Algorithm Implementation of Equivalent Expansive Force in Strength Reduction FEM and Its Application in the Stability of Expansive Soil Slope

Abstract

Loss of matric suction during rain has an important effect on the instability of expansive soil slope. A new device was designed for testing the expansive force in order to propose and determine the equivalent expansive force corresponding to the matric suction. First, the internal relation between the matric suction and the corresponding equivalent expansive force of unsaturated expansive soil was analyzed. Then, numerical algorithm implementation of the equivalent expansive force was discussed, and the equivalent expansive force was introduced into the strength reduction finite element method (FEM). Finally, the equivalent effects of the matric suction and the equivalent expansive force were compared and analyzed by evaluating the stability of an unsaturated expansive soil slope. The results show that the new testing device significantly improves the accuracy of the expansive force test and shortens the testing time. The relation between the matric suction and the equivalent expansive force with the change in initial water content is obvious. The equivalent expansive force can reflect the macro contribution of matric suction to unsaturated expansive soil, and the developed strength reduction FEM based on the equivalent expansive force can be used to evaluate the rainfall-induced instability of an expansive soil slope caused by the decrease in matric suction resulting from the rainfall infiltration.

1. Introduction

Rainfall-induced landslides often occur in expansive soil regions and always lead to regional geological disasters [1]. Expansive soil is a special type of soil with an obvious swell–shrink characteristic, which is distributed extensively [2,3]. The rainfall-induced instability of unsaturated expansive soil slopes is a problem that urgently needs to be solved in geotechnical engineering [4,5]. For one thing, rainfall-induced landslide hazard and risk analyses should be evaluated from a macroscopic perspective at the regional scale, and some probabilistic methods are used to assess regional susceptibility to landslides. For another, the triggering mechanism and kinematic process of landslides caused by rainfall infiltration should be researched [6].

Plenty of natural expansive soil landslides are often accompanied by rainfall infiltration, excavation, or earthquake action. Under the condition of rainfall infiltration, the water content of an expansive soil slope increases and the matric suction decreases [7], which indicates that the safety factor of an expansive soil slope also decreases gradually [8,9]. It is noticed that the measurement of matric suction is time-consuming, and the technique is complex and intricate [10,11,12]. For this reason, it is necessary to propose a simple, reasonable, and relatively advanced test method that can not only reflect the characteristics of unsaturated expansive soil but can also avoid the difficulties in the measurement of matric suction. In the view of Khan et al. [13], matric suction has a relation with the swell–shrink characteristic of unsaturated expansive soil. The expansive force can be used to effectively avoid the trouble in the measurement of matric suction.

At the same time, LU Zhaojun’s unsaturated expansive soil strength [14,15] disclosed the correlation between the matric suction and the expansive force. Therefore, the computational method for research on the rainfall-induced instability of unsaturated expansive soil, based on the equivalent substitute of matric suction by the expansive force, is also worthy of attention.

Owing to landslides of unsaturated soil slopes often occur during rainfall infiltration, the mechanics of unsaturated soil is also worthy of research. It is important to note that a reasonable evaluation of landslides should be based on the reasonable strength theory of unsaturated soil. Bishop et al. proposed the effective stress strength formula for unsaturated soil [16]. Based on the effective stress strength formula for unsaturated soil proposed by Bishop et al., Fredlund et al. suggested the double stress strength formula for unsaturated soil [17].

Through Bishop’s effective stress strength formula and Fredlund’s double stress strength formula, along with the analysis aspect of matric suction, Bishop’s slices method was applied to study the influence of matric suction on unsaturated expansive soil slope stability [18,19]. A plastic upper bound limit analysis was also conducted to analyze the influence of swelling deformation [20,21]. Yao et al. [22] showed that rainfall duration could also strongly affect the transient flows and reduce the stability of unsaturated expansive soil slopes. In the studies by Noorany Iraj [23] and Diyaljee et al. [24], the lateral extension in expansive soil slopes was also discussed. Based on a model test, Xiao et al. [25] evaluated and compared the reinforcement effect of geogrid in expansive soil slopes, and some failure mechanisms of expansive soil slopes were discussed from the aspects of strength characteristics [26,27]. These studies were advantageous to research on the stability principles of unsaturated expansive soil slopes. However, these methods only considered matric suction on sliding surfaces, whereas the actual influence of matric suction out of the sliding surface in unsaturated soil slope was underestimated, that is, these studies have not been involved in the non-uniform distribution of matric suction. However, it is worth noting that the strength reduction finite element method (FEM) based on Bishop’s effective stress strength formula and Fredlund’s double stress strength formula was developed and applied in the stability analysis of unsaturated slopes according to a non-uniform distribution of matric suction in the slopes [28].

Matric suction has a relation with the swell–shrink characteristic of unsaturated expansive soil [14,29], so the amount of swelling resulting from expansibility transforms into an expansive force when its bulk is restricted or its volume remains constant. Thus, the suctional strength contributed by matric suction should have a relationship with the equivalent expansive force (EEF), which is defined as the maximum expansive force for the expansive soil sample from a soaking state to a strictly saturated state with a constant volume under a certain initial water content. The inner correlation between the matric suction and the equivalent expansive force could provide a new analysis perspective to study the strength characteristic of unsaturated expansive soil. Lu et al. proposed a comprehensive formula for the strength of unsaturated expansive soil [15].

Aiming at the slope engineering of expansive soil, the strength reduction FEM based on Lu’s comprehensive formula, which will act as a new stability analysis method of unsaturated expansive soil slopes, will effectively widen the scope and thought for the stability analysis of unsaturated slopes.

In this study, the theoretical equivalent relationship between matric suction and the expansive force in the shear strength of unsaturated expansive soil is discussed. Then, some tests of matric suction and the expansive force as well as direct shear tests on expansive soil from Hanyin County in China are conducted. Studies on the correlation between matric suction and the equivalent expansive force with varying initial water contents of unsaturated expansive soil are also performed. The numerical algorithm implementation of the equivalent expansive force is discussed, and the equivalent expansive force is introduced into the strength reduction FEM. Equivalent effect research on evaluating the stability of unsaturated expansive soil slopes should be verified further based on the relation between matric suction and the equivalent expansive force for the sake of actual applications in engineering.

2. Equivalent Relation of Matric Suction and Expansive Force

This study focused on analyzing the theoretical equivalent relationship between matric suction and the expansive force in the shear strength of unsaturated expansive soil. Based on the deductive analysis of the relation between matric suction and the expansion force, the internal equivalent relation between them in the strength of unsaturated expansive soil can be reflected, and it is the theoretical basis of the core algorithm analysis for the stability analysis of expansive soil slopes.

Bishop’s effective stress strength formula for unsaturated soil is as follows,

$${\tau }_f=c^{\prime }+\left(\sigma -ua\right)\tan {\phi }^{\prime }+\chi \left(ua-uw\right)\tan {\phi }^{\prime }\hspace{0.17em}\hspace{0.17em}$$

(1)

where the shear strength of unsaturated soil is τ_f, the effective cohesion is c′, the angle of frictional resistance is φ’, the normal stress is σ, the pore water pressure is u_w, the pore air pressure is u_a, and the parameter of χ is determined by the saturation degree, S_r.

Fredlund’s double stress strength formula for unsaturated soil is as follows,

$${\tau }_f=c^{\prime }+(\sigma -ua)\tan {\phi }^{\prime }+(ua-uw)\tan {\phi }^b\hspace{0.17em}$$

(2)

where ${\phi }^{{}_b}$ is the suctional internal friction angle.

In further discussion under the condition of equal effective cohesion and equal effective stress, it can be seen that, in view of Formulas (1) and (2), Fredlund et al. proposed the equivalent relation of matric suction to the unsaturated soil strength as follows [30]:

$$\chi \left(ua-uw\right)\tan {\phi }^{\prime }=(ua-uw)\tan {\phi }^b\hspace{0.17em}$$

(3)

The simplified expression is Formula (4):

$$\chi =\frac{\tan {\phi }^b}{\tan {\phi }^{\prime }}$$

(4)

There is an inner correlation between the matric suction and the equivalent expansive force, as reported by Lu’s comprehensive formula for the strength of unsaturated expansive soil.

$$\hspace{0.17em}\tau f=c^{\prime }+(\sigma -ua)\tan {\phi }^{\prime }+m{p}_s\tan {\phi }^{\prime }\hspace{0.17em}\hspace{0.17em}$$

(5)

where m was the effective coefficient of the expansive force. p_s was the expansive force of the expansive soil under a constant volume during saturation.

For Formulas (1), (2) and (5), under the condition of equal effective cohesion and equal effective stress, the equivalent relation of matric suction and expansion force to unsaturated soil strength are proposed in this paper as follows:

$$\chi \left(ua-uw\right)\tan {\phi }^{\prime }=(ua-uw)\tan {\phi }^b=m{p}_s\tan {\phi }^{\prime }\hspace{0.17em}$$

(6)

Therefore, the simplified expressions are Formulas (7) and (8),

$$\chi =m\frac{p_s}{\left(ua-uw\right)}$$

(7)

$$m=\left(\frac{\tan {\phi }^b}{\tan {\phi }^{\prime }}\right)/\left(\frac{p_s}{(ua-uw)}\right)$$

(8)

Through the above deductive analysis of the relation between matric suction and the expansion force, it can be concluded that there is a certain internal equivalent relation between them in the strength of unsaturated expansive soil, and this internal equivalent relation is the theoretical basis of the core algorithm analysis for the expansive soil slope in this paper.

3. Methodology

The testing methods for the traditional expansive force are numerous. Common traditional methods include the direct expansion-back pressure, preloading expansion, and gradually balancing pressure methods.

The direct expansion-back pressure method applies loads on a soil sample to return to its initial volume after fully absorbing water and freely expanding. The soil sample is generally stressed while expanding to its maximum limit during the experiment until it returns to its initial volume. To a certain extent, the force measured by the expansion-back pressure method should be the consolidation pressure rather than the true expansion pressure. When applying a huge pressure on the soil sample in the preloading expansion method, expansive soil is compressed, and its microstructure is disturbed and changed. Thus, the test value of the expansion force and the actual value often exhibit some differences. In the gradually balancing pressure method, loads should be applied continuously on the soil sample to return the pointer of the dial indicator to its initial position, i.e., during soaking and expanding in the consolidation apparatus. The test processes are complicated, and the values of the applied loads cannot be predicted. Moreover, complete saturation of the soil sample cannot be controlled under this method.

In order to consider the equivalent expansive force according to the requirements of physical significance, the microstructure of soil mass should be maintained in its original state during the test, and the corresponding measurements should be simple, accurate, and highly efficient. Based on these requirements, a new device that can control the saturation and constant volume in the test for expansive force was developed in the present work. The device had received a patent for invention [31] and could be adopted in research on the equivalent expansive force from a soaking state to complete saturation under a constant volume.

The structure of the testing device is presented in Figure 1. The rigid cutting ring system mainly consisted of a screw, a beam, a plunger, a porous disc, a cutting ring, and a cutting pedestal. The porous disc, cutting ring, and cutting pedestal were rigid. The positions of the beam and the cutting pedestal were fixed. The top cutting ring secured the position of the porous disc by fastening a screw to form a rigid cutting ring system with a fixed position. The main function of this system was to maintain the constant volume of the soil sample. Simultaneously, the screw, beam, plunger, and porous disc constituted the preloading system. The cutting ring and its surrounding parts as well as the rubber mat constituted the sealing system of the device. The vacuum saturation system was composed of the vacuum pump and the connecting pipe. Its main function was to control the saturation state of the soil sample in the cutting ring through the exhaust process of the vacuum pump. The measurement acquisition system consisted of an earth pressure sensor and an externally connected digital display. The former could directly measure the pressure on the earth pressure cell, and the data could be directly read by an externally connected digital display. The actual expansive force of the expanding soil could be calculated.

This study simulated the instability of filling an embankment with an expansive soil in the G316 National Highway. The G316 National Highway in Shaanxi Province was 603 km long. The length of expansive soil along this highway was approximately 200 km, whereas that in Hanyin County of Ankang was 64 km. Most of soil in the Hanjiang River basin and Ankang basin was basically typical expansive soil with a slope. The expansive soil sample for the test was collected from Hanyin County. Hanyin County belongs to the northern subtropical monsoon humid climate zone. The annual average temperature is 15.1 °C, the frost-free period is 258 days, precipitation is 782 mm, and the climate is mild and humid. For the equivalent expansive force test experiments under different initial water contents by the developed testing device, the change curves of the expansive force with time from a soaking state to a strictly saturated state are illustrated in Figure 2.

Further analysis of Figure 2, based on the subsequent experiment analyses on the unsaturated expansive soil, the relation between the equivalent expansive force, p_s, and the initial water content, w, was found to follow a power function,

$$p_s=A{w}^{\lambda }\hspace{0.17em}$$

(9)

where A and λ were the fitting parameters according to the experimental data of the expansive soil.

The testing device had several advantages, including its simple structure, convenient operation, and highly efficient measurement process. It could solve problems on large disturbances in soil samples and the uncontrollable saturation in the direct expansion-back pressure, preloading expansion, and gradually balancing pressure methods.

4. Algorithm Implementation of Equivalent Expansion Force in Strength Reduction FEM

Effective strength parameters of the cohesion (c′) and angle of friction (φ′) are set to be the variables in the Mohr–Coulomb criterion in the strength reduction FEM. The reduced strength parameters ${c^{\prime }}_F$ and ${{\phi }^{\prime }}_F$ can be defined as

$${c^{\prime }}_F=c^{\prime }/F$$

(10)

$${{\phi }^{\prime }}_F={\tan }^{-1}\left(\tan {\phi }^{\prime }/F\right)$$

(11)

The reduced strength parameters, respectively, replaced the strength parameters of c′ and φ′ of Mohr–Coulomb criterion. The strength reduction FEM was used to calculate the stress and strain of soil. First, the initial reduction coefficient, F, was set to be small so that the soil was in an elastic state. Then, the value of F was gradually increased until the destruction of the slope occurred, which showed that the strength reduction FEM calculation diverged under a physically real convergence criterion. The safety factor at failure was the value of F of the previous step.

In the strength reduction FEM, the shape of the sliding surface was not necessary to be assumed in this computation, so the potential sliding surface and the corresponding safety factor could be acquired by the calculation and its post-treatment. The strength reduction FEM was gradually becoming an indispensable tool in the stability analysis of slope. The details of the application of this method in the stability analysis of unsaturated expansive soil slopes is elaborated in this study based on the equivalent expansive force and Formula (3).

In fact, because of the uneven distribution of the matric suction, the corresponding distribution of the equivalent expansive force was also non-uniform, and according to Formula (3), the unsaturated strength of the expansive soil should also be non-uniform in matching with the distribution of the equivalent expansive force. By this token, the new non-uniform characteristic of effective stress could be produced by the non-uniform distribution of the equivalent expansive force whether the soil itself was homogeneous or not.

According to Formula (3), it could be found that the strength parameters of c′ and φ′ were not changed, and the Mohr–Coulomb criterion was employed as the strength failure surface in effective stress space. Because the matric suction was the effective stress, the equivalent expansive force that was the equivalent effect of the matric suction also had its significant contribution to the effective stress. In strength reduction FEM calculation, the macro effect of the equivalent expansive force could be obtained from the product of m and p_s in every element, and then a new effective stress state could be formed when the macro effect mp_s of the equivalent expansive force was added to the gravitational effective stress in every element, which included the effects of the equivalent expansive force, as shown in Figure 3 within the three-dimensional effective stress space and Figure 4 within the meridian plane. Seen from that in the effective stress space, the stress state of the point A represented the initial or gravitational stress state in a saturated state, and mp_s represented the contribution of the equivalent expansive force to the increment of the effective stress. The stress state of the point B was a new effective stress state that integrated the gravitational effective stress with the mp_s in the unsaturated soil. The algorithm principle of the computational method on the stability of unsaturated expansive soil may indirectly reflect the contribution of matric suction using the corresponding equivalent expansive force, in which the matric suction is equivalently substituted by the equivalent expansive force.

In an analysis of strength reduction FEM, each element in the slope was assigned a value of the equivalent expansive force, so as to simulate its non-uniform distribution within the slope engineering. For the sake of effectively performing the implementation of the analysis on an unsaturated expansive soil slope in the strength reduction FEM, two distinctively different cases should be considered and treated separately:

(1) For those soil elements without the equivalent expansive force, as with the traditional strength reduction FEM, the effective stress strength parameters (c′ and φ′) were selected as the variables in the Mohr–Coulomb criterion.

(2) For those soil elements in which the equivalent expansive force existed, a new effective stress state could be formed when the mp_s was added to the gravitational effective stress in each element, while the effective stress parameters (c′ and φ′) were still set to be the variables in the Mohr–Coulomb criterion in the computation of strength reduction FEM.

Since the displacement in the strength reduction FEM calculation was only of relative significance, the displacement increment field was obtained by the difference between the two displacement fields and the adjacent value, F. It was assumed that the position of the potential sliding surface was determined according to the position of the maximum gradient of the displacement increment contour. In this program, it was one of the prominent advantages of this method that the position of a potential sliding surface could be determined by the contour density of the displacement increment.

According to the above discussion, through the algorithm implementation, which the equivalent expansive force introduced into the strength reduction FEM, the corresponding FORTRAN codes were developed and implemented for the stability analysis of unsaturated expansive soil slopes.

5. Results and Discussion

The expansive soil sample for the test was collected from Hanyin County of Shaanxi Province. The soil was brown, and its colour was deeper when it was under humid conditions and lighter when it was dry. The mineral contents of the expansive soil were identified via X-ray diffraction. The contents were mainly montmorillonite, quartz, and illite, including 51% montmorillonite and 31% quartz (Figure 5). The test employed a liquid–plastic limit combined device to measure the liquid limit (w_L = 38.2), plastic limit (w_P = 14.7), and shrinkage limit (w_S = 12.3), respectively, and to calculate the plasticity index (I_P = 23.5). The expansive soil had a high liquid limit and plasticity index. Based on the laboratory routine test measurement, the natural water content (w) of the expansive soil was 17.90%, soil natural density (ρ) was 1.94 g/cm³, and dry density (ρ_d) was 1.65 g/cm³.

According to the measured expansive force under different initial water contents in Figure 2, the equivalent expansion force was obtained by finding and determining the maximum expansion force, and the relation curve between the equivalent expansive force (p_s)and initial water content (w) is illustrated in Figure 6.

As seen in Figure 6, the equivalent expansion force of the expansive soil sample decreased with the increase in the initial water content at the same dry density, which is consistent with the conclusions of some studies [14,15]. According to a theoretical deduction, if the initial water content was sufficiently large, then its expansive force would be zero. When the equivalent expansive force and initial water content data were matched, it was found that a certain power function relationship existed between the equivalent expansive force and initial water content, as shown in Formula (12). The equivalent expansive force (p_s) can be described as follows:

$$\hspace{0.17em}{p}_s=A_1{w}^{{\lambda }_1}$$

(12)

where A₁ and λ₁ are the fitting parameters. In this study, these parameters of the expansive soil could be determined as: A₁ = 108492 and λ₁ = −2.8339.

The effect parameters of the equivalent expansive force were determined by the slow direct shear tests on unsaturated and saturated expansive soil (

Table 1. The parameters of m.

Parameter	Water Content of 17.9%	Water Content of 23.48%	Saturated Soil
c/kPa	31.9	22.8	13.1
φ/(°)	16.1	15.4	15.0
ps/kPa	31	16	—
m	2.1	2.2	—
Mean value of m	2.15

) according to Formula (13),

$$m=\frac{c_T-c^{\prime }}{p_s\tan \phi }$$

(13)

By contrast, the effective cohesion force, c′, and the effective internal friction angle, φ′, were measured by the direct shear test under a completely saturated state of the expansive soil sample. Meanwhile, the total cohesion force, c_T, and the equivalent expansive force, p_s, were measured by the direct shear and expansive force tests under an unsaturated state of the expansive soil sample.

An effective segment of the soil and water characteristic curve of the expansive soil under humid conditions is shown in Figure 7. Therefore, a section of the fitting curve between the water content and matric suction is shown as follows:

$$S=A_2{w}^{{\lambda }_2}$$

(14)

where A₂ and λ₂ are the fitting parameters. In this study, these parameters of expansive soil could be determined as: A₂ = 8 × 10¹⁴ and λ₂ = −10.644.

Comparing Figure 6 and Figure 7, the expansive force could comprehensively reflect the macro contributions of unsaturated matric suction. In the calculation of the unsaturated expansive soil slope, the range of matric suction in the soil and water characteristic curve is generally considered for parts below 120 kPa, which affects the macroscopic effect of expansive soil slopes. Therefore, the equivalent expansive force and matric suction can reflect the principal mechanical properties of unsaturated expansive soil. In previous research, the equivalent expansive force was introduced into the strength reduction FEM. Moreover, this research quantitatively demonstrated the relevance and effect of the expansive force and matric suction to maintain the stability of an actual unsaturated expansive soil slope.

The G316 National Highway in Shaanxi Province was 603 km long. The length of expansive soil along this highway was approximately 200 km, whereas that in Ankang Hanyin was 64 km. Most of soil in the Hanjiang River basin and Ankang basin was basically typical expansive soil with a slope. The following landslide example simulated the instability of filling an embankment with an expansive soil along the G316 National Highway. The slope consisted of two parts: the lower part of the slope was 10.0 m high and had a gradient of 1.0:1.5, whereas the upper part was 3.0 m high and had a gradient of 1:10 (Figure 8). The foundation depth was 13 m, and the underground water table was located at a depth of 7.0 m.

The corresponding finite element meshes are seen in Figure 9. The elements considered were 4-node quadrilateral elements. There were 1157 elements and 1236 nodes. The boundary conditions of the displacement were as follows. (1) The bottom was a fixed boundary. (2) The normal displacement of the left-to-right lateral boundary in the horizontal direction was set to zero. The constitution of the slope soil materials adopted the Mohr–Coulomb elastic–plastic strength criterion.

Table 2. Parameters of the soil.

Soil Parameter	Soil Above the Water Level	Soil Below the Water Level
γ/(kN/m3)	21.30	19.40
E/MPa	20.38	18.40
ν	0.30	0.35
c′/kPa	13.10	13.10
φ′/(º)	15.00	15.00
φb/(º)	10.60	15.00
χ	0.70	1.00
m	2.15	-
A1	108492	-
λ1	−2.8339	-
A2	8 × 1014	-
λ2	−10.644	-

provides the mechanical parameters of the soil measured during the test. m was the effective coefficient of the equivalent expansive force, χ was the parameter of suction, and φ^b was the internal friction angle of suction.

The calculation of Case 1 applied the non-uniform distribution of matric suction and the equivalent expansive force. Meanwhile, the non-uniform distribution scheme of matric suction was used for comparative analysis. To compare the differences and characteristics of the calculation results, Case 1-1 and Case 1-2 adopted the computing method of the strength reduction FEM, respectively, based on Bishop’s unsaturated strength formula and Fredlund’s unsaturated strength formula [28], while Case 1-3 adopted the new method of the strength reduction FEM based on the equivalent expansive force in this paper, as shown in

Table 3. Description of cases and safety factors.

Case	Matric Suction	Equivalent Expansive Force	Calculation Method	Safety Factor Fs
Case 1-1	Non-uniform distribution	-	Strength Reduction FEM based on Bishop’s unsaturated strength	1.64
Case 1-2	Non-uniform distribution	-	Strength Reduction FEM based on Fredlund’s unsaturated strength	1.63
Case 1-3	-	Non-uniform distribution	Strength Reduction FEM based on the equivalent expansive force	1.60
Case 2	Suction loss	-	Strength Reduction FEM	1.08

. The stability of the expansive soil slope after rainfall was simulated and analyzed using the matric suction loss in case 2.

The distribution of the actual initial water content was obtained from the actual expansive soil slope (Figure 10). According to corresponding relationship of the equivalent expansive force curve as well as the characteristic curve of soil and water in unsaturated expansive soil in Figure 6 and Figure 7, the distributions of matric suction and the equivalent expansive force were provided, respectively (Figure 11 and Figure 12).

For the non-uniform distribution of matric suction or the equivalent expansive force, Figure 13a shows the contours of the displacement increment for Case 1-1. The safety factor (F_s = 1.64) was calculated by the strength reduction FEM based on Bishop’s unsaturated strength formula. The displacement increment contours for Case 1-2 are shown in Figure 13b. The safety factor (F_s = 1.63) was calculated by the strength reduction FEM based on Fredlund’s unsaturated strength formula. The contours of the displacement increment for Case 1-3 are illustrated in Figure 13c. The safety factor (F_s = 1.60) was calculated by the strength reduction FEM based on the equivalent expansive force.

On one hand, the slope safety factors based on three kinds of unsaturated soil strength were almost identical when the calculation results and the contours of the displacement increment in Cases 1-1, 1-2, and 1-3 were compared. On the other hand, the smaller the water content of the slope is, the larger the matric suction or the equivalent expansive force is, which leads to a greater strength of the expansive soil, placing the slope in a stable state. The safety factors were 1.64, 1.63, and 1.60 in Cases 1-1, 1-2, and 1-3, respectively.

The calculation results indicated that the strength reduction FEM based on the equivalent expansive force was successful. Therefore, the calculation results completely demonstrated the equivalence between the equivalent expansive force and matric suction in evaluating the stability of the unsaturated expansive soil slope.

After matric suction or the equivalent expansive force was lost in rainfall, the contours of the displacement increment and the potential sliding surface in Case 2 are shown in Figure 14. The safety factor was 1.08, so the slope stability was almost in a critical sliding state, and the sliding surface passed through the toe of the slope. Compared with Cases 1-1, 1-2, and 1-3, the calculated safety factor was reduced significantly. The shallow sliding mass near the slope surface is usually unsaturated. With increasing water content during rainfall conditions, its volume will change significantly, and the matric suction will decrease. As reported in the literature [8], the variation in the safety factor of unsaturated expansive soil slopes over time is studied by taking account of the shear strength reduction related to the loss of matric suction.

In summary, by means of introducing the equivalent expansive force of expansive soil into the strength reduction FEM, a calculation method was yielded for the stability of unsaturated expansive soil slopes based on the equivalent expansive force and can be used to equivalently evaluate the rainfall-induced instability of expansive soil slopes caused by the rainfall infiltration instead of using matric suction.

6. Conclusions

For three kinds of unsaturated soil strength, through the deductive analysis of the relation between matric suction and expansion force, it can be seen that there is a certain internal equivalent relation between them in the strength of unsaturated expansive soil, and this internal equivalent relation is the theoretical basis of the core algorithm analysis for the stability analysis of expansive soil slopes.

The testing device for the equivalent expansive force, which can achieve a completely saturated state under a constant volume, was developed for unsaturated expansive soil. The equivalent expansive force and matric suction tests for expansive soil were investigated under different water contents. The correlations between the equivalent expansive force and matric suction with the change in initial water content were observed, and the equivalent expansive force was found to reflect the macroscopic effects of matric suction.

Through the algorithm implementation, the equivalent expansive force was introduced into the strength reduction FEM, and the algorithm principle of the computational method on the stability of unsaturated expansive soil may indirectly reflect the contribution of matric suction using the corresponding equivalent expansive force, in which the matric suction was equivalently substituted by the equivalent expansive force.

Thus, the stability of a practical unsaturated expansive soil slope was evaluated and the equivalence between the equivalent expansive force and matric suction was demonstrated in evaluating the stability of an unsaturated expansive soil slope under the condition of non-uniform distribution of matric suction or equivalent expansive force.

According to the measurement of expansive soil expansive force by the new test device, the described method of suction-equivalent expansive force was established. Aiming at expansive soil slope engineering, the strength reduction FEM based on the equivalent expansive force, which acts as a new stability analysis method of unsaturated expansive soil slopes, can be used to evaluate the rainfall-induced instability of expansive soil slopes caused by the decrease in matric suction resulting from the rainfall infiltration, and may widen the scope and thought for the stability analysis of unsaturated slopes. Based on the established new evaluation method of the disaster-causing mechanism of expansive soil slopes with suction-equivalent expansion force, it provides certain technical support for changing slope-to-terrace construction in southern Shaanxi, so as to improve and perfect the construction technology of slope terraces in this region.