Stochastic Finite Element Analysis of Plate Structures Considering Spatial Parameter Random Fields
Round 1
Reviewer 1 Report
1. It is not customary to use abbreviations in abstracts and conclusions.
2- A subsection should be added to the introduction and the main goals and main achievements of the article should be mentioned in it.
3- Punctuation marks should be checking through the paper, especially after the equations and at the end of statements.
4- English is generally good, I think it needs to be polished further and some typos need to be revised.
Foe example:
Line 17:", and the computational efficiency" change to ". The computational efficiency"
Line 17: "greatly" change to "signicantly"
Line 18: "was verified" change to "were verified"
Line 23: "mean value of response" change to "mean value of the response"
5- A direction of future work should be mentioned in it.
Editorial Office of Mathematics
Title: Stochastic Finite Element Analysis of Plate Structures Considering Spatial Parameter Random Fields
In this manuscript, Authors study thE plate theory considering a two-dimensional random field based on an efficient strategy called Karhunen-Loeve (KL) 12
expansion-based stochastic finite element method (SFEM). The work is interesting due to authors clarify the main motivation of this paper. The work brings novelties and information that holds the
mathematics reader's attention. The presentation of paper is acceptable. The article is especially a guide for other researchers with its new
methodology. The results are correct in terms of mathematically and the paper includes creative ideas.
The result of this submission is useful for the interested reader of the current journal. But, there are some issues in
the paper that should be fixed.
The authors should prepare a revised
version of the manuscript. So, I advise the following comments for improvement.
1. It is not customary to use abbreviations in abstracts and conclusions.
2- A subsection should be added to the introduction and the main goals and main achievements of the article should be mentioned in it.
3- Punctuation marks should be checking through the paper, especially after the equations and at the end of statements.
4- English is generally good, I think it needs to be polished further and some typos need to be revised.
Foe example:
Line 17:", and the computational efficiency" change to ". The computational efficiency"
Line 17: "greatly" change to "signicantly"
Line 18: "was verified" change to "were verified"
Line 23: "mean value of response" change to "mean value of the response"
5- A direction of future work should be mentioned in it.
Author Response
1# Reviewer:
General comment: In this manuscript, Authors study thE plate theory considering a two-dimensional random field based on an efficient strategy called Karhunen-Loeve (KL) expansion-based stochastic finite element method (SFEM). The work is interesting due to authors clarify the main motivation of this paper. The work brings novelties and information that holds the mathematics reader's attention. The presentation of paper is acceptable. The article is especially a guide for other researchers with its new methodology. The results are correct in terms of mathematically and the paper includes creative ideas. The result of this submission is useful for the interested reader of the current journal. But, there are some issues in the paper that should be fixed. The authors should prepare a revised version of the manuscript. So, I advise the following comments for improvement.
Response: We would like to thank you for your careful reading, helpful comments, and constructive suggestions, which has significantly improved the presentation of our manuscript.
Comment 1. It is not customary to use abbreviations in abstracts and conclusions.
Response 1: Thanks for your comment. We have revised the abstracts and conclusions. Please see them in the revised manuscript.
Comment 2- A subsection should be added to the introduction and the main goals and main achievements of the article should be mentioned in it.
Response 2: Thanks for your comment. We added these parts in the last paragraph of the introduction: “The purpose of this paper is to explore an efficient and accurate stochastic finite element analysis method considering multidimensional random field for solving the problems of plate structures. In order to obtain the mathematical expression of random field,…”
Please see Lines 88-90 in the revised manuscript for more details.
Comment 3: Punctuation marks should be checking through the paper, especially after the equations and at the end of statements.
Response 3: Thanks for your careful review, we have revised it.
Comment 4: English is generally good, I think it needs to be polished further and some typos need to be revised.
Response 4: We appreciate the reviewer's affirmation and encouragement, and will make persistent efforts to make further achievements in this field.
Comment 5: Line 17:", and the computational efficiency" change to ". The computational efficiency"
Response 5: Thanks for your careful review, we have revised it.
Comment 6: Line 17: "greatly" change to "signicantly"
Response 6: Thanks for your careful review, we have revised it.
Comment 7: Line 18: "was verified" change to "were verified"
Response 7: Thanks for your careful review, we have revised it.
Comment 8: Line 23: "mean value of response" change to "mean value of the response"
Response 8: Thanks for your careful review, we have revised it.
Comment 9: A direction of future work should be mentioned in it.
Response 9: Thanks for your comment. We have added the direction of in the conclusion: “…The plate structures problem considering the spatial random field includes not only the static behavior, but also the dynamic behaviors, nonlinear behaviors and temperature transmission behaviors, which are the directions that can be concerned in the future work.”
Please see Lines 337-339 in the revised manuscript for more details.
Author Response File: 
Author Response.docx
Reviewer 2 Report
In the presented manuscript, the Authors solve a structural mechanical problem involving a multidimensional random field using the stochastic finite element method. At first, the finite element method for modeling plate structures is presented. In the next step, the Authors presented the stochastic finite element method calculation strategy based on the Karhunen-Loeve expansion and point estimate method. The presented calculation strategy was validated and then the stochastic response for different boundary conditions and different plate types with different parameters are discussed.
The manuscript is well-written and can be interesting for scientist who conduct research at this field. In my opinion, the concise and clear description of the mathematical modeling is particularly noteworthy. The "Numerical Examples" section is quite extensive, although some readers may miss the example with "mixed" boundary conditions.
The results are presented clearly and transparently, and I have no major recommendations, although a few minor corrections would be needed:
- The abstract should be a concise summary of the paper, and some sentences seem more appropriate for an Introduction or Conclusion.
- All shorthand designation should be described, e.g. what does it mean “KLE-PEM”?
- Before Equation (1) is written "vertical w", isn't should be rather "vertical z"? If not, what does it "z" in this Equation?
- All variables appearing in the formulas should be described immediately after the equation in which they occur, e.g. what does "U" mean in Equation (5), and does "theta" described only after Equation (12) mean the same in Equation (11), ...
- According to the requirements of the Mathematics journal, the manuscript should contain a "Discussion" Section, which is theoretically missing, but some elements of the discussion are in the "Numerical Examples" Section, so I suggest changing the name of this Section.
I am not a specialist but in my opinion, minor editing of English language required, e.g. in line 18 should be "were verified" instead "was verified"; in line 160 should be "denote" instead "denotes", etc.
Author Response
2# Reviewer:
General comment: In the presented manuscript, the Authors solve a structural mechanical problem involving a multidimensional random field using the stochastic finite element method. At first, the finite element method for modeling plate structures is presented. In the next step, the Authors presented the stochastic finite element method calculation strategy based on the Karhunen-Loeve expansion and point estimate method. The presented calculation strategy was validated and then the stochastic response for different boundary conditions and different plate types with different parameters are discussed.
The manuscript is well-written and can be interesting for scientist who conduct research at this field. In my opinion, the concise and clear description of the mathematical modeling is particularly noteworthy. The "Numerical Examples" section is quite extensive, although some readers may miss the example with "mixed" boundary conditions.
The results are presented clearly and transparently, and I have no major recommendations, although a few minor corrections would be needed.
Response: We would like to thank you for your careful reading, helpful comments, and constructive suggestions, which has significantly improved the presentation of our manuscript.
Comment 1: The abstract should be a concise summary of the paper, and some sentences seem more appropriate for an Introduction or Conclusion.
Response 1: Thanks for comment. We have revised the abstract: “For plate structures, their random parameters can be regarded as a two-dimensional random field in the plane. To solve the plate theory considering a two-dimensional random field, an efficient strategy for stochastic finite element method was adopted. Firstly, stochastic finite element method was used to establish the plate structural model, in which the random field characteristics of the parameter were considered, and the mathematical expression of its random field was obtained through the Karhunen-Loève expansion; secondly, the point estimate method was applied to calculate the statistics of random structures. The computational efficiency can be significantly improved through the reference point selection strategy. The accuracy and efficiency of the calculation strategy were verified, and the influences of correlation length and coefficient of variation of parameter on the random response of plate structures under different plate types (including Kirchhoff plate and Mindlin plate) and boundary conditions (including simply supported and clamped supported) were discussed. The proposed method can provide some help for solving statics problems of plate structures.”
Please see Lines 10-22 in the revised manuscript for more details.
Comment 2: All shorthand designation should be described, e.g. what does it mean “KLE-PEM”?
Response 2: Thank you very much for your comment. We have added all shorthand designation in the revised manuscript.
Comment 3: Before Equation (1) is written "vertical w", isn't should be rather "vertical z"? If not, what does it "z" in this Equation?
Response 3: Thank you very much for your comment. Yes, it was "vertical z", we have revised it in the revised manuscript. Please see Line 108 for more details.
Comment 4: All variables appearing in the formulas should be described immediately after the equation in which they occur, e.g. what does "U" mean in Equation (5), and does "theta" described only after Equation (12) mean the same in Equation (11), ...
Response 4: Thanks for comments, we have added the description of variables in the revised version. "theta" described only after Equation (12) mean the same in Equation (11), we have made adjustments to the description: “…and denotes a group of uncorrelated random variables, and when is a Gaussian random field, it follows the standard normal distribution.”
Please see Lines 127-129 for more details.
Comment 5: According to the requirements of the Mathematics journal, the manuscript should contain a "Discussion" Section, which is theoretically missing, but some elements of the discussion are in the "Numerical Examples" Section, so I suggest changing the name of this Section.
Response 5: Thanks for comments. We have replaced “Numerical Examples” with “Discussion”.
Comment 6: I am not a specialist but in my opinion, minor editing of English language required, e.g. in line 18 should be "were verified" instead "was verified"; in line 160 should be "denote" instead "denotes", etc.
Response 6: Thanks for comments. We have revised them.
Author Response File: Author Response.docx
Reviewer 3 Report
This paper proposes a Karhunen-Loeve expansion-based stochastic finite element method (SFEM) to analyze the random characteristics of plate structures in engineering. The method considers the random field characteristics of the parameter and applies the point estimate method to calculate the statistics of random structures with improved computational efficiency. The method is verified to be accurate and efficient, and its influences on plate structures under different plate types and boundary conditions are analyzed, showing that the mean value of response is not sensitive to parameter changes, but the standard deviation and corresponding coefficient of variation are relatively sensitive. However, the following comments should be addressed before considering for publication.
1. In the introduction, the authors introduce the random field problem of structures but only provide limited information on the stochastic finite element method (SFEM) and its capabilities, advantages, and applicability to engineering problems. The reviewer suggests that the authors provide more comprehensive background information on SFEM in their paper to better contextualize their proposed approach.
2. In Section 2.1, the authors stated that "for Kirchhoff plates, each element has three nodes, and each node has three degrees of freedom". However, this description is not entirely accurate as Kirchhoff plate elements can also have four nodes.
3. In line 116, the authors stated that "according to Eq. (5), the stiffness matrix of Kirchhoff plate can be written as…". However, the reviewer suggests that the authors provide more detailed information about the equations and their interrelationships between Eq. (5) and Eq. (10) to make it easier to follow the analytical development of the model.
4. The authors extensively describe the theories behind the SFEM, including derivation and solution methods. However, the paper does not provide specific details on the implementation of the SFEM, such as whether existing software was used or whether it was implemented in a scientific computing environment like MATLAB. The reviewer suggests that the authors provide more information on the SFEM implementation to help readers better understand how the model was applied in practice.
5. The reviewer expresses concern regarding the novelty and scientific contribution of this study, stating that the proposed approach relies on existing theory applied to plate structures for analysis, resulting in limited novelty and contribution to the field. The reviewer suggests that the authors provide more detailed information about the specific contributions of their study to better contextualize its value and potential impact.
6. In conclusion, the authors need to acknowledge any limitations of the proposed method and suggest potential directions for future research. This will help readers better understand the model's applicability and identify areas where further development and refinement may be needed. The reviewer suggests that the authors provide a clear discussion of any limitations and suggestions for future work to improve the proposed approach.
Author Response
3# Reviewer
General comment: This paper proposes a Karhunen-Loeve expansion-based stochastic finite element method (SFEM) to analyze the random characteristics of plate structures in engineering. The method considers the random field characteristics of the parameter and applies the point estimate method to calculate the statistics of random structures with improved computational efficiency. The method is verified to be accurate and efficient, and its influences on plate structures under different plate types and boundary conditions are analyzed, showing that the mean value of response is not sensitive to parameter changes, but the standard deviation and corresponding coefficient of variation are relatively sensitive. However, the following comments should be addressed before considering for publication.
Response: We would like to thank you for your careful reading, helpful comments, and constructive suggestions, which has significantly improved the presentation of our manuscript.
Comment 1: In the introduction, the authors introduce the random field problem of structures but only provide limited information on the stochastic finite element method (SFEM) and its capabilities, advantages, and applicability to engineering problems. The reviewer suggests that the authors provide more comprehensive background information on SFEM in their paper to better contextualize their proposed approach.
Response 1: Thank you very much for your comments. We have provided more comprehensive background information on SFEM, Due to the large number and scattered, please see Lines 41-63 in the Introduction in the revised manuscript.
Comment 2: In Section 2.1, the authors stated that "for Kirchhoff plates, each element has three nodes, and each node has three degrees of freedom". However, this description is not entirely accurate as Kirchhoff plate elements can also have four nodes.
Response 2: Thank you very much for your professional comments. Yes, Kirchhoff plate elements can also have four nodes, and in the manuscript, the number of node we discussed was three too. We have revised it.
Comment 3: In line 116, the authors stated that "according to Eq. (5), the stiffness matrix of Kirchhoff plate can be written as…". However, the reviewer suggests that the authors provide more detailed information about the equations and their interrelationships between Eq. (5) and Eq. (10) to make it easier to follow the analytical development of the model.
Response 3: Thank you very much for your professional comments. Yes, there are indeed many steps between Eq.5 and Eq.10, but this is the traditional plate finite element theory, so it will not be repeated in the paper, and we explained it this in our manuscript: “The content of this sub-section is the traditional plate finite element theory, it will not be repeated, more details can be found in the work of Ferreira and Fantuzzi [18].”
Please see Lines 134-135 in the revised for more details.
Comment 4: The authors extensively describe the theories behind the SFEM, including derivation and solution methods. However, the paper does not provide specific details on the implementation of the SFEM, such as whether existing software was used or whether it was implemented in a scientific computing environment like MATLAB. The reviewer suggests that the authors provide more information on the SFEM implementation to help readers better understand how the model was applied in practice.
Response 4: Thank you very much for your comment. We have explained it this in the revised manuscript: “Combining the theories introduced in section 2 and 3, the corresponding numerical model can be established using the scientific computing environment MATLAB.”
Please see Lines 181-182 in the revised for more details.
Comment 5: The reviewer expresses concern regarding the novelty and scientific contribution of this study, stating that the proposed approach relies on existing theory applied to plate structures for analysis, resulting in limited novelty and contribution to the field. The reviewer suggests that the authors provide more detailed information about the specific contributions of their study to better contextualize its value and potential impact.
Response 5: Thank you very much for your comment. We have provided more detailed information about the contribution in the conclusion: “This study provides a new approach for the SFEM solution of plate structures, Because of its efficiency and accuracy, it can be used to solve some plate structure problems considering spatial random field, such as track slab deformation, floor slab deformation, etc. …”
Please see Lines 333-336 in the revised for more details.
Comment 6: In conclusion, the authors need to acknowledge any limitations of the proposed method and suggest potential directions for future research. This will help readers better understand the model's applicability and identify areas where further development and refinement may be needed. The reviewer suggests that the authors provide a clear discussion of any limitations and suggestions for future work to improve the proposed approach.
Response 6: Thank you very much for your comment. We have acknowledged the limitations of the proposed method and suggest potential directions for future research: “…However, the plate structures problem considering the spatial random field includes not only the static behavior, but also the dynamic behaviors, nonlinear behaviors and temperature transmission behaviors, which are the directions that can be concerned in the future work.”
Please see Lines 336-339 in the revised for more details.
Author Response File: Author Response.docx
