pavlenko МКЭ.pdf

5/19/2018 pavlenko .pdf

1/148

..

-

2006


2/148

2

539.3

12

30.121.2

:

. . , . .. ;

. . , . ..

..

12

: . : - , 2006.147.

.

ISBN 966-657-087-4

-

.

- .

,

,

.

30.121.2

ISBN 966-657-087-4 .., 2006

- , 2006


3/148

3

.

....................................................................................................... 5

1 ........................................... 7

1.1 ...................................................... 71.2 .............. 19

2 .................... 25

2.1 ......................................................................... 25

2.2 ......................................................................... 26

2.3 ............................................................................. 27

2.4 ................................................................ 29

2.5 - ...... 30

3

....................................... 35

3.1 ........................................................................................ 35

3.2 ............................................................. 36

3.3 ........................................................... 37

3.4 ....................................................................... 49

3.5 ............... 62

3.6 ........................................................................... 76

3.7 .......................................................... 102


4/148

4

4

.............................................................. 109

4.1 ...................................................... 109

4.2 .............................................. 121

......................................................................... 147


5/148

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8/148

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9/148

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10/148

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11/148

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12/148

12

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13/148

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:

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k

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14/148

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15/148

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16/148

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17/148

17

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18/148

18

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19/148

19

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20/148

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21/148

21

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(1.77)

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22/148

22

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(1.72)

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23/148

23

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(1.72)

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k

k RXGX (1.89)

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k

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k

k (1.93)

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24/148

24

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25/148

25

2

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26/148

26

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(2.3)

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27/148

27

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;;

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(2.7)

T, -

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2.3

, -

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28/148

28

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(2.9)

u, v, w x,y,z-

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(2.11)

:

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29/148

29

D :

.

0

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yz

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30/148

30

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2.5 - -

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2.4).

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31/148

31

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dux (2.21)

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(2.22)

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2.5.2

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( 2.5).


32/148

32

2.5

,

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33/148

33

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2

100

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2.5.3

, ( 2.6).


34/148

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35/148

35

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36/148

36

3.2

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37/148

37

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FF


38/148

38

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(3.5)

3.3.2

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39/148

39

3.3

,

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2

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22

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40/148

40

(3.14)

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KKK (3.15)

(3.16)

, -

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,

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3.3.3 -

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41/148

41

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42/148

42

, ij,

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43/148

43

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1

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44/148

44

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45/148

45

3.2

2 1, 3, 4.

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46/148

46

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= 0;

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47/148

47

3.3

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U;0

41

2

uu

U

;12

2

cuu

U

;42

2

2

cu

U

;32

2

cuu

U

;242

2

cuu

U

;013

2

uu

U;

23

2

c

uu

U

;

2

3

2

c

u

U

;0

43

2

uu

U

;014

2

uu

U;2

24

2

cuu

U

;034

2

uu

U.2

2

4

2

cu

U

,

3.2.


48/148

48

3.4

,

3.1.

-

(1= 4= 0):

.22

12

2

1

2

132

2

3

2

32

2

2 uPuP

,0232

;023

32

3

32

2

Pcucuu

Pcucuu

.

232

23

3

2

P

P

u

u

cc

cc

-

(. 3.1).


49/148

49

3.4

3.4.1

, -

( 3.4).

3.4

l, -

.

(2.24)

,)( CxEA

Pxu (3.18)

,

.

= 0:

,0)0( CEA

Puu i (3.19)

.iuC (3.20)

(3.18)

.)( iuxEA

Pxu (3.21)

= l:

,)( ij ulEA

Pulu (3.22)

.l

uu

EA

P ij (3.23)


50/148

50

(3.23) (3.21)

,)( iij

uxl

uuxu (3.24)

.1)( ji ul

xu

l

xxu

(3.25)

(3.25)

.

,1l

uuu

l

xu

l

x

dx

d

dx

du ijji

(3.26)

,Bu (3.27)

-

,

j

i

u

uu (3.28)

.11

ll

B (3.29)

(2.20)

.ij uulE

EBuE (3.30)

:

,A

F (3.31)

F , .

, ucuulEA

AF ij (3.32)


51/148

51

;l

EAc (3.33)

.ij uuu (3.34)

, .

,

.

11

11

l

EA

lEA

lEA

l

EA

l

EA

K (3.35)

.11

11

j

i

j

i

F

F

u

u

l

EA(3.36)


52/148

52

3.4

2 1, 3.

1:

2:

.110154

044

2

321

21

uuu

l

EA

KKK ee

u1 = 0;

u3= 0.

F2= .

-


53/148

53

,52

2 Pul

EA

.5

2

2 EA

Plu

.

5

1

5

4

05

20

110

044

23

1

P

P

EA

Pl

l

EA

F

F

.05

1

5

4321

PPPFFF

1 2:

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2

5

2011

1A

P

EA

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Ee

.5

1

05

211

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PEA

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llEe

:

;5

2

2

5

4

2

11

A

P

A

P

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.5

15

1

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A

P

A

Fe

, 1 , 2

.


54/148

54

3.5

. -

.

1:

2:

3:


55/148

55

u1

= 0;

u4= .

F2= 0;

F3= 0.

-

,0

0

231

0133

2

u

u

l

EA

.4

3

;4

1

3

2

u

u

:

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3

4

1

;32

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.

2

2

4

34

10

2200

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1

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EAl

l

EA

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F


56/148

56

.02

002

4321

EAlEAlFFFF

1, 2 3:

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4

1011

1 Elll

Ee

;2

4

34

111

2 Elll

Ee

.4

4

311

3 Elll

Ee

:

;42

11 E

lA

Fe

.4

3

2 ElA

Fe

, .


57/148

57

3.4.2

, ,

,

.

q

-

, -

.

l( 3.5 ),

-

q.

3.5

dF = qdx (), -

(3.25),

.1)( qdxul

xul

xdFxuW ji

(3.38)


58/148

58

.22

022

122

0

jiji

l

ji uql

uqll

ul

xu

l

xxqqdxu

l

xu

l

xdWW

(3.39)

FiFj, ij(-

3.5 ). -

.jjii uFuFW (3.40)

(3.39) (3.40),

,22

jijjii uql

uql

uFuF (3.41)

.2

;2

qlF

qlF

j

i

(3.42)


59/148

59

3.6

, -

.

1:

2:


60/148

60

u1

= 0;

u3= 0.

.

;2

3

;2

1

32

2

11

gAlRF

gAlF

gAlRF

-

,2

33 2 gAlu

l

EA

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2

2EA

gAlu

,32 gAlgAlgAlP

,6

2EA

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.3

2

;3

2

1

l

Pq

l

Pq


61/148

61

:

;

02

0

220

0112

1 2

3

1

EA

gAl

l

EA

gAlR

gAlR

;

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16

1

06

10

220

011

3

16

1

3

1

P

PP

P

P

R

R

.

3

23

1

3

1

P

P

RR

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2

3

2

33

13211 Pl

l

Pl

l

PPRlqlqR

1 2:

;66

0111 A

P

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.60

611

2A

PEA

Pl

llEe

, 1, 2 .


62/148

62

3.5

,

.

, -

(-

3.6).

3.6

:

, ;

.

ij

.sin

;cos

i

i

i

i

uv

uu(3.43)

(3.43) cos, sin

.sincossincos 22 iii

i uuvu (3.44)


63/148

63

-

(cos2

+sin2

= 1), .sincos i

ii vuu (3.45)

.sincos j

jj vuu (3.46)

, ,

, :

.cossin iii vuv (3.47)

.cossin j

jj vuv (3.48)

(3.45) (3.48) -

,

cossin00

sincos00

00cossin

00sincos

j

j

i

i

j

j

i

i

v

u

v

u

v

u

v

u

(3.49)

,uTu (3.50)

- -

;

.

cossin00

sincos00

00cossin

00sincos

T (3.51)

-

,TFF (3.52)

F

-


64/148

64

.

jy

jx

iy

ix

F

F

F

F

F (3.53)

,FKu (3.54)

.

0000

00

0000

00

jjii vuvu

l

EA

l

EA

l

EA

l

EA

K

(3.55)

(3.50) (3.52) (3.54) -

. TFKTu (3.56)

,

,ITTT (3.57)

:

,T FKTuT (3.58)

, FuK (3.59)

.

sinsincossinsincos

sincoscossincoscos

sinsincossinsincos

sincoscossincoscos

22

22

22

22

l

EAKTTK T (3.60)


65/148

65

-

,sincos00

00sincos1111

j

j

i

i

j

i

v

u

v

u

llE

u

u

llEE

(3.61)

.sincossincos

j

j

i

i

v

u

v

u

l

E (3.62)


66/148

66

3.7

,

.

1( ):

2:


67/148

67

:

u1 = 0;v1 = 0;

u3 = 0;

v3= 0.

:

Fx2= 0;

Fy2= P.

,0

11

13

2

2

Pv

u

l

EA

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;

2

2

EA

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EA

Plu


68/148

68

.0

0

0

3

00

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111100

000000

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3

3

1

1

P

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l

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y

x

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1 2:

;

3

0

0

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P

EA

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Pl

l

Ee

.

0

0

3

2

1

2

1

2

1

2

1

22

A

P

EA

PlEA

Pl

l

Ee

, 1, 2 .


69/148

69

3.8

2 3, .

1:

.

1111

11111111

1111

2

2211

1

vuvu

lEAKe

2:

.

11111111

1111

1111

2

3322

2

vuvu

l

EAKe

3:

.

0000

00

0000

00

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3

cc

cc

vuvu

Ke


70/148

70

.

222200

22220

220

22

220

22

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02222

332211

l

EA

l

EA

l

EA

l

EAl

EAc

l

EA

l

EA

l

EAc

l

EA

l

EA

l

EA

l

EA

l

EAl

EA

l

EA

l

EA

l

EA

l

EAl

EA

l

EA

l

EA

l

EA

cl

EA

l

EA

l

EAc

l

EAvuvuvu

K

:

.0

0

;0

3

1

1

v

v

u

:

.0

;

;0

3

2

2

x

y

x

F

PF

F

.l

EAk


71/148

71

,

0

0

222

20

20

3

2

2

P

u

v

u

ckkk

kk

kk

.2

;4

4

;4

3

2

2

c

Pu

kcckPv

c

Pu

.4

22

;2 23

c

P

c

P

uu

.

( -

).

.

2

12

1

0

02

4

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0

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222200

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0

2222

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1

1

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kkkk

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72/148

72

3.9

2 3, .

1:

.

1010

0000

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0000

2211

1

vuvu

l

EAKe

2:

.

0000

0101

0000

0101

3322

2

vuvu

l

EAK

e

3:

.

1111

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1111

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3311

3

vuvu

l

EAKe


73/148

73

.

110011

130211

002020

020200

112031

110011

2

332211

vuvuvu

l

EAK

:

,0

;0

;0

;0

3

2

1

1

v

v

v

u

v3 3

33 ee :

,02

45cos45sin 33333 uvvuv

.33 uv

:

,0

;

3

2

x

x

F

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45sin45cos33

333

yx

yx

x

FFFFF

.33 xy FF


74/148

74

.2

;322

;0

;222

;2

;2

333

3332

2

32

133

133

x

x

y

y

x

Fuul

EA

Fuuul

EA

F

Puul

EA

Fuul

EA

Fuul

EA

,

2

1

;2

1

;2

3

3

3

2

EA

Plv

EA

Plu

EA

Plu

.2

1

;

2

1

;0

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1

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1

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2

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F

PF

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y

x

y

y

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75/148

75

:

.002

1

2

1

;02

10

2

1

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21

211

3

1

3

321

3

1

321

3

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yyx

i

i

yyy

i

yi

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i

xi


76/148

76

3.6

3.6.1

,

, ( 3.7).

3.7

l, I -

.

( 3.7)

ij, : v

z.

.dx

dv (3.63)

F -

z.

).()(

2

2

xMdx

xvdEI (3.64)

.2

2

ii MxFdx

vdEI (3.65)


77/148

77

,

,26

)( 21

23

CxCxMxFxvEI ii (3.66)

1, 2 , -

.)0(

;)0(

j

ivv

(3.67)

,

;

2

1

i

i

vEIC

EIC (3.68)

;26

)(23

iiii vEIxEIx

Mx

FxvEI (3.69)

.2

)(2

iii EIxMx

FxEI (3.70)

, (3.64)

, 121

2

jj MxFdx

vdEI (3.71)

,1 xlx

( 3.8).

3.8


78/148

78

(3.71) ,

,26

)( 413

2

1

3

11 CxCxMxFxvEI jj (3.72)

3, 4 , -

.)0(

;)0(

j

jvv

(3.73)

,

;

4

3

j

j

vEIC

EIC (3.74)

;26

)( 1

2

1

3

11 jjjj vEIxEI

xM

xFxvEI (3.75)

.

2

)( 1

2

11 jjj EIxM

xFxEI (3.76)

(3.69), (3.70) vjj:

;26

23

iiiij vEIlEIl

Ml

FvEI (3.77)

.2

2

iiij EIlMl

FEI (3.78)

(3.75), (3.76) vii:

;26

23

jjjji vEIlEIl

Ml

FvEI (3.79)

.2

2

jjji EIlMl

FEI (3.80)


79/148

79

(3.77) (3.80)

Fi,Mi,Fj,Mj,

,4626

;612612

;2646

;612612

22

2323

22

2323

jjiij

jjiij

jjiii

jjiii

l

EIv

l

EI

l

EIv

l

EIM

l

EIv

l

EI

l

EIv

l

EIF

l

EIv

l

EI

l

EIv

l

EIM

l

EIv

l

EI

l

EIv

l

EIF

(3.81)

,FuK (3.82)

-

;

j

j

i

i

u

u

u

(3.83)

F -

;

j

j

i

i

M

F

M

F

F (3.84)

.

233

3636

323

3636

2

22

22

3

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ll

llll

ll

l

EIK (3.85)


80/148

80

3.6.2

-

( 3.9).

.)( 332

210 xaxaxaaxv (3.86)

3.9

,

.32)(

2

321 xaxaax

(3.87) 0, 1, 2, 3

.)(

;)(

;)0(

;)0(

j

j

i

i

l

vlv

vv

(3.88)

.32

;

;

;

2

321

3

3

2

210

1

0

lalaa

lalalaav

a

av

j

j

i

i

(3.89)

(3.89)


81/148

81

.12122

;132332

;0010

;0001

2323323

2222

1

0

ulllll

vv

la

ulllll

vv

la

ua

uva

jiji

jiji

i

i

(3.90)

;

232)( 3

32

2

2x

l

vv

lx

l

vv

lxvxv

jijijiji

ii

(3.91)

,

23

322)(

2

322x

l

vv

lx

l

vv

lx

jijijiji

i

(3.92)

;)( uxv (3.93)

,)( udx

dx (3.94)

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.3263416 222

lldx

d(3.96)

.lx (3.97)

3.6.3

(3.85) -

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.


82/148

82

-

,

, . , -

:

,2

,

ji

jiqq

UK

(3.98)

U ;

q ( ...;;;;24231211

qvqqvq ).

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.)(

2

1

0

2

2

2

l

dxdx

xvdEIU (3.99)

,

(3.86).

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1 323

2

32

2

2

0

2

3

3

2

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2

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2

32

2

232

2

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, -

-

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, (3.98)


83/148

83

,

4626

612612

2646

612612

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2323

22

2323

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EI

l

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l

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l

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l

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l

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l

EIl

EI

l

EI

l

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l

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EI

l

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l

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l

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(3.85).


84/148

84

3.10

.

.

233

3636

323

3636

2

22

22

2211

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ll

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362

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2

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v

ll

l

l

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85/148

85

.

;2

2

2

2

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EIMlv

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2

0

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36362 2

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Mx

dx

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86/148

86

3.11

.

1:

2:

:

.0

;0

2

1

v

v


87/148

87

:

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;

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3

2

1

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88/148

88

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6

53

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22

1

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EI

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dx

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89/148

89

2

.6

12

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1;0

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32

2

323

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20

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l

vv

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l

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xlxEI

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e

e

e

e


90/148

90

3.6.4

,

, , -

.

q, ( 3.10 ), -

, - ( 3.10 ).

3.10


91/148

91

, -

d:

.qdxdF (3.105)

v():

.)()( qdxuqdxxvdFxvW (3.106)

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221

00

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92/148

92

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93/148

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94/148

94

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95/148

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96/148

96

3.6.5

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3.11 ),

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,

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97/148

97

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98/148

98

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3.13 ), -

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99/148

99

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100/148

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101/148

101

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102/148

102

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.cossin j

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, ij -

:


103/148

103

;ii (3.121)

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(3.117) (3.122)

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104/148

104

(3.127)

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105/148

105

3.14

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106/148

106

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107/148

107

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108/148

108

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109/148

109

4

4.1

4.1.1

( 4.1 ).

4.1


110/148

110

(-

4.1 ):

;),( 321 yxyxu (4.1)

.),( 321 yxyxv (4.2)

1,2,3,1,2,3 :

;),(

;),(

;),(

kkk

jjj

iii

uyxu

uyxu

uyxu

(4.3)

,),(

;),(

;),(

kkk

jjj

iii

vyxv

vyxv

vyxv

(4.4)

, (4.1) (4.2):

;

;

;

321

321

321

kkk

jjj

iii

uyx

uyx

uyx

(4.5)

,

;

;

321

321

321

kkk

jjj

iii

vyx

vyx

vyx

(4.6)

:

;

1

1

1

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2

1

k

j

i

kk

jj

ii

u

u

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(4.7)

.

1

1

1

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2

1

k

j

i

kk

jj

ii

v

v

v

yx

yx

yx

(4.8)

(4.7) (4.8) 1,2,3,

1,2,3 (4.1) (4.2), :

;2

1),( kkkkjjjjiiii uycxbauycxbauycxbayxu

(4.9)


111/148

111

,2

1),( kkkkjjjjiiii vycxbavycxbavycxbayxv

(4.10)

:

;;;jj

ii

k

ii

kk

j

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i yx

yxa

yx

yxa

yx

yxa (4.11)

;1

1;

1

1;

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i y

yb

y

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;1

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1

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k

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i

x

xc

x

xc

x

xc (4.13)

.

1

1

1

2

1

kk

jj

ii

yx

yx

yx

(4.14)

:

.

(2.31), A (4.15)

(2.12)

,uD (4.16)

D :

.0

0

xy

y

x

D (4.17)

(4.9) (4.10)


112/148

112

,000

000

2

1

321

321

k

k

j

j

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i

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u

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v

u

v

u(4.18)

1, 2, 3 ( 4.2):

;),(1 ycxbayx iii (4.19)

;),(2 ycxbayx jjj (4.20)

.),(3 ycxbayx kkk (4.21)

, 1 1 i

0 j, k; 2 1 j 0

k, i; 3 1 k 0

i,j.


113/148

113

4.2

(4.18) (4.16)

,000

000

2

10

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321

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i

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x

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v

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xy

y

x

(4.22)


114/148

114

,000

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i

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x

v

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bcbcbc

ccc

bbb

(4.23)

,uB (4.24)

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1

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kji

kji

bcbcbc

ccc

bbb

B (4.25)

,

.

,

dUU (4.26)

- ;

U ( -

),

.21

21 TxyxyyyxxU (4.27)

U

. h

.2

Th

hUU

(4.28)

-


115/148

115

, FhuvhFuhFvhFuhFvhFuhFV Tkykkxkjyjjxjiyiixi (4.29)

Fih,Fjh,Fkh , -

;

F -

.

yk

xk

yj

xj

yi

xi

F

F

F

F

F

F

F (4.30)

-

-

( )

,2

1VU (4.31)

(4.28) (4.29) :

,22

Fuhh TT

(4.32)

. FuTT (4.33)

(4.15)

, FuA TT (4.34)

( ). FuA TT (4.35)

(4.24)

, FuBuABu TT (4.36)

, FuuABBu TTT (4.37)

,FuABBT (4.38)

,FuK (4.39)


116/148

116

K

. ABBKT

(4.40)

4.1.2

,

-

( ). -

. -

,

.

( 4.3)

.

4.3

- :

;),( 2652

4321 yxyxyxyxu (4.41)


117/148

117

.),( 2652

4321 yxyxyxyxv (4.42)

1, 2, ,6 1, 2, ,6

:

;),(

...

;),(

;),(

nnn

jjj

iii

uyxu

uyxu

uyxu

(4.43)

,),(

...

;),(

;),(

nnn

jjj

iii

vyxv

vyxv

vyxv

(4.44)

(4.41) (4.42):

;

..................................................................

;

;

2

65

2

4321

2

65

2

4321

2

65

2

4321

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ijjjjjj

iiiiiii

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(4.45)

,

..................................................................

;

;

2

65

2

4321

2

65

2

4321

2

65

2

4321

nnnnnnn

ijjjjjj

iiiiiii

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vyyxxyx

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(4.46)

:

;......

1

..................

1

1

2

1

22

22

22

n

j

i

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u

u

u

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(4.47)

.......

1

..................

1

1

2

1

22

22

22

n

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(4.48)

(4.47), (4.48)

1,2, ,61,2, ,6 (4.41), (4.42), :


118/148

118

;.....

),(

22

22

22

nnnnnnn

jjjjjjj

iiiiiii

uyrxyqxpycxbauyrxyqxpycxba

uyrxyqxpycxbayxu

(4.49)

,...

...

),(

22

22

22

nnnnnnn

jjjjjjj

iiiiiii

vyrxyqxpycxba

vyrxyqxpycxba

vyrxyqxpycxbayxv

(4.50)

,

...000000

000000

),(

),(

654321

654321

n

n

j

j

i

i

v

u

v

u

v

u

yxv

yxu(4.51)

1, 2, ,6 ( 4.4)

;),(

.........................................................................................

;),(

;),(

2

6,66,5

2

6,46,36,26,16

2

2,62,5

2

2,42,32,22,12

21,61,5

21,41,31,21,11

yMxyMxMyMxMMyx

yMxyMxMyMxMMyx

yMxyMxMyMxMMyx

(4.52)

p,q (p,q=1,2,,6)

.

1

..................

1

11

22

22

22

nnnnnn

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iiiiii

yyxxyx

yyxxyx

yyxxyx

M (4.53)


119/148

119

4.4

.0...00

0...000

0

),(621

621

xy

y

x

yxB (4.54)

,

,. - .


120/148

120

(4.26) (4.27), (4.15) (4.24)

,2

1

2

1

2

1

2

1

2

1

2

1

udSABBuhudABBudBuABu

dAdAddUU

S

TTTTT

TTT

(4.55)

.

-

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F

.

...

yn

xn

yj

xj

yi

xi

F

F

F

F

F

F

F (4.57)

(4.31)

,2

1

2

1FuhudSABBuh T

S

TT

(4.58)

,FuK (4.59)

K

.S

T dSABBK (4.60)


121/148

121

4.2

4.2.1

( 4.5). -

-

;),( 4321 xyyxyxu (4.61)

.),( 4321 xyyxyxv (4.62)

4.5

1, 2, 3, 4 1, 2, 3, 4

:

;),(

;),(

;),(

;),(

mmm

kkk

jjj

iii

uyxu

uyxu

uyxu

uyxu

(4.63)

,),(

;),(

;),(

;),(

mmm

kkk

jjj

iii

vyxv

vyxv

vyxv

vyxv

(4.64)


122/148

122

(4.61) (4.62):

;

;

;;

4321

4321

4321

4321

mnnnn

kkkkk

ijjjj

iiiii

uyxyx

uyxyx

uyxyxuyxyx

(4.65)

,

;

;

;

4321

4321

4321

4321

mnnnn

kkkkk

ijjjj

iiiii

vyxyx

vyxyx

vyxyx

vyxyx

(4.66)

;

1

1

1

1

m

k

j

i

m

k

j

i

mmmm

kkkk

jjjj

iiii

u

u

u

u

yxyx

yxyx

yxyx

yxyx

(4.67)

.

1

11

1

m

k

j

i

m

k

j

i

mmmm

kkkk

jjjj

iiii

v

vv

v

yxyx

yxyxyxyx

yxyx

(4.68)

(4.67), (4.68)

1, 2, 3, 4 1, 2, 3, 4 (4.61), (4.62), -

:

;...

...),(

mmmmmkkkkk

jjjjjiiiii

uxydycxbauxydycxba

uxydycxbauxydycxbayxu

(4.69)

,...

...),(

mmmmmkkkkk

jjjjjiiiii

vxydycxbavxydycxba

vxydycxbavxydycxbayxv

(4.70)

:


123/148

123

,0000

0000

),(

),(

4321

4321

m

m

k

k

j

j

i

i

v

u

v

u

v

u

v

u

yxv

yxu(4.71)

1, 2, 3,4 ( 4.6)

;),(

;),(

;),(

;),(

4,44,34,24,14

3,43,33,23,13

2,42,32,22,12

1,41,31,21,11

xyMyMxMMyx

xyMyMxMMyx

xyMyMxMMyx

xyMyMxMMyx

(4.72)

p,q (p,q=1,2,3,4)

.

1

1

1

11

mmmm

kkkk

jjjj

iiii

yxyx

yxyx

yxyx

yxyx

M (4.73)


124/148

124

4.6


125/148

125

.0000

0000

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),(

44332211

4321

4321

4321

4321

x

y

x

y

x

y

x

y

y

y

y

y

x

x

x

x

xy

y

x

yxB

(4.74)

h

(4.55)

,2

1udSABBuhU

S

TT

(4.75)

.

-

,FuhV T (4.76)

F

.

ym

xm

yk

xk

yj

xj

yi

xi

F

F

F

F

F

FF

F

F (4.77)

(4.31)

,2

1

2

1FuhudSABBuh T

S

TT

(4.78)


126/148

126

,FuK (4.79)

K

.S

T dSABBK (4.80)

4.2.2

( 4.7)

.

;),( 383

7

2

65

2

4321 yxyxyxyxyxu (4.81)

.),( 383

7

2

65

2

4321 yxyxyxyxyxv (4.82)

4.7


127/148

127

1, 2, , 81,2, ,8 -

:

;),(

...

;),(

;),(

sss

jjj

iii

uyxu

uyxu

uyxu

(4.83)

,),(...

;),(

;),(

sss

jjj

iii

vyxv

vyxv

vyxv

(4.84)

(4.81) (4.82):

;

.........................................................................................

;

;

3

8

3

7

2

65

2

4321

3

8

3

7

2

65

2

4321

3

8

3

7

2

65

2

4321

ssssssss

jjjjjjjj

iiiiiiii

yxyyxxyx

yxyyxxyx

yxyyxxyx

(4.85)

,

.........................................................................................

;

;

3

8

3

7

2

65

2

4321

3

8

3

7

2

65

2

4321

3

8

3

7

2

65

2

4321

ssssssss

jjjjjjjj

iiiiiiii

yxyyxxyx

yxyyxxyx

yxyyxxyx

(4.86)

:

;......

1

........................

1

1

8

2

1

3322

3322

3322

s

j

i

ssssssss

jjjjjjjj

iiiiiiii

u

u

u

yxyyxxyx

yxyyxxyx

yxyyxxyx

(4.87)

.......

1

........................

1

1

8

2

1

3322

3322

3322

s

j

i

ssssssss

jjjjjjjj

iiiiiiii

v

v

v

yxyyxxyx

yxyyxxyx

yxyyxxyx

(4.88)

-

1, 2, , 8,1,2, ,8 (4.81)

(4.82), :


128/148

128

;.....

),(

3322

3322

3322

sssssssss

jjjjjjjjj

iiiiiiiii

uytxhygxyfxdycxbauytxhygxyfxdycxba

uytxhygxyfxdycxbayxu

(4.89)

,..

...

),(

3322

3322

3322

sssssssss

jjjjjjjjj

iiiiiiiii

vytxhygxyfxdycxba

vytxhygxyfxdycxba

vytxhygxyfxdycxbayxv

(4.90)

,

...0...00

0...00

),(

),(

821

821

s

s

j

j

i

i

v

u

v

u

v

u

yxv

yxu(4.91)

1, 2, , 8 ( 4.8):

.),(

.........................................................................................................................

;),(

;),(

3

88

3

78

2

6858

2

483828188

3

82

3

72

2

6252

2

423222122

381

371

26151

2413121111

yMxMyMxyMxMyM

yMxMyMxyMxMyM

yMxMyMxyMxMyM

(4.92)

.

1

........................

1

11

3322

3322

3322

ssssssss

jjjjjjjj

iiiiiiii

yxyyxxyx

yxyyxxyx

yxyyxxyx

M (4.93)


129/148

129

4.8

.

...

0...0

0...0

0...0

0...00

0

),(

8811

81

81

81

81

x

y

x

y

y

y

x

x

xy

y

x

yxB (4.94)

h

(4.55)


130/148

130

,

2

1udSABBuhU

S

TT

(4.95)

.

-

,FuhV T (4.96)

F

.

...

8

8

2

2

1

1

y

x

y

x

y

x

F

F

F

F

FF

F (4.97)

(4.31)

,2

1

2

1FuhudSABBuh T

S

TT

(4.98)

,FuK (4.99)

K

.

S

T dSABBK (4.100)


131/148

131

4.1

.

. :

.2

01

1

001

2

1

1

1

1

2

12

33

22

11a

a

aa

yx

yx

yx

:

;0

2

33

22

1 aa

aa

yx

yxa

;000

0

11

332

a

yx

yxa

;000

22

11

3 aayx

yxa

;01

1

1

1

3

2

1 a

a

y

yb

;

10

1

1

1

1

3

2 aa

y

yb

;1

10

1

1

2

1

3 aay

yb


132/148

132

;01

1

1

1

3

2

1 aa

x

xc

;001

01

1

1

1

3

2 x

xc

.1

01

1

1

2

1

3 aax

xc

.

111001

100010

0101001

00

0000

0000

22

1000

000

21 2

332211

321

321

a

aaaa

aa

aa

abcbcbc

ccc

bbb

B

.

2

100

01

01

1 2

EA

.

311221

131221

111001

220220

220220

111001

14

2

EBABK TT

:

01u ; 01v ; 03u ; 03v .

:

02 xF ; PFy 2 .


133/148

133

,

0

10

02

14 2

2

2

Pv

uE

.14

;0

2

2

E

Pv

u

= 0,3

.2,5

;0

2

2

E

Pv

u

.0

14

0

1212

02

10

14 2

3

3

2

1

PP

P

E

PE

FF

F

F

y

x

y

x

.0;00

;00

13

321

321

aPaPaPaFFm

PPFFF

PPFFF

xk

yyy

xxx


134/148

134

.

14

0

14

0

0

14

0

00

111001

100010

0101001

aE

P

aE

P

E

Pa

xy

y

x

= 0,3

.2,5

;0

;2,5

aE

P

aE

P

xy

y

x

.

2

1

4

14

14

0

14

2

100

01

01

1 2

a

P

a

P

aP

aE

P

aE

P

E

xy

y

x

= 0,3

.2

;714,5

;714,5

a

P

a

P

a

P

xy

y

x


135/148

135

4.2

.

1 2.

:

;2

1

01

001

2

1 2

1

a

aa

ae

.2

01

1

001

2

1 2

2

a

a

aae

1:

211 aa e ; 012 ea ; ;013 ea

ab e 11 ; abe 12 ; 0

1

3 eb ;

011 ec ; ab e 12 ; ab

e 13 .

2:

22

1 aae ; 022

ea ; ;023 ea

021 eb ; abe 22 ; abe 23 ;

ac e 21 ; 02

2 eb ; abe 23 .


136/148

136

1:

.

011110

1010000001011

1

a

Be

2:

.

111001

100010

0101001

2

a

Be

.

2

100

01

01

1 2

EA

1:

.

202202

011110

2131112

211312

011110

202202

14

332211

21

vuvuvu

EKe

2:

.

311221

131221

111001

220220

220220

111001

14

443311

22

vuvuvu

EKe


137/148

137

.

31120021

13120021

11302201

22031110

00213112

00211312

22011130

11102203

14

44332211

2

vuvuvuvu

EK

:

01u ; 01v ; ;04 u 04 v .

:

02 xF ; 02 yF ; 03 xF ; PFy 4 .

,0

0

0

3022

0311

2131

2113

14

3

3

2

2

2

Pv

u

v

u

E


138/148

138

.27

514

;27

114

;27

414

;27

14

2

2

3

2

2

3

2

2

2

2

2

2

E

Pv

E

Pu

E

Pv

E

Pu

= 0,3

.14,3

;63,0

;92,2

;46,0

3

3

2

2

E

Pv

E

Pu

E

Pv

E

Pu

.

27

25

27

12

1200

1200

0111

1022

14

2

2

2

2

3

3

2

2

2

4

4

1

1

P

P

P

P

v

u

v

u

E

F

F

F

F

y

x

y

x

= 0,3

.76,0

;

;24,0

;

4

4

1

1

PF

PF

PF

PF

y

x

y

x

.


139/148

139

1:

.

27

18

27

14

2714

011110

101000

0001011

2

2

2

2

2

2

3

3

2

2

1

1

1

1

1

aE

P

aE

P

aEP

v

u

v

u

vu

ae

xy

e

y

e

x

= 0,3

.8,1

;49,0

;49,0

1

1

1

aE

P

aE

P

aE

P

e

xy

e

x

e

x

2:

.

27

1540

27

114

111001

100010

0101001

2

2

2

2

4

4

3

3

1

1

2

2

2

aE

P

aE

P

v

u

v

u

v

u

ae

xy

e

y

e

x

= 0,3

.4,3

;0

;63,0

2

2

2

aE

P

aE

P

e

xy

e

x

e

x


140/148

140

1:

.

27

1427

142714

2

100

01

01

1

2

2

2

1

1

1

2

1

1

1

a

Pa

Pa

P

E

e

xy

e

y

e

x

e

xy

e

y

e

x

= 0,3

.69,0

;69,0

;69,0

1

1

1

a

P

aP

a

P

e

xy

e

x

e

x

2:

.

27

522714

27

14

2

100

01

01

1

2

2

2

2

2

2

2

2

2

2

2

a

Pa

P

a

P

Ee

xy

ey

e

x

e

xy

ey

e

x

= 0,3

.31,1

;21,0

;69,0

2

2

2

a

P

a

P

a

P

e

xy

e

x

e

x


141/148

141

4.3

.

.

400

20

2

444400

04

022

0

400

10

3

40

4110

000001

400

201

2421

444221

0001

1

000001

222

2222

222

1

2

222

222

2

222

aaa

aaaa

aaa

aaa

aaaa

aa

aaa

aa

aaaaaaa

aaaaa

M

:

;23

1, 221

ya

;21

, 222

xa

x

;24211

, 222

2

23 y

axy

ax

ay

ax

a

;44

,24

xya

xa


142/148

142

;44

,2

2

25 xy

ax

a

.4444

, 2226

ya

xya

ya

xa

.

448440

8440

44

4840

40

48

4440

40

44

4414410

4410

441

4100

0041

043

0

4300

,

222

222

222

222

22

22

2222

2222

2

2

2

2

T

yaa

ya

xaa

ya

xaa

yaa

ya

xa

xa

xa

ya

xa

yaa

xa

xa

yaa

ya

xaa

ya

xaa

ya

xaa

ya

xaa

xaa

xaa

yaa

yaa

yxB

k=BTAB:

;1

432

14

2

1,1a

Eyak

;02,1 k

;03,1 k

;

1

434

2

1

44,1 a

yaxak


143/148

143

;

2

1

8411,12

a

Eyxaayk

.1

298410348

42

2222

12,12a

xyxyaxyaaEk

.0

,, a a

x

jiji dykdxK

:

01u ; 01v ; 03u ; 03v ; 06 u ; 06 v .

:

02 xF ; PFy 2 ; 04 xF ; 04 yF ; 05 xF ; 05 yF .


144/148

144

.

0

0

0

0

0

38141614148

14381418148

1614381408

14181438140

1414014130

888006

112

5

5

4

4

2

2

2

P

v

u

v

u

v

u

E

= 0,3

.19,5

;20,4

;51,4

;26,1

;40,19

;87,5

5

5

4

4

2

2

E

Pv

E

Pu

E

Pv

E

Pu

E

Pv

E

Pu


145/148

145

.

014181400

140141600

1480012

1480012

0008020014010

112

5

5

4

4

2

2

2

6

6

3

3

1

1

v

u

v

u

vu

E

F

F

F

F

FF

y

x

y

x

y

x

= 0,3

.288,0

;176,2

;688,0

;088,2

;600,0;088,0

6

6

3

3

1

1

PF

PF

PF

PF

PFPF

y

x

y

x

y

x

.0176,22

02

02

0

088,02

2

2

2

;0288,000688,0600,0

;0176,200088,20088,0

6544213

654321

654321

aPaaa

aPaPa

Fa

Fa

Fa

FaFaFFm

PPPPFFFFFFF

FFFFFFF

xyyxyxk

yyyyyyk

xxxxxxkx

.

21898

112

98

124

298

938112

...

,

2

2

2

2

6

6

2

2

1

1

ayxaE

P

xaE

P

ayxaE

P

v

u

v

u

v

u

yxB

xy

y

x


146/148

146

.

98

218

6

298

211812

212

2

100

01

01

1

2

2

2

2

2

ayx

a

P

ayxa

P

ayxaP

E

xy

y

x

xy

y

x

.0218:0

;0:0;02:0

ayx

xayx

xy

y

x

.218:0

;0298

2118:0

;02:0

2

ayx

ayx

ayx

xy

y

x


147/148

147

1. .., .. . -

: . .: , 1979.

2. .. . .:

, 1970.288.

3. .-., .-.

. . . .: , 1989.190.

4. Gallagher R.H. Finite Element Analysis. New Jersey: Prentice-

Hall, 1975.

5. , , : : 3 . .:

, 1968. . 1.567.

6. .., .., .. ANSYS

: . .: , 2004.

7. .., .. -

. ., 1964.608., .


148/148

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..

..

.

6084/16. . .

. . . .-. . . .

350 . . 175.

.

40007, ., . .-, 2

2365 08.12.2005 .

40007, ., . .-, 2

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