Simplified Calculation Method for the Natural Frequencies of the Offshore Wind Turbine Structures with Monopile Foundations Based on Euler–Bernoulli Beam Theory

Jiang, Yanpei; Zhou, Xiaomin; Liu, Yong

doi:https://doi.org/10.1155/2023/8262901

Advances in Civil Engineering

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Abstract Introduction Conclusion Appendix Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 8262901 | https://doi.org/10.1155/2023/8262901

Simplified Calculation Method for the Natural Frequencies of the Offshore Wind Turbine Structures with Monopile Foundations Based on Euler–Bernoulli Beam Theory

Yanpei Jiang,¹Xiaomin Zhou,¹and Yong Liu¹

Academic Editor: Agathoklis Giaralis

Received14 Feb 2023

Revised03 Jun 2023

Accepted06 Jun 2023

Published28 Jul 2023

Abstract

The wind and wave loads on the offshore wind-turbine (OWT) structures with monopile foundations occur at specific frequencies. When these excitation frequencies are close to the natural frequencies of OWT structures, resonance can disturb the proper operation of the power-generation equipment and shorten the service life of the structural system. Therefore, to ensure safe and efficient operation, the natural frequencies of the OWT structures with monopile foundations must be determined. To this end, a calculation method for the natural frequencies of the OWT structures with monopile foundations is proposed. This method, which considers the mass ratio (the ratio of the lumped mass on the top of the tower to the total mass of the tower, the transition piece, and the monopile above the mudline) as well as the nonuniform moment of inertia of the structures above the mudline and the soil–structure interaction (SSI), is derived using the Euler–Bernoulli beam differential equation and slope-displacement equation of a monopile according to the flexibility matrix, stiffness equivalent principle, and virtual work principle. Finally, the natural frequency calculation method is compared with finite element simulation and other calculation methods.

1. Introduction

At the Paris Climate Conference in 2015, more than 190 countries negotiated an agreement to curtail climate change, hoping to replace fossil fuels with green energy worldwide [1]. Moreover, sustainable development policies promote the transition from traditional energy to new energy [2]. In addition, the gradual depletion of hydrocarbon reserves is pushing the energy market toward a clean and sustainable path [3]. As a potential clean energy alternative, wind-energy technology is gradually shifting from onshore to offshore. Compared with similar onshore technologies, offshore wind energy technology has certain advantages [4], including higher wind velocity, larger wind turbines, and broader installation areas. In recent years, as the costs have decreased and generator dimensions and power have increased, offshore wind power consumption has increased significantly [5]. Based on the price in 2017, the cost per MW-hour of offshore wind farms is lower than that of nuclear power plants, and it can become competitive to natural gas and other energy sources in the near future [6].

Among offshore wind power support structures, the monopile option is the most prevalent, accounting for approximately 80% of the installed support structures [7]. The monopile foundation is an economical option for offshore wind power [8]. A monopile is a simple pipe segment driven into the seabed, and its advantages include easy production and installation as well as low-construction cost and risk [9]. The main loads on offshore wind turbines (OWTs) are of dynamic or cyclic nature, which makes the support structure highly sensitive to dynamic loads [10]. Therefore, it is important to fully understand the structural frequency and adjust the natural frequency of the structure and its components during the design stage [11]. By adjusting the natural frequency, the resonance caused by the wind, waves, and wind turbine operation can be avoided, and the service life of the structure can be prolonged. The wind, wave, and operating frequencies of the wind turbines are shown in Figure 1.

Currently, the structural designs are either soft–soft, soft–stiff, or stiff–stiff. The soft–soft design is used for frequencies below 1P, the soft–stiff design is used for frequencies between 1 and 3P, and the stiff–stiff design is used for frequencies above 3P [11]. With the soft–soft design, the structure tends to be excessively deformed, which can affect the operation of the OWT, and the natural frequency of the structure can become close to the frequency of the external wind and wave loads, resulting in resonance. With the stiff–stiff design, more structural steel is used, which increases the costs. Most of monopile OWTs adopt soft–stiff design to reduce the construction costs [12]. When monopiles are used for large-size turbines, it is difficult for the designer to avoid the soft–soft design [13]. Another design strategy for OWTs, which are becoming increasingly larger in size (with a consequent shift of the natural frequency toward the resonance range), is the implementation of vibration control devices [14–35].

The frequency corresponding to the first-order bending vibration type is closer to the 1P frequency than to the 3P frequency. Moreover, under the action of wind, wave, and tide loads, offshore wind turbine structures swing back and forth and sideways, exhibiting a swinging pattern similar to that of the first-order bending vibration type. That is, the wind, wave, and tide loads stimulate the first-order bending vibration of offshore wind turbine structures. Therefore, in this study, the frequency corresponding to the first-order bending vibration of the structure was considered.

Natural frequency calculation methods include numerical and simplified calculation methods. Owing to their overall complexity, numerical calculations require computer programing. However, in terms of a simplified calculation method for the natural frequency, a calculator or spreadsheet program can be used.

Using an elastically supported Euler–Bernoulli beam, Adhikari et al. [36] derived the characteristic equation that controls the natural frequency of the structure and proposed a simplified calculation method for the frequency of OWTs considering the monopile. Arany et al. [37] studied the frequency of the OWTs with monopile foundations using mechanical and mathematical models and provided an approximation calculation formula for the natural frequency. Similarly, Arany et al. [10] proposed a simple calculation method for the natural frequency of the OWTs with monopile foundations based on the dimensions of the tower and monopile as well as the soil properties. Darvishi-Alamouti et al. [1] established a mathematical model for the soil–pile interaction based on Winkler’s method and the concept of elastic foundation beams derived a simplified calculation method for the natural frequency of the OWTs with monopile foundations using the Rayleigh method of total energy conservation of the system. Ko [38] derived a closed-form solution for the natural frequency of the wind turbine structures with tapered towers based on Rayleigh’s method.

There are also deficiencies in the methodologies reported in the literature [1, 10, 36–38]. Many researchers have used a simplified treatment of the upper tower [10, 36, 37], together with a large number of parameters and a narrow application scope. Darvishi-Alamouti et al.’s methodology [1] is applicable to cohesionless soils. However, Ko’s methodology [38] is only applicable to tapered towers. To avoid such oversights, in this study, a natural frequency calculation method is established that reasonably considers the variation of upper tower diameter and wall thickness, has a wide application range and clearer dynamic principles, and is applicable to different soils.

2. Calculation Method for the Natural Frequency of the OWT Structures with Monopile Foundations

2.1. Natural Frequency for a Structure with a Fixed Support

First, a natural-frequency calculation method that does not consider monopiles and soil below the mudline is proposed. The ideal dynamic model of OWTs with monopile foundations is shown in Figure 2.

The undamped free vibration equation for the motion of Euler beams [39] is as follows:where is the mass per unit length, is the elastic modulus of the structure, is the moment of inertia, is time, and is the space coordinate.

Suppose that the solution of the above equation has the following form:

Substituting Equation (2) into Equation (1) gives

Separating the variables in the above equation yields

To make the above equation valid, the following equation must be satisfied as follows:

Using the above equation, two ordinary differential equations and circular frequency expressions are obtained as follows:

To determine the circular frequency from Equation (8), the differential Equation (7) must be solved, and the solution can be expressed as follows:

According to the boundary conditions of the displacement, shear force, bending moment, and slope in Figure 2, the following transcendental equation can be derived as follows:

By substituting and into Equation (10), the following simplified equation is derived as follows:where ; μ is the ratio of the lumped mass on the top of the tower (total mass of the nacelle, hub, and blade) to the total mass of the tower, transition piece, and monopile above the mudline. Substituting the into Equation (8) yields

The natural frequency of OWT structures with monopile foundations can be expressed as follows:

Here, the J value can be solved with the coefficient μ corresponding to different OWT structures using Equation (11). To obtain the natural frequency, the J value must be the first solution to the transcendental Equation (11). By substituting the J value into Equation (13), the structural natural frequency without considering the soil–structure interaction (SSI) can be derived. The J values corresponding to μ are shown in Figure 3.

A monopile-supported OWT structure usually has a nonuniform moment of inertia due to variations in tower diameter and tower wall thickness, the presence of the transition piece, and the monopile wall being thicker than the tower wall. As Equation (13) is applicable only for a uniform moment of inertia, it must be corrected.(1)Moment of inertia

The structure shown in Figure 4 is clearly divided into three segments: 0∼∼, and ∼. There is unnoticeable change in the diameter and wall thickness of segment ∼, whereas there is a noticeable change in the diameter and wall thickness of segment ∼. Therefore, it is necessary to consider the changes in the diameter and wall thickness of the tower based on the two sections when calculating the moment of inertia of the tower.

From Figure 4, the outer diameter function can be expressed as follows:and the inner diameter function can be expressed as

D and are functions of ; therefore, the moment of inertia can be expressed as a function of :

The weight number of the microelements in segment ∼ in Figure 4 is given byand the proportion of the moment of inertia of the microelement in segment ∼ is expressed as follows:

The moment of inertia of segment ∼ corresponds to the above integral and is expressed as follows:

When segment ∼ is discrete, as is segment , its moment of inertia is given bywhere is the inertia function of segment, is the lower limit value of segment, in the coordinate system, is the upper limit value of segment in the coordinate system, and is the length of segment ∼.

In the above formula, the analytical expression of the moment of inertia of segment ∼ contains an integral formula, which is not conducive to application; therefore, it can be simplified to the following expression:where is the discrete number of segment ∼ of the tower, is the moment of inertia of segment , and is the length of segment .

The equations for and have the same form as Equation (21). In practical engineering, the diameter and thickness of offshore wind towers change from bottom to top. The effect of this change on the structural stiffness through the moment of inertia is now considered.(2)Solution to equivalent natural frequency problem

In Figure 2, the moment of inertia is uniform, which is an ideal model. The moment of inertia varies among the diameter-varying segment of the tower (the tapering segment), the segment with constant tower diameter or the segment with mildly varying tower diameter, the transition piece, and the monopile. To apply Equation (13) for the calculation of the natural frequency of the OWT structures with monopile foundations, the moment of inertia of the diameter-varying segment of the tower, the segment with constant tower diameter or the segment with mildly varying tower diameter, the transition piece, and the monopile, must be transformed into equivalent measures. The calculation model for the natural frequency without considering the SSI is shown in Figure 5.

According to Crotti–Engesser’s theorem, the elastic displacement can be expressed as follows:where is the complementary energy of strain and represents forces.

Regarding the OWTs, when the wind turbine load is applied on the hub on the tower top, the displacement at the tower top can be expressed as followswhere is is the length of the diameter-varying segment of the tower, is the length of the segment with constant tower diameter or the segment with mildly varying tower diameter, is the length from the mudline to the tower bottom, is the flexural stiffness of the diameter-varying segment of the tower, is the flexural stiffness of the segment with constant tower diameter or the segment with mildly varying tower diameter, and is the flexural stiffness from the mudline to the tower bottom.

Substituting , and into the above equation yieldsand the structural displacement under unit load can be expressed as follows

According to the definition of stiffness, the structural stiffness is the reciprocal of Equation (25).

Another structural stiffness can be expressed in the following equation:

Now, we equate Equations (26) and (27). For an OWT with a monopile foundation made of steel, the elastic modulus of steel can be considered as the elastic modulus at all locations, and all the moduli are equal (). The elastic moduli at both sides of the equal sign can be reduced, yielding the following simplified equation:

If the structure above the mudline is a linearly tapered tower, can be expressed as follows:

The equivalent moment of inertia is introduced through the stiffness, which is approximately equal to the generalized stiffness. Therefore, the equivalent moment of inertia can be introduced through the generalized stiffness, and the equivalent mass can be introduced using the generalized mass. For a structure with one fixed support and one free end, the generalized mass [39] is expressed as follows:

With the equivalent mass introduced through the generalized mass, the coefficient in Equation (11) can be corrected toand Equation (13) can be corrected to

2.2. Natural Frequency with SSI

The following analysis focuses on the eigen solutions of a linear system; therefore, the nonlinear soil behavior is not modeled. The foundation systems generally do not go into a nonlinear regime, and therefore, a linear approximation is considered acceptable [10]. An OWT structure with a monopile foundation includes a nacelle, hub, blade, tower, transition piece, and monopile. The interaction between the monopile and soil below the mudline can be represented by three flexibility dimensions, as shown in Figure 6: horizontal flexibility , rotational flexibility , and coupling flexibility . Generally, the settlement at the monopile foundation root is not considered; therefore, the inclined spring model is used, as shown in Figure 6.

The horizontal force and bending moment on the mudline as well as the horizontal displacement and slope on the mudline of a monopile can be expressed using the following equations:where is the horizontal force on the mudline, is the bending moment on the mudline, is the monopile displacement on the mudline, and is the monopile slope on the mudline.

With a load on the tower top, the support displacement is . To calculate the displacement on the tower top, a virtual force system must be provided based on the virtual work principle. With a nonzero-force increment imposed on the tower top, the nonzero-force increment at the support is , as shown in Figure 7.

The increments of the bending moment and end reaction are expressed as follows:

The virtual force system performs virtual work in the deformation state, and its virtual force equation is given as follows:

Substituting Equations (34) and (35) into Equation (36) gives

Since the support displacement and support reaction are functions of , this yields

By substituting Equation (38) into Equation (37), the following simplified equation is obtained as follows:

The first integrand to the right of the equal sign in Equation (39) is the complementary strain energy density; therefore, Equation (39) can be expressed as follows:

The support displacement is given by and , whereas the support reaction is given by and −F_bL_t. Using and , the support reaction can be expressed as and . Substituting Equation (33) into Equation (40) yields

Figure 8 shows the deformation of the loaded tower top.

According to the deformation diagram of the loaded tower top, the total displacement of the tower top can be expressed as

Equations (41) and (42) are derived from the virtual work equation and geometric deformation conditions, respectively. With a smaller , Equations (41) and (42) become equal according to the concept of the equivalent infinitesimal, and therefore, the two equations share the same expression. Therefore, with a smaller , either Equations (41) or (42) can be used to represent the total displacement of the tower top after it is stressed. Equation (42) is selected to represent the total displacement at the tower top after the structural tower top is stressed.

Substituting Equation (33) into Equation (42) yields

With and , Equation (43) simplifies to

Considering the influence of the SSI on the natural frequency of the structure, an equivalent structure can be introduced to make the displacements at the top of the two structures equal when they are subjected to the same load. These two structures are illustrated in Figure 9.

The displacement of the tower top under stress is thus expressed as follows:

Under the same load, the top displacements of the two structures are equal:

Under a unit load, the displacements of both structures are equal and can be expressed as follows:

After converting the numerator and denominator at both sides of the equal sign in Equation (47), the equation remains valid. At this time, both sides of the equal sign represent the structural stiffness and can be further simplified as

The calculation equation of the structural natural frequency after considering the SSI is

The symbol explanations are given in Table 1. Elements and of the flexibility matrix are given in Appendix A.

The natural frequency of an OWT structure with a monopile foundation considering the SSI is then calculated using the following four steps:(1)Calculate the moment of inertia above the mudline for the structure.(2)Determine whether the monopile is rigid or slender. Depending on the soil properties, determine whether the monopile is rigid or slender using Equations (A3), (A4), (A8), and (A9). For a multilayer soil, this can be determined based on the percentage of the total thickness of the cohesive soil and the total thickness of the cohesionless soil in the embedded length of the monopile.(3)Calculate the flexibility matrix. According to the soil properties and depending on whether the monopile is rigid or slender, calculate the flexibility matrix using Equations (A5), (A6), (A10), and (A11).(4)Determine the natural frequency of the structure. First, substitute the elements in the flexibility matrix into Equation (48) to calculate the equivalent moment of inertia . Second, substitute the equivalent moment of inertia into Equation (49) to obtain the natural frequency of the structure.

3. Simulation Comparison

3.1. Parameter Selection for a Finite Element Model

Three finite element models are employed to enable differentiation and diversity of the study objects. In each case, the following assumptions are made: the total mass of the blade, hub, and nacelle is 243,000 kg. The values of , and are 35.877, 39.833, and 19 m, respectively. The monopile length is 74 m, and the monopile embedded length is 55 m. Poisson’s ratio is 0.3, Young’s modulus is 206 Gpa, and the density is 7,850 kg/m³. For Model 1, the outer diameter of the monopile is 5 m, and the wall thickness of the monopile is 0.06 m. For Model 2, the outer diameter of the monopile is 5.5 m, and the wall thickness of the monopile is 0.06 m. For Model 3, the outer diameter of the monopile is 5 m, and the wall thickness of the monopile is 0.08 m. The parameters of the upper tower section of the OWT are shown in Figure 10. Finite element models are shown in Figure 11. For the soil model, a hardened soil with a small-strain stiffness is employed. The parameters for the hardened soil with small-strain stiffness are given in Table 2.

In the finite element models, it is necessary to increase the elastic modulus and gravity density of the soil to reduce the horizontal displacement of the structural mudline. Using Model 1 as an example, the calculation results are shown in Figure 12.

As shown in Figure 12, with a gravity density of soil of 7.7E5 and elastic modulus of 2.06E12 kPa, the horizontal displacement of the structural mudline is quite small, close to zero. At that time, the position of the mudline is similar to that of a fixed support; therefore, the SSI is ignored. In the following finite element calculation, the elastic modulus and gravity density of the soil are 2.06E12 kPa and 7.7E5 kN/m³, respectively, without considering the SSI.

3.2. Comparative Analysis of Calculation Results

A horizontal load is applied to the tower top in the model, which is later removed to allow the structure to vibrate freely. The calculation results are shown Figure 13. The natural frequency of the structure can be obtained by evaluating the time required for each cycle.

According to Table 2, the total thickness of the sandy soil layer is 10.5 m, accounting for only 19.091% of the embedded length of the monopile (55 m), whereas the total thickness of the cohesive soil layer is 44.5 m, accounting for 80.909% of the embedded length of the monopile (55 m). Therefore, whether the monopile is rigid or slender is determined by the cohesive soil. According to Equations (A3) and (A4), the monopile embedded lengths in the three models are closer to the determination value of the rigid pile in Equation (A4); therefore, the monopiles in the three models are considered as rigid piles. For the layered soil, the element value in the flexibility matrix in Equation (48) is the weighted average value. To determine this, the elements in the flexibility matrix of each layer are first calculated, then multiplied by the thickness of the soil layer, and finally superimposed and divided by the monopile embedded length. The calculation results are shown in Table 3.

Different methods are used to calculate the natural frequency of the finite element model, which will be compared with the finite element calculation results. The calculation results are listed in Table 3.

According to Table 3, the calculation method for the natural frequency of the OWT structures with monopile foundations considering the SSI can consider the nonuniform moment of inertia above the mudline of the structure and the SSI into account when calculating the natural frequency of the structure. The natural frequencies of the three models are calculated and compared with the finite element results to validate this method. As shown in Table 3, the natural frequencies of the models are reduced after considering the SSI.

Compared with the calculation results of the finite elements model, the calculation results for the three models using the method proposed by Arany [10] are significantly different, with deviations ranging from 23.510% to 27.089%. The calculation results for the three models using the method proposed by Yang [40] are also significantly different, with deviations ranging from 3.974% to 22.985%. The reason for the significant deviation in the calculation results of the two methods is that the three models are beyond the scope of application of both approaches. Compared with the calculation results from the finite element model, the calculation results for the three models using the method proposed by Ko [38] are different, with deviations ranging from 2.318% to 12.392%.

To investigate the effect of μ on the frequency, only the mass of in Models 1, 2, and 3 is changed. In Figures 14–17, the coefficient μ in Equation (11) is employed.

The natural frequency decreases as μ increases. The deviation between the finite element calculation results and the calculation results of the simplified method is small, indicating that the simplified method used in this study meets the engineering accuracy requirement.

3.3. Other Simulation Comparison

The OWT parameters are obtained from relevant literature [41]. The bottom of the tower is fixed. The structure above the mudline is only a tapered tower; thus, Equation (29) is used to calculate . The calculation results are listed in Table 4.

According to the comparison of the calculation results between the four natural-frequency simplified calculation methods and the numerical calculation, the difference between the Arany method and the numerical calculation is large, indicating that the model with such geometric features is beyond the scope of application of the Arany method.

The OWT parameters are obtained from relevant literature [42]. It should be noted that the 1P range for the 10 MW DTU corresponds to 0.1–0.16 Hz, whereas the 3P range corresponds to 0.3–0.48 Hz [43]. Considering a safety margin of 10%, the “allowable” frequency range for the OWT is thus 0.176–0.273 Hz [42]. The calculation results are listed in Table 5.

Without considering the SSI, the calculation results of the Yang and Arany methods are significantly different from the finite element calculation results. This difference is significantly reduced when the SSI is considered. This is because the finite element calculation results considering the SSI are reduced by 11.106% compared with the finite element calculation results without the SSI, and the single-digit percentage is reduced by other methods. The results of the four simplified calculation methods are within the allowable range.

Whether or not the SSI is considered, the simplified calculation method proposed in this paper performs well for tapered and multisegment towers. Therefore, this simplified calculation method has a wide range of applications.

When the SSI is not considered, Ko’s method performs well for tapered towers. It is worth noting that there are errors in Equations (11) and (15) in Ko’s paper [38] and that the method can only be used after modifying the formulas. The Ko calculation method is suitable for models with tapered towers and substructures above the mudline and for models with only a tapered tower above the mudline. This method considers the changes in the moment of inertia and mass of the tapered tower.

In the Yang calculation method, it is assumed that the tower wall thickness is constant and the structure above the mudline is a tapered tower. Therefore, this method is suitable for models with only a tapered tower above the mudline.

In the Arany calculation method, it is assumed that a tapered tower with a variable wall thickness is ideally equivalent to a tower with a constant diameter and wall thickness and that the structure above the mudline comprises a tapered tower and substructure (monopile and transition piece). Therefore, this method is suitable for models with tapered towers and substructures above the mudline.

4. Conclusion

Based on the Euler–Bernoulli beam theory, an approximation algorithm was proposed to determine the natural frequencies of the OWT structures with monopile foundations. The algorithm considers the effects of changes in the diameter and wall thickness of the structure above the mudline on the natural frequency, using segmentation and a weighted average. The effect of the soil around the monopile on the natural frequency is also considered. Unlike existing numerical methods, this algorithm does not require programing. Compared with other approximation algorithms for the natural frequencies of the OWT structures with monopile foundations, the algorithm can consider not only the impact of the structural wall thickness changes on the natural frequency but also the impact of more complex cross-sectional changes; thus, the algorithm provides more accurate calculations and can be applied more widely.

Moreover, a coefficient μ was introduced to consider the influence of the changes in the lumped mass on the tower top and in the total mass of the tower, the transition piece, and the monopile above the mudline of the structure on the natural frequency. By solving the transcendental Equation (11), the relationship between μ and J can be established.

In addition, two finite element calculation models were established: without and with consideration of the SSI. According to the finite element calculations, the natural frequency calculated using the former model is higher than that calculated using the latter. In the calculation using the model without considering the SSI, it was found that the natural frequency of the OWT structure is overestimated because the model does not adequately include the displacement and slope of the position of the monopile mudline.

Appendix

A. Flexibility matrix

The calculation method for the flexibility matrix of rigid and slender piles is provided by Poulos and Davis [44] based on the constant or linear modulus of the subgrade reaction .(1)Cohesive soil

The modulus of the subgrade reaction of the cohesive soil is a constant, which is calculated as follows [45]:where is the pile diameter, is the elastic modulus of the soil, is the elastic modulus of the pile, is the moment of inertia of the pile, and is Poisson’s ratio of the soil.

The calculation method for the slenderness parameter β, is provided by Poulos and Davis [44].(2)Slender and rigid piles in cohesive soils

Poulos and Davis [44] proposed a determination method for slender and rigid piles under the condition of constant modulus of subgrade reaction. (a) Slender pileswhere is the embedded length of the pile. (b) Rigid piles(3)Stiffness matrix of cohesive soil

The displacement and slope equations of slender piles under the condition of constant modulus of subgrade reaction are expressed in Equation (A5), as proposed by Poulos and Davis [44].where β is the slenderness parameter.

The displacement and slope equations for rigid piles under the condition of constant modulus of the subgrade reaction are expressed in Equation (A6), as proposed by Poulos and Davis [44]. (4) Noncohesive soil

The noncohesive soil modulus of subgrade reaction is linear, and it is calculated as follows [44]:where is the coefficient of the subgrade reaction and is the depth below the mudline. (5) Slender and rigid piles in noncohesive soil

Poulos and Davis [44] proposed a determination method for slender and rigid piles under the condition of linear modulus of subgrade reaction. (a) Slender pile (b) Rigid pile (6) Stiffness matrix in noncohesive soil

The displacement and slope equations of a slender pile under the condition of linear modulus of subgrade reaction are expressed in Equation (A10), as proposed by Poulos and Davis [44].

The displacement and slope equations of a rigid pile under the condition of linear modulus of the subgrade reaction are expressed in Equation (A11), as proposed by Poulos and Davis [44].

Data Availability

The authors confirm that the data supporting the findings of this study are available within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

National Natural Science Foundation of China funded project (5167082378).

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Copyright © 2023 Yanpei Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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