Line theory: Calculation stage 2 bend moments

The bend moments are then calculated. Each node has bending spring-dampers at either side, spanning the angle $\alpha$ between the node's axial direction $\nz$ and the corresponding segment's axial direction $\sz$. Each of these spring-dampers applies to the node a bend moment that depends on $\alpha$. The axial directions are associated with the frames of reference of the node and the segments, as shown in the detailed line model diagram.

The node's frame of reference $\mat{N}_\mathrm{xyz}$ is a Cartesian set of axes that is fixed to (and so rotates with) the node. $\nz$ is in the axial direction; $\vec{n}_\mathrm{x}$ and $\vec{n}_\mathrm{y}$ are normal to the line axis and correspond to the end $x$- and $y$-directions that are determined by the gamma angle on the line data form.

Each segment has two frames of reference: $\mat{S}_\mathrm{x_1y_1z}$ at the end nearest end A, and $\mat{S}_\mathrm{x_2y_2z}$ at the other end. These two frames have the same $\sz$ direction, which was calculated in stage 1, so the bend angle $\alpha_2$ between $\nz$ and $\sz$ can now be calculated. The effective curvature vector $\vec{c}$ is then calculated: its direction is the binormal direction, i.e. the direction that is orthogonal to both $\sz$ and $\nz$, and its magnitude is $\alpha_2/(0.5\ l_0)$, where $l_0$ is the unstretched length of the segment.

Linear, isotropic bending stiffness

Isotropic bend stiffness means that the bending stiffness is the same in the $x$- and $y$-directions. For linear, isotropic bending stiffness then, the bend moment $\vec{m}_2$, generated by the bending spring-damper, is in the binormal direction, $\vec{b}_2$, where \begin{equation} \vec{b}_2 = \text{unit vector in direction } (\sz \times \nz) \end{equation} and has magnitude \begin{equation} \lvert \vec{m}_2 \rvert = EI\ \modc + d\ \ODt{\modc} \end{equation} where

$EI=$ bending stiffness, as specified on the line types form

$d = \frac{\lambda_\mathrm{b}}{100}\ d_\textrm{crit}$

$\lambda_\mathrm{b}=$ target bending damping, as specified on the general data form.

$d_\textrm{crit}=$ the bending critical damping value for a segment $= l_0 \sqrt{m\ EI\ l_0}$

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

The bend angle $\alpha_1$ and bend moment vector $\vec{m}_1$ relating to the segment on the other side of the node are calculated analogously. The node thus experiences two bend moments, one from each of the segments on either side of it.

Linear, non-isotropic bending stiffness

If the bend stiffnesses for the $x$- and $y$-directions differ, then the components of the bend moment in the $S_\mathrm{x_2}$ and $S_\mathrm{y_2}$ directions are given separately as \begin{equation} \begin{aligned} \text{$m_2$ component in the $S_\mathrm{x_2}$ direction} &= EI_x\ c_x + d_x \ODt{c_x} \\ \text{$m_2$ component in the $S_\mathrm{y_2}$ direction} &= EI_y\ c_y + d_y \ODt{c_y} \end{aligned} \end{equation} where

$EI_x, EI_y=$ bending stiffnesses of segment, as specified on the line types form

$c_x, c_y=$ components of the curvature vector $\vec{c}$ in the $S_\mathrm{x_2}$ and $S_\mathrm{y_2}$ directions

$d_x = \frac{\lambda_\mathrm{b}}{100} l_0 \sqrt{m\ EI_x\ l_0}$

$d_y = \frac{\lambda_\mathrm{b}}{100} l_0 \sqrt{m\ EI_y\ l_0}$

The curvature $\vec{c}$ used here is the value in the true plane of bending, taking full account of the 3D motions of the adjacent nodes.

Again, the bend angle $\alpha_1$ and bend moment vector $\vec{m}_1$ on the other side of the node are calculated similarly.

Nonlinear, isotropic bending stiffness

In the case of nonlinear isotropic bending stiffness, the magnitude of the bend moment is given by \begin{equation} \lvert \vec{m}_2 \rvert = B\!M(\modc) + d'\ \ODt{\modc} \end{equation} where

$B\!M(\modc)$ is the function relating curvature to bend moment.

$d' = \frac{\lambda_\mathrm{b}}{100} d'_\textrm{crit}$

$d'_\textrm{crit}=$ the bending critical damping value for a segment $= L_0 \sqrt{m\ EI_\textrm{nom}\ l_0}$

$EI_\textrm{nom}$ is the nominal bending stiffness, defined to be the bending stiffness at zero curvature.

The function $B\!M$ may be given by

• A tabular variable data source which defines bend moment as a function of curvature. This variable data source may be hysteretic or elastic (see below).
• A nonlinear stress-strain relationship for a homogeneous pipe. This approach results in a nonlinear elastic bend stiffness model.

Again, the bend angle $\alpha_1$ and bend moment vector $\vec{m}_1$ on the other side of the node are calculated similarly.

Nonlinear, non-isotropic bending stiffness

OrcaFlex does not permit the combination of both nonlinear and non-isotropic bending stiffness. If you wish to use nonlinear bending stiffness, then you may only do so isotropically; conversely, if you wish to apply non-isotropic bending stiffness, then it must be of linear form.

Elastic or hysteretic bending model

For nonlinear bend stiffness you must choose whether the curvature-moment data are interpreted hysteretically or not.

Non-hysteretic

Non-hysteretic means that the data are applied using a simple elastic model. In this case the bend moment function $B\!M$ used in the above equation is simply the specified function of the current curvature magnitude $\modc$ without any allowance made for the curvature history. So, if the curvature increases and then decreases, the bend moment follows the same nonlinear moment-curvature path. This is illustrated below, showing the bend moment $m$ that results when the non-hysteretic model is used and a simple sinusoidally varying curvature $c$ is applied.

Hysteretic

The hysteretic model includes hysteresis effects, i.e. it depends on the history of the curvature which has been applied. The function $B\!M$ in this case is interpreted as the bend moment that results when the line is bent with slowly-increasing curvature, and to explain hysteresis we must consider this function as being made up of a series of increments. If, having increased, the curvature subsequently decreases, the bend moment does not come back down the same path: instead, it comes down the path which is obtained by first undoing and reversing the first increment of curvature that was applied, then the second increment, and so on. In other words the hysteretic model treats curvature as being made up of a series of curvature increments and corresponding moment increments, and it undoes them on a first in first out basis, as opposed to the symmetric last in first out basis that non-hysteretic bend stiffness uses.

The effect of this hysteretic model is that the bend moment follows a classic hysteresis path, as shown below where the arrows show the direction of change of the curvature and moment. The left-hand diagram shows the bend moment that results from a sinusoidal curvature history; the right-hand diagram shows what happens if a small curvature cycle is followed by another of greater amplitude.

The hysteresis model is described in detail by Tan, Quiggin and Sheldrake (2007).