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-concretebeams. In these formulas, the effective moment of
inertiaIcis given by

(5.40)

where Igmoment of inertia of the gross concrete section, in^4 (mm^4 )
Mcrcracking moment, lb.in (K.Nm)
Mamoment for which deflection is being computed, lb.in (K.Nm)
Icrcracked concrete (transformed) section, in^4 (mm^4 )

Ifytis taken as the distance from the centroidal axis of thegross section, neglect-
ing the reinforcement, to the extreme surface in tension, the cracking moment may
be computed from

(5.41)

with the modulus of rupture of the concrete.
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 , where is the area of compression reinforcement and Asis
the area of tensionreinforcement.

ULTIMATE-STRENGTH DESIGN OF RECTANGULAR BEAMS
WITH TENSION REINFORCEMENT ONLY

Generally, the area Asof tension reinforcement in a reinforced-concrete beam is
represented by the ratio As/bd, where bis the beam width and dis the dis-
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 fyif the concrete does not fail in compression first. Total tension
in the steel then will be Asfyfybd. It is opposed, by an equal compressive
force:

(5.42)

wherefc28-day strength of the concrete, ksi (MPa)
adepth of the equivalent rectangular stress distribution
cdistance from the extreme compression surface to the neutral axis
 1 a constant

0.85fcba0.85fcb 1 c

2 1.2As/As0.6 As

fr7.5 2 fc

Mcr

frIg

yt

Ic

Mcr

Ma

3

Ig 1 

Mcr

Ma

3

IcrIg