Talat Lecture 2301: Design Of Members Example 4.1: Bending Moment Resistance Of Open Cross Section With Closed Part

TALAT Lecture 2301

Design of Members Bending Moment Example 4.1 : Bending moment resistance of open cross section with closed part 7 pages Advanced Level prepared by Torsten Höglund, Royal Institute of Technology, Stockholm

Date of Issue: 1999  EAA - European Aluminium Association

TALAT 2301 – Example 4.1

1

Example 4.1. Bending moment resistance of open cross section with closed part 300 . MPa

fo

γ 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 added to indicate shift in neutral axis Nodes, co-ordinates (y, z), thickness (t) and types (T)

i

0

b

h

o

o

1

c

h

t

2

2

c

h

t

o

3

b

h

t

1

4

c

h

o

o

5 6

y

o c

h

. mm z

c h

. mm t

d o

. mm T

6 MPa 10 . Pa

kN 1000 . newton

o o

7

o

h

c

d

o

8

o

o

1

d

o

9

o

h

d

o

10

b

h

o

o

11

b

h

t

1

300

200

100

The beam is composed by two extrusions. The weld is close to the neutral axis why HAZ softening does not influence the moment resistance.

z mm

Nodes Area of cross section elements

i

1 .. rows ( y )

100

rows ( y ) = 12

1

0 0

200

dAi

ti .

yi

yi 1

2

zi

zi 1

2 300

Cross section area

rows ( y )

1 dAi

A

A = 7.269 . 10 mm 3

i =1

TALAT 2301 – Example 4.1

2

2

200

100

0 y mm

100

200

First moment of area Gravity centre

Second moment of area

Elastic section modulus

rows ( y )

1

Sy

dAi zi 1 . 2

zi i =1

rows ( y )

A

z gc = 15.282 mm z gc.gr

1 2

zi

Iy

zi 1

dAi zi . zi 1 . 3

2

Iy

i =1

W el

Sy

z gc

I y.gr z gc > min z

if max z

z gc ,

Iy max z

z gc

,

z gc

Iy

A . z gc

2

I y = 3.344 . 10 mm 8

Iy

4

Iy min z

z gc W el = 1.31 . 10 mm 6

Torsion constant for open parts

rows ( y )

1

I tu

ti dAi . 3

3

2

I tu = 1.156 . 10 mm 5

4

i =1

Hollow part in top flange y1 yh

y2

yh=

y5

75 0

mm

rows y h

t1

240

z2

zh

240

zh =

z5

75

y6

Area within mid-line

z1

75

mm

165

th

240

z6

8

t2

8

th=

t5

4 4

t7

1 0.5 . y h i

A tu

yh

i

1

. z hi

zh

A tu = 5.625 . 10 mm 3

i

1

2

i =1

rows y h

Sum of l/t

1 if t i > 0 ,

Dn

yh

yh

i

Torsion constant It for closed part Torsion constant It whole section Torsion resistance

Wv

i

zh

1

i

th

i =1 4 .A

2

2 i

1

, 10

22

Dn = 71.783

i

2

tu

4

I t = 1.879 . 10 mm

4

6

I tu

2 . A tu . min t h

TALAT 2301 – Example 4.1

I t = 1.763 . 10 mm 6

Dn It

zh

min t h = 4 mm

3

W v = 4.5 . 10 mm 4

3

mm

Effective cross section. 5.4.3 (1)

(5.7 or 5.8)

Outstand Figure 5.2

5.4.3 (1) c)

5.4.4 (5)

εi

ψi

if zi 1

gi

if ψ i > 1 , 0.7

First iteration

z gc < zi

gi

if Ti > 0 , 1 , gi

βi

if t i > 0 , gi .

z gc ,

0.30 . ψ i ,

yi

yi 1

zi 1 zi

z gc

ψ

ρ c if Ti > 0 , if i

Effective thickness

t eff

β

i

2

ε

i

> 6 , 10 .

zi

zi 1

ε

i

β

i

24 .

for elements entirely on the tension side

Cross section area First moment of area Gravity centre

zi

i

z gc

max z

z gc

max z d

2

z gc , ε o . 2

ε

i

β

i

250 . MPa

ε o

,1

, 1.0 , if

β

i

ε

i

max z zi

max z

, ε o.

z gc

> 22 , 32 .

yi

yi 1

rows ( y )

2

zi

zi 1

1

ε

i

β

i

zi 1

220 .

i

β

i

ρ c= i

t eff

1

dAi zi 1 . 2

=

gi =

βi =

εi =

1 1 1 1 0.666 0.666 0.666 -0.109 0.064 1 1

1 1 1 1 0.9 0.9 0.9 0.667 0.719 1 1

9.375 18.75 9.375 1 23.862 1 23.862 27.696 42.968 1 37.5

0.973 9.635 0.779 6.235 0.973 19.271 1 8 0.973 9.635 0.779 6.235 0.973 1.028 1 0 0.973 24.524 0.939 3.756 0.973 1.028 1 0 0.973 24.524 0.939 3.756 99 0.28 1 4 99 0.434 1 4 99 0.01 1 0 99 0.379 1 8

i

mm

2

z gc

Sy A eff

z gc = 5.329 mm

4

i

ψi =

3

i =1

TALAT 2301 – Example 4.1

=

, 1.0

2

A eff = 6.952 . 10 mm zi

Sy

, 99

2

ε

3

i =1 rows ( y )

ε

z gc

A = 7.269 . 10 mm

dAi

A eff

ε o= 0.913

fo

βi

Comment: εi was given a large value (99)

i

zd

z gc

zi 1

z gc < zi

ρ .ct

t eff .

zi

i

ti

5.4.5 (3) a) or c) heat-treated, unwelded

,

0.8 1

if zi z gc . zi 1 z gc , if zi 1

Area of effective thickness elements dAi

z gc

2

Effective cross section.

Second iteration

Node 8 is moved to the neutral axis from the first iteration 5.4.3 (1)

(5.7 or 5.8)

Outstand element

5.4.3 (1) c)

5.4.4 (5)

εi

ψi

if zi 1

gi

if ψ i > 1 , 0.7

z gc < zi

gi

if Ti > 0 , 1 , gi

βi

if t i > 0 , gi .

zi

yi 1

z gc

,

zi

Effective thickness

t eff

β

i

ε

i

zi 1

zi

ε

i

β

i

zi

i

z gc

max z

max z d

i

zi 1

,1

z gc < zi

24 .

z gc = 5.329 mm

2

z8 = 5.339 mm

z gc , ε o .

2

ε

i

β

i

, 1.0 , if

β

i

ε

i

max z zi

z gc

> 22 , 32 .

ε

i

β

i

z9 = 240 mm max z

, ε o.

220 .

zi 1

i = Ti =ψ i =

β e if Ti > 0 , 0 , i

β

i

ε

i

1 2 3 4 5 6 7 8 9 10 11

β Imax max β e β Imax = 25.686 β e if Ti > 0 , i

β

i

ε

i

,0

β Omax max β e

gi =

βi =

εi =

ε

i

β

i

= i

ρ c= i

if β Imax 11 , 1 , if β Imax 16 , 2 , if β Imax 22 , 3 , 4

class I = 4

class O

if β Omax 3 , 1 , if β Omax 4.5 , 2 , if β Omax 6 , 3 , 4 if class I > class O , class I , class O

class O = 4

Area of effective thickness dAi elements

t eff . i

yi

TALAT 2301 – Example 4.1

yi 1

2

zi

zi 1

5

2

t eff

i

mm

=

2 1 1 9.375 0.933 10.044 0.758 6.062 0 1 1 18.75 0.933 20.088 1 8 1 1 1 9.375 0.933 10.044 0.758 6.062 0 1 1 1 0.933 1.071 1 0 0 0.68 0.904 23.974 0.933 25.686 0.912 3.65 0 0.68 0.904 1 0.933 1.071 1 0 0 0.68 0.904 23.974 0.933 25.686 0.912 3.65 06.263·10 -5 0.7 27.941 1.132 24.693 0.935 3.74 0 -1·10 30.01 0.613 0.913 0.672 1 4 0 1 1 1 99 0.01 1 0 1 1 1 37.5 99 0.379 1 8

class I class

, 99

, 1.0

β Omax = 10.044 Cross section class (5.15)

z gc

2

ε

βi

ρ .ct

z gc

max z = 245.329 mm

ψ

2

> 6 , 10 .

zd

z gc

0.8 1

zi 1 z gc , if zi 1

if zi z gc

ρ c if Ti > 0 , if i

Max slenderness - Outstands

z gc

ti

5.4.5 (3) a) or c) heat-treated, unwelded

Max slenderness - Internal elements

zi 1

z gc ,

0.30 . ψ i ,

yi

0.01 . mm

z gc

z8

class = 4

rows ( y ) Area of effective cross section

1 dAi

A eff

A = 7.269 . 10 mm 3

i =1

First moment of area. Gravity centre

2

A eff = 6.862 . 10 mm 3

rows ( y )

1

Sy

zi

dAi zi 1 . 2

2

Sy

z gc

A eff

i =1

z gc = 3.31 mm Second moment of area of effective cross section

rows ( y )

1

Iy

zi

2

zi 1

dAi zi . zi 1 . 3

2

I eff

i =1

Section modulus

zd

Compare gross section

zd

i

z gc

max z.eff

max z d

W eff

i

zi

z gc.gr

max z.gr

max z d

W gr

I eff max z.eff I y.gr max z.gr

ti .

6

W gr = 1.31 . 10 mm 6

3

yi 1

2

zi

zi 1

2

3

z8

A up

2

z7

2

z8 = 68.566 mm

t8 1

W pl

zi

zi 1 . t. i 2

yi

yi 1

2

zi

zi 1

2

i =1

W pl = 1.475 . 10 mm 6

W pl

= 1.126

W el

Shape factor (5.15)

3

A up = 3.249 . 10 mm

2

3

i =1

rows ( y ) Plastic section modulus

yi

4

W eff = 1.298 . 10 mm

A = 7.269 . 10 mm

7

A Move node 8 to plastic neutral axis

2

8

zi

A up

A eff . z gc

I eff = 3.158 . 10 mm

Plastic section modulus Cross section area of upper part

Iy

α 3I 1

22

β Imax W pl . 22 16 W el

α 3O 1

1

α 3 if α 3I < α 3O , α 3I , α 3O α

if class< 3 ,

TALAT 2301 – Example 4.1

W pl W el

, if class< 4 , α 3 ,

6

6

β Omax W pl . 6 4.5 W el

1

α 3= 0.661 W eff W el

class = 4

α = 0.991

3

Bending moment resistance (5.14)

M Rd

α . W el.

300

fo

γ M1

200

M Rd = 354 kN . m Cross section class

class = 4

100

0

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

100

200

300

TALAT 2301 – Example 4.1

7

200

100

0

100

200