.

1998

.......................................................................................................................... 3 ....................................................................................... 5 .................................................................................................... 5 .............................................................. 7 ................................................................................................................... 9 ..................................................................................................................... 10 Rayleigh-Ritz ....................................................................................................... 13 : q................................................... 14 Galerkin................................................................................................................ 20 : q ................................................................. 21 Rayleigh-Ritz ............... 25 1: ................................................................... 25 (1) ..................................................................................................... 26 (2) ..................................................................................................... 28 2 - ....................................................................................... 34 .................................................................... 38 ............... 39 ........................................... 40 .................................................. 42 ............................................. 44 ......................................................................................... 56 : ................................................................................................ 58 ................................. 60

2

.

, (energy methods),

(approximation theory),

CAD (Computer Aided Design).

- ,

, ,

.

,

Rayleigh-Ritz Galerkin,

.

.

, - - (pre- and post-processing)

. ,

CAD ,

.

,

(databases).

.

o 1o

.

.

.

,

3

. ,

, .,

.

(benchmark tests)

.

. ,

, .

.

.

(calculus of variations)

(variational principles).

..

, ,

.

.

Maple.

4

,

.

.

.

V

S,

.

.

.

:

+= v s iiiivirt dsuTdvuBW )( (1) - (.. ) iB

- ( S iT

Cauchy Gauss

:

( )( ) ( )

++=+=+=+

v v jiijijiji

v jiijv ii

s ijijv iiv s iiii

dvudvuB

dvudvuB

dsudvuBdsuTdvuB

,,

,

)(

(2)

(.

)

:

( ) ( ) ( ) ijijijijjiji uu +=+== ,, (3) 5

0= ijij :

( ) ijijjiij u = , (4) :

( ) ++=+ v v ijijiijijv s iiii dvdvuBdsuTdvuB ,)( (5)

)(iTiB

( ) 0, =+ ijij B , :

=+ v ijijs iiv ii dvdsuTdvuB )( (6) .

.

iB

iT

iu ij .

.

-.

.

iuij

:

=

+=

+=

v jiij

v ijijv jiijv

ijjiijv ijij

dvu

dvudvudvuu

dv

,

,,,,

)(

)(21)(

21

2 (7)

6

. :

( ) = v ijijv jiijv jiij dvudvudvu ,,,)( (8) (divergence theorem) :

==

v ijijs jiij

v ijijs jiijv jiij

dvudsu

dvudsudvu

,

,,

1

)( (9)

. :

( ) ( ) 01

)(, =++ s ijijiv iijij dsuTdvuB (10)

:

iu

0, =+ ijij B V

0)( = jijiT SjijiT =)( 1

,

Cauchy

(tractions).

-

.

(strain energy density) .

, :

U

7

ijij

U= (1)

U (positive definite) .

. (1)

:

=+ v ijijs iiv ii dvdsuTdvuB )( (2)

UUdvUdvdvUdsuTdvuBvvv ij

ijs iiv ii

)1()1()1()( ====+ (3)

.

V :

=1

)(

s iiv iidsuTdvuBV (4)

:

= 1 )()1( s jjiiv jjii dsuuuTdvu

uuBV

ijj

i

uu =

(Kronecker delta) :

= 1

)()1(

s iiv iidsuTdvuBV

(3) :

0)()1( =+ VU (5)

.

VU + :

8

0)1( = (6)

.

. (6)

Euler-Lagrange

Cauchy.

Shames I.H, Dym C.L. Energy and Finite Element Methods in Structural Mechanics Hemisphere

Pub. Co. 1985.

Courant, R., and Hilbert, D., Methods of Mathematical Physics, vol. 1, Interscience, New York, 1953.

Mikhlin, S. G., Variational Methods in Mathematical Physics, Macmillan, New York, 1964.

Kantorovich, L. V., and Krylov, V. I., Approximate Methods of Higher Analysis, Interscience, New York, 1964.

9

, .

.

.

.

, :

( ) ( ) ( ) ( )2322212 dxdxdxdxdxds ii ++== (1) .

:

ix

i

( )321 ,, = ii xx( )321 ,, xxxii = .

:

jj

iij

j

ii dxx

ddxdx

=

= (2)

:

( ) kmk

i

m

iii dd

xxdxdxds

==2 (3)

. i, m, k=1,2,3.

:

( ) kmk

i

m

iii dxdxxx

ddds

==2* (4)

10

.

:

( ) ( ) jiijj

i

i

i dxdxxx

dsds

= 22* (5)

( ) ( ) jij

i

i

iij dd

xxdsds

= 22* (6)

:

( ) ( ) jiij dxdxdsds = 222* (7) ( ) ( ) jiij dxdxdsds = 222* (8)

= ij

j

i

i

iij xx2

1 (9)

=

j

i

i

iijij

xx21

(10)

Green

Lagrange.

ij

ij Almansi

Euler.

:

iii xu = (11)

11

j

iij

j

i ux=

(12)

ijj

i

j

i

xu

x+

=

(13)

:

+

+=

j

k

i

k

i

j

j

iij x

uxu

xu

xu

21

(14)

+=

j

k

i

k

i

j

j

iij

uuuu21

(15)

11

Rayleigh-Ritz

.

,

:

)3,2,1(, =iui

lmnxxxu

xxxu

xxxu

n

mkkk

m

ljjj

l

iii

>>=

=

=

+=

+=

=

),,(

),,(

),,(

321

13

321

12

321

11

(1)

.

.

:

( r )= ,...,,, 321 (2) rii ,...,2,1, =.

,

.

:

rii

,...,2,10 ==

(3)

.

.

-,

. -

.

13

.

(2).

: q

Rayleigh-Ritz

.

. 8.

:

=

L

dxxqwdx

xwdEI

0

2

2

2

)()(2

(4)

:

( )( ) 0)(

0)0(0====

LwLwww

(5)

:

( ) 4433221 xxxxxw o ++++= (6) :

( ) 2432 432 xxxw ++= (7)

14

:

( ) 00 == ow ( ) 020 2 ==w (8)

:

( ) 44331 xxxxw ++= (9) ( ) 243 126 xxxw += (10)

:

( ) 44331 LLLLw ++= =0 ( ) 0126 243 =+= LLLw (11)

4 3 : 1

313

4 2 LL==

21

32L= (12)

:

+= 43321 12)( xLxLxxw (13)

Maple,

:

> w(x)=a1*L*(x/L-2*(x/L)^3+(x/L)^4); / 3 4 \ | x x | w(x) = a1 L |x/L - 2 ---- + ----| | 3 4 | \ L L /

15

> diff(",x,x); 2 / 2 \ d | x x | --- w(x) = a1 L |-12 ---- + 12 ----| 2 | 3 4 | dx \ L L / > int(0.5*E*II*(a1*L*(-12*x/L^3+12*x^2/L^4))^2-q*a1*L*(x/L-2*x^3/L^3+x^4/L^4),x=0..L); 3 a1 (12. E II a1 - 1. q L ) .2000000000 -------------------------- L > diff(",a1); 3 12. E II a1 - 1. q L a1 E II .2000000000 --------------------- + 2.400000000 ------- L L > simplify("); 3 24. E II a1 - 1. q L .2000000000 --------------------- L > solve(", a1); 3 q L .04166666667 ---- E II () ,

diff , int .

.

.

+= 4

4

3

34

224

)(Lx

Lx

Lx

EIqLxw (13)

16

( )Lx

EIqLxw =+= 42

4

224

)( (14)

.

( ) 554433221 xxxxxxw o +++++= (15) ,

:

20

( ) 5544331 LLLLLw +++= =0 ( ) 020126 35243 =++= LLLLw (16)

Maple

4 5 1 3 : > eq1:=a1*L+a3*L^3+a4*L^4+a5*L^5; 3 4 5 eq1 := a1 L + a3 L + a4 L + a5 L > eq2:=6*a3*L+12*a4*L^2+20*a5*L^3; 2 3 eq2 := 6 a3 L + 12 a4 L + 20 a5 L > solve({eq1, eq2}, {a4,a5}); 2 2 7 a3 L + 10 a1 2 a1 + a3 L {a4 = - 1/4 ---------------, a5 = 3/4 ------------} 3 4 L L :

( )4

231

531

23

4 423

,4

107L

LL

L +=+= (17)

1 : 3

17

>w(x):=a1*x+a3*x^3+(-1/4*(7*a3*L^2+10*a1)/L^3)*x^4+(3/4*(2*a1+a3*L^2)/L^4)*x^5; w(x) := 2 4 2 5 3 (7 a3 L + 10 a1) x (2 a1 + a3 L ) x a1 x + a3 x - 1/4 -------------------- + 3/4 ----------------- 3 4 L L x

:

> diff(",x,x); 2 2 2 3 (7 a3 L + 10 a1) x (2 a1 + a3 L ) x 6 a3 x - 3 -------------------- + 15 ----------------- 3 4 L L

(4) :

> int(0.5*E*II*(6*a3*x-3*(7*a3*L^2+10*a1)/L^3*x^2+15*(2*a1+a3*L^2)/L^4*x^3)^2-q*(a1*x+a3*x^3-1/4*(7*a3*L^2+10*a1)/L^3*x^4+3/4*(2*a1+a3*L^2)/L^4*x^5),x=0..L); 2 2 2 4 .003571428571 (1200. E II a1 + 360. E II a1 a3 L + 48. E II a3 L 3 5 - 70. q L a1 - 7. q L a3)/L

:

2 2 p := .003571428571 (1200. E II a1 + 360. E II a1 a3 L 2 4 3 5 + 48. E II a3 L - 70. q L a1 - 7. q L a3)/L

:

> equa1:=diff(p,a1); 2 3

18

2400. E II a1 + 360. E II a3 L - 70. q L equa1 := .003571428571 ------------------------------------------ L > equa2:=diff(p,a3); 2 4 5 360. E II a1 L + 96. E II a3 L - 7. q L equa2 := .003571428571 ------------------------------------------ L

:

> solve({equa1,equa2},{a1,a3}); 3 q L L q {a3 = -.08333333333 ----, a1 = .04166666667 ----} E II E II :

EIqL

EIqL

242,

24 33

1 == (18)

(17) :

( )0

423

,244

1074

231

531

23

4 =+==+= LL

EIq

LL

(19)

.

+= 4

4

3

34

224

)(Lx

Lx

Lx

EIqLxw (20)

Rayleigh-Ritz .

, ,

,

.

19

Galerkin Galerkin

.

..

- .

,

fLu = (1)

. L

(.. )(44

dxdEIL = ).

, :

=

=n

iiiu

1

~ (2)

.

= fuL~ (3)

.

n n

ii .

:

nidvfuLdvv iv i

,...,2,10)~( === (4) (4) n n

, nii ,...,2,1, =i i . ,

20

.

.

: q

. 8

:

EIq

dxxwd =4

4 )( (5)

:

( ) 554433221~ xxxxxxw o +++++= (6) ,

:

0)0(~0)0(~

==

ww

(7)

: 010 ==

( ) 55443322~ xxxxxw +++= (8) Galerkin

: 5432 ,,, xxxx

21

( ) 2,10)(~0

== idxfxwLL i

012024

012024

0

354

0

254

=

+

=

+

L

L

dxxEIqx

dxxEIqx

(9)

Maple

.

> w(x):=a2*x^2+a3*x^3+a4*x^4+a5*x^5; 2 3 4 5 w(x) := a2 x + a3 x + a4 x + a5 x > diff(",x,x,x,x); 24 a4 + 120 a5 x > int(((24*a4+120*a5*x-q/E/II)*x^2),x=0..L); 3 L (90 a5 L E II + 24 a4 E II - q) 1/3 ---------------------------------- E II > eq1:="=0; 3 L (90 a5 L E II + 24 a4 E II - q) eq1 := 1/3 ---------------------------------- = 0 E II > int(((24*a4+120*a5*x-q/E/II)*x^3),x=0..L); 4 L (96 a5 L E II + 24 a4 E II - q) 1/4 ---------------------------------- E II > eq2:="=0; 4 L (96 a5 L E II + 24 a4 E II - q) eq2 := 1/4 ---------------------------------- = 0 E II > solve({eq1,eq2},{a4,a5}); q {a5 = 0, a4 = 1/24 ----} E II

22

:

( ) 43322 24~ xEIqxxxw ++= (10)

.

4 2 3 q x w(x) := a2 x + a3 x + 1/24 ---- E II > diff(",x,x); 2 q x 2 a2 + 6 a3 x + 1/2 ---- E II > diff(",x); q x 6 a3 + ---- E II > equa1:=2*a2+6*a3*L+1/2*q/E/II*L^2=0; 2 q L equa1 := 2 a2 + 6 a3 L + 1/2 ---- = 0 E II > equa2:=6*a3+q/E/II*L=0; q L equa2 := 6 a3 + ---- = 0 E II > solve({equa1,equa2},{a2,a3}); 2 q L q L {a3 = - 1/6 ----, a2 = 1/4 ----} E II E II

23

:

( )( ) ( )4324

4322

4624

~

2464~

+=

+=

EIqLxw

xEI

qxEI

qLxEI

qLxw (11)

Lx=

.

24

Rayleigh-Ritz

1:

Rayleigh-Ritz

, .. L

0=x Lx = .

. ,

.

.

:

Lxxaxaxaxu ++= 0)( 33221 (1)

0)0( =u 0=x .

:

Lxxxaaxuxxxxaaxu

xxxaaxu

+=+=

+=

354

3232

210

)()(

0)( (2)

.

. :

0)0( =u0C

3543323

2322102

0

)()(

0)0(

xaaxaaxuxaaxaaxu

au

+=+=+=+=

== (3)

25

353231423120 )()(,)(,0 xaaxaaaxaaaa +=== :

Lxxxxaxxaxaxuxxxxxaxaxuxxxaxu

++=+==

33523321

322321

21

)()()()()(

0)( (4)

: cxq =

= L Lx dxxqudxEA0 02 )(21 (5) Rayleigh-Ritz

Maple :

(1) > u:=a1*x+a2*x^2+a3*x^3; 2 3 u := a1 x + a2 x + a3 x

> ex:=diff(u,x); 2 ex := a1 + 2 a2 x + 3 a3 x (5) :

> p:=int((0.5*E*A*ex^2-c*x*u),x=0..L); 2 5 4 3 p := .9000000000 E A a3 L + 1.500000000 E A a2 a3 L + E A L a1 a3 3 2 2 2 + .6666666667 E A L a2 + E A a1 a2 L + .5000000000 E A a1 L 5 4 - .2000000000 c a3 L - .2500000000 c a2 L 3 - .3333333333 c a1 L

:

26

> equa1:=diff(p,a1); equa1 := 3 2 3 E A L a3 + E A a2 L + 1.000000000 E A a1 L - .3333333333 c L > > equa2:=diff(p,a2); 4 3 2 equa2 := 1.500000000 E A a3 L + 1.333333333 E A L a2 + E A a1 L 4 - .2500000000 c L > equa3:=diff(p,a3); 5 4 3 equa3 := 1.800000000 E A a3 L + 1.500000000 E A a2 L + E A L a1 5 - .2000000000 c L

:

> solve({equa1,equa2,equa3},{a1,a2,a3}); c -9 L c {a3 = -.1666666670 ---, a2 = .6000000096 10 ---, E A E A 2 L c a1 = .4999999997 ----} E A .

:

02 =a

( 3236

)( xxLEAcxu = ) (3)

:

==

2222)(

222

2 xLEAcx

EAc

EAcLxx (4)

27

-

(2) > u1:=a1*x; u1 := a1 x > u2:=a1*L/3+a3*(x-L/3); u2 := 1/3 a1 L + a3 (x - 1/3 L) > u3:=a1*L/3+a3*L/3+a5*(x-2*L/3); u3 := 1/3 a1 L + 1/3 a3 L + a5 (x - 2/3 L)

:

> e1:=diff(u1,x); e1 := a1 > e2:=diff(u2,x); e2 := a3 > e3:=diff(u3,x); e3 := a5 :

> p1:=int(0.5*E*A*e1^2-c*x*u1,x=0..L/3); 2 3 p1 := .1666666667 E A a1 L - .01234567901 c L a1

28

> p2:=int(0.5*E*A*e2^2-c*x*u2,x=L/3..2*L/3); p2 := 2 3 .1666666667 E A a3 L - .03086419753 c a3 L 3 - .05555555556 c L a1 > p3:=int(0.5*E*A*e3^2-c*x*u3,x=2*L/3..L); 2 3 p3 := .1666666667 E A a5 L - .09259259259 c L a1 3 3 - .09259259259 c a3 L - .04938271605 c a5 L

:

> eq1:=diff(p1+p2+p3,a1); 3 eq1 := .3333333334 E A a1 L - .1604938272 c L > eq2:=diff(p1+p2+p3,a3); 3 eq2 := .3333333334 E A a3 L - .1234567901 c L > eq3:=diff(p1+p2+p3,a5); 3 eq3 := .3333333334 E A a5 L - .04938271605 c L

:

> solve({eq1,eq2,eq3},{a1,a3,a5}); 2 2 2 c L c L c L {a1 = .4814814815 ----, a5 = .1481481481 ----, a3 = .3703703702 ----} E A E A E A . :

29

LxLLxEAcL

EAcLxu

LxLLxEAcL

EAcLxu

LxxEAcLxu

+=

+=

=

3/2)3

2(1481481481.03

8518518517.0)(

3/23/)3

(3703703702.03

4814814815.0)(

3/04814814815.0)(

23

23

2

(2)

6=L

EAc

.

30

,

. ,

, ,

.

2C0C

3x4-3=9 .

.

.

s, .

:

3/03/)()(3/03/)(

3/0)(

5313

312

11

LssaLaasuLssaLasu

Lssasu

++=+==

(2)

:

3/03/2)(3/03/)(3/0)(

3

2

1

LscscLsqLscscLsqLscssq

+=+==

(2)

Maple

:

> u1:=a1*s; u1 := a1 s > u2:=a1*L/3+a3*s; > u2 := 1/3 a1 L + a3 s > u3:=(a1+a3)*L/3+a5*s; u3 := 1/3 (a1 + a3) L + a5 s

31

:

> q1:=c*s; q1 := c s > q2:=c*L/3+c*s; q2 := 1/3 c L + c s > q3:=2*c*L/3+c*s; q3 := 2/3 c L + c s :

> e1:=diff(u1,s); e1 := a1 > e2:=diff(u2,s); e2 := a3 > e3:=diff(u3,s); e3 := a5 :

> p1:=int(0.5*E*A*e1^2-q1*u1,s=0..L/3); 2 3 p1 := .1666666667 E A a1 L - .01234567901 c L a1 > p2:=int(0.5*E*A*e2^2-q2*u2,s=0..L/3); p2 := 2 3 .1666666667 E A a3 L - .03086419753 c a3 L 3 - .05555555556 c L a1 > p3:=int(0.5*E*A*e3^2-q3*u3,s=0..L/3); 2 3 p3 := .1666666667 E A a5 L - .04938271605 c a5 L 3 3 - .09259259259 c L a1 - .09259259259 c a3 L

32

:

> eq1:=diff(p1+p2+p3,a1); 3 eq1 := .3333333334 E A a1 L - .1604938272 c L > eq2:=diff(p1+p2+p3,a3); 3 eq2 := .3333333334 E A a3 L - .1234567901 c L > eq3:=diff(p1+p2+p3,a5); 3 eq3 := .3333333334 E A a5 L - .04938271605 c L

.

> solve({eq1,eq2,eq3},{a1,a3,a5}); 2 2 2 c L c L c L {a5 = .1481481481 ----, a1 = .4814814815 ----, a3 = .3703703702 ----} E A E A E A

x .

.

,

.

.

.

.

.

33

2 -

.

.

.

.

:

ji uu ,

saasu 21)( += (11) [ ]

=

2

11)(aa

ssu

0=s Ls = : ji uu ,

Laauaau

j

i

+=+=

21

21 0 { } (11)

=

2

1

101

aa

Luu

j

i [ ]{ }aAd = :

{ } [ ] { }dAa 1= (12)

=

=

j

i

j

i

uu

LLuu

Laa

/1/101

101

2

1

(11)

:

[ ]{ } [ ][ ] { } [ ]

=

===

j

i

j

i

uu

Ls

LsL

uu

LLsdAsassu

/1/101

111)( 1 (13)

[ ]

=

j

i

uu

NNsu 1211)( (14) [ ]

=

j

i

uu

Nsu )(

34

.

:

> S:=matrix(1,2,[1,s]); S := [1 s] > a:=matrix(2,1,[a1,a2]); [a1] a := [ ] [a2] > u(s):=multiply(S,a); u(s) := [a1 + s a2] > A:=matrix(2,2,[1,0,1,L]); [1 0] A := [ ] [1 L] > A1:=inverse(A); [ 1 0 ] A1 := [ ] [- 1/L 1/L] > N:=multiply(S,A1); N := [1 - s/L s/L]

:

34

2321)( sasasaasu +++=

(15) [ ]

=4

3

2

1

321)(

aaaa

ssssu

, .

35

0=s , , 3/Ls = 3/2Ls = Ls = :

ki uu ,

jl uu ,

34

2321

34

2321

34

2321

4321

)()(

)3/2()3/2(3/2

)3/()3/(3/

000

LaLaLaauLaLaLaau

LaLaLaauaaaau

j

l

k

i

+++=+++=

+++=+++=

(16)

:

> with(linalg); > S:=matrix(1,4,[1,s,s^2,s^3]); S := [ 2 3] [1 s s s ] > a:=matrix(4,1,[a1,a2,a3,a4]); [a1] [ ] [a2] a := [ ] [a3] [ ] [a4] > u(s):=multiply(S,a); u(s) := [ 2 3 ] [a1 + s a2 + s a3 + s a4] > A:=matrix(4,4,[1,0,0,0,1,L/3,L^2/9,L^3/27,1,2*L/3,4*L^2/9,8*L^3/27,1,L,L^2,L^3]); [1 0 0 0 ] [ ] [ 2 3] [1 1/3 L 1/9 L 1/27 L ] A := [ ] [ 2 3] [1 2/3 L 4/9 L 8/27 L ] [ ] [ 2 3 ] [1 L L L ] > A1:=inverse(A);

36

[ 1 0 0 0 ] [ ] [- 11/2 1/L 9 1/L - 9/2 1/L 1/L ] [ ] [ 1 1 1 1 ] [ 9 ---- - 45/2 ---- 18 ---- - 9/2 ----] A1 := [ 2 2 2 2 ] [ L L L L ] [ ] [ 1 1 1 1 ] [- 9/2 ---- 27/2 ---- - 27/2 ---- 9/2 ---- ] [ 3 3 3 3 ] [ L L L L ] > N:=multiply(S,A1); N := [ 2 3 [ s s [1 - 11/2 s/L + 9 ---- - 9/2 ---- , [ 2 3 [ L L 2 3 s s 9 s/L - 45/2 ---- + 27/2 ---- , 2 3 L L 2 3 2 3 ] s s s s ] - 9/2 s/L + 18 ---- - 27/2 ---- , s/L - 9/2 ---- + 9/2 ----] 2 3 2 3 ] L L L L ]

. 5

.

> L=5:evalm(N); [ 2 3 [ s s [1 - 11/2 s/L + 9 ---- - 9/2 ---- , [ 2 3 [ L L 2 3 s s 9 s/L - 45/2 ---- + 27/2 ---- , 2 3 L L 2 3 2 3 ] s s s s ] - 9/2 s/L + 18 ---- - 27/2 ---- , s/L - 9/2 ---- + 9/2 ----] 2 3 2 3 ] L L L L ]

37

> N;plot([1-11/2*s/5+9*s^2/5^2-9/2*s^3/5^3, 9*s/5-45/2*s^2/5^2+27/2*s^3/5^3, -9/2*s/5+18*s^2/5^2-27/2*s^3/5^3, s/5-9/2*s^2/5^2+9/2*s^3/5^3], s=0..5);

L/3 2L/3.

. ,

,

.

38

.

.

.

:

{ } [ ]{ } { } [ ]{ } { } { }{ } { } { } { } { } { }

+=

V S

TTi

T

V

TTTp

PDdsTudvBu

dvEE 0021

(1)

{ } [ ]u u v w T= , { } [ = x y z xy yz zx ]

]]

,

[ ]E =

{ } { } 0 0, = { } [ Tzyxi BBBB = , { } [ Tzyxi TTTT = , { }D = { }P =

39

, { }d{ } [ ]{ }dNu = (2)

. [ ]N

.

{ } [ ]{ }u= , { } [ ]{ }dB= , [ ] [ ][ ]NB = . (1) :

{ } [ ] { } { } { } { } { } = =

=N

n

TN

nne

Tnnn

Tnp PDrddkd

1 121

(3)

[ : ]k[ ] [ ] [ ][ ]= Ve T dVBEBk (4) { } [ ] [ ]{ } [ ] { }

[ ] { } [ ] { }

++=

Se iT

Ve iT

Ve Ve

TTe

dSTNdVBN

dVBdVEBr 00 (5)

{ }D .

:

{ } [ ]{ } { } { }RDDKD TTp = 21

(6)

[ ] [ ]=

=N

nnkK

1 { } (7) { } { }

=+=

N

nnerPR

1

.

40

, .

{ } { }0=

D

p (8)

:

[ ]{ } { }RDK = (9)

.

41

:

> X:=matrix(1,4,[1,x,x^2,x^3]); X := [ 2 3] [1 x x x ] > a:=matrix(4,1,[a1,a2,a3,a4]); [a1] [ ] [a2] a := [ ] [a3] [ ] [a4] > w(x):=multiply(X,a); w(x) := [ 2 3 ] [a1 + x a2 + x a3 + x a4] :

> A:=matrix(4,4,[1,0,0,0,0,1,0,0,1,L,L^2,L^3,0,1,2*L,3*L^2]); [1 0 0 0 ] [ ] [0 1 0 0 ] [ ] A := [ 2 3 ] [1 L L L ] [ ] [ 2] [0 1 2 L 3 L ]

-1> A :=inverse(A); [ 1 0 0 0 ] [ ] [ 0 1 0 0 ] [ ] [ 3 3 ] [- ---- - 2/L ---- - 1/L] A1 := [ 2 2 ] [ L L ] [ ] [ 2 1 2 1 ] [ ---- ---- - ---- ---- ] [ 3 2 3 2 ] [ L L L L ]

42

:

> N:=multiply(X,A1); N := [ 2 3 2 3 2 3 [ x x x x x x [1 - 3 ---- + 2 ---- , x - 2 ---- + ---- , 3 ---- - 2 ---- , [ 2 3 L 2 2 3 [ L L L L L 2 3 ] x x ] - ---- + ----] L 2 ] L ]

x :

> B:=map(diff,N,x,x); B := [ 6 x x 6 x [- ---- + 12 ---- , - 4/L + 6 ---- , ---- - 12 ---- , [ 2 3 2 2 3 [ L L L L L x ] - 2/L + 6 ----] 2 ] L ]

L :

43

> K:=EI*map(int,multiply(transpose(B),B),x=0..L); [ 12 6 12 6 ] [ ---- ---- - ---- ---- ] [ 3 2 3 2 ] [ L L L L ] [ ] [ 6 6 ] [ ---- 4/L - ---- 2/L ] [ 2 2 ] [ L L ] K := EI [ ] [ 12 6 12 6 ] [- ---- - ---- ---- - ----] [ 3 2 3 2 ] [ L L L L ] [ ] [ 6 6 ] [ ---- 2/L - ---- 4/L ] [ 2 2 ] [ L L ]

.

u

v . :

yaxaayxvyaxaayxu

654

321

),(),(

++=++=

(1)

> u:=a1+a2*x+a3*y; u := a1 + a2 x + a3 y > v:=a3+a4*x+a5*y; v := a3 + a4 x + a5 y > a:=matrix(6,1,[a1,a2,a3,a4,a5,a6]); [a1] [ ] [a2] [ ] [a3] a := [ ] [a4] [ ] [a5] [ ] [a6]

44

x y

:

> XY:=matrix(2,6,[1,x,y,0,0,0,0,0,0,1,x,y]); [1 x y 0 0 0] XY := [ ] [0 0 0 1 x y] :

>A:=matrix(6,6,[1,x1,y1,0,0,0,0,0,0,1,x1,y1,1,x2,y2,0,0,0,0,0,0,1,x2,y2,1,x3,y3,0,0,0,0,0,0,1,x3,y3]); [1 x1 y1 0 0 0 ] [ ] [0 0 0 1 x1 y1] [ ] [1 x2 y2 0 0 0 ] A := [ ] [0 0 0 1 x2 y2] [ ] [1 x3 y3 0 0 0 ] [ ] [0 0 0 1 x3 y3] :

-1> A :=inverse(A);

A-1 := [ x2 y3 - x3 y2 x1 y3 - x3 y1 x1 y2 - y1 x2 ] [- ------------- , 0 , ------------- , 0 , - ------------- , 0] [ %1 %1 %1 ] [y3 - y2 y3 - y1 -y1 + y2 ] [------- , 0 , - ------- , 0 , -------- , 0] [ %1 %1 %1 ] [ x3 - x2 x1 - x3 x1 - x2 ] [- ------- , 0 , - ------- , 0 , ------- , 0] [ %1 %1 %1 ] [ x2 y3 - x3 y2 x1 y3 - x3 y1 x1 y2 - y1 x2] [0 , - ------------- , 0 , ------------- , 0 , - -------------] [ %1 %1 %1 ] [ y3 - y2 y3 - y1 -y1 + y2] [0 , ------- , 0 , - ------- , 0 , --------] [ %1 %1 %1 ] [ x3 - x2 x1 - x3 x1 - x2]

45

[0 , - ------- , 0 , - ------- , 0 , -------] [ %1 %1 %1 ] %1 := -x2 y3 + y1 x2 + x1 y3 + x3 y2 - x3 y1 - x1 y2

:

> N:=multiply(XY,A1); N := [ x2 y3 - x3 y2 x (y3 - y2) y (x3 - x2) [- ------------- + ----------- - ----------- , 0 , [ %1 %1 %1 x1 y3 - x3 y1 x (y3 - y1) y (x1 - x3) ------------- - ----------- - ----------- , 0 , %1 %1 %1 x1 y2 - y1 x2 x (-y1 + y2) y (x1 - x2) ] - ------------- + ------------ + ----------- , 0] %1 %1 %1 ] [ x2 y3 - x3 y2 x (y3 - y2) y (x3 - x2) [0 , - ------------- + ----------- - ----------- , 0 , [ %1 %1 %1 x1 y3 - x3 y1 x (y3 - y1) y (x1 - x3) ------------- - ----------- - ----------- , 0 , %1 %1 %1 x1 y2 - y1 x2 x (-y1 + y2) y (x1 - x2)] - ------------- + ------------ + -----------] %1 %1 %1 ] %1 := -x2 y3 + y1 x2 + x1 y3 + x3 y2 - x3 y1 - x1 y2

,

:

> dN:=matrix(3,6,[0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,1,0]); [0 1 0 0 0 0] [ ] dN := [0 0 0 0 0 1] [ ] [0 0 1 0 1 0] > B:=multiply(dN,A1);

46

B := [y3 - y2 y3 - y1 -y1 + y2 ] [------- , 0 , - ------- , 0 , -------- , 0] [ %1 %1 %1 ] [ x3 - x2 x1 - x3 x1 - x2] [0 , - ------- , 0 , - ------- , 0 , -------] [ %1 %1 %1 ] [ x3 - x2 y3 - y2 x1 - x3 y3 - y1 x1 - x2 -y1 + y2 [- ------- , ------- , - ------- , - ------- , ------- , -------- [ %1 %1 %1 %1 %1 %1 ] ] ]

%1 := -x2 y3 + y1 x2 + x1 y3 + x3 y2 - x3 y1 - x1 y2

:

> EL:(E/(1-p^2))*matrix(3,3,[1,p,0,p,1,0,0,0,0.5-p]); [1 p 0 ] [ ] E [p 1 0 ] [ ] [0 0 .5 - p] -------------------- 2 1 - p Young p Poisson.

x y. EBB T

. EBB T :

47

> k:=multiply(transpose(B),EL,B); k := [ 2 2 [(y2 - y3) E (x2 - x3) E (.5 - p) [------------ + --------------------- , [ 2 2 2 2 [%1 (1 - p ) %1 (1 - p ) (y2 - y3) E p (x2 - x3) (x2 - x3) E (.5 - p) (y2 - y3) - ----------------------- - ------------------------------ , 2 2 2 2 %1 (1 - p ) %1 (1 - p ) (y2 - y3) E (y1 - y3) (x2 - x3) E (.5 - p) (x1 - x3) - --------------------- - ------------------------------ , 2 2 2 2 %1 (1 - p ) %1 (1 - p ) (y2 - y3) E p (x1 - x3) (x2 - x3) E (.5 - p) (y1 - y3) ----------------------- + ------------------------------ , 2 2 2 2 %1 (1 - p ) %1 (1 - p ) (y2 - y3) E (y1 - y2) (x2 - x3) E (.5 - p) (x2 - x1) --------------------- - ------------------------------ , 2 2 2 2 %1 (1 - p ) %1 (1 - p ) ] (y2 - y3) E p (x2 - x1) (x2 - x3) E (.5 - p) (y1 - y2)] ----------------------- - ------------------------------] 2 2 2 2 ] %1 (1 - p ) %1 (1 - p ) ] [ [ (y2 - y3) E p (x2 - x3) (x2 - x3) E (.5 - p) (y2 - y3) [- ----------------------- - ------------------------------ , [ 2 2 2 2 [ %1 (1 - p ) %1 (1 - p ) 2 2 (x2 - x3) E (y2 - y3) E (.5 - p) ------------ + --------------------- , 2 2 2 2 %1 (1 - p ) %1 (1 - p ) (x2 - x3) E p (y1 - y3) (y2 - y3) E (.5 - p) (x1 - x3) ----------------------- + ------------------------------ , 2 2 2 2 %1 (1 - p ) %1 (1 - p ) (x2 - x3) E (x1 - x3) (y2 - y3) E (.5 - p) (y1 - y3) - --------------------- - ------------------------------ , 2 2 2 2 %1 (1 - p ) %1 (1 - p ) (x2 - x3) E p (y1 - y2) (y2 - y3) E (.5 - p) (x2 - x1) - ----------------------- + ------------------------------ , 2 2 2 2

48

%1 (1 - p ) %1 (1 - p ) ] (x2 - x3) E (x2 - x1) (y2 - y3) E (.5 - p) (y1 - y2)] - --------------------- + ------------------------------] 2 2 2 2 ] %1 (1 - p ) %1 (1 - p ) ] [ [ (y2 - y3) E (y1 - y3) (x2 - x3) E (.5 - p) (x1 - x3) [- --------------------- - ------------------------------ , [ 2 2 2 2 [ %1 (1 - p ) %1 (1 - p ) (x2 - x3) E p (y1 - y3) (y2 - y3) E (.5 - p) (x1 - x3) ----------------------- + ------------------------------ , 2 2 2 2 %1 (1 - p ) %1 (1 - p ) 2 2 (y1 - y3) E (x1 - x3) E (.5 - p) ------------ + --------------------- , 2 2 2 2 %1 (1 - p ) %1 (1 - p ) (y1 - y3) E p (x1 - x3) (x1 - x3) E (.5 - p) (y1 - y3) - ----------------------- - ------------------------------ , 2 2 2 2 %1 (1 - p ) %1 (1 - p ) (y1 - y3) E (y1 - y2) (x1 - x3) E (.5 - p) (x2 - x1) - --------------------- + ------------------------------ , 2 2 2 2 %1 (1 - p ) %1 (1 - p ) ] (y1 - y3) E p (x2 - x1) (x1 - x3) E (.5 - p) (y1 - y2)] - ----------------------- + ------------------------------] 2 2 2 2 ] %1 (1 - p ) %1 (1 - p ) ] [ [(y2 - y3) E p (x1 - x3) (x2 - x3) E (.5 - p) (y1 - y3) [----------------------- + ------------------------------ , [ 2 2 2 2 [ %1 (1 - p ) %1 (1 - p ) (x2 - x3) E (x1 - x3) (y2 - y3) E (.5 - p) (y1 - y3) - --------------------- - ------------------------------ , 2 2 2 2 %1 (1 - p ) %1 (1 - p ) (y1 - y3) E p (x1 - x3) (x1 - x3) E (.5 - p) (y1 - y3) - ----------------------- - ------------------------------ , 2 2 2 2 %1 (1 - p ) %1 (1 - p ) 2 2 (x1 - x3) E (y1 - y3) E (.5 - p) ------------ + --------------------- , 2 2 2 2

49

%1 (1 - p ) %1 (1 - p ) (x1 - x3) E p (y1 - y2) (y1 - y3) E (.5 - p) (x2 - x1) ----------------------- - ------------------------------ , 2 2 2 2 %1 (1 - p ) %1 (1 - p ) ] (x1 - x3) E (x2 - x1) (y1 - y3) E (.5 - p) (y1 - y2)] --------------------- - ------------------------------] 2 2 2 2 ] %1 (1 - p ) %1 (1 - p ) ] [ [(y2 - y3) E (y1 - y2) (x2 - x3) E (.5 - p) (x2 - x1) [--------------------- - ------------------------------ , [ 2 2 2 2 [ %1 (1 - p ) %1 (1 - p ) (x2 - x3) E p (y1 - y2) (y2 - y3) E (.5 - p) (x2 - x1) - ----------------------- + ------------------------------ , 2 2 2 2 %1 (1 - p ) %1 (1 - p ) (y1 - y3) E (y1 - y2) (x1 - x3) E (.5 - p) (x2 - x1) - --------------------- + ------------------------------ , 2 2 2 2 %1 (1 - p ) %1 (1 - p ) (x1 - x3) E p (y1 - y2) (y1 - y3) E (.5 - p) (x2 - x1) ----------------------- - ------------------------------ , 2 2 2 2 %1 (1 - p ) %1 (1 - p ) 2 2 (y1 - y2) E (x2 - x1) E (.5 - p) ------------ + --------------------- , 2 2 2 2 %1 (1 - p ) %1 (1 - p ) ] (y1 - y2) E p (x2 - x1) (x2 - x1) E (.5 - p) (y1 - y2)] ----------------------- + ------------------------------] 2 2 2 2 ] %1 (1 - p ) %1 (1 - p ) ] [ [(y2 - y3) E p (x2 - x1) (x2 - x3) E (.5 - p) (y1 - y2) [----------------------- - ------------------------------ , [ 2 2 2 2 [ %1 (1 - p ) %1 (1 - p ) (x2 - x3) E (x2 - x1) (y2 - y3) E (.5 - p) (y1 - y2) - --------------------- + ------------------------------ , 2 2 2 2 %1 (1 - p ) %1 (1 - p ) (y1 - y3) E p (x2 - x1) (x1 - x3) E (.5 - p) (y1 - y2) - ----------------------- + ------------------------------ , 2 2 2 2 %1 (1 - p ) %1 (1 - p )

50

(x1 - x3) E (x2 - x1) (y1 - y3) E (.5 - p) (y1 - y2) --------------------- - ------------------------------ , 2 2 2 2 %1 (1 - p ) %1 (1 - p ) (y1 - y2) E p (x2 - x1) (x2 - x1) E (.5 - p) (y1 - y2) ----------------------- + ------------------------------ , 2 2 2 2 %1 (1 - p ) %1 (1 - p ) 2 2 ] (x2 - x1) E (y1 - y2) E (.5 - p)] ------------ + ---------------------] 2 2 2 2 ] %1 (1 - p ) %1 (1 - p ) ]

%1 := y1 x2 - y2 x1 - x2 y3 + x1 y3 - y1 x3 + x3 y2 t

,

(6x6).

.. yij=yi-yj,

FORTRAN :

> fortran(k); k(1,1) = (y2-y3)**2/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1- #p**2)+(x2-x3)**2/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p** #2)*(0.5E0-p) k(1,2) = -(y2-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p* #*2)*p*(x2-x3)-(x2-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/( #1-p**2)*(0.5E0-p)*(y2-y3) k(1,3) = -(y2-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p* #*2)*(y1-y3)-(x2-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1- #p**2)*(0.5E0-p)*(x1-x3) k(1,4) = (y2-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p** #2)*p*(x1-x3)+(x2-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1 #-p**2)*(0.5E0-p)*(y1-y3) k(1,5) = (y2-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p** #2)*(y1-y2)-(x2-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p #**2)*(0.5E0-p)*(x2-x1) k(1,6) = (y2-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p** #2)*p*(x2-x1)-(x2-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1 #-p**2)*(0.5E0-p)*(y1-y2) k(2,1) = -(y2-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p* #*2)*p*(x2-x3)-(x2-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/( #1-p**2)*(0.5E0-p)*(y2-y3) k(2,2) = (x2-x3)**2/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1- #p**2)+(y2-y3)**2/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p** #2)*(0.5E0-p)

51

k(2,3) = (x2-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p** #2)*p*(y1-y3)+(y2-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1 #-p**2)*(0.5E0-p)*(x1-x3) k(2,4) = -(x2-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p* #*2)*(x1-x3)-(y2-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1- #p**2)*(0.5E0-p)*(y1-y3) k(2,5) = -(x2-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p* #*2)*p*(y1-y2)+(y2-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/( #1-p**2)*(0.5E0-p)*(x2-x1) k(2,6) = -(x2-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p* #*2)*(x2-x1)+(y2-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1- #p**2)*(0.5E0-p)*(y1-y2) k(3,1) = -(y2-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p* #*2)*(y1-y3)-(x2-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1- #p**2)*(0.5E0-p)*(x1-x3) k(3,2) = (x2-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p** #2)*p*(y1-y3)+(y2-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1 #-p**2)*(0.5E0-p)*(x1-x3) k(3,3) = (y1-y3)**2/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1- #p**2)+(x1-x3)**2/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p** #2)*(0.5E0-p) k(3,4) = -(y1-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p* #*2)*p*(x1-x3)-(x1-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/( #1-p**2)*(0.5E0-p)*(y1-y3) k(3,5) = -(y1-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p* #*2)*(y1-y2)+(x1-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1- #p**2)*(0.5E0-p)*(x2-x1) k(3,6) = -(y1-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p* #*2)*p*(x2-x1)+(x1-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/( #1-p**2)*(0.5E0-p)*(y1-y2) k(4,1) = (y2-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p** #2)*p*(x1-x3)+(x2-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1 #-p**2)*(0.5E0-p)*(y1-y3) k(4,2) = -(x2-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p* #*2)*(x1-x3)-(y2-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1- #p**2)*(0.5E0-p)*(y1-y3) k(4,3) = -(y1-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p* #*2)*p*(x1-x3)-(x1-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/( #1-p**2)*(0.5E0-p)*(y1-y3) k(4,4) = (x1-x3)**2/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1- #p**2)+(y1-y3)**2/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p** #2)*(0.5E0-p) k(4,5) = (x1-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p** #2)*p*(y1-y2)-(y1-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1 #-p**2)*(0.5E0-p)*(x2-x1) k(4,6) = (x1-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p** #2)*(x2-x1)-(y1-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p #**2)*(0.5E0-p)*(y1-y2) k(5,1) = (y2-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p** #2)*(y1-y2)-(x2-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p #**2)*(0.5E0-p)*(x2-x1) k(5,2) = -(x2-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p* #*2)*p*(y1-y2)+(y2-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/( #1-p**2)*(0.5E0-p)*(x2-x1) k(5,3) = -(y1-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p* #*2)*(y1-y2)+(x1-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1- #p**2)*(0.5E0-p)*(x2-x1) k(5,4) = (x1-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p** #2)*p*(y1-y2)-(y1-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1 #-p**2)*(0.5E0-p)*(x2-x1) k(5,5) = (y1-y2)**2/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-

52

#p**2)+(x2-x1)**2/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p** #2)*(0.5E0-p) k(5,6) = (y1-y2)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p** #2)*p*(x2-x1)+(x2-x1)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1 #-p**2)*(0.5E0-p)*(y1-y2) k(6,1) = (y2-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p** #2)*p*(x2-x1)-(x2-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1 #-p**2)*(0.5E0-p)*(y1-y2) k(6,2) = -(x2-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p* #*2)*(x2-x1)+(y2-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1- #p**2)*(0.5E0-p)*(y1-y2) k(6,3) = -(y1-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p* #*2)*p*(x2-x1)+(x1-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/( #1-p**2)*(0.5E0-p)*(y1-y2) k(6,4) = (x1-x3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p** #2)*(x2-x1)-(y1-y3)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p #**2)*(0.5E0-p)*(y1-y2) k(6,5) = (y1-y2)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p** #2)*p*(x2-x1)+(x2-x1)/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1 #-p**2)*(0.5E0-p)*(y1-y2) k(6,6) = (x2-x1)**2/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1- #p**2)+(y1-y2)**2/(y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2*E/(1-p** #2)*(0.5E0-p)

:

> fortran(k,optimized); t1 = y2-y3 t2 = t1**2 t10 = (y1*x2-y2*x1-x2*y3+x1*y3-y1*x3+x3*y2)**2 t11 = 1/t10 t12 = t2*t11 t13 = p**2 t15 = 1/(1-t13) t16 = E*t15 t18 = x2-x3 t19 = t18**2 t20 = t19*t11 t21 = 0.5E0-p t22 = t16*t21 t25 = t1*t11 t26 = t25*E t27 = t15*p t30 = t18*t11 t31 = t30*E t32 = t15*t21 t35 = -t26*t27*t18-t31*t32*t1 t36 = y1-y3 t39 = x1-x3 t40 = t32*t39 t42 = -t25*t16*t36-t31*t40 t43 = t27*t39 t45 = t32*t36 t47 = t26*t43+t31*t45 t48 = y1-y2 t49 = t16*t48 t51 = x2-x1 t52 = t32*t51 t54 = t25*t49-t31*t52

53

t55 = t27*t51 t57 = t32*t48 t59 = t26*t55-t31*t57 t66 = t31*t27*t36+t26*t40 t70 = -t30*t16*t39-t26*t45 t71 = t27*t48 t74 = -t31*t71+t26*t52 t75 = t16*t51 t78 = -t30*t75+t26*t57 t79 = t36**2 t80 = t79*t11 t82 = t39**2 t83 = t82*t11 t86 = t36*t11 t87 = t86*E t89 = t39*t11 t90 = t89*E t92 = -t87*t43-t90*t45 t95 = -t86*t49+t90*t52 t98 = -t87*t55+t90*t57 t104 = t90*t71-t87*t52 t107 = t89*t75-t87*t57 t108 = t48**2 t109 = t108*t11 t111 = t51**2 t112 = t111*t11 t121 = t48*t11*E*t55+t51*t11*E*t57 k(1,1) = t12*t16+t20*t22 k(1,2) = t35 k(1,3) = t42 k(1,4) = t47 k(1,5) = t54 k(1,6) = t59 k(2,1) = t35 k(2,2) = t20*t16+t12*t22 k(2,3) = t66 k(2,4) = t70 k(2,5) = t74 k(2,6) = t78 k(3,1) = t42 k(3,2) = t66 k(3,3) = t80*t16+t83*t22 k(3,4) = t92 k(3,5) = t95 k(3,6) = t98 k(4,1) = t47 k(4,2) = t70 k(4,3) = t92 k(4,4) = t83*t16+t80*t22 k(4,5) = t104 k(4,6) = t107 k(5,1) = t54 k(5,2) = t74 k(5,3) = t95 k(5,4) = t104 k(5,5) = t109*t16+t112*t22 k(5,6) = t121 k(6,1) = t59 k(6,2) = t78 k(6,3) = t98

54

k(6,4) = t107 k(6,5) = t121 k(6,6) = t112*t16+t109*t22 ,

,

.

=2.18 kN/m2 Poisson p=0.3

:

# x yi i1 .5 .5 2 .7 .9 3 .6 1.1

k :=t*A/10* [ 7.211, 1.717, -20.260, -1.030, 13.049, -.686] [ 1.717, 3.090, -2.403, -2.403, .686, -.686] [ -20.26,-2.403, 62.156, -5.151,-41.895, 7.554] [ -1.030,-2.403, -5.151, 14.079, 6.181,-11.675] [ 13.049, .686, -41.895, 6.181, 28.846, -6.868] [ -.686, -.686, 7.554,-11.675, -6.868, 12.362]

. .

55

.

.

.

:

:

LL

LL 2

21

1 , == (1)

1 2 :

121 =+ (2)

x :

[ ] [ ]

=

=+=

2

1

2

1212211 x

xN

xx

xxx (3)

56

1 2 .

:

(4)

=

2

1

21

111xxx

(4) :

=

xxx

L1

111

1

2

2

1

(5)

x .

.

.

213222

211 ++= aa (6)

0,1 21 == 1,0 21 == 212211 , == , .

:

41

41

41

3213

22

11

aaa

aa

++===

(7)

2133 4 =a (8)

(6) :

( 21213222211 4 ++= ) (9)

( ) ( ) 32121221211 4 ++= (10) (2) :

57

( ) ( ) 321222111 41212 ++= (11) :

( ) ( )[

=3

2

1

212211 41212 ] (12)

21, 3 .

:

:

221121 ),( uuu += (13)

:

xu

xu

dxdu

x

+

== 2

2

1

1

(14)

(11) (5) :

+

=L

uL

ux11

21 (15)

[ ]{ }dBuu

LLx=

=2

111 (16)

:

> B:=matrix(1,2,[-1/L,1/L]); B := [- 1/L 1/L] > k:=AE*map(int,multiply(transpose(B),B),x=0..L); [ 1/L - 1/L] k := AE [ ] [- 1/L 1/L ]

58

:

( )!1!!

0 21 lklkLdL

L lk

++= (17)

,

.

59

, .

:

AA

AA

AA 3

32

21

1 ,, === (1)

:

1321 =++ (2)

x : y

332211 xxxx ++=

332211 yyyy ++= (3)

:

=

3

2

1

321

321

1111

yyyxxx

yx (4)

:

> A:=matrix(3,3,[1,1,1,x1,x2,x3,y1,y2,y3]); [1 1 1 ] [ ] A := [x1 x2 x3] [ ] [y1 y2 y3]

-1> A :=inverse(A);

60

[ x2 y3 - x3 y2 -y3 + y2 -x3 + x2 ] [- ------------- - -------- -------- ] [ %1 %1 %1 ] [ ] [ x1 y3 - y1 x3 -y3 + y1 -x3 + x1]

-1 A := [ ------------- -------- - --------] [ %1 %1 %1 ] [ ] [-x1 y2 + y1 x2 -y2 + y1 x2 - x1 ] [-------------- - -------- - ------- ] [ %1 %1 %1 ] %1 := -x2 y3 + x3 y2 + x1 y3 - x1 y2 - y1 x3 + y1 x2 %1 -

1,2,3 .

:

3,2,1,, == jixxx jiij (5)

3,2,1,, == jiyyy jiij (6)

( 321 ,, ) :

xxxx

+

+

=

33

2

2

1

1

(7)

yyyy

+

+

=

33

2

2

1

1

(7)

61

. 3 5 5 7 9 10 Rayleigh-Ritz 13: q 14 Galerkin 20: q 21 Rayleigh-Ritz 25 1: 25 (1) 26 (2) 28 2 - 34 38 39 40 42 44 56: 58 60

Rayleigh-Ritz: q

Galerkin: q

Rayleigh-Ritz 1: (1) (2)

2 - . :