## 1 7Avfbw dp

Vudp

8.7.4.3 Torsional Web Reinforcement

Meaningful additional torsional strength due to the addition of torsional reinforcement can be achieved only by using both stirrups and longitudinal bars. Ideally, equal volumes of steel in both the closed stirrups and the longitudinal bars should be used so that both participate equally in resisting the twisting moments. This principle is the basis of the ACI expressions for proportioning the torsional web steel. If s is the spacing of the stirrups, Al is the total cross-sectional area of the longitudinal bars, and At is the cross-section of one stirrup leg, the transverse reinforcement for torsion has to be based on the full external torsional moment strength value Tn, namely, Tu/f

s where A0 is the gross area enclosed by the shear flow path (sq. in.), At is the cross-sectional area of one leg of the transverse closed stirrups (sq. in.), fyv is the yield strength of closed transverse torsional reinforcement not to exceed 60,000 psi, and 6 is the angle of the compression diagonals (struts) in the space truss analogy for torsion.

Transposing terms in Equation 8.73, the transverse reinforcement area becomes

s 2A0fyvcot 6 v '

The area A0 is determined by analysis, except that the ACI 318 Code permits taking A0 = 0.85A0h in lieu of the analysis.

The factored torsional resistance f Tn must equal or exceed the factored external torsional moment Tu. All the torsional moments are assumed in the ACI 318-02 Code to be resisted by the closed stirrups and the longitudinal steel with the torsional resistance, Tc, of the concrete disregarded, namely Tc = 0. The shear Vc resisted by the concrete is assumed to be unchanged by the presence of torsion. The angle 6 subtended by the concrete compression diagonals (struts) should not be taken smaller than 30° nor larger than 60°. It can be obtained by analysis as detailed by Hsu [6]. The additional longitudinal reinforcement for torsion should not be less than

At fyv 2

where fyl is the yield strength of the longitudinal torsional reinforcement, not to exceed 60,000 psi.

The same angle 6 should be used in both Equations 8.73 and 8.74. It should be noted that as 6 gets smaller, the amount of required stirrups required by Equation 8.74 decreases. At the same time the amount of longitudinal steel required by Equation 8.75 increases. In lieu of determining the angle 6 by analysis, the ACI Code allows a value of 6 equal to

1. 45° for nonprestressed members or members with less prestress than in 2,

2. 37.5° for prestressed members with an effective prestressing force larger than 40% of the tensile strength of the longitudinal reinforcement.

The PCI recommends computing the value of 6 from the expression:

8.7.4.3.1 Minimum Torsional Reinforcement

It is necessary to provide a minimum area of torsional reinforcement in all regions where the factored torsional moment Tu exceeds the value given by Equation 8.69. In such a case, the minimum area of the required transverse closed stirrups is

50bws

The maximum spacing should not exceed the smaller of pn/8 or 12 in.

The minimum total area of the additional longitudinal torsional reinforcement should be determined by

fyl - fyl where At/s should not be taken less than 25bw/fv. The additional longitudinal reinforcement required for torsion should be distributed around the perimeter of the closed stirrups with a maximum spacing of 12 in. The longitudinal bars or tendons should be placed inside the closed stirrups and at least one longitudinal bar or tendon in each corner of the stirrup. The bar diameter should be at least

16 of the stirrup spacing but not less than a No. 3 bar. Also, the torsional reinforcement should extend for a minimum distance of (bt + d) beyond the point theoretically required for torsion because torsional diagonal cracks develop in a helical form extending beyond the cracks caused by shear and flexure. bt is the width of that part of cross-section containing the stirrups resisting torsion. The critical section in beams is at a distance d from the face of the support for reinforced concrete elements and at h/2 for prestressed concrete elements, d being the effective depth and h the total depth of the section.

### 8.7.4.4 SI-Metric Expressions for Torsion Equations

In order to design for combined torsion and shear using the SI (System International) method, the following equations replace the corresponding expressions in the PI (Pound-Inch) method:

ff ^YOL

20 Mu bw dp

Vu dp Mu

2A0Atfy

where fyv is in MPa, s is in millimeter, A0, At are in mm2, and Tn is in kN m

T Tn

At s 2A0fyv cot y Ai = A pj^ cot2 e s V/yl where fyv and fyl are in MPa, ph and s are in millimeters, and Al and At are in mm2

12fyl fyi

where At/s should not be taken less than 0.175 bw/fyv. Maximum allowable spacing of transverse stirrups is the smaller of 1ph or 300 mm, and bars should have a diameter of at least j! of the stirrups spacing but not less than No. 10 M bar size. Max. fyv or fy\ should not exceed 400 MPa. Min. Avt the smaller of

Avt 0.35bw Avt 1 rrJ bw\

whichever is larger, where bw, dp, and s are in millimeters

^ Apsfpu dp

Use the lesser of the two sets.

8.8 Camber, Deflection, and Crack Control 8.8.1 Serviceability Considerations

Prestressed concrete members are continuously subjected to sustained eccentric compression due to the prestressing force, which seriously affects their long-term creep deformation performance. Failure to predict and control such deformations can lead to high reverse deflection, that is, camber, which can produce convex surfaces detrimental to proper drainage of roofs of buildings, to uncomfortable ride characteristics in bridges and aqueducts, and to cracking of partitions in apartment buildings, including misalignment of windows and doors.

The difficulty of predicting very accurately the total long-term prestress losses makes it more difficult to give a precise estimate of the magnitude of expected camber. Accuracy is even more difficult in partially prestressed concrete systems, where limited cracking is allowed through the use of additional nonprestressed reinforcement. Creep strain in the concrete increases camber, as it causes a negative increase in curvature that is usually more dominant than the decrease produced by the decrease in prestress losses due to creep, shrinkage, and stress relaxation. A best estimate of camber increase should be based on accumulated experience, span-to-death ratio code limitations, and a correct choice of the modulus Ec of the concrete. Calculation of the moment-curvature relationships at the major incremental stages of loading up to the limit state at failure would also assist in giving a more accurate evaluation of the stress-related load deflection of the structural element.

The cracking aspect of serviceability behavior in prestressed concrete is also critical. Allowance for limited cracking in "partial prestressing'' through the additional use of nonprestressed steel is prevalent. Because of the high stress levels in the prestressing steel, corrosion due to cracking can become detrimental to the service life of the structure. Therefore, limitations on the magnitudes of crack widths and their spacing have to be placed, and proper crack width evaluation procedures used. The presented discussion of the state of the art emphasizes the extensive work of the author on cracking in pretensioned and posttensioned prestressed beams.

Prestressed concrete flexural members are classified into three classes in the new ACI 318 Code:

In this class, the gross section is used for section properties when both stress computations at service loads and deflection computations are made. No skin reinforcement needs to be used in the vertical faces.

This class is a transition between uncracked and cracked sections. For stress computations at service T loads, the gross section is used. The cracked bilinear section is used in the deflection computations. No skin reinforcement needs to be used in the vertical faces.

This class denotes cracked sections. Hence, a cracked section analysis has to be made for evaluation of the stress level at service and for deflection. Computation of Dfps or fs for crack control is necessary, where Dfps is the stress increase beyond the decompression state and fs is the stress in the mild reinforcement when mild steel reinforcement is also used. Prestressed two-way slab systems are to be designed as Class U.

Ideally, the load-deflection relationship is trilinear, as shown in Figure 8.9. The three regions prior to rupture are:

Region I — Precracking stage, where a structural member is crack free.

Region II — Postcracking stage, where the structural member develops acceptable controlled cracking in both distribution and width.

Region III — Postserviceability cracking stage, where the stress in the tensile reinforcement reaches the limit state of yielding.

The precracking segment of the load-deflection curve is essentially a straight line defining full elastic behavior, as in Figure 8.9. The maximum tensile stress in the beam in this region is less than its tensile strength in flexure, that is, it is less than the modulus of rupture fr of concrete. The flexural stiffness EI of the beam can be estimated using Young's modulus Ec of concrete and the moment of inertia of the uncracked concrete cross-section.

The precracking region ends at the initiation of the first crack and moves into region II of the load-deflection diagram in Figure 8.9. Most beams lie in this region at service loads. A beam undergoes varying degrees of cracking along the span corresponding to the stress and deflection levels at each section. Hence, cracks are wider and deeper at midspan, whereas only narrow, minor cracks develop near the supports in a simple beam.

The load-deflection diagram in Figure 8.9 is considerably flatter in region III than in the preceding regions. This is due to substantial loss in stiffness of the section because of extensive cracking and considerable widening of the stabilized cracks throughout the span. As the load continues to increase, the strain es in the steel at the tension side continues to increase beyond the yield strain ey with no additional stress. The beam is considered at this stage to have structurally failed by initial yielding of the tension steel. It continues to deflect without additional loading, the cracks continue to open, and the neutral axis continues to rise toward the outer compression fibers. Finally, a secondary compression failure develops, leading to total crushing of the concrete in the maximum moment region followed by rupture. Figure 8.10 gives the deflection expressions for the most common loading cases in terms of both load and curvature.

### 8.8.1.1 Strain and Curvature Evaluation

The distribution of strain across the depth of the section at the controlling stages of loading is linear, as is shown in Figure 8.11, with the angle of curvature dependent on the top and bottom concrete extreme fiber strains ect and ecb. From the strain distributions, the curvature at the various stages of loading can be expressed as follows:

1. Initial prestress:

Load deflection

Prestress camber

48EI Ac12

cgc cgs

384EI Ac 48

w |

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