Abstract
The aim of this study is to devive an analytical solution to predict the buckling load of the thin-walled arch under a point load at mid-span position. A deflection function and the energy method are adopted to build the nonlinear equilibrium formulae, by solving which, the analytical solution is expressed explicitly. Subsequently, a numerical simulation is established to track the load-displacement paths of equilibrium. The simulation results indicate the load drops significantly after its maxima (critical buckling load) and follows multiple branches characterized by load limits and displacement limits. A comparison is taken between the numerical and analytical results, and a good accordance is depicted. Moreover, parameters that may affect the buckling load are analyzed, with the inclusion of rotational stiffness supports, the central angle, as well as the normalized thickness on the load capacity. Finally, both the proposed theoretical formule and simulation results agree excellently with the test results and other closed-form expressions published elsewhere.
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References
ABAQUS (2013). User’s Manual: Version 6.12, Simulia, Providence, RI, USA.
Allen, H. G. and Bulson, P. S. (1980). Background to buckling, McGraw-Hill, London, UK.
Bažant, Z. P. and Cedolin, L. (2010). Stability of structures: Elastic, inelastic, fracture and damage theories, World Scientific, Hackensack, NJ, USA.
Belytschko, T. and Glaum, L.W. (1979). “Applications of higher order corotational stretch theories to nonlinear finite element analysis.” Computers & Structures, Vol. 10, Nos. 1–2, pp. 175–182, DOI: https://doi.org/10.1016/0045-7949(79)90085-3.
Boot, J. C. (1998). “Elastic buckling of cylindrical pipe linings with small imperfections subjected to external pressure.” Trenchless Technology Research, Vol. 12, Nos. 1–2, pp. 3–15.
Boot, J. C., Naqvi, M. M., and Gumbel, J. E. (2014). “A new method for the structural design of flexible liners for gravity pipes of egg-shaped cross-section: Theoretical considerations and formulation of the problem.” Thin-Walled Structures, Vol. 85, pp. 411–418, DOI: https://doi.org/10.1016/j.tws.2014.09.001.
Bradford, M. A., Uy, B., and Pi, Y. L. (2002). “In-plane elastic stability of arches under a central concentrated load.” Journal of Engineering Mechanics, ASCE, Vol. 128, No. 7, pp. 710–719, DOI: https://doi.org/10.1061/(ASCE)0733-9399(2002)128:7(710).
Chang, C. S. and Hodges, D. H. (2009). “Stability studies for curved beams.” Journal of Mechanics of Materials and Structures, Vol. 4, Nos. 7–8, pp. 1257–1270.
Dickey, R. W. and Roseman, J. J. (1996). “Symmetric and unsymmetric buckling of circular arches.” Quarterly of Applied Mathematics, Vol. 54, No. 4, pp. 759–775.
Dickie, J. F. and Broughton, P. (1971). “Stability criteria for shallow arches.” Journal of the Engineering Mechanics Division, ASCE, Vol. 97, No. 3, pp. 951–965.
Gambhir, M. L. (2004). Stability Analysis and Design of Structures. Springer, New York, NY, USA.
Gjelsvik, A. and Bodner, S. R. (1962). “Energy criterion and snap-through buckling of arches.” Journal of the Engineering Mechanics, ASCE, Vol. 88, pp. 87–134.
Kang, Y. J. and Yoo, C. H. (1994). “Thin-walled curved beams. Π: analytical solutions for buckling of arches.” Journal of Engineering Mechanics, ASCE, Vol. 120, No. 10, pp. 2102–2125, DOI: https://doi.org/10.1061/(ASCE)0733-9399(1994)120:10(2102).
Karnovsky, I. A. (2012). Theory of arches structures, Springer, New York, USA.
Li, Z., Tang, Y., Tang, F., Chen, Y., and Chen, G. (2018). “Elastic buckling of thin-walled polyhedral pipe liners encased in a circular pipe under uniform external pressure.” Thin-Walled Structures, Vol. 123, pp. 214–221, DOI: https://doi.org/10.1016/j.tws.2017.11.019.
Li, Z., Tang, F., and Chen, Y. (2019a). “Stability of the pipe-liner system with a grouting void surrounded by the saturated soil.” Engineering Structures, Vol. 196, p. 109284, DOI: https://doi.org/10.1016/j.engstruct.2019.109284.
Li, Z., Tang, F., Chen, Y., Tang, Y., and Chen, G. (2019b). “Elastic and inelastic buckling of thin-walled steel liners encased in circular host pipes under external pressure and thermal effects.” Thin-walled Structures, Vol. 137, pp. 213–223, DOI: https://doi.org/10.1016/j.tws.2018.12.044.
Li, Z., Tang, F., Chen, Y., and Zheng, J. (2019c). “Material distribution optimization of functionally graded arch subjected to external pressure under temperature rise field.” Thin-walled Structures, Vol. 138, 64–78, DOI: 10.1016/j.tws.2019.01.034.
Li, Z. and Huang, H. (2019). “Elastic and inelastic buckling of the confined liner with a non-uniformly annular gap subjected to a point load under a thermal rise field.” Engineering Failure Analysis, Vol. 105, pp. 1141–1153, DOI: https://doi.org/10.1016/j.engfailanal.2019.07.061.
Li, Z., Wang, L., Guo, Z., and Shu, H. (2012). “Elastic buckling of cylindrical pipe linings with variable thickness encased in rigid host pipes.” Thin-Walled Structures, Vol. 51, pp. 10–19, DOI: https://doi.org/10.1016/j.tws.2011.11.003.
Li, Z., Wang, R., and Chen, Y. (2019d). “Theoretical and numerical analysis of the structural stability of the pipe-grout-liner system with a crown void subjected to external pressure.” Composites Part B: Engineering, Vol. 173, p. 106944, DOI: https://doi.org/10.1016/j.compositesb.2019.106944.
Li, Z. and Zheng, J. (2019a). “Collapse mechanism of the thin-walled functionally graded cylinders encased in the saturated permeable mediums.” Engineering Structures, Vol. 198, p. 109472, DOI: https://doi.org/10.1016/j.engstruct.2019.109472.
Li, Z. and Zheng, J. (2019b). “Theoretical and numerical analyses on the confined functionally graded porous ring with graphene platelet reinforcement.” International Journal of Mechanical Science, Vol. 161–162, p. 105079, DOI: https://doi.org/10.1016/j.ijmecsci.2019.105079.
Li, Z., Zheng, J., and Chen, Y. (2019e). “Nonlinear buckling of thin-walled FGM arch encased in a rigid confinement subjected to external pressure.” Engineering Structures, Vol. 186, pp. 86–95, DOI: https://doi.org/10.1016/j.engstruct.2019.02.019.
Li, Z., Zheng, J., Chen, Y., Sun, Q., and Zhang, Z. (2019f). “Effect of temperature variations on the collapse mechanism of the confined functionally graded porous arch with nanocomposites reinforcement under mechanical loading.” Composites Part B: Engineering, Vol. 176, p. 107330, DOI: https://doi.org/10.1016/j.compositesb.2019.107330.
Li, Z., Zheng, J., Sun, Q., and He, H. (2019g). “Nonlinear structural stability performance of pressurized thin-walled FGM arches under temperature variation field.” International Journal of Non-linear Mechanics, Vol. 113, pp. 86–102, DOI: https://doi.org/10.1016/j.ijnonlinmec.2019.03.016.
Li, Z., Zheng, J., Meng, L., Zou, X., and Hu, X. (2019h). “Nonlinear stability analysis of thin-walled steel pipe confined in soft bilayer medium.” Engineering Structures, Vol. 196, p. 109318, DOI: https://doi.org/10.1016/j.engstruct.2019.109318.
Li, Z., Zheng, J., and Wang, R. (2019i). “Effects of grouting voids on the elastic buckling of confined pipe liners subjected to uniform pressure.” Thin-walled Structures, Vol. 137, pp. 502–514, DOI: https://doi.org/10.1016/j.tws.2018.12.045.
Moon, J., Yoon, K. Y., Lee, T. H., and Lee, H. E. (2009). “Out-of-plane buckling of arches with varying curvature.” KSCE Journal of Civil Engineering, KSCE, Vol. 13, No. 6, pp. 441–451.
Pi, Y. L., Bradford, M. A., and Uy, B. (2002). “In-plane stability of arches.” International Journal of Solids and Structures, Vol. 39, No. 1, pp. 105–125, DOI: https://doi.org/10.1016/S0020-7683(01)00209-8.
Pi, Y. L. and Bradford, M.A. (2010). “Nonlinear in-plane elastic buckling of shallow circular arches under uniform radial and thermal loading.” International Journal of Mechanical Sciences, Vol. 52, No. 1, pp. 75–88, DOI: https://doi.org/10.1016/j.ijmecsci.2009.10.011.
Pi, Y. L., Bradford, M. A., and Guo, Y. L. (2016). “Revisiting nonlinear in-plane elastic buckling and postbuckling analysis of shallow circular arches under a central concentrated load.” Journal of Engineering Mechanics, ASCE, Vol. 142, No. 8, p. 04016046, DOI: https://doi.org/10.1061/(ASCE)EM.1943-7889.0001098.
Pi, Y. L. and Trahair, N. S. (1998). “Nonlinear buckling and post-buckling of elastic arches.” Engineering Structures, Vol. 20, No. 7, pp. 571–579, DOI: https://doi.org/10.1016/S0141-0296(97)00067-9.
Raithel, A. and Franciosi, C. (1985). “The stability of arches in the Lagrangian approach.” International Journal of Solids and Structures, Vol. 21, No. 5, pp. 427–446, DOI: https://doi.org/10.1016/0020-7683(85)90007-1.
Rubin, M. B. (2002). “Buckling of elastic shallow arches using the theory of a cosserat point.” Journal of Engineering Mechanics ASCE, Vol. 130, No. 2, pp. 216–224, DOI: https://doi.org/10.1061/(ASCE)0733-9399(2004)130:2(216).
Ryu, H. J., Choi, J. Y., Yi, G. S., Lee, C. O., and Lim, N. H. (2012). “Elastic stability of circular arches with the open thin-walled monosymmetric section considering the prebuckling deformation.” The Open Civil Engineering Journal, Vol. 6, pp. 87–97.
Schmidt, R. and DaDeppo, D. A. (1972). “Buckling of clamped circular arches subjected to a point load.” Journal of Applied Mathematics and Physics, Vol. 23, pp. 146–148.
Schreyer, H. L. and Masur, E. F. (1966). “Buckling of shallow arches.” Journal of the Engineering Mechanics Division ASCE, Vol. 92, No. EM4, pp. 1–17.
Simitses, G. J. and Hodges, D. H. (2006). Fundamentals of structural stability. Elsevier Inc, Massachusetts, USA.
Timoshenko, S. P. and Gere, J. M. (1961). Theory of elastic stability, 2nd Ed, McGraw-Hill, New York, NY, USA.
Vlasov, V. Z. (1961). Thin-walled elastic beams, 2nd Ed., Israel Program for Scientific Translation, Jerusalem, Israel.
Xu, Y., Gui, X., Zhao, B., and Zhou, R. (2014). “In-plane elastic stability of arches under a radial concentrated load.” Engineering, Vol. 6, pp. 572–583, DOI: https://doi.org/10.4236/eng.2014.69058.
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Li, Z., Zheng, J. Nonlinear Buckling Mechanism of an Arch Subjected to a Symmetrically-placed Point Load. KSCE J Civ Eng 23, 4781–4789 (2019). https://doi.org/10.1007/s12205-019-5152-2
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DOI: https://doi.org/10.1007/s12205-019-5152-2