1. Reference problem#

1.1. Geometry#

A and D models:

\(t=x\)

B, C and E models:

\(\frac{\mathrm{\pi }}{4}=(\widehat{x},t)\) and \(t\mathrm{.}z=0\)

Beam length:

\(L=\mathrm{AB}=0.9m\)

Circular section:

Diameter: \(D=0.05m\)

Point coordinates ( \(m\) ) :

		Modeling A and D	Modeling B and C
	\(A\)	\(B\)	\(B\)
\(X\)	\(0.\)	\(0.9\)	\(0.9\mathrm{cos}(\mathrm{\pi }/4)\)
\(Y\)	\(0.\)	\(0.\)	\(0.9\mathrm{sin}(\pi /4)\)
\(Z\)	\(0.\)	\(0.\)	\(0.\)

Table 1.1-1 : Coordinates of the points \(A\) and \(B\)

1.2. Material properties#

\(E={2.10}^{11}\mathrm{Pa}\)

\(\rho =7800\mathrm{kg}/{m}^{3}\) except for C and E models.

For the C and E models, the density of the material is taken to be equal to zero. The mass is installed by means of discrete elements installed on each of the nodes.

1.3. Boundary conditions and loads#

Point \(A\): supported \(u=v=w={\mathrm{\theta }}_{x}=0\)

Point \(B\): supported \(u=v=w={\mathrm{\theta }}_{x}=0\)