Estimators for Structural Damage Detection Using Principal Component Analysis
, Alejo MartÃn 1, Yeudy F. Vargas-Alzate 1, Ramon Gonzalez-Drigo 2 and Jaume Clapés 1
Round 1
Reviewer 1 Report
In this paper, a nondestructive testing method for structural damage based on dynamic tests is investigated. The authors conducted through a large number of experiments, which basically can verify the conclusions of this paper. However, the innovation of this study is not clearly stated in the manuscript. The detailed review comments are as follows.
1. It is suggested that the significance of T2 and Q should be explained in the abstract.
2. The PCA-based nondestructive testing has been more mature, and it is suggested to state the research idea of this paper in the introduction, and explain where the innovation of this paper is compared with the previous literature.
3. pages 8 lines 358-366, it is suggested to present the situation that wants to be described in the form of a table.
4. The resolution of Figure 5 is low.
5. It is suggested to compare with other PCA-based assays to highlight the superiority of the method in this paper.
Author Response
Dear reviewer,
Thank you for your comments and suggestions.
- It is suggested that the significance of T2 and Q should be explained in the abstract.
We have changed the abstract to include a brief description of the significance of T2 and Q estimators maintaining the maximum abstract length of 200 words:
Abstract: Structural damaged detection is an important issue in conservation. In this research, principal component analysis (PCA) has been applied to the temporal variation of modal frequencies obtained from a dynamic test of a scaled steel structure subjected to different damages and different temperatures. PCA has been applied in order to reduce, as much as possible, the number of variables involved in the problem of structural damaged detection. The aim of the PCA study is to determine the minimum number of principal components necessary to explain all the modal frequency variation. Three estimators have been studied: T2 (the squared of the vector norm of the projection in the principal component plan), Q (the squared of the norm of the residual vector), and the variance explained. In the study, the results related to the undamaged structure need one principal component to explain the modal frequency variation. However, the high damaged configurations, need five principal components to explain the modal frequency. The T2 and Q estimators have been arranged in order of increasing damage for all the performed experimental tests. The results indicate that these estimators could be useful to detect damage and to distinguish among a range of intensity of structural damage.
- The PCA-based nondestructive testing has been more mature, and it is suggested to state the research idea of this paper in the introduction, and explain where the innovation of this paper is compared with the previous literature.
Thank you. This is an important point that we have not emphasized enough. For this reason, we have added two paragraph in the text. The first one at the end of the introduction “It is worth mentioning that the main idea of this article is that only knowing the minimum number of the principal component of the Q and the T2 to explain the modal frequency variation, it could be able to detect a high structural damage. This is an interesting idea because it is not necessary to know the undamaged configuration as most of the previous studies need.”
And the second paragraph in conclusions: “Moreover, it seems to be possible to predict a high damaged configuration, only knowing the minimum number of the principal component of the Q and the T2 to explain the modal frequency variation.”
- pages 8 lines 358-366, it is suggested to present the situation that wants to be described in the form of a table.
Thank you, the suggestion. We have added the table in the paper, actually it is clearer.
- The resolution of Figure 5 is low.
Thank you. We have improved all figure resolutions.
- It is suggested to compare with other PCA-based assays to highlight the superiority of the method in this paper.
This point is related with point 2. The superiority of this method is connected to the innovation of it.
Reviewer 2 Report
The paper analyzes the possibility to use principal components analysis to reduce the number of variables of the structural damage detection problem. To this aim, experimental tests on a reduced steel structure were performed. The structure was tested under different conditions of temperature and damage during which EMA were performed to obtain frequencies in each configuration. The data obtained were used for the PC analysis. The authors found that the estimators Q and T2 could be useful to detect and quantify the damage.
The paper is clear, and the main ideas are well developed. Nevertheless, some clarifications are required to improve the quality of the paper before publication.
Some comments are listed in the following:
- In the paper there are many typos, please reread it carefully. ( e.g. in the first paragraph of section 2.1:"non- linear" instead of "non-linear", "allow" instead of "allows", " "calculate" instead of "calculated", the phrase "is a type of synthesis of information" seem that need a point before instead of a comma)
-Please, improve also these figures:
-Figure 2: figure 2.a is difficult to read. I suggest to realized a photo composition of each element used: heaters, balls, structure ( I suggest to try to add the nodes number on it)
-Figure 3b: use different colors to distinguish the longitudinal by transversal accelerometers arrows, and please add a legend instead additional arrows
-Figure 4: Figure 4.a Please, add the temperature legend. Figure 4.b, improve the resolution, the text is illegible.
-Please, in figure 7a use the same thickness for all the curves
-Please, in Figure 8.a use the same formatting as other graphs.
-Please, when you recall an Equation wite Eq. "N°" and at a section with Section "n°". (e.g. in section 3.4 instead of "For each of the damage test, the T2 value is computed as explained above (5)) use "For each of the damage test, the T2 value is computed as explained in Section 2.2 Eq.(5)"
Author Response
Dear reviewer,
Thank you for your comments and suggestions.
- In the paper there are many typos, please reread it carefully. ( e.g. in the first paragraph of section 2.1:"non- linear" instead of "non-linear", "allow" instead of "allows", " "calculate" instead of "calculated", the phrase "is a type of synthesis of information" seem that need a point before instead of a comma)
Sorry for the mistakes. We have checked all the paper and we have corrected the mistakes.
-Figure 2: figure 2.a is difficult to read. I suggest to realized a photo composition of each element used: heaters, balls, structure ( I suggest to try to add the nodes number on it).
Thank you for the suggestion. The new added figure is clearer than the previous one and also the nodes number are added.
-Figure 3b: use different colors to distinguish the longitudinal by transversal accelerometers arrows, and please add a legend instead additional arrows
Thank you. This change made the picture easier to understand.
-Figure 4: Figure 4.a Please, add the temperature legend. Figure 4.b, improve the resolution, the text is illegible.
Sorry for forgetting the temperature legend. Thank you. We have added the legend. We have also improve the picture resolution.
-Please, in figure 7a use the same thickness for all the curves.
Sorry. The problem of figure 7a is the lack of resolution. The test 1 and 2 are overlapped (as it can be seen in figure 7b) and, we have solved the problem joining test 1 and 2 in the same line.
-Please, in Figure 8.a use the same formatting as other graphs.
Thank you. This change homogenizes the figures in the paper. We have also changed figure 5 and 6 for this reason.
Author Response File: 
Author Response.docx
Reviewer 3 Report
This study investigated the application of three estimators to structural damage detection using principal component analysis (PCA). Dynamic tests were conducted on a scaled steel structure under different damaged conditions and PCA was used to analyze the variation of modal frequencies. Variations of the three estimators with damaged conditions were displayed and discussed. It is a practical and interesting study. However, it is not clear how could the estimators be used in practical application. Some questions and comments are given as follows.
1. The main question of this manuscript was the quantitative relationship between the three estimators and the structural damage. From Figs. 5, 6, 7 and 8, it appeared that the three estimators did change with the damaged conditions. However, the principal component (PC) number was also involved in the relations. It is confusing how the proposed estimators can be applied to detect the damage of an existing structure.
2. Table 2 shows thirteen tests and ten modal frequencies. However, in Section 3.1, it explained there were thirteen modes, which was conflicted to the table lists.
3. In the manuscript, the decimal points were represented by comma. This may lead to misunderstanding.
4. In Table 1, it shows the temperature was changed within an around 10 degree range for the first four tests. Did the temperature change have any effect on the results shown in Fig. 5, 7, and 8? The temperature effect was not explained in the result sections.
5. The connections between the measured frequencies and the three estimators should be explained in more details. It is difficult to understand how the measured frequencies were used in the calculation of the three estimators.
Author Response
Dear reviewer,
Thank you for your comments and suggestions.
- The main question of this manuscript was the quantitative relationship between the three estimators and the structural damage. From Figs. 5, 6, 7 and 8, it appeared that the three estimators did change with the damaged conditions. However, the principal component (PC) number was also involved in the relations. It is confusing how the proposed estimators can be applied to detect the damage of an existing structure.
Thank you for the question. We have realized that we didn’t explain better enough the main idea of the paper. For this reason, we have added a paragraph in the introduction section:
“It is worth mentioning that the main idea of this article is that only knowing the minimum number of the principal component of the Q and the T2 to explain the modal frequency variation, it could be able to detect a high structural damage. With this methodology, damages are detected looking for the inflexion point of the graph be-tween estimators versus PC used (Caselles et al., 2021). As much principal components are needed as much damaged the structure is. This is an interesting idea because it is not necessary to know the undamaged configuration as most of the previous studies need.”.
We have also changed the sentence:” These steps are repeated for r=1, 2, … (one principal component, two principal com-ponents, ….)” for “These steps are repeated for r=1, 2, … (one principal component, two principal com-ponents, ….) in order to obtain a graph with the estimator values versus PC number used.”
And finally, we have added a sentence in conclusion:” Moreover, it seems to be possible to predict a high damaged configuration, only knowing the minimum number of the principal component of the Q and the T2 to explain the modal frequency variation¨
- Table 2 shows thirteen tests and ten modal frequencies. However, in Section 3.1, it explained there were thirteen modes, which was conflicted to the table lists.
Sorry in the old test was not clearly explained. We have changed the paragraph in order to clarify the table. In fact, for presentation reasons we don’t include the upper frequencies.
Old text: “The ESD of the different experiments presents similar pattern. In the spectrum, the peaks are selected to create the matrix X for each test where the PCA theory is ap-plied. Table 2 shows the frequencies for ten first principal components (PC) related to the test performed on the undamaged configuration of the structure. Only the first 13 modal frequencies were chosen because upper frequencies show clear non-linear behavior. For each damaged test, the selected modal frequencies are ordered in the same way for all temperature test. In the matrix X, each column are the frequencies modes from 1 to 13 and each row correspond to tests with different temperature.”
New text:¨ The ESD of the different experiments presents similar pattern. In the spectrum, the peaks are selected to create the matrix X for each test where the PCA theory is applied. In our study, 13 modal frequencies have been correctly detected by the peak picking method for all damaged and undamaged configurations. Table 2 shows the frequencies for only the ten first modes and the 13 temperature tests for the undamaged configuration of the structure. Only the first 13 modal frequencies were chosen because upper frequencies show clear non-linear behavior. For each damaged test, the selected modal frequencies are ordered in the same way for all temperature test. In the matrix X, each column are the frequencies modes from 1 to 13 and each row correspond to tests with different temperature.”
- In the manuscript, the decimal points were represented by comma. This may lead to misunderstanding.
Sorry for the mistake. They are all changed in the new text
- In Table 1, it shows the temperature was changed within an around 10 degree range for the first four tests. Did the temperature change have any effect on the results shown in Fig. 5, 7, and 8? The temperature effect was not explained in the result sections.
Thank you for the question. We have realized that we don’t explained enough the importance of assure enough range of temperature to properly apply the methodology. In order to clarify this point we have added a new sentence in section 3.2:” To properly apply the proposed methodology, it is needed enough temperature range in order to assure the frequency variations are higher than frequency resolution. Otherwise, the variance obtained by PC analysis mainly reflect the random error produced by the lack of resolution.”
- The connections between the measured frequencies and the three estimators should be explained in more details. It is difficult to understand how the measured frequencies were used in the calculation of the three estimators.
Thank you for the remark. We have added a new paragraph at the end of section 2.2 that clarify this question:
“In practice, X matrix is composed of all the modal frequencies (in rows) obtained for each temperatures test (in columns). Appling PCA to X matrix, it is obtained the eigenvectors and eigenvalues. P matrix is made with only the r first PC eigenvectors choose. For each dimension of the space (r) a P matrix and, Q and T2 estimator values are obtained. Where r varies from 1 to the maximum number of eigenvectors.”
Author Response File: Author Response.docx
Round 2
Reviewer 1 Report
This article addresses the review comments with reasonable improvements. Although there have been many studies on PCA-based damage detection, this article still has its innovative features. Therefore I would like to accept it.
Reviewer 3 Report
No other comments
