Structural Concrete - Hassoun

232 Chapter 6 Deflection and Control of Cracking
and the modulus of rupture of concrete as
and
f r = 7.5λ √ f ′ cpsi (0.623λ √ f ′ cMPa) (6.7)
M a = maximum unfactored moment in member at stage for which deflection is being computed
I g = moment of inertia of gross concrete section about centroidal axis, neglecting reinforcement
I cr = moment of inertia of cracked transformed section
= distance from centroidal axis of cross section, neglecting steel, to tension face
Y t
The following limitations are specified by the code:
1. For continuous spans, the effective moment of inertia may be taken as the average of the
moment of inertia of the critical positive- and negative-moment sections.
2. For prismatic members, I e may be taken as the value obtained from Eq. 6.5 at midspan for
simple and continuous spans and at the support section for cantilevers (ACI Code, Section
24.2.3.6 and 24.2.3.7).
Note that I e , as computed by Eq. 6.5, provides a transition between the upper and lower
bounds of the gross moment of inertia, I g , and the cracked moment of inertia, I cr , as a function
of the level of M cr /M a . Heavily reinforced concrete members may have an effective moment
of inertia, I e , very close to that of a cracked section, I cr , whereas flanged members may have
an effective moment of inertia close to the gross moment of inertia, I g .
3. For continuous beams, an approximate value of the average I e for prismatic or nonprismatic
members for somewhat improved results is as follows: For beams with both ends continuous,
For beams with one end continuous,
AverageI e = 0.70I m + 0.15(I e1 + I e2 ) (6.8)
AverageI e = 0.85I m + 0.15(I con ) (6.9)
where I m is the midspan I e , I e1 , I e 2 = I e at beam ends, and I con = I e at the continuous
end. Also, I e may be taken as the average value of the I e ’s at the critical positive- and
negative-moment sections. Moment envelopes should be used in computing both positive
and negative values of I e . In the case of a beam subjected to a single heavy concentrated
load, only the midspan I e should be used.
6.2.5 Properties of Sections
To determine the moment of inertia of the gross and cracked sections, it is necessary to calculate
the distance from the compression fibers to the neutral axis (x or kd).
1. Gross moment of inertia, I g (neglect all steel in the section):
a. For a rectangular section of width b and a total depth h, I g = bh 3 /12.
b. For a T-section, flange width b, web width b w , and flange thickness t, calculate y, the
distance to the centroidal axis from the top of the flange:
y = (bt2 ∕2)+b w (h − t)[(h + t)∕2]
bt + b w (h − t)
(6.10)