Chapter 21 Composite beams

Transcription

1 Chapter 21 Composite beams by MARK LAWSON and PETER WICKENS 21.1 Applications of composite beams In buildings and bridges, steel beams often support concrete slabs. Under load each component acts independently with relative movement or slip occurring at the interface. If the components are connected so that slip is eliminated, or considerably reduced, then the slab and steel beam act together as a composite unit (Fig. 21.1). There is a consequent increase in the strength and stiffness of the composite beam relative to the sum of the components. The slab may be solid in situ concrete or the composite deck slab considered in Chapter 20. It may also comprise precast concrete units with an in situ concrete topping. In buildings, steel beams are usually of standard UB scction, but UC and asymmetric beam sections are sometimes used where there is need to minimize the beam depth. A typical building under construction is shown in Fig Welded fabricated sections are often used for long-span beams in buildings and bridges. Design of composite beams in buildings is now covered by BS 5950: Part 3, 1 although guidance was formerly available in an SCI publication. 2 The design of composite beams incorporating composite slabs is affected by the shape and orientation of the decking, as indicated in Fig One of the advantages of composite construction is smaller construction depths. Services can usually be passed beneath, but there are circumstances where the beam depth is such that services can be passed through the structure, either by forming large openings, or by special design of the structural system. A good example of this is the stub-girder. 3 The bottom chord is a steel section and the upper chord is the concrete slab. Short steel sections or stubs are introduced to transfer the forces between the chords. Openings through the beam webs can be provided for services. Typically, these can be up to 70% of the beam depth and can be rectangular or circular in shape. Guidance on the design of composite beams with web openings is given in Reference 4. Examples of the above methods of introducing services within the structure are shown in Fig Economy Composite beam construction has a number of advantages over non-composite construction: 601

2 602 Composite beams force in support non composite beam composite beam Fig Behaviour of composite and non-composite beams Fig Composite building under construction showing decking and shear-connectors (1) savings in steel weight are typically 30% to 50% over non-composite beams. (2) the greater stiffness of the system means that beams can be shallower for the same span, leading to lower storey heights and savings in cladding, etc. It also shares the advantage of rapid construction. The main disadvantage is the need to provide shear-connectors at the interface between the steel and concrete. There may also be an apparent increase in complexity of design. However, design tables have been presented to aid selection of member sizes. 2

3 Guidance on span-to-depth ratios 603 deck overlap (a) (b) Fig Composite beams incorporating composite deck slabs: (a) deck perpendicular to beam, (b) deck parallel to beam The normal method of designing simply-supported beams for strength is by plastic analysis of the cross-section. Full shear connection means that sufficient shearconnectors are provided to develop the full plastic capacity of the section. Beams designed for full shear connection result in the lightest beam size. Where fewer shear-connectors are provided (known as partial shear connection) the beam size is heavier, but the overall design may be more economic. Partial shear connection is most attractive where the number of shear-connectors is placed in a standard pattern, such as one per deck trough or one per alternate trough where profiled decking is used. In such cases, the resistance of the shearconnectors is a fixed quantity irrespective of the size of the beam or slab. Conventional elastic design of the section results in heavier beams than with plastic design because it is not possible to develop the full tensile resistance of the steel section. Designs based on elastic principles are to be used where the compressive elements of the section are non-compact or slender, as defined in BS 5950 Part 1. This mainly affects the design of continuous beams (see section ) Guidance on span-to-depth ratios Beams are usually designed to be unpropped during construction. Therefore, the steel beam is sized first to support the self-weight of the slab before the concrete has gained adequate strength for composite action. Beams are assumed to be laterally restrained by the decking in cases where the decking crosses the beams (at an angle of at least 45 to the beam) and is directly attached to them. These beams can develop their full flexural capacity. Where simply-supported unpropped composite beams are sized on the basis of their plastic capacity it is normally found that span-to-depth ratios can be in the

4 604 Composite beams column,shear connector._mesh beam bolt E I. opening "sti1fener Q opening (a) Fig L1 0 services (b) (c) Different methods of incorporating services within the structural depth range of 18 to 22 before serviceability criteria influence the design. The depth in these cases is defined as the overall depth of the beam and slab. S355 steel is often specified in preference to S275 steel in composite beam design because the stiffness of a composite beam is often three to four times that of the non-composite beam, justifying the use of higher working stresses. The span-to-depth ratios of continuous composite beams are usually in the range of 22 to 25 for end spans and 25 to 30 for internal spans before serviceability criteria influence the design. Many continuous bridges are designed principally to satisfy the serviceability limit state.

5 Span conditions Types of shear connection The modern form of shear-connector is the welded headed stud ranging in diameter from 13 to 25 mm and from 65 to 125 mm in height. The most popular size is 19 mm diameter and 100 mm height before welding. When used with steel decking, studs are often welded through the decking using a hand tool connected via a control unit to a power generator. Each stud takes only a few seconds to weld in place. Alternatively, the studs can be welded directly to the steel beams in the factory and the decking butted up to or slotted over the studs. There are, however, some limitations to through-deck welding: the top flange of the beam must not be painted, the galvanized steel should be less than around 1.25 mm thick, the deck should be clean and free of moisture, and there should be no gap between the underside of the decking and the top of the beam.the minimum flange thickness must not be less than the diameter of the stud divided by 2.5 (typically, 19/2.5 = 7.6 mm). The power generator needs 415 V electrical supply, and the maximum cable length between the weld gun and the power control units should be limited to around 70 m to avoid loss of power. Currently, only 13, 16 or 19 mm diameter studs can be through-deck welded on site. Where precast concrete planks are used, the positions of the shear-connectors are usually such that they project through holes in the slab which are later filled with concrete. Alternatively, a gap is left between the ends of the units sitting on the top flange of the beam on to which the shear connectors are fixed. Reinforcement (usually in the form of looped bars) is provided around the shear-connectors. There is a range of other forms of welded shear-connector, but most lack practical applications. The bar and hoop and channel welded shear-connectors have been use in bridge construction. 5 Shot-fired shear-connectors may be used in smaller building projects where site power might be a problem. All shear-connectors should be capable of resisting uplift forces; hence the use of headed rather than plain studs. The number of shear-connectors placed along the beam is usually sufficient to develop the full flexural resistance of the member. However it is possible to reduce the number of shear-connectors in cases where the moment resistance exceeds the applied moment and the shear-connectors have adequate ductility (or deformation capacity). This is known as partial shear connection and is covered in section Span conditions In buildings, composite beams are usually designed to be simply-supported, mainly to simplify the design process, to reduce the complexity of the beam-to-column connections, and to minimize the amount of slab reinforcement and shear-connectors that are needed to develop continuity at the ultimate limit state. However, there are ways in which continuity can be readily introduced, in order to improve the stiffness of composite beams. Figure 21.5 shows how a typical connection detail at an internal column can be modified to develop continuity. The stub

6 606 Composite beams ram M bending moment M developed by shear connectors in this zone Fig tension in anchorage reinforcement lenath - mesh.,...., ;. -y I L çil-1i t It bolt compression stiffener Representation of conditions at internal column of continuous beam girder system also utilizes continuity of the secondary members (see Fig. 21.4(c)). Other methods of continuous design are presented in References 3 and 6. Continuous composite beams may be more economic than simply-supported beams where plastic hinge analysis of the continuous member is carried out, provided the section is plastic according to BS 5950: Part 1. However, where the lower flange or web of the beam is non-compact or slender in the negative (hogging) moment region, then elastic design must be used, both in terms of the distribution of moment along the beam, and also for analysis of the section. Lateral instability of the lower flange is an important design condition, although torsional restraint is developed by the web of the section and the concrete slab. 6 In bridges, continuity is often desirable for serviceability reasons, both to reduce deflections, and to minimize cracking of the concrete slab, finishes and wearing surface in road bridges. Special features of composite construction appropriate to bridge design are covered in the publication by Johnson and Buckby 7 based on BS 5400: Part Analysis of composite section Elastic analysis Elastic analysis is employed in establishing the serviceability performance of composite beams, or the resistance of beams subject to the effect of instability, for

7 Analysis of composite section 607 Be/ae Os D elastic neutral axis equivalent steel area (a) stress p Fig Elastic behaviour of composite beam. (a) Elastic stress distribution. (b) Transformed section example, in continuous construction, or in beams where the ductility of the shear connection is not adequate. The important properties of the section are the section modulus and the second moment of area. First it is necessary to determine the centroid (elastic neutral axis) of the transformed section by expressing the area of concrete in steel units by dividing the concrete area within the effective breadth of the slab, B e, by an appropriate modular ratio (ratio of the elastic modulus of steel to concrete). In unpropped construction, account is taken of the stresses induced in the noncomposite section as well as the stresses in the composite section. In elastic analysis, therefore, the order of loading is important. For elastic conditions to hold, extreme fibre stresses are kept below their design values, and slip at the interface between the concrete and steel should be negligible. The elastic section properties are evaluated from the transformed section as in Fig The term a e is the modular ratio. The area of concrete within the profile depth is ignored (this is conservative where the decking troughs lie parallel to the beam). The concrete can usually be assumed to be uncracked under positive moment. The elastic neutral axis depth, x e, below the upper surface of the slab is determined from the formula: x e = D s - Dp Ê r D ˆ + a e + Ds 2 Ë 2 ( 1 + a r) e (b) (21.1)

8 608 Composite beams where r = A/[(D s - D p )B e ], D s is the slab depth, D p is profile height (see Fig. 21.6) and A is the cross-sectional area of the beam of depth D. The second moment of area of the uncracked composite section is: 2 3 AD ( + Ds + Dp) Be( Ds - Dp) Ic = + + I (21.2) 41 ( + aer) 12ae where I is the second moment of area of the steel section. The section modulus for the steel in tension is: Z = I ( D+ D -x ) t c s e and for concrete in compression is: (21.3) Ze = Ic a e xe (21.4) The composite stiffness can be 3 to 5 times, and the section modulus 1.5 to 2.5 times that of the I-section alone Plastic analysis The ultimate bending resistance of a composite section is determined from its plastic resistance. It is assumed that the strains across the section are sufficiently high that the steel stresses are at yield throughout the section and that the concrete stresses are at their design strength. The plastic stress blocks are therefore rectangular, as opposed to linear in elastic design. The plastic moment resistance of the section is independent of the order of loading (i.e. propped or unpropped construction) and is compared to the moment resulting from the total factored loading using the load factors in BS 5950: Part 1. The plastic neutral axis of the composite section is evaluated assuming stresses of p y in the steel and 0.45f cu in the concrete. The tensile resistance of the steel is therefore R s = p y A, where A is the cross-sectional area of the beam. The compressive resistance of the concrete slab depends on the orientation of the decking. Where the decking crosses the beams the depth of concrete contributing to the compressive resistance is D s - D p (Fig. 21.6(a)). Clearly, D p is zero in a solid slab. Where the decking runs parallel to the beams (Fig. 21.6(b)), then the total cross-sectional area of the concrete is used. Taking the first case: R = 045. f ( D -D ) B c cu s p e (21.5) where B e is the effective breadth of the slab considered in section Three cases of plastic neutral axis depth x p (measured from the upper surface of the slab) exist. These are presented in Fig It is not necessary to calculate x p explicitly if the following formulae for the plastic moment resistance of I-section beams subject to positive (sagging) moment are used.the value R w is the axial resistance of the web and R f is the axial resistance of one steel flange (the section is

9 Analysis of composite section 609 O 451cu py Fig Plastic analysis of composite section under positive (sagging) moment (PNA: plastic neutral axis) assumed to be symmetrical). The top flange is considered to be fully restrained by the concrete slab. The moment resistance, M pc, of the composite beam is given by: Case 1: R c > R s (plastic neutral axis lies in concrete slab): È M R D D R s Ê Ds - Dpˆ pc = s + s - Á Î Í 2 Rc Ë 2 Case 2: R s > R c > R w (plastic neutral axis lies in steel flange): M R D Ê R D s + D pˆ ( R s - R c) pc = s + cá - 2 Ë 2 R NB the last term in this expression is generally small (T is the flange thickness). Case 3: R c < R w (plastic neutral axis lies in web): Ê M M R D s + D p + D ˆ R c 2 D pc = s + cá - Ë 2 R 4 (21.6) (21.7) (21.8) where M s is the plastic moment resistance of the steel section alone. This formula assumes that the web is compact i.e. not subject to the effects of local buckling. For this to be true the depth of the web in compression should not exceed 78te, where t is the web thickness (e is defined in BS 5950: Part 1). If the web is non-compact, a method of determining the moment resistance of the section is given in BS 5950: Part 3, Appendix B Continuous beams w f 2 T 4 Bending moments in continuous composite beams can be evaluated from elastic global analysis. However, these result in an overestimate of moments at the supports because cracking of the concrete reduces the stiffness of the section and

10 610 Composite beams permits a relaxation of bending moment. A simplified approach is to redistribute the support moment based on gross (uncracked) section properties by the amounts given in Table Alternatively, moments can be determined using the appropriate cracked and uncracked stiffnesses in a frame analysis. In this case, the permitted redistribution of moment is less. Table 21.1 Maximum redistribution of support moment based on elastic design of continuous beams at the ultimate limit state Classification of compression flange at supports Plastic Assumed section properties at Slender Semi-compact Compact Generally Beams (with nominal supports slab reinforcement) Gross uncracked 10% 20% 30% 40% 50% Cracked 0% 10% 20% 30% 30% The section classification is expressed in terms of the proportions of the compression (lower) flange at internal supports. This determines the permitted redistribution of moment. A special category of plastic section is introduced where the section is of uniform shape throughout and nominal reinforcement is placed in the slab which does not contribute to the bending resistance of the beam. In this case the maximum redistribution of moment under uniform loading is increased to 50%. A simplified elastic approach is to use the design moments in Table 21.2 assuming that: Table 21.2 Moment coefficients (multiplied by free moment of WL/8) for elastic design of continuous beams Classification of compression flange at supports Plastic Location Slender Semi-compact Compact Generally Beams (with nominal slab reinforcement) Middle 2 spans of end 3 or span more spans First 2 spans internal 3 or support more spans Middle of internal spans Internal supports (except first) Redistribution 10% 20% 30% 40% 50%

11 Analysis of composite section 611 (1) the unfactored imposed load does not exceed twice the unfactored dead load; (2) the load is uniformly distributed; (3) end spans do not exceed 115% of the length of the adjacent span; (4) adjacent spans do not differ in length by more than 25% of the longer span. An alternative to the elastic approach is plastic hinge analysis of plastic sections. Conditions on the use of plastic hinge analysis are presented in BS 5950: Part 3 1 and Eurocode 4 (draft). 8 However, large redistributions of moment may adversely affect serviceability behaviour (see section ). The ultimate load resistance of a continuous beam under positive (sagging) moment is determined as for a simply-supported beam. The effective breadth of the slab is based on the effective span of the beam under positive moment (see section ). The number of shear-connectors contributing to the positive moment capacity is ascertained knowing the point of contraflexure. The negative (hogging) moment resistance of a continuous beam or cantilever should be based on the steel section together with any properly anchored tension reinforcement within the effective breadth of the slab. This poses problems at edge columns, where it may be prudent to neglect the effect of the reinforcement unless particular measures are taken to provide this anchorage. The behaviour of a continuous beam is represented in Fig The negative moment resistance is evaluated from plastic analysis of the section: Case 1: R r < R w (plastic neutral axis lies in web): Ê M M R D ˆ Rq 2 D nc = s + s + Dr (21.9) Ë 2 - Rw 4 where R r is the tensile resistance of the reinforcement over width B e, R q is the capacity of the shear-connectors between the point of contraflexure and the point of maximum negative moment (see section ), and D r is the height of the reinforcement above the top of the beam. Case 2: R r > R w (plastic neutral axis lies in flange): M R D RD R R 2 ( s - r) T nc = s + r r - (21.10) 2 Rf 4 NB the last term in this expression is generally small. The formulae assume that the web and lower flange are compact i.e. not subject to the effects of local buckling. The limiting depth of the web in compression is 78te (where e is defined in Chapter 2) and the limiting breath : thickness ratio of the flange is defined in Table 11 of BS 5950: Part 1. If these limiting slendernesses are exceeded then the section is designed elastically often the situation in bridge design. The appropriate effective breadth of slab is used because of the sensitivity of the position of the elastic neutral axis and hence the zone of the web in compression to the tensile force transferred by the reinforcement. The elastic section properties are determined on the assumption that the concrete is cracked and does not contribute to the resistance of the section.

12 612 Composite beams point load udi B/L _ B Fig Basic design Effective breadths support mid span support Variation of effective breadth along beam and with loading The structural system of a composite floor or bridge deck is essentially a series of parallel T beams with wide thin flanges. In such a system the contribution of the concrete flange in compression is limited because of the influence of shear lag. The change in longitudinal stress is associated with in-plane shear strains in the flanges. The ratio of the effective breadth of the slab to the actual breadth (B e /B) is a function of the type of loading, the support conditions and the cross-section under consideration as illustrated in Fig The effective breadth of slab is therefore not a precise figure but approximations are justified. A common approach in plastic design is to consider the effective breadth as a proportion (typically 20% 33%) of the beam span. This is because the conditions at failure are different from the elastic conditions used in determining the data in Fig. 21.8, and the plastic bending capacity of a composite section is relatively insensitive to the precise value of effective breadth used. Eurocode 4 (in its ENV or pre-norm version) 8 and BS 5950: Part 3 1 define the effective breadth as (span/4) (half on each side of the beam) but not exceeding the actual slab breadth considered to act with each beam. Where profiled decking spans in the same direction as the beams, as in Fig. 21.3(b), allowance is made for the combined flexural action of the composite slab and the composite beam by limiting the effective breadth to 80% of the actual breadth. In building design, the same effective breadth is used for section analysis at both the ultimate and serviceability limit states. In bridge design to BS 5400: Part 5, 5 tabular data of effective breadths are given for elastic design at the serviceability

13 Basic design 613 Table 21.3 Modular ratios (a e ) of steel of concrete Duration of loading Type of concrete Short-term Long-term Office loading Normal weight Lightweight (density > 1750 kg/m 3 ) limit state. If plastic design is appropriate, the effective breadth is modified to (span/3). 5 In the design of continuous beams, the effective breadth of the slab may be based conservatively on the effective span of the beam subject to positive or negative moment. For positive moment, the effective breadth is 0.25 times 0.7 span, and for negative moment, the effective breadth is 0.25 times 0.25 times the sum of the adjacent spans. These effective breadths reduce to span and span respectively for positive and negative moment regions of equal-span beams Modular ratio The modular ratio is the ratio of the elastic modulus of steel to the creep-modified modulus of concrete, which depends on the duration of the load. The short- and long-term modular ratios given in Table 21.3 may be used for all grades of concrete. The effective modular ratio used in design should be related to the proportions of the loading that are considered to be of short- and long-term duration. Typical values used for office buildings are 10 for normal weight and 15 for lightweight concrete Shear connection The shear resistance of shear-connectors is established by the push-out test, a standard test using a solid slab. A typical load slip curve for a welded stud is shown in Fig The loading portion can be assumed to follow an empirical curve. 9 The strength plateau is reached at a slip of 2 3 mm. The shear resistance of shear-connectors is a function of the concrete strength, connector type and the weld, related to the diameter of the connector. The purpose of the head of the stud is to prevent uplift. The common diameter of stud which can be welded easily on site is 19 mm, supplied in 75 mm, 100 mm or 125 mm heights.the material properties, before forming, are typically: Ultimate tensile strength 450 N/mm 2 Elongation at failure 15%

14 614 Composite beams load per shear connector 0.6 u Fig <0.5 mm 2 to 3 mm >5 mm slip Load slip relationship for ductile welded shear-connector Higher tensile strengths (495 N/mm 2 ) are required in bridge design. 5 Nevertheless, the push-out strength of the shear-connectors is relatively insensitive to the strength of the steel because failure is usually one of the concrete crushing for concrete grades less than 40. Also the weld collar around the base of the shearconnector contributes to increased shear resistance. The modern method of attaching studs in composite buildings is by through-deck welding. An example of this is shown in Fig The common in situ method of checking the adequacy of the weld is the bend-test, a reasonably easy method of quality control which should be carried out on a proportion of studs (say 1 in 50) and the first 2 to 3 after start up. Other forms of shear-connector such as the shot-fired connector have been developed (Fig ). The strength of these types is controlled by the size of the pins used. Typical strengths are 30% 40% of the strengths of welded shear-connectors, but they demonstrate greater ductility. The static resistances of stud shear-connectors are given in Table 21.4, taken from BS 5400: Part 5 and also incorporated in BS 5950: Part 3. As-welded heights are some 5 mm less than the nominal heights for through-deck welding; for studs welded directly to beams the length after welding (LAW) is taken as the nominal height. The strengths of shear-connectors in structural lightweight concrete (density >1750 kg/m 2 ) are taken as 90% of these values. In Eurocode 4, 8 the approach is slightly different. Empirical formulae are given

15 Basic design 615 Fig Welding of shear-connector through steel decking to a beam based on two failure modes: failure of the concrete and failure of the steel. The upper bound strength is given by shear failure of the shank and therefore there is apparently little advantage in using high-strength concrete. In BS 5950: Part 3 and Eurocode 4, the design resistance of the shear-connectors is taken as 80% of the nominal static strength. Although this may broadly be considered to be a material factor applied to the material strength, it is, more correctly, a factor to ensure that the criteria for plastic design are met (see below). The design resistance of shear-connectors in negative (hogging) moment regions is conservatively taken as 60% of the nominal resistance. In BS 5400: Part 5, an additional material factor of 1.1 is introduced to further reduce the design resistance of the shear connectors.

16 616 Composite beams Fig Shot-fired shear-connector Table 21.4 Characteristic resistances of headed stud shearconnectors in normal weight concrete Dimensions of stud Characteristic strength shear-connectors (mm) of concrete (N/mm 2 ) Diameter Nominal As-welded height height For concrete of characteristic strength greater than 40 N/mm 2 use the values for 40 N/mm 2 For connectors of heights greater than tabulated use the values for the greatest height tabulated

17 Basic design 617 elastic shear flow actual shear flow at failure idealized plastic shear flow loading slip Fig Idealization of forces transferred between concrete and steel shear connector In plastic design, it is important to ensure that the shear-connectors display adequate ductility. It may be expected that shear-connectors maintain their design resistances at displacements of up to 5 mm. A possible exception is where concrete strengths exceed C 40, as the form of failure may be more brittle. For beams subject to uniform load, the degree of shear connection that is provided by uniformly-spaced shear-connectors (defined in section ) reduces more rapidly than the applied moment away from the point of maximum moment. To ensure that the shear connection is adequate at all points along the beam, the design resistance of the shear-connectors is taken as 80% of their static resistance. This also partly ensures that flexural failure will occur before shear failure. For beams subject to point loads, it is necessary to design for the appropriate shear connection at each major load point. In a simple composite beam subject to uniform load the elastic shear flow defining the shear transfer between the slab and the beam is linear, increasing to a maximum at the ends of the beam. Beyond the elastic limit of the connectors there is a transfer of force among the shear-connectors, such that, at failure, each of the shear-connectors is assumed to be subject to equal force, as shown in Fig This is consistent with a relatively high slip between the concrete and the steel. The slip increases as the beam span increases and the degree of shear connection reduces. For this reason BS 5400: Part 5 5 requires that shear-connectors in bridges are spaced in accordance with elastic theory. In building design, shear-connectors are usually spaced uniformly along the beam when the beam is subject to uniform load. No serviceability limit is put on the force in the shear-connectors, despite the fact that consideration of the elastic shear flow suggests that such forces can be high at

18 618 Composite beams working load.this is reflected in the effect of slip on deflection in cases where partial shear connection is used. When designing bridges 5 or structures subject to fatigue loading, a limit of 55% of the design resistance of the shear-connectors is appropriate for design at the serviceability limit state Partial shear connection In plastic design of composite beams the longitudinal shear force to be transferred between the concrete and the steel is the lesser of R c and R s. The number of shearconnectors placed along the beam between the points of zero and maximum positive moment should be sufficient to transfer this force. In cases where fewer shear-connectors are provided than the number required for full shear connection it is not possible to develop M pc. If the total capacity of the shear-connectors between the points of zero and maximum moment is R q (less than the smaller of R s and R c ), then the stress block method in section is modified as follows, to determine the moment resistance, M c : Case 4: R q > R w (plastic neutral axis lies in flange): M R D R D R 2 È q Ê Ds - Dpˆ Rs Rq T c = s + q s - Á (21.11) Î Í Rc Ë - ( - ) 2 2 Rf 4 NB the last term in this expression is generally small. Case 5: R q < R w (plastic neutral axis lies in web): È M M R D D R q Ê Ds - Dpˆ Rq 2 D c = s + q + s - Á (21.12) Î Í 2 Rc Ë 2 - Rw 4 The formulae are obtained by replacing R c by R q and re-evaluating the neutral axis position. The method is similar to that used in the American method of plastic design, 10 which predicts a non-linear increase of moment capacity with degree of shear connection K defined as: K = Rq Rs for Rs < Rc or K = R R for R < R q c c s An alternative approach, which has proved attractive, is to define the moment resistance in terms of a simple linear interaction of the form: Mc = Ms + K( Mpc -Ms) (21.13) The stress block and linear interaction methods are presented in Fig for a typical beam. It can be seen that there is a significant benefit in the stress block method in the important range of K = 0.5 to 0.7. In using methods based on partial shear connection a lower limit for K of 0.4 is specified. This is to overcome any adverse effects arising from the limited deformation capacity of the shear-connectors.

19 Basic design 619 moment 'ductile shear rigid shear connectors Fig K 3) degree of shear connection, K Interaction between moment capacity and degree of shear connection. (a) Stress block method. (b) Linear interaction method In BS 5950: Part 3, the limiting degree of shear connection increases with beam span (L in metres) such that: K ( L-6) (21.14) This formula means that beams longer than 16 m span should be designed for full shear connection, and beams of up to 10 m span designed for not less than 40% shear connection, with a linear transition between the two cases. Partial shear connection is also not permitted for beams subject to heavy off-centre point loads, except where checks are made, as below. A further requirement is that the degree of shear connection should be adequate at all points along the beam length. For a beam subject to point loads, it follows that the shear-connectors should be distributed in accordance with the shear force diagram. Comparison of the method of partial shear connection with other methods of design is presented in Table Partial shear connection can result in overall

20 620 Composite beams Table 21.5 Comparison of designs of simply-supported composite beams Plastic design Beam data Elastic design No connectors Partial Full shear (BS 5950: Part 1) connection connection Full depth (mm) Beam size (mm) UB UB UB UB Beam weight (kg/m) Number of 19mm diameter shear- (every trough) (alternate (every trough) connectors troughs) Imposed load deflection (mm) Self-weight deflection (mm) Beam span: 7.5m (unpropped) Beam spacing: 3.0m Slab depth: 130mm Deck height: 50mm Steel grade: 50 (p y = 355 N/mm 2 ) Concrete grade: 30 (normal weight) Imposed load: 5 kn/m 2 economy by reducing the number of shear-connectors at the expense of a slightly heavier beam than that needed for full shear connection. Elastic design is relatively conservative and necessitates the placing of shear-connectors in accordance with the elastic shear flow. Deflections are calculated using the guidance in section Influence of deck shape on shear connection The efficiency of the shear connection between the composite slab and the composite beam may be reduced because of the shape of the deck profile. This is analogous to the design of haunched slabs where the strength of the shear-connectors is strongly dependent on the area of concrete around them. Typically, there should be a 45 projection from the base of the connector to the core of the solid slab to transfer shear smoothly in the concrete without local cracking. The model for the action of a shear-connector placed in the trough of a deck profile is shown in Fig For comparison, also shown is the behaviour of a connector in a solid slab. Effectively, the centre of resistance in the former case moves towards the head of the stud and the couple created is partly resisted by bending of the stud but also by tensile and compressive forces in the concrete, encouraging concrete cracking, and consequently the strength of the stud is reduced. A number of tests on different stud heights and profile shapes have been performed. The following formula has been adopted by most standards worldwide. The

21 - av a. C a. 0 gii ;:. Cb II C- C 0 a Steel Designers' Manual - 6th Edition (2003) Basic design 621 Fig strength reduction factor (relative to a solid slab) for the case where the decking crosses the beams is: r p Model of behaviour of shear-connector. (a) Shear-connector in plain slab. (b) Shear-connector fixed through profile sheeting ba h Dp for n = 085. Ê - ˆ Á 10. = 1 n D Ë D 08. for n = 2 p p (21.15) where b a is the average width of the trough, h is the stud height, and n is the number of studs per trough (n < 3). The limit for pairs of studs is given in BS 5950 Part 3 and takes account of less ductile behaviour when n > 1. This formula does not apply in cases where the shear-connector does not project at least 35 mm above the top of the deck. A further limit is that h 2D p in evaluating r p. Where the decking is placed parallel to the beams no reduction is made for the number of connectors but the constant in Equation (21.15) is reduced to 0.6 (instead of 0.85). No reduction is made in the second case when b a /D p > 1.5. Further geometric limits on the placing of the shear-connectors are presented in Fig The longitudinal spacing of the shear-connectors is limited to a maximum of 600 mm and a minimum of 100 mm.

22 622 Composite beams 25 mm or mm II 1 looped bars : slab edge Fig minimum dimensions unless stated Geometric limits on location of shear-connectors Longitudinal shear transfer In order to transfer the thrust from the shear-connectors into the slab, without splitting, the strength of the slab in longitudinal shear should be checked. The strength is further influenced by the presence of pre-existing cracks along the beam as a result of the bending of the slab over the beam support. The design recommendations used to check the resistance of the slab to longitudinal shear are based on research into the behaviour of reinforced concrete slabs. The design longitudinal shear stress which can be transferred is taken as 0.9 N/mm 2 for normal weight and 0.7 N/mm 2 for lightweight concrete; this strength is relatively insensitive to the grade of concrete. 11 It is first necessary to establish potential planes of longitudinal shear failure around the shear-connectors. Typical cases are shown in Fig The top reinforcement is assumed to develop its full tensile resistance, and is resisted by an equal and opposite compressive force close to the base of the shear-connector. Both top and bottom reinforcement play an important role in preventing splitting of the concrete. The shear resistance per unit length of the beam which is equated to the shear force transferred through each shear plane (in the case of normal weight concrete) is: V = 0.9L s + 0.7A r f y 0.15 L s f cu (21.16) where L s is the length of each shear plane considered on a typical cross-section, which may be taken as the mean slab depth in Fig (a) or the minimum depth in Fig (b). The total area of reinforcement (per unit length) crossing the shear plane is A r. For an internal beam, the slab shear resistance is therefore 2V. The effect of the decking in resisting longitudinal shear is considerable. Where

23 a a Basic design 623 cover width of decking C b b a a overlap (a) (b) Fig Potential failure planes through slab in longitudinal shear the decking is continuous over the beams, as in Fig. 21.3(a), or rigidly attached by shear-connectors, there is no test evidence of the splitting mode of failure. It is assumed, therefore, that the deck is able to provide an important role as transverse reinforcement. The term A r f y may be enhanced to include the contribution of the decking, although it is necessary to ensure that the ends of the deck (butt joints) are properly welded by shear-connectors. In this case, the anchorage provided by each weld may be taken as 4f t s p y, where f is the stud diameter, t s is the sheet thickness, and p y is the strength of the sheet steel. Dividing by the stud spacing gives the equivalent shear resistance per unit length provided by the decking. Where the decking is laid parallel to the beams, longitudinal shear failure can occur through the sheet-to-sheet overlap close to the line of studs. Failure is assessed assuming the shear force transferred diminishes linearly across the slab to zero at a distance of B c /2. Therefore, the effective shear force that is transferred across the longitudinal overlaps is (1 cover width of decking/effective breadth) shear force. The shear resistance at overlaps excludes the effect of the decking. To prevent splitting of the slab at edge beams, the distance between the line of shear-connectors and the edge of the slab should not be less than 100 mm. When this distance is between 100 mm and 300 mm, additional reinforcement in the form of U bars located below the head of the shear-connectors is to be provided. 12 No additional reinforcement (other than that for transverse reinforcement) need be provided where the edge of the slab is more than 300 mm from the shearconnectors Interaction of shear and moment in composite beams Vertical shear can cause a reduction in the plastic moment resistance of a composite beam where high moment and shear co-exist at the same position along the beam

24 624 Composite beams moment Fig O5P shear force (i.e. the beam is subject to one or two point loads).where the shear force F v exceeds 0.5P v (where P v is the lesser of the shear resistance and the shear buckling resistance, determined from Part 1 of BS 5950), the reduced moment resistance is determined from: Mcv = Mc -( Mc -Mf )( 2Fv Pv - 1) 2 (21.17) where M c is the plastic moment resistance of the composite section, and M f is the plastic moment resistance of the composite section having deducted the shear area (i.e. the web of the section). The interaction is presented diagrammatically in Fig A quadratic relationship has been used, as opposed to the linear relationship in BS 5950: Part 1, because of its better agreement with test data, and because of the need for greater economy in composite sections which are often more highly stressed in shear than noncomposite beams Deflections Interaction between moment and shear Deflection limits for beams are specified in BS 5950: Part 1. Composite beams are, by their nature, shallower than non-composite beams and often are used in structures where long spans would otherwise be uneconomic. As spans increase, so tra-

25 Basic design 625 ditional deflection limits based on a proportion of the beam span may not be appropriate. The absolute deflection may also be important and pre-cambering may need to be considered for beams longer than 10 m. Elastic section properties, as described in section , are used in establishing the deflection of composite beams. Uncracked section properties are considered to be appropriate for deflection calculations. The appropriate modular ratio is used, but it is usually found that the section properties are relatively insensitive to the precise value of modular ratio. The effective breadth of the slab is the same as that used in evaluating the bending resistance of the beam. The deflection of a simple composite beam at working load, where partial shear connection is used, can be calculated from: 13 dc = dc ( -K)( ds - dc) for propped beams (21.18) dc = dc ( -K)( ds - dc) for unpropped beams where d c and d s are the deflections of the composite and steel beam respectively at the appropriate serviceability load; K is the degree of shear connection used in the determining of the plastic strength of the beam (section ). The difference between the coefficients in these two formulae arises from the different shearconnector forces and hence slip at serviceability loads in the two cases. These formulae are conservative with respect to other guidance. 10 The effect of continuity in composite beams may be considered as follows. The imposed load deflection at mid-span of a continuous beam under uniform load or symmetric point loads may be determined from the approximate formula: dcc = d c( ( M1 + M2) M0) (1.19) where d c is the deflection of the simply-supported composite beam for the same loading conditions; M 0 is the maximum moment in a simply-supported beam subject to the same loads; M 1 and M 2 are the end moments at the adjacent supports of the span of the continuous beam under consideration. To determine appropriate values of M 1 and M 2, an elastic global analysis is carried out using the flexural stiffness of the uncracked section. For buildings of normal usage, these support moments are reduced to take into account the effect of pattern loading and concrete cracking. The redistribution of support moment under imposed load should be taken as the same as that used at the ultimate limit state (see Table 21.1), but not less than 30%. For buildings subject to semi-permanent or variable loads (e.g. warehouses), there is a possibility of alternating plasticity under repeated loading leading to greater imposed load deflections. This also affects the design of continuous beams designed by plastic hinge analysis, where the effective redistribution of support moment exceeds 50%. In such cases a more detailed analysis should be carried out considering these effects (commonly referred to as shakedown ) as follows: (1) evaluate the support moments based on elastic analysis of the continuous beam under a first loading cycle of dead load and 80% imposed load (or 100% for semi-permanent load);

26 626 Composite beams (2) evaluate the excess moment where the above support moment exceeds the plastic resistance of the section under negative moment; (3) the net support moments based on elastic analysis of the continuous beam under imposed load are to be reduced by 30% (or 50% for semi-permanent loads) and further reduced by the above excess moment; (4) these support moments are input into Equation (21.19) to determine the imposed load deflection Vibration Shallower beams imply greater flexibility, and although the in-service performance of composite beams and floors in existing buildings is good, the designer may be concerned about the susceptibility of the structure to vibration-induced oscillations. The parameter commonly associated with this effect is the natural frequency of the floor or beams. The damping of the vibration by a bare steel composite structure is often low. However, when the building is occupied, damping increases considerably. The lower the natural frequency, the more the structure may respond dynamically to occupant-induced vibration. A limit of 4 Hz (cycles per second) is a commonly accepted lower bound to the natural frequency of each element of the structure. Clearly, vibrating machinery or external vibration effects pose particular problems and in such cases it is often necessary to isolate the source of the vibration. In practice, the mass of the structure is normally such that the exciting force is very small in comparison, leading to the conclusion that long-span structures may respond less than light short-span structures. Guidance is given in Reference 14 and Chapter Shrinkage, cracking and temperature It is not normally necessary to check crack widths in composite floors in heated buildings, even where the beams are designed as simply-supported, provided that the slab is reinforced as recommended in BS 5950: Part 4 or BS 8110 as appropriate. In such cases crack widths may be outside the limits given in BS 8110, but experience shows that no durability problems arise. In other cases additional reinforcement over the beam supports may be required to control cracking, and the relevant clauses in BS 8110: Part 2 15 and BS 5400: Part 5 5 should be followed. This is particularly important where hard finishes are used. Questions of long-term shrinkage and temperature-induced effects often arise in long-span continuous composite beams, as they cause additional negative (hogging) moments and deflections. In buildings these effects are generally neglected, but in bridges they can be important. 5,7

27 References 627 The curvature of a composite section resulting from a free shrinkage (or temperature induced) strain e s in the slab is: es( D+ Ds + Dp) A Ks = (21.20) 21 ( + a eri ) c where I c and r are defined in section and a e is the appropriate modular ratio for the duration of the action considered. The free shrinkage strain may be taken to vary between in external applications and in dry heated buildings. A creep reduction factor is used in BS 5400: Part 5 5 when considering shrinkage strains. This can reduce the effective strain by up to 50%. The central deflection of a simply-supported beam resulting from shrinkage strain is then 0.125K s L 2. References to Chapter British Standards Institution (1990) Structural use of steelwork in building. Part 3, Section 3.1: Code of practice for design of composite beams. BS 5950, BSI, London. 2. The Steel Construction Institute (SCI) (1989) Design of Composite Slabs and Beams with Steel Decking. SCI, Ascot, Berks. 3. Chien E.Y.L. & Ritchie J.K. (1984) Design and Construction of Composite Floor Systems. Canadian Institute of Steel Construction. 4. Lawson R.M. (1988) Design for Openings in the Webs of Composite Beams. The Steel Construction Institute, Ascot, Berks. 5. British Standards Institution (1979) Steel, concrete and composite bridges. Part 5: Code of practice for design of composite bridges. BS 5400, BSI, London. 6. Brett P.R., Nethercot D.A. & Owens G.W. (1987) Continuous construction in steel for roofs and composite floors. The Structural Engineer, 65A, No. 10, Oct. 7. Johnson R.P. & Buckby R.J. (1986) Composite Structures of Steel and Concrete, Vol. 2: Bridges, 2nd edn. Collins. 8. British Standards Institution (1994) Eurocode 4: Design of Composite steel and concrete structures. General rules and rules for buildings. DD ENV , BSI, London. 9. Yam L.C.P. & Chapman J.C. (1968) The inelastic behaviour of simply supported composite beams of steel and concrete. Proc. Instn Civ. Engrs, 41, Dec., American Institute of Steel Construction (1986) Manual of Steel Construction: Load and Resistance Factor Design. AISC, Chicago. 11. Johnson R.P. (1975 & 1986) Composite Structures of Steel and Concrete. Vol. 1: Beams. Vol. 2: Bridges, 2nd edn. Granada. 12. Johnson R.P. & Oehlers D.J. (1982) Design for longitudinal shear in composite L beams. Proc. Instn Civ. Engrs, 73, Part 2, March, Johnson R.P. & May I.M. (1975) Partial interaction design of composite beams. The Structural Engineer, 53, No. 8, Aug.,