a The values are intended as a preliminary guide for the design of building floors subjected to a uniformly distributed superimposed live load of 50 to J00 psf (2394 to 4788 Pa). For the final design, it is necessary to investigate for possible effects of camber, deflections, vibrations, and damping. The designer should verify that adequate clearance exists for proper placement of post-tensioning anchors.

a The values are intended as a preliminary guide for the design of building floors subjected to a uniformly distributed superimposed live load of 50 to J00 psf (2394 to 4788 Pa). For the final design, it is necessary to investigate for possible effects of camber, deflections, vibrations, and damping. The designer should verify that adequate clearance exists for proper placement of post-tensioning anchors.

The tendon profile is established based on the type and distribution of load with due regard to clear cover required for fire resistance and corrosion protection. Clear spacing between tendons must be sufficient to permit easy placing of concrete. For maximum economy, the tendon should be located eccentric to the center of gravity of the concrete section to produce maximum counteracting effect to the external loads. For members subjected to uniformly distributed loads, a simple parabolic profile is ideal, but in continuous structures parabolic segments forming a smooth reversed curve at the support are more practical. The effect is to shift the point of contraflecture away from the supports. This reverse curvature modifies the load imposed by post-tensioning from those assumed using a parabolic profile between tendon high points.

The post-tension force in the tendon immediately after releasing the hydraulic jack is less than the jacking force because of: 1) slippage of anchors; 2) frictional losses along tendon profile; and 3) elastic shortening of concrete. The force is reduced further over a period of months or even years due to change in the length of concrete member resulting from shrinkage and creep of concrete and relaxation of the highly stressed steel. The effective prestress is the force in the tendon after all the losses have taken place. For routine designs, empirical expressions for estimating prestress losses yield sufficiently accurate results, but in cases with unusual member geometry, tendon profile, and construction methods it may be necessary to make refined calculations.

Prestressing may be considered as a method of balancing a certain portion of the applied loads. This method, first developed by T. Y. Lin, is applicable to statically indeterminate systems just as easily as to statically determinate structures. Also, the procedure gives a simple method of calculating deflections by considering only that portion of the applied load not balanced by the prestress. If the effective prestress completely balances the applied load, the post-tensioned member will undergo no deflection and will remain horizontal, irrespective of the modulus of rigidity or flexural creep of concrete.

A question that usually arises in prestress design is how much of the applied load is to be balanced. The answer, however, is not simple. Balancing all the dead load often results in too much prestressing, leading to uneconomical design. On the other hand, there are situations in which the live load is significantly heavier than the dead load, making it more economical to prestress not only for full dead loads but also for a significant portion of the live load. However, in the design of typical floor framing systems, the prestressing force is normally selected to balance about 70 to 90% of the dead load and, occasionally, a small portion of the live load. This leads to an ideal condition with the structure having little or no deflection under dead loads.

Limiting the maximum tensile and compressive stresses permitted in concrete does not in itself assure that the prestressed member has an adequate factor of safety against flexural failure. Therefore, its nominal bending strength is computed in a procedure similar to that of a reinforced concrete beam. Underreinforced beams are assumed to have reached the failure load when the concrete strain reaches a value of 0.003. Since the yield point of prestressing steel is not well defined empirical relations based on tests are used in evaluating the strain and hence the stress in tendons.

The shear reinforcement in post-tensioned members is designed in a manner almost identical to that of nonprestressed concrete members, with due consideration for the longitudinal stresses induced by the post-tensioned tendons. Another feature unique to the design of post-tensioned members is the high stresses in the vicinity of anchors. Prestress-ing force is transferred to concrete at the tendon anchorages. Large stresses are developed in the concrete at the anchorages, which requires provision of well-positioned reinforcement in the region of high stresses. At a cross section of a beam sufficiently far away (usually 2 to 3 times the larger cross-sectional dimensions of the beam) from the anchor zone, the axial and bending stresses in the beam due to an eccentric prestressing force are given by the usual P/A and MC/I relations. But in the vicinity of stress application, the stresses are distributed in a complex manner. Of importance are the transverse tensile forces generated at the end blocks for which reinforcement is to be provided. The tensile stress has a maximum value at 90° to the axis of the prestressing force. Its distribution depends on the location of bearing area and its relative proportion with respect to the areas of the end face.

Because of the indeterminate nature and intensity of the stresses, the design of reinforcement for the end block is primarily based on empirical expressions. It usually consists of closely spaced stirrups tied together with longitudinal bars.

Certain rules of thumb such as span-to-depth ratios and the average value of post-tensioning stresses are useful in conceptual design. The span-to-depth for slabs usually works out between L/40 and L/50, whereas for joists it is between L/25 and L/35. Beams can be much shallower than joists, with a depth in the range of L /20 and L /30. Band beams, defined as those with a width-to-depth ratio in excess to 4, offer perhaps the least depth without using as much concrete as flat slab construction. Although a span-to-depth ratio approaching 35 is adequate for band beams from strength and serviceability considerations, clearance requirements for proper detailing of anchorages and for accessing stressing equipment may dictate a deeper section. As a rule of thumb, a minimum compression of 125 to 150 psi (862 and 1034kPa) is a practical and economical range for slabs. For beams, the range is 250 to 300 psi (1724 to 2068 kPa). Compression stresses as high as 500 psi (3447 kPa) have been used in band-beam systems. Even higher stresses may be required for transfer girders. Example Buildings

The first example shows a two-way post-tensioned flat plate system for a residential tower (Fig. 7.26). The tendons are V2-diameter (12.7 mm) stands that are banded in the north-south direction. Uniformly distributed tendons run from left to right across the building width. Additional tendons are used in the end panels to resist increased moments due to lack of continuity at one end.

As a second example, Fig. 7.27 shows the framing plan for a post-tensioned band-beam-slab system. Shallow beams only 16 in. (0.40 m) deep span across two exterior bays of 40 ft (12.19 m) and an interior bay of 21 ft (6.38 in). Post-tensioned slabs 8 in. (203 mm) deep span between the band beams, typically spaced at 30 ft (9.14 m) on center. In the design of the slab, additional beam depth is considered as a haunch at each end. Primary tendons for the slab run across the building width, while the tendons that control the temperature and shrinkage are placed in the north-south direction between the band beams.

7.3.4. Cracking Problems in Post-Tensioned Floors

Cracking caused by restraint to shortening is one of the biggest problems associated with post-tensioned floor systems. The reason is that shortening of a floor state is a time-dependent complex phenomenon. Only subjective empirical solutions exist to predict the behavior.

Shrinkage of concrete is the biggest contributor to shortening in both prestressed and nonprestressed concrete. In prestressed concrete, out of the total shortening, only about 15% is due to elastic shortening and creep. Therefore the problem is not in the magnitude of shortening itself, but in the manner in which it occurs.

Figure 7.26 Two-way post-tensioned flat plate system for a residential tower, the Museum Tower, in Los Angeles, CA. (Structural engineers John A. Martin & Assoc., Inc., Los Angeles, CA.)

When a nonprestressed concrete slab tries to shorten, its movement is resisted internally by the bonded mild steel reinforcement. The reinforcement is put into compression while the concrete is in tension. As the concrete tension builds up, the slab cracks at fairly regular intervals allowing the ends of the slab to remain in the same position in which they were cast. In a manner of speaking, the concrete has shortened by about the same magnitude as a post-tensioned system, but not in overall dimensions. Instead of the total shortening occurring at the ends, the combined widths of many cracks which occur across the slab make up for the total shortening. The reinforcement distributes the shortening throughout the length of the slab in the form of numerous cracks. Thus reinforced

Figure 7.27. Typical floor plan of one-way post-tensioned slab system.
Figure 7.28. Cracking in post-tensioned slab caused by restraint of perimeter walls.

concrete tends to take care of its own shortening problems internally by the formation of numerous small cracks, each small enough to be considered acceptable. Restraints provided by stiff vertical elements such as walls and columns tend to be of minor significance, since provision for total movement has been provided by the cracks in concrete.

This is not the case with post-tensioned systems in which shrinkage cracks, which would have formed otherwise, are closed by the post-tensioning force. Much less mild steel is present and consequently the restraint to the shortening provided is less. The slab tends to shorten at each end generating large restraining forces in the walls and columns particularly at the ends where the movement is greatest (Fig. 7.28). These restraining forces can produce severe cracking in the slab, walls, or columns at the slab extremities, causing problems to engineers and building owners alike. The most serious consequence is perhaps water leakage through the cracks.

The solution to the problem lies in eliminating the restraint by separating the slab from the restraining vertical elements. If a permanent separation is not feasible, cracking can be minimized by using temporary separations to allow enough of the shortening to occur prior to making the connection.

Cracking in a post-tensioned slab also tends to be proportional to initial pour size. Some general guidelines that have evolved over the years are as follows: 1) the maximum length between temporary pour strips (Fig. 7.29) is 150 ft (200 ft if restraint due to vertical elements is minimal); and 2) the maximum length of post-tensioned slab irrespective of the number of pour strips provided is 300 ft. The length of time for leaving the pour strips open is critical and can range anywhere from 30 to 60 days. A 30-day period is considered adequate for average restraint conditions with relatively centered, modest length walls, while a 60-day period is more the norm for severe shortening conditions with large pour sizes and stiff walls at the ends.

To minimize cracking caused by restraint to shortening, it is a good idea to provide a continuous mat of reinforcing steel in both directions of the slab. As a minimum, one-layer of #4 bars placed at mid-depth of slab, at 36 in. on centers both ways is recommended for typical conditions. For slab-pours in excess of 150 ft in length with relatively stiff walls at the ends, the minimum reinforcement should be increased to #4 bars at 24 in. on centers both ways.

7.3.5. Concept of Secondary Moments

In a prestressed statically determinate beam, such as a single-span simply supported beam, the moment Mp due to prestress is given by the eccentricity e of prestress multiplied by

Figure 7.29. Temporary pour strip: (a) at perimeter of building; (b) at interior of slab.

the prestress P. In prestressed design, the moment Mp = Pe is commonly referred to as the primary moment. In a simple beam or any other statically determinate beam, no support reactions can be induced by prestressing. No matter how much the beam is prestressed, only the internal stresses will be affected by the prestressing. The external reactions, being determined by statics, will depend on the dead and live loads, but are not affected by the prestress. Thus there are no secondary moments in a statically determinate beam. The total moment in the beam due to prestress is simply equal to the primary moment M0 = Pe.

The magnitude and nature of secondary moments may be illustrated by considering a two-span, continuous, prismatic beam that is not restrained by its supports but remains in contact with them. The beam is prestressed with a straight tendon with force P and eccentricity e. See Fig. 7.30.

When the beam is prestressed, it bends and deflects. The bending of the beam can be such that the beam will tend to deflect itself away from B. Because the beam is restrained from deflection at B, a vertical reaction must be exerted to the beam to hold it there. The induced reaction produces secondary moments in the beam. These are called secondary because they are by-products of prestressing and do not exist in a statically determinate beam. The term secondary is misleading because the moments are secondary in nature, but not necessarily in magnitude.

One of the principal reasons for determining the magnitude of secondary moments is because they are required in the computations of ultimate flexural strength. An elastic

Figure 7.30. Concept of secondary moment: (a) two-span continuous beam; (b) vertical upward displacement due to PT; (c) primary moment; (d) reactions due to PT; (e) secondary moment; (f) final moments.

analysis of a prestressed beam offers no control over the failure mode or the factor of safety. To assure that prestressed members will be designed with an adequate factor of safety against failure, ACI 318-02, like its predecessors, requires that Mu, the moment due to factored service loads including secondary moments, not exceed fMn, the flexural design strength of the member. The ultimate factored moment Mu is calculated by the following load combinations:

Since the factored load combination must include the effects due to secondary moments, its determination is necessary in prestress designs.

To further enhance our understanding of secondary moments, three numerical examples are given here:

1. A two-span continuous beam with a prestressd tendon at a constant eccentricity e.

2. The same beam as in the preceding example except the tendon is parabolic between the supports. There is no eccentricity of the tendon at the supports.

3. The same as in example 2, but the tendon has an eccentricity at the center support. Design Examples.

Example 1.

Given. A two-span prestressed beam with a tendon placed at a constant eccentricity e from the C.G. of the beam. The prestress in the tendon is equal to P (See Fig. 7.30.)

Required. Secondary moments in the beam due to prestress P

Solution. The beam is statically indeterminate to the first degree because it is continuous at the center support B. It is rendered determinate by removing the support at B. Due to the moments M0 = Pe at the ends, the beam bends and deflects upward. The magnitude of vertical deflection 8B due to moment M0 is calculated using standard beam formulas such as the one that follows.

Beam Deflection Formula

Type of ioad

Slope as shown

Maximum deflection

Deflection equation

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