Curse Stabilty f Structures Lecture ntes 2015.03.06 abut 3D beams, sme preliminaries (1:st rder thery) Trsin, 1:st rder thery 3D beams 2:nd rder thery Trsinal buckling Cupled buckling mdes, eamples Numerical calculatin methd /Per J. G. Pelarknäckning, i-planet knäckning, böjknäckning. In-plane clumn buckling. Vridknäckning. Trsinal buckling. Böjvridknäckning. Bending-trsinal buckling? Vippning, kantring Lateral-trsinal buckling. Tilting? 1
3D beams, sme preliminaries Crdinate system (right hand side rientatin): y z Stresses in beam (as cnsidered in gverning equatins): σ σ σ τ 0 y z yz 0, τy 0 and τz 0 (in general) Crss sectin reference pints and directins: y Shear centre Rtatin centre (y, z ) Principal directins z Centre f gravity 2
1:st rder gverning equatins fr 3D Bernulli/Euler/StVenant/Vlasv beam y, z, w m, u y, Principal directins z, Shear Rtatin centre centre (y, z ) ϕ, u Centre f gravity (EAu ) -q N (EIv) qy mz M (EIw ) qz my M (EIω ) (GKv) m z y bar-actin (1) bending in -y plane (2) bending in -z plane (3) trsin (4) The equatins are linear and uncupled. v=v(,y ) and w=(,y ). Thus v(,y)=v-(z-z )ϕ and w(,y)=w+(y-y )ϕ. 3
Trsin y, v, u Rtatin center (y, z ) z, w T L Kinematics: 1) In the y-z plane, the beam crss-sectin mves as a rigid bdy rtating arund the rtatin center ϕ T L 2) Displacement u is nt equal in different parts f a crss sectin: warping 4
Tw trsin defrmatin mdes: The magnitude f warping may be cnstant r vary alng the beam. These tw cases crrespnds t different stress and defrmatin perfrmance. a) ( St Venant thery ) The warping is cnstant alng the beam. This is the case if the trque is cnstant and there is nthing preventing warping at the ends f the beam. T T L In this case σ =ε =0. Twisting and warping is due t γ y 0 and γ z 0. (L) (0) L d d T GK v Fr distributed lad, equilibrium fr a part d gives dt+m d=0, i.e. T =-m : (GK v ) m (5) T be strict, m 0 shuld nt be allwed since it gives varying T and warping, and therefr cntradicts the thery. The case f cnstant warping is the St Venants trsin thery. Crss sectin gemetry parameter K v and shear stress distributin are in this thery determined by an Pissn s equatin, derived frm definitin f shear strain, τ=gγ and σ =0. 5
b) ( Vlasv thery ) The warping is varying alng the beam. This is the case if the trque is varying r warping is prevented at an end f the beam. T L In pure Vlasv trsin is γ y =0 and γ z =0, and twisting is due t ε 0. Preventin f warping is f great imprtance the trsinal stiffness f beams with certain thin-walled crss-sectins, eg: Fr mst ther crss-sectins is the influence (n trsinal stiffness) f prevented warping disregarded. The pure Vlasv trsin gives: (EI ) m (6) 6
An illustratin Prevented warping ( Vlasv ). Twist f beam due t nrmal stress σ and strain ε giving bending defrmatin f upper and lwer flanges. Free warping ( St Venant ). Twist f beam due t shear stress and strain giving spiral shape f the web and each f the tw flanges. Superpsitin f the tw stress-systems gives the gverning equatin fr mied trsin acc. t 1:st rder thery: (EI ) -(GK ) v m (7) Bundary cnditins if I ω =0 ϕ=0 (r ther knwn value) T knwn, giving ϕ =T/(GK v ) Bundary cnditins if I ω 0 ϕ=0 (r ther knwn value) ϕ =0 (prevented warping, clamped) ϕ =0 (free warping (r knwn nrmal stress giving sme ϕ 0 ) ) T knwn, giving -(EI ω ϕ ) +GK v ϕ =T 7
2:nd rder gverning equatins fr 3D Bernulli/Euler/- StVenant/Vlasv beam Assumptins Linear elastic material Cnstant EA, EI, GK v, EI ω and N alng beam N initial stress and n initial curvature (imperfectins) N distributed mment lads N knwn and n influence f shear n nrmal aial frce y z EAu EI z v' ''' EI y w'''' EI '''' GK '' ω v q q y q z m where (2:nd rder terms)= Ntatins: see net page. N(v'' z '') (M )'' N(w'' y '') (M )'' (2:nd rder terms) N(I /A) '' N(y w'' z v' ')... (M y v' )' 8 y z (V z v)' (M z w')' (V y w)'... 2(M 2(M y ')z z ')y... q z v q y w q z (z 1 z ) q y (y 1 y ) Reference: Byggnadsmekanik. Knäckning (Runessn, Samuelssn, Wiberg, 1992) bar-actin (8) bending in -y plane (9) bending in -z plane (10) trsin (11)
Ntatins y z (y,z ) is the lcatin f the rtatin center (y 1,z ) is the lcatin f lad q y (y,z 1 ) is the lcatin f lad q z I =I y +I z +(y 2 +z 2 )A is the plar mment f inertia with respect t the rtatin center V y, V z, M y, M z are sectin shear frces and mments acc. t 1:st rder thery 9
Outline f derivatin f gverning equatins fr 3D beam, 2:nd rder The three beam stress cmpnents σ, τ y and τ z, acting n a small vlume ddydz in the beam are cnsidered: y, v τ y σ τ z z, w Psitin when beam is unladed Psitin when beam is laded, u Equilibrium in y- and z-directins: τ τ y z U U y z (σ (σ v' ) 0 w') 0 (12a,b) U y and U z är lad/vlume in the y- and z-directins. 2:nd rder effects f the shear stresses are nt cnsidered, just as in the 2D beam analysis. The 2:nd rder equatins can btained frm the 1:st rder equatin by adding the secnd rder cmpnent f the nrmal stress t the lads in the y- and z- directins (lads/length q y and q z ) : replacing lad qy with lad replacing lad qz with lad (In case f trsin with prevented warping additinal cnsideratins are needed.) Nte: σ is the nrmal stress in directin (1, v, w ), that is in the directin alng a line in the beam which in the unladed beam was riented parallel with -ais. This directin f σ is thus nt the -directin and in general neither perpendicular t the surface n which σ acts. The latter is f relevance fr St Venant trsin. 10
Equatins (8-11) in the special case f duble symmetric and massive (thickwalled) beam crss sectins: Duble symmertric: y =z =0 Massive (thick walled): y z 0 EAu EI z v' ''' EI y w'''' EI '''' GK '' ω v q q y Nv' ' (M )'' q z Nw'' (M )'' m (2:nd rder terms) y z (13a,b,c,d) where (2:nd rder terms)= N(I /A) ''... (M (M y v' )' (V z v)' z w')' (V y w)'... q z v q y w q z z 1 q y y 1 Fr the I- beam sectin is I ω 0. Fr all ther is I ω 0. 11
If mrever n distributed lad (q =q y =q z =m =0): EAu 0 EI z v' ''' Nv'' (M y )'' EI y w'''' Nw'' (M z )'' EIω '''' GKv'' N(I '' (M /A) y v' )' (V z v)' (M z w')' (V y w)'..(14a,b,c,d) And fr stability analysis with respect nly t cmpressive frce N: EI EI EI z y v' ''' Nv' ' 0 w'''' Nw'' 0 '''' ( GK NI /A) '' 0 ω v (15a,b,c) Eq (15a,b,c) have the nice feature f being uncupled. 12
Trsinal buckling Only duble symmetric and/r thickwalled crss sectins cnsidered. Gverning equatin and slutin fr beams with I ω 0: EI ω '''' ( GKv NI /A) ' ' 0 (15c) C1cs(λ) C2sin( λ) C3 C4 fr (GKv NI /A) 0 (cmpressi n), λ -(GKv NI /A)/EIω C 1cs h(λ) C2sinh( λ) C3 C4 fr (GKv NI /A) 0 (tensinrsmall cmpr.), λ (GKv NI /A)/EI (16a,b) One slutin fr hmgeneus bundary cnditins is C=0. Frm the bundary cnditins: AC=0. By investigatin f det(a)=0 can pssible Ncr fr the actual bundary cnditins be determined. Gverning equatin and slutin fr beams with I ω =0: ( GK v NI /A) '' 0 (17) v C (18) 1 C 2 One slutin fr hmgeneus bundary cnditins (ϕ=0 and/r ϕ =0) is C=0. Other slutins are pssible nly if ( GK v NI /A) 0 (19) Thus the critical lad fr trsinal buckling is ω Ncr GK v A/I (20) The curve ϕ() may have any cntinuus shape in the case trsinal buckling f a beam with I ω =0! (Prvided that the bundary values at =0 and X=L are fulfilled.) 13
Further studies f trsinal buckling: Thin walled pipe σ Crss sectin area =A Radius = R Wall thickness = t<<r σ a σ h G h(1-csγ) γ h Ptential energy, π π σah(1 csγ) (1/2)τγV π σsinγ Gγ π σcsγ G where ah V and τ Gγ Fr γ=0 is π =0 (equilibrium) and π =0 (neutral equilibrium) fund fr σ cr = G (a simple and useful relatin!?) (21) Fr the pipe this means Ncr GA (22) Cmpare with Eq (20) fr a thin walled pipe: Ncr GK v A/I GI A/I GA 14
σ Crss sectin area =A Radius = R Wall thickness = t<<r σ a σ h G h(1-csγ) γ h The result σ cr = G shuld be pssible t derive als by means f equilibrium analysis. Hw? 15
Further studies f trsinal buckling: Derivatin f buckling lad at St Venant trsin (beams with I ω =0) σ A φ rφ r da L γ Ptential energy, π T π σ dal(1 csγ) 2 A where γ r/l, T GK /L v and cs γ1 γ 2 /2 2 2 2 (r/l) GK σ π σl da v 2 2L 2L A σ GK π I v L L σ GK π I v L L A r 2 GK da v 2L Fr ϕ=0 is π =0 (equilibrium) and π =0 (neutral equilibrium) fund fr GK σ v cr I giving Ncr GK v A/I which is Eq (20) The similar energy based derivatin f critical lad fr the case I ω 0 can be made, but then variatin f ϕ alng the beam must be cnsidered (by means f Eq (15a)). 2 16
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