3. AE Signal Analysis Method
This section describes in detail the acoustic emission analysis
methodology and the k-means clustering algorithm used in
this study.
3.1. Selection of AE Characteristic Parameters
This study, considering the inherent properties of concrete,
selected 14 AE feature parameters to analyze the damage
signals of concrete under tensile stress. These parameters
include ring count, energy, absolute energy, signal strength,
amplitude, average signal level (ASL), root-mean-square (RMS)
voltage value, rise time, duration, initial frequency, average
frequency, inverse frequency, center frequency, and peak fre-
quency. The AE feature parameters and their corresponding
meanings are shown in Table 2.
3.2. K-Means Clustering Analysis Classification Method
Cluster analysis automatically divides large amounts of data
into multiple categories based on their internal similarity. It
does so without relying on any specific classification criteria,
instead organizing the data based on inherent similarities [24].
The k-means clustering algorithm is designed to partition large
data samples according to their similarities. It offers advan-
tages such as good clustering results and fast convergence
[25], providing a new approach for finding the best method
to classify AE signals, as expressed in the following formulaic
representation.
First, the optimal number of clusters, K, for cluster analysis
is determined using the Davies-Bouldin (DB) criterion [23],
hereinafter referred to as the DB coefficient. The formulaic
definition of the DB coefficient is as follows: ​ ​
(1)​ DB =​ 1
K ​ ​
∑
i=1​
K ​ ​
max​
i≠j​ ​ ​ {​​
di​​​ +​ dj​​​ _​
D​ij​​ ​ }​​​
where​​
di​​​ ​ and ​​ ​ j​​​ represent the average intra-cluster distances for
cluster ​ ​ and cluster ​ ​ respectively, and ​ ​
Dij​​​ ​ indicates the average inter-cluster distance between
cluster ​ ​ and cluster ​ ​ .
Therefore, when the DB coefficient reaches its minimum
value, it indicates that the average intra-cluster distance is
minimized while the average inter-cluster distance is max-
imized, suggesting that the clustering result is optimal. The
corresponding number of clusters, ​ ​ is then regarded as the
optimal number of clusters.
Let the ​ ​ cluster centers be denoted as ​​ ​ 1​​​ ​​ ​ 2​​​ ...,​​ ​ k​​​ The
number of samples contained in each cluster is ​​ ​ 1​​​ ​​ ​ 2​​​ ...,​​ ​ k​​​ ,
and the clustering content in each cluster is ​​ ​ 1​​​ ​​ ​ 2​​​ ...,​​ ​ k​​​ Using
squared error ​ ​ as the objective function, the optimal clustering
result is obtained when ​ ​ is minimized. By calculating the sta-
tionary points of ​ ​ to minimize it, the clustering information ​​ ​ j​​​
for each cluster can be determined as follows: ​ ​
(2)​ J(​u​1​​, ​ ​ u​2​​, …,u​k​​)​ ​ =​ 1
2 ​ ​
∑
j=1​i​=1​
K ​ ​ ∑
N​j​​ ​ ​ ​ (​​ ​ x​i​​ − ​ u​i​​​)​​​​2​​​​ ​ ​
(3)​ ​ ∂ j
∂ ​ u​j​​​
=− 2​∑(​x​
i=1 ​ ​
N​j​​ ​ ​
i​​ − ​ u​i​​)​​​ ​ ​ ​
(4)​ uj​​​ =​ 1
N ​ ​
∑
i=1 ​ ​
N​j​​ ​ ​
x​i​​​​
According to the results of the two-dimensional clustering
of the 14 AE feature parameters in this study, it can be deter-
mined that the number of clustering combinations obtained is
105. To select the optimal clustering combination among them,
this study employs the coefficient of variation [26] to evaluate
the horizontal effectiveness of these cluster combinations, as
illustrated in Equation 5:
ME
|
AXIALTENSION
TA B L E 2
Acoustic emission (AE) characteristic parameters and their meanings
AE characterization parameters Meaning
Ring count Reflects the strength of the AE signal and can evaluate AE activity
Absolute energy, signal strength, energy Reflects the energy of the AE signal
Amplitude Reflects the strength of the AE signal and can help identify
different wave source types
ASL, RMS Evaluates the activity of AE signals and helps identify
the background noise level
Rise time Evaluates the activity of AE signals and helps identify
the background noise level
Duration Distinguishes between different types of AE wave source signals
Initial frequency, average frequency,
inverse frequency, center frequency Reflects the frequency of the AE signal
Peak frequency Distinguishes between different AE sources
44
M AT E R I A L S E V A L U AT I O N • M AY 2 0 2 5

(5)​ S =​ ​
√ _​_____________
​1
N ∑ ​i=1​ Nj​​​ ​ (x​ − ​ u ____________​​​​​j​​​​)​​j​​i​​​
uj​​​ ​ ​
where ​ ​
u​j​​​ represents the clustering center for each cluster, ​ ​
x​i​​​ represents the clustering content for each cluster, and ​ ​
Nj​​​​ represents the number of samples contained in each cluster.
When the intra-cluster coefficient of variation is minimized
while the inter-cluster coefficient of variation is maximized, it
suggests that the clustering combination has a high degree of
concentration within clusters and a high degree of dispersion
between clusters, making it an optimal clustering combination.
3.3. Correlation Analysis of AE Characteristic
Parameters
Correlation analysis is used to examine the relationship
between two variables, allowing for a self-similarity test of the
AE feature parameters discussed in this study [27]. This can
help verify the feasibility of the clustering analysis results. The
metric used to evaluate the correlation between two variables
is the correlation coefficient ​​ ​ xy​​​ as shown in Equation 6: ​ ​
(6)​ ​ rxy​​ ​ =​ ​ ∑ i=1​(​xi​​​ n ​ ​ − ​
_
​ )​​(​yi​​​ − ​
_
​ )​​ _______________​
√
___________________​
∑ i=1​(​xi​​​ n ​ ​ ​ − ​
_
​ )​​​ 2​​​ (​yi​​​ − ​
_
​ )​​​ 2​​​​​
When the absolute value of the correlation coefficient is
greater than 0.5, it indicates a strong correlation between the
two corresponding parameters, suggesting that the clustering
analysis results are reliable.
4. Results and Analysis
The analysis of axial tensile damage signals in concrete struc-
tures will be conducted in two distinct components.
4.1. K-Means Clustering Analysis of AE Signals
The concrete axial tension damage AE signals were analyzed
by k-means clustering as follows.
4.1.1. OPTIMIZATION OF CLUSTER NUMBER K
Based on experience, this study sets the value of the number of
clusters, ​ ​ to range from 2 to 10, and uses the Davies-Bouldin
(DB) criterion to evaluate the quality of each ​ ​ The DB coef-
ficients corresponding to different values of ​ ​ are shown in
Figure 4. When ​ ​ =3, the DB coefficient is at its lowest, indicat-
ing the best clustering effect. Therefore, the optimal number of
clusters for the concrete data is three.
4.1.2. EFFECTIVE CLUSTERING COMBINATIONS
AND RELATED ANALYSIS
Using ​ ​ =3 as the basis for classification, the k-means clus-
tering algorithm was applied to AE signals from axial tensile
damage in concrete. After data iteration and classification, as
described in Section 3.1, 14 parameters can be combined in
pairs to obtain 105 clustering results. By discriminating the
aggregation properties of the results, five sets of clustering
results with clear classification features were selected. The
effective cluster combinations are shown in Figure 5.
From Figure 5, it can be observed that the k-means cluster-
ing results for specimens C1, C2, and C3 exhibit a high degree
of similarity, both in terms of graphical distribution and the
delineation of each cluster, which confirms the accuracy of
the clustering results. Additionally, it is evident that, for the
inter-cluster parameters, there is little overlap between differ-
ent clusters, demonstrating clear differences between them.
As for the intra-cluster parameters, the clustering centers are
located at the center of the signals within the cluster, indicating
a strong clustering effect. Based on this, we can conclude that
these five sets of clustering results are quite satisfactory.
To explore the reasons why these five sets of parameter
clustering results are considered ideal, it can be seen that the
selected clustering parameters have strong correlations among
themselves, indicating inherent self-similarity within the data.
This provides a solid foundation for the cluster analysis. The
study conducts correlation analysis on the AE parameters
involved in the five clustering results. The absolute values of
the average correlation coefficients obtained from the correla-
tion analysis of C1, C2, and C3 are presented in Table 3.
From Table 3, it can be observed that the correlation coef-
ficients between the corresponding parameters in all five
clustering combinations are greater than 0.5, indicating the
self-similarity among the corresponding parameters in the
selected five clustering results. Through self-organizing and
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
2 3 4 5 6
Number of clusters K
7 8 9 10
Figure 4. The Davies-Bouldin (DB) coefficient of concrete under different
numbers of clusters (K).
TA B L E 3
Absolute value of average correlation coefficient among AE
characteristic parameters of C1–C3 specimens
Ring count ASL Center frequency Peak frequency
Ring count 1 0.745 0.858 0.598
ASL 0.745 1 0.546 0.624
Center frequency 0.858 0.546 1 0.142
Peak frequency 0.598 0.624 0.142 1
M AY 2 0 2 5 • M AT E R I A L S E V A L U AT I O N 45
DB
coefficient