Substituting, we get

3.142 x 2085 x 824

The moment magnification factor is given by

1 - (390/0.7 + 1176) Therefore, design Mu = 15 x 1.9 = 28.5 kip-ft (38.65 kN • m).

The equivalent column is designed for Pu = 390 kips (1735 kN) and Mu = 28.5 kip-ft (38.65 kN • m). The required reinforcement is obtained using a procedure conforming to the ACI code. For the present example, longitudinal reinforcing bars of ten #5 are found to be adequate to carry the design axial load and bending moment.

Computation of Number of Shear Studs. The shear studs between the stub pieces and the concrete slab form the backbone of composite stub girders. Their design is similar to composite beam design for which the shear connector formulas represent the horizontal shear at ultimate load divided by 2 to approximate conditions at working loads. The total horizontal shear resisted by the connectors between the point of maximum moment and each end of the stub girder is the smaller of the values obtained from the following equations:

AsFy 2

For the example problem, a value of Vh = 458 kips (2037 kN) obtained from the first equation governs the design. Using a value of 9.5 kips (42.26 kN) as the allowable shear load, the number of shear studs N = 458/9.5 = 48.2 ~ 50, giving 32 and 18 shear connectors at the exterior and interior stubs.

Check Exterior Stub W16 x 26, 6V2 ft (1.98 m) Long. The design check is performed for shear and bending stresses per the AISC specifications. The summation of shear forces in the six elements used in the computer model to represent the exterior stub gives the design shear. The design moment for the stub is obtained by multiplying the accumulated shear by the stub height. Assume for the example problem that the accumulated shear = 210 kips (934 kN). The design moment then is 210 x 16/12 or 280 kip-ft (380 kN • m). The shear stress is 210/78 x 25 or 10.76 ksi (74.25 MPa). The allowable shear stress is 0.4 x Fv = 0.4 x 36 = 14.4 ksi (99.3 MPa). Therefore, the stub is okay for shear.

To check the bending stresses, we calculate the moment of inertia and section modulus of the stub by including the contribution of the stiffener plates at the ends of stub. Without burdening the presentation with trivial calculations, let us assume that the section modulus of the stub piece and stiffener plate is equal to 300 in.3 (4.92 x 106 mm3). The bending stress is expressed as

280 x 12

b 300

This stress is checked against the allowable stresses per the AISC specifications.

A similar procedure is used to check the bending and shear stresses in the interior stub. Moment-Connected Stub Girder

The stub girder system, due to its large overall depth of approximately 3 ft (0.92 m), has a very large moment of inertia and can be used as part of a lateral-force-resisting system. The model used for analysis is a vierendeel truss, where the concrete slab and the bottom steel beam are simulated as linear elements and each stub piece is divided into several elements. The gravity and lateral load shear forces and moments are introduced as additional load cases in the computer analysis, and the combined axial forces and moments

in each section of the stub girder are obtained. all parts of the stub girder are checked for combined axial forces, shear, and moments as shown earlier. The controlling section for the slab is generally at the end of the first stub piece furthermost from the column. Particular care is required to transfer the moment at the column girder interfaces. If lateral moments are small, moment transfer can take place between the slab and the bottom steel beam. The slab needs to be attached to the column either by long deformed wire anchors or by welding reinforcing bars to the column. For relatively large moments, the solution for moment transfer is to extend the first stub piece to the column face. The top flange of the stub piece and the bottom flange of the W14 girder are welded to the column as in a typical moment connection. The design of the connection is, therefore, identical to welded beam-column moment connection. The girder should be checked along its full length for the critical combination of gravity and wind forces. Depending on the extent of stress reversals due to lateral load, bracing of the bottom chord may be necessary. Strengthening of Stub Girder

Strengthening of existing stub girders for tenant-imposed higher loads is more expensive than in conventional composite construction. A speculative type of investment building is usually designed for imposed loads of 50 psf (2.4 kN/m2) plus 20 psf (0.96 kN/m2) as partition allowance. For heavier loads, strengthening of local framing is required. The bottom girder, which is in tension and bending, is relatively easy to reinforce by welding additional plates or angles to the existing steel member. Reinforcing the top chord of the stub girder, which is in compression and bending, is somewhat tricky. The addition of structural steel angles using expansion anchors to the underside of metal deck and the welding of additional stub pieces to reduce the effective length of compression chord, which acts like a column, have been used with good results. From the point of view of ultimate load behavior, it is acceptable to strengthen the bottom chords to resist total load without the truss action. However, it is important to check the lateral bracing requirements for the top flange of the bottom chord.

7.4.6. Composite Columns

The term composite column in the building industry is taken to represent a unique form of construction in which structural steel is made to interact compositely with concrete. The structural steel section can be a tubular section filled with structural concrete or it can be a steel wide-flange section used as a core surrounded by reinforced concrete.

Historically, composite columns evolved from the concrete encasement of structural steel shapes primarily intended as fire protection. Although the increase in strength and stiffness of the steel members due to concrete used as fireproofing was intuitively known, it was not until the 1940s that methods to actually incorporate the increases were developed. In fact, in earlier days, the design of the steel column was penalized by considering the weight of concrete as an additional dead load on the steel column. Later developments took into account the increased radius of gyration of the column because of the concrete encasement, and allowed for some reduction in the amount of structural steel. In some earlier high-rise designs, the concrete encasement was ignored for strength considerations, but the additional stiffness of concrete was included in calculating lateral defections.

After the development of sprayed-on contact fireproofing in the 1950s and 1960s, use of concrete for fireproofing of structural steel was no longer an economical proposition. The high form-work cost of concrete could not be justified for fireproofing.

Over the last 20 years, the use of encased structural steel columns has found application in buildings varying from as low as 10 stories to as high as 70-story or even taller buildings. These columns have been incorporated in an overall construction known as the composite system, which has successfully captured the essential advantages associated with steel and concrete construction: the speed of steel with the stiffness and moldability of concrete. Concrete columns with small steel-core columns used as erection columns were perhaps the earliest applications. Later much heavier columns were used, serving the dual purpose for both steel erection and load resistance. The heavier steel columns were used essentially to limit the size of composite vertical elements.

Another version consists of exterior concrete columns acting compositely with steel-plate or precast cladding. Yet another version popular in some countries uses laced columns fabricated from light structural shapes such as angles, T-sections, and channels. The concrete enclosure provides both fireproofing qualities and also provides additional stiffness to the light structural shapes, inhibiting their local buckling tendencies. Additional conventional reinforcement can be accommodated in the concrete encasement, as in conventionally reinforced concrete columns.

The ACI building code encompasses the design of all types of composite column under one unified method using the same general principles as for conventionally reinforced concrete columns.

The ACI procedure is based on an ultimate concrete strain of 0.3%. As in conventionally reinforced concrete design, the tensile stress in the concrete is ignored. Either a parabolic or an equivalent uniform concrete strain can be assumed in the compression zone.

Figure 7.65. Comparison of interaction diagrams: (a) column detail; (b) load moment interaction diagram.

The axial load assigned to the concrete portion of the composite column is required to be developed by direct bearing through studs, lugs, plates, or reinforcing bars welded to the structural steel plate prior to the casting of concrete. In other words, the code requires a positive method for the transfer of axial load between the steel core and the concrete encasement for strength calculations. For calculation of stiffness, however, merely wrapping the concrete around the steel core will suffice. Axial loads induced in the concrete section of the composite column due to column bending need not be transferred in direct bearing.

Tied composite columns are required by the ACI code to have more lateral ties than ordinary reinforced concrete columns. In fact, the ACI code stipulates twice as many ties, but this is based on somewhat questionable assumptions. First, it assumes that concrete that is laterally contained by ties is thin. Second, it assumes that concrete has a tendency to spall out from the smooth faces of the steel core. To prevent this separation, the lateral ties are specified to be vertically spaced no more than half the least dimension of the composite member. The ACI code does not permit the use of longitudinal bars in the evaluation of stiffness of columns on the premise that the longitudinal bars are rendered ineffective because of separation of concrete at high strains. They may, however, be included in the calculation of strength. Finally, the yield strength of the steel core is limited to 52 ksi (359 MPa) to correspond to the yielding strain of concrete of 0.0018.

A practical approach to the design of composite columns is to assume that the steel wide-flange section behaves as reinforcing steel. With this assumption, interaction diagrams can be generated for various combinations of concrete columns size, structural steel shape, and reinforcing steel. Figure 7.65 shows an interaction diagram generated for a 36 x 36 in. (915 x 915 mm) column with twelve #18 (57-mm diameter) reinforcing bars and a W14 x 150 (378 x 394 mm x 2188 N/m) structural steel shape. For comparison purposes, the interaction diagram for the same concrete column without the embedded structural steel shape is given. It can be seen that large increases in column capacity occur when structural steel shapes are included within the concrete envelope.

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