工程力学 第13章 杆件的位移分析与刚度设计

113§13-1§13-22§13-3*§13-4§13-5§13-6313§FNxFQyFQzMxMyMzFQyFQz89FNxMyMzMxEAFxNNeyyyEIMr1zzzEIMr1PddGIMxxjeNFNxEAryMyEIyyrzMzEIzzxddjMxGIP(131)xuddNNexyydd1qrxzzdd1qr413lxEAFuxddNN131xEIMyyyddq132xEIMzzzddq133xGIMxddPj134duN—dxdqy—dxydqz—dxzdj—dx1312FNxMyMzMxn5132131321132abc()ΔlEAlFxEAFlxlxN0Nd∫=(135)∑=niiiixiAElFl1N(136)()∫lxxEAxFl0Nd(137)133131∫∫lxlxEAFxuddNNNe138132abcΔlnMyMzMMyMz133a133belasticcurvedeflectioncurve6deflectionwslopeqhorizontaldisplacementuuw133134abcABMFa132133AB1r134a134133bOxwqtandd=xw(139)qtanq»qq=xwdd(1310)wwxdeflectionequation1321337232dd11/⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛+′′xwwrEIMxwxw±=⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛+23222dd1dd/(1311)q=xwdd^1EIMxw±=22dd(1312)differencialequationforsmall135wdeflection135wx135bEIMxw-=22dd(1313)nrelativeangleoftwist134136abcP0PdGIlMxGIMxlxAB∫j(1314)∑=++niiixiDBCDACABGIlM1Pjjjj(1315)8()xGIxMlxABd0P∫j(1316)13613513813141316MMx1313§5()xqxF=ddQ()xFxMQdd=FQxMx(shearingforceequation)(bendingmomentequation)13132nn1321n0n()()axaxaxaxnn≥⎩⎨⎧--)(0(13-17)9xansingularfunctionn0120l2137abc0121370121317()()axaxaxnaxnxaxnnn≥⎩⎨⎧-----11)(0dd(13-18)()()axaxnaxnaxxaxnnn≥⎪⎩⎪⎨⎧+-+--++∫1)(01d11(13-19)nMi138axaix≥aiMMiMi〈xa〉01320iMaiiMix138nFPj138bxbix≥bi,MFPjFPj〈xbj〉1132110bjjFPjxnqk138cqk()221kkkcxqqM-1322ckkqkxnMiFPjqk13201322()∑∑∑-+-+-ikkijjiiicxqbxFaxMxM21P021132313201323MiFPjqk138132013231313131139EIBAC139131120lA3FP/4FP()()lxlxFxFxM≤≤---=040431P1P(a)xx=-10113(a)1313()()lxlxFxFxMxwEI≤≤-+-=-=0443dd1PP22(b)ClxFxFxwEIEI+-+-==2P2P42183ddq(c)DCxlxFxFEIw++-+-=3P3P46181(d)CD4ACx0w0xlw01D02ClllFlF+⎟⎠⎞⎜⎝⎛-+-=461810P3P2P1287lFC=5(c)(d)()⎥⎥⎦⎤⎢⎢⎣⎡+-+-=222P128742183llxxEIFxq(e)()⎥⎥⎦⎤⎢⎢⎣⎡+-+-=xllxxEIFxw233P128746181(f)BACEIlFwB3P2563=EIlFEIlFBA2P2P12851287==qq6()⎟⎠⎞⎜⎝⎛+-=≤22P12878340lxEIFxlxq()⎟⎠⎞⎜⎝⎛+-=xlxEIFxw23P128781()⎥⎥⎦⎤⎢⎢⎣⎡+⎟⎠⎞⎜⎝⎛-+-=≤≤222P1287421834llxxEIFxlxlq12()⎥⎥⎦⎤⎢⎢⎣⎡+⎟⎠⎞⎜⎝⎛-+-=xllxxEIFxw233P128746181()4§131superpositionmethod13101321321310aqqlql2EI11310bcd2131()321=iwCi13EIqlwEIqlwEIqlwCCC4342411614813845-===,,(a)BqBi,,,EIqlEIqlEIqlBBB33323131161241=-=-=qqq(b)3B,EIqlwwiCiC43138411-==∑=EIqliBiB3314811==∑=qq1331311aEI1413111331CAB1311b2611cd,,248112812813422241lEIqlEIqllwwEIqlwBBCC×--=×+==qEIqlEIqlCC323148161-==qq,3,EIqlwwiCiC42138441=∑=EIqliCiC321487==∑=qq*FPyzFPyFPzyzwywzwywzw1312131215zzzyyyEIlFwEIlFw3P3P3131==1312qq2223P22sincos3⎟⎟⎠⎞⎜⎜⎝⎛+=+=IyIEIlF⎟⎟⎠⎞⎜⎜⎝⎛==qatanarctanarctanyzyzIIww(1324)Iz≠Iya≠qnnn§redundantconstraintdegreeofstaticallyindeterminateproblem16compatibilityequation,constitutiveequations8n1341313aABCD1E1A1l123E2A2E3A3l2l3E1A1E2A2l1l2FP1.ABCD123FN1FN2FN31313b13131340=∑xF0=∑yF0sin-sinN2N3=aaFF0coscosPN3N2N1=-++FFFFaa⎭⎬⎫=+=PN2N1N3N1cos2FFFFFa(a)217AA′1313aA′aacoscos1132llll′==baa′3⎪⎪⎭⎪⎪⎬⎫==222N22111N11AElFlAElFlc4cbacos111N1222N2AElFAElF=d5ad⎪⎪⎪⎪⎭⎪⎪⎪⎪⎬⎫+==+=211122211122PN3N2211122PN12121lAElAElAElAEFFFlAElAEFFen1314181314aMeAMeB1314b1()0=∑FxM3eeeMMMBA==1314cn1315abcd345631315a1315bcd12313l5aBu=0u=01315cdFBx01315bc111315()()0=+=ByBBBFwqww()qwB()ByBFwqFBy1315a19B()EIqlqwB481=()EIlFFwByByB331-=qlFBy83=28185qlMqlFAAy==1315dFBx=0FByMB2qlFFByAy==MA=MBMBB()()()0=++ByBBBBFqMqqq()()()ByBBBBFqMqqqMBqFBy1315aB()EIlMMBBB=q()EIqlqB361=q()EIqlEIlFFByByB324121-=-=q122qlMB122qlMMBA=FNxFQyMzFNxMzFQyFQy0FNx0615eMCz()()0=+=qMCCzCCqqq20()CzCMq()qCqMCzqC()EIlMMCzCzC⎟⎠⎞⎜⎝⎛=2q()EilqqC3261⎟⎠⎞⎜⎝⎛-=q242qlMCz=1222qlMMqlFFBAByAy====()[]ww≤max(1325)[]qq≤max(1326)[]w[]q132[]jj≤max(1327)⎥⎦⎤⎢⎣⎡≤⎟⎠⎞⎜⎝⎛lljjmax(1328)[]j⎥⎦⎤⎢⎣⎡lj211351316FPFP20kNalml2mE=206GPaB[]q=0.5d1316135B1131EIlaFB3Pq2[]qq≤qrad[]q3-34964201012180m11110m111mm3p2060.510d×××××≥=×=××××1361317q=10kN/ml3mE=196GPas118MPa[]2501max//=lw2h2bbh2213171361[]ss≤=WMmaxmax(a)m45kNmN1045mN3101021213232max⋅=⋅×=⋅⎟⎠⎞⎜⎝⎛×××==qlM()32626322bbbbhW===amm1662mm083m10083m101182104533363≥==×=⎟⎟⎠⎞⎜⎜⎝⎛××××≥-bhb..2[]ww≤max⎥⎦⎤⎢⎣⎡≤lwlwmax(b)131EIqlw4max81EIqllw3max81(c)123bhI=(d)cdb⎥⎦⎤⎢⎣⎡≤lwEbqlmax4316323mm1792mm689m10689m101961625031010334933≥==×=⎟⎟⎠⎞⎜⎜⎝⎛××××××≥-bhb..3b89.6mmh179mm§nw0w0q01318l1318n1319131922413201320nnn1315dn1315eforcemethod25EAEIGIP134134231∞→11AE23011→AE23assemblestressthermalstressEIIl13211321EGIPG2627132[w]0.00030.0005l0.0002l0.010.03m0.020.05m[]radq0.0010.0050.0050.00250.00160.001m13ld36mmABCCDClCD282FP10ACl1lAElFlCP2=lABCDEC1311321321.2m110103m2150mmBCEs=200GPaEa70GPaFP60kNC133EIMxw-=22ddabcd133AbcBbaCbdDcd13413429135135136ABqlaEI136137EI()()3230224xlxlEIxqxw+-=137138413830139As≠0e≠0Bs≠0e0Cs0e0Ds0e≠01391310BECDd16mmABCh2.5mmABCABCBECDE200GPCCDABCC131013110.24mm1213113113122b2bb2bFPEs196GPaEi=98.0GPax1313ACAa200mm2Ea70GPaaa=26106/ºCDBAs80mm2Es190GPaas=18106/ºC20oC0.5mm140oC13121313131413141315ABCDBADG80GPaABCD32131513161316*1317ABBCBACEIFP40kNq20kN/m13171318**1318ABCDBCd0=1.2mmE=105GPaq30kN/mAD1319ABCD13191320FP=1.6kNd=32mmE200GPa[]w330.05mm132013211321W20kNE200GPaB[]q=051322q10kN/ml4m[]s=100MPa[]w=ll000E200GPa1322