes do not refer explicitly to the refined methods and instead only advise simple methods of elastic analysis. Most recent codes, e.g. Eurocode 3 [1], allow the use of all the well established methods of analysis. They therefore allow the analysis of practically all types of structures using different methods of analysis, depending on the available calculation tools. The approach adopted by the Eurocode 3 [1] is followed in this lecture. 2. EUROCODE 3 APPROACH TO ANALYSIS AND DESIGN 2.1 General Approach The Eurocode 3 [1] approach to analysis and design requires the following steps: 1. 2. 3. 4. 5. 6. Classification of the frame Assessment of imperfections Choice of the method of analysis Computation of internal forces and moments Global buckling check Checks of the members. The procedure is outlined in Figure 1. For the classification of frames, see Lecture 14.8. This Lecture 14.14 deals with imperfections, the different methods of analysis and the calculation of internal forces and moments. The global buckling check can be performed by means of exact or approximate methods. For member checks, see Lectures 7. 2.2 Second Order Effects Eurocode 3 allows a first order analysis in the cases outlined in Figure 2, i.e. in non-sway structures as defined in Lecture 14.8 or in sway structures when an indirect amplification is made of the moments. In particular, the indirect amplification can be made by means of the two methods known as the Sway Mode Buckling Length Method or Amplified Sway Moment Method. The procedures for applying the latter method is summarized in Figure 3. See also Section 3.4.4. 2.3 Imperfections 2.3.1 Common practice In common practice global frame imperfections are not included in the analysis of rigid jointed framed systems. In particular the geometrical imperfections of the frame are not usually taken into account. Other imperfections, i.e. cross-section and geometrical member imperfections and mechanical imperfections, can be considered part of the knowledge of the steel designer following the extensive research carried out in the 1960's. Therefore, whilst columns are immediately characterized by initial camber and out-of-straightness and crosssections are affected by residual stress patterns, it is not yet usually assumed that also the frame is affected by initial out-of-plumb, there is misalignment of the columns and girders and so on, due to erection procedures and fabrication processes. In common practice the effects of all these imperfections on the frame behaviour are assumed not to be significant and the safety coefficient takes into account the approximations of the analysis which assumes an ideal frame. 2.3.2 Definitions and Eurocode 3 provisions As already indicated, it is not common practice to assume global frame imperfections in the analysis. This approach is due partly to the fact that no extensive research has been carried out in this field. Some indications of frame imperfections can be found in the ECCS Recommendations [2], and in [3]. The concept of frame imperfections was introduced in Eurocode 3. A distinction has to be made between the allowable tolerances which can be accepted for erection purposes and the values which might be included in the analysis of the effects on internal forces and moments. For frame imperfections, see Figure 4, the Eurocode [1] provides in 5.2.4.2 the method of application at (1) and (2): "Imperfections shall be allowed for in the analysis by including appropriate additional quantities, comprising frame imperfections, member imperfections and imperfections for analysis of bracing systems. The effects of the frame imperfections given in 5.2.4.3 shall be included in the global analysis of the structure. The resulting forces and moments shall be used for member design". Clearly frame imperfections cannot be neglected. The engineer now needs to know their values and how to include them into design. Eurocode 3, at 5.2.4.3: Frame imperfections, provides the following indications on the values to be used: "The effects of imperfections shall be allowed for in frame analysis by means of an equivalent geometric imperfection in the form of an initial sway imperfection φ determined from: φ = k c k s φ0 where φ0 = 1/200 kc = but kc ≤ 1,0 but ks ≤ 1,0 ks = nc is the number of columns per plane ns is the number of storeys". Figure 5 shows the initial sway imperfection φ for the case nc = 2, i.e. for a one-bay multi-storey frame, together with a curve representing the erection tolerances [3] for the same frame. As an alternative to an analysis in which initial sway imperfections are explicitly taken into account, it is allowed to replace these imperfections by equivalent horizontal forces on each floor, see Figure 6. For more information on imperfections, see Lecture 14.9. 3. METHODS OF GLOBAL ELASTIC ANALYSIS 3.1 Premise When defining methods of analysis to be used for the calculation of internal forces, it is necessary to make some assumptions for the models to be adopted for the cross-section behaviour and for the material behaviour. The models can be extremely simple, e.g. elastic analysis which assumes the material, the cross-section and the structure to behave indefinitely in an elastic manner. Alternatively they can be more complicated up to the level of complete simulation of the real in-elastic behaviour of the structure, e.g. elasto-plastic analysis. The definitions, given below are extracted from the Eurocode 3 [1]. The first important division between the methods of analysis is the one which separates elastic and plastic methods. Whilst elastic methods can be used in all cases, the plastic methods may be used only in the case in which the material and the cross-sections satisfy specific requirements. The different types of plastic analysis are used depending on the assumptions made concerning the material and cross-section behaviour and the specific analytical procedure adopted for simulating the structure behaviour. In particular the most used types of analysis are RigidPlastic, Elastic-Perfectly Plastic and Elasto-Plastic methods. Another important distinction in the methods is between those which make allowance for the effects of deformations and those which neglect these effects. In common practice these methods are also referred to as first order and second order methods (Figure 7). It is clear that second order methods can be adopted in all cases since they do not make any simplification. Therefore they lead to a more accurate evaluation of the internal forces and moments than first order methods. First order methods in comparison have to be used with simplifying assumptions which guarantee that the actions in the structure as derived by the deformed configuration are negligible when compared to the ones computed on the undeformed structure. For a quantification of these requirements, see Section 3.4.3. 3.2 First Order Elastic Global Analysis First order elastic global analysis of rigid jointed steel frames can be carried out with the usual methods of structural analysis. The method commonly used is the displacement method which uses the equilibrium equations for the different internal actions in each node. The result is a system of equations where the displacements in each joint are the unknowns. In matrix form the problem is written: F = kδ (1) where F are the external actions on the nodes expressed in the global coordinate system δ are the unknown displacements of the nodes in the global coordinate system K is the stiffness matrix of the structure obtained by assembling the stiffness matrices of the different bars in the global coordinate system The solution of the system of equations in the unknown delta gives the global displacements. They enable, through the transfer matrix, the local displacements at each end of the elements to be derived. Hence the internal forces and moments are obtained through the element stiffness matrix. The stiffness matrix of the single element, written in the local coordinate system in the case of plane frames, is found in standard text books, see Section 7, Reference 6. For first order elastic global analysis of rigid jointed frames there are no coefficients to take account of relative rotations at the joints and of the change of the flexural stiffness terms due to axial loads. Since it is a first order elastic method, the solution is a one step process without any need for iteration of the external loads or updating of the matrices. 3.3. Second Order Elastic Global Analysis It is sometimes necessary to take into account that the deformations due to the external loads can considerably modify the structural response and therefore the value of the internal actions. Second order analysis is carried out by using the equilibrium equations for the deformed shape of the structure and the structural elements. In particular, for rigid jointed rectangular frames, the internal actions which cause the most modification of the response are the axial loads. With this assumption, the second order effects can be restricted to local second order effects (first non linearity), commonly referred to as P-delta effects, see e.g. Figure 8, and global second order effects (second non linearity), referred to as P-Delta effects, see e.g. Figure 9. Local second order effects arise in each element subjected to axial load (columns) due to the midspan deflection, whilst global second order effects arise in the frame due to relative displacements between the floors (drifts). 3.3.1 Local second order effects (P-delta) To take into account the local deformation of each element, it is necessary to rewrite the terms of the stiffness matrix of the single element. This is easily obtained if the flexibility coefficients of a simply supported beam under axial loads, where the effects of deformations of the beam itself are taken into account, are known. To derive the flexibility coefficients, the differential equation of the simply supported beam of Figure 8 is written (see standard text books on applied mechanics: -EIw" = (2) This equation shows that the bending moment includes also the effects of the axial load. By imposing the boundary conditions, the values of the flexibility coefficients are: α= where: ;β= (3) φ and ψ are coefficients depending on the parameter kL = L (4) kL represents the ratio between the design axial loads in the column and the Euler buckling load obtained assuming no interstorey drift. φ and ψ are found in standard text books. By adopting these modified flexibility coefficients, usual procedures for constructing the new beam stiffness matrix Ko can be followed. It is worth noting that, for values of kL less than 0,5, all the coefficients are approximately equal to 1 and therefore there is no need to modify the beam stiffness matrix. For kL = 0 the first order situation appears with the exact values φ = ψ = 1. 3.3.2 Global second order effects (P-Delta) To determine how the global frame stiffness matrix is affected by the effects of interstorey drifts in the case of rectangular frames, only some of the stiffness coefficients of the single element have to be reviewed. From Figure 9 it appears that a relative displacement u1 = 1 between the ends of the beam element leads to a modification of the end shears: ∆V1 = λ and ∆V2 = λ and, thus, to a modification of the matrix of the single element: ki = k0i (λ) - λk1i (5) where k0i is the stiffness matrix which takes into account the local deformation of the element, which is relevant also for non-sway frames k1i is the matrix which takes into account global displacements of nodes. The global stiffness matrix of the frame, which can be derived by assembling the single stiffness matrices with the usual procedure, has a form similar to matrix (5): k = K0 (λ) - λK1 (6) To compute the elastic critical load in the exact manner, the determinant of the matrix (6) where Ko is the stiffness matrix of the structure including the P-delta effects and K1, takes into account the P-Delta effects has to be zero. The eigenvalue problem to be solved is: |K0 (λ ) - λ K1| = 0 (7) The different solutions of this equation represent the eigenvalues λ and the smaller one represents the elastic critical load of the structure. Several numerical procedures are available in existing computer packages for finding these zeros. 3.3.3 Approximate evaluation of second order effects As mentioned earlier in Section 2.3, second order effects can be taken into account approximately by making use of equivalent horizontal forces applied at each storey. The procedure of second order analysis is an iterative one and makes use, in each step, of an elastic analysis. Equivalent horizontal forces can be applied as suggested by the Eurocode 3 [1] in lieu of the initial sway imperfections, see Figure 6. 3.4 Calculation of Internal Forces and Moments This section indicates, by reference to Eurocode 3 [1], strictly related to how to choose the level of elastic analysis of the structural behaviour in order to evaluate the internal forces and moments for the safety check of members. 3.4.1 Effects of deformation The internal forces and moments may generally be determined using either first order theory or second order theory in accordance with 5.2.6.2 of Eurocode 3, see Figure 2: • First order theory may be used for the global analysis in the following cases: a. non-sway frames b. sway frames, when design methods which make indirect allowances for second order effects are used • Second order theory may be used for the global analysis in all cases. When a first order elastic analysis is used based on the above assumptions, the calculated elastic moments may be redistributed by modifying the moments in any member by up to 15% of the peak moment in that member, provided that: a. the internal forces and moments in the frame remain in equilibrium with the applied loads b. all the members in which the moments are reduced have Class 1 or Class 2 sections 3.4.2 Braced frames The problem of choosing a first or second order elastic analysis can be solved by considering the detailed definitions and explanations given in Lecture 14.9. 3.4.3 Non-sway frames The definitions of the different systems (bracing systems and braced frames, sway and non-sway, etc.) are given in Lecture 14.9. Following the choice of a structural system, assumptions are made concerning the structural behaviour. These assumptions have to be verified. At this stage it is important to verify if the structure is a sway or non-sway type. For this purpose the following practical procedure, given in Eurocode 3, 5.2.5.2 [1] as an application rule, can be followed: "A frame may be classified as non-sway for a given load case if the elastic critical load ratio VSd/Vcr for that load case satisfies the criterion: VSd/Vcr ≤ 0,1 where VSd is the design value of the total vertical load and Vcr is the elastic critical value for failure in a sway mode". The problem is therefore shifted to the evaluation of the elastic critical load. Several procedures can be followed for this purpose: • • • • exact evaluation of the elastic critical load evaluation of the elastic critical load by means of a deformability check approximate evaluation of second order effects indirect evaluation of the elastic critical load by means of an application rule suggested in Eurocode 3 [1] for simple structures. For simple structures, 5.2.5.2 of Eurocode 3 provides the following application rule for the indirect evaluation of the critical load: "Beam-and-column type plane frames in building structures with beams connecting each column at each story level may be classified as non-sway for a given load case if the following criterion is satisfied. When first order theory is used, the horizontal displacements in each storey due to the design loads (both horizontal and vertical), plus the initial sway imperfection applied in the form of equivalent horizontal forces, should satisfy the criterion: ≤ 0,10 where δ is the horizontal displacement at the top of the storey, relative to the bottom of the storey h is the storey height H is the total horizontal reaction at the bottom of the storey V is the total vertical reaction at the bottom the storey". The simple interpretation of this criterion is that the second order moment equal to Vδ is at least ten times less than the first order moment H h, see Figure 7. 3.4.4 Design methods for the elastic analysis of sway frames (direct or indirect allowances) As already indicated, a rigid jointed frame can be considered as a non-sway frame when the ratio between the design vertical load Vsd and the critical load Vcr is less than 0,10. When this relation is not satisfied, it is necessary to include second order effects in the analysis. For this purpose Eurocode 3 [1] allows the adoption of the following procedures, see Figure 2: • Direct methods for second order elastic analysis 1a The first method, the most general, consists of checking the safety of elements in buckling and strength on the basis of the internal forces and moments computed with a second order elastic analysis as described in Section 2.3.2. For this purpose the safety check of single elements is carried out assuming values for effective length corresponding to the case of non-sway frames. 1b Alternatively for building structures, the approximate method known as the Equivalent Lateral Force Procedure can be used. This procedure assumes there are no axial deformations in the members and that the second order effects are due only to horizontal displacements. The procedure, which is an iterative one, evaluates the global storey moment, as given by the total axial load times the relative storey drift, and therefore the equivalent lateral forces which then enable a new horizontal displacement to be compared to the previous one. The procedure is stopped when the difference between two subsequent steps is small in terms of additional forces or displacements. For the strength and buckling safety checks the same considerations hold, as given in paragraph 1a above. • Design methods which make indirect allowances for second order effects (5.2.6.2 of Eurocode 3) 2a In the elastic analysis of sway frames it is possible, as an alternative to second order analysis carried out by following the procedures above, to carry out a first order elastic analysis by means of two different procedures in the safety check of members. The first one is defined as the Amplified Sway Moment Method which can be adopted when Vsd/Vcr is less than 0,25. The approximate evaluation of the second order effects is then based on the amplification of the bending moments associated with the loading conditions which produce lateral displacements of the frame, see Figure 3. The amplification factor for the moments is given by: C = 1/(1 - Vsd/Vcr ) or, approximately, by: C = 1/(1 - δV/hH) The Amplified Sway Moment Method requires the adoption of an effective length for the member buckling checks, equal to the one computed in the case of non-sway frames. 2b A second procedure, which still allows a first order elastic analysis, makes use of the effective length for the columns as computed for sway frames. This procedure is known as the Sway Mode Buckling Length Method. The well-known alignment charts or the Wood diagrams can be used for this purpose, see e.g. [4]. The calculation of internal forces and moments, amplified to take into account sway effects in the different ways described above, is followed by the strength and buckling safety checks on the different members as discussed in other lectures. 3.5 Cross-Section Requirements There are practically no limitations in choice of the cross-sections. Any class may be used, taking into account the limit on the resistance of cross-sections due to local buckling. 4. METHODS OF GLOBAL PLASTIC ANALYSIS 4.1 Rigid-Plastic Analysis The elastic methods described above enable a solution in terms of displacements from which the strains and therefore the stresses can then be derived. The classic methods of plastic analysis follow a different logical procedure. They are not concerned with the elastic deformations and the subsequent stresses. Plastic theory, in its first and classical derivation, in fact is only concerned with the derivation of the ultimate resistance of the structure and does not provide any information on the deformations of the structure itself. The methods which are commonly referred to as rigid plastic methods, and are described in Eurocode 3 [1], are the ones which adopt these assumptions. In fact, Eurocode 3 [1], when referring to "Rigid-Plastic" methods (Clause5.2.1.4(6)), states: "In Rigid-Plastic analysis, elastic deformations of the members and the foundations are neglected and plastic deformations are assumed to be concentrated at plastic hinge locations". The main difference between elastic and plastic methods is that whilst elastic analysis is concerned at the same time with equilibrium and compatibility, in plastic methods, only the equilibrium equations have to be satisfied since the structure can overcome the compatibility at different joints by allowing the formation of plastic hinges. It is clear that in some cases the application of the plastic concept is more simple and straightforward. If elastic analysis is used, the compatibility and equilibrium equations at the joints lead to the elastic distribution of internal forces; for plastic analysis, the plastic resistance of the cross-section of each member connected at the same joint immediately allows the ultimate resistance of that joint to be defined as the summation of all the plastic moments. The plastic method is sometimes simpler therefore and provides more information in terms of ultimate resistance which, in the limit state design philosophy, represents an important limit state. The safety of the structure is derived by defining the factored load and the factored resistance. Another advantage of the plastic method is that it is insensitive to geometrical and mechanical imperfections of the cross-sections and the frame since they affect the elastic distribution of stresses but have no effect on ultimate resistance. A disadvantage of the plastic method is however that it cannot provide any information relating to serviceability limit states since it has no concern with elastic deformations and loss of compatibility. It cannot be used alone therefore and has to be complemented with an elastic analysis of the structure for serviceability. 4.1.1 Assumptions, limitations and cross-section requirements The classical rigid-plastic methods are derived within the following main assumptions and limitations: • • • • the material and the cross-section are rigid-plastic, see Figure 10b. This assumption means that the value of the ultimate moment (plastic moment) for each cross-section is the only parameter which affects the analysis. There is no need to make any assumptions on the deformation capacity of the cross-section since the analysis does not lead to any information on the deformations of the structure. there is no modification of the plastic moment due to the effect of shear or of the axial loads. there is no allowance for any buckling phenomena either in the members or in the entire frame. the loads increase in proportion. In Eurocode 3 [1], to apply this method some requirements are made on the cross-section in order to guarantee that the fully plastic moment can be developed and sufficient rotations are developed within the joints in which plastic hinge locations are formed. Class 1 cross-sections have to be used. 4.1.2 Computation of collapse multiplier of loads In order to compute the collapse multiplier of external actions, the classic theorems of plastic analysis are usually adopted. The kinematic and static theorems which allow a set of unconservative (kinematic) and a set of conservative multipliers (static) to be defined which includes the collapse multiplier. These well-known theorems are: Static theorem: If a distribution of bending moments exists throughout a frame which is both safe and statically admissible with a set of loads λ, the value of λ must be less than or equal to the collapse load factor λc. Kinematic theorem: For a given frame subjected to a set of loads λ , the value of λ which corresponds to any assumed mechanism must be either greater than or equal to the collapse load factor λc. 4.2 Elastic-Perfectly Plastic Analysis The elastic-perfectly plastic methods are included in the category of plastic methods. They contain some improvements with respect to classical plastic theory and therefore with respect to the rigid-plastic methods. In particular, 5.2.1.5 of Eurocode 3 [1] states: "In Elastic-Perfectly Plastic analysis it is assumed that the cross-section remains fully elastic until the plastic resistance moment is reached and then becomes fully plastic. Plastic deformations are assumed to be concentrated at plastic hinge locations", see Figure 10c. The hypothesis and limitations are therefore practically the same as those already outlined for the rigid-plastic methods except for the introduction of elastic deformations of the cross-sections which allow not only the global collapse load but also the load-displacement history of the frame to be determined. To enable computation of the plastic rotations at all the joints, a further hypothesis is made that the material and the cross-sections are perfectly plastic, i.e. they can undergo indefinite deformations (rotations). In practice an elastic-perfectly plastic analysis is carried out by means of a step by step procedure. This method, even though it leads to the non-linear load-displacement curve of the frame, does not need any type of iteration. In fact it is simply made by the contribution of several linear steps each one characterized at its end by the formation of one or more plastic hinges which define the new structure to which further load is applied. The collapse multiplier obtained using the hypothesis of elastic-perfectly plastic analysis is the same as that obtained under the rigid-plastic hypothesis. This result arises, as indicated earlier, because the value of the multiplier is not affected by elastic redistribution of forces but only by the equilibrium equations. The only reason for adopting a step-by-step analysis to determine the collapse multiplier, is firstly to use a procedure which can easily be introduced in a computer program having a linear package, and secondly that the required rotations in all the sections are given as part of the output. 4.2.1 Cross-section requirements To apply this method some requirements are given in Eurocode 3 [1] for cross-sections in order to guarantee that the fully plastic moment can be developed and sufficient rotations are developed within the joints in which plastic hinge locations are formed. Class 1 cross-sections are required if no computation is made of the required rotation while at least Class 2 cross-sections have to be applied if they can provide the required rotation. 4.3 Elasto-Plastic Analysis The elasto-plastic methods remove the hypothesis of elastic-perfectly plastic material. Therefore: • the moment-rotation relationship defining the cross-section or the connection characteristic can be nonlinear even if the material is elasto-plastic due to the different shape factor of the cross-section itself, see Figure 10d. A generic non-linear relation has to be introduced therefore instead of the elasto-plastic law for the definition of the plastic capacity of the joints. To introduce a generic non-linear moment-rotation relationship in the analysis an iterative procedure has to be followed. In this case, some of the limitations of classic plastic analysis can be removed by making allowance for: • • • • making the plastic moment dependent upon the value of axial and shear load. local second order effects (beam-column buckling) and global second order effects (frame buckling). defining the ultimate value of the multiplier of external loads as the one which can be derived by excessive rotations of the plastic rotations. the possibility of having loads which do not increase proportionally in each section. These assumptions can be introduced in a complete non-linear program if an iterative procedure is introduced. In this case the analysis is carried out by imposing small increments of loads or displacements and then by searching through subsequent iterations to obtain the solution which satisfies equilibrium and compatibility in each step within predetermined accuracy. In each step the plastic moment can be adjusted in order to take into account the axial and shear effects. The stiffness matrices of the bar and the entire frame can be updated to take into account the local and global second order effects by means of the procedures specified in Section 3.3.2. Different well-known procedures can be used to obtain the best convergence of the iteration, i.e Newton, NewtonRaphson, Riks, etc. Eurocode 3 [1] refers to elasto-plastic analysis in Clause 5.2.1.4(8), as follows: "In Elasto-Plastic analysis, the bi-linear stress-strain relationship may be used. Alternatively a more precise relationship may be adopted. The cross-section remains fully elastic until the stress in the extreme fibres reaches the yield strength. As the moment continues to increase the section yields gradually as plasticity spreads across the cross-section and plastic deformations extend partially along the member" (see the dashed line in Figure 10d). In elasto-plastic zones rather than plastic hinges developing, as shown in Figure 11, failure occurs if a mechanism of fully plastified zones has developed. The number of fully plastified zones depends on the redundancy of the frame. The elasto-plastic analysis is a method which is not likely to be used by engineers in practice but rather by researchers. 4.3.1 Cross-section requirements There are no practical limitations in elastic-plastic analysis since the complete non-linear analysis allows all possible effects to be introduced into the simulation of the behaviour of the structure. To apply this method some requirements are given in Eurocode 3 [1] for the cross-sections in order to guarantee that the fully plastic moment can be developed and sufficient rotations are developed within the joints in which plastic hinge locations are formed. Class 1 sections are required if no computation is made of the required rotation while Class 2 cross-sections have to be applied if they can provide the required rotation. 4.4 Calculation of Internal Forces and Moments Section 3.4 described the general definitions and procedures for calculating internal forces and moments, that are not only valid for an elastic analysis but also for a global plastic analysis. Some additional requirements for global plastic analysis are given by 5.2.1.4 of Eurocode 3 [1]: a. when plastic global analysis is used, lateral restraint shall be provided at all plastic hinge locations at which plastic hinge rotation may occur under any load case. b. when rigid plastic methods are used, sway frames cannot be analyzed except as described in Section 4.4.1 below. c. when elasto-plastic analysis is carried out, it may be assumed to be sufficient, in the case of building structures, to apply the loads in a series of increments, stopping when the full design load is reached, and to use the resulting internal forces and moments to perform resistance and buckling checks. d. at the plastic hinge locations, the cross-sections of members containing plastic hinges shall have an axis of symmetry in the plane of loading. 4.4.1 Plastic analysis of sway frames When plastic analysis is used, Eurocode 3 (Clause 5.2.6.3) requires allowance to be made for the second order effects in the sway mode: "This should generally be done directly by using second order elastic-plastic analysis". However, as an alternative, rigid-plastic analysis with indirect allowance for second order effects may also be adopted for: a. frames one to two storeys high, provided that no plastic hinge locations occur in the columns, or else that columns satisfy Clause 5.2.7 of Eurocode 3 [1], Column requirements for plastic analysis; b. frames with fixed bases, in which the sway failure mode involves plastic hinge locations in the columns at the fixed bases only (Figure 11), and the design is based on an incomplete mechanism in which the columns are designed to remain elastic at the calculated plastic hinge moment. 5. CONCLUDING SUMMARY • • The methods of structural analysis have been reviewed in this lecture. Both the elastic and the plastic methods have been discussed since all are explicitly referred to in Eurocode 3 [1]. The design of rigid jointed steel frames, and in particular, how to use the internal forces derived within these frameworks in safety checks, and how to adopt simplified rules, even when using a simple elastic analysis, have been also discussed. 6. REFERENCES [1] Eurocode 3: "Design of Steel Structures": ENV 1993-1-1: Part1.1: General Rules and Rules for Buildings, CEN, 1992. [2] European Convention for Constructional Steelwork, European Recommendations for Steel Construction, Bruxelles, ECCS, (1978) 234. [3] De Luca, A., Faella, C., Mele, E., Advanced In-elastic Analysis: Numerical Results and Design Guidelines for Rigid and Semi-Rigid Sway Frames, SSRCW Shop in "Plastic Hinge Based Methods for Advanced Analysis and Design of Steel Frames" Pittsburgh, April 1992. [4] Ballio, G., Mazzolani, F. M., "Theory and Design of Steel Structures". Chapman and Hall, 1983. 7. ADDITIONAL READING 1. 2. 3. 4. 5. 6. 7. 8. Cosenza, E., DeLuca, A., Faella, C. In-elastic Buckling of Semi-rigid Sway Frames, Structural Connections: Stability and Strength, London, Elsevier Applied Science, 1989. Ballio, G. & Mazzolani, F.M. Theory and Design of Steel Structures, Chapman & Hall, London, 1983. Dowling, P.J., Knowles, P.R., Owens, G.W., Structural Steel Design, Butterworths, London, 1988. Galambos, T.V. Guide to Stability Design Criteria for Metal Structures, 4th Edition, John Wiley & Sons, New York, 1988. Neal, B. G., The Plastic Methods of Structural Analysis, Wiley, J. & Sons, 1977. Liverley, R. K., Matrix Methods of Structural Analysis, Oxford, Pergamon Press, 1975. Capurso, M., Introduzione al Calcolo Automatico delle Strutture, Ed. Sciendtifiche Cremonese, Roma. Massonnet, C., Save, M., Calcolo Plastico a Rottura delle Costruzioni, Milano, Clup, 1982. Lecture 14.15: Tall Building Design OBJECTIVE/SCOPE To describe the various structural systems for steel multi-storey buildings together with their economies, relative efficiencies and advantages. To discuss the primary design factors affecting the choice of system, and illustrate by examples of actual application. PREREQUISITES Lecture 4B.4: Practical Ways of Achieving Fire Resistance of Steel Structures Lecture 10.9: Composite Buildings RELATED LECTURES Lectures 1B.7: Introduction to Design of Multi-Storey Buildings Lecture 6.3: Elastic Instability Modes Lecture 14.7: Anatomy of Multi-Storey Buildings Lecture 14.8: Classification of Multi-Storey Frames Lecture 14.9: Methods of Analysis for Multi-Storey Frames Lecture 14.10: Simple Braced Non-Sway Multi-Storey Buildings Lecture 14.11: Influence of Connections on Behaviour of Frames Lecture 14.12: Simplified Method of Design for Low-Rise Frames Lecture 14.13 Design of Multi-Storey Frames with Partial Strength and Semi-Rigid Connections Lecture 14.14: Methods of Analysis of Rigid Jointed Frames Lectures 17: Seismic Design SUMMARY Numerous new structural systems for multi-storey buildings, especially in steel, have evolved since the mid-1960's. Many new forms which are efficient and economical for different ranges of heights have been developed, including plane frames, core truss systems, belt truss systems and varieties of tubular formations. Many of these systems have involved both steel and reinforced concrete elements working compositely. The evolution of the systems has also provided the impetus for many outstanding examples of structurally expressive architecture. In some instances it has been possible, by exploiting recent advances in fire engineering, to leave the steelwork exposed. 1. INTRODUCTION Historically, the development of high-rise buildings in steel owes much to the rapid expansion which took place in Chicago in the aftermath of the Great Fire of 1871. Determining factors were the scarcity of building land and the availability since the mid Nineteenth Century, of elevator technology. A further prerequisite was a construction system which would be sufficiently reliable, strong and stiff, to permit construction to great height and which would be fireproof. Structural steel systems for multi-storey buildings evolved from the traditional cast iron column-beam systems of the late Nineteenth Century. These systems involved massive floors of stone or brick masonry and structural members were sheathed and encased in masonry. While the steel provided the basic resistance for carrying gravity loads, the masonry encasements themselves provided considerable stiffening against wind forces in addition to fire protection. These systems led to steel tiered construction, which generally maintained the rectilinear configuration of the frame, but continuity of steel was now achieved through some form of semi-rigid, riveted or bolted, beam-column joints. The frames were arranged with regular column spacings in both directions and were again encased in concrete or masonry. Although the encasements contributed significantly to lateral stiffness, the frame itself was required to carry a larger share of wind forces. The architectural expression was still dominated by masonry with terra-cotta or brick or similar materials often providing adornments, bays and other decorations on the facades. The early Twentieth Century up to the 1930's saw a proliferation of such systems which culminated in the Empire State Building in 1932[1]. The massing involved variations in plan along rectilinear lines with setbacks often associated with zoning, with a pronounced tower form and generally an articulated and decorated pyramidal top. On the interior, the column spans were generally in the range of 6m to 7,5m. As the demand grew for lighter, taller and more rapidly constructed structures, the same frame 'vocabulary' was continued with bolted or welded rigid joints, often with diagonal wind bracing between some columns in the core. In time, masonry cladding gave way to lighter forms of cladding, heavy masonry partitions were replaced by light drywall construction and concrete fire protection was replaced by a light sprayed-on material. This trend is exemplified by the 25-storey Lever House Building in New York, built in 1952, which has a metal and glass facade and lighter non-structural materials. The trend in tall building design after the mid-1950's was to provide larger open spaces with longer spans and simpler facades with a clearly perceptible column grid. The influence of Mies van der Rohe, with respect to modularity of facades and structural frame expression, permeated tall building exterior architecture in the 1950's and 1960's. The frame expression was often simply and clearly depicted on the facade or in some cases, by special emphasis if highly articulated, as shown in Figure 1. The evolution of a modular metal and glass curtain wall not only gave rise to lighter enclosures, but also to non-integral window wall systems which were simply supported on the structure. The floor plan was divided into a grid pattern of regular bays in each direction and bay dimensions in the range of 7,5m to 10,5m were common. It should be noted that such Vierendeel frames were basically very inefficient for resisting wind forces since they could not rely on built-in encasements for some stiffening, and the structural premium for height was considerable. It was recognised that the shear-frame concept was uneconomical for structures over 25 storeys in height. The idea that steel components can be assembled in various forms to constitute an overall three-dimensional system which can efficiently resist wind forces as a total system began to emerge in the 1960's. The main proponent of this design trend was Fazlur Khan who systematically pursued a logical evolution of high-rise systems based on the premise that different height ranges demand a new composition of steel framing to maintain a tolerable steel premium for height [2]. Many systems were devised, such as Truss-Frame Interacting Systems, Tubular Systems, Bundled Tubular Systems, Mixed Systems, etc. These structural developments, combined with the pervasive Miesian approach to the architecture of the time, resulted in an explosion of structurally oriented architecture in the 1960's and 1970's, which can truly be identified as the Structuralist phase. In terms of massing, simple rectilinear prismatic forms with flat tops were used for the most part. Simple form modulations included transforming the plan into the shape of a cross or some other simple form. A trend in the last decade has been to design buildings that are shaped both in plan and profile in a variety of ways to respond to urban grid and subjective aesthetic criteria. Often the facade architecture has reference to historic buildings of the turn of the Century with their masonry expressions and profile setbacks. Varieties of stone cladding and combinations of stone and precast concrete and other materials have evolved. The facade architecture is often compositional in nature (with the slides, consider including Philip Johnson's 'Chippendale' architecture - AT & T Building, New York (1978-83)) with respect to colour, texture and adornments. Geometric forms have involved not only overall shaping, but also local undulations related to bays or projections. All these developments have suppressed the Structuralist form of facade architecture in all but a few special cases. The structural systems approach has taken on an aspect of "mix and match" where pieces or parts of structural systems are blended to create an overall system to fit the peculiarities of a certain form [3]. A brief discussion of the generic high-rise steel systems is given below followed by some consideration of the design of ultra high-rise systems. The system descriptions are conceptual in nature. The height ranges (number of storeys) referred to in this Lecture for the various framing options relate to American practice and American site dimensions. As slenderness is a significant parameter, the width of the building should be taken into account in assessing the suitability of a particular bracing option for a building of a given height. 2. SYSTEMS EVOLUTION The steel systems developed between 1960 to 1975, fit a logical evolutionary pattern with one development leading to another, each new system being a link in the process[4]. Although the primary motivation for these developments was structural efficiency, the systems offered great opportunities for Structuralist facade architecture. 2.1 Shear Frame Systems Shear frames or Vierendeel frames, in which beams and columns are rigidly connected to provide moment resistance at joints, are placed in two orthogonal directions to resist wind forces in each direction. Each frame is required to resist its proportion of the wind shear, which is determined on the basis of its relative stiffness compared to the total. The efficiency of development of lateral stiffness is dependent on bay span, number of bays in the frame, number of frames and the available depth in the floors for the frame girders. Bay dimensions in the range of 6m to 9m are commonly used. In these shear frames, the predominant contribution to sway deflection under wind load comes from the bending of beams and columns due to the wind shear and to a smaller extent, from column shortening or the cantilever component (Figure 2). The design of these frames is controlled therefore by the bending stiffnesses of individual members. The deeper the member, the more efficiently the bending stiffness can be developed. When the frames are regularly spaced in both directions, a rectilinear column grid is created which is suitable for rectilinear plan forms. The architecture of these buildings has centred on bay or frame expressions. An outstanding example of this expression is in the Business Mens Assurance Building, Kansas City - a 20-storey structure built in about 1960 (Figure 1). Frames can be placed at other angles or on an irregular basis to create various plan shapes to fill the area of an irregular site. Shear frames can also be placed only on the exterior faces instead of using a grid arrangement of frames. In current practice, buildings with pure shear frames are generally restricted to only a few storeys in height, since other more efficient forms are available. However, the uncluttered rectilinear Vierendeel form may still be preferred in cases where other forms involving diagonals or trusses may interfere with architecture and space planning. The basic inefficiency in shear frame buildings arises from the need for moment-connected rigid joints which are expensive to fabricate, in addition to the steel quantities involved. The optimization of frames in a practical sense has centred on minimizing the number of such joints, replacement of field welding by field bolting, and similar criteria. 2.2 Shear Truss and Frame System Shear trusses can be provided in the vertical direction, if the organisation of the core permits such an arrangement. In general, building core elements including elevator banks are centralized and will permit core trusses connecting columns in at least one direction. These trusses combined with perimeter shear frames produce a Shear Truss and Frame System (Figure 3). There may be other interior frames which participate as well. This type of interacting system has a wide range of application in structures from 10 to 40 storeys. In smaller buildings, the frame component may be eliminated, with the result that all wind forces are resisted by the core trusses only. In some isolated instances, braces have been provided in both directions. Particular coordination with respect to providing access into the core needs then to be considered. The combination of the Shear Frame and vertical braces produces interaction of two modes of behaviour, that of a Shear Frame and a Cantilever (Figure 3). This combination produces desirable results in the stiffness efficiency of the overall system. Vertical trusses resist wind forces as a cantilever and therefore provide lateral stiffness more efficiently than shear frames. However, the depth available to the truss, which is dependent on the planning of the core area, determines the overall effectiveness of the system. For the truss bracing K-forms, X-forms or single brace forms can be used. The K-form is commonest since the bracings do not participate extensively in carrying gravity load, and can thus be designed for axial forces due to wind without gravity axial forces being a major consideration. In the X and single brace forms, gravity axial forces may dominate in the design of braces. In general, Shear Truss and Frame Systems produce the most economical steel structural system for buildings up to 30 storeys in height. 2.3 Frames, Vertical Trusses, Belt and Outrigger Trusses The exterior fascia shear frames and the vertical trusses in the core can be tied together by a system of outrigger and belt trusses which are provided at plant room levels, where the trusses will not interfere with the interior space planning. Figure 4 shows the arrangement of trusses. The primary result of the outrigger trusses is the development of axial forces in the exterior columns due to wind action. This behaviour significantly improves the lateral stiffness under wind forces. The use of belt trusses on the facades, at the same level and perpendicular to the outrigger trusses, further enhances participation of exterior frames in the cantilever behaviour. The belt trusses transform the two-dimensional frame system into a three-dimensional frame system which resists wind action. The building sway under wind is significantly reduced by the introduction of these trusses. A review of the deflection curve indicates two stiffening effects: one related to the participation of the external columns in a total-building-width cantilever mode; the other related to the stiffening of the facade frame by the belt trusses. Improvements in overall stiffness of up to 25% can result as compared to the Shear Truss and Frame System without such outrigger-belt trusses (Figure 5). The effectiveness of the system depends on the number of trussed levels and the depth of the truss at each level. The accompanying slide set provides illustration of the first Wisconsin Centre, Milwaukee, a 42-storey building framed in this fashion. 2.4 The Framed Tube If the facade shear frame is made stronger by closer spacing of columns and larger member proportions and if such frames are continuous at corners, the overall frame is transformed into a cantilever Framed Tube fixed at the ground. The effectiveness of the cantilever depends on the minimization of the part of the sway deflection due to the shear frame. One basic objective is to reduce this component to less than 25% of the total sway so that the predominant deformation is that of a cantilever (Figure 6a). When such frames are provided on all four faces of a tower, one obtains a hollow tubular configuration. This 'silo' form containing small window perforations is most efficient in resisting wind forces. Figure 6b shows the distribution of column axial forces due to cantilever action. The more the distribution is similar to that of a fully rigid box with uniform axial stress on the flanges and triangular distribution on the webs, the more efficient the system will be as a cantilever. The Framed Tube system was first introduced in the mid-1960's in reinforced concrete. The dense grid exterior structure was readily formed, creating the appearance of a punched tube. This system was adopted later for steel buildings. The proportions of the Framed Tube require wide members for both beams and columns, and they need to be rigid at the joints. In concrete, the rigid joint is achieved by the use of in-situ concrete, whereas in structural steel, the joints need to be welded for rigidity and the members built-up for larger widths. The use of a prefabricated Framed Tube "Tree" element (Figure 7) where all welding can be done in the shop in a horizontal position has made the steel Framed Tube more practical and efficient. The "Trees" are then erected by bolting at mid-span of the beams. Considerable speed of construction of the order of 3 to 4 storeys per week can result if "Trees" are used. Steel Framed Tube buildings involve column spacings of 3m to 5m on the exterior which can be transferred or transitioned into wider spacings, if required, at the lower storeys to integrate street level activities. Such tubular systems have been used extensively for structures of 30 to 110 storeys in height. An outstanding example is the World Trade Centre in New York. Tubular systems are generally adaptable to prismatic vertical profiles. For varying vertical profiles and buildings involving significant fascia offsets, the discontinuity required in the tubular frame to achieve the shape introduces serious disadvantages. The system can however, be readily adapted to a variety of non-rectilinear plan forms. Figure 8 shows a particular plan configuration that has been used as a Framed Tube. Provided the proportionality of the elements of the tubes is maintained, any closed overall form can be used as a Tubular system. 2.5 The Diagonalized Tube The most efficient structure acting as a cantilever is the exterior Diagonalized system. This system was first introduced in the John Hancock Centre in Chicago, a multi-use, 100-storey structure (Figure 9a). The system is essentially a Trussed Tube with the fascia diagonals not only acting as a truss in the plane, but also interacting with the trusses on the perpendicular facades to develop the tube action (Figure 9b). A principal advantage of the Trussed Tube is that it eliminates the need for the closely spaced columns of a framed tube. In the John Hancock Centre, the column spacing on the broad face is 12,2m and on the short face, 7,62m. Another advantage of this Tube is that the interior is free from structure for resisting wind action. The Tube is therefore, most suitable for multi-function buildings. Additional interior columns, as required, and simple floor framing complete the system [5]. The 100-storey, 337,5m high John Hancock structure required only 141,8 kg of steel per square meter of gross area of floor. The design of most members was controlled by gravity forces rather than wind forces. Simpler fabrication and erection techniques resulted from shop fabrication of joints and field bolting. The clear disciplined structure set the basis for the overall exterior architecture of this building. The system is most efficient in a single exterior closed form, especially when in a rectilinear shape. It is not readily adaptable where interior tube frame lines are present, In some special cases, the interior diagonals can be organised to suit specific office layouts. The principal of fascia diagonalization can be readily used for partial tubular concepts. For instance, in long rectangular buildings, the end frames on the short face may be diagonalized, where the long face is a Shear Frame. The end diagonalized frame may be in the form of a channel or "C" shape to provide wind resistance in both directions. The diagonalization can also vary from a broad "X" form to smaller "X's", thus transforming each facade into a diagrid braced system. Many variations are possible, each having its own impact on the exterior architecture. 2.6 Bundled Tube or Modular Tube System The need for vertical modulation in a logical fashion has created a variation of the tubular structure based on clustering or bundling of smaller sized tubes, each of which rises to a different height. This variation is typified by the bundled tube system of the Sears Tower in Chicago, Figure 10. This building has brought about a new generic form of structure called the "Bundled Tube" [6]. In the Sears Tower, the Bundled Tube is composed of 22,86m square modules, and nine modules are lumped together to form the total system, as shown in Figure 10. These tubes rise to different heights and are terminated when they are no longer needed architecturally and structurally. The walls of the tube are formed by columns at 4,57m centres and deep frame beams at each floor. The introduction of Framed Tube lines on the interior greatly reduced the influence of the "shear lag" effect that is present in exterior tubes of large side dimensions (Figure 10). The intent of the system was not only to create a powerful structural system, but also to create vertical modulation in a logical fashion. The development of a variety of floor sizes and shapes in the same building is considered a positive asset from the point of view of marketing real estate. The modularity and the conceptual basis of the Bundled Tube have a broad application. The cells or tubes can be arranged in a variety of ways to create different massing. It can be applied to 30 storeys as well as to ultra-tall structures. Further, the shape of each tube itself can be changed to any other closed clustered shape. Triangular and hexagonal units have been used in some existing applications. Two recent examples of the application of the Bundled Tube principle are worth noting. One is the Crocker Centre in Los Angeles which involves two towers, 57 and 47 storeys in height. The site conditions together with considerations of the sculptured form resulted in a shape which included a square tube and a triangular tube, as shown in Figure 11a. A column spacing of 4,87m was selected for the Tubular system. A significant advantage of the Bundled Tube system is the enormous torsional resistance which is helpful in absorbing torsional lateral forces due to asymmetry. In this case, the torsional loads were generated both by wind and seismic forces. The Framed-Tube buildings that have hitherto been adopted in wind controlled environments are now being adapted to seismic designs, such as the Crocker Centre. Apart from meeting wind deflection and stress criteria, the detailing of members and connections for ductility and an appropriate sequence of plastic hinge formation is required. Higher tubular efficiencies result from deeper beam members. However, this advantage has to be balanced with the strongcolumn, weak-beam principle to assure plastic hinges in the beam element. Other items include panel zone areas which need to be reinforced with doubler plates. The 75 storey, 296m high Allied Bank Tower in Houston is another example of the Bundled Frame Tube application. The shape is formed by two quarter circles placed anti-symmetrically about the middle tubular line, see Figure 11b. The column spacings are 4,57m with the usual "tree" type construction. The system also uses two vertical trusses in the core which are connected to the exterior tube by outrigger and belt trusses. Significant improvement in tubular behaviour is obtained because of the participation of the trusses. This system, therefore, embodies elements from the Framed Tube, Bundled Tube and Truss systems with belt and outrigger trusses. 2.7 Mixed Steel-Concrete Systems Mixed Steel-Concrete systems are now a well established new system that can be used as readily as either steel or concrete systems for high-rise buildings [7]. Such mixed systems which involve reinforced concrete and structural steel components in forms that are generally applicable, such as the Composite Tubular system, and Concrete Core Braced systems, have been widely used. Composite structures in a true sense, they have liberated the traditionally rigid discipline of either steel or concrete elements. The properties of concrete that are most attractive are its rigidity and its ability to be cast into different types of structural elements. Therefore, most mixed systems rely on concrete for lateral load resistance. Shear wall elements and/or punched wall or framed-tube elements with monolithically cast beam-column joints are of higher strength concrete. Concrete has extended the application to structures of 50 to 80 storeys in height. Concrete strengths of 40N/mm2 to 55N/mm2 are commonly used, but in some cases, strengths up to 95N/mm2 have been used. The floor framing is of steel in mixed systems, which is advantageous because of the ability to span longer distances with lighter members. Thus larger column-free spaces are possible. 2.7.1 Composite tube systems The first Canadian Centre in Calgary, Canada consists of two towers and a 10-storey banking pavilion, located on an L-shaped site. The two towers are 64 and 43 storeys in height. A sculpted form which provides diagonal vistas to mountains and the city was highly desirable for this prominent corner site. Each tower is similarly shaped, basically involving a parallelogram with truncated and re-entrant corners. The structural concept is based on a tube-in-tube concept involving an exterior reinforced concrete framed-tube and an interior shear wall core tube. Structural steel floor framing and other interior steel columns complete the system, as shown in Figure 12. The Tube system on the exterior is a combination of Framed Tube of beams and columns with solid walls at the corners. 2.7.2 Core braced systems In contrast to exterior Tubular systems, Core Braced Systems resist wind forces by shear walls in the core. Core walls for wind resistance have been utilized quite extensively in concrete buildings. The closed tube shape is then in the interior with access penetrations into the core. This type of core element can be used in a steel frame to form the Core Braced Steel system. Since all wind forces are resisted by the concrete core, the steel components need only be connected non-rigidly to resist gravity forces. In conventional buildings, the overall size of the core wall tube is limited by the dimensional requirements of the core elements. However, it may be possible in some cases to evolve a larger wall system that will be suitable for buildings in the range of 50 to 80 storeys as illustrated below. Figures 13a and 13b show an arrangement of cores for a 75-storey structure which required considerable flexibility in exterior shaping with offsets and setbacks. A lighter, non-rigid structure in steel on the exterior allowed this flexibility. The Core Wall Tube system was planned with four inter-connected pods encasing an atrium in the lower part and an octagonal shaped core in the upper parts. This simple arrangement of walls and cores allowed maximum efficiency of the wall system for the structure and maximum flexibility for exterior architecture. 3. ULTRA HIGH-RISE STRUCTURES As the height of a building increases, the ability to resist lateral forces becomes the predominant structural design criterion. The major design criterion is the provision of enough lateral stiffness to limit wind sway and to limit occupant motion perception to acceptable levels. The motion of the building is essentially dynamic and all factors that affect this dynamic behaviour need to be considered. The structural parameters are stiffness, damping and mass. Among the non-structural parameters is the aspect of aerodynamic shaping of the building which will have a significant effect, especially in ultra-tall structures. Two wind effects are to be recognized. One is down wind behaviour, which is influenced by the drag parameters of the shape, and the other is the cross wind behaviour influenced by the uniformity of the vortex shedding inherent in the shape. While the basic design requirement of controlling lateral sway still predominates, overall shaping of the form becomes equally significant. The architecture of the shape must then be suitably integrated with its structural aspects to produce an optimum economic balance. 3.1 Superframe or Megaframe Superframes or Megaframes assume the form of a portal which is provided on the exterior of a building. The frames resist all wind forces as an exterior tubular structure. The portal frame of the Superframe is composed of vertical legs in each corner of the building which are linked by horizontal elements at about every 12 to 14 floors. Since the vertical elements are concentrated in the corner areas of the building, maximum efficiency is obtained for resisting wind forces. The vertical legs and the horizontal links are themselves frames with large dimensions in the plane of the frame [3]. Inherent in the concept of a Superframe is the ability of the system to accommodate varieties of spaces suitable for multiple functions. Historically, tall buildings which have multiple functions in a single building require flexibility of spaces. Structures, such as the John Hancock Centre in Chicago, have successfully accomplished this flexibility. Ultra high-rise Megastructures will need to further the idea of flexible space or modular spaces, where each space can be planned efficiently for its own use and inserted into a Megaframe. Superframes allow this modularization of space to occur with maximum freedom from structural encumbrance. Portal apertures on the exterior wall allow freedom of expression for each unit and also provide access for daylight. Figure 14b indicates an arrangement of interior spaces with an atrium in each space module. The structural efficiency is obtained from the concentration of material close to the corners. Each of these vertical legs is required to be stiff in its own plane. The legs then take the form of a diagonalized truss chord. The corner truss legs need strong horizontal connections at frequent modular intervals to make them function together like an equivalent cantilever. The horizontal members, therefore, will need to be equally stiff, and are diagonalized as well. The net effect of this combination would be to produce an equivalent cantilever structure as effective as a Tubular system. An appropriate diagonalization of structure is represented by an overall modularized diagrid frame, in which the structural members within the portals are removed (Figure 14a). Figure 14a also shows the adoption of the Superframe concept to an 80-storey structure in its simplest rectilinear form. The effect on the exterior architecture and the potential possibilities can be clearly observed. While the example illustrates a rectilinear form, the concept can be used for other forms where the principle of stiffnesses of horizontal and vertical elements and their interconnections are met. Figure 15 shows a study of a Superframe concept applied to a 170-storey, 655m tall tower involving 706.000 sq. metres of floor space. The multiple functions involved are indicated in Figure 16a. The tower form, step-tapered from 88m x 88m at the base to 44m x 44m at the top denotes the functional needs of space. The basic structure, Figure 15, involved telescoping Superframes, not only to fit the overall geometry of the shape, but also to augment the structural stiffness. The curtain wall was constructed over the general structural form and involved many more smaller offsets. A significant innovation involved the provision of through-apertures in the building to reduce aerodynamic oscillations. Their effectiveness was confirmed by wind tunnel testing. The equivalent dampening effect of this aerodynamic shaping illustrates the need for such design coordination in supertall structures. The significant aspects of this structural design were: i. Telescoping of the Superframes allowed an orderly transition of the structure and the needed lateral stiffness. ii. The aerodynamic behaviour was greatly assisted by the apertures which reduced the cross-wind accelerations and forces due to vortex shedding by as much as 25%. The overall tapered form assisted as well in reducing the wind 'sail' area gradually toward the top. iii. The structural system efficiency was enhanced by the sequential load transfer from the interior to the exterior, as shown in Figure 16, so that all gravity loads were carried by the Superframes. 3.2 Super-Trussed Tubes The general concept of the exterior Trussed Tube can be extended to ultra high-rise structures involving multiple functions. Figure 17 shows a 135-storey exterior Trussed Tube, which was submitted for a project competition in New York. This slender form, having an aspect ratio of 10 to 1, involved approximately 185.908 sq. metres of floor space. The arrangement of multiple functional spaces is indicated in the plans. The horizontal and vertical facets of the shape were modulized so that a triangular shaped framing could be used without discontinuities. This framing resulted from the concept of a tetrahedrally framed solid form where all the non-essential elements were eliminated to create the tube and the external shaping of it. In order to express the purity of the structure, the window wall was placed a certain distance behind the truss, creating an open truss trellis on the exterior. This open form has the effect of eliminating overall wind vortex formation and would significantly reduce cross-wind oscillations. This is another form of aerodynamic shaping to mitigate wind effects. The internal structure is supported on 3-storey tension truss elements which span across the tube thereby eliminating the need for internal columns (Figure 17). This arrangement allowed the use of gravity steel in the development of lateral stiffness and wind load resistance. It also made it possible to deal with the effects of temperature contraction of the exposed, but clad exterior columns. 4. EXPOSED STEEL SYSTEMS The beauty and essence of exposed steel can form the basis for exterior architecture as has been illustrated in the Hong Kong-Shanghai Bank in Hong Kong and the Pompidou Centre in Paris. Exposure of steel on the exterior needs to address the issues of fire and corrosion protection. Corrosion protection has been attained, in some cases, by the use of weathering steels and, in others, by the use of durable, long-life, fluor-carbon paint systems. For fire resistance, liquid filled members have been used on occasion. An outstanding example is the US Steel Building in Pittsburg. The analytical fire engineering methods now available permit exposed steel system to be more readily designed. Such an approach was used for Phase 11 of the Broadgate Project recently completed in London. The 10storey office building spans 78m over railroad tracks and its design was based on a tied arch solution (Figures 18a and 18b). This classic structure clearly expresses the steel through its articulated members and joints. It stands as a symbol for what is possible in exposed steel architecture. 5. CONCLUDING SUMMARY • • Numerous new structural systems for multi-storey buildings, especially in steel, have evolved since the mid-1960's. Efficient and economical forms suitable for different heights have been developed including: ⋅ Shear frame systems ⋅ Shear truss-frame interacting systems ⋅ Vertical trusses, and belt and outrigger trusses ⋅ Framed tubes ⋅ Diagonalised tubes ⋅ Bundled tubes ⋅ Modular tube systems ⋅ Mixed steel - concrete systems. • • For very high building structures, superframes and super-trussed tubes have been developed. Advances in corrosion protection and fire engineering have made it possible to use exposed steel systems giving impetus to structurally expressive architecture. 6. REFERENCES [1] Mujicha, Franciso, "History of the Skyscraper" Archaeology and Architectural Press, 1929, Copyright, 1930 by Mujicha, Property of the American Institute of Steel Construction. [2] Khan, Fazlur, R., "Structural Systems for Multi-Storey Steel Buildings". [3] Iyengar, H., "Steel Systems for High-Rise Buildings", International Conference on Steel Structures, Singapore, March 1984. [4] Iyengar, H., "preliminary Design and Optimization of Steel Building Systems", State of the Art Report No. 3, Technical Committee No. 14: Elastic Design, American Society of Civil Engineers-International Association for Bridge and Structural Engineering Joint Committee on Tall Buildings, August, 1972. [5] Iyengar, H., "Structural Systems for Two Ultra High-Rise Structures", Australian and New Zealand Conference on Planning and Design of Tall Buildings, Sydney, Australia, May, 1973. [6] Iyengar, S. H. and Khan, F., "Structural Steel Design of Sears Tower", Conference on Steel Developments, Australian Institute of Steel Construction, Newcastle, Australia, May, 1973. [7] Iyengar, Hal, "Recent Development sin Composite High-Rise Systems", Council on Tall Buildings and Urban Habitat, Monograph, Advances in Tall Buildings, 1986, Council on Tall Buildings and Urban Habitat, Meeting, Chicago, Illinois, October, 1982.