# 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 $$\mathbf{p}(L)$$.

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