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

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 h=200mm

Total width of flange in compression b₁ =74mm Total width of flange in tension b₂ =66mm Total width of edge fold c=20,8mm Internal radius r =3mm Nominal thickness t_nom =2mm Steel core thickness t=1,96mm

Basic yield strength 2

yb =350N mm f Modulus of elasticity _{E=210000N mm}2 Poisson’s ratio ν=0,3 Partial factor γ_M0 =1,00 EN1993-1-3 § 3.2.4(3) EN1993-1-3 § 2(3) The dimensions of the section centre line are:

Web height hp =h−tnom =200−2=198mm Width of flange in compression bp1 =b1−tnom =74−2=72mm Width of flange in tension bp2 =b2 −tnom =66−2=64mm

Width of edge fold cp =c−tnom 2=20,8−2 2=19,8mm

Checking of geometrical proportions

The design method of EN1993-1-3 can be applied if the following conditions are satisfied: 60 ≤ t b b₁ t=741,96=37,75 <60 – OK 50 ≤ t c c t=20,8 1,96=10,61 <50 – OK 500 ≤ t h h t=2001,96=102,04 <500 – 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 , 0 2 , 0 ≤ bc ≤ c b₁ =20,8 74=0,28 0,2<0,28<0,6 – OK 32 , 0 66 8 , 20 2 = = b c 0,2<0,32<0,6 – OK The influence of rounding of the corners is neglected if:

5 t r ≤ r t =31,96=1,53 <5 – OK 10 , 0 p ≤ b r r bp1 =3 72=0,04 <0,10 – OK 10 , 0 05 , 0 64 3 2 p = = < b r – OK EN1993-1-3 § 5.1(3)

Gross section properties

(

)

(

)

2 p p2 p1 p br =t 2c +b +b +h =1,96× 2×19,8+72+64+198 =732mm A

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

(

)

[

]

mm 88 , 96 2 2 2 2 p 2 p p p2 p p p b1 = + + + − = A t c h h b c h c z

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 (σ_com,Ed= f_yb/γM₀).

EN1993-1-3 § 5.5.3.2 (3)

Effective width of the compressed flange

The stress ratio: ψ =1 (uniform compression), so

the buckling factor is: k_σ=4 for internal compression element.

yb

235 f =

ε

The relative slenderness:

789 , 0 4 350 235 4 , 28 96 , 1 72 4 , 28 _σ p1 b p, = = _{× ×} = k t b ε λ

The width reduction factor is:

(

)

(

)

914 , 0 789 , 0 1 3 055 , 0 789 , 0 3 055 , 0 2 2 b p, b p, − + ₌ − × + ₌ = λ ψ λ ρ

The effective width is:

mm 8 65 72 914 0 p1 eff b , , b =ρ = × = mm 9 32 8 65 5 0 5 0 _eff e2 e1 b , b , , , b = = = × = EN1993-1-3 § 5.5.2 and EN1993-1-5 § 4.4

Effective width of the edge fold The buckling factor is:

if bp,c bp1≤0,35: kσ=0,5 if 0,35<bp,c bp1≤0,6: 3

(

)

2 p1 c p, σ=0,5+0,83 b b −0,35 k 35 , 0 275 , 0 72 8 , 19 p1 c p, b = = < b so k_σ=0,5 EN1993-1-3 § 5.5.3.2 (5a)

The relative slenderness:

614 , 0 5 , 0 350 235 4 , 28 96 , 1 8 , 19 4 , 28 _σ p p,c = × × = = k t c ε λ EN1993-1-5 § 4.4

The width reduction factor is:

13 , 1 614 , 0 188 , 0 614 , 0 188 , 0 2 2 c p, c p, − ₌ − ₌ = λ λ ρ but ρ ≤1 so 1ρ = The effective width is:

mm 8 , 19 8 , 19 1 p eff = c = × = c ρ

Effective area of the edge stiffener:

(

)

(

)

2 eff e2 s =tb +c =1,96× 32,9+19,8 =103,3mm A 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 EN1993-1-3

K is the spring stiffness per unit length: f p 2 1 3 1 p 2 1 2 3 5 , 0 1 ) 1 ( 4 b h b b b h k t E K + + ⋅ − = ν with: 1

b – 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) mm 73 , 61 96 , 1 ) 8 , 19 9 , 32 ( 2 9 , 32 96 , 1 9 , 32 72 ) ( 2 eff e2 e2 e2 p1 1 _{+ ×} = × × − = + − = t c b b t b b b 0 f =

k for bending about the y-y axis mm N 439 , 0 = K s

I is the effective second moment of area of the stiffener:

(

)

(

)

2 eff e2 2 eff eff eff 2 eff e2 2 eff e2 3 eff 3 e2 s _{12 12 2 2 2} ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + − + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + + = c b c c t c c b c t b t c t b I 4 s =3663mm I

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

2 s cr, _103,3 355,78N mm 3663 210000 439 , 0 2× × × ₌ = σ

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

992 , 0 78 , 355 350 s cr, yb d = σ = = λ f

The reduction factor will be: if λ_d ≤0,65 χ_d =1,0 if 0,65<λd <1,38 χd =1,47−0,723λd if λ_d ≥1,38 χ_d =0,66 λ_d 38 , 1 992 , 0 65 , 0 <λd = < so χd =1,47−0,723×0,992=0,753 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:

M0 yb d i Ed, com, χ γ σ = f and λ_p,red =λ_p χ_d 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 , 0 d = χ χ_d =χ_d,n =0,737 mm 9 , 32 e2= b be2= be2,n =35,9mm mm 8 , 19 eff = c ceff = ceff,n =19,8mm

mm 44 , 1 737 , 0 96 , 1 d red =tχ = × = t 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:

(

)

(

_{e2 eff}

)

_d e1 p p2 p d 2 eff 2 p p p2 p p p c 2 2 2 χ χ c b b h b c c h h b c h c h + + + + + + + + − = h_c=101,6mm

The stress ratio:

949 , 0 6 , 101 198 6 , 101 c p c− ₌ − ₌₋ = h h h ψ

The buckling factor: 2

σ=7,81−6,29ψ +9,78ψ

k k_σ =22,58

The relative slenderness:

914 , 0 58 , 22 350 235 4 , 28 96 , 1 198 4 , 28 _σ p h p, = × × = = k t h ε λ EN1993-1-5 § 4.4 (Table 4.1)

The width reduction factor is:

(

)

(

)

_0,959 914 , 0 949 , 0 3 055 , 0 914 , 0 3 055 , 0 2 2 h p, h p, − + ₌ − × − ₌ = λ ψ λ ρ

The effective width of the zone in compression of the web is: mm 5 , 97 6 , 101 959 , 0 c eff = h = × = h ρ

Near the flange in compression: mm 39 5 , 97 4 , 0 4 , 0 _eff e1= h = × = h

Near the neutral axis:

mm 5 , 58 5 , 97 6 , 0 6 , 0 _eff e2 = h = × = h

The effective width of the web is: Near the flange in compression:

mm 39

e1 1= h =

h

Near the flange in tension:

(

c e2

)

198

(

101,6 58,5

)

154,9mm

p

2 =h − h −h = − − =

h

Effective section properties

Effective cross-section area:

] ) ( [ p p2 1 2 e1 e2 eff d eff t c b h h b b c χ A = + + + + + +

(

)

[

19,8 64 39 154,9 32,9 35,9 19,8 0,737

]

96 , 1 eff = × + + + + + + × A 2 eff =689,2mm A

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

(

)

(

)

[

]

eff d 2 eff 2 1 2 p 2 p p2 p p p c 2 2 2 2 A c h h h h h b c h c t z = − + + − + + χ

Position of the neutral axis with regard to the flange in tension: mm 7 , 95 3 , 102 198 c p t =h −z = − = z

Second moment of area:

2 eff c d eff 2 c d e2 2 c e1 2 1 c 1 2 2 t 2 2 t 2 p 2 p t p d 3 eff 3 d e2 3 e1 3 p 3 p2 3 2 3 1 y eff, ) 2 )( ( ) ( ) 2 ( ) 2 ( ) 2 ( 12 ) ( 12 ) ( 12 12 12 12 12 c z t c z t b z t b h z t h h z t h tz b c z t c t c t b t b t c t b t h t h I − + + + + − + − + + − + + + + + + + + = χ χ χ χ 4 y eff, =4140000mm I

Effective section modulus:

- with regard to the flange in compression

3 c y eff, c y, eff, 40460mm 3 , 102 4140000 = = = z I W

- with regard to the flange in tension

3 t y eff, t y, eff, 43260mm 7 , 95 4140000 = = = z I W

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

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

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