J. Cent. South Univ. Technol. (2010) 17: 603-611

DOI: 10.1007/s11771-010-0529-x                                                                       

Influences of nonassociated flow rules on

three-dimensional seismic stability of loaded slopes

N. GANJIAN¹, F. ASKARI², O. FARZANEH1

1. Department of Civil Engineering, College of Engineering, University of Tehran, Tehran, Iran;

2. Department of Geotechnical Engineering, International Institute of Earthquake Engineering and Seismology,

Tehran, Iran

? Central South University Press and Springer-Verlag Berlin Heidelberg 2010

Abstract: The influences of soil dilatancy angle on three-dimensional (3D) seismic stability of locally-loaded slopes in nonassociated flow rule materials were investigated using a new rotational collapse mechanism and quasi-static coefficient concept. Extended Bishop method and Boussinesq theorem were employed to establish the stress distribution along the rupture surfaces that are required to obtain the rate of internal energy dissipation for the nonassociated flow rule materials in rotational collapse mechanisms. Good agreement was observed by comparing the current results with those obtained using the translational or rotational mechanisms and numerical finite difference method. The results indicate that the seismic stability of slopes reduces by decreasing the dilatancy angle for nonassociated flow rule materials. The amount of the mentioned decrease is more significant in the case of mild slopes in frictional soils. A nearly infinite slope under local loading, whether its critical failure surface is 2D or 3D, not only depends on the magnitude of the external load, but also depends on the dilatancy angle of soil and the coefficient of seismic load.

Key words: 3D slope stability; failure analysis; nonassociated flow rule

1 Introduction

A great variety of two-dimensional (2D) methods were developed for the stability analysis of slopes. Practically, all slope failures are three-dimensional (3D) in nature especially for those under local loading. The critical failure surface can be 3D depending on some parameters such as the magnitude of the external load in an infinite slope under local loading. Therefore, 3D stability analysis should be performed in most cases of slope stability problems. However, there are only a few 3D solutions in slope stability problems.

The 3D limit equilibrium methods of columns, which are analogous to the 2D methods of slices, have been the most common over the past several decades (e.g., HOVLAND [1], CHEN and CHAMEAU [2], UGAI [3], and HUNGR [4]). Limit analysis approach was developed based on the associated flow rule assumption and the principle of virtual work. Using this approach, true solutions can be bracketed (upper- and lower-bounds) by reducing the influence of the assumptions about inter-slice forces in limit equilibrium methods. Some 3D upper-bound solutions were presented assuming translational or rotational collapse mechanisms. GIGER and KRIZEK [5] used the upper-bound theorem of limit analysis to study the stability of a vertical corner cut subjected to a local load. LESHCHINSKY et al [6] presented a 3D analysis of slope stability based on a variational limiting equilibrium approach. MICHALOWSKI [7] introduced a rigorous 3D solution based on a translational collapse mechanism consisting of several rigid blocks. FARZANEH and ASKARI [8] improved the results of MICHALOWSKI [7] using several lateral surfaces for each block and extended his 3D analysis for nonhomogenous soils. BUHAN and GARNIER [9] evaluated the upper-bound bearing capacity of square foundations located near a slope using both rotational and punching collapse mechanisms.

Since most granular soils do not obey the associated flow rule, this assumption causes some overestimation of volumetric strain and consequently overestimation of the limit loads in the upper-bound methods. In this situation, DAVIS [10] proposed that consideration of the dilatancy angle could better predict the limit load, resulting in lower limit loads. Although the consideration of the nonassociated flow rule violates the basic assumption of the upper-bound theorem, applying the energy balance in an admissible velocity field enables us to study the effects of dilatancy angle in slope stability analysis. The influence of dilatancy angle on 2D stability of loaded slopes was studied by YIN et al [11], WANG et al [12], KUMAR [13], and YANG et al [14-15]. YANG et al [14] investigated the influence of nonassociated flow rules on the seismic bearing capacity of strip footings near slopes using a 2D multi-wedges translational mechanism. However, very few studies are available about the influences of nonassociated flow rule on the seismic failure in 3D stability problems.

In the present work, a 3D rotational collapse mechanism consisting of a log-spiral bottom surface and combination of several lateral surfaces is considered by taking into account the truly 3D geometry of the problem. It aims at improving the upper-bound solutions in the cases of seismic stability of slopes under surcharge loading in associated flow materials. In addition, 3D slope stability factors are computed using the proposed rotational collapse mechanism for soils with nonassociated flow rules by considering the energy balance equation. Since the rate of dissipation of internal energy for a nonassociated flow rule material depends on the normal stresses along the rotational rupture surface, the stress distribution is established on the basis of extended Bishop’s method.

2 Three-dimensional failure mechanism

The proposed rotational collapse mechanism, as an extension of 2D rotational mechanism, consists of a log-spiral cylindrical bottom surface (ABB′A′) and a combination of several lateral surfaces shown in Fig.1 as AEFD and BCFE. The equation of bottom surface is known in polar coordinates as:

                      (1)

where R₀ and θ₀ are the cylindrical coordinates of the start point of collapse on the surface, R and θ are the polar coordinates of an arbitrary point located on the bottom surface, and ψ is the dilatancy angle of soil that is equal to the internal angle of friction  according to the associated flow rule.

The lateral surfaces can be adapted by fulfilling the compatibility condition for a kinematically admissible velocity field through an equation of the following form:

z=f(r, θ)                                     (2)

where (r, θ, z) are the cylindrical coordinates of an arbitrary point P located on the lateral surface. Compatibility of strains means that at point P, the velocity vector across the surface makes an angle equal to the dilatancy angle ψ with the tangent of the slip surface, or an angle equal to π/2-ψ with the normal to the surface (Fig. 1). With this assumption, the following differential equation can be derived:

                   (3)

Here, a possible class of solutions for the above partial differential equation is examined as proposed by BUHAN and GARNIER [9]:

              (4)

where b and s are constants.

Considering a general point A(R₀, θ₀, a₀) on the lateral surface, parameter b can be obtained as a function of parameters R₀ (which is a function of the coordinates of point O), θ₀, a₀ and s. Consequently, there will be five independent geometrical variables (x₀, y₀, θ₀, a₀, s) that should be optimized to obtain the critical failure mechanism.

Fig.1 Proposed 3D collapse mechanism

BUHAN and GARNIER [9] used the rotational collapse mechanism with a lateral surface on each side to obtain bearing capacity of square foundations near the slopes. Performed analyses during the current work indicate that this collapse mechanism extends laterally in frictional soils, which results in overestimated limit loads.

To improve the results, a combination of several lateral surfaces with the same rotational velocity and center of rotation but different constant parameters (b_i and s_i) is proposed by taking into account the truly 3D geometry of the collapse mechanism. It is evident that an increase in the number of lateral surfaces leads to an increase in the number of variables that should be optimized. For each additional lateral surface, two independent variables will be added to the five existing ones. Considering an arbitrary angle θ₁ in Fig. 1, a point like E on the intersection of the bottom and first lateral surfaces will have the coordinate of (R₁, θ₁, z₁). By crossing the second lateral surface from point E (), b₂ will be known as a function of θ₁ and s₂. In other words, the independent- added variables to the problem are only θ₁ and s₂. Based on the current study, a combination of three lateral surfaces on each side (nine independent variables in  total) is sufficient to obtain the appropriate results.

Knowing the geometry of the collapse mechanism, the energy dissipation rate on discontinuity surfaces and the rate of work done by traction and body forces can be calculated. The most common definition of safety factor f_FS in geotechnical engineering is used here, which is based on the concept of the mobilized strength of the soil:

                     (5)

where c,  and ψ are the cohesion, internal angle of friction and dilatancy angle of materials, respectively; and c_d,  and ψ_d are the corresponding mobilized values.

The seismic failure mechanisms of loaded slopes are investigated here assuming quasi-static coefficient concept. In all of the current analyses, the inertia forces concern the soil mass weight and the surcharge load considering the same seismic coefficient.

Many approaches were developed to search for the critical failure mechanism in stability problems. Most procedures rely on traditional random generation techniques. The main limitation of these techniques is the uncertainty of the algorithm to find the global minimum limit load rather than the local minimum especially when the number of variable increases. Here, an alternative optimization method for determining the critical slip surface, called genetic algorithm (GA), is applied. This method is becoming increasingly popular in engineering optimization problems because it is very robust for a wide variety of problems [16].

3 Work and energy calculations

Eq.(6) mathematically expresses the energy balance equation for a collapse mechanism consisting of rigid blocks:

                (6)

where the first term (the left term) represents the rate of work dissipated by traction t_i over the velocity jumps ([v]_i is the relative velocity between two adjacent rigid blocks), and the second and the third terms respectively represent the rate of external work of traction T_i over velocity v_i at boundary S, and that of body force λ_i over velocity v_i in volume V.

Assuming the associated flow rule for a Mohr-Coulomb material means that the velocity jump becomes normal to the yield line and inclined to the velocity discontinuity at the angle of friction  as shown in Fig.2(a). Therefore, for the linear Mohr- Coulomb yield function, the internal energy dissipation is:

                             (7)

where [v] is the resultant velocity jump. Thus, the internal energy dissipation will be independent of the stresses along the rupture surfaces for associated materials.

The effect of dilatancy angle on the stability of slopes can be investigated using a kinematically admissible velocity field and balancing the rate of internal energy dissipation with the rate of work of external forces. As shown in Fig.2(b) for a nonassociated coaxial flow, [v] makes an angle of ψ with the shear plane. The normal and shear stresses on the rupture plane representing E lie on a fictitious yield line instead of the Mohr-Coulomb yield line. For this fictitious yield line we have:

                           (8)

where c* and can be related by the following expressions [10]:

                               (9)

                                    (10)

                            (11)

However, since [v] in the rupture plane is not normal to the assumed yield line, the internal energy dissipation will be obtained as:

          (12)

Fig.2 Tractions and direction of velocity jump on rupture plane for materials with associated (a) and nonassociated flow rule (b)

Hence, in general, the energy expression depends on the distribution of normal stresses along the rupture surface for materials with nonassociated flow rule.

DRESCHER and DETOURNAY [17] proved that for translational collapse mechanisms, the limit load did not depend on the orientation of the velocity jump. Therefore, a fictitious orientation and thus a fictitious flow rule ( can be used for which the specific energy dissipation is independent of σ_n:

                           (13)

However, in the case of rotational collapse mechanism, true orientation of the velocity jump should be used. Thus, determination of the stress distribution along the rupture surface is necessary according to Eq.(12).

Normal stress distribution along the rupture surfaces can be estimated using the limit equilibrium methods. Several 3D limit equilibrium approaches were presented, which essentially were extensions of the most general methods of slices. Among the proposed methods of  slices, Bishop’s simplified method [18] is shown to produce more accurate results compared to other techniques [4]. HUNGR [4] extended Bishop’s method into three dimensions without any additional assumptions that expected to exhibit as good performance as the original method.

Here, the distribution of normal stresses along the proposed slip surfaces is generated by extending the Bishop’s method into three dimensions. To apply the slice method, the proposed 3D collapse mechanism is divided into a series of soil columns, some of them are based on the lateral surfaces and the others on the bottom spiral cylinder, as shown in Fig.3(a).

Forces acting on a typical column are illustrated in Fig.3(b) neglecting the vertical inter-column shear. According to the Bishop’s assumptions, vertical shear forces acting on the lateral vertical faces of each column can be neglected in the vertical equilibrium equation. Therefore, the normal stress on the base of each column can be driven as [4]:

                    (14)

where m is the mass of column, and A is the true area of the column base derived as:

              (15)

Angle γ_x between normal force N and vertical axis x can be obtained from Eq.(16):

Fig.3 Establishment of normal stresses along slip surfaces: (a) Dividing proposed failure mechanism into columns; (b) Forces acting on typical column

        (16)

and c* and  can be related by Eqs.(9)-(11).

The described limit equilibrium method of columns does not consider the shear resistance along the vertical sides of the sliding mass [19]. In other words, the normal stresses generated by the earth pressure on the sides of the vertical columns along the vertical lateral surfaces are ignored. This will lead to an underestimation of the 3D safety factor. Since the proposed lateral surfaces of the applied rotational collapse mechanism can be degenerated into vertical planes in the case of a nonassociated soil with ψ=0, the removal of the mentioned limitation is imperative.

To include the normal stresses on vertical sides, at-rest earth pressure acting on the lateral vertical surface of the slide mass is considered as:

            (17)

where  is the average vertical stress over the depth, and k₀ is the coefficient of earth pressure at rest (k₀=   1-sin). The excess stresses on lateral surfaces due to surcharge load are also considered using the Boussinesq theorem.

4 Comparisons

4.1 Associated flow rule

To evaluate the accuracy of the current upper-bound results in the case of associated flow materials, they are compared with those of other investigations.

A few actual experimental tests were performed to assess the bearing capacity of rectangular foundations near the slopes. CANEPA and DESPRESLES [20] reported a series of full-scale experiments conducted on square foundations (B=1 m) placed at the top of a natural sandy slope having an angle of β=26.6? with the characteristics of c=0, 32.5? and γ=16 kN/m³. Table 1 presents the bearing capacities obtained using the proposed failure mechanism with different numbers of lateral surfaces compared with upper-bound results obtained by BUHAN and GARNIER [9] and the experimental results reported by CANEPA and DESPRESLES [20]. This comparison is conducted for three different values of distances between the edge of the foundation and the crest of the slope (a=B/2, B, ∞, where B is the loading width).

This comparison shows that the kinematic results overestimate the experimental results as long as the foundation remains distant from the slope. On the other hand, by increasing the number of lateral surfaces to three, the results will improve up to 15%. It can also be seen that the application of the proposed mechanism yields better (less) kinematic results that are closer to the experimental results. From this point of view, the current results are better than those of Ref.[9].

Table 1 Comparison of experimental and theoretical bearing capacities with those of proposed collapse mechanisms

WEI et al [21] studied the 3D stability and failure mode of locally-loaded slopes based on the strength reduction method (SRM) using a numerical finite difference code (Flac3D). Here, to compare the current results with those of numerical methods, a locally-loaded slope is considered. The slope geometry, dimensions of the rectangular area of the vertical loading and the soil properties are shown in Fig.4. Results of the analysis are compared in Table 2 at a loading q of 100 kPa and the ratio of the loading length L to loading width B between 0 and 10. The results are nearly the same as shown in this table though the current results are slightly less (better) for the slopes under rectangular foundations.

Fig.4 Geometry of slope under local loading used for verification of results at c=20 kPa, 20?, and γ=20 kN/m³

Table 2 Comparison of safety factors for locally-loaded slopes at loading of 100 kPa

4.2 Nonassociated flow rule

Considering different values of ψ for a nonassociated flow rule material, the values of 3D safety factors (f_3D) are obtained. Fig.5 shows the comparison of current results with those of the numerical finite difference code (Itasca, Flac3D [22]) and the results of a translational mechanism (TRASS^* [8]) using fictitious parameters c^* and  for a slope with β=45?, 18, d/H=1 and 32?, where the dimensionless parameter  is defined as:

                (18)

Fig.5 Comparison of current results for nonassociated flow rule material with results of Flac3D, and TRASS for slope with β= 45?, 18, d/H=1 and 32?

As discussed by DRESCHER and DETOURNAY [17] in the case of translational mechanisms, a fictitious flow rule can be selected for which the specific energy dissipation is independent of σ_n. Although this fictitious flow rule cannot be extended for a rotational collapse mechanism, the 3D safety factors on this basis are also presented in Fig.5 (as current^*) only to show that it is inaccurate. It can be seen from this figure that the current results are in a good agreement with those of the numerical method whereas lower safety factors are obtained using the fictitious flow rule (current^*).

5 Modes of failure due to local loading and nonassociativeness

To investigate the way that the critical failure mechanism is influenced by the dilatancy angle and earthquake forces when a slope is subjected to local loads, a slope with a height of 6 m and an angle of 45^° is considered (Fig.4). The unit weight, internal angle of friction and cohesion of the soil were assumed to be   20 kN/m³, 40^° and 20 kPa, respectively. For such a slope with a loading q of 100 kPa and L/B ratio of 2, three different cases were studied: case 1, an associated flow slope under vertical loading at ψ=40? and k_h=0; case 2, an associated flow slope under seismic loading at ψ=40? and k_h=0.2; and finally case 3, a nonassociated flow slope under seismic loading at ψ=0? and k_h=0.2.

The minimum safety factors for the mentioned cases were obtained as 2.339, 1.893 and 1.381, respectively, though the critical failure modes are different. Distinct 3D failures were formed in cases 1 and 2 while the assumption of nonassociated flow rule in case 3 resulted in a nearly 2D failure mechanism. Fig.6 illustrates the sections and plan views of critical failure mechanisms for the above-mentioned loading and flow rule conditions (the figure is not scaled). It can be concluded that a nearly infinite slope with local loading, both the external load and the dilatancy angle of soil as well as the coefficient of seismic load determine whether the critical failure surface is 2D or 3D.

Fig.6 Influence of dilatancy angle on failure mode of slopes under local loading, L/B=2, a/B=0.5 and d/H=3: (a)? Cases 1 and 2; (b) Case 3

6 Numerical results

The proposed 3D collapse mechanism can be used to estimate the seismic stability of locally-loaded slopes in associated and nonassociated flow rule materials based on the energy expression. To investigate the effects of dilatancy angle on the seismic stability or the seismic bearing capacity of slopes, some numerical analyses were performed.

Fig.7 shows the influences of dilatancy angle on the 3D safety factors for slopes with various friction angles, γH/c=2, d/H=3, and β=30^° and 60^°, in static (k_h=0) and seismic (k_h=0.2) conditions. As shown in Fig.7, reduction in ψ significantly decreases the safety factors up to about 25% in both static and seismic loading conditions. The effect of dilatancy angle is more important for the mild slopes in frictional soils. The results of analyses in the case of locally-loaded slopes in static or seismic conditions are presented in Fig.8. The loading surface is square-shaped with width B located at a distance of a=B/2 from the slope edge and subjected to the vertical load q (q=5c). The results confirm that the influence of dilatancy increases in considered locally-loaded slopes up to about 35%. This influence is also more significant in the case of slopes with smaller inclined angles.

The seismic bearing capacity of square foundations near the slopes and the effects of dilatancy angle on the bearing capacity of various slopes are also investigated. Fig.9 shows the bearing capacity factors, N_c, of square foundations near the slopes (a/B=0.5, d/H=3) with various friction angles assuming associated and nonassociated flow rules. Fig.10 demonstrates the influences of dilatancy angle on the 3D bearing capacity of footings near the slopes. is the bearing capacity factor corresponding to the associated flow slope with the same parameters. It can be seen that nonassociativeness leads to a decrease in seismic bearing capacity of mild slopes up to 70%.

The nonassociativeness also affects the shape of slip surfaces and the volume of critical failure mechanism in stability problems. According to the obtained results, in the bearing capacity problems, nonassociativeness leads to a smaller failure mass. Fig.11 illustrates the influences of dilatancy angle on the volume of critical collapse mechanisms for rectangular foundations near the slopes with the inclined angle of 30? and =7. In this figure,  represents the ratio of the volume of critical failure mass of the nonassociated flow slope to that of the associated flow slope in the same conditions. The results illustrate that the volume of failure mass decreases up to about 45% when the dilatancy angle is assumed to be 0.2 It is noteworthy that the influence of nonassociativeness on the volume of failure mass depends on the length of the foundation, which is more considerable in 3D analysis of square foundations under static loads.

Fig.7 3D safety factors for associated and nonassociated flow slopes with γH/c=2 and d/H=3: (a) k_h=0; (b) k_h=0.2

Fig.8 3D safety factors for associated and nonassociated flow slopes with q/c=5, a/B=0.5, γH/c=2 and d/H=3 under local loading:  (a) k_h=0, (b) k_h=0.2

Fig.9 Bearing capacity factors, N_c, of square foundations near nonassociated flow slopes with a/B=0.5 and d/H=3: (a) k_h=0; (b) k_h= 0.2

Fig.10 Influence of dilatancy angle on 3D bearing capacity factor of slopes with B/L=1.0, a/B=0.5 and d/H=3: (a) k_h=0; (b) k_h=0.2

Fig.11 Influences of dilatancy angle on volume of critical failure mass in bearing capacity of slopes with β=30?, 30?, a/B=1.0 and γH/c=7

7 Summary and conclusions

(1) A 3D method based on the upper-bound theorem of limit analysis approach is proposed to determine the seismic stability of slopes under local loading. Using the proposed 3D rotational collapse mechanism and applying the energy dissipation method, seismic stability factors for nonassociated slopes are determined, and then the effects of dilatancy angle on the stability of locally-loaded slopes are investigated. The distribution of normal stresses along the rupture surfaces, required for the stability analysis of nonassociated flow materials, is computed based on the extended Bishop’s method and the assumption of Boussinesq theory. To evaluate the results of the proposed mechanism, comparison with other analytical and numerical methods is performed.

(2) According to the obtained results, the effect of dilatancy angle is more important in 3D seismic analysis of locally-loaded slopes because the failure mode can also be affected by the nonassociativeness. The stability factors with nonassociated flow rule are smaller than those with the associated flow. These effects are more significant for the mild slopes in frictional soils.

(3) By increasing the local load on a slope, the effect of dilatancy angle on the seismic stability of slope will be enhanced. In other words, influences of nonassociated flow are more considerable when determining the bearing capacity of foundations located near slopes. In addition, nonassociativeness leads to a smaller critical failure mechanism particularly in bearing capacity analysis of square foundations under static loads.

References

[1] HOVLAND H J. Three-dimensional slope stability analysis [J]. Journal of Geotechnical and Geoenvironmental Engineering, 1977, 103(9): 971-986.

[2] CHEN R H, CHAMEAU J L. Three-dimensional limit equilibrium analysis of slopes [J]. Geotechnique, 1982, 32(1): 31-40.

[3] UGAI K. Three-dimensional slope stability analysis of vertical cohesive slopes [J]. Soils and Foundations, 1985, 25(3): 41-48.

[4] HUNGR O. An extension of Bishop’s simplified method of slope stability analysis to three dimensions [J]. Geotechnique, 1987, 37(1): 113-117.

[5] GIGER M W, KRIZEK R J. Stability analysis of vertical cut with variable corner angle [J]. Soils and Foundations, 1975, 15(2): 63-71.

[6] LESHCHINSKY D, BAKER R, SILVER M L. Three-dimensional analysis of slope stability [J]. International Journal for Numerical and Analytical Methods in Geomechanics, 1985, 9(3): 199-223.

[7] MICHALOWSKI R L. Three-dimensional analysis of locally loaded slopes [J]. Geotechnique, 1989, 39(1): 27-38.

[8] FARZANEH O, ASKARI F. 3D Analysis of nonhomogenous slopes [J]. Journal of Geotechnical and Geoenviromental Engineering, 2003, 129(2): 137-145.

[9] BUHAN P, GARNIER D. Three-dimensional bearing capacity analysis of a foundation near a slope [J]. Soil and Foundations, 1998, 38(3): 153-163.

[10] DAVIS E H. Theories of plasticity and the failure of soil masses [M]. London: Butteworks, 1968: 341-380.

[11] YIN J H, WANG Y J, SELVADURAI A P S. Influence of nonassociation on bearing capacity of a strip footing [J]. Journal of Geotechnical and Geoenviromental Engineering, 2001, 127(11): 985-989.

[12] WANG Y J, YIN J H, LEE C F. The influence of a non-associated flow rule on the calculation of the factor of safety of the soil slopes [J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2001, 25(13): 1351-1359.

[13] KUMAR J. Stability factors for slopes with nonassociated flow rule using energy consideration [J]. International Journal of Geomechanics, 2004, 4(4): 264-272.

[14] YANG Xiao-li, GUO Nai-zheng, ZHAO Lian-heng, ZOU Jin-feng. Influence of nonassociated flow rules on seismic bearing capacity factors of strip footing on soil slope by energy dissipation method [J]. Journal of Central South University of Technology, 2007, 14(6): 842-847.

[15] YANG Xiao-li, SUI Zhi-rong. Seismic failure mechanisms for loaded slopes with associated and nonassociated flow rules [J]. Journal of Central South University of Technology, 2008, 15(3): 276-279.

[16] GEN M, CHENG R. Genetic algorithms and engineering design [M]. New York: John Wiley & Sons, Inc., 1997.

[17] DRESCHER A, DETOURNAY C. Limit load in translational failure mechanisms for associated and nonassociated materials [J]. Geotechnique, 1993, 43(3): 443-456.

[18] BISHOP A W. The use of slip circle in the stability analysis of slopes [J]. Geotechnique, 1955, 5(1): 7-17.

[19] STARK T D, EID H T. Performance of three-dimensional slope stability methods in practice [J]. Journal of Geotechnical and Geoenviromental Engineering, 1998, 124(11): 1049-1060.

[20] CANEPA Y, DESPRESLES D. Catalogue des essays de chargement de fondations superficielles realises sur sites par les L.P.C. [R]. F.A.E.R., 1.17.02.9, Internal Report 8622. Melum, 1990. (in French)

[21] WEI W B, CHENG Y M, LI L. Three-dimensional slope failure analysis by the strength reduction and limit equilibrium methods [J]. Computers and Geotechnics, 2009, 36(1/2): 70-80.

[22] ITASCA 2000. FLAC3D [M]. Version 2.1. Houston: Itasca Consulting Group Inc., 2000.

Received date: 2009-07-02; Accepted date: 2009-10-16

Corresponding author: N. GANJIAN, PhD; Tel: +98-21-44404928; E-mail: nganjian@ut.ac.ir

(Edited by CHEN Wei-ping)