Abstract
Presented here is a general theory for the three-dimensional nonlinear dynamics of elastic anisotropic initially straight beams undergoing moderate displacements and rotations. The theory fully accounts for geometric nonlinearities (large rotations and displacements) by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvature and strain-displacement expressions that contain the von Karman strains as a special case. Extensionality is included in the formulation, and transverse shear deformations are accounted for by using a third-order theory. Six third-order nonlinear partial-differential equations are derived for describing one extension, two bending, one torsion, and two shearing vibrations of composite beams. They show that laminated beams display linear elastic and nonlinear geometric couplings among all motions. The theory contains, as special cases, the Euler-Bernoulli theory, Timoshenko's beam theory, the third-order shear theory, and the von Karman type nonlinear theory.
Similar content being viewed by others
References
Chree, C., ‘The equations of an isotropic elastic solid in polar and cylindrical coordinates, their solution and application’, Transactions of the Cambridge Philosophical Society 14, 1989, 250.
Cowper, G. R., ‘On the accuracy of Timoshenko's beam theory’, Journal of the Engineering Mechanics Division 94, 1968, 1447–1453.
Timoshenko, S. P. and Goodier, J. N., Theory of Elasticity, 3rd. Edn., McGraw-Hill, New York, 1970.
Shames, I. H. and Dym, C. L., Energy and Finite Element Methods in Structural Mechanics, McGraw-Hill, New York, 1985, pp. 197–204.
Timoshenko, S. P., ‘On the correction for shear of the differential equation for transverse vibrations of prismatic bars’, Philosophical Magazine 41, 1921, 744–746.
Timoshenko, S. P., ‘On the transverse vibrations of bars of uniform cross sections’, Philosophical Magazine Series 6, 43, 1922, 125–131.
Cowper, G. R., ‘The shear coefficient in Timoshenko's beam theory’, Journal of Applied Mechanics 33, 1966, 335–340.
Heyliger, P. R. and Reddy, J. N., ‘A higher order beam finite element for bending and vibration problems’, Journal of Sound and Vibration 126, 1988, 309–326.
Bauchau, O. A. and Hong, C. H., ‘Large displacement analysis of naturally curved and twisted composite beams’, AIAA Journal 25, 1987, 1469–1475.
Bauchau, O. A. and Hong, C. H., ‘Nonlinear composite beam theory’, Journal of Applied Mechanics 55, 1988, 156–163.
Stemple, A. D. and Lee, S. W., ‘Finite-element model for composite beams with arbitrary cross-sectional warping’, AIAA Journal 26, 1988, 1512–1520.
Kane, T. R., Ryan, R. R., and Banerjee, A. K., ‘Dynamics of a cantilever beam attached to a moving base’, Journal of Guidance, Control, and Dynamics 10, 1987, 139–151.
Krishna Murty, A. V., ‘Vibrations of short beams’, AIAA Journal 8, 1970, 34–38.
Sheinman, I. and Adan, M., ‘The effect of shear deformation on post-buckling behavior of laminated beams’, Journal of Applied Mechanics 54, 1987, 558–562.
Friedmann, P. P., ‘Recent developments in rotary-wing aeroelasticity’, Journal of Aircraft 14, 1977, 1027–1041.
Bolotin, V. V., The Dynamic Stability of Elastic Systems, translated by V. I. Weingarten et al., Holden-Day, Inc., San Francisco, 1964.
Moody, M. L., ‘The parametric response of imperfect columns’, Developments in Mechanics, Proceedings of the Tenth Midwestern Mechanics Conference, 4, 1967, 329–346.
Ho, C.-H., Scott, R. A., and Eisley, J. G., ‘Nonplanar, nonlinear oscillations of a beam-I. Foreed motions’, International Journal of Non-Linear Mechanics 10, 1975, 113–127.
Evan-Iwanowski, R. M., Sanford, W. F., and Kehagioglou, T., ‘Nonstationary parametric response of a nonlinear column’, Developments in Theoretical and Applied Mechanics 5, 1970, 715–743.
Busby, H. R.Jr. and Weingarten, V. I.: ‘Nonlinear response of a beam to periodic loading’, International Journal of Non-Linear Mechanics 7, 1972, 289–303.
Hodges, D. H. and Dowell, E. H., ‘Nonlinear equations of motion for the elastic bending and torsion of twisted nonuniform rotor blades’, NASA TN D-7818, 1974.
Hodges, D. H. and Peters, D. A., ‘On the lateral buckling of uniform slender cantilever beams’, International Journal of Solids and Structures 11, 1975, 1269–1280.
Dowell, E. H., Traybar, J., and Hodges, D. H., ‘An experimental-theoretical correlation study of nonlinear bending and torsion deformations of a cantilever beam’, Journal of Sound and Vibration 50, 1977, 533–544.
Crespo de Silva, M. R. M. and Glynn, C. C., ‘Nonlinear flexural-flexural-torsional dynanics of inextensional beams—I. Equations of motion’, Journal of Structural Mechanics 6, 1978, 437–448.
Alkire, K., ‘An analysis of rotor blade twist variables associated with different Euler sequences and pretwist treatments’, NASA TM 84394, 1984.
Maganty, S. P. and Bickford, W. B., ‘Large amplitude oscillations of thin circular rings’, Journal of Applied Mechanics 54, 1987, 315–322.
Rosen, A., Loewy, R. G. and Mathew, M. B., ‘Nonlinear analysis of pretwisted rods using principal curvature transformation. Part I: Theoretical derivation’, AIAA Journal 25, 1987, 470–478.
Rosen, A., Loewy, R. G., and Mathew, M. B., ‘Nonlinear analysis of pretwisted rods using principal curvature transformation, Part II: Numerical results’, AIAA Journal 25, 1987, 598–604.
Hodges, D. H., Crespo da Silva, M. R. M., and Peters, D. A., ‘Nonlinear effects in the static and dynamic behavior of beams and rotor blades’, Vertica 12, 1988 243–256.
Minguet, P. and Dugundji, J., ‘Experiments and analysis for composite blades under large deflections, Part 1 — Static behavior’, AIAA Journal 28, 1990, 1573–1579.
Pai, P. F. and Nayfeh, A. H., ‘Three-dimensional nonlinear vibrations of composite beams—I. Equations of motion’, Nonlinear Dynamics 1, 1990, 477–502.
Abarcar, R. B. and Cunniff, P. F., ‘The vibration of cantilever beams of fiber reinforced material’, Journal of Composite Materials 6, 1972, 504–517.
Kapania, R. K. and Raciti, S., ‘Nonlinear vibrations of unsymmetrically laminated beams’, AIAA Journal 27, 1989, 201–210.
Whitney, J. M., Structural Analysis of Laminated Anisotropic Plates, Technomic Publishing Company, Inc., Pennsylvania, 1987.
Krenk, S., ‘A linear theory for pretwisted elastic beams’, Journal of Applied Mechanics 50, 1983, 137–142.
Adams, R. D. and Bacon, D. G. C., ‘Measurement of the flexural damping capacity and dynamic Young's modulus of metals and reinforced plastics’, Journal of Physics D: Applied Physics 6, 1973, 27–41.
Rao, G. V., Raju, I. S., and Raju, K. K., ‘Nonlinear vibrations of beams considering shear deformation and rotary inertia’, AIAA Journal, Technical Notes, May 1976, 685–687.
Sathyamoorthy, M., ‘Nonlinear analysis of beams Part I: A survey of recent advances’, The Shock and Vibration Digest 14, 1982, 19–35.
Sathyamoorthy, M., ‘Nonlinear analysis of beams Part II: Finite element methods’, The Shock and Vibration Digest 14, 1982, 7–18.
Kapania, R. K. and Raciti, S., ‘Recent advances in analysis of laminated beams and plates, Part I: Shear effects and buckling’, AIAA Journal 27, 1989, 923–934.
Kapania, R. K. and Raciti, S. ‘Recent advances in analysis of laminated beams and plates, Part II: Vibrations and wave propagation’, AIAA Journal 27, 1989, 935–946.
Smith, C. E. Applied Mechanics — More Dynamics, John Wiley and Sons, New York, 1976.
MACSYMA, Symbolics, Inc., East Burlington, MA, November 1988.
Vlasov, V. Z., Thin-Walled Elastic Beams. Translated from Russian, National Technical Information Service, U.S. Department of Commerce, 1951.
Giavotto, V., Borri, M., Mantegazza, P., and Ghiringhelli, G., ‘Anisotropic beam theory and applications’, Computers & Structures 16, 1983, 403–413.
Wu, X. X. and Sun, C. T., ‘Vibration analysis of laminated composite thin-walled beams using finite elements’, AIAA Journal 29, 1991, 736–742.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pai, P.F., Nayfeh, A.H. A nonlinear composite beam theory. Nonlinear Dyn 3, 273–303 (1992). https://doi.org/10.1007/BF00045486
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00045486