Structural Concrete - Hassoun

140 Chapter 3 Flexural Analysis of Reinforced Concrete Beams
the steel ratio, ρ, in addition to the maximum and minimum values for ρ and R u . When the section
dimensions are known, R u is calculated; then ρ and A s are determined from tables. The values in
the tables are calculated based on tension-controlled sections with φ = 0.9. If φ is less than 0.9
(transition region), the values of R u should be multiplied by the ratio φ/0.9.
φM n = R u bd 2
R u = M u
bd 2 = φρf y
(
1 − ρf y
1.7f ′ c
A s = ρbd and ρ = A s
bd
For any given value of ρ, R u can be determined from tables. Then φ M n can be calculated. The
values of ρ and R u range between a minimum value of R u (min) when ρ minimum is used to a
maximum value as limited by the ACI Code, when ρ is equal to ρ (max), for tension controlled
sections with φ = 0.9.
The use of tables will reduce the manual calculation time. The next example explains the use
of tables.
Example 3.14
Calculate the design moment strength of the section shown in Example 3.2, Fig. 3.14, using tables. Use
b = 12 in., d = 21 in., f c ′ = 3ksi, f y = 60ksi, and three no. 9 bars.
Solution
1. Using three no. 9 bars, A s = 3.0 in. 2 , ρ = A s /bd = 3.0/(12 × 21) = 0.0119. From Table 3.2,
ρ max = 0.01356 >ρused. Therefore, φ = 0.9, and it is a tension-controlled section.
From Table A1, for ρ = 0.0119, I150 = 3ksiandf y = 60 ksi, get R u = 553 psi (by interpolation).
2. Calculate φ M n = R u bd 2 = 0.553 (12)(21) 2 = 2926 K⋅in. = 243.8 K⋅ft.
)
3.20 ADDITIONAL EXAMPLES
The following examples are introduced to enhance the understanding of the analysis and design
applications.
Example 3.15
Calculate the design moment strength of the precast concrete section shown in Fig. 3.39 using f c ′ = 4ksi
and f y = 60 ksi.
Figure 3.39 Example 3.15.