02_Elasticna Analiza II Reda Mii-separat 14_15

Zavod za konstrukcije Katedra za metalne konstrukcije

http://www.grad.unizg.hr/predmet/metkon3 Kolegij: Metalne konstrukcije 3 (diplomski studij)

Separat 2: Elastina analiza II reda Mii

1

1. Primjer: Elastina analiza 2. reda, metoda poveanog momenta uslijed bone pominosti

Svrha primjera: Potrebno je proraunati okvir prikazan na slici I.1. Primjer ilustrira elastinu analizu 2. reda indirektnom metodom nazvanom Metoda poveanog mo-menta uslijed bone pominosti.

Statiki sustav i raunska optereenja

Slika I.1. Statiki sustav i raunska optereenja

Nosa je pridrana u etvrtinama raspona (svaka 2 m).

Okvir: stupovi: HE 240 B c= y= 11260 cm4 A= 106,0 cm2

Wpl,y= 1053 cm3

nosai: IPE 240 b= y= 3892 cm4 A= 39,12 cm2

Wpl,y= 366,6 cm3

Kvaliteta elika S 355 Provjera okvira poduprt ili nepoduprt

Okvir je oito nepoduprt. Nije predvien vezni sustav koji bi reducirao horizontalni pomak. Za okvire utjecaj imperfekcija se moe zanemariti ako je:

EdEd V15,0H

kN8,4027215,0kN0,9

Potrebno je uzeti utjecaj imperfekcija.

EN 1993-1-1

5.3.2 (4)B

8,0 m

Hd=9 kN

6,0

m

qd=9 kN/m

IPE 240

HE

240 B

HE

240 B

Pd=100 kN Pd=100 kN

A

B

C

D




2

Imperfekcije okvira

mh0

0 1 200 /

0,1816,03

2816,0

6

2

h

2h

866,02/115,0m/115,0m

283

1866,0816,0

200

1

Ekvivalentna horizontalna sila

kN5,01002

0,89

283

1VH Edi

Kriterij pomian nepomian okvir

Uvjet za nepomian:

10F

F

Ed

crcr

Ed,HEd

Edcr

h

V

H.

Zbroj vertikalnog optereenja:

kN27210020,89P2LqV ddEd .

Zbroj horizontalnog optereenja:

kN105,029HHH idEd .

Konstanta za analizu okvira:

259,0800

600

26011

8923

L

hk

b

c

c

b

.

Pomak vrha stupa okvira

10259,0

1259,02

260110002112

600H

k

1k2

E12

h 3

Edc

3

Ed,H

cm46,4Ed,H

EN 1993-1-1

5.3.2(3)

Slika 5.2.

5.2.1.(3) 5.2.1.(4)




3

Uvjet:

1095,446,4

600

272

10cr

.

Okvir je klasificiran kao pomian. Uinke 2. reda treba uzeti u obzir.

Analiza okvira

95,4cr > 3,0

Elastina analiza 1. reda, metoda poveanog momenta uslijed bone pominosti. MEd, VEd i NEd izraunati elastinom analizom 1. reda, slika I.2.

Slika I.2. M, V i N dijagrami od pojedinanih djelovanja

EN 1993-1-1

5.2.2.(5)

38 38

MEd kNm

36

VEd kN 6,4 6,4

36 6

,4

NEd kN 36 36

6,4

+ tlak + +

+

30 30

MEd kNm

7,5

VEd kN 5,0 5,0

7,5

5,0

NEd kN 7,5 7,5

5,0

+ tlak +

+

30

vlak

MEd kNm VEd kN NEd kN 100 100

+ tlak + +

a) uslijed djelovanja qd=9 kN/m

b) uslijed djelovanja Hd=10,0 kN

c) uslijed djelovanja Pd=100 kN

34




4

Dimenzioniranje okvira

Faktor poveanja:

25,120,01

1

/11

1

cr

.

Stup CD (najnepovoljniji)

U toki D:

kNm5,750,3025,10,38MEd ,

kN1451005,725,10,36NEd ,

kN9,120,525,14,6VEd .

Izvijanje u ravnini okvira

Duina izvijanja za nepomian mod

111 12

K

K K Kc

c

221 22

K

K K Kc

c

3

c

cc cm8,18

600

26011

LK

3

b

bb11 cm4,2

800

89235,0

L5,0KK

89,04,28,18

8,181

2 10 , (zglob)

Oitano za 89,01 i 0,12 i nepomian okvir:

96,0L/Liy

cm57660096,0L96,0Liy .

Izvijanje izvan ravnine okvira

Duina izvijanja L L cmiz 600

EN 1993-1-1

5.2.2.(5)

.K.1

(VII.-100.)

(VII.-101.)

tablica VII.21.

slika VII.-83.




5

Klasifikacija presjeka

Hrbat

40,1610

164

10

212172240

t

r2t2h

t

c

w

f

73,263340,16 Klasa 1.

Pojasnica

53,517

94

172

21210240

t2

r2tb

t

c

f

w

29,7953,5 Klasa 1.

Presjek je svrstan u klasu 1.

Otpornost poprenog presjeka

Otpornost presjeka na tlak:

kN37630,1

5,350,106fAN

0M

yRd,c

0,104,03763

1450,1

N

N

Rd,c

Ed .

Otpornost presjeka na savijanje:

kNm374kNcm382370,1

5,350531fWM

0M

yy,plRd,y,pl

0,120,0374

5,750,1

M

M

Rd,y,pl

Ed,y .

Otpornost presjeka na poprenu silu:

kN681

0,1

3/5,3524,333/fAV

0M

yvRd,z,pl

7,11,220,17,1242106tr2ttb2AA fwfV

2cm24,33 > 2ww cm72,240,16,2020,1th

cm6,207,120,24t2hh fw

EN 1993-1-1

Tablica 5.2.

6.2.4.(2) 6.2.4.(1) 6.2.5.(2)

6.2.5.(1) 6.2.6.(2) 6.2.6.(3) 6.2.6.(1)




6

0,102,0681

9,120,1

V

V

Rd,z,pl

Ed,z .

Interakcija M, N i V:

kN9,12VkN341V5,0 Ed,zRd,z,pl .

Nema redukcije otpornosti zbog poprene sile.

kN145NkN941N25,0 EdRd,c ,

kN145NkN3660,1

5,350,16,205,0

fth5,0 Ed

0M

yww

Nema redukcije otpornosti zbog uzdune sile.

kNm374MM Rd,yRd,y,V,N

kNm374kNm5,75MM Ed,y,V,NEd,y .

Otpornost elementa

Otpornost elementa na izvijanje

Duine izvijanja: Liy=576 cm Liz=600 cm

Elastina kritina sila:

kN7034576

1126021000

L

EIN

2

2

2iy

y2

y,cr

kN2259600

392321000

L

EIN

2

2

2iz

z2

z,cr

731,07034

5,35106

N

fA

y,cr

yy

291,12259

5,23106

N

fA

z,cr

yz

Krivulja izvijanja oko osi y-y b 34,0

Krivulja izvijanja oko osi z-z c 49,0

22

1

EN 1993-1-1

6.2.8(2)

6.2.9.1(4)

6.2.9.1(2)

6.3.1.2(1) Tablica

6.2.

Tablica 6.1.

6.3.1.2(1)




7

2

2,015,0

857,0731,02,0731,034,015,0 2y 767,0

731,0857,0857,0

1

22y

601,1291,12,0291,149,015,0 2z

392,0

291,1601,1601,1

1

22z

kN88620,1

5,35106767,0

fAN

1M

yyRd,y,b

kN47510,1

5,35106392,0

fAN

1M

yzRd,z,b

0,105,02886

1460,1

N

N

Rd,y,b

Ed .

0,110,01475

1460,1

N

N

Rd,z,b

Ed .

Otpornost elementa na savijanje

5,0

z2

T2cr

z

w2cr

z2

1crEI

GIL

I

I

L

EICM

kNm886kNcm88617

392321000

7,1028077600

3923

486900

600

39232100077,1M

5,0

2

2

2

2

cr

65,088617

5,351053

M

fW

cr

yyLT

Krivulja izvijanja a 21,0LT

2LT

2LTLT

LT1

EN 1993-1-1

6.3.1.2(1)

6.3.1.1(3)

6.3.1.1(1)

.K. 1

Tablica VII.30.

EN 1993-1-1

6.3.2.2(1)

Tablica 6.4.

Tablica 6.3.

6.3.2.2(1)




8

2LTLTLTLT 2,015,0

759,065,02,065,021,015,0 2LT

869,065,0759,0759,0

1

22LT

kNm324kNcm484320,1

5,351053869,0

fWM

1M

yyLTRd,b

0,123,0324

5,750,1

M

M

Rd,b

Ed .

Interakcija M i N

0,1/M

Mk

/M

Mk

/N

N

1MRk,y

Ed,zyz

1MRk,yLT

Ed,yyy

1MRky

Ed

0,1/M

Mk

/M

Mk

/N

N

1MRk,y

Ed,zzz

1MRk,yLT

Ed,yzy

1MRkz

Ed

Odreivanje interakcijskih faktora kij (Annex B)

1MRky

Edmy

1MRky

Edymyyy

/N

N8,01C

/N

N2,01Ck

1MRkz

Ed

mLT

1MRkz

Ed

mLT

zzy

/N

N

25,0C

1,01

/N

N

25,0C

1,01k

Faktori ekvivalentnih momenata Cij

4,0M

M4,06,0C

2

1my

4,0M

M4,06,0C

2

1mLT

EN 1993-1-1

6.3.2.2(1)

6.3.2.1(3)

6.3.2.1(1) 6.3.3.(4) Tablica

B.2.

Tablica

B.3.




9

4,06,05,75

04,06,0Cmy

4,06,05,75

04,06,0CmLT

62,00,1/3763767,0

1458,016,0

61,00,1/3763767,0

1452,0731,016,0kyy

61,0kyy

97,0

0,1/3763392,0

145

25,060,0

1,01

96,00,1/3763392,0

145

25,060,0

291,11,01kzy

97,0kzy

0,119,000,1/374869,0

5,7561,0

0,1/3763731,0

145

.

0,132,000,1/374869,0

5,7597,0

0,1/3763392,0

145

.

Nosa BD

U toki D:

kNm5,750,3025,10,38MEd

kN7,120,525,14,6NEd

kN4,455,725,10,36VEd

U toki B:

kNm5,00,3125,10,38MEd

kN7,120,525,14,6NEd

kN6,265,725,10,36VEd

EN 1993-1-1




10

Klasifikacija presjeka

Hrbat

70,302,6

4,190

2,6

1528,92240

t

r2t2h

t

c

w

f

78,303870,30 klasa 2.

Pojasnica

28,48,9

9,41

8,9

2

1522,6120

t

2

r2tb

t

c

f

w

29,7928,4 klasa 1.

Presjek je svrstan u klasu 2. Otpornost poprenog presjeka

Otpornost presjeka na tlak:

kN13890,1

5,3512,39fAN

0M

yRd,c

0,101,01389

7,120,1

N

N

Rd,c

Ed .

Otpornost presjeka na savijanje:

kNm130kNcm014130,1

5,356,366fWM

0M

yy,plRd,y,pl

0,158,0130

5,750,1

M

M

Rd,y,pl

Ed,y

Otpornost presjeka na poprenu silu:

kN392

0,1

3/5,3515,193/fAV

0M

yvRd,z,pl

98,012212,39tr2ttb2AA fwfv

2cm15,1998,05,1262,0 >

EN 1993-1-1

Tablica

5.2. 6.2.4.(2) 6.2.4.(1) 6.2.5.(2) 6.2.5.(1) 6.2.6.(2) 6.2.6.(3)




11

> 2ww cm40,1662,004,2220,1th

cm04,2298,020,24t2hh fw

0,112,0392

4,450,1

V

V

Rd,z,pl

Ed,z

Interakcija M, N i V:

kN4,45VkN196V5,0 Ed,zRd,z,pl .

Nema redukcije otpornosti zbog poprene sile.

kN7,12NkN347N25,0 EdRd,c .

kN7,12NkN2430,1

5,3562,004,225,0

fth5,0 Ed

0M

yww

Nema redukcije otpornosti zbog uzdune sile.

kNm130MM Rd,yRd,y,V,N .

kNm130kNm5,75MM Rd,y,V,NEd,y .

Otpornost elementa

Otpornost elementa na izvijanje

Duine izvijanja: Liy =L/2=400 cm Liz= L/4=200 cm

Elastina kritina sila:

kN5042400

389221000

L

EIN

2

2

2iy

y2

y,cr

kN1469200

6,28321000

L

EIN

2

2

2iz

z2

z,cr

525,05042

5,3512,39

N

fA

y,cr

yy

972,01469

5,3512,39

N

fA

z,cr

yz

Krivulja izvijanja oko osi y-y a 21,0

Krivulja izvijanja oko osi z-z b 34,0

EN 1993-1-1

6.2.6.(1)

6.2.8.(2)

6.2.9.1(4)

6.2.9.1(1)

6.3.1.2(1)

Tablica 6.2.

Tablica 6.1.




12

22

1

2

2,015,0

672,0525,02,0525,021,015,0 2y

916,0

525,0672,0672,0

1

22y

104,1972,02,0972,034,015,0 2z

614,0

972,0104,1104,1

1

22z

kN12720,1

5,3512,39916,0

fAN

1M

yyRd,y,b

kN8530,1

5,3512,39614,0

fAN

1M

yzRd,z,b

0,101,01272

7,120,1

N

N

Rd,y,b

Ed .

0,102,0852

7,120,1

N

N

Rd,z,b

Ed .

Otpornost elementa na savijanje

z2

T2cr

z

w2cr

z2

1crEI

GIL

I

I

L

EICM , 77,1C1 (linearna apr.)

kNm370kNcm37025

6,28321000

88,128077200

6,283

37390

200

6,2832100077,1M

5,0

2

2

2

2

cr

593,037025

5,356,366

M

fW

cr

yyLT

EN 1993-1-1

6.3.1.2(1)

6.3.1.1(3)

6.3.1.1(1)

.K. 1

Tablica VII.30.

EN 1993-1-1

6.3.2.2(1)




13

Krivulja izvijanja a 21,0LT

2LT

2LTLT

LT1

2LTLTLTLT 2,015,0

717,0593,02,0593,021,015,0 2LT

893,0593,0717,0717,0

1

22LT

kNm116kNcm116220,1

5,356,366893,0

fWM

1M

yyLTRd,b

0,165,0116

5,750,1

M

M

Rd,b

Ed,y .

Interakcija M i N s problemom bonog izvijanja

(I) 0,1/M

Mk

/M

Mk

/N

N

1MRk,y

Ed,zyz

1MRk,yLT

Ed,yyy

1MRky

Ed

(II) 0,1/M

Mk

/M

Mk

/N

N

1MRk,y

Ed,zzz

1MRk,yLT

Ed,yzy

1MRkz

Ed

Odreivanje interakcijskih faktora kij (Annex B)

1MRky

Edmy

1MRky

Edymyyy

/N

N8,01C

/N

N2,01Ck

1MRkz

Ed

mLT

1MRkz

Ed

mLT

zzy

/N

N

25,0C

1,01

/N

N

25,0C

1,01k

Faktori ekvivalentnih momenata Cij

EN 1993-1-1

Tablica 6.4.

Tablica 6.3.

6.3.2.2(1)

6.3.2.1(2)

6.3.2.1(1) 6.3.3.(4) Tablica

B.2.




14

Razmatra se savijanje oko osi y-y izmeu oslonaca u ravnini:

05,75

5,0

M

M

;45,05,75

34

M

M

D

B

h

ss

4,08,01,0C smy

4,046,045,08,01,0Cmy

Razmatra se savijanje oko osi y-y izmeu oslonaca izvan ravnine:

4,08,02,0C

0

;48,05,75

36

M

M

smLTh

ss

58,048,08,02,0CmLT

46,00,1/1389916,0

7,128,0146,0

46,00,1/1389916,0

7,122,0525,0146,0kyy

0,1

0,1/1389614,0

7,12

25,058,0

1,01

99,00,1/1389614,0

7,12

25,058,0

972,01,01kzy

0,138,000,1/130717,0

5,7546,0

0,1/1389916,0

7,12

.

0,182,000,1/130717,0

5,750,1

0,1/1389614,0

7,12

.

EN 1993-1-1

Tablica

B.3.

Documents

02_Elasticna Analiza II Reda Mii-separat 14_15