Flexural members - Reinforced Concrete with FRP Bars: Mechanics and Design - page 117

cracks, a reduced or cracked moment of inertia results at various loca-
tions, as shown in the graph of Figure 4.12. It is therefore obvious that, for
the purpose of simplifying design, an effective moment of inertia must be
determined.
Effective moment of inertia for steel RC. To express the transition
between the gross moment of inertia, I g , and the cracked moment of inertia,
I cr , ACI 318-11 proposes to use the following equation (Branson's [26]
equation) to calculate an effective moment of inertia, I e , at any given cross
section along the beam:

3 3

=

M
M

+−

M
M

cr cr

I I 1 I ≤ I (4.74)

e g cr g

a a

where M a is the unfactored moment at the section where the deflection is
calculated.
As M a increases, the effective moment of inertia, I e , decreases. When M a
is close to the nominal flexural capacity of the cross section, I e becomes
close to the fully cracked moment of inertia. The relationship between the
sectional rigidity and the applied bending moment is illustrated in its gen-
eral fashion in Figure 4.13.
A single value of effective moment of inertia for the flexural member, I e,av ,
can be derived when the variable I results from the variation in the extent of
concrete cracking. For the case of a continuous beam such as the one shown
in Figure 4.12, ACI Committee 435 [27] recommends the following equation:

I e, ave = 0.70 I e. midspan + 0.15 ( I e, sup1 + I e, sup2 ) (4.75)

Flexural rigidity based
on gross section

E c I g

Flexural rigidity based
on cracked section

E c I cr

M cr M n

Applied Bending Moment

Figure 4.13 Flexural rigidity versus applied bending moment relationship.