This 3-sentence summary provides the high-level information about the document: The 7-page document presents Example 4.1 on calculating the bending moment resistance of an open cross-section aluminum member with a closed part, including determining the effective thickness of elements, cross-sectional properties, and section modulus. Nodes, coordinates, thicknesses, and other geometric properties of the beam are defined, and formulas are provided to calculate bending properties based on the effective cross section.

1. 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

2. Example 4.1. Bending moment resistance of open cross section
with closed part
fo 300 . MPa 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)
0 b h o o
1 c h t 2
2 c h t o
MPa 10 . Pa
6
3 b h t 1
4 c h o o kN 1000 . newton
5 o h c d o
i y . mm z . mm t . mm T
6 c h o o
7 o h c d o
8 o o 1 d o 300
9 o h d o
10 b h o o 200
11 b h t 1
100
The beam is composed by two extrusions. The weld is
close to the neutral axis why HAZ softening does not 0
z
influence the moment resistance. 0
mm
100
Nodes i 1 .. rows ( y ) 1 rows ( y ) = 12
Area of cross 200
ti .
section 2 2
dAi yi yi 1 zi zi 1
elements
300
200 100 0 100 200
Cross rows ( y ) 1
y
A = 7.269 . 10 mm
section 3 2
A dAi mm
area
i =1
TALAT 2301 – Example 4.1 2

3. First rows ( y ) 1
dAi Sy
moment Sy zi zi 1 . z gc
of area 2 A
Gravity centre i =1
z gc = 15.282 mm
z gc.gr z gc
Second rows ( y ) 1
dAi
zi . zi 1 . A . z gc
moment 2 2 2
Iy zi zi 1 Iy Iy
of area 3
i =1
I y = 3.344 . 10 mm
8 4
I y.gr Iy
Elastic section Iy Iy
modulus W el if max z z gc > min z z gc , ,
max z z gc min z z gc
W el = 1.31 . 10 mm
6 3
Torsion rows ( y ) 1 2
ti
dAi . I tu = 1.156 . 10 mm
constant for 5 4
I tu
open parts 3
i =1
Hollow part in top flange
y1 z1 t1
75 240 8
y2 75 z2 240 t2 8
yh yh= mm zh zh = mm th th= mm
y5 0 z5 165 t5 4
y6 75 z6 240 t7 4
rows y h 1
Area within
0.5 . y h . z A tu = 5.625 . 10 mm
3 2
mid-line A tu yh hi zh
i i 1 i 1
i =1
Sum of l/t rows y h 1 2 2
yh yh zh zh
i i 1 i i 1 22
Dn if t i > 0 , , 10 Dn = 71.783
th
i =1 i
4 .A
2
Torsion constant tu
I t = 1.763 . 10 mm
6 4
for closed part It
Dn
Torsion constant
I t = 1.879 . 10 mm
6 4
whole section It It I tu
Torsion
2 . A tu . min t h W v = 4.5 . 10 mm
4 3
resistance Wv min t h = 4 mm
TALAT 2301 – Example 4.1 3

4. Effective cross section. First iteration
zi 1 z gc zi z gc zd zi z gc
5.4.3 (1) ψi if zi 1 z gc < zi z gc , , i
zi z gc zi 1 z gc max z max z d
0.8
(5.7 or 5.8) gi if ψ i > 1 , 0.7 0.30 . ψ i ,
1 ψ i
Outstand
Figure 5.2 gi if Ti > 0 , 1 , gi
2 2
yi yi 1 zi zi 1 250 . MPa
5.4.3 (1) c) βi if t i > 0 , gi . ,1 ε o ε o 0.913
=
ti fo
max z max z
5.4.4 (5) εi if zi z gc . zi 1 z gc , if zi 1 z gc < zi z gc , ε o . , ε o. , 99
zi z gc zi 1 z gc
5.4.5 (3) 2 2
a) or c) β ε ε β ε ε
> 6 , 10 . 24 . > 22 , 32 . 220 .
i i i i i i
heat-treated, ρ c if Ti > 0 , if , 1.0 , if , 1.0
i ε β β ε β β
unwelded i i i i i i
Effective βi t eff
t eff ρ .c
t = i
thickness ε ρ c= =
ψi = gi = βi = εi = i i mm
1 1 9.375 0.973 9.635 0.779 6.235
1 1 18.75 0.973 19.271 1 8
Comment: 1 1 9.375 0.973 9.635 0.779 6.235
εi was given a large value (99) 1 1 1 0.973 1.028 1 0
for elements entirely on 0.666 0.9 23.862 0.973 24.524 0.939 3.756
the tension side 0.666 0.9 1 0.973 1.028 1 0
0.666 0.9 23.862 0.973 24.524 0.939 3.756
-0.109 0.667 27.696 99 0.28 1 4
0.064 0.719 42.968 99 0.434 1 4
1 1 1 99 0.01 1 0
1 1 37.5 99 0.379 1 8
Area of effective
thickness
t eff .
elements 2 2
dAi yi yi 1 zi zi 1
i
Cross rows ( y ) 1
A = 7.269 . 10 mm
3 2
section A eff dAi
area
A eff = 6.952 . 10 mm
3 2
i =1
First moment rows ( y ) 1
dAi Sy
of area Sy zi zi 1 . z gc
2 A eff
Gravity centre i =1
z gc = 5.329 mm
TALAT 2301 – Example 4.1 4

5. Effective cross section. Second iteration
Node 8 is moved to the neutral axis z8 z gc 0.01 . mm
from the first iteration
zi 1 z gc zi z gc zd zi z gc
5.4.3 (1) ψi if zi 1 z gc < zi z gc , , i
zi z gc zi 1 z gc max z max z d
0.8 max z = 245.329 mm
(5.7 or 5.8) gi if ψ i > 1 , 0.7 0.30 . ψ i ,
1 ψ i
Outstand gi if Ti > 0 , 1 , gi
element
yi yi 1
2
zi zi 1
2 z gc = 5.329 mm
5.4.3 (1) c) βi if t i > 0 , gi . ,1
ti z8 = 5.339 mm z9 = 240 mm
max z max z
5.4.4 (5) εi if zi z gc zi 1 z gc , if zi 1 z gc < zi z gc , ε o . , ε o. , 99
zi z gc zi 1 z gc
5.4.5 (3)
a) or c) 2 2
β ε ε β ε ε
> 6 , 10 . 24 . > 22 , 32 . 220 .
i i i i i i
heat-treated, ρ c if Ti > 0 , if , 1.0 , if , 1.0
unwelded i ε i
β i
β i
ε i
β i
β i
Effective βi t eff
t eff ρ .c
t = i
thickness ε ρ c= =
ψ
i = Ti = i = gi = βi = εi = i i mm
1 2 1 1 9.375 0.933 10.044 0.758 6.062
β i
Max slender- β e if Ti > 0 , 0 , 2 0 1 1 18.75 0.933 20.088 1 8
ness i ε i 3 1 1 1 9.375 0.933 10.044 0.758 6.062
- Internal 4 0 1 1 1 0.933 1.071 1 0
elements β Imax max β e
5 0 0.68
0.904 23.974 0.933 25.686 0.912 3.65
β Imax = 25.686 6 0 0.68
0.904 1 0.933 1.071 1 0
7 0 0.68
0.904 23.974 0.933 25.686 0.912 3.65
β i
8 06.263·10 -5 0.7 27.941 1.132 24.693 0.935 3.74
Max slender- β e if Ti > 0 , ,0 9 0 -1·10 30.01 0.613 0.913 0.672 1 4
ness i ε i 10 0 1 1 1 99 0.01 1 0
- Outstands
β Omax max β e 11 1 1 1 37.5 99 0.379 1 8
β Omax = 10.044
Cross section class I if β Imax 11 , 1 , if β Imax 16 , 2 , if β Imax 22 , 3 , 4 class I = 4
class
class O if β Omax 3 , 1 , if β Omax 4.5 , 2 , if β Omax 6 , 3 , 4 class O = 4
(5.15)
class if class I > class O , class I , class O class = 4
Area of effective
t eff .
thickness 2 2
dAi yi yi 1 zi zi 1
elements i
TALAT 2301 – Example 4.1 5

6. rows ( y ) 1
Area of
A eff dAi
A = 7.269 . 10 mm
effective 3 2
cross section i =1
A eff = 6.862 . 10 mm
3 2
First rows ( y ) 1
moment dAi Sy
of area. Sy zi zi 1 . z gc
2 A eff
Gravity centre i =1
z gc = 3.31 mm
Second rows ( y ) 1
moment of dAi
zi . zi 1 . A eff . z gc
2 2 2
area of effective Iy zi zi 1 I eff Iy
3
cross section i =1
I eff = 3.158 . 10 mm
8 4
Section I eff
W eff = 1.298 . 10 mm
6 3
modulus zd zi z gc max z.eff max z d W eff
i max z.eff
Compare I y.gr
W gr = 1.31 . 10 mm
6 3
gross zd zi z gc.gr max z.gr max z d W gr
i max z.gr
section
Plastic section modulus
A = 7.269 . 10 mm
3 2
Cross section 7
ti . A up = 3.249 . 10 mm
area of upper 2 2 3 2
A up yi yi 1 zi zi 1
part
i =1
A
Move node 8 A up
2
to plastic z8 z7 z8 = 68.566 mm
neutral axis t8
rows ( y ) 1
Plastic section zi zi 1
W pl . t. yi yi 1
2
zi zi 1
2
W pl = 1.475 . 10 mm
6 3
modulus 2 i
i =1 W pl
= 1.126
W el
Shape factor
22 β Imax W pl 6 β Omax W pl
(5.15) α 3I 1 . 1 α 3O 1 . 1
22 16 W el 6 4.5 W el
α 3 if α 3I < α 3O , α 3I , α 3O α 3 0.661
=
W pl W eff
α if class< 3 , , if class< 4 , α 3 , class = 4 α = 0.991
W el W el
TALAT 2301 – Example 4.1 6

7. Bending moment resistance 300
fo
(5.14) M Rd α . W el.
γ M1 200
M Rd = 354 kN . m
100
Cross section class = 4
class
Elastic neutral axis: 0
- first iteration dashed-dotted line
- second iteration dashed line
Plastic neutral axis dotted line
100
200
300
200 100 0 100 200
TALAT 2301 – Example 4.1 7