ec9 ex41 bending i section

7/30/2019 Ec9 Ex41 Bending I Section

1/5

Ec9_ex41_Bending_I.mcd Bending moment Page 1 of 5

Nodes, co-ordinates (y, z), thickness (t) and types (T)

MPa 106

Pa

kN 1000 newton

i

0

1

2

3

45

6

7

8

9

10

11

:= y

b

c

c

b

co

c

o

o

o

b

b

mm:= z

h

h

h

h

hh c

h

h c

o 1

h

h

h

mm:= t

o

t

t

t

od

o

d

d

d

o

t

mm:= T

o

2

o

1

oo

o

o

o

o

o

1

:=

200 100 0 100 200300

200

100

0

100

200

300

0z

mm

y

mm

The beam is composed by two extrusions. The weld isclose to the neutral axis why HAZ softening does notinfluence the moment resistance.

Nodes i 1 rows y( ) 1..:= rows y( ) 12=

Area of crosssectionelements

dAi

ti

yi

yi 1

( )2

zi

zi 1

( )2

+

:=

Crosssectionarea

A

1

rows y( ) 1

i

dAi

=

:= A 7.269 103 mm2=

Example 4.1. Bending moment resistance of cross section withclosed cross section part and outstands

0

7

6

5

4

321

9

8

1011

b

c

ct

d

2h

fo 300 MPa:= M1 1.1:=

Half flange width b 150:=Half section depth h 240:=

Half closed part c 75:=

Flange thickness t 8:=

Web thickness d 4:=

Zero o 0:=

Type of element- Internal element: T =0- Outstand: T >0

Comment: Node 8 is addedto indicate shift in neutral axis

2007-10-25 THD


2/5


Dn 71.783=

Torsion constantfor closed part

It4 Atu2

Dn:= It 1.763 10

6 mm

4=

Torsion constantwhole section

It It Itu+:= It 1.879 106

mm4

=

Torsionresistance

Wv 2 Atu min th( ):= min th( ) 4mm= Wv 4.5 104

mm3

=

Effective cross section. First iterationzd

iz

izgc:=

6.1.4.3 (1) i if zi 1zgc zi zgc 0.7 0.30 i+,

0.8

1 i,

:=

OutstandFigure 6.2

i if Ti0> 1, i,( ):=

6.1.4.3 (1) c) i if ti0> i

yi

yi 1

( )2

zi

zi 1

( )2

+

ti

, 1,

:= o250 MPa

fo:= o 0.913=

6.1.4.4 (4) i if zizgc( ) zi 1 zgc( ) if zi 1 zgc zi zgc< o

maxz

zi

zgc, o

maxz

zi 1

zgc,

, 99,

:=

6.1.5 (2) classA, no weld

ci

if Ti

0> ifi

i

6> 10i

i

24i

i

2

, 1.0,

, if

i

i

22> 32i

i

220i

i

2

, 1.0,

,

:=

Firstmomentof areaGravity centre

Sy

1

rows y( ) 1

i

zi

zi 1

+( )dA

i

2

=

:= zgcSy

A:= zgc 15.282mm=

zgc.gr zgc:=

Secondmomentof area

Iy

1

rows y( ) 1

i

zi( )

2z

i 1( )2

+ zi

zi 1

+ dA

i

3

=

:= Iy Iy A zgc2

:= Iy 3.344 108

mm4

=

Iy.gr Iy:=Elastic sectionmodulus

Wel if max z zgc( ) min z zgc( )>Iy

max z zgc( ),

Iy

min z zgc( ),

:= Wel 1.31 10

6 mm

3=

Torsionconstant foropen parts

Itu

1

rows y( ) 1

i

dAi

ti( )

2

3

=

:= Itu 1.156 105

mm4

=

Hollow part in top flange

yh

y1

y2

y5

y6

:= yh

75

75

0

75

mm= zh

z1

z2

z5

z6

:= zh

240

240

165

240

mm= th

t1

t2

t5

t7

:= th

8

8

4

4

mm=

Area withinmid-line

Atu

1

rows yh( ) 1

i

0.5 yhi

yhi 1

( ) zhi zhi 1+( )=

:= Atu 5.625 103

mm2

=

Sum of l/t

Dn

1

rows yh( ) 1

i

if ti

0>yh

iyh

i 1( )

2zh

izh

i 1( )

2+

thi

, 1022

,

=

:=

2007-10-25 THD


3/5


Node 8 is moved to the neutral axis

from the first iterationz

8zgc 0.01 mm+:=

zdi

zi

zgc:=6.1.4.3 (1) i if zi 1

zgc zi zgc 0.7 0.30 i+,0.8

1 i,

:=

Outstand

element

gi

if Ti

0> 1, gi

,

( ):=

zgc 5.329mm=6.1.4.3 (1) c) i if ti

0> gi

yi

yi 1

( )2

zi

zi 1

( )2

+

ti

, 1,

:=z

85.339mm= z

9240 mm=

6.1.4.4 (4) i if zizgc( ) zi 1 zgc( )+ if zi 1 zgc zi zgc< o

maxz

zi

zgc, o

maxz

zi 1

zgc,

, 99,

:=

6.1.5 (2)class A,no welds c

iif T

i0> if

i

i

6> 10i

i

24i

i

2

, 1.0,

, if

i

i

22> 32i

i

220i

i

2

, 1.0,

,

:=

Effectivethickness

teff c t( )

:=i

i

9.635

19.271

9.635

1.02824.524

1.028

24.524

0.28

0.434

0.01

0.379

=c

i

0.779

1

0.779

10.939

1

0.939

1

1

1

1

=teff

i

mm

6.235

8

6.235

03.756

0

3.756

4

4

0

8

=i

1

1

1

10.666

0.666

0.666

-0.109

0.064

1

1

= i1

1

1

10.9

0.9

0.9

0.667

0.719

1

1

= i9.375

18.75

9.375

123.862

1

23.862

27.696

42.968

1

37.5

= i0.973

0.973

0.973

0.9730.973

0.973

0.973

99

99

99

99

=

Comment:

i was given a large

value (99) for elements

entirely on the tension

side

Area of effectivethickness

elements dAi teffi yi yi 1( )2

zi zi 1( )2

+

:=

Crosssectionarea

A 7.269 103

mm2

=Aeff

1

rows y( ) 1

i

dAi

=

:=

Aeff 6.952 103

mm2

=

First momentof area

Gravity centre

Sy

1

rows y( ) 1

i

zi

zi 1

+( )dA

i

2

=

:= zgcSy

Aeff:=

zgc 5.329mm=

Effective cross section. Second iteration

2007-10-25 THD


4/5


class 4=

Area of effectivethicknesselements

dAi

teffi

yi

yi 1

( )2

zi

zi 1

( )2

+

:=

Area ofeffectivecross section

Aeff

1

rows y( ) 1

i

dAi

=

:= A 7.269 103

mm2

= Aeff 6.862 103

mm2

=

First momentof area.Gravity centre

Sy

1

rows y( ) 1

i

zi

zi 1

+( )dA

i

2

=

:= zgcSy

Aeff:= zgc 3.31mm=

Secondmoment ofarea of effectivecross section

Iy

1

rows y( ) 1

i

zi( )

2z

i 1( )2

+ zi

zi 1

+ dAi

3

=

:= Ieff Iy Aeff zgc2

:= Ieff 3.158 108

mm4

=

Sectionmodulus

zdi

zi

zgc:= maxz.eff max zd( ):= WeffIeff

maxz.eff:= Weff 1.298 10

6 mm

3=

Comparegrosssection

zdi

zi

zgc.gr:= maxz.gr max zd( ):= WgrIy.gr

maxz.gr:= Wgr 1.31 10

6 mm

3=

Effectivethickness

teff c t( )

:= i

i

10.04

20.09

10.041.07

25.69

1.07

25.69

24.69

2.1910 -3

0.01

0.38

=T

i

2

0

10

0

0

0

0

0

0

1

= gi

1

1

11

0.904

0.904

0.904

0.7

3.26110 -5

1

1

= ci

0.758

1

0.7581

0.912

1

0.912

0.935

1

1

1

=teff

i

mm

6.062

8

6.0620

3.65

0

3.65

3.74

4

0

8

=i

1

2

34

5

6

7

8

9

10

11

= i1

1

11

0.68

0.68

0.68

6.2610 -5

-2.45104

1

1

= i9.375

18.75

9.3751

23.974

1

23.974

27.941

210 -3

1

37.5

= i0.933

0.933

0.9330.933

0.933

0.933

0.933

1.132

0.913

99

99

=

Max slender-ness- Internalelements

Max slender-ness- Outstands

ei

if Ti

0> 0,i

i

,

:= e

iif T

i0>

i

i

, 0,

:=

Imax max e( ):= Omax max e( ):=

Imax 10.044= Omax 10.044=

Cross sectionclass

classI if Imax 11 1, if Imax 16 2, if Imax 22 3, 4,( ),( ),( ):= classI 1=

classO if Omax 3 1, if Omax 4.5 2, if Omax 6 3, 4,( ),( ),( ):= classO 4=6.1.4.4Table 6.2 class if classI classO> classI, classO,( ):=

2007-10-25 THD


5/5


Elastic neutral axis:- first iteration dashed-dotted line- second iteration dashed linePlastic neutral axis dotted line

class 4=Cross sectionclass

MRd 354kN m=

MRd Welfo

M1:=(6.25)

Bending moment resistance

200 100 0 100 200300

200

100

0

100

200

300

0.991=class 4= if class 3

ec9 ex41 bending i section

Documents