The moment capacity of the beam section f Mn may be expressed as the tension force multiplied by the moment arm of the force couple.

The strength reduction factor for flexure f is 0.9. Determination of Required Flexural Reinforcement Area

The maximum moment demand is determined from the structural analysis of the structure under the specified loads and load combinations, Mu. The nominal moment capacity Mn that the cross-section must supply is therefore

The beam cross-section dimensions, width b and thickness h, would be determined first or a first trial selected; the depth of the beam to the centroid of the tension reinforcement can be estimated by d = h — concrete cover — stirrup diameter — tension reinforcement bar radius (7.10)

A reasonable size of the stirrup and reinforcement bar can be assumed, if not known (a No. 4 or No. 5 bar size for stirrups is reasonable).

Rearranging the moment capacity equations presented in the previous section, the required flexural reinforcement is obtained by solving for As

The required tension reinforcement area As is obtained from the quadratic expression fyd ±J fy d)2- 4MnKm

where Km is a material constant:

Then, the sizes and quantity of bars are selected. Minimum requirements for reinforcement area and spacing must be satisfied (see the next two sections). Limits on Flexural Reinforcement Area

1. Minimum reinforcement area for beams:

3a if

2. Maximum reinforcement for beams: The maximum reinforcement As must satisfy the requirement that the net tensile strain et (extreme fiber strain less effects of creep, shrinkage, and temperature) is not less than 0.004. The net tensile strain is solved from the compatibility of strain (see Figure 7.4).

The neutral axis location c is related to the depth of the compression stress block a by the relationship (ACI

The factor b1 is dependent on the concrete strength as shown in Figure 7.5. Detailing of Longitudinal Reinforcement

Clear spacing between parallel bars should be large enough to permit the coarse aggregate to pass through to avoid honeycombing. The minimum clear spacing should be db, but it should not be less than 1 in. For crack control, center-to-center spacing of bars should not exceed

540 432

fs fs where fs (in ksi) is the stress in the reinforcement at service load, which may be assumed to be 60% of the specified yields strength. Typically, the maximum spacing between bars is about 10 in. The maximum bar spacing rule ensures that crack widths fall below approximately 0.016 in. For very aggressive exposure environments, additional measures should be considered to guard against corrosion, such as reduced concrete permeability, increased cover, or application of sealants.

If the depth of the beam is large, greater than 36 in., additional reinforcement should be placed at the side faces of the tension zone to control cracking. The amount of skin reinforcement to add need not exceed one half of the flexural tensile reinforcement and it should be spread out for a distance d/2. The spacing of the skin reinforcement need not exceed d/6, 12 in., and 1000Ab/(d — 30).

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