## Ei

RB, sec

48 x 263,774 innci. = = 10.05 kips (2 x 54)3

The secondary moments due to this redundant reaction are shown in Fig. 7.31C(h).

### 7.3.6. Step-by-Step Design Procedure

The aim of post-tension design is to determine the required prestressing force and hence the number, size, and profile of tendons for satisfactory behavior at service loads. The ultimate capacity must then be checked at critical sections to assure that prestressed members have an adequate factor of safety against failure.

The design method presented in this section uses the technique of load balancing in which the effect of prestressing is considered as an equivalent load. Take, for example, a prismatic simply supported beam with a tendon of parabolic profile, shown in Fig. 7.32. The tendon exerts a horizontal force equal to P cos d = P (for small values of d) at the ends along with vertical components equal to P sin d. The vertical component is neglected in design because it occurs directly over the supports. In addition to these loads, the parabolic tendon exterts a continuous upward force on the beam along its entire length. By neglecting friction between the tendon and concrete, we can assume that: 1) the upward pressure exerted is normal to the plane of contact; and 2) tension in tendon is constant. The upward pressure exerted by the tendon is equal to the tension in the tendon divided by the radius of curvature of the tendon profile. Due to the shallow nature of post-tensioned structures, the vertical component of the tendon force may be assumed constant. Considering one-half of the beam as a free body (Fig. 7.33b), the vertical load exerted by the tendon may be derived by summing moments about the left support. Thus the equivalent, load Wp = 8Pe/L2. Equivalent loads and moments produced by other types of tendon profile are shown in Fig. 7.32b through d.

The step-by-step procedure is as follows:

1. Determine preliminary size of prestressed concrete members using the values given in Table 7.6 as a guide.

2. Determine section properties of the member: area A, moment of inertia I, and section moduli St and Sb.

Figure 7.32. Equivalent loads and moments produced by prestressed tendons: (a) upward uniform load due to parabolic tendon; (b) constant moment due to straight tendon; (c) upward uniform load and end moments due to parabolic tendon not passing through the centroid at the ends; (d) vertical point-load due to sloped tendon.

Figure 7.32. Equivalent loads and moments produced by prestressed tendons: (a) upward uniform load due to parabolic tendon; (b) constant moment due to straight tendon; (c) upward uniform load and end moments due to parabolic tendon not passing through the centroid at the ends; (d) vertical point-load due to sloped tendon.

Figure 7.33. Load balancing concept: (a) beam with parabolic tendon; (b) free-body diagram.

3. Determine tendon profile with due regard to cover and location of mild steel reinforcement.

4. Determine effective span Le by assuming Lx = 1/16 to 1/19 of the span length for slabs, and L = 1/10 to 1/12 of the span length for beams. Lx is the distance between the center line of support and the inflection point. The concept of effective length will be explained shortly.

6. Determine the elastic moments for the total dead plus live loads (working loads). For continuous beams and slabs use a computer plane-frame analysis program, moment distribution method, or ACI coefficients, if applicable, in decreasing order of preference.

### 7. Reduce negative moments to the face of supports.

8. By proportioning the unbalanced load to the total load, determine the unbalanced moments at Mub at critical sections such as at the supports and at the center of spans.

9. Calculate the bending stresses fb and ft at the bottom and top of the cross section due to Mub at critical sections. Typically at supports the stresses ft and fb are in tension and compression, respectively. At center of spans the stresses are typically compression and tension at top and bottom, respectively.

10. Calculate the minimum required post-tension stress f by using the following equations.

For negative zones of one-way slabs and beams:

For positive moments in two-way slabs:

11. Find the post tension force P by the relation P = fp x A where A is the area of the cross section of the beam.

12. Calculate the balanced load Wp due to P by the relation

L where e = drape of the tendon Le = effective length of tendon between inflection points.

13. Compare the calculated value of Wp from step 12 with the value assumed in step 5. If they are about the same, the selection of post-tension force for the given loads and tendon profile is complete. If not, repeat steps 9-13 with a revised value of Wp = 0.75Wp1 + 0.25Wp2. Wp1 is the value of Wp assumed at the beginning of step 5, and Wp— is the derived value of Wp at the end of step 12. Convergence is fast requiring no more than three cycles in most cases.

Figure 7.34. Preliminary design: simple span beam.

7.3.6.1. Simple Span Beam

The concept of preliminary design discussed in this section is illustrated in Fig. 7.34 where a parabolic profile with an eccentricity of 12 in. is selected to counteract part of the applied load consisting of a uniformly distributed dead load of 1.5 kip-ft and a live load of 0.5 kip-ft.

In practice, it is rarely necessary to provide a prestress force to fully balance the imposed loads. A value of prestress, often used for building system, is 75 to 95% of the dead load. For the illustrative problem, we begin with an assumed 80% of the dead load as the unbalanced load.

First Cycle The load being balanced is equal to 0.80 x 1.5 = 1.20 kip-ft. The total service dead plus live load = 1.5 + 0.5 = 2.0 kip-ft, of which 1.20 kip-ft is assumed in the first cycle to be balanced by the prestressing force in the tendon. The remainder of the load equal to 2.0 - 1.20 = 0.80 kip-ft acts vertically downward, producing a maximum unbalanced moment Mub at center span given by

542 8

The tension and compression in the section due to Mub is given by fc = fb =

291.6 x 12

The minimum prestress required to limit the tensile stress to = 0.424 is given by fp = 1.55 - 0.424 = 1.13 ksi

Therefore, the required minimum prestressing force P = area of beam X 1.13 = 450 X 1.13 = 509 kips. The load balanced by this force is given by

and so Wp = 1.396 kip-ft compared to the value of 1.20 used in the first cycle. Since these two values are not close to each other, we repeat the above calculations starting with a more precise value for Wp in the second cycle.

Second Cycle We start with a new value of Wp by assuming a new value equal to 75% of the initial value + 25% of the derived value. The new value of

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