Condition Monitoring and Quantitative Evaluation of Railway Bridge Substructures Using Vehicle-Induced Vibration Responses by Sparse Measurement

Bridge substructure failure has been responsible for numerous recorded bridge collapses, particularly for small-and medium-span bridges, so it is crucial to efectively monitor the performance of the bridge substructures for efcient maintenance and management. Te current vibration-based approaches for quantitatively evaluating bridge substructures rely on in-situ experiments with a multitude of sensors or impact vibration test, making it challenging to implement long-term online monitoring. Tis paper proposes an accurate, low cost, and practicable method to achieve online quantitative monitoring of railway bridge substructures using only one vibration sensor and operational train-induced vibration responses. Te newly derived fexible-base Timoshenko beam models, along with the random decrement technique and Levenberg–Marquardt–Fletcher algorithm, are employed to identify the modal parameters and quantitatively assess the condition of bridge substructures. Te proposed method is numerically verifed through an established 3D train-bridge-foundation coupling system considering diferent damage scenarios. In addition, a real-world application is also conducted on the 2 nd Songhua River bridge in the Harbin–Dalian high-speed railway, aiming at examining the efectiveness and robustness of the method in condition monitoring of bridge substructure under a complete freeze-thaw cycle. Te results indicate that the proposed methodology is efective in extracting the modal parameters and monitoring the state evolution of the bridge substructures, which ofers an efcient and accurate strategy for condition monitoring and quantitative evaluation of railway bridge substructures.


Introduction
Te safety and reliability assessment of bridges is essential to ensure the smooth functioning of railways.Currently, there are approximately 92,000 railway bridges in China, including over 30,000 high-speed railway (HSR) bridges.However, several of these bridges have been in service for a substantial period, which increases the potential for uncertainty regarding their serviceability [1].Furthermore, newer bridges may experience a decline in service performance as a result of intricate external actions, such as train dynamic load, food erosion, debris fow, freeze-thaw cycles, vehicle collisions, and earthquakes.Consequently, railway bridge substructures are susceptible to various defects, including foundation settlement, pier tilting, and pier corrosion, leading to issues such as foundation damage, lower structural stifness, reduced durability, and insufcient bearing capacity.It is worth noting that a lot of bridge collapses can be attributed to the substructure failures [2,3].Terefore, there is an urgent requirement to evaluate and continuously monitor the service status of a vast number of bridge substructures.
Damages afecting the bridge substructures normally under the surface of ground or water, possessing high degrees of invisibility and suddenness.As such, they are challenging to detect and evaluate merely through visual inspections.To address this issue, many approaches have been explored to assess the condition of bridge substructures.Conventional inspection approaches, such as excavation and coring, are the simplest and most efcient methods, regardless of the foundation type, but at the same time they may afect the integrity and safety of the bridge substructure and are economically incompatible.To avoid these problems, many nondestructive testing (NDT) techniques have been deeply examined on bridge substructures and foundations.
Hossain et al. [4] evaluated unknown foundation depth using diferent NDT methods.Tey proved that the parallel seismic (PS) method and resistivity imaging (RI) method were efective to evaluate the foundation depth.Chen et al. [5] leveraged sonar imaging technology to detect the underwater foundation damage of an ancient stone arch bridge.Rashidyan et al. [6] investigated the practicability of the Induction Field Testing in determining the depth of steel and reinforced concrete foundations.Cardoso and Lopes [7] demented that electrical resistivity tomography (ERT) could be used to assess the bridge foundation depth.However, the practicality of these techniques is limited and infuenced by the characteristics of the substructure, soil characteristics, access restrictions, equipment costs, and experience of inspectors, and can only be used to determine the presence or absence of damage.Additionally, using buried sensors is a direct NDT approach to monitor bridge scour, such as the application of fber Bragg grating sensors [8,9], magnetbased smart rocks [10][11][12], and piezoelectric sensors [13].Of note, these sensors for direct measurement are generally expensive and can only for one-time use, and the installation and replacement process are challenging.
An alternative approach, called vibration-based method, adopts changes in modal parameters and their derived indicators to assess structural conditions [14], which shows great potential for detecting concealed damage and evaluating the overall service performance of structures.Recently, many researchers have reported methods for scour detection based on natural frequencies and mode shapes [15][16][17][18][19], which can preliminarily detect the presence of damage.To further quantitatively characterize the bridge substructure, system identifcation methods in conjunction with analytical model are leveraged to obtain calibrated fnite element model (FEM) or structural physical parameters.
Chen et al. [20] developed a foundation scour evaluation method using the ambient vibration measurements of the bridge superstructure, and the soil stifness and scour depth were estimated from a globally best ftted fnite element model.Davis and Sanayei [21] utilized live load dynamic strain and accelerations measured from substructure elements during operational loading to evaluate the bridge foundations.Mao et al. [22] employed experimental modal analysis and fnite element model updating to estimate the unknown foundation depth and conditions of the bridge substructure.A scaled bridge model experiment was also conduced to better understand diferent factors afecting structural identifcation technique for substructure characterization [2].Carbonari et al. [23] proposed a methodology for identifying physical parameters of soil-foundationbridge pier systems from identifed state-space models.Ghorbani et al. [3] developed an out-put only scour level quantifcation method by integrating an unscented Kalman flter, random decrement, and a continuous Euler-Bernoulli beam model.Zhan et al. [24] proposed a systematic procedure for HSR bridge substructure evaluation based on a simplifed model and fnite element model updating.Tey further used vibration measurements and fnite element model updating to quantitatively evaluate the scour depth of highway bridge piers [25].
Te abovementioned literatures demonstrate that the system identifcation methods can efectively localize and quantify the damage of bridge substructure.However, some of these methods require the application of impact loads and cannot be used for long-term online monitoring, and most approaches require the installation of many sensors, which can lead to difculties in management and processing of massive data.Tis paper aims to develop an accurate, lowcost, and practicable method to achieve online quantitative monitoring of railway bridge substructures using sparse measurements and operational train-induced vibrations.A newly derived fexible-base Timoshenko beam models, along with the random decrement technique and Levenberg-Marquardt-Fletcher algorithm, are integrated to identify the modal parameters and quantitatively assess the condition of bridge substructures.In summary, the main contributions of the present study can be briefy itemized as follows: (1) presenting a fexible-base Timoshenko beam model for bridge substructure to accurately interpret the transverse dynamics of diferent types of bridge substructures; (2) proposing a methodology that can quantitatively evaluate the condition of railway bridge substructures using only one vibration sensor and operational train-induced vibration responses; and (3) validating the feasibility and efectiveness of the methodology in a realworld application using online monitoring data during a complete freeze-thaw cycle.
Te paper is structured as follows: First, a 3D trainbridge-foundation coupling analytical model is established, which is suitable for interpreting the dynamics of the entire system.A fexible-base Timoshenko beam model is derived to characterize the transverse dynamics of the local pierbeam system, and the random decrement technique and Levenberg-Marquardt-Fletcher algorithm, are integrated to lead a novel framework for quantitatively assessing the condition of bridge substructures.Ten, the efectiveness of the proposed methodology is numerically investigated via the 3D train-bridge interaction model, considering single damage, multiple damages, and varying pier heights.Finally, feld tests that involve the impact vibration test and traininduced vibration test were conducted to verify the feasibility of the proposed modal analysis method for bridge substructures.Furthermore, monitoring of the bridge substructure during a complete freeze-thaw cycle was carried out to examine the efectiveness and robustness of the method in quantitatively monitoring the health condition of the bridge substructures.

Establishment of 3D Train-Bridge-Foundation Coupling System
Te dynamic interaction of a train-bridge-foundation system is a complicated and coupled problem that varies with time.Tis type of issue is usually solved through numerical simulation methods by establishing a dynamic interaction 2 Structural Control and Health Monitoring model for the train-bridge system.Te analytical model can be regarded as a large spatial dynamic system consisting of two subsystems, namely, the train and bridge subsystems.Each subsystem can be simulated as an elastic structure with unique vibration patterns.Te 3D HSR train-bridgefoundation system is depicted schematically in Figure 1.

Multibody Dynamic
where M v , C v , and K v denote the mass, damping, and stifness matrices of the train, respectively; x v , _ x v , and x v are the vector of train displacement, velocity, and acceleration, respectively.

Finite Element Model of Bridge.
Te HSR bridge is composed of tracks, girders, bearings, piers, and pile foundations, which is typically established by spatial FEM.It is assumed that there is no relative displacement between track and bridge deck, i.e. the elastic deformation of ballastless track system (including the fastener, track plate, base plate, and CA mortar layer) is neglected.Te motion equations of the bridge subsystem can be expressed as where M b , C b , K b are the bridge global mass, damping, and stifness matrices in order; x b , _ x b , and x b are the bridge displacement, velocity, and acceleration, respectively; F b is the force acting on the bridge deck by the wheel-sets.
Te equivalent stifness matrix is introduced into the bridge model to consider the soil-structure interaction, and then the stifness matrix can be given as where K bp and K pb are interaction stifness submatrices between bridge piers and piles, respectively; K piles is the equivalent stifness matrix of group-piles determined by [26,27], which mainly includes two procedures: determination of m (the scale coefcient of soil horizontal resistance) and calculation of equivalent stifness coefcient of group-piles.Te damping matrix C b adopts the Rayleigh damping expressed as follows [27]: where ξ is the bridge damping ratio; ω 1 and ω 2 are the frst and second-order transverse circular frequencies of the bridge in order.

Wheel-Rail Interaction Relationship and Track
Irregularity.Diferent wheel-rail contact models have their own superiorities.Since the overall responses of the trainbridge system are concerned in this study, the vertical wheelrail correspondence assumption and lateral simplifed Kalker linear creep theory are leveraged, which can provide sufcient calculation accuracy while do not need complicated analysis of wheel-rail contact geometry [27].
Te wheel-rail correspondence model assumes no relative movement between the wheel and the rail in the Zdirection of vehicle coordinates.Te displacements of the wheel-sets and the bridge couple together through the track irregularity with the following equations: where x wi is the position of the i th wheel-set (i � 1, 2, 3, 4); Z, _ Z, € Z are the displacement, velocity and acceleration, respectively; subscripts "wi", "b", and "s" denote the wheel-set, the bridge, and the track irregularities, respectively.
Te simplifed Kalker's linear creep theory is adopted to simulate the lateral wheel-rail relationship, and the lateral wheel-rail creep force can be calculated as where parameter S 22 r 2/3 can be found in [27]; N is the wheelrail normal connect force, which can be approximated as the static axle load G; _ y w and _ y r represent the velocity of the wheel-set and rail; V is the running speed of the train.

Structural Control and Health Monitoring
Te track irregularities are randomly generated by Power Spectral Density functions (PSDs) of German track irregularity spectra, which have the following expressions.
Track alignment irregularity Track vertical profle irregularity Track cross-level irregularity where Ω is the spatial frequency; Ω c , Ω r , Ω s are the cut-of frequencies; A a and A v are the roughness coefcients.For low disturbance irregularity, their values are On this basis, the samples of track irregularity are generated by the trigonometric series approach [27,28], and the In-Time-Step Iteration Method is leveraged to solve the vehicle-bridge interaction problem [27,29].

Flexible-Base Timoshenko Beam Model for Bridge
Substructure.In this section, an analytical model for interpreting the transverse dynamics of bridge substructures is theoretically derived.Te vibrations in longitudinal direction are not considered in this model since the assessment of bridge pier condition mainly focuses on the transverse direction in most studies.Te bridge substructure is modeled as a fexible-base Timoshenko beam with lumped mass as shown in Figure 3. Herein, two assumptions are adopted in the analytical model: (1) the transverse frst-order vibration of the pier-beam system is an overall lateral oscillation, and the lumped mass allocated on the pier top takes the mass of one girder (including Phase II dead load); (2) the superstructure vibrates synchronously with the pier due to strong pier-beam coupling efect in railway bridge.Te rationality of the hypothesis will be verifed in subsequent numerical simulation and feld tests.Te superstructure is regarded as a lumped mass on the pier top with mass of M (mass of one girder including Phase II dead load) and mass moment of inertia of J and d is the distance between the girder centroid and the pier top.Te pier, characterized by a Timoshenko beam, has a height of H, a density of ρ, a cross-sectional area of A, a moment of inertia of I, an elastic modulus of E, a shear modulus of G and a shear correction factor of K. Te soil-foundation interaction is represented by transverse and rotational springs with K t and K r , respectively.y(x, t) and ψ(x, t) are the transverse and rotational displacements of the pier at point x and time t.
Te equations that govern the free vibration of the bridge substructure are [30,31].
Te boundary conditions can be determined as Submitting the general solution form into the above boundary conditions yields the frequency eigenvalue equation as follows: Structural Control and Health Monitoring where By solving equation ( 12) using the False Position Iteration Method [32][33][34] developed in MATLAB [35], the theoretical transverse natural frequencies of the bridge substructure can be obtained.

Modal Identifcation Using Random Decrement
Technique.Te RD technique is a data processing method that extracts the free vibration response of a structure from its random response, which is based on the idea of sample averaging in statistics, and utilizes the characteristics that the structural response caused by stationary random excitation has a statistical mean of zero.
Specifcally, the RD function δ x (τ) for a single signal x(t) can be obtained as according to the derivation of Cole [36].
where N is the number of triggering points; τ � t − t n is the time past the triggering time t n ; C x : x(t n ) � x 0 is the triggering condition and x 0 is the trigger level as shown in Figure 4. Vandiver et al. [37] laid a theoretical basis of the RD function and equation ( 13) can be rewritten in a conditional expectation form as where the triggering condition is usually taken by the users in the range of (1∼2)σ x , and σ x is the standard deviation of the signal x(t).
Ten, the free vibration response can be processed by the Fast Fourier Transform (FFT) to yield the frequency response as where ω denote the frequency.

Levenberg-Marquardt-Fletcher Algorithm.
Using the RD method, the free vibration and modal parameters of the bridge substructure can be extracted.It is assumed that the pier body is intact and the transverse and rotational stifness decrease in the same proportion [24], the unknown condition indicator is defned as β � K e /K d to describe the weakening degree of the bridge foundation caused by scour and freeze-thaw cycle, where K e and K d denote the tested and designed stifness at pier bottom, respectively.To quantify the health condition of the substructures, herein, the LMF algorithm [38] is leveraged to achieve rapid system identifcation, which is more stable and efcient for solving nonlinear least squares problems.Tis system identifcation problem can be considered as an optimization problem where  f 1 is the tested frst-order transverse frequency of the substructure; f 1 is the theoretical frst-order transverse frequency calculated by the fexible-base Timoshenko beam model; r is the residual function.
β * that minimizes the objective function should satisfy the following necessary conditions where J � zr/zβ is called Jacobian matrix.Te update equation of the optimized parameter β at step (k + 1) can be described as where λ (k) is a scale parameter; D is a suitable diagonal matrix of scales, which is usually chosen as an identity matrix I or a diagonal of the matrix A � J T J.
Let the residuals r(β) are smooth functions, then its Taylor series expansion can be deduced as equation (19), and the Jacobian matrix J can be calculated as equation (20): Te entire optimization iteration process starts with a suitable guess β (0) , and the iteration procedure is continued until where ε is the admissible tolerance; N is the maximum number of iteration step.
3.4.Procedure of the Methodology.Figure 5 illustrates the overall procedure of the proposed methodology, which is composed of the RD technique and LMF algorithm with a newly developed fexible-base Timoshenko beam model for bridge substructure.Te RD method is leveraged to extract the free vibration of the bridge substructure, and the modal parameters can be further obtained by using the FFT.Te theoretical solution of modal parameters can be derived from the fexible-base Timoshenko beam model, and the spring stifnesses at pier bottom are constructed as an indicator to refect the foundation state.Te LMF continuously estimates the optimal parameters until the algorithm convergence is completed.Te proposed methodology is a model-based output-only approach using only one vibration transducer installed on the pier top and does not require any information about the unknown excitation.

Verifcation of the Proposed Analytical Model and Modal
Identifcation Method.To validate the efectiveness of the proposed method, three scenarios considering single condition degradation, multiple condition degradation, and varying pier heights have been set, as detailed in Table 3.Using Case 3 as an illustration, the modal analysis of the bridge-foundation system is conducted using the block Lanczos method [40].Te frst two transverse mode shapes and frequencies of the bridge-foundation system for Case 3 are shown in Figure 8.It can be observed that the frst two vibrations of the pier-beam system are in order manifested as an overall lateral oscillation and a lateral reverse vibration of adjacent piers, and the same pattern can be found for both Case 1 and Case 2, which preliminarily validates the model assumptions in Section 3.1.
Te measured acceleration responses on the pier top are contaminated by Gaussian white noise with 5% variance in root mean square (RMS) to consider the infuence of environmental noise where x n is the noisy acceleration signal, E p is the noise level, N 0 is the standard normal distribution vector with a mean value of zero and a unit standard deviation, x c is the clean acceleration signal, and σ(x c ) is its standard deviation.
A trigger point of 1.2σ(x(t)) and a number of triggering points of 5500 are selected for the RD implementation with a corresponding segment length equal to 5.5 s.Te original noisy responses on the pier top, the extracted free vibration responses, and the corresponding modal analysis results of diferent cases are shown in Figure 9.It can be seen that the frst order transverse frequency of the bridge substructure can be successfully identifed from the extracted free vibration response by picking up the frequency at the frst peak.However, the higher order frequencies are still diffcult to distinguish due to interference from multiple frequency peaks, which is related to the complexity of train lateral excitation and the spatial coupling vibration of multispan bridges.Table 4 presents the modes calculated by the Euler-Bernoulli beam model used in [3] and the proposed model in this paper.By comparing the results with the reference value extracted by random decrement-Fast Fourier Transform (RD-FFT), it can be concluded that the Timoshenko beam model has higher computational accuracy than the Euler-Bernoulli beam model, particularly for low piers.Fairly good agreement is obtained between the results of reference value identifed from the local pier beam system and those calculated by the Timoshenko beam analytical model.Terefore, the proposed fexible-base Timoshenko beam model is highly suitable for the modal analysis and quantitative evaluation of bridge substructures.

Verifcation of the Proposed Condition Evaluation
Method.In the implementation of LMF, the initial value of β (0) is 0.5, and its lower and upper bounds are set to 0 and 2 to ensure that the identifed parameters have physical meaning.Te termination tolerance ε � 10 − 9 , and the maximum number of iterations allowed N � 20.Furthermore, the trust region refective (TRR) Gaussian Newton algorithm adopted in [24] is also conducted to make a comparison with the LMF algorithm utilized in this study.
It can be seen from Figure 10(a) and Table 5 that both LMF and TRR can achieve satisfactory identifcation accuracy with an error of less than ±1%.Of note, the LMF algorithm has higher identifcation accuracy and smaller error than TRR, indicating that it is superior in condition quantifcation.In addition, Figure 10(b) shows the condition iteration process for pier 1.It can be seen that the LMF algorithm converges in just two steps under all setting scenarios, while TRR requires three or more steps, which demonstrates that the LMF needs less computation time.Good computational performance and efciency facilitate online system identifcation and quantitative assessment of railway bridge substructures.

Bridge Overview.
To investigate the performance of the proposed method in actual bridges, the 2 nd Songhua River bridge located on the Harbin-Dalian HSR is served as a realworld case study.Te Harbin-Dalian HSR traverses the coldest regions in China, experiencing a minimum air temperature of − 39.9 °C and a maximum soil freezing depth of 2.05 m.Compared to nonfreeze-thaw regions, the unique    11.Te three bridge pile caps are exposed to air, and the soil on the river side of the bridge pier is perennially eroded by the fowing water.Te moisture content of the foundation soil on both lateral sides of the bridge pier is notably diferent and is signifcantly infuenced by freeze-thaw phenomena.Te superstructure and piers adopt a 32.6 m length double line simply supported box girder and round-ended piers, respectively.Te strength grade of the concrete for the girder and pier are C50 and C35 in Chinese code, respectively (the corresponding elastic modulus are 3.45 × 10 4 MPa and 3.15 × 10 4 MPa).Te density of the concrete is 2500 kg/m 3 , and Poisson's ratio is 0.2.China's CRTS-I ballastless track system is used to consider the Phase II dead load.According to the design information, the total mass of one girder (Phase I + Phase II dead load) is calculated as 1290.74t.Te bored piles with a length of 42 m and a diameter of 1.25 m are symmetrically arranged about the center of the platform and have the same layout and sizes of group-piles as shown in Figure 6

Field Tests and Feasibility Verifcation.
Te impact vibration test and train-induced vibration test were conducted to verify the feasibility of the proposed modal analysis method for bridge substructures.In the tests, a INV3018C 24-bit data acquisition instrument equipped with 28 channels was utilized to collect the dynamic signals, each of which has a maximum sampling frequency of 102.4 kHz.Te 941B multifunctional vibration sensor was adopted to measure the structural response on the pier top.It comprises four gears, namely acceleration, microvelocity, medium velocity, and high velocity, designed to operate normally in temperatures ranging from − 35 °C to 70 °C.In the experiment, a microvelocity gear was utilized, featuring a sensitivity of 23 V•s/m, a maximum range of 0.125 m/s, a passband of 1∼100 Hz, and a resolution ratio of 4 × 10 − 8 m/s.Te sampling frequency was set to 512 Hz.
Te impact vibration test utilized a 30 kg cast iron heavy hammer to apply excitation, and the heavy hammer was wrapped in a hard rubber ring to safeguard the bridge  During the experiment, a rope can be utilized to secure the heavy hammer at the entrance ladder of the beam, and it can be struck from the middle of the pier top to one side to generate lateral vibration of the bridge pier.Te layout plan for the measurement points is shown in Figure 12.
Te time history of transverse velocity on the top of pier 455# (measurement point 3) under the impact of the heavy hammer is depicted in Figure 13(a).It can be seen that the heavy hammer can efectively stimulate the lateral vibration of the bridge pier, and the free decaying signal of the bridge pier is obvious.Figure 13(b) compares the lateral response of the pier top (measurement point 3) and the beam end (measurement point 12).From the observations made, it is evident that the measurement points located at the top of the pier and at the end of the beam exhibit synchronous vibration at the same phase, indicating that the pot rubber bearing has a notable lateral shear stifness, and the vibration coupling efect between the pier and beam is strong.Tis also confrms the rationality of the model assumption found in Section 3.1.
Te impact vibration test method (IVTM) [41] was utilized to analyze the frequency spectrum of pier 455#, and the results are depicted in Figure 14.Te fgure reveals that multiple peak points meet the phase angle condition, with values of 2.38 Hz, 2.88 Hz, 3.25 Hz, and 3.88 Hz, respectively.Furthermore, by combining the responses of additional measurement points, stochastic subspace identifcation (SSI) [42] was employed to determine the mode of the multispan simply supported beam system.Consequently, the stability diagram is obtained as Figure 15, where the stable axis, constituting a frequency of the system, is composed of various stable points of diferent modal orders.In the stabilization diagram, f, d, and v, respectively, indicate that the frequency, damping, and mode shapes are stable, and the symbol ⊕ denotes that all three vectors are stable simultaneously.Accordingly, the frst four modes dominated by lateral vibration of the bridge pier are depicted in Figure 16: (1) a lateral overall swing of the entire bridge (2.38 Hz); (2) lateral local vibration of piers (2.87 Hz); (3) lateral local vibration of piers in reverse direction (3.27 Hz); (4) a lateral staggered vibration of the entire bridge (3.94 Hz).It can be concluded that the modes identifed by IVTM and SSI are highly consistent, and both methods can efectively identify the modes of the substructure.By processing the measured train-induced response on the top of pier 455# using RD-FFT, the frst mode can be successfully extracted as shown in Figure 17.Te trigger point is set as ) and the number of triggering points is 6000.Te extracted modal frequency 2.39 Hz matches well with those identifed by IVTM and SSI, confrming the precision and viability of the suggested approach.
In addition, to further verify the applicability and accuracy of the proposed modal identifcation method for the railway simply supported beam bridge substructures, in-situ dynamic tests were conducted on bridge piers at various heights and locations.Te identifcation results of each central bridge pier are presented in Table 6.Te results indicate that the RD-FFT method has high accuracy and robustness in identifying the transverse fundamental    Structural Control and Health Monitoring frequency of the local pier-beam system and facilitates online system identifcation and condition evaluation on inservice bridge substructures by using only one vibration sensor.7, and the equivalent stifnesses at the pier bottom can be determined by [26] as presented in Table 8.

Condition Monitoring and
Currently, the Chinese code applicable to rating the operational performance of existing HSR bridge substructures is code [43], which prescribes the normal value of lateral amplitude at the pier top when multiple units pass through the bridge, as described in Table 9. Te normal values in the table are applicable to solid piers, hollow piers, and double cylindrical piers of double line bridges.H p is the total height of the pier, and B is lateral average width of pier shaft.It is noted that the code stipulates that when the lateral  18 Structural Control and Health Monitoring forced vibration frequency is close to the lateral natural frequency of the pier-beam system, the maximum transverse amplitude of the pier top should not be greater than twice the normal value.According to statistics, approximately 82∼86 trains pass through the bridge monitoring site every day.Figure 19 shows the daily statistical diagram of amplitude at the top of pier 494# on October 25 th .Apart from the high level of dispersion of the data, it is evident that the lateral forced vibration frequencies of over half of the trains closely resemble the lateral natural frequency of the pier-beam system, causing the amplitude of the pier top to surpass the normal value.At this point, the operational performance of the bridge substructure should be assessed utilizing twice the normal value specifed by the code.Te average value of the maximum amplitude at the pier top during the daily passage of all trains on pier 494# during the entire monitoring period has been plotted as shown in Figure 20.It can be seen that all statistical values fall below twice the normal value, indicating that the substructure is currently operating well.However, the amplitude results show signifcant fuctuations, high dispersion, and no notable trend of change throughout the entire monitoring period.Tis is because the waveform and amplitude of bridge piers under train excitation vary with diferent train speeds, types, axle loads, and track irregularities.In practical applications, analysis can only be based on a certain probability of exceedance.Terefore, it is diffcult to use amplitude indicators to refect the efect of freeze-thaw on the structure in the evaluation of the service performance of railway bridge piers in severe cold regions.
Te evolution diagram of the lateral natural frequencies of pier 494# extracted by RD-FFT during the entire monitoring period is shown in Figure 21.It can be seen from the fgure that the identifcation results are relatively concentrated, and the changes of structural frequency have a notable evolutionary pattern throughout the entire monitoring period, which is consistent with the temperature data changes recorded in Figure 22, and the evolution process can be generally divided into three stages.As the average temperature gradually decreases during the freezethaw development period, the soil layer begins to freeze and the foundation stifness continues to increase, ultimately leading to an increase in the structural natural frequency.As the frozen soil layer completed freezing, the natural frequency of the structure entered a stationary period, which approximately lasts from early January to the end of February.Since the beginning of March, the seasonal frozen soil layer has been gradually melting, causing a decrease in the structural stifness as the average temperature rises, and the lateral fundamental frequency of the substructure has been gradually returning to its presoilfreezing level.
Te results of quantitative system identifcation using LMF are depicted in Figure 23.As observed, the foundation condition indices during the monitoring period are larger than 1.0, indicating that the pier 494# is in good health condition, which agrees well with the amplitude-based evaluation results.Te proposed method possesses good capability in tracking the operational performance and quantitatively evaluating the HSR bridge substructures and ofers a novel solution for condition monitoring and quantitative evaluation of railway bridge substructures by utilizing daily operational train-induced vibration responses and only one transducer installed on the pier top.

Conclusions
Te current codes implemented in China mainly rely on the amplitude of the pier top to evaluate and monitor the health status of the railway bridge substructures.Te amplitudebased indicators can only be used to qualitatively determine the existence of damage, and they exhibit signifcant dispersion and randomness.Tis paper presents a novel methodology for quantitative evaluating railway bridge substructures online by utilizing operational train-induced responses and a single vibration sensor installed on the pier top.A fexible-base Timoshenko beam model that is capable of interpreting the traverse dynamics of both shallow foundation and pile foundation is derived theoretically.Te RD-FFT is employed to extract the modal parameters of the bridge substructure, and the spring stifnesses at pier bottom are constructed as an indicator.Te condition indicator is updated by the LMF algorithm to quantitatively refect the status of bridge foundation.Based on the preformed numerical and experimental investigations, the main conclusions can be drawn as follows: (1) Te proposed fexible-base Timoshenko beam model exhibits greater computational accuracy than the Euler-Bernoulli beam model, especially for low piers, making it suitable for the modal analysis and quantitative evaluation of bridge substructures.(2) Te extracted modal frequency of bridge substructures using the RD-FFT and train-induced vibrations shows good agreement with those identifed by IVTM and SSI, thus validating the proposed approach's efectiveness and good antinoise ability.
(3) Te LMF algorithm exhibits better identifcation accuracy compared to TRR, requiring less computation time to achieve satisfactory identifcation accuracy with an error of less than ±1%.
(4) Te proposed method shows good performance in quantitatively monitoring the service condition of the HSR bridge substructures under freeze-thaw cycles, which also has great potential to characterize the evolutionary process of the substructure in other application scenarios.
Ongoing eforts involve the development of more sophisticated analytical models and modal identifcation algorithms aimed at characterizing and extracting the highorder dynamic signatures of the bridge substructure.

Structural Control and Health Monitoring
Furthermore, a spatial comprehensive condition assessment by merging the three-dimensional structural responses and considering the infuence of environmental factors will also be conducted in our future study.

Figure 4 :
Figure 4: Schematic view of the RD method.

Figure 5 :
Figure 5: Procedure of the proposed methodology.

Table 3 :Figure 8 :
Figure 8: Te frst two transverse mode shapes and frequencies of the bridge-foundation system for case 3.

Figure 14 :
Figure 14: Te modal analysis results of pier 455# top measurement point using IVTM.

Figure 18 :
Figure 18: Schematic of pier 492#∼494# in the 2 nd Songhua River bridge and the SHM system.

Figure 19 :Figure 20 :
Figure 19: Daily statistical diagram of amplitude at the top of pier 494# (October 25 th ).

FrequencyFigure 21 :Figure 22 :
Figure 21: Evolution diagram of the lateral natural frequencies of pier 494# extracted by RD-FFT during the entire monitoring period.
Model of Train.Te train model is formulated based on the multi-body dynamic theory.A single train is composed of a car-body, two bogies and four wheel-sets, which are all assumed to be rigid and their elastic deformation during vibration are neglected.Te primary and secondary suspension systems are characterized by liner springs and viscous dampers.Te vibration of the car-body, bogie, and wheel-set along the longitudinal axis of the vehicle is not considered.Terefore, each car-body or bogie has 5 DOFs, namely, lateral displacements Y ci and Y tij , vertical displacements Z ci and Z tij , roll displacements θ ci and θ tij , pitch displacements φ ci and φ tij and yaw displacements ψ ci and ψ tij .Te wheels-set has 3 DOFs in directions of Y wijl , Z wijl , θ wijl , where i denotes the i th vehicle, j � 1, 2 represent the front and rear bogies, and l � 1, 2, 3, 4 represent four wheel-sets, respectively.Terefore, the train model is established as a four-axle vehicle with

Table 1 :
Equivalent spring stifness at pier bottom.

Table 2 :
Parameters for the train model.

Table 4 :
Comparison of results for the frst two modes using diferent models.

Table 5 :
Comparison of the identifed results using diferent algorithms.
structure from any damage inficted by the impact force.

Table 6 :
Comparison of results for the frst mode identifed by diferent methods.

Table 7 :
Results of the investigation of the in-situ geological environment.

Table 8 :
Equivalent spring stifness at the bottom of pier 494#.

Table 9 :
[43]al value of lateral amplitude at the top of HSR bridge piers in Chinese code[43].