## 1 Introduction
The latter establishes the relationship between the moving force and response via neural network training; hence the force could be inversely determined via response once the network reaches a certain accuracy. In terms of structural damage identification (SDI), dynamic fingerprint-based methods and dynamic signal-based methods have been studied. Since the dynamic fingerprints, such as frequencies, mode shapes, frequency response functions, will change when the structure is damaged, it is able to detect damages via monitoring the variation of the fingerprints. However, these parameters are not sensitive to minor damages. In order to effectively highlight the changes caused by minor damage, more types of fingerprints have been proposed on the basis of mode shapes, such as the modal confidence factor, curvature mode, modal strain energy etc. [15; 16; 17; 18]. However, errors in modal parameter measurement and noise pollution may inevitably make the change in these features difficult to evaluate. On the other hand, signal-based methods detect damages via distinguishing different dynamic responses. As the structural response is the result of the combined action of excitation and structural attribute, the variation of response only comes from the structural parameter change under similar excitation. The response signal can be processed in the time domain, frequency domain and frequency-time domain. The corresponding techniques include time series analysis techniques [19], power spectral density estimation [20], wavelet transformation [21] etc. It is worth noting that for signal-based methods, the environmental influence should not be overlooked; temperature changes, especially in long-term monitoring, seriously affect the accuracy of identification. [PERSON] et al. [22] proposed an autoregressive (AR) time series model with a two-step artificial neural network (ANN) to determine damage under temperature variations, which extracted the first three-ordercoefficients of the AR model as damage feature and utilized ANN models to compensate the detrimental temperature changes. It is apparent that most of the reviewed MFI methods require determined structure information to establish the correspondence between moving force and structural response, while the signal-based SDI methods need a similar excitation in order to guarantee the response variation solely caused by structural damage. These methods could be difficult to practice on bridges where unknown moving forces and structural damage coexist. The synchronized detection of both moving forces and structural damages has become vital. [PERSON] and [PERSON] [23] presented a method for damage detection of a simply supported bridge, in which the vehicles-bridge interaction forces and the structural damages were identified from the measured responses in a sequence of iteration without prior moving loads knowledge. [PERSON] and [PERSON] [24] assessed the condition of structures under unknown support excitation, which modeled the excitation as orthogonal polynomial approximations. A simultaneous identification method of moving masses and damages was presented via the virtual distortion method (VDM) [25], in which the damage was modified to virtual distortions and identified with moving load. Compared to other methods, the biggest advantage of VDM is that it unifies the solving path of SDI and MFI, which immensely simplifies the complexity of the method and greatly improves the computational efficiency [26]. Taking this advantage into account, an efficacious synchronized assessment method of moving force and structural damages is presented in this paper. Based on VDM, the synchronized identification of damage and moving forces are transformed to multiple forces identification and is solved using the Duhamel Integral. In order to reduce the singularity, a truncated load-shape function (TLSF) is developed to enhance the Duhamel Integral. Two numerical examples are presented to validate the proposed method.