Structural Concrete - Hassoun

100 Chapter 3 Flexural Analysis of Reinforced Concrete Beams
d t
d
ε c = 0.003
0.85f c ′
a/2
a
a
C = 0.85fc ′ ab
C b
A s
T = A s f y
b
ε y = f y /E s
Figure 3.11
Rectangular balanced section.
Step 2.
From the equilibrium equation,
C = T 0.85f ′ cab = A s f y (3.15)
a =
A s f y
0.85f cb
′ (3.16)
Here, a is the depth of the compressive block, equal to β 1 c, where β 1 = 0.85 for f c ′ ≤ 4000 psi
(27.6 MPa) and decreases linearly by 0.05 per 1000 psi (6.9 MPa) for higher concrete strengths
(Fig. 3.9). Because the balanced steel reinforcement ratio is used,
ρ b = A s(balanced)
= A sb
(3.17)
bd bd
and substituting the value of A sb in Eq. 3.15,
0.85f cab ′ = f y ρ b bd
Therefore,
ρ b = 0.85f c
′
f y d
a = 0.85f c
′
f y d (β 1c b )
Substituting the value of c b from Eq. 3.14, the general equation of the balanced steel ratio becomes
f ′ ( )( )
c 87 dt
ρ b = 0.85β 1 (3.18)
f y 87 + f y d
The value of d t is equal to d when only one single layer of steel is provided.
Step 3. The internal nominal moment, M n , is calculated by multiplying either C or T by the distance
between them:
M n = C(d − z) =T(d − z) (3.13)
For a rectangular section, the distance z = a/2 as the line of application of the force C lies at the
center of gravity of the area ab, where
a =
A s f y
0.85f cb
′
M n = C
(d − 1 )
2 a = T
(d − 1 )
2 a