6. Illustrations and validations

Typical soil mechanics problems are analyzed next to illustrate and validate the program.

6.3. Slope stability

Results of interest for this type of problem include mainly the slope safety factor and the displacement field. The displacement field is indeed illustrative of the failure mechanism and allows a clear identification of the slip surface as shown on figure 6.6.

Here again it is the localization of the strain field which is indicative of the failure. The associated safety factor corresponds to the one obtained by the C- reduction algorithm.

As compared to alternative approaches, the plasticity based approach appears to yield comparable results when a comparizon with classical methods is possible and it is more flexible when more general slip surfaces occur (fig. 6.8).

The slope safety factor is defined as the factor of reduction applied to the yield surface coefficients when failure occurs. Predictions of the slope stability factor obtained by use of Z_SOIL are compared to results obtained by the conventional method of slices in table 6.2.

Table 6.3 shows a similar comparizon with various approaches. Again slightly higher values are obtained using Z_SOIL.

Further comparizons of analyses of slopes in homogeneous and layered media are performed next.

Mohr-Coulomb material

For a homogeneous medium with material characteristics (c, ) and angle of dilatancy , the critical height of the cut can be determinated to be, under appropriate assumptions [Chen, 1990] :

^(6.1)

which is independant of . The orientation of (the velocity discontinuity) is given by a = 45° + /2, which is a function of dilatancy. Both the safety factor and the orientation of the velocity discontinuity are reproduced with a reasonable accuracy by the numerical simulation, as illustrated in table 6.4.

Deviatoric Plastic Flow SF = 1.25 c = 16 = 30 = 0
Associated Plastic Flow SF = 1.30 c = 16 = 30 = 30

Multilaminate material

The case of a vertical cut with a single lamina orientation at angle is considered first. An approximate analytical solution to this problem can be derived as follows (Eq. 6.2):

^(6.2)

A comparison of this approximate solution with the numerical solution is given in table 6.5.

Except for very steep lamina angle the numerical solution coincide with the approximate solution. For steep angles the discrepancy between approximate theory and numerical solution become more important, unless a tensile cut-off is assumed.

The next test concerns a vertical cut with two layers of material. Material properties are given in table 6.6. In this case, as can be expected from examination of stress states on Mohr circle, failure occurs almost simultaneously in both layers ; in the top layer we observe a matrix failure and in the bottom layer, a lamina failure, as velocity characteristics indicate in figure 6.9.

Slope stability

The stability of a slope with two sets of lamina oriented at ₁= 52.5° and ₂ = 90° is analyzed next and results are compared with those of (Sharma, 1988) (Fig. 6.10a and 6.10b). The simulation starts with an initial analysis state of the unexcavated medium, followed by a simulation of the excavation and finally by a stability analysis.

The following cases are analyzed :

a. associated plasticity in both sets of joints (c = 50, = 40, = 45), elastic matrix medium

b. non-associated plasticity in both sets of joints (c = 50, = 45, = 0), elastic matrix medium

The two results reproduce the results of [SHA, 1988] with a reasonable accuracy (SF = 1.18 for = 0, SF = 1.21 for = ), although quite different elements and algorithms are being used.