Page 194 - Civil Engineering Formulas

130 CHAPTER FIVE

DEFLECTION COMPUTATIONS AND CRITERIA FOR
CONCRETE BEAMS

The assumptions of working-stress theory may also be used for computing
deflections under service loads; that is, elastic-theory deflection formulas may be
used for reinforced-concrete beams. In these formulas, the effective moment of
inertia I is given by
c
3 M cr 3
M cr
I c I g 1 I (5.40)
cr I g
M a M a
4
4
where I moment of inertia of the gross concrete section, in (mm )
g
M cracking moment, lb.in (K.Nm)
cr
M moment for which deflection is being computed, lb . in (K . Nm)
a
4
4
I cracked concrete (transformed) section, in (mm )
cr
If y is taken as the distance from the centroidal axis of the gross section, neglect-
t
ing the reinforcement, to the extreme surface in tension, the cracking moment may
be computed from
f r I g
M cr (5.41)
y t
with the modulus of rupture of the concrete f r 7.5 2 c .
f
The deflections thus calculated are those assumed to occur immediately on
application of load. Additional long-time deflections can be estimated by multiply-
ing the immediate deflection by 2 when there is no compression reinforcement or
by 2 1.2A s /A s 0.6 , where A s is the area of compression reinforcement and A is
s
the area of tension reinforcement.
ULTIMATE-STRENGTH DESIGN OF RECTANGULAR BEAMS
WITH TENSION REINFORCEMENT ONLY
Generally, the area A of tension reinforcement in a reinforced-concrete beam is
s
represented by the ratio A /bd, where b is the beam width and d is the dis-
s
tance from extreme compression surface to the centroid of tension reinforce-
ment. At ultimate strength, the steel at a critical section of the beam is at its
yield strength f if the concrete does not fail in compression first. Total tension
y
in the steel then will be A f f bd. It is opposed, by an equal compressive
y
s y
force:
0.85 f c ba 0.85 f c b 1 c (5.42)
where f 28-day strength of the concrete, ksi (MPa)
c
a depth of the equivalent rectangular stress distribution
c distance from the extreme compression surface to the neutral axis
a constant
1