3.3. DIMENSION OF A COMPONENT UNDER STATIC LOADING 165
dimension design with the required reliability. Now we will use examples to show how to conduct
component dimension design under combined loading.
Example 3.15
On the critical section of a circular shaft, the resultant internal torsion and an internal resultant
bending moment of a shaft are T D 2:5 ˙ 0:18 (klb.in) and M D 4:6 ˙ 0:34 (klb.in), as shown
in Figure 3.7. e yield strength S
y
of the shaft’s material follows a normal distribution with the
mean
S
y
D 34:5 (ksi) and the standard deviation
S
y
D 3:12 (ksi). Use the distortion energy
theory with the modified Monte Carlo method to design the diameter d of the shaft with the
required reliability 0.99 when the diameter d has a tolerance ˙0:005.
(a) Schematic of Bending
and Torque
(b) Stress Element
at Point A
(c) Stress Element
at Point B
z
B
A
x
x
y
y
M
y
T
x
τ
T
τ
T
σ
M
σ
M
Figure 3.7: Schematic of a segment of a shaft under combined stress.
Solution:
In this example, there is no dimension-dependent parameter.
(1) e Von-Mises stress.
As shown in Figure 3.7, the critical points on this critical section will be the points A
and B because there are the maximum values of bending stress. Stress elements at points A and
B are shown in Figures 3.7b,c, where
M
and
T
are the bending stress due to the bending
moment M
y
and the shear stress due to the torque T
x
, respectively. e Von Mises stress [1, 2]
at points A and B for the loading case in this example are the same. e point A is used to run
the calculation. At point A, we have:
x
D
M
D
32M
y
d
3
;
y
D
z
D 0;
xy
D
T
D
16T
x
d
3
;
yz
D
zx
D 0: (a)
e Von Mises in this case is:
von
D
s
32M
y
d
3
2
C 3
16T
x
d
3
2
D
16
d
3
q
4M
2
y
C 3T
2
x
: (b)