384 x 29,000 x 612
Camber beam for 75% of calculated unshored condition. Therefore camber specified = 1.78 x 0.75 = 1.34, say, 1.25 in. Deflection under superimposed dead and live loads for
384 x 29,000 x 1995
Compared to L/360 = 40 x 12/360 = 1.33 in., the calculated deflection of 0.84 in. is small. Therefore the design is OK.
It should be noted that the lowest percentage of partial composite allowed by the AISC specifications is 25%. Some designers, however, will not allow partial composite action below 50%.
Composite haunch girders, although not often used as a floor framing system, merit mention because they help minimize the floor-to-floor height without requiring complicated fabrication. Fig. 7.58 shows a schematic floor plan in which composite haunch girders frame between exterior columns and interior core framing. The haunch girder typically consists of a shallow steel beam, 10- to 12-in. (254- to 305-mm) deep for spans in the 35- to 40-ft (10.6- to 12.19-m)-range. At each end of the beam a triangular haunch is formed by welding a diagonally cut wide-flange beam usually 24- to 27-in. (610- or 686-mm) deep (Fig. 7.59). The haunch is welded to the shallow beam and to the columns at each end of the girder. In this manner the last 8 or 9 ft (2.4 or 2.7 m) of the haunch girder at either end flares out toward the column with a depth varying from about 10 or 12 in. (254 or 305 mm) at the center to about 27 in. (686 mm) at the ends. This system uses less steel and provides greater flexibility for mechanical
ducts, which can be placed anywhere under the shallow central span. The reduction in floor-to-floor height further cuts costs of exterior cladding and of heating and cooling loads. This system, however, is not common because of higher fabrication costs.
A variation of the same concept shown in Fig. 7.60 uses nontapered haunches at each end. The square haunch girder can be fabricated using a shallow-rolled section in the center and two deep-rolled sections, one at each end. Another method of fabricating the girder is to notch the bottom portion of the girder at midspan and reweld the flange to the web. The method requires more steel but comparatively less fabrication work.
In comparison to a shallow girder of constant depth, a haunch girder is significantly stiffer. Figure 7.60b shows the moment diagram in a haunch girder subjected to combined gravity and lateral loads. The corresponding stiffness properties including the effect of composite action in the positive moment regions is also shown in Fig. 7.60c.
Figure 7.61 shows a typical floor-framing plan with composite trusses. To keep the fabrication simple, the top and bottom chords consist of T-sections to which double-angled web members are welded directly without the use of gusset plates. The top chord is made to act compositely with the floor system by using welded shear studs. The space between the diagonals is used for the passage of mechanical and air-conditioning ducts. When the space between the diagonals is not sufficient, vertical members may be welded between the chords to form a vierendeel panel.
In building design, maximum flexibility is achieved if structural, mechanical, electrical, and plumbing trades have their own designated space in the ceiling. This is achieved in a conventional system by placing HVAC ducts, lights, and other fixtures under the beams. Where deep girders are used, penetrations are made in the girder webs to accommodate the ducts. In an office building the typical span between the core and the exterior is about 40 ft (12.2 m), requiring 18- to 21-in. (457 to 533-mm)-deep beams. Usual requirements of HVAC ducts, lights, sprinklers, and ceiling construction result in depths of 4 to 4.25 ft (1.21 to 1.3 m) between the ceiling and top of the floor slab. The depth can, however, be decreased at a substantial penalty either by providing penetrations in relatively deep beams or by using shallower, less economical beam depths.
The stub girder system shown in Fig. 7.62, invented by engineer Dr. Joseph Caloco, attempts to eliminate some of these shortcomings while at the same time reducing the floor steel weight. The key components of the system are short stubs welded intermittently to the top flange of a shallow steel beam. Sufficient space is left between stubs to accommodate mechanical ducts. Floor beams are supported on top of, rather than framed into, the shallow steel beam. Thus the floor beams are designed as continuous members which results in steel savings and reduced deflections. The stubs consist of short wide flange beams placed perpendicular to and between the floor beams. The floor system consists of concrete topping on steel decking connected to the top of stubs. The stub girders are spaced at 25 to 35 ft (7.62 to 10.7 m) on center, spanning between the core and the exterior of the building.
The behavior of a stub girder is akin to a vierendeel truss; the concrete slab serves as the compression chord, the full-length steel beam as the bottom tension chord, and the steel stubs as vertical web members. From an overall consideration, the structure allows installation of mechanical system within the structural envelope, thus reducing floor-to-floor height; the mechanical ducts run through and not under the floor.
The primary action of a stub girder is similar to that of a vierendeel truss; the bending moments are resisted by tension and compression forces in the bottom and top chords of the truss and the shearing stresses by the stub pieces. The bottom chord is a steel wide flange and the top chord is the concrete slab. The effective width of the concrete slab varies
from 6 to 7 ft (1.83 to 2.13 m), requiring additional reinforcement to supplement the compression capacity of the concrete. Stub peices are welded to the top flange of the steel beam and are connected to the metal deck and concrete topping through shear connectors.
Because the truss is a vierendeel truss as opposed to a diagonalized truss, bending of the top and bottom chords is significant. Therefore, it is necessary to consider the interaction between axial loads and bending stresses in the design.
Figure 7.62a shows a typical floor plan with stub girders SG1, SG2, etc. Consider stub girder SG1, spanning 40 ft (12.19 m) between the exterior and interior of the building (Fig. 7.62b). The deck consists of a 2-in. (51-mm)-deep 19-gauge composite metal deck with a 3 if-in. (82.5-mm) lightweight structural concrete topping. A welded wire fabric is used as crack control reinforcement in the concrete slab.
The first step in the analysis is to model the stub girder as an equivalent vierendeel truss. This is shown in Fig. 7.63a. A 14-in. (356-mm)-wide flange beam is assumed as the continuous bottom chord of the truss. The slab and the steel beam are modeled as equivalent top and bottom chords. Note the beam elements representing these members are at the neutral axes of the slab and beam, as shown in Fig. 7.63.
The stub pieces are modeled as a series of vertical beam elements between the top and the bottom chords of the truss with rigid panel zones at the top and bottom.
The various steps of modeling of stub girder are summarized as follows:
1. Top chord of vierendeel truss. As shown in Fig. 7.64, the top chord consists of an equivalent transformed area of the concrete topping, which is obtained by dividing the effective width of concrete slab by the modular ration n = Ea/Ec. The mild steel reinforcement in the concrete slab can be used in calculating the transformed properties. Although for strength calculations, the modulus of elasticity of normal-weight concrete is used even for lightweight concrete slabs in composite beam design, in stub girders the lower value of n for lightweight concrete is used both for deflection and strength calculations.
The moment of inertia It of the top chord is obtained by multiplying the unit value of I of the composite slab, given in deck catalogs, by the effective width of the slab.
2. Bottom chord. The properties of the steel section are directly used for the bottom chord properties.
3. Stub pieces. The web area and moment of inertia of the stub in the plane of bending of the stub girder are calculated and apportioned to a finite number of vertical beam elements representing the stubs. The more elements employed to represent the stub pieces, the better will be the accuracy of the solution. As a minimum, the author recommends one vertical element for 1 ft (0.3 m) of in
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