entc 376 chapter 9 lecture notes-iii-stress transformation

5/27/2018 ENTC 376 Chapter 9 Lecture Notes-III-Stress Transformation

1/27

Chapter 9: StressTransformation

Finish Chapter 9

Work Problems

Chapter9: StressTransformation


2/27

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


3/27

9.6StressVariationsThroughoutaprismaticBeam


4/27

Direction(orientation)contours

whichgive

the

directions

of

principalstressesofequal

magnitude

Tensileprincipalstresscontourswhich

givethelocationsofidenticaltensile

principalstresses

StressTrajectories&Contours


5/27

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


6/27

)(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


7/27

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

==

=

==


8/27

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


9/27

9.7AbsoluteMaximumShearStress

General3D

stateof

stress Thru

3D

stress

transformation

and actingonanyskewedplanecanbedetermined.

Thereisanorientationhavingonly

principalstresses(min,int,max)actingontheelement Triaxial

Stress.

Atanother

orientation

the

absolutemaximumshear

stressabsmaxwilloccur.


10/27

2

2

minmaxavg

minmax

=

=maxabs

Howtofindabsmax

Circle with the largest R


11/27

Needtotakea3Dviewandfindabsmaxwhentheinplaneprincipalstresseshavethesame

sign bothtensileorbothcompressive

Aplanestresscasewith

principalstresseshavingthe

same

sign

22

0)( maxmaxmax'z'x

==maxabs

Givenplane

stress

case

Bothtensile


12/27

2)( minmaxmax'y'x

=maxabs

Noneedtotakea3Dviewwhentheinplane

principalstresseshavetheopposite one

tensileandone compressive

Oppositesign

Givenplanestresscase

Aplanestresscasewith

principalstresseshavingthe

opposite

sign


13/27

(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


14/27

Example:Duetotheappliedloading,theelementatthepointontheframeissubjectedtothestateof

stressshown. Determinetheprincipalstressesandtheabsolutemaximumshearstressatthepoint.

Oppositesign

max = -10+41.2=31.2min = -10-41.2 = -51.2

= 38.0o


15/27

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


16/27

Homework # 10Due Thursday April 16

9-699-73

9-75

9-889-97

Exam 2: Thursday 4/23/09 (not next Thursday)


17/27

Lets work some problems


18/27

Materialsafter

this

slide

are

extra

references


19/27

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


20/27

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


21/27

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


22/27

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


23/27

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


24/27

SpecialCases:BothPrincipalStressesHavetheSameSign

1>2>3=0(Botharetensile)1=0>2>3 (Botharecompressive)

.

)no(

)(2

1

max

StressincipalPrMinStressincipalPrMaxmax

thefindtowantweandelement

stressinitialanonactingstresses

principalminandmaxwithcase

stressuniaxialaasviewedbecanIt

PlaneStress


25/27

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


26/27

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


27/27

(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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