The project is situated on the right bank of the Yangtze River, where the wharf platform is designed as a high-pile beam-slab structure, measuring 461 m in length and 23 m in width. On the river-facing side, the front row consists of straight steel pipe piles with an outer diameter of 1200 mm and a wall thickness of 18 mm. On the bank-facing side, forked pile pairs are used, comprising prestressed high-strength concrete piles with a diameter of 1200 mm and a wall thickness of 150 mm, combined with steel pipe piles measuring 940 mm in diameter and 20 mm in wall thickness. These forked piles are installed at an inclination of 6:1. The pile foundations are embedded in fully weathered silty mudstone. The site’s topography includes both the riverbed and the first-level terrace of the Yangtze River. While the natural bank slope is relatively steep, the upper terrace is comparatively flat, with elevations ranging from approximately 30 to 32 m. To ensure construction success and maintain bank slope stability, steel sheet piles are adopted for slope protection. The relative layout of the wharf platform, excavation zones, and steel sheet piles is shown in Fig. 1.

The on-site construction follows the steel sheet pile support design parameters outlined in Scheme 5, utilizing a double-row steel sheet pile support system. The first row consists of 12-meter-long steel sheet piles, installed along the outer edge of the access road, while the second row features 18-meter-long steel sheet piles positioned 5 m from the first row along the riverbank. At the top outer edge of both rows, steel H20 beams are installed as waling beams. These beams are securely connected into a unified structure by welding 5# channel steel at 3 m intervals. The excavation outside the steel sheet piles slopes down to an elevation of 17.5 m to accommodate the draft requirements of the pile driving vessel. To assess the stability of the double-row steel sheet piles under the load exerted by the wharf platform pile driving and to evaluate potential damage to the structural integrity of the steel sheet piles, field vibration monitoring is conducted during the pile hammering process. The layout of the vibration testing points is shown in Fig. 5.

A total of 60 monitoring sessions were carried out, each recording the complete construction cycle of hammer-driven pile installation. Representative Peak Particle Velocity (PPV) data are presented in Table 4. The results indicate that PPV values progressively decrease as the monitoring points are located farther from the hammer impact location. Moreover, across all tests, the timing of PPV occurrences at the monitoring points remains highly consistent, with the corresponding number of hammer blows (N) being nearly identical.

To assess the stability of the monitored steel sheet piles under hammering operations, a numerical model is developed using the dynamic finite element software LS-DYNA to investigate their dynamic response characteristics under pile driving loads. The modeling is based on a typical cross-section of the project. A three-dimensional, full-scale (1:1) computational model of the bank slope is constructed using 8-node SOLID164 solid elements. The model dimensions are 65.2 m × 20.0 m × 32.7 m, comprising 3,134,322 elements generated through Lagrangian meshing. Automatic surface-to-surface contact definitions are applied between the steel sheet piles and the surrounding rock-soil layers. To replicate actual site conditions, the top surface of the model is treated as a free boundary, while all other boundaries are defined as non-reflecting. This setup effectively simulates wave propagation in an infinite or semi-infinite medium, thereby minimizing or eliminating artificial wave reflections at the model boundaries [26–28]. The dimensions of the numerical model and the boundary conditions are illustrated in Fig. 6.

Quantifying the complex properties of rock and soil media remains a significant challenge, as no explicit mathematical formulations currently exist to describe them precisely. Therefore, in this study, the rock and soil media are modeled as continuous, isotropic, elastic-plastic materials. The rock is assumed to be fully weathered silty mudstone. To simulate its behavior, the *MAT_PLASTIC_KINEMATIC material model provided by LSTC is employed [29,30]. The specific formulation of this material model is given in Eq. (2). (2)σy=[1+(ε˙c)1p](σ0+βEpεpeff)where p and c are constants from the Cowper-Symonds strain rate model. β represents the hardening parameter. σ_y is the yield stress. σ₀ is the initial yield stress. E_p denotes the plastic hardening modulus. ε˙ is the strain rate, and ε_p represents the effective plastic strain. εpeff is the effective plastic strain.

For the silty clay and muddy silty clay, the *MAT_DRUCKER_PRAGER [31,32] material model is used. This model is commonly applied to granular materials and incorporates options for simulating fracture and crushing. The parameters defined within the model include soil properties related to the yield surface, specifically the friction angle and viscosity coefficient.

The double-row steel sheet piles are influenced solely by soil pressure and pile driving force, with no thermal deformation involved. To model the material behavior of the steel sheet piles, a simplified Johnson-Cook model is employed. This model effectively captures the mechanical properties of the steel used in Larson steel sheet piles without requiring the definition of an equation of state. Consequently, the *MAT_SIMPLIFIED_JOHNSON_COOK material model is selected for both the Larson steel sheet piles and the metal connecting rods between them. The constitutive relationship for the Johnson-Cook model is expressed as: (3)σ=(A+Bεpn)(1+Cln⁡ε˙∗)where σ represents the flow stress. A is the static yield strength. B is the strain hardening modulus. C is the strain rate coefficient. n is the strain hardening exponent. ε_p is the equivalent plastic strain. ε˙∗ is the normalized effective strain rate.

In the numerical simulation, selecting appropriate parameters for rock and soil materials is essential to ensure the accuracy and reliability of the simulated results. In this study, the parameters for rock and soil are primarily derived from field and laboratory tests conducted on slope samples. The parameters for the steel sheet piles are obtained from empirical formulas and established literature [11,12]. The initial specific mechanical parameters are listed in Table 5.

The process of driving a pile into the soil is a highly complex nonlinear dynamic problem, characterized by significant soil deformation. Accurately determining the force exerted by the pile on the soil during the driving process, whether through numerical calculations or experiments, remains a significant challenge. At present, hammer force curves are predominantly estimated using equivalent load methods. By integrating insights from existing literature with numerical simulation results [33–35], the hammer force curve is derived, as illustrated in Fig. 7.

To validate the model parameters and simulation reliability, a parameter sensitivity analysis is conducted by comparing field measured PPV with simulated PPV under various parameter combinations, ultimately identifying the optimal parameter set. In this manuscript, there are several important parameters used for the model including soil parameters (elastic modulus E, Poisson’s ratio μ, initial yield strength σ₀), loading parameters (peak value P₀, rising duration t_r, total duration t_d) and boundary conditions (non-reflecting boundary, free boundary). Considering the monitoring points arranged on the silty clay layer, this study primarily conducts a sensitivity analysis on the material parameters of silty clay, while the material parameters of other soil layers and rock strata are estimated based on site investigation and design data. According to field survey and design documentation, the elastic modulus E of silty clay at this site ranges approximately from 8 to 15 MPa, Poisson’s ratio μ between 0.15 and 0.2, and yield strength σ₀ between 12 and 19 MPa. For the sensitivity analysis, three values are selected for each parameter of the silty clay: E (8, 11, 15 MPa), μ (0.15, 0.18, 0.2), and σ₀ (12, 15, 19 MPa). The load parameters are determined based on empirical formulas and relevant literature [33–35]. For the sensitivity analysis, three values are selected for each load-related parameter: peak load (6, 8, 10 MPa), rise time (263, 525, 788 μs), and total duration (1.35, 1.7, 2 ms).

The sensitivity analysis results are presented in Table 6. It is found that No. 2 demonstrates the smallest error between actual and simulated PPV values. The PPV waveforms at monitoring point P1 from the field test and numerical simulation of No. 2 shown in Fig. 8. Within the specified value ranges, variations in peak load significantly affect the PPV simulation results at monitoring points, while total load duration shows negligible influence. When the model boundary condition transitions from non-reflective to free boundary, the PPV waveform exhibits markedly increased oscillations. Furthermore, due to wave reflection at free boundaries, the peak PPV value at monitoring points becomes amplified. According to the simulated results, the parameters of No. 2 are chosen as the model parameters.

The results from the three monitoring points (P1, P2, P3) in the numerical simulation are compared with the field monitoring data. This comparison serves as a reliability benchmark for subsequent numerical studies. Table 7 presents the comparison of peak vibration velocity between the numerical simulation results and field measurements at each monitoring point. The field monitoring PPVs are slightly lower than those obtained from the simulations, with a maximum relative error of 9.8% and a minimum of 7.7%. This difference can be attributed to the various structural discontinuities present in the rock, which affect the propagation and frequency attenuation of the blasting stress waves [36]. These discrepancies fall within a reasonable range, confirming the reliability of the model.

Under continuous hammering loads, the double-row steel sheet pile structures undergo displacement, which may affect the overall stability of the bank slope. Therefore, analyzing the displacement behavior of the support structures under such dynamic loading is essential [37]. In the numerical model, hammering loads are applied at the midpoint of the soil layer behind the unexcavated slope. Twenty displacement monitoring points are arranged along the horizontal direction for both the upper and lower rows of the double-row sheet pile structures, as illustrated in Fig. 9. The distribution of peak displacements is presented in Fig. 10. Results indicate that horizontal displacement variation is minimal, suggesting uniform deformation across the structures. Some monitoring points near the model boundaries exhibit noticeable differences due to boundary effects, in contrast to those near the center. The peak displacements at central locations in both rows are comparable, demonstrating the effectiveness of the rigid connection between the upper and lower rows in resisting deformation. The high elastic modulus of the steel sheet piles also contributes to their strong deformation resistance. The maximum displacement recorded in the lower row is 3.14 cm, and in the upper row, 2.51 cm-both well within the allowable deflection limits of 30 and 45 mm, respectively. Moreover, the rigid connection enhances the overall deflection capacity beyond the theoretical limits. These results confirm that, under the applied hammering loads, the double-row steel sheet pile structures remain structurally sound and perform safely within acceptable displacement limits.

If the stress within a steel sheet pile exceeds its allowable limit, structural failure may occur, potentially compromising the stability of the entire bank slope system. Therefore, analyzing the dynamic stress response of the steel sheet pile is essential for understanding its overall dynamic behavior [38]. An evaluation of several monitoring points located at the midsection of the model reveals that the maximum principal stress in the steel sheet pile remains below 20 MPa. The stress values across these points show minimal variation and are substantially lower than the allowable stress limit of 218.52 MPa for Q239 steel, the material used for the piles. According to the maximum principal stress theory, the steel sheet pile structures remain intact throughout the pile driving process and is considered structurally safe. The maximum principal stress values at representative monitoring points are illustrated in Fig. 11.

When assessing the allowable stress in steel structures, reliance on maximum principal stress alone is insufficient. In numerical simulations, the von Mises stress is commonly employed to capture the combined effect of multiple principal stresses on the structural integrity of anisotropic materials. This comprehensive approach allows for a more accurate evaluation of the impact of dynamic loading on the stability of steel sheet pile structures by comparing both the maximum principal stress and the von Mises stress.

The von Mises stress accounts for stress components in all directions, offering a comprehensive measure of structural strength. Analyzing the maximum von Mises stress at each horizontal monitoring point provides a systematic understanding of the stress distribution within the structures. The monitoring points used for von Mises stress analysis correspond to the displacement monitoring points shown in Fig. 9, and the horizontal distribution of von Mises stress is illustrated in Fig. 12. Excluding outliers attributed to boundary effects (i.e., the first three and last three points), the von Mises stress is observed to increase as the monitoring points approach the vertical line connecting the hammering location and the steel sheet pile. The maximum von Mises stresses in the upper and lower rows of the steel sheet pile are 20.85 and 25.40 MPa, respectively—both significantly below the allowable stress limit of 218.52 MPa for Q239 steel. Based on the combined evaluation of maximum principal stress and von Mises stress, it can be concluded that the double-row steel sheet pile structures remain intact and operates safely within the allowable stress range under the impact of hammering loads.

In the context of slope-soil-pile system dynamics, time-frequency analysis plays a crucial role in optimizing dynamic slope protection design. Spectral analysis was conducted on vibration signals obtained from 20 monitoring points (Fig. 9) positioned along the upper and lower rows of steel sheet piles. Fig. 13 presents the vibration time histories for three representative monitoring points from each row. Using the Fast Fourier Transform (FFT) method, the vibration signals were processed to identify the dominant frequencies at each monitoring location. The corresponding dominant frequencies for all points are shown in Fig. 14. The results indicate that the dominant frequencies of the sheet piles fall within the range of 30–35 Hz. For the lower row, all monitoring points exhibit a consistent dominant frequency of 35 Hz, except for Point No. 1, which shows the lowest value at 30 Hz. In the upper row, monitoring Points No. 1 through No. 6 display a dominant frequency of 30 Hz, while Points No. 7 through No. 20 consistently register 35 Hz. According to relevant literature [39], the natural frequency of the steel sheet piles is 64 Hz. Therefore, under the current working conditions, resonance does not occur during pile hammering. To further illustrate the frequency characteristics, three representative monitoring points (No. 1, No. 10, and No. 20) were selected from each row for detailed spectral analysis. As shown in Fig. 15, the frequency spectra for all selected points span a bandwidth of 0–500 Hz.

The calculation method for slope stability using the limit equilibrium method is as follows [42,43]:

The stability coefficient of the bank slope is defined as the ratio of the resisting force to the sliding force: (4)F=τfτwhere F is the stability coefficient of any element in the bank slope. τ_f is the ultimate shear strength. τ is the shear stress.

The relationship of the shear stress and normal stress on the sliding surface can be described by the Mohr-Coulomb criterion: (5)τf=c+σntan⁡φ (6)σn=σx+sin2⁡α+σycos2⁡α−τxysin⁡2α (7)τ=12(σy−σx)sin⁡2α+τxycos⁡2αwhere c is the cohesion of the soil or rock mass. φ is the internal friction angle of the soil or rock mass. α is the angle between the tangent to the sliding surface and the x-axis. σ_x and σ_y are the radial forces. τ_xy is the tangential force.

By substituting Eqs. (6)~(7) into Eq. (5), the stability coefficient of the slope at a certain moment is obtained by integration: (8)FE=∫(c+σntan⁡φ)dl∫τdl

The radial forces (σ_x and σ_y) and tangential force (τ_xy) under the vibration load of the pile hammer vary with time. By substituting these forces into Eq. (8) for the entire bank slope, a time history analysis can be performed to derive the time history curve of the slope stability coefficient.

The acceleration time history curve obtained from field vibration monitoring is shown in Fig. 16. Using LS-DYNA dynamic finite element software, the acceleration time-history curve is applied to the model nodes via the keyword *BOUNDARY_PRESCRIBED_MOTION_SET. The application positions are aligned as closely as possible with the locations of the field monitoring. Stress time history data for model elements are recorded at 500 μs intervals within the time range of 0 to 105 μs, with as many elements as possible selected from the potential sliding surface.

The stress components (σ_x, σ_y and τ_xy) obtained are substituted into Eq. (4) through (8) for calculation, resulting in the stability coefficient time history curve of the slope, as shown in Fig. 17. In the figure, the red line represents the static stability coefficient of the slope (1.26), and the black curve represents the dynamic stability coefficient of the slope. Under the vibration load from pile driving, the fluctuation pattern of the stability coefficient mirrors the acceleration time history curve. The stability coefficient reaches a maximum value of 2.887 at 7.02 s and a minimum value of 1.011 at 7.29 s. The stability coefficient decreases immediately upon receiving the vibration load, showing an initial decrease, followed by an increase, then another decrease within one oscillation cycle. Finally, it stabilizes with slight fluctuations around 1.26, indicating a relatively stable state. This indicates that although the vibration load from pile driving impacts the slope’s stability, the slope regains stability once the load dissipates.

According to the safety evaluation criteria for slope stability outlined in the Technical Code for Building Slope Engineering, the stability coefficient F_st for temporary slope engineering with a safety level of Grade I is 1.25. Based on the above analysis, the slope stability coefficient F_E obtained from this time history analysis exceeds F_st. Therefore, it can be concluded that the slope remains in a stable condition.