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4,000 8,000 12,000 16,000

Concrete strength, psi

FIGURE 7.5 Relation between bi and concrete strength.

To ease reinforcement cage fabrication, a minimum of two top and two bottom bars should run continuously through the span of the beam. These bars hold up the transverse reinforcement (stirrups). At least one fourth of all bottom (positive) reinforcement should run continuously. If moment reversal is expected at the beam-column connection, that is, stress reversal from compression to tension, bottom bars must be adequately anchored into the column support to develop the yield strength.

The remaining top and bottom bars may be cut short. However, it is generally undesirable to cut bars within the tension zone (it causes loss of shear strength and ductility). It is good practice to run bars well into the compression zone, at least a distance d, 12db or l„/16 beyond the point of inflection (PI) (see Figure 7.6). Cut bars must also be at least one development length ld in length measured from each side of their critical sections, which are typically the point of peak moment where the yield strength must be developed. See Section 7.17 for development lengths.

To achieve structural integrity of the structural system, beams located at the perimeter of the structure should have minimum continuous reinforcement that ties the structure together to enhance stability, redundancy, and ductile behavior. Around the perimeter at least one sixth of the top (negative) longitudinal reinforcement at the support and one quarter of the bottom (positive) reinforcement should be made continuous and tied with closed stirrups (or open stirrups with minimum 135° hooks). Class A splices may be used to achieve continuity. Top bars should be spliced at the midspan, bottom bars at or near the support.

7.12.2.6 Beams with Compression Reinforcement

Reinforcement on the compression side of the cross-section (see Figure 7.4) usually does not increase in flexural capacity significantly, typically less than 5%, and for most design purposes its contribution to

strength can be neglected. The moment capacity equation considering the compression reinforcement area As located at a distance d' from the compression fiber is f Mn = A'sfy (d - d') + (As - 4)/y (d - 22) (7.18)

The above expressions assume the compression steel yield, which is typically the case (compression steel quantity is not high). For the nonyielding case, the stress in the steel needs to be determined by a stressstrain compatibility analysis.

Despite its small influence on strength, compression reinforcement serves a number of useful serviceability functions. It is needed for supporting the transverse shear reinforcement in the fabrication of the steel cage. It helps to reduce deflections and long-term creep, and it enhances ductile performance.

Stirrups

Hoops

Hoops

Spiral

Spiral

FIGURE 7.7 Typical types of transverse reinforcement.

7.12.2.7 Shear Capacity of Beams

Shear design generally follows after flexural design. The shear capacity f Vn of a beam consists of two parts: (1) the shear provided by the concrete itself Vc and (2) that provided by the transverse reinforcement Vs.

The strength reduction factor f for shear is 0.85. The nominal shear capacity of the concrete may be taken as the simple expression

which is in pound and inch units. An alternative empirical formula that allows a higher concrete shear capacity is

where Mu is the factored moment occurring simultaneously with Vu at the beam section being checked. The quantity Vud/Mu should not be taken greater than 1.0.

Transverse shear reinforcements are generally of the following types (see Figure 7.7): stirrups, closed hoops, spirals, or circular ties. In addition, welded wire fabric, inclined stirrups, or longitudinal bars bent at an angle may be used. For shear reinforcement aligned perpendicular to the longitudinal reinforcement, the shear capacity provided by transverse reinforcement is

When spirals or circular ties or hoops are used with this formula, d should be taken as 0.8 times the diameter of the concrete cross-section, and Av should be taken as two times the bar area.

When transverse reinforcement is inclined at an angle a with respect to the longitudinal axis of the beam, the transverse reinforcement shear capacity becomes

The shear formulas presented above were derived empirically, and their validity has also been tested by many years of design practice. A more rational design approach for shear is the strut-and-tie model, which is given as an alternative design method in ACI Appendix A. Shear designs following the strut-and-tie approach, however, often result in designs requiring more transverse reinforcement steel since the shear transfer ability of concrete is neglected.

7.12.2.8 Determination of Required Shear Reinforcement Quantities

The shear capacity must be greater than the shear demand Vu, which is based on the structural analysis results under the specified loads and governing load combination fVn > Vu (7.23)

Since the beam cross-section dimensions bw and d would usually have been selected by flexural design beforehand or governed by functional or architectural requirements, the shear capacity provided by the concrete Vc can be calculated by Equations 7.21a or 7.21b. From the above equations, the required shear capacity to be provided by shear reinforcement must satisfy the following:

Inserting Vs from this equation into Equation 7.22a, the required spacing and bar area of the shear reinforcement (aligned perpendicular to the longitudinal reinforcement) must satisfy the following:

For ease of fabrication and bending, a bar size in the range of No. 4 to No. 6 is selected, then the required spacing s along the length of the beam is determined, usually rounded down to the nearest 2 in.

In theory, the above shear design procedure can be carried out at every section along the beam. In practice, a conservative approach is taken and shear design is carried out at only one or two locations of maximum shear, typically at the ends of the beam, and the same reinforcement spacing s is adopted for the rest of the beam. Where the beam ends are cast integrally or supported by a column, beam, wall, or support element that introduces a region of concentrated compression, the maximum value of the shear demand need not be taken at the face of the support, but at a distance d away (see Figure 7.8).

Transverse reinforcement in the form of closed stirrups is preferred for better ductile performance and structural integrity. For beams located at the perimeter of the structure, ACI requires closed stirrups (or open stirrups within minimum 135° hooks). In interior beams, if closed stirrups are not provided, at least one quarter of the bottom (positive) longitudinal reinforcement at midspan should be made continuous over the support, or at the end support, detailed with a standard hook.

7.12.2.8.1 Minimum Shear Reinforcement and Spacing Limits

After the shear reinforcement and spacing are selected they should be checked against minimum requirements. The minimum shear reinforcement required is

J Jy

This minimum shear area applies in the beam where Vu > f V2/2. It does not apply to slabs, footings, and concrete joists. The transverse reinforcement spacing s should not exceed d/2 nor 24 in. These spacing limits become d/4 and 12 in. when Vs exceeds 4yf bw d.

When significant torsion exists, additional shear reinforcement may be needed to resist torsion. This is covered in Section 7.16.

■ Compression support

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