Zadatak 2. - grad.hr postupci_Slunjski.pdfZadatak 2. Za zadani sistem kombinacijom metoda Crossa i Werner-Csonke odrediti MT dijagrame. E = 3. 10 7 kN/m 2, =10 −5 K−1

GRAĐEVNA STATIKA II ____________________________________________________________________________________________________________

___________________________________________________________________________

GRAĐEVNA STATIKA II VEDRAN SLUNJSKI, 5228 ak.god.2006/07

1

Zadatak 2.

Za zadani sistem kombinacijom metoda Crossa i Werner-Csonke odrediti MT dijagrame.

E = 3 .

107 kN/m

2, 1510 −−= Ktα ;

grede: b/h = 48/96 [cm];

stupovi: b/h = 48/48 [cm].

Student Zadatak

[m] [m] [m] [m] [kN/m] [kN] [mm]/[m ]/[ C]

31 Vedran Slunjski 2 4.2 6 5.5 5 37.5 50 -18


___________________________________________________________________________


2

109

8765

4321

5,5

5

4,264,2

23

2,75

2,75

-18 C50 kN

50 kN

37,5 kN/m'

OGOS

S

G

EIEIEIEI

kNmEI

kNmEI

8

4,13271012

48,048,0103

2,106168312

96,048,0103

23

7

23

7

=⇒=

=

⋅⋅=

=

⋅⋅=

1 CROSS

OSNOVNI SISTEM


___________________________________________________________________________


3

55,5

4,2 6 4,2

0,99 mm

0,99 mm

6,75,6

9,10

PLAN POMAKA OD SKRAĆENJA ZBOG TEMPERATURE

410,956 102,4

9,9 −⋅−

==ψψ

467 100,6

9,9 −⋅=ψ

KOEFICJENTI KRUTOSTI ELEMENATA

5,548372615

OIEkkkk⋅

====

5

10,659OIEkk

⋅==

2,4

87856

OIEkk⋅

==

6

867

OIEk⋅

=

KOEFICJENTI KRUTOSTI ČVOROVA

O

EI,kkkk 2925956155

=++=

O

EIkkkkk 62,35610,667266

=+++=

OEIkkkk 42,37837677 =++=

OEIkkk 09,248788 =+=

OEIkkk 1,210,9599 =+=

OEIkkk 1,210,610,910 =+=


___________________________________________________________________________


4

MOMENTI UPETOSTI

kNmkM

kNmkM

kNmM

kNmM

kNmM

kNmM

kNmkM

kNmkM

kNmkM

kNmkM

kNmM

kNmM

5,3576

5,3576

0,365

3250

0,245

2350

1,5512

2,45,37

1,5512

2,45,37

7,287612

0,65,37

7,62612

0,65,37

5,3576

5,3576

4,348

5,550

4,348

5,550

56569,10

565610,9

2

2

95

2

2

59

2

87

2

78

6767

2

76

6767

2

67

565665

565656

51

15

=−=

=−=

−=⋅⋅−

=

=⋅⋅

=

−=⋅−

=

=⋅

=

−=−⋅−

=

−=−⋅

=

=−=

=−=

−=⋅−

=

=⋅

=

ψ

ψ

ψ

ψ

ψ

ψ

RAZDJELNI KOEFICJENTI

Čvor 5

09,028,20,5

1

83,028,22,4

8

08,028,25,5

1

5

5959

5

5656

5

1551

=⋅

==

=⋅

==

=⋅

==

k

k

k

k

k

k

µ

µ

µ

∑= 1

Čvor 6

37,0

62,30,3

4

05,062,35,5

1

6

6767

6

6262

=⋅

==

=⋅

==

k

k

k

k

µ

µ


___________________________________________________________________________


5

52,0

62,32,4

8

06,062,30,5

1

6

5665

6

10,6

10,6

=⋅

==

=⋅

==

k

k

k

k

µ

µ

∑= 1

Čvor 7

56,042,32,4

8

05,042,35,5

1

39,042,30,3

4

7

7878

7

3773

7

6776

=⋅

==

=⋅

==

=⋅

==

k

k

k

k

k

k

µ

µ

µ

∑= 1

Čvor 8

09,0

09,25,5

1

91,009,22,4

8

8

4884

8

7887

=⋅

==

=⋅

==

k

k

k

k

µ

µ

∑= 1

Čvor 9

91,0

1,22,4

8

09,01,20,5

1

9

10,9

10,9

9

5995

=⋅

==

=⋅

==

k

k

k

k

µ

µ

∑= 1

Čvor 10

91,0

1,22,4

8

09,01,20,5

1

9

10,9

9,10

10

696,10

=⋅

==

=⋅

==

k

k

k

k

µ

µ

∑= 1


___________________________________________________________________________


6

0,09

0,080,83

0,090,91

0,91 0,09

0,06

0,05

0,52

0,37

0,05

0,390,56

0,91 0,09

-36,0

-14,3

-15,3

-1,8

1,6

-0,6

0,8

-0,2

0,1

-0,1

-65,8

357,5

-162,7

-144,5

34,7

-17,6

4,9

-5,9

1,5

-2,1

0,5

-0,5

-0,1

0,1

65,8

____

0,1

0,3

-0,1

1,7

-0,3

3,3

-0,9

-30,6

-7,2

24,0

-9,9

-57,0

-34,4

-27,2

2,9

1,5

0,2

357,5

-325,3

-72,2

69,4

-8,8

9,9

-2,9

3,0

-1,0

1,0

-0,2

0,2

30,6

-32,2

-4,1

6,9

-2,1

1,0

-0,4

0,3

-0,1

0,1

-30,6

_____

-0,2

0,1

-0,8

0,5

-4,2

3,4

-8,3

-16,1

-25,6

357,5

-282,1

-35,7

30,4

-18,1

15,2

-3,1

2,7

-0,6

0,5

-0,1

0,1

66,7

____

0,1

0,7

1,5

-13,6

34,4

23,1

____

-0,1

-0,3

-1,7

-3,5

-5,6

7,2

____

0,1

0,5

6,6

____

-0,1

-0,2

-0,9

-1,2

357,5

-141,1

-71,5

15,2

-36,2

7,6

-6,3

1,3

-1,2

0,2

-0,2

125,3

-6,9

-3,5

-0,6

-0,1

-11,1

-62,7

-50,9

51,0

-25,7

4,1

-4,5

0,9

-0,8

0,2

-0,2

-88,6

-287,7

-25,5

102,0

-12,8

8,2

-2,2

1,9

-0,4

0,4

-0,1

0,1

-216,1

13,1

1,1

0,2

14,4

51,5

146,6

-8,2

11,7

-2,6

2,7

-0,6

0,6

-0,1

0,1

201,7

-55,1

73,3

-16,5

5,8

-5,3

1,3

-1,2

0,3

-0,3

2,3

-2,3

-1,7

-0,5

-0,1


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7

1. ITERACIJA Čvor 10

2,325,35709,0

3,3255,35791,0

5,357

−=⋅−

−=⋅−

=∆M

∑ −= 5,357 Čvor 9

3,148,15809,0

5,1448,15891,0

8,158

367,1625,357

−=⋅−

−=⋅−

=∆

−−=∆

M

M

∑ −= 8,158 Čvor5

2,279,33908,0

1,2829,33983,0

6,309,33909,0

9,339

4,345,357242,7

−=⋅−

−=⋅−

−=⋅−

=∆

−++−=∆

M

M

∑ −= 9,339 Čvor6

5,716,13752,0

9,66,13705,0

9,506,13737,0

3,86,13706,0

6,137

1,167,621,1415,357

−=⋅−

−=⋅−

−=⋅−

−=⋅−

=∆

−−−=∆

M

M

∑ −= 6,137 Čvor 7

0,1027,26139,0

1,137,26105,0

6,1467,26156,0

7,261

5,517,2875,25

=⋅

=⋅

=⋅

−=∆

+−−=∆

M

M

∑= 7,261 Čvor 8

7,12,1809,0

5,162,1891,0

2,181,553,73

−=⋅−

−=⋅−

=−=∆M

∑ −= 2,18

2. ITERACIJA

Čvor 10

9,63,7609,0

4,693,7691,0

3,761,42,72

=⋅

=⋅

−=−−=∆M

∑= 3,76 Čvor 9

8,14,1909,0

6,174,1991,0

4,193,157,34

−=⋅−

−=⋅−

=−=∆M

∑ −= 4,19 Čvor 5

9,26,3608,0

4,306,3683,0

3,36,3609,0

6,367,359,0

=⋅

=⋅

=⋅

−=−−=∆M

∑= 6,36 Čvor 6

2,366,6952,0

5,36,6905,0

7,256,6937,0

2,46,6906,0

6,694,3512,15

−=⋅−

−=⋅−

−=⋅−

−=⋅−

=++=∆M

∑ −= 6,69 Čvor 7

2,82139,0

1,12105,0

7,112156,0

212,88,12

=⋅

=⋅

=⋅

−=−−=∆M

∑= 21 Čvor 8

5,08,509,0

3,58,591,0

8,5

−=⋅−

−=⋅−

=∆M

∑ −= 8,5


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8

3. ITERACIJA

Čvor 10

0,19,1009,0

9,99,1091,0

9,101,28,8

=⋅

=⋅

−=−−=∆M

∑= 9,10 Čvor 9

6,05,609,0

9,55,691,0

5,66,19,4

−=⋅−

−=⋅−

=+=∆M

∑ −= 5,6 Čvor 5

5,14,1808,0

2,154,1883,0

7,14,1809,0

4,181,183,0

=⋅

=⋅

=⋅

−=−−=∆M

∑= 4,18 Čvor 6

3,62,1252,0

6,02,1205,0

5,42,1237,0

8,02,1206,0

2,126,71,45,0

−=⋅−

−=⋅−

−=⋅−

−=⋅−

=++=∆M

∑ −= 2,12 Čvor 7

9,18,439,0

2,08,405,0

7,28,456,0

8,46,22,2

=⋅

=⋅

=⋅

−=−−=∆M

∑= 8,4 Čvor 8

1,03,109,0

2,13,191,0

3,1

−=⋅−

−=⋅−

=∆M

∑ −= 3,1


3,03,309,0

0,33,391,0

3,34,09,2

=⋅

=⋅

−=−−=∆M

∑= 3,3 Čvor 9

2,03,209,0

1,23,291,0

3,28,05,1

−=⋅−

−=⋅−

=+=∆M

∑ −= 3,2 Čvor 5

2,02,308,0

7,22,383,0

3,02,309,0

2,31,31,0

=⋅

=⋅

=⋅

−=−−=∆M

∑= 2,3 Čvor 6

2,13,252,0

1,03,205,0

8,03,237,0

2,03,206,0

3,23,19,01,0

−=⋅−

−=⋅−

−=⋅−

−=⋅−

=++=∆M

∑ −= 3,2 Čvor 7

4,0139,0

0,0105,0

6,0156,0

0,16,04,0

=⋅

=⋅

=⋅

−=−−=∆M

∑= 1


___________________________________________________________________________


9

H1

M5,9

T9,5

T9,5

M9,5

M6,10

M10,6

T10,6

5

23

50 kN

T8,4T7,3

M8,4M7,3

M4,8M3,7M2,6

T6,2

M6,2

M1,5

T5,1

M5,1

H2

1

5,5

2,75

2,75

50 kN

50 kN


1,01,109,0

0,11,191,0

1,11,00,1

=⋅

=⋅

−=−−=∆M

∑= 1,1 Čvor 9

1,06,009,0

5,06,091,0

6,01,05,0

−=⋅−

−=⋅−

=+=∆M

∑ −= 6,0 Čvor5

06,008,0

5,06,083,0

1,06,009,0

6,0

=⋅

=⋅

=⋅

−=∆M

∑= 6,0 ODREĐIVANJE REAKCIJA U PRIDRŽANJIMA

kNH

HF

T

M

T

M

x

38,56

24,1114,450

24,115

6,256,30

0

14,455

1509,98,65

0

1

1

6,10

6

95

5

=

+=⇒=

−=−−

=

=

−=−−−

=

=

∑

∑

∑


___________________________________________________________________________


10

II

I

0 79,63

23,25

56,38T

23,25

56,38

MI,0

M0,IMI,II

79,6356,38

5,55

MII,I

kNHF

kNTM

kNTM

kNTM

kNTM

x 25,2364,093,304,316,315038,560

64,05,5

3,22,10

93,35,5

4,142,70

04,35,5

1,116,50

16,315,5

75,250571,230

2

844

733

622

511

=+−+++−=⇒=

−=−−

=⇒=

=+

=⇒=

−=−−

=⇒=

−=⋅−−

=⇒=

∑

∑

∑

∑

∑

2 WERNER – CSONKA

ZAMJENJUJUĆI POLUOKVIR

kNmMM

kNmMM

II

IIIIII

2192

5,563,79

95,1402

538,56

0,,0

,,

=⋅

==

=⋅

==


___________________________________________________________________________


11

KOEFICJENTI KRUTOSTI ELEMENATA

Stupova:

0,5

2

5,5

4

010,659,

0483726150

EIkkk

EIkkkkk

III

I

=+=

=+++=

Greda:

2,4

324

6,12

8,64444

010,9

0786756

EIkk

EIkkkk

IIG

IG

==

=++=

KOEFICJENTI KRUTOSTI ČVOROVA

0,

0,0

26,233

56,163

EIkkk

EIkkkk

IIIIIGII

IIIIIGI

=+=

=++=

RAZDJELNI KOEFICJENTI

Čvor 1

02,056,160,5

2

93,056,166,12

48,363

05,056,165,5

4

,

,

00

=⋅

==

=⋅

==

=⋅

==

I

III

III

I

IGIG

I

II

k

k

k

k

k

k

µ

µ

µ

∑= 1

Čvor 2

98,026,232,4

3233

02,026,230,5

2

,

,

,

,

=⋅

⋅==

=⋅

==

II

GII

GII

I

III

III

k

k

k

k

µ

µ

∑= 1


___________________________________________________________________________


12

1. ITERACIJA

Čvor I

1836005,0

8,33436093,0

2,736002,0

36021995,140

−=⋅−

−=⋅−

−=⋅−

=+=∆M

∑ −= 360

Čvor II

0,32,14802,0

2,1452,14898,0

2,1482,795,140

−=⋅−

−=⋅−

=+=∆M

∑ −= 2,148

2. ITERACIJA

Čvor I

2,0305,0

8,2393,0

0302,0

0,3

−=⋅−

−=⋅−

=⋅−

=∆M

∑ −= 3


___________________________________________________________________________


13

237,2

-337,6

200,8

_____ 0,2 18,0219,0

-334,8 -2,8

219,0 -18,0 -0,2

136,8_____ 3,0 -7,2141,0

-145,2

145,2

141,0 7,2 -3,0

-145,2

0,02

0,050,93

0,02

0,98

kNmMMM

kNmMMMMMMM

kNmMMM

kNmMMM

kNmMMMMM

kNmMMMMM

GII

GI

III

III

I

I

2,145

6,337

2,145

8,136

8,200

5,237

9,1010,9,

877876676556,

6,1095,

10,659,

847362510

483726150

−=+=

−=+++++=

=+=

=+=

=+++=

=+++=

( )( )

( )( )

( )( )

kNmMM

kNmkkk

kMM

kNmkkk

kMM

kNmkkk

kMM

kNmMM

kNmMM

kNmMMMM

kNmMMMM

6,722

2,145

5,626,3372

8,436,3372

5,626,3372

6,722

2,145

4,682

8,136

2,504

8,200

4,594

5,237

9,1010,9

786756

788778

786756

677667

786756

566556

6,1095

10,659

84736251

48372615

−=−

==

−=−⋅++

==

−=−⋅++

==

−=−⋅++

==

===

===

=====

=====


___________________________________________________________________________


14

72,5

-72,5

-70,3

0,09

0,080,83

0,090,91

0,91 0,09

0,06

0,05

0,52

0,37

0,05

0,390,56

0,91 0,09

72,6

-2,6

0,2

-0,1

0,1

0,1

70,3

-72,6

2,4

-0,7

0,7

-0,1

____

0,2

-0,3

0,1

-5,1

68,4

63,3

45,7

50,2

-4,5

-0,2

0,2

-72,6

1,2

-1,4

0,3

72,6

0,3

-0,1

-0,3

_____

-0,1

-0,6

0,7

68,4

68,4

-62,5

-46,5

2,8

-2,4

-2,4

2,0

-0,4

0,4

-109

____

0,1

-0,1

-2,3

59,4

57,1

____

-0,1

-0,2

0,3

59,4

59,4

60,9

____

0,1

1,4

59,4

____

-0,2

59,4

59,2

-62,5

-23,2

5,7

-1,2

-4,8

1,0

-0,7

0,2

-0,3

-85,8

50,2

0,5

-0,5

-0,1

50,1

-43,8

4,0

10,5

-3,4

0,5

-0,6

0,3

-0,2

-32,7

-43,8

2,0

21,0

-1,7

1,1

-0,3

0,6

-0,1

-21,2

50,2

2,7

0,1

0,1

53,1

-62,5

30,3

-1,2

1,7

-0,3

0,3

0,1

-31,6

-62,5

15,1

-2,5

0,8

-0,7

0,1

-0,1

-49,8

49,8

50,2

-0,3

-0,1


___________________________________________________________________________


15


5,41,5608,0

5,461,5683,0

1,51,5609,0

1,56

2,505,624,68

−=⋅−

−=⋅−

−=⋅−

=∆

+−=∆

M

M

∑ −= 1,56 Čvor 6

7,59,1052,0

5,09,1005,0

0,49,1037,0

7,09,1006,0

9,10

2,235,622,508,434,68

=⋅

=⋅

=⋅

=⋅

−=∆

−−+−=∆

M

M

∑= 9,10 Čvor 7

1,211,5439,0

7,21,5405,0

3,301,5456,0

1,54

5,622,5028,43

=⋅

=⋅

=⋅

−=∆

−++−=∆

M

M

∑= 1,54 Čvor 8

3,08,209,0

5,28,291,0

8,22,501,155,62

−=⋅−

−=⋅−

=++−=∆M

∑ −= 8,2 Čvor 9

2,06,209,0

4,26,291,0

6,2

=⋅

=⋅

−=∆M

∑= 6,2 Čvor 10

1,05,109,0

4,15,191,0

5,13,02,1

−=⋅−

−=⋅−

=+=∆M

∑ −= 5,1


___________________________________________________________________________


16

2. ITERACIJA

Čvor 5

2,09,208,0

4,29,283,0

3,09,209,0

9,28,21,0

−=⋅−

−=⋅−

−=⋅−

=+=∆M

∑ −= 9,2 Čvor 6

8,43,952,0

5,03,905,0

4,33,937,0

6,03,906,0

3,92,15,10

−=⋅−

−=⋅−

−=⋅−

−=⋅−

=−=∆M

∑ −= 3,9 Čvor 7

1,19,239,0

1,09,205,0

7,19,256,0

9,22,17,1

=⋅

=⋅

=⋅

−=−−=∆M

∑= 9,2 Čvor 8

1,08,009,0

7,08,091,0

8,0

−=⋅−

−=⋅−

=∆M

∑ −= 8,0 Čvor 9

1,08,009,0

7,08,091,0

8,07,01,0

=⋅

=⋅

−=−−=∆M

∑= 8,0


2,04,208,0

0,24,283,0

2,04,209,0

4,2

=⋅

=⋅

=⋅

−=∆M

∑= 4,2 Čvor 6

7,05,152,0

1,05,105,0

6,05,137,0

1,05,106,0

5,15,00,1

−=⋅−

−=⋅−

−=⋅−

−=⋅−

=+=∆M

∑ −= 5,1 Čvor 7

2,05,039,0

1,05,005,0

3,05,056,0

6,0

=⋅

=⋅

=⋅

−=∆M

∑= 6,0


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17

POPREČNE SILE – nakon 2 Cross-a

9,542,287,26'

2,285

4,685,72'

7,265

3,633,70'

,

6,10

95

=+=

=+

=

=+

=

IIIT

kNT

kNT

kNT

kNT

kNT

kNT

kNT

I 1,798,197,209,197,18'

8,195,5

8,492,59'

7,205,5

9,601,53'

9,195,5

4,591,50'

7,185,5

1,577,45'

0,

84

73

2,6

51

=+++=

=+

=

=+

=

=+

=

=+

=

( )

( ) ( )++=−=∆

+=−=∆

suobjeTTT

TTT

III

IIIIIIIII

5,0'

5,1'

0,0,0,

,,,

KOEFICJENT α

01,15,51,790,59,54

5,56,790,54,56

'' ,0,,,

0,0,,,=

⋅+⋅

⋅+⋅=

⋅+⋅

⋅+⋅=

IIIIIIIIII

IIIIIIII

hThT

hThTα


___________________________________________________________________________


18

KONAČNI MOMENTI

21 CRCR MMM α+=

kNmM

kNmM

kNmM

kNmM

kNmM

kNmM

kNmM

kNmM

kNmM

kNmM

2,53,7001,18,65

,5403,6301,19,9

0,488,4901,13,2

6,582,5901,12,1

0,681,5301,14,14

7,689,6001,12,7

5,391,5001,11,11

4,544,5901,16,5

8,107,4501,10,57

8,801,5701,11,23

95

59

84

48

73

37

62

26

51

15

=⋅+−=

=⋅+−=

=⋅+−=

=⋅+−=

=⋅+=

=⋅+=

=⋅+−=

=⋅+−=

−=⋅+−=

=⋅+=

kNmM

kNmM

kNmM

kNmM

kNmM

kNmM

kNmM

kNmM

kNmM

kNmM

6,425,7201,16,30

2,53,7001,18,65

0,488,4901,13,2

8,1696,3101,17,201

5,2372,2101,11,216

6,1217,3201,16,88

6,388,8501,13,125

4,4310901,17,66

6,425,7201,16,30

5,434,6801,16,25

9,10

10,9

87

78

76

67

65

56

6,10

10,6

−=⋅−=

−=⋅−=

−=⋅−=

=⋅−=

−=⋅−−=

−=⋅−−=

=⋅−=

−=⋅−=

=⋅+−=

=⋅+−=


___________________________________________________________________________


19

48,0

169,8

237,5

121,6

38,6

43,4

42,6

43,5

5,2

41,5

54,048,0

58,668,7

68,039,5

54,4

10,8

23,0

80,8

M

49,7

107,8

172,3

52,7

11,4

1,1

17,218,2

31,8

37,7

12,3

17,1 24,9 19,4

T

DIJAGRAMI


___________________________________________________________________________


20

RJEŠENJE U DIM-u

Nodal displacements: nd u_i v_i phi_i -------------------------------------------------------- 1: 0 0 0 2: 0 0 0 3: 0 0 0 4: 0 0 0 5: 0.00223916 2.75924e-10 -0.000324548 6: 0.00223916 -0.000990001 -0.000159187 7: 0.00223916 -5.12222e-09 0.000146206 8: 0.00223916 -9.40413e-10 -7.01765e-05 9: 0.00474139 4.52698e-10 -0.000213801 10: 0.00474139 -0.000990001 -0.000287 Element end forces: el N_ij T_ij M_ij N_ji T_ji M_ji ------------------------------------------------------------------------------------------------ 1: -15.0504 37.89 77.6536 15.0504 12.11 -6.7586 2: 66.8612 17.2428 51.2586 -66.8612 -17.2428 43.5765 3: 279.394 25.2815 65.9963 -279.394 -25.2815 73.052 4: 51.2953 19.5857 55.5541 -51.2953 -19.5857 52.1675 5: -10.6064 32.3323 57.8912 10.6064 17.6677 3.77019 6: 10.6064 17.6677 47.5617 -10.6064 -17.6677 40.7769 7: 44.4423 -4.44398 -51.1326 -44.4423 4.44398 32.4679 8: 44.8673 51.8107 -123.606 -44.8673 173.189 -240.529 9: 19.5857 106.205 167.477 -19.5857 51.2953 -52.1675 10: 17.6677 -10.6064 -3.77019 -17.6677 10.6064 -40.7769 Reactions: nd R_x R_y M -------------------------------------------------------- 1: -37.89 -15.0504 77.6536 2: -17.2428 66.8612 51.2586 3: -25.2815 279.394 65.9963 4: -19.5857 51.2953 55.5541


___________________________________________________________________________


21

M DIJAGRAM

T DIJAGRAM

Documents

Zadatak 2. - grad.hr postupci_Slunjski.pdfZadatak 2. Za zadani sistem kombinacijom metoda Crossa i Werner-Csonke odrediti MT dijagrame. E = 3. 10 7 kN/m 2, =10 −5 K−1