Dynamic response of single pile induced by the vibration of tunnel boring machine in hard rock strata (2025)

Due to its safety and efficiency, the Tunnel Boring Machine(TBM) is widely used in metro tunnel construction. More than half of China’s planned metro projects employ this method¹. As metro lines mostly traverse urban areas, new metro construction inevitably involves crossing existing underground pipelines, bridge piles, and building foundations. Therefore, during operation of the TBM, it is essential to ensure not only the safety of the tunnel itself but also the safety of surrounding buildings and their ancillary facilities.

The impact of the metro on buildings mainly involves surface settlement caused during construction and vibrations generated during the construction process^2–7. Numerous tunneling projects have demonstrated that tunnel excavation inevitably affects the usage status of nearby buildings, with vibrations generated by TBM in hard rock formations potentially inducing accompanying vibrations in adjacent buildings⁸. However, the mechanisms of how vibrations generated during TBM operation affect nearby buildings remain unclear. Therefore, analyzing the patterns of surface vibrations caused by TBM and the mechanisms of their impact on nearby building foundations is essential for mitigating vibration interference, which holds significant importance for future metro construction and the reinforcement of vibration-sensitive buildings.

A large amount of research has demonstrated that subway tunnel construction can generate vibration disturbances to surrounding buildings. For instance, studies have monitored vibrations from TBM during operation, with maximum spindle vibration acceleration amplitudes reaching up to 4g^9–12. Table1 summarizes some adverse effects of vibrations generated during TBM operation. It can be confirmed that vibrations from TBM pose common hazards to both the tunnel itself and nearby structures, although specific details remain unclear.

Nelson et al.²². monitored vibrations generated by TBM in hard rock during the construction of the Buffalo Light Rail Rapid Transit (LRRT) project in the United States. This marked the first instance where designers began to pay attention to and measure vibrations during tunneling. In recent years, with the rapid development of metro systems, the issue of tunnel construction vibrations has gradually gained attention. Subsequent to this, some scholars conducted studies on the propagation patterns and impact ranges of vibrations generated during tunnel construction²³. Flanagan monitored vibrations within a 60m radius of the ground surface during the operation of small-diameter TBM, roughly determining the peak acceleration of ground vibrations within this range²⁴. Wang et al.²⁵ used FLAC^3D to calculate the effects of vibrations generated by tunneling on sandy soil layers, proposing that the vibrations could induce liquefaction of saturated sandy soil and ultimately lead to surface settlement. Guo et al.²⁰ analyzed vibrations generated by cutterhead in gravelly sand formations, finding that ground vibrations were concentrated within the frequency range of 4–80Hz. These studies, employing methods such as field monitoring and numerical simulation, attempted to analyze the propagation patterns of vibrations generated by TBM operation in the strata and their interference with the strata, yielding rich research results.

In the construction of metro tunnels, scholars have increasingly focused on the impact of TBM vibrations on nearby structures. For instance, Xiao et al.²⁶ used finite element methods (FEM) to analyze vibrations of bridge main girders induced by cutterhead passing beneath high-speed railway bridges, suggesting that the tunneling machine could disrupt the original stress field of the bridge when approaching it. Wu et al.²⁷ combined on-site vibration monitoring with three-dimensional discrete element methods (DEM) to analyze the impact of cutterhead vibrations in hard rock formations on surface building structures, finding that buildings within 9m of the cutterhead experienced accompanying vibrations. Wang et al.¹⁴ conducted numerical simulations on a metro tunnel passing beneath an old bungalow, suggesting significant effects of construction vibrations on buildings ahead of the work face, with temporal fluctuations in building vibrations. Similar studies have attempted to analyze the mechanisms of the vibrations during TBM operation on surface buildings^28,29. As TBM advance underground in urban areas, surrounding underground structures (such as pipelines and building foundations) experience more intense vibration responses. Li et al.³⁰ analyzed vibrations of basement walls and floors collected when cutterhead passed beneath basements, noting that vibrations of walls perpendicular to the tunneling direction were most intense. Tang et al.³¹ studied vibration intensity and direction at different positions near metro tunnels during construction and operation, finding that vertical vibrations at the top of adjacent tunnels were most severe. A comprehensive analysis of existing research reveals that most researchers are increasingly concerned about the adverse effects of vibrations generated by cutterhead during metro tunnel construction on surface buildings, but there is currently limited research on interference with underground structures.

Pile foundations, which serve as the base for most urban buildings and municipal bridges, are often subjected to shield tunneling operations in close proximity. For these structures, piles are the first to be affected by vibrations and exhibit the most pronounced dynamic responses. Early studies on the mechanical response of piles due to shield tunneling focused mainly on static analyses, particularly on the horizontal deformation of piles in soft soil. Researchers have employed various approaches—including numerical simulations^2,32,33, theoretical analyses^34,35, and model tests—to investigate pile deformation³⁶. For example, Xu et al.³⁷ conducted numerical simulations of pile deformation as shield tunneling passed through a group of piles, determining the internal force distribution and horizontal displacements at different locations. Similarly, Wang et al.³⁸ used grid-based methods to analyze the deformation response of piles induced by large-diameter tunnel excavation, obtaining similar results. Research on the dynamic response of piles during shield tunneling has mainly focused on theoretical analyses and numerical simulations, while field monitoring data remain scarce. This scarcity is partly due to the difficulty of directly monitoring pile vibrations and partly because the complex ambient ground vibrations make it hard to isolate those generated by shield tunneling. Zhang et al.³⁹ carried out dynamic calculations using grid methods on a section of municipal bridge piles affected by shield tunneling, analyzing the principal stresses and deformation characteristics of the piles. Wang et al.⁴⁰ then constructed a vibration response prediction model for pile foundations during shield construction based on the Pasternak model. Jayasinghe et al.⁴¹ conducted field tests on the dynamic response of piles induced by ground vibrations; their work analyzed the instantaneous changes in pile axial forces and bending moments through controlled blasting. Although such tests were costly, they provided valuable monitoring data. In another study, Wang et al.⁴² monitored soil vibration responses at various burial depths caused by shield tunneling by installing sensors in steel pipes driven underground—a method that could also be applied to infer the dynamic response of pile foundations. It is evident that extensive research has been conducted on the effects of shield tunneling on pile foundations, although most of it has focused on static analyses. Research on the dynamic response of pile foundations remains limited. However, to ensure the safety of building foundations—especially the stability of municipal bridges when shield tunneling occurs in close proximity—it is necessary to analyze the dynamic response of pile foundations in hard rock strata induced by shield tunneling vibrations.

A synthesis of existing research indicates that, although considerable progress has been made in studying the deformation response of pile foundations during shield construction, dynamic analysis of this system has been rarely conducted. Furthermore, due to the numerous vibration sources at the construction site and the complex environment, monitoring data on shield construction vibrations are scarce, making it difficult to provide reliable reference data for similar studies. This study investigated a tunnel segment passing beneath an interchange bridge, conducting on-site vibration monitoring of the ground and bridge piers. Utilizing the discrete element method with finite difference method (DEM-FDM) coupling method, the dynamic response of pile foundations when TBM passed by was analyzed. The reliability of DEMFDM has been verified by numerous studies and is widely used in the analysis of tunnel construction technology; however, very little literature exists on the analysis of shield construction vibrations using this method. This paper combines field monitoring data with coupled DEMFDM analysis to determine the influence range of ground vibrations during TBM operation and to characterize the displacement and acceleration of pile foundations in both the time and frequency domains.

Vibration monitoring and analysis

Overview of the monitoring area

The Phase I project of Jinan Metro Line 4, Section No.7, has a total length of 4.82km, with the tunnel excavated along the main road of Jinan, Jingshi Road. This study focuses on the section from Yanshan Interchange Bridge East Station to Shanda Road Station, which spans 1.47km. The distance between the left and right lines in this section ranges from 5.8 to 15.2m, with a tunnel depth ranging from 8.66 to 20.24m. The tunnel segments in this section with an inner diameter of 5800mm and an outer diameter of 6400mm. The concrete strength grade of the segments is C50, with a segment width of 1500mm.

This section passes beneath the interchange bridge, as shown in Fig.1. The upper structure of the bridge consists of a continuous box girder, supported by pile-column steel pipe concrete piers. The foundation comprises reinforced concrete pile, with a top burial depth of 1m and a pile bottom burial depth of 22.8m. The horizontal distance between the pile and the ongoing construction of the right track is 3.84m, while the vertical distance between the pile bottom and the bottom of the tunnel is 6.34m. During construction, sensors were installed on the main shaft of the TBM to monitor the vibration of the cutterhead, and additional sensors were placed at a height of 2m above the ground to monitor the vibration of the bridge pier, as depicted in Fig.2. In addition, the vibration response of the first segment behind the cutterhead was monitored in the tunnel, with the sensor installed on the segment’s side wall. The collected vibration data is used to calculate the attenuation of the vibration from the cutterhead as it is transmitted into the tunnel, and it also serves as the input for tunnel vibration in numerical simulation analyses. The measurement points at the piers are primarily located at the bottom of the piers and 2m above ground level to analyze the vibration response of the top of the lower pile foundation, and they also serve as verification data for numerical simulation analyses.

Field surveys revealed that the tunnel was excavated through moderately weathered limestone (Fig.2). The rock mass exhibited a grayish-white color locally, with an intactness index (K_v) ranging from 0.55 to 0.60 and a saturated uniaxial compressive strength ranging from 100.5 to 115.4MPa. Consequently, the rock mass was relatively intact, classified as moderately hard to hard rock, with a basic quality grade of II to III. Microseismic measurements determined the intrinsic frequency of the limestone on site to be 4.6Hz. According to feedback from on-site construction personnel, noticeable vibrations were felt on the ground during the operation of the shield tunneling machine.

Monitoring methods and setting of monitoring points

To minimize background vibration interference, vibration monitoring was conducted from 1:00 a.m. to 3:00 a.m. Monitoring points were positioned as shown in Fig.3, with the tunnel depth at the monitoring section measuring 10.06m. Vibration monitoring points were established on the ground surface above the cutterhead. Along the tunnel axis, monitoring points (z1 to z8) were placed every 3m within a range of 9m in front of and 12m behind the cutterhead, to collect ground vibration signals at varying distances from the cutterhead. One of the z5 measurement points is located at the bottom of the bridge abutment, as shown in Fig.2. Additionally, two monitoring points (z9 and z10) were positioned on the ground 6m to the left and right sides of the cutterhead to gather vibration data from these areas. Vibration monitoring points were positioned on the curbstones at the edge of the road. Monitoring utilized the vibration sensor, equipped with Bluetooth signal transmission (see Fig.3). This sensor features digital filtering capabilities, automatically screening out background vibrations. Capable of simultaneously collecting triaxial displacement, velocity, and acceleration data, the sensor has displacement and velocity ranges of ± 30mm and ± 50mm/s, respectively, with a displacement accuracy of 1μm. Given that vibrations from cutterhead predominantly occur at low frequencies, the sensor’s sampling rate is set to 100Hz.

Vibration monitoring results

Vibration time-domain characteristics of ground surface and Bridge abutments

In hard rock formations during TBM opration, vibrations eventually propagate to the surface. This study monitored ground vibrations both ahead and behind the cutterhead, as well as on its sides. Figure4 illustrates the ground vibration signals. In actual construction, the vibration generated by the cutter comprises components along the three principal axes of the spatial coordinate system; however, for the pile on one side of the tunnel, the most impactful vibration is typically horizontal and directed toward the pile. However, onsite monitoring data indicate that among the measured vibration components, the intensity in the gravitational (zdirection) is the highest, the amplitude in the horizontal transverse (xdirection) is slightly lower, and the intensity in the tunnel’s axial (ydirection) is the lowest. Therefore, considering the requirements of subsequent modeling analyses and the relative contributions of the three directional components, the vibration data in the z and xdirections are ultimately combined to represent the vibration in the radial direction of the tunnel based on the measurement point locations. This is achieved by first triangulating the vibration data in both directions within the rightangled coordinate system of the tunnel crosssection, and then decomposing them into radial and circumferential components according to the measurement point location (right sidewall of the tunnel). In Fig.4, positive values indicate movement toward the surrounding rock, whereas negative values indicate movement toward the tunnel center. Monitoring data indicated that during TBM opration, the cutterhead’s vibration displacement could reach 1.5mm, with the peak displacement of the first segment being 0.3mm, which is 20% of the cutterhead’s peak vibration. When the depth was less than or equal to 10m, the peak vibration displacement on the ground directly above the cutterhead exceeded 0.08mm. Ground vibrations approximately 6m from the cutterhead measured around 0.03mm, which is 37.5% of the vibration peak at the top of the cutterhead’s vicinity. Monitoring results indicated that at a distance of 12m from the cutterhead, ground vibration displacement was already less than 0.01mm.

Taking the second derivative of ground vibration displacement data yields ground vibration acceleration. Figure5 illustrates ground vibration acceleration around the cutterhead, approximately 6m away, and at the cutterhead’s top. Monitoring results indicated that during operation of the TBM, the peak acceleration of cutterhead vibration reached 25m/s² (2.55g), with the peak acceleration of the ground at the cutterhead’s top measuring 0.05m/s² (0.005g), while the peak ground vibration acceleration approximately 6m from the cutterhead was all below 0.02m/s². Analysis of the monitoring data revealed that when the tunnel depth was 10m, the peak vibration acceleration of the ground at the cutterhead’s top was 0.2% of the cutterhead’s own vibration peak, while the peak ground vibration acceleration around the cutterhead, approximately 6m away, was 40% of the peak ground vibration acceleration at the cutterhead’s top.

During the passage of the cutterhead through the viaduct, vibration monitoring was conducted on the piers to observe their dynamic response. Figures6 and 7 depict the vibration displacement and acceleration of the piers at a distance of 2 m above the ground level. Displacement monitoring data indicated that the peak vibration of the piers was approximately 0.02mm when the cutterhead was 6m away from the piers both before and after passing through them, and increased to 0.05mm when the cutterhead reached the piers. Acceleration data for the piers showed that after the cutterhead reached the bottom of the piers, the peak vibration acceleration of the piers reached 0.05m/s². However, the peak vibration acceleration measured when the cutter plate is 6m from the abutment—both before and after passing it—is reduced. Specifically, the peak vibration acceleration recorded at the abutment when the cutter plate is 6m away is only 0.02m/s², and the overall vibration intensity is significantly diminished.

Vibration frequency domain characteristics of surface and piers

Figure 8 displays the Fourier-transformed spectra of the vibration accelerations measured at the cutterhead and the first segment. The results indicate that the vibration source frequencies during shield tunneling predominantly fall below 10Hz. Specifically, the cutterhead exhibited three peak frequencies at 2.5, 4.0, and 7.5Hz, whereas the segment showed four peak frequencies at 2.5, 5.5, 7.5, and 9.5Hz.

Performing Fourier transform on ground vibration acceleration yields frequency domain data of acceleration. Figure9(a) shows the acceleration spectrum, indicating that cutterhead induces low-frequency vibrations on the ground, with multiple observable peaks below 10Hz. The acceleration amplitude at the top of the cutterhead is relatively high. Figure9(b) presents the Power Spectral Density (PSD) at five monitoring positions, which aligns with the peak frequencies of Fourier acceleration amplitude. In the frequency domain, there are multiple peaks, with ground vibration energy mainly distributed between 6 and 8Hz. The vibration energy is strongest at the top of the cutterhead, followed by 6m ahead of the cutterhead, and least at 6m to the right of the cutterhead. Comparing with Fig.8, it can be found that the vibration of the surface is more complex, and compared with the cutterhead, the number of wave peaks is increased, but the vibration energy is lower.

To analyze the frequency domain characteristics of vibration waves induced by cutterhead on the bridge piers, Fig.10 presents the frequency domain data of pier vibration acceleration. This data was obtained by performing a Fourier transform on the acceleration monitoring data presented in Fig.7. Two monitoring points were established for the piers. One, the z5 measurement point, is located at the bottom of the pier and reflects the vibration pattern of the pile top; the other is located on the pier at a height of 2m above ground level (see Fig.2). The frequencydomain data in Fig.10 represent the vibration spectrum of the pier at a height of 2m above ground level. Monitoring results indicated that below 12Hz, pier vibrations could be roughly divided into four frequency bands: 0–3.5Hz, 3.5–7Hz, 7–10.6Hz, and 10.6–12Hz, suggesting the presence of four modalities in pier vibrations below 12Hz. The peak values of pier vibration acceleration amplitude were mainly distributed in the frequency band of 7–10.6Hz, indicating that during TBM operation, the vibration energy generated by the cutterhead primarily concentrated in the frequency components of 7–10.6Hz.

Numerical simulation

Modeling and parameter selection

The pile foundation of the overpass is a concealed structure buried underground, making it difficult to directly monitor the vibration signals of the piles. Therefore, numerical simulation analysis was conducted to analyze the dynamic response of adjacent piles during shield tunneling. Discrete Element Method (DEM) is particularly suitable for analyzing rock and soil bodies and has been widely used in mining and tunneling simulations. On the other hand, mesh methods such as Finite Difference Method (FDM) and Finite Element Method (FEM) are more suitable for analyzing continuous bodies like metals and concrete. Therefore, to better reflect the dissipation characteristics of vibration waves in rock and soil bodies, the coupled numerical simulation method of DEM-FDM was employed in this study to analyze the vibration response induced by cutterhead on a single pile. The advantage of using the DEM-FDM coupled method lies in its enhanced capability to accurately simulate the interactions during vibration processes among segmental linings, surrounding rock, and piles with adjacent soil. During cutterhead vibration, these heterogeneous media interfaces may separate, creating temporary gaps that can influence the displacement response of the piles. While purely mesh-based methods are more suitable for analyzing the transmission of vibrational energy through media, they present greater complexity in modeling contacts.

The model established based on the DEM-FDM coupling method is depicted in Fig.11, with dimensions of 48m in width and 26m in height. Segmental lining and the pile in the model are modeled using the mesh method (FDM) and set as elastic constitutive models. The tunnel excavation diameter is 6.4m, with a segment diameter of 6.2m and a thickness of 0.35m, resulting in a shield tail gap of 10cm. The pile length is 22.8m, with a diameter 0.6m (D = 0.6m), and it is located on one side of the tunnel at a depth of 10m. Five different pile-tunnel spacings were set to analyze variations in the pile’s dynamic response. The tunnel is positioned 9.8m from the model’s bottom boundary and 20.9m from both the left and right boundaries, minimizing the impact of reflected waves. To ensure computational accuracy and optimize computer performance, smaller particle sizes are set near the tunnel, and larger sizes are set away from it during the formation of the strata using particle packing. The smallest particle size is 18mm, the largest is 9mm, and the model comprises a total of 79,987 particles, with a Gaussian distribution of particle sizes.

Two types of monitoring points, M1 to M6, were established in the model from the top to the bottom of the pile to monitor pile displacement and acceleration. In addition, three monitoring points were set on the ground to compare the obtained data with field monitoring data, thereby validating the model’s effectiveness. To ensure pre-calculation balance, the model progressively established soil particle layers from the bottom to the top and balanced them layer by layer. During the modeling process, the model’s equilibrium was determined by observing the distribution characteristics of force chains and particle velocities. The model was considered balanced when the Ratio-aver was less than 1e-5 and the particle velocity was less than 0.1mm/s. The geological strata where the tunnel is located are moderately weathered limestone, and the mechanical parameters of the rock layers in the model, based on geotechnical test results, are shown in Table2. The linear parallel key contact parameters between Ball-ball were inherited attributes consistent with the particle’s own parameters (Table2), with k_n (\(\:\stackrel{-}{{k}_{n}}\)) and k_s (\(\:\stackrel{-}{{k}_{s}}\)) of weathered limestone being 1.72e10 Pa and 1.15e10 Pa respectively, with a critical damping of 0.6. Figure12 shows the simulation of triaxial tests conducted on rock samples obtained from the site using the calibrated parameters. The triaxial tests were performed under a confining pressure of 5MPa. The calibration results indicated that the linear parallel bond model effectively captured the mechanical properties of limestone.

The calculation scheme

In addressing the impact of cutterhead vibrations, this study analyzed the influence of three factors: pile-tunnel spacing, vibration source amplitude, and vibration source frequency on the dynamic response of adjacent pile. Table3 outlines the research design, which includes 16 numerical simulation cases. Among them, calculations were conducted for the three factors using a sinusoidal cyclic signal as model input (Fig.13(a)), with a vibration signal duration of 15s, where the amplitude gradually increased to the target level over the first 2.5s and then decreased to 0 over the subsequent 2.5s. The load is applied to the segment in the form of vibration displacement. To validate the model’s reliability, Case-16 incorporated a 600s vibration signal obtained from on-site monitoring, collected on the right side of the first ring segment after the TBM had started (Fig.13(b)). The vibration displacement for the sinusoidal curve is defined in the radial direction of the tunnel—positive values indicate movement toward the surrounding rock, and negative values indicate movement toward the tunnel center—while the vibration direction for Case 16 is determined as described in Sect.2.3. The logic of the numerical simulation scheme and its relationship with monitoring data are shown in Fig.14. Vibration data from the segment in the monitoring data were used as the source input at the corresponding position in the model, while ground surface vibration data were compared with simulated surface vibrations to validate the model’s accuracy. Frequency domain analysis of the monitoring data provided the basis for parameters in the simulation. Time domain and spatial pattern analyses of the monitoring data revealed the vibration propagation range and trends. The numerical simulation results reflected the dynamic response of nearby pile.

Validation and unbalanced forces of the model

Case-16 incorporated vibration signals collected from the field. The three vibration monitoring points set on the model’s surface corresponded to z9, z10, and z5 in Fig.3. Figure15 depicts the surface vibration acceleration. Comparative analysis revealed that the vibration accelerations at all three monitoring points in the numerical simulation data were below 0.05m/s². The distribution pattern indicated that the vibration response was most pronounced at the top of the cutterhead, followed by the left and right sides. A comparison with the field monitoring data in Fig.15 indicated that the results of the numerical simulation were generally reliable. Figure16 compares the acceleration time history at the pile top from the model with the data from the pier base. Since vibration data for bridge pile foundations are difficult to obtain directly on-site, the pier base data were approximated as the pile top vibration data. Figure16 shows that the peak acceleration and vibration intensity in the time domain from the simulation were lower than the on-site monitoring data. This discrepancy may result from the more complex environment surrounding the pier on-site, which is influenced by additional vibration sources. Furthermore, as the monitoring data were collected from the pier base, the pier, having more degrees of freedom than the underground pile foundation, is more prone to higher-intensity vibrations. Figure17 shows the surface settlement observed in the field and in the model. The field monitoring data points were generally close to the numerical simulation data curve. Moreover, both the field monitoring data and the numerical simulation results showed a settlement zone width ranging from 20 to 25m, further confirming the reliability of the model. Overall, the numerical simulation data generally reflected the actual on-site conditions.

The unbalanced force refers to the internal force in the model when it fails to reach a state of mechanical equilibrium during calculations. In the DEM, materials are assumed to consist of a large number of aggregated particles. Therefore, the unbalanced force in the calculation process primarily refers to the unbalanced contact forces acting on each particle at a certain moment. Various forces such as gravity, inertia, elasticity, and friction between particles may lead to the generation of unbalanced forces. Figure18 illustrates the unbalanced forces in model Case-15 during the initial 3.5s. It can be observed that the unbalanced forces in the particle portion synchronously change with the input sinusoidal load, exhibiting similar fluctuations to the load. Since the model established in this study was in equilibrium before the input vibration load, the unbalanced forces on the particles during dynamic calculations are mainly caused by the input vibration signal. Thus, the unbalanced forces in the model can qualitatively describe the dispersion of vibration waves. Figure18 demonstrates that the vibration waves in the model spread out from the center of the tunnel and transmit to the pile. Ultimately, the pile undergoes accompanying vibrations induced by the vibration waves. Additionally, Fig.18 indirectly confirms that the dimensions of the model are sufficient to allow vibrational waves to dissipate completely before reaching the boundaries, thereby preventing the pile’s response from being influenced by reflected waves.

Pile horizontal displacement

Figure 19 depicts the maximum horizontal displacement of the pile during vibration loading, the displacement distribution of the pile is shown in the Fig.19(b). Computational results indicate that within a range of 5 times the tunnel radius (15.5m), the difference in pile top displacement values is not significant with varying pile-tunnel spacing. Load frequency often has a significant impact on the dynamic response of structures near the vibration source. Therefore, only the M3 and M4 measurement points, which are closer to the tunnel, exhibit relatively greater horizontal displacement and demonstrate sensitivity to the distance between the pile and the tunnel. Although the M5 measurement point is approximately the same distance from the tunnel as M3, it does not follow this pattern. This discrepancy can be attributed to the increasing confinement from the surrounding rock at greater depths, resulting in less significant horizontal displacements at measurement points below the tunnel depth. In addition, as shown in Fig.19(a), once the pile-tunnel distance reached 12.4m (2D) and 15.5m (2.5D), the overall pile displacement was no longer significantly affected by vibration, indicating that the notable influence of vibration waves on the pile did not exceed 2D.

Figure 20 shows the peak displacements at various points along the pile under different load frequencies. The calculations reveal that low-frequency vibrations result in larger pile displacements. For instance, when the load frequency is 1Hz, the peak displacement at the pile top is 0.12mm, while at the pile base, it is 0.03mm, but the displacements of the M3 and M4 measuring points did not exceed the pile top, indicating that the pile body is mainly inclined. Moreover, the pile displacement response exhibits a non-linear negative correlation with the load frequency. After the load frequency exceeds 4Hz, the sensitivity of pile displacement to the load frequency decreases, this may be due to the large difference between the vibration frequency in this frequency band and the natural frequency of the rock formation. Additionally, when comparing the effects of distance between the pile and the tunnel, it becomes clear that the vibration frequency of the source exerts a greater influence on the pile’s vibration response than the distance itself.

Figure 21 illustrates the peak displacements at six measuring points under different load amplitudes, with Fig.21(b) depicting the horizontal displacement distribution of the pile. The computational results demonstrate that with increasing amplitude, the peak displacement at various points along the pile shows linear growth, with M3 and M4 being relatively more sensitive, exhibiting a linear coefficient of 0.027. At the pile base, the deformation was minimal yet increased linearly with amplitude at a rate of 0.014. From Fig.21(b), a slight bending of the pile was observed, which became more pronounced with increasing amplitude.

Based on a comprehensive analysis of the pile-tunnel distance, vibration source frequency, and vibration source amplitude, we found that tunnel-induced vibrations affect piles within a range of approximately 2D. Within this distance, low-frequency, high-amplitude waves exert a more pronounced influence on the pile. Monitoring data in Fig.8 show that the segment’s peak frequencies generally fall below 10Hz, although the first peak appears at 2.5Hz. These findings suggest that shield tunneling vibrations can still pose a potential risk to adjacent pile foundations.

Acceleration time-range response of pile

During TBM operation, nearby piles experience accompanying vibrations. Therefore, analyzing the acceleration at various points on the pile not only helps determine the motion trend of the pile but also reflects the additional inertial forces acting on them. Figure22 presents the acceleration time history data at the pile top for different pile-tunnel spacings, showing a distinct periodic vibration response at the pile top during the cutterhead vibrations. The acceleration peak at the pile top is significantly greater at a spacing of 3.1m compared to smaller spacings.

Figure23illustrates the relationship between the peak vibration acceleration at various points along the pile and the pile-tunnel spacing. The vibration acceleration of the pile base is negatively correlated with the pile-tunnel spacing, with greater acceleration at depth within the tunnel (M4). In vibration analyses, acceleration is often closely associated with inertial force. Consequently, Fig.23(b) shows relatively high inertial forces near M4, the measurement point closest to the tunnel. However, aside from piles within 0.5D of the tunnel, when the pile-tunnel distance reached at least 1D, differences in inertial force were not significant. This finding suggests that the additional vibration load on piles due to shield tunneling is predominantly concentrated within 1D.

Figure24 depicts the relationship between the peak vibration acceleration at various points along the pile and the vibration source frequency. Computational results indicate a strong negative exponential relationship between the acceleration peak at various points along the pile and the vibration source frequency. When the load frequency exceeds 4Hz, the acceleration peak at the pile top is less than 0.001m/s², while at a frequency of 1Hz, it reaches 0.0045m/s². Figure24(b) suggests that when the vibration source frequency is high, the vibration acceleration along the pile is consistent and relatively small, indicating insignificant additional loads from the surrounding soil.

Figure25 shows that the acceleration at various points along the pile increases linearly with the load amplitude, with the maximum acceleration occurring at M4. The linear coefficient of 0.0012 suggests that additional loads due to tunnel vibration become relatively significant at greater depths. Furthermore, Fig.25(b) indicates that increasing the vibration amplitude only slightly affects the top and bottom of the pile but exerts a pronounced impact on M4, which is located at the tunnel depth. This finding suggests that when the vibration source location and frequency are fixed, changes in amplitude alter the vibration mode of the adjacent pile, creating a shock-load-type vibration response at the point closest to the tunnel.

Frequency domain analysis of pile acceleration

During TBM operation, nearby piles experienced accompanying vibrations. Analysis of the pile displacement and acceleration responses indicated greater sensitivity to tunneling vibrations at points M1 and M4. Therefore, further Fourier transformation was conducted on the acceleration time history at M1 and M4. Figure26 illustrates the acceleration spectra at pile-tunnel spacings of 3.1, 9.3, and 15.5m, all with a vibration source frequency of 4Hz. Computational results revealed multiple peaks below 60Hz in the spectrum for smaller pile-tunnel spacings, with peak frequency above 2Hz distributed in 4Hz increments. As the pile-tunnel spacing increased, the number of peaks decreased. At a spacing of 15.5m, only five significant peaks were observed below 20Hz, occurring at 2, 6, 10, 14, and 18Hz. A comparison of the data from M1 and M4 revealed more complex vibration components at the pile top than at the midsection, with a broader frequency distribution range for pile top vibration. In addition, the acceleration amplitude at M4 exceeded that at M1, indicating more intense vibration responses at deeper tunnel depths. Figure27 depicts the PSD at the same locations, with energy concentration observed at 2Hz, 6Hz, and 10Hz. Moreover, as the pile-tunnel spacing decreased, the PSD significantly increased, indicating higher sensitivity.

Figure 28 displays the vibration spectra at loading frequencies of 2, 6, and 10Hz. The spectrum data indicate that at a source frequency of 2Hz, the peak acceleration amplitude in the M1 vibration response is concentrated at 1Hz, with multiple peaks incrementally increasing above 1Hz at 2Hz intervals, each peak decreasing in magnitude, suggesting a complex distribution of vibration components. In contrast, the peak acceleration amplitudes at source frequencies of 6Hz and 10Hz are relatively small, generally below 0.4m/s². Specifically, at a source frequency of 10Hz, only four significant peaks exist below 50Hz. Moreover, at source frequencies of 6Hz and 10Hz, the peak frequencies are 3Hz and 5Hz, respectively, indicating that the peak frequencies are roughly half of the source frequencies. A similar pattern is observed for the vibration acceleration amplitudes at M4, although the peaks in acceleration amplitude are relatively higher. Figure29 illustrates the PSD at corresponding locations, showing that the vibration energy at both M1 and M4 is concentrated in the low-frequency range below 10Hz when the vibration source is low-frequency, significantly greater than that from high-frequency sources.

Figure 30 presents the acceleration spectrum at points M1 and M4, while Fig.31 shows the PSD at the same location as Fig.30. When the vibration source amplitude is 1mm, the PSD peaks at M1 and M4 are 5 (m/s²)² and 7.8 (m/s²)² respectively. However, with a vibration source amplitude of 5mm, the PSD peaks at M1 and M4 increase to 120 and 195 (m/s²)² respectively, showing a growth rate of 24–25 times. The acceleration amplitudes at M1 and M4 change uniformly with the increase in vibration source amplitude, and they are relatively similar. Furthermore, comparing the acceleration amplitudes at different vibration source amplitudes reveals that as the vibration source amplitude increases, the frequency distribution of the accompanying pile vibration widens. When the vibration source amplitude is 5mm, there are significant peaks below 55Hz, whereas with a 1mm amplitude, only some peaks exist below 30Hz.

After obtaining the frequency domain data and power spectra at the two measurement points corresponding to the pile body, the kinetic energy and average power generated during the vibration process at these locations can be calculated. According to Parseval’s theorem, the energy of a signal in the time domain is equal to the integral of its energy in the frequency domain. When the collected data consist of N discrete samples, the energy can also be obtained by summing the squares of the magnitudes of the spectral components from the Discrete Fourier Transform (DFT) and multiplying the sum by the frequency resolution. For a discretetime signal x[n] (n = 0, 1, 2, …, N-1) of length N, its Fourier transform is defined as:

The total energy E in the time domain can then be expressed as:

According to Parseval’s theorem, under the employed Discrete Fourier Transform (DFT), the total energy E can be expressed as:

Based on the data from frequency domain analysis, Table4 compiles the total vibration energy and wave frequency at points M1 and M4 under three different parameter values. When comparing Case-1, Case-3, and Case-5, it was observed that increasing the pile-tunnel spacing by five times reduced the total vibration energy at the pile top by 4.5%, with the wave frequency at the pile top occurring at half the vibration source frequency, and at 1.5 times the frequency at the depth of the tunnel. Comparing Case-6, Case-8, and Case-9 reveals that the vibration energy at M4 was stronger than at M1, and with a fivefold increase in the vibration source frequency, the total vibration energy at the pile top decreased by 1.6%, with peak frequencies at M1 and M4 occurring at half the vibration source frequency. Comparing Case-9, Case-13, and Case-15 indicates that when the vibration source amplitude changed from 5mm to 1mm, the vibration energy at the pile top decreased by 4%. The results suggest that during tunnel construction, the pile top near the neighboring piles is the most sensitive to the vibration frequency of the vibration source. Therefore, adjusting construction parameters to minimize vibration frequency can significantly reduce pile vibrations.