Application of the Finite Fourier Sine Transform Method for the Flexural-Torsional Buckling Analysis of Thin-Walled Columns

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1 IOSR Journa of Mechanica and Civi Engineering (IOSR-JMCE) e-issn: ,p-ISSN: 3-334X, Voume 14, Issue Ver. I (Mar. - Apr. 17), PP Appication of the Finite Fourier Sine Transform Method for the Feura-Torsiona Bucking Anaysis of Thin-Waed Coumns Mama B.O. 1, Ike C.C., Nwoji C.U. 3, Onah H.N. 4 1 Dept of Civi Engineering University of Nigeria, Nsukka, Enugu State, Nigeria. Dept of Civi Engineering Enugu State University of Science & Technoogy, Enugu State, Nigeria. 3 Dept of Civi Engineering University of Nigeria, Nsukka, Enugu State, Nigeria. 4 Dept of Civi Engineering University of Nigeria, Nsukka, Enugu State, Nigeria. Abstract: In this work, the system of three couped differentia equations governing the generaised eastic thinwaed coumn bucking probem was soved by the finite Fourier sine transform method for the case of pinned ends. The boundary vaue probem was found to reduce to an agebraic eigenvaue eigenvector probem for which the characteristic bucking equation was found. Two cases were considered. For douby symmetric sections, it was found that the bucking modes are uncouped. For monosymmetric sections, it was found that one of the equations is uncouped, whie the other two are couped. The epressions for the bucking oad (eigenvaues) obtained in this study were found to be identica with soutions in iterature for the same probem obtained using the method of undetermined parameters. Keywords: finite Fourier sine transform method, monosymmetric cross-sections, eigenvaue probem, characteristic bucking equation, Euer feura bucking, torsiona bucking. I. Introduction Thin-waed structura eements with cross-sectiona shapes such as with isotropic or anisotropic materias are commony used as beams, coumns and beam coumns in engineering appications ranging from buidings, bridges, and aerospace structures. Due to their thin-waed, open crosssections, these eements are aways prone to torsion and instabiities [1]. The probem of bucking of thin-waed open-section coumns often require compe mathematica anaysis, and present probems of considerabe magnitude. The modes of faiure which form the basis of thin-waed coumn design are: feura bucking, oca bucking and torsiona bucking []. Feura bucking is a sudden deviation of the coumn from the initia configuration when the critica oad is attained. Loca bucking wi appear as a series of waves in the component parts of the cross-section; whie the coumn remains straight. It is virtuay independent of the coumn ength. Torsiona bucking occurs when the centra part of the coumn rotates bodiy reative to its ends. This occurs even if the ends are free to rotate []. These modes of bucking faiure do not necessariy occur independenty and interaction of the bucking modes frequenty occur []. Feura and torsiona bucking are often interreated whie feura and oca bucking interact in the post bucking stage. The bucking behaviour of thin waed coumns is very compe and compicated as a resut of the interaction effects of compression, feure and torsiona deformations [3]. In structura design, feura-torsiona bucking is an important consideration, particuary for thin-waed members. Feura-torsiona bucking mode invoves simutaneous bending and twisting of the cross-section. The cross-section undergoes transationa deformation in the two aes of the cross-section, as we as rotationa deformation about the shear center. The feura-torsiona bucking probems of thin waed coumns has been investigated by Asayed [4], Timoshenko and Gere [5], Aen and Buson [6], Chajes [7] and Wang et a [8] and used in the deveopment of the design criteria for stee design. Ike et a [9] have used Gaerkin variationa method to sove the thin-waed coumn bucking probem. II. Research aim and objectives The genera aim and objective of this work is to appy the finite Fourier sine transform method to the soution of the generaised feura-torsiona bucking probem of an eastic coumn with pinned supports at = and =, is the ength of the coumn. The specific objectives are: (i) to use the finite Fourier sine transform method to sove the generaised eastic coumn bucking probem defined by a system of three couped differentia equations in terms of three unknown dispacement functions v(), w() and ( ), for the case of pinned support at the ends =, and = Corresponding author DOI: 1.979/ Page

2 Appication of the Finite Fourier Sine Transform Method for the Feura-Torsiona Bucking.. (ii) to show that the generaised eastic coumn bucking equations reduce to an agebraic eigenvaueeigenvector probem and (iii) to derive the critica bucking oads for two specia cases; namey coumns with douby symmetric crosssections and coumns with singy symmetric cross-sections. III. Methodoogy The finite Fourier sine transform was first introduced by Doetsch [13] as a method of integra transformation of boundary vaue probems. Subsequenty, the method has been deveoped and generaised by severa researchers such as Kneit [14], Strandhagen [15], Roettinger [16] and Brown [17]. The genera phiosophy behind integra transformations, and the finite Fourier transform method is that they simpify boundary vaue probems by eiminating partia derivatives with respect to one of the independent variabes, hence the transformed equation has one ess independent variabe. Definitions: The finite Fourier sine transform S u( ) of a function u ( ) of, is defined as: m Sn Snu( ) u( )sin n m = 1,, 3, (1) Simiary, the finite Fourier sine transforms of the derivatives are given by u m m m sin u( a)sin m u( )sin u cos () m ucos is the finite Fourier cosine transform of u() Aso, m sin m ( 1) ( ) ( ) m ( )sin m u m u u u (3) m m 3 u m u w 3 sin ( 1) cos 3 m m m m 4 u 4 3 m 4 m u m m u u u (4) sin sin ( )( 1) ( ) m w w m ( 1) ( ) ( ) IV. Theoretica Framework feura-torsiona governing equations of bucking eastic coumns Consider an eastic coumn of ength, whose ongitudina ais is defined by the coordinate, and the pane of the cross-section is defined by the y Cartesian coordinate pane. The system of governing differentia equations that describe the feura-torsiona bucking behaviour of a generaied eastic coumn under an aia compressive oad N, acting through the centroid of the cross-section if the moments due to the transverse oads are ero, the appied torque vanishes; and the oad is appied such that the bi-moment is ero are given by the foowing [1, 11, 1]: 4 d v d v d EI N 4 N e (6) 4 d w d w d EI yy N 4 N ey (7) (5) DOI: 1.979/ Page

3 Appication of the Finite Fourier Sine Transform Method for the Feura-Torsiona Bucking.. 4 IEN w 4 y d d d v d w EC GJ e N e N A (8) v(), w() and ( ) are the dispacements E = Young s moduus of easticity G = shear moduus or moduus of rigidity C w = warping constant I = moment of inertia about the ais I yy = moment of inertia about the y ais e = coordinate of the shear centre e y = coordinate of the shear centre N = oad in the direction I E = poar moment of inertia about the shear centre J = St Venant torsiona stiffness of the section IE Iyy I ( ey e ) A (9) A area of cross-section. The governing equations are a system of three differentia equations in terms of the three dispacements v(), w() and ( ). V. Appication of the finite Fourier Sine Transform Method to the feura-torsiona bucking probem We seek to appy the finite Fourier Sine transform method to find soutions to the system of governing differentia equations for a coumn of ength with pinned ends at = and =. The unknown functions in the governing system of differentia equations are the three dispacement functions, namey v(), w() and ( ); and we seek a soution to the eastic coumn bucking probem for the case of pinned-pinned end supports (i.e.at = and = ). The boundary conditions for pinned-pinned supports at =, and = are: v( ) v( ) w( ) w( ) ( ) ( ) (1) v ( ) v ( ) w ( ) w ( ) ( ) ( ) d v d w v, w and d and the primes denote differentiation with respect to the coordinate variabe. The boundary conditions make the appication of the finite Fourier Sine transform method viabe for the probem for the case of pinned supports at =, and =. Hence, appying the finite Fourier sine transforms to the system of three differentia equations, we obtain: iv m EIv ( ) N v ( ) N e ( ) sin (11) iv m EI yyw ( ) N w ( ) N ey ( ) sin (1) iv IEN m ECw ( ) GJ ( ) N ev ( ) N eyw ( ) sin A (13) Simpifying, 4 d v( ) m d v( ) m d ( ) m EI sin N sin N e sin (14) 4 DOI: 1.979/ Page

4 Appication of the Finite Fourier Sine Transform Method for the Feura-Torsiona Bucking.. 4 d w m d w m d m EI sin N sin N e sin (15) yy 4 y 4 E w d m I N d m sin sin 4 A EC GJ d v m d w m e N sin e N sin (16) y Using integration by parts, we find: d v( ) m m m m sin v ( )sin V ( m, ) (17) 4 d v m m m m sin v( )sin V( m, ) (18) d m m m m sin ( )sin ( m, ) (19) d w m m m m sin w( )sin W( m, ) () 4 d w m m m m sin w( )sin W( m, ) (1) m V( m, ) v( )sin () m W( m, ) w( )sin (3) m (4) ( m, ) ( )sin and V( m, ) is the finite Fourier Sine transform of v(), W(m, ) is the finite Fourier Sine transform of w() and ( m, ) is the finite Fourier Sine transform of ( ). The system of governing differentia equations become: 4 m m m EI V( m, ) N V( m, ) N e ( m, ) 4 m m m EI yy W( m, ) N W( m, ) N ey ( m, ) 4 IEN m m ECw ( m, ) GJ ( m, ) A m m N e V( m, ) N ey W ( m, ) (5) (6) (7) DOI: 1.979/ Page

5 Appication of the Finite Fourier Sine Transform Method for the Feura-Torsiona Bucking.. We divide through by m to obtain m EI V( m, ) N V ( m, ) N e( m, ) m EI N V( m, ) N e ( m, ) m EI yy N W( m, ) N ey ( m, ) m IEN ECw GJ ( m, ) N ev ( m, ) N eyw ( m, ) A This is a system of homogeneous agebraic equations in terms of V(m, ), W(m, ) and ( m, ) written in matri form as: (8) (9) (3) (31) and can be m EI N N e V ( m, ) EI yy m N N e y W ( m, ) ( IEN) N e N ey ECw m GJ A ( m, ) (3) For nontrivia soutions, the determinant of the coefficient matri must vanish, and the requirement for the vanishing of the coefficient matri yieds the characteristic equation or stabiity equation for the probem. Hence, the stabiity equation is given by: m EI N N e EI yy m N N e y ( IEN) N e N ey ECw m GJ A (33) The characteristic bucking equation for the feura torsiona probem of eastic coumn with pinned ends at =, and = is obtained by the epansion of the determinanta equation (Equation (33)), and finding the roots or the eros of the resuting poynomia in N. Two particuar cases of this probem, which can be considered as simpifications of the genera feura-torsiona bucking probem are considered. Case 1: The cross section of the eastic coumn is douby symmetric about the y and coordinate aes. Some typica eampes of douby symmetric cross sections are symmetric I sections, and crucifi sections. For douby symmetric cross sections, e y = e = and the characteristic bucking equation simpifies to Equation (34). DOI: 1.979/ Page

6 EI Appication of the Finite Fourier Sine Transform Method for the Feura-Torsiona Bucking.. m N EI yy m N ( I ) EN EC w m GJ A (34) Epansion of the characteristic stabiity equation yieds the factoried form of the poynomia in N : m m m IEN EI N EI yy N ECw GJ (35) A The bucking equations are found to be uncouped as foows: EI EI EC yy w m N m N m IEN GJ A The three roots of the characteristic bucking equation are: m N EI P E m N EI yy PE yy A m N ECw GJ Pt I E (36) (37) (38) (39) (4) (41) P E is the Euer oad for feura bucking about the ais, P E is the Euer oad for feura yy t bucking about the yy ais and P t is the bucking oad in torsiona (twist) bucking. The stress ( ) in torsiona bucking is t 1 m Pt ECw GJ I p A I p = I E for douby symmetric cross-section. Case : In this case, the cross section is singy symmetric with respect to the coordinate aes. A typica eampe of singy symmetric cross-section is the channe section. If the ais is the ais of symmetry, then e y = and e. The characteristic bucking equation simpifies to become: (4) DOI: 1.979/ Page

7 Appication of the Finite Fourier Sine Transform Method for the Feura-Torsiona Bucking.. m EI N N e EI yy m N ( I ) EN N e ECw m GJ A Epansion of the equation yieds: m m m IEN EI yy N EI N ECw GJ N e (43) ( ) A Soving, we have The roots are: P m EI yy N m m IEN EI N ECw GJ ( N e ) A m N EI P N yy E yy Ae Pt PE P 4 1 t PE P E P t I E e A 1 I E (44) 1 N Pt PE P 4 t PE P E P t (49) E m EI Ae 1 I E A m Pt ECw GJ I E The epression for N given by Equation (48) represents the bucking oad for the couped feura toritiona bucking mode. A negative sign in Equation (48) woud yied smaer vaues for N. Hence, (45) (46) (47) (48) DOI: 1.979/ Page

8 Appication of the Finite Fourier Sine Transform Method for the Feura-Torsiona Bucking.. N P P P P P P Ae 1 I E Ae E 4 1 t t E t E I E 1 N Pt PE P 4 t PE P E P t (51) Critica bucking oads are obtained when m = 1 in the Equations (47) and (48). Critica bucking oads are thus obtained as: c N EI yy PE (5) and yy c c c c Ae Pt PE P 41 c c t PE P E P t I E N (53) e A 1 I E 1 N c t c c 4 c c Pt PE P t PE P E P t (54) the superscript c is used to denote critica. ft The critica bucking stress for the couped feura-torsiona bucking mode is obtained as: ft cr cr Ae 1 I E E T E T Ae E T 4 1 cr cr cr cr cr cr I E ft 1 E T E T E T 4 cr cr cr cr cr cr cr (56) E E cr r y T E J C cr w I E (1 ) VI. Discussion of Resuts This work has successfuy appied the finite Fourier Sine transform method to the feura-torsiona bucking anaysis of thin-waed open section coumns with both ends simpy supported. The finite sine transform was appied to the governing equations which were a system of three couped differentia equations in terms of the three dispacement functions v(), w() and ( ). After simpification, the system of transformed equations reduced to a system of homogeneous agebraic equations in terms of the three transformed DOI: 1.979/ Page (5) (55)

9 Appication of the Finite Fourier Sine Transform Method for the Feura-Torsiona Bucking.. dispacement functions V(m, ), W(m, ) and ( m, ) as presented in Equation (3). The bucking or characteristic equation is obtained from the requirement of vanishing of the determinant of the matri of coefficients and obtained as Equation (33). The eros of the resuting poynomia equation in N woud yied the bucking oads of the coumn. Two cases of the genera bucking probem were considered: namey case of douby symmetric crosssections and singy symmetric cross-sections. For douby symmetric cross-sections, the shear center coincide with the centroid of the cross-section, yieding e y = e =. The system of governing differentia equations become uncouped, resuting in bucking modes, that are uncouped. The characteristic bucking equation for douby symmetric sections found as equation (35); is uncouped, with three roots representing the eigenvaues of the bucking probem. The three roots are the Euer feura bucking oad in the ais, the Euer feura bucking oad in the y ais, and the oad in torsiona (twist) bucking. The critica bucking oad is the owest vaue of the three bucking oads, and determines how the coumn with douby symmetrica cross section wi fai. The second case considered coumns with monosymmetric cross-sections, ais is the ais of symmetry, e y =, and from the system of governing differentia equations, Equation (7) is uncouped from Equations (6) and (8). The characteristic bucking equation was found in this case as Equation (45) a poynomia equation in N with one feura bucking mode uncouped and the other feura bucking mode couped with the torsiona bucking mode. Thus such monosymmetric coumns can fai by Euer feura bucking mode in the yy direction and torsiona-feura bucking mode. The eros of the poynomia (equivaent to the eigenvaues) yied the vaues of the bucking oad. The poynomia woud yied three vaues of the bucking oad namey Euer feura bucking oad in the yy direction and two couped torsiona-feura bucking oads given by Equation (48). Evidenty, Equation (5) woud give ower vaues of the torsiona-feura bucking oad; and the bucking mode woud be governed by N in Equation (5) or the Euer feura bucking oad in the y-direction, whichever one is smaer. The epressions for the bucking oads obtained by finite Fourier Sine transform method agree eceenty with soutions by Det [11] and Wang et a [8]. VII. Concusion For this study, the foowing concusions can be adduced: (i) For douby symmetric thin-waed coumns, the system of governing differentia equations are uncouped resuting in bucking modes and bucking oads that are uncouped. (ii) For monosymmetric thin waed coumns, with ais being the ais of symmetry, feura bucking mode in the y direction is uncouped whie the feura bucking mode in the direction is couped with torsiona bucking. (iii) For thin waed couns without symmetry about any ais, the three bucking modes are couped and the two feura bucking modes interact with torsiona bucking mode. The finite Fourier Sine transform method has been shown to be an effective too for the anaysis of thin-waed coumns with pinned ends. The advantage of the method is that no apriori assumption about bucking modes was necessary to sove the system of differentia equations. The method yieded eact soutions. References [1] Zhu Shuai: Eastic Feura Torsiona Bucking Anaysis of Douby Symmetrica Web Tapered Beams. MSc Thesis, University of Pittsburgh, Oct 9, 8pp. [] Howett J.H.: An investigation into the structura behaviour of thin-waed auminum aoy weded battened struts. MSc Thesis, University of Durham, Juy 197, avaiabe at Durham E-Thesis onine http//ethesis.dur.ac.uk/193. [3] Trahair N.S.: Torsion Equations for atera bucking. Research Report R964, Juy 16, Schoo of Engineering, The University of Sydney. [4] Asayed S.H.: Ineastic behaviour of singe ange coumns. PhD Thesis, The University of Ariona, University Microfims Internationa. http//nd.hander.net/115/ [5] Timoshenko S.P. and Gere J.M.: Theory of Eastic Stabiity. McGraw Hi Kogakusha Ltd, New York, [6] Aen H.G. and Buson P.S.: Background to Bucking. McGraw Hi Book Co., 198. [7] Chajes A.: Principes of Structura Stabiity Theory. Prentice Ha, New Jersey, [8] Wang C.M., Wang C.Y. and Reddy J.N.: Eact Soution for Bucking of Structura Members. CRC Series in Computationa Mechanics and Appied Mechanics, CRC Press, USA, 5. [9] Ike C.C., Nwoji C.U., Ikwuee E.U. and Ofondu I.O.: Soution of the Generaised Eastic Coumn Bucking Probem by the Gaerkin Variationa Method. Internationa Journa of Research in Appied Science and Engineering Technoogy (IJRASET), Voume 5 Issue 1, pp , January 17. [1] Torsion in Structura Design. http//peope.virginia.edu/ttb/torsion.pdf [11] Det N.V.: Bucking Strength Anaysis of Bars and Frames, and Spherica Shes Cassification Notes No Norway, Apri 4. [1] Trahair N.S.: Feura-Torsiona Bucking of Structures. CRC Press Tokyo, [13] Doetsch G.: Integration von Differentiageichungen vermittes der endichen Fourier transformation Math. Ann, 11, pp 5-68, [14] Kneit K.: Lösung von Rondwert probeme bei systemem gewöhuicher Differentiageichungen vermittes der endichen Fourier transformation, Math. Zeit, 44, pp 66-91, [15] Strandhagen A.G.: Use of sine transform for non-simpy supported beams Quart. App. Math 1, pp , DOI: 1.979/ Page

10 Appication of the Finite Fourier Sine Transform Method for the Feura-Torsiona Bucking.. [16] Roettinger I.: An operationa approach to the soution of boundary vaue probems by generaised Fourier series. Bu American Math. Soc. 51 p.67, [17] Brown H.K.: Resoution of temperature probems by use of finite Fourier transformations. Bu. American Math. Soc., 5, pp , DOI: 1.979/ Page

Malaysian Journal of Civil Engineering 30(2): (2018)

Malaysian Journal of Civil Engineering 30(2): (2018) Maaysian Journa of Ci Engineering 3():331-346 (18) BUBNOV-GALERKIN METHOD FOR THE ELASTIC BUCKLING OF EULER COLUMNS Ofondu I.O. 1, Ikwueze E. U. & Ike C. C. * 1 Dept. of Mechanica and Production Engineering,