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Probabilistic Analysis
Slide now includes extensive Probabilistic Analysis capabilities
for the statistical analysis of slope stability using Monte
Carlo or Latin Hypercube simulation techniques. Virtually any
input parameter in the model can be defined as a random variable.
Any combination of the following can be used as random variables:
1. Material properties (e.g. cohesion, phi, etc…)
2. Support properties (e.g. out-of-plane spacing, anchor capacity,
etc…)
3. Seismic Load coefficients
4. Magnitudes of Line loads, and Distributed Loads
5. Location of the water table
6. Location of the tension crack
7. Tension crack properties (e.g. depth of water in the tension
crack, etc)
The random variables can be assigned any of the following distributions:
1. Normal
2. Uniform
3. Triangular
4. Beta
5. Exponential
6. Lognormal
7. Gamma
All distributions can
be truncated (e.g. minimum and maximum values). Correlation coefficients
can be specified (e.g. to correlate cohesion and phi for Mohr-Coulomb
materials).
Probabilistic results are displayed using Histogram,
Cumulative, Scatter plots.
All statistical data can be easily exported
to Excel or the clipboard for further analysis.
Slide can perform the probabilistic analysis using two different methods:
1. Assume that the failure surface is the deterministic global minimum
surface and do the sampling and safety factor calculations only on this
surface. The probability of failure and reliability index is then calculated for
this surface.
2. Or, Slide can search for a new global minimum for each set of random
variable samples -- with this option you can view the "band" of critical
surfaces, corresponding to different sets of random variables. The
probability of failure for the slope is determined by taking the number of
simulations with a global minimum less than one and dividing it by the
total number of simulations.
Advantages of Probabilistic Analysis
Depending on the uncertainty that may be present in your site conditions, two slopes with the same factor of safety can have different probabilities of failure. Performing a probabilistic analysis will determine the probability of failure for your slope, which will give you a much better representation of the level of safety in your design.
Whether your current work requires you to perform a probabilistic
analysis or you simply want more confidence in your design, performing
a probabilistic analysis will only improve your slope stability analyses.
For an introduction to the use of probabilistic analysis, including
an example of probabilistic analysis being applied to a slope stability
investigation, please refer to chapter 8 of Hoek's notes, Factor
of safety and probability of failure. Also see the Slide
Tutorial Manual - Part 2, for several examples of probabilistic
analysis using Slide.
Back-Analysis using Probabilistic Analysis
Probabilistic analyses can also be used for performing a back analysis
to determine material properties or groundwater conditions. If you
have a slope that has already failed, you can use the failure geometry
and the implicit factor of safety (<= 1.0, implied by the failure)
to determine material properties or groundwater conditions.
You simply specify a large range for the unknown properties (e.g.
friction angle), run the analysis, and see where the plot crosses
the factor of safety axis at a value of 1.0. For an example of how
a probabilistic analysis could be used in this manner, please refer
to the following article (published in a 2002 RocNews newsletter):
http://www.rocscience.com/roc/ProjectSpeight.htm
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Probabilistic
Analysis: Monte-Carlo and Latin-Hypercube Simulation
Slide provides two statistical sampling methods - Monte-Carlo and
Latin-Hypercube - for Probabilistic analysis. For both methods, you
can set the Number of Samples for the probability simulation. Latin-Hypercube
sampling allows you to get more accurate results with fewer samples
thus greatly improving the speed and accuracy of the solution.
Select the sampling method from the pull-down menu
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Probabilistic Analysis:
Statistical Distributions
In Slide, a wide range of statistical distributions are available
for defining the probability density functions of your random variables.
These include the statistical distributions which are most commonly
used in geotechnical engineering analysis.
To define a parameter as a random variable, first select one of
the seven possible choices for Statistical Distribution: normal,
uniform, triangular, beta, gamma, exponential or lognormal.
Select the Statistical Distribution from the pull-down
menu
Once a statistical distribution has been selected, you will then
be able to enter the mean, standard deviation (if required) and
relative minimum and maximum values. All distributions are automatically
checked to ensure unrealistic values (eg. negative) are not used.
Enter the mean and relative minimum and maximum values
for the exponential distribution
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Probabilistic Analysis:
Histogram Plot and Cumulative Plot
After a Probabilistic Analysis, detailed statistical results can
be viewed with the Histogram Plot, Cumulative Plot and Scatter Plot
options in the Statistics menu in the Slide Interpret program.
The Statistics drop down menu
Histogram Plot dialog to select the desired data to
plot
The histogram below represents a distribution of Safety Factor.
The red bars at the left of the distribution represent analyses
with Safety Factor less than 1.0. Notice that the actual number
of failed analyses is listed at the top of the plot, which gives
the Probability of Failure for the analysis when divided by the
total number of samples. The Reliabiltiy Index ( RI ) is also calculated
and listed on the plot.
If the Data Type selected is a results variable calculated in the
analysis (eg. Safety Factor), a Best Fit Distribution can be determined
for the variable, and displayed on the plot.
Safety Factor Histogram
In addition to Histograms, Cumulative distributions (S-curves) of
the statistical results can also be generated.
Cumulative Plot dialog
Cumulative distribution of Cohesion random variable
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Probabilistic Analysis: Scatter Plots
Scatter (Shotgun) plots allow the user to examine the relationship
between any two analysis variables. The Show Regression Line option
displays the best fit straight line through the data. The Correlation
Coefficient, alpha and beta values are listed at the bottom of
the plot. Alpha and beta represent the y-intercept and the slope,
respectively, of the best fit linear regression line to the scatter
plot data.
Scatter Plot dialog
Scatter plot between Safety Factor and Friction Angle for a Weak Soil Layer
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Probabilistic Analysis: Correlation Coefficients
The correlation between two different variables can be shown by plotting
a scatter plot of the two parameters with the
"Show Regression Line" option selected.
Select the Regression Line option to display the best fit straight line through the data
The Correlation Coefficient will be listed at the bottom of the plot.
The coefficient can vary between -1 and 1 where numbers close to zero
indicate a poor correlation, and numbers close to 1 or -1 indicate
a strong correlation.
Scatter plot between Safety Factor and Friction Angle for a Weak Soil Layer
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