材料力学第4章弯曲内力

1、第四章第四章 弯曲内力弯曲内力本章内容本章内容:1 弯曲的概念和实例弯曲的概念和实例2 受弯杆件的简化受弯杆件的简化3 剪力和弯矩剪力和弯矩4 剪力方程和弯矩方程、剪力图和弯矩图剪力方程和弯矩方程、剪力图和弯矩图5载荷集度、剪力和弯矩间的关系载荷集度、剪力和弯矩间的关系7平面曲杆的弯曲内力平面曲杆的弯曲内力4-1 4-1 弯曲的概念和实例弯曲的概念和实例车削工件车削工件4-1 4-1 弯曲的概念和实例弯曲的概念和实例 AB对称轴对称轴纵向对称面纵向对称面梁变形后的轴线与梁变形后的轴线与外力在同一平面内外力在同一平面内梁的轴线梁的轴线FRAF1F2FRB4. 2 受弯杆件的简化受弯杆件的简化1

2、支座的几种基本形式支座的几种基本形式u 固定铰支座固定铰支座9u 可动铰支座可动铰支座u 向心轴承向心轴承u 向心止推轴承向心止推轴承u 固定端约束固定端约束FAxFAy2 载荷的简化载荷的简化集中载荷集中载荷分布载荷分布载荷集中力偶集中力偶简支梁简支梁外伸梁外伸梁悬臂梁悬臂梁FAxFAyFByFAxFAyFByFAxFAyMA3、静定梁的基本形式、静定梁的基本形式 BAalFFFFRBABF , , () , RRR0000 xAxAByAyFFFaMFlF laFFlMFRAxFABFmmxRAyFSCFFRBFSCM() , , SRR00yAyCAyF laFFFlMMFx；mm+MM

3、mm（受压）（受压）MM0AMFRBBdEDAabclCFF1F2FRA021R bFaFlFB0BM0)()(21R blFalFlFAlblFalFFA)()(21R lbFaFFB21R BdEDAabclCFF1F2FRAAEcFSEFRA00SR EAyFF,F0, 0R cFMMAEEAEFFRS cFMAE RAEcFSEFRAa-cb-cCDl-cBEFSEF1F2 0yF021RS FFFFBE0ME0)()()(21R McbFcaFclFEB+cFMAE RAEFFRS BdEDAabclCFF1F2FRAFdBFSFMFFRB,SRR0000yFBFFBFFFMMFd-

4、+BFFFRS dFMBFR niiFF左（右）1S mkkiniiMaFM左（右）左（右）左（右）左（右）11F2=FACDBbacF1=FFF kN60RR FFFBAF2=FACDBbacF1=FFRBkN601SFFCmkN061 .bFMC060601RS FFFADmkN8 .13)(1R FacFacFMADC12MkN4RSS1 ACFFF左mkN41R1 ACFMM左kN4)4(RS2S BCFFF右mkN651)4()152(R2 .FMMBC右B1m2.5m10kNmAC12FF xFS(x)FS 图的坐标系图的坐标系OM 图的坐标系图的坐标系xOM(x)BAFlx)0(

5、)()0()(SlxFxxMlxFxF 0SFA左左FFA 右右SFSxFFlxM2RRqlFFBA lqFFABx)0(222)()0(2)(2RRSlxqxqlxxqxxFxMlxqxqlqxFxFAA )0(2)(SlxqxqlxF2SqlF 2SqlF +ql/2ql/2BlqFAxF)0(222)(2RlxqxqlxxqxxFxMA lqFABxF00 Mx，0, Mlx( )d0d2M xqlqxx2lx 822maxqlMMlx+82qll/2 lqFABxF+ql/2ql/2+82qll/282maxqlM2maxSqlFlFABCabFFlFbFA RlFaFB R xxlF

6、ABCabFRAFRB)2()0()()1()0()(SaxxlFbxMaxlFbxF )4()()()()()3()()()(SlxaxllFaaxFxlFbxMlxalFalblFFlFbxF )1()0()(SaxlFbxF)3()()(SlxalFaxF xxlFABCabFRAFRB+lFblFa)4()()()(lxaxllFaxM )2()0()(axxlFbxM+lFbaxxlFABCabFRAFRB+lFbalFblFaR AMFlR BMFl( )() ( )S01MFxxlllABCabFRAFRBM+MlxxlABCabFRAFRBM( ) MM xxl)0(ax (

7、)()MMM xx Mlxll)(lxa CMaMl左左 0M CMbMl右右+MalMbllABCabFRAFRBM+/M l+MalMblABFlxlxlFFA)(R lFxFB RxlxlFxFMAC)(R 0ddxMC0)2(xllF2lx 2lx xlxlFxFMAC)(R FlM41maxxyq(x)FMxyq(x)FMFS(x)M(x)FS(x)+dFS(x)M(x)+dM(x)mmnnq(x)C .nxmmn dx02d)d(d )()( )(d)( 0 S xxxqxxFxMxMxMMCFS(x)M(x)FS(x)+dFS(x)M(x)+dM(x)mmnnq(x)C0d)()

8、(d)()( 0SSS xxqxFxFxFFx )(d)(dSxqxxF)(d)(dSxFxxM)(d)(dSxqxxF)(d)(dSxFxxM)(d)(d22xqxxMxFS(x)OxOM(x)(d)(dSxqxxF)(d)(dSxFxxM)(d)(d22xqxxMxFS(x)OxOM(x)(d)(dSxqxxF)(d)(dSxFxxM)(d)(d22xqxxMOM(x)x)(d)(dSxqxxF)(d)(dSxFxxM)(d)(d22xqxxMFCMC或或q0FMC)(d)(dSxqxxF ( )( )2211SddxxxxFxq xx()()( )21S2S1dxxFxFxq xx)()

9、(2S1SxFxF)(d)(dSxFxxM( )()()21S21dxxFxxM xM xk623R.FA kN27R BFBACD2001151265FFFF231kN6 .23RS AAFF右23.61.727+BFRBACD2001151265FFFRA231kN27RS BDFF右kN0S右右BFkN27SmaxFkN7 . 1RS FFFAC右4.723.11+BACD2001151265FFFRAFRB231mkN11. 3115. 0R BDFM0MBmkN72. 4max MmkN72. 42 . 0R ACFM0MABACD2001151265FFFRAFRB2314.723

10、.11+23.61.727+ FEqABCD0.21.612kN806 . 11005 . 0RR BAFF+80kN80kNkN80RS AAFF右 kN80RS ACFFkN80RS BDFFkN80RS左 BBFFkN0S右右BF kN80maxS FFEqABCD0.21.6123m4mABCDE4m4mFRAFRBF1=2kNq=1kN/mM=10kNmF2=2kNkN7R AFkN5R BFkN7RS AAFF右kN34RS qFFAC左kN141RS FqFFAC右kN3R2S BDFFF3m4mABCDE4m4mFRAFRBF1=2kNq=1kN/mM=10kNmF2=2kNk

11、N3R2S BFFFkN22S FFB右kN7S 右右AFkN3S 左左CFkN1S 右右CFkN3S DF01RS FqxFFAx7kN1kN+3kN3kN2kNx=5m0MA 2R7164DBMFMF左左5 .20maxMMF647R2 BDFFM右632 FMB0 ME201666+20.53m4mABCDE4m4mFRAFRBF1=2kNq=1kN/mM=10kNmF2=2kN204242R qFMAC3m4mABCDE4m4mFRAFRBF1=2kNq=1kN/mM=10kNmF2=2kN7kN1kN+3kN3kN2kNx=5m201666+20.510.5113F=50kNM=5k

12、NmAECDKBFRAFRBMAq=20kN/m 81kN31kN29kN+10.5113F=50kNM=5kNmAECDKBFRAFRBMAq=20kN/m0RS qxFFBxm45. 1R qFxBx=1.4581kN31kN29kN+10.5113F=50kNM=5kNmAECDKBFRAFRBMAq=20kN/mkN mAAMM右右9 96 6. .5 5mkN5 .15181MMAE05 . 031MMECmkN31131MMCD96.515.53110.5113F=50kNM=5kNmAECDKBFRAFRBMAq=20kN/m R1KBFMMmkN345129.maxR2 45B

13、MFMmkN5545.122q5kN mBMM左左0MB右右96.53115.5x+5534510.5113F=50kNM=5kNmAECDKBFRAFRBMAq=20kN/m+10.5113F=50kNM=5kNmAECDKBFRAFRBMAq=20kN/mabcd18kN2kN14kN3m3m6m+( )SddFq xxkN18SFB左左kN2S FB右右CABDF=20kN( )SddFq xx( )SSd6dDCcq xxqFFkN/m26)2()14( qabcd18kN2kN14kN3m3m6m+q=2kNCABDF=20kN)(d)(dSxFxxM( )SdbBAaMMFxxmk

14、N543180abcd18kN2kN14kN3m3m6m+cbSBCxxFMMd)() 542348 kN m0Md48dab54c+40kNmabcd2m2m2m+)(d)(dSxFxxM( )SSd2cCBbFxxFMMS0 4020kN22CBMMFabcd20kNMe)(ddxqxFS0MA左左mkN40右右AM40KNmabcd2m2m2m+abcd20kNBCAD0SFB左左kN20SFB右右kN20SFB左左0SFB右右FF)()()(),(Sn21S11S21SnnFFFFFFFFFF (,)()()()121122nnnM F FFM FMFMFxF=ql/3qlxF=ql/

15、3ql2)(2qxFxxMxqxFxxMF )(2)(2qxxMq2)(2qxFxxMFxqxlqx+FxxMF)(Fl2)(2qxxMq22ql+-62ql62ql812qlBCF2F3aDEF1AaaaF1291acebdF2e122+acbd215acebdF3aaaa291acebdmkN1312152112229121)()(MCBCF2F3aDEF1Aaaa122+acebd215acebd291215131acebdABCalF1F2ABCM(x)FN(x)FS(x)CxF1xalF1F2ABCFS(x)CBaF1F2FN(x)M(x)x0 FN图图F1|CalF1F2ABFS图图F1+F2+CalF1F2ABF1aF1aF1a+F2l FOR ScosFF NsinFF sinMFR 0sin0N FFFn0cos0S FFFt0sin0CMMFR FtnC FNFS O+MFRq qmmxOFRAB( )()() ()cos1cos0MFxF RRFRqqqqqqqq ( ) ()S1sin0FFFqqqqqq ( ) ()N2cos0FFFqqqqqq O+FS图图ABOM图图2FROFN图图FF+xOFRq qmmAB( )()() ()cos1cos0MFx