第二章平面力系

1、第二章第二章 平面力系平面力系AF2F4F3F1FRF2F3F4FRFR1FR2F4F3F1FRF1AF2A F2F3F4FRF1A01niiF FF2F3FRF1AF4 PABCD ， FB FA P FBFAPPFB5 . 0tan PFPFBA2522 lhKMMN 2/sin sin2221PhPlFF Fh, 0 xABFOijy sincoscosFFFFFyx jiFFFyxyxFFjFiFyyxxFF,FFFFFFFyxyx/,(cos/),(cos22) jFiF AF2F4F3F1FRxy yiynyyRyxixnxxRxFFFFFFFFFF2121RRyRRRxRyixi

2、RyRxRFFFFFFFFF),cos(,),cos()()(2222jFiF0)()(2222 yixiRyRxRFFFFF 00yixiFF PABCD FBxy0sin, 00cos, 0 AByAxFFFFPF FAPFPFBA21,25 BMFACFBC0coscos, 00sin)(, 0 BABCyBCBAxFFFFFFF0cos, 0 CBMyFFF sin2FFFBABC cot2cosFFFCBM BCFCBFhFMO)(FOABA 2（Nm） AFxOyFyyxxyyOxOOyFxFMMM)()()(FFF)()()()()(21iOnOOOROMMMMMFFFFF)()

3、(xiiyiiROFyFxM FFnOrFrF hcos)(rFhFMnnnOFcos)()()()(rFMMMMnOOrOnOFFFFFFFFFdxFdxFMMMooo)()()(),(FFFF0F FF00FF01F2FABFdFDCF1F2MMMFF FF FF FF F / 2F / 2自由矢量自由矢量F44FF33F1FF1F22FFFdFdFM3111 dFdFM4222 3434,FFFFFF 2134)(MMdFFFdM 0 iM inMMMMM21？aaABCaBCABFBFCFABF02, 0 aFMMA22ABFFMa 22CBBFFFMa M1M2CABDM2CDM1A

4、B FB FA FC FD030cos, 01 MaFMBaFMB231 030sin, 02 aFMMCaFMC212 321 MM0, 0 aFMMAaMFFBA EDCBA FBDBC CFDFMDE FE FDCAE FC EF 045sin, 0 MaFMDaMFFED2 51sin045cossin, 0 ECyFFFaMFC5 EDCBA1F2F1F2FPRMoAMMBCABdABABdMFFFFF3F1F2FF1F2F3niiOOniiRMM11)(FFFFRnixiiyiiniiOOFyFxMM11)()(F22)()(yixiRFFFRyiRRxiRFFFF),cos(,)

5、,cos(jFiF与简化中心选择无关与简化中心选择无关与简化中心有关与简化中心有关niiOOMM1)(F00ORMF00ORMF00ORMF00ORMF00ORMFOFRFR ROFMd 00ORMFRF 00ORMFFRFRFR00ORMFOFRFR FRFRFR ldxxqdPP0)( lxdxxqxdPPh0)( lldxxqxdxxqh00)()( lqldxxqP0)(2)()(00ldxxqxdxxqhll xlqxq0)( PqPlqxdxlqdxxqPll000021)( 32)()(00ldxxqxdxxqhll 0)(, 0, 0111niiOniyinixiMFFF00O

6、RMF 几点说明：几点说明：0)(00FoyxMFF, 0 xF0 cos BAxFF, 0 yF0 sin21 BAyFFGFF , 0FAM 0 sin cos221 lFcFblFlGaFBB6060060 cos, 02 FFFAxx0)(FAM060 sin)(212112 llFlFMlFB, 0 yF060 sin21 FFFFBAykN 75. 0 AxFkN 56. 3 BFkN 261. 0 AyFqq 0224, 002, 00, 0MaaqaGaFMFGaqFFFFBABAyyAxxFqaGFqaGFFBAyAx2143 ,234 ,0 PABCDxy02, 0)(0,

7、 00, 0MaPaFMFFFPFFBABAyyAxxF.,PFPFPFBAyAx 020)(020)(0, 0MaPaFMaPMaFMPFFAyBBAAxxFFPFPFPFBAyAx PABCD02, 0)(02, 0)(02, 0)(MaFaFMMPaaFMPaMaFMBAxCAyBBAFFFPFPFPFBAyAx 0)(, 0)(, 0)(FFFCBAMMM0)(, 0, 0FAyxMFF0)(, 0)(, 0FFBAxMMFFRBAx0223, 0)(0223, 0)(023, 0)(aPMFaMaPMFaMMFaMBCABCAFFFaMFPaMFPaMFCBA3323332,3332

8、 FDECBAaaaACaaaB0)(0FoyMF0)(0)(FFBAMMF3F2F1Fn0 xF0, 00121, 0)(PFFFFFFPMNBNAyNABFN3750,N250 NBNAFFDABCFFFP321 qP0)(, 0)(12lPPebFbaPMNABF0)(, 0)(2bePbFaPMNBAFabePPbalPPe)(21 balPPeP 12abePP)(2 PABCFAFBFCPABFBFA(a)(b)kN3 . 0 AxF解得解得:kN2 . 1 CxF 例题例题8 已知：已知：P=0.4kN, Q=1.5kN, sin=4/5,求：支座求：支座A、C的反力。的反力。A

9、QCBPPABFAxFAyFCxFCyFBxFByFAxFAy) 3(0, 0)2(0sin2cos23cos2, 0)() 1 (0sin2cos2cos2, 0)(QFFFlQlPlFMlQlPlFMCxAxxAyCCyAFF解：解：(1)取整体为研究对象取整体为研究对象解上述方程，得解上述方程，得kN6 . 0,kN2 . 0 CyAyFF(2)取取AB为研究对象为研究对象0coscos2sin, 0)(AyAxBFlPlFMF代入（代入（3）式得）式得 045sin,0045 cos,002522,0GFFFFFFlGlFMEyAyExAxAEF81345 sin825GFGFGFAE

10、yA 02245 cos, 0lFlFlFMEyKDBCF823852GFGFGFDBExK DBFCyFCxF452G2G45, 0 xF045 cos2 GFBx, 0 yF0245 sin2 GGFByN 34745 cos2 GFBxN 847245 sin2 GGFBy0542 CBByFGF, 0 yF, 0 xF053 CBABBxFFFN 340 1 ABFN 660 1 CBF2GBAFBCFByFBxF45节点构造有节点构造有榫接（图榫接（图a）焊接（图焊接（图b）铆接（图铆接（图c）整浇（图整浇（图d）均可抽象简化为光滑铰链均可抽象简化为光滑铰链 FAxFAyFBy012

11、05, 0)(040, 0020, 0ByAByAyyAxxFMFFFFFFFAyFAxA20kNF1F2C10kN10kN10kN10kNAB12345678910111412131516171819212020kNC0, 0020, 012 FFFFyx045sin, 0045cos, 03134 FFFFFFFFAyyAxxkN24kN,16kN,20 ByAyAxFFFkN20, 021 FFkN36kN,21643 FF1FF3F4mn10kNA1234520kNCF6F7F8FAxFAyD0110121, 0)(, 012011, 0)(01045sin, 07867AxCAyDA

12、yyFFFMFFMFFFFFFAxFAyFBy10kN10kN10kN10kNAB12345678910111412131516171819212020kNCkN.42kN,26kN,36876 FFFkN24kN,16kN,20 ByAyAxFFFPEF2F3F4F5FAxFAyF1A2FF603, 0)(0, 00, 0aPaFMPFFFFFNBANBAyyAxxFaaaaaaP1ABECD05 . 0sin5 . 0cos5 . 0, 0)(22aFaFaPMEF0sin, 021 FFFFAyyFAxFAyFNB3/, 3/2, 0PFPFFNBAyAx 3/52PF PF 12EAB

13、CDFAyFAxFECDFDxFDy05 . 1, 0)(05 . 2, 0)(0, 0aqaaFMaqaaFMFFAyBEAAxxFFqaFqaFFEAyAx5 . 25 . 10 045sin5 . 0, 0)(aFaqaMCDFqaFC22 FCBCCAACBFAxFAyMACyFCxFFCxFCy FBFAxFAy FB01, 005 . 012, 0)(0, 0qFFFqFMFFBCyyBCCxxFkN5 . 1, 0,kN5 . 0 CyCxBFFF025 . 11, 0)(01, 00, 0CyAACyAyyCxAxxFqMMMqFFFFFFFmkN4kN,5 . 3, 0 AA

14、yAxMFF若不求若不求C处的力则可处的力则可选整体为研究对象选整体为研究对象DCEFExFEyFDxFDyAHDCGEBFAxFAy) 3(0, 0)2(025002, 0)() 1 (045002, 0)(ExDxxEyDDyEFFFFMFMFFN500,N1000 EyDyFF0650025004, 0)(BAFMFN1000 BFGEBExFEyF0224, 0)(EyExBGFFFMFN1500 ExFN1500 DxFN1000N1500 DyDxFFN500N1500 EyExFFDCEFExFEyFDxFDyAHDCGEBFAxFAyBDAFDyFDxFBxFByFAxFAy解

15、解:(1) 取整体为研究对象取整体为研究对象002, 0)(ByByCFaFMF(2) 取取DEF杆为研究对象杆为研究对象02, 0)(0, 0)( aPaFMaPaFMDxBDyEFF解得：解得：PFPFDxDy2, (3) 取取ADB杆为研究对象杆为研究对象0, 00, 002, 0)(ByDyAyyBxDxAxxDxBxAFFFFFFFFaFaFMF解得：解得：PFPFPFAyAxBx ,aBCDAFEPaaaFCxFCyFBxFByPDFEDxFDyFB例题例题16 求：求：A、D、B的约束反力。的约束反力。aBCDAFEPaaaaBCDAFEaaaAaBCDFEPaaaAaBCDFE

16、aaaPFBxFByFCyFCxBCByFFAyPBxFFAxAB) 3(0, 0)2(0, 0) 1 (02, 0)(CxBxxCyByyByCFFFPFFFaFPaMFPFFCyBy5 . 0 0, 0022, 0)(0, 0BxAxxAyAxBByAyyFFFPaaFaFMPFFFFPFPFPFBxAyAx ,5 . 1,PFCx PPABCDCD022, 0)(0, 00, 0aFaFMMFFFFFFCyCxDDCyDyyCxDxxFPaMPFPFDDyDx 5 . 0CxFCyFPFBxFByFCyFCxBCByFFAyPBxFFAxABPPABCDBCDEDABCDEHFNBFAx

17、FAyFCxFCyFNBDxFDyFFDxFDyFNEH解：解：(1) 取取DE杆为研究对象杆为研究对象kN110, 0322, 0)(DXDxHFqFMMF(2) 取取BDC杆为研究对象杆为研究对象kN3110, 031, 0)( NBNBDxCFFFMF(3) 取整体为研究对象取整体为研究对象0326, 0)(02, 00, 0qFMMMqFFFFFNBAAAyyNBAxxF0,100kNkN,3110 AAyAxMFF解得：解得：例题例题18 已知：已知：q=50kN/m, M=80kNm,求求:A、B的约束反力。的约束反力。02, 00, 021 GGGFFFFFFFByAyyBxAx

18、x 0) 4 2() 2 2() 8 2( 2 5) 2 8 2(, 012GGGGFFMByAF 0m 4m 10)m 2m 4(, 0EBxByCFGFFMF 0m 2m 4m 8, 021GGFMEDFkN 5 . 7,kN 5 .17kN, 5 .77,kN 5 .12 AxBxByEFFFFEFDF2GG1结论与讨论结论与讨论cosFx FjiFyxFFiRFFRRyRRRxRyixiRyRxRFFFFFFFFF),cos(,),cos()()(2222jFiF0iRFF00 yixiFFOABdFMO2)(F)()()()()(21iOnOOOROMMMMMFFFFF力力偶偶矩矩 FdM0 iMiMM nininiiRFF111jiFFyx nixiiyiiniiOOFyFxMM11)()(FROFMd 0)(0 FFFOOiRMM0)(, 0)(, 0)( FFFCBAMMM0)(, 0, 0 FAyxMFF0)(, 0)(, 0 FFBAxMMF力力 系系 名名 称称独立方程的数目独立方程的数目平