Chapter6

ForceAnalysisofStructuresandMachinesWangMing-luChapter6

ForceAnalysis2SpecializedVocabularyStructures:結構,構造,構[桁]架Truss:構架,桁架,鋼梁Joint:接合;榫;接點;聯軸節;鉸鏈Section:截口[面,線,點]Frame:框架,框子;【機】架,座身2SpecializedVocabularyStructu3ChapterObjectives.1.Usethemethodofjointstodeterminetheforceinaplanetruss.2.Usethemethodofsectionstodeterminetheforceinaplanetruss.3.Usemultiplefree-bodydiagramstodeterminetheforceinaframeormachine.3ChapterObjectives.1.Usethe46.1Introduction.46.1Introduction.5Structureoftrussinengineering

工程中的桁架結構6.1Introduction.5Structureoftrussinenginee6Structureoftrussinengineering

工程中的桁架結構6.1Introduction.6Structureoftrussinenginee7Structureoftrussinengineering

工程中的桁架結構6.1Introduction.7Structureoftrussinenginee8Structureoftrussinengineering

工程中的桁架結構6.1Introduction.8Structureoftrussinenginee9Structureoftrussinengineering

工程中的桁架結構6.1Introduction.9Structureoftrussinenginee10Truss:arigidstructuresystemcomposedofstraightelements,connectedattheirendsbypins,withoutdeformationundertheactionofforces.桁架：由桿組成，用鉸聯接，受力不變形的系統。6.2Simpleplanetrusses10Truss:arigidstructure11joint節點element桿件6.2Simpleplanetrusses11jointelement6.2Simpleplan12Theadvantageofthetruss:light,thecapabilityofthematerialcanexertsufficiently.桁架的優點：輕，充分發揮材料性能。Thecharactersofthetruss:桁架的特點：①straightmembers,theweightofthemselvescanbeneglected,theyarealldoubleforceequivalentrods;①直桿，不計自重，均為二力桿；②Thestraightelementsareconnectedattheirendsbyjoints.②桿端鉸接；③Allexternalloadsonthetrussactonlyatthejoints.③外力作用在節點上。6.2Simpleplanetrusses12Theadvantageofthetruss:13Themodeloftrussinmechanics力學中的桁架模型(basictriangle)Trianglesensurestability(基本三角形)三角形有穩定性

(a)(b)(c)6.2Simpleplanetrusses13Themodeloftrussinmechan14Thecommonsimplifiedmodelsfortrussinengineeringmechanics工程力學中常見的桁架簡化計算模型6.2Simpleplanetrusses14Thecommonsimplifiedmodels156.2Simpleplanetrusses156.2Simpleplanetrusses166.4MethodofjointsP=10kN,determinetheinternalforcesoneveryelements.已知：如圖

P=10kN，求各桿內力？[Example][例]Solution:

解：①studythewholestructure,determinethereactionforces:

①研究整體，求支座反力166.4MethodofjointsP=10kN,17②Studythejoints

A,CandDonebyone,determinetheinternalforcesofeverymember.②依次取A、C、D節點研究，計算各桿內力。Weget: (denotethattherodiscompressed)6.4Methodofjoints17②StudythejointsA,CandD18SubstitutingintoWeobtain:Substitutingintoweget:6.4Methodofjoints18SubstitutingintoWeobtain:S19TheanotherequationofthejointDcanbeusedtoverifytheresultofcalculation.節點D的另一個方程可用來校核計算結果ThesolutionofwhichisItequalsto,thecalculationisaccurate.恰與相等,計算準確無誤。6.4Methodofjoints19Theanotherequationofthe206.4.3Problems.Dotheproblems6.1-6.25byyourself,andthenIwilltellyoutheanswersofthem.6.4Methodofjoints206.4.3Problems.6.4Methodof21[Example]h,aandParegiven,determinetheinternalforcesontheelement4,5and6.[例]已知：如圖，h，a，P，求：4，5，6桿的內力。IISolution:

Studythewholestructure,determinethereactionforces.解：研究整體求支反力①①6.6Methodofsections21[Example]h,aandParegiv22②DrawacutI-I,studytheleft-handsectionofthetruss.選截面I-I，取左半部研究A'From6.6Methodofsections22②DrawacutI-I,studythel23Instruction:說明：MethodofIsolationjoints:Thisisconvenientwhenthestressesinallelementsofthetrusshavetobedetermined.節點法：用于設計，計算全部桿內力Methodofcuts:

Thisisconvenientindeterminingthestressesinindividualmembers,notablyforverifyingresults.截面法：用于校核，計算部分桿內力6.6Methodofsections23Instruction:6.6Methodofs24Beforethecalculation,assumefirstallelementstobeundertension,iftheresultisnegativeitmeanstheelementconcernediscompressed,oppositetothedirectionassumed.

先把桿都設為拉力,計算結果為負時,說明是壓力,與所設方向相反。6.6Methodofsections246.6Methodofsections253.Thejudgmentsoftheinternalforcesofspecialelements三、特殊桿件的內力判斷Ifnoloadactsonatwo-elementjointandthetwoelementsarenotcollinear,theinternalforcesofthemarezero,theyarecalledzeroelements.兩桿節點無載荷、且兩桿不在一條直線上時，該兩桿是零桿。①Ifnoloadactsonathree-elementjointandtwoelementsofthemarecollinear,thethirdisazeroelement.三桿節點無載荷、其中兩桿在一條直線上，另一桿必為零桿②6.6Methodofsections253.Thejudgmentsoftheinte26Ifnoloadactsonthefour-elementjointandtwopairsofthemarecollinear,theinternalforcesofeachpairareequalinmagnitude

andofthesamecharacter.四桿節點無載荷、其中兩兩在一條直線上，同一直線上兩桿內力等值、同性。③6.6Methodofsections26Ifnoloadactsonthefour-27[Example3]

Pdareknown,determinetheinternalforcesoftherodsa,b,candd.[例3]

已知Pd,求：a.b.c.d四桿的內力？Solution:fromthejudgmentsofthezeroelement解：由零桿判式6.6Methodofsections27[Example3]Pdareknown,28研究A點：StudythepointA:Again6.6Methodofsections28研究A點：StudythepointA:Again296.6.3Problems.Dotheproblems6.27-6.34byyourself,andthenIwilltellyoutheanswersofthem.6.6Methodofsections296.6.3Problems.6.6Methodof30[Example][例]

Equilibriumofabodysystem物體系統的平衡問題Bodysystem(BS):Asystemcomposedofseveralrigidbodiesconnectedtogetherbyconstraints.物體系統（物系）：由若干個物體通過約束所組成的系統。6.7Framesandmachines30[Example]Equilibriumofabo31Externalforces:forceswhichactonthebodysystemfromoutside.外力：外界物體作用于系統上的力叫外力。Internalforces:forcesamongthebodiesofthebodysystem.內力：系統內部各物體之間的相互作用力叫內力。6.7Framesandmachines31Externalforces:forceswhic32Thecharacteristicsoftheequilibriumofabodysystem:物系平衡的特點：

①Thebodysystemisstatic①物系靜止②Everybodyinthebodysystemisalsoinequilibrium.②物系中每個單體也是平衡的。

6.7Framesandmachines32Thecharacteristicsofthee33foreverybodythereare3equilibriumequations,sothewholesystemofthebodieshas3n

equations.(nisnumberof

bodiesinthebodysystem)每個單體可列3個平衡方程，整個系統可列3n個方程（設物系中有n個物體）Thegeneralmethodtosolvetheproblemsofthebodysystem:解物系問題的一般方法：fromwholeparts(usualway);frompartswhole(unusualway)

由整體局部（常用），由局部整體（用較少）6.7Framesandmachines33foreverybodythereare3e34[Example]LetOAhorizontal(OA=R,AB=l)andthepressureforcetobeP.[例]

已知：OA=R,AB=l,當OA水平時，沖壓力為P時，Determine:①M,②thereactionforcesofpointO,③theinternalforcesoftherodAB,④thesidepressureactingontheleadrail.求：①M=？②O點的約束反力？③AB桿內力？④沖頭給導軌的側壓力？ Solution:StudythepointB解：研究B

6.7Framesandmachines34[Example]LetOAhorizontal356.7Framesandmachines356.7Framesandmachines36Thenstudythewheel:再研究輪6.7Framesandmachines36Thenstudythewheel:6.7Fra37[thesignofminusmeansthedirectionoftheforcesisoppositetothedirectionindicatedinthediagram][負號表示力的方向與圖中所設方向相反]6.7Framesandmachines37[thesignofminusmeansthe38①②③④6.Thestepsandskillsofsolvingproblems:六、解題步驟與技巧

STEPS解題步驟

Selectabodytostudy;

選研究對象

Drawtheforcediagram(forceanalysis);

畫受力圖（受力分析）

Selectthecoordinates,acenter,andwritedownequilibriumequations;

選坐標、取矩點、列平衡方程。

Solvetheequations,determinetheunknownquantities.

解方程求出未知數

6.7Framesandmachines38①②③④6.Thestepsandskills39

[Example1]Allelementsareconnectedwitheachotherbyjoints,thebottomBisinsertedintotheearth,P=1000N,AE=BE=CE=DE=1m,theweightoftheelementsarenegligible.DeterminetheinternalforcesofthememberACandthereactionforcesofthepointB.

[例1]

已知各桿均鉸接，B端插入地內，P=1000N，AE=BE=CE=DE=1m，桿重不計。求AC桿內力？B點的反力？6.7Framesandmachines39[Example1]Allelement40Writedowntheequations

列方程為:④

Solvingthem,weobtain:解方程得③SelectthecoordinatesystemBxywiththepointBasthegivencentre;選坐標、取矩點、Bxy,B點②Drawtheforcediagram;受力如圖①Solution:解：Studythewholerigidstructure;選整體研究6.7Framesandmachines40Writedowntheequations④41①StudytheelementCD;再研究CD桿②Drawtheforcediagram;受力如圖6.7Framesandmachines41①StudytheelementCD;再研究CD桿42Solvingthem,wefind:解方程求未知數④③SelectpointEasthegivencenter,writedowntheequation:取E為矩心，列方程6.7Framesandmachines42Solvingthem,wefind:④③Sele43[Example2]LetP=100N,AC=1.6m,BC=0.9m,CD=EC=1.2m,AD=2mABishorizontal,EDisvertical,BDisperpendiculartotheincline.Determineandthereactionforceoftheconstraints.[例2]

已知:P=100N.AC=1.6m,BC=0.9m,CD=EC=1.2m,AD=2m且AB水平,ED鉛垂,BD垂直于斜面；

求

?和支座反力？Solution:解：

Studythewholerigidstructure;

研究整體

Drawtheforcediagram;

畫受力圖6.7Framesandmachines43[Example2]LetP=100N,AC=1.644Selectacoordinatesystem,writedowntheequations

選坐標列方程6.7Framesandmachines44Selectacoordinatesystem,45AgainstudytheelementAB,drawtheforcediagramasabove;再研究AB桿，受力如圖6.7Framesandmachines45AgainstudytheelementAB,466.7Framesandmachines466.7Framesandmachines47[Example4]Consideracontinuousbeam,P=10kN,Q=50kN,CEis

vertical,neglecttheweightofthebeam.DeterminethereactionforcesofthepointsA,BandD.(wefindthenumberoftheunknownquantitiesislargerthanthree,sowecannotsolveitbystudyingthewholerigidstructure,itmustbedividedintoseveralparts.)

[例4]

已知：連續梁上，P=10kN,Q=50kN,CE鉛垂,不計梁重求：A,B和D點的反力（看出未知數多余三個，不能先整體求出，要拆開）

6.7Framesandmachines47[Example4]Consideraconti48解：①研究起重機Solution:①studythecrane6.7Framesandmachines48解：①研究起重機Solution:①studythe49②Thenstudythebeam

CD②再研究梁CD6.7Framesandmachines49②ThenstudythebeamCD②再研50③再研究整體③Studythewholerigidstructure:6.7Framesandmachines50③再研究整體③Studythewholerig51

Aladderplacedonthesmoothhorizontalsurfaceisinequilibrium,F=60kN,l=3m,α=45o,computethereactionforcesonpinCandsurface[solution]Firstly,drawtheFBDsofthewholeandeachpart,findoutthenumberofunknowns.6.7FramesandmachinesSampleproblem551 Aladderplacedonthesmoo52FromtheFBDsabove,wecanmaketheprocedureofsolvingtheproblem,thatis:

FA,FBFcx,Fcyverify6.7Framesandmachines52FromtheFBDsabove,wecan536.7Framesandmachines536.7Framesandmachines546.7Framesandmachines546.7Framesandmachines55Verify:Ifthefollowingequationsaresatisfied,thenyourresultsarecorrect.6.7Framesandmachines55Verify:Ifthefollowingequa56Sampleproblem6

Thestructureshowninthefigureleft,AB=BC=1m,EK=KD,P=1732kN,Q=1000kN,neglectallthegravityforces,computethereactionforcesandforceinleverDC6.7Framesandmachines56Sampleproblem6 Thestruct57[solution]DrawtheFBDs,findoutthepathofsolution:[FBDofthewhole] Coplanarcommonforcesystem

3equations

5unknowns

cannotdeterminealltheunknowns6.7Framesandmachines57[solution]DrawtheFBDs,fi58[FBDofDEandCD] Coplanarcommonforcesystem

3equations

3unknowns

resolvableHeretheleverDCisaleversubjectedtwoforces.6.7Framesandmachines58[FBDofDEandCD] Coplanar59[FBDofDEandCD] Coplanarcommonforcesystem

3equations

4unknowns

cannotdeterminealltheunknownsFromtheFBDsabove,wecanfindoutthepathofsolutionasfollowing:6.7Framesandmachines59[FBDofDEandCD] Coplanar60[Pathofsolution]Fex,Fey,FDFAx,FAy,MAFBDofthewholeVerify6.7Framesandmachines60[Pathofsolution]Fex,Fey616.7Framesandmachines616.7Framesandmachines62Verify--omitted6.7Framesandmachines62Verify--omitted6.7Framesan63Sampleproblem7

Astructureshownasthefigureleft,l=2R,BD=2l,computetheexternalreactionforcesatpinAandB.6.7Framesandmachines63Sampleproblem7 Astructur64[solution]DrawtheFBDs,findoutthepathofsolution:[FBDofthewhole] Coplanarcommonforcesystem

3equations

4unknowns6.7Framesandmachines64[solution]DrawtheFBDs,fi65[FBDoftheBC-Wheel] Coplanarcommonforcesystem

3equations

4unknowns IfFBxhasbeenknown,then 3equations,3unknown.6.7Framesandmachines65[FBDoftheBC-Wheel] Copla66[FBDoftheACB] Coplanarcommonforcesystem

3equations

4unknowns IfFAxhasbeenknown,then 3equations,3unknown.6.7Framesandmachines66[FBDoftheACB] Coplanarc67[Pathofsolution]FAx,FBxFBy,FCy,FcxFByFAy6.7Framesandmachines67[Pathofsolution]FAx,FBx686.7Framesandmachines686.7Framesandmachines696.7Framesandmachines696.7Framesandmachines70Theend70TheendChapter6

ForceAnalysisofStructuresandMachinesWangMing-luChapter6

ForceAnalysis72SpecializedVocabularyStructures:結構,構造,構[桁]架Truss:構架,桁架,鋼梁Joint:接合;榫;接點;聯軸節;鉸鏈Section:截口[面,線,點]Frame:框架,框子;【機】架,座身2SpecializedVocabularyStructu73ChapterObjectives.1.Usethemethodofjointstodeterminetheforceinaplanetruss.2.Usethemethodofsectionstodeterminetheforceinaplanetruss.3.Usemultiplefree-bodydiagramstodeterminetheforceinaframeormachine.3ChapterObjectives.1.Usethe746.1Introduction.46.1Introduction.75Structureoftrussinengineering

工程中的桁架結構6.1Introduction.5Structureoftrussinenginee76Structureoftrussinengineering

工程中的桁架結構6.1Introduction.6Structureoftrussinenginee77Structureoftrussinengineering

工程中的桁架結構6.1Introduction.7Structureoftrussinenginee78Structureoftrussinengineering

工程中的桁架結構6.1Introduction.8Structureoftrussinenginee79Structureoftrussinengineering

工程中的桁架結構6.1Introduction.9Structureoftrussinenginee80Truss:arigidstructuresystemcomposedofstraightelements,connectedattheirendsbypins,withoutdeformationundertheactionofforces.桁架：由桿組成，用鉸聯接，受力不變形的系統。6.2Simpleplanetrusses10Truss:arigidstructure81joint節點element桿件6.2Simpleplanetrusses11jointelement6.2Simpleplan82Theadvantageofthetruss:light,thecapabilityofthematerialcanexertsufficiently.桁架的優點：輕，充分發揮材料性能。Thecharactersofthetruss:桁架的特點：①straightmembers,theweightofthemselvescanbeneglected,theyarealldoubleforceequivalentrods;①直桿，不計自重，均為二力桿；②Thestraightelementsareconnectedattheirendsbyjoints.②桿端鉸接；③Allexternalloadsonthetrussactonlyatthejoints.③外力作用在節點上。6.2Simpleplanetrusses12Theadvantageofthetruss:83Themodeloftrussinmechanics力學中的桁架模型(basictriangle)Trianglesensurestability(基本三角形)三角形有穩定性

(a)(b)(c)6.2Simpleplanetrusses13Themodeloftrussinmechan84Thecommonsimplifiedmodelsfortrussinengineeringmechanics工程力學中常見的桁架簡化計算模型6.2Simpleplanetrusses14Thecommonsimplifiedmodels856.2Simpleplanetrusses156.2Simpleplanetrusses866.4MethodofjointsP=10kN,determinetheinternalforcesoneveryelements.已知：如圖

P=10kN，求各桿內力？[Example][例]Solution:

解：①studythewholestructure,determinethereactionforces:

①研究整體，求支座反力166.4MethodofjointsP=10kN,87②Studythejoints

A,CandDonebyone,determinetheinternalforcesofeverymember.②依次取A、C、D節點研究，計算各桿內力。Weget: (denotethattherodiscompressed)6.4Methodofjoints17②StudythejointsA,CandD88SubstitutingintoWeobtain:Substitutingintoweget:6.4Methodofjoints18SubstitutingintoWeobtain:S89TheanotherequationofthejointDcanbeusedtoverifytheresultofcalculation.節點D的另一個方程可用來校核計算結果ThesolutionofwhichisItequalsto,thecalculationisaccurate.恰與相等,計算準確無誤。6.4Methodofjoints19Theanotherequationofthe906.4.3Problems.Dotheproblems6.1-6.25byyourself,andthenIwilltellyoutheanswersofthem.6.4Methodofjoints206.4.3Problems.6.4Methodof91[Example]h,aandParegiven,determinetheinternalforcesontheelement4,5and6.[例]已知：如圖，h，a，P，求：4，5，6桿的內力。IISolution:

Studythewholestructure,determinethereactionforces.解：研究整體求支反力①①6.6Methodofsections21[Example]h,aandParegiv92②DrawacutI-I,studytheleft-handsectionofthetruss.選截面I-I，取左半部研究A'From6.6Methodofsections22②DrawacutI-I,studythel93Instruction:說明：MethodofIsolationjoints:Thisisconvenientwhenthestressesinallelementsofthetrusshavetobedetermined.節點法：用于設計，計算全部桿內力Methodofcuts:

Thisisconvenientindeterminingthestressesinindividualmembers,notablyforverifyingresults.截面法：用于校核，計算部分桿內力6.6Methodofsections23Instruction:6.6Methodofs94Beforethecalculation,assumefirstallelementstobeundertension,iftheresultisnegativeitmeanstheelementconcernediscompressed,oppositetothedirectionassumed.

先把桿都設為拉力,計算結果為負時,說明是壓力,與所設方向相反。6.6Methodofsections246.6Methodofsections953.Thejudgmentsoftheinternalforcesofspecialelements三、特殊桿件的內力判斷Ifnoloadactsonatwo-elementjointandthetwoelementsarenotcollinear,theinternalforcesofthemarezero,theyarecalledzeroelements.兩桿節點無載荷、且兩桿不在一條直線上時，該兩桿是零桿。①Ifnoloadactsonathree-elementjointandtwoelementsofthemarecollinear,thethirdisazeroelement.三桿節點無載荷、其中兩桿在一條直線上，另一桿必為零桿②6.6Methodofsections253.Thejudgmentsoftheinte96Ifnoloadactsonthefour-elementjointandtwopairsofthemarecollinear,theinternalforcesofeachpairareequalinmagnitude

andofthesamecharacter.四桿節點無載荷、其中兩兩在一條直線上，同一直線上兩桿內力等值、同性。③6.6Methodofsections26Ifnoloadactsonthefour-97[Example3]

Pdareknown,determinetheinternalforcesoftherodsa,b,candd.[例3]

已知Pd,求：a.b.c.d四桿的內力？Solution:fromthejudgmentsofthezeroelement解：由零桿判式6.6Methodofsections27[Example3]Pdareknown,98研究A點：StudythepointA:Again6.6Methodofsections28研究A點：StudythepointA:Again996.6.3Problems.Dotheproblems6.27-6.34byyourself,andthenIwilltellyoutheanswersofthem.6.6Methodofsections296.6.3Problems.6.6Methodof100[Example][例]

Equilibriumofabodysystem物體系統的平衡問題Bodysystem(BS):Asystemcomposedofseveralrigidbodiesconnectedtogetherbyconstraints.物體系統（物系）：由若干個物體通過約束所組成的系統。6.7Framesandmachines30[Example]Equilibriumofabo101Externalforces:forceswhichactonthebodysystemfromoutside.外力：外界物體作用于系統上的力叫外力。Internalforces:forcesamongthebodiesofthebodysystem.內力：系統內部各物體之間的相互作用力叫內力。6.7Framesandmachines31Externalforces:forceswhic102Thecharacteristicsoftheequilibriumofabodysystem:物系平衡的特點：

①Thebodysystemisstatic①物系靜止②Everybodyinthebodysystemisalsoinequilibrium.②物系中每個單體也是平衡的。

6.7Framesandmachines32Thecharacteristicsofthee103foreverybodythereare3equilibriumequations,sothewholesystemofthebodieshas3n

equations.(nisnumberof

bodiesinthebodysystem)每個單體可列3個平衡方程，整個系統可列3n個方程（設物系中有n個物體）Thegeneralmethodtosolvetheproblemsofthebodysystem:解物系問題的一般方法：fromwholeparts(usualway);frompartswhole(unusualway)

由整體局部（常用），由局部整體（用較少）6.7Framesandmachines33foreverybodythereare3e104[Example]LetOAhorizontal(OA=R,AB=l)andthepressureforcetobeP.[例]

已知：OA=R,AB=l,當OA水平時，沖壓力為P時，Determine:①M,②thereactionforcesofpointO,③theinternalforcesoftherodAB,④thesidepressureactingontheleadrail.求：①M=？②O點的約束反力？③AB桿內力？④沖頭給導軌的側壓力？ Solution:StudythepointB解：研究B

6.7Framesandmachines34[Example]LetOAhorizontal1056.7Framesandmachines356.7Framesandmachines106Thenstudythewheel:再研究輪6.7Framesandmachines36Thenstudythewheel:6.7Fra107[thesignofminusmeansthedirectionoftheforcesisoppositetothedirectionindicatedinthediagram][負號表示力的方向與圖中所設方向相反]6.7Framesandmachines37[thesignofminusmeansthe108①②③④6.Thestepsandskillsofsolvingproblems:六、解題步驟與技巧

STEPS解題步驟

Selectabodytostudy;

選研究對象

Drawtheforcediagram(forceanalysis);

畫受力圖（受力分析）

Selectthecoordinates,acenter,andwritedownequilibriumequations;

選坐標、取矩點、列平衡方程。

Solvetheequations,determinetheunknownquantities.

解方程求出未知數

6.7Framesandmachines38①②③④6.Thestepsandskills109

[Example1]Allelementsareconnectedwitheachotherbyjoints,thebottomBisinsertedintotheearth,P=1000N,AE=BE=CE=DE=1m,theweightoftheelementsarenegligible.DeterminetheinternalforcesofthememberACandthereactionforcesofthepointB.

[例1]

已知各桿均鉸接，B端插入地內，P=1000N，AE=BE=CE=DE=1m，桿重不計。求AC桿內力？B點的反力？6.7Framesandmachines39[Example1]Allelement110Writedowntheequations

列方程為:④

Solvingthem,weobtain:解方程得③SelectthecoordinatesystemBxywiththepointBasthegivencentre;選坐標、取矩點、Bxy,B點②Drawtheforcediagram;受力如圖①Solution:解：Studythewholerigidstructure;選整體研究6.7Framesandmachines40Writedowntheequations④111①StudytheelementCD;再研究CD桿②Drawtheforcediagram;受力如圖6.7Framesandmachines41①StudytheelementCD;再研究CD桿112Solvingthem,wefind:解方程求未知數④③SelectpointEasthegivencenter,writedowntheequation:取E為矩心，列方程6.7Framesandmachines42Solvingthem,wefind:④③Sele113[Example2]LetP=100N,AC=1.6m,BC=0.9m,CD=EC=1.2m,AD=2mABishorizontal,EDisvertical,BDisperpendiculartotheincline.Determineandthereactionforceoftheconstraints.[例2]

已知:P=100N.AC=1.6m,BC=0.9m,CD=EC=1.2m,AD=2m且AB水平,ED鉛垂,BD垂直于斜面；

求

?和支座反力？Solution:解：

Studythewholerigidstructure;

研究整體

Drawtheforcediagram;

畫受力圖6.7Framesandmachines43[Example2]LetP=100N,AC=1.6114Selectacoordinatesystem,writedowntheequations

選坐標列方程6.7Framesandmachines44Selectacoordinatesystem,115AgainstudytheelementAB,drawtheforcediagramasabove;再研究AB桿，受力如圖6.7Framesandmachines45AgainstudytheelementAB,1166.7Framesandmachines466.7Framesandmachines117[Example4]Consideracontinuousbeam,P=10kN,Q=50kN,CEis

vertical,neglecttheweightofthebeam.DeterminethereactionforcesofthepointsA,BandD.(wefindthenumberoftheunknownquantitiesislargerthanthree,sowecannotsolveitbystudyingthewholerigidstructure,itmustbedividedintoseveralparts.)

[例4]

已知：連續梁上，P=10kN,Q=50kN,CE鉛垂,不計梁重求：A,B和D點的反力（看出未知數多余三個，不能先整體求出，要拆開）

6.7Framesandmachines47[Example4]Consideraconti118解：①研究起重機Solution:①studythecrane6.7Framesandmachines48解：①研究起重機Solution:①studythe119②Thenstudythebeam

CD②再研究梁CD6.7Framesandmachines49②ThenstudythebeamCD②再研120③再研究整體③Studythewholerigidstructure:6.7Framesandmachines50③再研究整體③Studythewholerig121

Aladderplacedonthesmoothhorizontalsurfaceisinequilibrium,F=60kN,l=3m,α=45o,computethereactionforcesonpinCandsurface[solution]Firstly,