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