J. Cent. South Univ. Technol. (2009) 16: 0292-0296

DOI: 10.1007/s11771-009-0050-2

Slope stability analysis considering joined influences of nonlinearity and dilation

YANG Xiao-li(杨小礼), HUANG Fu(黄 阜)

 (School of Civil and Architectural Engineering, Central South University, Changsha 410075, China)

Abstract: The soil masses of slopes were assumed to follow a nonlinear failure criterion and a nonassociated flow rule. The stability factors of slopes were calculated using vertical slice method based on limit analysis. The potential sliding mass was divided into a series of vertical slices as well as the traditional slice technique. Equating the external work rate to the internal energy dissipation, the optimum solutions to stability factors were determined by the nonlinear programming algorithm. From the numerical results, it is found that the present solutions agree well with previous results when the nonlinear criterion reduces to the linear criterion, and the nonassociated flow rule reduces to the associated flow rule. The stability factors decrease by 39.7% with nonlinear parameter varying from 1.0 to 3.0. Dilation and nonlinearity have significant effects on the slope stability factors.

Key words: nonlinear failure criterion; nonassociated flow rule; soil slope

1 Introduction

Though the development of numerical techniques such as finite element method, finite difference method and boundary element method, the traditional slice method based on limit equilibrium is still used for stability analysis of soil slopes. This is due to the fact that the slice method based on limit equilibrium is very simple, and that it is widely accepted by practical engineers. By dividing the soil mass into slices and making some assumptions regarding the interslice forces to satisfy the equilibrium conditions, the safety factor or stability factor can be easily obtained for soil slopes. However, none of the limit equilibrium methods can be considered rigorous due to the arbitrary static assumptions regarding the interslice forces, and no proof is given that a statically admissible stress field is satisfied within the slices, or that the assumed failure mechanism is kinematically admissible.

To avoid the shortcomings of slice method based on limit equilibrium, the slice method based on limit analysis is applied to slope stability problems. Using the limit analysis method, many efforts have been carried out to analyze the stability on the basis of a linear failure criterion and an associated flow rule. However, experiments have shown that the strength envelopes of geomaterials have the nature of nonlinearity [1]. The friction angle in most soils, particularly in granular soils, decreases with increasing confining pressures. The Mohr envelope is curved. Meanwhile, the adoption of associated flow rule, on which the limit analysis method is established, results in an over prediction of soil dilation. Consequently, a question, which frequently arises, is how to determine the slope stability using a nonlinear failure criterion in conjunction with nonassociated flow rule.

As for the effect of nonlinearity of failure criterion on the stability problems, BAKER and FRYDMAN [2] applied the variational calculus approach to formulating the governing equations for bearing capacity of a strip footing on the upper surface of a flat-sided slope. ZHANG and CHEN [3] converted the complex differential equations to an initial value problem, and presented an effective numerical procedure, called the inverse method, for solving the slope stability problems with a nonlinear failure criterion. DRESCHER and CHRISTOPOULOS [4] proposed a tangential technique to calculate the stability factors of an infinite, homogeneous, free-of-surcharge soil slope with a nonlinear failure criterion based on the upper bound theorem. They showed that upper bound limit analysis solutions could be obtained by means of a series of linear failure surfaces that are tangent to and exceed the actual nonlinear failure surface, together with utilizing the previously calculated linear stability factors given by CHEN [5]. Generalized tangential technique was proposed and extended in Refs.[6-10]. That is, instead of the actual nonlinear failure criterion, a tangential line   is used to derive the objective function, which is to be Minimized and corresponds to the dissipated power. This technique, however, does not need to utilize the previously calculated factors given by CHEN [5]. In Refs.[5-10] the effects of dilation were not considered.

In this work, the slice method based on limit analysis was employed to evaluate the slope stability with the nonlinear failure criterion and the nonassociated flow rule. The numerical results were presented. The effects of nonlinear parameter in the failure criterion and dilation angle in the nonassociated flow rule were investigated.

2 Nonlinear failure criterion

In many practical problems, a substantial amount of experimental evidence suggests that the Mohr envelopes of many geomaterials are not linear in the σ_n-τ stress space, where  σ_n and τ are the normal and shear stresses on the failure surface respectively. This departure from linearity is significant for stability calculation [6-10]. In general, a nonlinear failure criterion for the nonlinear envelope can be expressed as

τ=c₀(1+σ_n/σ_t)^1/m                               (1)

where the values of c₀, σ_t and m are determined by tests. When m=1, Eqn.(1) reduces to the well-known linear Mohr-Coulomb failure criterion. A mobilized internal friction angle φ_t as an intermediate variable is introduced as tan φ_t=dτ/dσ_n. Using Eqn.(1), the normal stress σ_n and shear stress τ of the nonlinear failure criterion can be obtained [3-4, 7-8]. The tangential line to the curve at the location of tangency point can be expressed as τ=c_t+σ_n tan φ_t, where c_t is the intercept of the tangential line on the τ-axis. c_t is expressed as

         (2)

In order to ensure that the tangential line always lies outside of the curve, and that the strength corresponding to the tangential line is more than or equal to that of the corresponding nonlinear curve, the requirement m≥1 is to be satisfied.

3 Upper bound analysis

The soil mass of slopes is idealized as perfectly plastic nonassociative material. The nonassociative material obeys a coaxial flow rule, which states that the axes of principal plastic strain increments agree well with the axes of principal stress during plastic flow, and that the dilation angle is introduced for a reasonable representation of the soil dilation characteristics [11-15]. Herein, a nonlinear failure criterion was incorporated into the nonassociative material so that the joined influences of nonlinearity and dilation on the slope stability could be investigated.

3.1 Nonassociated plasticity

The real deformation and failure property of soils can be better simulated using the nonassociated flow rule. Therefore, the introduction of the nonassociated flow rule in the upper bound analysis is necessary. DRESCHER and DETOURNAY [11] made a substantial contribution to determine the limit load in a translational failure mechanism for nonassociative geomaterials with a coaxiality of the principal directions stresses and deformation rates. They adopted the following relationships to consider the dilation:

                       (3)

                        (4)

where  ψ is a dilation angle varying from zero to the internal friction angle φ_t(0≤φ≤φ_t), and φ_t=ψ means that the soil follows the associated flow rule; c^* and φ^* are the modified cohesion and friction angle for the upper bound analysis, respectively. The dilative parameter n, which relates the dilation angle to the soil friction angle φ_t, is defined as n=ψ/φ_t (0≤n≤1). Eqns.(3) and (4) are valid for a situation where geomaterials are assumed to follow the tangential line failure criterion. Herein, the vertical slice method based on limit analysis was employed to evaluate the stability of slopes under the condition of plane strain. The joined effects of the nonlinear failure criterion and the nonassociated flow rule were investigated.

3.2 Compatible velocity

A kinematically admissible velocity field should satisfy the compatibility, the flow rule and velocity boundary conditions. The solution obtained from upper bound theorem is not less than the actual solution by a kinematically admissible velocity field. To obtain better solution, work should be done as many trial kinematically admissible velocity fields as possible. An admissible translational failure mechanism is used in this work. The plastic zone above the assumed slip surface is divided into a number of vertical slices as usual. Each of vertical slices moves as a block. The plastic energy dissipation only occurs at the interfaces of the two adjoining vertical slices and the bases of vertical slices. The failure mechanism is composed of k vertical slices. The geometry of the ith vertical slice is characterized by the length of the base d_i, the angles α_i with respect to horizontal line (i=1, ???, k), and the length of interface   L_{i, i+1} between the ith vertical slice and the (i+1)th (i=1, ???, k-1).

The reasonable velocity direction at the base of the vertical slice is shown in Fig.1. So, the right side vertical slice, i.e. the ith vertical slice, moves with velocity v_i, which inclines at φ^* with respect to the base. Similarly, the left side vertical slice, i.e. the (i+1)th vertical slice, moves with velocity v_i+1 at φ^*. In general, the direction of the relative velocity v_{i, i+1} of the ith vertical slice with respect to the (i+1)th vertical slice along the interface is upward, which inclines at φ^* with respect to the interface [15]. In order to ensure the velocities assigned to the translational failure mechanism to be kinematically admissible, the two adjoining vertical slices must not move to cause overlapping or separation. The velocity and relative velocity of the ith vertical slice with respect to the (i+1)th vertical slice along the interface are given as follows [15]:

                       (5)

                      (6)

In general, the absolute velocity of the first vertical slice is assumed to be unit. So, the velocities and relative velocities of all vertical slices can be calculated according to Eqns.(5) and (6) repeatedly.

Fig.1 Relative velocity between ith and (i+1)th vertical slices

3.3 Energy dissipation

With the velocities for the translational failure mechanism, the work rate done by the external load and the internal energy dissipation rates can be calculated. Since the soil mass is regarded as perfectly rigid and no general plastic deformation is permitted to occur, the internal energy is dissipated only along velocity discontinuity surface d_i (i=1, ???, k) between the material at rest and the material in motion, and along the relative velocity discontinuity interface L_{i, i+1} (i=1, ???, k-1) between two adjoining vertical slices. The two parts are calculated as follows:

 (7)

While the external work rate P_ext is done by external load such as soil self-weight, which is expressed as

                   (8)

where  G_i denotes the weight of the ith vertical slice. Equating the work rate of external load to the total internal energy dissipation rates P_int=P_ext the objective function can be obtained. By the upper bound theorem of limit analysis, the critical height H_c can be obtained by minimization of these coefficients with respect to the mechanism parameters and the location of tangential point. The stability factor N_s is defined as N_s=γH_c/c₀ [3] for slope problems with the nonlinear failure criterion. Thus, N_s is a dimensionless number, and the magnitude of N_s depends on not only the parameters of the nonlinear failure criterion including m, σ_t and c₀, but also the dilative parameter n.

4 Numerical results

The problem considered here is the slope stability with the nonlinear failure criterion and the nonassociated flow rule. The numerical results to this problem are obtained. Using the generalized tangential technique, the least upper bound for the stability factor is obtained by minimizing the objective function with respect to failure mechanism’s parameter α_i and tangential location parameter φ_t. The upper bound solution can be improved by increasing the number of vertical slices, when the influences of the nonlinearity and dilation are considered.

Numerical results are summarized in following analysis. Example problems including the effects of linear failure criterion and associated flow rule (comparisons are made for stability factors of slopes), and the effects of nonlinear failure criterion and nonassociated flow rule on the slope stability factors, are studied.

4.1 Comparison

Considering the influences of the linear failure criterion and the associated flow rule, the present stability factors using the vertical slice method based on limit analysis were calculated. From the numerical results, it is found that the present stability factors agree well with the previously published solutions when nonlinear criterion reduces to linear criterion (m=1) and nonassociated flow rule reduces to associated flow rule (n=1), according to Figs.2 and 3. As a result, the stability factors calculated by the vertical slice method based on limit analysis are effective.

4.2 Effects of nonlinearity

To investigate how the slope stability factor is influenced by the nonlinear failure criterion with the nonassociated flow rule, Fig.2 illustrates the effects of the nonlinear parameter m on the stability factors at the dilative prameter n=0.6, slope angle α=0?, slope angle β=75?, c₀=90 kPa and σ_t=247.3 kPa. The magnitudes of c₀ and σ_t are equal to those in Ref.[3]. It can be seen from Fig.2(a) that the nonlinear parameter m has a significant influence on the stability factors of soil slopes. With the increase of nonlinear parameter m, the stability factor decreases. The same phenomenon can also be found in Fig.2(b) corresponding to α=5?, β=75?, c₀=90 kPa and σ_t=247.3 kPa, with the dilative parameter n being 0.4. From Fig.2, it is found that the stability factors decrease by 39.7% when m increases from 1.0 to 3.0.

Fig.2 Effects of nonlinear parameter m on stability factors at n=0.6 (a) and n=0.4 (b)

4.3 Effects of dilation

MICHALOWSKI [15] presented the stability factors of soil slopes with the associated flow rule. To investigate how the slope stability factor is influenced by the nonassociated flow rule, Fig.3(a) illustrates the effects of the dilative parameter n on the stability factors at c₀=90 kPa, σ_t=247.3 kPa, m=1.2, α=5? and β=80?. It can be seen from Fig.3(a) that the stability factor with the nonassociated flow rule is less than that with the associated flow rule, and that stability factor increases with the increase of the dilative parameter n. The same phenomenon can also be found in Fig.3(b) corresponding to α=0?, β=75?, c₀=90 kPa and σ_t=247.3 kPa, with the nonlinear parameter m being 1.8.

Fig.3 Effects of parameter n on stability factors for m=1.2 (a) and m=1.8 (b)

5 Conclusions

(1) Incorporating the nonlinear failure criterion and the nonassociated flow rule, the vertical slice method based on limit analysis is employed to estimate the stability factors of slopes. Translational, kinematically admissible failure mechanism is used to calculate the stability factors of slopes.

(2) Nonlinear failure criterion and nonassociated flow rule have important effects on the slope stability. Therefore, it is advisable to use a proper value of dilation angle and nonlinear parameters to find the corresponding values of stability factors for engineering design.

(3) The work for calculation of the slope stability using a linear Mohr-Coulomb failure criterion and associated flow rule is extended to that using the nonlinear failure criterion and nonassociated flow rule.

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Foundation item: Project (200550) supported by the Foundation for the Author of National Excellent Doctoral Dissertation of China; Project (200631878557) supported by West Traffic of Science and Technology of China

Received date: 2008-07-05; Accepted date: 2008-09-17

Corresponding author: YANG Xiao-li, Professor; Tel: +86-731-2656248; E-mail: yxnc@yahoo.com.cn

(Edited by CHEN Wei-ping)