Line types: Equivalent line data

An equivalent line is one which is used, in a model simplification, to represent a collection of two or more line types, and whose properties are derived by OrcaFlex (as far as possible) from the constituent line types. Consider, for example, a pipe-in-pipe system: this can be modelled by combining the properties of both external and internal line types into a single equivalent line type. Representative values for the mass, diameters, stiffnesses, etc. of the equivalent line are calculated by OrcaFlex.

The input data for an equivalent line type are:

• A carrier line type: the name of an existing line type defined in the model.
• One or more secondary lines, also defined by referencing existing line types. Secondary lines can be either internal or external.
• Other data. Not all equivalent line data can be derived by OrcaFlex. The values which cannot, e.g. drag, lift, added mass, fluid inertia, must be given on the equivalent line data form.

Carrier line

One of the constituent line types is deemed to be the carrier line and is treated differently from the secondary lines:

• Any internal secondary lines are deemed to be inside the bore of the carrier line.
• The carrier line, together with the internal secondary lines, determines the cross sectional area associated with the line contents data.
• Stress results are reported for the carrier line.

Secondary lines

Multiple secondary lines can be defined, and each one may be either internal or external to the carrier. Each also defines a contents density, associated with the bore of the secondary line. The axial, bending and torsional stiffnesses of each secondary line may, individually, contribute or not to the equivalent line's stiffness.

Other data

Drag/lift coefficients, drag/lift diameters, added mass/inertia coefficients, centre of mass and allowable tension cannot be derived by OrcaFlex for the equivalent line: all must be given explicitly.

Modelling details

Here, we describe how OrcaFlex derives the representative values for the equivalent line type. These values can be viewed (but not edited) in the 'all' view mode or the line type properties report. We use the following subscript notation:

$\E$ indicates properties of the equivalent line, e.g. $\OD_\E$

$\C$ indicates properties of the carrier line, e.g. $\OD_\C$

$\iint$ indicates properties of the $i$^th internal secondary line, e.g. $\OD_\iint$

$\iext$ indicates properties of the $i$^th external secondary line, e.g. $\OD_\iext$

Geometry

$\OD_\E$ is calculated to give a displacement equal to the displacement of the carrier line together with all the external lines \begin{equation} \OD_\E = \left( {\OD_\C}^2 + \sum {\OD_\iext}\!^2 \right)^{\small{1/2}} \end{equation} $\ID_\E$ is calculated to give an internal cross sectional area equal to that of the carrier line less the external cross sectional area of all the internal lines \begin{equation} \ID_\E = \left( {\ID_\C}^2 - \sum {\OD_\iint}\!^2 \right)^{\small{1/2}} \end{equation}

Mass

Mass per unit length, $m_\E$, is the sum of the mass per unit length of the carrier line and all secondary lines \begin{equation} m_\E = m_\C + \sum m_\iint + \sum m_\iext \end{equation}

Structure

Axial stiffness, $\EA_\E$, is the sum of the axial stiffness of the carrier line and all contributing secondary lines \begin{equation} \EA_\E = \EA_\C + \sum_\textrm{contr} \EA_\iint + \sum_\textrm{contr} \EA _\iext \end{equation} where '$\textrm{contr}$' indicates that the summation is performed only over those secondary lines which contribute to axial stiffness.

Analogous formulae apply for bending stiffness and torsional stiffness.

Stress

Stress results are reported for the carrier line, so the stress diameters and the allowable stress for the equivalent line are simply the corresponding values for the carrier line. The tensile stress loading factor $C_{1\E}$ is defined as \begin{equation} C_{1\E} = C_{1\C}\ \frac{\EA_\C}{\EA_\E} \end{equation} Likewise, the torsional stress loading factor $C_{4\E}$ is defined as \begin{equation} C_{4\E} = C_{4\C}\ \frac{G\!J_\C}{G\!J_\E} \end{equation} The bending stress loading factor $C_{2\E}$ is defined similarly but with the minor complication that it takes a single value despite there being separate stiffness values, $\EI_\mathrm{x}$ and $\EI_\mathrm{y}$, for $x$ and $y$. In practice this is not a limitation, since the stress results derivation is predicated on the material being isotropic, but for the sake of completeness we take \begin{equation} C_{2\E} = C_{2\C}\ \max\left\{ \frac{\EI_{\mathrm{x}\C}}{\EI_{\mathrm{x}\E}},\ \frac{\EI_{\mathrm{y}\C}}{\EI_{\mathrm{y}\E}} \right\} \end{equation} Finally, the shear stress loading factor, $C_{3\E}$, is simply set equal to $C_{2\E}$.

Other data

The remaining line type data fall into two categories:

• Values which must be set explicitly, as listed above
• Values which are automatically set to the value of the carrier line. These are bulk modulus, compression is limited, minimum bend radius, Poisson ratio, contact diameters, clashing stiffness/damping, friction coefficients and Rayleigh damping coefficients.