# Probabilistic Analysis

This tutorial will familiarize the user with the Probabilistic Analysis features of **UnWedge**. In a Probabilistic Analysis, you can define statistical distributions for input parameters (e.g., Joint Properties, Field Stress, Water Pressure, Bolt Properties) to account for uncertainty in their values. The computed analysis results in a distribution of factors of safety for each wedge, from which Probabilities of Failure (PF) are calculated.

Topics Covered in this Tutorial:

- Project Settings
- Random Variables
- Fisher Distribution
- Mean Wedges
- Picked Wedges
- Histograms
- Scatter Plots
- Design Factor of Safety

Finished Product:

The finished tutorial can be found in the *Tutorial 07 Probabilistic Analysis.weg5* file located in the *Examples > Tutorials* folder in your **UnWedge** installation folder.

## 1.0 Model

If you have not already done so, run the **UnWedge **program by double-clicking the **UnWedge **icon in your installation folder or by selecting **Programs > Rocscience > UnWedge > UnWedge** in the Windows **Start **menu. When the program starts, a default model is automatically created.

If the **UnWedge **application window is not already maximized, maximize it now so that the full screen is available for viewing the model.

### 1.1 PROJECT SETTINGS

Let's start by configuring basic parameters in the **Project Settings** dialog.

- Select
**Projects Settings****Analysis**menu. - In the
**General**tab, set**Analysis Type = Deterministic**. - Make sure
**Units**is set to**Metric, stress as MPa**.

#### 1.1.1 Sampling and Random Numbers

- Select the
**Sampling**tab.

The**Sampling Method**determines how the statistical distributions for the random input variables will be sampled. - Leave the default
**Sampling Method**=**Latin Hypercube**and**Number of Samples**=**10,000**as is. - Select the
**Random Numbers**tab. - Do not make any changes in this tab but note that the default for
**Random Number Generation**is**Pseudo-Random**by default.

This allows you to obtain reproducible results for a Probabilistic Analysis, by using the same “seed” value to generate random numbers. We will discuss**Pseudo-random**versus**Random**sampling later in this tutorial.

#### 1.1.2 Design Standards

- Select the
**Design Standard**tab. - Leave all the settings as is.

Eurocode 7 is a design document that establishes rules and standards for geotechnical engineering design across Europe (BSI, 2004). Eurocode 7 represents a major change in design philosophy. Traditionally a single, lumped Factor of Safety accounts for all of the uncertainties in the problem. With Eurocode 7, partial factors of safety are applied to different components of the analysis. The partial factors are applied prior to the analysis to give design values that are used in the calculation. The final result is an over-design factor, which must be greater than 1 to ensure the serviceability limit state requirement is satisfied. For more information on using Eurocode 7 in geotechnical design, see Smith (2006) and Bond and Harris (2008). This tab allows the user to design using Eurocode 7 specifications.

#### 1.1.3 Project Summary

- Select the
**Project Summary**tab. - Enter
**UnWedge Probabilistic Analysis Tutorial**as the**Project Title**. - Select
**OK**to save your settings and close the**Project Settings**dialog.

**NOTE:** The **Project Summary** information can be displayed on printouts of analysis results using the **Page Setup** option in the **File** menu and defining a Header and/or Footer.

### 1.2 MODEL GEOMETRY

We will start by creating the two-dimensional cross-section of the excavation we want to analyze. The cross-section can either be imported as a DXF file or defined within the program using the **Add Opening Section** option. For this tutorial, we will use the **Add Opening Section** option.

#### 1.2.1 Set View Limits

First, we'll set **View Limits**.

- Make sure
**Opening Section**is selected in the**View**dropdown on the toolbar. - Select
**View > View Limits**.

The**View Limits**dialog appears. - Enter
**250, 300**for the**Minimum X, Y Coordinates**and**280, 300**for the**Maximum X, Y Coordinates**. - Select
**OK**.

#### 1.2.2 Add the Opening Section

Now we can add the **Opening Section**.

- Select
**Add Opening Section****Boundaries**menu. - In the
**Prompt Line**at the bottom right of the screen, type**t**and hit**ENTER**.

The**Enter Coordinates**dialog appears. - Enter coordinates as shown below.
- Select
**OK**. - In the
**Prompt**, type**a**and hit**Enter**.

The**Arc Options**dialog appears. - Under
**Arc Defintion Method**, select**3 points on arc**. - Under
**Arc to Polyline Conversion Method**, set**Number of Segments = 12**. - Select
**OK**. - In the
**Prompt**, type**271, 320**and hit**ENTER**. - In the
**Prompt**, type**264.5, 317**and hit**ENTER**. - In the
**Prompt**, type**c**to close the opening and hit**ENTER**.

Your screen should look like this.

The **Opening Section** boundary should be automatically zoomed to the center of the view. If it is not, select **Zoom All** (or press the **F2 **function key) to zoom the excavation to the center of the view.

## 2.0 Probabilistic Input Data and Distributions

Now we'll look at the default **Joint Orientations** and **Joint Properties** default input data in the **Input Data** dialog and then define the following as random variables using the **Statistics** options. All other model input parameters will be assumed to be “exactly” known (i.e., **Statistical Distribution = None**) and will not be involved in the statistical sampling.

- Joint 1 Orientation
- Joint 2 Orientation
- Joint 3 Orientation
- Phi of Joint Properties 1

### 2.1 VIEW DEFAULT INPUT DATA

- Select
**Input Data****Analysis**menu. - In the
**General**tab, note the**Design Factor of Safety**.

This option is used in probabilistic analyses for determining the probability of failure and required support pressure. The probability of failure is P(FS < Design FS). - View the default values in the
**Joint Orientations**and**Joint Properties**tabs but leave them as is. - Select
**Cancel**to close the**Input Data**dialog.

### 2.2 DEFINE JOINT ORIENTATION VARIABILITY

Now we'll define variability of the **Joint Orientations** using the **Joint Orientation Statistics** dialog. To carry out a Probabilistic Analysis with **UnWedge**, at least one input parameter must be defined as a random variable.

- Select
**Joint Orientations**on the**Statistics**menu.

The**Joint Orientation Statistics**dialog appears. - To define a random variable for the
**Joint 1 Orientation**, click on**Joint 1 (Joint Properties 1)**on the left.**NOTE**: There are TWO methods of defining the variability of joint orientation in an**UnWedge**Probabilistic Analysis:- Orientation Definition Method = Dip / Dip Direction
- Orientation Definition Method = Fisher Distribution

With the **Dip / Dip Direction** method, the **Dip** and **Dip Direction** are treated as independent random variables (i.e., you can define different statistical distributions for each one).

- Click the
**Add**button in the**Joint Orientations**dialog.

Notice that**Dip**has been added to the list of**Distributions**and that it has a**Normal Distribution**. By clicking on the**Distribution**dropdown, you can select various other distribution types. - Click the
**Add**button again to add**Dip Direction**to the list as another random variable. - Check the
**Use Fisher Distribution**checkbox.

The**Fisher Distribution**method generates a symmetric, three-dimensional distribution of orientations around the mean plane orientation. Only a single standard deviation is required. In general, a Fisher Distribution is recommended for generating random joint plane orientations because it provides more predictable orientation distributions and lessens the chance of input data errors. - Notice our previous
**Dip / Dip Direction**definitions are now overwritten. - Set the
**Standard Deviation**=**7**. - Repeat the steps for
**Joint 2**with a**Standard Deviation = 7**, and**Joint 3**with a**Standard Deviation = 10**. - Select
**OK**.

**NOTE:** Joint orientations can also be imported from Dips. This will be demonstrated in a future tutorial.

### 2.3 DEFINE JOINT PROPERTIES VARIABILITY

We will now define **Phi**, the angle of internal friction, for **Joint Properties 1** as a random variable.

- Select
**Joint Properties**on the**Statistics**menu.

The**Joint Properties Statistics**dialog appears.

Because all of our joints have the same properties,**Joint Properties 1**is the only property on the left of the dialog. - Click the
**Add**button.

The**Phi Property**is added to the list.

If you click on the**Property**dropdown, you can see all the other properties you can assign distributions to. - Leave
**Distribution = Normal**. - Set
**Std. Dev. = 5**and**Rel. Min = 15**and**Rel. Max = 15**. - Select
**OK**.

## 3.0 Compute

We are now ready to compute.

- Select
**Compute****Analysis**menu.

Using the **Latin Hypercube **sampling method, **UnWedge** generates **10,000** random input data samples for each random variable using the specified statistical distributions, and computes the probabilistic output for 10,000 possible wedge arrangements.

The calculation may take a few minutes. The progress of the calculation is indicated in the dialog.

## 4.0 Probabilistic Analysis Results

### 4.1 PROBABILITY VIEW

The results of the analysis can be studied in the **Probability View**.

- Select
**Probability View**in the**View**dropdown on the toolbar or on the**View > Select View**menu.

From the dropdown at the top of the **Sidebar**, we can see that the values in the **Wedge Info** panel represent the **Maximum Support Pressure**.

It is important to understand the significance of this cross-section. Due to the variability in our input data, 10,000 different wedge arrangements have been computed. **Support Pressure** is calculated for each segment from the 10,000 trials and the 10,000 possible wedges that may span this segment. *The maximum of these 10,000 possible values is represented by the number displayed on the segment.*

You can also use the **Percentile** option in the **Sidebar **to adjust which value of **Support Pressure** to show. The default **Percentile** is 100%, which is the maximum of all support pressure values computed for a particular segment. However, if you wanted to show the 95th percentile, where 95 percent of all support pressure values for this segment lie below this value, this is possible as well. Take note that **Maximum Support Pressure** and all probabilistic output is a function of location on the tunnel perimeter.

- In the
**Sidebar**dropdown, select**Probability of Failure**from the**Sidebar**.

Your screen should look as follows:

NOTES:

- Because the wedges on the sides of the tunnel have Factors of Safety greater than our
**Design Factor of Safety = 1**, their**Probability of Failure**is zero. - Similarly, because the
**Roof**wedge (8) had a**Factor of Safety**of 0.000, its**Probability of Failure**is 1.000.

### 4.2 WEDGE DISPLAY

- Select
**3D Wedge View**from the**View**dropdown on the toolbar.

The wedges initially displayed after a Probabilistic Analysis are based on the mean input values and are referred to as **Mean Wedges**. They will appear exactly the same as ones based on Deterministic **Input Data** and have the same **Factors of Safety**, as shown in the **Sidebar**.

### 4.3 HISTOGRAMS

To plot histograms of results after a Probabilistic Analysis:

- Select
**Plot Histogram****Statistics**menu.

The**Histogram Plot Parameters**dialog appears. - Select
**Factor of Safety**in from the**Data Type**dropdown. - Leave
**Location**as**Multiple Segments**. - Click
**Pick Segments**.

The cursor turns into a selection box. - Right-click on the screen and select
**Selection Window > Select Inside Only**. - Left-click and, keeping the button pressed, drag a box over just the roof section as shown.
- Release the mouse button and press
**ENTER**.

The resulting histogram represents the distribution of Factor of Safety for all valid wedges generated by the random sampling of the **Input Data**, for all segments comprising the roof of the excavation.

To view Failed Wedges:

- Right-click on the Histogram and select
**3D Histogram**from the popup menu. - Right-click again and select
**Show Failed Wedges**.

This option shows the failed wedges (FS < Design FS) along with the safe wedges (FS > Design FS) on the same Histogram. The failed wedges are depicted in red.

### 4.4 PICKED WEDGE

- Right-click on the Histogram and select
**3D Histogram**to turn the setting off**.** - Select
**New Window**on the toolbar or the**Window**menu.

This option opens a**3D Wedge View**and tiles the Histogram with all other open views. - Minimize the
**Probability View.** - Select
**Tile Vertically**from the toolbar such that you see the**3D Wedge View**and Histogram side-by-side as shown below.

A useful property of Histograms (as well as Cumulative Plots and Scatter Plots) is the following: If you double-click the LEFT mouse button anywhere on the plot, the nearest corresponding wedge is displayed in the **3D Wedge View** and results for the wedge are displayed in the **Sidebar**.

For example:

- Double-click on the Histogram somewhere where the Factor of Safety is about 5.

Notice that the wedge in reference is now displayed and has been highlighted. In the**Sidebar**, the analysis results are updated to display results for the wedge that you are viewing. The wedge has a Factor of Safety of about 5 in the**Sidebar**, as expected. - Double-click at various points along the Histogram, and notice the different wedges and analysis results that are displayed.

This feature allows you to view any wedge computation generated by the Probabilistic Analysis, corresponding to any point on a Histogram, Cumulative Plot, or Scatter Plot. In addition to the **3D Wedge View**, all other applicable views (for example, the **Info Viewer** and the **Stereonet View**) are also updated to display data for the currently Picked Wedge.

NOTES:

- This feature can be used on Histograms of any statistical data generated by
**UnWedge**, and not just the**Factor of Safety**Histogram - This feature also works on Scatter Plots and Cumulative Plots.

To reset the Mean Wedges display:

- Close the Histogram and
**3D Wedge View**. - Maximize the
**Probability View**. - Select
**View > Show Mean Wedges**.

### 4.5 HISTOGRAMS OF OTHER DATA

In addition to the **Factor of Safety**, you can also plot Histograms of:

- Other random output variables (e.g.,
**Wedge Weight**,**Support Pressure**) - Random input variables (i.e., any input data variable that was assigned a statistical distribution)

#### 4.5.1 Other Random Output Variables

- On the
**Sidebar**, change the display to**Maximum Wedge Depth**.

Again, note that due to the variability in our input data, 10,000 different wedge arrangements have been computed. Wedge Depth is calculated for each segment from the 10,000 trials. The maximum of these values is represented by the number displayed on the segment. It should also be noted that the Wedge Depth discussed here and the Apex Height which can be displayed in the**Sidebar**are synonymous. - Click on the roof segment on the left (with
**Maximum Wedge Depth = 12.01**).

The segment should turn red. - Select
**Plot Histogram**on the toolbar or the**Statistics**menu**.** - In the
**Histogram Plot Parameters**dialog, set**Data Type = Wedge Depth** - Ensure
**Single Segment**is selected. Set the**Single Segment = Segment 16: (265, 318) to (265, 317)**. - Select the
**Best Fit Distribution**checkbox. - Select
**OK**.

A histogram of the wedge depth and the best-fit distribution to the data is displayed. - Select the
**New Window**button from the toolbar or the**Window**menu.

You should now see a tiled view of the Histogram and the**3D Wedge View**. - Double-click on the Histogram where the
**Maximum Wedge Depth**is about**12 m**.

The wedge will be displayed on the **3D Wedge View**, as shown below.

We can now see from the **3D Wedge View** that the relatively large **Maximum Wedge Depth** for this segment and the adjoining segment to the right is due to a side wedge that intersects the upper left corner of the roof.

To reset the Mean Wedge display:

- Close the
**Wedge Depth**Histogram view and the**3D Wedge View**. - Maximize the
**Probability View**. - Select
**View > Show Mean Wedges**.

#### 4.5.2 Input Random Variables

Now let’s generate a Histogram of an input random variable.

- Select
**Plot Histogram**on the toolbar or the**Statistics**menu**.** - In the
**Histogram Plot Parameters**dialog, set**Data Type = Dip (Joint 1)**. - Select
**OK**.

The Histogram is depicted below.

For input random variables, the **Input Distribution** can be displayed on histograms. However, because the orientation of Joint 1 was generated using a Fisher Distribution, which is three-dimensional, the **Input Distribution** cannot be displayed on the Histogram, which is a two-dimensional plot of only one component (Dip) of the Joint 1 orientation.

- Return to the
**Probability View**by clicking on the**Probability View**tab on the bottom left of the screen.

### 4.6 SCATTER PLOTS

Scatter plots allow you to examine the relationship between any two analysis variables.

To generate a Scatter Plot:

- Select
**Plot Scatter****Statistics**menu. - In the
**Scatter Plot Parameters**dialog, select the variables you would like to plot on the X and Y axes. For example, let’s plot the**Factor of Safety**versus**Dip (Joint 1)**.- Set
**X Axis Dataset = Factor of Safety**. - Set
**Y Axis Dataset = Dip (Joint 1)**.

- Set
- Ensure
**Multiple Segments**is selected. - Select the
**Show Regression Line**option to display the best fit straight line through the data. - Click
**Pick Segments**. - Select all the roof segments using the mouse as before.
- Press
**ENTER**.

The following scatter plot should appear.

Note the **Correlation Coefficient** listed at the bottom of the plot, which indicates the degree of correlation between the two variables plotted. The **Correlation Coefficient** can vary between -1 and 1 where numbers close to zero indicate a poor correlation and numbers close to 1 or -1 indicate a good correlation.

**NOTE**: A negative **Correlation Coefficient** simply means that the slope of the best fit linear regression line is negative.

## 5.0 Compute (Random Sampling)

So far in this tutorial we have used the default **Pseudo-Random** sampling option. **Pseudo-Random** sampling allows you to obtain reproducible results for a Probabilistic Analysis by using the same “seed” value to generate random numbers. This is why you can obtain the exact values shown in this tutorial. We will now demonstrate how different outcomes can result from a Probabilistic Analysis by allowing a variable seed value to generate the random input data samples.

Before we start, let’s arrange the views as follows:

- Select
**the Tile Vertically**option from the toolbar or the**Window**menu to tile all of the open views.

You should have**Probability View**, Histogram Plot, and Scatter Plot open. - Click on the
**Probability View**and switch to the**3D Wedge View**.

You should see the following on your screen.

If your screen does not look similar to the figure (e.g. you have additional views open), then close all views except for the three noted above such that your screen resembles the figure.

To switch to Random Number Generation:

- Select
**Project Settings****Analysis**menu. - In the
**Project Settings**dialog, select the**Random Numbers**tab. - Change the
**Random Number Generation method**from**Pseudo-Random**to**Random**.**NOTE**: The**Random**option uses a different seed value to generate random numbers each time you re-run the Probabilistic Analysis. This will result in different sampling of your input random variables, and different analysis results (e.g.,**Probability of Failure**) each time you re-compute. - Select the
**Sampling**tab. - Decrease the
**Number of Samples**from**10,000**to**1,000**to make the change in results easier to see on the plots. - Select
**OK**. - Select
**Compute**on the toolbar.

Notice that the Histogram Plot and Scatter Plot are updated with new results.

**UnWedge** will only allow the user to select **Compute** if a change has been made to the input data. In order to see the way that data changes when using **Random Number Generation**, we want to select **Compute** repeatedly. To do this:

- Select
**File >****Preferences**and make sure the**Disable compute button when results are up to date**option is unchecked. - Select
**OK**. - Select
**Compute**repeatedly and observe how the windows are updated each time the analysis is re-run.

**NOTE:** The **3D Wedge View** does not change when you re-compute since, by default, the **Mean Wedges** are displayed, (i.e., the wedges based on the mean **Input Data**) which are not affected by re-running the analysis.

## 6.0 References

Bond, A. J. and Harris, A. J., 2008. *Decoding Eurocode 7*, Taylor & Francis.

British Standards Institution, 2004. *Eurocode 7: Geotechnical design – Part 1: General rules*, BS EN 1997-1, London, UK.

Smith, 2006. *Smith’s Elements of Soil Mechanics*, 8th Edition, Blackwell Publishing.

That concludes the **UnWedge** Probabilistic Analysis Tutorial.