The Approximate Solution of the Nonlinear Exact Equation of Deflection of an Elastic Beam with the Galerkin Method
Round 1
Reviewer 1 Report
Author Response
Thanks, we made some revisions.
Author Response File: 
Author Response.docx
Reviewer 2 Report
This paper aims to find the approximate solution of the nonlinear exact equation of deflection of an elastic beam with the Galerkin method. The manuscript is well provided but the following comments need to be considered in the revised manuscript.
1) Please put the reference numbers in [ ] in the context. Some of them are used without [ ].
2) There is a course in mechanical engineering as “Finite Element” where the response of a structure can be obtained by assembling local elements by diving the total structure into elements with finite dimensions. How the results of this study can use for formulating the response with Finite Element procedure? From my point of view, the proposed procedure of the current study is not practical and has many difficulties extending it to other types of structures with different boundary conditions.
3) The procedure is proposed here for a fixed-end beam condition. What is the reason for selecting this type of boundary condition? In a real structure, beams have pin-pin, fix-pin, and fix-fix conditions and fewer members have only one fixed end condition. How the proposed strategy can be extended for other boundary conditions? Please comment and extend the results for other types of boundary conditions that are common in real structures.
4) The considered beam is under the effect of a single concentrated load and the formulations are proposed for this condition. What will be happened if the input load change from a concentrated load to a distributed load (as is common in real structures)? Please solve an example for a fixed-end beam under uniform input load with intensity w, and solve an other example with considering uniform load along with a concentrated load and a single moment (that can be found in any books of mechanical engineering). I ask the author to do this example to show the capability of their proposed method to solve different examples. This will help to increase the impact of your paper.
5) Give more details of your findings in the abstract.
6) It is recommended to refer to some works published by MDPI specially “applied science” in the revised manuscript. As an example: https://www.mdpi.com/2076-3417/11/19/9307
Author Response
Please see the attachment.
Author Response File: Author Response.docx
Reviewer 3 Report
Report on the manuscript
Title: The approximate solution of the nonlinear exact equation of 2 deflection of an elastic beam with the Galerkin method 3 Chencheng Lian, Baochen Meng, Ji Wang * and Lihong Wang
Journal: Applied Sciences
Manuscript number: applsci-2080885-peer-review-v1
In the paper the authors study the nonlinear differential equation of the deflection of an elastic beam, solved by the Galerkin method with an optimal trial function, and the approximate solution in the trigonometric functions is obtained with excellent accuracy. The usefulness of the Galerkin method to nonlinear differential equations is further demonstrated. The results of the study can be of interest for the designer in the domain. I think that the results of this manuscript can be useful for the researcher. Academic speaking, the paper is well written, the material is judiciously divided and organized and correct from scientific point of view. Some changes are, however, necessary. For these reasons I can recommend the acceptance of this paper after some corrections, presented in the attached file.
Comments for author File: 
Comments.pdf
Author Response
Please see the attachment.
Author Response File: Author Response.docx
Round 2
Reviewer 2 Report
I have no further comments for the authors. The revised manuscript is well improved.
Reviewer 3 Report
The paper was improved. Now we consider that the level is enough high to be published.
