Structural Concrete - Hassoun

162 Chapter 4 Flexural Design of Reinforced Concrete Beams
2. The ACI Code indicates that for sections in the transition zone, φ 0.004
0.679
Or, alternatively, calculate a = 5.08 ×60/(0.85 ×4 ×15) = 5.976, c = a/0.85 = 7.03, d t = d = 17.5 in.
Then ε t = 0.003(d t − c)/c = 0.004467. Calculate
( ) 250
φ = 0.65 +(ε t − 0.002) = 0.856
3
φM n = 0.856(368.6) =315.4K ⋅ ft
3. It can be noticed that, despite an additional amount of steel, 5.08 − 4.67 = 0.41 in. 2 (or about 9%),
the design moment strength remained the same. This is because the strength reduction factor, φ,
was decreased. Therefore, the design of sections within the tension-controlled zone with φ = 0.9
gives a more economical design based on the ACI Code limitations.
4.4 RECTANGULAR SECTIONS WITH COMPRESSION REINFORCEMENT
A singly reinforced section has its moment strength when ρ max of steel is used. If the applied factored
moment is greater than the internal moment strength, as in the case of a limited cross section,
a doubly reinforced section may be used, adding steel bars in both the compression and the tension
zones. Compression steel will provide compressive force in addition to the compressive force in
the concrete area.
4.4.1 Assuming One Row of Tension Bars
The procedure for designing a rectangular section with compression steel when M u , f ′ c, b, d, and d ′
are given can be summarized as follows:
1. Calculate the balanced and the maximum steel ratio, ρ max , using Eqs. 3.18 and Eqs. 3.31:
f ′ ( )
c 87
ρ b = 0.85β 1
f y 87 + f y
Calculate A s,max = A s1 = ρ max bd (maximum steel area as singly reinforced).
2. Calculate R u,max using ρ max (φ = 0.9):
(
R u,max = φρ max f y 1 − ρ )
maxf y
1.7f c
′
(R u,max can be obtained from the tables in Appendix A or Table 4.1.)
3. Calculate the moment strength of the section, M u1 , as singly reinforced using ρ max and R u,max :
M u1 = R u,max bd 2