Civil Engineering Reference
In-Depth Information
Balanced Condition
The criterion for the balanced condition in the plasticity truss model was defined by
Nielsen and Braestrup (1975). Adding Equations (2.15) and (2.16) and noticing that
sin
2
cos
2
α
r
+
α
r
=
1gives:
ρ
f
+
ρ
t
f
t
=
σ
d
(2.18)
At the balanced condition both the steel yields (
f
=
f
y
,
f
t
=
f
ty
) while the concrete
f
c
. Equation (2.18) then becomes
crushes at an effective stress of
ζ
f
c
ρ
f
y
+
ρ
t
f
ty
=
ζ
(2.19)
Define
ρ
f
y
ζ
ω
=
=
longitudinal reinforcement index
f
c
ρ
t
f
ty
ζ
ω
t
=
=
transverse reinforcement index
f
c
The balanced condition Equation (2.19), can be written in a nondimensional form:
ω
+
ω
t
=
1
(2.20)
Under-reinforced Elements
When
ω
+
ω
t
<
1, we have an under-reinforced element where both the steel yields (
f
=
f
y
,
f
t
=
f
ty
) before the crushing of concrete. The shear stress
τ
t
becomes
τ
ty
. Substituting
sin
2
α
r
from Equation (2.15) and cos
2
α
r
from Equation (2.16) into Equation (2.17) results in
(
τ
ty
=
ρ
f
y
)(
ρ
t
f
ty
)
(2.21)
α
r
for this under-reinforced element can be obtained by dividing Equation (2.15)
by Equation (2.16):
The angle
ρ
y
f
y
ρ
ty
f
ty
tan
α
r
=
(2.22)
f
c
we obtain
Dividing both sides of Equation (2.21) by
ζ
τ
ty
ζ
f
c
=
√
ω
ω
t
(2.23)
f
c
, will be called the shear stress ratio. Dividing both the
numerator and denominator in the square root by
The nondimensional ratio,
τ
ty
/
ζ
f
c
ζ
Equation (2.22) can also be written as
ω
ω
t
tan
α
r
=
(2.24)
The three pairs of equations: Equations (2.7) and (2.8); Equations (2.21) and (2.22); and
Equations (2.23) and (2.24), are actually the same, except that they are expressed in terms of
different units. Equations (2.7) and (2.8) are in terms of force per unit length (
q
y
,
n
y
,
n
ty
).
