Page 195 - Civil Engineering Formulas

CONCRETE FORMULAS 131

Equating the compression and tension at the critical section yields
pf y
c d (5.43)
0.85 1 f c
The criterion for compression failure is that the maximum strain in the con-
crete equals 0.003 in/in (0.076 mm/mm). In that case,
0.003
c d (5.44)
f s /E s 0.003
where f steel stress, ksi (MPa)
s
E modulus of elasticity of steel
s
29,000 ksi (199.9 GPa)
Balanced Reinforcing

Under balanced conditions, the concrete reaches its maximum strain of 0.003
when the steel reaches its yield strength f . This determines the steel ratio for
y
balanced conditions:
0.85 1 f c 87,000
b (5.45)
f y 87,000 f y

Moment Capacity

For such underreinforced beams, the bending-moment capacity of ultimate
strength is
2
M u 0.90[bd f c (1 0.59 )] (5.46)

0.90 A s f y d a 2 (5.47)

where f y /f c and a A s f y /0.85f c .

Shear Reinforcement
The ultimate shear capacity V of a section of a beam equals the sum of the
n
nominal shear strength of the concrete V and the nominal shear strength pro-
c
vided by the reinforcement V ; that is, V V V . The factored shear force V
s n c s u
on a section should not exceed
V n (V c V s ) (5.48)
where capacity reduction factor (0.85 for shear and torsion). Except for
brackets and other short cantilevers, the section for maximum shear may be
taken at a distance equal to d from the face of the support.