Structural Concrete - Hassoun

3.9 Singly Reinforced Rectangular Section in Bending 103
For members subjected to flexure, the relationship between the steel ratio, ρ, was given in
Eq. 3.24:
or
ε t + 0.003 = 0.003 + f y∕E s
ρ∕ρ b
(3.24)
ρ
ρ b
= 0.003 + f y∕E s
0.003 + ε t
(3.29)
For f y = 60 ksi and E s = 29,000 ksi, f y /E s may be assumed to be 0.00207.
ρ
ρ b
= 0.00507
0.003 + ε t
(3.30)
The limit for tension to control is ε t ≥ 0.005 according to ACI. For ε t = 0.005, Eq. 3.30 becomes
ρ
= 0.005
ρ b 0.008 = 5 = 0.625 (3.30a)
8
or ρ ≤ 0.63375 ρ b for tension-controlled sections if ε t = 0.00507 = f y /E s . Both values can be used
for practical analysis and design. The small increase in ρ will slightly increase the moment capacity
of the section. For example, if f ′ c = 4ksiandf y = 60 ksi, ρ b = 0.0285 and ρ ≤ 0.01806 for tension
to control (as in the case of flexural members). The φ factor in this case is 0.9. This value is less
than ρ max = 0.75ρ b = 0.0214 allowed by the ACI Code for flexural members when φ = 0.9 can
be used.
Design of beams and other flexural members can be simplified using the limit of ε t = 0.005.
ρ
= 0.003 + f y∕E s
ρ b 0.008
In this case, ρ = ρ max = upper limit for tension-controlled sections.
( )
0.003 + fy ∕E s
ρ max =
ρ
0.008 b
(3.31)
(3.31a)
Note that when ρ used ≤ ρ max , tension controls and φ = 0.9. When ρ>ρ max , the section will be in
the transition region with φ