Design of a Thin-Walled, Cold-Formed C-Section According to EN 1993-1-3

Technical Article on the Topic Structural Analysis Using Dlubal Software

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Technical Article

Utilize the RF-/STEEL Cold-Formed Sections module extension to perform ultimate limit state designs of cold-formed sections according to EN 1993-1-3 and EN 1993-1-5. In addition to the cold-formed cross-sections from the cross-section database, you can design general cross-sections from SHAPE-THIN.

In the following example from the Steel Structures Yearbook 2009 [3], the cross-section design for a single-span beam with a thin-walled, cold-formed C-section is performed under normal force loading. The C-section is modeled in SHAPE-THIN and then designed in RF-/STEEL Cold-Formed Sections.

System

System and loading are shown in Figure 01.

Image 01 - System and Loading (Dimensions in mm, Force in kN)

Material

S 355 EN 10025-2
E = 210,000 N/mm²
G = 80,769 N/mm²
ν = 0.3
fy = fyb = 355 N/mm²
γM0 = γM1 = 1.00 (design according to CEN)

External Dimensions

The external dimensions of the cross-section are shown in Figure 02.

Image 02 - External Dimensions of the Cross-Section in mm

H = 102 mm (web height)
b = 120 mm (belt width)
c = 26 mm (lip length)
t = 2 mm (steel core thickness)

Notional Flat Widths

The notional flat widths are determined according to [1], 5.1. The notional flat widths are shown in Figure 03.

Image 03 - Notional Flat Widths in mm

Formula 1

rm =r  t2 =10  22 =11 mmgr =rm · tanϕ2 - sinϕ2 =11 · tan 45° - sin 45°  =3,22 mmhw =H - t - 2 · gr =102  - 2 - 2 · 3.22=93.56 mmbp =b - t - 2 · gr =120  - 2 - 2 · 3.22=111.56 mmbp,c =c - t2 -  gr =26 - 22 - 3.22=21.78 mm

Checking Width-Thickness Ratios

The width-thickness ratios are checked according to [1], 5.2 (1).

b/t = 120/2 = 60 ≤ 60

c/t = 26/2 = 13 ≤ 50

H/t = 102/2 = 51 ≤ 500

The width-thickness ratios are met.

Checking Stiffener Dimensions

The stiffener dimensions are checked according to [1], 5.2 (2).

0.2 ≤ c/b = 26/120 = 0.22 ≤ 0.6

The lips can be applied as stiffeners.

Checking the Angle Between the Stiffener and the Planar Element

The angle between the stiffener and the planar element is 90° within the limits of 45° and 135° mentioned in [1], 5.5.3.2 (1).

Determining the Effective Cross-Section

For steel sections that are not subjected to double symmetry and subjected to compression, the centroidal position of the effective cross-section shifts compared to the gross cross-section. The external compressive force acting centrically on the gross cross-section now acts eccentrically on the effective cross-section, and an additional bending moment is created. According to [1], the additional moments resulting from the shift of the centroid must be taken into account. Subsequently, in addition to the effective cross-section for pure compressive stress, the effective cross-section for pure bending stress must be determined.

Determining the Effective Cross-Section Under Pure Compression

According to [2], 4.4 (2), the coefficient results in:

Formula 2

ε = 235fyb = 235355 = 0.814

Web:

According to [2], Table 4.1, the buckling value results in:

${\mathrm k}_{\mathrm\sigma}\;=\;4$

According to [2], 4.4 (2), the buckling slenderness results in:

Formula 4

λ¯p = hw / t28.4 · ε · kσ = 93.56 / 228.4 · 0.814 · 4 = 1.012λ¯p = 1.012  0.5  0.085 - 0.055 · ψ =0.5  0.085 - 0.055 · 1  = 0.673

The slenderness ratio is greater than the limit value 0.673 according to [2], 4.4 (2). Thus, a reduction is required.

According to [2], 4.4 (2), the reduction factor results in:

Formula 5

ρ = λ¯p - 0.055 · 3  ψλ¯p2 = 1.012 - 0.055 · 3  11.0122 =0.773  1

According to [2], Table 4.1, the effective web height results from [2], Table 4.1 in:

Formula 6

heff = ρ · hw = 0.773 · 93.56 =72.3 mmhe1 = he2  = 0.5 · heff =0.5 · 72.3 = 36.17 mm

Flange with edge stiffener:

In the first step, a first approach for the effective cross-section of the stiffener is determined with the assumption that the edge stiffness acts as a rigid support and that σcom,Ed = fyb / γM0.

Flange:

According to [2], Table 4.1, the buckling value results in:

Formula 3

kσ = 4

According to [2], 4.4 (2), the buckling slenderness results in:

Formula 7

λ¯p = bp / t28.4 · ε · kσ = 111.56 / 228.4 · 0.814 · 4 = 1.207λ¯p = 1.207  0.5  0.085 - 0.055 · ψ =0.5  0.085 - 0.055 · 1  = 0.673

The slenderness ratio is greater than the limit value 0.673 according to [2], 4.4 (2). Thus, a reduction is required.

According to [2], 4.4 (2), the reduction factor results in:

Formula 8

ρ = λ¯p - 0.055 · 3  ψλ¯p2 = 1.207 - 0.055 · 3  11.2072 =0.678  1

According to [2], Table 4.1, the effective flange width results in:

Formula 9

beff = ρ · bp = 0.678 · 111.56 =75.6 mmbe1 = be2  = 0.5 · beff =0.5 · 75.6 = 37.79 mm

Edge stiffness:

According to [1], 5.5.3.2 (5), the buckling value results in:

$\frac{{\mathrm b}_{\mathrm p,\mathrm c}}{{\mathrm b}_{\mathrm p}}\;=\;\frac{21,78}{111,56}\;=\;0,195\;\leq\;0,35\\{\mathrm k}_{\mathrm\sigma}\;=\;0,5$

According to [2], 4.4 (2), the buckling slenderness results in:

${\overline{\mathrm\lambda}}_{\mathrm p}\;=\;\frac{{\mathrm b}_{\mathrm p,\mathrm c}\;/\;\mathrm t}{28,4\;\cdot\;\mathrm\varepsilon\;\cdot\;\sqrt{{\mathrm k}_{\mathrm\sigma}}}\;=\;\frac{21,78\;/\;2}{28,4\;\cdot\;0,814\;\cdot\;\sqrt{0,5}}\;=\;0,666\\{\overline{\mathrm\lambda}}_{\mathrm p}\;=\;0,666\;\leq\;0,748$

The slenderness ratio is smaller than the limit value 0.748 according to [1], 4.4 (2). Thus, no reduction is necessary, that is: ρ = 1.0.

According to [1], Eq. 5.13a, the first approach of the effective width results in:

${\mathrm c}_{\mathrm{eff}}\;=\:\mathrm\rho\;\cdot\;{\mathrm b}_{\mathrm p,\mathrm c}\;=\;1,0\;\cdot\;21,78\;=\;21,78\;\mathrm{mm}$

In the second step, the reduction factor for the shape instability of the cross-section is determined using the effective first approach for the cross-section, taking into account the elastic translation spring.

The effective cross-section values of the edge stiffening are calculated with SHAPE-THIN. The edge stiffness is shown in Figure 04.

Image 04 - Effective Edge Stiffness

As = 122.58 mm 2
Is = 7,130 mm 4
zs = 13.88 mm

The spring stiffness K of the edge stiffness is determined on the basis of a structural analysis for the entire cross-section. For this purpose, a unit distance load u, acting in the centroid of the effective stiffener, is applied to the cross-section, and the corresponding deformation δ of the stiffener is calculated. For a rectangular cross-section w/h = t/t = 2/2 mm, the deformation results in δ = 3.02 mm (Figure 05).

Image 05 - Determination of Spring Stiffness

The spring stiffness per unit length K can be calculated according to [1], Eq. 5.9 as follows:

Formula 13

K = uδ = 13.02 · 2 = 0.166 N/mm2

According to [1], Eq. 5.15, the critical stress of the edge stiffness results in:

Formula 14

σcr,s = 2 · K · E · IsAs = 2 · 0.166 · 210,000 · 7,130122.58 = 257 N/mm2

According to [1], Eq. 5.12d, the related slenderness ratio results in:

Formula 15

λ¯d = fybσcr,s = 355257 = 1.176 > 0.65< 1.38

According to [1], 5.5.3.1 (7), the reduction coefficient for the shape instability is calculated as follows:

Formula 16

χd =1.47 - 0.723 · λ¯d =1.47 - 0.723 · 1.176 =0.62

According to [1], Eq. 5.17, the reduced effective cross-sectional area of the edge stiffness is obtained, taking into account the flexural buckling:

Formula 17

As,red = χd · As · fyb / γM0σcom,Ed = 0.62 · 122.58 · 1.0 =76.01 mm2

Effective cross-section properties under pure compression:

The cross-section can be optimized by means of an iterative calculation. For two iterations, it results in the following effective cross-section values:

Area Aeff = 4.62 cm²
Centroidal distance from web zs,eff = 42.18 mm
Centroidal displacement eN,y = zs – zs,eff = 8.78 mm

Determining the Effective Cross-Section Under Pure Bending Stress

Web:

The web is subjected to tension and is thus fully effective.

Flange with edge stiffener:

In the first step, a first approach for the effective cross-section of the stiffener is determined with the assumption that the edge stiffness acts as a rigid support and that σcom,Ed = fyb / γM0.

Flange:

According to [2], Table 4.1, the buckling value results in:

Formula 18

ψ = σ2σ1 =-253.1336.2 = -0.753kσ = 7.81 - 6.29 · ψ  9.78 · ψ2 = 7.81 - 6.29 · (-0.753)  9.78 · (-0.753)2 =18.08

According to [2], 4.4 (2), the buckling slenderness results in:

Formula 19

λ¯p = bp / t28.4 · ε · kσ = 111.56 / 228.4 · 0.814 · 18.08 = 0.568λ¯p = 0.568  0.5  0.085 - 0.055 · ψ =0.5  0.085 - 0.055 · (-0.753)  = 0.856

The slenderness ratio is smaller than the limit value 0.856 according to [2], 4.4(2). Thus, no reduction is required.

According to [2], Table 4.1, the effective widths result in:

Formula 20

beff = ρ · bp / (1 - ψ) =1.0 · 111.56 / (1  0.753) = 63.65 mmbe1 = 0.4 · beff =0.4 · 63.65 = 25.46 mmbe2 = 0.6 · beff =0.6 · 63.65 = 38.19 mm

Edge stiffness:

According to [1], 5.5.3.2 (5), the buckling value results in:

Formula 10

bp,cbp = 21.78111.56 = 0.195  0.35kσ = 0.5

According to [2], 4.4 (2), the buckling slenderness results in:

Formula 11

λ¯p = bp,c / t28.4 · ε · kσ = 21.78 / 228.4 · 0.814 · 0.5 = 0.666λ¯p = 0.666  0.748

The slenderness ratio is smaller than the limit value 0.748 according to [1], 4.4 (2). Thus, no reduction is necessary, that is: ρ = 1.0.

According to [1], Eq. 5.13a, the first approach of the effective width results in:

Formula 12

ceff =ρ · bp,c = 1.0 · 21.78 = 21.78 mm

In the second step, the reduction factor for the shape instability of the cross-section is determined by using the effective first approach for the cross-section, taking into account the elastic translation spring.

The effective cross-section values of the edge stiffening are calculated with SHAPE-THIN. The edge stiffness is shown in Figure 06.

Image 06 - Effective Stiffening of the Cross-Section

As = 97.92 mm 2
Is = 6,271 mm 4
zs = 8.59 mm

The spring stiffness K of the edge stiffness is determined on the basis of a structural analysis for the entire cross-section. For this purpose, a unit distance load u, acting in the centroid of the effective stiffener, is applied to the cross-section and the corresponding deformation δ of the stiffener is calculated. For a rectangular cross-section w/h = t/t = 2/2 mm, the deformation results in δ = 3.4 mm (Figure 07).

Image 07 - Determination of the Spring Stiffness

The spring stiffness per unit length K can be calculated according to [1], Eq. 5.9 as follows:

Formula 21

K = uδ = 13,4 · 2 = 0.146 N/mm2

According to [1], Eq. 5.15, the critical stress of the edge stiffness results in:

Formula 22

σcr,s = 2 · K · E · IsAs = 2 · 0.146 · 210,000 · 6,27197.92 =283 N/mm2

According to [1], Eq. 5.12d, the related slenderness ratio results in:

Formula 23

λ¯d = fybσcr,s = 355283 = 1.120 > 0.65< 1.38

According to [1], 5.5.3.1 (7), the reduction coefficient for the shape instability is calculated as follows:

Formula 24

χd =1.47 - 0.723 · λ¯d =1.47 - 0.723 · 1.120 =0.66

According to [1], Eq. 5.17, the reduced effective cross-sectional area of the edge stiffness is obtained taking into account the flexural buckling:

Formula 25

As,red = χd · As · fyb / γM0σcom,Ed = 0,66 · 97,92 · 355 / 1,0312,2 =73,82 mm2

Effective cross-section properties under pure bending stress:

All the cross-section parts are fully effective, so iteration is not necessary.

Area Aeff = 6.86 cm²
Section modulus Weff, y = 17.01 cm³

Cross-Section Design of Combined Loading Due to Compression and Bending

The resistance to pure compression is calculated according to [1], 6.1.3 (1) as follows:

Formula 26

Nc,Rd = Aeff · fybγM0 =4.62· 35.51 = 164.16 kN

The resistance to pure bending is calculated according to [1], 6.1.4.1 (1) as follows:

Formula 27

Mcy,Rd,com = Weff,y · fybγM0 =17.01 · 35.51 = 6.04 kNm

The additional moment resulting from the shift of the centroid is determined according to [1], 6.1.9 (2) as follows:

Formula 28

My,Ed = NEd · eNy =130 · 8.78 · 10-3 = 1.14 kNm

According to [1], 6.1.9 (1), the design for combined loading from compression and bending results in:

Formula 29

NEdNc,Rd  My,EdMcy,Rd,com =130164.16  1.146.04 = 0.98  1

The design is thus performed.

Modeling a Cold-Formed C-Section in SHAPE-THIN

General cold-formed sections can be modeled in SHAPE-THIN. In the general data, activate the "c/t parts and effective cross-section properties" check box (Figure 08).

Image 08 - General Data

Then, select the "EN 1993-1-3 (Cold-formed cross-section)" option in the "c/t-parts and effective cross-section" tab (Figure 09) of the Calculation Parameters dialog box.

The effective cross-section must be determined separately for pure compression and pure bending. Therefore, select the "Neglect additional bending moments due to the shift of the centroid of the effective cross-section" check box.

We have calculated with two iterations in the example, so two iterations are also set in SHAPE-THIN.

The geometric conditions mentioned in [2], 5.2 for the applicability of the standard can be optionally checked. To do this, select the corresponding check boxes.

Image 09 - Calculation Parameters

The elements of the cross-section have to be entered first. The notional flat widths are usually generated automatically from the geometry conditions, but can also be created as user-defined in Table "1.7 Notional Flat Widths According to EN 1993-1-3" (Figure 10) or in the corresponding dialog box.

Image 10 - Table 1.7 Notional Flat Widths | EN 1993-1-3

The stiffeners can then be defined in Table "1.8 Stiffeners" or in the corresponding dialog box (Figure 11).

Image 11 - Table 1.8 Stiffeners

Furthermore, the buckling panel must be specified in Table "1.9 Panels" (Figure 12) or in the corresponding dialog box. To do this, select the elements of the buckling panel. The stiffeners located in the stiffened panel are identified automatically.

Image 12 - Table 1.9 Panels

Additionally, a load case for compression force and bending is created in Table "2.1 Load Cases" (Figure 13).

Image 13 - Table 2.1 Load Cases

Then, enter the internal forces in Table "3.1 Internal Forces" or in the corresponding dialog box (Figure 14).

Image 14 - Table 3.1 Internal Forces

The results of the effective cross-section are available with the "Effective Parts" button (Figure 15).

Image 15 - Effective Cross-Section Properties

Cross-Section Design of Cold-Formed C-Section in RF-/STEEL Cold-Formed Sections

Cold-formed sections can be designed according to [1] and [2] with the RF-/STEEL Cold-Formed Sections add-on module.

In the General Data, you must first select the member and load case to be designed. "CEN" is selected as the National Annex (Figure 16).

Image 16 - General Data

You can see and, if necessary, adjust the parameters of the National Annex in the "Cold-Formed (EN 1993-1-3)" tab of the corresponding window (Figure 17).

Image 17 - Parameters of the National Annex

In the Detail settings, activate the design check for cold-formed sections in the "Cold-Formed" tab (Figure 18).

Image 18 - Details, Cold-Formed Sections Tab

Only the cross-section design should be performed. Therefore, the "Perform stability analysis" check box in the "Stability" tab of the detail settings must be deactivated (Figure 19).

Image 19 - Details, Stability Tab

After the calculation, the corresponding output tables show, among other things, effective cross-section properties due to axial force N, bending moment My, bending moment Mz, internal forces, and the entire design (Figure 20).

Image 20 - Figure 20 - Results

Author

Sonja von Bloh, M.Sc.

Sonja von Bloh, M.Sc.

Product Engineering & Customer Support

Ms. von Bloh provides technical support for our customer and is responsible for the development of the SHAPE‑THIN program.

Keywords

Cross-section design Cold-formed section

Reference

[1]   Eurocode 3: Design of steel structures - Part 1‑3: General rules - Supplementary rules for cold-formed members and sheeting; EN 1993‑1‑3:2010‑12
[2]   Eurocode 3: Design of steel structures - Part 1-5: General rules - Plated structural elements; EN 1993-1-5:2006 + AC:2009
[3]   Kuhlmann, U.: Stahlbau-Kalender 2009 - Stabilität, Membrantragwerke. Berlin: Ernst & Sohn, 2009

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SHAPE-THIN 9
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Properties and stresses of thin-walled and cold-formed cross-sections

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