tensile tests
TRANSCRIPT
Lect
ure
Not
e 1.
1
Stre
ss a
nd S
trai
n
Firs
t Sem
este
r, A
cade
mic
Yea
r 201
2D
epar
tmen
t of M
echa
nica
l Eng
inee
ring
Chu
lalo
ngko
rn U
nive
rsity
Obj
ectiv
es #1
D
escr
ibe
basi
c fe
atur
es o
f ela
stic
ity in
Car
tesi
an c
oord
inat
es
Rel
ate
defo
rmat
ion
to s
train
s an
d lo
ads
to s
tress
esD
escr
ibe
beha
vior
sin
stre
ssst
rain
diag
ram
san
d
Des
crib
e be
havi
ors
in s
tress
-stra
in d
iagr
ams
and
dete
rmin
e st
ress
/stra
in h
isto
ries
D
escr
ibe
the
char
acte
ristic
s an
d pr
oper
ties
of s
tress
and
st
rain
com
pone
nts
as te
nsor
qua
ntiti
es
Rel
ate
stre
ss a
nd s
train
com
pone
nts
for l
inea
r ela
stic
m
ater
ials
in 2
D a
nd 3
D
2
T&C
Tes
ts T
ensi
le T
ests
Dia
gram
s
3
keyw
ords
:st
retc
hing
, lat
eral
con
tract
ion
neck
ing,
regi
on o
f nec
king
fract
ure,
regi
on o
f fra
ctur
e
Stre
ss –
Stra
in D
iagr
ams
Dia
gram
s
4
Stre
ss –
Stra
in D
iagr
ams
Mild
Ste
elD
iagr
ams
5
Stre
ss –
Stra
in D
iagr
ams
Load
ing
& U
nloa
ding
Dia
gram
s
6
Exam
ple
Ger
e 1.
4-4
#1
A c
ircul
ar b
ar o
f mag
nesi
um a
lloy
is 7
50 m
m lo
n g. T
he s
tress
–
Dia
gram
s
gst
rain
dia
gram
is s
how
n. T
he b
ar
is lo
aded
in te
nsio
n to
an
elon
gatio
n of
4.5
mm
, and
then
th
e lo
ad is
rem
oved
.a)
Wha
t is
the
perm
anen
t set
of
the
bar?
b)If
thb
il
dd
hti
7
b) I
f the
bar
is re
load
ed, w
hat i
s th
e pr
opor
tiona
l lim
it?
Exam
ple
Ger
e 1.
4-4
#2
69
8810
Pa
4410
Pa
000
2pr
E
Dia
gram
s
3 3
6 9
3
0.00
2
4.5
10 m
0.00
675
010
m16
010
Pa
0.00
644
10 P
a2.
3636
10 m
pr
A
AB
A
B
L
E
8
31.
7727
10 m
1.77
mm
Ans
()
160
MP
aA
ns
pB
pr
B
new
A
p
L
Stre
ss –
Stra
in D
iagr
ams
Com
mon
App
rox.
Dia
gram
s
Ela
stic
-Per
fect
ly P
last
ic
Ela
stic
-Stra
in H
arde
ning
9
Stre
ss A
rea
Elem
ent
0lim zA
F A F
zz
zˆ
ˆˆ
xy
z
xy
z
FF
FF
Fi
Fj
Fk
Gen
Sta
te
0 0
lim limz zA A
F A F A
xx
yyz
z
zz
xy
zj
10
: p
lane
, d
irect
ion
: p
lane
, d
irect
ion
: p
lane
, d
irect
ion
z zz x y
z
z x yz z
Stre
ss V
olum
e El
emen
tG
en S
tate
00
0,
, lim
limlim
AA
A
FF
FA
AA
x
yx
zx
yzx
11
00
0
00
0
00
0
, ,
limlim
lim
, ,
limlim
lim
xx
x
yy
y
zz
z
AA
A
AA
A
AA
A
AA
AF
FF
AA
A
FF
FA
AA
x
yx
zx
yz
y yx
zx
yz
xx
yy
yy
zz
zz
z
Stat
e of
Str
ess
Def
initi
on
9
stre
ss c
ompo
nent
sσ x
, σy,
σ z, τ
xy, τ
xz ,
τ yx , τ
yz ,
τ zx,
τ zy
Gen
Sta
te Fr
om c
ompl
emen
tary
prin
cipl
e of
she
arτ x
y=
τ yx
, τxz
=τ z
x, τ y
z=
τ zy
6
inde
pend
ent c
ompo
nent
sσ x
, σy,
σ z, τ
xy, τ
yz ,
τ zx
12
To
def
ine
a ge
nera
l sta
te o
f stre
ss
Mag
nitu
de o
f stre
ss c
ompo
nent
s
Dire
ctio
n of
stre
ss c
ompo
nent
s (+
ve,−
ve)
O
rient
atio
n of
the
volu
me
elem
ent (
x, y
, zax
es)
Stre
ss C
ompl
emen
tary
Prin
cipl
e of
She
ar
C
onsi
der a
vol
ume
elem
ent s
ubje
ct to
pur
e sh
ear
A
ll fo
ur s
hear
stre
sses
mus
t hav
e eq
ual m
agni
tude
and
be
dire
cted
eith
er to
war
d or
aw
ay fr
om e
ach
othe
r at o
ppos
ite
dft
hl
t
Gen
Sta
te
edge
s of
the
elem
ent.
1
2
13
Stat
e of
Str
ess
Nor
mal
Str
ess
Sign
Con
vent
ion
Gen
Sta
te
14
Stre
ss E
lem
ents
2DG
en S
tate
MP
a
MP
a
MP
a
x y xy
15
Stre
ss E
lem
ents
3DG
en S
tate
MP
ax
MP
a
MP
a
MP
a
y y xy
16
MP
a
MP
a
yz zx
Stat
e of
Str
ain
Def
initi
on
9
stra
in c
ompo
nent
sε x
, εy,
ε z, γ
xy, γ
xz ,
γ yx , γ
yz ,
γ zx,
γ zy
Gen
Sta
te Fr
om c
ompl
emen
tary
prin
cipl
e of
she
arγ x
y=
γ yx
, γxz
=γ z
x, γ y
z=
γ zy
6
inde
pend
ent c
ompo
nent
sε x
, εy,
ε z, γ
xy, γ
yz ,
γ zx
17
To
def
ine
a ge
nera
l sta
te o
f stra
in
Mag
nitu
de o
f stre
ss c
ompo
nent
s
Dire
ctio
n of
stre
ss c
ompo
nent
s (+
ve,−
ve)
O
rient
atio
n of
the
volu
me
elem
ent (
x, y
, zax
es)
Stat
e of
Str
ain
Volu
me
Elem
ent
Gen
Sta
te
,,
xxy
xzu
uv
uw
18
, ,
, ,
xxy
xz
yxy
yz
zxzy
z
xy
xz
xv
uv
vw
xy
yz
yw
uw
vw
xz
yz
z
Stat
e of
Str
ain
Mod
ern
nota
tion
Gen
Sta
te
11
1,
,2
22
xxxy
xzu
uu
vu
wx
xy
xz
x
19
22
2
11
1,
,2
22
11
1,
,2
22
yxyy
yz
zxzy
zz
xx
yx
zx
vu
vv
vw
xy
yy
zy
wu
wv
ww
xz
yz
zz
Pois
son’
s R
atio
Def
initi
onH
ooke
’s
lat
long
20
Tria
xial
Str
ess
Gen
eral
Hoo
ke’s
Law
#1
Hoo
ke’s
C
onsi
der n
orm
al s
train
in
dire
ctio
n ca
uses
xx
E
21
cau
ses
cau
ses
cau
ses
1(
)
EE E
E
xx
x
yx
y
zx
z
xx
xx
xy
z
Tria
xial
Str
ess
Gen
eral
Hoo
ke’s
Law
#2
1(
)E
xx
yz
Hoo
ke’s
(1
)(
)(1
)(12
)E
xx
yz
1(
)
1(
)
E E
yy
xz
zz
xy
22
(1)(1
2)
(1)
()
(1)(1
2)
(1)
()
(1)(1
2)
E E
yy
xz
zz
xy
Uni
axia
l Str
ess
Stra
in E
nerg
y #2
1(
)2
xx
yy
zz
u
Hoo
ke’s
22
2
2 11
()
()
22
1(
)2 1
()
()
2
yy
xx
yz
yy
xz
zz
xy
xy
zx
yy
zz
x
EE
E E
u u uE
23
22
2
2
(1)(
)2
()
2(1
)(12
)x
yz
xy
yz
zx
EE
E
u
Pure
She
ar G
ener
al H
ooke
’s L
awH
ooke
’s
Con
side
r she
ar s
train
sC
onsi
der s
train
ene
rgy
24
xyxy
yzyz
zxzx
G G G
1(
)2
xyxy
yzyz
zxzx
u
2(
1)
EG
Hoo
ke’s
Law
2D S
peci
al C
ases
#1
P
lane
stre
ss: σ
z=
τ yz
= τ z
x=
0
11
()
xx
yz
xx
yE
E
Hoo
ke’s
11
()
1(
)
xx
yz
xx
y
yy
zx
yy
x
zz
xy
zx
y
xyxy
xyxy
EE
EE
EE
GG
25
0 0
xyxy
yzyz
yz
zxzx
zx
GG
G G
Hoo
ke’s
Law
2D S
peci
al C
ases
#2
P
lane
stra
in: ε
z=
γ yz
= γ z
x=
0
11
()
()
xx
yz
xx
yz
EE
Hoo
ke’s
11
()
()
1(
)0
()
xx
yz
xx
yz
yy
zx
yy
zx
zz
xy
zx
y
xyxy
xyxy
EE
EE
E GG
26
00
00
xyxy
yzyz
yz
zxzx
zx
GG
G G
Ther
mal
Effe
cts
G
ener
ally
, a b
ody
expa
nds
or tr
ies
to e
xpan
d w
hen
tem
pera
ture
rise
s.
Hoo
ke’s
If th
e bo
dy c
an e
xpan
d, th
ere
will
be th
erm
al s
train
s ε T
=α(
ΔT) i
n th
e no
rmal
dire
ctio
ns.
If
the
body
can
not e
xpan
d fre
ely,
ther
e w
ill be
ther
mal
st
ress
es.
1
27
, , ,
1(
)
1(
)
1(
)
xM
xT
xy
z
yM
xT
yz
x
zM
xT
zx
y
TE
TE
TE
α=
coef
ficie
nt
of li
near
ther
mal
ex
pans
ion
Hoo
ke’s
Law
With
and
With
out T
emp
Effe
cts
11
()
()
xx
yz
xx
yz
TE
E
Hoo
ke’s
11
()
()
11
()
()
xx
yz
xx
yz
yy
zx
yy
zx
zz
xy
zz
xy
xyxy
xyxy
EE
TE
E
TE
E
GG
28
xyxy
yzyz
yzyz
zxzx
zxzx
GG
GG
GG
Obj
ectiv
es #2
D
escr
ibe
basi
c fe
atur
es o
f ela
stic
ity in
Car
tesi
an c
oord
inat
es
Giv
en a
sta
te o
f stre
ss o
r sta
in, d
eter
min
e th
e eq
uiva
lent
st
ate
atot
hero
rient
atio
nsst
ate
at o
ther
orie
ntat
ions
D
eter
min
e th
e pr
inci
pal a
nd m
axim
um s
hear
st
ress
es/s
train
s in
2D
by
Moh
r’s c
ircle
D
eter
min
e th
e pr
inci
pal a
nd m
axim
um s
hear
st
ress
es/s
train
s in
3D
by
linea
r alg
ebra
C
alcu
late
Tre
sca
and
von
Mis
esst
ress
es
Rev
iew
ofst
ress
and
defo
rmat
ion
inax
ially
load
ed
29
R
evie
w o
f stre
ss a
nd d
efor
mat
ion
in a
xial
ly lo
aded
m
embe
rs, t
hick
-wal
led
tors
ion
and
beam
ben
ding
Stat
e of
Str
ess
Sign
Con
vent
ion
& O
rient
atio
nTr
ansf
orm
=
30
posi
tive
dire
ctio
npo
sitiv
e an
gle
of ro
tatio
n θ
(cou
nter
cloc
kwis
e)
Incl
ined
Pla
neTr
ansf
orm
31
Incl
ined
Pla
neSt
ress
es o
n In
clin
ed P
lane
#1
Tran
sfor
m
0
(co
s)c
os(
cos
)sin
xx
xxy
FA
AA
32
22
(si
n)s
in(
sin
)cos
0
cos
sin
2si
nco
s
0
(co
s)s
in(
cos
)cos
(si
n)c
os(
sin
)sin
0
yxy
xx
yxy
yx
yx
xy
yxy
xy
AA
FA
AA
AA
22
()s
inco
s (
cos
sin
)y
xxy
Incl
ined
Pla
ne S
tres
ses
on In
clin
ed P
lane
#3
Tran
sfor
m
33
22
0
(si
n)s
in(
sin
)cos
(co
s)c
os(
cos
)sin
0
sin
cos
2si
nco
s
yy
xxy
yxy
yx
yxy
FA
AA
AA
Stre
ss T
rans
form
atio
n Eq
uatio
nsTr
ansf
orm
cos2
sin
22
2
sin
2co
s22
xy
xy
xxy
xy
xy
xy
34
2
cos2
sin
22
2x
yx
yy
xy
xy
xy
Exam
ple
Hib
bele
r 9-4
/5 #
1
The
stat
e of
stre
ss a
t a p
oint
in a
mem
ber i
s sh
own
on th
e el
emen
t. D
eter
min
e th
e st
ress
com
pone
nts
actin
g on
the
Tran
sfor
m
incl
ined
pla
ne A
B.
35
Exam
ple
Hib
bele
r 9-4
/5 #
2
Sta
te o
f stre
ss (
axe
s)3
ksi,
2 ks
i, 4
ksi
xy
xy
xy
Tran
sfor
m
,,
The
rota
tion
of
to
is
60
cos2
sin2
22
32
32
cos (
260
)(
4)si
n(2
60)
2.71
41 k
si2
2
xy
xy
xy
xy
xxy
x
xyx
y
36
()
()
()
22 sin2
cos2
2x
yx
yxy
x
32
sin(
260
)(
4)co
s(2
60)
4.16
51 k
si2
xy
Exam
ple
Hib
bele
r 9-4
/5 #
3Tr
ansf
orm
37
2.71
ksi
4.17
ksi
A
nsx xy
100
MP
a,
0x
yxy
Tran
sfor
m Stre
ss T
rans
Uni
axia
l Str
ess
#2
38
Stre
ss T
rans
Uni
axia
l Str
ess
#3
100
MP
a,
0x
yxy
Tran
sfor
m
39
Stre
ss T
rans
Bia
xial
Str
ess
100
MP
a,
50 M
Pa,
0
xy
xy
Tran
sfor
m
40
Stre
ss T
rans
Pur
e Sh
ear
0,
100
MP
ax
yxy
Tran
sfor
m
41
Stre
ss T
rans
Gen
eral
100
MP
a,
50 M
Pa,
100
MP
ax
y
xy
Tran
sfor
m
xy
42
Gen
eral
Str
ess
Max
imum
Nor
mal
Str
esse
s
'
'm
ax &
min
,
0x
y
Tran
sfor
m
43
Gen
eral
Str
ess
Max
imum
She
ar S
tres
ses
''
''
max
& m
in
, x
yx
y
Tran
sfor
m
44
Gen
eral
Str
ess
Prin
cipa
l Str
esse
sσ 1
& σ
2#1
Th
eσ 1
is th
e m
axim
um n
orm
al s
tress
and
σ2
is th
e m
inim
um
norm
al s
tress
in th
e pl
ane.
P
rinci
pal p
lane
s ar
e de
fined
by
prin
cipa
l ang
les
θ p1
& θ p
2.
Spec
ial
The
σ 1 o
ccur
s in
θp 1
plan
e an
d σ 2
occ
urs
in θ
p 2pl
ane.
max
& m
in
45
0x
y
Gen
eral
Str
ess
Prin
cipa
l Str
esse
sσ 1
& σ
2#2
cos2
sin
20
22
xy
xy
xxy
dd
dd
Spec
ial
2ta
n2
xyp
xy
12
90p
p
46
12
mut
ually
pe
rpen
dicu
lar
plan
es
pp
Gen
eral
Str
ess
Prin
cipa
l Str
esse
sσ 1
& σ
2#3
2ta
n2
xyp
xy
Spe
cial
22
1
2
cos2
2
xy
xy
xy
p
R
R2
47
1
2
sin
22xy
p
R R
22
1,2
22
xy
xy
xy
Gen
eral
Str
ess
Prin
cipa
l Str
esse
sσ 1
& σ
2#4
Spe
cial
3
plan
e st
ress
0
48
Gen
eral
Str
ess
Max
In-P
lane
She
ar S
tres
s τ m
ax #
1
''
''
max
& m
in
, x
yx
y
Spec
ial
49
Gen
eral
Str
ess
Max
In-P
lane
She
ar S
tres
s τ m
ax #
2
Max
imum
in-p
lane
she
ar s
tress
es τ
max
and
τ min
Th
e τ m
axis
the
max
imum
+ve
she
ar s
tress
and
τm
inis
the
max
imum
−ve
she
ar s
tress
.
Spec
ial
Pla
nes
of τ
max
and
τ min
are
defin
ed b
y an
gles
θs 1
and
θ s2.
max
& m
in
50
xy
Gen
eral
Str
ess
Max
In-P
lane
She
ar S
tres
s τ m
ax #
3
sin2
cos2
02
xy
xy
xy
dd
dd
Spe
cial
tan
22x
ys
xy
51
11
22
45 45s
p
sp
Gen
eral
Str
ess
Max
In-P
lane
She
ar S
tres
s τ m
ax #
4
22
12
,2
2x
ym
axm
inxy
xy
Spe
cial
22
2 2
xy
avg
xy
52
Gen
eral
Str
ess
Max
In-P
lane
She
ar S
tres
s τ m
ax #
5
M
axim
um in
-pla
ne s
hear
stre
sses
are
foun
d by
rota
ting
x 1y 1
z 1ax
es a
bout
z1
axis
thro
ugh
45.
Th
e ro
tatio
ns o
f 45
abou
t set
s of
two
prin
cipa
l axe
s gi
ves
lith
it
fl
ht
Spec
ial
plan
es w
ith m
axim
um o
ut-o
f-pla
ne s
hear
stre
sses
.
11
1
21
12
()
, (
),
()
22
2m
axx
max
ym
axz
53
Exam
ple
Hib
bele
r 9-4
/5 #
4
Det
erm
ine
the
prin
cipa
l stre
sses
, the
max
imum
in-p
lane
she
ar
stre
ss a
nd a
vera
ge n
orm
al s
tress
as
wel
l as
thei
r cor
resp
ondi
ng
orie
ntat
ions
.
Spec
ial
oe
tato
s
54
Exam
ple
Hib
bele
r 9-4
/5 #
5S
peci
al
Sta
te o
f stre
ss0
(ax
es)
xy
55
0 (
axe
s)3
ksi
2 ks
i4
ksi
x y xy
xy
Exam
ple
Hib
bele
r 9-4
/5 #
6Pr
inci
pal S
tress
esS
peci
al
1
22(
4)ta
n21.
63
22
57.9
95, 5
7.99
518
028
.997
, 61
.003
At
28.9
97xyp
xy
pp
x
56
1
11
1
1 1
cos2
sin2
22
32
32
cos(
228
.997
)(
4)si
n(2
28.9
97)
22
5.21
70 k
si
xy
xy
xx
x
x
yx
x
Exam
ple
Hib
bele
r 9-4
/5 #
7
2A
t 61
.003
cos2
sin
24
2170
ksi
x
xy
xy
Prin
cipa
l Stre
sses
Spec
ial
22
2
21
21
21
1
12
1
cos2
sin
24.
2170
ksi
22
, ,
,
4.22
ksi
, 61
.0
522
ksi
290
Ans
xx
xyx
xp
xx
px
p
57
22
5.22
ksi
,29
.0
Ans
p
Exam
ple
Hib
bele
r 9-4
/5 #
9M
ax S
hear
Stre
sses
32
tan
20.
625
22(
4)x
ys
xy
Spec
ial
3
33
33
232
.225
, 32
.225
180
16.0
03,
73.9
97
At
73.9
97 si
n2
cos2
4.71
70 k
si2
xy
ss
x
xy
xy
xxy
x
58
4
44
44
At
16.0
03 sin
2co
s24.
7170
ksi
2
x
xy
xy
xxy
x
Max
She
ar S
tress
esEx
ampl
e H
ibbe
ler 9
-4/5
#10
12
33
44
At
&
, 2
0.5
ksi
ss
xy
xy
avg
avg
Spe
cial
31
34
24
1 2
, ,
,
4.72
ksi
,74
.0
4.72
ksi
,16
.0
0.5
ksi
in b
oth
plan
esA
ns
max
xs
xm
inx
sx
max
s
min
s
avg
59
pav
g
Moh
r’s C
ircle
Gra
phic
al M
etho
d
From
stre
ss tr
ansf
orm
atio
n eq
uatio
ns
Tran
sfor
m
22
22
cos2
sin
2...
(1)
22
sin
2co
s2...
(2)
2
xy
xy
xxy
xy
xy
xy
60
22
22
(1)
(2)
cons
t2
2x
yx
yx
xy
xy
Moh
r’s C
ircle
Con
stru
ctio
n
22
2(
)x
avg
xy
R
Tran
sfor
m
22
2 2
xy
avg
xy
xyR
61
Moh
r’s C
ircle
Prin
cipa
l & S
hear
Str
esse
s
min
1
Spec
ial
2
62 m
ax
Exam
ple
Hib
bele
r 9-4
/5 #
12
The
stat
e of
stre
ss a
t a p
oint
in a
mem
ber i
s sh
own
on th
e el
emen
t. U
sing
the
Moh
r’s c
ircle
, det
erm
ine
the
stre
ss c
ompo
nent
s ac
ting
on th
e in
clin
ed p
lane
AB
. The
n, fi
nd th
e pr
inci
pal s
tress
es,
Tran
sfor
m
act
go
te
ced
pa
ee
,d
te
pc
past
esse
s,th
e m
axim
um in
-pla
ne s
hear
stre
ss a
nd a
vera
ge n
orm
al s
tress
as
wel
l as
thei
r cor
resp
ondi
ng o
rient
atio
ns.
St
tf
tt
0(
)
63
Sta
te o
f stre
ss a
t 0
( a
xes)
3 ks
i, 2
ksi,
4 ks
i
The
rota
tion
of
to
is
60
xy
xyxy
xyx
y
Exam
ple
Hib
bele
r 9-4
/5 #
13
0
.5 k
si2
(0)
(0
50)
xy
avg
CC
Tran
sfor
m
(,0
)(
0.5,
0)
(,
)(
3,
4)
avg
xxy
CC
AA
64
22
(3
0.5)
(4)
4.71
70 k
si4
tan
, 57
.995
2.5
R R
Exam
ple
Hib
bele
r 9-4
/5 #
14
60
Rot
ate
260
CC
Wto
G
Tran
sfor
m
Rot
ate
260
CC
W to
2
6057
.99
62.0
1
Coo
rdin
ate
of
(
2.71
, 4.
17)
G
GG
65
Exam
ple
Hib
bele
r 9-4
/5 #
15Tr
ansf
orm
66
2.71
ksi
4.17
ksi
Ans
x xy
Exam
ple
Hib
bele
r 9-4
/5 #
16Pr
inci
pal S
tress
esM
ax S
hear
Stre
sses
14.
22 k
si
Spe
cial 1 22
61.0
CW
5.22
ksi
29.0
CC
W
4.72
ksi
740
p p max
67
1 2
74.0 4.
72 k
si16
.0 0.5
ksi
Ans
s min
s avg
Ex
ampl
e H
ibbe
ler 9
-4/5
#17
Spe
cial
68
Prin
cipa
l Stre
sses
Max
She
ar S
tress
es
Plan
e St
rain
P
lane
stra
in in
the
xypl
ane
0z
zxzy
69
D
oes
plan
e st
ress
occ
urs
in a
pla
ne s
train
bod
y?
Plan
e St
rain
Sig
n C
onve
ntio
n an
d O
rient
atio
n
70
posi
tive
dire
ctio
npo
sitiv
e an
gle
of ro
tatio
n θ
(cou
nter
cloc
kwis
e)
Stra
in T
rans
form
atio
n N
orm
al S
trai
n #1
71
Stra
in T
rans
form
atio
n N
orm
al S
trai
n #2
cos
sin
cos
cos
sin
cos
xy
xy
xx
yxy
ddx
dydy
ddx
dydy
1 1
22
cos
sin
cos
As
cos
, si
n,
cos
sin
sin
cos
xx
yxy
xx
yxy
dsds
dsds
dxdy
dsds
72
1
22
Rem
embe
r co
ssi
n 2
sin
cos
?x
xy
xy
Stra
in T
rans
form
atio
n Sh
ear S
trai
n #1
73
Stra
in T
rans
form
atio
n Sh
ear S
trai
n #2
11
12
3
xy
γ
2
sin
cos
sin
()s
inco
ssi
n
xy
xy xy
xy
dxdy
dsds
dy ds
2(
)i
(90
)(
90)
(90
)
74
11
2
2
22
()s
in(
90)c
os(
90)
cos
(90
)
()s
inco
sco
s
2()s
inco
s(c
ossi
n)
xy
xy
xy
xy
xy
xy
xy
11
22
()s
inco
s (
cos
sin
)x
yy
xxy
Stra
in T
rans
form
atio
n Eq
uatio
ns
co
s2si
n22
22xy
xy
xy
x
sin2
cos2
2
cos2
sin2
2
22
22
xy
xx
y
xy
xy
y xy
x
y
xy
y
75
Prin
cipa
l Str
ains
ε 1
is th
e m
axim
um n
orm
al s
train
and
ε2
is th
e m
inim
um n
orm
al
stra
in in
the
plan
e.
22
tan2
22
x
x
yp
y
Th
e pr
inci
pal s
tress
es a
nd p
rinci
pal s
train
s oc
cur i
n th
e i
tti
1,2
22
2x
yx
yxy
P
rinci
pal p
lane
s ar
e θ p
1an
d θ p
2,
76
sam
e or
ient
atio
n.
In p
rinci
pal p
lane
s γ x
’y’=
0.
For p
lane
stra
in, t
hird
prin
cipa
l stra
in ε
3 =
0.
Max
imum
In-p
lane
She
ar S
trai
ns
γ max
is th
e m
axim
um p
ositi
ve s
hear
stra
in a
nd γ m
inis
the
max
imum
neg
ativ
e sh
ear s
train
.
22
In
pla
ne o
f max
imum
she
ar s
train
s,
O
ccur
in p
lane
s θ s
1an
d θ s
2,
12
,
22
22
xy
xym
axm
in
tan
2x
ys
xy
77
2x
yx
yav
g
Moh
r’s C
ircle
Pla
ne S
trai
n
22
2(
)2x
yx
avg
R
2
22
2 22
xy
avg
xy
xyR
78
Moh
r’s C
ircle
Str
ain
Tran
sfor
mat
ion
/2
min
1
2
79
/2m
ax
Stra
in R
oset
te
S
train
gau
ges
can
only
mea
sure
nor
mal
stra
ins.
C
lust
ers
of s
train
gau
ges,
stra
in ro
sette
s, a
re u
sed
to
mea
sure
stat
esof
plan
est
rain
mea
sure
sta
tes
of p
lane
stra
in.
80
22
22
22
cos
sin
sin
cos
cos
sin
sin
cos
cos
sin
sin
cos
ax
ay
axy
aa
bx
by
bxy
bb
cx
cy
cxy
cc
Stra
in R
oset
te R
ecta
ngul
ar R
oset
te
0,
45,
90a
ba
, ,
2(
)x
ay
c
xyb
ac
81
W
hat i
s th
e re
stric
tion
in s
peci
fyin
g an
gles
in s
train
rose
tte?
No
two
gaug
es a
re a
rran
ged
180°
to e
ach
othe
r
Exam
ple
Hib
bele
r 10-
8 #1
The
stat
e of
stra
in a
t the
poi
nt o
n th
e sp
anne
r wre
nch
has
com
pone
nts
of ε
x =
260(
10–6
), ε y
= 3
20(1
0–6 )
, and
γ xy
= 18
0(10
–
6 ) .
Use
the
stra
in-tr
ansf
orm
atio
n eq
uatio
ns to
det
erm
ine
(a)
)U
set
est
ata
so
ato
equa
tos
tode
tee
(a)
the
in-p
lane
prin
cipa
l stra
ins
and
(b) t
he m
axim
um in
-pla
ne
shea
r stra
in a
nd a
vera
ge n
orm
al s
train
. In
each
cas
e, s
peci
fy
the
orie
ntat
ion
of th
e el
emen
t and
sho
w h
ow th
e st
rain
s de
form
the
elem
ent w
ithin
the
x-y
plan
e.
Sta
te o
f stra
in
82
6 6 6
0 (
axe
s)26
010
320
10
180
10
x y xy
xy
Exam
ple
Hib
bele
r 10-
8 #2
Prin
cipa
l Stra
in
635
.783
180
10ta
n2
xy
1
11
1
6
6
tan
2
54
.218
(260
320)
10
At
35.7
83
cos2
sin
219
5.13
102
22
pp
xy
x
xy
xy
xyx
xx
83
2
22
2
6
At
54.2
18
cos2
sin
238
4.87
102
22
x
xy
xy
xyx
xx
Exam
ple
Hib
bele
r 10-
8 #3
Prin
cipa
l Stra
in
12
21
1
61
,
385
1054
.2
px
px
p
84
2
62
195
1035
.8A
nsp
Exam
ple
Hib
bele
r 10-
8 #4
Max
She
ar S
train
1
145
54.2
1845
9.21
8
4554
218
4599
218
sp
21
12
6
4554
.218
4599
.218
: 2
238
4.87
195.
1310
22
max
,min
sp
max
,min
85
6
6
189.
7410
290
102
max
,min
xy
avg
Exam
ple
Hib
bele
r 10-
8 #5
Max
She
ar S
train
19.
22s
1 2
6
6
190
10
99.2 19
010
s max
s min
86
6
On
both
pla
ne29
010
Ans
avg
Exam
ple
Hib
bele
r 10-
8 #6
Moh
r’s C
ircle
At
(/2
)A
6 6 6
At
(,
/2)
0 (
axe
s)26
010
320
10
180
10
xxy
x y xy
A
xy
87
6
At
(,0
)(
)/2
290
10
avg
avg
xy
C
6
61
6
94.8
6810 90
10ta
n71
.565
(290
260)
10
R
Exam
ple
Hib
bele
r 10-
8 #7
Moh
r’s C
ircle
1
61
385
1054
.2p
88
2
62
195
1035
.8A
nsp
Exam
ple
Hib
bele
r 10-
8 #8
Moh
r’s C
ircle
1 2
6
6
9.22 19
010
99.2 19
010
s max
s
89
6
6
190
10
On
both
pla
ne29
010
Ans
min
avg
Stre
ss &
Str
ain
Tens
or 3D
Tens
or
xxy
xzxx
xyxz
/2/2
/2/2
yxy
yzyx
yyyz
zxzy
zzx
zyzz
xxxy
xzxx
xyxz
90
/2/2
/2/2
yxyy
yzyx
yyyz
zxzy
zzzx
zyzz
Stre
ss T
rans
form
atio
n 2D
#1
Tens
or
ˆ
ˆˆ
ˆx
yx
yF
Fi
Fj
Fi
Fj
cos
sin
cos(
)co
sco
ssi
nsi
nsi
n()
sin
cos
cos
sin
cos
sin
cos
sin
x y x yFF
FF
FF
FF
FF
FF
FF
F
F
91
cos
sin
cos
sin
cos
sin
sin
cos
xx
y
yy
x
FF
FF
FF
x yF F
F
aF
Stre
ss T
rans
form
atio
n 2D
#2
Tens
or
T
aa
22
22
cos
sin
cos
sin
sin
cos
sin
cos
cos
cos
2si
nco
s
2i
xx
xy
xxxy
yx
yy
yxyy
xx
yy
xy
92
22
cos
cos
2si
nco
s
sin
cos
sin
cos
2y
xx
yxy
xy
xy
xyxy
2
cos
Stre
ss T
rans
form
atio
n 2D
#3
22
cos
cos
2si
nco
s
1co
s21
cos2
2si
nco
s2
2
xx
yy
xy
xy
yxy
Tens
or
2
22
cos2
cos2
2si
nco
s2
22
2
sin
c
cos2
sin
22
2os
sin
cos
2cos
yx
xy
yxy
xy
xy
xyxy
xy
xy
xxy
93
()s
inx
y
2
cos
(2c
sin
2co
os1)
s22
xy
xy
xy
xy
2 2
sin
22
sin
cos
sin
(1co
s2)/
2co
s(1
cos2
)/2
xy
xy
Stre
ss T
rans
form
atio
n 3D
#1
Tens
or
Ta
a
94
cos
cos
cos
cos
cos
cos
cos
cos
cos
T
xx
xy
xz
yx
yy
yz
zx
zy
zz
aa
a
Stre
ss T
rans
form
atio
n 3D
#2
Tran
sfor
mat
ion
mat
rix
cos
cos
cos
xx
xy
xz
Tens
or
cos
cos
cos
cos
cos
cos
cos
cos
cos9
0co
sco
sco
s90
cos9
0co
s90
cos0
xx
xy
xz
yx
yy
yz
zx
zy
zz
xx
xy
yx
yy
aa
95
cos
sin
0si
nco
s0
00
1a
Stre
ss T
rans
form
atio
n 3D
Prin
cipa
l Str
ess
#1Te
nsor Th
e pr
inci
pal v
alue
s an
d di
rect
ions
of t
he s
tress
and
stra
in
tens
ors
can
be o
btai
ned
in th
e sa
me
met
hod
of c
alcu
latin
g ei
genv
alue
san
d ei
genv
ecto
rsof
a m
atrix
and
equ
ival
ent t
o e
gea
ues
ad
ege
ecto
so
aat
ad
equ
ae
tto
mat
rix d
iago
naliz
atio
n.
ˆˆ
From
ˆR
earra
ngin
g(
)0
Non
-triv
ial s
olut
ion
if de
t()
0
Tn
n
TI
n
TI
96
1 2 3
Equ
atio
ns fo
r 3 e
igen
valu
es
det(
)0
de
t()
0
de
t()
0
TI
TI
TI
Stre
ss T
rans
form
atio
n 3D
Prin
cipa
l Str
ess
#2Te
nsor
For e
ach
eige
nval
ue
, ˆˆ
Det
erm
ine
eige
nvec
tor
from
()
0i
ii
in
TI
n
12
3
11
1
22
2
33
3
g(
)ˆ
ˆˆ
Then
, ass
embl
e tra
nsfo
rmat
ion
mat
rix fr
om
, a
nd
as u
nit n
orm
al d
irect
ions
ii
i
xy
z
px
yz
xy
z
nn
n
nn
na
nn
nn
nn
97
y
1
2
3
Form
nor
mal
tran
sfor
mat
ion
00
00
00
T
pp
pa
a
Stre
ss T
rans
form
atio
n 3D
Prin
cipa
l Str
ess
#3
0Fr
om0
00
xxy
xyy
z
Tens
or
20
Eig
enva
lues
00
00
Yiel
ds[(
)()
]()
0
z
xxy
xyy
z
xy
xyz
98
2
3
Or
()(
)0
(1)
and
0 (2
)
For (
2)
yy
xy
xy
z
z
Stre
ss T
rans
form
atio
n 3D
Prin
cipa
l Str
ess
#4
22
22
For (
1)(
)0
()
()
4()
xy
xy
xy
Tens
or
1,2
22
2
1,2
22
2
1,2
()
()
4()
2 24
4(
)2
2(
)2
42
4
xy
xy
xy
xy
xx
yy
xy
xyx
y
xy
xx
yy
xy
99
2
21,
21,
2
24
()
()
22
xy
xy
xy
Prin
cipa
l Stre
sses
=Ei
genv
alue
s
Stre
ss T
rans
form
atio
n 3D
Prin
cipa
l Str
ess
#5
0E
igen
vect
ors
00
xxy
x
xyy
yn n
Tens
or
33
33
3
33
33
3
00
0Fo
r ,
00
00
Yiel
ds(
)0
(3.1
)
xyy
y
zz
xxy
x
zxy
yy
zz
xx
xyy
n
n n nn
n
100
33
3
33
()
0(3
.2)
()
0(3
.3)
Sol
ve (3
.1)-(
3.3)
axyx
yy
zz
nn
n
22
23
33
3
12
3
nd
1
0,
0,
1 x
yz
zz
z
nn
nn
nn
n
Stre
ss T
rans
form
atio
n 3D
Prin
cipa
l Str
ess
#6
2
21
()
()
For
22
xy
xy
x y
Tens
or
11
11
11
11
1
11
1
22
0E
iven
vect
ors
00
00
Yiel
ds(
)0
(1.1
)(
)0
(1.2
)
y
xxy
x
xyy
y
zz
xx
xyy
xyx
yy
n n nn
n
nn
101
11
1
11
11
()
0(1
.2)
()
0(1
.3)
Solv
e (1
.1)-(
1.3)
and
1
to fi
nd
,
xyx
yy
zz
x
nn
n nn
11
2
,
Sim
inar
pro
cedu
re fo
r
yz
nn
Exam
ple
Hib
bele
r 9-4
/5 #
1
The
stat
e of
stre
ss a
t a p
oint
in a
mem
ber i
s sh
own
on th
e el
emen
t. D
eter
min
e th
e st
ress
com
pone
nts
actin
g on
the
Tens
or incl
ined
pla
ne A
B. T
hen,
det
erm
ine
the
prin
cipa
l stre
sses
and
th
e co
rresp
ondi
ng o
rient
atio
ns.
102
Exam
ple
Hib
bele
r 9-4
/5 #
2Te
nsor
34
04
20
ksi
00
0
xxy
zx
xyy
yz
zxyz
z
103
1/2
3/2
0co
ssi
n0
sin
cos
03
/21/
20
00
10
01
a
Exam
ple
Hib
bele
r 9-4
/5 #
3
T
xx
yx
z
aa
Tens
or
1/2
3/2
01/
23
/20
34
03
/21/
20
42
03
/21/
20
00
10
00
00
1
yx
yy
z
zx
zy
z
104
2.71
414.
1651
04.
1651
1.71
410
ksi
00
0A
t 60
,
2.71
ksi
, 1.
71 k
si,
4.17
ksi
,
0A
nsx
yx
y
zy
zx
z
Exam
ple
Hib
bele
r 9-4
/5 #
4
34
04
20
ksi
00
0
xxy
zx
xyy
yz
zxyz
z
Tens
or
00
0 34
04
20
00
00
()
(3
)(2
)16
0
189
189
zxyz
z
xxy
zx
xyy
yz
zxyz
z
105
12
3
189
189
0,,
22
189
4.21
699
ksi,
0,
21
895.
2169
9 ks
i2
Ans
Exam
ple
Hib
bele
r 9-4
/5 #
5
1tra
nsfo
rm20
0
00
x
xyzx
xyy
yz
Tens
or
3
trans
form
00
34
04.
2169
90
04
20
ksi
0
00
ksi
00
00
05.
2169
9
zxyz
z
106
0 0 0
xxy
zxx
xyy
yzy
zxyz
zzn n n
Exam
ple
Hib
bele
r 9-4
/5 #
6
11
89Fo
r 4.
2169
9 ks
i2
34
00
Tens
or
11
11
11
11
34
00
42
00
00
00
589
()
40
22
Sol
vesi
mul
tane
ousl
yw
ith5
89
x y z
xy
n n n
nn
107
22
21
11
11
1
1
Sol
ve s
imul
tane
ousl
y w
ith5
894
()
01
22
189
()
02
2 0.
xy
xy
z
z
xnn
nn
n
n
n
1
148
477,
0.
8746
4,
0y
zn
n
Exam
ple
Hib
bele
r 9-4
/5 #
7
2Fo
r 0
34
00
n
Tens
or
22
22
22
22
22
22
22
22
34
00
42
00
00
00
34
0S
olve
sim
ulta
neou
sly
with
42
01
00
x y z
xy
xy
xy
z
n n n
nn
nn
nn
nn
108
2
22
2
00 0,
0,
1z
xy
z
n nn
n
Exam
ple
Hib
bele
r 9-4
/5 #
8
31
89Fo
r 5.
2169
9 ks
i2
34
00
Tens
or
33
33
33
33
34
00
42
00
00
00
589
()
40
22
Sol
vesi
mul
tane
ousl
yw
ith5
89
x y z
xy
n n n
nn
109
22
23
33
33
3
3
Sol
ve s
imul
tane
ousl
y w
ith5
894
()
01
22
189
()
02
2 0.xy
xy
z
z
xnn
nn
n
n
n
3
187
464,
0.
4847
7,
0y
zn
n
Exam
ple
Hib
bele
r 9-4
/5 #
9
11
1
22
2
0.48
477
0.87
464
00
01
Ans
xy
z
px
yz
nn
na
nn
n
Tens
or
33
30.
8746
40.
4847
70
0.48
477
0.87
464
03
40
0.48
477
00
00
14
20
0.87
464
0.48
477
00
00
xy
z
T
pp
p
p
nn
n
aa
.874
640.
8746
40
0.48
477
01
0
110
4.21
699
00
00
0 k
si0
05.
2169
9p
3D S
tate
of S
tres
s3
Moh
r C
ircle
s
111
3D S
tate
of S
tres
s Tr
iaxi
al S
tres
s
If
ther
e is
no
shea
r stre
ss, σ
x', σ
y', σ
z'ar
e pr
inci
pal s
tress
es.
3 M
ohr
Circ
les
12
3
max
int
min
112
Tria
xial
Str
ess
Max
imum
She
ar S
tres
s3
Moh
r C
ircle
s
113
abso
lute
max
imum
shea
r stre
ss
1
3(
)2
abs
max
Yiel
d C
riter
ia &
Yie
ld S
urfa
ces
If th
e bo
dy is
sub
ject
ed to
a u
niax
ial s
tress
, the
yie
ld
stre
ss c
an b
e di
rect
ly o
btai
ned
from
the
stan
dard
test
ing.
Equi
vale
nt S
tress
How
ever
, if t
he b
ody
is s
ubje
cted
to a
mul
tiaxi
al s
tate
of s
tress
, th
e th
eorie
s of
failu
rear
e us
ed to
det
erm
ine
whe
ther
poi
nts
in
the
body
hav
e re
ache
d th
e yi
eld.
A yi
eld
crite
rion
is a
hyp
othe
sis
conc
erni
ng th
e lim
it of
el
astic
ity u
nder
any
com
bina
tion
of s
tress
es. Y
ield
sur
face
is
desc
ribed
thro
ugh
stat
esof
stre
ssTh
est
ates
ofst
ress
insi
de
114
desc
ribed
thro
ugh
stat
es o
f stre
ss. T
he s
tate
s of
stre
ss in
side
th
e yi
eld
surfa
ce a
re e
last
ic w
hile
the
surfa
ce s
igni
fies
the
yiel
d po
ints
.
Tres
ca S
tres
sEq
uiva
lent
Stre
ss
13
T
The
Tres
ca c
riter
ion
(max
imum
she
ar s
tress
theo
ry) a
ssum
es
that
yie
ld o
ccur
s w
hen
the
max
imum
she
ar s
tress
exc
eeds
the
max
imum
shea
rstre
ssin
asi
mpl
ete
nsile
case
max
imum
she
ar s
tress
in a
sim
ple
tens
ile c
ase.
115
Von
Mis
es S
tres
sEq
uiva
lent
Stre
ss
The
von
Mis
es c
riter
ion
(max
imum
dis
torti
on e
nerg
y th
eory
) as
sum
es th
at y
ield
occ
urs
the
dist
ortio
n co
mpo
nent
of t
he to
tal
stra
in e
nerg
y U
sex
ceed
s th
at fr
om th
e si
mpl
e te
nsile
test
.
26
sy
UG
116
22
21
22
33
1(
)(
)(
)2
v