Geometric Parameter Effects on Bandgap Characteristics of Periodic Pile Barriers in Passive Vibration Isolation

Abstract

To investigate the impact of the geometric parameters of periodic pile barriers on bandgap characteristics in passive vibration isolation, a two-dimensional, three-component unit cell was developed using the finite element method (FEM). This study analyzed the bandgap properties of periodic pile barriers and validated the effectiveness of the FEM through model testing. The FEM was then methodically applied to evaluate the effects of pipe pile thickness, periodic constant, arrangement pattern, and cross-sectional shape on the bandgap characteristics, culminating in the proposition of a novel H-shaped cross-section for the piles. The results demonstrated that the FEM-calculated bandgap frequency range, featuring steel piles arranged in a square pattern, closely aligned with the attenuation zone in the model tests. The lower band frequency (LBF) was primarily influenced by the pipe pile’s inner radius, while the upper band frequency (UBF) was predominantly affected by its outer radius. As the periodic constant increased, the LBF, UBF, and the width of band gap (WBG) all decreased. Conversely, changing the arrangement pattern from square to hexagonal led to increases in UBF and WBG, while the LBF diminished. Notably, the WBG of the H-section steel piles, possessing the same cross-sectional area, was 1.31 times greater than that of the steel pipe piles, indicating an enhanced vibration isolation performance. Additionally, the impact of transverse and vertical characteristic dimensions of the H-shaped pile on the band gap distribution was assessed, revealing that the transverse characteristic dimensions exerted a more significant influence than the vertical dimensions.

1. Introduction

As economic development advances, environmental vibrations resulting from machinery operation, transportation activities, and construction increasingly affect precision instruments, buildings, laboratories, and human habitations. This escalating issue demands targeted attention and mitigation strategies [1,2,3,4,5,6,7]. Currently, the primary approaches to controlling environmental vibrations involve managing their generation, propagation, and interference. Among these, controlling vibration propagation pathways has emerged as a critical isolation method [8]. Pile barriers, in particular, have attracted significant attention due to their rapid deployment, adaptability to various sites, cost-effectiveness, and efficient isolation performance, prompting extensive research into their capabilities [9,10,11,12,13,14,15,16,17].

Building on a series of in situ tests, Woods et al. [18] proposed basic guidelines for designing vibration isolation using a row of piles and introduced an amplitude attenuation coefficient to analyze isolation performance. It was suggested that effective isolation occurs when the pile radius is at least one-sixth of the wavelength being shielded and the spacing between piles is at most one-fourth of that wavelength. Gao et al. [19] developed a vibration isolation method using multiple rows of small-section holes or piles, thereby achieving effects comparable to continuous barriers and overcoming the limitations on pile radius suggested by Woods et al. Through model testing, Sun et al. [20] demonstrated that reducing the spacing between closely positioned piles significantly enhances vibration isolation. Liu et al. [21,22] evaluated the isolation performance of single and multiple rows of concrete piles under multi-point excitation through model testing, finding that increasing the pile length, radius, and quantity, as well as reducing the spacing between them, all enhance the isolation effectiveness. Employing the integral equation of Rayleigh wave scattering, Li et al. [8] conducted a three-dimensional analysis of passive vibration isolation using multiple rows of pile barriers. Their results indicated that the number of pile rows significantly affected isolation effectiveness, with more rows providing better results. They also found that the pile’s cross-sectional size and spacing had minimal impact. The high-precision three-dimensional indirect boundary element method has been utilized for multi-domain scattering based on Biot’s theory of two-phase media.

Inspired by the periodic structure theory from solid-state physics, the band gap characteristics of periodic structures have been initiated to artificially adjust targeted frequency bands for the isolation or attenuation of specific vibration frequencies. Jia et al. [23] suggested applying periodic structures to railway traffic vibration reduction, using a model with concrete as the matrix and lead as the pile core material. This model produced low-frequency band gaps between 2.49 Hz and 3.72 Hz, which align well with the frequency range relevant in seismic engineering. Through model testing and the finite element method (FEM), Liu et al. [24] verified the attenuation characteristics of periodic pile barriers, designing conditions with hollow and steel pipe piles, and hexagonal and square lattices. The results showed that periodic pile barriers provide better vibration attenuation horizontally than vertically. Among the different lattice arrangements, hexagonal arrangements outperformed square ones. In hexagonal arrangements, filled soil reduced the first-order band gap and bandwidth. In contrast, for square arrangements, filled soil increased both. Huang et al. [25,26] analyzed the attenuation zones of two-dimensional periodic pile barriers under a plane wave using the plane wave expansion method. They found that softer soil and denser piles reduced the lower band frequency (LBF). Additionally, changing the pile arrangement to a triangular pattern or increasing the filling fraction significantly enlarged the width of band gap (WBG). Meng et al. [27] discussed the impact of different pile types and arrangements on vibration isolation. They found that solid piles provided better vibration reduction than hollow piles. Additionally, triangular arrangements resulted in a higher upper band frequency (UBF), LBF, and WBG compared to square arrangements.

In recent years, the propagation characteristics of surface waves in periodic structures have garnered widespread attention. Sun et al. [28] employed the finite-difference time-domain method (FDTD) to analyze the propagation of surface waves in two-dimensional steel structures. Zhou et al. [29] proposed the surface wave Bloch modal synthesis (SW-BMS) method to expedite the computation of surface waves in elastic periodic structures. Palas Mandal et al. [30] investigated the propagation of seismic surface waves in two-dimensional periodic pile barriers by altering the geometric and mechanical parameters of the piles. Piles typically exhibit either circular or square geometries [25,31,32], but with advancements in geotechnical engineering, it has become necessary to modify these typical pile shapes to enhance their performance.

Wang et al. [33] introduced topologically optimized periodic pile barriers, primarily for mid-frequency vibration isolation, revealing the influence of material distribution on the width of the attenuation zone. Jiang Bolong et al. [34] used the plane wave expansion method to study the effects of periodic constant and filling fraction on periodic pile barriers with honeycomb, orthogonal, and X-shaped cross-sections. The results indicated that periodic pile barriers with orthogonal and X-shaped cross-sections possess more complex geometric control dimensions and richer bandgap tuning capabilities. Gudapati et al. [35] established a 3D finite element model to study the frequency–amplitude response of Y-shaped piles and compared them with conventional circular piles of the same cross-sectional area, demonstrating the superior performance of the Y-shaped piles.

Research on vibration reduction using periodic pile barriers has predominantly focused on theoretical analysis and numerical 47simulation [26,36,37,38,39,40,41,42,43]. Experimental studies, particularly on periodic pile barriers, are relatively scarce, and there has been limited systematic investigation into the influence of geometric parameters on the bandgap characteristics of periodic pile barriers. To address this gap, this study proposed establishing a two-dimensional tri-component fundamental unit using the FEM. The study aimed to analyze the bandgap characteristics of periodic pile barriers and to validate the effectiveness of the FEM through model experiments. Furthermore, the FEM systematically considered the effects of pipe pile thickness, periodic constant, arrangement pattern, and cross-sectional shape on bandgap features. Additionally, a novel H-shaped pile was proposed, based on existing complex cross-sectional periodic piles. Its vibration isolation performance was compared with that of traditional circular piles with the same cross-sectional area, to elucidate the advantages of the new pile type.

2. Bandgap Calculation Method

2.1. Theoretical Framework

In the absence of damping, the governing equation for elastic wave propagation in periodic media is as follows [44,45]:

$$\nabla [(\lambda (\mathit{r})+2G(\mathit{r}))(\nabla \cdot \mathit{u})]-\nabla \times [G(\mathit{r})\nabla \times \mathit{u}]=-\rho (\mathit{r}){\omega }^2\mathit{u}$$

(1)

$$\lambda ={\displaystyle \frac{\nu E}{(1+\nu )(1-2\nu )}}$$

(2)

$$\mu ={\displaystyle \frac{E}{2\left(1+\nu \right)}}$$

(3)

where ∇ is the differential operator; u is the displacement vector; r denotes the coordinate vector; ρ is the mass density; λ and μ represent the first and second Lame constants, respectively; ω denotes the angular frequency of the wave; and ν is the Poisson’s ratio.

Exploiting the symmetry inherent in periodic structures, Bloch utilized quantum mechanics to describe the wave displacement field u(r, t) in periodic media by

$$\mathit{u}(\mathit{r},t)={\mathit{u}}_{\mathit{k}}(\mathit{r})e^{i(\mathit{k}\cdot \mathit{r}-\omega \cdot t)}$$

(4)

where t is the time parameter; k is the wave vector of irreducible Brillouin zone; and u_k(r) is a periodic function relative to the periodic constant vector a:

$${\mathit{u}}_{\mathit{k}}(\mathit{r})={\mathit{u}}_{\mathit{k}}(\mathit{r}+\mathit{a})$$

(5)

Substituting Equation (5) into Equation (4) yields the periodic boundary conditions, which can be expressed as

$$\mathit{u}(\mathit{r}+\mathit{a},t)=\mathit{u}(\mathit{r},t)e^{i\mathit{k}\cdot \mathit{a}}$$

(6)

Using the periodic boundary conditions [46,47], the periodic structure can be effectively represented by a single unit cell. Such a tri-component unit cell comprising a pile core, pipe pile, and matrix is shown in Figure 1. u_B represents the bottom boundary, u_T the top boundary, u_L the left boundary, and u_R the right boundary. Considering that surface waves diminish rapidly with depth, the analysis can be simplified by employing a unit cell with considerable depth and a fixed bottom boundary, effectively approximating a unit cell of infinite thickness. This approach facilitates the analysis of wave propagation and bandgap characteristics within the periodic structure [48].

Combining Equations (1) and (6) allows for the derivation of the discrete form of the characteristic frequency equation for a single unit cell [26,44]:

$$(\mathbf{K}-{\omega }^2\mathbf{M}\mathit{u})=0$$

(7)

where K and M represent the stiffness matrix and mass matrix, respectively. Considering both the periodic boundary condition Equation (6) and the fixed boundary condition of the unit cell (u = 0), one can determine that the eigenvalues of Equation (7) can be obtained by a finite element software (COMSOL Multiphysics 6.1, hereinafter referred to as COMSOL) for each given reduced wave vector k.

In this work, FEM was utilized to investigate the propagation characteristics within periodic structures, which resulted in a dispersion relation including both surface and bulk wave modes. To accurately delineate the bandgap distribution for surface waves, it is essential to extract the surface wave modes from mixed modes [41,46].

2.2. Bandgap Calculation

The computational method for periodic pile barriers posits that elastic waves propagate within a plane in an elastic semi-infinite space, with the pile length considered to be infinitely large. In this model, both the piles and the soil are treated as continuous homogeneous materials, ensuring a fully coupled interaction at the pile–soil interface [49]. To illustrate this, square and hexagonal arrangements of periodic pile barriers are depicted in Figure 2. In these arrangements, R denotes the radius of the scatterer, r represents the radius of the filler, and a is the periodic constant.

Leveraging the principles of periodic theory, the wave vector is simplified to lie within the irreducible Brillouin zone, which is the shaded area depicted in Figure 3. Consequently, all propagation modes can be derived by scanning along the boundaries of this irreducible Brillouin zone, specifically along the paths M-Γ-X-M and M-Γ-K-M. As this investigation concentrated exclusively on vibration isolation within the x-y plane, the task involved identifying all relevant surface modes. After distinguishing the required surface modes, the dispersion relation of the periodic structure can be graphically characterized, with the wave vector k as the horizontal axis and frequency f as the vertical axis.

3. Model Test and Validation of FEM

3.1. Model Test

To validate the accuracy of the FEM, model tests were conducted using artificially replaced sandy soil at a site measuring 4 m in length, 4 m in width, and 1.5 m in depth. Throughout the testing process, the soil’s moisture content was consistently maintained between 8% and 10%, with a density ranging from 1.7 g/cm³ to 1.8 g/cm³. Rubber bags densely packed around the site mitigated the impact of boundary effects on vibration waves, as illustrated in Figure 4.

The testing apparatus included a dynamic signal acquisition device (model DH5922H) and a force hammer (model LC02) with a sensitivity of approximately 4 pC/N at (23 ± 5 °C), a maximum range of 5 kN, and a resonance frequency exceeding 22 kHz, as depicted in Figure 5. Two triaxial acceleration sensors (model 1A315EB) were also employed, featuring a frequency response (±10%) from 0.5 to 7000 Hz in the X/Y/Z directions, and a resolution of 0.0004 g, as shown in Figure 6. These sensors were part of a system designed to excite surface waves by vertically striking the soil surface with the force hammer.

Tsai [50] investigated the vibration isolation performance of steel pipe piles, concrete hollow piles, concrete solid piles, and timber piles using the boundary element method. The study demonstrated that concrete hollow piles lost their vibration isolation effectiveness due to inadequate stiffness. In contrast, steel pipe piles provided superior vibration isolation compared to concrete solid piles. In response to these findings, the model test utilized steel pipe piles, each with a 0.5 m length and arranged in a square pattern. A rectangular area of 1.2 m in length, 1.2 m in width, and 0.5 m in height at the site center was selected. Initially, the soil in this area was excavated, followed by the placement of steel pipe piles according to predetermined reference points. Sandy soil was then filled around and within the steel pipe piles, compacted in five layers, and left to settle for three days to ensure integration with the surrounding soil. Finally, response points were established, and the test was conducted. The square arrangement of steel pipe piles is shown in Figure 7a, with the experimental geometric parameters and the layout of excitation and acceleration sensors detailed in Figure 7b, and material parameters listed in

Table 1. Material parameters of the piles and soil.

Components	Modulus (MPa)	Density (kg/m3)	Poisson’s Ratio
Steel piles	206,000	7800	0.3
Sandy soil	20	1710	0.25

.

The specific material parameters utilized in the model test are systematically listed in Table 1.

Impact loads were delivered using a handheld force hammer, generating multiple hammer strikes per operational condition with impact force amplitudes around 450 N. Typical time curves of the impact force and the acceleration at the measurement points are displayed in Figure 8. Acceleration frequency response amplitudes with and without piles at measurement point 3 were recorded to assess the attenuation effects using acceleration frequency response functions, calculated as follows:

$$FRF=20\log \left(a_i/a_0\right)$$

(8)

where FRF is the acceleration frequency response function expressed in dB; a_i and a₀ are the acceleration amplitudes at point 3 with and without pile barriers, respectively. An FRF less than zero signifies effective attenuation, with smaller FRF values signifying greater attenuation by the pile barriers.

Notably, at 0.03 s and beyond, although the measured force values gradually approached 0 N, the acceleration remained significant. This persistence is attributable to the system’s inertial effects and the viscoelastic lag in the material’s response. As the external force was reduced to nearly 0 N, the pile–soil system, driven by internal inertial forces and free vibration response, maintained a residual level of acceleration. The phenomenon in which the decline in acceleration lagged behind the reduction in force resulted from the combined effects of material lag and the system’s dynamic characteristics.

The recorded acceleration amplitudes at measurement point 3, both with and without piles, are shown in Figure 9a. The corresponding horizontal acceleration frequency response functions are depicted in Figure 9b, where the shaded area represents the bandgap range. In Figure 9a, the acceleration in certain frequency bands is lower at the site without piles compared to the site with piles. This occurred because the presence of piles complicated the propagation paths of the elastic waves. Multiple reflections and interferences of these waves between piles can cause localized amplification of wave energy in certain frequency bands, leading to increased acceleration. In contrast, without piles, waves propagate in a more straightforward manner, which may prevent energy concentration in these frequency bands, resulting in a lower acceleration.

The experimental findings demonstrated that an FRF value of −10.46 corresponded to a 70% attenuation level, while an FRF value of −19.4 correlated with an 89% attenuation level. As illustrated in Figure 9b, for the square arrangement of steel pipe piles, the frequency range where −19.4 < FRF < −10.46 lay between 196.9 and 207.2 Hz. Within this specific frequency range, the vibrations were significantly attenuated, with the level of attenuation exceeding 70% across the board and peaking at 89% at certain points. This highlights the outstanding performance of the pile barriers in vibration isolation, particularly within the identified frequency range.

3.2. Validation

To verify the accuracy of the FEM in solving the dispersion relations of periodic structures, an analysis using the finite element software COMSOL was performed on the steel–soil periodic pile system outlined in this experiment. The periodic pile structure, illustrated in Figure 10a, had a lattice constant of a = 0.3 m, an outer pile radius of R = 0.073 m, and an inner radius of r = 0.053 m. The material parameters for the sand and steel pipe piles are provided in Table 1. To streamline the calculations, Floquet periodic boundary conditions were imposed on the upper and lower, as well as the left and right, boundaries of the model, as shown in Figure 1. The mesh was then discretized, and the wave vector k was defined to traverse the irreducible Brillouin zone. Eight characteristic frequencies were selected, and the corresponding results were used to plot the dispersion relation curve, as illustrated in Figure 10b.

Each frequency band represented in the figure corresponds to a specific vibration mode; the frequency bands within the shaded frequency ranges signify the absence of corresponding vibration modes, implying that vibrations within these frequency bands were impeded from propagating through the periodic structure, thereby resulting in the formation of bandgaps. In the realm of periodic theory, frequency ranges without a corresponding characteristic frequency are identified as complete bandgaps. A complete bandgap theoretically has the capacity to obstruct the propagation of elastic waves within its frequency range, while directional bandgaps can prevent the propagation of elastic waves in specific directions. However, in practical scenarios, elastic waves within a complete bandgap can still traverse through a periodic structure but will undergo significant attenuation.

The comparative analysis of the experimental results and FEM revealed a strong correlation of attenuation effects and bandgap ranges. The experimental data indicated that within the frequency range of 196.9 to 207.2 Hz, the vibration attenuation exceeded 70%, peaking at 89%. The complete bandgap identified by FEM spanned from 192.8 to 207.5 Hz, closely corresponding with the experimental frequency range where attenuation surpassed 70%. The slight discrepancy between these findings was likely due to the experiment’s exclusive focus on surface wave attenuation in soil, without accounting for the influence of pile length on the attenuation zone. Moreover, considering the presence of directional bandgaps in periodic pile barriers, certain discrepancies might be observed when comparing the bandgap ranges determined through the FEM with the results of the model test. Despite these potential variations, the majority of the bandgap distributions calculated (as depicted by the gray areas in the Figure 10b) aligned with the frequency range where the experimental attenuation exceeded 70%. Therefore, the results exhibited a relatively high level of consistency. This underscores the efficacy of the FEM in analyzing the dispersion curves of periodic structures.

4. Bandgap Influencing Factor Analysis

Building on the validation of the FEM, this section delves deeper into the impact of geometric parameters such as the pipe pile thickness, periodic constant, arrangement pattern, and cross-sectional shape on the bandgap characteristics. For periodic pile barriers, a square-arranged tri-component model composed of an outer matrix, pipe steel, and pile core was employed, utilizing sandy soil for both the pile core and the outer matrix. Detailed specifications are provided in Table 1.

4.1. Pipe Pile Thickness

The bandgap characteristics were characterized by three key parameters of LBF, UBF, and WBG [43]. To examine the influence of pipe pile thickness on the bandgap characteristics within the periodic pile barriers, an analysis was conducted with a periodic constant of 1.2 m and an inner radius of 0.35 m, while the outer radius of the pipe pile was varied from 0.38 m to 0.47 m.

The results, depicted in Figure 11, revealed a slight decrease in the LBF as the pipe pile’s outer radius increased; specifically, an increase from 0.38 m to 0.47 m (an increase of 23.68%) resulted in a reduction in the LBF from 52.58 Hz to 45.33 Hz (a reduction of 13.79%). Conversely, the UBF showed an upward trend with the enlargement of the pipe pile’s outer radius, escalating from 58.76 Hz to 79.85 Hz, which represents a 35.89% increase. These observations indicated that the outer radius of the pipe pile predominantly influenced the bandgap through its effect on the UBF. Furthermore, the WBG expanded as the pipe pile’s outer radius increased, suggesting that modifications of pipe pile thickness can significantly impact the bandgap characteristics of periodic pile barriers. Increasing the outer radius of the pile enhanced the stiffness of the pile barriers, thereby improving the vibration isolation effect. Consequently, the bandgap width increased with the outer radius of the pipe pile.

Figure 12 demonstrates the influence of the inner radius on the bandgap characteristics, keeping the periodic constant at 1.2 m and the outer radius at 0.4 m. The inner radius of the pipe pile was varied from 0.20 m to 0.35 m. The results revealed that the LBF increased with the inner radius; specifically, as the inner radius was augmented from 0.20 m to 0.35 m (a 75% increase), the LBF increased from 41.16 Hz to 49.73 Hz (an increase of 20.82%). Conversely, the UBF of the bandgap experienced a slight decline from 65.38 Hz to 64.47 Hz, with a reduction of merely 1.39%. These observations suggest that the inner radius of the pipe pile predominantly impacted the bandgap via its effect on the LBF, and this influence was comparatively less pronounced than that of the outer radius. Moreover, the width of the bandgap narrowed as the inner radius of the pipe pile expanded. Because the increase in the inner diameter reduced the stiffness and decreased the vibration isolation effect, thus the bandgap width decreased with the increase in the inner radius.

Consequently, for the isolation of low-frequency vibrations, a reduction in the inner radius could prove advantageous. On the other hand, to target higher and broader frequency vibrations, strategies involving an increase in the outer radius of the pipe pile coupled with a decrease in the inner radius may be considered.

4.2. Periodic Constant

To examine the influence of the periodic constant on the bandgap characteristics within periodic pile barriers, it was essential to study the effect of the filling fraction on the bandgap. For square-arranged periodic pile barriers, the filling fraction can be defined as f = π(R² − r²)/a². The outer radius remained fixed at 0.4 m and the inner radius at 0.37 m, while the periodic constant was varied from 0.9 m to 1.4 m.

As depicted in Figure 13, the results demonstrated that as the periodic constant increased, corresponding to a decrease in the filling fraction, the LBF, UBF, and WBG all exhibited a corresponding decrease. Specifically, as the periodic constant increased from 0.9 m to 1.4 m, there was a notable decrease in the LBF from 79.82 Hz to 45.57 Hz, with a reduction of 42.91%. Similarly, the UBF diminished from 95.87 Hz to 46.87 Hz, translating to a decrease of 51.11%; and the WBG contracted from 16.05 Hz to 1.30 Hz, a significant reduction of 91.90%. This indicates that the periodic constant affected both the LBF and UBF, thus influencing the WBG. These trends underscore that the periodic constant exerts a more pronounced effect on the bandgap distribution compared to factors such as the pipe pile outer and inner radius. This is because an increase in the periodic constant reduces the density of the piles, thereby decreasing the destructive interference of elastic waves and reducing the vibration isolation effect.

4.3. Arrangement Pattern

The arrangement pattern manifests as lattice types, including square and hexagonal lattices. To examine the influence of these lattice types on the bandgap characteristics within periodic pile barriers, the outer radius was maintained at 0.4 m and the inner radius at 0.37 m. Various filling fractions were selected: 11.22%, 12.07%, 13.01%, 14.07%, 15.27%, 16.63%, 18.17%, 19.95%, 21.99%, 24.37%, and 27.15%. The comparisons between the two lattice types are illustrated in Figure 14.

For identical radius r and periodic constant a, the hexagonal lattice exhibited a higher filling fraction compared to the square lattice. From the results, it is evident that the square and hexagonal lattices exhibited a comparable trend in influencing the bandgap distribution, and both the LBF and UBF increased with the filling fraction. Specifically, as the filling fraction expanded from 11.22% to 27.15%, the LBF increased by 75.30% for square lattices and 58.79% for hexagonal lattices. Meanwhile, the UBF ascended by 172.00% for square lattices and 146.36% for hexagonal lattices, indicating that increasing the filling fraction had a greater effect on the UBF, with the square lattices showing greater variability in response. The WBG for both lattice types initially increased and then decreased as the filling fraction was increased.

Moreover, at the same filling fraction, the WBG and UBF for the hexagonal lattice surpassed those for the square lattice, while the LBF was approximately the same. Therefore, for isolating broader frequency vibrations, a hexagonal arrangement may be more suitable.

4.4. Cross-Sectional Shape

4.4.1. Periodic H-Shaped Pile Barriers

To assess the impact of different pile types on the bandgap distribution within periodic pile barriers, a novel H-shaped steel pile was introduced as a vibration isolation barrier. Its cross-sectional dimensions are depicted in Figure 15. This section compared the H-shaped piles with traditional steel pipe piles under a square lattice arrangement, ensuring both types maintained the same cross-sectional area and periodic constant, to accurately gauge the effect of the cross-sectional shape on the bandgap characteristics.

Previous research on the influence of the inner radius on pipe pile bandgap distribution showed that the characteristics remain largely unchanged when the inner radius is less than 0.17 m. Consequently, for this comparative analysis, an inner radius of 0.17 m was chosen, alongside a periodic constant of 1.2 m and an outer radius of 0.4 m. In keeping with the principle of an equal cross-sectional area, the vertical dimensions n and q for the H-shaped steel piles were set at 0.75 m and 0.25 m, respectively, with a horizontal dimension of 0.30 m. Given the identical periodic constants, the filling fraction for both cross-sectional styles also remained consistent. Calculations of the bandgap distributions for both pipe piles and H-shaped piles using the FEM are illustrated in Figure 16.

The results revealed that the difference in LBF between the two types was minimal, at just 3.1%. A more significant variance was observed in the UBF, where the UBF of the H-shaped steel piles exceeded that of the steel pipe piles by 13.46%, resulting in a 30.79% increase in the WBG. These findings indicate that H-shaped steel piles offer a more effective vibration isolation capability compared to steel pipe piles, particularly in terms of achieving a wider WBG and higher UBF.

4.4.2. H-Shaped Pile Geometric Configuration

As the vibration isolation performance of the H-shaped piles was superior to that of the pipe piles, this section considers the influence of the geometric configuration of H-shaped steel piles on the bandgap distribution, focusing on four crucial factors: the filling fraction periodic constant, and the three characteristic dimensions n, q, and p. H-shaped steel piles were arranged within a matrix of sandy soil, with the selected lattice type being a square lattice. Meanwhile, it was evident that the bandgap distribution was subject to simultaneous influences from multiple factors, thereby resulting in a more complex tuning pattern for H-shaped steel piles. The geometric parameters used for calculations in this section are listed in

Table 2. Geometric configuration parameters of H-shaped steel piles. (a) Periodic constant varies, (b) vertical dimension varies, (c) horizontal dimension varies, (d) vertical dimension varies.

(a)
Number	Periodic Constant a (m)	Vertical Dimension n (m)	Horizontal Dimension q (m)	Vertical Dimension p (m)	Filling Fraction f (%)
1	0.6	0.48	0.15	0.25	39.17
2	0.8	0.64	0.20	0.33	39.17
3	1.0	0.80	0.25	0.42	39.17
4	1.2	0.96	0.30	0.50	39.17
5	1.4	1.12	0.35	0.58	39.17
(b)
Number	Periodic constant a (m)	Vertical dimension n (m)	Horizontal dimension q (m)	Vertical dimension p (m)	Filling fraction f (%)
1	1.2	0.55	0.25	0.25	19.97
2	1.2	0.65	0.25	0.25	25.17
3	1.2	0.75	0.25	0.25	30.38
4	1.2	0.85	0.25	0.25	35.59
5	1.2	0.95	0.25	0.25	40.80
(c)
Number	Periodic constant a (m)	Vertical dimension n (m)	Horizontal dimension q (m)	Vertical dimension p (m)	Filling fraction f (%)
1	1.2	0.90	0.15	0.25	22.92
2	1.2	0.90	0.20	0.25	30.56
3	1.2	0.90	0.25	0.25	38.19
4	1.2	0.90	0.30	0.25	45.83
5	1.2	0.90	0.35	0.25	53.47
(d)
Number	Periodic constant a (m)	Vertical dimension n (m)	Horizontal dimension q (m)	Vertical dimension p (m)	Filling fraction f (%)
1	1.2	0.90	0.25	0.20	39.93
2	1.2	0.90	0.25	0.25	38.19
3	1.2	0.90	0.25	0.30	36.46
4	1.2	0.90	0.25	0.35	34.72
5	1.2	0.90	0.25	0.40	32.99

. The material parameters are listed in Table 1.

Figure 17 offers insight into how the periodic constant affected the bandgap distribution, showcasing the changes in the LBF, UBF, and the WBG as the periodic constant varied. The analysis revealed that, with the filling fraction held constant, an increase in the periodic constant led to a decrease in the LBF, UBF, and WBG for the first complete bandgap. This is consistent with the variation pattern of steel pipe piles.

The filling fraction, which plays a crucial role in determining the bandgap distribution of H-shaped steel piles, is intricately influenced by the three characteristic dimensions n, q, and p, spanning both vertical and horizontal directions. For H-shaped steel piles, this filling fraction can be mathematically represented as f = (3qn − 2pq)/a². Given the complex interplay between these dimensions and their differential impact on the LBF and UBF, discussing the impact of the filling fraction on the bandgap distribution in isolation is not reasonable. The following discussion delves into the separate effects of n, q, and p on the bandgap distribution, shedding light on how each dimension influences the vibration isolation capabilities of H-shaped steel pile barriers.

Figure 18 explores how the vertical characteristic dimension n affects the bandgap characteristics within periodic pile barriers, presenting the LBF and UBF along with the corresponding WBG as n varies. When all parameters except n were held constant, as n increased from 0.55 m to 0.95 m, a rise of 72.73%, induced only a modest change in LBF, of approximately 3.93%. The UBF experienced a larger increase of 29.60% when n was below 0.75 m, and a greater increase of 5.90% when n exceeded this value. The WBG first increased and then diminished as n grew. Thus, alterations in n predominantly affect the UBF, and within specific parameters, amplifying n can bolster the vibration isolation capabilities of periodic pile barriers.

Figure 19 delves into the impact of the horizontal characteristic dimension q on the bandgap characteristics within periodic pile barriers, detailing the LBF and UBF, along with the corresponding WBG as q changes. When all other parameters were fixed and q was altered, an increase in q from 0.15 to 0.35, a 133.33% rise, resulted in a relatively small overall variation in the LBF of 7.83%. The UBF ascended sharply by 102.80% when q remained under 0.3 m, and only marginally by 0.35% when q exceeded 0.3 m. The WBG displayed a pattern of increase followed by a decrease with q. These findings suggest that changes in q significantly impact UBF, and expanding q within a defined range can enhance the vibration isolation performance of periodic pile barriers.

Figure 20 shows the effect of the vertical characteristic dimension p on the bandgap characteristics within periodic H-shaped piles, presenting the LBF, UBF, and corresponding WBG as p varies. When all other parameters were fixed and p was altered, an increase in p from 0.2 to 0.4, a 100% rise, resulted in negligible changes in the LBF, UBF, and WBG. Specifically, LBF altered by merely 1.44%, and the UBF by just 0.28%. This minimal range of variation in p indicates that p does not significantly influence the vibration isolation characteristics of periodic pile barriers.

Therefore, when designing H-shaped steel pile barriers for vibration isolation purposes, adjustments to p may not be as critical for tuning the bandgap characteristics as changes to other geometric dimensions such as n or q.

Analysis of the variations in the vertical and horizontal characteristic dimensions n, p, and q revealed that the modifications predominantly affected the UBF. The dimensions n and q significantly influenced the bandgap tuning, while p had a lesser effect. Relative to dimensions n and p, q is more responsive to adjustments in bandgap distribution. The specific results are shown in

Table 3. Variations in characteristic dimensions of H-shaped steel piles.

Characteristic Dimensions	n/p/q	LBF	UBF
n	72.73%	3.93%	37.36%
q	133.33%	7.83%	103.52%
p	100%	1.44%	0.28%

. Thus, adjusting the lateral characteristic dimension q proved to be more effective in enhancing the vibration isolation performance of the H-type pile barriers.

5. Conclusions

A two-dimensional, three-component unit cell was developed using the FEM to investigate the bandgap characteristics of periodic pile barriers in this paper. The accuracy of the FEM was supported by the model test, affirming its effectiveness in this context. Using the validated FEM model, the influence of the main factors, including the pipe pile thickness, periodic constant, arrangement pattern, and cross-sectional shape, on the bandgap characteristics were analyzed. Finally, an innovative type of pile with an “H” cross-sectional profile was proposed. Based on the results of the analysis, the following conclusions were drawn:

(1)

The bandgap frequency ranges determined through the FEM for square-arranged piles showed a strong correlation with the attenuation domains observed in the model testing. This concordance underscored the effectiveness of using the FEM to analyze the bandgap characteristics of periodic pile barriers.

(2)

The FEM revealed that the LBF was predominantly affected by the outer radius of the steel pipe piles and the periodic constant. Specifically, changes of 23.68% and 55.56% in the outer radius and periodic constant, respectively, resulted in LBF variations of 13.79% and 42.91%. Periodic pile barriers with larger periodic constants, larger outer radius, and smaller inner radius yielded a lower LBF. Furthermore, employing a hexagonal lattice proved beneficial for attaining a larger WBG.

(3)

With identical cross-sectional areas, the WBG of the H-shaped steel piles was 1.31 times greater than that of steel pipe piles. H-shaped steel piles outperformed steel pipe piles in modulating the bandgap distribution. This was particularly evident in the wider WBG and higher UBF. Analysis of the bandgap distribution adjustments for H-shaped steel piles indicated that the LBF, UBF, and the WBG were particularly sensitive to variations in the horizontal characteristic dimension q, while showing minimal sensitivity to changes in the vertical characteristic dimensions p and n.