tutorial 02 probabilistic analysis

7/30/2019 Tutorial 02 Probabilistic Analysis

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Probabilistic Analysis Tutorial 2-1

Probabilistic Analysis Tutorial

This tutorial will familiarize the user with the Probabilistic Analysisfeatures of Swedge .

In a Probabilistic Analysis, you can define statistical distributions forinput parameters (e.g. joint orientation, shear strength, water level), toaccount for uncertainty in their values. When the analysis is computed,this results in a distribution of safety factors, from which a probability of failure (PF) is calculated.

The finished product of this tutorial can be found in the Tutorial 02Probabilistic.swd file, located in the Examples > Tutorials folder inyour Swedge installation folder.

Topics Covered in this Tutorial

Project Settings Random Variables Fisher Distribution Tension Crack Mean Wedge Picked Wedges Histograms Scatter Plots Stereonet View Show Failed Wedges

Swedge v.5.0 Tutorial Manual


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If you have not already done so, run the Swedge program by double-clicking on the Swedge icon in your installation folder. Or from the Startmenu, select Programs Rocscience Swedge 5.0 Swedge.

If the Swedge application window is not already maximized, maximize itnow, so that the full screen is available for viewing the model.

When the Swedge program is started, a default model is automaticallycreated, allowing you to begin defining your model immediately. If you doNOT see a wedge model on your screen:

Select: File New

Whenever a new file is created, the default input data will form a validwedge.

Project Settings

The Project Settings option allows you to configure the main analysisparameters for your model (i.e. Analysis Type, Units, Sampling Methodetc). Select Project Settings from the toolbar or the Analysis menu.

Select: Analysis Project Settings

You will see the Project Settings dialog.

Analysis Type

By default a Deterministic Analysis is selected for a new file. Select theGeneral tab in the Project Settings dialog, and change the Analysis

Type to Probabilistic .



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NOTE: the Analysis Type can also be selected from the drop-list in thetoolbar.

Units

For this tutorial we will be using Metric units, so make sure the Metric option is selected for Units .

Sampling and Random Numbers

Select the Sampling tab in the Project Settings dialog. The SamplingMethod determines how the statistical distributions for the random inputvariables will be sampled. The default Sampling Method = LatinHypercube, and the default Number of Samples = 10,000. See theSwedge help topics for more information about the sampling options.

Select the Random Numbers tab. Note that Pseudo-Random samplingis in effect by default. This allows you to obtain reproducible results for aprobabilistic analysis, by using the same seed value to generate randomnumbers. We will discuss Pseudo-Random versus Random sampling laterin this tutorial.

Do not make any changes to these settings, we will use the defaults.

Project Summary

Select the Project Summary tab in the Project Settings dialog.

Enter Swedge Probabilistic Analysis Tutorial as the Project Title.

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

Select OK to close the Project Settings dialog.



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Probabilistic Input Data

Select Input Data from the Analysis menu or the toolbar.

Select: Analysis Input Data

For a Probabilistic analysis, the Input Data dialog is organized underseveral tabs as shown below.

To carry out a Probabilistic Analysis with Swedge , at least one inputparameter must be defined as a random variable . To define a randomvariable, select a statistical distribution (e.g. Normal, Lognormal, Fisher,etc) for the variable, and enter appropriate statistical parameters for thedistribution (e.g. standard deviation, min and max values).

For more information about statistical input see the Swedge help system.

For this example, we will be defining the following input parameters asrandom variables:

Joint 1 orientation Joint 1 shear strength Joint 2 orientation Joint 2 shear strength Tension Crack orientation

All other model input parameters will be assumed to be exactly known(i.e. Statistical Distribution = None) and will not be involved in thestatistical sampling.



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Slope

Select the Slope tab in the Input Data dialog. We will assume that theorientation of the slope plane is constant for the probabilistic analysis, sowe will not enter statistical data (i.e. Statistical Distribution = None).

Use the default orientation values (Dip = 65, Dip Direction = 185)and Unit Weight = 2.6.

Enter a Slope Height = 20 meters.

Select the Length checkbox and enter a Slope Length = 60meters (see note below).

Slope Length

NOTE: for a Probabilistic analysis, it is usually a good idea to define aSlope Length . This will limit the size of wedges according to thisdimension. If you leave the slope length undefined, then, depending onyour joint orientation distributions, very large wedges can be generatedparallel to the slope, which may give unrealistic or misleading analysisresults.

Upper Face

Select the Upper Face tab in the Input Data dialog. We will assumethat the orientation of the upper face is constant for the probabilisticanalysis, so we will not enter statistical data (i.e. Statistical Distribution= None).

Use the default orientation values (Dip = 12, Dip Direction =185).

Select the Bench Analysis checkbox and enter a Bench Width =15 meters (see note below).

Bench Width

NOTE: for a Probabilistic analysis, it is usually a good idea to define aBench Width . This will limit the size of wedges according to thisdimension. If you leave the Bench Width undefined, then, depending onyour joint orientation distributions, very large wedges can be generatedperpendicular to the slope, which may give unrealistic or misleading

analysis results.



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Joint 1 Orientation

Select the Joint 1 tab in the Input Data dialog.

Note that there are TWO methods of defining the variability of jointorientation in an Swedge Probabilistic analysis:

Orientation Definition Method = Dip / Dip Direction

Orientation Definition Method = Fisher Distribution

With the Dip / Dip Direction method, the Dip and Dip Direction aretreated as independent random variables (i.e. you can define differentstatistical distributions for Dip and Dip Direction).

The Fisher Distribution method generates a symmetric, 3-dimensionaldistribution of orientations around the mean plane orientation. Only asingle standard deviation is required. In general, a Fisher Distribution is

recommended for generating random joint plane orientations, because itprovides more predictable orientation distributions, and lessens thechance of input data errors.

For more information about the Orientation Definition Method see theSwedge Help system.

We will use the Fisher Distribution option. Select Orientation DefinitionMethod = Fisher Distribution . Enter Mean Dip = 45, Mean DipDirection = 105, and Standard Deviation = 7.



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Joint 2 Orientation

Select the Joint 2 tab in the Input Data dialog.

Select Orientation Definition Method = Fisher Distribution .

Enter Mean Dip = 70, Mean Dip Direction = 235, and StandardDeviation = 7.



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Joint 1 Strength

Select the Strength 1 tab in the Input Data dialog.

Note that there are TWO methods of defining the statistical variability of joint shear strength in an Swedge Probabilistic analysis:

Random Variables = Parameters

Random Variables = Strength

With the Parameters method, the individual strength criterionparameters (e.g. cohesion and friction angle) can each be assigned astatistical distribution.

With the Strength method, the shear strength variability is defined withrespect to the mean strength envelope. This method has the advantage of only requiring a single parameter (coefficient of variation) to define the

shear strength variability.

For more information about the probabilistic joint shear strength optionsin Swedge , see the Swedge Help system.

Select Random Variables = Strength .

Select Statistical Distribution = Lognormal.

Enter Coefficient of Variation = 0.25, Cohesion = 2, Phi = 20.



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NOTE:

The Coefficient of Variation is defined as the Standard Deviation(of the shear strength) divided by the Mean (shear strength).

Only Lognormal (and Gamma) distributions are allowed for

defining shear strength as a random variable, because Lognormaland Gamma distributions are only defined for positive values.This ensures that the randomly generated values of shearstrength will always be positive (negative shear strength has nophysical meaning in Swedge ).

Joint 2 Strength

Select the Strength 2 tab in the Input Data dialog.

Select Random Variables = Strength .

Select Statistical Distribution = Lognormal.

Enter Coefficient of Variation = 0.25, Cohesion = 0, Phi = 30.



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Tension Crack

Lets include a Tension Crack for this model, and define the orientationas a random variable.

1. Select the Tension Crack tab in the Input Data dialog.

2. Select the Tension Crack Exists checkbox.

3. Select Orientation Definition Method = Fisher Distribution .

4. Enter Mean Dip = 70, Mean Dip Direction = 165, and StandardDeviation = 7.

5. For the Tension Crack Location, select the Use Bench Width toMaximize option.

NOTE: the Use Bench Width to Maximize option will automatically

locate the Tension Crack to create the maximum possible wedge size forthe specified Bench Width. A Tension Crack will NOT be included if itdecreases the wedge size.



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Compute

Select OK in the Input Data dialog to Compute the Swedge Probabilisticanalysis.

Using the Latin Hypercube sampling method, Swedge will generate10,000 random input data samples for each random variable, using thespecified statistical distributions, and compute the safety factor for10,000 possible wedges.

The calculation should only take a few seconds. The progress of thecalculation is indicated in the status bar.

TIP: you can also select the Apply button in the Input Data dialog toCompute the analysis without closing the dialog. This allows you toeasily test different input parameters and re-compute the results.



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Probabilistic Analysis Results

The primary result of interest from a Probabilistic analysis is theProbability of Failure . This is displayed in the toolbar at the top of thescreen.

For this example, if you entered the Input Data correctly, you shouldobtain a Probability of Failure (PF) of about 9% (PF = 0.0882).

Sidebar Information Panel

A summary of analysis results is displayed in the Sidebar informationpanel at the right of the screen.

Notice that the Probability of Failure is equal to the Number of Failed Wedges (i.e. safety factor < 1), divided by the Number of Samples (entered in the Project Settings dialog) = 882 / 10000.

NOTE: for a discussion of the Probability of Failure see the Swedge helpsystem.



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Wedge Display

The wedge initially displayed after a Probabilistic analysis, is based onthe mean input values, and is referred to as the Mean Wedge . It willappear exactly the same as one based on Deterministic input data withthe same orientation as the mean Probabilistic data.

The safety factor of the Mean Wedge = 1.366 as shown in the Sidebar.

Figure 1: Mean Wedge displayNote that the Tension Crack for the Mean Wedge is located to create themaximum wedge size for the given bench width. Remember that the UseBench Width to Maximize option is in effect for the Tension Crack.

You can also view the wedge with the Minimum safety factor generatedby the Probabilistic analysis. Right-click in the Wedge View and selectShow Min FS Wedge from the popup menu. The minimum safety factorwedge will be displayed, and the Sidebar now displays analysisinformation for the Min FS Wedge (Safety Factor = 0.544).

To restore the Mean Wedge display and information, right-click in the

Wedge View and select Show Mean FS Wedge .



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Histograms

To plot histograms of results after a Probabilistic Analysis, select PlotHistogram from the toolbar or the Statistics menu:

Select: Statistics Plot Histogram

Select OK to plot a histogram of Safety Factor. The histogram representsthe distribution of Safety Factor for all valid wedges generated by therandom sampling of the Input Data. The red bars at the left of thedistribution represent wedges with Safety Factor less than 1.0.

Right-click on the histogram and select 3D Histogram from the popupmenu. This will display the histogram bars in 3D.

Figure 2: Safety Factor histogram.



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Mean Safety Factor

At the bottom of the histogram plot, notice the mean, standard deviation,min and max values.

Note that the mean Safety Factor from a Probabilistic Analysis (i.e. the

average of all of the Safety Factors generated by the Probabilistic Analysis) will in general, be slightly different from the Safety Factor of the Mean Wedge (i.e. the Safety Factor of the wedge corresponding tothe mean Input Data values).

In this case:

From the histogram, the mean safety factor = 1.424.

In the Sidebar, the safety factor of the Mean Wedge = 1.366.

Theoretically, for an infinite number of samples, these two values shouldbe equal. However due to the random nature of the statistical sampling,the two values will usually be slightly different, for a typical probabilisticanalysis with a finite number of samples.



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Selecting Random Wedges

Now tile the Histogram and Wedge views, so that both are visible.

Select: Window Tile Vertically

Figure 3: Safety Factor histogram and wedge view.

A useful property of Histograms (and also Scatter Plots) is the following:

If you double-click the LEFT mouse button anywhere on the plot,the nearest corresponding wedge will be displayed in the Wedgeview, and results for the wedge will be displayed in the Sidebar.

For example:

1. Double-click at any point along the histogram.

2. Notice that a different wedge is now displayed.

3. In the Sidebar, the analysis results are updated to display resultsfor the wedge that you are viewing, which is referred to as a

Picked Wedge .

4. Double-click at various points along the histogram, and notice thedifferent wedges and analysis results which are displayed. Forexample, double-click in the red Safety Factor region, to viewwedges with a Safety Factor < 1.



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This feature allows you to view any wedge generated by the Probabilistic Analysis, corresponding to any point on a histogram or scatter plot.

In addition to the Wedge View, all other applicable views (for example,the Info Viewer and the Stereonet View) are also updated to display datafor the currently Picked Wedge .

Note:

this feature can be used on histograms of any statistical datagenerated by Swedge , and not just the Safety Factor histogram

this feature also works on Scatter plots.

Right-click in the wedge view and select Show Mean FS Wedge fromthe popup menu, to reset the mean wedge display.

Histograms of Other Data

In addition to Safety Factor, you can also plot histograms of:

other random output variables (e.g. wedge weight, normal stresson joint planes, driving force etc),

random input variables (i.e. any input data variable which wasassigned a statistical distribution).

For example:


In the dialog, select Data Type = Wedge Weight , select the Best FitDistribution checkbox, and select OK. A histogram of the wedge weightand the best-fit distribution to the data will be displayed.

In this case the Best Fit distribution is a Normal distribution,with parameters listed at the bottom of the plot. The Best Fitdistribution can be displayed for analysis output variables.

The features described above for the Safety Factor histogram,also apply to other Data Types. For example, if you double-clickon the Wedge Weight histogram, the nearest correspondingwedge will be displayed in the Wedge View.

Close the Wedge Weight histogram view, and the Safety Factorhistogram view, by selecting the X in the upper right corner of each view.

Right-click in the wedge view and select Show Mean FS Wedge fromthe popup menu, to reset the mean wedge display.



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Now lets generate a histogram of an input random variable.


Select Data Type = Dip of Joint 1 .

NOTE: for input random variables, the Input Distribution can bedisplayed on histograms. However, because the orientation of Joint 1 wasgenerated using a Fisher Distribution, which is 3-dimensional, the InputDistribution cannot be displayed on the histogram, which is a 2-dimensional plot of only one component (Dip) of the Joint 1 orientation.

Show Failed Wedges

Lets demonstrate one more feature of Histogram plots, the Show Failed Wedges option. Right-click on the Histogram and select Show FailedWedges from the popup menu.

The distribution of failed wedges (i.e. wedges with Safety Factor < 1) isnow highlighted on the Histogram. The Show Failed Wedges optionallows you to see the relationship between wedge failure, and thedistribution of any input or output variable.

In this case, there is not a strong correlation between wedge failure andJoint 1 dip angle. However, there appears to be some bias towards failureat higher dip angles, as might be expected.

Figure 4: Joint 1 Dip Angle failed wedge distribution is also displayed.



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Scatter Plots

Scatter plots allow you to examine the relationship between any twoanalysis variables. To generate a Scatter Plot:

Select: Statistics Plot Scatter

In the Scatter Plot dialog, select the variables you would like to plot onthe X and Y axes. For example, lets plot the normal stress versus shearstrength for one of the joint planes. Select the Show Regression Lineoption to display the best fit straight line through the data.

Select OK to generate the plot. Right-click on the plot and select ShowFailed Wedges , to highlight the data corresponding to failed wedges.

From the failed wedge data, it can be readily seen that wedge failurecorresponds to low values of normal stress and shear strength, as wewould expect.

Since we used the Mohr-Coulomb strength criterion, the best fit linearregression line for the Scatter plot corresponds (approximately) to themean strength envelope. We can verify this from the parameters listed atthe bottom of the plot.

The alpha value (2.029) represents the y-intercept of the linearregression line on the Scatter plot. For Joint 1, recall that wedefined the cohesion = 2 tonnes/m2. For the Mohr-Coulombcriterion, cohesion is the y-intercept of the strength envelope.

The beta value (0.362) represents the slope of the linearregression line. For the Mohr-Coulomb criterion, the slope of thestrength envelope is equal to tan(phi). For Joint 1 we defined phi= 20 degrees. Arctan(0.362) = 19.9.



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Figure 5: Normal stress versus shear strength for Joint 1.

Also note the Correlation Coefficient, listed at the bottom of the plot,which indicates the degree of correlation between the two variablesplotted. The Correlation Coefficient can vary between -1 and 1 wherenumbers close to zero indicate a poor correlation, and numbers close to 1or 1 indicate a good correlation. Note that a negative correlationcoefficient simply means that the slope of the best fit linear regressionline is negative.

The Correlation Coefficient is related to the Coefficient of Variationwhich we defined for the shear strength of Joint 1. To demonstrate this:

1. Select Input Data and select the Strength 1 tab.

2. Enter Coefficient of Variation = 0.1 and select Apply in the dialogto re-compute the analysis.

3. Notice that the scatter of data around the mean strengthenvelope is much narrower, and the Correlation Coefficient hasincreased to 0.898.

4. Enter Coefficient of Variation = 0.01 and select Apply.

5. The scatter of data is very narrow, and the Coefficient of Variation = 0.998.

6. Select Cancel in the Input Data dialog to restore the originalstrength data (or re-enter Coefficient of Variation = 0.25 andselect OK).



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Stereonet View

The Stereonet View in Swedge displays a stereographic projection of thewedge planes (great circles) and corresponding poles. For a Probabilisticanalysis, the stereonet can display the poles of all randomly generatedplane orientations, and the joint intersections. Orientations

corresponding to failed wedges can be highlighted.

Select: Analysis Stereonet

Right-click on the Stereonet View and make sure that the Show Planes ,Show All Poles , Show Intersections and Show Failed options are allselected. Your screen should look like the following figure.

Figure 6: Stereonet view showing random poles, intersections and failed data.

Notice the three sets of data (poles) corresponding to Joint 1, Joint 2 andthe Tension Crack orientations. The set of data in the lower half of theplot are the joint intersections. The poles and intersections correspondingto failed wedges are highlighted in red.



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Compute (Random Sampling)

So far in this tutorial we have used the default Pseudo-Random samplingoption. Pseudo-Random sampling allows you to obtain reproducibleresults for a Probabilistic analysis, by using the same seed value togenerate 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 aProbabilistic analysis, by allowing a variable seed value to generate therandom input data samples.

Before we start, lets arrange the views as follows:

1. Select the Tile option from the toolbar or the Window menu, totile all of the open views.

2. If you have followed the instructions in this tutorial, you shouldhave four views open as shown in the following figure (Wedge

View, Stereonet View, Joint 1 Dip Histogram, and Scatter Plot).

Figure 7: Tiled views of probabilistic analysis results.

If your screen does not look similar to the above figure (e.g. you haveadditional views open), then close all views except for the four notedabove, and re-tile the views.

Now go to the Project Settings dialog.

Select: Analysis Project Settings



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1. Select the Random Numbers tab, and change the RandomNumber Generation method from Pseudo-Random to Random .The Random option will use a different seed value to generaterandom numbers, each time you re-run the Probabilistic analysis.

This will result in different sampling of your input randomvariables, and different analysis results (e.g. Probability of Failure) each time you re-compute.

2. Select the Sampling tab in the Project Settings dialog, anddecrease number of samples from 10,000 to 1000. (This will make

the change in results easier to see on the plots).

3. Select OK in the Project Settings dialog.

4. Select the Compute option from the toolbar.

5. Notice that the Histogram plot, Scatter plot, Stereonet view, andProbability of Failure, are updated with new results.

6. Select Compute repeatedly, and observe how the plots and theprobability of failure are updated each time the analysis is re-run.

7. Note that the Wedge view does not change when you re-compute,since by default the Mean Wedge is displayed, (i.e. the wedgebased on the mean Input Data), which is not affected by re-running the analysis.



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8. For this example, if you re-run the analysis several times, youwill find that the Probability of Failure will vary between about 7and 11%.

Selecting Random Wedges

To conclude this tutorial, we will again demonstrate the ability to pickrandom wedges by double-clicking on either Histograms or Scatter plots,and we will also note the effect on the Stereonet view.

Double-click on the Histogram or Scatter plots repeatedly and observethe following:

1. The Sidebar displays results for the Picked Wedge (i.e. thewedge which corresponds to the data location at which youclicked on the plot).

2. The Wedge View is updated to display the Picked Wedge .

3. The great circles on the Stereonet are updated to display theplanes representing the Picked Wedge .

Figure 8: New random sampling, 1000 samples, picked wedge.

You are encouraged to experiment further with the probabilistic analysisfeatures of Swedge (e.g. Export Dataset in the Statistics menu, Chartin Excel in the right-click menu for charts, etc).

That concludes the Swedge Probabilistic Analysis Tutorial.

tutorial 02 probabilistic analysis

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