4. Lateral-Torsional Buckling of Members in Bending

Transcription

1 4. Laeral-Torsional Buckling of Members in Bending This chaper is devoed laeral-orsional buckling of members in bending. A firs paragraph is devoed o a phenomenological descripion of he phenomenon. The second one collecs a series of compleel heoreical models derived for elasic beams of diverse cross secion. Finall, a comprehensive discussion of he main code provisions is repored; boh he Ialian and he European codes are analzed and applied. 4.1 Definiions and phenomenological descripion Sabili wihin he Euler definiion has been alread exposed in chaper wih reference o he case of members in compression. Under he wo basic hpoheses of i) small displacemens and ii) no imperfecion, he behaviour of members in compression appeared as bifurcaion problem whose soluion is he rivial one under low level of axial load, swiching o an adjacen one as he axial force approaches a paricular criical value. The same behaviour can be observed for members in bending; even in hese cases, a criical value of he ransverse load resuls in a bifurcaion phenomenon: saring from he plane configuraion characerizing he response of members in bending, an adjacen equilibrium configuraion is assumed under ha criical value. The possible adjacen configuraion is usuall described b ou-of-plane displacemens hroughou he beam axis and a wising angle (Figure 4.1). Figure 4.1: Laeral-orsional buckling for a canilever beam. As well as momens of ineria pla a relevan role in affecing he buckling phenomenon under axial forces, oher geomeric properies of he ransverse secion resul in geing he member more or less suscepible of laeral-orsional buckling. Laeral-orsional buckling of members in bending is also affeced b load condiion, being he phenomenon sensiive of he poin of applicaion of he ransverse force hroughou he secion. All hese aspecs of he laeral-orsional buckling phenomenon will be analzed in he following. Firs of all a raher comprehensive explanaion of he elasic heor will be given wih some insighs abou he soluion mehods which can be uilized for obaining approximae soluion of he problem, being closed-form ones onl available in few simple cases. Finall, boh Ialian Code and Eurocode 3 provisions for laeral-orsional buckling will be examined, commened and applied in some worked examples. 4. Theoreical insighs The heoreical sud of he menioned phenomenon sars from members wih recangular secion, considering, in paricular, he case of deep beams in which he wo principal momens of ineria are significanl differen; he case of more general secions will be also addressed wih paricular Dr. Enzo MARTINELLI 59 Draf Version 18/1/01

2 focus on he I-shaped beams which are raher common in seel srucures. The wa in which he poin of applicaion of loads affecs he behaviour of flexural members in erms of laeral-orsional buckling occurrence will be also examined; in paricular, he difference in behaviour deriving b appling he ransverse load eiher above, beneah or direcl in he secion cenroid, will be also addressed. Influence of iniial imperfecion is also considered wih he aim of undersanding he behaviour of he so-called indusrial member, wih respec o he ideal case of perfecl sraigh ideal members Laeral buckling of beams wih recangular secion Le us consider a beam of recangular secion whose principal radii of graion are significanl differen; in paricular, he shorer one is referred o he ou-of-plane bending as shown in Figure 4.: x G ϕ Figure 4.: Adjacen laeral-orsional buckled configuraion described b he wising angle ϕ The wising roaion j which can possibl arise during he loading process resuls in a non perpendicular condiion beween he bending momen vecor M (orhogonal o he bending plane) and he -axis of he ransverse secion; consequenl, he exernal bending momen is characerized b a non-zero componen (approximael equal o Mϕ) along he -axis and, hence, an ou-of-plane bending. If I is he momen of ineria wih respec o he -axis, he following Navier relaionship can be saed beween he bending momen and he resuling ou-of-plane curvaure: d u EI = M ϕ (4.1) dz where u=u(z) is he displacemen componen in x direcion of he cenroid G of he secion a abscissa z. Msinϕ Mϕ M G Figure 4.3: Bending momen in he wo direcions Verical plane Laeral displacemens possibl arising in he beam have also an influence on is orsional behaviour. Le us be GJ he orsional siffness of he beam and consider a segmenal par of he beam whose lengh is dz subjeced o a bending momen M and a orsional one M. According o he smbols inroduced in Figure 4.3 he equilibrium equaion o roaion in longiudinal direcion can be easil wrien as follows: M + M + dm M dε = (4.) ( ) 0 Dr. Enzo MARTINELLI 60 Draf Version 18/1/01

3 x M M dε/ z M+dM Mdε M dε/ M+dM M +dm Figure 4.4: Relaionship beween orque and bending momen Horizonal view On he conrar, under he kinemaical poin of view, a simple relaionship can be saed beween he elemen lengh dz (which can be approximaed b he arc of lengh ds) and he radius of curvaure ρ u of he beam axis in he horizonal plane: dz d u dε = dz (4.3) ρu dz The following equaion can be obained b subsiuing he (4.3) in (4.) and expressing all quaniies in erms of derivaives: dm d u M, (4.4) dz dz which, inroducing equaion (4.1) and ex pressing he orque M in erms of he uni angle of orsion according o he De Sain-Venan heor, leads o he following relaionship beween he wis angle ϕ(z) and he corresponding bending momen M(z): ( ) d dϕ M z + GJ ϕ. (4.5) dz dz EI In he overwhelmingl common case of uniform ransverse secion, he above formula can be ransformed in he following one, which is a second-order differenial equaion: ( ) d ϕ M z + ϕ. (4.6) dz GJ EI The above equaion is he well-known PRANDTL-MICHELL equaion, who derived i in he nineeenh cenur. 4.. Soluion of some significan cases. Equaion (4.6) can be solved following differen approaches, depending also b he ransverse load applied hroughou he beam lengh and he resuling bending momen funcion M(z) s Case: Simpl-suppored beam under uniform bending momen M(z)=M 0. In his case he equaion (4.6) has consan coefficiens and, consequenl, an eas closed-form soluion can be found following he usual analical mehods for linear differenial equaions: Dr. Enzo MARTINELLI 61 Draf Version 18/1/01

4 d ϕ M0 + ϕ. (4.7) dz GJ EI According o he menioned end resrain condiions he general funcion describing he wising roaion ϕ(z) can be assumed in he following sinusoidal form: πz ϕ( z) = a0 sin L, (4.8) which can be properl regarded as he firs erm of he Fourier series of he real (and unknown) funcion ϕ(z) (cosine erms are null as a resul of he odd naure of he given funcion). Since he origin of he axes is placed a one of he wo ends and, consequenl, z [ 0, L], he equaion compl wih he resrain condiions in he case of simpl-suppored beams wih roaional fixiies a boh ends. Subsiuing equaion (4.7) in (4.6), he following relaionship can be obained: π M 0 πz a0 sin, (4.9) L GJ EI L and is lef member vanishes for he non-rivial configuraion (namel, if a 0 0) if and onl if he firs facor does; consequenl, he following criical value M 0,cr for he bending momen can be evacuae: π M0,cr = GJ EI. (4.10) L 4... nd Case: Simpl suppored beam under uniforml disribued load. In his case, assuming a reference ssem cenred in he mid-span poin, he bending momen can be easil expressed as follows: ql M ( ζ ) = ( 1 ζ ), (4.11) 8 being ζ [ 1,1] and he equaion (4.6) has variable coefficiens. Consequenl, no-closed form soluions o be sough b analical procedures are available and an alernaive approximae mehod is needed. In paricular, he Undeermined Coefficien Mehod can be uilized b inroducing a powerseries approximaion of he unknown funcion considering separael even and odd exponens wihin he wo erms ϕ 0 e ϕ 1 reflecing he naure of is componen funcions. A cerain (even) number n of erms can be chosen for he power series and he expression of ϕ(z) can be placed in he following form: n n/ n/ 1 i j j i j j i j j + ( ) = ( ) + ( ) = A = A + A ϕ ζ ϕ ζ ϕ ζ ζ ζ ζ, (4.1) 0 1 In he curren case he funcion mus be even in naure due o he srucural smmer and, hence, onl he powers wih even exponen can be considered: ( ) ( ) = = / A 0 n j j j ϕ ζ ϕ ζ ζ. (4.13) The non-dimensional abscissa ζ can be easil inroduced in erms as a funcion of he dimensional one z as follows: z L/ L ζ = and dz = dζ. (4.14) L/ and he equaion (4.6) can be wrien as follows: 3 ( ql 16 ) ( ) d ϕ + 1 ζ ϕ, (4.15) dζ GJ EI And inroducing he following definiion, Dr. Enzo MARTINELLI 6 Draf Version 18/1/01

5 resuls in he final form repored below: d k ( ql 3 16 ) = GJ EI ϕ + ( ζ ) ϕ = dζ k 1 0, (4.16), (4.17) Inroducing equaion (4.13) in (4.17), he maximum exponen of he various erms in ζ is n+4. Collecing he erms corresponding characerized b he same power o he ζ variable he following se of linear equaion can be wrien in erms of he consans uilized for defining he above power series. In paricular, a homogeneous se of n/+1 equaions wih he unknown A j (j=0 n/) repored below (in he considered case n=0) can be obained: A0k + A A0k A k 1A4 A0k Ak + A4k + 30A6 Ak A4k + A6k + 56A8 A4k A6k + A8k + 90A10 A6k A8k + A10k + 13A1. (4.18) A8k A10k + A1k + 18A14 A10k A1k + A14k + 40A16 A1k A14k + A16k + 306A18 A14k A16k + A18k + 380A0 A16k A18k + A0k B he above se of n/ linear independen equaions, he unknowns A j can be deermined as a funcion of A 0, and can be subsiued in equaion (4.19) (in which ζ has been replaced b z) wih he aim of obaining ϕ: k ζ k ζ k ζ k ζ 7k ζ k ζ 13k ζ k ζ φ ( ζ ) = A (4.19) A suiable boundar condiion has o be imposed o his expression of ϕ(ζ) considering ha he wising roaion vanishes a ζ= 1, due o he resraining effec of he roaional fixiies: φ ( 1) ; (4.0) he above condiion is alwas rue for he rivial soluion hroughou all he beam axis and can be imposed o he soluion (4.19) describing he laerall buckled configuraion. In his second case, a criical value k cr of he consan k can be deermined obaining he following approximae soluion: k cr ( q 3 16) crl = = 1,7697 GJ EI, (4.1) from which he corresponding criical value q cr of he uniforml disribued load q can be easil deermined: Dr. Enzo MARTINELLI 63 Draf Version 18/1/01

6 16 kcr 8, 315 qcr = GJ 3 EI = GJ 3 EI. (4.) L L This soluion has been assuming an approximaed expression of he funcion ϕ(z) involving he firs 11 power erms of even exponen repored in equaion (4.13). A wider sensiivi analsis has been carried ou abou he level of approximaion which can be achieved b he numerical soluion involving up o 0 power erms. Figure 4.5 shows he convergence of he numerical soluion oward a sable value k cr confirming he good approximaion of he soluion repored in equaion (4.). k cr 30,0 9,5 9,0 8,5 8,0 7,5 7,0 6,5 6, Figure 4.5: Value of k cr as a funcion of he order n of approximaion of he numerical soluion. The same figure shows ha he same convergence process is generall non monoonic as he order of he approximaing soluion varies from 3 o rd Case: Simpl-suppored beam under a poin load Q a mid-span. In his case he analical expression of he bending momen, assuming ha he non dimensional abscissa sars form he mid-span poin, can be placed in he following form: wih [ 1,1 ] QL 4 M ( ζ ) = ( 1 ζ ), (4.3) ζ and equaion (4.6) has once more variable coefficiens. The soluion is sough following he same numerical approach uilized above assuming for per ϕ(ζ) a similar polnomial expression repored in (4.13). Following a similar procedure, he criical value Q cr of he concenraed load Q can be derived as expressed below: 8 kcr 11, 8041 Qcr GJ EI GJ EI L L = =. (4.4) h Case: Canilever beam under uniform bending momen M(z)=M 0. In his case, equaion (4.6) has consan coefficien 0 d ϕ M 0 + ϕ =, (4.5) dz GJ EI and, hence, a sinusoidal expression can be assumed for he general soluion according o he imposed boundar condiions (roaional fixi a z=0): ϕ (z) = ϕ1 πz sin L, (4.6) which can be, as usual, inended as he firs erm of he Fourier series of he unknown soluion ϕ(z) (cosine coefficiens are once more zero due o he paricular boundar condiion). Since he origin of Dr. Enzo MARTINELLI 64 Draf Version 18/1/01

7 he reference ssem is a he fixed end of he beam. Inroducing equaion (4.7) in (4.6) he following relaionship can be easil obained: π M 0 πz a0 sin L GJ EI L, (4.7) which is rue as a 0 =0, or, in he non-rivial case, as he firs facor in he square parenheses is zero and, consequenl, he criical value M 0,cr of he bending momen M can be derived: π M0, cr = GJ EI, (4.8) L h Case: Canilever under uniforml disribued load. Assuming he origin f he z axis a he free end of he canilever, he funcion describing he bending momen can be expressed as follows: q ql M( z) = z M( ζ) = ζ, (4.9) in which he non dimensional abscissa ζ = z L has been convenienl inroduced. The equaion (4.6), can be, hence, wrien in he following form: or, equivalenl, under he hpohesis ha: 3 ( ql ) ( ζ ) φ 4 φ ζ + GJ EI = d d 1 0, (4.30) d φ 4 k ( 1 ζ ) φ 0 dζ + =, (4.31) k 3 ( ql ) = GJ EI. (4.3) Also in his case, a power series can be inroduced for approximaing he unknown soluion; based on he naure of he problem and he choice of he relevan variables, he following approximaion is inroduced: ( ) ( ) 0 n/ φ ζ φ ζ A ζ j j j = =. (4.33) Inroducing (4.6) in (4.3), collecing he coefficiens of he similar erms and imposing ha a null produc can be obained if one of is facors is zero, he following homogeneous se of linear equaion can be wrien (in he case of n=8): Dr. Enzo MARTINELLI 65 Draf Version 18/1/01

8 A 1A4 A0k + 30A6 Ak + 56A8 A4k + 90A10 A6k + 13A1 A8k 18A = A10k + 40A16 A1k + 306A18 A14k + 380A0 A16k + 46Ak A18k + 55A4k A0k + 650A6k A1k + 756A8k A4k. The variables A j can be derived as a funcion of A 0 and inroducing hem in equaion (4.6) he following expression of he wising roaion ϕ(ζ) can be obained: k ζ k ζ k ζ k ζ φ ( ζ ) = A (4.35) As a boundar condiion, he firs derivaive of ϕ(ζ) a he free end of he beam (ζ=1) have o be forced o zero. Consequenl, a criical value k cr of k can be deduced in he non rivial configuraion, obaining he following relaionship: and or, in erms of global load: k cr ( q 3 ) crl (4.34) = = 6, 478, (4.36) GJ EI kcr 1,8557 qcr = GJ 3 EI = GJ 3 EI, (4.37) L L 1,8557 Qcr = qcr L = GJ EI, (4.38) L or bending momen M qcr L 6, 477 = = GJ EI, (4.39) L cr 4..3 Influence of he ransverse secion shape: smmeric I-shaped profiles Le us consider he smmeric I-shaped secion whose deph is h and le ϕ(z) be he wising angle around he z-axis. The laeral displacemen u (in x-direcion) of he above flange can be relaed o he local value of he wising angle as follows: Dr. Enzo MARTINELLI 66 Draf Version 18/1/01

9 h u( z) = φ( z). (4.40) G x G Figure 4.6: Smmeric I-shaped secion. Consequenl, a curvaure d u/dz arises hroughou he axis of he flange and he corresponding bending momen in -direcion can be relaed o ha curvaure according o he usual linear relaionship involving he relevan flexural siffness: d u dz M1, EI, f = ; (4.41) B equilibrium, he shear force V 1, orhogonal o he -direcion corresponds o he above bending momen M 1, : V1, dm1, dz = (4.4) and, hence, a relaionship beween his shear force and he funcion describing he wising angle ϕ can be easil saed considering equaions (4.8) and (4.9): V 3 h d φ = EI f. (4.43) dz 1,, 3 Since he force V 1, acs on he above flange in he direcion parallel o is axis and, for he same reasons, an opposie one is applied on he boom flange, a secondar orque arises on he ransverse secion, whose sign is opposie o he one covered b he De Sain-Venan heor and menioned above for he recangular secion: 3 1 d 1, 1,, f 3 φ M = V h = EI h. (4.44) dz Consequenl, he relaionship beween he orque and he wising roaion ϕ can be obained b summing he De Sain-Venan conribuion inroduced in equaion (4.5) and he furher one described b equaion (4.44): 3 1 d φ, f 3 dφ M = EI h + GJ. (4.45) dz dz Since he orque M sems ou b he laerall buckle configuraion and is consequenl relaed o he bending momen, he following relaionship repored in equaion (4.4), can be saed beween he wo couples: dm dz d u M, (4.46) dz Dr. Enzo MARTINELLI 67 Draf Version 18/1/01

10 and, inroducing equaion (4.1) and (4.1), he following fourh-order differenial equaion in erms of ϕ can be derived: and, finall, 4 d φ d, f 4 [ M( z) ] 1 φ EI h + GJ + φ, (4.47) dz dz EI [ M( z) ] 1 EI + + φ. (4.48) GJ 4, f d φ d φ h 4 dz dz GJ EI In a more general sense, he warping momen of ineria (or secorial momen of ineria) I ω can be inroduced, being in he case of smmeric I-shaped secion: and he general form of equaion (4.45): h EI, fh EIω = EI, f = Soluion of some significan cases. [ M( z) ], (4.49) d φ d φ EIω + GJ φ =. (4.50) dz dz EI s Case: Simpl-suppored beam in uniform bending momen M(z)=M 0. Equaion (4.45) has generall variable coefficiens due o he fac he bending momen M ( z) is relaed o he abscissa z. Far from being a pracical represenaive case, uniform bending momen is a cas d école in which he coefficien of he above equaion are consan. Consequenl, if M ( z) = M0 equaion (4.45) akes he following expression 4 EIω d φ d φ M0 + + φ GJ 4 dz dz GJ EI, (4.51) and, as a sinusoidal expression is assumed for (he firs erms of he Fourier series) he unknown funcion ϕ(z) π φ ( z ) = φ1 sin z, (4.5) L equaion (4.46) can be ransformed as follows: 4 EIω π π M 0 1 sin πz + φ = GJ 0 L L GJ EI L. (4.53) A criical value M 0,cr of he uniform bending momen M 0 resuling n laeral-orsional buckling of he considered beam can be defined as follows: π EIω π M0, cr = GJ EI 1+ L GJ L. (4.54) I is sraighforward o observe ha equaion (4.54) reduces o (4.10) for recangular secions in which EIω GJ L 0; indeed, he laer equaion provides a lower bound for he criical bending momen resuling, and equaion (4.54) poins ou he sabilizing conribuion of he flanges (and more generall of he erms EIω GJ L ) o laeral-orsional buckling occurrence. In he case of a canilever beam, equaion (4.5) has o be replaced b he following expression which ake accouns of he diverse resrain condiion, forcing wising roaion o vanish a he fixed end for z=0: πz φ ( z ) = φ1 sin. (4.55) L The above equaion, inroduced in (4.50), and ransformed as repored above wih reference o a simpl-suppored beam, resuls in he following expression of he criical bending momen: Dr. Enzo MARTINELLI 68 Draf Version 18/1/01

11 π EIω 1 π M0, cr = GJ EI 1+, (4.56) L GJ 4 L even reducing o he corresponding equaion (4.6) in he case of recangular secion, namel wih EIω GJ L 0. Finall, comparing equaions (4.54) and (4.56) he role of he differen resrain condiion on he wo menioned beams wih respec o he reference case reproduced b equaions (4.10) and (4.6) deermined for he cases of recangular secion for simpl-suppored and canilever beams, respecivel nd Case: Trave semplicemene appoggiaa con carico uniformemene disribuio q. The equaion (4.50) can be ransformed in he corresponding non-dimensional shape in erms of he non dimensional abscissa defined above: where.. k = ( ζ ) 4 φ d φ k 4 d C φ, (4.57) dζ dζ 3 ( ql 16) ω C = 4 GJ L GJ EI EI. (4.58) 4..5 Influence of he posiion of he poin load on he ransverse secion Wihin he previous paragraph, he loads are inended o be applied on he cenroid G of he ransverse secion. Neverheless, load can be also applied in oher poins of he secion, above or beneah he cenroid G. Figure 4.7 shows he case of a load applied in a poin C, on he -axis, a he disance d* from he cenroid G (assuming posiive disances according o he direcion of he -axis). u G C x G C d* qdz Figure 4.7: Smmeric I-shaped secion wih load applied in a general poin C The eccenrici d* resuls in a variaion dm * of he of he global orque acing on he ransverse secion which has o be added o he wising momen expressed in equaion (4.): dm * = qdz φd *. (4.59) Dr. Enzo MARTINELLI 69 Draf Version 18/1/01

12 Coming back on he ransformaions leading form he equilibrium equaion (4.) o he final differenial ones repored in equaion (4.6) or (4.15) he effec induced b his momen variaion can be easil undersood. In paricular, considering equaion (4.50), he following second-order differenial equaion can be derived aking accoun of he eccenrici d* beween G and C: [ M z ] 4 EIω d φ d φ ( ) qd * + + φ GJ 4 dz dz GJ EI GJ. (4.60) Simpl suppored beam under an uniforml disribued load q. As described in secion 4..., he expression of he bending momen in erms of a nondimensional abscissa ζ, whose origin is a mid-span of he beam, can be expressed as follows: being [ 1,1 ] ql ( ) ( 1 ζ ) M ζ =, (4.61) 8 ζ and he equaion (4.6) has variable coefficiens. The simple case of recangular secion is considered ( EIω GJ L 0) wih he aim of poining ou he role of he disance d*; in ha case equaion (4.6) akes he following form: d φ [ M( z) ] qd * + φ. (4.6) dz GJ EI GJ and he following non-dimensional expression can be easil derived corresponding o equaion (4.15) on he basis of he relaionship (4.14): 3 d φ ( ql 16 ) qd * + ( 1 ζ ) φ. (4.63) dζ GJ EI GJ or, equivalenl: 3 ( ql ) φ EI ( ζ ) GJ 3 ζ EI ql d 16 4 d * h + 1 φ. (4.64) d h L and he following parameer has been alread inroduced k 3 ( ql 16) = GJ EI. (4.65) A furher non-dimensional parameer k 1 can be inroduced considered depending on he quaniies which conrols he insabili phenomenon, such as he EI /ql 3 raio, relaed o he flexural srain induced b he exernal load, he d*/h and h/l raios: 4 EI d * h k1 3 ql h L and, finall, he following expression can be obained =. (4.66) ( ζ ) d φ + k 1 k 1 φ d. (4.67) ζ Equaion (4.67) is formall similar o (4.17) and can be, hence, solved b considering a power series approximaion of he unknown soluion, considering n erms and obaining n-1 independen equaions which can be used o express he value of he n-1 consan as a funcion of A 0. In he presen case he criical value k cr depends also on he parameer k 1 ; alhough no closed-form soluions can be found for k cr or q cr in he general case, he following one deals wih he case of k1=0, namel, d*=0: γ = =. (4.68) 16 kcr cr qcr GJ 3 EI GJ 3 EI L L Dr. Enzo MARTINELLI 70 Draf Version 18/1/01

13 Neverheless, Figure 4.8 shows he values of he coefficien γ cr deermined hrough a numerical procedure as a funcion of k 1 ; i is eas o verif ha for k 1 =0 he same value repored in (4.68) can be found. On he conrar, he γ cr value (namel he criical load q cr ) hugel increases for k 1 >0 and, hence, for load applied beneah he cenroid G, while decreases (alhough in a less sensiive wa) for d*<0. This apparen asmmer of he problems is acuall ver limied, since he values of k 1 of pracical ineres are in he neighbours of zero, ranging from -0,10 o 0,10, as one can easil check b considering (4.66) and he pical values of he parameers involved wihin he definiion of k 1. γ ,315 -,50 -,00-1,50-1,00-0,50 0,00 0,50 1, Figure 4.8: Values of k cr as a funcion of k 1 defined b (4.66) 4.3 Code Provisions The above heoreical models dealing wih laeral-buckling for elasic members, can be usefull uilized for beer undersanding he provision of wo codes of sandards devoed o seel srucures. As proposed in he previous secion for flexural buckling under axial load, boh Ialian and European provisions will be discussed. Some worked examples will be also proposed for poining ou some pracical aspecs of hose provisions Ialian Code (CNR 10011, DM96) The menioned Ialian code proposes wo alernaive mehods for checking flexural members agains laeral-orsional sabili: - he ω 1 -mehod; - he isolaed flange mehod. The firs one is based on he definiion of an amplificaion coefficien ω 1 for he normal sress in he compressed flange whose value is defined in erms of he beam deph-o-widh and span-lengh-oflange-hickness raios. No deails is given herein abou ha mehod for he sake of brevi and he ineresed reader can find furher informaion on he code ex. On he conrar, he so-called isolaed flange mehod is described in he following because of is significan meaningfulness under he mechanical sandpoin. Indeed, he flange in compression is considered ad is slenderness is defined as he raio beween he effecive span lengh L 0 (defined according o he same rules described in he previous chapers depending on he end resrain condiions) and he ransverse radius of ineria ρ which can be easil evaluaed as follows: L0 λ =. (4.69) ρ Dr. Enzo MARTINELLI 71 Draf Version 18/1/01

14 The corresponding value of he ω coefficien can be deermined as a funcion of he ransverse slenderness b uilizing eiher he sabili curve c or d defined in secion for profiles respecivel hinner or hicker han 40 mm. The axial sress acing on he flange in compression has o be amplified b he above menioned ω coefficien as follows: Meq S x I M x eq (4.70) σ = ω ω G fad. A I f wih he following meaning of he involved smbols: - A f is he ransverse area of he flange in compression; - I x is he momen of ineria of he profile around he x-axis; - S x is he saic momen of he flange in compression wih respec o he same axis; - G is he disance beween he cenroid G and he secion edge in compression. Furhermore, M eq is he equivalen bending momen which depends on he shape of he bending momen diagram hroughou he beam axis and can be assumed according o he same rules repored in secion Finall, i is worh noicing ha, according o equaion (4.70), he laeral-orsional buckling occurrence can be inended as he ou-of-plane flexural buckling of he secion flange in compression European Code (EC 3) The EC3 formulaion of he laeral-orsional sabili check of members in bending is more general as well as much more formall complex. The exernal relaionship o be checked for verifing he member in bending agains he onse of laeral-orsional buckling phenomenon is repored below: fa Mb,Rd = χltβwwpl,. (4.71) γm1 where furher smbols have been inroduced. In paricular, W pl, is he plasic modulus around he ransverse axis (according o he EC3 assumpion - and z-z are he wo main cenroidal axis as represened in Figure.19). Furhermore, he β w facor can be defined as follows depending on he class of he ransverse secion: - Class 1 and : β w = 1; - Class 3: β w = Wel, Wpl, ; - Class 4: β w = Weff, Wpl, ; wih he same meaning of he smbols alread inroduced in chaper. The reducion facor χ LT akes ino accoun he effec of laeral slenderness on he occurrence of he laeral-orsional buckling. A similar definiion o ha provided b equaion (.75) for he reducion facor χ, can be uilized for χ LT : 1 χ LT =, (4.7) Φ + Φ λ LT LT LT where λ LT is he relaive slenderness which even conrol he value of Φ LT according o he following funcion Φ LT.5 1 LT ( LT 0.) + α λ + λlt. (4.73) The parameer α LT represens an imperfecion coefficien and is value can be assumed as follows: - ho-rolled secion α LT.1; - ho-rolled secion α LT.49. The relaive (or non-dimensional) slenderness LT λ is defined as he square roo of he raio beween he ulimae flexural srengh of he ransverse secion (namel, he plasic momen for profiles Dr. Enzo MARTINELLI 7 Draf Version 18/1/01 x

15 in classes 1 and, he ielding momen for profiles in class 3 and he effecive one for profiles in class 4) and he elasic laeral-buckling momen M cr : βw Wpl, fa λ LT =. (4.74) Mcr The parameers affecing he criical momen resuling in laeral-orsional buckling for elasic members have been poined ou in he firs secion on he presen paragraph. The following general relaionship can be assumed he ransverse secion of whichever shape: πei z k I ( ) W kl GJ Mcr = C1 ( Czg ) Cz + + g, (4.75) ( kl ) kw Iz π EI z where he smbols have he meaning described below: - z g is he disance beween he secion cenroid G and he acual poin of applicaion of he loads. I is assumed being posiive if he load is applied above he cenroid and, hence, he following relaionship can be recognized wih one of he parameers inroduced in secion 4..5: zg = d* ; - he facor k W is he coefficien relaed o he effecive lengh of he beam wih respec o he resrains o warping (namel, he possible deformaion of he ransverse secion wih respec o is plane). The value 0.5 is assumed for he double fix ends, while he value 1.0 is given in he case of free warping a he end. Finall, 0.7 can be assumed if warping is resrained on one end and free on he oher one; - he facor k akes accoun of he end resrains in erms of wising roaion. The value 0.5 corresponds o full resrains, 1.0 o compleel free scheme and finall 0.7 can be aken in he inermediae siuaion; - he parameer J is orsional siffness; - he parameer I z is momen of ineria around he weak axis (I hroughou he previous secion devoed o he heoreical bases); - he parameer I W is he so-called warping consan which is a sor of second momen of he momens of ineria. For he doubl smmeric I-shaped secions, he following relaionship exiss beween I W and I z : I ( ) f I h z W =. (4.76) 4 - L is he beam span lengh; - he consans C 1 and C can be aken b Table 4.1 and Table 4. depending on he shape of he bending momen diagram hroughou he beam axis. Wih he aim of beer clarifing he meaning of he facors k and k W, Figure 4.9 shows differen resraining condiions regarding wising roaion and warping. Figure 4.9: Differen resrain condiion on wising roaion and warping Dr. Enzo MARTINELLI 73 Draf Version 18/1/01

16 The general equaion (4.75) can be simplified in some cases of pracical ineres such as: - ransversal load applied on he cenroidal poin G (which for doubl smmeric profiles coincides wih he shear cenre): πei z k IW ( kl ) GJ Mcr = C1 +, (4.77) ( kl ) kw Iz π EI z - full fixed beam wih ransversal load applied in G: πei z IW ( kl ) GJ Mcr = C1 +, (4.78) ( kl ) Iz π EIz Table 4.1: Coefficiens of equivalen uniform momen for linear diagrams (end forces). Dr. Enzo MARTINELLI 74 Draf Version 18/1/01

17 Table 4.: Coefficiens of equivalen uniform momen for general cases (disribued loads). 4.4 Worked examples The following examples are proposed for poining ou some pracical aspecs relaed o laeralorsional sabili checks in seel beams in bending Laeral orsional sabili check simpl suppored beam Le us consider he simpl suppored beam in Figure 4.10 under he following uniforml disribued loads: - Dead Load g k =9.0 kn/m; - Live Load q k =6.0 kn/m; Figure 4.10: Simpl suppored beam wih orsional end resrains. The ransverse secion of he beam is realized b a sandard IPE70 profile whose ke geomeric properies are lised below (seel grade S35): - deph h 70 mm; - widh b 135 mm; - flange hickness f 10. mm; - web hickness w 6.6 mm; - chord radius r 15 mm; Dr. Enzo MARTINELLI 75 Draf Version 18/1/01

18 - secion area A cm ; - self-weigh per uni lengh g k 0.35 kn/m; - momen of ineria around he srong axis I 5790 cm 4 ; - momen of ineria around he weak axis I z 40.0 cm 4 ; - orsional consan J cm 4 ; - warping consan I W cm 6 ; - plasic modulus around he srong axis W pl, cm 4 ; - plasic modulus around he weak axis W pl,z cm 4 ; - span lengh L 6.00 m Laeral-orsional sabili check according o he Ialian Code provisions. The isolaed-flange mehod can be uilized for carring ou he safe check agains laeralorsional buckling according o he Ialian Code provisions. The procedure follows he seps repored below: Sep #1: evaluaion laeral slenderness of he flange in compression: Since roaional resrains are considered a boh ends he effecive lengh (evaluaed aking onl accoun of he roaional resrains) is L 0 =L/. Moreover, he ou-of-plane radius of graion of he profile flange can be easil evaluaed b 135 ρ = = = mm, (4.79) 1 1 and he corresponding slenderness is: L λ = = = (4.80) ρ Finall, he plasic slenderness can be evaluaed as follows: Ea λp = π = π = (4.81) f 35 a Sep #: evaluaion of he ω facor: Since he flange siffness is smaller han 40 mm he curve c menioned in secion.8.1 has o be considered. The non-dimensional slenderness λ λ 0.81 leads o he following value for he ω facor: p = ω =1.540, (4.8) Sep #3: evaluaion of he compressive sress in he flange: The maximum bending momen hroughou he beam axis can be easil evaluaed as follows: ( g ( k k ) q k ) ( ( ) ) γ g + g' + γ q L Mmax = = = 99.4 knm, (4.83) 8 8 and he following mean value can be assumed because of he sinusoidal shape of he bending momen diagram: Mm = Mmax = 99.4 = 66.6 knm, (4.84) 3 3 and he equivalen momen is Meq = 1.3 Mm = = knm, (4.85) which is included wihin he range [0.75 M max, M max ]. Since he flange hickness is negligible wih respec o he beam deph, he approximaion in equaion (4.70) is accepable (being on he conservaive side) and he following value of he compressive sress can be evaluaed. Meq x 7 x S I σ = ω = 1, A f ( ) and he sabili check agains laeral-orsional buckling is no saisfied. = MPa fad = 35., (4.86) Dr. Enzo MARTINELLI 76 Draf Version 18/1/01

19 Laeral-orsional sabili check according o he EC3 provisions. The same exercise can be also faced wihin he framework of he EC3 provisions which can be applied following sep-b-sep he formulae repored in secion Sep #1: classifing he ransverse secion: Since he adoped seel grade is f =35 MPa, he value ε=1 can be assumed for he parameer menioned in Table.3 and Table.4. The following values of he lengh-o-hickness raios can be evaluaed for flange and web: - flange c/ f =(135/)/10.= Class 1; - web d/ w =( )/6.6= Class 1 Finall, he profile IPE70 made ou of seel S35 is in class 1 if loaded in bending. Sep #: evaluaing he criical momen for he elasic beam: According o he assumed hpoheses on he ransverse secion and he end resrain condiions, he following values can be assumed for he parameers involved in equaion (4.77) which can be properl considered because he ransverse load is applied of he shear cenre (namel, he cenroid G) of he ransverse secion: - C 1 =0.97; - k=0.5; - k W =1.0; and he criical momen can be derived b he menioned relaionship: 10 π ( ) M = + cr 0.97 = ( ) π = Nmm = 14.41kNm, (4.87) Sep #3: evaluaing he non dimensional laeral slenderness: According o he definiion provided in equaion (4.74), he value of he non-dimensional slenderness can be easil quanified: βw Wpl, fa λ LT = =.956. (4.88) 6 M cr Sep #4: evaluaing he reducion facor χ LT : The erm Φ LT has o be firsl deermined as a funcion of he non-dimensional slenderness and he imperfecion coefficien α LT.1: ( ) Φ LT.5 1+ αlt λlt 0. + λ LT =. (4.89) = ( ) = Consequenl, he following value of he reducion facor can be derived according o equaion (4.7): 1 χ LT =.696, (4.90) Sep #5: evaluaing design value of he resising momen M b,rd : The bending momen resuling in he occurrence of he laeral-orsional buckling phenomenon can be easil evaluaed as a design value according o equaion (4.71): fa 35 Mb,Rd = χltβ wwpl, = γ (4.91) 6 M1 = Nmm = knm Dr. Enzo MARTINELLI 77 Draf Version 18/1/01

20 which is greaer han he corresponding exernal momen MSd = Mmax = 99.4 knm. Consequenl, he laeral-orsional sabili check is no saisfied neiher according o he EC3 provisions. 4.5 Unworked examples The following exercises are lef o he readers: 1) for he same beam repored in paragraph , wha is he maximum value of he live loads q k for he beam o compl wih he laeral-orsional sabili check according o he Ialian Code provisions?; ) for he same beam repored in paragraph , wha is he maximum span lengh for he beam o compl wih he laeral-orsional sabili check according o he Ialian Code provisions?; 3) for he same beam repored in paragraph , design he ransverse secion o compl wih he laeral-orsional sabili check according o he Ialian Code provisions; 4) for he same beam repored in paragraph , evaluae he resising momen M b,rd agains he laeral-orsional buckling phenomenon considering ha he same loads acs eiher a he boom (case 1) or he op (case ) of he secion; 5) for he same beam repored in paragraph , perform he same laeral sabili check wih reference o a seel grade S355; 6) for he same beam repored in paragraph , perform he same laeral sabili check considering a canilever scheme wih lengh L=3.00 m. Dr. Enzo MARTINELLI 78 Draf Version 18/1/01