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CHAPTER 1(week 1-3)
Strain Analysis
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Wh at is plane strain?• Perpendicular f rce applied n a
surface• W it h a !ery lar"e #-a$is di%ensi n• plane strain
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Wh at is plane stress?• T h in plate l aded parallel t t h e plane
• &nif r% !er t h e t h ickness• ' stress al n" #-a$is
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GENERAL EQUATION OF PLANE STRAIN
TRANSFORMATION
Transf r% n r%al s h earstrain$ y c %p nent t $* y*S+,' C '.E'T+ '//strain $ and y are p siti!eif cause el n"ati n al n" $and y a$isSh ear strain $y is p siti!eif t h e interi r an"le A 0
ec %e s%aller t h an 2 4
will e p siti!ec untercl ckwise
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• +n 5i" a 6
$
7 $8d$• P siti!e $ ccur line d $
el n"ated $ d $ • wh ich cause line d$* t
el n"ated $ d $ c s 4
Normal Strains ,$
, y
sincos
'
'
dxdydxdx
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• Si%iliar f r y y 7 y8dy
• cause line dy* tel n"ated
y d y sin
Normal Strains, $ , y
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• Sh ear strain is an"le in radian
• 5 r s%all an"le tan • Sh ear strain $y sincetan
• tan $y $y 7 $y 8dy• dy displaced $y dy t t h e ri" h t4
• S l!e al n" $* d$* el n"ate $y dy c s
Shear strain,$y
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• Addin" all t h e el n"ati ns
cossinsincos
dx
cos)sindx(sin)sindx(cos)cosdx(dx
x
xy2
y2
xx
'
'xy
'y
'x
x
'
'
x
'
'
'
cossincos' dydydx xy y x x
Normal and Shear Strains (cont.
sin
cos'
'
dxdy
dxdx
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cossinsincos xy2y2xx '
cossin22sin2/)2cos1(cos
1sincos
2
22
2sin2
2cos22
2sin2
2cos22
'
'
xy y x y x
y
xy y x y x
x
hence
Normal and Shear Strains (cont.
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Normal and Shear Strains (cont.
2sin2
2cos22
xyyxyx
x '
2sin22cos22xyyxyx
y '
Using trigonometric identities:
2cos
22sin
22
xyyxyx ''
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• Principal Strains9 nly n r%al strains9 ' s h ear strain
• :irecti n a$is f principlestrain6
• ;a$ in Plane s h ear strain
• A!e s h ear strain
• :irecti n a$is f s h earstrain
2
xy
2
yx planeinmax,
222
2yx
ave
)(2tan
yx
xy
p
xy
yxs2tan
2
xy
2
yxyx
2,1 222
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C nstructi n f t h e ; h r*s Circle
• (h ri# ntal) represents t h en r%al strain with p siti!e tt h e ri" h t
• (!ertical) represents h alf t h e!alue f t h e s h ear strain /2with p siti!e d wnward 4
• center f t h e circle C is nt h e a$is a!" 7 ( $ < y)8= fr %t h e ri"in4
• Pl t p int A 7 ( $ $y8=)4Represents 7 4• R 7 distance etween A t C
• nce R h as een deter%inedsketc h t h e circle centred C
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• Principal Strains
9 nly n r%al strains9 ' s h ear strain
• :irecti n a$is f principlestrain6
• ;a$ in Plane s h ear strain
• A!e s h ear strain
• :irecti n a$is f s h earstrain
2
xy
2
yx planeinmax,
222
2yx
ave
)(2tan
yx
xy
p
xy
yxs2tan
2
xy
2
yxyx
2,1
222
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• Principal strain 7 n s h earstrain
• 0 : 7• An"le n ; h rs circle 7 =• critical an"le 7 (an"le
etween CA t h ri#antal)8=• :irecti n f plane sa%e as
circle•
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•At p ints E and 54 i4e7 =R
•An"le is h al!es t h esa%e directi n as ncircle
•E4" line CA t CE
Ma"im#m In Plane Shear Strain
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• Si%ilar as !alue at ar itraryplane ( th er plane an"le)
• C rresp nd n r%al andsh ear strain at P >
Strains on Ar$itrar% Plane
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Strain Rosettes
• T h e a$es f t h e t h ree"au"es are arran"ed at t h ean"les f a c4
• +f t h e readin" f a ctaken $ y $y can e
defined4• .alue f $ y $y aredeter%ined y s l!in"t h ese e uati ns4
ccxyc2
yc2
xc
b bxy b2
y b2
x b
aaxya2
ya2
xa
cossinsincos
cossinsincoscossinsincos
Notgiv en
45 o or Rectanlar Rosette
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45 o or Rectanlar Rosette
0
0
0
90
45
0
c
b
a
Th e equation become:
ca bxycy
ax
2
Example of 45 o strain rosette
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60 0 Strain Rosette
c bxy
ac by
ax
32
2231
0
c
0
b
0
a
120
60
0
Th e equation become:
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E$a%ple
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S S i R l i hi!
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Stress Strain Relationshi!• +f a %aterial su @ect t tria$ial
stress ( $ y #) ass ciatedn r%al stress( $ y #)de!el ped in t h e %aterial4
• Wh en $ is applied in $-directi n th e ele%ent el n"atedwith $ in $ directi n4
• Applicati n n y cause t h eele%ent t c ntract wit h astrain $ in t h e $ directi n4
• Applicati n f # cause t h eele%ent t c ntract wit h astrain ***$ in t h e $ directi n4
z
x
y x
x x
'''
''
'
Notgiv en
giv en
St St i R l ti hi! ( t
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• T h e sa%e result can ede!el ped f r t h e n r%alstrain in t h e y and #directi n4
• 5inal results can e
written asB44
yxzz
zxyy
zyxx
1
1
1
Stress Strain Relationshi! (cont.
Notgiv en
giv en
Stress Strain Relationshi! (cont.
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A!!l%in& onl% shear stress, % to the element.+f t apply s h ear stress y t t h e ele%ent4
$y will nly cause def r%ati n t $y4 $y will n t cause def r%ati n t y#4and $# y# and $# will nly cause def r%ati n t
y# and $# respecti!ely4H ke aw f r sh ear stress and s h earstrain written as6
x z x z
y z y z
xy xy
G
G
G
1
1
1
Element subjected to normalstresses only
S h ear stress applied to t h eelements
(
Notgiv en
giv en
Stress Strain Relationshi! (cont.
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• ; dulus f elasticity E isrelated t s h ear % dulus ,4
• :ilatati n (t h e c h an"e in! lu%e per unit ! lu%e r
D! lu%etric strain* e 4
• 0ulk ; dulus (! lu%e% dulus f elasticity) k 4
12
E G
zyxE21
e
(
213 E
k
Notgiv en
giv en
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pr essu r e=s t r ess
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