Symplectic Transfer-Matrix Method for Bending of Nonuniform Timoshenko Beams on Elastic Foundations
Publication: Journal of Engineering Mechanics
Volume 146, Issue 6
Abstract
This paper proposes a new symplectic transfer-matrix method (STMM) to solve the bending problem of nonuniform beams. The STMM is a hybrid method that combines the symplectic dual solution system and the traditional transfer-matrix method (TMM). It is first necessary to obtain the benchmark solution of uniform beams in a symplectic system. On this basis, this paper derives the transfer matrices under various nonuniform conditions and discusses the mathematical structure of these transfer matrices. After obtaining the global transfer equation, the equation is solved by changing its rows and columns. Three numerical examples are given to verify the correctness and applicability of the STMM. This paper proves that the transfer matrix in the symplectic system is a symplectic matrix in mathematics, whether it is a field transfer matrix, a point transfer matrix, or a global transfer matrix. The STMM reveals the mathematical property of the transfer matrix and provides a theoretical basis for the standardized application of the “transfer type” method in structural analysis.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
All data, models, or code generated or used during the study are available from the corresponding author by request.
References
Arici, M., M. F. Granata, and P. Margiotta. 2013. “Hamiltonian structural analysis of curved beams with or without generalized two-parameter foundation.” Arch. Appl. Mech. 83 (12): 1695–1714. https://doi.org/10.1007/s00419-013-0772-3.
Arnol’d, V. I. 1992. Ordinary differential equations. 3rd ed. New York: Springer-Verlag.
Li, R., X. Ni, and G. Cheng. 2015. “Symplectic superposition method for benchmark flexure solutions for rectangular thick plates.” J. Eng. Mech. 141 (2): 04014119. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000840.
Li, X., F. Xu, and Z. Zhang. 2017. “Symplectic eigenvalue analysis method for bending of Timoshenko beams resting on elastic foundations.” J. Eng. Mech. 143 (9): 04017098. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001315.
Li, X., F. Xu, and Z. Zhang. 2018. “Symplectic method for natural modes of beams resting on elastic foundations.” J. Eng. Mech. 144 (4): 04018009. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001427.
Lim, C. W., and X. S. Xu. 2010. “Symplectic elasticity: Theory and applications.” Appl. Mech. Rev. 63 (5): 050802. https://doi.org/10.1115/1.4003700.
Lim, C. W., W. A. Yao, and S. Cui. 2008. “Benchmark symplectic solutions for bending of corner-supported rectangular thin plates.” IES J. Part A Civ. Struct. Eng. 1 (2): 106–115. https://doi.org/10.1080/19373260701646407.
Myklestad, N. O. 1944. “A new method of calculating natural modes of uncoupled bending vibration of airplane wings and other types of beams.” J. Aeronaut. Sci. 11 (2): 153–162. https://doi.org/10.2514/8.11116.
Onu, G. 2000. “Shear effect in beam finite element on two-parameter elastic foundation.” J. Struct. Eng. 126 (9): 1104–1107. https://doi.org/10.1061/(ASCE)0733-9445(2000)126:9(1104).
Razaqpur, A. G., and K. R. Shah. 1991. “Exact analysis of beams on two-parameter elastic foundations.” Int. J. Solids Struct. 27 (4): 435–454. https://doi.org/10.1016/0020-7683(91)90133-Z.
Ren, C. B., W. M. Yu, and D. Z. Yun. 1997. “A new method for bending analysis of Timoshenko beams.” [In Chinese.] Mech. Eng. 19 (4): 63–64.
Shirima, L. M., and M. W. Giger. 1992. “Timoshenko beam element resting on two-parameter elastic foundation.” J. Eng. Mech. 118 (2): 280–295. https://doi.org/10.1061/(ASCE)0733-9399(1992)118:2(280).
Tan, C. A., and W. Kuang. 1994. “Distributed transfer function analysis of cone and wedge beams.” J. Sound Vib. 170 (4): 557–566. https://doi.org/10.1006/jsvi.1994.1084.
Thomson, W. T. 1950. “Matrix solution for the vibration of nonuniform beams.” J. Appl. Mech. Trans. 17 (3): 337–339.
Wang, X. 1993. “General solution of longitudinal-transverse bending differential equation of stepped beam on Winkler’s foundation.” [In Chinese.] J. Hydraul. Eng. 1993 (4): 23–32. https://doi.org/10.13243/j.cnki.slxb.1993.04.003.
Xu, X., Y. Ma, C. W. Lim, and H. Chu. 2006. “Dynamic buckling of cylindrical shells subject to an axial impact in a symplectic system.” Int. J. Solids Struct. 43 (13): 3905–3919. https://doi.org/10.1016/j.ijsolstr.2005.03.005.
Yao, W. A., W. X. Zhong, and C. W. Lim. 2009. Symplectic elasticity. Singapore: World Scientific.
Yokoyama, T. 1991. “Vibrations of Timoshenko beam-columns on two-parameter elastic foundations.” Earthquake Eng. Struct. Dyn. 20 (4): 355–370. https://doi.org/10.1002/eqe.4290200405.
Zhong, W. X. 2004. Duality system in applied mechanics and optimal control. Boston: Kluwer Academic Publishers.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 146 • Issue 6 • June 2020
Copyright
©2020 American Society of Civil Engineers.
History
Received: Aug 29, 2019
Accepted: Jan 14, 2020
Published online: Apr 7, 2020
Published in print: Jun 1, 2020
Discussion open until: Sep 7, 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.