Structural Concrete - Hassoun

230 Chapter 6 Deflection and Control of Cracking
6.2.4 Moment of Inertia
The moment of inertia, in addition to the modulus of elasticity, determines the stiffness of the
flexural member. Under small loads, the produced maximum moment will be small, and the tension
stresses at the extreme tension fibers will be less than the modulus of rupture of concrete; in this
case, the gross transformed cracked section will be effective in providing the rigidity. At working
loads or higher, flexural tension cracks are formed. At the cracked section, the position of the neutral
axis is high, whereas at sections midway between cracks along the beam, the position of the neutral
axis is lower (nearer to the tension steel). In both locations only the transformed cracked sections
are effective in determining the stiffness of the member; therefore, the effective moment of inertia
varies considerably along the span. At maximum bending moment, the concrete is cracked, and its
portion in the tension zone is neglected in the calculations of moment of inertia. Near the points
of inflection the stresses are low, and the entire section may be uncracked. For this situation and in
the case of beams with variable depth, exact solutions are complicated.
Figure 6.2a shows the load–deflection curve of a concrete beam tested to failure. The beam
is a simply supported 17-ft span and loaded by two concentrated loads 5 ft apart, symmetrical
about the centerline. The beam was subjected to two cycles of loading: In the first (curve cy 1),
the load–deflection curve was a straight line up to a load P = 1.7 K when cracks started to occur
in the beam. Line a represents the load–deflection relationship using a moment of inertia for the
uncracked transformed section. It can be seen that the actual deflection of the beam under loads less
than the cracking load, based on a homogeneous uncracked section, is very close to the calculated
deflection (line a). Curve cy 1 represents the actual deflection curve when the load is increased
to about one-half the maximum load. The slope of the curve, at any level of load, is less than the
slope of line a because cracks developed, and the cracked part of the concrete section reduces
the stiffness of the beam. The load was then released, and a residual deflection was observed at
midspan. Once cracks developed, the assumption of uncracked section behavior under small loads
did not hold.
In the second cycle of loading, the deflection (curve c) increased at a rate greater than that of
line a, because the resistance of the concrete tension fibers was lost. When the load was increased,
the load–deflection relationship was represented by curve cy 2. If the load in the first cycle is
increased up to the maximum load, curve cy 1 will take the path cy 2 at about 0.6 of the maximum
load. Curve c represents the actual behavior of the beam for any additional loading or unloading
cycles.
Line b represents the load–deflection relationship based on a cracked transformed section; it
can be seen that the deflection calculated on that basis differs from the actual deflection. Figure 6.2c
shows the variation of the beam stiffness EI with an increase in moment. ACI Code, Section
24.2.3.5, presents an equation to determine the effective moment of inertia used in calculating
deflection in flexural members. The effective moment of inertia given by the ACI Code (Eq.
24.2.3.5a) is based on the expression proposed by Branson [3] and calculated as follows:
I e =
(
Mcr
M a
) 3
I g +
[
1 −
(
Mcr
M a
) 3
]
I cr ≤ I g (6.5)
where I e is the effective moment of inertia, the cracking moment is given as
( ) fr l g
M cr =
Y t
(6.6)