Example Calculation of Effective Section Properties for a Cold-Formed Lipped Channel Section in Bending

Example: Calculation of effective section properties for a cold-formed lipped channel section in bending

CALCULATION SHEET

Document Ref:

SX022a-EN-EU

1

Title

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

Sheet

of

8

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 coldformed 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

Created on Monday, October 25, 2010 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Access Steel Licence Agreement

The dimensions of the cross-section and the material properties are: h = 200 mm Total height Total width of flange in compression

b1 = 74 mm

Total width of flange in tension

b2 = 66 mm

Total width of edge fold

c = 20,8 mm

Internal radius

r = 3 mm

Nominal thickness

t nom = 2 mm

Steel core thickness

t = 1,96 mm

Basic yield strength

f yb = 350 N mm 2

Modulus of elasticity

E = 210000 N mm2

Poisson’s ratio

ν = 0,3

Partial factor

γ M0 = 1,00

The dimensions of the section centre line are:

Web height

hp = h − t nom = 200 − 2 = 198 mm

Width of flange in compression

bp1 = b1 − t nom = 74 − 2 = 72 mm

Width of flange in tension

bp2 = b2 − t nom = 66 − 2 = 64 mm

EN1993-1-3 § 3.2.4(3)

EN1993-1-3 § 2(3)

Example: Calculation of effective section properties for a cold-formed lipped channel section in bending

CALCULATION SHEET

Document Ref:

SX022a-EN-EU

2

Title

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

Sheet

of

8

cp = c − t nom 2 = 20,8 − 2 2 = 19,8 mm

Width of edge fold

Checking of geometrical proportions The design method of EN1993-1-3 can be applied if the following conditions EN1993-1-3 are satisfied: § 5.2

b t ≤ 60

b1 t = 74 1,96 = 37,75 < 60 – OK

c t ≤ 50

c t = 20,8 1,96 = 10,61 < 50 – OK

h t ≤ 500

h t = 200 1,96 = 102,04 < 500 – OK

Created on Monday, October 25, 2010 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Access Steel Licence Agreement

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:

0,2 ≤ c b ≤ 0,6

c b1 = 20,8 74 = 0,28

0,2 < 0,28 < 0,6 – OK

c b2 = 20,8 66 = 0,32

0,2 < 0,32 < 0,6 – OK

The influence of rounding of the corners is neglected if:

r t≤5

r t = 3 1,96 = 1,53 < 5 – OK

r bp ≤ 0,10

r bp1 = 3 72 = 0,04 < 0,10 – OK r bp 2 = 3 64 = 0,05 < 0,10 – OK

Gross section properties Abr = t (2cp + bp1 + bp2 + hp ) = 1,96 × (2 × 19,8 + 72 + 64 + 198 ) = 732 mm 2

Position of the neutral axis with respect to the flange in compression: z b1

[c (h = p

p

]

− cp 2 ) + bp2 hp + hp2 2 + cp2 2 t Abr

= 96,88 mm

EN1993-1-3 § 5.1(3)

Example: Calculation of effective section properties for a cold-formed lipped channel section in bending

CALCULATION SHEET

Document Ref:

SX022a-EN-EU

Title

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

Sheet

3

of

8

Effective section properties of the flange and lip in compression The general (iterative) procedure is applied to calculate the effective EN1993-1-3 properties of the compressed flange and the lip (plane element with edge § 5.5.3.2 stiffener). The calculation should be carried out in three steps: Step 1: Obtain an initial effective cross-section for the stiffener using effective widths EN1993-1-3 of the flange determined by assuming that the compressed flange is doubly § 5.5.3.2 (3) supported, the stiffener gives full restraint ( K = ∞ ) and that design strength is not reduced ( σ com,Ed = f yb / γ M 0 ). Effective width of the compressed flange Created on Monday, October 25, 2010 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Access Steel Licence Agreement

The stress ratio: ψ = 1 (uniform compression), so the buckling factor is: kσ = 4 for internal compression element.

ε = 235 f yb The relative slenderness:

λp,b =

bp1 t 28,4 ε k σ

=

72 1,96 = 0,789 28,4 × 235 350 × 4

The width reduction factor is:

ρ=

λp,b − 0,055(3 + ψ ) 0,789 − 0,055 × (3 + 1) = = 0,914 0,789 2 λp,b 2

The effective width is: beff = ρ bp1 = 0 ,914 × 72 = 65,8 mm

be1 = be2 = 0,5beff = 0,5 × 65,8 = 32,9 mm

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 bending

CALCULATION SHEET

Document Ref:

SX022a-EN-EU

Title

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

Sheet

Effective width of the edge fold The buckling factor is:

2

bp, c bp1 = 19,8 72 = 0,275 < 0,35

so

kσ = 0,5 EN1993-1-5 § 4.4

The relative slenderness:

Created on Monday, October 25, 2010 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Access Steel Licence Agreement

cp t

28,4 ε kσ

=

8

kσ = 0,5 + 0,83 3 (bp, c bp1 − 0,35)

if 0,35 < bp, c bp1 ≤ 0,6 :

λp,c =

of

EN1993-1-3 § 5.5.3.2 (5a)

kσ = 0,5

if bp, c bp1 ≤ 0,35 :

4

19,8 1,96 = 0,614 28,4 × 235 350 × 0,5

The width reduction factor is:

ρ=

λp,c − 0,188 0,614 − 0,188 = = 1,13 0,614 2 λp,c 2

but ρ ≤ 1

so

ρ =1

The effective width is:

EN1993-1-3 § 5.5.3.2 (5a)

ceff = ρ cp = 1 × 19,8 = 19,8 mm

Effective area of the edge stiffener:

(

)

(

)

As = t be2 + ceff = 1,96 × 32,9 + 19,8 = 103,3 mm

§ 5.5.3.2 (6) 2

Step 2:

Use the initial effective cross-section of the stiffener to determine the EN1993-1-3 reduction factor, allowing for the effects of the continuous spring restraint. § 5.5.3.2 (3) The elastic critical buckling stress for the edge stiffener is

σ cr ,s = where:

2 K E Is As

EN1993-1-3 § 5.5.3.2 (7)

Example: Calculation of effective section properties for a cold-formed lipped channel section in bending

CALCULATION SHEET

Document Ref:

SX022a-EN-EU

Title

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

Sheet

5

of

K is the spring stiffness per unit length: K=

EN1993-1-3 § 5.5.3.1(5)

1 E t3 ⋅ 2 3 2 4(1 − ν ) b1 hp + b1 + 0,5 b1 b2 hp kf

with:

b1 – distance from the web to the centre of the effective area of the stiffener in compression (upper flange) b1 = bp1 −

be2t be2 2 32,9 × 1,96 × 32,9 2 = 61,73 mm = 72 − (be2 + ceff )t (32,9 + 19,8) × 1,96

Created on Monday, October 25, 2010 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Access Steel Licence Agreement

kf = 0 for bending about the y-y axis K = 0,439 N mm I s is the effective second moment of area of the stiffener: 2

3 2 ⎤ ⎡ ceff ⎤ ⎡ ceff 2 be2 t 3 ceff t ceff − Is = + + be2 t ⎢ ⎥ ⎥ + ceff t ⎢ 12 12 ⎣ 2 2(be2 + ceff )⎦ ⎣ 2(be2 + ceff ) ⎦

I s = 3663 mm 4

so, the elastic critical buckling stress for the edge stiffener is

σ cr,s =

2 × 0,439 × 210000 × 3663 = 355,78 N mm 2 103,3

8

2

Example: Calculation of effective section properties for a cold-formed lipped channel section in bending

CALCULATION SHEET

Document Ref:

SX022a-EN-EU

6

Title

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

Sheet

of

Thickness reduction factor χd for the edge stiffener The relative slenderness:

λd =

The reduction factor will be:

Created on Monday, October 25, 2010 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Access Steel Licence Agreement

EN1993-1-3 § 5.5.3.1 (7)

if λd ≤ 0,65

χ d = 1,0

if 0,65 < λd < 1,38

χ d = 1,47 − 0,723 λd

if λd ≥ 1,38

χ d = 0,66 λd so

EN1993-1-3 § 5.5.3.2 (3) Figure 5.10d

f yb σ cr,s = 350 355,78 = 0,992

0,65 < λd = 0,992 < 1,38

8

EN1993-1-5 § 4.4 (2)

χ d = 1,47 − 0,723 × 0,992 = 0,753 EN1993-1-3 § 5.5.3.2 (3)

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. Figure 5.10e The iterations are carried out based on modified values of ρ obtained using:

σ com, Ed, i = χ d f yb γ M0 and

λp, red = λp χ d

The iteration stops when the reduction factor χ converges. Initial values (iteration 1):

Final values (iteration n):

χ d = 0,753

χ d = χ d, n = 0,737

be2 = 32,9 mm

be2 = be2, n = 35,9 mm

ceff = 19,8 mm

ceff = ceff, n = 19,8 mm

Final values of effective properties for flange and lip in compression are:

χd = 0,737

be2 = 35,9 mm

and be1 = 32,9 mm

ceff = 19,8 mm

EN1993-1-3 § 5.5.3.2 (10)

Example: Calculation of effective section properties for a cold-formed lipped channel section in bending

CALCULATION SHEET

Document Ref:

SX022a-EN-EU

7

Title

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

Sheet

tred = tχ d = 1,96 × 0,737 = 1,44 mm

of

8

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:

cp (hp − cp 2 ) + bp2 hp + hp 2 + ceff χ d 2 2

hc =

2

cp + bp2 + hp + be1 + (be2 + ceff )χ d

hc = 101,6 mm

The stress ratio:

Created on Monday, October 25, 2010 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Access Steel Licence Agreement

ψ=

hc − hp hc

=

101,6 − 198 = −0,949 101,6

The buckling factor: kσ = 7,81 − 6,29ψ + 9,78ψ 2 The relative slenderness:

λp, h =

hp t 28,4 ε kσ

=

198 1,96 = 0,914 28,4 × 235 350 × 22,58

The width reduction factor is:

ρ=

λp, h − 0,055(3 + ψ ) 0,914 − 0,055 × (3 − 0,949) = = 0,959 0,9142 λp, h 2

kσ = 22,58

EN1993-1-5 § 4.4 (Table 4.1)

Example: Calculation of effective section properties for a cold-formed lipped channel section in bending

CALCULATION SHEET

Document Ref:

SX022a-EN-EU

Title

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

Sheet

The effective width of the zone in compression of the web is:

heff = ρ hc = 0,959 × 101,6 = 97,5 mm Near the flange in compression:

he1 = 0,4heff = 0,4 × 97,5 = 39 mm Near the neutral axis:

he2 = 0,6heff = 0,6 × 97,5 = 58,5 mm The effective width of the web is:

Created on Monday, October 25, 2010 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Access Steel Licence Agreement

Near the flange in compression:

h1 = he1 = 39 mm Near the flange in tension: h2 = hp − (hc − he2 ) = 198 − (101,6 − 58,5) = 154,9 mm

Effective section properties

Effective cross-section area: Aeff = t[cp + bp 2 + h1 + h2 + be1 + (be 2 + ceff ) χ d ]

Aeff = 1,96 × [19,8 + 64 + 39 + 154,9 + 32,9 + (35,9 + 19,8) × 0,737] Aeff = 689,2 mm 2

Position of the neutral axis with regard to the flange in compression: zc =

[

t cp (hp − cp 2 ) + bp2 hp + h2 (hp − h2 2 ) + h1 2 + ceff χ d 2

zc = 102,3 mm

2

Aeff

2

]

8

of

8

b g

x

a

m

p

l

:

C

a

l

c

SX022a-EN-EU

Title

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

of

8

y

c

t

o

Document Ref:

Sheet

9

u

i

CALCULATION SHEET

e

r

E

p

O

Position of the neutral axis with regard to the flange in tension:

l

o

n

i

d

s

a

y

c

,

o

z t = hp − z c = 198 − 102,3 = 95,7 mm

Second moment of area:

i

M a

3

3 3 3 cp t be1t 3 be2 ( χ d t )3 ceff 3 ( χ d t ) h1 t h2 t bp2t I eff,y = + + + + + + + 12 12 12 12 12 12 12 2 + cp t ( z t − cp 2) 2 + bp 2tz t + h2t ( z t − h2 2) 2 + h1t ( zc − h1 2) 2 +

+ be1t zc + be2 ( χ d t ) zc + ceff ( χ d t )( zc − ceff 2) 2 2

n r

2

o e

I eff, y = 4140000 mm 4

- with regard to the flange in compression I eff,y zc

=

4140000 = 40460 mm3 102,3

- with regard to the flange in tension Weff,y,t =

C T

r h

e i

a s

e m

Weff,y,c =

t

d a

t

Effective section modulus:

I eff, y zt

=

4140000 = 43260 mm3 95,7

l

a

t

Example: Calculation of effective section properties for a cold-formed lipped channel section in bending

Example: Calculation of effective section properties for a cold-formed lipped channel section in bending SX022a-EN-EU.doc

Quality Record

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

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

G W Owens

SCI

23/08/06

Created on Monday, October 25, 2010 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Access Steel Licence Agreement

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