Beam-like Bridge Research Articles

Accurate identification of moving vehicle loads on beam-like bridge structures integrating novel PCA-based dictionary with grouping and weighting strategy

Most moving load identification (MLI) methods incorporate the dictionary theory as a regularization technique to address the ill-posedness of the problem. However, the selection of atoms in the dictionary often inadequately represents the actual vehicle loads in existing methods, and the prior information from these atoms is disregarded in the subsequent MLI computations. Therefore, a novel MLI framework is proposed for beam-like bridge structures based on a novel dictionary derived from principal component analysis (PCA) and newly grouping and weighting strategy in this study. At first, a vehicle-bridge coupling system (VBCS) is established to obtain the interaction forces between the moving vehicles and bridge deck. The system matrices between the interaction forces and structural responses are derived in the time domain. The PCA technique is then employed to extract information from these interaction forces and to subsequently construct a PCA-based dictionary. Based on the eigenvalues of each principal component in the dictionary, a weighted group sparse model, incorporating the newly grouping and weighting strategy, is defined to obtain the coefficients of each atom. The solution to this model is obtained using the alternating direction method (ADM). Finally, the proposed method is validated in numerical simulations comparing with some existing methods. The effects of noise levels, road surface roughness, number of training data, response combinations and the tolerances in ADM were also studied. Similarity law for the VBCS is conducted in experimental verifications to construct the PCA-based dictionary, allowing for a reasonable identification of gross vehicle weights. The results indicate that the MLI accuracy has been enhanced by the proposed method, and the similarity law employed in experiments is reasonable. Settingthe tolerance in ADM as ε = 1 × 10−5 can lead to a higher accuracy and reduce the computation cost in both numerical simulations and experimental verifications.

Structural response reconstruction of beam-like bridge based on equivalent loads under moving forces

Response reconstruction has advantages in enriching monitoring data with high quality and it is of great importance to bridge structural health monitoring. However, there still are many current difficulties with response reconstruction due to the complex operation of bridges, e.g., hard to deal with time-varying positions of vehicles and inefficient reconstruction for cases with a long time. According to the theory of equivalent load identification, this paper proposes a model-based method for reconstructing responses induced by moving forces. The proposed method introduces moving time windows to extract the measured responses in an orderly manner and re-organizes them into a matrix form. The measured responses, expressed in matrix form, then serve as the input data to identify both equivalent loads and structural initial conditions, but not to estimate the actual moving forces. Herein, the equivalent loads represent some concentrated loads with time-invariant acting positions and are employed to approximately re-express the main information of moving forces. Matrix regularization with a sparse penalty term is used to ensure the robustness of identified results. Since the structural inputs have been estimated, the structural responses happening at the validation sensors then can be calculated as a forward problem. In this process, the role of identified structural inputs is an intermediate variable connecting the measuring sensors and the validation sensors so that, the transmissibility function (TF) discussed in the proposed method is expressed in an implicit pattern. Compared to the response reconstruction methods developed from moving force identification (MFI), an obvious characteristic is that information on the actual moving forces is no longer necessary. Therefore, it can bring convenience in practical application such as the time-varying positions of moving vehicles are not needed in advance. In addition, the application of matrix regularization has the potential for efficient computation. To verify the feasibility and effectiveness of the proposed method, both numerical simulations and experimental studies are finally carried out. The illustrated results indicate that the proposed method can accurately reconstruct the structural response. In addition, some related issues are also discussed.

Applying matrix regularization to identification of traffic-induced equivalent loads and multi-vehicle weight on beam-like bridge

Identification of traffic-induced forces is an important topic in the field of bridge structural health monitoring (B-SHM). It can provide reliable data to support the great demand for monitoring and evaluating the working conditions of bridges. Bridges may serve under consistent moving vehicles and this requires that the methods of moving force identification (MFI) should be applicable under circumstances such as multiple-vehicle loading and long-term monitoring. Centering on cases containing multiple vehicles, this study proposed a matrix regularization-based method for the identification of traffic-induced equivalent loads and multi-vehicle weights. Herein, multiple moving forces in the spatial domain are converted into a certain number of equivalent loads which virtually act at fixed locations on the bridge as a kind of coordinate transformation. The MFI problem then can be transformed into tasks of identifying some concentrated forces. To solve the problem of equivalent load identification, the structural input-output relationship is firstly reorganized in a matrix-matrix-matrix form with the help of segmentation via time windows and consideration of unknown structural initial conditions. Then the matrix regularization with a sparse penalty term is introduced to formulate an identified equation so that, it can ensure the strong noise robustness of the identified results. Finally, the total weight of multiple vehicles can be empirically estimated from the sum of the identified equivalent loads. Numerical simulations and model experiments are performed to verify the proposed method's performance. The effects of vehicle speeds, measuring noise, number of modal orders, number of vehicles, and vehicle spacings are investigated. The illustrated results show that the proposed method turns out to be effective and precise in identifying the traffic-induced equivalent loads and the weight of multiple vehicles.

Unsupervised structural damage assessment method based on response correlations

Structural damage assessment for civil engineering structures has become an important research topic and has gained much attention for decades in the field of structural health monitoring (SHM). Current unsupervised structural damage assessment methods can achieve damage detection for practical civil engineering structures. However, these methods cannot fully mine the response correlations among sensors, thus are difficult to localize damage in practical engineering applications. Therefore, this study proposes an unsupervised structural damage assessment method with bidirectional long short-term memory (BiLSTM) networks to establish the response correlations among sensors. Numerical studies of a beam-like bridge model, an experimental study of Yellow Frame, and an in-field study of Yonghe Bridge were conducted to validate the accuracy and reliability of the proposed approach. The results show that unsupervised structural damage identification and localization can be achieved even in the case of sparse sensor deployment due to the fact that the response correlations among sensors can be well extracted. The proposed approach can be applied to damage identification of practical engineering structures under operational and ambient excitations and can automatically provide damage location warnings to ensure structural health and prevent catastrophic events.

Baseline-free structural damage detection using PCA- Hilbert transform with limited sensors

Conventional vibration-based damage identification methods need a large number of sensors on the bridge to identify the structural modal information and rely on the response data of bridges under a non-damage state for comparison, which restricts their implementations in the health monitoring of bridge structures in service. This paper proposes a baseline-free structural damage detection method based on limited sensor information. The proposed method uses the Hilbert transform combined with principal component analysis (PCA) to locate the damage. Firstly, PCA is performed on the displacement response data matrix of the bridge under a moving load to obtain the principal component matrix. According to the physical interpretation of PCA of the responses of a beam-like bridge under a moving load, each column of the principal component matrix, namely the principal component, composes of a mode shape and the dynamic information at this modal frequency of the bridge. The dynamic information of the modal frequency is indirectly obtained by filtering out the mode shape by low pass filter. Then the Hilbert transform is used to process the dynamic component information to receive the instantaneous frequency which is regarded as the damage index. When the structure is damaged, the first instantaneous frequency at the damaged location changes abruptly. To demonstrate the effectiveness and performance of the proposed method, numerical simulations and experimental validations on beam bridge models are conducted. The results show that the method is free from the requirement of the information of bridges under the undamaged state, and only limited sensors are needed to locate the damage.

Nonlinear vibration analysis of beam-like bridges with multiple breathing cracks under moving vehicle load

The beam-like bridges with small and medium spans are designed to mainly sustain the moving vehicle. Hence, the dynamic behavior of bridges under vehicular loading can provide important information for the bridge design, assessment, and maintenance. Cracks, as one of the most common damage in the bridge structures, alter the bridge dynamic characteristics and therefore responses to the moving vehicle. This paper proposes a novel semi-analytical approach for nonlinear vibration analysis of the beam-like bridges with breathing cracks subjected to vehicular loading, taking into account the dynamic interactions between the vehicle and cracked bridge. The Dirac’s delta function is utilized to represent the local stiffness change at the crack location, leading to closed-form solutions of mode shapes of the cracked beam. The breathing process of the crack, causing eigenproperties of the beam to change over time, is characterized by the time-varying curvature of the bridge. Subsequently, the nonlinear dynamic equation of the vehicle-cracked bridge interaction system is established, which is solved through a formulated double-iterative numerical scheme. Eigenvalues and dynamic responses of the bridge with open crack presented in the literature are compared with those obtained from the proposed approach for purposes of validation. The effects of various crack parameters, vehicle speed, and bridge surface irregularity are investigated through a parametric study. In addition, the detection of breathing cracks is also discussed. The results show that the proposed approach can be used to effectively obtain the nonlinear vibration characteristics and dynamic responses of the beam-like bridge with general boundary conditions and different crack numbers, locations, depths, and initial status under moving vehicle. Moreover, the approach has the potential to be incorporated into the vibration-based bridge damage detection.

Vehicle weight identification based on equivalent loads reconstructed from responses of beam-like bridge

Moving force identification (MFI) is an effective way for closely monitoring vehicle weight in a dynamic state. Time-varying position of vehicle is usually needed. This applying condition may limit the applicability because the measurement of time-varying position has certain difficulty. To tackle this problem, this study proposes an equivalent load-based method for identifying the gross weight of a vehicle moving on a beam-like bridge. The proposed method is developed based on an innovative concept of equivalent load and the application of force reconstruction. Herein, the equivalent loads are expressed as some concentrated loads whose acting positions are time invariant. Modal forces induced by the equivalent loads are equal to the ones caused by the vehicle-induced moving forces. Under the crucial assumption of equivalent loads, the first key step is to reconstruct the equivalent loads by using a force identification method that is implemented via matrix regularization. Since the equivalent loads have been estimated, the second step aims to introduce a practical strategy for identifying the vehicle weight from the sum of the identified equivalent loads rather than the sum of moving forces. Because of this, a highlight is that time-varying position of vehicle is not needed. To verify the effectiveness and feasibility of the proposed method, beam structures with single span are carried out for numerical simulations and experimental validations. Illustrated results show that the proposed method can be applied for identifying the gross weight of a vehicle moving on a beam-like bridge with high accuracy and strong robustness. Some related issues are discussed as well.

Output-only complete mode shape identification of bridges using a limited number of sensors

Mode shape is an important structural property that needs be identified in structural condition monitoring and analysis. The conventional modal identification methods usually can only provide sparse and low-spatial resolution mode shape measurements because of the limited number of sensors, which could seriously affect the effectiveness and accuracy of the mode shape-based structural damage detection methods. In this paper, a new output-only method is proposed for identifying the complete mode shapes for beam-like bridge structures under a moving load using a limited number of sensors based on proper orthogonal decomposition (POD). Firstly, the physical interpretation of the POD of the responses of a beam-like bridge under a moving load is derived. It is proved that each component of the POD consists of a mode shape of the bridge and a dynamic component with the frequency corresponding to this mode. The contribution ratio of the component in the POD gives an excellent estimation of the modal contribution ratio (CR) in terms of the vibration energy. The mode shape can be identified by filtering out the dynamic information in each component of the POD. The identified mode shapes have a high spatial resolution that they can be regarded as the complete mode shapes of the bridge. To demonstrate the proposed method, a beam bridge with different boundary conditions is numerically modelled and a simply supported beam bridge model is tested in the lab. Both the numerical and experimental results demonstrate that this method can successfully identify the complete mode shapes with high resolution and modal contribution ratios with a limited number of sensors. This method breaks through the challenge on the requirement of a large number of sensors in vibration tests for full mode shape identifications in traditional methods.

Feasibility Study of Tractor-Test Vehicle Technique for Practical Structural Condition Assessment of Beam-Like Bridge Deck

The tractor-test vehicle technique of non-destructive testing for indirect measurement of the modal properties of a bridge deck is revisited in this paper with several improvements for possible practical application to the structural condition assessment of a beam-like bridge deck. The effect of damping of the vehicle-bridge system is considered and the modal properties from only the first vibration mode of the structure will be used for a quick and simple assessment. The two test vehicles are designed to have the same modal frequency and damping ratio but with parameters in the follower No.2 test vehicle proportional to those in the follower No.1 test vehicle. This effectively removes the effect of road surface roughness in the response of an equivalent vehicle such that the error in the subsequent condition assessment is reduced. Through data collected on-sitetransmitted to theremote computer platform, a simple technique based on the moment-curvature relationship acceptable to practical engineers is adopted for the condition assessment with improvements in the estimation of the element bending stiffness of the deck. Scenarios with different damping, vehicle speed, road surface roughness, and local damages in the bridge structure are studied with or without temperature effect in the measurement. Through numerical simulations and field tests, the tractor-test vehicle technique of non-destructive testing with the proposed modifications and improvements has been demonstrated to give consistently accurate estimates of the element bending stiffness of the bridge deck but with a small error close to the end of the deck.

RETRACTED ARTICLE: Reality of virtual damage identification based on neural networks and vibration analysis of a damaged bridge under a moving vehicle

Presented here is a reality of virtual damage detection and vibration behaviour study of a discrete beam-like bridge with one or several non-propagating edge cracks subjected to a moving vehicle. In this model, the simply supported beam elements are replaced by a range of rigid bars, which are connected by transverse and rotational springs, while the mass and rotational moment of inertia may be lumped at various points along the beam. The adopted vehicle model here is a four degrees-of-freedom, two axes half-vehicle model with tires flexibility and linear suspensions. Damage can be modelled by altering the spring stiffness equation at the crack position according to predictions, which allows the inclusion of simple or complex damage. To simplify, damage is represented here by an open crack, and stiffness of a given element with damage is calculated by fracture mechanics. Both the discrete element and finite element methods are used to investigate vibration analysis of a discrete beam model subjected to a moving vehicle to confirm model feasibility in vibration analysis under a moving vehicle. Besides, some dynamic response laws are obtained. Considering an irregular road profile, the effects of the moving vehicle velocity, the moving vehicle mass, the crack location and the crack depth on dynamic response of a beam-like bridge are analysed by a numerical example, combining a vehicle–bridge coupled vibration MATLAB program with ANSYS. In addition, the neural network is used to identify the damage of the structure. Numerical results of the numerical model predictions, compared with those obtained from the continuous elements beam, support the accuracy of the discrete elements beam model in both cases of undamaged beam and damaged one. The evidence for condition assessment and damage identification of bridge is obtained from this simulation as obtaining the vibrational characteristics of the damaged beam structure subjected to a moving vehicle. And the inversion results show that the neural network method can identify the injury location and injury size of the structure accurately.

Experimental and Numerical Investigation of the Effect of Standing People on Dynamic Properties of a Beam-Like Bridge

This paper studies the vertical dynamic properties of a beam-like bridge attached with standing people. A purpose-built lively bridge was constructed. Model properties of the empty structure are obtained using ambient vibration testing method. Experimental tests of the bridge attached with standing people were also conducted covering a variety of densities of occupants and different postures. The considerable number of participants and repetitions makes it possible to take into account inter- and intrasubject variability. To illustrate the variations of dynamic properties of the structure, a mathematic model of standing people-structure interaction system is developed employing the single degree-of-freedom (SDOF) human body model. It is shown that the model developed herein can effectively illustrate the experimental observations. Based on the model, the effect of damping and natural frequency of the human body on dynamic properties of the occupied structure is scrutinized. Results show that the modal properties of the human body contribute remarkably to the structural damping but little to the structural natural frequencies.

Dynamic Analysis of a Cracked Beam-Like Bridge Subjected to Earthquake and Moving Vehicle

This paper presents the influence of a crack on the dynamic behavior of a beam-type bridge subjected to separately or simultaneously a moving vehicle and earthquake excitation. In this study, the bridge is modelled as a 3D beam by the finite element method. The stiffness matrix of a cracked beam element with rectangular section adopted from fracture mechanics is presented. The influences of crack appearance time, crack location, crack depth, and vehicle speed on the dynamic response of the bridge are investigated. When a crack is induced during external excitation the stiffness of the structure is changed leading to a change in natural frequencies during vibration. This change in the frequency is analysed by wavelet spectrum, a time-frequency analysis which can examine locally a signal in both time and frequency domains. The relationship between the crack depth and the instantaneous frequency (IF) of the structure is established which may prove useful for assessment of the crack depth. Numerical simulation results are presented to investigate the efficiency of the method.

Crack detection of a beam-like bridge using 3D mode shapes

In this paper, mode shapes of a 3D cracked beam with a rectangular cross section are analyzed for crack detection. The influence of coupling mechanism between horizontal and vertical bending vibrations due to the 3D crack model on the mode shapes is investigated. Due to the coupling mechanism the mode shapes of a beam are twisted in space. They change from plane curves to space curves. This phenomenon can be used for crack detection. The existence of the crack can be detected when the mode shapes are space curves. Also, the mode shapes of a cracked beam bridge have distortions or sharp changes at the crack position. Therefore, the position of the crack can be determined as a position at which the mode shapes exhibit such distortions or sharp changes. Moreover, using the mode shapes in 3D crack model, a crack with depth as small as 1% of the beam height can be detected, while in previous studies using 2D crack model, distortions in the mode shapes caused by a small crack cannot be detected. These results are new and can be used for crack detection of a beam bridge. The stiffness matrix of a 3D cracked element obtained from fracture mechanics is presented and numerical simulations are provided in this paper.

Monitoring a sudden crack of beam-like bridge during earthquake excitation

This paper presents a wavelet spectrum technique for monitoring a sudden crack of a beam-like bridge structure during earthquake excitation. When there is a sudden crack caused by earthquake excitation the stiffness of the structure is changed leading to a sudden change in natural frequencies during vibration. It is difficult to monitor this sudden change in the frequency using conventional approaches such as Fourier transform because in Fourier transform the time information is lost so that it is impossible to analyse short time events. To overcome this disadvantage, wavelet spectrum, a time-frequency analysis, is used for monitoring a sudden change in frequency duringearthquake excitation for crack detection. In this study, a model of 3D crack is applied. The derivation of the stiffness matrix of a 3D cracked beam element with rectangular section adopted from fracture mechanics is presented. Numerical results showed that the sudden occurrence of the crack during earthquake excitation can be detected by the sudden change in frequency using wavelet power spectrum. When the crack depth increases, the instantaneous frequency (IF) of the structure is decreased.

Monitoring breathing cracks of a beam-like bridge subjected to moving vehicle using wavelet spectrum

In this paper a wavelet spectrum technique for monitoring the breathing crack phenomenon of a beam-like bridge subjected to moving vehicle is presented. The stiffness of element with a breathing crack is modeled as a time dependent stiffness matrix using the finite element method. The stiffness matrix of the structure at each moment depends on the curvature of the structure at the crack position. The breathing crack phenomenon can be detected by analysing the instantaneous frequency (IF) of the system using the wavelet spectrum. When the crack depth is large, the crack area might be determined by the significant peak in the IF. The simulation results show that when the crack “breaths” the amplitude of the vibration obtained from the vehicle is smaller than in the case of an open crack. This is a warning when using the amplitude of the dynamic response to estimate the crack depth when there is a breathing crack in the structure. Therefore, it is important to distinguish the open crack and breathing crack to obtain a more accurate estimation of the crack depth. The results showed that crack with a depth as small as 10% of the beam height can be detected by the method. The proposed method can be applied with a noise level up to 10%.

Comparison studies of open and breathing crack detections of a beam-like bridge subjected to a moving vehicle

In this paper, based on the comparison between open and breathing crack detections of a vehicle–bridge system subjected to a moving vehicle, a wavelet spectrum technique for detection of breathing crack phenomenon is presented. The stiffness of element with an open crack is calculated from fracture mechanics and the stiffness of element with a breathing crack is modeled as a time dependent stiffness matrix using the stiffness of the element with an open crack. When there is a breathing crack, the stiffness matrix of the structure at each moment depends on the curvature of the structure at the crack position. The simulation results show that when the crack “breaths” the amplitude of the vibration of the beam is smaller than in the case of an open crack. This is a warning for crack detection by using the amplitude of the dynamic response when there is a breathing crack in the structure. The open and breathing cracks can be distinguished by monitoring the instantaneous frequency (IF) of the system using the wavelet spectrum. While the IF in the case of open cracks remains unchanged during vibration, it is varying when there is a presence of breathing cracks. It is interesting that peaks in the wavelet transform of the response used to determine crack positions in case of breathing crack(s) are much larger in comparison with the case of open crack(s). The cracks can be detected with a noise level up to 10% for the case of breathing cracks while it is only 5% for the case of open cracks [28]. These imply that the wavelet-based method for crack detection is much more efficient when breathing crack(s) are present in comparison with the case of open crack(s).