Quasi static catenary line

The catenary line model implemented in FRyDoM is based on the elastic catenary theory [GRECO], under the assumptions of uniformly distributed load. It is a quasi static approach, in which we look for the equilibrium configuration. Let us consider a line, of unstretched length \(L\), with line ends located respectively at \(\mathbf{p}(0)\) and \(\mathbf{p}(L)\).

Equilibrium configuration

Fig. 13 Unstrained configuration (in solid black) and strained configuration (in dashed blue), under uniform distributed load.

The equilibrium equation for a segment \([0,s]\), with \(s \leq L\), is given by:

\[\mathbf{t}(s) = \mathbf{t}_0 - s \mathbf{q}\]

where

  • \(\mathbf{t}(s)\) is the line tension applied by the segment \([0,s]\) to the rest of the line,
  • \(\mathbf{t}_0\) is the tension force at the origin.
  • \(\mathbf{q} = q \mathbf{u}\) is the uniformly distributed load

The position of the line at the abscisse \(s\) is given by:

\[\mathbf{p}(s) = \mathbf{p}(0) + \mathbf{p}_c(s) + \mathbf{p}_{\epsilon}(s)\]

where

\[\begin{split}\mathbf{p}_{\epsilon}(s) &=& \frac{q.s}{EA} \left(\frac{\mathbf{t}_0}{q} - \frac{s}{2} \mathbf{u} \right)\\ \mathbf{p}_c(s) &=& \left(\mathbb{I} - \mathbf{u} \mathbf{u}^T \right) \frac{\mathbf{t}_0}{q} \ln\left[\frac{\rho(s)}{\rho(0)} \right] - \mathbf{u} \left( \left\| \frac{\mathbf{t}_0}{q} - s \mathbf{u} \right\| - \left\| \frac{\mathbf{t}_0}{q} \right\| \right)\\ \rho(s) &=& \left\| \mathbf{t}(s) \right\| - \mathbf{u}^T \mathbf{t}(s)\end{split}\]

REFERENCES

[GRECO]Greco, L., Impollonia, N., Cuomo, M., A procedure for the static analysis of cables structures following elastic catenary theory, International Journal of Solids and Structures, pp 1521-1533, 2014