第七章应力状态与应变状态分析

CHAPTER7

ANALYSISOFSTRESSANDSTRAIN

MechanicsofMaterials第七章应力状态与应变状态分析材料力学CHAPTER7ANALYSISOFTHESTATEOFSTRESSANDSTRAIN

§7–4

PRINCEPALSTRESSESANDTHEIRTRAJECTORIESOFTHEBEAM§7–5

ANALYSISOFTRIAXIALSTRESSEDSTATE—METHODOFSTRESSCIRCLE§7–6

ANALYSISOFSTRAININAPLANE§7–7

RELATIONBETWEENSTRESSANDSTRAINUNDERCOMPLEXSTRESSEDSTATE—（GENERALIZEDHOOKE’SLAW）

§7–8STRAIN-ENERGYDENSITYUNDERCOMPLEX

STRESSEDSTATE

§7–1CONCEPTSOFTHESTATEOFSTRESS§7–2ANALYSISOFTHESTATEOFPLANESTRESS—ANALYTICALMETHOD§7–3ANALYSISOFTHESTATEOFPLANESTRESS—

GRAPHYCALMETHOD

第七章应力状态与应变状态分析

§7–1应力状态的概念§7–2平面应力状态分析——解析法§7–3平面应力状态分析——图解法§7–4

梁的主应力及其主应力迹线§7–5

三向应力状态研究——应力圆法§7–6

平面内的应变分析§7–7

复杂应力状态下的应力--应变关系

——（广义虎克定律）§7–8

复杂应力状态下的变形比能问题引入：杆件在几种基本变形（拉伸、压缩、扭转、弯曲）的强度问题，建立了只用正应力或切应力作用时的强度条件，而工程实际中，几种基本变形组合在一起，称为组合（叠加）变形（小变形情况下）对应构件横截面上某点存在正应力和切应力时，能否分别对正应力和切应力建立独立的强度条件进行计算？（X）前面讨论构件基本变形的强度问题时，是用横截面上的危险点处的应力建立强度条件进行强度计算，而有些破坏没有发生在试件横截面上，而是斜截面上。需研究斜截面上的应力状态。应力状态（概念）：指构件内过一点处沿不同方向斜截面上的应力状态。应力状态和强度理论为研究杆件在复杂变形时的强度问题提供了理论基础。。。。研究一点处的应力状态，引入单元体（P193）§7–１

CONCEPTSOFTHESTAFEOFSTRISS1、Forward1)、Investigationonthetensile,compressiveandtorsionaltestofcastironandlow-carbonsteelMLow-carbonsteelCastironPPCastironintension

PCastironincompression2)、Howwillthememberruptureincombineddeformations？MPANALYSISOFSTRESSANDSTRAIN§7–１应力状态的概念应力状态与应变状态一、引言1、铸铁与低碳钢的拉、压、扭试验现象是怎样产生的？M低碳钢铸铁PP铸铁拉伸P铸铁压缩2、组合变形杆将怎样破坏？MP4、Expressionofstressesingeneralcase

3、Element：Element—Delegateofapointinthemember.Itisainfinitesimalgeometricbodyenvelopingthestudiedpoint.Incommonuseitisacorrectitudecubicbody.Propertiesofanelement—a、Stressesaredistributed uniformlyinthesections；

b、Thestressesintwoplanesthatare paralleltoeachotherareequal.2、Stateofstressatapoint：

Therearecountlesssectionsthroughapoint.Thegatheringofstressesinallsectionsiscalledthestateofstressatthispoint.xyzs

xsz

s

ytxyANALYSISOFSTRESSANDSTRAIN四、普遍状态下的应力表示

三、单元体：单元体——构件内的点的代表物，是包围被研究点的无限小的几何体，常用的是正六面体。单元体的性质——a、平行面上，应力均布；

b、平行面上，应力相等。二、一点的应力状态：

过一点有无数的截面，这一点的各个截面上应力情况的集合，称为这点的应力状态（StateofStressataGivenPoint）。xyzs

xsz

s

y应力状态与应变状态txyxyzs

xsz

s

ytxy5、Theoremofconjugateshearingstress:

Shearingstressesonperpendicularplanesareequalinmagnitudeandhavedirectionssuchthatbothstressespointtoward,orbothpointawayform,thelineofintersectionofthefaces.ANALYSISOFSTRESSANDSTRAINProvement:Theelementisinequilibrium.xyzs

xsz

s

y应力状态与应变状态txy五、剪应力互等定理（TheoremofConjugateShearing

Stress):

过一点的两个正交面上,如果有与相交边垂直的剪应力分量,则两个面上的这两个剪应力分量一定等值、方向相对或相离。6、Originalelement（knownelement）：Example1

PlottheknownelementsofpointA、B、Cshowninthefollowingfigures.ANALYSISOFSTRESSANDSTRAINtzx

PPAAsxsxMPxyzBCsxsxBtxztxytyxtzx六、原始单元体（已知单元体）：[例1]

画出下列图中的A、B、C点的已知单元体。

应力状态与应变状态PPAAsxsxMPxyzBCsxsxBtxztxytyx7、Principalelement、principalplanes、principalstresses：

Principalelement：

Theelementinwhichtheshearingstressesinsideplanesareallzero.

PrincipalPlanes：

Theplanesonwhichtheshearingstressesarezero.

Principalstresses：

Normalstressesactingontheprincipleplanes.conventionoftheorderforthreeprincipalstresses:

Inmagnitudeofthealgebraicvalue，s1s2s3xyzsxsyszANALYSISOFSTRESSANDSTRAIN七、主单元体、主平面、主应力：主单元体(Principalbidy)：各侧面上剪应力均为零的单元体。主平面(PrincipalPlane)：剪应力为零的截面。主应力(PrincipalStress

）：主平面上的正应力。主应力排列规定：按代数值大小，应力状态与应变状态s1s2s3xyzsxsysz补充：

从受力构件内的某一点处，取出一个单元体。一般来讲，其侧面上既有正应力，也有切应力。但是可以证明：在该点以不同方位截取的诸单元体中，有一个特殊的单元体，在这个单元体的测面上只有正应力而无切应力。这样的单元体称为该点处的主单元体。主单元体的侧面称为主平面。主平面的正应力为主应力，主应力是正应力的极值。主平面的法线方向为主方向，即主应力方向。在一般情况下，过一点所去主单元体的6个侧面上有3对主应力，3个主应力皆不为零，该点的应力状态称为3向应力状态，有2个不为零，称为两向应力状态，。。单向应力状态。

Stateoftheuniaxialstress:

Stateofstressthatoneprincipalstressisnotequaltozero.

Stateofthebiaxialstress：

Stateofstressthatoneprincipalstressisequaltozero.

StateofthetriaxialStress:Stateofstressthatallthethreeprincipalstressesarenotequaltozero.AsxsxtzxsxsxBtxzANALYSISOFSTRESSANDSTRAIN单向应力状态（UnidirectionalStateofStress）：一个主应力不为零的应力状态。

二向应力状态（PlaneStateofStress）：一个主应力为零的应力状态。应力状态与应变状态三向应力状态（Three—DimensionalStateof

Stress）：三个主应力都不为零的应力状态。AsxsxtzxsxsxBtxz

§7–2

ANALYSISOFTHESTATEOFPLANESTRESS—ANALYTICALMETHOD

equivalentsxtxysyxyzxysxtxysyOANALYSISOFSTRESSANDSTRAIN§7–2

平面应力状态分析——解析法应力状态与应变状态sxtxysyxyzxysxtxysyOStipulate：

ispositiveifitsdirectionisthesamewithoneoftheexternalnormallineofthesection；taispositiveifitmaketheelementrotateclockwise；

Acountclockwiseangleaisconsideredtobepositive.Fig.1AssumethatareaoftheinclinedsectionisS.Accordingtotheequilibriumofthefreebodyweget：1、StressesactinginarbitraryinclinedplanexysxtxysyOsytyxsxsataaxyOtnFig.2ANALYSISOFSTRESSANDSTRAIN规定：截面外法线同向为正；

ta绕研究对象顺时针转为正；

a逆时针为正。图1设：斜截面面积为S，由分离体平衡得：一、任意斜截面上的应力应力状态与应变状态xysxtxysyOsytyxsxsataaxyOtn图2Fig.1xysxtxysyOsytyxsxsataaxyOtnFig.2Consideringconjugateofshearingstressesandtrigonometricidentitiesweget：Similarly：ANALYSISOFSTRESSANDSTRAIN图1应力状态与应变状态xysxtxysyOsytyxsxsataaxyOtn图2考虑剪应力互等和三角变换，得：同理：2、TheextremevaluesforthestressxysxtxysyOANALYSISOFSTRESSANDSTRAINandtwoextremumsThuswecangettwostationarypointsExtremenormalstressesareprincipalstresses.二、极值应力应力状态与应变状态xysxtxysyOisinthequadrantfortheshearingstresstopointandleantothelargerofbothxandy222xyyxminmaxtsstt+-±=îíì

）（xysxtxysyOMainelemeutANALYSISOFSTRESSANDSTRAINThatistheanglebetweentheplanesinwhichshearingstressesreachextremumsandtheprincipalplanesis.在剪应力相对的象限内，且偏向于x

及y较大的一侧。应力状态与应变状态222xyyxminmaxtsstt+-±=îíì

）（xysxtxysyO

主单元体Example2Analyzethefailureofthecircularshaftintorsion.Solution：Determinethecriticalpointandplottheoriginalelement.Determinetheextreme-valuestresstxyCtyxMCxyOtxytyxANALYSISOFSTRESSANDSTRAIN[例2]

分析受扭构件的破坏规律。解：确定危险点并画其原始单元体求极值应力应力状态与应变状态txyCtyxMCxyOtxytyxAnalysisoffailure:Low-carbonsteelCastironANALYSISOFSTRESSANDSTRAINLow-carbonsteel:Castiron:破坏分析应力状态与应变状态低碳钢铸铁

§7–3

ANALYSISOFTHESTATEOFSTRESS— GRAPHYCALMETHOD

Eliminatingtheparameter2

fromtheaboveequation，weget：1、StressCircle

xysxtxysyOsytxyxsxsataaxyOtncurveofthisequationisacircle—stresscircle（orMohr’scircle，introducedbyGermanengineerOttoMohr)ANALYSISOFSTRESSANDSTRAIN§7–3

平面应力状态分析——图解法对上述方程消去参数（2），得：一、应力圆（

StressCircle）应力状态与应变状态xysxtxysyO此方程曲线为圆—应力圆（或莫尔圆，由德国工程师：OttoMohr引入）sytxyxsxsataaxyOtnSetupastresscoordinatesystemasshowninthefollowingfigure.（Payattentiontotheselectionofthescale）2、MethodtoplotthestresscirclePlotthepoint

A(x，xy)andthepointB(y，yx)inthecoordinatesystem.

Inclinedline

ABintersectstheaxis

saatthepointC.Thispointisthecenterofthestresscircle.Plotacircle—stresscirclewiththecenterCandtheradiusAC.sxtxysyxyOnsataaOsataCA(sx,txy)B(sy,tyx)x2anD(sa,

ta)ANALYSISOFSTRESSANDSTRAIN建立应力坐标系，如下图所示，（注意选好比例尺）二、应力圆的画法在坐标系内画出点A(x，xy)和B(y，yx)

AB与sa

轴的交点C便是圆心。以C为圆心，以AC为半径画圆——应力圆；应力状态与应变状态sxtxysyxyOnsataaOsataCA(sx,txy)B(sy,tyx)x2anD(sa,

ta)sxtxysyxyOnsataaOsataCA(sx,txy)B(sy,tyx)x2anD(sa,

ta)3、CorrespondingrelationbetweentheelementandstresscircleStress(，)inplane

Apoint(，)onthestresscircumferenceNormallineofplaneRadiusofthestresscircleAnglebetweentwosections

Angle2betweentworadiuses；Andthedirectionofrotationisthesame.ANALYSISOFSTRESSANDSTRAIN应力状态与应变状态sxtxysyxyOnsataaOsataCA(sx,txy)B(sy,tyx)x2anD(sa,

ta)三、单元体与应力圆的对应关系面上的应力(，)

应力圆上一点(，)面的法线应力圆的半径两面夹角两半径夹角2

；且转向一致。4、MarkextremestressesonthecircumferenceofthestresscircleOCsataA(sx,txy)B(sy,tyx)x2a12a0s1s2s3ANALYSISOFSTRESSANDSTRAIN四、在应力圆上标出极值应力应力状态与应变状态OCsataA(sx,txy)B(sy,tyx)x2a12a0s1s2s3s3Example3

Determineprincipalstressesandorientationofprincipalplanesoftheelementasshowninthefigure.(unit：MPa)AB

12Method1-graphicalmethod:Stresscoordinatesystemisshowninthefigure.IntersectionCoftheperpendicularbisectionlineofABandtheaxis

saisthecenterofthecircle.PlotthecirclewiththecenterCandthearadiusAC—stress

circle.0s1s2BAC2s0sata(MPa)(MPa)O20MPaPlotthepointand

ANALYSISOFSTRESSANDSTRAINs3例3

求图示单元体的主应力及主平面的位置。(单位：MPa)AB

12解法1——图解法：主应力坐标系如图AB的垂直平分线与sa

轴的交点C便是圆心，以C为圆心，以AC为半径画圆——应力圆0应力状态与应变状态s1s2BAC2s0sata(MPa)(MPa)O20MPa在坐标系内画出点s3s1s2BAC2s0sata(MPa)(MPa)O20MPaPrincipalstressesandprincipalplanesasshowninthefigure

102ABANALYSISOFSTRESSANDSTRAINs3应力状态与应变状态s1s2BAC2s0sata(MPa)(MPa)O20MPa主应力及主平面如图

102ABMethod2—analyticalmethod：Analysis—setupthecoordinateasshowninthefigure.60°xyOANALYSISOFSTRESSANDSTRAIN解法2—解析法：分析——建立坐标系如图60°应力状态与应变状态xyO§7–4

PRINCIPALSTRESSESANDTHEIR TRAJECTORIESOFTHEBEAM12345P1P2qElement：ANALYSISOFSTRESSANDSTRAINAsshowninthefigure，thebeamproducedtheshearbending（transversalbending）,where.M、Q>0inthebeam.Trytodeterminethemagnitudeoftheprincipalstressesandthepositionoftheprincipalplanesofeachpointinthesection.§7–4

梁的主应力及其主应力迹线应力状态与应变状态12345P1P2q如图，已知梁发生剪切弯曲（横力弯曲），其上M、Q>0，试确定截面上各点主应力大小及主平面位置。单元体：21s1s3s33s1s34s1s1s35a0–45°a0stA1A2D2D1COsA2D2D1CA1Ot2a0stD2CD1O2a0=–90°sD2A1Ot2a0CD1A2stA2D2D1CA1OANALYSISOFSTRESSANDSTRAIN应力状态与应变状态21s1s3s33s1s34s1s1s35a0–45°a0stA1A2D2D1COsA2D2D1CA1Ot2a0stD2CD1O2a0=–90°sD2A1Ot2a0CD1A2stA2D2D1CA1OTensileforceCompressiveforcePrincipalstresstrajectories：

Envelopesofthedirectionlinesofprincipalstresses—tangentateachpointonthecurveindicatestheorientationofthetensile(orcompressive)principalstressatthesamepoint.Solidlinesexpressthetensileprincipalstresstrajectories；dashedlinesexpressthecompressiveprincipalstresstrajectories.1313ANALYSISOFSTRESSANDSTRAIN拉力压力主应力迹线（StressTrajectories)：主应力方向线的包络线——曲线上每一点的切线都指示着该点的拉主应力方位（或压主应力方位）。实线表示拉主应力迹线；虚线表示压主应力迹线。应力状态与应变状态1313xyMethodtoplotprincipalstresstrajectories：11

22

33

44

ii

nn

bacdANALYSISOFSTRESSANDSTRAINq1331Sectionsectionsectionsectionsectionsection

qxy主应力迹线的画法：11截面22截面33截面44截面ii截面nn截面bacd13应力状态与应变状态31§7–5

ANALYSISOFTRIAXIALSTRESSEDSTATE— METHODOFSTRESSCIRCLEs2s1xyzs31、SpatialstressedstateANALYSISOFSTRESSANDSTRAIN§7–5

三向应力状态研究——应力圆法应力状态与应变状态s2s1xyzs31、空间应力状态2、Analysisofthetriaxialstress

Fig.aFig.bThemaximumshearingstressinsidethewholeelementis：tmaxs2s1xyzs3ANALYSISOFSTRESSANDSTRAINElastictheoryprovedthat

stressesonanyplanepassingthroughapointinbytheelementshowninFig.amaybe

correspondingto

thecoordinatesofapointonthecircumferenceofthestresscircleorintheshadowregion.2、三向应力分析弹性理论证明，图a单元体内任意一点任意截面上的应力都对应着图b的应力圆上或阴影区内的一点。图a图b整个单元体内的最大剪应力为：tmax应力状态与应变状态s2s1xyzs3Example4Determinetheprincipalstressesandthemaximumshearingstressoftheelementshowninthefigure.(MPa)Solution:

Fromtheelementsketchweknowplaneyzisaprincipalplane.Setupstresscoordinatesasshowninthefigure.Plotthestresscircleandlocatethepoint1,get:5040xyz3010

(MPa)sa（MPa）taABCABs1s2s3tmaxANALYSISOFSTRESSANDSTRAIN[例4]

求图示单元体的主应力和最大剪应力。（MPa）解：由单元体图知：yz面为主平面建立应力坐标系如图，画应力圆和点1,得:应力状态与应变状态5040xyz3010(M

Pa)sa（M

Pa）taABCABs1s2s3tmax§7–6

ANALYSISOFSTRAININAPLANExyO

1、DeterminetheexpressionofstrainanalysisbythemethodofsuperpositionabcdaAOBShearingstrain：Incrementalquantityoftherightangle！（OnlylikethisthecontentsintheprecedingsectionandthefollowingsectionCancorrespondtoeachother.）DD1EE1ANALYSISOFSTRESSANDSTRAIN§7–6

平面内的应变分析xyO一、叠加法求应变分析公式abcdaAOB剪应变：直角的增大量！（只有这样，前后才对应）应力状态与应变状态DD1EE1xyOabcdaAOBDD2EE2ANALYSISOFSTRESSANDSTRAIN应力状态与应变状态xyOabcdaAOBDD2EE2DD3EE3xyOabcdaAOBANALYSISOFSTRESSANDSTRAINDD3EE3应力状态与应变状态xyOabcdaAOBANALYSISOFSTRESSANDSTRAIN应力状态与应变状态2)、Plotthestraincircleaccordingtostrain()ofapoint.2、Graphicalmethodofstrainanalysis—straincircle1)、AnalogicrelationofstresscircleandstraincircleSetupstraincoordinatesasshowninthefigurelocatethepoint

A(x，xy/2)and

B(y，-yx/2)inthecoordinatesystemintersectionofABandaxisa

isthecenterofthecircle.plotcirclebythecenterCwitharadiusAC—straincircle.eaga/2ABCANALYSISOFSTRESSANDSTRAIN2、已知一点A的应变（），画应变圆二、应变分析图解法——应变圆（StrainCircle）1、应变圆与应力圆的类比关系建立应变坐标系如图在坐标系内画出点

A(x，xy/2)

B(y，-yx/2)AB与a

轴的交点C便是圆心以C为圆心，以AC为半径画圆——应变圆。应力状态与应变状态eaga/2ABCeaga/23、Correspondingrelationbetweenthestrainindirectionandstraincirclemaxmin20D(，/2)2nStrainindirection(，/2)Apointonthestraincircle(，/2)Thedirectionlineof

RadiusofthestraincircleAnglebetweentwodirections

Angle2betweenthetworadiusesandthedirectionofrotationisthesame.ABCANALYSISOFSTRESSANDSTRAINeaga/2三、方向上的应变与应变圆的对应关系maxmin20D(，/2)2n应力状态与应变状态方向上的应变(，/2)

应变圆上一点(，/2)

方向线应变圆的半径两方向间夹角两半径夹角2

；且转向一致。ABC4、ValuesandorientationofprincipalstrainsANALYSISOFSTRESSANDSTRAIN四、主应变数值及其方位应力状态与应变状态Example5Knowingthreestrains1、2and3indirections1、2and3atapointinsomeplane,determinetheprincipalstrainsinthisplane.Solution：Firstfindout

x，y，xy

bysolvingtheabovethreeequationsthendeterminetheprincipalstrains.ANALYSISOFSTRESSANDSTRAIN[例5]

已知一点在某一平面内的1、2、3

方向上的线应变分别为1、2、3，，求该面内的主应变。解：由i=1,2,3这三个方程求出

x，y，xy；然后再求主应变。应力状态与应变状态Example6Determinetheprincipalstrainsofthepointafterthreelinearstrainsatthispointaretestedbythe

strainfoilof45°.xyu45o0maxANALYSISOFSTRESSANDSTRAIN[例6]用45°应变花测得一点的三个线应变后，求该点的主应变。xyu45o0max应力状态与应变状态

§7–7

STRESS—STRAINRELATIONUNDERTHECOMPLEX STRESSEDSTATE—（GENERALIZEDHOOKE’SLAW）

1、Stress-strainrelationinuniaxialtension

2、Stress-strainrelationinpureshearxyzsxxyz

x

yANALYSISOFSTRESSANDSTRAIN§7–7

复杂应力状态下的应力--应变关系

——（广义虎克定律）一、单拉下的应力--应变关系二、纯剪的应力--应变关系应力状态与应变状态xyzsxxyz

x

y3、Stress-strainrelationincomplexstressedstate

Accordingtosuperpositionweget:

xyzszsytxysxANALYSISOFSTRESSANDSTRAIN三、复杂状态下的应力---应变关系依叠加原理,得:应力状态与应变状态

xyzszsytxysxPrincipalstress—principalstrainrelation4、Stress-strainrelation

underthestateofplanestressThedirectionsarethesames1s3s2ANALYSISOFSTRESSANDSTRAIN主应力---主应变关系四、平面状态下的应力---应变关系:方向一致应力状态与应变状态s1s3s2Thedirectionsoftheprincipalstressandtheprincipalstrainarethesame.ANALYSISOFSTRESSANDSTRAIN主应力与主应变方向一致。应力状态与应变状态5、Relationbetweenthevolumetricstrainandstresscomponents:Volumetricstrain：Relationbetweenthevolumetricstrainandstresscomponents:s1s3s2a1a2a3ANALYSISOFSTRESSANDSTRAIN五、体积应变与应力分量间的关系体积应变：体积应变与应力分量间的关系:应力状态与应变状态s1s3s2a1a2a3Example7Astructurememberissubjectedtosomeforces.Twoprincipalstrainsatapointonthefreesurfaceofthememberare1=24010-6

2=–16010-6，modulusofelasticityisE=210GPa，Possion’sratiois=0.3.Trytodeterminetheprincipalstressesandanotherprincipalstrainatthispoint.,ThenthispointisinthestatANALYSISOFSTRESSANDSTRAINSolution:OnthefreesurfaceOfplanestress..[例7]已知一受力构件自由表面上某一点处的两个面内主应变分别为：1=24010-6，

2=–16010-6，弹性模量E=210GPa，泊松比为=0.3，试求该点处的主应力及另一主应变。所以，该点处为平面应力状态应力状态与应变状态ANALYSISOFSTRESSANDSTRAINe3342.-=×10-6应力状态与应变状态e3342.-=×10-6Example

8

Athin-walledcontainersubjectedtoinsidepressureisshowninFig.a.Inordertodeterminethevalueoftheinsidepressurethehoopstraintestedonthesurfaceofthecontainerwithstrainfoilist

=350×l06.IfthemeandiameterofthecontainerisD=500mm，thicknessofitswallis=10mm，E=210GPa，=0.25.Tryto:1.derivetheexpressionsofstressinthelateralandlongitudinalsectionsofthecontainer.2.calculatetheinsidepressureofthecontainer.pppxstsmLpODxAByFig.aANALYSISOFSTRESSANDSTRAIN[例8]

图a所示为承受内压的薄壁容器。为测量容器所承受的内压力值，在容器表面用电阻应变片测得环向应变t

=350×l06，若已知容器平均直径D=500mm，壁厚=10mm，容器材料的E=210GPa，=0.25，试求:1.导出容器横截面和纵截面上的正应力表达式；2.计算容器所受的内压力。应力状态与应变状态pODxABy图apppxstsmL1)LongitudinalstressesSolution：Expressionsofthecircumferentialandlongitudinalstressofthecontainerreservoir

CuttingthecontaineralongasectionshownandinFig.bconsideringtheequilibriumoftherightpart，wegetpsmsmxDFig.bANALYSISOFSTRESSANDSTRAIN1、轴向应力:(longitudinalstress)解：容器的环向和纵向应力表达式用横截面将容器截开，受力如图b所示,根据平衡方程应力状态与应变状态psmsmxD图b

Imaginecutoffthecontaineralonglongitudinalsection，andtaketheupperpantasthestudyobject.ThefreebodydiagramisshowninFig.c2、Circumferentialstress3、Determinetheinsidepressure（bystress-strainrelation）t

mEx