The cantilever beam model

We are interested in the 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.

beam geometry

The beam geometry

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 intertia 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)

Load the use case

We can load this classical model from the use cases module as follows :

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

API documentation

See CantileverBeam.