Closed-Form Analysis of Thin-Walled Composite Beams Using Mixed Variational Approach

Round 1

Reviewer 1 Report

The paper “Closed-form Analysis of Thin-walled Composite Beams using Mixed Variational Approach” submitted to Aerospace deals with a variational approach for thin-walled composite beams.

The paper is well-written although small corrections in the language should be made. Please, revise the English. Furthermore, as is a mostly fully analytical paper, an improved explanation regarding ALL variables must be made as well as some equations must be revised (please, see Eq. (7)-(8b) and Eq. (17)). It is of my understanding that this paper is an improvement of the authors’ theory. Still, the entire analytical procedure shall be available inside the paper OR a direct citing in each explanation of variables should be made.

Regarding the quality of the paper: it has been seen sadly fewer and fewer papers that deal with analytical approaches. For this reason, I strongly suggest publishing since the paper deals with composite tubes, which are being increasingly used in Industry, and the approach of the paper could be applied at a pre-design, design, and evaluation of tubulations.

I have some questions regarding the applicability of the variational approach:

·         As I understood, there is no constraint concerning the cross-section shape but its thickness. The examples considered rectangular (filled or hollow) sections. Is the approach applicable to, for example, C-shapes (known to have a strong warping)?

·         Figure 5 shows some warping shapes. Are they comparable to the results of other researchers?

·         The use of k=5/6 is not always a good approach, in my opinion. There is the work of Jemielita [1] that shows how the accurate correction can deviate from 5/6 depending on the stacking configuration (although this would be only a cosmetic modification on the paper).

·      By considering hoop stress resultant Nss=0, composite tubes under pressure could not be evaluated, or?

[1] Jemielita, G. (2002). Coefficients of shear correction in transversely nonhomogeneous moderately thick plates. Journal of Theoretical and Applied Mechanics, 40(1), 73–84.

Author Response

1. The paper is well-written although small corrections in the language should be made. Please, revise the English.

Ans.) The paper has been reviewed thoroughly and revised accordingly.

2. Furthermore, as is a mostly fully analytical paper, an improved explanation regarding ALL variables must be made as well as some equations must be revised (please, see Eq. (7)-(8b) and Eq. (17)). It is of my understanding that this paper is an improvement of the authors’ theory. Still, the entire analytical procedure shall be available inside the paper OR a direct citing in each explanation of variables should be made.

Ans.) A nomenclature section is newly added in the bottom (pages 14 – 15) of the manuscript. Furthermore, relevant references are inserted in the text whenever applicable so that the variables are clearly explained.

3. As I understood, there is no constraint concerning the cross-section shape but its thickness. The examples considered rectangular (filled or hollow) sections. Is the approach applicable to, for example, C-shapes (known to have a strong warping)?

Ans.) Certainly, it is. The present theory is so general and applies to thin-walled beams with either closed or open contours. For open cross-sections (including C-shaped section), the solution procedure is significantly reduced as compared with that of the closed sections. Specifically, the undetermined constants in Eq. (19) become zeros assuming that there are no applied surface tractions, which means that there is no need to solve the constraint conditions in Eq. (21) to find the unknown constants. The remaining parts remain the same for both open and closed cross-sections.

4. Figure 5 shows some warping shapes. Are they comparable to the results of other researchers?

Ans.) Yes, it is. Dhadwal et al. (Ref. [36]) displayed all warping modes for the same thin-walled box cross-section. When comparing the warping modes predicted between the two approaches, there is a close correlation while capturing the coupled warping deformation shapes due to the elastic couplings (extension-shear and bending-torsion) of composites. The only difference comes from the in-plane warping deformation which has been neglected in the present analytical beam theory.

5. The use of k=5/6 is not always a good approach, in my opinion. There is the work of Jemielita [1] that shows how the accurate correction can deviate from 5/6 depending on the stacking configuration (although this would be only a cosmetic modification on the paper).

[1] Jemielita, G. (2002). Coefficients of shear correction in transversely nonhomogeneous moderately thick plates. Journal of Theoretical and Applied Mechanics, 40(1), 73–84.

Ans.) The authors are aware that the introduction of the shear correction factor bears limitations in the description of the wall shear behavior (not the transverse shear of the cross-section!). As mentioned by the reviewer, the values can vary depending on the stacking sequence of the laminate. Even though the effect should be small for thin walls, there exists certain limitation of the present work since the theory describes the wall behavior based on the first-order shear deformation theory. A higher-order shear deformation theory might be needed, especially for thick-walled sections, to enhance the prediction capability of the beam theory, which could be a future work.

6. By considering hoop stress resultant N_ss=0, composite tubes under pressure could not be evaluated, or?

Ans) No. Since we assume that the hoop stress resultant is negligible (N_ss = 0), the beam theory in the current version cannot analyze thin-walled tubes under pressure.

Reviewer 2 Report

Comments:

1. In the paper, a 4-layer laminated beam under concentrated tensile forces in X direction was used to validate the normal stresses of the analytical model. Is it appropriate to analyze composite box beam there are with out-of-plan deformation.

2. All poisson's ratio were set to be 0.42 in line 333. Is it appropriate? The mechanical properties in plane yz should be isotropic. 

Author Response

1. In the paper, a 4-layer laminated beam under concentrated tensile forces in X direction was used to validate the normal stresses of the analytical model. Is it appropriate to analyze composite box beam there are with out-of-plan deformation.

Ans.) The present theory can analyze thin-walled composite beams of arbitrary cross-sectional shapes. The sections include rectangular solid, thin-walled open, and closed contours. As can be seen in Figure 5, all out-of-plane warping modes of a thin-walled box section are captured clearly. The comparison of in-plane stresses in Figure 6 indicate also that the present predictions are highly accurate for thin-walled, closed cross-section beams which essentially contain out-of-plane warping deformations.

2. All poisson's ratio were set to be 0.42 in line 333. Is it appropriate? The mechanical properties in plane yz should be isotropic. 

Ans.) The authors agree with the reviewer’s comment. The values were adopted (without corrections) from Ref. [36]. In fact, there is no agreement on the value of ν₂₃ between the references cited (Refs. [34-36] in the manuscript). Since the Poisson's ratio in plane yz has no contribution in the development of CLPT as well as the present beam formulation, the corresponding value is removed from the material properties to avoid unnecessary confusion. The relevant reference is inserted in the revised manuscript.