67071_05.pdf

7/24/2019 67071_05.pdf

1/25

C H A P T E R

5

D E S I G N O F M A C H I N E E L E M E N T S

F O R S T R E N G T H

S Y M B O L S 5 , 6

A

b

B

Csz

e sT

CsT-

E

F

F;

G

h

Ksz

K~

K~

Ky,

Mb

M;m

M ,

M;m

n

n a

n d

q

qy

t"

t

XO

Ymax

Zb

Z t

0

a r e a o f c r o s s - s e c ti o n ,

m 2 ( i n 2 )

a s h a p e f a c t o r ( b > 0 )

a c o n s t a n t

s ize coeff ic ient

s u r f a c e c o e f f i c i e n t i n c a s e o f t e n s i o n a n d b e n d i n g

sur face coef f i c i en t in case of to rs ion

Y o u n g ' s m o d u l u s , G P a ( M p s i)

n o r m a l l o a d ( a l s o w i t h s u f fi x es a n d p r i m e s ) , k N ( l b f)

s t a ti c e q u i v a l e n t o f cy c l ic l o a d , k N ( l b f)

m o d u l u s o f r ig i di ty , G P a ( M p s i )

t h i c k n e s s , m ( i n )

s i ze fac tor

s u r f a c e f a c t o r

t h e o r e ti c a l n o r m a l s t r e s s - c o n c e n t ra t i o n f a c t o r

t h e o r e t i c a l s h e a r s t r e s s - c o n c e n t r a t i o n f a c t o r

f a t ig u e n o r m a l s t r e s s -c o n c e n t r a t io n f a c t o r

f a t i g u e s h e a r s t r e s s - c o n c e n t r a t i o n f a c t o r

b e n d i n g m o m e n t ( a l so w i t h s uf fi xe s a n d p r i m e s ) , N m ( l b f i n )

s t at ic e q u i v a l e n t o f cy c li c b e n d i n g m o m e n t , N m ( l b f i n )

t w i st in g m o m e n t ( a l so w i t h s u f fi xe s a n d p r i m e s ) , N m ( l b f i n )

s t a ti c e q u i v a l e n t o f c y c li c t w i s t in g m o m e n t , N m ( l b f i n )

s a f e t y f a c t o r

a c o n s t a n t

ac tua l s a fe ty fac tor ( a l so wi th suf f ixes )

des ign sa fe ty fac tor ( a l so wi th suf f ixes )

i n d e x o f s e n s it i v it y

i n d e x o f n o t c h s e n s it i v it y f o r a l t e r n a t i n g s t re s s e s

n o t c h r a d iu s , m m ( in )

t i m e , h

t h e g u a r a n t e e d v a l u e o f x ( x 0 >_ 0 )

m a x i m u m d e fl ec ti on

3 3 3

f le x u ra l s ec ti on m o d u l u s , m o r c m ( i n )

3 3 .~

p o l a r s e ct io n m o d u l u s , m o r c m ( i n )

chara c te r i s t i c or s ca le va lue (0 _> x0)

7/24/2019 67071_05.pdf

2/25

5 . 2 C H A P T E R F I V E

O"

o- 0

O ' s u

~e

O d

O"

ey

O ' n o m

O ' m a x

I

O"

//

O"

T

"re

7-xy

7-no m

~y

c

I

c

c t

CO

u 0

n o r m a l s t re s s ( a ls o w i t h su f fi x es a n d p r i m e s ) , M P a ( p s i)

in i t i a l s t re s s , MPa (ps i )

u l t i m a t e s tr e n g t h , M P a ( p s i)

e l a st i c l i m i t f o r s t a n d a r d s p e c i m e n f o r 1 2 . 5 m m ( n ) , M P a ( p s i)

des ign s t res s ( a l so wi th suf f ixes ) , MPa (ps i )

n o r m a l s t r e ss in x d i r e c t io n , M P a ( p s i)

y ie ld s t res s , MPa (ps i )

n o r m a l s t r e s s i n y d i re c t i o n , M P a ( p s i)

n o m i n a l n o r m a l s tr es s, M P a ( p si )

m a x i m u m n o r m a l s tr es s, M P a ( p si)

e l a st i c li m i t f o r a n y t h i c k n e s s h b e t w e e n 1 2 .5 m m ( n ) a n d

7 5 m m ( 3 i n) , M P a ( p si )

e l a st i c li m i t fo r 7 5 m m ( 3 i n ) s p e c im e n , M P a ( p s i)

e n d u r a n c e l im i t i n b e n d i n g , M P a ( p s i)

shea r s t res s ( a l so wi th suf f ixes and p r im es ) , M Pa (ps i )

e l as t i c l im i t i n shea r , M Pa (ps i )

y ie ld s t rength in shea r , MPa (ps i )

shea r s t res s in xy p l a n e , M P a ( p s i )

n o m i n a l s h e a r s t re s s, M P a ( p s i)

e n d u r a n c e l im i t i n t o r s i o n , M P a ( p s i )

e n g i n e e r i n g o r a v e r a g e s t r a i n , l a m / m ( l ai n /i n )

t rue s t ra in , l am /m ( l a in /in )

to ta l c reep , a f t e r a t im e t , l am /m ( l a in /in )

in i t ia l c reep , l am /m ( l a in / in )

c reep ra t e ( l a /m ) /h [ ( l a in / in ) /h ]

a c o n s t a n t

S u f f ix e s

s

u

y

6'

a

b

m

t

m a x

m i n

f

o

f o r

s t a t i c s t r e n g t h ( a , o r a y )

u l t im a t e s t r e n g t h

yie ld s t rength

e las t i c lim i t

a m p l it u d e

b e n d i n g

m e a n

t e n s i o n

m a x i m u m

m i n i m u m

e n d u r a n c e l im i t ( a l s o u se d f o r r e v e r s e d c y cl e )

e n d u r a n c e l im i t r e p e a t e d c y c le

P r i m e s

' ( s i n g l e )

" ( d o u b l e )

f o r

s t a t i c equiva len t

c o m b i n e d s tr e ss

7/24/2019 67071_05.pdf

3/25

D E S I G N O F M A C H I N E E L E M E N T S F O R S T R E N G T H

Part icu lar

Formula

S T A T I C L O A D S

I n f l u e n c e o f s i z e

13.

D

E _

.o

UJ

A

B

t E

(Y"e

1 9 ~ 9 ~ r~n 7 ~

S i z e o f s e c t i o n , m m

F I G U R E 5- 1 C ha n ge o f e l a s t ic l i m i t

wi th s ize of sec t ion .

1 2 0 0

l o o o ~

SA H 3 240 ... OH

" E 8 0 0 \ - - , - ~ - - - . . ~ ~ 0 . . o /. ,

o E ~ ; 6 0 0 ~ - , v

1) t -

L U - ~ 4 0 0 ~ ~ ~ ' ~ S A E 1 0 45 C

-' ~ ~ ~ -

2 0 0 == . .~ ~ = --- SAE 1015 BE

o I

0 2 0 4 0 6 0

S i z e o f s e c ti o n , m m

F I G U R E 5 -2 I n f l ue nc e o f s i ze on e la s t i c l i m i t s .

~ . _

OH

M~'AL - - "

BESSEMER

8 0 1 0 0 1 2 0

300

2 5O

2 00

g_

1 5 0 N -

g o

100

- 5o

0

140

T h e s i z e c o e f f i c i e n t ( F i g . 5 - 1 , F i g . 5 - 2 , a n d T a b l e 5 - 1 )

e ~ z = l - 0 . 0 1 6 1 - ( h - 1 2 . 5 )

o-

w h e r e o- = e l a s ti c l im i t f o r 1 2 . 5 m m ( 0 .5 i n )

t

ae = e l a st i c l im i t f o r 7 5 m m ( 3 .0 i n )

( 5

T A B L E 5 -1

S t r e n g t h r a t i o s o f v a r i o u s m a t e r i a l s f o r u se in E q s . ( 5 - 1 ) a n d ( 5 - 2 )

Values of

'e / Ore

Ma ter ial Natural s tate Annealed

D r aw n a t

650C

D r aw n a t

535C

D r aw n a

4 2 5 C

A l u m i n u m , s t ro n g , w r o u g h t 0 .9 3

T o b i n b r o n z e 0 .9 0

M o n e l m e t a l , f o r g e d 0 .8 0

D uc t i l e i r on 0 . 8 0

Lo w- c a r bo n s t e el , C < 0 . 2 0% 0 . 8 4

M e d i u m - c a r b o n s te e l, 0. 3 0 t o 0 . 5 0 % C

Ni c ke l s t e e l , SAE 2 34 0

C r - Ni s t e e l , SAE 314 0

C a s t i r on , C l a s s no . 2 0 0 . 55

C a s t i r on , C l a s s no . 2 5 0 . 73

C a s t i r on , C l a s s no . 35 0 . 60

W r o u g h t i r o n 0 .5 5

0.98

0.85

0.86

0.86

0 . 72

0.80

0.75

0.59

0.74

0.70

m

0.53

0.65

7/24/2019 67071_05.pdf

4/25


P a rt i cu l a r

F o r m u l a

T h e s i z e f a c t o r

T h e r e l a t i o n b e t w e e n s i z e c o e f f ic i e n t a n d s i z e f a c t o r

T h e e l a st ic li m i t f o r a n y t h i c k n e s s h b e t w e e n 1 2 .5 m m

a n d 7 5 m m c a n b e d e t e rm i n e d f r o m t h e r e l at io n ( F i g.

5-])

K S Z - - "

2 5 0

l

O"

3 0 0 - 4 h + - - ( 4 h - 5 0 )

O e

1

e s z - - k s - - 7

l

(7- - - (7-

( 0 " e - - o ' ~ ) ( h - 1 2 . 5 )

( 7 5 - 1 2 . 5 )

( 5

( 5

( 5

I N D E X O F S E N S I T I V I T Y

T h e i n d e x o f s e n s i t i v i t y

T h e a c t u a l o r r e a l s t r e s s - c o n c e n t r a t i o n f a c t o r

g ~ a - 1

q = ~ ~ - 1

g a a - - 1

- t- q(K ,~ - 1)

( 5

( 5

S U R F A C E C O N D I T I O N ( Fig . 5 -3 )

T h e s u r f a ce f a c t o r f o r t h e ca s e o f t e n s i o n a n d b e n d i n g

T h e s u r f a c e c o e f f i c ie n t i n c a s e o f t o r s i o n

1

S T - - - - -

e S T

es~- = 0. 42 5 + 0.575e~.,-

( 5

(

1.o , 'b

~' ~'

1.o

POLISHED

~-'w ~--~...~.._zG ROUND

2 0 9 C I 0 95

~ - . 2 % . ~ I

o 8 I

~ o . 9 o - ~

~ "1~Oo~ : ~ ~ O o o . 8 o

- < J 0 . 7 5

0 -4 - 0 . 7 0 8

o 0-3 6o ~ {e4 ) ~- 0 . 6 5

~ 0"2 " ~ ~ O i m m ' ~~%~-~

~ ~ 0 .6 0

0.1 (]URVE -i d

0 " 5 5

2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0 1 4 0 0

U l t i m a te ten s i l e s t ren g th , s ta t i c , %u , M P a

F IG U R E 5 - 3 Re c i p r oc a l s o f st r e s s - c onc e n t r a ti on f a c to r s c a us e d by s u rf a c e c ond i t i ons .

7/24/2019 67071_05.pdf

5/25


Part i cular

Formula

S A F E T Y F A C T O R

T h e g e n e r a l e q u a t i o n f o r d e s i g n s a f e t y f a c t o r ( T a b l e

5 - 2 )

T A B L E

5 - 2

A c t u a l s a f e t y f a c t o r a

n = k l k 2 k 3 k 4 . . . k m n a

w h e r e k l =

Ksz

= s i z e f a c t o r

k2 = Ks,-

= s u r f a c e f a c t o r

k 3 = K I = l o a d f a c t o r ( T a b l e 1 4 -3 )

k 4 = m a t e r i a l f a c t o r

na

= a c t u a l s a f e t y f a c t o r ( T a b l e 5 - 2 ) .

(

Circumstance

Actua l fac tor o f sa fe

or re l i ab i li ty fac tor ,

St r e ng t h p r ope r t i e s o f m a t e r i a l w e ll know n , l oa d a c c u r a t e l y p r e d ic t a b l e , pa r t s p r oduc e d w i t h c l os e

d i m e ns i ona l c on t r o l a n d b r oug h t t o c l o s e to l e r a nc e s pe c i fi c a ti ons, a nd l ow - w e i gh t c r it e r ia

Loa d a c c u r a t e l y p r e d i c ta b l e a nd l ow - w e i gh t a nd l ow - c os t c r i te r i a

Loa d a c c u r a t e l y p r e d i c ta b l e a nd l ow - c os t c r i te r i a ( l ow - w e i gh t - no c r i t e ri a )

O ve r l oa ds e xpe c t e d , m a t e r i a ls o r d i na r y bu t r e l i a b il i ty i m por t a n t

S t r e ng t h p r ope r t i e s no t w e l l de f ine d , l oa d i ng un c e r t a i n , hu m a n l if e a t s t a ke i f f ai l u re oc c u r r e d ,

h i g h m a i n t e n a n c e a n d s h u t d o w n c o s t

1

1.1-1 .5

1 . 5 - 2

2 - 3

> 3

a 1 '1, , . . . . . . 1 . .. .

1 , , ~ v a , u c~ a r e r e c o m m e n d e d f o r u s e in d e s ig n , i n t h e a b s e n c e o f s p ec i fi c re l i a b il i t y d a ta .

T h e d e s i g n s a f et y fa c t o r b a s e d o n u l t i m a t e s t r e n g t h

T h e r e l a t i o n s h i p s b e t w e e n a l l o w a b l e s t r e s s a n d s p e c i -

f ie d m i n i m u m y ie ld s t re n g t h u s in g t h e A I S C C o d e a r e

g i v e n h e r e :

T e n s i o n

S h e a r

B e a r i n g

B e n d i n g

T h e e x p r e s s i o n f o r f o r c e s o r l o a d s u s e d t o f i n d s t re s s e s

i n m a c h i n e m e m b e r s o r s t ru c t u re s a s p e r A I S C C o d e .

rlud -- KszKo-anua ( 5 -

0.45Crsy

7/24/2019 67071_05.pdf

6/25


Particular

Formula

T A B L E 5 - 3

AISC service factor K for use in Eq. (5-15)

Particular

For suppor t o f e leva tor s

For cab-operated traveling-crane suppor t girders and their connections

For pendant-operated traveling-crane suppor t girders and their connections

Fo r suppo r t of l ight m achinery, shaf t- or m otor -dr iven

For sup por ts of reciprocating m achinery or power-dr iven units

For hangers suppor t ing f loors and balconies

2

1.25

1.10

> 1.20

> 1.50

1.33

T h e v a l u e o f d e s ig n s h e a r s t r es s

T h e d e s i g n s a f e t y f a c t o r

T h e e q u a t i o n f o r d e s i g n s a f e t y f a c t o r

T h e r e a l i z e d s a f e t y f a c t o r

T h e d e s i g n s a f e ty f a c t o r b a s e d o n e l a s ti c l im i t

T h e d e s i g n s a f e t y f a c t o r b a s e d o n y i e l d s t r e n g t h

T h e d e s i g n s a fe t y f a c to r b a s e d o n e n d u r a n c e l i m i t o n

b e n d i n g

D e s i g n s t r e s s b a s e d o n e l a s t i c l i m i t

D e s i g n s t re s s b a s e d o n u l t i m a t e s t r e n g t h

D e s i g n s t r e s s b a s e d o n y i e l d s t r e n g t h

D e s i g n s t r e s s b a s e d o n y i e l d s t r e n g t h i n s h e a r

S t a t i c d e s i g n s t r e s s

7-d < % (5 -1

s t r e n g t h

n d = ~ = n s n L ( 5 -

s t r es s

w h e r e n~ = s a f et y f a c t o r t o t a k e i n t o a c c o u n t t

u n c e r t a i n t y o f s t r en g t h

nL - - s a f e t y f a c t o r t o t a k e i n t o a c c o u n t t

u n c e r t a i n t y o f l o a d .

s t r e n g t h i n f o r c e u n i t s ( 5 -

n d = a p p l i e d f o r c e o r l o a d

as rs (5

n r

- - - o r ? '/r - -

o" T

n e d - - g s z g a a n e a

( 5 -

n y d - - K~z g a a n y a

( 5 -

nfd = K~zK~TKldnfa ( 5

w h e r e K t d = l o a d f a c t o r

O"e

O 'e d - - ~ ( 5 -

n e d

Osu

(5-

O'ud --~

?tud

CrYd = Osy ( 5 -

n y d

"ryd = % (5-

n y d

O'su O'sy

cr~ d = ~ o r ~ a s t h e c a s e m a y b e ( 5 -

n u d r l y d

7/24/2019 67071_05.pdf

7/25


P a rt i cu l a r Fo rmu l a

D e s i g n s tr e ss b a s e d o n e n d u r a n c e l i m i t

T h e c o r r e c t e d f a t i g u e s t r e n g t h o r d e s i g n f a t i g u e

s t r e n g t h

T h e c o r r e c t e d e n d u r a n c e l i m i t o r d e s ig n e n d u r a n c e

l im i t

T h e s iz e f a c t o ksz f o r b e n d i n g o r t o r s i o n o f r o u n d b a r s

m a d e o f d u c t i le m a t e r i a ls a c c o r d i n g t o J u v i n a l l

T h e s i z e f a c t o r f o r a x i a l f o r c e

T h e s i ze f a c t o r a s s u g g e st e d b y th e A S M E n a t i o n a l

s t a n d a r d o n " D e s i g n o f T r a n sm i s s io n S h a f ti n g "

T h e s u r f a c e f a c t o r

F o r a r e c t a n g u l a r c r o s s - se c t i o n in b e n d i n g

T h e e f f ec ti v e d i a m e t e r o f r o u n d - s e c t i o n c o r r e s p o n d -

i n g t o a n o n r o t a t i n g s o l i d o r h o l l o w r o u n d - s e c t i o n

T h e e ff e ct iv e d i a m e t e r o f a r e c t a n g u l a r s e c ti o n o f

d i m e n s i o n s h x b w h i c h h a s A 0 .9 5c r - - 0 . 0 5 b h

crfd = Crsf ( 5 -

njd

O 's f = K s r K s z K l d K R K T K m e o ' t s f

( 5 - 2

O'se - - k s r k s z k l d k R k r k m e o ' t s e (5-2

w h e r e ~r~e = e n d u r a n c e l i m i t o f te s t s p e c i m e n

a ~ f = f a t i g u e s t r e n g t h o f te s t sp e c i m e n

Ks,. = s u r f a c e f a c t o r

K~z = s i ze f a c t o r

Kid = l o a d f a c t o r

K R = r e l i a b i l it y f a c t o r

K r = t e m p e r a t u r e f a c to r

Kme = m i s c e l l a n e o u s - e f f e c t f a c t o r a l s o

k n o w n a s f a t i g u e s t r e n g t h r e d u c t i o n

f a c t o r ~ 1 /K ~ , u ( 5 - 2

K~z

1

0 .9

0.8

0 . 7

d < 1 0 m m ( 0 .4 i n )

1 0 m m ( 0 .4 i n) < d < 5 0 m m ( 2 i n)

5 0 m m ( 2 i n ) < d < 1 00 m m ( 4 i n )

1 0 0 m m ( 4 i n) < d < 1 5 0 ra m ( 5 i n)

( 5 - 2

K~z = 0 .7 to 0 .9

( 5 - 2

I d 0"19 2 < d < 10 in

Ksz =

1 . 85 d - 19 5 0 < d < 2 5 0 m m

( 5 - 2

1 .0 0 f o r l o n g i t u d i n a l h a n d p o l i s h

0 .9 0 f o r h a n d b u r n i s h

K s '. = 0 . 8 7 f o r s m o o t h m i l l c u t

0 .7 9 f o r r o u g h m i ll c u t

A l s o r e f e r t o F i g . 5 - 3 f o r s u r f a c e c o e f f i c i e n t

1 1

e s ' . - K s r

o r

Ksr=--esr

( 5 - 2

d - - 0 . 8 1 x / A

w h e r e A - - a r e a o f t h e c ro s s s e c t i o n

d e - - 0 . 3 7 0 D

w h e r e D - - d i a m e t e r

d e - -- 0 . 8 0 8 ( h b ) l / 2

(5-2

(5-

(5-

7/24/2019 67071_05.pdf

8/25

5 . 8 C H A P T E R FI V E

Particular

Formula

T h e e q u i v a le n t d i a m e t e r r o t a t i n g - b e a m s p e c im e n fo r

a n y c r o s s - s e c t i o n a c c o r d i n g t o S h i g l e y a n d M i t c h e l l

T h e l o a d f a c t o r a c c o r d i n g t o S h i g l e y

T h e f a t i g u e s t r e s s c o n c e n t r a t i o n f a c t o r w h i c h i s u s e d

h e r e a s t h e f a t i g u e s t r e n g t h r e d u c t i o n f a c t o r a t e n d u r -

ance l im i t 106 cyc les

T h e f a t i g u e s t r e n g t h r e d u c t i o n f a c t o r f o r l i v e s l e s s

tha n N = 106 cyc les i s K~f and i s g iven by

Fo r r e l i a b i l i t y f a c t o r K R

/

A 9 , .

d e q = ~/ (5 -

V 0 . 0 7 6 6

wh ere A95 i s the po r t io n of the c ros s s ec t iona l

a r e a o f th e n o n r o u n d p a r t t h a t i s s t r es s e

b e tw e e n 9 5 % a n d 1 0 0% o f t h e m a x i m u m

stress .

0.923 axial loading as,,, 1520 M Pa (2 20 kp

k m = 1 b e nd in g

0.577 tors ion and shear

( 5 - 2

K ~ , f = 1 +

q ( k o , -

1) (5-

w h e r e K~f, K~t a n d q h a v e th e s a m e m e a n i n g a

g i v e n i n C h a p t e r 4 .

K ~ f = a N b ( 5 - 2

( 1 ) 1 1 (5_2

w h e r e a = ~ a n d b = - s l o g

K 'U = 1 at 103 cycles.

R e f e r t o T a b l e 5 - 3 A .

T A B L E 5 - 3 A

R e l i a b i l i t y c o r r e c t i o n f a c t o r b a s e d o n a s t a n d

d e v i a t i o n e q u a l t o 8 % o r t h e m e a n f a t i g u e l i m i t .

Reliabil ity , % KR

50 1.000

90 0.897

99 0.814

99.9 0.743

99.999 0.659

T h e t e m p e r a t u r e f a c t o r a s su g g e s te d b y S h i g le y a n d

M i t c h e l l

Fo r t y p i c a l f r a c t u r e s u r f a c e s f o r l a b o r a t o r y t e s t

s p e c i m e n s s u b j e c t e d t o r a n g e o f d i ff e r e n t l o a d i n g

c o n d i t i o n s

1 for T < 450C (840

K T = 1 - 0 . 0 0 5 8 ( T - 4 5 0) f o r 4 5 0 C < T < 5 5 0

1 - 0 . 0 0 3 2 ( T - 8 40 ) f o r 8 4 0 F < T < 10 2

( 5 - 2

T h e s e e q u a t i o n s a r e a p p l i c a b l e t o s t e e l . T h e s e c a n

b e u s e d f o r A 1 , M g , a n d C u a l l o y s .

Refe r to Fig . 5 -3A.

7/24/2019 67071_05.pdf

9/25

H i g h N o m i n a l S t r e s s L o w N o m i n a l S t r e s s

_ N o S t e r es s _ _ M i ld S t r es s _ _ Sev e r S t r es s _ _ N o S t r es s _ _ M i l d S t r es s _ _ Sev e r e S t r es s _

C o n c en t r a t i o n C o n c en t r a t i o n C o n c en t r a t i o n C o n c en t r a t io n C o n c en t r a t i o n C o n c en t r a t i o n

T e n s i o n - T e n s i o n o r T e n s i o n - C o m p r e s s i o n

NiN NiN NiN NiB

U n d i m c t i o n a B e n d i n g

.......... i ?:~. J

i'ilil NiN NiN

R e v e r s e d B e n d i n g

R o t a t i on a l B e n d i n g

T o r i s o n

r - ] S t r e s s C o n c e n t r a t i o n

F I G U R E 5 . 3 A T y p i c a l f r a c tu r e s u r f a ce s fo r l a b o r a t o r y t e s t s p e c i m e n s s u b j e c t e d to a r a n g e o f d i f fe r e n t l o a d i n g c o n d i t i

Courtesy:

R e p r o d u c e d f ro m

Metals Handbook,

V o l . 1 0 , 8 t h e d i t io n , p . 1 0 2 , A m e r i c a n S o c i e t y f o r M e t a l s , M e t a l s P a r k , O h i o , 1

7/24/2019 67071_05.pdf

10/25

5 . 1 0 C H A P T E R F IV E

Particular

Formula

T H E O R I E S O F F A I L U R E

T h e m a x i m u m n o r m a l s tr es s t h e or y o r R a n k in e ' s

t h e o r y

c r e : 1 [ ( c r x .q l_C r y ) + V / ( O ~ ; _ c r y ) 2 ._ _

4TZy

(5 -

T h e m a x i m u m s h e a r st re s s t h e o r y o r G u e s t ' s t h e o r y

T h e s h e a r - e n e r g y t h e o r y o r c o n s t a n t e n e r g y - o f -

d i s t o r ti o n o r H e n c k y - v o n M i s e s t h e o r y

T h e m a x i m u m s t ra i n th e o r y o r S a i n t V e n a n t ' s t h e o r y

T h e b e a r i n g s t re s s w h i c h c a u s e s f a i l u r e f o r n o f r i c t io n

a t t h e s u r f a c e o f c o n t a c t

The bea r ing s t r e ss wh ich cause s f a i l u re fo r t he f r i c t i on

a t t h e s u r f a c e o f c o n t a c t

C r e - - V / ( c r x - c r y ) 2

-+-47"2

O " - -- V / ( c r X - O ' y ) 2 --1 -3 7 - x 2 y

ere -- [(1 - u)(crx + Cry)

+ 1 - a t - / . / ) g / c r x - c r y )2

+ 4-r2y]

crb -- 1. 81 cre

crb = 2cre

(5 -

(5-

( 5 -

(5 -

(5 -

C Y C L I C L O A D S ( F ig s . 5 - 4 a nd 5 - 5)

T h e f a ti g u e s tr e s s - c o n c e n t r a t i o n f a c t o r f o r n o r m a l

st ress

The f a t i gue s t r e ss -concen t ra t i on f ac to r fo r shea r s t r e ss

T h e e m p i r i c al f o r m u l a f o r n o t c h s e n s it i v it y f o r a lt e r -

na t i ng s t r e ss o f s t e e l

N o t c h s e n s it i v it y c u r v e s f o r st ee l a n d a l u m i n u m a l l o y s

T h e e m p i r i c al f o r m u l a f o r n o t c h s e n s i ti v it y f o r

a l t e r n a t i n g s t r es s f o r h i g h - s t r e n g t h a l u m i n u m a l l o y s

having cru = 415 t o 550 M Pa (60 t o 80 kps i )

E n d u r a n c e s t r e n g t h f o r f i n i t e l i f e

T h e e m p i r i c a l r e l a t i o n b e t w e e n u l ti m a t e s t r e n g t h a n d

e n d u r a n c e l i m i ts f o r v a r i o u s m a t e r i a l s

K f ~ , = q f K ~ - 1) + 1

K f ~ _ = q f K ~ _ - 1) + 1

q f = 1 - e x p [ -

rcr2

0.904 x 106

Refe r t o F ig . 5 -6 .

( - r )

qf = 1 - exp

t ( 1 0 6 ) 0 "0 9

~ s = ~ y W -

where N = requ i red l i f e i n cyc l e s .

Re fe r t o Tab le s 5 -4 and 5 -5 .

(5-

( 5 -

(5 -

(5-

( 5 -

7/24/2019 67071_05.pdf

11/25

D E S I G N O F M A C H I N E E L E M E N T S F O R S T R E N G T H 5

~

W -

n ,

1 3 .

0

o

/

a )

R E V E R S E D S T R ES S

t

k _ q O ' r

l Y l

G m i n

= 0

b ) R E P E A T E D

S T R E S S

, ; ; : / % . . . . . m t J

I - ~ / ~ ~ ~ a - - - ~ '~ ,~ a x

C )

A L T E R N A T IV E S T R E S S

t

d ) F L U C T U A T I N G S T R E S S

FIG UR E 5-4 Typ es of fat igue s t ress var iat ions .

+ B

( ~ u / /

% . / / ,

O'max '3 .~0~( ,

~t ~ - ~ - - -

+ o ,

~ o . 6

/ o ,

. ~

/ ~ . - "

(Ymin

. . . . . . ~

0-4

a:: /

Z

S T E E L S

0 C~m o'y % 0 2

"~

ALUMINUM

' ALLOY

E ~ '~ 0 I I

M e a n

stress,

(~m

0 0 . 5 1 - 0 1 - 5 2 - 0 2 , 5 3 0 3 5

O Notch radius r , mm

- - O ' f

1 k g f /m m 2 = 9 . 8 0 6 6 N / m m 2

F I G U R E 5 -5 M o d i f i ed G o o d m a n d ia g r am .

FIG UR E 5-6 Notch- sens i t iv i ty curves for s t eel and alumin um all

7/24/2019 67071_05.pdf

12/25

5 . 1 2 C H A P T E R F I V E

T A B L E 5 - 4

E m p i r i c a l r e la t i o n s h i p b e twe e n u l t i m a te s tr e n g th a n d e n d ur a n c e l i m i t s f o r v a r i o us m a te r i a l s (a p p r o x i m a te )

Tension , compression, and bending Torsion

M aterial (reversed or repeated cycle) a (reversed or repeated cycle

G r a y c a s t i r o n oft = 0.6Crib to 0 .7a fb z = 0.75 afb to 0 .9afb

orb

= 1.2afb to 1 .5afb z =

1.2rf

t o

1.3TU

C a r b o n s t e e l s a o t - 1 .6a lb "7-0 = 1.8 TU t o 2Zf

aob

= 1.5afb

S t e e ls ( g e n e r a l ) oft - - 0 .7a f b t o 0 .8a f b Tf - - 0 .55af b t o 0 .58af b

Oft = 0 . 3 6 a ~ ; Oot = 0 . 5 a u Yf = 0 . 2 2 a u

af b = 0 .46 a~ ; aob = 0 . 6 a ~ % = 0 . 3 a u

A l l o y s t e e l s oft = 0 .95af b % - - 1.8Tf t o 2r f

aot-- 1.5crft t o 1.6aft

aob = 1 .6a~ ,

A l u m i n u m a l l oy s aot - - 0 .7a f b rf = 0.55~-fb to 0.58~-fb

aob = 1.8afb % = 1.4rf

to 2 Tf

C o p p e r a l l o y s -if = 0 . 5 8 a ~

To = 1.4~-f t

2r f

o.t ( 1 0 6 ) 0 .09

E n d u r a n c e s t r e n g t h f o r f i n it e l if e f =

a f

a f - - - e nsura nc e l im i t ( a l so for r e ve rse d cyc le ) ; o - - e ndura nc e for r e pe ate d c ycle; t - - t e ns ion; b- -be nding; u- -u l t im a te ; N- - nu m be r of c ycles

T A B L E 5 - 5

T h e e m p i r i c a l r e la t i o n f o r e n d ur a n c e l i m i t

Endurance limit,

a f

Material Bending Axial Torsion

F o r s t e e l a n d o t h e r f e r r o u s m a t e r i a l s [ f o r or. < 1 3 7 4 M P a ( 1 9 9 .5 k p s i ) ]

F o r n o n f e r r o u s m a t e r i a ls

1/2 -5 /8 au 7 /20-5 /8 au 7 /80-5 /32C

1/4 -1 /3au 7 /40-1 /3au 7/160-1 /12

S T R E S S - S T R E S S A N D S T R E S S - L O A D

R E L A T I O N S

A x i a l l o a d

T h e m a x i m u m s t r es s

T h e m i n i m u m s t r es s

T h e l o a d a m p l i t u d e

T h e m e a n l o a d

m a x

O'max A

F m i n

O'min ---

A

F m a x - - F m i n

F a - - 2

F m a x q - F m i n

F m = 2

( 5

( 5

( 5

( 5

7/24/2019 67071_05.pdf

13/25


P a r t i c u l a r

F o r m u l a

T h e s t r e ss a m p l i t u d e ( F i g s. 5 - 4 a n d 5 - 5 )

Fa (5 -

O ' a - - A

T h e m e a n s tr es s

O-m =

Fm

( 5 -

A

T h e r a t i o o f a m p l i t u d e s t re s s t o m e a n s tr e ss

O-__2_a= Fa

( 5 -

O-m Fm

T h e s t a t i c e q u i v a l e n t o f c y cl i c l o a d Fm + Fa

F'm = Fm + O-ja Fa ( 5 -

O-fd

T h e s t a t i c e q u i v a l e n t o f m e a n s t r e ss O-m i o"

' F~m (5 -

O ' m - - A

T h e G e r b e r p a r a b o l i c r e l a t i o n ( F i g . 5 - 7 )

)2

-a O-m

+ ~ = 1 (5 -

O'f O-ud

A._..... G

o" \ --< co ~,e

"o

= ~ ,~, ~ O o x

< ( e - < ~ e . < . . \

O ' a \ N

m

O (Ym ._ ~ Oy (Yut

n2

M ean s t re s s , Om

FIG UR E 5-7 Graph ica l representa t ion of s teady and variable s t resses .

T h e G o o d m a n r e l a t i o n ( F i gs . 5 -5 , 5 -7 , a n d 5 -9 )

T h e S o d e r b e r g r e l a t i o n ( F i g s. 5 - 7 a n d 5 - 8)

B e n d i n g l o a d s

T h e m a x i m u m s tr es s

T h e m i n i m u m s tr es s

T h e b en d in g m o m e n t a m p l it u de

O-a O-m

+ - - = 1

O- d O-ud

O-a O-m

- - - t - ~ - - 1

crfd ayd

O'max =

O-m in =---

Mba -~-

Mb(max)

Zb

M b ( m i n )

Zb

M b ( m a x ) - - M b ( m i n )

( 5

( 5 -

( 5 -

(5

( 5

7/24/2019 67071_05.pdf

14/25

5 . 1 4 C H A P T E R F IV E

Part i cular Formula

~ f

o'f

a

--ff-

Q .

E

Kfo .

or)

~ P (om ,K fa ,C a ~ ~

G m Gy Oy

n

Mean stress, G m

F I G U R E 5-8 Representat ion of safe l im i t of m ean s t ress and s tress am pl itude by Sod erberg cri ter ion.

T h e m e a n b en d in g m o m e n t

T h e b e n d i n g s tr e ss a m p l i t u d e

T h e m e a n b e n d i n g s tr e ss

T h e r a t i o o f s tr e ss a m p l i t u d e t o m e a n s tr e ss

T h e s ta t ic e q u i v a l e n t o f cy c li c b e n d i n g m o m e n t

M b m M b a

Th e s t a t i c equ iva le n t o f cyc l ic s tres s

T h e G e r b e r p a r a b o l i c r e l a t i o n ( F i g . 5 - 7 )

m b m

Mb a

O-ba -- Zb

bm

O'bm Zb

O'ba Mba

p

O' bm M bm

Mb(max) + Mb(m in)

M tbm - - M bm

+ O s-- - -~

M b a

aya

Zb

O'ba 0"2m

~ + - S - f - = 1

O'f grud

Th e G oo dm an s t ra ight - l ine re l a t ion (Figs . 5 -5 , 5 -7 , Crba

a n d 5 - 9 )

Ofd

The Soderberg s t ra ight - l ine re l a t ion (Figs . 5 -7 and 5-8)

T o r s i o n a l m o m e n t s

T h e m a x i m u m s h e ar st re ss

- ~ - O'bm 1

O'ud

O'ba O'bm

~ - ~ -

oyd Crya

= 1

7 ma x -.~

Mt(max)

Z t

( 5

(5

(5

( 5

(5 -

(5-

(5-

( 5 -

(5-

(5-

7/24/2019 67071_05.pdf

15/25


P a r t i c u l a r F o r m u l a

(~f

6

"o

O f

a . -fi-

E

K f d O a

O0

~ 0 / , % . O

. . . . .

O'm ~y O 'u O'y

n n

M e a n s t r e s s , 0 m

(Yu

FIGUR E 5-9 Representa t ion of safe l im i t of m ean st ress and st ress am pl i tude by Goo dm an cr i te r ion.

T h e m i n i m u m s h e ar st re ss

Trnin --

T h e l o a d a m p l i tu d e M t a - -

T h e m e a n l o a d m t m - -

T h e s h e a r s t r e ss a m p l i t u d e

T h e m e a n s h e a r s tr e ss

T h e r a t i o o f s tr e ss a m p l i t u d e t o m e a n s t re s s

T h e s t a ti c e q u i v a l e n t o f cy c li c t w i s ti n g m o m e n t

Mtm i Mta

T h e s t a t ic e q u i v a l e n t o f c y c li c s tr e s s

T h e G e r b e r p a r a b o l i c r e l a t i o n ( F i g . 5 -7 )

Mt(min)

Z t

m t a

7 " a = Z t

m t m

7 " m = Z t

Mt(m ax) - Mt(m in)

Mt(max) + Mt(min)

Ta Mta

7"m M t m

M~m = Mtm -+- Tsd Mtd

zfd

,

7 m =

Z t

Ta +L-T- = 1

7-fd Tud

T h e G o o d m a n s t r a i g h t - l in e r e l a t i o n ( F i g s . 5 - 5, 5 - 7 , %

and 5 -9 ) Tfd

T h e S o d e r b e r g s t r a i g h t - l in e r e l a t i o n ( F i g s . 5 - 7 a n d 5 - 8)

_ _ + r m = 1

rud

Ta +_ T m = l

rfd ryd

( 5 -

(5 -

(5 -

(5 -

(5 -

(5 -

( 5 -

( 5 -

( 5 -

(5 -

(5 -

7/24/2019 67071_05.pdf

16/25

5 . 1 6 C H A P T E R F IV E

Particular

Formula

T H E C O M B I N E D S T R E S S E S

M e t h o d 1

The s t a t i c equ iva l en t o f O"m "n- O"

Th e sta t ic e qu iva len t o f 7_m i %

T h e m a x i m u m n o r m a l s tr es s t h e o ry o r R a n k i n e 's

t h e o r y

T h e m a x i m u m s h e ar t he o r y o r C o u l o m b ' s o r T re s c a

c r i t e r i a o r G u e s t ' s t h e o r y

T h e d i s t o r t i o n e n e r g y t h e o r y o r H e n c k y - v o n M i s e s

t h e o r y

T h e m a x i m u m s t ra i n th e o r y o r S a i n t V e n a n t ' s t h e o r y

M e t h o d 2

T h e c o m b i n e d m a x i m u m n o r m a l st re ss

T h e c o m b i n e d m i n i m u m n o r m a l s tr es s

T h e co m b i n e d m a x i m u m s h e ar st re ss

T h e co m b i n e d m i n i m u m s h ea r st re ss

T h e c o m b i n e d m a x i m u m n o r m a l s tr es s a c c or d in g t o

s t r a i n t h e o r y

T h e co m b i n e d m i n i m u m n o r m a l s tr es s a c c or d in g t o

s t r a i n t h e o r y

T h e c o m b i n e d m a x i m u m o c t a h e d r a l s h ea r s tr es s

T h e c o m b i n e d m i n i m u m o c t a h e d r a l s h ea r s tr es s

T h e c o m b i n e d m e a n s t re s s

t Osd

O m = O m -I -~ O a

crfa

, rsa

7 - m = 7 - m + m %

7-Ja

e r e =

1 [ o " + V / d 2 + 4 r ~

7_,, = (O-'m2 + 4T ~

a , , = ~//Oam2+ 3r'm

. . . - ~ [ ~ 1 - - , - 'm + ~' + . , ; < + 4 m']

. .

[ 2 + 4 . 2 ]

.ma x __ 1 O.ma x _~ O.ma x

. .

2 l

.mi n = O'min -Jr - m in - I -4~ n i n

tt i v / 2 + 4 7 _ 2

7"max - O 'max m ax

,, V

_min _ / 2 9

O'mi n -I- 47"m i n

O ' ~ a x - - - l [ ( 1 - - L / / ) ' m a x + (1 + u)V/crZax + 47-ma x

" [ ; -

1

.mi n _ / (1 - / / ) O 'm i n + ( 1 + u) many + 4 rm i2

" V/a

m a x - 1 m a x - I- 3 T 2 a x

~m,n [ ~ /

_ 2 2

O'mi n -+- 37-m i n

I I I

tt O'max ~ O 'min

O-m - - -

2

(5 -

(5 -

(5 -

(5 -

(5

(s-

( 5 -

(5 -

(5-

(5-

( 5 -

(5 -

(5 -8

(5 -8

(s-8

7/24/2019 67071_05.pdf

17/25

DESIGN OF MACHINE ELEMENTS FOR STRENGTH 5

P a r t i c u l a r

F o r m u l a

The com bined s tress am pl i tude

The Gerber parabo l ic re la t ion (Fig . 5 -7 )

The G ood m an s t ra igh t - line re la tion (Figs . 5 -5 , 5 -7 ,

and 5-9 )

The Soderberg straight- l ine relation (Figs. 5-7 and 5-8)

C O M B I N E D S T R E S S E S I N T E R M S O F

L O A D S

M e t h o d 1

Max im u m sh ear s tr es s th eo r y

The shear energy theory

M e t h o d 2

Ma x im u m sh ear s tr es s t h eo r y

The shear energy theory

l/ I I

tt O ' m a x - - O ' m i n

(7" ~--

2

,, ( , , ) 2

O- O-m

~ +

~ = 1

O'f O'ud

l l

O" O"

~ - ~ - ~ = 1

Orf O'ud

I I I I

(7" O"

~ - ~ - ~ =

1

O'fd Oyd

(5-S

(5-8

(s-8

(5-8

' e - ~ l M t b m Y t m l 2 ( M ;m ) 2

F l e d - - - ~ b --~ ---~ + 4 - ~ t

O'e

~ M 1 b m

F m ) 2

f M ; m ) 2

= - ~ b + -- ~- + 3 \

Z t

ed

where

7rd3

z b = - ~ -

~-d2

A - ~

4

for solid shafts

7rd3

an d

Z t -

1 6

(5-8

(5-S

Mb(m ax) Fm ax Mt(max) 1 1

~ - ~ + ~

Zb Z t Tfa T

"-~-[~(Mb(min)-~-~-~12-'~-4(gt(min))2t

x - - - + - - = 2 ( 5-9

Tfd Td

[~(Mb(max ) -+-~-~ )2 -~ -3 (g t (max ) ) 2 ]t I ~ f d l J

+ [ ~ / ( M b ( m i n ) - k - ~ ) 2 + B ( M t ( m i n ) ) 2t

1 1

- - ~ + ~

~fa

= 2 (5-9

7/24/2019 67071_05.pdf

18/25

5 . 1 8 C H A P T E R F I V E

Particular

Formula

C R E E P

C r e e p i n t e n s i o n

W h e n t h e c u r v e fo r t o t a l c re e p e l is a p p r o x i m a t e d a s a

s t r a i g h t l i n e i t s e q u a t i o n i s

T h e c r e e p r a te ~ c a n b e a p p r o x i m a t e d b y t h e e q u a t i o n

C r e e p r a t e ~, w h e n e x t r a p o l a t e d i n t o t h e r e g io n

o f l o w e r s tr e ss e s, c a n b e d e t e r m i n e d w i t h g r e a t e r

a c c u r a c y b y t h e h y p e r b o l i c s i n e t e r m

T r u e s t r a i n

C r e e p l ife o f a l u m i n u m

T i m e f o r t h e st r e ss t o d e c r e a s e f r o m a n i n i t ia l v a l u e o f

a 0 t o a v a l u e o f a

e t = e o + e t ( 5 - 9

--- O o "n

( 5 - 9

R e f e r t o T a b l e 5 - 6 f o r c r e e p c o n s t a n t s B a n d n .

osin -

e ' = l n ( 1 + e )

1

g cr - - -~

1

t =

E B ( n -

1 ) 4 - l

a0 - 1

(7

( 5 - 9

( 5 - 9

( 5 -

( 5 -

C r e e p i n b e n d i n g

T h e m a x i m u m s t re s s a t th e e x t r e m e f ib e rs in ca s e o f

b e n d i n g o f b e a m is g i v e n b y t he r e l a t io n

T h e m a x i m u m d e fl e ct io n o f a c a n ti le v e r b e a m l o a d e d

a t f r e e e n d b y a l o a d F

C ) 1 / n

o r = ~ M b

( 5 -

Y m a x =

t F n I n + 2

D ( n + 2 )

( 5 -

h ) 2 n + 1

l ( 2 b ) "

w h e r e D = -

l .

B ( 2 + n

C r e e p c o n s t a n t s B a n d n a r e t a k e n f r o m T a b l e 5 -

T A B L E 5 - 6

Creep constants for va rious steels for use in Eqs. (5-91 b) to (5-9 5)

Steel Temperature

C

0 .39% C 400

0 .30% C 400

0 .45% C 475

2 % N i , 0 .8% Cr , 0 .4% Mo 450

2 % N i , 0 . 3% C, 1 .4% M n 450

1 2 % C r , 3 % W , 0 . 4 % M n 5 50

N i - C r - M o 5 0 0

N i - C r - M o 5 0 0

12 % Cr 455

14 x 1 0 - 3 6

44

1 0 - 3 0

10 x 1 0 - 1 9

2 1 1 0 - 2 2

2 4 1 0 - 1 4

12 x 1 0 - 1 6

16 x 10 -12

12 x 10 -22

8

6

6

3

4

1

2

1

4

7/24/2019 67071_05.pdf

19/25


Particular Formula

R E L I A B I L I T Y

T h e p r o b a b i l i t y f u n c t i o n o r f r e q u e n c y f u n c t i o n

T h e c u m u l a t i v e p r o b a b i l i t y f u n c t i o n

T h e s a m p l e m e a n o r a r it h m e t ic m e a n o f a s a m p l e

T h e p o p u l a t i o n m e a n o f a p o p u l a t i o n c o n s i s t i n g o f n

e l e m e n t s

T h e s a m p l e v a r i a n c e

A s u i t a b le e q u a t i o n f o r v a r i a n c e f o r u s e i n a c a lc u l a t o r

T h e s a m p l e s t a n d a r d d e v i a t i o n ( t h e s y m b o l u se d f o r

t r u e s t a n d a r d d e v i a t i o n is ~ )

A s u i t a b l e e q u a t i o n f o r s t a n d a r d d e v i a t i o n f o r u s e i n a

c a l c u l a t o r

T h e c o e f f ic i e n t o f v a r i a t i o n

T h e n o r m a l , o r G a u s s i a n , d i s t r ib u t i o n ( F i g . 5 - 10 )

T h e n o r m a l d i s t r i b u t io n a s d e f in e d b y p a r a m e t e r s , t h e

m e a n # a n d s t a n d a r d d e v i a ti o n ~ a c c o r d i n g t o t h e

r e l a t i o n f o r t h e r e l a t i v e f r e q u e n c y f ( t ) , w h i c h i s

t h e o r d i n a t e a t t

p = f ( x ) ( 5

F ( x j ) - - Z f ( x i ) ( 5 -

xi

7/24/2019 67071_05.pdf

20/25

5 . 2 0 C H A P T E R F IV E

Part i cular

Formula

x

>,,

to

t

(D

:3

Er

(D

_

ii

G 1

/

G3 > G2 > (~1 [

(3'2~~.

P X l X 2

X

F I G U R E 5 - 10 T h e s h a p es o f n o r m a l d i s tr i b u ti o n c u r ve s

f o r va r i ous a a nd c ons t a n t # .

T h e a r e a u n d e r n o r m a l d i s t r i b u t i o n c u r v e t o th e r ig h t

o f t ( F i g . 5 - 1 1 )

E r r o r f u n c t i o n o r p r o b a b i l i ty i n t e g r a l

T h e r e s u l ta n t m e a n o f a d d i n g t h e m e a n s o f tw o

p o p u l a t i o n s ( F i g . 5 - 1 2 )

o')

v

"o

i-

v

o

r-

E

II

I I

80 140

Strength (S) and Stressf(o)

F I G U R E 5 - 1 2 D i s t ri b u t io n c u r v es fo r tw o m e a n s o f

popu l a t i ons .

T h e r e s u l t a n t m e a n o f su b t r a c t in g t h e m e a n s o f t w o

p o p u l a t i o n s

T h e r e s u l t a n t s t a n d a r d d e v i a t i o n f o r b o t h s u b t r a c t io n

a n d a d d i t io n o f tw o s t a n d a r d d e v i a t io n s # s a n d ~

0 t

F I G U R E 5 - 1 1 T h e G a u s s i a n ( n o r m a l ) d i s tr i b u ti o n cu rv e .

R e f e r to T a b l e 5 - 8 f o r a r e a u n d e r t h e s t a n d a r d n o r m

d i s t r i b u t i o n c u r v e .

B ( t ) = 1 - A ( t )

( 5 - 1

w h e r e A ( t ) i s t h e a r e a t o t h e l e f t o f t .

T h e a r e a u n d e r t h e e n ti r e n o r m a l d i s t r i b u t i o n c u r v

A ( t ) + B ( t )

a n d is e q u a l t o u n i t y . T h e t e r m B ( t ) c a n

f o u n d f r o m T a b l e 5 -8 o r b y i n t e g r a t in g t h e a r e a u n

t h e c u r v e .

e r f ( x ) = - ~ e - ?

d t

( 5 - 1

R e f e r t o T a b l e 5 - 9 f o r e rf ( x ) f o r v a r i o u s v a l u e s o

/z = # s + #~ ( 5 - 1

# = # s - # ~ ( 5 - 1

c~ = 4 6 -2 + c~ (5 -1

7/24/2019 67071_05.pdf

21/25

DESIGN OF MACHINE ELEMENTS FOR STRENGTH 5

T A B L E 5 - 7

S t a n d a r d n o r m a l c u r ve o r d i n a t e s

1 - ? / 2

y = - - . ~ e

0 t

t 0 1 2 3 4 5 6 7 8 9

0.0 .3989 .3989 .3989 .3988 .3986 .3984 .3982 .3980 .3977 .3

0.1 .3970 .3965 .3961 .3956 .3951 .3945 .3939 .3932 .3925 .3

0.2 .3910 .3902 .2894 .3885 .3876 .3867 .3857 .3847 .3836 .3

0.3 .3814 .3802 .3790 .3778 .3765 .3752 .3739 .3725 .3712 .3

0.4 .3683 .3668 .3653 .3637 .3621 .3605 .3589 .3572 .3555 .3

0.5 .3521 .3503 .3485 .3467 .3448 .3429 .3410 .3391 .3372 .3

0.6 .3332 .3312 .3292 .3271 .3251 .3230 .3209 .3187 .3166 .3

0.7 .3123 .3101 .3079 .3056 .3034 .3011 .2989 .2966 .2943 .2

0.8 .2897 .2874 .2850 .2827 .2803 .2780 .2756 .2932 .2709 .2

0.9 .2661 .2637 .2613 .2589 .2565 .2541 .2516 .2492 .2468 .2

1.0 .2420 .2396 .2371 .2347 .2323 .2299 .2275 .2251 .2227 .2

1.1 .2179 .2155 .2131 .2107 .2083 .2059 .2036 .2012 .1989 .1

1.2 .1942 .1919 .1895 .1872 .1849 .1826 .1804 .1781 .1758 .1

1.3 .1714 .1691 .1669 .1647 .1626 .1604 .1528 .1561 .1539 .1

1.4 .1497 .1476 .1456 .1435 .1415 .1394 .1374 .1354 .1334 .1

1.5 .1295 .1276 .1257 .1238 .1219 .1200 .1i82 .1163 .1145 .1

1.6 .1109 .1092 .1074 .1057 .1040 .1023 .1006 .0989 .0973 .0

1.7 .0940 .0925 .0909 .0893 .0878 .0863 .0848 .0833 .0818 .0

1.8 .0790 .0775 .0761 .0748 .0734 .0721 .0707 .0694 .0681 .0

1.9 .0656 .0644 .0632 .0620 .0608 .0596 .0584 .0573 .0562 .0

2.0 .0540 .0529 .0519 .0508 .0498 .0488 .0487 .0468 .0459 .0

2.1 .0440 .0431 .0422 .0413 .0404 .0396 .0387 .0379 .0371 .0

2.2 .0355 .0347 .0339 .0332 .0325 .0317 .0310 .0303 .0297 .0

2.3 .0283 .0277 .0270 .0264 .0258 .0252 .0246 .0241 .0235 .0

2.4 .0224 .0219 .0213 .0208 .0203 .0198 .0194 .0189 .0184 .0

2.5 .0175 .0171 .0167 .0163 .0158 .0154 .0151 .0147 .0143 .0

2.6 .0136 .0132 .0129 .0126 .0122 .0119 .0116 .0113 .0110 .0

2.7 .0104 .0101 .0099 .0096 .0093 .0091 .0088 .0086 .0084 .0

2.8 .0079 .0077 .0075 .0073 .0071 .0069 .0067 .0065 .0063 .0

2.9 .0060 .0058 .0056 .0055 .0053 .0051 .0050 .0048 .0047 .0

3.0 .0044 .0043 .0042 .0040 .0039 .0038 .0037 .0036 .0035 .0

3.1 .0033 .0032 .0031 .0030 .0029 .0028 .0027 .0026 .0025 .0

3.2 .0024 .0023 .0022 .0022 .0021 .0020 .0020 .0019 .0018 .0

3.3 .0017 .0017 .0016 .0016 .0015 .0015 .0014 .0014 .0013 .0

3.4 .0012 .0012 .0012 .0011 .0011 .0010 .0010 .0010 .0009 .0

3.5 .0009 .0008 .0008 .0008 .0008 .0007 .0007 .0007 .0007 .0

3.6 ,0006 .0006 .0006 .0005 .0005 .0005 .0005 .0005 .0005 .0

3.7 .0004 .0004 .0004 .0004 .0004 .0004 .0003 .0003 .0003 .0

3.8 .0003 .0003 .0003 .0003 .0003 .0002 .0002 .0002 .0002 .0

3.9 .0002 .0002 .0002 .0002 .0002 .0002 .0002 .0002 .0001 .0

,

7/24/2019 67071_05.pdf

22/25

5.22 CHAPTER FIVE

TABLE 5-8

Areas under the standard normal distribution curve

Co 1 e_ ? 2

( t )= ~ at

0 + t

t 0 1 2 3 4 5 6 7 8 9

0.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0

0.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0

0.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1

0.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1

0.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1

0.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2

0.6 .2258 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2518 .2

0.7 .2580 .2612 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2

0.8 .2881 .2910 .2939 .2967 .2996 .3023 .3051 .3078 .3106 .3

0.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3

1.0 .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .3

1.1 .3643 .3665 .3686 .3708 .3729 .3749 .3770 .3790 .3810 .3

1.2 .3849 .3869 .3888 .3907 .3925 .3944 .3962 .3980 .3997 .4

1.3 .4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147 .4162 .4

1.4 .4192 .4207 .4222 .4236 .4251 .4265 .4279 .4292 .4306 .4

1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4

1.6 .4452 .4463 .4474 .4484 .4495 .4506 .4515 .4525 .4535 .4

1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4

1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4

1.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4

2.0 .4772 .4778 .4783 .4788 .4793 .4798 .4803 .4808 .4812 .4

2.1 .4821 .4826 .4830 .4834 .4838 .4842 .4846 .4850 .4854 .4

2.2 .4861 .4864 .4868 .4871 .4875 .4878 .4881 .4884 .4887 .4

2.3 .4893 .4896 .4898 .4901 .4904 .4906 .4909 .4911 .4913 .4

2.4 .4918 .4920 .4922 .4925 .4927 .4929 .4931 .4932 .4934 .4

2.5 .4938 .4940 .4941 .4943 .4945 .4946 .4948 .4949 .4951 .4

2.6 .4953 .4955 .4956 .4957 .4959 .4960 .4961 .4962 .4963 .4

2.7 .4965 .4966 .4967 .4968 .4969 .4970 .4971 .4972 .4973 .4

2.8 .4974 .4975 .4976 .4977 .4977 .4978 .4979 .4979 .4980 .4

2.9 .4981 .4982 .4982 .4983 .4984 .4984 .4985 .4985 .4986 .4

3.0 .4987 .4987 .4987 .4988 .4988 .4989 .4989 .4989 .4990 .4

3.1 .4990 .4991 .4991 .4991 .4992 .4992 .4992 .4992 .4993 .4

3.2 .4993 .4993 .4994 .4994 .4994 .4994 .4994 .4995 .4995 .4

3.3 .4995 .4995 .4995 .4996 .4996 .4996 .4996 .4996 .4996 .4

3.4 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .4

3.5 .4998 .4998 .4998 .4998 .4998 .4998 .4998 .4998 .4998 .4

3.6 .4998 .4998 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .4

3.7 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .4

3.8 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .4

3.9 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5

7/24/2019 67071_05.pdf

23/25

DESIGN OF MACHINE ELEMENTS FOR STRENGTH

5

T A B L E 5 - 9

E r r o r f u n c t i o n o r p r o b a b i l i t y i n t e g r a l

2 J .

rf(x) = ~ e - t 2 d t

x 0 1 2 3

4 5 6 7 8 9

0.0 .01128 .02256 .03384

0.1 .11246 .12362 .13476 .14587

0.2 .22270 .23352 .24430 .25502

0.3 .32863 .33891 .34913 .35928

0.4 .42839 .43797 .44747 .45689

0.5 .52050 .52924 .53790 .54646

0.6 .60386 .61168 .61941 .62705

0.7 .67780 .68467 .69143 .69810

0.8 .74210 .74800 .75381 .75952

0.9 .7 96 91 .80188 .80677 .81156

1.0 .84270 .84681 .85084 .85478

1.1 .88021 .88353 .88679 .88997

1.2 .91 03 1 .91296 .91553 .91805

1.3 .93401 .93606 .93807 .94002

1.4 .95229 .95385 .95538 .95686

1.5 .96 61 1 .96728 .96841 .96952

1.6 .97635 .97721 .97804 .97884

1.7 .98379 .9 84 41 .98500 .98558

1.8 .98909 .98952 .98994 .99035

1.9 .99279 .99309 .99338 .99366

2.0 .99532 .99552 .99572 .99591

2.1 .99702 .99715 .99728 .99741

2.2 .99814 .99822 .99831 .99839

2.3 .99886 .99891 .99897 .99902

2.4 .99 93 1 .99935 .99938 .99941

2.5 .99959 .99961 .99963 .99965

2.6 .99976 .99978 .99979 .99980

2.7 .99987 .99987 .99988 .99989

2.8 .99992 .99993 .99993 .99994

2.9 .99996 .99996 .99996 .99997

3.0 .99998

. . . .

.04511 .05637 .06762 .07886 .09008 .10

.15695 .16800 .17901 .18999 .20094 .21

.26570 .27633 .28690 .29742 .30788 .31

.36936 .37938 .38933 .39921 .40901 .41

.46623 .47548 .48466 .49375 .50275 .51

.55494 .56332 .57162 .57982 .58792 .59

.63459 .64203 .64938 .65663 .66378 .67

.70468 .71116 .71754 .72382 .73001 .73

.76514 .77067 .77610 .78144 .78669 .79

.81627 .82089 .82542 .82987 .83243 .83

.85865 .86244 .86614 .86977 .87333 .87

.89308 .89612 .89910 .90200 .90484 .90

.92051 .92290 .92524 .92751 .92973 .93

.94191 .94376 .94556 .94731 .94902 .95

.95830 .95970 .96105 .96237 .96365 .96

.97059 .97162 .97263 .97360 .97455 .97

.97962 .98038 .98110 .98181 .98249 .983

.98613 .98667 .98719 .98769 .98817 .988

.99074 .99111 .99147 .99182 .99216 .992

.99392 .99418 .99443 .99466 .99489 .995

.99609 .99626 .99642 .99658 .99673 .996

.99753 .99764 .99775 .99785 .99795 .998

.99846 .99854 .99861 .99867 .99874 .998

.99906 .99911 .99915 .99920 .99924 .999

.99944 .99947 .99950 .99952 .99955 .999

.99967 .99969 .99971 .99972 .99974 .999

.99981 .99982 .99983 .99984 .99985 .999

.99989 .99990 .99991 .99991 .99992 .999

.99994 .99994 .99995 .99995 .99995 .999

.99997 .99997 .99997 .99997 .99997 .999

The standard variable tR (deviation multiplication

factor) in order to determine the probability of failure

or the reliability

The reliability associated with tR

tR ---

(5-1

where subscripts s and ~ refer to strength and

stress, respectively.

R - O . 5 + A ( t R )

(5-1

where

A(tR)

is the area under a s tandard norma

distr ibution curve.

Refer to Tabl e 5-10 for typical values of R a

function of s tandar dized variable tR.

7/24/2019 67071_05.pdf

24/25

5 . 2 4 C H A P T E R F IV E

Part icu lar Formula

T A B L E 5 - 1 0

R e l i a b i l i t y R a s a f u n c t i o n o f t R

Survival rate

(R ) % t R

5O 0

90.00 1.288

95.00 1.645

98.00 2.050

99.00 2 .330

99.90 3.080

99.99 3.700

T h e f a t i g u e s t r e n g t h r e d u c t i o n f a c t o r b a s e d o n

r e l i a b i l i t y

I f a f a c t o r o f s a f e t y n ' i s t o b e s p e c i f i e d t o g e t h e r w i t h

r e l i a b il i ty , t h e n E q . ( 5 - 1 1 2 ) i s r e w r i t t e n t o g i v e a n e w

e x p r e s s i o n f o r t g

T h e e x p r e s s i o n f o r s a f e t y f a c to r n ' f r o m E q . ( 5 - 1 1 5 )

T h e b e s t - f i t t i n g s t r a i g h t l i n e w h i c h f it s a s e t o f

s c a t t e r e d d a t a p o i n t s a s p e r l i n e a r r e g r e s s i o n

T h e e q u a t i o n s f o r r e g r e s s i o n

T h e c o r r e l a t i o n c o e f f i c i e n t

A s a f e t y f a c t o r o f 1 i s t a k e n i n t o a c c o u n t i n d e t e r m

i n g t h e r e l ia b i l i ty f r o m E q . ( 5 - 1 1 3 ) .

C R - - 1 - 0 . 0 8 ( t R ) ( 5 - 1

w h e r e t R i s a l s o c a l l e d t h e d e v & t i o n m u l t i p l i c a t i

f a c t o r ( D M F ) , t a k e n f ro m T a b l e 5 -1 0.

I

#s - n t /za #s - n #a

t ~ = = ~ ( 5 - 1

' [ J l

' = - - # s - t R ~ 2 + # 2 ( 5- 11

1

= - - ( # s - t R # ) ( 5 - 1 1

y = m x + b ( 5 - 1

w h e r e m is th e s l o p e a n d b is t h e i n t e r c e p t o n t h

a x i s

E x E y

xy -

m = n ( 5 - 1 1

(Ex )

b = E y - m E x ( 5 - 1 1

n

ms x

r = ~ ( 5 - 1

Sy

w h e r e r l ie s b e t w e e n - 1 a n d + 1 .

I f r i s n e g a t i v e , i t i n d i c a t e s t h a t t h e r e g r e s s i o n l i n e

a n e g a t i v e s l o p e .

I f r = 1 , t h e r e i s a p e r fe c t c o r r e l a t i o n , a n d i f r - -

t h e r e i s n o c o r r e l a t i o n .

7/24/2019 67071_05.pdf

25/25

D E S IG N O F M A C H I N E E L E M E N T S F O R ST R EN G T H 5

T he e qua t ion f o r f r e que nc y o r de ns i ty f unc t ion

a c c o r d ing to W e ibu l l

: b ( X - - x o ) b- l ( e x p

f ( x ) O - x o o xo

( 5-1

T h e c u m u l a ti v e d i s t r ib u t i o n f u n c t io n

L

(x ) = f ( x ) dx = 1 - exp

0

( 5-1

E qu a t ion ( 5 -12 1 ) a f t e r s im p l i fi c a ti on

F ( x ) = 1 - exp

0)

R E F E R E N C E S

1 . M a le e v , V . L . , a nd J . B . H a r tm a n , Machine Des ign , I n t e r n a t i o n a l T e x t b o o k C o m p a n y , S c r a n t

Pennsylvania , 1954.

2 . Shig ley , J . E . , and L. D . M itche l l , Mechanic al Engineering Design , M c G r a w - H i l l B o o k C o m p a n y , N e w Y o

1983.

3. Faires, V. M. ,

Design of Mac hine Elem ents ,

T h e M a c m i ll a n C o m p a n y , N e w Y o r k , 1 96 5.

4 . L inga i a h , K . , a nd B . R . N a r a ya na Iye nga r ,

Machine Desig n Data Handbook,

E ng ine e r ing C o- ope r a

Soc ie ty , Banga lore , India , Banga lore , India , 1962 .

5 . L inga i a h , K . , a nd B . R . N a r a ya na Iye nga r ,

Machi ne Design Data Handbook,

Vol. I

(SI and Custom ary U n

S um a P ub l i she r s , B a nga lo r e , I nd i a , 1986 .

6 . L inga i a h , K . ,

Machine Design Data Handbook,

Vol. II

(SI and Custom ary Metr ic U n i t s ) ,

S u m a P u b l i s h

Banga lore , India , 1986.

7. Juvinall , R. C. , Fundamentals of Machine Com ponent Des ign , J ohn W i le y a nd S ons , N e w Y or k , 1983 .

8 . D e u t sc h m a n , A . D . , W . J . M ic he ls , a nd C . E . W i l son , Machine Design--Theory and Prac t i ce , M a c m i

P u b l i sh i n g C o m p a n y , N e w Y o r k , 1 97 5 .

9 . E dw a r ds , J r . , K . S . , a nd R . B . M c K e e , Fundamen ta ls o f Mechan i ca l Com ponen t Des ign , M c G r a w - H

P u b l i sh i n g C o m p a n y , N e w Y o r k , 1 99 1 .

10 . Nor ton , R . L . , Machine Design--An In tegrated Approach, P r e n t i c e H a l l I n t e r na t iona l , I nc . , U ppe r S a d

River , New Je r sey , 1996.

11 . Lin ga iah , K . Machine Design Data Handbook, M c G r a w - H i l l P u b l i sh i n g C o m p a n y , N e w Y o r k , 1 99 4.

12. Metals Handbook, A m e r i c a n S oc i e ty f o r M e ta l s , V o l . 10 , 8 th e d i ti on , p . 1 02 , M e ta l s P a r k , O h io , 1975 .

67071_05.pdf

Documents