C

.

:

, 2013

2013/80

C

1

,

.

,

,

.

,

.

C

2

C

3

.............................................................7

1.1 ....................................................................................................................................... 7

1.2 ..................................................... 8

1.3 ...................................................................................................................... 10

1.4 .................................................................................................................. 12

1.5 ................................................................................... 14

1.6 .............................................. 14

1.7 .................................................................................................................................. 15

................................................................. 18

2.1 .......................................................................................................................... 18

2.2 ........................................................... 19

2.3 .................................................................................................. 20

2.4 ............................................................................................. 21

2.5 .......................................................................................................... 23

2.6 ................................................................................................ 25

............................................................................ 26

3.1 ................................................................................................................................................ 26

3.2 ................................................................................... 26

3.3 ...................................................................................... 28

3.4 ................................................................................................... 31

3.5 ........................................................................................ 32

3.5.1 ................................................................................................................................. 32

3.5.2 ........................................................................................... 32

3.5.3 ................................................................................................ 35

3.5.3.1 ...................................................................................................................................... 35

3.5.3.2 ................................................................................................................. 39

3.6 ........................................................................................ 42

3.6.1 ........................ 42

3.6.2 .......... 44

.............................................................. 45

4.1 ......................................................................................... 45

C

4

4.2 ............................................................................ 47

4.2.1 C ....................................................................................................... 47

4.2.2 C .......................................................... 51

........................................................................................................... 57

5.1 ......................................................................................... 57

5.2 ......................................................................................................... 57

5.3 C3. ......... 58

. 59

6.1 ................................................................................................................................................ 59

6.2 ........................................................................................................................ 59

6.3 FEMAP v10. ......................................... 60

6.3.1 (modeling) .................................................................................... 60

6.3.2 ....................................................................................................................... 63

6.3.2.1 ................................................................................ 65

6.3.3 ( property ) .................................................................................... 65

6.3.4 (mesh generation) ........................................................................................ 67

6.3.5 .................................................................................................... 71

6.3.6 ( loads ).............................................................................................................. 74

6.3.7 ( nalyze ) .......................................................................................... 77

6.4 FEMAP v10 ................................................................ 81

................................................................................. 83

7.1 ............................................................................................................................................ 83

7.2 C ........................................................................................................ 83

7.3 C ............................................................ 87

7.4 C C

........................................................................................................................................... 91

7.5 ........................................................................................................ 93

7.6 ................................................................................... 95

7.7

. .................................................................................................................... 99

................................................................................................................... 101

............................................................................................................................... 102

C

5

2013/80 .. ( : .)

,

C .

C

, . ,

,

.

, .

28 , C ,

, ,

.

,

, ,

.

.

,

,

3 V 1993-1-3.

3.

.

FEMAP v10,

.

, ,

.

.

C

6

National Technical University of Athens

Faculty of Civil Engineering

Institute of Steel Structures

Diploma Thesis

2013/80

Bending of thin-walled beams C and Z section with web perforation

Oikonomou D.E. ( supervised by Raftoyiannis I.)

Abstract

The present study is a preliminary research concerning the detrimental effects of a web

whole, to the load-carrying capacity of lightweight cold-formed steel C and Z section under

bending.

Lightweight steel C and Z-sections are extensively used in constructing light industrial

steel buildings. However, local instabilities as distortional and local buckling can significantly

influence the load carrying capacity of such members, especially in areas of large deformations

and perforations. Due to the limited awareness of the static behavior of perforated cold-formed

steel sections, the research of this phenomenon is essential. In the present study, 28 models

were analyzed of C and Z shaped beams, both with and without stiffeners, with circular and

quadrangular web perforations of three sizes.

In the first part of this study there is a general presentation of cold-formed steel

elements, describing their features, the cold-forming techniques, the usual instabilities and

applications in the construction industry. In the second part the materials are presented.

Following the third part where the types of buckling of lightweight steel members are

presented, focusing on local instabilities based on ENV 1993.1.3.

In the fourth part we present the computation of the moment of one axis bending based

on ENV 1993.1.3. Afterwards we decide which static model will be used to convert the moment

to loads. In the sixth part there is a presentation of FEMAP v10, a finite element analysis

program that has been used for a nonlinear static analysis defining the maximum bending load

of the members.

Finally the results of the above methods are presented along with comparisons and

comments and some final conclusions.

C

7

1

1.1

C Z

.

, ,

.

.

C Z,

, .

.

.

.

.

3, 1993-1-3 ,

,

.

1.1 : C Z

C

8

1.2

.

, ,

,

, .

1.2:

,

. ,

(,

), .

,

(

, , ).

.

, , ,

.

. , .

C

9

,

.

1.3 : Zayed National Museum

1.4 :

C

10

1.3

.

:

1. , ,

2. .

1.5.

,

. (

),

. 50 300mm

1 8mm.

1.5:

C

11

1.6.

.

40 200mm 0,5 2mm.

.

1.6:

,

.

.

1.7 :

1.8 :

C

12

1.4

:

1.

2.

3.

( 6 ).

.

90.

( 1.9) .

.

1.9 : ( folding )

,

.

.

( 90 )

( 1.10).

14 . (extruders)

. ,

, .

C

13

, ,

.

.

1.10 : (press banking)

1.11 : (cold rolling)

C

14

1.5

: 1) , 2)

3) . ,

.

,

.

:

1.

2.

3.

4.

, . . . . . , , , .

1.6

:

,

.

C

15

, .

, , .

.

.

.

1.7 :

1.12 :

C

16

1.13 :

1.14 :

C

17

1.15 :

1.16 :

C

18

2

2.1

.

.

.

2.1 :

fyb(N/mm2) fu(N/mm

2)

. 2:

EN 10025: Part 2 S 235

S 275

S 355

235

275

355

360

430

510

.

3:

/

,

EN 10025: Part 3 S 275 N

S 355 N

S 420 N

S 460 N

S 275 NL

S 355 NL

S 420 NL

S 460 NL

275

355

420

460

275

355

420

460

370

470

520

550

370

470

520

550

C

19

2.1 :

.

4:

,

EN 10025: Part 4 S 275 M

S 355 M

S 420 M

S 460 M

S 275 ML

S 355 ML

S 420 ML

S 460 ML

275

355

420

460

275

355

420

460

360

450

500

530

360

450

500

530

2.2

. , ,

,

.

fya, :

fya = fyb + (fu fyb) 2

g

k n t

A

fya 2

u ybf f

( 2.1)

:

g k k = 7 k = 5 n 90 ri 5t ( 90 n) t

C

20

eff g eff fyb. fya :

i) ii)

iii) , 580C .

2.3

. , b , h r ( ) :

2.1 : , bd , . t .

C

21

2.4 . , . : r 5t r 0.10bp . . r , :

Ag Ag,sh (1 - ) (2.2) Ig Ig,sh (1 2) (2.2)

Iw Iw,sh (1 4) (2.2) :

m

1i

ip,

jn

1j

j

b

90r

43,0

: Ag Ag,sh Ag bp,i i Ig Ig,sh Ig Iw Iw,sh Iw

m n rj j

C

22

2.2 : (2.2) eff , Iy,eff , Iz,eff Iw,eff . , r > 0,04tE/fy .

X

P2

2

rt

bpgr

() X P

2/m trr

)

2sin()

2tan(mr

rg

2.3 : bp ,

C

23

cbp,c

bb

p

cbp,c

bp,d

d

b

bp

() bp

hhwb

p = s

w

sw

() bp (bp = sw)

bp

bp

() bp

bp

bp

() bp

2.3 : bp ,

2.5

,

.

.

.

C

24

2.2: -

b / t 50

b / t 60

c / t 50

b / t 90

c / t 60

d / t 50

b / t 500

45 90

h / t 500 sin

, :

0,2 c / b 0,6

0,1 d / b 0,3

C

25

b , c d 2.2. c/b < 0,2

d/b

C

26

3

3.1

. , , . , , . , . .

3.2

, . , ( thin walled beams) . , . . . . , , . , . , :

C

27

( Local buckling ) ( ). , , . , . ( Distorsional buckling ) , .

,

. ,

.

, , .

.

. ,

.

3.1 :

3.2 : ) ) , )

C

28

, . , , , . . .

3.3

.

( 3.3). ,

x. ,

.

3.3 :

C

29

:

= 1

1 cosn y n y

l ln n

n

n y n yA e B e

l l

(3.1)

n n

. :

2

2x y

, 2

2y x

, 2

xyx y

(3.2)

:

2

2 2 2

1

0 0

2 2 2(1 )2

l

y y y

hdxdy

E

(3.3)

(3.1) (3.2) :

2 23 3

1 21

22 2

nn n n n

n

B A B Anh

l E G G

(3.4)

:

1 2

2cos cos ...o

x xM M M M

l l

(3.5)

= w + Mf

(w) (Mf). Mf = e :

0

2 0N h dy

(3.6)

0

2w xM h e dy M

:

2 2 22

2

0 02 2

l l

wM dxN dx

AE EI (3.7)

C

30

:

1 2 (3.8)

:

0n n

n n.

2 ,

x f.

2 , . c c ,

l, :

wl c

M e

I

, 2 0l ch A

2

42

(3 2 )

l

(3.9)

.

.

.

,

,

beffb .

Von Karman :

cr

y

beff

bp f

(3.10)

Winter :

1 0,22cr cr

y y

beff

bp f f

(3.11)

cr .

C

31

,

.

.

,

,

. ,

,

.

.

. ,

, .

,

:

eff y

M

W fR

, Weff

, 2

1

,

, .

3.4

,

.

.

.

3.4 : ) ) )

C

32

3.5

3.5.1

1993-1-3

, :

.

, .

,

.

,

.

com,Ed,ser

, .

.

3.5.2

:

2

212 1cr

p

t EK

b

: bp t Poisson K , 3.1 3.2

y

p

cr

f

:

C

33

0,673p 1 0,22

1p p

0,673p =1

3.1 :

C

34

3.2 :

max , ,

:

C

35

max1

yf

, 3.1 3.2.

max1

yf

, max p

:

max,max

1

p pyf

. ,

:

p

yf

,

.

:

1. 1 2,

.

2. = 2/1 3.1 3.2 .

3. p

4. , p

5. beff.

6. beff bp.

7. M beff , p t ,

, 5.5.2 EC.3.1.1.

3.5.3

3.5.3.1

,

. .

C

36

, ,

.

3.5 :

.

1993-1-3,

,

, .

,

, .

.

C

37

) C

3.6 :

, ,

. ,

.

,

u :

u

K

,

.

C, C1, C2

,

.

:

C

38

233

12 1

3

p

p

ubb

t

p

ub

C

C

3.6(). 1

1 : 3

1 2 2 3

1 1 1 2

1

4(1 ) 0,5w w f

t

b h b bb h k

:

b1 -

1

( be2 ),

b2 2 ( ).

hw

kf = 0 2 (.. y-y)

2

1

f

Aeffk

Aeff 2 .

eff1 eff2 1 2

.

,

.

d

d :

d = 1,0 0,65d

d = 1,47-0,723 d 0,65 1,38d

d = 0,66

d 1,38d

:

,

yb

d

cr s

f

( cr,s )

C

39

3.5.3.2

( ). , . bp. : 1. 45 135. 2. c : c > 0,2bp 3. bp/t 60 bp/t 90 . , . : .

3.7 :

1. . , n nn-1 (n< n-1), n-1 (n-1) . : i) , com,Ed

=0

yf

. , bp 3.7

C

40

3.5.2 be1 be2. ceff deff : ) : ceff = bp,c

3.5.2

:

, 0,35p c

p

b

b , = 0,5

,0,35 0,6p c

p

b

b , = , 230,5 0,83 ( 0,35)p c

p

b

b

) :

Ceff = bp,c

3.5.2,

( 3.1).

,eff p dd b

3.5.2

. ( 3.2)

ii)

,

.

s :

2( )s e effA t b c (3.12)

2 1 2( )s e e e effA t b c c d (3.13)

,cr s

:

,2 S

cr s

S

KEI

A (3.14)

:

S

iii) 1

3.5.3.1 .

2 ,

com,Ed cr,s.

C

41

3.8 :

, :

C

42

S,red

:

0,

,

yb

s red d s

com Ed

f

A x A

com,Ed

.

s,red tred

s.

t.

3.6

3.6.1

c,Rd

( 3.9) :

el0

Wpl f

yb

Wel f

yb

0

Mc,R

Weff

fyb

3.9 :

effW

elW

M0ybeffRdc, /fWM (3.15)

C

43

effW

Wel

M0elelplelybRdc, /)/1(4)( WWWfM Wplfyb/0 (3.16)

el/

p )3(055,025,05,0el

.

p 673,0el .

d 65,0el .

3.9.

(3.16) :

a) .

b) , .

c) 60.

, :

M0yaelRdc, /fWM (3.17)

3.10 :

Weff , max,Ed

fyb / M0 ,

C

44

, . , .

= 2 / 1 , 3.10.

3.6.2

,

com,Ed = fyb / M0.

. , Wpp,eff

.

C

45

4

4.1

. C ,

, .

l= 1200mm. C

.

.

4.1 :

h (mm) b (mm) c (mm) r (mm) t (mm)

C 100 50 2 2

C 100 50 20 2 2

100 50 2 2

100 50 20 2 2

:

h,

b,

c,

r,

t,

C

46

) )

4.1 : C ) )

4.2 :

C

47

4.3 :

4.2

4.2.1 C

C

( 4.4) S355MC.

S355MC : fyb = 355 N/mm2 fu=430 N/mm2

4.4 : C

C

48

1. .

r=2mm , 5t=10mm r

C

49

2.

,

,

.

,

.

1 = 2 :

= 2/1 = 1

, = 0,43

p :

p = 549 355

1,052 1,052 1,616 0,6732 2,1 10 0,43

p ybb f

t E K

=0,22 1 0,22 1

(1 ) (1 ) 0,53461,616 1,616

p p

beff=bp=0,534549=26,19mm ( 4.6)

4.6 :

3.

,

z ( )

G , :

26,19 2 98 98 2 4942,55

(26,19 98 49) 2Gz

mm

4.

:

C

50

= 2

1

42,550,7674

98 42,55

, 0>>-1 2 27,81 6,92 9,78 7,81 6,92 ( 0,7674) 9,78 0,7674 18,88

:

5

49 3551,052 1,052 0,24 0,673

2 2,1 10 18,88

p yb

p

b f

t E K

= 1,00 ( )

:

9855,45

(1 ) 1 0,7674

p

eff c

hh h mm

1 0,4 22,18eff effh h mm

2 0,6 33,27eff effh h mm

5.

:

21 2 346,38eff eff p eff eff p cA b b h h h h t mm ( < g )

:

,

26,19 2 13,095 49 2 24,58,91

346,38G effY mm

,

26,19 2 98 22,18 2 86,91 33,27 2 59,185 42,55 2 21,28

346,38G effZ

= 42,55mm

:

, 12,25 8,91 3,34YN G G effe Y Y mm ( )

:

, 49 42,55 6,45ZN G G effe Z Y mm ( )

6.

:

, ,

yb

Y Rd eff com

fM W

, Weff=

eff

comz

:

, 1 43,55G effz z mm

100 43,55 56,45comz h z mm

:

C

51

IY,eff=3 3 3 3 32 22,18 2 33,27 2 42,55 49 2 26,19 2

12 12 12 12 12

2 2 226,19 2 98 42,55 49 2 42,55 22,18 2(86,91 42,55)

2

2 442,5522,18 2(86,91 42,55) 42,55 2 503.551,712

mm

Weff,com =3503.551,71 8.920,31

56,45

eff

com

Imm

z

MY,Rd = ,0

3.166.711,39yb

eff com

fW Nmm

=3.166,71kNmm

4.2.2 C

.

.

.

, .

, .

1. .

: r = 2mm , 5t = 10mm r

C

52

4.7 : C

:

22 ( ) 98 2 (48 19) 2 464g p p pA h b c t mm

2 (48 2 24) 2 (19 2 48)17,79

464Gy mm

3 3 32 22 98 48 2 2 192 2 2 (48 2 49 ) 2 (19 2 39,5 )

12 12 12yI

=738.786,67mm4

3 3 32 298 2 2 48 19 22 2 98 2 17,79 2 (48 2 (24 17,79) )

12 12 12zI

+

2 42 (19 2 (48 17,79) ) 175.750,81mm

2.

:

i)45135

= 90

ii)cp>0,2bp c = 20mm > 0,248 = 9,6mm

iii) 60pb

t

4824 60

2

C

53

. ,

, 4.8.

4.8 :

1 : .

) :

1=2 =2/1=1.

= 4

0

355yb

com

fMPa

5

48 3551,052 0,519 0,673

2 2,1 10 4p

= 1,00

beff = 48mm

beff1=beff2=0,548= 24mm

) :

ceff :

Ceff=cp

i) 0,35p

p

c

b =0,5

ii) 0,35 0,6p

p

c

b : 230,5 0,83 ( / 0,35)p pK c b

19

0,39648

p

p

c

b 0,35

C

54

5

19 3551,052 1,052 0,529 0,673

2 2,1 10 0,603

p yb

p

c f

t E K

= 1,00

Ceff=cp=19 mm

As = (beff2+ ceff)t = (24 + 19)2=86mm2

( beff,2 , ceff)

:

Z =24 2 1 19 2 10,5

5,2086

mm

- :

IS()=

3 32 224 2 2 1924 2 (4,20) 19 2 (1 9,5 5,20)

12 12

3.129,49mm4

2 : : 2 3

20,423 0,31 1,5 0,043

p yb ps

s p

h f bI

A b E t

= 0,5

3 : .

S,red ,

:

S,red= 1

,

yb

s

com Ed

f

A

, S,red As

com,Ed .

: s,red=0,586 = 43 mm2

, s,red

:

, 0,5 2 1,00s redreds

t At mm

A

: 2

, , 1 43 2 24 91eff s red effA t b mm

3. :

,

, ,

.

:

24 (2 98 1 98,5) 19 1 86 98 2 49 19 2 10

24 2 24 1 19 1 98 2 46 2 19 2GZ

=44,81mm

C

55

= 2

1

44,810,842

53,19

.

EC3 -1

C

56

Iy,eff=3 32 76,563 2 21,72

12 12

3 3 3 346 2 23 1 1 19 2 19

12 12 12 12

2 276,563 2 (44,43 38,28) 21,72 2(87,284 44,43) 2 223 1 (98,5 44,43) 19 2(44,43 10) 219 1 (88 44,43) 246 2 44,43

2 423 1 (98,5 44,43) 645.788,76mm

Weff,com=, 3645788,76 11.830,67

54,57

y eff

com

Imm

z

MY,Rd=Weff,com0

ybf

11830,67

355

1,00 =4.199.886,58 Nmm = 4.199,89kNmm

4.2 : EC3

EC3(kNmm)

C , Z 3.166,71

C , Z 4.199,89

C

57

5

5.1

(

).

,

Rd.

, ,

- . ,

, .

5.2

5.1.

5.1 :

.

C

58

5.3

C3.

.

1200l mm

h=100mm t=2mm. :

t