Chapter 3 Failure theories and material strengths

plantcalicobeansUrban and Civil

Nov 29, 2013 (3 years and 8 months ago)

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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⸱

周Tories映 ail畲e

(r敶iew⁢y⁹o畲s
敬昩

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.

Loads and stresses



Classification of loads

Force, bending moment, torque

Static load, impact load, repeated load



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



gradual development



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

爽r
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
ombined radius of
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