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16.STRESSATAPOINT
I MainTopics
AStressvector(trac=on)onaplane
BStressatapoint
CPrincipalstresses
D Transforma=onofprincipalstressestotrac=onsin
2D
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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
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= 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
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16.STRESSATAPOINTIIIStressatapointC Dimensionsofstress:
force/unitarea
D Conven=onforstresses
1 Tensionisposi=ve2 Compressionis
nega=ve
3 Followsfromon-inconven=on
4 Consistentwithmostmechanicsbooks
5 Countertomostgeologybooks
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16.STRESSATAPOINTIII Stressatapoint
C
D
E
Innature,thestateofstresscan(andusuallydoes)varyfrompointto
point
F Forrota=onalequilibrium,xyyx,xzzx,yzzy
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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.
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16.STRESSATAPOINT
IVPrincipalStresses(cont.)
F Principalstresses
representthestress
statemostsimply
G
H
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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
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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
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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
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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
