Matrična analiza konstrukcija

8/10/2019 Matrina analiza konstrukcija

1/17

PRIMER:

Sraunati nepoznata pomeranja i sile u presecima sistema datog na slici, a zatim nacrtati

dijagrame presenih sila M, T i N. Sistem je izloen uticaju optereenja.

Napomena: promena krivine greda prema kvadratnoj paraboli.

7 23 10E kN m= , 51 10 1t C =

Proraunska ema konstrukcije:

Koordinate sistema:


2/17

Lokalne koordinate elemenata:

Koordinate vorova:

vor x y

1 12,0 8,5

2 32,0 8,5

3 44,0 8,5

4 0,0 8,5

5 3,5 0,06 40,5 0,0

Geometrija tapova:

( ) ( )2 2

k i k il x x y y= + , ( ) ( )sin , cosk i k is y y l c x x l = = = =

Tabela 1

tapkraj tapa

xk-xi yk-yil

[m]

c

(cos)

s

(sin)i k

1 1 4 -12,0 0,0 12,0 -1 0

2 1 2 20,0 0,0 20,0 1 0

3 2 3 12,0 0,0 12,0 1 0

4 1 5 -8,5 -8,5 12,0208 -0,7071 -0,7071

5 2 6 8,5 -8,5 12,0208 0,7071 -0,7071


3/17

- MATRICE KRUTOSTI TAPOVA U LOKALNIM KOORDINATAMA:

Matrica krutosti tapa i-k sa promenljivim poprenim presekom:

( ) ( )

( ) ( )

2 2

2 2

0 0 0 0

0 0

0 0

0 0 0 0

0 0

0 0

1 1

1 1

ik ik

ik ki ik ik ki ki

ik ik ik ik

ik ik

ik ki ik ik ki ki

ki ik ki ki

c c l c l c c l c l

c l a c l b

c c l c l c c l c l

c l b c l a

+ + =

+ +

k

Matrica krutosti tapa i-g sa promenljivim poprenim presekom :

2 2

2 2

1 0 0 1 0

0 0

0 01 0 0 1 0

0 0

ig ig

ig ig ig

ig ig ig g

ig ig

ig ig ig

d l d l d l

d l d d l

d l d l d l

=

k

Matrica krutosti tapa i-g sa konstantnim poprenim presekom :

3 2 3

2 2

3 2 3

0 0 0

0 3 3 0 3

0 3 3 0 3

0 0 0

0 3 3 0 3

ig ig ig ig

ig ig ig ig ig ig

ig ig ig ig ig ig ig

ig ig ig ig

ig ig ig ig ig ig

EF l EF l

EI l EI l EI l

EI l EI l EI l

EF l EF l

EI l EI l EI l

=

k

- Proraun matrica krutosti:

Matrica krutosti tapova 1 i 3:

Aksijalno naprezanje - reenje numerikom integracijom:

10

00 0 0

1 1 1 1 1,3

l l l

ig x m m

mx x x

dx dx dxEF bE h bE bE h=

= = = = =


4/17

x hx =1/hx

0,0 0,600 1,666667 1 1,66667

1,2 0,609 1,642036 4 6,56814

2,4 0,636 1,572327 2 3,14465

3,6 0,681 1,468429 4 5,87372

4,8 0,744 1,344086 2 2,68817

6,0 0,825 1,212121 4 4,84848

7,2 0,924 1,082251 2 2,16450

8,4 1,041 0,960615 4 3,84246

9,6 1,176 0,85034 2 1,70068

10,8 1,329 0,752445 4 3,00978

12,0 1,500 0,666667 1 0,66667

= 36,17393

10

0

1 1 1,2 36,17393 36,173933 0,4 3

ig m

m

EbE E=

= = =

3

11 14 41 44

127,644220 10

36,17393ig

Ek k k k E

= = = = = =

Aksijalno naprezanje teoretsko reenje:12

2

0 0

1 1 1 1 1 20 6 6 36,173947arctan ,

0,4 0,00625 0,6 0,4 3 2 E

l

ig

x

dx dxbE h E x E

= = = = +

Popreno savijanje numerikom integracijom:

( )12

i x

x xM

l= =

( )

2 2

( ) ( )

( )3

( )0 0 0

2

( )

3

( )

12 12

12,

3

x

l l li x i x

ig x

x

ki x

ig m m

m i x

M Mdx dx dx

EI bE bEh

M

bE h=

= = = =

= =

x hx Mix = (Mix)2/( hx)3

0,0 0,600 0,00000 0,00000 1 0,000001,2 0,609 0,10000 0,04427 4 0,17710

2,4 0,636 0,20000 0,15549 2 0,31097

3,6 0,681 0,30000 0,28497 4 1,13989

4,8 0,744 0,40000 0,38851 2 0,77702

6,0 0,825 0,50000 0,44522 4 1,78089

7,2 0,924 0,60000 0,45634 2 0,91268

8,4 1,041 0,70000 0,43435 4 1,73742

9,6 1,176 0,80000 0,39351 2 0,78702

10,8 1,329 0,90000 0,34507 4 1,38029

12,0 1,500 1,00000 0,29630 1 0,29630

= 9,299565


5/17

12 12 1,2 111,5947849,299565

3 0,4 3

k

ik m

m ibE E E =

= = =

318,960992 10ig

ig

d E= =

Popreno savijanje teoretsko reenje:

( )

2 2 12 2( ) ( )

3 2 3

( )0 0 0

12 12 ( 12)

0,4 (0,00625 0,6)x

l lg x g x

ig

x

M M xdx dx dx

EI bE h E x = = =

+

12 125 6 6 10 1arctan 111,594298

0,4 81 2 27ig

E E

= + =

318,961031 10ig

ig

d E= =

Sada su, prema Error! Reference source not found., matrice krutosti za tapove 1 i 3,

3

1 3 10 E

= =

k k

27,644220 0 0 -27,644220 0

0 0,062229 0,746749 0 -0,062229

0 0,746749 8,960992 0 -0,746749

-27,644220 0 0 27,644220 0

0 -0,062229 -0,746749 0 0,062229

Matrica krutosti tapa 2 :

Aksijalno naprezanje - reenje numerikom integracijom:

( )

( ) ( )0 0 0

1 1 1l l l

ik x

x x

dxdx dx

EF bE h bE = = =

( )

1 1,

3

k

ik m m

m i xbE h=

= =


6/17

x hx =1/hx

0,0 1,500 0,66667 1 0,66667

2,0 1,320 0,75758 4 3,03030

4,0 1,180 0,84746 2 1,69492

6,0 1,080 0,92593 4 3,70370

8,0 1,020 0,98039 2 1,96078

10,0 1,000 1,00000 4 4,00000

12,0 1,020 0,98039 2 1,96078

14,0 1,080 0,92593 4 3,70370

16,0 1,180 0,84746 2 1,69492

18,0 1,320 0,75758 4 3,03030

20,0 1,500 0,66667 1 0,66667

= 26,112746

10

0

1 1 226,112746 43,5212433 0,4 3

ik m

mEbE E=

= = =

3

11 14 41 44

122,977285 10

43,521243ik

Ek k k k E

= = = = = =

Aksijalno naprezanje teoretsko reenje:20

2

0 0

1 1 1 1 1 2 43,52098820 2 arctan

0,4 0,005 0,1 1,5 0,4 2 E

l

ik

x

dx dxbE h E x x E

= = = = +

Popreno savijanje numerika integracija Simpsonovim pravilom:

( ) 120

i x

xM =

( )

20

k x

xM =

( )

2 2

( ) ( )

( )3

( )

12 12

x

i x i x

ik x

x

M Mdx dx dx

EI bE h bE = = =

( )

2

( )

3

12,

3x

ki x

ik m m

m i

M

bE h=

= =

Zbog simetrije tapa je ik = ki

( )

( ) ( ) ( ) ( )

( )3

( )

12 12

x

i x k x i x k x

ik x

x

M M M Mdx dx dx

EI bE h bE = = =

( )

( ) ( )

3

12,3

x

ki x k x

ik ki m m

m i

M M

bE h=

= = =


7/17

m x hm Mi Mk m m m

0 0,0 1,50 1,0 0,0 0,29630 0,00000 1

1 2,0 1,32 0,9 0,1 0,35218 0,03913 4

2 4,0 1,18 0,8 0,2 0,38952 0,09738 2

3 6,0 1,08 0,7 0,3 0,38898 0,16670 4

4 8,0 1,02 0,6 0,4 0,33924 0,22616 2

5 10,0 1,00 0,5 0,5 0,25000 0,25000 4

6 12,0 1,02 0,4 0,6 0,15077 0,22616 2

7 14,0 1,08 0,3 0,7 0,07144 0,16670 4

8 16,0 1,18 0,2 0,8 0,02435 0,09738 2

9 18,0 1,32 0,1 0,9 0,00435 0,03913 4

10 20,0 1,50 0,0 1,0 0,00000 0,00000 1

m =6,371847 3,940839

12 12 2 127, 4369426,371847

3 0, 4 3

k

ik ki m

m ibE E E =

= = = =

12 12 2 78,816784

3,9408393 0, 4 3

k

ik ki m

m ibE E E =

= = = =

( )2 2 2 22 21 1

127,436942 78,816784 10.028,088794ik ik

E E = = =

23127,436942 12,707999 10

10.028,088794

kiik ki

Ea a E

E

= = = =

2

378,816784 7,859602 1010.028,088794

ikik ki

Eb b E

E

= = = =

3 3 312, 707999 10 7, 859602 10 20, 567601 10ik ki ik ik c c a b E E E = = + = + =

Popreno savijanje teoretsko reenje:

( )

2 2 20 2( ) ( )

3 2 3

( )0 0 0

12 12 (1 0,05 )

0, 4 (0, 005 0,1 1,5)x

l li x i x

ik

x

M M xdx dx dx

EI bE h E x x

= = =

+

12 25 2 2 55 127, 435185arctan0, 4 8 2 36

ikE E

= + =

( )

8( ) ( ) ( ) ( )

3 2 3

( )0 0 0

12 12 (1 0,005 )(0, 005 )

0, 3 (0, 005 0,1 1, 5)x

l li x k x i x k x

ik

x

M M M M x xdx dx dx

EI bE h E x x

= = =

+

12 5 2 2 25 78,820370arctan

0,3 8 2 12ik

E E

= + =

312,709108 10ik kia a E= =

37,860754 10ik kib b E

= = 320,569861 10ikc E

=


8/17

Sada je, prema Error! Reference source not found., matrica krutosti za tap 2,

2

=

k

022,977285 0 -22,977285 0 0

0 0,102838 1,028380 0 -0,102838 1,028380

0 1,028380 12,707999 0 -1,028380 7,859602

-22,977285 0 0 22,977285 0 0

0 -0,102838 -1,028380 0 0,102838 -1,0283800 1,028380 7,859602 0 -1,028380 12,707999

E

-310

Matrica krutosti tapova 4 i 5: 12,0208l m= , 20,24F m= , 40,0072I m=

4 5 E

= =

k k-3

19,965368 0 0 -19,965368 0

0 0,012435 0,149481 0 -0,012435

100 0,149481 1,796883 0 -0,149481

-19,965368 0 0 19,965368 0

0 -0,012435 -0,149481 0 0,012435

- MATRICE KRUTOSTI TAPOVA U GLOBALNIM KOORDINATAMA:

- Globalne koordinate tapova

- Matrice transformacije

Prema Error! Reference source not found.a, Error! Reference source not found.itabeli 1, imamo:

1

=

T

-1 0 0 0 0

0 -1 0 0 0

0 0 1 0 0

0 0 0 -1 0

0 0 0 0 -1

, 2

=

T

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

, 3

=

T

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

,


9/17

4

=

T

-0,70711 -0,70711 0 0 0

0,70711 -0,70711 0 0 0

0 0 1 0 0

0 0 0 -0,70711 -0,70711

0 0 0 0,70711 -0,70711

5

=

T

0,70711 -0,70711 0 0 0

0,70711 0,70711 0 0 0

0 0 1 0 0

0 0 0 0,70711 -0,70711

0 0 0 0,70711 0,70711

- Matrice krutosti u globalnom koordinatnom sistemu

* 3

1 1 1 1 10T E

= =

k T k T

2 5 7 1 10

2

5

7

1

10

27,644220 0 0 -27,644220 0

0 0,062229 -0,746749 0 -0,062229

0 -0,746749 8,960992 0 0,746749

-27,644220 0 0 27,644220 0

0 -0,062229 0,746749 0 0,062229

* 3

2 2 2 2 10T E

= =k T k T

2 5 7 3 6 8

22,977285 0 0 -22,977285 0 0

0 0,102838 1, 028380 0 -0,102838 1, 028380

0 1,028380 12,707999 0 -1,028380 7,859602

-22,977285 0 0 22,977285 0 0

0 -0,102838 -1,028380 0 0,102838 -1,0283800 1,028380 7,859602 0 -1,02 8

2

5

7

3

6

8380 12,707999

* 3

3 3 3 3 10T E

= =

k T k T

3 6 8 4 9

3

6

8

4

9

27,644220 0 0 -27,644220 0

0 0,062229 0,746749 0 -0,062229

0 0,746749 8,960992 0 -0,746749

-27,644220 0 0 27,644220 0

0 -0,062229 -0,746749 0 0,062229

* 3

4 4 4 4 10T

E= =k T k T

2 5 7 11 12

9,988902 9,976466 0,105699 -9,988902 -9,9764669,976466 9,988902 -0,105699 -9,976466 -9,988902

0,105699 -0,105699 1,796883 -0,105699 0,105699

-9,988902 -9,976466 -0,105699 9,988902 9,976466

-9,976

2

5

7

11

12466 -9,988902 0,105699 9,976466 9,988902

* 3

5 5 5 5 10T E

= =k T k T

3 6 8 13 14

9,988902 -9,976466 0,105699 -9,988902 9,976466

-9,976466 9,988902 0,105699 9,976466 -9,988902

0,105699 0,105699 1,796883 -0,105699 -0,105699

-9,988902 9,976466 -0,105699 9,988902 -9,976466

9,97646

3

6

8

13

146 -9,988902 -0,105699 -9,976466 9,988902


10/17

- MATRICA KRUTOSTI SISTEMA:* *

*

* *

ss so

os oo

=

K KK

K K

*

ss E=K

1 2 3 4 5 6 7 8

-3

27,64422 -27,644220 0 0 0 0 0 0-27,64422 60,610407 -22,977285 0 9,976466 0 0,105699 0

0 -22,977285 60, 610407 -27,64422 0 -9,976466 0 0,105699

0 0 -27,644220 27,64422 0 0 0 010

0 9,976466 0 0 10,153969 -0,102838 0,175932 1,

6

7

8

1

2

3

4

5028380

0 0 -9, 976466 0 -0,102838 10,153969 -1,028380 -0,175932

0 0,105699 0 0 0,175932 -1,028380 23,465874 7,859602

0 0 0,105699 0 1,028380 -0,175932 7,859602 23,465874

*

0 0 0 0 00 0 0

0 0 0

0 0 0 0 0

0 0

0 0 0

0 0

0 0 0

so E

=

K

9 10 11 12 13 14

-3

00 -9,988902 -9,976466

0 -9,988902 9,976466

010

0 -0,062229 -9,976466 -9,988902

-0,062229 9,976466 -9,988902

0 0,746749 -0,105699 0,105699

-0,746749 -0,105699 -0,105699

6

7

8

1

2

3

4

5

*

os E

=

K

1 2 3 4 5 6 7 8

-3

0 0 0 0 0 -0,062229 0 -0,746749

0 0 0 0 -0,062229 0 0,746749 0

0 -9,988902 0 0 -9,976466 0 -0,105699 010

0 -9,976466 0 0 -9,988902 0 0,105699 0

0 0 -9,988902 0 0 9,976466 0 -0,105699

0 0 9,976466 0 0 -9,988902 0 -0,105699

9

10

11

12

13

14

*

oo E

=

K

9 10 11 12 13 14

9

10

11

12

13

14

-3

0,062229 0 0 0 0 0

0 0,062229 0 0 0 0

0 0 9,988902 9,976466 0 0

10 0 0 9,976466 9,988902 0 0

0 0 0 0 9,988902 -9,976466

0 0 0 0 -9,976466 9,988902

* 1

ssE

=K3

2,057554 2,021380 1,999633 1,999633 -1,974472 1,952773 0,068042 0,069374

2,021380 2,021380 1,999633 1,999633 -1,974472 1,952773 0,068042 0,069374

1,999633 1,999633 2,021380 2,021380 -1,952773 1,974472 0,069374 0,

10

068042

1,999633 1,999633 2,021380 2,057554 -1,952773 1,974472 0,069374 0,068042

-1,974472 -1,974472 -1,952773 -1,952773 2,027603 -1,905999 -0,065594 -0,072383

1,952773 1,952773 1,974472 1,974472 -1,905999 2,027603 0,072383 0

,065594

0,068042 0,068042 0,069374 0,069374 -0,065594 0,072383 0,050610 -0,013846

0,069374 0,069374 0,068042 0,068042 -0,072383 0,065594 -0,013846 0,050610


11/17

A) UTICAJ ZADATOG OPTEREENJA:

- Vektori ekvivalentnog optereenja tapova:

Proraun momenata punog ukljetenja

tap 1:

( )

2

( )

1( )

( )

3,

160 5

,12

112

2

x

x

o x

xh

xM

M p x x

= +

=

=

2 212 12 101( ) 1( )

11 30( ) ( )0 0

2

1

3

12 12,

3

1,2

x x

m

mx x

mm

m

M ME dx dx

I b bh

Mm

h

=

= = =

= =

12 12 101( ) ( ) 1( ) ( )

10 30( ) ( )0 0

1

3

12 12

3

x o x x o x

m

mx x

m om

m

m

M M M ME dx dx p

I b bh

M M

h

=

= = =

=

m x hm M1 Mo m m m

0 0,00 0,600 0,0 0,00 0,00000 0,00000 1

1 1,20 0,609 0,1 6,48 0,04427 2,86895 42 2,40 0,636 0,2 11,52 0,15549 8,95594 2

3 3,60 0,681 0,3 15,12 0,28497 14,36256 4

4 4,80 0,744 0,4 17,28 0,38851 16,78359 2

5 6,00 0,825 0,5 18,00 0,44522 16,02805 4

6 7,20 0,924 0,6 17,28 0,45634 13,14253 2

7 8,40 1,041 0,7 15,12 0,43435 9,38205 4

8 9,60 1,176 0,8 11,52 0,39351 5,66657 2

9 10,80 1,329 0,9 6,48 0,34507 2,48452 4

10 12,00 1,500 1,0 0,00 0,29630 0,00000 1

m = 9,299565 269,601769

10

11

0

12 12 1,29,299565 111,594784

3 0, 4 3m

m

Eb

=

= = =

10

10

0

12 12 1,0269,601769 3.235,221228

3 0,4 3m

m

E p pb

=

= = =

2

1 14

3.235, 22122828,990793 1,6106 463,852681

111,594784 8

p plX M p kNm= = = = =

2 2

4 14

1 1 16 12463,852681 57,34561

2 12 2

plV M kN

l

= + = =

1 4 16 12 57,34561 134,65439V pl V kN = = =


12/17

1

kN

kN

kNm

kN

kN

=

Q

0

134,65439

463,85268

0

57,34561

*

1 1 1

T

= =

Q T Q

2

5

7

1

10

0

-134,65439

463,85268

0

-57,34561

tap 2:

2

( )

1( )

2( )

3, 2,0

200 10 2

1 ,20

20

x

x

x

x xh m

x

M

xM

= + =

=

=

220 101( )

11 30( )0

12 12

3

x

m

mx

ME dx

b bh

=

= =

20 101( ) 2( )

12 30( )0

12 12

3

x x

m

mx

M ME dx

b bh

=

= =

220 102( )

22 3 0( )0

12 12

3

x

mmx

ME dx

b bh

=

= =

20 101( ) ( )

10 30( )0

12 12

3

x o x IV

m

mx

M ME dx

b bh

=

= =

20 102( ) ( )

20 30( )0

12 12

3

x o x V

m

mx

M ME dx

b bh

=

= =

m x hm M1 M2 Mo m m = m IV V

m m = m

0 0,0 1,50 0,0 1,0 0 0,00000 0,00000 0,00000 1

1 2,0 1,32 0,1 0,9 40 0,00435 0,03913 1,73915 4

2 4,0 1,18 0,2 0,8 80 0,02435 0,09738 9,73809 2

3 6,0 1,08 0,3 0,7 120 0,07144 0,16670 28,57796 4

4 8,0 1,02 0,4 0,6 160 0,15077 0,22616 60,30863 2

5 10,0 1,00 0,5 0,5 200 0,25000 0,25000 100,00000 4

6 12,0 1,02 0,6 0,4 160 0,33924 0,22616 90,46294 2

7 14,0 1,08 0,7 0,3 120 0,38898 0,16670 66,68191 4

8 16,0 1,18 0,8 0,2 80 0,38952 0,09738 38,95238 2

9 18,0 1,32 0,9 0,1 40 0,35218 0,03913 15,65239 4

10 20,0 1,50 1,0 0,0 0 0,29630 0,00000 0,00000 1

m = 6,371847 3,940839 1249,529748


13/17

10

11 22

0

12 12 26,371847 127,436942

3 0, 4 3m

m

E Eb

=

= = = =

10

12

0

12 12 23,940839 78,816784

3 0, 4 3m

m

Eb

=

= = =

10

10 20

0

12 12 21249,529748 24.990,594953

3 0, 4 3

IV

m

m

E Eb

=

= = = =

1

2

127,436942 78,816784 24.990,5949530

78,816784 127,436942 24.990,594953

X

X

+ =

1 12 121,164332X M= =

2 21 121,164332X M= =

2

kN

kN

kNm

kN

kN

kNm

=

Q

0

-20,00000

-121,16433

0

-20,00000

121,16433

*

2 2 2

8

T

= =

Q T Q

2

5

7

3

6

0

-20,00000

-121,16433

0

-20,00000

121,16433

- Vektor ekvivalentnog optereenja sistema tapova:

* * *

*

* * *

0

s s s

o o

= = +

S Q RS

S Q R

Ovde je: - * 0s =R vektor sila zadatih u vorovima, u pravcima slobodnih koordinata sistema,

- *oR vektor sila u pravcima vezanih koordinata, odnosno reakcije oslonaca i ukljetenja

{ }* * *1 2 3 4 5 6 7 8

0 0 0 0 154,65439 20 342,68835 121,16433 T

s s s= + = S Q R

{ }*10 11 12

0 57,34561 0 0 0 0 T

o = Q

9 13 14

- Odreivanje pomeranja u pravcima slobodnih koordinata sistema:

Polazei od sistema uslovnih jednaina

* * *=K q S ,

odnosno* * * *

* * * *

ss so s s

os oo o o

=

K K q S

K K q S,


14/17

dobijamo* * * * *

ss s so o s+ =K q K q S ,

odakle, uz uslov da su pomeranja u pravcima vezanih stepeni slobode jednaka nuli, tj. * 0o =q , imamo

* * *

ss s s=K q S .

Sada je vektor pomeranja u pravcima slobodnih koordinata sistema

* * 1 *

s ss s

=q K S ,

odnosno,

*

sE

=q3

2,057554 2,021380 1,999633 1,999633 -1,974472 1,952773 0,068042 0,069374

2,021380 2,021380 1,999633 1,999633 -1,974472 1,952773 0,068042 0,069374

1,999633 1,999633 2,021380 2,021380 -1,952773 1,974472 0,069374 0,068

10

042

1,999633 1,999633 2,021380 2,057554 -1,952773 1,974472 0,069374 0,068042

-1,974472 -1,974472 -1,952773 -1,952773 2,027603 -1,905999 -0,065594 -0,072383

1,952773 1,952773 1,974472 1,974472 -1,905999 2,027603 0,072383 0,06

0

0

0

0

154,65439

20,00000

342,68835

121,16

5594

0,068042 0,068042 0,069374 0,069374 -0,065594 0,072383 0,050610 -0,013846

0,069374 0,069374 0,068042 0,068042 -0,072383 0,065594 -0,013846 0,050610 433

3* 10s

E

= =

q

1

2

3

4

298, 02802 9, 934 mm

298, 02802 9, 934 mm

294,53343 9,818 mm

294,53343 9,818 mm

-306,70618 -10,224 mm

286,97138 9,566 mm

24,36251 0,00081 rad

11,26959 0,00038 rad

6

7

8

5

- Odreivanje reakcija oslonaca i oslonakih ukljetenja *oR :

Iz sistema uslovnih jednaina imamo, * * * * * * *0os s oo o o o+ = = +K q K q S Q R ,

Odakle je, za * 0o =q ,* * * *

o os s o= R K q Q

*

o

E

=

R 3

0 0 0 0 0 -0,062229 0 -0,746749

0 0 0 0 -0,062229 0 0,746749 0

0 -9,988902 0 0 -9,976466 0 -0,105699 0

0 -9,976466 0 0 -9,988902 0 0,105699 010

0 0 -9,988902 0 0 9,976466 0 -0,105699

0 0 9,976466 0 0 -9,988902 0 -0,105699

310

E

298,02802

298,02802 0

294,53343 -57,346

294,53343 0

-306, 70618 0

286,97138 0

24,36251 0

11,26959

*

o

= = =

= =

=

=

R

9

10

11

12

13

14

3

4

5

5

6

6

V -26,2735 kN

V 94,6243 kN

H 80,2963 kN

V 92,9664 kN

H -80,2963 kN

V 70,6828 kN


15/17

- Vektori generalisanih pomeranja tapova u lokalnom koordinatnom sistemu:

3 3*

1 1 1

10 10

E E

= = =

q T q

-1 0 0 0 0 298,02802 -298,02802

0 -1 0 0 0 -306,70618 306,70618

0 0 1 0 0 24,36251 24,36251

0 0 0 -1 0 298,02802 -298,028020 0 0 0 -1 0 0

2

5

7

1

10

3 3*

2 2 2

8

10 10

E E

= = =

q T q

1 0 0 0 0 0 298,02802 298,02802

0 1 0 0 0 0 -306,70618 -306,70618

0 0 1 0 0 0 24,36251 24,36251

0 0 0 1 0 0 294,53343 294,53343

0 0 0 0 1 0 286,97138 286,9713

0 0 0 0 0 1 11,26959

2

5

7

3

6

8

11,26959

3 3*

3 3 3

10 10

E E

= = =

q T q

1 0 0 0 0 294,53343 294,53343

0 1 0 0 0 286,97138 286,971380 0 1 0 0 11,26959 11,26959

0 0 0 1 0 294,53343 294,53343

0 0 0 0 1 0 0

3

6

8

4

9

3 3*

4 4 4

10 10

E E

= = =

q T q

-0,7071 -0,7071 0 0 0 298,02802 6,13639

0,7071 -0,7071 0 0 0 -306,70618 427,61165

0 0 1 0 0 24,36251 24,36251

0 0 0 -0,7071 -0,7071 0 0

0 0 0 0,7071 -0,7071 0 0

2

5

7

11

12

3 3*

5 5 5

10 10

E E

= = =

q T q

0,7071 -0,7071 0 0 0 294,53343 5,34717

0,7071 0,7071 0 0 0 286,97138 411,18599

0 0 1 0 0 11,26959 11,26959

0 0 0 0,7071 -0,7071 0 0

0 0 0 0,7071 0,7071 0 0

3

6

8

13

14

- Vektori generalisanih sila na krajevima tapova u lokalnom koordinatnom sistemu:

Za tap j , j j j j= R k q Q

3

1 3

10

10

EI

EI

=

R

27,6442 0 0 -27,6442 0 -298,02802 00 0, 0622 0, 7467 0 -0, 0622 306, 70618 134,6544

0 0,7467 8,9610 0 -0,7467 24,36251 463,8

-27,6442 0 0 27,6442 0 -298,02802

0 -0,0622 -0,7467 0 0,0622 0

=

0-97,376

527 -16,508

0 0

57,3456 -94,624

3

2 3

10

10

EI

EI

=

R

022,9773 0 -22,9773 0 0 298,02802

0 0,1028 1,0284 0 -0,1028 1,0284 -306,706

0 1, 0284 12, 7080 0 -1, 0284 7, 8596

-22,9773 0 0 22,9773 0 0

0 -0,1028 -1, 0284 0 0,1028 -1, 0284

0 1, 0284 7, 8596 0 -1, 0284 12, 7080

=

0 80,296

18 -20,0000 -4, 409

24,36251 -121,1643 -91,189

294,53343 0 -80,296

286,97138 -20,0000 44,409

11,26959 121,1643 -396,997


16/17

3

3 3

10

10

EI

EI

= =

R

27,6442 0 0 -27,6442 0 294,53343 0

0 0,0622 0,746749 0 -0,0622 286,97138 26,274

0 0,7467 8,9610 0 -0,7467 11,26959 315,282

-27,6442 0 0 27,6442 0 294,53343

0 -0,0622 -0,7467 0 0,0622 0

0

-26,274

3

4 3

10

10

EI

EI

= =

R

19,9654 0 0 -19,9654 0 6,13639 122,515

0 0,0124 0,1495 0 -0,0124 427,61165 8,959

0 0,1495 1,7969 0 -0,1495 24,36251 107,696

-19,9654 0 0 19,9654 0 0 -122,51

0 -0,0124 -0,1495 0 0,0124 0

5

-8,959

3

5 3

10

10

EI

EI

= =

R

19,9654 0 0 -19,9654 0 5,34717 106,758

0 0,0124 0,1495 0 -0,0124 411,18599 6,798

0 0,1495 1,7969 0 -0,1495 11,26959 81,715

-19,9654 0 0 19,9654 0 0 -106,758

0 -0,0124 -0,1495 0 0,0124 0

-6,798

Sile na krajevima tapova:

- Pomeranja vorova, reakcije oslonaca:


17/17

- Dijagrami presenih sila:

Matrična analiza konstrukcija

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