1C8 Advanced design of steel structures

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

2. List of lessons • Lateral-torsional instability of beams. • Buckling of plates. • Thin-walled steel members. • Torsion of members. • Fatigue of steel structures. • Composite steel and concrete structures. • Tall buildings. • Industrial halls. • Large-span structures. • Masts, towers, chimneys. • Tanks and pipelines. • Technological structures. • Reserve.

3. 1. Buckling of plates Introduction (plate stability and strength). Buckling due to direct stresses. Effective width method. Shear buckling. Buckling under local loading. Objectives Introduction Buckling due to direct stresses Effective width method Shear buckling Buckling under local loading Interaction N+M+F Assessment Notes 3

4. Introduction Objectives Introduction Buckling due to direct stresses Effective width method Shear buckling Buckling under local loading Interaction N+M+F Assessment Notes various loading various boundary conditions Stability of an ideal (flat) plate: Solution is based on linearized relation of a plate with „large deflections": + relevant boundary conditions Thereof infinitely many solutions: • critical stresses * (or N*) – take the lowest • respective shapes of deflection w(modes of buckling) 4

5. Introduction Objectives Introduction Buckling due to direct stresses Effective width method Shear buckling Buckling under local loading Interaction N+M+F Assessment Notes k = 4 k = 23,9 k = PE b PE 1 or critical stress factor Euler stress Critical stresses are given as: "Euler stress" Auxiliary value, for a compression strut of width "1": Critical stress factor: (depends on loading and boundaryconditions, see literature) 5

6. Objectives Introduction Buckling due to direct stresses Effective width method Shear buckling Buckling under local loading Interaction N+M+F Assessment Notes Introduction Plate imperfections stability (buckling modes) initial deflections residual stresses due to welding cr,1 b Equations of a plate with „large deflections“ (Karman’s equations): a idealized w0 = b/200 real cr,1 (1) w0 weld (2) cr,1 w0 Strength of an actual (imperfect) plate: 6

7. Introduction Objectives Introduction Buckling due to direct stresses Effective width method Shear buckling Buckling under local loading Interaction N+M+F Assessment Notes t beff/2 b beff/2 initial deflection max = fy Example of a compression plate with initial deflections and residual stresses: Resulting strengths are used inthe form of reduction (buckling) factors  : 7

8. Buckling due to direct stresses Objectives Introduction Buckling due to direct stresses Effective width method Shear buckling Buckling under local loading Interaction N+M+F Assessment Notes a) Aeff, Ieff b) A, I x x, z, w eM Eurocode 1993-1-5: Plated structural elements 1. Buckling due to direct stress (loading N, M): Verification of class 4 cross sections: • 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: Note: b) does not include stressredistribution after bucklingamong individual parts ofcross section!!! 8

9. Objectives Introduction Buckling due to direct stresses Effective width method Shear buckling Buckling under local loading Interaction N+M+F Assessment Notes Buckling due to direct stresses Reduction (buckling) factors: Internal elements:  = 2/1 For outstand compression elements similarly: For k see next tables or Eurocode. 9

10. Objectives Introduction Buckling due to direct stresses Effective width method Shear buckling Buckling under local loading Interaction N+M+F Assessment Notes 1 2 • internal elements:  = 1/2 bc bt be1 be2 1 > ≥ 0: beff = b be1 be2 be2 = beff - be1  < 0: beff =  bc = b / (1-) be1 = 0,4 beff be2 = 0,6 beff Factors k Effective width method Effective width method The effectivep area of the compression zone of a plate: 10

11. Objectives Introduction Buckling due to direct stresses Effective width method Shear buckling Buckling under local loading Interaction N+M+F Assessment Notes bt bc beff 1 1 2 2 c beff beff beff 1 1 2 2 bc bt c Effective width method • outstand elements:  = 1/2 1 > ≥ 0: beff = c < 0: beff = bc =  c/(1- ) 1 > ≥ 0: beff = c < 0: beff = bc =  c/(1- ) Factors k 11

12. Objectives Introduction Buckling due to direct stresses Effective width method Shear buckling Buckling under local loading Interaction N+M+F Assessment Notes axial compression moment eM eM eN this eccentricity invokes additional moment from the axial force due to shift of neutral axis in interaction of M - N Effective width method Effective cross sections (class 4 cross sections): Effective parameters of class 4 cross sections (Aeff, Weff) are determined by common way. Verification of cross section in ULS: (in stability checks: introduce , LT) 12

13. Objectives Introduction Buckling due to direct stresses Effective width method Shear buckling Buckling under local loading Interaction N+M+F Assessment Notes Ac,eff,loc b1,edge,eff b3,edge,eff b1 b2 b3 middle part edges global buckling reduction factor (approx. given by reduction factor of the effective stiffener - possible to calculate asa strut in compression) Effective width method Stiffened plates: Examples: - stiffened flange of a box girder, - web of a deep girder. [For more details see course: Stability of plates] 13

14. Objectives Introduction Buckling due to direct stresses Effective width method Shear buckling Buckling under local loading Interaction N+M+F Assessment Notes Effective width method Example of buckling of longitudinally and transversally stiffened flange of a box girder: 14

15. Shear buckling Objectives Introduction Buckling due to direct stresses Effective width method Shear buckling Buckling under local loading Interaction N+M+F Assessment Notes tf t hw tf bf 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):  = 1,2 up to steels S460 contribution from the flanges (can be ignored) contribution from the web Verification of ULS: 15

16. Objectives Introduction Buckling due to direct stresses Effective width method Shear buckling Buckling under local loading Interaction N+M+F Assessment Notes Phase 1 Beam behaviour Vcr Forming of tension diagonals in panels: Vcr Phase 2 Truss behaviour Vt Vt Shear buckling may be ignored for web slenderness: unstiffened webs stiffened webs (transverse, longitudinal) Phase 3 frame behaviour (influence of several %) (i.e. 60 for S235) Vf Vf Shear buckling 16

17. Objectives Introduction Buckling due to direct stresses Effective width method Shear buckling Buckling under local loading Interaction N+M+F Assessment Notes 1,2 Rigid end post 1 difference 22% Non-rigid end post 1 2 Shear buckling Contribution from the web Factor w for the contribution of the webto the shear buckling resistance may be (in acc. to tests) increased for rigid end post and internal panels:   w Reason: anchorage of panels → 17

18. Objectives Introduction Buckling due to direct stresses Effective width method Shear buckling Buckling under local loading Interaction N+M+F Assessment Notes Shear buckling Web slenderness • unstiffened webs (with the exception at the beam ends): • webs with transverse stiffeners in distance a: hw n a Critical stress factor k: [For webs with longitudinal stiffenerssee course: Stability of plates] 18

19. Objectives Introduction Buckling due to direct stresses Effective width method Shear buckling Buckling under local loading Interaction N+M+F Assessment Notes Buckling under local loading Type (a) Type (b) Type (c) Fs Fs Fs ss V1,s V2,s hw ss c ss Vs V2,s a Local design resistance: reduction factor due to local buckling (governed by critical stress) effective length of web Leff = Fℓy effective loaded length (governed by ss) 3. Buckling under local loading 3 types of loading are distinguished: a) through the flange , b) through the flange and transferred directly to the other one, c) through the flange adjacent to an unstiffened end. [In detail see Eurocode, or course: Stability of plates] 19

20. Objectives Introduction Buckling due to direct stresses Effective width method Shear buckling Buckling under local loading Interaction N+M+F Assessment Notes Buckling under local loading Example of local web buckling: 20

21. Interaction N + M + F Objectives Introduction Buckling due to direct stresses Effective width method Shear buckling Buckling under local loading Interaction N+M+F Assessment Notes Verification for local buckling: Interaction N + M + F: i.e.: 21

22. Objectives Introduction Buckling due to direct stresses Effective width method Shear buckling Buckling under local loading Interaction N+M+F Assessment Notes Assessment • Ideal and actual plate – differences. • 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.

23. Objectives Introduction Buckling due to direct stresses Effective width method Shear buckling Buckling under local loading Interaction N+M+F Assessment Notes Notes to users of the lecture • This session requires about 90 minutes of lecturing. • 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 non-linear 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 of plates, ideal plate, real plate, buckling due to direct stresses, buckling under shear, local buckling, interaction formulas for buckling.

24. Objectives Introduction Buckling due to direct stresses Effective width method Shear buckling Buckling under local loading Interaction N+M+F Assessment Notes Notes for lecturers • Subject: Buckling of plates. • 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.