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Probabilistic Analysis
To enable Probabilistic analysis with RocSupport, first set
the Analysis Type to Probabilistic in the Project Settings dialog.
A Probabilistic Analysis allows you to enter statistical properties
for the tunnel and rock parameters. Using Monte Carlo or Latin Hypercube
sampling, statistical samples are generated for each probabilistic
input parameter. The RocSupport analysis is repeated for
each set of samples, and this results in a statistical distribution
of calculated safety factors for the support system. A probability
of failure for the support system is then calculated.
Project Settings dialog – Probabilistic
Analysis selected.
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Probabilistic Analysis: Pseudo-Random versus
Random
The Pseudo-Random checkbox in the Project Settings dialog allows you
to obtain reproducible results from a Probabilistic Analysis. If this
checkbox is ON, then you will see the same results each time you run
the analysis because the same series of random numbers is generated.
To generate different random results with each Compute, turn OFF
the Pseudo-Random Sampling checkbox. If you do this, then each time
you select Compute you will obtain different probabilistic results,
due to different random sampling of the input distributions.
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Probabilistic Analysis: Input
Data
For a Probabilistic Analysis, all tunnel and rock parameters can be
defined as random variables. The Tunnel and Rock Input Data dialog
will appear as shown below. The variables are presented in a spreadsheet
style format, which allows you to easily select a statistical distribution
and enter the mean, standard deviation, and minimum and maximum values
for each parameter.
Probabilistic Input Data – Tunnel
and Rock Parameters.
Several different statistical distributions are available in RocSupport,
including Normal, Uniform, Triangular, Beta, Exponential, Lognormal
and Gamma distributions. If a variable is assumed to be exactly known,
then set the Distribution = None.
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Probabilistic Analysis: Graphing
Results
The results of a Probabilistic Analysis can be plotted as:
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Histograms
Cumulative
Distributions
Scatter Plots
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The random variables which can be plotted include:
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calculated (output) variables (e.g. Factor of Safety)
input variables (e.g. In-Situ Stress)
user-defined variables
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The Histogram in the following figure shows the best fit statistical
distribution displayed over the raw output data (Factor of Safety).
Note that the results with Factor of Safety less than 1 are highlighted
in red. The Probability of Failure is 8.9 % for this example. The
Probability of Failure is given by the area of the red bars divided
by the total area under the histogram.
Histogram plot of Safety Factor (output random variable).
Histograms of probabilistic input variables can also be plotted. The
following plot shows a histogram of In-Situ Stress. For input variables,
the input distribution you have defined (in the Tunnel and Rock Parameters
dialog) can be displayed over the histogram – this shows how
well your theoretical input distribution was sampled. In this case,
the failed (red) bars indicate that failure of the support system
occurs for higher values of In-Situ Stress, as would be expected.
Histogram plot of In-Situ Stress (input random variable).
Statistical results can also be plotted as cumulative probability
distributions.
Cumulative distribution (S-Curve) of Safety Factor.
Scatter plots allow you to plot any two random variables against each
other so you may view the correlation (or lack of correlation) between
the two variables.
Scatter plot of Young's modulus (rock mass) versus Safety
Factor.
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Probabilistic Analysis: User Defined
Variables
An advanced feature of RocSupport is the ability to define
and plot user defined variables. With the following dialog, you may
define any function of the analysis input and/or output variables.
Simply define an equation, assign a name to the variable, and it can
then be plotted on histogram, cumulative or scatter plots, just like
the standard input and output variables.
Dialog for defining user variable.
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