4. Application of the Virtual Work Principle

If damping is not taken into account, the virtual work of all the forces applied to a cable section during virtual displacement \(\delta u\) is:

\(W(u,\delta u)={W}_{\text{int}}(u,\delta u)-{W}_{\text{iner}}(\ddot{u},\delta u)-{W}_{\text{ext}}(u,\delta u)\) [3]

distinguishing the work of internal forces, inertial forces and external forces. According to [eq]:

\({W}_{\text{int}}(u,\delta u)={\int }_{{s}_{1}}^{{s}_{2}}(N\text{.}\delta g)\mathrm{ds}={\int }_{{s}_{1}}^{{s}_{2}}\left\{N\text{.}\left[\frac{\partial (x+u)}{\partial {s}_{o}}\text{.}\frac{\partial \delta u}{\partial {s}_{o}}\right]\right\}\mathrm{ds}={\int }_{{s}_{1}}^{{s}_{2}}(N\text{.}B\delta u)\text{ds}\) [4]

where:

\(B={\left\{\left[\frac{\partial }{\partial {s}_{o}}1\right](x+u)\right\}}^{T}\left[\frac{\partial }{\partial {s}_{o}}1\right]\) [5]

by designating the transpose of a matrix with the upper index \(T\).

\({W}_{\text{iner}}(\ddot{u},\delta u)=-{\int }_{{s}_{1}}^{{s}_{2}}(\rho A\ddot{u}\text{.}\delta u)\mathrm{ds}\) [6]

In any case, we consider work \({W}_{\text{ext}}\) to be independent of \(u\) in a time step, because:

• or it really is, in the case of conservative forces such as gravity;

• or, in the case of Laplace forces, the force applied to a cable element depends not only on the movement of this element (conventional follower force), but also on the movements of all the cables. It is then considered that, during a time step, the force is constant and equal to its value at the end of the previous time step.