An Iterative Solution Method for Dynamic Response of Bridge-Vehicles Systems

Abstract

SUMMARYAn iterative solution method is presented and illus1 rated to analyse the dynamic response of bridge-vehicle systems. The method consists in dividing the whole system into 2 subsystems at the interface of the bridge and vehicles; these 2 subsystems are solved separately; their compatibility at the interface is achieved by an iterative procedure with under-relaxation or with Aitken acceleration. The characteristics of this method are explained on a simplified system with 2 degrees of freedom (DOF). The numerical results for a simple example demonstrate the high performances of the proposed method: good convergence rate and high accuracy. Finally, the method is applied to a practical example: the linear dynamic response of the Yangtze-River Bridge at Wuhan under a moving train with 2 locomotives and 4 freight cars. The efficiency is attained because neither formation nor factorisation of the coefficient matrices for the equations of the system are needed at every time step in linear analysis. The Aitken acceleration technique is more efficient in systems with multi-degrees of freedom than the relaxation technique. The proposed method will be even more efficient in non-linear dynamic response because, in this case, the iterations are necessary whether the system is solved as a whole or not.

A bridge-monitoring tool based on bridge and vehicle accelerations

“…It is necessary to solve both subsystems while ensuring compatibility at the contact points (i.e., displacements of the bridge and the vehicle being the same at the contact point of the wheel with the roadway) (González, 2010). Several approaches on implementing VBI simulations are available in the literature (Green et al 1995, Wang et al 1996, Yang and Fonder 1996, Green and Cebon 1997, Zhu and Law 2002, Yang and Lin 1995, Henchi et al 1998, Yang et al 2004b, Kim et al, 2005 This paper uses the iterative approach described by Cebon 1994, Green andCebon 1997) to ensure compatibility between the two sub-systems. This approach has shown good agreement with other techniques available to model the interaction between the vehicle and the bridge (González, 2010).…”

An investigation into the acceleration response of a damaged beam-type structure to a moving force

“…A planar vehicle-bridge interaction simulation model is implemented using the iterative approach described in [23,38,39]. The vehicle represents a 2-axle rigid truck with four degrees of freedom: the pitch and vertical displacement of the sprung mass and the displacement of the two unsprung masses.…”

Vehicle-Bridge Dynamic Interaction Using Finite Element Modelling

“…Planar vehicle models have been found to provide a reasonable bridge response for ratios bridge width to www.intechopen.com vehicle width greater than 5 (Moghimi & Ronagh, 2008a). A single-DOF model can be used for a preliminary study of the tyre forces at low frequencies due to sprung mass bouncing and pitching motion (Chatterjee et al, 1994b;Green & Cebon, 1997) and a two-DOF model (i.e., a quarter-car) can be employed to analyse main frequencies corresponding to bodybounce and axle hop modes (Green & Cebon, 1994;Chompooming & Yener, 1995;Yang & Fonder, 1996;Cebon, 1999). If the influence of axle spacing was investigated, then a rigid walking beam (Hwang & Nowak, 1991;Green & Cebon, 1994;Chompooming & Yener, 1995) or an articulated multi-DOF model (Veletsos & Huang, 1970;Hwang & Nowak, 1991;Green et al, 1995;Harris et al, 2007) will become necessary.…”

“…It is necessary to solve both subsystems while ensuring compatibility at the contact points (i.e., displacements of the bridge and the vehicle being the same at the contact point of the wheel with the roadway). The algorithms to carry out this calculation can be classified in two main groups: (a) those based on an uncoupled iterative procedure where equations of motion of bridge and vehicle are solved separately and equilibrium between both subsystems and geometric compatibility conditions are found through an iterative process (Veletsos & Huang, 1970;Green et al, 1995;Hwang & Nowak, 1991;Huang et al, 1992;Chatterjee et al, 1994b;Wang et al, 1996;Yang & Fonder, 1996;Green & Cebon, 1997;Zhu & Law, 2002;Cantero et al, 2009), and (b) those based on the solution of the coupled system, i.e., there is a unique matrix for the system that is formed by eliminating the interaction forces appearing in the equations of motion of bridge and vehicle, and updated at each point in time (Olsson, 1985;Yang & Lin, 1995;Yang & Yau 1997;Henchi et al, 1998;Yang et al, 1999Yang et al, , 2004aKim et al, 2005;Cai et al, 2007;Deng & Cai, 2010;Moghimi & Ronagh, 2008a). The use of Lagrange multipliers can also be found in the solution of VBI problems (Cifuentes, 1989;Baumgärtner, 1999;González et al, 2008a).…”