Flexural members - Reinforced Concrete with FRP Bars: Mechanics and Design - page 120

4.9.2.2 Elastic immediate deflections according to Bischoff
As demonstrated by Bischoff [28], Branson's equation overestimates mem-
ber stiffness when the I g / I cr ratio of the member is greater than about 3
or 4. This corresponds to most FRP reinforced concrete beams, which typi-
cally have an I g / I cr ratio between 5 and 25. It is for this reason that past
research on deflection of FRP reinforced concrete beams has shown that
Branson's equation underestimates deflection, particularly for lightly rein-
forced members with a high I g / I cr ratio.
Bischoff [28] and, more recently, Bischoff and Gross [29] proposed an
alternative section-based expression for the effective moment of inertia I e
that works equally well for both steel and FRP reinforced concrete mem-
bers without the need for empirical correction factors. Branson's original
expression represents a weighted average of the uncracked and cracked
member stiffness ( EI ), while Bischoff's proposed approach represents a
weighted average of flexibility (1/ EI ). The approach using a weighted aver-
age of flexibility represents better the deflection response of members with
discrete cracks along their length [30].
The section-based expression proposed by Bischoff [28] is modified to
include an additional factor γ to account for the variation in stiffness along
the length of the member. The modified expression for the effective moment
of inertia is given by

I

cr

I = ≤ I (4.83)

e g

2

−γ

M
M

I
I

cr cr

1 1 −

a g

This approach provides reasonable estimates of deflection for FRP RC
beams and one-way slabs. The factor γ is dependent on load and bound-
ary conditions and accounts for the length of the uncracked regions of the
member and for the change in stiffness in the cracked regions. In lieu of
a more comprehensive analysis, the value γ = 1.72 − 0.72 ( M cr / M a ) is sug-
gested, which is the result from integrating the curvature over the length of
a beam with a uniformly distributed load.
Unless stiffness values are obtained by more comprehensive analysis,
immediate deflections can be computed with the effective moment of iner-
tia given by Equation (4.83) using the maximum service load moment M a
in the member.
When M a > M cr , the assumed M cr value has a significant effect on com-
puted values of deflection. A lower M cr can be used to account for the ten-
sile stresses that develop in the concrete from restraint to shrinkage [30].
The use of a reduced M cr also accounts for cases where the calculated