# The cantilever beam model¶

We are interested in the vertical deviation of a diving board created by a child diver. We consider a child whose weight generates a force approximately equal to 300N (i.e. almost 30 kg). Because of the uncertainties in the weight of the person, we consider that the force is a random variable. The length of the diving board is between 2.5 m and 2.6 m. The Young modulus is uncertain and between 65 and 75 GPa, which corresponds to the fiberglass material, a material often used for diving boards. Uncertainties in the production of the material are taken into account in the Young modulus and the section modulus of the board.

We consider a cantilever beam defined by its Young’s modulus , its length and its section modulus . One end of the cantilever beam is built in a wall and we apply a concentrated bending load at the other end of the beam, resulting in a deviation .

## Inputs¶

• : Young modulus (Pa), Beta(, , a = , )

• : Length of beam (m), Uniform(min=2.5, max= 2.6)

• : Moment of inertia (), Beta(, , , ).

In the previous table and are the mean and the standard deviation of .

We assume that the random variables , , and are dependent and associated with a gaussian copula which correlation matrix is:

In other words, we consider that the variables and are negatively correlated: when the length increases, the moment of inertia decreases.

## Output¶

The vertical displacement at free end of the cantilever beam is:

A typical event of interest is when the beam deviation is too large which is a failure:

## API documentation¶

class CantileverBeam

Data class for the cantilever beam example.

Examples

>>> from openturns.usecases import cantilever_beam
>>> # Load the cantilever beam model
>>> cb = cantilever_beam.CantileverBeam()
Attributes:
dimThe dimension of the problem

dim=4.

EBeta distribution

ot.Beta(0.9, 3.5, 65.0e9, 75.0e9)

FLogNormal distribution

ot.LogNormalMuSigma()([300.0, 30.0, 0.0])

LUniform distribution

ot.Uniform(2.5, 2.6)

IBeta distribution

ot.Beta(2.5, 4.0, 1.3e-7, 1.7e-7)

modelSymbolicFunction, the physical model of the cantilever beam.
RCorrelationMatrix

Correlation matrix used to define the copula.

copulaNormalCopula

Copula of the model.

distributionJointDistribution

The joint distribution of the parameters.

independentDistributionJointDistribution

The joint distribution of the parameters with independent copula.

## Examples based on this use case¶

Export a metamodel

Export a metamodel

Create a polynomial chaos metamodel by integration on the cantilever beam

Create a polynomial chaos metamodel by integration on the cantilever beam

Compute Sobol’ indices confidence intervals

Compute Sobol' indices confidence intervals

Kriging : cantilever beam model

Kriging : cantilever beam model

Kriging: choose an arbitrary trend

Kriging: choose an arbitrary trend

Kriging the cantilever beam model using HMAT

Kriging the cantilever beam model using HMAT

Kriging: choose a polynomial trend on the beam model

Kriging: choose a polynomial trend on the beam model

Analyse the central tendency of a cantilever beam

Analyse the central tendency of a cantilever beam

Use a randomized QMC algorithm

Use a randomized QMC algorithm

Use the Adaptive Directional Stratification Algorithm

Use the Adaptive Directional Stratification Algorithm

Use the Directional Sampling Algorithm

Use the Directional Sampling Algorithm

Use the Importance Sampling algorithm

Use the Importance Sampling algorithm

Use the FORM - SORM algorithms

Use the FORM - SORM algorithms