Introduction of Finite Element Method-Lesson3

Lesson3:DisplacementandShapeFunctions何昊哲/游景皓1.1有限元的基本思路分析步骤221l/2l/2l/2123P,Fδ,Fδ,Fδ,Fδ1l/212231.2基本力学量以矩阵表示xyijmqsqv{}Tsysxsysxsqqqqq==][{}Tvyvxvyvxvqqqqq==][(1-1)(1-2)(1-3)4.{}Txyyxγεεε=][{}Txyyxτσσσ=][xux∂∂=εyvy∂∂=εxvyuxy∂∂+∂∂=γTxvyuyvxu∂∂+∂∂∂∂∂∂=][ε(1-4)(1-5)(1-6)−−=xyyxxyyxEγεεµµµτσσ21001112()xyxxEµεεµσ+−=21{}[]{}εσD=(1-7)(1-8)21µ−Eµµµ−−1−−=xyyxxyyxEγεεµµµτσσ21001112{}[]{}εσD=Dxyijm1.3位移函数(displacementfunction)与形函数(Shapefunction)位移函数设定n=i,j,m(1-10)[]Tnnnvu=}{δ{}{}{}{}[]Tmmjjiimjivuvuvu==δδδδ(1-11)yaxaau321++=yaxaav654++=选取位移函数应考虑的问题例题:1234yaxaau321++=yaxaav654++=123456(1-12)形函数(ShapeFunction)iiiyaxaau321++=jjjyaxaau321++=mmmyaxaau321++=iiiyaxaav654++=jjjyaxaav654++=mmmyaxaav654++=mmmjjjiiiyxuyxuyxuAa211=mmjjiiyuyuyuAa111212=mmjjiiuxuxuxAa111213=mmmjjjiiiyxvyxvyxvAa214=mmjjiiyvyvyvAa111215=mmjjiivxvxvxAa111216=(1-14)mmjjiiyxyxyxA11121=xyijm(1-15)将1-14代入1-12])()()[(21mmmmjjjjiiiiuycxbauycxbauycxbaAu++++++++=])()()[(21mmmmjjjjiiiivycxbavycxbavycxbaAv++++++++=(1-16)mjimjijmmjixxcyybyxyxa+−=−=−=(1-17)ijm])()()[(21mmmmjjjjiiiiuycxbauycxbauycxbaAu++++++++=])()()[(21mmmmjjjjiiiivycxbavycxbavycxbaAv++++++++=NimmjjiiuNuNuNu++=mmjjiivNvNvNv++=(1-19)(1-18)进一步简化=mmjjiimjimjivuvuvuNNNNNNvu000000{}[]{}δNf=(1-20)[]mjiNNNN=][100100INNNNNiiiii===(1-21)(1-22)形函数性质1),(=iiiyxN0),(=jjiyxN0),(=mmiyxN1),(),(),(=++yxNyxNyxNmjiijiixxxxyxN−−−=1),(jijjxxxxyxN−−=),(0),(=yxNm3AdxdyNAi=∫∫ijdlNli21=∫{}[]{}δNf=