Structural Concrete - Hassoun

754 Chapter 19 Introduction to Prestressed Concrete
Top fibers, loaded condition:
e ≥ 9.4 + 7336
306.2 − 2.25(372)(9.4) ≥ 7.7in.
306.2
Bottom fibers, loaded condition:
e ≥ −8.6 + 7336
306.2 − 0.424(372)(8.6) ≥ 11.0in.
306.2
Minimum e = 11.0 in. controls.
c. Consider a section 16 ft from midspan (section 3, Fig. 19.6a): M D (self−weight) = 745 K ⋅ in.,
M a = 3840 K ⋅ in., and M T = 4585 K ⋅ in.
• Top fibers, unloaded condition, e ≤ 13.3 in. (max) controls.
• Bottom fibers, unloaded condition, e ≤ 14.4 in.
• Top fibers, loaded condition, e ≥ − 1.3 in.
• Bottom fibers, loaded condition, e ≥ 1.9 in. (min) controls.
d. Consider a section 3 ft from the end (anchorage length): M D = 314 K ⋅ in., M a = 1620 K ⋅ in.,
and M T = 1934 K ⋅ in.
• Top fibers, unloaded condition, e≤ 12.1 in. (max) controls.
• Bottom fibers, unloaded condition, e ≤ 13.3 in.
• Top fibers, loaded condition, e ≥ − 10 in.
• Bottom fibers, loaded condition, e≥ − 6.7 in. (min) controls.
4. The tendon profile is shown in Fig. 19.6. The eccentricity chosen at midspan is e = 14.5 in., which
is adequate for section B at 8 ft from midspan. The centroid of the prestressing steel is horizontal
between A and B and then harped linearly between B and the end section at E. The eccentricities at
sections C and D are calculated by establishing the slope of line BE, which is 14.5/16 = 0.91 in./ft.
The eccentricity at C is 7.25 in. and at D it is 2.72 in. The tendon profile chosen satisfies the upper
and lower limits of the eccentricity at all sections.
Harping of tendons is performed as follows:
a. Place the 20 tendons ( 7 diameter) within the middle third of the beam at spacings of 2 in., as
16
shown in Fig. 19.6a. To calculate the actual eccentricity at midspan section, take moments for
the number of tendons about the baseline of the section:
Distance from base = 1 (16 × 5 + 4 × 11) =6.2in.
20
e (midspan) =y b − 6.2in.
= 20.8 − 6.2 = 14.6in.
which is close to the 14.5 in. assumed. If the top two tendons are placed at 3 in. from the row
below them, then the distance from the base becomes 1 (16 × 5 + 2 × 10 + 2 × 13) =6.3in.
20
The eccentricity becomes 20.8 − 6.3 = 14.5 in., which is equal to the assumed eccentricity.
Practically, all tendons may be left at 2 in. spacing by neglecting the difference of 0.1 in.
b. Harp the central 12 tendons only. The distribution of tendons at the end section is shown in
Fig. 19.6a. To check the eccentricity of tendons, take moments about the centroid of the concrete
section for the 12 tendons at top and the 8 tendons left at bottom:
e = 1 (8 × 14.5 − 12 × 9.2) =0.28 in.
20
This value of e is small and adequate. The actual eccentricity at 3 ft from the end section is
e = 3 (14.5 − 0.28)+0.28 = 2.95 in. (3in.)
16
The actual eccentricity at 8 ft from the end section is
e = 1 (14.5 − 0.28)+0.28 = 7.4in.
2