Job No. Sheet 1 of 7 Rev A. Made by ER/EM Date Feb Checked by HB Date March 2006

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1 Job No. Sheet of 7 Rev A Design Example Design of a lipped channel in a Made by ER/EM Date Feb 006 Checked by HB Date March 006 DESIGN EXAMPLE DESIGN OF A LIPPED CHANNEL IN AN EXPOSED FLOOR Design a simply supported beam with a lipped channel in an. The material is stainless steel grade.440 CP500, i.e. cold worked to a yield strength of 500 N/mm. The beam is simply supported with a span, l of 4 m. The distance between adjacent beams is m. As the load is not applied through the shear centre of the channel, it is necessary to check the interaction between the torsional resistance of the cross-section and the lateral torsional buckling resistance of the member. However, this example only checks the lateral torsional buckling resistance of the member. Factors Partial factor γm0, and γm, Table. Load factor γg,35 (permanent loads) and γq,5 (variable loads) Section.3. Actions Permanent actions (G): kn/m Variable actions (Q): 3 kn/m Load case to be considered in the ultimate limit state: q * γ G + γ Q 7, kn / m Eq..3 j G,j k,j Q, k, Structural Analysis Reactions at support points (Design shear force) * q 4 V Ed 4, 4 kn Design bending moment * 4 q M Ed 4, 4 knm 8 Material Properties Yield strength fy 500 N/mm Modulus of elasticity E N/mm Shear modulus G N/mm Table 3.5 Section 3..4 Section 3..4 Cross-section Properties The influence of rounded corners on cross-section resistance may be neglected if the internal radius r 5t and r 0,0b p and the cross section may be assumed to consist of plane elements with sharp corners. For cross-section stiffness properties the influence of rounded corners should always be taken into account. Section

2 Job No. Sheet of 7 Rev A Design Example Design of a lipped channel in a Made by ER/EM Date Feb 006 Checked by HB Date March 006 z b h y r c t y h 60 mm b 5 mm c 30 mm t 5 mm r 5 mm z b b t g 5,6 mm Figure 4.5 p r [ tan( φ ) sin( φ ) ], mm gr rm r m r + t 7,5 mm r 5 mm 5t 5 mm r 5 mm 0,0bp,56 mm The influence of rounded corners on section properties may be taken into account with sufficient accuracy by reducing the properties calculated for an otherwise similar cross-section with sharp corners, using the following approximations: Notional flat width of the flange, b b t g 5,6 mm Notional flat width of the web, Notional flat width of the lip, Ag,sh 6 mm Iyg,sh 9, mm 4 n φ m j δ 0,43 r / b o 90 j p,i j i p,f r bp,w h t gr 50,6 mm bp,l c t / gr 0,0 5,30 mm Eq 4. Ag Ag,sh ( - δ) 9 mm Eq 4.8 I g I g,sh ( - δ) 8, mm 4 Eq 4.9 Classification of the cross-section Section E ε f y ,5 Table 4. 0,669 Flange: Internal compression parts. Part subjected to compression. c bp b t gr 5,6 mm c/t3, > 30,7 ε, therefore the flanges are Class 4 Web: Internal compression parts. Part subjected to bending. 84

3 Job No. Sheet 3 of 7 Rev A Design Example Design of a lipped channel in a Made by ER/EM Date Feb 006 Checked by HB Date March 006 ch-t-gr50,6 mm c/t30, 56 ε, therefore the web is Class Lip: Outstand flanges. Part subjected to compression, tip in compression, c c t / gr 5,30 mm c/t5,06 0 ε, therefore the lip is Class Calculation of the effective section properties Section 4.4. Flange effective width: Internal compression elements. Part subjected to compression. Table 4.3 b bp b t g r 5,6 mm Assuming uniform stress distribution in the compression flange: ψ and the buckling factor k 4 b t 8,4ε k p 0,608 Eq 4. 0,77 0,5 Cold formed internal elements: ρ 0,93 < p b eff ρb 07,64 mm, be 0,5beff 53,8 mm, be 0,5beff 53,8 mm p Effects of shear lag Section 4.4. Shear lag in flanges may be neglected if b 0 < L e /50, where b 0 is taken as the flange outstand or half the width of an internal element and L e is the length between points of zero bending moment. For internal elements: bo(b-t)/60 mm The length between points of zero bending moment: Le4000 mm, L e /5080 mm Therefore shear lag can be neglected Flange curling Section b a s u,55 mm -3, clause E t z 5.4 bs4 mm is the distance between webs t5mm z77,5 mm is the distance of the flange under consideration from neutral axis a is mean stress in the flanges calculated with gross area (fy500 N/mm is assumed) Flange curling can be neglected if it is less than 5% of the depth of the profile crosssection: u,55 mm < 0,05h8 mm, therefore flange curling can be neglected. 85

4 Job No. Sheet 4 of 7 Rev A Design Example Design of a lipped channel in a Made by ER/EM Date Feb 006 Checked by HB Date March 006 Stiffened elements. Edge stiffeners Distortional buckling. Plane elements with edge stiffeners Section 4.5. and pren , clause Step : Initial effective cross-section for the stiffener For flanges (as calculated before) b5 mm bp5,6 mm beff07,65 mm be0,5beff53,8 mm be0,5beff53,8 mm For the lip, the effective width ceff should be calculated using the corresponding buckling factor k, p and ρ expressions as follows: bp,cc-t/-gr5,30 mm bp5,6 mm -3, clause bp,c/bp0, < 0,35 then k 0,5-3, Eq. 5.3b b t 8,4ε k p 0,45 ( b 30 mm ) Eq 4. 0,3 Cold formed outstand elements: ρ,08 > then ρ Eq 4.b p p c eff ρb p,c 5,30 mm -3, Eq. 5.3a Step : Reduction factor for distortional buckling Calculation of geometric properties of effective edge stiffener section be 53,8 mm ceff5,30 mm As(be+ceff)t395,64 mm 86

5 Job No. Sheet 5 of 7 Rev A Design Example Design of a lipped channel in a Made by ER/EM Date Feb 006 Checked by HB Date March 006 b e g r t/ t/ g r -3, Fig. 5.9 y a ceff y b ya4,0 mm yb8,7 mm Is,8 mm 4 Calculation of linear spring stiffness K 3 Et 3 4( υ ) b hw + b + 0,5b bhwk f,487 N/mm -3, Eq. 5.0b bb-yb-t/04,3 mm (the distance from the web-to-flange junction to the gravity center of the effective area of the edge stiffener, including the efficient part of the flange be) kf0 (flange is in tension) hw50 mm is the web depth Elastic critical buckling stress for the effective stiffener section cr,s KEI A s s 59,95 N mm Reduction factor χd for distortional buckling d f 0,98 yb cr, s -3, Eq ,65< d <,38 then χ d,47 0,73 d 0, 76-3, Eq. 5.d Reduced area and thickness of effective stiffener section f yb γ M0 A s,red χdas 300,88 mm -3, Eq. 5.7 com,ed tredtas,red/as3,8 mm Calculation of effective section properties with distortional buckling effect Aeff,sh08 mm n φ m j δ 0,43 rj / b o p,i j 90 i 0,0 Eq 4. Aeff Aeff,sh ( - δ) 987 mm Eq

6 Job No. Sheet 6 of 7 Rev A Design Example Design of a lipped channel in a Made by ER/EM Date Feb 006 Checked by HB Date March 006 zg68,98 mm (distance from the bottom fibre to the neutral axis) Iy,eff,sh8, mm 4 I y,eff I y,eff,sh ( - δ) 7, mm 4 Eq 4.9 Wy,eff, sup9, mm 3 Wy,eff, inf5, 0 3 mm 3 Resistance of cross-section Section 4.7 Cross-section subject to bending moment Section W f / γ 4,97 knm for Class 4 cross-section Eq. 4.9 Mc,Rd y,eff,min y M0 Design bending moment M Ed 4,4 knm Cross-section moment resistance is Ok Cross-section subject to shear Section A f 3 / γ 09,95 Eq ( ) kn Vpl,Rd v y M0 Av800 mm is the shear area Design shear force V Ed 4,4 kn Cross-section shear resistance is Ok Cross-section subjected to combination of loads Section VEd4,4 kn > 0,5Vpl,Rd04,97 kn There is no interaction between bending moment and shear force Flexural members Section 5.4 Lateral-torsional buckling Section 5.4. M χ W f γ for Class 4 cross-section Eq 5.8 b,rd LT y,eff,sup y M χ LT Eq 5.9 ϕlt ϕ 0,5 Eq 5.0 0,5 + [ ϕlt LT ] ( + α ( 0, ) + ) LT LT LT 4 cr LT Wy,eff f y LT Eq 5. M αlt0,34 for cold formed sections Determination of the elastic critical moment for lateral-torsional buckling M cr / π EI z k + ( ) z I w C g 3 j g 3 j kzl π k w Iz EIz ( kzl) GIt + ( C z C z ) ( C z C z ) 88 Appendix B, Section B.

7 Job No. Sheet 7 of 7 Rev A Design Example Design of a lipped channel in a Made by ER/EM Date Feb 006 Checked by HB Date March 006 For simply supported beams with uniform distributed load: C,, C0,45 and C30,55. Assuming normal conditions of restraint at each end: kzkw zj0 for equal flanged section zg za-zsh/80 mm za is the co-ordinate of point load application zs is the co-ordinate of the shear centre yg45,34 mm (distance from the central axis of the web to the gravity centre) Iz,sh4, mm 4 It,sh8,0 0 3 mm 4 Iw,sh3,9 0 9 mm 6 IzIz,sh ( - δ) 4, mm 4 It It,sh ( - δ) 7, mm 4 Iw Iw,sh ( - 4δ), mm 6 Note: The expression used to determine the warping torsion is obtained from Wei- Wen You, Cold-Formed Steel Design, Appendix B-Torsion Then, / π EI z k z I w Mcr C g g ( k L) k z + + w Iz π EIz ( k L) GI z t ( C z ) ( C z ) 33,74 knm Wy,eff,supf y LT,7 (Wy,eff, sup9, mm 3, compression flange) M ϕ LT χ cr ( + α ( 0,4) + ), 35 0,5 LT LT LT [ ϕlt LT ] LT 0,5 ϕlt + 0,5 Mb,Rd χltwy,eff,supf y γ M,9 knm Design moment M Ed 4,4 knm, therefore lateral torsional buckling resistance Ok Note: As the load is not applied through the shear centre of the channel, it is also necessary to check the interaction between the torsional resistance of the cross-section and the lateral torsional buckling resistance of the member. Shear buckling resistance Section The shear buckling resistance only requires checking when unstiffened web. The recommended value for η,0 h w / t 8, 5 ε η 8, 99, therefore no further check required. h w / t 5ε η for an 89

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Made by SMH Date Aug Checked by NRB Date Dec Revised by MEB Date April 2006

Made by SMH Date Aug Checked by NRB Date Dec Revised by MEB Date April 2006 Job No. OSM 4 Sheet 1 of 8 Rev B Telephone: (0144) 45 Fax: (0144) 944 Made b SMH Date Aug 001 Checked b NRB Date Dec 001 Revised b MEB Date April 00 DESIGN EXAMPLE 9 - BEAM WITH UNRESTRAINED COMPRESSION