Pile foundations are crucial for supporting large-scale structures like high-rise buildings [1], offshore bridges [2], and ocean platforms [3]. Therefore, understanding their response to complex dynamic loads is essential. While pile foundations are generally subjected to static and dynamic vertical and horizontal loads, wind and machine-induced vibrations can cause torsional loads that significantly affect their dynamic performance [4–6]. Accordingly, assessing the impact of dynamic torsional loads is vital to ensure the stability and safety of pile foundations. To address this, researchers have proposed various methods to investigate torsional vibration characteristics of piles.

Cai et al. [7] performed a comprehensive study on the torsional vibration of elastic piles in a uniform poroelastic medium. Based on this fundamental research, Chen et al. [8] introduced the concept of transverse isotropy to examine the vibration characteristics of piles embedded in saturated soils under transient loading. Their study emphasized the impact of soil transverse isotropy and the pile slenderness ratio on the vibration characteristics of the piles. Subsequent research further illustrated the effects of soil properties on end-bearing piles [9, 10] and pipe piles [11, 12]. Given the inherently layered and vertically non-uniform nature of soil profiles, Zou et al. [13] explored the mechanical behavior of single piles in a two-layer vertically non-uniform subgrade and subjected to axial-torsional combined loads. Liu and Zhang [14] expanded upon these findings by investigating the transient torsional vibration behavior of heterogeneous piles in multi-layered poroelastic media, with a focus on the influence of typical pile defects. To advance this field of study, Li et al. [15], Zhang and Pan [16] assessed the influences of construction disturbances on the surrounding soil, analyzing the impact of radial inhomogeneity induced by such disturbances on the vibration behavior of piles in layered media. Further research has addressed open-ended pipe piles [17–19], examining the effects of construction-induced inhomogeneity in radial direction on the torsional impedance of piles. Additionally, studies have investigated the impact of pile end soil [20] and variation in cross-sectional dimensions [21] on the torsional response of piles. Typically, these investigations model soil as a composite material composed of pore water and soil particles. However, this assumption does not always hold true in practical applications, particularly in surface or shallow soils where unsaturated conditions are prevalent. Unsaturated soils, characterized by incomplete pore saturation, exhibit markedly different mechanical behaviors compared to their saturated counterparts. Research concerning unsaturated soils primarily focused on the effects of transverse isotropy [22, 23], radial inhomogeneity [24, 25], vertical inhomogeneity of soil [25, 26], and pile end soil [27]. Despite the significant influence of torsional loads on pile performance, this factor has not been received adequate consideration in studies of nodular piles.

In soft soil regions, traditional piles, such as cast-in-situ and precast piles often face challenges such as low skin friction and construction-related defects in the pile body. To address these problems, static drill-rooted nodular (SDRN) piles, an innovative foundation type initially developed in Japan, have been increasingly used in engineering practice [28, 29]. As for SDRN pile, the prefabricated nodular pile is embedded into cemented soil via static drilling, effectively overcoming the limitations of conventional pile installation techniques and providing a more efficient, cost-effective scheme. Due to the construction similarities, SDRN piles are often evaluated in comparison to cast-in-situ piles. Research by Zhou et al. [30–33] has demonstrated, through extensive field studies and finite element simulations, that SDRN piles exhibit significantly superior skin friction and compressive bearing capacity compared to cast-in-situ piles, making them particularly advantageous for use in soft soil conditions. Li et al. [34] estimated the vibration characteristics of SDRN piles in layered soil profiles and under vertical loads, affirming the substantial benefits of SDRN piles in mitigating vertical deformation and vibration relative to cast-in-situ piles. Additionally, Wu et al. [35] investigated the impact of various cross-sectional geometries on vertical vibration. Despite considerable research on the performance of SDRN piles under vertical loads [36–38], the effects of torsional loads on SDRN piles remain underexplored.

The literature review above reveals that studies on the dynamic performance of SDRN piles under torsional loading are limited. Therefore, this article aims to examine the torsional vibration characteristics of SDRN piles in elastic soils using an analytical method. It should be pointed out that the analytical solutions presented here provides engineers with a precise, convenient, and efficient tool for quickly assessing the performance of SDRN piles. Specifically, it allows for the evaluation of factors such as pile inner and outer radii, node width, vertical node spacing, and variations in properties of the surrounding cemented soil on the overall performance of SDRN piles. The layout is as follows: Section 2 establishes the mathematical model. Section 3 derives the analytical solutions for the dynamic response problem. Numerical examples and the corresponding analysis and discussion are presented in Section 4. Section 5 applies the present solution into the vibration characteristic of pile-supported foundation. The main conclusions are summarized in Section 6.

As depicted in Figure 1A, a SDRN pile in elastic soil, undergoing a time-harmonic torque is considered. This study considers the hardening surrounding cemented soil, a radially semi-infinite natural soil deposit and the nodular pile itself, but neglects the effect of the soil within the SDRN pile. Given the large slenderness ratio of the pile, it is approximated as an elastic rod with a fixed inner radius r _in and a series of equally spaced, enlarged nodes (see Figure 1B). The external (outer) radius of pile is r ₀ at non-noded sections, while the maximum external radius at nodes is r _oumax. The vertical spacing between two adjacent nodes is l _b , and the radial protrusion thickness and length at each node are w _b . Based on the node distribution, the pile-soil interaction model is divided into N segments (elements), numbered sequentially from the bottom upward. Each segment has equal thickness in both the pile section and the surrounding soil section. The thickness and the radius of segment i are denoted by h _i and r _i , respectively. For node sections, sufficient segmentation is required to simulate the continuous variation in the pile’s outer radius. The soil around the pile is split into two distinct zones: (1) the inner zone, composed of cemented soil, extends to a distance r _c from the pile’s centroid, and (2) the outer zone, consisting of radially semi-infinite natural soil, lies beyond the cemented soil layer.

Previous studies [16, 39] have shown that neglecting the gradient of shear stress σ _θz along the z-axis (i.e., assuming a plane strain model) has a minimal effect on the dynamic impedance of the pile. Therefore, following the plane strain assumption, the wave equation for any soil layer under torsional load is given by [39]. u θ , r r r , t + r − 1 u θ , r r , t − r − 2 u θ r , t = ρ s G s u θ , t t r , t (1) where the subscript, “j” denotes the partial derivative to the variable j (j = r, t); u _θ (r, t), G _s and ρ _s represents the tangential displacement, shear modulus and density of the soil, respectively.

The dynamic equilibrium equation for an elastic pile under torsional load is given by G p I p ϕ , z z z , t + 2 π r 2 τ z , t = ρ p I p ϕ , t t z , t (2) where ϕ(z, t), G _p , ρ _p and I _p represent the twist angle, shear modulus, density and polar moment of inertia of the pile, respectively; τ(z, t) denotes the tangential shear stress applied by the soil around the pile.

The boundary condition for the outer natural soil deposit can be specified as u θ i j r → ∞ , t = 0 , j = 2 (3) where the subscript ij represents the component in the j-th (where j = 1, 2) radial region corresponding to the i-th (where i = 1-N) pile segment; Specifically, j = 1 and j = 2 refer to the inner and outer radial zones, respectively; In the subsequent sections, the subscript ij will maintain this definition, unless specified otherwise.

The boundary conditions at top and bottom of the i-th (i = 1-N) pile element are ϕ i , z z = 0 , t = − T i t G p i I p i (4) ϕ i , z z = h i , t + ϕ i z = h i , t Θ i G p i I p i = 0 (5) where I p i = 0.5 π r i 4 − r in 4 ; ϕ _i (z, t) denotes the twist angle in the i-th pile element; Θ _i represents the complex impedance at the lower end of the i-th pile element; T _i (t) represents the torque exerted on the top of the i-th pile element; For the N-th pile element (i.e., the top element), T _N (t) = T ₀(t) holds; When i = 1, the impedance from the pile bottom soil is Θ 1 = 16 G s u r 0 3 − r in 3 / 3 as described by Li et al. [17].

The continuity conditions at r = r _c can be given by u θ i 2 r = r c , t = u θ i 1 r = r c , t (6) τ r θ i 2 r = r c , t = τ r θ i 1 r = r c , t (7)

The continuity conditions at r = r _i can be formulated as u θ i j r = r i , t = ϕ i z , t r i (8) τ i z , t = τ r θ i r = r i , t = G s i u θ i 1 , r r i , t − r − 1 u θ i 1 r i , t (9)

The time-harmonic solution of Equation 1 can be given by u θ r = A K 1 q r + B I 1 q r (10) where q = iω(ρ _s /G _s )^0.5; i=(−1)^0.5; ω is the angular frequency.

Substituting the boundary condition from Equation 3 into Equation 10, it follows that B = 0. Consequently, the field quantities in the natural soil deposit corresponding to the i-th element can be given by u θ i 2 r = A i 2 K 1 q i 2 r (11a) τ r θ i 2 r = − A i 2 G s i 2 q i 2 K 2 q i 2 r (11b) where q _i2 = iω(ρ _si2/G _si2)^0.5.

The field quantities in the cemented soil layer can be formulated as u θ i 1 r = A i 1 K 1 q i 1 r + B i 1 I 1 q i 1 r (12a) τ r θ i 1 r = − A i 1 G s i 1 q i 1 K 2 q i 1 r + B i 1 G s i 1 q i 1 I 2 q i 1 r (12b) where q _i1 = iω(ρ _si1/G _si1)^0.5.

Substituting Equations 11, 12 into continuity conditions given by Equations 6, 7 results in B i 1 = κ i 1 A i 1 (13) where κ i 1 = G s i 1 q i 1 K 2 q i 1 r c K 1 q i 2 r c − G s i 2 q i 2 K 2 q i 2 r c K 1 q i 1 r c G s i 1 q i 1 I 2 q i 1 r c K 1 q i 2 r c + G s i 2 q i 2 K 2 q i 2 r c I 1 q i 1 r c (14)

Making use of Equation 13, Equation 12 can be rewritten as u θ i 1 r = A i 1 K 1 q i 1 r + κ i 1 A i 1 I 1 q i 1 r (15a) τ r θ i 1 r = − A i 1 G s i 1 q i 1 K 2 q i 1 r + κ i 1 A i 1 G s i 1 q i 1 I 2 q i 1 r (15b)

In the case of a time-harmonic load, Equation 2 can be reformulated as (for i-th element) G p i I p i d 2 ϕ i z d z 2 + 2 π r i 2 τ i z = − ρ p i I p i ω 2 ϕ i z (16)

Combining Equations 8, 9, 15, 16 yields d 2 ϕ i z d z 2 + γ i 2 ϕ i z = 0 (17) where γ i 2 = δ i 1 + ρ p i I p i ω 2 / G p i I p i (18) δ i 1 = 2 π r i 3 G s i 1 q i 1 − K 2 q i 1 r i + κ i 1 I 2 q i 1 r i K 1 q i 1 r i + κ i 1 I 1 q i 1 r i (19)

The solution of Equation 17 can be expressed as ϕ i z = C i ⁡ cos γ i z + D i ⁡ sin γ i z (20)

Combining Equation 5 and Equation 20 results in D i = η i C i (21) where η i = γ ¯ i ⁡ sin γ ¯ i − Θ ¯ i ⁡ cos γ ¯ i γ ¯ i ⁡ cos γ ¯ i + Θ ¯ i ⁡ sin γ ¯ i ;   Θ ¯ i = Θ i h i G p i I p i ;   γ ¯ i = γ i h i (22)

Substituting Equation 20, 21 into Equation 4 yields k T i = T i ϕ i z = 0 = − G p i I p i γ i η i (23) where k _Ti denotes the top-end torsional impedance of the i-th pile element.

The principle of impedance function recursion has been effectively used in past studies to address the vibration behavior of piles in layered soils [40]. According to this principle, the torque and twist angle at the interface between adjacent pile elements are continuous, meaning that the torsional impedance at the interface (i.e., torque/twist angle) is also continuous. That is to say, the impedance at the top of the i-th element is equal to the impedance at the bottom of (i+1)-th element (i.e., k _Ti = Θ _i+1). Hence based on Equation 23, the torsional impedance function at the top end of the pile can be determined via a step-by-step recursion from the first element to the N-th element, which can be written as follows k T N = − G p N I p N γ N η N (24)

Following Militano and Rajapakse [39], the normalized torsional impedance can be defined as k T ′ = − 3 G p N I p N γ N η N 16 G 0 r 0 r 3 (25) where k T ′ denotes the normalized torsional impedance at the pile top; G ₀ and r _0r denote the reference shear modulus and radius, respectively.

First, to confirm the correctness of the developed model, a comparison is made with the analytical results from Militano and Rajapakse [39]. For this comparison, the SDRN pile is reduced a solid cylindrical rod and the surrounding soil is assumed to be a uniform elastic material. The used parameters are ρ _p = 2,500 kg/m³, G _p = 12.1 or 16.5 GPa, r ₀ = 0.3 m, r _in = 0 m, r _oumax = 0.3 m, r _c = 0.5 m, H = 10 m, w _b = 0 m, ρ _s1 = ρ _s2 = 1800 kg/m³, G _s1 = G _s2 = G _su = G _s0 = 20 MPa, β _s1 = β _s2 = 0. For analysis purposes, the normalized frequence is defined as a ₀ = ωr _0r (ρ ₀/G ₀)^0.5, in which reference density ρ ₀ = 1800 kg/m³ and G ₀ = 20 MPa. In Figure 2, Re ( ) and Im ( ) denote respectively the real and imaginary components of the torsional impedance. Figure 2 demonstrates that an increase in excitation frequency leads to a decrease in real stiffness (real part) and an increase in damping (imaginary part). Additionally, the torsional impedance in the current solution corresponds well with those in the existing solution.

After verifying the present solution, the effect of key parameters on the torsional impedance at the pile top is examined here. To account for soil damping, G _sj is replaced by G _sj (1+2iβ _sj ), where β _sj denotes the damping ratio. Unless specified differently, the properties for SDRN pile and composite soil layers are outlined in Table 1. Additionally, each node is divided into 10 elements to simulate the continuous variation of cross-sectional dimension (see Figure 1B), where the length of the first pile segment h ₁ fixed at 0.425 m.

Figure 3 depicts the impact of inner radius (r _in) of the SDRN pile on the torsional impedance across different frequencies. The real stiffness initially decreases with an increase in frequency, then begins to increase at higher frequencies. In contrast, the imaginary part consistently increases as frequency rises. Moreover, both the real and imaginary parts decrease as r _in increases, indicating that the thinner wall (t _w = r ₀−r _in) reduces the stiffness and damping of the pile, thereby decreasing its torsional resistance. This behavior is mainly attributed to the fact that a larger r _in decreases the polar moment of inertia of the pile, thereby reducing its dynamic resistance.

Figure 4 demonstrates the impact of outer radius (r ₀) of the SDRN pile on the torsional impedance at various frequencies. For consistency, the radial width of cemented soil (r _c −r ₀) is fixed at 0.2 m. As shown in Figure 4, r ₀ has a greater impact on the torsional impedance than the inner radius. Both the real and imaginary components increase significantly as r ₀ increases.

The effects of the node width (w _b ) and vertical spacing (l _b ) on the torsional impedance at different frequencies are presented in Figures 5, 6. It should be noted that for a fixed pile length, a smaller l _b corresponds to a greater number of nodes along the pile body. Figure 5 shows that increasing w _b leads to a rise in both the real and imaginary parts, revealing that a larger node width enhances the torsional resistance of the pile. This improvement is due to the increases in the polar moment of inertia at the node location, which enhances the dynamic resistance. It is evident from Figure 6 that the real part decreases as l _b increases, while the imaginary part is insensitive to l _b .

Figures 7, 8 describe the effects of the radial width and shear modulus of cemented soil on the torsional impedance at different frequencies. The radial width is represented by the outer radius of the cemented soil, with a larger r _c corresponding to a broader enhanced range. According to Figure 7, increasing the radial width leads to an increase in both the real and imaginary parts. In comparison, the effect of the shear modulus G _s1 is relatively smaller. As depicted in Figure 8, the components of torsional impedance increase as G _s1 increases, although the rate of increase slows down as G _s1 continues to rise. This suggests that enhancing the radial width of the cemented soil is more effective than increasing the shear modulus in improving the torsional impedance of the pile.

Figure 9 describes the effect of the damping ratio of the outer soil layer on the torsional impedance for different frequencies. The focus here is to present the impact of the damping ratio of the outer natural soil layer due to minor effect of damping ratio of the inner soil. As shown in Figure 9, at lower frequencies, the damping ratio has little effect on the real part. However, at higher frequencies, the torsional impedance increases with increasing damping ratio.

When a pile-supported rigid foundation is subjected to dynamic torsional loads (see Figure 10), the dynamic equilibrium of the foundation is defined by the equation given below I f d 2 φ t d t 2 + d φ t d t + k f φ t = T f t (26) where I _f and φ(t) represent the mass moment of inertia and angle of rotation of the foundation, respectively; k _f and c _f denote the stiffness and equivalent damping coefficient of the supported pile; T _f (t) denotes the external time-dependent torsional load.

To obtain the solution for the foundation, Equation 24 can be reformulated as k T N = k f + i ω c f (27)

Assuming a time-harmonic external load on the foundation and making use of Equation 27, the solution to Equation 26 can be given by φ = T f / k f − I f ω 2 + i ω c f (28)

Making use of Equation 28, the normalized twist angle amplitude can be formulated as A φ = 16 G 0 r 0 3 φ 3 T f = 16 G 0 r 0 3 3 k f − I f ω 2 2 + ω c f 2 (29)

The following parts discuss the impact of key parameters on the twist angle amplitude, with the parameters listed in Table 1. To clearly observe the changing trends in resonant frequency and amplitude, we use the module of the twist angle amplitude instead of the real and imaginary part. In the analysis, I _f is fixed at 4,144 kg m². Figure 11 illustrates the effect of inner radius (r _in) and outer radius (r ₀) of the SDRN pile on the twist angle amplitude at varying frequencies. The data in Figure 11A indicate that the twist angle amplitude initially ascends with frequency, reaches a peak, and then gradually approaches zero. The resonant peak shows obvious increase with an increase in r _in, while the resonant frequency slightly decreases as r _in increases. As shown in the Figure 11B, r ₀ has a greater effect on the twist angle amplitude than the inner radius. The twist angle amplitude increases significantly as r ₀ decreases. Furthermore, the resonant frequency increases markedly with an increase in r ₀. This indicates that increasing the outer radius can substantially alter the natural frequency and reduce the twist angle of the whole system.

Figure 12 describes the effect of the node width (w _b ) and vertical spacing (l _b ) of the nodes on the twist angle amplitude for different frequencies. As observed in Figure 12A, the twist angle amplitude decreases significantly with an increase in w _b , revealing that a larger node width enhances the dynamic torsional resistance of the system. Besides, the resonant frequency rises with increasing w _b . Therefore, it can be concluded that increasing the node dimension is an effective strategy for improving the vibration behavior of the system in engineering practice. It can be observed in Figure 12B that contrary to expectations, the vertical spacing of the nodes has a relatively small impact on the twist angle amplitude. Additionally, the resonant peak shows a slight increase with increasing l _b .

Figure 13 describes the impact of the shear modulus and radial width of cemented soil on the twist angle amplitude at different frequencies. As observed in Figure 13A, the resonant peak decreases as G _s1 increases, while the resonant frequency increases with higher G _s1. At high frequencies, the influence of G _s1 becomes negligible. The impact of r _c is more pronounced than of the shear modulus of the cemented soil. The resonant peak significantly decreases as r _c increases, while the resonant frequency shows the opposite trend (see Figure 13B).

Figure 14 shows the effect of the damping ratio in the outer soil layer on the twist angle amplitude for different frequencies. The twist angle amplitude is notably affected by the damping ratio. The twist angle amplitude decreases substantially with increasing damping ratio, indicating that materials with a higher damping ratio are more effective in reducing vibrations.

A closed-form solution is developed to address the time-harmonic torsional vibration of a SDRN pile embedded in elastic soils. The wave equations corresponding to the pile and surrounding soils are initially established. In the case of a time-harmonic load, the general solutions for the composite soil layers are derived. The pile and soil are partitioned into multiple elements, and interface continuity conditions of the pile and soil for each element are applied to derive the torsional impedance at the top end of each pile element. Using the principle of impedance function recursion, the torsional impedance of the pile and the twist angle amplitude are ultimately determined. The correctness of the proposed solution is meticulously checked, and detailed numerical analysis is conducted. The key findings are outlined as follows:

(1) Increasing the inner radius of the pile significantly reduces the torsional impedance of the pile and enhances the resonant peak of the system. In contrast, enlarging the outer radius markedly improves the torsional impedance of the pile, reduces the twist angle amplitude and raises the resonant frequency.