Structural Concrete - Hassoun

764 Chapter 19 Introduction to Prestressed Concrete
(
M n = 0.836(228)
33.7 − 7.5
2
= 15, 102 K ⋅ in. = 1258.5K⋅ ft
φM n = 0.9(1258.5) =1132.7K⋅ ft
Check the reinforcement index for the flanged section:
ρ pw
(web) = A pw
b w d p
= 0.836
6 × 33.7 = 0.00413
)
(
+ 1.342 × 228 33.7 − 6 )
2
ω pw
(web) =ρ pw
f ps
f ′ c
≤ 0.32β 1 = 0.32 × 0.8 = 0.256
(β 1 = 0.8 for f c ′ = 5ksi.) ω pw = 0.00413 228 = 0.188 < 0.256 φ = 0.9
5
3. Calculate the external factored moment due to dead and live loads:
Dead load = self − weight + additional dead load
= 0.388 + 0.9 = 1.29 K∕ft
Live load = 1.1K∕ft
U = 1.2D + 1.6L
M u = (48)2 [1.2(1.29)+1.6(1.1)] = 952.7K⋅ ft
8
This external moment is less than the factored moment capacity of the section of 1132.7 K ⋅
ft; therefore, the section is adequate.
4. The cracking moment (Eq. 19.48) is
M cr = I (
7.5λ √ f c ′ + F y b A + (Fe) y )
b
I
From Example 19.4 F = 306.2 K, A = 372 in. 2 , e = 14.5 in., y b = 20.8 in., I = 66, 862 in. 4 ,
f c ′ = 5ksi,and7.5λ√ f c ′ = 7.55000 = 530 psi.
[
66, 862
M cr = 0.53 + 306.2
]
(306.2) (14.5)(20.8)
+
20.8 372 66, 862
Check that 1.2 M cr ≤ φM n .
= 8790 K ⋅ in. = 732.5K⋅ ft
1.2M cr = 1.2(732.5) =879 K ⋅ ft
This value is less than φM n = 1132.7 K⋅ft. Thus, the beam is adequate against cracking.
19.7 DEFLECTION
Deflection of a point in a beam is the total movement of the point, either downward or upward,
due to the application of load on that beam. In a simply supported prestressed concrete beam, the
prestressing force is usually applied below the centroid of the section, causing an upward deflection