Example Calculation of Effective Section Properties for a Cold-Formed Lipped Channel Section in...
Transcript of Example Calculation of Effective Section Properties for a Cold-Formed Lipped Channel Section in...
Document Ref: SX022a-EN-EU Sheet 1 of 8 Title
CALCULATION SHEET
Example: Calculation of effective section properties for a cold-formed lipped channel section in bending
Eurocode Ref EN 1993-1-3 Made by V. Ungureanu, A. Ruff Date Dec 2005 Checked by D. Dubina Date Dec 2005
Example: Calculation of effective section properties for a cold-formed lipped channel section in bending This example deals with the effective properties calculation of a cold-formed lipped channel section subjected to bending about its major axis.
For practical design of light gauge sections to EN1993, designers will normally use software or refer to manufacturers’ data. This example is presented for illustrative purposes
Basic Data
The dimensions of the cross-section and the material properties are: Total height mm200=h
Total width of flange in compression mm741 =b
Total width of flange in tension mm662 =b Total width of edge fold mm8,20=c Internal radius mm3=r
Nominal thickness mm2nom =t
Steel core thickness mm96,1=t
Basic yield strength 2yb mmN350=f
Modulus of elasticity 2mmN210000=E Poisson’s ratio 3,0=ν
Partial factor 00,1M0 =γ
EN1993-1-3 § 3.2.4(3) EN1993-1-3 § 2(3)
The dimensions of the section centre line are:
Web height mm1982200nomp =−=−= thh
Width of flange in compression mm72274nom1p1 =−=−= tbb
Width of flange in tension mm64266nom2p2 =−=−= tbb
Example: Calculation of effective section properties for a cold-formed lipped channel section in bendingC
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Document Ref: SX022a-EN-EU Sheet 2 of 8 Title
CALCULATION SHEET
Example: Calculation of effective section properties for a cold-formed lipped channel section in bending
Eurocode Ref EN 1993-1-3 Made by V. Ungureanu, A. Ruff Date Dec 2005 Checked by D. Dubina Date Dec 2005
Width of edge fold mm8,19228,202nomp =−=−= tcc
Checking of geometrical proportions
The design method of EN1993-1-3 can be applied if the following conditions are satisfied:
60≤tb 6075,3796,1741 <==tb – OK
50≤tc 5061,1096,18,20 <==tc – OK
500≤th 50004,10296,1200 <==th – OK
EN1993-1-3 § 5.2
In order to provide sufficient stiffness and to avoid primary buckling of the stiffener itself, the size of stiffener should be within the following range:
6,02,0 ≤≤ bc 28,0748,201 ==bc 0,60,280,2 << – OK
32,0668,202 ==bc 0,60,320,2 << – OK
The influence of rounding of the corners is neglected if:
5tr ≤ 553,196,13 <==tr – OK
10,0p ≤br 10,004,07231p <==br – OK
10,005,06432p <==br – OK
EN1993-1-3 § 5.1(3)
Gross section properties
( ) ( ) 2pp2p1pbr mm73219864728,19296,12 =+++××=+++= hbbctA
Position of the neutral axis with respect to the flange in compression:
( )[ ]mm88,96
222
br
2p
2ppp2ppp
b1 =+++−
=A
tchhbchcz
Example: Calculation of effective section properties for a cold-formed lipped channel section in bendingC
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Document Ref: SX022a-EN-EU Sheet 3 of 8 Title
CALCULATION SHEET
Example: Calculation of effective section properties for a cold-formed lipped channel section in bending
Eurocode Ref EN 1993-1-3 Made by V. Ungureanu, A. Ruff Date Dec 2005 Checked by D. Dubina Date Dec 2005
Effective section properties of the flange and lip in compression
The general (iterative) procedure is applied to calculate the effective properties of the compressed flange and the lip (plane element with edge stiffener). The calculation should be carried out in three steps:
EN1993-1-3 § 5.5.3.2
Step 1:
Obtain an initial effective cross-section for the stiffener using effective widths of the flange determined by assuming that the compressed flange is doubly supported, the stiffener gives full restraint ( ∞=K ) and that design strength is not reduced ( 0ybEdcom, / Mf γσ = ).
EN1993-1-3 § 5.5.3.2 (3)
Effective width of the compressed flange
The stress ratio: 1=ψ (uniform compression), so
the buckling factor is: for internal compression element. 4σ =k
yb235 f=ε
The relative slenderness:
789,043502354,28
96,1724,28 σ
p1bp, =
××==
ktb
ελ
The width reduction factor is:
( ) ( ) 914,0789,0
13055,0789,03055,022
bp,
bp, =+×−
=+−
=λ
ψλρ
The effective width is: mm865729140p1eff ,,bb =×== ρ
mm9328655050 effe2e1 ,,,b,bb =×===
EN1993-1-3 § 5.5.2
and
EN1993-1-5 § 4.4
Example: Calculation of effective section properties for a cold-formed lipped channel section in bendingC
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Document Ref: SX022a-EN-EU Sheet 4 of 8 Title
CALCULATION SHEET
Example: Calculation of effective section properties for a cold-formed lipped channel section in bending
Eurocode Ref EN 1993-1-3 Made by V. Ungureanu, A. Ruff Date Dec 2005 Checked by D. Dubina Date Dec 2005
Effective width of the edge fold The buckling factor is:
if 35,0p1cp, ≤bb : 5,0σ =k
if 6,035,0 p1cp, ≤< bb : ( )3 2p1cp,σ 35,083,05,0 −+= bbk
35,0275,0728,19p1cp, <==bb so 5,0σ =k
EN1993-1-3 § 5.5.3.2 (5a)
The relative slenderness:
614,05,03502354,28
96,18,194,28 σ
pp,c =
××==
ktc
ελ
EN1993-1-5 § 4.4
The width reduction factor is:
13,1614,0
188,0614,0188,022
cp,
cp, =−
=−
=λ
λρ
but 1≤ρ so 1=ρ
The effective width is:
mm8,198,191peff =×== cc ρ
Effective area of the edge stiffener:
( ) ( ) 2effe2s mm3,1038,199,3296,1 =+×=+= cbtA
EN1993-1-3 § 5.5.3.2 (5a)
§ 5.5.3.2 (6)
Step 2:
Use the initial effective cross-section of the stiffener to determine the reduction factor, allowing for the effects of the continuous spring restraint.
EN1993-1-3 § 5.5.3.2 (3)
The elastic critical buckling stress for the edge stiffener is
s
sscr A
IEK2, =σ
where:
EN1993-1-3 § 5.5.3.2 (7)
Example: Calculation of effective section properties for a cold-formed lipped channel section in bendingC
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s an
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Document Ref: SX022a-EN-EU Sheet 5 of 8 Title
CALCULATION SHEET
Example: Calculation of effective section properties for a cold-formed lipped channel section in bending
Eurocode Ref EN 1993-1-3 Made by V. Ungureanu, A. Ruff Date Dec 2005 Checked by D. Dubina Date Dec 2005
K is the spring stiffness per unit length:
fp213
1p2
12
3
5,01
)1(4 khbbbhbtEK
++⋅
−=
ν
with:
1b – distance from the web to the centre of the effective area of the stiffener in compression (upper flange)
EN1993-1-3 § 5.5.3.1(5)
mm73,6196,1)8,199,32(
29,3296,19,3272)(2
effe2
e2e2p11 =
×+××
−=+
−=tcb
btbbb
0f =k for bending about the y-y axis
mmN439,0=K
sI is the effective second moment of area of the stiffener:
( ) ( )
2
effe2
2effeff
eff
2
effe2
2eff
e2
3eff
3e2
s 2221212 ⎥⎦
⎤⎢⎣
⎡+
−+⎥⎦
⎤⎢⎣
⎡+
++=cb
cctccb
ctbtctbI
4s mm3663=I
so, the elastic critical buckling stress for the edge stiffener is
2scr, mmN78,355
3,1033663210000439,02
=×××
=σ
Example: Calculation of effective section properties for a cold-formed lipped channel section in bendingC
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Document Ref: SX022a-EN-EU Sheet 6 of 8 Title
CALCULATION SHEET
Example: Calculation of effective section properties for a cold-formed lipped channel section in bending
Eurocode Ref EN 1993-1-3 Made by V. Ungureanu, A. Ruff Date Dec 2005 Checked by D. Dubina Date Dec 2005
Thickness reduction factor χd for the edge stiffener
The relative slenderness:
992,078,355350scr,ybd === σλ f
The reduction factor will be:
if 65,0d ≤λ 0,1d =χ
if 38,165,0 d << λ dd 723,047,1 λχ −=
if 38,1d ≥λ dd 66,0 λχ =
38,1992,065,0 d <=< λ so 753,0992,0723,047,1d =×−=χ
EN1993-1-3 § 5.5.3.2 (3)
Figure 5.10d
EN1993-1-3 § 5.5.3.1 (7)
EN1993-1-5 § 4.4 (2)
Step 3:
As the reduction factor for buckling of the stiffener is χd < 1, iterate to refine the value of the reduction factor for buckling of the stiffener.
EN1993-1-3 § 5.5.3.2 (3)
Figure 5.10e
The iterations are carried out based on modified values of ρ obtained using:
M0ybdiEd,com, γχσ f= and dpredp, χλλ =
The iteration stops when the reduction factor χ converges.
EN1993-1-3 § 5.5.3.2 (10)
Initial values (iteration 1): Final values (iteration n):
753,0d =χ 737,0nd,d == χχ
mm9,32e2 =b mm9,35ne2,e2 == bb
mm8,19eff =c mm8,19neff,eff == cc
Final values of effective properties for flange and lip in compression are:
737,0d =χ mm9,35e2 =b mm8,19eff =c
and mm9,32e1 =b
Example: Calculation of effective section properties for a cold-formed lipped channel section in bendingC
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Document Ref: SX022a-EN-EU Sheet 7 of 8 Title
CALCULATION SHEET
Example: Calculation of effective section properties for a cold-formed lipped channel section in bending
Eurocode Ref EN 1993-1-3 Made by V. Ungureanu, A. Ruff Date Dec 2005 Checked by D. Dubina Date Dec 2005
mm 44,1737,096,1dred =×== χtt EN1993-1-3 § 5.5.3.2 (12)
Effective section properties of the web
The position of the neutral axis with regard to the flange in compression:
( )( ) deffe2e1pp2p
d2
eff2
ppp2pppc
222χχ
cbbhbcchhbchc
h+++++
+++−= mm6,101c =h
The stress ratio:
949,06,1011986,101
c
pc −=−
=−
=h
hhψ
The buckling factor: 2σ 78,929,681,7 ψψ +−=k 58,22σ =k
The relative slenderness:
914,058,223502354,28
96,11984,28 σ
php, =
××==
kth
ελ
EN1993-1-5 § 4.4
(Table 4.1)
The width reduction factor is:
( ) ( ) 959,0914,0
949,03055,0914,03055,022
hp,
hp, =−×−
=+−
=λ
ψλρ
Example: Calculation of effective section properties for a cold-formed lipped channel section in bendingC
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ed o
n M
onda
y, O
ctob
er 2
5, 2
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Thi
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Document Ref: SX022a-EN-EU Sheet 8 of 8 Title
CALCULATION SHEET
Example: Calculation of effective section properties for a cold-formed lipped channel section in bending
Eurocode Ref EN 1993-1-3 Made by V. Ungureanu, A. Ruff Date Dec 2005 Checked by D. Dubina Date Dec 2005
The effective width of the zone in compression of the web is:
mm5,976,101959,0ceff =×== hh ρ
Near the flange in compression:
mm395,974,04,0 effe1 =×== hh
Near the neutral axis:
mm5,585,976,06,0 effe2 =×== hh
The effective width of the web is:
Near the flange in compression:
mm39e11 == hh
Near the flange in tension:
( ) ( ) mm9,1545,586,101198e2cp2 =−−=−−= hhhh
Effective section properties
Effective cross-section area:
])([ deff2e1e212ppeff χcbbhhbctA ++++++=
( )[ ]737,08,199,359,329,15439648,1996,1eff ×++++++×=A
2eff mm2,689=A
Position of the neutral axis with regard to the flange in compression:
( ) ( )[ ]eff
d2
eff2
12p2pp2pppc
2222A
chhhhhbchctz
χ++−++−=
mm3,102c =z
Example: Calculation of effective section properties for a cold-formed lipped channel section in bendingC
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Document Ref: SX022a-EN-EU Sheet 9 of 8 Title
CALCULATION SHEET
Example: Calculation of effective section properties for a cold-formed lipped channel section in bending
Eurocode Ref EN 1993-1-3 Made by V. Ungureanu, A. Ruff Date Dec 2005 Checked by D. Dubina Date Dec 2005
Position of the neutral axis with regard to the flange in tension:
mm7,953,102198cpt =−=−= zhz
Second moment of area:
2effcdeff
2cde2
2ce1
21c1
22t2
2t2p
2ptp
d3
eff3
de23
e13
p3
p23
23
1yeff,
)2)(()(
)2()2()2(12
)(12
)(1212121212
cztcztbztb
hzthhzthtzbcztc
tctbtbtctbththI
−+++
+−+−++−+
+++++++=
χχ
χχ
4yeff, mm4140000=I
Effective section modulus:
- with regard to the flange in compression
3
c
yeff,cy,eff, mm 40460
3,1024140000
===z
IW
- with regard to the flange in tension
3
t
yeff,ty,eff, mm 43260
7,954140000
===z
IW
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Example: Calculation of effective section properties for a cold-formed lipped channel section in bending
SX022a-EN-EU.doc
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RESOURCE TITLE Example: Calculation of effective section properties for a cold-formed lipped channel section in bending
Reference(s)
ORIGINAL DOCUMENT
Name Company Date
Created by V. Ungureanu, A. Ruff BRITT Ltd. Timisoara, Romania
05/12/2005
Technical content checked by D. Dubina BRITT Ltd. Timisoara, Romania
08/12/2005
Editorial content checked by
Technical content endorsed by the following STEEL Partners:
1. UK G W Owens SCI 12/4/06
2. France A Bureau CTICM 12/4/06
3. Sweden B Uppfeldt SBI 11/4/06
4. Germany C Müller RWTH 11/4/06
5. Spain J Chica Labein 12/4/06
Resource approved by Technical Coordinator
G W Owens SCI 23/08/06
TRANSLATED DOCUMENT
This Translation made and checked by:
Translated resource approved by:
Example: Calculation of effective section properties for a cold-formed lipped channel section in bendingC
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