1. Reference problem

1.1. Geometry

\(L=\mathrm{50m}\); \({R}_{f}=\mathrm{1m}\); \({R}_{i}=\frac{5}{3}m\); \({R}_{e}=\mathrm{3m}\); \({k}_{1}={10}^{9}{\mathrm{N.m}}^{-1}\);; \({k}_{2}=0.5{10}^{7}{\mathrm{N.m}}^{-1}\); \({\rho }_{\mathrm{fluide}}=1000{\mathrm{kg.m}}^{-3}\); \({\rho }_{s}=\mathrm{7800 }{\mathrm{kg.m}}^{-3}\); shell thickness: \(50\mathrm{cm}\).

1.2. Material properties

Fluid: density \({\rho }_{1}=1000{\mathrm{kg.m}}^{-3}\); \({\rho }_{2}=750{\mathrm{kg.m}}^{-3}\).

Structure: \({\rho }_{s}=7800\mathrm{kg}/{m}^{3}\); \(E=2.1{10}^{11}\mathrm{Pa}\);; \(\nu =0.3\) (steel).

1.3. Boundary conditions and loads

The outer cylinder is on the one hand connected to a fixed frame via the four springs of unitary stiffness \({k}_{1}\), on the other hand connected to the middle cylinder by four springs of unitary stiffness \({k}_{2}\). Both structures are rigid in this reference solution.