CIVL 432 COMPOSITE BEAMS R.G. DRIVER REFERENCES S16: §17 (Sub-clauses are specified in context below.) Handbook (11th Ed.): pp. 5-74–5-125; Tables 5-5, 5-6, 5-7. KEY DEFINITIONS Cover Slab: the concrete above the flutes of the steel deck. Full Composite Action: the state where the interconnection between the steel and the concrete is sufficient for the two materials to behave as if monolithic. Partial Composite Action: the state where the interconnection between the steel and the concrete permits some degree of horizontal slip at the interface. Shear Stud: a “headed” solid cylindrical steel bar that is welded to the steel beam top flange and encased in the concrete slab to permit composite action to occur. Steel Deck: a light-gauge fluted galvanized steel stay-in-place form that is often designed to carry the weight of the wet concrete without shoring. INTRODUCTION Engineers are always striving to optimise the use of materials. In both steel buildings and steel bridges, concrete is placed as a floor or deck on the steel framework to provide a loading surface. Particularly in positive moment regions, it makes sense to interconnect the two materials to take advantage of the concrete compressive capacity to increase the moment resistance of the section. This way, the concrete is not merely more weight from the point of view of the steel beam, but rather it contributes significantly to the strength of the system after it has cured. If the materials are interconnected sufficiently so as to act together, it leads to what we call composite action. Depending on whether the job is of a sufficient scale (and the spans of sufficient length) to justify bringing the specialised welding equipment onto the site and introducing another operation into the construction sequence, this often leads to great economic advantages, in part by reducing the sizes of the steel beams required. The composite system is also very stiff— reducing the likelihood of vibration or deflection problems—and ductile. We will focus on composite beams typical for building applications, although the principles apply equally to composite bridge girders. Moreover, we will focus on rolled steel W-shapes as the steel section, but composite flexural systems using other rolled sections, welded plate girders, trusses, or joists are also used. –1– CIVL 432 COMPOSITE BEAMS R.G. DRIVER SHEAR CONNECTION Since shear connection is the whole reason that composite action can be attained, it is useful to begin by discussing why it is called “shear” connection, what it achieves, and how it is accomplished. To understand this, think back to your Strength of Materials course and the development of shear stresses within a beam under a moment gradient. Not only do vertical shear stresses develop to equilibrate the applied transverse loads, but horizontal (longitudinal) shear stresses are also required for complete equilibrium of the beam. It is implicit in a monolithic beam made of a single material that these shear stresses can be resisted by the material, and for most structural materials used for beams this does not represent a problem. However, if the beam is made of individual pieces—say, the flanges and web (could be either the same material or different)—the pieces have to be connected by a means that provides enough strength to resist these longitudinal shear stresses. In the case of a built-up welded steel I-girder, the welds between the web and flanges are checked to ensure sufficient longitudinal shear capacity. In the case of a composite beam, it is the steel section and the concrete slab that must be connected for these shear stresses. This is the reason we refer to this connection as a shear connection and the means by which it is achieved as shear connectors. Consider the simple composite beam in the figure below, where it is assumed that the concrete carries all of the compression and the steel all of the tension. Technical Stuff The shear stresses arise due to the moment gradient. Within a constant-moment region, therefore, theoretically no interconnection is required. Furthermore, under a uniformly-distributed load, the shear connectors theoretically should be spaced closer together near the pinned supports or points of inflection than near the maximum moment. Why? Note that longitudinal shear stresses must develop at the interface between the two materials to maintain equilibrium of the two components between the point of maximum moment and the point of zero moment. (If no shear transfer occurs on this interface, the materials act as two individual beams and the behaviour is then non-composite.) To resist the longitudinal shear effectively, some form of interconnection is required. –2– CIVL 432 COMPOSITE BEAMS R.G. DRIVER Typically, the interconnection between the steel and concrete is made mechanically by welding shear connectors to the beam top flange so that they extend up into the concrete after it is cast. Although conceptually nearly any shape of connector could be used, certain types are more common and have been proven in-service, as well as in the lab. The following figure shows three methods of interconnection, with the headed shear studs (or simply shear studs) in Figure (b) being by far the most common (often with only one line of studs down the centre of the beam) and the only kind discussed in detail in this course. Another possible, but less common, type is similar to those shown in Figure (b) except that instead of a formed “head”, they are simply hooked at the top. Headed or hooked shear connectors are referred to in S16 as “end-welded studs” (although this term is rarely used in practice). Tip “Shear connector” is a generic term and “shear stud” refers to a specific type of shear connector. Please never call shear studs “bolts”! Headed shear studs are commonly referred to by the proprietary name Nelson® studs; however, they really should be referred to generically unless they really are made by Nelson Stud Welding, Inc. These studs are welded to the top flange using a specialised stud-welding “gun” that allows very rapid installation by a single worker, either in the shop or in the field. Studs are available in several diameters, but 19 mm (3/4 inch) and 22 mm (7/8 inch) are most commonly used, although S16 [§17.6.5] requires that the stud diameter, d, not exceed 2.5 times the If the flange is flexible as thickness of the part to which it is welded (normally compared to the force that the the top flange). It also stipulates [§17.9.1] that the attached stud is expected to width of the part to which it is welded not be less resist (assumed proportional than 1.4d + 20 mm. This latter requirement is really to the diameter), it can for truss chords, since the width of the top flange of a standard wide-flange section would provide plenty deform (or even tear), reducing the effectiveness of of room for the stud. the shear transfer. –3– Technical Stuff CIVL 432 COMPOSITE BEAMS R.G. DRIVER Often it is more economical to use partial shear connection (as opposed to full shear connection), where the number of shear studs provided is fewer than that which would develop the full flexural capacity of the composite cross-section. This can be because the design is controlled by deflection criteria and the full capacity is not required, or simply that the savings in shear studs and associated labour is greater than the additional cost of providing a slightly larger steel section. This latter situation occurs frequently because the number of shear studs provided is not directly proportional to the composite section capacity. In fact, as a rule-of-thumb, providing about 50% of the shear studs required for full composite action results in about 80% of the flexural capacity of the fully composite beam. SOLID SLABS VS. RIBBED SLABS WITH STEEL DECK Composite beams can incorporate solid slabs atop the steel section (left figure below) or ribbed slabs (right figure below). The former provides more concrete area for the same overall depth, but has a drawback in that the concrete usually needs to be shored (i.e., temporarily supported by posts at a close spacing) during curing due to the flat (traditional) formwork required. t Ribbed slabs are very common, primarily because they arise from the use of galvanized corrugated—or fluted—steel deck (with a variety of available proprietary profiles and depths). The steel deck acts as the concrete formwork and spans between steel section top flanges. It can be designed with enough flexural stiffness to support the wet concrete without shoring, keeping the work area below free of obstructions. Composite deck is used that incorporates embossments that improve the interconnection between the concrete –4– Tip Deck manufacturers provide catalogues that indicate allowable deck spans for a given slab thickness and under a variety of conditions. CIVL 432 COMPOSITE BEAMS R.G. DRIVER and the deck itself and allow the deck to act as slab reinforcement, with a similar effect to reinforcing bars. The deck flutes can run either parallel or perpendicular to the beam, a difference that is accounted for in the design procedures (discussed below). Tip According to S16 [§17.3.4], the maximum deck The use of ribbed slabs and steel depth, hd , is 80 mm and the minimum average deck interconnected with steel flute width, wd , is 50 mm. The thicknesses of beams is sometimes referred to as steel deck material used most commonly are “hollow composite” construction. 0.76 mm (22 gauge), 0.91 mm (20 gauge), and 1.21 mm (18 gauge), although other thicknesses are available (these thicknesses exclude coatings). If the material is too thin it can be damaged during construction. The shear studs can be welded to the steel top flange right through up to two layers of overlapping deck sheets as long as neither sheet is more than 1.52 mm thick (16 gauge) excluding the zinc coating [§17.6.3]. Otherwise, holes must be cut in the deck with a clearance sufficient for welding the stud directly to the flange. SLAB REINFORCEMENT Although the composite steel deck provides some reinforcement to the slab, S16 [§17.5] has other miscellaneous requirements where standard reinforcing bars need to be provided, primarily for crack control. This section is simply to make you aware that these provisions exist. We will not focus on these clauses in this course. Usually welded wire mesh (WWM) is also provided in the slab for crack control. DESIGN Design of composite beams for flexure is dramatically different from the procedures for either steel or concrete beams because the influence of both materials must be accounted for and the appropriate interconnection must be provided. By contrast, in the design of composite beams for shear the steel beam is assumed to resist the full shear force [§17.3.2]. Therefore, the design method for shear (described in CIVL 331) for steel beams applies also to composite beams and it won’t be repeated here. Effective Concrete Width, Thickness, and Area Similar to what we did for concrete T-beams, an effective concrete slab width is used in the design of composite beams. For slabs extending on both sides of the steel section, the effective slab width, b, is the lesser of [§17.4.1]: 0.25 times the composite beam span; and a distance extending half way to the next parallel support on each side. –5– CIVL 432 COMPOSITE BEAMS R.G. DRIVER For slabs extending on one side only of the steel section, the effective slab width, b, is the width of the top flange of the steel section plus the lesser of [§17.4.2]: 0.10 times the composite beam span; and half of the clear distance to the next parallel support. If there is a small slab overhang on the other side of the steel section, which is commonly the case, the overhang is included in the overall effective slab width, b. Normally only the concrete above the flutes (the effective cover slab thickness, t) is considered effective in contributing to the resistance of the design moments in ribbed slabs. The minimum effective cover slab thickness permitted by S16 [§17.2] is 65 mm. The full thickness, t, is effective in solid slabs. The maximum effective concrete area is simply the effective width, b, multiplied by the effective thickness, t. The contribution of any concrete in tension is neglected. [§17.9.2] Trap Flexural Capacity The factored moment resistance of a composite Don’t forget that until the concrete beam is based upon the ultimate capacity of the gains its strength it does not provide cross-section; the slab is considered to provide lateral support. However, the steel full lateral support to the steel section (i.e., no deck, when properly fastened to the LTB!). The method for determining the flexural steel section, also provides lateral capacity of a composite beam depends on the support if the flutes run perpendicular location of the plastic neutral axis. Three cases to the axis of the steel member. are addressed in S16 [§17.9.3]: 1. Full shear connection with the plastic neutral axis in the slab; 2. Full shear connection with the plastic neutral axis in the steel section; and 3. Partial shear connection (plastic neutral axis is always in the steel section). Let’s consider solid slabs first. In each case, the location of the neutral axis can be determined from an assessment of horizontal equilibrium. The moment capacity is then determined from the force resultant magnitudes and their respective moment arms. Note in advance that the maximum compressive force possible in the slab is 1cfcbt, where c = 0.65, and the maximum tensile force possible in the steel section is AsFy , where = 0.9. As was the case with conventional reinforced concrete members, 1 = 0.85–0.0015fc 0.67. Now, consider three composite beams with solid slabs that correspond to the three cases enumerated above. –6– CIVL 432 COMPOSITE BEAMS R.G. DRIVER Case 1—Full shear connection with the plastic neutral axis in the slab The figure below depicts this situation, which occurs when 1cfcbt > AsFy . Since the steel section is entirely in tension, the factored tensile force, Tr , is simply equal to AsFy . However, the factored compressive force in the concrete, Cr, must be less than 1cfcbt. 1 c fc Tip The forces act at the centroids of the areas over which they are distributed (i.e., a uniform stress is assumed). Denoting the depth of the compressive stress block as “a”, the compressive force in the slab, Cr, and the tensile force in the steel, Tr , are: Tip Cr 1 c f cba [1] [2] Tr A s Fy From horizontal equilibrium and Equations [1] and [2], the depth of the compressive stress block is: A s Fy a t [3] 1 c f cb The “prime” symbol is used with Cr when the force is in the concrete. [§17.9.3(a)] The factored moment capacity, Mrc , of the composite section is therefore: M rc Tr e A s Fy e [4] [§17.9.3(a)] Finally, in order to ensure full composite action the longitudinal shear force that develops between the two materials, Vh , must be resisted by shear connectors. Therefore, the required sum of the factored resistances of all shear connectors between the points of maximum and zero moment, Qr , is determined as follows: Q r Tr A s Fy Vh [5] [§17.9.3(a); §17.9.5] –7– CIVL 432 COMPOSITE BEAMS R.G. DRIVER Case 2—Full shear connection with the plastic neutral axis in the steel section This case occurs when AsFy > 1cfcbt , resulting in a part of the total compressive force being resisted by the steel section. The figure below depicts this situation. Since the steel section is not entirely in tension, the tensile force in the steel, Tr , is less than AsFy and the compressive force in the concrete, Cr, is equal to 1cfcbt. 1 c fc The compressive force in the concrete, Cr, is: Cr 1 c f cbt [6] [§17.9.3(b)] From equilibrium, the compressive force in the steel, Cr , is: [7] C r Tr Cr A s Fy C r Cr A s Fy Cr 2 [§17.9.3(b)] The factored moment capacity, Mrc , of the composite section is therefore: M rc C r e Cr e [8] [§17.9.3(b)] Finally, in order to ensure full composite action, the required sum of the factored resistances of all shear connectors between the points of maximum and zero moment, Qr , is determined as follows: Q r Cr 1 c f cbt Vh [9] [§17.9.3(b); §17.9.5] Case 3—Partial shear connection (plastic neutral axis always in the steel section) Partial shear connection implies that only some fraction of the number of shear studs required for full composite action is provided. S16 permits this fraction to be as low as 40% if the design is governed by strength requirements and 25% if it is governed by serviceability considerations. If the shear transfer capacity falls below these levels, the behaviour is assumed to be non-composite. [§17.9.4] –8– CIVL 432 COMPOSITE BEAMS R.G. DRIVER The figure below depicts partial composite action. Since the steel section is not entirely in tension, the tensile force in the steel, Tr , is less than AsFy . Due to the partial composite action, the compressive force in the concrete, Cr, is less than 1cfcbt. 1 c fc Technical Stuff Actually, the entire slab depth is in compression, but at a stress lower than 1cfc. The stress distribution shown is idealised. Because of the partial composite action, the part of the total concrete slab strength that can develop is directly related to the capacity of the shear studs actually provided, Qr . Therefore: Cr Q r Vh 1 c f cbt [10] [§17.9.3(c); §17.9.6] The depth of the compressive stress block is calculated as: Cr a t [11] 1 c f cb [§17.9.3(c)] From equilibrium, the compressive force in the steel, Cr , is (same as Case 2): [12] C r Tr Cr A s Fy C r Cr A s Fy Cr 2 [§17.9.3(c)] The factored moment capacity, Mrc , of the partially composite section is therefore: [13] M rc C r e Cr e [§17.9.3(c)] Composite construction using corrugated steel deck (hollow composite constr.) The flexural requirements discussed above for Cases 1 to 3 apply equally to composite beams constructed with corrugated steel deck. The only difference is that the concrete within the flutes is considered to be ineffective, so the maximum depth of the compressive stress block, t, is taken as being from the top of the slab to the top of the steel deck. No force (neither tension nor compression) is assumed to be present within the depth of the deck flutes. –9– CIVL 432 COMPOSITE BEAMS R.G. DRIVER Negative moment capacity Under a negative moment, the concrete is in tension and therefore considered to be ineffective. Although S16 has provisions to include rebar (if properly anchored and if sufficient shear studs are provided) in negative moment capacity calculations [§17.9.7], normally the steel section is considered to act alone and the rebar provided merely for crack control. Class of Steel Section for Use in Composite Construction (Local Buckling) You may have noticed that the composite flexural capacity equations presented above are all based on the steel section reaching the fully plastic condition. Since local buckling checks are required only when the flanges and web are at least partially in compression, no such check is required as long as the neutral axis is in the slab. Furthermore, if the neutral axis is in the top flange of the steel section, the web is still fully in tension and the flange is only partially in compression with the strains remaining very low. If the neutral axis is in the web of the steel section, the chance of local buckling occurring increases, although the presence of the slab reduces the compressive strains that will develop as compared to those in a non-composite beam. S16 does not address local buckling of steel sections used in composite beams head-on, but it is considered prudent to limit your choice to Class 1 and 2 sections whenever the neutral axis falls within the steel section. (If the neutral axis is within the top flange, checking the web is unnecessary.) Design of Shear Studs For headed shear studs to qualify for use, they must have a height, h, to diameter, d, ratio of at least 4.0 and, in the case of hollow composite construction, they must project above the deck flutes by at least two stud diameters, 2d, (heights are taken prior to welding) in order to engage the concrete effectively Tip [§17.7.2.1]. Moreover, the longitudinal spacing of studs along the steel flange must be at least six stud diameters, 6d, No minimum clear to ensure that each stud is surrounded by sufficient concrete cover above the stud [§17.7.2.5], and the spacing should be at most 1000 mm, to head is specified in ensure that the slab and steel section do not separate S16. I recommend significantly under factored loads [§17.7.2.5; §17.8]. The 20 mm as a minimum average stud spacing in a span should not exceed 600 mm for interior exposure. [§17.8]. Finally, the transverse spacing of studs must be at least four stud diameters, 4d, when more than one stud is needed at a cross-section [§17.7.2.5]. (All stud spacing requirements above are given centre-to-centre.) The factored capacity of a single shear stud, qr , has been defined mainly from test results and it depends on several factors, including the environment of concrete within which it is embedded. S16 addresses three cases [§17.7.2]: – 10 – CIVL 432 COMPOSITE BEAMS R.G. DRIVER A. Solid slabs; B. Ribbed slabs with ribs running parallel to the beam; and C. Ribbed slabs with ribs running perpendicular to the beam. Case A—Solid slabs This is the “best-case scenario”. In fact, the other two cases (B and C) include a check to ensure that the strength calculated is not greater than the capacity of the same stud in a solid slab. The factored capacity of a single shear stud in a solid slab, qrs (second subscript stands for “solid”), is: [14] q rs 0.50 sc A sc f c E c sc A sc Fu where Asc is the cross-sectional area of the shear connector, Fu is the nominal ultimate tensile strength of the shear connector (most commonly 450 MPa), and the other variables are well known to you from the concrete part of the course. The resistance factor used in the design of the shear connectors, sc , is 0.80 [§17.7.1]. [§17.7.2.2] Tip Unless a better value is available, Ec = 21,000 MPa is generally conservative. Alternatively, equations are available in A23.3 §8.6.2.2 and §8.6.2.3. The second part of Equation [14] is from the observation that if shear studs do not fail by crushing the concrete around the stud, they will eventually bend over and fail in tension. Case B—Ribbed slabs with ribs running parallel to the beam In this case, the factored capacity of a single shear stud, qrr (second subscript stands for “ribbed”), is: wd 1.5 q rs hd [15] if 3.0 > wd/hd ≥ 1.50 (“wide rib profile”), q rr q rs 0.75 0.167 [§17.7.2.3(a)] [16] if wd/hd < 1.50 (“narrow rib profile”), q rr sc 0.92 wd 0.8 0. 2 dh f c 11sd f c 0.75q rs hd [§17.7.2.3(b)] where wd is the average (they are usually trapezoidally shaped) width of the deck flutes that are filled with concrete, hd is the depth of the flutes, s is the longitudinal stud spacing, and the other variables have been defined previously. Case C—Ribbed slabs with ribs running perpendicular to the beam In this case, the factored capacity of a single shear stud, qrr , is: – 11 – CIVL 432 COMPOSITE BEAMS R.G. DRIVER [17] if hd = 75 mm, q rr 0.35scA p f c q rs [§17.7.2.4(a)] [18] if hd = 38 mm, q rr 0.61scA p f c q rs [§17.7.2.4(b)] where = 1.0 for normal density and = 0.85 for semi-low-density concrete, and Ap is the concrete pull-out area taking the deck profile and stud welding burn-off into account, and the other variables have been defined previously. Trap Note that the deck profile depths, hd , that distinguish between Equations [17] and [18] might be misleading. Steel deck is very often provided as 3 inch (76.2 mm) or 1.5 inch (38.1 mm) deck. This is still well within the spirit of the clause. Calculating Ap is tedious. The following figure from the S16 Commentary depicts the intent (lines are at 45°). Thankfully, the Handbook provides assistance with these calculations in Tables 5-3 and 5-4. I will not make you do Ap calculations. If studs are provided in pairs (i.e., two studs at the same cross-section, and therefore in a single flute), Ap is determined for the stud pair, with a ridge extending between the tops of the two studs [§17.7.2.4]. In this case, qrr is the capacity of the stud pair (which cannot be greater than qrs for two studs). Distribution of shear studs The required sum of the factored resistances of all shear studs between the points of maximum and zero moment, Vh , was specified in the section entitled “Flexural Capacity” above. The number of shear studs, n, required within this region, therefore, is: V n h [19] qr – 12 – Tip It is not unusual to have a combination of single studs and stud pairs on a beam in Case C. Trap The total number of studs on the beam will be 2n, not n! [§17.9.8] CIVL 432 In general, the ductility of the system allows the shear studs to be spaced uniformly (which is especially convenient when the deck ribs are perpendicular to the beam axis!). However, in a region of positive moment, the number of shear studs, n, between any concentrated load and the nearest point of zero moment must not be less than: [20] COMPOSITE BEAMS R.G. DRIVER Tip When deck flutes are perpendicular to the beam axis and there are no concentrated loads, it is common practice to specify that one stud be placed in every second flute starting from each end of the beam until all “2n” studs have been installed. If there are any studs remaining after reaching mid-span, they are filled in from the ends of the beam in the alternate flutes. If there are still studs left over when every flute has a stud, they can be “doubled up” where they are needed most (near the ends where the moment gradient is steepest). If doubling-up is necessary, recall that stud pairs may have a lower capacity than the combined capacity of two single studs. M Mr n n f 1 M M r f where Mf1 = factored moment at the concentrated load; Mr = factored moment resistance of the steel; and Mf = maximum positive factored moment. Since the capacity of the steel section alone, Mr , does not depend on the presence of shear studs, it is subtracted from both Mf1 and Mf . [§17.9.8] Tip Even if Equation [20] needs to be checked, a uniform stud spacing may still turn out to be adequate. Trap If n results in a wider Additional Shear Check on Vertical Planes for Solid Slabs spacing in the region where and Ribbed Slabs with Ribs Running Parallel to the Beam the BMD is steepest, use a As discussed previously, the compressive force present in uniform spacing instead. the slab at the point of maximum moment must be equilibrated entirely by the point of zero moment through shear at the interface between the steel and the concrete, and the requirements for selecting the shear studs ensure that sufficient capacity is provided for this force transfer to take place. For the same reason, certain other planes within the slab need to be checked to ensure sufficient shear capacity. Note that the general forms of the equations in S16 that are used to check these shear planes account for the fact that rebar may be present in the slab in the two directions. However, it is always conservative to neglect the rebar in the demand and/or capacity equations presented below. – 13 – CIVL 432 COMPOSITE BEAMS R.G. DRIVER Consider the following figure of a composite beam with a ribbed slab extending from the point of maximum moment to the point of zero moment. Technical Stuff You might think that qr in the Vu equation should really be Vh (i.e., the actual shear force rather than the shear strength). However, the original research recommends that the capacity of all of the studs be developed prior to failing the concrete shear plane, even if more studs are provided than required, in order to ensure a ductile failure mode. Tip This check is also required for solid slabs. The planes marked with the area Acv are the ones being checked. Any force that is applied within the shaded area Ac does not need to be transferred across these planes. Take Ac to be the area of the rectangular compressive stress block with the depth “a” between the vertical planes marked Acv in the figure. (Using the depth “a” is conservative, since more of the shaded area would attract compressive force if the vertical planes actually started to fail.) The total shear force, Vu , to be resisted on these vertical planes is: [21] Vu q r 1 c f cA c r A r Fyr – 14 – [§17.9.10] CIVL 432 COMPOSITE BEAMS R.G. DRIVER where Ar (Arl in the figure) is the total area of the longitudinal reinforcing steel within the shaded area Ac , Fyr is the nominal yield strength of these bars, and the other variables have been defined previously. The resistance factor for the reinforcing steel, r , is (as always) 0.85. For normal-density concrete (equations for semi-low-density are available in the literature), the factored shear resistance along the vertical planes marked Acv is: Vr 0.80r A r Fyr 2.76c A cv 0.50c f cA cv [§17.9.10] [22] Technical Stuff where Ar (Art in the figure) is the total Note that no benefit is considered for area of the transverse reinforcing steel the shear resistance of the steel deck since crossing the shear planes that constitute no research is available to substantiate it. Acv (i.e., counting two planes in this Neglecting it is conservative. case), and the other variables have been defined previously. For determining the area Acv , use the full depth of concrete at that section with the length equal to the distance from between the points of maximum and zero moment. For a solid slab, check the vertical planes at the edges of the top flange of the steel section. While planes closer to the studs theoretically carry slightly more force, the zone around the studs themselves is considered to be accounted for by the procedures used to select the studs. Use of the depth “a” is also quite conservative. As long as Vr ≥ Vu , the shear capacity along these planes is adequate. Unshored Beams Although beams can be shored until the concrete gains its design strength to ensure that all loads are carried by the composite section, it is common to omit shoring to keep the work areas below unobstructed and to avoid the associated direct and indirect expenses. In unshored construction, S16 requires that the stresses in the tension flange of the steel section be checked to ensure that they do not exceed the nominal yield stress, Fy , under service loads [§17.11]. The purpose of this check is to limit deflections, so it is a serviceability check. To perform this check, the loads are separated into those that are applied prior to the slab reaching 75% of its nominal design capacity (usually about 7 days) and those applied thereafter. The former are assumed to be applied to the bare steel section and the latter to the composite section. Although no equation is provided in S16, this requirement is satisfied by the following: – 15 – CIVL 432 [23] COMPOSITE BEAMS M1 M 2 Fy Ss St R.G. DRIVER Trap Don’t apply this to shored construction. where M1 = moment from specified loads that are applied prior to the concrete reaching 0.75fc (excludes loads that are present only during construction, such as construction LL, plywood formwork, etc.); M2 = moment from any specified loads that are applied after the concrete reaches 0.75fc; Ss = elastic section modulus of the steel section (to the bottom fibre); and St = elastic section modulus of the composite section (to the bottom fibre) determined using the transformed area method (from your Strength of Materials course) to obtain an equivalent steel section. Beam Design for Construction Loads Since wet concrete has no strength, during Construction live loads should be construction the bare steel section must be estimated as closely as possible for designed to carry all factored construction the case considered, but a common loads—including the weight of the wet concrete nominal value for light construction itself—until the slab reaches its design strength, taking into account the lateral bracing and shoring is 1 kPa (picture one 225 lb dude on conditions furnished during construction [§17.12]. every square metre of floor area). The design procedures for non-composite beams were all covered in CIVL 331. Tip Deflections A part of the total deflection of a composite beam occurs prior to the beam becoming composite. This deflection is computed based on the stiffness of the steel section alone. The remainder of the deflection, due to loads applied after the concrete has cured, is calculated assuming composite action. Deflections of composite beams under specified loads are affected by many things including creep and shrinkage of the concrete and the potential for interfacial slip between the two materials in the case of partial shear connection. The latter effect is accounted for by determining an effective moment of inertia, Ie , as follows: [24] Ie Is 0.85p 0.25 I t Is where Is = moment of inertia of the steel section; p = fraction of full shear connection; and It = transformed moment of inertia of composite beam (equivalent steel section) based on the modular ratio, n = E/Ec . – 16 – [§17.3.1(a)] Technical Stuff Even for full composite action (p = 1.0), there is still a reduction in the moment of inertia because there is some flexibility in the shear studs that can lead to slip, increasing deflections. CIVL 432 COMPOSITE BEAMS R.G. DRIVER To account for creep, the elastic deflections caused by dead loads and long-term live loads using the effective moment of inertia defined by Equation [24] are amplified by 15% [§17.3.1(b)]. The concrete in a composite beam is not able to shrink freely since it is connected to the steel section. The resulting restrained shrinkage tends to induce curvature in the composite beam that has the effect of increasing deflections, as indicated in the figure below. Figure (a) indicates how the slab would shrink if unrestrained, while Figure (b) shows the results of the shrinkage when restrained by connection to the steel by the use of studs. The force applied to the section due to the concrete shrinkage can be considered to act at the centroid of the slab. This, in turn, leads to a moment acting at a distance y above the elastic neutral axis of the composite section, as shown in the figure below. – 17 – CIVL 432 COMPOSITE BEAMS R.G. DRIVER f A c L2 y 8n s I es [§17.3.1(c)] The resulting deflection is: [25] s c where c = empirical coefficient to adjust theoretical relationship to better match with test results (accounts for things such as concrete cracking and the non-linear stress vs. strain relationship of concrete) f = free shrinkage strain of the concrete (use 580 in the absence of a more accurate value); Ac = effective area of the concrete slab; L = beam span; y = distance from the centroid of the effective area of the concrete slab to the elastic neutral axis of the composite beam; ns = modular ratio suitable for shrinkage calculations, E/Ec; E = modulus of elasticity of steel; Ec = age- and creep-adjusted effective modulus of concrete in tension; and Ies = effective moment of inertia (Equation [24]), with the transformed moment of inertia of the composite beam (equivalent steel section) based on the modular ratio ns instead of n. The effective modulus of concrete, Ec , is time-dependent, but since the modular ratio and the effective moment of inertia (denominator of Equation [25]), as well as the distance y (numerator of Equation [25]), vary with it, the net result is that it has a relatively small effect on the deflection. Annex H of S16 provides some guidance for determining Ec . If deflections are calculated assuming simple supports, even when simple shear connections are used at the beam ends this can overestimate the deflection by a significant amount due to the continuity at the supports provided by the composite system. It has been suggested that applying negative end moments equal to 25% of the midspan moment of a uniformly loaded, simply-supported beam will give a good estimate of the true deflection. – 18 – CIVL 432 COMPOSITE BEAMS R.G. DRIVER SUMMARY OF IMPORTANT CONCEPTS Composite beams take advantage of benefits of both steel and concrete by connecting them together so they act (more-or-less) as one. Composite action is achieved through the use of shear connectors (usually headed shear studs). The longitudinal shear at the interface between the two materials must be transferred between the point of maximum moment and the point of zero moment (for equilibrium) and there are special stud distribution requirements when there are concentrated loads. A designer can select either full composite action or partial composite action. For full composite action, whether the neutral axis falls within the slab or the steel section depends on the relative proportions and material strengths of the two components. If only a fraction of the shear connectors required for full composite action is provided (must be ≥ 40% for strength and ≥ 25% for serviceability), partial composite action results, and the neutral axis will always be in the steel section. Slabs can be either solid or ribbed, the latter resulting from the use of corrugated steel deck forms. Ribbed slabs may have the ribs parallel or perpendicular to the steel section, and the rib orientation affects the capacity of the shear studs. In solid slabs and ribbed slabs with the ribs parallel to the steel section, the slab could fail on vertical shear planes adjacent to the steel section. These planes must be checked to ensure adequate capacity. In unshored construction, a serviceability check is required to ensure that the maximum tensile stress in the steel under unfactored loads does not exceed Fy . Deflection calculations (under service loads) for composite beams must account for interfacial slip, as well as creep and shrinkage of the concrete. Credits: The figures used in this document are © Kulak, G.L. and Grondin, G.Y. (2002) Limit States Design in Structural Steel, Canadian Institute of Steel Construction, Toronto, ON, © CISC (2000) Handbook of Steel Construction, Canadian Institute of Steel Construction, Toronto, ON, or © Chien, E.Y.L. and Ritchie, J.K. (1984) Design and Construction of Composite Floor Systems, , Canadian Institute of Steel Construction, Toronto, ON. steelcentre.ca – 19 –