1C8 Advanced design of steel structures prepared b Josef achacek
List of lessons 1) Lateral-torsional instabilit of beams. ) Buckling of plates. 3) Thin-walled steel members. 4) Torsion of members. 5) Fatigue of steel structures. 6) Composite steel and concrete structures. 7) Tall buildings. 8) Industrial halls. 9) Large-span structures. 10) asts, towers, chimnes. 11) Tanks and pipelines. 1) Technological structures. 13) Reserve.
1. Lateral-torsional buckling (stabilit and strength).. the. Interaction of moment and axial force. Eurocode approach. 3
Stabilit of ideal (straight) beam under bending impulse L segment laterall supported in bending and torsion cr Strength of real beam (with imperfections 0, 0 ) W f cr,1 bifurcation under bending b,rd W f 1 strength 0, 0 initial, reduction factor depends on: W f cr 4
h f Stabilit of ideal beam under bending (determination of cr ) F S=G S "Basic beam" - with - axis of smmetr (simpl supported in bending and torsion, loaded onl b ) Two equations of equilibrium (for lateral and torsional buckling) ma be unified into one equation: EI w 4 d GI 4 dx t d dx EI 0 The first non-trivial solution gives = cr : EI L GI t cr 1 EI L GI w t cr EI L GI t where EI w cr 1 L GI t 1 wt wt L EI GI w t 5
Generall (EN 1993-1-1) for beams with cross-sections according to picture: C g C3 j C g C3 j C1 cr 1 wt k wt k w L EI GI w t g k L g EI GI t j k L j EI GI t F aa g g s s F S G S G (C) h f = h s S F G S F G S F G S F G F S G (T) smmetr about - smmetr about k -, loading through shear centre C 1 represents mainl shape of bending moment, C comes in useful onl if loading is not applied in shear centre, C 3 comes in useful onl for cross-sections non-smmetrical about -. 6
Procedure to determine cr : 1. Divide the beam into segments of lengths L according to lateral segment 3 support: segment segment 1 e.g. segment 1: (L,1 ) lateral support (bracing) in bending and torsion (lateral support "near" compression flange is sufficient). Define shape of moment in the segment: from table factor C 1 1: 1 ~1,77 ~,56 e.g. segment : a) usuall linear distribution b) almost never ~1,13 ~1, 35 (because loading here creates continuous support) 7
3. Determine support of segment ends: (actuall ratio of "effective length") k = 1 (pins for lateral bending) k w = 1 (free warping of cross section) Other cases of k : k = 1 k w = 1 angle of torsion is ero k = 1 k w = 0,5 stiffener non-rigid in torsion possible lateral buckling stiffener rigid in torsion (½ tube) possible lateral buckling
k = 0,7 k w = 1 conservativel konervativní (theor. k hodnot = k w = 0,5) k = 1 k w = 0,7 (conservativel 1) structure torsionall rigid torsionall non-rigid possible lateral buckling torsionall rigid Cantilever: - onl if free end is not laterall and torsionall supported (otherwise concerning cr this case is not a cantilever but normal beam segment), - for cantilever with free end: k = k w =. 9
4. Formula for cr depends also on position of loading with respect to shear centre ( g ): F Come in useful for lateral loading (loading b end moments is considered in shear centre). S - lateral loading acting to shear centre S ( g > 0) is destabiliing: it increases the torsional moment S - lateral loading acting from shear centre S ( g < 0) is stabiliing: it decreases the torsional moment F Factor C for moment shape : (valid for I cross-section) el 0,46 0,55 1,56 1,63 0,88 1,15 pl (plast. hinges) 0,98 1,63 0,70 1,08 10
5. Cross-sections non-smmetrical about - F a a gg s s S (C) S G h f = h s h f For I cross section with unequal flanges: warping constant parameter of asmmetr I w ( f )I( hs / f I I 1 ) fc fc I + I ft ft second mom. of area of compress. and tens. flange about - (T) 0, 5 j s ( ) da 0, 45 f hf I A Factor C 3 greatl depends on f and moment shape (below for k = k w = 1): cr =+1 cr = 0 cr = -1 f = -1 1,00 1,47,00 0,93 f = 0 1,00 1,00 0,00 0,53 f = 1 1,00 1,00 -,00 0,38 11
Cross sections with imposed axis of rotation suck S often V (imposed axis) cr is affected b position of imposed axis of rotation ( cr is alwas greater, holding is favourable) For a simple beam with doubl smmetric cross section and general imposed axis: g G S v axis cr E I w 1 v E Iv kwl g v G I t For suck loading applied at tension flange: h E Iw E I G It kwl cr h 1 coefficients for shape of : 1,00 0,00 0,93 0,81 0,60 0,81 1
Approximate approach for lateral-torsional buckling impulse roughl h w /6 In buildings, the reduction factor for lateral buckling corresponding to "equivalent compression flange" (defined as flange with 1/3 of compression web) ma be taken instead: f L i f, 1 E 1 93, 9 f Note: According to Eurocode the reduction factor is taken from curve c, but for cross sections with web slenderness h/t w 44 from curve d. The factor due to conservatism ma be increased b 10%. 13
Practical case of a continuous beam (or a rafter of a frame): The top beam flange is usuall laterall supported b cladding (or decking). Instead of calculating stabilit to imposed axis a conservative approach ma be used, considering loading at girder shear centre and neglecting destabiliing load location (C = 0): el C 1 =,3 pl C 1 = 1,1 el C 1 =,58 pl C 1 = 1,3 according to distribution (for different ma be used graphs b A. Bureau) Note: For suck stabiliing effect should be applied (C, g < 0). It is desirable to secure laterall the dangerous ones of bottom free compression flanges against instabilit: b bracing in the level of bottom flanges, or b diagonals (sufficientl strong) between bottom flange and crossing beams (e.g. purlins). Length L then corresponds to the distance of supports). 14
Beams that do not lose lateral stabilit: 1. Hollow cross sections Reason: high I t high cr. Girders bent about their minor axis Reason: high I t high cr 3. Short segment ( 0 4 ) - all cross sections, e.g. Reason: 1, 4. Full lateral restraint: "near" to the compression flange is sufficient ( approx. within h/4 ) compression flange loaded tension flange loaded v 0,47 g v 0,47 g g or higher v v g or anwhere higher 15
the ( b,rd ) Similarl as for compression struts: actual strength b,rd < cr (due to imperfections) Eurocode EN 1993: e.g. DIN: b, Rd pl,rd 1 n 1/n n =,0 (rolled) =,5 (welded) The procedure is the same as for columns: acc. to is determined with respect to shape of the cross section (see next slide). Note: For a direct. order analsis the imperfections e 0d are available. f b,rd W... W is section modulus acc. to cross section class 1 1 but 1,0 1,51,0 0 16
the ( b,rd ) Similarl as for compression struts: actual strength b,rd < cr (due to imperfections) Eurocode EN 1993: e.g. DIN: b, Rd pl,rd 1 n 1/n n =,0 (rolled) =,5 (welded) The procedure is the same as for columns: acc. to is determined with respect to shape of the cross section (see next slide). Note: For a direct. order analsis the imperfections e 0d are available. b,rd W... W is section modulus acc. to cross section class 1 f 1 but 1 0 1, 1,0 0, 5 For common rolled and welded cross sections:, 0 0, 4 = 0,75 For non-constant the factor ma be reduced to,mod (see Eurocode). 17
the ( b,rd ) For common rolled and welded cross sections:, 0 0, 4 = 0,75 For non-constant the factor ma be reduced to,mod (see Eurocode). Choice of buckling curve: rolled I sections shallow h/b (up to IPE300, HE600B) b high h/b > c welded I sections h/b c greater residual stresses due to welding rigid cross section h/b > d 18
the ( b,rd ) In plastic analsis (considering redistribution of moments and "rotated" plastic hinges) lateral torsional buckling in hinges must be prevented and designed for,5 % N f,ed : strut providing support girder in location of pl 0,05 N f,ed force in compression flange Complicated structures (e.g. haunched girders) ma be verified using "stable length" L m (in which = 1) - formulas see Eurocode. h s h h L h L 19
Interaction + N ("beam columns") Alwas must be verified simple compression and bending in the most stressed cross section see common non-linear relations. In stabilit interaction two simultaneous formulas should be considered: N N Usual case + N : N Ed k NRd N Ed k N Ed 1 N N Ed 1 Rk Rk Rd k k,rk,ed,ed,rd,ed 1,Rk,Ed 1,Rd,Ed,Ed 1 1 k k,ed,ed Note: historical unsuitable linear relationship (without nd order factors k, k ),Rk 1 1,Ed,Rk,Ed 1 1 factors k 1,8; k 1,4 for class 4 onl (for relations see EN 1993-1-1, Annex B) N N Ed b, Rd,Ed b,rd 1 N 0
Interaction + N ("beam columns") Complementar note: Generall FE ma be used (complicated structures, non-uniform members etc.) to analse lateral and lateral torsional buckling. First analse the structure linearl, second critical loading. Then determine: ult,k cr,op - minimum load amplifier of design loading to reach characteristic resistance (without lateral and lateral-torsional buckling); - minimum load amplifier of design loading to reach elastic critical loading (for lateral or lateral torsional buckling). ult,k op op = min (, ) cr,op Resulting relationship: N N Rk Ed 1,Rk,Ed 1 op 1
Ideal and differences. Procedure for determining of critical moment. Destabiliing and stabiliing loading. Approximate approach for lateral torsional buckling.. according to Eurocode.
to users of the lecture This session requires about 90 minutes of lecturing. Within the lecturing, design of beams subjected to lateral torsional buckling is described. Calculation of critical moment under general loading and entr data is shown. Finall resistance of and design under interaction of moment and axial force in accordance with Eurocode 3 is presented. Further readings on the relevant documents from website of www.access-steel.com and relevant standards of national standard institutions are strongl recommended. Kewords for the lecture: lateral torsional instabilit, critical moment, ideal beam, real beam, destabiliing loading, imposed axis, beam resistance, stabilit interaction.
for lecturers Subject: Lateral torsional buckling of beams. Lecture duration: 90 minutes. Kewords: lateral torsional instabilit, critical moment, ideal beam, real beam, destabiliing loading, imposed axis, beam resistance, stabilit interaction. Aspects to be discussed: Ideal beam and real beam with imperfections. Stabilit and strength. and factors which influence its determination. Eurocode approach. After the lecturing, calculation of critical moments under various conditions or relevant software should be practised. Further reading: relevant documents www.access-steel.com and relevant standards of national standard institutions are strongl recommended. Preparation for tutorial exercise: see examples prepared for the course. 4