2CO8 Advanced design of steel and composite structures Lectures: Machacek (M), Netusil (N), Wald (W), Ungureanu (U)

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1 CO8 Advanced design of seel and composie srucures Lecures: Machacek (M), Neusil (N), Wald (W), Ungureanu (U) 1 (M) Global analysis. Torsion of seel members (M) Buil-up seel members 3 (M) aigue 4 (M) Laeral-orsional buckling 5 (M) Buckling of plaes 6 (M) Composie seel and concree members 7 (M) Large-span srucures 1 8 (M) Large-span srucures 9 (M) Sainless seel srucures 1 10 (M) Sainless seel srucures 11 (N) Membrane srucures 1 1 (N) Membrane srucures 13 (W) Aluminium srucures 1 14 (W) Aluminium srucures 15 (W) Aluminium srucures 3 16 (U) Shell srucures 1 17 (U) Shell srucures 18 (U) Ligheigh seel srucures 1 19 (U) Ligheigh seel srucures 0 (U) Ligheigh seel srucures 3 C08-1 1

2 1. Global analysis. Torsion. Global analysis, imperfecions, classificaion of srucures according o sabiliy and mehods of analysis. Torsion of open and closed secions. Mehods of analysis: linear analysis (1 s order, elasic) elasic plasic analysis GMNIA fibre plasiciy srengh (resisance) collapse e geomerically nonlinear analysis GNIA (or simplified nd order analysis) δ imperfecion e: 1. member imperfecion (expressed by iniial deflecion e 0 ),. frame imperfecion (expressed by iniial say of he frame φ). e 0 φ C08-1

3 Imperfecions: 1. Member imperfecions: covered by equivalen geomerical imperfecion e 0 e 0g e.g. for I cross secion + σ r f y e.g. imperfec hinge geomerical maerial srucural (shape imperf.) (residual sresses) (boundary condiions) + e 1 Effec of all hese imperfecions, i.e. iniial deflecions e 0g, residual sresses in cross secion σ r, srucural imperfecions e 1, are covered in he analysis by a single equivalen geomerical imperfecion ih ampliude e 0.. rame imperfecion (iniial frame say): covered by angle φ (or equivalen horizonal forces) V 1 φ V V 1 1 in Eurocode: φ = φ0α hα m V V here φ0 = 1/ 00 φ V φ αh = bu αh 1. 0 h 3 1 h.. oal heigh [m] α = m m.. number of loaded m columns per plane e 0 C08-1 3

4 Models according o effec of deflecion: 1. Geomerically nonlinear (in seel srucures he nd order analysis is sufficien). GNA is required alays for sabiliy analysis and non-rigid elemens (cables). Analysis mus cover influence of imperfecions (GNIA): - in compression members (using equivalen geomerical imperfecion e 0 ), - in sorey frames (using frame say φ, or equivalen horizonal forces), - plaes in buckling (imperfecions according o Eurocode ). Drabacks: superposiion is no valid, sofare necessary (oherise difficul).. Linear (1 s order heory) analysis is approximae only. In srucures ih rigid members he effec of GNIA need o be covered by reducion facors for compression (χ), bending (χ LT ), buckling (ρ), derived from nd order heory. Models according o use of maerial: 1. Elasic alays possible, bu only up o achieving seel yield poin f y. No economical, if plasic analysis is possible.. Plasic may be used provided he folloing is rue: a) maerial: he seel is ducile (min. 15 % mee by all common seels), b) cross secion: cross secions in plasic hinges are plasic (Class 1 cross secions), c) member: laeral orsional buckling of he members in plasic hinges is prevened). C08-1 4

5 Design for buckling in compression N N N L L cr = L L cr = L L cr = 0,5L Basic cases (for 1 s commonly λ = λ λ L L = β L = cr / i [bachelor sudy revision] buckling mode): N N N cr y y = = = 1 λ1 Ncr Ncr E cr Af χ Ed In general: for n h criical mode of he frame (use sofare): α cr,n = cr,n Ed i.e. (n = 1 up o ) cr, n = α Criical member forces (o deermine slenderness ): cr,n Ed λ member i N cr, n = α N α cr,n Ed cr,1 N Ed (1 s mode is conservaive) or member i he corresponding mode n is decisive (bu using mode n = 1 is conservaive). C08-1 5

6 rame classificaion The classificaion depends of boh geomery and loading i.e. i need o be deermine for each loading combinaion!! 1. rames hich may be solved by 1 s order heory (α cr 10): cr for plasic Noe: for arbirary Ed he α cr resuls from αcr = 10 analysis 15 sabiliy analysis by EM (various sofare). design loading Ed for given loading + frame imperfecion φ ΣV Ed V δ H, Ed H 1 H V 1 V h Approximaely for say mode according o picure: H Ed h α cr = VEd δ H,Ed or regular building frames his formula may be used for each sorey; decisive is he loes value. Noe: This approximae value of α cr is valid provided: λ 0. 3 (oherise individual members may buckle in heir sysem lenghs) Design of all frame members for his loading may be performed for heir sysem buckling lenghs (beeen he frame nodes) and is very conservaive (according o Eurocode for α cr > 5 even he value χ = 1 may be used). Af N y Ed C08-1 6

7 Explanaion: for α cr = cr Ed he effec of nonlineariy is negligible: 10 cr Ed δ somehere here is a resisance. rames o be solved by nd heory (α cr < 10): In general 3 mehods are used: a) Geomerically nonlinear analysis of compleely imperfec srucure. (GNIA = geomerically nonlinear analysis ih imperfecions). nd order effecs and boh member and frame imperfecions are covered in resuling inernal forces and verificaion of individual compression members is hen for simple compression only, ih reducion facors χ = 1. design Ed for given loading φ e 0(i) e 0(i) e 0(i) This analysis is demanding for sofare, inroducion of imperfecions and evaluaion. C08-1 7

8 b) Geomerically nonlinear analysis ih frame imperfecion only (i.e. by inroducion of iniial say bu commonly ih equivalen horizonal forces). Individual members are hen verified for buckling lengh equal o sysem lengh (i.e. beeen he frame nodes). Thus he nd order of frame effecs are covered by he iniial say and of member effecs by he relevan member reducion facors. E.g. for he frame column he buckling lengh h (possibly ih reducion for buil-in end), for frame rafer L/. h cr h ~ L cr In his case he sabiliy check suppose ficive say suppor, as global say buckling is covered by he nd order analysis ih say imperfecion. Noe: for negligible rafer slope (or horizonal rafer, bu in heory for < 15º) is L cr equal o spacing of columns. Insead of such simplified GNIA a simplified analysis is possible for 3 α cr < 10: To solve a frame ih say mode of buckling he 1 s order heory may be used, bu ih all horizonal forces H Ed (including imperfecions V Ed φ) increased by nd order facor: α cr C08-1 8

9 c) requenly (classical mehod) is used 1 s order heory ihou any imperfecions and members are checked ih equivalen global buckling lenghs (using relevan reducion coefficiens χ): Global buckling lengh may be deermined from criical loading corresponding o buckling of he analysed member and depends on rigidiy of he frame members and loading (see nex page)!! Neverheless, global buckling lengh are given in formulas and ables in many references. E.g. for frame columns he global buckling lengh may be of many-muliple sysem lengh. or rafers he same procedure may be used, or o ake sysem lengh and increase momens from horizonal loading for abou 0%: h cr = β h given in many references δ L cr L/ L cr deermined similarly as for columns or o use sysem lengh and increase momens resuling from horizonal loadings for abou ~ 0%. ensure sabiliy of free flange!! C08-1 9

10 Typical global buckling lenghs (for say buckling mode): rafer Valid for symmerical loading h if I raf = 0 if I raf = rafer if I raf = h h if I raf = 0 Preferably he global buckling lengh may be deermined from criical loading N cr by common sofare from corresponding α cr,i (corresponding o buckled member i ) as follos: L cr,i = π EI N cr,i i = π EI α N cr,i i Ed,i (N Ed,i is compression of he relevan member i under given loading Ed and E I i is rigidiy). Noe: 1) Using α cr from approximae formula (i.e. for say buckling mode), he minimum buckling lengh equals he sysem lengh. ) When using α cr,min (valid in general for buckling of anoher member) he resul ill be safe, bu uneconomical. 3) Mind he modificaion of cross secions afer check: resuls in differen α cr and hence also L cr. C

11 Torsion Common is elasic soluion. Plasic soluion (ih recangular sress disribuion - see Srelbickaja, Trahair). Eurocode combine elasic orsion ih plasic bending. Disinguish: - simple orsion: arise shear sresses only, - arping orsion: arise boh shear and direc (normal) sresses. 1. Open cross secions (e.g. I, U, L) a) Simple (Sain Venan s) orsion (occurs only excepionally, see laer) T τ max T 3 b τ b τ i i = 1 i = i = 3 Only shear arises (max. in max ): τ (i) T = I (i) Torsion consan: I 1 = α bi 3 i 3 i f y / γ M0 3 (influence of rounding of rolled secions, oherise = 1) C

12 b) Warping orsion (acc. o Vlasov s heory) Assumpions: 1. Rigid cross secion (righ angles remain righ).. Null shear deformaion (i.e. shear lag is ignored). - One par of a orsion momen T is ransmied by simple orsion: T - Oher par by bending orsion: T - In oal: T = T + T T in arping orsion everyhing is relaed o cenral line S shear cenre (bending cenre) τ τ = + + Inernal forces: T T B momen of simple orsion + momen of bending orsion - σ + cross-secion arping - bimomen bending orsion C08-1 1

13 Sresses in orsion: Shear sresses: simple orsion bending orsion e.g. shear sresses in upper flange of I : Direc sresses: σ τ T B = = I TS = I B W firs secorial momen arping consan + = τ,max τ secorial secion modulus τ 0 sress hrough hickness Noe: Neglecing rigidiy in simple orsion ( ), he bending analogy for bending par of orsion may be used (differenial equaion of bending and bending orsion is he same): Bending: verical uniform loading q Bending orsion: uniform orsion loading m = q e Bending momen M (např. 1/8 ql ) Bimomen B (analog. 1/8 ml ) Direc sress σ = M/W Direc sress σ = B/W Shear force V τ =VS/(I ) Momen of bending orsion T τ =T S /(I ) Deflecion δ (např. 5/384 ql 4 /(EI) ) Torsion angle θ (např. 5/384 ml 4 /(EI ) Roaion δ Warping θ oal C

14 Warping characerisics or rolled secions in ables. In general from secorial coordinae: e.g. cross-secion I: h b z r G S U cross secion: In his posiion no orsion!! S S Posiion of main cenre : 1 da = 0 Posiion of shear cenre S: y da = A (secorial produc of ineria) The main secorial coordinae for I : hb = r ds = 4 The firs secorial momen for I : h b S = da = 16 The second secorial momen for I : (arping consan) I s A = da= A z da = 0 A h I 4 z A S G, S, I... see ables a = a C

15 Deerminaion of inernal forces due o orsion: Soluion of Vlasov s differenial equaions, or direcly from relevan formulas. Based on bending analogy (see page 13). Acc. o Eurocode (EN Czech NA: see Complemenary noes) as follos: Disribuion of orsion inernal forces: e Simple suppor in orsion (= couple of forces). If arping is zero, bimomen is no zero (requires bending A e.g. closed siffeners a girder eb). V Disribuion of orsional momen due o eccenrical force M corresponds o disribuion of ransverse force a eccenriciy. orsion T = Ve B Me Par ransmied by simple orsion is se aside: T V e κ T V e 1 κ ( ) B M e ( 1 κ ) Superposiion for more complex loading is necessary: e -e 3 κ e 1 see able of Eurocode Czech NA C

16 Simplified (conservaive) soluion neglecs simple orsion: T T T/h bending of flanges only = bimomen 0 h (ofen adequae: conservaive for σ ) Imporan noes: T/h 1. Direc sresses are big, hey can no be ignored!!. Direc sresses (from bending orsion) do no arise: a) or loading by shear sresses τ only (roughly also due o end T orsion (simple orsion arises only): b) In secions composed of radial ousands (because = I = 0): (shear cenre S is in cross poin) 3. In pracice usually occurs orsion abou enforced axis (V): T T S V S e analysis abou original shear cenre S is uneconomical!!! moreover, orsion is usually resriced by cladding rigidiy orsion may ofen be ignored. C

17 . Closed secions a) Simple orsion (shear sresses only, usual for design) T τ i d i Bred s shear flo (τ ) = cons. τ = = A s A s Conrary o open cross secion he maximal shear τ is in he hinnes plae and along hickness consan!! Torsion consan: b) Warping orsion: - Umanskij s heory (uses rigid cross secion), (area enclosed by cenre line) - Vlasov s heory (uses non-rigid cross secion), - EM (including influence of cross secion disorion, gives also ransversal bending momens in all plaes). The sresses are he same as in open cross secions: τ, τ, σ. Hoever, τ, σ are very small, commonly ignored even for bridges. (i) T Ω (i) Ω = ds C I

18 Ineracion of bending and orsion ( M y + T ) In general, bending and orsion sresses may be summed and von Mises crierion applied: σ x,ed fy / γ Direc sresses (open cross secions only): i.e. M0 σ = σ + σ f / γ M χ LT y,ed W My y B + W Shear sresses: VEd 1 Vpl,T,Rd σ z,ed + fy / γ M0 Ed f or open secions I and U or closed secions σ x,ed fy / γ y y M0 / γ σ z,ed fy / γ M1 V M1 V pl,t,rd...design plasic shear resisance of he cross secion. V pl, T,Rd M0 + 3 f M0 τ = 1 1, 5 y τ,ed 1 fy / 3 / γ M,Ed,Ed Vpl, Rd ( f / 3 )/ γ ( f / 3 )/ γ M0 Vpl, Rd ( ) pl, T,Rd = 0 y τ Ed / γ 1 C y τ e M0 in U secions only

19 Complemenary noes C

20 Inernal forces due o orsion according o he EN (Czech NA) Bimomen, momen of simple orsion and momen of bending orsion: B Ed = M Ed e (1 κ) T,Ed = V Ed e κ T,Ed = V Ed e (1 κ) here correcing facor: κ = 1/[ β + ( α / K ) ] and dimensionless facor: K = L (GI T / EI ) 0,5 The coefficiens α, β according o ype of loading and boundary condiions: Boundary orsion condiions Torsion loading α β Girder (boh sides) simple suppors (free arping) buil-in suppor (zero arping) ull uniform General ull uniform or inernal forces in suppor or maximum ihin span General Canilever buil-in suppor General (for inernal forces in suppor) C08-1 0