Line theory: Pipe stress matrix

$\newcommand{\us}{_\textrm{stress}}$ $\newcommand{\as}{a\us}$ $\newcommand{\IDs}{I\!D\us}$ $\newcommand{\ODs}{O\!D\us}$

The pipe stress matrix at point $\vec{p}$ can be written as \begin{bmatrix} \sigma_\textrm{RR} & \sigma_\textrm{RC} & \sigma_\textrm{RZ} \\ \sigma_\textrm{RC} & \sigma_\textrm{CC} & \sigma_\textrm{CZ} \\ \sigma_\textrm{RZ} & \sigma_\textrm{CZ} & \sigma_\textrm{ZZ} %use \textrm to get smaller subscripts than with \mathrm \end{bmatrix} These stress components are calculated as follows. For terminology see pipe stress calculation.

Note: If homogeneous pipe additional bending stiffness is in use, then while the stress loading factors used below must each have their default value of 1, the input loads to these calculations of stress will have been automatically scaled to represent the load share between the original bending stiffness and the additional stiffness.

Diagonal terms

The three diagonal entries of the stress matrix, $\sigma_\textrm{RR}$, $\sigma_\textrm{CC}$ and $\sigma_\textrm{ZZ}$, are the radial, circumferential (or hoop) and axial (or longitudinal) stresses, respectively.

Radial and hoop stresses

$\sigma_\textrm{RR}$ and $\sigma_\textrm{CC}$ are due to the internal and external pressure. They are calculated using Lamé's equation for a thick-walled cylinder whose internal and external diameters are $\IDs$ and $\ODs$, as specified on the line types form. This gives \begin{align} \sigma_\textrm{RR} &= \text{ radial stress } = a - b/r^2 \\ \sigma_\textrm{CC} &= \text{ hoop stress } = a + b/r^2 \end{align} where a and b satisfy \begin{align} a - \frac{b}{(0.5\ \IDs)^2} &= -p_\mathrm{i} \\ a - \frac{b}{(0.5\ \ODs)^2} &= -p_\mathrm{o} \end{align}

Notes: $\IDs$ and $\ODs$ are by default equal to the geometric ID and OD on the line type form. They can, however, be set to different values, in which case the above calculation assumes that the material between ID and $\IDs$, and between $\ODs$ and OD, is transparent to pressure. That is, the internal pressure applies right through to the $\IDs$ and the external pressure applies right through to $\ODs$.
If $\IDs$ is zero, then OrcaFlex assumes that external pressure applies throughout the structure – i.e. that $\sigma_\textrm{RR} = \sigma_\textrm{CC} = -p_\mathrm{o}$.

Axial stress (linear stress-strain)

The axial stress $\sigma_\textrm{ZZ}$ is given by \begin{equation} \sigma_\textrm{ZZ} = \text{direct tensile stress} + \text{bending stress} \end{equation} The direct tensile stress is the contribution due to wall tension; the bending stress is the contribution due to bend moment. The wall tension is assumed to be uniformly distributed across the stress area, so its contribution is $T_\mathrm{w}/\as$, while the contribution due to bend moment varies across the cross section and is given by $C_2 r (m_\mathrm{x}\sin\theta - m_\mathrm{y}\cos\theta) / I_\mathrm{xy}$ where $C_2$ is the bending stress loading factor.

The axial stress in the linear case is therefore \begin{equation} \sigma_\textrm{ZZ} = \frac{T_\mathrm{w}}{\as} + \frac{C_2 r}{I_\mathrm{xy}} (m_\mathrm{x}\sin\theta - m_\mathrm{y}\cos\theta) \end{equation}

Axial stress (nonlinear stress-strain)

For a homogeneous pipe with nonlinear stress-strain, the nonlinearity means that $\sigma_\textrm{ZZ}$ cannot be split into tensile and bending components. Instead, it is calculated directly from the supplied stress-strain relationship\begin{equation} \sigma_\textrm{ZZ} = \sigma(\epsilon_\textrm{ZZ}) \end{equation} where $\epsilon_\textrm{ZZ}$ is the axial strain given by \begin{equation} \epsilon_\textrm{ZZ} = \epsilon_t + r(c_\mathrm{x}\sin\theta + c_\mathrm{y}\cos\theta) \end{equation} and $\epsilon_t$ is the direct tensile strain.

Off-diagonal terms

The six off-diagonal terms are the shear stresses, but there are in fact only three independent terms since the matrix is symmetric. They are given by \begin{align} \sigma_\textrm{RC} &= 0 \\ \sigma_\textrm{RZ} & = \frac{C_3}{\as} (s_\mathrm{x}\cos\theta + s_\mathrm{y}\sin\theta) \label{srz} \\ \sigma_\textrm{CZ} & = C_4\frac{\tau r}{I_\mathrm{z}} + \frac{C_3}{\as} (s_\mathrm{y}\cos\theta - s_\mathrm{x}\sin\theta) \label{scz} \end{align} where $C_3$ and $C_4$ are the shear and torsional stress loading factors.

The contribution $C_4\frac{\tau r}{I_\mathrm{z}}$ in equation (\ref{scz}) is the shear stress contribution due to torque. If torsion is not included in the model, then it is zero. The remaining $C_3$ contribution in (\ref{scz}), and that in (\ref{srz}), is due to the shear force, which is assumed to be uniformly distributed across the stress area.

Finally, for two useful references on this subject, see Sparks (1980) and Sparks (1984).