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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