Moving internal node element method for dynamic analysis of beam structure under moving vehicle

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Abstract

This paper presents a new Moving Internal Node Element (MINE) method for the dynamic analysis of bridges/beam structure under a moving load/vehicle. By introducing a moving internal node within the standard beam element, the resulting MINE method eliminates the need for interpolation for the dynamic responses at the contact point between the moving load/vehicle and the beam structure. The governing equation of the moving internal node element is derived from the resulting bi-element with variable lengths but yet keeping the total length of the bi-element a constant. With this bi-element, the moving load/vehicle can be tracked easily as it is coincident with the moving internal node. This allows the reduction of the number of elements when compared to the conventional FEM needed to accurately predict the dynamic responses (e.g., displacement, velocity, and acceleration) of the moving load/vehicle (including at the contact point). Accuracy and computational efficiency are enhanced due to the smaller number of elements needed in the MINE method. The illustrative moving load/vehicle problems solved herein demonstrated the accuracy of the MINE method as well as its versatility and easy handling of any vehicle model.

Introduction

Vibration-based methods are commonly used for monitoring the operational and damage conditions of bridges [1]. Most of these methods require installing many sensors on the bridge for measuring the modal properties such as frequencies, mode shapes and damping coefficients. The measured modal properties are then compared with either theoretically obtained modal properties of the designed bridge or the historical measured data over the years for deviation to ascertain the current damage or the structural condition of the bridge.

The vibration-based methods may be divided into direct and indirect approaches. In the direct approach, the modal properties are retrieved from vibration data sets measured directly from the bridge [2]. However, one drawback with the direct approaches is that they usually require numerous sensors to be installed on the bridge along with data acquisition systems that are costly to install and maintain [3]. Moreover, such a monitoring system can hardly be transferred to another bridge for use [4] and the vast amount of data generated requires a tailored monitoring system to process effectively [5].

In the indirect approach, the bridge's dynamic properties are extracted directly from the measured responses of a passing vehicle that is instrumented with sensors. This indirect approach is called the vehicle scanning method (VSM) [6] or a drive-by damage detection technique [7]. As the vehicle passes over a bridge, the moving vehicle acts as both an exciter and a receiver of the vehicle-bridge interaction (VBI) responses [8]. From the acceleration data collected from a passing vehicle, one can extract useful information (such as the bridge frequency and modal shape) for health monitoring purposes [9].

We review notable studies conducted on VSM. Siringoringo and Fujino [10] estimated the first natural frequency of a bridge, that spans across a dam lake in Tokyo, from the finite element method (FEM) and compared to the bridge's measured frequency. They found the frequencies are in agreement as long as the vehicle driving velocity is below 30 km/hr. When the vehicle speed exceeds 30 km/hr, the vehicle frequencies appear as multiple peaks in the frequency spectrum. These peaks are higher than the bridge frequencies, thereby making the identification of the bridge frequencies challenging. In overcoming the multiple frequency peaks of the passing vehicle, Yang et al. [11] combined the band-pass filter and singular spectrum analysis to filter out the vehicle frequencies using the VBI element method. In order to verify the damage assessment of VSM, the estimated bridge mode shapes from the acceleration of a passing vehicle are adopted. One predicted the bridge's damaged state by comparing it with its intact state [12,13]. More recently, OBrien et al. [14] used the Empirical Mode Decomposition (EMD) to decompose the signal and Intrinsic Mode Functions (IMFs) to detect the damaged location from the frequencies obtained. Despite the noise (due to road roughness and numerical simulation errors) in the dynamic responses of a moving vehicle, Keenahan and OBrien [15] were able to identify the location of damage by utilizing the time-shifted curvature that was derived from the measured displacements.

Although much progress has been made in data interpretation in the VSM, there is still room for improvement in the determination of a passing vehicle's dynamic responses. Owing to FEM versatility in spatial discretization, it has been used to solve beam and plate problems under a moving load [16,17]. Based on FEM, Yang and Yau [18] proposed the Vehicle-Bridge-Interaction (VBI) element method by simplifying the conventional FEM assembly process to improve the computational efficiency. Zhang [19] and Chang [20] further developed the VBI element method to allow for random traffic and road roughness.

Besides the VBI element method, Koh et al. [21] proposed the Moving Element Method (MEM) for dynamic analysis of train-tracks resting on Winkler foundation under moving train loads. Considering the infinitely long nature of the rail beam, the MEM reduces the computational cost dramatically by using a moving coordinate system. Ang and Dai [22] extended the MEM to handle the phenomenon of the “jumping wheel” due to the abrupt change of foundation stiffness, while Tran et al. [23] used MEM to study the resonance of a rail system for high-speed trains traveling with non-uniform speed. Dai et al. [24] extended the MEM for a discretely supported track. They found that such a track design produces more severe vibrations than a continuously supported track of an equivalent foundation stiffness.

More recently, researchers have developed more advanced methods for tackling practical moving load problems. Xiao and Ren [25] developed a versatile 3D vehicle-track-bridge (VTB) element to consider the lateral contact forces and geometric shapes of the wheel and rails and even the occasional jumping of wheels from the rails. Xie et al. [26] suggested applying the plane continuum finite element model for determining train and track vibration because it has superiorities in capturing higher frequencies from the rail-wheel interaction. Zhang et al. [27] used the discrete singular convolution (DSC) method to analyze functionally graded beams under a moving load because of its simplicity and stability for handling complex geometries and boundary conditions. However, these aforementioned methods have not attempted to improve the dynamic response solution (especially for the acceleration) at the contact point between the moving vehicle and the structure.

In this paper, we propose a moving internal node that tracks the moving load/vehicle's location for implementation in FEM (or VBI element method) and MEM. By introducing a moving internal node within an element in FEM, the finite element becomes a bi-element essentially. Therefore, the degree of freedom (DOF) of the moving load/vehicle can be easily implemented in the model. We have named this new element as the Moving Internal Node Element (or MINE for brevity). It will be shown herein that the MINE method improves the accuracy of velocity and acceleration responses of structures under a moving vehicle (or load).

Section snippets

Formulation of moving internal node element (MINE)

Consider a beam structure of length L which is divided equally into N number of MINEs. MINE is a bi-element comprising two beam elements e1 and e2 with variable lengths ξ and (L/Nξ), respectively, joined together as shown in Fig. 1. The lengths of elements e1 and e2 change according to the location of the moving vehicle. When the mass or vehicle moves from point A to point C, the internal node B of the two elements shifts according to the moving vehicle, as shown in Fig. 1. In order to

Procedure of MINE method

In summary, the MINE method involves the following procedural steps:

  • Step 1.

    Input all structure and vehicle data.

  • Step 2.

    Start with time t=0 and assemble hibernated elements (e.g., KMINEh and MMINEh in Eq. (27)) to set up the beam's initial conditions. Form the MINE function to calculate the MINE matrices, e.g., KMINEband MMINEb in each time step. Select a proper time increment Δt for the Newmark integration scheme.

  • Step 3.

    For the incremental step, let ts+1(j+1)=ts+1(j)+Δts+1(j) and use j=1 for the first iteration.

Numerical verifications

Three example problems of moving mass/vehicle will be solved to verify and illustrate the present MINE method as applied to dynamic interactions of beam structures under a moving mass/vehicle. The first example involves the dynamic analysis of a simply supported beam subjected to a moving mass. The beam's dynamic response at the contact point is to be determined by using the MINE method. The MINE results will be compared to those from two other approaches, namely, the FEM and a semi-analytical

Conclusion

Presented herein is a new moving internal node element (MINE) method for determining accurate dynamic responses of a moving load/mass/vehicle across a beam structure. Unlike the conventional FEM for moving load problem, the MINE method has the following advantages:

  • the vertical force from the moving load/mass/vehicle can be imposed on the internal node without any interpolation or having to deal with a singular function (e.g., Dirac delta function)

  • the dynamic responses (i.e., deflection,

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by the Department of Transport and Main Roads (TMR) Chair Professor Grant in The University of Queensland and the Ministry of Science and Technology in Taiwan through Grant 107-2211-E-032-002-MY2 and 109-2221-E-032-003.

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