## Line theory: Calculation stage 4 torsion moments |

The next stage, if torsion has been included, is to calculate the torque moment due to torsion. To do so, we must first determine the directions $\s{x_1}$, $\s{y_1}$, $\s{x_2}$ and $\s{y_2}$, since so far only the segment axial direction $\s{z}$ has been calculated.

The directions $\s{x_2}$ and $\s{y_2}$ at the end of the segment are determined from the orientation $\mat{N}_\mathrm{xyz}$ of the adjacent node, by rotating $\mat{N}_\mathrm{xyz}$ until its $z$-direction is aligned with $\s{z}$. With reference to the line model diagram, we can see that this rotation must be through angle $\alpha_2$ about the binormal direction (the direction orthogonal to both $\vec{n}_\mathrm{z}$ and $\s{z}$). (Note that rotations about the binormal direction are bending-only rotations: they involve no twisting.) The directions $\s{x_1}$ and $\s{y_1}$ at the other end of the segment are derived in the same way, but starting from the orientation of the node at that other end of the segment.

The twist angle $\tau$ in the segment can then be calculated – it is the angle between the directions $\s{x_1}$ and $\s{x_2}$ (and must also be the angle between $\s{y_1}$ and $\s{y_2}$). In other words the orientations $\mat{S}_\mathrm{x_1y_1z}$ and $\mat{S}_\mathrm{x_2y_2z}$, at the two ends of the segment, differ by just a twist through angle $\tau$.

In the case of linear torsional stiffness the torque generated by the torsion spring-damper is a moment vector $\vec{m}$ about the segment axial direction $\s{z}$ with magnitude \begin{equation} \lvert \vec{m} \rvert = k\ \frac{\tau}{l_0} + k_\mathrm{tt}\ \epsilon + c\ \ODt{\tau} \end{equation} where

$k=$ torsional stiffness, as given on the line types form

$\tau=$ segment twist angle (in radians), between directions $\s{x_1}$ and $\s{x_2}$

$l_0=$ unstretched length of segment

$\ODt{\tau}=$ rate of twist (in radians per second)

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

$\epsilon=$ total mean axial strain

$c=$ torsional damping coefficient of the line (this is defined below)

This torque moment is then applied (with opposite signs) to the nodes at each end of the segment.

If the torsional stiffness is nonlinear, then torque is calculated as a moment vector $\vec{m}$ about the segment axial direction $\s{z}$ with magnitude \begin{equation} \lvert \vec{m} \rvert = T(\tau/l_0) + k_\mathrm{tt}\ \epsilon + c\ \ODt{\tau} \end{equation} where

$T$ is the function relating twist per unit length to torque as specified by the variable data source defining torsional stiffness.

The variable data source providing $T$ can support either nonlinear elastic or hysteretic behaviour. For hysteresis, the model used is directly analogous to the one used for hysteretic bending. If you need to implement a different relationship between torque and twist, $T$ may be defined using an external function.

The damping coefficient $c$ is calculated using the formula \begin{equation} c = \frac{\lambda_\mathrm{t}}{100}\ c_\textrm{crit} \end{equation} where

$\lambda_\mathrm{t}=$ target torsion damping, as given on the general data form

$c_\textrm{crit} = \sqrt{\frac{2 I_\mathrm{z} k}{l_0}}$ is the critical damping value for a segment

$I_\mathrm{z}=$ rotational moment of inertia of the segment about its axis

Here, $I_\mathrm{z}$ allows only for the structural mass of the line, not the mass of any contents: we assume that the contents of a pipe do *not* twist with the pipe itself.

Notes: | The numerical damping term is only included when using the explicit integration scheme. The implicit integration scheme includes in-built numerical damping. |

If the torsional stiffness is nonlinear or externally calculated then we use the nominal torsional stiffness $k_\mathrm{nom}$ in the formula for $c$. External functions are given a specific opportunity to provide a nominal stiffness value. For torsional stiffness defined using a variable data source, we define $k_\mathrm{nom}$ to be the stiffness at zero twist per unit length. |