Slope Stability Analysis Under Combined Failure Criteria

2.1. Strength Reduction Finite Element Method

The Factor of Safety (FOS) of the slope is defined here as the factor by which the original shear strength parameters must be divided in order to bring the slope to the failure [2-4, 16]. And the factored shear strength parameters C' and φ' are defined as follows, respectively.

(1)

(2)

Where SRF is “Strength Reduction Factor”, φ is the original friction angle (in degree), C is the original cohesion. To find the exact Factor of Safety (FOS), it is necessary to initiate a systematic search for the SRF value that will just cause the slope failure, and FOS is determined as the corresponding SRF value, FOS=SRF.

2.2. Slope Failure Definition

There are several popular definitions of failures, such as some tests of bulging of the slope profile; limiting of the shear stresses on the potential failure surface and non-convergence of the solution. When the solution cannot converge with a user-specified maximum number of iterations, this implication is that no stress distribution can be found to match both the failure criterion and global equilibrium [3]. And most solution chose an iteration ceiling of 1000, which most depends on the user’s experience and brings quite high requirement for geology and civil engineers. Before non-convergence of the solution appearance, there will be a running-through zone of plastic strain along a sliding surface from slope toe to the top [17], which can be easily found beyond primary operation experience of common commercial software. This running-through moment is assumed as the slope failure time and FOS is achieved as the corresponding SRF at this moment in this paper.

2.3. Mohr-Coulomb Plastic Criterion

The Mohr-Coulomb criterion assumes that yield occurs when the shear stress on any point in a material reaches a value that depends linearly on the normal stress in the same plane (Fig. 1). The yield line is the best straight line that touches these Mohr's circles [18].

Therefore, the Mohr-Coulomb model is defined by

(3)

Where σ is positive in compression. From Mohr's circle,

(4)

(5)

Substituting for τ and σ, multiplying both sides by cos φ, and reducing, the Mohr-Coulomb model can be written as

(6)

Where S is half of the difference between the maximum principal stress, σ₁, and the minimum principal stress, σ₃,

;

σ_m is the average of the maximum and minimum principal stresses,

;

and φ is the friction angle.

2.4. Geometric and Material Parameters

Three typical slopes are employed in this paper (geometric parameters are listed in Table 1) while the material properties, including unit weight, deformation modulus, Poisson's ratio and original shear strength parameters C (Cohesion) and φ (Friction angle), are presented in Table 2. And a value of the dilation angle, ψ=0, is used in this paper, corresponding to a non-associated flow rule with zero volume change during yield. Maximum element length, L, is set as one sixtieth of slope height, H.

Table 1.

Model	H, m	AH/H	TH/H	DH/H	BH/H	Foot gradient α,°
M1	10	2.00	2.00	2.00	6.00	26.6
M2	20	1.5	2.75	2.00	5.25	45.0
M3	100	1.00	1.00	1.7	2.2	78.7

4.1. Combined Failure Criterion

Most experimental results show that the linear failure criterion is a special case of nonlinear failure criteria while the strength envelopes of almost all geo-materials are characteristically nonlinear. And most of these nonlinear failure methods are relativeby a complicated procedure, walking farer way from the simple computation, beyond most engineers’ capacities and not suitable to wider application [8, 12, 13]. In our paper, a simple combined failure criterion with linear Mohr-Coulomb failure criterion and maximum principal tensile stress criterion are employed to determine the shear failure zone and tensile failure zone as follows (Eqn.7 & Fig. 5).

(7)

Where σ₁, σ_t are the maximum principal stress and tensile strength of slope media, respectively, and a value of σ_t=0 is used in this paper.

4.2. Combined Failure Zone and FOS Determination

During the solution progress, we can see that the tensile failure zone developed from the slope top while the shear failure zone developed from the slope toe, as shown in (Fig. 6). When the shear failure zone and tensile failure zone goes through each other, the slopes will lose the stability and receive the whole failure (Fig. 7).

Given the combined failure zone as the definition of the slope failure, we can find that FOS under combined failure criterion is 1.592(M1), 1.202(M2) and 1.162(M3), which are less than 1.602(M1), 1.217(M2) and 1.263(M3) under linear Mohr-Coulomb failure criterion, respectively. Furthermore, we can find that the foot gradient has dramatical influence on the tensile failure zone, and that the total height of failure zone, divided by slope height H, developed from less than one tenth, one third time to eight fifths times when foot gradient increases from 26.6^o, 45.0^o to 78.7^o.