Bucklinganalysisofnon-prismaticcolumnsbasedonmodiedvibrationmodesA.R.Rahai*,S.KazemiDepartmentofCivilEngineering,AmirkabirUniversityofTechnology(TehranPolytechnic),No.424HafezAvenue,Tehran15875-4413,IranReceived19February2006;receivedinrevisedform16July2006;accepted12September2006Availableonline22December2006AbstractIn this paper, a new procedure is formulated for the buckling analysis of tapered column members. The calculation ofthe buckling loads was carried out by using modied vibrational mode shape (MVM) and energy method. The change ofstiness within a column is characterized by introducing a tapering index. It is shown that, the changes in the vibrationalmodeshapesofataperedcolumncanberepresentedbyconsideringalinearcombinationofvariousmodesofuniform-section columns. As a result, by making use of these modied mode shapes (MVM) and applying the principle of station-ary total potential energy, the buckling load of tapered columns can be obtained. Several numerical examples on taperedcolumnsdemonstratetheaccuracyandeciencyoftheproposedanalyticalmethod.2006ElsevierB.V.Allrightsreserved.Keywords: Buckling;Taperedcolumn;Dynamicequationofmotion;Energymethod;Bucklingeigen-vector;Vibrationaleigen-vector1.IntroductionAsubstantial increaseincritical bucklingloadsofcolumnsmaybeobtainedbyappropriatemomentofinertiadistributionalongitslength. Thesetaperedcolumnsarewidelyusedasslenderstructural membersof high-risebuilding, bridges, oshorestructures, marine, aerospaceindustriesandetc. Theincreasinguseis due to appropriate distribution of strength and weight, which helps to achieve an ecient and reliable designandsometimessatisfythefunctionalrequirementssothatthesteelconsumptioncanbereducedasmuchaspossible. In the viewpoint of structural and optimal design, evaluation of buckling loads of stepped columns,whichhavethesamevolumewithspecicspanlength,isofgreatimportance.Inordertoavoidthecatastrophicfailureinsteppedcolumnsitisdesirabletoinvestigatetheirbucklingloads. Thesetypesofcolumnsareoftensubjectedtoaxial compressiveforces, whichmakesthempronetolocal instability. Presence of stepped parts in these types of columns will result in changes in the stress distri-butionwithinthememberandconsequentlyareductioninthebucklingcapacityofthecolumn.1007-5704/$-seefrontmatter 2006ElsevierB.V.Allrightsreserved.doi:10.1016/j.cnsns.2006.09.009*Correspondingauthor.Tel.:+982166468055;fax:+982166413969.E-mailaddresses:Rahai@aut.ac.ir(A.R.Rahai),s_kazemi@aut.ac.ir(S.Kazemi).Available online at www.sciencedirect.comCommunicationsinNonlinearScienceandNumericalSimulation13(2008)17211735www.elsevier.com/locate/cnsnsOntheotherhand,existenceofnon-uniformsectionwithinthesestructuresleadstochangesindynamiccharacteristicsof thestructuresuchasthevibrationresponses, modeshapes, andthenatural frequencies.Therefore, using stiness changes within a column and inuence of higher vibration modes of uniform-sectioncolumns canresult inthevibrationmodeshapes of taperedcolumn. Basedonthesimilarities that existbetween the buckling and vibrational modes, a procedure is given to use vibrational modes instead of bucklingmodeshapes. Thus, thebucklingloadof taperedcolumncanbeachievedbysubstitutingthevibrationalmodesintheenergymethod.Fortheestimationofexactbucklingloadsofcolumnsseveral theoriesareproposedoverthepastyears[1,2]. Theimplementationof energymethodinthestabilityof columns is discussedbyChenandLui [2]andthedynamicbehaviourofcolumnsaredescribedbyMeirovitch[3].Some researches have been directed towards the study of initial local buckling of tapered columns subjectedtocompressiveloads.Girijavallabhanusedthenitedierencemethodforevaluatingthebucklingloadsofnon-uniformcolumns[4]. Just[5] andKarabalis[6] useddirect integral approachtoestablishthestinessequationofataperedbeamelementforstabilityanalysisofstructurescomposedoftaperedbeams.CoulterandMillerdevelopedthebucklingandvibrationoftaperedplanebeamssubjectedtodistributedaxialforceNomenclatureA areaofcross-sectionCi,Biarbitrarycoecientd(EI) taperingfunction,variationinprimarystinessinvariouspartofcolumndm changeinprimarymassdistributioninsteppedpartofcolumndyitaper-inducedchangeinithmodeshapeofauniformsectioncolumnE Youngsmodulusofelasticityf(x, i) externalforceG shearmodulusofthematerialI0,I momentofinertiaofcross-sectionalareaL lengthofcolumnLilengthofsteppedpartsm massfunctionperunitlengthofcolumnM numberofsuperimposedmodesPcrcriticalbucklingloadofcolumnr radiusofgyrationofthecolumncross-sectiont timeU strainenergyWeexternalworkdonebyedgeloadingx,y Cartesiancoordinatey(x, t) exuraldeectionyi, yj, yktheith,jth,andkthmodeshapeofuniform-sectioncolumn(yt)itheithmodeshapeoftaperedcolumnXi, Xj, Xktheith,jth,andkthnaturalfrequenciesofuniform-sectioncolumnq massdensityperunitvolumeaijeect ofjth mode shape of uniform section column on theith mode shape of the correspondingtaperedcolumndijKroneckersymboldP rstvariationoftotalpotentialenergyf taperedsizeparameterw stinessratioj constantdependsonthecross-sectionalshapeP totalpotentialenergy1722 A.R.Rahai,S.Kazemi/CommunicationsinNonlinearScienceandNumericalSimulation13(2008)17211735[7]. Bradford and Cuk used general cubic polynomial shape function approach to achieve elastic buckling loadof tapered monosymmetric I-beams [8]. Frieman and Kosmatka presented an exact stiness matrix of a non-uniformbeambasedontheexibilitystinesstransformationapproach. Theyincludedsheardeformationeects in bending stiness matrix of tapered members, so that it could be applied to BernoulliEuler and Tim-oshenko beam [9,10]. Rajasekaran studied the instability of tapered thin-walled beams of generic open sectionusing the updated Lagrangian approach [11,12]. Kim et al. developed an inelastic buckling analysis of taperedmembers using the virtual forces approach [13]. Al-Gahtani in his research program utilized boundary integralapproachtoobtaintheexactstinessoftaperedmembers[14].DubeandDumiremployedthepolynomialseriesapproachforbucklingandvibrationalanalysisoftaperedthinopensectionbeamsonelasticfounda-tion. Theyalsopresentedexactsolutionsforthecoupledexuraltorsional axial bucklingloadsoftaperedbeamswithathin-walledopensectionrestingonanelasticfoundation[15,16].Gupta et al. focused on lateraltorsional buckling of non-prismatic I-beam. They also developed a methodtoobtainthefundamentalfrequenciesofthetapered cantileverbeamsbyusingvirtualforcesapproach[17].Kim and Kim presented a consistent nite element formulation for the free vibration and spatial stability anal-ysisofthin-walledtaperedbeamsandspaceframes.Theyalsoderivedthekineticandpotentialenergiesofthesemembersbyapplyingtheextendedvirtual workprincipleandincludingsecondordertermsofnitesemi-tangentialrotations.Theyconcludedthattheniteelementdevelopedintheirstudyiswellapplicablefor those analyses of thin-walled tapered beam-columns and space frames, especially for the inelastic stabilityanalysisandtheconnectionproblembetweentheniteelementsthatthesectionalpropertiesarevarieddis-continuously [18]. An equilibrium dierential equation considering both the eects of constant axial force andthe shear deformation of tapered beams (referred to as tapered TimoshenkoEuler beam) is established by Liand Li [19]. Their studies show that the stiness of a tapered member will be reduced signicantly due to theaxial compression and shear deformation in certain cases. They used Chebyshev polynomial approach to solvethe second-order dierential equation with variable coecients. To develop a theoretical approach for second-order inelastic analysis of steel frames of tapered members with slender web, a concentrated plasticity model isproposed by Li et al. Their study included signicant eects as residual stresses, initial geometric imperfection,gradualsection yielding at the element ends, distributed plasticity within the element and local webbuckling[20].Madhusudanetal.hasrecentlyanalyzedthepost-bucklingofnon-uniformcantilevercolumnsunderacombined load consisting of a tip-concentrated load and distributed axial load based on the dynamic formu-lation[21].Thepurposeofthepresentstudyistoinvestigatethebucklingloadsoftaperedcolumnsusingmodiedvibrational mode shapes (MVM). The changes along the length can be represented in various forms including,continuous, piecewise, andsingle-valued; andmaytakepositiveornegativevalues. Althoughcolumnsareonlyconsideredhere,theproceduremaybereadilyextendedtosolveothercasessuchasplatewherevibra-tionalmodeshapesareknown.The main advantage of this method is that the proposed method can be applied in various form of taperingfunction and dierent regionof columns. Furthermore, thismethoddoes not require any complexand time-consuminganalysis;Consequently,itneedsonlyaminimalinputdatatocarryoutthenumericalcomputa-tions. Inordertoillustratetheaccuracyandpractical usefulnessofthismethod, manynumerical solutionsarepresentedandcomparedwiththeresultsofotherresearchers.2.Theoryandformulation2.1.GeneralConsider an abruptly varying thickness column of length L and a modulus of elasticity E (Fig. 1). The col-umn is formed by two parts. The stepped part of column has a stiness w EI0 and length fL, where the rest ofthecolumnhasastinessofEI0. Thestinessratiow = (EI)stepped/(EI)restrepresenttheratioofstinessatsteppedparttotherestofcolumn. Byassumingtheexpressionw < 1, itisconsideredthatthecolumnhasareducedstinesspart of lengthsfLandlocalizedatL2 1 f < x 1and w < 1indicatesrespectivelytheincreaseanddecreaseinstinessofcolumnatdierentpart.Inthisstudy, itisassumedthattherearenochangesontheboundariesofcolumn. Thus, theboundaryconditions applied to a column in the uniform-section state can be equally applied to the column in the taperedFig.1. 3-Dviewofabruptlyvaryingthicknesscolumn.1724 A.R.Rahai,S.Kazemi/CommunicationsinNonlinearScienceandNumericalSimulation13(2008)17211735state. Furthermore, the tapering in a column is assumed to be uniform through the thickness of column (i.e.,thickness-throughtapering).BasedonFoxformulation,thechangesintheithmodeshapeofasteppedcol-umn can be expressed as a linear combination of natural modes of the corresponding uniform section column[22]:dyi

Mj1aijyji; j 1; 2; 3; . . . ; M 4SubstitutingEq.(4)indynamicequationofmotionEq.(3)fortheithmodeshapeyields:dEIo4yiox4 EI

Mj1aijo4yjox4 dmo2yiot2 m

mj1aijo2yjot2 0 5MultiplyingEq. (5)ontheleftsidebyyk, that, i 5kandusingym X2mymandintegratingforthewholelengthofcolumnyields:_L0dEI yko4yiox4dx

Mj1aij_L0EI o4yjox4 ykdx_ _ X2i_L0dm yi yk dx

Mj1aij X2j_L0m yj ykdx 06Theorthogonalitypropertyimpliesthat:_L0mymyndx dmn7_L0EI ymo4ynox4 dx X2m dmn8whereXmarethenatural frequenciesfortheuniform-sectioncolumnanddmnisthekroneckersymbol anddenedasfollows:dmn 1 m n0 m 6 n_SubstitutingEqs.(7)and(8)intoEq.(6),theaboveequationisexpressedas_L0dEIo4yiox4 ykdx aikX2k X2i_L0dm yi ykdx aikX2i 0 9From the above equation, the eect of jth mode shape on changes in ith mode shape can be derived as follows:aik 1X2i X2k_L0dEIo4yiox4 ykdx X2i_L0dm yi ykdx_ _10IntegrationbypartsofthersttermofEq.(10);theaboveequationcanbewrittenasfollows:aik 1X2i X2k_L0dEIo2ykox2o2yiox2dx X2i_L0dm yi ykdx_ _11Dierentiatingtheorthogonalitypropertyofmodeshapeswithrespecttomasswouldyields:_L0dm y2i dx 2_L0myidyidx 0 12A.R.Rahai,S.Kazemi/CommunicationsinNonlinearScienceandNumericalSimulation13(2008)17211735 1725It can be assumed that the eect of changes in the mass distribution function are negligible in comparison withthe signicant changes in the stiness of the structure. Therefore, the rst term in the left side of Eq. (12) canberemoved.SubstituteEq.(4)intothesecondpartofaboveequationgives:2_L0m yi

aij yjdx 0 13Byusingorthogonalproperty,Eq.(7),andapplyingtheKroneckerproperties:2 aii_L0m y2i dx 0 14Itcanbeshown:_L0m y2i dx 1Thusaii 0 15Generally,thevibrationalmodeshapesofataperedcolumnaregivenbyyti yi dyi16(yi)iaretheithvibrational modeshapesoftaperedcolumnsanddyiindicatestaper-inducedchangesinithmode shape of a uniform-section column, yi refer to various mode shapes of uniform-section columns. Substi-tuting,dyifromEq.(4)intoEq.(16),mayyields:yti yi

Mj1aijyj; i; j 0; 1; 2; . . . ; M 17where yi and yj are the natural modes satisfying the eigen-value problem of uniform-section columns, aij showsthe eect of jth mode shape of uniform-section column on theith mode shape of tapered column and are de-nedasfollows:aij 1X2i X2j_L0dEI o2yiox2o2yjox2dx X2i_L0dm:yi:yjdx_ _i 6 j0 i j___i; j 0; 1; 2; . . . ; M18In Eq. (18) Xi and Xj are the natural frequencies of uniform section column and d(EI) is tapering function andshowstheincreaseordecreaseofstinessindierentregionsofcolumnandcanberepresentedinvariousforms; suchas, continuous, piecewise, single-valued; andmaytakepositiveforstienedornegativevaluesfor reduced stiness columns. The main advantage of this method is that the tapering function can be appliedto any form of stepped columns, having various boundary conditions. That is, of course, if the mode shapes ofuniform-sectioncolumnareknown.2.3.TheenergymethodAsindicatedatthebeginning, thevibrational modesofcolumnmaybeusedinsteadofbucklingmodeshapes. Thisimpliesthat thesolutionobtainedforvibrationandbucklingdierential equationof columnare similar. By using the vibrational and buckling governing dierential equation, and applying the boundarycondition, it can be shown that the eigen-vectors are the same as for both buckling and vibration analysis of acolumn except for magnitude. This constant can be vanished in the process of determining buckling loads in away that vibrational modes satisfy the buckling dierential equation; and may thus be used as deected shapeofcolumnforthebucklinganalysis.1726 A.R.Rahai,S.Kazemi/CommunicationsinNonlinearScienceandNumericalSimulation13(2008)17211735In this part, the calculus of variation in conjunction with the principle of stationary total potential energy isused to determine the conditions that should be satised by the column. By applying the principle of station-ary total potential energy, we need to evaluate the strain energy U, the external work We, and the total poten-tialenergy Pofthecolumn.Foran elasticsystem,the workdoneby the external forceson thesystemis stored as strain energyin thesystem. Thus, if a virtual displacement is introduced to system, the external virtual work done by the externalforcesonthesystemisstoredasvirtualstrainenergyinthesystem. ThestrainenergyforacolumncanbewrittenasU _L0EI2o2ytio2x_ _2dx 19Forthecentrallyloadedcolumn,Fig.2theexternalworkdonebyedgeloadingcanbewrittenasWe P2_L0oytiox_ _2dx 20The sum of the strain energy and potential energy of the system is the total potential energy. Using the symbolPtodenotethetotalpotentialenergyofasystemandtherefore,canbewritteninthefollowingform:P U We21In view of the inextensibility of the middle surface, total potential energy can be used to investigate the buck-lingloadofcolumnandcanbewritteninthefollowingform:P _L0EI2o2ytio2x_ _2dx P2_L0oytiox_ _2dx 22Forequilibriumcondition,therstvariationoftotalpotentialenergymustvanish,ityields:dP 0 23The last expression is the mathematical statement of the principle of stationary total potential energy. A sta-tionaryvaluemaycorrespondtoaminimumormaximumvalueofthetotal potential energy. Aminimumvalue indicates that the equilibrium is stable and a maximum value indicates that the equilibrium is unstable,fromthisprocedure;thebucklingload(theleastvalue)ofcolumnscanbeachieved.3.Numericalresults3.1.GeneralEq. (17) provides a general equation for evaluating the buckling load of columns. This method (MVM) usesthe buckled shape of the uniform-section column, and the modication coecient aij from Eq. (18) to providebuckling mode shape and the corresponding load for tapered column. The selected examples are for compres-sion purpose only and do not represent the full capabilities of the proposed method. The numerical procedurebased on the aforementioned developed formulation are then programmed on a desktop computer and numer-icalresultedarepresentedformanycasesinthefollowingsections.Fig.2. Ananalyticalmodelforbucklinganalysisofapinned-endedcolumn.A.R.Rahai,S.Kazemi/CommunicationsinNonlinearScienceandNumericalSimulation13(2008)17211735 17273.2.ComparisonofresultsInordertodemonstratetheaccuracy,convergency,andapplicabilityofthepresentmethod(MVM),sev-eral buckling problem of columnswith variable thicknesses are analyzed and numerical results are tabulatedand comparedwith the values available in the literature. Generallyspeaking, thismethodis applicable to allcolumnswhosemodeshapescanbeexpressedbyanalyticalfunctions,eithertrigonometricorpolynomial.Thenaturalvibrationmodecorrespondingtonaturalvibrationfrequencyxiforaprismatic,simplysup-portedcolumnFig.2becomes:yix C SinipxL_ _i 1; 2; 3 . . . ; M 24whereyi(x) is theith natural vibration mode of the uniform-section column andC is an arbitrary coecient,these mode shapes are unique, except for the magnitude. The magnitude of the eigen-vectors can be renderedunique by a process known as normalization; thus using the orthogonal property, the following equation canbewritten:_L0m yix yjx dx diji; j 1; 2; 3; . . . ; M 25whereMindicatesthenumberofnormalmodessuperposedintheanalysisand dijisthekroneckerdelta.By substituting Eq. (24) into Eq. (25), and applying the Kronecker properties, the following expression canbeobtainedasm C2_L0SinipxL_ _ _ _2dx 1 i 1; 2; 3; . . . ; M 26Introducingtheconstant,C,intoEq.(24),themass-normalizedmodeshapecanbeachievedasfollows:yi 2m L_ SinipxL_ _i 1; 2; 3; . . . ; M 27By substituting Eq. (27), and the natural frequency of simply supported intact column, Xi ip2 EI=mL4_,into Eq. (18), the modied vibrational mode shape of the corresponding stepped column can be achieved fromEq.(17).Basedontheprincipleofstationarytotalpotentialenergy,amongallthepossiblemodeshapes(thethirdoneisshowninFig. 3)theonethatminimizestheenergyfunctional isgovernedanditshouldbeusedasdeectedshapeofcolumnforbucklinganalysis.According to the numerical studies on buckling load of several tapered columns it is found that in the caseof 1 < w < 2 the rst mode shape is governed. However, in order to identify the lowest buckling load for w > 2Fig.3. Firstthreevibrational/bucklingmodeshapesofasimplysupported,taperedcolumns.1728 A.R.Rahai,S.Kazemi/CommunicationsinNonlinearScienceandNumericalSimulation13(2008)17211735higher order values of i and j may be required to be taken into account. This is because during the evaluationprocess of (yt)i, there is no prior guarantee that the rst mode shape is always induces the least buckling load.Numerical studies shows that in most cases, the accurate buckling loads can be estimated with 7 up to 11 modeshapesandthereisnosignicantimprovementwithanincreasednumberofmodeshapes.Herein the study focused on nding the modied rst mode shape of such a column with w < 2.5. Hence toobtainthemodiedrstmodeshapeofsuchtaperedcolumns,onecansubstitutei = 1inEq.(17),ityields:yt1 y1

Mj1a1j yjj 1; 2; 3; . . . ; M 28where(yt)1andy1representtherstmodeshapeoftaperedandtheuniform-sectioncolumnrespectively.Itcan be seen that the manner of convergence of present method (MVM) results is desirable. Based on precedingformulation,aFORTRANcomputerprogramwaswrittentoobtainthebucklingloadofsuchacolumn.3.2.1.Bucklingloadofsimplysupportedsteppedcolumn;middleportionfattedAnexaminationofthebending-momentdiagramforthesimplysupportedbuckledcolumnshowsthatacolumnofuniformcross-sectionisnotthemostoptimal formtocarrycompressiveload. Forexample, inthe case of a compressed bar with hinged ends, the stability can be increased by removing a portion of materialfromtheendsandincreasingthecross-sectionoverthemiddlesection[1].Fig. 4 shows a simply supported column of length L, Youngs modulus of elasticity E = 210 (GPa), exuralmomentofinertiaI0 = 2.1644e9m4. Thiscolumnhasanabruptchangesincross-sectionoflengthfL,thestiness fL ratio w = 2 and are subjectedto axial compressionPcr. Themodiedmode shape of thiscolumnthat is used in Eq. (22), is shown in Fig. 5. The obtained results have been comparedwith those obtained byFEM method. To develop a model of the stepped column under buckling analysis, the general purpose niteelementprogramANSYS[24] wasusedinthisstudy. TheaforementionedcolumnhasbeenmodeledwithBEAM3. BEAM3isauniaxial elementwithtension, compression, andbendingcapabilities. Eachnodeoftheelementhasthreedegreesof freedom: translationsinthenodal xandydirectionsandrotationaboutFig.4. Abruptlyvaryingthicknesscolumn,with fLasthelengthofstienedregion.Fig.5. Modiedvibration/bucklingmodeshapesofasimplysupportedsteppedcolumn.A.R.Rahai,S.Kazemi/CommunicationsinNonlinearScienceandNumericalSimulation13(2008)17211735 1729thenodal z-axis.Table1presentthebucklingloadcomputedbythepresentmethod(MVM)andarecom-paredtothoseobtainedbytheniteelementmethod[24], andexact expressionproposedbyTimoshenkoandGere[1].The variation of buckling load of the simply supported tapered column subjected to axial compression load(seeFig.4),versustaperedsizeparametersfispresentedinFig.6.Forcomparison,thesamegureshowsresultsobtained byTimoshenko and Gere[1]. Asitis seen,good agreementexistsbetweenresults.It shouldbenotedthattheexpressionproposedbyTimoshenko, canbeusedifthenumberoftaperedpartisunity.However, themethodproposedbyTimoshenkocanbeusedifthenumberofchangesinthecross-sectionisgreaterthanone,butthederivationofexpressionbecomemorecomplicated.3.2.2.MultiplesteppedcolumnssubjectedtocompressiveloadsAging steel structures are prone to suer various types of damages such as corrosion, cracks, and dents andthese members should be free from disastrous structural failures due to these sectional non-uniformity. Hence,it is of interest to investigate the buckling load of such columns. Fig. 7 shows a simply supported columns oflength L, Youngs modulus E = 210 (GPa), exural moment of inertia I0 = 4.2231e9m4, stiness ratiow = 0.5, andaresubjectedtoaxial compressionPcr. Present method(MVM) canbeappliedtodeterminethe bucklingloadof suchstiness-reducedcolumn. Table 2provides some informationonthe accuracyandconvergenceofthepresentsolutionwiththoseobtainedbyFEManalysis[24]. Thetablealsoincludesanothercaseofsimplysupportedcolumnwithmanyreduced-stinessparts(seeFig. 8), subjectedtoaxialTable1Comparisonofthepresentanalysisresults(MVM)withtheotherresultsonthebucklingloadsofasimplysupportedsteppedcolumn(seeFig.4)Specimendimension Pcr(N)L(m) f Timoshenko[1] Presentanalysis(FEM)[24] Presentanalysis(MVM)5 0.125 195.023 195.027 199.3255 0.25 222.914 222.913 226.3155 0.375 254.589 254.576 258.6596.25 0.3 154.976 154.971 154.5026.25 0.5 183.902 183.902 184.1276.25 0.7 210.125 210.116 210.9347.5 0.4 116.094 116.080 117.4107.5 0.6 138.126 138.117 138.7357.5 0.8 150.420 150.429 150.623Fig.6. Variationoftheelasticcriticalbucklingloadofthetaperedcolumnversustaperedsizeparameter, f.1730 A.R.Rahai,S.Kazemi/CommunicationsinNonlinearScienceandNumericalSimulation13(2008)17211735compression. The results obtained by the present methodare in very good agreement with thoseof obtainedbytheniteelementprogramANSYS[24].3.3.Bucklingofsimplysupportedtaperedcolumn,sidepartfattedBabcock and Waas [23] have conducted tests in which the column is buckled and loaded into post-buckledrange (see Fig. 9). The specimen ends are ground to form a V-shape, and are supported on V-notched blockstosimulatesimplysupportedendconditions[23].Fig.9showsasimplysupportedcolumnsoflengthLandexuralstinessEI0 = .6658 (N m2)withabruptchangeincross-sectionoflength fLandthestinessratiow = 2.4095, subjectedtoaxial compressionPcr. TheresultsarereportedatTable3forvariousvalueoff.Fig.7. Abruptlyvaryingthicknesscolumn.Table2Comparisonofthepresentanalysisresults(MVM)withtheFEMresultsonthebucklingloadsofsimplysupportedmultiplesteppedcolumns(seeFigs.7and8)Multipletaperedcolumn Pcr(N)Model No.oftaperedpart Length(m) Presentanalysis(FEM)[24] Presentanalysis(MVM)Fig.7 2 6 157.60 162.914Fig.8 3 6 224.91 225.621Fig.8. Multiplesteppedcolumn;stinessreductionatseveralpartofcolumn.Fig.9. Geometry,coordinatessystem,andappliedin-planeloadsoftaperedcolumnwithabruptly varyingsection.Table3Comparisonofthepresentanalysisresults(MVM)withthepublishedtestresultsonthebucklingloadsofasimplysupportedsteppedcolumn(seeFig.9)Specimendimensions Pcr(N)L(m) f Testresults[23] Madhusudanetal.[21] Presentanalysis(FEM)[24] Presentanalysis(MVM)0.2032 0.38 169.032 185.490 185.513 187.1420.2286 0.33 146.791 139.674 140.221 145.7860.1905 0.40 204.618 215.738 215.680 220.0640.1842 0.41 209.066 233.531 233.442 234.9150.1778 0.43 262.445 256.662 256.493 261.5780.1715 0.44 289.134 279.348 279.334 284.673A.R.Rahai,S.Kazemi/CommunicationsinNonlinearScienceandNumericalSimulation13(2008)17211735 1731The results obtained by present method (MVM) are compared by those reported by Babcock and Waas [23],Madhusudanetal.[21]andFEM[24].The buckling analysis of this stepped column is carried out to study the convergence behavior of the presentmethod. Fig. 10 shows the variation of buckling load with respect to the number of imposed modes. The num-ber of imposed modes are varied from one to seven in this study. For the case, w < 2.5 it is observed that thebuckling loads are found to converge with seven modes and increasing the number of higher modes does notshowanyappreciableimprovement.Forthecolumns,havingmultiplesteppedpartsandw > 2.5thehighermodeshapes,elevenmodes,mayresultinagoodresults.3.4.BucklingofcantilevertaperedcolumnFig. 11showsasteel taperedcantilevercolumnoflengthL, primaryexural stinessI0 = 24.3e 6 m4,Youngs modulus E = 200 GPa, the stiness ratio w = 2. The vibrational mode shape of a prismatic cantilevercolumncanbeexpressedasyix Ccosaix coshaix pcosaix coshaixPi cosaiL coshaiLsinaiL sinhaiL; a4i x2imEI29Fig.10. Convergenceofbucklingloadwithanincreasingnumberofimposedmode.Fig.11. Cantilevercolumnwithabruptlyvaryingthickness;geometryandcoordinate.1732 A.R.Rahai,S.Kazemi/CommunicationsinNonlinearScienceandNumericalSimulation13(2008)17211735Moreover,thecorrespondingnaturalfrequencycanbedenedasfollows:xi aL2iEImL4_; aLi p22i 1; i 1; 2; 3; . . . ; M 30Table4Comparison of the present analysis results (MVM) with the other results on the buckling loads of a cantilever stepped column (see Fig. 11)Specimendimension Pcr(kN)L(m) f Timoshenko[1] Presentanalysis(FEM)[24] Presentanalysis(MVM)5 0.125 545.488 545.482 545.6125 0.25 623.491 623.497 624.1305 0.375 712.065 712.060 713.0547 0.3 335.663 335.659 337.2217 0.5 410.072 410.067 411.1217 0.7 468.533 468.530 469.1209 0.4 225.473 225.472 227.5059 0.6 268.294 268.291 269.2319 0.8 292.209 292.206 292.976Fig.12. Modiedvibrational/bucklingmodeshapeoftaperedcantilevercolumn.Fig.13. Geometry,coordinatessystem,andappliedin-planeloadsoftaperedcolumnoflinearlyvaryingsection.A.R.Rahai,S.Kazemi/CommunicationsinNonlinearScienceandNumericalSimulation13(2008)17211735 1733The modied mode shape of this column can be obtained by using Eq. (17) which is shown in Fig. 12. UsingtheaforementionedprocedureusedforsimplysupportedsteppedcolumnandusingEqs. (29)and(30)willresultinTable4,whichimpliestheagreementbetweenpresentmethod(MVM)and thoseobtained byTim-oshenko[1],andniteelementanalysisperformedbyANSYS[24].And nally, Fig. 13 shows a steel tapered cantilever column used by Karabalis [6], and Li [19] as a numer-ical example for calculating the elastic critical axial load. The results obtained by Karabalis [6], and Li and Li[19] are compared with those obtained by present study (MVM). Table 5 presents the buckling loads of such acantilever column and conrms that the present method is an eective one for buckling analysis of cantilevercolumn.Asitisseeninthemostexamples,thebucklingloadobtainedbypresentmethod(MVM)ishigherthanthoseobtainedbyexperimentandresultsdevelopedbyotherresearchers.4.ConclusionIn this study, a new procedure using energy method based on modied vibrational mode shapes (MVM) isformulated for the buckling analysis of columns. This method is applicable for columns of various regions oftapering, including dierent stiness function along their length. It is shown that the vibrational mode shapesofataperedcolumnisinfactalinearcombinationofvariousmodeshapesoftheuniform-sectioncolumns.Thisphenomenonisusedtoestimatethevibrational modeshapesof tapercolumns. Inturn, thesemodeshapes are incorporated to evaluate their buckling loads. The method enables to solve for the buckling loadsofanypolynomialthicknessvariationthatcanberepresentedasafunctioninx-direction.Manynumericalanalysesarecarriedouttorepresenttheaccuracyoftheproposedmethodbycomparingtheresultstotheworkspresentedbyotherresearchers.References[1] TimoshenkoSP,GereJM.Theoryofelasticstability.2nded. NewYork: McGraw-Hill;1961.[2] ChenWF,LuiEM.Structuralstability;theoryandimplementation. NewYork: Elsevier;1987.[3] MeirovitchL.Principlesandtechniquesofvibrations. NewJersey: Prentice-Hall,Inc.;1997.[4] GirijavallabhanVC.Bucklingloadsofnonuniformcolumns.JStructDiv1969;95(11):241931.[5] JustDJ.PlaneframeworksoftaperingboxandI-section.JStructDiv1977;103(1):7186.[6] KarabalisDL.Staticdynamicandstabilityanalysisofstructurescomposedoftaperedbeams.JComputStruct1983;16(6):73148.[7] CoulterBA, MillerRE. Vibrationandbucklingofbeam-columnssubjectedtonon-uniformaxial loads. 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