GBT (GENERALISED BEAM THEORY)-BASED ELASTIC - PLASTIC POST-BUCKLING ANALYSIS OF STAINLESS STEEL THIN-WALLED MEMBERS

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1 4 TH INTERNATIONAL EXERTS SEMINAR STAINLESS STEEL IN STRUCTURES ASCOT, UK, 6-7 DECEMBER 2012 (GENERALISED BEAM THEORY)-BASED ELASTIC - LASTIC OST-BUCKLING ANALYSIS OF STAINLESS STEEL THIN-WALLED MEMBERS TECHNICAL UNIVERSITY OF LISBON MIGUEL ABAMBRES, DINAR CAMOTIM, NUNO SILVESTRE

2 MOTIVATION STAINLESS STEEL HAS SEVERAL ATTRACTIVE FEATURES WHEN COMARED WITH CARBON STEEL: BETTER AEARANCE HIGHER CORROSION RESISTANCE MORE COST-EFFECTIVE AND LONGER LIFE CYCLE DISSEMINATION OF EASY-TO-USE DESIGN TOOLS IS NEEDED IN ORDER TO INCREASE THE STRUCTURAL USE OF STAINLESS STEEL SINCE SHELL FINITE ELEMENT ANALYSES ARE OFTEN RONE TO DATA INUT ERRORS, YIELD DIFFICULT-TO-INTERRET RESULTS, AND INVOLVE HIGH COMUTATIONAL COSTS CLOUD GATE, CHICAGO (USA) ADRE ARRUE FOOTBRIDGE, BILBAO (SAIN), HTT://

3 MOTIVATION A -BASED FORMULATION TO REDICT THE BEHAVIOUR AND COLLASE OF THIN-WALLED MEMBERS CONSTITUTED BY HIGHLY NON-LINEAR MATERIALS (e.g., STAINLESS STEEL) IS ROOSED SUMMARY BRIEF OVERVIEW OF THE FORMULATION ILLUSTRATIVE EXAMLES: NON-LINEAR BEHAVIOUR OF TWO LEAN DULEX SHS COLD-FORMED COLUMNS (THEOFANOUS & GARDNER 2009) 1999 mm mm 400 mm FIXED-ENDED LONG IN-ENDED STUB COLUMN

4 FORMULATION - ASSUMTIONS : BEAM THEORY INCLUDING CROSS - SECTION IN - LANE DEFORMATION q ds s (v) z (w) d t (u) σ = 0, ε = 0 zz iz ε ε ss γ s σ E Eν ν 1-ν e Eν E D = σ ss ν 1-ν σ s 0 0 G

5 FORMULATION DISLACEMENT FIELD LINEAR COMBINATION OF DEFORMATION MODES WHITH AMLITUDES VARYING ALONG THE MEMBER LENGTH ds d CROSS-SECTION ANALYSIS s (v) z (w) (u) u(,s) = u (s) ζ () k k, v(,s) = w(,s) = v (s) k w (s) k ζ () k ζ () k MEMBER ANALYSIS

6 FORMULATION - EQUILIBRIUM ds s (v) z (w) d (u) VIRTUAL WORK δw +δw = = 0 + σ δγ dv int FE AROXIMATION: HERMITE CUBIC et σ δε σssδεs s + s s L e b t ζ = Ψ d v, w = 0 k, H k γ s =u ε,s + v= u,, - 2zw -, +,s +u u,, +,s v, +, v w,s +, w, w,s ζ 1 2 = Ψ d v, w 0 k H k e INCREMENTAL-ITERATIVE SCHEME: CYLINDRICAL ARC-LENGTH Ktan j Δd = Δλf σ σ ε σ εss σmn γ = + + d ε d ε d γ d mn mn mn s pl pl ss pl s pl

7 LASTICITY - OVERVIEW THEORY: J 2 WITH ASSOCIATED FLOW RULE NUMERICAL INTEGRATION: MEAN-NORMAL STRESS - STRAIN CURVE: QUACH ET AL. (2008) BASED ON ONLY THREE ARAMETERS n σ σ ,0 σ σ0.2 E σ0.2 n2 σ * σ* ε = ε + + ε *, σ < σ σ 2p E0.2 σ* 1 σ - a, σ 2 < σ σu b σ

8 ILLUSTRATIVE EXAMLES STRESS-STRAIN CURVE RAMBERG - OSGOOD ARAMETERS OBTAINED FROM FLAT COUON TESTS (CORNER STRENGTH ENHANCEMENT NOT MODELLED) σ (N/mm 2 ) Fied-ended column ( ) in-ended column (80804) % 0% 0.2% 5% 0.4% 10% 0.6% 15% 0.8% 1.0% 20% ε

9 LONG IN-ENDED COLUMN LEAN DULEX (EN ): σ 0 y = N/mm 2, σ 0.2 =657 N/mm 2, E= N/mm 2, ν=0.3 z y F = λ N [mm] : S4 SFEs mm = d.o.f : 41 / 124 MODES + 48 FEs = 3909 d.o.f (18%) 2.5 δ y δ y t = KNIFE EDGES (THEOFANOUS & GARDNER, 2009 ) INITIAL GEOMETRICAL IMERFECTIONS 2.0 Δ = L/ λ b,glob = Δ = mm λ b,loc =5.76 L

10 EQUILIBRIUM ATH λ B Test (Theofanous and Gardner) A % & MODAL ARTICIATION λ u, = 1.04 λ u,test = 1.00 DIF 1.5% OTHERS 6 δ y z y F = λ δ y EX. OTHERS 75% 3 93% δ y 1

11 DISLACEMENTS λ z y A δ y LOCAL BUCKLING OVERSHADOWED 90 δ y BY GLOBAL BENDING A RECALL THE INITIAL IMERFECTION... δ y L L

12 STRESSES z y W3 λ B A W2 W1 δ y WHEREVER σ MISES > σ 0 y, THE DIAGRAM NON-LINEARITY REFLECTS THE NEUTRAL AXIS CROSSES THE CROSS-SECTION AT THE A STATE STAINLESS TENSILE STRESSES STEEL NON-LINEAR ONLY DEVELO HARDENING AFTER THE EAK LOAD σ (N / mm 2 ) σ MISES (N / mm 2 ) B A W1 W2 W A B W1 W2 W3

13 STRESSES z y W3 λ B A W2 W1 δ y WHEREVER σ MISES > σ 0 y, THE DIAGRAM NON-LINEARITY REFLECTS THE STAINLESS STEEL NON-LINEAR HARDENING A STATE () σ MISES (N / mm 2 ) A COMRESSION SIDE 600 LASTICITY ALSO IN TENSION ( BUT TO A LESSER EXTENT ) TENSION SIDE B W1 W2 W3

14 FIXED-ENDED STUB COLUMN LEAN DULEX (EN ): σ 0 y = N/mm 2, σ 0.2 =560 N/mm 2, E= N/mm 2, ν=0.3 z F = λ N t = [mm] : S4 SFEs 8 8 mm = d.o.f : 28 / 204 MODES + 16 FEs = 865 d.o.f (6%) INITIAL GEOMETRICAL IMERFECTION δ z Δ = t / L λ b,loc =

15 EQUILIBRIUM ATH & MODAL ARTICIATION λ B Test (Theofanous and Gardner) % CORNER STRENGTH ENHANCEMENT SHOULD BE MODELLED z 77.3% 166, DIF 4.9% 28% λ u, = λ u,test = % OTHERS 9, 14 19% A Δ Δ TWO DISTINCT BEHAVIOURS: EX GLOBAL DEFORMATION GOVERNS U TO THE EAK LOAD LOCAL BUCKLING EMERGES IN 1 THE OST-EAK STAGE F = λ Δ 9 14 OTHERS

16 MISES STRESSES λ AT THE EAK (), THE WHOLE COLUMN MID-SURFACE IS ALREADY YIELDED: σ MISES > N/mm A Δ THE HIGHEST A STRESSES OCCUR AT THE MID-SAN CORNER REGIONS (RED), DUE TO FROM TO A, THE COLUMN EXERIENCES UNLOADING (BLUE + GREEN) AND ALSO THE FURTHER REDISTRIBUTION LASTIC DEFORMATION CAUSED BY THE (YELLOW LOCAL + DEFORMATION RED) (EFFECTIVE WIDTH CONCET)

17 DISLACEMENTS z δ z λ B A δ 400 Δ AT AT OST-COLLASE, (A), THE THREE THE AXIAL HALF-WAVES STIFFNESS RETURN DRO AT DUE MID-SAN TO BOTH ROFILES DISLAY FIVE HALF-WAVES. THE INITIAL IMERFECTION HAD THREE... LOCALISED ASSOCIATED LASTIC WITH DEFORMATION THE INCREASING AT MID-SAN LOCAL(ISED) + ELASTIC DEFORMATION UNLOADING 1.2 δ z B L δ z A δ B A L L

18 CONCLUDING REMARKS A FORMULATION TO ERFORM OST-BUCKLING ANALYSES OF THIN-WALLED MEMBERS MADE OF ISOTROIC MATERIALS EXHIBITING NON-LINEAR HARDENING WAS DEVELOED AND IMLEMENTED THE ALICATION AND OTENTIAL OF THE FORMULATION WERE ILLUSTRATED THROUGH THE NON-LINEAR BEHAVIOUR AND COLLASE OF TWO LEAN DULEX SHS COLD-FORMED COLUMNS TESTED AT IMERIAL COLLEGE BY THEOFANOUS & GARDNER (2009) THE RESULTS WERE VALIDATED AGAINST SFEA VALUES - QUITE GOOD AGREEMENT WAS FOUND, EVEN IF THE ANALYSES INVOLVE ONLY 18% AND 6% OF THE D.O.F. NUMBERS REQUIRED BY THE SFEA

19 CONCLUDING REMARKS THE MODAL NATURE MAKES IT OSSIBLE TO ACQUIRE IN-DETH INSIGHT ON THE MECHANICAL FEATURES OF THE COLUMN BEHAVIOUR CORNER STRENGTH ENHANCEMENT AND MEMBRANE RESIDUAL STRESS EFFECTS WILL BE FAIRLY SOON INCORORATED IN THE ANALYSES (ALSO FOR OEN SECTIONS AND DISTORTIONAL BUCKLING ROBLEMS) ADDITIONAL SIMULATIONS, INVOLVING TESTS CARRIED OUT AT THE UNIV. OF SYDNEY, WILL BE REORTED IN THE NEAR FUTURE (FIFTH SEMC CONFERENCE SETEMBER 2013, CAE TOWN, S. AFRICA)