## Line results: Stress / strain |

Stress and strain results are available at mid-segment points and at line ends. The stress calculations make the following assumptions (for terminology see the pipe stress calculation topic):

- At each point along the line all the loads are taken by a single simple cylinder of the specified stress OD and stress ID and made of a homogeneous material.
- The stresses included are those due to tension, bending, shear and hoop stress.
- The loads (tension, bend moment, shear and torque) which are used in stress calculations are scaled before being used. Scaling is done either by the stress loading factors, or will be done automatically if homogeneous pipe additional bending stiffness is in use.
- Internal pressure in the line generates wall tension in the line as it would do in a sealed cylinder.
- For the purpose of calculating the components of the stress matrix, the shear stress is assumed to be uniformly distributed across the cross section. Although this is not strictly the case, the shear stress is normally negligible so this simplifying assumption is reasonable.
- The hoop stress due to static internal and external pressure at the current $Z$-level is included, and is calculated using the standard Lamé equation for thick walled cylinders. However the effect of dynamic variations in pressure, for example from the passage of the wave, are not included.

These assumptions mean that the stress calculations are only valid for pipes such as steel or titanium risers, not for composite flexible risers, ropes chains, etc.

If the pipe has nonlinear stiffness then OrcaFlex cannot, in general, accurately calculate pipe stresses. We use the same formulae for stress calculation as for linear stiffness. For example, the bending stress is calculated as $|\vec{m}|r/I_\mathrm{xy}$. An exception to this is made for a homogeneous pipe with nonlinear stress-strain, where stress results are calculated using the stress-strain data. This approach is valid provided that the direct tensile stress remains within the linear elastic region of the stress-strain curve.

If the line type stress diameters differ from the outer and inner diameters, then see the note on the line pressure effects page.

OrcaFlex does not, and indeed cannot, allow for the complex stress concentrations that can occur at joints or at the top and bottom of a riser.

Available at mid-segment points. The ratio of line segment stretched length to unstretched, unpressured length. The stretched length includes contributions from all possible sources of strain. This result corresponds to the total mean axial strain presented in the theory describing OrcaFlex line tension calculations.

Available at mid-segment points and line ends. The component of axial strain due to wall tension, $T_\mathrm{w}$ (which includes the effects of internal pressure, external pressure, and tension torque coupling). In the case of linear axial stiffness, $\EA$, it is calculated as $T_\mathrm{w}/\EA$. When the axial stiffness is nonlinear, the variable data source is linearly interpolated and the strain corresponding the value of wall tension is reported.

Note: | The direct tensile strain result is not available at arc lengths where the line type uses hysteretic or externally calculated axial stiffness. Derived results Worst ZZ strain, ZZ strain and Max pipelay von Mises strain are also unavailable. |

Available at mid-segment points and line ends. The axial strain due to bending at the outer fibre on the outside of the bend.

Available at mid-segment points and line ends. This is whichever of direct tensile strain $\pm$ max bending strain has the larger absolute value.

Available at mid-segment points and line ends. Calculated as direct tensile strain + bending strain. ZZ strain varies across the cross section, and so its value is reported at a specified cross section position, $(R,\theta)$.

Available at mid-segment points and line ends. A simplified equivalent strain measure commonly used in S-lay analysis, given by \begin{equation} \epsilon_\mathrm{vm} = \sqrt{ \epsilon_\mathrm{zz}^2 + \epsilon_\mathrm{cc}^2 - \epsilon_\mathrm{zz}\epsilon_\mathrm{cc} } \end{equation} where $\epsilon_\mathrm{zz}$ is the axial strain, due to direct tensile strain and bending strain, and $\epsilon_\mathrm{cc}$ is the hoop strain, calculated as \begin{equation} \epsilon_\mathrm{cc} = \sigma_\mathrm{cc} / E_\textrm{eff} \end{equation} Here, $\sigma_\mathrm{cc}$ is the hoop stress and $E_\textrm{eff}$ is an effective Young's modulus. For a homogeneous pipe $E_\textrm{eff}$ is equal to the user-specified $E$ when $E$ is constant. When nonlinear Young's modulus is used, $E_\textrm{eff}$ is equal to the stress-strain curve gradient at zero strain.

For line types without an explicitly defined Young's modulus, $E_\textrm{eff}$ is calculated as \begin{equation} E_\textrm{eff} = C_2 EI_\textrm{nom} / I_\mathrm{xy} \end{equation} where $C_2$ is the bending stress loading factor and $EI_\textrm{nom}$ the nominal bending stiffness, defined to be the bending stiffness at zero curvature.

The pipelay von Mises strain is evaluated at four points in the plane of bending: the inner and outer pipe fibres on each side of the neutral axis of bending. The maximum of these values is reported.

Note: | The pipelay von Mises strain result is formulated in a way commonly used in the pipelay industry, and relies upon a number of simplifying assumptions which make it inconsistent with more formal definitions of equivalent strain. The pipe axes are assumed to be the principal axes of the strain tensor, therefore shear strains are not included. The pipe radial strain is assumed to be small and is neglected. A Poisson ratio of zero is assumed in the derivation of this result; to be completely consistent, a Poisson ratio of zero may be given for the line type. |

Available at mid-segment points and line ends. The internal and external static pressures, $p_\mathrm{i}$ and $p_\mathrm{o}$, as defined in line pressure effects.

Pressures in OrcaFlex are gauge pressures, not absolute pressures. That is, they are relative to atmospheric pressure and so can be as low as minus 1 atmosphere (-101.325 kPa).

Available at mid-segment points and line ends. Defined as $p_\mathrm{i} - p_\mathrm{o}$.

Available at mid-segment points and line ends. The axial stress due to wall tension (which includes the effects of internal and external pressure). It is constant across the cross section and calculated as $T_\mathrm{w}/a_\textrm{stress}$. A positive value indicates tension; a negative value indicates compression.

Available at mid-segment points and line ends. The maximum value that the bending stress takes anywhere in the cross section, given by \begin{equation} \textrm{max bending stress} = \frac{C_2 |\vec{m}| O\!D_\textrm{stress}}{2 I_\mathrm{xy}} \end{equation} where $C_2$ is the bending stress loading factor. This maximum occurs at the extreme fibre on the outside of the bend.

For a homogeneous pipe with nonlinear stress-strain\begin{align} \textrm{max bending stress} &= \sigma(\epsilon_\mathrm{zz}) - \textrm{direct tensile stress} \\ &= \sigma(\epsilon_\mathrm{zz}) - \frac{T_\mathrm{w}}{a_\textrm{stress}} \end{align} where $\sigma(\epsilon)$ is the given stress-strain relationship and $\epsilon_\mathrm{zz}$ is the axial strain at the extreme fibre on the outside of the bend.

Available at mid-segment points and line ends. The value of zz stress that has the greatest magnitude anywhere in the cross section.

Available at mid-segment points and line ends. The hoop stress $\sigma_\mathrm{cc}$ is due to internal and external pressure. It varies across the section and can be positive (tension) or negative (compression); this result reports the hoop stress of greatest magnitude. It is obtained by finding the point in the cross section where the unsigned magnitude of the hoop stress is largest: this must be at either the inside or the outside fibre of the stress area.

Available at mid-segment points and line ends. This value is based upon an assumption that the pipe is thin walled, and is calculated as \begin{equation} \textrm{max xy-shear stress} = \frac{C_4 |\tau|\, O\!D_\textrm{stress}}{2I_\mathrm{z}} + \frac{2\, C_3 |\vec{s}|}{a_\textrm{stress}} \end{equation} where $C_3$ and$C_4$ are the shear and torsional stress loading factors respectively.

Available at mid-segment points and line ends. The von Mises stress, $\sigma_\mathrm{vm}$, is often used as a yield criterion. It is a combination of all the components of the stress matrix; in terms of principal stresses it is given by \begin{equation} \sigma_\mathrm{vm} = \sqrt{ \frac{(\sigma_1-\sigma_2)^2 + (\sigma_2-\sigma_3)^2 + (\sigma_3-\sigma_1)^2}{2} } \end{equation} where $\sigma_1,\sigma_2,\sigma_3$ are the principal stresses, i.e. the eigenvalues of the $3{\times}3$ stress matrix.

The **von Mises stress** varies across the cross section, so its value is reported at a specified cross section position, $(R,\theta)$.

The **max von Mises stress** is an estimate of the maximum value of the von Mises stress over the cross section. The way in which it is calculated depends on whether the line includes torsion or not, as follows.

- If torsion is not included, then we assume that the torque is zero. In this case the maximum value of the von Mises stress must occur in the plane of bending. We also assumes that the maximum occurs at either the inner or outer fibre. (This is a commonly-used assumption that is almost always valid: if the internal pressure stress contribution is dominant then the maximum will be at the inner fibre, whereas if bending stress is dominant then it will occur at the outer fibre.) OrcaFlex therefore calculates the von Mises stress at 4 points ($R = \pm I\!D_\textrm{stress}/2$ and $\pm O\!D_\textrm{stress}/2$, in the plane of bending) and reports the largest value.
- If torsion
*is*included, then the maximum value of the von Mises stress can, in general, occur anywhere in the pipe wall, so OrcaFlex calculates the von Mises stress at a grid of points across the pipe wall and reports the largest value found. The grid comprises 36 $\theta$-values (i.e. every 10° around the pipe circumference) at each of 5 $R$-values across the pipe wall.

Available at mid-segment points and line ends. These are the individual stress components at a specified cross section position, $(R,\theta)$. See the pipe stress calculation and pipe stress matrix topics for details.

Available at mid-segment points and line ends. Calculated as zz stress $-$ direct tensile stress. Bending stress varies across the cross section and so its value is reported at a specified cross section position, $(R,\theta)$.

Available at mid-segment points and line ends. These are the primary membrane stress and primary bending stress, respectively. The results are defined in terms of the ZZ stress, $\sigma_\textrm{ZZ}(R, \theta)$ as follows \begin{equation} \begin{aligned} P_\mathrm{m}(\theta) &= \sigma_\textrm{ZZ}(R_\textrm{mid}, \theta) \\ P_\mathrm{b}(\theta) &= \sigma_\textrm{ZZ}(R_\textrm{outer}, \theta) - P_\mathrm{m}(\theta) \\ R_\textrm{mid} &= (O\!D + I\!D)/4 = \textrm{the mid-wall radius} \\ R_\textrm{outer} &= O\!D/2 = \textrm{the outer radius} \\ \end{aligned} \end{equation}