-
5/20/2018 Unit1 From Mechanics of Materials by b c Punmia
1/1
Me
ha
n
i
c
al
Pr
o
per
t
ies
o
f
M
a
ter
i
a
l
s
1
.
1
.
I
M
PORTAN
T M
E
C
HA
N
ICAL
PR
O
PERTIES
Th
e
following
ar
e
th
e
mo
s
t
important
m
ec
hanicalpr
o
p
e
rti
e
sof
e
n
g
in
ee
ring
mat
e
rial
s
:
(i)El
a
stici
t
y (ii)
Pl
a
s
t
ici
t
y
(
i
i
i
)
D
u
c
t
il
it
y
(i
v
)
B
rit
t
l
e
n
e
ss
(
v
)
Malleabil i
t
y
(
vi)
T
oughness
(v
ii
)
H
ar
d
ne s
s,
and(v
ii
i)Str
en
gt
h
S
o
m
e
oft
he
abo
v
ep
ro
pe
rti
e
s
cann
o
tbe
m
u
t
ua
ll
y
r
e
co
nc
il
e
d
;hen
c
e
no
m
ate
ri
a
l
can
pos
se
ss
themall
s
imulta
n
eou
s
ly.
Th
ec
riteri
a
of
s
uitability
(or
otherwise)of an
e
n
gineerin
g
material,forming
part
ofeither
a
mahineorastructure,
is
dependentupon
the
possession
ofon
e
or
mo
re
of th
e
above
prope
r
t
i
es.
T
h
e
abov
e
p
r
ope
r
t
i
es
ar
e
a
ssessed ,w
i
th
a
f
ai
r
degre
e
o
f
a
c
cu
r
acy,by
r
e
sorti
ng
t
ome
c
han
i
ca
l
tests .
12
.
EL
A
S
TI
C
ITY
.
W
hen
ex
t
e
r
na
l
f
or
ce
s
a
r
e
app
li
edonabody,
m
ade
of
eng
i
nee
r
ing
m
ate
ri
a
l
s,
the
e
x
t
e
r
n
a
l
f
o
r
c
e
s
t
e
nd
t
o
d
efo
r
m
t
h
e
b
od
y
w
h
i
l
e
t
h
e
m
o
l
e
c
u
l
a
r
f
o
r
c
e
s
a
c
t
i
n
g
be
t
w
een
t
he
mol
e
cu
l
e
s
offe
rresis
tan
c
e
ag
a
i
n
s
td
e
fo
rm
at
i
on
.
Th
e
defo
r
ma
ti
on
o
r
d
is
p
l
a
c
emen
t
of
th
e
pa
rticles
c
on t
i
nu
es
t
i
ll
,
fu
llr
e
sis
tan
c
e
to
t
h
eex
te
r
na
l
fo
rc
e
sisset
up
.
I
f
th
e
fo
rces
a
re
no
w
g
ra
dua
lly
d
i
m
i
n
is
hed
,
t
hebodyw
ill
r
etu
r
n,who
ll
y
or
pa
rtl
yto
its
o
ri
g
i
na
l
shape .E
l
ast
i
c
i
ty
i
s
t
he
pr
o
p
e
r
tybyv
irt
ue
of
whicha
m
at
e
rial deform
e
dund
e
r th
e
load i
s
e
nabl
e
d to r
e
tur
n
t
o
itsoriginal
dim
e
nsionwh
e
n
the load
i
s
remo
ve
d.Ifabod
y
rega
i
ns
completely
itsoriginal
s
hape
,
it
issaid
tobeperfe
c
tly
elastic .Foranypartiularmaterial ,a riticalvalueof theload ,
nown astheelastic l
i
m itmars
the part
i
a
l
break
downof
el
ast
i
c
i
ty b
e
yond wh
i
c
h
remova
l
of
l
oa
d
r
e
su
l
ts
in
a
deg
r
e
e
of p
e
rman
e
nt
defo
r
ma
ti
on
o
r
pe
rm
anent
s
e
t(
F
i
g.1
.
1
)
.
St
ee
l
,
a
l
um
i
n
i
um ,
coppe
r
,
s
to
ne ,
con
r
e
t
e
e
t
c.may
be
con
si
de
r
ed
t
o
bepe
rf
ect
ly
e
l
ast
i
c,
wi
th
i
nce
rt
a
i
n
li
m
its
.
S
t
r
ess
-S
t
r
a
i
n
r
e
l
at
i
o
n
sh
i
p:
The
l
oadpe
r
un
i
ta
r
ea,no
rm
a
l
totheapp
li
ed
l
oad
is
k
nown
a
s
s
t
r
e
s
s
(
p
)
.
S
i
m
i
l
a
r
l
y
,
t
h
e
d
e
f
or
m
a
ti
o
n
pe
r
u
n
i
t
l
eng
t
h
i
n
t
he
d
i
r
e
c
tio
n
of
d
ef
or
m
a
t
i
o
n
i
s
k
n
o
w
n
asstrain(e).
Theelasticprope
r
tiesof materialsused
in
engineering
aredetermined
by
tests
pe
r
fo
r
med
o
nsmal
l
spe
i
mens o
f
ma
t
e
ri
al .
T
hetes
t
s a
r
econduc
t
ed
in
ma
t
e
r
ia
l
s-
t
es
t
ing
-
lab
or
a
t
o
ri
es
equ
i
p
p
ed w
i
t
h
t
e
st
i
n
g
mach
i
nes
c
apa
ble
of
l
oad
i
n
g
th
e
s
p
e
c
i
men
s
i
n g
r
adua
lly
app
li
ed
i
nc
r
ement
s.
and
t
he
r
esu
lti
ngs
tr
e
ss
e
s
andst
r
a
i
n
s
a
r
emeasu
r
ed at a
ll
such
l
oad
i
nc
r
e
m
ents,
till (
he
s
pec
im
en
fai
l
s .F
i
g .
1
.
1shows
o
nesuchs
tr
ess
-
st
r
aind
i
ag
r
am
(
shema
t
i
)
.
I
nF
i
g.1.1
(
a
)
,
t
hespe
i
men
is
lo
aded
o
n
l
yup
t
opo
i
nt
A
,we
ll
w
i
th
i
nthee
l
as
ti
c
li
m
i
t
E
.Whenthe
l
oad,c
orr
esp
o
nd
i
ng
to
point
A,is
g
ra
dual
l
y
r
emo
v
ed
th
e
c
u
rve
fo
ll
o
wst
h
e
s
am
e
path
AO
and
the
s
tr
ai
nc
o
m
p
l
ete
l
y
disappears.
S
uh
a
behaviour
is
knownas
theelasticbehaviour .
In
F
ig.1.1(b),
thespeimen
(i
)
-
5/20/2018 Unit1 From Mechanics of Materials by b c Punmia
2/1
2
M
E
CH
A
N IC
S
OF M
A
T
E
R I
A
L
S
(a)
(
b
)
F IG
.
1.1
.
E
LA
S
T I
C
IT
Y
A
N
D
P
LA
S
T
I
C
IT
Y
i
s
l
oadedupt
o
po
i
nt5
,
beyondt
h
ec
l
ast
i
c
li
m
i
tE.Whenth
e
spec
i
me
n
i
sg
r
adua
ll
yun
l
oad
e
d
,
th
e
c
ur
ve
fo
ll
ows
pathBC ,
res
u
l
t
i
n
g
i
n
a
resi
dua
l
s
tra
i
n
(i
OQ
or
pe
r
manent
s
tra
i
n.Suc
h
a
b
eh
a
v
i
ou
r
of th
e
mate
ri
a
l,
l
oaded
bey
ond th
e
e
las
t
i
c
li
m
i
t
,
is
knowna
s
pa
r
t
i
a
lly
e
l
a
s
t
i
cbehav
i
ou
r.
A
mor
e
detail
e
ddisus
s
ion
of
s
tres
s
-
s
trai
n
urv
e
is
gi
v
e
n
in
2.4 .
Homogene
it
yandIso
tr
opy:
A
mate
ri
al
i
sh
o
mogeneousi
f
ithassame om
p
os
i
t
i
on
throughoutth
e
body .For
s
u
c
h
a
mate
ri
a
l,
th
e
el
a
s
t
i
c pr
o
p
er
t
ie
s a
re
th
e
s
a
meatea
ch
and
e
ver
y
po
i
nt
in
th
e
body
.
It
i
s
i
nte
r
e
s
t
i
ng tonot
e
th
a
t fo
r
a
homogen
e
o
u
smate
ri
a
l
th
e
e
l
a
s
t
ic
p
r
ope
r
t
ies
n
e
e
d
notb
e
t
h
e
s
a
m
e
in
a
ll
t
h
e
d
ir
e
c
t
ion s.I
f
a
m
ate
ria l
is
e
q
ua
llyel
a
stic
i
n
all
th
e
d
ir
e
cti
o
n
s
,
i
tis
sa
id
to
b
e
i
sot
ro
p
i
c .If,howeve
r
,
i
t
i
s
n
ot
equa
l
ly
e
la
st
i
c
in
a
l
l
d
ir
ect
i
ons,L
e
.
i
tpossess
e
s
d
i
ff
e
r
en
t
e
l
as
t
i
c
p
r
o
pe
r
t
i
e
s
i
n
d
i
f
fe
r
en
t
d
i
r
e
c
t
i
o
ns
,
i
t
i
s
ca
l
l
ed
a
n
i
s
o
t
r
o
p
i
c
.
A
t
h
e
o
r
e
t
i
c
a
l
l
y
i
d
ea l
mat
eri
a
lc
ou
l
db
e
e
qua
llyclasticin
a
ll
d
ir
e
c
t
i
on
s,
Le
.
is
ot
r
op
ic.M
an
ystruct
u
r
a
l
mate
ri
a
ls
me
e
t
th
e
r
equ
ir
e
m
ent
s
of
homogene
i
t
y
a
n
d
is
ot
r
op
y .
W
e
s
h
all
b
e
dea
li
n
g
w
i
thon
ly
the
homo
g
eneo
us
and
is
ot
r
op
i
cmate
ri
a
ls
i
n
th
i
s
b
ook
.
1
3
.
PLASTICITY
P
la
st
i
c
i
ty
i
stheconvers
e
of e
l
ast
i
city
.
A
m
ate
ri
a
l
i
n
p
l
ast
ic
stat
e
i
s
pe
rm
anent
l
ydeform
e
d
by
the app
li
cat
i
on
o
f
l
o
a
d
,
and
i
tha
s
notenden
cy
to
re
cover .Eve
ry
e
l
ast
i
cmate
rial
po sses
se
s
t
h
ep
r
o
p
e
rty
o
f
p
la
s
ti
c
it
y
.U
n
d
e
r t
he
a
tion of
larg
e
for
c
es
,
mo
ste
n
gi
n
ee
r
i
n
g
m
at
e
r
i
a
lsbec
o
me
p
l
a
s
t
ic
and
beha
v
e
i
n
a
m
anne
rsimil
a
r
to
a
visc
ou
s
l i
qu
i
d
.
Th
ec
ha
r
a
c
te
ris
t
ic
of
th
e
m
ate
ri
a
l
by
which
i
tundergoes
in
e
lasticstrains
beyo
n
d
tho s
e
atth
e
el
a
stic
limitisknown
a
splasticity .
W
he
n
l
a
r
g
e
d
e
f
o
r
m
a
t
i
o
n
s
o
c
cu
r
i
n
a
d
u
c
t
i
l
e
m
a
t
e
ri
a
l
l
o
aded
in
t
h
e
p
l
as
t
i
c
r
e
g
i
o
n
,
th
e
m
ate
r
i
al
i
s
sai
dtounde
r
go p
l
a
s
t
ic
f
l
ow
.
Th
ep
r
ope
r
t
y
is
pa
r
t
ic
u
l
a
rly
u
s
efu
l
in
th
e
ope
r
a
ti
on
s
of p
res
s
in
g
and
f
orging.,P
l
ast
i
c
i
ty ,
i
s
a
l
sousefu l
in
th
e
d
e
s
i
g
n
of structu
r
almembe
r
s
,
utilisingits
ulti
m
ate
str
en
gt
h.
1
.
4
.
D UCT IL IT
Y
Duct
ilit
y
is
th
e
cha
r
acter
i
st
i
cwh
i
ch
p
e
r
m
i
ts
a
mate
ri
a
l
tobedrawnout
l
ong
i
tud
i
na
ll
y
toa
r
e
d
uced
s
ect
i
on,
unde
rt
he
a
c
t
i
on
o
f
a
t
ens
il
e
f
o
r
ce.
In
a
du
t
il
e
mate
ri
a
l
,
t
he
r
e
for
e,
l
a
r
ge
defo
rm
a
ti
on
is
po
ssi
b
l
e
befo
r
e
ab
s
o
l
ut
e
fa
il
u
re
o
rru
ptu
re
ta
k
e
s
p
lac
e
.
Adu
c
t
ile
m
ate
ri
a
l
m
u
st
,
of
necessity,possess ah
i
ghdeg
r
eeof
pl
astic
i
tyand
st
r
ength.
D
u
ri
ng
d
u
t
i
l
eex
t
ens
i
on,ama
t
e
ri
a
l
show
s
ace
r
ta
i
ndeg
r
eeo
f
e
l
as
ti
i
ty,togethe
r
w
it
hac
o
ns
i
de
r
ab
l
edeg
r
eeo
f
p
l
as
ti
c
i
t
y
.
D
ut
il i
ty
is m
e
a
s
ured i
n
t
h
e
ten
s
il
e
te
s
t of
s
p
ec
ime
n
of th
e
materi
a
l
,
e
ith
e
r
in t
e
r
m
s of p
e
r
c
en
tage e long
a
ti
o
n
-
5/20/2018 Unit1 From Mechanics of Materials by b c Punmia
3/1
M
E
CHA
N
ICAL P
R
OP
E
RT I
ES
O
F
M
ATE
RI
A
L
S
o
ri
n
te
r
m
s
o
f
pe
rc
en t
a
g
e
r
e
du
cti
o
n
i
n
th
e
cr
o
ss-s
e
cti
ona
l
a
r
ea
of th
e
icsis
pe
ci
men
.T
h
e
p
r
ope
r
t
y
ofduct
ili
t
y
i
s
ut
ili
se
d
i
nw
ire
d
r
aw
i
ng .
1
.5.
BR ITT LENESS
B
rittl
eness
i
mp
li
es
l
ac
of
duct
ilit
y.
A
ma
t
e
ri
a
l
is
s
a
id
t
ob
e
b
r
i
t
t
l
e
w
hen
i
t
c
an
no
t
b
e
d
r
aw
n
o
u
t
by
t
ension
to
smal
l
e
r
sec
ti
on
.I
nab
r
i
ttl
ema
t
e ria
l
,
fa
il
ur
e
tak
e
splac
e
underloa
d
w
i
thouts
i
gnif
i
cant
def
orm
a
ti
o
n
.
Bri
tt
l
efra
t
u
r
e
st
a
k
e
pl
ace
wi
t
h
o
ut
w
a
r
n
i
ngand
t
he
p
ro
pe
r
t
y
is
gene
r
a
ll
y
h
i
gh
ly
un
-
des
ir
ab
l
e.
E
xa
m
p
l
e
s
of
b
rit
t
l
e
ma
t
e
ri
a
ls
a
r
e
(i)
cast
iron(ii)
high
carbon
s
t
ee l
,
(iii
)
concrc
t
e
(iv
)
s
t
one ,
(
v
)
g
l
ass ,
(vi)
ce
r
am
i
c
mate
r
ia
l
s,
and
(
v
ii
)many com-
mo
n
m
e
t
a
lli
ca
ll
oys .F
i
g.2.2show
s
a typ
i
ca
l
s
tr
ess-
s
tr
a
i
n
cu
rv
e
f
o
r
a
t
y
pi
ca
l
b
r
i
ttl
e
m
a
t
e
ri
a
l
wh
i
ch
f
a
il
withon
l
ylittleelongationafterproportiona
l
lim it
(
p
o
i
n
t
A
)
i
s
e
xc
eede
d
,
an
d
t
h
e
fr
a
c
tu
r
e
s
tr
e
s
s
(
po
i
n
t
F
)is
the
s
a
m
ea
s
u
l
t
im
a
t
e
s
tr
e
s
s .
O
r
d
i
na
ry
g
l
a
ss
isanearly idea lbrittlemate r
i
alinwhichthe stress-
str
a
inc
u
rv
e
i
n
ten
si
on
is
e
ss
en t
i
a
lly
as
t
r
a
i
gh t
li
ne
,
w
i
th
fa
il
u
re
o
cc
u
rri
n
g
befo
re
an
y
yi
e
l
d
i
ng
takesplace.Thus,
glass
exhibits
almostnodutilitywhatsoever .
1
.
6
.
M ALL
E
AB IL IT
Y
M
a
ll
eab
ili
t
yis
a
p
r
ope
r
t
y
o
f
a
mate
ri
a
lw
h
ic
h
pe
rmi
t
s
th
e
m
ate
ri
a
ls
to
b
e
e
x
tended
i
n
a
ll
d
ir
e
tio
nsw
it
hou
t
r
uptu
r
e.
A
ma
ll
eab
l
ema
t
e
ri
a
l
p
ossessesah
i
ghdeg
r
eeof
pl
ast
i
i
ty,but
notnece
s
sa
ri
lyg
r
ea
t
st
r
eng
t
h .
T
h
is
p
r
ope
r
ty
is
u
til is
ed
i
n
m
any
o
pe
r
a
tio
n
s
suh
a
sf
o
r
g
i
ng,
h
o
t
r
oll ing ,
drop
-
stampingetc
.
1
.
7
.
TOU
G
H N
E
S
S
Toughness
is
thepropertyofa
mate
r
ia
l
whih enables it to absorb energy without fracture.
T
h
i
sp
r
ope
rt
y
i
sve
r
ydesi
r
ab
l
e
i
ncomponents
s
ub
j
e
c
tto
cy
c
lic
or
s
hock
l
o
a
d
i
n
g.T
oughn
e
ss
is
mea
s
u
r
ed
i
n te
rmsof
ene
r
gy
r
e
q
u
ir
ed pe
r
un
i
t
volum
e
of th
e
material ,to
c
aus
e
ruptur
e
under
thea
ti
on
of
g
r
adua
ll
y
i
nc
r
eas
i
ng
t
ens
il
e
l
oad.
T
h
is
ene
r
gy
i
n
l
udesthe
w
o
r
kd
o
ne
upto
t
he
ela
s
ti
c
lim i
t
whi
c
h
is
s
mal
l
in
c
omparisonwith
t
h
e
ene
r
gy subse
q
u
e
n
t
l
y
e
xp
an
d
e
d
.
F
i
g
.
1
.3
s
ho
w
s
t
h
e
s
t
r
e
s
s-st
rai
n
cu
r
ve
s
,
bo
thf
o
r
mil
dst
ee
l
as
wel
l
ashighcarbon steel .Thetoughness is rep-
resented byth
e
a
rea
underth
e
stress-st
r
a
i
ncur
v
e
f
or
the
mate
ri
a
l
.
A
comm
oncompa
r
a
tiv
e
t
est
for toughnes
s
is
th
e
bend te
s
t
i
n wh
ic
h
a
mater
i
a
l
isex
pe
c
te
d
to
sus
ta
i
n
angu
l
a
r
bendin
g
w
ithout
fai
l
u
r
e
.
1
.
8
.
HARDNESS
H
a
r
dne
ssist
h
e
ab
ili
t
y
of
a
mate
ri
a
l
to
r
e
sis
t
i
ndentat
i
o
n
o
rs
u
r
fa
ce
ab
rasi
on
.
S
i
n
ce
t
hese
r
es
i
s
t
ances
a
r
e
notnecessa
r
i
l
y synonym
o
us ,
it
isusua
l
t
o
base
t
he
est
i
mat
io
n
of
t
he
ha
r
dness
F IG
.
1.2 .
ST
R
ESS
STRA IN
CUR V
EF
OR
A
BR ITTLE MAT ER I
A
L
H
I
GHC
A
RBON
U
NI
T
STRA IN-
F IG .
1.
3 .M E
A
SUR E
O
F
T
O
U
G
HN ESS.
-
5/20/2018 Unit1 From Mechanics of Materials by b c Punmia
4/1
4
M ECH
A
N IC S O
F
M
A
T
E
R I
A
L
S
of
a
mate
r
ia
l
on
resis
tan
ce
to
i
nden tat
i
onon
l
y .
T
e
s
t
s
on
ha
r
dnes
s
m
ay
b
e
c
lassi
f
ie
d
i
nto
(i)
sc
r
a
t
ch
tes
t
,and
(i
f
)
i
nden
t
at
i
on
t
est .
T
hesc
r
a
t
h
t
estcons
i
st
s
o
f
p
r
e
s
s
i
nga
l
oaded
d
i
a
mo
nd
i
nto
th
e
s
u
r
fa
ce
of
th
e
s
pe
cim
en
,
and
t
he
n
pu
lli
n
g
t
he
d
i
a
m
ond
s
o
as1
0
m
a
k
e
a
scr
at
c
h
.
T
h
e
h
ar
d
ness
nu
m
b
e
ris
t
h
en
d
ete
rm
i
n
e
d
o
n
t
h
e
ba
sisof
(
/
)lo
ad
r
e
qu ir
ed
to
m
a
k
e
a
scra
t
c
h
of
a
givenw
i
dth ,or(i
i
)the
w
i
dthof
th
e
sc
r
atchmad
e
w
i
thag
i
ven
l
oad .Th
e
i
nd
e
ntat
i
o
n
t
e
st
c
on
s
ist
s
o
f
p
re
s
si
n
g
a
bo
dy
of
s
t
a
n
d
a
r
d
s
h
a
p
e
i
n
t
o
t
h
e
s
u
r
f
a
c
e
o
f
t
he
t
est
s
p
e
c
i
m
e
n
.
I
n
t
he
c
ommon
ly
u
s
e
dB
ri
nne
ll
ha
r
dnesste
s
t
a
harden
e
d
s
tee
l
ba
ll
of
a
g
i
vend
i
ameter
i
s
s
quee
z
e
d
i
n
to
th
e
s
u
rfa
ce
oft
es
t
s
pe
c
ime
n
,
unde
r
a
fixed
standard
load
and
then
surface
area
o
f
the
i
ndent
is
m
e
a
s
u
r
ed
.
B
ri
ne
ll'
s
ha
r
dn
ess
numbe
r(
B
.
H.N .)
i
sthe
n
g
iv
e
n
b
y:
B
.
H
.
N
.
=
-- -
w
h
e
r
e
P
=
Standardload(
N
);
D
=
diameterofsteelball(m
m
)
d
=
d
i
amete
r
ofthe
i
ndent
(
mm
)
1
.
9
.
STR
E
NGTH
This
isthemo
s
timportant
propert
y
ofamaterial
,from
des
i
gn
po
i
nt
o
f
v
i
ew.
T
he
s
t
re
n
g
tho
f
a
m
at
e
r
i
a
l
e
nab
les
i
tto
r
es
i
st
fr
actu
re
unde
r
load
.T
h
el
o
a
d
r
equired
tocau
s
e
f
r
a
c
tu
r
e
,
d
i
v
i
d
e
dbyth
e
a
r
e
a
of
th
e
tes
t
spec
i
men
,
i
ste
r
medasth
e
u
l
t
i
mat
e
st
re
ngthofth
e
mater
i
a
l
,
an
d
is
e
x
p
re
s
se
d
i
nth
e
un
i
tof
s
t
r
es
s
.An
i
mportan t
c
on s
i
de
r
at
i
o
n
in
eng
i
nee
ri
n
g
de
si
gn
is
t
h
e
c
apa
ci
t
y
o
f
t
heob
j
e
c
t
(s
u
c
h
a
s
bu
il
d
i
n
g
str
u
c
tu
r
e
,m
a
c
h
i
ne
,
a
i
r
cr
aft
,
veh
i
c
l
e ,
s
h
ip
et
c
.
)
,
usua
ll
y
r
e
f
e
rr
ed
to
as
st
r
uc
t
u
r
e ,
t
o
supp
or
to
r
tr
ansm
i
tloads.
If
st
r
u
t
u
r
a
l
fa
i
lu
r
e
i
s
t
o be
avo
i
ded
,
th
e
lo
ad
sth
at
a
structur
e
a
ct
u
ally ca
n
sup
p
or
t
mu
s
t
be
g
r
e
a
te
rth
an
th
e
l
o
a
dsit w
il
l
be
r
e
qu
i
r
ed
tosu
s
ta
i
nwhen
i
nse
r
v
ic
e
.
S
i
n
ce
;h
e
ab
ili
tyof
a
st
r
u
c
tu
re
to
r
e
sis
t
l
oads
is
c
a
lle
dst
re
ngth
,
the go
v
e
r
n
i
n
g
cr
i te
ri
on
i
s thatth
e
a
c
tua
l
s
t
re
ngt
h
of
a
s
t
r
uctu
re
mu
st
c
x
c
e
c
dth
e
r
equ
ire
d
s
t
r
ength
.
T
he
r
at
i
o
o
f
t
h
e
a
c
tu
a l
s
t
r
e
n
gthtot
h
e
r
e
qu
i
r
e
d
s
t
r
en
g
th
is call
ed
t
h
e
f
ac
to
rof
s
a
f
e
ty.How
e
ver
,
f
a
i
l
u
r
e
m
a
y
o
c
cu
r
u
n
d
e
r
t
h
e
a
c
t
i
o
n
of
t
e
ns
i
l
e
l
o
a
d
,
c
o
m
p
r
e
s
s
i
v
e
l
o
a
d
o
r
shea
rl
o
a
d
.
H
ence
itisessentialtoknowthe ult
i
mate
s
tr
e
ngthof themateria
l
inea
c
hof thesethree
c
ondition
s
,
an
dt
h
e
th
r
e
e
u
l
t
im
ate
s
t
r
ength
s
a
res
epa
r
ate
ly
dete
rmi
ne
d
e
x
pe
rim
enta
lly .
1
.
10.
ME
CH
A
NI
CS (O
R
STREN
G
T
H
)O
F
MATE
RI
A
L
S
T
h
r
ee
f
undamen ta
l
a
r
e
as
ofeng
i
nee
ri
n
g
me
c
han
ics
(
o
r
app
li
ed
me
c
han
ics
)
a
rc
(i)
S
t
at
ics(ii)
D
y
na
mics
and
(i
ii
)
M
echan
i
cs
(or
st
r
ength)
of
ma
t
e
ri
a
l
s.
St
a
ti
csanddynam
i
cs a
r
edevo
t
edp
ri
ma
ril
y
t
o
t
hestudy
of
the ex
t
endedeffectso
f
f
o
r
ces
on
ri
g
i
dbod
i
e
s,
i
e.th
e
bod
i
e
s
fo
r
wh
ich
the chang
e
in
s
hap
e
(
o
r
defo
r
mat
i
on
s)
c
a
n
be neg
l
ected .
In
c
o
n
t
r
a
s
t
t
o
th
is,m
e
c
h
a
n
icsof m
at
e
ri
a
ls ,c
o
mmon ly
k
n
ow
n
a
s s
t
r
e
ngth
o
f
ma
t
erials
d
ea
ls
wi
t
h
t
h
e
r
e
l
a
t
i
o
n
be
t
w
een
e
x
t
e
r
n
a
ll
y
a
pp
l
i
e
d
l
o
ad
s
a
n
d
t
he
i
r
i
n
t
e
r
n
a
l
e
f
f
e
c
t
s
o
n
s
o
l
i
d
b
od
i
e
s
.T
h
e
solid bod
i
e
s
in
c
ludea
x
iail
y
loadedmember
s,
shaftintor
s
ion
,t
h
i
n
and
t
h
i
c cy
li
nde
rs
and she
ll
s,
beams,
and
c
ol
umns,
a
s
,
w
e
ll
a
s
s
tr
u
t
u
r
e
s
t
hat a
r
ea
s
se
m
b
li
e
s of t
heseco
m
ponen
t
s.
T
he
s
ebod
i
e
s
a
r
e
no
l
onge
r
ass
umed
to
b
eri
g
i
d
;
thedefo
rm
at
i
on
,
howe
v
e
rs
ma
ll,
a
re
o
f
maj
o
ri
nte
r
e
s
t
.
In
ac
tua
l
de
si
gn
,
th
e
eng
i
n
e
e
r
m
u
s
t
c
on
si
de
r
bo
th
d
i
men
si
on
s
and
m
ate
ri
a
l
p
r
ope
r
t
ies
to
s
a
tisfy
th
e
l
equ
ir
ement
s
of
s
t
re
ngth
and
ri
g
i
d
ity
.A
m
a
ch
i
n
e
pa
rt
orst
r
u
c
tureshou
ld
ne
i
ther
b
re
a
k
nor
defor
m
exc
es
s
ively.Th
e
purpos
e
of
s
tudyin
g
str
e
ngt
hof
mat
e
rials istoen
s
ur
e
tha
t
t
h
e
s
t
r
u
ct
u
r
e
u
s
e
d
w ill
b
e
s
afe
aga
i
n
s
t
m
a
xim
um
i
nte
r
na
l
eff
e
c
t
st
ha
t
m
a
y
b
e
p
r
odu
c
ed
b
y
an
y
co
m
bina ti
o
nofl
o
a
d
i
n
g .
-
5/20/2018 Unit1 From Mechanics of Materials by b c Punmia
5/1
2
S
im
p
l
eS
t
resses
a
ndS
t
rain
s
2
.
1
.
SI
M
PLE
S
TR
ESSES
W
hena
body(Le.
strutura
l
element)
i
s
a
c
t
e
d
u
p
o
n
b
y
e
x
t
e
r
n
a
l
f
o
r
c
e
o
r
l
o
a
d
,
i
n
t
e
r
n
a
l
r
es
i
s
t
ing f
or
ce
i
ssetup .
S
uchabody
i
s then
said t
o
beinast
a
te
o
f s
t
r
e
ss ,whe restress
is t
h
e
r
esist
an
ce o ff e
r
ed by t
h
e
body to d
e
fo
r
ma
ti
on
.
F
or further understanding of this interna
l
resis-
tance,considerapr
i
smaticba
r
ABsubjeted
toax
i
a lf
or
cesa
t
theendsasshown
i
nF
i
g .
2
.
1
(
a
)
.
A
p
ri
sma
ti
c
ba
r
is
as
tr
aigh
t
st
r
uctu
r
a
l
member
ofu
n
iform
cross-s
e
ction{A)
t
h
r
ough
ou
titslen
g
th
(
L ).In
o
rd
e
r
t
o
know
theinte
rn
al stressesproducedinthep
ri
smatic
b
a
r,
ta
e a
s
e
t
i
o
n
m
n
n
or
m
a
l
t
o
t
h
e
l
on
g
i
t
u
d
i
n
a
l
a
x
isof t
h
e
ba
r
;
s
u
c
h
a
s
e
ctio
n
is
k
nownas
a
c
r
os
s
-
sec
t
i
on.If w
e
c
on
si
d
e
rthe
e
qu
ili
b
ri
um of c
i
therth
e
l
eftpar
t
o
r
th
e
rig
htparta
t
se
ct
i
o
n
mn
,
taken
as
a f
r
ee
b
o
dy,
t
h
ei
n
tern
a
lresista
n
ce
o
r
the
stress (p
)
of feredbythe
m
o
lec
u
les
against
th
e
e
x
terna
l
for
c
ema
y
be
as
s
umedto
beuniform l
y
distr
i
butedoverthewholearea
o
f
cro
s
s-
s
e
ct
i
o
n
.
T
hen
whe
re
p
=
Inte
r
na
lr
e
sis
tan
ce
= str
e
ss
=
i
nten
si
t
y
o
f
fo
rc
e
.
A = A reaofcro
s
s
-
sectionno r
m
altotheaxis.
A
s
t
hest
r
ess pa
ts
i
nad
ir
ec
tio
npe
r
pend
i
u
l
a
r
t
o
t
he utsu
r
face
it
i
s
r
e
f
e
rr
edto
as
a
n
o
r
m
al
s
tr
e
ss.Sin
ce
th
e
norma l
st
ressp
is
obtain
e
d
b
ydividin
g
th
ea
xial
for
ce
b
yt
he
ros
s
-s
ect
ionalar
e
a,ith
a
sth
e
unit
s
of
for
c
e
perunitarea
,s
u
ch
askN/m
2
o
r
N
/mm
i
SaintVenant'spr
i
nciple
We
have a
s
sumed
above
t
hat
t
he d
i
s
tri
bu
t
i
o
n
of
s
tr
e
s
s
ove
rt
he
c
r
oss -section
m
n
is
un
i
fo
rm .
T
h
is
a
ss
u
m
pt
i
o
n is
b
ase
d
o
nS
aint
V
enant ,
s
p
ri
n
c
ip
l
e.
T
h
is
p
r
in
ci
p
les
tate
s
that
exc
ept
in
th
e
r
eg
i
on of
extr
em
e
end
s
of
a
b
a
r c
a
rryi
n
g
d
ir
e
c
t
l
o
a
d
i
ng
,
th
e
s
t
ress
d
is
t
ri
bu t
i
o
n
o
ver
t
h
e
cr
o
ss-s
e
cti
o
n
i
s
un
i
fo
rm .
L
m
-
B
l
o>
B
t >In
D
l
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fig
:2.
1.S
T
A
TE
OF
S
T RE
SS
(
5
)
-
5/20/2018 Unit1 From Mechanics of Materials by b c Punmia
6/1
6
Cons
i
de
r
asqua
r
eba
r
(Fig
.
2.2
a
)of se
c
tionbxb
,
sub
j
eted
t
o ax
i
a
l
fo
r
ce
P
.
T
he
s
t
res
sd ist
r
ibu t io
n
at
se
ctio
n
min i
,
dist
a
n
t
b
/2f
r
o
m
the
end
i
s shownin
F
ig.2.2 (b),
w
her
e
th
e
m
a
ximu
m
normal
st
r
ess(p a**)
i
s
f
oundto
b
e
e
q
u
al
to
1
.
3
8
7t
i
me
s
t
h
e
a
v
erage
s
tress (pa
v
) . The
s
tress
d
i
s
tri
bu
ti
o
nat
s
e
c
ti
o
n
min
i ,
d
i
s
ta
n
t
b
f
r
o
mt
h
e
e
nd
i
s shown
in
F
i
g .
2.2
(
)
, whe
r
e
p
ma
is found to be
1
.
02 7
p
.Last ly,
at
section
m
3
/
ij
,
d
i
s
t
a
n
t
3
b
/
2
f
r
om
t
he
end (Fig
.
2.2 d)
,
pm*xisfound
tobeequal
topa
v.
Thi
s
il-
lust
r
atesSain tV enan t,s
fam
o
u
s
p
ri
n
i
pa
lof
r
a
pi
dd
i
s-
si
pa
tion
o
f
l
o
calisedst
r
esses.
Hen
cei
na
ll
p
r
a
c
t
ic
a
lc
a
ses
o
f
stress ana
l
ysis,
S
t .
Venan
t
,
s
p
r
iniplecanbesa
f
e
l
y
fo
l-
lo
we
d,
an
d
t
h
e
n
o
rma
l
st
r
ess
distributio
n
given b
y
Eq.2.1
c
anbeas
su
me
d .
2
.
2
.
K
I
NDS OF S
T
R
ESSES
T
he
r
e
a
r
e
t
he
f
ol
low
i
ng
kind
s
o
f
s
tr
esse
s
(
1
)Normal
st
resses
(
i
)T
e
nsile
stress
(ii)Compr
e
s
s
iv
es
tress
(2)
S
hear
s
tre
s
sortangentia
l
s
tress
(
3
)
Bendingstress
(
4
)
Tw
i
st
i
ng
ort
o
rsio
na
l
s
tr
e
ss
(
5
)B
ea
ri
ngs
tr
e
ss
N
orma
l
s
t
resses
M ECH
A
N IC
S
O
F
M
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TER I
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L S
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>
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it
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o
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)STRESSCc)STRESS
(d
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R
ESS
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IST
RI
B
U
T
IO
N
DI
ST
RI
BUT
IO
N
DI
ST
RI
B
U
T
I
O
N
A
T
m
|
n,
A
T
n
jA
T
n
,
(
a
)
F
IG
.
2.2
.
St .
V
E
NANTS
PRINC IPLE
When a
s
tress ats inadire
c
tionperpendi
c
ular tothe
c
utsurfa
c
e
,
it isknown as normal
st
r
ess
or
d
ir
ectst
r
ess.No
r
ma
l
st
r
esse
s
a
r
eof two
t
ype
s
:
(/)
t
ens
il
est
r
ess,and
(ii)
c
o
mp
r
ess
i
ve
str
es
s .
Ten
s
ilestres
s
W
hen
a
bod
y
is
str
et
c
hed
b
y
th
e
fo
rc
e
P ,
a
ss
hown
in
F
i
g
.
2.
1,
th
eres
u
lti
n
g
str
e
ss
e
s
are
t
ensiles
t
resses.
T
hus
t
ensiles
t
ressexistsbe
t
ween
t
wopar
t
so
f
abodywheneahdraws
theothe rt
o
wardsits
e
lf .Suc
h
a
s
tat
e
o
f
s
tressiss
h
o
wn
i
n
Fig .
2
.1
w
h
e
re
-
5/20/2018 Unit1 From Mechanics of Materials by b c Punmia
7/1
SIMPLE
ST
R
E
SSES
AN
D
S
TRA IN
S
7
P
~
=P= %
-
.
(
2.
1)
C
ompress ive
s
t
ress
I
f
t
h
e
f
or
c
e
sa
re
r
e
v
e
rse
d
i
n
dir
e
ct ion ,
ca
u
s
i
n
g
t
he
b
o
dy
to
b
e
c
o
m
p
r
e
ss
e
d
,
w
e
o
b
t
a
i
n
comp
r
e
s
s
i
ve
s
tr
es
s
es .
T
hus,
comp
r
ess
i
ve st
r
ess
e
xis
ts betw
e
e
n
twopa
r
tsof abod
y
wh
en
e
a
ch
p
u
she st
h
e
o
t
h
e
r
f
r
o
m
it .S
u
ch a stateof stress
is
shownin
F
ig .2.3 .where
P
=P=
J
...(2.2)
Shea
r
st
re
ss
S
hea
r
s
tr
e
ss jst
he
o
ne
w
h
i
h
ac
tsp
a
r
a
ll
e
l
o
r
t
angen
ti
a
lt
o
t
hesu
rf
ace .
T
hus,shea
rs
t
r
es
s
exis
t
s
between
th
e
pa
r
t
s
of
a
bod
y
when
thetwo
pa
r
t
s
e
x
e
r
t
e
qua
l
and
oppo
si
t
e
fo
rc
e
s
on
e
a
c
h
o
t
he
r
l
a
t
e
r
a
ll
y
i
n
a
d
i
r
e
c
tio
n
ta
n
g
en
ti
a
l
t
o
t
h
e
i
r
s
u
r
fa
c
e
i
n
c
on
t
a
c
t
.
F
i
g . 2.4
{a
) shows a
ri
ve
t
ed connec
tio
n, whe
r
e the
ri
vet
r
es
i
s
t
s
t
he
s
hea
r
a
r
os
s
i
ts c
r
oss
-s
et
i
ona
l
a
r
ea
(/I)
,when
s
ub
j
ected
to
p
u
llsP
app
li
ed
to
t
hep
l
a
t
e
s
s
o
joi
nted.
U
nde
rt
heac
tio
n
of
t
he
pu
llsP
,
t
he
t
w
op
l
a
t
e
s
will
p
r
e
ss
aga
i
n
st
the
riv
et
i
n
bea
r i
ng,
and
c
o
n
t
a
t
st
r
e
s
se
s
,
ca
ll
ed
b
e
a
ri
ng
s
t
resses
w
ill
b
e
de
v
e
l
op
e
d
a
g
ai
n
s
t
t
he
riv
et
.
A
f
re
e
-
bod
y
d
ia
g
r
amofth
e
riv
et
(
F
i
g
.
2.
4
a
i
i
)
showsthesebea
r
ing s
tr
esses.
T
h
i
s
fr
eebodyd
i
ag
r
amshows
t
hat
t
he
r
e
i
sa
t
endency
t
o
shea
r
the
r
iveta
l
ongc
r
os
s
sect
i
onmn.
F
ro
m
t
he
fr
eebodyd
i
ag
r
am
of
t
hesect
io
nmnof
the
r i
vet
(
F
i
g.2.4a
ii
i
)
,wesee
t
hat
s
hea
r
fo
r
ce
V
at
s
ove
r
the ut su
r
face.In
th
is
pa
r
t
i
c
ul
a
r
c
a
s
e
(
nowna
s
the
ca
s
eo
f
s
i
ng
l
e
shea
r)
,the
shea
r
f
orc
e
V is
equa
l
to
P
.
T
h
is
shea
r
f
or
ce
i
s
,
infact,
t
he
r
esultant
ofthe
s
hea
r
str
esses
di
st
ributed
ove
rt
he
c
ros
s-
sectional
a
r
ea
of
t
he
r
i
ve
t
,
s
h
o
w
n
i
n
F
i
g
.
2
.
4
(
a
i
v
).
m
B
(
n
)
m
P
I
lb
)
M
O .
2.3
.
COMP
R
LSS I
V
h S
TR IP
S
I
,
I
i
11
}
V
m
n
v
O
V
.
l
i)
(ii
)
m
(o )SHEARSTR
E
SS INARIVE
T
EDCONNEC
T
ION
B
,
rm ,
I
"
i
;
"
i
, ,
\
-
>
i
I
D
c
tr
m
. xi n
(
iv )
in
V
(III ) (IV )
i
)
(ii
)
(
b )S
H
EARST
R
ESSI
N
ABOLTE
D
C
O
N
NE
C
TION
FIG .
2.4 .
EXAMPL
ES
O
FD
I
R
E
CT SH
EAR
O
R
SI
M
PL
E
SH
EAR
T
h
e
a
v
e
r
a
g
e
s
hea
rs
t
r
e
sso
n
th
e
cr
o
ss-s
e
c
t
i
o
n
of
t
he
r
i
v
et
is
obt
ain
ed
b
ydiv
i
d
in
g
th
e
sh
e
arfo
r
ce
V
b
yth
e
are
a
ofcross-sect
i
o
n
(A)ofth
e
ri
vet:
-
5/20/2018 Unit1 From Mechanics of Materials by b c Punmia
8/1
8
MECH
A
NICS O
F
M
A
TER I
A
LS
V
P
*
=
1 = 1
.. .(2.3 )
Anot
h
er
p
ra
c
ti
c
al
e
xam
p
l
e
of
shea
r
s
tr
es
sist
he
bolt
e
d
c
on
n
etio
n
sh
ownin
F
ig
.
2.4
(
b
)
,
c
onsi
s
ting
of aflat
bar
A
,
a
c
levis
C
,
and
abolt
B
that
passesthroughholesinthe
bar
and
t
h
e
l
e
v
i
s
.
U
n
d
e
r
t
he
a
c
t
i
o
n
of
p
u
l
l
s
P
,
t
h
e
ba
r
and
t
h
e
c
l
ev
i
s
w
il
l
p
r
e
ss
aga
i
n
s
t
t
he
bo
lt
inbear
i
ng,resultinginthedevel
o
pmentof bea
r
ingstressesagainstthebolt ,asshown
inFig .
2
.
4
(
b
i
i
)
.
T
hebo
l
tw
ill
ha
v
e
t
he
t
enden
c
y
t
o
getshea
r
eda
r
o
ss
se
ti
on
s
m
i
ni
andm2
n
2.
F
i
g2.4
(piii
)
s
howsth
e
free bod
y
d
i
ag
r
am of the port
i
o
n
mj
/
i j
-
m
2n2 of th
e
bo
l
t ,wh
ic
h
s
ug
g
e
s
t
s
thatshearfo
r
ces
V must
a
ctover
th
e
cut
surfaces
m
ini
and min
i
ofthe bo
l
t.I
n
th
i
s
part
i
cu
l
ar
c
a
s
e
(
knowna
s
the
c
aseof
doub
l
eshea
r)
,eachshea
r
f
o
rc
e
is
equa
lt
o
P/
2.
T
he
s
eshea
r
f
o
r
c
e
s
are
in
f
a
c
t
t
he
re
s
ul
t
an
tsoft
he
s
hear
st
re
ss
e
s
d i
str
ibu
t
ed
o
v
er
t
he
c
ro
ss
-
s
e
ct
ion a
l
area
o
f
the
b
o
l
t
,atse
ct
i
o
ns
mini
a
nd
m2
n2(
F
i
g
.2.
4
b
iv).
T
heave
r
ageshea
r
s
tr
es
s
onthec
r
oss
-
se
tio
n
of
thebo
l
t
is
o
bta
i
nedbyd
i
v
i
d
i
ngthe
shear
f
o
rceVbyt
h
ea
r
ea
o
ft
h
ecr
o
ss-secti
o
n{A )ofthebolt :
r
"
=
1
=
Ta
- (2.'
a
)
Th
e
examplesshown
i
n
F
i
g.2.
4
ar
e
th
e
examp
le
sof dir
e
ct
sh
e
ar
or
si
m
pl
e
sh
e
ar.
S
uch
d
ir
ectst
re
ss
e
s a
ri
s
e
i
nbo
l
t
e
d
,
p
i
nned ,
ri
veted ,we
l
d
e
dorg
l
ued
j
o
i
nts,where
i
n
th
e
shea
r
stress
e
s
a
r
e
cau
s
edby
a
d
ir
ectac
ti
on
of
the
f
o
r
ce
tryi
ng
t
oa
t
t
h
r
ough
t
he
m
a
t
e
ri
a
l
.
S
hea
r
s
tr
e
ss
e
s
a
r
e
a
ls
o
de
v
e
l
oped
i
n
an
i
nd
ir
ect
m
anne
rw
hen
m
e
m
be
rs
a
r
e
sub
j
ec
t
ed
t
o
bend
i
ng
o
rt
o
r
s
i
on.
Bending
s
tre
s
se
s,
tor
s
iona l
s
tre
s
sesand
bearing
s
tresses have
been d
i
s
us
s
edin later
hapter
s
.
Un its of stress
(i)
SI sys
te
m
:
Si
nce no
rm
a
l
s
tr
e
ss
p
is
ob
t
a
i
ned by d
ivi
d
i
ng
t
he ax
i
a
l
f
o
r
ce by the c
r
oss
-
se
ti
ona
l
area
,
ithasunit
s
of
for
ce
perunitof
area .
InS.I.
unit
s,
t
he unitof
for
c
e is
newtone
x
pres
se
d
b
yt
he
s
ymb
o lN,a
nd
t
he
a
r
ea
is
ex
pr
e
ss
ed
i
n
squ
a
r
e
me
tr
e
s(
m
3).H
e
nc
e
t
he
un
it
ofstr
e
ss
is
new tons
persquare
metre(N/m2
)
o
r
P
ascals (
P
a
)
.
Ho
w
ever,
new ton
i
ssuch
asmallunit
ofs
t
r
e
ss
t
h
at
i
t
b
e
c
o
mesn
e
c
e
ss
a
ry
to
workw
i
th
l
a
r
g
e
mul
t
i
p
les.D
u
e
to
t
h is,
f
orc
e
is
g
e
n
e
rally
exp
re
ss
e
d
i
nt
e
rms
o
f
k
il
o-n
e
wtona
n
dm
e
gan
e
wton
,
wh
ere
:
1ki
lo
-
n
e
wto
n
(kN
)
=
10
JN
1
mega
-
ne
wto
n
(MN
)
=
10
fN
1
giga-ne
wt
on
1
(
G
N
)= 10?
N
Si
m
i
la
rly ,t
he
str
e
ss
u
nit
Pa
sc
a
l(l
e
.N/
m2)
issuc
h a
s
ma
ll
un
it
of str
e
ss
tha
t
i
t
b
e
co
mes
ne
c
e
ss
a
r
y
t
o
w
o
r
k
w
i
t
h
l
a
r
g
e
m
u
l
t
i
p
l
e
s
.
H
enc
e
s
t
r
es
s
i
s
gen
e
r
a
ll
y
e
x
p
r
e
s
s
e
d
in
t
e
r
m
s
o
f
kN/
m
2,MN/
m
2 ,GN/
mZ
and
N/
m
m
;
(MPa) .
A
sa
n
exam
ple,
a
ty
p
i
c
a
lt
e
ns
i
l
e
str
e
ss
in
a
s
teelbarm ighthaveamagnitudeof
1
50N
/
m
m
Z
(150
MPa)wh
i
c
h
is
150
x1
0f
Pa .A
m
or
e
commo
n
f
ormof
u
ni
t
of
s
t
r
e
ss(whic
h
isno
t
r
e
com
m
e
n
d
e
d
i
n
S
I)isN/mm
2
,
w
hi
c
h
isa
unitidentical
to
megapasa
l
(M
P
a)Thus,
wehave
1
N/mm 2= 10
f
N/mZ = 10A
P
a
=1
MP
a
(ii)M.
K.S
.sys
t
em:
I
n
M.K.
S
.
sys
t
em ,
t
heunit
o
f
f
o
rce
i
s
in
the gr
a
vi
t
a
t
ionalunit,i.e.
kil
og
r
am
fo
rc
e
k
g
(f),c
o
mm
on
ly
e
x
p
r
e
ss
ed
b
yk
gon
ly.W
hen
fo
rc
e
isl
a
r
g
e
,i
t
is
e
x
p
r
e
ss
ed
b
y
t
onnes ,whe
r
e
1 1 =1000
k
g .
T
heunito
f
s
tr
es
s
is
u
s
ual
ly
e
xpr
e
ss
ed
a
s
k
g/c
mZ
.
-
5/20/2018 Unit1 From Mechanics of Materials by b c Punmia
9/1
S
IM PLE
ST
R
ESS
E
S
AN
D
S
T R
AI
N
S
T
h
e
fo
ll
o
wi
n
g
r
e
l
at
i
on
s
h
i
p
s
e
xist
:
T
a
k
ing
A
l
s
o .
Hen
c
e
1 g(
f
)=
g
new
t
ons
g=9.807
,
l
g
(
f
)
=
9
.
8
0
7
N
I
ON
1
t
onne
= 9 .807
k
N ~ 10
k
N
I
O
N
or
1
kg/c
m2
1
g
/
nr
10<
m
2
10
5
N
10
'
1
0
, N
/
m
2
1
0
S Pa
~
0
.
1
N
/
mm
2
mm
*
I
d
)
i
-
4
-
*
P
.
.
J
2
3
.STRA IN
-
W
h
e
n
a
p
rism
a
tic
b
a
riss
u
b
j
e
c
te
d
to
a
xi
a
llo
a
d
,itundergoes
a
change
in length,
as
i
n
d
i
c
a
t
e
d
i
n
F
i
g
.
2.5
.
T
h
i
s
c
ha
n
g
e
i
n
l
en
g
t
h
is
usua
lly
ca
ll
ed de
f
o
rm
at
i
on.
If
the
ax
i
a
l
f
orc
e
istensile
,
thelengthoftheba
r
isinc
r
ea sed
,
wh
ile
i
fth
e
axial
fo
rce
isc
o
m
p
r
e
ssiv
e
,
there
i
ssho
rt
en
i
ng
of
the
l
eng
t
h
of
t
heba
r
.
T
h
i
s
e
l
ongat
i
on
(or
sh
ort
en
i
ng
)
,as
t
hecasemay
be
,
is th
e
c
u
m
ul
a
tiv
e
re
s
ultof th
e
s
tret
c
hing
(or
c
ompres
s
ing)ofthema teria
l
throughout
thelength L of thebar.Thedeformation (Le.
elonga
t
ion
or
shor
t
ening)pe
r
unit
l
engthof
t
he
ba
r is t
e
r
med
a
s
s
tr
a
i
n
(
e
or
e
)
.Ingene
r
a
l
,
s
t
r
a
i
n
i
s
t
h
e
m ea
s
u
r
e
o
f
t
h
e
d
e
f
o
r
ma
t
i
on
c
au
s
e
d
due to
ex
t
erna l
loading
.
If
theba r
i
s
i
n
t
ens
i
on,
t
he
r
esul
t
ing s
tr
ain
is
nowna
s
tens
i
lest
r
a
i
n .
S
im
il
a
rl
y,the
s
tr
a
i
n
r
esu
lti
ng
from
aco
m
p
r
ess
i
ve
f
o
r
ce
i
s
k
nowna
s
com
pr
ess
i
vest
r
a
in
.
I
ngene
r
a
l
,s
tr
a
i
n
s
ass
o
ciated
withn
o
r
m
a
lst
r
esses
a
r
ekn
o
wn
as
n
o
r
m
a
lstrai
n
s .S imil
a
rly, - the
strainass
o
ciated
w
it
hs
h
e
arstr
e
ss
i
s
knownassh
e
ar
s
tra in .
S
i
n
ce
s
t
r
a
i
n
ist
hedefo
rm
at
i
onpe
r
un
i
t
l
ength
,
i
t
is
a
d
ime
n
si
on
less
quant
ity .
T
hu
s,it
hasnounits,and
t
he
r
ef
o
re,
i
t
i
sexp
r
essedaspu
r
enumbe
r
.Fo
r
examp
l
e, i
f
thedef
or
mat
i
on
of
aba rof 1.6
m
l
eng
this1.2m
m,
t
h
estrain
e
= A/
L
=
1
.2
mm
/1
.6
x
10
00
m
m
=0.0
00
75
= 750 x10
f
.
S
o
m
e
ti
mes,
i
n p
r
ac
tic
e ,
s
tr
a
i
n
is r
eco
r
ded
i
n
f
o
rmss
u
h
a
s m
m
/
m
o
r
p
m/m
e
t
c .
T
h
u
s
,
s
tr
a
i
n
o
f
t
h
e
a
b
ov
e
e
x
a
m
p
l
e
i
s
0
.7
5
m
m
/
m
o
r
750
u
m
/
m
.
Examp
l
e2.1.Apr
i
smaticbar hasa
cross-sectiono
f
25 mm
x50 mm
andalength
o
f
2
m
.
U
nde
r
an
a
xi
a
l
ten
sil
e
f
o
r
ce
o
f
90
kN,
the
mea
s
u
r
ed
e
l
onga
ti
on
o
f
the
ba
ris
1.5
m
m .
C
om
put
e
t
he
ten
sile
s
tress
an
d
st
r
a
i
n
i
ntheba r
.
l
b)
F
IG .25 .DE
F
ORM AT IO
N
A
N
D
S
T
R
A
I
N
S
olu
t
ion
_
_
P
90
x
1000
St
f'JJ
"=i=
i
90
2m
=
72
N/
mm
2
(
or
M
Pa
)
F
IG
.
2
.
6
-
5/20/2018 Unit1 From Mechanics of Materials by b c Punmia
10/
1
0
M
E
CHA
N
ICS O
F
M
A
T
E
RIALS
S
trai
n
=
=
1
.
5
mm
L(
2
x
1000
) m
m
=
0
.
00075
= 750
x
10
-
6
2
.
4
.
STRESS
-S
T
RA IN
D I
AGRA
M
Themechan
i
ca
l
proper-
tie
s o
f
a
materia
l,
di
sc
u
s
se
d
in
hap
t
e
r
1,a
r
cde
t
e
r
m
i
nedin
th
e
laboratory
b
yperform
i
ng
t
e
sts
onsma
ll
spec
i
men
s
of the
ma
t
e
r
ia
l
,
i
n
t
he
m
ate
ri
a
l
s
t
es
ti
ng
labora
t
ory.Themostcomm
o
n
m
at
erials
t
e
s
t is
t
h
e
te
n
s
i
on
t
e
st
pe
r
f
or
med
o
nac
y
l
i
n
dri
a
l
s
peime
n
o
f
th
e
material.Th
e
load
s
ar
c m
ea
s
ured
on
t
he mai
n
d
i
a
l
of
t
he
m
a
c
h
i
n
e
w
h
i
l
e
t
h
e
elong ation
s
arc measu
r
ed with
thehelpof exten
s
ometers.The
cy
li
ndricalspec
i
menhasen -
l
a
r
gedend
s
s
othat
t
hey
can
f
i t
i
n
t
he g
ri
p
s
of
t
he
ma
hine. .
T
hi
s
en
s
ure
s t
ha t
failure
w il
l
occurir.thecen tralun iform
region,wherethestressiseasy
t
o
beca l
c
u la
t
ed
ra
t
her
t
han
at
o
rnearendswhere
t
hes
t
ress
d
i
st
r
i
b
u
t
i
o
ni
s
n
o
t
u
n
i
fo
r
m
.
W
hensuchaspec
im
enof
a
du
c
tile m
a
terialis
s
u
b
jet
e
d t
o
agradua llyincreasing
pul
l
in
ate
n
siont
e
s
t
m
a
c
h
in
e,
i
t
is
foundth
a
tth
e
resu l
t
ant
s
t
rain
A
=
P
r
op o
r
t
i
on
a
l
L
i
m
i
t
B
=
E
l
as
t i
cL
imi
t
C =
Y
i
e
l
d
Po
i
n
t
C
,
=LowerY ield
P
oint
E =U lti
m
at
e
Str
en
gt
h
F =Ruptu
r
eSt
r
ength
P
O
a
=
L
i
n
e
a
r
D
e
fo
r
m
a
t i
o
n
Ob
= E
l
as
t
i
c
Deform
a
t
i
o
n
bd
=
Per
f
ec
t
P lasticYieldi
ng
d
e
=
S
tra
in
H
a
r
d
e
ni
n
g
ef
=
N
e
ck
i
ng
-ST
RA I
N
F
I
G.
2.
7.TEN
SI
LE
TE
S
T
DIAGRAM(NOT
TO
S
CALE)
isproportion
a
l
to
t
he
c
o
rr
e
spondin
g
s
t
r
e
ssup
t
o
a
limi
t
on ly
a
n
d
b
e
yon
d
t
h
a
t
,
th
e
r
e
la
t
i
o
n
is
n
o
t
li
nea
r
.
I
n
i
nves
ti
gat
i
ng
t
he
m
echan
i
ca
l
p
r
ope
rti
es
o
f
t
hema
t
e
ri
a
l
bey
o
nd
t
h
i
s
li
m
i
t ,the
r
e
l
a
ti
onsh
i
p
between
t
he
s
tr
a
i
n
and
t
he
c
orr
espond
i
ng
st
r
e
ss
is
usua
ll
y
r
ep
r
e
s
en
t
ed
g
r
aph
i
ca
ll
y
by
a
tensile
testd
i
agramorstress
strain
diagram.
A
st
r
e
s
s
-
s
tr
a
i
n
d
i
ag
r
a
m
f
o
r
a
t
y
p
i
c
a
ls
t
r
ut
u
r
a
ls
t
ee
l
i
n
t
e
ns
i
o
n
i
s
s
h
o
w
n
i
n
Fi
g
.
2
.
7
(
no
t
tos
c
a
l
e)
,
where
t
h
e
s
t
r
a
i
n
i
sp
l
otted a
l
on
g
th
e
ho
riz
on t
al
a
x
i
swh
ile
s
t
r
es
s
i
sp
l
ott
e
donth
e
ve
r
t
ic
a
l
ax
i
s
.
T
he d
ia
g
r
am b
e
g
i
n
s
w
i
th
a
stra
i
ght
lin
e
O
to A
y
in
wh
ic
hthe
s
t
r
ess
s
t
r
a
i
n
r
e
l
at
i
o
n
sh
ip
i
s
li
n
e
a
r,
ue
.
s
t
r
es
s
and
s
t
r
a
in
a
r
ed
ir
e
c
t
ly
p
r
opo
r
t
i
ona
l
.Po
i
nt
A
ma
r
k
s
th
e
lim i
t
o
f p
r
opo
r
t
i
ona
lit
y
beyondwh
i
hthecurvebecomesslight
l
y urved,unti
l
point B,thee
l
astic limitofthe material,
i
s reached .Region AB
i
sthenon-linearreg ion inwhihthestress is notproportiona
l
tostrain.
andthee
l
onga
tio
n
i
nc
r
ea
s
e
smor
e
r
ap
i
d
l
y.
H
oweve
r
,up
t
othep
oi
nt
Byt
he
r
e
m
ova
l
o
fl
oad
would
resultincompleterecoveryby
the
specimen
ofitsorigina
l
dimensions .Ifthe
l
oadis
increased
f
ur
t
her,yielding
t
akesplace;pointC
i
sthepoint
of
sudden
largeex
t
ensi
o
n, nown
a
s
th
e
y
iel
d po int .Afte
r
th
e
y
i
e
ld
po
i
nt
s
t
r
ess
i
s
r
e
a
ched
,
th
e
du ct
ile
e
x
t
e
n
si
on
s
takep
lac
e
,
thes
tr
a
i
n
si
nc
r
eas
i
nga
t
anacce
l
e
r
a
ti
ng
