3. Mechanical hypotheses

Cables are considered to be perfectly flexible wires, which cannot withstand any moment, neither bending nor twisting, and are only the seat of normal tension. This tension plays the role of a generalized constraint.

We want to calculate the displacement field \(u({s}_{o},t)\) at time \(t\) with respect to the reference situation. This is a static configuration of the cable subjected, for example, to gravity and to temperature \({T}_{o}\); it is defined by the position vector field \(x({s}_{o})\).

Figure 1: Cable section in reference and displaced situations

Like [bib1], we take as deformation the Green measure of the relative elongation compared to the reference situation [Figure]:

. \(g=\frac{{\text{ds}}^{2}-{\text{ds}}_{o}^{2}}{2{\text{ds}}_{o}^{2}}\)

\(g\) should stay small. The second member is suitable for calculation because it only includes squares of elementary lengths. We can see in the [Figure] that:

\(g=\frac{\partial x}{\partial {s}_{o}}\text{.}\frac{\partial u}{\partial {s}_{o}}+\frac{1}{2}{(\frac{\partial u}{\partial {s}_{o}})}^{2}\) [1]

The behavioral relationship is:

\(N={E}_{a}A\left[g-\alpha (T-{T}_{o})\right]\) [2]

with:

\({E}_{a}=\{\begin{array}{c}E\text{si}N>0\\ {E}_{c}\text{si}N\le 0\end{array}\)