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Chapter 9: StressTransformation
Finish Chapter 9
Work Problems
Chapter9: StressTransformation
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9.5StressinShaftsDuetoAxialLoadandTorsion
Example:Anaxialforceof900Nandatorqueof2.5Nmareappliedtothe
shaftasshown. Iftheshafthasdiameterof40mm,determinetheprincipal
stressesatapointPonitssurface.
InitialstresselementatP
1
2x
x
kPa7.409
)9.198()2
2.7160
()2(R
kPa1.3582
kPa2.7160
2
2222y
y
avg
xy
==
=
===
x
x
3
Combinedloadings(Ch.8)C (avg, 0)
C (358.1,0)
A (x , xy)
A (0, 198.9)
1 = 767.7 kPa
2 = -51.5 kPa
Clockwise angle = 14.5o
= Tc/J
=N/A
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9.6StressVariationsThroughoutaprismaticBeam
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Direction(orientation)contours
whichgive
the
directions
of
principalstressesofequal
magnitude
Tensileprincipalstresscontourswhich
givethelocationsofidenticaltensile
principalstresses
StressTrajectories&Contours
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2=x=Longitudinal
1=y=Hoop25 CCW
Example AtankistobemadebyrollingflatsheetofAISI1040colddrawnsteelintothespiralshapeasshown,wherethespiralmakesanangleof65withthe
horizontalaxisofthetank. P=1.75MPa,Di=900mm.
(a)Specifyathickness,t,toprovideN=4basedonsyorN=6basedons
u.
(b)Determinethestressconditiononanelementalignedwiththeweld.
InitialStresselement
=PrincipalStressElement
2=x=Longitudinal
1=y=Hoop
StressElementalong
WeldLine
25
vw wv
wv
True3DViewofInitialStressElement
(notusedinthisexample)
2=x=Longitudinal
1=y=Hoop
3=z=Radial
Xaxis
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)(205.1138
908
)(9088900
8,
06.7)5.111(2
)900)(75.1(
222
5.1116
669
6
3.1414
565
4
900nominal&
669,565:1040
75.1
.sonbased6Norsonbased4Nprovidetot,,thicknessaSpecify(a)
min
uy
donecorrectisassumptionwallthint
D
meanmmtDD
assumptionwallthinofvaliditycheckmmtspecifysizepreferredFor
mmMPa
mmMPapDpDt
t
pD
twotheofsmallerMPas
MPas
mmDgivenDcylinderwalledthinAssume
MPasMPassteelCDAISI
MPap
m
im
allowable
m
Hoop
mmHoop
allowableud
y
d
im
uy
>==
=+=+=
=
=====
=====
===
==
==
=
==
Q
Contd
MPa66.492
MPa31.99)mm8(2
)mm908)(MPa75.1(
t2
pD
)b(
1alLongitudinx2
m
TangentialorHoopy1
==
=elementstressprincipaltheassametheiselementstressinitialthecasethisIn
:s)coordinateY-Xtheonbasedelement(stresselementstressinitialtheConstruct
weld.thewithalignedelementanonconditionstresstheDetermine
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2=x=Longitudinal=49.66MPa
1=y=Hoop=99.31MPa
InitialStressElement=PrincipalStress
Element,
andaspecialcaseinMohrscircle
Contd
Xaxis25 CCW
v=90.44MPa
w=58.52MPawvvw=19.02MPa
StressElementalongWeldLine
(CW)
+12O=avg
Xaxis
50 CCW
vw
vw
wv
0
R
(v,vw)
(w,wv)
MPa44.90)50cos(83.2448.74)50cos(R
MPa02.19)50sin(83.24)50sin(R
MPa52.58)50cos(83.2448.74)50cos(R
.weldthealongconditionstresstherepresent
thatcircletheonsintpothefindto5025x2byCCWRotate
MPa83.242/)66.4931.99(2/)(RRadius
MPa48.742/)66.4931.99(2/)(Center
avgv
wv
avgw
21
21avg
==
=
==
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2=x=Longitudinal
1=y=Hoop25
vw w
v
wv
InitialStresselement
=PrincipalStressElement
StressElementalong
WeldLine
2=x=Longitudinal=49.66MPa
1=y=Hoop=99.31MPa
25 CCW
v=90.44MPa
w=58.52MPawvvw=19.02MPa
Summary
Xaxis
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9.7AbsoluteMaximumShearStress
General3D
stateof
stress Thru
3D
stress
transformation
and actingonanyskewedplanecanbedetermined.
Thereisanorientationhavingonly
principalstresses(min,int,max)actingontheelement Triaxial
Stress.
Atanother
orientation
the
absolutemaximumshear
stressabsmaxwilloccur.
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2
2
minmaxavg
minmax
=
=maxabs
Howtofindabsmax
Circle with the largest R
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Needtotakea3Dviewandfindabsmaxwhentheinplaneprincipalstresseshavethesame
sign bothtensileorbothcompressive
Aplanestresscasewith
principalstresseshavingthe
same
sign
22
0)( maxmaxmax'z'x
==maxabs
Givenplane
stress
case
Bothtensile
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2)( minmaxmax'y'x
=maxabs
Noneedtotakea3Dviewwhentheinplane
principalstresseshavetheopposite one
tensileandone compressive
Oppositesign
Givenplanestresscase
Aplanestresscasewith
principalstresseshavingthe
opposite
sign
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(cw)
+1=0
23
+
(cw)
12
3=0
Key:maxistheradiusofthelargestcircle.
SpecialCase1inTextbookSpecialCase2inTextbook
Initial
Stress
Elements
Principal
Stress
Elements
+
(cw)
12=03
1>2=0 >3 1>2>3 =01=0>2>3
Need3D
View Need
3D
View
Review:MohrsCircle
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Example:Duetotheappliedloading,theelementatthepointontheframeissubjectedtothestateof
stressshown. Determinetheprincipalstressesandtheabsolutemaximumshearstressatthepoint.
Oppositesign
max = -10+41.2=31.2min = -10-41.2 = -51.2
= 38.0o
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Example:Thepointonthesurfaceofthecylindricalpressurevesselissubjectedtothestateofstress.
Determinetheabsolutemaximumshearstressatthepoint.
Anorientationofanelement45 withintheplanecontainingmax=32MPaandmin=0yieldsthestateofabsolutemaximumshear
stressand
the
associated
average
normal
stress.
Thisisacasethatprincipalstresseshavethesamesign,soneedto
thinkthe
stress
state
in
3D.
MPa162032
2
MPa162
032
2
minmaxavg
minmax
===
===maxabs
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Homework # 10Due Thursday April 16
9-699-73
9-75
9-889-97
Exam 2: Thursday 4/23/09 (not next Thursday)
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Lets work some problems
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5/27/2018 ENTC 376 Chapter 9 Lecture Notes-III-Stress Transformation
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Materialsafter
this
slide
are
extra
references
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InitialStateof
Stress
InitialStressElement(basedonxyzcoordinates)
Stress
Analysis
Design
Project
MinPrincipal
Stress
MaxShear
Stress
Chapter11: TheGeneralCaseofCombinedStressandMohrsCircle
Goals:
Combined
Stress
xA xyyx
y
x
y
MohrsCircle
directoruniaxialdirecttorsionalbending
v(beam)
MaxPrincipal
Stress
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MoreaboutInitialStressElementforaGeneralCaseofCombinedStresses
DirectNormal
and/or
BendingStress
TorsionalShear
and/or
VerticalShear
xPoint
of
Interest xy
yx
y
y
x
xy
yx
(Combined)NormalStress (Combined)ShearStress
Whataretheresultants(calledmaxprincipalstressandminprincipalstress)
and
max(maximumshear)duetoallstressescombined?
x
y
Plane
Stress
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StressTransformation:UniaxialStress
=
=
cossincossinA
P
cos
A
sinP
A
V
coscosAP
cos
AcosP
AN
x
2x
2
A
sinPV
cosPN
Forces
=cos
AA
PA
Px=
P
V
N
S=P
P
x
y
xmax
xmax
2
1,45
,0
When
When
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StressTransformation:BiaxialStress(noshearstress)
cossin)(
cos)tanA(sin)A()secA(
:
sincos
sin)tanA(cos)A()secA(
:
yx
yx
2
y
2
x
yx
ofdirectiontheinmequilibriuForce
ofdirectiontheinmequilibriuForce
'
yx'
2sin)(2
1
2cos)(21)(
21
:
yx
'
yxyx'
'' for2
substitute,andfindTo
2cos)(2
1)(
2
1yxyx
2sin)(2
1yx
stressesprincipal
calledarestressnormalofvaluesminandmaxSuch
smallest.theisothertheandlargesttheisone
stressestheseAmong;tofromvaries yx 2cos)(
21)(
21 yxyxQ
x
y
x
y
x
y
A
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StressTransformation:GeneralCaseofPlaneStress
2cos2sin)(2
1
2sin2cos)(2
1
)(2
1
xyyx
xyyxyx
:equationsstressshearandnormalGeneral
planes)principaltheonstressesshearnoareThere
stressprincipalmin
stressprincipalmax
.,e.i(0
)](2
1[)(
2
1
)](
2
1[)(
2
10
d
d
2
xy
2
yxyx2min
2
xy
2
yxyx1max
=
))(
2
1
()](2
1[
dd
avgmaxavgyx
2
xy
2
yxmax
=
withdaccompanieis(i.e.,
planes)principaltheto45atoccur
0 Rememberthemax=((x/2)2+xy2) inCh.10?
0
)2
()
avg
2
xy
2yx22
avg
=
=
andatcenterwithcircleaofequationanisThis
(
:givesandCombine
x
y
xyyx
x
y
x
y
xy
yxxy
yx
MohrsCircle
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SpecialCases:BothPrincipalStressesHavetheSameSign
1>2>3=0(Botharetensile)1=0>2>3 (Botharecompressive)
.
)no(
)(2
1
max
StressincipalPrMinStressincipalPrMaxmax
thefindtowantweandelement
stressinitialanonactingstresses
principalminandmaxwithcase
stressuniaxialaasviewedbecanIt
PlaneStress
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Case1:1>2>3=0(BothAreTensile)
2
StressincipalPrMinTrue0Circles'Mohrst1thefromStressincipalPrMin
Circles'Mohrst1thefromStressincipalPrMax
1max
3
2
1
==
==
.stressshearimummaxtruetheis
MPa3.1082
6.216
2)(
2
1
and
,stressprincipalimummintruethenowis0.,e.i
.caseD3aconsidertoneedwe,Thus
!signsamethehaveand
MPa4.1036.56160R
MPa6.2166.56160R
MPa6.564040baR
MPa40b
MPa40)120200(2
1)(
2
1a
MPa160)120200(2
1)(
2
1o,Center
:Circles'MohrFirst
131max
3
21
avg2
avg1
2222
max
xy
yx
yxavg
==
=
==
==
=
=
Q
SameSign
1
2
x-y planey-z plane
x-z plane
x-y plane
x
y
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Case2:1=0>2>3 (BothareCompressive)
2
circles'Mohrst1thefromStressincipalPrMin
circles'Mohrst1thefromStressincipalPrMax
StressincipalPrMaxTrue0
3max
3
2
1
===
=
.stressshearimummaxtruetheis
MPa3.9326.186
2)(
21
and
)principalmintrue(MPa6.186and,MPa4.43
,stressprincipalimummaxtruethenowis0.,e.i
.caseD3aconsidertoneedwe,Thus!signsamethehaveand
MPa6.1866.71115R
MPa4.436.71115R
MPa6.713065baR
MPa30b
MPa65)18050(2
1)(
2
1a
MPa115)18050(2
1)(
2
1o,Center
:Circles'MohrFirst
331max
32
1
21
avg2
avg1
2222max
xy
yx
yxavg
==
=
==
=
Q
3
=186.6MPa
2=43.4MPa
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(cw)
+1=0
23
+
(cw)
12
3=0
Key:maxistheradiusofthelargestcircle.
SpecialCase1inTextbookSpecialCase2inTextbook
Initial
Stress
Elements
Principal
Stress
Elements
+
(cw)
12=03
1>2=0 >3 1>2>3 =01=0>2>3
Need3D
View Need
3D
View
Review:MohrsCircle