Elasticity Solutions for In-Plane Free Vibration of FG-GPLRC Circular Arches with Various End Conditions

Round 1

Reviewer 1 Report

This paper is well written in English and the

results appear to be correct.

The methods used to find the

results provide innovations in the scientific field used.

The paper is accepted for publication in the journal.

Author Response

Special thanks for your good comments.

Reviewer 2 Report

This is a generally well-written manuscript (MS), although the presentation is rather odd, in particular the alignment of embedded equations, and a change in font size in Conclusions.  On reading through the MS the following points were noted.

1. Title: Should read “arches”, not “arche”.
2. Figure 1 caption: “curvilinear” should be “polar”. Within the Figure, r(z) is misleading; would be better if it was just r.  Below the figure, r – θ  is not a Cartesian coordinate system.
3. There appears to be errors in the Appendix. In particular, different end conditions are abbreviated as S-S (Simply supported), C-F (Clamped-Free), which is fine.  But for the C-F and C-C arches, the text states S-S.
4. In Conclusions, “promotion of natural frequency” should read “increase of natural frequency”.
5. This reviewer would not agree with the statement in the Abstract that the solution is “completely exact”. Exactitude can only be defined within a defined model – for example, something can be exact within the spirit of the linear theory of elasticity.  Moreover, the law of fractions is probably only exact for density; for other properties it is an approximation.

Author Response

1. Title: Should read “arches”, not “arche”.

Re: Thanks for your comments. We are very sorry for our incorrect writing, and the title has been checked.

2. Figure 1 caption: “curvilinear” should be “polar”. Within the Figure, r(z) is misleading; would be better if it was just r.  Below the figure, r – θ  is not a Cartesian coordinate system.

Re: Thanks again for the useful suggestion.  We have made correction according to the Reviewer’s comments. The z-axis is replaced as a new variable to avoid the misleading.

3. There appears to be errors in the Appendix. In particular, different end conditions are abbreviated as S-S (Simply supported), C-F (Clamped-Free), which is fine.  But for the C-F and C-C arches, the text states S-S.

Re: It is really sorry for our mistake. The Appendix has been rewritten.

4. In Conclusions, “promotion of natural frequency” should read “increase of natural frequency”.

Re: Once again, thank you very much for your comments and suggestions. The statement of “promotion of natural frequency” is corrected as “increase of natural frequency”.

5. This reviewer would not agree with the statement in the Abstract that the solution is “completely exact”. Exactitude can only be defined within a defined model – for example, something can be exact within the spirit of the linear theory of elasticity.  Moreover, the law of fractions is probably only exact for density; for other properties it is an approximation.

Re: We totally agree with the reviewer. The statement of “completely exact” is not exact, and we have corrected it. As we known, boundary conditions for simply supported beam or arch can be assumed as trigonometric series, and the stress and displacement could satisfy these automatically. So it can be called 'exact solution'.

Reviewer 3 Report

As a computational/theoretical chemist, this referee has reviewed with interest the proposed two-dimensional in-plane free vibrations analysis of functionally graded graphene platelets reinforced nanocomposites (FG-GPLRCs).

There is nothing particularly wrong with the proposed mathematical development. Unfortunately, I feel the paper is more suitable for other audience rather than for the expected contribution in Applied Sciences.

I strongly recommend transferring this manuscript for a more suitable journal, e.g., Mathematical and Computational Applications.

Author Response

Thanks so much for your kind consideration and suggestion on our paper. In our opinion, the topic of our paper belongs to Applied Mechanics in Civil Engineering, and is quite suitable for publishing in Applied Sciences. Of course, it also can be concern with applied mathematics or computational applications. Thus, another new paper on this will be finished soon and is considered to submit to the journal of Mathematical and Computational Applications.

Thanks one more time.

Round 2

Reviewer 3 Report

As there are minor (style) changes in the original manuscript only, my recommendation is consequently the same: resubmitting to Mathematical and Computational Applications.

I am sorry for being so picky, though I am positive the manuscript better fits to this other MDPI journal. Indeed, the present contribution and the new manuscript the be submitted soon might be thus published as a dyad in the same journal, which seems more suitable for correctly interpreting the proposed theoretical strategy.

That must be said that if Editor/other referees feel the paper is suitable for Applied Sciences, I have no further changes to recommend.