Experimental Study on Strength and Liquefaction Characteristics of Sand under Dynamic Loading

Abstract

To investigate the influence of moisture content, density, and loading frequency on the dynamic strength and liquefaction characteristics of sandy soil, a series of vibration triaxial tests were designed and conducted. The tests were divided into incremental loading tests and cyclic loading failure tests. The results of the incremental loading tests indicated that the shear modulus decreases with increasing moisture content and density, while it increases with increasing loading frequency. The damping ratio of unsaturated samples showed no significant correlation with moisture content, whereas samples with a density above 50% exhibited an increasing trend in damping ratio with increasing density. The damping characteristics of the sandy soil were found to be related to the loading frequency, exhibiting the characteristics of viscous damping. The results of the cyclic loading tests revealed that the dynamic failure mode of the sand soil is the ultimate equilibrium failure mode. Increasing moisture content and decreasing density make the samples more susceptible to failure. During the process of cyclic loading leading to dynamic failure, the shear modulus of unsaturated samples remains constant, while the shear modulus of saturated samples gradually decreases. The damping ratio of saturated soil is significantly higher than that of unsaturated soil. During the process of cyclic loading leading to dynamic failure, the damping ratio of saturated soil shows no apparent correlation with loading frequency, but it decreases with increasing density.

1. Introduction

When earthquakes occur, liquefaction is a major threat to the safety of civil engineering structures constructed on sandy ground. Liquefaction of saturated soils may result in loss of shear strength causing the soil to behave like an aqueous medium [1]. Soil liquefaction is one of the primary causes of secondary seismic hazards [2,3]. Previous seismic damage investigations have indicated that liquefaction of saturated sandy soil layers can lead to significant ground deformations, resulting in the destruction of critical infrastructure and lifeline systems in earthquake-stricken areas, further escalating the loss of life and property and increasing the difficulty of disaster relief efforts [4]. It is of great significance to study the law of property evolution in the liquefaction process of sand, especially for understanding and solving the problem of large deformation of liquefied sand [5].

The dynamic shear modulus and damping ratio of sandy soil are crucial parameters for analyzing the seismic response of soil layers and structures embedded within them, and studying the dynamic properties of soil is fundamental to determining design seismic input parameters [6]. Generally, the strength of sandy soil during vibration is measured by the vibration triaxial test [7]. Youssef et al. [8] and Hardin et al. [9] summarized the influences of factors such as number of load cycles, confining pressure, and effective mean principal stress on the variation of dynamic shear modulus and damping ratio with shear strain based on a large number of resonant column tests. Li Ruishan et al. [10] investigated the influence of loading frequency on the dynamic characteristics of sandy and clayey soils. Experimental results have demonstrated that the factors affecting the dynamic shear modulus and damping ratio of soil are numerous, including soil density, void ratio, confining pressure, cyclic strain amplitude, saturation, and loading history [11,12,13]. Among these factors, saturation, density, and loading frequency are the primary influencing factors. These parameters not only affect the dynamic constitutive relationship of sand, but also change the shear modulus and damping ratio of sand during liquefaction.

Liquefaction happens due to increasing excess pore water pressure and a similar decreasing in effective stress in a soil deposit. Olson and Stark [14] summarize many other cases of liquefaction flow failures. Soil undrained shear strength mobilized during liquefaction is an important parameter in undrained stability analysis for evaluating the occurrence of flow deformation. Forrest and Noorany [15] suggested that the sites with medium sand as a foundation are more prone to liquefaction. The liquefaction of saturated soil is more obvious under earthquake action. Earthquake shaking can result in the buildup of excess pore pressure, which can reduce the stiffness and the shear strength of the soils [16]. Research has shown that saturation liquefaction of saturated sandy soil is a behavior and process in which a solid transforms into a liquid. It is a continuous transformation process during which the internal structure and properties of saturated sandy soil undergo continuous changes, gradually transitioning from a solid to a liquid state [17]. The transformation process from solid to liquid and the properties of the solid and liquid phases during the liquefaction process are key issues in describing the entire liquefaction process of saturated sandy soil [18]. The main factors influencing soil liquefaction include saturation, density, stress history, and initial shear stress [19,20,21,22,23,24]. Kong et al. [25,26,27] investigated the dynamic response characteristics of molten quartz sand through resonant column tests and torsional shear tests, and discussed the influence of pore fluid on dynamic elastic modulus and damping ratio, preliminarily exploring the variations in effective stress paths and dynamic pore pressure. Konstadinou et al. [28] analyzed the rate of excess pore pressure growth during liquefaction, dividing the development process of excess pore pressure into three stages and proposing a predictive model for excess pore pressure growth. Pan et al. [29] analyzed the large deformation characteristics of liquefied sand and gravel soil, revealing that the stress–strain relationships of liquefied sand–gravel composite materials and fine sand in Nanjing can be divided into three stages: low-intensity stage, super-linear strength recovery stage, and sub-linear strength recovery stage. However, the study on the dynamic shear modulus and damping ratio of sand during liquefaction is still lacking.

At present, the research on dynamic liquefaction of saturated sand still has the following problems: the influence of moisture content and loading frequency on the dynamic shear modulus and damping ratio of sandy soil is not yet clear, and there is a lack of systematic studies on the shear modulus and damping ratio of samples during the liquefaction failure process. In order to investigate the relationship between the dynamic strength and liquefaction characteristics of susceptible sandy soil and moisture content, density, and loading frequency, this paper conducted dynamic triaxial tests on the sandy soil. Initially, the effects of different moisture content, density, and loading frequency on the dynamic shear modulus and damping ratio of the soil were studied. The dynamic constitutive curves of shear modulus and damping ratio were obtained according to these results. The pore water pressure growth pattern and strength variation of the soil during the cyclic loading to liquefaction process were also investigated.

2. Dynamic Triaxial Testing

The experiments were conducted using a GCTS (Geotechnical Consulting & Testing Systems) triaxial testing system, as shown in Figure 1. The maximum confining pressure range is 2 MPa. The pore pressure sensor had a range of 0~2 MPa with an accuracy of 0.1 kPa. The displacement sensor had a range of 100 mm with an accuracy of 0.01 mm. The loading frequency ranged from 0 to 5 Hz. The loading waveforms included sine waves, square waves, and triangular waves.

The test sand used in the experiment was obtained from a river sand deposit in Beijing, which belongs to medium sand. The particle size distribution curve of the sand is shown in Figure 2. From the particle size distribution curve, it can be observed that the proportion of medium sand with particle sizes ranging from 0.25 mm to 0.5 mm was 87.1%, and the content of fine sand with particle sizes smaller than 0.075 mm was 2.7%. The maximum dry density of the test sand was 1.61 g/cm³, and the minimum dry density was 1.43 g/cm³.

To investigate the effects of water content, compaction density, and loading frequency on the dynamic shear modulus, damping ratio, and liquefaction performance of the soil, a series of experimental conditions were designed, as shown in

Table 1. Loading condition table.

Number	Loading Frequency	Water Content	Degree of Density
1A/1B	1 Hz	Saturated (20%)	50%
2A/2B		15%
3A/3B		12%
4A/4B		9%
5A/5B		6%
6A/6B	1 Hz	Saturated (20%)	40%
7A/7B			60%
8A/8B			70%
9A/9B			80%
10A/10B	0.5 Hz	Saturated (20%)	50%
11A/11B	2 Hz
12A/12B	3 Hz
13A/13B	4 Hz

. Each experimental condition was subjected to Test A and Test B. In Test A, the specimen was subjected to incremental loading using a staged approach. In Test B, the specimen was subjected to a fixed deviatoric stress under cyclic loading until failure. Conditions 1 to 5 were selected to study the influence of water content by conducting tests with different water content. Conditions 6 to 9 were chosen to study the effect of compaction density by conducting tests with different densities. Conditions 10 to 13 were selected to study the effect of loading frequency by conducting tests with different frequencies.

Solid cylindrical specimens were used in the experiments, as shown in Figure 3. The specimens had a diameter of 50 mm and a height of 100 mm. According to Table 1, the specimens were categorized into saturated and unsaturated specimens. All specimens were prepared in the same way, and all parameters are consistent except those listed in Table 1. The compactness of the sample is expressed by the relative density, and the quality of dry sand required for each layer is calculated according to the density and sample size. The preparation of the specimens was carried out directly in the triaxial saturation apparatus. The specimens were prepared in three layers with uniform thickness, approximately 33.3 mm per layer. The saturated specimen was prepared by the combined method of soaking and back pressure saturation The saturated specimens were prepared with a water content of 10% based on the desired compaction density. After preparation, the specimens, together with the saturation apparatus, were immersed in water for at least 24 h to achieve full saturation. After the specimen was installed on the test system, the reverse pressure saturation was turned on. When the pore pressure coefficient B reached 0.95, it was considered saturated. The unsaturated specimens were prepared directly based on the desired compaction density and water content. The required dry sand quantity and water quantity were directly weighed and then stirred evenly. Once the specimen was made, it was divided into three layers in the triaxial saturator for production, and was directly installed on the test system to start the test.

The applied confining pressure for the experiments was fixed at 50 kPa. The cyclic loading was applied in the form of sinusoidal waves with a constant amplitude. Each experimental condition was subjected to Test A and Test B. In Test A, the specimen was subjected to incremental loading using a staged approach, with each stage incrementing the deviatoric stress by 10 kPa and applying 20 cycles of loading per stage. In Test B, the specimen was subjected to a fixed deviatoric stress of 100 kPa under cyclic loading until failure. In the sand liquefaction test, the initial liquefaction failure criterion (stress failure criterion) or strain failure criterion are generally selected. The initial liquefaction failure criterion is that when the pore water pressure of the sample increases to equal to the confining pressure value under cyclic load, the sample is considered to have reached liquefaction failure. The criterion of strain failure is that when the axial strain value of the sample under cyclic load reaches a certain limit value, it is considered that the sample has reached liquefaction failure. However, when the content of silt or clay is large, the pore water pressure still cannot reach the confining pressure value even if the sample has a large deformation, and it is not appropriate to use the initial failure standard at this time. In view of this situation, it is considered more reasonable to select the strain failure criterion. Therefore, the experiment was terminated when the axial strain reached 5% [30] or the number of loading cycles reached 2000, indicating failure according to the predetermined criteria. Before the start of the experiments, the specimens were first subjected to a consolidation period of 30 min under a confining pressure of 50 kPa. During the experiment, the applied stresses, strains, number of loading cycles, and pore water pressure were recorded.

3. Dynamic Shear Modulus and Damping Ratio of the Sand

3.1. Data Processing Method

As shown in Figure 4 [31], the ideal stress–strain hysteresis loop is used to analyze the dynamic properties of the soil. The average slope of the hysteresis loop reflects the magnitude of the dynamic elastic modulus (E_d) under the current cyclic loading conditions, while the size of the hysteresis loop area reflects the energy dissipation during the loading and unloading processes. This energy dissipation is represented by the damping ratio of the specimen. The damping ratio of the soil reflects the lag exhibited by the stress–strain hysteresis loop under periodic dynamic loading conditions. The damping ratio and the dynamic elastic modulus (E_d) were calculated using the following formulas:

$${\lambda }_d=W/4\pi W_t$$

(1)

$$E_d=({\sigma }_{d\max }-{\sigma }_{d\min })/({\epsilon }_{d\max }-{\epsilon }_{d\min })$$

(2)

In the equation, W represents the area enclosed by the hysteresis loop A, B, C, D, which corresponds to the energy dissipation during one cycle. W_t represents the area of the triangle AOE, representing the elastic strain energy. σ_dmax and σ_dmin represent the maximum and minimum axial dynamic stress, respectively, within the same cycle of loading, while ε_dmax and ε_dmin represent the maximum and minimum axial dynamic strain, respectively. Based on this, the dynamic shear modulus and dynamic shear strain of the soil can be determined as follows:

$$\left.\begin{array}{c}{\gamma }_d={\epsilon }_d(1+\mu )\\ G_d=\frac{E_d}{2(1+\mu )}\end{array}\right\}$$

(3)

In the equation, G_d represents the dynamic shear modulus, γ_d represents the dynamic shear strain and µ represents the Poisson’s ratio.

3.2. Variation Characteristics of Dynamic Shear Modulus

The dynamic shear modulus and damping ratio of the sand were analyzed based on the results of the graded loading tests. The average values of shear modulus and damping ratio were obtained by averaging the middle 10 hysteresis loops from the 20 cycles of each loading level. Figure 5a–c present the relationships between dynamic shear modulus and shear strain for different moisture contents, densities, and loading frequencies. The horizontal coordinate indicates shear strain γ_d and the vertical coordinate indicates shear modulus G_d. The results indicate that within the strain range of 10⁻⁴ to 10⁻², the dynamic shear modulus of the sand decreases with increasing shear strain. The decrease is more pronounced within the strain range of 10⁻⁴ to 10⁻³, while it becomes slower within the strain range of 10⁻³ to 10⁻². The results are consistent for different water content, compactness and loading frequency. It suggests that the strength of the sand rapidly decreases with increasing deformation at lower strains, while it tends to stabilize at higher strains. Figure 5a reveals that the dynamic shear modulus of the sand gradually decreases with increasing moisture content (with a moisture content of 20% for saturated specimens), indicating that an increase in moisture content enhances the strength of the soil. Figure 5b shows that an increase in density leads to a decrease in the dynamic shear modulus. Figure 5c demonstrates that the dynamic shear modulus of the specimens increases with an increase in loading frequency, indicating that the strength of the sand is lower under low-frequency seismic waves compared to high-frequency seismic waves.

A hyperbolic model was used to fit the dynamic shear modulus data of model soil. The fitting equation is expressed as follows:

$$\left.\begin{array}{c}1/G_d=a+b{\gamma }_d\\ G_d/G_{\max }=1/(1+{\gamma }_d/{\gamma }_r)\end{array}\right\}$$

(4)

In the equation, a = 1/G_max, b = 1/τ_max, γ_r = a/b = τ_max/G_max is called the reference strain. Plotting the scatter diagram of 1/G_d-γ_d, the slope of the fitted straight line is b, and the reciprocal of the ordinate intercept at γ_d = 0 is G_max.

The experimental results of dynamic shear modulus and strain under different conditions in Figure 5 were fitted, and the values of G_max and γ_r at different confining pressures are presented in

Table 2. Maximum dynamic shear modulus.

Different Water Content Conditions
Condition Code	Water Content	Gmax/MPa	γr	R2
1A	Saturated soil	26.04	0.0090	0.9341
2A	15%	25.00	0.0073	0.9628
3A	12%	23.47	0.0116	0.9490
4A	9%	22.57	0.0083	0.9674
5A	6%	23.09	0.0093	0.9105
Different Compaction Density Conditions
Condition Code	Compaction Density	Gmax/MPa	γr	R2
6A	40%	29.15	0.0056	0.9707
1A	50%	26.04	0.0090	0.9341
7A	60%	28.65	0.0090	0.9806
8A	70%	26.95	0.0071	0.9615
9A	80%	24.39	0.0141	0.9314
Different Loading Frequency conditions
Condition Code	Loading Frequency	Gmax/MPa	γr	R2
10A	0.5Hz	25.71	0.0066	0.9956
1A	1Hz	26.04	0.0090	0.9341
11A	2Hz	24.51	0.0095	0.9599
12A	3Hz	29.24	0.0040	0.9215
13A	4Hz	28.65	0.0039	0.9410

. The results show that the maximum dynamic shear modulus decreases with increasing water content and compaction density, while it increases with increasing loading frequency. Specifically, the maximum dynamic shear modulus of the sample with 6% water content is 11.3% lower than that of the saturated sample, the maximum dynamic shear modulus of the sample with 80% compaction density is 16.3% lower than that of the sample with 40% compaction density, and the maximum dynamic shear modulus of the sample subjected to a loading frequency of 4 Hz is 11.4% higher than that of the sample subjected to a loading frequency of 0.5 Hz. Meanwhile, under the same loading conditions, the decrease in shear modulus leads to increased deformation, which can contribute to higher susceptibility to failure when structures are present in the site. Therefore, in the analysis of site response and soil–structure interaction, the effects of water content, compaction density, and seismic frequency should be considered when considering the dynamic shear modulus of the soil.

3.3. Variation Characteristics of Damping Ratio

Figure 6a–c depict the relationship between the damping ratio and shear strain under different conditions of moisture content, compaction density, and loading frequency. The horizontal coordinate indicates shear strain γ_d and the vertical coordinate indicates the damping ratio λ_d. The results indicate that the trends of the damping ratio–shear strain (${\lambda }_{\mathrm{d}}-{\gamma }_{\mathrm{d}}$) relationship are consistent across different conditions. Specifically, the damping ratio increases with increasing shear strain, reaching a minimum value of approximately 0.05 and a maximum value of 0.24. From the perspective of moisture content, the saturated samples exhibit lower damping ratios compared to unsaturated samples, while there is no significant difference in damping ratios among different moisture content conditions for the unsaturated samples. Regarding the compaction density of the samples, the highest damping ratio is observed in the sample with a density of 40%, while the sample with a density of 50% exhibits the lowest damping ratio. The damping ratio of samples with densities exceeding 50% increases with increasing compaction density. In terms of loading frequency, the damping ratio initially decreases and then increases with increasing frequency, with the minimum damping ratio occurring at a loading frequency of 1 Hz. There are notable differences in the damping ratios among samples subjected to different loading frequencies, indicating the correlation between the damping characteristics of the sand and the loading frequency, consistent with the characteristics of viscous damping.

In summary, the dynamic shear modulus decreases with increasing moisture content and compaction density, while it increases with increasing loading frequency. The damping ratio of unsaturated samples is independent of moisture content, whereas the damping ratio of saturated samples increases with increasing compaction density. Additionally, the damping ratio initially decreases and then increases with increasing loading frequency, exhibiting characteristics of viscous damping.

4. Sand Liquefaction Performance

4.1. Pore Water Pressure and Dynamic Failure Modes

Figure 7 shows the variation of pore water pressure ($P_n$) with increasing cyclic loading cycles ($n$). The numbers in the figure indicate the magnitudes of pore water pressure at the point of dynamic failure. It can be observed that the growth curve of pore water pressure approximates a parabolic shape, indicating a gradual slowdown in pore water pressure increase with the number of cycles. The magnitudes of pore water pressure at the point of dynamic failure did not reach the applied confining pressure of 50 kPa, indicating that the sand did not experience complete liquefaction failure. During the cyclic loading process, the pore water pressure in the specimens continuously develops. According to the Mohr–Coulomb criterion, the effective stress in the specimen decreases, and the stress path moves toward the radial strength envelope. When the pore water pressure reaches a critical value, the specimen reaches a state of ultimate equilibrium, at which point failure occurs. The different magnitudes of pore water pressure at failure for each specimen indicate the existence of variations in the critical pore water pressure threshold.

4.2. Number of Cyclic Loading Cycles at Specimen Failure

Samples under different conditions, labeled as 1B to 13B, were subjected to cyclic loading until failure under fixed confining pressure and deviatoric stress. The number of cycles at failure reflects the liquefaction resistance of the specimens. Figure 8 illustrates the variation of the number of cycles ($n$) at 5% axial strain for different combinations of water content ($w$), relative density ($Dr$), and loading frequency ($f$). The numbers in the figure represent the number of cycles at which the axial strain reaches 5%. It should be noted that, in the case of 6% water content and the case of 80% density, the axial strain did not reach 5% even after 2000 cycles, indicating no occurrence of failure. From the perspective of water content, an increase in water content leads to a gradual reduction in the number of cycles, indicating that higher water content makes the sand more prone to liquefaction. It is worth noting that the number of cyclic loadings under 9% moisture content condition is lower than 12%, which requires further analysis of the reasons, but it does not affect the overall pattern. It is difficult to achieve liquefaction conditions where the sand has only 6% water content. With increasing compactness, the number of cycles gradually increases, indicating that increasing compactness can increase the liquefaction resistance of sand. At a density of 80%, liquefaction no longer occurs. When the loading frequency is 0.5 Hz, the number of cycles is 142 times to achieve liquefaction failure, while the minimum number of cycles required for liquefaction failure was 59 at a frequency of 1 Hz. For loading frequencies exceeding 1 Hz, the number of cycles increased with increasing frequency. Therefore, a frequency of 1 Hz is the most susceptible to dynamic failure. In summary, the number of cycles at specimen failure decreases with increasing water content, increases with increasing density, and is minimized at a loading frequency of 1 Hz. Consequently, to enhance the liquefaction resistance of sandy ground, measures such as lowering the water content and increasing the density through dewatering and grouting can be employed.

4.3. Dynamic Shear Modulus and Damping Ratio during Liquefaction Process

The dynamic shear modulus and damping ratio of sand also change during the process of cyclic loading and liquefaction failure. The shear modulus and damping ratio for each cycle during specimen loading can be calculated using Equations (1) and (2). Figure 9 and Figure 10 depict the variation of shear modulus ($G_n$) and damping ratio (${\lambda }_n$) with increasing cycle numbers ($n$) under different conditions.

From Figure 9, it can be observed that the shear modulus rapidly increases in the initial cycles. This is attributed to the vibration-induced densification process that occurs in the early stage of loading. Consequently, the shear modulus in the early stages of loading does not reflect the true strength of the specimen. Figure 9a shows that for unsaturated specimens, the shear modulus reaches a stable state after the rapid increase, indicating that the shear modulus remains constant during the cyclic loading process until dynamic failure. In contrast, Figure 9b,c demonstrate that the shear modulus gradually decreases after reaching its peak value for saturated specimens. This indicates that the shear modulus of saturated specimens decreases gradually during cyclic loading until dynamic failure. Therefore, during the cyclic dynamic loading process, the dynamic shear modulus of saturated specimens gradually decreases, leading to an increase in deformation, which is an important factor accelerating specimen failure. Additionally, higher water content is associated with lower shear modulus during cyclic loading, while higher density is related to higher shear modulus. The shear modulus exhibits no clear correlation with loading frequency, with the lowest values observed at a frequency of 1 Hz.

Figure 10 demonstrates that the initial damping ratio of the sand specimens under various conditions range from 0.26 to 0.3. With increasing cycle numbers, the damping ratio gradually decreases at a decelerating rate until reaching a stable value. However, the trend for specimens with 40% relative density is different. After the initial loading, the damping ratio exhibits a slow increase, but due to the low relative density, liquefaction occurs quickly. From Figure 10a it can be observed that, for unsaturated sand specimens, the damping ratio rapidly decreases with increasing cycle numbers. After 50 cycles, the damping ratio drops below 0.15 and gradually stabilizes. The final damping ratios of specimens with different water contents are relatively close, ranging from 0.08 to 0.13. Among them, the specimen with 6% water content has the lowest damping ratio, with a final value of 0.084. From Figure 10b, it can be seen that for specimens with different relative densities, the damping ratio decreases rapidly within the first 50 cycles, but there are significant differences in the final damping ratios. The final damping ratios for specimens with 50%, 60%, 70%, and 80% relative densities are 0.244, 0.198, 0.151, and 0.080, respectively. Therefore, the damping ratio during the vibration-induced liquefaction process of saturated sand is significantly influenced by the relative density. Figure 10c shows that, for specimens with different loading frequencies, the trend of damping ratio variation with cycle numbers is similar to other conditions. The damping ratio in the later stages of liquefaction ranges from 0.16 to 0.22, and its magnitude may increase or decrease with increasing loading frequency, without showing a clear correlation. Hence, during the cyclic loading until liquefaction process, the damping ratio of saturated sand is significantly higher than that of unsaturated sand. The damping ratio of unsaturated sand is independent of water content, and the damping ratio of saturated sand does not exhibit a clear correlation with loading frequency but decreases with increasing relative density.

Contrasting the variations of dynamic shear modulus and damping ratio during the liquefaction process with the results under incremental loading reveals that the changes in shear modulus and damping ratio relative to water content, relative density, and loading frequency are different. This discrepancy arises because the results under incremental loading represent the dynamic properties of the soil itself, while cyclic loading and failure reflect the strength development during the soil’s failure process. Therefore, it is not appropriate to directly apply the influence patterns of water content, relative density, and loading frequency on the dynamic characteristics of the soil to represent the influence patterns during the dynamic failure process. In the analysis and seismic design of structures in site response calculations and on-site conditions, it is necessary to consider the dynamic parameters of the soil comprehensively. When analyzing the liquefaction failure process of the site, the strength development during the cyclic loading and failure process of the soil at the site should be taken into account to enhance the credibility of characterizing the relationship between soil strength and liquefaction behavior. This will provide scientific support for seismic disaster prevention in underground engineering structures.

5. Conclusions

To investigate the relationship between the dynamic strength and liquefaction behavior of sandy soil relative to water content, relative density, and loading frequency, dynamic triaxial tests were conducted in this study, and the following main conclusions were drawn:

(1)

The results of incremental loading tests revealed that the shear modulus decreases with an increase in water content and relative density, while it increases with an increase in loading frequency. The damping ratio of unsaturated samples showed no significant correlation with water content, whereas for samples with relative density above 50%, the damping ratio increased with an increase in relative density. The damping characteristics of the sand were found to be dependent on the loading frequency, exhibiting viscous damping behavior.

(2)

The dynamic failure mode of the sand was observed to be governed by the ultimate equilibrium failure mode. The number of cyclic loading cycles until failure decreased with an increase in water content and increased with an increase in relative density. The fewest number of cyclic loading cycles until failure was observed at a loading frequency of 1 Hz.

(3)

The shear modulus of unsaturated samples remained constant during the cyclic loading until failure, while the shear modulus of saturated samples gradually decreased during the cyclic loading until failure. The damping ratio of saturated sand was significantly higher than that of unsaturated sand. During the process of cyclic loading leading to dynamic failure, the damping ratio of saturated sand showed no clear correlation with loading frequency but decreased with an increase in relative density.

(4)

In the analysis of site response and seismic design of structures in field conditions, it is not appropriate to directly apply the influence patterns of water content, relative density, and loading frequency on the dynamic properties of the soil to represent the influence patterns during the dynamic failure process. It is recommended that the dynamic parameters of the site soil should be considered comprehensively in engineering design. When analyzing the liquefaction failure process of the site, it is necessary to consider the strength development during the cyclic loading and the failure process of the soil at the site to enhance the credibility of characterizing the relationship between soil strength and liquefaction behavior.