Talat Lecture 2301: Design Of Members Example 4.2: Hollow Cross Section (polygon)

TALAT Lecture 2301

Design of Members Bending Moment Example 4.2 : Hollow cross section (polygon) 5 pages Advanced Level prepared by Torsten Höglund, Royal Institute of Technology, Stockholm

Date of Issue: 1999  EAA - European Aluminium Association

TALAT 2301 – Example 4.2

1

Example 4.2 Hollow cross section (polygon) Half width

b

50 . mm

Thickness

s

1.2 . mm

Half width of flat parts

Co-ordinates and thickness

Comment: Nodes 4 and 9 are added to indicate shift in neutral axis

y

1

0 . mm

O

b

a

a = 20.711 mm

b

a

s

a

b

s

a

b

s

b

a

s

fo

b

O

s

γ M1 1.1 Nodes

z

b

a

t

s

a

b

s

a

b

s

b

a

s

b

O

s

b

a

s

dAi

ti .

yi

2

zi

zi 1

6 MPa 10 . Pa

kN 1000 . newton

z

rows ( y )

0

2 50 0 mm

dAi

A = 397.6 mm

i =1 First moment of area. Gravity centre

Second moment of area

rows ( y )

dAi zi 1 . 2

zi

z gc.gr

1 zi

2

zi 1

2

dAi

zi . zi 1 . 3

Torsion resistance

yi 1 . zi

4

1

yi

yi 1

2

zi

zi 1

A v = 8.284 . 10 mm 3

zi 1

Check: ( 2 .b )

Dn

Wv

2

1 0.5 . yi

4 .A v

A . z gc 5

Av

Iv

Iy

z gc

I y = 5.255 . 10 mm

i =1 Torsion constant

Iy

i =1

i =1

Sum of l/t

A

z gc = 0 mm

Iy

rows ( y )

2

Sy

z gc

i =1

rows ( y ) Area within mid-line

1

Sy

rows ( y )

50

y

1

A

2

2 .( b

2

a ) = 8.284 . 10 mm 2

3

2

ti

Dn = 276.142

2

I v = 9.941 . 10 mm 5

Dn 2 . A v . min ( t )

TALAT 2301 – Example 4.2

1

0

50

Cross section area

1 .. rows ( y )

i

0

mm

yi 1

300 . MPa

50

Gross cross section Area of cross section elements

2

min ( t ) = 1.2 mm

2

4

W v = 1.988 . 10 mm 4

3

2

Effective cross section. ψ

5.4.3 (1)

(5.7 or 5.8)

εi

5.4.4 (5)

if t i > 0 , gi .

βi

if zi > z gc

5.4.5 (3) c) heat-treated, unwelded

ρ c if i

Effective thickness

t eff

Area of effective thickness dAi elements

yi

yi 1

zi 1 zi

i

ε

i

> 22 , 32 .

z gc

ε

i

β

i

i

,

zi

zd

z gc

zi 1

2

z gc

max z

zi 1

2

z gc < zi

z gc , ε o .

2

ε

i

β

i

yi 1

2

zi

zi 1

1 dAi

A eff

1 zi

Sy

zi 1

2

z gc

ε

=

ρ c= i

εi =

0.414 1 0.414 0 0 0.414 1 0.414 0 0

0.824 1 0.824 0.7 0.7 0.824 1 0.824 0.7 0.7

28.452 34.518 28.452 12.081 12.081 28.452 34.518 28.452 12.081 12.081

0.913 31.167 0.8 0.96 0.913 37.812 0.692 0.831 0.913 31.167 0.8 0.96 1.418 8.518 1 1.2 99 0.122 1 1.2 99 0.287 1 1.2 99 0.349 1 1.2 99 0.287 1 1.2 99 0.122 1 1.2 1.418 8.518 1 1.2

dAi zi 1 . 2

2

i

A = 397.645 mm z gc

2

Sy A eff

0 z mm

2

0 0

if zi > z gc . zi 1 < z gc , z gc , zi

z GC1 mm

i

1 .. rows ( y ) zi

Check

1

if zi > z gc . zi 1 < z gc , z gc , zi

z4 = 4.046 mm

z9 = 4.046 mm

50

OK !

50

0 y mm

TALAT 2301 – Example 4.2

3

=

βi =

50

zi

i

gi =

z gc = 4.046 mm

0 .. rows ( y )

t eff

ψi =

The nodes in the vertical webs are moved to the neutral axis from first iteration i

, 99

Comment: εi is given a large value (99) for elements entirely on the tension side

, 1.0

i =1

Gravity centre

max z d

max z

, ε o.

z gc

zi

A eff = 362.498 mm rows ( y )

z gc

ε o= 0.913

fo

max z

i =1

First moment of area.

250 . MPa

ε o

,1

βi

yi

zi

i

i

zi

220 .

ρ c 0

0

max z = 50 mm

ψ

ρ .ct

t eff .

z gc

zd

0.8 1

zi 1 > z gc , if zi 1

β

Zero position

ti

rows ( y )

Cross section area

z gc ,

0.30 . ψ i ,

if ψ i > 1 , 0.7

gi

5.4.3 (1) c)

z gc < zi

if zi 1

i

First iteration

,

50

y GC1 mm

mm

Second iteration ψi

5.4.3 (1)

(5.7 or 5.8)

gi

βi

5.4.3 (1) c)

5.4.4 (5)

εi

if zi > z gc

ρ c if i

Effective thickness

t eff

Cross section class Area of effective thickness elements Cross section area First moment of area.

β e i

class

dAi

yi

if t i > 0 , gi .

z gc ,

0.30 . ψ i , 1

if ψ i > 1 , 0.7

5.4.5 (3) c) heat-treated, unwelded

Max slenderness - Internal elements

z gc < zi

if zi 1

yi 1

2

zi 1 z gc zi z gc , zi z gc zi 1 z gc

i

ε

> 22 , 32 .

i

β

i

ψ

max z

i

zi

220 .

i

zi 1

i

ε

i

z gc < zi

z gc , ε o .

max z zi

t eff . i

max z

, ε o.

z gc

zi 1

yi

yi 1

rows ( y )

2

i

β

i

, 1.0

max β e

zi 1

=

ρ c= i

t eff

βi =

εi =

ε

0.458 1 0.458 0 0 0.363 1 0.363 0 0

0.837 1 0.837 0.7 0.7 0.809 1 0.809 0.7 0.7

28.906 34.518 28.906 14.441 9.721 27.918 34.518 27.918 9.721 14.441

0.913 0.913 0.913 1.349 99 99 99 99 99 1.349

31.665 0.791 0.949 37.812 0.692 0.831 31.665 0.791 0.949 10.707 1 1.2 0.098 1 1.2 0.282 1 1.2 0.349 1 1.2 0.282 1 1.2 0.098 1 1.2 10.707 1 1.2

2

i

mm

dAi i =1 rows ( y )

A eff = 361.596 mm

1

dAi zi 1 . 2

zi

Sy i =1

2

Sy

z gc

A eff

z gc = 4.144 mm

Gravity centre

50 0

Gross cross section

z

I y.gr

Iy

mm

I y.gr = 5.255 . 10 mm 5

4

z GC1 mm

0 0

z GC2 mm

50 50

0 y mm

TALAT 2301 – Example 4.2

4

=

gi =

1

A eff

i

ψi =

class = 4

zi

, 99

z gc

2

ε

β max = 37.812

if β max 22 , 3 , 4

max z d

,1

βi β max

z gc

max z = 54.046 mm

2

ρ .ct β

zi

i

ti

ε

zd

0.8

zi 1 > z gc , if zi 1

β

z gc = 4.046 mm

,

50

y GC1 mm

Second moment of area of effective I y cross section

rows ( y )

1 2

zi

zi 1

I eff

dAi

zi . zi 1 . 3

2

zd

Compare gross section

zd

Moment resistance

M Rd

2

5

i

zi

z gc

max z.eff

max z d

W eff

i

zi

z gc.gr

max z.gr

max z d

W gr

W eff .

A . z gc

I eff = 5.187 . 10 mm

i =1

Section modulus

Iy

I eff

4

W eff = 9.579 . 10 mm 3

max z.eff I y.gr

W gr = 1.051 . 10 mm 4

max z.gr

fo

3

3

M Rd = 2.613 kN . m

γ M1

Axial force Move node close zi to GC1 to nearest node 5.4.3 (1) c)

βi

ρ c if i

Effective thickness

t eff

Area of effective thickness elements dAi

First moment of area. Gravity centre

yi

if t i > 0 ,

5.4.5 (3) c) heat-treated, unwelded

Cross section area

z GC1 < 1 . mm , zi 1 , zi

if zi

β

yi 1

2

zi

2

zi 1

,1

ti i

ε o

> 22 , 32 .

ε o β

220 .

i

ε o β

2

ε o= 0.913

, 1.0

i

βi

ρ .ct

t eff .

yi

i

yi 1

rows ( y )

2

zi

zi 1

2

1 dAi

A eff

A eff = 275.335 mm

2

i =1 rows ( y )

1

Sy

zi

dAi zi 1 . 2

=

i = βi =

ε o

1 2 3 4 5 6 7 8 9 10

37.812 37.812 37.812 0 37.812 37.812 37.812 37.812 0 37.812

34.518 34.518 34.518 0 34.518 34.518 34.518 34.518 0 34.518

ρ c= i 0.692 0.692 0.692 1 0.692 0.692 0.692 0.692 1 0.692

Sy

z gc

A eff

i =1 z gc = 0 mm

50 0 z mm 0

Axial force resistance

N Rd

A eff .

0

fo

z GC

γ M1

mm

N Rd = 75.091 kN

50 50

0 y GC

y , mm mm

TALAT 2301 – Example 4.2

5

50

t eff

i

mm

=

0.831 0.831 0.831 1.2 0.831 0.831 0.831 0.831 1.2 0.831