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Probabilistic Analysis
In a Probabilistic Analysis, you can define statistical distributions for input parameters (e.g. joint orientation, shear strength, water level), to account for uncertainty in their values. When the analysis is computed, this results in a safety factor distribution from which a Probability of Failure (PF) is calculated.
Safety factor histogram and mean wedge
The Probabilistic Analysis option can be selected in the Project Settings dialog or from the drop-list in the toolbar.
Probabilistic Analysis selected in Project Settings
Probabilistic Analysis selected in toolbar
The Probability of Failure (PF) is displayed in the toolbar, as shown above (e.g. PF = 0.048 means that there is a 4.8 % probability of failure).
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Probabilistic Analysis: Input Data
The Input Data dialog for a Probabilistic Analysis is organized under 10 tabs - Slope, Upper Face, Joint1, Strength1, Joint2, Strength2, Tension Crack, Water, Seismic and Forces, as shown below.
Probabilistic Input Data dialog
In order to carry out a Probabilistic Analysis, one or more input parameters must be defined as random variables. To define a random variable you must assign a statistical distribution to the variable. The following statistical distributions are available in Swedge:
Most input parameters in Swedge can be defined as random variables (e.g. joint orientations, shear strength, water pressure, etc).
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Probabilistic Analysis: Joint Orientation
There are two methods of defining Joint Orientation as a random variable in Swedge:
Dip / Dip Direction
Fisher Distribution
The Dip/DipDirection method allows you the flexibility to define independent distributions for the Dip and Dip Direction (i.e. they are treated as independent random variables). This can be useful if the actual distribution of orientations you are trying to model, is asymmetric (e.g. an elliptical distribution on the stereonet).
Dip and Dip Direction of joint as independent random variables
If your orientation distribution is approximately circular, then you should use the Fisher Distribution option. The Fisher Distribution allows you to define three-dimensional variability of joint orientation around the mean, with a single parameter (Fisher K or standard deviation). This has several advantages - it is easier and more intuitive to define, provides more predictable orientation distributions, and you do not have to worry about angular transitions such as Dip Direction near 0/360.
Fisher Distribution method for generating random joint orientations
Effect of Fisher K on a randomly generated joint set of 500 samples
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Probabilistic Analysis: Joint Strength
There are two methods of defining Joint Strength as a random variable in Swedge:
The Parameters method allows you the flexibility to define independent distributions for each parameter in the shear strength criterion (e.g. cohesion and friction angle for the Mohr-Coulomb criterion).
Defining cohesion and friction angle as independent random variables
The Strength method allows you to specify a variability of shear strength about the mean strength envelope, as illustrated below. This is easier and more intuitive to define, and provides more predictable shear strength distributions. This option is particularly useful for Barton-Bandis or Power Curve strength models, since it eliminates the need for defining statistical distributions for individual strength parameters, and the issue of statistical correlation of individual strength parameters is avoided.
Defining variability of the mean strength envelope (Mohr-Coulomb)
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Probabilistic Analysis: Sampling
The Sampling Method determines how the statistical input distributions for the random variables you have defined for a Probabilistic Analysis will be sampled. Two Sampling Methods are available in Swedge:
Monte Carlo
Latin Hypercube
Latin Hypercube is the default sampling method, due to its ability to more smoothly sample the input distributions, with a fewer number of samples, as illustrated below.
Latin Hypercube sampling of Normal distribution (1000 samples)
Monte Carlo sampling of Normal distribution (1000 samples)
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Probabilistic Analysis: Random Numbers
Sequences of random numbers are utilized in conjunction with the Sampling Method, to generate random input data samples for a Probabilistic Analysis. The generation of random numbers is controlled by the Random Numbers option in the Project Settings dialog. Two options are available:
Random Numbers option in Project Settings
With the Pseudo-Random option, a constant seed value is used to generate the random numbers. This allows you to obtain reproducible results for the probabilistic analysis. This is useful for demonstration purposes etc.
With the Random option, you will obtain different results each time you re-compute the probabilistic analysis, because a different seed value is used each time to generate a new sequence of random numbers, and therefore different input data samples are generated.
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Probabilistic Analysis: Probability of Failure / Reliability Index
The primary result from a Probabilistic Analysis is the Probability of Failure (PF). This is displayed in the toolbar, the Sidebar and the Info Viewer. The Probability of Failure is defined as the number of failed wedges, divided by the total number of samples.
Probability of failure summary in sidebar
Another commonly used probabilistic measure of safety is the Reliability Index. The Reliability Index is listed in the Swedge Info Viewer, and represents the number of standard deviations which separate the mean Factor of Safety from the critical Factor of Safety ( = 1 ).
Reliability index (beta) for a normal distribution of safety factor
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Probabilistic Analysis: Histogram Plots
Histograms plots of probabilistic analysis results (output variables) or input variables can be easily plotted. Data corresponding to failed wedges can be highlighted in red. For example, the safety factor histogram (shown below) gives a visual representation of the probability of failure (i.e. the proportion of red histogram bars divided by the total area of the histogram).
Safety factor histogram
Useful properties of Histogram plots include:
Plots and data can be exported to Excel with a single mouse click
If you double-click at any point on a Histogram plot, the wedge
corresponding to that location will be displayed in the Wedge View and
the selected wedge data will be displayed in the Sidebar
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Probabilistic Analysis: Cumulative Plots
Probabilistic results can also be displayed in the form of cumulative distribution plots. A cumulative distribution gives the probability that a random variable will be less than or equal to a specified value. For example, if you plot the safety factor cumulative distribution, the value of the function at safety factor = 1 is equal to the probability of failure.
Cumulative distribution plot of safety factor
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Probabilistic Analysis: Scatter Plots
Scatter plots allow you to plot any two random variables from the probabilistic analysis against each other, to see the correlation (or lack of correlation) between the two variables. This includes input and output random variables. Data corresponding to failed wedges can be highlighted in red, and the best fit (linear) curve through the data can be plotted to obtain the correlation coefficient between the two variables.
Scatter plot of normal stress versus shear strength on joint plane
Note:
Scatter plots and the corresponding data can be exported to Excel with a
single mouse click
If you double-click at any point on a Scatter plot, the wedge corresponding
to that location will be displayed in the Wedge View and the selected
wedge data will be displayed in the Sidebar
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Probabilistic Analysis: Stereonet View
For a Probabilistic Analysis, the Stereonet view will display all randomly generated joint orientations (poles), if joint orientations were defined as random variables. In the following example, both joints and the tension crack were assigned Fisher orientation distributions. The stereonet can also display the intersections of Joint 1 and Joint 2, and orientations corresponding to failed wedges can be highlighted in red.
Stereonet view showing random joint poles and intersections
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Probabilistic Analysis: Selecting Random Wedges
After a Probabilistic Analysis is computed, the Mean Wedge is initially displayed. If desired you can view any random wedge generated by the Probabilistic Analysis by double-clicking on Histogram or Scatter plots. For example:
Double-click the left mouse button anywhere on a Histogram or Scatter
plot, and the nearest corresponding wedge from the plot will be displayed
in the Wedge View. This is referred to as a Picked Wedge.
When a Picked Wedge is displayed, the Sidebar and all views of the
current document will be updated to display the results for the Picked
Wedge.
Double-click at any point on a histogram to display the corresponding wedge
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Probabilistic Analysis: Export to Excel
Probabilistic data and plots (e.g. histograms, scatter plots) can be exported to Excel with a single mouse click, allowing you to carry out customized post-processing and filtering of data if required.
Select Chart in Excel from right-click menu
Data and chart is automatically exported to Excel
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