Page 196 - Civil Engineering Formulas

132 CHAPTER FIVE

The shear strength V carried by the concrete alone should not exceed
c
2 f c b w d, where b is the width of the beam web and d, the depth of the centroid
w
of reinforcement. (As an alternative, the maximum value for V may be taken as
c
V u d
V c 1.9 f c 2500 w b w d 3.5 f c b w d (5.49)
M u
where A /b d and V and M are the shear and bending moment, respec-
w
s
u
u
w
tively, at the section considered, but M should not be less than V d.)
u
u
When V is larger than V , the excess shear has to be resisted by web rein-
c
u
forcement.
2
2
The area of steel required in vertical stirrups, in (mm ), per stirrup, with a
spacing s, in (mm), is
V s S
A v (5.50)
f y d
where f is the yield strength of the shear reinforcement and A is the area of
y
v
the stirrups cut by a horizontal plane. V should not exceed 8 f c b w d in sec-
s
tions with web reinforcement, and f should not exceed 60 ksi (413.7 MPa).
y
Where shear reinforcement is required and is placed perpendicular to the axis
of the member, it should not be spaced farther apart than 0.5d, or more than
24 in (609.6 mm) c to c. When V exceeds 4 f c b w d , however, the maximum
s
spacing should be limited to 0.25d.
Alternatively, for practical design, to indicate the stirrup spacing s for the
design shear V , stirrup area A , and geometry of the member b and d,
w
v
u
A v f y d
s (5.51)
V u 2 f c b w d
The area required when a single bar or a single group of parallel bars are all
bent up at the same distance from the support at angle with the longitudinal
axis of the member is
V s
A v (5.52)
f y sin
in which V should not exceed 3 f c b w d . A is the area cut by a plane normal to
v
s
the axis of the bars. The area required when a series of such bars are bent up at
different distances from the support or when inclined stirrups are used is
V s s
A v (5.53)
(sin cos )f y d
A minimum area of shear reinforcement is required in all members, except
slabs, footings, and joists or where V is less than 0.5V .
u
c