4. Torsion Open and closed cross sections, simple St. Venant and warping torsion, interaction of bending and torsion, Eurocode approach.

Transcription

1 4. orsion Open and closed cross secions, simple S. enan and arping orsion, ineracion of bending and orsion, Eurocode approach. Common is elasic soluion (nonlinear plasic analysis e.g. Srelbickaja) Eurocode 3 enables combinaion of plasic bending momen and elasic orsion. Should be disinguished: - simple orsion: only shear sresses arise, - arping orsion: boh shear and direc (normal) sresses arise. 1. Open cross secions (e.g. I, U, L) a) Simple (Sain enan s) orsion (occurs only excepionally, see laer) τ b τ i i = 1 i = i = 3 Only shear arises: τ I (i) = I (i) 1 = α bi 3 i f y 3 i 3 (maximal in max ) (influence of rounding of rolled secions, OK3 oherise = 1) 1

2 b) Warping orsion (according o lasov s heory) ssumpions: 1. Rigid cross secion,. Null shear deformaion (shear lag ignored). one par of a orsion momen is ransmied by simple orsion, oher par by bending orsion : = + in arping orsion everyhing is relaed o cenral line S shear cenre (bending cenre) τ σ inernal forces: B momen of simple orsion momen of bending orsion bimomen τ = bending orsion OK3

3 Resuling sresses: Shear sresses: simple orsion τ firs secorial momen bending orsion τ S = I arping consan e.g.: τ,max τ sress hrough hickness Direc (normal) sress: σ = B = I B W secorial secion modulus pplies bending analogy : B M or σ σ or τ τ OK3 3

4 Secorial characerisics Rolled secions see ables. In general from secorial coordinae: Posiion of S: y d = z d = 0 (produc secorial momens) I cross secion: h z b r G S U cross secion: in his posiion no orsion!! S S he main secorial coordinae: hb = r ds = 4 s Firs secorial momen: h b S = d = 16 Second secorial momen (arping consan): h I = d = Iz 4 S G, S, I... see ables a = a OK3 4

5 Deerminaion of inernal forces due o orsion: soluion of lasov s differenial equaions, or direcly from formulas, based on bending analogy. Disribuion of orsion momen: e F simple suppor in orsion (couple of forces) bending M orsion Disribuion of orsional momen due o eccenrical force corresponds o disribuion of ransverse force a eccenriciy. Par ransmied by simple orsion is se aside: = e e κ ( ) e 1 κ κ... see able of Eurocode Czech N B Me B M e ( 1 κ ) Superposiion for more complex loading is necessary: e 1 e -e3 OK3 5

6 Simplified (conservaive) soluion neglecs simple orsion: 0 /h /h h bending of flanges only = bimomen (ofen adequae: i is conservaive from σ poin of vie) Imporan noes: 1. Large direc sresses, hey can no be ignored!!. Direc sresses (arping orsion) do no arise: a) for loading by sresses τ, roughly also due o end loading (simple orsion arises only): b) in secions composed of radiaing ousands (because of = I = 0): (shear cenre S is in cross poin) 3. In pracice usually occurs orsion abou enforced axis (): S S e analysis abou original shear cenre S is uneconomical!!! moreover, orsion is usually resriced by cladding rigidiy orsion may ofen be ignored. OK3 6

7 . Closed cross secions (e.g. ) a) Simple orsion (shear sresses only, usual design) Bred s shear flo (τ ) = cons. τ i d i τ = (i) Ω (i) = s s Conrary o open cross secion he maximal shear τ is in he hinnes plae and along hickness consan!! b) Warping orsion: - Umanskij s heory (rigid cross secion), - lasov s heory ih non rigid cross secion, - FEM (including influence of bevelled cross secion, gives also ransversal bending momens in plaes). he sresses are he same as in open cross secions: τ, τ, σ. Hoever, τ, σ are very small, commonly ignored even for bridges. OK3 7

8 3. Ineracion of bending and orsion ( M y + ) e In general, bending and orsion sresses may be summed and von Mises crierion applied: Direc sresses (open cross secions only): σ = σ + σ f i.e. M χ My L Shear sresses: Ed pl,,rd y,ed W y 1 B + W Ed For open secions I and U For closed secions y f y pl,,rd M1 M1 is design plasic shear resisance of he cross secion. pl,,rd σ x,ed fy σ z,ed + fy τ = 1 1, 5 y τ,ed 1 fy / 3 M,Ed,Ed pl, Rd ( f / 3 )/ γ ( f / 3 )/ γ pl, Rd ( ) pl,,rd = 0 σ x,ed fy σ z,ed fy OK3 8 y τ + 3 τ Ed fy 1 in U secions only