_ 112.5 in.3


_ 12 in. :

x 7.5

_ 90 in.2

A 1-ft width of slab is analyzed as a continuous beam. The effect of column stiffness is ignored.

The moment diagram for a service load of 226 plf is shown in Fig. 7.39.

Moments at the face of supports have been used in the design instead of center line moments. Negative center line moments are reduced by a "Va/3" factor (V = shear at that support, a = total support width), and positive moments are reduced by Va/6 using average adjacent values for shear and support widths. A frame analysis may be course be used to obtain more accurate results.

The design of continuous strands will be based on the negative moment of 10.6 kip-ft. The additional prestressing required for the negative moment of 16.8 kip-ft will be provided by additional tendons in the end bays only.

Determination of Tendon Profile. Maximum tendon efficiency is obtained when the cable drape is as large as the structure will allow. Typically, the high points of the tendon over the supports and the low point within the span are dictated by concrete cover requirements and the placement of mild steel.

(Inlfection points at L/16)

Figure 7.40. Example 1: one-way post-tensional slab tendon profile, interior bay.

(Inlfection points at L/16)

Figure 7.40. Example 1: one-way post-tensional slab tendon profile, interior bay.

Figure 7.41. Dimensions for determining tendon drape.

The high and low points of tendon in the interior bay of the example problem are shown in Fig. 7.40. Next, the location of inflection points are determined. For slabs, the inflection points usually range within 1/16 to 1/19 of the span. The fraction of span length used is a matter of judgment, and is based on the type of structure. For this example, we choose 1/16 of span which works out to 1'-10 Jf.

An interesting property useful in determining the tendon profile shown in Fig. 7.41 is that, if a straight line (chord) is drawn connecting the tendon high point over the support and the low point midway between, it intersects the tendon at the inflection point. Thus, the height of the tendon can be found by proportion. From the height, the bottom cover is subtracted to find the drape.

Referring to Fig. 7.41,

Slope of the chord line =

This simplifies to h3 =

The drape hd is obtained by subtracting h2 from the foregoing equation. Note that notation e is also used in these examples to denote drape hd.

In this case, the height of the inflection point is exact for symmetrical layout of the tendon about the center span. If the tendon is not symmetrical, the value is approximate but sufficiently accurate for preliminary design.

Returning to our example problem we have ht = 6.5'', h2 = 1'', Lx = 1.875', and L2 = 13.125'.

Height of tendon at the inflection point:

(1.875 + 13.125) Drape hd = e = 5.813 - 1 = 4.813" in.

Allowable stresses from ACI 318-99 are as follows: f = tensile stress 6^/f" fc = compressive stress = 0.45f C For fC = 4000 psi concrete:

Design of Through Strands. The design procedure is started by making an initial assumption of the equivalent load produced by the prestress. A first value of 65% of the total dead load is used.

First Cycle Assume Wp = 0.65 Wd where

Wp = equivalent upward load due to post-tensioning, also denoted as Wpt Wd = total dead load

The balancing moment caused by the equivalent load is calculated from

"s where

Mpt = balancing moment due to equivalent load

(also indicated by notation Mb) Ms = moment due to service load, D + L Ws = total applied load, D + L

In our example, Ms = 10.6 kip-ft for the interior span. 82

Next, Mpt is subtracted from Ms to give the unbalanced moment Mub. The flexural stresses are then obtained by dividing Mub by the section moduli of the structure's cross section at the point where Ms is determined. Thus f = Mub ft =

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