## 2 Principles
### Smart Aggregate-Based Structural Health Monitoring System  
Piezoelectric materials, exhibiting piezoelectricity that can convert mechanical strain to electrical signals and vice versa, have been widely used as ultrasonic transmitters and receivers in many applications. Lead Zirconate Titanate (PZT) is one of the most popular materials among piezoelectric materials. Since PZT material is fragile and easily damaged, a smart aggregate (SA) was designed by embedding a waterproofed PZT patch with lead wires into a concrete block [39]. The schematic structure and a photo of smart aggregate are shown in Figure 1. The detailed parameters of the rectangular PZT patch is provided in Table 1. Due to the stable physical properties of concrete, the designed SAs are capable of acting as embedded transducers to perform the structural health monitoring in real-time. When SAs are installed in the reinforcing cage before concrete casting, we used wires to especially fix the SAs with the reinforcing cage. Just like an aggregate in concrete, our smart aggregate will be surrounded by the liquid cement. Both a chemical and physical bond will form between the cement and the SA during the curing process and therefore good bonding is ensured. Figure 2 gives an example to illustrate the smart aggregate-based active sensing approach for multi-crack detection in a concrete column. As shown in Figure 2, one smart aggregate functions as an actuator to generate guided waves propagating in the concrete structure. Other distributed \begin{table}
\begin{tabular}{c c c c} \hline \hline
**PZT Type** & **Parameters** & **Values** & **Unit** \\ \hline \multirow{5}{*}{PZT-5H} & Dimension & \(10\times 10\times 1\) & mm \\  & Density & 7800 & kg\(\cdot\)m\({}^{-3}\) \\  & [PERSON]’s modulus & \(46\times 10^{9}\) & Pa \\ \cline{1-1}  & Poisson’s ratio & 0.3 & - \\ \cline{1-1}  & Piezoelectric strain coefficients d\({}_{33}\) & 4.95 & \(10^{-10}\) m\(\cdot\)V\({}^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The detailed parameters of the lead zirconate titanate (PZT) patch. Figure 1: Structure of a smart aggregate. smart aggregates are used as sensors to detect the wave responses. If a crack appears in the structure, it acts as a stress relief in the wave propagation path. Therefore, the energy of the received signal from corresponding SA sensors will attenuate. Note that the energy attenuation ratio is highly dependent on the severity of the cracks. In order to detect the crack severity, a wavelet packet-based structural damage index was developed in this paper. In the wavelet decomposition process, a sensor signal \(X\) can be decomposed with \(n\)-level wavelet packets into \(2^{n}\) signal sets {\(X_{1}\), \(X_{2}\),..., \(X_{2}\)\({}^{n}\)} [44]. \(Xj\) is the decomposed sensor signal and \(j\) is the frequency band (\(j=1\)... \(2^{n}\)). \(Xj\) can be expressed in Equation (1). \[X_{j}=[X_{j,1},\,X_{j,2},\,\ldots\,\,,X_{j,m}] \tag{1}\]  
where \(m\) is the number of sampling data. Additionally, the energy of the decomposed signal \(E_{j}\) is defined by Equation (2):  
\[E_{j}=X_{j,1}{}^{2}+X_{j,2}{}^{2}+\cdots X_{j,m}{}^{2} \tag{2}\]  
According to the data measured at time \(i\), the signal energy vector at time \(i\) is given by Equation (3):  
\[E_{ij}=[E_{i,1},\,E_{i,2},\,\ldots\,\,E_{i,2}{}^{m}] \tag{3}\]  
Therefore, by using root-mean-square deviation (RMSD), a wavelet packet-based structural damage index \(I\) is given by Equation (4) [39]:  
\[I=\sqrt{\sum_{j=1}^{2^{n}}\left(E_{i,j}-E_{1,j}\right)^{2}\Bigg{/}\sum_{j=1}^{ 2^{n}}E_{1,j}{}^{2}} \tag{4}\]  
where \(E_{1,j}\) is considered as the health status of the structure. From the structural damage index, it can be found that when the values of subsequent energy vectors are close to \(E_{1,j}\), corresponding to minor or no change of the structure, the value of the structural damage index approaches 0. In contrast, when the values of subsequent energy vectors approach 0, corresponding to serious structure damage, the value of the structural damage index approaches 1.