Lateral Dynamic Response of Helical Pile in Viscoelastic Foundation Considering Shear Deformation

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Helical pile is a new type of pile with good application prospects. Due to the existence of helixes, the dynamic interaction between helical pile and soil is an important and difficult topic in geotechnical engineering. This paper derived analytical solutions for the lateral dynamic displacement, bending moment and shear force of helical pile and conducted a parametric study based on these solutions. In general, the work in this paper is nice and should be interested by the readers of Applied Sciences. If the authors addressed the following comments, the paper can be accepted for publication in AS.

(1) As shown in Fig. 3, the soil medium used in this article is a uniform single layer soil, but the actual working condition is not single layer soil, but composed of multiple layers of soil. How does this affect the lateral dynamic response of helical pile?

(2) The vertical load caused by the upper structure of the pile body has not been considered, which is inconsistent with the actual situation. Please discuss and explain.

(3) In Eq. (6), how to obtain the value of βs?

(4) In table 1, why is the length of helical pile 4.5m?

(5) As shown in Fig. 6, what is the difference between Euler beam theory and Timoshenko beam theory?

Author Response

Helical pile is a new type of pile with good application prospects. Due to the existence of helixes, the dynamic interaction between helical pile and soil is an important and difficult topic in geotechnical engineering. This paper derived analytical solutions for the lateral dynamic displacement, bending moment and shear force of helical pile and conducted a parametric study based on these solutions. In general, the work in this paper is nice and should be interested by the readers of Applied Sciences. If the authors addressed the following comments, the paper can be accepted for publication in AS.

Answer: Thank you very much for your positive comments.

1. As shown in Fig. 3, the soil medium used in this article is a uniform single layer soil, but the actual working condition is not single layer soil, but composed of multiple layers of soil. How does this affect the lateral dynamic response of helical pile?

Answer: Thank you very much for your comments. In Fi. 3, in order to highlight the variable cross-section characteristics of the helical pile, the pile surrounding soil is set as a single layer. In fact, the present solutions can be applied to the lateral vibration problem of helical piles in layered soil. For layered soil, only the soil parameters in Eqs. (5) and (6) need to be set according to the corresponding soil layer parameters. Please check the sentences with red words in revised manuscript.

2. The vertical load caused by the upper structure of the pile body has not been considered, which is inconsistent with the actual situation. Please discuss and explain.

Answer: Thank you very much for your comments. As pointed out by the reviewer, the vertical load caused by the upper structure would have a significant impact on the lateral dynamic response of the helical pile. However, this article mainly considers helical piles in marine engineering structures, so it is assumed that the influence of the upper structure on helical piles is not considered, please see assumption 5. If the readers want to extend our solution to consider the influence of upper structure loads, it is only needed to introduce the inertia term of the upper structure in the governing equation of the helical pile, please see Eq. (7). Please check the sentences with red words in revised manuscript.

3. In Eq. (6), how to obtain the value of βs?

Answer: Thank you very much for your comments. Βs is the damping ratio of pile surrounding soil, and it can be obtained by dynamic sounding or seismic exploration methods.

4. In table 1, why is the length of helical pile 4.5m?

Answer: Thank you very much for your comments. In order to compare the present solutions in this article with those in Ref. 45, the parameters of helical pile are set according to those in Ref. 45. Meanwhile, the length of helical piles in engineering is around 4~10m, and the 4.5m selected in table 1 meets the scope of the project.

5. As shown in Fig. 6, what is the difference between Euler beam theory and Timoshenko beam theory?

Answer: Thank you very much for your comments. Compared with the Euler beam theory, the biggest advantage of the Timoshenko beam theory is that it can consider the shear deformation of the column structure, which is in line with the stress characteristics of helical piles. From Fig. 6, it can also be seen that when considering the shear deformation of helical pile, the lateral dynamic response of helical pile is larger than those obtained by Euler beam theory, which is more in line with engineering practice.

Reviewer 2 Report

Comments and Suggestions for Authors

The followings should be addressed:

1-      The paper miss related studies on the topic. Thus, the followings should be discussed to improve the quality of the submission.

 

2013. Dynamic response of vertically loaded helical and driven steel piles. Canadian Geotechnical Journal, 50(5), pp.521-535.

2019, October. Experimental evaluation of dynamic response of helical piles in dry sand using 1g shaking table tests. In 7th International Conference on Earthquake Geotechnical Engineering.

2022, August. Recent Advances in Helical Piles for Dynamic and Seismic Applications. In Conference on Performance-based Design in Earthquake. Geotechnical Engineering (pp. 24-49). Cham: Springer International Publishing.

2023. Response of Pipe Piles Embedded in Sandy Soils Under Seismic Loads. Transportation Infrastructure Geotechnology, https://doi.org/10.1007/s40515-023-00318-x

2-      The novelty of the work should be stressed in the introduction.

3-      Section 4, it is not reasonable to use a single value for the modulus of elasticity of the soil bearing in mind that the problem considered by the authors requires an accurate capture to the stiffness of the soil along the pile shaft.

4-      Section 4. It is not reasonable to verify the solution you proposed by comparing with other solutions. You need to compare either with field results or finite element modelling results to provide that your solution is valid.

5-      Section 5.1.2: what does the dimensionless frequency represent in reality. How the results you produced in this section are useful. You need to demonstrate this in the writing.

6-      The paper is silent about the effect of the l/D.

 

7-      What are the practical implications of the work. You need to discuss this in a section before the conclusion. 

Author Response

1. The paper miss related studies on the topic. Thus, the followings should be discussed to improve the quality of the submission.

Dynamic response of vertically loaded helical and driven steel piles. Canadian Geotechnical Journal, 50(5), pp.521-535.

2019, October. Experimental evaluation of dynamic response of helical piles in dry sand using 1g shaking table tests. In 7th International Conference on Earthquake Geotechnical Engineering.

2022, August. Recent Advances in Helical Piles for Dynamic and Seismic Applications. In Conference on Performance-based Design in Earthquake. Geotechnical Engineering (pp. 24-49). Cham: Springer International Publishing.

2023. Response of Pipe Piles Embedded in Sandy Soils Under Seismic Loads. Transportation Infrastructure Geotechnology, https://doi.org/10.1007/s40515-023-00318-x

Answer: Thank you very much for your comments. I have carefully read these four papers. Definitely, these four papers are very consistent with the topic of this article, and we have cited them and made corresponding description in the revised manuscript. Please check the sentences with red words in revised manuscript.

2. The novelty of the work should be stressed in the introduction.

Answer: Thank you very much for your comments. As pointed out by the reviewer, the novelty of the paper needs to be emphasized in the introduction. Therefore, the last paragraph of the introduction is rewritten as ‘In order to investigate the lateral vibration characteristics of helical pile more realistically, this paper takes the shear deformation effect of helical pile into account, and develops an equivalent stiffness method to establish the dynamic governing equations of helical pile embedded in viscoelastic foundation based on the Winkler foundation theory and Timoshenko beam theory. Then, the analytical solutions for the lateral dynamic displacement, bending moment and shear force of helical pile are strictly de-rived, and the rationality of the analytical solutions is also verified by comparing with existing solutions. Based on the obtained solutions, the influence of pile and soil parameters on the lateral vibration characteristics of helical pile with space and time response is investigated in detailed.’

3. Section 4, it is not reasonable to use a single value for the modulus of elasticity of the soil bearing in mind that the problem considered by the authors requires an accurate capture to the stiffness of the soil along the pile shaft.

Answer: Thank you very much for your comments. Since the existence of helixes would have important influence on the lateral dynamic response of helical pile, this paper mainly discusses the influence of the relevant parameters of helixes on the lateral dynamic response of helical pile. Therefore, in order to highlight the influence of pile body parameters, the pile surrounding soil is set as a single layer. In fact, the present solutions can be applied to the lateral vibration problem of helical piles in layered soil. For layered soil, only the soil parameters in Eqs. (5) and (6) need to be set according to the corresponding soil layer parameters.

Certainly, as pointed out by the reviewer, the soil in engineering practice is layered, and the layered characteristics of the soil should be considered in analyses. Our group will carefully investigate this issue in near future.

4. Section 4. It is not reasonable to verify the solution you proposed by comparing with other solutions. You need to compare either with field results or finite element modelling results to provide that your solution is valid.

Answer: Thank you very much for your comments. Since the objective of this paper is to derive analytical solutions for the lateral dynamic response of helical pile considering the influence of shear deformation, we utilized the existing solutions to verify the present solutions. This meets the general requirements of scientific research methods.

Certainly, to fully verify the rationality of the solutions in this article, it is definitely more reasonable to compare them with experimental results. However, to be honest, we have conducted several model tests on the lateral vibration of helical piles under small-scale conditions before, but the results obtained were not ideal due to the small size. Conducting large-scale on-site experiments is time-consuming and labor-intensive, and we do not have sufficient financial support. Of course, using finite element method for verification is also a very reasonable means, and our group already has a doctoral student conducting relevant research. However, the results of finite element analysis are not only used for comparative verification, but also for exploring the interaction between pile, helixes and soil. Our group will carefully investigate this issue in near future.

5. Section 5.1.2: what does the dimensionless frequency represent in reality. How the results you produced in this section are useful. You need to demonstrate this in the writing.

Answer: Thank you very much for your comments. In order to highlight the relationship between various physical quantities of helical pile more prominently, it is usually necessary to make some physical quantities dimensionless. Based on this, this article utilizes dimensionless frequency to analyze the lateral vibration characteristics of helical pile. The dimensionless frequency is equal to the circular frequency multiplied by the pile diameter and then divided by the wave velocity.

According to the comments of the reviewer, we have explained the specific application of the results in this article, as shown in the last paragraph of Introduction, ‘The relevant results can be utilized to guide the dynamic design of helical piles under lateral loads, especially suitable for helical piles in marine engineering.’

6. The paper is silent about the effect of the l/D.

 Answer: Thank you very much for your comments. In Pile Foundation Dynamics, the aspect ratio (l/D) is a very important factor that affects the vibration response of pile foundations. Therefore, when conducting parametric study in this article, we also analyzed the influence of aspect ratio on the lateral dynamic characteristics of helical, but found that this influence pattern is basically consistent with the influence of conventional cylindrical piles, so it was not included in this article.

7. What are the practical implications of the work. You need to discuss this in a section before the conclusion. 

 Answer: Thank you very much for your suggestions. We added some description in the revised manuscript, such as, ‘The relevant results can be utilized to guide the dynamic design of helical piles under lateral loads, especially suitable for helical piles in marine engineering.’; ‘Therefore, in the lateral dynamic design of helical pile, it is necessary to set a reasonable pile-soil stiffness ratio, rather than blindly increasing the pile body stiffness.’; and ‘Therefore, in the lateral dynamic design of helical pile, the correlation between the natural frequency of the helical pile-soil system and the load frequency should be considered carefully to avoid the generation of resonance frequency.’

Reviewer 3 Report

Comments and Suggestions for Authors

The manuscript has presented an analytical method developed based on the Winkler foundation model and Timoshenko beam theory to establish the lateral vibration model of helical piles considering shear deformation. Based on the developed analytical method, the manuscript has investigated the properties of helical piles and the soil mass surrounding the pile according to the lateral dynamic response of the helical pile. The conclusions drawn from this investigation are: the load excitation frequency and pile-soil stiffness ratio have a significant influence on the lateral dynamic displacement, bending moment, and shear force of helical pile with space and time response. This is an interesting manuscript and this manuscript aligns perfectly with the journal's scope.

However, there are some unclear problems in the manuscript, which need a major revision before it can be published. Consider my comments for improvement:

1, The equivalent conversion when calculating helical piles to equivalent cylindrical piles of special diameter needs to be clarified. According to pages 3 and 4, lines 93 to 133, the helical pile has been used the equivalent stiffness method to transform helical pile to cylindrical pile of special diameter. This is not really convincing because the performance of helical piles does not only depend on the stiffness of the pile;

2, In this manuscript, to solve the governing equation more conveniently and ensure the universality of the results, the following assumptions are adopted during the analysis:

(1) The pile surrounding soil is homogeneous, isotropic and viscoelastic medium. Therefore, the stiffness and damping coefficients of the soil layers are both constants along the vertical direction.

(2) The pile body is a combination of circular and rectangular section, and only the bending deformation of the pile is considered after simplification.

(3) The pile-soil system is subjected to small deformations and strains during lateral vibration, and the longitudinal displacement of pile surrounding soil is ignored.

(4) No relative sliding occurs at the pile-soil interface.

(5) The influence of pile cap is not considered.

(6) The harmonic excitation acts horizontally on the pile top.

According to the reviewer, simplifying the calculation model mentioned above will make the research results less convincing. Normally, helical piles are used to increase project stability and work in different layers of soil and rock with varying properties. Converting the pile's surrounding environment into a uniform and consistent soil and rock, and assuming it works solely in a viscoelastic environment, will not accurately represent the actual working condition of the pile.

It should also be mentioned that the phenomenon of relative slippage between the pile and the soil cannot be ignored. When studying dynamic loads, the relative slippage between the pile and the surrounding soil must be separately taken into account;

3, The graphs presented the influence of element division accuracy on lateral dynamic response of helical pile appearing in the pile corresponding to cases where the number of element division is different (S = 20, 40, 60 and 80) do not clearly indicate the difference (Fig. 4);

4, Similarly, the graphs in Fig.5 also need to be scaled to clarify the differences in calculation cases;

5, It is necessary to include units for quantities in equations and Figures.

6, The results in this manuscript are quite interesting. However, it's important to have actual field data or simulations using numerical software (Plaxis2D, 3D, or FLAC3D) that simulate the working of helical piles. This comparison will validate the accuracy of the research results published in this manuscript.

Comments on the Quality of English Language

Minor editing of English language required

Author Response

The manuscript has presented an analytical method developed based on the Winkler foundation model and Timoshenko beam theory to establish the lateral vibration model of helical piles considering shear deformation. Based on the developed analytical method, the manuscript has investigated the properties of helical piles and the soil mass surrounding the pile according to the lateral dynamic response of the helical pile. The conclusions drawn from this investigation are: the load excitation frequency and pile-soil stiffness ratio have a significant influence on the lateral dynamic displacement, bending moment, and shear force of helical pile with space and time response. This is an interesting manuscript and this manuscript aligns perfectly with the journal's scope.

However, there are some unclear problems in the manuscript, which need a major revision before it can be published.

Answer: Thank you very much for your positive comments.

1. The equivalent conversion when calculating helical piles to equivalent cylindrical piles of special diameter needs to be clarified. According to pages 3 and 4, lines 93 to 133, the helical pile has been used the equivalent stiffness method to transform helical pile to cylindrical pile of special diameter. This is not really convincing because the performance of helical piles does not only depend on the stiffness of the pile.

Answer: Thank you very much for your comments. The dynamic interaction between pile and soil is a complex problem, and the existence of helixes makes the dynamic interaction between helical and soil more complex. It is not easy to establish a reasonable model for the interaction between pile, helixes and soil. Therefore, this paper utilized the equivalent stiffness method to transform helical pile to cylindrical pile of special diameter, which is proved by many scholars (see: Zhang, X.C.; Bai, Y.C.; He, Z.Q.; Zhu, A. Research on the dynamical response characteristics of steel screw pile under lateral vibration. Chin. J. Constr. Mach. 2019, 17, 547-553.; Wang, C.Z.; Wu, J.; Wang, L.X.; Liu, H.; Wu, W.B. Horizontal vibration characteristics of screw pile in viscoelastic foundation. Journal of Central South University (Science and Technology), 2022, 53, 2279-2289.).

Certainly, as the reviewer pointed out, if we want to investigate the lateral vibration characteristics of helical pile more realistically, it is needed to consider the dynamic interaction between pile, helixes and soil. This is a very meaningful scientific issue, and our group will continue to study it in future by means of experiment and Finite Element Method.

2. In this manuscript, to solve the governing equation more conveniently and ensure the universality of the results, the following assumptions are adopted during the analysis:

(1) The pile surrounding soil is homogeneous, isotropic and viscoelastic medium. Therefore, the stiffness and damping coefficients of the soil layers are both constants along the vertical direction.

(2) The pile body is a combination of circular and rectangular section, and only the bending deformation of the pile is considered after simplification.

(3) The pile-soil system is subjected to small deformations and strains during lateral vibration, and the longitudinal displacement of pile surrounding soil is ignored.

(4) No relative sliding occurs at the pile-soil interface.

(5) The influence of pile cap is not considered.

(6) The harmonic excitation acts horizontally on the pile top.

According to the reviewer, simplifying the calculation model mentioned above will make the research results less convincing. Normally, helical piles are used to increase project stability and work in different layers of soil and rock with varying properties. Converting the pile's surrounding environment into a uniform and consistent soil and rock, and assuming it works solely in a viscoelastic environment, will not accurately represent the actual working condition of the pile.

It should also be mentioned that the phenomenon of relative slippage between the pile and the soil cannot be ignored. When studying dynamic loads, the relative slippage between the pile and the surrounding soil must be separately taken into account.

Answer: Thank you very much for your comments. Since the existence of helixes would have important influence on the lateral dynamic response of helical pile, this paper mainly discusses the influence of the relevant parameters of helixes on the lateral dynamic response of helical pile. Therefore, in order to highlight the influence of pile body parameters, the pile surrounding soil is set as a single and viscoelastic layer. In fact, the present solutions can be applied to the lateral vibration problem of helical piles in layered soil. For layered soil, only the soil parameters in Eqs. (5) and (6) need to be set according to the corresponding soil layer parameters.

As shown in Ref. 1 and 18 (see: 1.   El Naggar, M.H.; Youssef, M.A.; Ahmed, M. Monotonic and cyclic lateral behaviour of helical pile specialized connectors. Eng. Struct. 2007, 29, 2635-2640.; 18. El Naggar, M.H. Recent advances in helical piles for dynamic and seismic applications[C]. Proceedings of the 4th International Conference on Performance Based Design in Earthquake Geotechnical Engineering, 2022, 24-49.), when a helical pile undergoes horizontal cyclic dynamic loads, a gap effect will occur on the pile side, resulting in lateral void of the interface of pile and soil, instead of upward and downward sliding. This phenomenon was discovered through experiments, but it is difficult to establish physical and mathematical models for it. Just as it is difficult to establish a theoretical model for pile-soil interaction problems under most large deformation conditions, it is also difficult to establish a theoretical model for the interaction problem between helical and soil under large deformation conditions. Therefore, this paper makes the corresponding assumptions, namely, ‘(3) The pile-soil system is subjected to small deformations and strains during lateral vibration, and the longitudinal displacement of pile surrounding soil is ignored.’ and ‘(4) No relative sliding occurs at the pile-soil interface.’. These assumptions apply to the design of dynamic foundations on helical piles.

In view of this, a description of the limitations of this article is added in the revised manuscript. Please check the sentences with red words in revised manuscript.

3.The graphs presented the influence of element division accuracy on lateral dynamic response of helical pile appearing in the pile corresponding to cases where the number of element division is different (S = 20, 40, 60 and 80) do not clearly indicate the difference (Fig. 4).

Answer: Thank you very much for your comments. When S changes in a small range, it will have a significant impact on the lateral dynamic response of the helical pile. However, as S further increases, the calculation results of the pile-soil dynamic response will gradually converge, as shown in Figure 4 with S=80. Similar results were also found in Ref. 50 (see: Wang, C.Z.; Wu, J.; Wang, L.X.; Liu, H.; Wu, W.B. Horizontal vibration characteristics of screw pile in viscoelastic foundation. Journal of Central South University (Science and Technology), 2022, 53, 2279-2289.). This phenomenon also conforms to the general laws of most pile-soil dynamic interactions.

4. Similarly, the graphs in Fig.5 also need to be scaled to clarify the differences in calculation cases.

Answer: Thank you very much for your comments. To verify the rationality of the present solutions, we compared the present solutions with the solutions presented by Hu et al. [49]. Due to the same parameters taken by the two analytical solutions, the obtained pile-soil vibration characteristics should be relatively close, which also proves the rationality of the present solutions in this paper.

5. It is necessary to include units for quantities in equations and Figures.

Answer: Thank you very much for your comments. We already made corresponding revision in the revised manuscript, please check the sentences with red words in revised manuscript.

6. The results in this manuscript are quite interesting. However, it's important to have actual field data or simulations using numerical software (Plaxis2D, 3D, or FLAC3D) that simulate the working of helical piles. This comparison will validate the accuracy of the research results published in this manuscript.

Answer: Thank you very much for your comments. Since the objective of this paper is to derive analytical solutions for the lateral dynamic response of helical pile considering the influence of shear deformation, we utilized the existing solutions to verify the present solutions. This meets the general requirements of scientific research methods.

Certainly, to fully verify the rationality of the solutions in this article, it is definitely more reasonable to compare them with experimental results. However, to be honest, we have conducted several model tests on the lateral vibration of helical piles under small-scale conditions before, but the results obtained were not ideal due to the small size. Conducting large-scale on-site experiments is time-consuming and labor-intensive, and we do not have sufficient financial support. Of course, using finite element method for verification is also a very reasonable means, and our group already has a doctoral student conducting relevant research. However, the results of finite element analysis are not only used for comparative verification, but also for exploring the interaction between pile, helixes and soil. Our group will carefully investigate this issue in near future.

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors refused to provide rational validation for their results. Thus, I cannot recommend this paper for publication and it should be rejected.

Author Response

The authors refused to provide rational validation for their results. Thus, I cannot recommend this paper for publication and it should be rejected.

Answer: Thank you very much for your comments. Comparing the theoretical solution with existing theoretical solutions to verify the rationality of the theoretical solution is a common method commonly used by scientific and technological investigators. Therefore, the author believes that the comparative method used in this article is reasonable.

Certainly, if the obtained solution can be compared with the experimental results, the theoretical solution will be verified from the perspective of measured data, which is definitely a reasonable verification method. However, as this article mainly focuses on the theoretical study of the lateral vibration characteristics of helical pile, it may be more reasonable to use a comparison method with the theoretical solution, for this can more accurately set the comparison conditions and parameters. Meanwhile, the model test and on-site test of helical piles are valuable research directions, and our group will raise research funds to conduct in-depth research on this issue.

Reviewer 3 Report

Comments and Suggestions for Authors

After reviewing the edited document again, the reviewer confirmed that the requested revisions had been made and found them satisfactory for publishing in the Journal. The manuscript has been accepted for publication.

Author Response

After reviewing the edited document again, the reviewer confirmed that the requested revisions had been made and found them satisfactory for publishing in the Journal. The manuscript has been accepted for publication.

Answer: Thank you very much for your positive comments.