Open AccessArticle

Evaluation of Dynamic Compaction Load Conversion Methods and Vibration Reduction Treatments

Jixuan Li

¹,

Wenli Wang

^1,*,

Longping Luo

¹,

Xiaoliang Yao

¹ and

Jiangang Hu

State Key Laboratory of Eco-hydraulics in Northwest Arid Region, Xi’an University of Technology, Xi’an 710048, China

Regional Geological Survey Academe of Shaanxi Geology and Mining Group Co., Ltd., Xianyang 712099, China

Author to whom correspondence should be addressed.

Buildings 2025, 15(1), 111; https://doi.org/10.3390/buildings15010111

Submission received: 18 December 2024 / Revised: 30 December 2024 / Accepted: 30 December 2024 / Published: 31 December 2024

(This article belongs to the Section Construction Management, and Computers & Digitization)

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Figure 1
Schematic diagram of site vibration monitoring points: (a) plan view of the field test; (b) cutaway view of the field test. "> Figure 2
DC calculation geometry model. "> Figure 3
Single tamping attenuation diagram. "> Figure 4
Load application method. "> Figure 5
Comparison of soil vibration velocity: (a) 30 m from the tamping point; (b) 75 m from the tamping point. "> Figure 6
Comparison chart of tamping settlement. "> Figure 7
The monitoring point sketch map. "> Figure 8
Vibration velocity outside the isolation trench: (a) east–west direction; (b) north–south direction. "> Figure 9
Vibration velocity and ARF of a 2 m deep isolation trench: (a) east–west direction; (b) north–south direction. "> Figure 10
Vibration velocity and ARF of an isolation trench located 60 m away from the tamping point: (a) east–west direction; (b) north–south direction. "> Figure 11
Vibration velocity outside the isolation trench for filling materials with different unit weights: (a) east–west direction; (b) north–south direction. "> Figure 12
Vibration velocity outside the isolation trench for different damping ratios of the filling material: (a) east–west direction; (b) north–south direction. "> Figure 13
Vibration velocity outside the isolation trench for different cohesion and friction angles of the filling material: (a) east–west direction; (b) north–south direction. "> Figure 14
Vibration velocity outside the isolation trench for different stiffness and spacing of spring dampers: (a) east–west direction; (b) north–south direction. ">

Versions Notes

Abstract

This study aims to evaluate the accuracy of different dynamic compaction (DC) load equivalent conversion methods in DC vibration calculations. It also investigates the effect of vibration isolation treatments on the vibration reduction performance of loess foundations, with the goal of optimizing vibration control during DC construction. Five classical methods were used to convert the DC loads into time-dependent surface loads, which were subsequently fed into Plaxis’s dynamic multiplier table for the numerical implementation of DC tests. By comparing the numerical simulation results with in situ monitoring data from a loess site, the accuracy of the five DC load equivalent conversion methods was evaluated. The momentum theorem method was identified as the most precise for both vibration velocity and settlement. Subsequently, the momentum theorem method was utilized to investigate the influence of depth and distance of vibration isolation trench, as well as the properties of vibration isolation materials on vibration reduction effect. It is indicated that the optimal depth for the vibration isolation trench of the loess site is 2 m, beyond which the improvement in vibration reduction effects is not notable. The excavation distance of the vibration isolation trench should be set as close as possible to the boundary of the construction site to achieve the best vibration reduction effect. As for the properties of vibration isolation materials, it is shown that the unit weight and damping ratio of the filling material have a significant effect on the vibration reduction effect, while the influence of the shear strength of the filling material is negligible. Besides the vibrating reduction influence of filling materials, utilizing spring dampers has a better vibration reduction effect. Increasing the stiffness of the spring dampers and reducing their spacing can significantly enhance the vibration reduction effect. In practical engineering applications, it is essential to consider both the effects and economic costs to select the optimal vibration reduction treatment and its parameters. This study provides a scientific basis for vibration control during DC construction, contributing to ensuring construction safety and efficiency while minimizing the impact on the surrounding environment.

Keywords:

DC-induced vibration; in situ monitoring; loess foundation; DC load; method evaluation

1. Introduction

In the foundation treatment of buildings, DC is widely applied in foundation engineering under different geotechnical conditions due to its high efficiency and simple operation [1,2,3,4]. It is used to enhance the bearing capacity of soft soil layers and reduce settlement [5,6,7]. However, the vibrations resulting from DC during construction activities may raise potential risks to the safe operation of surrounding structures and various production activities. Subsequently, the safe implementation of DC relies heavily on accurately assessing the influence of compaction vibrations on the surrounding site and adopting reasonable vibration reduction treatments. Conducting practical DC tests can yield parameter values that are most in line with engineering reality to evaluate vibration influence, while it usually leads to large economic costs and time-consuming methods. Consequently, engineering designers usually prefer to select more economical and efficient numerical methods to assess the influence of DC and obtain optimal vibration reduction treatment parameters before the practical DC activities.

At present, there is a substantial amount of research on numerical calculations related to DC activities. Generally, researchers combine the elastoplastic constitutive model of soil with practical engineering dynamic propagation characteristics and boundary conditions and use the dynamic calculation module in numerical software to describe the dynamic response of soil. For instance, Huang et al. [8] studied the deformation patterns of soil columns under impact loads based on stress wave propagation and the Duncan–Chang model. The results showed that the soil column ends deformed into a mushroom shape, with both axial and lateral deformations increasing with impact velocity. Yao et al. [9] modeled the nonlinear dynamic behavior of soil during DC using a non-hardening shear yield surface and a hardening cap model and analyzed the progressive improvement of soil relative density under continuous tamping. Li et al. [10] used the hysteretic damping model in discrete element software to simulate the dynamic behavior of soil under impact loads. Jia et al. [11] employed a three-dimensional discrete element-finite difference coupling method to simulate the softening process during DC. Tashakori et al. [12] used the cap plastic model to consider the plastic strain hardening behavior of soil and employed equivalent simplified methods to simulate the contact process between the tamper and soil. Kundu et al. [13] used the Drucker–Prager model to study the settlement effect of DC on solid waste landfills with different compressibility. It shows that DC can effectively increase the capacity and service life of landfills. IZRA et al. [14] showed that the Drucker–Prager analysis model based on the infinite element method can accurately simulate the mechanical behavior during DC. From the above elaboration, it is clear that the constitutive relationship, as the core theory in DC numerical analysis, has been well developed and can generally provide a good description of the dynamic behavior of different projects under DC conditions.

In the calculation of DC, the precision of the equivalent conversion method for DC loads is another crucial factor. Based on this consideration, researchers have proposed various conversion methods to obtain dynamic boundary conditions in DC calculations. For example, Miller [15] derived field integral expressions for sinusoidal time-dependent stress on the surface of a semi-infinite isotropic solid. Scott [16], based on structural dynamics theory, took DC as a deceleration impact process of soil and proposed a converting formula for impact contact stress. Mayne [17] proposed an impact stress calculation formula based on the principle of momentum. Qian et al. [18] optimized the contact stress formula proposed by Scott [16] and obtained the dynamic response and reinforcement depth of soil. Peng et al. [19] studied the contact stress under various soil conditions and found that contact stress decreased exponentially with increasing soil moisture. Liu et al. [20] modified the stress peak and stress function in the loading and unloading elastic model proposed by Qian et al. [18] and simplified the complex formulas into sinusoidal impact load functions. Xie et al. [21] established nonlinear finite element equations for soil based on the assumption of large deformation and used the energy conservation principle to establish an optimization model. Zhang et al. [22] proposed a density-dependent constitutive model for coarse-grained soil and a rigid–flexible contact algorithm to simulate the dynamic contact between the tamper and soil. Zhao et al. [23] proposed a simplified dynamic response model based on the deformation and dynamic stress response characteristics of soil under impact loading, analyzing the mechanical response of soil in DC foundation treatment. Wang et al. [24] employed the FEM-SPH coupling method to simulate the large deformation issues of soil during the DC process, optimized the tamper scheme through parametric analysis, and improved the efficiency of foundation treatment. Ran et al. [25] obtained dynamic stress data through field DC tests and used sensitivity analysis and regression analysis to establish a linear regression equation for the impact stress at the compaction point. By introducing a vertical dynamic stress transfer index, they modified the ideal elastic static theory formula and proposed a calculation expression for the impact stress in soil–gravel mixed fill. This led to an analysis and determination of the effective reinforcement depth for DC, providing theoretical foundations and technical support for related engineering practices. Summarizing the available research works, the equivalent conversion methods for DC loads can be classified into five categories: the Scott method, the Miller method, the improved sine method, the momentum theorem method, and the equivalent contact static method. The accuracy of these methods has been verified by researchers under different engineering conditions; however, a comprehensive and effective comparative analysis of their accuracy across various engineering conditions remains absent. Therefore, evaluating the accuracy of different conversion methods is of great significance for further improving the precision of DC calculations and estimating the effects of DC vibration reduction treatments.

In order to evaluate the accuracy of different impact load conversion methods in DC vibration calculations and to analyze the effect of vibration isolation treatments on the vibration reduction performance of loess foundations. The dynamic analysis module of Plaxis software V20.1.0.98 is employed to compare numerical calculation results with in situ monitoring data. The calculation accuracy of five classical DC load conversion methods is analyzed. The conversion method with the highest accuracy is then adopted to further investigate the influence of different vibration reduction treatment parameters on the vibration reduction effect, such as the depth and distance of the vibration isolation trench and the mechanical and physical properties of its filling materials.

2. In Situ DC Tests

The in situ DC test site is located within a school foundation pit in Xianyang City, Shaanxi Province. The length, width, and depth of the foundation pit are 190 m, 160 m, and 6.57 m, respectively. During the preliminary geotechnical investigation phase of this engineering project, we unexpectedly discovered dense ancient tomb clusters. As a result, it became necessary to first carry out rescue excavation work on the tombs before performing site backfilling and subsequent foundation treatment. Considering the large porosity and collapsibility of the loess in this area, which could adversely affect the stability and bearing capacity of the foundation, we decided to reinforce the foundation using the dynamic compaction method to ensure the safety and stability of the project. Physical and mechanical parameters of each soil layer within the foundation pit were obtained based on the field investigation report, as shown in Table 1 and Table 2. By analyzing the data in Table 1 and Table 2, it can be seen that the soil belongs to silty clay with a low liquid limit. The excavation depth of the foundation pit is greater than the burial depth of the building foundation (approximately 5.5 m), and there is a thick layer of fill soil beneath the foundation. The site is characterized by Level III self-weight collapsible loess. The groundwater level at the DC test site ranges from 18.70 to 21.90 m in depth, which is much deeper than the foundation burial depth; therefore, the influence of groundwater is not considered in this study.

Based on DC construction experience in the research area, the primary dynamic compaction energy is 4000 kN·m, with a total of 9 blows. The average settlement for the final two blows must be controlled within 50 mm. The full-coverage compaction energy is 1500 kN·m. The initial elevation for dynamic compaction is 445.40 m, and the effective influence depth is no less than 5 m. The primary compaction tamper has a mass of 22.5 tons, while the full-coverage tamper weighs 10 tons. The lifting height for the primary tamper is 17.8 m, and for the full-coverage tamper, it is 15 m. The tamper diameter is 2.4 m, and four vent holes are provided. A YL-VMI vibration monitoring instrument produced by Shanghai Yanlian Engineering Technology Co., Ltd. (Shanghai, China) was used to monitor the vibration velocity at different monitoring points. To obtain the changing trend of the vibration velocity with the increase in tamping number, vertical vibration velocity was monitored in the north and east directions of the tamping point and at distances of 30 m and 75 m, respectively. The peak velocity at the tamping points during the test process was recorded. The positions of the tamping and monitoring points are shown in Figure 1.

3. Numerical Simulation Methods for DC

3.1. DC Loads Equivalent Conversion Method

Equations (1)–(5) are the specific formulas currently used in DC load conversion calculations. The details of each method are shown as follows:

(1) Scott method:

σ = V ρ c

(1)

where σ is the contact stress,

V = \sqrt{2 g h}

is the impact velocity, ρ = 1.26 kg/m³ is the soil density, and c = 83.94 m/s is the dilatation velocity.

(2) Miller method:

σ = \frac{S V}{π r^{2} ω} e^{- (R / 2 M) t} \cos (ω t - c o s^{- 1} \frac{R ω}{S})

(2)

where R = 1.2 m is the contact radius,

R = 0.6 π r^{2} ρ c

is the damping constant,

S = 2 r E / (1 - μ^{2})

is the elastic constant, M = 22.5 t is the tamper mass,

ω = \sqrt{S / M - R^{2} / 4 M^{2}}

is the angular frequency of loading, and t = 0.138 s is the tamping action time.

(3) Improved sine method:

σ_{m a x} = - \frac{M \ddot{u}}{π r^{2}} = \frac{V S}{π r^{2} ω}

(3)

where

ω = \sqrt{S / M}

is the angular frequency of loading.

(4) Momentum theorem method:

σ = F / s = \frac{M g}{π r^{2}} (1 + \sqrt{\frac{2 h}{g}} \frac{1}{Δ t})

(4)

where F is the contact impact force, S is the hammer bottom area, g = 9.8 N/kg, and h = 17.8 m is the tamper drop distance.

(5) Equivalent contact static method:

σ_{m a x} = \frac{1}{2} η σ_{s t} (1 + \sqrt{1 + \frac{4 h}{η s_{c}}})

(5)

where

η

is the proportion of volume increase of tamping pit to total volume change after tamping, Q is the tamper gravity, B is the tamper bottom area,

σ_{s t} = \frac{Q}{B}

, and

s_{c}

is the vertical displacement of soil column under compaction gravity.

By reviewing the existing research work, Table 3 outlines the advantages and disadvantages of the above-mentioned five categories of methods for calculating the contact stress in DC calculations. Subsequent numerical calculations corresponding to engineering conditions will be conducted using these contact stress calculation methods. Furthermore, the applicability of each method in the DC calculations for loess foundation pits will be compared with in situ monitoring results.

3.2. Numerical Model Parameters and Boundary Conditions

In this study, a classical elastoplastic constitutive model based on the Mohr–Coulomb yield criterion is employed in the numerical analysis. The Rayleigh damping in Plaxis software is used to simulate the effects of viscous damping. Since the shape of the DC test site is a regular rectangle, the geometric model established for numerical analysis is based on the principle of symmetrical boundary conditions, and the geometric model is built according to 1/4 of the DC test site. The geometric dimensions of the model are set to 95, 80, and 20 m along the X, Y, and Z directions, respectively, as shown in Figure 2. The maximum vibration frequency induced by actual DC construction generally does not exceed 15 Hz, with the corresponding maximum wavelength of 3.8 m. Therefore, the geometric model uses 10-node tetrahedral elements to divide the grid elements, with a refinement factor of 0.13 set in the DC reinforcement area. After mesh generation, the minimum element size is 0.12 m. This dimension meets the mesh discretization accuracy requirements. According to the DC test conditions, the compaction energy is 4000 kN·m, with a total of 9 tamping cycles, and each tamping number is set with a 3 s interval. Corresponding to the in situ monitoring results of DC tests, the vertical peak vibration velocity under each tamping number condition in the Plaxis calculation is adopted for comparative analysis with the measured results. Therefore, the velocity mentioned in the following text refers to the vertical peak velocity generated under a single tamping number load, as shown in Figure 3.

In numerical simulations, accurately capturing the dynamic response of the tamper is challenging. Therefore, a simplified approach was adopted in this study. The impact load exerted by the tamper on the soil during its drop was calculated using Equations (1)–(5), converting the instantaneous contact stress applied by the tamper to the ground into surface loads. These loads were then simplified into triangular loads and input into the dynamic multiplier table of the Plaxis dynamic calculation module. By applying the loads repeatedly in a tabular format, the process of continuous tamping was simulated, as illustrated in Figure 4. Under the actual test conditions where the tamper contacts the ground, the contact area is a circular region with a radius of 1.2 m. A circular contact area with a radius of 1.2 m is established using Plaxis’s polyline command, and the corresponding load is added to it for a total of 9 tamping cycles.

The foundation soil layer is divided into three layers, with a total thickness of 20 m extracted. The excavation depth of the foundation pit is 6.57 m. The physical and mechanical parameters of each soil layer, as shown in Table 4, are obtained from an in situ investigation report. The rest of the same parameters are shown in Table 1. The damping coefficients α and β are employed in the Plaxis dynamic calculation module, and the compression (V_p) and shear (V_s) wave velocities are calculated with Equations (6) and (7).

V_{p} = \sqrt{\frac{E_{o e d}}{ρ}}

(6)

V_{s} = \sqrt{\frac{G}{ρ}}

(7)

where

E_{oed}

is the compression modulus; G is the shear modulus, and ρ is the soil density.

As shown in Figure 2, the division surfaces of the geometric model are set as free-field boundaries. The bottom and the two outside surfaces are equipped with energy absorption boundaries to absorb the stress increments caused by dynamic loads at the boundaries, thereby minimizing the reflection of waves within the soil body to the greatest extent.

4. Results and Analysis

4.1. Comparative Analysis of DC Loads Equivalent Conversion Methods

Figure 5 shows the monitoring results of vibration velocity at 30 and 75 m from the tamping point, as well as the calculation results obtained using various DC loads equivalent conversion methods. Overall, the calculation results from all conversion methods show the same trend as the monitoring results with the increase in tamping number. The vibration velocity at 30 and 75 m both exhibit a rapid increase tendency during the first three tamping processes. As the number of tamping strikes continues to increase, the increase in vibration velocity at each monitoring point gradually decreases, gradually approaching a certain value. This indicates that the significant influence of DC on the engineering site mainly occurs during the initial few tamping processes. With the increase in the tamping number, the reinforcement of the DC loads on the soil layers gradually decreases.

The numerical calculation results of vibration velocity for each conversion method and their errors compared with the monitoring results are shown in Table 5. By comparing the calculation results of different DC load conversion methods with the monitoring results, it can be found that among the five conversion methods, the momentum theorem method has the highest precision, followed by the equivalent contact static method, the Miller method, and the Scott method, while the improved sine method shows the largest discrepancy.

Figure 6 shows the duration curves of the tamping settlement obtained by the five conversion methods, where negative values indicate the direction of settlement is downward. It can be observed that the settlement patterns of the different conversion methods are similar, all showing an increase in settlement as the tamping time increases. According to practical experience from heavy tamping projects in loess soil layers, the maximum settlement of 1 m can be achieved under tamping energy of 4000 kN·m. Obviously, the final settlement calculated by the momentum theorem method is the closest to the practical engineering experience value. While for the calculation results of other conversion methods, the final settlement of the Miller method is less than 1 m; the calculated final settlements of the equivalent contact static method, the Scott method, and the improved sine method are all much larger than 1 m. This is obviously not in line with reality.

Based on the analysis of the vibration velocity and the settlement results, it can be concluded that the calculation results with the momentum theorem method are in the best agreement with actual engineering conditions. Conversely, the improved sine method and the Scott method generally overestimate the values of vibration velocity and settlement. Although the Miller method and the equivalent contact static method have smaller deviations from the monitoring data, their performance is still not as good as the momentum theorem method. Thus, the momentum theorem method will be used to estimate the influence of different parameters under different vibration reduction treatments.

4.2. Analysis of Vibration Reduction Treatment Parameters

This section will further analyze the influence of vibration reduction treatment parameters on the vibration reduction effect of the DC site using the momentum theorem method. The factors considered for vibration reduction treatments mainly include the depth of the isolation trench, the distance of the isolation trench from the tamping point, and the basic physical and mechanical parameters of the filling material in the isolation trench. The basic calculation parameters for the vibration reduction treatments carried out are the same as those in the previous section.

4.2.1. Depth of Vibration Isolation Trench

As shown in Figure 7, empty trenches are excavated at a distance of 35 m from the tamping point, with a uniform opening width of 2 m. The X-axis direction is east–west, and the Y-axis direction is north–south. Vibration isolation trenches are set in the east–west and south–north directions. Here, four different vibration isolation trench depths are considered, namely, 1, 1.5, 2, and 3 m. To analyze the influence of trench depth on the vibration reduction effects, the vibration velocity calculations under different trench depths are compared with the results without isolation trenches.

Figure 8 shows the duration curves of vibration velocity outside the east–west and south–north vibration isolation trenches. It can be observed that, compared to the results without isolation trenches, as the depth of the isolation trenches increases, the reduction in vibration velocity becomes more apparent. This indicates that the vibration reduction effect is proportional to the depth of the isolation trench. With the isolation trench depth decreasing further, the rate of decrease in velocity shows an attenuation tendency. Under the conditions of 1.5 m and 2 m isolation trench depths, the curves of vibration velocity are getting close to each other. When the depth of the vibration isolation trench exceeds 2 m, the additional vibration reduction effect caused by further increase of the depth is negligible, and there is no significant difference in vibration velocity between 2 m and 3 m isolation trenches.

To further clarify the influence of the vibration isolation trench depth on the vibration velocity amplitude attenuation at the DC construction site, the ratio of the vibration velocity amplitude with and without the vibration isolation trench is defined as the Amplitude Reduction Factor (A_RF). It is a key index showing the vibration reduction effects. A smaller the A_RF means the better vibration reduction effects. The average A_RF obtained under different depths of vibration isolation trenches is shown in Table 6, where the average A_RF is the average of the A_RF after 9 tamping cycles. It can be observed that the average A_RF for a 2 m deep vibration isolation trench is approximately 64%, which is about 12% lower than that of a 1 m deep trench. In contrast, the average A_RF for the 2 m and 3 m deep vibration isolation trenches shows a small difference, both being around 63%.

Figure 9 shows the duration curves of A_RF and vibration velocity outside and inside a 2 m deep vibration isolation trench in the east–west and south–north directions. It can be seen that the 2 m deep vibration isolation trench has the best vibration reduction effect. The difference between the inside and outside vibration velocity after the ninth tamping cycle is approximately 0.6 cm/s, and the A_RF reaches a constant value after the third tamping cycle. This indicates that once the depth of the vibration isolation trench reaches 2 m, the vibration reduction effect no longer increases significantly with greater depth. This further demonstrates that there exists an optimal depth for the vibration isolation trench in actual engineering sites; beyond this optimal depth, the vibration isolation effect does not significantly improve.

In summary, increasing the depth of the vibration isolation trench can improve the vibration isolation effect, but there is an optimal depth beyond which the enhancement becomes negligible. For the loess foundation soil layer studied in this paper, the optimal depth of the vibration isolation trench was found to be 2 m.

4.2.2. Distance of Vibration Isolation Trench

The vibration reduction effect of the isolation trench depth at 35 m from the tamping point has been verified in the above section. At the same time, the distance of the vibration isolation trench to the vibration source also has a significant influence on the vibration reduction effects. This section will further compare the vibration reduction effects of the isolation trenches at 35 m and 60 m from the tamping point, both of which have a depth of 2 m. The results of vibration velocity and A_RF in the east–west and south–north directions, 60 m away from the tamping point outside the isolation trench, are shown in Figure 10. Compared with Figure 8, it can be seen that the A_RF at the isolation trench located 60 m from the tamping point is higher than that observed at 35 m. This indicates that the vibration reduction effect is better when the isolation trench is 35 m away from the tamping point.

The average A_RF obtained under different vibration isolation trench distance conditions is shown in Table 7. Regardless of the direction, the average A_RF of the vibration isolation trench at a distance of 60 m from the tamping point is always greater than that at 35 m, increasing by about 5%. Considering that during the actual construction process, the tamping point needs to cover all positions in the foundation pit range, the distance between the moving tamping point and the fixed vibration isolation trench changes during the actual DC process. Therefore, in order to satisfy the vibration reduction effect for all tamping points, the excavation position of the vibration isolation trench should be set as close as possible to the boundary of the construction site.

4.2.3. Physical and Mechanical Parameters of Isolation Trench Filling Materials

In this section, the vibration reduction effects of different materials filled in the vibration isolation trenches are further analyzed, considering different unit weights, damping ratios, and strength parameters. Additionally, the influence of spring dampers installed in the isolation trench at different spacing intervals and stiffness levels is also investigated.

(1) Unit weight of different materials

Figure 11 shows the duration curves of vibration velocity outside the east–west and south–north vibration isolation trenches for different filling materials with varying unit weights. It can be seen that as the unit weight of the filling material increases, the vibration velocity outside the isolation trench increases continuously. The increase in amplitude is proportional to the increase in unit weight. For instance, in both the east–west and south–north directions, when the soil weight density reaches 30 kN/m³, its vibration velocity has already exceeded the vibration velocity in the unfilled state.

The average A_RF for different filling materials compared to an empty trench is shown in Table 8. When the unit weight of the filling material reaches 5 and 10 kN/m³, the average A_RF is less than that of the unfilled state. However, when the unit weight of the filling material is 20 kN/m³, the vibration reduction effect is close to that of the empty trench. For the loess foundation soil layer studied in this paper, the saturated dry unit weight is approximately 20 kN/m³. This indicates that when the unit weight of the filling material in the vibration isolation trench is higher than the surrounding soil layer, the filling material within the isolation trench will not have a vibration reduction effect. For materials unit weight less than that of the surrounding soil layer, the most economical and readily available material in actual engineering applications is water. Therefore, it is possible to consider filling the isolation trench with water to further enhance the vibration isolation effect.

(2) Damping Ratio

Figure 12 shows the duration curves of the vibration velocity outside the isolation trench in the east–west and south–north directions for different damping ratios of the filling material. It can be seen that, as the damping ratio of the material increases, the vibration velocity outside the isolation trench decreases continuously. This reduction tendency is inversely proportional to the increase in the damping ratio. However, the vibration velocity under all four different conditions with damping ratios is smaller than that of the empty trench. This indicates that the damping ratio has a fairly good vibration reduction effect.

The average A_RF for different damping ratio fillings and the empty trench is shown in Table 9. As the damping ratio increases, the average A_RF in both directions continuously decreases, leading to a significant improvement in the vibration reduction effect. For instance, when the damping ratio increases to 40%, the average A_RF for the east–west and south–north directions of the isolation trenches reach 47.55% and 53.51%, respectively, which overall exceeds the vibration reduction effect of the empty trench. Therefore, for engineering cases with higher vibration tolerance requirements, this type of vibration isolation method has a significant advantage.

(3) Cohesion and internal friction angle

Figure 13 shows the duration curves of the vibration velocity outside the isolation trench in the east–west and south–north directions under different cohesion and friction angles. It can be observed that adjusting the cohesion and friction angle of the filling material does not significantly affect the vibration velocity of the soil outside the isolation trench. The difference in vibration velocity for both the east–west and south–north directions remains below 0.1%, indicating that the cohesion and friction angle of the damping material have no significant influence on the vibration reduction effect in terms of wave propagation. In summary, when selecting filling materials, there is no need to focus on the influence of the material shear strength on the vibration isolation effect.

(4) Spring dampers

As mentioned above, the damping ratio of the soil is crucial to the vibration reduction effect of materials, and materials with high damping ratios can significantly reduce the influence of tamper vibrations. Based on this observation, high-damping materials can be filled into the empty trench. Additionally, spring dampers with damping ratios far exceeding those of the soil layers at the construction site can be considered. Here, point-to-point anchor rods in Plaxis software are used to simulate spring connections, which are set as spring elements in the software, requiring only the specification of the axial stiffness parameter, equivalent to the spring coefficient. Stiffness parameters of 500 and 1000 kN are set to investigate their correlation with the vibration reduction effect. Furthermore, different spring spacing distances are considered to analyze the influence of the spacing between spring dampers on the vibration reduction effect, and the results are shown in Figure 14.

It can be observed that the vibration reduction effect of the spring dampers is positively correlated with their stiffness and spacing distance. For both the east–west and south–north orientations, as the stiffness increases and the spacing decreases, the vibration velocity continuously decreases. However, the decrease in vibration velocity due to increased stiffness is less notable compared to that caused by a spacing decrease. Especially under the condition of a stiffness of 500 kN and a spacing of 2 m, the vibration reduction effect is significantly higher than that observed under a stiffness of 1000 kN and a spacing of 5 m. This indicates that in practical applications, it is necessary to consider the optimization of both stiffness and spacing to achieve the best vibration reduction effect.

The average A_RF of different vibration reduction treatments is shown in Table 10. It can be seen that the average A_RF of the spring dampers with a stiffness of 500 kN and a spacing of 2 m installed in the isolation trench is much lower than that of other types of filling materials, with an A_RF of less than 43%. Therefore, spring dampers can provide the greatest vibration reduction effect and reduce the potential impact on surrounding structures. However, in actual engineering, it is also necessary to consider the effects and construction costs of various vibration reduction treatments and select the optimal treatments and parameters.

5. Conclusions

This paper uses five classic methods for converting DC loads into time-dependent surface loads during the DC process and inputs them into the dynamic multiplier table of Plaxis to simulate the multiple tamping processes of the DC test. By comparing the monitoring results of the loess foundation DC test, the accuracy of different DC loads equivalent conversion methods is analyzed. Based on this, the momentum theorem method, which has the highest calculation precision, is used to analyze the impact of various vibration reduction treatment parameters on the vibration reduction effect of the construction site, and the following conclusions are drawn:

(1) By comparing the vibration velocity and overall vibration settlement at 30 and 75 m from the tamping point between in situ monitoring data and numerical simulation results, the accuracy of different DC load equivalent conversion methods is verified. Among them, the momentum theorem method has the highest accuracy in terms of vibration velocity and settlement, followed by the equivalent contact static method, the Miller method, and the Scott method, with the improved sine method having the largest accuracy error. Therefore, the most suitable DC loads equivalent conversion method for loess areas is the momentum theorem method.

(2) The increase in the depth of the vibration isolation trench can significantly improve the vibration reduction effect, but there is an optimal depth. Once the depth of the vibration isolation trench reaches this optimal value, the improvement in the vibration reduction effect is negligible. For the loess foundation soil layer studied in this paper, the optimal depth of the vibration isolation trench is 2 m. In addition, reducing the distance between the vibration isolation trench and the tamping point can effectively enhance the vibration reduction effect. Considering that the distance between the moving tamping points and the fixed vibration isolation trenches during the actual DC construction process is changing, the excavation position of the vibration isolation trench should be set as close as possible to the boundary of the construction site.

(3) The damping ratio and unit weight of the filling material in the vibration isolation trench both have a significant impact on the vibration isolation effect. Generally, a smaller unit weight and higher damping ratio of the filling material yield a better vibration reduction effect. In practical construction engineering, water can be injected into the vibration isolation trench to further enhance the vibration isolation effect. The vibration reduction effect of filling materials with different shear strength parameters is relatively weak, so the shear strength parameters of the filling materials do not need to be considered when selecting the filling materials for the vibration isolation trench. The average A_RF of the spring dampers with a stiffness of 500 kN and a spacing of 2 m is reduced to below 43%, which is significantly better than other vibration isolation treatments, demonstrating the best vibration reduction performance. A higher stiffness and smaller spacing of the spring damper would lead to a better vibration reduction effect. In actual engineering, it is necessary to comprehensively consider the effects and construction costs to select the optimal vibration reduction treatment and parameters.

(4) The study is primarily based on a specific loess foundation soil layer, and the conclusions may not be applicable to other types of soils. Although various vibration reduction treatments have been proposed, their economic costs and construction feasibility have not been fully considered. Future work will involve a more extensive parameter study to identify more optimal vibration reduction treatments. A cost–benefit analysis of various vibration reduction measures will be conducted to identify the most economically viable solution. Through these future efforts, the research findings of this study can be further expanded and deepened, providing more comprehensive and effective solutions for vibration control in DC construction.

Author Contributions

Conceptualization, J.L. and W.W.; methodology, W.W. and J.H.; software, J.L., W.W. and L.L.; validation, J.L. and L.L.; data curation, J.L. and W.W.; writing—original draft preparation, J.L., L.L. and W.W.; writing—review and editing, X.Y. and J.H.; supervision, W.W. and J.H.; project administration, X.Y.; funding acquisition, X.Y. All authors have read and agreed to the published version of the manuscript.

Funding

This research was partly supported by the National Natural Science Foundation of China (No. 42272319).

Data Availability Statement

The data presented in this study are all available in the article.

Conflicts of Interest

Author Jiangang Hu was employed by the company Regional Geological Survey Academe of Shaanxi Geology and Mining Group Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Figure 1. Schematic diagram of site vibration monitoring points: (a) plan view of the field test; (b) cutaway view of the field test.

Figure 2. DC calculation geometry model.

Figure 3. Single tamping attenuation diagram.

Figure 4. Load application method.

Figure 5. Comparison of soil vibration velocity: (a) 30 m from the tamping point; (b) 75 m from the tamping point.

Figure 6. Comparison chart of tamping settlement.

Figure 7. The monitoring point sketch map.

Figure 8. Vibration velocity outside the isolation trench: (a) east–west direction; (b) north–south direction.

Figure 9. Vibration velocity and A_RF of a 2 m deep isolation trench: (a) east–west direction; (b) north–south direction.

Figure 10. Vibration velocity and A_RF of an isolation trench located 60 m away from the tamping point: (a) east–west direction; (b) north–south direction.

Figure 11. Vibration velocity outside the isolation trench for filling materials with different unit weights: (a) east–west direction; (b) north–south direction.

Figure 12. Vibration velocity outside the isolation trench for different damping ratios of the filling material: (a) east–west direction; (b) north–south direction.

Figure 13. Vibration velocity outside the isolation trench for different cohesion and friction angles of the filling material: (a) east–west direction; (b) north–south direction.

Figure 14. Vibration velocity outside the isolation trench for different stiffness and spacing of spring dampers: (a) east–west direction; (b) north–south direction.

Table 1. Particle composition of soil.

Soil Layer	Particle Composition (% by Mass)					d₆₀ (mm)	d₁₀ (mm)	d₃₀ (mm)	C_u	C_c
Soil Layer	0.25~0.074 (mm)	0.074~0.05 (mm)	0.05~0.005 (mm)	0.005~0.002 (mm)	<0.002 (mm)	d₆₀ (mm)	d₁₀ (mm)	d₃₀ (mm)	C_u	C_c
#1 loess	0	2.95	91.75	1.98	2.95	0.025	0.011	0.015	2.27	0.82
#2 pale-osol	0	27.69	65.78	1.94	3.95	0.034	0.013	0.019	2.62	0.82
#3 loess	0	8.79	88.86	0.85	1	0.029	0.013	0.018	2.23	0.86

Table 2. Soil layer distribution and geotechnical parameters of the site.

Soil Layer	Soil Thickness (m)	Water Content ω (%)	Liquid Limit ω_L (%)	Plasticity Index I_P	Unit Weight of Soil γ (kN/m³)	Poisson’s Ratio e	Compression Modulus E_s1–2 (MPa)	Cohesion c (kPa)	Internal Friction Angle φ (°)
#1 loess	0~9.7	15.90	30.5	12.1	12.60	0.96	9.05	26.20	20.30
#2 paleosol	9.7~12.7	18.30	31.1	12.4	13.70	0.99	10.32	26.80	21.10
#3 loess	12.7~20	22.00	31.5	12.5	14.90	0.83	10.43	28.60	22.10

Table 3. The advantages and disadvantages of the different impact load conversion methods.

Method Name	Advantages and Disadvantages of each Method
Scott method	The stress boundary conditions that w ≠ 0 and σ = 0 when t→∞ are not satisfied.
Miller method	Considering the reflection and damping factors of soil internal stress wave, it is close to the actual situation.
Improved sine method	Displacement is negative when t = 0 is eliminated, and the problem that the impact load is too large in the loading stage is optimized, but the damping factor of soil mass is not considered.
Momentum theorem method	Based on the momentum theorem, it is rigorous in theoretical description.
Equivalent contact static method	Ignoring tamper ground bounce and energy loss during the impact process.

Table 4. Physical and mechanical properties of soils.

Soil Layer	Soil Thickness (m)	Damping Coefficient α	Damping Coefficient β	Young’s Modulus E (MPa)	Shear Modulus G (MPa)	$Compression Wave Velocity V_{p}$ (m/s)	$Shear Wave Velocity V_{s}$ (m/s)
#1 loess	0~9.7	3.11	0.80 × 10⁻³	5.644	2.09	83.94	40.32
#2 paleosol	9.7~12.7	3.11	0.80 × 10⁻³	6.43	2.38	85.96	41.30
#3 loess	12.7~20	3.11	0.80 × 10⁻³	6.50	2.41	82.87	39.81

Table 5. Vibration velocity and measured error value of tamping point.

Method Name	Vibration Velocity 30 m from the Tamping Point/cm/s	Vibration Velocity 100 m from the Tamping Point/cm/s	The Error of 30 m from the Tamping Point	The Error of 75 m from the Tamping Point
Test result	2.60	0.49	0%	0%
Improved sine method	4.09	0.81	57.31%	65.31%
Scott method	3.40	0.67	30.77%	36.73%
Equivalent contact static method	3.17	0.62	21.92%	26.53%
Momentum theorem method	2.42	0.50	6.92%	2.04%
Miller method	1.98	0.36	23.84%	26.53%

Table 6. The average A_RF of the vibration isolation trench at 35 m for various working conditions.

Depth of Vibration Isolation Trench/m	East–West Vibration Isolation Trench A_RF	North–South Vibration Isolation Trench A_RF
1.0	76.29%	76.64%
1.5	66.38%	66.42%
2.0	64.92%	63.30%
3.0	64.54%	62.99%

Table 7. The average A_RF of the vibration isolation trench at different distances with a 2 m depth.

Distance/m	East–West A_RF	North–South A_RF
35	64.92%	63.30%
60	71.39%	68.97%

Table 8. The average A_RF reduction of different filling materials and empty trench.

Vibration Reduction Treatment	East–West Vibration Isolation Trench A_RF	North–South Vibration Isolation Trench A_RF
Empty trench	64.92%	63.30%
Unit weight 5 kN/m³	54.32%	62.15%
Unit weight 10 kN/m³	56.55%	63.74%
Unit weight 20 kN/m³	61.12%	67.93%
Unit weight 30 kN/m³	64.98%	71.01%

Table 9. The average A_RF reduction for damping ratio fillers and the empty trench.

Vibration Reduction Treatment	East–West Vibration Isolation Trench A_RF	North–South Vibration Isolation Trench A_RF
Empty trench	64.92%	63.30%
Damping ratio 5%	59.15%	66.02%
Damping ratio 10%	57.31%	63.94%
Damping ratio 20%	53.58%	60.11%
Damping ratio 40%	47.55%	53.51%

Table 10. The average A_RF reduction for different vibration reduction treatments.

Vibration Reduction Treatment	East–West Vibration Isolation Trench A_RF	North–South Vibration Isolation Trench A_RF
Empty trench	64.92%	63.30%
Unit weight 5 kN/m³	54.32%	62.15%
Damping ratio 40%	47.55%	53.51%
Cohesion 200 kPa	47.88%	65.96%
Friction angle 45°	47.81%	66.02%
Stiffness 500 kN, Spacing 2 m	42.66%	34.31%

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Li, J.; Wang, W.; Luo, L.; Yao, X.; Hu, J. Evaluation of Dynamic Compaction Load Conversion Methods and Vibration Reduction Treatments. Buildings 2025, 15, 111. https://doi.org/10.3390/buildings15010111

AMA Style

Li J, Wang W, Luo L, Yao X, Hu J. Evaluation of Dynamic Compaction Load Conversion Methods and Vibration Reduction Treatments. Buildings. 2025; 15(1):111. https://doi.org/10.3390/buildings15010111

Chicago/Turabian Style

Li, Jixuan, Wenli Wang, Longping Luo, Xiaoliang Yao, and Jiangang Hu. 2025. "Evaluation of Dynamic Compaction Load Conversion Methods and Vibration Reduction Treatments" Buildings 15, no. 1: 111. https://doi.org/10.3390/buildings15010111

APA Style

Li, J., Wang, W., Luo, L., Yao, X., & Hu, J. (2025). Evaluation of Dynamic Compaction Load Conversion Methods and Vibration Reduction Treatments. Buildings, 15(1), 111. https://doi.org/10.3390/buildings15010111

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Evaluation of Dynamic Compaction Load Conversion Methods and Vibration Reduction Treatments

Abstract

1. Introduction

2. In Situ DC Tests

3. Numerical Simulation Methods for DC

3.1. DC Loads Equivalent Conversion Method

3.2. Numerical Model Parameters and Boundary Conditions

4. Results and Analysis

4.1. Comparative Analysis of DC Loads Equivalent Conversion Methods

4.2. Analysis of Vibration Reduction Treatment Parameters

4.2.1. Depth of Vibration Isolation Trench

4.2.2. Distance of Vibration Isolation Trench

4.2.3. Physical and Mechanical Parameters of Isolation Trench Filling Materials

5. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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