This paper presents the practical results of the evaluation of the data obtained by using ground-based radar interferometer during measurements carried out on bridge structures. Due to the nature of the objects studied, the authors proposed a comprehensive method of data analysis, which identifies whether the passage of the vehicle did not damage the bridge. The effective use of vehicles as a source of bridge excitation allowed us to first develop a method for determining the damping parameters resistant to potentially occurring beating frequencies. As a result, it is possible to determine these subsets of data registered with radar, for which it is possible to assume compliance with linear systems. This type of data, often omitted in other works, forms the basis for the second important element of the research—an algorithm based on the ARMA model supporting defect detection. The optimization of the performed calculations, in particular the proposed optimal ARMA model order, the method of fault identification based on the DSF parameter, or fault identification based on a nonmetrical Cook’s distance leads to a robust and scalable method. The method’s low computational complexity allows for implementation in real-time solutions. In addition, the distribution of errors and the sensitivity of classifiers based on the DSF parameter and Cook’s distances leaving them will enable the automation of the classification process using machine learning. The proposed method is universal; in particular, it can be used for radar interferometry methods because it is resistant to potentially variable environmental conditions.

The inspection measurements of important and nonstandard engineering structures and related studies are the basis for assessing their safety. In the group of objects that must be monitored during load tests and require monitoring are, among others, bridge structures and buildings exposed to the influence of seismic factors. The monitoring of such facilities and their examination under test loads should provide a basis for assessing the safety of the structure at the time of its commissioning and in the future.

Many failures can be identified based on the analysis of observations carried out using various measuring devices. In this group, attention should be paid to the radar interferometry technique that allows simultaneous observation of many elements representing the tested structure. Its important advantage is the ability to conduct measurements in a noncontact manner and that there is no requirement to install any devices on the object.

The use of radar observations of many points on the site to analyze the health of the structure is quite wide. This type of research is carried out for both bridge and high-rise buildings by determining the vibration parameters based on dynamic displacement monitoring [

Research on the condition of structures based on radar observations has also been carried out for high objects. Hu et al. [

Another example is a historic masonry bell tower examined by Castellano et al. [

Damage detection is one of the most important applications of SHM systems and algorithms. Modern computational technologies based on digital signal processing, the evaluation of patterns by means of machine learning, or the evaluation of patterns by the analysis of statistical characteristics of signals can be used to assess the safety of building objects [

Recently, an important trend in this field is the use of machine learning to identify potential problems. While the use of AI methods is widely known for systems based on the analysis of dynamic data, it is also worth noting that it is possible to effectively analyze data from high-resolution measurement systems using deep machine learning methods [

Finally, the applications of the measured displacement data for SHM are reviewed, including examples of structural modal property identification, structural model updating, damage detection, and cable force estimation [

As important as research work that focuses on system reliability, there are those that aim to reduce computational complexity while maintaining damage detection efficiency. In particular, unlike conventional strategies employing a frequency response function or response data, a damage detection methodology is addressed by employing transmissibility functions that retain a strong interrelation with structural damage or deterioration in order to avoid the measurement of excitation, together with the principal component analysis that leads to a reduction in computational costs [

Studies on SHM algorithms concern, just like in this paper, the identification of damages occurring on bridge structures. These tests, which are crucial for practical applications, must also include an analysis of the interaction between the bridge and the vehicles that constitute the source of disturbances and vibrations [

An interesting solution, presented by Krishnan et al. [

On the contrary, the health analysis of the structure based on data representing the stationary parts of measurement signals has been presented, among others, by Sohn et al. and Nair et al. [

This work presents an algorithm for conducting research under testing or operational load that allows simultaneous observation which will allow for identifying any structural damage that may occur during testing and are the basis for building a reference database of the system based on SHM.

An algorithm for detecting and automatically identifying the defects of buildings and structures is applied. It is particularly useful for engineering structures susceptible to dynamic excitations such as bridges, viaducts, flyovers, masts, and towers, as well as free-standing chimneys (single and multi-flue) based on tests carried out under testing or operational load.

The computational technique is that the measurement signals, which are variable in time, are measured, and the results are delivered to the computing unit in the form of time series and spectrograms, and analyses are carried out for the stationary fragments of the time series.

In the first stage, when measuring a given structure, measuring devices, such as the accelerometers, interferometric radar, or GNSS receivers, are positioned in such a way that the following can be performed:

It is possible to accurately identify the mode shapes resulting from the modal analysis of the structure

They are located in the places that are subject to damage during tests under test loads and operational loads

In the second stage, the values of the identified amplitudes obtained from those parts of the time series, which represent free vibrations, are compared with the results of the modal analysis in the range of values in the frequency domain and in the range of the logarithmic decrement of damping calculated from the Hilbert transform of free vibration [

Then, in the parts of the time series that represent the stationary signal, the ARMA model is fitted (a linear model of autoregressive moving average) [

In the third stage, it is calculated if the distance of the given calculated coefficient, on the basis of the given time series after the crossing of a vehicle, changes the coefficients of the regressive lines fitted into the previous realizations of the DSF with the use of Cook’s distance. In this way, the dynamic behavior of the bridge, which deviates from the norm, is identified.

The signal recorded during the load testing can be divided into three parts in the time domain (Figure

The data represent the stationary signal. This is the basis for finding the structure’s features representing its condition prior to possible damage and in the parts representing the condition of the object after free vibration has expired. The second part is the basis for evaluating whether or not the force damaged the object.

The data represent the deflection of the construction. The standard procedure may be used to calculate other parameters such as the coefficients of the dynamic amplification factor (DAF).

The data represent the free vibration. The correctly filtered and standardization process allows for the calculation of an amplitude spectrum and also may determine if the design is acting in accordance with the damping based on the values of the logarithmic decrement of damping.

The decomposition of the measurement signal (reproduced from [

It is common practice to use free-damping data to verify FEA (finite element analysis) models. Usually, data are used to obtain information about the amplitude spectra (Figure

Example of time series and its amplitude spectrum (reproduced from [

If a Hilbert transform was calculated for such a signal, the envelope of the damped oscillator was obtained as a result (Figure

Hilbert transform calculated for signal obtained from damping vibrations with beat frequencies (reproduced from [

Taking into account equation (

Directly from the definition by fitting the exponential function into the result of the Hilbert transform

By fitting the linear function into the logarithm of the Hilbert transform

A classic logarithmic decrement of damping is calculated upon the basis of the following equation:

The direct use of equation (

Logarithmic representation of Hilbert transform (blue) and linear estimation (red) (reproduced from [

The standard equation describing the vibration is exponential. It is by its nature difficult to be analyzed by regression algorithms. The proposed solution is based on linearizing the equation before estimating the parameters. The logarithmic representation of the Hilbert transform can be easily estimated using linear regression or generalized linear regression with selected cost function (the authors present the use of LSF as a cost function). It is a more effective way and a more robust solution.

Such an approach allows for the estimation of the damping coefficients to be based on a robust estimation. In addition, while the estimation is being determined, the entire data acquisition of the measuring signal is being utilized, rather than an arbitrarily chosen amplitude (Figure

If during a load test, damage to the construction occurred, it would change the statistical characteristics of the measured data. There exists a group of methods which has been developed for the identification of the damage. They are based on the congruency of the ARMA (autoregressive moving average) models into the given data. The general form is as follows:

The algorithms of the group are discussed in detail [

Selected and implemented ARMA algorithm framework.

The structure of the proposed algorithm is discussed in more detail below. The assumption is to answer the question whether the condition of the structure before the vehicle’s approach during the loading of the bridge structure and after that has changed. The algorithm operates on data portions—called batch or data intervals. Batch data processing is an efficient way of processing high volumes of data where a group of transactions is collected over a period of time. Data are collected, entered, and processed, and then the results are produced. Since we would like to be able to compare the results between measurements, the intervals of the data must be standardized in the beginning of the process, as is shown in Figure

After the standardization, the time series is entered into the model of ARMA in accordance with equation (

Therefore, the effect of the action of the algorithm will be the result of the damage sensitive feature of the DSF parameters calculated for the specific data vectors representing the engineering structure before and after the potential damage (Figure

DSF obtained from example data (reproduced from [

The classical approach to identify the damage in a given structure is that one must take all the obtained DSF coefficients prior to the test (marked in Figure

Hence, for both groups of data, the mean values have to be estimated. Upon this basis, it may be concluded that there will be a substantial difference between the groups, using the standard

This type of approach has two characteristic shortcomings:

It is crucial to take a sufficient number of samples representative of the structures behavior after force has been applied to the construction, in order for the statistical significance from the given test to be properly kept at accordingly a high level.

Limiting the possibility of using the calculation techniques of bridge structures while under operation being subjected to continual use, there may not be a suitable length of time between the impact of the structure to gather the proper amount of data to run a

The abovementioned limitations may be solved by using a different criterion than the statistical difference estimated between the two groups of data. A dataset was considered in which after the excitation and damping of the object and before the next excitation, a limited amount of data can be registered. It means that two consecutive forces are applied to the structure in a short time. In the case of such data, it is possible to calculate a limited number of the DSF coefficients (in Figure

DSF obtained from object with limited data after extraction (reproduced from [

Such a situation may be encountered when research is being carried out in bridge structures that are in current use, especially those with a large variety of vehicles that are not standard and are oversized. The question at hand is whether or not a given vehicle may be the cause of damage to a structure even during minimal intervals between the impacts.

In order to verify whether the limited number of DSF parameters that were registered are significantly different from the average realization, the formula that may be used in such a regression analysis is based upon Cook’s distance given by the following equation:

There are several reasons why Cook’s distance has been chosen as a tool to detect changes in the DSF coefficients value. First of all, the use of this method allows for the diagnosis of the object’s state immediately after the load has been removed which is crucial for the algorithm. Thus, the potential damage to a bridge object can be detected on the basis of a small amount of data. Second, there are unambiguous, objective criteria for assessing whether Cook’s persistence is statistically significant [

Figure

Cook’s distance calculated for the linear model of DSF coefficients [

The dashed line in Figure

The object on which the test was carried out was a tram viaduct. The ground-based interferometric radar IBIS-S was used to acquire the data (Figure

IBIS-S radar unit under the tested bridge span.

The design specifications on the phase accuracy applied on the radar system, which was used in the presented research, are suitable for measuring short-term displacements with a range accuracy better than 0.1 mm [

In the conducted research, it was assumed that the impact of the atmospheric disturbance and the multipath signal effect is negligible. This is possible because, during the observation, the atmospheric conditions did not change and the configuration of the measurement system and the object remained unchanged. In addition, taking into account the fact that the precision of the measurement result is more important to the performed tests than its accuracy, it may be assumed that the record of 0.01 mm displacement is an actual observation.

The time series subjected to further analysis is shown in Figure

Data input for time-series-based damage detection algorithm (searching for stationary signals using selected parameters of object damping).

In the proposed algorithm, the frequency spectrum analysis is not a key but an auxiliary element of the solution. The essence of the algorithm is based on the transformations of stationary signals. However, the proposed application of the method is monitoring bridges that will be subjected to vehicle traffic. Therefore, in order to correctly analyze the data, it is necessary to verify when after the excitation the construction vibration has been damped.

To determine the parameters of damping the structure, the observation intervals marked with arrows were used (Figure

According to the proposed algorithm, the Hilbert transform was used to determine the damping of the structure. The signal from the observation is marked in blue, while the envelope of the vibration (i.e., the graph of the Hilbert transform) is shown in red (Figure

Hilbert transform of selected intervals.

Then, in the logarithmic representation of the Hilbert transform, the linear function was fitted (Figure

Logarithmic representation of Hilbert transform.

The algorithm of the structure damage detection based on the autoregressive moving average model has several parameters that can be adjusted adequately to the analyzed building objects. Among them are the

Distribution of DSF values depending on the AR process order: (a) version order of AR = 4, order of MA = 3; (b) version order of AR = 5, order of MA = 3; (c) version order of AR = 6, order of MA = 3.

Distribution of DSF values depending on the MA process order: (a) version order of AR = 4, order of MA = 2; (b) version order of AR = 4, order of MA = 3; (c) version order of AR = 4, order of MA = 4.

In the following figures, different symbols were used for marking the DSF values obtained as a result of the analysis of the signal recorded before the occurrence of the load and after the load termination and related effects (like the damped vibrations). The solid lines in the corresponding colors represent the regression lines fitted into the set of DSF values determined for the adopted number of the analyzed data vectors.

The values of the model orders of the AR and MA processes have a range that makes them appropriate for the analysis [

In the next stage, the effect of the length of the vector containing the data to determine the DSF parameter and the number of analyzed vectors was verified. Observation data were divided into two ways: (1) 21 vectors with 200 elements and (2) 11 vectors with 400 elements (Figure

Analyzed cases of vector length: (a) version order of AR = 4, order of MA = 3; (b) version order of AR = 4, order of MA = 3.

In the proposed algorithm, Cook’s distance was used to determine if the implementation of a limited number of DSF parameters are significantly different from the average realization. The analyses were made on the basis of the DSF datasets, as shown in Figure

Distribution of the DSF values depending on the order of the MA process.

It should be noted that the DSF values that would indicate a change of the structure state (vectors no. 9–13 in Figure

The presented algorithm comprehensively discusses the methods of prototyping engineering structures, in particular, examining bridges under testing and operational loads. Its basic assumptions and features are the following:

The decomposition of the recorded signal represents the vibration of a given bridge structure into three groups in the time domain. The first group contains data before an impact and after free vibration, and technically, it is the group of stationary signals of linear systems. The second group is the response of the construction (i.e., the deflection of the span occurred). The third group is the part of the signal which represents the free vibration in a structure that is excited.

The decomposition of a signal in a frequency spectrum, especially with band-pass filters, allows for the more effective spectral analysis. The band width is the results of the FEM analysis.

The amplitude spectrum is comparable to the analysis made with the finite elements method through the calculation of the fast Fourier transform.

Construction damping of an object is represented by the logarithmical decrement. The calculation of its values is not dependent upon the implementation of the direct definition but on the calculation of the Hilbert transform. Furthermore, for the logarithm of the envelope, the linear regression with the robust least squares fitting method is calculated. The calculated coefficients of the linear estimation allow for an estimation of logarithmic decrement of damping in the entire signal, even when the structure experiences beat frequencies.

The identification of the potential damage to a structure as a result of impact is based on the DSF coefficients. The answer to the question if the damage occurred is based on Cook’s distance rather than the comparison of the average values of the tests is obtained as follows: the effect of such an examination is when in real time the conclusion may be drawn whether or not the data from the tested object indicate the damage, even in the cases when the damage occurs during the operation of the tested object.

It is of utmost importance that the data supporting the algorithm in the field of stationary signals are analyzed properly. The important parameters are as follows: the order of the ARMA model, the length of the data windows, and the test if the residuals obtained are normal, impeded, and identically distributed. Verification of the construction condition has to be based on the proper baseline (the same environmental conditions). In addition, the proposed solution presents current and modern approaches to solving the problem. In particular, it offers the following:

The effective separation of stationary and nonstationary signals

Optimal ARMA model parameters

Implementation possibilities supporting online solutions by limiting computational complexity

Effective input data for the analyzes conducted using the AI method, in particular for classifying the DSF parameters and Cook’s distances assigned to them

A methodology of using Hilbert transforms for oversize excitations

The use of observation methods based on interferometric radars, which facilitate the location of potential damage, because the input data are uniform in the time domain and strictly defined to the location; due to the easy coverage of the tested object with multiple observations, the analysis of data, and consequently the location of the damage, is easier

Input data from radar systems which allow, due to the frequency and accuracy of the displacement measurements, the use of the most algorithms that were developed for the analysis of the measurements performed with the accelerometers

No influence of weather conditions variability on the possibility of inference about the state of the object for the dynamic issues.

Further research is the technological implementation of machine learning which will allow for the automatic classification of the DSF coefficients.

The PDF data used to support the findings of this study are included within the supplementary information file(s).

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This publication was prepared within statutory research fund no. 11.11.150.005 of the Department of Engineering Surveying and Civil Engineering, Faculty of Mining Surveying and Environmental Engineering of AGH University of Science and Technology in Kraków.

Raw radar measurement data have been provided as a supplementary material in the form of a PDF file.