计算力学第4课(2015)

1、 2 矩形单元矩形单元 平面等参元平面等参元 坐标变换坐标变换 3 ( )( )( )Tiijjijiju xN x uNx uNNuu(0)1,( )0(0)0,( )1iijjNN lNN l12( )(, )kNxxki j 4 1211,l ( )1ixN xl ( )jxNxl( ) ( )( )eiijju xNdN x uNx u( ) eu xNd1 22 1( )1111 Txiijjijijedu xN uN uuuuuBddxllll 12( )(, )kNxxki j (0)1,( )0(0)0,( )1iijjNN lNN l 5 DE222211 11eeTeeee

2、EAEAdVdVllllKBD B dVEEAEAdVdVllll12233456uxvxxx1232345610000001uxvxxx 6 02 16 12 6 d udv 123456 T02310000001xxxx,iiijjju vu v6 66 16 1 ed 16 66 16 1 ed 1006 62 16 12 62 66 16 1 ueevHddA dH 2310000010uvHxHxxx 7 22 xueeyvduHdxA dBdyHd vydx eDD Bd TeeKBD B dV 8 12345678( , )( , )u x yxyxyv x yxyxy 9 ,x

3、yab( , ); 11, 11 ( 11)( 11)xayb 12345678( , )( , )uv 12345678( , )( , )u x yxyxyv x yxyxy 10 1010 ( ,)1(,)(,)(,)0iiiijjikkimmNNNN (1)(1)(1)(1)(1)(1)(1)(1)ijkmNNNN1111,4444abcd (1)(1)(1)(1)(1)(1)(1)(1)ijkmNaNbNcNd 11 1(1)(1)(, , ,)4rrrNri j k m 111 11xyxyuuuubxaxvvvvayybabuvuvuvuvabbayxyx1(1)(1)41(1)

4、(1)41(1)(1)41(1)(1)4ijkmNNNN ( 11)( 11)xayb 12 ijkmeeBBBBdBd00(1)0011000(1)04(1)(1)iiiiiiiiiiiiiiiNbxbNNBaaNyabababNNabyx eeijkmeDD BdSdSSSSd2(1)(1)(1)(1)4(1)11(1)(1)22iiiiiiiiiiiiibaESbaabab 13 eeeKdF8 33 33 88 8 iiijikimjijjjkjmTeekikjkkkmmimjmkmmKKKKKKKKKBD B dVKKKKKKKK1111 TTTrsrsersrsKBD B dVBS

5、 hdxdyhabBS d d 21111(1)(1)323211114(1)(1)(1)2323rsrsrsrsrsrsrsrsrsrsrsrsbaEhababba 2,11EE 14 eeeFTF eeedTd TeeeeKTKT 15 ,1ijkeijkLLL ,jikijkeeeLLL 16 1()2iiiiieLab xc yN11111()()() ()2221ijjjkkjjkjkiiikkxyxyx yx yyyxxxyab xc yxy 11(),()22jjjjkkkkeeLab xc y Lab xc y112iiiijjjjekkkkLabcLabcxLabcy 17

6、11,11jjjjiiikkkkxyyxabcxyyx 112iiiiejjjjkkkkabcLxabcLyabcL 1111iijkjijkkLxxxxLyyyyL 18 1()21()2jikijkijkeijkjikijkijkeijkLLLbbbxLxLxLxLLLLLLcccyLyLyLyLLL 19 ! ! !2,( , ,)(2)!epqrijkep q rL L L dxdyp q rpqr 为常整数! !,( ,)(1)!pqijlp qL L dSlp qpq为常整数 20 1kijLLL 1100ieLijdxdyJ dLdL 2iiejjxyLLJxyLL 11002(

7、1)ieLpqrpqrijkeijijijL L L dxdyL LLLdLdL 1111000! !2(1)2(1)(1)!iLpqrpq reijijjieiiiq rLLLLdL dLLLdLqr 2! !(1)! ! !212(1)!(2)!(2)!p q reeq rp qrp q rqrpqrpqr 21 22 4141( , )( , )iiiijjkkmmijjiijjkkmmjxNxN xN xN xN xyNyN yN yN yN y 1(1)(1) (, , ,) ( ,)(1,2,3,4)4rrriiNri j k mi 23 iijjkkmmiijjkkmmuN uN

8、 uN uN uvN vN vN vN viijjkkmmiijjkkmmxN xN xN xN xyN yN yN yN y 24 2222123456782222910111213141516uv 25 8811,rrrrrruN u vN v21(1)(1)(5,7)2irNi 21(1)(1)(4,8)2irNi 8811,rrrrrrxN xyN y1(1)(1)(1)(1,2,3,4)4irrrrNi 26 112233,NL NL NL412523613(21)(1,2,3)4,4,4rrrNLLrNL L NL L NL L 27 28 edNd24223366114488II

9、IISIIIIISdNdN dNddN dNdN d213468,IIIIIIIIIdddddd213468,IIIIIIIIIdddddd213468111111,NNNNNN24IIISSdd1(1)(1)(1)(1,2,3,4)4irrrrNi 29 30 iijjkkmmiijjkkmmuN uN uN uN uvN vN vN vN v eijkmeuxvBdBBBBdyuvyx edNd00iiiiiNxNByNNyx 31 rrrrrrNNNxyxyNNNxyxy ,iiNNxy rrrrrrNxyNNxxJNNNxyyy ,xyxy,iiNNxy1 rrrrNNxJNNy( ,

10、 )( , ),iiNNxy 32 eeijkmeD BdSdSSSSd211122iiiiiiiiNNxyNNESD BxyNNyx 33 ,dd TeeKBD B dVdrdxidyj xxdxddyydydd()()xxyydrddiddj 34 xxxxdddAdxdyd dJ d dyyyydd 1111 eTTTeeKBD B dVhBD B dAhBD B J d d 35 12345 TTTTTTddddddrrrduv12345 TTTTTTPPPPPP rrxryPPP 36 (1)(1)(1)11112131(1)(1)(1)22122232(1)(1)(1)331323

11、331 FKKKdFKKKdFKKKd(1)(1)(1)11111213(1)(1)(1)22212223(1)(1)(1)333132331 00000000000000000000FdKKKFdKKKFdKKK111 FKd 37 313233343313233343xyXXXXPYYYYP3 1323 3343 FFFFP KdP41eeKK 38 1()()2exxyyxyxyeeiieiiejjejjekkekkdVX uY vX uY vX uY v 1 2TTTeeeeeedBD Bd dVdF 1 2TTeeeedVdF 44111 ( ) 2TTTeeeeeeeedBD B dVddF 411( )2TTeeeeeedKddF 39 444111 , , eeee