Transcript Slide 1
1C8 Advanced design of steel structures prepared by Josef Machacek List of lessons 1) Lateral-torsional instability of beams. 2) 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) Masts, towers, chimneys. 11) Tanks and pipelines. 12) Technological structures. 13) Reserve. 2 Objectives Introduction 1. Buckling of plates Buckling due to direct stresses Effective width method Shear buckling Buckling under local loading Interaction N+M+F Assessment Notes • Introduction (plate stability and strength). • Buckling due to direct stresses. • Effective width method. • Shear buckling. • Buckling under local loading. 3 Objectives Introduction Introduction Buckling due to direct stresses Effective width method Stability of an ideal (flat) plate: Shear buckling Buckling under local loading Interaction N+M+F Assessment Notes various loading various boundary conditions Solution is based on linearized relation of a plate with „large deflections": 2 2 2 4w 4w 4w * w * w * w Nx D 2 2 2 2N xy Ny 0 4 x y x y y 4 x 2 y 2 x + relevant boundary conditions Thereof infinitely many solutions: • critical stresses * (or N*) – take the lowest • respective shapes of deflection w (modes of buckling) 4 Objectives Introduction Introduction Buckling due to direct stresses Effective width method Critical stresses are given as: cr k σ E Shear buckling Buckling under local loading critical stress factor Interaction N+M+F Assessment "Euler stress" Notes PE b 1 PE or cr k E Euler stress E Auxiliary value, for a compression strut of width "1": 2 PE 2D 2E t t E 189800 1 t 1 t b 2 12 1 2 b b Critical stress factor: k = 4 (depends on loading and boundary conditions, see literature) k = 23,9 2 b k = 5,34 4 a 2 for a 1 b 5 Objectives Introduction Introduction Buckling due to direct stresses Effective width method Strength of an actual (imperfect) plate: Shear buckling Equations of a plate with „large deflections“ (Karman’s equations): Buckling under local loading (1) Interaction N+M+F Assessment (2) 4w 2 2w 4w 4w 2 2w 2 2w 0 D 4 2 2 2 Et 2 2 4 2 2 2 x y y x y x 2 x y x y x y 4 4 4 2 2 w w 2w 0 2 2 2 x 4 x y y 4 x 2 y 2 xy Notes Plate imperfections stability (buckling modes) b initial deflections residual stresses due to welding cr,1 a idealized real w0 = b/200 cr,1 w0 cr,1 w0 weld 6 Objectives Introduction Introduction Buckling due to direct stresses Effective width method Example of a compression plate with initial deflections and residual stresses: Shear buckling t Buckling under local loading Interaction N+M+F beff/2 b beff/2 Assessment initial deflection Notes max = fy Resulting strengths are used in the form of reduction (buckling) factors : fy beff b b db 0 7 Objectives Introduction Buckling due to direct stresses Buckling due to direct stresses Effective width method Eurocode 1993-1-5: Plated structural elements Shear buckling 1. Buckling due to direct stress (loading N, M): Buckling under local loading Verification of class 4 cross sections: Interaction N+M+F Assessment Notes a) effective width method, in which the buckling parts of plates are excluded, b) reduced stress method, in which the stresses of full cross section are determined and limited by buckling reduction factors x, z, w: Ieff a) aAeffA,effI,eff b)b A,A,I I x x eMe x x, zz, w M Note: b) does not include stress redistribution after buckling among individual parts of cross section!!! 8 Objectives Introduction Buckling due to direct stresses Buckling due to direct stresses Effective width method Reduction (buckling) factors: Shear buckling Internal elements: Buckling under local loading Interaction N+M+F p 0,055 3 Assessment 2 p 1,0 p fy cr b/t 28 ,4 k σ = 2/1 Notes For outstand compression elements similarly: p 0,188 2 p 1,0 For k see next tables or Eurocode. 9 Objectives Effective width method Introduction Buckling due to direct stresses Effective width method Effective width method The effectivep area of the compression zone of a plate: Shear buckling Interaction N+M+F internal elements: = 1/2 • Buckling under local loading Ac,eff Ac 1 2 Assessment be1 1 > ≥ 0: beff = b be2 bc 2 beff 5 be2 = beff - be1 b Notes be1 bt < 0: be1 be2 beff = bc = b / (1-) be1 = 0,4 beff be2 = 0,6 beff b Factors k 1 1>>0 0 0 > > -1 -1 -1 > > -3 k 4,0 8,2/(1,05+) 7,81 7,81-6,29+9,782 23,9 5,98(1-)2 10 Objectives Introduction Effective width method Buckling due to direct stresses Effective width method • outstand elements: beff Shear buckling Buckling under local loading 1 > ≥ 0: 1 2 = 1/2 beff = c c bt bc 1 2 < 0: beff = bc = c /(1- ) beff Interaction N+M+F Assessment Notes 1 0 -1 1 ≥ ≥ -3 k 0,43 0,57 0,85 0,57-0,21+0,0782 beff 1 > ≥ 0: 1 2 beff = c beff < 0: 1 c bc bt 2 beff = bc = c /(1- ) Factors k 1 1>>0 0 0>>-1 -1 k 0,43 0,578/(+0,34) 1,70 1,7-5+17,12 23,8 11 Objectives Introduction Effective width method Buckling due to direct stresses Effective width method Shear buckling Effective cross sections (class 4 cross sections): axial compression moment Buckling under local loading eM eM Interaction N+M+F Assessment eN Notes this eccentricity invokes additional moment from the axial force due to shift of neutral axis in interaction of M - N Effective parameters of class 4 cross sections (Aeff, Weff) are determined by common way. Verification of cross section in ULS: 1 NEd M NEd eN Ed 1,0 f y Aeff f y Weff M0 (in stability checks: introduce , LT) M0 12 Objectives Introduction Effective width method Buckling due to direct stresses Effective width method Stiffened plates: Ac,eff,loc b1,edge,eff b3,edge,eff Shear buckling Buckling under local loading Interaction N+M+F Examples: - stiffened flange of a box girder, - web of a deep girder. Assessment 2 b 2 2 b 3 b1 b2 2 3 2 2 2 b1 Notes middle part 1 b2 b3 edges Ac,eff c Ac,eff,loc bedge,eff t global buckling reduction factor (approx. given by reduction factor of the effective stiffener - possible to calculate as a strut in compression) [For more details see course: Stability of plates] 13 Objectives Introduction Effective width method Buckling due to direct stresses Effective width method Example of buckling of longitudinally and transversally stiffened flange of a box girder: Shear buckling Buckling under local loading Interaction N+M+F Assessment Notes 14 Objectives Introduction Shear buckling Buckling due to direct stresses Effective width method Shear buckling Buckling under local loading Interaction N+M+F 2. Shear buckling (loading by shear force V): Rotating stress field theory is used. Influence of stiffeners is included proportionally to higher critical stress – after modification agrees with tests. Design resistance to shear (including shear buckling): Assessment Notes f y hw t = 1,2 up to steels S460 235 3 M1 fy contribution from the flanges (can be ignored) contribution from the web Vb,Rd Vbw,Rd Vbf,Rd tf Verification of ULS: VEd 3 1,0 Vb,Rd t hw tf bf 15 Objectives Introduction Shear buckling Buckling due to direct stresses Effective width method Shear buckling Buckling under local loading Shear buckling may be ignored for web slenderness: hw 72 (i.e. 60 for S235) unstiffened webs t stiffened webs (transverse, longitudinal) hw 31 k t Interaction N+M+F Assessment Notes Forming of tension diagonals in panels: Vcr Vt Vf Vcr Phase 1 Beam behaviour Phase 2 Vt Truss behaviour Vf Phase 3 frame behaviour (influence of several %) 16 Objectives Introduction Shear buckling Buckling due to direct stresses Effective width method Contribution from the web Vbw,Rd w fyw hw t 3 M1 Shear buckling Buckling under local loading Factor w for the contribution of the web to the shear buckling resistance may be (in acc. to tests) increased for rigid end post and internal panels: Interaction N+M+F Slenderness Rigid end post Non-rigid end post w 0,83 / Assessment Notes 0,83 / w 0 ,83 / w 1,08 1,37 / 0,7 w w 1,08 w 1,2 1 Rigid end post difference 22% 0,83 / w 0,83 / w Reason: anchorage of panels → Non-rigid end post 1 2 w 17 Objectives Introduction Shear buckling Buckling due to direct stresses Effective width method Shear buckling Web slenderness w • unstiffened webs (with the exception at the beam ends): Buckling under local loading w Interaction N+M+F Assessment fy / 3 cr hw 86,4 t • webs with transverse stiffeners in distance a: Notes hw w hw 37,4 t na k Critical stress factor k: k 5 ,34 4 ,00 hw / a 2 k 4 ,00 5 ,34 hw / a 2 as far as a / hw 1 as far as a / hw 1 [For webs with longitudinal stiffeners see course: Stability of plates] 18 Objectives Introduction Buckling under local loading Buckling due to direct stresses Effective width method 3. Buckling under local loading 3 types of loading are distinguished: Shear buckling a) through the flange , Buckling under local loading b) through the flange and transferred directly to the other one, c) through the flange adjacent to an unstiffened end. Interaction N+M+F Type (a) Fs Assessment Notes ss V1,s Type (b) Fs V V2,s 2,s hw Type (c) Fs ss c ss Vs a Local design resistance: FRd Leff t w fy M1 reduction factor due to local buckling (governed by critical stress) effective length of web Leff = Fℓy effective loaded length (governed by ss) [In detail see Eurocode, or course: Stability of plates] 19 Objectives Introduction Buckling under local loading Buckling due to direct stresses Effective width method Example of local web buckling: Shear buckling Buckling under local loading Interaction N+M+F Assessment Notes 20 Objectives Introduction Interaction N + M + F Buckling due to direct stresses Effective width method Shear buckling Buckling under local loading Verification for local buckling: FEd 2 FRd FEd Leff t w Interaction N+M+F Assessment Notes fy 1,0 M1 Interaction N + M + F: 2 0,8 1 1,4 i.e.: FEd Leff t w fy M1 M NEd eN N 0,8 Ed Ed 1,4 fy Aeff fy Weff M0 M0 21 Objectives Introduction Buckling due to direct stresses Effective width method Assessment • Ideal and actual plate – differences. Shear buckling Buckling under local loading Interaction N+M+F Assessment Notes • Eurocode approaches concerning buckling effects. • Verification of class 4 sections. • Design resistance to shear. • Behaviour of webs under shear. • Types of local loading. • Verification for local loading. 22 Objectives Introduction Notes to users of the lecture Buckling due to direct stresses Effective width method • This session requires about 90 minutes of lecturing. Shear buckling • Within the lecturing, buckling of plates under direct, shear and local loading is described. The lecture starts with linear theory of buckling and resulting critical stress, followed with nonlinear theory of buckling of actual imperfect plate and its resistance. The buckling resistances under direct stress, shear and local loading in accordance with Eurocode are commented. • Further readings on the relevant documents from website of www.access-steel.com and relevant standards of national standard institutions are strongly recommended. • Keywords for the lecture: Buckling under local loading Interaction N+M+F Assessment Notes buckling of plates, ideal plate, real plate, buckling due to direct stresses, buckling under shear, local buckling, interaction formulas for buckling. Objectives Introduction Notes for lecturers Buckling due to direct stresses Effective width method • Subject: Buckling of plates. Shear buckling • Lecture duration: 90 minutes. • Keywords: buckling of plates, ideal plate, real plate, buckling due to direct stresses, buckling under shear, local buckling, interaction formulas for buckling. • Aspects to be discussed: Ideal plate, critical stress, real plate, reduction factor. Behaviour of plates under shear loading. Behaviour of plates under local loading. Eurocode approach. • After the lecturing, determination of effective cross section parameters (class 4 effective parameters) should be practised. • Further reading: relevant documents www.access-steel.com and relevant standards of national standard institutions are strongly recommended. • Preparation for tutorial exercise: see examples prepared for the course. Buckling under local loading Interaction N+M+F Assessment Notes 24