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B/H= 0.5, 0.75, 1.0
ρ
s
= 0., 0.002, 0.01
f
c
= 30, 40, 50 MPa
M
n
/
M
n
= 1.1, 1.2, 1.3, …
ρ
s
= 0.0045, 0.00875, 0.013
f
y
= 350, 450, 550 MPa
GFRP
E
GFRP
= 45 GPa
f
GFRP
= 400 MPa
CFRP
E
CFRP
= 400 GPa
f
CFRP
= 3000 MPa
Determine
C
n
and ρ
f
FIGURE 5.14
Variation of design variables in the parametric study of Rasheed and Motto
(2010). (First published by Engineers Australia. Reprinted with permission.)
reinforcement force ratio is
λ=
ρ
ρ
0.9
f
f
fu
M
M
M
M
, while the strengthening ratio is
=
n
n
u
f
sy
if
φ= :
0.9
M
M
n
λ=
0.9626
−
0.976
(5.85)
n
In a follow-up study, Saqan, Rasheed, and Hawileh (2013) derived a similar statis-
tically accurate linear relationship while considering Ψ
f
= 0.85, as per ACI 440.2R-08.
However, their definition of the ρ=
f
A
bd
differs from the definition of the same vari-
f
f
A
bd
able
ρ=
f
f
in this textbook. Accordingly, the same relationship, developed by
Saqan, Rasheed, and Hawileh (2013), is rederived here using the latter definition
of the FRP reinforcement ratio. This linear relationship correlates 177 data points
of beam section designs performed in accordance to ACI 440.2R-08 and yield-
ing an
R
2
= 0.9973, as seen in Figure 5.15. The
x
-axis h
a
fs the FRP effective force
(
)
×
ρ
ρ
f
f
d
d
f
, while the
y
-axis has the strengthening ratio
M
n
:
ff
sy
M
M
ρ
ρ
×+
f
f
d
d
n
f
f
f
=
0.7815
1
(5.86)
n
sy
To prove the accuracy of this linear relationship, the formulation developed by
Saqan, Rasheed, and Hawileh (2013) will be followed here.
Moment equilibrium:
−
β
c
−
β
c
1
1
MAfd
=
+ψ
Af d
(5.87)
n
s
y
f
f
f
f
2
2
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