飞行器结构有限元练习题答案

13Ni(xj,yj)=δijNi+Nj+Nm=13NiNjNm1()2iiiNabxc=++∆iy(,,)ijm1211iijjmmxyxyxy∆=jimmjxyaxy=11jimyy=−b11jimxcx=(,,)ijm1111(,)()211122jjjjiiiiimmmmxyyxNxyxyxyyx=−+=⋅∆∆∆=1111(,)()001122jjjjijjjjmmmmxyyxNxyxyxyyx=−+=∆∆⋅=1111(,)()01122jjjjimmmmmmmmxyyxNxyxyxyyx=−+=∆∆0⋅=j(,)ijjiNxyδ=1()21[()()()]21(200)12ijmiiijjjmmmijmijmijmNNNabxcyabxcyabxcyaaabbbxcccyxy++=++++++++∆=++++++++∆=∆+⋅+⋅=∆1a123a223tEν=0NBSKe3NiNjNm12Na∆,)m()iiiibxcy=++(,ij11iijjmm21xyxyxy∆=jimmjxyaxy=11jimyy=−b11jimxcx=(,,)ijm∵1(,0)a2(0,)a3(0,0)∴11Na=x21Na=y3111Nxaa=−−y∵[]{}[]{}eeijmuNNNvδδ==IIIN∴0001000xyaxy[]xyaa−−=xy−−N∵{}0001000{}[]{2xijmeeyijmxyiijjmmbbbccccbcbcbε}εεδγ===∆Bδ∴1000101]000101011011a−[=−−−B∵{}{}[][][]{}[]{}eeDDσεδ===BSδ∵101010010100(1)/2001/2EDEνννν==−−0[]∴100010[][]00010101/21/201/21/2EDa−[]==−−−SB200020011011011011[][][][]0002024211031011213TEtDt−−−−−=∆=−−−−−−−eKBB231111215161121222526135152555633616265663xyxyfkkkkufkkkkvfkkkkufkkkkv=11u=230uu==1230vvv===11x1fk=2x31fk=35x1fk=1y21fk=24y1fk=36y1fk=x1230xxxfff++=y1230yyyfff++=1121314151610kkkkkk+++++=0[K]=[K]T034iijjmmxxLxLxL=++iijjmmyyLyLyL=++1()2iiiiLabx∆==++∆∆icy1()2jjjjLabx∆==++∆∆jcy1()2mmmmLabx∆==++∆∆mcyaibiciajbjcjambmcm2∆1211iijjmmxyxyxy∆=1[()()()]21[()()()]21[020]2iijjmmiiiijjjjmmmmiijjmmiijjmmiiiiiixLxLxLabxcyxabxcyxabxcyxaxaxaxbxbxbxxcxcxcxyxyx++=++++++++∆=++++++++∆=+∆⋅+⋅=∆iijjmmyLyLyLy++=4j(6,3)i(2,2)m(5,6)q1q2xy555NiNjNmB12iiNa=+∆()iibxcy+(,,)ijm1211iijmmjxyxyxy∆=jjimmxyaxy=11jimyby=−11jimxcx=(,,)ijm∵(2,2)i(6,3)j(5,6)m∴12221631156∆==33421312461iiijjjmmmabcabcabc==−=−=−==−=−=−=∴1(213)13i=−−Nxy1(243)13jNx=−+−y1(64)13mNx=−−+y∴0003040111000010304213133441ijmijmiijjmmbbbccccbcbcb−−0[]==−−∆−−−−B57∵iijjmmuNuNuNu=++iijjmmvNvNvNv=++0ijmuuuu===0ijmvvvv===∴00()1iijjmmijmuNuNuNuNNNuuu=++=++=⋅=0000()1iijjmmijmvNvNvNvNNNvvv=++=++=⋅=∴000uxvyuxyxε∂∂{}∂==∂∂∂+∂∂∴{}0[]{}00Dσε==∴6(x3,y3)654321(x2,y2)(x1,y1)qxy88142qtL1L2L31112333)(21)NLL=−22(21)NLL=−(21NLL=−414NL=⋅2L3L1L524NL=⋅634NL=⋅L3=01423560NNN===∴3300xyff=5500xyff=6600xyff=142142ξ0-11142-101142124)11(1)2ϕξξ=+1(1)ϕξξ=−−(1)(1ϕξξ=−+∴1111126lfqtdqltϕξ−==∫1441223lfqtdqltϕξ−==∫1221126lfqtdqltϕξ−==∫l142∴112cos11sinsin66yyylqtqtffxxlθθθ−===−11cosxffθ211442442coscos22sinsin33xyf11fyylqtqtffxxlθθθθ−===−222222coscos11sinsin66xyf11fyylqtqtffxxlθθθθ−===−θq7N1N6∑Ni=1L1L2L39i=16111222333(21)NLL=−(21)NLL=−(21)NLL=−414NL=⋅524NL=⋅634NL=⋅)(221)4LLL+−2L3L1L∴11223312233(211)(44iNLLLLLLLLL=−−++⋅+⋅+∑1⋅3)22212312233112222444(LLLLLLLLLLLL=+++⋅+⋅+⋅−++21233211232[()(22)]()LLLLLLLLL=++++−++∵1231LLL++=22[(1)(2NLLL=−+−∑∴33312)]()iLL−++3L2111=×−=80xy1180°S2iii∆i,)jm1()Nabxcy=++(,i1211iijjmmxyxyxy∆=jjijmmmxyaxyxy==−mjxy11jijmybyy=−=−my11jimmxcxxjx==−(,,)ijm001[]0002ijmijiijjmmbbbccccbcbcb0m=∆B[][][][]TDt=∆eKBB[][][]D=SB∵bimcibjcjbcm∴[Ke][S][B]∴bicibjcj∵[Ke][S]bmcm∴180bicibjcjbm∵180[Ke][S]cm∴91i,j,m1KS80662−−−16160120413.54.57.54.513.51.51.59.51.55.5K−−−−=−−1003030000301201.51.50.51.5S−=−−−−−KS180mjmiimj18018066261601204()13.51.51.55.5K−−−−−−0030300301201.51.50.51.5−−−13.54.57.54.5′=1()00S′=−−9.51.5−jimmjiijjmmi29.51.5267.54.55.5041.51.58066160124.513.5K−−−−−−−−=−23000300100030.51.5201.51.5S−=−−13.5mjjii1012∂N1/∂x∂N2/∂y(1/2,1/2)Q41(1)(1)(1)/4Nξηξη=+++−82(1)(1)(1)/4Nξηξη=−+−+−(1)(1)(1)/4Nξηξη=−−−−−34(1)(1)(1)/4Nξηξη=+−−−226(1)(1)/Nηξ=−−3.1255(1)(1)/2Nξη=−+2227(1)(1)/2Nξη=−−2(1)(1)/Nηξ=−+∵81iiixNx==∑yQ(1/2,1/2)81iiiyN==∑1∴8,,115.625iiixNxξξ===∑8,,15iiiyNyξξ===∑38,,1iiixNxηη===∑8,,115iiiyNyηη===∑∴,,,,15.6255[]3.1255xyJxyξξηη==∴11,1,11,1,15.62550.56250.0257[].125150.56250.0321xyNNJNNξη−−==⋅=∴12,2,12,2,15.62550.18750.0114[]3.125150.06250.0018xyNNJNNξη−−==⋅=(10,10)0567812xyηξQ1234(30,20)(5,40)(40,50)1113∂N4/∂x∂N4/∂yP(1.5,2.0)L1L2L3111(21)L=−222(21)NLL=−333(21)NLL=−∆NL414NL=⋅2L3L1L524NL=⋅634NL=⋅14∴44142122144NNLNLbbLL∂∂∂∂∂+1211224(22)(2)bb=∆+∆∆∆1222xLxLx=+=∂∂∂∂∂∆∴4414224NNLNLL1212121244(2222(2)ccLcyLyLy∂∂∂∂∂+=∆+∆∂∂∂∂∂∆∆∆12)c=+=∵1002142114∆==111.52.0214211==2100211.52.04114∆==45112214b=−=−210414b==114311c==−21011c=−=−∴421122244(22)[4(2)54]0.245(2)14Nbbx∂=∆+∆=×−+×=∂∆421122244(22)[4(3)5(1)]0.347(2)14Nccy∂=∆+∆=×−+×−=−∂∆6542(4,2)1(0,0)xy3(1,4)P(1.5,2.0)ijm131215Alρ20012llEAudxAudxρ′Π=−∫∫00ll0EAuudxAudxρ′′δΠ=⋅δ−⋅δ∫∫x00|()llEAuuEAuAudxρ′′′=δ−+⋅δ=∫0EAuAρ′′+=0=xlEAu′==0000xuuu==δ′=221()22luxxlxEEEρρρ=−=−15xρA1316P1/4yxxyyx1/42d2a2b2c1614