Civil Engineering Reference
In-Depth Information
The position of the neutral axis can be determined by taking the static moments of the
shaded areas (Figure 3.3d), about the centroidal axis (same as neutral axis):
1
2
b
(
kd
)
2
mA
s
(
kd
d
)
+
−
=
nA
s
(
d
−
kd
)
(3.28)
Dividing Equation (3.28) by
bd
2
ρ
=
A
s
/
and denoting
ρ
=
A
s
/
bd
and
bd
results in:
2
n
ρ
d
d
k
2
ρ
)
k
+
ρ
+
−
ρ
+
=
2(
n
m
m
0
(3.29)
Let
ρ
m
β
c
=
(3.30)
n
ρ
and solving Equation (3.29) by the quadratic equation formula gives:
1
d
d
k
=
(
n
ρ
)
2
(1
+
β
c
)
2
+
2
n
ρ
+
β
c
−
n
ρ
(1
+
β
c
)
(3.31)
Equation (3.31) determines the location of the neutral axis for doubly reinforced cracked
beams. For the special case of singly reinforced cracked beams,
β
c
=
0, and Equation (3.31)
degenerates into Equation (3.23).
The moment resistance of the cross-section
M
can be calculated from the stress diagram in
Figure 3.3(c) by taking moments about the tensile force:
f
c
k
1
bd
2
1
2
k
3
A
s
f
s
(
d
d
)
−
+
−
M
=
(3.32)
From Hooke's law and Bernoulli's strain compatibility, (Figure 3.3b), we can express the
compressive rebar stress
f
s
as
(
mE
c
)
d
d
ε
c
kd
−
m
(
E
c
ε
c
)
kd
−
f
s
E
s
ε
s
=
=
=
(3.33)
kd
kd
Substituting
f
s
from Equation (3.33),
A
s
=
ρ
bd
and
m
ρ
=
n
ρβ
c
into Equation (3.32)
gives:
bd
2
(
E
c
ε
c
)
1
2
k
1
k
1
d
d
d
d
k
3
ρβ
c
1
k
M
=
−
+
n
−
−
(3.34)
The curvature can be written according to Bernoulli's hypothesis (Figure 3.3b), as
ε
c
kd
φ
=
(3.35)
/φ
, the cracked moment of inertia
I
cr
for
doubly reinforced sections can be derived according to Equations (3.34) and (3.35):
Since the bending rigidity
E
c
I
cr
is defined as
M
bd
3
1
2
k
2
1
ρβ
c
k
1
k
3
d
d
d
d
I
cr
=
−
+
n
−
−
(3.36)
