Out-of-Plane Dynamic Response of Elliptic Curved Steel Beams Based on the Precise Integration Method
Round 1
Reviewer 1 Report
The research on static and dynamic mechanics mechanism and performance of curved girder bridge is of great significance. However, the research on analytical mechanics calculation method of response of curved girder bridge is relatively weak at present. In this study, the formula of dynamic response under moving load is derived according to the basic principle of the precise integration method. The stiffness matrix of a variable-curvature beam is obtained using the idea of matrix inversion, and the mass matrix of the structure is obtained by the concentrated mass method. According to numerical analysis verification and dynamic experiment research, the analytical calculation method in this paper has been proved to have good accuracy and convenience, and it is suitable for bridge type selection and rapid dynamic analysis. However, the paper needs to be properly modified and supplemented before publication. The specific problems are as follows:
1. Eq.(3) is composed of three formulas, and its expression form is not standardized enough;
2. Some parameter symbols are expressed in regular font and should be expressed in italics, the symbols in Figure 4-7 have similar problems;
3. In the paper, the variable curvature beam is divided into 6 elements according to the way of dividing one element for every 10° change of curvature on the beam. The stiffness matrix generated by this way of uniform division by angle is uneven. If the elements are divided by the way of stiffness quality, or more elements are divided, can the accuracy of the analytical method be improved?
4. In Figures 13 and 14, the torsion angle is written as Twist angle, it is suggested to correct this error.
5. In the paper, it is written that as shown in Figure 19 and 20, a rolling track was set up on the curved beam and 0.75 kg balls were released at the left end at different speeds. However, the additional rolling track cannot be seen in the figure. It is recommended to further modify the figure or clearly indicate the names of each part.
6. For how to apply the analytical method in actual steel structural engineering, the authors should give further supplement and elaboration, for example, how to apply them in the optimization of bridge design schemes or comparative analysis of bridge responses.
Author Response
Response 1,2,4: The issues raised by the reviewers in points 1, 2 and 4 are corrected in the article
Response 3: The method of dividing elements by mass affects the selection of points on the beam and the calculation results do not change much. If the number of divided elements is increased, the calculation results will slightly improve the accuracy of the calculation, but it will lead to an increase of the calculation volume.
Response 5: The images in the text have been updated and a description of the experimental model has been added to the text.
Response 6: Prospects for the application of the methods in this paper in steel engineering, such as preventing damage such as bridge overturning and controlling beam vibration, have been added in the paper.
Reviewer 2 Report
The article addresses an important and very interesting topic of out-plane dynamic response of elliptic curved steel beams based on precise integration method, which is appreciated. The study includes the numerical and experimental research. The aim of this paper was analyzed the statics theory of beams with variable curvature outside the structural plane based on existing research. The Reviewer has some concerns regarding to the language, introduction, results, discussion of results, conclusions and references. Generally, in this paper the English language should be improved and checked by the Native Speaker. Some sentences are not clear (please use a passive voice) and too long. In opinion of Reviewer this paper should be subjected to major revision.
Other comments:
1. What is novelty in this paper? Based on description of your research in introduction it can be concluded, that isn't any novelty in this paper. In addition, the Authors wrote “The study of curved beams is not a new topic, the mechanical analysis theory of out-of-plane loads is relatively complete, and there are analytical solutions”.
2. What is mean “very coarse finite element mesh”?
3. In my opinion Authors should show the description of problem on the earthquake loads based on the Time History analysed (according line 82 – 110). Below you can find few papers about these researches:
· https://doi.org/10.1016/j.engstruct.2017.11.042
· https://doi.org/10.12989/scs.2022.42.6.747
· https://doi.org/10.1016/j.compgeo.2021.104245
4. Please add the properties and approach of numerical analysis.
5. What do these Figures mean e.g. Figure 9-11? Where are the comments on the Figures?
6. Line 310 – 326 should be added in the last part of this paper.
7. Where is discussion of results?
8. The conclusions are really poor. The Reviewer cannot see the most important conclusion from research. The conclusions are really theoretical approach. This is end of your research? Please improved this point.
9. References are too small. Please add the literature from whole World. This is good Journal for your research? The Reviewer cannot see any paper from this Journal. Why Master Thesis were cited?
Finally, I hope that my comments will be helpful for Authors.
Author Response
This article has been touched up on the official MDPI website.
Response 1:The study of dynamic response of beams with variable curvature is a gap in the field of curved beam research and a novelty of this study. Although the finite element method is more commonly used in curved rod members, more optimal solutions can be considered. The finite element modeling is limited by the variation of curvature when we study curved beams with variable curvature, and it is not possible to use the sparse cell division method for curved beams with large curvature, otherwise it will cause large errors. The method proposed in this paper is based on the idea of structural mechanics to construct the model, and the numerical results can be obtained by using the efficient and accurate precise integration method.
Response 2: The very coarse finite element mesh has been changed to a sparse mesh division
Response 3: These papers you mentioned have been added to the references in this article
Response 4: The method of fine integration is essentially a high-precision method for calculating partial differential equations, which uses the basic principle of multi-scale wavelet transform to construct a multi-scale wavelet interpolation operator, and then uses the operator to discretize the partial differential equation into a system of ordinary differential equations. Finally, the system of equations is solved by the fine integration method.
Response 5: Figures 9-11 show the flexibility matrix, stiffness matrix and mass array of the numerical model respectively. These three figures show the solution results of the design model using the semi-analytical method, which provide data support for the subsequent fine integration method calculation.
Response 6-8: The concluding part of this paper has been trimmed and revised and has been touched up in the official MDPI channel.
Response 9: Because the two mentioned master's thesis are the derivation of geometric relations for variable curvature beams previously studied by our research team and the semi-analytic method applied in this study, on which this paper is based for further research.
Reviewer 3 Report
1. In the real design of bridges and other structures where curvilinear rods are used, FEM is used successfully and with high precision. At the same time, the area of constraints and types of span structures is much wider than in the proposed approach. Therefore, it is not clear why these developments were made. The authors should highlight the advantages of the method, if any, for example, in comparison with step methods of integration of equation (1). As the authors rightly point out, the precise solution for the dynamic response of the curvilinear rod is not given. Perhaps this is because it is not required.
2. As a rule, curvilinear rods of bridge spans are complex steel and reinforced concrete structures. The proposed analytical dependences, in particular (6)-(15) in our opinion, can be effectively applied only for isotropic materials like steel. Therefore, there is a question about the practical relevance of this paper.
3. The moving load on curvilinear sections of bridges usually has a varying intensity. This change occurs in time depending on the length of the curve and is associated with acceleration and deceleration of transport. This phenomenon is not addressed in the article.
4. In Fig. 21, for each position, the experimental values of vertical displacements exceed the theoretical ones. This indicates the imperfection of the theoretical model proposed by the authors. Perhaps it is related to the damping of oscillations. One way or another, this fact can lead to an emergency situation in the real design of the structure using the proposed methodology.
5. Nothing is said about the method of precise integration, no data is given about the intermediate calculations. This could be done for a simple structure, which is considered in the article.
Author Response
Response 1: Although the finite element method is more commonly used in curved bars, more optimal solutions can be considered. In our study of curved beams with variable curvature, the finite element modeling is limited by the curvature variation, and it is not possible to use the sparse cell division method for curved beams with large curvature, otherwise it will cause large errors. The method proposed in this paper is based on the idea of structural mechanics to construct the model, and the numerical calculation results can be obtained by using the efficient and accurate fine integration method.
Response 2：Steel-hybrid bridges are now more constructed, applied and studied, but when we explore their macroscopic dynamic properties, all of them are simplified to isotropic materials, and pure steel bridges with small spans also exist in cities. At the same time, this paper adopts more structural mechanics approach to modeling, also in the macroscopic sense to study its dynamic response.
Response 3: Since the study of the dynamics of variable-curvature steel beams is now a relatively blank stage, the change of acceleration and deceleration is the next research focus to be tackled by our research team based on this paper, which involves factors such as curvature change, friction force due to speed change, and centripetal force change.
Response 4: Considering the experimental errors and other factors, the theoretical calculation results of this study are compatible with the experimental data. Both the finite element verification mentioned in the previous section and the experimental results here are proofs of the correctness of the theory in this paper, which can be multiplied by the amplification factor to ensure structural safety in practical applications.
Response 5: The construction parameters of the model in this paper are given in the form of tabular data, and since the theoretical derivation is given in Chapter 2, the medium-term computational data and the computational results are shown here directly in the form of images.
Reviewer 4 Report
Dear authors,
The paper deals with out-plane dynamic response of elliptic curved steel beams based on precise integration method. In the reviewers' opinion, the paper can be considered for further process. However, the following points need to be clarified:
Major comments
1. The finite element model used for the verification is recommended to describe. Currently it is very hard to work out the force loads. Where are the loads exactly applied?
2. How were the number elements of the new design obtained? Is there any influence to the structural flexibility?
Minor comments
1. Check missing legend in Figure 21, unit bracket of moment of inertia in Table 1
2. Avoiding using word “we” in the manuscript
Author Response
Response 1: The relevant statements describing the finite element model have been added to this paper. The loads studied in this paper are set to constant values, and the external loads are distributed proportionally to the nodes at each end of the cell as they move over the cell.
Response 2: Since the curvature of the elliptic linear shape varies irregularly, the location of the points on the beam in this study is determined according to the angle between its normal and the negative direction of the Y-axis, so the division of the beam into cells and the number of cells is also determined according to its angle, which has no effect on the flexibility of the structure.
Response 3: Minor corrections to 1 & 2 have been made in the article.
Round 2
Reviewer 2 Report
Thank you for your improving, but I don't see your improvement in the manuscript. Please highlighted in the manuscript what has been corrected.
Author Response
Response: The modified parts of the article have been highlighted in yellow, such as the files uploaded in the attachment: Highlights. The redacted parts were adjusted according to the revised manuscripts of the editors.
Author Response File: 
Author Response.docx
Reviewer 3 Report
The authors state: "Since the mass of the small ball set in this study was large, the damping effect was not considered in the theoretical analysis".
In reality, the damping effect is present in the operation of bridges and must be taken into account in the design. The question arises about the correctness of application of the model to the real calculations.
Moreover, if damping is not considered in theoretical analysis, then why C value is given in equations (1) and further?
Perhaps it was meant that the damping was not taken into account in the theoretical verification of the experiment because of its scale?
Author Response
Response: There is no doubt that the damping effect must be considered in the practical bridge, because damping is one of the important factors affecting the bridge vibration. However, in the theoretical analysis, we need to first consider the time-history response under the undamped state, because considering the damping effect is also the dynamic response under the undamped state to carry out corresponding changes. However, for the laboratory model we set, we cannot obtain the influence brought by the damping, so we apply Vaseline on the sliding track to reduce the influence brought by friction. As one of the factors affecting vibration, damping is also taken into account by the fine integral method in this paper. Therefore, the setting of the fine integral method can undoubtedly be applied to the dynamic analysis of beams with variable curvature.
Round 3
Reviewer 2 Report
Thank you for your improving.
