# Least Squares Weighted Residual Method for Solving the Generalised Elastic Column Buckling Problem

Least Squares Weighted Residual Method for Solving the Generalised Elastic Column Buckling Problem

Charles C. IkeClifford U. Nwoji Benjamin O. Mama Hyginus N. Onah

Department of Civil Engineering, Faculty of Engineering, Enugu State University of Science and Technology, Enugu, 400001, Enugu State, Nigeria

Department of Civil Engineering, University of Nigeria, Nsukka, Enugu State, Nigeria

Corresponding Author Email:
ikecc2007@yahoo.com
Page:
78-85
|
DOI:
https://doi.org/10.18280/ti-ijes.630111
19 January 2019
|
Accepted:
15 March 2019
|
Published:
31 March 2019
| Citation

OPEN ACCESS

Abstract:

In this work, the least squares weighted residual method (LSWRM) was used to solve the generalised elastic column buckling problem for the case of pinned ends. Mathematically, the problem solved was a boundary value problem (BVP) represented by a system of three coupled linear ordinary differential equations (ODEs) in terms of three unknown displacement functions and subject to boundary conditions. The least squares residual method used formulated the problem as a variational problem, and reduced it to an algebraic eigenvalue problem which was solved to obtain the characteristic buckling equation. The characteristic stability equation was found to be a cubic polynomial for the general asymmetric sectioned column. The buckling modes were found as coupled flexural – torsional buckling modes. Two special cases of the problem were studied namely: doubly symmetric and singly symmetric sections. For doubly symmetric sections, the buckling loads and the buckling mode were found to be decoupled and the buckling mode could be flexural or flexural – torsional. For singly symmetric section columns, one of the bucking modes becomes decoupled while the others are coupled. The buckling equation showed the column could fail by either pure flexure or coupled flexural – torsional buckling mode. The results of the present work agree with Timoshenko’s results, and other results from the technical literature.

Keywords:

Least squares weighted residual method, generalised elastic column buckling problem, asymmetric section, singly symmetric section, doubly symmetric section, characteristic buckling equation, algebraic eigenvalue eigenvector problem

1. Introduction
2. Theoretical Framework
3. Methodology
4. Results
5. Discussion
6. Conclusion
References

[1]    Torsion in Structural Design. http//people.virginia. edu/ttb/torsion.pdf. pp.73, accessed on 01/01/2017.

[2]    Onah HN, Ike CC, Nwoji CU. (2017). Flexural – torsional buckling analysis of thin walled columns using the Fourier series method. International Journal of Advanced Engineering Research and Science (IJAERS) 4(3): 292-298. https//dx.doi/10.22161/ijaers.4.3.45

[3]    Mama BO, Ike CC, Nwoji CU, Onah HN. (2017). Application of the finite Fourier sine transform method for the flexural – torsional buckling analysis of thin-walled columns IOSR. Journal of Mechanical and Civil Engineering (IOSR-JMCE) 14(2): 51-60. https://doi.org/10.9790/1684-1402015160

[4]    Ike CC, Nwoji CU, Ikwueze EU, Ofondu IO. (2017). Solution of the generalised elastic column buckling problem by the Galerkin variational method. International Journal for Research in Applied Science and Engineering Technology (IJRASET) 5(1): 468-475.

[5]    Timoshenko SP, Gere JM. (1961). Theory of elastic stability. McGraw Hill Koga Kusha Ltd New York.

[6]    Al-Sheikh AMS. (1985). Behaviour of thin-walled structures under combined loads. PhD Thesis Loughborough University of Technology.

[7]    Riley CE. (2003). Elastic buckling loads of slender columns with variable cross-sections by the Newmark method. MSc Thesis, Department of Civil Engineering Colorado State University.

[8]    Euler L. (1759). Sur la force des colonnes. Momoires de l’académie des sciences de Berlin 13: 252-282. in Opera Omnia set 2 17: 89-118.

[9]    St Venant AJCB. (1855). Memoire sur la torsion des Prismes Mem Divers Savants 14: 233-560.

[10]    Michell AGM. (1899). Elastic stability of long beams under transverse forces. Philos Mag 48 5th Series 5 48(292): 298-309. https://doi.org/10.1080/14786449908621336

[11]    Prandtl L. (1899) Kipperscheinungen. Doctoral Dissertation des Universitat München.

[12]    Timoshenko SP. (1905). On the stability in pure bending of a T beam. Bull Pol. Ins St Petersburg, pp. 4-5.

[13]    Timoshenko SP. (1936). Theory of Elastic Stability. McGraw Hill, New York.

[14]    Timoshenko SP. (1945). Theory of bending torsion and buckling of thin-walled members of open cross-section. Journal Franklin Institute 239(3): 201-219. https://doi.org/10.1016/0016-0032(45)90093-7

[15]    Wagner H. (1936). Torsion and Buckling of open sections. Technical Memorandum No 807 US National Advisory Committee for Aeronautics, p. 18.

[16]    Vlasov VZ. (1961). Thin walled elastic beams English translation. National Science Foundation, Washington DC London Oldbourne Press.

[17]    Alsayed SH. (1987). Inelastic behaviour of single angle columns. PhD Thesis, The University of Arizona University Microfilms International http//hdl.handlev.net/10150/184041.

[18]    Zlatko TZ. (2012). Stress and strain deflection of an open profile thin walled beam at constrained torsion by boundary element method. Journal of Theoretical and Applied Mechanics Sophia 42(2): 43-54. https://doi.org/10.2478/v10254-012-0007-y

[19]    Trahair NS. (1993). Flexural – Torsional Buckling of Structures. CRC Press Ann Arbor.

[20]    Trahair NS. (2016). Torsion equations for lateral buckling Research report R 964 July 2016 School of Engineering, The University of Sydney.

[21]    Allen HG, Bulson PS. (1980). Background to Buckling. McGraw Hill Book Company, London.

[22]    Chajes A. (1974). Principles of structural stability theory. Prentice Hall New Jersey.

[23]    Avcar M. (2014). Elastic buckling of steel columns under axial compression. American Journal of Civil Engineering 2(3): 102-108. https://doi.org/10.11648/j.ajce.20140203.17

[24]    Nwakali JA. (1990). The collapse behaviour of double layer space trusses incorporating eccentrically loaded tee-section members. Ph.D Thesis Department of Civil Engineering University of Surrey, November 1990.

[25]    Howlett JN. (1972). An investigation into the structural behaviour of thin walled aluminium alloy welded battered struts. MSc Thesis University of Durham July 1972 available at Durham E. Thesis online http//ethesis.dur.ac.uk/10293

[26]    Wang CM, Wang CY, Reddy JN. (2005). Exact solution for buckling of structural members. CRC series in Computational Mechanics and Applied Mechanics CRC Press, USA.

[27]    Det Norske Veritas (2004). Buckling strength analysis of bars and frames, and spherical shells. Classification Notes 30: 15.

[28]    Zhu S. (2009). Elastic flexural torsional buckling analysis of doubly symmetrical web tapered beams. MSc Thesis University of Pittsburg Oct. 2009, p. 208.