Simultaneous identification of bridge structural parameters and vehicle loads Dongming Feng a,⇑ , Hao Sun b , Maria Q. Feng a a Department of Civil Engineering & Engineering Mechanics, Columbia University, New York, NY 10027, United States b Department of Civil & Environmental Engineering, MIT, Cambridge, MA 02139, United States article info Article history: Received 7 December 2014 Accepted 11 May 2015 Keywords: Vehicle axle loads Moving forces Vehicle–bridge interaction Output-only identification Bayesian regularization Damped Gauss–Newton optimization abstract Most of the existing methods for identification of vehicle axle loads are based on a model with known system parameters. In this study, a new method is proposed to simultaneously identify bridge structural parameters and vehicle dynamic axle loads of a vehicle–bridge interaction system from a limited number of response measurements. As an inverse output-only identification problem, the estimation of unknown axle loads is incorporated in the framework of an iterative parametric optimization process, wherein the objective is to minimize the error between the measured and predicted system responses. A Bayesian inference regularization is presented to solve the ill-posed least squares problem for input axle loads. Numerical analyses of a simply-supported single-span bridge and a three-span continuous bridge are conducted to investigate the accuracy and efficiency of the proposed method. Effects of the vehicle speed, the number of sensors, the measurement noise, and initial estimates of structural parameters on the accuracy of the identification results are investigated, demonstrating the robustness and efficiency of the proposed algorithm. Finally, it is shown that the bridge dynamic response can be accurately predicted using the identified axle load histories and structural parameters. Ó 2015 Elsevier Ltd. All rights reserved. 1. Introduction Repeated operational traffic loads over time can lead to the deterioration of bridge structures. It is therefore important to mea- sure the operational vehicle axle loads or the moving dynamic forces for evaluation and maintenance bridge structures. From bridge design point of view, it is also desirable to measure the vehi- cle axle loads [1]. Weigh-in-motion (WIM) systems [2] have been developed, which measure the bridge strains caused by moving vehicles to estimate the equivalent static axle loads [3,4]. However, the accuracy of the WIM systems often depends on the road surface condition and the results are reliable only when the vehicle speed is low and bridge pavement surface is smooth [5]. In fact, the dynamic effects of the moving vehicle cause much lar- ger bridge responses, especially when unfavorable road roughness exists, by increasing the average surface damage two to four times compared to that from the static axle forces [5]. In view of this, it would be beneficial if the dynamic axle forces of the operational traffic vehicles as well as the structural parameters could be identified. Such information is of particular importance for the pur- pose of deck and pavement design, management, maintenance, safety assessment as well as fatigue estimation [5,6]. Consequently, there has been a considerable amount of research efforts towards identification of dynamic vehicle axle loads or moving forces from measured bridge responses including strain, displacement, acceleration, and bending moment [1–13]. For example, Law et al. [7] introduced a regularization method in the ill-posed inverse problem to provide bounds to the identified moving forces. Based on the finite element formulation, a general- ized orthogonal function approximation was developed by Zhu and Law [8] to obtain the derivatives of the bridge modal responses, and moving forces were identified using the regularized least-squares method in the time domain. Zhu and Law [12] fur- ther studied the effects of vertical and rotational support stiffness on the identification results based on an elastically supported multi-span continuous bridge deck. González et al. [4] developed a moving force identification method using the first-order Tikhonov regularization applied to a two dimensional orthotropic plate bridge from 21 strain measurements. Deng and Cai [5] pre- sented a moving force identification technique using the superpo- sition principle and the influence surface to deal with the bridge structure, and effects of factors such as the vehicle speed, the road http://dx.doi.org/10.1016/j.compstruc.2015.05.017 0045-7949/Ó 2015 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. Tel.: +1 917 834 4717. E-mail addresses: df2465@columbia.edu (D. Feng), haosun@mit.edu (H. Sun), mqf2101@columbia.edu (M.Q. Feng). Computers and Structures 157 (2015) 76–88 Contents lists available at ScienceDirect Computers and Structures journal homepage: www.elsevier.com/locate/compstruc