Page 275 - Handbook of Civil Engineering Calculations, Second Edition

2.60 REINFORCED AND PRESTRESSED CONCRETE ENGINEERING AND DESIGN

Calculation Procedure:

1. Compute the beam-weight stresses at B
In the composite stress diagram, Fig. 35b, the difference between an ordinate to EFG and
the corresponding ordinate to AHJ represents the value of f ti at the given section. It is ap-
parent that if AE does not exceed HF, then f ti does not exceed HE in absolute value any-
where along the span. Therefore, for the center interval BC, the critical stresses at transfer
occur at the boundary sections B and C. Analogous observations apply to Fig. 35c.
Computing the beam-weight stresses at B yields f bw ( /4)( 374) 281 lb/sq.in.
3
( 1937.5 kPa); f tw 281 lb/sq.in. ( 1937.5 kPa).
2. Tentatively set the critical stresses equal to their allowable
values to secure the allowable unit superimposed load
Thus, at B: f bi f bp 281 2400; f ti f tp 281 190; f bp 2681 lb/sq.in.
( 18,485.5 kPa); f tp 471 lb/sq.in. ( 3247.5 kPa).
At M: f bf 0.85(2681) 374 f bs 425; f tf 0.85( 471) 374 f ts 2250;
f bs 2330 lb/sq.in. ( 16,065.4 kPa); f ts 2277 lb/sq.in. ( 15,699.9 kPa). The latter
value controls.
Also, w s 83(2277/374) 505 lb/lin ft (7369.9 N/m); 505/452 1.12. The capacity
is increased 12 percent.
When the foregoing calculations are compared with those in the previous calculation
procedure, the effect of deflecting the tendons is to permit an increase of 281 lb/sq.in.
(1937.5 kPa) in the absolute value of the prestress at top and bottom. The accompanying
increase in f ts is 0.85(281) 239 lb/sq.in. (1647.9 kPa).
3. Find the minimum prestressing force and the eccentricity e 1
Examination of Fig. 34 shows that f cai is not affected by the form of trajectory used.
Therefore, as in the previous calculation procedure, F i 85,920 lb (382,172.2 N); f tp
1074 85,920e 1 /133 471; e 1 2.39 in. (60.706 mm).
Although it is not required, the value of f bp 1074 1074 ( 471) 2619
lb/sq.in. ( 18,058kPa), or f bp 2681 53/0.85 2619 lb/sq.in. ( 18,058kPa).
4. Establish the allowable range of values of e 2
At the supports, the tendons may be placed an equal distance above or below the center.
Then e 2,max 1.96 in. (23.44 mm); e 2,min 1.96 in. ( 23.44 mm).

BEAM WITH CURVED TENDONS

The beam in the second previous calculation procedure is to be prestressed by tendons
lying in a parabolic arc. Evaluate the allowable unit superimposed load, the magnitude of
the prestressing force, the eccentricity of this force at midspan, and the increase in capac-
ity accruing from the use of curved tendons.

Calculation Procedure:

1. Tentatively set the initial and final stresses at midspan
equal to their allowable values to secure the allowable unit
superimposed load
Since the prestressing force has a parabolic trajectory, lines EFG in Fig. 35b and c will be
parabolic in the present case. Therefore, it is possible to achieve the full allowable initial