Seismic Structural Resonance Observation Based on Recent Estimation of Earthquake Predominant Frequency

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper investigates the phenomenon of seismic structural resonance by analyzing the correspondence between the newly proposed displacement-based predominant frequency of near-fault earthquakes and the natural frequencies of multi-degree-of-freedom (MDOF) building models. The author aims to confirm that the structural resonance can occur when these two frequencies match.

However, the study presents significant methodological and conceptual limitations that must be addressed before it can be considered for publication. The current version oversimplifies the physical behavior of real structures, relies heavily on previously published results, and lacks quantitative or statistical rigor.

1. The author deliberately restricts the analysis to a linear-elastic MDOF system, arguing that resonance in nonlinear systems is complex. While this simplifies the numerical treatment, it severely limits the physical realism of the results. In real buildings, nonlinear behavior (e.g., stiffness degradation, yielding) alters the resonance condition, often mitigating amplification. At least one nonlinear analysis should be included to demonstrate the difference between resonance in linear and nonlinear systems.
2. The damping coefficient is given directly, but the damping ratio (ξ) is not. What damping ratio (2%, 5%, etc.) does this value correspond to? Underdamping will dramatically increase resonance amplification, significantly impacting the system's resonant response. Therefore, the model's realistic nature is debatable.
3. The author states that if the difference between the resonant frequency and the dominant frequency is less than 10%, "the existence of a predominant frequency is proven." However, it should be stated how the 10% limit was determined, which literature source it was taken from, or what the statistical justification is.
4. The author does not assess the "clarity" of the resonance with quantitative criteria. Numerical measures such as magnification ratio, Q-factor, and bandwidth can be used. Such interpretations fall short of scientific objectivity.
5. The literature is mostly focused on studies from 1990 to 2019, with no mention of studies after 2020. New studies from current sources should be added.

The manuscript addresses an interesting and underexplored subject in earthquake engineering. However, it requires substantial revision to enhance methodological rigor, independence from previous work, and scientific robustness.

Author Response

Response to Reviewer 1:

1) We agree with the reviewer in that there is some limitation in the physical realism of the results as in real buildings nonlinear analysis is necessary; in fact, we have added a full paragraph that starts with this same wording, please see it highlighted in cyan (which is the color for enhancements associated to all Reviewer 1 comments); moreover, we have added the limitation or words “linear” and “elastic” in the Abstract and elsewhere (e.g. lines 88 and 105). Now, we would like to refer to two important facts and points: A) a literature review on “seismic structural resonance” indicates that there are no papers in which the original resonance concept is studied and, more importantly, observed clearly; that is, where a concrete match of input predominant frequency and structure natural frequency is analyzed, and here lies the novelty of this work because it applies very recent predominant frequencies without which resonance analysis is not possible. B) Apart from the fact that there is no single resonance in nonlinear systems, and there is the jump phenomenon, L. Meirovitch writes that “resonance is not possible for mass-nonlinear spring systems” [21], and A. Chopra writes that “the natural period is not defined for inelastic systems” [22]. That is, vibrations resonance (which is the original and actual one) is hard or impossible to study in its mathematical or original terms for nonlinear structures because there is no absolute and unequivocal structural period. Therefore, considering points and facts A and B, it is believed that a very first work on the subject should be logically for linear systems, where resonance can be studied in its original (vibrations) or mathematical terms, which absolutely imply a frequency match, which in turn it has been only possible once a seismic predominant frequency has been defined. Of course, future work must be to investigate the problem in nonlinear structures, this is discussed in the revised paper (last paragraph). Finally, fact A can be confirmed in search systems (Web of Science or Scopus).

2) The damping ratio z is given now, this is in line 175 of the revised paper (highlighted in cyan)

3) The paragraph where the relative difference is presented has been enhanced and corrected according to this correct comment by the reviewer. First, instead of a strong “this will confirm” there is now “this can be considered a confirmation”. Moreover, it is written now that the 10% was chosen arbitrarily although a basis is given now, associated to resonance experiments in the ideal case of perfect harmonic excitation where differences of less than 5% are common. These enhancements are highlighted in cyan in page 5.

4) Resonance clarity is assessed now with the quantitative criteria suggested by the reviewer; as a matter of fact, we consider that out of all comments by all reviewers this is the one that has improved the paper the most. Indeed, Table 2 now has the bandwidth and transmissibility (magnification) for each case, these are columns 5 and 6. Moreover, a full presentation is given in the paragraph below that table of this important matter, and in the text these two measures are added to the discussion of the cases with very clear resonance and of the case in which there is no clarity; again, these improved case comments are a great enhancement.

5) The literature has now more references from 2020 onwards as suggested; this can be noted in the reference list where the additions are highlighted (one is in purple because a similar comment was given by Reviewer 2). It is stressed, nevertheless, that a literature review was carried out again, and it has been confirmed that papers that deal with seismic structural resonance, in the original or mathematical sense of a concrete spectral match, are very scarce (as discussed in point 1 above).   

Reviewer 2 Report

Comments and Suggestions for Authors

The examined manuscript studies the resonance effect in engineering structures using a five-story MDOF oscillator. The research investigates several real seismic records and provides the corresponding response spectrums.  

The author must consider carefully the following comments to improve his manuscript.

1) It is not clear how the predominant frequency of the examined seismic signals is calculated. More information must be provided on this essential component of the examined study. 

2) More documentation must be added about the novel information which the examined study provided to an earthquake or structural engineer.

3) The mixed "Discussion and Conclusions" can confuse and discourage a reader. For this purpose, a separate Conclusions section must be included with a brief and comprehensive summary of the research, the main outcomes of the study and recommendations for future research. The "Discussion" part, as well as their figures, could be incorporated in the "Results" section, which could be renamed as the "Results and Discussion" section. 

4) The plotting of the different seismic records' spectra with multiple colored lines in the same figure could provide a clearer comparison between the different MDOF building responses.

5) Figures inside the text seem not to be referred to in the appropriate order. Fig. 9 is cited after Fig. 10, and Fig. 8 is not referred to inside the text. 

6) Also, a limitation which must be highlighted seems to be the analysis of only a 5-storey building configuration with variable stiffnesses. How could the different building configuration influence the results? How the resonance effect could occur in a real building which may suffer from damage and degradation of its stiffness under a strong earthquake. Is this a potential direction for future research?

7) For a more comprehensive state-of-the-art review, I propose the following recent and strongly related publications, which estimate the structural response using advanced computational methods, be included in the introduction section.
https://doi.org/10.1016/j.istruc.2024.106524
https://doi.org/10.3390/app12083845  

8) Lines 35-36: The reference style must be aligned to the publisher template.

9) Lines 45-46: The reference number is more convenient to be after the authors' names. Also, the "et al." must be added after more than two authors' publications, and both authors' names must be included inside the text in case of two authors.

10) Line 88: Double "that", could possibly be corrected with "that the".
Line 135: A letter symbol is missing.
Line 216: Typo in "the seism predominant frequency".

Author Response

Response to Reviewer 2:

1) A full new paragraph has been added to explain better how the predominant frequency was calculated. This is at the end of Section 2, highlighted in purple (as all improvements associated to Reviewer 2 comments), note that the new paragraph has 10 lines, so surely there is enough information now about that methodology (of a previous paper).

2) We have also added a new paragraph with information regarding what this study can provide to earthquake and structural engineers (it is the second paragraph of the new Conclusion section, highlighted in purple).

3) There are now separate Discussion and Conclusion sections, as the reviewer suggested.

4) For a clearer comparison among the responses, the inclusion of the mathematical bandwidth and magnification (transmissibility) are added in Table 2 for each case, this in the new columns 5 and 6 of the table; this enhancement is presented in text in page 7.

5) Fig. 9 is cited before Fig. 10 in the revised paper (see page 8); now, Fig. 8 is not referred to individually but collectively as “Figs. 7-10” in page 8; the reason is that we analyzed several earthquakes so it is not possible to analyze each plot individually although most have been, as can be confirmed.

6) A discussion on resonance in more real or inelastic buildings has been added, it is in fact a long new paragraph at the end of the Discussion section (highlighted in cyan because a similar comment was raised by Reviewer 1). Additionally, in the new Conclusion section, in its future work portion or paragraph (which is the last), this subject is presented as precisely potential direction for future research, as the reviewer suggested or hinted.

7) Publication doi.org/10.3390/app12083845 has been included as proposed by the reviewer, it is ref. 23.

8) The referencing in those lines has been improved or corrected, as the reviewer indicated (highlighted in purple).

9) Referencing in these lines has been checked and corrected when necessary.

10) The double that was corrected with “that this” and the symbol has been added, as can be confirmed in the revised manuscript; finally, even though “seism” was not a typo it has been taken out as it could be confused as one (this can be confirmed in line 242 now, also highlighted).

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Your revisions have strengthened the manuscript, particularly through the inclusion of quantitative resonance metrics, clarification of the damping ratio, and the addition of more recent literature. While the exclusive use of a linear-elastic model still limits the physical realism of the study, your explanation that this work aims to serve as an initial, mathematically focused exploration of resonance makes the approach acceptable at this stage. Although including a simple nonlinear example would have enhanced the study, the manuscript in its revised form is now sufficiently improved to move forward, with the acknowledged limitations clearly stated.

Author Response

We do thank this consideration that, after the enhancements to the original manuscript, there is nothing else to be improved (as shown in the report table), and that the approach or method is acceptable now.

Reviewer 2 Report

Comments and Suggestions for Authors

Some improvements have been made by the author based on reviewers' comments. I highlight below the main omissions based on the comments of my previous report and the added parts.

As a general comment in Lines 313-316: The resonance is a basic concept of the structural dynamics, and is well known among civil engineers for its potential destructiveness.

Previous report Comment 1: How do you contribute to the avoidance of this effect (resonance) during an earthquake? No specific or complete information was provided about the examined calculation.

Previous report Comment 2: The utility of the study must also be established in the abstract and at the end of the introduction.

Previous report Comment 7: Not completely addressed.

Author Response

General comment. Near those lines (second paragraph of Section 5) the novelty is stressed now with the addition of 7 more lines (highlighted in purple); that is, it is emphasized that before there were no clear predominant frequencies of seismic action without which there is no resonance study (new lines 319-321); moreover, it is written now that the results of this work can be considered a double confirmation for engineering applications.

Previous comment 1. More and specific information has been provided about the predominant frequency calculation of ref. [15], in lines 169-171 (highlighted in purple); in fact, the specific information on the damping ratio employed in the ref. is given. If even more details are required, the reader is referred (as commonly) to the original reference. Regarding the contribution to the avoidance of resonance, the second paragraph of the Conclusion (which was a suggestion of the reviewer in Round 1) has been extended to clarify the novelty and contribution. Indeed, in lines 321-325 it is stated now that one contribution is confirmations, deemed necessary, related to seismic resonance in buildings.

Previous comment 2. The utility of the paper has been added at the end of the Introduction (highlighted in purple, the only color used as Reviewer 1 had no additional comments)

Previous comment 7. We have really revised all references suggested by both reviewers in Round 1, and we could add only those that had a very clear and direct relationship with the work presented. For example, in the case of suggestions of Reviewer 2, the first paper deals with nonlinear response of complex structures through advanced computational methods so it was included, as resonance is a special type of response, but although the second one dealt also with advanced computational methods, these were more or specifically for damage prediction rather than general seismic response. Here we followed the advice of the Journal which reads “If a reviewer recommends references, critically analyze them to ensure that their inclusion would really enhance the manuscript…”