, 2009

:: . ...

i

5 10

ii

.............................................................. 1 1: ................................................. 5 1.1 .................................................................................... 6 1.1.1 .................................................................................. 6 1.1.2 ............................................................................. 8

1.1.2.1 .............................................. 8 1.1.2.2 ................................... 11 1.1.2.3 .............................................. 14 1.1.2.4 ...................................... 15 1.1.2.5 ........................16

1.2 .................................................................................................. 17 1.2.1 ..................................................................... 17 1.2.2 Cauchy............................................................... 18 1.2.3 .................................................................................. 19 1.2.4 Piola Kirchhoff............................21 1.2.5 ................................................................................ 23 1.2.6 Piola Kirchhoff......................... 26 1.2.7 ......................................................28 1.2.8 Cauchy ....................................... 29

2: ............................................ 30 2.1 .................................. 31 2.2 ......................................................................................32

3: ............................................................... 37 3.1 ..................................................................................................................38 3.2

.....................................................................................................43

iii

3.2.1 ......................................................................................43 3.2.2 .....................................................................................43 3.2.3 ............................................................................... 47 3.2.4 ............................................................................................... 49 3.2.5 -

....................................................... 51 3.2.6 ( )mu x,t , ( )x x,t - .................................................................... 61 3.2.7

.................................................................................67 3.2.8

................................................................. 70

4: .... 72 4.1 ( )mu x,t , ( )x x,t ..73 4.2.1

PS .................................................................................................................... 80 4.2.2

SS ......................................................................................................................80 4.2.3

, , , , , , , , , , , , , P PP n t S y z yy zz yz w wy wzA I I I I C S S I I I A A A ..................................85

5: ..................... 87 5.1 ........................................................................................................... 88 5.2 .....................................................................................88 5.2.1 1 .............................................................................. 88 5.2.2 2 .............................................................................. 95 5.2.3 3 .............................................................................. 99 5.3 ................................................................................................. 104 ......................................................................................... 106

2

, , . Coulomb (1784) . . St. Venant (1855) .

St. Venant , , ( ). (). , . . / . - St. Venant- . , .

: , St. Venant. . Wagner (1929), Kappus (1937) Marguerre (1940) Benscoter (1954) Vlasov (1963) St. Venant.

3

, . Schulz & Filippou (1998), Ladevze & Simmonds (1998) Sapountzakis & Mokos (2003) , .

, . , . Young (1807), . Weber (1921) Cullimore (1949). , Attard (1986), .

, . Gere (1954) . Petersen (1991) . Sapountzakis (2005) .

30 . , - , - - . Rozmarynowski & Szymczak (1984) ( - ). .

, . , . -

4

, . , , , ( Pai & Nayfeh, 1994) .

. , 3, . . , . , . . . , , , . , , .

. . , , , . , , . , 2009

1 :

6

1.1

Novozhilov (1953), Bathe (1996) (1998).

1.1.1

, t 0= , Q . 1 2 3Ox x x . P 1x , 2x , 3x . t , Q. P,

( )1 2 3P , , ,t . P :

( )1 1 1 2 3x ,x ,x ,t = (1.1.1) ( )2 2 1 2 3x ,x ,x ,t = (1.1.1) ( )3 3 1 2 3x ,x ,x ,t = (1.1.1)

1 , 2 , 3 . . , 1-1 P P , . , :

( )1 1 1 2 3x x , , ,t = (1.1.2) ( )2 2 1 2 3x x , , ,t = (1.1.2) ( )3 3 1 2 3x x , , ,t = (1.1.2)

. (1.1.1) Lagrange, . (1.1.2) Euler. Lagrange ( )1 2 3P x ,x ,x

. Euler ( )1 2 3P , , t .

1 :

7

t .= , . P

1 2 3u u u= + + 1 2 3u e e e (1.1.3) , ,1 2 3e e e 1Ox , 2Ox ,

3Ox 1u , 2u , 3u

1 1 1u x= (1.1.4) 2 2 2u x= (1.1.4) 3 3 3u x= (1.1.4)

( )1 1 1 2 3u u x ,x ,x= , ( )2 2 1 2 3u u x ,x ,x= , ( )3 3 1 2 3u u x ,x ,x= 1u , 2u , 3u . . 1-1 P, P,

1 1 1

1 2 3

2 2 2

1 2 3

3 3 3

1 2 3

x x x

J 0x x x

x x x

=

1 1 1

1 2 3

2 2 2

1 2 3

3 3 3

1 1 1

u u u1x x x

u u uJ 1 0x x xu u u1x x x

+ = + +

(1.1.5)

t 0= , ,

1u 0= , 2u 0= , 3u 0= . ( )J t 0 1= = (1.1.6)

J , . (1.1.5), (1.1.6) J 0> , t 0 (1.1.7) . (1.1.7) 1-1 P, P / .

[ ]X . (1.1.5), J , (deformation gradient).

1 :

8

1.1.2

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

1.1.2.1

PA , ds . P 1 2 3P( x ,x ,x ) A 1 1 2 2 3 3A( x dx ,x dx ,x dx )+ + + .

( ) ( ) ( )2 2 21 1 1 2 2 2 3 3 3ds x dx x x dx x x dx x= + + + + + 2 2 2 2

1 2 3ds dx dx dx = + + (1.1.8) , PA , P A ,

( )1 2 3P , , ( )1 1 2 2 3 3A d , d , d + + + . PA . P A

( ) ( ) ( )2 2 21 1 1 2 2 2 3 3 3ds d d d = + + + + + 2 2 2 2

1 2 3ds d d d = + + (1.1.9) (1.1.2), (1.1.4), :

1 1 1 1 1 11 1 2 3 1 1 2 3

1 2 3 1 2 3

u u ud dx dx dx d 1 dx dx dxx x x x x x = + + = + + +

(1.1.10)

2 2 2 2 2 22 1 2 3 2 1 2 3

1 2 3 1 2 3

u u ud dx dx dx d dx 1 dx dxx x x x x x = + + = + + +

(1.1.10)

1 :

9

3 3 3 3 3 33 1 2 3 3 1 2 3

1 2 3 1 2 3

u u ud dx dx dx d dx dx 1 dxx x x x x x = + + = + + +

(1.1.10)

( )

2 2 2 2 211 1 22 2 33 3

12 1 2 13 1 3 23 2 3

1 ds ds dx dx dx2

2 dx dx 2 dx dx 2 dx dx

= + + ++ + +

(1.1.11)

. (1.1.11) (strains)

2 2 231 1 2

111 1 1 1

uu u u1x 2 x x x

= + + +

(1.1.12)

2 2 232 1 2

222 2 2 2

uu u u1x 2 x x x

= + + +

(1.1.12)

2 2 23 31 2

333 3 3 3

u uu u1x 2 x x x

= + + +

(1.1.12)

3 32 1 1 1 2 212

1 2 1 2 1 2 1 2

u uu u u u u u1 12 x x 2 x x x x x x

= + + + + (1.1.12)

3 3 31 1 1 2 213

1 3 1 3 1 3 1 3

u u uu u u u u1 12 x x 2 x x x x x x

= + + + + (1.1.12)

3 3 32 1 1 2 223

2 3 2 3 2 3 2 3

u u uu u u u u1 12 x x 2 x x x x x x

= + + + + (1.1.12)

Green

11 12 13

21 22 23

31 32 33

= G (1.1.13)

ij Green . (1.1.12). 21 , 31 , 32 . (1.1.12--) .

21 12 = (1.1.14) 31 13 = (1.1.14) 32 23 = (1.1.14)

1 :

10

T = G G (1.1.15) Green .

PA (magnification factor of the extension of line element PA ),

2

A 21 dsMF 12 ds

= (1.1.16)

PA , 11dxnds

= , 22 dxn ds= , 3

3dxnds

= , , 2ds . (1.1.11), AMF

2 2 2A 11 1 22 2 33 3 12 1 2 13 1 3 23 2 3MF n n n 2 n n 2 n n 2 n n = + + + + +

(1.1.17) AMF PA ( Ae )

( )A Ads dse ds 1 e dsds = = + (1.1.18)

( )2 2 22 2 2 AA 2 2 2

1 e ds ds1 ds 1 ds ds 1MF 12 2 2ds ds ds

+ = = =

( )2A A A A1MF 1 e 1 e 1 2 MF 12 = + = + (1.1.19) . , 1-1 , ds 0 > . . (1.1.18) Ae 1> . (1.1.19) A

1MF2

> . . (1.1.19) Taylor

2A A A

1e MF MF ...2

= + (1.1.20)

. (1.1.20) AMF : AMF 1

1 :

11

AMF () PA . , . (1.1.12), (1.1.12), (1.1.12). , 1Ox , 1n 1= , 2n 0= , 3n 0= . , . (1.1.17) 2Ax1 11 1 Ax1 11MF n MF = = , . (1.1.20)

2Ax1 11 11

1e ...2

= + (1.1.21) , ( )211 111 0

1 :

12

31 1 1 1 2 1 1 1 1 11 2 3

1 2 3 1 2 3

dxd u dx u dx u d u u u1 1 n n nds x ds x ds x ds ds x x x = + + + = + + +

(1.1.26)

32 2 1 2 2 2 1 2 2 21 2 3

1 2 3 1 2 3

dxd u dx u dx u d u u u1 n 1 n nds x ds x ds x ds ds x x x = + + + = + + +

(1.1.26)

3 3 3 3 3 3 3 3 31 21 2 3

1 2 3 1 2 3

d u u u dx d u u udx dx 1 n n 1 nds x ds x ds x ds ds x x x = + + + = + + +

(1.1.26) . (1.1.18)

( ) ( 1.1.19 )AA A

ds 1 ds 1ds 1 e dsds 1 e ds 1 1 2 MF 1

= + = = + + +

A

ds 1ds 1 2 MF

= + (1.1.27) . (1.1.26), (1.1.27) ()

11 1

31 2A1 1 2 3

A A A

uu u1xx xn n n n

1 2 MF 1 2 MF 1 2 MF

+ = + + + + + (1.1.28)

22 2

31 2A2 1 2 3

A A A

uu u1xx xn n n n

1 2 MF 1 2 MF 1 2 MF

+ = + + + + + (1.1.28)

33 3

31 2A3 1 2 3

A A A

uu u 1xx xn n n n

1 2 MF 1 2 MF 1 2 MF

+ = + + + + + (1.1.28)

12 , 13 , 23 2 PA , PB . , , ( P A , P B ).

cos 1 1 cos cos = = = A B A B A Bn n n n n n A1 B1 A2 B2 A3 B3cos n n n n n n = + + (1.1.29)

1 :

13

: PA , PB 1Ox , 2Ox . ( ) ( )A1 A2 A3n ,n ,n 1,0,0= , ( ) ( )B1 B2 B3n ,n ,n 0,1,0= , A 11MF = ,

22MF = 90 = . . (1.1.28)

1

1A1

11

u1xn

1 2

+ = + , 1

2B1

22

uxn

1 2

= + (1.1.30)

2

1A2

11

uxn

1 2

= + ,

2

2B2

22

u1xn

1 2

+ = + (1.1.30)

3

1A3

11

uxn

1 2

= + ,

3

2B3

22

uxn

1 2

= + (1.1.30)

. (1.1.29) , :

3 32 1 1 1 2 2

1 2 1 2 1 2 1 2

11 22

u uu u u u u ux x x x x x x xcos

1 2 1 2

+ + + + = + + (1.1.31) 12 (. (1.1.12)),

12

11 22

2cos1 2 1 2

= + + (1.1.32)

: , . (1.1.19), (1.1.21)

1111

11 11 2

1 :

14

12 2 1Ox , 2Ox . ij ( i j ) () 2 iOx jOx .

ij ij2 = , i j (1.1.34) ij () .

1.1.2.3

, () . 1Ox , 2Ox , 3Ox . 1 2 3dV dx dx dx= . dV , : ( )T 1dx ,0,0=OA O

( ) 31 21 2 3 1 1 11 1 1

d ,d ,d dx , dx , dxx x x

= = O (1.1.35) (1.1.10) 3 . 2 , ( )T 20,dx ,0=OB ( )30,0,dx =O

( ) 31 21 2 3 2 2 22 2 2

d ,d ,d dx , dx , dxx x x

= = O (1.1.35)

( ) 31 21 2 3 3 3 33 3 3

d ,d ,d dx , dx , dxx x x

= = O (1.1.35) dV

( )dV = O O O (1.1.36)

1 :

15

. , dV J dV = (1.1.37) J (. (1.1.5)).

1.1.2.4

dA n . dA n . ,

( )1 2 3dx ,dx ,dx =1n .

( )dV dA= 1n n (1.1.38) ( )

1n , (, ) .

( )dV dA = 1n n (1.1.39)

( ) ( )1 2 3d ,d ,d =1n . . (1.1.5) (1.1.10) [ ] () .

[ ] = 1 1n n (1.1.40) . (1.1.37)

( ) ( ) ( 1.1.40 )dV J dV dA J dA = = 1 1n n n n [ ] ( ) ( )dA J dA = 1 1 n n n n (1.1.41)

1n ,

[ ] ( )dA J dA = n n (1.1.42)

1 :

16

.

1.1.2.5

Green . ,

ij 1

1 :

17

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

1.2

1.2.1

, . , .. 1 1t t= .

() 1t t= . 1Q , 2Q

1d A

( )1 2 3P , , , 1n 1Q

1 n 2Q . (traction vector) ( )1 1t n 1n , P ,

( ) 1 11 1 1d A 0 dlim d A= pt n (1.2.1) 1d p ()

1d A . ,

1Q ( 1d A , P )

2Q . (1.2.1) ( ) ( )1 1 1 1= t n t n (1.2.2)

Cauchy.

1 :

18

, . . , , .

(traction forces) , (body forces), . , . . 1d V ( )1 2 3P , , ,

1

11

1d V 0

dlimd V

= fpf (1.2.3) 1d fp

1d V . , 1f ( 1t ).

1.2.2 Cauchy

. 1d A 1Ox .

( )1 T T 1,0,0= =1n e , 1e 1Ox .

11 j , j 1,2,3=

( )1 1t e ,

( ) ( )T1 1 1 111 12 13, , =1t e (1.2.4) , 2 . ( )1 11 12 13 = + + 1 1 2 3t e e e e (1.2.5) ( )1 21 22 23 = + + 2 1 2 3t e e e e (1.2.5)

1 :

19

( )1 31 32 33 = + + 3 1 2 3t e e e e (1.2.5) Cauchy (true or Cauchy stress tensor) 1

11 12 13

121 22 23

31 32 33

= (1.2.6)

ij . ij , i 1,2,3= , j 1,2,3= jOx

1d A () ie . Cauchy ( ).

1.2.3

, Cauchy . () ( )1 1t n Cauchy 1 , 1e , 2e , 3e 1n . , , , 1 1 1 1

1 11 1 21 2 31 3t n n n = + + (1.2.7) 1 1 1 1

2 12 1 22 2 32 3t n n n = + + (1.2.7) 1 1 1 1

3 13 1 23 2 33 3t n n n = + + (1.2.7)

( ) ( )T1 1 1 1 11 2 3t , t , t=t n ( )1 T 1 1 11 2 3n , n , n=n . . (1.2.7)

( ) T1 1 1 1 = t n n (1.2.8)

1 :

20

.

1V ( 1t t= ), 1 1A V= . 1V , .

1t ( 1A ) 1f ( 1V ) : ( )1 1

1 1 1 1 1

A Vd A d V+ = t n f 0 (1.2.9)

. (1.2.8)

1 1

T1 1 1 1 1

A Vd A d V + = n f 0 (1.2.10)

Gauss (Gauss divergence theorem) , . (1.2.10)

1 1 1

T T1 1 1 1 1 1 1

V V Vd V d V d V + = + = div f 0 div f 0 (1.2.11)

1V , . (1.2.11)

T1 1 + = div f 0 (1.2.12) . (1.2.12) . :

13111 211

1 2 3f 0

+ + + = (1.2.13) 13212 22

21 2 3

f 0 + + + = (1.2.13)

113 23 333

1 2 3f 0

+ + + = (1.2.13) ( 1 ,

1 :

21

2 , 3 ) ( 1x , 2x , 3x ). , 1d , 2d , 3d .

K : ( )1 1

1 1 1 1 1

A Vd A d V + = t n KE f KB 0 (1.2.14)

, 1E A 1B V 1V . . (1.2.9), (1.2.13), (1.2.14) ,

T1 1 = (1.2.15) Cauchy .

21 12 = (1.2.16) 31 13 = (1.2.16) 32 23 = (1.2.16)

.

1.2.4 Piola Kirchhoff

1 Piola Kirchhoff ( t 0= ). . (1.2.9) 1t t=

( )1 1 0 0

1 11 1 1 1 1 1 0 1 0

0 0A V A V

d A d Vd A d V d A d Vd A d V

+ = + = t n f 0 t f 0 (1.2.17) , , ( )1 00 t n 10f

( ) 0 11 00 0d A 0 dlim d A= pt n (1.2.18)

1 :

22

0

110 0d V 0

dlimd V

= fpf (1.2.18) (1.2.1) (1.2.18)

( ) ( ) ( ) ( ) 11 0 0 1 1 1 1 0 1 10 0 0d Ad A d A d A= =t n t n t n t n (1.2.19) (1.2.3) (1.2.18)

11 10 0

d Vd V

=f f 1 10 J=f f (1.2.19) . (1.1.27) 1 0d V Jd V= . (1.2.17) ( )0 0

1 0 0 1 00 0

A Vd A d V+ = t n f 0 (1.2.20)

(1.2.19)

( ) ( ) ( )1 1T1 0 1 1 1 0 1 10 00 0d A d Ad A d A = = t n t n t n n (1.2.21) . (1.1.42) ,

( ) T1 0 1 1 00 J = t n n ( )1 0 1 1 00 J = t n n (1.2.22) Cauchy .

1 Piola Kirchhoff 10 P

1 1 10 J

= P (1.2.23) . (1.2.22) 1 Piola Kirchhoff 0n () 0d A ( )1 00 t n , ( )1 0 1 00 0 = t n P n (1.2.24) . (1.2.20) 10 P

1 :

23

0 0

1 0 0 1 00 0

A Vd A d V + = P n f 0 (1.2.25)

Gauss ( ) ( )0 0 0

1 0 1 0 1 1 00 0 0 0

V V Vd V d V d V + = + = div P f 0 div P f 0 (1.2.26)

0V , () ( )1 10 0 + = div P f 0 (1.2.27) , ( 10 ijP )

11311 120 1

1 2 3

PP P f 0x x x

+ + + = (1.2.28) 12321 220 2

1 2 3

PP P f 0x x x

+ + + = (1.2.28) 131 32 330 3

1 2 3

P P P f 0x x x

+ + + = (1.2.28) 1 Piola Kirchhoff Lagrange. . (1.2.23) ( Cauchy) .

1.2.5

2 Piola Kirchhoff , .

. , .

1 :

24

1t t= . 1V , (. (1.2.21)):

( )1 1

T1 1 1 1 1 1

V Vd V d V + = + = div f 0 div f 0 (1.2.29)

( )

( ) ( )T1 1 1 11 2 3u , u , u =r (1.2.30) , (1.2.29)

( )1

T1 1 1 1

Vd V 0 + = div f r (1.2.31)

( ) (calculus of variations). (1.2.31) .

( ) T T1 1 1 1 1 1 1 = + div r div r tr r (1.2.32) . (1.2.31) :

( )1

T1 1 1 1 1 1 T 1 1

Vd V 0 + = div r tr r f r

( ){ }1 1

T1 1 1 T 1 1 1 1 1 1

V Vd V d V + = div r f r tr r (1.2.33)

Gauss ,

( )1 1 1

T T1 1 1 1 1 T 1 1 1 1 1 1

A V Vd A d V d V + = n r f r tr r

(1.2.34) (1.2.8)

1 :

25

( )1 1 1

T T1 1 1 1 1 T 1 1 1 1 1 1

A V Vd A d V d V + = t n r f r tr r (1.2.34)

, (spatial virtual work equation), (spatial external virtual work) 1 extW (spatial internal virtual work) int

1W . ,

. , 1 , 2 , 3 ( Euler).

1 1 r

1 1 11 1 1

1 2 31 1 1

1 1 2 2 2

1 2 31 1 1

3 3 3

1 2 3

u u u

u u u

u u u

=

r (1.2.35)

, , . . (1.2.34)

T1 1 = ,

T1 1 1 31 211 22 33

1 2 3

3 31 2 1 212 13 23

2 1 3 1 3 2

uu u

u uu u u u

= + + + + + + + +

tr r

(1.2.36)

.

( )int1

T1 1 1 1

VW d V = tr (1.2.37)

1 :

26

1 1t t=

31 1 2 1

1 2 1 3 1

1 31 2 2 2

2 1 2 3 2

3 3 31 2

3 1 3 2 3

uu u u u1 12 2

uu u u u1 12 2

u u uu u1 12 2

+ + = + + + +

(1.2.38)

1 , 1 (work conjugate) () .

1.2.6 Piola Kirchhoff

1x , 2x , 3x , Piola Kirchhoff . , (1.2.34) 1.2.4 :

( )0 0 0

T T1 0 1 0 1 T 1 0 1 1 00 0 0

A V Vd A d V d V + = t n r f r tr P (1.2.39)

int

1W ,

int0

T1 1 1 00

VW d V = tr P (1.2.40)

10 P 1 .

Piola Kirchhoff, , Green 10 G . Green ( ) Green . , , (

1 :

27

) ( ), .

Piola Kirchhoff, 0d p

1d p

10 1 1d d = p p (1.2.41)

(1.2.24) (1.2.18)

10 1 1 0 00d d A

= p P n (1.2.42) 10 P (. (1.2.23))

10 1 1 1 0 0d J d A = p n (1.2.42)

, 2 Piola Kirchhoff 10 S 0 0d An

0d p .

11 1 1 10 J

= S 11 1 1

0 0 = S P (1.2.43-)

, 10 S ( 10 P ),

T1 10 0 = S S (1.2.44)

. (1.2.40)

[ ]1 1 10 12 =

G I (1.2.45)

[ ]I 3 3 , :

1 :

28

int0

T1 1 1 00 0

VW d V =

Gtr S (1.2.46)

10 S , 10 G ,

( )0 0 0

T T1 0 1 0 1 T 1 0 1 1 00 0 0 0

A V Vd A d V d V + =

Gt n r f r tr S (1.2.47)

10 S . ( )1 00 t n ext

1W 1 Piola Kirchhoff 10 P 10 S (1.2.24).

1.2.7

Cauchy . 1 11 ()

1d A 1Ox

1d A . 1 Piola Kirchhoff

. 10 11P 1Ox

1d A 0d A . 0d A 1Ox

, 1d A .

2 Piola Kirchhoff . 1 Piola Kirchhoff . , 10 11S

() 1d A 0d A . 1Ox

1 :

29

, 1d A .

, 2 Piola Kirchhoff ( ) , Cauchy 1 Piola Kirchhoff . 2 Piola Kirchhoff .

1.2.8 Cauchy

, . , ( 1x , 2x , 3x ) ( 1 , 2 , 3 ) , Cauchy . , (1.2.13)

3111 210 1

1 2 3f 0

x x x + + + = (1.2.48)

3212 220 2

1 2 3f 0

x x x + + + = (1.2.48)

13 23 330 3

1 2 3f 0

x x x + + + = (1.2.48)

(. (1.2.7)) 0 1 11 0 1 21 0 2 31 0 3t n n n = + + (1.2.49) 0 2 12 0 1 22 0 2 32 0 3t n n n = + + (1.2.49) 0 3 13 0 1 23 0 2 33 0 3t n n n = + + (1.2.49) Cauchy , .

2 :

31

2.1

Novozhilov (1953). : 6 (. (1.1.12)

Green . (1.1.45) )

3 (. (1.2.13) Cauchy . (1.2.28) 1 Piola Kirchhoff)

( ) :

(

)

( )

t : 3 1u , 2u , 3u 6 11 , 22 , 33 , 12 21 = , 13 31 = ,

23 32 = 6 , 2 Piola Kirchhoff 11S ,

22S , 33S , 12 21S S= , 13 31S S= , 23 32S S= . [ ]X . , . 1.2, .

15 9 ,

. 6 ( ). . . .

2 :

32

. .

. . . , .

2.2

, , . , .

. () (ideally elastic) () . : ,

. .

, . , , .

,

, . . (hyperelastic) Green (Green elastic) ( )t t t t t t t t t0 11 0 22 0 33 0 21 0 12 0 31 0 13 0 32 0 23 1 2 3F F , , , , , ,x ,x ,x = = = = , Green ( )1 2 3x ,x ,x , : ( )t t t t t t t 00 11 0 22 0 33 0 12 0 13 0 23 1 2 3d W F , , , , , ,x ,x ,x d V = (2.2.1) 0 1 2 3d V d x d x d x = , t0 ij

2 :

33

Green td W () () 0d V . F (strain energy function per unit initial volume). F ( )t t t t t t t 00 11 0 22 0 33 0 12 0 13 0 23d W F , , , , , d V = (2.2.2) , (homogeneous). . F (2.2.1) . (2.2.2) .

. (1.2.46) 2 Piola Kirchhoff Green ( )d , ( ) ( )t t t t t t t t t 00 11 0 11 0 22 0 22 0 23 0 23 0 32 0 32d d W S d S d ... S d S d d V = + + + + (2.2.3) . (2.2.2) ( )d ,

( )t t t t t 00 11 0 22 0 23 0 32t t t t0 11 0 22 0 23 0 32

F F F Fd d W d d ... d d d V = + + + +

(2.2.4) . (2.2.3), (2.2.4) t

0 ij t0 ij

FS = , i, j 1,2,3 = (2.2.5-)

6 (2.2.5-) . F (15 15 ).

F ,

2 :

34

( ) ( )( )

3 3 3 3 3 3t t t

ij 0 ij ijkl 0 ij 0 kli 1 j 1 i 1 j 1 k 1l 1

3 3 3 3 3 3t t t

ijklmn 0 ij 0 kl 0 mni 1 j 1 k 1l 1 m 1 n 1

F A B

C ...

= = = = = =

= = = = = =

= + +

+ +

(2.2.6)

ijA , ijklB , ijklmnC , . . (2.2.6)

t0 ij 0 td W . , . (2.2.5) , . . (2.2.6) . (2.2.5)

( ) ( )3 3 3 3 3 3t t t t0 rs rsij 0 ij rsijkl 0 ij 0 kli 1 j 1 i 1 j 1 k 1 l 1

S A B C ... = = = = = =

= + + + (2.2.7) , . (2.2.7) A 0= (2.2.8)

( ) ( )3 3 3 3 3 3t t t t0 rs rsij 0 ij rsijkl 0 ij 0 kli 1 j 1 i 1 j 1 k 1 l 1

S B C ... = = = = = =

= + + (2.2.9)

, t0 ij 1

2 :

35

eD : ijd , 6 6 . ijd .

Hooke (generalized Hookes law). Hooke

= t e t0 0 D (2.2.11) :

( ) ( )t t t t t t0 11 0 22 0 33 0 12 0 13 0 23, , , , , =t0 : Cauchy. .

( ) ( )t t t t t t t t t0 11 0 22 0 33 0 12 0 12 0 13 0 13 0 23 0 23, , , 2 , 2 , 2 = = = =t0 : .

eD 6 6 36 =

ijk . , . , (anisotropic). 3 (orthotropic) , (isotropic). ijk .

. , , eD :

ij jik k= , i, j 1,2,3 = (2.2.12) 2

2 :

36

( )( )

( )

11 12 12

12 11 12

12 12 11

11 12

11 12

11 12

k k k 0 0 0k k k 0 0 0k k k 0 0 00 0 0 1 0 0k k

20 0 0 0 1 0k k

20 0 0 0 0 1 k k

2

=

eD

(2.2.13)

12k = (2.2.14) 11 12k k 2 = (2.2.14)

() Lam , (Lam constants). :

( )t t t t0 11 0 11 0 22 0 331 S S SE = + (2.2.15) ( )t t t t0 22 0 22 0 11 0 331 S S SE = + (2.2.15) ( )t t t t0 33 0 33 0 11 0 221 S S SE = + (2.2.15)

t t t0 12 0 12 0 12

12 SG

= = , t t t0 13 0 13 0 1312 SG = = , t t t

0 23 0 23 0 2312 SG

= = (2.2.15--) :

( )3 2E + = + : Young (elastic modulus,

Young s modulus) G = : (shear modulus) ( )2

= + : Poisson (Poisson s ratio) G, E,

( )EG

2 1 = + (2.2.16)

3 :

38

3.1

, 15 15 (3 , 6 , 6 ). , , . , .

: ,

. , , , . , .

, , .

St. Venant . : , , .

(, ). , St. Venant , , . , , .

(torsional loading) . : Mt (torque), . Mt

3 :

39

, . .

. 3.1.1, - : .

. 3.1.2, : , . , , ( ) . .

() () 3.1.1 () . () .

3 :

40

()

() ()

3.1.2 () . () . () .

Coulomb (1784), . . Coulomb . .

, . St. Venant (1855)

3 :

41

(. 3.1.3, 3.1.4). , .

3.1.3 .

t

()

3.1.4 Saint-Venant, .

St. Venant

St. Venant (uniform torsion, St. Venant torsion) (. 3.1.4). . St. Venant (St. Venant shear stresses, primary shear stresses). ( ) , (non uniform torsion). St. Venant (. 3.1.4, 3.1.5). St. Venant

3 :

42

(secondary shear stresses, warping shear stresses). , (. 3.1.6).

3.1.5 (,

) ().

(A)

(A)

dx

x+dx

x

x+dx

x

S2

k, Gk

k, Gk

j, Gj S1

S

() () ()

3.1.6 (, ) () .

St. Venant . (), . , , . , , ( ) . . .

3 :

43

3.2

3.2.1

. , , . : ,

( ). .

: . .

: . , . , - - () .

: , Green ij 1

3 :

44

2006, Timoshenko & Goodier, 1951) () . ) ) ( ), .

l , . = (. 3.2.1). E, G ( Poisson ) . C (centroids) . . S , S C . S . (center of gravity) . Sxyz , x , y z (. 3.2.1). x t ( )x x,t , .

n t

y

z

2

1

CS ()

s

P

q

( ) = +

()

3 :

45

x,u

z,w

l

CS y,v ( ),tm x t

( ),wm x t

: :

SC

( ),n x t ()

3.2.1 () ().

,

Sx . , , v w ( v Sy w Sz ) . (. 3.2.2): ( ) ( ) ( ) ( ) ( )x xv x, y,z,t PP sin MP x,t sin z x,t = = = (3.2.1) ( ) ( ) ( ) ( ) ( )x xw x, y,z,t PP cos MP x,t cos y x,t = = = (3.2.1)

2 0t =3 0t =

+1

()

1 2

11+= = Kj j

3u2u

x (x, t)

n

t

s

S

z

y

3.2.2

.

3 :

46

, , 2 3u u (Chen & Trahair, 1992): ( ) ( ) ( )( )x xv x, y,z,t z sin x,t y 1 cos x,t = (3.2.2) ( ) ( ) ( )( )x xw x, y,z,t y sin x,t z 1 cos x,t = (3.2.2)

,

, 1u . St. Venant, ( ), ( ) ( ) ( )Px Su x, y,z x y,z = (3.2.3)

( )PS y,z - - (primary warping function). S Sxyz . , . (3.2.3) ( ) ( ) ( ) ( ) ( )P Sm x S Su x, y,z,t u x,t x,t y,z x, y,z,t = + + (3.2.4)

( )mu x,t - - x ( )SS x, y,z,t - - (secondary warping function).

mu . (3.2.4) 3.2.1 (3 ). mu . .. , y,z C . (3.2.4) , . mu . , . (3.2.4) . , .

3 :

47

3.2.3

. (. . (1.1.45), (1.1.46)). . (3.2.1), (3.2.3) :

SP S

xx m x Su ux x

= = + + (3.2.5)

yyv 0y

= = (3.2.5)

zzw 0z

= = (3.2.5) P SS S

xy xu v zy x y y

= + = + (3.2.5)

P SS S

xz xu w yz x z y

= + = + + (3.2.5)

yz x xv w 0z y

= + = + = (3.2.5)

Green (. . (1.1.12) (1.1.34)). . : ,

x . , . ( ) . , , () .

, , ( )u x, y,z,t x , . , , .

3 :

48

Green :

2 2 2 2 2

xx xxu 1 u v w u 1 v wx 2 x x x x 2 x x

= + + + = + + (3.2.6)

2 2 2 2 2

yy yyv 1 u v w v 1 v wy 2 y y y y 2 y y

= + + + = + + (3.2.6)

2 2 2 2 2

zz zzw 1 u v w w 1 v wz 2 z z z z 2 z z

= + + + = + + (3.2.6)

xyv u u u v v w wx y x y x y x y

= + + + +

xyv u v v w wx y x y x y

= + + + (3.2.6)

xzw u u u v v w wx z x z x z x z

= + + + +

xzw u v v w wx z x z x z

= + + + (3.2.6)

yzw v u u v v w wy z y z y z y z

= + + + +

yzw v v v w wy z y z y z

= + + + (3.2.6) . (3.2.2), (3.2.4)

( ) ( )S 2P 2 2Sxx m x S x1u y zx 2 = + + + + (3.2.7) yy 0 = (3.2.7) zz 0 = (3.2.7)

P SS S

xy x zy y = +

(3.2.7)

P SS S

xz x yz z = + +

(3.2.7)

yz 0 = (3.2.7) (3.2.5), (3.2.7) :

,

3 :

49

, xx .

xx .

( ) ( )22 2 x1 y z2 + , x Wagner (Wagner term) (Chen & Trahair, 1992).

3.2.4

. . , , Green 2 Piola Kirchhoff . (2.2.15). ( . (3.2.7))

( )( ) ( ) ( )xx xx yy zzES 11 1 2 = + + + ( )( )S 2P 2 2Sxx m x S x1S E u y zx 2 = + + + + (3.2.8)

( )( ) ( ) ( )yy yy xx zzES 1

1 1 2 = + + +

( )( )S 2P 2 2Syy m x S x1S E u y zx 2 = + + + + (3.2.8) ( )( ) ( ) ( )zz zz xx yyES 11 1 2 = + + +

( )( )S 2P 2 2Szz m x S x1S E u y zx 2 = + + + + (3.2.8) P SS S

xy xy xy xS G S G z Gy y = = +

(3.2.8)

P SS S

xz xz xz xS G S G y Gz z = = + +

(3.2.8)

yz yz yzS G S 0= = (3.2.8) ( )

( )( )E 1

E1 1 2

= + , ( )( )EE

1 1 2

= +

3 :

50

.

Cauchy . , . (3.2.8), . (3.2.5)

SP S

xx m x SE u x = + +

(3.2.9)

SP S

yy m x SE u x = + +

(3.2.9)

SP S

zz m x SE u x = + +

(3.2.9)

P SS S

xy xG z Gy y = +

(3.2.9)

P SS S

xz xG y Gz z = + +

(3.2.9)

yz 0 = (3.2.9)

Poisson ( 0 = ), E E = , E 0 = . . E ( E ), yyS , zzS yy , zz , . , 0 , , . (3.2.8, 3.2.8), ( , - ).

(Green et al.,1967). , ( ). , ,

3 :

51

(Davi, 1990). St. Venant (Ladevze & Simmonds, 1998), . , , (Parker, 1984). , .

3.2.5 -

, 4 , ( )mu x,t , ( )x x,t , ( )PS y,z ,

( )SS x, y,z,t 12 15 (6 - 6 - ). 4 ( - ). PS , SS (. 1.2.47)). , Lagrange (. . (1.2.47)). . (1.2.46) , ( )int xx xx xy xy xz xz yy yy zz zz yz yz

VW S S S S S S dV = + + + + + (3.2.10)

V () . 3 . (3.2.7, , ), ( )int xx xx xy xy xz xz

VW S S S dV = + + (3.2.11)

(1.2.47)

int mass extW W W + = (3.2.12)

( )massV

W u u v v w w dV = + + (3.2.13)

3 :

52

( )ext x y zF

W t u t v t w dF = + + (3.2.13) F () , xt , yt , zt ( ), ( ) ( t ). . (3.2.12) ( 3.2.1). . (1.2.47) massW . (3.2.13) DAlembert (Bathe, 1996).

, . (3.2.12) . (. (3.2.11))

P PP S S

1 xx x S x xy xzV V

I S dV S S dVy z

= + + (3.2.14) Washizu (1975), x

( ) ( ) ( )( )( )

l lP P

x xx S x xx Sx 0 x 0

lP

x xx Sx 0

S d dx S d dxx

S d

= =

=

= +

(3.2.15)

y,z . (3.2.14),

( )

P Pl lxy PS S xz

x xy xz x Sx 0 x 0

Px xy y xz z S

S SS S d dx dy z y z

S n S n ds dx

= =

+ = + + + +

(3.2.16)

yn cos = , zn sin = ( , = y n ) (. 3.2.1). . (3.2.14),

3 :

53

( )

lxy P Pxx xz

1 x S x xy y xz z Sx 0

lP

x xx Sx 0

SS SI d S n S n ds dxx y z

S d

=

=

= + + + + +

(3.2.17)

2 3I ,I

massW , extW

lP P

2 x S x SV x 0

I u dV u d dx

=

= = (3.2.18)

0 l lat

P P P P3 x x S x x S x x S x x S

F FI t dF t d t d t dF

= = + + (3.2.18)

0 , l ( x 0,l= ) latF ( lat 0 lF F = ). PS , , , . (3.2.17), (3.2.18)

xyxx xzSS S u 0x y z

+ + = , ( )x 0,l (3.2.19)

xy y xz z xS n S n t+ = , ( )x 0,l (3.2.19) PS (3.2.19). PS ( )x 0,l x 0,l= . . (3.2.17), (3.2.18) -

( )l

Pres x x xx S

x 0

I t S d

=

= (3.2.20) , ( ), PS . PS . (3.2.19) , . , , , x 0 = resI .

3 :

54

SS , . (3.2.17), (3.2.18, )

( )

lxy S Sxx xz

1 S xy y xz z Sx 0

lS

xx Sx 0

SS SI d S n S n ds dxx y z

S d

=

=

= + + + + +

(3.2.21)

lS S

2 S SV x 0

I u dV u d dx

=

= = (3.2.21)

0 l lat

S S S S3 x S x S x S x S

F FI t dF t d t d t dF

= = + + (3.2.21)

SS , , . (3.2.21) SS PS , . (3.2.19). , - SS .

, , , (. (3.2.19)) (. (3.2.19)). , u . ,

yxxx zx u 0x y z

+ + = , ( )x 0,l (3.2.22) xy y xz z xn n t + = , ( )x 0,l (3.2.22)

, 1 (. . (1.2.48), (1.2.49)), ( )x 0,l . , , x a priori. , x ( ) , . ,

3 :

55

.

( y,z ) ( Poisson ). , (St. Venant, 1855) . , y,z ( ) , ( ).

, . (3.2.19) (1.2.28), (1.2.24) 1 Piola-Kirchhoff 2 Piola-Kirchhoff. , 1 . , . .

. (3.2.19)

( )( )

2 SP 2 2 2 P 2 SS

m x S x x x S S2

P Sm x S S

E u y z G Gx

u 0

+ + + + + + + + =

(3.2.23)

P S P SS S S S

x y x z xG z G n G y G n ty y z z + + + + =

(3.2.23)

2 2 2 2 2/ y / z = + Laplace. . , : ( SS ) - - x

( 2 S

S2x

). .

resI (. . (3.2.20)):

3 :

56

SS , . (3.2.20). , .

(3.2.23) . St. Venant , . , PS

2 PS 0 = (3.2.24)

PS

y zzn ynn = (3.2.24)

( )n

n . PS , . (3.2.8, 3.2.8, 3.2.9, 3.2.9). . (3.2.24) . (3.2.23) SS

( )2 S P 2 2S m m x x S x xE E Eu u y zG G G G G = + + (3.2.25) SS xtn G = (3.2.25)

. (3.2.25) PS . , . (3.2.8, 3.2.8, 3.2.9, 3.2.9) .

(3.2.19).

PS (. (3.2.24)) SS

3 :

57

2 S PS m m x x S

E Eu uG G G G

= + (3.2.26)

SS xtn G = (3.2.26)

, . (3.2.25) (3.2.26). ( mu 0 = ) (. (3.2.26)) Sapountzakis & Mokos (2003). , SS (. (3.2.26)) . (3.2.26) . xt 0= , . (3.2.26) . , xt 0 .

PS , SS , .

SSEx

(. . (3.2.9)) . . (3.2.2), (3.2.4) , , Oxyz ( Ox ). , ( x x , y y , z z ). , (Armenakas, 2003), ( )

( )Y xx CM z z d = (3.2.27)

( )Z xx CM y y d = (3.2.27)

Cy , Cz Oxyz .

3 :

58

(3.2.9)

SSEx

P PY x O C OM E zd z d

= (3.2.28)

P PZ x O C OM E yd y d

= (3.2.28)

. ( Px O ),

PS d 0

= (3.2.29)

PS yd 0

= (3.2.29)

PS zd 0

= (3.2.29)

S Oxyz , . (3.2.29) PO .

xxN d = (3.2.30)

(3.2.9),

Pm x ON EA u E d

= + (3.2.31)

. (3.2.29). (3.2.29, 3.2.29) , . (3.2.29) (3.2.24) PS Neumann (, 1999) . . (3.2.29).

3 :

59

Oy ,Oy Oz Oz . PS , PO . . (3.2.24)

( ) ( )P PS O S Sy,z y,z y z z y c = + + (3.2.32) Sy , Sz S Oxyz ( Sy y y= , Sz z z= ) c . . (3.2.29) . (3.2.32)

y S z S wS y S z Ac A + = (3.2.33) yy S yz S y wzI y I z S c A + = (3.2.33) yz S zz M z wyI y I z S c A + = (3.2.33)

A d

= yS zd

= zS yd

= (3.2.34,,)

yzI yzd

= 2yyI z d

= 2zzI y d

= (3.2.34,,) P

w OA d = Pwy OA yd

= Pwz OA zd

= (3.2.34,,)

,

(. . (3.2.28))

( )Y xx CM S z z d

= (3.2.35) ( )Z xx CM S y y d

= (3.2.35)

xxN S d

= (3.2.35) 2 Piola-Kirchhoff , . (3.2.8) PS ,

( ) ( )2P 2 2m x O x1N EA u E d E y z d2 = + + + (3.2.36)

3 :

60

( ) ( )( )2P P 2 2Y x O C O x C1M E zd z d E z z y z d2 = + + (3.2.36)

( ) ( )( )2P P 2 2Z x O C O x C1M E yd y d E y y y z d2 = + (3.2.36)

, . , , (. . (3.2.29)) . , . . (3.2.36) , ( ( )2x ) . , (Attard, 1986).

. , S Sxyz ,

SS d 0

= (3.2.37)

. (3.2.29). . (3.2.4)

P Sm x S Sud u A d d

= + + (3.2.38)

m

udu

A

=

(3.2.39)

. (3.2.29), (3.2.37).

3 :

61

mu . (3.2.37). , . (3.2.39)

mu . ( ).

3.2.6 ( )mu x,t , ( )x x,t -

( )mu x,t , ( )x x,t

( (3.2.11)-(3.2.13)) ( 3.2.5). ,

P PP P PS St xy xzM S z S y dy z

= + + (3.2.40) P

w xx SM S d

= (3.2.40) PtM St. Venant (primary twisting moment, St. Venant twisting moment) (Sapountzakis & Mokos, 2003), P Pxy xzS ,S (primary shear stresses)

. (3.2.8), (3.2.8) wM (warping moment). , 2 Piola-Kirchhoff Cauchy. . (3.2.35)

( )2Pm xI1N EA u 2 A = + (3.2.41)

Pt t xM GI = (3.2.41) w S xM EC = (3.2.41)

( )2 2PI y z d

= + (3.2.42)

3 :

62

P P2 2 S S

tI y z y z dz y = + + (3.2.42)

( )2PS SC d

= (3.2.42) (polar moment of inertia), St. Venant (torsion constant, St. Venant torsion constant) (warping constant). . (3.2.41) .

(3.2.8) (3.2.7) (3.2.2), (3.2.4) , mu , x . Euler-Lagrange

( )m NA u n x,tx = (3.2.43)

( )( ) ( )

P 22t w

P x S x PP x x P m x P m x2

t w

M M 3I C EI EI u EI ux 2x

m x,t m x,tx

= = +

(3.2.43)

PPI

( )22 2PPI y z d

= + (3.2.44) ( )n x,t , ( )tm x,t , ( )wm x,t extW ( ) ( )xn x,t t x,t ds

= (3.2.45)

( ) ( ) ( ) ( ) ( )t y x x z x xm x,t t x,t z cos y sin t x,t y cos z sin ds

= + (3.2.45) ( ) ( ) Pw x Sm x,t t x,t ds

= (3.2.45)

3 :

63

. , . (1.2.24) 1 Piola-Kirchhoff. , . (3.2.43) xx xy xzS ,S ,S . .

Euler-Lagrange ( )x 0,l .

x 0,l= ( ) mN N u 0 = (3.2.46) ( ) ( )3Pw t PP x P m x t x1M M EI EI u M 02 + + + = (3.2.46) ( )w w xM M 0 + = (3.2.46)

xN t d

= (3.2.47) ( ) ( )t y x x z x xM t z cos y sin t y cos z sin d

= + (3.2.47)

Pw x SM t d

= (3.2.47)

extW . (3.2.46) ( x 0,l= ) massW .

. (3.2.41), ( ( )x 0,l )

( )m m P x xA u EA u EI n x,t = (3.2.48) ( )

( ) ( )

2P x S x t x S x PP x x P m x P m x

t w

3I C GI EC EI EI u EI u2

m x,t m x,tx

+ = = +

(3.2.48)

( . (3.2.46)) ( x 0,l= )

3 :

64

1 2 m 3a N u + = (3.2.49) 1 t 2 x 3M + = 1 w 2 x 3M + = (3.2.49, )

tM (twisting moment)

( )3t t x S x P m x PP x1M GI EC EI u EI2 = + + (3.2.50) N , wM (. (3.2.41, )) . . ia , i , i =( i 1,2,3 ) . (3.2.49) . . (3.2.49) () - . (3.2.41), (3.2.50) . , . . (3.2.48) . (3.2.48) . - (initial boundary value problem) , . (3.2.48). mu , x ( . (3.2.48) ) ( ( ) t 0, x 0,l= )

( ) ( )m m0u x,0 u x= ( ) ( )m m0u x,0 u x= (3.2.51, ) ( ) ( )x x0x,0 x = ( ) ( )x x0x,0 x = (3.2.51, )

.

, SS . ( ) .

SS ( PS , SS

3 :

65

3.2.5). SS massW ( 3.2.5), intW . SS intW ( S1I ) xy xzS ,S (

SS SxyS G y

= , S

S SxzS G z

= ) ( )S P S PS1 xy xy xz xz

VI S S dV = + (3.2.52)

P

P Sxy x zy

= ,

PP Sxz x yz

= + .

, . (3.2.52)

( )Pl S 2 P S SS1 S S x S y z xx 0

I G d zn yn ds dxn

=

= + (3.2.53) . (3.2.24)

S1I 0= (3.2.54) (!).

SS intW ( S 2I ) xxS (

SS SxxS E x

= )

S PS 2 xx xx

VI S dV= (3.2.55) ( )P P 2 2xx m x S x xu y z = + + + SxxS .

( )( )S P 2 2SS 2 m x S x xV

I E u y z dVx = + + + (3.2.56)

SS (. (3.2.25)) . S 2I

3 :

66

x . , , S 2I . (

SSx ) 3.2.5

2 S

S2 0x = .

. SS : mu , x - (3.2.48) - (3.2.51), SS (3.2.25). , SS , . (3.2.7), (3.2.8), (3.2.4).

, SSx

( ), .

(. . Fatmi, 2007, Machado & Cortinez, 2007) (independent warping parameter). , . (3.2.4) ( ) ( ) ( ) ( )Pm x Su x, y,z,t u x,t x,t y,z = + (3.2.57)

x x x = + x x x , .

SS 0x

. . (3.2.57) .

3 :

67

St. Venant. x . Timoshenko . (

SSx )

.

3.2.7

- . (axial inertia) mA u . (3.2.48). , . . , P x xEI . (3.2.48). .

. ( ( )n x,t 0= ) . (3.2.48)

Pm x x

IuA

= (3.2.58) x

( )2Pm xI1 Nu 2 A EA = +

, [ ]x 0,l (3.2.59) N - - . (axially immovable ends),

3 :

68

( )mu 0,t 0= ( )mu l,t 0= (3.2.60, ) N ,

( )l 2P x0

EI1N x,t dx2 l

= (3.2.61) ( )N l,t ,

( )mu 0,t 0= ( ) ( )N l,t N l ,t= (3.2.62, )

( )N N l,t= (3.2.63)

. (3.2.58), (3.2.59) . (3.2.48)

( )( ) ( )

2PP x S x t x S x n x x

t w

I 3I C GI N EC EIA 2

m x,t m x,tx

+ + = = +

(3.2.64)

nI

2P

n PPII IA

= (3.2.65) - , . (3.2.51, ), , . (3.2.49, ). (3.2.50)

( )3Pt t x S x n xI 1M GI N EC EIA 2 = + +

(3.2.66) . (3.2.64), (3.2.66) :

nI . (

3 :

69

) ( Taylor )

, . (3.2.64) Rozmarynowski & Szymczak (1984) . N 0 .

, . (3.2.64) . () ( ). , nI .

, ,

, . (3.2.24). ( mA u 0 , ( )n x,t 0= ) . (3.2.58) . (3.2.25), ( ( )SS red . )

( ) ( ) ( )2 S 2 2 PPS x x x x Sred . IE Ey z y,zG A G G = + (3.2.67) ( )SS red . 0n

= (3.2.67)

3.2.6 xt 0= ( ),

xt 0 . ,

( )2 S PS li n x x SE y,zG G = (3.2.68)

SS li n 0n

= (3.2.68)

3 :

70

3.2.8

: 1. Oxyz

(3.2.24) ( )PO y,z .

2. 3 3 . (3.2.33), . (3.2.34) Oxyz . S Sy ,z Oxyz Sxyz . . .

3. (3.2.24) ( )PS y,z ( Sxyz ) . (3.2.29).

4. P t S PP nI ,I ,C ,I ,I . (3.2.42), (3.2.44), (3.2.65).

5. - . (3.2.48) - (3.2.51) ( )mu x,t , ( )x x,t . 4 - - , - mu , x . 3.2.7, - . (3.2.64), (3.2.51, ), (3.2.49, ) ( )x x,t . ( )mu x,t x . (3.2.59). - . ( ) .

6. ( )SS x, y,z,t ( y,z ) (3.2.25) . (3.2.29). 3.2.7,

3 :

71

. (3.2.67). (3.2.26) . (3.2.68) 3.2.7.

7. . (3.2.8).

3.2.6, SSx

. , . 8. . (3.2.7).

3.2.6, SSx

. 9. . (3.2.2), (3.2.4).

4 :

73

4.1 ( )mu x,t , ( )x x,t

3,

( )mu x,t , ( )x x,t - . (3.2.48) - (3.2.51). , . ( 0 x l ) ( 0 t < ). (MAE) (Analog Equation Method (AEM), Katsikadelis (2002)) () (Boundary Element Methods, BEM).

( ) ( )1 mu x,t u x,t= , ( ) ( )2 xu x,t x,t= . x 1u ,

2u ,

( )2 1 12u q x,tx = ( )

42

24u q x,tx

= (4.1.1, ) ( ),iq x t , ( )i 1,2= / (fictitious loads / fictitious load distributions). . (4.1.1) . . (4.1.1) (Sapountzakis & Katsikadelis, 2000)

( )x l*l

* * 1 11 1 1 1 1

0 x 0

u duu x,t q u dx u ux dx

=

=

= (4.1.2)

( )l3 * 2 2 * 3 *l

* * 2 2 2 2 2 22 2 2 2 23 2 2 3

0 0

u du u d u u d uu x,t q u dx u udx xx x dx dx

= + (4.1.2) *1u ,

*2u (fundamental solutions)

*1

1u r2

= 3 2

* 32

1 r ru l 2 312 l l

= + (4.1.3, )

r x = , x , . *1u , *2u (particular singular solutions)

4 :

74

( )2 *12d u xdx = ( )4 *

24

d u xdx

= (4.1.4, ) ( )x ( ) Dirac (, 1999). . (4.1.3) (4.1.2)

( )ll

11 1 2 2 1 1

0 0

u1 1u x,t q ( r ) l dx ( r ) l ( r )u2 2 x

= + + + (4.1.5)

( )l3 2l 2 2 2

2 2 4 4 3 2 1 23 200

u u uu x,t q ( r )dx ( r ) ( r ) ( r ) ( r )uxx x

= + + + (4.1.5) (kernels) , j ( r ) ( j 1,2,3,4 ) = (Sapountzakis, 2000)

11 r( r ) sgn2 l

= (4.1.6)

21 r( r ) l 12 l

= (4.1.6) 2

31 r r r( r ) l 2 sgn4 l l l

= (4.1.6) 3 2

34

1 r r( r ) l 2 312 l l

= + (4.1.6)

. (4.1.5) r x = , x , ( )0,l , r x = , ( )x 0,l , 0,l = . 1 2u ,u -

( ),iq x t . , x . (4.1.5) iu

( ) x ll1 11 1 10

x 0

u x,t uq ( r )dx ( r )x x

=

= = (4.1.7)

( ) ( )2 1 12u x,t q x,tx = (4.1.7)

( ) l3 2l2 2 2 22 3 3 2 13 20

0

u x,t u u uq ( r )dx ( r ) ( r ) ( r )x xx x

= + + (4.1.7)

4 :

75

( ) l2 3 2l2 2 22 2 2 12 3 20

0

u x,t u uq ( r )dx ( r ) ( r )x x x

= + (4.1.7)

( ) l3 3l2 22 1 13 30

0

u x,t uq ( r )dx ( r )x x

= (4.1.7) ( ) ( )4 2 24u x,t q x,tx

= (4.1.7)

, . ( )0,l L ( ) iq - x - ( , , , ) (. . (4.1.1)). , iq (nodal point) . .

4.1.1 l .

/ . (4.1.5), (4.1.7) , ,

( ) ( ){ }Ti i i u 0,t u l ,t=u , ( )i 1,2= (4.1.8) ( ) ( ) Ti i

i xu 0,t u l ,t ,

x x =

u , ( )i 1,2= (4.1.8) ( ) ( ) T2 22 2

2 xx 2 2u 0,t u l ,t ,

x x

= u (4.1.8)

4 :

76

( ) ( ) T3 32 22 xxx 3 3

u 0,t u l ,t ,x x

= u (4.1.8)

{ }Ti i ii 1 2 Lq q ... q=q (4.1.8) , ,

{ }T1 1 1 1 x ,=d q u u (4.1.9) { }T2 2 2 2 x 2 xx 2 xxx , , ,=d q u u u u (4.1.9)

2L 12+ - 1d ( L 4+ ) 2d ( L 8+ ).

( )iu ,t , ( )i xu , ,t , ( )2 xxu , ,t ( )2 xxxu , ,t ( 0,l = )

iq , (4.1.5, ), (4.1.7) x 0,l= , (. . (3.2.49)) 0,l .

1

1 12 13 1nl2122 23 1 x

3 35 36 37 38 2

4 46 47 48 2nl nl55 56 58 2 x 52 53

66 67 2 xx

2 xxx

,

, , ,

+

q0 0 0F E E 0 0 0 0 0 u

0 0D0 D D 0 0 0 0 0 u0 0 0 F E E E E q 0 0 00 0 0 F 0 E E E u 0 0 00 0 0 0 D D 0 D u 0 D D0 0 0 0 0 D D 0 u 0 0 0

u

2 32

32

1 23

3

=

0u0

u0

u u

(4.1.10) 22D , 23D , 55D , 56D , 58D , 66D , 67D ,

nl21D ,

nl52D ,

nl53D 2 2

j j ja , , ( j 1,2= ) , 3 , 3 , 3 2 1 , , 3 3 3a , 12E , 13E , 35E ,

36E , 37E , 38E , 46E , 47E , 48E 2 2 j ( r ) ( j 1,2,3,4 )= , , 1 3 4F F F 2 L . ,

4 :

77

, j ( r ) ( j 1,2,3,4 ) = (, 2007) . j j ja , , ( j 1,2,3= ) , , . (4.1.10) . , . (4.1.10)

( ) ( ) T2 2

2 222

u 0,t u l ,t

x x

= u (4.1.11)

( ) ( ) T3 3

2 232

u 0,t u l ,t

x x

= u (4.1.11)

( ) ( ) ( ) ( ) T

1 2 1 21 2

u 0,t u 0,t u l ,t u l ,t x x x x

= u u (4.1.11)

. (4.1.5), (4.1.7) (3.2.48) . ,

01 1 1 0 1 1 1 x ,= + +u A q C u C u (4.1.12)

11 x 1 1 0 1 x, ,= +u A q C u (4.1.12) 1 xx 1, =u q (4.1.12)

02 2 2 0 2 1 2 x 2 2 xx 3 2 xxx , , ,= + + + +u A q C u C u C u C u (4.1.12)

12 x 2 2 0 2 x 1 2 xx 2 2 xxx , , , ,= + + +u A q C u C u C u (4.1.12)

22 xx 2 2 0 2 xx 1 2 xxx , , ,= + +u A q C u C u (4.1.12)

32 xxx 2 2 0 2 xxx, ,= +u A q C u (4.1.12) 2 xxxx 2, =u q (4.1.12)

, ji1 2A A ( i 0, 1)= , ( j 0, 1, 2, 3 )= L L ; 0C , 1C , 1C ,

2C , 3C L 2 iu , i x,u , i xx,u , i xxx,u , i xxxx,u ( )iu x,t L . . (4.1.12)

01 1 1=u H d (4.1.13)

4 :

78

11 x 1 1, =u H d (4.1.13)

02 2 2=u H d (4.1.13)

12 x 2 2, =u H d (4.1.13)

22 xx 2 2, =u H d (4.1.13)

32 xxx 2 2, =u H d (4.1.13)

, ji1 2H H ( i 0, 1)= , ( j 0, 1, 2, 3 )= ( )L L 4 + ( )L L 8 + i1A ,

j2A , 0C , 1C , 1C , 2C , 3C .

(3.2.48) - L ( ) . (4.1.13) 2L (semidiscretized equations of motion)

( ), , , = 1 jnl i1 1 1 2222

+ + dd

M K k H H d d fdd

(4.1.14)

nlk , , M K f ,

01

0 2P 2 S 2

A

I C

= H 0

M0 H H

(4.1.15)

2t 2 S

EA

GI EC

= + I0 0

K0 H I0

(4.1.15)

{ } { }( ) { }( )nl 1 2P 2 2 2 2i i iEI= k H d H d (4.1.15) { } { } { }( ) { }( ) { }( )

{ } { }( ) 2

nl 1 2 1 2PP 2 2 2 2 P 1 1 2 2L i i i i i

1P 1 2 2i i

3 EI EI2

EI

+ =

k H d H d H d H d

d H d (4.1.15)

= + t wn

fm m

(4.1.15)

, {}i () i

... ( i 1, 2, , L )= , I0 ( )L L 8 + [ ]=I0 I 0 I 0 L L L 8 . , , t wn m m L (collocation points), .

4 :

79

wm ( )wm x,tx

( )wm x,t (Sapountzakis & Mokos, 2004). .

, (4.1.13) (3.2.51, ) 2L 1d , 2d t 0=

( )01 1 0 = m0H d u ( )02 2 0 = 0H d (4.1.16, )

, (4.1.10) t 0= , 2L 12+ ( ) ( )1 0d , ( )2 0d . , (4.1.13) . (3.2.51, ) - - 2L 1d , 2d t 0=

( )01 1 0 = m0H d u ( )02 2 0 = 0H d (4.1.17, ) , ( t 0= ) t . (4.1.10), 2L 12+ ( ) ( )1 0d , ( )2 0d . ( )1 0d , ( )2 0d ( )1 0d , ( )2 0d . (4.1.16) . (4.1.17).

( )1 0d , ( )2 0d , ( )1 0d , ( )2 0d (4.1.14) . Petzold - Gear (Brenan et al., 1989) . 2L 12+ ( ) (Bazant & Cedolin, 1991). . (4.1.10) , (Brenan et al., 1989). - . (3.2.49, ), (3.2.51, ), (3.2.64)

4 :

80

. , .

4.2.1 PS

3, PS . (3.2.24). . ( (1999), Sapountzakis (2000), (2007), (2007)), (, 1999). ( ) (. 4.2.1). - - PS .

j

y

z

S ()

s

2

j

1

j

K

4.2.1. .

4.2.2 SS

3, SS

4 :

81

. (3.2.25) ( xt 0= )

( )2 S P 2 2S m m x x S x xE E Eu u y zG G G G G = + + (4.2.1) SS 0n = (4.2.1)

. (4.2.1) Poisson . (, 1999)

S S SS S p S L = + (4.2.2)

SS p Poisson (4.2.1)

SS p m m 1 x x p x x 1

E E Eu u F B HG G G G G

= + (4.2.3)

SS L ()

2 SS L 0 = (4.2.4)

SSS pS L

n n = (4.2.4)

1 1F ,H . (4.2.3)

2 2

1y zF

4+=

4 4

1y zH

12+= (4.2.5, )

pB

2 Pp SB = (4.2.6)

pB 0n

= (4.2.6) . SS . (4.2.2)

4 :

82

(4.2.1). . (4.2.3), . (4.2.4)

SS L 1 1

m m x xF HE Eu u

n G G n G n = + (4.2.7)

pB Poisson (4.2.6) , . (4.2.6) . (4.2.7). , , . (4.2.6) SC

SS .

. (4.2.6) (Sapountzakis & Mokos, 2003, , 2007) .

SS () (4.2.4) . - - Sapountzakis (2000). PS

[ ]{ } [ ] SS S LS LH G n = (4.2.8)

{ } ( ) ( ) ( ){ }TS S S SS L S L S L S L1 2 N... = (4.2.9) TS S S S

S L S L S L S L

1 2 N

...n n n n

= (4.2.9)

SS L

SS Ln

(. . (4.2.7))

( N ), . , . (4.2.8) [ ]H [ ]G N N . (4.2.27) Neumann (, 1999), . , Lutz et. al. (1998) (Aimi & Diligenti, 2008).

4 :

83

, (, 1999, Strese, 1984) . (3.2.24) , , Lutz et. al. (1998) . (Lutz et. al., 1998, Aimi & Diligenti, 2008), ( Poisson) Laplace. (4.2.6), Laplace PS . , Poisson . (4.2.6) (, 1999) . , SS .

SS (. . (4.2.2)) . (3.2.37)

SS d 0

= (4.2.10)

(. . (4.2.2)) SS L

S SS L S pdA dA

= (4.2.11)

SS L SS L (4.2.8), SS L

,

S SS L S L c = +

(4.2.12)

c . . (4.2.11), c

S SS p S L

1c dA dAA

= +

(4.2.13)

. (4.2.3), . (4.2.13)

4 :

84

SS p m m 1 x x 1

E EdA u u F dA H dAG G G

= + (4.2.14)

pB dA 0

= (4.2.15) . (4.2.6) Neumann (4.2.14). , pB SC . , . (4.2.14)

( )3 31 P y z1 1F dA I y n z n ds4 12 = = + (4.2.16) ( ) ( )3 2 2 3 5 51 PP y z y z1 1 1H dA I y z n y z n ds y n z n ds12 36 60 = + = + (4.2.16)

Green (, 1999), . (4.2.13) ( ) ( ) S S S 2 2 S 2 SS L S L S L 1 1 S L 1 S LdA 1dA F F dA F dA = = + =

SS S L1

S L 1F ds F dsn n

=

(4.2.17)

, . (4.2.7)

S S 1 1 1

S L S L 1 m m x xF F HE EdA ds F u u dsn G G n G n

= + (4.2.18)

c . . . (4.2.12) (4.2.2). . (3.2.67) (3.2.26) (3.2.68) .

4 :

85

4.2.3 , , , , , , , , , , , , , P PP n t S y z yy zz yz w wy wzA I I I I C S S I I I A A A

3,

. , , , , , , , , P PP t y z yy zz yzA I I I S S I I I Gauss (Gauss divergence theorem) (, 1999) :

( )y z1 y z 1A d d yn zn ds2 y z 2 = = + = + (4.2.19)

( ) ( ) ( ) ( )3 32 2 3 3P y zy z1 1I y z d d y n z n ds3 y z 3 = + = + = +

(4.2.19)

( ) ( )( ) ( ) ( ) ( )

22 2 4 4 2 2PP

5 5 3 2 2 3

5 3 2 5 2 3y z

I y z d y z 2y z d

y z y z y z1 1d d5 y z 3 y z

1 1 1 1y y z n z y z n ds5 3 5 3

= + = + + =

= + + + = = + + +

3 2 2 3 2 2PP y z

1 1 1 1I y y z n z z y n ds5 3 5 3

= + + + (4.2.19)

( ) ( )2 P 2 PP P S S2 2 S St

yz z y z yI y z y z d d

z y y z

+ = + + = +

( ) ( )2 P 2 Pt S y S zI yz z n y z y n ds

= + + (4.2.19) ( )2 2

y z

z1 1S zd d z n ds2 z 2

= = = (4.2.19) ( )2 2

z y

y1 1S yd d y n ds2 y 2

= = = (4.2.19) ( )32 3

yy zz1 1I z d d z n ds

3 z 3 = = = (4.2.19)

( )32 3zz y

y1 1I y d d y n ds3 y 3

= = = (4.2.19)

4 :

86

( ) ( ) ( )2 2 2 2yz y zy z yz1 1I yzd d y zn yz n ds4 y z 4 = = + = +

(4.2.19) nI . (3.2.65)

, , , S w wy wzC A A A Green (, 1999) :

( ) ( ) P2P P 2 2 P SS S S p p S pC d B B d B dsn = = = (4.2.20) ( )P P P 2 2 Pw S S S 1 1 SA d 1d F F d = = =

PP S1

w S 1FA F dsn n

= (4.2.20)

( ) PyP P 2 2 P P Swy S S y y S S yGA yd G G d G dsn n = = = (4.2.20)

( ) PP P 2 2 P P Szwz S S z z S S zGA zd G G d G dsn n = = = (4.2.20)

y 11G zF2

= , z 11G yF2= . , .

5 : -

88

5.1

, / FORTRAN 90/95, .

( ) . ( ( )mu x,t , ( )x x,t ) , . , , . () .

. ( ) , . . 3 (2003), .

5.2

5.2.1 1

l 40,0m= ( 6 2E 3,23184 10 kN / m= , 6 2G 1,3466 10 kN / m= ,

2 42,5484kN sec / m = ) ,

5 : -

89

( )N l,t 100000kN= ( ). 41 700 . 5.2.1 5.2.1, . - (. 3.2.7) x ( , . 3.2.7). (. 5.2.2) Sapountzakis & Mokos (2006). () ( )x0 l / 2 1,3rad =

( )x0 x . () .

0.3m

y

z

3.7m 3.0m

1.0m

()

l 40m=

00

= =w

0M 0 =

=

N 100000kN=mu 0=

()

5.2.1 () () 5.2.1.

5.2.1 5.2.1. ( )2A m 2,800

( )4PI m 3,848 ( )6PPI m 8,496 ( )6nI m 3,209 ( )4tI m 0,977 ( )6SC m 0,200

5 : -

90

0 5 10 15 20 25 30 35 40x(m)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

x(r

ad)

5.2.2 x

5.2.1.

5.2.3 ( & ) ( )SS red . ( (3.2.67))

SS lin (

(3.2.68)), ( )0,0 x 38,05 m t 0,1353 sec= . ( )x l / 2,t ( )0,0 t 0,15 sec ( ( )xmax l / 2,t 1,305rad = ,

( )0,0 t 0,15 sec ). ( t 0,1353 sec= ), 5.2.4 ( ) ( ) ( )S S Sxn xy y xz zred . red . red .S S n S n= + ( ( )SS red . ) , S S Sxnlin xylin y xzlin zS S n S n= + ( SS lin ), ( )0,0 x 38,05 m . ( )38,05 x 40,0 m , , 5.2.2. 5.2.3, 5.2.4 ( )SS red . , SS lin , ( )Sxn red .S , SxnlinS . , 5.2.2 ,

5 : -

91

. (3.2.67) ( )x l ,t 0 = . , 5.2.3 5.2.2 S SS Smin max = ( ) SS lin ( )SS red . .

0 5 10 15 20 25 30 35 40x(m)

-0.002

-0.0015

-0.001

-0.0005

0

0.0005

0.001

0.0015

0.002

(m)

Max Max Min Min

5.2.3 ( )SS red . , SS lin , ( ( )0,0 x 38,05 m ) 5.2.1.

5 : -

92

0 5 10 15 20 25 30 35 40x(m)

0

10000

20000

30000

40000

50000

60000

70000

80000

90000S x

nS(k

Pa)

Max Max

5.2.4 ( )Sxn red .S ( ( )SS red . )

, SxnlinS ( SS lin ),

( ( )0,0 x 38,05 m ) 5.2.1. 5.2.2 ( )SS red . , SS lin ( )Sxn red .S , SxnlinS

5.2.1. ( )x m 38,5 39,5 40,0

( ) ( )red.

max mSS -37,55510 -23,27310 -25,39810 ( )linmax mSS -37,54810 -23,30110 -25,39810 ( ) ( )

red.min mSS -3-8,14610 -2-3,44110 -2-5,39810

( )linmin mSS -3-7,54810 -2-3,30110 -2-5,39810 ( ) ( )S 2xn red.max S kN/m 53,89310 61,65810 62,57610 ( )S 2xnlinmaxS kN/m 53,60310 61,57510 62,57610

5 : -

93

, 5.2.5, 5.2.6 ( )SS red . , SS lin ( )Sxn red .S , SxnlinS , ( x 34,6 m= ) ( t 0,1353 sec= ). , , S ( ) SS lin ,

SxnlinS y zS ,S . ,

5.2.7 ( )Sxn red .S , SxnlinS . . 5.2.3 5.2.5 - 5.2.7 ( - ). ( ) , Vlasov (Vlasov, 1963) .

-0.00035-0.0003-0.00025-0.0002-0.00015-0.0001-5E-00505E-0050.00010.000150.00020.000250.00030.000350.00040.00045

()

-0.00014-0.00012-0.0001-8E-005-6E-005-4E-005-2E-00502E-0054E-0056E-0058E-0050.00010.000120.00014

() 5.2.5 ( )SS red . (a)

, SS lin (), x 34,6 m= ( t 0,1353 sec= ) 5.2.1.

5 : -

94

0100020003000400050006000700080009000100001100012000

()

0500100015002000250030003500400045005000550060006500

() 5.2.6 ( )Sxn red .S (a) SxnlinS () x 34,6 m= ( t 0,1353 sec= )

5.2.1.

(a) (b) 5.2.7 ( )Sxn red .S

(a) SxnlinS () x 34,6 m= ( t 0,1353 sec= ) 5.2.1.

5 : -

95

5.2.3 x 34,6 m= ,

t 0,1353 sec= 5.2.1 ( - ).

( )-1x radm -2-9,53210 ( )-2x radm -3-2,41110 ( )-3x radm -45,25410

( )-1 -2x radm sec 40,996

5.2.2 2

l 4,0m= 5.2.8 ( f wt t 0,01m= = , h b 0,20m= = ).

8 2E 2,1 10 kN / m= , 7 2G 8,1 10 kN / m= , 2 48,002kN sec / m = 21 ( ) 600 ( ). 5.2.4. ( ) ( )N l,t 3000kN= ( ) ( , . 3.2.7). , ( ( )x0 x 0 = , ( )m0u x 0= ) ( ( )x0 x 50rad / sec = ) ( ) ( ) ( )m0u x N l,0 / EA x = - . - (. 3.2.6, 3.2.7) ( ) .

5 : -

96

5.2.8 5.2.2, 5.2.3. 5.2.4 5.2.2,

5.2.3. ( )2A m -35,80010 ( )4PI m -55,43410 ( )6PPI m -76,72210 ( )6nI m -71,63110 ( )4tI m -72,08010 ( )6SC m -71,20010

5.2.9, 5.2.10 ( )x l / 2,t ( )mu l ,t ,

( )0,0 t 0,20 sec , ( )0,0 t 0,04 sec , , . ( , , ) ( mA u - ) .

tf=0.01m

tw=0.01m

b=0.20m

tf=0.01m

h=0.20m

z

yS

5 : -

97

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2t(sec)

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x(r

ad)

x x=l/2 - -

5.2.9 ( )x l / 2,t

5.2.2 ( )0,0 t 0,20 sec .

-0.013

-0.0125

-0.012

-0.0115

-0.011

-0.0105

-0.01

-0.0095

um(m

)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04t(sec)

um x=l - -

5.2.10 ( )mu l ,t

5.2.2 ( )0,0 t 0,04 sec .

5 : -

98

, 5.2.11, 5.2.12 SS ( () (3.2.25)), ( )SS red . ( () (3.2.67)) SxnS , ( )Sxn red .S , ( x 1,05m= ) ( ( )0,019 t 0,020 sec ). . ( , ).

0.019 0.0192 0.0194 0.0196 0.0198 0.02Time (sec)

-5E-005

-4E-005

-3E-005

-2E-005

-1E-005

0

1E-005

2E-005

3E-005

4E-005

(m)

Max - Max - Min - Min -

5.2.11 SS ( )SS red . x 1,05m= 5.2.2

5 : -

99

0.019 0.0192 0.0194 0.0196 0.0198 0.02Time (sec)

5000

10000

15000

20000

25000

30000

35000

40000

45000

50000

55000

60000S x

nS(k

Pa)

Max . . - Max . . -

5.2.12

SxnS , ( )Sxn red .S x 1,05m= 5.2.2

5.2.3 3

l 4,0m= 5.2.8 (

f wt t 0,01m= = , h b 0,20m= = ). 8 2E 2,1 10 kN / m= , 7 2G 8,1 10 kN / m= ,

2 48,002kN sec / m = 21 ( ) 600 ( ). 5.2.4. ( )

5 : -

100

( )N l,t 2500kN= ( ) ( , . 3.2.7). tM 20,0kNm= 40m / sec = . t 0= , ( )0,0 t 0,10 sec

( )t 0,10 sec> . ( ) ( ) ( )m0u x N l,0 / EA x = - . - (. 3.2.6, 3.2.7) .

5.2.13, 5.2.14 ( )x l / 2,t ( )mu l ,t , ( )0,0 t 0,15 sec , , . .

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

t(sec)

-1

-0.5

0

0.5

1

1.5

2

2.5

x(ra

d)

x x=l/2 - -

5.2.13 ( )x l / 2,t

5.2.3 ( )0,0 t 0,15 sec .

5 : -

101

-0.02

-0.0175

-0.015

-0.0125

-0.01

-0.0075um

(m)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14t(sec)

um x=l - -

5.2.14 ( )mu l ,t

5.2.2 ( )0,0 t 0,15 sec .

- ,

5.2.5 ( x 1,9m= ) t 0,0588 sec= ( )x l / 2,t ( ( )xmax l / 2,t 1,181rad = ) ( )0,0 t 0,15 sec . 5.2.5, (3.2.25) (3.2.26) SS ( SS lin ) 5.2.15. S SS lin ( yS ).

SS SS lin . 5.2.16 SxnS ,

SxnlinS ( x 1,9m= , t 0,0588 sec= )

.

5 : -

102

- . 5.2.17 SxnS ,

SxnlinS ( x 1,9m= , t 0,0588 sec= )

SxnS

( SxnlinS ). 5.2.5 x 1,9m= ,

t 0,0588 sec= 5.2.3 ( - ). ( )-1mu m -31,19910

( )-2mu msec 4-1,53010 ( )-1x radm 0,237 ( )-2x radm -0,803 ( )-3x radm -0,842

( )-1 -2x radm sec 6-1,58510

-8E-005

-6E-005

-4E-005

-2E-005

0

2E-005

4E-005

6E-005

8E-005

()

-7E-005-6E-005-5E-005-4E-005-3E-005-2E-005-1E-00501E-0052E-0053E-0054E-0055E-0056E-0057E-005

()

5.2.15 SS (a) , SS lin (), x 1,9m= ( t 0,0588 sec= ) 5.2.3.

5 : -

103

() ()

5.2.16 SxnS (a)

SxnlinS () x 1,9m= ( t 0,0588 sec= ) 5.2.3.

0100002000030000400005000060000700008000090000100000

()

0

10000

20000

30000

40000

50000

60000

70000

80000

() 5.2.17 SxnS

(a) SxnlinS () x 1,9m= ( t 0,0588 sec= ) 5.2.3.

,

( ) , 5.2.6 ( x 1,9m= , t 0,0588 sec= ).

5 : -

104

. , ( ) . 5.2.6

( x 1,9m= , t 0,0588 sec= ) 5.2.3. ( )Px Smax m -32,34410 ( )Px Smin m -3-2,34410 ( )SSmax m -55,33010 ( )SSmin m -5-7,33810 ( )SS linmax m -56,82310 ( )SS linmin m -5-6,82310

( )SxnmaxS MPa 96,971 ( )SxnlinmaxS MPa 81,106 ( )PxnmaxS MPa 191,346 ( )xnmaxS MPa 244,468

5.3

: 1.

/ , , .

2. ( ) .

3. .

4. , .

5. , .

5 : -

105

6. .

7. .

107

: 1. .. (1999),

- & , , . 2. , . . (2007),

, , ..., .

3. , . (1998), , ..., .

4. . (2007), , , ..., .

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38. Sapountzakis E.