1

緒論

緒論LeonardoDaVincisaid:“mechanicsisamathematicparadise,becauseweacquiredmathematics'sfruithere."2LeonardoDaVinciGalileoGalileiHemadeadetailedstudyonthebasicconceptsofmovementincludingthecenterofgravity,speedandaccelerationandcameupwiththerigidmathematicformulas.Especiallytheconceptofaccelerationisthemilestoneinthehistoryofmechanics.

Galileo(1564~1642)isaGermanastronomer,mechanistandphilosopher.HewasborninPisaonFeb151564anddiedonJan.81642atthesameplace.INTRODUCTION達芬奇說：“力學是數學的樂園，因為我們在這里獲得了數學的果實。”3緒論達芬奇伽利略4INTRODUCTIONHeonceinformallyproposedlawofinertia,whichestablishedthefoundationforNewtontoproposeformallythefirstlawandthesecondlaw.ItcanbesaidthatGalileoisthePioneerofNewtoninthesettlementoftheclassicalmechanics.Galileoalsobroughtupwiththelawofresultantandtheruleoftheparabolicmotionandsetuptheprincipleofrelativity.Heisthefirstscientisttomakeatotofachievementsbythetelescope.Hekeptonfightingwiththeidealismandchurchphilosophyandsuggestedthatweshouldstudythelawofnaturebyspecificexperimentsandthoughtthatexperiencesarethesourceoftheory.GalileoGalilei達芬奇說：“力學是數學的樂園，因為我們在這里獲得了數學的果實。”5緒論達芬奇伽利略6Introduction

Herearehiscontributionstomechanics:hemadeafurtherstudyonthebasisofGalileoGalileiandotherpeopleandconcludedthethreeprinciplesofobjects’movementandmadeafirmfoundationformechanics.Heisthediscovererofthegravitationlawandsetupthetheoreticalsystemoftheclassicalmechanics.Healsomadeprofoundcontributionstothefieldofmathematics,opticsandastronomy.<Themaththeoryofnaturalphilosophy>ishismostimportantwork.Heconcludedmanyimportantdiscoveriesandstudyresultsinallhislifeinthebook.IsaacNewton(1642~1727)isagreatBritishphysicist,mathematicianandastronomer.HewasborninafamilyofpeasantsinLincolnonDec251624anddiedofKidneystoneinLondononMar201727.7緒論8Introduction

Hebeganhiscreativeworkbetween1903and1906anddidresearchintheUniversityinGermanyeveryyearsupervisedbyfamousscholars.Hewasaprofessorofcollegesbetween1907and1917.HecametoAmericain1922andengagedinthestudyofmechanics.In1928,hefound“themechanicsdepartmentofASME”andheld,variouskindsofmechanicsseminarsperiodically.Hehasmanyworksonappliedapplyingmechanics.Especiallysincetheyearoflatetwentieshehaswrittenabout20booksappliedsuchas<mechanicsofMaterials>,<AdvancedMechanicsofMaterials>and<MechanicsofStructures>exceptthathehasdonesomeworkinteachingandtrainingmaters.S.PisaRussiandynamicistwithAmericannationality.HewasborninUkraineonDec.231878anddiedinGermanonMay.291972.9緒論10IntroductionHemakesgreatcontributionstotheresearchoflocaldamageandtheengineeringapplicationoffracturedynamics.Hehastrainedsomeresearchersinmechanics.Heisactiveinacademicactivitiesandtakessomepositionsbothathomeandabroad.Heisacouncilmemberofthefirstsessioncouncilconference.Hecouncilchairpersonofthesecondsessioncouncilconferenceandtheexecutivemanagerofthethirdsessioncouncilconference.HewasborninfamilyinShoucity,Anhuiprovince.NowheistheprofessorofSouthwestJiaoTongUniversity.SunXunfangisanengineeringmechanistandmechanicseducationist.Hehasengagedintheresearchoffracture,damage,fatigueandcreepofsolidmechanics.Heisthefirsttoapplyfracturedynamicstopracticeanddevelopedthemethodofanalysisinelasto-plasticfracturedynamicswithsurfacecracksandassessmentinintegrity.11緒論12§0-1THERESEARCHOBJECTSOFMECHANICSOFMATERIALS1、Structuremembers2、Classificationof structuremembersINTRODUCTIONplateOrshellbarorrodclumpbody13§0-1材料力學的研究對象1、構件2、構件分類緒論塊體BeamsandrodsarethemainresearchobjectsofmechanicsofmaterialsINTRODUCTION材料力學以“梁、桿”為主要研究對象緒論MoststructuresaremadeupfromthebeamsandrodsINTRODUCTION工程中多為梁、桿結構緒論§0-2THETASKSOFMECHANICSOFMATERIALS ANDITSRELATIONWITHENGINEERING

18Strength、rigidity、stabilityINTRODUCTION§0-2材料力學的任務及與工程的聯系19強度、剛度、穩定性緒論20INTRODUCTION21緒論

Thetasksofmechanicsofmaterials22Undertherequestthatthestrength,rigidity,stabilityaresatisfied,offerthenecessarytheoreticalfoundationandcalculationmethodfordeterminingreasonableshapesanddimensions,choosingpropermaterialsforthecomponentsatthemosteconomicprice.,INTRODUCTION

材料力學的任務23在滿足強度、剛度、穩定性的要求下，以最經濟的代價，為構件確定合理的形狀和尺寸，選擇適宜的材料，而提供必要的理論基礎和計算方法。緒論1、Strength:Capacitytoresistfailure

ofacomponentoranelement.24

INTRODUCTION1、強度：25構件的抗破壞能力緒論INTRODUCTION緒論28INTRODUCTION29緒論30INTRODUCTION31緒論2、Rigidity:Capacitytoresistdeformationsofacomponentoranelement.32INTRODUCTION2、剛度：構件的抗變形能力。33緒論34INTRODUCTION35緒論36INTRODUCTION37緒論38StrengthandrigidityINTRODUCTION39強度和剛度緒論40

ProblemsaboutthestrengthandrigidityofengineeringcomponentsINTRODUCTION41

工程構件的強度、剛度問題緒論

3Stability:CapacitytoremaintheoriginalstateinequilibriumofacomponentoranelementINTRODUCTION構件保持原有平衡狀態的能力3、穩定性：緒論44Problemsaboutthestrength,rigidityandstabilityofengineeringstructuresProblemsaboutstabilitystrengthrigidityINTRODUCTION45

工程結構的強度、剛度和穩定問題穩定問題強度剛度緒論46

Thereareproblemsaboutthestrength,rigidityandstabilityinabicyclestructuretooINTRODUCTION47

自行車結構也有強度、剛度和穩定問題緒論48ThespacestationandspaceimplementsINTRODUCTION49空間站和航天器緒論50AirplanesandguidedmissilesintheweaponindustryINTRODUCTION51兵器工業、飛機與導彈緒論52weaponindustryINTRODUCTION53兵器工業緒論54civilaviationINTRODUCTION55民用航空緒論56vehicleandroadINTRODUCTION57車輛與道路緒論Problemsaboutthestrength,rigidityandstabilityinlargebridges.58INTRODUCTION大型橋梁的強度、剛度、穩定問題59緒論60MacaobridgeNanpubridgeinShanghaiSomefamousbridgesinourcountryNanjingYangtzeRiverbridgeINTRODUCTION61澳門橋上海南浦大橋我國著名橋梁南京長江大橋緒論62INTRODUCTION63緒論64INTRODUCTION65緒論66MakeholeplatformforoceanpetroleumINTRODUCTION67海洋石油鉆井平臺緒論

§0-3THEPROPERTIESANDTHEFUNDAMENTAL ASSUMPTIONOFTHESOLIDDEFORMABLEBODIES

681.Continuity:Thematerialofasoliddeformablebodyiscontinuouslydistributedoveritsvolumesothattherearenotanycracks,defectsorholesetc.

2.Homogeneity:Thematerialofthesoliddeformablebodyishomogeneouslydistributedoveritsvolumesothatthesmallestelementcutfromthebodypossessesthesamespecificmechanicalpropertiesasthebody.

3.Isotropy:Themechanicalpropertiesarethesameinalldirectionsatapoint.materialwiththispropertyiscalledisotropymaterial.Materialthatthemechanicalpropertiesaredifferentinalldirectionsatapointiscalledanisotropymaterial.

4.Smalldeformations:Thedeformationsforasoliddeformablebodycausedbyexternalforcesareverysmallcomparedwiththedimensionsofthebody.Thuswhenwestudytheequilibriumandmotionofthesoliddeformablebody,thedeformationofthebodymaybeneglected.

INTRODUCTION§0-3可變形固體的性質及其基本假設

69

一、連續性假設：物質密實地充滿物體所在空間，毫無空隙。

（可用微積分數學工具）

二、均勻性假設：物體內，各處的力學性質完全相同。

三、各向同性假設：組成物體的材料沿各方向的力學性質完全

相同。（這樣的材料稱為各向同性材料；沿各方向的力學

性質不同的材料稱為各向異性材料。）

四、小變形假設：材料力學所研究的構件在載荷作用下的變形

與原始尺寸相比甚小，故對構件進行受力分析時可忽略其

變形。

緒論70§0-4

BASICTYPESOFDEFORMATIONSOFRODINTRODUCTION

CombinedLoadinganddeformation

Content

Types

Loadingcharacteristics

Deformationcharacteristics

AxialTension

Shear

Torsion

Bending

71§0-4桿件變形的基本形式緒論組合受力與變形

內容種類

外力特點

變形特點軸向拉伸

及壓縮剪切扭轉平面彎曲2

軸向拉伸和壓縮§1–1

CONCEPTSANDPRACTICALEXAMPLESOF AXIALTENSIONANDCOMPRESSIONCharacteristicoftheexternalforce：Theactinglineoftheresultantofexternalforcesiscoincidedwiththeaxisoftherod.1、ConceptsCharacteristicofthedeformation：Deformationoftherod

ismainlyelongationorcontractionalongtheaxisoftherodandcompaniedwithlateralreductionorenlargement.Axialtension：Deformationoftherodisaxialelongationandlateralshortening.Axialcompression：Deformationoftherodisaxialshorteningandlateralenlargement.AXIALTENSIONANDCOMPRESSION574拉壓§1–1軸向拉壓的概念及實例軸向拉壓的外力特點：外力的合力作用線與桿的軸線重合。一、概念軸向拉壓的變形特點：桿的變形主要是軸向伸縮，伴隨橫向縮擴。軸向拉伸：桿的變形是軸向伸長，橫向縮短。軸向壓縮：桿的變形是軸向縮短，橫向變粗。Inaxialcompression，thecorrespondingforceiscalledcompressiveforce.Inaxialtension，thecorrespondingforceiscalledtensileforce.MechanicalmodelsareshowninthefiguresAXIALTENSIONANDCOMPRESSION776拉壓軸向壓縮，對應的力稱為壓力。軸向拉伸，對應的力稱為拉力。力學模型如圖Practicalexamplesinengineering2、AXIALTENSIONANDCOMPRESSION978拉壓工程實例二、AXIALTENSIONANDCOMPRESSION1180拉壓1、Internalforce

Internalforceistheresultantofinternalforces,whichisactingmutuallybetweentwoneighbourpartsinsidethebody,causedbytheexternalforces.

§1–2

INTERNALFORCE、METHODOFSECTION、AXIAL FORCEANDITSDIAGRAMAXIALTENSIONANDCOMPRESSION1382拉壓一、內力

指由外力作用所引起的、物體內相鄰部分之間分布內力系的合成（附加內力）。§1–2內力·截面法·軸力及軸力圖2、Methodofsection·axialforce

Calculationoftheinternalforcesisthefoundationtoanalyzetheproblemsofstrength、rigidity、stabilityetc.Thegeneralmethodtodetermineinternalforcesisthemethodofsection.1).Basicstepsofthemethodofsection：①Cutoff：Assumetoseparatetherodintotwodistinctpartsinthesectioninwhichtheinternalforcesaretobedetermined.②Substitute：Takearbitrarypartandsubstitutetheactionofanotherpartonitbythecorrespondinginternalforceinthecut-offsection.③Equilibrium：Setupequilibriumequationsfortheremainedpartanddeterminetheunknowninternalforcesaccordingtotheexternalforcesactedonit.（Heretheinternalforcesinthecut-offsectionaretheexternalforcesfortheremainedpart）AXIALTENSIONANDCOMPRESSION1584拉壓二、截面法·軸力

內力的計算是分析構件強度、剛度、穩定性等問題的基礎。求內力的一般方法是截面法。1.截面法的基本步驟：①截開：在所求內力的截面處，假想地用截面將桿件一分為二。②代替：任取一部分，其棄去部分對留下部分的作用，用作用在截開面上相應的內力（力或力偶）代替。③平衡：對留下的部分建立平衡方程，根據其上的已知外力來計算桿在截開面上的未知內力（此時截開面上的內力對所留部分而言是外力）。2.Axialforce—internalforceoftherodinaxialtensionorcompression，designatedbyN.Suchas：DetermineNbythemethodofsection.APPPANSimplesketchAPPCutoff:：Substitute：Equilibrium：AXIALTENSIONANDCOMPRESSION1786拉壓2.軸力——軸向拉壓桿的內力，用N表示。例如：截面法求N。

APP簡圖APP截開：代替：平衡：PAN①Reflectedthevarietyrelationbetweenthecorrespondingaxialforceandthepositionofthesection.②Findoutvalueofthemaximumaxialforceandthepositionofthesectioninwhichthemaximumaxialforceact.Thatistodeterminethepositionofthecriticalsectionandsupplytheinformationforthecalculationofstrength.3、Diagramoftheaxialforce—sketchexpressionofN(x)3).Signconventionsfortheaxialforce:axialforceN(tensileforce)ispositivewhenitsdirectionpointtotheoutwarddirectionofthenormallineofthesection,(compressiveforce)negativeinward

N>0NNN<0NNxNP+meaningAXIALTENSIONANDCOMPRESSION1988①反映出軸力與橫截面位置變化關系，較直觀；②確定出最大軸力的數值及其所在橫截面的位置，即確定危險截面位置，為強度計算提供依據。拉壓三、軸力圖——N(x)的圖象表示。3.軸力的正負規定:

N

與外法線同向,為正軸力(拉力)N與外法線反向,為負軸力(壓力)N>0NNN<0NNNxP+意義Example1Theforceswithmagnitudes

5P、8P、4PandPact

respectivelyatpointsA、B、C、Doftherod.Theirdirectionsareshowninthefigure.Trytoplotthediagramoftheaxialforceoftherod.Solution：DeterminetheinternalforceN1insegmentOA.Takethefreebodyasshowninthefigure.ABCDPAPBPCPDOABCDPAPBPCPDN1AXIALTENSIONANDCOMPRESSION2190拉壓[例1]圖示桿的A、B、C、D點分別作用著大小為5P、8P、4P、

P

的力，方向如圖，試畫出桿的軸力圖。解：求OA段內力N1：設置截面如圖ABCDPAPBPCPDOABCDPAPBPCPDN1Similarly，wegettheinternalforcesinsegmentAB、BC、CD.Theyarerespectively：N2=–3P

N3=5PN4=PThediagramoftheaxialforceisshownintherightfigure.BCDPBPCPDN2CDPCPDN3DPDN4Nx2P3P5PP++–AXIALTENSIONANDCOMPRESSION2392拉壓同理，求得AB、BC、CD段內力分別為：N2=–3P

N3=5PN4=P軸力圖如右圖BCDPBPCPDN2CDPCPDN3DPDN4Nx2P3P5PP++–Simplemethodtoplotthediagramofaxialforce：Fromthelefttotheright:Characteristicofthediagramoftheaxialforce：Valueofsuddenchange=concentratedloadIfmeetingtheforcePtotheleft

，theincreaseoftheaxialforceNispositive；Ifmeetingtheforcetotheright

，theincreaseoftheaxialforceNisnegative.5kN8kN3kN+–3kN5kN8kNAXIALTENSIONANDCOMPRESSION2594拉壓軸力(圖)的簡便求法：自左向右:軸力圖的特點：突變值=集中載荷遇到向左的P

，軸力N增量為正；遇到向右的P

，軸力N增量為負。5kN8kN3kN+–3kN5kN8kNSolution：Thefreeendoftherodistheoriginofthecoordinateandcoordinatextotherightispositive.Takethesegmentoflengthxontheleftofpointx,itsinternalforceis

qK

LxOExample2Lengthoftherodshowninthefigureis

L.Distributedforceq=kxisactedonit,directionoftheforceisshowninthefigure.Trytoplotthediagramofaxialforceofthetherod.Lq(x)N(x)xq(x)NxO–AXIALTENSIONANDCOMPRESSION2796拉壓解：x坐標向右為正，坐標原點在自由端。取左側x段為對象，內力N(x)為：qq

LxO[例2]圖示桿長為L，受分布力q=kx

作用，方向如圖，試畫出桿的軸力圖。Lq(x)NxO–N(x)xq(x)1、Conceptofstress

§1–3

STRESSESONTHESECTIONANDSTRENGTHCONDITIONSBringforwardtheproblem：1).Themagnitudeoftheinternalforcecannotscalethestrengthofthestructuremember.2).Strength：①Intensityofthedistributedinternalforcesinthesectionstress；

②Theload-bearingcapacityofthematerial.1).Definition：Intensityoftheinternalforceduetotheexternalforces.AXIALTENSIONANDCOMPRESSION29PPPP98拉壓一、應力的概念

§1–3截面上的應力及強度條件問題提出：1.內力大小不能衡量構件強度的大小。2.強度：①內力在截面的分布集度

應力；

②材料承受荷載的能力。1.定義：由外力引起的內力集度。PPPP

Undermostcasesdistributionoftheinternalforceinsideengineeringmembersisnotuniform.Definitionofintensityisneitheraccurateandimportantbecausebreakageorfailureoftenbeginsfromthepointatwhichintensityoftheinternalforceismaximum.

P

AM①Averagestress：②Wholestress（sumstress）：2).Expressionofstress：AXIALTENSIONANDCOMPRESSION31100拉壓

工程構件，大多數情形下，內力并非均勻分布，集度的定義不僅準確而且重要，因為“破壞”或“失效”往往從內力集度最大處開始。

P

AM①平均應力：②全應力（總應力）：2.應力的表示：③Wholestressmaybedecomposedinto:p

M

a.Stressperpendiculartothesectioniscalled“normalstress”b.Stresslyinginthesectioniscalled“shearingstress”

AXIALTENSIONANDCOMPRESSION33102拉壓③全應力分解為：p

M

a.垂直于截面的應力稱為“正應力”

(NormalStress)；b.位于截面內的應力稱為“剪應力”(ShearingStress)。Beforedeformation1).Experimentonthelawofdeformationandthehypothesisofplanesection：Hypothesisofplanesection：Crosssectionsremainplanesbeforeandafterdeformations.Deformationsoflongitudinalfibersarethesame

abcdAfterloadingPPd′a′c′b′2、StressinthecrosssectionoftherodintensionorcompressionAXIALTENSIONANDCOMPRESSION35104拉壓變形前1.變形規律試驗及平面假設：平面假設：原為平面的橫截面在變形后仍為平面。縱向纖維變形相同。abcd受載后PPd′a′c′b′二、拉（壓）桿橫截面上的應力Thematerialishomogeneousand,itsdeformationisuniform,sotheinternalforceisdistributeduniformly。2.Tensilestress：Normalstressduetotheaxialforces—

：distributesuniformlyinthecrosssection.Criticalsection：Thesectioninwhichinternalforceismaximumandofwhichthedimensionissmallest.Criticalpoint：Thepointatwhichthestressismaximum.3.Criticalsectionandthemaximumworkingstress：AXIALTENSIONANDCOMPRESSION37sN(x)P106拉壓均勻材料、均勻變形，內力當然均勻分布。2.拉伸應力：軸力引起的正應力——

：在橫截面上均布。危險截面：內力最大的面，截面尺寸最小的面。危險點：應力最大的點。3.危險截面及最大工作應力：sN(x)P

Straightrod、crosssectionoftherodiswithoutsuddenchange、thereisacertaindistancefromthesectiontothepointatwhichtheloadacts.4).Applicationconditionsoftheformula：6).Stressconcentration：

Stressincreasesabruptlynearthecrosssectionwithasuddenchangeindimension

5).Saint-Venantprinciple：

Distributionandmagnitudeofthestressinthesectionatacertaindistancefromthepointatwhichtheloadisactedarenotaffectedbytheactingformofexternalloads.AXIALTENSIONANDCOMPRESSION39108拉壓

直桿、桿的截面無突變、截面到載荷作用點有一定的距離。4.公式的應用條件：6.應力集中（StressConcentration）：

在截面尺寸突變處，應力急劇變大。5.Saint-Venant原理：

離開載荷作用處一定距離，應力分布與大小不受外載荷作用方式的影響。SketchofSaint-Venantprincipleandstressconcentrations(Redreallinesdenotethelinebeforedeformationandreddashedlinesdenotetheshapeafterdeformation.)Sketchofdeformation：abcPPSketchofthestressdistribution：AXIALTENSIONANDCOMPRESSION41110拉壓Saint-Venant原理與應力集中示意圖(紅色實線為變形前的線，紅色虛線為紅色實線變形后的形狀。)變形示意圖：abcPP應力分布示意圖：7).Criterionofthestrengthdesign：where：[

]—allowablestress，

max–themaximumworkingstress atthecriticalpoint.②Designthedimensionofthesection：Threekindsofcalculationofstrengthmaybedoneaccordingtothecriterionofstrength:

Thatstructuremembersareensurednottobewreckedandhavecertainsafedegree.①Checkthestrength：③Determinetheallowableload：

AXIALTENSIONANDCOMPRESSION43112拉壓7.強度設計準則（StrengthDesign）：其中：[

]--許用應力，

max--危險點的最大工作應力。②設計截面尺寸：依強度準則可進行三種強度計算：

保證構件不發生強度破壞并有一定安全余量的條件準則。①校核強度：③許可載荷：

Example3Acircularrodissubjectedtoatensileforce

P=25kN.Itsdiameterisd=14mmanditsallowablestressis[

]=170MPa.Trytocheckthestrengthoftherod.Solution：①Axialforce：N=P

=25kN②Stress：③Checkthestrength：④Conclusion：Thestrengthoftherodsatisfiesrequest.Therodcanworknormally.AXIALTENSIONANDCOMPRESSION45114拉壓[例3]已知一圓桿受拉力P=25kN，直徑d=14mm，許用應力

[

]=170MPa，試校核此桿是否滿足強度要求。解：①軸力：N=P

=25kN②應力：③強度校核：④結論：此桿滿足強度要求，能夠正常工作。Example4

Athree-pinhouseframeonwhichaverticaluniformload,withtheindensityintensityisq

=4.2kN/misappliedisshowninthefigure.Diameterofthesteeltensilerodintheframeisd=16mmanditsallowablestressis[

]=170MPa.Trytocheckthestrengthoftherod.AXIALTENSIONANDCOMPRESSION47Tiebar4.2m8.5m116拉壓[例4]

已知三鉸屋架如圖，承受豎向均布載荷，載荷的分布集度為：q

=4.2kN/m，屋架中的鋼拉桿直徑d=16mm，許用應力[

]=170MPa。試校核鋼拉桿的強度。鋼拉桿4.2m8.5m①DeterminethereactionsfirstaccordingtotheglobalequilibriumSolution：AXIALTENSIONANDCOMPRESSION49Tiebar8.5m4.2mRARBHA118拉壓①整體平衡求支反力解：鋼拉桿8.5m4.2mRARBHA③Stress：④Strengthcheckandconclusion：Thisrodsatisfiestherequestofstrength.Itissafe.②Determinetheaxialforceaccordingtothepartialequilibrium：

AXIALTENSIONANDCOMPRESSION51RAHARCHCN120拉壓③應力：④強度校核與結論：

此桿滿足強度要求，是安全的。②局部平衡求軸力：

HCRAHARCHCNExample5A

simplecraneisshowninthefigure.ACisarigidbeam，sumweightofthehoistandheavybodythatisliftedisP.Whatshouldbetheangle

sothattherodBDhastheminimumweight？

Theallowablestressoftherod[

]isknown.Analysis：xLhqPABCDAXIALTENSIONANDCOMPRESSION53122拉壓[例5]簡易起重機構如圖，AC為剛性梁，吊車與吊起重物總重為P，為使BD桿最輕，角

應為何值？已知BD

桿的許用應力為[

]。分析：xLhqPABCD

Thecross-sectionareaAoftherodBD：Solution：

Internalforce

N((q)oftherodBD：

TakeACasourstudyobjectasshowninthefigure.YAXAqNBDxLPABCAXIALTENSIONANDCOMPRESSION55124拉壓

BD桿橫截面面積A：解：

BD桿內力N(q)：取AC為研究對象，如圖YAXAqNBDxLPABCYAXAqNBDxLPABC③DeterminetheminimumvalueofVBD

：AXIALTENSIONANDCOMPRESSION67126拉壓YAXAqNBDxLPABC③求VBD

的最小值：3、StressesintheinclinedsectionoftherodintensionorcompressionPPkka

Aa:Areaoftheinclinedsection；Pa：Internalforceintheinclinedsection.

：PkkaPaFromgeometricrelationSubstitutingitintotheaboveformulawegetSolution:Adoptthemethodofsection.Accordingtotheequilibriumequation：Pa=PthenAssumeastraightrodissubjectedtoatensileforceP.Determinethestressintheinclinedsectionk-k.

Wholestressintheinclinedsection：AXIALTENSIONANDCOMPRESSION59128拉壓三、拉(壓)桿斜截面上的應力設有一等直桿受拉力P作用。求：斜截面k-k上的應力。PPkka解：采用截面法由平衡方程：Pa=P則：Aa：斜截面面積；Pa：斜截面上內力。由幾何關系：代入上式，得：斜截面上全應力：PkkaPa

Decomposition：pa=Itindicatesthechangeofstressesindifferentsectionsthroughapoint.As

=90°，As

=0,90°，As

=0°，(Themaximumnormalstressexists inthecrosssection)As

=±45°，(Themaximumshearingstressexistsintheinclinedsectionof45°)Wholestressintheinclinedsection：AXIALTENSIONANDCOMPRESSION61PPkkaPkkapatasaa130拉壓斜截面上全應力：分解：pa=反映：通過構件上一點不同截面上應力變化情況。當

=90°時，當

=0,90°時，當

=0°時，(橫截面上存在最大正應力)當

=±45°時，(45°斜截面上剪應力達到最大)PPkkaPkkapa

atasaa2、Element：

Element—delegateofapointinsidethemember,infinitesimalgeometricbodywhichenvelopsthestudypoint.Theelementincommonuseisjusthexahedron

propertiesofanelement—a、stressisdistributeduniformlyinanarbitraryparallelarbitraryplane；b、stressesintheparallelplaneoppositeplane areequal.3、stresselementatapointMintherodintensionorcompression:

1.Stateofstressatapoint：Therearecountlesssectionsthroughapoint.Sumofstressesinthedifferentsectionthroughapointiscalledthestateofstressatthispoint.Complementary：sPMssssAXIALTENSIONANDCOMPRESSION631322、單元體：

單元體—構件內的點的代表物，是包圍被研究點的無限小的幾何體，常用的是正六面體。

單元體的性質—a、平行面上，應力均布；

b、平行面上，應力相等。3、拉壓桿內一點M

的應力單元體:

1.一點的應力狀態：過一點有無數的截面，這一點的各個截面上的應力情況，稱為這點的應力狀態。補充：拉壓sPMssssTakeafreebodyasshownintheFig.3.a

ispositive

ifitisalongcountclockwise；ta

ispositiveifitmakesthefreebodyrotateclockwise.Fromtheequilibriumofthefreebodyweget：4、Stressintheinclinedsectionoftherodintensionorcompressionssss

tasaxs0Fig.3AXIALTENSIONANDCOMPRESSION65134取分離體如圖3，a逆時針為正；ta繞研究對象順時針轉為正；由分離體平衡得：拉壓4、拉壓桿斜截面上的應力ssss

tasaxs0圖3Example6

Arod,whichthediameterd=1cmissubjectedtoatensileforceP=10kN.Determinethemaximumshearingstress,thenormalstressandshearingstressintheinclinedsectionofanangle30°aboutthecrosssection.Solution：Stressesintheinclinedsectionoftherodintensionorcompressioncanbedetermineddirectlybytheformula：AXIALTENSIONANDCOMPRESSION67136[例6]

直徑為d=1cm

桿受拉力P=10kN的作用，試求最大剪應力，并求與橫截面夾角30°的斜截面上的正應力和剪應力。解：拉壓桿斜截面上的應力，直接由公式求之：拉壓Example7

Atensilerodasshowninthefigureismadefromtwopartsgluedmutuallytogetheralong

mn.Itissubjectedtotheactionofforce

P.Assumethattheallowablenormalstressis[

]=100MPaandallowableshearingstressis[

]=50MPafortheadhesive.AreaofcrosssectionoftherodisA=4cm2.Ifstrengthoftherodiscontrolledbytheadhesivewhatistheangle

(

:between00~600)togetthelargesttensileforce?Combine(1)、(2)andget：PPmnaSolution：AXIALTENSIONANDCOMPRESSION69Pa600300B00138[例7]圖示拉桿沿mn由兩部分膠合而成,受力P，設膠合面的許用拉應力為[

]=100MPa

；許用剪應力為[

]=50MPa

，并設桿的強度由膠合面控制,桿的橫截面積為A=4cm2，試問:為使桿承受最大拉力,

角值應為多大?(規定:

在0~60度之間)。聯立(1)、(2)得：拉壓PPmna解：Pa600300B00Thecurvesofformula(1)and、(2)areshownintheTig.(2).ObviouslythestrengthoftherodontheleftofpointBiscontrolledbythenormalstress，thatontherightofpointBiscontrolledbytheshearingstress.Asa=60°，fromformula(2)wecangetSolution:Atthepointofintersectionofcurves(1)and(2)：Discussion：AsAXIALTENSIONANDCOMPRESSION71Pa600300B100140(1)、(2)式的曲線如圖(2)，顯然，B點左側由正應力控制桿的強度，B點右側由剪應力控制桿的強度，當a=60°時，由(2)式得解(1)、(2)曲線交點處：拉壓討論：若Pa600300B100

1)、Thewholelongitudinal deformationoftherod：

3)、Averagestain：

2)、Strain：lineardeformationperunitlength.1、Deformationandstrainoftherodintensionorcompression

§1－4

DEFORMATIONOFTHERODINAXIALTENSION ANDCOMPRESSIONLAWOFELASTICITY

AXIALTENSIONANDCOMPRESSION73abcdLCrosssection1421、桿的縱向總變形：3、平均線應變：2、線應變：單位長度的線變形。一、拉壓桿的變形及應變§1－4拉壓桿的變形

彈性定律拉壓abcdL4、Longitudinalstrainatpointx：6、Lateralstrainatpointx：5、Lateraldeformationoftherod：PPd′a′c′b′L1AXIALTENSIONANDCOMPRESSION751444、x點處的縱向線應變：6、x點處的橫向線應變：5、桿的橫向變形：拉壓PPd′a′c′b′L12、Elasticlawoftherodintensionorcompression

1)、Caseofequalinternalforces

2)、Caseofvariableinternalforces

Wheninternalforcesinnsegmentsareconstant

※“EA”iscalledtheaxialrigidityoftherodintensionorcompression.

PPN(x)dxxAXIALTENSIONANDCOMPRESSION77146二、拉壓桿的彈性定律1、等內力拉壓桿的彈性定律2、變內力拉壓桿的彈性定律內力在