Structural Concrete - Hassoun

172 Chapter 4 Flexural Design of Reinforced Concrete Beams
42"
14" 14"
18.5"
in.
A 2
s A s1
14" 14"
A st = 3.57 in 2
Figure 4.9 Analysis of Example 4.8.
3. Find the portion of the design moment taken by the overhanging portions of the flange (Fig. 4.9).
First calculate the area of steel required to develop a tension force balancing the compressive force
in the projecting portions of the flange:
A sf = 0.85f c ′(b − b w )t 0.85 × 3 ×(42 − 14)×3
= = 3.57in 2 .
f y 60
Here, φ M n = M u1 + M u2 , that is, the sum of the design moment of the web and the design
moment of the flanges:
(
M u2 = φA sf f y d − 1 )
2 t (
= 0.9 × 3.57 × 60 18.5 − 3 )
= 3277K ⋅ in.
2
4. Calculate the design moment of the web (as a singly reinforced rectangular section):
M u1 = M u − M u2 = 5080 − 3277 = 1803K ⋅ in.
R u =
M u1 1, 803, 000
=
(b w d 2 ) 14 ×(18.5) = 376psi
2
From Eq. 4.2 or the tables in Appendix A, for R u = 376 psi, ρ 1 = 0.0077:
A s1 = ρ 1 b w d = 0.0077(14)(18.5) =1.99in. 2
Total A s = A sf + A s1 = 3.57 + 1.99 = 5.56in. 2 (uses ix no.9 bars in two rows)
Total h 5= 18.5 + 3.5 = 22 in. Calculate A s,max for T-sections using Eq. 3.72:
MaxA s = 7.02in. 2 > 5.56in. 2
5. Check ε t : a = 1.99 ×60/(0.85 ×3 ×14) = 3.34 in., c = 3.93 in., and d t = 19.5 in. Then ε t = 0.003
(d t − c)/c = 0.0119 > 0.005, tension-controlled section (φ = 0.9).
Example 4.9
In a slab–beam system, the flange width was determined to be 48 in., the web width was b w = 16 in., and
the slab thickness was t = 4 in. (Fig. 4.10). Design a T-section to resist an external factored moment of
M u = 812 K ⋅ ft. Use f c ′ = 3ksi and f y = 60 ksi.
Solution
1. Because the effective depth is not given, let a = t and calculate A sft for the whole flange:
A sft = 0.85f c ′ bt = 0.85(3)(48)(4) = 8.16in. 2
f y 60