Experimental Study on the Coefficient of Earth Pressure at Rest for Sand

Abstract

The coefficient of earth pressure at rest K₀ is a state soil variable correlated with relative density D_r. As previously conducted K₀ tests could not guarantee zero lateral deformation in the sand specimens, significant errors occurred in the test results. In this paper, a centrifugal model test method is used to study the K₀ of sand with varying densities. The sand specimens with varying relative densities are prepared by sand pluviation, and a 50 g-ton centrifugal force is applied. Subsequently, the relationship of K₀ and D_r with different densities is analyzed. The test results show that for the same type of sand, the value of K₀ gradually increased with D_r. Based on the meso-evolution characteristics of sand particle recombination, various relationships between K₀, the displacement deflection angle, and the friction offset angle between particles are analyzed. Furthermore, the relationship between particle volume fraction and K₀ is derived, the assumption of increasing the coefficient K₀ with the increase in D_r is verified, and the effect of D_r of sand on the force transfer behavior of the internal particle fabric is briefly discussed. The research results could significantly improve the current earth pressure theories.

1. Introduction

The coefficient of earth pressure at rest K₀ is defined as the ratio of the horizontal effective stress σ_H to the vertical effective stress σ_V of the soil under zero lateral deformation [1,2]. The effective horizontal stress is an important design parameter and is often expressed as a proportion of the effective vertical stress relative to K₀. Thus, the correct determination of the static earth pressure coefficient K₀ significantly impacts the earth pressure distribution, engineering cost, and the safety reliability analyses acting on retaining structures [3,4,5].

Several studies have shown that the value of K₀ is related to factors such as soil type [6], relative density [7,8,9], stress history [10], soil structure [11], and anisotropy [12]. For sandy soil, the relative density D_r is one of the most significant influencing factors. However, research shows that the relationship between K₀ and relative density is inconsistent. Some scholars consider that K₀ increases with D_r [13], and some hold opposite views [14,15,16]. Therefore, it is necessary to further study the variation of K₀ for sandy soils with varying relative densities.

The coefficient K₀ of soil is often determined by empirical formulae and tests such as in situ tests (e.g., pressure tests [17], flat dilatometer tests [18], and multi-functional cone penetration tests [19]) and laboratory tests (e.g., triaxial tests [20] and lateral pressure meter tests [21]). However, due to the inaccuracies inherent in the sample acquisition procedures, large margins of error exist when determining the K₀ value using the tests mentioned above. The empirical formulae for the coefficient K₀ are based on the relationship between K₀ and indicators such as the effective internal friction angle of the soil φ’ [22], plasticity index I_p [23], and over-consolidation ratio OCR [24,25]. The most commonly used empirical formula is the Jaky formula, as shown in Equation (1):

$$K_0=1-\sin {\phi }^{\prime }$$

(1)

However, this formula is only suitable for loose sand and gives a uniquely determined value for a sand type, irrespective of D_r.

In general, the most accurate way to obtain the K₀ value of the soil is restoring the lateral limit state of the soil during measurement to mimic the in situ soil conditions, which can be problematic. The soil will inevitably be disturbed during the in situ test, and measuring the static soil pressure coefficient of the soil after lateral deformation will provide inaccurate results. The triaxial instrument method attempts to counteract this by continuously adjusting the confining pressure and vertical stress during loading so that the soil sample does not produce lateral deformation, thus obtaining a more realistic K₀ value of the soil sample. However, this type of feedback control has significant hysteresis, resulting in the final K₀ value being inaccurate. The consolidation instrument method measures the water pressure in the closed pressure chamber to determine the horizontal stress of the soil and calculate its K₀. This method can result in issues such as uneven lateral deformation of the soil, rubber film embedded in the sand, and water compression of the soil, so the K₀ value obtained also has a certain margin of error. Ensuring that the soil sample does not undergo lateral deformation during the test is problematic in these methods.

However, this problem can be solved using the centrifugal model test. Lee et al. [26] analyzed the horizontal earth pressure on a buried box culvert. The authors used centrifugal model tests to determine the earth pressure coefficient for sand with a single relative density. Cho et al. [27] used the shear wave velocity Vs method to study the V_s–K₀ relationship at different locations in centrifugal model tests. Abdoun et al. [28] studied the relationship between the coefficient K₀ and the over-consolidation ratio OCR through centrifugal model tests. Using centrifugal model tests, Gaudin et al. [29] studied the coefficient K₀ of dense Fontainebleau sand. Most of them studied K₀ at different depths for sand with the same relative density and lacked meso-mechanism analysis of K₀. The relationship between sand K₀ value and the relative density of sand samples and its meso-mechanism analysis need to be further studied.

In this study, the K₀ centrifugal model test is carried out on Fujian standard sand for varying relative densities. The correlation between the coefficient K₀ of the sand sample with its D_r is analyzed. According to the results, a particle microstructure generalization model is proposed to describe the meso-stressed force characteristics of sand. The relationship between the coefficient K₀, its microstructure particle contact angle, the friction offset angle is derived, and the influence of D_r on the internal force transmission mechanism of sand is reviewed. Finally, the influence of D_r on the coefficient K₀ is analyzed from a macro perspective.

2. Materials and Methods

2.1. Test Sand

Fujian standard sand (D₅₀ = 0.183 mm) was used in the centrifugal tests; the size distribution curve of a sand particle is shown in Figure 1, and the physical properties are shown in

Table 1. Basic physical properties of sand.

Specific Gravity Gs	Min. Mass Density ρmin/(g/cm3)	Max. Mass Density ρmax/(g/cm3)	Median Size D50/mm	Coefficient of Uniformity Cu	Coefficient of Curvature Cc
2.65	1.36	1.60	0.183	1.58	0.99

.

2.2. Test Equipment

The K₀ centrifugal model tests of sand in this study were conducted using a 60 g-ton geotechnical centrifuge from the Nanjing Hydraulic Research Institute, as shown in Figure 2, with a simplified diagram of the model test box depicted in Figure 3a. The model box dimensions were 700 mm × 350 mm × 450 mm (length × width × height). BM-3 interface earth pressure sensors were used to measure the horizontal earth pressure [30] in Figure 3b. In contrast, laser displacement sensors were used to measure the soil sample’s compression during centrifuge. The displacement sensors were model YP11MGVL80 non-contact high-precision laser sensors produced by Wenglor in Germany in Figure 3c, with a resolution of up to 20 μm.

2.3. Test Program

The earth pressure sensor was calibrated before the coefficient K₀ tests [31]. The model box was placed between two prefabricated metal wall panels, with dimensions of 450 mm × 350 mm × 20 mm (height × width × thickness). The earth pressure sensors’ measuring surface flushes with the wall panels’ surface. Two rows of earth pressure sensors were arranged in parallel on each wall panel, with a horizontal spacing of 50 mm and a vertical spacing of 60 mm. During the test, the wall panels were fixed on the left and right sides of the model box to ensure that the sand samples did not undergo lateral displacement.

Before the test, the model box was placed flat on the ground, the inside was cleaned, and silicone grease was applied to the side walls. Then, a layer of plastic wrap was pasted on the side walls of the model box to reduce the friction between the sand particles and the pressure box and minimize any pressure sensor test error.

The sand samples were tested using the sand pluviation method with mesh and duck-bill type outlets, as shown in Figure 4a,b. The duck-bill sand spout was 3 mm wide and 100 mm long, and the mesh sand spout had a mesh diameter of 3 mm and a 100 mm outlet diameter. The calibration tank, as shown in Figure 4c, had an inner diameter of 211 mm, a height of 110 mm, and a net weight of 3.7 kg. The soil samples were evenly and slowly sprinkled into the calibration tank using the self-made sample preparation device and varying drop distances to obtain the relative densities due to the two sand outlets at varying drop distances. The test continued until the sand thickness in the calibration tank increased by 20 mm. The weight of the sand in the calibration tank was weighed, and the relative density of the sand sample was calculated from the known volume of the calibration tank. Finally, the relationship between the two types of sand outlet, drop distance, and relative sample density was obtained.

The mesh and duck-bill sand spouts were used to prepare sand samples with different relative densities based on the sand pluviation test results by controlling the drop distance. The specimen preparation results are shown in

Table 2. Relative density D_r of sand specimens.

Sand Outlet Type	Drop Height H/cm	Relative Density Dr	Sand Outlet Type	Drop Height H/cm	Relative Density Dr
duck-bill spout	60	0.30	mesh spout	40	0.77
	70	0.38
	85	0.46		100	0.89
	100	0.56

. The sand thickness in the model box was approximately 320 mm, and four earth pressure sensors were buried, as shown in Figure 3a. After sand placement, a laser displacement sensor was fixed on the model box to measure the soil sample compression during the centrifuge operation.

The model box with the sand specimen was placed in the centrifuge, and the centrifugal acceleration was applied in stages. Once the acceleration reached 50 g, it was maintained for 30 min. Figure 5 shows the K₀ centrifugal model test process. The horizontal stress of the specimen at different depths was obtained from the earth pressure sensors, and the vertical stress relative to each earth pressure sensor was obtained using Equation (2). The coefficient K₀ of the sand samples under the corresponding conditions was obtained by linear fitting the horizontal stress at varying depths and the corresponding vertical stress:

$${\sigma }_z=\gamma \cdot z\cdot n$$

(2)

where σ_z is the vertical stress, kPa; γ is the soil unit weight, kN/m³; z is the depth of the soil layer, m; and n is the centrifuge acceleration, m/s².

3. Results

3.1. The Relationship between Drop Distance and Relative Density from the Sand Pluviation Test

Figure 6 shows the samples’ relative density variation based on drop height and spout type obtained from the sand pluviation test. For both spout types, the relative density of the sand samples increased with drop distance, although the growth rate gradually decreased. In the region of relatively low compaction, the drop distance change significantly impacted the sample’s relative density. Where the compaction was relatively high, the pore volume between the particles was small. The compaction was relatively stable, resulting in little influence of drop height on the relative density.

Furthermore, due to the large sand outflow per unit time of the mesh sand spout, the sand particle collisions were more forceful at the sand layer’s surface, making it easier to obtain relatively dense specimens. The relationship between drop height and relative density varied between the spout types. When the mesh sand spout was used, the compactness increased rapidly with drop height and gradually stabilized. However, the change in compaction was relatively slow, with an increase in the drop height when the duck-bill sand spout was used. Thus, it was easier to prepare loose sand samples from this spout type.

3.2. Analysis of Soil Sample Settlement

The vertical sand stress can be determined from the centrifugal acceleration at the measurement point location and the corresponding sand density. Since six initial relative density sand models were used in this centrifugal test corresponding to loose, medium, and compact density conditions, it was necessary to consider the vertical settlement of the varying initial D_r sand models. With a centrifugal acceleration of 50 g, the maximum settlement of the sand layer was 1.74 mm with a low initial relative density (D_r = 0.3), as indicated in Figure 7. The sand compaction increased by 0.54% from the initial state. Once the compaction became dense, the vertical settlement variation was close to zero. Therefore, it was concluded that the compression level of the sand specimens had a negligible effect on D_r of the sand.

Figure 8 shows the sand specimens’ lateral earth pressure distribution curve at varying depths and relative densities D_r. At the same depth, the lateral soil pressure increased with D_r, with the overall pattern showing an increasing trend. The lateral soil pressure at a depth of 50 mm showed marginal changes with an increase in D_r of the sand sample due to the loose upper soil layer, allowing the particles to slide easily at the earth pressure sensor interface.

3.3. Effect of Relative Density on the Coefficient of Earth Pressure at Rest

The relationship between the coefficient K₀ of the medium sand and the relative density D_r has a positive correlation, as shown in Figure 9. As given in Equation (3), the correlation formula was derived by linear fitting of the test results, such that the maximum value of K₀ was 0.442, and the minimum was 0.367. Therefore, when calculating the coefficient K₀, it is first necessary to establish the D_r of the sand.

$$K_0=0.129D_{\mathrm{r}}+0.331$$

(3)

4. Analysis and Discussion

4.1. Mechanism Analysis of the Coefficient of Lateral Pressure at Rest

From a micro perspective, sand can be seen as an aggregate of many discrete solid particles [32,33,34]. The particle system has a particular structure, and its macroscopic deformation and mechanical properties are closely related to the microstructure [35,36]. Therefore, the microstructure forces of the mesoscopic sand particle system were analyzed based on the results from the centrifugal model test. Moreover, the vibration of vertical and horizontal stresses of the internal microstructure particles under different relative densities of sand was studied.

According to the particle size distribution curve of the sand in the centrifugal test, the particle size was regarded as a single particle size. Thus, the sand particle system was simplified to a two-dimensional single-particle-size system, as shown in Figure 10a. In the soil particle system, gravity or external loads cause the soil particles to squeeze each other. Since the soil particles transmit these loads through a force chain, the transfer direction is connected along the center of mass of the particles.

The displacement deflection angle θ is the angle between the centroidal line of two adjacent particles and the vertical direction. As shown in Figure 10a, as the angle θ increases, the force chain formed among the particles will develop horizontally, i.e., the stress component of the force chain will increase in the horizontal direction and is represented by an increase in the horizontal stress in the particle system. The friction angle μ is the angle between the perpendicular connection of adjacent particles and the force chain, as shown in Figure 10b. When no friction is present at the contact point, the force chain is transmitted along the direction of the particle’s center of mass. Consequently, the direction of the chain will be deflected due to the friction among the particles [37].

If particles in the particle system are uniformly arranged and the horizontal and vertical stresses at the same depth are evenly distributed, these stresses can be quantified by the partial differential Equation (4):

$$\{\begin{array}{l}\frac{\partial {\sigma }_x}{\partial x}=0\\ \frac{\partial {\sigma }_y}{\partial y}=0\end{array}$$

(4)

where σ_x and σ_y are the horizontal and vertical effective stresses, respectively.

The rectangular area with lengths L_x and L_y indicated in Figure 10b was selected as the microelement force structure to explore the relationship between the coefficient K₀, the displacement deflection angle θ, and the friction angle μ. Substituting these values into Equation (4), the total horizontal and vertical stresses of the particle microelement stress structure are, respectively:

$$\{\begin{array}{l}{F}_x=L_y{\sigma }_x\cos \theta \\ F_y=L_x{\sigma }_y\sin \theta \end{array}$$

(5)

where F_x and F_y are the horizontal and vertical forces, respectively, and L_x and L_y are the horizontal and vertical lengths.

The intergranular contact force at contact point G, shown in Figure 10b, is designated as f₀, with f_x and f_y being the horizontal and vertical components of f₀, respectively. Parameters f_x and f_y can be expressed by applying the force balance of the particles on both sides of point G as follows:

$$\{\begin{array}{l}2f_0\cos \left(\theta -\mu \right)=f_y\\ 2f_0\sin \left(\theta -\mu \right)=f_x\end{array}$$

(6)

Since f_x = F_x and f_y = F_y, these can be written as follows:

$$\frac{f_x}{f_y}=\frac{2r{\sigma }_x\cos \theta }{2r{\sigma }_y\sin \theta }=\tan \left(\theta -\mu \right)$$

(7)

where r is the radius of the particles.

The coefficient K₀ of the particle structure in the sand model can thus be expressed as follows:

$$K_0=\frac{{\sigma }_x}{{\sigma }_y}=\tan \left(\theta -\mu \right)\tan \theta$$

(8)

Since the friction angle μ between the particles is constant, it can be deduced from Equation (8) that the coefficient K₀ increases with the displacement deflection angle θ. Moreover, the displacement deflection angle θ in the particle system increases with the particle density D_r, resulting in a larger share of the force being distributed to the sidewalls in the system. When the particle system is sufficiently dense, and the force chain is sufficiently developed, the K₀ value will tend to be saturated.

The particle accumulation system is composed of particles and pores, and the relative density of the accumulation is closely related to the proportion of these. In other words, the denser the system is, the greater proportion of particle volume. Assuming that the particles in Figure 10a are regularly arranged, it can be seen that the particle stacking body is composed of multiple rhombic unit structures.

A single-cell structure is taken as the research object. The volume fraction ψ in the cell can be generalized to the volume fraction in the overall particle system. As shown in Figure 11, r is the radius of particles A, B, C, and D, and β is the angle between the diamond-shaped AD edge and the horizontal line AC. The volume fraction of the cell is equal to the ratio of the sum of the areas of the four sectors, AEH, BEF, CFG, and DGH, to the area of the diamond ABCD. Therefore:

$$S_{\mathrm{sector}-\mathrm{AEH}}=S_{\mathrm{sector}-\mathrm{CFG}}=\frac{2\beta \pi r^2}{{360}^{\circ }}$$

(9)

$$S_{\mathrm{sector}-\mathrm{BEF}}=S_{\mathrm{sector}-\mathrm{DGH}}=\frac{\left({180}^{\circ }-2\beta \right)\pi r^2}{{360}^{\circ }}$$

(10)

$$S_{\mathrm{diamond}-\mathrm{ABCD}}=4r^2\sin 2\beta$$

(11)

According to Equations (9)–(11), the volume fraction of particles in the cell can be written as:

$$\psi =\frac{S_{\mathrm{sector}-\mathrm{AEH}}+S_{\mathrm{sector}-\mathrm{CFG}}+S_{\mathrm{sector}-\mathrm{BEF}}+S_{\mathrm{sector}-\mathrm{DGH}}}{S_{\mathrm{diamond}-\mathrm{ABCD}}}=\frac{\pi }{4\sin 2\beta }$$

(12)

The displacement deflection angle θ is

$$\theta =\frac{\pi }{2}-\frac{\mathrm{arc}\sin \frac{\pi }{4\psi }}{2}$$

(13)

The following expression can be obtained by substituting Equation (13) into Equation (8):

$$K_0=\tan \left(\frac{\pi }{2}-\frac{\arcsin \frac{\pi }{4\psi }}{2}-\mu \right)\cdot \left(\frac{\pi }{2}-\frac{\arcsin \frac{\pi }{4\psi }}{2}\right)$$

(14)

where μ is a constant. When there is zero friction, the friction angle μ = 0, and Equation (14) can be further simplified to

$$K_0=\tan {\left(\frac{\pi }{2}-\frac{\arcsin \frac{\pi }{4\psi }}{2}\right)}^2$$

(15)

In Equation (15), K₀ increases with the volume fraction ψ. When the volume fraction of particles in the cell increases, the pore volume among the particles will decrease, and the relative density of the corresponding overall particle system will increase. Therefore, as the relative density D_r increases, the particle volume fraction ψ and the corresponding K₀ will increase. This observation correlates with the trend of the K₀ centrifugal model tests.

4.2. Discussion

Based on the accumulation volume composed of soil particles, Jaky used the parabolic interpolation method to propose the following K₀ calculation formula [5,38]:

$$K_0=\frac{1-\sin {\phi }^{\prime }}{1+\sin {\phi }^{\prime }}\left(1+\frac{2}{3}\sin {\phi }^{\prime }\right)$$

(16)

Later, Jaky [7] simplified the formula to Equation (1); these two formulae were derived on the premise that the accumulation and internal friction angles were equal. The accumulation angle is formed by the accumulation of particulate matter in the loosest state. It is an inherent mechanical property of granular material and, therefore, independent of the particle state. The internal friction angle is a static variable related to soil density, soil particle size, and other factors; the accumulation angle can thus be regarded as a value of the internal friction angle.

Therefore, according to the derivation assumptions of Equations (16) and (1), Jaky’s formula is only suitable for calculating sand’s lower stationary horizontal pressure coefficient K₀ value in the loosest state. Since the result is a fixed value for the same type of sand, it cannot reflect the variation of K₀ under different relative densities of the same sand. This was reported by Gaudin et al. [29], who found that the coefficient of earth pressure at rest increased with relative density through three sets of centrifugal models of sand with different relative densities.

To this end, according to experimental model results, Sherif et al. [13] proposed a formula to calculate K₀ for sand under varying D_r values:

$$K_0=K_{0\mathrm{J}}+K_{01}=\left(1-\sin {\phi }^{\prime }\right)+5.5\cdot \left(\frac{{\gamma }_{\mathrm{d}}}{{\gamma }_{\mathrm{d}\min }}-1\right)$$

(17)

where K_0J is Jaky’s formula (Equation (1)), and γ_d and γ_dmin correspond to the sand’s dry weight and minimum dry weight, respectively. For the same type of sand, K_0J and γ_dmin in Equation (17) will be fixed values, and the only unknown variable will be γ_d. Therefore, when the D_r of the sand increases, the coefficient K₀ also increases, confirming the rationality of the results from the side. When the sand is loosest, the above formula can be simplified to Equation (1). Thus, the Jaky formula can be regarded as a particular case of Equation (17) in the loosest state of sand.

From the macroscopic perspective of the soil mass, when the soil is in a consolidated and stable state, it is assumed that the soil at this time is an ideal elastomer, and the relationship between the pressure coefficient K₀ on the stationary side and the Poisson’s ratio ν can be derived. In other words, when the soil mass is in zero lateral deformation, according to Hooke’s law, the lateral strain of the soil mass can be expressed by the effective horizontal stress and the effective vertical stress, as shown in Equation (18):

$$\{\begin{array}{l}{\epsilon }_x=\frac{1}{E_s}\left[{\sigma }_x-\nu \left({\sigma }_y+{\sigma }_z\right)\right]=0\\ {\epsilon }_y=\frac{1}{E_s}\left[{\sigma }_y-\nu \left({\sigma }_x+{\sigma }_z\right)\right]=0\end{array}$$

(18)

where ε_x and ε_y are the horizontal and vertical strains, σ_x and σ_y are the minor and major principal stresses, σ_z is the effective vertical stress, E_s is the elastic modulus of the soil, and ν is Poisson’s ratio.

According to the definition of K₀, namely σ_x = σ_y = K₀ · σ_z, substituting σ_x and σ_y into Equation (18) and rearranging, it can be obtained that:

$$K_0=\frac{\nu }{1-\nu }$$

(19)

Poisson’s ratio ν is often higher for dense sands than loose sands, leading to a greater value of K₀ for denser sand [39,40]. Thus, from the macroscopic perspective, this verifies that the coefficient K₀ of sand increases with D_r.

5. Conclusions

In this paper, the influence of relative density D_r on the distribution of pressure coefficients on the sand at rest was studied through the K₀ centrifugal model tests, and the following conclusions were obtained:

(1)

It is simpler to prepare soil samples with relatively loose densities using the duck-bill sand spout, while the mesh sand spout is more suited for denser soil samples. Moreover, when the drop height of the two sand spouts is minimal, the relative density of the soil samples gradually increases with drop height. Once the drop height reaches a certain level, the relative density of the soil samples shows minimal changes.

(2)

In the centrifugal model test, where the K₀ value of sand was determined, the maximum settlement of the sand layer was 1.74 mm. For the sand specimens with a height of 320 mm, the change in the self-weight stress of the sand caused by the centrifugal acceleration had a negligible effect on its vertical deformation and could be disregarded.

(3)

The results of the centrifugal model test showed that the coefficient K₀ demonstrated a good linear increasing trend with an increase in the initial D_r of sand. Therefore, the following formula was proposed for the relationship between K₀ and relative density D_r: K₀ = 0.129 D_r + 0.331.

(4)

The meso-evolution analysis model of the pressure coefficient on the stationary side of sand was improved, and the relationship between K₀, the displacement deflection angle θ, and the friction offset angle μ was obtained. The relationship between the coefficient K₀ and the volume fraction of meso-particles ψ was established. Lastly, the influence mechanism of D_r on the coefficient K₀ was shown from the meso-level.