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.

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.