## Modelling nonlinear homogeneous pipes |

A nonlinear stress-strain relationship is most commonly used to model nonlinear behaviour of elastomeric bend stiffeners or plastic deformation of steel pipes during installation.

This may be used in conjunction with a line type of category either general or homogeneous pipe. If the homogeneous pipe category is applicable, it offers a number of advantages over the general category:

- The data are entered directly, without the need to use the line type wizard for homogeneous pipe or the plasticity wizard, which makes data checking simpler.
- The effects of a diameter profile and a nonlinear material (e.g. bend stiffener) are handled automatically by OrcaFlex.
- The effects of direct tensile strain variation are again handled automatically (see below).
- Stress results are derived using the stress-strain relationship.

If, on the other hand, you wish to model hysteretic bending behaviour, then you must use a line type of the general category.

A nonlinear stress-strain relation may be set up to define nonlinear elastic material properties for homogeneous pipes.

The relationship between stress and strain can be specified by either **Ramberg-Osgood curve** or **Stress-strain table**.

These data define the relationship between stress $(\sigma)$ and strain $(\epsilon)$ in terms of a Ramberg-Osgood curve, as

\begin{equation} \epsilon(\sigma) = \begin{cases} \cfrac{\sigma}{E} + K\left(\frac{\sigma}{\sigma_\mathrm{y}}\right)^n & \text{for $\sigma{\geq}0$} \\ -\epsilon(-\sigma) & \text{for $\sigma{\lt}0$} \end{cases} \end{equation}

We denote the reference stress parameter by $\sigma_\mathrm{y}$ (since it is usually taken to be the yield stress).

Note that an alternative parameterisation of the Ramberg-Osgood equation exists. It is straightforward to convert between the two forms of the equation, but please take care that the data you use correspond to the parameterisation given here.

This table directly defines the relationship between stress and strain. Values for positive strain must be entered; those for negative strain are determined by reflection so that $\epsilon(\sigma) = -\epsilon(-\sigma)$. The table is interpolated linearly; for values of strain outside the table linear extrapolation will be used.

OrcaFlex uses the stress-strain relationship to generate a table of bend moment against curvature, using the same algorithm as the plasticity wizard. Each segment in the OrcaFlex model uses a distinct bend moment / curvature table – clearly necessary if the line type has a non-uniform diameter profile.

The use of distinct bend moment / curvature tables also allows OrcaFlex to account for the variation of direct tensile strain within a line. The bend moment / curvature relationship depends upon:

- The stress-strain relationship (as above)
- The line type inner and outer diameter
- The direct tensile strain

The direct tensile strain can have a significant effect on the nonlinear bending behaviour if it is large. To see why this is so, consider a steel pipe under tension such that the direct tensile strain equals the yield strain. When the pipe is in this state, any small amount of curvature will yield the pipe outer fibres. On the other hand consider an unstressed steel pipe, where the direct tensile strain is zero. In this state the pipe can withstand significant curvature before the outer fibres yield.

In principle the bend moment / curvature relationship could be recalculated at each time step of an OrcaFlex calculation. However this would incur a significant performance cost. Instead we make the assumption that the effect of dynamic variation of direct tensile strain on the bend moment / curvature relation is small.

This allows us to use a constant value of direct tensile strain *for the purpose of deriving the bend moment / curvature relation*. Note that each segment in the model has a distinct bend moment / curvature relationship based on a distinct value of direct tensile strain. By constant, we mean here that we do not update the bend moment / curvature relationship during the dynamic simulation. Note that this discussion of direct tensile strain only applies to the generation of bend moment / curvature relationships: OrcaFlex does, of course, account for dynamic variation of direct tensile strain when calculating wall tension, effective tension etc.

This constant (for each segment) value of direct tensile strain is chosen by first performing a statics calculation assuming zero direct tensile strain. Nonlinear bend moment / curvature relationships are created for each segment under this assumption. The direct tensile strain values from this initial static solution are then used to update the nonlinear bend moment / curvature relationships. Finally, the statics calculation is repeated with these updated relations, to obtain a solution which does account for the effects of direct tensile strain.

The nonlinear stress-strain relationship is also used to calculate certain stress results from strain values. Nonlinear stress-strain leads to a nonlinear elastic *bend* stiffness, but the *axial* and *torsional* stiffnesses are assumed to remain linear. This means that the only stress results affected are those that depend on bending: max bending stress, von Mises stress, max von Mises stress and zz stress.