Structural Concrete - Hassoun

3.11 Adequacy of Sections 109
accommodate a concrete strength higher than 5 ksi. The two minimum ratios are equal when f ′ c =
4440 psi. This indicates that
⎧
200
⎪
⎪ f y
ρ min = ⎨3 √ f c
⎪
′
⎪ f y
⎩
for f ′ c < 4500 psi
for f ′ c ≥ 4500 psi
For example, if f y = 60 ksi, ρ min = 0.00333 when f c ′ < 4500 psi, whereas ρ min = 0.00353 when
f c ′ = 5000 psi and 0.00387 when f c ′ = 6000 psi.
In the case of a rectangular section, use b = b w in the preceding expressions. For statically
determinate T-sections with the flange in tension, as in the case of cantilever beams, the value of
A s,min must be equal to or greater than following equation:
( √ )
3 f
′
c
A s,min = (x)(d) ≥ 200xd
f y f y
where
x = 2 b w or b f whichever is smaller
b w = width of web
b f = width of flange
For example if b f = 48 in., b w = 16 in., d = 20 in., f ′ c = 4000 psi, and f y = 60,000 psi, then
(
3 √ )
4000
A s,min =
(32)(20) =2.02 in. 2
60,000
200(32)(20)
= 2.13 in. 2 (controls)
60,000
A s,min = 2.13 in. 2
3.11 ADEQUACY OF SECTIONS
A given section is said to be adequate if the internal moment strength of the section is equal to
or greater than the externally applied factored moment, M u ,orφM n ≥ M u . The procedure can be
summarized as follows:
1. Calculate the external applied factored moment, M u .
M u = 1.2M D + 1.6M L
2. Calculate φM n for the basic singly reinforced section:
a. Check that ρ min ≤ ρ ≤ ρ max .