Simultaneous Identification of Moving Vehicles and Bridge Damages Considering Road Rough Surface

Abstract

A method for the simultaneous identification of moving vehicles and the damages of the supporting structure from measured responses is presented. A two-axle vehicle model with two degrees of freedom (DOF) is adopted. The extent of the damage and the vehicle parameters were chosen as the optimisation variables, which allow ill conditioning to be avoided and decrease the number of sensors required. The identification is performed by minimising the distance between the measured responses and the computed responses to given optimisation variables. The virtual distortion method (VDM) was used, such that the response of the damaged structure can be computed from comparison with the intact structure subjected to the same vehicle excitation and to the response-coupled virtual distortions. These are related to the optimisation variables by the system impulse response matrix and are expressed by a linear system, which allowed both types of optimisation variables to be treated in a unified way. The numerical cost is reduced by using a moving influence matrix. The adjoint variable method is used for fast sensitivity analysis. A three-span bridge numerical example is presented, where the identification was verified with 5% root mean square (RMS) measurement, and model, error whilst also considering the surface roughness of the road.

1. Introduction

In structural health monitoring, accurate load and damage identification are indispensable to the maintenance of structural integrity, as well as providing the evidence for forensic engineering. Particularly, moving loads or moving vehicle parameter identification is important not only for prediction and analysis of the dynamic responses of bridges, but also in traffic studies, in design code calibration, for traffic control, and so forth. In recent decades, many investigations have been performed on either load identification or damage identification. However, in practice, both unknown system damage and unknown (moving) excitations usually coexist and together influence the system response in a coupled way. Therefore, their simultaneous identification is worthy of further exploration.

As opposed to local, high-frequency ultrasonic scanning, this paper considers only global or low-frequency damage identification methods which are based on the structural vibrations. These methods are mainly categorised into two groups: frequency domain methods or time domain methods. In the frequency domain, damage is detected, located, and identified through measured changes in the related modal parameters or dynamic signature [1] such as natural frequency and mode shape. However, the modal parameters are obtained from the responses to a certain kind of excitation and these methods are limited to time-invariant systems. Under the action of a moving vehicle, a coupled bridge-vehicle system is time variant, and the system parameters are changing as the vehicles move [2]. The vibration caused by traffic is a nonstationary process that strengthens with decreasing span [3]. In this case, the analysis is most often performed in the time domain by the direct comparison of the simulated and measured responses [4]. However, this usually requires a known moving load or other such vehicle parameters. Sieniawska et al. [5] identify parameters of a linear structure from its responses to a moving load using a static substitution of the equation of motion, where the moving load is a known constant magnitude. Chang et al. [3, 6] present, and experimentally verify, a methodology for damage identification in bridges using a pseudostatic formulation based on coupling vibration with a moving vehicle, where the moving force is provided as a vector or calculated using the measured acceleration responses of the axles. Ling and Haldar [7] propose a linear system identification approach with unknown input, but they require a known dynamic response of all the degrees of freedoms; this is hard to perform on real-life, complex structures.

Moving load identification has been studied extensively in the past two decades [8]. Techniques of indirect identification from measured responses have been investigated and can be performed more easily and at lower cost compared to methods requiring direct measurements of moving loads. Chan et al. and Law et al. have proposed four methods for indirect identification: the time-domain method (TDM) [9], the frequency-time domain method (FTDM) [10], interpretive method I (IM-I) [11], and interpretive method II (IM-II) [12]. All of them require, a priori, known model parameters for the bridge. Each method has its merits and limitations, which are compared elsewhere [13]. Load identification is an inverse problem, and numerical ill-conditioning seems to be the main factor that influences the accuracy of the identification results. The accuracy can be improved by some regularisation methods, such as Tikhonov regularisation [14]. However, the determination of the optimal value of the regularisation parameter is numerically costly and requires lengthy computation. Pinkaew and Asnachinda [15, 16] point out that the regularisation parameter turns out to be sensitive to properties of both the vehicle and bridge and is difficult to be precisely assigned: they propose, and experimentally verify, an iterative regularisation method called the updated static component (USC) technique to decrease the sensitivity of the regularisation parameter. Moving forces are usually treated as unknown quantities to be identified. This requires a number of sensors equal to or exceeding the number of unknown moving forces to obtain a unique solution. Jiang et al. [17] present a moving vehicle parameter identification method based on genetic algorithms, where each moving vehicle is modelled as a two-DOF system that comprises mass, spring stiffness, and damping. Zhang et al. [18] provide an approach of identifying moving vehicle parameters based on VDM using a dynamic influence matrix. In general, the identification of moving forces or vehicle(s) requires a known, well-defined, bridge model to establish the relationship between load and response.

For unknown coexistent load and structural damage, it is generally difficult to decouple the related identification problems and solve any one of them independently. Since the two factors have essentially different natures, Zhang et al. [19] present a method to identify load and damage simultaneously using Chebyshev polynomials to parameterise the unknown force, and thus all parameters related to the damage and excitation can be updated simultaneously in each iteration. Zhang et al. [20] present, and experimentally verify, a method for the simultaneous identification of nonmoving excitations and damage, which uses a VDM [21] to model structural damage and thus improve the identification efficiency thereof. In case of unknown moving loads, the vehicle-bridge system is a coupled time-varying system. Zhu and Law [22] present a two-step iteration procedure to simultaneously identify the moving loads and the damage caused to an Euler-Bernoulli simply supported beam; the number of the sensors is one less than the number of the beam elements.

Zhang et al. [23] present a method for the simultaneous identification of structural damage and moving masses; moving masses and damage extents are used as its optimisation variables, which avoids the ill-conditioning problem present in traditional moving force identification and decreases the number of sensors required; damage is modelled using virtual distortions, and a dynamic influence matrix is introduced to reduce numerical analysis cost. This paper simulates the vehicle using a two-axle, two-DOF, vehicle model, and the related parameters including mass, rotary inertia, spring stiffness, and damping are treated as unknowns together with the structural damage induced. The damaged bridge is assumed to be modelled by the decay of its mechanical properties and is often represented as a decreasing stiffness [24]. Based on VDM, the response of the damaged structure is modelled by the intact structure subjected to the same vehicle excitation and to the response-coupled virtual distortions. Through the related system impulse response matrix (dynamic influence matrix in the terminology of VDM), the moving loads and virtual distortions are connected with the optimisation variables in a linear system. In this way, the optimisation variables related to the vehicle parameter, and to the damage, are treated in a unified way. Then, they can be optimised using any standard optimisation algorithm. The numerical cost is reduced by using the moving dynamic influence matrix. Moreover, rapid sensitivity analysis is performed using the adjoint variable method.

Sections 2 and 3 discuss the equation of motion of the system and simulation of the system dynamic response by moving vehicle and virtual distortions; Section 4 discusses the inverse identification problem. Section 5 verifies the proposed approach using a numerical example of a three-span frame structure considering road surface roughness. The approach and the results are discussed in Section 6.

2. Dynamic Analysis of a Coupled Vehicle-Beam System

2.1. Equation of Motion for a Moving Vehicle on a Bridge

A popular two-axle, two-DOF, vehicle model [3] was adopted; see Figure 1, where𝑢^𝑣 and𝜃^𝑣, respectively, denote the bounce and pitching motions of the vehicle. Exactly speaking,𝑢^𝑣is the vertical displacement of the vehicle, which is measured vertically upwards from its vertical static equilibrium position before the vehicle reached the bridge and𝜃^𝑣_𝑖are its rotary displacement relative to the static position. Denoted by 𝑚 and 𝐽, respectively, are the vehicle mass and the rotary inertia; denoted by 𝑘 and 𝑐, respectively, are the spring constant and damping coefficient of the vehicle.

Figure 1 

A coupled vehicle-bridge system.

For the𝑖th moving vehicle on the bridge,  the additional elongation of the left and right spring except the elongation caused by the weight is respectively denoted by Δ_𝑖𝑙, and Δ_𝑖𝑟, then Δ_𝑖𝑙=𝑢^𝑣_𝑖+𝜃_𝑖1𝑒_𝑖𝑙−𝑥_𝑖𝑙−𝑟_𝑖𝑙;Δ_𝑖𝑟=𝑢^𝑣_𝑖−𝜃_𝑖1𝑒_𝑖𝑟−𝑥_𝑖𝑟−𝑟_𝑖𝑟, (1) where𝑒_𝑖𝑙,  𝑒_𝑖𝑟denote, respectively, the distance from the left and right axle to the vehicle’s centre of mass;𝑥_𝑖𝑙,  𝑥_𝑖𝑟, respectively, denote the vertical displacements of the bridge at the current position of the vehicle’s left and right axle;𝑟_𝑖𝑙,  𝑟_𝑖𝑟denote the roadway surface roughness at the current respective position, which is also measured vertically upwards with reference to the deck level.

The forces of the𝑖th moving vehicle are shown in Figure 2, which includes the inertial forces. Based on the dynamic balance, the equation of motion of the𝑖th moving vehicle can be formulated as 𝐦^𝑣_𝑖𝑌̈^𝑣_𝑖(𝑡)+𝑐_𝑖𝐞_𝑖𝑘𝑌̇^𝑣_𝑖(𝑡)+𝑘_𝑖𝐞_𝑖𝑘𝑌^𝑣_𝑖(𝑡)=𝑘_𝑖𝑆_𝑖𝑥_𝑖(𝑡)+𝑟_𝑖(𝑡)+𝑐_𝑖𝑆_𝑖𝑥̇_𝑖(𝑡)+𝑟̇_𝑖(𝑡), (2) where𝐦^𝑣_𝑖=^𝑚𝑖0_0𝐽𝑖,𝑌^𝑣_𝑖(𝑡)=^𝑢𝑣𝑖_𝜃^(𝑡)_{𝑣𝑖(𝑡)},𝑟_𝑖(𝑡)=^𝑟𝑖𝑙_𝑟^(𝑡)_{𝑖𝑟(𝑡)},𝑠_𝑖=_𝑒¹¹_{𝑖𝑙−𝑒𝑖𝑟},𝑥_𝑖(𝑡)=^𝑥𝑖𝑙_𝑥^(𝑡)_{𝑖𝑟(𝑡)},𝐞_𝑖𝑘=^{2𝑒𝑖𝑙−𝑒𝑖𝑟}_{𝑒𝑖𝑙−𝑒𝑖𝑟𝑒2𝑖𝑙+𝑒2𝑖𝑟}.

Figure 2 

Forces of the𝑖th moving vehicle.

In this study, the random roughness of the road surface𝑟(𝑥)is simulated by a zero-mean, real-value, stationary Gaussian process as described in the following [17, 25]: 𝑟(𝑥)=^𝑁𝑇_𝑘=1𝛼_𝑘cos2𝜋𝑛_𝑘𝑥+𝜑_𝑘, (3) where𝛼_𝑘is the amplitude of the cosine wave,𝑛_𝑘is the spatial frequency (in cycle/m),𝜑_𝑘is a random phase angle with uniform probability distribution in the interval [0,2𝜋],𝑥is the position measured from the left end of the bridge, and𝑁_𝑇is the total number of the cosine waves which are used to construct the roughness. Parameter𝛼_𝑘is computed by 𝛼²_𝑘=4𝐺_𝑑𝑛_𝑘Δ𝑛, (4) where𝐺_𝑑(𝑛_𝑘)is the power spectral density function of the road surface and related to the ground flatness coefficient𝐺_𝑑(𝑛₀)in the form as [26] 𝐺_𝑑𝑛_𝑘=𝐺_𝑑𝑛₀𝑛_𝑘𝑛₀⁻²,𝑛_𝑘=𝑛_𝑙𝑛+𝑘Δ𝑛,Δ𝑛=^𝑢−𝑛_𝑙𝑁_𝑇, (5) where𝐺_𝑑(𝑛₀)depends on the classification of road surface condition.𝑛_𝑙and𝑛_𝑢, are respectively, lower and upper cut-off spatial frequencies, and the power spectral density function𝐺_𝑑(𝑛_𝑘)is defined within the interval of[𝑛_𝑙,𝑛_𝑢]. For the vehicle velocity between 70 km/h and 120 km/h on expressway, the power spectral density is most meaningful in the frequency interval of [0.0221 m⁻¹, 1.4142 m⁻¹].

2.2. Equation of Motion for a Bridge under Moving Vehicles

The bridge is idealized as a beam, which suffers the traffic excitations. Denoted by𝑝_𝑖𝑙(𝑡)and𝑝_𝑖𝑟(𝑡), respectively, are the interaction forces of the𝑖th moving vehicle at its left and right wheel, and through the dynamic balance analysis, they can be expressed using the additional elongation of the spring and the gravity of the vehicle, 𝑝_𝑖𝑙(𝑡)=𝑘_𝑖Δ_𝑖𝑙(𝑡)+𝑐_𝑖Δ̇_𝑖𝑙(𝑡)+𝑚_𝑖𝑔𝑒_𝑖𝑟𝑒_𝑖𝑙+𝑒_1𝑟,𝑝_𝑖𝑟(𝑡)=𝑘_𝑖Δ_𝑖𝑟(𝑡)+𝑐_𝑖Δ̇_𝑖𝑟(𝑡)+𝑚_𝑖𝑔𝑒_𝑖𝑙𝑒_𝑖𝑙+𝑒_𝑖𝑟. (6)

The equation of motion for the bridge can be formulated as follows: 𝐌𝐮̈^𝑏(𝑡)+𝐂𝐮̇^𝑏(𝑡)+𝐊𝐮^𝑏(𝑡)=^𝑛𝑚_𝑖=1𝐛_𝑖(𝑡)𝐩_𝑖(𝑡), (7) where𝐌,𝐂, and𝐊are, respectively, the mass, stiffness, and damping matrices of the bridge,𝐮^𝑏represents the displacement vector, and 𝐩_𝑖(𝑡)=𝑝_𝑖𝑙(𝑡)𝑝_𝑖𝑟(𝑡)^T,𝐛_𝑖(𝑡)=𝑏_𝑖𝑙(𝑡)𝑏_𝑖𝑟(𝑡),𝑏_𝑖𝑙(𝑡),  𝑏_𝑖𝑟(𝑡), are respectively, the time-varying load distribution vector in global coordinates with regard to the left and right wheels. Thus,𝑏_𝑖𝑙(𝑡),  𝑏_𝑖𝑟(𝑡)can be formulated using the shape function of the beam element in finite element (FE) analysis, and their elements are zeroed when the corresponding wheel is off the bridge during the sampling time.

2.3. Dynamic Response of the Coupled Vehicle-Bridge System

In the equation of motion for the moving vehicle and bridge ((2) and (7)), it can be seen that the system is coupled: the vibration of the bridge influenced the vibration of the moving vehicle (2), and the vehicle in return interactively influenced the vibration of the bridge (7). The equation of motion for the coupled system can be expressed as follows: 𝐌𝐦^𝑣₁⋱𝐦^𝑣_𝑛𝑚𝐮̈^𝑏(𝑡)𝑌̈^𝑣₁(𝑡)⋮𝑌̈^𝑣_𝑛𝑚(𝑡)+𝐊+Δ𝐊(𝑡)−𝑏₁(𝑡)𝑆^T₁𝑘₁⋯−𝑏_𝑛𝑚(𝑡)𝑆^T_𝑛𝑚𝑘_𝑑−𝑆₁𝑏^T₁(𝑡)𝑘₁𝑘₁𝐞_1𝑘⋮⋱−𝑆_𝑛𝑚𝑏^T_𝑛𝑚(𝑡)𝑘_𝑛𝑚𝑘_𝑛𝑚𝐞_𝑛𝑚𝑘×𝐮^𝑏(𝑡)𝑌^𝑣₁(𝑡)⋮𝑌^𝑣_𝑛𝑚(𝑡)+𝐂+Δ𝐂(𝑡)−𝑏₁(𝑡)𝑆^T₁𝑐₁⋯−𝑏_𝑛𝑚(𝑡)𝑆^T_𝑛𝑚𝑐_𝑑−𝑆₁𝑏^T₁(𝑡)𝑐₁𝑐₁𝐞_1𝑘⋮⋱−𝑆_𝑛𝑚𝑏^T_𝑛𝑚(𝑡)𝑐_𝑛𝑚𝑐_𝑛𝑚𝐞_𝑛𝑚𝑘×𝐮̇^𝑏(𝑡)𝑌̇^𝑣₁(𝑡)⋮𝑌̇^𝑣_𝑛𝑚(𝑡)=^𝑖𝑏_𝑖𝑚_𝑖𝑒_𝑖𝑔−𝑘_𝑖𝑟_𝑖+𝑐_𝑖𝑟̇_𝑖𝑘₁𝑆₁𝑟₁+𝑐₁𝑆₁𝑟̇₁⋮𝑘_𝑛𝑚𝑆_𝑛𝑚𝑟_𝑛𝑚+𝑐_𝑛𝑚𝑆_𝑛𝑚𝑟̇_𝑛𝑚, (8) whereΔ𝐊(𝑡)=∑_𝑖𝑘_𝑖𝑏_𝑖(𝑡)𝑏^T_𝑖(𝑡),Δ𝐂(𝑡)=∑_𝑖𝑐_𝑖𝑏_𝑖(𝑡)𝑏^T_𝑖(𝑡), 𝑒_𝑖=𝑒_𝑖𝑙/(𝑒_𝑖𝑙+𝑒_𝑖𝑟)𝑒_𝑖𝑟/(𝑒_𝑖𝑙+𝑒_𝑖𝑟)^T.

Equation (8) shows that the system is time variant with the vehicle’s motion, and this indicated that the vibration of the bridge caused by the traffic was nonstationary. Therefore, the performance of damage identification in this case may have similar limitations to those pertaining to methods subjected to a stationary assumption, for example, modal based methods in the frequency domain. In contrast, time-domain analysis for identification is more accurate and appropriate. Then, the key procedure is the estimation of the system response to damage. However, if the response is computed in the traditional way by using the system motion equation, it will need to reassemble the varying system parameter matrix at each sampling time-step, which will be time-consuming, especially for bridges with complex structures. Aiming at solving this problem, the following section presents a fast reanalysis, VDM-based method.

3. Fast Reanalysis of the Damaged System

3.1. Dynamic Response of the Damaged Structure under Known Excitation

Denoted by𝜇_𝑖the damage extent of the𝑖th finite element, which is quantified by the ratio between its original stiffness matrix𝐊_𝑖and the damaged stiffness matrix𝐊_𝑖, 𝐊_𝑖=𝜇_𝑖𝐊_𝑖. (9)

Then, the stiffness matrix of the damaged bridge can be expressed by 𝐊=_𝑖𝜇_𝑖𝐊_𝑖. (10)

Substitute (10) into the equation of motion of the damaged beam to obtain 𝐌𝐮̈^𝑏(𝑡)+𝐂𝐮̇^𝑏(𝑡)+_𝑖𝜇_𝑖𝐊_𝑖𝐮^𝑏(𝑡)=^𝑛𝑚_𝑖=1𝐛_𝑖(𝑡)𝐩_𝑖(𝑡). (11)

Compared with the intact structure, remove the modified part of the stiffness matrix on the right-hand side of (11); then the equation is transformed into the equation of motion of the distorted structure (in the terminology of VDM); that is, the intact structure was subjected to the same external excitation and to a certain response coupled pseudoload𝐩⁰(𝑡), 𝐌𝐮̈^𝑏(𝑡)+𝐂𝐮̇^𝑏(𝑡)+𝐊𝐮^𝑏(𝑡)=^𝑛𝑚_𝑖=1𝐛_𝑖(𝑡)𝐩_𝑖(𝑡)+𝐩⁰(𝑡), (12) where the pseudo-load𝐩⁰(𝑡)is related to the extent of the damage by 𝐩⁰(𝑡)=_𝑖1−𝜇_𝑖𝐊_𝑖𝐮(𝑡). (13)

In VDM, virtual distortion is more often adopted to simulate stiffness-degrading damage, which is related to the pseudo-load as follows: 𝐩⁰_𝑒,𝑖(𝑡)=𝐊_{𝑒,𝑖𝑗}𝜅⁰_𝑖𝑗(𝑡)𝜑_𝑖𝑗, (14) where𝐩⁰_𝑒,𝑖is the local pseudo-load applied on the𝑖th element expressed in the local DOFs,𝐊_𝑒,𝑖is the stiffness matrix of the𝑖th element in its local coordinate,𝜅⁰_𝑖𝑗(𝑡)𝜑_𝑖𝑗is the𝑗th virtual distortion of the  𝑖th element,𝜑_𝑖𝑗is the𝑗th nonzero eigenvector to the𝑗th nonzero positive eigenvalue𝝀_𝑖𝑗of the stiffness matrix𝐊_𝑒,𝑖=∑_𝑗𝝀_𝑖𝑗𝜑_𝑖𝑗𝜑^T_𝑖𝑗, and𝜑_𝑖𝑗represents the𝑗th local unit distortion,𝜅⁰_𝑖𝑗(𝑡)=(1−𝜇_𝑖)𝜑^T_𝑖𝑗𝐮_𝑒,𝑖(𝑡)is the time-varying combination coefficient of the corresponding𝑗th local unit distortion𝜑_𝑖𝑗.𝐮_𝑒,𝑖(𝑡)is the nodal displacement of the𝑖th element in its local coordinate system, which shows that𝜅⁰_𝑖𝑗(𝑡)is coupled with the system vibration.

Denoted by𝐩_𝑒,𝑖is the local nodal load applied on the𝑖th element; then it can be similarly expressed by 𝐩_𝑒,𝑖(𝑡)=𝐊_{𝑒,𝑖𝑗}𝜅_𝑖𝑗(𝑡)𝜑_𝑖𝑗, (15) where𝜅_𝑖𝑗(𝑡)𝜑_𝑖𝑗is the𝑗th actual distortion of the𝑖th damaged element and the combination coefficient𝜅_𝑖𝑗(𝑡)=𝜑^T_𝑖𝑗𝐮_𝑒,𝑖(𝑡)has the relation to the coefficient𝜅⁰_𝑖𝑗(𝑡)as follows [23]: 𝜅⁰_𝑖𝑗(𝑡)=1−𝜇_𝑖𝜅_𝑖𝑗(𝑡). (16)

Based on (12), by virtual distortions, the response of the damaged bridge under moving vehicles can be expressed by 𝑦_𝛼(𝑡)=_𝑖^𝑡₀𝐷^𝑚_𝛼𝑖(𝑡,𝜏)𝑝_𝑖(𝜏)𝑑𝜏+_𝑖,𝑗^𝑡₀𝐷^𝜅_𝛼𝑖𝑗(𝑡−𝜏)𝜅⁰_𝑖𝑗(𝜏)𝑑𝜏. (17)

Here, it assumed zero value initial conditions and𝑦_𝛼(𝑡)was the response of the𝛼th measurement, which is modelled as the sum of the linear responses of the intact structure to a known excitation𝑝_𝑖(𝑡)and to certain virtual distortions.𝐷^𝑚_𝛼𝑖(𝑡,𝜏)denoted the impulse response of the intact structure at time 𝑡 to unit impulsive excitation applied at time𝜏at the respective locations of the excitation𝑝_𝑖(𝑡). It is worth noticing that the excitation location is changing with the vehicles.𝐷^𝜅_𝛼𝑖𝑗(𝑡)denotes the impulse response (in the scope of the VDM, it is called the dynamic influence matrix) to an impulse unit distortion𝜑_𝑖𝑗of the𝑖th element which is equivalent to a local impulsive load𝐊_𝑒,𝑖𝜑_𝑖𝑗.

Similarly to (17), the distortion response𝜅_𝑖𝑗(𝑡)of the damaged structure can be formulated as 𝜅_𝑖𝑗(𝑡)=_𝑘^𝑡₀𝐷^𝜅𝑚_𝑖𝑗𝑘(𝑡,𝜏)𝑝_𝑘(𝜏)𝑑𝜏+_𝑘,𝑙^𝑡₀𝐷^𝜅𝜅_{𝑖𝑗𝑘𝑙}(𝑡−𝜏)𝜅⁰_𝑘𝑙(𝜏)𝑑𝜏, (18) where𝐷^𝜅𝑚_𝑖𝑗𝑘(𝑡,𝜏)is the𝑗th distortion of the𝑖th damaged element at time𝑡to unit impulsive excitation applied at time𝜏at the respective location of the excitation𝑝_𝑘(𝑡), and𝐷^𝜅𝜅_{𝑖𝑗𝑘𝑙}(𝑡)is the𝑗th distortion of the𝑖th damaged element to an impulse unit distortion𝜑_𝑘𝑙of the𝑘th damaged element.

Substitute (16) into (18) to obtain 1−𝜇_𝑖𝑘^𝑡₀𝐷^𝜅𝑚_𝑖𝑗𝑘(𝑡,𝜏)𝑝_𝑘(𝜏)𝑑𝜏=𝜅⁰_𝑖𝑗(𝑡)−1−𝜇_{𝑖𝑘,𝑙}^𝑡₀𝐷^𝜅𝜅_{𝑖𝑗𝑘𝑙}(𝑡−𝜏)𝜅⁰_𝑘𝑙(𝜏)𝑑𝜏. (19)

If combined, all the damaged elements𝑖and the distortions𝑗 form a system of Volterra integral equations of the second kind, which is always well posed and thus uniquely solvable (Kress 1989 [27]). Then, the response𝑦_𝛼(𝑡)can be computed by (17) with the obtained virtual distortions.

3.2. Response of the Damaged Structure under Moving Vehicles

In practice, the interaction between the moving vehicle and the bridge, (see (6)), is time varying and coupled with the system vibration, which is usually hard to measure in advance for monitoring and identification of the extent of damage in a bridge. While the vehicle parameters, such as mass, stiffness, and damping, can be obtained more easily through traffic count statistics on the bridge, in this case, the gain of the moving excitation, besides the virtual distortion, is the key procedure, by which the response of the damaged structure in (17) is computed. Equation (6) shows that the interaction was related to the vibration of the moving vehicle; that is,𝑢^𝑣_𝑖(𝑡), 𝜃^𝑣_𝑖(𝑡), and the vertical response of the bridge𝑥_𝑖(𝑡)at the contact location of the vehicle wheels, as well as their corresponding vibration velocity: the solutions are discussed in the following paragraphs.

For the response𝑥_𝑖(𝑡), it can be formulated in a manner similar to 𝑥_𝑖(𝑡)=_𝛽^𝑡₀𝐷^𝑚𝑚_𝑖𝛽(𝑡,𝜏)𝑝_𝛽(𝜏)𝑑𝜏+_𝑗,𝑘^𝑡₀𝐷^𝑚𝜅_𝑖𝑗𝑘(𝑡−𝜏)𝜅⁰_𝑗𝑘(𝜏)𝑑𝜏, (20) where𝛽=2𝑙+𝑙_𝑤−2is the number of the contact point of the𝑙_𝑤th wheel(𝑙_𝑤=1,2)of the𝑙th moving vehicle.𝐷^𝑚𝑚_𝑖𝛽(𝑡,𝜏)is the impulse response at the location of contact point𝛽to unit moving impulsive excitation applied at time𝜏at the location of𝑝_𝛽(𝜏)and to an impulse unit distortion𝜑_𝑗𝑘of the𝑗th damaged element. It is worth noticing that position of𝑥_𝑖(𝑡)changes with vehicle move. Matrix𝐃^𝑚𝑚_𝑖𝛽 consisting of𝐷^𝑚𝑚_𝑖𝛽(𝑡,𝜏)is thus called moving dynamic influence matrix.𝐷^𝑚𝜅_𝑖𝑗𝑘(𝑡)is the impulse response corresponding to an impulse unit distortion𝜑_𝑗𝑘of the𝑗th damaged element.

Similarly to (20), the velocity response of the bridge𝑥̇_𝑖(𝑡)can be expressed as follows: 𝑥̇_𝑖(𝑡)=_𝛽^𝑡₀𝐷̇^𝑚𝑚_𝑖𝛽(𝑡,𝜏)𝑝_𝛽(𝜏)𝑑𝜏+_𝑗,𝑘^𝑡₀𝐷̇^𝑚𝜅_𝑖𝑗𝑘(𝑡−𝜏)𝜅⁰_𝑗𝑘(𝜏)𝑑𝜏, (21) where𝐷̇^𝑚𝑚_𝑖𝛽(𝑡,𝜏)and𝐷̇^𝑚𝜅_𝑖𝑗𝑘(𝑡)are the respective velocity impulse responses.

Using the equation of motion of the moving vehicle (2), the dynamic response of the moving vehicle can be expressed by the convolution integral between the impulse response and the excitation (the right-hand side of (2)) 𝐘^𝑣_𝑖(𝑡)=^𝑡₀𝐇^𝑣_𝑖(𝑡−𝜏)𝐬_𝑖𝑘_𝑖𝐱_𝑖(𝜏)+𝐫_𝑖(𝜏)+𝑐_𝑖𝐱̇_𝑖(𝜏)+𝐫̇_𝑖(𝜏)𝑑𝜏, (22) where𝐇^𝑣_𝑖(𝑡)=^{𝐡𝑢𝑢,𝑖(𝑡)𝐡𝑢𝜃,𝑖}_𝐡^(𝑡)_{𝜃𝑢,𝑖(𝑡)𝐡𝜃𝜃,𝑖(𝑡)}consists of the impulse response with respect to each DOF of the vehicle. For example,𝐡_{𝑢𝜃,𝑖}(𝑡)is the impulse response along the vertical direction to unit impulsive excitation along the rotational direction: the velocity of the vehicle vibration𝐘^𝑣_𝑖(𝑡)is formulated as 𝐘̇^𝑣_𝑖(𝑡)=^𝑡₀𝐇̇^𝑣_𝑖(𝑡−𝜏)𝐬_𝑖×𝑘_𝑖𝐱_𝑖(𝜏)+𝐫_𝑖(𝜏)+𝑐_𝑖𝐱̇_𝑖(𝜏)+𝐫̇_𝑖(𝜏)𝑑𝜏, (23) where matrix𝐇̇^𝑣_𝑖(𝑡)consists of the corresponding velocity impulse response.

Then, given damage extents and the moving vehicle parameters, the moving excitation𝑝_𝑖(𝑡), and virtual distortions𝜅⁰_𝑖𝑗(𝑡), can be computed by solving the combined equations (6), (19), (20), (21), (22), and (23), where the relevant impulse matrices of the intact bridge and of the vehicles are required. Then, substitute excitation𝑝_𝑖(𝑡)and virtual distortions𝜅⁰_𝑖𝑗(𝑡)into (17); the responses of the damaged structure can be rapidly computed.

3.3. Discretization

In applications, the responses are usually measured or obtained by numerical simulation and are thus discrete. Collect the dynamic responses𝑦_𝛼(𝑡)for all the time steps and arrange them in a certain sequence, and the discrete matrix form of (17) becomes 𝑦=𝐃^𝑚𝐩+𝐃^𝜅𝜿⁰, (24) where vectors𝐲,  𝐩,  𝜿⁰, respectively, consist of the discrete responses (of all the considered sensors), discrete excitations, discrete virtual distortions (of all the damaged elements) for all time steps, and thus they are of respective lengths𝑛_𝑎𝑛_𝑡, 2𝑛_𝑚𝑛_𝑡, and 𝑛_𝑑𝑛_𝑡, where𝑛_𝑡denotes the measured time step, and𝑛_𝑎denotes the number of sensors, and𝑛_𝑑 denotes the considered number of virtual distortions. Matrices𝐃^𝑚and𝐃^𝜅are block matrices and, respectively, consist of the corresponding discrete integral operators in (17) with lower-triangular𝑛_𝑡×𝑛_𝑡blocks. Thereinto, the submatrix of𝐃^𝜅is Toeplitz matrix.

Similarly, the discrete response𝐱,𝐱̇and the discrete distortions𝜿depend on the discrete excitation𝐩and discrete virtual distortions𝜿⁰in the following way: 𝐱𝐱̇𝜿=𝐃^𝑚𝑚𝐃^𝑚𝜅𝐃̇^𝑚𝑚𝐃̇^𝑚𝜅𝐃^𝜅𝑚𝐃^𝜅𝜅𝐩𝜿⁰, (25) which is an aggregated discrete version of (20), (21), and (18).

The discrete version of vehicle vibration in (22) and (23) is as follows: 𝐘^𝑣=𝐇^𝑣𝐒(𝐤()(̇𝐫̇,𝐘̇𝐱+𝐫+𝐜𝐱+))_𝑣=𝐇̇^𝑣𝐒(𝐤(𝐱+𝐫)+𝐜(𝐱̇+𝐫̇)), (26) where𝐇^𝑣=∑^𝑛𝑚_𝑖=1𝐈_𝑖⊗𝐇^𝑣_𝑖, 𝐒=∑^𝑛𝑚_𝑖=1𝐈_𝑖⊗𝐒_𝑖, 𝐒_𝑖=𝐒_𝑖⊗𝐈_{𝑛𝑡×𝑛𝑡},𝐤=∑^𝑛𝑚_𝑖=1𝑘_𝑖𝐈_𝑖̂,𝐜=∑^𝑛𝑚_𝑖=1𝑐_𝑖𝐈_𝑖̂,𝐈_𝑖̂=𝐈_𝑖⊗𝐈_{2𝑛𝑡,2𝑛𝑡},𝐈_{2𝑛𝑡,2𝑛𝑡}is a unit matrix with dimension2𝑛_𝑡.𝐈_𝑖is a square matrix with dimension𝑛_𝑚, of which only element𝐈_𝑖(𝑖,𝑖)has value 1, and all other elements have zero values.

Equation (27) shows the expression of discrete excitation𝐩: 𝐩=𝐤𝐒^T𝐘^𝑣+𝐜𝐒^T𝐘̇^𝑣−𝐤(𝐱+𝐫)−𝐜(𝐱̇+𝐫̇)+𝐦𝐞𝐠, (27) where𝐦=∑^𝑛𝑚_𝑖=1𝑚_𝑖𝐈_𝑖̂, and𝐠is the vector of Earth’s gravities𝑔.𝐞=[𝑒^T₁,…𝑒^T_𝑛𝑚]^T⊗𝟏_𝑛𝑡×1, 𝐞_𝑖=𝑒_𝑖𝑟/(𝑒_𝑖𝑟+𝑒_𝑖𝑙)𝑒_𝑖𝑙/(𝑒_𝑖𝑟+𝑒_𝑖𝑙)^T.

Further, the discrete version of (16) is as follows: 𝜿⁰=(𝐈−𝝁)𝜿, (28) where𝝁is a block diagonal matrix of respective dimensions with diagonal blocks𝜇_𝑖𝐈_{2𝑛𝑡,2𝑛𝑡}.

Finally, the previous discrete equations (25), (26), and (27) are combined and generate the following large linear system: 𝐈−𝟎𝐇^𝑣_𝑐𝐒Θ₁𝟎𝐃^𝑚𝑚_allΘ₂−𝐃^𝑚𝑚_allΘ₁𝐃^𝑚𝜅_all()𝐃𝐈−𝝁_𝜅𝑚Θ₂−𝐃_𝜅𝑚Θ₁𝐃_𝜅𝜅()×𝐘𝐈−𝝁_all𝐱_all𝜿=𝟎𝐃^𝑚𝑚_all𝐃_𝜅𝑚𝐦𝐇𝐞𝐠+^𝑣𝑐𝐒−𝐃^𝑚𝑚_all−𝐃_𝜅𝑚Θ₁𝐫_all𝑑𝑦, (29) where𝐇^𝑣_𝑐=(𝐇^𝑣)^T(𝐇̇^𝑣)^TT,Θ₁=𝐤𝐜,𝐘^𝑣_all=(𝐘^𝑣)^T(𝐘̇^𝑣)^TT,𝐃^𝑚𝑚_all=(𝐃^𝑚𝑚)^T(𝐃̇^𝑚𝑚)^TT,𝐃^𝑚𝜅_all=(𝐃^𝑚𝜅)^T(𝐃̇^𝑚𝜅)^T,Θ₂=𝐤𝐒^T𝐜𝐒^T,𝐱_all=𝐱^T𝐱̇^TT,𝐫_all=𝐫^𝑇𝐫̇^TT.

The block matrices in (29) store all the necessary information about the dynamics of the systems and are independent of the damage. Thus, given the values of the moving vehicle parameters𝑚_𝑖, 𝑣_𝑖, 𝑐_𝑖and the extent of the damage𝜇_𝑖, the system (29) can be quickly assembled and then solved to obtain the moving excitations and virtual distortions, which can then be used in (24) to compute the response of the damaged structure to the moving vehicles.

However, in application there are different vehicles passing over any given bridge, and it was necessary to set up the corresponding impulse frequency matrix of the vehicle(s),𝐇^𝑣_𝑐in (29), repeatedly. Based on the reanalysis concept of VDM, the problem can be improved. Denote𝑚_𝑖, 𝐽_𝑖̃, 𝑘_𝑖, 𝑐_𝑖̃as the initial vehicle system parameters of the𝑖th moving vehicle, and denote𝜇^𝑚_𝑖, 𝜇^𝐽_𝑖, 𝜇^𝑘_𝑖, 𝜇^𝑐_𝑖, respectively, as the ratios between the actual, and initial, values; that is, 𝑚_𝑖=𝜇^𝑚_𝑖𝑚_𝑖,𝐽_𝑖=𝜇^𝐽_𝑖𝐽_𝑖̃,𝑘_𝑖=𝜇^𝑘_𝑖𝑘_𝑖,𝑐_𝑖=𝜇^𝑐_𝑖𝑐_𝑖̃. (30)

Substitute (30) into (2), and shift the modified counterpart compared to the initial vehicle value to obtain 𝐦^𝑣_𝑖𝑌̈^𝑣_𝑖(𝑡)+𝑐_𝑖̃𝐞_𝑖𝑘𝑌̇^𝑣_𝑖(𝑡)+𝑘_𝑖𝐞_𝑖𝑘𝑌^𝑣_𝑖(𝑡)=𝜇^𝑘_𝑖𝑘_𝑖𝑆_𝑖𝑥_𝑖(𝑡)+𝑟_𝑖(𝑡)+𝜇^𝑐_𝑖𝑐_𝑖̃𝑆_𝑖𝑥̇_𝑖(𝑡)+𝑟_𝑖̇(𝑡)+𝐈−𝜇_𝑖^𝑚,𝐽𝐦^𝑣_𝑖𝑌̈^𝑣_𝑖(𝑡)+1−𝜇^𝑐_𝑖𝑐_𝑖̃𝐞_𝑘𝑖𝑌̇^𝑣_𝑖(𝑡)+1−𝜇^𝑘_𝑖𝑘_𝑖𝐞_𝑘𝑖𝑌^𝑣_𝑖(𝑡), (31) where𝜇_𝑖^𝑚,𝐽=diag𝜇^𝑚_𝑖𝜇^𝐽_𝑖.

Then, through the use of the impulse response matrix of the initial vehicle system𝐇^𝑣, the vibration of the vehicle can be reformulated as follows: 𝐘^𝑣=𝐇^𝑣𝐒𝝁^𝑘𝐤(𝐱+𝐫)+𝝁^𝑐𝐜̃(̇𝐫̇)+𝐇𝐱+_𝑣𝐈−𝝁^𝑚,𝐽𝐦_𝑚,𝐽𝐘̈^𝑣+𝐇^𝑣𝐒_𝑒𝐈−𝝁^𝑘𝐤𝐘^𝑣+𝐈−𝝁^𝑐𝐜̃𝐘̇^𝑣, (32) where matrices𝐤,𝐜̃,𝐦have a similar form to matrices𝐤,𝐜,𝐦, as did𝝁^𝑘,𝝁^𝑐, and𝝁^𝑚.𝐦_𝑚,𝐽=∑^𝑛𝑚_𝑖=1𝐈_𝑖⊗𝐦^𝑣_𝑖, 𝐦^𝑣_𝑖=𝐦^𝑣_𝑖⊗𝐈_{𝑛𝑡,𝑛𝑡}, 𝐒_𝑒=∑^𝑛𝑚_𝑖=1𝐈_𝑖⊗𝐒_𝑒𝑖, 𝐒_𝑒𝑖=𝐞_𝑖𝑘⊗𝐼_{𝑛𝑡𝑛𝑡}, 𝝁^𝑚,𝐽=∑^𝑛𝑚_𝑖=1𝐈_𝑖⊗𝜇_𝑖^𝑚,𝐽.

Meanwhile, if𝐇^𝑣, in (32), is replaced bẏ𝐇^𝑣, it yields the expression for𝐘̇^𝑣and using the reformulated expression for the vehicle vibration𝐘^𝑣and𝐘̇^𝑣, (29) becomes 𝐇𝐈−^𝑣all𝐈₂𝐒_𝑒𝑚𝐈−𝜇^𝑘𝑐𝑚𝐇^𝑣_𝑐𝐒Θ₁𝝁^𝑘𝑐0𝐃^𝑚𝑚_allΘ₂𝝁^𝑘𝑐𝑚−𝐃^𝑚𝑚_allΘ₁𝝁^𝑘𝑐𝐃^𝑚𝜅_all()𝐃𝐈−𝝁_𝜅𝑚Θ₂𝝁^𝑘𝑐𝑚−𝐃_𝜅𝑚Θ₁𝝁^𝑘𝑐𝐃_𝜅𝜅()×𝐘𝐈−𝝁_𝑣all𝐱_all𝜿=0𝐃^𝑚𝑚_all𝐃_𝜅𝑚𝜇^𝑚𝐦𝐇𝐞𝐠+^𝑣𝑐𝐒−𝐃^𝑚𝑚_all−𝐃_𝜅𝑚Θ₁𝝁^𝑘𝑐𝐫_all, (33) where𝐇^𝑚𝑚_all=diag𝐇^𝑣𝐇̇^𝑣,  𝐒_𝑒𝑚=diag[𝐒_𝑒𝐤𝐒_𝑒𝐜̃𝐦_𝑚,𝐽],  𝝁^𝑘𝑐𝑚=diag𝝁^𝑘𝝁^𝑐𝝁^𝑚,𝐽,𝝁^𝑘𝑐=diag𝝁^𝑘𝝁^𝑐,𝐈₂=one(2)⊗𝐈_{2𝑛𝑚𝑛𝑡}.

Equation (33), like (29), is a large linear system with full rank, and its solution is well conditioned. There, the block matrices, such as the related influence matrix or initial system matrices, are generated by the FE model, which is noiseless and only needs to compute once for each set of moving vehicle and damage parameters. Therefore, it provides a fast reanalysis approach to the estimation of the dynamic responses and the moving excitation, which makes the optimisation of the structural damage identification and vehicle parameters feasible.

4. Identification of Moving Vehicles and Damages

The analysis in Section 2.3 shows that the coupled vehicle-bridge system is time variant, and the response is nonstationary. Time-domain analysis is the proper identification of unknown structural systems and moving vehicles.

4.1. Objective Function and the Optimization Variables

Equation (24) shows that the dynamic response of the damaged bridge was a function of the moving excitation and virtual distortions, which were in a one-to-one correspondence to the moving vehicle parameters and structural damage. Thus, here the inverse problem of identification of unknown vehicle and damage extent was stated as an optimisation problem involving the minimisation of the normalised mean-square distance between the measured structural response𝐲^𝑀and the computed response y. The optimisation variables are the modifying factors of vehicle parameters𝜇^𝑚_𝑖, 𝜇^𝐽_𝑖, 𝜇^𝑘_𝑖, 𝜇^𝑐_𝑖, and damage extent𝜇_𝑖which takes advantage of two aspects: on one hand, the magnitudes of the adopted optimisation variables were much less than the magnitudes between the vehicle parameters and bridge damage extents, for the latter can seriously impair the accuracy of many optimisation procedures: the damage extents𝜇_𝑖belong to the interval[0,1], but the vehicle parameters might be as large as tens of tonnes, and so forth. Moreover, this makes the direct adoption of (33) to obtain the moving excitation and dynamic response of the damaged bridge to given optimisation variables with the prerequisite of the initial system matrices which only are computed once.

Therefore, the initial vehicle parameters were firstly computed using the measured responses of the damaged structure while assuming that the bridge was intact and was under the action of moving excitations which equal the weight forces of the vehicles with initial mass𝑚_𝑖,  for which the corresponding responses equal the measured responses, therefore giving rise to the following expression: 𝐲^𝑀=𝐃^𝑚𝐦𝐠, (34) where𝐲^𝑀and𝐠are known, and matrix𝐃^𝑚can be constructed using the intact bridge system, matrix vector𝐦consisting of𝑚_𝑖is the unknown to be solved. For the number of the required sensors is bigger than the number of vehicles, (34) is an overdetermined system and the initial mass𝑚_𝑖can be computed via the least-square method.

Then, the rest of the initial parameters for the vehicles are determined by their empirical relationships to the vehicles’ masses. Given the initial parameters, the optimisation variables can be represented by the following dimensionless variables𝜇^⋆_𝑖(𝑖=1,…,4𝑛_𝑚+𝑛_𝑒), 𝑛_𝑒is the number of the potential damaged elements. Denote𝑖=4(𝑗−1)+𝑗_𝑚, there is 𝜇^⋆_𝑖=𝜇^𝑚_𝑗if𝑗≤𝑛_𝑚,𝑗_𝑚𝜇=1_𝐽𝑗if𝑗≤𝑛_𝑚,𝑗_𝑚𝜇=2_𝑘𝑗if𝑗≤𝑛_𝑚,𝑗_𝑚𝜇=3_𝑐𝑗if𝑗≤𝑛_𝑚,𝑗_𝑚𝜇=4_{𝑖−4𝑛𝑚}if𝑗>𝑛_𝑚. (35)

All the optimization variables𝜇^⋆_𝑖are of the same magnitude and have the natural initial value of 1 for the optimization. The objective function is thus built as minimize𝑓𝜇^⋆₁,…,𝜇^⋆_{4𝑛𝑚+𝑛𝑒}=12𝐲^𝑀−𝐲²𝐲^𝑀2subjectto𝜇^⋆_𝑖≥0,𝑖=1,…,4𝑛_𝑚+𝑛_𝑒, (36) where𝐲is the computed response of the structure to the given optimization variables by (35), (33), (27), and (28).

4.2. Sensitivity Analysis

For the optimization of the objective function (36), it can be performed quickly using the gradient-based algorithms provided that the gradient can be computed at a reasonable cost. The formulation based on (24), (27), (28), and (33) allows the discrete adjoint method to be used, which is quicker by one order of magnitude compared with the standard differentiation method [28].

Denote𝐃=𝐃^𝑚𝑚𝐃^𝜅,Λ=𝐏^T(𝜅⁰)^TT; then the first derivative of (36) with respect to the variable𝜇^⋆_𝑖is stated as 𝑓_𝑖𝐲=−^𝑀−𝐲^T𝐲^𝑀2𝐃Λ_𝑖. (37)

Here and below, the subscript𝑖in each variable stands for its first derivative to the variable𝜇^⋆_𝑖.

In order to obtain the expression of Λ_𝑖, (27) is firstly reformulated as follows: 𝐩=Γ₁𝐳₁+𝐰, (38) whereΓ₁=𝐤𝐒^T𝐜𝐒^T−𝐤−𝐜,  𝐳₁=(𝐘^𝑣)^T(𝐘̇^𝑣)^T𝐱^T𝐱̇^TT, 𝐰=−𝐤𝐫−𝐜𝐫̇+𝐦𝐞𝐠.

Then 𝐩_𝑖=Γ₁𝐳_1𝑖+Γ_1𝑖𝐳₁+𝐰_𝑖. (39)

Equation (28) is differentiated to obtain 𝜅⁰_𝑖=Γ₂𝜿_𝑖+Γ_2𝑖𝜿, (40) whereΓ₂=𝐈−𝝁.

The combination of (39) and (40) generates the following expression: Λ_𝑖=Γ𝐳_𝑖+Ω_𝑖, (41) where𝐳=(𝐘^𝑣_all)^T𝐱^T_all𝜿^TT,Γ=diagΓ₁Γ₂,Ω_𝑖=[Γ_1𝑖𝐳₁](Γ_1𝑖𝐳₁+𝐰_𝑖)^T(Γ_2𝑖𝜿)^TT.

Vector𝐳  represents the parameters in (33) required to be solved. By reformulating (33) as the form𝐀𝐳=𝐛, the first derivative𝐳_𝑖satisfies the following relationship: 𝐀𝐳_𝑖=𝐛_𝑖−𝐀_𝑖𝐳. (42)

Substitute (41) into (37); there is 𝑓_𝑖𝐲=−^𝑀−𝐲^T𝐲^𝑀2𝐃Γ𝐳_𝑖−𝐲^𝑀−𝐲^T𝐲^𝑀2𝐃Ω_𝑖 (43) which includes the first derivatives𝐳_𝑖. The discrete adjoint method adds the scalar product of the adjoint vector𝝀with (42) to (43), to obtain 𝑓_𝑖=𝝀^T𝐛_𝑖−𝐀_𝑖𝐳−𝐲^𝑀−𝐲^T𝐲^𝑀2𝐃Ω_𝑖+𝝀^T𝐲𝐀−^𝑀−𝐲^T𝐲^𝑀2𝐃Γ𝐳_𝑖. (44)

In this way, the gradient of the objective function can be expressed as 𝑓_𝑖=𝝀^T𝐛_𝑖−𝐀_𝑖𝐳−𝐲^𝑀−𝐲^T𝐲^𝑀2𝐃Ω_𝑖, (45) where the adjoint vector𝝀is computed only at the cost of a single solution of the adjoint equation 𝐀^T𝝀=Γ^T𝐃^T𝐲^𝑀−𝐲𝐲^𝑀2. (46)

4.3. Remarks

If a small number of time steps are used, then the system matrix in (33) can be computed and used directly. However, for offline identification, in the case of a dense time discretization or a longer sampling time, the system can become prohibitively large and computationally hardly manageable. To reduce the numerical costs, the system matrix, which is a block matrix composed of lower triangular matrices, can be rearranged into a lower triangular block form; then the system can be solved by a specialized linear solver, like block forward substitution [29]. Due to the fact that all the data in the system (33) are computed based on the ideal FEM model and are away from any measurement noise pollution, the unknown vectors can be computed stepwise precisely.

If the bridge is undamaged, the virtual distortions in (24) vanish; that is, the value of damage extent equals one in related Equations, and the proposed method can be also used for identification of moving vehicles only.

5. Numerical Example

A multispan frame was used to illustrate and verify the proposed methods for simultaneous identification of moving vehicles and damage. Measurement error and model error were taken into account to test the robustness of the method.

5.1. Structural and Moving Vehicles

The frame, shown in Figure 3, is made of steel with Young’s modulus 2.15 × 10¹¹ Pa, and a density of 7.8 × 10³ kg/m³. It has a uniform mass distribution of 15.3 × 10³ kg/m³. The beam is 200 m long with a simplified rectangular cross-section of𝑏×ℎ=0.89 m × 2.21 m, so that the inertial moment of the area is 0.8 m⁴; each of the two side spans is 50 m. Each pier is 20 m high with inertial moment of 0.16 m⁴.

Figure 3 

The vehicle-bridge coupled system.

The road surface condition is considered to be good, and the roughness coefficient𝐺_𝑑(𝑛₀) is 16 × 10⁻⁶m³. According to the vehicle velocity,𝑛_𝑙=0.0221 m⁻¹,𝑛_𝑢=1.4142 m⁻¹; then the length of the shortest effective pavement is1/𝑛_𝑙=45.25 m, so the effective pavement can be the whole length of the bridge. In this way,Δ𝑛=1/𝐿=0.01m⁻¹,𝑁_𝑇=fix((𝑛_𝑢−𝑛_𝑙)/Δ𝑛)= 278. Figure 4 shows the simulated road surface roughness via (3).

Figure 4 

The road surface roughness.

Two moving vehicles passed through the bridge from opposite direction, with respective constant velocities𝑣₁=34 m/s,𝑣₂=−30 m/s. A two-axle, two-DOF, vehicle model was adopted. The vehicle masses were𝑚₁=61.2×10³ kg, 𝑚₂=53×10³ kg. The other vehicle parameters such as the rotary inertia, stiffness, and damping are provided by multiplying the vehicle mass by a certain coefficient according to the literature [30]. The rest of the parameters of the two vehicles were defined, respectively, as 𝑘=396𝑚 N/m, 𝑐=1.4𝑚 N⋅s/m; the rotary inertia𝐽=9𝑚kg⋅m². The distance between the centre of mass and either of the two axles was 3 m. Denoted by𝑥_1,0=−3 m,𝑥_2,0=204 m, respectively, were the initial centroidal points of each vehicle’s mass.

Two sensors are employed: S1 at the location of 65.2 m and S2 at the location 145.2 m as shown in Figure 3. The sensors are placed at bottom surface of the beam.

5.2. Identification Cases

For the identification purpose, the beam is divided into 20 elements, and each of the piers is divided into two elements. Measurements are simulated with an uncorrelated Gaussian noise. Besides the measurement error, the influence of the model error is tested by using a different finite element (FE) model of the structure; that is, the theoretical FE model of the intact structure is modified as the actual intact structure.

Two following identification cases are discussed.

Case 1. Measurement error is simulated at 5% RMS level. No model error is assumed. Assume that two pier elements (Nos. 21 and 23) are damaged with the damage extents𝜇₂₁=0.4and𝜇₂₃=0.7.

Case 2. As in Case 1, but the model error is additionally simulated. It considers the modification of the stiffness of all the elements. More precisely, uncorrelated Gaussian modifications with mean −2% and standard deviation 5% are used; see Figure 5, for the stiffness modifications. Then, the actual damage extents are slightly different from that listed in case 1, which include the model error besides the real damage.

Figure 5 

Stiffness reduction levels of the elements.

In both of the two cases, it was assumed that the damage location was limited to the four pier elements. Two moving vehicles were identified simultaneously as being causally linked to the extent of the damage. Four stiffness modification coefficients were used in the optimisation, besides the eight variables related to the masses, rotary moments, stiffness, and damping of the two vehicles. In this way, the damage extent and its location, as well as vehicle parameters, were treated as unknowns and thus identified. Meanwhile, the dynamic interaction of the coupled vehicle-bridge system was established.

5.3. Moving Vehicle and the Damage Identification

Responses of the two sensors are used for vehicle and damage identification. In each case, the dynamic responses are calculated using the discrete FE model and the Newmark integration method with the parameters𝛼=0.25and𝛽=0.5. The first two damping ratios are 0.01 and 0.015. The integration time step equals 0.01 s (100 Hz sampling frequency). A total of 100 time steps is used; hence, the measured time is 1 s. The simulated responses with measurement noise in Case 1 are show in Figure 6. Table 1 shows the natural frequencies of the original and damaged bridge model.

Table 1 

The first eight natural frequencies of the original and damaged bridge model (Hz).

Figure 6 

Simulated strain responses of the damaged and the intact structure in Case 1 with measurement noise at 5% RMS level.

The damage was limited to the two piers, that is, to the four pier elements. Together with two known vehicles (eight modification coefficients), there were twelve variables to be optimised by minimising the objective function (36). Responses from two sensors were used for this purpose. The initial trial mass values were estimated using (34). Then, the other initial vehicle parameters were provided based on their masses.

The identification results for the two vehicle parameters are listed, respectively, in Table 2 (Case 1) and Table 3 (Case 2).

Table 2 

Identified parameters of moving vehicles in Case 1.

Table 3 

Identified parameters of moving vehicles in Case 2.

The identified extents of the structural damages in the two cases are listed, respectively, in Table 4 (Case 1) and Table 5 (Case 2).

Table 4 

Identified extents of the damages in Case 1.

Table 5 

Identified extents of the damages in Case 2.

The parameter identification results were assessed by their relative accuracy, while the damage identification results were more naturally assessed in terms of their absolute accuracy. It showed that, even with all the simulated errors, the parameters of the two vehicles and four potential damage identifiers could be identified acceptably. Since only two damages were actually assumed, the optimisation allowed their number and location (limited to the four considered pier elements) to be identified as well.

With the identified vehicles and structural damage, the moving loads can be computed meanwhile by using (27). Figure 7 shows the estimated results which have reasonable accuracy. The relative error was 2.6% even under both 5% RMS measurement noise and model error in Case 2.

(a) Moving loads of vehicle 1

(b) Moving loads of vehicle 2

Figure 7 

Identified moving loads.

6. Conclusion

This paper presented an effective method for the simultaneous identification of moving vehicles and structural damage based on a virtual distortion method. The vehicle parameters and the structural damage were treated as optimisation variables. Through the use of a moving dynamic influence matrix, the response of the damaged system to given optimisation variables could be computed quickly without the need of a numerical simulation and the repeated assembly of the time-variant system parameter matrix at each time-step. This reduced the numerical cost significantly. The adjoint method was adopted for the fast sensitivity analysis of the objective function. A numerical example of a three-span beam was presented to verify the proposed method, where two vehicles (eight variables) and two damaged piers (four damage instances) were identified using only two sensors. Even with both the pollution of 5% RMS measurement error and a certain model error, the maximum identification error was 13.82%. Moreover, the moving excitation effects were estimated accurately.

Acknowledgments

The authors gratefully acknowledge the support of National Science Foundation of China (NSFC) (51108066, 51108067), of the Fundamental Research Funds for the Central Universities (China) (DC13010316), of the Dr. Start-up fund of Dalian Nationalities University (20116207), and of the Project of Housing and Urban-Rural Development (K2201243, 2013-k2-7). Financial support of Structural Funds in the Operational Programme-Innovative Economy (IE OP) financed from the European Regional Development Fund-Project “Modern material technologies in aerospace industry” (POIG.0101.02-00-015/08) is gratefully acknowledged.