Civil Engineering Reference
In-Depth Information
If
V
u
is divided by the effective beam area
b
w
d
, the result is what is called an
average
shearing stress
. This stress is not equal to the diagonal tension stress but merely serves as
an indicator of its magnitude. Should this indicator exceed a certain value, shear or web
reinforcing is considered necessary. In the ACI Code the basic shear equations are pre-
sented in terms of shear forces and not shear stresses. In other words, the average shear
stresses described in this paragraph are multiplied by the effective beam areas to obtain
total shear forces.
For this discussion
V
n
is considered to be the nominal or theoretical shear strength of
a member. This strength is provided by the concrete and by the shear reinforcement.
V
n
V
c
V
s
The design shear strength of a member,
V
n
, is equal to
V
c
plus
V
s
, which must at
least equal the factored shear force to be taken,
V
u
:
V
u
V
c
V
s
The shear strength provided
by
the concrete,
V
c
, is considered to equal an average
shear stress strength (normally ) times the effective cross-sectional area of the mem-
ber,
b
w
d
, where
b
w
is the width of a rectangular beam or of the web of a T beam or an I
beam.
c
2
f
c
b
w
d
V
c
2
f
(ACI Equation 11-3)
c
Or in SI units with
f
in MPa
c
f
V
c
b
w
d
6
Beam tests have shown some interesting facts about the occurrence of cracks at dif-
ferent average shear stress values. For instance, where large moments occur even though
appropriate longitudinal steel has been selected, extensive flexural cracks will be evident.
As a result, the uncracked area of the beam cross s
ec
tion will be greatly reduced and the
nominal shear strength
V
c
can be as low as . On the other hand, in regions
where the moment is small, the cross section will be either uncracked or slightly cracked
and a large portion of the cr
oss
section is available to resist shear. For such a case, tests
show that a
V
c
of about can be developed before shear failure occurs.
1
Based on this information, the Code (11.3.1.1) suggests that, conservatively,
V
c
(the
she
ar
force that the concrete can resist without web reinforcing) can go as high as
. As an alternative, the following shear force (from Section 11.3.2.1 of the Code)
may be used, which takes into account the effects of the longitudinal reinforcing and the
moment and shear magnitudes. This value must be calculated separately for each point
being considered in the beam.
c
b
w
d
1.9
f
c
b
w
d
3.5
f
c
b
w
d
2
f
c
2500
w
V
u
d
M
u
c
b
w
d
V
c
1.9
f
b
w
d
3.5
f
(ACI Equation 11-5)
1
ACI-ASCE Committee 326, 1962, “Shear and Diagonal Tension,” part 2,
Journal ACI
, 59, p. 277.
