Civil Engineering Reference
In-Depth Information
The stress in the diagonal concrete struts
σ
d
can be related to the thickness
t
d
and the shear
flow
q
by inserting
τ
t
=
q
/
t
d
into the equilibrium Equation (7.3):
q
σ
d
=
(7.72)
t
d
sin
α
r
cos
α
r
σ
d
in Equation (7.72) reaches the maximum
At failure
σ
d
,
max
, while the torsional moment
=
T
n
/
reaches the nominal capacity
T
n
. Substituting
q
2
A
o
at failure into Equation (7.72)
gives:
T
n
2
A
o
σ
d
,
max
sin
t
d
=
(7.73)
α
r
cos
α
r
t
d
in Equation (7.46) is neglected, and
A
o
can be
If
t
d
is assumed to be small, the last term
ξ
expressed by the thin-tube approximation:
1
2
p
c
t
d
A
o
=
A
c
−
(7.74)
A
c
Substituting
A
o
from Equation (7.74) into (7.73) and multiplying all the terms by 2
p
c
/
result in:
p
c
A
c
t
d
2
2
p
c
A
c
t
d
T
n
p
c
A
c
1
σ
d
,
max
sin
−
+
=
0
(7.75)
α
r
cos
α
r
Define:
t
do
=
A
c
/
p
c
A
c
τ
n
,
max
=
σ
d
,
max
sin
τ
n
=
T
n
p
c
/
α
r
cos
α
r
Equation (7.75) becomes
t
d
t
do
2
2
t
d
t
do
τ
n
τ
n
,
max
=
−
+
0
(7.76)
When
t
d
/
τ
n
/τ
n
,
max
in Figure 7.11, Equation (7.76) represents a parabolic
curve. Solving
t
d
from Equation (7.76) gives:
t
do
is plotted against
t
do
1
1
τ
n
τ
n
,
max
t
d
=
−
−
(7.77)
This approach of determining the thickness of the shear flow zone was first proposed by
Collins and Mitchell (1980) and was later adopted by the 1984 Canadian Standard (CAN3-
A23.3-M84); which gives:
1
1
tan
A
1
p
1
T
n
p
1
1
tan
t
d
=
−
−
α
r
+
(7.78)
.
ϕ
c
f
c
A
1
α
r
0
7
In Equation (7.78)
A
c
and
p
c
are replaced by
A
1
and
p
1
, respectively, since the concrete
cover is considered to be ineffective.
ϕ
c
f
c
, in which the material
σ
d
,
max
isassumedtobe0
.
7
reduction factor
ϕ
c
can be taken as 0.6.
