Page 197 - Civil Engineering Formulas

CONCRETE FORMULAS 133

Development of Tensile Reinforcement
At least one-third of the positive-moment reinforcement in simple beams and one-
fourth of the positive-moment reinforcement in continuous beams should extend
along the same face of the member into the support, in both cases, at least 6 in
(152.4 mm) into the support. At simple supports and at points of inflection, the
diameter of the reinforcement should be limited to a diameter such that the develop-
ment length l satisfies
d
M n
l d l a (5.54)
V u
where M computed flexural strength with all reinforcing steel at section
n
stressed to f y
V applied shear at section
u
l additional embedment length beyond inflection point or center of
a
support
At an inflection point, l is limited to a maximum of d, the depth of the centroid
a
of the reinforcement, or 12 times the reinforcement diameter.

Hooks on Bars
The basic development length for a hooked bar with f 60 ksi (413.7 MPa) is
y
defined as
1200d b
l hb (5.55)
f c
where d is the bar diameter, in (mm), and f c is the 28-day compressive
b
2
strength of the concrete, lb/in (MPa).
WORKING-STRESS DESIGN OF RECTANGULAR BEAMS
WITH TENSION REINFORCEMENT ONLY

From the assumption that stress varies across a beam section with the distance
from the neutral axis, it follows that
k
nf c
(5.56)
f s 1 k
where n modular ratio E /E
s c
E modulus of elasticity of steel reinforcement, ksi (MPa)
s
E modulus of elasticity of concrete, ksi (MPa)
c
f compressive stress in extreme surface of concrete, ksi (MPa)
c
f stress in steel, ksi (MPa)
s