Civil Engineering Reference
In-Depth Information
I
1
I
2
I
em
=
(14.15)
M
r
M
2
I
2
+
1
−
0.5
(
I
1
−
I
2
)
For a cracked member subjected to a normal force
N
combined with
moment
M
, the mean axial strain
ψ
m
can be
calculated by Equations (7.36) and (7.40). Validity of these equations, again
with
ε
om
and the mean curvature
β
1
β
2
replaced by
β
=
0.5, for prestressed members using carbon FRP has
ed
3
by comparison with published experimental data of several
been veri
fi
beams.
14.6 Relationship between deflection, mean
curvature and strain in reinforcement
Consider a straight non-prestressed concrete member having steel or FRP
reinforcement. The de
fl
ection,
D
centre
at mid-length section varies almost lin-
early with the strain
ε
f
in the reinforcement at the same section. The
symbol
D
centre
represents the transverse de
ε
s
or
ection measured from the chord
joining the two ends of a continuous or simply-supported member. The linear
relationship of
D
centre
versus
fl
ε
f
is demonstrated below for a simply supported
beam reinforced with FRP and will be used in the following section in
developing an equation for recommended span to thickness ratio.
Using virtual work, the de
fl
ection
D
centre
can be expressed as:
1
2
l
/2
1
l
l/2
ψ
D
centre
=
ψ
xdx
+
2
(
l
−
x
)
dx
(14.16)
0
This is a geometric relationship applicable for any straight member. Here
the symbol
ψ
stands for
ψ
m
in the cracked part of the length and for
ψ
1
in the
uncracked part;
l
is length of member. When
ψ
is assumed to vary as a
second-degree parabola between
ψ
end1
,
ψ
centre
and
ψ
end 2
(curvature at the two
ends and at mid-length), Equation (14.16) gives:
l
2
D
centre
=
96
(
ψ
end 1
+
10
ψ
centre
+
ψ
end 2
)
(14.17)
Equation (14.16) is employed to calculate
D
centre
for a simple beam
reinforced with glass FRP and having the cross-section shown in Fig. 14.2(a).
The integrals in the equation are evaluated numerically assuming
to vary
linearly over short segments of length 0.025
l
. Figure 14.2(b) is a plot of the
de
ψ
ψ
m centre
at the centre of a simple beam
subjected to uniform load, whose intensity
q
is varied between zero and 4
q
r
;
where
q
r
is the value just su
fl
ection,
D
centre
versus the curvature,
cient to produce cracking at mid-span section.
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