Civil Engineering Reference
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A
B
A
w
A
B
A
M
1
,neg
M
2
,neg
M
cr
M
cr
M
pos
I
I
g
I
e.ave
I
cr
0
ℓ
Figure 4.12
Moment of inertia for a continuous beam.
where
M
0
is the midspan moment,
M
1
and
M
2
are the end moments,
l
is the
span length,
E
c
is the modulus of elasticity of concrete, and
I
e
is the effective
moment of inertia.
Equation (4.73) is based on the assumptions of linear distribution of the
strains over the member cross section. For a flexural member subjected to
various load cases, the deflection for each case is calculated separately and
then algebraically added to the others to obtain the total.
Moment of inertia.
When computing deflections of a flexural member,
the magnitude of the flexural rigidity (defined as the product of the concrete
modulus of elasticity and the moment of inertia,
E
c
I
) must be determined.
E
c
I
is not constant throughout the length of the element as
I
(the moment
of inertia) is a section property that depends on the applied moment and
the resulting cracking.
The span of a continuous reinforced concrete beam subjected to a uni-
formly distributed load is shown in Figure 4.12. If the applied load is such
that bending moments do not exceed the cracking moment, the flexural
rigidity is constant throughout the beam and can be computed using the
uncracked or gross moment of inertia,
I
g
. As the load increases and the
induced bending moments exceed the cracking moment, cracking occurs
at the supports first and, eventually, at midspan. When a beam section
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