Analytical Solution for Initial Postbuckling Deformation of the Sandwich Beams Including Transverse Shear
Publication: Journal of Engineering Mechanics
Volume 139, Issue 8
Abstract
This paper presents analytical approximate solutions for the initial postbuckling deformation of sandwich beams including transverse shear. The approximate procedure is based on the nonlinear beam equation (with transverse shear included) by coupling the well-known Maclaurin series expansion and orthogonal Chebyshev polynomials; the governing differential equation with sinusoidal nonlinearity can be reduced to form a cubic-nonlinear equation. Analytical approximations to the resulting boundary condition problem are established by combining the methods of Newton linearization and harmonic balance. Unlike the classical method of harmonic balance, the linearization is performed prior to proceeding with harmonic balancing thus resulting in a set of linear algebraic equations instead of one of nonlinear algebraic equations, thereby establishing analytical approximate solutions. Illustrative examples are presented for a few typical sandwich construction configurations, and it is shown that the proposed approximate solutions are in excellent agreement with the numerical solution obtained via the shooting method for both small and large angles of rotation.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
This work was supported by the Program for Deep Continental Scientific Drilling Equipment Development-SinoProbe-09-05 (Project No. 201011082), Innovative Project of Scientific Forefront and Interdisciplinary of Jilin University (Grant No. 201103137), and China Postdoctoral Science Foundation (Grant No. 20110491290).
References
Allen, H. G. (1969). Analysis and design of structural sandwich panels, Pergamon Press, Oxford, NY.
Atanackovic, T. M. (1997). Stability theory of elastic rods, World Scientific, Singapore.
Beléndez, A., Álvarez, M. L., Fernández, E., and Pascual, I. (2009). “Linearization of conservative nonlinear oscillators.” Eur. J. Phys., 30(2), 259–270.
Cveticanin, L. J., and Atanackovic, T. M. (1998). “Leipholz column with shear and compressibility.” J. Eng. Mech., 124(2), 146–151.
El Fatmi, R. (2007). “A non-uniform warping theory for beams.” C. R. Mec., 335(8), 467–474.
Frostig, Y., Baruch, M., Vilnay, O., and Sheinman, I. (1992). “High-order theory for sandwich-beam behavior with transversely flexible core.” J. Eng. Mech., 118(5), 1026–1043.
Ghazouani, N., and El Fatmi, R. (2010). “Extension of the non-uniform warping theory to an orthotropic composite beam.” C. R. Mec., 338(12), 704–711.
Huang, H. Y., and Kardomateas, G. A. (2002). “Buckling and initial post-buckling behavior of sandwich beams including transverse shear.” AIAA J., 40(11), 2331–2335.
Hunt, G. W., Da Silva, L. S., and Manzocchi, G. M. E. (1988). “Interactive buckling in sandwich structures.” Proc. R. Soc. A., 417(1852), 155–177.
Hunt, G. W., and Da Silva, L. S. (1990). “Interactive bending behaviour of sandwich beams.” J. Appl. Mech. Trans. ASME, 57(1), 189–196.
Hunt, G. W., and Wadee, M. A. (1998). “Localization and mode interaction in sandwich structures.” Proc. R. Soc. A, 454(1972), 1197–1216.
Ji, W., and Waas, A. M. (2007). “Global and local buckling of a sandwich beam.” J. Eng. Mech., 133(2), 230–237.
Ji, W., and Waas, A. M. (2008). “Wrinkling and edge buckling in orthotropic sandwich beams.” J. Eng. Mech., 134(6), 455–461.
Jonckheere, R. E. (1971). “Determination of the period of nonlinear oscillations by means of Chebyshev polynomials.” Z. Angew. Math. Mech., 51(5), 389–393.
Kim, S., and Sridharan, S. (2005). “Analytical study of bifurcation and nonlinear behavior of sandwich columns.” J. Eng. Mech., 131(12), 1313–1321.
Léotoing, L., Drapier, S., and Vautrin, A. (2002). “Nonlinear interaction of geometrical and material properties in sandwich beam instabilities.” Int. J. Solids Struct., 39(13), 3717–3739.
Li, P. S., Sun, W. P., and Wu, B. S. (2008). “Analytical approximate solutions to large amplitude oscillation of a simple pendulum.” J. Vib. Shock, 27, 72–74 (in Chinese).
Lönnö, A. (1998). “Experiences from using carbon fiber composites/sandwich construction in the Swedish navy.” Sandwich Construction 4, EMAS UK, Stockholm, Sweden.
Luongo, A. (1991). “On the amplitude modulation and localization phenomena in interactive buckling problems.” Int. J. Solids Struct., 27(15), 1943–1954.
Lyckegaard, A., and Thomsen, O. T. (2006). “Nonlinear analysis of a curved sandwich beam joined with a straight sandwich beam.” Compos., Part B Eng., 37(2–3), 101–107.
Mirmiran, A., and Wolde-Tinsae, A. M. (1993). “Stability of prebuckled sandwich elastica arches: Parametric study.” J. Eng. Mech., 119(4), 767–785.
Nayfeh, A. H., and Mook, D. T. (1979). Nonlinear oscillations, Wiley, New York.
Shariyat, M. (2011). “A double-superposition global––local theory for vibration and dynamic buckling analyses of viscoelastic composite/sandwich plates: A complex modulus approach.” Arch. Appl. Mech., 81(9), 1253–1268.
Wadee, M. A. (2000). “Effects of periodic and localized imperfections on struts on nonlinear foundations and compression sandwich panels.” Int. J. Solids Struct., 37(8), 1191–1209.
Wadee, M. A., and Simões da Silva, L. A. P. (2005). “Asymmetric secondary buckling in monosymmetric sandwich struts.” J. Appl. Mech. Trans ASME, 72(5), 683–690.
Wadee, M. A., Yiatros, S., and Theofanous, M. (2010). “Comparative studies of localized buckling in sandwich struts with different core bending models.” Int. J. Nonlinear Mech., 45, 111–120.
Williams, D., Leggett, D. M. A, and Hopkins, H. G. (1941). “Flat sandwich panels under compressive end loads.” ACR Tech. Rep. and Memoranda No. 1987, H. M. Stationery Office, London.
Wu, B. S., Lim, C. W., and Sun, W. P. (2006a). “Improved harmonic balance approach to periodic solutions of non-linear jerk equations.” Phys. Lett. A, 354(1–2), 95–100.
Wu, B. S., Sun, W. P., and Lim, C. W. (2006b). “An analytical approximate technique for a class of strong non-linear oscillators.” Int. J. Nonlinear Mech., 41, 766–774.
Wu, B. S., Yu, Y. P., and Li, Z. G. (2007). “Analytical approximations to large post-buckling deformation of elastic rings under uniform hydrostatic pressure.” Int. J. Mech. Sci., 49, 661–668.
Yiatros, S., and Wadee, M. A. (2011). “Interactive buckling in sandwich beam-columns.” IMA J. Appl. Math., 76(1), 146–168.
Yu, Y. P., Lim, C. W., and Wu, B. S. (2008). “Analytical approximations to large hygrothermal buckling deformation of a beam.” J. Struct. Eng., 134(4), 602–607.
Yu, Y. P., Wu, B. S., and Lim, C. W. (2012). “Numerical and analytical approximations to large post-buckling deformation of MEMS.” Int. J. Mech. Sci., 55(1), 95–103.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 139 • Issue 8 • August 2013
Pages: 1084 - 1090
Copyright
© 2013 American Society of Civil Engineers.
History
Received: Sep 22, 2011
Accepted: May 2, 2012
Published online: Jul 15, 2013
Published in print: Aug 1, 2013
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.