276.47 x 12

fp = 1.475 - 0.454 = 1.021 ksi P = 1.021 x 450 = 459.3 kips 1 x 8

compared to 1.24 assumed at the beginning of third cycle. The value of 1.26 kip-ft is considered close enough for design purposes.

The above example illustrates the salient features of load balancing. Generally, the pre-stressing force is selected to counteract or balance a portion of dead load, and under this loading condition the net stress in the tension fibers is limited to a value = 6^/f If it is desired to design the member for zero stress at the bottom fiber at center span (or any other value less than the code allowed maximum value of ), it is only necessary to adjust the amount of post-tensioning provided in the member.

There are some qualifications to the foregoing procedure that should be kept in mind when applying the technique to continuous beams. Chief among them is the fact that it is not usually practical to install tendons with sharp break in curvature over supports, as shown in Fig. 7.35a. The stiffness of tendons requires a reverse curvature (Fig. 7.35b) in the tendon profile with a point of contraflexure some distance from the supports. Although this reverse curvature modifies the equivalent loads imposed by post-tensioning from those assumed for a pure parabolic profile between the supports, a simple revision to the effective length of tendon, as will be seen shortly, yields results sufficiently accurate for preliminary designs.

Consider the tendon profiles shown in Figs. 7.36a,b for a typical exterior and an interior span. Observe three important features.

Figure 7.35. Tendon profile in continuous beams: (a) simple parabolic profile; (b) reverse curvature in tendon profile.

1. The effective span Le, the distance between the inflection points which is considerably shorter than the actual span.

2. The sag or drape of the tendon is numerically equal to average height of inflection points, less the height of the tendon midway between the inflection points.

3. The point midway between the inflection points is not necessarily the lowest point on the profile.

The upward equivalent uniform load produced by the tendon is given by

p l2

where

Wp = equivalent upward uniform load due to prestress P = prestress force e = cable drape between inflection points Le = effective length between inflection points

Note that relatively high loads acting downward over the supports result from the sharply curved tendon profiles located within these regions (Fig. 7.37).

Since the large downward loads are confined to a small region, typically 1/10 to 1/8 of the span, their effect is secondary as compared to the upward loads. Slight differences occur in the negative moment regions between the applied load moments and the moment due to prestressing force. The differences are of minor significance and can be neglected in the design without losing meaningful accuracy.

As in simple spans the moments caused by the equivalent loads are subtracted from those due to applied loads, to obtain the net unbalanced moment that produces the flexural stresses. To the flexural stresses, the axial compressive stresses from the prestress are added to obtain the final stress distribution in the members. The maximum compressive and tensile stresses are compared to the allowable values. If the comparisons are favorable, an acceptable design has been found. If not, either the tendon profile or the force (and very rarely the cross-sectional shape of the structure) is revised to arrive at an acceptable solution.

In this method, since the moments due to equivalent loads are linearly related to the moments due to applied loads, the designer can bypass the usual requirement of determining the primary and secondary moments.

Section

Figure 7.38. Example 1: one-way post-tensioned slab.

Section

Figure 7.38. Example 1: one-way post-tensioned slab.

7.3.6.2.1. Example 1: One-Way Post-Tensioned Slab.

Given a 30'-0" column grid layout, design a one-way slab spanning between the beams shown in Fig. 7.38.

Slab and beam depths:

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