1 Introduction

Performance-based engineering (PBE) is concerned with the design, evaluation, and construction of engineering facilities whose future performance can meet predefined objectives under multi-hazard levels [1–3]. While PBE has been mostly utilized by earthquake engineers over the past several years, there is great potential for PBE to impact other fields including life-cycle assessment, multi-hazard engineering, and Structural Health Monitoring (SHM) [4–6] for better design and assessment of structural systems.

SHM is a multidisciplinary concept, which is used for proactive performance assessment and management of various types and kinds of structures including civil, mechanical, and electrical infrastructures [7–10]. SHM sensor network designs for performance-based structural evaluation also need to be developed for the best of use of experimental data [13]. In order to support the PBE with reliable information, a given structure has to be monitored over a long term so that the associated uncertainty in the measured data is minimized [14, 15]. The data collected and analyzed through SHM can be beneficial in terms of understanding the responses of structures under different events and hazards, and can be ultimately employed for decision-making related to operation, maintenance, and retrofit. As such, continuous monitoring of the structural performance provides valuable information within a PBE context to assure promised performance levels [11, 12].

Long-term monitoring of the civil infrastructure has not been practical until recently due to the lack of cost-effective sensors and reliable data acquisition systems generating large amounts of data, especially if these are to be deployed in harsh environments and for extended periods of time [16, 17]. Lately, however, with rapid advances in the sensing technology, monitoring potential hotspots over extended periods of time is no longer a prohibitive challenge [18–20]. However, the next challenge for civil infrastructure monitoring is the absence of efficient methodologies for interpreting the massive amounts of data that are accumulated over long periods of time [21–23]. Real-time signal/data processing, data analysis, and interpretation are becoming even more crucial considering the proliferation of intelligent infrastructure networks and the emerging area of smart cities [24, 25]. It is ideal to have SHM systems that can provide some practical, rapid, and also insightful information for decision-making using the well-established data analysis and interpretation techniques such as machine learning, statistical pattern recognition, as well as advanced geometric modeling based on physics-based parameters.

Fundamentally, the data interpretation approaches for SHM can be classified into two main categories, namely, parametric (physics-based or model-based) and non-parametric (model-free or data-driven) [26–29]. The parametric approach is preferred in cases where conceptualization and prediction are the main concerns [30, 31]. There are some responses or attributes of a structure that can be measured using various sensors such as strain, displacement, acceleration, etc. These measurements are either used for parametric models involving experimental data or used in conjunction with advanced geometric models such as finite element models for a physical interpretation of the current status of the structure. These parametric models can also be employed for predictive evaluation of future performance under various scenarios. In most cases, the parametric models can be easily developed for PBE with an established limit state identifying the healthy status of the structure. For example, responses, capacity, reliability indices, etc. can be obtained at the element and also at the structural system level. The integration of a SHM system along with sensing and data analysis, modeling, and performance evaluation is presented within a structural identification framework [32, 33]. In fact, such an approach employing structural identification techniques can be viewed as a translation of “raw” measured parameters into “actionable” information with a PBE perspective.

Alternatively, non-parametric or data-driven approaches are superior in the circumstances wherein creating a behavioral model is either time consuming or expensive [34–36]. Data-driven approaches are model-free and can be utilized to process data without access to geometrical and material information, making this a main advantage over parametric approaches [37–39]. Fundamentally, statistical models are generated based on data collected from a baseline condition, and any variations from the baseline are then reported as abnormal behavior. These abnormal behaviors can be due to structural damage or a structural change due to some transient effect [40–43]. When it comes to long-term monitoring data especially for ordinary structures, dealing with such statistical models is much more convenient and in most cases is preferable to a complex mechanical model.

Although the non-parametric approach is more practical for real-time processing of large amounts of SHM data, a thorough interpretation of the measurements in terms of assessing the root causes of abnormal behavior or damage cannot be achieved. In other words, detecting the potential abnormal behavior in the structural system is possible using a non-parametric approach; however, assessing and classifying the identified damage is typically not achievable through the existing non-parametric techniques.

As a result, the two main categories of data interpretation approaches, parametric (physics-based or model-based) and non-parametric (model-free or data-driven) for SHM applications, have their distinct advantages and disadvantages. A hybrid approach mainly capitalizing on the advantages of these techniques would be desirable for certain applications.

1.1 Objective and scope of the paper

The objective of this paper is to present a hybrid data interpretation framework for SHM data so that the extracted information can be employed for decision-making purposes including performance-based evaluation of infrastructure systems. As such, this framework leverages the benefits of both parametric and non-parametric approaches while mitigating their shortcomings. The proposed approach can be employed not only to detect the damage but also to assess the identified abnormal behavior. Furthermore, this framework allows for determining the number of sensors, as well as their corresponding locations, required to effectively monitor a structure. In particular, the sensor network is optimized so that the collected information can be ultimately used for PBE. The proposed framework is categorized as a supervised classification approach where an algorithm is initially trained utilizing the data generated through Monte-Carlo simulation techniques with advanced finite element models. The data are processed using Moving Principal Component Analysis (MPCA) to extract relevant features. Next, statistical learning methods are implemented to estimate the underlying distributions of the features during a training phase and a decision rule is developed using hypothesis testing. Once the training phase is completed, the real-time SHM data are analyzed and classified using MPCA and hypothesis testing, respectively, to identify the performance status of the structure with the possible root causes of the identified abnormal behavior or structural changes. The hybrid framework is evaluated by means of analytical and experimental studies. An experimental structure (UCF 4-Span Bridge) is employed along with an in-house developed Fiber Bragg Grating (FBG) system for simulating common bridge damage cases and collecting the SHM data. The experimental and numerical results underscore the efficiency of the proposed approach for sensor network design and for continuous evaluation to structural performance.

2 Integrated hybrid framework for automated performance monitoring

As discussed, the hybrid framework for continuous performance assessment is expected to not only detect damage or structural change, but also to effectively classify them based on the integration of non-parametric (data-driven or model free) and parametric (model-based) approaches. For demonstration purposes, some common damage scenarios for bridge-type structures are identified and implemented in the laboratory environment. A comprehensive study is conducted to identify the most common and critical damage scenarios that a given structure is likely to undergo in the course of its lifetime. This can be accomplished through several different sources including engineering judgment, feedback from experts, long-term performance of a specific type of structure, etc. The experimental test setup, damage scenarios, sensing network are detailed in the later sections. Before providing the details of the experiments, we describe the steps followed for developing the hybrid framework.

2.1 Simulation of selected damage scenarios and considering uncertainties using a Monte-Carlo technique

In general, a wide range of computational algorithms that were developed based on the idea of repeated random sampling to reach numerical results are referred to as Monte-Carlo methods or simulations. Some of the main applications of Monte-Carlo simulations include optimization, numerical integration, and sampling from a distribution. In this study, the Monte-Carlo technique is employed for the generation of samples from distributions associated with the uncertainty of the critical parameters (parameters which are affecting the desired responses of the structure). In other words, in order to construct an appropriate predictive model, large sets of parent and offspring FEMs are generated from the probability distributions related to sensitive parameters using Monte-Carlo simulation.

Due to the possible structural configurations, damage as well as uncertainties in loading, material properties, boundary and continuity condition, time-variant behavior of some parameters, a Monte-Carlo simulation is carried out in order to simulate and predict the performance/responses of a given structure under these scenarios. Obviously, utilization of Monte-Carlo simulations provides the opportunity to incorporate the uncertainty associated with the governing parameters in the mathematical model of the structure and eventually in the predicted responses. First, a calibrated Finite Element Model (FEM) of the structure, termed as the parent model, is generated for the baseline condition. Upon obtaining a calibrated FEM that represents an accepted performance as a baseline, the selected damage scenarios are individually simulated in order to develop FEMs for each individual scenario to serve as the parent model for that condition. Therefore, there would be a representative calibrated FEM for each structural condition (including baseline and damage conditions), as shown in Fig. 1. Finally, a Monte-Carlo simulation is performed to produce offspring FEMs based on the calibrated (parent) FEMs and the associated uncertainty of the governing parameters. More detailed discussion of the parent and offspring model (family of models) utilized in this paper can be found in the literature [44, 45].

Fig. 1
figure 1

Monte-Carlo simulation for generating offspring FEMs from the parent model

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Having established a predictive model for each damage scenario, one can forecast the performance of the structure under a particular damage condition. One of the important issues here is to determine a sufficient number of offspring models for the population considering the uncertainties. Uncertainty plays an important role in engineering. Based on the literature, the uncertainty has two main categories—aleatoric and epistemic (although there may have several sources) [46]. In this study, the uncertainties associated with material properties, section properties, and traffic load factor are considered in order to generate offspring FEMs as illustrated in Fig. 1. The number of offspring FEMs that is required for each calibrated (parent) model is derived from the following equation:

$$N = \frac{{(1 - P_{\text{f}} )}}{{V_{p}^{2} \times P_{\text{f}} }},$$
(1)

where P f is the estimated probability of failure while N indicates the number of required simulations. In this study, it is assumed that probabilities as low as 0.01 are estimated, and the coefficient of variation, V p , of the estimates is kept at or below 10 %. This yields a total of 9900 samples. In this study, 10,000 offspring FEMs are generated from each parent FEM. Equation (1) is used to determine the number of simulations needed for each condition (baseline and individual damage scenarios) in such a way that the uncertainty associated with response prediction is less than a certain value. The generated offspring FEMs are analyzed under a simulated traffic load in order to compute the range of responses corresponding to each damage condition.

2.2 Moving principal component analysis (MPCA)

After developing and analyzing the offspring FEMs under the simulated traffic loads, the structural responses are stored in individual matrices called matrices of raw data. Having 10,000 offspring FEMs for each damage condition will eventually result in generating 10,000 matrices of raw data, as illustrated in Fig. 2. These matrices are then processed using advanced statistical analysis methods in order to derive the damage-sensitive indices as discussed in the following section.

Fig. 2
figure 2

Matrices of raw data and the procedure for conducting MPCA

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Principal Component Analysis (PCA) is one of the most effective dimensionality reduction techniques that can significantly reduce the dimension of large data sets while retaining the informative part of the measurements [47]. Recently, PCA has been implemented in a form known as Moving PCA (MPCA) for effectively segmenting large data and for determining the changes, as data are continuously collected [42]. For this study, the MPCA algorithm is employed (based on the great potential that it has shown for SHM applications) for processing the data and extracting the performance features, which are subsequently used to classify the damages. The reader is referred to [41, 42] for detailed information about the MPCA algorithm. The implementation of MPCA in this paper is outlined in the following steps and illustrated in Fig. 2.

The subsequent steps provide details about developing the matrices of raw data and calculating the damage indices as illustrated in Fig. 2.

  1. 1.

    Generate the matrices of raw data by assigning the time history of the measurements from each variable (sensor) to individual columns, as illustrated in Fig. 2. It should be also noted that each matrix contains data from a baseline as well as a corresponding damage scenario, i.e., the data consist of history before and after a given damage scenario. The notations that are used in Fig. 2 are

    • \(U_{{D_{i} }}\): Matrix of normalized raw data corresponding to damage D i after subtracting the temporal means.

    • \(u_{{s,D_{i} }} (t)\): Measurement of sensor s under condition D i at time t.

    • \(s \in \{ 1,2 \ldots N_{s} \}\): Refers to the sensor index where N s is the total number of sensors.

    • \(D \in \left\{ {B,D_{1} , \ldots D_{4} } \right\}\): Performance condition of the structure (B refers to the baseline condition while D i represents damage scenario i).

    • N s : Total number of sensors installed on the structure.

    • N m : The total number of the measurements/observations from a sensor under a baseline and a given damage condition.

    • N w : The size of the fixed-window.

    • \(U_{{D_{i} ,k}}\): (N w  + 1) × N s matrix whose rows \(U_{{D_{i} ,k}} \left( {t_{j} } \right), \,j = k, \ldots , k + N_{w}\) correspond to the sensor data at time t j under damage scenario D i . k is the index of the moving window k ∊ {1, 2, …, N m  − N w }.

    • \(e \in \{ 1,2 \ldots 10000\}\): Number of simulations.

  2. 2.

    Since the matrices of raw data may include different measurements, which are collected with different types of sensors (e.g., strain gages, accelerometers, microphones, etc.), the measurements have to be scaled before implementation. In this study, the data were scaled in such way that all the variables have zero mean and unit variance.

  3. 3.

    Selecting the size of the moving window (N w ) is a key parameter for performing MPCA. There is a fundamental tradeoff associated with the choice of the size of the window since it should be selected large enough so that it is not influenced by periodicity in data, but also small enough in order to timely detect any abnormal behavior.

  4. 4.

    Computing the covariance matrix C k and subsequently the corresponding eigenvectors and eigenvalues for the kth time step (data within the kth moving window) using the following equation:

    $$C_{k} = \mathop \sum \limits_{j = k}^{{k + N_{w} }} U_{{D_{i,k} }} (t_{j} )^{T} U_{{D_{i} ,k}} (t_{j} ).$$
    (2)
  5. 5.

    Extracting the eigenvectors ϕ i and eigenvalues \(\lambda_{i} , i = 1, \ldots , N_{s}\), of the covariance matrix as shown in Fig. 2 and Eq. 3.

    $$(C_{k} - \lambda_{i} I)\phi_{i} = 0.$$
    (3)

2.3 Damage indices and damage-sensitive features

A critical objective of SHM is to timely detect and identify abnormal behavior so that catastrophic failures can be avoided. In order to achieve such an objective, one has to select sensitive damage indices that reflect the corresponding malfunctions. The damage indices that are adopted for this study are based on the first principal component (PC) of the data, which in most cases contain the most informative portion (largest variance) of the data. Different entries of the first principal component, as visualized in Fig. 3, are computed individually and introduced as the damage index. This damage index is derived based on Eq. 4.

$${\text{Damage}}\;{\text{index}}_{i} \left( k \right) = \phi_{i1 } (k),$$
(4)

where ϕ ij (k) is used to denote the ith entry of the jth PC in the kth moving window. Note that this is repeated for every offspring \(e \in \{ 1,2 \ldots, 10000\}\).

Fig. 3
figure 3

The procedure for deriving the damage index using family of models

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The first principal component indicates the direction in which the data have the most variance. An assumption for the MPCA algorithm is that if abnormal behavior occurs, it should affect the direction in which the data have its most variance. As a result, it should be detected by monitoring the corresponding time histories of individual entries (Fig. 3).

It should also be noted that the smooth transition from the baseline to the damage condition in Fig. 3 is due to limited data points generated through simulations. In the transition phase, the data within the moving window belong to both the baseline and the damage scenarios. Once the sliding window passes through all the baseline data, it is completely in the damage region, hence the damage index stabilizes as observed in Fig. 3.

The effect and reliability of these damage indices in detecting different types of damage have been tested and evaluated using data from both laboratory and real-life practice [38–41].

It is also important to evaluate and assess the damage after detection. Thus, a feature that is sensitive to the damage of the structure is needed. Therefore, after deriving the damage indices, the subsequent step is to extract damage-sensitive features that can intuitively reflect the performance level of the structure. Herein, the damage-sensitive features are considered as the separation of the extracted damage indices due to different scenarios.

These features are derived through the following equation:

$$X = {\text{DI}}_{\text{before damage}} - {\text{ DI}}_{\text{after damage}},$$
(5)

where DIbefore damage and DIafter damage represent the average damage indices before and after change, respectively. Note that (5) is used to calculate the damage-sensitive feature X for different damage indices i computed in (4), where i corresponds to the ith entry of the principal component. The premise is that different types and levels of damage induce different levels of separation in damage indices, and as a result this can be a potential feature to classify the damage.

2.4 Learning the distributions of the damage-sensitive features

Once the corresponding damage-sensitive features are extracted for each individual damage level (scenario) using Eq. 5, the following step is carried out to learn the underlying distributions. In fact, by learning the underlying distributions for individual scenarios, the proposed approach can be used to predict and classify the new features into relevant categories and identify the current performance status of the structure. Herein, we used hypothesis testing to classify the features in order to identify the damage scenarios. For completeness, the main problem of binary hypothesis testing is briefly described. The goal is to decide between two hypotheses, H 0 (termed the null hypothesis), and H 1 (the alternative hypothesis). Given an observation X, the goal is to decide which of the two hypotheses has occurred. We consider Gaussian observation models, where under each hypothesis the observation is drawn from a given Gaussian distribution as shown in Eq. (6).

$$\left\{ {\begin{array}{*{20}c} {H_{0} :X\sim N(m_{0} ,K_{0} )} \\ {H_{1} :X\sim N(m_{1} ,K_{1} )} \\ \end{array} } \right.$$
(6)

where m 0 and K 0 are the mean vector and covariance matrix under the null hypothesis H 0, respectively, and m 1 and K 1 denote the mean vector and covariance matrix under H 1 . In other words, under hypothesis H i , i ∊ {0,1}, the observation X has a Gaussian distribution with mean m i and covariance matrix K i . In this study, X (observation) is the extracted features obtained from Eq. (5). Since we do not have access to the true distributions, we learn them from the training data. The optimal decision rule [46] is to compare a sufficient statistic S(X) to a threshold, i.e.,

$$S\left( X \right){ \gtrless }_{{H_{0} }}^{{H_{1} }} \eta,$$
(7)

where S(X) is some function of the data. If the sufficient statistic S(X) exceeds the threshold η, we decide in favor of H 1, otherwise we choose H 0. For the binary hypothesis testing problem with Gaussian observations in (6), the sufficient statistic of the optimal likelihood ratio test is given by [46]

$$S\left( X \right) = \frac{1}{2}X^{T} \left( {K_{0}^{ - 1} - K_{1}^{ - 1} } \right)X + X^{T} (K_{1}^{ - 1} m_{1} - K_{0}^{ - 1} m_{0} ) ,$$
(8)

where X = [X 1, X 2 ,… X n ]T is a multidimensional data vector with entries X i. In our study, X i corresponds to the ith damage-sensitive feature.

The optimal threshold η is given by

$$\eta = \frac{1}{2}\left[ {m_{1}^{T} K_{1}^{ - 1} m_{1} - m_{0}^{T} K_{0}^{ - 1} m_{0} + \ln \left( {\frac{{\left| {K_{1} } \right|}}{{\left| {K_{2} } \right|}}} \right)} \right],$$
(9)

where |K| denotes the determinant of the matrix K. Thus, Eq. 7 is a decision criterion with which one can decide whether the observation belongs to hypothesis H 0 or H 1. Note that these concepts can be naturally extended to more than 2 hypotheses, i.e., multiple hypothesis testing.

For binary hypothesis testing, the Receiver Operating Characteristic (ROC) curve [46] is used to evaluate the performance of the decision rule. The ROC curve presents the fraction of true positive out of positives [true positive rate (TPR)] against the fraction of false positive out of negatives [false positive rate (FPR)] at various threshold settings (Eq. 7 through Eq. 9). The usage of ROC curves will be further explained later with simulation data.

The objective is to first extract the feature from the live SHM data (using MPCA) and use a hypothesis test to decide between H 0 (the structure is in baseline condition and maintenance action is not required), and H 1 (the structure is experiencing damage and consequently, the corresponding maintenance action has to be taken. Note that in this study, we have more than 2 hypotheses corresponding to the different damage scenarios. However, since multiple hypothesis testing uses pairwise comparisons between pairs of hypotheses, we described the binary test for ease of exposition. Furthermore, we use binary hypothesis testing to first decide whether a damage has occurred and then use successive comparisons between pairs of hypotheses to identify the type of damage and easily visualize the performance of our approach.

3 The proposed hybrid algorithm with optimized sensor network configuration

Based on the aforementioned procedures, the proposed hybrid algorithm is summarized in Fig. 4. As mentioned earlier, the goal of the designed algorithm is to first detect the damage and then to classify the damage. In addition, using the proposed algorithm, one is also able to identify the optimal sensor configuration for the structure. In fact, the hybrid framework enables the designer to identify the minimum number of sensors to be installed on the structure so that the damage can be quickly detected and efficiently classified.

Fig. 4
figure 4

The proposed hybrid framework for continuous performance monitoring of civil infrastructure

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3.1 Identifying the optimized sensor network

As shown in Fig. 4, Phase II of the hybrid algorithm is dedicated to design the network of sensors. Basically, Phase II is a trial-and-error process whereby the locations and the number of required sensors are identified. It should be also noted that since the MPCA algorithm is fundamentally a multivariate technique, the minimum number of required sensors is two. Prior to designing the sensor network, the candidate locations have to be determined. The critical locations of the structures have to be identified and chosen as candidate locations for installing the sensors. Given N s sensors and N loc candidate locations, the total number of candidate configurations is given by

$$N_{C} = \left( {\begin{array}{*{20}c} {N_{\text{loc}} } \\ {N_{S} } \\ \end{array} } \right) = \frac{{\left( {N_{\text{loc}} } \right)!}}{{\left( {N_{\text{loc}} - N_{S} } \right)!\left( {N_{S} } \right)!}}.$$
(10)

For instance, if the number of candidate locations (critical locations of the structure) is 10, then the possible number of configurations would be equal to 45. These networks are compared against each other based on the ROC curve. A better ROC curve means a better tradeoff between the probability of detection and the probability of false alarm. For the first trial, we start with a number of sensors (N s ) equal to 2.

If a given configuration results in a perfect ROC curve then we stop the search and select this configuration. However, if the performance can be further improved, then we increase Ns and the search process continues. This procedure is illustrated through the second phase of the algorithm in Fig. 4. Having identified the optimized network, the matrices of raw data are generated and the underlying distributions of features are learnt (due to different structural conditions) through the Monte-Carlo simulation as described earlier.

Finally, in Phase III, we process the SHM data and extract the features needed to classify the damage using hypothesis testing. This phase is conducted in real-time and the owner can be updated regarding the real-time performance of the infrastructure. Furthermore, in the case of damage, the relevant maintenance action is taken with the minimum possible delay.

It should also be mentioned that for complex and large civil infrastructures, the number of candidate locations, and consequently the number of configurations in Eq. 10, can dramatically increase, which may pose a computational challenge. To address this challenge, the complex structure may be initially divided into multiple substructures. The critical damage scenarios associated with each substructure should be carefully identified and simulated, and a local sensor network can be designed for each of these individual substructures to reduce the computational burden.

4 Evaluation of the proposed approach through comparison of the experimental and analytical studies

4.1 Experimental configuration and damage scenarios

A laboratory bridge model (UCF 4-Span Bridge Model) was utilized in order to evaluate the efficiency of the proposed framework. The structure consists of two 120-cm approach (end) spans and two 304.8-cm main spans with a 3.18 mm thick, 120-cm wide steel deck supported by two HSS 25 × 25 × 3 girders separated 60.96 cm from each other. It should be pointed out that even though the structure is not a scaled down model of a specific bridge, its responses are representative of typical values for medium-span bridges. A Radio-controlled vehicle is crawled over the deck to represent a traffic load on the bridge as shown in Fig. 5. It should be noted that the same type of load was applied to the structure for the baseline condition and the different damage scenarios.

Fig. 5
figure 5

Experimental configuration (4-Span Bridge, data acquisition, and instrumentation)

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Using the 4-span bridge model in the UCF Structural Laboratory (Fig. 5), it is feasible to simulate and test a variety of damage scenarios that are commonly observed in bridge-type structures including both local and global damage scenarios. It is possible to simulate most of the common boundary conditions, including rollers, pin, and fixed support. In addition to these, the bolts connecting the girders and deck can be loosened or removed at different locations to modify the stiffness of the system and to simulate damage. In other words, changing the boundary conditions simulates the global damage scenarios; change in connectivity is desirable for local damage simulations. For the numerical phase of this study (Monte-Carlo simulation in Phase II of Fig. 4), an FE model of the 4-span bridge is developed, which consists of 1084 frames, 720 shells, 32 solids, 963 rigid links, using the nominal structural parameters and material properties.

Based on discussions with the Department of Transportation (DOT) engineers, several critical and common damage scenarios are identified and simulated on the 4-Span Bridge model. A crucial type of damage observed in bridges is alterations in boundary conditions. These types of alterations may cause stress redistributions and in most cases may result in additional load in different elements. Therefore, two cases are devoted to this type of damage by shifting from pinned to fixed or roller conditions or vice versa. Missing bolts and section stiffness reductions are also cases observed in existing bridges. The last two damage cases (Case 3 and case 4) are designed to simulate a mixed issue, namely the loss of connectivity between the girder and the deck, as well as boundary deficiencies. The damage scenarios implemented in this study are demonstrated in Fig. 6.

Fig. 6
figure 6

Implemented damage scenarios for 4-Span Bridge

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4.2 Demonstration of the proposed hybrid framework

In this section, first the optimization of the network (Phase II of Fig. 4) using the hybrid algorithm is demonstrated. Afterward, the efficiency of the algorithm in terms of assessing the performance of the structure is presented utilizing the SHM data collected from the 4-Span Bridge.

The first step (Phase I) as described in Fig. 4 is to identify the critical damage scenarios associated with a given structure. The critical damage scenarios are demonstrated in the previous section and in Fig. 6. Before starting the trial-and-error process of Phase II to design the optimized network configuration, the candidate locations or critical regions of the structure (N loc) are identified based on the positions and types of the expected damage scenarios.

These candidate locations are the potential locations for installing the sensors. It should be also highlighted that optimizing the network in terms of locations and number of sensors will have a significant impact on the budget of the SHM project, in particular when it comes to large size civil infrastructures. With respect to the 4-Span Bridge, ten individual sections were initially selected as candidate positions to mount the FBG sensors shown in Fig. 5. Having selected the candidate locations, the next step is to decide the minimum number of sensors that should be distributed over those locations so that the predefined objectives are met. The initial value for N s (Phase II of Fig. 4) is 2 and as a consequence the number of possible network configurations is 45.

In order to pinpoint the optimized network (out of the 45 possible arrangements), one should identify which of the arrangements provide the most informative data to detect and classify the different abnormal behaviors. The designer requires a criterion whereby all the networks can be compared against each other and subsequently adopt the optimized configuration. The approach for decision-making that is utilized herein is based on the study of the ROC curves corresponding to pairs of damage conditions. An ROC curve is an efficient tool to visualize and assess the performance of a decision rule as it shows the tradeoff between the probability of detection and the probability of false alarm. If the ROC curves corresponding to different pairs of damage scenarios cannot be significantly improved, we increase the number of sensors and the process is repeated. It is worth mentioning that we could also formulate the problem as a multihypothesis test, however, we adopt the proposed approach to visualize the performance improvements in detecting all pairs of hypotheses using ROC curves.

In order to demonstrate the consequences associated with miscalculation of sensor network design, three different types of designs are reviewed. The designs are categorized in one of three levels, namely, unacceptable, poor, or unsatisfactory, and finally optimized. The first design that is being discussed here is an unacceptable design for the sensor network. This design, which is depicted in Fig. 7, includes two sensors at locations 1 and 4. Additionally, the ROC curves corresponding to damage case 2 against 3 for all 45 networks are presented. As shown, the ROC curve for this design is fairly close to the 45-degree line indicating that this arrangement (locations 1 and 4) is unacceptable as it leads to an unfavorable tradeoff between the probability of detection and the probability of false alarm. We point out that in our analysis we are considering multidimensional distributions for the damage-sensitive features to conduct the hypothesis tests; however, for illustration we plot marginal distributions for specific features corresponding to given entries of the PCs. As shown in Fig. 7, the underlying distributions of the extracted performance-sensitive features for cases 2, 3, and 4 are almost indistinguishable, which leads to unfavorable tradeoff curves. However, the corresponding distribution of case 1 is still well separated from the other distributions. This is due to the fact that the level of induced damage in case 1 is much lower than the damage induced in cases 2, 3, and 4.

Fig. 7
figure 7

Unaceptable design of a sensor network for the 4-Span Bridge

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As shown in Fig. 7 (right), the underlying distributions of the first and second performance-sensitive features for cases 2, 3, and 4 (shown in green, black, and red, respectively) are indistinguishable. As mentioned earlier, this will lead to an unfavorable tradeoff curve, which is almost diagonal as shown in Fig. 7. The inseparability of the distributions is further shown in the zoomed plots (shown in yellow) in Fig. 7 (right).

The second type of design is the one that is labeled as poor or unsatisfactory. The layout as well as the related outcomes is shown in Fig. 8. This design (one sensor at location 3 and the other at location 7) is preferred over the first one (unacceptable type) because of having a better tradeoff curve, i.e., higher probability of detection for the same probability of false alarm. It is easy to see from Fig. 8 that the underlying distributions for cases 1 and 4 are well distinguishable in comparison to cases 2 and 3. This indicates that there is a significant risk of misclassification between cases 2 and 3, while the chance of misclassification is negligible when considering case 1 against case 4. Hence, it is concluded that while changing the arrangement of the sensors from the previous configuration could indeed lead to an improved tradeoff curve, it is still hard to distinguish between the underlying distributions of features for cases 2 and 3. The underlying distributions of the first and second performance-sensitive features for cases 1, 2, 3, and 4 are shown in Fig. 8 in blue, green, black, and red, respectively. The zoomed plots (highlighted in yellow) illustrate the corresponding distributions for cases 2 and 3. It is clear that it is quite impossible to distinguish between the distributions of the second feature for cases 2 and 3, while there is minor difference with respect to the distributions of the first feature.

Fig. 8
figure 8

Poor or unsatisfactory design of a sensor network for the 4-Span Bridge

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Finally, an example of an optimized network, which includes two sensors at locations 2 and 8, is pictured in Fig. 9. The associated ROC curve is almost similar to a right-angle curve, which demonstrates a favorable tradeoff between the detection and the false alarm probabilities. In this design, as shown in Fig. 9, the underlying distributions of the first features for cases 1, 2, 3, and 4 are well separated, which results in an almost perfect ROC curve. After discussing different examples of sensor network design in the next sections, the ROC curves are studied with respect to their performance in classifying different binary combinations of the implemented damage scenarios.

Fig. 9
figure 9

Optimized design of sensor network for the 4-Span Bridge

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4.2.1 ROC curves for case 1 versus cases 2, 3, and 4

The underlying distributions of the performance-sensitive features associated with case 1 are clearly classifiable from the ones corresponding to case 2, 3, and 4. As a consequence, the relevant ROC curves, displayed in Fig. 10, are all in right-angle shape. This demonstrates that if we are only concerned about detecting and classifying case 1 from the other cases, then no further arrangement of sensors is required. Therefore, all the different sensor configurations (45 arrangements) would be acceptable in this case. This can also be realized from column A in Table 1.

Fig. 10
figure 10

The corresponding ROC curves for individual binary combinations of implemented damages

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Table 1 Performance of individual design of sensor networks with respect to different binary combinations of damage scenarios
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4.2.2 ROC curves for case 2 versus case 3

Considering Fig. 10 and column B of Table 1, it is clear that only 4 (out of 45) configurations are reliable. In fact, any other design will significantly increase the risk of misclassification between cases 2 and 3. Based on Table 1 (column B), the optimized configurations are (2,8), (3,8), (5,8), (6,8). Note that configuration (a 1, a 2) indicates that the first sensor is installed at location a 1 while the second one is mounted at location a 2 (Table 1).

4.2.3 ROC curves for case 2 versus case 4

There are 17 individual configurations (17 out of 45) that can be implemented for detecting and differentiating between case 2 and case 4 (Fig. 10). Column C in Table 1 shows the appropriate designs for this case.

4.2.4 ROC curves for case 3 versus case 4

Finally, there are 12 configurations, out of 45 possible designs, that are acceptable with respect to case 3 and case 4. These configurations are presented in Table 1 and column D. The difference between cases 3 and 4 is in the number of bolts removed where 4 bolts are removed for case 3 and 6 bolts for case 4.

4.2.5 Optimized sensor network for the 4-Span Bridge

In the previous sections, the optimized networks of sensors were discussed for different binary combinations of damage scenarios. However, since the objective is to design a reliable network of sensors and a decision rule to distinguish among all damage scenarios, the selected sensor network should have a reliable performance in terms of all binary combinations of the implemented damages. In other words, the optimized network would be the one that is common among all binary combinations in Table 1. As concluded from Table 1, with respect to the 4-Span Bridge, there is only one optimized design or configuration (utilizing two sensors) that can meet this criterion. The optimized design is achieved by positioning the first sensor at location 2 and the second sensor at location 8, as highlighted in Table 1. Thus, this arrangement is the only option available for a designer to distribute two sensors over the bridge in such a way that not only the critical and common abnormal behaviors are detected, but also those can be classified appropriately. Other configurations will inevitably result in misclassification and may result in taking inappropriate maintenance actions. It is worth noting that since a reliable configuration for the studied damage scenarios was obtained using two sensors, we no longer need to consider a larger number of sensors for this particular demonstration. In practice, it may be useful to add redundancy with few more sensors to account for various uncertainties.

4.3 Phase III: real-time interpretation of SHM data directly from measurements

In the previous section (Phase II of the algorithm), the optimized network configuration was identified. Using the optimized network and Monte-Carlo simulations, we learn the underlying distributions of the performance-sensitive features under common damage scenarios. This is then used to effectively classify new incoming live data. Therefore, Phase III is dedicated to real-time processing of the SHM data using statistical analysis to classify the extracted features. Having classified the new incoming features, the current status of the structure is identified and subsequently the corresponding maintenance action can be taken in real-time. The proposed methodology is categorized as a supervised classification algorithm since training data from the FEM model is used to learn the underlying distributions using a simulation study (Phase I and II), then the newly extracted features from live SHM data are classified using a statistical decision rule.

For the training part (Phase I and Phase II), the FE model of the 4-span Bridge is utilized to simulate the selected damages. Monte-Carlo simulations are conducted to learn the responses of the 4-Span Bridge under common damages. In order to experimentally demonstrate the proposed framework in Phase III, an experimental study is designed to collect the SHM data from the 4-Span Bridge using the in-house developed FBG system. The data are collected under the same damage scenarios that were utilized in analytical Phases I and II (training phase). Figure 11 illustrates the classification outcome for the different network configurations, where 1 is used to denote correct classification and 2 to denote misclassification. As it was discussed earlier, all the candidate configurations are acceptable when the objective is to classify case 1 from all other cases. However, as observed from Fig. 11, there are only 4 networks, which can classify case 2 from case 3.

Fig. 11
figure 11

Classification outcome based on simulation data (Table 1)

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Moreover, in order to be consistent with the training phases, the 4-Span Bridge was instrumented with all possible design scenarios for the sensor network (45 different arrangements). The SHM data collected under different damages scenarios are then used in the monitoring phase (Phase III) in order to interpret and classify the damage. The outcomes of the classifier (hypothesis testing) are presented in Fig. 12 for individual scenarios. The horizontal axis indicates different designs of the sensor network (which were also shown in Table 1) and the vertical axis represents different damage scenarios. The test was repeated 15 times for each configuration and damage scenario. However, since the outcomes of the classifier were very similar for all the tests, only the results corresponding to one of the tests is presented in Fig. 12.

Fig. 12
figure 12

Experimental evaluation of Phase III (monitoring phase) of the proposed algorithm for Damage Scenarios 1, 2, 3, and 4

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The top left plot in Fig. 12 shows the outcomes of classification when the SHM data are collected under the first damage scenario. As evidenced by this plot, regardless of the implemented sensor network, there is no misclassification in the outcomes. The fact that there is no misclassification in this case is due to the difference in the level of the damage induced in this case compared to other scenarios. As it was also shown in Phase II (training phase), the underlying distribution of features extracted from this case is well separated from the ones associated with others cases, which ensures correct classification. The experimental outcomes validate the analytical results presented in the previous sections as both indicate negligible misclassification for Case 1.

However, the optimized network (2, 8), which was selected through Phase II, exhibits faultless performance in the sense that there are no associated misclassifications. This shows that using the proposed methodology, we are able to detect and classify the common damages and also design an optimized network of sensors with a minimum probability of misclassification.

5 Summary and conclusions

A hybrid framework was proposed in this study for continuous real-time performance assessment of a structure. The framework integrates non-parametric and model-based data interpretation approaches. The main contribution of this study is that this framework leverages the benefits of both approaches while mitigating their shortcomings. In contrast to existing algorithms, which were either parametric or non-parametric, the proposed algorithm can be employed not only to detect the damage (non-parametric algorithms) but also to assess the identified abnormal behavior (parametric algorithm). Furthermore, this framework allows for determining the number of sensors, as well as their corresponding locations, required to effectively monitor a structure. In particular, the sensor network is optimized so that the collected information can be ultimately used for PBE.

A supervised classification algorithm consisting of three phases, Phase I (study phase), Phase II (training phase), and finally Phase III (monitoring phase) is proposed. During Phase I, a comprehensive study is conducted on a given structure to decide the corresponding critical damage scenarios. Upon identifying the corresponding damages scenarios, an FE model of the structure is developed to extract and learn the relevant performance criteria of the structure using Monte-Carlo simulations (Phase II) under these different scenarios. In addition to detecting and classifying the damage, another objective is to optimize the number of sensors and their locations. A data-driven method, Moving Principal Component Analysis (MPCA), is implemented to process and extract the corresponding performance-sensitive features for individual damage scenarios. Then, the underlying distributions of the extracted features are determined to design a classifier based on hypothesis testing. In phase III (the monitoring phase), features are extracted from live SHM data and fed into this classifier. With this approach, the binary hypothesis testing is carried out with the Receiver Operating Characteristic (ROC) curve to evaluate the performance of the decision rule. The ROC curve presents the fraction of true positive out of positives (TPR = true positive rate) against the fraction of false positive out of negatives (FPR = false positive rate) at various threshold settings, providing a better characterization of the findings. The real-time classification of these features determines the current performance condition of the structure, which potentially leads to appropriate maintenance actions.

The efficiency of the proposed methodology is verified through an experimental study on a 4-Span Bridge and an in-house developed FBG system. The SHM data are generated under the same damage scenarios used in the analytical part, and fed into the third phase of the algorithm to study the classification performance of the proposed algorithm. The results are shown to be consistent with the results of the analytical phase. The optimized sensor network configuration, which was designed through Phase II, demonstrates perfect performance for classifying the different damage scenarios. As a result, it is shown that practical and efficient data-driven methods can be utilized to interpret data using the proposed hybrid approach. This framework leverages the benefits of data-driven methods, while mitigating their limitation with regard to data interpretation for more detailed understanding of structural conditions for decision-making.

As part of the research development, the authors are planning to employ the proposed hybrid framework for real-life applications and further investigate the efficiency of the proposed algorithm. Although the efficiency of the family model technique (parametric part of the framework) and that of the MPCA algorithm (the non-parametric part of the framework) have been individually tested and reported in previous studies using real-life data, there is need for testing the hybrid framework utilizing real data.