tutorial_06_persistence_analysis.pdf

Persistence Analysis Tutorial 6-1

Swedge v.6.0 Tutorial Manual

Persistence Analysis Tutorial

In a standard Swedge analysis, the program always tries to determine the maximum size wedge (i.e. Swedge will automatically scale a wedge to fit the dimensions of the slope). Joint persistence, joint trace length, wedge volume, or wedge weight may also be used to scale the wedge size. In all cases, the program will generate 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 will limit 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 will result in valid wedges. In the case of persistence, a wedge will not form if the joint persistence is below some particular 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.



Topics Covered in this Tutorial

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

In this tutorial we’ll 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’ll compare these results with the results of a standard Swedge analysis that uses maximum wedge size.

Model

Select Project Settings from the toolbar or the Analysis menu.

Select: Analysis Project Settings

1. Select the General tab in the dialog. Select the Probabilistic Analysis Type.

2. Select Metric, stress as MPa Units.

3. Select the Sampling tab in the dialog. Make sure the Sampling Method is set to Latin Hypercube and Number of Samples is 10000.

4. Select the Random Numbers tab in the dialog. Make sure the Random Number Generation method = Pseudo-random, and the Specify Seed check box is unchecked.

5. Press the OK button to exit the Project Settings dialog.

Input Data

Now let’s define the slope and joint properties in the Input Data dialog.

Select: Analysis Input Data

1. Select the Slope tab in the Input Data dialog. Enter Dip= 65, Dip Direction=185, and Height=30m for the Slope. Make sure the Unit Weight=0.026 MN/m3. The statistical distributions for the Dip and Dip Direction of the Slope should be set as None.

2. Select the Slope Length checkbox and enter a length of 50m.



3. Select the Upper Face tab. Define Dip= 0, Dip Direction=185. The statistical distributions for the Dip and Dip Direction of the Upper Face should be set as None.

4. Select the Bench Width option and enter a width of 30m.



5. Select the Joint1 tab. Change the Orientation Definition Method to a Fisher Distribution. Define Mean Dip=45 and Mean Dip Direction=125. Enter a Standard Deviation of 10. Keep the Waviness = 0 and Distribution = None.

6. Select the Joint2 tab. Change the Orientation Definition Method to a Fisher Distribution. Define Mean Dip=70 and Mean Dip Direction=225. Enter a Standard Deviation of 10. Keep the Waviness = 0 and Distribution = None.



7. Select the Strength1 tab. Change the Strength Model to Barton-Bandis. Make sure the Random Variables option is set to Parameters. Enter Mean JRC=6, JCS=50, and Phir=25. All the statistical distributions should be set to None.

8. Select the Strength2 tab and enter the same parameters as joint 1. Change the Strength Model to Barton-Bandis. Make sure the Random Variables option is set to Parameters. Enter Mean JRC=6, JCS=50, and Phir=25. All the statistical distributions should be set to None.

9. Press the OK button to save your changes, compute the probability of failure, and exit the Input Data dialog.

Analysis Results

After closing the Input Data dialog, computation of 10000 Latin Hypercube samples will occur and the probability of failure will be calculated. Figure 1 illustrates the results of this computation. Some of the notable results are:

The only Random variables are the orientation of the joints.

The results of the probabilistic analysis are summarized in the Sidebar information panel. The wedge computed using the mean values of all the input parameters is displayed along with its factor of safety results.



The probability of failure is 0.0995 or 9.995%. Out of the 10000 samples, 9969 produced valid wedges. Of these 9969, 995 had a factor of safety less than 1.0 (failed). #failed/total #samples=probability of failure =995/10000=0.0995.

The mean wedge has a factor of safety=1.27

It is important to remember that Swedge will do its best to fit a wedge inside the slope dimensions. This means it tries to determine a maximum size wedge for any slope geometry. It will adjust 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 is not accounted for and the maximum size wedge is computed.

Figure 1: Analysis results.

Persistence Analysis

As mentioned previously, 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 of probability of failure, Swedge allows you to define statistics for both the joint location and joint length (in terms of either persistence or trace length).



Furthermore, two different Joint Spacing options are available for a persistence analysis:

1. Large Joint Spacing

2. Small Joint Spacing (Ubiquitous Joints)

Large Joint Spacing

With the Large Joint Spacing option, it is assumed that 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.

It is important to 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 will limit the formation of wedges.

Small Joint Spacing

Now consider if there is spacing (repeated joints) associated with the two joint sets joint 1 and joint 2. No longer is there one wedge but a number of possible wedges that can form on the slope. If a wedge cannot form due to the persistence not being large enough, then a wedge higher up the slope face, which meets the persistence conditions, can form. If this is the case, then the Small or Ubiquitous joint spacing model is more applicable. This model will automatically scale down the wedge size until the persistence conditions are met. So 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 will always create a wedge independent 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.

Select: Analysis Persistence Analysis



1. Select the Persistence Analysis check box.

2. Make sure the Large Joint Spacing option is selected.

By selecting the Large joint spacing 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.

3. Make sure the Persistence option is selected for Joint Length.

4. Define statistical distributions for the persistence of Joint 1 and Joint 2. Toggle on the Joint 1 Persistence and Joint 2 Persistence checkboxes.

5. For both joints 1 and 2, enter an exponential distribution with a mean of 20m, and a relative minimum and maximum of 20m and 40m. This means that the joint persistence varies exponentially between 0 and 60m with a mean of 20m.

6. Press the OK button to save your changes, compute the probability of failure, and exit the Persistence Analysis dialog.



Persistence Analysis Results

After closing the Persistence Analysis dialog, computation of 10000 Latin Hypercube samples will occur and the probability of failure will be calculated. For every one of the 10000 simulations, a value of joint persistence is sampled. Due to the selection of the Large joint spacing, if the wedge which 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.

Figure 2 illustrates the results of this computation. Some of the notable results are:

The random variables now include the height of the wedge, and the maximum joint persistence.

The probability of failure has reduced from 0.0995 to 0.0035 (i.e. 28 times smaller).

The number of valid wedges has 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 is not sufficient to produce a valid wedge, then 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. 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.



Figure 2: Persistence Analysis results.

Histogram Plots

A number of useful features exist for the display of data from the probabilistic simulation.

To get an idea of the relative distribution of failed to stable samples, we can plot a histogram of Factor of Safety.

Select: Statistics Plot Histogram

Leave the Data Type as Safety Factor and press the OK button. A histogram of Safety Factor is displayed. Right-click in the histogram window, a menu will appear. Choose the 3D Histogram menu option.



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


Use the Data Type combo box to select Maximum Persistence Joint1. Do NOT select the Persistence Joint1 data type which is near the top of the list. Check the Plot Input Distribution check box. Press the OK button to generate the histogram.

Right-click on the plot and select Relative Frequency from the popup menu. This will change the y-axis of the histogram from Relative Frequency to the actual Frequency (i.e. number of samples).



This is a plot of the sampled maximum persistence for Joint 1 for all valid wedges. As mentioned above, for every one of the 10000 simulations, a value of maximum joint persistence is sampled. If the wedge which 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 we defined an exponential distribution of joint persistence that varied between 0m and 60m with a mean of 20m. Note that the histogram does not match this input distribution. 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 window, and select the Include Invalid Wedges option. The plot now includes all the sampled data (10000 samples), and the histogram bars match the input distribution.



To see a plot of the actual persistence values for all the valid wedges:


Use the Data Type combo box to select Persistence Joint1. Toggle on the display of the best fit distribution. Press the OK button.

Note that most of the wedges have small persistence values but a few have larger persistence 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 Persistence Analysis tutorial.

tutorial_06_persistence_analysis.pdf

Documents

tutorial persistence

persistence analysis

joint persistence

standard swedge analysis

analysis menu

case of persistence

analysis input data

analysis project settings