﻿ Line types: Plasticity wizard

# Line types: Plasticity wizard

Line type bend stiffness may be nonlinear, in the form of a table of bend moment against curvature. These tables are sometimes derived from a provided nonlinear stress-strain relationship.

The OrcaFlex homogeneous pipe line type category can accept a stress-strain relationship directly. However, if you are modelling a uniform, homogeneous pipe using a general category line type, then instead of setting up the bend stiffness table from scratch, you can use the plasticity wizard to create the table for you.

The plasticity wizard is opened by the plasticity wizard button on the variable data form.

 Note: Before you can open the plasticity wizard you must have created and selected a bend stiffness variable data source.

The wizard requires the following data:

### Stress diameters

The inside and outside diameters of the load-bearing cylinder.

### Direct tensile strain

The plasticity wizard calculates bend moment curvature relationship by integrating the stress profile across the pipe cross section. This calculation requires a direct tensile strain to be specified – this data item serves that purpose.

### Stress-strain relationship

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

### Material E, Reference Stress, K, n (Ramberg-Osgood curve only)

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

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

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.

### Stress, Strain (stress-strain table only)

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.

Once you have entered appropriate values for these data, click the calculate button: the curvature / bend moment relationship is generated, and you are returned to the variable data form where the corresponding table is filled out. This bend stiffness variable data source is initialised to be hysteretic.

### Maximum curvature for derived data source

The bend stiffness variable data source is defined for curvature values between 0 and $c_\textrm{max} = \epsilon_\textrm{max} / r_\mathrm{o}$ where $r_\mathrm{o}$ is the radius to the outer fibre. The outer fibre strain corresponding to $c_\textrm{max}$ is denoted $\epsilon_\textrm{max}$ and is defined as follows:

• For a Ramberg-Osgood curve $\epsilon_\textrm{max} = \max\{0.05,\ 5\epsilon(\sigma_\mathrm{y})\}$. So the value used for $\epsilon_\textrm{max}$ will be 5 times the strain corresponding to the reference stress or 5%, whichever is larger.
• For a stress-strain table $\epsilon_\textrm{max}$ is simply the largest value of strain specified in the table.