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Learn more: PMC Disclaimer | PMC Copyright Notice Sci Rep . 2026 Mar 3;16:11895. doi: 10.1038/s41598-026-39773-0 Search in PMC Search in PubMed View in NLM Catalog Add to search Frequency domain analysis of torsional vibration of single pile in orthotropic viscoelastic layered foundation Zixin Lian Zixin Lian 1 School of Intelligent Construction and Architectural Engineering, Zhongyuan University of Technology, Zhengzhou, 450007 Henan China Find articles by Zixin Lian 1 , Yanzhi Zhu Yanzhi Zhu 1 School of Intelligent Construction and Architectural Engineering, Zhongyuan University of Technology, Zhengzhou, 450007 Henan China 2 Architectural Design and Research Institute, Zhongyuan University of Technology, Zhengzhou, 450007 Henan China Find articles by Yanzhi Zhu 1, 2, ✉ , Yongzhi Jiu Yongzhi Jiu 1 School of Intelligent Construction and Architectural Engineering, Zhongyuan University of Technology, Zhengzhou, 450007 Henan China Find articles by Yongzhi Jiu 1 Author information Article notes Copyright and License information 1 School of Intelligent Construction and Architectural Engineering, Zhongyuan University of Technology, Zhengzhou, 450007 Henan China 2 Architectural Design and Research Institute, Zhongyuan University of Technology, Zhengzhou, 450007 Henan China ✉ Corresponding author. Received 2025 Dec 9; Accepted 2026 Feb 6; Collection date 2026. © The Author(s) 2026 Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/ . PMC Copyright notice PMCID: PMC13065807 PMID: 41775780 Abstract The present paper establishes a theoretical analytical framework based on a three-parameter solid model (standard linear solid model) for the torsional vibration of a single pile in an orthotropic viscoelastic layered foundation. The three-dimensional partial differential form of the pile-soil coupling equations is reduced to a set of axisymmetric frequency-domain ordinary differential equations by introducing Hankel integral transformations. In order to construct a layered transfer matrix model that accounts for viscoelastic energy dissipation, the following steps are taken. Firstly, transverse anisotropic soil constitutive relations and interlayer continuity conditions are combined. An explicit frequency-domain solution for the torsional complex stiffness at the pile top was derived. The study systematically analysed the effects of soil anisotropy coefficients, viscoelastic parameters, and soil layer distribution on the system’s dynamic response. The present study proposes a theoretical framework for the design of pile foundations in complex soil conditions, with particular reference to torsional loading. Keywords: Orthotropy, Viscoelastic layered ground, Torsional vibration of single pile, Hankel integral transform, Three-parameter solid model, Dynamic response analysis Subject terms: Engineering, Materials science, Mathematics and computing Introduction In the context of dynamic loads, such as those occasioned by earthquakes, it is well documented that building structures, with foundations being of particular concern, are often subject to damage which can ultimately result in structural collapse. Given the extensive utilisation of pile foundations in civil engineering, marine engineering, and associated domains, research into their vibration characteristics has become imperative. As key structural components bearing dynamic torsional forces, the torsional vibration behaviour of pile foundations under complex conditions, such as wind loads and seismic actions, directly impacts the overall structural safety. Natural sedimentary strata characteristically manifest orthotropic anisotropy, manifesting layered isotropic behaviour in the horizontal direction and pronounced anisotropy in the vertical direction. Concurrently, the inherent viscoelastic behaviour of soil induces energy dissipation effects, further complicating the dynamics of pile-soil interaction. In the context of early pile foundation vibration analysis, researchers primarily utilised simplified soil interaction models, including the dynamic Winkler model and the plane strain simplification model 1 – 3 . The utilisation of virtual springs within these models serves to reduce computational complexity, whilst the simplification of wave propagation paths is another salient feature. However, they neglect interlayer soil connections and vertical stress gradient variations, thus failing to accurately reflect the three-dimensional wave effects and energy transfer mechanisms in the soil surrounding the pile. This has the potential to result in substantial inaccuracies in practical applications. Conversely, models grounded in three-dimensional viscoelastic soil media have been shown to offer a more authentic simulation of pile-soil interaction, thereby providing substantial value in the clarification of vibration mechanisms and the enhancement of the applicability of theoretical solutions. However, a review of extant research reveals notable limitations: In the field of geotechnical engineering, research has been primarily focused on the longitudinal and lateral vibration analysis of pile foundations in homogeneous isotropic soils 4 . However, a lack of a systematic theoretical framework exists for pile-top torsional vibration in orthotropic layered foundations. Furthermore, traditional homogeneous models have been found to struggle to characterise the anisotropy and inter-layer wave reflection coupling effects in layered soils 5 . This has resulted in prediction errors in torque transmission patterns. In recent years, significant progress has been made in layered foundation theory, though the majority of research has focused on specific conditions or simplified models. In their seminal paper, Liu et al. 5 proposed a torsional vibration analysis model for piles in layered unsaturated viscoelastic soil. The model’s innovative approach is in capturing the layered nature of unsaturated soil, a feat achieved by developing a multi-layer pile-soil system. However, this approach did not fully account for the directional dependence of the anisotropic stiffness matrix. Liu Hongbo et al. 6 investigated the longitudinal vibration characteristics of reinforced composite piles in viscoelastic unsaturated foundations based on elastic dynamics theory, emphasizing the non-flowing viscosity of the soil skeleton. However, their study was confined to longitudinal vibration modes. Zheng et al. 7 conducted theoretical research on the horizontal vibration response of pipe piles in viscoelastic foundations using a one-dimensional Euler-Bernoulli beam model. Nevertheless, their torsional vibration analysis based on the three-dimensional wave equation for soil remains inadequate. Yao et al. 8 experimentally validated the superiority of the three-parameter solid model in characterising soil delayed elastic response, demonstrating higher accuracy than the traditional Kelvin model. However, they failed to extend it to anisotropic conditions. Chen 9 applied an improved Sugeno fuzzy integral model to optimize composite foundation treatment schemes and derived viscoelastic solutions for settlement. However, the focus remained on static problems. In their study, Yu et al. 10 investigated the torsional vibrations of partially exposed pipe piles in saturated soil. The soil plug effect and fractional-order viscoelasticity were taken into consideration, but anisotropic coupling effects were neglected. Regarding analytical methods for layered foundations, literature 11 – 21 provides a crucial foundation, yet significant limitations persist. Gan 11 investigated the vertical dynamic response of pile foundations based on Hankel integral transforms and developed a dimension-reduction solution method, but focused solely on vertical vibration without addressing torsional modes; Zhang 12 analyzed torsional vibrations of pipe piles in radially heterogeneous saturated soil, but assumed isotropic soil behavior and failed to incorporate orthotropic constitutive models; Guan 13 investigated the longitudinal coupled vibration characteristics of large-diameter piles and soil by considering three-dimensional wave effects in the pile body, but this study was limited to longitudinal vibrations and did not incorporate viscoelastic energy dissipation mechanisms; Manh 14 investigated the torsional dynamic response of a single circular cross-section pile embedded in multi-layered soil. Miaojun et al. 15 investigated torsional vibration of pipe piles in radially heterogeneous, laterally isotropic saturated soil, but failed to integrate the constitutive relationship with a three-parameter viscoelastic model; Ma et al. 16 investigated the torsional dynamic response of end-supported piles in homogeneous unsaturated isotropic soil, but did not extend the analysis to layered foundations; Cui et al. 17 derived frequency-domain impedance solutions for pile torsional vibration in bidirectionally heterogeneous viscous damping soil, but the influence of anisotropy coefficients was not systematically quantified; Hu et al. 18 investigated the torsional vibration characteristics of pile foundations considering pile-soil interaction, but their model assumed homogeneous soil and could not reflect layered effects; Gupta et al. 19 analyzed the torsional behavior of medium-bearing piles in layered anisotropic foundations, but did not incorporate viscoelastic parameters, neglecting energy dissipation effects; Shuai et al. 20 employed a fractional-order Merchant model to investigate the transient response of partially embedded piles in layered viscoelastic saturated cross-anisotropic soil, but focused primarily on longitudinal vibrations; Wang Boyu et al. 21 , 22 investigated the longitudinal vibration response of helical piles considering three-dimensional soil wave effects and examined the influence of vertical soil reactions on pile sides in layered soils, but did not address torsional loading conditions. Although these studies have advanced the theoretical development of dynamic layered foundations, significant gaps remain. Firstly, the coupled effects of anisotropy and viscoelasticity have not been systematically incorporated into theoretical frameworks, particularly with regard to the influence of directional and frequency-dependent shear modulus characteristics on impedance responses. Secondly, the energy transfer mechanisms at the interfaces between layers lack quantitative characterisation, resulting in an inability to accurately predict wave reflection and transmission. Thirdly, the regulatory mechanism of viscoelastic parameters (such as the viscosity coefficient) on the characteristics of the impedance frequency response has not been systematically elucidated, which limits the accuracy of damping design under complex stratigraphic conditions. These limitations mean that existing models struggle to accurately reflect dynamic coupling behaviour in actual working conditions when predicting the torsional vibration of pile foundations in orthotropic viscoelastic layered foundations. This study innovatively combines the three-parameter solid model with orthotropic constitutive theory, building on existing theoretical foundations, particularly the analytical frameworks of Hankel integral transforms and state vector transfer methods. Embedding the standard linear solid model in a transversely isotropic stiffness matrix establishes a constitutive equation that characterises both the directional anisotropy and the rate dependency of soil simultaneously. This precisely describes the frequency-dependent behaviour of the shear modulus and the coupling mechanism of the anisotropy. Using Hankel integral transforms to reduce three-dimensional partial differential equations to a system of axisymmetric, frequency-domain ordinary differential equations and employing the transfer matrix method to solve the layered interface wave problem yields an explicit, frequency-domain solution for the torsional complex stiffness at the pile top. Compared to prior research, this study makes breakthroughs by systematically analysing the influence of soil anisotropy coefficients, viscoelastic parameters and stratigraphic distribution on the dynamic response, quantitatively revealing the regulatory mechanism of interlayer impedance matching on energy transfer and filling the theoretical gap in the coupling effects between orthotropic anisotropy and viscoelastic energy dissipation. This framework overcomes the limitations of homogeneous models in layered soils and provides a universal analytical tool for the torsional design of pile foundations in complex strata. It significantly enhances the accuracy with which safety margins can be predicted under dynamic torque conditions. Orthotropic viscoelastic constitutive models Three-parameter solid model The soil stress-strain relationship employs a standard linear solid model (Figure 1 ), which consists of a Kelvin element (spring in parallel with viscous element ) connected in series with an elastic spring . Its constitutive equation can be expressed in differential form: 1 Where is stress, is the first derivative of stress with respect to time (stress rate), is strain (dimensionless); represents the first derivative of strain with respect to time (strain rate), denotes the time constant (characterizing delayed response), is the elastic parameter (related to the instantaneous modulus), is the viscoelastic parameter (related to dissipation), . Fig. 1. Open in a new tab Three parameters solid viscoelastic model. Material parameters meet: 2 Here represents the viscosity coefficient (characterizing energy dissipation), denotes the elastic shear modulus (instantaneous response), signifies the shear modulus of the Kelvin model (delayed response), . In the frequency domain, the analytical expression for the complex shear modulus is: 3 Where is the complex shear modulus (characterizing the viscoelasticity of soil in the frequency domain), is the angular frequency (excitation frequency), , and is the imaginary unit (denoting complex numbers) (dimensionless). At the physical mechanism level: The elastic spring characterizes the transient elastic response; the Kelvin element ( in parallel with ) describes the delayed strain recovery behavior; and the viscous pot reflects energy dissipation effects. This model can simultaneously simulate two typical time-dependent behaviors of soil: creep (continuous deformation growth under constant loading) and stress relaxation (stress decay over time under constant strain). Constitutive model innovation: Verified through creep experiments (Table 1 ), the three-parameter solid model significantly improves accuracy compared to traditional models. Table1. Accuracy comparison of constitutive models. Model type Instantaneous modulus error Delay modulus error Relaxation time error Three-parameter solid 5.2% 7.8% 9.1% Kelvin 18.7% 23.5% 31.2% Maxwell 32.9% 41.6% 57.3% Open in a new tab Orthotropic stiffness matrix Horizontally layered soil exhibits transverse isotropy and vertical anisotropy, allowing the stiffness matrix to be simplified as: 4 where denotes the orthotropic stiffness matrix of the -th soil layer, Torsional vibration requires only the radial-circumferential shear modulus and axial-circumferential shear modulus , which expand into complex stiffness in the frequency domain: 5 is defined by Equation ( 3 ), representing the direction-dependent viscoelasticity of the soil. denotes the radial-circumferential complex shear modulus (horizontal direction),Unit: ; denotes the axial-circumferential complex shear modulus (vertical direction), denotes the horizontal complex shear modulus, ; denotes the vertical complex shear modulus, . Determination of model parameters for real soils To address the practical determination of the key parameters ( , , ) in the three-parameter solid model for real soil materials, a combination of laboratory tests, in-situ tests, and empirical correlations is recommended. This ensures the model’s parameters are grounded in measurable soil properties. Laboratory testing Creep test: A constant shear stress is applied to a soil sample, and the time-dependent strain response is recorded. The instantaneous elastic strain upon loading is used to determine the instantaneous shear modulus . The delayed elastic strain component and its asymptotic value are used to fit the Kelvin model shear modulus . The rate of the delayed strain development allows for the calculation of the viscosity coefficient . Stress relaxation test: A constant shear strain is applied to the soil sample, and the decay of shear stress over time is monitored. The relaxation time constant derived from the stress decay curve, together with the initial stress (related to ), provides another dataset for solving the parameters , , and . The accuracy of the three-parameter solid model fitted via these tests has been validated against experimental data, as shown in Table 1 , demonstrating superior performance over traditional Kelvin and Maxwell models. In-situ testing and empirical correlations For preliminary design or when laboratory data is scarce, the model parameters can be estimated based on standard in-situ test results and soil classification. The representative ranges of mechanical parameters for typical soil layers, as referenced from the Highway Engineering Geological Investigation Code (JTG C20-2011) and presented in Table 2 of this study, serve as a reliable guide. Table 2. Distribution characteristics of mechanical parameters of soil layer around pile. Soil layer types Density kg/m 3 clay 45±8 0.62±0.15 300–800 1800 pink soil 80±15 0.75±0.10 500–1200 1950 sandy soil 120±25 0.83±0.12 200–600 2100 Open in a new tab The horizontal shear modulus (which informs the parameters within and ) can be correlated with Standard Penetration Test (SPT) N-values or Cone Penetration Test (CPT) tip resistance qcusing established empirical relationships for different soil types. The values of Gh for clay, silty soil, and sandy soil provided in Table 2 are typical examples. The anisotropy ratio can be estimated based on soil deposition history and type. For instance, naturally sedimentary clays often exhibit a less than 1 (e.g., 0.62±0.15 for clay in Table 2 ), while sandy soils may have a ratio closer to 1 (e.g., 0.83±0.12 in Table 2 ). The viscosity coefficient exhibits a wide range depending on soil type and water content (e.g., 300–800 MPa·s for clay vs. 200–600 MPa·s for sandy soil in Table 2 ). Empirical values or back-analysis from dynamic field tests (e.g., spectral analysis of surface waves) can be used for estimation. This multi-faceted approach ensures that the innovative three-parameter solid model is not only theoretically sound but also practically applicable, with its key parameters determinable through standardized geotechnical investigation practices. Control equations and analytical solutions Pile-soil system modeling The geological exploration data and the range of typical soil layer parameters used in this study refer to the code for geological investigation of Highway Engineering (JTG c20-2011) 23 . The specification is an industry standard for highway engineering geological survey, providing detailed geotechnical classification, test methods and parameter guidelines. The specification text can be publicly accessed through official channels, and the official website of the Ministry of transport. The data are extracted from Chapter 3 (technical requirements for engineering geological survey) and chapter 5–8 (special geotechnical and unfavorable geological survey) of the specification, ensuring the representativeness and reliability of the parameters. The three-dimensional pile-soil interaction model established by this research institute is shown in Figure 2 . The system comprises: Concrete pile body: (radius , shear modulus ) Embedded in n layers of orthotropic viscoelastic soil (layer thickness , density ) Boundary conditions are set as follows: Pile top subjected to harmonic torsional moment Surface free Pile bottom fixed. Fig. 2. Open in a new tab Three position pile-soil interaction model. The model employs a layered soil structure comprising three strata with distinct mechanical properties: A 2-meter-thick clay layer at the top (light gray), a 3-meter-thick silty soil layer in the middle (light green), and a 5-meter-thick sandy soil layer at the bottom (light blue). The pile body consists of a 1-meter-diameter cylinder (dark gray) penetrating all soil layers. The coordinate system is defined with the pile centerline as the z-axis (depth direction), while the radial r-axis and tangential -axis form the horizontal plane. The principal material axes are explicitly marked within the clay layer, indicating the anisotropic characteristics of the soil’s mechanical properties. Model parameters are calibrated using actual engineering data, accurately reflecting the geometric features of typical pile foundation projects. Derivation of the torsional vibration control equation Fundamental assumptions and coordinate system Based on the pile-soil system model shown in Figure 2 , a cylindrical coordinate system is adopted. The pile body is modeled as a homogeneous linear elastic cylinder (radius , shear modulus ). The foundation consists of n layers of orthotropic viscoelastic soil, with each layer’s mechanical properties described by a three-parameter solid model and a transverse anisotropic stiffness matrix. Boundary conditions: free surface , fixed pile base , and pile top subjected to harmonic torsional moment . Soil motion equation Considering the circumferential motion equilibrium of a soil element, the three-dimensional wave equation neglecting body forces is: 6 In the equation, represents the radial-circumferential shear stress, denotes the vertical-circumferential shear stress, is the soil density, ; and is the circumferential displacement, Unit: m. Orthotropic viscoelastic constitutive relations From Equations ( 4 ) and ( 5 ) and the definition of the frequency-domain complex modulus: 7 Among them 8 The shear modulus and are given by the three-parameter solid model (Equation 3 ). Frequency domain form of the control equation Substituting the constitutive relationship into the motion equation ( 6 ) and applying the frequency domain resonance condition yields: 9 where is the density of the i-th soil layer, is the frequency-domain circumferential displacement, Unit: Hankel integral transformation for dimension reduction Applying a Hankel transformation of order to equation ( 9 ) : 10 This transformation converts the original partial differential equation into an ordinary differential equation with respect to depth : 11 Equation of motion for the pile body Treating the pile body as a linear elastic rod, the governing equation for torsional vibration is: 12 Where is the shear modulus of the pile body, is the polar moment of inertia of the pile, is the rotation angle of the pile, is the density of the pile body; is the radius of the pile, is the shear stress at the pile-soil interface, , determined by the soil solution at the location : 13 State vector transfer solution The general solution of the frequency-domain expression for the control equation can be represented as a combination of exponential functions: 14 Among these, the repetition factors , are undetermined coefficients, Unit: . Define the state vector: 15 The state vector precisely characterizes the coupled relationship between displacement and stress within soil layers. Through the continuity condition at interfaces, interlayer transfer relationships are established: 16 To derive the transfer matrix , the general solution (Eq. 14 ) is substituted into the state vector (Eq. 15 ). The derivation proceeds as follows: Express the state vector at the top of the layer ( ) and at the bottomin ( ) terms of the constants and . At 17 At 18 Eliminate and by solving for these constants from . For simplicity, shift the coordinate system so that (relative coordinates). Then: 19 Solve for and : 20 Substitute and into to express in terms of . This yields a linear relationship: 21 Substitute the general solution into the state vectors at and , then eliminate matrix operations to derive the transfer matrix form : 22 The transfer matrix precisely characterizes the interlayer wave coupling effect, where hyperbolic functions describe wave attenuation and phase changes within each layer. This solution employs a recursive algorithm to achieve a global solution for multi-layer foundations, avoiding the layered superposition calculations of traditional methods. After Hankel integral transformation, the pile torsional vibration equation couples with the soil solution at the interface: 23 Using the inverse transform formula, we obtain the frequency-domain impedance relationship:Using the inverse transform formula, we obtain the frequency-domain impedance relationship: 24 Where is the torsional complex stiffness at the pile top (frequency-domain impedance), . Parameter influence analysis This section systematically analyzes the effects of soil anisotropy coefficients, viscous parameters, soil layer distribution, and interface slip on the torsional stiffness of pile tops, revealing the regulatory mechanisms of each parameter in dynamic response. Through frequency-domain analytical solutions, the contributions of each factor to the impedance spectrum are quantified, providing a theoretical basis for engineering design. Influence of anisotropy coefficients The degree of soil anisotropy is characterized by the ratio of horizontal to vertical shear modulus, , which directly influences the directional dependency of the soil. Parameter analysis indicates that the anisotropy coefficient significantly influences the torsional stiffness of the pile cap. In the low-frequency range ( ), increasing from 1.0 to 2.5 elevates the real part of impedance by 23%. This reflects that enhanced horizontal shear stiffness effectively suppresses torsional deformation of the pile body, thereby improving the system’s torsional bearing capacity. Simultaneously, the resonance frequency shifts noticeably from 18 Hz to 22 Hz, representing a 22% displacement. This phenomenon is attributed to the enhanced directional dependency within the soil stiffness matrix altering the system’s wave propagation characteristics, facilitating greater dissipation of high-frequency energy. Figure 3 visually illustrates the variation of the real part of impedance with frequency at different values. As shown, increasing causes the curve to shift upward overall, with the resonance peak migrating toward higher frequencies and exhibiting increased peak amplitude. A critical inflection point occurs in the range, where the resonance peak transitions from a sharp to a broadened shape.The term "critical inflection point" refers to the specific frequency at which the system’s dynamic response undergoes a qualitative change due to the anisotropy coefficient . At this point, the rate of change of the real part of the torsional impedance with respect to frequency reaches a local maximum or minimum, indicating a transition in the wave energy distribution mechanism. Specifically, when exceeds a threshold value (approximately for the given stratum), the enhanced horizontal shear stiffness alters the wave interference pattern between soil layers. This shifts the resonance peak from a sharp, high-Q factor response (indicating concentrated energy dissipation) to a broadened, lower-Q factor response (indicating distributed energy dissipation). The criticality arises because this transition signifies a shift from stiffness-dominated to damping-influenced resonant behavior, which directly impacts the risk of resonance amplification in practical engineering. Engineers must identify and avoid operating near this inflection point to prevent excessive vibrational energy concentration. This indicates that enhanced anisotropy suppresses low-frequency deformation but may intensify high-frequency energy concentration. Fig. 3. Open in a new tab Influence of anisotropy coefficient on real part of torsional impedance at pile top. The anisotropy coefficient regulates wave propagation paths through the directional dependence of the stiffness matrix. The increase in horizontal shear modulus directly enhances radial confinement around the pile, suppressing torsional deformation and thereby boosting the real part of impedance. The shift in resonance frequency to higher values stems from the shortened wave wavelength due to increased equivalent soil stiffness, elevating the system’s natural frequency. Furthermore, anisotropy amplifies the difference between vertical and radial wave velocities, inducing constructive interference from interlayer reflections and further altering resonance characteristics. The anisotropy coefficient is a critical parameter for adjusting system resonance to mitigate risks. Engineering designs must control (where the critical value is defined in Equation 20 ) through soil improvement measures (e.g., incorporating graded aggregates) to avoid overlapping excitation frequency bands. It is recommended to prioritize optimizing in seismic designs to enhance low-frequency torsional bearing capacity. Effects of viscosity parameters The viscosity coefficient is a key parameter governing energy dissipation in soil. When increases from 200 MPa·s to 2000 MPa·s, the peak of the reactive impedance component decreases significantly by 40%, indicating that viscous dissipation effectively attenuates vibrational energy. Concurrently, the resonance peak exhibits broadening, with the half-width increasing from 4.2 Hz to 5.1 Hz, representing a broadening rate of 21.4%. Phase analysis further reveals that viscous delay induces a 25° phase lag, significantly altering the system’s dynamic response characteristics. Figure 4 compares the imaginary impedance spectra corresponding to different values. The blue shaded regions indicate enhanced energy dissipation bands. It is evident that under high operating conditions, the resonance peak amplitude significantly decreases, the curve broadens, and phase delay intensifies. The inflection point occurs at , where the change in imaginary slope reflects the maximization of the viscoelastic delay effect. Fig. 4. Open in a new tab Influence of viscosity parameters on imaginary part of torsional impedance at pile top. The viscous parameter dissipates vibrational energy through viscous pot elements in the three-parameter solid model. Increasing prolongs stress relaxation time, causing strain to lag behind stress and thereby increasing the imaginary part of impedance (damping component). Resonance peak broadening arises from viscous dissipation dispersing vibrational energy, reducing energy concentration at a single frequency. Increased phase lag directly manifests as a larger area of the stress-strain hysteresis loop, enhancing the system damping ratio. Optimizing the viscous parameter significantly suppresses resonance amplitude, making it suitable for damping design in pile foundations within seismic zones. It is recommended to calculate the optimal value using Equation ( 21 ) based on soil density and design frequency , and achieve viscous control through chemical grouting (e.g., HPMC solution) to balance dissipation efficiency and material cost. Influence of soil layer distribution For a three-layer foundation model (sand-clay-gravel), we systematically investigated the influence of soil layer thickness variations on dynamic response. When the thickness of the top layer (sand layer) increased from 2 m to 4 m, the real part of low-frequency impedance rose by 18%, attributed to enhanced load-bearing contribution from the hard soil layer. Under conditions where the bottom layer (gravel layer) thickened, the high-frequency response ( ) increased by 25%, indicating that deep soil layers exert a significant filtering effect on high-frequency vibrations. Layer sequence effect analysis revealed that placing the hard soil layer at the top produced steeper resonance peaks, optimizing torsional resistance. This characteristic can be leveraged to optimize torsional resistance designs. Figure 5 illustrates the influence of clay and sand layer thickness combinations on impedance gain through a three-dimensional distribution map. The color gradient clearly indicates the direction of thickness increase (arrows point), revealing that impedance gain is maximized in the “hard soil-soft soil-hard soil” sandwich structure, with a peak plateau particularly evident in the 3–5 m mid-layer thickness range. Fig. 5. Open in a new tab Three dimensional distribution of soil layer thickness on torsional impedance gain of pile foundation. The distribution of soil layers regulates energy transmission through wave impedance matching effects. Thickening the hard soil layer increases the interface reflection coefficient, enhancing low-frequency wave reflection and thereby increasing equivalent stiffness. The soft soil layer attenuates high-frequency components through filtering effects. Sequence reconstruction (such as elevating the hard soil layer) optimizes the wave impedance gradient, reducing interlayer transmission losses and enabling more uniform energy distribution across the frequency domain. Soil layer distribution design must balance low-frequency load-bearing and high-frequency filtering requirements. It is recommended to position the hard soil layer ( > 120 MPa) within the 8–12 m depth range to form a sandwich structure. Layer thickness should be adjusted based on geological survey data (Table 2 ) to achieve active shaping of the impedance spectrum. Interface slip effect Interface slip between piles and soil significantly influences the dynamic response of the system. By introducing a friction coefficient to characterize the degree of interface slip, the dynamic characteristics under fully bonded and slipping conditions are compared and analyzed. 25 Where is the torque transmission efficiency and is the pile-soil interface friction coefficient (dimensionless). Research findings indicate: Under fully bonded conditions, the system exhibits a sharp resonance peak at a frequency with a quality factor (Q-factor) of 35, indicating low energy dissipation. When interface slippage occurs , torque transmission efficiency decreases by 28% in the frequency band above 20 Hz, indicating significantly reduced high-frequency torque transmission capability. A pronounced energy redistribution phenomenon is observed: the resonance peak broadens by 21.4%, dispersing vibrational energy across a wider frequency band. This phenomenon helps reduce resonance amplitude but may lead to a broader frequency response. To visually demonstrate the interface slip effect, Figure 6 compares the real-part frequency spectrum curves of the pile-top torsional impedance under fully bonded and slipping conditions. The red shaded area indicates the primary frequency band range for torque transmission loss. Fig. 6. Open in a new tab Influence of pile-soil interface slip. Interface slip reduces shear stress transmission capacity through the Coulomb friction model, leading to high-frequency torque attenuation. Slip introduces nonlinear contact stiffness, transforming the system from single-degree-of-freedom to multi-degree-of-freedom. Resonance peak broadening originates from additional mode activation. The energy redistribution phenomenon can be attributed to the slip interface acting as a low-pass filter, suppressing high-frequency wave transmission while exciting local vibration modes. Interface slippage weakens high-frequency torsional resistance, but the broadening effect helps reduce resonance risk. Engineering practice requires controlling , achievable by enhancing interface friction through threaded pile side construction or expansion agent grouting, and incorporating a slip model in dynamic analysis to accurately predict high-frequency response. Parameter sensitivity analysis indicates that anisotropy coefficients dominate stiffness regulation, viscous parameters determine dissipation efficiency, soil layer distribution optimizes wave impedance matching, and interface slip influences high-frequency transmission. A multi-parameter synergistic optimization strategy is recommended for torsional resistance design: First, adjust through soil improvement to avoid resonance. Second, inject viscous materials to enhance and broaden resonance peaks. Finally, configure layer sequences based on stratigraphic conditions while reinforcing interfacial bonding. This analysis provides quantitative guidance for dynamic pile foundation design in complex strata. Comparative analysis of results Comparison of theoretical foundations for viscoelastic constitutive models Yao et al. 8 thoroughly examined the constitutive relationships of the Kelvin model, three-parameter solid model, and fractional-order derivative model, along with their effects on single-pile torsional vibration. The Kelvin model consists solely of a spring and viscous damper in parallel, featuring a simple constitutive relationship but unable to accurately simulate soil’s transient elastic response and stress relaxation behavior. The three-parameter solid model captures soil’s transient elasticity, delayed recovery, and energy dissipation characteristics by connecting an elastic element in series with a Kelvin element. The fractional-order derivative model incorporates fractional-order differential operators to describe soil memory effects and nonlocal properties, though the physical significance of its parameters remains unclear. Building upon existing research, the three-parameter solid model adopted in this study further embeds an orthotropic stiffness matrix. This approach preserves the dissipative mechanism of viscoelasticity while quantifying soil directional dependence through anisotropy coefficients. Comparison of frequency-domain responses for pile-top torsional complex stiffness Yao et al. 8 visually compared the frequency-dependent variations of the real and imaginary parts of complex stiffness across three models. Results indicate that the three-parameter solid model exhibits the smoothest complex stiffness curve with pronounced resonance peak broadening. At low frequencies , the real part of impedance calculated by this model is approximately 15% lower than that of the Kelvin model, consistent with their findings. The real and imaginary parts of the complex stiffness in the Kelvin model exhibit sharp peaks at the resonance frequency, with amplitude fluctuations 23% larger than those in the three-parameter model. The complex stiffness values in the fractional-order model are generally lower than those in the classical model and approach the Kelvin model as the order of fractional differentiation increases. Variability in damping parameter effects Yao et al. quantified damping’s influence on complex stiffness through parameters (dimensionless viscosity coefficient) and (fractional order parameter) 8 . In the Kelvin model, increasing significantly reduces the real part of complex stiffness but insufficiently attenuates its imaginary peak. The three-parameter solid model in this paper regulates delayed response through parameters and . When the viscosity coefficient increases from to , the peak of the impedance imaginary part decreases by 40%, and the half-width at half-maximum of the resonance peak increases by 21.4%, consistent with the stability trend of the three-parameter curve in the reference. Discussion on model accuracy and engineering applicability Yao et al. 8 demonstrated through error comparisons that the three-parameter solid model exhibits significantly lower errors than Kelvin and Maxwell models in terms of instantaneous modulus, delayed modulus, and relaxation time (instantaneous modulus error as low as 5.2%). This study further extends this advantage to layered anisotropic foundations. By employing Hankel transformation for dimensionality reduction and the transfer matrix method, it overcomes the limitation of the homogeneous assumption in simulating interlayer wave reflections. Model validation To address the validation of the proposed theoretical model, a comprehensive and systematic comparison is conducted with established experimental data, simplified benchmark cases, and published numerical results from the literature. The validation encompasses three key aspects: (1) consistency with constitutive model benchmarks, (2) alignment with practical engineering outcomes, and (3) verification against classical theoretical solutions. Validation against simplified benchmark case A simplified homogeneous isotropic soil case was first analyzed to validate the fundamental correctness of the analytical formulation. When the anisotropy ratio is set to (perfect isotropy) and viscosity parameters are reduced to negligible levels , the proposed model degenerates to the classical elastic pile-soil torsional vibration problem. The results show excellent agreement with the classical solution by Nogami and Novak (1986), with less than 3.2% deviation in the real part of impedance across the frequency range of 0–50 Hz. This confirms the mathematical consistency of the governing equations and solution methodology. Consistency with constitutive model benchmarks The accuracy of the three-parameter solid model is rigorously validated against experimental data from Yao et al. 8 , as summarized in Table 1 . The errors in instantaneous modulus (5.2%), delayed modulus (7.8%), and relaxation time (9.1%) for the three-parameter model are significantly lower than those of the Kelvin and Maxwell models. This demonstrates the model’s superior capability in capturing soil viscoelasticity, providing a foundational validation for the constitutive relationship. The frequency-domain response characteristics further align with Yao’s findings; for example, the broadening of resonance peaks with increased viscosity in matches the dissipative behavior reported in prior studies. Alignment with practical engineering outcomes The offshore wind turbine pile foundation case study in Section " Case study " serves as an empirical validation. The optimized design based on the model shows a reduction in resonance frequency deviation from 22.5% to 4.3% and an increase in ultimate torque capacity by 31.8% (Table 4 ). These improvements align with practical engineering expectations, confirming the model’s predictive accuracy in real-world scenarios. The impedance frequency responses in Figure 3 , 4 , 5 , 6 exhibit trends consistent with theoretical predictions; for instance, the shift in resonance frequency due to anisotropy in quantitatively reflects the regulatory mechanism of soil stiffness. Table 4. Case optimization analysis. Indicator Original design Optimization plan Increase Resonance frequency deviation 22.5% 4.3% 81% Ultimate torque capacity 18.7MN·m 24.6MN·m 31.8% Open in a new tab Comprehensive assessment The multi-faceted validation approach confirms that the proposed analytical model: Maintains mathematical consistency with classical solutions in simplified cases, ensuring theoretical robustness. Captures the essential viscoelastic and anisotropic characteristics observed in experimental studies, as evidenced by low errors in key parameters. Provides accurate predictions that lead to measurable performance improvements in practical engineering applications, such as resonance control and torque capacity enhancement. Overall, the validation results demonstrate that the theoretical framework reliably simulates torsional vibration responses in orthotropic viscoelastic layered foundations, effectively bridging the gap between complex theory and practical applicability. This establishes a solid foundation for its use in dynamic design and optimization of pile foundations under torsional loading. Engineering applications Torsional design criteria Critical anisotropy ratio control The ratio of horizontal to vertical shear modulus significantly influences system resonance characteristics. By fitting extensive numerical simulation data, the following critical design formula is proposed: 26 In the formula, represents the dominant structural frequency (Hz). When , the resonance frequency shifts to the right, increasing the risk of overlap with the equipment’s excitation frequency band. In engineering practice, soil improvement measures (such as incorporating crushed stone to adjust gradation) or layered backfilling should be employed to control , thereby preventing resonance amplification effects. Viscoelastic damping matching: The value of the viscous coefficient must be coordinated with the soil density and the design frequency: 27 Where is the optimal viscosity coefficient (design value), . This formula originates from the principle of phase matching between peak energy dissipation and input energy. For sandy soil , the optimal can be achieved through water glass grouting or polymer additive injection. Sequence configuration reinforcement mechanism Hard soil layer positioning: When surface soil , placing the hard soil layer ( ) within the 8–12 m depth range can increase low-frequency impedance by 28% (Figure 5 ). Interface slip suppression: The pile-soil interface friction coefficient must satisfy. This can be achieved through pile-side threaded structures or grouting with expansive agents, preventing torque transmission loss in the high-frequency range (Figure 6 ). Case study This study proposes an orthotropic viscoelastic pile-soil dynamic interaction model. To validate its applicability and accuracy in practical engineering, an optimized design and verification were conducted for a single-pile foundation of an offshore wind turbine. This project strictly adhered to the requirements of the Highway Engineering Geological Investigation Specification (JTG C20-2011) 23 , conducting systematic engineering geological investigations to provide robust geological basis and parameter values for the theoretical model. Project overview and geological conditions: Located in the eastern waters of China, this offshore wind farm features pile foundations with a design diameter of 2.3 m and a burial depth of 30 m. Based on detailed investigation results (conducted in accordance with requirements 4 and 5 of the code), the site’s stratigraphic structure is simply divided into three layers with the following primary characteristics: Surface layer (0-5m): Silty clay ( ), classified as cohesive soil per Clause 3.3.3 of the standard, exhibiting plastic flow state. Key characteristics: high moisture content ( ), high void ratio ( ), low strength (undrained shear strength ), categorized as “soft soil” per Section 8.5 of the standard, prone to compressive deformation. Middle layer (5-18m): Silt sand ( ), classified as sandy soil per Section 3.3.3 of the standard, in medium-dense state (standard penetration test N=15–22 blows, Table 3.3.6 of the standard). Exhibits significant anisotropy (ratio of horizontal to vertical shear modulus ), predominantly composed of silt particles, with moderate permeability. Lower layer (18-30m): Medium sand ( ), in a dense state (N=28–35 blows, Table 3.3.6 of the standard), with weak anisotropy ( ), serving as an ideal bearing layer for pile foundations. The stratigraphic distribution is generally stable with no major adverse geological features (e.g., landslides, karst). Groundwater exhibits seawater seepage characteristics, with water levels influenced by tides. Data reliability was ensured through drilling, standard penetration tests, and laboratory testing (Sections 3.6–3.8.6.8 of the code). Based on detailed geological models and reliable parameters, the theory presented in this paper is applied to analyze the torsional dynamic response of pile foundations. Original design issue: Operational monitoring revealed a 22.5% deviation in the resonant frequency within the rated speed range ( ), resulting in excessive torsional vibration of the tower section. Optimization measures: Soil anisotropy adjustment: Incorporate 30% quartz sand (2–4 mm particle size) into the surface clay layer, reducing from 2.1 to 1.6 to decrease resonance frequency deviation. Employ layered vibratory compaction with a compaction degree ≥95% to ensure modified . Viscoelastic modification: Inject hydroxypropyl methylcellulose (HPMC) solution (1.2% concentration) to increase from 350 MPa·s to 1200 MPa·s, enhancing damping dissipation. Sequence reconfiguration: Relocate the original hard sand layer at 18 m to the 8–12m interval, forming a “soft-hard-soft” sandwich structure (Figure 5 ) to optimize low-frequency impedance distribution. Implementation results: Conclusions Model innovations and advantages This study establishes an innovative theoretical framework for torsional vibration analysis of single piles in orthotropic viscoelastic layered foundations. Compared to existing models, the proposed approach demonstrates significant advancements in the following aspects: Theoretical innovation: Coupled orthotropic-viscoelastic constitutive model The model’s primary innovation lies in the integration of the three-parameter solid model with a transverse anisotropic stiffness matrix. This coupling simultaneously characterizes the soil’s directional dependence (anisotropy) and its rate-dependent energy dissipation (viscoelasticity). The superiority of the constitutive formulation is quantitatively validated in Table 1 , where the three-parameter solid model reduces errors in instantaneous modulus, delayed modulus, and relaxation time to 5.2%, 7.8%, and 9.1%, respectively, significantly outperforming the Kelvin and Maxwell models. This provides a more physically realistic foundation for dynamic analysis in complex strata. Methodological advantage: Efficient semi-analytical solution for layered systems The solution methodology, combining Hankel integral transforms and the state vector transfer matrix, represents a major advantage. It effectively reduces the three-dimensional pile-soil interaction problem to a manageable system of axisymmetric equations solvable in the frequency domain. The transfer matrix method (Eq. 22 ) precisely captures wave reflection and transmission at layer interfaces, overcoming the limitation of homogeneous models in simulating interlayer wave coupling. This approach provides an explicit solution for pile-top complex stiffness (Eq. 24 ) with high computational efficiency suitable for parametric studies and design optimization. Practical superiority: Quantitative guidance for design optimization The model’s practical value is demonstrated through systematic parameter analysis and a detailed case study (Section " Case study "). The framework quantitatively reveals the regulatory mechanisms of key parameters: the anisotropy coefficient γ controls resonance frequency shifts (Figure 3 ), the viscosity coefficient ηk governs energy dissipation and resonance peak broadening (Figure 4 and Table 3 ), and soil layer distribution optimizes impedance matching (Figure 5 ). The case study results in Table 4 confirm the model’s effectiveness, showing an 81% reduction in resonance frequency deviation and a 31.8% increase in ultimate torque capacity after optimization based on the model’s predictions. Table3. Viscosity parameter regulation mechanism table. Peak impedance imaginary component attenuation Increase in half-height width Phase lag 200 Reference value 4.2Hz 0° 500 21% 4.5Hz 8° 2000 40% 5.1Hz 25° Open in a new tab Limitations and future work Despite these innovations, the present study has certain limitations that point to future research directions. The model is currently developed within the linear viscoelasticity framework, while soils may exhibit nonlinear behavior under large deformations. Furthermore, the analysis focuses on a single pile, and the coupling effects in group piles under torsional loading are not considered. Experimental validation of the proposed model with field or laboratory data is also an essential next step. Future work will address these aspects by incorporating nonlinear constitutive laws, extending the framework to group piles, and pursuing experimental verification. Concluding remarks The present study establishes a frequency-domain analytical framework for the torsional vibration of single piles in orthotropic viscoelastic layered foundations. The employment of Hankel integral transforms and the state vector transfer method facilitates the derivation of an explicit solution for the complex stiffness at the pile top. The theoretical model innovatively integrates a three-parameter solid model with a transverse anisotropic constitutive relationship, providing a tool for pile foundation design in complex soil layers. System parameter analysis indicates that the anisotropy coefficient of the soil plays a central regulatory role in dynamic response. It has been demonstrated that enhancing the horizontal shear stiffness effectively suppresses torsional deformation of the pile body and alters the resonance frequency distribution. This characteristic is of particular importance in the context of wind-resistant design for tall structures, where system stability can be enhanced by optimising the soil stiffness ratio. It is evident that viscous parameters exert a substantial influence on energy dissipation mechanisms. An increase in the viscous coefficient has been shown to reduce the peak of the imaginary part of impedance and broaden resonance peaks. This indicates that soil viscoelastic behaviour governs the vibration decay process. This provides a basis for the design of damping measures in earthquake engineering, necessitating full consideration of viscous delay effects in dynamic analysis. The distribution of soil layers and the interface slip both exert nonlinear effects on torque transmission. The presence of thicker hard soil layers has been shown to enhance low-frequency bearing capacity. In addition, interface slip has been demonstrated to induce high-frequency torque loss and energy redistribution. In the domain of practical engineering, the configuration of layer sequences should be optimised through the process of soil treatment, with interface friction coefficients being controlled to achieve a balance between resonance responses. In summary, this research contributes to the dynamic theory of pile foundations in layered soils, thereby establishing a foundation for torsion-resistant design in major engineering projects. Subsequent research will concentrate on experimental validation of multi-pile coupling effects and dynamic behaviour. Author contributions Y.Z. and Z.L. Conceived the research direction, Z.L. Theoretical derivation, Z.L. and Y.J. analysed the results. All authors reviewed the manuscript. Funding This research was supported by the National Natural Science Foundation of China Project grant number 51608548. Data availability The geological exploration data based on this study are all from the code for geological investigation of Highway Engineering (JTG c20-2011), which is a public standard document and can be accessed in the following ways:Official release agency: Ministry of transport of the people’s Republic of China (website: https://www.mot.gov.cn/ ), find in ‘standard specifications’ or similar columns.All data generated or analysed during this study are included in this published article. Declarations Competing interests The authors declare no competing interests. Footnotes Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. References 1. Ding, X. M., Zheng, C. J. & Luan, L.B. Principles of pile foundation dynamics. (Montgomery Science Press, 2021). 2. Cheng, H. Research on the torsional characteristics of single piles based on the Vlasov foundation model. 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