# Chapter 3 Failure theories and material strengths

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29 Νοε 2013 (πριν από 5 χρόνια και 2 μήνες)

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1

Chapter
3

Failure theories and material strengths

K
ey

points:

Fatigue strength;

Miner

s hypothesis;

Contact strength.

Difficulties:

Fatigue strength at regularly varying stresses.

S
lim

——
Design stress

——
Allowable stress

lim

——
Limit stress

S
——
Allowable s
afety factor

1.

Static strength

1)

For ductile materials such as steel

2
.
2
~
2
.
1
s

s

——

yield s
trength

2)

For brittle materials such as cast iron, some ceramics, some plastics

S
b

b

——

U
ltimate strength

2. Fa
tigue strength

Fluctuating stress

?
lim

2

§
3⸱

(r敶iew⁢y⁹o畲s

1.

The maximum
-
normal
-
stress theory ( the first strength theory)

2.

The maximum
-
normal
-
strain theory ( the second strength theory)

3.

The maximum
-
shear
-
stress theory ( the third strength theory)

4.

The distortion
-
energy theory
( the fou
r
th strength theory)

3.2
Bulk strengths of the machine components

Bulk strengths

——

tensile
strength
, compressive
strength
, bending
strength
, shear
strength

Surface
strengths

——

contact strength, extrusion strength, wear
strength,

1.

Force, bending moment, torque

Classification of
stresses

T
ension/tensile stress, compression/compressive stress, shear

stress,

Static
stress
,
varying/fluctuating stress

3

Maximum stress
max

Minimum stress
min

Mean stress

2
max
min

m

Alternating stress/stress amplitude
2
min
max

a

Stress ratio

max
min

r

σ

N

σ
max

σ
m

σ
min

σ
a

N

σ

r=1 static stress

σ

N

r=
-
1

completely

reversed stress

4

2.

Stress
-
strength design method
/factor
-
of
-
safety method of design

The maximum stress condition

S
lim
max

S
lim
max

The safety factor condition

S
S

lim

S
S

lim

3.

Strength design under static stre
s
ses

(review by yourself)

4.

Strength design under varying stresses

Fatigue failure

Small crack

stress concentration

fracture

suddenly

Fatigue
zone

——

beach mark
s( 0.1~1
μ
m

intervals
)
, smooth
surface
;

Final fracture zone

——

rough

crystalline

Stress
-
life definitions

N

σ

Repeated
pulsan
t

stress

5

First test

Specimen

σ
-
N
diagra
m

fatigue curve
,

Stress
-
cycle diagram

fatigue

strength
N
1

Rely on the test only.

Endurance
/fatigue l
imit
1

C
ritical number of c
ycle

0
N

Number of cycles
N

Static strength

3
10

N

Low
-
cycle

fatigue

3
10

N

High
-
cycle

fatigue

F
atigue curve equation

S
1

lgb

lg
σ
b

lgN

m
1

lgN
0

6

log
-
log plot

b
kx
y

b
N
m
N
lg
lg
1
lg
1

b
N
m
N
lg
lg
lg
1
1

b
N
m
N

1
1

C
b
N
m
m
rN

C
N
N
m
m
N

0
1
1

1
0
1
1

N
m
N
K
N
N

Set

m
N
N
N
K
0

S
K
N
1

here

N
K
——
Life coefficient

m
——
material constant

For steels

m=9
, for
bending, tension or compression

stresses

m=
6, for contact stresses

For bronze

m=9, for bending stresses

m=8, for c
ontact stresses

0
N
——

C
ritical number of c
ycle

7

D
k

7
6
0
10
~
10

N
, for steels with hardnes
s

350HB

7
7
0
10
25
~
10
10

N
, for steels with hardnes
s
>
350HB

7
0
10
25

N
, for
nonferrous metals

S
implified fatigue diagrams

Second test

1
max
min

r

Haign’s diagram

Broken line simplifie
d diagram

Allowable fatigue design diagrams

k
k
D

combined influence

factor

k
——

stress concentration factor

——

size factor

——

surface

condition

factor

F1

F2

F2

(
σ
s,0)

A

σ
a

B
)
2
,
2
(
0
0

)
2
,
2
(
0
0

(0,
σ
-
1
)

45
°

σ
m

45
°

S

A

E

)
2
,
2
(
0
0
D
N
N
k
K
K

)
2
,
2
(
0
0

σ
a

)
2
,
2
(
0
0

)
2
,
2
(
0
0

(0,
σ
-
1
)

)
,
0
(
1
D
N
k
K

45
°

8

L
ine A
E
'
s equation

0
2
2
0
0
1
0
'
1
'

N
D
N
D
N
me
D
N
ae
K
k
K
k
K
k
K

0
1
0
'
1
'
2

me
N
ae
D
K
k

Set

0
0
1
2

1
'
'

N
me
ae
D
K
k

Material property
——

F
or carbon steel
2
.
0
~
1
.
0

For alloyed steel
3
.
0
~
2
.
0

Line
E
'
S

s equation

s
me
ae

'
'

Fatigue life prediction with stable alternating stresses

S
me
ae
'
'
max

S
S
me
ae

max
'
'

a
ae
a
a
S

'

a
a
ae
a
S
S

'

r=c For most gyro shafts

9

'
1
1
2
)
(
2
)
(
min
max
min
max
c
r
r
tg
m
a

I
n

O
E
'
A
'

Fatigue strength

(2)

(1)

1
'
'
'
'

N
me
ae
D
me
ae
m
a
K
k

From (1)

a
m
ae
me

'
'

1
'
'

N
a
m
ae
ae
D
K
k

a
m
D
N
ae
k
K

1
'

a
m
a
D
N
a
ae
a
S
k
K
S

1
'

I
n

O
E
'
S

Static strength

S
S
a
m
s

c
m

For springs in vibration

r=c

σ
a

σ
m

p

E

α

', σ

')

O

A

σ
a

σ
m

p

E

O

Q

c
m

A

10

in

OA
'
E
'
Q

Fatigue strength

(2)

(1)

1
'
'
'

N
me
ae
D
me
m
K
k

1
'

N
m
ae
D
K
k

D
m
N
ae
k
K

1
'

a
a
D
m
N
a
ae
a
S
k
K
S

1
'

in

QE
'
S

S
tatic strength

S
S
a
m
s

c

min

For some bolts

I
n OA
'
M Seldom

in

O
M
'
E
'
P

Fatigue strength

11

A

σ
a

σ
m

S

E

o

P

-
min

(2)

(1)

1
'
'
'
'
min

N
me
ae
D
ae
me
K
k

'
min
'
ae
me

1
'
min
'

N
ae
ae
D
K
k

min
1
'

N
ae
D
K
k

D
N
ae
k
K
min
1
'

a
a
D
N
a
ae
a
S
k
K
S

min
1
'

in

PE
'
S

Static strength

S
S
a
s

2

min

Prediction of fatigue life with regularly
varying
stresses

Miner’s
rule/
Hypothesis
, Palmgren

s rule,

Hypothesis of
linear
c
umulati
ve
-
damage

rule

M

12

1
3
3
2
2
1
1

N
n
N
n
N
n

1

i
i
N
n

Miner’s Hypothesis

2
.
2
~
7
.
0

i
i
N
n

§
3.3

surface
strengths

of the machine components

1.

Surface contact stresses and strength
s

Hertz formula

For higher pair

H
ertz stress

H
H
E
E
b
F

2
2
2
1
2
1
2
1
1
1
1
1

E
——
Elastic modul
u

——
Poission ratio

Effect factors

N
ormal load acted on unit contact length

b
F

C
curvature

2
1
1
1
1

Materials

Discussion

13

2.

Surface
extrusion s
tresses and strength
s

For low pair without sliding

p
p
A
F

3.

Surface
Wear

strength

For low pair with sliding

p
p

for low velocity

pv
pv

for mid
-
high velocity

v
v

for high velocity