Structural Concrete - Hassoun

106 Chapter 3 Flexural Analysis of Reinforced Concrete Beams
The area of steel reinforcement to provide a balanced condition is
A sb = ρ b bd = 0.0285 × 16 × 25.5 = 11.63 in. 2
b. For a tension-controlled section, ρ max = 0.63375, ρ b = 0.63375 × 0.0285 = 0.01806 or, from
Eq. 3.32,
A s max = ρ max bd = 0.01806 × 16 × 25.5 = 7.37 in. 2 for φ = 0.9
For the transition region, ρ max t = 0.724 ρ b = 0.0206. For the case of ε t = 0.004, A s max t =
0.0206(16 × 25.5) = 8.41 in. 2 for φ = 0.817
c. The depth of the equivalent compressive block using A s max is
a max = A s max f y
0.85f ′ c b
7.37 × 60
= = 8.13 in.
0.85 × 4 × 16
The distance from the top fibers to the neutral axis is c = a/β 1 . Because f c ′ = 4000 psi, β 1 = 0.85;
thus,
c = 8.13 = 9.56 in.
0.85
or c/d = 0.375 and c = 0.375(25.5) = 9.56 in.
Example 3.2
Determine the design moment strength and the position of the neutral axis of the rectangular section
showninFig.3.14ifthereinforcementusedisthreeno.9bars.Given:f c ′ = 3ksiandf y = 60 ksi.
ε c
ksi
21
18.06
ε t = 0.0061
Figure 3.14 Example 3.2.
Solution
1. Theareaofthreeno.9barsis3.0in. 2
ρ = A s
bd = 3.0
12 × 21 = 0.0119