J. Cent. South Univ. (2019) 26: 1769-1779

DOI: https://doi.org/10.1007/s11771-019-4132-5

Energy analysis of geosynthetic-reinforced slope in unsaturated soils subjected to steady flow

XU Jing-shu(许敬叔), DU Xiu-li(杜修力)

College of Architecture and Civil Engineering, Beijing University of Technology, Beijing 100124, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: Soils are actually unsaturated in nature. In the present study, a stability analysis of a geosynthetic-reinforced slope in unsaturated soils subjected to various steady flow conditions is conducted based on limit analysis. Work rate by apparent cohesion due to matric suction is calculated based on the effective stress-based equation. Analytical expression of the required cohesion/stability number of slope is derived from the energy balance equation. An optimization code is programmed to capture the optimized solution of the stability number. Comparison is made to verify the present work and a parametric analysis is conducted to investigate the effects of soil type, infilitration rate, reinforcement strength and soil suction on slope stability afterwards. A set of numerical solutions is presented at the end of the paper for preliminary design purposes.

Key words: unsaturated soils; geosynthetic-reinforced slope; stability analysis; steady flow

Cite this article as: XU Jing-shu, DU Xiu-li. Energy analysis of geosynthetic-reinforced slope in unsaturated soils subjected to steady flow [J]. Journal of Central South University, 2019, 26(7): 1769-1779. DOI: https://doi.org/ 10.1007/s11771-019-4132-5.

1 Introduction

Soils are assumed to be dry or saturated in most slope stability analyses. However, due to rainfall and many other reasons, soils are actually unsaturated, leading to a distinct cohesion strength and consequently quite a different stability condition of slopes in unsaturated soils from stability of slopes in dry/saturated soils [1-4]. As an effective and money-saving measure, geosynthetic has been frequently used to conserve soil and to improve stability of slopes [5-7]. It is important for geotechnical engineers to be aware of the stability of geosynthetic-reinforced slopes in unsaturated soils.

Many efforts have been devoted to investigating both the stability of unsauturated slopes and geosynthetic-reinforced slopes. With regard to the investigations on stability of slopes in unsaturated soils, a set of linear and nonlinear equations has been proposed by scholars to estimate the shear strength of unsaturated soils [8-12], and to estimate slope stability. One of the most representative equations is the extended Mohr-Coulomb formula presented by FREDLUND et al [8], which states that the shear strength of unsaturated soil increases linearly with matric suction at the rate of tanφ^b (φ^b=15° commonly). Based on the equations proposed by scholars, ZHANG et al [13] conducted a stability analysis of a slope in unsaturated soils based on the numerical limit equilibrium method, the effects of matric suction and water table on slope stability were analyzed. However, more recently, it was found that the relationship between shear strength and matric suction was nonlinear,and a effective stress-based equation was proposed by LU et al [14] to take into account the contribution of matric suction on shear strength of unsaturated soils. Based on the equation proposed by LU et al [14], a limit-equilibrium analysis of an unsaturated slope was performed by VAHEDIFARD et al [15], the impacts of suction stress, soil types and infiltration rate on slope stability was investigated. LI et al [16] conducted a stability analysis of an unsaturated slope using limit analysis, the effects of 3D character of slope, soil type and infiltration rate on slope stability were analyzed.

With regard to investigations on stability of geosynthetic-reinforced slopes, MICHALOWSKI [5] performed a stability analysis of a geosynthetic- reinforced slope in purely frictional soils based on limit analysis, a set of design charts concerning the required reinforcement strength was proposed for preliminary design purposes. GAO et al [6] extended the work by Michalowski to three dimension, and calculated both the required reinforcement strength and length of slope. XU et al [17] performed a stability analysis of a reinforced slope in nonhomogeneous soils, and analyzed the effects of reinforcement distribution patterns and strength, 3D character of slope and cohesion nonhomogeneity on slope stability. A stability analysis of geosynthetic-reinforced earth structures subjected to seismic excitation was performed by VAHEDIFARD et al [18] to investigate the impacts of soil cohesion on slope stability. Formulas with closed-form solutions to estimate the seismic active earth pressure coefficient for c-φ soils were derived, and a parametric study was performed to explore the effects of factors such as soil cohesion and tension crack on slope stability.

To the authors’ best knowledge, no research on stability of a geosynthetic-reinforced slope in different unsaturated soils subjected to various steady flow conditions has been conducted using the limit analysis method before. Therefore, in the present study, based on the kinematical approach of limit analysis, a stability analysis of a geosynthetic- reinforced slope in four different kinds of unsaturated soils subjected to various steady flow conditions is conducted, the work rate by apparent cohesion due to matric suction is calculated and consequently the energy balance equation is built to derive the analytical expression of the required cohesion/stability number of slope. Comparison is made to verify the present study and a parametric analysis is thereafter carried out to investigate the effects of reinforcement strength, soil type and infiltration rate on slope stability. A set of numerical solutions is presented at the end of the present work for preliminary design purposes.

2 Limit analysis

Upper bound theorem of limit analysis is a valid method to investigate stability of geotechnical structures such as tunnels, foundations, and slopes [19-26]. The upper bound theorem states that the materials deform plastically according to the normality rule, and the internal energy dissipations is not less than the work rate by external forces:

             (1)

where σ^*_ij denotes the effective stress tensor associated with strain rate; denotes the kinematically admissible velocity field; T_i denotes surcharge loads; S denotes the surface of failure block; v_i denotes the velocity of the failure block; V denotes the failure block volume; and X_i denotes the distributed forces vector.

3 Apparent cohesion due to matric suction

In virtue of the suction stress characteristic curve (SSCC), the effective stress σ′ of soil can be expressed as [14]:

                           (2)

where σ denotes the total stress, u_a denotes the pore-air pressure. With regard to unsaturated soils, the suction stress σ^s can be expressed as:

             (3)

where u_w denotes the pore-water pressure and (u_a-u_w) denotes matric suction; 1/α denotes the air-entry pressure of which α varies with in the range 0.001 kPa^-1≤α≤0.5 kPa^-1, and n=1.1-8.5 is a fitting parameters related to the breadth of the soil’s pore size, namely concerning soil type. Values of α and n of soils are listed in Table 1.

Four kinds of soils, i.e., clay, silt, loess, and sand, are considered in the present work. Parameters α and n of the soils are as shown in Table 2.

Table 1 Values of α and n concerning various soils [15]

Table 2 Strength parameters concerning different kinds of soils [15]

Based on Darcy’s law and Gardner’s model [27], matric suction (u_a-u_w) of unsaturated soil subjected to steady flow can be expressed as follows [15, 28]:

         (4)

where γ_w denotes water unit weight, z denotes the vertical distance between a point in the failure mechanism and water table, and q denotes the vertical specific discharge.

Combining Eq. (3) and Eq. (4), apparent cohesion c_app of unsaturated soil can be expressed as:

 (5)

where φ is the effective internal friction angle of soil.

4 Stability of geosynthetic-reinforced slope

4.1 Failure mechanism of slope

The rotational failure mechanism of a geosynthetic-reinforced slope is shown in Figure 1. BC is the failure surface of slope, the angle of slope is denoted by β, and the height of slope and length of slope crest are H and L, respectively; the vertical distance between the ith reinforcement layer and slope crest is denoted by x. Meanings of the other symbols are as shown in Figure 1.

Figure 1 Rotational failure mechanism of geosynthetic- reinforced slope

It is known from the geometrical relationship in Figure 1 that H/r₀ and L/r₀ can be expressed as:

          (6)

                 (7)

4.2 Reinforcement patterns of slope

It has been proved by XU et al [17] that the downwardly-strong triangular distribution (DTD) pattern is the most effective choice to improve slope stability, compared with the uniformly distribution pattern and the upwardly-strong triangular distribution pattern, as shown in Figure 2. Therefore, only the DTD pattern is considered in the present work to investigate the stability of a geosynthetic- reinforced slope.

Figure 2 Reinforcement patterns of slope:

4.3 Energy balance equation

The energy balance equation of a geosynthetic-reinforced slope is built equating the external work rate by soil weight to the sum of the internal energy dissipation taken place along the failure surface BC, the energy dissipation rate by apparent cohesion and the work rate dissipated by geosynthetics. The external work rate by soil weight W_γ and the internal energy dissipation D_c taken place along the failure surface BC can be found in Ref. [29], and the work rate dissipated be geosynthetics is calculated as follows. The energy dissipation rate by apparent cohesion is:

                       (8)

where c_app is the apparent cohesion due to matric suction. And symbol ‘z’ in Eq. (5) is:

                  (9)

Reinforcement strength of the DTD pattern is:

    (10)

where k₀ is the average required strength of reinforcement and k_t-DTD is the reinforcement strength of the DTD pattern as a function of the rotation angle θ. Work rate dissipated by geosynthetics can consequently be expressed as:

              (11)

where v is the linear velocity, and r is the rotating radius corresponding to θ.

4.4 Energy balance equation

Based on the upper bound theorem of limit analysis, the energy balance equation of a geosynthetic-reinforced slope can be expressed as:

                       (12)

The analytical expression of the required cohesion c of soil can be derived by solving the energy balance equation. An optimization code is programmed to capture the optimized solution of the stability number c′/(γH) of slope (c′ is the optimized maximum cohesion of soil), following

5 Comparisons

Comparison is first made in Table 3 to verify the present study. Maximum required cohesion c′ of un-reinforced slopes in different soils are calculated under four different kinds of steady flow conditions, i.e., infiltration (q=-3.14×10^-8 m/s), evaporation (q=1.15×10^-8 m/s), the no flow condition (q=0), and the no-suction condition. The effective frictional angle φ of the soils are 20°, 25°, 28° and 30°, respectively, as listed in Table 3.

It is clear from Table 3 that the present solutions of required cohesion are in good agreement with the results by VAHEDIFARD et al [15]. Consequently, the present study can be validated.

6 Parametric analysis

A parametric analysis is conducted to investigate the effects of soil type, infiltration rate, reinforcement strength, slope geometry, and parameters α and n on slope stability.

6.1 Effects of reinforcement strength on slope stability

Effects of reinforcement strength and slope height on stability of slope in different soils subjected to various steady flow conditions are illustrated in Figure 3, the slope angle β=90°, the infiltration rate q=0 (no flow condition), the reinforcement coefficient n_r varies from 0 to 5.0, and slope height H are 1, 4, 7, 10 and 20 m.

Table 3 Comparison of required cohesion (kPa) with solutions by VAHEDIFARD et al [15] under β=90°, H=5 m, γ=20 kN/m³ and z₀=0

Figure 3 Effects of reinforcement strength coefficient n_r on stability of slope in different soils:

It is shown in Figure 3 that reinforcement strength has a significant positive impact on slope stability. The stability number decreases rapidly with the increase of n_r, especially for a slope under the no suction condition. Besides, it is clear from Figure 3 that the height of slope also has a tremendous effect on stability of slopes in silt and loess, for that the stability number increases with the increase of H. However, stability number of a slope in clay or sand barely changed by the height of slope.

6.2 Effects of slope angle on slope stability

Effects of slope angle on stability of slopes in different soils are investigated in Figure 4. In Figure 4, the slope angle β varies from 40° to 90°, the effective frictional angle φ=15°, and the reinforcement coefficients n_r are 0 (un-reinforced condition), 1.0, 3.0 and 5.0.

Figure 4 illustrated the stability number of slope under different reinforcement strength coefficient n_r versus slope angle β. It is seen that the stability number increases with the increase of slope angle and the decrease of the reinforcement strength coefficient n_r, meaning that the increase of slope angle has an unfavorable effect on slope stability while the increase of n_r has a positive effect. Besides, it is clear that when β increases from 30°to 90°, the stability number increases by 995.3% and 596.4% for a slope in clay under n_r=0 and 5.0, respectively, which means the increase of n_r will effectively reduce the extinction effect of slope angle on slope stability.

6.3 Effects of α and n on slope stability

Effects of parameters α and n on slope stability are illustrated in Figure 5. In Figure 5, the saturated hydraulic conductivity k_s=5.0×10^-8 m/s, and parameter n varies from 1.1 to 8.5. Besides, in Figure 5(a), the slope angle β=90°, slope height H=5 m, the reinforcement strength coefficient n_r=3.0, and the effective frictional angle φ=20°; in Figure 5(b), the slope angle β=60°, slope height H=10 m, the reinforcement strength coefficient n_r=3.0, and the effective frictional angle φ=15°.

Figure 4 Effects of slope angle and reinforcement strength on slope stability

Figure 5 Effects of parameters α and n on slope stability

It is seen from Figure 5 that parameters α and n both have significant impacts on slope stability. For a slope with a given n, the stability number decreases rapidly with the increase of 1/α first, then asymptotically reached a constant value. For a slope of H=5 m with a given α, within the range of 0<1/α≤45 kPa, the stability number increases with the increase of n, thereafter when 1/α>45 kPa, the stability number decreases with the increase of n. It is also clear that the greater parameter n is, the more vulnerable of the stability number curves be affected by the change of slope height H, the later the stability number gets stable along with the increase of 1/α.

6.4 Effects of slope height versus stability number on slope stability

Figure 6 is presented to investigate the effects of slope height H versus stability number on stability of a vertical geosynthetic-reinforced slope (β=90°) in different soils under various steady flow conditions with n_r=3.0.

It is clear from Figure 6 that for slopes under the no suction condition, the stability number is not related with slope height H. For slopes under infiltration, evaporation and the no-flow conditions, the stability number increase with the increase of H. Besides, stability number of a slope with a given H under infiltration, evaporation and the no-flow condition is lower than that of a slope under the no suction condition, meaning that the apparent cohesion has a favorable impact on slope stability. In addition, the effects of steady flow conditions on slope stability are different, more specifically, infiltration rate affects the stability of slopes in clay and silt significantly, but barely influence the stability of slopes in loess and sand, namely the infiltration rate should be carefully measured for slopes in clay and silt.

7 Numerical solutions

A set of numerical solutions of the stability number of slopes in clay, silt, loess, and sand subjected to various steady flow conditions is presented for design purposes, as shown in Table 4 to Table 7, respectively. In the tables, the slope angle β varies from 30° to 90°, and the reinforcement strength coefficient n_r varies from 0 (un-reinforced condition) to 5.0.

Figure 6 Slope height of slope versus stability number of slope in various soils under different steady flow conditions:

Table 4 Stability number N_s of a geosynthetic-reinforced slope in clay subjected to various steady flow

Table 5 Stability number N_s of a geosynthetic-reinforced slope in silt subjected to various steady flow conditions

Table 6 Stability number N_s of a geosynthetic-reinforced slope in loess subjected to various steady flow conditions

Table 7 Stability number N_s of a geosynthetic-reinforced slope in sand subjected to various steady flow conditions

8 Conclusions

In the present study, the kinematical approach of limit analysis is employed to investigate the stability of a geosynthetic-reinforced slope in unsaturated soils subjected to steady flow. The external work rate by apparent cohesion due to matric suction is taken into account, consequently, an optimization code is programmed to capture the required cohesion/stability number of slope derived from the energy balance equation. Comparison is made to verify the present study. Then a parametric analysis is conducted to investigate the effects of soil type, infiltration rate, reinforcement strength, slope geometry and parameters α and n on slope stability. A set of numerical solutions of stability number of slope is proposed for preliminary design purposes. Main conclusions of the present work can be drawn:

1) Reinforcement strength has a significant impact on slope stability, especially for a slope under the no suction condition. Meanwhile, increase of the reinforcement strength coefficient will reduce the extinction effect of the increase of slope angle on slope stability.

2) Parameters α and n both have tremendous impacts on slope stability. For a slope with a given n, the stability number decreases rapidly with the increase of 1/α first, then asymptotically reached a constant value. Stability number curve is more vulnerable to be affected by the change of slope height H, with the increase of n.

3) Infiltration rate affects the stability of slopes in clay and silt significantly, but barely influence the stability of slopes in loess and sand. For a slope under the no suction condition, the stability number is not related with slope height, however, for slopes under infiltration, evaporation and the no-flow conditions, the stability number increases with the increase of slope height.

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(Edited by HE Yun-bin)

中文导读

稳定渗流作用下加筋土加固非饱和土边坡能量分析

摘要：实际状态下的土体一般是非饱和的。基于极限分析理论，本文开展了稳定渗流作用下加筋土加固非饱和土边坡的稳定性分析。借助有效应力法对基质吸力引起的表观黏聚力功率进行了计算，从边坡稳定的能量平衡方程中推导出了维持边坡稳定的所需黏聚力和稳定性系数，并编制了最优化程序来捕捉稳定性系数最优解。文章进行对比分析以验证本文所采用方法的有效性，其后采用参数响应分析的手段以探讨土体类型、渗流速率、加固强度以及土体基质吸力对边坡稳定性的影响。为方便初步设计使用，文章最后给出了一系列的稳定性系数最优解。

关键词：非饱和土；加筋土边坡；稳定性分析；稳定渗流

Foundation item: Project(2019M650011) supported by China Postdoctoral Science Foundation; Project(51421005) supported by the Science Fund for Creative Research Groups of the National Natural Science Foundation of China; Project(2015CB057902) supported by the National Basic Research Program of China

Received date: 2018-12-27; Accepted date: 2019-03-15

Corresponding author: DU Xiu-li, PhD, Professor; Tel: +86-13801139161; E-mail: duxiuli2015@163.com; ORCID: 0000-0003-2523- 3575