## Info

Axial compression due to post-tension = ——- = 0.238 ksi

Total stress at bottom = 0.228 - 0.238 = -0.10 ksi (Compression) This is less than allowable tension of 0.380 ksi. Therefore, design OK.

### 7.3.7. Strength Design for Flexure

In the design of prestress members it is not enough to limit the maximum values of tensile and compressive stresses within the permitted values at various loading stages. This is because although such a design may limit deflections, control cracking, and prevent crushing of concrete, an elastic analysis offers no control over the ultimate behavior or the factor-of-safety of a prestressed member. To ensure that prestressed members will be designed with an adequate factor-of-safety against failure, ACI 318-02, similar to its predecessors, requires that Mu, the moment due to factored service loads, not exceed fMn, the flexural design strength of the member.

The nominal bending strength of a prestressed beam with bonded tendons is computed in nearly the same manner as that of a reinforced concrete beam. The only difference is in the method of stress calculation in the tendon at failure. This is because the stress-strain curves of high-yield-point steels used as tendons do not develop a horizontal yield range once the yield strength is reached. It continues upward at a reduced slope. Therefore, the final stress in the tendon at failure fps must be predicted by an empirical relationship.

The method of computing the bending strength of a prestressed beam given in the following section applies only to beams with bonded tendons. The analysis is performed using strain compatibility. Because by definition there is no strain compatibility between the tendon and concrete in an unbonded prestressed beam, this method cannot be used for prestressed beams with unbonded tendons; the empirical approach given in ACI 318-02, Section 18.7 is the recommended method.

The procedure for bonded tendons consists of assuming the location of the neutral axis, computing the strains in the prestressed and non-prestressed reinforcement, and establishing the compression stress block. Knowing the stress-strain relationship for the reinforcement, and assuming that the maximum strain in concrete is 0.003, the forces in the prestressed and nonprestressed reinforcement are determined and the sum of compression and tension forces are computed. If necessary, the neutral axis location is adjusted on a trial-and-error basis until the sum of the forces is zero. The moment of these forces is then computed to obtain the nominal strength of the section. To compute the stress in the prestressing strand, the idealized curve shown in Fig. 7.48 (adapted from PCI Design Handbook, 5th Edn.) is used.

The analysis presented here follows a slightly different procedure. Instead of assuming the location of the neutral axis, we assume a force in the prestressing strand, and compare it to the derived value. The analysis is continued until the desired convergence is reached.

7.3.7.1. Examples

Given. A rectangular prestressed concrete beam, as shown in Fig. 7.48a: fc = 5000 psi

Mild steel reinforcement = 4 #5 bars at bottom, fy = 60 ksi Prestressed strands = 4 - 1/2 f, fps = 270 ksi

Figure 7.48. Typical stress-strain curve with seven-wire low-relaxation prestressing strand. These curves can be approximated by the following equations:

250 ksi 270 ksi eps < 0.0076 : f = 28,500 e (ksi) e < 0.0086 : f = 28,500 eps(ksi)

Figure 7.48. Typical stress-strain curve with seven-wire low-relaxation prestressing strand. These curves can be approximated by the following equations:

250 ksi 270 ksi eps < 0.0076 : f = 28,500 e (ksi) e < 0.0086 : f = 28,500 eps(ksi)

em > 0.0076 : f = 250 --(ksi) e > 0.0086 : f = 270 --(ksi)

ps ps

Figure 7.48a. Example 1: beam section.

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