Explicit Solution to Large Deformation of Cantilever Beam by Improved Homotopy Analysis Method II: Vertical and Horizontal Displacements

Round 1

Reviewer 1 Report

This paper is complete. The method is sound. The review would like to raise the authors' attention to the two references on the theories of nonlinear elasticity and deformable body dynamics.

1. Novozhilov, V.V., 1953, Foundations of the nonlinear theory of elasticity, Graylock Press.
2. Luo, A.C.J., 2010, Nonlinear deformable-body dynamics, Higher Education Press, Beijing, and Springer-Verlag Berlin Heidelberg.

Author Response

According to the comments you provide, we have added the two articles mentioned by the reviewer to the references and adjust the number of other references in the text.

Reviewer 2 Report

The work is devoted to the development of methods for solving nonlinear problems of bending of rods. The proposed method of homotopy analysis (IHAM) is an improvement of the previously known HAM by introducing an auxiliary nonlinear operator that increases the rate of convergence and expands the range of convergence.

An essential advantage of the method is that it does not depend on small or large physical parameters. Thus, it is applicable not only for weakly, but also for strongly nonlinear problems and does not have the limitations inherent in standard perturbation methods. It is important to note that the proposed method is well combined with other computational methods used to solve nonlinear problems, such as, for example, the Padé approximation method.

Notes on the article

There are no significant comments. However, the following should be noted.

1. Many numerical algorithms are used; therefore, the proposed IHAM method can hardly be entirely attributed to analytical methods. It would be correct to call this method semi-analytical.
2. The Authors' assertion that "the solution according to the proposed IHAM method is more explicit and simple compared to the exact solution obtained using elliptic integrals" is controversial. It depends on the type of considered problem.
3. Models of nonlinear rod bending are very often used in engineering practice. However, it is doubtful that such large deformations can occur during the operation of buildings. Hence, there is hardly a need to use the models considered in the manuscript to describe them.

However, these remarks do not diminish the scientific significance of the article. I believe that the article can be published in the journal.

Author Response

Please see the attachment.

Author Response File: Author Response.doc

Reviewer 3 Report

The novelty is weak. 

Author Response

As mentioned by the third reviewer, the novelty is weak. However, the present work introduced a semi-analytical method which is especially applicable to strongly nonlinear problems. We think the method itself is a contribution to the nonlinear analysis.