Parametric Study on the Nonlinear Dynamics of a Three-Stay Cable Network under Stochastic Free Vibration Gian Felice Giaccu 1 ; Bernardo Barbiellini 2 ; and Luca Caracoglia, A.M.ASCE 3 Abstract: Crossties and cable networks are used on cable-stayed bridges to mitigate wind-induced stay vibration. A nonlinear free-vibration analysis of a cable network with random oscillation amplitudes is presented in this study. Specifically, a nonlinear restoring-force spring model is introduced at the crosstie to simulate in-plane network free vibration at large amplitudes. The current constitutive model of the crosstie is also combined with taut-cable theory to simulate the dynamics in the stays and is solved by the equivalent linearization method. Stochastic functions are introduced in the model because the measure of the amplitudes produced by wind- and rain-wind–induced vibrations in the network can be affected by various uncertainties. The stochastic approximation (SA) algorithm is applied to find the roots of the characteristic polynomials associated with a stochastic vibration amplitude parameter. Brute force Monte Carlo methods are also used to analyze the SA convergence properties. DOI: 10.1061/(ASCE)EM.1943-7889.0000887. © 2014 American Society of Civil Engineers. Introduction On a cable-stayed bridge, in-plane cable networks are obtained by connecting two or more stays by transverse restrainers, designated as crossties (Ehsan and Scanlan 1990; Yamaguchi and Nagahawatta 1995). These are often used as devices for the mitigation of stay- cable vibration, induced by wind or wind and rain (Bosdogianni and Olivari 1996; Matsumoto et al. 2003b). Vibration in the stays of a cable-stayed bridge have often been observed (Matsumoto et al. 2003a; Phelan et al. 2006; Zuo and Jones 2010; Zuo et al. 2008); their suppression by using a number of countermeasures, including the use of crossties, has been examined by several researchers. For example, taut-cable linear theory has been used for predicting the dynamic response in a cable network, based on observations of existing systems (Caracoglia and Zuo 2009). Even though experi- mental or full-scale data on the performance of cable networks are limited, the study and optimization of cable networks have received renewed attention by engineers, particularly in recent years (Ahmad and Cheng 2013, 2014; Bosch and Pagenkopf 2013; Bosch and Park 2005). In most of the existing dynamic models for cable networks, non- linear behavior in the restrainers is not simulated. Recently, the non- linear dynamic response of an in-plane cable network has gained attention. A new algorithm was proposed (Giaccu and Caracoglia 2012) to analyze the dynamics of cable networks in the presence of internal nonlinear interactions, which have been observed experi- mentally but have been thoroughly investigated in the case of very simple systems only (Yamaguchi and Alauddin 2003). The algo- rithm for examination of nonlinear cable-crosstie systems operates by linearization; it solves an equivalent eigenvalue/eigenvector prob- lem to obtain linearized frequencies and modes as a function of a de- terministic vibration amplitude parameter l (Giaccu and Caracoglia 2013). Nonlinearity is concentrated in the restoring-force mecha- nism in each restrainer; the internal force transmitted through the crosstie is simulated by a power-law stiffness spring model with a positive integer exponent. Nonlinearity is used to replicate large vibration in the restrainer in an attempt to analyze extreme con- ditions in the cables. A more detailed discussion on the motivations, which led to the choice of this specific constitutive law for the restrainers, is given in a subsequent section of this study. Considering the current state of the research, the reasoning be- hind this study is as follows: • Crossties are primarily an energy redistribution mechanism (Caracoglia and Zuo 2009). The two important parameters in the design of a cable network are a beneficial frequency shift, which is obtained by connecting two or more stays, and the increment in the modal mass, at least for the fundamental global modes, which inhibits the vibration. Accurate estimation of the frequencies is therefore necessary to evaluate the sensitivity of the system to the frequency range of typical wind-induced vibrations. • Accurate evaluation of the frequency shift is hindered by non- linear effects in the crossties; the solution is also influenced by the complexity and, on occasion, by insufficient knowledge of various oscillation regimes, synoptically represented by the am- plitude parameter l. Therefore, the objectives of this study are • To investigate the dynamics of the system by examining the free- vibration response and by including nonlinear effects; this ap- proach is meaningful because the oscillations are predominantly aeroelastic, bearing in mind that the ultimate goal is to provide simple solutions for engineering design. The aim is not to examine aeroelastic loads and to explain trigger mechanisms in a phenom- enological way, but to promote simple methods, based on frequency studies, to analyze or to design cable networks (Bosch 2007). • To examine the problem of inadequate knowledge of the vibration mechanism, definitely influenced by irregular wind-load features. This lack of knowledge can be included by a stochastic pertur- bation operating on the vibration amplitude parameter l. This stochastic model provides a simple yet robust analysis approach. • To observe the frequency behavior of higher network modes (e.g., the second one); this is relevant because these modes can 1 Assistant Professor, Dept. of Architecture, Design and Urban Planning, Univ. of Sassari, 07041 Alghero, Italy. E-mail: gf.giaccu@uniss.it 2 Senior Research Scientist, Dept. of Physics, Northeastern Univ., Bos- ton, MA 02115. E-mail: bba@neu.edu 3 Associate Professor, Dept. of Civil and Environmental Engineering, Northeastern Univ., 400 Snell Engineering Center, Boston, MA 02115 (corresponding author). E-mail: lucac@coe.neu.edu Note. This manuscript was submitted on June 9, 2014; approved on September 30, 2014; published online on October 28, 2014. Discussion period open until March 28, 2015; separate discussions must be submitted for individual papers. This paper is part of the Journal of Engineering Mechanics, © ASCE, ISSN 0733-9399/04014166(10)/$25.00. © ASCE 04014166-1 J. Eng. Mech. J. Eng. Mech. 2015.141. Downloaded from ascelibrary.org by Northeastern University Library on 06/03/15. Copyright ASCE. For personal use only; all rights reserved.