beskonacna greda na winklerovoj podlozi

7/21/2019 Beskonacna Greda Na Winklerovoj Podlozi

1/14

10 IAA EDAAJA I EDEA

AA, G F BICA

3.

, () ()

( 3.1). A () =() , 1/() ( 3.1). , (), () (), () ( 3.1).

3.1, () :

( )4

4( ) ( )b

d w xE I p x q x

dx= (3.1)

: 2

()

() /

() /

(), () :

)()( xwBkxq = (3.2)

: ()

( ), /3

3.1 ) , )

M M

d

RR

a)

2

2

( ) 1

( )( ) 1

( )

b

M x

E I R xd w x

dx R x

=

x

M M+dM

T+dT

T

p(x)

q(x)

dx

b)

( )( )

( )( ) ( )

dM xT x

dxdT x

q x p xdx

=

=


2/14

IAA EDAAJA I EDEA 11

AA, G F BICA

A , , (3.2) (3.1), :

( ) ( ) ( )4 4

4 4( ) ( )b b

d w x d w xE I p x k B w E I k B w x p x

dx dx= + = (3.3)

, ( ), :

)(...)()(),...,,( n21n21 PwPwPwPPPw ++=

(3.3), :

( ) ( ) = + + +x x1 2 3 4w ( x ) e C cos x C sin x e C cos x C sin x (3.4)

:4

b

kB

4 E I=

1,2,3, I

K (),

(), () (), :

( )( )

( )( )

( ) ( )2 3b b2 3

dw x d w x dM x d w x( x ) , M x E I , T x E I

dx dx dx dx = = = = (3.5)

1

, ,

. 1/

. , .

K 1, 2, 3, () . A , ( , ). , :

.


3/14


AA, G F BICA

, . H , .

G (1932), G (1970), D (1976) ., , .K , .

4.

, 1, 2, 3,

(3.4), , , , , .

A , =0 =0 . A , , =0 =0. ,

=0, =0.

,

.

, , 0 .

K , , .


4/14


AA, G F BICA

4.1

(3.4) , ,

, , , ( 4.1).

P

z

x

Konvencija za + znak

P

z

x

M M

T

T

w

4.1 ,

D , = = , 1 2

(3.4) , 1=2=0, (3.4) :

[ ]x 3 4w e C cos x C sin x

= + (4.1)

I , =0, , =0. :

[ ] [ ]x x3 4 3 4dw

e C cos x C sin x e C sin x C cos x 0dx

= = + + + =

( ) ( )0 3 40 e C cos 0 sin 0 C sin 0 cos 0 0

= + + + =

( ) ( )3 4 3 4C 1 0 C 0 1 0 C C + + + = =

3 (4.1), :

( ) ( )x3 3w C e cos x sin x C A x

= + = (4.2)

(4.2) :

( )x3 3dw

2 C e s in x 2 C B xdx

= = = (4.3)

( ) ( )

22 x 2

3 32

d w

C e cos x sin x C C xdx

= = (4.4)


5/14


AA, G F BICA

( )3

3 x 3

3 33

d w4 C e c os x 4 C D x

dx

= = (4.5)

F(), (), (), () ,

, 4.2.

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0 4.0 5.0

Koeficije

nti:A,B,C,D

Parametar x

A

C D

B

4.2 B

. ||2.5 2% || 5 0.5% . I, || 2.5 11% || 5 0.8% .

, .

||2.5, .

, ( ), ( ) +( ), /2 /2. (3.5), 3, :

( ) ( ) ( ) ( ) ( )l d lP P

T dx T 0 , T dx T 0 T 0 P 2 2

= = = = =

3

3 xb b 33d wT E I E I 4 C e c os xdx

= =


6/14


AA, G F BICA

( ) 3d b 3 3 3b

P PT 0 E I 4 C C

2 8 E I

= = =

K 3 (4.2) (4.3),

, , :

( ) ( )

= = =

3

43

3

b b

P kB Pw A x , w A x

8 E I 4E I 2Bk (4.6)

( ) ( )

224

2

3

b b

P kB P2 B x , B x

8 E I 4E I Bk

= = =

(4.7)

(3.5), (4.4) (4.5), , :

( ) ( )

2

42 2

b 3

b b

P kB PE I C x , M C x

8 E I 4E I 4

= = =

(4.8)

( ) ( )

3

43 3

b 3

b b

P kB PT E I 4 C x , T D x

8 E I 4E I 2

= = =

(4.9)

D , , ( 1 +1, ),

5 5.

1.0

0.8

0.6

0.4

0.2

0.0

0.2

0.4

0.6

0.8

1.0

5 4 3 2 1 0 1 2 3 4 5

/

'

4.3 ,


7/14


AA, G F BICA

4.2

0 , 0/( 4.4).

z

x


M0

z

x

M M

T

T

w

M0

4.4 ,

, , 0, , :

( ) ( )( )0P

w( x,M ) w( x,P ) w( x x, P ) A x A x x2Bk

= + = +

( ) ( ) ( )0

A x A x x xMP xw

2Bk x 2Bk x

+ = =

( ) ( )( )

2

0 0 0

x 0

A x dA xM M Mw lim B x

2Bk x 2Bk dx Bk

= = = (4.10)

(4.10), ,

0, , :

( )2 30 0

dB xM MdwC( x )

dx Bk dx Bk

= = = (4.11)

( )32 2

0 0b b2 2

M Md w d A( x )M E I E I D x

dx Bk dx 2

= = = (4.12)

( )33 3

0 0b b3 3

M Md w d A( x )T E I E I A x

dx Bk dx 2

= = = (4.13)

D , ( 1 +1, ),

5 5.


8/14


AA, G F BICA

1.0

0.8

0.6

0.4

0.2

0.0

0.2

0.4

0.6

0.8

1.05 4 3 2 1 0 1 2 3 4 5

/

'

4.5 ,


9/14


AA, G F BICA

4.3

J , ,

.

z

x


z

x

M M

T

w

p

r s

T

p

4.6 ,

4.6, , :

( ) ( ) ( ) ( )

( ) ( )d xdP pdx p p

dw A x A x A x A x d x2Bk 2Bk 2Bk 2Bk

= = = = (4.14)

, , , :

( ) ( ) ( ) ( )r

0 0

sp p

w A x d x A x d x2Bk 2Bk

= +

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

r

0 0

s

p p pw A x d x A x d x 2 D r D s

2Bk 2Bk 2 Bk = + = (4.15)

(4.15), , , , :

( ) ( )dw p

r A sdx 2Bk

= =

(4.16)

( ) ( )2

b 2 2

d w pM E I B r B s

dx 4

= =

(4.17)

( ) ( )3

b 3

d w pT E I C r C s

dx 4

= = (4.18)


10/14


AA, G F BICA

A ( 4.6),

, , , , :

( ) ( ) ( ) ( ) ( ) ( )s

0 0

rp p p

w A x d x A x d x D r D s2Bk 2Bk 2Bk

= = (4.19)

z

x

p

s


z

x

M M

T

w

T

p

r

4.6 ,

( ) ( )dw p

r A sdx 2Bk

= =

(4.20)

( ) ( )2

b 2 2

d w pM E I B r B s

dx 4

= =

(4.21)

( ) ( )3

b 3d w pT E I C r C sdx 4

= = (4.22)

A (4.6), ,

, , , :

z

x

p

s


z

x

M M

T

w

T

p

r

4.6 ,

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

r s

0 0

p p pw A x d x A x d x D r D s

2Bk 2Bk 2Bk = = (4.23)

( ) ( )

dw p

r A sdx 2Bk

= = (4.24)


11/14


AA, G F BICA

( ) ( )2

b 2 2

d w pM E I B r B s

dx 4

= =

(4.25)

( ) ( )3

b 3

d w pT E I C r C s

dx 4

= = (4.26)

, , , .

4.7 , .

: : :

/ , / ,

/ , / .

( )xC4

PM

= ( )xA

Bk2

Pw

=

( )xD2

PT = ( )xB

Bk

P 2

=

( )xD2

MM o m= ( )xB

Bk

Mw

2o

m=

( )xA2

MT o

= ( )xC

Bk

M 3

o

=

( ) ( )[ ]sBrB4

pM

2

+= ( ) ( )[ ]sDrD2

Bk2

pw =

( ) ( )[ ]sCrC4pT

= ( ) ( )[ ]sArABk2

p

=

( ) ( )[ ]sBrB4

pM

2

= m ( ) ( )[ ]sDrD

Bk2

pw =

( ) ( )[ ]sCrC4

pT

= ( ) ( )[ ]sArA

Bk2

p

=

4.1

x P

xP

TM

TM

x

Mo

x

Mo

TM

TM

p

r s

TM

pr

s

pr

sTM

TM


12/14


AA, G F BICA

4.4 1

( , , ) ,

, , 4.7. . , .

z

xA B

P=1,3 MN

p=50,0 kN/m

3,0 2,0 4,0 1,0

L=10,0

B=1,5

0,4

1,0

0,4

E =21,0 GPaI=0,159 m

b4

k=30,0 MN/m3

4.7 A B

:

(), :

144

b

kB 30,0 1,50,2409 m4 E I 4 21000 0,159

= = =

, , , :

( ) ( ) ( )AP p

w A 3 D 5 D 92Bk 2Bk

= +

( ) ( ) ( )[ ] m104821692D2051D30000512

0507230A

30000512

240901300w 3A

=

+

= ,,,

,

,,

,

,

( ) ( ) ( )2

A

P pB 3 A 5 A 9

Bk 2Bk

= +

( ) ( ) ( )[ ] rad1058601692A2051A30000512

024090507230B

3000051

240901300 32

A

=

+

= ,,,,

,,

,

,

( ) ( ) ( )A 2P p

M C 3 B 5 B 94 4

=


13/14


AA, G F BICA

( ) ( ) ( )A1300,0 50,0

C 0,723 B 1,205 B 2,169 18,028 kNm0,964 0,232

= =

( ) ( ) ( )AP p

T D 3 C 5 C 92 4

= +

( ) ( ) ( )A1300,0 50,0

T D 0,723 C 1,205 C 2,169 235,914 kN 2 0,964

= + =

, , , :

( ) ( ) ( )BP p

w A 7 D 5 D2Bk 2Bk

= +

( ) ( ) ( )[ ] m109302410D2051D30000512

0506871A

30000512

240901300w

3

B

=

+

= ,,,,

,,

,

,

( ) ( ) ( )2

B

P pB 7 A 5 A

Bk 2Bk

= +

( ) ( ) ( )[ ] rad1038402410A2051A30000512

024090506871B

3000051

240901300 32

B

=

+

= ,,,,

,,

,

,

( ) ( ) ( )B 2P p

M C 7 B 5 B4 4

= +

( ) ( ) ( )B1300,0 50,0

C 1,687 B 1,205 B 0,241 257 ,049 kNm0,964 0,232

= + =

( ) ( ) ( )

= + BP p

T D 7 C 5 C 2 4

( ) ( ) ( ) = + = B1300,0 50,0

T D 1,687 C 1,205 C 0,241 24,930 kN 2 0,964

:

( ) ( ) ( )P p

w A 0 D 2 D 6 2Bk 2Bk

=

( ) ( )[ ] m107734451D4820D30000512050001

30000512240901300w 3=+ =

,,,,

,,,

,


14/14


AA, G F BICA

( ) ( ) ( )2P p

B 0 A 2 A 6Bk 2Bk

= +

( ) ( )[ ] rad10076504451A4820A3000051202409050

003000051

240901300 32

=

+

= ,,,,,

,,

,

18.03

314.09

747.04

1337.66

796.47

419.67

167.47

5.80

123.29

208.17 257.05

400

200

0

200

400

600

800

1,000

1,200

1,400

1,600

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

()

235.91

360.50

508.96

674.15

625.85

457.26

298.91

208.98

141.4697.41 75.98

24.93

700

600

500

400

300

200

100

0

100

200

300

400

500

600

700

800

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

()

4.8 AB

beskonacna greda na winklerovoj podlozi

Documents