Simultaneous Buckling, Contact, and Load-Carrying Capacity
Publication: Journal of Engineering Mechanics
Volume 147, Issue 5
Abstract
This paper considers the case of a relatively large number of parallel columns that buckle simultaneously. The close proximity between columns results in the possibility of contact between adjacent columns as buckling proceeds, and this brings with it some interesting observations on load-carrying capacity. Some experimental results verify the theoretical development based on the versatility of high-fidelity 3D printing, but also highlights the difficulty of applying purely axial loading. The sensitive nature of initial geometric imperfections (slight lack of straightness) and load eccentricity strongly influence postbuckled contact, load-carrying capacity, and, as the number of columns is increased, a statistically based evaluation of anticipated behavior is explored and ultimately found to be unrealistic.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
Some or all data, models, or code generated or used during the study are available online at https://doi.org/10.5061/dryad.rxwdbrv5w.
Acknowledgments
The author appreciates the assistance of Peter Galindez and Eric Stach with 3D printing at Duke University, and Elliot Cartee at Cornell University with some of the statistical development.
References
Allen, H. G., and P. S. Bulson. 1980. Background to buckling. New York: McGraw-Hill.
Arbocz, J., and M. A. M. Hoi. 1991. “Collapse of axially compressed cylindrical shells with random imperfections” AIAA J. 29 (12): 2247–2256. https://doi.org/10.2514/3.10866.
Arnold, V. I. 1991. Bifurcation theory and catastrophe theory. New York: Springer.
Bazant, Z. P., and L. Cedolin. 2010. Stability of structures. Singapore: World Scientific.
Britvec, S. J. 1973. The stability of elastic systems. Oxford, UK: Pergamon.
Calladine, C. R. 1983. Theory of shell structures. Cambridge, UK: Cambridge University Press.
Chai, H. 1998. “The post-buckling response of a bi-laterally constrained column.” J. Mech. Phys. Solids 46 (7): 1155–1181. https://doi.org/10.1016/S0022-5096(98)00004-0.
Chajes, A. 1974. Principles of structural stability theory. Englewood Cliffs, NJ: Prentice Hall.
Champneys, A. R., T. J. Dodwell, R. M. Groh, G. W. Hunt, R. M. Neville, A. Pirrera, A. H. Sakhaei, M. Schenk, and M. A. Wadee. 2019. “Happy catastrophe: Recent progress in analysis and exploitation of elastic instability.” Front. Appl. Math. Stat. 5 (34): 1–30. https://doi.org/10.3389/fams.2019.00034.
Chen, J.-S., and L.-C. Wang. 2020. “Contact between two planar buckled beams pushed together transversely.” Int. J. Solids Struct. 199 (Aug): 181–189. https://doi.org/10.1016/j.ijsolstr.2020.04.029.
Cox, B. S., R. M. J. Groh, D. Avitabile, and A. Pirrera. 2018. “Modal nudging in nonlinear elasticity: Tailoring the elastic post-buckling behaviour of engineering structures.” J. Mech. Phys. Solids 116 (Jul): 135–149. https://doi.org/10.1016/j.jmps.2018.03.025.
Deng, H., L. Cheng, X. Liang, D. Hayduke, and A. C. To. 2020. “Topology optimization for energy dissipation design of lattice structures through snap-through behavior.” Comput. Methods Appl. Mech. Eng. 358 (Jan): 112641. https://doi.org/10.1016/j.cma.2019.112641.
Denoel, V., and E. Detournay. 2011. “Eulerian formulation of constrained elastic.” Int. J. Solids Struct. 48 (3–4): 625–636. https://doi.org/10.1016/j.ijsolstr.2010.10.027.
Domokos, G., P. Holmes, and B. Royce. 1997. “Constrained Euler buckling.” J. Nonlinear Sci. 7: 281–314. https://doi.org/10.1007/BF02678090.
Franzoni, F., R. Degenhardt, J. Albus, and M. A. Arbelo. 2019. “Vibration correlation technique for predicting the buckling load of imperfection-sensitive isotropic cylindrical shells: An analytical and numerical verification.” Thin Walled Struct. 140 (Jul): 236–247. https://doi.org/10.1016/j.tws.2019.03.041.
Healey, T. J. 1988. “A group-theoretic approach to computational bifurcation problems with symmetry.” Comput. Methods Appl. Mech. Eng. 67 (3): 257–295. https://doi.org/10.1016/0045-7825(88)90049-7.
Holmes, P., G. Domokos, J. Schmitt, and I. Szeberenyi. 1999. “Constrained Euler buckling: An interplay of computation and analysis.” Comput. Methods Appl. Mech. Eng. 170 (3–4): 175–207. https://doi.org/10.1016/S0045-7825(98)00194-7.
Hu, N., and R. Burgueno. 2017. “Harnessing seeded geometric imperfection to design cylindrical shells with tunable elastic postbuckling behavior.” J. Appl. Mech. 84 (1): 011033. https://doi.org/10.1115/1.4034827.
Hunt, G. W. 1982. “Symmetries of elastic buckling.” Eng. Struct. 4 (1): 21–28. https://doi.org/10.1016/0141-0296(82)90020-7.
Keller, J. B., and J. E. Flaherty. 1973. “Contact problems involving a buckled elastica.” SIAM J. Appl. Math. 24 (2): 215–225. https://doi.org/10.1137/0124022.
Kochmann, D., and K. Bertoldi. 2017. “Exploiting microstructural instabilities in solids and structures: From metamaterials to structural transitions.” Appl. Mech. Rev. 69 (5): 050801. https://doi.org/10.1115/1.4037966.
Manning, R. S., and G. B. Bulman. 2005. “Stability of an elastic rod buckling into a soft wall.” Proc. R. Soc. London, Ser. A 461: 2423–2450. https://doi.org/10.1098/rspa.2005.1458.
Miller, J., T. Su, E. V. Dussan, J. Pabon, N. Wicks, K. Bertoldi, and P. Reis. 2015. “Buckling-induced lock-up of a slender rod injected into a horizontal cylinder.” Int. J. Solids Struct. 72 (Oct): 153–164. https://doi.org/10.1016/j.ijsolstr.2015.07.025.
Papachristou, K. S., and D. S. Sophianopoulos. 2013. “Buckling of beams on elastic foundation considering discontinuous (unbonded) contact.” Int. J. Mech. Appl. 3 (1): 4–12. https://doi.org/10.5923/j.mechanics.20130301.02.
Plaut, R. H., J. E. Sidbury, and L. N. Virgin. 2005. “Analysis of buckled and pre-bent fixed-end columns used as vibration isolators.” J. Sound Vib. 283 (3–5): 1216–1228. https://doi.org/10.1016/j.jsv.2004.07.029.
Poston, T., and I. Stewart. 2012. Catastrophe theory and its applications. New York: Dover.
Reis, P. 2015. “A perspective on the revival of structural (in)stability with novel opportunities for function: From buckliphobia to buckliphilia.” J. Appl. Mech. 82 (11): 111001. https://doi.org/10.1115/1.4031456.
Singer, J. M. 2011. “Central limit theorems.” In International encyclopedia of statistical science, edited by M. Lovric. Berlin: Springer.
Tarantino, M. G., and K. Danas. 2019. “Programmable higher-order Euler buckling modes in hierarchical beams.” Int. J. Solids Struct. 167 (Aug): 170–183. https://doi.org/10.1016/j.ijsolstr.2019.03.009.
Thomopoulos, N. T. 2013. Essentials of Monte Carlo simulation: Statistical methods for building simulation models. Berlin: Springer.
Thompson, J. M. T., and G. W. Hunt. 1973. A general theory of elastic stability. New York: Wiley.
Timoshenko, S. P., and J. M. Gere. 1961. Theory of elastic stability. New York: McGraw-Hill.
Vaillette, D. P., and G. G. Adams. 1983. “An elastic beam contained in a frictionless channel.” J. Appl. Mech. 50 (3): 693. https://doi.org/10.1115/1.3167118.
Virgin, L. N. 2007. Vibration of axially loaded structures. Cambridge, UK: Cambridge University Press.
Virgin, L. N. 2017. “On the flexural stiffness of 3D printer thermoplastic.” Int. J. Mech. Eng. Educ. 45 (1): 59–75. https://doi.org/10.1177/0306419016674140.
Virgin, L. N. 2018a. “Enhancing the teaching of elastic buckling using additive manufacturing.” Eng. Struct. 174 (Nov): 338–345. https://doi.org/10.1016/j.engstruct.2018.07.059.
Virgin, L. N. 2018b. “Tailored buckling constrained by adjacent members.” Structures 16 (Nov): 20–26. https://doi.org/10.1016/j.istruc.2018.08.005.
Virgin, L. N., and R. B. Davis. 2003. “Vibration isolation using buckled structures.” J. Sound Vib. 260 (5): 965–973. https://doi.org/10.1016/S0022-460X(02)01177-X.
Virgin, L. N., Y. Guan, and R. H. Plaut. 2017. “On the geometric conditions for multiple stable equilibria in clamped arches.” Int. J. NonLinear Mech. 92 (Jun): 8–14. https://doi.org/10.1016/j.ijnonlinmec.2017.03.009.
Virgin, L. N., and R. H. Plaut. 2004. “Postbuckling and vibration of linearly elastic and softening columns under self-weight.” Int. J. Solids Struct. 41 (18–19): 4989–5001. https://doi.org/10.1016/j.ijsolstr.2004.03.023.
von Karman, T., and H.-S. Tsien. 1941. “The buckling of thin cylindrical shells under axial compression.” J. Aeronaut. Sci. 8 (8): 303. https://doi.org/10.2514/8.10722.
Wang, C. M., C. Y. Wang, and J. N. Reddy. 2004. Exact solutions for buckling of structural members. Boca Raton, FL: CRC Press.
Zucco, G., and P. M. Weaver. 2020. “The role of symmetry in the post-buckling behaviour of structures.” Proc. R. Soc. London, Ser. A 476 (2233): 20190609. https://doi.org/10.1098/rspa.2019.0609.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 147 • Issue 5 • May 2021
Copyright
© 2021 American Society of Civil Engineers.
History
Received: Sep 24, 2020
Accepted: Dec 30, 2020
Published online: Mar 8, 2021
Published in print: May 1, 2021
Discussion open until: Aug 8, 2021
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.