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06 - Probabilstic Analysis

1.0 Introduction

In a Probabilistic Analysis, statistical input data can be entered to account for uncertainty in material unit weight, joint shear strength and joint water pressure. The result is a distribution of factors of safety, from which probability of failure is calculated for any given block formed from a single realization of the joint geometry (i.e., geometry of the blocks are deterministic across all sample runs; only material and joint property parameters are used as random variables).

Finished Product

The finished product of this tutorial can be found in the Probabilistic Analysis.rocslope_model file. All tutorial files installed with RocSlope can be accessed by selecting File > Recent Folders > Tutorials Folder from the RocSlope main menu.

2.0 Opening the Starting File

  1. Select File > Recent > Tutorials Folder.
  2. Go to the Probabilistic Analysis folder, and open the file Probabilistic Analysis - starting file.rocslope_model.

This model already has the following defined and provides a good starting point to start computing blocks:

  • External Geometry
  • Measured Joints

In the starting model, Measured Joints are defined over the west extents of open pit.

Starting model
3D View showing the Measured Joints over the West Extents of an Open Pit

2.1 Project Settings

Our first step is to configure the Statistics settings for the model in Project Settings.

  1. Select Analysis > Project Settings project settings icon
  2. Select the Units tab. Ensure Units are Metric, stress as MPa.
    project settings dialog
    Units tab in Project Settings dialog
  3. Select the Analysis tab.
    1. Set Design Factor of Safety = 1.5.
    2. Ensure Successive Failure is ON.
      project settings dialog
      Analysis tab in Project Settings dialog
  4. Select the Statistics tab.
    1. Select Probabilistic Analysis to turn on probabilistic analysis mode.
    2. Leave the default Sampling Method = Latin-Hypercube and default Number of Samples = 1000
  5. Click OK to save the settings and close the dialog.

2.2 External Geometry

The External is of a pit shell and composed of one volume assigned with the Schist material property. The Schist material property is currently defined as deterministic.

2.3 Measured Joints

Review the Measured Joints.

  1. Select Joints > Define Measured Joints define measured joints icon
  2. 20 Measured Joints are defined are listed in order of Dip, Dip Direction, X, Y, Z, Radius, and Joint Property:

    DipDip DirectionXYZRadiusJoint Property
    59 48 -390 1 1018 500 Rough
    77 131 -319 2 1004 500 Rough
    78 123 -313 4 1003 500 Rough
    78 313 -486 2 1037 500 Rough
    61 13 -793 4 1099 500 Rough
    89 255 -272 -3 994 500 Rough
    51 341 -375 -5 1015 500 Rough
    67 107 -548 47 1050 500 Rough
    57 86 -281 91 996 500 Rough
    82 105 -758 33 1092 500 Rough
    53 62 -546 -1 1049 500 Smooth
    46 84 -665 -2 1073 500 Smooth
    55 79 -772 -4 1094 500 Smooth
    29 90 -798 -4 1100 500 Smooth
    50 82 -305 193 1001 500 Smooth
    34 83 -699 -5 1080 500 Smooth
    39 87 -348 -3 1010 500 Smooth
    81 36 -258 -5 992 500 Smooth
    58 314 -544 -1 1049 500 Smooth
    39 91 -744 -6 1089 500 Smooth
    define measured joints dialog
    20 Joints defined in Define Measured Joints dialog

    The Rough and Smooth joint properties are currently defined as deterministic.

  3. Click Cancel to exit the dialog.

3.0 Random Variables

In order to run a Probabilistic Analysis, at least one random variable must be defined and used in the model. In RocSlope, the following can be set as a random variable:

  • Material Property parameters (e.g., Unit Weight, Ru Value)
  • The availability of random variables depends on the Groundwater Method selected.
  • Joint Property parameters (e.g., Cohesion, Phi, Waviness, Uniform Water Pressure)
  • The availability of random variables depends on the Strength Type and Water Pressure Method selected.

For this model, we will be modelling the variability in unit weight of the material and the shear strength of the joints.

3.1 Material Property Statistics

To assign statistics to Material Properties:

  1. Select: Materials > Define Materials define materials properties icon. The Define Materials dialog shows the mean parameter values.
  2. Select the Statistics statistics icon button.
  3. The Define Material Statistics dialog allows users to add any applicable inputs as random variables. Only one Material Property exists (i.e., Schist).

    The Define Material Statistics dialog can also be accessed in Statistics > Define Material Statistics material statistics icon

  4. Click the Add add icon button.
  5. In the Add Random Variables dialog:
    1. Select Unit Weight (MN/m3).
    2. Click OK. The Unit Weight Property is added to the grid with a Normal Distribution.
      add random variables dialog
      Add Random Variables dialog
  6. In order for the Unit Weight to be considered a valid random variable, a non-zero Standard Deviation, and Relative Minimum and/or Relative Maximum must be set for the Schist Material Property's Unit Weight:
    • Distribution = Normal
    • Mean = 0.026
    • Std. Dev. = 0.002
    • Rel. Min. = 0.005
    • Rel. Max. = 0.005
    • The Rel. Min. and Rel. Max is relative to the Mean. They are not the absolute min and max values.
      Define material statistics dialog
      Define Material Statistics dialog
  7. Click OK to close the Define Material Statistics dialog.
  8. Click OK again to close the Define Materials dialog.

3.2 Joint Property Statistics

To assign statistics to Joint Properties:

  1. Select Joints > Define Joint Properties joint properties icon. The Define Joint Properties dialog shows the mean parameter values.
  2. Click the Statistics statistics icon button.
  3. The Define Joint Property Statistics dialog allows users to add any applicable inputs as random variables. There are two Joint Properties.

    The Define Joint Statistics dialog can also be accessed in Statistics > Define Joint Property Statistics statistics joints icon

  4. For the Smooth Joint Property:
    1. Click the Add add icon button.
    2. In the Add Random Variables dialog:
      • Select Phi and Cohesion.
      • add random variables dialog
        Add Random Variables dialog
      • Click OK. The Cohesion and Phi properties are added to the grid with a Normal Distribution.
    3. For Cohesion:
      • Distribution = Normal
      • Mean = 0
      • Std. Dev. = 0.002
      • Rel. Min. = 0
      • Rel. Max. = 0.006
    4. For Phi:
      • Distribution = Normal
      • Mean = 10
      • Std. Dev. = 2
      • Rel. Min. = 6
      • Rel. Max. = 6
      • define joint property statistics dialog
        Smooth Joint Property statistics in Define Joint Property Statistics dialog
  5. For the Rough Joint Property:
    1. Select the Add add icon button.
    2. In the Add Random Variables dialog:
      • Select Phi and Waviness.
      • add random variables dialog
        Add Random Variables dialog
      • Click OK. The Phi and Waviness properties are added to the grid with a Normal Distribution.
    3. For Phi:
      • Distribution = Normal
      • Mean = 10
      • Std. Dev. = 2
      • Rel. Min. = 6
      • Rel. Max. = 6
    4. For Waviness:
      • Distribution = Normal
      • Mean = 3
      • Std. Dev. = 1
      • Rel. Min. = 3
      • Rel. Max. = 3
      • smooth joint property statistics
        Smooth Joint Property statistics in Define Joint Property Statistics dialog
  6. Click OK to close the Define Joint Statistics dialog.
  7. Click OK again to close the Define Joint Properties dialog.

4.0 Compute

RocSlope has a two-part compute process.

4.1 Compute Blocks

The first step is to compute the blocks which may potentially be formed by the intersection of joints with other joints and the intersection of joints with the free surface.

To compute the blocks:

  1. Navigate to the Compute workflow tab compute workflow
  2. Select Analysis > Compute Blocks compute blocks icon

As compute is run, the progress bar reports the compute status. Once compute is finished, the Results node is added to the Visibility Tree and All Valid Blocks are blocks are shown in the viewport. The Results node consists of the collection of valid blocks and the socketed slope. The original External and Measured Joints visibility is turned off.

compute blocks 3d view
3D View of all Valid Blocks

Once compute is finished, the blocks are coloured according to the Block Color option (Random Colors) set in the Results node's Property pane.

Compute Blocks only determines the geometry of the blocks. In order to obtain other information such as the factor of safety, Compute Kinematics needs to be run.

4.2 Compute Kinematics

The second and final compute step is to compute the removability, forces, and factor of safety for each of the valid blocks.

To compute the block kinematics:

  1. Ensure that the Compute workflow tab compute workflow is the active workflow.
  2. Select Analysis > Compute Kinematics compute kinematics icon

As compute is run, the progress bar reports the compute status. By default, after Compute Kinematics is run, only Removable Blocks are shown.

compute kinematics 3d view
3D View of Removable Blocks only

In this Probabilistic Analysis, for each block, the kinematics are computed 1,000 times, each time with statistically sampled inputs for Unit Weight of the Schist Material Property, Cohesion and Phi for the Smooth Joint Property, and Phi and Waviness for the Rough Joint Property, according to their respective random variables' distributions.

The number of the times the kinematics are computed is determined by the Number of Samples set in the Project Settings: Statistics tab. In this example, it is set to default value of 1000, as specified in Section 2.1 above.

5.0 Interpreting Results

Since block geometry does not change between probabilistic runs (i.e., deterministic joint geometry), the visualization of the Results in the 3D CAD View is representative of all possible block geometries.

Once both blocks and kinematics are computed, all block results can be viewed in a grid format.

    5.1 Block Information

    To view all block results:

    1. Navigate to the Results workflow tab results workflow
    2. Select Interpret > Block Information block information icon

    Visualizing blocks can be difficult when the slope extents are large compared to the block extents.

    To zoom into all blocks:

    • Select Interpret > Zoom To All Blocks zoom all blocks icon

    The Block Information pane shows the collection of blocks according to the Results Set settings. The Results Set shown can be selected in the Results tab of the Display Options, or the Properties pane for the Results Node. In this case, only Removable Blocks are coloured and listed in Block Information.

    In the case of a Probabilistic Analysis, for any given block, the Factors of Safety are affected by the random variables being sampled in each run. This then affects which blocks are considered "Failed" (i.e., Factor of Safety < Design Factor of Safety). In the case of Successive Removal, this also impacts the Removability of blocks which can only be removed if key blocks are removed. For these reasons, the definitions of the Results Set displayed are modified for a Probabilistic Analysis as follows:

    • All Valid Blocks: Identical as Deterministic Analysis since block formation is independent of the random variables for material and joint properties.
    • Removable Blocks: For a given block, if any sample run results in the block being Removable, then the block is included in the Results Set.
    • Failed Blocks (FS < Design FS): For a given block, if any sample run results in the Factor of Safety < Design Factor of Safety, then the block is included in the Results Set.
    block information pane
    Block Information pane showing probabilistic block results

      For Probabilistic Results, only the critical values among all sample runs are reported for each block:

      • Minimum Factor of Safety
      • Maximum Weight
      • Maximum Required Support Pressure
      • Maximum Failure Depth
      • Probability of Failure
      • Total Removable
      • Total Failed

      5.2 Contour Blocks

      In RocSlope, blocks can be contoured by several metrics. In a Probabilistic Analysis, blocks can be contoured by any of the critical values.

      To show block contours:

      1. Select Interpret > Contour Blocks contour blocks icon
      2. From the Legend pane to the right, select Probability of Failure. The blocks are contoured by the Probability of Failure = Total Failed / Number of Samples.
        blocks contoured by probability of failure
        3D View of Removable Blocks contoured by Probability of Failure
      3. From the Legend pane to the right, select Max. Support Pressure.
        blocks contoured by max support pressure
        3D View of Removable Blocks contoured by Max. Support Pressure

      Note that probabilistic results, like deterministic results, are location-specific.

      6.0 Statistical Plots

      Input or output distributions can be charted in the following forms:

      • Histogram Plot
      • Scatter Plot
      • Cumulative Plot
      Any random variables can be plotted in addition to computed block metrics such as Factor of Safety, Weight Required Support Pressure, Failed Depth, and Slope Face Area.

      For each type of plot, the blocks considered in the plot data can be one of the three:

      1. Current Result Set (blocks considered in All Valid Blocks, Removable Blocks, or Failed Blocks, as selected in Display Options)
      2. Single Block (a single block is considered, identified by Block ID)
      3. Filtered Blocks

      6.1 Histogram Plot

      Histogram Plots allow users to see the ordered frequency of a set of block data.

      To plot a histogram plot:

      1. Select Statistics > Histogram Plot histogram icon
      2. In the Histogram Plot dialog:
        1. Set Data to Plot = Factor of Safety
        2. Set Number of Bins = 30
        3. Set Blocks to Plot = Single Block and Block ID = 56
        4. Select the Highlight Data checkbox and set the Factor of Safety < 1.5 to highlight any values less than the Design Factor of Safety.
        5. Select the Best Fit Distribution checkbox to plot the fitted distribution
        6. Select Relative Frequency to scale the histogram Frequency axis such that the area under then distribution = 1. Otherwise, the Frequency is simply the count.
          histogram plot dialog
          Histogram Plot dialog
        7. Click Plot to generate the histogram.
      histogram plot of block id 56
      Factor of Safety Histogram Plot of Block ID 56

      Relative Frequency is plotted on the y-axis, while Factor of Safety is plotted on the x-axis, lumped into 30 bins. The Best Fit Distribution shows the distribution type and parameters which best fit the data. The highlighted bars indicate the Relative Frequency of Factor of Safety < 1.5 (blocks which are kinetically unstable); all sample runs for this block results in a Factors of Safety < 1.5.

      6.2 Scatter Plot

      Scatter Plots allow users to see the correlation between two sets of block data.

      To plot a scatter plot:

      1. Navigate back to the 3D Geometry View by selecting the tab below the toolbar at the top of the screen.
      2. Select Statistics > Scatter Plot scatter icon
      3. In the Scatter Plot dialog:
        1. Set Horizontal Axis = Factor of Safety
        2. Set Vertical Axis = Smooth: Phi
        3. Set Blocks to Plot = Single Block and Block ID = 56
        4. Select the Highlight Data checkbox and set the Factor of Safety < 1.5 to highlight any values less than the Design Factor of Safety.
        5. Select the Regression Line checkbox to plot the line of best fit.
        6. scatter plot dialog
          Scatter Plot dialog
        7. Click Plot to generate the scatter plot.
      scatter plot of factor of safety
      Factor of Safety Scatter Plot of Block ID 56

      The Regression Line and tight clustering of the scatter plot data points around that line indicates that there is a strong correlation between Factor of Safety and Friction Angle of the Smooth joint property.

      6.3 Cumulative Plot

      Cumulative Plots allow users to see the cumulative probability of a set of block data.

      To plot a cumulative plot:

      1. Navigate back to the 3D Geometry View by selecting the tab below the toolbar at the top of the screen.
      2. Select Statistics > Cumulative Plot cumulative icon
      3. In the Cumulative Plot dialog:
        1. Set Data to Plot = Weight
        2. Set Number of Bins = 30
        3. Set Blocks to Plot = Current Result Set
          cumulative plot dialog
          Cumulative Plot dialog
        4. Click Plot to generate the cumulative plot.
      cumulative plot dialog
      Weight Cumulative Plot of Removable Blocks

      The cumulative plot shows the cumulative distribution of block Weight values for the Removable Results Set. At any given Weight value, the Cumulative Probability is the percentile (as a fraction) of blocks sampled which have a Weight less than or equal to that value. Vice versa, looking at some percentile we can get the corresponding Weight value (e.g., 90th percentile would correspond to approx. Weight = 3000 MN).

      This concludes Tutorial 06.

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