# 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 Overview of 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 2 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 **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. Select **Project Settings** on the toolbar or on the **Analysis** menu.

** Select:** Analysis > Project Settings

### 3.1 ANALYSIS TYPE

BY DEFAULT A DETERMINISTIC ANALYSIS IS SELECTED FOR A NEW FILE.

- Select the
**General**tab. - Change the
**Analysis Type**to**Probabilistic**.

### 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 SAMPLING AND RANDOM NUMBERS

THE **SAMPLING METHOD** DETERMINES HOW THE STATISTICAL DISTRIBUTION FOR THE RANDOM INPUT VARIABLES WILL BE SAMPLED. THE DEFAULT SETTINGS ARE **SAMPLING METHOD** = **LATIN HYPERCUBE** AND **NUMBER OF SAMPLES** = **5,000**. FOR MORE HELP, SEE Sampling in CPillar Project Settings.

### 3.4 PROJECT SETTINGS

Select the **Project Summary** tab and enter **CPillar** Elastic Analysis of Rectangular Roof Beam 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.

- Click
**OK**to close the**Project Settings**dialog.

## 4.0 Probabilistic 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 **Pillar Information**. Let's disable it for now to save on computing time when entering input data.

**Select:**Analysis > Autocompute- Uncheck
**Autocompute for probability analysis** - Click
**OK**.

In **CPillar**, the entirety of the input parameters are entered in the **Pillar Information** section of the Sidebar. The **Pillar Information** section 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:

- Analysis Method =
**Elastic** - Analysis Type =
**Probabilistic**

**4**.2 GEOMETRY

Enter the following mean data for the pillar geometry parameters:

- Mean Pillar Length =
**10 m** - Mean Pillar Width =
**100 m** - Mean Pillar Height =
**1 m** - Mean Rock Unit Weight =
**0.027 MN/m3** - Mean Overburden Unit Weight = 0.027 MN/m3
- Water Unit Weight = 0.0098 MN/m3
- Is Pillar Permeable? =
**No**

### 4.3 LATERAL STRESS

Enter the following mean data for the stress parameters:

- Stress Type = Constant
- Mean Water Height =
**0 m** - Mean Overburden Thickness =
**1 m** - Mean Horizontal Sigma x =
**2 MPa** - Mean Horizontal Sigma y =
**2 MPa**

### 4.4 STRENGTH

Enter the following mean data for the strength parameters:

- Strength Type =
**Hoek-Brown** - Mean Intact UCS
**= 50 MPa** - Mean Rock Mass m Value = 0.3
- Mean Rock Mass s Value =
**0.0001** - Mean Rock Mass Modulus =
**5000 MPa**

### 4.5 RANDOM VARIABLES

In order to perform a Probabilistic analysis, at least one random variable must be assigned a statistical distribution. The **Statistical Distribution** is changed by clicking on the distribution icon to the left of the parameter value.

Let's model the following parameters as random variables:

- Click the distribution icon to the left of
**Mean Horizontal Sigma x**, and set: - Click the distribution icon to the left of
**Mean Horizontal Sigma y**, and set: - Click the distribution icon to the left of
**Intact UCS**, and set:

Distribution = **Normal**

Std. Dev. = **0.5 MPa**

Rel. Min and Rel Max = **3x Std. Dev.** (Check **3x std. dev.**)

Distribution = **Normal**

Std. Dev. = **0.5 MPa**

Rel. Min and Rel Max = **3x Std. Dev.** (Check **3x std. dev.**)

Distribution = **Normal**

Std. Dev. = **5**

Rel. Min and Rel Max = **3x Std. Dev.** (Check **3x std. dev.**)