Chapter 7. Shafts
1
Chapter 7 - Shafts
2
We’ll consider
stress based
shaft design
Fluctuating stresses due to bending and torsion
For fully reversed bending and steady torsion
Shaft Stresses
J
cTK
J
cTK
I
cMK
I
cMK
mfsm
afsa
mfm
afa
,
,
0
0
a
m
T
M
3
Circular Shafts
33
33
16,
16
32,
32
d
TK
d
TK
d
MK
d
MK
mfsm
afsa
mfm
afa
Substituting
gives
32,
64,
2
44 dJ
dI
dc
4
Distortional Energy
Recall that the distortional energy theory states
that we can compute a factor of safety by
comparing the von Mises effective stress, ′, to
the tensile yield stress.
Similarly, under fatigue loading, we can compare
′ to the appropriate fatigue strengths.
5
von Mises Stress in Rotating Shafts
2/1
2
3
2
3
2/122
2/12
3
2
3
2/122
163
323
163
323
d
TK
d
MK
d
TK
d
MK
mfsmf
mmm
afsaf
aaa
6
Fatigue Failure Criteria for
Non-zero Mean Stress
DE - Goodman Fatigue Failure Theory
DE – Goodman:
defining
we can derive
And
See text for similar expressions for other fatigue failure theories
8
Yield Criteria
9
Example 7-1
10
Example 7-1 (cont.)
11
Example 7-1 (cont.)
12
Example 7-1 (cont.)
13
Example 7-1 (cont.)
Slides 15 - 26
Reference materials from Chapter 8 and Appendix A
14
Appendix A-15 Stress Concentration Factors
15
Appendix A-15 Stress Concentration Factors
16
Notch Sensitivity
17
Notch Sensitivity (shear)
18
Fatigue Stress Concentration Factors
)1(1
)1(1
1
1
1
1
shear
tsshearfs
tf
ts
fs
t
f
KqK
KqK
K
Kq
K
Kq
Solving for Kf and Kfs gives
Relation between Endurance Limit and
Tensile Strength (cont.)
ksi200ksi,100
ksi200,5.0
ut
utute
S
SSS
MPaSMPa
MPaSSS
ut
utute
1400,700
1400,5.0
in SI units
20
Endurance Limit Modifying Factors
f
e
d
c
b
a
efedcbae
k
k
k
k
k
k
SkkkkkkS
- Surface condition modifying factor
- Size modification factor
- Load modification factor
- Temperature modification factor
- Reliability factor
- Miscellaneous effects modification factor
These are
called “Marin”
factors
21
Surface Factor, ka
buta Sak
22
Size Factor, kb
For bending and torsion
For axial loads
kb = 1
mm2545151.1
mm5179.224.1
in10291.0
in211.0879.0
157.0
107.0
157.0
107.0
dd
dd
dd
dd
kb
23
Loading Factor, kc
torsion59.0
axial85.0
bending1
ck
24
Temperature Factor, kd
RT
Td
S
Sk
25
Reliability Factor, ke
Accounts for scatter in fatigue data and desired reliability
26
Miscellaneous Effects Factor, kf
Corrosion
Electrolytic plating
Metal spraying
Cyclic frequency
Frettage corrosion
(corrosion at contact surfaces)
27
