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Figure 7.54. Effective width concept as defined in the AISC specifications.

Figure 7.54. Effective width concept as defined in the AISC specifications.

assumes transverse compatibility at the concrete and steel interface. The unit stress in each material is equal to the strain times its modulus of elasticity. Because of strain compatibility, the stress in steel is n times the stress in concrete, where n is the modular ratio Esi Ec. A unit area of steel is, therefore, mathematically equivalent to n times the concrete area. Thus, the effective area of concrete Ac = bt can be replaced by an equivalent steel area Ac in.

Concrete is neither linearly elastic nor ductile and its stress-strain curve exhibits a constantly changing slope with a sudden brittle failure. In spite of these characteristics, concrete is considered elastic within a stress-strain range of up to 0.50/c, and the modulus of elasticity in pounds per square inch can be approximated by the relation Ec = W15 x 33^/f7, where Wc is the unit weight of concrete in pounds per cubic foot and fc is the compressive strength of concrete in pounds per square inch. The compressive strength fc of concrete normally used in floor construction is in the range of 3000 to 5000 psi (20.7 to 34.4 MPa) giving a value of Ec for normal weight concrete of 3.12 x 106 < Ec < 4.03 x 106 psi (21,512 < Ec < 27, 787 MPa), compared to Es of steel at 29 x 106 psi (199,955 MPa). The value of n = Es IEc, therefore, lies between 9.3 and 7.2 and is usually approximated to the whole number in recognition of the error in the formula for Ec when compared to actual performance.

For strength calculations, the AISC specification uses the value of n for normal-weight concrete of the specified strength. However, for deflection computations, n depends not only on the specified strength but also on the unit weight of concrete, Therefore, in computing deflections, especially for beams subjected to heavy sustained loads, it is necessary to account for the effects of creep by using an appropriate value of n. This is even more important in shored construction when the dead load of the concrete is resisted by the composite action. Creep effect is accounted for in computing deflections by using a higher modular ratio, n. A factor of 2 for creep effects is typically adequate in building designs. Live loads are always resisted by the composite section. If they are of short duration, the deflections are computed using the short-term modular ratio.

The transformed steel section can be conveniently considered as the original steel beam with an added cover plate to the top flange of thickness t equal to slab thickness, and an equivalent width bin. The composite properties of the transformed section are calculated by locating the neutral axis and the transformed moment of inertia Ir. The maximum bending stress in the steel beam bottom flange is given by f = MYl fbs ~ , tr where M is the total bending moment, Ytr is the distance of the extreme bottom steel fibers from the neutral axis, and Itr is the transformed moment of inertia. The maximum com-pressive stress in the concrete is given by f - MCl

Jbc — , tr where Ct is the distance from the neutral axis to the extreme concrete fibers and n is the modular ratio. The value a — Itr Str — Y

r is called the transformed section modulus of the beam referred to the bottom flange.

For construction without temporary shores, concrete compressive stress is based upon the load applied after it has reached 75% of the required strength. This compressive stress is limited to 0.45/', just as in the working stress design of reinforced concrete beams.

The total horizontal shear to be resisted between the point of maximum positive moment and point of zero moment is the smaller of the two values as determined by

085fAc 2

where fc — specified compressive strength of concrete Ac — actual area of effective concrete flange As — area of steel beam Fy — specified yield stress of steel beam

Note that the formula Vh — 0.85fcAc /2 assumes that there is no longitudinal reinforcing steel in the compression zone of composite beam. If the compressive zone is designed with mild steel reinforcement, the formula for horizontal shear is to be modified as follows:

where

A's — area of the longitudinal compressive steel Fyr — yield stress of the reinforcing steel

AISC permits averaging of horizontal shear flow; that is, the total number of connectors between the point of maximum moment and point of zero moment must be sufficient to satisfy the total shear flow within that length. The shear connector formulas represent the horizontal shear at ultimate load divided by 2 to approximate conditions at working loads.

The number of shear connectors required for full composite action is determined by dividing the smaller value of Vh by the shear capacity of one connector. The number of connectors obtained represents the shear connectors required between the point of maximum positive moment and point of zero moment. For example, in a simply supported, uniformly loaded beam, this represents half the span; and in a simply supported beam with two equidistant concentrated loads, this represents the distance between the point load to the support point. The total number of connectors required for the entire span is thus double the number obtained earlier.

A composite beam subject to negative bending moment experiences tensile stresses in the concrete zone and loses much of its advantage. However, when reinforcement is placed parallel to the beam within the effective width of slab, and is anchored adequately to develop the tensile forces, the advantage of continuous construction is restored. The steel used in the tensile zone is included in computing the property of the composite section. Similarly, when the compressive stress in concrete subject to positive moment exceeds the allowable stress, it is permissible to use compressive steel in the effective width zone to reduce stresses.

Consider a continuous composite beam shown in Fig. 7.55. The total horizontal shear to be resisted by shear connectors between an interior support and each adjacent point of contraflexure (regions a, b, and c in Fig. 7.55c) is given as

where

Asr = area of reinforcing steel provided at the interior support within the effective flange width

Fyr = yield stress of the reinforcing steel

AISC permits uniform spacing of connectors between the points of maximum positive moment and the point of zero moment. Also, the connectors required in the region of negative bending can be uniformly distributed between the point of maximum moment and each point of zero moment. For concentrated loads, the numbers of shear connectors N2 required between any concentrated load and the nearest point of zero moment is determined by the AISC formula

where

M = moment at concentrated load point (less than the maximum moment) N1 = number of connectors required between point of maximum moment and point of zero moment b = ratio of transformed section modulus to steel section modulus

### This relation is schematically shown in Fig. 7.56.

In the design of composite beams it is often unnecessary to develop the full composite action. A partial composite action with fewer studs is all that may be necessary to achieve the required strength and stiffness. AISC permits designs of less than 100% composite

action by introducing the concept of effective section modulus as determined by the relation

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