An Analytical Algorithm for Estimating the Deck’s Maximum Deflection and Deck-End Rotation Angle of a Suspension Bridge under Live Load
Publication: Journal of Bridge Engineering
Volume 27, Issue 7
Abstract
With the appearance of large-span and heavy-duty high-speed railway suspension bridges, their deck deflections and deck-end rotation angles (DERA) under the live load became very topical indicators of bridge safety and stability. In contrast to multiple studies of the deck deflection, quite a few were devoted to the DERA. Based on the deflection theory, the continuous functions of deck deflection and DERA with the uniformly distributed load of any length acting at any position were introduced in this study. The method of finding the maximum value by derivative was performed to find the maximum DERA, maximum deck deflection, and its position. The proposed analytical algorithm was applied to a calculation example. The results were compared against those obtained by a trial-and-error method based on the finite-element method, which verified the feasibility and accuracy of the proposed analytical algorithm. Besides, the effects of several design parameters on the maximum DERA and maximum deflection of the deck, including dead load, span length, bending stiffness of the deck, and axial stiffness of the main cable were analyzed via the proposed method. It revealed that adjusting the main span length, dead load, and deck bending stiffness value was the most effective way to control the maximum deck deflection and maximum DERA.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
All data, models, and code generated or used during the study appear in the published article.
Acknowledgments
The work described in this paper was financially supported by the National Natural Science Foundation of China under Grant Nos. 52078134 and 51678148, the Natural Science Foundation of Jiangsu Province (Grant No. BK20181277), and the National Key R&D Program of China (Grant No. 2017YFC0806009), which are gratefully acknowledged.
Notation
The following symbols are used in this paper:
- EbIb
- bending stiffness of the deck;
- EcAc
- axial tensile stiffness of the main cable;
- f
- sag of the main cable;
- Hp
- horizontal force of the main cable;
- Hq
- increment of the horizontal component of force in the main cable under the live load;
- h(x)
- suspender force under the dead and live loads;
- L
- span length;
- p
- dead load with an intensity;
- q(x)
- live load with an intensity;
- t
- application length of the live load;
- v(x)
- deflection;
- x1
- distance from the starting point of live load distribution to the left endpoint of the deck; and
- φ
- deck-end rotation angle.
References
AASHTO. 2010. LRFD bridge design specification. Washington, DC: AASHTO.
Cheng, J., and Y. Li. 2015. “Simplified method for predicting the deflections of cable-stayed suspension bridges considering live loads.” KSCE J. Civ. Eng. 19 (5): 1413–1419. https://doi.org/10.1007/s12205-014-1036-7.
Choi, D. H., S. G. Gwon, H. Yoo, and H. S. Na. 2013. “Nonlinear static analysis of continuous multi-span suspension bridges.” Int. J. Steel Struct. 13 (1): 103–115. https://doi.org/10.1007/s13296-013-1010-0.
Del Arco, D. C., and A. C. Aparicio. 2001. “Preliminary static analysis of suspension bridges.” Eng. Struct. 23 (9): 1096–1103. https://doi.org/10.1016/S0141-0296(01)00009-8.
Dickey, R. W. 1968. “An improved method for numerical solution of suspension bridge deflection equations.” Math. Comput. 22 (102): 298–303. https://doi.org/10.2307/2004660.
Gimsing, N. J., and C. T. Georgakis. 2011. Cable supported bridges: Concept and design. 3rd ed. New York: Wiley.
Grigorjeva, T., A. Juozapaitis, and Z. Kamaitis. 2006. “Simplified engineering method of suspension bridges with rigid cables under action of symmetrical and asymmetrical loads.” Balt. J. Road Bridge Eng. 1 (1): 11–20.
Grigorjeva, T., A. Juozapaitis, and Z. Kamaitis. 2010. “Static analysis and simplified design of suspension bridges having various rigidity of cables.” J. Civ. Eng. Manage. 16 (3): 363–371. https://doi.org/10.3846/jcem.2010.41.
Jia, L., Z. Lin, R. Xiao, and Y. Jiang. 2018. “Parameter effects on the mechanical performance of triple-tower four-span suspension bridges.” Adv. Struct. Eng. 21 (2): 256–269. https://doi.org/10.1177/1369433217717115.
Li, Y., X. Li, M. Gao, and T. He. 2013. “Comparison of Germany, Japan and China specifications of limit value standards of vertical rotation angles at beam ends of railway bridges.” [In Chinese.] J. China Railway Soc. 35 (4): 84–89.
Liang, P., X. Wu, W. Li, and Y. Xu. 2011. “Research on mechanical behaviours of three-tower suspension bridges from the perspective of influence line.” Adv. Mat. Res. 163–167: 1466–1473.
Ohshima, H., K. Sato, and N. Watanabe. 1984. “Structural analysis of suspension bridges.” J. Eng. Mech. 110 (3): 392–404. https://doi.org/10.1061/(ASCE)0733-9399(1984)110:3(392).
Park, K. J., D. Y. Kim, and E. S. Hwang. 2018. “Investigation of live load deflection limit for steel cable stayed and suspension bridges.” Int. J. Steel Struct. 18 (4): 1252–1264. https://doi.org/10.1007/s13296-018-0108-9.
Qin, S. Q., and Z. Y. Gao. 2017. “Developments and prospects of long-span high-speed railway bridge technologies in China.” Eng. 3 (6): 787–794. https://doi.org/10.1016/j.eng.2017.11.001.
Semper, B. 1993. “A mathematical model for suspension bridge vibration.” Comput. Math. Appl. 26 (5): 77–91. https://doi.org/10.1016/0898-1221(93)90076-8.
Shin, S. U., M. R. Jung, J. Park, and M. Y. Kim. 2015. “A deflection theory and its validation of earth-anchored suspension bridges under live loads.” KSCE J. Civ. Eng. 19 (1): 200–212. https://doi.org/10.1007/s12205-014-0641-9.
Ulstrup, C. C. 1993. “Rating and preliminary analysis of suspension bridges.” J. Struct. Eng. 119 (9): 2653–2679. https://doi.org/10.1061/(ASCE)0733-9445(1993)119:9(2653).
Wollmann, G. P. 2001. “Preliminary analysis of suspension bridges.” J. Bridge Eng. 6 (4): 227–233. https://doi.org/10.1061/(ASCE)1084-0702(2001)6:4(227).
Wu, X. N., Y. Xu, W. H. Li, and P. Liang. 2011. “Geometrical nonlinear analysis on three-tower suspension bridges under live load.” Adv. Civ. Eng. 255–260: 1209–1213.
Zhang, W. M., C. Y. Yang, and J. Q. Chang. 2021. “Cable shape and construction parameters of triple-tower double-cable suspension bridge with two asymmetrical main spans.” J. Bridge Eng. 26 (2): 04020127. https://doi.org/10.1061/(ASCE)BE.1943-5592.0001674.
Zhang, W. M., C. Y. Yang, Z. W. Wang, and Z. Liu. 2019. “An analytical algorithm for reasonable central tower stiffness in the three-tower suspension bridge with unequal-length main spans.” Eng. Struct. 199: 109595. https://doi.org/10.1016/j.engstruct.2019.109595.
Zhou, Y., Y. Xia, B. Chen, and Y. Fujino. 2020. “Analytical solution to temperature-induced deformation of suspension bridges.” Mech. Syst. Sig. Process. 139: 106568. https://doi.org/10.1016/j.ymssp.2019.106568.
Zhou, Y., Y. Xia, Y. Fujino, and K. Yamaguchi. 2021. “Analytical formulas of thermal deformation of suspension bridges.” Eng. Struct. 240: 112366. https://doi.org/10.1016/j.engstruct.2021.112366.
Information & Authors
Information
Published In
Journal of Bridge Engineering
Volume 27 • Issue 7 • July 2022
Copyright
© 2022 American Society of Civil Engineers.
History
Received: Aug 14, 2021
Accepted: Mar 7, 2022
Published online: May 11, 2022
Published in print: Jul 1, 2022
Discussion open until: Oct 11, 2022
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.