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    16.STRESSATAPOINT

    I MainTopics

    AStressvector(trac=on)onaplane

    BStressatapoint

    CPrincipalstresses

    D Transforma=onofprincipalstressestotrac=onsin

    2D

    10/16/11 GG303 1

    16.STRESSATAPOINTI Stressvector(trac=on)onaplane

    A

    B Trac=onvectorscanbeaddedasvectors

    C Atrac=onvectorcanberesolvedintonormal(n)andshear(s)components

    1 Anormaltrac=on(n)actsperpendiculartoaplane

    2 Asheartrac=on(s)actsparalleltoaplane

    D Localreferenceframe

    1 Then-axisisnormaltotheplane

    2 Thes-axisisparalleltotheplane

    10/16/11 GG303 2

    = limA0

    F / A

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    16.STRESSATAPOINT

    III Stressatapoint(cont.)

    A Stressesrefertobalancedinternal"forces(perunitarea)".Theydifferfromforcevectors,which,ifunbalanced,causeaccelera=ons

    B"On-inconven=on":Thestresscomponent ijactsontheplanenormaltothei-direc=onandactsin

    thej-direc=on1Normalstresses:ij

    2Shearstresses:ij

    10/16/11 GG303 3

    16.STRESSATAPOINTIIIStressatapointC Dimensionsofstress:

    force/unitarea

    D Conven=onforstresses

    1 Tensionisposi=ve2 Compressionis

    nega=ve

    3 Followsfromon-inconven=on

    4 Consistentwithmostmechanicsbooks

    5 Countertomostgeologybooks

    10/16/11 GG303 4

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    16.STRESSATAPOINTIII Stressatapoint

    C

    D

    E Innature,thestateofstresscan(andusuallydoes)varyfrompointto

    point

    F Forrota=onalequilibrium,xyyx,xzzx,yzzy

    10/16/11 GG303 5

    ij =

    xx xy xz

    yx yy yz

    zx zy zz

    3-D

    9components

    ij =

    xx xy

    yx

    yy

    2-D

    4components

    16.STRESSATAPOINTIVPrincipalStresses(thesehave

    magnitudesandorienta=ons)

    A Principalstressesactonplaneswhichfeelnoshearstress

    B Theprincipalstressesarenormalstresses.

    C Principalstressesactonperpendicularplanes

    D Themaximum,intermediate,andminimumprincipalstressesareusually

    designated1,2,and3,respec=vely.

    E Principalstresseshaveasinglesubscript.

    10/16/11 GG303 6

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    16.STRESSATAPOINT

    IVPrincipalStresses(cont.)

    F Principalstresses

    representthestress

    statemostsimply

    G

    H

    10/16/11 GG303 7

    ij =

    xx

    xy

    xz

    yx yy yz

    zx zy zz

    3-D

    3components

    ij =

    1

    0

    0 2

    2-D

    2components

    16.STRESSATAPOINT

    10/16/11 GG303 8

    V Transforma=onofprincipalstressestotrac=onsin2D

    A Descrip=onofterms

    1 ThreeplanesA,A1,andA2formthesidesofatriangularprism;thesehavenormalsinthen-,1-,and-2-direc=ons,respec=vely.

    2 PlaneA1isactedonbyknownnormalstress1.

    3 PlaneA2isactedonbyknownnormalstress2.

    4 Then-direc=onisatangle1

    ()withrespecttothe1-direc=on,andatangle2withrespecttothe2-direc=on.

    5 Thes-direc=onisatacounter-clockwise90anglerela=vetothen-direc=on(likeyandx).

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    16.STRESSATAPOINT

    10/16/11 GG303 9

    VTransforma=onofprincipalstressestotrac=onsin2D

    B Approach

    Findweigh=ngfactorsthatdeterminecontribu=onsofknownstressestodesiredtrac=onsandsumcontribu=ons

    1 nwn11+wn222 sws11+ws22

    16.STRESSATAPOINT

    10/16/11 GG303 10

    C Contribu=onof1(onfaceA1ofareaA1)ton

    (onfaceAofareaA)

    Startwiththedefini=onoftrac=on:

    1 n(1)Fn

    (1))/A

    FindunknownsFn(1)andAfromknowns2and.

    FirstfindtheforceF1associatedwith1

    2 F11A1Force(stress)(area)

    FindFn(1),thecomponentofF1inthen-direc=on

    3 Fn(1)F1cos1

    FindAintermsofA1

    A1Acos1(seediagramatright)

    4 AA1/cos1

    Contribu=onof1to

    n:

    5a n(1)Fn

    (1)/AF1cos1/(A1/cos1)

    5b n(1)(F1/A1)cos1cos11cos1cos1

    Weigh=ngfactorwn1

    6 wn1cos1cos1coscos

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    16.STRESSATAPOINT

    10/16/11 GG303 11

    D Contribu=onof2(onfaceA2ofareaA2)ton

    (onfaceAofareaA)

    Startwiththedefini=onoftrac=on:

    1 n(2)Fn

    (2))/A

    FindunknownsFn(2)andAfromknowns2and.

    FirstfindtheforceF2associatedwith2

    2 F22A2Force(stress)(area)

    FindFn(2),thecomponentofF2inthen-direc=on

    3 Fn(2)F2cos2

    FindAintermsofA2

    A2Acos2(seediagramatright)

    4 AA2/cos2

    Contribu=onof2ton:

    5a n(2)Fn(2)/AF2cos2/(A2/cos2)5b n

    (2)(F2/A2)cos2cos22cos2cos2

    Weigh=ngfactorwn2

    6 wn2cos2cos2sinsin

    16.STRESSATAPOINT

    10/16/11 GG303 12

    E Contribu=onof1(onfaceA1ofareaA1)tos(on

    faceAofareaA)

    Startwiththedefini=onoftrac=on:

    1 s(1)Fs

    (1))/A

    FindunknownsFs(1)andAfromknowns1and.

    FirstfindtheforceF1associatedwith1

    2 F11A1Force(stress)(area)

    FindFs(1),thecomponentofF1inthes-direc=on

    3 Fs(1)-F1cos2

    FindAintermsofA2

    A1Acos1(seediagramatright)

    4 AA1/cos1

    Contribu=onof1to

    nof

    1:

    5a s(1)Fs

    (1)/A-F1cos2/(A1/cos1)

    5b s(1)-(F1/A1)cos2cos1-1cos2cos1

    Weigh=ngfactorws1

    6 ws1-cos2cos1-sincos

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    16.STRESSATAPOINT

    10/16/11 GG303 13

    F Contribu=onof2(onfaceA1ofareaA1)tos

    (onfaceAofareaA)

    Startwiththedefini=onoftrac=on:

    1 s(2)Fs

    (2))/A

    FindunknownsFs2)andAfromknowns2and.

    FirstfindtheforceF2associatedwith2

    2 F22A2Force(stress)(area)

    FindFs(2),thecomponentofF2inthes-direc=on

    3 Fs(2)F2cos1

    FindAintermsofA1

    A1Acos1(seediagramatright)

    4 AA2/cos2

    Contribu=onof1tonof1:

    5a s(2)Fs(2)/AF2cos1/(A2/cos2)5b s

    (2)(F2/A2)cos1cos22cos1cos2

    Weigh=ngfactorws2

    6 ws2cos1cos2cossin

    16.STRESSATAPOINT

    10/16/11 GG303 14

    VTransforma=onofprincipalstressestotrac=onsin2D

    G Originalequa=ons

    1 nwn11+wn12

    2 sws11+ws12H Originalequa=ons

    1 ncoscos1

    +sinsin22 s-sincos1+sincos2 Weigh=ngfactorsareproducts

    oftwodirec=oncosines