Ft A md

377 187 5 3,340

= +691 .2 — 745 .7 = -55 psi (C), no tension at transfer, OK.

377 187 5 3,750

- -2501.3 + 664.2 - -1837 psi (C) < fci - 2,250 psi, OK .

Analysis of stresses at service load. From Equation 8.25a ft - - Pi fi _ - Ml f _ AcV r2 J St

Msd + Ml - (100 + 1,100)(65)2 x 12/8 - 7,605,000 in . lb

Total moment MT = MD + MSD + ML = 2,490,638 + 7,605,000 = 10,095,638 in. lb (1,141 kN m)

377 187 5 3,340

= +566.5 - 3022 . 6 = -2,456 psi (C) >f = -2,250 psi

Hence, either enlarge the depth of the section or use higher strength concrete. Using f = 6,000 psi

10,095,638

Check support section stresses. Allowable fc0i _ 0 75 x 6000 _ 4500 psi fci _ 0 60 x 4500 _ 2700 psi fi _ 3vfi _ 201 psi for midspan fti _ 6 fc0i _ 402 psi for support f _ 0 . 45fc' _ 2700 psi

3,750

1. At transfer. Support section compressive fiber stress

377 187.5

so that

Accordingly, try ee = 12.49 in.

377 187.5

Thus, use mild steel at the top fibers at the support section to take all tensile stresses in the concrete, or use a higher strength concrete for the section, or reduce the eccentricity. 2. At service load

fb =---- 1 H--+ 0 = —1,844 psi (C) < —2,700 psi, OK.

Hence, adopt the 40-in. (102-cm)-deep I-section prestressed beam of f0 equal to 6,000 psi (41.4 MPa) normal-weight concrete with thirteen 1-in. strands tendon having midspan eccentricity ec = 15.0 in. (381 mm) and end section eccentricity ee = 12.5 in. (318 m).

An alternative to this solution is to continue using f0 = 5000 psi, but change the number of strands and eccentricities.

8.6.3 Development and Transfer Length in Pretensioned Members and Design of their Anchorage Reinforcement

As the jacking force is released in pretensioned members, the prestressing force is dynamically transferred through the bond interface to the surrounding concrete. The interlock or adhesion between the prestressing tendon circumference and the concrete over a finite length of the tendon gradually transfers the concentrated prestressing force to the entire concrete section at planes away from the end block and toward the midspan. The length of embedment determines the magnitude of prestress that can be developed along the span: the larger the embedment length, the higher is the prestress developed.

As an example for 1-in. seven-wire strand, an embedment of 40 in. (102 cm) develops a stress of 180,000 psi (1,241 MPa), whereas an embedment of 70 in. (178 cm) develops a stress of 206,000 psi (1,420 MPa). The embedment length ld that gives the full development of stress is a combination of the transfer length 4 and the flexural bond length lf. These are given, respectively, by

3000 b v 7

where fps is the stress in prestressed reinforcement at nominal strength (psi), fpe is the effective prestress after losses (psi), and db is the nominal diameter of prestressing tendon (in.). Combining Equations 8.29b and 8.29c gives

Equation 8.29d gives the minimum required development length for prestressing strands. If part of the tendon is sheathed toward the beam end to reduce the concentration of bond stresses near the end, the stress transfer in that zone is eliminated and an increased adjusted development length ld is needed.

8.6.3.1 Design of Transfer Zone Reinforcement in Pretensioned Beams

Based on laboratory tests, empirical expressions developed by Mattock et al. give the total stirrup force F as

lt where h is the pretensioned beam depth and lt is the transfer length. If the average stress in a stirrup is taken as half the maximum permissible steel fs, then F=2 Atfs. Substituting this for F in Equation 8.30 gives

Pi h

where At is the total area of the stirrups and fs < 20,000 psi (138 MPa) for crack-control purposes.

8.6.4 Posttensioned Anchorage Zones: Strut-and-Tie Design Method

The anchorage zone can be defined as the volume of concrete through which the concentrated pre-stressing force at the anchorage device spreads transversely to a linear distribution across the entire cross-section depth along the span. The length of this zone follows St Venant's principle, namely, that the stress becomes uniform at an approximate distance ahead of the anchorage device equal to the depth, h, of the section. The entire prism that would have a transfer length, h, is the total anchorage zone.

This zone is thus composed of two parts:

1. General zone: The general extent of the zone is identical to the total anchorage zone. Its length extent along the span is therefore equal to the section depth, h, in standard cases.

2. Local zone: This zone is the insert prism of concrete surrounding and immediately ahead of the anchorage device and the confining reinforcement it contains.

After significant cracking is developed, compressive stress trajectories in the concrete tend to congregate into straight lines that can be idealized as straight compressive struts in uniaxial compression. These struts would become part of truss units where the principal tensile stresses are idealized as tension ties in the truss unit with the nodal locations determined by the direction of the idealized compression struts. Figure 8.4 sketches standard strut-and-tie idealized trusses for concentric and eccentric cases both for solid and flanged sections as given in ACI 318-02 Code.

Simplified equations can be used to compute the magnitude of the bursting force, Tburst, and its centroid distance, dburst, from the major bearing surface of the anchorage [9]. The member has to have

Rectangular section Concentric P T ~ 0.25P

Flanged section Concentric P T ~ 0.50P

Rectangular section Concentric P T ~ 0.25P

Flanged section Concentric P T ~ 0.50P

Flanged section Eccentric P T ~ 0.50P

FIGURE 8.4 Strut-and-tie idealized trusses in standard concentric and eccentric cases, ACI 318-02 [8].

a rectangular cross-section with no discontinuities along the span. The bursting force, Tburst and its distance, dburst, can be computed from the following expressions:

where ^ Psu is the sum of the total factored prestress loads for the stressing arrangement considered (lb), a is the plate width of anchorage device or single group of closely spaced devices in the direction considered (in.), e is the eccentricity (always taken positive) of the anchorage device or group closely spaced devices with respect to the centroid of the cross-section (in.), and h is the depth of the cross-section in the direction considered (in.).

The ACI 318 Code requires that the design of confining reinforcement in the end anchorage block of posttensioned members be based on the factored prestressing force Psu for both the general and local zones. A load factor of 1.2 is to be applied to an end anchorage stress level fpi = 0.80fpu for low-relaxation strands at the short time interval of jacking, which can reduce to an average value of 0.70fpu for the total group of strands at the completion of the jacking process. For stress-relieved strands, a lower fpi = 0.70fpu is advised. The maximum force Psu stipulated in the ACI 318 Code for designing the confining reinforcement at the end-block zone is as follows for the more widely used low-relaxation strands:

The AASHTO Standard for the case where Psu acts at an inclined angle a in the direction of the beam span adds the term [0.5^(Psusin a)]to Equation 8.32a and 5e(sin a) to Equation 8.32b. For horizontal Psu sin a = 0.

8.6.4.1 Allowable Bearing Stresses

The maximum allowable bearing stress at the anchorage device seating should not exceed the smaller of the two values obtained from Equations 8.33a and 8.33b as follows:

where fb is the maximum factored tendon load, Pu, divided by the effective bearing area Ab; fci is the concrete compressive strength at initial stressing; A is the maximum area of the portion of the supporting surface that is geometrically similar to the loaded area and concentric with it, contained wholly within the section, with the upper base being the loaded surface area of the concrete and sloping sideway with a slope of 1 vertical to 2 horizontal; and Ag is the gross area of the bearing plate.

Equations 8.33a and 8.33b are valid only if general zone reinforcement is provided and if the extent of concrete along the tendon axis ahead of the anchorage device is at least twice the length of the local zone.

8.6.5 Example 4: End Anchorage Design by the Strut-and-Tie Method

Design an end anchorage reinforcement for the posttensioned beam in Example 3, giving the size, type, and distribution of reinforcement. Use f0 = 5,000 psi (34.5 MPa) normal-weight concrete. fpu = 270,000 psi low-relaxation steel.

Assume that the beam ends are rectangular blocks extending 40 in. (104 cm.) into the span beyond the anchorage devices which then transitionally reduce to the 6-in. thick web.

1. Establish the configuration of the tendons to give eccentricity ee = 12.49 in. (317 mm). From Example 3, cb = 18.84 in.; hence, distance from the beam fibers = cb - ee = 6.35 in. (161 mm). For a centroidal distance of the 132-in. size strands = 6.35 in. from the beam bottom fibers, try the following row arrangement of tendons with the indicated distances from the bottom fibers:

first row: five tendons at 2.5 in. second row: five tendons at 7.0 in. third row: three tendons at 11.5 in.

Distance of the centroid of tendons =-= 6 . 35 in. , OK.

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