The post-buckling of an axially loaded elastic beam resting on linearly elastic medium is investigated in this paper from a geometrically exact analysis. It is known that the elastic foundation increases the bifurcation limit, but it may have a destabilizing effect on the post-buckling behavior associated to imperfection sensitivity. This unstable nature of the post-buckling behavior may lead to drastic softening phenomena, as already investigated for plasticity or Continuum Damage Mechanics media. It is suggested in this paper to study the influence of gradient terms in the interaction foundation model on the post-buckling behavior of this structural system. The gradient elasticity foundation model of Pasternak is used and introduced by variational arguments in a geometrically exact framework. A general nonlinear fourth-order differential equation is obtained, and numerically solved with a nonlinear boundary value solver. The post-buckling behavior is analyzed from an asymptotic method. The gradient elasticity constitutive law significantly affects the post-localization process.
Accepted:
Published online:
Noël Challamel 1
@article{CRMECA_2011__339_6_396_0, author = {No\"el Challamel}, title = {On the post-buckling of elastic beams on gradient foundation}, journal = {Comptes Rendus. M\'ecanique}, pages = {396--405}, publisher = {Elsevier}, volume = {339}, number = {6}, year = {2011}, doi = {10.1016/j.crme.2011.04.003}, language = {en}, }
Noël Challamel. On the post-buckling of elastic beams on gradient foundation. Comptes Rendus. Mécanique, Volume 339 (2011) no. 6, pp. 396-405. doi : 10.1016/j.crme.2011.04.003. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2011.04.003/
[1] Beams on Elastic Foundation, The University of Michigan Press, Ann Arbor, 1958
[2] Elastic Analysis of Soil–Foundation Interaction, Elsevier, Amsterdam, 1979
[3] Die Lehre von der Elasticität und Festigkeit, Dominicus, Prague, 1867
[4] Exact asymptotic solution for the initial post-buckling of a strut on linear elastic foundation, ZAMM Z. Angew. Math. Mech., Volume 54 (1974), pp. 677-683
[5] Elastic stability and post-buckling (R.E. Langer, ed.), Proc. Symp. “Nonlinear Problems”, University of Wisconsin Press, 1963
[6] On the propagation of localization in the plasticity collapse of hardening–softening beams, Internat. J. Engrg. Sci., Volume 48 (2010) no. 5, pp. 487-506
[7] Buckling of elastic beams on nonlocal foundation: a revisiting of Reissner model, Mech. Res. Comm., Volume 37 (2010), pp. 472-475
[8] A variationally-based nonlocal damage model to predict diffuse microcracking evolution, Int. J. Mech. Sc., Volume 52 (2010), pp. 1783-1800
[9] P.L. Pasternak, On a new method of analysis of an elastic foundation by means of two foundation constants, Moscow, Russia, 1954.
[10] A generalized model of elastic foundation based on long-range interactions: integral and fractional model, Int. J. Solids Struct., Volume 46 (2009) no. 17, pp. 3124-3137
[11] Post-buckling behavior of beam on two-parameter elastic foundation, Int. J. Struct. Stab. Dyn., Volume 4 (2004), pp. 21-43
[12] Influence of local wrinkling on membrane behaviour: a new approach by the technique of slowly variable Fourier coefficients, J. Mech. Phys. Solids, Volume 58 (2010), pp. 1139-1153
[13] Asymptotic analyses of the buckling of imperfect columns on nonlinear elastic foundations, Int. J. Solids Struct., Volume 6 (1970), pp. 1341-1356
[14] Post-buckling problems for long elastic beams, Acta Mech., Volume 164 (2003), pp. 189-198
[15] Postbuckling of nano rods/tubes based on nonlocal beam theory, Int. J. Appl. Mech., Volume 1 (2009) no. 2, pp. 259-266
[16] Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates, Internat. J. Engrg. Sci., Volume 48 (2010), pp. 1507-1518
[17] Force cycles and force chains, Phys. Rev. E, Volume 81 (2010), p. 011302
[18] Force-chain buckling in granular media: a structural mechanics perspective, Philos. Trans. R. Soc. Ser. A, Volume 13 (2010) no. 368, pp. 249-262
[19] Non-linear and buckling analysis of bars lying on an elastic foundation, Int. J. Nonlinear Mech., Volume 24 (1989) no. 4, pp. 295-307
[20] Theory of Elastic Stability, McGraw-Hill, New York, 1961
[21] Stability of Elastic Structures, Springer-Verlag, Berlin, 2000
[22] Stability of Structures – Elastic, Inelastic, Fracture, and Damage Theories, Dover Publications, Inc., New York, 2003
[23] Exact Solutions for Buckling of Structural Members, CRC Ser. Comput. Mech. Appl. Anal., CRC, Boca Raton, FL, 2005
[24] A General Theory of Elastic Stability, John Wiley & Sons, 1973
[25] Softening branches of a two-degree-of-freedom system induced by spatial buckling, Int. J. Struct. Stab. Dyn., Volume 6 (2006) no. 4, pp. 493-512
Cited by Sources:
Comments - Policy