Civil Engineering Reference
In-Depth Information
14.5 Curvature and deflection of flexural members
In this section we consider the moment-mean curvature relationship for
members reinforced with FRP; the mean curvatures calculated at various
sections can be used to give de
ections in service (e.g. by the equations in
Appendix C). For simplicity, the subscript 'service' is omitted in Sections 14.5
to 14.8, which are concerned with de
fl
ections or curvatures in service condi-
tion. Section 7.4 presents equations that give the mean curvature,
fl
ψ
m
of a
member reinforced with non-prestressed steel bars subjected to bending
moment,
M
>
M
r
, with
M
r
being the moment that produces
rst cracking.
The equations are repeated here with the symbols adjusted for use when FRP
is employed:
fi
ψ
m
=
(
1
−
ζ
)
ψ
1
+
ζ
ψ
2
(14.10)
M
E
c
I
1
M
E
c
I
2
ψ
1
=
;
ψ
2
=
(14.11)
M
r
M
2
ζ
=
1
−
β
(14.12)
or
f
ct
σ
1max
2
ζ
=
1
−
β
(14.13)
where
I
1
is the second moment of areas of a transformed area consisting of
A
c
plus
A
f
, with
A
c
and
A
f
being the cross-sectional areas of concrete and FRP
(ignoring the bars in compression) and
α
E
f
/
E
c
; where
E
f
and
E
c
are the
moduli of elasticity of FRP and concrete, respectively;
f
ct
is tensile strength of
concrete;
α
=
σ
1 max
is stress at extreme
fi
bre in state 1, where cracking is ignored.
The coe
cient
β
replaces the product of
β
1
and
β
2
which account for the bond
quality and the e
=
0.5 is recommended. This recommendation is based on comparisons of pub-
lished experimental de
ff
ect of repetitious loading; with FRP bars a value of
β
fl
ections of numerous beams reinforced with di
ff
erent
FRP types with the values of de
fl
ections calculated from curvatures using the
above equations.
2
Equation (14.10) can be rewritten in the form:
M
EI
em
ψ
m
=
(14.14)
where
I
em
is an e
ective second moment of area for use in calculation of mean
curvature in members subjected to bending moment, without axial force,
ff
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