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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

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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

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VTransforma=onofprincipalstressestotrac=onsin2D

B Approach

Findweigh=ngfactorsthatdeterminecontribu=onsofknownstressestodesiredtrac=onsandsumcontribu=ons

1 nwn11+wn222 sws11+ws22

16.STRESSATAPOINT

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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)


1 n(2)Fn

(2))/A

FindunknownsFn(2)andAfromknowns2and.



FindFn(2),thecomponentofF2inthen-direc=on

3 Fn(2)F2cos2

FindAintermsofA2


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)


1 s(1)Fs

(1))/A

FindunknownsFs(1)andAfromknowns1and.



FindFs(1),thecomponentofF1inthes-direc=on

3 Fs(1)-F1cos2

FindAintermsofA2


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)


1 s(2)Fs

(2))/A

FindunknownsFs2)andAfromknowns2and.



FindFs(2),thecomponentofF2inthes-direc=on

3 Fs(2)F2cos1

FindAintermsofA1


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

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VTransforma=onofprincipalstressestotrac=onsin2D

G Originalequa=ons

1 nwn11+wn12

2 sws11+ws12H Originalequa=ons

1 ncoscos1

+sinsin22 s-sincos1+sincos2 Weigh=ngfactorsareproducts

oftwodirec=oncosines

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Lec.16.pptx