Home > Theory > Equations of Motion > Virtual Works Principle

Virtual work principle

At the local scale, the equilibrium is given by the following relation :

\[ Div (\sigma) + f = \rho \frac{d^2 x}{dt^2} \]

where \(\sigma\) is the stress tensor, \(f\) the volumic loading and \(\rho \frac{d^2 x}{dt^2}\) the inertia forces.

The virtual work principle is another way to express the equilibrium of the structure. According to this principle, the equilibrium is reached when any allowed virtual displacement generates no work in the structure. Each term from the local equation is then multiplied by a virtual displacement \(\delta x\), and is integrated over the volume. Finally the structure state is given by a function \(G\) and the equilibrium writes :

\[ G(\vec{x},\vec{\dot{x}},\vec{\ddot{x}},\delta \vec{x})=G_{internal}(\vec{x},\vec{\dot{x}},\delta \vec{x})+G_{external}(\vec{x},\vec{\dot{x}},\vec{\ddot{x}},\delta \vec{x})+G_{inertia}(\vec{x},\vec{\dot{x}},\vec{\ddot{x}},\delta \vec{x}) \]

where \(G_{internal}\) , \(G_{external}\) and \(G_{inertia}\) stand for the contribution of the internal efforts, the external loads and inertia to the virtual work.

The framework of DeepLines is the slender structures : flexible risers, rigid risers, umbilicals and mooring lines, as well as optical fibres, onshore pipelines...

For such structures, one dimension, called the axis, is much larger than the other ones, defining the section. To study the global response of these structures, the deformations of the section are assumed negligible. In the WVP, a first integration over the section is performed since the physical quantities only depend on arc length. The working scale is no longer the local one but the section scale.