ch 6 fatigue
DESCRIPTION
Materi kuliah fatigueTRANSCRIPT
Bud
ynas
−Nis
bett:
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gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
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hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
260
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
6Fa
tigue
Failu
re R
esultin
gfr
om
Vari
able
Loadin
g
Chapte
r O
utlin
e
6–1
Intro
duct
ion
to F
atig
ue in
Met
als
25
8
6–2
App
roac
h to
Fat
igue
Fai
lure
in A
naly
sis a
nd D
esig
n264
6–3
Fatig
ue-Li
fe M
etho
ds265
6–4
The
Stre
ss-Li
fe M
etho
d265
6–5
The
Stra
in-Li
fe M
etho
d268
6–6
The
Linea
r-Ela
stic
Frac
ture
Mec
hani
cs M
etho
d270
6–7
The
Endu
ranc
e Lim
it274
6–8
Fatig
ue S
treng
th275
6–9
Endu
ranc
e Lim
it M
odify
ing
Fact
ors
278
6–1
0St
ress
Con
cent
ratio
n an
d N
otch
Sen
sitiv
ity287
6–1
1C
hara
cter
izin
g Fl
uctu
atin
g St
ress
es292
6–1
2Fa
tigue
Fai
lure
Crit
eria
for F
luct
uatin
g St
ress
295
6–1
3To
rsio
nal F
atig
ue S
treng
th u
nder
Flu
ctua
ting
Stre
sses
309
6–1
4C
ombi
natio
ns o
f Loa
ding
Mod
es309
6–1
5Va
ryin
g, F
luct
uatin
g St
ress
es;
Cum
ulat
ive
Fatig
ue D
amag
e313
6–1
6Su
rface
Fat
igue
Stre
ngth
319
6–1
7St
ocha
stic
Ana
lysis
322
6–1
8Ro
ad M
aps
and
Impo
rtant
Des
ign
Equa
tions
for t
he S
tress
-Life
Met
hod
336
25
7
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
261
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
25
8M
echa
nica
l Eng
inee
ring
Des
ign
In C
hap.
5 w
e co
nsid
ered
the
ana
lysi
s an
d de
sign
of
part
s su
bjec
ted
to s
tatic
loa
ding
.T
he b
ehav
ior
of m
achi
ne p
arts
is
entir
ely
diff
eren
t w
hen
they
are
sub
ject
ed t
o tim
e-va
ryin
g lo
adin
g. I
n th
is c
hapt
er w
e sh
all e
xam
ine
how
par
ts f
ail u
nder
var
iabl
e lo
adin
gan
d ho
w to
pro
port
ion
them
to s
ucce
ssfu
lly r
esis
t suc
h co
nditi
ons.
6–1
Intr
oduct
ion t
o F
atigue
in M
etals
In m
ost
test
ing
of t
hose
pro
pert
ies
of m
ater
ials
tha
t re
late
to
the
stre
ss-s
trai
n di
agra
m,
the
load
is
appl
ied
grad
ually
, to
giv
e su
ffici
ent
time
for
the
stra
in t
o fu
lly d
evel
op.
Furt
herm
ore,
the
spec
imen
is te
sted
to d
estr
uctio
n, a
nd s
o th
e st
ress
es a
re a
pplie
d on
lyon
ce.
Test
ing
of t
his
kind
is
appl
icab
le,
to w
hat
are
know
n as
sta
tic
cond
itio
ns;
such
cond
ition
s cl
osel
y ap
prox
imat
e th
e ac
tual
con
ditio
ns t
o w
hich
man
y st
ruct
ural
and
mac
hine
mem
bers
are
sub
ject
ed.
The
cond
ition
freq
uent
lyar
ises
,ho
wev
er,
inw
hich
the
stre
sses
vary
with
tim
e or
they
fluct
uate
betw
een
diff
eren
t le
vels
.For
exam
ple,
apa
rtic
ular
fiber
onth
esu
rfac
eof
aro
tatin
gsh
afts
ubje
cted
toth
eac
tion
ofbe
ndin
glo
ads
unde
rgoe
sbo
thte
nsio
nan
dco
m-
pres
sion
for
each
revo
lutio
nof
the
shaf
t.If
the
shaf
tis
part
ofan
elec
tric
mot
orro
tatin
gat
1725
rev/
min
,the
fiber
isst
ress
edin
tens
ion
and
com
pres
sion
1725
times
each
min
ute.
If,i
nad
ditio
n,th
esh
afti
sal
soax
ially
load
ed(a
sit
wou
ldbe
,for
exam
ple,
bya
helic
alor
wor
mge
ar),
anax
ialc
ompo
nent
ofst
ress
issu
perp
osed
upon
the
bend
ing
com
pone
nt.
Inth
isca
se,s
ome
stre
ssis
alw
ays
pres
enti
nan
yon
efib
er,b
utno
wth
ele
velo
fst
ress
isflu
ctua
ting.
The
sean
dot
her
kind
sof
load
ing
occu
rrin
gin
mac
hine
mem
bers
prod
uce
stre
sses
that
are
calle
dva
riab
le,r
epea
ted,
alte
rnat
ing,
orflu
ctua
ting
stre
sses
.O
ften
, mac
hine
mem
bers
are
fou
nd t
o ha
ve f
aile
d un
der
the
actio
n of
rep
eate
d or
fluct
uatin
g st
ress
es;
yet
the
mos
t ca
refu
l an
alys
is r
evea
ls t
hat
the
actu
al m
axim
umst
ress
es w
ere
wel
l bel
ow th
e ul
timat
e st
reng
th o
f th
e m
ater
ial,
and
quite
fre
quen
tly e
ven
belo
w th
e yi
eld
stre
ngth
. The
mos
t dis
tingu
ishi
ng c
hara
cter
istic
of
thes
e fa
ilure
s is
that
the
stre
sses
hav
e be
en r
epea
ted
a ve
ry la
rge
num
ber
of ti
mes
. Hen
ce th
e fa
ilure
is c
alle
da
fati
gue
fail
ure.
Whe
nm
achi
nepa
rts
fail
stat
ical
ly,
they
usua
llyde
velo
pa
very
larg
ede
flect
ion,
beca
use
the
stre
ssha
sex
ceed
edth
eyi
eld
stre
ngth
,and
the
part
isre
plac
edbe
fore
frac
ture
actu
ally
occu
rs.T
hus
man
yst
atic
failu
res
give
visi
ble
war
ning
inad
vanc
e.B
uta
fatig
uefa
ilure
give
sno
war
ning
!Iti
ssu
dden
and
tota
l,an
dhe
nce
dang
erou
s.It
isre
lativ
ely
sim
-pl
eto
desi
gnag
ains
tast
atic
failu
re,b
ecau
seou
rkno
wle
dge
isco
mpr
ehen
sive
.Fat
igue
isa
muc
hm
ore
com
plic
ated
phen
omen
on,o
nly
part
ially
unde
rsto
od,a
ndth
een
gine
erse
ek-
ing
com
pete
nce
mus
tacq
uire
asm
uch
know
ledg
eof
the
subj
ecta
spo
ssib
le.
A f
atig
ue f
ailu
re h
as a
n ap
pear
ance
sim
ilar
to a
bri
ttle
frac
ture
, as
the
frac
ture
sur
-fa
ces
are
flat a
nd p
erpe
ndic
ular
to th
e st
ress
axi
s w
ith th
e ab
senc
e of
nec
king
. The
fra
c-tu
re f
eatu
res
of a
fat
igue
fai
lure
, how
ever
, are
qui
te d
iffe
rent
fro
m a
sta
tic b
rittl
e fr
actu
rear
isin
g fr
om th
ree
stag
es o
f de
velo
pmen
t. St
age
Iis
the
initi
atio
n of
one
or
mor
e m
icro
-cr
acks
due
to
cycl
ic p
last
ic d
efor
mat
ion
follo
wed
by
crys
tallo
grap
hic
prop
agat
ion
exte
ndin
g fr
om t
wo
to fi
ve g
rain
s ab
out
the
orig
in. S
tage
I c
rack
s ar
e no
t no
rmal
ly d
is-
cern
ible
to th
e na
ked
eye.
Sta
ge I
Ipr
ogre
sses
fro
m m
icro
crac
ks to
mac
rocr
acks
for
min
gpa
ralle
l pla
teau
-lik
e fr
actu
re s
urfa
ces
sepa
rate
d by
long
itudi
nal r
idge
s. T
he p
late
aus
are
gene
rally
sm
ooth
and
nor
mal
to th
e di
rect
ion
of m
axim
um te
nsile
str
ess.
The
se s
urfa
ces
can
be w
avy
dark
and
ligh
t ban
ds r
efer
red
to a
s be
ach
mar
ksor
clam
shel
l mar
ks,a
s se
enin
Fig
. 6–
1. D
urin
g cy
clic
loa
ding
, th
ese
crac
ked
surf
aces
ope
n an
d cl
ose,
rub
bing
toge
ther
, an
d th
e be
ach
mar
k ap
pear
ance
dep
ends
on
the
chan
ges
in t
he l
evel
or
fre-
quen
cy o
f lo
adin
g an
d th
e co
rros
ive
natu
re o
f th
e en
viro
nmen
t. St
age
III
occu
rs d
urin
gth
e fin
al s
tres
s cy
cle
whe
n th
e re
mai
ning
mat
eria
l can
not s
uppo
rt th
e lo
ads,
res
ultin
g in
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
262
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g25
9
Figure
6–1
Fatig
ue fa
ilure
of a
bol
t due
tore
peat
ed u
nidi
rect
iona
lbe
ndin
g. T
he fa
ilure
sta
rted
atth
e th
read
root
at A
,pr
opag
ated
acr
oss
mos
t of
the
cros
s se
ctio
n sh
own
byth
e be
ach
mar
ks a
t B,
befo
refin
al fa
st fra
ctur
e at
C.
(Fro
mA
SMH
andb
ook,
Vol
. 12
:Fr
acto
grap
hy,
ASM
Inte
r-na
tiona
l, M
ater
ials
Park
, OH
4407
3-00
02, fi
g 50
, p. 1
20.
Repr
inte
d by
perm
issi
on o
fA
SM In
tern
atio
nal ®
,w
ww
.asm
inte
rnat
iona
l.org
.)
1 See
the
ASM
Han
dboo
k, F
ract
ogra
phy,
ASM
Int
erna
tiona
l, M
etal
s Pa
rk, O
hio,
vol
. 12,
9th
ed.
, 198
7.
a su
dden
, fa
st f
ract
ure.
A s
tage
III
fra
ctur
e ca
n be
bri
ttle,
duc
tile,
or
a co
mbi
natio
n of
both
. Qui
te o
ften
the
beac
h m
arks
, if t
hey
exis
t, an
d po
ssib
le p
atte
rns
in th
e st
age
III f
rac-
ture
cal
led
chev
ron
lines
,poi
nt to
war
d th
e or
igin
s of
the
initi
al c
rack
s.T
here
is
a go
od d
eal
to b
e le
arne
d fr
om t
he f
ract
ure
patte
rns
of a
fat
igue
fai
lure
.1
Figu
re 6
–2 s
how
s re
pres
enta
tions
of
failu
re s
urfa
ces
of v
ario
us p
art
geom
etri
es u
nder
diff
erin
g lo
ad c
ondi
tions
and
lev
els
of s
tres
s co
ncen
trat
ion.
Not
e th
at,
in t
he c
ase
ofro
tatio
nal b
endi
ng, e
ven
the
dire
ctio
n of
rot
atio
n in
fluen
ces
the
failu
re p
atte
rn.
Fatig
ue f
ailu
re is
due
to c
rack
for
mat
ion
and
prop
agat
ion.
A f
atig
ue c
rack
will
typ-
ical
ly i
nitia
te a
t a
disc
ontin
uity
in
the
mat
eria
l w
here
the
cyc
lic s
tres
s is
a m
axim
um.
Dis
cont
inui
ties
can
aris
e be
caus
e of
:
•D
esig
n of
rap
id c
hang
es i
n cr
oss
sect
ion,
key
way
s, h
oles
, etc
. whe
re s
tres
s co
ncen
-tr
atio
ns o
ccur
as
disc
usse
d in
Sec
s. 3
–13
and
5–2.
•E
lem
ents
that
roll
and/
orsl
ide
agai
nst
each
othe
r(b
eari
ngs,
gear
s,ca
ms,
etc.
)un
der
high
cont
actp
ress
ure,
deve
lopi
ngco
ncen
trat
edsu
bsur
face
cont
acts
tres
ses
(Sec
.3–1
9)th
atca
nca
use
surf
ace
pitti
ngor
spal
ling
afte
rm
any
cycl
esof
the
load
.
•C
arel
essn
ess
in lo
catio
ns o
f st
amp
mar
ks, t
ool m
arks
, scr
atch
es, a
nd b
urrs
; poo
r jo
int
desi
gn; i
mpr
oper
ass
embl
y; a
nd o
ther
fab
rica
tion
faul
ts.
•C
ompo
sitio
n of
the
mat
eria
l its
elf a
s pr
oces
sed
by ro
lling
, for
ging
, cas
ting,
ext
rusi
on,
draw
ing,
hea
t tre
atm
ent,
etc.
Mic
rosc
opic
and
sub
mic
rosc
opic
sur
face
and
sub
surf
ace
disc
ontin
uitie
s ar
ise,
suc
h as
incl
usio
ns o
f fo
reig
n m
ater
ial,
allo
y se
greg
atio
n, v
oids
,ha
rd p
reci
pita
ted
part
icle
s, a
nd c
ryst
al d
isco
ntin
uitie
s.
Var
ious
cond
ition
sth
atca
nac
cele
rate
crac
kin
itiat
ion
incl
ude
resi
dual
tens
ilest
ress
es,
elev
ated
tem
pera
ture
s,te
mpe
ratu
recy
clin
g,a
corr
osiv
een
viro
nmen
t,an
dhi
gh-f
requ
ency
cycl
ing.
The
rate
and
dir
ectio
n of
fatig
ue c
rack
pro
paga
tion
is p
rim
arily
con
trol
led
by lo
cal-
ized
str
esse
s an
d by
the
str
uctu
re o
f th
e m
ater
ial
at t
he c
rack
. H
owev
er,
as w
ith c
rack
form
atio
n, o
ther
fac
tors
may
exe
rt a
sig
nific
ant
influ
ence
, su
ch a
s en
viro
nmen
t, te
m-
pera
ture
, and
fre
quen
cy. A
s st
ated
ear
lier,
crac
ks w
ill g
row
alo
ng p
lane
s no
rmal
to
the
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
263
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
260
Mec
hani
cal E
ngin
eerin
g D
esig
n
Figure
6–2
Sche
mat
ics
of fa
tigue
frac
ture
surfa
ces
prod
uced
in s
moo
than
d no
tche
d co
mpo
nent
s w
ithro
und
and
rect
angu
lar c
ross
sect
ions
und
er v
ario
us lo
adin
gco
nditi
ons
and
nom
inal
stre
ssle
vels.
(Fro
m A
SM H
andb
ook,
Vol.
11: F
ailu
re A
naly
sis a
ndPr
even
tion,
ASM
Inte
rnat
iona
l,M
ater
ials
Park
, OH
4407
3-00
02, fi
g 18
, p. 1
11.
Repr
inte
d by
per
miss
ion
ofA
SM In
tern
atio
nal®
,w
ww
.asm
inte
rnat
iona
l.org
.)
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
264
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g261
Figure
6–3
Fatig
ue fr
actu
re o
f an
AIS
I43
20 d
rive
shaf
t. Th
e fa
tigue
failu
re in
itiat
ed a
t the
end
of
the
keyw
ay a
t poi
nts
Ban
dpr
ogre
ssed
to fi
nal r
uptu
re a
tC
. The
fina
l rup
ture
zon
e is
smal
l, in
dica
ting
that
load
sw
ere
low
. (Fr
omA
SMH
andb
ook,
Vol
.11:
Fai
lure
Ana
lysis
and
Pre
vent
ion,
ASM
Inte
rnat
iona
l, M
ater
ials
Park
,O
H 4
4073
-000
2, fi
g 18
, p.
111
. Rep
rinte
d by
perm
issio
n of
ASM
Inte
rnat
iona
l ®,
ww
w.a
smin
tern
atio
nal.o
rg.)
Figure
6–4
Fatig
ue fr
actu
re s
urfa
ce o
f an
AIS
I 864
0 pi
n. S
harp
cor
ners
of th
e m
ism
atch
ed g
reas
eho
les
prov
ided
stre
ssco
ncen
tratio
ns th
at in
itiat
edtw
o fa
tigue
cra
cks
indi
cate
dby
the
arro
ws.
(Fro
mA
SMH
andb
ook,
Vol
.12:
Frac
togr
aphy
,ASM
Inte
rnat
iona
l, M
ater
ials
Park
,O
H 4
4073
-000
2, fi
g 52
0,p.
331
. Rep
rinte
d by
perm
issio
n of
ASM
Inte
rnat
iona
l ®,
ww
w.a
smin
tern
atio
nal.o
rg.)
max
imum
ten
sile
str
esse
s. T
he c
rack
gro
wth
pro
cess
can
be
expl
aine
d by
fra
ctur
em
echa
nics
(se
e Se
c. 6
–6).
A
maj
or
refe
renc
e so
urce
in
th
e st
udy
of
fatig
ue
failu
re
is
the
21-v
olum
e A
SMM
etal
s H
andb
ook.
Fig
ures
6–1
to
6–8,
rep
rodu
ced
with
per
mis
sion
fro
m A
SMIn
tern
atio
nal,
are
but
a m
inus
cule
sam
ple
of e
xam
ples
of
fatig
ue f
ailu
res
for
a gr
eat
vari
ety
of c
ondi
tions
incl
uded
in th
e ha
ndbo
ok. C
ompa
ring
Fig
. 6–3
with
Fig
. 6–2
, we
see
that
fai
lure
occ
urre
d by
rot
atin
g be
ndin
g st
ress
es,
with
the
dir
ectio
n of
rot
atio
nbe
ing
cloc
kwis
e w
ith r
espe
ct t
o th
e vi
ew a
nd w
ith a
mild
str
ess
conc
entr
atio
n an
d lo
wno
min
al s
tres
s.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
265
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
262
Mec
hani
cal E
ngin
eerin
g D
esig
n
Figure
6–5
Fatig
ue fr
actu
re s
urfa
ce o
f afo
rged
con
nect
ing
rod
of A
ISI
8640
ste
el. T
he fa
tigue
cra
ckor
igin
is a
t the
left
edge
, at t
hefla
sh li
ne o
f the
forg
ing,
but
no
unus
ual r
ough
ness
of t
he fl
ash
trim
was
indi
cate
d. T
hefa
tigue
cra
ck p
rogr
esse
dha
lfway
aro
und
the
oil h
ole
atth
e le
ft, in
dica
ted
by th
ebe
ach
mar
ks, b
efor
e fin
al fa
stfra
ctur
e oc
curre
d. N
ote
the
pron
ounc
ed s
hear
lip
in th
efin
al fr
actu
re a
t the
righ
t edg
e.(F
rom
ASM
Han
dboo
k,
Vol.
12: F
ract
ogra
phy,
ASM
Inte
rnat
iona
l, M
ater
ials
Park
,O
H 4
4073
-000
2, fi
g 52
3,
p. 3
32. R
eprin
ted
bype
rmiss
ion
of A
SMIn
tern
atio
nal ®
,w
ww
.asm
inte
rnat
iona
l.org
.)
Figure
6–6
Fatig
ue fr
actu
re s
urfa
ce o
f a 2
00-m
m (8
-in) d
iam
eter
pist
on ro
d of
an
allo
yste
el s
team
ham
mer
use
d fo
r for
ging
. Thi
s is
an e
xam
ple
of a
fatig
ue fr
actu
reca
used
by
pure
tens
ion
whe
re s
urfa
ce s
tress
con
cent
ratio
ns a
re a
bsen
t and
a cr
ack
may
initi
ate
anyw
here
in th
e cr
oss
sect
ion.
In th
is in
stanc
e, th
e in
itial
crac
k fo
rmed
at a
forg
ing
flake
slig
htly
bel
ow c
ente
r, gr
ew o
utw
ard
sym
met
rical
ly, a
nd u
ltim
atel
y pr
oduc
ed a
brit
tle fr
actu
re w
ithou
t war
ning
.(F
rom
ASM
Han
dboo
k, V
ol.1
2: F
ract
ogra
phy,
ASM
Inte
rnat
iona
l, M
ater
ials
Park
, OH
4407
3-00
02, fi
g 57
0, p
. 342
. Rep
rinte
d by
per
miss
ion
of A
SMIn
tern
atio
nal ®
, ww
w.a
smin
tern
atio
nal.o
rg.)
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
266
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g263
Figure
6–7
Fatig
ue fa
ilure
of a
n A
STM
A18
6 ste
el d
oubl
e-fla
nge
traile
r whe
el c
ause
d by
sta
mp
mar
ks. (
a) C
oke-
oven
car
whe
el s
how
ing
posit
ion
ofsta
mp
mar
ks a
nd fr
actu
res
in th
e rib
and
web
. (b)
Sta
mp
mar
k sh
owin
g he
avy
impr
essio
n an
d fra
ctur
e ex
tend
ing
alon
g th
e ba
se o
f the
low
erro
w o
f num
bers
. (c)
Not
ches
, ind
icat
ed b
y ar
row
s, c
reat
ed fr
om th
e he
avily
inde
nted
sta
mp
mar
ks fr
om w
hich
cra
cks
initi
ated
alo
ng th
e to
pat
the
fract
ure
surfa
ce. (
From
ASM
Han
dboo
k, V
ol.1
1: F
ailu
re A
naly
sis a
nd P
reve
ntio
n,A
SM In
tern
atio
nal,
Mat
eria
ls Pa
rk, O
H 4
4073
-00
02, fi
g 51
, p. 1
30. R
eprin
ted
by p
erm
issio
n of
ASM
Inte
rnat
iona
l ®, w
ww
.asm
inte
rnat
iona
l.org
.)
Alu
min
um a
lloy
7075
-T73
Roc
kwel
l B 8
5.5
Ori
gina
l des
ign
10.2
00
A
Lug
(1 o
f 2)
25.5
4.94
Frac
ture
3.62
dia
Seco
ndar
yfr
actu
re
1.75
0-in
.-di
abu
shin
g,0.
090-
in. w
all
Prim
ary-
frac
ture
surf
ace
Lub
rica
tion
hole
1 in
Lub
rica
tion
hole
Impr
oved
des
ign
Det
ail A
(a)
Figure
6–8
Alu
min
um a
lloy
7075
-T73
land
ing-
gear
torq
ue-a
rmas
sem
bly
rede
sign
to e
limin
ate
fatig
ue fr
actu
re a
t a lu
bric
atio
nho
le. (
a) A
rm c
onfig
urat
ion,
orig
inal
and
impr
oved
des
ign
(dim
ensio
ns g
iven
in in
ches
).(b
) Fra
ctur
e su
rface
whe
rear
row
s in
dica
te m
ultip
le c
rack
orig
ins.
(Fro
mA
SMH
andb
ook,
Vol
. 11:
Fai
lure
Ana
lysis
and
Pre
vent
ion,
ASM
Inte
rnat
iona
l, M
ater
ials
Park
,O
H 4
4073
-000
2, fi
g 23
, p.
114
. Rep
rinte
dby
perm
issio
n of
ASM
Inte
rnat
iona
l ®,
ww
w.a
smin
tern
atio
nal.o
rg.)
Med
ium
-car
bon
stee
l(A
STM
A18
6)
(a)
Cok
e-ov
en-c
ar w
heel
Web
30 d
ia
Flan
ge(1
of
2)Fr
actu
re
Tre
adFr
actu
re
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
267
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
264
Mec
hani
cal E
ngin
eerin
g D
esig
n
6–2
Appro
ach
toFa
tigue
Failu
rein
Analy
sis
and
Des
ign
As
note
d in
the
pre
viou
s se
ctio
n, t
here
are
a g
reat
man
y fa
ctor
s to
be
cons
ider
ed, e
ven
for
very
sim
ple
load
cas
es. T
he m
etho
ds o
f fa
tigue
fai
lure
ana
lysi
s re
pres
ent
a co
mbi
-na
tion
of e
ngin
eeri
ng a
nd s
cien
ce. O
ften
sci
ence
fai
ls to
pro
vide
the
com
plet
e an
swer
sth
at a
re n
eede
d. B
ut th
e ai
rpla
ne m
ust s
till b
e m
ade
to fl
y—sa
fely
. And
the
auto
mob
ilem
ust b
e m
anuf
actu
red
with
a r
elia
bilit
y th
at w
ill e
nsur
e a
long
and
trou
blef
ree
life
and
at t
he s
ame
time
prod
uce
profi
ts f
or t
he s
tock
hold
ers
of t
he i
ndus
try.
Thu
s, w
hile
sci
-en
ce h
as n
ot y
et c
ompl
etel
y ex
plai
ned
the
com
plet
e m
echa
nism
of
fatig
ue, t
he e
ngin
eer
mus
t stil
l des
ign
thin
gs th
at w
ill n
ot f
ail.
In a
sen
se th
is is
a c
lass
ic e
xam
ple
of th
e tr
uem
eani
ng o
f en
gine
erin
g as
con
tras
ted
with
sci
ence
. Eng
inee
rs u
se s
cien
ce to
sol
ve th
eir
prob
lem
s if
the
sci
ence
is
avai
labl
e. B
ut a
vaila
ble
or n
ot, t
he p
robl
em m
ust
be s
olve
d,an
d w
hate
ver
form
the
solu
tion
take
s un
der
thes
e co
nditi
ons
is c
alle
d en
gine
erin
g.In
thi
s ch
apte
r, w
e w
ill t
ake
a st
ruct
ured
app
roac
h in
the
des
ign
agai
nst
fatig
uefa
ilure
. As
with
sta
tic f
ailu
re, w
e w
ill a
ttem
pt to
rel
ate
to te
st r
esul
ts p
erfo
rmed
on
sim
-pl
y lo
aded
spe
cim
ens.
How
ever
, be
caus
e of
the
com
plex
nat
ure
of f
atig
ue,
ther
e is
muc
h m
ore
to a
ccou
nt f
or. F
rom
this
poi
nt, w
e w
ill p
roce
ed m
etho
dica
lly, a
nd in
sta
ges.
In a
n at
tem
pt to
pro
vide
som
e in
sigh
t as
to w
hat f
ollo
ws
in th
is c
hapt
er, a
bri
ef d
escr
ip-
tion
of th
e re
mai
ning
sec
tions
will
be
give
n he
re.
Fatigue-
Life
Met
hods
(Sec
s. 6
–3 t
o 6
–6)
Thr
ee m
ajor
app
roac
hes
used
in d
esig
n an
d an
alys
is to
pre
dict
whe
n, if
eve
r, a
cycl
ical
lylo
aded
mac
hine
com
pone
nt w
ill f
ail
in f
atig
ue o
ver
a pe
riod
of
time
are
pres
ente
d. T
hepr
emis
es o
f ea
ch a
ppro
ach
are
quite
dif
fere
nt b
ut e
ach
adds
to o
ur u
nder
stan
ding
of
the
mec
hani
sms
asso
ciat
ed w
ith f
atig
ue. T
he a
pplic
atio
n, a
dvan
tage
s, a
nd d
isad
vant
ages
of
each
met
hod
are
indi
cate
d. B
eyon
d Se
c. 6
–6,
only
one
of
the
met
hods
, th
e st
ress
-lif
em
etho
d, w
ill b
e pu
rsue
d fo
r fu
rthe
r de
sign
app
licat
ions
.
Fatigue
Stre
ngth
and t
he
Endura
nce
Lim
it (
Secs
. 6–7
and 6
–8)
The
stre
ngth
-lif
e(S
-N)
diag
ram
prov
ides
the
fatig
uest
reng
thS
fve
rsus
cycl
elif
eN
ofa
mat
eria
l.T
here
sults
are
gene
rate
dfr
omte
sts
usin
ga
sim
ple
load
ing
ofst
anda
rdla
bora
tory
-co
ntro
lled
spec
imen
s.T
helo
adin
gof
ten
isth
atof
sinu
soid
ally
reve
rsin
gpu
rebe
ndin
g.T
hela
bora
tory
-con
trol
led
spec
imen
sar
epo
lishe
dw
ithou
tge
omet
ric
stre
ssco
ncen
tra-
tion
atth
ere
gion
ofm
inim
umar
ea.
For
stee
l and
iron
, the
S-N
diag
ram
bec
omes
hor
izon
tal a
t som
e po
int.
The
str
engt
hat
this
poi
nt is
cal
led
the
endu
ranc
e lim
itS′ e
and
occu
rs s
omew
here
bet
wee
n 10
6an
d 10
7
cycl
es. T
he p
rim
e m
ark
on S
′ ere
fers
to th
e en
dura
nce
limit
of th
eco
ntro
lled
labo
rato
rysp
ecim
en.
For
nonf
erro
us m
ater
ials
tha
t do
not
exh
ibit
an e
ndur
ance
lim
it, a
fat
igue
stre
ngth
at a
spe
cific
num
ber o
f cyc
les,
S′ f, m
ay b
e gi
ven,
whe
re a
gain
, the
pri
me
deno
tes
the
fatig
ue s
tren
gth
of th
e la
bora
tory
-con
trol
led
spec
imen
.T
he s
tren
gth
data
are
bas
ed o
n m
any
cont
rolle
d co
nditi
ons
that
will
not
be
the
sam
eas
tha
t fo
r an
act
ual
mac
hine
par
t. W
hat
follo
ws
are
prac
tices
use
d to
acc
ount
for
the
diff
eren
ces
betw
een
the
load
ing
and
phys
ical
con
ditio
ns o
f th
e sp
ecim
en a
nd th
e ac
tual
mac
hine
par
t.
Endura
nce
Lim
it M
odif
yin
g F
act
ors
(Se
c. 6
–9)
Mod
ifyi
ng f
acto
rs a
re d
efine
d an
d us
ed t
o ac
coun
t fo
r di
ffer
ence
s be
twee
n th
e sp
eci-
men
and
the
act
ual
mac
hine
par
t w
ith r
egar
d to
sur
face
con
ditio
ns, s
ize,
loa
ding
, tem
-pe
ratu
re, r
elia
bilit
y, a
nd m
isce
llane
ous
fact
ors.
Loa
ding
is s
till c
onsi
dere
d to
be
sim
ple
and
reve
rsin
g.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
268
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g265
Stre
ss C
once
ntr
ation a
nd N
otc
h S
ensi
tivi
ty (
Sec.
6–1
0)
The
actu
alpa
rtm
ayha
vea
geom
etri
cst
ress
conc
entr
atio
nby
whi
chth
efa
tigue
beha
v-io
rde
pend
son
the
stat
icst
ress
conc
entr
atio
nfa
ctor
and
the
com
pone
ntm
ater
ial’s
sens
i-tiv
ityto
fatig
ueda
mag
e.
Fluct
uating S
tres
ses
(Sec
s. 6
–11 t
o 6
–13)
The
se s
ectio
ns a
ccou
nt f
or s
impl
e st
ress
sta
tes
from
fluc
tuat
ing
load
con
ditio
ns th
at a
reno
t pur
ely
sinu
soid
ally
rev
ersi
ng a
xial
, ben
ding
, or
tors
iona
l str
esse
s.
Com
bin
ations
of
Loadin
g M
odes
(Se
c. 6
–14)
Her
e a
proc
edur
e ba
sed
on th
e di
stor
tion-
ener
gy th
eory
is p
rese
nted
for
ana
lyzi
ng c
om-
bine
d flu
ctua
ting
stre
ss s
tate
s, s
uch
as c
ombi
ned
bend
ing
and
tors
ion.
Her
e it
isas
sum
ed th
at th
e le
vels
of
the
fluct
uatin
g st
ress
es a
re in
pha
se a
nd n
ot ti
me
vary
ing.
Vary
ing, Fl
uct
uating S
tres
ses;
Cum
ula
tive
Fatigue
Dam
age
(Sec
.6–1
5)
The
fluc
tuat
ing
stre
ss l
evel
s on
a m
achi
ne p
art
may
be
time
vary
ing.
Met
hods
are
pro
-vi
ded
to a
sses
s th
e fa
tigue
dam
age
on a
cum
ulat
ive
basi
s.
Rem
ain
ing S
ections
The
rem
aini
ng t
hree
sec
tions
of
the
chap
ter
pert
ain
to t
he s
peci
al t
opic
s of
sur
face
fatig
ue s
tren
gth,
sto
chas
tic a
naly
sis,
and
roa
dmap
s w
ith im
port
ant e
quat
ions
.
6–3
Fatigue-
Life
Met
hods
The
thr
ee m
ajor
fat
igue
lif
e m
etho
ds u
sed
in d
esig
n an
d an
alys
is a
re t
he s
tres
s-li
fem
etho
d,th
est
rain
-lif
e m
etho
d,an
d th
e li
near
-ela
stic
frac
ture
mec
hani
cs m
etho
d. T
hese
met
hods
atte
mpt
to p
redi
ct th
e lif
e in
num
ber
of c
ycle
s to
fai
lure
, N, f
or a
spe
cific
leve
lof
loa
ding
. L
ife
of 1
≤N
≤10
3cy
cles
is
gene
rally
cla
ssifi
ed a
s lo
w-c
ycle
fat
igue
,w
here
ashi
gh-c
ycle
fati
gue
is c
onsi
dere
d to
be
N>
103
cycl
es. T
he s
tres
s-lif
e m
etho
d,ba
sed
on s
tres
s le
vels
onl
y, i
s th
e le
ast
accu
rate
app
roac
h, e
spec
ially
for
low
-cyc
leap
plic
atio
ns. H
owev
er, i
t is
the
mos
t tra
ditio
nal m
etho
d, s
ince
it is
the
easi
est t
o im
ple-
men
t for
a w
ide
rang
e of
des
ign
appl
icat
ions
, has
am
ple
supp
ortin
g da
ta, a
nd r
epre
sent
shi
gh-c
ycle
app
licat
ions
ade
quat
ely.
The
str
ain-
life
met
hod
invo
lves
mor
e de
taile
d an
alys
is o
f th
e pl
astic
def
orm
atio
n at
loca
lized
reg
ions
whe
re t
he s
tres
ses
and
stra
ins
are
cons
ider
ed f
or l
ife
estim
ates
. Thi
sm
etho
d is
esp
ecia
lly g
ood
for
low
-cyc
le f
atig
ue a
pplic
atio
ns. I
n ap
plyi
ng t
his
met
hod,
seve
ral
idea
lizat
ions
mus
t be
com
poun
ded,
and
so
som
e un
cert
aint
ies
will
exi
st i
n th
ere
sults
. Fo
r th
is r
easo
n, i
t w
ill b
e di
scus
sed
only
bec
ause
of
its v
alue
in
addi
ng t
o th
eun
ders
tand
ing
of th
e na
ture
of
fatig
ue.
The
fra
ctur
e m
echa
nics
met
hod
assu
mes
a c
rack
is a
lrea
dy p
rese
nt a
nd d
etec
ted.
It
is th
en e
mpl
oyed
to p
redi
ct c
rack
gro
wth
with
res
pect
to s
tres
s in
tens
ity. I
t is
mos
t pra
c-tic
al w
hen
appl
ied
to l
arge
str
uctu
res
in c
onju
nctio
n w
ith c
ompu
ter
code
s an
d a
peri
-od
ic in
spec
tion
prog
ram
.
6–4
The
Stre
ss-L
ife
Met
hod
Tode
term
ine
the
stre
ngth
ofm
ater
ials
unde
rth
eac
tion
offa
tigue
load
s,sp
ecim
ens
are
subj
ecte
dto
repe
ated
orva
ryin
gfo
rces
ofsp
ecifi
edm
agni
tude
sw
hile
the
cycl
esor
stre
ssre
vers
als
are
coun
ted
tode
stru
ctio
n.T
hem
ostw
idel
yus
edfa
tigue
-tes
ting
devi
ce
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
269
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
266
Mec
hani
cal E
ngin
eerin
g D
esig
n
7 163
0.30
in
in
9in
R.
7 8
Figure
6–9
Test-
spec
imen
geo
met
ry fo
r the
R. R
. Moo
re ro
tatin
g-be
am m
achi
ne. T
he b
endi
ng m
omen
t is
unifo
rm o
ver t
hecu
rved
at t
he h
ighe
st-str
esse
d po
rtion
, a v
alid
test
ofm
ater
ial,
whe
reas
a fr
actu
re e
lsew
here
(not
at t
he h
ighe
st-str
ess
leve
l) is
grou
nds
for s
uspi
cion
of m
ater
ial fl
aw.
100
50100
101
102
103
104
105
106
107
108
Num
ber
of s
tres
s cy
cles
, N
S e
S ut
Fatigue strength Sf, kpsi
Low
cyc
leH
igh
cycl
e
Fini
te li
feIn
fini
te
life
Figure
6–1
0
An
S-N
diag
ram
plo
tted
from
the
resu
lts o
f com
plet
ely
reve
rsed
axi
al fa
tigue
tests
.M
ater
ial:
UN
S G
4130
0ste
el, n
orm
aliz
ed;
S ut=
116
kpsi;
max
imum
S ut=
125
kpsi.
(Dat
a fro
mN
AC
ATe
ch. N
ote
3866
,D
ecem
ber 1
966.
)
isth
eR
.R.M
oore
high
-spe
edro
tatin
g-be
amm
achi
ne.T
his
mac
hine
subj
ects
the
spec
imen
topu
rebe
ndin
g(n
otr
ansv
erse
shea
r)by
mea
nsof
wei
ghts
.T
hesp
ecim
en,
show
nin
Fig.
6–9,
isve
ryca
refu
llym
achi
ned
and
polis
hed,
with
afin
alpo
lishi
ngin
anax
ial
dire
ctio
nto
avoi
dci
rcum
fere
ntia
lsc
ratc
hes.
Oth
erfa
tigue
-tes
ting
mac
hine
sar
eav
ail-
able
for
appl
ying
fluct
uatin
gor
reve
rsed
axia
lst
ress
es,t
orsi
onal
stre
sses
,or
com
bine
dst
ress
esto
the
test
spec
imen
s.To
est
ablis
h th
e fa
tigue
str
engt
h of
a m
ater
ial,
quite
a n
umbe
r of
test
s ar
e ne
cess
ary
beca
use
of th
e st
atis
tical
nat
ure
of f
atig
ue. F
or th
e ro
tatin
g-be
am te
st, a
con
stan
t ben
d-in
g lo
ad is
app
lied,
and
the
num
ber o
f rev
olut
ions
(str
ess
reve
rsal
s) o
f the
bea
m re
quir
edfo
r fa
ilure
is r
ecor
ded.
The
firs
t tes
t is
mad
e at
a s
tres
s th
at is
som
ewha
t und
er th
e ul
ti-m
ate
stre
ngth
of
the
mat
eria
l. T
he s
econ
d te
st i
s m
ade
at a
str
ess
that
is
less
tha
n th
atus
ed in
the
first
. Thi
s pr
oces
s is
con
tinue
d, a
nd th
e re
sults
are
plo
tted
as a
n S-
Ndi
agra
m(F
ig. 6
–10)
. Thi
s ch
art m
ay b
e pl
otte
d on
sem
ilog
pape
r or
on
log-
log
pape
r. In
the
case
of f
erro
us m
etal
s an
d al
loys
, the
gra
ph b
ecom
es h
oriz
onta
l af
ter
the
mat
eria
l ha
s be
enst
ress
ed f
or a
cer
tain
num
ber
of c
ycle
s. P
lotti
ng o
n lo
g pa
per
emph
asiz
es t
he b
end
inth
ecu
rve,
whi
ch m
ight
not
be
appa
rent
if
the
resu
lts w
ere
plot
ted
by u
sing
Car
tesi
anco
ordi
nate
s.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
270
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g267
80 70 60 50 40 35 30 25 20 18 16 14 12 10 8 7 6 5 103
104
105
106
107
108
109
Lif
eN
, cyc
les
(log
)
Peak alternating bending stress S, kpsi (log)
Sand
cas
tPerm
anen
t mol
d ca
st
Wro
ught
Figure
6–1
1
S-N
band
s fo
r rep
rese
ntat
ive
alum
inum
allo
ys, e
xclu
ding
wro
ught
allo
ys w
ithS u
t<
38kp
si.(F
rom
R. C
.Ju
vina
ll,En
gine
erin
gC
onsid
erat
ions
of S
tress
,St
rain
and
Stre
ngth
.Cop
yrig
ht©
196
7 by
The
McG
raw
-Hill
Com
pani
es, I
nc. R
eprin
ted
bype
rmiss
ion.
)
The
ord
inat
e of
the
S-N
diag
ram
is
calle
d th
e fa
tigu
e st
reng
thS
f;
a st
atem
ent
ofth
is s
tren
gth
valu
e m
ust a
lway
s be
acc
ompa
nied
by
a st
atem
ent o
f th
e nu
mbe
r of
cyc
les
Nto
whi
ch it
cor
resp
onds
.So
on w
e sh
all l
earn
that
S-N
diag
ram
s ca
n be
det
erm
ined
eith
er f
or a
test
spe
cim
enor
for
an
actu
al m
echa
nica
l el
emen
t. E
ven
whe
n th
e m
ater
ial
of t
he t
est
spec
imen
and
that
of
the
mec
hani
cal
elem
ent
are
iden
tical
, th
ere
will
be
sign
ifica
nt d
iffe
renc
esbe
twee
n th
e di
agra
ms
for
the
two.
In t
he c
ase
of t
he s
teel
s, a
kne
e oc
curs
in
the
grap
h, a
nd b
eyon
d th
is k
nee
failu
rew
ill n
ot o
ccur
, no
mat
ter
how
gre
at t
he n
umbe
r of
cyc
les.
The
str
engt
h co
rres
pond
ing
to th
e kn
ee is
cal
led
the
endu
ranc
e li
mit
S e, o
r th
e fa
tigue
lim
it. T
he g
raph
of
Fig.
6–1
0ne
ver
does
bec
ome
hori
zont
al f
or n
onfe
rrou
s m
etal
s an
d al
loys
, and
hen
ce th
ese
mat
e-ri
als
do n
ot h
ave
an e
ndur
ance
lim
it. F
igur
e 6–
11 s
how
s sc
atte
r ban
ds in
dica
ting
the
S-N
curv
es f
or m
ost
com
mon
alu
min
um a
lloys
exc
ludi
ng w
roug
ht a
lloys
hav
ing
a te
nsile
stre
ngth
bel
ow 3
8 kp
si. S
ince
alu
min
um d
oes
not h
ave
an e
ndur
ance
lim
it, n
orm
ally
the
fatig
ue s
tren
gth
Sf
is r
epor
ted
at a
spe
cific
num
ber
of c
ycle
s, n
orm
ally
N=
5(10
8)
cycl
es o
f re
vers
ed s
tres
s (s
ee T
able
A–2
4).
We
note
that
a s
tres
s cy
cle
(N=
1)co
nstit
utes
a s
ingl
e ap
plic
atio
n an
d re
mov
al o
fa
load
and
the
n an
othe
r ap
plic
atio
n an
d re
mov
al o
f th
e lo
ad i
n th
e op
posi
te d
irec
tion.
Thu
sN
=1 2
mea
ns t
he l
oad
is a
pplie
d on
ce a
nd t
hen
rem
oved
, whi
ch i
s th
e ca
se w
ithth
e si
mpl
e te
nsio
n te
st.
The
bod
y of
kno
wle
dge
avai
labl
e on
fat
igue
fai
lure
fro
m N
=1
toN
=10
00cy
cles
is g
ener
ally
cla
ssifi
ed a
s lo
w-c
ycle
fati
gue,
as in
dica
ted
in F
ig. 6
–10.
Hig
h-cy
cle
fati
gue,
then
, is
conc
erne
d w
ith f
ailu
re c
orre
spon
ding
to
stre
ss c
ycle
s gr
eate
r th
an 1
03
cycl
es.
We
also
dis
tingu
ish
a fin
ite-
life
reg
ion
and
an i
nfini
te-l
ife
regi
onin
Fig
. 6–1
0. T
hebo
unda
ry b
etw
een
thes
e re
gion
s ca
nnot
be
clea
rly
defin
ed e
xcep
t for
a s
peci
fic m
ater
ial;
but i
tlie
s so
mew
here
bet
wee
n 10
6an
d10
7cy
cles
for
ste
els,
as
show
n in
Fig
. 6–1
0.A
s no
ted
prev
ious
ly,
it is
alw
ays
good
eng
inee
ring
pra
ctic
e to
con
duct
a t
estin
gpr
ogra
m o
n th
e m
ater
ials
to b
e em
ploy
ed in
des
ign
and
man
ufac
ture
. Thi
s, in
fac
t, is
are
quir
emen
t, no
t an
opt
ion,
in
guar
ding
aga
inst
the
pos
sibi
lity
of a
fat
igue
fai
lure
.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
271
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
268
Mec
hani
cal E
ngin
eerin
g D
esig
n
Bec
ause
of
this
nec
essi
ty f
or t
esti
ng,
it w
ould
rea
lly
be u
nnec
essa
ry f
or u
s to
pro
ceed
any
furt
her
in th
e st
udy
of fa
tigu
e fa
ilur
e ex
cept
for
one
impo
rtan
t rea
son:
the
desi
re to
know
why
fat
igue
fai
lure
s oc
cur
so t
hat
the
mos
t ef
fect
ive
met
hod
or m
etho
ds c
an b
eus
ed t
o im
prov
e fa
tigu
e st
reng
th.
Thu
s ou
r pr
imar
y pu
rpos
e in
stu
dyin
g fa
tigue
is
toun
ders
tand
why
fai
lure
s oc
cur
so t
hat
we
can
guar
d ag
ains
t th
em i
n an
opt
imum
man
-ne
r. Fo
r th
is r
easo
n, t
he a
naly
tical
des
ign
appr
oach
es p
rese
nted
in
this
boo
k, o
r in
any
othe
r bo
ok, f
or th
at m
atte
r, do
not
yie
ld a
bsol
utel
y pr
ecis
e re
sults
. The
res
ults
sho
uld
beta
ken
as a
gui
de, a
s so
met
hing
tha
t in
dica
tes
wha
t is
im
port
ant
and
wha
t is
not
im
por-
tant
in d
esig
ning
aga
inst
fat
igue
fai
lure
.A
s st
ated
ear
lier
, the
str
ess-
life
met
hod
is t
he l
east
acc
urat
e ap
proa
ch e
spec
iall
yfo
r lo
w-c
ycle
app
lica
tion
s. H
owev
er,
it i
s th
e m
ost
trad
itio
nal
met
hod,
wit
h m
uch
publ
ishe
d da
ta a
vail
able
. It
is
the
easi
est
to i
mpl
emen
t fo
r a
wid
e ra
nge
of d
esig
nap
plic
atio
ns a
nd r
epre
sent
s hi
gh-c
ycle
app
lica
tion
s ad
equa
tely
. For
the
se r
easo
ns t
hest
ress
-lif
e m
etho
d w
ill
be
emph
asiz
ed
in
subs
eque
nt
sect
ions
of
th
is
chap
ter.
How
ever
, car
e sh
ould
be
exer
cise
d w
hen
appl
ying
the
met
hod
for l
ow-c
ycle
app
licat
ions
,as
the
met
hod
does
not
acc
ount
for
the
tru
e st
ress
-str
ain
beha
vior
whe
n lo
cali
zed
yiel
ding
occ
urs.
6–5
The
Stra
in-L
ife
Met
hod
The
bes
t app
roac
h ye
t adv
ance
d to
exp
lain
the
natu
re o
f fat
igue
failu
re is
cal
led
by s
ome
the
stra
in-l
ife
met
hod.
The
app
roac
h ca
n be
use
d to
est
imat
e fa
tigue
str
engt
hs, b
ut w
hen
it is
so
used
it
is n
eces
sary
to
com
poun
d se
vera
l id
ealiz
atio
ns, a
nd s
o so
me
unce
rtai
n-tie
s w
ill e
xist
in th
e re
sults
. For
this
rea
son,
the
met
hod
is p
rese
nted
her
e on
ly b
ecau
seof
its
valu
e in
exp
lain
ing
the
natu
re o
f fa
tigue
.A
fat
igue
fai
lure
alm
ost
alw
ays
begi
ns a
t a
loca
l di
scon
tinui
ty s
uch
as a
not
ch,
crac
k, o
r ot
her
area
of
stre
ss c
once
ntra
tion.
Whe
n th
e st
ress
at t
he d
isco
ntin
uity
exc
eeds
the
elas
tic l
imit,
pla
stic
str
ain
occu
rs. I
f a
fatig
ue f
ract
ure
is t
o oc
cur,
ther
e m
ust
exis
tcy
clic
pla
stic
str
ains
. Thu
s w
e sh
all
need
to
inve
stig
ate
the
beha
vior
of
mat
eria
ls s
ub-
ject
to c
yclic
def
orm
atio
n.In
1910
,Bai
rsto
wve
rifie
dby
expe
rim
entB
ausc
hing
er’s
theo
ryth
atth
eel
astic
lim-
itsof
iron
and
stee
lcan
bech
ange
d,ei
ther
upor
dow
n,by
the
cycl
icva
riat
ions
ofst
ress
.2
Inge
nera
l,th
eel
astic
limits
ofan
neal
edst
eels
are
likel
yto
incr
ease
whe
nsu
bjec
ted
tocy
cles
ofst
ress
reve
rsal
s,w
hile
cold
-dra
wn
stee
lsex
hibi
tade
crea
sing
elas
ticlim
it.R
. W. L
andg
raf
has
inve
stig
ated
the
low
-cyc
le f
atig
ue b
ehav
ior
of a
lar
ge n
umbe
rof
ver
y hi
gh-s
tren
gth
stee
ls, a
nd d
urin
g hi
s re
sear
ch h
e m
ade
man
y cy
clic
str
ess-
stra
inpl
ots.
3Fi
gure
6–1
2 ha
s be
en c
onst
ruct
ed to
sho
w th
e ge
nera
l app
eara
nce
of th
ese
plot
sfo
r th
e fir
st f
ew c
ycle
s of
con
trol
led
cycl
ic s
trai
n. I
n th
is c
ase
the
stre
ngth
dec
reas
esw
ith s
tres
s re
petit
ions
, as
evid
ence
d by
the
fac
t th
at t
he r
ever
sals
occ
ur a
t ev
er-s
mal
ler
stre
ss l
evel
s. A
s pr
evio
usly
not
ed,
othe
r m
ater
ials
may
be
stre
ngth
ened
, in
stea
d, b
ycy
clic
str
ess
reve
rsal
s.T
he S
AE
Fat
igue
Des
ign
and
Eva
luat
ion
Stee
ring
Com
mitt
ee r
elea
sed
a re
port
in
1975
in
whi
ch t
he l
ife
in r
ever
sals
to
failu
re i
s re
late
d to
the
str
ain
ampl
itude
�ε/2 .
4
2 L. B
airs
tow
, “T
he E
last
ic L
imits
of
Iron
and
Ste
el u
nder
Cyc
lic V
aria
tions
of
Stre
ss,”
Phi
loso
phic
alTr
ansa
ctio
ns,S
erie
s A
, vol
. 210
, Roy
al S
ocie
ty o
f L
ondo
n, 1
910,
pp.
35–
55.
3 R. W
. Lan
dgra
f, C
ycli
c D
efor
mat
ion
and
Fati
gue
Beh
avio
r of
Har
dene
d St
eels
,Rep
ort n
o. 3
20, D
epar
tmen
tof
The
oret
ical
and
App
lied
Mec
hani
cs, U
nive
rsity
of
Illin
ois,
Urb
ana,
196
8, p
p. 8
4–90
.4 Te
chni
cal R
epor
t on
Fati
gue
Pro
pert
ies,
SAE
J10
99, 1
975.
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ynas
−Nis
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hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
272
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g269
4th 2d
1st r
ever
sal
3d 5th
A
B
Δ�
Δ�p
Δ�e
Δ�
�
�
Figure
6–1
2
True
stre
ss–t
rue
strai
n hy
stere
sislo
ops
show
ing
the
first
five
stres
s re
vers
als
of a
cyc
lic-
softe
ning
mat
eria
l. Th
e gr
aph
is sli
ghtly
exa
gger
ated
for
clar
ity. N
ote
that
the
slope
of
the
line
AB
is th
e m
odul
us o
fel
astic
ityE.
The
stre
ss ra
nge
is�
σ,�
εp
is th
e pl
astic
-stra
inra
nge,
and
�εe
is th
eel
astic
strai
n ra
nge.
The
tota
l-stra
in ra
nge
is�
ε=
�ε
p+
�εe.
The
rep
ort c
onta
ins
a pl
ot o
f th
is r
elat
ions
hip
for
SAE
102
0 ho
t-ro
lled
stee
l; th
e gr
aph
has
been
rep
rodu
ced
as F
ig.
6–13
. To
expl
ain
the
grap
h, w
e fir
st d
efine
the
fol
low
ing
term
s:
•Fa
tigu
e du
ctil
ity
coef
ficie
nt ε
′ Fis
the
tru
e st
rain
cor
resp
ondi
ng t
o fr
actu
re i
n on
e re
-ve
rsal
(po
int A
in F
ig. 6
–12)
. The
pla
stic
-str
ain
line
begi
ns a
t thi
s po
int i
n Fi
g. 6
–13.
•Fa
tigu
e st
reng
th c
oeffi
cien
t σ
′ Fis
the
tru
e st
ress
cor
resp
ondi
ng t
o fr
actu
re i
n on
ere
vers
al (
poin
t Ain
Fig
. 6–1
2). N
ote
in F
ig. 6
–13
that
the
elas
tic-s
trai
n lin
e be
gins
at
σ′ F/
E.
•Fa
tigu
e du
ctil
ity
expo
nent
cis
the
slo
pe o
f th
e pl
astic
-str
ain
line
in F
ig. 6
–13
and
isth
e po
wer
to
whi
ch t
he l
ife
2Nm
ust
be r
aise
d to
be
prop
ortio
nal
to t
he t
rue
plas
tic-
stra
in a
mpl
itude
. If
the
num
ber
of s
tres
s re
vers
als
is 2
N,
then
Nis
the
num
ber
ofcy
cles
.
100
10–
4
10–3
10–2
10–1
100
101
102
103
104
105
106
Rev
ersa
ls to
fai
lure
, 2N
Strain amplitude, Δ�/2
�' F
c
1.0
b1.
0
�' F E
Tota
l str
ain
Plas
tic s
trai
n
Ela
stic
str
ain
Figure
6–1
3
A lo
g-lo
g pl
ot s
how
ing
how
the
fatig
ue li
fe is
rela
ted
toth
etru
e-str
ain
ampl
itude
for
hot-r
olle
d SA
E 10
20 s
teel
.(R
eprin
ted
with
per
miss
ion
from
SA
E J1
099_
2002
08
© 2
002
SAE
Inte
rnat
iona
l.)
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ynas
−Nis
bett:
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gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
273
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
270
Mec
hani
cal E
ngin
eerin
g D
esig
n
•Fa
tigu
e st
reng
th e
xpon
entb
is th
e sl
ope
of th
e el
astic
-str
ain
line,
and
is th
e po
wer
tow
hich
the
life
2Nm
ust b
e ra
ised
to b
e pr
opor
tiona
l to
the
true
-str
ess
ampl
itude
.
Now
, fro
m F
ig. 6
–12,
we
see
that
the
tota
l str
ain
is th
e su
m o
f the
ela
stic
and
pla
stic
com
pone
nts.
The
refo
re th
e to
tal s
trai
n am
plitu
de is
hal
f th
e to
tal s
trai
n ra
nge
�ε 2
=�
εe
2+
�ε
p
2(a
)
The
equ
atio
n of
the
plas
tic-s
trai
n lin
e in
Fig
. 6–1
3 is
�ε
p
2=
ε′ F(2
N)c
(6–1
)
The
equ
atio
n of
the
elas
tic s
trai
n lin
e is
�ε
e
2=
σ′ F E(2
N)b
(6–2
)
The
refo
re, f
rom
Eq.
(a)
, we
have
for
the
tota
l-st
rain
am
plitu
de
�ε 2
=σ
′ F E(2
N)b
+ε
′ F(2
N)c
(6–3
)
whi
ch i
s th
e M
anso
n-C
offin
rel
atio
nshi
p be
twee
n fa
tigue
lif
e an
d to
tal
stra
in.5
Som
eva
lues
of
the
coef
ficie
nts
and
expo
nent
s ar
e lis
ted
in T
able
A–2
3. M
any
mor
e ar
ein
clud
ed in
the
SAE
J10
99 r
epor
t.6
Tho
ugh
Eq.
(6–
3) is
a p
erfe
ctly
legi
timat
e eq
uatio
n fo
r ob
tain
ing
the
fatig
ue li
fe o
fa
part
whe
n th
e st
rain
and
oth
er c
yclic
cha
ract
eris
tics
are
give
n, i
t ap
pear
s to
be
of l
it-tle
use
to
the
desi
gner
. The
que
stio
n of
how
to
dete
rmin
e th
e to
tal
stra
in a
t th
e bo
ttom
of a
not
ch o
r dis
cont
inui
ty h
as n
ot b
een
answ
ered
. The
re a
re n
o ta
bles
or c
hart
s of
str
ain
conc
entr
atio
n fa
ctor
s in
the
liter
atur
e. I
t is
poss
ible
that
str
ain
conc
entr
atio
n fa
ctor
s w
illbe
com
e av
aila
ble
in r
esea
rch
liter
atur
e ve
ry s
oon
beca
use
of t
he i
ncre
ase
in t
he u
se o
ffin
ite-e
lem
ent a
naly
sis.
Mor
eove
r, fin
ite e
lem
ent a
naly
sis
can
of it
self
app
roxi
mat
e th
est
rain
s th
at w
ill o
ccur
at a
ll po
ints
in th
e su
bjec
t str
uctu
re.7
6–6
The
Linea
r-El
ast
ic F
ract
ure
Mec
hanic
s M
ethod
The
first
phas
eof
fatig
uecr
acki
ngis
desi
gnat
edas
stag
eI
fatig
ue.
Cry
stal
slip
that
exte
nds
thro
ugh
seve
ralc
ontig
uous
grai
ns,i
nclu
sion
s,an
dsu
rfac
eim
perf
ectio
nsis
pre-
sum
edto
play
aro
le.S
ince
mos
toft
his
isin
visi
ble
toth
eob
serv
er,w
eju
stsa
yth
atst
age
Iin
volv
esse
vera
lgr
ains
.The
seco
ndph
ase,
that
ofcr
ack
exte
nsio
n,is
calle
dst
age
IIfa
tigue
.The
adva
nce
ofth
ecr
ack
(tha
tis,
new
crac
kar
eais
crea
ted)
does
prod
uce
evi-
denc
eth
atca
nbe
obse
rved
onm
icro
grap
hsfr
oman
elec
tron
mic
rosc
ope.
The
grow
thof
5 J. F
. Tav
erne
lli a
nd L
. F. C
offin
, Jr.,
“E
xper
imen
tal S
uppo
rt f
or G
ener
aliz
ed E
quat
ion
Pred
ictin
g L
ow C
ycle
Fatig
ue,’’
and
S. S
. Man
son,
dis
cuss
ion,
Tra
ns. A
SME
, J. B
asic
Eng
.,vo
l. 84
, no.
4, p
p. 5
33–5
37.
6 See
also
, Lan
dgra
f, I
bid.
7 For
furt
her
disc
ussi
on o
f th
e st
rain
-lif
e m
etho
d se
e N
. E. D
owlin
g, M
echa
nica
l Beh
avio
r of
Mat
eria
ls,
2nd
ed.,
Pren
tice-
Hal
l, E
ngle
woo
d C
liffs
, N.J
., 19
99, C
hap.
14.
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ynas
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ign,
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hth
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on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
274
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g271
the
crac
kis
orde
rly.
Fina
lfra
ctur
eoc
curs
duri
ngst
age
IIIf
atig
ue,a
lthou
ghfa
tigue
isno
tin
volv
ed.
Whe
nth
ecr
ack
issu
ffici
ently
long
that
KI=
KIc
for
the
stre
ssam
plitu
dein
volv
ed,t
hen
KIc
isth
ecr
itica
lst
ress
inte
nsity
for
the
unda
mag
edm
etal
,and
ther
eis
sudd
en,
cata
stro
phic
failu
reof
the
rem
aini
ngcr
oss
sect
ion
inte
nsile
over
load
(see
Sec.
5–12
).St
age
III
fatig
ueis
asso
ciat
edw
ithra
pid
acce
lera
tion
ofcr
ack
grow
thth
enfr
actu
re.
Cra
ck G
row
thFa
tigu
e cr
acks
nuc
leat
e an
d gr
ow w
hen
stre
sses
var
y an
d th
ere
is s
ome
tens
ion
inea
ch s
tres
s cy
cle.
Con
side
r th
e st
ress
to
be fl
uctu
atin
g be
twee
n th
e li
mit
s of
σm
inan
dσ
max
, w
here
the
str
ess
rang
e is
defi
ned
as �
σ=
σm
ax−
σm
in.
Fro
m E
q. (
5–37
) th
est
ress
inte
nsit
y is
giv
en b
y K
I=
βσ√ π
a.T
hus,
for
�σ,
the
stre
ss in
tens
ity
rang
e pe
rcy
cle
is
�K
I=
β(σ
max
−σ
min)√ π
a=
β�
σ√ π
a(6
–4)
To d
evel
op f
atig
ue s
tren
gth
data
, a n
umbe
r of
spe
cim
ens
of th
e sa
me
mat
eria
l are
test
edat
var
ious
lev
els
of �
σ.
Cra
cks
nucl
eate
at
or v
ery
near
a f
ree
surf
ace
or l
arge
dis
con-
tinui
ty. A
ssum
ing
an i
nitia
l cr
ack
leng
th o
f a i
,cr
ack
grow
th a
s a
func
tion
of t
he n
um-
ber
of s
tres
s cy
cles
Nw
ill d
epen
d on
�σ,
that
is, �
KI.
For �
KI
belo
w s
ome
thre
shol
dva
lue
(�K
I)th
a cr
ack
will
not
gro
w.
Figu
re 6
–14
repr
esen
ts t
he c
rack
len
gth
aas
afu
nctio
n of
N
for
thre
e st
ress
le
vels
(�
σ) 3
>(�
σ) 2
>(�
σ) 1
, w
here
(�
KI)
3>
(�K
I)2
>(�
KI)
1. N
otic
e th
e ef
fect
of
the
high
er s
tres
s ra
nge
in F
ig. 6
–14
in t
he p
ro-
duct
ion
of lo
nger
cra
cks
at a
par
ticul
ar c
ycle
cou
nt.
Whe
n th
e ra
te o
f cr
ack
grow
th p
er c
ycle
, da/
dN
in F
ig. 6
–14,
is p
lotte
d as
sho
wn
in F
ig.
6–15
, th
e da
ta f
rom
all
thre
e st
ress
ran
ge l
evel
s su
perp
ose
to g
ive
a si
gmoi
dal
curv
e. T
he t
hree
sta
ges
of c
rack
dev
elop
men
t ar
e ob
serv
able
, and
the
sta
ge I
I da
ta a
relin
ear
on l
og-l
og c
oord
inat
es,
with
in t
he d
omai
n of
lin
ear
elas
tic f
ract
ure
mec
hani
cs(L
EFM
) va
lidity
. A g
roup
of
sim
ilar
curv
es c
an b
e ge
nera
ted
by c
hang
ing
the
stre
ssra
tioR
=σ
min/σ
max
of th
e ex
peri
men
t.H
ere
we
pres
ent
a si
mpl
ified
pro
cedu
re f
or e
stim
atin
g th
e re
mai
ning
lif
e of
a c
ycli-
cally
str
esse
d pa
rt a
fter
dis
cove
ry o
f a c
rack
. Thi
s re
quir
es th
e as
sum
ptio
n th
at p
lane
str
ain
Log
N
Stre
ss c
ycle
s N
Crack length a
a a i
(ΔK
I)3
(ΔK
I)2
(ΔK
I)1
da
dN
Figure
6–1
4
The
incr
ease
in c
rack
leng
th a
from
an
initi
al le
ngth
of a
ias
a fu
nctio
n of
cyc
le c
ount
for
thre
e str
ess
rang
es, (
�σ
) 3>
(�σ
) 2>
(�σ
) 1.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
275
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
272
Mec
hani
cal E
ngin
eerin
g D
esig
n
cond
ition
s pr
evai
l.8A
ssum
ing
a cr
ack
is d
isco
vere
d ea
rly
in s
tage
II,
the
crac
k gr
owth
inre
gion
II
of F
ig. 6
–15
can
be a
ppro
xim
ated
by
the
Pari
s eq
uatio
n,w
hich
is o
f th
e fo
rm
da
dN
=C
(�K
I)m
(6–5
)
whe
reC
and
mar
e em
piri
cal
mat
eria
l co
nsta
nts
and
�K
Iis
giv
en b
y E
q. (
6–4)
.R
epre
sent
ativ
e, b
ut c
onse
rvat
ive,
val
ues
of C
and
mfo
r va
riou
s cl
asse
s of
ste
els
are
liste
d in
Tab
le 6
–1. S
ubst
itutin
g E
q. (
6–4)
and
inte
grat
ing
give
s
∫ Nf
0d
N=
Nf
=1 C
∫ a f a i
da
(β�
σ√ π
a)m
(6–6
)
Her
ea i
is t
he i
nitia
l cr
ack
leng
th, a
fis
the
fina
l cr
ack
leng
th c
orre
spon
ding
to
failu
re,
and
Nf
is t
he e
stim
ated
num
ber
of c
ycle
s to
pro
duce
a f
ailu
re a
fter
the
initi
al c
rack
is
form
ed. N
ote
that
βm
ay v
ary
in th
e in
tegr
atio
n va
riab
le (
e.g.
, see
Fig
s. 5
–25
to 5
–30)
.
Log
ΔK
Log
da dN
Incr
easi
ngst
ress
rat
ioR
Cra
ckpr
opag
atio
n
Reg
ion
II
Cra
ckin
itiat
ion
Reg
ion
I
Cra
ckun
stab
le
Reg
ion
III
(ΔK
) th
Kc
Figure
6–1
5
Whe
nda
/d
Nis
mea
sure
d in
Fig.
6–1
4 an
d pl
otte
d on
logl
og c
oord
inat
es, t
he d
ata
for d
iffer
ent s
tress
rang
essu
perp
ose,
givi
ng ri
se to
asig
moi
d cu
rve
as s
how
n.(�
KI) t
his
the
thre
shol
d va
lue
of�
KI,
belo
w w
hich
a c
rack
does
not
gro
w. F
rom
thre
shol
dto
rupt
ure
an a
lum
inum
allo
yw
ill sp
end
85--9
0 pe
rcen
t of
life
in re
gion
I, 5
--8 p
erce
nt in
regi
on II
, and
1--2
per
cent
inre
gion
III.
Table
6–1
Con
serv
ativ
e Va
lues
of
Fact
orC
and
Expo
nent
min
Eq.
(6–5
) for
Vario
us F
orm
s of
Ste
el(R
. =0)
Mate
rial
C,
m/c
ycl
e( M
Pa
√ m) m
C,
in/c
ycl
e( k
psi
√ in) m
m
Ferri
tic-p
earli
tic s
teel
s6.
89(1
0−12
)3.
60(1
0−10
)3.
00M
arte
nsiti
c ste
els
1.36
(10−
10)
6.60
(10−
9)
2.25
Aus
teni
tic s
tain
less
ste
els
5.61
(10−
12)
3.00
(10−
10)
3.25
From
J.M. B
arsom
and S
.T. Ro
lfe,F
atigu
e and
Frac
ture C
ontro
l in St
ructur
es, 2
nd ed
.,Pre
ntice
Hall,
Uppe
r Sad
dle Ri
ver, N
J, 19
87,
pp. 2
88–2
91, C
opyri
ght A
STM
Intern
ation
al. Re
printe
d with
perm
ission
.
8 Rec
omm
ende
d re
fere
nces
are
: Dow
ling,
op.
cit.
; J. A
. Col
lins,
Fai
lure
of M
ater
ials
in M
echa
nica
l Des
ign,
John
Wile
y &
Son
s, N
ew Y
ork,
198
1; H
. O. F
uchs
and
R. I
. Ste
phen
s, M
etal
Fat
igue
in E
ngin
eeri
ng,J
ohn
Wile
y &
Son
s, N
ew Y
ork,
198
0; a
nd H
arol
d S.
Ree
msn
yder
, “C
onst
ant A
mpl
itude
Fat
igue
Lif
e A
sses
smen
tM
odel
s,”
SAE
Tra
ns. 8
2068
8,vo
l. 91
, Nov
. 198
3.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
276
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g273
If t
his
shou
ld h
appe
n, t
hen
Ree
msn
yder
9su
gges
ts t
he u
se o
f nu
mer
ical
int
egra
tion
empl
oyin
g th
e al
gori
thm
δa j
=C
(�K
I)m j(δ
N) j
a j+1
=a j
+δa j
Nj+
1=
Nj+
δN
j(6
–7)
Nf=∑
δN
j
Her
eδa j
and
δN
jar
e in
crem
ents
of
the
crac
k le
ngth
and
the
num
ber
of c
ycle
s. T
he p
ro-
cedu
re i
s to
sel
ect
a va
lue
of δ
Nj,
usi
ng a
ide
term
ine
βan
d co
mpu
te �
KI,
dete
rmin
eδa j
, and
then
find
the
next
val
ue o
f a.
Rep
eat t
he p
roce
dure
unt
il a
=a
f.
The
fol
low
ing
exam
ple
is h
ighl
y si
mpl
ified
with
βco
nsta
nt i
n or
der
to g
ive
som
eun
ders
tand
ing
of th
e pr
oced
ure.
Nor
mal
ly, o
ne u
ses
fatig
ue c
rack
gro
wth
com
pute
r pro
-gr
ams
such
as
NA
SA/F
LA
GR
O 2
.0 w
ith m
ore
com
preh
ensi
ve t
heor
etic
al m
odel
s to
solv
e th
ese
prob
lem
s.
EXA
MPLE
6–1
The
bar
sho
wn
in F
ig. 6
–16
is s
ubje
cted
to
a re
peat
ed m
omen
t 0
≤M
≤12
00 l
bf·in
.T
he b
ar i
s A
ISI
4430
ste
el w
ith S
ut=
185
kpsi
, Sy=
170
kpsi
, and
KIc
=73
kpsi√ in
.M
ater
ial
test
s on
var
ious
spe
cim
ens
of t
his
mat
eria
l w
ith i
dent
ical
hea
t tr
eatm
ent
indi
cate
wor
st-c
ase
cons
tant
s of
C=
3.8(
10−1
1) (
in/c
ycle
)�(k
psi√ in
)man
dm
=3.
0. A
ssh
own,
a n
ick
of s
ize
0.00
4 in
has
bee
n di
scov
ered
on
the
botto
m o
f th
e ba
r. E
stim
ate
the
num
ber
of c
ycle
s of
life
rem
aini
ng.
Solu
tion
The
str
ess
rang
e �
σis
alw
ays
com
pute
d by
usi
ng th
e no
min
al (
uncr
acke
d) a
rea.
Thu
s
I c=
bh2
6=
0.25
(0.5
)2
6=
0.01
042
in3
The
refo
re, b
efor
e th
e cr
ack
initi
ates
, the
str
ess
rang
e is
�σ
=�
M
I/c
=12
00
0.01
042
=11
5.2(
103)
psi
=11
5.2
kpsi
whi
ch i
s be
low
the
yie
ld s
tren
gth.
As
the
crac
k gr
ows,
it
will
eve
ntua
lly b
ecom
e lo
ngen
ough
suc
h th
at th
e ba
r will
com
plet
ely
yiel
d or
und
ergo
a b
rittl
e fr
actu
re. F
or th
era
tioof
S y/
S ut
it is
hig
hly
unlik
ely
that
the
bar
will
rea
ch c
ompl
ete
yiel
d. F
or b
rittl
e fr
actu
re,
desi
gnat
e th
e cr
ack
leng
th a
s a
f.
If β
=1,
then
fro
m E
q. (
5–37
) w
ith K
I=
KIc
, w
eap
prox
imat
ea
fas
af
=1 π
( KIc
βσ
max
) 2. =
1 π
( 73
115.
2
) 2 =0.
1278
in
Figure
6–1
6
MM
Nic
k
in1 2
in1 4
9 Op.
cit.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
277
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
274
Mec
hani
cal E
ngin
eerin
g D
esig
n
From
Fig
. 5–2
7, w
e co
mpu
te th
e ra
tio a
f/
has
af h
=0.
1278
0.5
=0.
256
Thu
sa
f/
hva
ries
from
nea
r zer
o to
app
roxi
mat
ely
0.25
6. F
rom
Fig
. 5–2
7, fo
r thi
s ra
nge
βis
nea
rly
cons
tant
at a
ppro
xim
atel
y 1.
07. W
e w
ill a
ssum
e it
to b
e so
, and
re-
eval
uate
af
as
af
=1 π
(73
1.07
(115
.2)
) 2 =0.
112
in
Thu
s, f
rom
Eq.
(6–
6), t
he e
stim
ated
rem
aini
ng li
fe is
Nf=
1 C
∫ a f a i
da
(β�
σ√ π
a)m
=1
3.8(
10−1
1)
∫ 0.11
2
0.00
4
da
[1.0
7(11
5.2)
√ πa]
3
=−
5.04
7(10
3)
√ a
∣ ∣ ∣ ∣0.11
2
0.00
4
=64
.7(1
03)
cycl
es
6–7
The
Endura
nce
Lim
itT
hede
term
inat
ion
ofen
dura
nce
limits
byfa
tigue
test
ing
isno
wro
utin
e,th
ough
ale
ngth
ypr
oced
ure.
Gen
eral
ly,s
tres
ste
stin
gis
pref
erre
dto
stra
inte
stin
gfo
ren
dura
nce
limits
.Fo
r pr
elim
inar
y an
d pr
otot
ype
desi
gn a
nd f
or s
ome
failu
re a
naly
sis
as w
ell,
a qu
ick
met
hod
of e
stim
atin
g en
dura
nce
limits
is
need
ed. T
here
are
gre
at q
uant
ities
of
data
in
the
liter
atur
e on
the
resu
lts o
f ro
tatin
g-be
am te
sts
and
sim
ple
tens
ion
test
s of
spe
cim
ens
take
n fr
om th
e sa
me
bar
or in
got.
By
plot
ting
thes
e as
in F
ig. 6
–17,
it is
pos
sibl
e to
see
whe
ther
the
re i
s an
y co
rrel
atio
n be
twee
n th
e tw
o se
ts o
f re
sults
. The
gra
ph a
ppea
rs t
osu
gges
t th
at t
he e
ndur
ance
lim
it ra
nges
fro
m a
bout
40
to 6
0 pe
rcen
t of
the
ten
sile
stre
ngth
for
ste
els
up to
abo
ut 2
10 k
psi (
1450
MPa
). B
egin
ning
at a
bout
Su
t=
210
kpsi
(145
0 M
Pa),
the
sca
tter
appe
ars
to i
ncre
ase,
but
the
tre
nd s
eem
s to
lev
el o
ff,
as s
ug-
gest
ed b
y th
e da
shed
hor
izon
tal l
ine
at S
′ e=
105
kpsi
.W
ew
ish
now
topr
esen
ta
met
hod
for
estim
atin
gen
dura
nce
limits
.Not
eth
ates
ti-m
ates
obta
ined
from
quan
titie
sof
data
obta
ined
from
man
yso
urce
spr
obab
lyha
vea
larg
esp
read
and
mig
htde
viat
esi
gnifi
cant
lyfr
omth
ere
sults
ofac
tual
labo
rato
ryte
sts
ofth
em
echa
nica
lpr
oper
ties
ofsp
ecim
ens
obta
ined
thro
ugh
stri
ctpu
rcha
se-o
rder
spec
ifi-
catio
ns.S
ince
the
area
ofun
cert
aint
yis
grea
ter,
com
pens
atio
nm
ustb
em
ade
byem
ploy
-in
gla
rger
desi
gnfa
ctor
sth
anw
ould
beus
edfo
rst
atic
desi
gn.
For
stee
ls, s
impl
ifyi
ng o
ur o
bser
vatio
n of
Fig
. 6–1
7, w
e w
ill e
stim
ate
the
endu
ranc
elim
it as
S′ e=⎧ ⎨ ⎩0.
5Su
tS u
t≤
200
kpsi
(140
0M
Pa)
100
kpsi
S ut>
200
kpsi
700
MPa
S ut>
1400
MPa
(6–8
)
whe
reS u
tis
the
min
imum
tens
ile s
tren
gth.
The
pri
me
mar
k on
S′ e
in th
is e
quat
ion
refe
rsto
the
rota
ting
-bea
m s
peci
men
itsel
f. W
e w
ish
to re
serv
e th
e un
prim
ed s
ymbo
l Se
for t
heen
dura
nce
limit
of a
ny p
artic
ular
mac
hine
ele
men
t su
bjec
ted
to a
ny k
ind
of l
oadi
ng.
Soon
we
shal
l lea
rn th
at th
e tw
o st
reng
ths
may
be
quite
dif
fere
nt.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
278
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g275
Stee
ls t
reat
ed t
o gi
ve d
iffe
rent
mic
rost
ruct
ures
hav
e di
ffer
ent
S′ e/S u
tra
tios.
It
appe
ars
that
the
mor
e du
ctile
mic
rost
ruct
ures
hav
e a
high
er r
atio
. Mar
tens
ite h
as a
ver
ybr
ittle
nat
ure
and
is h
ighl
y su
scep
tible
to fa
tigue
-ind
uced
cra
ckin
g; th
us th
e ra
tio is
low
.W
hen
desi
gns
incl
ude
deta
iled
he
at-t
reat
ing
spec
ific
atio
ns
to
obta
in
spec
ific
mic
rost
ruct
ures
, it
is p
ossi
ble
to u
se a
n es
timat
e of
the
end
uran
ce l
imit
base
d on
tes
tda
ta fo
r the
par
ticul
ar m
icro
stru
ctur
e; s
uch
estim
ates
are
muc
h m
ore
relia
ble
and
inde
edsh
ould
be
used
.T
he e
ndur
ance
lim
its f
or v
ario
us c
lass
es o
f ca
st i
rons
, po
lishe
d or
mac
hine
d, a
regi
ven
in T
able
A–2
4. A
lum
inum
allo
ys d
o no
t ha
ve a
n en
dura
nce
limit.
The
fat
igue
stre
ngth
s of
som
e al
umin
um a
lloys
at
5(10
8 ) cy
cles
of
reve
rsed
str
ess
are
give
n in
Tabl
eA
–24.
6–8
Fatigue
Stre
ngth
As
show
n in
Fig
. 6–
10,
a re
gion
of
low
-cyc
le f
atig
ue e
xten
ds f
rom
N=
1to
abo
ut10
3cy
cles
. In
this
reg
ion
the
fati
gue
stre
ngth
Sf
is o
nly
slig
htly
sm
alle
r th
an th
e te
n-si
le s
tren
gth
S ut.
An
anal
ytic
al a
ppro
ach
has
been
giv
en b
y M
isch
ke10
for
both
020
4060
8010
012
014
016
018
020
026
030
022
024
028
0
Tens
ile s
tren
gth S u
t, kp
si
020406080100
120
140
Endurance limit S'e, kpsi
105
kpsi
0.4
0.5
S'e Su=
0.6
Car
bon
stee
ls
Allo
y st
eels
Wro
ught
iron
s
Figure
6–1
7
Gra
ph o
f end
uran
ce li
mits
ver
sus
tens
ile s
treng
ths
from
act
ual t
est r
esul
ts fo
r a la
rge
num
ber o
f wro
ught
irons
and
ste
els.
Rat
ios
of S
′ e/S u
tof
0.6
0, 0
.50,
and
0.4
0 ar
e sh
own
by th
e so
lid a
nd d
ashe
d lin
es.
Not
e al
so th
e ho
rizon
tal d
ashe
d lin
e fo
r S′ e=
105
kpsi.
Poi
nts
show
n ha
ving
a te
nsile
stre
ngth
gre
ater
than
210
kps
i hav
e a
mea
n en
dura
nce
limit
of S
′ e=
105
kpsi
and
a sta
ndar
d de
viat
ion
of 1
3.5
kpsi.
(Col
late
d fro
m d
ata
com
pile
d by
H. J
. Gro
ver,
S. A
. Gor
don,
and
L.R
. Jac
kson
in F
atig
ue o
f Met
als
and
Stru
ctur
es,B
urea
u of
Nav
al W
eapo
ns D
ocum
ent N
AVW
EPS
00-2
5-53
4, 1
960;
and
from
Fat
igue
Des
ign
Han
dboo
k,SA
E, 1
968,
p. 4
2.)
10J.
E. S
higl
ey, C
. R. M
isch
ke, a
nd T
. H. B
row
n, J
r., S
tand
ard
Han
dboo
k of
Mac
hine
Des
ign,
3rd
ed.,
McG
raw
-Hill
, New
Yor
k, 2
004,
pp.
29.2
5–29
.27.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
279
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
276
Mec
hani
cal E
ngin
eerin
g D
esig
n
high
-cyc
le a
nd l
ow-c
ycle
reg
ions
, re
quir
ing
the
para
met
ers
of t
he M
anso
n-C
offi
neq
uati
on p
lus
the
stra
in-s
tren
gthe
ning
exp
onen
t m
. E
ngin
eers
oft
en h
ave
to w
ork
wit
h le
ss i
nfor
mat
ion.
Figu
re6–
10in
dica
tes
that
the
high
-cyc
lefa
tigue
dom
ain
exte
nds
from
103
cycl
esfo
rst
eels
toth
een
dura
nce
limit
life
Ne,
whi
chis
abou
t106
to10
7cy
cles
.The
purp
ose
ofth
isse
ctio
nis
tode
velo
pm
etho
dsof
appr
oxim
atio
nof
the
S-N
diag
ram
inth
ehi
gh-
cycl
ere
gion
,whe
nin
form
atio
nm
aybe
assp
arse
asth
ere
sults
ofa
sim
ple
tens
ion
test
.E
xper
ienc
eha
ssh
own
high
-cyc
lefa
tigue
data
are
rect
ified
bya
loga
rith
mic
tran
sfor
mto
both
stre
ssan
dcy
cles
-to-
failu
re.E
quat
ion
(6–2
) ca
n be
use
d to
det
erm
ine
the
fatig
uest
reng
th a
t 10
3cy
cles
. D
efini
ng t
he s
peci
men
fat
igue
str
engt
h at
a s
peci
fic n
umbe
r of
cycl
es a
s (S
′ f) N
=E
�ε
e/2 ,
wri
te E
q. (
6–2)
as
(S′ f) N
=σ
′ F(2
N)b
(6–9
)
At1
03cy
cles
,
(S′ f) 1
03=
σ′ F(2
.103
)b=
fSu
t
whe
ref
is th
e fr
actio
n of
Su
tre
pres
ente
d by
(S′ f
) 103
cycl
es. S
olvi
ng f
or f
give
s
f=
σ′ F
S ut(2
·103
)b(6
–10)
Now
, fro
m E
q. (
2–11
), σ
′ F=
σ0ε
m, w
ith ε
=ε
′ F.
If th
is tr
ue-s
tres
s–tr
ue-s
trai
n eq
uatio
nis
not
kno
wn,
the
SAE
app
roxi
mat
ion11
for
stee
ls w
ith H
B≤
500
may
be
used
:
σ′ F
=S u
t+
50kp
sior
σ′ F
=S u
t+
345
MPa
(6–1
1)
To fi
nd b
, su
bstit
ute
the
endu
ranc
e st
reng
th a
nd c
orre
spon
ding
cyc
les,
S′ e
and
Ne,
resp
ectiv
ely
into
Eq.
(6–
9) a
nd s
olvi
ng f
or b
b=
−lo
g( σ
′ F/
S′ e)lo
g(2
Ne)
(6–1
2)
Thu
s, t
he e
quat
ion
S′ f=
σ′ F(2
N)b
is k
now
n. F
or e
xam
ple,
if
S ut=
105
kpsi
and
S′ e=
52.5
kpsi
at f
ailu
re,
Eq.
(6–
11)
σ′ F
=10
5+
50=
155
kpsi
Eq.
(6–
12)
b=
−lo
g(15
5/52
.5)
log( 2
·106) =
−0.0
746
Eq.
(6–
10)
f=
155
105
( 2·1
03) −0.0
746
=0.
837
and
for
Eq.
(6–
9), w
ith S
′ f=
(S′ f) N
,
S′ f=
155(
2N
)−0.
0746
=14
7N
−0.0
746
(a)
11Fa
tigu
e D
esig
n H
andb
ook,
vol.
4, S
ocie
ty o
f Aut
omot
ive
Eng
inee
rs, N
ew Y
ork,
195
8, p
. 27.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
280
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g277
The
pro
cess
giv
en f
or fi
ndin
g f
can
be r
epea
ted
for
vari
ous
ultim
ate
stre
ngth
s.Fi
gure
6–1
8 is
a p
lot o
f ffo
r70
≤S u
t≤
200
kpsi
. To
be c
onse
rvat
ive,
for
S ut<
70kp
si,
letf
� 0
.9.
For
an a
ctua
l m
echa
nica
l co
mpo
nent
, S′ e
is r
educ
ed t
o S e
(see
Sec
. 6–9
) w
hich
is
less
tha
n 0.
5 S u
t. H
owev
er,
unle
ss a
ctua
l da
ta i
s av
aila
ble,
we
reco
mm
end
usin
g th
eva
lue
of f
foun
d fr
om F
ig. 6
–18.
Equ
atio
n (a
), fo
r the
act
ual m
echa
nica
l com
pone
nt, c
anbe
wri
tten
in th
e fo
rm
Sf=
aN
b(6
–13)
whe
reN
is c
ycle
s to
fai
lure
and
the
con
stan
ts a
and
bar
e de
fined
by
the
poin
ts10
3,( S
f) 10
3an
d10
6,
S ew
ith( S
f) 10
3=
fSu
t. S
ubst
itutin
g th
ese
two
poin
ts i
n E
q.(6
–13)
giv
es
a=
(fS
ut)
2
S e(6
–14)
b=
−1 3
log
( fSu
t
S e
)(6
–15)
If a
com
plet
ely
reve
rsed
str
ess
σa
is g
iven
, se
tting
Sf=
σa
in E
q. (
6–13
), t
he n
umbe
rof
cyc
les-
to-f
ailu
re c
an b
e ex
pres
sed
as N=( σ a a
) 1/b(6
–16)
Low
-cyc
le f
atig
ue is
oft
en d
efine
d (s
ee F
ig. 6
–10)
as
failu
re th
at o
ccur
s in
a r
ange
of1
≤N
≤10
3cy
cles
. On
a lo
glog
plo
t suc
h as
Fig
. 6–1
0 th
e fa
ilure
locu
s in
this
rang
eis
nea
rly
linea
r be
low
103
cycl
es. A
str
aigh
t lin
e be
twee
n 10
3,
fSu
tan
d 1,
Su
t(t
rans
-fo
rmed
) is
con
serv
ativ
e, a
nd it
is g
iven
by
Sf
≥S u
tN(l
ogf)
/3
1≤
N≤
103
(6–1
7)
7080
9010
011
012
013
014
015
016
017
020
018
019
0
S ut,
kpsi
f
0.76
0.780.
8
0.82
0.84
0.86
0.880.
9Fi
gure
6–1
8
Fatig
ue s
treng
th fr
actio
n, f,
ofS u
tat
103
cycl
es fo
rS e
=S′ e
=0.
5S u
t.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
281
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
278
Mec
hani
cal E
ngin
eerin
g D
esig
n
EXA
MPLE
6–2
Giv
en a
105
0 H
R s
teel
, est
imat
e(a
) th
e ro
tatin
g-be
am e
ndur
ance
lim
it at
106
cycl
es.
(b)
the
endu
ranc
e st
reng
th o
f a
polis
hed
rota
ting-
beam
spe
cim
en c
orre
spon
ding
to
104
cycl
es to
fai
lure
(c)
the
expe
cted
life
of
a po
lishe
d ro
tatin
g-be
am s
peci
men
und
er a
com
plet
ely
reve
rsed
stre
ss o
f 55
kps
i.
Solu
tion
(a)
From
Tab
le A
–20,
Su
t=
90kp
si. F
rom
Eq.
(6–
8),
Ans
wer
S′ e=
0.5(
90)=
45kp
si
(b)
From
Fig
. 6–1
8, f
or S
ut=
90kp
si,f
. =0.
86. F
rom
Eq.
(6–
14),
a=
[0.8
6(90
)2]
45=
133.
1kp
si
From
Eq.
(6–
15),
b=
−1 3
log
[ 0.86
(90)
45
] =−0
.078
5
Thu
s, E
q. (
6–13
) is
S′ f=
133.
1N
−0.0
785
Ans
wer
For
104
cycl
es to
fai
lure
, S′ f
=13
3.1(
104)−0
.078
5=
64.6
kpsi
(c)
From
Eq.
(6–
16),
with
σa
=55
kpsi
,
Ans
wer
N=( 55
133.
1
) 1/−0.
0785
=77
500
=7.
75(1
04)c
ycle
s
Kee
p in
min
d th
at th
ese
are
only
est
imat
es.S
o ex
pres
sing
the
answ
ers
usin
g th
ree-
plac
eac
cura
cy is
a li
ttle
mis
lead
ing.
6–9
Endura
nce
Lim
it M
odif
yin
g F
act
ors
We
have
see
n th
at t
he r
otat
ing-
beam
spe
cim
en u
sed
in t
he l
abor
ator
y to
det
erm
ine
endu
ranc
e lim
its i
s pr
epar
ed v
ery
care
fully
and
tes
ted
unde
r cl
osel
y co
ntro
lled
cond
i-tio
ns. I
t is
unre
alis
tic to
exp
ect t
he e
ndur
ance
lim
it of
a m
echa
nica
l or
stru
ctur
al m
em-
ber
to m
atch
the
valu
es o
btai
ned
in th
e la
bora
tory
. Som
e di
ffer
ence
s in
clud
e
•M
ater
ial:
com
posi
tion,
bas
is o
f fa
ilure
, var
iabi
lity
•M
anuf
actu
ring
:m
etho
d, h
eat
trea
tmen
t, fr
ettin
g co
rros
ion,
sur
face
con
ditio
n, s
tres
sco
ncen
trat
ion
•E
nvir
onm
ent:
corr
osio
n, te
mpe
ratu
re, s
tres
s st
ate,
rel
axat
ion
times
•D
esig
n:si
ze, s
hape
, lif
e, s
tres
s st
ate,
str
ess
conc
entr
atio
n, s
peed
, fre
tting
, gal
ling
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
282
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g279
Mar
in12
iden
tified
fac
tors
that
qua
ntifi
ed th
e ef
fect
s of
sur
face
con
ditio
n, s
ize,
load
ing,
tem
pera
ture
, and
mis
cella
neou
s ite
ms.
The
que
stio
n of
whe
ther
to a
djus
t the
end
uran
celim
it by
sub
trac
tive
corr
ectio
ns o
r m
ultip
licat
ive
corr
ectio
ns w
as r
esol
ved
by a
n ex
ten-
sive
sta
tistic
al a
naly
sis
of a
434
0 (e
lect
ric
furn
ace,
air
craf
t qu
ality
) st
eel,
in w
hich
aco
rrel
atio
n co
effic
ient
of
0.85
was
fou
nd f
or t
he m
ultip
licat
ive
form
and
0.4
0 fo
r th
ead
ditiv
e fo
rm. A
Mar
in e
quat
ion
is th
eref
ore
wri
tten
as
S e=
k ak b
k ck d
k ek f
S′ e(6
–18)
whe
rek a
=su
rfac
e co
nditi
on m
odifi
catio
n fa
ctor
k b=
size
mod
ifica
tion
fact
or
k c=
load
mod
ifica
tion
fact
or
k d=
tem
pera
ture
mod
ifica
tion
fact
or
k e=
relia
bilit
y fa
ctor
13
k f=
mis
cella
neou
s-ef
fect
s m
odifi
catio
n fa
ctor
S′ e=
rota
ry-b
eam
test
spe
cim
en e
ndur
ance
lim
it
S e=
endu
ranc
e lim
it at
the
criti
cal l
ocat
ion
of a
mac
hine
par
t in
the
geom
-et
ry a
nd c
ondi
tion
of u
se
Whe
nen
dura
nce
test
sof
part
sar
eno
tav
aila
ble,
estim
atio
nsar
em
ade
byap
plyi
ngM
arin
fact
ors
toth
een
dura
nce
limit.
Surf
ace
Fact
or
ka
The
surf
ace
ofa
rota
ting-
beam
spec
imen
ishi
ghly
polis
hed,
with
afin
alpo
lishi
ngin
the
axia
ldi
rect
ion
tosm
ooth
out
any
circ
umfe
rent
ial
scra
tche
s.T
hesu
rfac
em
odifi
catio
nfa
ctor
depe
nds
onth
equ
ality
ofth
efin
ish
ofth
eac
tual
part
surf
ace
and
onth
ete
nsile
stre
ngth
ofth
epa
rtm
ater
ial.
Tofin
dqu
antit
ativ
eex
pres
sion
sfo
rco
mm
onfin
ishe
sof
mac
hine
part
s(g
roun
d,m
achi
ned,
orco
ld-d
raw
n,ho
t-ro
lled,
and
as-f
orge
d),t
heco
ordi
-na
tes
ofda
tapo
ints
wer
ere
capt
ured
from
apl
otof
endu
ranc
elim
itve
rsus
ultim
ate
tens
ilest
reng
thof
data
gath
ered
byL
ipso
nan
dN
oll
and
repr
oduc
edby
Hor
ger.
14T
heda
ta c
an b
e re
pres
ente
d by
k a=
aSb u
t(6
–19)
whe
reS u
tis
the
min
imum
tens
ile s
tren
gth
and
aan
db
are
to b
e fo
und
in T
able
6–2
.
12Jo
seph
Mar
in, M
echa
nica
l Beh
avio
r of
Eng
inee
ring
Mat
eria
ls,P
rent
ice-
Hal
l, E
ngle
woo
d C
liffs
, N.J
.,19
62, p
. 224
.13
Com
plet
e st
ocha
stic
ana
lysi
s is
pre
sent
ed in
Sec
. 6–1
7. U
ntil
that
poi
nt th
e pr
esen
tatio
n he
re is
one
of
ade
term
inis
tic n
atur
e. H
owev
er, w
e m
ust t
ake
care
of
the
know
n sc
atte
r in
the
fatig
ue d
ata.
Thi
s m
eans
that
we
will
not
car
ry o
ut a
true
rel
iabi
lity
anal
ysis
at t
his
time
but w
ill a
ttem
pt to
ans
wer
the
ques
tion:
Wha
t is
the
prob
abili
ty th
at a
kno
wn
(ass
umed
) st
ress
will
exc
eed
the
stre
ngth
of
a ra
ndom
ly s
elec
ted
com
pone
ntm
ade
from
this
mat
eria
l pop
ulat
ion?
14C
. J. N
oll a
nd C
. Lip
son,
“A
llow
able
Wor
king
Str
esse
s,”
Soci
ety
for
Exp
erim
enta
l Str
ess
Ana
lysi
s,vo
l.3,
no. 2
, 194
6, p
. 29.
Rep
rodu
ced
by O
. J. H
orge
r (e
d.),
Met
als
Eng
inee
ring
Des
ign
ASM
E H
andb
ook,
McG
raw
-Hill
, New
Yor
k, 1
953,
p. 1
02.
Bud
ynas
−Nis
bett:
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gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
283
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
280
Mec
hani
cal E
ngin
eerin
g D
esig
n
EXA
MPLE
6–3
A s
teel
has
a m
inim
um u
ltim
ate
stre
ngth
of
520
MPa
and
a m
achi
ned
surf
ace.
Est
imat
ek a
.
Solu
tion
From
Tab
le 6
–2, a
=4.
51 a
nd b
=−0
.265
. The
n, f
rom
Eq.
(6–
19)
Ans
wer
k a=
4.51
(520
)−0.2
65=
0.86
0
Surf
ace
Fact
or
aEx
ponen
tFi
nis
hS u
t,k
psi
S ut,
MPa
b
Gro
und
1.34
1.58
−0.0
85M
achi
ned
or c
old-
draw
n2.
704.
51−0
.265
Hot
-rolle
d14
.457
.7−0
.718
As-f
orge
d39
.927
2.−0
.995
Table
6–2
Para
met
ers
for M
arin
Surfa
ce M
odifi
catio
nFa
ctor
, Eq.
(6–1
9)
From
C.J. N
oll an
d C. L
ipson
, “All
owab
le Wo
rking
Stre
sses,”
Soc
iety f
or Ex
perim
ental
Stre
ss An
alysis
,vol.
3,
no. 2
, 194
6 p.
29. R
eprod
uced
by O
.J. H
orger
(ed.)
Meta
ls En
ginee
ring D
esign
ASME
Han
dboo
k, Mc
Graw
-Hill,
New
York.
Copy
right
© 1
953
by Th
e McG
raw-Hi
ll Com
panie
s, Inc
. Rep
rinted
by pe
rmiss
ion.
Aga
in, i
t is
im
port
ant
to n
ote
that
thi
s is
an
appr
oxim
atio
n as
the
dat
a is
typ
ical
lyqu
ite s
catte
red.
Fur
ther
mor
e, t
his
is n
ot a
cor
rect
ion
to t
ake
light
ly. F
or e
xam
ple,
if
inth
e pr
evio
us e
xam
ple
the
stee
l was
for
ged,
the
corr
ectio
n fa
ctor
wou
ld b
e 0.
540,
a s
ig-
nific
ant r
educ
tion
of s
tren
gth.
Size
Fact
or
kb
The
siz
e fa
ctor
has
bee
n ev
alua
ted
usin
g 13
3 se
ts o
f da
ta p
oint
s.15
The
res
ults
for
ben
d-in
g an
d to
rsio
n m
ay b
e ex
pres
sed
as
k b=
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩(d/0.
3)−0
.107
=0.
879d
−0.1
070.
11≤
d≤
2in
0.91
d−0
.157
2<
d≤
10in
(d/7.
62)−
0.10
7=
1.24
d−0
.107
2.79
≤d
≤51
mm
1.51
d−0
.157
51<
d≤
254
mm
(6–2
0)
For
axia
l loa
ding
ther
e is
no
size
eff
ect,
so k b=
1(6
–21)
but s
ee k
c.O
ne o
f th
e pr
oble
ms
that
ari
ses
in u
sing
Eq.
(6–
20)
is w
hat t
o do
whe
n a
roun
d ba
rin
ben
ding
is
not
rota
ting,
or
whe
n a
nonc
ircu
lar
cros
s se
ctio
n is
use
d. F
or e
xam
ple,
wha
t is
the
size
fac
tor
for
a ba
r 6
mm
thic
k an
d 40
mm
wid
e? T
he a
ppro
ach
to b
e us
ed
15C
harl
es R
. Mis
chke
, “Pr
edic
tion
of S
toch
astic
End
uran
ce S
tren
gth,
”Tr
ans.
of A
SME
, Jou
rnal
of V
ibra
tion
,A
cous
tics
, Str
ess,
and
Rel
iabi
lity
in D
esig
n,vo
l. 10
9, n
o. 1
, Jan
uary
198
7, T
able
3.
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ynas
−Nis
bett:
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gley
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hani
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ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
284
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g281
here
em
ploy
s an
effe
ctiv
e di
men
sion
de
obta
ined
by
equa
ting
the
volu
me
of m
ater
ial
stre
ssed
at
and
abov
e 95
per
cent
of
the
max
imum
str
ess
to t
he s
ame
volu
me
in t
hero
tatin
g-be
am s
peci
men
.16It
tur
ns o
ut t
hat
whe
n th
ese
two
volu
mes
are
equ
ated
,th
ele
ngth
s ca
ncel
, and
so
we
need
onl
y co
nsid
er th
e ar
eas.
For
a r
otat
ing
roun
d se
ctio
n,th
e 95
per
cent
str
ess
area
is th
e ar
ea in
a r
ing
havi
ng a
n ou
tsid
e di
amet
er d
and
an in
side
diam
eter
of
0.95
d. S
o, d
esig
natin
g th
e 95
per
cent
str
ess
area
A0.
95σ
, we
have
A0.
95σ
=π 4
[d2−
(0.9
5d)2
]=
0.07
66d
2(6
–22)
Thi
s eq
uatio
n is
als
o va
lid f
or a
rot
atin
g ho
llow
rou
nd. F
or n
onro
tatin
g so
lid o
r ho
llow
roun
ds, t
he 9
5 pe
rcen
t st
ress
are
a is
tw
ice
the
area
out
side
of
two
para
llel
chor
ds h
av-
ing
a sp
acin
g of
0.9
5d, w
here
dis
the
diam
eter
. Usi
ng a
n ex
act c
ompu
tatio
n, th
is is
A0.
95σ
=0.
0104
6d2
(6–2
3)
with
d ein
Eq.
(6–
22),
set
ting
Eqs
. (6–
22)
and
(6–2
3) e
qual
to e
ach
othe
r en
able
s us
toso
lve
for
the
effe
ctiv
e di
amet
er. T
his
give
s
d e=
0.37
0d(6
–24)
as th
e ef
fect
ive
size
of
a ro
und
corr
espo
ndin
g to
a n
onro
tatin
g so
lid o
r ho
llow
rou
nd.
A r
ecta
ngul
ar s
ectio
n of
dim
ensi
ons
h×
bha
sA
0.95
σ=
0.05
hb.
Usi
ng t
he s
ame
appr
oach
as
befo
re,
d e=
0.80
8(hb
)1/2
(6–2
5)
Tabl
e 6–
3 pr
ovid
es A
0.95
σar
eas
of c
omm
on s
truc
tura
l sh
apes
und
ergo
ing
non-
rota
ting
bend
ing.
EXA
MPLE
6–4
A s
teel
sha
ft lo
aded
in b
endi
ng is
32
mm
in d
iam
eter
, abu
tting
a fi
llete
d sh
ould
er 3
8m
min
dia
met
er.
The
sha
ft m
ater
ial
has
a m
ean
ultim
ate
tens
ile s
tren
gth
of 6
90 M
Pa.
Est
imat
e th
e M
arin
siz
e fa
ctor
kb
if th
e sh
aft i
s us
ed in
(a) A
rot
atin
g m
ode.
(b) A
non
rota
ting
mod
e.
Solu
tion
(a)
From
Eq.
(6–
20)
Ans
wer
k b=( d
7.62
) −0.1
07
=( 32 7.
62
) −0.1
07
=0.
858
(b)
From
Tab
le 6
–3,
d e=
0.37
d=
0.37
(32)
=11
.84
mm
From
Eq.
(6–
20),
Ans
wer
k b=( 11
.84
7.62
) −0.1
07
=0.
954
16Se
e R
. Kug
uel,
“A R
elat
ion
betw
een
The
oret
ical
Str
ess
Con
cent
ratio
n Fa
ctor
and
Fat
igue
Not
ch F
acto
rD
educ
ed f
rom
the
Con
cept
of
Hig
hly
Stre
ssed
Vol
ume,
” P
roc.
AST
M,v
ol. 6
1, 1
961,
pp.
732
–748
.
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ynas
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bett:
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gley
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hani
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ng
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ign,
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hth
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on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
285
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
282
Mec
hani
cal E
ngin
eerin
g D
esig
n
Loadin
g F
act
or
kc
Whe
nfa
tigue
test
sar
eca
rrie
dou
twith
rota
ting
bend
ing,
axia
l(pu
sh-p
ull)
,and
tors
ion-
allo
adin
g,th
een
dura
nce
limits
diff
erw
ithS u
t.T
his
isdi
scus
sed
furt
her
inSe
c.6–
17.
Her
e,w
ew
illsp
ecif
yav
erag
eva
lues
ofth
elo
adfa
ctor
as
k c={ 1
bend
ing
0.85
axia
l0.
59to
rsio
n17(6
–26)
Tem
per
atu
re F
act
or
kd
Whe
n op
erat
ing
tem
pera
ture
s ar
e be
low
roo
m t
empe
ratu
re,
britt
le f
ract
ure
is a
str
ong
poss
ibili
ty a
nd s
houl
d be
inv
estig
ated
firs
t. W
hen
the
oper
atin
g te
mpe
ratu
res
are
high
-er
tha
n ro
om t
empe
ratu
re,
yiel
ding
sho
uld
be i
nves
tigat
ed fi
rst
beca
use
the
yiel
dst
reng
th d
rops
off
so
rapi
dly
with
tem
pera
ture
; se
e Fi
g. 2
–9.
Any
str
ess
will
ind
uce
cree
p in
a m
ater
ial o
pera
ting
at h
igh
tem
pera
ture
s; s
o th
is fa
ctor
mus
t be
cons
ider
ed to
o.
A0.9
5σ
={ 0.
05ab
axis
1-1
0.05
2xa
+0.
1tf(
b−
x)ax
is2-
2
12
2
1a
bt f
x
A0.9
5σ
={ 0.
10at
fax
is1-
1
0.05
bat f
>0.
025a
axis
2-2
1
22
1a
bt f
A0.9
5σ
=0.
05hb
d e=
0.80
8√hb
b
h
2 2
11
A0.
95σ
=0.
0104
6d2
d e=
0.37
0dd
Table
6–3
A0.
95σ
Are
as o
fC
omm
on N
onro
tatin
gSt
ruct
ural
Sha
pes
17U
se th
is o
nly
for
pure
tors
iona
l fat
igue
load
ing.
Whe
n to
rsio
n is
com
bine
d w
ith o
ther
str
esse
s, s
uch
asbe
ndin
g, k
c=
1an
d th
e co
mbi
ned
load
ing
is m
anag
ed b
y us
ing
the
effe
ctiv
e vo
n M
ises
str
ess
as in
Sec.
5–5.
Not
e:Fo
r pu
re to
rsio
n, th
e di
stor
tion
ener
gy p
redi
cts
that
(k c
) tors
ion=
0.57
7.
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ynas
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ign,
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hth
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on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
286
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g283
Fina
lly, i
t may
be
true
that
ther
e is
no
fatig
ue li
mit
for
mat
eria
ls o
pera
ting
at h
igh
tem
-pe
ratu
res.
Bec
ause
of
the
redu
ced
fatig
ue r
esis
tanc
e, t
he f
ailu
re p
roce
ss i
s, t
o so
me
exte
nt, d
epen
dent
on
time.
The
lim
ited
amou
nt o
f da
ta a
vaila
ble
show
tha
t th
e en
dura
nce
limit
for
stee
lsin
crea
ses
slig
htly
as
the
tem
pera
ture
ris
es a
nd th
en b
egin
s to
fall
off
in th
e 40
0 to
700
°Fra
nge,
not
unl
ike
the
beha
vior
of
the
tens
ile s
tren
gth
show
n in
Fig
. 2–9
. For
this
rea
son
it is
pro
babl
y tr
ue th
at th
e en
dura
nce
limit
is r
elat
ed to
tens
ile s
tren
gth
at e
leva
ted
tem
-pe
ratu
res
in th
e sa
me
man
ner a
s at
room
tem
pera
ture
.18It
see
ms
quite
logi
cal,
ther
efor
e,to
em
ploy
the
sam
e re
latio
ns to
pre
dict
end
uran
ce li
mit
at e
leva
ted
tem
pera
ture
s as
are
used
at r
oom
tem
pera
ture
, at l
east
unt
il m
ore
com
preh
ensi
ve d
ata
beco
me
avai
labl
e. A
tth
e ve
ry l
east
, th
is p
ract
ice
will
pro
vide
a u
sefu
l st
anda
rd a
gain
st w
hich
the
per
for-
man
ce o
f va
riou
s m
ater
ials
can
be
com
pare
d.Ta
ble
6–4
has
been
obt
aine
d fr
om F
ig. 2
–9 b
y us
ing
only
the
tens
ile-s
tren
gth
data
.N
ote
that
the
tabl
e re
pres
ents
145
test
s of
21
diff
eren
t car
bon
and
allo
y st
eels
. A f
ourt
h-or
der
poly
nom
ial c
urve
fit t
o th
e da
ta u
nder
lyin
g Fi
g. 2
–9 g
ives
k d=
0.97
5+
0.43
2(10
−3)T
F−
0.11
5(10
−5)T
2 F
+0.
104(
10−8
)T3 F−
0.59
5(10
−12)T
4 F(6
–27)
whe
re70
≤T
F≤
1000
◦ F.
Two
type
s of
pro
blem
s ar
ise
whe
n te
mpe
ratu
re i
s a
cons
ider
atio
n. I
f th
e ro
tatin
g-be
am e
ndur
ance
lim
it is
kno
wn
at r
oom
tem
pera
ture
, the
n us
e
k d=
S T S RT
(6–2
8)
Tem
per
atu
re, °C
S T/S
RT
Tem
per
atu
re, °F
S T/S
RT
201.
000
701.
000
501.
010
100
1.00
810
01.
020
200
1.02
015
01.
025
300
1.02
420
01.
020
400
1.01
825
01.
000
500
0.99
530
00.
975
600
0.96
335
00.
943
700
0.92
740
00.
900
800
0.87
245
00.
843
900
0.79
750
00.
768
1000
0.69
855
00.
672
1100
0.56
760
00.
549
*Data
sourc
e: Fig
. 2–9
.
Table
6–4
Effe
ct o
f Ope
ratin
gTe
mpe
ratu
re o
n th
eTe
nsile
Stre
ngth
of
Stee
l.* (S
T=
tens
ilestr
engt
h at
ope
ratin
gte
mpe
ratu
re;
S RT=
tens
ile s
treng
that
room
tem
pera
ture
; 0.
099
≤σ̂
≤0.
110)
18Fo
r m
ore,
see
Tab
le 2
of A
NSI
/ASM
E B
106.
1M
-198
5 sh
aft s
tand
ard,
and
E. A
. Bra
ndes
(ed
.), S
mit
hell
’sM
etal
s R
efer
ence
Boo
k,6t
h ed
., B
utte
rwor
th, L
ondo
n, 1
983,
pp.
22–
134
to 2
2–13
6, w
here
end
uran
ce li
mits
from
100
to 6
50°C
are
tabu
late
d.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
287
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
284
Mec
hani
cal E
ngin
eerin
g D
esig
n
from
Tab
le 6
–4 o
r Eq.
(6–2
7) a
nd p
roce
ed a
s us
ual.
If th
e ro
tatin
g-be
am e
ndur
ance
lim
itis
not
giv
en,
then
com
pute
it
usin
g E
q. (
6–8)
and
the
tem
pera
ture
-cor
rect
ed t
ensi
lest
reng
th o
btai
ned
by u
sing
the
fact
or f
rom
Tab
le 6
–4. T
hen
use
k d=
1 .
EXA
MPLE
6–5
A 1
035
stee
l has
a te
nsile
str
engt
h of
70
kpsi
and
is to
be
used
for
a p
art t
hat s
ees
450°
Fin
ser
vice
. Est
imat
e th
e M
arin
tem
pera
ture
mod
ifica
tion
fact
or a
nd (
S e) 4
50◦
if(a
) T
he r
oom
-tem
pera
ture
end
uran
ce li
mit
by te
st is
(S′ e)
70◦=
39.0
kpsi
.(b
) O
nly
the
tens
ile s
tren
gth
at r
oom
tem
pera
ture
is k
now
n.
Solu
tion
(a)
Firs
t, fr
om E
q. (
6–27
),
k d=
0.97
5+
0.43
2(10
−3)(
450)
−0.
115(
10−5
)(45
02)
+0.
104(
10−8
)(45
03)−
0.59
5(10
−12)(
4504
)=
1.00
7
Thu
s,
Ans
wer
(Se)
450◦
=k d
(S′ e)
70◦=
1.00
7(39
.0)=
39.3
kpsi
(b)
Inte
rpol
atin
g fr
om T
able
6–4
giv
es
(ST/
S RT) 4
50◦=
1.01
8+
(0.9
95−
1.01
8)45
0−
400
500
−40
0=
1.00
7
Thu
s, th
e te
nsile
str
engt
h at
450
°F is
est
imat
ed a
s
(Su
t)45
0◦=
(ST/
S RT) 4
50◦ (
S ut)
70◦=
1.00
7(70
)=
70.5
kpsi
From
Eq.
(6–
8) th
en,
Ans
wer
(Se)
450◦
=0.
5(S
ut)
450◦
=0.
5(70
.5)=
35.2
kpsi
Part
agi
ves
the
bette
r es
timat
e du
e to
act
ual t
estin
g of
the
part
icul
ar m
ater
ial.
Rel
iabili
ty F
act
or
ke
The
dis
cuss
ion
pres
ente
d he
re a
ccou
nts
for
the
scat
ter
of d
ata
such
as
show
n in
Fig.
6–17
whe
re th
e m
ean
endu
ranc
e lim
it is
sho
wn
to b
e S′ e/
S ut
. =0.
5,or
as
give
n by
Eq.
(6–
8).
Mos
t en
dura
nce
stre
ngth
dat
a ar
e re
port
ed a
s m
ean
valu
es.
Dat
a pr
esen
ted
byH
auge
n an
d W
irch
ing19
show
sta
ndar
d de
viat
ions
of e
ndur
ance
str
engt
hs o
f les
s th
an8
perc
ent.
Thu
s th
e re
liabi
lity
mod
ifica
tion
fact
or to
acc
ount
for
this
can
be
wri
tten
as
k e=
1−
0.08
z a(6
–29)
whe
rez a
is d
efine
d by
Eq.
(20
–16)
and
val
ues
for
any
desi
red
relia
bilit
y ca
n be
det
er-
min
ed f
rom
Tab
le A
–10.
Tab
le 6
–5 g
ives
rel
iabi
lity
fact
ors
for
som
e st
anda
rd s
peci
fied
relia
bilit
ies.
For
a m
ore
com
preh
ensi
ve a
ppro
ach
to r
elia
bilit
y, s
ee S
ec. 6
–17.
19E
. B. H
auge
n an
d P.
H. W
irsc
hing
, “Pr
obab
ilist
ic D
esig
n,”
Mac
hine
Des
ign,
vol.
47, n
o. 1
2, 1
975,
pp.1
0–14
.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
288
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g285
Mis
cella
neo
us-
Effe
cts
Fact
or
kf
Tho
ugh
the
fact
or k
fis
inte
nded
to a
ccou
nt f
or th
e re
duct
ion
in e
ndur
ance
lim
it du
e to
all
othe
r ef
fect
s, i
t is
rea
lly i
nten
ded
as a
rem
inde
r th
at t
hese
mus
t be
acc
ount
ed f
or,
beca
use
actu
al v
alue
s of
kf
are
not a
lway
s av
aila
ble.
Res
idua
l st
ress
esm
ay e
ither
im
prov
e th
e en
dura
nce
limit
or a
ffec
t it
adve
rsel
y.G
ener
ally
, if
the
resi
dual
str
ess
in th
e su
rfac
e of
the
part
is c
ompr
essi
on, t
he e
ndur
ance
limit
is im
prov
ed. F
atig
ue f
ailu
res
appe
ar to
be
tens
ile f
ailu
res,
or
at le
ast t
o be
cau
sed
by t
ensi
le s
tres
s, a
nd s
o an
ythi
ng t
hat
redu
ces
tens
ile s
tres
s w
ill a
lso
redu
ce t
he p
ossi
-bi
lity
of a
fatig
ue fa
ilure
. Ope
ratio
ns s
uch
as s
hot p
eeni
ng, h
amm
erin
g, a
nd c
old
rolli
ngbu
ild c
ompr
essi
ve s
tres
ses
into
the
surf
ace
of th
e pa
rt a
nd im
prov
e th
e en
dura
nce
limit
sign
ifica
ntly
. Of
cour
se, t
he m
ater
ial m
ust n
ot b
e w
orke
d to
exh
aust
ion.
The
end
uran
ce l
imits
of
part
s th
at a
re m
ade
from
rol
led
or d
raw
n sh
eets
or
bars
,as
wel
l as
part
s th
at a
re f
orge
d, m
ay b
e af
fect
ed b
y th
e so
-cal
led
dire
ctio
nal c
hara
cter
-is
tics
of t
he o
pera
tion.
Rol
led
or d
raw
n pa
rts,
for
exa
mpl
e, h
ave
an e
ndur
ance
lim
itin
the
tran
sver
se d
irec
tion
that
may
be
10 to
20
perc
ent l
ess
than
the
endu
ranc
e lim
it in
the
long
itudi
nal d
irec
tion.
Part
sth
atar
eca
se-h
arde
ned
may
fail
atth
esu
rfac
eor
atth
em
axim
umco
rera
dius
,de
pend
ing
upon
the
stre
ssgr
adie
nt.F
igur
e6–
19sh
ows
the
typi
calt
rian
gula
rst
ress
dis-
trib
utio
nof
aba
run
der
bend
ing
orto
rsio
n.A
lso
plot
ted
asa
heav
ylin
ein
this
figur
ear
eth
een
dura
nce
limits
S efo
rthe
case
and
core
.For
this
exam
ple
the
endu
ranc
elim
itof
the
core
rule
sth
ede
sign
beca
use
the
figur
esh
ows
that
the
stre
ssσ
orτ,
whi
chev
erap
plie
s,at
the
oute
rco
rera
dius
,is
appr
ecia
bly
larg
erth
anth
eco
reen
dura
nce
limit.
S e(c
ase)
� o
r �
S e(c
ore)
Cas
e Cor
e
Figure
6–1
9
The
failu
re o
f a c
ase-
hard
ened
part
in b
endi
ng o
r tor
sion.
Inth
is ex
ampl
e, fa
ilure
occ
urs
inth
e co
re.
Rel
iabili
ty, %
Transf
orm
ation V
ari
ate
za
Rel
iabili
ty F
act
or
ke
500
1.00
090
1.28
80.
897
951.
645
0.86
899
2.32
60.
814
99.9
3.09
10.
753
99.9
93.
719
0.70
299
.999
4.26
50.
659
99.9
999
4.75
30.
620
Table
6–5
Relia
bilit
y Fa
ctor
s k e
Cor
resp
ondi
ng to
8Pe
rcen
t Sta
ndar
dD
evia
tion
of th
eEn
dura
nce
Limit
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
289
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
286
Mec
hani
cal E
ngin
eerin
g D
esig
n
Of
cour
se,
if s
tres
s co
ncen
trat
ion
is a
lso
pres
ent,
the
stre
ss g
radi
ent
is m
uch
stee
per,
and
henc
e fa
ilure
in th
e co
re is
unl
ikel
y.
Cor
rosi
onIt
is to
be
expe
cted
that
par
ts th
at o
pera
te in
a c
orro
sive
atm
osph
ere
will
hav
e a
low
ered
fatig
ue r
esis
tanc
e. T
his
is, o
f co
urse
, tru
e, a
nd i
t is
due
to
the
roug
heni
ng o
r pi
tting
of
the
surf
ace
by t
he c
orro
sive
mat
eria
l. B
ut t
he p
robl
em i
s no
t so
sim
ple
as t
he o
ne o
ffin
ding
the
endu
ranc
e lim
it of
a s
peci
men
that
has
bee
n co
rrod
ed. T
he r
easo
n fo
r th
is is
that
the
corr
osio
n an
d th
e st
ress
ing
occu
r at
the
sam
e tim
e. B
asic
ally
, thi
s m
eans
that
intim
e an
y pa
rt w
ill f
ail
whe
n su
bjec
ted
to r
epea
ted
stre
ssin
g in
a c
orro
sive
atm
osph
ere.
The
re is
no
fatig
ue li
mit.
Thu
s th
e de
sign
er’s
pro
blem
is to
atte
mpt
to m
inim
ize
the
fac-
tors
that
aff
ect t
he f
atig
ue li
fe; t
hese
are
:
•M
ean
or s
tatic
str
ess
•A
ltern
atin
g st
ress
•E
lect
roly
te c
once
ntra
tion
•D
isso
lved
oxy
gen
in e
lect
roly
te
•M
ater
ial p
rope
rtie
s an
d co
mpo
sitio
n
•Te
mpe
ratu
re
•C
yclic
fre
quen
cy
•Fl
uid
flow
rat
e ar
ound
spe
cim
en
•L
ocal
cre
vice
s
Ele
ctro
lyti
c P
lati
ngM
etal
lic c
oatin
gs, s
uch
as c
hrom
ium
pla
ting,
nic
kel p
latin
g, o
r cad
miu
m p
latin
g, re
duce
the
endu
ranc
e lim
it by
as
muc
h as
50
perc
ent.
In s
ome
case
s th
e re
duct
ion
by c
oatin
gsha
s be
en s
o se
vere
tha
t it
has
been
nec
essa
ry t
o el
imin
ate
the
plat
ing
proc
ess.
Zin
cpl
atin
g do
es n
ot a
ffec
t th
e fa
tigue
str
engt
h. A
nodi
c ox
idat
ion
of l
ight
allo
ys r
educ
esbe
ndin
g en
dura
nce
limits
by
as m
uch
as 3
9 pe
rcen
t bu
t ha
s no
eff
ect
on t
he t
orsi
onal
endu
ranc
e lim
it.
Met
al S
pray
ing
Met
al s
pray
ing
resu
lts i
n su
rfac
e im
perf
ectio
ns t
hat
can
initi
ate
crac
ks.
Lim
ited
test
ssh
ow r
educ
tions
of
14 p
erce
nt in
the
fatig
ue s
tren
gth.
Cyc
lic F
requ
ency
If,
for
any
reas
on,
the
fatig
ue p
roce
ss b
ecom
es t
ime-
depe
nden
t, th
en i
t al
so b
ecom
esfr
eque
ncy-
depe
nden
t. U
nder
nor
mal
con
ditio
ns,
fatig
ue f
ailu
re i
s in
depe
nden
t of
fre
-qu
ency
. But
whe
n co
rros
ion
or h
igh
tem
pera
ture
s, o
r bo
th, a
re e
ncou
nter
ed, t
he c
yclic
rate
bec
omes
im
port
ant.
The
slo
wer
the
fre
quen
cy a
nd t
he h
ighe
r th
e te
mpe
ratu
re, t
hehi
gher
the
crac
k pr
opag
atio
n ra
te a
nd th
e sh
orte
r th
e lif
e at
a g
iven
str
ess
leve
l.
Fre
ttag
e C
orro
sion
The
phe
nom
enon
of
fret
tage
cor
rosi
on i
s th
e re
sult
of m
icro
scop
ic m
otio
ns o
f tig
htly
fittin
g pa
rts
or s
truc
ture
s. B
olte
d jo
ints
, be
arin
g-ra
ce fi
ts,
whe
el h
ubs,
and
any
set
of
tight
ly fi
tted
part
s ar
e ex
ampl
es. T
he p
roce
ss in
volv
es s
urfa
ce d
isco
lora
tion,
pitt
ing,
and
even
tual
fat
igue
. The
fre
ttage
fac
tor
k fde
pend
s up
on t
he m
ater
ial
of t
he m
atin
g pa
irs
and
rang
es f
rom
0.2
4 to
0.9
0.
Bud
ynas
−Nis
bett:
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gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
290
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g287
6–10
Stre
ss C
once
ntr
ation a
nd N
otc
h S
ensi
tivi
tyIn
Sec.
3–13
itw
aspo
inte
dou
tth
atth
eex
iste
nce
ofir
regu
lari
ties
ordi
scon
tinui
ties,
such
asho
les,
groo
ves,
orno
tche
s,in
apa
rtin
crea
ses
the
theo
retic
alst
ress
essi
gnifi
-ca
ntly
inth
eim
med
iate
vici
nity
ofth
edi
scon
tinui
ty.
Equ
atio
n(3
–48)
defin
eda
stre
ssco
ncen
trat
ion
fact
orK
t(o
rK
ts),
whi
chis
used
with
the
nom
inal
stre
ssto
obta
inth
em
axim
umre
sulti
ngst
ress
due
toth
eir
regu
lari
tyor
defe
ct.I
ttur
nsou
ttha
tsom
em
ate-
rial
sar
eno
tfu
llyse
nsiti
veto
the
pres
ence
ofno
tche
san
dhe
nce,
for
thes
e,a
redu
ced
valu
eof
Ktca
nbe
used
.For
thes
em
ater
ials
,the
max
imum
stre
ssis
,in
fact
,
σm
ax=
Kfσ
0or
τ max
=K
fsτ 0
(6–3
0)
whe
reK
fis
a r
educ
ed v
alue
of
Kt
and
σ0
is th
e no
min
al s
tres
s. T
he f
acto
r K
fis
com
-m
only
cal
led
a fa
tigu
e st
ress
-con
cent
rati
on f
acto
r,an
d he
nce
the
subs
crip
t f.
So i
t is
conv
enie
nt t
o th
ink
of K
fas
a s
tres
s-co
ncen
trat
ion
fact
or r
educ
ed f
rom
Kt
beca
use
ofle
ssen
ed s
ensi
tivity
to n
otch
es. T
he r
esul
ting
fact
or is
defi
ned
by th
e eq
uatio
n
Kf
=m
axim
umst
ress
inno
tche
dsp
ecim
en
stre
ssin
notc
h-fr
eesp
ecim
en(a
)
Not
ch s
ensi
tivi
ty q
is d
efine
d by
the
equa
tion
q=
Kf−
1
Kt−
1or
q she
ar=
Kfs
−1
Kts
−1
(6–3
1)
whe
req
is u
sual
ly b
etw
een
zero
and
uni
ty. E
quat
ion
(6–3
1) s
how
s th
at i
f q
=0,
then
Kf
=1,
and
the
mat
eria
l ha
s no
sen
sitiv
ity t
o no
tche
s at
all.
On
the
othe
r ha
nd,
ifq
=1,
then
Kf=
Kt,
and
the
mat
eria
l ha
s fu
ll no
tch
sens
itivi
ty. I
n an
alys
is o
r de
sign
wor
k, fi
nd K
tfir
st, f
rom
the
geom
etry
of
the
part
. The
n sp
ecif
y th
e m
ater
ial,
find
q, a
ndso
lve
for
Kffr
om th
e eq
uatio
n
Kf=
1+
q(K
t−
1)or
Kfs
=1
+q s
hear(K
ts−
1)(6
–32)
For
stee
ls a
nd 2
024
alum
inum
allo
ys, u
se F
ig. 6
–20
to fi
nd q
for
bend
ing
and
axia
llo
adin
g. F
or s
hear
loa
ding
, use
Fig
. 6–2
1. I
n us
ing
thes
e ch
arts
it
is w
ell
to k
now
tha
tth
e ac
tual
tes
t re
sults
fro
m w
hich
the
cur
ves
wer
e de
rive
d ex
hibi
t a
larg
e am
ount
of
00.
020.
040.
060.
080.
100.
120.
140.
16
00.
51.
01.
52.
02.
53.
03.
54.
0
0
0.2
0.4
0.6
0.8
1.0
Not
ch r
adiu
s r ,
in
Not
ch r
adiu
s r,
mm
Notch sensitivity q
Sut=
200
kpsi
(0.4
)
60100
150
(0.7
)
(1.0
)
(1.4
GPa
)
Stee
ls
Alu
m. a
lloy
Figure
6–2
0
Not
ch-se
nsiti
vity
cha
rts fo
rste
els
and
UN
S A
9202
4-T
wro
ught
alu
min
um a
lloys
subj
ecte
d to
reve
rsed
ben
ding
or re
vers
ed a
xial
load
s. F
orla
rger
not
ch ra
dii,
use
the
valu
es o
f qco
rresp
ondi
ngto
the
r=
0.1
6-in
(4-m
m)
ordi
nate
.(Fr
om G
eorg
e Si
nes
and
J. L.
Wai
sman
(eds
.),M
etal
Fat
igue
,McG
raw
-Hill,
New
Yor
k. C
opyr
ight
©19
69by
The
McG
raw
-Hill
Com
pani
es, I
nc. R
eprin
ted
bype
rmiss
ion.
)
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
291
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
288
Mec
hani
cal E
ngin
eerin
g D
esig
n
scat
ter.
Bec
ause
of
this
sca
tter
it is
alw
ays
safe
to
use
Kf
=K
tif
the
re i
s an
y do
ubt
abou
t the
true
val
ue o
f q.
Als
o, n
ote
that
qis
not
far
fro
m u
nity
for
larg
e no
tch
radi
i.T
he n
otch
sen
sitiv
ity o
f th
e ca
st i
rons
is
very
low
, va
ryin
g fr
om 0
to
abou
t 0.
20,
depe
ndin
g up
on th
e te
nsile
str
engt
h. T
o be
on
the
cons
erva
tive
side
, it i
s re
com
men
ded
that
the
valu
e q
=0.
20be
use
d fo
r al
l gra
des
of c
ast i
ron.
Figu
re 6
–20
has
as it
s ba
sis
the
Neu
ber
equa
tion
,whi
ch is
giv
en b
y
Kf=
1+
Kt−
1
1+
√ a/r
(6–3
3)
whe
re√ a
is d
efine
d as
the
Neu
ber
cons
tant
and
is a
mat
eria
l co
nsta
nt.
Equ
atin
gE
qs.(
6–31
) an
d (6
–33)
yie
lds
the
notc
h se
nsiti
vity
equ
atio
n
q=
1
1+
√ a √ r
(6–3
4)
For
stee
l, w
ith S
utin
kpsi
, th
e N
eube
r co
nsta
nt c
an b
e ap
prox
imat
ed b
y a
thir
d-or
der
poly
nom
ial fi
t of
data
as
√ a=
0.24
579
9−
0.30
779
4(10
−2)S
ut
+0.
150
874(
10−4
)S2 ut−
0.26
697
8(10
−7)S
3 ut
(6–3
5)
To u
se E
q. (
6–33
) or
(6–
34)
for
tors
ion
for
low
-allo
y st
eels
, in
crea
se t
he u
ltim
ate
stre
ngth
by
20 k
psi i
n E
q. (
6–35
) an
d ap
ply
this
val
ue o
f √ a.
00.
020.
040.
060.
080.
100.
120.
140.
16
00.
51.
01.
52.
02.
53.
03.
54.
0
0
0.2
0.4
0.6
0.8
1.0
Not
ch r
adiu
s r,
in
Not
ch r
adiu
s r,
mm
Notch sensitivity qshear
Alu
min
um a
lloysA
nnea
led
stee
ls (
Bhn
< 2
00)
Que
nche
d an
d dr
awn
stee
ls (
Bhn
> 2
00)
Figure
6–2
1
Not
ch-se
nsiti
vity
cur
ves
for
mat
eria
ls in
reve
rsed
tors
ion.
For l
arge
r not
ch ra
dii,
use
the
valu
es o
f qsh
ear
corre
spon
ding
to r
=0.
16 in
(4 m
m).
EXA
MPLE
6–6
A s
teel
sha
ft in
ben
ding
has
an
ultim
ate
stre
ngth
of
690
MPa
and
a s
houl
der
with
a fi
l-le
t ra
dius
of
3 m
m c
onne
ctin
g a
32-m
m d
iam
eter
with
a 3
8-m
m d
iam
eter
. Est
imat
e K
f
usin
g:(a
) Fi
gure
6–2
0.(b
) E
quat
ions
(6–
33)
and
(6–3
5).
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
292
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Solu
tion
From
Fig.
A–1
5–9,
usin
gD
/d
=38
/32
=1.
1875
,r/
d=
3/32
=0.
093
75,
we
read
the
grap
hto
find
Kt
. =1.
65.
(a) F
rom
Fig
. 6–2
0, fo
r S u
t=
690
MPa
and
r=
3m
m,q
. =0.
84. T
hus,
from
Eq.
(6–3
2)
Ans
wer
Kf=
1+
q(K
t−
1). =
1+
0.84
(1.6
5−
1)=
1.55
(b)
From
Eq.
(6–
35)
with
Su
t=
690
MPa
=10
0 kp
si, √ a
=0.
0622
√ in=
0.31
3√ mm
.Su
bstit
utin
g th
is in
to E
q. (
6–33
) w
ith r
=3
mm
giv
es
Ans
wer
Kf=
1+
Kt−
1
1+
√ a/r
. =1
+1.
65−
1
1+
0.31
3√ 3
=1.
55
For
sim
ple
load
ing,
itis
acce
ptab
leto
redu
ceth
een
dura
nce
limit
byei
ther
divi
ding
the
unno
tche
dsp
ecim
enen
dura
nce
limit
byK
for
mul
tiply
ing
the
reve
rsin
gst
ress
byK
f.H
owev
er,i
nde
alin
gw
ithco
mbi
ned
stre
sspr
oble
ms
that
may
invo
lve
mor
eth
anon
eva
lue
offa
tigue
-con
cent
ratio
nfa
ctor
,the
stre
sses
are
mul
tiplie
dby
Kf.
EXA
MPLE
6–7
Con
side
r an
unn
otch
ed s
peci
men
with
an
endu
ranc
e lim
it of
55
kpsi
. If
the
spe
cim
enw
as n
otch
ed s
uch
that
Kf
=1.
6 , w
hat w
ould
be
the
fact
or o
f sa
fety
aga
inst
fai
lure
for
N>
106
cycl
es a
t a r
ever
sing
str
ess
of 3
0 kp
si?
(a)
Solv
e by
red
ucin
g S′ e.
(b)
Solv
e by
incr
easi
ng th
e ap
plie
d st
ress
.
Solu
tion
(a)
The
end
uran
ce li
mit
of th
e no
tche
d sp
ecim
en is
giv
en b
y
S e=
S′ e
Kf
=55 1.
6=
34.4
kpsi
and
the
fact
or o
f sa
fety
is
Ans
wer
n=
S e σa
=34
.4
30=
1.15
(b)
The
max
imum
str
ess
can
be w
ritte
n as
(σa) m
ax=
Kfσ
a=
1.6(
30)=
48.0
kpsi
and
the
fact
or o
f sa
fety
is
Ans
wer
n=
S′ e
Kfσ
a=
55 48=
1.15
Up
to t
his
poin
t, ex
ampl
es i
llust
rate
d ea
ch f
acto
r in
Mar
in’s
equ
atio
n an
d st
ress
conc
entr
atio
ns a
lone
. Let
us
cons
ider
a n
umbe
r of
fac
tors
occ
urri
ng s
imul
tane
ousl
y.
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g289
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
293
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
290
Mec
hani
cal E
ngin
eerin
g D
esig
n
EXA
MPLE
6–8
A 1
015
hot-
rolle
d st
eel
bar
has
been
mac
hine
d to
a d
iam
eter
of
1 in
. It
is t
o be
pla
ced
inre
vers
ed a
xial
loa
ding
for
70
000
cycl
es t
o fa
ilure
in
an o
pera
ting
envi
ronm
ent
of55
0°F.
Usi
ng A
STM
min
imum
pro
pert
ies,
and
a r
elia
bilit
y of
99
perc
ent,
estim
ate
the
endu
ranc
e lim
it an
d fa
tigue
str
engt
h at
70
000
cycl
es.
Solu
tion
From
Tab
le A
–20,
Sut
=50
kps
i at
70°
F. S
ince
the
rot
atin
g-be
am s
peci
men
end
uran
celim
it is
not
kno
wn
at r
oom
tem
pera
ture
, we
dete
rmin
e th
e ul
timat
e st
reng
th a
t th
e el
e-va
ted
tem
pera
ture
firs
t, us
ing
Tabl
e 6–
4. F
rom
Tab
le 6
–4,
( S T S RT
) 550◦
=0.
995
+0.
963
2=
0.97
9
The
ulti
mat
e st
reng
th a
t 550
°F is
then
(Su
t)55
0◦=
(ST/
S RT) 5
50◦(S
ut)
70◦=
0.97
9(50
)=
49.0
kpsi
The
rot
atin
g-be
am s
peci
men
end
uran
ce li
mit
at 5
50°F
is th
en e
stim
ated
fro
m E
q. (
6–8)
as
S′ e=
0.5(
49)=
24.5
kpsi
Nex
t, w
e de
term
ine
the
Mar
in f
acto
rs.
For
the
mac
hine
d su
rfac
e, E
q. (
6–19
) w
ithTa
ble
6–2
give
s
k a=
aSb u
t=
2.70
(49−0
.265
)=
0.96
3
For
axia
l loa
ding
, fro
m E
q. (
6–21
), th
e si
ze fa
ctor
kb=
1, a
nd f
rom
Eq.
(6–
26)
the
load
-in
g fa
ctor
is k
c=
0.85
. The
tem
pera
ture
fac
tor
k d=
1, s
ince
we
acco
unte
d fo
r th
e te
m-
pera
ture
in
mod
ifyi
ng t
he u
ltim
ate
stre
ngth
and
con
sequ
ently
the
end
uran
ce l
imit.
For
99 p
erce
nt r
elia
bilit
y, f
rom
Tab
le 6
–5,
k e=
0.81
4. F
inal
ly,
sinc
e no
oth
er c
ondi
tions
wer
e gi
ven,
the
mis
cella
neou
s fa
ctor
is
k f=
1. T
he e
ndur
ance
lim
it fo
r th
e pa
rt i
s es
ti-m
ated
by
Eq.
(6–
18)
as
Ans
wer
S e=
k ak b
k ck d
k ek f
S′ e
=0.
963(
1)(0
.85)
(1)(
0.81
4)(1
)24.
5=
16.3
kpsi
For
the
fatig
ue s
tren
gth
at 7
0 00
0 cy
cles
we
need
to
cons
truc
t th
e S-
Neq
uatio
n. F
rom
p. 2
77, s
ince
Su
t=
49<
70kp
si,t
hen
f�
0.9.
Fro
m E
q. (
6–14
)
a=
(fS
ut)
2
S e=
[0.9
(49)
]2
16.3
=11
9.3
kpsi
and
Eq.
(6–
15)
b=
−1 3
log
( fS u
t
S e
) =−
1 3lo
g
[ 0.9(
49)
16.3
] =−0
.144
1
Fina
lly, f
or th
e fa
tigue
str
engt
h at
70
000
cycl
es, E
q. (
6–13
) gi
ves
Ans
wer
Sf=
aN
b=
119.
3(70
000)
−0.1
441
=23
.9kp
si
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
294
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g291
EXA
MPLE
6–9
Figu
re 6
–22a
show
s a
rota
ting
shaf
t si
mpl
y su
ppor
ted
in b
all
bear
ings
at
Aan
dD
and
load
ed b
y a
nonr
otat
ing
forc
e F
of 6
.8 k
N. U
sing
AST
M “
min
imum
” st
reng
ths,
est
imat
eth
e lif
e of
the
part
.
Solu
tion
From
Fig
. 6–2
2bw
e le
arn
that
fai
lure
will
pro
babl
y oc
cur
at B
rath
er th
an a
t Cor
at t
hepo
int
of m
axim
um m
omen
t. Po
int
Bha
s a
smal
ler
cros
s se
ctio
n, a
hig
her
bend
ing
mom
ent,
and
a hi
gher
str
ess-
conc
entr
atio
n fa
ctor
than
C, a
nd th
e lo
catio
n of
max
imum
mom
ent h
as a
larg
er s
ize
and
no s
tres
s-co
ncen
trat
ion
fact
or.
We
shal
l sol
ve th
e pr
oble
m b
y fir
st e
stim
atin
g th
e st
reng
th a
t poi
nt B
, sin
ce th
e st
reng
thw
ill b
e di
ffer
ent e
lsew
here
, and
com
pari
ng th
is s
tren
gth
with
the
stre
ss a
t the
sam
e po
int.
From
Tab
le A
–20
we
find
S ut=
690
MPa
and
Sy
=58
0M
Pa. T
he e
ndur
ance
lim
itS′ e
is e
stim
ated
as
S′ e=
0.5(
690)
=34
5M
Pa
From
Eq.
(6–
19)
and
Tabl
e 6–
2,
k a=
4.51
(690
)−0.
265
=0.
798
From
Eq.
(6–
20),
k b=
(32/
7.62
)−0.1
07=
0.85
8
Sinc
ek c
=k d
=k e
=k f
=1 ,
S e=
0.79
8(0.
858)
345
=23
6M
Pa
To fi
nd th
e ge
omet
ric
stre
ss-c
once
ntra
tion
fact
or K
tw
e en
ter
Fig.
A–1
5–9
with
D/d
=38
/32
=1.
1875
and
r/d
=3/
32=
0.09
375
and
read
K
t. =
1.65
.S
ubst
itut
ing
S ut=
690/
6.89
=10
0kp
si i
nto
Eq.
(6–
35)
yiel
ds √ a
=0.
0622
√ in=
0.31
3√ mm
.Su
bstit
utin
g th
is in
to E
q. (
6–33
) gi
ves
Kf=
1+
Kt−
1
1+
√ a/r
=1
+1.
65−
1
1+
0.31
3/√ 3
=1.
55
(a)
(b)
AB
MB
MC
Mm
ax
CD
3030
3235
38
1010
AB
CD
6.8
kN
250
125
100
75
R2
R1
Figure
6–2
2
(a) S
haft
draw
ing
show
ing
all
dim
ensio
ns in
milli
met
ers;
all
fille
ts 3-
mm
radi
us. T
he s
haft
rota
tes
and
the
load
issta
tiona
ry; m
ater
ial i
sm
achi
ned
from
AIS
I 105
0co
ld-d
raw
n ste
el. (
b) B
endi
ng-
mom
ent d
iagr
am.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
295
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
292
Mec
hani
cal E
ngin
eerin
g D
esig
n
The
nex
t st
ep i
s to
est
imat
e th
e be
ndin
g st
ress
at
poin
t B
. The
ben
ding
mom
ent
is
MB
=R
1x
=22
5F
550
250
=22
5(6.
8)
550
250
=69
5.5
N·m
Just
to th
e le
ft o
f B th
e se
ctio
n m
odul
us is
I/c
=π
d3/32
=π
323/32
=3.
217
(103
)mm
3.
The
rev
ersi
ng b
endi
ng s
tres
s is
, ass
umin
g in
finite
life
,
σ=
Kf
MB
I/c
=1.
5569
5.5
3.21
7(1
0)−6
=33
5.1(
106)
Pa=
335.
1M
Pa
Thi
s st
ress
is
grea
ter
than
Se
and
less
tha
n S y
. Thi
s m
eans
we
have
bot
h fin
ite l
ife
and
no y
ield
ing
on th
e fir
st c
ycle
. Fo
r fin
ite l
ife,
we
will
nee
d to
use
Eq.
(6–
16).
The
ulti
mat
e st
reng
th,
S ut=
690
MPa
= 1
00 k
psi.
From
Fig
. 6–1
8, f
=0.
844 .
Fro
m E
q. (
6–14
)
a=
(f
S ut)
2
S e=
[0.8
44(6
90)]
2
236
=14
37M
Pa
and
from
Eq.
(6–
15)
b=
−1 3
log
( fS u
t
S e
) =−
1 3lo
g
[ 0.84
4(69
0)
236
] =−0
.130
8
From
Eq.
(6–
16),
Ans
wer
N=( σ a a
) 1/b=( 33
5.1
1437
) −1/0.
1308
=68
(103
)cy
cles
6–11
Chara
cter
izin
g F
luct
uating S
tres
ses
Flu
ctua
ting
str
esse
s in
mac
hine
ry o
ften
take
the
form
of
a si
nuso
idal
pat
tern
bec
ause
of th
e na
ture
of
som
e ro
tati
ng m
achi
nery
. How
ever
, oth
er p
atte
rns,
som
e qu
ite
irre
g-ul
ar,
do o
ccur
. It
has
bee
n fo
und
that
in
peri
odic
pat
tern
s ex
hibi
ting
a s
ingl
e m
axi-
mum
and
a s
ingl
e m
inim
um o
f fo
rce,
the
sha
pe o
f th
e w
ave
is n
ot i
mpo
rtan
t, bu
t th
epe
aks
on b
oth
the
high
sid
e (m
axim
um)
and
the
low
sid
e (m
inim
um)
are
impo
rtan
t.T
hus
Fm
axan
dF
min
in a
cyc
le o
f fo
rce
can
be u
sed
to c
hara
cter
ize
the
forc
e pa
tter
n.It
is
also
tru
e th
at r
angi
ng a
bove
and
bel
ow s
ome
base
line
can
be
equa
lly
effe
ctiv
ein
cha
ract
eriz
ing
the
forc
e pa
tter
n. I
f th
e la
rges
t fo
rce
is F
max
and
the
smal
lest
for
ceis
Fm
in,
then
a s
tead
y co
mpo
nent
and
an
alte
rnat
ing
com
pone
nt c
an b
e co
nstr
ucte
das
fol
low
s:
Fm
=F
max
+F
min
2F
a=∣ ∣ ∣ ∣F
max
−F
min
2
∣ ∣ ∣ ∣w
here
Fm
is t
he m
idra
nge
stea
dy c
ompo
nent
of
forc
e, a
nd F
ais
the
am
plitu
de o
f th
eal
tern
atin
g co
mpo
nent
of
forc
e.
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ynas
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hani
cal E
ngin
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ng
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ign,
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hth
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on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
296
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g293
Figu
re 6
–23
illus
trat
es s
ome
of th
e va
riou
s st
ress
-tim
e tr
aces
that
occ
ur. T
he c
om-
pone
nts
of s
tres
s, s
ome
of w
hich
are
sho
wn
in F
ig. 6
–23d
, are
σm
in=
min
imum
str
ess
σm
=m
idra
nge
com
pone
nt
σm
ax=
max
imum
str
ess
σr
=ra
nge
of s
tres
s
σa
=am
plitu
de c
ompo
nent
σs
=st
atic
or
stea
dy s
tres
s
The
ste
ady,
or
stat
ic, s
tres
s is
not
the
sam
e as
the
mid
rang
e st
ress
; in
fac
t, it
may
hav
ean
y va
lue
betw
een
σm
inan
dσ
max
. The
ste
ady
stre
ss e
xist
s be
caus
e of
a fi
xed
load
or p
re-
load
app
lied
to th
e pa
rt, a
nd it
is u
sual
ly in
depe
nden
t of
the
vary
ing
port
ion
of th
e lo
ad.
A h
elic
al c
ompr
essi
on s
prin
g, f
or e
xam
ple,
is
alw
ays
load
ed i
nto
a sp
ace
shor
ter
than
the
free
leng
th o
f th
e sp
ring
. The
str
ess
crea
ted
by th
is in
itial
com
pres
sion
is c
alle
d th
est
eady
, or
stat
ic, c
ompo
nent
of
the
stre
ss. I
t is
not t
he s
ame
as th
e m
idra
nge
stre
ss.
We
shal
l hav
e oc
casi
on to
app
ly th
e su
bscr
ipts
of
thes
e co
mpo
nent
s to
she
ar s
tres
s-es
as
wel
l as
norm
al s
tres
ses.
The
fol
low
ing
rela
tions
are
evi
dent
fro
m F
ig. 6
–23:
σm
=σ
max
+σ
min
2
σa
=∣ ∣ ∣ ∣σ
max
−σ
min
2
∣ ∣ ∣ ∣(6
–36)
Tim
e
Tim
e
Tim
e
Tim
e
Tim
e
Tim
e
Stress
Stress Stress
Stress
Stress Stress
(a)
(b)
(c)
�a
�a
�r
�m
�m
in
�m
ax
�a
�a
�r
�a
�a
�r
�m
�m
ax
O O
O
(d)
(e)
(f)
�m
in =
0
�m
= 0
+
Figure
6–2
3
Som
e str
ess-t
ime
rela
tions
:(a
)fluc
tuat
ing
stres
s w
ith h
igh-
frequ
ency
ripp
le; (
ban
dc)
nons
inus
oida
l fluc
tuat
ing
stres
s; (d
) sin
usoi
dal fl
uctu
atin
gstr
ess;
(e) r
epea
ted
stres
s;(f)
com
plet
ely
reve
rsed
sinus
oida
l stre
ss.
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ynas
−Nis
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hani
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ngin
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ng
Des
ign,
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hth
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on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
297
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
294
Mec
hani
cal E
ngin
eerin
g D
esig
n
In a
dditi
on to
Eq.
(6–
36),
the
stre
ss r
atio
R=
σm
in
σm
ax(6
–37)
and
the
ampl
itud
e ra
tio
A=
σa
σm
(6–3
8)
are
also
defi
ned
and
used
in c
onne
ctio
n w
ith fl
uctu
atin
g st
ress
es.
Equ
atio
ns (
6–36
) ut
ilize
sym
bols
σa
and
σm
as t
he s
tres
s co
mpo
nent
s at
the
loc
a-tio
n un
der
scru
tiny.
Thi
s m
eans
, in
the
abse
nce
of a
not
ch, σ
aan
dσ
mar
e eq
ual
to t
heno
min
al s
tres
ses σ
aoan
dσ
mo
indu
ced
by lo
ads
Fa
and
Fm
, res
pect
ivel
y; in
the
pres
ence
of a
not
ch t
hey
are
Kfσ
aoan
dK
fσ
mo,
resp
ectiv
ely,
as
long
as
the
mat
eria
l re
mai
nsw
ithou
t pl
astic
str
ain.
In
othe
r w
ords
, th
e fa
tigue
str
ess
conc
entr
atio
n fa
ctor
Kf
isap
plie
d to
bot
hco
mpo
nent
s.W
hen
the
stea
dy s
tres
s co
mpo
nent
is h
igh
enou
gh to
indu
ce lo
caliz
ed n
otch
yie
ld-
ing,
the
des
igne
r ha
s a
prob
lem
. T
he fi
rst-
cycl
e lo
cal
yiel
ding
pro
duce
s pl
astic
str
ain
and
stra
in-s
tren
gthe
ning
. T
his
is o
ccur
ring
at
the
loca
tion
whe
re f
atig
ue c
rack
nuc
le-
atio
n an
d gr
owth
are
mos
t lik
ely.
The
mat
eria
l pro
pert
ies
(Sy
and
S ut)
are
new
and
dif
-fic
ult t
oqu
antif
y. T
he p
rude
nt e
ngin
eer
cont
rols
the
conc
ept,
mat
eria
l and
con
ditio
n of
use,
and
geom
etry
so
that
no
plas
tic s
trai
n oc
curs
. T
here
are
dis
cuss
ions
con
cern
ing
poss
ible
way
s of
qua
ntif
ying
wha
t is
occu
rrin
g un
der
loca
lized
and
gen
eral
yie
ldin
g in
the
pres
ence
of
a no
tch,
ref
erre
d to
as
the
nom
inal
mea
n st
ress
met
hod,
resi
dual
str
ess
met
hod,
an
d th
e lik
e.20
The
no
min
al
mea
n st
ress
m
etho
d (s
et σ
a=
Kfσ
aoan
dσ
m=
σm
o)
give
s ro
ughl
y co
mpa
rabl
e re
sults
to th
e re
sidu
al s
tres
s m
etho
d, b
ut b
oth
are
appr
oxim
atio
ns.
The
re is
the
met
hod
of D
owlin
g21fo
r du
ctile
mat
eria
ls, w
hich
, for
mat
eria
ls w
ith a
pron
ounc
ed y
ield
poi
nt a
nd a
ppro
xim
ated
by
an e
last
ic–p
erfe
ctly
pla
stic
beh
avio
rm
odel
, qua
ntita
tivel
y ex
pres
ses
the
stea
dy s
tres
s co
mpo
nent
str
ess-
conc
entr
atio
n fa
ctor
Kfm
as
Kfm
=K
fK
f|σ m
ax,o|<
S y
Kfm
=S y
−K
fσ
ao
|σ mo|
Kf|σ m
ax,o|>
S y
Kfm
=0
Kf|σ m
ax,o
−σ
min
,o|>
2Sy
(6–3
9)
For
the
purp
oses
of
this
boo
k, f
or d
uctil
e m
ater
ials
in f
atig
ue,
•A
void
loca
lized
pla
stic
str
ain
at a
not
ch. S
et σ
a=
Kfσ
a,o
and
σm
=K
fσ
mo.
•W
hen
plas
tic s
trai
n at
a n
otch
can
not b
e av
oide
d, u
se E
qs. (
6–39
); o
r co
nser
vativ
ely,
setσ
a=
Kfσ
aoan
d us
e K
fm=
1 , th
at is
, σm
=σ
mo.
20R
. C. J
uvin
all,
Stre
ss, S
trai
n, a
nd S
tren
gth,
McG
raw
-Hill
, New
Yor
k, 1
967,
art
icle
s 14
.9–1
4.12
; R. C
.Ju
vina
ll an
d K
. M. M
arsh
ek, F
unda
men
tals
of M
achi
ne C
ompo
nent
Des
ign,
4th
ed.
, Wile
y, N
ew Y
ork,
200
6,Se
c.8.
11; M
. E. D
owlin
g, M
echa
nica
l Beh
avio
r of
Mat
eria
ls, 2
nd e
d., P
rent
ice
Hal
l, E
ngle
woo
d C
liffs
,N
.J.,
1999
, Sec
s. 1
0.3–
10.5
.21
Dow
ling,
op.
cit.
, p. 4
37–4
38.
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ynas
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ign,
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hth
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on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
298
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g295
6–12
Fatigue
Failu
re C
rite
ria f
or
Fluct
uating S
tres
sN
ow th
at w
e ha
ve d
efine
d th
e va
riou
s co
mpo
nent
s of
str
ess
asso
ciat
ed w
ith a
par
t sub
-je
cted
to
fluct
uatin
g st
ress
, w
e w
ant
to v
ary
both
the
mid
rang
e st
ress
and
the
str
ess
ampl
itude
, or
alte
rnat
ing
com
pone
nt, t
o le
arn
som
ethi
ng a
bout
the
fatig
ue r
esis
tanc
e of
part
s w
hen
subj
ecte
d to
suc
h si
tuat
ions
. Thr
ee m
etho
ds o
f pl
ottin
g th
e re
sults
of
such
test
s ar
e in
gen
eral
use
and
are
sho
wn
in F
igs.
6–2
4, 6
–25,
and
6–2
6.T
hem
odifi
ed G
oodm
an d
iagr
amof
Fig
. 6–2
4 ha
s th
e m
idra
nge
stre
ss p
lotte
d al
ong
the
absc
issa
and
all
othe
r co
mpo
nent
s of
str
ess
plot
ted
on t
he o
rdin
ate,
with
ten
sion
in
the
posi
tive
dire
ctio
n. T
he e
ndur
ance
lim
it, f
atig
ue s
tren
gth,
or
finite
-lif
e st
reng
th,
whi
chev
er a
pplie
s, is
plo
tted
on th
e or
dina
te a
bove
and
bel
ow th
e or
igin
. The
mid
rang
e-st
ress
line
is a
45◦
line
from
the
orig
in to
the
tens
ile s
tren
gth
of th
e pa
rt. T
he m
odifi
edG
oodm
an d
iagr
am c
onsi
sts
of t
he l
ines
con
stru
cted
to
S e(o
rS
f)
abov
e an
d be
low
the
orig
in. N
ote
that
the
yiel
d st
reng
th is
als
o pl
otte
d on
bot
h ax
es, b
ecau
se y
ield
ing
wou
ldbe
the
crite
rion
of
failu
re if
σm
axex
ceed
ed S
y.
Ano
ther
way
to d
ispl
ay te
st r
esul
ts is
sho
wn
in F
ig. 6
–25.
Her
e th
e ab
scis
sa r
epre
-se
nts
the
ratio
of
the
mid
rang
e st
reng
th S
mto
the
ulti
mat
e st
reng
th, w
ith t
ensi
on p
lot-
ted
to t
he r
ight
and
com
pres
sion
to
the
left
. The
ord
inat
e is
the
rat
io o
f th
e al
tern
atin
gst
reng
th t
o th
e en
dura
nce
limit.
The
lin
e B
Cth
en r
epre
sent
s th
e m
odifi
ed G
oodm
ancr
iteri
on o
f fa
ilure
. Not
e th
at th
e ex
iste
nce
of m
idra
nge
stre
ss in
the
com
pres
sive
reg
ion
has
little
eff
ect o
n th
e en
dura
nce
limit.
The
ver
y cl
ever
dia
gram
of
Fig.
6–2
6 is
uni
que
in th
at it
dis
play
s fo
ur o
f th
e st
ress
com
pone
nts
as w
ell
as t
he t
wo
stre
ss r
atio
s. A
cur
ve r
epre
sent
ing
the
endu
ranc
e lim
itfo
r va
lues
of
Rbe
ginn
ing
at R
=−1
and
endi
ng w
ith R
=1
begi
ns a
t S e
on th
e σ
aax
isan
d en
ds a
t S u
ton
the
σm
axis
. Con
stan
t-lif
e cu
rves
for
N=
105
and
N=
104
cycl
es
Figure
6–2
4
Mod
ified
Goo
dman
dia
gram
show
ing
all t
he s
treng
ths
and
the
limiti
ng v
alue
s of
all
the
stres
s co
mpo
nent
s fo
r apa
rticu
lar m
idra
nge
stres
s.
S u S y S e
S yS u
�m
�m
ax Max
. stre
ss
�m
in
Midr
ange
stres
s
Min. stres
s
Stress
0
S e
Mid
rang
e st
ress
Parallel
45°
�a
�a
�a
�r
+
Bud
ynas
−Nis
bett:
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gley
’s
Mec
hani
cal E
ngin
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ng
Des
ign,
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hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
299
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
296
Mec
hani
cal E
ngin
eerin
g D
esig
n
have
bee
n dr
awn
too.
Any
str
ess
stat
e, s
uch
as th
e on
e at
A, c
an b
e de
scri
bed
by th
e m
in-
imum
and
max
imum
com
pone
nts,
or
by th
e m
idra
nge
and
alte
rnat
ing
com
pone
nts.
And
safe
ty is
indi
cate
d w
hene
ver t
he p
oint
des
crib
ed b
y th
e st
ress
com
pone
nts
lies
belo
w th
eco
nsta
nt-l
ife
line.
Figure
6–2
6
Mas
ter fa
tigue
dia
gram
crea
ted
for A
ISI 4
340 s
teel
havi
ngS u
t=
158
and
S y=
147
kpsi. Th
e stre
ssco
mpo
nent
s at
Aar
eσ
min
=20
,σ
max
=12
0,
σm
=70
, an
d σ
a=
50, al
l in
kpsi.
(Sou
rce:
H. J.
Gro
ver,
Fatig
ue o
f Airc
raft
Stru
ctur
es,
U.S
. G
over
nmen
t Prin
ting
Offi
ce, W
ashi
ngto
n, D
.C.,
1966, pp
. 317, 322. Se
eal
so J.
A. C
ollin
s, F
ailu
re o
fM
ater
ials in
Mec
hani
cal
Des
ign,
Wile
y, N
ewYo
rk,
1981, p.
216.)
–120
–100
–80
–60
–40
–20
020
4060
8010
012
014
016
018
0
20406080100
120
A =
�R
= –
1.0
20
40
60
80
100
120
140
160
180
Midr
ange
stres
s �m, k
psi
20
40
60
80
100
120
Altern
ating
stres
s �a, k
psi
RA
4.0
–0.
62.
33–
0.4
1.5
–0.
2A
= 1
R=
00.
67 0.2
0.43 0.4
0.25 0.6
0.11 0.8
0 1.0
Maximum stress �max, kpsi
Min
imum
str
ess
�m
in, k
psi
105 10
6
104
cycl
es
A
S ut
S e
Figure
6–2
5
Plot
of f
atig
ue fa
ilure
s fo
r mid
rang
e str
esse
s in
bot
h te
nsile
and
com
pres
sive
regi
ons.
Nor
mal
izin
gth
eda
ta b
y us
ing
the
ratio
of s
tead
y str
engt
h co
mpo
nent
to te
nsile
stre
ngth
Sm/S u
t, s
tead
y str
engt
hco
mpo
nent
to c
ompr
essiv
e str
engt
h S m
/S u
can
d str
engt
h am
plitu
de c
ompo
nent
to e
ndur
ance
lim
itS a
/S′ e
enab
les
a pl
ot o
f exp
erim
enta
l res
ults
for a
var
iety
of s
teel
s. [D
ata
sour
ce: T
hom
as J.
Dol
an,
“Stre
ss R
ange
,” S
ec.6
.2 in
O. J
. Hor
ger (
ed.),
ASM
E H
andb
ook—
Met
als
Engi
neer
ing
Des
ign,
McG
raw
-Hill,
New
Yor
k, 1
953.
]
–1.2
–1.0
–0.
8–
0.6
–0.
4–
0.2
00.
20.
40.
60.
81.
0
0.2
0.4
0.6
0.8
1.0
1.2
Mid
rang
e ra
tio
Com
pres
sion
S m/S
uc
Tens
ion
S m/S
ut
Amplitude ratio Sa/Se
AB
C
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
300
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g297
Whe
n th
e m
idra
nge
stre
ss i
s co
mpr
essi
on,
failu
re o
ccur
s w
hene
ver
σa
=S e
orw
hene
ver σ
max
=S y
c, a
s in
dica
ted
by th
e le
ft-h
and
side
of
Fig.
6–2
5. N
eith
er a
fat
igue
diag
ram
nor
any
oth
er f
ailu
re c
rite
ria
need
be
deve
lope
d.In
Fig
. 6–2
7, t
he t
ensi
le s
ide
of F
ig. 6
–25
has
been
red
raw
n in
ter
ms
of s
tren
gths
,in
stea
d of
str
engt
h ra
tios,
with
the
sam
e m
odifi
ed G
oodm
an c
rite
rion
toge
ther
with
fou
rad
ditio
nal
crite
ria
of f
ailu
re.
Such
dia
gram
s ar
e of
ten
cons
truc
ted
for
anal
ysis
and
desi
gn p
urpo
ses;
they
are
eas
y to
use
and
the
resu
lts c
an b
e sc
aled
off
dir
ectly
.T
he e
arly
vie
wpo
int e
xpre
ssed
on
a σ
aσ
mdi
agra
m w
as th
at th
ere
exis
ted
a lo
cus
whi
chdi
vide
d sa
fe f
rom
uns
afe
com
bina
tions
of
σa
and
σm
. E
nsui
ng p
ropo
sals
inc
lude
d th
epa
rabo
la o
f Ger
ber (
1874
), th
e G
oodm
an (1
890)
22(s
trai
ght)
line
, and
the
Sode
rber
g (1
930)
(str
aigh
t) li
ne. A
s m
ore
data
wer
e ge
nera
ted
it be
cam
e cl
ear
that
a f
atig
ue c
rite
rion
, rat
her
than
bei
ng a
“fe
nce,
” w
as m
ore
like
a zo
ne o
r ban
d w
here
in th
e pr
obab
ility
of f
ailu
re c
ould
be e
stim
ated
. We
incl
ude
the
failu
re c
rite
rion
of
Goo
dman
bec
ause
•It
is a
str
aigh
t lin
e an
d th
e al
gebr
a is
line
ar a
nd e
asy.
•It
is e
asily
gra
phed
, eve
ry ti
me
for
ever
y pr
oble
m.
•It
rev
eals
sub
tletie
s of
insi
ght i
nto
fatig
ue p
robl
ems.
•A
nsw
ers
can
be s
cale
d fr
om th
e di
agra
ms
as a
che
ck o
n th
e al
gebr
a.
We
also
caut
ion
that
itis
dete
rmin
istic
and
the
phen
omen
onis
not.
Itis
bias
edan
dw
eca
nnot
quan
tify
the
bias
.Iti
sno
tcon
serv
ativ
e.It
isa
step
ping
-sto
neto
unde
rsta
ndin
g;it
ishi
stor
y;an
dto
read
the
wor
kof
othe
reng
inee
rsan
dto
have
mea
ning
fulo
rale
xcha
nges
with
them
,iti
sne
cess
ary
that
you
unde
rsta
ndth
eG
oodm
anap
proa
chsh
ould
itar
ise.
Eith
er th
e fa
tigue
lim
it S e
or th
e fin
ite-l
ife
stre
ngth
Sf
is p
lotte
d on
the
ordi
nate
of
Fig.
6–2
7. T
hese
val
ues
will
hav
e al
read
y be
en c
orre
cted
usi
ng t
he M
arin
fac
tors
of
Eq.
(6–1
8). N
ote
that
the
yiel
d st
reng
th S
yis
plo
tted
on th
e or
dina
te to
o. T
his
serv
es a
sa
rem
inde
r th
at fi
rst-
cycl
e yi
eldi
ng r
athe
r th
an f
atig
ue m
ight
be
the
crite
rion
of
failu
re.
The
mid
rang
e-st
ress
axi
s of
Fig
. 6–
27 h
as t
he y
ield
str
engt
h S y
and
the
tens
ilest
reng
thS u
tpl
otte
d al
ong
it.
Figure
6–2
7
Fatig
ue d
iagr
am s
how
ing
vario
us c
riter
ia o
f fai
lure
. For
each
crit
erio
n, p
oint
s on
or
“abo
ve”
the
resp
ectiv
e lin
ein
dica
te fa
ilure
. Som
e po
int A
on th
e G
oodm
an li
ne, f
orex
ampl
e, g
ives
the
stren
gth
S mas
the
limiti
ng v
alue
of σ
m
corre
spon
ding
to th
e str
engt
hS a
, whi
ch, p
aire
d w
ith σ
m, i
sth
e lim
iting
val
ue o
fσa.
Alternating stress �a
Mid
rang
e st
ress
�m
0S m
A
S ut
S y0S aS eS y
Sode
rber
g lin
e
Mod
ifie
d G
oodm
an li
ne
ASM
E-e
llipt
ic li
ne
Loa
d lin
e, s
lope
r =
Sa/S
m
Ger
ber
line
Yie
ld (
Lan
ger)
line
22It
is d
iffic
ult t
o da
te G
oodm
an’s
wor
k be
caus
e it
wen
t thr
ough
sev
eral
mod
ifica
tions
and
was
nev
erpu
blis
hed.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
301
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
298
Mec
hani
cal E
ngin
eerin
g D
esig
n
Five
crite
ria
offa
ilure
are
diag
ram
med
inFi
g.6–
27:
the
Sode
rber
g,th
em
odifi
edG
oodm
an,
the
Ger
ber,
the
ASM
E-e
llipt
ic,
and
yiel
ding
.The
diag
ram
show
sth
aton
lyth
eSo
derb
erg
crite
rion
guar
dsag
ains
tany
yiel
ding
,but
isbi
ased
low
.C
onsi
deri
ng t
he m
odifi
ed G
oodm
an l
ine
as a
cri
teri
on, p
oint
Are
pres
ents
a l
imit-
ing
poin
t with
an
alte
rnat
ing
stre
ngth
Sa
and
mid
rang
e st
reng
th S
m. T
he s
lope
of t
he lo
adlin
e sh
own
is d
efine
d as
r=
S a/
S m.
The
cri
teri
on e
quat
ion
for
the
Sode
rber
g lin
e is
S a S e+
S m S y=
1(6
–40)
Sim
ilarl
y, w
e fin
d th
e m
odifi
ed G
oodm
an r
elat
ion
to b
e
S a S e+
S m S ut
=1
(6–4
1)
Exa
min
atio
n of
Fig
. 6–2
5 sh
ows
that
bot
h a
para
bola
and
an
ellip
se h
ave
a be
tter
oppo
rtun
ity to
pas
s am
ong
the
mid
rang
e te
nsio
n da
ta a
nd to
per
mit
quan
tifica
tion
of th
epr
obab
ility
of
failu
re. T
he G
erbe
r fa
ilure
cri
teri
on is
wri
tten
as
S a S e+( S m S u
t
) 2 =1
(6–4
2)
and
the
ASM
E-e
llipt
ic is
wri
tten
as ( S a S e
) 2 +( S m S y
) 2 =1
(6–4
3)
The
Lan
ger
firs
t-cy
cle-
yiel
ding
cri
teri
on i
s us
ed i
n co
nnec
tion
wit
h th
e fa
tigu
ecu
rve:
S a+
S m=
S y(6
–44)
The
str
esse
s nσ
aan
dnσ
mca
n re
plac
e S a
and
S m, w
here
nis
the
desi
gn f
acto
r or
fac
tor
of s
afet
y. T
hen,
Eq.
(6–
40),
the
Sode
rber
g lin
e, b
ecom
es
Sode
rber
gσ
a S e+
σm S y
=1 n
(6–4
5)
Equ
atio
n (6
–41)
, the
mod
ified
Goo
dman
line
, bec
omes
mod
-Goo
dman
σa S e
+σ
m S ut
=1 n
(6–4
6)
Equ
atio
n (6
–42)
, the
Ger
ber
line,
bec
omes
Ger
ber
nσa
S e+( nσ
m
S ut
) 2 =1
(6–4
7)
Equ
atio
n (6
–43)
, the
ASM
E-e
llipt
ic li
ne, b
ecom
es
ASM
E-e
llipt
ic( nσ
a
S e
) 2 +( nσ
m
S y
) 2 =1
(6–4
8)
We
will
em
phas
ize
the
Ger
ber
and
ASM
E-e
llipt
ic f
or f
atig
ue f
ailu
re c
rite
rion
and
the
Lan
ger
for
first
-cyc
le y
ield
ing.
How
ever
, con
serv
ativ
e de
sign
ers
ofte
n us
e th
e m
odifi
edG
oodm
an c
rite
rion
, so
we
will
con
tinue
to
incl
ude
it in
our
dis
cuss
ions
. T
he d
esig
neq
uatio
n fo
r th
e L
ange
r fir
st-c
ycle
-yie
ldin
g is
Lan
ger
stat
ic y
ield
σa+
σm
=S y n
(6–4
9)
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
302
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g299
Inte
rsec
ting E
quations
Inte
rsec
tion C
oord
inate
s
S a S e+( S m S u
t
) 2 =1
S a=
r2S2 ut
2S e
⎡ ⎣ −1
+√ 1
+( 2
S erS
ut
) 2⎤ ⎦
Load
line
r=
S a S mS m
=S a r
S a S y+
S m S y=
1S a
=rS
y
1+
r
Load
line
r=
S a S mS m
=S y
1+
r
S a S e+( S m S u
t
) 2 =1
S m=
S2 ut
2S e
⎡ ⎣ 1−√ 1
+( 2
S e S ut
) 2(1
−S y S e
)⎤ ⎦S a S y
+S m S y
=1
S a=
S y−
S m,r
crit
=S a
/S m
Fatig
ue fa
ctor
of s
afet
y
n f=
1 2
( S ut
σm
) 2 σa S e
⎡ ⎣ −1
+√ 1
+( 2σ
mS e
S utσ
a
) 2⎤ ⎦σ
m>
0
Table
6–7
Am
plitu
de a
nd S
tead
yC
oord
inat
es o
f Stre
ngth
and
Impo
rtant
Inte
rsec
tions
in F
irst
Qua
dran
t for
Ger
ber
and
Lang
er F
ailu
reC
riter
ia
Inte
rsec
ting E
quations
Inte
rsec
tion C
oord
inate
s
S a S e+
S m S ut
=1
S a=
rSeS
ut
rSut
+S e
Load
line
r=
S a S mS m
=S a r
S a S y+
S m S y=
1S a
=rS
y
1+
r
Load
line
r=
S a S mS m
=S y
1+
rS a S e
+S m S u
t=
1S m
=( S y
−S e) S u
t
S ut−
S eS a S y
+S m S y
=1
S a=
S y−
S m,r
crit
=S a
/S m
Fatig
ue fa
ctor
of s
afet
y
n f=
1σ
a S e+
σm S ut
Table
6–6
Am
plitu
de a
nd S
tead
yC
oord
inat
es o
f Stre
ngth
and
Impo
rtant
Inte
rsec
tions
in F
irst
Qua
dran
t for
Mod
ified
Goo
dman
and
Lan
ger
Failu
re C
riter
ia
The
fai
lure
cri
teri
a ar
e us
ed i
n co
njun
ctio
n w
ith a
loa
d lin
e, r
=S a
/S m
=σ
a/σ
m.
Prin
cipa
l in
ters
ectio
ns a
re t
abul
ated
in
Tabl
es 6
–6 t
o 6–
8. F
orm
al e
xpre
ssio
ns f
orfa
tigue
fac
tor
of s
afet
y ar
e gi
ven
in th
e lo
wer
pan
el o
f Ta
bles
6–6
to 6
–8. T
he fi
rst r
owof
eac
h ta
ble
corr
espo
nds
to t
he f
atig
ue c
rite
rion
, th
e se
cond
row
is
the
stat
ic L
ange
rcr
iteri
on,
and
the
thir
d ro
w c
orre
spon
ds t
o th
e in
ters
ectio
n of
the
sta
tic a
nd f
atig
ue
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
303
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
300
Mec
hani
cal E
ngin
eerin
g D
esig
n
crite
ria.
The
firs
t co
lum
n gi
ves
the
inte
rsec
ting
equa
tions
and
the
sec
ond
colu
mn
the
inte
rsec
tion
coor
dina
tes.
The
re a
re tw
o w
ays
to p
roce
ed w
ith
a ty
pica
l ana
lysi
s. O
ne m
etho
d is
to a
ssum
eth
at f
atig
ue o
ccur
s fi
rst
and
use
one
of E
qs. (
6–45
) to
(6–
48)
to d
eter
min
e n
or s
ize,
depe
ndin
g on
the
tas
k. M
ost
ofte
n fa
tigu
e is
the
gov
erni
ng f
ailu
re m
ode.
The
nfo
llow
wit
h a
stat
ic c
heck
. If
stat
ic f
ailu
re g
over
ns th
en th
e an
alys
is is
rep
eate
d us
ing
Eq.
(6–4
9).
Alte
rnat
ivel
y, o
ne c
ould
use
the
tabl
es. D
eter
min
e th
e lo
ad li
ne a
nd e
stab
lish
whi
chcr
iteri
on th
e lo
ad li
ne in
ters
ects
firs
t and
use
the
corr
espo
ndin
g eq
uatio
ns in
the
tabl
es.
Som
e ex
ampl
es w
ill h
elp
solid
ify
the
idea
s ju
st d
iscu
ssed
.
EXA
MPLE
6–1
0A
1.5
-in-
diam
eter
bar
has
bee
n m
achi
ned
from
an
AIS
I 10
50 c
old-
draw
n ba
r. T
his
part
is to
with
stan
d a
fluct
uatin
g te
nsile
load
var
ying
fro
m 0
to 1
6 ki
p. B
ecau
se o
f th
e en
ds,
and
the
fille
t ra
dius
, a
fatig
ue s
tres
s-co
ncen
trat
ion
fact
or K
fis
1.8
5 fo
r 10
6or
lar
ger
life.
Fin
d S a
and
S man
d th
e fa
ctor
of
safe
ty g
uard
ing
agai
nst
fatig
ue a
nd fi
rst-
cycl
eyi
eldi
ng, u
sing
(a)
the
Ger
ber
fatig
ue li
ne a
nd (
b) th
e A
SME
-elli
ptic
fat
igue
line
.
Solu
tion
We
begi
n w
ith s
ome
prel
imin
arie
s. F
rom
Tab
le A
–20,
Su
t=
100
kpsi
and
Sy
=84
kpsi
.N
ote
that
Fa
=F
m=
8ki
p. T
he M
arin
fac
tors
are
, det
erm
inis
tical
ly,
k a=
2.70
(100
)−0.
265
=0.
797:
Eq.
(6–
19),
Tab
le 6
–2, p
. 279
k b=
1(a
xial
load
ing,
see
k c)
Inte
rsec
ting E
quations
Inte
rsec
tion C
oord
inate
s
( S a S e
) 2 +( S m S y
) 2 =1
S a=√ √ √ √
r2S2 e
S2 y
S2 e+
r2S2 y
Load
line
r=
S a/S m
S m=
S a rS a S y
+S m S y
=1
S a=
rSy
1+
r
Load
line
r=
S a/S m
S m=
S y1
+r
( S a S e
) 2 +( S m S y
) 2 =1
S a=
0,2
S yS2 e
S2 e+
S2 y
S a S y+
S m S y=
1S m
=S y
−S a
,rcr
it=
S a/S m
Fatig
ue fa
ctor
of s
afet
y
n f=√ √ √ √
1
(σa/S e
)2+( σ
m/S y) 2
Table
6–8
Am
plitu
de a
nd S
tead
yC
oord
inat
es o
f Stre
ngth
and
Impo
rtant
Inte
rsec
tions
in F
irst
Qua
dran
t for
ASM
E-El
liptic
and
Lan
ger
Failu
re C
riter
ia
Bud
ynas
−Nis
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gley
’s
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hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
304
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g301
k c=
0.85
:E
q. (
6–26
), p
. 282
k d=
k e=
k f=
1
S e=
0.79
7(1)
0.85
0(1)
(1)(
1)0.
5(10
0)=
33.9
kpsi
:Eqs
. (6–
8), (
6–18
), p
. 274
, p. 2
79
The
nom
inal
axi
al s
tres
s co
mpo
nent
s σ
aoan
dσ
mo
are
σao
=4
Fa
πd
2=
4(8)
π1.
52=
4.53
kpsi
σm
o=
4F
m
πd
2=
4(8)
π1.
52=
4.53
kpsi
App
lyin
gK
fto
bot
h co
mpo
nent
s σ
aoan
dσ
mo
cons
titut
es a
pre
scri
ptio
n of
no
notc
hyi
eldi
ng:
σa
=K
fσ
ao=
1.85
(4.5
3)=
8.38
kpsi
=σ
m
(a)
Let
us
calc
ulat
e th
e fa
ctor
s of
saf
ety
first
. Fro
m th
e bo
ttom
pan
el f
rom
Tab
le 6
–7 th
efa
ctor
of
safe
ty f
or f
atig
ue is
Ans
wer
nf
=1 2
( 100
8.38
) 2(8.
38
33.9
)⎧ ⎨ ⎩−1+√ 1
+[ 2(
8.38
)33.
9
100(
8.38
)
] 2⎫ ⎬ ⎭=3.
66
From
Eq.
(6–
49)
the
fact
or o
f sa
fety
gua
rdin
g ag
ains
t firs
t-cy
cle
yiel
d is
Ans
wer
ny
=S y
σa+
σm
=84
8.38
+8.
38=
5.01
Thu
s, w
e se
e th
at f
atig
ue w
ill o
ccur
firs
t an
d th
e fa
ctor
of
safe
ty i
s 3.
68. T
his
can
bese
en in
Fig
. 6–2
8 w
here
the
load
line
inte
rsec
ts th
e G
erbe
r fa
tigue
cur
ve fi
rst a
t poi
nt B
.If
the
plot
s ar
e cr
eate
d to
true
sca
le it
wou
ld b
e se
en th
at n
f=
OB/
OA
.Fr
om th
e fir
st p
anel
of
Tabl
e 6–
7, r
=σ
a/σ
m=
1,
Ans
wer
S a=
(1)2
1002
2(33
.9)
⎧ ⎨ ⎩−1+√ 1
+[ 2(
33.9
)
(1)1
00
]2⎫ ⎬ ⎭=30
.7kp
si
Stress amplitude �a, kpsi
Mid
rang
e st
ress
�m
, kps
i
030
.78.
38
8.38
4250
6484
100
020
33.9
30.74250
Loa
d lin
e
Lan
ger
line
r crit
84100
Ger
ber
fatig
ue c
urve
A
B
C
D
Figure
6–2
8
Prin
cipa
l poi
nts
A,B
,C, a
ndD
on th
e de
signe
r’s d
iagr
amdr
awn
for G
erbe
r, La
nger
, and
load
line
.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
305
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
302
Mec
hani
cal E
ngin
eerin
g D
esig
n
Ans
wer
S m=
S a r=
30.7 1
=30
.7kp
si
As
a ch
eck
on t
he p
revi
ous
resu
lt, n
f=
OB/
OA
=S a
/σ
a=
S m/σ
m=
30.7
/8.
38=
3.66
and
we
see
tota
l agr
eem
ent.
We
coul
d ha
ve d
etec
ted
that
fat
igue
fai
lure
wou
ld o
ccur
firs
t with
out d
raw
ing
Fig.
6–28
by
calc
ulat
ing
r cri
t. F
rom
the
thir
d ro
w th
ird
colu
mn
pane
l of T
able
6–7
, the
inte
r-se
ctio
n po
int b
etw
een
fatig
ue a
nd fi
rst-
cycl
e yi
eld
is
S m=
1002
2(33
.9)
⎡ ⎣ 1−√ 1
+( 2(
33.9
)
100
) 2(1
−84 33.9
)⎤ ⎦ =64
.0kp
si
S a=
S y−
S m=
84−
64=
20kp
si
The
cri
tical
slo
pe is
thus
r cri
t=
S a S m=
20 64=
0.31
2
whi
ch is
less
than
the
actu
al lo
ad li
ne o
f r=
1 . T
his
indi
cate
s th
at fa
tigue
occ
urs
befo
refir
st-c
ycle
-yie
ld.
(b)
Rep
eatin
g th
e sa
me
proc
edur
e fo
r th
e A
SME
-elli
ptic
line
, for
fat
igue
Ans
wer
nf=√
1
(8.3
8/33
.9)2
+(8
.38/
84)2
=3.
75
Aga
in, t
his
is l
ess
than
ny
=5.
01an
d fa
tigue
is
pred
icte
d to
occ
ur fi
rst.
From
the
firs
tro
w s
econ
d co
lum
n pa
nel
of T
able
6–8
, w
ith r
=1 ,
we
obta
in t
he c
oord
inat
es S
aan
dS m
of p
oint
Bin
Fig
. 6–2
9 as
Stress amplitude �a, kpsi
Mid
rang
e st
ress
�m
, kps
i
08.
3831
.442
5060
.584
100
0
8.38
31.4
23.54250
Loa
d lin
e
Lan
ger
line
84100
ASM
E-e
llipt
ic li
ne
A
B
C
D
Figure
6–2
9
Prin
cipa
l poi
nts
A,B
,C, a
ndD
on th
e de
signe
r’s d
iagr
amdr
awn
for A
SME-
ellip
tic,
Lang
er, a
nd lo
ad li
nes.
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ynas
−Nis
bett:
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gley
’s
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hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
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on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
306
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g303
EXA
MPLE
6–1
1A
flat
-lea
f sp
ring
is
used
to
reta
in a
n os
cilla
ting
flat-
face
d fo
llow
er i
n co
ntac
t w
ith a
plat
e ca
m. T
he f
ollo
wer
ran
ge o
f m
otio
n is
2 in
and
fixe
d, s
o th
e al
tern
atin
g co
mpo
nent
of f
orce
, ben
ding
mom
ent,
and
stre
ss i
s fix
ed, t
oo. T
he s
prin
g is
pre
load
ed t
o ad
just
to
vari
ous
cam
spe
eds.
The
pre
load
mus
t be
inc
reas
ed t
o pr
even
t fo
llow
er fl
oat
or j
ump.
For
low
er s
peed
s th
e pr
eloa
d sh
ould
be
decr
ease
d to
obt
ain
long
er l
ife
of c
am a
ndfo
llow
er s
urfa
ces.
The
spr
ing
is a
ste
el c
antil
ever
32
in lo
ng, 2
in w
ide,
and
1 4in
thic
k,as
see
n in
Fig
. 6–3
0a. T
he s
prin
g st
reng
ths
are
S ut=
150
kpsi
,Sy
=12
7kp
si, a
nd S
e=
28kp
si f
ully
cor
rect
ed. T
he t
otal
cam
mot
ion
is 2
in.
The
des
igne
r w
ishe
s to
pre
load
the
spri
ng b
y de
flect
ing
it 2
in f
or lo
w s
peed
and
5 in
for
hig
h sp
eed.
(a)
Plot
the
Ger
ber-
Lan
ger
failu
re li
nes
with
the
load
line
.(b
) W
hat a
re th
e st
reng
th f
acto
rs o
f sa
fety
cor
resp
ondi
ng to
2 in
and
5 in
pre
load
?
Solu
tion
We
begi
n w
ith p
relim
inar
ies.
The
sec
ond
area
mom
ent o
f th
e ca
ntile
ver
cros
s se
ctio
n is
I=
bh3
12=
2(0.
25)3
12=
0.00
260
in4
Sinc
e, f
rom
Tab
le A
–9, b
eam
1, f
orce
Fan
d de
flect
ion
yin
a c
antil
ever
are
rel
ated
by
F=
3E
Iy/
l3 ,th
en s
tres
s σ
and
defle
ctio
n y
are
rela
ted
by
σ=
Mc I
=32
Fc
I=
32(3
EIy
)
l3
c I=
96E
cy
l3=
Ky
whe
reK
=96
Ec
l3=
96(3
0·1
06)0
.125
323
=10
.99(
103)
psi/i
n=
10.9
9kp
si/in
Now
the
min
imum
s an
d m
axim
ums
of y
and
σca
n be
defi
ned
by
y min
=δ
y max
=2
+δ
σm
in=
Kδ
σm
ax=
K(2
+δ)
Ans
wer
S a=√ (1
)233
.92(8
4)2
33.9
2+
(1)2
842
=31
.4kp
si,
S m=
S a r=
31.4 1
=31
.4kp
si
To v
erif
y th
e fa
tigue
fac
tor
of s
afet
y, n
f=
S a/σ
a=
31.4
/8.
38=
3.75
.A
s be
fore
, le
t us
cal
cula
te r
crit.
From
the
thi
rd r
ow s
econ
d co
lum
n pa
nel
ofTa
ble
6–8, S a
=2(
84)3
3.92
33.9
2+
842
=23
.5kp
si,
S m=
S y−
S a=
84−
23.5
=60
.5kp
si
r cri
t=
S a S m=
23.5
60.5
=0.
388
whi
ch a
gain
is le
ss th
an r
=1 ,
ver
ifyi
ng th
at f
atig
ue o
ccur
s fir
st w
ith n
f=
3.75
.T
he G
erbe
r an
d th
e A
SME
-elli
ptic
fat
igue
fai
lure
cri
teri
a ar
e ve
ry c
lose
to
each
othe
r an
d ar
e us
ed i
nter
chan
geab
ly.
The
AN
SI/A
SME
Sta
ndar
d B
106.
1M–1
985
uses
ASM
E-e
llipt
ic f
or s
haft
ing.
Bud
ynas
−Nis
bett:
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gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
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hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
307
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
304
Mec
hani
cal E
ngin
eerin
g D
esig
n
The
str
ess
com
pone
nts
are
thus
σa
=K
(2+
δ)−
Kδ
2=
K=
10.9
9kp
si
σm
=K
(2+
δ)+
Kδ
2=
K(1
+δ)=
10.9
9(1
+δ)
Forδ
=0,
σa
=σ
m=
10.9
9=
11kp
si
2 in
32 in
(a)
� =
2 in
� =
5 in
� =
2 in
pre
load
� =
5 in
pre
load
1 4in
+ + +
Figure
6–3
0
Cam
follo
wer
reta
inin
g sp
ring.
(a) G
eom
etry
; (b)
des
igne
r’sfa
tigue
dia
gram
for E
x. 6
–11.
Amplitude stress component �a, kpsi
Stea
dy s
tres
s co
mpo
nent
�m
, kps
i
(b)
1133
5065
.910
011
5.6
127
150
050100
150
AA'
A"
Ger
ber
line
Lan
ger
line
Bud
ynas
−Nis
bett:
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gley
’s
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hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
308
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g305
Forδ
=2
in,
σa
=11
kpsi
,σ
m=
10.9
9(1
+2)
=33
kpsi
Forδ
=5
in,
σa
=11
kpsi
,σ
m=
10.9
9(1
+5)
=65
.9kp
si
(a)
A p
lot
of t
he G
erbe
r an
d L
ange
r cr
iteri
a is
sho
wn
in F
ig. 6
–30b
. The
thr
ee p
relo
adde
flect
ions
of
0, 2
, an
d 5
in a
re s
how
n as
poi
nts
A,
A′ ,
and
A′′ .
Not
e th
at s
ince
σa
isco
nsta
nt a
t 11
kps
i, th
e lo
ad l
ine
is h
oriz
onta
l an
d do
es n
ot c
onta
in t
he o
rigi
n. T
hein
ters
ectio
n be
twee
n th
e G
erbe
r lin
e an
d th
e lo
ad li
ne is
fou
nd f
rom
sol
ving
Eq.
(6–
42)
for
S man
d su
bstit
utin
g 11
kps
i for
Sa:
S m=
S ut√ 1
−S a S e
=15
0√ 1−
11 28=
116.
9kp
si
The
int
erse
ctio
n of
the
Lan
ger
line
and
the
load
lin
e is
fou
nd f
rom
sol
ving
Eq.
(6–
44)
for
S man
d su
bstit
utin
g 11
kps
i for
Sa:
S m=
S y−
S a=
127
−11
=11
6kp
si
The
thre
ats
from
fat
igue
and
firs
t-cy
cle
yiel
ding
are
app
roxi
mat
ely
equa
l.(b
) Fo
r δ
=2
in,
Ans
wer
nf=
S m σm
=11
6.9
33=
3.54
ny
=11
6
33=
3.52
and
for δ
=5
in,
Ans
wer
nf=
116.
9
65.9
=1.
77n
y=
116
65.9
=1.
76
EXA
MPLE
6–1
2A
ste
el b
ar u
nder
goes
cyc
lic lo
adin
g su
ch th
at σ
max
=60
kpsi
and
σm
in=
−20
kpsi
. For
the
mat
eria
l, S u
t=
80kp
si,
S y=
65kp
si,
a fu
lly c
orre
cted
end
uran
ce l
imit
of S
e=
40kp
si, a
nd f
=0.
9 . E
stim
ate
the
num
ber
of c
ycle
s to
a f
atig
ue f
ailu
re u
sing
:(a
) M
odifi
ed G
oodm
an c
rite
rion
.(b
) G
erbe
r cr
iteri
on.
Solu
tion
From
the
give
n st
ress
es,
σa
=60
−(−
20)
2=
40kp
siσ
m=
60+
(−20
)
2=
20kp
si
From
the
mat
eria
l pro
pert
ies,
Eqs
. (6–
14)
to (
6–16
), p
. 277
, giv
e
a=
(fS
ut)
2
S e=
[0.9
(80)
]2
40=
129.
6kp
si
b=
−1 3
log
( fSu
t
S e
) =−
1 3lo
g
[ 0.9(
80)
40
] =−0
.085
1
N=( S
f a
) 1/b=( S
f
129.
6
) −1/0.
0851
(1)
whe
reS
fre
plac
edσ
ain
Eq.
(6–
16).
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
309
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
306
Mec
hani
cal E
ngin
eerin
g D
esig
n
(a)
The
mod
ified
Goo
dman
lin
e is
giv
en b
y E
q. (
6–46
), p
. 29
8, w
here
the
end
uran
celim
itS e
is u
sed
for
infin
ite l
ife.
For
fini
te l
ife
at
Sf>
S e,
repl
ace
S ew
ithS
fin
Eq.
(6–4
6) a
nd r
earr
ange
giv
ing
Sf=
σa
1−
σm S ut
=40
1−
20 80
=53
.3kp
si
Subs
titut
ing
this
into
Eq.
(1)
yie
lds
Ans
wer
N=( 53
.3
129.
6
) −1/0.
0851
. =3.
4(10
4)
cycl
es
(b)
For
Ger
ber,
sim
ilar
to p
art (
a), f
rom
Eq.
(6–
47),
Sf
=σ
a
1−( σ
m S ut
) 2=40
1−( 20 80
) 2=42
.7kp
si
Aga
in, f
rom
Eq.
(1)
,
Ans
wer
N=( 42
.7
129.
6
) −1/0.
0851
. =4.
6(10
5)
cycl
es
Com
pari
ng t
he a
nsw
ers,
we
see
a la
rge
diff
eren
ce i
n th
e re
sults
. Aga
in,
the
mod
ified
Goo
dman
cri
teri
on i
s co
nser
vativ
e as
com
pare
d to
Ger
ber
for
whi
ch t
he m
oder
ate
dif-
fere
nce
in S
fis
then
mag
nifie
d by
a lo
gari
thm
ic S
,Nre
latio
nshi
p.
For
man
y br
ittl
em
ater
ials
, the
firs
t qu
adra
nt f
atig
ue f
ailu
re c
rite
ria
follo
ws
a co
n-ca
ve u
pwar
d Sm
ith-D
olan
locu
s re
pres
ente
d by
S a S e=
1−
S m/
S ut
1+
S m/
S ut
(6–5
0)
or a
s a
desi
gn e
quat
ion,
nσa
S e=
1−
nσm/
S ut
1+
nσm/
S ut
(6–5
1)
For
a ra
dial
loa
d lin
e of
slo
pe r
, we
subs
titut
e S a
/r
for
S min
Eq.
(6–
50)
and
solv
e fo
rS a
, obt
aini
ng
S a=
rSu
t+
S e2
[ −1+√ 1
+4r
S utS
e
(rS u
t+
S e)2
](6
–52)
The
fatig
uedi
agra
mfo
rabr
ittle
mat
eria
ldif
fers
mar
kedl
yfr
omth
atof
adu
ctile
mat
eria
lbe
caus
e:
•Y
ield
ing
is n
ot in
volv
ed s
ince
the
mat
eria
l may
not
hav
e a
yiel
d st
reng
th.
•C
hara
cter
istic
ally
, th
e co
mpr
essi
ve u
ltim
ate
stre
ngth
exc
eeds
the
ulti
mat
e te
nsile
stre
ngth
sev
eral
fold
.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
310
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g307
•Fi
rst-
quad
rant
fat
igue
fai
lure
loc
us i
s co
ncav
e-up
war
d (S
mith
-Dol
an),
for
exa
mpl
e,an
d as
flat
as
Goo
dman
. Bri
ttle
mat
eria
ls a
re m
ore
sens
itive
to m
idra
nge
stre
ss, b
eing
low
ered
, but
com
pres
sive
mid
rang
e st
ress
es a
re b
enefi
cial
.
•N
ot e
noug
h w
ork
has
been
don
e on
bri
ttle
fatig
ue to
dis
cove
r in
sigh
tful
gen
eral
ities
,so
we
stay
in th
e fir
st a
nd a
bit
of th
e se
cond
qua
dran
t.
The
mos
t lik
ely
dom
ain
of d
esig
ner
use
is i
n th
e ra
nge
from
−S u
t≤
σm
≤S u
t. T
helo
cus
in th
e fir
st q
uadr
ant i
s G
oodm
an, S
mith
-Dol
an, o
r som
ethi
ng in
bet
wee
n. T
he p
or-
tion
of t
he s
econ
d qu
adra
nt t
hat
is u
sed
is r
epre
sent
ed b
y a
stra
ight
lin
e be
twee
n th
epo
ints
−Su
t,S u
tan
d 0,
Se,
whi
ch h
as th
e eq
uatio
n
S a=
S e+( S e S u
t−
1) S m−
S ut≤
S m≤
0(f
orca
stir
on)
(6–5
3)
Tabl
e A
–24
give
s pr
oper
ties
of g
ray
cast
iro
n. T
he e
ndur
ance
lim
it st
ated
is
real
lyk a
k bS′ e
and
only
cor
rect
ions
kc,
k d,k
e, a
nd k
fne
ed b
e m
ade.
The
ave
rage
kc
for
axia
lan
d to
rsio
nal l
oadi
ng is
0.9
.
EXA
MPLE
6–1
3A
gra
de 3
0 gr
ay c
ast
iron
is
subj
ecte
d to
a l
oad
Fap
plie
d to
a 1
by
3 8-i
n cr
oss-
sect
ion
link
with
a 1 4
-in-
diam
eter
hol
e dr
illed
in
the
cent
er a
s de
pict
ed i
n Fi
g. 6
–31a
. The
sur
-fa
ces
are
mac
hine
d. I
n th
e ne
ighb
orho
od o
f th
e ho
le, w
hat i
s th
e fa
ctor
of
safe
ty g
uard
-in
g ag
ains
t fai
lure
und
er th
e fo
llow
ing
cond
ition
s:(a
) T
he lo
ad F
=10
00lb
f te
nsile
, ste
ady.
(b)
The
load
is 1
000
lbf
repe
ated
ly a
pplie
d.(c
) T
he lo
ad fl
uctu
ates
bet
wee
n −1
000
lbf
and
300
lbf
with
out c
olum
n ac
tion.
Use
the
Smith
-Dol
an f
atig
ue lo
cus.
S a =
18.
5 kp
si
S a =
7.6
3
S e
S ut
–S u
t–
9.95
7.63
010
2030
S ut
Mid
rang
e st
ress
�m
, kps
i
Alte
rnat
ing
stre
ss, �
a
1 4in
D. d
rill
F F1 in
S m
r =
1
r =
–1.
86
(b)
(a)
3 8in
Figure
6–3
1
The
grad
e 30
cas
t-iron
par
t in
axia
l fat
igue
with
(a) i
ts ge
omet
ry d
ispla
yed
and
(b) i
ts de
signe
r’s fa
tigue
dia
gram
for t
heci
rcum
stanc
es o
f Ex.
6–13
.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
311
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
308
Mec
hani
cal E
ngin
eerin
g D
esig
n
Solu
tion
Som
epr
epar
ator
yw
ork
isne
eded
.Fr
omTa
ble
A–2
4,S u
t=
31kp
si,
S uc=
109
kpsi
,k a
k bS′ e
=14
kpsi
.Sin
cek c
for
axia
llo
adin
gis
0.9,
then
S e=
(kak b
S′ e)k c
=14
(0.9
)=
12.6
kpsi
.Fro
mTa
ble
A–1
5–1,
A=
t(w
−d)=
0.37
5(1
−0.
25)=
0.28
1in
2,d
/w
=0.
25/1
=0.
25,a
ndK
t=
2.45
.The
notc
hse
nsiti
vity
for
cast
iron
is0.
20(s
eep.
288)
,so
Kf
=1
+q(K
t−
1)=
1+
0.20
(2.4
5−
1)=
1.29
(a)
σa
=K
fF
a
A=
1.29
(0)
0.28
1=
0σ
m=
KfF
m
A=
1.29
(100
0)
0.28
1(1
0−3)=
4.59
kpsi
and
Ans
wer
n=
S ut
σm
=31
.0
4.59
=6.
75
(b)
Fa
=F
m=
F 2=
1000 2
=50
0lb
f
σa
=σ
m=
KfF
a
A=
1.29
(500
)
0.28
1(1
0−3)=
2.30
kpsi
r=
σa
σm
=1
From
Eq.
(6–
52),
S a=
(1)3
1+
12.6
2
[ −1+√ 1
+4(
1)31
(12.
6)
[(1)
31+
12.6
]2
] =7.
63kp
si
Ans
wer
n=
S a σa
=7.
63
2.30
=3.
32
(c)
Fa
=1 2|30
0−
(−10
00)|
=65
0lb
fσ
a=
1.29
(650
)
0.28
1(1
0−3)=
2.98
kpsi
Fm
=1 2
[300
+(−
1000
)]=
−350
lbf
σm
=1.
29(−
350)
0.28
1(1
0−3)=
−1.6
1kp
si
r=
σa
σm
=3.
0
−1.6
1=
−1.8
6
From
Eq.
(6–
53),
Sa
=S e
+(S
e/S u
t−
1)S m
and
S m=
S a/
r . I
t fol
low
s th
at
S a=
S e
1−
1 r
( S e S ut−
1) =12
.6
1−
1
−1.8
6
( 12.6
31−
1) =18
.5kp
si
Ans
wer
n=
S a σa
=18
.5
2.98
=6.
20
Figu
re 6
–31b
show
s th
e po
rtio
n of
the
desi
gner
’s f
atig
ue d
iagr
am th
at w
as c
onst
ruct
ed.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
312
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g309
6–13
Tors
ional Fa
tigue
Stre
ngth
under
Fluct
uating S
tres
ses
Ext
ensi
ve t
ests
by
Smith
23pr
ovid
e so
me
very
int
eres
ting
resu
lts o
n pu
lsat
ing
tors
iona
lfa
tigue
. Sm
ith’s
firs
t re
sult,
bas
ed o
n 72
tes
ts,
show
s th
at t
he e
xist
ence
of
a to
rsio
nal
stea
dy-s
tres
s co
mpo
nent
not
mor
e th
an t
he t
orsi
onal
yie
ld s
tren
gth
has
no e
ffec
t on
the
tors
iona
l end
uran
ce li
mit,
pro
vide
d th
e m
ater
ial i
s du
ctil
e,po
lish
ed,n
otch
-fre
e,an
dcy
lind
rica
l.Sm
ith’s
sec
ond
resu
lt ap
plie
s to
mat
eria
ls w
ith s
tres
s co
ncen
trat
ion,
not
ches
, or
surf
ace
impe
rfec
tions
. In
thi
s ca
se,
he fi
nds
that
the
tor
sion
al f
atig
ue l
imit
decr
ease
sm
onot
onic
ally
with
tor
sion
al s
tead
y st
ress
. Si
nce
the
grea
t m
ajor
ity o
f pa
rts
will
hav
esu
rfac
es th
at a
re le
ss th
an p
erfe
ct, t
his
resu
lt in
dica
tes
Ger
ber,
ASM
E-e
llipt
ic, a
nd o
ther
appr
oxim
atio
ns
are
usef
ul.
Joer
res
of A
ssoc
iate
d Sp
ring
-Bar
nes
Gro
up,
confi
rms
Smith
’s r
esul
ts a
nd r
ecom
men
ds t
he u
se o
f th
e m
odifi
ed G
oodm
an r
elat
ion
for
puls
at-
ing
tors
ion.
In
cons
truc
ting
the
Goo
dman
dia
gram
, Joe
rres
use
s
S su
=0.
67S u
t(6
–54)
Als
o, f
rom
Cha
p. 5
, S s
y=
0.57
7Syt
from
dis
tort
ion-
ener
gy t
heor
y, a
nd t
he m
ean
load
fact
or k
cis
giv
en b
y E
q. (
6–26
), o
r 0.
577.
Thi
s is
dis
cuss
ed f
urth
er in
Cha
p. 1
0.
6–14
Com
bin
ations
of
Loadin
g M
odes
It m
ay b
e he
lpfu
l to
thin
k of
fat
igue
pro
blem
s as
bei
ng in
thre
e ca
tego
ries
:
•C
ompl
etel
y re
vers
ing
sim
ple
load
s
•Fl
uctu
atin
g si
mpl
e lo
ads
•C
ombi
nati
ons
of l
oadi
ng m
odes
The
sim
ples
t ca
tego
ry i
s th
at o
f a
com
plet
ely
reve
rsed
sin
gle
stre
ss w
hich
is
han-
dled
with
the
S-N
diag
ram
, re
latin
g th
e al
tern
atin
g st
ress
to
a lif
e. O
nly
one
type
of
load
ing
is a
llow
ed h
ere,
and
the
mid
rang
e st
ress
mus
t be
zero
. The
nex
t cat
egor
y in
cor-
pora
tes
gene
ral
fluct
uatin
g lo
ads,
usi
ng a
cri
teri
on t
o re
late
mid
rang
e an
d al
tern
atin
gst
ress
es (
mod
ified
Goo
dman
, G
erbe
r, A
SME
-elli
ptic
, or
Sod
erbe
rg).
Aga
in,
only
one
type
of
load
ing
is a
llow
ed a
t a
time.
The
thi
rd c
ateg
ory,
whi
ch w
e w
ill d
evel
op i
n th
isse
ctio
n, in
volv
es c
ases
whe
re th
ere
are
com
bina
tions
of
diff
eren
t typ
es o
f lo
adin
g, s
uch
as c
ombi
ned
bend
ing,
tors
ion,
and
axi
al.
In S
ec. 6
–9 w
e le
arne
d th
at a
loa
d fa
ctor
kc
is u
sed
to o
btai
n th
e en
dura
nce
limit,
and
henc
e th
e re
sult
is d
epen
dent
on
whe
ther
the
loa
ding
is
axia
l, be
ndin
g, o
r to
rsio
n.In
this
sec
tion
we
wan
t to
answ
er th
e qu
estio
n, “
How
do
we
proc
eed
whe
n th
e lo
adin
gis
a m
ixtu
reof
, say
, axi
al, b
endi
ng, a
nd to
rsio
nal l
oads
?” T
his
type
of l
oadi
ng in
trod
uces
a fe
w c
ompl
icat
ions
in
that
the
re m
ay n
ow e
xist
com
bine
d no
rmal
and
she
ar s
tres
ses,
each
with
alte
rnat
ing
and
mid
rang
e va
lues
, and
sev
eral
of
the
fact
ors
used
in d
eter
min
-in
g th
e en
dura
nce
limit
depe
nd o
n th
e ty
pe o
f lo
adin
g. T
here
may
als
o be
mul
tiple
stre
ss-c
once
ntra
tion
fact
ors,
one
for
eac
h m
ode
of lo
adin
g. T
he p
robl
em o
f ho
w to
dea
lw
ith c
ombi
ned
stre
sses
was
enc
ount
ered
whe
n de
velo
ping
sta
tic f
ailu
re t
heor
ies.
The
dist
ortio
n en
ergy
fai
lure
the
ory
prov
ed t
o be
a s
atis
fact
ory
met
hod
of c
ombi
ning
the
23Ja
mes
O. S
mith
, “T
he E
ffec
t of
Ran
ge o
f St
ress
on
the
Fatig
ue S
tren
gth
of M
etal
s,”
Uni
v. o
f Ill
. Eng
. Exp
.
Sta.
Bul
l. 33
4, 1
942.
Bud
ynas
−Nis
bett:
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gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
313
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
310
Mec
hani
cal E
ngin
eerin
g D
esig
n
mul
tiple
str
esse
s on
a s
tres
s el
emen
t int
o a
sing
le e
quiv
alen
t von
Mis
es s
tres
s. T
he s
ame
appr
oach
will
be
used
her
e.T
he fi
rst s
tep
is to
gen
erat
e tw
ost
ress
ele
men
ts—
one
for t
he a
ltern
atin
g st
ress
es a
ndon
e fo
r th
e m
idra
nge
stre
sses
. App
ly th
e ap
prop
riat
e fa
tigue
str
ess
conc
entr
atio
n fa
ctor
sto
eac
h of
the
stre
sses
; i.e
., ap
ply
(Kf) b
endi
ngfo
r th
e be
ndin
g st
ress
es, (
Kfs
) tor
sion
for
the
tors
iona
l st
ress
es, a
nd (
Kf) a
xial
for
the
axia
l st
ress
es. N
ext,
calc
ulat
e an
equ
ival
ent
von
Mis
es s
tres
s fo
r ea
ch o
f th
ese
two
stre
ss e
lem
ents
, σ
′ aan
dσ
′ m.
Fina
lly,
sele
ct a
fat
igue
failu
re c
rite
rion
(m
odifi
ed G
oodm
an, G
erbe
r, A
SME
-elli
ptic
, or
Sode
rber
g) to
com
plet
eth
e fa
tigue
ana
lysi
s. F
or t
he e
ndur
ance
lim
it, S
e, u
se t
he e
ndur
ance
lim
it m
odifi
ers,
k a,k
b, a
nd k
c, f
or b
endi
ng. T
he to
rsio
nal l
oad
fact
or, k
c=
0.59
shou
ld n
ot b
e ap
plie
d as
it is
alr
eady
acc
ount
ed f
or i
n th
e vo
n M
ises
str
ess
calc
ulat
ion
(see
foo
tnot
e 17
on
page
282)
. The
load
fac
tor
for
the
axia
l loa
d ca
n be
acc
ount
ed f
or b
y di
vidi
ng th
e al
tern
atin
gax
ial
stre
ss b
y th
e ax
ial
load
fac
tor
of 0
.85.
For
exa
mpl
e, c
onsi
der
the
com
mon
cas
e of
a sh
aft
with
ben
ding
str
esse
s, t
orsi
onal
she
ar s
tres
ses,
and
axi
al s
tres
ses.
For
thi
s ca
se,
the
von
Mis
es s
tres
s is
of
the
form
σ′ =( σ
x2+
3τx
y2) 1/2
. Con
side
ring
that
the
bend
ing,
tors
iona
l, an
d ax
ial
stre
sses
hav
e al
tern
atin
g an
d m
idra
nge
com
pone
nts,
the
von
Mis
esst
ress
es f
or th
e tw
o st
ress
ele
men
ts c
an b
e w
ritte
n as
σ′ a={ [ (K
f) b
endi
ng(σ
a) b
endi
ng+
(Kf) a
xial
(σa) a
xial
0.85
] 2 +3[ (K
fs) t
orsi
on(τ
a) t
orsi
on] 2} 1/2 (6
–55)
σ′ m
={ [ (K
f) b
endi
ng(σ
m) b
endi
ng+
(Kf) a
xial
(σm
) axi
al] 2 +
3[ (K
fs) t
orsi
on(τ
m) t
orsi
on] 2} 1
/2
(6–5
6)
For fi
rst-
cycl
e lo
caliz
ed y
ield
ing,
the
max
imum
von
Mis
es s
tres
s is
cal
cula
ted.
Thi
sw
ould
be
done
by
first
add
ing
the
axia
l and
ben
ding
alte
rnat
ing
and
mid
rang
e st
ress
es to
obta
inσ
max
and
addi
ng th
e al
tern
atin
g an
d m
idra
nge
shea
r st
ress
es to
obt
ain
τ max
. The
nsu
bstit
uteσ
max
and
τ max
into
the
equa
tion
for t
he v
on M
ises
str
ess.
A s
impl
er a
nd m
ore
con-
serv
ativ
e m
etho
d is
to a
dd E
q. (
6–55
) an
d E
q. (
6–56
). T
hat i
s, le
t σ′ m
ax. =
σ′ a+
σ′ m
If th
e st
ress
com
pone
nts
are
not i
n ph
ase
but h
ave
the
sam
e fr
eque
ncy,
the
max
ima
can
be fo
und
by e
xpre
ssin
g ea
ch c
ompo
nent
in tr
igon
omet
ric
term
s, u
sing
pha
se a
ngle
s,an
d th
en fi
ndin
g th
e su
m. I
f tw
o or
mor
e st
ress
com
pone
nts
have
dif
feri
ng f
requ
enci
es,
the
prob
lem
is
diffi
cult;
one
sol
utio
n is
to
assu
me
that
the
tw
o (o
r m
ore)
com
pone
nts
ofte
n re
ach
an in
-pha
se c
ondi
tion,
so
that
thei
r m
agni
tude
s ar
e ad
ditiv
e.
EXA
MPLE
6–1
4A
rot
atin
g sh
aft
is m
ade
of 4
2- ×
4-m
m A
ISI
1018
col
d-dr
awn
stee
l tu
bing
and
has
a6-
mm
-dia
met
er h
ole
drill
ed tr
ansv
erse
ly th
roug
h it.
Est
imat
e th
e fa
ctor
of
safe
ty g
uard
-in
g ag
ains
t fat
igue
and
sta
tic fa
ilure
s us
ing
the
Ger
ber a
nd L
ange
r fai
lure
cri
teri
a fo
r the
follo
win
g lo
adin
g co
nditi
ons:
(a)
The
sha
ft is
sub
ject
ed to
a c
ompl
etel
y re
vers
ed to
rque
of
120
N · m
in p
hase
with
aco
mpl
etel
y re
vers
ed b
endi
ng m
omen
t of
150
N · m
.(b
) T
he s
haft
is
subj
ecte
d to
a p
ulsa
ting
torq
ue fl
uctu
atin
g fr
om 2
0 to
160
N ·
m a
nd a
stea
dy b
endi
ng m
omen
t of
150
N · m
.
Solu
tion
Her
e w
e fo
llow
the
proc
edur
e of
est
imat
ing
the
stre
ngth
s an
d th
en th
e st
ress
es, f
ollo
wed
by r
elat
ing
the
two.
Bud
ynas
−Nis
bett:
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gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
314
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g311
From
Tab
le A
–20
we
find
the
min
imum
str
engt
hs t
o be
Su
t=
440
MPa
and
Sy
=37
0M
Pa. T
he e
ndur
ance
lim
it of
the
rot
atin
g-be
am s
peci
men
is
0.5(
440)
=22
0 M
Pa.
The
sur
face
fac
tor,
obta
ined
fro
m E
q. (
6–19
) an
d Ta
ble
6–2,
p. 2
79 is
k a=
4.51
S−0.2
65u
t=
4.51
(440
)−0.2
65=
0.89
9
From
Eq.
(6–
20)
the
size
fac
tor
is
k b=( d
7.62
) −0.1
07
=( 42 7.
62
) −0.1
07
=0.
833
The
rem
aini
ng M
arin
fac
tors
are
all
unity
, so
the
mod
ified
end
uran
ce s
tren
gth
S eis
S e=
0.89
9(0.
833)
220
=16
5M
Pa
(a)
The
oret
ical
stre
ss-c
once
ntra
tion
fact
ors
are
foun
dfr
omTa
ble
A–1
6.U
sing
a/D
=6/
42=
0.14
3an
dd/
D=
34/42
=0.
810 ,
and
usin
glin
ear
inte
rpol
atio
n,w
eob
tain
A=
0.79
8an
dK
t=
2.36
6fo
rbe
ndin
g;an
dA
=0.
89an
dK
ts=
1.75
for
tors
ion.
Thu
s,fo
rbe
ndin
g,
Zne
t=
πA
32D
(D4−
d4)=
π(0
.798
)
32(4
2)[(
42)4
−(3
4)4]=
3.31
(103
)mm
3
and
for
tors
ion
J net
=π
A
32(D
4−
d4)=
π(0
.89)
32[(
42)4
−(3
4)4]=
155
(103
)mm
4
Nex
t, us
ing
Figs
. 6–2
0 an
d 6–
21, p
p. 2
87–2
88, w
ith a
not
ch r
adiu
s of
3 m
m w
e fin
d th
eno
tch
sens
itivi
ties
to b
e 0.
78 f
or b
endi
ng a
nd 0
.96
for
tors
ion.
The
tw
o co
rres
pond
ing
fatig
ue s
tres
s-co
ncen
trat
ion
fact
ors
are
obta
ined
fro
m E
q. (
6–32
) as
Kf=
1+
q(K
t−
1)=
1+
0.78
(2.3
66−
1)=
2.07
Kfs
=1
+0.
96(1
.75
−1)
=1.
72
The
alte
rnat
ing
bend
ing
stre
ss is
now
fou
nd to
be
σxa
=K
fM Zne
t=
2.07
150
3.31
(10−
6)
=93
.8(1
06)P
a=
93.8
MPa
and
the
alte
rnat
ing
tors
iona
l str
ess
is
τx
ya=
Kfs
TD
2J n
et=
1.72
120(
42)(
10−3
)
2(15
5)(1
0−9)
=28
.0(1
06)P
a=
28.0
MPa
The
mid
rang
e vo
n M
ises
com
pone
nt σ
′ mis
zer
o. T
he a
ltern
atin
g co
mpo
nent
σ′ a
is g
iven
by
σ′ a=( σ
2 xa+
3τ2 xya
) 1/2=
[93.
82+
3(28
2)]
1/2
=10
5.6
MPa
Sinc
eS e
=S a
, the
fat
igue
fac
tor
of s
afet
y n
fis
Ans
wer
nf
=S a σ
′ a
=16
5
105.
6=
1.56
Bud
ynas
−Nis
bett:
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gley
’s
Mec
hani
cal E
ngin
eeri
ng
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ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
315
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
312
Mec
hani
cal E
ngin
eerin
g D
esig
n
The
firs
t-cy
cle
yiel
d fa
ctor
of
safe
ty is
Ans
wer
ny
=S y σ
′ a
=37
0
105.
6=
3.50
The
re is
no
loca
lized
yie
ldin
g; th
e th
reat
is f
rom
fat
igue
. See
Fig
. 6–3
2.(b
) T
his
part
ask
s us
to fi
nd th
e fa
ctor
s of
saf
ety
whe
n th
e al
tern
atin
g co
mpo
nent
is d
ueto
pul
satin
g to
rsio
n, a
nd a
ste
ady
com
pone
nt i
s du
e to
bot
h to
rsio
n an
d be
ndin
g. W
eha
ve T
a=
(160
−20
)/2
=70
N· m
and
Tm
=(1
60+
20)/
2=
90N
· m. T
he c
orre
-sp
ondi
ng a
mpl
itude
and
ste
ady-
stre
ss c
ompo
nent
s ar
e
τx
ya=
Kfs
T aD
2J n
et=
1.72
70(4
2)(1
0−3)
2(15
5)(1
0−9)
=16
.3(1
06)P
a=
16.3
MPa
τx
ym=
Kfs
T mD
2J n
et=
1.72
90(4
2)(1
0−3)
2(15
5)(1
0−9)
=21
.0(1
06)P
a=
21.0
MPa
The
ste
ady
bend
ing
stre
ss c
ompo
nent
σxm
is
σxm
=K
fM
m
Zne
t=
2.07
150
3.31
(10−
6)
=93
.8(1
06)P
a=
93.8
MPa
The
von
Mis
es c
ompo
nent
s σ
′ aan
dσ
′ mar
e
σ′ a=
[3(1
6.3)
2]1/
2=
28.2
MPa
σ′ m
=[9
3.82
+3(
21)2
]1/2
=10
0.6
MPa
From
Tab
le 6
–7, p
. 299
, the
fat
igue
fac
tor
of s
afet
y is
Ans
wer
nf=
1 2
( 440
100.
6
) 2 28.2
165
⎧ ⎨ ⎩−1+√ 1
+[ 2(
100.
6)16
5
440(
28.2
)
] 2⎫ ⎬ ⎭=3.
03
Von Mises amplitude stress component �a, MPa '
Von
Mis
es s
tead
y st
ress
com
pone
nt �
m, M
Pa'
305
500
440
165
100
105.
6
85.5
200
300
400 0
Ger
ber
r =
0.2
8
Figure
6–3
2
Des
igne
r’s fa
tigue
dia
gram
for
Ex. 6
–14.
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ynas
−Nis
bett:
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gley
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Mec
hani
cal E
ngin
eeri
ng
Des
ign,
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hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
316
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g313
From
the
sam
e ta
ble,
with
r=
σ′ a/σ
′ m=
28.2
/10
0.6
=0.
280 ,
the
str
engt
hs c
an b
esh
own
to b
e S a
=85
.5M
Paan
dS m
=30
5M
Pa. S
ee th
e pl
ot in
Fig
. 6–3
2.T
he fi
rst-
cycl
e yi
eld
fact
or o
f sa
fety
ny
is
Ans
wer
ny
=S y
σ′ a+
σ′ m
=37
0
28.2
+10
0.6
=2.
87
The
re i
s no
not
ch y
ield
ing.
The
lik
elih
ood
of f
ailu
re m
ay fi
rst
com
e f
rom
firs
t-cy
cle
yiel
ding
at t
he n
otch
. See
the
plot
in F
ig. 6
–32.
6–15
Vary
ing,
Fluct
uating S
tres
ses;
Cum
ula
tive
Fatigue
Dam
age
Inst
ead
of a
sin
gle
fully
rev
erse
d st
ress
his
tory
blo
ck c
ompo
sed
of n
cycl
es, s
uppo
se a
mac
hine
par
t, at
a c
ritic
al lo
catio
n, is
sub
ject
ed to
•A
ful
ly r
ever
sed
stre
ss σ
1fo
rn 1
cycl
es, σ
2fo
rn 2
cycl
es, .
..,
or
•A
“w
iggl
y” ti
me
line
of s
tres
s ex
hibi
ting
man
y an
d di
ffer
ent p
eaks
and
val
leys
.
Wha
t st
ress
es a
re s
igni
fican
t, w
hat
coun
ts a
s a
cycl
e, a
nd w
hat
is t
he m
easu
re o
fda
mag
ein
curr
ed?
Con
side
r a
fully
rev
erse
d cy
cle
with
str
esse
s va
ryin
g 60
, 80,
40,
and
60kp
si a
nd a
sec
ond
fully
rev
erse
d cy
cle
−40 ,
−60 ,
−20 ,
and
−40
kpsi
as
depi
cted
inFi
g.6–
33a.
Fir
st, i
t is
cle
ar t
hat
to i
mpo
se t
he p
atte
rn o
f st
ress
in
Fig.
6–3
3aon
a p
art
it is
nec
essa
ry t
hat
the
time
trac
e lo
ok l
ike
the
solid
lin
e pl
us t
he d
ashe
d lin
e in
Fig
.6–
33a.
Fig
ure
6–33
bm
oves
the
sna
psho
t to
exi
st b
egin
ning
with
80
kpsi
and
end
ing
with
80
kpsi
. Ack
now
ledg
ing
the
exis
tenc
e of
a s
ingl
e st
ress
-tim
e tr
ace
is to
dis
cove
r a
“hid
den”
cyc
le s
how
n as
the
dash
ed li
ne in
Fig
. 6–3
3b. I
f th
ere
are
100
appl
icat
ions
of
the
all-
posi
tive
stre
ss c
ycle
, th
en 1
00 a
pplic
atio
ns o
f th
e al
l-ne
gativ
e st
ress
cyc
le,
the
100 50 0
–50
100 50 0
–50
(a)
( b)
Figure
6–3
3
Varia
ble
stres
s di
agra
mpr
epar
ed fo
r ass
essin
gcu
mul
ativ
e da
mag
e.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
317
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
314
Mec
hani
cal E
ngin
eerin
g D
esig
n
hidd
en c
ycle
is
appl
ied
but
once
. If
the
all-
posi
tive
stre
ss c
ycle
is
appl
ied
alte
rnat
ely
with
the
all-
nega
tive
stre
ss c
ycle
, the
hid
den
cycl
e is
app
lied
100
times
.To
ens
ure
that
the
hid
den
cycl
e is
not
los
t, be
gin
on t
he s
naps
hot
with
the
lar
gest
(or s
mal
lest
) str
ess
and
add
prev
ious
his
tory
to th
e ri
ght s
ide,
as
was
don
e in
Fig
. 6–3
3b.
Cha
ract
eriz
atio
n of
a c
ycle
tak
es o
n a
max
–min
–sam
e m
ax (
or m
in–m
ax–s
ame
min
)fo
rm.
We
iden
tify
the
hidd
en c
ycle
firs
t by
mov
ing
alon
g th
e da
shed
-lin
e tr
ace
inFi
g.6–
33b
iden
tifyi
ng a
cyc
le w
ith a
n 80
-kps
i m
ax,
a 60
-kps
i m
in,
and
retu
rnin
g to
80kp
si. M
enta
lly d
elet
ing
the
used
par
t of
the
tra
ce (
the
dash
ed l
ine)
lea
ves
a 40
, 60,
40cy
cle
and
a −4
0 ,−2
0 ,−4
0cy
cle.
Sin
ce f
ailu
re lo
ci a
re e
xpre
ssed
in te
rms
of s
tres
sam
plitu
de c
ompo
nent
σa
and
stea
dy c
ompo
nent
σm
, we
use
Eq.
(6–
36)
to c
onst
ruct
the
tabl
e be
low
:
The
mos
t dam
agin
g cy
cle
is n
umbe
r 1.
It c
ould
hav
e be
en lo
st.
Met
hods
for
cou
ntin
g cy
cles
incl
ude:
•N
umbe
r of
tens
ile p
eaks
to f
ailu
re.
•A
ll m
axim
a ab
ove
the
wav
efor
m m
ean,
all
min
ima
belo
w.
•T
he g
loba
l m
axim
a be
twee
n cr
ossi
ngs
abov
e th
e m
ean
and
the
glob
al m
inim
abe
twee
n cr
ossi
ngs
belo
w th
e m
ean.
•A
ll po
sitiv
e sl
ope
cros
sing
s of
lev
els
abov
e th
e m
ean,
and
all
nega
tive
slop
e cr
oss-
ings
of
leve
ls b
elow
the
mea
n.
•A
mod
ifica
tion
of t
he p
rece
ding
met
hod
with
onl
y on
e co
unt
mad
e be
twee
n su
cces
-si
ve c
ross
ings
of
a le
vel a
ssoc
iate
d w
ith e
ach
coun
ting
leve
l.
•E
ach
loca
l m
axi-
min
exc
ursi
on i
s co
unte
d as
a h
alf-
cycl
e, a
nd t
he a
ssoc
iate
d am
pli-
tude
is h
alf-
rang
e.
•T
he p
rece
ding
met
hod
plus
con
side
ratio
n of
the
loca
l mea
n.
•R
ain-
flow
cou
ntin
g te
chni
que.
The
met
hod
used
her
e am
ount
s to
a v
aria
tion
of th
e ra
in-fl
ow c
ount
ing
tech
niqu
e.T
hePa
lmgr
en-M
iner
24cy
cle-
rati
o su
mm
atio
n ru
le,
also
cal
led
Min
er’s
rul
e,is
wri
tten
∑n i N
i=
c(6
–57)
whe
ren i
is th
e nu
mbe
r of
cyc
les
at s
tres
s le
vel σ
ian
dN
iis
the
num
ber
of c
ycle
s to
fail-
ure
at s
tres
s le
vel σ
i. T
he p
aram
eter
cha
s be
en d
eter
min
ed b
y ex
peri
men
t; it
is u
sual
lyfo
und
in th
e ra
nge
0.7
<c
<2.
2w
ith a
n av
erag
e va
lue
near
uni
ty.
Cycl
e N
um
ber
�m
ax
�m
in�
a�
m
180
�60
7010
260
4010
503
�20
�40
10�
30
24A
. Pal
mgr
en, “
Die
Leb
ensd
auer
von
Kug
ella
gern
,” Z
VD
I, v
ol. 6
8, p
p. 3
39–3
41, 1
924;
M. A
. Min
er,
“Cum
ulat
ive
Dam
age
in F
atig
ue,”
J. A
ppl.
Mec
h.,v
ol. 1
2, T
rans
. ASM
E,v
ol. 6
7, p
p. A
159–
A16
4, 1
945.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
318
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g315
Usi
ng th
e de
term
inis
tic f
orm
ulat
ion
as a
line
ar d
amag
e ru
le w
e w
rite
D=∑
n i Ni
(6–5
8)
whe
reD
is th
e ac
cum
ulat
ed d
amag
e. W
hen
D=
c=
1 , f
ailu
re e
nsue
s.
EXA
MPLE
6–1
5G
iven
a p
art w
ith S
ut=
151
kpsi
and
at t
he c
ritic
al lo
catio
n of
the
part
, Se=
67.5
kpsi
.Fo
r th
e lo
adin
g of
Fig
. 6–3
3, e
stim
ate
the
num
ber
of r
epet
ition
s of
the
stre
ss-t
ime
bloc
kin
Fig
. 6–3
3 th
at c
an b
e m
ade
befo
re f
ailu
re.
Solu
tion
From
Fig
. 6–1
8, p
. 277
, for
Su
t=
151
kpsi
,f
=0.
795 .
Fro
m E
q. (
6–14
),
a=
(fS
ut)
2
S e=
[0.7
95(1
51)]
2
67.5
=21
3.5
kpsi
From
Eq.
(6–
15),
b=
−1 3
log
( fSu
t
S e
) =−
1 3lo
g
[ 0.79
5(15
1)
67.5
] =−0
.083
3
So,
Sf
=21
3.5
N−0
.083
3N
=( S
f
213.
5
) −1/0.
0833
(1),
(2)
We
prep
are
to a
dd tw
o co
lum
ns to
the
prev
ious
tabl
e. U
sing
the
Ger
ber f
atig
ue c
rite
rion
,E
q. (
6–47
), p
. 298
, with
Se=
Sf, a
nd n
=1 ,
we
can
wri
te
Sf={
σa
1−
(σm/
S ut)
2σ
m>
0
S eσ
m≤
0(3
)
Cyc
le1:
r=
σa/σ
m=
70/10
=7 ,
and
the
stre
ngth
am
plitu
de f
rom
Tab
le 6
–7, p
. 299
, is
S a=
7215
12
2(67
.5)
⎧ ⎨ ⎩−1+√ 1
+[ 2(
67.5
)
7(15
1)
] 2⎫ ⎬ ⎭=67
.2kp
si
Sinc
eσ
a>
S a, t
hat i
s, 7
0>
67.2
, lif
e is
red
uced
. Fro
m E
q. (
3),
Sf
=70
1−
(10/
151)
2=
70.3
kpsi
and
from
Eq.
(2)
N=( 70
.3
213.
5
) −1/0.
0833
=61
9(10
3)
cycl
es
Cyc
le2:
r=
10/50
=0.
2 , a
nd th
e st
reng
th a
mpl
itude
is
S a=
0.22
1512
2(67
.5)
⎧ ⎨ ⎩−1+√ 1
+[ 2(
67.5
)
0.2(
151)
] 2⎫ ⎬ ⎭=24
.2kp
si
FatigueStrength
LifeCycles
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
319
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
316
Mec
hani
cal E
ngin
eerin
g D
esig
n
Sinc
eσ
a<
S a,
that
is
10
<24
.2,
then
S
f=
S ean
d in
defin
ite
life
follo
ws.
Thu
s,
N∞
�.
Cyc
le3:
r=
10/−3
0=
−0.3
33, a
nd s
ince
σm
<0 ,
Sf
=S e
, ind
efini
te li
fe f
ollo
ws
and
N∞
From
Eq.
(6–
58)
the
dam
age
per
bloc
k is
D=∑
n i Ni
=N
[1
619(
103)
+1 ∞
+1 ∞] =
N
619(
103)
Ans
wer
Setti
ngD
=1
yiel
dsN
=61
9(10
3)
cycl
es.
Cycl
e N
um
ber
S f,
kpsi
N,
cycl
es
170
.361
9(10
3 )2
67.5
∞3
67.5
∞
←←
To f
urth
er il
lust
rate
the
use
of th
e M
iner
rul
e, le
t us
choo
se a
ste
el h
avin
g th
e pr
op-
ertie
sS u
t=
80kp
si,
S′ e,0
=40
kpsi
, and
f=
0.9 ,
whe
re w
e ha
ve u
sed
the
desi
gnat
ion
S′ e,0
inst
ead
of th
e m
ore
usua
l S′ e
to in
dica
te th
e en
dura
nce
limit
of th
e vi
rgin
,or
unda
m-
aged
,mat
eria
l. T
he lo
g S–
log
Ndi
agra
m f
or th
is m
ater
ial i
s sh
own
in F
ig. 6
–34
by th
ehe
avy
solid
lin
e. N
ow a
pply
, say
, a r
ever
sed
stre
ss σ
1=
60kp
si f
or n
1=
3000
cycl
es.
Sinc
eσ
1>
S′ e,0,
the
endu
ranc
e lim
it w
ill b
e da
mag
ed,
and
we
wis
h to
find
the
new
endu
ranc
e lim
it S′ e,
1of
the
dam
aged
mat
eria
l usi
ng th
e M
iner
rul
e. T
he e
quat
ion
of th
evi
rgin
mat
eria
l fai
lure
line
in F
ig. 6
–34
in th
e 10
3to
106
cycl
e ra
nge
is
Sf=
aN
b=
129.
6N
−0.0
8509
1
The
cyc
les
to f
ailu
re a
t str
ess
leve
l σ1
=60
kpsi
are
N1
=( σ
1
129.
6
) −1/0.
085
091
=( 60
129.
6
) −1/0.
085
091
=85
20cy
cles
72 60 4038
.6 103
104
105
106
65
43
4.9
4.8
4.7
4.6
4.5
LogS
Sokpsi
N
Log
N
�1
0.9S
ut
n 1 =
3(1
03 )
n2
= 0
.648
(106 )
S f,0
S f,2
S e,0
S e,1
S f,1
N1
= 8
.52(
103 )
N1
– n 1
= 5
.52(
103 )
Figure
6–3
4
Use
of t
he M
iner
rule
topr
edic
t the
end
uran
ce li
mit
ofa
mat
eria
l tha
t has
bee
nov
erstr
esse
d fo
r a fi
nite
num
ber o
f cyc
les.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
320
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g317
Fig
ure
6–34
sho
ws
that
the
mat
eria
l ha
s a
life
N1
=85
20cy
cles
at
60 k
psi,
and
con-
sequ
entl
y, a
fter
the
app
lica
tion
of
σ1
for
3000
cyc
les,
the
re a
re N
1−
n 1=
5520
cycl
es o
f li
fe r
emai
ning
at σ
1. T
his
loca
tes
the
fini
te-l
ife
stre
ngth
Sf,
1of
the
dam
aged
mat
eria
l, as
sho
wn
in F
ig. 6
–34.
To
get
a se
cond
poi
nt, w
e as
k th
e qu
esti
on: W
ith
n 1an
dN
1gi
ven,
how
man
y cy
cles
of
stre
ss σ
2=
S′ e,0
can
be a
ppli
ed b
efor
e th
e da
mag
edm
ater
ial
fail
s?
Thi
s co
rres
pond
s to
n2
cycl
es o
f st
ress
rev
ersa
l, an
d he
nce,
fro
m E
q. (
6–58
), w
e ha
ve
n 1 N1
+n 2 N
2=
1(a
)
or
n 2=( 1
−n 1 N
1
) N2
(b)
The
n
n 2=[ 1
−3(
10)3
8.52
(10)
3
] (106
)=
0.64
8(10
6)
cycl
es
Thi
s co
rres
pond
s to
the
finite
-lif
e st
reng
th S
f,2
in F
ig. 6
–34.
A li
ne th
roug
h S
f,1
and
Sf,
2
is t
he l
og S
–log
Ndi
agra
m o
f th
e da
mag
ed m
ater
ial
acco
rdin
g to
the
Min
er r
ule.
The
new
end
uran
ce li
mit
is S
e,1
=38
.6kp
si.
We
coul
d le
ave
it at
thi
s, b
ut a
litt
le m
ore
inve
stig
atio
n ca
n be
hel
pful
. W
e ha
vetw
opo
ints
on
the
new
fat
igue
loc
us,
N1−
n 1,σ
1an
dn 2
,σ2.
It i
s us
eful
to
prov
e th
atth
e sl
ope
of th
e ne
w li
ne is
stil
l b. F
or th
e eq
uatio
n S
f=
a′ Nb′ , w
here
the
valu
es o
f a′
and
b′ are
est
ablis
hed
by tw
o po
ints
αan
dβ
. The
equ
atio
n fo
r b′ i
s
b′ =lo
gσ
α/σ
β
log
Nα/
Nβ
(c)
Exa
min
e th
e de
nom
inat
or o
f E
q. (
c):
log
Nα
Nβ
=lo
gN
1−
n 1n 2
=lo
gN
1−
n 1(1
−n 1
/N
1)N
2=
log
N1
N2
=lo
g(σ
1/a)
1/b
(σ2/a)
1/b
=lo
g
( σ1
σ2
) 1/b=
1 blo
g
( σ1
σ2
)Su
bstit
utin
g th
is in
to E
q. (
c) w
ith σ
α/σ
β=
σ1/σ
2gi
ves
b′ =lo
g(σ
1/σ
2)
(1/b)
log(
σ1/σ
2)
=b
whi
ch m
eans
the
dam
aged
mat
eria
l lin
e ha
s th
e sa
me
slop
e as
the
vir
gin
mat
eria
l lin
e;th
eref
ore,
the
lin
es a
re p
aral
lel.
Thi
s in
form
atio
n ca
n be
hel
pful
in
wri
ting
a co
mpu
ter
prog
ram
for
the
Palm
gren
-Min
er h
ypot
hesi
s.T
houg
h th
e M
iner
rul
e is
qui
te g
ener
ally
use
d, i
t fa
ils i
n tw
o w
ays
to a
gree
with
expe
rim
ent.
Firs
t, no
te th
at th
is th
eory
sta
tes
that
the
stat
ic s
tren
gth
S ut
is d
amag
ed, t
hat
is,
decr
ease
d, b
ecau
se o
f th
e ap
plic
atio
n of
σ1;
see
Fig.
6–3
4 at
N=
103
cycl
es.
Exp
erim
ents
fai
l to
veri
fy th
is p
redi
ctio
n.T
he M
iner
rul
e, a
s gi
ven
by E
q. (
6–58
), d
oes
not a
ccou
nt f
or th
e or
der
in w
hich
the
stre
sses
are
app
lied,
and
hen
ce ig
nore
s an
y st
ress
es le
ss th
an S
′ e,0. B
ut it
can
be
seen
inFi
g. 6
–34
that
a s
tres
s σ
3in
the
ran
ge S
′ e,1
<σ
3<
S′ e,0
wou
ld c
ause
dam
age
if a
pplie
daf
ter
the
endu
ranc
e lim
it ha
d be
en d
amag
ed b
y th
e ap
plic
atio
n of
σ1.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
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ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
321
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
318
Mec
hani
cal E
ngin
eerin
g D
esig
n
Man
son’
s25ap
proa
ch o
verc
omes
bot
h of
the
defi
cien
cies
not
ed f
or t
he P
alm
gren
-M
iner
met
hod;
his
tori
cally
it
is a
muc
h m
ore
rece
nt a
ppro
ach,
and
it
is j
ust
as e
asy
tous
e. E
xcep
t for
a s
light
cha
nge,
we
shal
l use
and
reco
mm
end
the
Man
son
met
hod
in th
isbo
ok. M
anso
n pl
otte
d th
e S–
log
Ndi
agra
m i
nste
ad o
f a
log
S–lo
gN
plot
as
is r
ecom
-m
ende
d he
re.
Man
son
also
res
orte
d to
exp
erim
ent
to fi
nd t
he p
oint
of
conv
erge
nce
ofth
eS–
log
Nlin
es c
orre
spon
ding
to th
e st
atic
str
engt
h, in
stea
d of
arb
itrar
ily s
elec
ting
the
inte
rsec
tion
of N
=10
3cy
cles
with
S=
0.9S
ut
as is
don
e he
re. O
f co
urse
, it i
s al
way
sbe
tter
to u
se e
xper
imen
t, bu
t ou
r pu
rpos
e in
thi
s bo
ok h
as b
een
to u
se t
he s
impl
e te
stda
ta to
lear
n as
muc
h as
pos
sibl
e ab
out f
atig
ue f
ailu
re.
The
met
hod
of M
anso
n, a
s pr
esen
ted
here
, con
sist
s in
hav
ing
all l
og S
–log
Nlin
es,
that
is,
lin
es f
or b
oth
the
dam
aged
and
the
vir
gin
mat
eria
l, co
nver
ge t
o th
e sa
me
poin
t,0.
9Su
tat
103
cycl
es. I
n ad
ditio
n, th
e lo
g S–
log
Nlin
es m
ust b
e co
nstr
ucte
d in
the
sam
ehi
stor
ical
ord
er in
whi
ch th
e st
ress
es o
ccur
.T
he d
ata
from
the
prec
edin
g ex
ampl
e ar
e us
ed f
or il
lust
rativ
e pu
rpos
es. T
he r
esul
tsar
e sh
own
in F
ig.6
–35.
Not
e th
at t
he s
tren
gth
Sf,
1co
rres
pond
ing
to
N1−
n 1=
5.52
(103
)cy
cles
is f
ound
in th
e sa
me
man
ner
as b
efor
e. T
hrou
gh th
is p
oint
and
thro
ugh
0.9S
ut
at10
3cy
cles
, dra
w th
e he
avy
dash
ed li
ne to
mee
t N
=10
6cy
cles
and
defi
ne th
een
dura
nce
limit
S′ e,1
of t
he d
amag
ed m
ater
ial.
In t
his
case
the
new
end
uran
ce l
imit
is34
.4 k
psi,
som
ewha
t les
s th
an th
at f
ound
by
the
Min
er m
etho
d.It
is n
ow e
asy
to s
ee f
rom
Fig
. 6–3
5 th
at a
rev
erse
d st
ress
σ=
36kp
si, s
ay, w
ould
not h
arm
the
endu
ranc
e lim
it of
the
virg
in m
ater
ial,
no m
atte
r ho
w m
any
cycl
es it
mig
htbe
app
lied.
How
ever
, if σ
=36
kpsi
sho
uld
be a
pplie
d af
ter
the
mat
eria
l was
dam
aged
byσ
1=
60kp
si, t
hen
addi
tiona
l dam
age
wou
ld b
e do
ne.
Bot
h th
ese
rule
s in
volv
e a
num
ber
of c
ompu
tatio
ns, w
hich
are
rep
eate
d ev
ery
time
dam
age
is e
stim
ated
. Fo
r co
mpl
icat
ed s
tres
s-tim
e tr
aces
, th
is m
ight
be
ever
y cy
cle.
Cle
arly
a c
ompu
ter
prog
ram
is u
sefu
l to
perf
orm
the
task
s, in
clud
ing
scan
ning
the
trac
ean
d id
entif
ying
the
cycl
es.
72 60 40
103
104
105
106
65
43
4.9
4.8
4.7
4.6
4.5
LogS
Sokpsi
N
Log
N
�1
0.9S
ut
n 1 =
3(1
03 )
S f,0
S'e,
0
S'e,
1
S f,1
N1
= 8
.52(
103 )
N1
– n 1
= 5
.52(
103 )
34.4
Figure
6–3
5
Use
of t
he M
anso
n m
etho
d to
pred
ict t
he e
ndur
ance
lim
it of
a m
ater
ial t
hat h
as b
een
over
stres
sed
for a
fini
tenu
mbe
r of c
ycle
s.
25S.
S. M
anso
n, A
. J. N
acht
igal
l, C
. R. E
nsig
n, a
nd J
. C. F
resc
he, “
Furt
her
Inve
stig
atio
n of
a R
elat
ion
for
Cum
ulat
ive
Fatig
ue D
amag
e in
Ben
ding
,” T
rans
. ASM
E, J
. Eng
. Ind
.,se
r. B
, vol
.87,
No.
1, p
p. 2
5–35
,Fe
brua
ry 1
965.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
322
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g319
Col
lins
said
it w
ell:
“In
spite
of
all t
he p
robl
ems
cite
d, th
e Pa
lmgr
en li
near
dam
age
rule
is
freq
uent
ly u
sed
beca
use
of i
ts s
impl
icity
and
the
exp
erim
enta
l fa
ct t
hat
othe
rm
ore
com
plex
dam
age
theo
ries
do
not
alw
ays
yiel
d a
sign
ifica
nt i
mpr
ovem
ent
in f
ail-
ure
pred
ictio
n re
liabi
lity.
”26
6–16
Surf
ace
Fatigue
Stre
ngth
The
surf
ace
fatig
uem
echa
nism
isno
tde
finiti
vely
unde
rsto
od.
The
cont
act-
affe
cted
zone
,in
the
abse
nce
ofsu
rfac
esh
eari
ngtr
actio
ns,
ente
rtai
nsco
mpr
essi
vepr
inci
pal
stre
sses
.R
otar
yfa
tigue
has
itscr
acks
grow
nat
orne
arth
esu
rfac
ein
the
pres
ence
ofte
nsile
stre
sses
that
are
asso
ciat
edw
ithcr
ack
prop
agat
ion,
toca
tast
roph
icfa
ilure
.The
rear
esh
ear
stre
sses
inth
ezo
ne,w
hich
are
larg
estj
ustb
elow
the
surf
ace.
Cra
cks
seem
togr
owfr
omth
isst
ratu
mun
tilsm
allp
iece
sof
mat
eria
lare
expe
lled,
leav
ing
pits
onth
esu
r-fa
ce.B
ecau
seen
gine
ers
had
tode
sign
dura
ble
mac
hine
rybe
fore
the
surf
ace
fatig
ueph
e-no
men
onw
asun
ders
tood
inde
tail,
they
had
take
nth
epo
stur
eof
cond
uctin
gte
sts,
obse
rvin
gpi
tson
the
surf
ace,
and
decl
arin
gfa
ilure
atan
arbi
trar
ypr
ojec
ted
area
ofho
le,
and
they
rela
ted
this
toth
eH
ertz
ian
cont
act
pres
sure
.T
his
com
pres
sive
stre
ssdi
dno
tpr
oduc
eth
efa
ilure
dire
ctly
,bu
tw
hate
ver
the
failu
rem
echa
nism
,w
hate
ver
the
stre
ssty
peth
atw
asin
stru
men
tal
inth
efa
ilure
,th
eco
ntac
tst
ress
was
anin
dex
toits
mag
nitu
de.
Buc
king
ham
27co
nduc
ted
a nu
mbe
r of
tes
ts r
elat
ing
the
fatig
ue a
t 10
8cy
cles
to
endu
ranc
e st
reng
th (
Her
tzia
n co
ntac
t pre
ssur
e). W
hile
ther
e is
evi
denc
e of
an
endu
ranc
elim
it at
abo
ut 3
(107
)cy
cles
for c
ast m
ater
ials
, har
dene
d st
eel r
olle
rs s
how
ed n
o en
dura
nce
limit
up t
o 4(
108)
cycl
es.
Subs
eque
nt t
estin
g on
har
d st
eel
show
s no
end
uran
ce l
imit.
Har
dene
d st
eel e
xhib
its s
uch
high
fatig
ue s
tren
gths
that
its
use
in r
esis
ting
surf
ace
fatig
ueis
wid
espr
ead.
Our
stu
dies
thu
s fa
r ha
ve d
ealt
with
the
fai
lure
of
a m
achi
ne e
lem
ent
by y
ield
ing,
by f
ract
ure,
and
by
fatig
ue. T
he e
ndur
ance
lim
it ob
tain
ed b
y th
e ro
tatin
g-be
am t
est
isfr
eque
ntly
cal
led
the
flexu
ral e
ndur
ance
lim
it,b
ecau
se it
is a
test
of
a ro
tatin
g be
am. I
nth
is s
ectio
n w
e sh
all s
tudy
a p
rope
rty
of m
atin
g m
ater
ials
calle
d th
e su
rfac
e en
dura
nce
shea
r. T
he d
esig
n en
gine
er m
ust f
requ
ently
sol
ve p
robl
ems
in w
hich
two
mac
hine
ele
-m
ents
mat
e w
ith o
ne a
noth
er b
y ro
lling
, slid
ing,
or
a co
mbi
natio
n of
rol
ling
and
slid
ing
cont
act.
Obv
ious
exa
mpl
es o
f su
ch c
ombi
natio
ns a
re th
e m
atin
g te
eth
of a
pai
r of
gea
rs,
a ca
m a
nd f
ollo
wer
, a w
heel
and
rai
l, an
d a
chai
n an
d sp
rock
et. A
kno
wle
dge
of th
e su
r-fa
ce s
tren
gth
of m
ater
ials
is
nece
ssar
y if
the
des
igne
r is
to
crea
te m
achi
nes
havi
ng a
long
and
sat
isfa
ctor
y lif
e.W
hen
two
surf
aces
rol
l or
rol
l an
d sl
ide
agai
nst
one
anot
her
with
suf
ficie
nt f
orce
,a
pitti
ng f
ailu
re w
ill o
ccur
aft
er a
cer
tain
num
ber
of c
ycle
s of
ope
ratio
n. A
utho
ritie
s ar
eno
t in
com
plet
e ag
reem
ent o
n th
e ex
act m
echa
nism
of
the
pitti
ng; a
lthou
gh th
e su
bjec
tis
qui
te c
ompl
icat
ed, t
hey
do a
gree
that
the
Her
tz s
tres
ses,
the
num
ber o
f cyc
les,
the
sur-
face
fini
sh, t
he h
ardn
ess,
the
degr
ee o
f lu
bric
atio
n, a
nd th
e te
mpe
ratu
re a
ll in
fluen
ce th
est
reng
th.
In S
ec.
3–19
it
was
lea
rned
tha
t, w
hen
two
surf
aces
are
pre
ssed
tog
ethe
r, a
max
imum
she
ar s
tres
s is
dev
elop
ed s
light
ly b
elow
the
cont
actin
g su
rfac
e. I
t is
post
ulat
edby
som
e au
thor
ities
tha
t a
surf
ace
fatig
ue f
ailu
re i
s in
itiat
ed b
y th
is m
axim
um s
hear
stre
ss a
nd th
en is
pro
paga
ted
rapi
dly
to th
e su
rfac
e. T
he lu
bric
ant t
hen
ente
rs th
e cr
ack
that
is f
orm
ed a
nd, u
nder
pre
ssur
e, e
vent
ually
wed
ges
the
chip
loos
e.
26J.
A. C
ollin
s, F
ailu
re o
f Mat
eria
ls in
Mec
hani
cal D
esig
n,Jo
hn W
iley
& S
ons,
New
Yor
k, 1
981,
p. 2
43.
27E
arle
Buc
king
ham
, Ana
lyti
cal M
echa
nics
of G
ears
,McG
raw
-Hill
, New
Yor
k, 1
949.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
323
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
320
Mec
hani
cal E
ngin
eerin
g D
esig
n
To d
eter
min
e th
e su
rfac
e fa
tigue
str
engt
h of
mat
ing
mat
eria
ls, B
ucki
ngha
m d
esig
ned
a si
mpl
e m
achi
ne f
or t
estin
g a
pair
of
cont
actin
g ro
lling
sur
face
s in
con
nect
ion
with
his
inve
stig
atio
n of
the
wea
r of
gea
r te
eth.
Buc
king
ham
and
, lat
er, T
albo
urde
t gat
here
d la
rge
num
bers
of
data
fro
m m
any
test
s so
tha
t co
nsid
erab
le d
esig
n in
form
atio
n is
now
avai
labl
e. T
o m
ake
the
resu
lts u
sefu
l fo
r de
sign
ers,
Buc
king
ham
defi
ned
a lo
ad-s
tres
sfa
ctor
,als
o ca
lled
a w
ear
fact
or,w
hich
is
deri
ved
from
the
Her
tz e
quat
ions
. Equ
atio
ns(3
–73)
and
(3–
74),
pp.
118
–119
, for
con
tact
ing
cylin
ders
are
fou
nd to
be
b=√ 2
F
πl
( 1−
ν2 1
) /E
1+( 1
−ν
2 2
) /E
2
(1/d 1
)+
(1/d 2
)(6
–59)
p max
=2
F
πbl
(6–6
0)
whe
reb
=ha
lf w
idth
of
rect
angu
lar
cont
act a
rea
F=
cont
act f
orce
l=
leng
th o
f cy
linde
rs
ν=
Pois
son’
s ra
tio
E=
mod
ulus
of
elas
ticity
d=
cylin
der
diam
eter
It is
mor
e co
nven
ient
to u
se th
e cy
linde
r ra
dius
, so
let 2
r=
d. I
f w
e th
en d
esig
nate
the
leng
th o
f th
e cy
linde
rs a
s w
(for
wid
th o
f ge
ar, b
eari
ng, c
am, e
tc.)
ins
tead
of
lan
dre
mov
e th
e sq
uare
roo
t sig
n, E
q. (
6–59
) be
com
es
b2=
4F
πw
( 1−
ν2 1
) /E
1+( 1
−ν
2 2
) /E
2
1/r 1
+1/
r 2(6
–61)
We
can
defin
e a
surf
ace
endu
ranc
e st
reng
thS C
usin
g
p max
=2
F
πbw
(6–6
2)
as
S C=
2F
πbw
(6–6
3)
whi
ch m
ay a
lso
be c
alle
d co
ntac
t str
engt
h,th
eco
ntac
t fat
igue
str
engt
h,or
the
Her
tzia
nen
dura
nce
stre
ngth
. T
he s
tren
gth
is t
he c
onta
ctin
g pr
essu
re w
hich
, af
ter
a sp
ecifi
ednu
mbe
r of
cyc
les,
will
cau
se f
ailu
re o
f th
e su
rfac
e. S
uch
failu
res
are
ofte
n ca
lled
wea
rbe
caus
e th
ey o
ccur
ove
r a
very
lon
g tim
e. T
hey
shou
ld n
ot b
e co
nfus
ed w
ith a
bras
ive
wea
r, ho
wev
er. B
y sq
uari
ng E
q. (
6–63
), s
ubst
itutin
g b2
from
Eq.
(6–
61),
and
rea
rran
g-in
g, w
e ob
tain
F w
( 1 r 1+
1 r 2
) =π
S2 C
[ 1−
ν2 1
E1
+1
−ν
2 2
E2
] =K
1(6
–64)
The
left
exp
ress
ion
cons
ists
of
para
met
ers
a de
sign
er m
ay s
eek
to c
ontr
ol in
depe
nden
tly.
The
cen
tral
exp
ress
ion
cons
ists
of
mat
eria
l pr
oper
ties
that
com
e w
ith t
he m
ater
ial
and
cond
ition
spe
cific
atio
n. T
he t
hird
exp
ress
ion
is t
he p
aram
eter
K1, B
ucki
ngha
m’s
loa
d-st
ress
fac
tor,
dete
rmin
ed b
y a
test
fixt
ure
with
val
ues
F,w
,r 1
,r 2
and
the
num
ber
of
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
324
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g321
cycl
es a
ssoc
iate
d w
ith t
he fi
rst
tang
ible
evi
denc
e of
fat
igue
. In
gea
r st
udie
s a
sim
ilar
Kfa
ctor
is u
sed:
Kg
=K
1 4si
nφ
(6–6
5)
whe
reφ
is t
he t
ooth
pre
ssur
e an
gle,
and
the
ter
m [
(1−
ν2 1)/
E1+
(1−
ν2 2)/
E2]
isde
fined
as
1/(π
C2 P) ,
so
that
S C=
CP
√ F w
( 1 r 1+
1 r 2
)(6
–66)
Buc
king
ham
and
oth
ers
repo
rted
K1
for
108
cycl
es a
nd n
othi
ng e
lse.
Thi
s gi
ves
only
one
poin
t on
the
S CN
curv
e. F
or c
ast m
etal
s th
is m
ay b
e su
ffici
ent,
but f
or w
roug
ht s
teel
s, h
eat-
trea
ted,
som
e id
ea o
f th
e sl
ope
is u
sefu
l in
mee
ting
desi
gn g
oals
of
othe
r th
an 1
08cy
cles
.E
xper
imen
ts s
how
tha
t K
1ve
rsus
N,
Kg
vers
us N
, and
SC
vers
us N
data
are
rec
ti-fie
d by
logl
og tr
ansf
orm
atio
n. T
his
sugg
ests
that
K1
=α
1N
β1
Kg
=a
Nb
S C=
αN
β
The
thre
e ex
pone
nts
are
give
n by
β1
=lo
g(K
1/
K2)
log(
N1/
N2)
b=
log(
Kg1
/K
g2)
log(
N1/
N2)
β=
log(
S C1/
S C2)
log(
N1/
N2)
(6–6
7)
Dat
a on
ind
uctio
n-ha
rden
ed s
teel
on
stee
l gi
ve (
S C) 1
07=
271
kpsi
and
(S C
) 108
=23
9kp
si, s
o β
, fro
m E
q. (
6–67
), is
β=
log(
271/
239)
log(
107/10
8)
=−0
.055
It m
ay b
e of
inte
rest
that
the
Am
eric
an G
ear
Man
ufac
ture
rs A
ssoc
iatio
n (A
GM
A)
uses
β�
�0.
056
betw
een
104
<N
<10
10if
the
des
igne
r ha
s no
dat
a to
the
con
trar
ybe
yond
107
cycl
es.
A lo
ngst
andi
ng c
orre
latio
n in
ste
els
betw
een
S Can
dH
Bat
108
cycl
es is
(SC) 1
08={ 0.
4H
B−
10kp
si2.
76H
B−
70M
Pa(6
–68)
AG
MA
use
s0.
99(S
C) 1
07=
0.32
7H
B+
26kp
si(6
–69)
Equ
atio
n (6
–66)
can
be
used
in d
esig
n to
find
an
allo
wab
le s
urfa
ce s
tres
s by
usi
nga
desi
gn f
acto
r. Si
nce
this
equ
atio
n is
non
linea
r in
its
str
ess-
load
tra
nsfo
rmat
ion,
the
desi
gner
mus
t de
cide
if
loss
of
func
tion
deno
tes
inab
ility
to
carr
y th
e lo
ad. I
f so
, the
nto
find
the
allo
wab
le s
tres
s, o
ne d
ivid
es th
e lo
ad F
by th
e de
sign
fac
tor
n d:
σC
=C
P
√F
wn d
( 1 r 1+
1 r 2
) =C
P√ n d
√ F w
( 1 r 1+
1 r 2
) =S C √ n d
and
n d=
(SC/σ
C)2
. If
the
loss
of
func
tion
is f
ocus
ed o
n st
ress
, the
n n d
=S C
/σ
C. I
t is
reco
mm
ende
d th
at a
n en
gine
er
•D
ecid
e w
heth
er lo
ss o
f fu
nctio
n is
fai
lure
to c
arry
load
or
stre
ss.
•D
efine
the
desi
gn f
acto
r an
d fa
ctor
of
safe
ty a
ccor
ding
ly.
•A
nnou
nce
wha
t he
or s
he is
usi
ng a
nd w
hy.
•B
e pr
epar
ed to
def
end
his
or h
er p
ositi
on.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
325
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
In t
his
way
eve
ryon
e w
ho i
s pa
rty
to t
he c
omm
unic
atio
n kn
ows
wha
t a
desi
gn f
acto
r(o
rfa
ctor
of
safe
ty)
of 2
mea
ns a
nd a
djus
ts, i
f ne
cess
ary,
the
judg
men
tal p
ersp
ectiv
e.
6–17
Stoch
ast
ic A
naly
sis2
8
As
alre
ady
dem
onst
rate
d in
this
cha
pter
, the
re a
re a
gre
at m
any
fact
ors
to c
onsi
der
ina
fati
gue
anal
ysis
, muc
h m
ore
so th
an in
a s
tati
c an
alys
is. S
o fa
r, e
ach
fact
or h
as b
een
trea
ted
in a
det
erm
inis
tic
man
ner,
and
if n
ot o
bvio
us, t
hese
fac
tors
are
sub
ject
to v
ari-
abil
ity
and
cont
rol
the
over
all
reli
abil
ity
of t
he r
esul
ts. W
hen
reli
abil
ity
is i
mpo
rtan
t,th
en fa
tigu
e te
stin
g m
ust c
erta
inly
be
unde
rtak
en. T
here
is n
o ot
her
way
. Con
sequ
entl
y,th
e m
etho
ds o
f st
ocha
stic
ana
lysi
s pr
esen
ted
here
and
in
othe
r se
ctio
ns o
f th
is b
ook
cons
titu
te g
uide
line
s th
at e
nabl
e th
e de
sign
er t
o ob
tain
a g
ood
unde
rsta
ndin
g of
the
vari
ous
issu
es i
nvol
ved
and
help
in
the
deve
lopm
ent
of a
saf
e an
d re
liab
le d
esig
n.In
this
sec
tion,
key
sto
chas
tic m
odifi
catio
ns to
the
dete
rmin
istic
fea
ture
s an
d eq
ua-
tions
des
crib
ed in
ear
lier
sect
ions
are
pro
vide
d in
the
sam
e or
der
of p
rese
ntat
ion.
Endura
nce
Lim
itTo
beg
in,
a m
etho
d fo
r es
timat
ing
endu
ranc
e lim
its,
the
tens
ile
stre
ngth
cor
rela
tion
met
hod,
is p
rese
nted
. T
he r
atio
�=
S′ e/S̄ u
tis
cal
led
the
fati
gue
rati
o.29
For
ferr
ous
met
als,
mos
t of
whi
ch e
xhib
it an
end
uran
ce l
imit,
the
end
uran
ce l
imit
is u
sed
as a
num
erat
or. F
or m
ater
ials
that
do
not s
how
an
endu
ranc
e lim
it, a
n en
dura
nce
stre
ngth
at
a sp
ecifi
ed n
umbe
r of
cyc
les
to f
ailu
re i
s us
ed a
nd n
oted
. G
ough
30re
port
ed t
he s
to-
chas
tic n
atur
e of
the
fat
igue
rat
io �
for
seve
ral
clas
ses
of m
etal
s, a
nd t
his
is s
how
n in
Fig.
6–36
. The
firs
t ite
m t
o no
te i
s th
at t
he c
oeffi
cien
t of
var
iatio
n is
of
the
orde
r 0.
10to
0.1
5, a
nd th
e di
stri
butio
n va
ries
for
cla
sses
of
met
als.
The
sec
ond
item
to n
ote
is th
atG
ough
’s d
ata
incl
ude
mat
eria
ls o
f no
int
eres
t to
eng
inee
rs.
In t
he a
bsen
ce o
f te
stin
g,en
gine
ers
use
the
corr
elat
ion
that
�re
pres
ents
to e
stim
ate
the
endu
ranc
e lim
it S′ e
from
the
mea
n ul
timat
e st
reng
th S̄
ut.
Gou
gh’s
dat
a ar
e fo
r en
sem
bles
of
met
als,
som
e ch
osen
for
met
allu
rgic
al i
nter
est,
and
incl
ude
mat
eria
ls t
hat
are
not
com
mon
ly s
elec
ted
for
mac
hine
par
ts.
Mis
chke
31
anal
yzed
dat
a fo
r 13
3 co
mm
on s
teel
s an
d tr
eatm
ents
in
vary
ing
diam
eter
s in
rot
atin
gbe
ndin
g,32
and
the
resu
lt w
as �=
0.44
5d−0
.107
LN
(1,0.
138)
whe
red
is th
e sp
ecim
en d
iam
eter
in in
ches
and
LN
(1,0.
138)
is a
uni
t log
norm
al v
ari-
ate
with
a m
ean
of 1
and
a s
tand
ard
devi
atio
n (a
nd c
oeffi
cien
t of v
aria
tion)
of 0
.138
. For
the
stan
dard
R. R
. Moo
re s
peci
men
,
�0.
30=
0.44
5(0.
30)−0
.107
LN
(1,0.
138)
=0.
506L
N(1
,0.
138)
322
Mec
hani
cal E
ngin
eerin
g D
esig
n
28R
evie
w C
hap.
20
befo
re r
eadi
ng th
is s
ectio
n.29
From
this
poi
nt, s
ince
we
will
be
deal
ing
with
sta
tistic
al d
istr
ibut
ions
in te
rms
of m
eans
, sta
ndar
dde
viat
ions
, etc
. A k
ey q
uant
ity, t
he u
ltim
ate
stre
ngth
, will
her
e be
pre
sent
ed b
y its
mea
n va
lue,
S̄u
t. T
his
mea
ns th
at c
erta
in te
rms
that
wer
e de
fined
ear
lier
in te
rms
of th
e m
inim
um v
alue
of
S utw
ill c
hang
e sl
ight
ly.
30In
J. A
. Pop
e, M
etal
Fat
igue
,Cha
pman
and
Hal
l, L
ondo
n, 1
959.
31C
harl
es R
. Mis
chke
, “Pr
edic
tion
of S
toch
astic
End
uran
ce S
tren
gth,
” Tr
ans.
ASM
E, J
ourn
al o
f Vib
rati
on,
Aco
usti
cs, S
tres
s, a
nd R
elia
bili
ty in
Des
ign,
vol.
109,
no.
1, J
anua
ry 1
987,
pp.
113
–122
.32
Dat
a fr
om H
. J. G
rove
r, S.
A. G
ordo
n, a
nd L
. R. J
acks
on, F
atig
ue o
f Met
als
and
Stru
ctur
es, B
urea
u of
Nav
al W
eapo
ns, D
ocum
ent N
AV
WE
PS 0
0-25
0043
5, 1
960.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
326
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g323
Als
o, 2
5 pl
ain
carb
on a
nd lo
w-a
lloy
stee
ls w
ith S
ut>
212
kpsi
are
des
crib
ed b
y
S′ e=
107L
N(1
,0.
139)
kpsi
In s
umm
ary,
for
the
rota
ting-
beam
spe
cim
en,
S′ e=
⎧ ⎪ ⎨ ⎪ ⎩0.50
6S̄u
tLN
(1,0.
138)
kpsi
orM
PaS̄ u
t≤
212
kpsi
(146
0M
Pa)
107L
N(1
,0.
139)
kpsi
S̄ ut>
212
kpsi
740L
N(1
,0.
139)
MPa
S̄ ut>
1460
MPa
(6–7
0)
whe
reS̄ u
tis
the
mea
nul
timat
e te
nsile
str
engt
h.E
quat
ions
(6–
70)
repr
esen
t the
sta
te o
f in
form
atio
n be
fore
an
engi
neer
has
cho
sen
a m
ater
ial.
In c
hoos
ing,
the
des
igne
r ha
s m
ade
a ra
ndom
cho
ice
from
the
ens
embl
e of
poss
ibili
ties,
and
the
stat
istic
s ca
n gi
ve th
e od
ds o
f di
sapp
oint
men
t. If
the
test
ing
is li
m-
ited
to fi
ndin
g an
est
imat
e of
the
ulti
mat
e te
nsile
str
engt
h m
ean
S̄ ut
with
the
cho
sen
mat
eria
l, E
qs. (
6–70
) ar
e di
rect
ly h
elpf
ul. I
f th
ere
is t
o be
rot
ary-
beam
fat
igue
tes
ting,
then
sta
tistic
al i
nfor
mat
ion
on t
he e
ndur
ance
lim
it is
gat
here
d an
d th
ere
is n
o ne
ed f
orth
e co
rrel
atio
n ab
ove.
Tabl
e 6–
9 co
mpa
res
appr
oxim
ate
mea
n va
lues
of
the
fatig
ue r
atio
φ̄0.
30fo
r se
vera
lcl
asse
s of
fer
rous
mat
eria
ls.
Endura
nce
Lim
it M
odif
yin
g F
act
ors
A M
arin
equ
atio
n ca
n be
wri
tten
as S e=
k ak b
k ck d
k fS′ e
(6–7
1)
whe
re t
he s
ize
fact
or k
bis
det
erm
inis
tic a
nd r
emai
ns u
ncha
nged
fro
m t
hat
give
n in
Sec.
6–9.
Als
o, s
ince
we
are
perf
orm
ing
a st
ocha
stic
ana
lysi
s, th
e “r
elia
bilit
y fa
ctor
” k e
is u
nnec
essa
ry h
ere.
The
sur
face
fac
tor
k aci
ted
earl
ier
in d
eter
min
istic
for
m a
s E
q. (
6–20
), p
. 28
0, i
sno
w g
iven
in s
toch
astic
for
m b
y
k a=
aS̄b u
tLN
(1,C
)(S̄
utin
kps
i or
MPa
)(6
–72)
whe
re T
able
6–1
0 gi
ves
valu
es o
f a,
b, a
nd C
for
vari
ous
surf
ace
cond
ition
s.
0.3
0.4
0.5
0.6
0.7
05
Probability density
Rot
ary
bend
ing
fatig
ue r
atio
�b
2
1
34
5
1 2 3 4 5
All
met
als
Non
ferr
ous
Iron
and
car
bon
stee
ls
Low
allo
y st
eels
Spec
ial a
lloy
stee
ls
380
152
111 78 39
Cla
ssN
o.Fi
gure
6–3
6
The
logn
orm
al p
roba
bilit
yde
nsity
PD
F of
the
fatig
ue ra
tioφ
bof
Gou
gh.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
327
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
324
Mec
hani
cal E
ngin
eerin
g D
esig
n
Table
6–9
Com
paris
on o
fA
ppro
xim
ate
Valu
es o
fM
ean
Fatig
ue R
atio
for
Som
e C
lass
es o
f Met
als
Mate
rial C
lass
φ0.3
0
Wro
ught
ste
els
0.50
C
ast s
teel
s0.
40Po
wde
red
steel
s0.
38G
ray
cast
iron
0.35
Mal
leab
le c
ast i
ron
0.40
Nor
mal
ized
nod
ular
cas
t iro
n0.
33
Table
6–1
0
Para
met
ers
in M
arin
Surfa
ce C
ondi
tion
Fact
or
ka
�aS
b utLN
(1,C
)Su
rface
aCoef
fici
ent
of
Finis
hk
psi
MPa
bV
ari
ation,
C
Gro
und∗
1.34
1.
58−0
.086
0.12
0M
achi
ned
or C
old-
rolle
d 2.
67
4.45
−0
.265
0.05
8H
ot-ro
lled
14.5
58.1
−0.7
190.
110
As-f
orge
d39
.827
1−0
.995
0.14
5
*Due
to th
e wide
scatt
er in
groun
d surf
ace d
ata, a
n alte
rnate
functi
on is
ka
�0.
878L
N(1,
0.1
20).
Note:
Sut
in kp
si or
MPa.
EXA
MPLE
6–1
6A
ste
el h
as a
mea
n ul
timat
e st
reng
th o
f 52
0 M
Pa a
nd a
mac
hine
d su
rfac
e. E
stim
ate
k a.
Solu
tion
From
Tab
le 6
–10,
k a=
4.45
(520
)−0.2
65L
N(1
,0.
058)
k̄ a=
4.45
(520
)−0.2
65(1
)=
0.84
8
σ̂ka
=C
k̄ a=
(0.0
58)4
.45(
520)
−0.2
65=
0.04
9
Ans
wer
sok a
=L
N(0
.848
,0.
049)
.
The
load
fac
tor
k cfo
r ax
ial a
nd to
rsio
nal l
oadi
ng is
giv
en b
y
(kc)
axia
l=
1.23
S̄−0.0
778
ut
LN
(1,0.
125)
(6–7
3)
(kc)
tors
ion
=0.
328S̄
0.12
5u
tL
N(1
,0.
125)
(6–7
4)
whe
reS̄ u
tis
in k
psi.
The
re a
re f
ewer
dat
a to
stu
dy f
or a
xial
fat
igue
. Equ
atio
n (6
–73)
was
dedu
ced
from
the
data
of
Lan
dgra
f an
d of
Gro
ver,
Gor
don,
and
Jac
kson
(as
cite
d ea
rlie
r).
Tors
iona
l da
ta a
re s
pars
er,
and
Eq.
(6–
74)
is d
educ
ed f
rom
dat
a in
Gro
ver
et a
l.N
otic
e th
e m
ild s
ensi
tivity
to
stre
ngth
in
the
axia
l an
d to
rsio
nal
load
fac
tor,
so k
cin
thes
e ca
ses
is n
ot c
onst
ant.
Ave
rage
val
ues
are
show
n in
the
last
col
umn
of T
able
6–1
1,an
d as
foo
tnot
es t
o Ta
bles
6–1
2 an
d 6–
13. T
able
6–1
4 sh
ows
the
influ
ence
of
mat
eria
lcl
asse
s on
the
loa
d fa
ctor
kc.
Dis
tort
ion
ener
gy t
heor
y pr
edic
ts (
k c) t
orsi
on=
0.57
7fo
rm
ater
ials
to w
hich
the
dist
ortio
n-en
ergy
theo
ry a
pplie
s. F
or b
endi
ng, k
c=
LN
(1,0)
.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
328
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Table
6–1
3
Aver
age
Mar
in L
oadi
ngFa
ctor
for T
orsio
nal L
oad
S̄ ut,
kpsi
k* c
500.
535
100
0.58
315
00.
614
200
0.63
6
*Ave
rage e
ntry 0
.59.
Table
6–1
4
Aver
age
Mar
in T
orsio
nal
Load
ing
Fact
or k
cfo
rSe
vera
l Mat
eria
ls
Mate
rial
Range
nk̄
cσ̂
kc
Wro
ught
ste
els
0.52
–0.6
931
0.60
0.03
Wro
ught
Al
0.43
–0.7
413
0.55
0.09
Wro
ught
Cu
and
allo
y 0.
41–0
.67
70.
560.
10W
roug
ht M
g an
d al
loy
0.49
–0.6
02
0.54
0.08
Tita
nium
0.37
–0.5
73
0.48
0.12
Cas
t iro
n0.
79–1
.01
90.
900.
07C
ast A
l, M
g, a
nd a
lloy
0.71
–0.9
15
0.85
0.09
Sourc
e: Th
e tab
le is
an ex
tensio
n of P
. G. F
orres
t, Fa
tigue
of M
etals,
Perga
mon P
ress,
Lond
on, 1
962,
Table
17,
p. 1
10,
with
stand
ard de
viatio
ns es
timate
d from
rang
e and
samp
le siz
e usin
g Tab
le A–
1 in
J. B.
Kenn
edy a
nd A.
M. N
eville
, Ba
sic S
tatist
ical M
ethod
s for
Engin
eers
and S
cienti
sts,3
rd ed
., Ha
rper &
Row,
New
York,
198
6, pp
. 54–
55.
Table
6–1
1
Para
met
ers
in M
arin
Load
ing
Fact
or
kc
�αS u
t− β
LN(1
, C
)M
ode
of
αA
vera
ge
Loadin
gk
psi
MPa
βC
kc
Bend
ing
11
00
1A
xial
1.23
1.43
−0.0
778
0.12
50.
85To
rsio
n0.
328
0.25
80.
125
0.12
50.
59
Table
6–1
2
Aver
age
Mar
in L
oadi
ngFa
ctor
for A
xial
Loa
d
S̄ ut,
kpsi
k* c
500.
907
100
0.86
015
00.
832
200
0.81
4
*Ave
rage e
ntry 0
.85.
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g325
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
329
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
326
Mec
hani
cal E
ngin
eerin
g D
esig
n
EXA
MPLE
6–1
7E
stim
ate
the
Mar
in lo
adin
g fa
ctor
kc
for
a 1–
in-d
iam
eter
bar
that
is u
sed
as f
ollo
ws.
(a)
In b
endi
ng. I
t is
mad
e of
ste
el w
ith S
ut=
100L
N(1
,0.
035)
kpsi
, and
the
des
igne
rin
tend
s to
use
the
corr
elat
ion
S′ e=
�0.
30S̄ u
tto
pre
dict
S′ e.
(b)
In b
endi
ng, b
ut e
ndur
ance
test
ing
gave
S′ e=
55L
N(1
,0.
081)
kpsi
.(c
) In
pus
h-pu
ll (a
xial
) fa
tigue
, Su
t=
LN
(86.
2,3.
92)
kpsi
, and
the
desi
gner
inte
nded
tous
e th
e co
rrel
atio
n S′ e
=�
0.30
S̄ ut.
(d)
In to
rsio
nal f
atig
ue. T
he m
ater
ial i
s ca
st ir
on, a
nd S
′ eis
kno
wn
by te
st.
Solu
tion
(a)
Sinc
e th
e ba
r is
in b
endi
ng,
Ans
wer
k c=
(1,0)
(b)
Sinc
e th
e te
st is
in b
endi
ng a
nd u
se is
in b
endi
ng,
Ans
wer
k c=
(1,0)
(c)
From
Eq.
(6–
73),
Ans
wer
(kc)
ax
=1.
23(8
6.2)
−0.0
778L
N(1
,0.
125)
k̄ c=
1.23
(86.
2)−0
.077
8(1
)=
0.87
0
σ̂kc
=C
k̄ c=
0.12
5(0.
870)
=0.
109
(d)
From
Tab
le 6
–15,
k̄c=
0.90
,σ̂kc
=0.
07, a
nd
Ans
wer
Ckc
=0.
07
0.90
=0.
08
The
tem
pera
ture
fac
tor
k dis
k d=
k̄ dL
N(1
,0.
11)
(6–7
5)
whe
rek̄ d
=k d
, giv
en b
y E
q. (
6–27
), p
. 283
.Fi
nally
, kfis
, as
befo
re, t
he m
isce
llane
ous
fact
or th
at c
an c
ome
abou
t fro
m a
gre
atm
any
cons
ider
atio
ns, a
s di
scus
sed
in S
ec. 6
–9, w
here
now
sta
tistic
al d
istr
ibut
ions
, pos
-si
bly
from
test
ing,
are
con
side
red.
Stre
ss C
once
ntr
ation a
nd N
otc
h S
ensi
tivi
tyN
otch
sen
sitiv
ity q
was
defi
ned
by E
q. (
6–31
), p
. 287
. The
sto
chas
tic e
quiv
alen
t is
q=
Kf−
1
Kt−
1(6
–76)
whe
reK
tis
the
the
oret
ical
(or
geo
met
ric)
str
ess-
conc
entr
atio
n fa
ctor
, a
dete
rmin
istic
quan
tity.
A s
tudy
of
lines
3 a
nd 4
of
Tabl
e 20
–6, w
ill r
evea
l th
at a
ddin
g a
scal
ar t
o (o
rsu
btra
ctin
g on
e fr
om)
a va
riat
e x
will
aff
ect o
nly
the
mea
n. A
lso,
mul
tiply
ing
(or
divi
d-in
g) b
y a
scal
ar a
ffec
ts b
oth
the
mea
n an
d st
anda
rd d
evia
tion.
With
this
in m
ind,
we
can
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
330
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g327
rela
te th
e st
atis
tical
par
amet
ers
of th
e fa
tigue
str
ess-
conc
entr
atio
n fa
ctor
Kf
to th
ose
ofno
tch
sens
itivi
ty q
. It f
ollo
ws
that
q=
LN( K̄
f−
1
Kt−
1,
CK̄
f
Kt−
1
)
whe
reC
=C
Kf
and
q̄=
K̄f−
1
Kt−
1
σ̂q
=C
K̄f
Kt−
1(6
–77)
Cq
=C
K̄f
K̄f−
1
The
fatig
uest
ress
-con
cent
ratio
nfa
ctor
Kf
has
been
inve
stig
ated
mor
ein
Eng
land
than
inth
eU
nite
dSt
ates
.Fo
r K̄
f,
cons
ider
a m
odifi
ed N
eube
r eq
uatio
n (a
fter
Hey
woo
d33),
whe
re th
e fa
tigue
str
ess-
conc
entr
atio
n fa
ctor
is g
iven
by
K̄f
=K
t
1+
2(K
t−
1)
Kt
√ a √ r
(6–7
8)
whe
re T
able
6–1
5 gi
ves
valu
es o
f √ a
and
CK
ffo
r st
eels
wit
h tr
ansv
erse
hol
es,
shou
lder
s, o
r gr
oove
s.O
nce
Kf
isde
scri
bed,
qca
nal
sobe
quan
tifi
edus
ing
the
set
Eqs
.(6–
77).
The
mod
ified
Neu
ber
equa
tion
give
s th
e fa
tigue
str
ess
conc
entr
atio
n fa
ctor
as
Kf
=K̄
fL
N( 1,
CK
f
)(6
–79)
Table
6–1
5
Hey
woo
d’s
Para
met
er√ a
and
coef
ficie
nts
ofva
riatio
nC
Kffo
r ste
els
√ a(√ in
) ,√ a
(√ mm
) ,
Notc
h T
ype
S utin
kpsi
S utin
MPa
Coef
fici
ent
of
Vari
ation C
Kf
Tran
sver
se h
ole
5/S u
t17
4/S u
t0.
10Sh
ould
er4/
S ut
139/
S ut
0.11
Gro
ove
3/S u
t10
4/S u
t0.
15
EXA
MPLE
6–1
8E
stim
ate
Kf
and
qfo
r th
e st
eel s
haft
giv
en in
Ex.
6–6
, p. 2
88.
Solu
tion
From
Ex.
6–6
, a
stee
l sh
aft
with
Su
t=
690
Mpa
and
a s
houl
der
with
a fi
llet
of 3
mm
was
fou
nd t
o ha
ve a
the
oret
ical
str
ess-
conc
entr
atio
n-fa
ctor
of
Kt
. =1.
65.
From
Tab
le6–
15,
√ a=
139
S ut
=13
9
690
=0.
2014
√ mm
33R
. B. H
eyw
ood,
Des
igni
ng A
gain
st F
atig
ue, C
hapm
an &
Hal
l, L
ondo
n, 1
962.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
331
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
From
Eq.
(6–
78),
Kf=
Kt
1+
2(K
t−
1)
Kt
√ a √ r
=1.
65
1+
2(1.
65−
1)
1.65
0.20
14√ 3
=1.
51
whi
ch is
2.5
per
cent
low
er th
an w
hat w
as f
ound
in E
x. 6
–6.
From
Tab
le 6
–15,
CK
f=
0.11
. Thu
s fr
om E
q. (
6–79
),
Ans
wer
Kf=
1.51
LN
(1,0
.11)
From
Eq.
(6–
77),
with
Kt=
1.65
q̄=
1.51
−1
1.65
−1
=0.
785
Cq
=C
KfK̄
f
K̄f−
1=
0.11
(1.5
1)
1.51
−1
=0.
326
σ̂q
=C
qq̄
=0.
326(
0.78
5)=
0.25
6
So,
Ans
wer
q=
LN
(0.7
85,0.
256)
EXA
MPLE
6–1
9T
he b
ar s
how
n in
Fig
. 6–
37 i
s m
achi
ned
from
a c
old-
rolle
d fla
t ha
ving
an
ultim
ate
stre
ngth
of
S ut=
LN
(87.
6,5.
74)
kpsi
. T
he a
xial
loa
d sh
own
is c
ompl
etel
y re
vers
ed.
The
load
am
plitu
de is
Fa
=L
N(1
000,
120)
lbf.
(a)
Est
imat
e th
e re
liabi
lity.
(b)
Ree
stim
ate
the
relia
bilit
y w
hen
a ro
tatin
g be
ndin
g en
dura
nce
test
sho
ws
that
S′ e=
LN
(40,
2)kp
si.
Solu
tion
(a)
From
Eq.
(6–
70),
S′ e=
0.50
6S̄u
tLN
(1,0.
138)
=0.
506(
87.6
)LN
(1,0.
138)
=44
.3L
N(1
,0.
138)
kpsi
From
Eq.
(6–
72)
and
Tabl
e 6–
10,
k a=
2.67
S̄−0.2
65u
tL
N(1
,0.
058)
=2.
67(8
7.6)
−0.2
65L
N(1
,0.
058)
=0.
816L
N(1
,0.
058)
k b=
1(a
xial
load
ing)
3 4in
D.
3 16in
R.
in
1 42
in1 2
1in
1 4
1000
lbf
1000
lbf
Figure
6–3
7
328
Mec
hani
cal E
ngin
eerin
g D
esig
n
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
332
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g329
From
Eq.
(6–
73),
k c=
1.23
S̄−0.0
778
ut
LN
(1,0.
125)
=1.
23(8
7.6)
−0.0
778L
N(1
,0.
125)
=0.
869L
N(1
,0.
125)
k d=
kf=
(1,0)
The
end
uran
ce s
tren
gth,
fro
m E
q. (
6–71
), is
S e=
k ak b
k ck d
kfS′ e
S e=
0.81
6LN
(1,0
.058
)(1)
0.86
9LN
(1,0
.125
)(1)
(1)4
4.3L
N(1
,0.1
38)
The
par
amet
ers
of S
ear
e
S̄ e=
0.81
6(0.
869)
44.3
=31
.4kp
si
CSe
=(0
.058
2+
0.12
52+
0.13
82)1/
2=
0.19
5
soS e
=31
.4L
N(1
,0.
195)
kpsi
.In
com
putin
g th
e st
ress
, th
e se
ctio
n at
the
hol
e go
vern
s. U
sing
the
ter
min
olog
yof
Tabl
e A
–15–
1 w
e fin
d d/w
=0.
50,
ther
efor
e K
t. =
2.18
. Fr
om
Tabl
e 6–
15,
√ a=
5/S u
t=
5/87
.6=
0.05
71an
dC
kf=
0.10
. Fr
om E
qs.
(6–7
8) a
nd (
6–79
) w
ithr
=0.
375
in,
Kf
=K
t
1+
2 (K
t−
1 )
Kt
√ a √ r
LN( 1,
CK
f
) =2.
18
1+
2 (2.
18−
1 )
2.18
0.05
71√ 0.
375
LN
(1,0.
10)
=1.
98L
N(1
,0.
10)
The
str
ess
at th
e ho
le is
�=
Kf
F A=
1.98
LN
(1,0.
10)10
00L
N(1
,0.
12)
0.25
(0.7
5)
σ̄=
1.98
1000
0.25
(0.7
5)10
−3=
10.5
6kp
si
Cσ
=(0
.102
+0.
122)1/
2=
0.15
6
so s
tres
s ca
n be
exp
ress
ed a
s �
=10
.56L
N(1
,0.
156)
kpsi
.34
The
end
uran
ce l
imit
is c
onsi
dera
bly
grea
ter
than
the
loa
d-in
duce
d st
ress
, in
dica
t-in
g th
at fi
nite
lif
e is
not
a p
robl
em. F
or i
nter
feri
ng l
ogno
rmal
-log
norm
al d
istr
ibut
ions
,E
q.(5
–43)
, p. 2
42, g
ives
z=
−ln
( S̄ e σ̄
√ 1+
C2 σ
1+
C2 S e
)√ ln
[( 1+
C2 S e
)( 1+
C2 σ
)]=−
ln
⎛ ⎝31.
4
10.5
6
√ 1+
0.15
62
1+
0.19
52
⎞ ⎠√ ln
[(1
+0.
1952
)(1
+0.
1562
)]=
−4.3
7
From
Tab
le A
–10
the
prob
abili
ty o
f fa
ilure
pf
=�
(−4.
37)=
.000
006
35,
and
the
relia
bilit
y is
Ans
wer
R=
1−
0.00
000
635
=0.
999
993
65
34N
ote
that
ther
e is
a s
impl
ifica
tion
here
. The
are
a is
not
a de
term
inis
tic q
uant
ity. I
t will
hav
e a
stat
istic
aldi
stri
butio
n al
so. H
owev
er n
o in
form
atio
n w
as g
iven
her
e, a
nd s
o it
was
trea
ted
as b
eing
det
erm
inis
tic.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
333
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
(b)
The
rot
ary
endu
ranc
e te
sts
are
desc
ribe
d by
S′ e=
40L
N(1
,0.
05)
kpsi
who
se m
ean
isle
ssth
an th
e pr
edic
ted
mea
n in
par
t a. T
he m
ean
endu
ranc
e st
reng
th S̄
eis
S̄ e=
0.81
6(0.
869)
40=
28.4
kpsi
CSe
=(0
.058
2+
0.12
52+
0.05
2)1/
2=
0.14
7
so t
he e
ndur
ance
str
engt
h ca
n be
exp
ress
ed a
s S e
=28
.3L
N(1
,0.
147)
kpsi
. Fr
omE
q.(5
–43)
,
z=
−ln
⎛ ⎝28.
4
10.5
6
√ 1+
0.15
62
1+
0.14
72
⎞ ⎠√ ln
[(1
+0.
1472
)(1
+0.
1562
)]=
−4.6
5
Usi
ngTa
ble
A–1
0,w
ese
eth
epr
obab
ility
offa
ilure
pf=
�(−
4.65
)=
0.00
000
171
,an
d
R=
1−
0.00
000
171
=0.
999
998
29
anin
crea
se!
The
redu
ctio
nin
the
prob
abili
tyof
failu
reis
(0.0
0000
171
−0.
000
006
35)/
0.00
000
635
=−0
.73 ,
are
duct
ion
of73
perc
ent.
We
are
anal
yzin
gan
exis
ting
desi
gn,s
oin
part
(a)
the
fact
orof
safe
tyw
asn̄
=S̄/
σ̄=
31.4
/10
.56
=2.
97.I
npa
rt(b
)n̄
=28
.4/
10.5
6=
2.69
,ade
crea
se.T
hise
xam
ple
give
syou
the
oppo
rtun
ityto
see
the
role
ofth
ede
sign
fact
or.G
iven
know
ledg
eof
S̄ ,C
S,σ̄
,Cσ,a
ndre
liabi
lity
(thr
ough
z),t
hem
ean
fact
orof
safe
ty(a
sa
desi
gnfa
ctor
)sep
arat
esS̄
and
σ̄so
that
the
relia
bilit
ygo
alis
achi
eved
.K
now
ing
n̄al
one
says
noth
ing
abou
tth
epr
obab
ility
offa
ilure
.Loo
king
atn̄
=2.
97an
dn̄
=2.
69sa
ysno
thin
gab
outt
here
spec
tive
prob
abili
ties
offa
ilure
.The
test
sdi
dno
tred
uce
S̄ esi
gnifi
cant
ly,b
utre
duce
dth
eva
riat
ion
CS
such
that
the
relia
bilit
yw
asin
crea
sed.
Whe
n a
mea
n de
sign
fac
tor
(or
mea
n fa
ctor
of
safe
ty)
defi
ned
as S̄
e/σ̄
is s
aid
tobe
sile
nton
mat
ters
of
freq
uenc
y of
fai
lure
s, i
t m
eans
tha
t a
scal
ar f
acto
r of
saf
ety
by i
tsel
f do
es n
ot o
ffer
any
inf
orm
atio
n ab
out
prob
abil
ity
of f
ailu
re.
Nev
erth
eles
s,so
me
engi
neer
s le
t th
e fa
ctor
of
safe
ty s
peak
up,
and
the
y ca
n be
wro
ng i
n th
eir
conc
lusi
ons.
330
Mec
hani
cal E
ngin
eerin
g D
esig
n
As
reve
alin
g as
Ex.
6–1
9 is
con
cern
ing
the
mea
ning
(an
d la
ck o
f m
eani
ng)
of a
desi
gn f
acto
r or
fac
tor
of s
afet
y, le
t us
rem
embe
r th
at th
e ro
tary
test
ing
asso
ciat
ed w
ithpa
rt (
b) c
hang
ed n
othi
ngab
out
the
part
, bu
t on
ly o
ur k
now
ledg
e ab
out
the
part
. T
hem
ean
endu
ranc
e lim
it w
as 4
0 kp
si a
ll th
e tim
e, a
nd o
ur a
dequ
acy
asse
ssm
ent
had
tom
ove
with
wha
t was
kno
wn.
Fluct
uating S
tres
ses
Det
erm
inis
tic f
ailu
re c
urve
s th
at li
e am
ong
the
data
are
can
dida
tes
for
regr
essi
on m
od-
els.
Inc
lude
d am
ong
thes
e ar
e th
e G
erbe
r an
d A
SME
-elli
ptic
for
duc
tile
mat
eria
ls, a
nd,
for
britt
le m
ater
ials
, Sm
ith-D
olan
mod
els,
whi
ch u
se m
ean
valu
es in
thei
r pr
esen
tatio
n.Ju
st a
s th
e de
term
inis
tic f
ailu
re c
urve
s ar
e lo
cate
d by
end
uran
ce s
tren
gth
and
ultim
ate
tens
ile (
or y
ield
) st
reng
th, s
o to
o ar
e st
ocha
stic
fai
lure
cur
ves
loca
ted
by S
ean
d by
Su
t
orS
y.
Figu
re 6
–32,
p. 3
12,
show
s a
para
bolic
Ger
ber
mea
n cu
rve.
We
also
nee
d to
esta
blis
h a
cont
our
loca
ted
one
stan
dard
dev
iatio
n fr
om t
he m
ean.
Sin
ce s
toch
astic
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
334
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g331
curv
es a
re m
ost l
ikel
y to
be
used
with
a r
adia
l loa
d lin
e w
e w
ill u
se th
e eq
uatio
n gi
ven
in T
able
6–7
, p. 2
99, e
xpre
ssed
in te
rms
of th
e st
reng
th m
eans
as
S̄ a=
r2S̄2 u
t
2S̄e
⎡ ⎣ −1
+√ 1
+( 2S̄
e
rS̄u
t
) 2⎤ ⎦(6
–80)
Bec
ause
of
the
posi
tive
corr
elat
ion
betw
een
S ean
dS u
t, w
e in
crem
ent
S̄ eby
CSe
S̄ e,S̄
ut
byC
SutS̄
ut,
and
S̄a
byC
SaS̄ a
, sub
stit
ute
into
Eq.
(6–
80),
and
sol
ve f
or C
Sato
obt
ain
CSa
=(1
+C
Sut)
2
1+
CSe
⎧ ⎨ ⎩−1+√ 1
+[ 2S̄
e(1
+C
Se)
rS̄u
t(1
+C
Sut)
] 2⎫ ⎬ ⎭⎡ ⎣ −
1+√ 1
+( 2S̄
e
rS̄u
t
) 2⎤ ⎦−
1(6
–81)
Equ
atio
n (6
–81)
can
be
view
ed a
s an
inte
rpol
atio
n fo
rmul
a fo
r CSa
, whi
ch fa
lls b
etw
een
CSe
and
CSu
tde
pend
ing
on lo
ad li
ne s
lope
r. N
ote
that
Sa
=S̄ a
LN
(1,C
Sa) .
Sim
ilarl
y, t
he A
SME
-elli
ptic
cri
teri
on o
f Ta
ble
6–8,
p. 3
00, e
xpre
ssed
in
term
s of
its m
eans
is
S̄ a=
rS̄yS̄ e
√ r2S̄2 y
+S̄2 e
(6–8
2)
Sim
ilarl
y, w
e in
crem
ent
S̄ eby
CSe
S̄ e,
S̄ yby
CS
yS̄ y
, an
d S̄ a
byC
SaS̄ a
, su
bstit
ute
into
Eq.
(6–8
2), a
nd s
olve
for
CSa
:
CSa
=(1
+C
Sy)(
1+
CSe
)√ √ √ √r2
S̄2 y+
S̄2 e
r2S̄2 y
(1+
CS
y)2
+S̄2 e
(1+
CSe
)2−
1(6
–83)
Man
y br
ittl
em
ater
ials
fol
low
a S
mith
-Dol
an f
ailu
re c
rite
rion
, wri
tten
dete
rmin
isti-
cally
as
nσa
S e=
1−
nσm/
S ut
1+
nσm/
S ut
(6–8
4)
Exp
ress
ed in
term
s of
its
mea
ns,
S̄ a S̄ e=
1−
S̄ m/
S̄ ut
1+
S̄ m/
S̄ ut
(6–8
5)
For
a ra
dial
load
line
slo
pe o
f r,
we
subs
titut
e S̄ a
/r
for
S̄ man
d so
lve
for
S̄ a, o
btai
ning
S̄ a=
rS̄u
t+
S̄ e2
⎡ ⎣ −1
+√ 1
+4r
S̄ utS̄
e
(rS̄ u
t+
S̄ e)2
⎤ ⎦(6
–86)
and
the
expr
essi
on f
or C
Sais
CSa
=rS̄
ut(
1+
CSu
t)+
S̄ e(1
+C
Se)
2S̄a
·{ −1+√ 1
+4r
S̄ utS̄
e(1
+C
Se)(
1+
CSu
t)
[rS̄ u
t(1
+C
Sut)
+S̄ e
(1+
CSe
)]2
} −1
(6–8
7)
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
335
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
332
Mec
hani
cal E
ngin
eerin
g D
esig
n
EXA
MPLE
6–2
0A
rot
atin
g sh
aft
expe
rien
ces
a st
eady
tor
que
T=
1360
LN
(1,0.
05)
lbf·
in,
and
at a
shou
lder
with
a 1
.1-i
n sm
all
diam
eter
, a
fatig
ue s
tres
s-co
ncen
trat
ion
fact
or K
f=
1.50
LN
(1,0.
11) ,
Kfs
=1.
28L
N(1
,0.
11) ,
and
at
that
loc
atio
n a
bend
ing
mom
ent
ofM
=12
60L
N(1
,0.
05)
lbf·
in. T
he m
ater
ial o
f whi
ch th
e sh
aft i
s m
achi
ned
is h
ot-r
olle
d10
35 w
ith S
ut=
86.2
LN
(1,0.
045)
kpsi
and
Sy
=56
.0L
N(1
,0.
077)
kpsi
. Est
imat
e th
ere
liabi
lity
usin
g a
stoc
hast
ic G
erbe
r fa
ilure
zon
e.
Solu
tion
Est
ablis
h th
e en
dura
nce
stre
ngth
. Fro
m E
qs. (
6–70
) to
(6–
72)
and
Eq.
(6–
20),
p. 2
80,
S′ e=
0.50
6(86
.2)L
N(1
,0.
138)
=43
.6L
N(1
,0.
138)
kpsi
k a=
2.67
(86.
2)−0
.265
LN
(1,0.
058)
=0.
820L
N(1
,0.
058)
k b=
(1.1
/0.
30)−
0.10
7=
0.87
0
k c=
k d=
kf=
LN
(1,0)
S e=
0.82
0LN
(1,0.
058)
0.87
0(43
.6)L
N(1
,0.
138)
S̄ e=
0.82
0(0.
870)
43.6
=31
.1kp
si
CSe
=(0
.058
2+
0.13
82)1/
2=
0.15
0
and
so S
e=
31.1
LN
(1,0.
150)
kpsi
.
Stre
ss(i
n kp
si): σ
a=
32K
fM
a
πd
3=
32(1
.50)
LN
(1,0.
11)1
.26L
N(1
,0.
05)
π(1
.1)3
σ̄a
=32
(1.5
0)1.
26
π(1
.1)3
=14
.5kp
si
Cσ
a=
(0.1
12+
0.05
2)1/
2=
0.12
1
�m
=16
Kfs
Tm
πd
3=
16(1
.28)
LN
(1,0.
11)1
.36L
N(1
,0.
05)
π(1
.1)3
τ̄ m=
16(1
.28)
1.36
π(1
.1)3
=6.
66kp
si
Cτ
m=
(0.1
12+
0.05
2)1/
2=
0.12
1
σ̄′ a=( σ̄
2 a+
3τ̄2 a
) 1/2=
[14.
52+
3(0)
2]1/
2=
14.5
kpsi
σ̄′ m
=( σ̄
2 m+
3τ̄2 m
) 1/2=
[0+
3(6.
66)2
]1/2
=11
.54
kpsi
r=
σ̄′ a
σ̄′ m
=14
.5
11.5
4=
1.26
Stre
ngth
: Fro
m E
qs. (
6–80
) an
d (6
–81)
,
S̄ a=
1.26
286
.22
2(31
.1)
⎧ ⎨ ⎩−1+√ 1
+[ 2(
31.1
)
1.26
(86.
2)
] 2⎫ ⎬ ⎭=28
.9kp
si
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
336
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
CSa
=(1
+0.
045)
2
1+
0.15
0
−1+√ 1
+[
2(31
.1)(
1+
0.15
)
1.26
(86.
2)(1
+0.
045)
] 2
−1+√ 1
+[ 2(
31.1
)
1.26
(86.
2)
] 2−
1=
0.13
4
Rel
iabi
lity
:Si
nce
S a=
28.9
LN
(1,0.
134)
kpsi
an
d �
′ a=
14.5
LN
(1,0.
121)
kpsi
,E
q.(5
–44)
, p. 2
42, g
ives
z=
−ln
( S̄ a σ̄a
√ 1+
C2 σ
a
1+
C2 S a
)√ ln
[( 1+
C2 S a
)( 1+
C2 σ
a
)]=−
ln
⎛ ⎝28.9
14.5
√ 1+
0.12
12
1+
0.13
42
⎞ ⎠√ ln
[(1
+0.
1342
)(1
+0.
1212
)]=
−3.8
3
From
Tab
le A
–10
the
prob
abili
ty o
f fa
ilure
is
pf=
0.00
006
5 , a
nd t
he r
elia
bilit
y is
,ag
ains
t fat
igue
,
Ans
wer
R=
1−
pf=
1−
0.00
006
5=
0.99
993
5
The
cha
nce
of fi
rst-
cycl
e yi
eldi
ng i
s es
timat
ed b
y in
terf
erin
g S
yw
ith�
′ max
. T
hequ
antit
y�
′ max
is f
orm
ed f
rom
�′ a+
�′ m
. T
he m
ean
of �
′ max
isσ̄
′ a+
σ̄′ m
=14
.5+
11.5
4=
26.0
4kp
si.
The
coe
ffici
ent
of v
aria
tion
of t
he s
um i
s 0.
121,
sin
ce b
oth
CO
Vs
are
0.12
1, t
hus
Cσ
max
=0.
121 .
We
inte
rfer
e S
y=
56L
N(1
,0.
077)
kpsi
with
�′ m
ax=
26.0
4LN
(1,0.
121)
kpsi
. The
cor
resp
ondi
ng z
vari
able
is
z=
−ln
⎛ ⎝56
26.0
4
√ 1+
0.12
12
1+
0.07
72
⎞ ⎠√ ln
[(1
+0.
0772
)(1
+0.
1212
)]=
−5.3
9
whi
ch r
epre
sent
s, f
rom
Tab
le A
–10,
a p
roba
bilit
y of
fai
lure
of
appr
oxim
atel
y 0.
0735
8[w
hich
rep
rese
nts
3.58
(10−8
) ] o
f fir
st-c
ycle
yie
ld in
the
fille
t.T
he p
roba
bilit
y of
obs
ervi
ng a
fat
igue
fai
lure
exc
eeds
the
pro
babi
lity
of a
yie
ldfa
ilure
, som
ethi
ng a
det
erm
inis
tic a
naly
sis
does
not
for
esee
and
in
fact
cou
ld l
ead
one
to e
xpec
t a
yiel
d fa
ilure
sho
uld
a fa
ilure
occ
ur.
Loo
k at
the
�′ aS a
inte
rfer
ence
and
the
�′ m
axS
yin
terf
eren
ce a
nd e
xam
ine
the
zex
pres
sion
s. T
hese
con
trol
the
rel
ativ
e pr
oba-
bilit
ies.
A d
eter
min
istic
ana
lysi
s is
obl
ivio
us t
o th
is a
nd c
an m
isle
ad.
Che
ck y
our
sta-
tistic
s te
xt f
or e
vent
s th
at a
re n
ot m
utua
lly e
xclu
sive
, bu
t ar
e in
depe
nden
t, to
qua
ntif
yth
e pr
obab
ility
of
failu
re:
pf=
p(yi
eld)
+p(
fatig
ue)−
p(yi
eld
and
fatig
ue)
=p(
yiel
d)+
p(fa
tigue
)−
p(yi
eld)
p(fa
tigue
)
=0.
358(
10−7
)+
0.65
(10−4
)−
0.35
8(10
−7)0
.65(
10−4
)=
0.65
0(10
−4)
R=
1−
0.65
0(10
−4)=
0.99
993
5
agai
nst e
ither
or
both
mod
es o
f fa
ilure
.
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g333
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
337
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
334
Mec
hani
cal E
ngin
eerin
g D
esig
n
Exa
min
e Fi
g. 6
–38,
whi
ch d
epic
ts th
e re
sults
of
Ex.
6–2
0. T
he p
robl
em d
istr
ibut
ion
ofS e
was
com
poun
ded
of h
isto
rica
l exp
erie
nce
with
S′ e
and
the
unce
rtai
nty
man
ifes
tatio
nsdu
e to
fea
ture
s re
quir
ing
Mar
in c
onsi
dera
tions
. The
Ger
ber
“fai
lure
zon
e” d
ispl
ays
this
.T
he i
nter
fere
nce
with
loa
d-in
duce
d st
ress
pre
dict
s th
e ri
sk o
f fa
ilure
. If
addi
tiona
l in
for-
mat
ion
is k
now
n (R
. R
. M
oore
tes
ting,
with
or
with
out
Mar
in f
eatu
res)
, th
e st
ocha
stic
Ger
ber
can
acco
mm
odat
e to
the
inf
orm
atio
n. U
sual
ly, t
he a
ccom
mod
atio
n to
add
ition
alte
st in
form
atio
n is
mov
emen
t and
con
trac
tion
of th
e fa
ilure
zon
e. I
n its
ow
n w
ay th
e st
o-ch
astic
fai
lure
mod
el a
ccom
plis
hes
mor
e pr
ecis
ely
wha
t th
e de
term
inis
tic m
odel
s an
dco
nser
vativ
e po
stur
es in
tend
. Add
ition
ally
, sto
chas
tic m
odel
s ca
n es
timat
e th
e pr
obab
ility
of f
ailu
re, s
omet
hing
a d
eter
min
istic
app
roac
h ca
nnot
add
ress
.
The
Des
ign F
act
or
in F
atigue
The
desi
gner
,in
envi
sion
ing
how
toex
ecut
eth
ege
omet
ryof
apa
rtsu
bjec
tto
the
impo
sed
cons
trai
nts,
can
begi
nm
akin
ga
prio
ride
cisi
ons
with
out
real
izin
gth
eim
pact
onth
ede
sign
task
.Now
isth
etim
eto
note
how
thes
eth
ings
are
rela
ted
toth
ere
liabi
lity
goal
.T
he m
ean
valu
e of
the
desi
gn f
acto
r is
giv
en b
y E
q. (
5–45
), r
epea
ted
here
as
n̄=
exp
[ −z√ ln
( 1+
C2 n
) +ln√ 1
+C
2 n
] . =ex
p[C
n(−
z+
Cn/2)
](6
–88)
in w
hich
, fro
m T
able
20–
6 fo
r th
e qu
otie
nt n
=S/
�,
Cn
=√ C
2 S+
C2 σ
1+
C2 σ
whe
reC
Sis
the
CO
V o
f th
e si
gnifi
cant
str
engt
h an
d C
σis
the
CO
V o
f th
e si
gnifi
cant
stre
ss a
t th
e cr
itic
al l
ocat
ion.
Not
e th
at n̄
is a
fun
ctio
n of
the
rel
iabi
lity
goa
l (t
hrou
ghz)
and
the
CO
Vs
of th
e st
reng
th a
nd s
tres
s. T
here
are
no
mea
ns p
rese
nt, j
ust m
easu
res
of v
aria
bili
ty.
The
nat
ure
of C
Sin
a f
atig
ue s
itua
tion
may
be
CSe
for
full
y re
vers
edlo
adin
g, o
r C
Saot
herw
ise.
Als
o, e
xper
ienc
e sh
ows
CSe
>C
Sa>
CSu
t, s
o C
Seca
n be
used
as
a co
nser
vativ
e es
tim
ate
of C
Sa. I
f th
e lo
adin
g is
ben
ding
or
axia
l, th
e fo
rm o
f
50 40 30 20 10 00
1020
3040
5060
7080
90
Mea
n Lange
r curv
e
+1Si
gma
curv
e
Mea
nG
erbe
r cur
ve
–1Si
gma
curv
e
Stea
dy s
tres
s co
mpo
nent
�m
, kps
i
Amplitude stress component �a, kpsi
Loa
d lin
e� S
a
� �a
S a_ �a_
Figure
6–3
8
Des
igne
r’s fa
tigue
dia
gram
forE
x. 6
–20.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
338
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g335
�′ a
mig
ht b
e
�′ a
=K
fM
ac
Ior
�′ a
=K
fF A
resp
ectiv
ely.
Thi
s m
akes
the
CO
V o
f �
′ a, n
amel
y C
σ′ a, e
xpre
ssib
le a
s
Cσ
′ a=( C
2 Kf+
C2 F
) 1/2ag
ain
a fu
nctio
n of
var
iabi
litie
s. T
he C
OV
of
S e, n
amel
y C
Se, i
s
CSe
=( C
2 ka+
C2 kc
+C
2 kd+
C2 k
f+
C2 Se
′) 1/2ag
ain,
a f
unct
ion
of v
aria
bilit
ies.
An
exam
ple
will
be
usef
ul.
EXA
MPLE
6–2
1A
stra
pto
bem
ade
from
aco
ld-d
raw
nst
eels
trip
wor
kpie
ceis
toca
rry
afu
llyre
vers
edax
ial
load
F=
LN
(100
0,12
0)lb
fas
show
nin
Fig.
6–39
.C
onsi
dera
tion
ofad
jace
ntpa
rts
esta
blis
hed
the
geom
etry
assh
own
inth
efig
ure,
exce
ptfo
rth
eth
ickn
ess
t.M
ake
ade
cisi
onas
toth
em
agni
tude
ofth
ede
sign
fact
orif
the
relia
bilit
ygo
alis
tobe
0.99
995
,th
enm
ake
ade
cisi
onas
toth
ew
orkp
iece
thic
knes
st.
Solu
tion
Let
us
take
eac
h a
prio
ri d
ecis
ion
and
note
the
cons
eque
nce:
The
se e
ight
a p
rior
i de
cisi
ons
have
qua
ntifi
ed t
he m
ean
desi
gn f
acto
r as
n̄=
2.65
.Pr
ocee
ding
det
erm
inis
tical
ly h
erea
fter
we
wri
te
σ′ a=
S̄ e n̄=
K̄f
F̄
(w−
d)t
from
whi
ch
t=
K̄fn̄
F̄
(w−
d)S̄
e(1
)
A P
riori
Dec
isio
nConse
quen
ce
Use
101
8 C
D s
teel
S̄ ut�
87.
6kp
si,C
Sut�
0.0
655
Func
tion:
Car
ry a
xial
load
CF�
0.1
2,C
kc�
0.1
25R
≥0.
999
95z
� �
3.89
1M
achi
ned
surfa
ces
Cka
� 0
.058
Hol
e cr
itica
l C
Kf�
0.1
0,C
�� a�
(0.1
02�
0.1
22 )1/
2=
0.15
6A
mbi
ent t
empe
ratu
reC
kd�
0C
orre
latio
n m
etho
dC
S� e�
0.13
8H
ole
drille
dC
Se�
(0.0
582
+0.
1252
+0.
1382
)1/2
=0.
195
Cn
� √ √ √ √C
2 Se+
C2 σ
′ a
1+
C2 σ
′ a
=√ 0.
1952
+0.
1562
1+
0.15
62=
0.24
67
n̄�
exp[ −
(−3.
891)√ ln
(1+
0.24
672)+
ln√ 1
+0.
2467
2]
=2.
65
3 8in
D. d
rill
Fa
= 1
000
lbf
Fa
= 1
000
lbf
3 4in
Figure
6–3
9
A s
trap
with
a th
ickn
ess
tis
subj
ecte
d to
a fu
lly re
vers
edax
ial l
oad
of 1
000
lbf.
Exam
ple
6–21
con
sider
s th
eth
ickn
ess
nece
ssar
y to
atta
in a
relia
bilit
y of
0.9
99 9
5 ag
ains
ta
fatig
ue fa
ilure
.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
339
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
To e
valu
ate
the
prec
edin
g eq
uatio
n w
e ne
ed S̄
ean
dK̄
f. T
he M
arin
fac
tors
are
k a=
2.67
S̄−0.2
65u
tL
N(1
,0.
058)
=2.
67(8
7.6)
−0.2
65L
N(1
,0.
058)
k̄ a=
0.81
6
k b=
1
k c=
1.23
S̄−0.0
78u
tL
N(1
,0.
125)
=0.
868L
N(1
,0.
125)
k̄ c=
0.86
8
k̄ d=
k̄ f=
1
and
the
endu
ranc
e st
reng
th is
S̄ e=
0.81
6(1)
(0.8
68)(
1)(1
)0.5
06(8
7.6)
=31
.4kp
si
The
hol
e go
vern
s. F
rom
Tab
le A
–15–
1 w
e fin
d d/w
=0.
50, t
here
fore
Kt=
2.18
. Fro
mTa
ble
6–15
√ a=
5/S̄ u
t=
5/87
.6=
0.05
71,
r=
0.18
75in
. Fr
om E
q. (
6–78
) th
efa
tigue
str
ess
conc
entr
atio
n fa
ctor
is
K̄f
=2.
18
1+
2(2.
18−
1)
2.18
0.05
71√ 0.
1875
=1.
91
The
thic
knes
s t
can
now
be
dete
rmin
ed f
rom
Eq.
(1)
t≥
K̄fn̄
F̄
(w−
d)S
e=
1.91
(2.6
5)10
00
(0.7
5−
0.37
5)31
400
=0.
430
in
Use
1 2-i
n-th
ick
stra
p fo
r th
e w
orkp
iece
. The
1 2-i
n th
ickn
ess
atta
ins
and,
in th
e ro
undi
ngto
ava
ilabl
e no
min
al s
ize,
exc
eeds
the
relia
bilit
y go
al.
The
exam
ple
dem
onst
rate
sth
at,f
ora
give
nre
liabi
lity
goal
,the
fatig
uede
sign
fact
orth
atfa
cilit
ates
itsat
tain
men
tis
deci
ded
byth
eva
riab
ilitie
sof
the
situ
atio
n.Fu
rthe
rmor
e,th
ene
cess
ary
desi
gnfa
ctor
isno
taco
nsta
ntin
depe
nden
tof
the
way
the
conc
eptu
nfol
ds.
Rat
her,
itis
afu
nctio
nof
anu
mbe
rofs
eem
ingl
yun
rela
ted
apr
iori
deci
sion
sth
atar
em
ade
ingi
ving
defin
ition
toth
eco
ncep
t.T
hein
volv
emen
tof
stoc
hast
icm
etho
dolo
gyca
nbe
limite
dto
defin
ing
the
nece
ssar
yde
sign
fact
or.I
npa
rtic
ular
,in
the
exam
ple,
the
desi
gnfa
ctor
isno
tafu
nctio
nof
the
desi
gnva
riab
let;
rath
er,t
follo
ws
from
the
desi
gnfa
ctor
.
6–18
Road M
aps
and I
mport
ant
Des
ign E
quations
for
the
Stre
ss-L
ife
Met
hod
As
stat
ed i
n Se
c. 6
–15,
the
re a
re t
hree
cat
egor
ies
of f
atig
ue p
robl
ems.
The
im
port
ant
proc
edur
es a
nd e
quat
ions
for
det
erm
inis
tic s
tres
s-lif
e pr
oble
ms
are
pres
ente
d he
re.
Com
ple
tely
Rev
ersi
ng S
imple
Loadin
g1
Det
erm
ine
S′ eei
ther
fro
m te
st d
ata
or
p. 2
74S′ e
=
⎧ ⎪ ⎨ ⎪ ⎩0.
5Su
tS u
t≤
200
kpsi
(140
0M
Pa)
100
kpsi
S ut>
200
kpsi
700
MPa
S ut>
1400
MPa
(6–8
)
336
Mec
hani
cal E
ngin
eerin
g D
esig
n
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
340
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g337
2M
odif
yS′ e
to d
eter
min
e S e
.
p. 2
79S e
=k a
k bk c
k dk e
k fS′ e
(6–1
8)
k a=
aSb u
t(6
–19)
Rel
iabili
ty,
%Tr
ansf
orm
ation V
ari
ate
za
Rel
iabili
ty F
act
or
ke
500
1.00
090
1.28
80.
897
951.
645
0.86
899
2.32
60.
814
99.9
3.09
10.
753
99.9
93.
719
0.70
299
.999
4.26
50.
659
99.9
999
4.75
30.
620
Table
6–5
Relia
bilit
y Fa
ctor
s k e
Cor
resp
ondi
ng to
8Pe
rcen
t Sta
ndar
dD
evia
tion
of th
eEn
dura
nce
Limit
Surf
ace
Fact
or
aEx
ponen
tFi
nis
hS u
t,k
psi
S ut,
MPa
b
Gro
und
1.34
1.58
−0.0
85M
achi
ned
or c
old-
draw
n2.
704.
51−0
.265
Hot
-rolle
d14
.457
.7−0
.718
As-f
orge
d39
.927
2.−0
.995
Table
6–2
Para
met
ers
for M
arin
Surfa
ce M
odifi
catio
nFa
ctor
, Eq.
(6–1
9)
Rot
atin
g sh
aft.
For
bend
ing
or to
rsio
n,
p. 2
80k b
=
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩(d/0.
3)−0
.107
=0.
879d
−0.1
070.
11≤
d≤
2in
0.91
d−0
.157
2<
d≤
10in
(d/7.
62)−0
.107
=1.
24d
−0.1
072.
79≤
d≤
51m
m
1.51
d−0
.157
51<
254
mm
(6–2
0)
For
axia
l,
k b=
1(6
–21)
Non
rota
ting
mem
ber.
Use
Tab
le 6
–3,
p. 2
82,
for
d ean
d su
bsti
tute
int
o E
q. (
6–20
)fo
rd.
p. 2
82k c
=
⎧ ⎪ ⎨ ⎪ ⎩1be
ndin
g
0.85
axia
l
0.59
tors
ion
(6–2
6)
p.28
3U
se T
able
6–4
for
kd,
or
k d=
0.97
5+
0.43
2(10
−3)T
F−
0.11
5(10
−5)T
2 F
+0.
104(
10−8
)T3 F−
0.59
5(10
−12)T
4 F(6
–27)
pp.2
84–2
85,
k e
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
341
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
338
Mec
hani
cal E
ngin
eerin
g D
esig
n
pp.2
85–2
86,
k f
3D
eter
min
e fa
tigue
str
ess-
conc
entr
atio
n fa
ctor
, Kf
orK
fs. F
irst
, find
Kt
orK
tsfr
omTa
ble
A–1
5.
p. 2
87K
f=
1+
q(K
t−
1)or
Kfs
=1
+q
(Kts
−1)
(6–3
2)
Obt
ain
qfr
om e
ither
Fig
. 6–2
0 or
6–2
1, p
p. 2
87–2
88.
Alte
rnat
ivel
y, f
or r
ever
sed
bend
ing
or a
xial
load
s,
p.28
8K
f=
1+
Kt−
1
1+
√ a/r
(6–3
3)
For
S ut
in k
psi, √ a
=0.
245
799
−0.
307
794(
10−2
)Su
t
+0.1
5087
4(10
−4)S
2 ut−
0.26
697
8(10
−7)S
3 ut
(6–3
5)
For
tors
ion
for
low
-allo
y st
eels
, inc
reas
e S u
tby
20
kpsi
and
app
ly to
Eq.
(6–
35).
4A
pply
Kf
orK
fsby
eith
erdi
vidi
ng S
eby
it
orm
ulti
plyi
ng i
t w
ith
the
pure
lyre
vers
ing
stre
ss n
otbo
th.
5D
eter
min
e fa
tigu
e li
fe c
onst
ants
aan
db.
If
S ut≥
70kp
si,
dete
rmin
e f
from
Fig.
6–1
8, p
. 277
. If
S ut<
70kp
si, l
et f
=0.
9.
p.27
7a
=(
fS u
t)2/
S e(6
–14)
b=
−[lo
g(f
S ut/
S e)]
/3
(6–1
5)
6D
eter
min
e fa
tigu
e st
reng
th S
fat
Ncy
cles
, or,
Ncy
cles
to
fail
ure
at a
rev
ersi
ngst
ress
σa
(Not
e: th
is o
nly
appl
ies
to p
urel
y re
vers
ing
stre
sses
whe
re σ
m=
0 ).
p. 2
77S
f=
aN
b(6
–13)
N=
(σa/a)
1/b
(6–1
6)
Fluct
uating S
imple
Loadin
gFo
r S e
,Kf
orK
fs, s
ee p
revi
ous
subs
ectio
n.
1C
alcu
late
σm
and
σa. A
pply
Kf
to b
oth
stre
sses
.
p.29
3σ
m=
(σm
ax+
σm
in)/
2σ
a=
|σ max
−σ
min|/2
(6–3
6)
2A
pply
to a
fat
igue
fai
lure
cri
teri
on, p
. 298
σm
≥0
Sode
rbur
gσ
a/
S e+
σm/
S y=
1/n
(6–4
5)
mod
-Goo
dman
σa/
S e+
σm/
S ut=
1/n
(6–4
6)
Ger
ber
nσa/
S e+
(nσ
m/
S ut)
2=
1(6
–47)
ASM
E-e
llipt
ic(σ
a/
S e)2
+(σ
m/
S ut)
2=
1/n2
(6–4
8)
σm
<0
p.29
7σ
a=
S e/
n
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
342
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g339
Tors
ion.
Use
the
sam
e eq
uatio
ns a
s ap
ply
for σ
m≥
0 , e
xcep
t re
plac
e σ
man
dσ
aw
ithτ m
and
τ a,
use
k c=
0.59
for
S e,
repl
ace
S ut
with
S su
=0.
67S u
t[E
q. (
6–54
), p
. 30
9],
and
repl
ace
S yw
ithS s
y=
0.57
7Sy
[Eq.
(5–
21),
p. 2
17]
3C
heck
for
loca
lized
yie
ldin
g.
p.29
8σ
a+
σm
=S y
/n
(6–4
9)
or, f
or to
rsio
n,τ a
+τ m
=0.
577S
y/
n
4Fo
r fin
ite-l
ife
fatig
ue s
tren
gth
(see
Ex.
6–1
2, p
p. 3
05–3
06),
mod
-Goo
dman
Sf
=σ
a
1−
(σm/
S ut)
Ger
ber
Sf
=σ
a
1−
(σm/
S ut)
2
If d
eter
min
ing
the
finite
lif
e N
with
a f
acto
r of
saf
ety
n, s
ubst
itute
Sf/
nfo
rσ
ain
Eq.
(6–1
6). T
hat i
s,
N=( S
f/
n
a
) 1/b
Com
bin
ation o
f Lo
adin
g M
odes
See
prev
ious
sub
sect
ions
for
ear
lier
defin
ition
s.
1C
alcu
late
von
Mis
es s
tres
ses
for
alte
rnat
ing
and
mid
rang
e st
ress
sta
tes,
σ′ a
and
σ′ m.
Whe
n de
term
inin
g S e
, do
not u
se k
cno
r div
ide
by K
for
Kfs
. App
ly K
fan
d/or
Kfs
dire
ctly
to
each
spe
cifi
c al
tern
atin
g an
d m
idra
nge
stre
ss. I
f ax
ial
stre
ss i
s pr
esen
tdi
vide
the
alt
erna
ting
axi
al s
tres
s by
kc=
0.85
. For
the
spe
cial
cas
e of
com
bine
dbe
ndin
g, to
rsio
nal s
hear
, and
axi
al s
tres
ses
p.31
0
σ′ a={ [ (K
f) b
end
ing(σ
a) b
end
ing+
(Kf) a
xial
(σa) a
xial
0.85
] 2 +3[ (K
fs) t
orsi
on(τ
a) t
orsi
on] 2} 1/2
(6–5
5)
σ′ m
={ [ (K
f) b
end
ing(σ
m) b
end
ing+
(Kf) a
xial
(σm
) axi
al] 2 +
3[ (K
fs) t
orsi
on(τ
m) t
orsi
on] 2} 1
/2
(6–5
6)
2A
pply
str
esse
s to
fat
igue
cri
teri
on [
see
Eq.
(6–
45)
to (
6–48
), p
. 338
in
prev
ious
subs
ectio
n].
3C
onse
rvat
ive
chec
k fo
r lo
caliz
ed y
ield
ing
usin
g vo
n M
ises
str
esse
s.
p.29
8σ
′ a+
σ′ m
=S y
/n
(6–4
9)
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
343
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
PRO
BLE
MS
Prob
lem
s 6–
1 to
6–3
1 ar
e to
be
solv
ed b
y de
term
inis
tic m
etho
ds. P
robl
ems
6–32
to
6–38
are
to
be s
olve
d by
sto
chas
tic m
etho
ds. P
robl
ems
6–39
to 6
–46
are
com
pute
r pr
oble
ms .
Det
erm
inis
tic
Pro
ble
ms
6–1
A1 4
-in
drill
rod
was
hea
t-tr
eate
d an
d gr
ound
. The
mea
sure
d ha
rdne
ss w
as fo
und
to b
e 49
0 B
rine
ll.E
stim
ate
the
endu
ranc
e st
reng
th if
the
rod
is u
sed
in r
otat
ing
bend
ing.
6–2
Est
imat
eS′ e
for
the
follo
win
g m
ater
ials
:(a
) AIS
I 10
20 C
D s
teel
.(b
) AIS
I 10
80 H
R s
teel
.(c
) 20
24 T
3 al
umin
um.
(d) A
ISI
4340
ste
el h
eat-
trea
ted
to a
tens
ile s
tren
gth
of 2
50 k
psi.
6–3
Est
imat
e th
e fa
tigue
str
engt
h of
a ro
tatin
g-be
am s
peci
men
mad
e of
AIS
I 102
0 ho
t-ro
lled
stee
l cor
-re
spon
ding
to
a lif
e of
12.
5 ki
locy
cles
of
stre
ss r
ever
sal.
Als
o, e
stim
ate
the
life
of t
he s
peci
men
corr
espo
ndin
g to
a s
tres
s am
plitu
de o
f 36
kps
i. T
he k
now
n pr
oper
ties
are
S ut=
66.2
kpsi
,σ0
=11
5kp
si,m
=0.
22, a
nd ε
f=
0.90
.
6–4
Der
ive
Eq.
(6–1
7).
For
the
spec
imen
ofPr
ob.
6–3,
estim
ate
the
stre
ngth
corr
espo
ndin
gto
500
cycl
es.
6–5
For
the
inte
rval
103
≤N
≤10
6cy
cles
, de
velo
p an
exp
ress
ion
for
the
axia
l fa
tigue
str
engt
h(S
′ f) a
xfo
r th
e po
lishe
d sp
ecim
ens
of 4
130
used
to
obta
in F
ig.
6–10
. T
he u
ltim
ate
stre
ngth
is
S ut=
125
kpsi
and
the
endu
ranc
e lim
it is
(S′ e)
ax
=50
kpsi
.
6–6
Est
imat
e th
e en
dura
nce
stre
ngth
of
a 32
-mm
-dia
met
er r
od o
f AIS
I 10
35 s
teel
hav
ing
a m
achi
ned
finis
h an
d he
at-t
reat
ed to
a te
nsile
str
engt
h of
710
MPa
.
6–7
Two
stee
lsar
ebe
ing
cons
ider
edfo
rm
anuf
actu
reof
as-f
orge
dco
nnec
ting
rods
.One
isA
ISI
4340
Cr-
Mo-
Nis
teel
capa
ble
ofbe
ing
heat
-tre
ated
toa
tens
ilest
reng
thof
260
kpsi
.The
othe
ris
apl
ain
car-
bon
stee
lAIS
I104
0w
ithan
atta
inab
leS u
tof
113
kpsi
.Ife
ach
rod
isto
have
asi
zegi
ving
aneq
uiva
-le
ntdi
amet
erd e
of0.
75in
,is
ther
ean
yad
vant
age
tous
ing
the
allo
yst
eelf
orth
isfa
tigue
appl
icat
ion?
6–8
A s
olid
rou
nd b
ar, 2
5 m
m in
dia
met
er, h
as a
gro
ove
2.5-
mm
dee
p w
ith a
2.5
-mm
rad
ius
mac
hine
din
to i
t. T
he b
ar i
s m
ade
of A
ISI
1018
CD
ste
el a
nd i
s su
bjec
ted
to a
pur
ely
reve
rsin
g to
rque
of
200
N ·
m. F
or th
e S-
Ncu
rve
of th
is m
ater
ial,
let
f=
0.9 .
(a)
Est
imat
e th
e nu
mbe
r of
cyc
les
to f
ailu
re.
(b)
If th
e ba
r is
als
o pl
aced
in a
n en
viro
nmen
t with
a te
mpe
ratu
re o
f 45
0◦ C, e
stim
ate
the
num
ber
of c
ycle
s to
fai
lure
.
6–9
A s
olid
squ
are
rod
is c
antil
ever
ed a
t on
e en
d. T
he r
od i
s 0.
8 m
lon
g an
d su
ppor
ts a
com
plet
ely
reve
rsin
g tr
ansv
erse
loa
d at
the
oth
er e
nd o
f ±1
kN. T
he m
ater
ial
is A
ISI
1045
hot
-rol
led
stee
l.If
the
rod
mus
t su
ppor
t th
is l
oad
for
104
cycl
es w
ith a
fac
tor
of s
afet
y of
1.5
, w
hat
dim
ensi
onsh
ould
the
squ
are
cros
s se
ctio
n ha
ve?
Neg
lect
any
str
ess
conc
entr
atio
ns a
t th
e su
ppor
t en
d an
das
sum
e th
at f
=0.
9 .
6–1
0A
rec
tang
ular
bar
is
cut
from
an
AIS
I 10
18 c
old-
draw
n st
eel
flat.
The
bar
is
60 m
m w
ide
by10
mm
thi
ck a
nd h
as a
12-
mm
hol
e dr
illed
thr
ough
the
cen
ter
as d
epic
ted
in T
able
A–1
5–1.
The
bar
isco
ncen
tric
ally
load
ed in
pus
h-pu
ll fa
tigue
by
axia
l for
ces
Fa, u
nifo
rmly
dis
trib
uted
acr
oss
the
wid
th.
Usi
ng a
des
ign
fact
or o
f n d
=1.
8 , e
stim
ate
the
larg
est
forc
e F
ath
at c
an b
e ap
plie
dig
nori
ng c
olum
n ac
tion.
6–1
1B
eari
ng r
eact
ions
R1
and
R2
are
exer
ted
on t
he s
haft
sho
wn
in t
he fi
gure
, w
hich
rot
ates
at
1150
rev/
min
and
sup
port
s a
10-k
ip b
endi
ng f
orce
. U
se a
109
5 H
R s
teel
. Sp
ecif
y a
diam
eter
dus
ing
a de
sign
fac
tor
of n
d=
1.6
for
a lif
e of
3 m
in. T
he s
urfa
ces
are
mac
hine
d.
340
Mec
hani
cal E
ngin
eerin
g D
esig
n
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
344
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g341
6–1
2A
bar
of
stee
l ha
s th
e m
inim
um p
rope
rtie
s S e
=27
6M
Pa,
S y=
413
MPa
, and
Su
t=
551
MPa
.T
he b
ar is
sub
ject
ed to
a s
tead
y to
rsio
nal s
tres
s of
103
MPa
and
an
alte
rnat
ing
bend
ing
stre
ss o
f17
2 M
Pa. F
ind
the
fact
or o
f sa
fety
gua
rdin
g ag
ains
t a s
tatic
fai
lure
, and
eith
er th
e fa
ctor
of
safe
-ty
gua
rdin
g ag
ains
t a f
atig
ue f
ailu
re o
r th
e ex
pect
ed li
fe o
f th
e pa
rt. F
or th
e fa
tigue
ana
lysi
s us
e:(a
) M
odifi
ed G
oodm
an c
rite
rion
.(b
) G
erbe
r cr
iteri
on.
(c) A
SME
-elli
ptic
cri
teri
on.
6–1
3R
epea
t Pro
b. 6
–12
but w
ith a
ste
ady
tors
iona
l str
ess
of 1
38 M
Pa a
nd a
n al
tern
atin
g be
ndin
g st
ress
of 6
9 M
Pa.
6–1
4R
epea
t Pr
ob.
6–12
but
with
a s
tead
y to
rsio
nal
stre
ss o
f 10
3 M
Pa,
an a
ltern
atin
g to
rsio
nal
stre
ssof
69
MPa
, and
an
alte
rnat
ing
bend
ing
stre
ss o
f 83
MPa
.
6–1
5R
epea
t Pro
b. 6
–12
but w
ith a
n al
tern
atin
g to
rsio
nal s
tres
s of
207
MPa
.
6–1
6R
epea
t Pro
b. 6
–12
but w
ith a
n al
tern
atin
g to
rsio
nal s
tres
s of
103
MPa
and
a s
tead
y be
ndin
g st
ress
of 1
03 M
Pa.
6–1
7T
he c
old-
draw
n A
ISI
1018
ste
el b
ar s
how
n in
the
figu
re i
s su
bjec
ted
to a
n ax
ial
load
fluc
tuat
ing
betw
een
800
and
3000
lbf
. E
stim
ate
the
fact
ors
of s
afet
y n
yan
dn
fus
ing
(a)
a G
erbe
r fa
tigue
failu
re c
rite
rion
as
part
of
the
desi
gner
’s f
atig
ue d
iagr
am, a
nd (
b) a
n A
SME
-elli
ptic
fat
igue
fai
l-ur
e cr
iteri
on a
s pa
rt o
f th
e de
sign
er’s
fat
igue
dia
gram
.
Prob
lem
6–1
7
1 4in
D.
3 81 in in
Prob
lem
6–2
0
16 in
3 8in
D.
Fm
ax =
30
lbf
Fm
in =
15
lbf
6–1
8R
epea
t Pro
b. 6
–17,
with
the
load
fluc
tuat
ing
betw
een
−800
and
3000
lbf.
Ass
ume
no b
uckl
ing.
6–1
9R
epea
t Pro
b. 6
–17,
with
the
load
fluc
tuat
ing
betw
een
800
and
−300
0lb
f. A
ssum
e no
buc
klin
g.
6–2
0T
he fi
gure
sho
ws
a fo
rmed
rou
nd-w
ire
cant
ileve
r sp
ring
sub
ject
ed t
o a
vary
ing
forc
e. T
he h
ard-
ness
tes
ts m
ade
on 2
5 sp
ring
s ga
ve a
min
imum
har
dnes
s of
380
Bri
nell.
It
is a
ppar
ent
from
the
mou
ntin
g de
tails
that
ther
e is
no
stre
ss c
once
ntra
tion.
A v
isua
l ins
pect
ion
of th
e sp
ring
s in
dica
tes
dd
d/1
0 R
.
d/5
R.
1.5d 1
in
R1
R2
F =
10
kip
12 in
6 in
6 in
Prob
lem
6–1
1
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
345
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
that
the
sur
face
fini
sh c
orre
spon
ds c
lose
ly t
o a
hot-
rolle
d fin
ish.
Wha
t nu
mbe
r of
app
licat
ions
is
likel
y to
cau
se f
ailu
re?
Solv
e us
ing:
(a)
Mod
ified
Goo
dman
cri
teri
on.
(b)
Ger
ber
crite
rion
.
6–2
1T
he f
igur
e is
a d
raw
ing
of a
3-
by 1
8-m
m la
tchi
ng s
prin
g. A
pre
load
is o
btai
ned
duri
ng a
ssem
-bl
y by
shi
mm
ing
unde
r th
e bo
lts
to o
btai
n an
est
imat
ed i
niti
al d
efle
ctio
n of
2 m
m. T
he l
atch
-in
g op
erat
ion
itse
lf r
equi
res
an a
ddit
iona
l de
flec
tion
of
exac
tly
4 m
m. T
he m
ater
ial
is g
roun
dhi
gh-c
arbo
n st
eel,
bent
the
n ha
rden
ed a
nd t
empe
red
to a
min
imum
har
dnes
s of
490
Bhn
. The
radi
us o
f th
e be
nd i
s 3
mm
. E
stim
ate
the
yiel
d st
reng
th t
o be
90
perc
ent
of t
he u
ltim
ate
stre
ngth
.(a
) Fi
nd th
e m
axim
um a
nd m
inim
um la
tchi
ng f
orce
s.(b
) Is
it li
kely
the
spri
ng w
ill f
ail i
n fa
tigue
? U
se th
e G
erbe
r cr
iteri
on.
342
Mec
hani
cal E
ngin
eerin
g D
esig
n
Prob
lem
6–2
1D
imen
sions
in m
illim
eter
s
100
3
18
Sect
ion
A–A
AA
F
6–2
2R
epea
t Pro
b. 6
–21,
par
t b, u
sing
the
mod
ified
Goo
dman
cri
teri
on.
6–2
3T
hefig
ure
show
sth
efr
ee-b
ody
diag
ram
ofa
conn
ectin
g-lin
kpo
rtio
nha
ving
stre
ssco
ncen
trat
ion
atth
ree
sect
ions
.The
dim
ensi
ons
are
r=
0.25
in,d
=0.
75in
,h=
0.50
in,w
1=
3.75
in,a
ndw
2=
2.5
in.
The
forc
esF
fluct
uate
betw
een
ate
nsio
nof
4ki
pan
da
com
pres
sion
of16
kip.
Neg
lect
colu
mn
actio
nan
dfin
dth
ele
astf
acto
rof
safe
tyif
the
mat
eria
lis
cold
-dra
wn
AIS
I10
18st
eel.
Prob
lem
6–2
3F
F
h
w1
w2r
A Ad
Sect
ionA–A
6–2
4T
he to
rsio
nal c
oupl
ing
in th
e fig
ure
is c
ompo
sed
of a
cur
ved
beam
of
squa
re c
ross
sec
tion
that
isw
elde
d to
an
inpu
t sha
ft a
nd o
utpu
t pla
te. A
torq
ue is
app
lied
to th
e sh
aft a
nd c
ycle
s fr
om z
ero
toT
. T
he c
ross
sec
tion
of t
he b
eam
has
dim
ensi
ons
of 5
by
5 m
m,
and
the
cent
roid
al a
xis
of t
hebe
am d
escr
ibes
a c
urve
of
the
form
r=
20+
10θ/π
, w
here
ran
dθ
are
in m
m a
nd r
adia
ns,
resp
ectiv
ely
(0≤
θ≤
4π).
The
cur
ved
beam
has
a m
achi
ned
surf
ace
with
yie
ld a
nd u
ltim
ate
stre
ngth
val
ues
of 4
20 a
nd 7
70 M
Pa, r
espe
ctiv
ely.
(a)
Det
erm
ine
the
max
imum
allo
wab
le v
alue
of
Tsu
ch th
at th
e co
uplin
g w
ill h
ave
an in
finite
life
with
a f
acto
r of
saf
ety,
n=
3 , u
sing
the
mod
ified
Goo
dman
cri
teri
on.
(b)
Rep
eat p
art (
a) u
sing
the
Ger
ber
crite
rion
.(c
) U
sing
Tfo
und
in p
art (
b), d
eter
min
e th
e fa
ctor
of
safe
ty g
uard
ing
agai
nst y
ield
.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
346
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g343
6–2
5R
epea
t Pro
b. 6
–24
igno
ring
cur
vatu
re e
ffec
ts o
n th
e be
ndin
g st
ress
.
6–2
6In
the
figur
esh
own,
shaf
tA,m
ade
ofA
ISI1
010
hot-
rolle
dst
eel,
isw
elde
dto
afix
edsu
ppor
tand
issu
bjec
ted
tolo
adin
gby
equa
land
oppo
site
forc
esF
via
shaf
tB.A
theo
retic
alst
ress
conc
entr
atio
nK
tsof
1.6
isin
duce
dby
the
3-m
mfil
let.
The
leng
thof
shaf
tAfr
omth
efix
edsu
ppor
tto
the
con-
nect
ion
atsh
aftB
is1
m.T
helo
adF
cycl
esfr
om0.
5to
2kN
.(a
) Fo
r sh
aft A
, find
the
fact
or o
f sa
fety
for
infin
ite li
fe u
sing
the
mod
ified
Goo
dman
fat
igue
fai
l-ur
e cr
iteri
on.
(b)
Rep
eat p
art (
a) u
sing
the
Ger
ber
fatig
ue f
ailu
re c
rite
rion
.
Prob
lem
6–2
4
T
T
20
5
60
(Dim
ensi
ons
in m
m)
Prob
lem
6–2
6
F
F
25 m
m
Shaf
tBSh
aftA
10 m
m
20 m
m
25 m
m
3 m
mfi
llet
6–2
7A
sch
emat
ic o
f a
clut
ch-t
estin
g m
achi
ne i
s sh
own.
The
ste
el s
haft
rot
ates
at
a co
nsta
nt s
peed
ω.
An
axia
l lo
ad i
s ap
plie
d to
the
sha
ft a
nd i
s cy
cled
fro
m z
ero
to P
. The
tor
que
Tin
duce
d by
the
clut
ch f
ace
onto
the
shaf
t is
give
n by
T=
fP
(D
+d)
4w
here
Dan
dd
are
defin
ed in
the
figur
e an
d f
is th
e co
effic
ient
of
fric
tion
of th
e cl
utch
fac
e. T
hesh
aft i
s m
achi
ned
with
Sy
=80
0M
Pa a
nd S
ut=
1000
MPa
. The
theo
retic
al s
tres
s co
ncen
trat
ion
fact
ors
for
the
fille
t are
3.0
and
1.8
for
the
axia
l and
tors
iona
l loa
ding
, res
pect
ivel
y.(a
) Ass
ume
the
load
var
iatio
n P
is s
ynch
rono
us w
ith s
haft
rot
atio
n. W
ith f
=0.
3 , fi
nd th
e m
ax-
imum
allo
wab
le lo
ad P
such
that
the
shaf
t will
sur
vive
a m
inim
um o
f 10
6cy
cles
with
a f
acto
rof
saf
ety
of 3
. U
se t
he m
odifi
ed G
oodm
an c
rite
rion
. D
eter
min
e th
e co
rres
pond
ing
fact
or o
fsa
fety
gua
rdin
g ag
ains
t yie
ldin
g.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
347
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
(b)
Supp
ose
the
shaf
t is
not
rot
atin
g, b
ut t
he l
oad
Pis
cyc
led
as s
how
n. W
ith f
=0.
3 , fi
nd t
hem
axim
um a
llow
able
load
Pso
that
the
shaf
t will
sur
vive
a m
inim
um o
f 10
6cy
cles
with
a f
ac-
tor
of s
afet
y of
3. U
se t
he m
odifi
ed G
oodm
an c
rite
rion
. Det
erm
ine
the
corr
espo
ndin
g fa
ctor
of s
afet
y gu
ardi
ng a
gain
st y
ield
ing.
344
Mec
hani
cal E
ngin
eerin
g D
esig
n
Prob
lem
6–2
7P
Fric
tion
pad
D =
150
mm
d =
30
mm
R=
3�
6–2
8Fo
r th
e cl
utch
of
Prob
. 6–2
7, th
e ex
tern
al lo
ad P
is c
ycle
d be
twee
n 20
kN
and
80
kN. A
ssum
ing
that
the
shaf
t is
rota
ting
sync
hron
ous
with
the
exte
rnal
load
cyc
le, e
stim
ate
the
num
ber
of c
ycle
sto
fai
lure
. Use
the
mod
ified
Goo
dman
fat
igue
fai
lure
cri
teri
a.
6–2
9A
flat
lea
f sp
ring
has
fluc
tuat
ing
stre
ss o
f σ
max
=42
0M
Pa a
nd σ
min
=14
0M
Pa a
pplie
d fo
r5
(104 )
cycl
es. I
f th
e lo
ad c
hang
es to
σm
ax=
350
MPa
and
σm
in=
−200
MPa
, how
man
y cy
cles
shou
ld t
he s
prin
g su
rviv
e? T
he m
ater
ial
is A
ISI
1040
CD
and
has
a f
ully
cor
rect
ed e
ndur
ance
stre
ngth
of
S e=
200
MPa
. Ass
ume
that
f=
0.9 .
(a)
Use
Min
er’s
met
hod.
(b)
Use
Man
son’
s m
etho
d.
6–3
0A
mac
hine
par
t w
ill b
e cy
cled
at ±4
8kp
si f
or 4
(10
3 ) cy
cles
. The
n th
e lo
adin
g w
ill b
e ch
ange
dto
±38
kpsi
for
6 (
104 )
cycl
es. F
inal
ly, t
he l
oad
will
be
chan
ged
to ±
32kp
si. H
ow m
any
cycl
esof
ope
ratio
n ca
n be
exp
ecte
d at
this
str
ess
leve
l? F
or th
e pa
rt,
S ut=
76kp
si,
f=
0.9 ,
and
has
afu
lly c
orre
cted
end
uran
ce s
tren
gth
of S
e=
30kp
si.
(a)
Use
Min
er’s
met
hod.
(b)
Use
Man
son’
s m
etho
d.
6–3
1A
rota
ting-
beam
spe
cim
en w
ith a
n en
dura
nce
limit
of 5
0 kp
si a
nd a
n ul
timat
e st
reng
th o
f 100
kps
iis
cyc
led
20 p
erce
nt o
f th
e tim
e at
70
kpsi
, 50
perc
ent
at 5
5 kp
si, a
nd 3
0 pe
rcen
t at
40
kpsi
. Let
f=
0.9
and
estim
ate
the
num
ber
of c
ycle
s to
fai
lure
.
Stoch
ast
ic P
roble
ms
6–3
2So
lve
Prob
. 6–
1 if
the
ulti
mat
e st
reng
th o
f pr
oduc
tion
piec
es i
s fo
und
to b
e S u
t=
245L
N(1
,0.
0508
) kps
i.
6–3
3T
he s
ituat
ion
is s
imila
r to
that
of
Prob
. 6–1
0 w
here
in th
e im
pose
d co
mpl
etel
y re
vers
ed a
xial
load
Fa
=15
LN
(1,0.
20)
kN i
s to
be
carr
ied
by t
he l
ink
with
a t
hick
ness
to
be s
peci
fied
by y
ou, t
hede
sign
er.
Use
the
101
8 co
ld-d
raw
n st
eel
of P
rob.
6–1
0 w
ith S
ut=
440L
N(1
,0.
30)
MPa
and
Syt
=37
0LN
(1,0.
061)
. T
he r
elia
bilit
y go
al m
ust
exce
ed 0
.999
. U
sing
the
cor
rela
tion
met
hod,
spec
ify
the
thic
knes
s t.
6–3
4A
sol
id r
ound
ste
el b
ar is
mac
hine
d to
a d
iam
eter
of
1.25
in. A
gro
ove
1 8in
dee
p w
ith a
rad
ius
of1 8
in i
s cu
t in
to t
he b
ar.
The
mat
eria
l ha
s a
mea
n te
nsile
str
engt
h of
110
kps
i. A
com
plet
ely
reve
rsed
ben
ding
mom
ent
M=
1400
lbf· i
n is
app
lied.
Est
imat
e th
e re
liabi
lity.
The
siz
e fa
ctor
shou
ld b
e ba
sed
on th
e gr
oss
diam
eter
. The
bar
rot
ates
.
Bud
ynas
−Nis
bett:
Shi
gley
’s
Mec
hani
cal E
ngin
eeri
ng
Des
ign,
Eig
hth
Editi
on
II. F
ailu
re P
reve
ntio
n6.
Fat
igue
Fai
lure
Res
ultin
g fr
om V
aria
ble
Load
ing
348
© T
he M
cGra
w−H
ill
Com
pani
es, 2
008
Fatig
ue F
ailu
re R
esul
ting
from
Var
iabl
e Lo
adin
g345
6–3
5R
epea
t Pro
b. 6
–34,
with
a c
ompl
etel
y re
vers
ed to
rsio
nal m
omen
t of
T=
1400
lbf· in
app
lied.
6–3
6A
11 4-i
n-di
amet
er h
ot-r
olle
d st
eel b
ar h
as a
1 8-i
n di
amet
er h
ole
drill
ed tr
ansv
erse
ly th
roug
h it.
The
bar
is n
onro
tatin
g an
d is
sub
ject
to a
com
plet
ely
reve
rsed
ben
ding
mom
ent o
f M
=16
00lb
f· in
inth
e sa
me
plan
e as
the
axis
of t
he tr
ansv
erse
hol
e. T
he m
ater
ial h
as a
mea
n te
nsile
str
engt
h of
58
kpsi
.E
stim
ate
the
relia
bilit
y. T
he s
ize
fact
or s
houl
d be
bas
ed o
n th
e gr
oss
size
. Use
Tab
le A
–16
for
Kt.
6–3
7R
epea
tPro
b.6–
36,w
ithth
eba
rsub
ject
toa
com
plet
ely
reve
rsed
tors
iona
lmom
ento
f240
0lb
f·in
.
6–3
8T
he p
lan
view
of
a lin
k is
the
sam
e as
in
Prob
. 6–
23;
how
ever
, th
e fo
rces
Far
e co
mpl
etel
yre
vers
ed, t
he r
elia
bilit
y go
al is
0.9
98, a
nd th
e m
ater
ial p
rope
rtie
s ar
e S u
t=
64L
N(1
,0.
045)
kpsi
and
Sy
=54
LN
(1,0.
077)
kpsi
. Tre
at F
aas
det
erm
inis
tic, a
nd s
peci
fy th
e th
ickn
ess
h.
Com
pute
r Pro
ble
ms
6–3
9A
1 4by
11 2
-in
stee
l ba
r ha
s a
3 4-i
n dr
illed
hol
e lo
cate
d in
the
cen
ter,
muc
h as
is
show
n in
Tabl
eA
–15–
1. T
he b
ar is
sub
ject
ed to
a c
ompl
etel
y re
vers
ed a
xial
load
with
a d
eter
min
istic
load
of 1
200
lbf.
The
mat
eria
l has
a m
ean
ultim
ate
tens
ile s
tren
gth
of S̄
ut=
80kp
si.
(a)
Est
imat
e th
e re
liabi
lity.
(b)
Con
duct
a c
ompu
ter
sim
ulat
ion
to c
onfir
m y
our
answ
er to
par
t a.
6–4
0Fr
om y
our
expe
rien
ce w
ith P
rob.
6–3
9 an
d E
x. 6
–19,
you
obs
erve
d th
at f
or c
ompl
etel
y re
vers
edax
ial a
nd b
endi
ng f
atig
ue, i
t is
poss
ible
to
•O
bser
ve th
e C
OV
s as
soci
ated
with
a p
rior
i des
ign
cons
ider
atio
ns.
•N
ote
the
relia
bilit
y go
al.
•Fi
nd t
he m
ean
desi
gn f
acto
r n̄ d
whi
ch w
ill p
erm
it m
akin
g a
geom
etri
c de
sign
dec
isio
n th
atw
ill a
ttain
the
goal
usi
ng d
eter
min
istic
met
hods
in c
onju
nctio
n w
ith n̄
d.
Form
ulat
ean
inte
ract
ive
com
pute
rpr
ogra
mth
atw
illen
able
the
user
tofin
dn̄ d
.Whi
leth
em
ater
-ia
lpro
pert
ies
S ut,
Sy,a
ndth
elo
adC
OV
mus
tbe
inpu
tby
the
user
,all
ofth
eC
OV
sas
soci
ated
with
�0.
30,k
a,k
c,k
d,a
ndK
fca
nbe
inte
rnal
,and
answ
ers
toqu
estio
nsw
illal
low
Cσ
and
CS,a
sw
ell
asC
nan
dn̄ d
,to
beca
lcul
ated
.Lat
eryo
uca
nad
dim
prov
emen
ts.T
est
your
prog
ram
with
prob
-le
ms
you
have
alre
ady
solv
ed.
6–4
1W
hen
usin
gth
eG
erbe
rfa
tigue
failu
recr
iteri
onin
ast
ocha
stic
prob
lem
,Eqs
.(6–
80)
and
(6–8
1)ar
eus
eful
.The
yar
eal
soco
mpu
tatio
nally
com
plic
ated
.Iti
she
lpfu
lto
have
aco
mpu
ters
ubro
utin
eor
proc
edur
eth
atpe
rfor
ms
thes
eca
lcul
atio
ns.
Whe
nw
ritin
gan
exec
utiv
epr
ogra
m,
and
itis
appr
opri
ate
tofin
dS a
and
CS
a,a
sim
ple
call
toth
esu
brou
tine
does
this
with
am
inim
umof
effo
rt.
Als
o,on
ceth
esu
brou
tine
iste
sted
,iti
sal
way
sre
ady
tope
rfor
m.W
rite
and
test
such
apr
ogra
m.
6–4
2R
epea
t Pro
blem
. 6–4
1 fo
r th
e A
SME
-elli
ptic
fat
igue
fai
lure
locu
s, im
plem
entin
g E
qs. (
6–82
) an
d(6
–83)
.
6–4
3R
epea
tPro
b.6–
41fo
rthe
Smith
-Dol
anfa
tigue
failu
relo
cus,
impl
emen
ting
Eqs
.(6–
86)a
nd(6
–87)
.
6–4
4W
rite
and
test
com
pute
r su
brou
tines
or
proc
edur
es th
at w
ill im
plem
ent
(a)
Tabl
e 6–
2, r
etur
ning
a,b
,C, a
nd k̄
a.
(b)
Equ
atio
n (6
–20)
usi
ng T
able
6–4
, ret
urni
ng k
b.
(c)
Tabl
e 6–
11, r
etur
ning
α,β
,C, a
nd k̄
c.
(d)
Equ
atio
ns (
6–27
) an
d (6
–75)
, ret
urni
ng k̄
dan
dC
kd.
6–4
5W
rite
and
tes
t a
com
pute
r su
brou
tine
or p
roce
dure
tha
t im
plem
ents
Eqs
. (6
–76)
and
(6–
77),
retu
rnin
gq̄
,σ̂q, a
nd C
q.
6–4
6W
rite
and
tes
t a
com
pute
r su
brou
tine
or p
roce
dure
tha
t im
plem
ents
Eq.
(6–
78)
and
Tabl
e 6–
15,
retu
rnin
g√ a
,CK
f, a
nd K̄
f.