Classical Solution of Post-buckled nonlinear Beam on an Elastic Foundation

A. S. M. Z. Hasan, B. Ahmed, M. M. Rana

Abstract

This paper is concerned with the buckling problem of flexible beams on an elastic foundation for free vibration. An exact solution for the post-buckled geometric nonlinear beam with clamped-clamped and clamped-hinged end conditions is presented in this paper. The cubic nonlinearity of the governing equation of motion is induced due to the mid-plane stretching, which is considered in the analysis. The critical buckling forces are obtained as 4.015, 4.061 and 4.122 with respect to nondimensional foundation stiffness 1,10 and 20, respectively, with the fundamental mode of vibration. The critical buckling load, associated mode shape, the effect of foundation stiffness, and vibration behavior are obtained. The optimum locations of an internal hinge and the optimum buckling force are also investigated for various foundation stiffness of the nonlinear beam on an elastic foundation. The bifurcation diagrams and the internal hinge locations are useful for practical application of axially loaded nonlinear beam on an elastic foundation restrained by Clamped-Clamped and Clamped-Hinge beam.

Conclusion

Classical solution is presented to solve the nonlinear vibrations of post-buckled beam on an elastic foundation with C-C and C-H end conditions. The effect of foundation stiffness, critical buckling force and interesting vibration behaviors are investigated. The exact vibration solutions for axially loaded nonlinear beams on an elastic foundation with an internal hinge are obtained. The optimum non-dimensional buckling forces are investigated corresponding to the different foundation stiffness for C-C and C-H beam. The result shows that the foundation stiffness is greatly influenced to the buckling force of the beam with an internal hinge. The result obtained from the bifurcation diagrams and the internal hinge locations are useful for practical application with such kind of axially loaded nonlinear beam on an elastic foundation.

References

[1] Nayfeh A.H. Emam S.A., “Exact solution and stability of post buckling configurations of beams”, Nonlinear Dyn, vol. 54, pp. 395-408, 2008.

[2] Fang W. Wickert J.A., “Post buckling of micromachined beams”, J. Miromech. Microeng, vol. 4, pp. 116-122, 1994.

[3] Nayfeh A.H. Kreider W. Anderson T.J., “Investigation of natural frequency and mode shapes of buckled beam”, AIAA Journal, vol. 33, no. 6, pp. 1121-1126, 1995.

[4] Addessi A. Lacarbonara W. Paolone A., “Free in-plane vibrations of highly buckled beams carrying a lumped mass”, Acta Mechanica, vol. 180, pp. 133-156, 2005.

[5] Zhang Y. Murphy K.D., “Secondary buckling and tertiary states of a beam on a non-linear elastic foundation”, International Journal of Non-linear Mechanics, vol. 40, pp.795-805, 2005.

[6] Li S.R. Batra R.C., “Thermal buckling and post buckling of Euler-Bernoulli beams supported on nonlinear elastic foundations”, AIAA Journal, vol. 45, No. 3, pp. 712-720, 2007.

[7] Emam S.A. Nayfeh A.H., “Post buckling and free vibrations of composite beams”, Composite Structures, vol. 88, pp.636-642, 2009.

[8] Xia W. Wang L. Yin L., “Nonlinear non-classical microscale beams: Static bending, post buckling and free vibration”, International Journal of Engineering Science, vol. 48, pp. 2044-2053, 2010.

[9] Chen H.P., “Free vibration of parbuckled and post buckled plates with delamination”, Composites Science and Technology, vol. 51, pp. 451-462, 1994.

[10] Librescu L. Lin W., “Post buckling and vibration of shear deformable flat and curved panels on a non-linear elastic foundation”, International Journal of Non-linear Mechanics, vol. 32, no. 2, pp. 211-225, 1997.

[11] Whiting A.I.M., “A Galerkin procedure for localized buckling of a strut on a nonlinear elastic foundation”, International Journal of Solids and Structures, vol. 34, no. 6, pp. 727-739, 1997.

[12] Wang C.Y., “Post-buckling of a clamped-simply supported elastica”, International Journal of Non-linear Mechanics, vol. 32, no. 6, pp. 1115-1122, 1997.

[13] Li S. Zhou Y.H. Zheng X., “Thermal post buckling of a heated elastic rod with pinned-fixed ends”, Journal of Thermal Stresses, vol. 25, pp. 43-56, 2002.

[14] Jiao P. Borchani W. Lajnef N., “Large deformation solutions to post-buckled beams confined by movable and flexible constraints: A static and dynamic analysis”, International Journal of Solids and Structures, vol. 128, pp. 85-98, 2017.

[15] Rammerstorfer F.G., “Buckling of elastic structures under tensile loads”, Acta Mech, vol. 229, pp. 881-900, 2018.

[16] Zare M. Asnaf A., “Nonlinear pre- and post-buckled analysis of curved beams using differential quadrature element method”, International Journal of Mechanical and Materials Engineering, vol. 14, no. 15, pp. 1-10.

[17] Neukirch S. Yavari M. Challamel N. Thomas O., “Comparison of the Von Kármán and Kirchhoff models for the post-buckling and vibrations of elastic beams”, Journal of Theoretical, Computational and Applied Mechanics, INRIA, vol. 10, pp.1-18, 2021.

[18] Wang C.Y. Wang C.M., “Vibration of a beam with an internal hinge”, International Journal of Structural Stability and Dynamics, vol. 1, no. 1, pp. 163-167, 2001.

[19] Lee Y.Y. Wang C.M. Kitipornchai S., “Vibration of Timoshenko beams with internal hinge”, Journal of Engineering Mechanics, vol. 129, no. 3, 293-301, 2003.

[20] Cheng, G.F. Lee Y.Y. Mei C., “Nonlinear random response of internally hinged beams”, Finite Elements in Analysis and Design, vol. 39, no. (5-6), pp. 487-504, 2003.

[21] Xiang Y. Wang C.M. Wang C.Y., “Buckling of rectangular plates with internal hinge”, International Journal of Structural Stability and Dynamics, vol. 1, no. 2, pp. 169-179, 2001.

[22] Xiang Y. Wang C.M. Wang C.Y. Su G.H., “Ritz buckling analysis of rectangular plates with internal hinge”, Journal of Engineering Mechanics, vol. 129, no. 6, 683-688, 2003.

[23] Gupta P.R. Reddy J.N., “Buckling and vibration of orthotropic plates with an internal line hinge”, International Journal of Structural Stability and Dynamics, vol. 2, no. 4, pp. 457-486, 2002.

[24] Wang C.M. Wang C.Y. Reddy J.N., “Exact solutions for buckling of structural members”, CRC Series in Computational Mechanics and Applied Analysis, 2005.

[25] Wang C.Y., “Optimum location of an internal hinge of a uniform column on an elastic foundation”, Journal of Applied Mechanics, vol. 75, pp. 034501(1-4), 2008.

[26] Wang C.Y., “Buckling of a weakened infinite beam on an elastic foundation”, Journal of Engineering Mechanics, vol. 136, No. 4, 534-537, 2010.

[27] Barari A. Kaliji H.D. Ghadimi M. Domairry G., “Non-linear vibration of Euler-Bernoulli beams”, Latin American Journal of Solids and Structures, vol. 8, pp. 139-148, 2011.

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