1 . : , . 1, 2007

1

1.1 1

(), , A . , F , , ( )AB 2. . xN N= , , (-) x , . . , x ,

0 :xF N F+ = =

(1.1)

. , .. , ( )F F t= . , , . (1.1).

1 . stress 2 .

2 . : , . 1, 2007

- x , . ( , )P y z+ ( , )P y z . P+ dA P+ , dN+ . dN . 3

dN dN+ = (=) (1.2)

dN dN dN+ = = (1.3)

dNdA

= (1.4)

(1.4) . dN , 3. . , ( ),.

( , , , )x y z t = (1.5) dN ,

3 . normal stress

3 . : , . 1, 2007

( ) ( )A AN dN dA= = (1.6)

, , ,

( )A

NN dA AA

= = = (1.7) E. (1.7) . . (1.1) (1.7) ,

.FA

= = (1.8)

(1.7) . , z y . , dN dA= , ,

( )0z

Ay dA = (1.9)

y(A)

M z dA 0= (1.10) . 6, (1.9), (1.10), xN N= zM yM z y . , :

4 . : , . 1, 2007

( ) ( )0 0 , 1,2,n n

A Ay dA z dA n = = = (1.11)

:

, ,

0 0N > > (1.12) , , ,

0 0N < < (1.13) . , . (1.12) (1.13), .

,. (1.4), . ,

2[ ] FL = (1.14)

2Pa Nm= , 310kPa Pa= , 310MPa kPa= ...

, 10F kN= , , 21A cm= . , . (1.8) ,

( )5

2 2 4 2 22

5

10 10 10 101 1010

10 100 0.1

F kN kN kN kNA cm m mm

kPa MPa GPa

= = = = =

= = =

5 . : , . 1, 2007

1.2

( ). n, x . nA ,

cosnAA = (1.15)

( 0 = ). 4

n nx xt t e=G G (1.16) , x ,

xN N= , , 0nx n nyt A N t= = (1.17)

cos/ cosnx n

N NtA A

= = = (1.18)

nt n , n , . 5.

4 . traction vector. 5 . shear stress.

6 . : , . 1, 2007

, ( , )x y . , :

0 : cos

0 : sin

n n nx n

n n nx n

F A t A

F A t A

= =

= = (1.19)

. (1.19) . Error! Reference source not found. ,

2cos

sin cos

n

n

=

= (1.20)

, . n ={ , ,x y zn n n }

. , n .

( 0 = ) , , n n . . (1.20) . ( , )x y ( 1, 0x yn n= = ) ,

, 0nx nyt t= = (1.21)

( cos , sinx yn n = = ) 2cos , sin cosn nt t = = (1.22)

7 . : , . 1, 2007

, . (1.20) . . 2. . (1.20) ,

cos 22 2

sin 22

n

n

= +

= (1.23)

( ),n n ,

( )2 22 2 2cos 2 sin 22 2n n + = + 2 2

22 2n n + = (1.24)

. 2, E.(1.23) , 2 ., Mohr .

, . ( ) : ) , ) .

:

) . (1.20) ,

0ndd =

8 . : , . 1, 2007

,

1

2

02 cos sin 0

2

= = = =

,

21 2

0 ( 0) :0 : , 0 , 2

0 ( 0) :n

n ndd

< >= = = = = > >

22 2

0 ( 0) :: 0 , 0 , 2

0 ( 0) :2n

n ndd

> >= = = = = + < = > >

9 . : , . 1, 2007

4

2

2

/ 4 : / 2 , / 2

0 ( 0) :2

0 ( 0) :

n n

ndd

= = = =

< >= > >

45D . . , .

. (.. ) , , , , . , , . 6. (.. ) , l 45D , , , . ( 7).

6 . Mode I 7 . Mode I.

10 . : , . 1, 2007

1.3 8

(), . (AB) A . = + A A A . 9 ,

= AA (1.25)

, ,

[ ] 1 = (1.26)

:

, ( 0 >A ), . ( 0

11 . : , . 1, 2007

0.5mstrain = .

, . (1.25), 11, , 1

12 . : , . 1, 2007

( )x

dux x dx u x dxdx

= + + + (1.28)

x , ,

: ( )d x dx x dx= + =A :

( )( ) ( )x x

du dud x dx u x dx x u x dx dxdx dx

= + + + + = + A

: x

dud d d dxdx

= = A A A

x , . (1.25), ,

x

d dud dx

= = AA (1.29)

, . x ,

. (0)du u u xdx

= = = + (1.30)

. . , ,

(0) 0u = (1.31) ,

( )u x u = = = = AA A A A (1.32)

13 . : , . 1, 2007

. (Machu Picchu, Peru, . . 2006).

, . (1.25), 13. . (1.29) , dA x , . , . , .

13 . engineering strain

14 . : , . 1, 2007

1.5 -

1.5.1 Hooke

14..

, , , . 15 . ,

( ) = (1.33) . . (1.33), 16. -, - -. , -

14 . , . , . . (2002). : . , (. - & . , .) . , . 187-210. 15 . 16 . constitutive equation

15 . : , . 1, 2007

0

d Ed =

= + " (1.34)

-, Hooke17. E. (1.34) E - Young. , ,

2[ ]E FL= (1.35) 2/kN mm , ,

( )6 6

22 231 1 10 10 1

10

kN kN kN kPa GPamm mm

= = = = (1.36)

.. Young 75GPa .

E , Young, . Thomas Young (1807) ( )EA , A , /E , 18. Young , , -.

17 Robert Hooke (1635-1703) : ut tensio sic vis. 18 J.F. Bell, 1973, The Experimental Foundations of Solid Mechanics, Springer, Sect. 3.7, p. 186.

16 . : , . 1, 2007

. - , . A , Y , Hooke, = ( ). . . Y ( ). . 19. , F . . ( / ) F =A A , , .

19 , . hardening

17 . : , . 1, 2007

, ( ) , . (. ). , () , , (), (;). ,

(0 1)G = < , ( 0 > ). Hooke (1.34) (1.40) - , . -,

20 . shear modulus

18 . : , . 1, 2007

( ) ( )

( )T E TE = + = + = (1.41)

Young .

2kN

mm

210

74

70

30

10

6 o110C

11

3 9

22

12

3 9

, ,

0 0T = = (1.42) 21. , . .

(), . - (;). , ,

21 . workless thermal strain

19 . : , . 1, 2007

. 0 = =A ( ( )= A ). , . - . (1.41) ,

0 T E TE = + =

0 > ( 0 < ), ( 0 < ) ( 0 > ). .. ,

2210. /kN mm = 6 111. 10 oC = , 6 6

2 3 21 1210 11 10 2310 10 2.31

(10 )o o okN kN MPaT T T

mm C m C C = = =

22 St, 220MPa = , , < , ( max, < ) ,

omax,o o

MPa 220MPa2.31 T 220MPa 95.2 CC 2.31MPa / C

= < = = =

L.F. Coffin (1979) & S.S. Mason (1960)23 , , - , - .

22 . , . , , , , . 23 S.S. Manson , Interpretive report on cumulative fatigue damage in the low cycle range. Weld. J. Res. Suppl. 43 (1964), pp. S344S352..

20 . : , . 1, 2007

- . .

, () () , -, . T 0 > , . Maxwell () C.R. Calladine (1978)24. Maxwell , 3 n = + , , n 25. Calladine, 3 n s m = + + , s m .

24 C.R. Calladine, Buckminster Fullers Tensegritty structures and Clerk Maxwells rules for the construction of stiff frames, Int. J. Solids Structures, 14 (1978), pp. 161-172. 25 . . . , , . 3, . , 2004.

21 . : , . 1, 2007

1.5.3

A , 26 S . , A , E . ,

SA

= (1.43)

A ,

= AA (1.44)

Hooke,

E = (1.45) ,

( )S EA = AA (1.46)

( )EA 27 . E. (1.45) ,

TE = + (1.47)

E. (1.46) ,

( )S EA T = AA (1.48)

26 S , , . 27 , stiffness.

22 . : , . 1, 2007

1.6

1.6.1

. (;). , RA, R 28 S1 S2 . ,

1 2S S S= = (1.49)

2 cos 0

12 cos

S FFS

= = (1.50)

. (1.46) ,

SEA

=A A (1.51)

, + A A , .

:

2

( )cos

1 12 cos

FEA

=

=

A

A (1.52)

28 , , . , stress. . , . (1.43).

23 . : , . 1, 2007

:

SA

= 12 cos

FS =

(),

:

S

: 3m=A , o30 = , F 30kN= , 2210 /E kN mm= , 220MPa = . :

4 2 2

32

1 30 1 0.787 10 0.7872 cos30 220 10

okN m cmkN

m

= =

.. , D ,

4 0.01001 10.01D D A m mm = = =

12.5D mm= ,

( )2 2 -4 2 20.0125 1.227 10 m 1.2274 4

DA m cm= = = =

:

2

1 1 1 1 30 141140.2kPa=141.1MPa2 cos 2 cos30 0.0001227mo

S F kNA A

= = = =

,

141.1MPa

24 . : , . 1, 2007

( )

22

2-3

223

141140.2

210 0.0001227m

141140.20.0055=5.5 10

210 0.0001227m10

kPakNA

mmkNm

kNm

= = =

= =

AA

( 1

25 . : , . 1, 2007

( )* 3 2 31 1 1cos tan tan6 2 3

= = + +"A (1.53)

() ,

( )22 2 2( ) 2 cos2

A = + = + A A A A (1.54)

( )* *2 2 211 2 sin 12

= = + AA (1.55)

Hooke29, ,

( )S EA = AA (1.56)

,

*

( )SS

EA= (1.57)

Hooke ( ) ,

*S = (1.58) , , , ,

11 12 sin 2 6

F FS = + (1.59)

/ 2 = . (1.50) (. ), . (1.59) (. ). (1.50) ,

29 .

26 . : , . 1, 2007

(1.59) . , . (1.55) (1.59) - 30 . ,

( )FEA

= (1.60)

. (1.59) ,

* 11 12 sin 2 6

S = + (1.61)

. (1.58), (1.55) (1.61) , - ,

2 2( ) (1.62)

. (1.62), , : (0), ,

: 0 = = (1.63) , . , , . (1.62), (1) ,

30 - .

27 . : , . 1, 2007

3max

1 2:3 3 3

= = = (1.64)

(1) , , (1) . (1.62) , (1-2-1) . , 0 = , (1-2-1), 31 , . , 32 ( 1) ( 1) , (1-1). (1-1) , 33 . 34 35 .

1.6.3

() () ( 0 = ). F , , . (1.50) ( / 2 ) . , , . , 0 = .(1.53) (1.62) ,

3 *3 * 3 (1.65)

31 , . softening branch. 32 . . 33 , . snap-through. . Durchschlagen. Richard von Mises. 34 von Mises R. (1923). ber die Stabilittsprobleme der Elastiyittatheorie, ZAMM, 3, 406-422. von Mises R. und Ratzerdorfer, J. (1925). Die Knicksicherheit von Fachwerken, ZAMM,5, 218-235. 35 Huang, N. C. (1972) Dynamic buckling of a some elastic shallow structure subjected to periodic loading with high frequency, International Journal of Solids and Structures, 8, 315326.

28 . : , . 1, 2007

. (1.61) ,

* *212

S (1.66)

, , ,

1/3

( )FEA

A (1.67)

2 /31( )2 ( )

FS EAEA

(1.68)

, , - .. .

1.6.4

, ' :

1 (0 1)= + <

29 . : , . 1, 2007

(1) (2) (3) :

1( ') 0( ) ( ) 0OOOO OO

= > = = . ,

1 2N N N= = (2),

0 1T T T = +

(1): 1d NL EA

= =

(2): 2 0 1 1d N d NT T

L EA L EA += = + = +

,

32 . : , . 1, 2007

112

NL NLd T N EA TEA EA

= = + =

(1) (2)

1 112

N E TA

= = ( )

2 112

N E TA

= = ( )

10

3/ 10 . (1) 36. (1) 1 = . ,

31, 1, 5

1 1 1 1 12 2 10 2010 2 10 10 10

E T CC

= = = =

DD

(2)

0 1 100 20 120T T T = + = + =D D D . 2) () ()

T . ().

36 . .

33 . : , . 1, 2007

:

:

2 2

cos

a ba

= +=

A

A

():

0 :yF+ = 1 2( )sin ( )sin ) 0S S + = 1 2S S S= = (1.78) 0 :xF

+ = 1 2 1 2( ) ( ) cos ( )cos 0N N S S + = 2 1

1cos ( )2

S N N = (1.79)

-, :

TE = +

:( )

S TEA

= + AA

: 1( )

Naa EA =

: 2( )

Ncc EA =

34 . : , . 1, 2007

:

) :

:

c c a a c a+ + + = + c a = Hooke ,

2 1( ) ( )

N Nc aEA EA

= 2 1aN Nc= (1.80)

. (1.79) :

2 1 11 1cos ( ) (1 )2 2

aS N N Nc

= = + 1 2 cos1 /N Sa c = + (1.81)

) ():

1 cos( ) ( )

N Sa TEA EA

= + A (1.82)

35 . : , . 1, 2007

. (1.81) (1.82) ,

2( )2 1cos ( )

1 / ( ) cosEAS S EA T

a c EA

= + +

3

( )( )2 cos

1 / ( )

EA TS EA

a c EA

= ++

S , . (1.81) (1.80).

3) , 1 3( ) ( ) ( )EA EA EA= = , 2( ) 2( )EA EA= .

.

4) , () , (). F. .

36 . : , . 1, 2007

5) ( ) (F = 0), (2) C. .

6) , , . = 30C. , , , . : , = 70 GPa, = 23 .10-6 /oC = 3 cm2.

7) Ac , () As. () () . : ) c s . ) c s , (-) . , Ec = 25 GPa , ESt = 200 GPa. , Ac = 400cm2 , ASt = 4 cm2. ac=10 .10-6/oC St= 12 .10-6/oC. , = 200kN, , = 20C.

37 . : , . 1, 2007

:

Z ( )s D ( )c :

Z D= 0L ,

0 0 0c sL L L= =

100Z P kN= =

( )s : 0 2 2100. 250

4. (10 )St

St

P kN MPaA m

= = =

( )c : 0 2 2100. 2.5

400. (10 )c

c

P kN MPaA m

= = =

Z D 20T C = D .

Z D P = =

P P P = +

38 . : , . 1, 2007

( )s : / StSt StSt

P A TE

= +

( )c : / cc cc

P A TE

= +

o ,

0 (1 )c St st cL L L = = + = = :

/ /St cSt c

St c

P A P AT TE E

= + = + ( )1 1St c

St St c c

TP

E A E A

= +

( )s : 492.6Stst

P MPaA

= =

( )c : 4.9cc

P MPaA

= =

8) (), . ,

1 2 3, 2 , 3= = =A A A A A A . , a , 2a 3a , . , Fx b= , F . : ,

5 22.1 10 /StE E N mm= = , , 21.2A mm= , , 2.5F kN= , 1.5a m=

0.25m=A . : 1) . 2) , ( 2b a= ).

39 . : , . 1, 2007

1.8

. , , ( )x = .

1.8.1

', . '. x

0N dN dB NdN dB+ = = (1.83)

37 ( )A x

x .

1( ) , [ ]A x FL = = (1.84) dB ,

( )dB x dx= (1.85) , . (1.83), :

( )dN xdx

= (1.86)

. (1.84) (1.86) ( )N x ( )A x = x :

37 .. , 325 /kN m =

40 . : , . 1, 2007

( )dN A xdx

= (1.87)

,

0 0

0

( )

( ) (0) ( ) ; ( ) ( )

x x

x

dN A d

N x N V x V x A d

=

= =

x ,

( ) ( )B x V x=

( ) (0) ( )N x N B x= + (1.88) AV , , . . (1.88) x H= ,

( ) (0)(0) ( ) 0 ( )

N H F N BN B F

= = + = + < (1.89)

( ) ( ( ))N x F B B x= ( ) ( ( ))N x F V V x= (1.90) . :

0( ) (0) ( )

xdu u x u ddx

= = + (1.91) ( )u x . F A F, F.

41 . : , . 1, 2007

( ) , .(1.45),

NEA

= (1.92)

( )N N x= , , . .. . :

0

( )( )

h N xH dxEA x

= (1.93)

1.8.2

x ,

x BN 1A

= A (1.94)

x , :

N x B1A A

= = A (1.95)

:

0

0

(x) dxE

B x 1 B1 dxEA 2 EA

=

= =

A

A

A

AA

(1.96)

, , , , .

42 . : , . 1, 2007

1.8.3

, x , . F .

:

(1) (2) = = A A A (1.97) :

(1) (2)(1) (2)

1 2F N N (EA) (EA) = + = +A AA A

V v 1 2F (EA) , (EA) (EA) (EA) (Voigt)= = +AA (1.98)

V(EA) , Voigt38, . (1.98).

1.8.4

(1) (2) " . F .

:

(1) (2)N N F= = (1.99) :

38 (.. ). ( ), Voigt.

43 . : , . 1, 2007

(1) (2)(1) (1)

1 21 2

N N(EA) (EA)

= + = +A A A A A

1 2R

1 2

R 1 2

F ,(EA)

1 1 1 (Re uss)(EA) (EA) (EA)

= = +

= +

AA A A A

A AA A

(1.100)

, 1 2= +A A A , R(EA) , Reuss39, . (1.100), .

: (1) , (2) . .

T . ( ( ) 22.52 /( )StE N mm C = D . 1.8.5

, 0R R R(x)= . F. , R R(x)= , . x ,

N(x)A(x)

= (1.101)

39 Reuss.

44 . : , . 1, 2007

( )2 20A(x) R (x) R= (1.102)

. ( 0) = = > (1.103) . (1.101) ,

N A(x) dN dA = = (1.104)

N dN dB N 0+ + = (1.105) . (1.104) (1.105)

dA dB 0 + = (1.106) dB

dB A(x)dx= (1.107) :

dA Adx 0 + = dA dxA = (1.108)

:

45 . : , . 1, 2007

H 0 , [H] L= > = (1.109)

:

dA dxA H

= (1.110)

y ln z= , y 1/ z = , dzdyz

=

. (1.110)

dA dx xln A c , c .A H H

= = + = (1.111)

lnx cA He e+=

xH

0A A e= (1.112) x 0= , :

20 0, [ ]

FA A L= = = (1.113)

H . (1.112) :

01(0)2 (0)

A xR RR H + (1.114)

:

10.h m= , 0 0.05R m= , 325 /kN m = , 1MPa = , 100F kN=

: ( ) ( )( )2 2 20(0) 0.2 0.2 0.05m =0.1178R m A m m= = : 2

0

100 848.8 0.850.1178

F kN kPa MPaA m

= = = = <

46 . : , . 1, 2007

: 3

2

1 10 40.25 /

kPaH mkN m

= = =

20( ) exp 0.12m exp 40x xA x AH m

= =

R 0.2m+0.0023 x : 10. 0.225x m R m= =

1.8.6

. , ( )N x , . : () ,

( )dN n xdx

= (1.115)

() ,

dudx

= (1.116)

() -

N TEA

= + (1.117)

(1.115) . ( )N x ( )x . (1.115) (1.117) () :

( )( ) ( )d du dEA x n x E Tdx dx dx

= + (1.118)

47 . : , . 1, 2007

h , . 40 .

( )d x x ,

( )1 1 2 xd D D D h= x ,

( ) 21 1 24xA D D Dh

=

. ,

( ) 21 1 2( ) 4xn A x D D Dh

= =

, . (1.115),

0( ) ( ) (0) ( )

xdN n x N x N n ddx

= = + , ( 0)A x = . , . (1.118). . (1.118) ,

40 , . , .

48 . : , . 1, 2007

( ) ( )d duE A x n xdx dx =

,

( ) ( )2 1 21 1 2 1 1 22 2 0D Dx d u du xD D D D D Dh h dx E hdx + =

,

( ) 21

1 1 , 0 1Dxh D

= < = < (1.119)

uvh

= (1.120)

,

2

2 2 0d v dv

dd + + = (1.121)

,

( )1

1h

E = (1.122)

. (1.121) ,

21 2

1 16

v C C = + (1.123)

(1.119) (1.120). , ,

0 : 0 1 , 0x u v= = = =

: 0 , 0x h u v = = = = ,

49 . : , . 1, 2007

( )2 2 21 1 16v = + + + + (1.124) ( )u x . (1.120) ( )x . (1.116). , Hooke,

( )N EA = .

1) :

1 20.50 , 0.2 , 3.5D m D m h m= = = , 325 /kN m = 230 /E kN mm= .

, , . : .

2) ( 0x = ) 1 50T C= D , ( x h= ) 2 20T C= D

50 . : , . 1, 2007

1.9

1) , (2) 2 mm. , ; GPa200= 23cm= . : ! ( ).

2) (1) (2) . ().

:

31 SS = , 42 SS = , 212 SS = ,

=aS17

3) . .

( )2 PLu 3 33= , ( ) PLv 3u 2 3 3= =

( )1 2S 3 3 P3= , ( )2

2

3 1 3S P

2

=

35 3 6S P

3=

51 . : , . 1, 2007

4) . .

:

Fu L= , 0v = , 4F L

a =

6) (3). .

:

: 1 3 2S S 2 S 2= =

: u

v =

: 3ua

+ = , 2 va = , 1 u v2a

=

:

3 33 3

S uS (EA)EA a

= = =

52 . : , . 1, 2007

2 22 2

S vS (EA)EA a

= = =

1 11 1

S 1 u vS (EA)EA 2 a

= = =

( ) ( ) ( )u v u 2 v 2 v2 2 2 1 = = = + , ( )2 2 1u 2 2 1+= +

( )3 1S (EA)a2 2 1 = + , ( )2 1S (EA)a2 2 1 = + , ( )1 1S (EA)a2 2 = +

6) . .

:

1dS F

c d= + , 4

cS Fc d

= + , 2S F= , 3S F=

53 . : , . 1, 2007

7) , mm1 . , . , , =200GP 21cm= . , =100GP 22cm= .

:

1 2S S= , 3 1S S 2=

:

4 5S S= , 3 5S S 2=

:

1 2( )

10 = =

4 5( )

4 = =

( )3 0.001m ( ) ( ) / 3 = :

1 2 4 5S S S S 2.377k= = = = , 3S 3.36k=

1 2 11.8 Pa = = , 4 5 23.6 Pa = = , 3 33.6 Pa =

54 . : , . 1, 2007

8) ( )0 a < 0.

21 2= 12 2= . :

: v 2v = : 1 22S S=

: 1va

= , 2 va=

55 . : , . 1, 2007

: 111

S = , 22

2 =S

: 1 1 21 11 1 1 1 1

2S S = + = + = +

+=+=+=

11

2

22

22

2

22

SS

: 21 1

2Sv a = + , 2

1 1

Sv a = +

2 2 1 12

1 1 1 1

225

S S S + = + =

1 125

= , 2 215 =

10) , (1) . (2) (3) 2 3=A A , (1) 1A . () . .

:

: 132 SSS == :

1 11

1

Sv = = + = + A

2 22 3

2

Sv / 2 = = = = A

:

56 . : , . 1, 2007

12

1 2

S ( )2

= +A

A A ,

11 2

1 2

S S ( )2

= = +A

A A

11) (1) 1 (0 )= <

57 . : , . 1, 2007

:

: 2 10 / 2 0xF S S F+ = + = (1)

02/0 43 ==+ FSSFy (2)

: ul

SSlu

1

111

11

1

11

=== (3)

2 2 22 2

2 2 2 2

Sv S vl l

= = = (4) 3 3 3

3 33 3 3 3

Su S ul l

= = = (5)

4 4 44 4

4 4 4 4

Sv S vl l

= = = (6)

(1)+(3)+(5) 3 31 11 3

02

Fu ul l

+ + =1

3 31 1

1 32Fu

l l

= +

(2)+(4)+(6) 2 2 4 42 4

02

Fv vl l

=1

2 2 4 4

2 42Fv

l l

= +

11 1 3

33 3 1

12

lFSl

= + ,

14 4 2

22 2 4

12

lFSl

= + ..

: ( ) ( )4321 ,,,min,max llllu

58 . : , . 1, 2007

13) (1) (2) .

:

:

1 230 2 02

A aWa F S a S a = + + + = 022

321 =+++ SSFW (1)

: 2v v = :

1 11 1

v S S vh h

= = = = (2)

2 22 2

2v S S v vh h h

= = = = = (3)

(1)+(2)+(3): 3 22 02FW v v

h h + = 3

2 5F hv W = +

+=2

351

1FWS , 12 2SS =

59 . : , . 1, 2007

14) , . .

:

: 03320 321 =+++= aFaSaSaS 0332 321 =+++ FSSS (1)

:

2v v = , 3v v = :

1 11 1

v S S vh h

= = = = (2)

2 22 2

2v S S v vh h h

= = = = = (3)

2 23 3

2 62

v S S v vh h h

= = = = = (4)

(1)+(2)+(3)+(4): 2 62 3 3 0v v v Fh h h + = 3

23Fhv =

233

1FS = ,

236

2FS = ,

2318

3FS =

60 . : , . 1, 2007

15) . . (1), (2) (3).

:

: 2 3 10 2 2 0S a S a F a S a = + + =

2 3 12 2 0S S F S+ + = (1) :

v v = , 2v v = :

1 11 1

v S S vh h

= = = = (2)

2 22 2

2 2/ 2

v S S v vh h h

= = = = = (3)

3 33 3

2 4/ 2

Sv S v vh h h

= = = = = (4)

(1)+(2)+(3)+(4): 2 42 2 0v v F vh h h + =

= 112 lF

112

1FS = ,

114

2FS = ,

118

3FS = ( 1S , 2S 3S )

61 . : , . 1, 2007

16) 2 . . (1) (2) 2. . .

:

: 1 2 1 20 2 0 2 0S b S b S S = + = + = (1)

: 2v v = :

1 11 12 2

v S S vl l

= = + = = (2)

2 22 22 2 2 2

v S S vl l

= = + = + = (3)

(1)+(2)+(3): 2 2 2 2 02

v vl l

= 9 105 02 9v lvl

= =

== 94

95

11 SS

== 922

920

22 SS

( =1S , =2S , : =2200 1 mmCC )

62 . : , . 1, 2007

17) . , . , 2mm . . GPa200= . 23cm 22cm . .

:

: 1 20 8 5 100 4 20 8 0S m S m m m = + + + =

31 28 5 (400 160) 10 0S S+ + + = 31 28 5 560 10 0S S + + = (1)

: == uu

uu

8558 (2)

: ( )muu ,, ( ) , 21 SS 1 1 1

1 4 2 9 214 2 10 200 10 /u S Sm m m

= = = =

7117 104 4 10

u S S u = = (3)

63 . : , . 1, 2007

32 2 2

2 4 2 9 22

2 101 3 10 200 10 /

u m S Sm m m

= = = =

( )3 72 2 10 6 10S u = (4) (3)+(2): 8

5u u = , 71 8 105S u= (5)

(1)+(4)+(5):

( )7 3 7 488 10 5 2 10 6 10 56 10 05 u u + + = 37.3 10 7.3u m mm = =3

1 117 10 117S k = = , 3 72 (2 7.3) 10 6 10 318S k= =

18) , 500T C = D , . . . : ) GPa100= , Ca 06 /10= ,

220cm= . ) , , : GPa200= , 210cm= . , : GPa100= , 220cm= . : Ca 06 /10= .

:

. .

: 41 SS = , 222 11

2 SSS ==

64 . : , . 1, 2007

: 53 SS = , 222 33

2 SSS ==

=

===212

5431

SS

SSSS (1)

:

1 11 4

/ 22.5 2

S vm

= = + = 1 5vS = ( ) v m (2)

3 33 5

/ 21 2

S vm

= = + =

= 53S ( ) v m (3)

2 22 1.5

S v vm

+= + = 2 1.5v vS + = (4)

(1)+(2)+(4): 21.5 5

v v v + = ( )1.5 5 2 0v v v + + =

( ) =++ 5.7525 (5) (1)+(2)+(3):

5 2v v = 2 5v v = (6)

(5)+(6): ( )5 2 2 7.5v v a + + =

637.5 7.5 10 500 0.411 10 0.411

7 2 7 2av m mm

= = = =+ +

30.164 10 0.164v m mm = = :

34 9

1 3 4 5 20.411 1010 10 200 10 16.44

5mS S S S m k

m

= = = = =

= kS 2.232

65 . : , . 1, 2007

v b+

2bu +

v b

2bu +

v b

2bu

19) , , . . (2) T . .

:

3: , ,u v . :

:

:

:

66 . : , . 1, 2007

v b

2bu

:

:

1: 1 11/ 2

2Su b u

b b += = + = = (1)

2: 2 22 2 2 2Su b u

b b = = = + = + (2)

3: 3 331 / 2 3

2 42 2 2Sv b u b u v

bb + = + = = = (3)

4: 4 441 / 2 3

2 42 2 2Sv b u b v u

bb + = = = = (4)

:

341 1 4 30 : 0 22 2x

SSF S S S S+ = + = + (5)

1S

2S

2b

y

67 . : , . 1, 2007

3 42 2 3 40 : 0 2 02 2y

S SF S S S S+ = = + + = (6)

1 2 1 20 : 2 0 52 2b bS S b S S = + + = = (7)

(1)+(2)+(3): 3 22 2S S= (5)

24 23 SS = (6) (5), (6), (7) (1), (2), (3) (4):

52 2 2

u vb b

+ =

3 2 22 4 2 2

v u vb b

+ =

3 3 22 4 2 2

v u vb b

=

:

u Xab= , v Ya

b= , Za = ZYX ,, :

5 3 52

X Y Z+ + =

( ) 31 2 2 2 2 4 22X Y Z + + + = ( ) 31 3 2 3 2 6 22X Y Z + + + =

68 . : , . 1, 2007