Calculation Procedures:- Compute the beam properties
Thus, A = 12(18) = 216 in^2 (1393.6 cm^2 ); Sb = St = (Y 6 )(^)(IS)^2 = 648 in^3 (10,620.7 cm^3 );
ww = (216/144)(150) = 225 Ib/lin ft (3,283.6 N/m). - Calculate the prestress in the top and bottom fibers
Since the section is rectangular, apply fbp = (FJA)(I + 6elh) = (230,000/216)(1 + 6 x
3.3/18) = + 2236 IMn^2 (+15,417.2 kPa);^ = (FJA)(I - 6e/h) = -106 lb/in^2 (-730.9 kPa).
For convenience, record the stresses in Table 3 as they are obtained. - Determine the stresses at midspan due to gravity loads
Thus M 5 = (%)(840)(30)^2 (12) = 1,134,000 in-lb (128,119.32 N-m);/^ = -1,134,000/648 =
-1750 lb/in^2 (-12,066.3 kPa);/, = + 1750 lb/in^2 (12,066.3 kPa). By proportion,/^ =
-1750(225/840) = -469;/^ = +469 lb/in^2 (+ 3233.8 kPa). - Compute the initial and final stresses at the supports
Thus,/w = + 2236 lb/in
2
(+15,417.2 kPa);/,, = -106 lb/in
2
(-730.9 kPa);^ 7 = 0.85(2236) =
- 1901 lb/in
2
(+13,107.4 kPa);^-= 0.85(-106) = -90 lb/in
2
(-620.6 kPa).
- Determine the initial and final stresses at midspan
Thusfbi = + 2236 - 469 = + 1767 lb/in
2
(+12,183.5 kPa);/tf = -106 + 469 = +363 lb/in
2
(+2502.9 kPa);/^= + 1901 - 469 - 1750 = -318 lb/in
2
(-2192.6 kPa);/^-= -90 + 469 +
1750 = +2129 lb/in
2
(+14,679.5 kPa).
- Construct the initial-stress diagram
In Fig. 320, construct the initial-stress diagram AfA^C at the support and the initial-stress
diagram MtM^f)E 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
(a) Initial prestress,
lb/in^2 (kPa)
(b) Final prestress,
lb/in^2 (kPa)
(c) Stress due to
beam weight,
lb/in^2 (kPa)
(d) Stress due to
superimposed
load, lb/in^2
(kPa)
Initial stress:
(a) +(c)
Final stress: (b) +
(c) + (d)
At support
Bottom fiber Top fiber
+2,236 (+15,417.2) -106 (-730.9)+1,901 (+13,107.4) -90 (-620.6)+2,236 (+15,417.2) -106 (-730.9)+1,901 (+13,107.4) -90 (-620.6)At midspan
Bottom fiber Top fiber
+2,236 (+15,417.2) -106 (-730.9)+1,901 (+13,107.4) -90 (-620.6)-469 (-3,233.8) +469 (3,233.8)-1,750 (-12,066.3) +1,750 (+12,066.3)+1,767 (+12,183.5) +363 (+2,502.9)-318 (-2,192.6) +2,129 (+14,679.5)