1. Reference problem#

1.1. Geometry#

We consider a hollow pipe with a cross-section varying homothetically, with an external radius varying from \(10\mathit{mm}\) to \(5\mathit{mm}\), with a thickness varying from \(5\mathit{mm}\) to \(2.5\mathit{mm}\) and with a length \(2m\).

_images/100002010000020200000072517CB7DC8A612B8C.png

1.2. Material properties#

The material is isotropic elastic whose properties are:

  • \(E=210\mathit{GPa}\)

  • \(\nu =0.3\)

  • \(\mathrm{{\rm P}}=7800\mathit{kg}/{m}^{3}\)

  • \(\alpha =1.2E-5{K}^{-1}\)

Ramberg Osgood’s law used for post-treatments is written as:

\({ϵ}_{p}=K{(\frac{\sigma }{E})}^{1/n}\)

With:

  • \(K=0.01\)

  • \(n=2\)

1.3. Boundary conditions and loads#

A set of elastic calculations are performed independently on the beam-like structure.

_images/100002010000019C000000785C63C96ABAD54C29.png

1.3.1. Net weight#

A \(g=9.81m/{s}^{2}\) gravity field is imposed in the \(-Z\) direction. Point \(A\) is embedded.

1.3.2. Imposed displacement#

Point \(A\) is embedded. The structure is torsified at point \(B\) via an imposed angle \(\theta =0.02\mathit{rad}\) along the axis \(X\).

1.3.3. Thermal expansion#

Point \(A\) is embedded. Point \(B\) is stuck along the \(X\) axis. A variation in the temperature field \(\mathrm{\Delta }T=100°K\) generates compression in the beam.

1.3.4. Pressure imposed#

The pressure inside the pipe \(P=1\mathit{MPa}\) is taken into account.

1.3.5. Spectral seismic#

Point \(A\) is embedded. Point \(B\) is stuck along the \(Z\) axis. Spectral acceleration is imposed along the 3 axes \(X\), \(Y\) and \(Z\) with the respective ratios \(1.\), \(1.\) and \(0.5\):

_images/1000020100000280000001E0D438E8C97734A253.png

The first 4 dynamic natural modes and the static correction are taken into account via pseudo-modes along the 3 axes. Reduced amortization is set to \(2\text{\%}\).