1C8 Advanced design of steel structures. prepared by Josef Machacek

Transcription

1 1C8 Advanced design of steel structures prepared b Josef Machacek

2 List of lessons 1) Lateral-torsional instabilit of beams. ) Buckling of plates. 3) Thin-alled steel members. ) 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, toers, chimnes. 11) Tanks and pipelines. 1) Technological structures. 13) Reserve.

3 1. Buckling of plates (plate stabilit and strength)

4 Stabilit of an ideal (flat) plate: various boundar conditions various loading Solution is based on linearized relation of a plate ith large deflections": * * * D x N N x N x x x x + relevant boundar conditions 0 Thereof infinitel man solutions: critical stresses * (or N*) take the loest respective shapes of deflection (modes of buckling)

5 Critical stresses are given as: cr k σ E critical stress factor Euler stress "Euler stress" b 1 P E P E E Critical stress factor: (depends on loading and boundar conditions, see literature) or cr k Auxiliar value, for a compression strut of idth "1": PE D E t E t 1 t b E 1 1 k = k = 3,9 b b a k = 5,3 for 1 a b t b 5

6 6 0 x x x x E t x x D 0 x x x x Equations of a plate ith large deflections (Karman s equations): (1) () Strength of an actual (imperfect) plate: Plate imperfections a b cr,1 cr,1 cr, = b/00 stabilit (buckling modes) initial deflections residual stresses due to elding eld idealized real Objectives

7 Example of a compression plate ith initial deflections and residual stresses: b t initial deflection max = f b eff / b eff / Resulting strengths are used in the form of reduction (buckling) factors : f b b eff b db 0 7

8 Eurocode : Plated structural elements 1. direct stress (loading N, M): Verification of class cross sections: a) effective idth, in hich the buckling parts of plates are excluded, b) reduced stress, in hich the stresses of full cross section are determined and limited b buckling reduction factors x, z, : a) aa A, I b A, I eff, effi eff b) A, I e M e M x x x x, z z, Note: b) does not include stress redistribution after buckling among individual parts of cross section!!! 8

9 Reduction (buckling) factors: Internal elements: p 0, 055 p 3 1, 0 p f cr b / t 8, k σ = / 1 For outstand compression elements similarl: p 0, 188 p 10, For k see next tables or Eurocode. 9

10 The effective p area of the compression zone of a plate: A c, eff Ac internal elements: = 1 / 1 b e1 b e b 1 > 0: b eff = b be1 beff 5 b e = b eff - b e1 b c b t b e1 b e b < 0: b eff = b c = b / (1-) b e1 = 0, b eff b e = 0,6 b eff Factors k 1 1 > >0 0 0> > > >-3 k,0 8,/(1,05+) 7,81 7,81-6,9+9,78 3,9 5,98(1-) 10

11 outstand elements: = 1 / b eff 1 1 > 0: b eff = c c b t b c 1 b eff k 0,3 0,57 0,85 0,57-0,1+0,078 < 0: b eff = b c = c /(1- ) b eff 1 c 1 > 0: b eff = c b eff 1 b c b t < 0: b eff = b c = c /(1- ) Factors k 1 1 > >0 0 0>>-1-1 k 0,3 0,578/(+0,3) 1,70 1,7-5+17,1 3,8 11

12 Effective cross sections (class cross sections): axial compression moment e M e M e N this eccentricit invokes additional moment from the axial force due to shift of neutral axis in interaction of M - N Effective parameters of class cross sections (A eff, W eff ) are determined b common a. Verification of cross section in ULS: N M0 1 0 Ed Ed Ed N 1, f Aeff f Weff M0 M N e (in stabilit checks: introduce, LT ) 1

13 Stiffened plates: Examples: - stiffened flange of a box girder, - eb of a deep girder. A b c,eff,loc 1,edge,eff b3,edge,eff b 1 b 3 1 b b3 b 1 b b 3 Ac, eff ca middle part c,eff,loc b edges edge, eff t global buckling reduction factor (approx. given b reduction factor of the effective stiffener - possible to calculate as a strut in compression) [For more details see course: Stabilit of plates] 13

14 Example of buckling of longitudinall and transversall stiffened flange of a box girder: 1

15 . (loading b shear force V): Rotating stress field theor is used. Influence of stiffeners is included proportionall to higher critical stress after modification agrees ith tests. Design resistance to shear (including shear buckling): V b,rd V b,rd V bf,rd f h 3 t = 1, up to steels S60 35 M1 f contribution from the flanges (can be ignored) contribution from the eb t f Verification of ULS: V Ed 3, Vb,Rd 1 0 t h t f b f 15

16 ma be ignored for eb slenderness: h unstiffened ebs 7 t (i.e. 60 for S35) stiffened ebs (transverse, longitudinal) h t 31 k Forming of tension diagonals in panels: V cr V cr Phase 1 Beam behaviour V t V t Phase Truss behaviour V f V f Phase 3 frame behaviour (influence of several %) 16

17 Contribution from the eb V b, Rd Factor for the contribution of the eb to the shear buckling resistance ma be (in acc. to tests) increased for rigid end post and internal panels: f 3 Slenderness Rigid end post Non-rigid end post 0,83 / 0, 83 / 108, 108, h M1 0, 83 / 0, 83 / 1, 37 / 0, 7 t 0, 83 / 1, 1 Rigid end post difference % Non-rigid end post Reason: anchorage of panels 1 17

18 Web slenderness unstiffened ebs (ith the exception at the beam ends): ebs ith transverse stiffeners in distance a: n a h f / cr h 37, t 3 h 86, t k Critical stress factor k : k k 5, 3, 00, 00 5, 3 h / a as far as a / h 1 h / a as far as a / h 1 [For ebs ith longitudinal stiffeners see course: Stabilit of plates] 18

19 3. 3 tpes of loading are distinguished: a) through the flange, b) through the flange and transferred directl to the other one, c) through the flange adjacent to an unstiffened end. Tpe (a) Tpe (b) Tpe (c) F s F s F s V 1,s V,s,s h a s s s s c s s V s Local design resistance: F L Rd eff t f M1 effective length of eb L eff = F l [In detail see Eurocode, or course: Stabilit of plates] reduction factor due to local buckling (governed b critical stress) effective loaded length (governed b s s ) 19

20 Example of local eb buckling: 0

21 Interaction N+M+F Interaction N + M + F Verification for local buckling: F Interaction N + M + F: F Ed Ed, F f Rd Lefft M1 0 1,, i.e.: L eff FEd NEd MEd NEd en 0, 8 1, f f Aeff f W eff t M1 M0 M0 1

22 Ideal and actual plate differences. Eurocode approaches concerning buckling effects. Verification of class sections. Design resistance to shear. Behaviour of ebs under shear. Tpes of. Verification for.

23 to users of the lecture This session requires about 90 minutes of lecturing. Within the lecturing, buckling of plates under direct, shear and is described. The lecture starts ith linear theor of buckling and resulting critical stress, folloed ith nonlinear theor of buckling of actual imperfect plate and its resistance. The buckling resistances under direct stress, shear and in accordance ith Eurocode are commented. Further readings on the relevant documents from ebsite of.access-steel.com and relevant standards of national standard institutions are strongl recommended. Keords for the lecture: buckling of plates, ideal plate, real plate, buckling due to direct stresses, buckling under shear, local buckling, interaction formulas for buckling.

24 for lecturers Subject: Buckling of plates. Lecture duration: 90 minutes. Keords: buckling of plates, ideal plate, real plate, buckling due to, 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. Eurocode approach. After the lecturing, determination of effective cross section parameters (class effective parameters) should be practised. Further reading: relevant documents.access-steel.com and relevant standards of national standard institutions are strongl recommended. Preparation for tutorial exercise: see examples prepared for the course.