MohrCircleforstress In2Dspace(e.g.,onthe12,13,or23plane),thenormalstress(n)andtheshearstress(s),couldbegivenbyequations(1)and(2)inthenextslides

Note:Theequationsaregivenhereinthe12 plane,where1 isgreaterthan2.

Ifweweredealingwiththe23plane,thenthetwoprincipalstresseswouldbe2and3

NormalStressThenormalstress,n

n=(1+2)/2+(12)/2cos2

Inparametricformtheequationbecomes:n=c+rcos

Where

c =(1+2)/2isthe center,whichliesonthenormalstressaxis(xaxis)

r=(12)/2 istheradius =2

SignConventionsn iscompressivewhenitis+,i.e.,whenn>0n istensilewhenitis-,i.e.,whenn

ResolvedNormalandShearStressnormal to plane

ShearStressTheshearstress

s =(12)/2sin2 Inparametricformtheequationbecomes:

s=rsin where =2s>0+ shearstressrepresentsleftlateralshears

ConstructionoftheMohrCirclein2D

Plotthenormalstress,n,vs.shearstress,s,onagraphpaperusingarbitraryscale(e.g.,mmscale!)

Calculate: Centerc=(1+2)/2 Radiusr=(12)/2

Note:Diameteristhedifferentialstress (12)

Thecircleintersectsthen (xaxis)atthetwoprincipalstresses(1 and2)

ConstructionoftheMohrCircle

Multiplythephysicalangleby2 Theangle2 isfromthec linetoanypointonthecircle

+2 (CCW)anglesarereadabovethexaxis 2 (CW)anglesbelowthexaxis,from the1 axis

Thenandsofapointonthecirclerepresentthenormalandshearstressesontheplanewiththegiven2angle

NOTE:TheaxesoftheMohrcirclehavenogeographicsignificance!

MohrCircleforStress

.Max s

MohrCirclein3D

Maximum&MinimumNormalStressesThenormalstress

n=(1+2)/2+(12)/2cos2 inphysicalspaceistheanglefrom1 tothenormal

totheplane

When then cos2and n=(1+2)/2+(12)/2whichreducestoamaximumvalue:n=(1+2+12)/2 n=21/2 n=1When then cos2and n=(1+2)/2 (12)/2whichreducestoaminimumn=(1+2 1+2)/2 n=2/2 n=

SpecialStatesofStress UniaxialStress

UniaxialStress (compressionortension) Oneprincipalstress(1 or3)isnonzero,andtheothertwoareequaltozero

UniaxialcompressionCompressivestressinonedirection:1 >2=3 =0

|a 0 0||0 0 0||0 0 0|

TheMohrcircleistangenttotheordinateattheorigin(i.e.,2=3=0)onthe+(compressive)side

SpecialStatesofStress

UniaxialTension

Tensioninonedirection:1 =2 >3

|0 0 0||0 0 0||0 0-a|

TheMohrcircleistangenttotheordinateattheoriginonthe (i.e.,tensile)side

SpecialStatesofStress AxialStress

Axial(confined)compression:1 >2=3>0|a 0 0||0 b 0||0 0 b|

Axialextension(extension):1 =2 >3>0|a 0 0||0 a 0||0 0 b|

TheMohrcircleforbothofthesecasesaretotherightoftheorigin(nontangent)

SpecialStatesofStress BiaxialStress BiaxialStress:

Twooftheprincipalstressesarenonzeroandtheotheriszero

PureShear:1 =3 andisnonzero(equalinmagnitudebutoppositeinsign)

2 =0 (i.e.,itisabiaxialstate) Thenormalstressonplanesofmaximumsheariszero(pureshear!)|a 0 0 ||0 0 0 ||0 0 -a|

TheMohrcircleissymmetricw.r.t.theordinate(centerisattheorigin)

SpecialStatesofStress

SpecialStatesofStress TriaxialStress

TriaxialStress: 1,2, and3havenonzerovalues 1>2 >3 andcanbetensileorcompressive

Isthemostgeneralstateinnature|a 0 0 ||0 b 0 ||0 0 c |

TheMohrcirclehasthreedistinctcircles

TriaxialStress