Energy Equation for Beam Lateral Buckling
Publication: Journal of Structural Engineering
Volume 118, Issue 6
Abstract
This paper presents a derivation of the classical energy equation for the lateral buckling of doubly symmetric thin‐walled beams. This is based on the use of second‐order rotation components to obtain the nonlinear relationship between the longitudinal normal strain and the member deformations. The classical energy equation is compared with an alternative energy equation, and found to be significantly different in terms used to represent the work done by centroidal loads during buckling. The difference is attributed to the omission of some nonlinear components from the longitudinal displacements used for the alternative equation. Comparisons of finite element predictions based on the two energy equations demonstrate some substantial differences with the predictions by the alternative energy equation being higher for simply supported and continuous beams, and lower for cantilevers. Comparisons show that the available experimental evidence agrees well with the classical predictions, and they provide independent evidence that the alternative energy equation is incorrect.
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References
1.
Anderson, J. M., and Trahair, N. S. (1972). “Stability of monosymmetric beams and cantilevers.” J. Struct. Div., ASCE, 98(1), 269–287.
2.
Attard, M. A. (1986). “Nonlinear theory of non‐uniform torsion of thin‐walled open beams.” Thin‐Walled Struct., 4(2), 101–134.
3.
Barsoum, R. S., and Gallagher, R. H. (1970). “Finite element analysis of torsional and torsional‐flexural stability problems.” Int. J. Numer. Methods Engrg., 2, 335–352.
4.
Bild, S., Chen, G., and Trahair, N. S. (1988). “Out‐of‐plane strengths of steel beams.” Research Report No. R560, School of Civ. and Mining Engrg., Univ. of Sydney, Australia.
5.
Bleich, F. (1952). Buckling strength of metal structures. McGraw‐Hill, Inc., New York, N.Y.
6.
Brown, P. T., and Trahair, N. S. (1968). “Finite integral solution of differential equations.” Civ. Engrg. Trans., Institution of Engineers, 10(2), 193–196.
7.
Chajes, A. (1974). Principles of structural stability theory. Prentice‐Hall, Inc., Englewood Cliffs, N.J.
8.
Chan, S. L. (1990). “Buckling analysis of structures composed of tapered members.” J. Struct. Engrg., ASCE, 116(7), 1893–1906.
9.
Chen, W. F., and Liu, E. M. (1987). Structural stability: Theory and implementation. Elsevier, New York, N.Y.
10.
Clark, J. W., and Jombock, J. R. (1957). “Lateral buckling of I‐beams subjected to unequal end moments.” J. Engrg. Mech. Div., ASCE, 83(3), 1291:1–1291:19.
11.
DeJong, H. (1990). “An approach to more complicated lateral‐buckling problems.” J. Constr. Steel Res., 16(3), 231–246.
12.
Donald, G. S., and Kleeman, P. W. (1971). “A stiffness matrix for the large deflection analysis of space frames.” Proc. 3rd Australasian Conference on the Mech. of Struct. and Mater., Univ. of Auckland, New Zealand.
13.
Flint, A. R. (1950). “The stability and strength of slender beams.” Engrg., 170, Dec., 545–549.
14.
Flint, A. R. (1952). “The lateral stability of unrestrained beams.” Engrg., 173, Jan., 65–67.
15.
Gellin, S., and Lee, G. C. (1988). “Finite elements available for the analysis of noncurved thin‐walled structures.” Finite element analysis of thin‐walled structures, J. W. Bull, Ed., Elsevier Applied Science, London, U.K., 1–45.
16.
Guide to stability design criteria for metal structures. (1988). 4th Ed., T. V. Galambos, Ed., John Wiley & Sons, New York, N.Y.
17.
Hancock, G. J. (1975). “The behaviour of structures composed of thin‐walled members,” PhD thesis, Univ. of Sydney, Sydney, Australia.
18.
Hancock, G. J., and Trahair, N. S. (1978). “Finite element analysis of lateral buckling of continuously restrained beam‐columns.” Civ. Engrg. Trans., Institution of Engineers, 20(2), 120–127.
19.
Handbook of structural stability. (1971). Corona Publishing Co., Tokyo, Japan.
20.
Hechtman, R. A., Hattrup, J. S., Styer, E. F., and Tiedemann, J. C. (1955). “Lateral buckling of rolled steel beams.” Proc. Engrg. Mech. Div., ASCE, 81, 797:1–797:21.
21.
Ings, N. L., and Trahair, N. S. (1987). “Beam and column buckling under directed loading.” J. Struct. Engrg., ASCE, 113(6), 1251–1263.
22.
Kitipornchai, S., and Trahair, N. S. (1975). “Inelastic buckling of simply supported steel I‐beams.” J. Struct. Div., ASCE, 101(7), 1333–1347.
23.
Kitipornchai, S., and Chan, S. L. (1987). “Nonlinear finite element analysis of angle and tee beam‐columns.” J. Struct. Engrg., ASCE, 113(4), 721–739.
24.
Love, A. E. H. (1944). A treatise on the mathematical theory of elasticity. 4th Ed., Dover Publications, Inc., New York, N.Y.
25.
Michell, A. G. M. (1899). “Elastic stability of long beams under transverse forces.” Phil. Mag., 48, 298–309.
26.
Nethercot, D. A. (1983). “Elastic lateral buckling of beams.” Beams and beam columns—Stability and strength, R. Narayanan, Ed., Applied Science Publishers Ltd., Essex, U.K., 1–33.
27.
Pandey, M. D., and Sherbourne, A. N. (1990). “Elastic, lateral‐torsional stability of beams: General considerations.” J. Struct. Engrg., ASCE, 116(2), 317–335.
28.
Poowannachaikul, T., and Trahair, N. S. (1976). “Inelastic buckling of continuous steel I‐beams.” Civ. Engrg. Trans., Institution of Engineers, 18(2), 134–139.
29.
Pi, Y. L., Trahair, B. S., and Rajasekaran, S. (1991). “Energy equation for beam lateral buckling.” Research Report No. R635, School of Civ. and Mining Engrg., Univ. of Sydney, Australia.
30.
Prandtl, L. (1899). “Kipperscheinungen,” thesis, Munich, Germany (in German).
31.
Rajasekaran, S. (1977). “Finite element method for plastic beam‐columns.” Theory of beam‐columns, Vol. 2, space behaviour and design, W. F. Chen and T. Atsuta, Eds., McGraw‐Hill, Inc., New York, N.Y., 539–608.
32.
Roberts, T. M. (1983). “Instability, geometric non‐linearity and collapse of thin‐walled beams.” Beams and beam columns—stability and strength, R. Narayanan, Ed., Applied Science Publishers, London, U.K., 135–160.
33.
Timoshenko, S. P. (1953a). “Einige Stabilitaetsprobleme der Elatizitaetstheorie.” Collected papers of Stephen P. Timoshenko, McGraw‐Hill, Inc., New York, N.Y. 1‐50 (in German).
34.
Timoshenko, S. P. (1953b). “Sur la Stabilite des Systeme Elastiques.” Collected papers of Stephen P. Timoshenko, McGraw‐Hill, Inc., New York, N.Y., 92‐224 (in French).
35.
Timoshenko, S. P., and Gere, J. M. (1961). Theory of elastic stability. 2nd Ed., McGraw‐Hill, Inc., New York, N.Y.
36.
Trahair, N. S. (1969). “Elastic stability of continuous beams.” J. Struct. Div., ASCE, 95(6), 1295–1312.
37.
Trahair, N. S., and Bradford, M. A. (1988). The behavior and design of steel structures. 2nd Ed., Chapman and Hall, London, U.K.
38.
Van Erp, G. M., Menken, C. M., and Veldpaus, F. E. (1988). “The nonlinear flexural‐torsional behaviour of straight slender elastic beams with arbitrary cross sections.” Thin‐Walled Struct., 6(5), 385–404.
39.
Vlasov, V. Z. (1961). Thin‐walled elastic beams. 2nd Ed., Israel Program for Scientific Translation, Jerusalem, Israel.
40.
Whittaker, E. T. (1927). A treatise on the analytical dynamics of particles and rigid bodies. 3rd Ed., Cambridge Univ. Press, London, U.K.
41.
Woolcock, S. T., and Trahair, N. S. (1974). “Post‐buckling behaviour of determinate beams.” J. Engrg. Mech. Div., ASCE, 100(2), 151–171.
42.
Yang, Y.‐B., and McGuire, W. (1986). “Stiffness matrix for geometric nonlinear analysis.” J. Struct. Engrg., ASCE, 112(4), 853–877.
43.
Zienkiewicz, O. C., and Taylor, R. L. (1989). The finite element method. 4th Ed., McGraw‐Hill, Inc., Maidenhead, U.K.
Information & Authors
Information
Published In
Journal of Structural Engineering
Volume 118 • Issue 6 • June 1992
Pages: 1462 - 1479
Copyright
Copyright © 1992 ASCE.
History
Published online: Jun 1, 1992
Published in print: Jun 1992
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