Stability of an ideal (flat) plate. = k. critical stresses σ* (or N*) take the. Thereof infinitely many solutions: Critical stresses are given as:

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1 . Buckling of plaes Linear and nonlinear heor of uckling, uckling under direc sresses (class secions), uckling under shear, local loading and Eurocode approach. Saili of an ideal (fla) plae various loading Soluion is ased on linearized relaion of a plae ih large deflecions": various oundar condiions * * * D N N + N + relevan oundar condiions = 0 Thereof infiniel man soluions: criical sresses σ* (or N*) ake he loes respecive shapes of deflecion (modes of uckling) Criical sresses are given as: σ cr = k σ σ E or τ cr = k τ σ E criical sress facor Euler sress OK3 1 Prof. Ing. Josef Macháček, DrSc.

2 Prof. Ing. Josef Macháček, DrSc. OK3 "Euler sress" E σ P E P E 1 ( ) E E = = = = E D P ν π π σ Criical sress facor: (depends on loading and oundar condiions, see lieraure) Auiliar value, for a compression sru of idh "1": k σ = k σ = 3,9 k τ = 1 for 5,3 + a a Srengh of an acual (imperfec) plae 0 = E D Φ Φ Φ 0 = Φ Φ Φ Equaions of a plae ih large deflecions (Karman s equaions): (1) ()

3 Plae imperfecions saili (uckling modes) iniial deflecions residual sresses due o elding a σ cr,1 0 = /00 σ cr,1 0 τ cr,1 0 Eample of a compression plae ih iniial deflecions and residual sresses: iniial deflecion σ ma = f eff / eff / Resuling srenghs are used in he form of reducion (uckling) facors ρ : ρ = σ = f eff σ = σ d 0 OK3 3 Prof. Ing. Josef Macháček, DrSc.

4 Eurocode : Plaed srucural elemens 1. Buckling due o direc sress (loading N, M): λ p ρ = 0, 055 λ p ( 3 + ψ ) 1, 0 f / λ p = = σ 8, ε k cr σ ψ = σ /σ 1 For ousand compression elemens similarl: Verificaion of class cross secions: λ p ρ = 0, , λ p (for k σ see Eurocode) a) effecive idh mehod, in hich he uckling pars of plaes are ecluded, ) reduced sress mehod, in hich he sresses of full cross secion are deermined and limied uckling reducion facors ρ, ρ z, χ : a) aa A, I A, I eff, effi eff ) A, I e M e M ρ ρ, ρ z z, χ ρ, ρ, χ Noe: ) does no include sress redisriuion afer uckling among individual pars of cross secion!!! OK3 Prof. Ing. Josef Macháček, DrSc.

5 Effecive idh mehod The effecive p area of he compression zone of a plae: A c, eff = ρ Ac inernal elemens: ψ = σ 1 /σ σ 1 σ e1 e 1 > ψ 0: eff = ρ e1 = eff e = eff - e1 5 ψ c ψ < 0: eff = ρ c = ρ / (1-ψ) e1 = 0, eff e1 e e = 0,6 eff Facors k σ ψ 1 1 > ψ >0 0 0>ψ > > ψ >-3 k σ,0 8,/(1,05+ψ) 7,81 7,81-6,9ψ+9,78ψ 3,9 5,98(1-ψ) OK3 5 Prof. Ing. Josef Macháček, DrSc.

6 ousand elemens: ψ = σ 1 /σ eff σ 1 1 > ψ 0: c ψ < 0: σ c eff = ρ c σ eff σ 1 eff = ρ c = ρ c /(1- ψ) ψ ψ -3 k σ 0,3 0,57 0,85 0,57-0,1ψ+0,078ψ eff σ 1 σ 1 > ψ 0: eff = ρ c eff σ 1 σ ψ < 0: eff = ρ c = ρ c /(1- ψ) c c Facors k σ ψ 1 1 > ψ >0 0 0>ψ>-1-1 k σ 0,3 0,578/(ψ+0,3) 1,70 1,7-5ψ+17,1ψ 3,8 OK3 6 Prof. Ing. Josef Macháček, DrSc.

7 Effecive cross secions (class cross secions) aial compression momen e M e M e N Effecive parameers of class cross secions (A eff, W eff ) are deermined common a. Verificaion of cross secion in ULS: his eccenrici invokes addiional momen from he aial force due o shif of neural ais in ineracion of M - N η N Ed Ed Ed N 1 = +, f Aeff f Weff γ M0 M + N γ M0 e 1 0 (in saili checks: o inroduce χ, χ LT ) OK3 7 Prof. Ing. Josef Macháček, DrSc.

8 Siffened plaes: 1,edge,eff A c,eff,loc 3,edge,eff Eamples: - siffened flange of a o girder, - e of a deep girder. ρ ρ ρ 1 ρ middle par edges A = A + c, eff ρ c c,eff,loc edge, eff gloal uckling reducion facor (appro. given reducion facor of he effecive siffener - possile o calculae as a sru in compression) [For more deails see sujec: Saili of plaes] OK3 8 Prof. Ing. Josef Macháček, DrSc.

9 Eample of uckling of longiudinall and ransversall siffened flange of a o girder: OK3 9 Prof. Ing. Josef Macháček, DrSc.

10 . Shear uckling (loading shear force V): Roaing sress field heor is used. Influence of siffeners is included proporionall o higher criical sress afer modificaion agrees ih ess. Design resisance o shear (including shear uckling): V,Rd = V,Rd + V Verificaion of ULS: f,rd η f h 3 γ M1 conriuion from he flanges (can e ignored) conriuion from he e η Shear uckling ma e ignored for e slenderness: h unsiffened es 7 ε (i.e. 60 for S35) η V Ed 3 =, V,Rd 1 0 η = 1, up o seels S60 35 ε = f f f h f siffened es (ransverse, longiudinal) h 31 ε η k τ OK3 10 Prof. Ing. Josef Macháček, DrSc.

11 Forming of ension diagonals in panels: Phase 1 Beam ehaviour Phase Truss ehaviour Phase 3 frame ehaviour (influence of several %) OK3 11 Prof. Ing. Josef Macháček, DrSc.

12 Conriuion from he e χ f h V,Rd = 3 γ M1 Facor χ for he conriuion of he e o he shear uckling resisance ma e (in acc. o ess) increased for rigid end pos and inernal panels: Slenderness Rigid end pos Non-rigid end pos λ < 0,83 / η 0, 83 / η λ < 108, λ 108, η η 0, 83 / λ 0, 83 / λ ( 0, 7 + ) 1, 37 / λ 0, 83 / λ χ 1, 1 Rigid end pos difference % Reason: anchorage of panels 1 λ Non-rigid end pos OK3 1 Prof. Ing. Josef Macháček, DrSc.

13 We slenderness λ unsiffened es (ih he ecepion a he eam ends): λ f / 3 = = τ cr h 86, ε es ih ransverse siffeners in disance a: n a h λ = h 37, ε k τ Criical sress facor k τ : k k τ τ = 5, 3 +, 00 =, , 3 ( h / a) as far as a / h 1 ( h / a) as far as a / h < 1 [For es ih longiudinal siffeners see course: Saili of plaes] OK3 13 Prof. Ing. Josef Macháček, DrSc.

14 3. Buckling under local loading 3 pes of loading are disinguished: a) hrough he flange, ) hrough he flange and ransferred direcl o he oher one, c) hrough he flange adjacen o an unsiffened end. Tpe (a) Tpe () Tpe (c) F s F s F s s s V 1,s V,s,s h s s c s s V s a Local design resisance: f F Rd = Leff γ M1 reducion facor due o local uckling (governed criical sress) effecive lengh of e L eff = χ F l effecive loaded lengh (governed s s ) [In deail see Eurocode, or course: Saili of plaes] OK3 1 Prof. Ing. Josef Macháček, DrSc.

15 Eample of local e uckling: OK3 15 Prof. Ing. Josef Macháček, DrSc.

16 Verificaion for local uckling: η F F Ed Ed = =, F f Rd Leff γ M1 Ineracion N + M + F: η +, 8 η 1 0 1, 10 i.e.: L eff FEd NEd MEd + NEd en + 0, 8 1, f + f Aeff f W eff γ M1 γ M0 γ M0 OK3 16 Prof. Ing. Josef Macháček, DrSc.