Research Papers

The Postbuckling Behavior of Planar Elastica Constrained by a Deformable Wall

[+] Author and Article Information
Shmuel Katz

Faculty of Mechanical Engineering,
Technion—Israel Institute of Technology,
Haifa 32000, Israel
e-mail: katzshm@technion.ac.il

Sefi Givli

Faculty of Mechanical Engineering,
Technion—Israel Institute of Technology,
Haifa 32000, Israel
e-mail: givli@technion.ac.il

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received January 2, 2017; final manuscript received February 16, 2017; published online March 8, 2017. Assoc. Editor: Yihui Zhang.

J. Appl. Mech 84(5), 051001 (Mar 08, 2017) (15 pages) Paper No: JAM-17-1004; doi: 10.1115/1.4036018 History: Received January 02, 2017; Revised February 16, 2017

Attributed to its significance in a wide range of practical applications, the post-buckling behavior of a beam with lateral constraints has drawn much attention in the last few decades. Despite the fact that, in reality, the lateral constraints are often flexible or deformable, vast majority of studies have considered fixed and rigid lateral constraints. In this paper, we make a step toward bridging this gap by studying the post-buckling behavior of a planar beam that is laterally constrained by a deformable wall. Unfortunately, the interaction with a compliant wall prevents derivation of closed-form analytical solutions. Nevertheless, careful examination of the governing equations of a simplified model reveals general properties of the solution, and let us identify the key features that govern the behavior. Specifically, we construct universal “solution maps” that do not depend on the mode number and enable simple and easy prediction of the contact conditions and of the mode-switching force (the force at which the system undergoes instantaneous transition from one equilibrium configuration (or mode) to another). The predictions of the mathematical model are validated against finite element (FE) simulations.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.


Feodosyev, V. I. , 1977, Selected Problems and Questions in Strength of Materials, Mir Publishers, Moscow, Russia.
Link, H. , 1954, “ Über den Geraden Knickstab mit Begrenzter Durchbiegung,” Ing.-Arch., 22(4), pp. 237–250. [CrossRef]
Miersemann, E. , Mittelmann, H. D. , and Törnig, W. , 1986, “ A Free Boundary Problem and Stability for the Nonlinear Beam,” Math. Methods Appl. Sci., 8(1), pp. 516–532. [CrossRef]
Holmes, P. , Domokos, G. , Schmitt, J. , and Szeberényi, I. , 1999, “ Constrained Euler Buckling: An Interplay of Computation and Analysis,” Comput. Methods Appl. Mech. Eng., 170(3–4), pp. 175–207. [CrossRef]
Adan, N. , Sheinman, I. , and Altus, E. , 1994, “ Post-Buckling Behavior of Beams Under Contact Constraints,” ASME J. Appl. Mech., 61(4), pp. 764–772. [CrossRef]
Pocheau, A. , and Roman, B. , 2004, “ Uniqueness of Solutions for Constrained Elastica,” Phys. D, 192(3–4), pp. 161–186. [CrossRef]
Ro, W.-C. , Chen, J.-S. , and Hong, S.-Y. , 2010, “ Vibration and Stability of a Constrained Elastica With Variable Length,” Int. J. Solids Struct., 47(16), pp. 2143–2154. [CrossRef]
Doraiswamy, S. , Narayanan, K. R. , and Srinivasa, A. R. , 2012, “ Finding Minimum Energy Configurations for Constrained Beam Buckling Problems Using the Viterbi Algorithm,” Int. J. Solids Struct., 49(2), pp. 289–297. [CrossRef]
Wang, T. Y. , Koh, C. G. , and Liaw, C. Y. , 2010, “ Post-Buckling Analysis of Planar Elastica Using a Hybrid Numerical Strategy,” Comput. Struct., 88(11–12), pp. 785–795. [CrossRef]
Borchani, W. , Lajnef, N. , and Burgueño, R. , 2015, “ Energy Method Solution for the Postbuckling Response of an Axially Loaded Bilaterally Constrained Beam,” Mech. Res. Commun., 70, pp. 114–119. [CrossRef]
Denoel, V. , and Detournay, E. , 2011, “ Eulerian Formulation of Constrained Elastica,” Int. J. Solids Struct., 48(3–4), pp. 625–636. [CrossRef]
Adams, G. G. , and Benson, R. C. , 1986, “ Postbuckling of an Elastic Plate in a Rigid Channel,” Int. J. Mech. Sci., 28(3), pp. 153–162. [CrossRef]
Roman, B. , and Pocheau, A. , 1999, “ Buckling Cascade of Thin Plates: Forms, Constraints and Similarity,” EPL (Europhys. Lett.), 46(5), p. 602. [CrossRef]
Roman, B. , and Pocheau, A. , 2002, “ Postbuckling of Bilaterally Constrained Rectangular Thin Plates,” J. Mech. Phys. Solids, 50(11), pp. 2379–2401. [CrossRef]
Manning, R. S. , and Bulman, G. B. , 2005, “ Stability of an Elastic Rod Buckling Into a Soft Wall,” Proc. R. Soc. A, 461(2060), pp. 2423–2450. [CrossRef]
Villaggio, P. , 1979, “ Buckling Under Unilateral Constraints,” Int. J. Solids Struct., 15(3), pp. 193–201. [CrossRef]
Soong, T. C. , and Choi, I. , 1986, “ An Elastica That Involves Continuous and Multiple Discrete Contacts With a Boundary,” Int. J. Mech. Sci., 28(1), pp. 1–10. [CrossRef]
Miersemann, E. , and Mittelmann, H. D. , 1991, “ Stability and Continuation of Solutions to Obstacle Problems,” J. Comput. Appl. Math., 35(1–3), pp. 5–31. [CrossRef]
Chen, J.-S. , and Hung, S.-Y. , 2014, “ Deformation and Stability of an Elastica Constrained by Curved Surfaces,” Int. J. Mech. Sci., 82, pp. 1–12. [CrossRef]
Chen, J.-S. , and Li, C.-W. , 2007, “ Planar Elastica Inside a Curved Tube With Clearance,” Int. J. Solids Struct., 44(18–19), pp. 6173–6186. [CrossRef]
Lu, Z.-H. , and Chen, J.-S. , 2008, “ Deformations of a Clamped–Clamped Elastica Inside a Circular Channel With Clearance,” Int. J. Solids Struct., 45(9), pp. 2470–2492. [CrossRef]
Wang, Z. , Ruimi, A. , and Srinivasa, A. R. , 2015, “ A Direct Minimization Technique for Finding Minimum Energy Configurations for Beam Buckling and Post-Buckling Problems With Constraints,” Int. J. Solids Struct., 72, pp. 165–173. [CrossRef]
Chen, J.-S. , and Ro, W.-C. , 2012, “ Vibration Method in Stability Analysis of Planar Constrained Elastica,” Advances in Computational Stability Analysis, S. B. Coşkun , ed., InTech, Rijeka, Croatia.
Chateau, X. , and Nguyen, Q. S. , 1991, “ Buckling of Elastic Structures in Unilateral Contact With or Without Friction,” Eur. J. Mech. A, Solids, 10(1), pp. 71–89.
Chai, H. , 2002, “ On the Post-Buckling Behavior of Bilaterally Constrained Plates,” Int. J. Solids Struct., 39(11), pp. 2911–2926. [CrossRef]
Liu, C.-W. , and Chen, J.-S. , 2013, “ Effect of Coulomb Friction on the Deformation of an Elastica Constrained in a Straight Channel With Clearance,” Eur. J. Mech. A/Solids, 39, pp. 50–59. [CrossRef]
Vaz, M. A. , and Patel, M. H. , 1995, “ Analysis of Drill Strings in Vertical and Deviated Holes Using the Galerkin Technique,” Eng. Struct., 17(6), pp. 437–442. [CrossRef]
Thompson, J. M. T. , Silveira, M. , Heijden, G. H. M. V. D. , and Wiercigroch, M. , 2012, “ Helical Post-Buckling of a Rod in a Cylinder: With Applications to Drill-Strings,” Proc. R. Soc. A, 468(2142), pp. 1591–1614.
Lubinski, A. , and Althouse, W. S. , 1962, “ Helical Buckling of Tubing Sealed in Packers,” J. Pet. Technol., 14(6), pp. 655–670. [CrossRef]
Tan, X. C. , and Forsman, B. , 1995, “ Buckling of Slender String in Cylindrical Tube Under Axial Load: Experiments and Theoretical Analysis,” Exp. Mech., 35(1), pp. 55–60. [CrossRef]
Fang, J. , Li, S. Y. , and Chen, J. S. , 2013, “ On a Compressed Spatial Elastica Constrained Inside a Tube,” Acta Mech., 224(11), pp. 2635–2647. [CrossRef]
Miller, J. T. , Su, T. , Dussan, V. E. B. , Pabon, J. , Wicks, N. , Bertoldi, K. , and Reis, P. M. , 2015, “ Buckling-Induced Lock-Up of a Slender Rod Injected Into a Horizontal Cylinder,” Int. J. Solids Struct., 72, pp. 153–164. [CrossRef]
Leijnse, N. , Oddershede, L. B. , and Bendix, P. M. , 2015, “ Helical Buckling of Actin Inside Filopodia Generates Traction,” Proc. Natl. Acad. Sci. U.S.A., 112(1), pp. 136–141. [CrossRef] [PubMed]
Pronk, S. , Geissler, P. L. , and Fletcher, D. A. , 2008, “ The Limits of Filopodium Stability,” Phys. Rev. Lett., 100(25), p. 258102. [CrossRef] [PubMed]
Daniels, D. R. , and Turner, M. S. , 2013, “ Islands of Conformational Stability for Filopodia,” PLoS One, 8(3), p. e59010. [CrossRef] [PubMed]
Hajianmaleki, M. , and Daily, J. S. , 2014, “ Advances in Critical Buckling Load Assessment for Tubulars Inside Wellbores,” J. Pet. Sci. Eng., 116, pp. 136–144. [CrossRef]
Mattila, P. K. , and Lappalainen, P. , 2008, “ Filopodia: Molecular Architecture and Cellular Functions,” Nat. Rev. Mol. Cell Biol., 9(6), pp. 446–454. [CrossRef] [PubMed]
Faix, J. , and Rottner, K. , 2006, “ The Making of Filopodia,” Curr. Opin. Cell Biol., 18(1), pp. 18–25. [CrossRef] [PubMed]
Latson, L. , and Qureshi, A. , 2010, “ Techniques for Transcatheter Recanalization of Completely Occluded Vessels and Pathways in Patients With Congenital Heart Disease,” Ann. Pediatr. Cardiol., 3(2), pp. 140–146. [CrossRef] [PubMed]
Schneider, P. A. , 2005, Endovascular Skills: Guidewire and Catheter Skills for Endovascular Surgery, Marcel Dekker, New York.
Alderliesten, T. , Konings, M. K. , and Niessen, W. J. , 2007, “ Modeling Friction, Intrinsic Curvature, and Rotation of Guide Wires for Simulation of Minimally Invasive Vascular Interventions,” IEEE Trans. Biomed. Eng., 54(1), pp. 29–38. [CrossRef] [PubMed]
Katz, S. , and Givli, S. , 2015, “ The Post-Buckling Behavior of a Beam Constrained by Springy Walls,” J. Mech. Phys. Solids, 78, pp. 443–466. [CrossRef]
Lagrange, R. , and Averbuch, D. , 2012, “ Solution Methods for the Growth of a Repeating Imperfection in the Line of a Strut on a Nonlinear Foundation,” Int. J. Mech. Sci., 63(1), pp. 48–58. [CrossRef]
Li, R. , Li, M. , Su, Y. W. , Song, J. Z. , and Ni, X. Q. , 2013, “ An Analytical Mechanics Model for the Island-Bridge Structure of Stretchable Electronics,” Soft Matter, 9(35), pp. 8476–8482. [CrossRef]
Tzaros, K. A. , and Mistakidis, E. S. , 2011, “ The Unilateral Contact Buckling Problem of Continuous Beams in the Presence of Initial Geometric Imperfections: An Analytical Approach Based on the Theory of Elastic Stability,” Int. J. Nonlinear Mech., 46(9), pp. 1265–1274. [CrossRef]
Papachristou, K. S. , and Sophianopoulos, D. S. , 2013, “ Buckling of Beams on Elastic Foundation Considering Discontinuous (Unbonded) Contact,” Int. J. Mech. Appl., 3(1), pp. 4–12.
Chai, H. , 1998, “ The Post-Buckling Response of a Bi-Laterally Constrained Column,” J. Mech. Phys. Solids, 46(7), pp. 1155–1181. [CrossRef]
Genna, F. , and Bregoli, G. , 2014, “ Small Amplitude Elastic Buckling of a Beam Under Monotonic Axial Loading, With Frictionless Contact Against Movable Rigid Surfaces,” J. Mech. Mater. Struct., 9(4), pp. 441–463. [CrossRef]
Benichou, I. , and Givli, S. , 2011, “ The Hidden Ingenuity in Titin Structure,” Appl. Phys. Lett., 98(9), p. 091904.
Benichou, I. , and Givli, S. , 2013, “ Structures Undergoing Discrete Phase Transformation,” J. Mech. Phys. Solids, 61(1), pp. 94–113. [CrossRef]
Benichou, I. , and Givli, S. , 2015, “ Rate Dependent Response of Nanoscale Structures Having a Multiwell Energy Landscape,” Phys. Rev. Lett., 114(9), p. 095504.
Benichou, I. , Zhang, Y. , Dudko, O. K. , and Givli, S. , 2016, “ The Rate Dependent Response of a Bistable Chain at Finite Temperature,” J. Mech. Phys. Solids, 95, pp. 44–63. [CrossRef]


Grahic Jump Location
Fig. 2

As Δ is increased, several configurations might take place: (a) a region (length) of contact with the WF, and point contacts with the rigid wall. (b) Line-contact regions with the rigid wall. (c) Once the line-contact configuration becomes unstable, the beam switches to a new configuration with two additional folds. (d) Line-contact segments with the rigid wall for the four-folds configuration. On the right, results of FE simulations, where vertical displacement is denoted by different colors.

Grahic Jump Location
Fig. 1

Schematic illustration of the system, consisting of a clamped–clamped beam that is laterally constrained by a rigid wall at the bottom and a WF located at distance D0 above the beam

Grahic Jump Location
Fig. 7

Summary of predicted behavior, showing domains of point-contact configuration, line-contact configuration, and mode switching associated with the symmetric case. Results of the FE simulations are shown with arrows in (a). Filled circles indicate transitions from point-contact to line-contact configuration based on the FE results. Occurrence of mode switching, based on FE results, is indicated by the departure of the arrows from the original vertical branch.

Grahic Jump Location
Fig. 3

Transition from point-contact to line-contact configuration. (a) Domains of point-contact and line-contact configurations. (b) Same as (a) but for a larger range of stiffness. The solid line, indicating transition from point-contact to line-contact configuration, starts at (3π, 5/2) at the left end, and approaches ζ/n=2 for large values of β1/4L0/n. The second solid line shows the so-called run away of the beam deflection, where the magnitude of the deflection becomes extremely large. Lines of constant γ are illustrated by dashed lines. (c) H̃2 at the onset of transition from point-contact to line-contact configuration (red line in (a) and (b)). Linear approximation is given by Eq. (2.22). (Color images can be viewed online).

Grahic Jump Location
Fig. 4

In line-contact configuration, H̃1 and H̃2 are directly related to γ through Π1

Grahic Jump Location
Fig. 6

Switching from mode n to mode n+1 for the symmetric case. (a) Mode switching for n≥2 and for n=1 are illustrated by a solid and a broken lines, respectively. (b) Same as (a) but for a larger range of stiffness. The solid straight line shows the so-called run away of the beam deflection, where the magnitude of the deflection becomes extremely large. Lines of constant γ are illustrated by dashed lines. (c) H̃2 at the onset of mode switching. Linear approximation is given by Eq.(2.28).

Grahic Jump Location
Fig. 8

Force–displacement response. A comparison between results of the theoretical model and FE analysis for: (a) β1/4L0=17.8, (b) β1/4L0=8.4, and (c) ϕ=2. Results of the mathematical model are shown with continuous lines, where each line corresponds to a different mode. Dashed and solid lines distinguish between point-contact and line-contact configurations, respectively. Results of the FE simulations are illustrated with filled diamonds, where transition from point-contact to line-contact configuration is indicated with a circle.

Grahic Jump Location
Fig. 5

Mode switching is governed by the length of the longest line segment, c. Three cases are of special interest: (i) the lengths of all line-contact segments, except one, are very small; (ii) all line-contact segments have the same length; and (iii) the symmetric case where the beam is comprised of n identical and symmetric parts. (a) Case (ii) and case (iii) are identical for n=1. This not the case for n≥2, as illustrated in (b) for n=2.



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In