# Material Statistics

In order to define material properties as Random Variables, select the Materials option from the Statistics menu. This will display the Material Statistics dialog.

The Material Statistics dialog allows you to do the following:

- Choose the material properties that you wish to define as Random Variables.
- Enter the statistical parameters of material property Random Variables.
- (Optional) - Define a Correlation Coefficient between Cohesion and Friction angle, for any material with a Mohr-Coulomb strength type.
- (Optional) - For advanced users, the Equate and Advanced Correlation options are also available.
- (Optional) - Define spatial variability for random variables.

These options are described below.

## Probabilistic Shear Strength

## Add Random Variables

The first step in defining material properties as Random Variables, is to use the Add option in the Material Statistics dialog, to select the desired materials and properties. This is easily done as follows:

- Select the Add button in the Material Statistics dialog.
- You will see a series of three dialogs, in "wizard" format, which allow you to quickly select the desired materials and properties.
- The first dialog is the
**Add Random Variables > Select Materials**dialog. Use the checkboxes to select the MATERIALS, for which you will be defining random variables. When the desired materials are selected, select the Next button. - The second dialog is the
**Add Random Variables > Select Properties**dialog. Use the checkboxes to select the PROPERTIES, for the selected MATERIALS. When the desired properties are selected, select the Next button. (See the note below about multiple shear strength models). - The third dialog is the
**Add Random Variables > Select Distribution**dialog. Select the desired Statistical Distribution, and select the Finish button. - You will be returned to the Material Statistics dialog. All selected PROPERTIES for all selected MATERIALS will now appear in the Material Statistics dialog, in a spreadsheet format.
- Each row in the dialog represents a MATERIAL PROPERTY that you have chosen to define as a Random Variable for a Probabilistic Analysis.
- The selected DISTRIBUTION is in effect for all variables, however, this can be changed for individual variables, if desired.
- You can now enter the statistical parameters which define the statistical distribution for each Random Variable. This is described in the following sections.

## Add Random Variables (Multiple Shear Strength Models)

It is important to be aware of the following, if you are using the Add option, and some of your selected materials use different strength models.

- If some of your selected materials (in Step 3) use different shear strength models, then (in Step 4), you will see ALL properties for ALL strength models, for all selected materials. This means, for example, that you could see a list which contains strength parameters for both Mohr-Coulomb and Hoek-Brown strength models. This could be quite confusing. If this occurs, the user should be aware of this, and the following alternative procedures may be helpful.
- The process of Adding Random Variables, as described above in Steps 1 – 5, can be repeated at any time, even if some variables have previously been defined. If you have materials with different strength models, you could repeat Steps 1 – 5, for each different strength model, so that in Step 4, you would only see the parameters of materials with the same strength model.
- Another alternative is to use the Edit option, rather than the Add option. The Edit option is simply an alternative method of adding random variables. See below for details.

## Delete Random Variables

To delete Random Variables that you have added in the Material Statistics dialog, you can use the Delete option.

- First, use the mouse to select the Random Variables you wish to delete, by selecting the desired row(s) in the dialog. When rows are selected, they will be highlighted.
- Select the Delete button, and the selected variables will be deleted.
- When you delete a Random Variable, it simply means that the material property will no longer be treated as a Random Variable in the Probabilistic Analysis.

## Edit Random Variables

The Edit option in the Material Statistics dialog, is an alternative method of adding or deleting random variables. Rather than using the "wizard" format of adding random variables (with the Add option), the materials and properties are presented in the form of a tree-structured list of checkboxes, which allows you to select (or delete) any material property as a random variable, by selecting (or clearing) the desired checkboxes.

- The Edit option is simply an ALTERNATIVE method of ADDING or DELETING Random Variables from the Material Statistics dialog. It is entirely equivalent to using the Add or Delete options.
- In general, the Add option for adding random variables is more efficient than using the Edit option, especially if you only want to specify certain properties for each material, as Random Variables (for example, if you want to specify the Cohesion and Friction Angle for several materials as random variables, but NOT the Unit Weight, then it is much easier to do this with the Add option than the Edit option).
- However, you may find the Edit option useful at times, especially if your materials use different strength models. Since the Edit option allows you to see the available properties for EACH material individually, this avoids the possible confusion of using the Add option for this purpose.

## Statistical Distribution

A Statistical Distribution must be chosen for each Random Variable in *Slide2*. The Statistical Distribution, together with the mean, standard deviation and minimum/maximum values, determines the shape and extent of the probability density function you are defining for the Random Variable.

- When you initially add Random Variables to the Material Statistics dialog, a Statistical Distribution will automatically be assigned to each variable.
- In most cases this will be the Normal Distribution, unless you have selected a different distribution, using the third dialog of the Add option (see above).
- To select a different distribution for any Random Variable, use the mouse to select a Distribution from the drop-down list available for each variable.
- You may then enter the Standard Deviation (if applicable) and Relative Minimum Maximum values for the Random Variable.

There are several different Statistical Distributions available in Slide2, for defining Random Variables, including Normal, Uniform and others. For more information, see the Statistical Distributions Overview topic.

## Mean

The Mean represents the average value of the Random Variable. The Mean value of a Random Variable in the Material Statistics dialog, is the same as the value of the variable that you have entered in the Define Material Properties dialog. This value is displayed in the Material Statistics dialog, in order to make it easier to define the other statistical parameters of the variable (standard deviation and relative min/max values).

- You MAY CHANGE the MEAN value of a Random Variable, in the Material Statistics dialog.
- If you change the MEAN value in the Material Statistics dialog, THIS WILL ALSO CHANGE THE VALUE OF THE PROPERTY, IN THE Material Properties DIALOG.
- And vice versa (if you change a property in the Material Properties dialog, the mean value will change in the Material Statistics dialog).

## Standard Deviation

The Standard Deviation of a Random Variable, is a measure of the variance or scatter of the variable about the Mean value. The larger the Standard Deviation, then the wider the range of values which the Random Variable may assume. NOTE:

- The Standard Deviation is applicable for Normal, Lognormal, and Beta distributions.
- It is NOT APPLICABLE for Uniform, Triangular, or Exponential distributions. If you are using one of these distributions, then you will NOT be able to enter a Standard Deviation.
- For tips on estimating values of Standard Deviation, see the Random Variables topic.

## Relative Minimum/Maximum Values

For each Random Variable, you must define a Minimum and Maximum allowable value. The Minimum/Maximum values are specified in the Material Statistics dialog as RELATIVE quantities (i.e. as distances from the Mean), rather than as absolute values.

During the analysis, the Relative Minimum and Maximum values are converted to the actual Minimum and Maximum values, when the statistical sampling is carried out for each Random Variable, as follows:

MINIMUM = MEAN – Relative MINIMUM

MAXIMUM = MEAN + Relative MAXIMUM

EXAMPLE: If the Mean Friction Angle = 35, and the Relative Minimum = Relative Maximum = 10, then the actual Minimum = 25 degrees, and the actual Maximum = 45 degrees.

Specifying the Minimum and Maximum values for each Random Variable, as RELATIVE numbers, rather than as ABSOLUTE numbers, simplifies the data input for the user, and is much less prone to error.

For each Random Variable, you must always specify non-zero values for the Relative Minimum and/or the Relative Maximum. If BOTH the Relative Minimum and Relative Maximum are equal to zero, no statistical samples will be generated for that variable, and the value of the variable will always be equal to the Mean.

The Relative Minimum and Maximum values, are applicable for ALL Statistical Distributions in Slide2.

## Automatic Calculation of Relative Minimum/Maximum Values

The Normal Statistical Distribution is the most commonly used distribution for most probabilistic analyses in geotechnical engineering. In most cases, you will probably use the Normal Distribution for most of your Random Variables.

For a Normal distribution, it can be shown that:

- about 68% of samples should fall within ONE standard deviation of the mean
- about 95% of samples should fall within TWO standard deviations of the mean
- about 99% of samples should fall within THREE standard deviations of the mean

This means that for practical purposes, a complete Normal distribution is defined by setting the Relative Minimum and Maximum values, equal to 3 Standard Deviations.

As a convenience, a shortcut option has been provided for this purpose in the Material Statistics dialog. If you are using a Normal Distribution, then you can use these options as follows:

- First enter the Standard Deviation for all applicable Random Variables.
- Select all rows in the dialog, for which you want to automatically calculate the Relative Minimum and Maximum values.
- Select the Auto Min / Max button at the right of the dialog.
- The Relative Minimum and Maximum will AUTOMATICALLY be set equal to 3 times the Standard Deviation, for each selected Random Variable.

This option is provided as a convenience and may save time when entering data for Normal Distributions. The use of this is entirely optional, and the user can always manually enter the Relative Minimum and Maximum values in the Material Statistics dialog.

## Correlation Coefficient

See the Mohr-Coulomb Correlation and Advanced Correlation topics for details.

## Other Dialog Features

The Material Statistics dialog also has the following useful features:

- Sorting – the rows of random variable data can be sorted according to Material Name, Property, or Distribution, by clicking the mouse on the title of the desired column. The rows can be sorted in ascending or descending alphabetical order, by re-clicking on a column title.
- Copy, Paste – data can be copied from the dialog or pasted into the dialog, by using the Copy and Paste buttons at the right of the dialog.
- Export to Excel – all of the data in the dialog can be exported to the Microsoft Excel program with a single mouse click, by selecting the Excel button at the right of the dialog.
- Re-size dialog – if some of the data in the dialog is not visible, select the Re-size button at the right of the dialog to automatically re-size the dialog to fit all of the data. You can also re-size the dialog by using the mouse to click and drag the lower right corner of the dialog.

## Sensitivity Analysis

The Material Statistics dialog is also used to define the material property variables that you would like to use in a Sensitivity Analysis. It is important to note that the following options:

- Statistical Distribution
- Standard Deviation
- Correlation Coefficient

are NOT applicable for a Sensitivity Analysis. A Sensitivity Analysis only requires that you define the Minimum and Maximum values for a variable.

- If you are performing a Sensitivity Analysis, but you are NOT performing a Probabilistic Analysis, then the Statistical Distribution and Standard Deviation columns, will NOT appear in the Material Statistics dialog. All other options in the dialog, behave as described for the Probabilistic Analysis. See this topic for details.
- If you are performing BOTH a Sensitivity Analysis AND a Probabilistic Analysis, then the Sensitivity Analysis will simply use the Minimum and Maximum values you have defined for each Random Variable. It will ignore the parameters which are not relevant to the Sensitivity Analysis (i.e. Statistical Distribution, Standard Deviation, Correlation Coefficient).

For more information about Sensitivity Analysis with Slide2, see the Sensitivity Analysis topic.

## Saturated Unit Weight as a Random Variable

If you have defined different unit weights for a material, above and below the Water Table, using the Saturated Unit Weight option, and you are defining Unit Weight as one of your Random Variables, it is important to note the following:

- In the Material Statistics dialog, only the UNSATURATED unit weight, is defined as a Random Variable.
- The SATURATED Unit Weight will not appear in the list of Random Variables.

HOWEVER, during the analysis, the value of the SATURATED unit weight will be directly correlated to the value of the UNSATURATED unit weight, such that the difference between the deterministic (mean) values, will be maintained. For example, if the mean UNSATURATED unit weight = 20 kN/m3, and the mean SATURATED unit weight = 21 kN/m3, then a difference of 1 kN / m3 will be maintained for all random sample values of the unit weight (e.g. if a random value of 19.5 is generated for the UNSATURATED unit weight, then the SATURATED unit weight will automatically be set to 20.5).

Therefore the distribution of the SATURATED unit weight will be identical to the distribution of the UNSATURATED unit weight, only shifted over to the right, by the difference in the mean values.

## Non-Random Material Properties

Most of the material properties available in *Slide2* for the various strength models, etc., can be defined as Random Variables in a Probabilistic Analysis. However, NOT ALL material properties can be defined as Random Variables, for the purposes of a Probabilistic Analysis in Slide2. These include the following:

- Shear / Normal Strength Function
- Anisotropic Strength Functions
- Discrete Strength Function
- Hu Coefficient (for Water Table)
- Hydraulic Properties (Groundwater Analysis)

However, the first three on the list can be defined with random shear strength using the Probabilistic Shear Strength option.

## Spatial Variability

See the spatial variability topic for details.