# 3 - Voussoir Analysis of Rectangular Roof Plate

In Tutorial 01 - CPillar Quick Start (Rigid Analysis of Square Pillar), you learned about the Rigid analysis method in **CPillar**. In Tutorial 02 - Elastic Analysis of Rectangular Roof Beam in CPillar, you learned about the Elastic analysis method. In this tutorial, you will learn about the Voussoir analysis method.

Topics Covered in this Tutorial:

- Project Settings
- Random Variables
- Rectangular Roof Plate Model
- Voussoir Analysis
- Mohr-Coulomb Strength Criterion
- Probabilistic Analysis
- Arch Snap-thru Theory

Finished Product:

The finished product of this tutorial can be found in the *Tutorial 3 **Voussoir Analysis of Rectangular Roof Plate**.cpil5* file, located in the *Examples > Tutorials* folder in your **CPillar** installation folder.

## 1.0 Introduction

This model represents a rectangular roof plate (**30 x 20 m**) with a thickness of **2.5 m**.

The **Voussoir** analysis method will be used. The Voussoir method is applicable for low confining stresses. A span to depth ratio smaller than 3 is not recommended. External lateral stress is not an option for a Voussoir analysis. Lateral stress exists, but it is induced by the ‘arching’ action of the rock, it is not an input parameter.

The **Mohr-Coulomb** strength criterion will be used. Cohesion is automatically set to zero for a Voussoir analysis when using the Mohr-Coulomb strength criterion. Only a friction angle is entered.

## 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 SUMMARY

Select the **Project Summary** tab and enter **CPillar** Voussoir Analysis of Rectangular Roof Plate 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** - Select
**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 =
**Voussoir** - Analysis Type =
**Probabilistic**

**4**.2 GEOMETRY

Enter the following mean data for the pillar geometry parameters:

- Mean Pillar Length =
**30 m** - Mean Pillar Width =
**20 m** - Mean Pillar Height =
**2.5 m** - Mean Rock Unit Weight =
**0.03 MN/m3** - Mean Overburden Unit Weight = 0 MN/m3
- Mean Face Dip =
**0 deg**

**NOTE:** Water is not included in a Voussoir analysis.

### 4.3 LATERAL STRESS

Enter the following mean data for the stress parameters:

- Mean Overburden Thickness = 0 m
- Mean Support Pressure =
**0 MPa**

### 4.4 STRENGTH

Enter the following mean data for the strength parameters:

- Strength Type =
**Mohr-Coulomb** - Mean Intact UCS
**= 45 MPa** - Mean Rock Mass Friction Angle =
**30 deg** - Mean Rock Mass Modulus =
**3500 MPa** - Mean Poisson Ratio =
**0.25**

### 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
**Intact UCS**, and set: - Click the distribution icon to the left of
**Mean Rock Mass Friction**, and set: - Click the distribution icon to the left of
**Mean Rock Mass Modulus**, and set: - Click the distribution icon to the left of
**Mean Poisson Ratio**, and set:

Distribution = **Normal**

Std. Dev. = **5**

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

Distribution = **Normal**

Std. Dev. = **5 deg**

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

Distribution = **Normal**

Std. Dev. = **500 MPa**

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

Distribution = **Normal**

Std. Dev. = **0.05 MPa**

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

The **Pillar Information** side panel should look as follows:

**NOTE:** When using the Mohr-Coulomb Strength Criterion with a Voussoir analysis, the cohesion is automatically set to zero, since by definition the roof is supporting itself through frictional resistance only.

Your model should look like this:

## 5.0 Analysis Results

Since we have **Autocompute** turned OFF, we have to run **Compute** from the toolbar or on the **Analysis** menu in order to get the latest analysis results for the current inputs.

** Select**: Analysis > Compute

The primary result from an **CPillar** Probabilistic analysis are:

- The pillar Mean Factor(s) of Safety (FoS).
- Standard Deviation of the Factor(s) of Safety
- Probability of Failure (expressed as a percentage)

Depending on the Analysis Method (i.e. **Rigid**, **Elastic**, or **Voussoir**), additional data is also presented. The FoS(s) are displayed in the **Results **section. The **Results **section appears in the **Sidebar** on the left side of the **CPillar** application window and displays a summary of analysis results.

Your results should looks as shown:

Note that since this is a **Voussoir** Analysis, three failure modes are considered:

**Shear**(vertical slippage at the abutments),**Arch Snap Thru**, and**Compression**.

The probability of failure is 0.6% under Shear and 0% under Compression. The probability of failure under Arch Snap Thru is **Low**.

The driving force for Arch Snap-Thru buckling is the self-weight of the rock (i.e. pillar). For a **Voussoir Analysis**, the **Mean Buckling Parameter **and Mean Midspan Displacement are also displayed under Arch Snap Thru. The Mean Buckling Parameter represents the percentage of unstable arch configurations for a given geometry and rock mass modulus. A buckling parameter greater than or equal to 35 indicates that the roof is unstable. The 35 corresponds to a midspan deflection of 10% of the beam thickness. This means that arch stability can also be assessed by monitoring the displacement at midspan, relative to the undeflected state.

Note also the Probability of Failure defined as Low. The failure probabilities correspond to mean buckling parameter ranges as indicated below:

Failure Probability | Mean Buckling Parameter |

Low | 0 to 10 |

Medium | 10 to 25 |

High | 25 to 50 |

Very High | greater than 50 |

Now we will try to predict the critical plate thickness for the collapse of the arch.

## 6.0 Collapse of the Arch

Now let’s gradually collapse our arch by decreasing the plate thickness.

### 6.1 PILLAR HEIGHT = 1 METER

In the P**illar Information** side-panel change the pillar height to **1 m**. Your results should look as follows:

Note that the Shear factor of safety increases, while the Snap-Thru and Compression factors of safety decrease. Note also the increase in the arch Midspan Displacement, from **10.4 mm** to **63.5 mm**. The Probability of Arch Snap-Thru Failure has increased from **Low** to **Medium**.

### 6.2 PILLAR HEIGHT = 0.8 METER

In the Pillar Information side-panel change the pillar height to **0.8 m**. Your results should look as follows:

Again, Shear factor of safety increases, while Snap-Thru and Compression stability decreases. Arch midspan displacement is now **102 mm**. Recall that when the midspan displacement reaches about 10% of the plate or beam height, arch collapse is imminent.

### 6.3 PILLAR HEIGHT = 0.5 METERS

Finally, reduce the pillar height to **0.5 m**. Your results should look as follows:

The arch is completely unstable, and probability of failure is **100%**.

This concludes the tutorial. You can now proceed to Tutorial 04: Empirical Scaled Span Approach in CPillar.