Civil Engineering Reference
In-Depth Information
Accordingly, Equation (5.75) may be further simplified by considering two constant
values of β for each range of ε
cf
specified for α. The constants are selected to be the
average of the two end values of each strain range (Rasheed and Motto 2010):
β=
β+β
0
0.0015
=
0.3447
0
≤ε<
0.0015
cf
2
β=
β β
0.0015
0.003
=
0.3864
0.0015
≤ε<
0.003
(5.76)
cf
2
Substituting into the moment equilibrium Equation (5.62) for the first range of ε
cf
values 0 ≤ ε
cf
< 0.0015, this leads to the following cubic equation:
2
3
c
d
c
d
c
d
(5.77)
A
+
B
+
D
+=
Q
0
3
3
3
2
f
f
f
where if ε
cf
< 0.0015,
A
=
0.3447
Q
(1
−Ψ− − Ψ
)
Q
0.0417
(5.78)
3
1
f
2
f
−Ψ− Ψε
max
(5.79)
B
= Ψ−
0.0561
0.3447
Q
(1
)
366.67
3
f
1
f
f
fu
max
D
=
126.3911
Ψ ε− Ψ
0.0144
(5.80)
3
f
fu
f
Similarly, if 0.0015 ≤ ε
cf
≤ 0.003, Equation (5.77) holds with:
−Ψ− − Ψ
(5.81)
A
=
0.3864
Q
(1
)
Q
0.4042
3
1
f
2
f
max
B
= Ψ−
0.5604
0.3864
Q
(1
−Ψ− Ψε
)
125
(5.82)
3
f
1
f
f
fu
max
=Ψε− Ψ
D
48.3
0.1562
(5.83)
3
f
fu
f
d
f
is determined, ρ
f
can be calculated from the force equilibrium:
Once
f
c
d
f
c
y
ρ=α
− ρ
(5.84)
f
s
0.9
f
0.9
f
fu
fu
5.4.3.4 Linear Regression Solution for Rupture Failure Mode
Alternatively, Rasheed and Motto (2010) derived a statistically accurate linear rela-
tionship between the strengthening ratio and the reinforcement force ratio based on
a parametric study of singly and doubly-reinforced strengthened rectangular sec-
tions, as seen in Figure 5.14. This parametric study had 516 data points yielding an
R
2
= 0.9994 which represents perfect linearity. However, Ψ
f
was not considered in
that equation. This linear equation may be equally used in analysis and design. The
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