Structural Concrete - Hassoun

3.9 Singly Reinforced Rectangular Section in Bending 107
2. ρ max = 0.01356 >ρ, tension-controlled section, φ = 0.9 or check ε t :
a =
A s f y
0.85f c ′ b = 3(60)
= 5.88 in.
0.85 × 3 × 12
c =
a = 6.92 in.
0.85
or
d t = d = 21 in.
( ) 21 − 6.92
ε t =
0.003
6.92
= 0.0061 > 0.005, φ = 0.9
c
d t
= 0.33 < 0.375
3. φM n = φA s f y
(
d − 1 2 a )
a =
A s f y
0.85f c ′ b = 3.0 × 60
0.85 × 3 × 12
(
φM n = 0.9 × 3.0 × 60
21 − 5.88
2
(OK)
= 5.88 in.
)
= 2926 K ⋅ in. = 243.8K⋅ ft
Discussion
In this example, the section is tension controlled, which implies that the steel will yield before the
concrete reaches its maximum strain. A simple check can be made from the strain diagram (Fig. 3.14).
From similar triangles,
ε c
=
c
ε y d − c
and
ε y = f y
E s
= 60
29,000 = 0.00207
6.92
ε c = × 0.00207 = 0.00102
21 − 6.92
which is much less than 0.003. Therefore, steel yields before concrete reaches its limiting strain of 0.003.
Example 3.3
Repeat Example 3.2 using three no. 10 bars as the tension steel (Fig. 3.15).
ε c
ksi
21
8.75
7.47
3.735
17.265”
228.6 k
= 0.04168 ε t
228.6 k
Figure 3.15 Example 3.3.