Talat Lecture 2301: Design Of Members Example 9.1: Tension Force And Bending Moment

TALAT Lecture 2301

Design of Members Axial force and bending moment Example 9.1 : Tension force and bending moment 6 pages Advanced Level prepared by Torsten Höglund, Royal Institute of Technology, Stockholm

Date of Issue: 1999  EAA - European Aluminium Association

TALAT 2301 – Example 9.1

1

Example 9.1. Tension force and bending moment Dimensions and material properties Flange height:

h

200 . mm

Flange depth:

b

140 . mm

Web thickness:

tw

Flange thickness:

tf

Length:

L

12 . mm 6 . mm 2 .m

Width of web plate: bw

2 .t f

h

b w = 188 mm

[1] Table 3.2b Alloy: EN AW-6082 T6 EP/O t > 5 mm 260 .

f 0.2

[1] (5.4), (5.5) f o [1] (5.6)

fv

newton mm

2

f 0.2

310 .

fa

fu

newton mm

2

fo f v = 150 MPa

3 E

fu

70000 . MPa

27000 . MPa

G

γ M11.10

Partial safety factors:

γ M2 1.25

Moment and force 140 . kN

Axial tension force and excentricity

N Ed

Bending moment

M y.Ed

Web-flange corner radius

r

kN 1000 . newton

S.I. units

N Ed . exc

exc

114.286 . mm

M y.Ed = 16 kNm

4 . mm

kNm kN . m

MPa 1000000 . Pa

Classification of the cross section in bending. Effective thickness Web [1] Tab. 5.1 [1] 5.4.5

bw β w 0.40 . tw

β w= 6.267

β 1w 9 . ε class w

Local buckling: ρ cw if

250 . MPa

ε

β 2w 13 . ε

ε = 0.981

fo

β 3w 18 . ε

if β w > β 1w , if β w > β 2w , if β w > β 3w , 4 , 3 , 2 , 1

β w ε

18 , 1.0 ,

29

198

β w

β w

ε

TALAT 2301 – Example 9.1

ε

2

2

t w.ef

if class w 4 , t w . ρ cw, t w

class w = 1

ρ cw= 1 t w.ef = 12.0 mm

Flanges [1] 5.4.3 [1] Tab. 5.1

β f

b

tw

2.5 . ε

β 1f

tf

β 3f 5 . ε

β =f 21.333

if β >f β 1f , if β f > β 2f , if β f > β 3f , 4 , 3 , 2 , 1

class f

[1] 5.4.5 Local buckling: ρ cf if

β f

18 , 1.0 ,

ε

29

198

β f

β f

ε t f.ef

β 2f 4 . ε

class f = 4

ρ cf= 0.915

2

ε

if class f 4 , t f . ρ cf, t f

t f.ef = 5.49 mm

Classification of the total cross-section: class

if class f > class w , class f , class w

class = 4

Cross weld [1] Tab. 5.2

ρ haz 0.65 t f.haz ρ haz . t f

HAZ softening factor Effective thickness, flange t f.ef

if t f.ef < t f.haz , t f.ef , t f.haz

t f.ef = 3.9 mm

Effective thickness, web t w.ef [1] Fig. 5.6

t w.haz

ρ haz. t f

t w.haz = 3.9 mm

if t w.ef < t w.haz , t w.ef , t w.haz

Extent of HAZ i web (MIG-weld) b haz

t f.haz = 3.9 mm

t w.ef = 3.9 mm 0.5 . t w

t1

t 1 = 9 mm

tf

if t 1 > 6 . mm , if t 1 > 12 . mm , if t 1 > 25 . mm , 40 . mm , 35 . mm , 30 . mm , 20 . mm

b haz = 30 mm

Bending moment resistance [1] 5.6.2

Elastic modulus of gross cross sectionWel: A gr

2 .b .t f

I gr

1. . 3 bh 12

W el

2 .t f .t w

h b

tw . h

A gr = 3.936 . 10 mm 3

2 .t f

I gr = 2.246 . 10 mm

3

7

I gr . 2

4

W el = 2.246 . 10 mm

3

W ple = 2.69 . 10 mm

3

5

h

2

Plastic modulus W ple

1. 4

b .h

2

b

tw . h

2 .t f

2

5

Elastic modulus of the effective cross sectionWeffe: t f = 6 mm

t f.ef = 3.9 mm

t w = 12 mm

t w.ef = 3.9 mm

Allowing for local buckling and HAZ: A effe

A gr

b. t f

TALAT 2301 – Example 9.1

t f.ef

b haz . t w

3

t w.ef

A effe = 3.399 . 10 mm 3

2

Shift of gravity centre: e ef

b. t f

tf

h t f.ef . 2

2

h t w.ef . 2

b haz . t w

tf

b haz 2

1

.

e ef = 14.038 mm

A effe

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

b. t f

I gr

h t f.ef . 2

tf

2

3

b haz

. t w 12

2

t w.ef

b haz . t w

h t w.ef . 2

tf

b haz

2

2

I effe = 1.816 . 10 mm

4

I effe = 1.749 . 10 mm

4

7

Second moment of area with respect to centre of effective cross section: I effe W effe

7

I effe h 2

[1] Tab. 5.3

2 e ef . A effe

I effe

W effe = 1.533 . 10 mm 5

e ef

Shape factor α for welded, class 4 cross-section

W effe

α

α = 0.683

W el

Design moment and axial force resistance of the cross sectionM( y,Rd and NRd) [1] (5.14)

M y.Rd

f o . α . W el

N Rd

γ M1

f o . A effe

γ M1

M y.Rd = 36.2 kNm

N Rd = 803.4 kN

Section check [1] (5.42a-c)

[1] (5.40)

η 0 1

Class 4 cross section: N Ed N Rd

η0

M y.Ed M y.Rd

TALAT 2301 – Example 9.1

γ0

= 0.616

4

γ 0

1

ξ 0

1

3

Lateral-torsional buckling [1] 5.9.4.3 Lateral stiffness constant

Iz

3 2 .b .t f

h .t w

12

12

h [1] Figure J.2

Varping constant:

Iw

Torsional constant:

It

3

I z = 2.773 . 10 mm

2 t f .I z

h .t w

3

Moment relation

ψ

[1] H.1.2(6)

C1 - constant

C1

I t = 1.354 . 10 mm 5

3

[1] 5.6.6.3(2)

[1] 5.6.6.3(1)

λ LT

6

4

ψ =1

1 1.4 . ψ

1.88

0.52 . ψ

2

C1=1 G = 2.7 . 10 MPa 4

Shear modulus

[1] 5.6.6.3(3)

mm

L=2 m

[1] H.1.2

M cr

I w = 2.609 . 10 3

Length

[1] H.1.3(3)

4

10

4 2 .b .t f

6

2 C 1 .π .E .I z I w . 2 Iz L

2 L .G .I t

π

2.

Wy

E .I z

W el

α .W y .f o

M cr = 62.517 kN . m

λ LT= 0.799

M cr

α LT if ( class> 2 , 0.2 , 0.1 )

α LT= 0.2

λ 0LT if ( class> 2 , 0.4 , 0.6 )

λ 0LT= 0.4

φ LT 0.5 . 1 χ LT

α LT . λ LT 1

φ LT

φ LT

2

λ 0LT

λ LT

λ LT

2

φ LT= 0.859 χ LT = 0.851

2

Design moment and axial force resistance of the cross section(no HAZ) [1] (5.14)

M c.Rd

f .W . o el χ LT γ M1

TALAT 2301 – Example 9.1

N Rd

f o . A gr

γ M1

5

M c.Rd = 45.2 kNm

N Rd = 930.3 kN

Lateral-torsional buckling check [1] 5.9.3.1

ψ vec 0.8

W com

W el A gr = 3.936 . 10 mm 3

[1] (5.38)

M y.Ed σ com.Ed W com

N Ed ψ vec . A gr

N Ed = 140 kN

σ com.Ed= 42.8 MPa [1] (5.39)

M eff.Ed

2

M y.Ed = 16 kNm

W com . σ com.Ed

M eff.Ed = 9.61 kNm M c.Rd = 45.196 kNm M eff.Ed < M c.Rd

TALAT 2301 – Example 9.1

6

OK!