parabolic in the present case. Therefore, it is possible to achieve the full allowable initial
stresses at midspan. Thus,/w =fb - 374 = +2400;/^ =ftp, + 374 - -190;/^ = +2774 lb/in
2
(+19,126.7 kPa);^ = -564 lb/in* (-3888.8 kPa);/^= 0.85(2774) - 374 +fbs = -425; ftf=
0.85(-564) + 374 +fts = + 2250; fbs = -2409 lb/in
2
(-16,610.1 kPa);/ 5 = +2356 lb/in
2
(+16,244.6 kPa). The latter value controls.
Also, w, = 83(2356/374) = 523 Ib/lin ft (7632.6 N/m); 523/452 =1.16. Thus the capac-
ity is increased 16 percent.
When the foregoing calculations are compared with those in the earlier calculation
procedure, the effect of using parabolic tendons is to permit an increase of 374 lb/in
2
(2578.7 kPa) in the absolute value of the prestress at top and bottom. The accompanying
increase in/, is 0.85(374) = 318 lb/in
2
(2192.6 kPa).
- Find the minimum prestressing force and its eccentricity
at midspan
As before, F 1 = 85,920 Ib (382,172.2 N); ^ = 1074 - 85,920^/133 = -564; e = 2.54 in
(64.516mm).
DETERMINATION OF SECTION MODULI
A beam having a cross-sectional area of 500 in^2 (3226 cm^2 ) sustains a beam-weight mo-
ment equal to 3500 in-kips (395.4 kN-m) at midspan and a superimposed moment that
varies parabolically from 9000 in-kips (1016.8 kN-m) at midspan to O at the supports. The
allowable stresses are: initial, +2400 and-190 lb/in^2 (+16,548 and-1310.1 kPa);/ma/, +
2250 and -200 lb/in^2 (+15,513.8 and -1379 kPa). The member will be prestressed by ten-
dons deflected at the quarter points. Determine the section moduli corresponding to bal-
anced design, the magnitude of the prestressing force, and its eccentricity in the center in-
terval. Assume that the calculated eccentricity is attainable (i.e., that the centroid of the
tendons will fall within the confines of the section while satisfying insulation require-
ments).
Calculation Procedure:
- Equate the critical initial stresses, and the critical final stresses,
to their allowable values
Let Mw and Ms denote the indicated moments at midspan; the corresponding moments at
the quarter point are three-fourths as large. The critical initial stresses occur at the quarter
point, while the critical final stresses occur at midspan. After equating the stresses to their
allowable values, solve the resulting simultaneous equations to find the section moduli
and prestresses. Thus: stresses in bottom fiber, fbi -fbp - Q.15MJSb = +2400;/^ = 0.85/J,p
-MJSh-M/Sh = -200. Solving gives Sb = (Ms + 0.3625MJ/2240 - 4584 in^3 (75,131.7
cm^3 ) mdfbp = + 2973 lb/in^2 (+20,498.8 kPa); stresses in top fiber, fti =ftp + Q.I 5(MJSt) =
-l9Q;ftf = 0.854 + M*ISt+ MA = +2250. Solving yields St = (Ms + 0.3625MJ/2412 =
4257 in
3
(69,772.2 cm
3
) andftp = -807 lb/in
2
(-5564.2 kPa). - Evaluate F 1 - and e
In this instance, e denotes the eccentricity in the center interval. Thusfbp = FJA + F^/Sb =
+2973; f^ = F 1 IA - FtelSt = -807; F 1 = (2973S 6 - 8Q7St)A/(Sb + S 1 ) = 576,500 Ib
(2,564,272.0 N); e = 2973 S 1 JF 1 - S 1 JA = 14.47 in (367.538 mm).