Abstract
To compensate for the deficiencies of the conventional Euler–Bernoulli beam-dynamic Winkler foundation model in analyzing the horizontal dynamic response of a single pile, in this paper, the Timoshenko–Pasternak model, which considers both the soil shear effect and pile shear deformation, is established to reveal the pile-soil dynamic interaction accurately. Based on Timoshenko beam theory and Pasternak foundation theory, the pile horizontal vibration control equation considering pile shear deformation and the soil shear effect is derived based on D’Alembert’s principle. On this basis, the transfer matrix method is applied to consider the layering characteristics of the soil, and the dynamic complex impedance frequency domain analytical solution of the pile top is obtained by using the initial parameter method combined with the pile bottom boundary conditions. A comparison with the existing model solution is carried out to verify the correctness and reasonableness of the model in this paper; finally, the parameters of the pile characteristics and soil properties around the pile are analyzed. The results show that (1) the pile bottom boundary condition and pile length have almost no effect on the pile top impedance after the single pile reaches the critical value of the length-to-diameter ratio. (2) The trend of pile top impedance becomes increasingly evident with increasing dimensionless frequency. (3) The pile top impedance increases with the decrease in the pile-soil elastic modulus ratio, and the changing trend becomes increasingly apparent; meanwhile, the influence of the elastic modulus of the surface soil on the pile top impedance is much more significant than the influence of the underlying soil on it.
1. Introduction
With the rapid development of engineering construction, pile foundation engineering has received more and more attention. Pile foundations are subjected to horizontal dynamic loads in complex environments, such as seismic effects [1], wave loads, and wind loads [2]. Pile-soil interaction analysis is the key to the problem of the horizontal dynamic response of pile foundations. The analysis methods are usually classified into three categories: finite element model [3], continuous medium model [4, 5], and dynamic Winkler foundation beam model [6]. Among them, the dynamic Winkler foundation beam model has been widely used because it simply and effectively describes the mechanical state of the soil on the pile side and the force performance of the pile [7]. However, the dynamic Winkler foundation beam model still has defects and ignores the influence of pile shear deformation on pile foundation deformation, and the soil shear effect is an important factor of soil dynamics that cannot be ignored [8]. Therefore, it is very important to establish a reasonable pile-soil dynamic interaction model in a complex dynamic environment.
Earlier, scholars used the dynamic Winkler foundation beam model [9] to model the foundation soil as springs and dampers, which is intuitive and straightforward but does not reflect the continuity of the soil well. Later, Nogami and Novak [10] studied the horizontal vibration characteristics of end-bearing piles by treating the soil as a three-dimensional continuous medium and characterizing the gradient variation of soil displacement and stress along the longitudinal direction. In fact, this method applies only to homogeneous soils and exaggerates the depth and diffusion capacity of the foundation. In response to the defects of the above two models, the Pasternak foundation model [11] uses two mutually independent parameters to reflect the characteristics of the soil, satisfying the engineering requirements of clear mechanical concepts and simple calculations while considering the continuity of the soil. Therefore, many scholars have studied the effects of the soil shear effect [12], foundation soil stratification [13], and soil shear stiffness [14] on the horizontal dynamic response of pile foundations based on the Pasternak foundation model.
However, to meet the pile foundation horizontal bearing capacity requirements, the pile diameter of special projects such as offshore platforms and onshore wind power is often larger, and its horizontal vibration characteristics are very different from the horizontal vibration characteristics of slender piles. Ji [15] concluded from model tests that the shear deformation of the pile is a nonnegligible factor when the pile diameter is too large. The calculation error caused by the pile shear deformation also increases with increasing pile diameter. Hu and Xie [16, 17] also came to a similar conclusion: neglecting the shear deformation during horizontal vibration of the pile would lead to significant errors. These conclusions show that it is necessary to consider the pile as a Timoshenko beam and to consider the effect of its shear deformation on the vibration response of a horizontally loaded pile.
In summary, this paper establishes the Timoshenko–Pasternak model (hereafter referred to as the T-P model, i.e., the dual shear model considering both pile shear deformation and soil shear effect) based on Timoshenko beam theory and Pasternak foundation theory for horizontal dynamic loading in layered foundations to study the effects of various influencing factors (pile bottom boundary conditions, vertical load, pile length-to-diameter ratio, dimensionless frequency, soil shear effect, and pile-soil elastic modulus ratio) on the horizontal vibration characteristics of monopiles. To this end, the established T-P model is used to derive the monopile horizontal vibration control equation based on D’Alembert’s principle. On this basis, the transfer matrix method is used to consider the layered characteristics of the soil, and the displayed expression of the force-displacement relationship at the top of the pile is obtained using the initial parameter method. Finally, the arithmetic examples are compared with the existing model solutions to verify the correctness and reasonableness of the model solutions in this paper.
2. Basic Assumptions
To facilitate the study of the problem, the pile and foundation soil properties are simplified by introducing the following basic assumptions:(1)The foundation soil is a layered, transversely isotropic, linear viscoelastic medium; the pile is regarded as a uniform, isotropic linear elastomer, and the monopile is equivalent to a circular equal-section Timoshenko beam.(2)The action of the soil around the pile on the pile is simplified to a series of distributed springs, dampers, and pure shear units. The pure shear units are connected to independent springs and dampers and can only undergo shear deformation but not compression.(3)Under the action of an external load, the pile-soil system produces steady-state simple harmonic excitation, the pile-soil close contact does not occur during the vibration process of relative slip and detachment, and its deformation is limited to the range of linear elasticity.
3. Double Shear Modeling
To study the horizontal dynamic response of a single pile under the combined action of vertical load , horizontal simple harmonic load , and sway simple harmonic load in a layered foundation, a dual shear model of the pile-soil system is established, as shown in Figure 1. The pile is equated to a Timoshenko beam and divided into N-segment pile units from top to bottom according to the layering of foundation soil; the foundation is a Pasternak foundation, and the action of the soil around the pile on the pile is simplified to a series of distributed springs, dampers, and pure shear units, where imaginary number is the dynamic load circular frequency; and is the time. The pile length and diameter are L and d, respectively, and the thickness of the ith layer of soil is .
The pile is equated to a Timoshenko beam, and the pile is divided into N-segment units according to the foundation soil stratification. Considering the pile shear deformation and rotational inertia, the force of the ith segment pile unit is shown in Figure 2. A Pasternak foundation is used for the foundation, and the action of the soil around the pile on the pile is simplified into a series of distributed springs, dampers, and pure shear units. Based on a previous study, the stiffness coefficient , damping coefficient , and soil shear coefficient [18 –20] of the ith layer of soil are expressed aswhere , , and are the elastic modulus, density, hysteresis damping ratio, and shear wave velocity of layer i soil, respectively; is the pile diameter; is the dimensionless frequency, is the horizontal dynamic excitation circular frequency, where , and is Poisson’s ratio of layer i soil; is the foundation shear coefficient ratio of layer i soil. Fwa et al. [21] suggested the value of foundation shear coefficient ratio through theoretical and experimental comparison as .
4. Analytical Solution of the Horizontal Vibration of a Single Pile
4.1. Establishing the Transfer Matrix of the Single Pile Horizontal Vibration Control Equation
According to D’Alembert’s principle, the Pasternak foundation theory is used to establish the equilibrium differential equation of horizontal vibration of the pile body unit in the foundation soil of layer i:where Qi, Mi and upi are the Section i pile unit shear force, bending moment, and horizontal displacement, respectively; , Ap and Ip are the pile density, cross-sectional area, and section moment of inertia, respectively.
According to Timoshenko beam theory, there arewhere is the section angle of the ith section of the pile unit, Ep and Gp are the elastic modulus and shear modulus of the pile, respectively, and the section shear shape factor , where is Poisson’s ratio of the pile.
Substituting (4) and (5) into (2) and (3), respectively, we obtain
The pile makes steady-state simple harmonic excitation motion, and the displacement and rotation angle can be expressed as is omitted in the following for writing convenience.
Combine (8) and (9) to reduce Equation (8) to a function on the horizontal displacement and substitute into the first-order derivative equation of (7), after finishing to obtain the fourth-order differential equation containing only the displacement variable as
4.2. Establishing the Monopile Horizontal Vibration Transmission Matrix
To overcome the numerical instability in the calculation process, according to the initial parameter method, (10) is simplified and transformed by the initial parameter method to obtainwhere , , , , .
When , the Pasternak foundation of the double shear model can be degraded to the Winkler foundation; when , , the Timoshenko beam of the dual shear model can be stained to the Euler–Bernoulli beam.
Equation (11) is a fourth-order constant-coefficient chi-square differential equation whose equation has a general solution of the initial parameters aswhere , and , , and are coefficients to be determined for the boundary conditions.
Substituting (6) into (7), the pile section turning angle can be obtained as
The relationship between the pile bending moment and shear force with displacement and turning angle is obtained from Timoshenko beam theory:
Combining equations (14)∼(17), the relationship between the deformation, internal force, and the coefficient to be determined for the ith section of the pile unit can be obtained aswhere the transfer matrix is expressed aswhere , ; , , ; , , , and .
According to the initial parameter method, the local coordinates of the top of the ith segment pile unit and the local coordinates of the bottom are substituted into (15) to obtain the relationship between the displacement, turning angle, shear force, and bending moment of the top and bottom of the ith segment pile unit as
To reduce the effect of instability of numerical inversion and eliminate the pending coefficients , , and the matrix transformation of (17) yields
Equation (18) expresses the relationship between the top and bottom displacements, turning angles, shear forces, and bending moments of the ith section of the pile unit.
The pile is divided into N pile units from top to bottom according to the stratification of the foundation soil. According to the pile continuity condition, the displacement, turning angle, shear force, and bending moment at the bottom of the unit in Section i are equal to the corresponding physical quantities at the top of the unit in Section i + 1, that is,
Using the transfer matrix method, the relationship between the displacement at the top of the pile, the rotation angle, the shear force, and the bending moment and the corresponding physical quantities at the bottom of the pile can be obtained aswhere , decomposes the transfer matrix in Equation (20) into 4 submatrices, that is,
Equation (21) is substituted into (22) to obtain
The relationship between the force and displacement at the boundary of the pile bottom is given bywhere Novak et al. [22] suggest taking the value ofwhere is the shear modulus of the soil at the base of the pile.
Combining (24) into (22) and (23), the explicit expression of the relationship between the force and displacement at the top of the pile can be obtained:where Pile top impedance .
For the end-bearing pile, the boundary conditions in the case of embedded solidity at the bottom of the pile are , and the shear force and bending moment at the end of the pile can be neglected. In this case, the pile top impedance can be degraded to .
4.3. Pile Top Horizontal Impedance and Sway Impedance
According to the definition of horizontal dynamic complex impedance at the top of the pile, we can obtainwhere and are the dynamic stiffness of the monopile horizontal vibration complex impedance and dynamic damping, respectively.
The horizontal dynamic complex impedance at the top of the pile can be expressed as the dimensionless form of horizontal dynamic stiffness and dynamic damping as follows:
According to the definition of the complex dynamic impedance of pile top sway, it is obtained thatwhere are the dynamic stiffness and dynamic damping of the monopile rocking vibration complex impedance, , respectively.
The pile top sway vibration complex impedance can be expressed as a dimensionless form of dynamic sway stiffness and dynamic damping:
5. Model Validation and Analysis
5.1. Correctness and Reasonableness of the Foundation Model
Gazetas and Dobry [20] proposed a simplified method for estimating the lateral dynamic stiffness and damping of a single pile based on the dynamic Winkler foundation model, analyzing an example in the San Francisco area of the United States. The outer diameter of the steel pipe concrete pile is 1.4 m, the wall thickness of the steel pipe is 85 mm, and the pile length is 34 m. The characteristic parameters of elastic modulus, density, shear wave velocity, Poisson's ratio, and damping ratio of foundation soil are derived from relevant papers Gazetas and Dobry [20], as shown in Table 1.
The monopile stiffness coefficients and damping coefficients are calculated using the double shear model, as shown in Figure 3, where , and the foundation is the Pasternak foundation when the foundation is degraded to the Winkler foundation. To facilitate the verification analysis, the horizontal vibration impedance of the pile is rewritten as follows: , where the monopile horizontal impedance relative to dynamic stiffness, the monopile horizontal impedance relative to dynamic damping , and the monopile static stiffness .
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Figure 3 shows that, in the low-frequency range (), the calculation results of the degraded Winkler model are different from the solution of the dynamic Winkler model of Gazetas because the dynamic Winkler model of Gazetas considers the fundamental frequency of the site soil () on the horizontal impedance. In contrast, the fundamental frequency of the site soil is not considered in this paper. When the low-frequency range is exceeded, the calculated results of the degraded Winkler model are in good agreement with the solution of the dynamic Winkler model of Gazetas, which shows the correctness of the Pasternak foundation in the double shear model. The slight error that still exists between the two in the figure is caused by the dynamic Winkler model of Gazetas not considering the pile shear deformation when the length-diameter comparison is tiny.
In addition, the relative dynamic stiffness and relative dynamic damping of the degraded Winkler model are smaller than the relative dynamic stiffness and relative dynamic damping of the T-P model solution. This difference gradually becomes more prominent as the frequency increases because the soil shear coefficient of the degraded Winkler model tends to zero, ignoring the soil shear effect. In contrast, the double shear model considers the soil shear effect fully, and the comparison results of the two model solutions illustrate the use of the double shear model. The comparative results of the two model solutions indicate the reasonableness of using the double shear model.
5.2. Correctness and Rationality of the Monopile Model
Xiong [21] obtained the horizontal vibration impedance of Timoshenko model piles in layered foundations by considering the natural layering characteristics of the soil layers. The calculated parameters are as follows: pile length , diameter , density , modulus of elasticity , and pile end embedded in bedrock. The characteristic parameters of elastic modulus, density, shear wave velocity, Poisson's ratio, and damping ratio of foundation soil are derived from relevant papers Xiong [21], as shown in Table 2.
The variation trend of the horizontal impedance at the top of the pile with the dimensionless frequency is shown in Figure 4. The degenerate Euler–Bernoulli model degenerates from the double shear model without considering the pile shear deformation (, ), and its calculated results are consistent with the trend of the Hui solution, which shows the correctness of the Timoshenko beam in the double shear model. The smaller error in the figure is caused mainly by neglecting the soil shear effect in the Hui solution.
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However, the horizontal vibration complex impedance of the degenerate Euler-Bernoulli model has the same trend as the horizontal vibration complex impedance of the T-P model. The calculation results of the two models also have some differences: the horizontal dynamic impedance and dynamic damping at the top of the pile of the degenerate Euler–Bernoulli model are larger than the calculation results of the T-P model because the degenerate Euler–Bernoulli model considers only the bending deformation and ignores the shear deformation of the pile, which makes the pile displacement smaller and the stiffness larger, while the T-P model considers both the bending deformation and shear deformation of the pile, and the calculation results are more accurate, reflecting the reasonableness of the double shear model.
6. Parameter Analysis
6.1. Influence of Pile Bottom Boundary Conditions on Pile Top Vibration Impedance
In this section, the effect of different vertical loads and extreme boundary conditions on the pile top impedance is calculated and analyzed using a double shear model, as shown in Figure 5. The calculated parameters of the pile-soil system are pile elastic modulus , density , Poisson’s ratio , and pile-soil elastic modulus ratio of each soil layer in order from top to bottom . The thickness of the first and second soil layers is 2 m, pile-soil density ratio , soil Poisson's ratio , and foundation viscous damping ratio , .(1)Figure 5 shows that there exist different effective pile lengths for the horizontal and swaying impedance dimensionless dynamic stiffness of the pile top. When the dimensionless frequency is small, the pile top horizontal dynamic stiffness depends on the static bearing capacity in the horizontal direction, while the pile top sway dynamic stiffness depends on the static bending resistance.(2)When and , the horizontal and swaying dimensionless dynamic stiffness of the pile top gradually decreases with increasing pile length-to-diameter ratio in the case of the embedded pile bottom, while the horizontal and swaying dimensionless dynamic stiffness of the pile top gradually increases with increasing pile length-to-diameter ratio in the case of the free pile bottom. At this time, it is not appropriate to simply regard the boundary conditions of the pile bottom as free and embedded, and the boundary conditions of the pile bottom should be treated according to the rigid disk theory on viscoelastic semi-infinite space. When with , the influence of the length-diameter ratio of the pile body and the different boundary conditions of the pile bottom on the horizontal and swaying dimensionless dynamic stiffness of the pile top can be ignored, and the pile can be treated as an infinitely long pile.
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6.2. Influence of the Length-to-Diameter Ratio and Dimensionless Frequency on the Complex Impedance of Pile Top Vibration
For the different length-to-diameter ratios of different single piles in actual projects, the variation of pile top impedance with pile length-to-diameter ratio for different dimensionless frequencies is studied in this section using a double shear model, as shown in Figure 6. The calculated parameters of the pile-soil system are elastic modulus , density , Poisson’s ratio , and dimensionless frequency ; the pile-soil elastic modulus ratios of each soil layer from top to bottom are , the first soil layer and the second soil layer are both 2 m thick, the pile-soil density ratio , the soil Poisson’s ratio , the foundation viscous damping ratio , and the pile bottom is embedded in bedrock.(1)The effect of the pile body length-diameter ratio on the dynamic stiffness of the pile top is shown in Figures 6(a) and 6(c). At that , time, the horizontal and swaying dimensionless dynamic stiffness decreases with the increase of length-diameter ratio; when afterward, the horizontal and swaying dimensionless dynamic stiffness decreases or is even almost unaffected by the change of the length-diameter ratio. After the pile length reaches the critical value of the length-diameter ratio, the horizontal vibration at the bottom of the pile-soil system decreases, and the effective vibration part of the pile-soil system is not affected by the change in the length-diameter ratio of the pile at this time. Figures 6(b) and 6(d) reflect that, at that time , , the horizontal, sway dimensionless dynamic damping increases with increasing length-diameter ratio. The longer the pile body is, the larger the pile-soil contact area is, and the more the pile energy is dissipated when transferred to the soil body. When afterward, the horizontal and sway dimensionless dynamic damping is not significantly affected by the change in the length-diameter ratio.(2)Figures 6(a) and 6(c) show the effect of the pile length-to-diameter ratio on the dynamic damping of the pile top. When the ratio exceeds the low dimensionless frequency range (), the horizontal and sway dimensionless dynamic stiffness decreases significantly with the increase of dimensionless frequency, and the changing trend is accelerated; however, in the low dimensionless frequency range (), the horizontal and sway dimensionless dynamic stiffness is very little affected by the change of dimensionless frequency. In the low-frequency range, the monopile impedance dynamic stiffness depends mainly on the static bearing capacity, and when the dynamic load frequency is smaller, the horizontal and sway dimensionless dynamic stiffness is not affected by the frequency change.
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In Figures 6(b) and 6(d), when beyond the low dimensionless frequency range (), the horizontal and swing dimensionless dynamic damping rises significantly with the increase of the dimensionless frequency, and the changing trend is accelerated; however, in the low dimensionless frequency range (), the horizontal and swing dimensionless dynamic damping is very little affected by the change of the dimensionless frequency because in the low-frequency range, energy dissipation is caused mainly by material damping, not energy radiation; when the dimensionless frequency increases, energy dissipation is caused mainly by energy radiation.
In addition, the effective pile length and shows that when horizontal vibration occurs in a single pile, it has different effective pile lengths in terms of horizontal bearing and energy dissipation.
6.3. Influence of the Soil Shear Effect and Pile-Soil Elastic Modulus Ratio on the Complex Impedance of Pile Top Vibration
In practical engineering, the dynamic Winkler foundation beam model is widely used, but the model suffers from the defect of discontinuous soil variation. To remedy this defect, this section investigates the effect of the soil shear effect on the pile top impedance under different pile-soil elastic modulus ratios using a double shear model considering soil shear, as shown in Figure 7. The values of the first two layers of the soil foundation coefficient ratio are not the same, and they are independent of each other from 0 to 1, i.e., and . The calculated parameters are as follows: dimensionless frequency , pile aspect ratio , elastic modulus , density , Poisson’s ratio , pile bottom free. The thickness of both the first and second soil layers is 2 m, pile-soil density ratio , soil Poisson’s ratio , and foundation viscous damping ratio . When the pile-soil elastic modulus ratios of the surface soil to vary, ; when the pile-soil elastic modulus ratios of the second layer of soil to vary, , the bottom pile-soil elastic modulus ratio .(1)From the trend (slope) of pile top impedance with foundation coefficient ratio in Figure 7 and Figure 8, the influence of soil shear effect on the dimensionless dynamic stiffness of pile top (slope of the curve) is significantly greater than the effect on the dimensionless dynamic damping of pile top (slope of the curve); with the decrease of pile-soil elastic modulus ratio (the elastic modulus of pile does not change while the elastic modulus of soil increases), the shear effect of the soil body affecting the impedance of the pile top influence tends to increase. However, in essence, the second foundation parameter of the Pasternak foundation does not change the form of monopile horizontal vibration. However, the dimensionless dynamic stiffness of the pile top in the double shear model ( in range) is significantly larger than the dimensionless dynamic stiffness of the pile top under the Winkler foundation model (when ) because the Winkler foundation model ignores the soil continuity. In contrast, the double shear model theoretically compensates. The double shear model is theoretically more reasonable. The results show that when the soil modulus of elasticity is large, it is necessary to consider the soil shear effect to accurately calculate the pile top impedance.(2)The variation in the pile top horizontal impedance and swaying impedance with the pile-soil elastic modulus ratio is shown in Figures 7 and 8. The pile top impedance is significantly influenced by the pile-soil elastic modulus ratio when the pile-soil elastic modulus ratio changes regardless of the surface soil or the submerged soil layer; as the pile-soil elastic modulus ratio decreases, the pile top dimensionless dynamic stiffness gradually increases, the dynamic damping gradually decreases, and the trend of both increases. In addition, comparing Figures 7 and 8, from the comparative analysis of the changing trend of the pile top impedance when the topsoil and second-layer soil elastic modulus ratios change, we can conclude that the pile top dynamic stiffness increases when the topsoil pile-soil elastic modulus ratio decreases. This trend is much larger than the upward trend of the dynamic stiffness of the pile top when the pile-soil elastic modulus ratio of the underlying soil layer decreases, which indicates that the elastic modulus of the topsoil has a more critical influence on the pile top impedance. Therefore, in engineering practice, it is necessary to accurately determine the soil properties of the topsoil, and it is not reasonable to simplify the layered foundation to a homogeneous foundation.
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7. Conclusions
In this paper, based on Pasternak foundation theory and Timoshenko beam theory, the Timoshenko–Pasternak model of multidirectional loaded piles in layered foundations is established. The analytical solution of the horizontal vibration response of a single pile is derived by the matrix transfer method, the complex dynamic impedance of multidirectional loaded piles in layered viscoelastic foundations is given, and the degraded model is verified and compared to verify the correctness and reasonableness of the T-P model solutions in this paper. The following conclusions are obtained from the parametric analysis.(1)For different pile bottom boundary conditions, an effective pile length exists; when , the pile top impedance will not be affected by the pile bottom boundary conditions and the change in pile length. When , the pile bottom embedded makes the pile top horizontal dynamic stiffness decrease with the increase in the length-to-diameter ratio, while the pile bottom free will make the pile top horizontal dynamic stiffness increase gradually with the length-to-diameter ratio.(2)The pile top impedance dynamic stiffness and will decrease with increasing dimensionless frequency , while the pile top impedance dynamic damping and will increase with increasing dimensionless frequency, and this trend is increasingly obvious with increasing dimensionless frequency.(3)The pile top impedance gradually increases with decreasing pile-soil elastomeric modulus ratio, and the variation trend increases. However, the influence of the pile-soil elastic modulus ratio on the pile top impedance of the submerged soil layer is much smaller than the influence of the pile-soil elastic modulus ratio on the surface soil.(4)The pile top dynamic stiffness in the Pasternak foundation is significantly greater than the pile top dynamic stiffness under the Winkler foundation model. The discrepancy (slope of pile top impedance to foundation coefficient ratio ) increases as the pile-soil elastic modulus ratio decreases.
Data Availability
The data used to support the findings of this study are included in the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research was supported by the National Natural Science Foundation of China (Grant no. 52068004), Natural Science Foundation of Guangxi Province (Grant no. 2018JJA160134), and Key Research Projects of Guangxi Province (Grant no. AB19245018).