Lateral torsional buckling of rectangular beams using variational iteration method

Transcription

1 Scieific Research ad Essas Vol. 6(6), pp , 8 March, Available olie a hp:// ISSN Academic Jourals Full egh Research Paper aeral orsioal bucklig of recagular beams usig variaioal ieraio mehod Seval Piarbasi Deparme of Civil Egieerig, Kocaeli Uiversi, Umuepe Campus, 438, Kocaeli, Turke. sevalp@gmail.com. Tel: Acceped 5 March, aeral orsioal bucklig is he mai failure mode ha corols he desig of sleder beams; ha is, he beams which have greaer major ais bedig siffess ha mior ais bedig siffess or he beams which have cosiderabl large laerall usuppored leghs. Sice he bucklig equaios for beams are usuall much more comple ha hose for colums, mos of he aalical sudies i lieraure o beam bucklig are coceraed o simple cases. This paper shows ha comple beam bucklig problems, such as laeral orsioal bucklig of arrow recagular cailever beams whose mior ais fleural ad orsioal rigidiies var epoeiall alog heir leghs, ca successfull be solved usig variaioal ieraio mehod (VIM). The paper also ivesigaes he effeciveess of hree VIM algorihms, wo of which have bee proposed ver recel i solvig laeral orsioal bucklig equaios. Aalsis resuls show ha all ieraio algorihms ield eacl he same resuls i all sudied problems. As far as he compuaio imes ad spaces are cocered, however, oe of hese algorihms, called variaioal ieraio algorihm II, is foud o be superior ha he ohers especiall i laeral orsioal bucklig problems where he beam rigidiies var alog he beam legh. Ke words: aeral orsioal bucklig, arrow recagular beam, apered beam, variable rigidi, variaioal ieraio algorihms, variaioal ieraio mehod. INTRODUCTION Beams are mai srucural elemes which primaril suppor rasverse loads, ha is, loads perpedicular o heir aes. The are also called fleural members sice he are picall desiged o carr fleural loads, ha is, rasverse shear forces ad bedig momes. As he ormal force i a beam is geerall egligibl small, he desig of a laerall-braced beam is usuall sraighforward: selec he mos ecoomical compac cross secio which has adequae major ais secio modulus o safel carr maimum bedig mome i he beam wih adequae web area o carr maimum shear force. However, similar o a sleder colum which buckles uder compressive loads, a laerall-ubraced sleder beam ca also buckle uder fleural loads. This occurs due o he fac ha whe subjeced o bedig momes, compressive sresses develop a oe par of he beam Abbreviaios: VIM, Variaioal ieraio mehod; VIA, variaioal ieraio algorihm. cross secio which eds o buckle i laeral (ou-of plae) direcio. I such a bucklig mode, he beam o ol displaces laerall bu also wiss due o he fac ha he remaiig par of he beam cross secio, which is subjeced o esile sresses, resiss agais bucklig. For his reaso, his bucklig mode is commol called laeral orsioal bucklig. aeral orsioal bucklig, or simpl laeral bucklig, is he mai limi sae ha has o be cosidered i he desig of sleder beams, ha is, he beams which have greaer siffess i i-plae (major ais) bedig ha i ou-of-plae (mior ais) bedig or he beams which have cosiderabl large laerall usuppored leghs. Uless properl braced agais laeral deflecio ad/or orsio, a sleder beam will buckle prior o he aaime of is major ais bedig capaci. Sice deermiig he laeral orsioal bucklig load or mome of a sleder beam is crucial i heir desig, ma sudies have bee coduced o beam bucklig. However, due o he fac ha he beam bucklig equaios are much more comple ha he colum bucklig equaios, mos of he aalical sudies i

2 446 Sci. Res. Essas lieraure are coceraed o simple cases. The soluios for simple bucklig problems ca be obaied from wellkow srucural sabili books, such as Timosheko ad Gere (96), Chajes (974), Wag e al. (5) ad Simises ad Hodges (6). I rece ears, he adve of compuer-aided umerical echiques has eabled he researchers o obai soluios for more comple laeral bucklig problems. The variaioal ieraio mehod (VIM) is a kid of oliear aalical echique which was proposed b He (999) ad developed full i he followig ear. The echique was successfull applied o various kids of oliear problems (He,, 7; He e al., 7) sice 999. Accordig o He e al. (), he umber of publicaios o VIM has reached 3 i Ocober, 9, which clearl verifies he effeciveess of he echique i solvig oliear problems. Ver recel, VIM is also applied o he bucklig problems. Cosku ad Aa (9), Aa ad Cosku (9), Cosku () ad Oka e al. () aaled he elasic sabili of Euler colums wih variable cross secios uder differe loadig ad boudar codiios ad verified ha VIM is a ver efficie ad powerful mehod i aalsis of bucklig problems of colums wih variable cross secios. I his paper, his powerful aalical echique is applied o wo fudameal beam bucklig problems: laeral bucklig of (a) simpl suppored arrow recagular beams wih variable rigidiies uder uiform mome ad (b) arrow recagular cailever beams wih variable rigidiies carrig coceraed load a heir free eds. Boh liear ad epoeial variaios are cosidered i mior ais fleural ad orsioal rigidiies of he beams. Eac soluios o hese problems, some of which are cosiderabl comple, are available i lieraure ol for beams of cosa rigidiies ad some paricular cases of liearl apered beams. To verif he effeciveess of VIM i solvig laeral bucklig problems, bucklig loads/ momes for uiform beams wih cosa rigidiies are sudied firs. The, he ouiform cases are sudied for each problem separael. I he paper, he effeciveess of he hree VIM algorihms, wo of which have recel bee proposed b He e al. (), i solvig laeral orsioal bucklig equaios is also ivesigaed. ATERA TORSIONA BUCKING OF BEAMS I his secio of he paper, firs, he basic laeral bucklig heor as give i Timosheko ad Gere (96) will be summaried ver shorl ad he oaio ha will be used i he sud will be iroduced. The, he differeial equaios for laeral orsioal bucklig of beams wih wo fudameal loadig ad resrai codiios will be derived. Geeral beam bucklig equaios I order o derive geeral laeral orsioal bucklig equaios for arrow recagular beams, we firs cosider a recagular beam subjeced o arbirar loadig i - plae causig is srog-ais bedig. The fied,, coordiae ssem defies he udeformed cofiguraio of he beam, as show i Figure a. Similarl, ξ,,, coordiae ssem locaed a he ceroid of he cross secio a a arbirar secio of he beam alog is legh defies he deformed cofiguraio of he beam. As show i Figure b, he deformaio of he beam ca be defied b laeral (u) ad verical (v) displacemes of he ceroid of he beam, which are posiive i he posiive direcios of ad, respecivel, ad agle of wis () of he cross secio, which is posiive abou posiive ais, obeig he righ had rule. Hece, he displacemes illusraed i Figure b, are boh egaive. For small deformaios, he curvaures i ad plaes ca be ake, respecivel, as d u/d ad d v/d ad he cosies of he agles bewee he aes ca be ake as lised i Table. Sice he warpig rigidi of a arrow recagular beam ca realisicall be ake as ero, usig posiive direcios of ieral momes defied i Figure, he equilibrium equaios for he buckled beam ca be wrie as follows: EI d v ξ d M, ξ EI d u d d M ad GI φ M () ζ d represeig he major ais bedig, mior ais bedig ad wisig of he beam, respecivel. I Equaio, EI ξ ad EI deoe he srog ais ad weak ais fleural rigidiies of he beam, respecivel. Similarl, GI deoe he orsioal rigidi of he beam. aeral bucklig of simpl suppored recagular beams i pure bedig Cosider a simpl suppored recagular beam wih variable GI alog is legh (Figure 3). rigidiies ( ) ξ EI, EI ( ) ad ( ) If he beam is subjeced o equal ed momes M o abou -ais, he bedig ad wisig momes a a cross secio ca be foud b deermiig he compoes of M o abou ξ, σ, aes. Cosiderig he sig coveio defied i Figure, ad usig Table, hese compoes ca be wrie as: M ξ M, du o M φmo ad Mζ Mo d The, he equilibrium equaios for he buckled beam (Figure b) become d v ( ) o, ( ) EI ξ M d dφ du GI ( ) M d d o () d u φ o ad EI M d I is appare from Equaio 3 ha v is idepede from u ad, which are coupled. Thus, i his problem, i is sufficie o cosider ol he coupled equaios. Differeiaig he las equaio i Equaio 3 wih respec o ad usig he resulig equaio o elimiae u i he secod equaio i Equaio 3, he followig differeial equaio for he agle of wis of he beam is obaied: ( ) d φ dgi dφ Mo + + φ d d GI ( ) d GI ( ) EI ( ) (3) (4)

3 Piarbasi 447 m m Side View Secio m- Top View (a) -u -v C ζ C Side View C ξ -v ξ -u C φ ζ Top View Secio m- (b) Figure. (a) Udeformed ad (b) buckled shapes of a double smmeric beam loaded o bed abou is major ais. Sice he eds of he simpl suppored beam are resraied agais roaio abou ais, he boudar codiios for he problem are ξ a boh ad. aeral bucklig of recagular cailever beams wih verical ed load Cosider a arrow recagular cailever beam of legh wih variable rigidiies EIξ ( ), EI ( ) ad GI ( ). If he beam is subjeced o a verical load P passig hrough is ceroid a is free ed, as show i Figure 4, he compoes of he momes of he load a a arbirar secio m- abou,, aes are 3 ( ) M P, M ad M P ( u u) + (5) where u is he laeral displaceme of he loaded ed of he beam as show i Figure 4b. Cosiderig he sig coveio defied i Figure, he bedig ad wisig momes a his arbirar secio ca be wrie as M ξ P ( ), M P ( ) φ du Mζ P P u u d ( ) ( ) ad d v (6) ( ) ( ) EI ξ P d The, he equilibrium equaios for he buckled beam become

4 448 Sci. Res. Essas Μ ξ Μ Sice he fied ed of he beam is resraied agais roaio ad sice he wisig mome a he free ed is ero, he boudar codiios for his problem are a ad d /d a. Μ ζ Μ ξ Μ Figure. Posiive direcios for ieral momes. Μ ζ VIM FORMUATIONS FOR THE BUCKING PROBEMS I a rece paper, He e al. () proposed hree variaioal ieraio algorihms for solvig various pes of differeial equaios. The firs algorihm is he classical VIM algorihm defied i He (999). For a geeral homogeeous oliear differeial equaio, ( ) φ ( ) φ + N (9) M o m M o Where is a liear operaor ad N is a oliear operaor, ad he correcio fucioal is ( ) ( ) ( ) ( ) ( ) φ+ φ + λ ξ φ ξ + Nφ ξ d ξ () I Equaio, λ ( ξ ) is a geeral agrage muliplier ha ca be ideified opimall via variaioal heor, φ is he -h Secio m- Figure 3. Simpl suppored recagular beam uder pure bedig. approimae soluio ad φ δφ deoes a resriced variaio, ha is,. The ieraio algorihm i origial VIM, called variaioal ieraio algorihm I (VIA I), is as follows: ( ) ( ) ( ){ ( ) ( )} φ+ φ + λ ξ φ ξ + Nφ ξ d ξ () Table. Cosie of agles bewee aes (Timosheko ad Gere, 96). ξ -du/d - -dv/d du/d dv/d d u ad ( ) φ ( ) EI P d dφ du GI P P u u d d ( ) ( ) ( ) Similar o he pure bedig case, v is idepede from u ad. Differeiaig he las equaio i Equaio 7 wih respec o ad usig he resulig equaio o elimiae u i he secod equaio i Equaio 7, he followig differeial equaio is obaied for : ( ) d φ dgi dφ P + + ( ) φ d d GI ( ) d GI ( ) EI ( ) (7) (8) 4 I he secod algorihm proposed i He e al. (), called variaioal ieraio algorihm II (VIA II), he ieraio formula is much simpler: + ( ) ( ) + ( ) ( ) φ φ λ ξ Nφ ξ d ξ () The mai shorcomig of his algorihm is saed o be he requireme ha φ be seleced o saisf he iiial/boudar codiios. The hird algorihm i He e al. (), called variaioal ieraio algorihm III (VIA III), has he followig ieraio formula: ( ) ( ) + ( ) ( ) ( ) φ φ λ ξ Nφ ξ Nφ ξ d ξ (3) As summaried i He e al. (), for a secod order differeial equaio such as he equaios of he problems cosidered i his paper, ha is, Equaio 4 ad Equaio 8, λ ( ξ ) simpl equals o λ ( ξ ) ξ (4) Thus, he hree VIM ieraio algorihms for Equaio 4 are:

5 Piarbasi 449 m P Secio m- (a) -u -v Side View ζ C ξ C -v φ -u ζ -u Secio m- Top View (b) Figure 4. (a) Udeformed ad (b) buckled shapes of a arrow recagular cailever beam carrig coceraed load a is free ed. ( ) ( ) + ( ) ( ) ( ) + ( ) + ( ) ( ) φ φ ξ φ ξ F ξ φ ξ FM ξ φ ξ d ξ + { } φ ( ) φ ( ) ( ξ ) ( ξ ) φ + + F ( ξ ) + FM ( ξ ) φ ( ξ ) d ξ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) φ+ φ+ + ξ F ξ φ+ ξ φ ξ + FM ξ φ+ ξ φ ξ d ξ (5) ad hose for Equaio 8 are φ ( ) φ ( ) ( ξ ) φ ( ξ ) ( ξ ) φ F ( ξ ) + FP ( ξ )( ξ ) φ ( ξ ) d ξ ( ) ( ) + ( ) ( ) ( ) + ( )( ) ( ) φ φ ξ F ξ φ ξ FP ξ ξ φ ξ d ξ + φ ( ) φ ( ) ( ξ ) ( ξ ) φ ( ξ ) φ ( ξ ) F + + FP ( ξ )( ξ ) φ + ( ξ ) φ ( ξ ) d ξ (6) I Equaio 5 ad Equaio 6, 5

6 45 Sci. Res. Essas Table. Normalied bucklig momes () for he firs hree modes cosa rigidiies. Mode # VIA I VIA II VIA III Eac ( ), ( ) GI F ( ) GI Mo FM ( ) GI ( ) ( ) EI ad P FP ( ) GI EI ( ) ( ) (7) ad prime deoes he differeiaio of fucios wih respec o heir variables. ANAYSIS OF RESUTS Criical mome for pure bedig case Beams wih cosa rigidiies If he mior ais fleural ad orsioal rigidiies of he EI EI ad GI ( ) GI, beam are cosa; ha is, ( ) he Equaio 4 reduces o he followig simpler equaio: Eac values for he secod ad hird modes ca be obaied b deermiig larger roos of Equaio, which leads o four ad ie imes he firs mode criical mome, respecivel. I order o compare he efficiec of he hree variaioal ieraio algorihms meioed earlier, his case of he problem is solved usig all hree VIM algorihms. For easier compuaios, he odimesioal form of he differeial equaio ad he relaed boudar codiios are wrie: d φ + αφ d where, φ ( ) Mo α () wih φ ( ) ad ad is he odimesioal criical mome. For all hree algorihms, he iiial approimaio is chose as a liear fucio wih ukow coefficies as follows: where /, φ φ φ A + B (3) d φ + λ φ d Where Mo λ The soluio of Equaio 8 is i he form ( ) cos( ) (8) φ C si λ + C λ (9) where C ad C are cosas o be deermied from boudar codiios. Whe he relaed boudar codiios are used, Equaio 9 leads o he followig characerisic equaio ( ) si λ () whose smalles roo ields he firs mode criical mome M cr as M cr π () 6 The coefficies A ad B are deermied b imposig he boudar codiios o he approimae soluio obaied a he ed of he ieraios, which leads o a characerisic equaio whose roos give he bucklig momes i differe modes. I order o see he covergece of he approimae soluios o he eac soluios, seveee ieraios are coduced for each algorihm ad criical momes for he firs hree modes are compued. I is also worh oig ha he fucio give i Equaio 3 cao saisf boh of he boudar codiios simulaeousl. I is surprisig ha for his special case of he beam bucklig problem, all ieraio algorihms ield eacl he same resuls for he ormalied bucklig momes, as lised i Table, where eac resuls are also abulaed. Variaio of perce errors wih ieraios ploed i Figure 5 shows how VIM soluios coverge o he eac soluio. Sice all hree VIM algorihms give ideical resuls i all ieraios, ol oe plo is preseed i Figure 5. As show i Table, VIM soluios are i ecelle agreeme wih he eac resuls. I is sufficie o perform ol five ieraios o obai he eac value of he smalles criical mome. For higher mode values, he umber of ieraios has o be icreased (Figure 5). The ecelle mach of VIM soluios wih he eac resuls verifies ha VIM is a powerful echique i predicig bucklig momes of simpl suppored recagular beams wih cosa rigidiies uder uiform mome ad also ecourages is use i much more comple beam

7 Piarbasi 45 Error (%) Mode Mode Mode Ieraio Number Figure 5. Variaio of perce errors i VIM ieraios for bucklig momes of he firs hree modes - cosa rigidiies. bucklig problems, like laeral bucklig of beams wih variable rigidiies. Beams wih liearl varig rigidiies ( ( )) si kl + b (7) The smalles roo of which ields he firs mode criical mome M cr as If boh he mior ais fleural ad orsioal rigidiies of he beam chages i liear form, ha is, if GI ( ) GI + b ad ( ) EI EI + b (4) where b is a cosa deermiig he sharpess of he chages i rigidiies alog he legh of he beam, he, he bucklig equaio give i Equaio 4 akes he followig form: d φ b dφ M d b d b o + + φ + + ( / ) (5) whose eac soluio (Wag e al., 5) is i he followig form: ( ) ( ) φ C si k l + C cos k l (6) where, + b / ad k Mo b Whe he relaed boudar codiios are used, he characerisic equaio is obaied as follows: 7 M cr πb l + ( b) (8) Eac values for he secod ad hird mode criical momes ca be obaied b deermiig he larger roos of he characerisic equaio, which leads o four ad ie imes he firs mode value, respecivel. The odimesioal form of Equaio 5 ca be wrie as: d φ b dφ + + α φ d + b d + b ( ) where /, φ φ (9) ad is he odimesioal criical mome, as defied i Equaio. Sice i he case of cosa rigidiies, all VIM algorihms lead o he same resuls, he effeciveess of each algorihm is ivesigaed furher i his problem. For his purpose, usig all VIM algorihms, criical momes for he firs hree modes are compued for b.3. I order o see he covergece of he approimae soluios o he eac soluios, hiree ieraios are coduced for each algorihm. Similar o he case of cosa rigidiies, he ieraios are sared wih he liear approimaio give i Equaio 3. To simplif he iegraio processes, variable coefficies i he ieraio iegrals are

8 45 Sci. Res. Essas Table 3. Normalied bucklig momes () for he firs hree modes liearl varig rigidiies (b.3). Mode # VIA I VIA II VIA III Eac Error (%) Mode Mode Mode Ieraio Number Figure 6. Variaio of perce errors wih ieraios for he firs hree bucklig loads liearl varig rigidiies (b.3). epaded i series usig ie erms. Similar o he case of cosa rigidiies, he differe VIM algorihms lead o he same resuls for bucklig momes whe b.3, as lised i Table 3. Variaio of perce errors wih ieraios are also ploed i Figure 6 o show how VIM soluios coverge o he eac soluio. Sice all hree VIM algorihms give ideical resuls, ol oe plo is preseed. As show i Table 3, he agreeme bewee he VIM resuls ad he eac resuls is cosiderabl good. Eve hough all VIM algorihms ield eacl he same resuls, some impora differeces are observed bewee he ieraio algorihms i his case as far as he compuaio ime ad space are cocered. VIA II is observed o complee he same umber of ieraios i much smaller amou of ime ha VIA I ad VIA III, for which he compuaio imes are almos ideical. VIA II is superior o VIA I ad VIA III also i ha, is oupu file akes up less space. The sies of he oupu files creaed b VIA I ad VIA III are almos four imes larger ha ha b VIA II. Thus, i ca be cocluded ha VIA II is more effecive i solvig his kid of differeial equaios ha he oher wo ieraio algorihms. Such differeces i compuaio ime ad space ma occur due o he fac ha VIA I ecessiaes, i each ieraio, secod order differeiaio of he soluio obaied i he previous ieraio ad VIA III uses, i each ieraio, he precedig wo ieraio resuls. Furhermore, i VIA I ad VIA III, he iegral erm has o be added o he soluio obaied i he previous ieraio whereas i is added o he iiial approimaio i VIA II. Usig VIA II, he ormalied bucklig momes for oher values of b are also compued ad ploed i Figure 7, where eac resuls are also show. As i is see from he figure, VIM soluios ad eac resuls are i ver good agreeme. I is worh oig ha he small differeces bewee he resuls as b icreases occurs due o he fac ha i is ecessar o epad he variable coefficies i he ieraio iegrals i series usig more erms whe b is close o oe. As he umber of erms i series is icreased, VIM resuls are observed o coverge o eac resuls. Beams wih epoeiall varig rigidiies If he mior ais bedig ad orsioal rigidiies of he beam chages i he followig epoeial form: 8

9 Piarbasi 453 Nodimesioal Criical Mome (α) VIA II Eac b α b VIA II Eac Figure 7. Variaio of odimesioal criical mome wih b values. Nodimesioal criical mome () a α a-values Figure 8. Variaio of odimesioal criical mome wih a values. ( / ) GI ( ) GI e a ad ( ) ( / ) a EI EI e (3) where a is a posiive cosa deermiig he sharpess of he chages i rigidiies alog he legh of he beam, he, he odimesioalied form of he bucklig equaio i Equaio 4 become d φ dφ a a + αe φ d d (3) 9 where / ad is he odimesioal criical mome as defied i Equaio., φ φ Sice i he liear case, VIA II is foud o be more effecive ha VIA I ad VIA III, Equaio 3 is solved usig VIA II for various a values, ad he smalles odimesioal criical momes i he firs bucklig modes are obaied. The resuls are ploed i Figure 8, which shows how criical mome of a simpl suppored recagular beam decreases as a icreases.

10 454 Sci. Res. Essas Table 4. Normalied bucklig loads (β) for he firs hree modes cosa rigidiies. Mode # VIA I VIA II VIA III Eac Eve hough i ca be raher difficul o obai eac aalical soluios for his paricular case of he beam bucklig problem, VIM ca effecivel be used o deermie bucklig momes i a kid of variaios i rigidiies, such as he epoeial variaio sudied i his secio. Criical load for cailever case Beams wih cosa rigidiies If he mior ais bedig ad orsioal rigidiies of he beam are cosa, he Equaio 8 reduces o he followig simpler equaio: d φ + λ ( ) φ d Where, P λ (3) Iroducig a ew variable s, Timosheko ad Gere (96) obaied he soluio of Equaio 3 as ( / ) ( / ) φ sa λ + λ J/ 4 s AJ / 4 s (33) Where J / 4 ad J / 4 represe Bessel fucios of he firs kid of order /4 ad /4, respecivel. Whe he relaed boudar codiios are used o deermie he cosas A ad A, he followig characerisic equaio is obaied: ( λ ) J / 4 / (34) The smalles roo of his equaio is λ /.63 leadig o he firs mode criical load P cr P cr 4.6 (35) observed ha he umerical value i Equaio 35 chages o.46 ad i he secod ad hird mode criical load epressios, respecivel. This bucklig problem is also solved usig all hree VIM algorihms. For easier compuaios, he odimesioal form of he differeial equaio ad he relaed boudar codiios are wrie: d φ + β ( ) φ d where 4 P β dφ d wih φ ( ) ad ( ) (36) where /, φ φ ad β is he odimesioal criical load. For all hree algorihms, he iiial approimaio is chose as give i Equaio 3. The coefficies A ad B are obaied as defied i pure bedig case. Agai, i order o see he covergece of he approimae soluios o he eac soluios, seveee ieraios are coduced for each algorihm ad criical momes for he firs hree modes are compued. Similar o he pure bedig case, eacl he same soluios are obaied for criical loads from hree differe VIM algorihms, as lised i Table 4. Variaio of perce errors wih ieraios are also ploed i Figure 9 o show how VIM soluios coverge o he eac soluio. As show i Table 4 ad Figure 9, VIM soluios are i ecelle agreeme wih he eac resuls. I is o be oed ha while i is raher difficul o obai he criical load from he characerisic equaio give i Equaio 34, which eeds fidig he roos of a Bessel fucio, i is much easier o solve he characerisic equaio derived usig VIM ieraios which are i he form of polomials. This is oe of he advaages of usig VIM i his problem, eve i he case of cosa rigidiies. Beams wih liearl varig rigidiies If boh he mior ais fleural ad orsioal rigidiies of he beam chages i liear form, ha is, if GI ( ) GI b ad ( ) EI EI b (37) where b is a posiive cosa ha ca ake values bewee ero ad oe, he, he bucklig equaio give i Equaio 8 akes he followig form: ( / ) ( / ) d φ b dφ P + φ d b d b (38) Whe larger roos of Equaio 34 are obaied, i is

11 Piarbasi 455 Error (%) Mode Mode Mode Ieraio Number Figure 9. Variaio of perce errors wih ieraios for he firs hree bucklig loads cosa rigidiies. The odimesioal form of Equaio 38 ca be wrie as: ( ) ( ) d φ b dφ + β φ d b d b (39) where /, φ φ ad β is he odimesioal criical mome, as defied i Equaio 36. Sice i he case of cosa rigidiies, all VIM algorihms lead o he same resuls, he effeciveess of each algorihm o solvig Equaio 39 is also ivesigaed. For b.5, he firs mode criical mome is compued usig each ieraio algorihms separael. The ieraios are sared wih he liear approimaio give i Equaio 3. To simplif he iegraio processes, variable coefficies i he ieraio iegrals are epaded i series usig we oe erms. Similar o he earlier sudied cases, all algorihms lead o he same resul. However, as i he pure mome case, VIA II is more effecive ha VIA I ad VIA III wih regard o he compuaio ime ad space. As boh he mior ais fleural rigidi ad orsioal rigidi of a recagular beam is direcl proporioal o he heigh of he beam cross secio, a special case of liear variaio i beam rigidiies occurs whe he heigh of he beam is liearl apered. iearl apered seel beams have wide applicaios i ma egieerig applicaios. The lead o ecoomical desigs i laerallbraced cailever beams wih ed loadig, sice uder such a loadig, he major ais bedig mome decreases liearl from he fied ed of he beam o he free ed. However, as show i Figure, which is ploed usig VIA II, he bucklig load of a laerallubraced cailever beam decreases cosiderabl as b, i oher words, he slope of he aperig, icreases. Beams wih epoeiall varig rigidiies If he mior ais fleural ad orsioal rigidiies of he beam chages i epoeial form as give i Equaio 3, he odimesioalied form of he bucklig equaio i Equaio 8 becomes d φ dφ a a + βe ( ) φ d d (4) where /, φ φ ad β is he odimesioal criical mome, as defied i Equaio 36. Sice i has alread bee verified ha VIA II is he mos effecive ieraio algorihm for he bucklig equaios sudied i his paper, Equaio 4 is solved usig VIA II for various values of a. The variaio of ormalied firs mode bucklig loads wih a is show i Figure. As show i he figure, he bucklig load of a ouiform cailever beam wih ed loadig ca drop as low as he quarer of is uiform value whe a is equal o. Coclusio I his paper, wo fudameal beam bucklig problems; laeral orsioal bucklig of (a) simpl suppored arrow recagular beams uder uiform mome ad (b) arrow

12 456 Sci. Res. Essas Nodimesioal criical load (β) b-values b β Figure. Variaio of odimesioal criical load wih b values. Nodimesioal criical load (β) a-values a β Figure. Variaio of odimesioal criical load wih a values. recagular cailever beams carrig coceraed load a heir free eds, are sudied usig variaioal ieraio mehod (VIM). Eac soluios o hese problems are available i lieraure ol for beams of cosa rigidiies ad some paricular cases of liearl apered beams. I order o verif he effeciveess of VIM o solvig beam bucklig equaios ad o show he applicaio of he mehod; firsl, he problems wih cosa rigidiies are sudied. The ecelle mach of he VIM resuls wih he eac resuls verifies he efficiec of he echique i he aalsis of laeral orsioal bucklig problems. The, he bucklig problems i which he mior ais fleural ad orsioal rigidiies of he beams var alog heir leghs are sudied. Boh liear ad epoeial variaios are cosidered i ouiform beams. For he case of variable rigidiies, he differeial equaios of he sudied bucklig problems have variable coefficies, which hider he derivaio of eac soluios for hese pes of problems. However, as show i he paper, i is relaivel eas o wrie he variaioal ieraio algorihms for hese differeial equaios, which lead o he bucklig load/mome of he beam afer a few ieraios eve whe he rigidiies of he beam chage alog is legh, i epoeial form. I he paper, he effeciveess of hree VIM algorihms, wo of which have bee proposed ver recel b He e al. (), i solvig laeral bucklig equaios was also ivesigaed. Aalsis resuls show ha all ieraio algorihms ielded eacl he same resuls i all sudied problems. As far as he compuaio imes ad spaces are cocered, however, Variaioal

13 Piarbasi 457 Ieraio Algorihm II (VIA II) is foud o be superior ha he ohers especiall i problems where he beam rigidiies var alog he beam legh. REFERENCES Aa MT, Cosku SB (9). Elasic sabili of Euler colums wih a coiuous elasic resrai usig variaioal ieraio mehod. Compu. Mah. Appl., 58: Chajes A (974). Priciples of srucural sabili heor. Preice Hall, Eglewood Cliffs. pp Cosku SB, Aa MT (9). Deermiaio of criical bucklig load for elasic colums of cosa ad variable cross-secios usig variaioal ieraio mehod, Compu. Mah. Appl., 58: Cosku SB (). Aalsis of il-bucklig of Euler colums wih varig fleural siffess usig homoop perurbaio mehod. Mah. Model. Aal., 5(3): He JH (999). Variaioal ieraio mehod - a kid of oliear aalical echique: some eamples, I. J. No iear Mech., 34(4): He JH (). A review o some ew recel developed oliear aalical echiques, I. J. Noliear Sci. Numer. Simul., (): 5-7. He JH (7). Variaioal ieraio mehod some rece resuls ad ew ierpreaios., J. Compu. Appl. Mah., 7 (): 3-7. He JH, Wawa AM, Xu (7). The variaioal ieraio mehod: reliable, efficie ad promisig, Compu. Mah. Appl., 54(7-8): He JH, Wu GC, Ausi F (). The variaioal ieraio mehod which should be followed, Noliear Sci. e. A, (): -3. Oka F, Aa MT, Cosku SB (). Deermiaio of bucklig loads ad mode shapes of a heav verical colum uder is ow weigh usig he variaioal ieraio mehod, I. J. Noliear Sci. Numer. Simul., (): Simises GJ, Hodges DH (6). Fudameals of srucural sabili. Elsevier. pp Timosheko SP, Gere JM (96). Theor of elasic sabili. Secod Ediio, McGraw-Hill Book Compa, New York. Wag CM, Wag CY, Redd JN (5). Eac soluios for bucklig of srucural members. CRC Press, Florida. pp