Load transformation from the yielding part of the soil to the adjacent part is known as the soil arching effect, which plays an important role in the design of various geotechnical infrastructures. Terzaghi’s trapdoor test was an important milestone in the development of theories on soil arching. The research on earth pressure of the trapdoor problem is presented in this paper using the three-dimensional (3D) discrete element method (DEM). Five 3D trapdoor models with different heights are established by 3D DEM software PFC 3D. The variation of earth pressure on the trapdoor with the downward movement of the trapdoor, the distribution of vertical earth pressure along the horizontal direction, the distribution of vertical earth pressure along the vertical direction, the distribution of lateral earth pressure coefficient along the depth direction, the magnitude and direction of contact force chain are studied, respectively. Related research results show that the earth pressure on the trapdoor decreases rapidly after the downward movement of the trapdoor, and then reaches the minimum earth pressure. After that, the earth’s pressure will rise slightly, and whether this phenomenon occurs depends on the depth ratio. For the bottom soil, due to the stress transfer caused by the soil arching effect, the ratio of earth pressure in the loose area decreases, while the ratio of earth pressure in the stable area increases. With the trapdoor moving down, the vertical earth pressure along the depth in the stable zone is basically consistent with the initial state, which shows an approximate linear distribution. After the trapdoor moves down, the distribution of earth pressure along with the depth in the loose area changes, which is far less than the theoretical value of vertical earth pressure of its self-weight. Because of the compression of the soil on both sides, the lateral earth pressure coefficient of most areas on the central axis of the loose zone is close to the passive earth pressure coefficient _{p}. The existence of a ‘soil arch’ can be observed intuitively from the distribution diagram of the contact force chain in the loose zone.

Soil arching effect refers to the phenomenon of relative displacement and stress redistribution of soil, which widely exists in geotechnical engineering. Lai et al. [

In terms of model test, Terzaghi [

Some researchers use numerical simulation to study the soil arching effect. Lai et al. [

In theoretical studies, Liang et al. [

It can be found that many scholars have used different research methods to study the soil arching effect, but there are relatively few studies on the soil arching effect in trapdoor problems by using the three-dimensional (3D) DEM. In order to better reveal the change of earth pressure caused by the soil arching effect in the sand and the failure mode in loose areas, it is necessary to carry out a numerical simulation based on meso-structure. In this paper, a trapdoor model is established through the 3D DEM software PFC3D. The relative displacement of the particles is caused by moving the trapdoor down, which results in the soil arching effect. Further, the magnitude and distribution of the loose earth pressure and the distribution of the force chain between the particles are studied.

The discrete element method was firstly introduced by Cundall et al. [_{i} is the unit normal vector in _{n} is the normal force to the normal displacement ^{n}. _{s} is the shear force to the shear displacement ^{s}. _{n} is normal stiffness and _{s} is shear stiffness.

During each calculation cycle, all contact information is updated according to the position of the particles and the wall. According to the force-displacement law, the contact force between the contact particles is calculated. After the contact force is obtained, the force and moment acting on the particles are calculated. According to Newton’s second law, the velocity and acceleration of each particle are obtained. At the end of each calculation cycle, the displacement and rotation angle of the particles are obtained, and then the position of the particles and the wall is updated to form new contacts, and then the next calculation cycle is carried out.

In this paper, spheres are used in 3D DEM modeling for granular materials. Additionally, the linear contact model is chosen for DEM simulations in PFC3D. The linear contact model provides the relation between contact force and relative displacement of the particles as shown in _{n}, _{s}, and

Referring to the typical trapdoor experimental model, a 3D trapdoor model is established in PFC3D. Typical calculation model diagram shows in

In the DEM model, the soil is simulated by spherical particles. The particle micro parameters used in this simulation are mainly obtained by reference, and do not rely on specific projects or tests. The particle density is _{s }= 2600 kg/m^{3}, the particle radius is 12 to 30 mm evenly distributed, the particle normal stiffness is _{n }= 1.5 × 10^{8} N/m, and the particle shear stiffness is _{s }= 1.0 × 10^{8} N/m, the friction coefficient is ^{2}) and the particles are balanced under its self-weight stress.

The calibration method of macro parameters corresponding to particle micro parameters refers to the method described by Chen et al. [

Therefore, the unit weight is 16.31 kN/m^{3}. Jaky [

The lateral earth pressure acting on the wall per unit length

The lateral earth pressure acting on the wall per unit length can be measured by measuring spheres at different depths. The variation of lateral earth pressure per unit length at different depths in calibration is shown in

Taking the effect of depth ratio (^{−4} m. In each cycle, the unbalanced force is generated after the trapdoor is lowered and then eliminated by equilibrium calculation. The calculation is carried out on a computer with dual Intel Xeon E5-2678 v3 CPUs and 128 G RAM. The calculation time varies from several hours to hundreds of hours depending on the number of particles. The earth pressure at a different place of soil is measured by measurement spheres arranged above the trapdoor during the process of the trapdoor moving down. The diameter of each measurement sphere is 0.2 m. Five measurement spheres are arranged in the length direction of the model, and 3 measurement spheres are arranged in the width direction. The number of measurement spheres in the height direction is determined according to the height of the soil, and the typical measurement spheres arrangement schematic is shown in

Equilibrium calculation of the particles is carried out after the generation of particle, and then the earth pressure acting on the trapdoor is recognized. Then moving down the trapdoor, the displacement of each movement is 1.0× 10^{−4} m, and after the movement, the unbalanced force is eliminated by equilibrium calculation, and then the soil earth pressure acting on the trapdoor is recorded.

The displacement of each movement is 1.0 × 10^{−4} m, 500 cycles in total, and the downward movement of the trapdoor is 0.05 m, which is 8.33% of the trapdoor width. The simulation results of five groups under of different buried depth ratios are shown in

The minimum value of earth pressure is related to the buried depth ratio. In the case of

From the simulation results, it can also be found that the minimum earth pressure occurs at the moment when the vertical displacement reaches 1.72%, 1.44%, 2.22%, 2.42% and 2.40% of the trapdoor width for 5 different depth ratio, respectively. Dewoolkar et al. [

It can be seen from the results that the downward movement of the trapdoor causes the soil arching effect in the loose area, and the ratio of earth pressure in the middle loose area is significantly lower than the initial ratio of 1. The ratio of earth pressure measured in the stable area on both sides will be higher than the initial ratio as the trapdoor moves down, mainly due to the soil arching effect, and the earth pressure in the middle loose area will transfer to the stable area on both sides. For

In addition, it can be seen from the results that with the increase of the displacement of the trapdoor, the soil arching effect becomes more obvious. The more stress transfer occurs, the ratio of earth pressure decreases in the loose area, and the ratio of earth pressure in the stable area increases. Taking

The calculation results of vertical earth pressure distribution of lines (II), (III) and (IV) along the vertical direction with the downward movement of the trapdoor are shown in

The vertical earth pressure results located on lines (II), (III) and (IV) can be seen from

For the soil in the loose zone, with the moving down of the trapdoor, the variations of the earth pressure distribution along the depth on line (III) and line (IV) are shown in

In addition, the results of

The simulation results of lateral earth pressure coefficient on the central axis (IV) of loose zone are shown in

According to the simulation results of five groups of different buried depth ratios, the distribution of contact force chain can be observed by cross sections. The front section view is shown in

The 3D DEM software PFC3D is used to study the trapdoor problem, the change of earth pressure and the lateral ratio of earth pressure in loose and stable areas is studied, and the preliminary conclusions and suggestions were as follows:

Due to the presence of the soil arching effect, the earth pressure on the trapdoor decreases rapidly after the simulation begins, and then reaches the minimum earth pressure. The earth pressure on the trapdoor then remains basically unchanged for the large buried depth ratio, while the earth pressure will rise slightly for shallow burial depth. The larger the depth ratio is, the smaller the earth pressure in the loose zone relative to the initial earth pressure will be.

For the bottom soil, due to stress transfer caused by the soil arching effect, the ratio of earth pressure in the loose area decreases, while the ratio of earth pressure in the stable area increases.

As the trapdoor moves down, the vertical earth pressure in the stable area is basically the same along the depth distribution as the initial state, showing an approximate linear distribution. After the trapdoor moves down, the earth pressure in the loose area changes along the depth distribution; unlike the linear distribution of the initial state, the deep earth pressure will be much smaller than the theoretical value of its self-weight earth pressure.

When the soil arching effect occurs in the loose area, the lateral earth pressure coefficient of most areas on the axis of the loose area is close to the passive earth pressure coefficient _{p} due to the extrusion of the soil on both sides.

The contact force chain in the loose zone shows that the soil stress is transferred to the stable zone on both sides of the loose zone through the distribution diagram of the contact force.