Keywords: dynamic stability, parametric resonance, bowstring arches.

Most failures of engineering structures are due to structural instability or buckling. Less frequent but also well known to engineering professionals are some problems related to dynamic loading as in ordinary resonance. A different problem, and perhaps not so well known, is that of parametric resonance or dynamic instability which is produced by compressive dynamic loading. A systematic introduction to the theory of parametric stability of structures under both deterministic and stochastic loading can be found in the book by Xie [1].

Dynamic stability is characterized by exponential growth of the response amplitudes even in the presence of damping. As a result, parametric resonance may be more dangerous than ordinary resonance. While in ordinary resonance the loading appears as the forcing term, in parametric resonance it appears as a parameter which changes the system structural properties in the governing differential equation of motion.

Dynamic stability of arches has been the research subject of many works in recent years. Pi and Bradford [2] considered the dynamic buckling of shallow pin-ended arches under an in-plane sudden central concentrated load and investigated the effect of static preloading. Mallon et al. [3] examined the influence of the initial curvature of thin shallow arches in the dynamic pulse buckling. Yi et al. [4], and Wang et al. [5] investigated the dynamic stability of arches with geometrical imperfections. Finally, Blekherman [6] has presented some very interesting and practical results for parametric resonance in the Solferino Bridge at Paris which is a pedestrian steel arch bridge.

This paper investigates the dynamic stability of a bowstring arch subjected to dynamic loads. A non-linear dynamic analysis with integration in the time domain has been performed. An out-of-plumb initial imperfection is introduced in the geometry. The arch is loaded with a uniform distributed load. To simulate dead loading and moving live loading, 2/3 of the distributed load remains constant and 1/3 is harmonically variable. The results show that this type of arches develops parametric resonance. The maximum amplification factor is obtained when the frequency of the applied variable load is approximately double than its first order natural frequency of vibration. The response is exponentially amplified so that for a load equal to only 1/5 of the buckling load the initial imperfection is amplified by a factor of 9.25 after 32 cycles of vibration.

1

W.C. Xie, "Dynamic Stability of Structures", Cambridge University Press, 2006.

2

Y.L. Pi, M.A. Bradford, "Dynamic buckling of shallow pin-ended arches under a sudden central concentrated load", Journal of Sound and Vibration, 317(3-5), 898-917, 2008. doi:10.1016/j.jsv.2008.03.037

3

N.J. Mallon, R.H.B. Fey, H. Nijmeijer, G.Q. Zhang, "Dynamic buckling of a shallow arch under shock loading considering the effects of the arch shape", International Journal of Non-Linear Mechanics, 41(9), 1065-1075, 2006. doi:10.1016/j.ijnonlinmec.2006.10.017

4

Z. Yi, Y. Zhao, K. Zhu, "The dynamic stability analysis of shallow arches with geometrical imperfections", Jisuan Lixue Xuebao/Chinese Journal of Computational Mechanics, 25(6), 932-938, 2008.

5

L. Wang, Z. Yi, H. Zhang, "Dynamic stability of arch with the geometrical imperfection subject to the periodic load", Hunan Daxue Xuebao/Journal of Hunan University Natural Sciences, 34(11), 16-19, 2007.

6

A.N. Blekherman, "Autoparametric resonance in a pedestrian steel arch bridge: Solferino bridge, Paris", Journal of Bridge Engineering, 12(6), 669-676, 2007. doi:10.1061/(ASCE)1084-0702(2007)12:6(669)

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