Civil Engineering Reference
In-Depth Information
(
)
2
1
2
bh y
yh
+
(
)
0
Nbhy
=
+
σ
=
σ
[4.13a]
c
01
3
(
)
2
−
0
1
6
2
σ
⎛
⎝
⎞
⎠
h
∫
0
(
)
3
NdNb h
=
=
−
y
3
−
σ
[4.13b]
t
t
0
3
k
y
The internal moment
M
c
caused by
N
c
and the internal moment
M
t
caused
by
N
t
may be derived and expressed as follows:
(
)
2
(
)
1
6
bh y
+
h y
hy
2
−
(
)
(
)
0
0
Mb h
=−
+
y
2
h
−
y
σ
=
σ
[4.14a]
c
0
0
1
3
(
)
6
−
0
1
12
σ
h
⎡
⎢
⎤
⎥
∫
0
(
)
3
(
)
MydNb h
=
=
−
y
42
h
+−
y
3
hy
+
σ
[4.14b]
t
t
0
0
0
3
k
y
Based on the equilibrium condition of the section, the following equa-
tions may be established:
NN
c
=
[4.15a]
t
NL
MM M
=+=
4
[4.15b]
c
t
where
M
is the external bending moment.
From Eqs [4.12]-[4.15], the effective stress
˜
3
and degree of damage
D
of the extreme tension fi bre at the bottom of the beam may be derived
and expressed as follows:
σ
˜
1
,
σ
(
)
6
hy h y
hy
+
0
0
σ
1
=
k
[4.16]
(
)
3
−
0
hy
hy
−
6
0
σ
3
=
k
[4.17]
(
)
2
−
0
==
−
−
σ
3
hy
hy
6
0
D
[4.18]
(
)
2
k
0
As cracks form in the tensile zone of a concrete beam, failure occurs
and the load bearing capacity of the beam will depend on the effective
tensile stress
˜
3
. When the effective tension stress of concrete
˜
3
reaches
σ
σ
k
⎛
⎝
⎞
⎠
the critical cracking stress
σ
σσ
==
, then a beam without
cr
3
cr
2
reinforcement will fail. From Eqs [4.13] and [4.14], the displacement of
IGNA (
y
0
)
cr
and
N
cr
may also be described as follows:
y
()
=− .
0 102
h
[4.19]
0
cr
0 944
.
kbh
L
2
N
=
[4.20]
cr
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