# 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 RocSlope3 can be accessed by selecting **File > Recent Folders > Tutorials Folder** from the RocSlope3 main menu.

## 2.0 Opening the Starting File

- Select
**File > Recent > Tutorials Folder**. **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.

### 2.1 Project Settings

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

- Select
**Analysis > Project Settings** - Select the
**Units**tab. Ensure**Units**are**Metric, stress as MPa**.Units tab in Project Settings dialog - Select the
**Analysis**tab.- Set
**Design Factor of Safety = 1.5**. - Ensure
**Successive Failure**is**ON**.Analysis tab in Project Settings dialog

- Set
- Select the
**Statistics**tab.- Select
**Probabilistic Analysis**to turn on probabilistic analysis mode. - Leave the default
**Sampling Method = Latin-Hypercube**and default**Number of Samples = 1000**

- Select
- 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**.

- Select
**Joints > Define Measured Joints** - Click
**Cancel**to exit the dialog.

20 Measured Joints are defined are listed in order of **Dip, Dip Direction, X, Y, Z, Radius**, and
**Joint Property**:

Dip | Dip Direction | X | Y | Z | Radius | Joint 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 |

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

## 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 RocSlope3, the following can be set as a random variable:

**Material Property**parameters (e.g., Unit Weight, Ru Value)**Joint Property**parameters (e.g., Cohesion, Phi, Waviness, Uniform Water Pressure)

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:

- Select:
**Materials > Define Materials**. The**Define Materials**dialog shows the mean parameter values. - Select the
**Statistics**button. - Click the
**Add**button. - In the
**Add Random Variables**dialog:- Select
**Unit Weight (MN/m3)**. - Click
**OK**. The**Unit Weight**Property is added to the grid with a**Normal**Distribution.Add Random Variables dialog

- Select
- 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 - Click
**OK**to close the**Define Material Statistics**dialog. - Click
**OK**again to close the**Define Materials**dialog.

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**

### 3.2 Joint Property Statistics

To assign statistics to Joint Properties:

- Select
**Joints****> Define Joint Properties**. The**Define Joint Properties**dialog shows the mean parameter values. - Click the
**Statistics**button. - For the
**Smooth**Joint Property: - Click the
**Add**button. - In the
**Add Random Variables**dialog: - Select
**Phi**and**Cohesion**. - Click
**OK**. The**Cohesion**and**Phi**properties are added to the grid with a**Normal**Distribution. - For
**Cohesion**: **Distribution = Normal****Mean = 0****Std. Dev. = 0.002****Rel. Min. = 0****Rel. Max. = 0.006**- For
**Phi**: **Distribution = Normal****Mean = 10****Std. Dev. = 2****Rel. Min. = 6****Rel. Max. = 6**- For the
**Rough**Joint Property: - Select the
**Add**button. - In the
**Add Random Variables**dialog: - Select
**Phi**and**Waviness**. - Click
**OK**. The**Phi**and**Waviness**properties are added to the grid with a**Normal**Distribution. - For
**Phi**: **Distribution = Normal****Mean = 10****Std. Dev. = 2****Rel. Min. = 6****Rel. Max. = 6**- For
**Waviness**: **Distribution = Normal****Mean = 3****Std. Dev. = 1****Rel. Min. = 3****Rel. Max. = 3**- Click
**OK**to close the**Define Joint Statistics**dialog. - Click
**OK**again to close the**Define Joint Properties**dialog.

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**

## 4.0 Compute

**RocSlope3 **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:

- Navigate to the
**Compute**workflow tab - Select
**Analysis > Compute Blocks**

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.

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:

- Ensure that the
**Compute**workflow tab is the active workflow. - Select
**Analysis > Compute Kinematics**

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

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.

**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:

- Navigate to the
**Results**workflow tab - Select
**Interpret > Block Information**

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**

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.

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 RocSlope3, 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:

- Select
**Interpret > Contour Blocks** - 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.3D View of Removable Blocks contoured by Probability of Failure - From the
**Legend**pane to the right, select**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**

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

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

- Select
**Statistics >****Histogram Plot** - In the
**Histogram Plot**dialog:- Set
**Data to Plot = Factor of Safety** - Set
**Number of Bins = 30** - Set
**Blocks to Plot = Single Block**and**Block ID = 56** - Select the
**Highlight Data**checkbox and set the**Factor of Safety < 1.5**to highlight any values less than the Design Factor of Safety. - Select the
**Best Fit Distribution**checkbox to plot the fitted distribution - 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 - Click
**Plot**to generate the histogram.

- Set

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:

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

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:

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

- Set

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.