10 Slope Stability Analysis of a Rock Slope

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1 Slope Stability Analysis of a Rock Slope Slope Stability Analysis of a Rock Slope 10.1 Problem Statement Limit equilibrium methods are commonly used to evaluate the stability of slopes in rock or soil. For materials in which well-defined structural patterns (e.g., features such as bedding planes or joints) are not present, rupture surfaces develop naturally along planes or curved surfaces within the slope. In soils, the rupture surface is commonly observed to be circular: this is the basis for many limit-equilibrium stability theories. One common method is the Bishop simplified method of slices (Bishop 1955). This method is one of several used by Hoek and Bray (1981) to produce series of slope stability charts for circular failure. A limiting equilibrium condition for the development of a tensile crack can be included with the circular failure, as well as the influence of water pressure due to the presence of a phreatic surface in the slope. The charts developed by Hoek and Bray can be used to determine the location of the critical failure circle and tension crack, and find the associated factor of safety for stability of the slope. An example application of Bishop s method is given by Hoek and Bray for the case of a slope excavated in highly weathered granitic rock. The slope contains three 15 m high benches with two 8 m wide beams. The bench faces are inclined at 75 to the horizontal, and the top of the slope is cut at 45 from the top of the third bench to the ground surface. Figure 10.1 illustrates the geometry of the slope: critical center for φ = 45 tension crack a R 5 m Scale m a/r = Figure 10.1 Failure surface solution from Bishop s method for a rock slope (Hoek and Bray 1981)

2 10-2 Example Applications The rock mass is classified as a Hoek-Brown material with strength parameters of: m = 0.13 s = σ c = 150 MPa σc m = sσ c = 0.47 MPa The tensile strength is estimated to be MPa. For the Bishop s method, a tangent to the curved Hoek-Brown failure envelope is drawn at a normal stress level estimated from the slope geometry. Mohr-Coulomb properties for friction angle and cohesive strength are then estimated to be (see HOEK.FIS in Section 3 in the FISH volume): φ = 45 c = 0.14 MPa The mass density of the dry rock mass is 2500 kg/m 3, and the mass density of the saturated rock mass is 2800 kg/m 3. The phreatic surface is located as shown in Figure 10.1, and the mass density of water is 1000 kg/m 3. Based upon these parameters, Hoek and Bray report that the Bishop method produces a location for the circular failure surface and tension crack, as shown in Figure 10.1, and a factor of safety of Modeling Procedure In FLAC, the failure surface can evolve during the calculation in a way that is representative of the natural evolution of the physical failure plane in the slope. It is not necessary to make an estimate for the location of the circular failure line when beginning an analysis, as it is with limit-equilibrium methods. FLAC will find the failure plane and the failure mechanism by simulating the material behavior directly. Run the data file ROCKSL.DAT in Section 10.5 to perform this analysis. A reasonably fine grid should be selected to ensure that the failure plane will be well-defined as it develops. It is best to use the finest grid possible when studying problems involving localized failure (see Section in the User s Guide). For the bench-cut slope, a model grid is created as shown in Figure 10.2.

3 Slope Stability Analysis of a Rock Slope 10-3 JOB TITLE : SLOPE STABILITY ANALYSIS OF A ROCK SLOPE FLAC (Version 6.00) LEGEND Jun-08 9:54 step E+01 <x< 1.500E E+01 <y< 1.200E+02 Grid plot 0 5E 1 Water Table Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 10.2 FLAC model grid with water table FLAC can perform a factor-of-safety calculation by using the strength reduction method as described in note 12 in Section 3.8 in the User s Guide. The method is implemented by invoking the SOLVE fos command. In this approach, the strength of the material is reduced until a failure of the slope occurs. Both friction angle and cohesion are reduced simultaneously by a constant factor, and FLAC runs are automatically made with each new pair of strength parameters using a bracketing approach until a safety factor is found. We begin this analysis at the strength parameters selected by Hoek and Bray to characterize the slope (φ =45, c = 0.14 MPa and t f = MPa). The model is first brought to an equilibrium state assuming an unsaturated condition. We use the SOLVE elastic command to minimize plastic yield during the initial development of gravitational stresses. We then introduce the water table and adjust the mass density below the phreatic surface to its saturated value (using the FISH function wet den). Note that the total stresses are adjusted automatically (by specifying CONFIG ats at the start) to correspond to the increase in pore pressure resulting from the WATER table command. The model is stepped to equilibrium again, and the resulting stress state is almost entirely within the elastic range for the imposed conditions, as can be seen from the failure envelope plot shown in Figure 10.3.

4 10-4 Example Applications JOB TITLE : SLOPE STABILITY ANALYSIS OF A ROCK SLOPE FLAC (Version 6.00) LEGEND 20-Jun-08 9:54 step 5193 Failure Surface Plot Shear Stress vs Normal Stress Zone Stress States Mohr-Coulomb Fail. Surf. Friction = E+01 Cohesion = E+05 Tension = E (10 ) Itasca Consulting Group, Inc. Minneapolis, Minnesota USA 05 (10 ) Figure 10.3 Failure surface and initial zone stresses The factor-of-safety calculation is then begun by specifying SOLVE fos. With the strength reduction approach, the factor of safety, f s, adjusts the friction and cohesion as follows. The reduced friction angle, φ r,is and the reduced cohesion, c r,is φ r = arctan(tan φ/f s ) c r = c/f s Note also that we simulate a loss in tensile strength when the tensile strength limit is reached. We assume that the tensile strength drops to zero instantaneously; this is prescribed by default with the Mohr-Coulomb model. (Tensile softening as a function of plastic tensile strain can be prescribed with the strain-softening model.)

5 Slope Stability Analysis of a Rock Slope Results While FLAC is executing the SOLVE fos command, the bracketing values for f s are printed continuously to the screen. When completed, the final value for f s is displayed. In this case, the calculated f s is The failure surface is identified by the plots in Figures 10.4 and These plots are generated after restoring the file ROCKSL FOS.SAV. In Figure 10.4, the contours of shear strain rate indicate the plane of shear failure, while the plot of zero tensile strength within zones denotes the region in which tensile failure occurs (and the tensile strength drops to zero). In Figure 10.5, the plot of velocity vectors also indicates the pattern of motion at the initiation of failure. Compare Figures 10.4 and 10.5 to Figure The failure surface in FLAC closely resembles that produced from the Bishop solution. However, the tensile failure extends farther up the slope in the FLAC solution. It is important to recognize that the limit equilibrium solution only identifies the onset of failure, whereas the FLAC solution includes the effect of stress redistribution and progressive failure after failure has been initiated. In this problem, tensile failure continues up the slope as a result of the tensile softening. The resulting factor of safety allows for this weakening effect. This rock slope example is also run using FLAC/Slope (see Section in the FLAC/Slope User s Guide). The factor of safety calculated in the FLAC/Slope simulation is 1.38, and the failure pattern is slightly different. For example, compare Figure 10.5 to Figure 1.98 in the FLAC/Slope User s Guide. This difference is related to the effect of the loading path on the development of tensile failure in the model. In FLAC/Slope, the SOLVE fos solution is invoked from an initial stress state of zero in the model. In this example, the model is brought to an equilibrium stress state, with only a minor amount of tensile failure, before the SOLVE fos solution is performed. The approach in FLAC/Slope produces slightly more tensile failure in the model. Note that if the tensile strength is set to a high value (e.g., 0.14 MPa), then no tensile failure occurs, and the calculated factor of safety is the same for both simulations: More detailed comparisons of FLAC to limit equilibrium solutions have been made by others (e.g., Thompson 1993 and Dawson et al. 1999).

6 10-6 Example Applications JOB TITLE : SLOPE STABILITY ANALYSIS OF A ROCK SLOPE FLAC (Version 6.00) LEGEND 27-Jun-08 15:38 step E+01 <x< 1.500E E+01 <y< 1.200E+02 Factor of Safety 1.41 tension 0.00E E E E E E E+04 Contour interval= 2.00E+03 Extrap. by averaging Max. shear strain increment Contour interval= 5.00E-01 Minimum: 0.00E+00 Maximum: 3.50E+00 Boundary plot E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 10.4 Failure surface in slope indicated by shear strain contours and zero tension zones JOB TITLE : SLOPE STABILITY ANALYSIS OF A ROCK SLOPE FLAC (Version 6.00) LEGEND 20-Jun-08 9:54 step E+01 <x< 1.500E E+01 <y< 1.200E+02 Factor of Safety 1.41 User-defined Groups rock mass Water Table Velocity vectors max vector = 5.434E E -2 Plasticity Indicator * at yield in shear or vol. o at yield in tension Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure Failure surface in slope indicated by velocity vectors and plasticity indicators

7 Slope Stability Analysis of a Rock Slope References Bishop, A. W. The Use of the Slip Circle in the Stability Analysis of Earth Slopes, Géotechnique, 5, 7-17 (1955). Dawson, E. M., W. H. Roth and A. Drescher. Slope Stability Analysis by Strength Reduction, Géotechnique, 49(6), (1999). Hoek, E., and J. Bray. Rock Slope Engineering. London: IMM, Thompson, R. J. The Location of Critical Slip Surfaces in Slope-Stability Problems, J. S. Afr. Inst. Min. Metall., 93(4), (1993).

8 10-8 Example Applications 10.5 Data File ROCKSL.DAT ;Project Record Tree export ;... State: rocsl 1.sav... ; ; rocksl.dat rock slope stability analysis ; comparison to solution in Hoek & Bray, 1981 ; config ats grid 60,40 gen -40.0, , , ,-40.0 i=1,61 j=1,41 model elastic ; Add top surface table 1 ( ) ( ) ( ) (0 0) ( ) (11.88 & 15.00) ( ) ( ) ( ) ( ) (80.00 & 76.00) ( ) ( ) ( ) gen table 1 model null region ; fix x y j=1 fix x i=1 fix x i=61 ; group rock mass reg tab 1 model mohr group rock mass notnull prop density= bulk=1e8 shear=3e7 cohesion= friction=45.0 & dilation=0.0 tension= group rock mass notnull ; set gravity=9.81 history 999 unbalanced solve elastic save rocsl 1.sav ;... State: rocsl 2.sav... ; Add water table table 2 ( ) (0 0) ( ) ( ) ( ) & ( ) ( ) ( ) ( ) ( ) & ( ) def wet den loop i (1,izones) loop j (1,jzones) if model(i,j)>1 then xa=(x(i,j)+x(i+1,j)+x(i+1,j+1)+x(i,j+1)) xc=0.25*xa ya=(y(i,j)+y(i+1,j)+y(i+1,j+1)+y(i,j+1))

9 Slope Stability Analysis of a Rock Slope 10-9 yc=0.25*ya if yc < table(2,xc) then density(i,j) = 2800 endif endif endloop endloop end wet den water density= water table=2 solve save rocsl 2.sav ;... State: rocksl fos.fsv... solve fos no restore file rocksl fos.fsv ;*** plot commands **** ;plot name: grid plot hold grid water iwhite ;plot name: fail Mohr set pltc pltf 45.0 pltt plot hold fail normal ;plot name: fos - ssi plot hold fos tension block inv ssi iwhite bound ;plot name: fos - vel. vectors plot hold fos group water velocity plasticity no past

10 10-10 Example Applications

1 Slope Stability for a Cohesive and Frictional Soil

Slope Stability for a Cohesive and Frictional Soil 1-1 1 Slope Stability for a Cohesive and Frictional Soil 1.1 Problem Statement A common problem encountered in engineering soil mechanics is the stability