1 Introduction

Concrete is a heterogeneous composition due to cement, water, fine and coarse aggregates. Most global structures are constructed with concrete, either with plain concrete, reinforced concrete, or pre-stressed concrete. As a result, it is crucial to keep track of the performance of concrete, which is used to construct critical civil structures like bridges and buildings. Due to environmental conditions, deficient design, overloading, etc., concrete deteriorates with time. Because concrete is a heterogeneous compound, it is difficult to conduct damage and strength estimations. In general, structural engineers are unconcerned about a structure's post-construction behaviour. But, the monitoring of structures through appropriate instrumentation is necessary to perceive adverse changes in the performance or state of structural elements. Structural health monitoring (SHM) deals with recording the responses of a structure and evaluating the damage or deterioration that may affect serviceability and safety based on the recorded responses [1]. Non-destructive testing (NDT) methods like rebound hammer, penetration techniques, ultrasonic pulse velocity technique, resonant frequency technique, etc. are in use for the quality control of concrete structures. These methods have become popular because such tests do not affect the appearance, strength, and functionality of structures. Such tests continuously monitor the structures and determine possible variations during the time. NDT techniques have the objective of detecting the existence of damage to the structure’s well-being [2,3,4]. Even though NDT methods are often used to test concrete structures without damaging them, they have some major drawbacks, such as equipment that is expensive and big, wiring systems that are hard to set up, and limited access to some parts of structures, like foundations.

The researchers have carried out many NDT techniques under SHM to monitor and assess structural strength during its lifetime. One promising SHM method is the EMI technique, which is based on smart materials such as piezoelectric transducers (PZT). The superlative thing about PZTs as sensors is that they can be used to check on concrete structures anywhere, even in places that are hard to get to. The EMI technique is typically less expensive than many NDT tests. The merits of the PZTs are that they are small, lightweight, low cost, stable over the long term, quick to respond, and easy to apply. The EMI method works on the principle that surface charges are formed in piezoelectric materials as a result of mechanical deformations, and they also experience mechanical deformations when electric fields are applied across them. Therefore, PZT materials can function as both an actuator and a sensor. The EMI technique is a method that can detect changes in structural properties. Due to this ability, its range of implementation is extensive. The EMI method has been implemented in various areas, including the detection of damage on concrete structures [5, 6], corrosion of metallic structures [7, 8], and concrete strength predictions [9,10,11]. Liang et al. proposed the electro-mechanical interaction of a one-dimensional (1D) structure, surface bonded by a PZT patch [12]. The "effective impedance" approach allowed the EMI approach to be expanded from one-dimensional (1D) to two-dimensional (2D) structures [13]. The EMI methodology was first used on single-lap joints constructed of aluminium that were spot welded [14]. Previously, the EMI's implementation was limited to small structures [15]. Using an equivalent stiffness parameter (ESP)-based approach, the EMI technique was implemented to access damage in considerably larger real-life steel joints. The double-lap shear joints were made with high-strength friction grip (HSFG) bolts, and the researchers used surface-bonded PZT sensors to assess the joints. The test pieces were made of two mild steel plates with measurements of 100 × 200 × 5 mm3 that were attached to one another on either side by a 16-mm-diameter HSFG bolt of grade 10.9 that had been pretensioned to 0.4 fy of stress. The plates overlapped by 60 mm and were joined at their far ends by 75 × 75 × 10 mm angles to base plates that were 16 mm thick. In order to accurately measure fatigue-induced damage in bolted double-lap joints and estimate the joint's remaining serviceable life, a novel method using the equivalent stiffness determined by surface-bonded PZT patches was presented in the article [16]. EMI has proven to detect cracks and local damage to various structures. With the aid of a PZT sensing system based on the EMI approach, cracks in RC beams were detected [17, 18]. A network of PZT sensors was used to assess the damage of a concrete beam under shear/flexural monotonic and cyclic stress, as highlighted by researchers. The damages were assessed statistically using the RMSD statistic method [19]. The damage severity of a concrete beam was determined by means of the cross-correlation coefficient (CC) index based on the EMI data [20]. The electro-mechanical-based damage detection was carried out with the test data attained from a RC model of a two-storeyed portal frame subjected to base vibrations [21]. Researchers conducted concrete reinforcing bar damage evaluation experiments using piezoelectric transducers [22]. To assess the artificial damage to the longitudinal steel bars, an integrated strategy based on guided wave propagation and electrometrical admittance methodology was studied. RMSD index for quantification of the damage was used [23]. Damage detection of concrete slabs was carried out with piezoelectric patches attached to their surfaces. RMSD and correlation coefficient deviation (CCD) damage indices were used to detect the structural damage [24]. PZT transducers were attached to the RC beam’s steel reinforcement bars. The tested beams were then subjected to flexural monotonic and cyclic loadings, and damage assessment was done using the EMI approach at different loading levels. The sensitivity of the monitoring system was demonstrated by comparisons of the response signals from the PZTs in the undamaged and damaged stages [25]. SHM of RC beams was performed using embedded transducers as smart aggregates in concrete. RC beams were tested to failure and examined for cracks. According to test results, transducers near the key diagonal crack produced audible differences between admittance responses at undamaged and damaged stages. Using RMSD and CC calculated from measurements taken by the transducers, the damage was assessed [26]. One of the most studied applications of the EMI technique is the detection of structural cracks. Experimental investigations have also been carried out to detect damage of different materials like metal, and composites [27,28,29,30,31,32].

PZT actuators emit waves that carry information about the host structure and can thus be used to assess the presence of any damage. The waves are generated by PZT at high frequencies ranging from 30 to 400 kilohertz (kHz) [10, 33], to provide high piezoelectric sensitivity to incipient structural damage. The generated stress wave's wavelength is smaller than the normal size of the defects to be discovered [34,35,36]. The use of high frequencies is necessary because the ambient noise and vibrations are immune to high frequencies that are commonly present in practical situations. The identification of changes occurring within the host structures is ensured by a wide range of frequencies that are used to excite the PZT. But it also restricts the sensing area to the vicinity of the PZT. The PZT that is excited at high frequencies induces ultrasonic vibrational modes of the structure that are essentially local, thus leading to a constrained sensing zone [37].

The EMI method makes use of high frequencies with a magnitude of kHz in its range, due to which it possesses great damage sensitivity for localised damage of various structures and systems [14, 38,39,40,41]. The EMI technique's frequency range for activating the PZT has an impact on the PZT transducer's sensitivity and sensing range. A selection of a high range of frequencies makes the actuation and sensing ability of the PZT localized. Some researchers worked on the sensitivity of the PZT on different structures [42,43,44]. A PZT patch's sensing range has been found to range from 0.4 m in composite materials to 2 m in metallic materials [45]. A surface-bonded PZT patch on concrete materials had a sensing radius of 0.7–0.9 m, while an embedded PZT patch in concrete had a sensing radius of 0.4 m [46]. The sensing zone changes depending on the host structure's material characteristics, boundary constraints, and geometries. Identifying the real sensing region of a PZT patch on concrete is still a challenge [47, 48]. More research needs to be done to find out where and how many PZTs are optimal for each structure.

The RMSD, CC, and CCD are the metrics that determine whether two impedance spectra are similar for a specific frequency range. After receiving the response of PZT in undamaged and damaged stages, the responses are compared using damage statistics (RMSD, CC, and CCD) to detect the existence of damage. These damage indices have an interlinking relationship with the damage, so the damage indices obtained from the EMI data can be considered to evaluate the sensing zone of the sensors. It should be noted that the location of the PZT sensor has a significant impact on the PZT impedance signature. The PZT’s sensitivity reduces as the distance between the damage and the PZT widens. Various researchers worked and studied the implementation of the EMI method to assess damage to various structures. However, research into the determination of the sensing area of a PZT has gotten little attention. Because there has been little research on the sensitivity of a PZT, it is required to determine the sensing area of a PZT transducer before implementing a PZT-based EMI approach. The research in this paper deploys three PZT transducers at three distinct points along the RC beam. The beam is subjected to loading through a Universal Testing Machine (UTM). The response from all three PZT patches was recorded. After the quantitative analysis of the recorded signatures based on RMSD index, the effective sensing range and sensitivity of the PZT patch towards crack were determined using sensitivity analysis based on the variance decomposition (VD) model.

2 Theoretical Background of Coupled Electromechanical Admittance

EMI is based on the dual effect of PZT, where it converts electrical signals into mechanical actuations and vice versa. Typically, the PZT patch of length l and thickness h, is surface bonded to the host structure. It is adhered to the surface with an epoxy adhesive. The PZT is excited under harmonic voltage, and surface waves are generated radially outwards from the patch because of the excitations. The waves then return to the PZT and are recorded by an impedance analyser or an impedance, capacitance, and resistance (LCR) metre in the form of electrical admittance. Electromechanical interaction between the PZT and the structure generates the admittance signature. The signature contains information about the mechanical impedance of the host structure. Any change in the host structure's mechanical impedance changes the signature. The expression of electromechanical admittance (\(\overline{Y}\)) across the electrodes of a surface bonded PZT has been proposed [49] to be complex in nature, that is formed by a real part called conductance (G) and an imaginary part called susceptance (B), is as follows:

$$\begin{aligned} \bar{Y} &= \frac{I}{V}= G + iB \\ & = 4\omega j\frac{{l^{2} }}{h}\left[ {\overline{{\varepsilon _{{33}}^{T} }} - ~\frac{{2~d_{{31~~}}^{2} \overline{{Y^{E} }} }}{{\left( {1 - \mu } \right)}} + ~\frac{{2~d_{{31~~}}^{2} \overline{{Y^{E} }} }}{{\left( {1 - \mu } \right)}}~\left( {\frac{{Z_{{a,{\text{eff}}}} }}{{Z_{{s,{\text{eff}}}} + ~Z_{{a,{\text{eff}}}} }}} \right)\bar{T}~} \right] \end{aligned}$$
(1)

where ω = angular frequency of excitation; \(\overline{{\varepsilon _{{33}}^{T} }} =\) complex dielectric permittivity, \(\overline{{Y^{E} }}\) = complex young’s modulus, \(\mu\) = Poisson’s ratio of the PZT patch, Za,eff = impedance of the patch, Zs,eff = impedance of the structure.

Zs,eff considers 2D relations of the PZT with the structure. Zs,eff comprises of the structural parameters of the structure, the stiffness (k), the mass (m), and the damping (c). Damage to the structure in any form and during the life of the structure causes variations in the k, m, and c parameters, i.e., alters the Zs,eff of the host structure. The local loss of stiffness that develops close to the damage's vicinity is detected by EMI response, which is utilized to detect the damage. In Eq. (1), keeping the properties of PZT constant, any variation in Zs,eff will modify the EMI response, and alteration in the response indicates modification to the mechanical parameters of the host structure like k, m, and c. Any damage to the structure causes the conductance part to deviate more than the susceptibility part [2, 50]. Because of the greater sensitivity to mechanical changes and reduced vulnerability to environmental impacts, the G signatures of the admittance are the most frequently used to detect the damage in the structure. The G signatures show variations according to structural damage, and the B component varies with the PZT fault [51]. This research paper considers the shifts observed in the G spectrum of the electro-mechanical interaction in the identification of damage to the RC beam.

3 Quantification Index for Damage Detection

The structural characteristics of the host structure get modified due to the presence of damage, leading to variations in the conductance spectrum. Therefore, metrics are needed to quantify the damage after acquiring the conductance signatures. Use of damage indices is recommended, which compares the conductance signature in the undamaged state with the signatures in the damaged state. Researchers have proposed many metrics. The most common damage quantification index is the RMSD. The RMSD index [52] is defined as:

$${\text{RMSD}}\left( \% \right) = \sqrt {\frac{{\mathop \sum \nolimits_{{i = 1}}^{N} \left( {G_{i}^{1} - {\text{~}}G_{i}^{0} } \right)^{2} }}{{\mathop \sum \nolimits_{{i = 1}}^{N} \left( {{\text{~}}G_{i}^{0} } \right)^{2} }}}$$
(2)

where, \(G_{i}^{0}\) = Conductance signature in undamaged state, \(G_{i}^{1}\) = Conductance signature in damaged state.

RMSD is based on a formulation that sees the two conductance signatures over a given frequency range as being similar or not.

4 Sensitivity Analysis Based on the Variance Decomposition (VD) Approach

VD is a statistical approach that is used to analyse the relative importance of input variables on the output of a model. The most important input variables are identified, and their contributions to the model’s output are measured using the VD approach. The basic idea behind the VD is to break down the variance of the model’s output into parts that are linked to input variables or groups of input variables. This is done by estimating the partial variances of the output for each input variable and the interactions between the different input variables.

VD is founded on a probabilistic outline that divides the variance of the experimental model's output into portions that can be assigned to inputs. A model with three inputs and one output, for example, might reveal that the deviation in the first input contributes 70% of the variance in the output, the deviation in the second input contributes 20%, and the remaining 10% comes from the third input. These fractions can be used to calculate sensitivity directly. The VD method is based on the idea of variance, and statistical techniques are used to estimate the input variable’s sensitivity indices. With this approach, sensitivity indices can be obtained that can be used to prioritize input variables for further analysis or optimization in the model.

This method is widely used in various fields, including engineering, finance, environmental science, etc. [53,54,55,56]. The VD method can be used with any type of model, including empirical models, computer simulations, and physical experiments.

5 Experimental Set Up

An RC beam of M35 grade [57], 700 mm in length, 150 mm wide, and 150 mm in height, was cast for the experimental investigation. The beam was cast with top bars of 8 φ 2 no.s, bottom bars of 10 φ 3 no.s, and stirrups of 8 φ @ 160 mm c/c. High yielding strength deformed (HYSD) steel was used to make the bars [58]. The details and properties of concrete, steel, and PZT are given in Table 1. Figure 1a, b, and c show the geometrical characteristics, cross section of RC beams showing reinforcement, and details of a RC beam with three surface-bonded PZTs, respectively. The RC beams were divided into square grids of 50 mm × 50 mm that go from A to N. RC beams are often divided into 50 mm by 50 mm grids and marked outside the beam for experimental purposes, such as load testing or measuring deflection. This is because the grid helps to accurately measure the behaviour of the beam under load and provides a reference for comparing the results of the experiment [59]. Figure 2 displays the full experimental setup. For the flexural test of the RC beams, the standard four-point bending setup and scheme were used as seen in Fig. 2. Simply edge-supported, the beam was placed on roller supports that were 600 mm apart from one another. In the middle of the beam’s midspan, the load was applied at two points that were 200 mm apart from one another. The LCR meter was used to extract signatures from the PZTs by connecting the positive and negative probes of the LCR meter to the positive and negative electrodes of the PZTs through wires to record the electrical response as conductance and susceptance signatures. The data were accessed on the personal computer (PC) via the LCR.

Table 1 Details and material properties
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Fig. 1
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RC beam details a geometrical characteristics b cross section of RC beam showing reinforcement. c RC beam with three PZTs

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Fig. 2
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Complete experimental setup

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Three square PZTs of grade PIC 151 [60] were surface bonded to the RC beam with epoxy adhesive. Figure 1c shows the placement of three PZTs, P1, P2, and P3. In a four-point bending test, a peak bending moment is produced along a region of the beam between the loading points, and shear force is produced between the support and the loading pin on both sides of the beam. Taking this behaviour of the four-point bending test on the RC beam into account, the P1 and P3 were placed near the support, and the P2 was placed near the centre of the beam. P1, P2, and P3 were surface bonded at the junction of grid C–D, grid G–H, and grid K–L, respectively, at a height of 50 mm from the bottom surface of the beam. At a temperature of 25 °C, the configuration of the RC beam with three PZTs was left alone for 24 h. Before starting the experiment, the LCR metre [61], extracted conductance signatures from P1, P2, and P3. The frequency ranges for P1, P2, and P3 were chosen as 130–300 kHz, 180–300 kHz, and 130–300 kHz, respectively. The obtained conductance signatures through the interaction of PZTs with the undamaged RC beam were termed as baseline signatures. The conductance signatures of the damaged beam were compared with baseline signatures. Figures 3a, b, and c show the baseline conductance signatures of P1, P2, and P3, respectively, in the undamaged state. The frequency range of P1, P2, and P3 was selected manually by seeking the peak and selecting that specific range for experimental investigation. Each impedance test was conducted using a frequency interval of 500 Hz and a voltage of 1 V. A frequency range below 400 kHz was found to be ideal for the experimental investigation of the EMI technique [62].

Fig. 3
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a P1, b P2, and c P3 baseline conductance signatures

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The RC beam was loaded using a UTM machine of model WEW-1000B, with a capacity of 1000 kN and a resolution of 0.1 kN [63]. The experiments were conducted in a load-control environment. The load started at 0 kN and was taken to 30 kN at a loading rate of 2 kN/s. At the 30 kN load level, conductance signatures of P1, P2, and P3 were measured using the LCR meter. The conductance signatures were recorded at 30 kN, 40 kN, 50 kN, 60 kN, 70 kN, 80 kN, 90 kN, 100 kN, 110 kN, 120 kN, 130 kN, 140 kN, and 150 kN, i.e., at the interval of 10 kN up to the failure load of 150 kN. Before the EMI experimental procedure, a RC beam without any PZTs bonded to it was tested to determine the ultimate load failure of the beam, and this beam was termed the baseline beam. The RC beam was tested under the standard four-point bending test. The load was gradually increased at a rate of 2 kN/s, and the RC beam was loaded until the beam’s ultimate failure. The baseline beam's failure load was around 146 kN.

The RC beam with PZTs underwent damage as cracks propagated up to the ultimate failure load level. Many flexural and shear cracks propagated through the bottom of the beam and travelled upwards vertically and diagonally. Along with the RMSD% of P1, P2, and P3, the cracking patterns of the RC beams at 70 kN, 90 kN, 110 kN, 130 kN, and 150 kN load levels are also shown in Fig. 4. Table 2 shows the experimental data on cracks propagating at load levels and in specific grids. Figure 5 shows the damaged RC beam at failure load with cracks as B, C, D, E, F, H, I, L, and M markings. The beam had shear cracks propagating from the bottom surface of the beam at load levels from 90 kN up to 150 kN near the supports and some flexural cracks propagating at load levels of 70 kN to 150 kN in the centre span of the beam. P1 and P3 were placed near the supports, closer to the shear cracks. Flexural cracks occurred at the centre where P2 was set.

Fig. 4
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RMSD% of P1, P2, and P3 and cracking patterns at different load levels of the tested RC beam

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Table 2 Experimental data of cracks along with the horizontal distance of the crack from the centre of the PZT
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Fig. 5
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Damaged RC beam at failure load with flexural and shear cracks

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Flexural cracks were observed at a load level of 70 kN propagating vertically in grids H and F, respectively. At load level 80 kN, crack I was marked in grid I. The B crack was observed to occur at 90 kN and propagated diagonally upwards. At 100 kN, C, D, and L1 all simultaneously appeared from the base of the beam. The path of the cracks was diagonal. L2 and M were observed to appear at 110 kN. Flexural cracks expanded vertically upward until they reached the failure load. Shear cracks grew diagonally from the base and extended towards the top, up to 150 kN.

In Eq. (1), as the structural impedance of the structure (Zs,eff) is linked with the \(\overline{Y}\), thus any variation in Zs,eff will modify the EMI response, and alteration in the EMI response indicates modification to the mechanical parameters of the host structure. When a host structure undergoes mechanical stress, such as from the application of a load or the formation of a crack, the piezoelectric impedance signal changes. In the case of crack formation, the change in impedance can be attributed to the altered mechanical properties of the material around the crack. As cracks propagate rapidly, it leads to large changes in the piezoelectric impedance signal.

The conductance signatures obtained from P1, P2, and P3 at various load levels were plotted as shown in Fig. 6a, b, and c, respectively. The conductance signatures, when plotted, showed variations from the baseline signatures in lateral and vertical directions, indicating a change in the host structure’s characteristics that may have occurred due to the presence of damage. Throughout the experiment, the PZTs were intact, and they had not undergone any damage. It indicates the efficiency of PZTs in the detection of damage to the beam through the variations in obtained signatures. For good health monitoring, the PZT sensors must work well. This means that they must not be damaged, as this could cause them to be misinterpreted [34].

Fig. 6
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Plot of conductance signatures obtained at various load levels from a P1, b P2, and c P3

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The damage is not quantified by the shifts in conductance signatures. The damage was quantified using the RMSD index as a damage index. Figure 7a, b, and c show the RMSD% values of P1, P2, and P3, respectively, at various load levels. Due to the symmetrical placement of P1 and P3 near the supports, both showed higher RMSD values at loads of 100 kN, indicating the identification of shear cracks. P2 being placed at the centre detected flexural cracks at 70–80 kN, as at these load levels, RMSD% values showed a sudden increase in the magnitude.

Fig. 7
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a P1, b P2, and c P3 RMSD% values at various load levels

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Experimentally, flexural cracks propagated at 70–80 kN near P2 and shear cracks at 100–110 kN near P1 and P3. EMI technique-based PZTs accurately detected cracks at 70 kN, 80 kN, 90 kN, 100 kN, and 110 kN. P1’s RMSD% values showed an increasing trend up to 90 kN and a sudden increase at 100 kN where shear cracks propagated at 90–100 kN. P2’s RMSD% values showed a sudden increase at 70 kN and gradually increased up to 100 kN, indicating the detection of flexural cracks. P3’s RMSD% values showed an increasing trend up to 90 kN. A sudden increase at 100 kN indicated the detection of a shear crack in Grid M close to P3.

P1, P2, and P3 accurately detected the presence of cracks up to load levels of 110 kN, 100 kN, and 120 kN, respectively. The EMI technique accurately detected damage growing to an average load of 110 kN. But based on RMSD% values, the EMI technique does not effectively detect damage at higher load values because, at higher load values, the corresponding RMSD% values showed irregular behaviour. The EMI approach is measured over a wide frequency range (higher than 30 kHz). This is the main distinction between the impedance method and traditional vibration-based damage detection methods. Because the wavelength of the excitation is so short at such high frequencies, it can detect incipient to moderate changes in structural integrity [64].

In a similar way, the remaining eight RC beam specimens were tested experimentally. The propagation of cracks in various grids was observed and marked. The EMI responses of all three PZTs were recorded. The failure load of RC beams of grade M35 varied from 142 to 156 kN. The cracks propagated in grids from B to M. All nine beams showed consistent results with shear cracks propagating in grids B to E and J to M near the supports and flexural cracks propagating in grids F to I. The horizontal distances of cracks from P1, P2 and P3 were noted. All the data for each crack and EMI responses from all nine beams were considered to carry out the sensitivity analysis to find the contribution of P1, P2, and P3 in detecting that crack using the variance decomposition method as explained in the next section.

6 Sensitivity Analysis-Based VD Model of PZTs in Detecting Damage

The basis of sensitivity analysis (SA) is the ability to identify, classify, and attribute various sources of uncertainty in an input to the output of a mathematical model or system. Due to the EMI model's extreme complexity, it is possible that the linkages between inputs and outputs are not well understood. There are many methods for carrying out a SA, including variance decompositions, partial derivatives, elementary effects, etc. [65,66,67,68]. This research focuses on SA based on variance decomposition. The variance-based decomposition method has several advantages over other sensitivity analysis methods. Firstly, it can handle non-linear and non-additive models, which are common in real-world applications. Secondly, it runs a comprehensive analysis of the sensitivity of the model to each input variable, which can help in understanding the behaviour of the model under various scenarios [56].

In this study, P1, P2, and P3 detected cracks propagating in grids B, C, D, E, F, H, I, L, and M. Some of the detected cracks were near P1; some were near P2, and a few were detected near P3. The crack closer to a PZT significantly changes its conductance spectrum, leading to a significant change in RMSD% value compared to the other two PZTs. The closer the PZT is to the crack, the more sensitive it will be to that crack.

As cracks spread along the beams, the conductance signatures of the P1, P2, and P3 sensors changed, and as a result, the RMSD% values of the sensors also changed. All four variables—the RMSD% values of P1, P2, and P3 and the distance of cracks from the sensors—are interlinked. This interlinking relationship between cracks and RMSD% values of sensors was considered in this research to evaluate the sensing zone of the sensors, i.e., the range up to which the sensors can detect a crack effectively. In the analysis, four variables were considered: crack, PZT patch P1, PZT patch P2, and PZT patch P3. The horizontal distance of the crack from each sensor was used as the output variable and the RMSD values of P1, P2, and P3 as the input variables. Using VD methodology, the contribution of P1, P2, and P3 in the determination of the crack was found. Variance-based analysis was carried out on data derived from nine beams in a software called RStudio. In the analysis, parameters were evaluated and run through the methodology as shown in Fig. 8. A VD contribution factor was calculated to establish the contribution of sensors to the detection of the cracks. The libraries that were used in the code to carry out the analysis are shown in Table 3 along with the function of each library.

Fig. 8
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Methodology—VD model

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Table 3 The libraries used in the formation of code to carry the variance-based sensitivity analysis
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The analysis uses a multivariate linear model called vector autoregression (VAR), in which the endogenous variables in the system are functions of the lagged values of every endogenous variable. This enables a straightforward and adaptable replacement for the conventional structural system of equations. There is one equation for each model variable. Each equation's right side contains a constant (c) as well as the lags for each variable in the model. A two-variable, one-lag VAR is considered to keep things straightforward. A 2-dimensional VAR is written as:

$$y1,\;t = c1 + \phi 11,\;1y1,\;t - 1 + \phi 12,\;1y2,\;t - 1 + e1,\;t$$
(3)
$${\text{y2}},{\text{t }} = {\text{ c2 }} + \phi {\text{21}},\;{\text{1y1}},{\text{t}} - {\text{1}} + \phi {\text{22}},\;{\text{1y2}},{\text{t}} - {\text{1}} + {\text{e2}},{\text{t}}$$
(4)

where e1,t and e2,t are white noise processes that may be contemporaneously correlated. The coefficient ϕii, ℓ captures the influence of the ℓth lag of output variable yi on itself, while the coefficient ϕij, ℓ captures the influence of the ℓth lag of variable yj on yi [69, 70].

Using VAR model, the role of PZT patches P1, P2 and P3 in the detection of crack has been established based on the data of each individual crack obtained from the testing of nine RC beams. It was quite evident from the results that the PZT patch closest to beam had higher contribution in crack detection compared to other two PZT patches. Table 4 shows the calculated VD contribution factors of P1, P2, and P3 in detecting cracks.

Table 4 The VD contribution factors of P1, P2, and P3 in detecting cracks
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From Table 4, it was deduced that cracks in grids E, F, G, H, I, and J were nearer to the P2 patch, for which its VD contribution factor was more significant than P1 and P3. As the cracks started to propagate towards the supports, i.e., towards P1 and P3 in grids B, C, D, and K, L, M, the VD contribution factor of P1 and P3 began to increase. P2's sensitivity in detecting cracks at the centre of the beam is greater than P1 and P3. The sensitivity of P1 and P3 is greater than that of P2 in detecting cracks near the support.

Cracks in grids B, C, and D situated at 54, 34, and 28 mm respectively of P1 were efficiently detected by a VD factor of 57%, 69%, and 69% respectively. Cracks in grids E, F, G, H, and I located within 56–290 mm of P1 were detected by a factor of 49%, 49%, 34%, and 20%, respectively. This is because as the crack distance increases from the patch, the ability to detect the crack decreases, thus its VD factor decreases. P2 detected cracks at the centre span in grids E, F, G, H, I, and J, which were up to 177 mm from P2 with a factor of 51%, 51%, 59%, 49%, 48%, and 55%, respectively. P2’s sensitivity in detecting E, F, G, H, I, and J was greater than P1 and P3 because P2’s placement was near those cracks. The cracks, B, C, D, K, L, and M, were away from P2. Hence, its VD contribution factor was lower than P1 and P3. P3 efficiently measured the cracks in grids K, L, and M with 60%, 66%, and 61% VD factors, respectively. P1 and P3’s symmetrical placement showed a similar trend as cracks simultaneously propagated near P1 and P3 at 100–110 kN. As the distance of cracks increases from PZT, the ability to detect the cracks decreases. The PZTs effectively detect the cracks in their vicinity up to a 200–300 mm radius. Figure 9 shows the pictorial representation of the contribution of each PZT in the identification of each crack based on the VD contribution factor. The fundamental goal of the VD model is to decompose the variance of the output of the model into components that are related to individual input variables. Individual variations in the RMSD% of the sensors account for the overall variance in the horizontal distance of the crack. The VD contribution factor determined the relative contributions of P1, P2, and P3 to the total variance of the crack parameter. This methodology helped in establishing the effective sensing zone of P1, P2, and P3 for detecting cracks based on the relation between the horizontal distance of the crack and the RMSD% values of the sensors.

Fig. 9
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Pictorial representation based on the VD contribution factor of P1, P2, and P3 in the detection of cracks

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7 Conclusions

The EMI technique effectively detects damage at the incipient level when cracks start to propagate. P1 and P3 showed an increasing trend of RMSD% values up to 110 kN and 120 kN, respectively, whereas P2 showed a growing trend of RMSD% values up to 100 kN. P1, P2, and P3 could not detect the severe damage at higher load values because the corresponding RMSD% values showed an irregular pattern beyond 120 kN and up to 156 kN. The impedance-based approach is extremely sensitive to detecting early to moderate changes in a PZT’s near field because of the high frequency range used. The EMI's sensitivity is restricted to moderate alterations. PZTs efficiently detected cracks near their sensing zone. According to VD contribution factors, P1’s contribution in detecting B, C, and D cracks was the highest. P2 detected E, F, G, H, I, and J cracks with the highest VD factors, and P3 efficiently detected K, L, and M cracks with the highest contribution than P1 and P2. The VD factor was higher when the cracks were near the PZT. As the crack moved away from the PZT, the VD factor was reduced. PZTs can efficiently detect damage up to 0.300 m in concrete. The VD model successfully calculated the influence of P1, P2, and P3 in detecting cracks lying in grids from B to M. The use of PZTs and the EMI methodology to find cracks in RC beams has been demonstrated. The PZTs were able to detect concrete cracks under the various damage levels examined. The VD model was carried out using the observed cracks and RMSD values of P1, P2, and P3 as the variables. The VD contribution factor of the more sensitive PZTs is higher than the VD contribution factor of the less sensitive PZTs when they are placed near where the fracture in the beams occurs. The benefit of the SA is to possibly get an idea of the extent of the ability to detect cracks by the PZTs. VD indices are useful because they help researchers identify the most significant sources of variability in a variable. By breaking down the total variance into its different components, researchers can gain a better understanding of the factors that are driving the variability of the variable. This knowledge can be used in future research to make more accurate predictions and develop more effective policies and interventions. The VD method can be used with any type of model, including empirical models, computer simulations, and physical experiments. The suggested method could be used to analyse RC structures at a low cost without the need for additional heavy equipment. The ability to achieve continuous monitoring with this active sensing method is another benefit of EMI sensing technology. After installing the PZT and the controlling system, this active sensing method does not require human intervention in order to continue collecting the signatures. Additionally, continuous monitoring can be accomplished using this method in conjunction with wireless technology.