Civil Engineering Reference
In-Depth Information
τ
t
/ζ
f
c
) and steel stress indices (
ω
,ω
t
)
Figure 2.4
Circular relationship between shear stress ratio (
2.1.2.4 Design Limitations
f
c
The balanced condition can also be expressed graphically by a semicircular curve in a
τ
ty
/
ζ
vs
ω
t
diagram, as shown in Figure 2.4. This diagram is derived in the following manner.
Squaring both sides of Equation (2.27) gives
τ
ty
ζ
2
t
+
ω
−
ω
t
=
0
(2.31)
f
c
5
2
Adding 0
.
to both sides of Equation (2.31) results in
τ
ty
ζ
2
5)
2
5
2
+
(
ω
t
−
0
.
=
0
.
(2.32)
f
c
f
c
When
τ
ty
/
ζ
is plotted against
ω
t
in Figure 2.4, Equation (2.32) represents a circle with
radius 0.5 and center located on the
ω
t
axis at
ω
t
=
0.5. This circle gives the nondimensional
relationship between the shear stress
τ
t
and the transverse steel stress,
ρ
t
f
ty
. When the
longitudinal steel is chosen to satisfy the balanced condition,
ω
+
ω
t
=
1, an axis pointing to
the left is also drawn for
ω
.
Substituting
ω
from Equation (2.23) into Equation (2.24) gives
f
c
)
(
τ
ty
/ζ
tan
α
r
=
(2.33)
ω
t
α
r
represents the slope of a straight line through
the origin in Figure 2.4. In the ACI Code, the angle
Equation (2.33) shows that the angle
α
r
is limited to a range of 30
◦
to 60
◦
,
represented by the straight line OC and OA. At these limits, the maximum shear stress ratio
τ
ty
/
f
c
is 0.433.
Design of reinforcement within the semicircle will give an under-reinforced element, while
the region outside the semicircle represents over-reinforcement. Design within the fan-shaped
ζ
