lect strain transformation

7/26/2019 Lect Strain Transformation

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ME 2406: Strength of Materials

Strain Transformation

Dr. Faraz Junejo


2/29

Oje!ti"es

Apply the stresstransformation methodsderived in previous chapteri.e. chapter 9 to similarlytransform strain


3/29

#lain Strain

As explained earlier, general state ofstrain in a body is represented by acombination of 3components of normalstrain ( x, y, z), and 3 components of

shear strain (xy, xz, yz).

A plane-strained element is subjected

to two components of normal strain (x,y) and one component of shear strain,

xy.


4/29

#lain Strain $%ont&.' he deformations are shown graphically

below. !ote that the normal strains are produced

by changes in length of the element in thex

and y directions, while shear strain isproduced by the relative rotation of twoadjacent sides of the element.


5/29

#lain Strain $%ont&.' !ote that plane stress does not always cause

plane strain.

"n general, unless # $, the %oisson e&ectwill prevent the simultaneous occurrence of

plane strain and plane stress. 'ince shear stress and shear

strain not a&ected by %oissons

ratio, condition of xz# yz# $reuires xz# yz# $.


6/29

#lain Strain: Sign %on"ention

o use the same convention asde*ned in %ha(ter 2.

+ith reference to di&erential

element shown, normal strainsxand yare positive if they

cause elongation along the xandyaxes respectively.

'hear strain xyis positive if the interior

angle Abecomes smaller than 9$.


7/29

#lain Strain Transformation

eneral /uations for %lain 'trainransformation are0

( )5102sin22cos22' -

xyyxyx

x +

+

+

=

( )6-102cos22sin22

''

xyyxyx

+

=


8/29

"f normal strain in the ydirection isreuired, it can be obtained from E)uation

*0+, by substituting (1 9$) for .heresult is( )7-102sin

22cos

22'

xyyxyx

y

+=


9/29

#rin!i(al Strains

+e can orientate an element at a ptsuch that the elements deformation isonly represented by normal strains,with no shear strains.

he material must be isotropic, and theaxes along which the strains occur mustcoincide with the axes that de*ne the

principal axes.

hus from /uations 9-2 and 9-3,( )8-102tan

yx

xy

p

=


10/29

#rin!i(al strains

Ma-imum in+(lane shear strain

4sing /uations 9-5, 9-6 and 9-7, we

get

( )910222

22

2,1 -

+

+

=

xyyxyx

( )1110222

22plane-in

max

-

+

= xyyx

( )10102tan -

=xy

yx

s


11/29

#rin!i(al Strains $!ont&.'

Ma-imum in+(lane shear strain

4sing /uations 9-5, 9-6 and 9-7, weget

( )12102 -avgyx

+=


12/29

Summar

8ue to Poisson efect, the state of

plane strain is not a state of plane

stress, and vice versa.

A point on a body is subjected to planestress when the surface of the body is

stress-ree.

+hen the state of strain is representedby the (rin!i(al strains, no shear

strain will act on the element.


13/29

Summar $!ont&.'

he state of strain at the point canalso be represented in terms of the

ma-imum in+(lane shear strain."n this case, an average normal

strain will also act on the element.

he element representing themaximum in-plane shear strain and

its associated average normal strainsis 4, from the element representingtheprincipal strains.


14/29

E-am(le: *A di&erential element ofmaterial at a point is

subjected to a state ofplane strain x /

,00$*0+6' y /

100$*0+6' and xy =

200$*0+6' which tends

to distort the element as

shown. Determine the

e)ui"alent strains

acting on an element

oriented at the point,

clockwise 30 rom the

original position.


15/29

E-am(le: * $!ont&.' 'ince is clocwise, then / 10, use

strain-transformation /uation :$-3 toobtain

( ) ( )( ) ( ) ( )( )

( )( )( )

( )6

'

6

6

6

'

10213

302sin2

10200

302cos102

300500

102

300500

2sin2

2cos22

=

+

+

+=

+

++

=

x

xyyxyxx


16/29

E-am(le: * $!ont&.'

'ince is clocwise, then / 10

, usestrain-transformation /uation :$-5 toobtain

( )( )( )

( ) ( )( )

( )6''

6

''

10793

302cos2

10200

302sin2

300500

2cos22sin22

=

+

=

+

=

yx

xyyxyx


17/29

E-am(le: * $!ont&.' 'train in theydirection can be obtained from

/uation :$-6 with # ;


18/29

E-am(le: * $!ont&.'

he results obtained tend to deform the

element as shown below.


19/29

E-am(le: 2A differential element of material at a point is subjected

to a state of plane strain defined by x= 350(0-6!"y= #00(0-6!" xy= $0(0-6!" %&ic& tends to distort t&e

element as s&o%n' etermine t&e principal strains at

t&e point and associated orientation of t&e element'


20/29

E-am(le: 2 $!ont&.'>rom /uation :$-7, we have rientation of the

element

/ach of these angles is measuredpositive counterclocwise, from thexaxis to the outward normals oneach face of the element.

( )

==+=

=

=

9.8514.4,17218028.828.82

)10(200350

)10(802tan

6

6

andt&atsoand)&us

p

p

yx

xyp


21/29

E-am(le: 2 $!ont&.'

#rin!i(al strains can be obtained >rom

/uation :$-9,

( )( ) ( )

( ) ( )( ) ( )6261

66

6226

22

2,1

1035310203

109.277100.75

102

80

2

200350

2

10200350

222

== =

+

+=

+

+=

xyyxyx


22/29

E-am(le: 2 $!ont&.'

#rin!i(al strains

+e can determine which of these twostrains deforms the element in the xdirection by applying /uation :$-3 with

# ; 2.:2. hus

( ) ( ) ( )

( )( )

( )6

'

6

66

'

10353

14.42sin2

1080

14.4cos10

2

20035010

2

200350

2sin2

2cos22

=

+

+

+=

+++=

x

xyyxyxx


23/29

E-am(le: 2 $!ont&.'

%rincipal strains

=ence x# ?. +hen subjected to the

principal strains, the element is distortedas shown.


24/29

E-am(le: 1

A di&erential element of material at a point is

subjected to a state of plane strain de*ned by x# ;


25/29

E-am(le: 1 $!ont&.'

rientation of the element can beobtained >rom /uation :$-:$,

!ote that this orientation is 4, from thatshown in previous /xample i.e. /xample ?as expected.

( )( )

( )

==+=

=

=

9.1309.40

,72.26118072.8172.8121080

102003502tan

6

6

and

t&atsoand)&us"

s

s

xy

yx

s


26/29

E-am(le: 1 $!ont&.'@aximum in-plane shear strain

Applying /n :$-::,

he proper sign of can be obtained by

applying /uation :$-5 with s/ 40.3.

( )

( )6

622

22

10556

10280

2200350

222

plane-in

max

plane-in

max

=

+

=

+

=

xyyx

plane-in

max

l $ & '


27/29

E-am(le: 1 $!ont&.'

@aximum in-plane shear strain

hus tends to distort the element so that theright angle between dx and dy is decreased

(positive sign convention).

( ) ( )

( )( )

( )6''

6

6

''

10556

9.402cos2

1080

9.402sin102

200350

2cos2

2sin22

=

+

=

+=

yx

xyyxyx

plane-in

max


28/29

E-am(le: 1 $!ont&.'here are associated average normal strains

imposed on the element determined from/uation :$-:?0

)&ese strains tend to

cause t&e element tocontract'

( ) ( )66 1075102

200350

2

=+=+

= yx

avg


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%ourse or5 E-er!ises

*ec&anics of *aterials $t&+dition by ,'' .ibbeler

/ 0' / 0'6

/ 0'3 / 0'

lect strain transformation

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