Line theory: Calculation stage 1 tension forces

$\newcommand{\Te}{T_\mathrm{e}}$ $\newcommand{\Tw}{T_\mathrm{w}}$ $\newcommand{\po}{p_\mathrm{o}}$ $\newcommand{\peei}{p_\mathrm{i}}$ $\newcommand{\ao}{a_\mathrm{o}}$ $\newcommand{\ai}{a_\mathrm{i}}$

Firstly the tensions in the segments are calculated. To do this, OrcaFlex calculates the distance (and its rate of change) between the nodes at the ends of the segment, and the segment axial direction $\vec{s}_\mathrm{z}$, the unit vector in the direction joining the two nodes.

The tension in the axial spring-damper at the centre of each segment is the vector, in direction $\vec{s}_\mathrm{z}$, whose magnitude is given by the effective tension $\Te$ \begin{equation} \Te = \Tw + (\po\ao - \peei\ai) \end{equation} where

$\Tw=$ wall tension, as defined below

$\peei, \po=$ internal pressure and external pressure, respectively

$\ai, \ao=$ internal and external cross sectional stress areas, respectively.

Linear axial stiffness

In the case of linear axial stiffness, the wall tension $\Tw$ is made up of the following contributions \begin{align} \Tw &= \EA\ \epsilon & &\text{(axial stiffness)} \\ &- 2\ \nu\ (\po\ao - \peei\ai) & &\text{(external and internal pressure via the Poisson ratio effect)} \\ &+ k_\mathrm{tt}\ \frac{\tau}{l_0} & &\text{(torque coupling)} \\ &+ \EA\ c\ \ODt{l}\ \frac{1}{l_0} & &\text{(axial damping)} \label{C_linear} \end{align} Here,

$\EA=$ axial stiffness of the line, as given on the line types form (= effective Young's modulus $\times$ cross section area)

$\epsilon = \text{total mean axial strain} = (l-\lambda l_0)/(\lambda l_0)$

$l=$ instantaneous length of segment

$\lambda=$ expansion factor of segment

$l_0=$ unstretched length of segment

$\nu=$ Poisson ratio

$k_\mathrm{tt}=$ tension / torque coupling

$\tau=$ segment twist angle (in radians)

$c=$ damping coefficient, in seconds, as defined below

$\ODt{l}=$ rate of increase of length.

Notes: The effective tension, $\Te$, can be negative, indicating effective compression.
The tension / torque coupling term is only included when the line includes torsion.

Nonlinear axial stiffness

When the axial stiffness is nonlinear then those components of the wall tension involving axial stiffness differ accordingly \begin{align} \Tw &= \Tw^\textrm{air}(\epsilon) & &\text{(axial stiffness)} \\ &- 2\ \nu\ (\po\ao - \peei\ai) & &\text{(external and internal pressure via the Poisson ratio effect)} \\ &+ k_\mathrm{tt}\ \frac{\tau}{l_0} & &\text{(torque coupling)} \\ &+ \EA_\textrm{nom}\ c\ \ODt{l}\ \frac{1}{l_0} & &\text{(axial damping)} \label{C_nonlinear} \end{align} where we introduce the terms

$\Tw^\textrm{air}(\epsilon)$, the function relating the total mean axial strain to wall tension in air, without twist, i.e. the contribution from axial stiffness alone, as specified by the variable data source defining axial stiffness

$\EA_\textrm{nom}$ is the nominal axial stiffness

The variable data source providing $\Tw^\textrm{air}(\epsilon)$ can optionally be interpreted hysteretically. In the non-hysteretic case, the data are applied using a simple elastic model and $\Tw^\textrm{air}(\epsilon)$ relates the tension to instantaneous strain, without any allowance made for the strain history. In the hysteretic case, the model includes hysteresis effects, i.e. it depends on the history of the strain which has been applied. The model used is directly analogous to the one used for hysteretic bending. The independent variable in the hysteretic bending model is curvature, whereas for axial stiffness the independent variable is strain; similarly, the dependent variable in the bending model is bend moment, whereas for axial stiffness it is tension.

For nonlinear elastic or nonlinear hysteretic axial stiffness, $\EA_\textrm{nom}$ is defined to be the axial stiffness at zero strain.

Alternatively, $\Tw^\textrm{air}(\epsilon)$ can be defined using an external function. In this case, the external function is given the explicit opportunity to provide a nominal stiffness.

Tension force

For any case within the linear or nonlinear axial stiffness behaviour, the calculated effective tension force vector is then applied (with opposite signs) to the nodes at each end of the segment. Each mid-node therefore receives two tension forces, one from each of the segments on either side of it.

Damping coefficient $c$

The damping coefficient $c$ in the contributions (\ref{C_linear}) and (\ref{C_nonlinear}) above represents the numerical damping in the line. It is calculated from the formula \begin{equation} c = \frac{\lambda_\mathrm{a}}{100}\ c_\textrm{crit} \end{equation} where

$\lambda_\mathrm{a}=$ target tension damping, as specified on the general data form

$c_\textrm{crit} = \sqrt{\frac{2ml_0}{\EA}}$ is the critical damping value for a segment

$m=$ segment mass, including contents but not the mass of any attachments.

Notes: This numerical damping term is only included when using the explicit integration scheme. The implicit integration scheme includes in-built numerical damping.
If the axial stiffness is nonlinear then we use the nominal axial stiffness $\EA_\textrm{nom}$ in the formula for $c_\textrm{crit}$.