# 6 - Joint Persistence Analysis

In this tutorial, we look at the Probabilistic analysis of a slope, taking into account the persistence distribution of the joints and the spatial variation of the location of the joints. We then compare these results with the results of a standard **SWedge** analysis that uses maximum wedge size.

Topics Covered in this Tutorial:

- Persistence Analysis
- Limiting Wedge Size
- Slope Length
- Bench Width
- Persistence Distribution
- Exponential Distribution
- Wedge Spatial Location

Finished Product:

The finished product of this tutorial can be found in the *Tutorial 06 Persistence Analysis.swd7* file, located in the *Examples > Tutorials* folder in your **SWedge** installation folder.

## 1.0 Introduction

In a standard **SWedge** analysis, the program always tries to determine the maximum size wedge (i.e., it automatically scales a wedge to fit the dimensions of the slope). Joint **Persistence**, joint **Trace Length**, wedge **Volume**, and wedge **Weight** can also be used to scale the wedge size. In all cases, the program generates a wedge if possible.

In a Probabilistic analysis, this has a conservative effect on the Probability of Failure. In many cases, the spatial location and persistence of the joints limits the existence of valid wedges. In the case of a slope with a particular **Length**, **Height**, and **Bench Width**, only certain spatial locations of the joint planes result in valid wedges. In the case of **Persistence**, a wedge does not form if the joint **Persistence** is below a certain value.

**SWedge** has the ability to take these factors into consideration in a Probabilistic analysis. Using the **Persistence Analysis** option, you can both spatially vary the location of the joint planes and define statistical distributions of either **Persistence** or **Trace Length**. Using this option, you can refine your calculation of the Probability of Failure.

## 2.0 Creating a New File

If you have not already done so, run the **SWedge** program by double-clicking the **SWedge** icon in your installation folder or by selecting **Programs > Rocscience > SWedge > SWedge** in the Windows **Start** menu.

When the program starts, a default model is automatically created. If you do NOT see a model on your screen:

**Select:** File > New

Whenever a new file is created, the default input data forms valid slope geometry, as shown in the image below.

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

Notice the four-pane, split screen format of the display, which shows **Top**, **Front**, **Side** and **Perspective** views of the model. This view is referred to as the **Wedge View**. The **Top**, **Front** and **Side** views are orthogonal with respect to each other (i.e., viewing angles differ by 90 degrees.)

## 3.0 Model

### 3.1 PROJECT SETTINGS

Let's start by setting up the project in the **Project Settings** dialog. Select **Project Settings** on the toolbar or on the **Analysis** menu.

**Select:** Analysis > Project Settings

- Select the
**General**tab. - Change the
**Analysis Type**to Probabilistic. - Set Units =
**Metric, stress as MPa**. - Select the
**Sampling**tab. - Make sure
**Sampling Method**=**Latin Hypercube**and**Number of Samples**=**10000**. - Select the
**Random Numbers**tab. - Make sure
**Random Number Generation method**=**Pseudo-random**and that the**Specify Seed**check box is UNCHECKED. - Click
**OK**.

### 3.2 INPUT DATA

Now let’s define the slope and joint properties in the **Input Data** dialog. To open the dialog, select **Input Data** on the toolbar or on the **Analysis** menu.

**Select:** Analysis > Input Data

- Select the
**Slope**tab. - Set
**Dip Mean Value**=**65**,**Dip Direction Mean Value**=**185**, and**Slope Properties****Height**=**30**. - Make sure
**Statistical Distribution**for both**Dip**and**Dip Direction**are set to**None**. - Make sure
**Slope Properties****Unit Weight (MN/m****3)**=**0.026**. - Select the
**Slope Properties Length (m)**check box and enter**50**. - Select the
**Upper Face**tab. - Set
**Dip Mean Value**=**0**,**Dip Direction Mean Value**=**185**. - Make sure
**Statistical Distribution**for both**Dip**and**Dip Direction**are set to**None**. - Select the
**Bench Width**option and set**Width (m) = 30**. - Select the
**Joint1**tab. - Change the
**Orientation Definition Method**to**Fisher Distribution**. - Set
**Mean Dip Value**=**45**and**Mean Dip Direction Value**=**125**. - Enter a
**Standard Deviation**of**10**. - Keep
**Waviness Mean Value = 0**and**Statistical Distribution = None**. - Select the
**Joint 2**tab. - Change the
**Orientation Definition Method**to**Fisher Distribution**. - Set
**Mean Dip Value**=**70**and**Mean Dip Direction Value**=**225**. - Enter a
**Standard Deviation**of**10**. - Keep
**Waviness Mean Value = 0**and**Statistical Distribution = None**. - Select the
**Strength1**tab. - Change the
**Strength Model**to**Barton-Bandis**and make sure**Random Variables**=**Parameters**. - Set
**JRC Mean Value**=**6**,**JCS Mean Value**=**50**, and**Phir Mean Value**=**25**. - Make sure all
**Statistical Distribution = None**. - Select the
**Strength 2**tab and enter the same parameters as**Strength 1** - Click the
**OK**button to save your changes, compute the probability of failure, and close the**Input Data**dialog.

## 4.0 Analysis Results

Computation of 10000 Latin Hypercube samples occurs and the Probability of Failure is calculated, as shown in the figure below.

Note the following:

- The only
**Random Variables**are the orientation of the joints. - The results of the Probabilistic analysis are summarized in the
**Wedge Information Panel**in the application**Sidebar**. The wedge computed using the mean values of all the input parameters is displayed with its Factor of Safety results. - The
**Probability of Failure**is 0.0995 or 9.95%. Out of the 10000 samples, 9969 produced**Valid Wedges**. Of these 9969, 995 had a Factor of Safety of less than 1.0 (failed), calculated as #failed/total #samples = Probability of Failure =995/10000=0.0995. - The
**Mean Wedge**has a**Factor of Safety**of 1.27 - It's important to remember that
**SWedge**does its best to fit a wedge inside the slope dimensions, i.e., tries to determine a maximum size wedge for any slope geometry. It adjusts the location of the joints in order to determine this maximum size wedge. - The
**Probability of Failure**is conservative (upper bound solution) since positional information and joint length are not accounted for and the maximum size wedge is computed.

## 5.0 Persistence Analysis

As just noted, both the joint length and the positional location of the joints are not accounted for by default. This results in a conservative upper bound solution for the Probability of Failure. To refine the computation, **SWedge** allows you to define statistics for both the joint location and joint length (in terms of either **Persistence** or **Trace Length**).

In addition, two different **Joint Spacing** options are available for a Persistence analysis:

- Large Joint Spacing
- Small Joint Spacing (Ubiquitous Joints)

### 5.1 LARGE JOINT SPACING

With the **Large Joint Spacing** option, it is assumed there is only one trace of **Joint 1** and one trace of **Joint 2** on the slope face. The point of intersection of the two joint planes on the slope face is randomly located somewhere between the toe and crest of the slope, resulting in a uniform distribution of wedge height (measured vertically from this intersection point to the slope crest).

Once the intersection point has been selected, orientations for each joint are determined. If a valid wedge forms, the required joint length (JR) is calculated for each joint and compared to the deterministic or sampled joint length from user inputs (JS). If the required length exceeds the sampled length, the wedge is considered “invalid”. Otherwise the wedge is valid and the Factor of Safety is calculated.

Note that, if the spatial location of the joints is not uniformly random, the wedge height should NOT be uniformly varied, as this could lead to an under- or over-estimation of the Probability of Failure.

The **Large Joint Spacing** option is a lower bound solution when it comes to Probability of Failure, as the spacing and persistence condition limits the formation of wedges.

### 5.2 SMALL JOINT SPACING

Now consider the case where there is spacing (repeated joints) associated with the two joint sets **Joint 1** and **Joint 2**. There is no longer one wedge but a number of possible wedges that can form on the slope. If a wedge cannot form because the **Persistence** is not large enough, a wedge that meets the **Persistence** conditions can form higher up the slope face. If this is the case, the **Small** or **Ubiquitous Joint Spacing** option is more applicable. This option automatically scales down the wedge size until the **Persistence** conditions are met. Therefore, a wedge is almost always formed in each simulation if the geometry of the joints and slope creates a kinematically feasible wedge. Its size is dependent on the sampled Persistence and the geometry of the bench.

The **Small Joint Spacing** option is an upper bound solution for Probability of Failure because the program always creates a wedge independently of any spatial location of the joints on the slope face. The only thing that limits the size of the wedge is the geometry of the bench and the **Persistence** of the joints.

### 5.3 PERSISTENCE ANALYSIS

We'll now perform a Persistence analysis using the **Large Joint Spacing** option.

- Select
**Analysis > Persistence Analysis**to open the**Persistence Analysis**dialog. - Select the
**Persistence Analysis**check box. - Make sure the
**Persistence**option is selected for**Joint Length**. - Make sure
**Large**is selected for the**Joint Spacing**option.

By selecting this option, the location of the joint planes varies uniformly throughout the slope face. The point of intersection of the two joint planes on the slope face is randomly located somewhere between the toe and crest of the slope. The height of the wedge, measured vertically from this intersection point to the slope crest, is a uniform distribution varying between zero and the slope height. - Select both the
**Joint 1 Persistence**and**Joint 2 Persistence**check boxes and set the**Statistical Distribution**for both to**Exponential**. - For both
**Joint 1 Persistence**and**Joint 2 Persistence**, enter a**Mean Value**of**20**, a**Relative Minimum**of**20**, and a**Relative Maximum**of**40**.

This results in the**Joint Persistence**varying exponentially between 0 and 50 m with a mean of 20 m. - Click
**OK**to save your changes, compute the Probability of Failure, and close the**Persistence Analysis**dialog.

## 6.0 Persistence Analysis Results

Computation of 10000 Latin Hypercube samples occurs and the Probability of Failure is calculated. For every one of the 10000 simulations, a value of **Joint Persistence** is sampled. Because the **Large Joint Spacing** option is selected, if the wedge that is formed based on the random location of the joint planes has **Joint Persistence** values that exceed the sampled values, the wedge is flagged as invalid.

The figure below illustrates the results of this computation.

Note the following:

- The random variables now include the height of the wedge and the
**Maximum Persistence**of the joints. - The
**Probability of Failure**is reduced from 0.0995 to 0.0035 (28 times smaller). - The
**Number of Valid Wedges**has been reduced from 9969 to 1874. This is because the program no longer tries to fit a wedge inside the slope dimensions by adjusting the location of the joints. If the location and length of the joint planes are not sufficient to produce a valid wedge, the wedge is flagged as invalid. - The
**Number of Failed Wedges**is reduced from 995 to 35 for the same reasons. - In general, the Probability of Failure from a Persistence analysis is more accurate because it accounts for the spatial location and length of the joint planes.

**NOTE:** If the spatial location of the joints is not uniformly random and more than one possible wedge can form on the bench, do NOT select the **Large Joint Spacing** option. This could lead to an underestimation of the Probability of Failure. For example, if the joints have a small spacing such that many repeating joints for a particular joint set exist on a bench face, you should use the **Small (Ubiquitous) Joint Spacing** option.

## 7.0 Histogram Plots of Results

A number of useful features exist for the display of data from a Probabilistic analysis. For example, to get an idea of the relative distribution of failed to stable samples, we can plot a histogram of the Factor of Safety. To open the **Histogram Plot Parameters** dialog, select **Plot Histogram** on the toolbar or on the **Statistics** menu.

**Select:** Statistics > Plot Histogram

- Leave the
**Data Type**as**Safety Factor**. - Click the
**OK**button.

A**Safety Factor**histogram is displayed. - Right-click in the histogram and select
**3D Histogram**from the popup window.

Now let’s plot the maximum **Persistence** of **Joint 1** that was statistically generated as part of the Persistence analysis.

- Re-open the
**Histogram Plot Parameters**dialog. - In the
**Data Type**drop-down, select**Maximum Persistence Joint1**at the bottom of the list. - Select the
**Plot Input Distribution**check box. - Click the
**OK**button to generate the histogram. - Right-click in the plot and deselect
**Relative Frequency**from the popup menu.

This changes the y-axis of the histogram from**Relative Frequency**to the actual**Frequency**(i.e., the number of samples). - Right-click in the plot again and select
**Input Distribution**from the popup menu.

This is a plot of the sampled Maximum Persistence for Joint 1 for all VALID wedges. As mentioned above, for each one of the 10000 simulations, a **Maximum Persistence** value of the joint is sampled. If the wedge that is formed based on the random location of the joint planes has an actual **Joint Persistence** value that exceeds the sampled value, the wedge is flagged as invalid.

Remember that we defined an exponential distribution of **Joint Persistence** that varied between 0 m and 60 m with a mean of 20 m. Note that the histogram does not match this input distribution (the green line). This is because it does not include the invalid wedges. Only the data from valid wedges is plotted by default.

To see the distribution of all 10000 sampled **Persistence** values, right-click in the plot and select **Include Invalid Wedges** in the popup menu. The plot now includes all the sampled data (10000 samples), and the histogram bars match the input distribution.

Finally, let's look at a plot of the actual **Persistence** values for all the valid wedges.

- Re-open the
**Histogram Plot Parameters**dialog. - In the
**Data Type**drop-down, select**Persistence Joint 1**. - Select the
**Best Fit Distribution**check box. - Click
**OK**.

Note that most of the wedges have small **Persistence** values but a few have larger values. The **Best Fit Distribution** is also exponential. This is what one would expect for a joint set with exponentially varying Persistence.

This concludes the tutorial.