Example: FOS Calculated for Jointed Rock Slopes

Problem Statement

Note

To view this project in FLAC2D, use the menu command Help ► Examples…. Choose “FLAC/ ExampleApplications/ JointedSlopes” and select “jointedslopes.prj” to load. The data files used are shown at the end of this example.

Factor-of-safety calculations using the strength reduction method in FLAC2D can determine both the safety factor and the mode of failure of a slope in a jointed rock mass. Several models are run in this section to illustrate the different types of failure modes that can be identified from a factor-of-safety calculation. Modes of failure include rock mass failure in a homogeneous and unjointed rock slope, plane failure of slopes containing either daylighting or non-daylighting discontinuities, and block and flexural toppling failure involving either forward or backward toppling of blocks. (These slope models and modes of failure are also described in detail by Lorig and Varona [2004].)

A simple slope geometry is used for all of the stability analysis cases described in this section. The slope has a height of 260 m and slope angle of 55°. The rock in the model is represented as deformable Mohr-Coulomb material, and the discontinuities behave as Coulomb joint material. A maximum zone size of 5 m is assigned for the deformable blocks in all models. The model slope geometry used for all cases is shown in Figure 1.

Figure 1: Slope geometry.

Six slope stability cases are analyzed. The cases include one model with no joint structure, three models with one joint set, and two models with two joint sets. The rock properties and joint properties for the six cases are listed in Table 1.

Table 1: Slope Stability Cases

Soil Property	Case 1	Case 2	Case 3	Case 4	Case 5	Case 6
Rock Density (kg/m3)	2660	2660	2660	2660	2660	2660
Rock Bulk Modulus (GPa)	6.3	6.3	6.3	6.3	6.3	6.3
Rock Shear Modulus (GPa)	3.6	3.6	3.6	3.6	3.6	3.6
Rock Cohesion (kPa)	675	675	675	675	675	1010
Rock Tension (kPa)	0	0	0	0	0	1010
Rock Friction (degrees)	43	43	43	43	43	43
Joint Set 1 Dip (degrees)	–	35	70	-70	-70	55
Joint Set 1 Spacing (m)	–	20	20	20	20	10
Joint Set 1 Friction (degrees)	–	40	40	40	40	40
Joint Set 1 Cohesion (kPa)	–	100	0	0	0	0
Joint Set 1 Stiffness (GPa/m)	–	1	1	1	1	1
Joint Set 2 Dip (degrees)	–	–	–	–	20	0
Joint Set 2 Spacing (m)	–	–	–	–	30	40
Joint Set 2 Friction (degrees)	–	–	–	–	40	40
Joint Set 2 Cohesion (kPa)	–	–	–	–	0	0
Joint Set 2 Stiffness (GPa/m)	–	–	–	–	1	1

The cases illustrate six different failure conditions. They are discussed separately below. The command listing for the six cases is given below.

Model Building

Each model is built using Sketch. Edges are created to represent the joints and they are assigned group names. The grouped edges are then turned into Zone Joints using the procedures described below.

Case 1 - Unjointed Slope

• Create a new Sketch Set. Select the Slope Wizard and create a slope as shown in Figure 2. Note that the Slope angle is automatically calculated when the Rise and Run are entered.

Figure 2: Slope Wizard entries for creating the basic slope geometry.

• Delete the vertical line at x=398 and the point on the bottom at 398,0. Also delete the vertical line at x=582 and the point at 582,0. Now redraw the bottom horizontal line.

The internal lines are useful for creating a structured mesh for a simple slope, but since most of our slopes will be jointed, these internal lines aren’t necessary.

Figure 3: Edges making up the unjointed slope.

• Specify a zone length of 15 and Mesh All Polygons.

Figure 4: Unjointed slope after zoning.

• Create the zones and save the state. It is also a good idea to go to the State Record tab, right-click and Save to file as data file.

Case 2 - Daylighting Joints

• Start with the Sketch from Case 1. Remove the existing mesh by selecting the Clean Up Meshes tool.

• For Case 2, joints are dipping at 35° with a spacing of 20 m. To create the joints, we will start by drawing a single long line at the correct orientation, and then create an array of parallel lines.

We want to draw the first line outside of the slope so we are not duplicating the intersection points when we make the joint set(s). We need the lines to be larger than the slope, so first draw a line from 200,600 to 1200,-100

Figure 5: First joint

• Select the line and open the tool to Create Array of Edges. The spacing is 20 m perpendicular to the joints, so if we are making new lines to the left, we need to enter a horizontal distance of 20/sin35° = 34.867. Thirty lines are enough to cover the entire slope.

Figure 6: Create an array of joints to the left.

• The Sketch should appear as in Figure 7

Figure 7: Array of joints covering the whole slope.

• You will now need to delete the joint segments outside of the slope. First click on a point, so FLAC2D knows that you wish to select points. Now simply draw a box around sets of points that lie outside of the block and hit the delete key on your keyboard. This will automatically delete the dangling edges along with the unwanted points. You will have to delete some of the edges one by one at the toe of the slope where there are no unwanted points.

Figure 8: Jointed slope with dangling edges deleted.

• Now set the zone length to 10, then go to Set meshing parameters … and check the box for Create unstructured meshes only. Then click Mesh Now.

Figure 9: Zoned slope model.

• Create the zones and save the state. If you are building on top of Case 1, you will need to delete the existing zones.

Cases 3 to 6

The procedure for creating the other jointed slopes is similar to that shown above. Coordinates for the first joint, horizontal spacings, and the number of joints are shown for each case are shown in Table Table #jointedslopes2-2d.

Note that for Cases 4 and 5, some nodes end up being very close together. When you generate zones, you will get an error. Go back and look at the Sketch and search for orange regions that indicate bad zones. Turn off grid snapping and drag nodes on top of each other if they are very close together.

Running the Factor of Safety Analysis

Once each model is constructed, a data file is created. The saved state is restored, properties and boundary conditions are assigned with commands as shown in the data files at the end of this example.

To create the Zone Joints, first the model is “skinned” so that a group name is assigned to each edge between different zone groups. Since zone groups are assigned automatically to each block in the Sketch, each edge that corresponds to a joint will get a group name in the slot “skin”. The faces are then separated, and zone joints are created between the separated faces.

; skin internal faces using blocks that were assigned in Sketch
zone face skin internal on slot 'Block'

; separate faces
zone separate by-slot 'skin'

; create joints
zone joint config
zone joint create skinned

Properties are then assigned to the joints.

During the factor of safety analysis, you need to explicitly include the joints so that their cohesion and friction is reduced in the same way as the cohesion and friction of the zones. e.g.

model f-o-s joint include 'cohesion' joint include 'friction'  ...
  filename 'case2'

Results

Case 1: Unjointed, homogeneous rock – rock mass failure

For Case 1, the slope is a homogeneous rock without joints. Failure of the slope primarily involves shearing though the rock mass, and the shear failure surface is approximately circular as shown in Figure 10. For the Case 1 rock properties listed in Table 1, a factor of safety of 1.67 is calculated. Note that the failure surface in a FLAC2D model can usually be most clearly identified from a plot of velocity vectors, and either a displacement or velocity contour plot. In Figure 10 velocity vectors and velocity contours clearly show the failure surface:

Figure 10: Case 1 — rock mass failure.

Case 2: Daylighting joint structure — plane failure

In the Case 2 simulation, a single joint set is added to the model. The joints dip at 35° and are spaced at 20 m. The failure mechanism that develops combines sliding along joints near the slope toe with tensile failure of the blocks near the top of the slope. Figure 11 shows the failure surface. The calculated factor of safety is 1.27 for this case. Compare to the analytical solution of 1.32 for rigid blocks (Lorig and Varona, 2004).

Figure 11: Case 2 — plane failure in slope with daylighting joints.

Case 3: Non-daylighting joint structure — plane failure

In the third case, the dip angle of the single joint set is set to 70° in the same direction as the slope. This produces non-daylighting joints along the slope face. The joint spacing is 20 m. The failure mode that develops in this case involves sliding along the discontinuities, and shearing through the rock blocks at the toe of the slope. Figure 12 illustrates the failure mechanism. The resulting factor of safety is 1.57 for the given problem conditions.

Figure 12: Case 3 — plane failure in slope with non-daylighting joints.

Case 4: Joints dipping into the slope — flexural toppling failure

The joint set is oriented at a dip angle of 70° into the slope and spaced at 20 m, in Case 4. This results in joints dipping steeply into the slope face. The joints form columns that tend to bend out of the slope, and result in a flexural toppling failure mode. Figure 13 shows the failure surface, and Figure 14 illustrates the flexural toppling mode from a magnified view of the block deformation. The calculated factor of safety is 1.24.

Figure 13: Case 4 — flexural toppling failure for joints dipping into the slope.

Figure 14: Case 4 — flexural toppling mode identified from magnified block deformation.

Case 5: Two orthogonal joint sets — forward block toppling failure

The slope contains two orthogonal joint sets, in Case 5. One set dips at 70° with a spacing of 20 m, and a cross-joint set dips at −20° with a spacing of 30 m. The cross-joints provide release surfaces for rotation of the blocks. The blocks, driven by self-weight, rotate forward out of the slope. Figure 15 shows the failure surface for the Case 5 conditions. The calculated factor of safety is 1.11. The magnified block deformation plot in Figure 16 illustrates the forward block rotation out of the slope.

Figure 15: Case 5 — forward block toppling failure for a slope with two joint sets.

Figure 16: Case 5 — forward block toppling mode identified from magnified block deformation.

Case 6: Two orthogonal joint sets — reverse (backward) block toppling failure

Backward or reverse block toppling failure of a slope can occur when joints parallel to the slope face and flatter cross-joints are particularly weak. In Case 6, one joint set is oriented at 55° (i.e., parallel to the slope face) with a spacing of 10 m. A cross-joint set is horizontal and spaced at 40 m. Note that in this case, in order to highlight the failure mode, elastic material behavior is prescribed for the rock blocks. Figure 17 displays the reverse toppling failure mode. The calculated factor of safety is 1.85. The backward block toppling mode is clearly seen in the magnified block deformation plot in Figure 18.

Figure 17: Case 6 — reverse block toppling failure for a slope with two joint sets.

Figure 18: Case 6 — reverse block toppling mode identified from magnified block deformation.

Data Files

case1-run.dat

; Unjointed, homogeneous rock mass
model new
program call 'case1-sketch'

zone cmodel assign mohr-coulomb
zone property density 2660 bulk 6.3e9 shear 3.6e9 cohesion 675e3 friction 43

zone face skin
zone face apply velocity-x 0 range group 'West' or 'East1'
zone face apply velocity 0 0 range group 'Bottom'

model gravity 9.81

model large-strain off
model solve elastic

zone gridpoint initialize displacement 0,0
zone gridpoint initialize velocity 0,0

model f-o-s filename 'case1'

case2-run.dat

model new
program call 'case2-sketch'

; skin internal faces using blocks that were assigned in Sketch
zone face skin internal on slot 'Block'

; separate faces
zone separate by-slot 'skin'

; create joints
zone joint config
zone joint create skinned

; joint properties
contact property stiff-norm 1e9 stiff-shear 1e9 ...
  fric 40 coh 1e5 coh-resid 1e5

; zone properties
zone cmodel assign mohr-coulomb
zone property density 2660 bulk 6.3e9 shear 3.6e9 cohesion 675e3 friction 43

; boundary conditions
zone face apply velocity-x 0 range position-x 0
zone face apply velocity-x 0 range position-x 702
zone face apply velocity 0 0 range position-y 0

model gravity 9.81

model large-strain off
model solve elastic

zone gridpoint initialize displacement 0,0
zone gridpoint initialize velocity 0,0

model f-o-s joint include 'cohesion' joint include 'friction'  ...
  filename 'case2'

case3-run.dat

model new
program call 'case3-sketch'

; skin internal faces using blocks that were assigned in Sketch
zone face skin internal on slot 'Block'

; separate faces
zone separate by-slot 'skin'

; create joints
zone joint config
zone joint create skinned

; joint properties
contact property stiff-norm 1e9 stiff-shear 1e9 ...
  fric 40 coh 0 

; zone properties
zone cmodel assign mohr-coulomb
zone property density 2660 bulk 6.3e9 shear 3.6e9 cohesion 675e3 friction 43

; boundary conditions
zone face apply velocity-x 0 range position-x 0
zone face apply velocity-x 0 range position-x 702
zone face apply velocity 0 0 range position-y 0

model gravity 9.81

model large-strain off
model solve elastic

zone gridpoint initialize displacement 0,0
zone gridpoint initialize velocity 0,0

model f-o-s joint include 'cohesion' joint include 'friction'  ...
  filename 'case3'

case4-run.dat

model new
program call 'case4-sketch'

; skin internal faces using blocks that were assigned in Sketch
zone face skin internal on slot 'Block'

; separate faces
zone separate by-slot 'skin'

; create joints
zone joint config
zone joint create skinned

; joint properties
contact property stiff-norm 1e9 stiff-shear 1e9 ...
  fric 40 coh 0 

; zone properties
zone cmodel assign mohr-coulomb
zone property density 2660 bulk 6.3e9 shear 3.6e9 cohesion 675e3 friction 43

; boundary conditions
zone face apply velocity-x 0 range position-x 0
zone face apply velocity-x 0 range position-x 702
zone face apply velocity 0 0 range position-y 0

model gravity 9.81

model large-strain off
model solve elastic

zone gridpoint initialize displacement 0,0
zone gridpoint initialize velocity 0,0

model f-o-s joint include 'cohesion' joint include 'friction'  ...
  filename 'case4'

case5-run.dat

model new
program call 'case5-sketch'

; skin internal faces using blocks that were assigned in Sketch
zone face skin internal on slot 'Block'

; separate faces
zone separate by-slot 'skin'

; create joints
zone joint config
zone joint create skinned

; joint properties
contact property stiff-norm 1e9 stiff-shear 1e9 ...
  fric 40 coh 0 

; zone properties
zone cmodel assign mohr-coulomb
zone property density 2660 bulk 6.3e9 shear 3.6e9 cohesion 675e3 friction 43

; boundary conditions
zone face apply velocity-x 0 range position-x 0
zone face apply velocity-x 0 range position-x 702
zone face apply velocity 0 0 range position-y 0

model gravity 9.81

model large-strain off
model solve elastic

zone gridpoint initialize displacement 0,0
zone gridpoint initialize velocity 0,0

model f-o-s joint include 'cohesion' joint include 'friction'  ...
  filename 'case5'

case6-run.dat

model new
program call 'case6-sketch'

; skin internal faces using blocks that were assigned in Sketch
zone face skin internal on slot 'Block'

; separate faces
zone separate by-slot 'skin'

; create joints
zone joint config
zone joint create skinned

; joint properties
contact property stiff-norm 1e9 stiff-shear 1e9 ...
  fric 40 coh 0 

; zone properties - elastic
zone cmodel assign mohr-coulomb
zone property density 2660 bulk 6.3e9 shear 3.6e9 cohesion 1e20 tension 1e20

; boundary conditions
zone face apply velocity-x 0 range position-x 0
zone face apply velocity-x 0 range position-x 702
zone face apply velocity 0 0 range position-y 0

model gravity 9.81

model large-strain off
model solve elastic

zone gridpoint initialize displacement 0,0
zone gridpoint initialize velocity 0,0

model f-o-s joint include 'cohesion' joint include 'friction'  ...
  filename 'case6'