ec9 ex91 tension bending interaction

7/30/2019 Ec9 Ex91 Tension Bending Interaction

1/4

Ec9_ex91_Interaction_Tension Tension force and bending moment Page 1 of 4

Moment and force

Axial tension force and eccentricity NEd 300 kN:=

Bending moment My.Ed 40 kNm:=

S.I. units kN 1000 newton kNm kN m MPa 1000000 Pa

Classification of the cross section in bending. Effective thickness

Web w 0.40bw

tw:= w 5.6=

250 MPa

fo:= 0.981=

The cross welds at the centre of the beam does not influence the buckling class (see 6.1.4.4 (3))

[1] Table 6.21w 11 := 2w 16 := 3w 22 :=

BC "A"=classw if w 1w> if w 2w> if w 3w> 4, 3,( ), 2,( ), 1,( ):= classw 1=

[1] 6.1.5 (2)

Local buckling: cw ifw

22 1.0,32

w

220

w

2

,

:= tw.ef if classw 4 tw cw, tw,( ):= cw 1=

tw.ef 12.0 mm=

Example 9.1. Tension force and bending momentUnder the load at the centre of the beam there is a transverse web stiffener which means that there are crosswelds on both the flanges and the web.

The beam is restrained laterally at the load application point.

tf

tw

h bw

y

z

bf

Dimensions and material properties

Flange height: h 200 mm:=

Flange depth: b 140 mm:=

Web thickness: tw 12 mm:=

Flange thickness: tf 16 mm:=

Length: L 2 m:=

Width of web plate:

bw h 2 tf:= bw 168 mm=

L L

FEd

Lateral buckling

Bending moment

NNEd

[1] Table 3.2b Alloy:EN AW-6082 T6 EP/O t >5 mm

fo 260 MPa:= fu 310 MPa:=

Buckling class BC "A":=

E 70000 MPa:= G 27000 MPa:=

Partial safety factors: M1 1.10

M2

1.25

T Hglund aluMATTER 2007-07-19


2/4


Extent of HAZ i web (MIG-weld) t1 0.333 tw 2 tf+( ):= t1 14.652 mm=

bhaz if t1 6 mm> if t1 12 mm> if t1 25 mm> 40 mm, 35 mm,( ), 30 mm,( ), 20 mm,( ):= bhaz 35 mm=

Bending moment resistance

te,f

te,w

bhaz

[1] 6.2.5.1 Elastic modulus of gross cross section Wel:

Agr 2 b tf h 2 tf( ) tw+:= Agr 6.496 103

mm2

=

Igr1

12b h

3 b tw( ) h 2 tf( )

3:= Igr 4.276 10

7 mm

4=

Wel

Igr 2

h:= Wel 4.276 10

5 mm

3=

Plastic modulus

Wple1

4b h

2 b tw( ) h 2 tf( )

2:= Wple 4.968 10

5 mm

3=

Elastic modulus of the effective cross section Weffe:

tf 16 mm= tf.ef 13.842 mm= tw 12 mm= tw.ef 12 mm=

Allowing for local buckling:

Aeffe Agr b tf tf.ef( ) bhaz tw tw.ef( ):= Aeffe 6.194 103

mm2

=

Shift of gravity centre:

eef b tf tf.ef( )h

2

tf

2

bhaz tw tw.ef( )h

2

tfbhaz

2

+

1

Aeffe

:= eef 4.487 mm=

Second moment of area with respect to centre of gross cross section:

t2

b3

b2

Flangesf

b tw

tf:= 1f 3 := 2f 4.5 := 3f 6 := f 8=

[1] 6.1.4.3

[1] Table 6.2 classf if f 1f> if f 2f> if f 3f> 4, 3,( ), 2,( ), 1,( ):= classf 4=[1] 6.1.5 (2)

Local buckling: cf

iff

6 1.0,

10

f

24

f

2,

:= cf

0.865=

tf.ef if classf 4 tf cf, tf,( ):= tf.ef 13.84 mm=

Classification of the total cross-section:

class if classf classw> classf, classw,( ):= class 4=

Net section at the cross weld

[1] Table 6.2 HAZ softening factor u.haz 0.60:=

[1] 6.2.5.1 (2) Effective thickness, flange tf.haz u.haz tf:= tf.haz 9.6 mm= tf 16 mm=

Effective thickness, web tw.haz u.haz tw:= tw.haz 7.2 mm=

[1] 6.1.6.3 (3)



3/4


Design moment and axial force resistance of the net section (My,Rd.net andNRd.ner)

[1] (5.14) My.Rd.netfu Wnet

M2

:= NRd.netfu Anet

M2

:= My.Rd.net 63.5 kNm= NRd.net 966.6 kN=

Bending and axial force [1] 6.2.9

At effective section

[1] 6.2.9.1 Class 4 cross section: 0 1:= 0 1:= 0 1:=

[1] (5.40)NEd

NRd

0My.Ed

My.Rd

0

+ 0.646= in section without weld

At net section

0 1:= 0 1:= 0 1:=

[1] (5.40)NEd

NRd.net

0My.Ed

My.Rd.net

0

+ 0.94= in section with weld

Ieffe Igr b tf tf.ef2 2

12

tw tw.ef bhaz tw tw.ef2

tf2

:=

Ieffe 4.02 107

mm4

=Second moment of area with respect to centre of effective cross section:

Ieffe Ieffe eef2

Aeffe:= Ieffe 4.007 107

mm4

=

Weffe

Ieffe

h

2eef+

:= Weffe 3.835 105 mm3=

[1] Tab. 5.3 Shape factor for welded, class 4 cross-section Weffe

Wel:= 0.897=

Design moment and axial force resistance of the cross section (My,Rd andNRd)

[1] (5.14) My.Rdfo Wel

M1

:= NRdfo Aeffe

M1

:= My.Rd 90.7 kNm= NRd 1.5 103

kN=

Net section with HAZ softeningAnet 2 b tf.haz tw.haz bw+:= Anet 3.898 10

3 mm

2=

Inet 0.5 b tf.haz h tf( )2

tw.haz bw3

1

12+:= Inet 2.56 10

7 mm

4=

Wnet

Inet 2

h:= Wnet 2.56 10

5 mm

3=



4/4


[1] I.1.2 kwt

kw L

E Iw

G It:= kwt 0.894=

[1] (I.3) crC1

kz1 kwt

2+:= cr 2.409=

Mcr crE Iz G It

L

:=Wy Wel:=

[1] 6.3.2.3 (1) LT Wy fo

Mcr:= LT 0.563=

[1] 6.3.2.2 (2) LT if class 2> 0.2, 0.1,( ):= LT 0.2=

0LT if class 2> 0.4, 0.6,( ):= 0LT 0.4=

[1] 6.3.2.1 (1) LT 0.5 1 LT LT 0LT( )+ LT2

+:= LT 0.675=

LT1

LT LT2 LT2+

:= LT 0.955=

Design moment and axial force resistance of the cross section (no HAZ)

[1] (5.14) Mb.Rd LTfo Wel

M1

:= NRdfo Agr

M1

:= Mb.Rd 96.5 kNm= NRd 1.5 103

kN=

Lateral-torsional buckling check with influence of axial tensile force

In combination with bending moment, reduce axial tension force with a factor vec 0.8:=

My.Ed

Mb.Rdvec

NEd

NRd 0.258=

Lateral-torsional buckling [1] 6.3.2

Lateral stiffness constant Iz2 b

3 tf

12

h tw3

12+:= Iz 7.346 10

6 mm

4=

[1] Figure J .2 Varping constant:Iw

h tf( )2

Iz

4:=

Iw 6.218 10

10

mm

6=

Torsional constant: It2 b tf

3 h tw

3+

3:= It 4.975 10

5 mm

4=

Length L 2 m=

[1] I.1.2 Moment relation and 0:=

ky 1:= kz 1:= kw 1:=Support conditions

[1] I.1.2 (6) C1 - constant C1 0.310 0.428 + 0.262 2

+( )0.5

:= C1 1.796=

Mono symmetry parameter zj 0:=

Load application parameter zg 0:=

Shear modulus G 2.7 104

MPa=


ec9 ex91 tension bending interaction

Documents