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

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1 Maaysian Journa of Ci Engineering 3(): (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, Enugu State Unersity of Science and Technoogy, Enugu State, Nigeria. Dept. of Ci Engineering, Enugu State Unersity of Science and Technoogy, Enugu State, Nigeria. *Corresponding Author: ikecc7@yahoo.com Abstract: In this work the Bubnov-Gaerkin variationa method was appied to determine the critica bucking oad for the eastic bucking of coumns with fixed-pinned ends. Coordinate shape functions for Euer coumn with fixed-pinned ends are used in the Bubnov-Gaerkin variationa integra equation to obtain the unknown parameters. One parameter and two parameter shape functions were used. In each case, the Bubnov-Gaerkin method reduced the boundary vaue probem to an agebraic eigen-vaue probem. The soution of the characteristic homogeneous equations yieded the bucking oads. One parameter coordinate shape function yieded reate error of 4% compared with the exact soution. Two parameter coordinate shape function gave a reate error of.77%, which is negigibe. Keywords: Bubnov-Gaerkin method, eastic bucking, Euer coumn bucking, bucking oad 1. Introduction Artice history: Receed 3 August 17 Receed in revised form 1 Dec 17 Accepted 15 Apri 18 Pubished onine 3 June Background/Literature Review Coumns are ong sender bars under axia compresse forces. They can be horizonta, vertica or incined. They are cassified as short coumns, ong coumns or intermediate coumns. When a sender member is subjected to an axia compresse force, it may fai due to bucking (Rao, 16; homepages, 16; Lagace, 9). Bucking is a geometric instabiity in which the atera dispacement of the axiay compressed coumn can suddeny become very arge (Rao, 16; Lagace, 9; Punmia et a., ; Jayaram, 7). Short coumns fai by crushing or compresse yieding of the materia. Long coumns fai by bucking or fexura bucking which is a geometric faiure or instabiity. Intermediate coumns fai by a combination of crushing and fexura bucking faiures. A rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Penerbit Unersiti Teknoogi Maaysia

2 33 Maaysian Journa of Ci Engineering 3(): (18) Intermediate coumns fai by both compresse yieding and fexura bucking of the coumn. Thin structures subject to compression oads that have not achieved the materia strength imits can fai by bucking. (Beeman, 14). Bucking is characterized by a sudden faiure of a structura member subjected to high compresse stress where the actua compresse stress at the imminence of faiure is ess than the utimate materia compresse strength (Beeman, 14; Novoseac et a., 1). The critica bucking oad for an axiay compressed coumn, determined using a inear eastic bucking anaysis of an ideaized perfect structure does not necessariy correspond to the oad at which instabiity of the rea structure takes pace (Fernandez, 13). The cacuated critica bucking oad does not provide sufficient information about when faiure due to the instabiity of the structure as a whoe wi occur. This depends on other factors ike initia geometrica imperfections, eccentricities of oading, and the noninear defection behaviour of the structure (Yao and Lee, 11). Bucking can be anayzed using inear bucking anaysis (eigenvaue) or non inear bucking anaysis (Eryimaz et a., 13). The objecte of inear bucking anaysis is to determine the bucking oad factor and the critica bucking oad (Digita Engineering, 17). Critica bucking oads in inear (eigenvaue) bucking anaysis may be determined using any of the foowing methods: i. by exact mathematica methods of soving the governing differentia equations of equiibrium subject to the boundary conditions. This yieds exact vaues of the critica bucking oads. ii. by using approximate methods, which may be based on energy principes, variationa methods or discrete approximations of the governing differentia equations of equiibrium and the boundary conditions. Mathematicay rigourous techniques of soving the boundary vaue probem of coumn bucking which consists of soving the governing differentia equation of equiibrium on the probem domain subject to the prescribed boundary conditions presents considerabe difficuties and can ony be achieved for simpe bucking probems for structures with ow degrees of freedom. Such probems which are difficut to sove in cosed anaytica form are usuay soved using the approximate methods based on discretization of the governing equations, variationa methods and energy principes. Approximate methods have been used to sove the coumn bucking probem by Zdravkovic et a., 13; Li et a., 11; Huang and Li, 11; Kaakowski et a., 16; Reddy, 14; Yuan and Wang, 11; Atay, 9; and Okay et a., 1). Basebuk et a., (14) used the Hemotopy Anaysis Method (HAM) to find the critica bucking oad of a coumn under end oad dependent on direction. Eryimaz et a. (13)

3 Maaysian Journa of Ci Engineering 3(): (18) 333 impemented the HAM to determine the bucking oads of Euer coumns with a continuous eastic restraint. Atay (9) determined the critica bucking oads for variabe stiffness Euer coumns using homotopy perturbation method. Okay et a. (1) used the variationa iteration method (VIM) to determine bucking oads and bucking moda shapes of coumns. Yuan and Wang (11) used the differentia quadrature method (DQM) to carryout bucking and post bucking anaysis of beamcoumns. Reddy (14) impemented bucking anaysis of cracked stepped coumn using the Finite Eement Method (FEM). Zdravkovic et a. (13) used the energy method for the efficient estimation of the eastic bucking oad of axiay compressed three-segment stepped coumn. In this paper the Bubnov-Gaerkin method is used to determine the critica bucking oad of prismatic Euer coumn of ength, with fixed-pinned ends at x, and x respectey, where x is the ongitudina coordinate axis of the coumn. 1. Euer Theory of Bucking Euer considered an eastic coumn of ength that is pinned at the ends x and x and subjected to an axia compresse force P. The coumn undergoes a atera defection denoted by v(x). Moment equiibrium of a section of the defected coumn cut at an arbitrary point, x, from the end, x, and the appication of the moment curvature equation resuts in (homepages, 16; Lagace, 9; Megson, 5; Lowe, 1971): d v( x) Pv( x) M( x) EI (1) dx Or, d v P vx ( ) dx + EI () where E is the Young s moduus of easticity, I is the moment of inertia and M(x) is the bending moment variation aong the ongitudina axis of the coumn, and x denotes the ongitudina axis coordinate of the coumn. 1.3 Genera Equation for coumn bucking The second order differentia Equation () appies to coumns with simpy supported ends x and x. The more genera coumn bucking equation uses the formuation simiar to the bending of a beam, but incuding the axia forces (Lowe, 1971; homepages, 16). The forces and moments acting on an eementa part of a coumn under axia compresse forces are shown in Figure 1.

4 334 Maaysian Journa of Ci Engineering 3(): (18) Figure 1: Forces and moments acting on an eementa coumn For vertica equiibrium, dq qx ( ) dx (3) where Q(x) is the shear force. For horizonta equiibrium dp P( x) P( x + x) P( x) + dx (4) dx P(x) is constant For moment equiibrium, dm dv + P Q dx dx (5) dv The equation for moment equiibrium contains an extra term P dx which is absent in the equation for Euer Bernoui beam fexure theory. By differentiation of Equation (5), with respect to x, we obtain: d dm dv dq P + q( x) dx dx dx dx (6)

5 Maaysian Journa of Ci Engineering 3(): (18) 335 Simpifying, d M d v + P q( x) (7) dx dx Appication of the moment curvature reation in Equation (7) yieds: d d v d v EI P q( x) + dx dx dx (8) For the prismatic coumns, EI is constant and the Equation (8) simpifies to (Megson, 5): 4 d v d v EI + P q( x) (9) 4 dx dx 1.4 Assumptions of the Euer s Theory of coumn bucking Euer s theory of coumn bucking is based on the foowing assumptions (Punmia et a., 1; Jayaram, 7; Megson, 5; Lowe, 1971): (i) (ii) (iii) () (v) (vi) (vii) The coumn is straight in the ongitudina direction before the appication of oad. The coumn has a uniform cross section throughout its ongitudina axis. The coumn materia is isotropic and homogeneous. The sef-weight of the coumn materia is disregarded. The ine of appication of axia compresse oad is coincident with the ongitudina axis of the coumn. The reduction in ength of the coumn due to axia compression is very sma and disregarded. The coumn fais due to bucking aone. 1.5 Research Aim and Objecte The research aim and objecte is to use the Bubnov-Gaerkin variationa method to determine the eastic bucking oads of Euer coumns with fixed pinned ends at x, and x; respectey.

6 336 Maaysian Journa of Ci Engineering 3(): (18). Theoretica Framework and Methodoogy The governing differentia equation for the eastic bucking of prismatic Euer coumns under axia compresse oad P when transverse oads are absent is gen by the fourth order ordinary differentia equation (ODE) with constant coefficients gen by: EIv + Pv (1) where v(x) is the defection, E is the Young s Moduus of the coumn materia. I is the moment of inertia. The fourth order ordinary differentia Equation (1) is soved subject to the boundary conditions of the coumn. The mathematica probem of the coumn bucking is an eigenvaue probem (Lowe, 1971). In the Bubnov-Gaerkin method, the defection function v(x) is chosen in terms of shape functions that automaticay satisfy the boundary conditions and undetermined parameters caed generaized dispacement parameters that are sought such that the Bubnov-Gaerkin variationa integra woud vanish. This impies that the weighted error or weighted residua where the shape functions serve as the weighting functions woud vanish over the entire coumn. For one parameter dispacement fied v( x) c N ( x) (11) 1 1 where c 1 is the unknown dispacement parameter N 1(x) is the dispacement shape function that satisfies the boundary conditions. The Bubnov-Gaerkin integra for a one parameter dispacement fied is gen by: ( EIv + Pv ) N1( x) dx (1) P ( ) 1 v + v N ( x) dx (13) EI P ( ) c N N + c N N dx (14) EI P ( ) c1 N1 N1 + N1N1 dx (15) EI

7 Maaysian Journa of Ci Engineering 3(): (18) 337 P c1 N1 N1 dx + N1 N1 dx EI (16) ( ) c k + k (17) g where, k N N dx (18) k 11g 1 1 N N dx (19) and is the bucking oad factor. P () EI For non-tria soutions, c1 and the characteristic bucking equation becomes the agebraic eigen-vaue eigen vector probem k 11 k11 g + (1) Expanding, and soving, k11 () k 11 g The critica bucking oad can then be determined. For a two parameter dispacement fied, v(x) c 1N 1(x) + c N (x) (3) where c 1, c are the two unknown parameter and N 1(x), N (x) are the dispacement shape functions which are chosen to satisfy the boundary conditions. The Bubnov-Gaerkin variationa integra is:

8 338 Maaysian Journa of Ci Engineering 3(): (18) Expanding, ( c N + c N ) + ( c N + c N ) N dx (4) EI P ( c N + c N ) + ( c N + c N ) N dx (5) EI P P ( ) P c1 ( N1 + N1 ) N1 dx + c N + N N1 dx EI EI (6) P P ( ) ( ) (7) c1 N1 + N1 N dx + c N + N N dx EI EI P P ( ) ( ) (8) c1 N1 N1dx + N1N1dx + c N N1dx + NN1dx EI EI P P ( ) ( ) (9) c1 N1 Ndx + N1Ndx + c N Ndx + NNdx EI EI Let 1 1 Then, we have: k N N dx (3) k 1g 1 N N dx (31) 1 k N N dx (3) k 1g 1 N N dx (33) k N N dx (34) k g N N dx (35) c k + k + c k + k (36) 1( ) ( 1 1 ) g g c k + k + c k + k (37) 1( 1 1 ) ( ) g g

9 Maaysian Journa of Ci Engineering 3(): (18) 339 k11 + k11 k g 1 + k1g c1 k + k k + k c 1 1g g (38) For non-tria soutions, c1 c The bucking equation becomes: k + k k + k 11 11g 1 1g k + k k + k 1 1g g (39) Expanding, ( k + k )( k + k ) ( k + k )( k + k ) (4) g g g g This yieds a quadratic equation in terms of, from which the two roots of yied two vaues of bucking oads. 3. Appication to the Bucking of fixed - Hinged or Camped Pinned Coumns 3.1 Probem considered The eastic bucking of Euer coumn with fixed-pinned ends at x and x was considered as shown in Figure. The governing differentia equation is gen by Equation (1).

10 34 Maaysian Journa of Ci Engineering 3(): (18) Figure : Eastic bucking of Euer coumn with fixed-pinned ends The boundary conditions are: v(x ) θ(x ) v(x ) v( x ) v( x ) (41) A suitabe dispacement coordinate shape function that satisfies the boundary conditions at x, and x is: x x x ( ) ( ) ( ) 4 3 v( x) c for a one parameter choice of v(x) where N 1(x) is the dispacement shape function, c 1 is the generaized dispacement parameter: x x x ( ) ( ) ( ) 4 3 (4) N1( x) (43) For a two parameter choice of v(x) Equation (3) is used where N 1(x) is gen by Equation (43) and N x 7 x 4 x ( ) ( ) ( ) (44) 3 3 c 1 and c are the two unknown generaized dispacement parameters.

11 Maaysian Journa of Ci Engineering 3(): (18) One term (Parameter) Bubnov-Gaerkin Soution The Bubnov-Gaerkin variationa integra is gen by Equation (15). By differentiation, 1x 15x 3 N1 + (45) 4 3 N 4 (46) 1 4 Therefore, the Bubnov-Gaerkin variationa integra becomes: ( ) ( ) ( ) x 15x 3 EIc * x. x. x 1 + Pc Integrating and simpifying, c dx (47) P (48) EI The differentia equation simpifies to an agebraic eigen-vaue eigen-vector probem gen by Equation (48). For non-tria soution, c 1 Then the characteristic bucking equation becomes the homogeneous equation Soving 1. 8 P (49) 3 EI P cr EI 1 (5) 3.3 Two term (parameter) Bubnov-Gaerkin Soution For the dispacement (shape) coordinate functions considered Equations (45) and (44), differentiation yieds:

12 34 Maaysian Journa of Ci Engineering 3(): (18) 3 x 8x 8x N + (51) N 1x 56 (5) 5 4 Whie N1 and N 1 are gen by Equations (45) and (46) respectey By integration, N N dx (53) 18. N 1 N1dx (54) 3 N N dx (55) N Ndx (56) 3 8. N N1dx (57) 3 N N dx (58) N N dx (59) 8. N 1 Ndx (6) 3 The Gaerkin variationa integras then reduce to the system of agebraic equations in terms of the unknown generaized parameters, c 1 c as foows:

13 Maaysian Journa of Ci Engineering 3(): (18) P P c 1 c EI EI P P c 1 c EI EI (61) (6) In matrix form, c c (63) P where (64) EI ( ) ( ) c1 ( ) ( ) c (65) For non-tria coumns c1 c The characteristic bucking equation then becomes: (66) By expansion, we obtain after simpification, (67) Soving, P (68) EI

14 344 Maaysian Journa of Ci Engineering 3(): (18) then, P cr EI (69) The exact soution for P cr is P cr EI. 197 (7) 4. Discussion The Bubnov-Gaerkin variationa method has been successfuy appied in this work to determine the critica bucking oad of Euer coumns under axia compresse oad P when the ends are fixed at x and pinned at x. Coordinate shape functions that satisfy a the boundary conditions at the fixed-pinned ends were empoyed in a one parameter and a two parameter dispacement shape functions as Equations (43) and (44). The Gaerkin variationa integra statement for a one-parameter soution was obtained as Equation (15). For two parameter soution, the Gaerkin variationa integra statement was obtained as the system of Equations (61) and (6) upon evauation of the integras. For a one-parameter Bubnov-Gaerkin method, the probem reduced to an agebraiceigen vaue probem expressed by Equation (48). The characteristic bucking equation was obtained as Equation (49) and the soution gave the critica bucking oad for one parameter Bubnov-Gaerkin soution as Equation (5). Simiary, the two parameter Bubnov-Gaerkin soution reduced to the agebraic eigen-vaue probem expressed by Equation (63) or (65). The corresponding characteristic bucking equation was found as Equation (66). Equation (66) yieded a quadratic equation in terms of, which was soved to obtain the critica bucking oad for a two parameter Bubnov-Gaerkin soution as Equation (69). Comparison of the one term and two term Bubnov-Gaerkin soutions with the exact soutions gen as Equation (7) shows the one term soution has a reate error of 4% whie the two terms Bubnov- Gaerkin soution has a reate error of.77%. This shows the effecteness of the Bubnov-Gaerkin method in the eastic bucking anaysis of Euer coumns with fixed-pinned ends. 5. Concusions From the study, the foowing concusions can be made: i. The Bubnov-Gaerkin variationa method simpifies the boundary vaue probem of eastic bucking of Euer-coumn to an agebraic eigenvaue probem.

15 Maaysian Journa of Ci Engineering 3(): (18) 345 ii. As the number of undetermined generaized dispacement parameters increase, the accuracy of the Bubnov-Gaerkin method increases provided the coordinate (shape) functions satisfy a the boundary conditions at the ends. References Atay, M.T. (9): Determination of critica bucking oads for variabe stiffness Euer coumns using homotopy perturbation method. Internationa Journa of Noninear Sciences and Numerica Simuation 9 1() Beeman, A. (14): Coumn Bucking Anaysis. M.Eng Thesis Mechanica Engineering. Graduate Facuty Rensseaer Poytechnic Institute Gronton Apri, 14. Digita Engineering. Linear and Noninear Bucking in FEA. and_noninear-bucking_in_fea/ Accessed on 18/11/17. Eryimaz, A.. Atay, M.T., Coskun, S.B., Basbuk, M. (13): Bucking of Euer coumns with a constant eastic constraint via homotopy anaysis method. Journa of Appied Mathematics 13; 13; 8 coi /13/ [Cross Ref]. Fernandez, P. (13): Practica methods for critica oad determination and stabiity evauation of stee structures MSc Ci Engineering Thesis.Graduate Schoo Unersity of Coorado pp. 19. digita.auraria.edu/content/aa///1/11/aa 1_1.pdf. homepages, engineering. auckand.ac.n/ /7Easticity _ Appications_5_Bucking pdf. Huang, Y. and Li, X.F. (11): Bucking anaysis of nonuniform and axiay grade coumns with varying fexura rigidity. Journa of Engineering Mechanics voume 137, Issue 1, January, 11 (ASCE). Jayaram M.A. (7): Mechanics of Materias with Programs in C, Prentice Ha of India Prate Ltd, New Dehi. Koakowski, Z., Kowa-Michaska, K. and Mania, R.J. (16): Static bucking of FML Coumns in Eastic Pastic Range. Mechanics and Mechanica Engineering vo, No, (16) pp Lodz Unersity of Technoogy. Lagace P.A. (9): Unit M.4.7, The Coumn and Bucking, 16.3/4, Unified Engineering, Department of Aeronautics and Astronautics, Massachusetts Institute of Technoogy web mit.edu/16. unified/ Unified 9. pdf. Li, X.F.; Xi, L.Y. and Huang, Y. (11): Stabiity anaysis of composite coumns and parameter optimization against bucking. Composites Part B, vo. 4, No. 6, pp , 11. Lowe P. G (1971): Cassica Theory of Structures Based on the Differentia Equation,. Cambridge Unersity Press, Cambridge. Megson T.H.G. (5): Structura and Stress Anaysis, Second Edition, Esevier, Butterworth, Heinemann, Amsterdam. Novoseac, S; Ergic T., Baicevic, P. (1): Linear and Noninear bucking and post bucking anaysis of a bar with the infuence of imperfections. Tehnicki vjesnik 19, 3 (1) pp Okay, I.; Atay, M.T., Cockun, C.B. (1): Determination of bucking oads and mode shapes of a heavy vertica coumn under its own weight using the variationa iteration method. Internationa Journa of Noninear Sciences and Numerica Simuation 1. 11(1)

16 346 Maaysian Journa of Ci Engineering 3(): (18) Punmia B.C, Jain A.K., Jain A.K. (): Mechanics of Materias, Laxmi Pubication (P) Ltd, New Dehi. https//.books.googe.com/books? isbn Rao P.V. Unit IV Theory of Coumns %theory%of%coumns%. Rao pdf. Reddy, P.S. (14): Vibration and Bucking Anaysis of a Cracked Stepped Coumn using Finite Eement Method. M.Tech Ci Engineering (Structura Engineering) Thesis. Department of Ci Engineering, Nationa Institute of Technoogy, Ronkea, May, 14. Yao, C.H. and Lee, S.C. (11): Stabiity of Structures Principes and Appications. Butterworth Heinemann, New York 11. Yuan, Z.; Wang, X. (11): Bucking and post bucking anaysis of extense beam coumns by using the differentia quadrature method. Computers and Mathematics with appications 11: 6(1): doi:1.116/j.camwa [Cross Ref]. Zdravkovic Nebojsa, Gasic Miomir, Savkovic Mie. (13): Energy method in efficient estimation of eastic bucking oad of axiay oaded three-segment stepped coumn FME Transactions (13) vo 41, No 3, -, p 9.

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

Application of the Finite Fourier Sine Transform Method for the Flexural-Torsional Buckling Analysis of Thin-Walled Columns IOSR Journa of Mechanica and Civi Engineering (IOSR-JMCE) e-issn: 78-1684,p-ISSN: 3-334X, Voume 14, Issue Ver. I (Mar. - Apr. 17), PP 51-6 www.iosrjournas.org Appication of the Finite Fourier Sine Transform