Civil Engineering Reference
In-Depth Information
area OABC further satisfies the ACI provisions that limit the range of
α
r
. On the semicircular
curve which expresses the balanced condition, point B represents case (1) of the balanced
condition, while the arcs AB and BC give case (2) and case (3), respectively.
In summary, the design of reinforcement in 2-D elements to resist shear is subjected to two
limitations. First, the range of 30
◦
≤
α
r
≤
60
◦
is specified to prevent excessive cracking. This
purpose will be further discussed in Section 5.2.4, after the strain compatibility conditions are
introduced. Second, the balanced condition is imposed to ensure the yielding of steel in both
directions at failure. To achieve this purpose, a more rigorous method will be elaborated in
Section 5.4.7.
2.1.2.5 Minimum Reinforcement
Before cracking, an element subjected to shear stress
τ
t
will produce a principal tensile stress
=
σ
1
) and inclined at an angle of 45
◦
. At cracking,
σ
1
, which is equal in magnitude (
τ
t
σ
1
is assumed to reach the tensile strength of concrete
f
t
. Substituting
f
t
into Equations
τ
t
=
(2.12) and (2.13) we have
f
t
tan
ρ
f
=
α
r
(2.34)
f
t
cot
ρ
t
f
t
=
α
r
(2.35)
Summing Equations (2.34) and (2.35) gives
f
t
ρ
f
+
ρ
t
f
t
=
(2.36)
sin
α
r
cos
α
r
A minimum amount of reinforcement to ensure that the steel will not yield immediately
at cracking can be obtained by assuming the yielding of the steel
f
=
=
f
t
f
y
, and by
α
r
is in the vicinity of 45
◦
. Then Equation
noticing that sin
α
r
cos
α
r
is very close to 0.5 when
(2.36) becomes:
2
f
t
f
y
ρ
+
ρ
t
=
(2.37)
For a typical case when
f
y
=
413 MPa (60 000 psi) and
f
t
=
2
.
07 MPa (300 psi), Equation
(2.37) gives the minimum total reinforcement (
ρ
+
ρ
t
)
min
of 1%. If the total reinforcement is
equally distributed in the longitudinal and transverse directions, each direction will require a
minimum steel percentage of 0.5%.
In the ACI Code the minimum horizontal reinforcement in walls is specified to be in
the range of 0.20-0.25%, and the vertical reinforcement in the range of 0.12-0.15%. These
ACI requirements provide only about 1/4 to 1/2 the minimum reinforcement required by
Equation (2.37). Because most walls in use are designed with steel percentages close to
the ACI minimum requirement, it is obvious that these walls will fail in a brittle manner
upon cracking. The common notion that shear failure of walls is brittle stems primarily
from this practice of supplying insufficient amount of steel in the walls. If a minimum steel
requirement was provided according to Equation (2.37), then the wall is expected to behave in a
ductile manner.
