Line theory: Pipe stress calculation

OrcaFlex provides stress results that apply only to simple pipes. More precisely, the stress calculation assumes that the loads on the line are taken by a simple cylinder whose inside and outside diameters are given by the stress diameters specified on the line-types form. It also assumes that the cylinder is made of a uniform material. The pipe stress results are therefore only valid for things like steel or titanium pipes – they do not apply to composite-structure flexible pipes.

Consider a cross section through a mid-segment point, as shown in the following diagram. The diagram shows the frame of reference used for the cross section, which has origin $\vec{O}$ at the pipe centreline, $z$-axis along the pipe axis (positive towards end B) and $x$- and $y$-axes normal to the pipe axis (and so in the plane of the cross section).

OrcaFlex calculates, at the cross section, the following values:

• internal $(p_\mathrm{i})$ and external $(p_\mathrm{o})$ pressures
• effective tension and resulting wall tension, vectors in the axial direction with magnitudes $T_\mathrm{e}$ and $T_\mathrm{w}$ respectively
• curvature, a vector $\vec{c}$ in the cross section plane with components $(c_\mathrm{x},c_\mathrm{y},0)$
• bend moment, a vector $\vec{m}$ in the cross section plane with components $(m_\mathrm{x},m_\mathrm{y},0)$
• shear force $\vec{s}$, also a vector in the cross section plane, with components $(s_\mathrm{x},s_\mathrm{y},0)$
• torque, a vector in the axial direction with magnitude $\tau$.

In addition we define the following terminology:

$\ODs$ and $\IDs$ are the stress diameters given on the line types form

$\as=$ cross sectional stress area $= \frac{\pi}{4} \left(\ODs^2 - \IDs^2\right)$

$I_\mathrm{xy}=$ second moment of stress area about $x$ or $y=$ $\frac{\pi}{64}\left(\ODs^4 - \IDs^4\right)$

$I_\mathrm{z}=$ second moment of stress area about $z=$ $2I_\mathrm{xy}$

$C_1, C_2, C_3,C_4=$ stress loading factors ($C_1=$ tensile, $C_2=$ bending, $C_3=$ shear, $C_4=$ torsional), as specified on the line types form.

The stress generated by the above loads varies across the cross section. Consider a point $\vec{p}$ within the annulus, shown as a black dot in the diagram, which can be identified by its polar coordinates $(r,\theta)$. At $\vec{p}$, we define a local set of axes $(R,C,Z)$ where $R$ is radially outwards, $C$ is in the circumferential direction (positive in the direction of $\theta$ increasing) and $Z$ is in the axial direction $z$.

With respect to these axes, the stress at $\vec{p}$ is a symmetric $3{\times}3$ matrix of stress components. OrcaFlex calculates this matrix and then derives stress results from it.