Civil Engineering Reference
In-Depth Information
1.5
Precise Computer Analysis for
Beam B1 to B5 with Increasing Percentage
of Reinforcement (Hsu, 1968)
B5
B4
B3
1.0
Parabolic Curve
Not Applicable
when t
d
/t
do
> 1
B2
τ
n
τ
n, max
B1
Parabolic Curve
0.5
τ
n
τ
n, max
t
d
t
do
t
d
t
do
(
)
(
)
2
=
2
τ
n
τ
n, max
or t
d
= t
do
1
1
0
0
0.5
1.0
t
d
t
do
Figure 7.11
Graphical presentation of Equations (7.76) or (7.77)
Equations (7.78) and (7.77) clearly show that the thickness ratio
t
d
/
t
do
is primarily a function
of the shear stress ratio
τ
n
/
f
c
. The thickness ratio
t
d
/
t
do
is also a function of the cracking angle
α
r
varies in the vicinity of 45
◦
.
α
r
, but is not sensitive when
τ
n
>τ
n
,
max
means over-reinforcement. The case of over-rei
nfo
rcement can not be expressed by Equation
In Equation (7.77),
τ
n
≤
τ
n
,
max
represents the case of under-reinforcement, while
(7.77), because it gives a complex number
√
−
1. Figure 7.11 shows that Equation (7.77)
is applicable when
τ
n
,
max
,
t
d
is increasing unreasonably fast. This problem reflects the difficulty in using the thin-tube
approximation for
A
o
(Equation 7.74) to find
t
d
. When
t
d
exceeds about 0
τ
n
is less than about 0
.
9
τ
n
,
max
. However, when
τ
n
exceeds 0
.
9
.
7
t
do
, the tube
t
d
in Equation (7.46) cannot be neglected.
To avoid this weakness, a different approach is adopted. Using the RA-STM presented in
Section 7.1, a computer program was written to analyze the torsional behavior of reinforced
concrete members and to calculate the thickness
t
d
(Hsu and Mo, 1985b). This computer
program was used to analyze the 61 eligible torsional members available in the literature.
becomes so thick that the third term
ξ
