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# 5 - Rigid Analysis of Polygonal Pillar

In Tutorial 01 - CPillar Quick Start (Rigid Analysis of Square Pillar), you learned about the Rigid analysis method in CPillar for a rectangular pillar with vertical abutments. In this tutorial, you will learn how to define the geometry and lateral stress orientations for a polygonal pillar with non-vertical abutments.

Topics Covered in this Tutorial:

• Project Settings
• Rigid Analysis for Polygonal Pillar
• Non-vertical Abutments
• Principal Stresses
• Deterministic Analysis
• Input Data
• Analysis Results
• Mohr-Coulomb Strength Criterion
• Water Pressure
• Sensitivity Analysis

Finished Product:

The finished product of this tutorial can be found in the Tutorial 5 Rigid Analysis of Polygonal Pillar.cpil5 file, located in the Examples > Tutorials folder in your CPillar installation folder.

## 1.0 Introduction

This model represents a polygonal pillar with a thickness of 4 meters. There is 1 meter of overburden and 1 meter of free water (6 – (4+1) = 1).

The lateral stress is defined by gravity. Lateral stresses will be calculated based on the Horizontal / Vertical stress ratios.

## 2.0 Creating a New File

If you have not already done so, run the CPillar program by double-clicking the CPillar icon in your installation folder or by selecting Programs > Rocscience > CPillar > CPillar in the Windows Start menu.

When the program starts, a default model is automatically created. If you do NOT see a model on your screen:

Select: File > New

Whenever a new file is created, the default input data forms valid pillar geometry, as shown in the image below.

If the CPillar application window is not already maximized, maximize it now so that the full screen is available for viewing the model. You will have a 3D pillar displayed on the screen in isometric orientation.

## 3.0 Project Settings

The Project Settings dialog allows you to configure the main analysis parameters for your model, such as Units, Analysis Type, and Sampling Method . To open the dialog, select Project Settings on the toolbar or on the Analysis menu.

Select: Analysis > Project Settings

### 3.1 ANALYSIS TYPE

There are two main analysis types in CPillar: Deterministic, and Probabilistic. The default choice for new files is Deterministic.

### 3.2 UNITS

In this tutorial we are using metric (MPa) units, so make sure the Metric, stress as MPa option is selected for Units in the Project Settings General tab (default setting).

### 3.3 PROJECT SUMMARY

• Select the Project Summary tab and enter CPillar Rigid Analysis of Polygonal Pillar Tutorial as the Project Title.

NOTE: You can have Project Summary information appear on analysis results printouts by setting up a header or footer through Page Setup on the File menu.

• Leave all other settings as is and click OK to close the Project Settings dialog.

## 4.0 Input Data

In CPillar, the entirety of the input parameters are entered in the Pillar Information panel in the Sidebar.

The Pillar Information panel is organized under four headings: Analysis, Geometry, Lateral Stress, and Strength. To change a parameter, click on the value and enter the new value or select from the dropdown as necessary. The model will reflect any changes, immediately.

### 4.1 ANALYSIS

Set up the Analysis:

1. Analysis Method = Rigid - Polygon
2. Analysis Type = Deterministic

### 4.2 GEOMETRY

1. Select the Edit Coordinates button in the Pillar Information panel under Geometry.
The Polygon Coordinates dialog appears allowing you to enter the polygon vertex coordinates. By default, the polygon coordinates for a 10 x 10 m pillar are preset.
2. Enter the following coordinates:
3.  Point Easting (X) Northing (Y) 1 0 0 2 -1 1 3 -2 5 4 6 8 5 7 4 6 3 1 7 2 -3 8 1 -4

The Polygon Coordinates dialog should look as follows:

4. Click OK.
5. The pillar should looks as follows:

• Enter the following data for the pillar Geometry parameters:
1. Abutment Trend = 0 deg
2. Abutment Plunge = 60 deg
3. Pillar Height = 4 m
4. Rock Unit Weight = 0.027 MN/m3
5. Overburden Unit Weight = 0.02 MN/m3
6. Water Unit Weight = 0.0098 MN/m3
7. Is Pillar Permeable? = No

### 4.3 LATERAL STRESS

• Enter the following mean data for the Stress parameters:
1. Stress Type = Gravity
2. Water Height = 6 m
3. Overburden Thickness = 1 m
4. Locked In Stress = 0.5 MPa
5. Trend 3 = 0 deg
6. Horizontal/Vertical K1 = 0.8
7. Horizontal/Vertical K3 = 0.8

NOTE: The Water Height is measured from the base of the pillar. The Overburden Thickness is measured from the top of the pillar.

### 4.4 STRENGTH

• Enter the following mean data for the Strength parameters:
1. Strength Type = Mohr-Coulomb
2. Rock Mass Cohesion = 0 MPa
3. Rock Mass Friction Angle = 30 deg

Your Pillar Information panel should look as follows:

Your model should look like this:

Notice that the abutments are no longer vertical. The slippage along the abutments will occur at a 60 degree angle from the horizontal.

## 5.0 Analysis Results

To ensure the latest analysis results are always displayed, CPillar automatically computes an analysis whenever:

• a file is opened, or
• input data is entered or modified in the Pillar Information section.

The primary result from a CPillar Deterministic analysis is the pillar Factor of Safety (FoS). The FoS is displayed in the Results section. The Results section appears in the Sidebar below and displays a summary of analysis results.

Your results should looks as shown:

Note that since this is a Rigid Analysis, Shear is the only failure mode; as such Shear is the only failure result displayed.

## 6.0 Sensitivity Analysis

The final section of the tutorial demonstrates the CPillar Sensitivity Analysis feature. In a Sensitivity Analysis, individual variables can be varied among user-defined minimum and maximum values while all other input parameters remain constant. This allows you to determine the effect of individual variables on the FoS.

We will use Sensitivity Analysis to show the effects of abutment and stress orientation on the FoS.

1. Select Sensitivity Analysis on the toolbar or on the Analysis menu.
2. In the Sensitivity Analysis Input dialog, select the Add button in the Sensitivity Analysis Input dialog then select Abutment Trend from the drop-down list.
3. Enter From = 0 and To = 360 deg.
4. In the Sensitivity Analysis Input dialog, select the Add button in the Sensitivity Analysis Input dialog then select Abutment Plunge from the drop-down list.
5. Enter From = 45 and To = 90 deg.
6. In the Sensitivity Analysis Input dialog, select the Add button in the Sensitivity Analysis Input dialog then select Trend 3 from the drop-down list.
7. Enter From = -90 and To = 90 deg.
8. Click Plot.

You should see the following sensitivity plot.

Notice that the FoS follows a sinusoidal relationship with respect to Abutment Trend with a period of 180 degrees. This is due to the fact the shear strength is the same when the abutment is slanting in the opposite trend direction. The FoS decreases to the minimum FoS as the Abutment Plunge approaches 90 degrees (i.e. vertical). Trend 3 orientation has no impact on the factor of safety since K1 = K3 (i.e. isotropic soil stress behaviour).

This concludes the tutorial.