Applied Mathematics and Mechanics >
Deformation and stability of a circular-arc arch compressed by a rigid plate: incorporating tension, shear, and bending
Received date: 2025-07-18
Revised date: 2025-10-17
Online published: 2025-12-30
Supported by
Project supported by the National Natural Science Foundation of China (Nos. 124B2043, U2241267, 12172155, and 12302278), the Science and Technology Leading Talent Project of Gansu Province of China (No. 23ZDKA0009), and the Natural Science Foundation of Gansu Province of China (Nos. 24JRRA473 and 24JRRA489)
Copyright
The contact deformation and buckling of elastic rods against rigid surfaces represent a prevalent phenomenon in applications such as oil drilling, arterial stents, and energy harvesting. This has attracted widespread attention from researchers. In this paper, the deformation and buckling behaviors of a circular arch subject to compression by a rigid plate are investigated with a planar elastic rod model that incorporates tension, shearing, and bending. In comparison with the existing models that solely consider the bending energy, the deflection curve, the internal force distribution, and the critical load of the present model show good agreement with the finite element results. Through the dimensional analysis and order-of-magnitude estimation, we examine the factors influencing the critical load. The study reveals that the semi-central angle of the arch has the most significant effect. The dimensionless geometric parameter describing arch slenderness becomes prominent when the semi-central angle is less than 30
Key words： circular-arc arch; elastic rod; finite deformation; buckling; critical load
Yunkai TANG , Shengyi TANG , Kai LING , Donghui LIU , Huadong YONG , Youhe ZHOU . Deformation and stability of a circular-arc arch compressed by a rigid plate: incorporating tension, shear, and bending[J]. Applied Mathematics and Mechanics, 2026 , 47(1) : 19 -38 . DOI: 10.1007/s10483-026-3330-9
| [1] | LUBINSKI, A. K. and BLENKARN, A. Buckling of tubing in pumping wells, its effects and means for controlling it. Transactions of the AIME, 210, 73–88 (1957) |
| [2] | THOMPSON, J. M. T., SILVEIRA, M., VAN DER HEIJDEN, G. H. M., and WIERCIGROCH, M. Helical post-buckling of a rod in a cylinder: with applications to drill-strings. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 468, 1591–1614 (2012) |
| [3] | LATSON, L. A. and QURESHI, A. M. Techniques for transcatheter recanalization of completely occluded vessels and pathways in patients with congenital heart disease. Annals of Pediatric Cardiology, 3, 140–146 (2010) |
| [4] | ALDERLIESTEN, T., KONINGS, M. K., and NIESSEN, W. J. Modeling friction, intrinsic curvature, and rotation of guide wires for simulation of minimally invasive vascular interventions. IEEE Transactions on Biomedical Engineering, 54, 29–38 (2007) |
| [5] | LAJNEF, N., BURGUE?O, R., BORCHANI, W., and SUN, Y. A concept for energy harvesting from quasi-static structural deformations through axially loaded bilaterally constrained columns with multiple bifurcation points. Smart Materials and Structures, 23, 055005 (2014) |
| [6] | JIAO, P. C., BORCHANI, W., ALAVI, A. H., HASNI, H., and LAJNEF, N. An energy harvesting and damage sensing solution based on postbuckling response of nonuniform cross-section beams. Structural Control and Health Monitoring, 25, e2052 (2018) |
| [7] | LIANG, K., ZHOU, S. J., LUO, Y. J., ZHANG, X. P., and KANG, Z. Topology optimization design of recoverable bistable structures for energy absorption with embedded shape memory alloys. Thin-Walled Structures, 198, 111757 (2024) |
| [8] | JIN, T., ZHAO, J., and ZHANG, Y. Postbuckling analyses of ribbon-type 3D structures assembled on cylindrical substrates. Acta Mechanica Sinica, 40, 424130 (2024) |
| [9] | PRONK, S., GEISSLER, P. L., and FLETCHER, D. A. Limits of filopodium stability. Physical Review Letters, 100, 258102 (2008) |
| [10] | SIPOS, A. A. and VáRKONYI, P. L. A unified morphoelastic rod model with application to growth-induced coiling, waving, and skewing of plant roots. Journal of the Mechanics and Physics of Solids, 160, 104789 (2022) |
| [11] | RIKS, E. The application of Newton’s method to the problem of elastic stability. ASME Journal of Applied Mechanics, 39, 1060–1065 (1972) |
| [12] | RIKS, E. An incremental approach to the solution of snapping and buckling problems. International Journal of Solids and Structures, 15, 529–551 (1979) |
| [13] | SIMO, J. C., WRIGGERS, P., SCHWEIZERHOF, K. H., and TAYLOR, R. L. Finite deformation post-buckling analysis involving inelasticity and contact constraints. International Journal for Numerical Methods in Engineering, 23, 779–800 (1986) |
| [14] | KANTO, Y. and YAGAWA, G. A dynamic contact buckling analysis by the penalty finite element method. International Journal for Numerical Methods in Engineering, 29, 755–774 (1990) |
| [15] | SEO, K. and MAN, K. Post-buckling analysis of nonfrictional contact problems using linear complementarity formulation. Computers & Structures, 57, 783–794 (1995) |
| [16] | DENO?L, V. and DETOURNAY, E. Eulerian formulation of constrained elastica. International Journal of Solids and Structures, 48, 625–636 (2011) |
| [17] | WANG, J., DEBOEUF, S., ANTKOWIAK, A., and NEUKIRCH, S. Constrained Euler buckling: the von Kármán approximation. International Journal of Solids and Structures, 313, 113279 (2025) |
| [18] | ADAN, N., SHEINMAN, I., and ALTU, E. Post-buckling behavior of beams under contact constraints. ASME Journal of Applied Mechanics, 61, 764–772 (1994) |
| [19] | ADAMS, G. G. and BENSON, R. C. Postbuckling of an elastic plate in a rigid channel. International Journal of Mechanical Sciences, 28, 153–162 (1986) |
| [20] | DOMOKOS, G., HOLMES, P., and ROYCE, B. Constrained euler buckling. Journal of Nonlinear Science, 7, 281–314 (1997) |
| [21] | CHAI, H. The post-buckling response of a bi-laterally constrained column. Journal of the Mechanics and Physics of Solids, 46, 1155–1181 (1998) |
| [22] | RO, W., CHEN, J., and HONG, S. Vibration and stability of a constrained elastica with variable length. International Journal of Solids and Structures, 47, 2143–2154 (2010) |
| [23] | CHEN, J. and LI, C. Planar elastica inside a curved tube with clearance. International Journal of Solids and Structures, 44, 6173–6186 (2007) |
| [24] | LU, Z. and CHEN, J. Deformations of a clamped-clamped elastica inside a circular channel with clearance. International Journal of Solids and Structures, 45, 2470–2492 (2008) |
| [25] | VILLAGGIO, P. Buckling under unilateral constraints. International Journal of Solids and Structures, 15, 193–201 (1979) |
| [26] | ROMAN, B. and POCHEAU, A. Buckling cascade of thin plates: forms, constraints and similarity. Europhysics Letters, 46, 602 (1999) |
| [27] | WANG, Z., RUIMI, A., and SRINIVASA, A. R. A direct minimization technique for finding minimum energy configurations for beam buckling and post-buckling problems with constraints. International Journal of Solids and Structures, 72, 165–173 (2015) |
| [28] | WU. J. and JUVKAM-WOLD, H. C. The effect of wellbore curvature on tubular buckling and lockup. ASME Journal of Applied Mechanics, 117, 214–218 (1995) |
| [29] | HUANG, W. and GAO, D. Helical buckling of a thin rod with connectors constrained in a torus. International Journal of Mechanical Sciences, 98, 14–28 (2015) |
| [30] | HUANG, W., GAO, D., and LIU, Y. A study of tubular string buckling in vertical wells. International Journal of Mechanical Sciences, 118, 231–253 (2016) |
| [31] | LIU, J., ZHONG, X., CHENG, Z., FENG, X., and REN, G. Buckling of a slender rod confined in a circular tube: theory, simulation, and experiment. International Journal of Mechanical Sciences, 140, 288–305 (2018) |
| [32] | LIU, C. and CHEN, J. Effect of Coulomb friction on the deformation of an elastica constrained in a straight channel with clearance. European Journal of Mechanics-A/Solids, 39, 50–59 (2013) |
| [33] | TIKHONOV, V. S. and SAFRONOV, A. I. Analysis of postbuckling drillstring vibrations in rotary drilling of extended-reach wells. ASME Journal of Energy Resources Technology, 133, 043102 (2011) |
| [34] | HORVáTH, M. G., SIPOS, A. A., and VáKONYI, P. L. Shape of an elastica under growth restricted by friction. International Journal of Solids and Structures, 156-157, 137–147 (2019) |
| [35] | PLAUT, R. H., SUHERMAN, S., DILLARD, D. A., WILLIAMS, B. E., and WATSON, L. T. Deflections and buckling of a bent elastica in contact with a flat surface. International Journal of Solids and Structures, 36, 1209–1229 (1999) |
| [36] | CHEN, J. and HUNG, S. Deformation and stability of an elastica constrained by curved surfaces. International Journal of Mechanical Sciences, 82, 1–12 (2014) |
| [37] | MORIMOTO, T. and SATO, K. Contact responses between a semi-circular ring and a rigid plane. International Journal of Solids and Structures, 264, 112122 (2023) |
| [38] | JUDAH, N. and GIVLI, S. The post-buckling behavior of a beam constrained by nonlinear springy walls. ASME Journal of Applied Mechanics, 91, 061004 (2024) |
| [39] | CHAI, H. On the post-buckling behavior of bilaterally constrained plates. International Journal of Solids and Structures, 39, 2911–2926 (2002) |
| [40] | LIU, Z. and JU, J. Developing a scalable analytical framework for predicting the contact behavior of elastic rings against rigid surfaces. International Journal of Solids and Structures, 301, 112967 (2024) |
| [41] | ENGESSER, F. Die Knickfestigkeit Gerader Stabe, W. Ernst & Sohn, Berlin (1891) |
| [42] | PISANTY, A. and TENE, Y. Equilibrium equations for plane slender bars undergoing large shear deformations. Acta Mechanica, 17, 263–275 (1973) |
| [43] | SCHMIDT, R. and DADEPPO, D. A. Nonlinear theory of arches and beams with shear deformation. ASME Journal of Applied Mechanics, 39, 1144–1146 (1972) |
| [44] | TIMOSHENKO, S. P. and GERE, J. M. Theory of Elastic Stability, McGraw-Hill, New York (1961) |
| [45] | DADEPPO, D. A. and SCHMIDT, R. Large deflections of elastic arches and beams with shear deformation. The Journal of the Industrial Mathematics Society, 22, 17–34 (1972) |
| [46] | HARINGX, J. A. On highly compressible helical springs and rubber rods, and their application for vibration—free mountings, part I, Philips Res. report 3. Philips Gloeilampenfabrieken, Eindhoven, the Netherlands (1948) |
| [47] | ANTMAN, S. S. and ROSENFELD, G. Global behavior of buckled states of nonlinearly elastic rods. SIAM Review, 20, 513–566 (1978) |
| [48] | ROGERS, R. C. Derivation of linear beam equations using nonlinear continuum mechanics. Zeitschrift für angewandte Mathematik und Physik, 44, 732–754 (1993) |
| [49] | ATANACKOVIC, T. M. Stability Theory of Elastic Rods, World Scientific Publishing Co. Pte. Ltd., Singapore (1997) |
| [50] | YOSHIAKI, G., TOMOO, Y., and MAKOTO, O. Elliptic integral solutions of plane elastica with axial and shear deformations. International Journal of Solids and Structures, 26, 375–390 (1990) |
| [51] | REISSNER, E. On one-dimensional finite-strain beam theory: the plane problem. Zeitschrift für angewandte Mathematik und Physik, 23, 795–804 (1972) |
| [52] | PLAUT, R. H., DALRYMPLE, A. J., and DILLARD, D. A. Effect of work of adhesion on contact of an elastica with a flat surface. Journal of Adhesion Science and Technology, 15, 565–581 (2001) |
| [53] | MAJIDI, C. Remarks on formulating an adhesion problem using Euler’s elastica (draft). Mechanics Research Communications, 34, 85–90 (2007) |
| [54] | GLASSMAKER, N. J. and HUI, C. Y. Elastica solution for a nanotube formed by self-adhesion of a folded thin film. Journal of Applied Physics, 96, 3429–3434 (2004) |
| [55] | LI, J., ZHANG, X. Y., QIN, J. Q., MA, M. Z., and LIU, R. P. Strength and grain refinement of Ti-30Zr-5Al-3V alloy by Fe addition. Materials Science and Engineering: A, 691, 25–30 (2017) |
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