BUCKLING OF FRP BEAMS AND COLUMNS

Transcription

1 BUCKLIG OF FRP BEMS D COLUMS Lásló Kollár n Ákos Spkás Bupes Universi o Technolog n Economics, Deprmens o Mechnics, Merils n Srucures BSTRCT The pper presens he sbili nlsis o hin lle n close oss secion orhoropic composie bems subjece o xil or rnsverse los. In he nlsis boh he rnsverse sher n he resrine rping inuce sher eormions re ken ino ccoun. Explici expressions re presene or he buckling lo o xill loe composie columns n or he lerl orsionl buckling lo o rnsversel loe composie bems. Simple expressions re lso presene o eermine he pproxime reucion in he buckling lo ue o sher eormions. The locl buckling nlsis o composie bems is lso iscusse.. ITRODUCTIO Pulrue iber reinorce plsic (FRP) hin-lle members re imporn srucurl elemens. Since he siness o he composie is relivel lo, buckling is mjor consierion in esign. Slener members usull buckle globll []. The shorer he member n he hinner he lls, he more likel h locl buckling ill occur irs. For inermeie-spn members inercion beeen locl n globl buckling moes is n imporn phenomenon []. In bem or column he xil sresses re he highes, hence mos o he ibers re oriene in he xil irecion. Consequenl, he siness o he lls perpeniculr o he xis o he bem n he sher siness re signiicnl loer hen he xil siness. This hs o consequences: he locl buckling o he segmens is ver imporn in he esign o composie members n he sher eormion pls n imporn role in he globl buckling nlsis [,, 5].. PROBLEM STTEMET We consier n close hin-lle secion prismic members consising o l recngulr ll segmens. The lup o ech ll segmen is orhoropic (hich is he cse o pulrue members). The meril n he lup m be ieren in ech ll segmen, bu mus be uniorm long he ih o ech ll segmen o he member. The oss secion o he bem hs one xis o smmer. The ens o he composie member m be simpl suppore, buil in or ree. The composie bem m be subjece o n xil lo (Fig. ) or o rnsverse lo shon in Fig. b o e. We ssume h he meril behves in linerl elsic mnner n he eormions re smll. We ill eermine he los hich resul in globl or locl buckling o he member. When column is subjece o n xil lo i m resul in globl lexurl-orsionl buckling, hich, or oubl smmericl oss secions simpli o o in-plne buckling n o pure orsionl buckling (Fig. ). This ill be ree in Secion. When he bem is subjece o rnsverse lo i m resul in lerl-orsionl buckling (Fig. b), hich ill be ree in Secion 4. Boh xil lo n rnsverse lo (i.e. bening) inrouce xil compressive sresses, hich m cuse locl buckling in hin-lle members. The locl buckling ill be iscusse in Secion 5.

2 () (b) M M (c) q () P (e) q () P Fig. : The ieren lo cses n suppor coniions. BUCKLIG OF COMPOSITE COLUMS SUBJECTED TO XIL LODS Buckling o secion bems s ree in eil in [6], n onl he inl resuls re presene belo. For n secion composie bem he bening sinesses ( EI n EI ), he orsionl siness ( GI ), n he rping siness ( EI ) cn be eine s given in [5].

3 ) b) q x à x x x Fig. : Buckling moes o xil lo (), n lerl-orsionl buckling (b) One o he mjor ierences beeen n isoropic n composie bem is h in he ler cse he eec o sher eormions mus be ken ino ccoun. The sher sinesses ( S ij ; i, j =,, ) o ieren oss secion bems re lso given in [5]. oe h in orsion he sher eormions m pl n imporn role, hich is represene b he rping sher siness ( S ). When buckling occurs in he x- or x- plnes he buckling los re π = EI + S ( kl), π = EI + S ( kl) () When pure orsionl buckling occurs bou he sher cener he buckling lo is π ψ = EI S GI + + i ( kl) i i () here i is he polr rius o grion, n kl is he eecive lengh, here k =,.5, or simpl suppore, buil in n or cnilever columns, respecivel. These los re ienicl o he buckling los o bems ih oubl smmericl oss secions. When he secion is monosmmericl n is he xis o smmer he buckling lo cn be clcule b he expressions given in [6]. s n pproximion e m neglec he eec o he oss erm ( S ) in he nlsis n obin n expression n secon orer equion or he buckling lo = :, ( i ) ( + ψ ) i + ψ i =, () sc here, n ψ re eine b Eqs. () n (). In [5] onl secion bems ere consiere. For isoropic close secion bems, s rule, he eec o resrine rping is negligible. Hoever, or composie bems, hen he sinesses o he jcen lls re signiicnl ieren, he eec o rping sher eormions cnno be neglece. Here e presen n engineering pproch o obin pproximel he buckling lo. We clcule irs he buckling lo b o conservive mehos, Meho n Meho B (Fig. ). In Meho e neglec he rping siness o he oss secion hus e hve close GI, EI =, S =. (4)

4 GI close G h G h G h G h { neglece EI = cus S = GI op en EI = S = lls Fig. : Propose pproxime clculion meho or close oss secions In Meho B e ssume one or more cus in he longiuinl irecion in he ll(s) (or even e neglec one or more lls, Fig. ) n hence e obin n secion bem ih sinesses GI, EI, S, (5) close here GI, GI, EI n S re eermine in ppenix. Firs e clcule he iicl lo ih sinesses given b Eq.(4) hen ih he sinesses given b Eq.(5). The iicl los re enoe b n B, respecivel. Boh re conservive esimes, hence e m pproxime he buckling lo b B ( c ) r, = mx. (6) 4. BUCKLIG OF COMPOSITE BEMS SUBJECTED TO TRSVERSE LODS When monosmmericl secion bems re loe in he plne o smmer cerin level o he pplie lo he bem m buckle lerll, hile he oss secions o he bem roe simulneousl bou he bem s xis. This phenomenon is clle lerl-orsionl buckling. I e neglec he eec o he oss erm ( S ) in he nlsis, he buckling lo o simpl suppore or cnilever composie bems cn be clcule in he orm o [] M ψ = C + + ( + ) + C Cβ C Cβ i, (7) C C here,, C re consns, n M is he iicl vlue o he mximum bening momen, hich is rele is o he los (see Tble ). n ψ re clcule ccoring o Eqs. () n (). The vlue o prmeer k epens on he en coniions o he bem n i is given lso in Tble. The eile nlsis cn be oun in []. In cse o close secion bems he sme soluion meho cn be pplie o obin he buckling lo ( ) s i s inrouce or he xill loe columns: or Meho he M

5 Tble : Prmeers in Eq.(7) C C C Simpl suppore bem (k=) En momens (Fig. b).5 Uniorml isribue lo M = ql / 8 (Fig. c) Concenre lo he mispn M = P l / 4 (Fig. ) Cnilever bem (k=) Disribue lo M = ql / (Fig. e).5 n. n. Concenre orce he en M = P l (Fig. ).8 n. n. rping siness is neglece ( EI = ), n e clcule M, hile in Meho B he B M. orsionl siness ( GI ) o n e secion is ken ino ccoun, hich resuls in The greer o hese is he buckling lo B ( M ) M M mx, =. (8) 5. LOCL BUCKLIG Locl buckling nlsis o hin-lle FRP composie members is o gre prcicl impornce n is ell explore re [, 7, 8,,, 4]. In mos o he cses he ormulions re rher complice; he soluions re no usull in simple orm. Explici expressions cn be erive b moelling he ll segmens s orhoropic ples n b ssuming h eges common o o or more ples remin srigh. Then he buckling lo is eermine, b consiering he ll segmens s iniviul ples, hich re elsicll resrine b he jcen lls [4]. Here e o no presen he locl buckling nlsis o composie members, rher e reer o [7, 5], here explici expressions re presene or he buckling lo. 6. UMERICL EXMPLES umericl exmples or buckling o secion composie columns n bems cn be oun in [5] n [] hereore e onl presen exmples or close secion members. We consier bem me o uniirecionl crbon iber reinorce epox. The Young moulus in he iber irecion is E = 48 k/mm n he sher moulus is G = k/mm. The oss secion o he bem is given in Fig. 4. The hickness o he op n he boom lls is 5 mm, hile he hickness o he vericl lls is mm. The bem is simpl suppore he ens, is lengh is l =6 mm. The bem subjece o xil orces he ens. We eermine he iicl lo ( ψ ) belo. The bem is oubl smmeric. The nonero sinesses ere clcule ccoring o he expressions given in ppenix. The oss-secionl properies re presene in Fig. 4. Firs e clcule he orsionl buckling lo ccoring o Meho. In his cse e consier close-secion n neglec he rping siness ( EI ) n he rping sher siness S ) o he bem. The orsionl siness ( GI ( close ) is clcule b Eq.(.6) n is given in

6 mm 5 mm mm C SC 5 mm 4 mm mm 6 mm mm EI 7.7 x 6 EI 5.6 x 9 GI 9.86 x close GI 6.84 x 5 S S.85 x i 7 Fig. 4: Cross secion n sinesses o he close secion bem Fig. 4. Wih he bove esibe sinesses Eq.() gives ψ = 5. k. ex he orsionl buckling lo ccoring o Meho B is clcule. In his cse he originll close-secion is e ih our cus he eges. The rping siness ( EI ) n he rping sher siness ( S ccoring o Eqs.(.8) n (.9), n he orsionl siness ( GI Eq.(.7). Wih hese sinesses Eq. () gives B ψ = 4.7 k. The clcule nlicl buckling lo is he higher vlue ) o he oss secion re clcule B ( ψ, ψ ) 5. ψ = mx = k. ) is clcule b The sme bem s lso invesige ih he i i inie elemen progrm SYS. Four noe elsic shell elemens ere use ih he mximum elemen sie o.5 mm. The resul is SYS ψ = 98. k. We lso invesige he eec o he lengh o he bem; he buckling lo versus he bem lengh is shon in Fig. 5, op. We lso presen he resuls or bems subjece o rnsverse los (Fig. 5, boom). The clcule buckling los gree surprisingl ell ih he resuls o numericl clculions perorme b he FE progrm. 7. COCLUSIO In he pper e presene simple expressions or he globl buckling lo o composie members, king he eec o sher eormion ino ccoun. The shorer he bem he more imporn is he sher eec.

7 . [km] Meho (close) Meho B () SYS Lengh [mm] 5. M [km] 4... Meho (close) Meho B () SYS Lengh [mm] Fig. 5: Criicl los o oubl smmeric simpl suppore close secion bems The quesion rises, hen he eec o sher eormions mus be ken ino ccoun. We inrouce prmeer α hich is he reucion in he buckling lo ue o he sher eormion. For n I or box bem, hich buckles bou he (horionl) xis, α cn be pproxime b [9] (see Fig. 6) α = =, (9) l S l + + π EI π b ( ) ( ) here n re eine b Eq.(.), subsip n reer o he lnge n he eb, respecivel.

8 ) Correcion cor, b) Correcion cor, ( ( 4 ) ) isoropic 5 =.5 isoropic ( ( 4 ) = 4 ) Fig. 6: The correcion cors in Eqs.(9) n () l b b l b b For n I-bem, hich buckles bou he (vericl) xis or orsionll [] i cn be pproxime b (see Fig. 6b) α = =. () l S l + + π EI π b ( ) ( ) oe h even or slener composie bems he eec o sher eormions is signiicn. CKOWLEDGEMET This ork s suppore b he Hungrin Science Founion (OTK no. T5) hich is highl pprecie. PPEDIX. ELSTIC PROPERTIES OF BOX BEM In he olloing e presen he elsic properies o hin-lle recngulr close-secion bem illusre in Fig. 7. We ssume h he lups o he ll re smmericl n orhoropic. The ensile siness ( E ) n he bening sinesses ( EI n EI ) re E b b = +, ( ) ( ) EI b b = + +, (.) ( ) ( ) ( ) b

9 b C SC c b Fig. 7: Cross secion o hin-lle recngulr close secion bem EI b b b = + +, (.) ( ) ( ) ( ) here subsips n reer o he lnges n he ebs, respecivel. ij n ij re he elemens o he complince mrices o he lmine, n re clcule s = D D, = D D (.) D here n D re he elemens o he siness mrices o lmine n mus be ij ij clcule or he op lnges ( ), n or he eb ( ). The lerl sher siness is S =, (.4). ( ) n he polr rius o grion o he oss secion bou he sher cener is clcule s EI EI i =. (.5) E The orsionl siness in Meho or close secion bem is s ollos GI + =. (.6) close ( ) + ( ) In Meho B e cu he close secion ino o prs (o ebs n o lnges ) he corners (see Fig. ). The sepre prs re o oubl smmeric I-secions hich ebs re missing. The orsionl siness is clcule s GI ( ) ( ) = 8 +, (.7)

10 hile he rping siness is. (.8) 4 EI = + ( ) 4( ) Boh he ebs n he lnges re oubl smmeric, hence S n ( ) ( ) S = +. (.9).. When he lnges n he eb re me o single orhoropic ler he expressions o,, n (Eq. B.) simpli o =, E h =, =, G h E h = G h, here E is he Young moulus in he irecion o he bem s xis, G is he in-plne sher moulus, n h is he hickness o he lmine. Reerences [] Bnk, L.C., Sher Coeiciens or Thin-lle Composie Bems, Composie Srucures, Vol. 8. (987), [] Brbero, E. J., Preicion o Buckling - Moe Inercion in Composie Columns, Mechnics o Composie Merils n Srucures, Vol. 7 (), [] Brbero, E. J. n Roinnis, I., Locl Buckling o FRP Bems n Columns, Journl o Merils in Civil Engineering, Vol. 5() (99), [4] Bleich, F., Buckling o Mel Srucures, McGr-Hill, e York, (95) [5] Kollár, L.P. n Springer, G. S., Mechnics o Composie Srucures, Cmbrige Universi Press, () [6] Kollár, L.P, Flexurl-orsionl Buckling o Open Secion Composie Columns ih Sher Deormion, Inernionl Journl o Solis n Srucures, 8 (), [7] Kollár, L.P, Locl Buckling o FRP Composie Srucurl Members ih Open n Close Cross Secions, Journl o Srucurl Engineering, Vol. 9 (), 5-5. [8] Lee, D. J., The Locl Buckling Coeicien or Orhoropic Srucurl Secions, eronuicl Journl, Vol. 8(7) (978), -. [9] Plusik,, n Kollár L.P., Eec o Sher Deormion n Resrine Wrping on he Displcemens o Composie Bems, Journl o Reinorce Plsics n Composies, Vol. (), [] Qio, P., Dvlos, J.F. n Wng, J., Locl Buckling o Composie FRP Shpes b Disee Ple nlsis, SCE Journl o Srucurl Engineering, 7 (), [] Spkás, Á n Kollár L.P., Lerl-orsionl Buckling o Composie Bems ih Sher Deormion, Inernionl Journl o Solis n Srucures, 9 (), [] Webber, J.P.H, Hol, P.T, n Lee, D.., Insbili o Crbon Fibre Reinorce Flnges o I secion Bems n Columns, Composie Srucures, Vol. 4 (985), [] Zureick,. n Sco, D.W., Shor-Term Behvior n Design o Fiber-Reinorce Polmeric Slener Members uner xil Compression, Journl o Composies or Consrucion, Vol. (4) (997), [4] Zureick,. n Shih, B., Locl Buckling o Fiber-Reinorce Polmeric Srucurl Members uner Linerl-Vring Ege Loing - Pr. Theoreicl Formulion, Composie Srucures, Vol. 4(4) (998),