4.8. RELIABILITY OF A SHAFT UNDER TORSION 199
where S
sy
is material shear yield strength. K
s
is the shear stress concentration factor under tor-
sion. T and d
o
have the same meaning as those in Equation (4.26).
e limit state function of a hollow shaft with circular cross-section under torsion will be:
g
S
sy
; K
s
; d
o
; d
i
; T
D S
sy
K
s
T d
o
=2
J
D S
sy
K
s
T d
o
=2
d
4
o
d
4
i
=32
D S
sy
K
s
16T d
o
d
4
o
d
4
i
D
8
ˆ
<
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure;
(4.28)
where d
i
is the inner diameter of a hollow shaft. e rests of parameters in Equation (4.28) have
the same meaning as those in Equation (
4.27).
When the component’s torsion is described by a PMF, we can use the total probability
theorem Equation (2.24) in Chapter 2 to calculate the reliability of the components. In this case,
the torque is expressed by PMF as:
P
.
T D T
i
/
D p
i
i D 1; 2; : : : ; n; (4.29)
where p
i
is the PMF when the torque T is equal to T
i
. Since it is a PMF, we have:
R D
n
X
iD1
.
p
i
R
i
/
; (4.30)
where R is the reliability of a shaft under such torsion. R
i
is the reliability of a shaft when the
torque T is equal to T
i
. Even T
i
can be a constant value, we still need to use Equations (4.27)
or (4.28) to calculate R
i
because other variables in the limit state function are still random
variables.
Equations (4.27), (4.28), and (4.30) can be used to calculate the reliability of a shaft by the
H-L method, R-F method, or Monte Carlo method which have been discussed in Chapter 3.
Two examples will demonstrate how to calculate the reliability of a shaft under torsion.
Example 4.14
e stepped shaft is with a smaller shaft diameter d
1
D 0:750 ˙ 0:005
00
and a larger diameter
d
2
D 1:000 ˙ 0:005
00
as shown in Figure 4.12. e fillet radius is 1=16
00
. e stepped shaft is
subjected to a torque T D 1000 ˙ 160 (lb/in). e shear yield strength of the shaft material
follows a normal distribution with a mean
S
sy
D 32,200 (psi) and the standard deviation
S
sy
D
3630 (ksi). Use the H-L method to calculate the reliability of the shaft under the torsion.