Snap-Through of Shallow Circular Arches with Variable Horizontal Supports under Unilateral Displacement Control
Publication: Journal of Engineering Mechanics
Volume 146, Issue 4
Abstract
This study proposes a numerical method to solve the snap-through problem of elastic shallow inextensible circular arches. Specifically, the snap-through behavior of arches subjected to a downward load at a point along the span was systematically evaluated for variable elastic horizontal supports under unilateral displacement control. Through theoretical analysis based on a dimensionless formulation, the critical state of snap-through can be determined by embedding ordinary differential equations into the nonlinear unconstrained optimization. Then, critical stiffness lines for horizontal springs and snap regions with varying loading position were systematically analyzed to judge whether a snap-through phenomenon will occur. Parametric analysis was further carried out to investigate the influence of variable horizontal spring stiffness and different arch length on the overall deformation and the critical displacement of critical states. The results show that the critical stiffness increases with the decrease of arch length and the decrease of horizontal stiffness and arch length can both expand the snap region. This study highlights the important role of stiffness of supports in snap-through behavior control and provides a numerical method for more accurate evaluation of the snap-through behavior of arches under various supports and loading conditions in engineering applications.
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 from the corresponding author by request:
•
Codes for computation of ordinary differential equations and roots searching of nonlinear equations;
•
Data for critical stiffness, snap region, and fixed points; and
•
Data for parametric analysis of variable horizontal spring stiffness and different arch lengths.
Acknowledgments
The work presented in this study was supported by the National Natural Science Foundation of China (Grant Nos. 51822805, 51878147, and U1937202), the Natural Science Foundation of Jiangsu Province (Grant No. BK20170083), the Six Top Talent Peak Projects of Jiangsu Province (Grant No. JZ-001), and the Priority Academic Program Development of Jiangsu Higher Education Institutions. W. Xia acknowledged the support from the Department of Civil and Environmental Engineering and College of Engineering at North Dakota State University.
References
Bertoldi, K., V. Vitelli, J. Christensen, and H. M. Van. 2017. “Flexible mechanical metamaterials.” Nat. Rev. Mater. 2 (11): 17066. https://doi.org/10.1038/natrevmats.2017.66.
Bradford, M. A. 2006. “In-plane nonlinear behaviour of circular pinned arches with elastic restraints under thermal loading.” Int. J. Struct. Stab. Dyn. 6 (2): 163–177. https://doi.org/10.1142/S0219455406001897.
Bradford, M. A., B. Uy, and Y.-L. Pi. 2002. “In-plane elastic stability of arches under a central concentrated load.” J. Eng. Mech. 128 (7): 710–719. https://doi.org/10.1061/(ASCE)0733-9399(2002)128:7(710).
Bradford, M. A., T. Wang, Y.-L. Pi, and R. I. Gilbert. 2007. “In-plane stability of parabolic arches with horizontal spring supports. I: Theory.” J. Struct. Eng. 133 (8): 1130–1137. https://doi.org/10.1061/(ASCE)0733-9445(2007)133:8(1130).
Cai, J., and J. Feng. 2010. “Effect of support stiffness on stability of shallow arches.” Int. J. Struct. Stab. Dyn. 10 (5): 1099–1110. https://doi.org/10.1142/S0219455410003919.
Cai, J., J. Feng, Y. Chen, and L. Huang. 2009. “In-plane elastic stability of fixed parabolic shallow arches.” Sci. China Ser. E: Technol. Sci. 52 (3): 596–602. https://doi.org/10.1007/s11431-009-0057-9.
Cai, J., Y. Xu, J. Feng, and J. Zhang. 2012. “In-plane elastic buckling of shallow parabolic arches under an external load and temperature changes.” J. Struct. Eng. 138 (11): 1300–1309. https://doi.org/10.1061/(ASCE)ST.1943-541X.0000570.
Camescasse, B., A. Fernandes, and J. Pouget. 2013. “Bistable buckled beam: Elastica modeling and analysis of static actuation.” Int. J. Solids Struct. 50 (19): 2881–2893. https://doi.org/10.1016/j.ijsolstr.2013.05.005.
Cazottes, P., A. Fernandes, J. Pouget, and M. Hafez. 2009. “Bistable buckled beam: Modeling of actuating force and experimental validations.” J. Mech. Des. 131 (10): 101001. https://doi.org/10.1115/1.3179003.
Chen, J. S., and S. Y. Hung. 2011. “Snapping of an elastica under various loading mechanisms.” Eur. J. Mech. A. Solids 30 (4): 525–531. https://doi.org/10.1016/j.euromechsol.2011.03.006.
Dancygier, A. N., Y. S. Karinski, and A. Chacha. 2016. “A model to assess the response of an arched roof of a lined tunnel.” Tunnelling Underground Space Technol. 56 (Jun): 211–225. https://doi.org/10.1016/j.tust.2016.03.009.
Fargette, A., S. Neukirch, and A. Antkowiak. 2014. “Elastocapillary snapping: Capillarity induces snap-through instabilities in small elastic beams.” Phys. Rev. Lett. 112 (13): 137802. https://doi.org/10.1103/PhysRevLett.112.137802.
Harvey, P. S., and L. N. Virgin. 2015. “Coexisting equilibria and stability of a shallow arch: Unilateral displacement-control experiments and theory.” Int. J. Solids Struct. 54 (Feb): 1–11. https://doi.org/10.1016/j.ijsolstr.2014.11.016.
Hsu, C. S. 1968. “Stability of shallow arches against snap-through under timewise step loads.” J. Appl. Mech.-Trans. ASME 35 (1): 31–39. https://doi.org/10.1115/1.3601170.
Mallon, N. J., R. H. B. Fey, H. Nijmeijer, and G. Zhang. 2006. “Dynamic buckling of a shallow arch under shock loading considering the effects of the arch shape” Int. J. Non Linear Mech. 41 (9): 1057–1067. https://doi.org/10.1016/j.ijnonlinmec.2006.10.017.
Moon, J., K.-Y. Yoon, T.-H. Lee, and H.-K. Lee. 2007. “In-plane elastic buckling of pin-ended shallow parabolic arches.” Eng. Struct. 29 (10): 2611–2617. https://doi.org/10.1016/j.engstruct.2007.01.004.
Pandey, A., D. E. Moulton, D. Vella, and D. P. Holmes. 2014. “Dynamics of snapping beams and jumping poppers.” Europhys. Lett. 105 (2): 24001. https://doi.org/10.1209/0295-5075/105/24001.
Pi, Y.-L., and M. A. Bradford. 2010a. “In-plane thermoelastic behaviour and buckling of pin-ended and fixed circular arches.” Eng. Struct. 32 (1): 250–260. https://doi.org/10.1016/j.engstruct.2009.09.012.
Pi, Y.-L., and M. A. Bradford. 2010b. “Nonlinear in-plane elastic buckling of shallow circular arches under uniform radial and thermal loading.” Int. J. Mech. Sci. 52 (1): 75–88. https://doi.org/10.1016/j.ijmecsci.2009.10.011.
Pi, Y.-L., and M. A. Bradford. 2010c. “Nonlinear thermoelastic buckling of pin-ended shallow arches under temperature gradient.” J. Eng. Mech. 136 (8): 960–968. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000134.
Pi, Y.-L., and M. A. Bradford. 2012. “Non-linear buckling and postbuckling analysis of arches with unequal rotational end restraints under a central concentrated load.” Int. J. Solids Struct. 49 (26): 3762–3773. https://doi.org/10.1016/j.ijsolstr.2012.08.012.
Pi, Y.-L., M. A. Bradford, and F. Tin-Loi. 2007. “Nonlinear analysis and buckling of elastically supported circular shallow arches.” Int. J. Solids Struct. 44 (7–8): 2401–2425. https://doi.org/10.1016/j.ijsolstr.2006.07.011.
Pi, Y.-L., M. A. Bradford, and F. Tin-Loi. 2008. “Nonlinear in-plane buckling of rotationally restrained shallow arch under a central concentrated load.” Int. J. Non Linear Mech. 43 (1): 1–17. https://doi.org/10.1016/j.ijnonlinmec.2007.03.013.
Plaut, R. H. 1979. “Influence of load position on the stability of shallow arches.” Z. Angew. Math. Phys. 30 (3): 548–552. https://doi.org/10.1007/BF01588902.
Plaut, R. H. 2015a. “Snap-through of arches and buckled beams under unilateral displacement control.” Int. J. Solids Struct. 63 (Jun): 109–113. https://doi.org/10.1016/j.ijsolstr.2015.02.044.
Plaut, R. H. 2015b. “Snap-through of shallow extensible arches under unilateral displacement control.” J. Appl. Mech.-Trans. ASME 82 (9): 094503. https://doi.org/10.1115/1.4030741.
Plaut, R. H. 2018. “Snap-through of shallow reticulated domes under unilateral displacement control.” Int. J. Solids Struct. 148 (Sep): 24–34. https://doi.org/10.1016/j.ijsolstr.2017.10.008.
Plaut, R. H., and L. N. Virgin. 2017. “Snap-through under unilateral displacement control with constant velocity.” Int. J. Non Linear Mech. 94 (Sep): 292–299. https://doi.org/10.1016/j.ijnonlinmec.2017.01.015.
Ren, W.-X., T. Zhao, and I. E. Harik. 2004. “Experimental and analytical modal analysis of steel arch bridge.” J. Struct. Eng. 130 (7): 1022–1031. https://doi.org/10.1061/(ASCE)0733-9445(2004)130:7(1022).
Riks, E. 1979. “An incremental approach to the solution to the solution of buckling and snapping problems.” Int. J. Solids Struct. 15 (7): 529–551. https://doi.org/10.1016/0020-7683(79)90081-7.
Rubin, M. B. 2004. “Buckling of elastic shallow arches using the theory of a Cosserat point.” J. Eng. Mech. 130 (2): 216–224. https://doi.org/10.1061/(ASCE)0733-9399(2004)130:2(216).
Schreyer, H. L., and E. F. Masur. 1966. “Buckling of shallow arches.” J. Eng. Mech. Div. 92 (4): 1–20.
Simitses, G. J. 1976. An introduction to the elastic stability of structures. Englewood Cliffs, NJ: Prentice Hall.
Timoshenko, S. P., and J. M. Gere. 1961. Theory of elastic stability. 2nd ed. New York: McGraw-Hill.
Wang, T., M. A. Bradford, R. I. Gilbert, and Y.-L. Pi. 2007. “In-plane stability of parabolic arches with horizontal spring supports. II: Experiments.” J. Struct. Eng. 133 (8): 1138–1145. https://doi.org/10.1061/(ASCE)0733-9445(2007)133:8(1138).
Wicks, P. J. 1991. “General equations for buckling of thin, shallow arches of any shape.” J. Eng. Mech. 117 (2): 225–240. https://doi.org/10.1061/(ASCE)0733-9399(1991)117:2(225).
Younis, M. I., H. M. Ouakad, F. M. Alsaleem, R. Miles, and W. Cui. 2010. “Nonlinear dynamics of MEMS arches under harmonic electrostatic actuation.” J. Microelectromech. Syst. 19 (3): 647–656. https://doi.org/10.1109/JMEMS.2010.2046624.
Zhou, Y., Z. Yi, and I. Stanciulescu. 2019. “Nonlinear buckling and postbuckling of shallow arches with vertical elastic supports.” J. Appl. Mech.-Trans. ASME 86 (6): 061001. https://doi.org/10.1115/1.4042572.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 146 • Issue 4 • April 2020
Copyright
©2020 American Society of Civil Engineers.
History
Received: Jun 11, 2019
Accepted: Oct 7, 2019
Published online: Jan 29, 2020
Published in print: Apr 1, 2020
Discussion open until: Jun 29, 2020
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.