## Vu

bw d

After satisfying these criteria, the torsional moment capacity is determined by frn = f 2AoAtfyvcot e (7 . 64)

The shear flow area Ao may be taken as 0.85Aoh, where Aoh is the area enclosed by the closed hoop (see Figure 7.28). The angle e may be assumed to be 45°. More accurate values of Ao and e may be used from analysis of the space truss analogy.

To determine the additional transverse torsional reinforcement required to satisfy ultimate strength, that is, f Tn > Tu, the transverse reinforcement area At and its spacing s must satisfy the following:

s f2Aofyvcot e v '

The area At is for one leg of reinforcement. This torsional reinforcement area should then be combined with the transverse reinforcement required for shear demand Av (see Section 7.11). The total transverse reinforcement required for the member is thus

The above expression assumes that the shear reinforcement consists of two legs. If more than two legs are present, only the legs adjacent to the sides of the cross-section are considered effective for torsional resistance.

Torque

Longitudinal reinforcement

Torque

Longitudinal reinforcement

Aob shear flow area

FIGURE 7.28 Torsional reinforcement and shear flow area.

The total transverse reinforcement must exceed the following minimum amounts:

Jyv Jyv

A minimum amount of longitudinal reinforcement is also required:

The reinforcement area Al is additional to that required for resisting flexure and axial forces and should not be less than

5Vf0Acy _ MA f fyl fyl

where At/s should not be less than 25bw/fv. The torsional-longitudinal reinforcement should be distributed around the section in a uniform manner.

### 7.16.2 Detailing of Torsional Reinforcement

The spacing of closed transverse reinforcement under torsion must not exceed ph/8 or 12 in. Torsion reinforcement should be provided for a distance of at least (bt + d) beyond the point theoretically required. Torsional stresses cause unrestrained corners of the concrete to spall off. Transverse torsion reinforcement needs to be anchored by 135° hooks. In hollow cross-sections, the closed hoops should be placed near the outer surface of the wall. The distance from the centerline of the hoop reinforcement to the inside wall face should not be less than 0.5Aoh/ph.

The longitudinal torsion reinforcement should be distributed so that its centroid is near the centroid of the cross-section. It should be distributed around the perimeter and be positioned inside the closed hoop with a maximum spacing of 12 in. There should be at least one longitudinal bar at each corner of the hoop. The longitudinal reinforcement must have a diameter of at least 0.042 times the hoop spacing. The ends of the longitudinal reinforcement must be fully developed for yielding. It is permitted to reduce the area of the longitudinal reinforcement by an amount equal to Mu/(0.9dfj) since flexural compression offsets the longitudinal tension due to torsion.

7.17 Reinforcement Development Lengths, Hooks, and Splices

The various ultimate capacity formulas presented in the previous sections are premised on the assumption that the reinforcement will reach its yield strength fy. This is not assured unless the reinforcement has (1) sufficient straight embedment length on each side of the point of yielding, (2) a hook of sufficient anchorage capacity, or (3) a qualified mechanical anchor device.

7.17.1 Tension Development Lengths

The ACI development length equation for bars in tension ld is expressed in terms of a multiple of the bar diameter db (inch unit):

where the transverse reinforcement index Ktr = Atrfyt/1500sn, which may be assumed to be zero for simplicity. Table 7.13 gives the development length for the case of normal weight concrete (1 = 1.0) and uncoated reinforcement (b = 1.0). Development lengths need to be increased under these conditions: beam reinforcement positioned near the top surface, epoxy coating, lightweight concrete, and bundling of bars (see ACI Section 12.2.4).

TABLE 7.13 Development Lengths in Tension

0 0