第十三章能量法材料力学教学课件

1、(chapter thirteen)(energy method) )d(d11lfw ealfl11d)d( pf lf lfo l l1d l1df1f1lfealffealfwwf22dd2101 lflfwu2121n ealfeaflln ealfealfu222n2 niiiiiaelfu12n2 lxeaxxfu02n)(2)d(allfvuu2121 e 222122eeu 或或p2p2epeee222121giltgilmgilmmmwu lxxgixtud)(2)(p2 niiiiiigltu1p2n2 eilmeilmmmwu221212eeee xxeixmuld)(2

2、)(2e xxeixmxxgixtxxeaxfullld)(2)(d)(2)(d)(2)(2p22n dxdydzxyzabd dxdydzxyzabd zyxwvwvuu21dddddddd )dd(d21)d)(dd(21dzyxxzyw u21 ggu2222 lavxauvuudddggu2222 p222p2p22d)(2dd2dd2giltaitglxag)it(xagual al a p22gilmu ppgimlgitl 2p2 lgiu 123fu21 f3bcf2af1)(233221123322112ffccffcffcfcfc f3abcf1f2123 if3abcf1

3、f212333221121fffu iabflxxfxm )(eilfxeifxxeixmull6d2)(d2)(32022 b21vfw eiflv33b abcfx1x2ableilbafbeilafaeilbfxeixlfaxeixlfbxeixmubal63232d2)(d2)(d2)(22232223222202210212 c21vfw eilbfav322c eirfreifrreimul8d2)sin(d2)(322022 sin)(frm bv21 fweifr43bv abcabf1f2abcabf1f2eaafb11 eaafbfw22121111 eabafcfw2)(

4、2122222 eabafc2)(22 abcabf1f2eaafb22 eaaffbfw21213 eaaffeabafeaafbfcfbfwu2122212122112)(22121 abcabf1f2eabaf2)(22 eaaf221abcabf1f2eaaf1eaaff21eaaffeabafeaafbfcfbfwu2122212122112)(22121 meeilmeifl16482e31 eilmeifl316e22 acbfl/2l/2)16696(121212e2e322e1flmlmlfeimfu eifl4831 eiflffw48212131 eilm3e eilmm

5、mw32121eee2 acbfl/2l/2acbfl/2l/2meeilm162e3 eilmffw162e33 acbl/2l/2eilmmeilmfeiflfu321164821ee2e3 acbfl/2l/2me 1 2f1f2f1f2 1 2f3 3 4f422112121ff 44332121ff f1f2 1 2f3 3 4f42 1 12ff2211 ffffffu221144332211121212121 f1f2 1 2 3 4f4f3ffffffu443344332211221212121 21uu ffff44332211 4 3 ff3311 31 f1f2a a l

6、xeixmud2)(2 lxeixmud2)(2afuuu 11af1f2=1f0af1f2fa)()(xmxm lxeixmxmud2)()(2212uu laxeixmxmfuud2)()(120 llllaxeixmxmxeixmxeixmxeixmxmfuud2)()(d2)(d2)(d2)()(12220 laxeixmxmfd2)()( lxeixmxmd2)()( lxeixmxmd2)()( niiiiealff1nn lllxeixmxmxgixtxtxeaxfxfd2)()(d)()(d)()(pnnaqbcll/2ql/2ql/222)(2qxxqlxm )(0lx aa

7、b11/21/2cxxxm21)( )2(0lx qbcll/2ql/2ql/2)(3845)d22(22)d()(4220 eiqlxqxxqleixeixxmxmfl/lcql/2aab11/l1/lx11)( xlxm)(0lx qcll/2eiqlxqxxqllxeieixxmxmlla24)d221)(1)d()(320 ql/2bacqf=qaa2abaabca2a1xcqf=qaa2arax1/22qara 22)(2qxxqaxm 2)(xxm xqaxm )(xxm )()(32)d)()d2)(22(140202 eiqaxxqaxxxqxxqaeifaac)(32)d)(

8、)d2)(22(140202 eiqaxxqaxxxqxxqaeifaacbaabca2acqf=qaa2ara1/2xx1ba1xxabca2axcqf=qaa2arax1/222)(2qxxqaxm axxm2)( xqaxm )(1)( xmeiqaxqaxxaxqxxqaeiaac65)(1)d()d2)(22(130202 aabcflei1ei2aabcflei1ei2xxfxxm )(faxm )(xxm )(axm )(abc1lei1ei2xxa)(3)d( )(1)d( )(1d)()(d)()(221302010201 eilpaeipaxafaeixxfxeixeixm

9、xmxeixmxmlalay0)( xm1)( xmfxxm )(faxm )(abcflei1ei2xxaabclei1ei2xxa202010201(1)d)(1(0)d)(1d)()(d)()(eifalxfaeixfxeixeixmxmxeixmxmlalab 1cfababfab1abxxabbaccfxxm )(xxm )(0)(n xf0)(n xfxxfaxm )(axm )(fxf )(n1)(n xf llcxxfxfeaxxmxmeih)d()(1)d()(1nn)(323 eafbeibfaeifaxafaeixxfxeibad )(1d )(100 xfeabd1)(

10、10 fab1abxxabbaccxxabcfabxxfxxm )(xxm )(0)( xt0)( xtabcfababcabxxfxxm )(xxm )(fbxt )(bxt )(xxabcfababcabxx lnlcxxmxmgixxmxmeiv)d()(1)d()(1nnxxfxeixxfxeibad )(1d )(100 xbfbgiapd )(10 pgifabbaeif233)(3 )( a1a2bcllffxlxm )(flxm )(xxfxxm )(xxm )(fxxm )(xxm )(a1a2bcllffxxxa1a2bcll11) (35)d)()d)(21300 eif

11、lx-l-flx-x-fxeillabcllqxxxx2)(2qxxm 2)(2qlxm xxm )(lxm )(abcllqabcllq)(85)d2d2(140022 eiqlxlqlxxqxeilly2)(2qxxm 2)(2qlxm 0)( xmxxm )()(4)d2d02(140022 eiqlxxqlxqxeillxxxxxabcllqabcllqxx2)(2qxxm 1)( xm2)(2qlxm 1)( xmeiqlxqlxqxeilla32)d12d12(130022 xxabcllqabcllq1faafabcde132456789afaaabcde132456789afa

12、afabcde132456789a11 niiiiealff1nnifnifniliiilffnnf2f221 / 21 / 21 / 21 / a2a22fa/2fa/2fa/eafa.eafa)(ealffiiiiac12423291nn baorff(a)barpf d (b)barp1(c)eifrreifrreimmab302203d)cos(12d)()(2 )cos(1)( frm)cos(1)( fmsdoo 2 1 3f1f2f3 332211212121fffwu,321 2 1 3f1f2f3iif21 iifff2211 iiiiffffu212211iif21 iif

13、ffu2211iiiiffff 2211iifu iifu iifu iifu xfxfeaxfeaxxfffuiiiid)()(2)d(nn2n xfxtgixtgixxtffuiiiid)()(2)d(pp2 xfxmeixmeixxmffuiiiid)()(2)d(2 ijnjjjiiffealffu n1niifu 2)d(2)d(2)d(2p22n lllieixxmgixxteaxxffxfxmeixmxfxtgixtxfxfeaxfiiid)()(d)()(d)()(pnn fabcmelae1e11)()(mxlfalmxm 111)(xlafxm 1)(1e11 lxmxm2

14、22)(fxxm 0)(e22 mxm222)(xfxm abclafx1x2lfalmra eme1011d)()(xfxmeixmflc abclafx1x22022221011d)()(d)()(xfxmeixmxfxmeixmala 202222d)()(xfxmeixma )( )363(13e2 falamflaei)63(1eflalmei meabcdaa2ame)(21cerrmmafffayd frdfraxfrayffax rxmmafxm)(21)(ce xfxm )(axmxm2)(c ammafxm2)(21)(ce xafxm 2)(0)(c mxmxfxm )(

15、0)(c mxmxxabcdaa2amexfxm cfxxm )(frdfraxfray2axxabcdaame00c fmxfu00cc fmcmu axxei0d01 axxamei0e)d2(1 axxaxmei20ed21)(6172e eiam0d00dd2212000ee aaaxxmxaxaxmeieiam32e frdfraxfraybcllqxfxm )(2)(2qxfxxm blqmbxbcllqfxxqlfq 22bqlflm xqlfqxxm)()( xfxm )(2)(2bqlflmxt lfxt )(0 fifu644di 324dip lplxfxtgixtxfx

16、meixmxfxmeixm00d)()(d)()(d)()( lpllxlqlgixxqlxeixxqxei02002d21d1)d)(2(1)(241144 pgiqleiqlmef1xffabcdll2 le)(mfxxm xfxm )(1)(e mxmff1abcf2flf1xffabcdll2 lmefx1)(e mxm1)(e mxmlfxm2)( lfxm2)( e2)(mflxm xfmflxm1e2)( ffabcdll2 lxf2flf1xeifl2132 d1)2(d121001 llxxfflxfleiffmdmu 1e0e lxfxei20d11memexxmxmeil

17、)d()(1 xeaffldnn xgittlpd xxmxml)d()( ldxxcxcmmxxmxml)d()( eimxeixmxmlcd)()( xxmxl)d(tg ctgmxc xccm(x)xx)(xmlbxaxm )(xxmxbxxmaxxm)bxa(xxmxmllll)d()d()d(d )()(000 lxxm0)d(xxml 0)d(m(x)xlx)(xmxcc lxxmx0)d(clxxxm 0)d(ccclllmxbaxbaxxmxbxxmaxxmxm )()d()d(d )()(00cxba m(x)xlx)(xmxcceimxxmxmeilcd )()(1 eim

18、c balh三角形三角形cclh顶点顶点二次抛物线二次抛物线3bl 3al 2hl 85l83llh 32lh顶点顶点cn 次抛物线次抛物线lh顶点顶点c二次抛物线二次抛物线3hl 1 nhllnn2)1( 2 nlcqfc4lc11c221cm2cm82ql825l/ 2428323221qllql llmc 325485cqcf)(38453252424321 eiqllqleieimeimfccc4lc11c221cm2cm82ql825l/ fcabalqfcabaalqmql2/8fa0)212322322(132 aql-afaafaleivc)(83alaqlf 1abalcm abbafa/2a/2eifaeifaab32)212123181(233 0 ab a)(838531324144 eiqa)(eiqaahc5a300b500=1f01020 c5a300b500=1020 x1x1xxxeixmxmxgixmxmllpnnbd)()(d)()(01101 fxxmab )(xxmab )(0f.xmca30)(1n 30)(1n0.xmca xeifxxgi.f.pdd303030021500 34333102021040325030603010105210