## Line theory: Analytic catenary |

The analytic catenary equations used in OrcaFlex account for simple properties of a line such as its weight, buoyancy and axial stiffness. More complicated effects, such as bend stiffness, are ignored. In what follows, we sketch a brief outline of the analytic catenary equations solved by OrcaFlex. These equations vary depending upon whether the line is suspended above the seabed or laid down on the seabed.

For a line with unstretched length $l$, hanging freely in a vertical plane between its bottom end $(\xb,\zb)$ and top end $(\xt,\zt)$, the following equations must hold \begin{align} \tt &= \tb + \frac{w l}{\Th} \\ \xt &= \xb + l \left\{ \frac{1}{\tt - \tb} \ln \left( \frac{\tt + \sqrt{1+\tt^2}}{\tb + \sqrt{1+\tb^2}} \right) + \frac{\Th}{\EA} \right\} \\ \zt &= \zb + l \left\{ \frac{\sqrt{1+\tt^2} - \sqrt{1+\tb^2}}{\tt - \tb} + \frac{\Th}{\EA} \frac{\tb + \tt}{2} \right\} \end{align} where

$\tb$ is the tangent $\OD{z}{x}$ to the line at the bottom end

$\tt$ is the tangent $\OD{z}{x}$ to the line at the top end

$w$ is the effective weight of the line per unit length, including any buoyancy contributions

$\Th$ is the horizontal component of tension in the suspended line, which is constant along its length

$\EA$ is the axial stiffness of the line

For a line laid down on the seabed, the following relations must hold \begin{align} \xt &= \xb + \sD \cos \theta \\ \zt &= \zb + \sD \sin \theta \end{align} Here, $\theta$ is the seabed slope angle ($\theta{=}0$ corresponds to a flat seabed) and $\sD$ is the stretched length laid down on the seabed, given by \begin{equation} \sD = l + \frac{(\Tt - \Tb) (2 \EA + \Tb + \Tt)}{2 \, \EA \, w (\sin \theta + \mu \cos \theta)} \end{equation} where

$\Tb$ is the bottom tension

$\Tt$ is the top tension

$\mu$ is the axial coefficient of seabed friction

$\Tb$ and $\Tt$ are further related by the expression \begin{equation} \Tt = \Tb + w (\sin \theta + \mu \cos \theta) l \end{equation} There is one further caveat: the tension in the laid down section is never permitted to drop below zero (i.e. the line cannot support compression). If this is predicted, then the tension in the relevant part of the line is set identically to zero, provided that there is sufficient friction to support the weight of the line on the sloping seabed.

The simple friction model of the analytic catenary representation differs from the more complicated one used in the finite element representation. In particular, the finite element friction model accounts for the past history of the line (i.e. its previous positions). Such effects cannot be captured in the analytic catenary equations because there is no notion of history. Instead, we assume that the friction force acts towards the anchor, so that the tension reduces in the vicinity of the anchor.

Clump attachments are treated as point masses, defined at given arc lengths on the line. How they are handled depends on whether or not they are in contact with the seabed; in either case, for a given clump we define

$\wc$ is the weight of the clump (including its buoyancy)

$\tminus$, $\tplus$ are the tangents $\OD{z}{x}$ of the line on the bottom and top side, respectively, at the arc length of the line where the clump is connected

Then for a clump that is not in contact with the seabed, the tangents $\tminus$ and $\tplus$ are related by \begin{equation} \tplus = \tminus + \frac{\wc}{\Th} \end{equation} and for a clump that is in contact with the seabed, we instead have \begin{equation} \frac{\ts \tplus + 1}{\sqrt{\tplus^2 + 1}} \Tplus - \frac{\ts\tminus + 1}{\sqrt{\tminus^2 + 1}} \Tminus = \ts \wc \end{equation} where

$\ts \equiv \tan \theta$, the seabed slope

$\Tminus$ is the tension at the clump on the bottom side

$\Tplus$ is the tension at the clump on the top side

In general, a line will be composed of multiple sections. Some of these sections may be net heavy, others may be net buoyant. Solutions are parameterised by two values: $\Th$, the horizontal component of the tension at the closest point to the bottom end (the anchor) that is not in contact with the seabed; and $\Delta z$, the vertical offset of the top end (the fairlead) above the datum level defined by its position in reset state. A given solution may include both laid down and suspended sections. The analytic catenary equations are solved iteratively – by varying tangent values and the lengths of laid down sections – for a given value of $\Th$, until a solution is found that places the fairlead at its correct vertical offset. Once this solution is found, the horizontal and vertical coordinates of the fairlead, $(x, z)$, are added to the solution grid that will be used to compute (via interpolation) the load applied at the fairlead during the simulation.

The analytic catenary solver cannot handle the following scenarios:

- Buoyant lines that float on the sea surface.
- Laydown of line on any part of the seabed that is out of the water.