Civil Engineering Reference
In-Depth Information
7.3.3 i
nteraCtion
D
iagramS
For
r
eCtangular
C
olumnS
Since the derivation of the concrete contribution of force and moment equations
for points B and C requires the integration of simpler integrands with a constant
section width, these expressions will be evaluated analytically in closed form by
integrating the stress-strain expressions perpendicular to the centroidal
x
-axis.
The addition of the longitudinal steel bar contribution will have to be accounted
for numerically anyway. However, the steel bar contribution is also easy enough
to consider by hand.
7.3.3.1 Contribution of Concrete
Determining the pure axial compression point A is straightforward. To develop the
equations for determining points B and C on the interaction diagram, the following
expressions are derived in closed form:
c
y
c
t
(
)
2
φ−
−
EE
f
∫
∫
∫
(
)
(
)
(
)
2
c
2
Pbfd
=
=
bEy
φ
yd bf
+ +φ
Ey
d
cn
cy
cs
s
y
c
2
s
y
4
c
0
0
y
t
y
c
(
)
2
t
2
3
2
bE
y
EE
f
−
y
y
c
2
2
=φ −
b
φ
+ +φ
bf ybE
sc
s
c
2
s
2
4
3
2
c
y
0
t
(
)
2
E
yb
EE
f
−
b
E
b
E
c
2
c
2
23
2
2
2
2
=φ −
b
φ+ +
y fc
c
φ −
bfy
−
φ
y
s
t
st
c
s
ct
st
2
12
2
2
c
while the section curvature
φ=
ε
ccu
, the
P
cn
expression may be written as
s
c
2
(
)
2
EE
f
−
ε
yb
EE
+
−ε
bE
c
2
ccu
c
2
ccu
2
3
2
P
=−
b
y fy
−
+ +ε
bf c
c
cn
t
t
c
t
c
ccu
12
c
2
c
2
c
(7.85)
Similarly, the expression for
M
n
can be derived as follows:
c
c
h
fbdy
h
∫
∫
Mf bd
=
yhc
+ −− =
+
−
c
cn
c
y
c
y
2
2
0
0
y
c
t
(
)
2
EE
f
−
h
h
∫
∫
[
]
c
2
22
=
bEy
φ −
φ
yd y
+ −+ +φ +−
cbf
Eyd
y
c
cs
s
y
c
2
s
y
4
2
2
c
0
y
t
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