Page 270 - Handbook of Civil Engineering Calculations, Second Edition

PRESTRESSED CONCRETE 2.55

Calculation Procedures:

1. Compute the beam properties
2
2
3
Thus, A 12(18) 216 sq.in. (1393.6 cm ); S b S t ( /6)(12)(18) 648 in (10,620.7
1
3
cm ); w w (216/144)(150) 225 lb/lin ft (3,283.6 N/m).
2. Calculate the prestress in the top and bottom fibers
Since the section is rectangular, apply f bp (F i /A)(1 6e/h) (230,000/216)(1 6
3.3/18) 2236 lb/sq.in. ( 15,417.2 kPa); f tp (F i /A)(1 6e/h) 106 lb/sq.in.
( 730.9 kPa).
For convenience, record the stresses in Table 1 as they are obtained.
3. Determine the stresses at midspan due to gravity loads
2
Thus M s ( /8)(840)(30) (12) 1,134,000 in.·lb (128,119.32 N·m); f bs 1,134,000/
1
648 1750 lb/sq.in. ( 12,066.3 kPa); f ts 1750 lb/sq.in. (12,066.3 kPa). By propor-
tion, f bw 1750(225/840) 469; f tw 469 lb/sq.in. ( 3233.8 kPa).
4. Compute the initial and final stresses at the supports
Thus, f bi 2236 lb/sq.in. ( 15,417.2 kPa); f ti 106 lb/sq.in. ( 730.9 kPa); f bf
0.85(2236) 1901 lb/sq.in. ( 13,107.4 kPa); f tf 0.85( 106) 90 lb/sq.in.
( 620.6 kPa).
5. DetermIne the initial and final stresses at midspan
Thus f bi 2236 469 1767 lb/sq.in. ( 12,183.5 kPa); f ti 106 469
363 lb/sq.in. ( 2502.9 kPa); f bf 1901 469 1750 318 lb/sq.in. ( 2192.6
kPa); f tf 90 469 1750 2129 lb/sq.in. ( 14,679.5 kPa).
6. Construct the initial-stress diagram
In Fig. 32a, construct the initial-stress diagram A t A b BC at the support and the initial-stress
diagram M t M b DE at midspan. Draw the parabolic arcs BD and CE. The stress diagram at
an intermediate section Q is obtained by passing a plane normal to the longitudinal axis.

TABLE 3. Stresses in Prestressed-Concrete Beam
At support At midspan
_____________________________ _________________________________
Bottom fiber Top fiber Bottom fiber Top fiber
(a) Initial prestress, 2,236 ( 15,417.2) 106 ( 730.9) 2,236 ( 15,417.2) 106 ( 730.9)
lb/sq.in. (kPa)
(b) Final prestress, 1,901 ( 13,107.4) 90 ( 620.6) 1,901 ( 13,107.4) 90 ( 620.6)
lb/sq.in. (kPa)
(c) Stress due to ..... ..... 469 ( 3,233.8) 469 (3,233.8)
beam weight,
lb/sq.in. (kPa)
(d) Stress due to ..... ..... 1,750 ( 12,066.3) 1,750( 12,066.3)
superimposed
load, lb/sq.in.
(kPa)
Initial stress: 2,236 ( 15,417.2) 106 ( 730.9) 1,767 ( 12,183.5) 363 ( 2,502.9)
(a) (c)
Final stress: (b) 1,901 ( 13,107.4) 90 ( 620.6) 318 ( 2,192.6) 2,129( 14,679.5)
(c) (d)