Loosening earth pressure above shallow trapdoor in unsaturated soil with different groundwater level

Abstract

The consideration of unsaturated conditions is infrequently addressed in current Terzaghi’s soil arching research. A modified analytical method for calculation of unsaturated loosening earth pressure above shallow trapdoor is proposed in this paper. By assuming the existence of a vertical slip surface above the trapdoor, the stress state of the soil in the loosening area are delineated in the extended Mohr–Coulomb circle. To account for the non-uniform distribution of vertical stress at arbitrary points along the horizontal differential soil trip, a virtual rotation circle trajectory of major principal stress is employed. Subsequently, the average vertical stress acting on the soil trip is determined through integral approach. Taking into account the influence of matric suction on soil weight and apparent cohesion, the differential equation governing the soil trip is solved analytically for cases of uniform matric suction distribution and alternatively using the finite difference method for scenarios involving non-uniform matric suction distribution. The proposed method’s validity is confirmed through comparison with published results. The parameter analysis indicates that the loosening earth pressure initially decreases and subsequently increases with the increase of the soil saturation. With the rise of groundwater level, the normalized effective loosening earth pressure shows a decreasing trend.

Graphical abstract

Keywords

soil arching effect / unsaturated soil / trapdoor test / loosening earth pressure / groundwater level variation

1 Introduction

The soil arching effect is a universal phenomenon in engineering practice and plays an important role in the design of underground infrastructures. Due to the relative displacement generated in the soil mass, load transfer and stress redistribution take place, leading to a significant reduction of the earth pressure acting on the structures. It is of great significance to consider the soil arching effect scientifically and reasonably in the construction of buried structures.

Terzaghi [1] confirmed the existence of the soil arching effect through trapdoor tests and provided an analytical solution for loosening earth pressure by assuming a vertical slip surface based on limit equilibrium theory. Since then, this loosening earth pressure solution and its modified versions considering the soil arching effect have found extensive application in the design and analysis of various structures, including shield tunnels, underground pipes, retaining walls and pile-supported embankments [2–5]. Many researchers have investigated the progressive failure mechanism of the soil arching effect and methods for calculating loosening earth pressure through trapdoor experiments, numerical simulations and theoretical analyses. In the realm of trapdoor experiments, Iglesia et al. [6,7] conducted centrifuge tests and observed that as the relative displacement of trapdoor increased, the stress acting on the trapdoor decreased rapidly at first and then researched a minimum value when the soil arching effect was fully functioning. Soon afterward, the earth pressure incremented slightly and ultimately trended toward stability. The slip surface initially exhibited a curved arch shape, which subsequently transitioned into a triangular form before finally becoming a vertical slip surface. Al-Naddaf et al. [8] studied the influence of surface loading and geosynthetic reinforcement on soil arching, noting that surface loading and geosynthetic reinforcement tends to degrade and mobilize the soil arching effect, respectively. Zhao et al. [9] employed a particle image velocimetry system to conduct a series of trapdoor tests in transparent granular materials, aiming to explore the progressive failure of soil arching. They delineated the arching phenomenon into distinct phases: the initial arching stage, the maximum arching stage, the load recovery stage, and the ultimate stage, as the relative displacement increases. Fattah et al. [10] conducted a comprehensive series of total 42 model tests to examine the behavior of embankments reinforced with stone columns. These tests revealed that the ratio of the embankment height to the clear spacing between columns is a critical parameter governing the extent of soil arching development. On the aspect of numerical approach, Lai et al. [11] found that the shape of the sliding surface could be vertical, trapezoid, spiral or arching, depending on the soil strength parameters and cover ratio by adopting an adaptive finite element limit analysis method. Fattah et al. [12] developed a finite element model to study the arching phenomenon in reinforced embankment, observing that the bearing capacity is enhanced when a horizontal layer of geogrid is introduced at the interface. Liang and Xu [13] employed a discrete element method (DEM) to analyze stress distribution within the loosening zone, concluding that the vertical stress distribution in this region exhibits a concave pattern. Rui et al. [14] established multi-trapdoor DEM numerical models to explore the development of soil arching and found that the contact forces between particles created a robust force chain network that encased triangular and tower-shaped deformation regions. In the domain of theoretical analysis, Hewlett and Randolph [15] proposed a hemispherical three-dimensional model grounded in limit equilibrium theory, considering the failure conditions at the crown of the arch. van Eekelen et al. [16,17] categorized soil arching into three distinct types: limit equilibrium models, frictional models, and rigid models. Most of the modified models focus on the improvement of lateral earth pressure coefficient and the slip surface shape [11,18]. Considering the rotation of principal stress, both minor and major principal stress trajectories were utilized to determine the lateral earth pressure coefficient [13,18–20]. Xu et al. [21] highlighted that the variation in soil stress values due to different trajectory shapes is generally less than 10%. The form of the failure slip surface significantly affects the analytical model, with various shapes such as vertical, triangular, basin, trapezoid, spiral and arched slip surfaces being encountered in different engineering scenarios [11,22–24].

From the literatures review above, it can be seen that the soil arching effect has been studied and obtained notable achievements. However, the majority of current studies have concentrated on dry soil, saturated sandy and clay soil conditions, with the unsaturated state being relatively overlooked, which is commonly observed in practical settings. Cui et al. [25] conducted trapdoor experiments in sand with low water content and observed that the loosening earth pressure in moist sand is significantly smaller than in dry sand. Song et al. [26] studied the effect of groundwater level variation on soil arching, revealing that the zone influenced by groundwater level is twice the width of trapdoor. Xu et al. [27] performed 2D trapdoor tests incorporating the influence of seepage flow and found that seepage flow significantly increases the vertical stress acting on the trapdoor door while having a minimal effect on the development of the slip surface. Lin et al. [28] modified the Terzaghi’s solution by accounting for the effect of matric suction, determining that the loosening earth pressure reaches its minimum when the matric suction equals the air entry value. However, in Ref. [28], the lateral pressure coefficient is treated as a constant and the effect of soil saturation on the soil parameters such as unit weight are not considered. Fang et al. [29] extended Terzaghi’s model to an unsaturated condition by incorporating matric suction and found that the scale effect was significant in unsaturated condition. However, the principal stress rotation phenomenon is not considered in their study.

It can be seen that the loosening earth pressure in unsaturated soil has garnered significant interest from researchers. However, given the inherent complexity of unsaturated soil behavior, the research in this field remains somewhat limited. The presence of matric suction has a great influence on the evolution of soil arching effect. Moreover, the apparent cohesion attributed to matric suction is not stable and easily decreases, or even vanish due to factors such as groundwater level variation, rainfall, and leakage. This behavior can lead to potential risks to engineering construction if not properly considered. This study presents an analytical approach for calculating loosening earth pressure above a shallow trapdoor when the slip surface has reached a ground condition. The method integrates the rotation trajectory of the major principal stress with the shear strength theory of unsaturated soil, incorporating the soil–water characteristic curve to account for the effect of matric suction on soil weight and apparent cohesion. The loosening earth pressure of sandy or clay soil can be obtained by solving an equilibrium differential equation in the vertical direction by the finite difference method. The accuracy of the proposed method is confirmed through its alignment with results from existing trapdoor tests, numerical simulation results, and analytical solutions. Then the influence of groundwater level variation on loosening earth pressure is studied.

2 Unsaturated soil theory

The shear strength of unsaturated soil is closely associated with matric suction [30–32]. So far, the shear strength of unsaturated soil is predominantly described in two ways. One approach is proposed by Bishop et al. [33]:

(1)\tau =c^{\prime }+(\sigma -u_{\text{a}}+\chi (u{}_{\text{a}}-u_{\text{w}}))\mathrm{t}\mathrm{a}\mathrm{n}{\phi }^{\prime },

where \tau is the shear stress, c^{\prime } is the effective cohesion, \sigma is the total normal stress, u_{\text{a}} is the pore air pressure, which is typically zero, (u{}_{\text{a}}-u_{\text{w}}) is the matric suction, u_{\text{w}} is the pore water pressure, {\phi }^{\prime } is the effective internal friction angle, and \chi is the effective stress coefficient relating to the degree of saturation S_{\text{r}}. According to previous literatures [34–37], \chi is set equal to S_r in this study. \chi (u{}_{\text{a}}-u_{\text{w}})\tan {\phi }^{\prime } is usually called apparent cohesion.

Another method for expressing the shear strength of unsaturated soil is that established by Fredlund et al. [38], which is formulated using two independent stress state variables:

(2)\tau =c^{\prime }+(u_{\text{a}}-u_{\text{w}})\mathrm{t}\mathrm{a}\mathrm{n}{\phi }^{\text{b}}+(\sigma -u_{\text{a}})\mathrm{t}\mathrm{a}\mathrm{n}{\phi }^{\prime },

where {\phi }^{\text{b}} is the suction friction angle varying with matric suction.

As can be seen from Eqs. (1) and (2), the shear strength of unsaturated soil comprises the effective cohesion, the frictional strength associated with external loads, and the apparent cohesion related to matric suction. If:

(3)\chi \tan {\phi }^{\prime }=\tan {\phi }^{\text{b}}.

Then both expressions can be unified into an extended Mohr–Coulomb form:

(4)\tau =c+(\sigma -u_{\text{a}})\tan {\phi }^{\prime },

where c represents the total cohesion and is expressed as c^{\prime }+\chi (u_{\text{a}}-u_{\text{w}})\tan {\phi }^{\prime }.

The soil–water characteristic curve (SWCC) is usually employed to describe the relationship between matric suction u{}_{\text{a}}-u_{\text{w}} and the degree of saturation S_r. The V–G model [39] is adopted in this context:

(5)S_{\text{e}}=[1+(\mathit{\text{α}}(u_{\text{a}}-u_{\text{w}}{)}^{\text{d}}){]}^{-\text{m}},

where S_e is the effective saturation, expressed as S_{\text{e}}=(S_{\text{r}}-S_0)/(1-S_0). S_0 is the irreducible saturation. The constants \text{α}, m and d serve as fitting constants, with the conventional assumption that m=1-1/d.

In addition, the effective stress {\sigma }^{\mathrm{\prime }} in this study is also expressed in the Bishop form:

(6){\sigma }^{\prime }=\sigma -\chi u_{\text{w}}-(1-\chi )u{}_{\text{a}}.

3 Analysis model

The trapdoor test analysis model considering the effect of groundwater level used in this study is shown in Fig.1. The vertically downward direction is taken as the z axis, with the origin positioned at the ground surface. The active door width is denoted as D and the covering soil height is H. The groundwater level height is indicated as H_{\text{w}}. A uniform load q is applied to the ground surface.

Several main assumptions are adopted here: 1) the soil is considered saturated below the groundwater level and unsaturated above it, fulfilling the extended Mohr–Coulomb failure criterion; 2) once the trapdoor has descended by a specific distance, the slip surface is assumed to be vertical, as suggested by Refs. [13,29,40]; 3) within the loosening area abcd, the soil is considered to be in a limit state.

The stress state evolution of the soil mass in the loosening area as the trapdoor descends can be described in the Mohr stress circle as shown in Fig.2. When the trapdoor is stationary, the soil pressure is in a state of at-rest or called K_0 state. As the trapdoor continues to move downward, the ground loss increases, and the soil arching effect progressively develops until it reaches the limit state. Points A and B correspond to the stress state at the vertical slip surface and the center line, respectively. Point C is an arbitrary point situated along the horizontal differential soil trip. According to the maximum principal stress trajectory theory, {\sigma }_{\text{v}}-u_{\text{a}} and {\sigma }_{\text{h}}-u_{\text{a}} represent the total vertical normal stress and horizontal normal stress, respectively. \theta is the angle between the maximum principal stress and the horizontal direction, ranging from 0 to {\theta }_0=\text{π}/\phantom{\text{π}4}4+{\phi }^{\prime }/\phantom{{\phi }^{\prime }2}2.

Considering an arbitrary point along the horizontal differential soil trip, the vertical normal stress {\sigma }_{\text{v}}-u_{\text{a}}, horizontal normal stress {\sigma }_{\text{h}}-u_{\text{a}}, and shear stress \tau can be expressed as follows:

\begin{array}{}\text{(7a)}& {\sigma }_{\text{v}}-u_{\text{a}}=({\sigma }_3-u_{\text{a}}){\cos }^2\theta +({\sigma }_1-u_{\text{a}})\text{si}{\text{n}}^2\theta ,\end{array}

\begin{array}{}\text{(7b)}& {\sigma }_{\text{h}}-u_{\text{a}}=({\sigma }_1-u_{\text{a}}){\cos }^2\theta +({\sigma }_3-u_{\text{a}})\text{si}{\text{n}}^2\theta ,\end{array}

\begin{array}{}\text{(7c)}& \tau =(({\sigma }_1-u_{\text{a}})-({\sigma }_3-u_{\text{a}}))\cos \theta \sin \theta ,\end{array}

where {\sigma }_1-u_{\text{a}} and {\sigma }_3-u_{\text{a}} are the maximum and minimum principal stresses, respectively.

The coefficient of active earth pressure, denoted as K_{\text{a}}, can be expressed as:

(8)K_{\text{a}}={\tan }^2({45}^{\circ }-{\phi }^{\prime }/2)=\frac{{\sigma }_3-u_{\text{a}}+c\cot {\phi }^{\prime }}{{\sigma }_1-u_{\text{a}}+c\cot {\phi }^{\prime }}.

Then, the relationship between the horizontal normal stress {\sigma }_{\text{h}}-u_{\text{a}} and {\sigma }_{\text{v}}-u_{\text{a}} can be described as:

(9)\overline{K}=\frac{{\sigma }_{\text{h}}-u_{\text{a}}+c\cot {\phi }^{\prime }}{{\sigma }_{\text{v}}-u_{\text{a}}+c\cot {\phi }^{\prime }}=\frac{{\cos }^2\theta +K_{\text{a}}\text{si}{\text{n}}^2\theta }{K_{\text{a}}{\cos }^2\theta +\text{si}{\text{n}}^2\theta }.

As for the circular curve, the governing equation of the trajectory is formulated within the local coordinate system xo{\mathit{\text{z}}}^{\prime } as depicted in Fig.2, illustrating the virtual circular curve based on the rotation trajectory of the major principal stress theory, as referenced in Ref. [41]:

(10)x^2+({\mathit{\text{z}}}^{\prime }{)}^2=\frac{D^2}{4{\sin }^2{\theta }_0}.

Then, the x axis can be written as:

(11)x=\frac{D}{2\sin {\theta }_0}\sin \theta .

The average vertical stress {\sigma }_{\mathrm{a}\mathrm{v}}-u_{\mathrm{a}} applied to the horizontal differential soil trip can be calculated in the following:

(12)\begin{array}{rl}& ({\sigma }_{\text{av}}-u_{\text{a}})+c\cot {\phi }^{\prime }\\ =& \frac{2}{D}{\int }_0^{D/2}(({\sigma }_3-u_{\text{a}}+c\cot {\phi }^{\prime }){\cos }^2\theta +({\sigma }_1-u_{\text{a}}+c\cot {\phi }^{\prime })\text{si}{\text{n}}^2\theta )\text{d}x\\ =& \frac{2}{D}{\int }_0^{{\theta }_0}(({\sigma }_1-u_{\text{a}}+c\cot {\phi }^{\prime })(K_{\text{a}}{\cos }^2\theta +\text{si}{\text{n}}^2\theta ))\frac{D}{2\sin {\theta }_0}\cos \theta \text{d}\theta .\end{array}

Upon performing the integral, Eq. (12) can be simplified to:

(13){\sigma }_{\text{av}}-u_{\text{a}}+c\cot {\phi }^{\prime }=\left(\frac{1}{3}(1-K_{\text{a}}){\sin }^2{\theta }_0+K_{\text{a}}\right)({\sigma }_1-u_{\text{a}}+c\cot {\phi }^{\prime }).

Thereafter, the relationship between the vertical normal stress {\sigma }_{\text{v}}-u_{\text{a}} and average vertical normal stress {\sigma }_{\text{av}}-u_{\text{a}} at arbitrary point can be described as:

(14)m=\frac{{\sigma }_{\text{v}}-u_{\text{a}}+c\cot {\phi }^{\prime }}{{\sigma }_{\text{av}}-u_{\text{a}}+c\cot {\phi }^{\prime }}={\displaystyle \frac{K_{\text{a}}{\cos }^2\theta +\text{si}{\text{n}}^2\theta }{{\displaystyle \frac{1}{3}}(1-K_{\text{a}}){\sin }^2{\theta }_0+K_{\text{a}}}}.

Then the following equation can be obtained:

(15){\sigma }_{\text{av}}-u_{\text{a}}=\frac{{\sigma }_{\text{vA}}-u_{\text{a}}+c\cot {\phi }^{\prime }}{m_{\text{A}}}-c\cot {\phi }^{\prime },

where {\sigma }_{\text{vA}}-u_{\text{a}} and m_{\text{A}} are the values of vertical normal stress and m at point A, respectively.

The equivalent lateral earth pressure coefficient K_A at the slip surface can be determined using Eqs. (9) and (15):

(16)K_{\text{A}}=\frac{{\sigma }_{\text{hA}}-u_{\text{a}}+c\cot {\phi }^{\prime }}{{\sigma }_{\text{vA}}-u_{\text{a}}+c\cot {\phi }^{\prime }}=\overline{K}{m}_{\text{A}}=\frac{{\cos }^2{\theta }_0+K_{\text{a}}{\sin }^2{\theta }_0}{{\displaystyle \frac{1}{3}}(1-K_{\text{a}}){\sin }^2{\theta }_0+K_{\text{a}}},

where {\sigma }_{\text{hA}}-u_{\text{a}} is the horizontal or normal stress at point A.

Upon conducting a stress analysis on the horizontal differential soil trip, the following equation can be derived:

(17)Dd({\sigma }_{\text{av}}-u_{\text{a}})+2\tau \text{d}\mathit{\text{z}}=D\gamma (\mathit{\text{z}})\text{d}\mathit{\text{z}},

where \gamma (\mathit{\text{z}}) is the wet unit weight of soil at the depth z.

The shear stress at the slip surface can be expressed as:

(18)\tau =\overline{K}({\sigma }_{\text{vA}}-u_{\text{a}}+c\cot {\phi }^{\prime })\tan {\phi }^{\prime }.

Substituting Eqs. (15), (16), and (18) into Eq. (17), the following equation can be obtained:

(19)\frac{\text{d}({\sigma }_{\text{av}}-u_{\text{a}})}{\text{d}\mathit{\text{z}}}+\frac{2K_{\text{A}}({\sigma }_{\text{av}}-u_{\text{a}})\tan {\phi }^{\prime }}{D}+\frac{2c{K}_{\text{A}}}{D}=\gamma (\mathit{\text{z}}).

For soil located above the groundwater level, the matric suction u_{\text{a}}-u_{\text{w}} at a specific depth z can be expressed as:

(20)u_{\text{a}}-u_{\text{w}}={\rho }_{\text{w}}g(H_{\text{w}}-\mathit{\text{z}}),

where {\rho }_{\text{w}} is the density of water, and g is the gravitation acceleration.

The wet density of soil {\rho }_{\text{t}} at depth z is given by:

(21){\rho }_{\text{t}}=\frac{G_{\text{s}}+e{S}_{\text{r}}(\mathit{\text{z}}){\rho }_{\text{w}}}{1+e},

where G_{\text{s}} is the specific gravity of the soil mass, e is the void ratio, S_{\text{r}}(\mathit{\text{z}}) is a distribution function of the degree of saturation that can be obtained through SWCC.

Substituting Eqs. (20) and (21), and the expression of total cohesion C into Eq. (19), the following relationship can be derived:

(22)\begin{array}{rl}& \frac{\text{d}({\sigma }_{\text{av}}-u_{\text{a}})}{\text{d}\mathit{\text{z}}}+\frac{2K_{\text{A}}({\sigma }_{\text{av}}-u_{\text{a}})\tan {\phi }^{\prime }}{D}\\ +& \frac{2(c^{\prime }+S_{\text{r}}(\mathit{\text{z}})({\rho }_{\text{w}}g(H_{\text{w}}-\mathit{\text{z}}))\tan {\phi }^{\prime })K_{\text{A}}}{D}=\frac{G_{\text{s}}+e{S}_{\text{r}}(\mathit{\text{z}}){\rho }_{\text{w}}}{1+e}g.\end{array}

The boundary condition is:

(23){({\sigma }_{\text{av}}-u_{\text{a}})|}_{\mathit{\text{z}}=0}=q.

Subsequently, the loosening earth pressure above the trapdoor can be determined by solving Eq. (22) subject to the boundary condition in Eq. (23).

Here, a finite difference method is adopted to solve Eq. (22), the difference quotient equation can be expressed by the incremental depth \mathrm{\Delta }\mathit{\text{z}}:

(24){\sigma }_{\mathit{\text{z}}}(\mathit{\text{z}}+\mathrm{\Delta }\mathit{\text{z}})-{\sigma }_{\mathit{\text{z}}}(\mathit{\text{z}})=\frac{\text{d}{\sigma }_{\mathit{\text{z}}}}{\text{d}\mathit{\text{z}}}\mathrm{\Delta }\mathit{\text{z}}.

For the soil mass situated below the groundwater level, the loosening earth pressure q_1 at depth H_{\text{w}} can be obtained first through Eq. (22). Then taking q_1 as the boundary condition acting on the saturation soil surface, the expression of Eq. (19) can be rewritten as follows:

(25)\frac{\text{d}({\sigma }_{\text{av}}-u_{\text{a}})}{\text{d}\mathit{\text{z}}}+\frac{2K_{\text{A}}({\sigma }_{\text{av}}-u_{\text{a}})\tan {\phi }^{\prime }}{D}+\frac{2c^{\prime }{K}_{\text{A}}}{D}=\frac{G_{\text{s}}+e{\rho }_{\text{w}}}{1+e}g,

with the boundary condition:

(26){({\sigma }_{\text{av}}-u_{\text{a}})|}_{\mathit{\text{z}}=H_{\text{w}}}=q_1.

Up to now, the loosening earth pressure above the trapdoor at any depth can be obtained. Particularly, if the soil mass above the trapdoor is homogeneous, indicating that the matric suction is constant with depth, an analytical solution of Eq. (19) can be obtained by combing it with the boundary condition in Eq. (23), yielding:

(27)\begin{array}{rl}{\sigma }_{\text{av}}=& \left(\frac{D(G_{\text{s}}+e{S}_{\text{r}}{\rho }_{\text{w}})g}{2(1+e)K_{\text{A}}\tan {\phi }^{\prime }}-\frac{c}{\tan {\phi }^{\prime }}\right)\\ & \cdot (1-e^{-\frac{2\mathit{\text{z}}{K}_{\text{A}}\tan {\phi }^{\prime }}{D}})+q{e}^{-\frac{2\mathit{\text{z}}{K}_{\text{A}}\tan {\phi }^{\prime }}{D}}.\end{array}

In fact, the aforementioned method can be applied to any distribution of matric suction along the depth direction.

4 Validation of the method

In this section, the accuracy and reliability of the proposed method will be assessed through a comparative analysis against existing trapdoor tests, numerical simulation results and analytical solutions.

4.1 Compared with trapdoor tests and analytical solutions

Liang et al. [42] and Iglesia et al. [7] conducted trapdoor tests of dry sand soil in a conventional gravity device and a centrifugal machine, respectively. If the groundwater level is exceptionally deep, the solution presented in this study can be simplified to represent dry conditions. Herein, the analytical solution proposed in this paper is contrasted with the results of their trapdoor tests. Meanwhile, for the sake of comparison, Terzaghi’s theory and the analytical solution proposed by Liang and Xu [13] are also utilized.

Fig.3 shows the comparison of normalized average loosening earth pressure, {\sigma }_{\text{av}}/(\gamma H), for various approaches at different relative depths. It can be seen that with the relative depth increases, the loosening earth pressure does not decrease linearly due to the soil arching effect, and it tends to stabilize at a certain value. The solution proposed in this paper shows a satisfactory match with the trapdoor tests results when the relative depth is less than 2. However, when the relative depth is larger than 2, the solution deviates from the test results, primarily because the slip surface is not perfectly vertical in such cases. In comparison, for the trapdoor test results in a normal gravity device conducted by Liang et al. [42], Terzaghi’s solution [1], and Liang and Xu’s solution [13] are found to slightly underestimate the trapdoor test outcomes. For the centrifugal tests conducted by Iglesia et al. [7], Terzaghi’s solution and our solution slightly overestimate the results, while Liang and Xu’s solution underestimates the trapdoor test results. As can be seen from Fig.3, the solution proposed by this paper yields results that are higher than those obtained from Terzaghi’s solution and Liang and Xu’s solution for dry sandy soil. Terzaghi’s solution does not account for the rotation phenomenon, leading to a distinct expression for the lateral earth pressure coefficient (as given in Eq. (16) of this paper). In contrast, Liang and Xu’s solution incorporates an arc-shaped trajectory for the major principal stress, which diverges from the approach of Eq. (10) utilized in this paper.

4.2 Compared with numerical result

A numerical trapdoor model was developed under plane strain conditions for further validation using the ABAQUS finite element software, as shown in Fig.4. Within ABAQUS framework, the pore fluid in unsaturated soil is treated as multiphase flow, including less compressible fluid water and compressible fluid air. The seepage flow adheres to the Forchheimer law [43]. The model is 6 m high in the vertical direction and 30 m wide in the horizontal direction. The width of active trapdoor D is 6 m and the thickness of overlying soil layer H is 6 m. The groundwater level H_{\text{w}} is set as 3 m. For the lateral boundaries on either side, horizontal displacement constraints are imposed. The lower boundary is rigidly fixed, whereas the upper boundary is left unrestricted. The active door is defined by a displacement boundary condition characterized by a constant downward speed.

The unsaturated soil is simulated as a CPE4P liquid–solid coupling element, which adheres to the extended Mohr–Coulomb failure criterion. According to Chen et al. [44], the elasticity modulus E has a slight influence on the calculation results of loosening earth pressure when the soil arching effect fully developed. Herein, the elasticity modulus E of unsaturated and saturated soil layers are set at 30 and 20 MPa, respectively. The Poisson ratio is 0.3. The values of other parameters are detailed in Tab.1. Following the establishment of initial ground stress equilibrium, the trapdoor is moved downward in the vertical direction with a speed of 1\times {10}^{-5} m per step. The numerical simulations are terminated once the displacement of the active door reaches a specified threshold or when the loosening earth pressure against the active door attains a steady-state. Two distinct matric suction distribution patterns are calculated to verify the accuracy of the analytical method proposed by this research. The first pattern represents a uniform distribution without the groundwater, while the second depicts a linear distribution in the region above the groundwater level.

Fig.5 displays the results of the normalized loosening earth pressure, {\sigma }_{\text{av}}/(\gamma H), for various degrees of saturation under a uniform distribution of matric suction. It can be seen that with the increase of saturation degree, the normalized loosening earth pressure initially decreases with increasing saturation, reaches a minimum at a specific saturation degree, and then increases gradually. The solutions proposed in this paper demonstrate a high degree of agreement with the numerical results.

Fig.6 shows the loosening earth pressure results for a linear distribution of matric suction above the groundwater level (H_{\text{w}} = 3 m) along the depth direction. It can be seen that the rate of increase in loosening earth pressure is lower in the saturated soil layer compared to the unsaturated soil layer. This can be attributed to the presence of matric suction in unsaturated soil, which enhances the shear strength through the contribution of apparent cohesion. The solutions proposed by this paper are in good agreement with the numerical results and align with the theoretical solution proposed by Fang et al. [29].

5 Parameter analysis

In this section, the influence of saturated degree S_{\text{r}} and groundwater level H_{\text{w}} on the loosening earth pressure are discussed. The parameters adopted are shown in Tab.1.

5.1 Influence of saturation degree

Examples with a uniform distribution of matric suction are adopted here to analyze the effect of saturation degree. Fig.7 shows the curve of normalized loosening earth pressure above the trapdoor varying with degree of saturation for different soil types. It can be seen that the overall trend of variation is similar for different soil types. For fine silty sand soil and sand silt soil, the normalized loosening earth pressure initially decreases gradually as the saturation degree increases. The curve starts to increase gradually once it reaches a minimum value at a certain saturation degree. The corresponding saturation degree of the minimum normalized loosening earth pressure value is about 0.4 in this example. In the case of silty clay soil examined in this study, as the active door descends, self-stabilization may occur as a result of the strong cohesion between clay particles. When the soil’s degree of saturation falls within the range of 0.2 to 0.7, the apparent cohesion is also significant, which increases the likelihood of self-stabilization. Consequently, the clay soil layer re-mains stationary and does not move downward in concert with the descending trapdoor within this saturation range. As a result, the normalized loosening earth pressure above the trapdoor is considered negligible and is therefore set to zero. As can be seen from Eq. (4), the shear strength is related to matric suction. According to Fredlund et al. [38] and Lin et al. [45], the apparent cohesion initially increases with the matric suction and then exhibits a declining trend, indicating the presence of a critical saturation degree. When the saturated degree of soil is below this critical value, the apparent cohesion rises with the saturated degree, thereby mobilizing the soil arching effect, which results in a decrease in the loosening earth pressure acting on the trapdoor decreases. Conversely, when the saturated degree of soil exceeds the critical value, the apparent cohesion diminishes with the saturated degree, weakening the soil arching effect, and leading to an increase in the loosening earth pressure. At the critical saturation degree, the loosening earth pressure reaches its minimum when the soil arching effect is fully developed. Compared to sandy soil, the loosening earth pressure in clay soil exhibits a more pronounced and broader range of variation with changes in saturation degree. From the discussion above, it can be seen that the degree of saturation plays an important influence in the development of soil arching effect and must be considered in engineering practice.

5.2 Influence of groundwater level location

Examples with a linear distribution of matric suction and fine silty sand soil are adopted here to study the effect of groundwater level. The value of H/D is equal to 1.5 for the whole example. Define the value of H_{\text{w}}/H ranging from 0 to 1 to represent the variation of groundwater level, where a higher value of H_{\text{w}}/H signifies a lower groundwater level. Fig.8 shows the curve of normalized effective loosening earth pressure along the depth direction varying with the groundwater level variation. It can be seen that with the increased value of H_{\text{w}}/H, the normalized effective loosening earth pressure shows a decreasing tendency at the same depth, with a gradual and significant rate of decline near the shallow surface and above the trapdoor, respectively. Notable inflection points appear when the values of H_{\text{w}}/H are equal to 0.5 and 1, indicating the transition between saturated soil and unsaturated soil. The normalized effective loosening earth pressure shows an increasing trend in the unsaturated area and a decreasing tendency along the depth direction, respectively, which can be attributed to the enhanced shear strength due to the matric suction in the unsaturated area and the reduced shear strength caused by the groundwater level variation in the saturated area. The degree of saturation at the same location is lower than that for larger values of H_{\text{w}}, indicating a higher apparent cohesion at that location. Consequently, the shear strength of soil along the vertical slip surface increases in the unsaturated area. However, in the saturated area below the groundwater, the total shear strength diminishes with the rise in groundwater level, leading to a weakening of the soil arching effect. Therefore, it should be paid great attention on the variation of groundwater level in engineering practice.

6 Conclusions

Based on the rotation trajectory of major principal stress and the shear strength theory of unsaturated soil, a modified analytical method for the calculation of unsaturated loosening earth pressure above shallow trapdoor is proposed. The influence of groundwater level variation on loosening earth pressure is studied. The main conclusions can be shown as follows.

1) The analytical method presented in this paper can be used in the calculation of loosening earth pressure above shallow trapdoors in unsaturated soil effectively.

2) As the saturation degree increases, the normalized loosening earth pressure decreases first and then increases gradually. At the critical saturation degree, the loosening earth pressure gets the minimum value and the soil arching effect is fully developed. The loosening earth pressure in clay soil exhibits a more significant and broader range of variation compared to that in sandy soil.

3) With the increased value of H_{\text{w}}/H, the normalized effective loosening earth pressure shows a decreasing tendency at the same depth location. Along the depth direction, the normalized effective loosening earth pressure demonstrates an increasing trend in the unsaturated area and a decreasing tendency in the saturated zone.

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