Generalized Anisotropic Material

1. Introduction

This tutorial describes how to simulate an anisotropic material in Slide2. There are actually five different ways to do this, but the emphasis of this tutorial will be on using the Generalized Anisotropic option, which allows you to specify different material types in different directions. The tutorial will also explain how to perform a probabilistic analysis with this type of material.

The finished product of this tutorial can be found in the Tutorial 20 Generalized Anisotropic.slmd data file. All tutorial files installed with Slide2 can be accessed by selecting File > Recent Folders > Tutorials Folder from the Slide2 main menu.

2. Model

For this tutorial, we will read in a file.

Select File > Recent Folders > Tutorials Folder from the Slide2 main menu, and open the Tutorial 20 Generalized Anisotropic start.slmd file.

MATERIAL PROPERTIES

For the slope material, we will use the Generalized Anisotropic option. This allows you to specify different material types in different directions. (Note: there are other anisotropic material models available in Slide2, see the next section for information).

Before we set up the Generalized Anisotropic material, let’s look at the sub-materials that make up the generalized material. For this example, we will assume the soil has weak, subhorizontal bedding. Two materials have been set up: one that represents the direction parallel to bedding and one that represents all other orientations.

1. Select Define Materials from the Properties menu.
   
   Look at the Soil Mass and Bedding materials. So in this case, both sub-materials use Mohr-Coulomb strength but have different values of cohesion and friction angle. These two materials will be assigned to a Generalized Anisotropic material as described below.
   
   
2. Click on Material 3. Change the Strength Type to Generalized Anisotropic.
   
   Notice the Input Type dropdown which contains the options “Angle range” and “Angle or surface, A, B”. We will look at both Input Types in this tutorial. For now, keep Input Type as Angle range.
   
   
3. Click the pencil icon next to the Generalized Function input.
   
   You will see a dialog in which you can specify different materials over different angular ranges. We want the Bedding material to be active within plus/minus 10° of horizontal and the Soil Mass to be active in all other directions. Angles in the dialog are measured from horizontal, so 90° represents vertical.
   
   
4. Enter −10 in the Angle To column. Leave the Material as Soil Mass.
5. In the next row, enter 10 in the Angle To column and change the Material to Bedding.
6. Finally, in the next column, set the Angle To = 90 and leave the Material as Soil Mass. Select OK.
   
   Material 3 is now assigned the Generalized Function ‘User Defined 1’. The material is currently named “Material 3.” Rename it to “Angle Range.”
   
7. Click OK to close the Define Material Properties Dialog.

ANISOTROPIC MATERIAL MODELS IN SLIDE2

There are actually five different anisotropic material strength models available in Slide2. These are:

• Anisotropic Strength
• Anisotropic Function
• Anisotropic Linear
• Generalized Anisotropic
• Modified Anisotropic Linear

For this tutorial, you could create the same model using the Anisotropic Function strength type, which allows you to define discrete angular ranges with Mohr-Coulomb properties. However, the Generalized Anisotropic strength type offers the following advantages:

• Any material type can be assigned to each angular range. All of the materials do not have to have a Mohr-Coulomb failure criterion. You can mix and match any material types you wish (e.g. Hoek-Brown and Mohr-Coulomb).
• The Generalized Anisotropic strength type allows for probabilistic analysis whereas the Anisotropic Function strength type does not. This will be explored further later in the tutorial.
• You have the option of defining the anisotropic direction as an anisotropic surface. This too is explored further in the tutorial.

See the Slide2 Help system for details about the different anisotropic strength models which are available.

ASSIGN MATERIALS

By default, the slope is assigned Material 1 (Soil Mass). We need to set the slope material to the Generalized Anisotropic material that you defined above. Right-click inside the model and select Assign Material > Angle Range. The model will look like this.

3. Compute

1. Save the model by selecting File > Save As in the menu.
2. Select Analysis > Compute in the menu to perform the analysis.
3. Then select Analysis > Interpret to view the results.

4. Interpret

The Interpret program shows the results of the Spencer analysis. You can see that the factor of safety is about 1.48.

The limit equilibrium method calculates the stability of each possible failure surface by dividing up the circular area into slices and comparing the shear stress and strength on the base of each slice. With the Generalized Anisotropic material model, the angle of the base of each slice determines which material is used to calculate the strength.

1. Go to Query > Query Slice Data.
2. Click on a slice about halfway down the slope.
3. In the Slice Data dialog, scroll down to the bottom so that you can see the Base Material. For this slice, it should be 1 (Soil Mass).
   
4. Now click on a slice near the toe of the slope.
   
   In this region, the base of each slice is almost horizontal, so the base material is Material 2 (Bedding). This demonstrates how the applied strength model depends on the orientation of the slice base, for the Generalized Anisotropic model.
   
   If you scroll back up in the Slice Data dialog, you will see that the base friction and cohesion reflect the values entered for Material 2 (c = 0 and φ = 20°).
5. Close the Slice Data dialog.
   
6. Now right click on the slip surface and select Graph Query from the popup menu.
7. In the Graph Slice Data dialog choose Primary Data = Base Cohesion.
   
   You should see the following plot, which shows the Bedding cohesion (=0) for the low base angle slices near the toe, and the soil mass cohesion (=5) for the higher base angle slices.
   

5. Non-Circular Failure Surface

Because of the weak bedding plane, it is likely that portions of the failure surface would tend to follow the bedding in a sub-horizontal direction. By forcing a circular failure surface, we are probably over-estimating the factor of safety. We can easily test this by specifying a non-circular failure surface and observing the results.

1. Go back to the Slide2 Model program.
2. Select Surfaces > Surface Options.
3. Under Surface Type, choose Non-Circular.
4. Under Search Method, choose Particle Swarm Search.
   
   This method will automatically search for the critical non-circular surface. See the Surface Options help page for more information.
   
   
5. Click OK to close the dialog.

6. Compute

1. Save the model and then select Analysis > Compute.
2. Select Analysis > Interpret to view the results.

7. Interpret

You can see that the factor of safety is about 1.24. Notably less than the value of 1.48 calculated assuming a circular surface. It is also interesting to observe the shape of the critical surface – a section of sub-horizontal slip connected to the ground surface by a steep incline. If you query the slices that make up the sub-horizontal section, you will see that the base material for each slice is Material 2 (Bedding).

This shows the importance of using a non-circular failure surface in anisotropic models since the failure surface ‘seeks out’ the weak bedding orientation to yield a lower factor of safety.

8. Input Type = Angle or Surface, A, B

One assumption of the Input Type = Angle range option, is that the strength of applied to the base of the slice is discrete, meaning that if the angle is in one range it takes the strength of one material, and in the other it takes the strength of the other material. However, there may be cases where you would want to consider an interpolation of the strengths of the Soil Mass and Bedding materials.

1. Return to the Slide2 modeler.
2. Right-click on Group 1 – Master Scenario in the Document Viewer and select Add Scenario.
3. Rename this child scenario “Input = Angle, A, B” using the right-click option.
4. Ensure you are in the child scenario. Select Material 4 and name it “Angle, A, B”. Now select the following:
• Strength Type = Generalized Anisotropic
• Input Type = Angle or surface, A, B
5. Click on the pencil to define the function. Users of Slide3 will recognize this dialog as the same on used in Slide3’s Generalized Anisotropic function.
6. Define the following:
• Name = Angle, A, B
• Base Material = Soil Mass
• Anisotropy Definition = Angle
7. And input the following row in the grid:
• Angle = 0
• A = 10
• B = 10
• Material = Bedding

The dialog should look as follows:

The way this is interpreted is as follows. At an angle of “Angle +/- A” (i.e. 0 +/- 10), the strength of Bedding will be applied. Otherwise, Base Material (i.e. Soil Mass) will be applied. This is identical to the function we defined previously using the Input Type = Angle Range option. (Note that due to the algorithmic difference, the results between this definition of anisotropy and the previous may exhibit a negligible numerical difference).

8. Now set B to 40.
   
   This means that in the ranges listed below, the strengths of Soil Mass and Bedding will be interpolated:
• [Angle + A, Angle + B] = [0 + 10, 0 + 40] = [10, 40]
• [Angle – A, Angle – B] = [0 – 10, 0 – 40] = [-10, -40]
9. Click OK, and click OK again in the Define Material Properties dialog.
10. Right-click on the slope and select Assign Material > Angle, A, B. Compute the model and select Analysis > Interpret to view the results.
    
    Notice that the FS has gone down considerably, since we have gradually expanded the anisotropic range.
    
    
11. Select Query > Query Slice Data and click on a slice towards the middle.
    
    Notice that the base cohesion is a value between the 0 and 5 kPa of the Bedding and Soil Mass materials respectively; it has been interpolated.
    
    In the Slice Data dialog, you can also see that the “Angle of Slice Base” falls in the [10, 40] range as expected.
    
    
12. Return to the modeler and define another child scenario named “Input = Surface, A, B.” Ensure you have the scenario selected.
    
    We will define the same case with an anisotropic surface.
    
    
13. Select Boundaries > Add Anisotropic Surface.
14. Input the points: (-2, 20) and (132, 20), pressing Enter after each one. Press Enter again to finish.
15. Double-click on the slope to open the Define Material Properties dialog and now select Material 5. Name it “Surface, A, B”.
16. Now select the following:
• Strength Type = Generalized Anisotropic
• Input Type = Angle or surface, A, B
17. Click on the pencil to define a new function. In the Define Generalized Strength Function dialog, select the green plus button in the bottom left to add a new function.
18. Define the following:
• Name = Surface, A, B
• Base Material = Soil Mass
• Anisotropy Definition = Surface
19. And input the following row in the grid:
• Surface = Anisotropic Surface 1
• A = 10
• B = 40
• Material = Bedding

The dialog should look as follows:

This can be interpreted in the same way as our previous function. Instead of inputting the angle ourselves, the program will locate the point on the anisotropic surface that is closest to each slice base, and use the angle of the surface instead. A and B will then be used in the same way as before. Our anisotropic surface is horizontal, so the angle will be 0 for all slice bases.

20. Click OK. In the Define Material Properties dialog, ensure the Generalized Function is set to “Surface, A, B”. Click OK.
21. Right-click on the slope and select Assign Material > Surface, A, B.
22. Compute the model and select Analysis > Interpret to view the results.

As expected, the slip surface and factor of safety are identical to the previous scenario.

9. Probabilistic Analysis

Go back to the Slide2 Model program and select the master scenario in the Document Viewer (the orange slope). We now assume that the friction angle of the bedding orientation is not well known, and we will determine the probability of failure for a given distribution of friction angles for the bedding.

1. Open the Project Settings dialog from the Analysis menu.
2. On the left side click on Statistics. Check the box for Probabilistic Analysis. Leave the analysis type as Global Minimum.
   
   This will find a critical surface deterministically and then will calculate the probability of failure using this surface with varying material properties. To re-compute the critical failure surface for each randomization of material properties you could choose Overall Slope. Since this takes a longer time to compute, it will be left as an additional exercise.
   
   
   
3. Click OK to close the dialog.

We are now going to define the statistical distribution of the strength of the bedding layer.

1. Go to Statistics > Materials.
2. We are going to vary the Bedding strength so click on Bedding, then click on the Add button.
3. The cohesion of the bedding is 0, so we will only alter the friction angle. Check the box for Phi.
   
4. Click Next for Statistical Distribution, choose Normal.
5. Now click Finish.
6. You now need to enter the Mean and Standard Deviation for the distribution. The Mean is automatically set to a deterministic value from the previous analysis (20o), so we only need to set the standard deviation. Enter 5. Now you can automatically set the maximum and minimum to 3 standard deviations by clicking the 3x Std Deviation button on the right. The dialog should look like this:
   
   
7. Click OK to close the dialog.

10. Compute

1. Save the model and then select Analysis > Compute.
2. Select Analysis > Interpret to view the results.

11. Interpret

You will now see the deterministic global minimum failure surface along with some statistical data.

You can see that the mean factor of safety (1.25) and deterministic factor of safety (1.24) are nearly the same and that the probability of failure (PF) is 10.5%.

You can look at the distribution of safety factors by going to:

1. Statistics > Histogram Plot.
2. Set the Data to Plot = Factor of Safety – spencer.
3. Select the Highlight Data checkbox.
4. As the highlight criterion, select “Factor of Safety – spencer” and set the criterion to < 1.
   
5. Select the Plot button, and the Histogram will be generated as shown.

You can see a normal distribution of safety factors with about 10% of the area shown in red (safety factor less than 1). Because the Latin Hypercube method samples the input distributions smoothly (compared to Monte Carlo), the output, in this case, is also a relatively smooth normal distribution.

Right-click on the plot and select Change Plot Data. Plot the Bedding friction angle. As expected, the low safety factors (< 1) correspond to low sampled values of the bedding friction angle.

This concludes the tutorial.