stres solid mechanic

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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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