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 $E$ , its length $L$ and its section modulus $I$ . One end of the cantilever beam is built in a wall and we apply a concentrated bending load $F$ at the other end of the beam, resulting in a deviation $Y$ .

Inputs¶

$E$ : Young modulus (Pa), Beta( $\alpha = 0.9$ , $\beta = 3.5$ , a = $65.0 \times 10^9$ , $b = 75.0 \times 10^9$ )

$F$ : Loading (N), Lognormal( $\mu_F=300.0$ , $\sigma_F=30.0$ , shift=0.0)

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

$I$ : Moment of inertia ( $m^4$ ), Beta( $\alpha = 2.5$ , $\beta = 4.0$ , $a = 1.3 \times 10^{-7}$ , $b = 1.7 \times 10^{-7}$ ).

In the previous table $\mu_F=E(F)$ and $\sigma_F=\sqrt{V(F)}$ are the mean and the standard deviation of $F$ .

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

$R = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -0.2 \\ 0 & 0 & -0.2 & 1 \end{pmatrix}$

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

Output¶

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

$Y = \dfrac{F\, L^3}{3 \, E \, I}$

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

$Y \ge 0.30 (m)$

API documentation¶

class CantileverBeam

Data class for the cantilever beam example.

Attributes:

dimint: The dimension of the problem, dim=4.
EBeta: Young’s modulus distribution Beta(0.9, 3.5, 65.0e9, 75.0e9)
FLogNormal: Load distribution LogNormalMuSigma()([300.0, 30.0, 0.0])
LUniform: Length distribution Uniform(2.5, 2.6)
IIBeta: Moment of inertia distribution 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

>>> from openturns.usecases import cantilever_beam
>>> # Load the cantilever beam model
>>> cb = cantilever_beam.CantileverBeam()

Examples based on this use case¶

Export a metamodel

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: choose an arbitrary trend

Gaussian Process Regression : cantilever beam model

Kriging: choose a polynomial trend on the beam model

Analyse the central tendency of a cantilever beam

Use a randomized QMC algorithm

Use the Adaptive Directional Stratification Algorithm

Use the Directional Sampling Algorithm

Use the Importance Sampling algorithm

Use the FORM - SORM algorithms

Estimate Sobol’ indices for the beam by simulation algorithm

Estimate Sobol' indices for the beam by simulation algorithm

OpenTURNS

An Open source initiative for the Treatment of Uncertainties, Risks'N Statistics

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The cantilever beam model¶

Inputs¶

Output¶

API documentation¶

Examples based on this use case¶