Soil Dynamics and Earthquake Engineering 30 (2010) 769–781
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Soil Dynamics and Earthquake Engineering
journal homepage: www.elsevier.com/locate/soildyn
Linear vertical vibrations of suspension bridges: A review of continuum models and some new results ´ J. Enrique Luco a, Jose Turmo b,n
Department of Structural Engineering, University of California, San Diego, La Jolla, California, USA Civil Engineering Department, University of Castilla-La Mancha, Ciudad Real, Spain
a r t i c l e in fo
Article history: Received 10 July 2009 Accepted 30 October 2009 Keywords: Suspension bridge Suspension cable Vibrations Continuum model Linear response
The classical continuum model for the linear vertical vibrations of a suspension bridge (Bleich et al., 1950 ) is re-examined. The primary objective of the study is to extend the deﬁnitive analytical and numerical results of Irvine and Caughey (1974) , Irvine and Grifﬁn (1976)  and Irvine (1980, 1981) [4,5] for the natural frequencies, mode shapes, and modal participation factors for an extensible suspension cable, which depend on one dimensionless parameter related to the elasticity of the cable, to the case of a stiffened suspension bridge in which the response depends also on a second dimensionless parameter related to the stiffness of the girder. The continuum suspension bridge model is also used to understand the pattern of variation of mode shapes as a function of cable elasticity and girder stiffness, which has been shown by West et al. (1984)  to be considerably more complex than that for a suspension cable. Finally, the threshold amplitudes of free vibrations that would result in the incipient slackening of the hangers are determined. & 2009 Elsevier Ltd. All rights reserved.
1. Introduction With the advent of powerful computational methods to analyze the dynamic response of suspension bridges (e.g. [7–12]), the emphasis has changed from simple continuum models that could be used to study a class of bridges to extremely detailed discretized models used to analyze particular bridges. The objective of this paper is to return to some of the earlier continuum models  that have not been fully analyzed. Although not as general, the simpler continuum models are useful to identify the key dimensionless parameters that control the dynamic response of the bridge and to conduct extensive parametric analyses. These models also lead to approximate formulae suitable for preliminary designs, and to benchmark results that can be used to test the accuracy of numerical models. The study of the dynamics of suspended cables, pertinent to suspension bridges but without inclusion of the stiffening girder, was initiated by Poisson in 1820 with his equations of motion for a cable element subjected to general forces. Solutions for the free vertical vibrations of inextensible cables have been presented by Rohrs , Routh , Rannie and von Karman , Pugsley , Saxon and Cahn  and Goodey . The effects of the elasticity of the cable were introduced in 1945 by Vincent , and later by Bleich et al. . The deﬁnitive analytical work on the linear theory of a suspended elastic cable, including an explanation for the
Corresponding author. E-mail address: email@example.com (J. Turmo).
transition from a taut string to an inextensible suspended cable of small sag, was presented by Irvine and Caughey  and Irvine . These authors introduced a dimensionless parameter that reﬂects the effect of the elasticity of the cable on the natural frequencies of vibration. This parameter also affects the mode shapes, and, in particular, the number of internal nodes in a given mode. West et al.  studied the free vertical vibrations of a suspended cable by representing the cable as a discrete set of axially deformable linkages and conﬁrmed the mode shape transitions described by...