ec9 ex44 bending trapez class 4

7/30/2019 Ec9 Ex44 Bending Trapez Class 4

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Ec9_ex44_Bending_Trapez_Class Bending moment resistance Page 1 of 7

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

hzbhaz

c b( )2

h2

+

h:=

Material EN-AW 6005A-T6, t


2/7


th

6

6

8

8

8

mm=

Area withinmid-line

Atu

1

rows yh( ) 1

i

0.5 yhi

yhi 1

( ) zhi zhi 1+( )=

:= Atu 1.98 104

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

,

=

:= Dn 86.694=

Torsion constantfor closed part It4

Atu

2

Dn:= It 1.809 107 mm4=

Torsion constantwhole section

It It Itu+:= It 1.818 107

mm4

=

Torsionresistance

Wv 2 Atu min th( ):= min th( ) 6mm= Wv 2.376 105

mm3

=

zgc 39.211 mm=

Effective cross section. First iteration

zdi zi zgc:=6.1.4.3 (1) i 1:= j 9 11..:= jz8 zgc

z11

zgc:= j 4+ j:=

maxz max zd( ):=

Crosssectionarea

A

1

rows y( ) 1

i

dAi

=

:= tg t:=

A 5.388 103

mm2

=

Firstmomentof areaGravity centre

Sy

1

rows y( ) 1

i

zi

zi 1

+

( )

dAi

2

=:= z

gc

Sy

A:=

zgc 39.211 mm=

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.gr Iy:= Iy 9.842 106

mm4

=

Elastic sectionmodulus

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

max z zgc( ),

Iy

min z zgc( ),

:=

Wel 1.646 105

mm3

=

Torsionconstant foropen parts

Itu

1

rows y( ) 1

i

dAi

ti( )

2

3

=

:= Itu 9.255 104

mm4

=

Hollow part, (nodes 2, 5, 11, 12 and 15)

yh

y2

y5

y11

y12

y15

:= yh

120

120

80

80

120

mm= zh

z2

z5

z11

z12

z15

:= zh

0

0

99

99

0

mm= th

t1

t4

t11

t12

t14

:=

T Hglund aluMATTER 2007-07-23


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C2.O

20:=

6.12( )

ci

if Ti

0> ifi

i3.O> C1.O

i

i

C2.Oi

i

2

, 1.0,

, if

i

i3.I> C1.I

i

i

C2.Ii

i

2

, 1.0,

,

:=

Effectivethickness

teff c t( )

:=

Heat Affected Zone

Flange thaz.f o.haz t2:= thaz.f 3.18mm=

:= (to give in table)

Web thaz.w

o.haz

t9

:= thaz.w

4.24mm=

n 2 3..:= teffn

if teffn

thaz.f> thaz.f, teffn

,( ):= teff8 n teffn:=teff

9if teff

9thaz.w> thaz.w, teff

9,( ):= teff15 teff9:=

Give results in a table g

c

teff

mm

Table augment g, , , , c,teff

mm,

:=

Comment:i was given a large value (99)

for elements entirely on

the tension side

Table

0 1 2 3 4 5 6

0

1

2

34

5

6

7

8

9

10

11

12

1314

15

0 0 0 0 0 0 0

1 1 12.667 1.332 9.513 0.725 4.351

1 1 12.667 1.332 9.513 0.725 3.18

1 1 38.667 1.332 29.039 0.764 3.181 1 38.667 1.332 29.039 0.764 4.583

1 1 38.667 1.332 29.039 0.764 3.18

1 1 12.667 1.332 9.513 0.725 3.18

1 1 12.667 1.332 9.513 0.725 4.351

1 1 0 1.332 0 1 0

-0.656 0.503 6.717 1.332 5.044 1 4.24

-0.656 0.503 6.717 99 0.068 1 8

-0.656 0.503 6.717 99 0.068 1 8

1 1 0 99 0 1 8

-0.656 0.503 6.717 99 0.068 1 8-0.656 0.503 6.717 99 0.068 1 8

-0.656 0.503 6.717 1.332 5.044 1 4.24

=

(6.4 or 6.5) gi

if i 1> 0.7 0.30 i+,0.8

1 i,

:= Outstand

Figure 6.2g

iif T

i0> 1, g

i,( ):=

6.1.4.3 (1) c) k 1 2..:= k

y2

y0

0.5t13

t2

:= k 5+ k:= y2 120 mm=

y0

200 mm=

l 3 5..:= l

y5

y2

t13

t4

:= y2 y0

t2

13.333=

j if t110> g

j

y11

y8

( )2

z11

z8

( )2

+

t11

, 1,

:= j 4+ j:=

6.1.4.4 (4) o250 MPa

fo:= i o

maxz

zgc:= i if zi zgc( ) zi 1 zgc( ) i, 99,:= o 1.078=

6.1.5 (2) and Table 6.3 - for internal part T =0 3.I 18:= C1.I 29:= C2.I 198:=

BC "A"= - for outstand parts T >1 3.O

5:= C1.O

9:=



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k 1 2..:= k

y2

y0

0.5t13

t2

:= k 5+ k:= l 3 5..:= l

y5

y2

t13

t4

:=

j if t110> g

j

y11

y8

( )2

z11

z8

( )2

+

t11

, 1,

:= j 4+ j:=

6.1.4.4 (4) o250 MPa

fo:= i o

maxz

zgc:= i if zi zgc( ) zi 1 zgc( ) i, 99,:= o 1.078=

6.1.5 (2) and Table 6.3 - for internal part T =0 3.I 18= C1.I 29= C2.I 198=

BC "A"= - for outstand parts T >1 3.O

5= C1.O

9= C2.O

20=

6.12( )

ci

if Ti

0> ifi

i

3.O> C1.Oi

i

C2.Oi

i

2

, 1.0,

, if

i

i

3.I> C1.Ii

i

C2.Ii

i

2

, 1.0,

,

:=

Effectivethickness

teff c t( )

:=

Area of effectivethicknesselements dA

iteff

iy

iy

i 1( )

2z

iz

i 1( )

2+

:=

Crosssectionarea Aeff

1

rows y( ) 1

idAi

=:=

A 5.388 103

mm2

=

Aeff 4.404 103

mm2

=First momentof area

Gravity centre

Sy

1

rows y( ) 1

i

zi

zi 1

+( )dA

i

2

=

:=zgc

Sy

Aeff:= zgc 47.259 mm=

Effective cross section. Second iteration

Node 10 and 13 is moved to the neutral

axis from the first iteration

z10

zgc 0.01 mm+:= z13 zgc 0.01 mm+:=

y10

y5

z10

z11

y5

y11

( ):= y13 y10:=

zdi

zi

zgc:=6.1.4.3 (1) i 1:= j 9 11..:= j

z8

zgc

z11

zgc:= j 4+ j:=

maxz max zd( ):=

(6.4 or 6.5) gi

if i 1> 0.7 0.30 i+,0.8

1 i,

:= Outstand

Figure 5.2g

iif T

i0> 1, g

i,( ):=

6.1.4.3 (1) c)



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ei

if Ti

0>i

i

, 0,

:= Omax max e( ):= Omax 11.747=

Cross section

class

classI if Imax 7 1, if Imax 11 2, if Imax 15 3, 4,( ),( ),( ):= classI 4=

classO if Omax 2 1, if Omax 3 2, if Omax 4 3, 4,( ),( ),( ):= classO 4=6.1.4.4

class if classI classO> classI, classO,( ):= 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 5.388 10

3 mm

2=

Aeff 4.224 103

mm2

=First

momentof area.Gravity centre

Sy

1

rows y( ) 1

i

zi

zi 1

+( )dAi

2

=

:= zgcSy

Aeff:= zgc 49.274 mm=

Heat Affected Zone

teffn

if teffn

thaz.f> thaz.f, teffn

,( ):= teff8 n teffn:=

:= (to give in table)

teff9

if teff9

thaz.w> thaz.w, teff9

,( ):= teff15 teff9:=

Table augment g, , , , c, teffmm

,

:= g

c

teff

mm

Comment:

i was given a large value (99)

for elements entirely on

the tension side

Table

0 1 2 3 4 5 6

0

1

2

3

4

5

67

8

9

10

11

12

13

14

15

0 0 0 0 0 0 0

1 1 12.667 1.078 11.747 0.621 3.727

1 1 12.667 1.078 11.747 0.621 3.18

1 1 38.667 1.078 35.858 0.655 3.18

1 1 38.667 1.078 35.858 0.655 3.929

1 1 38.667 1.078 35.858 0.655 3.18

1 1 12.667 1.078 11.747 0.621 3.181 1 12.667 1.078 11.747 0.621 3.727

1 1 0 1.078 0 1 0

-0.913 0.426 5.686 1.078 5.273 1 4.24

-0.913 0.426 5.686 1.078 5.273 1 8

-0.913 0.426 5.686 1.078 5.273 1 8

1 1 0 1.078 0 1 8

-0.913 0.426 5.686 1.078 5.273 1 8

-0.913 0.426 5.686 1.078 5.273 1 8

-0.913 0.426 5.686 1.078 5.273 1 4.24

=

Max slender-ness- Internalelements

ei

if Ti

0> 0,i

i

,

:= Imax max e( ):= Imax 35.858=

Max slender-ness- Outstands



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Sy

1

rows y( ) 1

i

zi

zi 1

+( )dA

i

2

=

:= zgc.eSy

Aele:=

zgc.e 43.145 mm=Secondmoment ofarea of effectivecross section

Iy

1

rows y( ) 1

i

zi( )2

zi 1( )2

+ zi zi 1+

dAi

3=:= Iele Iy Aele zgc.e

2

:=

Iele 9.088 106

mm4

=

Sectionmodulus

zdi

zi

zgc.e:= maxz.ele max zd( ):= WeleIele

maxz.ele:= Wele 1.627 10

5 mm

3=

Wel 1.646 105

mm3

=

Plastic section modulus for section with reduced thickness in HAZ

A 5.388 103

mm2

=

Area of grosscross section

Apl

1

rows y( ) 1

i

ti

yi

yi 1

( )2 zi zi 1( )2+

=

:=Apl 4.824 10

3 mm

2=

Cross sectionarea of upperpart

Aup

1

7

i

ti

yi

yi 1

( )2

zi

zi 1

( )2

+

=

2 t9

y9

y8

( )2

z9

z8

( )2

+

+:=

Aup 2.316 103

mm2

=z

927.815 mm=

Move node 10to plasticneutral axis

z10

z9

A

2Aup

t11

t13

+:= z

13z

10:= z

1051.453 mm=

y10 y5

z10

z11

y5 y11( ):= y13 y10:= y10 99.211mm=

Secondmoment ofarea of effectivecross section

Iy

1

rows y( ) 1

i

zi( )

2z

i 1( )2

+ zi

zi 1

+ dA

i

3

=

:= Ieff Iy Aeff zgc2

:=

Ieff 7.812 106

mm4

=

Section

modulus

zdi

z

i

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

maxz.eff

:= Weff 1.571 105

mm3

=

Comparegrosssection

zdi

zi

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

maxz.gr:= Wgr 1.646 10

5 mm

3=

Elastic section modulus for section with reduced thickness in HAZ

Gross thickness t tg:=

Reduced thickness in HAZ h 2 3..:= th

o.haz th:= t

h 3+th

:= t9

o.haz t9:= t

15o.haz t15

:=

Area dAi

ti

yi

yi 1

( )2

zi

zi 1

( )2

+

:=A 5.388 10

3 mm

2=

Area ofeffectivecross section

Aele

1

rows y( ) 1

i

dAi

=

:= Aele 4.824 103

mm2

=

Firstmomentof area.Gravity centre



7/7


200 150 100 50 0 50 100 150 200110

50

10

class 4=Cross section

Ieff 7.812 106 mm4=Aeff 4.224 103 mm2=My.Rd 30.7kN m=

Wple 2.081 105

mm3

=Wel 1.646 105

mm3

= 0.954=My.Rd Welfo

M1:=(6.25)

Bending moment resistance

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

200 150 100 50 0 50 100 150 200100

49

0

0.954=class 4=

Plastic sectionmodulus

Wple

1

rows y( ) 1

i

zi

zi 1

+

2ti

yi

yi 1

( )2

zi

zi 1

( )2

+

=

:= Wple 2.081 105

mm3

=

Wple

Wel1.264=

Shape factor

3.I 18 o:= 2.I 13 o:= 3.O 5 o:= 2.O 4 o:=

(6.27) 3IWele

Wel1

3.I Imax

3.I 2.I

Wple

Wele1

+

:= 3O

Wele

Wel1

3.O Omax

3.O 2.O

Wple

Wele1

+

:=

3I 0.146= 3O 0.639=

3 if 3I 3O< 3I, 3O,( ):= 3 0.639=

if class 3

ec9 ex44 bending trapez class 4

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