# 2 - Elastic Analysis of Rectangular Roof Beam

In Tutorial 01 - CPillar Quick Start (Rigid Analysis of Square Pillar), you learned about the two main analysis types in **CPillar**: Deterministic, and Probabilistic, and how to run a Deterministic analysis (the default). In this tutorial, you will learn how to run a Probabilistic analysis.

In a Probabilistic analysis, statistical input data can be entered to account for uncertainty in geometry, lateral stress, and strength values. The result is a distribution of Factors of Safety, from which a Probability of Failure is calculated. To learn more about Probabilistic Analysis, see Probabilistic Analysis in CPillar.

**Topics Covered in this Tutorial:**

- Project Settings
- Random Variables
- Roof Beam Model
- Probabilistic Analysis
- Elastic Analysis for Rectangular Pillar
- Hoek-Brown Strength Criterion
- Constant Lateral Stress
- Failure Modes
- Histogram Plot

**Finished Product:**

The finished product of this tutorial can be found in the *Tutorial 02 Elastic Analysis of Rectangular Roof Beam (Long Excavation).cpil5* file, located in the** Examples > Tutorials** folder in your CPillar installation folder.

## 1.0 Introduction

This model represents a roof beam (**10 x 100 m**) with a thickness of **1 m**. There is also **1 m** of overburden having the same density as the beam.

The lateral stress is defined as a constant. Lateral effective stresses of **2 ± 0.5 MPa** in both the **x** and **y** directions will be used.

## 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.

### 3.1 General

- Select
**Project Settings**from the toolbar or the**Analysis**menu. - Navigate to the
**General**tab. - Ensure
**Units = Metric, stress as MPa**(default setting). - Set
**Analysis Type****=****Probabilistic**. - Set
**Pillar Shape = Rectangular Pillar**.The

NOTE:**Elastic**analysis is formulated for a rectangular thin plate. Therefore, the**Pillar Shape**must be set to**Rectangular Pillar**. - Set
**Analysis Method = Elastic**.

**Polygonal Pillar**is only applicable to

**Rigid**analysis.

### 3.2 Sampling

The **Sampling Method** determines how the statistical distribution for the random input variables will be sampled. For each sample, the computed Factor of Safety is compared against the Design Factor of Safety to determine the Probability of Failure for all samples. Any computed **Factor of Safety < Design Factor of Safety** is considered a failed pillar for the applicable failure mode.

- Navigate to the
**Sampling**tab of the**Project Settings**dialog. - Leave the default values.
**Sampling Method = Latin HyperCube****Number of Samples = 5000****Design Shear Factor of Safety = 1****Design Elastic Factor of Safety = 1**

**Shear Factor of Safety**and

**Elastic Buckling Factor of Safety**are considered in an

**Elastic**analysis.

**Compression Factor of Safety**and

**Buckling Parameter**are only applicable to

**Voussoir**analysis.

For more information, see Sampling.

### 3.3 Random Numbers

- Navigate to the
**Random Numbers**tab of the**Project Settings**dialog. - Note that the
**Random Number Generation = Pseudo-random**with a constant seed value. This ensures that Probabilistic results are reproducible for the exact same set of inputs every time you run**Compute**.

### 3.4 Project Summary

- Navigate to the
**Project Summary**tab of the**Project Settings**dialog. - Enter
**CPillar Elastic Analysis of Rectangular Roof Beam Tutorial**as the**Project Title**. - Click
**OK**to close the**Project Settings**dialog.

**Project Summary**information appear on analysis results printouts by setting up a header or footer through

**Page Setup**on the

**File**menu.

## 4.0 Mean Input Data

**Autocompute** is turned ON by default when performing a **Probabilistic** analysis. Autocompute** **will automatically compute the model every time a change is made in Input Data.

In CPillar, the input parameters are entered in the **Input Data **dialog. The Input Data** **dialog is organized under four tabs: **Geometry**, **Stresses**, **Material Properties**, and **Rock Mass and Abutment 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. Mean values are entered directly in the edit controls in the Input Data dialog.

### 4.1 Geometry

To set up the analysis in the **Input Data** dialog:

- Select
**Input Data**from the toolbar or**Analysis**menu. - Navigate to the
**Geometry**tab. - Enter the following mean data for the
**Pillar Dimension**parameters:- Mean
**Pillar Height = 1 m** - Mean
**Pillar Length = 10 m** - Mean
**Pillar Width = 100 m**

- Mean
- Enter the following mean data for the
**Unit Weight**parameter:- Mean
**Rock Unit Weight = 0.027 MN/m3**

- Mean

### 4.2 Stresses

Enter the following mean data for the stress parameters:

- Navigate to the
**Stresses**tab of the**Input Data**dialog. - Enter the following mean data for the
**Lateral Stress**parameters:**Stress Type = Constant**.- Mean
**Horizontal Sigma x = 2 MPa** - Mean
**Horizontal Sigma y = 2 MPa**

- Enter the following mean data for the
**Overburden**parameters:- Mean
**Overburden Thickness = 1 m** - Mean
**Overburden Unit Weight = 0.027 MN/m3**

- Mean
- Enter the following mean data for the
**Water**parameters:- Mean
**Water Height =****0 m** - Mean
**Water Unit Weight = 0.0098 MN/m3** **Pillar Is Permeable = No**

- Mean

### 4.3 Material Properties

Enter the following mean data for the shear strength parameters:

- Navigate to the
**Material Properties**tab of the**Input Data**dialog. - Enter the following mean data for the
**Material Properties 1:****Strength Type = Hoek-Brown**- Mean
**Rock Mass s Value = 0.0001** - Mean
**Rock Mass m Value = 0.3**

### 4.4 Rock Mass and Abutment Strength

Enter the following mean data for the rock mass parameters:

- Navigate to the
**Rock Mass and Abutment Strength**tab of the**Input Data**dialog. - Enter the following mean data for the
**Rock Mass**parameters:- Mean
**Intact UCS = 50 MPa** - Mean
**Rock Mass Modulus = 5000 MPa**

- Mean
- Click
**OK**to apply the changes and close the dialog.

The **3D Pillar View** shows an isometric perspective of a 3D pillar model with principle stresses and dimensions labeled:

**Mean Pillar Data**(entered as mean values in the

**Input Data**dialog).

## 5.0 Random Variables

In order to perform a Probabilistic analysis, at least one random variable must be assigned a statistical distribution. The **Statistical Distribution** can be assigned by either:

- Clicking on the distribution icon to the left of the mean parameter value in the Input Data dialog; or
- In the
**Statistics**dialogs.

To model the parameters as random variables in the **Geometry and Stress Statistics **dialog:

- Select
**Geometry and Stress**from the**Statistics**menu. - Click the
**Add**button and set the following:**Property = Sigma x****Distribution = Normal****Std. Dev. = 0.5 MPa****Rel. Min**and**Rel. Max****= 1.5 MPa**.

- Click the
**Add**button and set the following:**Property = Sigma y****Distribution = Normal****Std. Dev. = 0.5 MPa****Rel. Min**and**Rel. Max****= 1.5 MPa**.

- Click the
**Add**button and set the following:**Property =****Intact UCS****Distribution = Normal****Std. Dev. = 5 MPa****Rel. Min**and**Rel. Max****= 15 MPa**.

- Select
**OK**.

**TIP**: Click the

**3X Std. Dev.**button to quickly set the

**Rel. Min**and

**Rel. Max**values as 3 times the

**Std. Dev**.

The **3D Pillar View** shows an isometric perspective of a probabilistic 3D pillar model with principle stresses and dimensions labelled:

The list of **Random Variables** defined are listed under **Pillar Information** in the Sidebar.

## 6.0 Analysis Results

Since we have **Autocompute **turned ON, the model is automatically computed and shows the latest analysis results for the current inputs.

The primary result from an CPillar** **Probabilistic analysis is the pillar Probability of Failure. The Probability of Failure is displayed at the top-center of the **3D Pillar View** and in the **Pillar Information** pane. The **Pillar Information** pane appears in the Sidebar on the right side of the CPillar application window and displays a summary of analysis results.

Depending on the **Analysis Method **(i.e., **Rigid**, **Elastic**, or **Voussoir**), additional data is also presented. Note that since this is an **Elastic **analysis, two failure modes are considered:

- Shear (vertical slippage at the abutments), and
- Elastic Buckling.

Both the **Shear Probability of Failure** and **Elastic Buckling** **Probability of Failure** are** 0.0000 or 0%**.

**TIP**: Optionally show Probability of Failure as fraction or percent by toggling the

**Show PF as Percentage**checkbox in the

**Display Options**dialog.

For an **Elastic **analysis, the **Tension Count** is also displayed. The **Tension Count** indicates the number of cases where the shear strength was lowered due to bending induced tensile stresses. See Elastic Analysis for more information on how the Factor of Safety is computed. Notice that a majority of the samples have corrected shear strength (**Tension Count = 4766/5000**). This is due to the fact that we are modelling a long pillar, with **Pillar Width** much greater than **Pillar Length**.

**Rigid**analysis will give the same Shear Factor of Safety. In general, the Shear Factor of Safety from an

**Elastic**analysis will be between 50 and 100% of that determined from a rigid analysis, depending on the magnitude of the calculated bending stresses, and the lateral stresses.

**Elastic Buckling** will generally not be the governing mode of failure for an Elastic analysis. The mean Elastic Buckling Factor of Safety will generally be very large. Elastic buckling will only be an issue if:

- Lateral stresses are high, and/or
- Span/depth ratio is high, and/or
- Rock mass modulus is low.

## 7.0 Probability of Failure Histogram

Now let's reduce the **Mean Intact UCS** value and see how it affects the Shear Probability of Failure.

- Select
**Input Data**from the toolbar or**Analysis**menu. - Navigate to the
**Rock Mass and Abutment Strength**tab. - Set the Mean
**Intact UCS = 3 MPa**. - Click the distribution icon to the left of
**Intact UCS**, and change:**Std. Dev. = 0.5 MPa****Rel. Min**and**Rel. Max****= 1.5 MPa**(3x Std. Dev.)- Click
**OK**to close the**Distribution**popup.

- Click
**OK**to apply the changes and close the**Input Data**dialog.

The **Shear Probability of Failure** is now **0.005** or **0.5%**:

It is helpful to plot a histogram of the data, in order to see the meaning of the Probability of Failure value.

To plot a probability of failure histogram:

- Select
**Plot Histogram**from the toolbar or the**Statistics**menu. - Set
**Failure Mode = Shear**. - Set
**Data to Plot = Factor of Safety**. - Leave the default
**Number of Bins = 30**. - Click
**Plot**.

The histogram plot of **Shear Factor of Safety vs. Relative Frequency** is shown:

Notice the red shaded area on the left side; this represents the 0.5% Probability of Failure value. That is, this area represents the 0.5% of the 5000 samples (entered in the **Project Settings** dialog) which resulted in a Factor of Safety less than 1 (**Design Shear Factor of Safety**).

This concludes the tutorial. You can now proceed to Tutorial 03: Voussoir Analysis of Rectangular Roof Plate in CPillar.