Upload
aap1
View
12
Download
1
Embed Size (px)
DESCRIPTION
Mohr Circle for stress
Citation preview
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
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