ADVANCED DESIGN OF STEEL AND COMPOSITE STRUCTURES

Transcription

1 ADVACED DESG OF STEEL AD COPOSTE STRUCTURES Luís Simões da Silva Leture : 0//04 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events CZ-ERA UDUS-EC

2 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events COTETS odule A Design of non-uniform members ntrodution on-uniform members approahes and problems 3 Tapered olumns 3. Differential equation 3. Elasti ritial load 3.3 Ayrton-Perry formulation 4 Design resistane of tapered olumns and beams 4. Derivation 4. Tapered olumns 4.. Design methodology 4.. Eample 4.3 Tapered beams 4.3. Elasti ritial moment 4.3. Design methodology Eample 5 Beam-olumns odule A Design of non-uniform members

3 ntrodution

4 European Erasmus undus aster Course ntrodution Sustainable Construtions under atural Hazards and Catastrophi Events q Tapered steel members are used in steel strutures q Strutural effiieny à optimization of ross setion apaity à saving of material q Aesthetial appearane ulti-sport omple Coimbra Portugal odule A Design of non-uniform members Constrution site in front of the Central Station Europaplatz Graz Austrial

5 European Erasmus undus aster Course ntrodution Sustainable Construtions under atural Hazards and Catastrophi Events Tapered members are ommonly used in steel frames: industrial halls warehouses ehibition enters et. Adequate verifiation proedures are then required for these types of strutures! odule A Design of non-uniform members

6 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events ntrodution However there are several diffiulties in performing the stability verifiation of strutures omposed of non-uniform members; Guidelines are ineistent or not lear for the designer Due to this reason simplifiations that are not mehanially onsistent are adopted These may be either too onservative or even Unonservative! odule A Design of non-uniform members

7 on-uniform members Approahes and Problems

8 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events on-uniform members approahes and problems Prismati members Clauses 6.3. to Developed for prismati members Sinusoidal imperfetions δ e 0 0 π sin L Column δr δ'r δ''r L Ayrton-Perry type equation: s maimum at mid span: ε + Rk Constant yrk π Eδ sin L Sinusoidal OK! odule A Design of non-uniform members

9 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events on-uniform members approahes and problems on-uniform members Clauses 6.3. to apply??? Ed Cross setion utilization due to applied first order fores is not onstant anymore. Rk ot Constant odule A Design of non-uniform members

10 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events on-uniform members approahes and problems on-uniform members Clauses 6.3. to apply???. 0.8 Column δr δ'r δ e 0 0 π sin L 0.6 δ''r L π Eδ sin L Ayrton-Perry type equation: s it maimum at mid span??? ε + Rk yrk KO! odule A Design of non-uniform members

11 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events on-uniform members approahes and problems on-uniform members Clauses 6.3. to apply??? Position of the ritial ross-setion not at mid span Aount for nd order effets; iterative proedure not pratial; st order ritial ross setion is onsidered! odule A Design of non-uniform members

12 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events on-uniform members approahes and problems on-uniform members Clauses 6.3. to apply??? Variation of ross setion lass Class 3 Class Class yed Definition of an equivalent lass for the member Ed <<< Af y odule A Design of non-uniform members

13 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events on-uniform members approahes and problems on-uniform members nd order analysis with imperfetions Definition of loal imperfetions: Same problem: e 0 /L alibrated for prismati members with sinusoidal imperfetions Bukling urve a. Elasti analysis Plasti analysis to EC3-- Table 6. e 0 /L e 0 /L a 0 /350 /300 a /300 /50 b /50 /00 /00 /50 d /50 /00 odule A Design of non-uniform members

14 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events on-uniform members approahes and problems q on-uniform members nd order analysis with imperfetions q Definition of loal imperfetions? Barajas Airport adrid Spain Auvent de la Gare Routière Ermont taly pavilion World Epo 00 Shanghai odule A Design of non-uniform members

15 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events on-uniform members approahes and problems on-uniform members GEERAL ETHOD lause n-plane resistane n-plane GA alulations Out-of-plane elasti ritial load LEA alulations ultk rop λ op ult k r op / Bukling urve 3 χ op inimum χ χ χ χ χ op nterpolated χ χ χ op ult k / odule A Design of non-uniform members

16 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events on-uniform members approahes and problems on-uniform members GEERAL ETHOD lause % / ultk should aount for loal seond order effets? Φ Rolled f so: again à problem with definition of imperfetions Prismati olumn vs λ_z PE 360 PE 00 HE 550 B HE 500 A HE 450 B HE 400 B HE 360 B HE 340 B HE 300 B HE 00 A Higher in-plane seond order effets lead to a derease in the out-ofplane redution fator Even for the ase of beam-olumns this effet is not as restritive. odule A Design of non-uniform members

17 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events on-uniform members approahes and problems on-uniform members GEERAL ETHOD lause Definition of a proper bukling urve Curve a Curve d Curve b Curve For some ases eisting bukling urves may even be unonservative! odule A Design of non-uniform members

18 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events on-uniform members approahes and problems on-uniform members GEERAL ETHOD lause Redution fator minimum or interpolation inimum ay be too onservative Does not follow bukling mode orretly nterpolation Provides a transition between FB y 0 and B 0 What type of funtion for a orret mode transition? odule A Design of non-uniform members

19 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events on-uniform members approahes and problems on-uniform members FE numerial analysis Problem with definition of imperfetions Requires a high eperiene in FE modeling from the user in order to ahieve reliable results Limited guidelines For the most simple ases it is preferable to provide simple rules whih inlude as muh as possible the real behavior of the member odule A Design of non-uniform members

20 Tapered olumns

21 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Tapered olumns Differential equation Equilibrium n on dq Q + d d dδ d + d d + d d B δ n d d ξ dδ η δ B n d Q A A on L + nξ dξ odule A Design of non-uniform members

22 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Tapered olumns Differential equation Equilibrium on L + nξ dξ + neglet nd order terms dq Q + d d dδ d + d d + d d B d Eq. oments in B d d. dy + Qd + d + n d 0 d ξ η ξ $!# 0!" 0 d dy Q d d A δ n Q d Eq. Horizontal Q Q + dq d d dq d 0 dq d 0 d d d d dδ d odule A Design of non-uniform members

23 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Tapered olumns Differential equation Equilibrium d δ E + d d 0 d d d dδ d dq Q + d d dδ d + d d + d d B d E d d d δ + d d d dδ 0 d δ n Q d E δ + δ 0 r Ed n r ned δ δ r A odule A Design of non-uniform members

24 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Tapered olumns Elasti ritial load Differential equation E δ + δ 0 Proposal for determination of major ais ritial load of tapered olumns arques et al 04 r Tap y.ma A r min y.min A tan odule A Design of non-uniform members

25 odule A Design of non-uniform members European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Equilibrium First yield riterion 0 + δ δ E 0 n r Ed r δ δ δ δ δ E 0 b r b Ed b δ δ 0 E E b r b δ δ 0 E R b r b R Ed b R R Ed b + + δ ε Tapered olumns Ayrton-Perry formulation

26 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Tapered olumns Ayrton-Perry formulation Assumption for imperfetion δ δ e 0 r 0 b ε R Ed + R b R Ed + 0 E $! # δ!!" b δ r e0 r b R ntroduing slenderness and redution fator definitions λ R / r Ed χ R b / Ed ε χ + χ b e 0 r R R δ E r Ed r R R R R At the ritial loation ε odule A Design of non-uniform members χ χ + e0 λ χ E. δ r r. Ed R R λ 0. EC3

27 Design resistane of tapered olumns and beams

28 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams aterial: Perfet elasti-plasti Boundary onditions: Fork supports End ross setions remain straight f y E 0 GPa f y 36 Pa ember imperfetions: With amplitude e 0 L/000 Same shape as the bukling mode εy ε L/000 aterial imperfetions: 0.b fy 0.8h 0.5 f y 0.5 fy 0.5 f y 0.5 f y Calulations: LBA GA odule A Design of non-uniform members

29 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams Columns ε χ + χ b e 0 r R R δ E r Ed r R R R R δ Rk E Rk Rk Beams ε χ χ + λ χ e 0 A ξ λ W z λz δ rh min rztap + Ez + rztap rtap rtap rztap h h min A W z W z λ λz A λz λ y yrk z zrk ϖrk E z zrk v Eωφ ϖ ϖrk odule A Design of non-uniform members

30 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams Columns ε χ + χ b e 0 r R R EC3 λ 0. δ E r Ed r R R R R η uniform η non uniform Beams ε χ χ + λ χ e 0 A ξ λ W z λz δ rh min rztap + Ez + rztap rtap rtap rztap h h min A W z W z λ λz A λz λ λz 0. odule A Design of non-uniform members

31 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams Columns ε χ + χ b e 0 r R R δ E r Ed r R R R R ε χ Beams χ + λ χ e χ 0 + χ λ A ξ λ W z λz χ δ rh min rztap EC3 + Ez + λ 0. β rztap rtap rtap rztap h h min A W z W z λ λz A λz λ ε χ + λ χ χ λ λz 0. β λ z odule A Design of non-uniform members

32 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams nterpretation of and β eample olumn h /L β λ y λ y h odule A Design of non-uniform members

33 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams Simplifiation of the analytial models β; β lim ; lim χ Columns Beams β 0; χ χ lim λ + χ lim λ χ χlim lim + λ χ z lim λ βlim lim lim λ λ lim 0.! βlim λ lim λz lim 0. #"! odule A Design of non-uniform members

34 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams Simplifiation of the analytial models β; β lim ; lim χ Columns Beams β 0; χ χ lim Transformation of variables ϕ ultk ultk λ + χ lim λ χ χlim lim + λ χ z lim λ βlim lim λ lim lim lim λ λ lim 0.! βlim λ lim λz lim 0. #"! ϕ λ χ χ / ϕ lim odule A Design of non-uniform members

35 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams COLUS DESG ETHODOLOGY. Required data Calulate utilization ε Ed / Rk based on Semi-Comp approah Calulate ϕ Calulate r Rk ilt k Ed odule A Design of non-uniform members

36 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams COLUS DESG ETHODOLOGY. Appliation of the method 3. Verifiation η ϕ λ 0. λ ultk r χ ultk b φ χ ϕ η λ + ϕ λ φ + φ ϕ ϕ λ Ed b Ed odule A Design of non-uniform members

37 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams COLUS DESG ETHODOLOGY eessary parameters y.ma Critial load multiplier r ay be numerial y.min From the literature For in-plane loading a formula was developed onsidering Rayleigh-Ritz ethod : 0.04 tan 0.56 r Tap A r min A Diff % H&C HEB300 H&C PE00 H&C 000 EQU HEB300 EQU PE00 EQU 000 L&al. HEB300 L&al. PE00 L&al odule A Design of non-uniform members

38 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams COLUS DESG ETHODOLOGY eessary parameters mperfetion fator y or z FB out-of-plane FB in-plane Hot-rolled: Welded: Hot-rolled: Welded: η χ y EC Best fit η Euler λ y odule A Design of non-uniform members

39 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams COLUS DESG ETHODOLOGY eessary parameters Overstrength fator φ y or φ z + FB out-of-plane FB in-plane ht A + 4 h h 0 h h + w min w h min Amin h + t odule A Design of non-uniform members

40 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams COLUS DESG ETHODOLOGY Results Out-of-plane hh ma /h min.8 n-plane hh ma /h min 6 b.0 b EC3-0.4 EC3-b 0. Euler GA 0. Euler GA 0.0 Proposal Proposal λ z λ y odule A Design of non-uniform members

41 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams COLUS DESG ETHODOLOGY Possible problems Web bukling ritial loation varies φ was alibrated onsidering ritial loation without loal bukling effets! ay lead to unsafe results! odule A Design of non-uniform members

42 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams COLUS - EXAPLE L5m h wmin 70 mm h wma 60 mm.59º Ed tf 5 mm b 06 mm tw 5 mm S35 w /t w <33εà Class <38εà Class <4ε à Class4 Ed 00 k L 5 m 3.65m 4.55m Flange lass: f /t f 06/-5//53.8 <9ε à Class Flange thikness in vertial plan t f ' t f / os 5. 0mm h min 0.05 mm h ma mm h /0.053 odule A Design of non-uniform members

43 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams COLUS - EXAPLE λ inor ais verifiation Calulation of slenderness at Rk / r Ed r hmin Overstrength-fator φ Rk / / Ed Ed 30./00 30./00 r hmin π E L hmin h wmin 70 mm h wma 60 mm b 06 mm tf 5 mm tw 5 mm S35 Ed 00 k L 5 m h ϕz + A t min + 4 h h 0 h Determination of imperfetion η min w. ϕ λ > η η odule A Design of non-uniform members

44 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams COLUS - EXAPLE inor ais verifiation Redution fator φ 0.5 χ + η + ϕλ φ + ϕ φ ϕ λ h wmin 70 mm h wma 60 mm b 06 mm tf 5 mm tw 5 mm S35 Ed 00 k L 5 m Verifiation b z Rd χ Pl k > 00 k umerial analysis GA odule A Design of non-uniform members b z Rd k 0.5% diff

45 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams COLUS - EXAPLE ajor ais verifiation Critial load rytap A 0.56 r y Tap hma hmin 3365m 047m 0.04 tan A rmin k k umerial bukling analysis LBA: r y Tap 356k à diff. 3.9% h wmin 70 mm h wma 60 mm b 06 mm tf 5 mm tw 5 mm S35 Ed 00 k L 5 m odule A Design of non-uniform members

46 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams COLUS - EXAPLE λ ajor ais verifiation Calulation of slenderness at Rk / r Ed r y Tap / / Overstrength-fator φ Rk Ed Ed 30./ / h wmin 70 mm h wma 60 mm b 06 mm tf 5 mm tw 5 mm S35 Ed 00 k L 5 m ϕ y h A t min w h + min 0.05mm 5mm mm 3+ h Determination of imperfetion η.8 ϕ λ < η η odule A Design of non-uniform members

47 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams COLUS - EXAPLE ajor ais verifiation Redution fator χ φ 0.5 φ + + η + ϕλ ϕ φ ϕ λ > χ h wmin 70 mm h wma 60 mm b 06 mm tf 5 mm tw 5 mm S35 Ed 00 k L 5 m Verifiation b y Rd χ Pl k > 00 k umerial analysis GA odule A Design of non-uniform members b z Rd 30. k 0% diff

48 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams BEAS DESG ETHODOLOGY. eessary data Calulate utilization ε Ed / Rk based on Semi-Comp approah Calulate ϕ Calulate lim Calulate r Rk ilt k Ed odule A Design of non-uniform members

49 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams BEAS DESG ETHODOLOGY. Appliation of the method W W y el z el r π E z Rk L 3. Verifiation χ ult k b Taras λ z η λ z 0. ult k λ r b Ed φ λ ϕ η + ϕ λ λ z Ed χ φ + φ ϕ ϕ λ Ψ Ed odule A Design of non-uniform members

50 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams BEAS DESG ETHODOLOGY eessary parameters Critial load multiplier r ay be numerial From the literature mperfetion fator odel onsistent with reently developed proposals for prismati beams Hot-rolled: Welded: W y el lim W z el lim 0. W W y el z el lim lim W 0.64 yel lim η - 0.ψ 0.3ψ W zel lim odule A Design of non-uniform members

51 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams BEAS DESG ETHODOLOGY eessary parameters lim lim /L 0.5 h 0.4 h h h.6 0. h3. 0. h3.6 h ψ 0.5 For ψ ψ 0.07ψ ψ 0.006ψ 0.06 f ψ < 0 and ψ w +.4 For UDL h h 0 / L h h lim h odule A Design of non-uniform members

52 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams BEAS DESG ETHODOLOGY eessary parameters Overstrength fator φ.3 φ ψ w w.5 w w.5 w 3 w 3.5 w 4 w 4.5 w 5 w 5.5 w 6 w ψlim ψlim w- odule A Design of non-uniform members

53 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams BEAS DESG ETHODOLOGY eessary parameters Overstrength fator φ UDL: 0.005a a Ψ: A ψ + B ψ + C a ψ lim φ A B C a a 0.44 a a a a w w a a 0 a + 3 a + 40 a a ψ ψ lim < ψ lim ψ ψ lim a.9 a 5.4 a a.355 a 5. a a a.8 a a a.34 a a 058 a a a 8050 a a 0.33 a a a 0 a 0.0 a 0.4 a a 3 w a w ψ >ψ lim a 0.08 a a a 0.03 a a a a odule A Design of non-uniform members

54 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams BEAS DESG ETHODOLOGY Results χ Euler 0. odel φ.35 GA λ Possible problems: Web bukling ψ - h 3 nfluene of shear! χ 0.8 UDL h Shear interation Euler 0. odel φ GA λ.8 odule A Design of non-uniform members

55 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams BEAS - EXAPLE L5m.59º w /t w <4εà Class Flange lass: f /t f 06/-5//53.8 <9ε à Class yed h wmin 70 mm h wma 60 mm b 06 mm tf 5 mm tw 5 mm S35 yed 300 km L 5 m Flange thikness in vertial plan t f ' t f / os 5. 0mm h min 0.05 mm h ma mm h /0.053 V Ed 300/5 k 60 k V plrdmin f y k > V Ed h wma /t w 60/54<7ε odule A Design of non-uniform members

56 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams BEAS - EXAPLE Elasti ritial moment numerial analysis r Tap m km ε y Ed y Rd h wmin 70 mm h wma 60 mm b 06 mm tf 5 mm tw 5 mm S35 yed 300 km L 5 m Série ε L 0.38 odule A Design of non-uniform members

57 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams BEAS - EXAPLE Calulation of slenderness at λ ult k r / / Seond order failure loation lim w W W ψ 0 y elma y elmin... 3 h 400.7mm mm h wmin 70 mm h wma 60 mm b 06 mm tf 5 mm tw 5 mm S35 yed 300 km L 5 m ψ 0.07ψ ψ 0.006ψ lim / L h odule A Design of non-uniform members

58 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams a BEAS - EXAPLE ψ 0 ψ Over-strength fator φ w W W ψ 0 y el ma y el min... 3 h η lim Determination of imperfetion η 0. odule A Design of non-uniform members 400.7mm mm w ψ lim A B C w Wy el lim λ 0. 0.ψ 0.3ψ W W y el z el lim lim W 0.64 z el lim w w ϕ.957 A ψ + B ψ + C λ η h wmin 70 mm h wma 60 mm b 06 mm tf 5 mm tw 5 mm S35 yed 300 km L 5 m.053 A π E < f y L.50

59 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Design resistane of tapered olumns and beams BEAS - EXAPLE Redution fator φ χ Verifiation 0.5 λ + ϕ η z λ φ + φ ϕ lim ϕ λ + ϕ λ h wmin 70 mm h wma 60 mm b 06 mm tf 5 mm tw 5 mm S35 yed 300 km L 5 m b Rd χ y Rd k > y Ed 300 km umerial analysis GA odule A Design of non-uniform members Rd b k 3.3% diff

60 Beam-olumns

61 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Beam-olumns How to solve the problem nd order effets P- Δ + Global imperfetions nd order effets P- δ + Loal imperfetions aterial nonlinearity Chek Bukling length φ e 0y e 0z 0 YES YES YES YES a. Load fator GA - YES YES YES O Cross setion - a YES YES O O Out- of- plane ember stability proedures L b YES O O O ember stability proedures Global Lrz 3 O O O O ember stability proedures Global Lrz Lry odule A Design of non-uniform members

62 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Beam-olumns Diffiulties for eah approah nd order effets P- Δ + Global imperfetions nd order effets P- δ + Loal imperfetions aterial nonlinearity Chek Bukling length φ e 0y e 0z 0 YES YES YES YES a. Load fator GA - YES YES YES O Cross setion - a YES YES O O Out- of- plane ember stability proedures L b YES O O O ember stability proedures L 3 O O O O ember stability proedures Global Lrz Lry? Bukling urve a. Elasti Plasti?? to EC3-- Table analysis analysis 6. e 0 /L e 0 /L a 0 /350 /300 a /300 /50 b /50 /00 /00 /50 d /50 /00 eed to alibrate e0/l a. to new imperfetion fators for welded members! odule A Design of non-uniform members

63 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Beam-olumns Diffiulties for eah approah nd order effets P- Δ + Global imperfetions nd order effets P- δ + Loal imperfetions aterial nonlinearity Chek Bukling length φ e 0y e 0z 0 YES YES YES YES a. Load fator GA - YES YES YES O Cross setion - a YES YES O O Out- of- plane ember stability proedures L b YES O O O ember stability proedures L 3 O O O O ember stability proedures Global Lrz Lry Calibrate e0/l for new imperfetion fators odule A Design of non-uniform members eed to develop à n-plane à Out-of-plane Approahes: à nteration à Generalized slenderness

64 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Beam-olumns Diffiulties for eah approah nd order effets P- Δ + Global imperfetions nd order effets P- δ + Loal imperfetions aterial nonlinearity Chek Bukling length φ e 0y e 0z 0 YES YES YES YES a. Load fator GA - YES YES YES O Cross setion - a YES YES O O Out- of- plane ember stability proedures L b YES O O O ember stability proedures L 3 O O O O ember stability proedures Global Lrz Lry Fous on approah a: à Develop out-of-plane verifiation proedure; à To aount for B in the in-plane stability redue pl by χ in the ross setion hek odule A Design of non-uniform members

65 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Beam-olumns Adaptation of the nteration formula χ z Ed Rk + k zy χ y Ed y Rk.0 k zy fator determined from ethod Anne B Beause yy and zz modes are learly separated Cmy fators determined with EBER UTLZATO y / yrk ε ε s 0.97 ψ ε.ε h ε h s C C mi mi 0. /L ψ < 0.4 C ψ ε ε mi s odule A Design of non-uniform members

66 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Beam-olumns Adaptation of the nteration formula Results yed [km] nteration GA Ed yed [k] [km] 0 Low slenderness: χ ov ethod nteration nteration GA ross setion Ed [k] χ ov GA odule A Design of non-uniform members

67 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Beam-olumns χ ethod Adaptation of the nteration formula n-plane failure mode % -5% yed [km] Csetion GA λy 0.8 GA λy 0.6 GA λy yed [km] Ed [k] Ed [k] ross setion GA λy 00.8 GA λy GA λy.39 χ 6.6 GA Also up to 0% onservative odule A Design of non-uniform members

68 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Beam-olumns Adaptation of the nteration formula n-plane and out-of-plane failure mode χ ethod or % -6% L y L 0.6 L y / L z L/ L z / χ GA Also up to 0% onservative odule A Design of non-uniform members

69 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Beam-olumns BEA-COLUS - EXAPLE ember lass 0.6 L5m.59º yed Ed h wmin 70 mm h wma 60 mm b 06 mm tf 5 mm tw 5 mm S35 Ed 00 k yed 300 km L 5 m 0.5 ε Class Class Class /L odule A Design of non-uniform members

70 odule A Design of non-uniform members European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events nteration formulae Data: BEA-COLUS - EXAPLE.0 + Rk y Ed y yy Rk y Ed k χ χ.0 + Rk y Ed y zy Rk z Ed k χ χ el y Rk y Rk Rk Ed y Ed km k km k?? zy yy y y y z z k k χ χ χ χ χ χ Beam-olumns

71 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Beam-olumns BEA-COLUS - EXAPLE C my C m ψ el 0 ε s C my el C s / ε el h 0.36 / m s k yy k zy C my 0.05 C 0.05 Ed ϕ λ y y $!!#!!" χ y Rk 0.6 odule A Design of non-uniform members ϕ λ z z m 0 $!!#!!" χ z Ed Rk 0.05 $!#!" $!!!#!!!"

72 odule A Design of non-uniform members European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Verifiation BEA-COLUS - EXAPLE Rk y Ed y yy Rk y Ed k χ χ Rk y Ed y zy Rk z Ed k χ χ zy yy el y Rk y Rk Rk y y y z z Ed y Ed k k km k km k χ χ χ χ χ χ umerial analysis GA Edma k yedma km LF.6 LF / % diff Beam-olumns

73 Researh hallenges

74 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events Researh hallenges Generalization of the over-strength fators to any shape of rosssetion / loading Utilization of the ompressed flange Calibrate equivalent imperfetions e 0 /L for new imperfetion fators and also for other types of ross setion: Properly aount for loal bukling due to shear / bending Development of diret redution fator approah for beam-olumns Other boundary onditions Partial restraints odule A Design of non-uniform members

75 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events REFERECES Simões da Silva L Rimões R and Gervásio H - Design of Steel Strutures Ernst & Sohn and ECCS Press Brussels 00. arques L. Taras A. Simões da Silva L. Greiner R. and Rebelo C. Development of a onsistent design proedure for tapered olumns" Journal of Construtional Steel Researh 7 pp arques L. Simões da Silva L. Greiner R. Rebelo C. and Taras A. Development of a onsistent design proedure for lateral-torsional bukling of tapered beams Journal of Construtional Steel Researh 89 pp odule A Design of non-uniform members

76 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events REFERECES arques L. Simões da Silva L. and Rebelo C. Rayleigh-Ritz proedure for determination of the ritial loads of tapered olumns Steel and Composite Strutures 6 pp arques L. Simões da Silva L. Rebelo C. and Santiago A. Etension of the interation formulae in EC3-- for the stability verifiation of tapered beam-olumns Journal of Construtional Steel Researh 04 in print. Simões da Silva L. arques L. and Rebelo C. 00 umerial validation of the general method in EC3--: lateral lateral-torsional and bending and aial fore interation of uniform members Journal of Construtional Steel Researh 66 pp odule A Design of non-uniform members

77 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events SOFTWARE Beam - Elasti Lateral Torsional Bukling of Beams v.0. CTC deslog.asp?idrub&lng SemiComp ember Design Design resistane of prismati beam-olumns Greiner et al RFCS 0. ECCS Steel ember Calulator Design of olumns beams and beam-olumns to EC3-- Silva et al ECCS odule A Design of non-uniform members

78 European Erasmus undus aster Course Sustainable Construtions under atural Hazards and Catastrophi Events ACKOWLEDGEETS This leture was prepared for the Edition of SUSCOS 03/5 by LUÍS SÕES DA SLVA and LLAA ARQUES UC. The SUSCOS powerpoints are overed by opyright and are for the elusive use by the SUSCOS teahers in the framework of this Erasmus undus aster. They may be improved by the various teahers throughout the different editions. odule A Design of non-uniform members

79 Gab. SF 5.