第6章 単純支持の矩形板を解く Navier の方法

1 1 051027 060120 6 Navier 6 Navier 1 I 1 II 5 6 9 4 9 2 4 13 3 4 14 4 4 15 Navier 16 Navier 18 1 6 Navier Navier Fourier 2 3 I byaxpp sinsin0= 0p ( ) ( )

2 NyaxDpywyxwxw sinsin204422422= + + 4 0,0 y 00,0 0========yxMwbyMwaxx 0,0 y 00,0 02222= ==== ===ywwbyxwwaxx ( ) ( ) ( ) b a x y 0 2 x ( ) + =2222ywxwDMx =0 w=0 y w=0 w y L,0,022= = ywyw ( ) ( ) ( ) byaxAw sinsin= ( ) ( ) A ( ) . byaxbaDpw sinsin1122240 += ( ) ( ) ( ) ( ) ( ),( ) byaxabbapyxwDTTbyaxbabapxwywDMbyaxbabapy wxwDMyxyx coscos111)1()1(sinsin111sinsin1112222022 222220222222222202222 + = = = + += + = + += + = w yxMM, 2/,2/byax== 22240max11 +=baDpw + += + +=2222220max2222220max111)(,111)(babapMb abapMyx ( ),( ) ba= 220maxmax240max4)1()()(,4 apMMDapwyx+=== ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 22044,sindxwdEIMaxEIpdxwd == 220max apM= ( ) 30% ( ),( ) byaxbabpwyDSbyaxbaapwxDSyx cossin11sincos1122022202 += = += = ax= ( )

3 BybabaapyTSVaxxxx sin2111222220 + += + == xV x=a by sin ( ) 0=x 2/ax= by= Vy axabbabpxTSVbyyyy sin2111222220 + += + == ( ) ( ) ba, a xw p0 axp sin0 ( ) ( ) ( ( ),( ),( )) ab RRRVy Vx Rx y ( ) 4 4 R 4 yxVV, 4 P 2222020222222222200011)1(8421121111422 + += ++ + +=+= baabpabpabbbaabaabpdxVdyVPbayx ( ) Q 200004sinsin abpdxdybyaxpQba== P Q 4 R QRP=+4 R 2222011)1(2 + =baabpR R ( ) 4 4 R ba> ( ) maxmax)()(xyMM> ( ) 2 a>b 222211baba +>+ 22221111abab < ( ) a ( ) x ( ) ( ) ( ) ( ) ( ) 5 + +==22222220max2max1116)(6)(babahpMhyy ba= )(,2)1(3)()(2220maxmaxbahapxy=+== ( ) ( ) 2/,2/axby== + += + +=222220max222220max2111)(,2111)(abbabpVbabaapVyx ba> +> +2222211211baaabb )(,)()(maxmaxbaVVxy>> ( ) a>b 0})

4 ({2112112332222>++ = + +babbaabababaaabb II ( ) ),3,2,1,(,sinsin0L==nmbynaxmpp nm, ),3,2,1(sin)(L==maxmxf m=1,2,3, ( ) ( ) ( ) ( ) ( ) ( ) 000xxxa a/2a/3a/3ax sin ax 2sin ax 3sin axmxf sin)(= 6 ( ) bynaxmDpywyxwxw sinsin204422444= + + ( ) ( ) bymaxmbnamDpw sinsin2222240 += ( ) ( ) ba/1,/1 bnam/,/ nm, ( ) 4 ),(yxfp= ),(yxf byax<<<<0,0 ),(yxf = ==11sinsin),(nmmnbynaxmAyxf mnA nm, ( ) 2a 2b x=a x=b ( ) ( ) ( ) ( ) ff f f f f fff f f f f f f f f f f f f f f f f 0 a a 2a 2a 3a 3b 2b b b 2b ( ) f=f(x,y) y x 7 f(x,y) ( ) Amn )(,2sinsin)(,0sinsin0201mmadxaxmaxmImmdx axmaxmIaa == = = = ( ) m m + = axmmaxmmaxmaxm )cos()cos(21sinsin 1 I1 0)(sin)()(sin)(2101= + + =aaxmmmmaaxmmmmaI mm = == axmaxmaxmaxm 2cos121sinsinsin2 ( ) 2 I2 22sin2212cos121002aaxmmaxdxaxmIaa= = = ( ) m dxaxm sin 0 a ( ) = = 10sin2sin),(nnmabynAadxaxmyxf dybyn sin 0 b nmbaAabdxdybynaxmyxf = 4sinsin),(00 )}

5 ,(yxf byax<<<<0 0 ),(yxf ( ) mnA dxdybynaxmyxfabAbamn sinsin),(400 = ( ) ),(yxf Fourier ( ) Fourier . ( ),( ) 0p m n , mnA nm, ( ) p ( ) ( ) bynaxmabnamADwnmmn sinsin111222224 = = += ( ) 4 1820 Navier ( ) ( ) ( ) 8 ( ) ( ),( ) + = = + += + += = = = = = =11222221122222222211222222222coscos)1(1 sinsin1sinsin1nmmnyxnmmnynmmnxbynaxmabmn bnamATTbynaxmbnambnamAMbynaxmbnambnamAM += += = = = =1122211222cossin1sincos1nmmnynmmnxbynax mbnamnAbSbynaxmbnammAaS 0=x ax= 0=y by= + += + += = = = = )2(1,sin)2(1nmmnynmmnxaxmbnambnamAbVbynb nambnamAaV ( ) 2 ( ))

6 2 m n 2 ( ),( ) w 3 ( ) ( ) ( ) 9 4 4 0p ( ) 0),(pyxfp== p0= ( ) Fourier )1)(cos1(cos4sinsin4200 == nmmnpdxdybynaxmabpAmn nm, 2,4,6,.. 0 nm, 1,3,5,.. 0 ),5,3,1,(1620L==nmmnpAmn ( ) , w ),5,3,1,(sinsin162222260L= += nmbnammnbynaxmDpw ( ),(6,32),( ) ( ) Fourier ( ) Fourier a b X Y x a-xY X P P 0 C 4 p0y x ( ) ( ) ( ) 10 (1) (2) ( ) a=b )/( ( ) 2 (3) ( ) xM . a=b.

7 (4) 2 2 r 2 4 . (1) ( ) m n 6 C 2 XX YY YY P P 2 P ),6,4,2(sin),5,3,1(sinsin)(sinLL= === = maxmmaxmaxmmxaam XX YY m n (2) ( ) x=a/2 y=b/2 == +=LL,5,3,1,5,3,122222602sin2sin16)(mnmzx bnammnnmDpw ( ) m n 2sin m2sin n2sin2sin nm12 +nm 1 1 1 1 1 0 1 3 1 -1 -1 1 3 1 -1 1 -1 1 1 5 1 1 1 2 3 3 -1 -1 1 2 5 1 1 1 1 2 1 7 1 -1 -1 3 3 5 -1 1 -1 3 5 3 1 -1 -1 3 7 1 -1 1 -1 3 2sin2sin nm ( )

8 11 == + + =LL,5,3,1,5,3,1222221260)1(16)(mnnmmzxbn ammnDpw a=b ()DapDapDapnmmnDapwmnnmmzx4040640,5,3,1, 5,3, ) (6708175081916216901115014116)1(16)(=+ + = + + ++ =+ = == +LLLL ( ) ( ) ( ) 2 )(== ( ) + (3) ( ) Mx ( ) Amn ( ) 1 x=a/2 y=b/2 + + = == +22225,3,15,3,1222221240max)1(16)(bnambn ammnpMmnnmx LL a=b ()() == +++ =LL5,3,15,3,12222212420max)1(16)(mnnmxnm nmmnapM ( ) 1) 20420max)1( ) ()1(16)(apapMx +=+ + + += 1) 6 )1()75(751)93(931)111(1111)1(222222 += + ++ ++ + ( ) ( ) ( ) ( ) ( ) ( ) ( ) 12 )(2020maxapapMx== ( ) ( ) ( ) m n 4 ( ) (4) 2 2 r 2 4 2 16 max4maxwrW= ( ) 222221 +=bnamc 222221 +=BnAmC kBAba== A=ra B=rb crnbambrnkmbrnBAmBC422224422224422224)/( )/(= += += += ( ) ( ) r(=A/a) (a) P P=p0ab (b) P P=p0AB=r2p0ab=r2P r2 r2 P r2 ( ) ( ) wmax Wmax a b B A (a) (b) 13 2 4 0p ),(yxfp= +<< +<< ==22,22,)

9 ,(0 bxbaxapyxfp p=0 ( ) ( ) Fourier x 3 + + ++=222200sin),( aaaaaadxdxdxdxaxmyxf 2 0),(pyxf= 1 3 0 Fourier ),5,3,1,(,2sin2sin)1(162sin2sin2sin2sin1 6sinsin412202022220L= === ++ + nmbnammnpbnamnmmnpdxdybynaxmpabAnmbbaamn 12)1(2sin2sin + =nmnm m n ( ) Fourier ( ) w b a p0 p0 x y ( ) ( ) ( ) 14 == + + =LL,5,3,1,5,3,1222221260sinsin2sin2sin)1 (16mmnmbynaxmbnambnammnDpw ba== , ( ) ( ) ( ) a=b 2/,2/ba== ( ) ()DapDapnsmnmmnDapwmm40640,5,3,1,5,3, ) (164sin4sin116=+ =+= ==LLL ( ) 1/4 1/2 3 4 4 P ( ) 0,0 P 0pP= 0,0 Taylor bnbnamamamam22sin,22!)

10 3122sin3 ==+ =L ( ) ( ) P == + + =LL,5,3,1,5,3,122222124sinsin)1(4mnnmbyn axmbnamDabPw a=b x=a/2, y=b/2 DPaDPanmDPawmn242,5,3,1,5,3, ) (4)(14=+++++=+= ==LLL ( ) 60Pa2/D m n 2 ( ) ( ) ( ) ( ) ( ) 15 4 4 ),( A P 4 . P f(x,y) <<++<<<<= <<++<<<<=)(0)()0(0),()(0)()0(0),(byddypy yxfaxddxpxyxf