Advanced stability analysis and design of a new Danube archbridge. DUNAI, László JOÓ, Attila László VIGH, László Gergely

Transcription

1 Advanced stability analysis and design of a new Danube archbridge DUNAI, László JOÓ, Attila László VIGH, László Gergely

2 Subject of the lecture Buckling of steel tied arch Buckling of orthotropic steel plates ANALYSIS DESIGN METHODS APPLICATION

3 Contents About the bridge Global buckling of tied arch - Experimental buckling analysis - Evaluation of classical and advanced design methods - Application Orthotropic plate buckling - Overview of design methods - FE simulation based stability analysis and design - Application Concluding remarks

4 Dunaújváros Danube bridge

5 Location c c Budapest Dunaújváros

6 Geometry Total length of the bridge: Main span; tied arch bridge: 1780 m m arch height: 48 m steel box: 2 x 3.8 m

7 Current stage Webcam:

8 Preliminary phase: Tasks of the Department advisor for designer Design phase: research on design methods model test arch stability wind tunel test on section model analysis and design stability, fatigue, earthquake, aerodynamic Construction phase: erection method structural design for erection

9 Model test on arch stability

10 Purpose Experimental test design aims equivalent global arch stability 1) check the safety of the standard design methods 2) verify advanced numerical model and calibrate imperfection sizes

11 Bridge model M=1:34

12 Loading system 15 load cases

13 Total loading: Σq=220 kn Self-weight + 75 x 40 t trucks deflections [mm] deflections [mm] numerical test

14 partial half-sided loading: Σq=50 kn Lehajlások [mm] deflections [mm] test numerical

15 Failure test 1 total loading: 320 kn out-of-plane buckling local plate buckling out-of-plane buckling of the arch

16 Failure test 1

17 half-sided loading: 110 kn Failure test 2 in-plane buckling of the arch

18 Numerical model Model data Element type Ansys beam model BEAM44 LINK10 Ansys shell model SHELL181 LINK10 number of elements ~6 000 ~ number of nodes ~ ~ Analysis Linear Instability Geometrically nonlinear Virtual experiment material and geometrical linearity Block Lanczos buckling analysis geometrical nonlinearity, imperfect model material and geometrical nonlinearity, imperfect model

19 Verification material and geometrical nonlinearity imperfection: e 0 = 2 mm on the half side of the model non-symmetrical behaviour e 0 load 350 [kn] total loading half-sided loading measured calculated deflection [mm] load [kn] a b total loading measured calculated a b arch horizontal deformation [mm]

20 Hungarian Standard: HS Design methods (1) (2) N N e N N ke, z + ψ y 1 M M y e, y + ψ z M M z e, z 1 strength check + buckling check Japanese Standard: JSHB Eurocode: EC3 (3) (4) N N ke, z N N ke, y N N + ψ + k y yy M M M M y e, y y e, y + ψ + k z yz M M M M z e, z z e, z 1 1 y z (5) + k + k 1 zy zz ke, z M M e, y M M e, z linear interaction combined interaction

21 Experimental test Solution method ultimate load Numerical model Internal forces: N, M y, M z from analyses: 1. linear + second order modification factor 2. geometrically nonlinear equivalent imperfection 1 3. geometrically nonlinear equivalent imperfection 2 standard ultimate load experimental ultimate load

22 Equivalent imperfection size 1 Eurocode 3 Part 1.1: shape of the elastic critical buckling mode: η cr e 0, d α 2 χλ 1 M Rk γ 1 ( λ 0.2) 2 = M N Rk 1 χλ η second order analysis init = e 0, d 2 λ N EIη Rk " cr,max η cr in-plane buckling mode: out-of-plane buckling mode: In-plane Out-of-plane 3.44 mm mm

23 Equivalent imperfection size 2 Eurocode 3 Part 2: (Design of bridges) η 0, z = l 500 in-plane η 0, y = l 250 out-of-plane in-plane buckling mode: out-of-plane buckling mode: In-plane Out-of-plane mm mm

24 Comparison of classical design methods total load half-sided load HS JSHB EC experimental ultimate load / standard ultimate load

25 Comparison of Eurocode approaches total load half-sided load EC3 linear EC3 eqv. geom. imp EC3 eqv. geom. imp experimental ultimate load / standard ultimate load

26 Arch bridge erection

27 Finite element model Model data Ansys Ansys beam model shell model Element type BEAM44 LINK10 SHELL181 LINK10 number of elements number of nodes ~ ~ ~ ~ DOF ~ ~

28 Erection phases 1. Bridge is on the riverbank on a rack system 2. The cables are stressed to the self weight 3. Additional bars are built in the bridge 4. The bridge is palced on barges

29 Stress analysis 1. Load case: self-weight 2. Load case: ship reaction forces 3. Load case: 1. load case + 2. load case

30 Stiffening bar buckling Instability analysis Out-of-plane buckling In-plane buckling α cr = 3.97 α cr = α cr =

31 Orthotropic plate buckling

32 Orthotropic plates N + M V N (+M) V

33 Design methods - Hungarian Standard allowable stress design f y = 460 MPa σ e = 300 MPa factor ~1.47 dominantly compressed stiffened plates or plate parts (1) buckling of fictive column stub stiffened plate subject to complex stress field (2) orthotropic plate check irregular configuration and stress field -??? no rule given (3) generalized plate check

34 (1) Buckling of fictive column stub fictive column = stiffener + adjacent plating column slenderness (λ) and reduction factor (φ) a) flexural buckling b) torsional buckling λ = L φ 1 1 kt I b + b e Ab t f Lkt Lkt b s 16 bs 4 λ 2 = L kt I 0.04 L 2 2 = β β ( ) ( λ / ) 2 1+ λ2 / λ E 1 1 λ E φ = 1 5 u 2 kt + I I T 0.4 η + I η h 2 φ σ e (allowable stress) check: N A b b φ σ e

35 Actual calculations on the Danube bridge stiffeners t p b Case t p b a stiffener Nr. [mm] [m] [m] x x x T x x x t p plate thickness; b plate width; a plate length between transverse stiffeners or diaphragms

36 (2) Orthotropic plate subject to complex stress field plate-type behaviour plate slenderness (λ 0 ) and reduction factor (φ b ) 3.3 b λ 0 = t k red k red : buckling coefficient, e.g. Klöppel-Scheer- Möller (overall plate buckling of horizontally and longitudinally stiffened plates) φb reduction factor for plates check: σ red = σ 2 + 3τ 2 φ b σ e

37 Actual calculations on the Danube bridge B) SUBMODELS - FEM A) TYPICAL PLATES energy method

38 Actual calculations on the Danube bridge axial stresses [MPa] vertical stresses [MPa] α cr = shear stresses [MPa] buckling shape

39 (3) Irregular configuration and stress field -??? no rule given assume plate-type behaviour generalized plate slenderness (λ 0 ) and reduction factor (φ b ) σ = α σ α cr : critical load factor red, cr cr red,max λ = 0 π σ 2 E red,cr φb reduction factor for plates check: σ red = σ 2 + 3τ 2 φ b σ e

40 Actual calculations on the Danube bridge t = 0 f t a = 50 t g = IVCSP2

41 Design methods - Eurocode 3 Part 1-5 (1) basic procedure for stiffened plates in complex stress fields (no use of numerical models) (2) partial use of FEM: plate slenderness from bifurcation analysis (3) reduced stress method (4) finite element analysis based design (full numerical simulation)

42 (1) Basic procedure (no use of numerical models) consideration of both plate-type and column-like buckling a) plate-type: b) column-like: λ = p ρ β A, c σ σ cr, p f cr, p y λ = crit. stress for overall buckling σ cr, c = e.g. from orthotropic plate theory A sl, 1 a χ c check: c β A, c σ f cr, c y π 2 EI sl,1 σ cr, p c) interpolation: ρ c = ( ρ χ c ) ξ (2 ξ ) + χ c ξ = 1 (0 ξ 1) σ cross-section resistance with effective area N c, Rd = A c, eff γ f M 0 y cr, c N Ed N c, Rd 2

43 (1) Basic procedure (no use of numerical models) consideration of both plate-type and column-like buckling a) plate-type: b) column-like: λ = p ρ β A, c σ σ cr, p f cr, p y proposal for modification β A, c f y λ c = σ cr, c (Maquoi, Skaloud): 2 EI sl,1 crit. stress for overall buckling σ cr, c = e.g. from orthotropic ρ c = χ c but plate not theory smaller than ρ p, A sl, 1 a χ c σ cr, p c) interpolation: ρ c = ( ρ χ c ) ξ (2 ξ ) + χ c ξ = 1 (0 ξ 1) σ cross-section resistance with effective area N c, Rd = A c, eff γ f M 0 y check: cr, c π N Ed N c, Rd 2

44 (4) Finite element analysis based design geometrical and material non-linearity equivalent geometric imperfections non-linear simulation

45 geometrical and material model E t = E/10000 = 21 N/mm 2 Case t p b a stiffener Nr. [mm] [m] [m] x x x T x x x t p plate thickness; b plate width; a plate length between transverse stiffeners or diaphragms

46 equivalent geometric imperfections e w = min( a / 400, / 400) min( a / 400, / 400) 0 b e w = φ 1 / 50 0 b 0 = a) global imperfection of stiffener b) imperfection of subpanel c) local imperfection of stiffener ~ alternatively, relevant buckling shapes, i.e. a) overall buckling, b) local buckling of subpanels, c) torsion mode of the stiffener

47 equivalent geometric imperfections combination of the imperfections: leading (100%) + others (70%) PROBLEM when using buckling shapes as imperfections: overall/local plate buckling usually accompanied by the torsion of stiffener the requirements for the imperfection amplitudes are difficult to satisfy

48 Actual calculations on the Danube bridge 1 a = 4.56 m; b = 2 m; t p = 40 mm 2 x σ cr = 930,9 MPa 500 normal imperfection large imperfection e e α α 0w,1 0 x,2 0,1 0,2 = min ( a / 400, b / 400) = hbφ0 = 280 5,00 = 2, ,6 = 0, = 1932 = = 5,6mm = / 400 = 5mm Load [N/mm 2 ] FEM EC3-1-5 Hungarian Standard (allowable stress) 1,47 x Hungarian Standard Lateral deflection of panel center [mm]

49 3 a = m; b = 2 m; t p = 50 mm 2 x T σ cr = 3560 MPa 500 normal imperfection large imperfection e e α α 0w,1 0 x,2 0,1 0,2 = min ( a / 400, b / 400) = hbφ0 = 270 5,00 = 0, ,4 = 0, = 6614 = = 5,4mm = / 400 = 5mm Load [N/mm 2 ] FEM EC3-1-5 Hungarian Standard (allowable stress) 1,47 x Hungarian Standard Lateral deflection of panel center [mm]

50 4 a = 3.9 m; b = 3.8 m; t p = 20 mm 5 x σ cr = 745 MPa σ cr = 881 MPa 400 normal imperfection 350 e e e α α α 0w,1 0 x,2 0w,3 0,1 0,2 0,3 = min ( a / 400, b / 400) 1 = hbφ0 = 280 = 5,6mm 50 = 650 / 400 = 1,625mm 9,5 = 2, ,6 = 0, ,625 = 0, = 4170 = 9705 = 1999 = 3800 / 400 = 9,5mm Load [N/mm 2 ] large imperfection FEM EC ,47 x Hungarian Standard Hungarian Standard (allowable stress) Lateral deflection of panel center [mm]

51 Comparison Flanges Web plates σmax calc / σmax FEM EC3/1-1 ι = δ b t 4 γ f f 0.2 EC3/1-2 KH ι

52 Concluding remarks Tied arch bridge project Studies on global stability of tied arch - Model test - Evaluation of classical and advanced design methods in comparison to the test ultimate loads. Studies on the buckling of orthotropic plates - Design methods classical and advanced - Comparison of different design methods to FE simulation based results. Application

53 Thank you for your attention! 6th European Solid Mechanics Conference 28 August - 1 September, Budapest, Hungary, 2006