Civil Engineering Reference
In-Depth Information
where
C
m
=
(2
m
1
m
2
). For under-reinforced members,
m
1
varies from 0.55 to 0.85, while
m
2
varies from 0.13 to 0.22. These values are obtained from the Appendix of Hsu and Mo's
report (1983b). The low values of
m
2
are due to the softening of concrete. For an increasing
amount of reinforcement,
m
2
increases while
m
1
decreases. Therefore, the product
m
1
m
2
can
be taken approximately as a constant 0.125, making
C
m
a constant of 4. Then
1
/
4
T
n
A
c
f
c
t
d
=
(7.82)
Equation (7.82) is also plotted in Figure 7.12. Comparison of Equations (7.82) and (7.79)
shows the difference to be small. Actually, Equation (7.82) can be considered as a simplification
of Equation (7.79) by neglecting the small first term with constant 0.082, and increasing the
constant in the second term from 3.405 to 4. The small effect of
α
r
is also neglected by taking
45
◦
.
Inserting Equation (7.82) into the thin-tube expression of
A
o
in Equation (7.74) and
p
o
in
Equation (7.49) gives:
sin 2
α
r
=
1, which is the exact value when
α
r
=
1
2
p
c
t
d
=
2
T
n
p
c
A
c
f
c
A
o
=
A
c
−
A
c
−
(7.83)
16
T
n
A
c
f
c
p
o
=
p
c
−
4
t
d
=
p
c
−
(7.84)
The lever arm area
A
o
in Equation (7.83) is used in conjunction with Equation (7.68) to
calculate the torsional strength
T
n
for the 61 beams available in the literature. The calculated
values are plotted in Figures 7.13(a) and (b) and compared with the test values. The average
T
n
,
test
/
T
n
,
calc
value is 1.013 and the standard deviation is 0.055.
The maximum thickness of the shear flow zone
t
d
,
max
to ensure that the beam remains under-
reinforced can be obtained by examining the results of six beams in series B of PCA tests
70
250
60
200
50
40
150
Hsu (1968)
Leonhardt & Shelling (1974)
McMullen & Rangan (1978)
Mitchell & Collins (1974)
Bradburn (1968)
McMullen & Warwaruk (1967)
Lampert & Thurlimann (1969)
30
Hsu (1968)
Leonhardt & Shelling (1974)
McMullen & Rangan (1978)
Mitchell & Collins (1974)
Bradburn (1968)
McMullen & Warwaruk (1967)
100
20
50
10
0
0
0
10
20
30
40
50
60
70
80
0
50
100
150
200
250
300
T
n
, calc (kN-m)
T
n
, calc (kN-m)
(a)
(b)
Figure 7.13
Comparison of test strengths with calculated strengths using proposed
t
d
(Equation 7.82);
(b) is an expanded scale for the lower portion of (a)
