Civil Engineering Reference
In-Depth Information
Substituting Equation (5.59) into Equation (5.61),
(
)
0.9
f
M
fbd
d
d
−
β
c
d
f
c
d
f
d
d
−
β
c
fu
u
c
y
=
Q
1
+Ψ α
− ρ
1
1
f
s
2
φ
0.9
f
0.9
f
f
d
c
f
f
fu
fu
c
f
f
M
fbd
d
d
−
β
c
d
+Ψ
α
c
−
β
c
−
β
c
u
=
Q
1
1
−Ψ
Q
1
(5.62)
1
f
f
1
2
φ
d
d
d
c
f
f
f
f
f
where
f
f
d
d
y
Q
=ρ
(5.63)
1
s
cf
Recalling the α expression from Equations (2.20) and substituting the strain com-
patibility expression for ε
cf
,
2
2
max
max
α=
ε
ε
1
3
ε
ε
c
dc
ε
1
3(
c
dc
ε
cf
cf
fu
fu
(5.64)
−
=
−ε
−
(
)
−ε
)
c
c
f
c
f
c
Introducing the β expression for the centroid location of the Hognestad's parab-
ola, with respect to the top extreme fiber, and substituting the strain compatibility
expression for ε
cf
,
max
1
3 2(
c
dc
c
dc
ε
1
3 2
1
ε
fu
cf
−
−
−ε
)
ε
f
c
c
β=
=
(5.65)
−
ε
max
ε
cf
fu
1
−
3
ε
3(
−ε
)
c
f
c
Substituting Equations (5.64) and (5.65) into Equation (5.62) and rearranging
the terms,
2
3
4
5
c
d
c
d
c
d
c
d
c
d
3
(5.66)
A
+
B
+
D
+
E
+ +ε=
F
9
Q
0
2
c
f
f
f
f
f
where
(
)
3
2max
3
A
=− ε− εε +ε−Ψ
27
Q Q
3
3
Q
1
2
c
2
c
fu
1
c
f
(
)
(
)
(5.67)
3
2max
=
31
Q
−Ψ− ε− εε
9
Q
3
Q
1
f
2
c
2
cfu
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