COMSOL 6.4 - Strength Reduction Method for Slope Stability

Strength Reduction Method for Slope Stability

The strength reduction method in combination with finite element analysis is a tool to find the factor of safety (FOS) in geomechanics, particularly for the stability of slopes and embankments. In the strength reduction method, the characteristic material properties are gradually reduced until failure occurs. Although the definition of an FOS depends on the context, in geotechnical problems it is defined with respect to the strength parameters of the soil, as discussed in Ref. 1.

The strength reduction method is applicable to linear failure criteria, like the Mohr–Coulomb criterion. When the Mohr–Coulomb criterion is used with the strength reduction method, the cohesion, the angle of internal friction, and the dilatation angle are simultaneously reduced until mechanical equilibrium is lost. Decreasing the material parameters results in a reduction of the shear strength of the soil, which eventually becomes unstable. This phenomena produces a collapse of the slope for a certain combination of loads, material parameters, and boundary conditions. The ratio between the initial cohesion and the cohesion at failure gives the FOS. More details of the method are given in Ref. 1 and Ref. 2.

The geometry, boundary conditions, loading conditions, and material parameters in this example are the same as discussed in Ref. 1. A similar example model can be found in Slope Stability in an Embankment Dam.

Model Definition

Figure 1 shows a cross section of the soil embankment. The lengths L1 and L2 are 85 m, and 20 m, respectively, and the heights of the embankment H1 and H2 are 20 m and 10 m, respectively. The slope angle α varies from 15° to 45°.

Figure 1: Geometry of the cross-section of an embankment.

The material properties for both associative and nonassociative plasticity are summarized in Table 1, and taken from Ref. 1.

Table 1: Material Properties.
Property	Material case 1	Material case 2
E	20 MPa	20 MPa
ν	0.3	0.3
c	20 kPa	20 kPa
ϕ	25°	25°
ψ	0°	25°
ρ	1940 kg/m3	1940 kg/m3

A plane strain approximation is used to model the soil embankment in 2D. The effect of gravity is included. The material properties for the Mohr–Coulomb model are parameterized with respect to a factor of safety parameter, FOS. A parametric study increases the FOS parameter, thereby reducing the strength of the soil with every parameter step. The actual factor of safety is the value of the FOS parameter at which the model no longer converges, which is an indication of the collapse of the slope.

The Mohr–Coulomb yield function F and plastic potential Q are

(1)

(2)

where I1 is the first stress invariant and J2 is the second deviatoric stress invariant. The parameterized cohesion C, parameterized angle of internal friction Φ, and parameterized dilatation angle Φ are given in terms of the FOS,

(3)

where c is the cohesion, ϕ is the angle of internal friction, and ψ is the dilatation angle. Note that c, ϕ, and ψ are initial, unreduced material parameters. For the associative flow rule, ϕ = ψ.

For the nonassociative flow rule, ψ is kept constant as long as it is smaller than Φ, see Ref. 2 for details. However, in this example, ψ is zero when using the nonassociative flow rule, so no special treatment is needed, and Equation 3 is applicable to the associative as well as the nonassociative flow rule.

For the nonassociative flow rule, the strength reduction method might trigger numerical instabilities, which in turn can result in a nonunique failure surface and corresponding FOS. To avoid potential instabilities and convergence issues, the Davis procedure B approach, as suggested in Ref. 1 and Ref. 2, is used. In this approach, the associative flow rule is applied with reduced values of the cohesion and the angle of internal friction to capture the effects of the nonassociative flow rule. The reduced cohesion c' and the reduced angle of internal friction ϕ' are given by

(4)

where the reduction factor β is

Hence, Equation 3 is rewritten in terms of the reduced cohesion c' and the reduced angle of internal friction ϕ' for the associative and nonassociative flow rules, as the reduction factor β is unity for the associative flow rule.

Results and Discussion

The factor of safety (FOS) for different slope angles is shown in Figure 2. The FOS decreases as the slope angle increases, which is expected. The FOS for the same slope inclination with the nonassociative flow rule (material case 1) is always smaller than for the associative flow rule (material case 2), but the influence of the nonassociative flow rule is marginal for the material parameters chosen. The results presented in Figure 2 are in good agreement with the results presented in Figure 4 of Ref. 1. As a verification that we have reached failure, the maximum displacement of the soil embankment is shown in Figure 3 as a function of the FOS for all analyzed cases.

Figure 2: Factor of safety versus slope angle.

Figure 3: Maximum displacement in the soil embankment versus the factor of safety for material case 1 (solid) and material case 2(dashed).

The equivalent plastic strain for different slope angles just before collapse is shown in Figure 4 and Figure 5 for the two material sets. The localization of plastic strains in the figures gives an indication of the failure surface for different slope angles. It is evident that for lower slope angles, multiple failure surfaces develop in the embankment.

A 3D visualization of the displacement for a slope angle equal to 45° is shown in Figure 6. The results are qualitatively in good agreement with the results presented in Figure 2 of Ref. 1. The figures also clearly show how parts of the embankment outside the slip surface start sliding once the material becomes unstable.

Figure 4: Equivalent plastic strain just before collapse for material case 1 (nonassociative flow).

Figure 5: Equivalent plastic strain just before collapse for material case 2 (associative flow).

Figure 6: Displacement magnitude in the soil embankment.

Notes About the COMSOL Implementation

Two stationary study steps are added. The first study step computes the in-situ stresses due to gravity. The Mohr–Coulomb criterion is added in the second study to compute the elastoplastic failure due to the combined effect of gravity and the reduction in strength, where the initial stresses generated in the first step are incorporated in the analysis with the help of an node.

Two outer parametric sweeps are added to change the slope angle and the set of material parameters. A is added to the continuation parameter sweep such that the solution stops when the maximum displacement in the embankment exceeds a tenth of the length of the slope.

References

1. H.F. Schweiger “Strength reduction technique with finite element method for slopes without stabilization measures,” Benchmark, International Magazine for Engineers, Designers and Analysts from NAFEMS, pp. 51–58, 2020.

2. S. Oberhollenzer, F. Tschuchnigg, and H.F. Schweiger “Finite element analysis of slope stability problems using non-associated plasticity,” Journal of Rock Mechanics and Geotechnical Engineering, vol. 10, pp. 1091–1101, 2018.

Geomechanics_Module/Verification_Examples/strength_reduction_method

Modeling Instructions

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Model Wizard

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Parameters for the model geometry, material, and solver are available in the appended text files.

Global Definitions

Geometry and Solver Parameters

In the window, under click .

In the window for , type Geometry and Solver Parameters in the text field.

Locate the section. Click

Browse to the model’s Application Libraries folder and double-click the file strength_reduction_method_parameters.txt.

Material Parameters

In the toolbar, click

and choose > .

In the window for , type Material Parameters in the text field.

Locate the section. Click

Browse to the model’s Application Libraries folder and double-click the file strength_reduction_method_material_parameters1.txt.

In the toolbar, click

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file strength_reduction_method_material_parameters2.txt.

Definitions

Define the parameterized cohesion and angle of internal friction for the Mohr–Coulomb criterion.

Variables 1

In the window, expand the > node.

Right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
beta_f	cos(atan(tan(phi)/FOS))cos(atan(tan(psi)/FOS))/(1-sin(atan(tan(phi)/FOS))sin(atan(tan(psi)/FOS)))		Reduction factor
c_r	beta_f*c	Pa	Reduced cohesion
phi_r	atan(beta_f*tan(phi))	rad	Reduced friction angle
c_p	c_r/FOS	Pa	Parameterized cohesion
phi_p	atan(tan(phi_r)/FOS)	rad	Parameterized friction angle

Geometry 1

Polygon 1 (pol1)

In the toolbar, click

In the window for , locate the section.

From the list, choose .

In the text field, type 0, L1,L1,L2+Lslope,L2,0.

In the text field, type 0, 0, H1+H2,H1+H2,H1,H1.

Click

Solid Mechanics (solid)

Linear Elastic Material 1

In the window, under > click .

In the window for , locate the section.

Select the checkbox.

Soil Plasticity 1

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and choose .

In the window for , locate the section.

From the Ff list, choose .

Click to expand the section. From the list, choose .

In the lint text field, type 0.1.

Linear Elastic Material 1

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Initial Stress and Strain 1

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and choose .

Add two study steps in order to account for the in situ stresses due to gravity. Add the in situ stresses computed in the first study step as initial stresses for the second study step. You can access these stresses using the withsol operator as follows:

In the window for , locate the section.

Specify the S0 matrix as


withsol('sol2',solid.sxx)	withsol('sol2',solid.sxy)	withsol('sol2',solid.sxz)
withsol(’sol2’,solid.sxy)	withsol('sol2',solid.syy)	withsol('sol2',solid.syz)
withsol(’sol2’,solid.sxz)	withsol(’sol2’,solid.syz)	withsol('sol2',solid.szz)

Gravity 1

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and choose .

Roller 1

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Select Boundaries 1 and 6 only.

Fixed Constraint 1

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Select Boundary 2 only.

Materials

Soil Material

In the window, under right-click and choose .

In the window for , type Soil Material in the text field.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Young’s modulus	E	E_soil	Pa	Young’s modulus and Poisson’s ratio
Poisson’s ratio	nu	nu_soil	1	Young’s modulus and Poisson’s ratio
Density	rho	rho_soil	kg/m³	Basic
Initial cohesion	cohesion0	c_p	Pa	Soil material
Friction angle	phis	phi_p	rad	Soil material