# Cantilever Beam (Reliability Analysis)

This page shows how to perform Reliability Analysis of the Cantilever Beam with COSSAN-X.

# Prerequisites

This analysis requires that a Physical Model is already defined in COSSAN-X as shown in the page Cantilever Beam (Uncertainty Quantification).

# Additional objects

## Input

### Parameters

In addition to the object defined in Cantilever Beam (Uncertainty Quantification) an addition parameter that define the maximum allowed displacement of the Beam is required.

A parameter named MaxDisplacement with a value equal to 0.010 is defined.

## Evaluator

An evaluator is used in a definition of a model. If a model could be seen as a black box that returns output values given the input values, an evaluator is actually the content of such a black box. Thus, an evaluator is a low-level definition of the functional relation between the inputs and outputs of a model.

See also Category:Evaluators

### Performance Function

The Performance function is used to define the domain of definition of the model into two sets, the safe set and the failure set. For more information see Performance function definition.

The new Performance Function is created using the wizard.

Then the Performance Function editor will appear. The simplest way to define a performance function is to use the concept of demand and capacity.

- The Capacity represents the maximum allowed displacement of the Beam. Therefore, the object Parameter MaxDisplacement is added to the field Capacity.
- The Demand represents the actual displacement of the Beam. The output of the Matlab script (i.e. w) is added to the field Demand
- Output name. A name for the value of the performance function must be edited. The default value vg is used.

## Model

To perform reliability analysis is it necessary to define a Probabilistic Model. |

### Probabilitic Model

A new Probabilistic Model can be created invoking the wizard by moving the mouse pointer to "Probabilistic Model" and clicking the right mouse button and selecting "Add Probabilistic Model":

A wizard pops up showing default names which can be modified. In the following the default values are accepted by pressing the 'Finish' button.

Then, the editor appears, in which a Physical Model and the Performance Function are combine.

Then the input and output associated with the Physical Model and the Performance Function are shown at the right in the windows 'Inputs' and 'Outputs' as shown. If this doesn't happen automatically, then manual selection can be achieved by clicking the plus button and selecting the physical model and performance function from their respective lists. If the inputs and outputs still do not appear in the 'Input' and 'Output' boxes, saving the file using 'ctrl s' will amend this.

# Reliability Analysis

After saving the data, the simulation is ready to start. Different simulations can be started by pressing at the small white triangle within a green circle at top right. This is still carried out in the probabilistic model tab. |

The next window allows to choose among different analysis types such as: 'Design of Experiments', 'Sensitivity Analysis', 'Uncertainty Quantification' and 'Userdefined Analysis'. In this tutorial 'Reliability Analysis' is selected.

After pressing 'Next' in the above window, various options for performing reliability analysis are provided. Here the aim of the reliability analysis is to estimate the failure probability associated to the Probabilistic Model.

## Monte Carlo simulation

Then, the simulation method used to estimate the failure probability is selected. The well known Monte Carlo sampling procedure is selected using 10000 samples. At the right, the number of independent samples can be declared. It is possible to split the computation into several batches which might be processed by different computers using parallel computing. In this simple example with little computational effort, only a single batch is used. It is possible to use the Coefficient of Variation of estimator of the failure probability as a termination criteria.

Pressing Next it is possible to specify the Grid Settings (i.e. how to perform the analysis). In this example the analysis will be performed on the local machine and it is not necessary to change the default values in the mask.

The actual computations start after pressing the 'Finish' button. Then, the following window appears.

When completed, a message of successful completion is provided.

## Line Sampling simulation

The Line Sampling procedure is selected. To do this, repeat the previous steps starting from 'Reliability Analysis' sub-section. In the next page the setting for Line Sampling procedure are available (see Line Sampling (wizard)). The default parameters are used in this case:

Then, it is necessary to define the sensitivity analysis method used to calculate the important direction:

The actual computations starts after pressing the 'Finish' button.

When completed, a message of successful completion is provided.

In order to reduce further the computational cost, the Adaptive method of Line Sampling can be selected as show in the following figure. By doing so the points along the lines are automatically computed by the method.

## Show the Results

All results are accessible in the section "Analysis". The results are stored in sub-folders according to the performed analysis. In the present case, Reliability Analyses have been performed. This can be recognised by the value '(2)' in the folder 'Reliability Analysis'. Expanding the folder 'Results' of the Reliability Analysis shows the following sub folders and all the involved variables defined in the input. For more information about the structure of the results (outputs), please refer to the following page Category:Outputs. |

#### Failure Probability

##### Monte Carlo Simulation

In the Result folder the object EngineOutput contains now two different items, the MonteCarlo_output and the Timer. Doble-clicking on the object EngineOutput it can be open in the editor. Then selecting MonteCarlo_output it is possible to see the value of the estimated failure probability (pfhat) and the standard deviation of the estimator.

Selection CossanTimer it is possible to visualize the computational time required by the analysis.

##### Line Sampling

Doble-clicking on the object EngineOutput present in the result folder of the Reliability Analysis it can be open in the editor. The object EngineOutput contains now two different items, the LineSampling_output and the Timer.

Then selecting LineSampling_output it is possible to see the value of the estimated failure probability (pfhat) and the standard deviation of the estimator.

And selectin the CossanTimer it is possible to visualize the execution time required by the LineSampling.

It is possible to recogize the failure probability can be estimated with a fraction of the samples required by the Monte Carlo Simulation. Furthermore, the adaptive methods allow reducing further the number of model evaluations.#