<< Click to Display Table of Contents >> Navigation:  Flexcom > Theory > Model Building > Geometric Properties > Geometric Properties in Flexible Riser Format > Material Models > Non-Linear Elastic >
Curvature Slippage

theory

It is possible to change material bending properties from linear to non-linear between successive analysis stages. Furthermore, the user-defined (non-linear) moment-curvature relationship used in a restart stage can be adjusted such that it is offset by moment-curvature values obtained from the preceding solution. This modelling behaviour is termed curvature slippage.

In some respects, curvature slippage is similar to Hysteretic Bending, but bending reversals do not cause the structure to alternate between pre-slip and post-slip conditions. Rather, a single slip is modelled, and there is no loop behaviour from that point onwards.

The following figure shows some sample linear and non-linear material properties, each of which define some relationship between bending moment and curvature. In the context of curvature slippage, the elastic/unpressurised state is represented by the linear relationship, while the friction/pressurised state is represented by the non-linear relationship.

Sample Moment-Curvature Relationships

The following figure shows some actual material properties used in a notional Flexcom analysis. The left image shows the initial linear moment-curvature relationship, while the right image shows the modified non-linear moment-curvature relationship which is used in a restart analysis. The intersection point of both lines corresponds to the state of slippage {Ks, Ms} values.

 

Curvature Slippage Stages

User Inputs

Initial Static Analysis

Contrary to conventional Flexcom models, both linear and non-linear material properties are defined in the initial static stage. Specifically, the *GEOMETRIC SETS keyword should contain numerical inputs which represent the bending stiffness about the local y and z axes, and the name (or names) of non-linear moment-curvature relationships for these same degrees of freedom. Any non-linear relationships which are referenced should be defined under the *MOMENT-CURVATURE keyword. Finally, the *NONLINEAR MODEL keyword should be included, and the EI input set to LINEAR.

Restart Analysis

The *NONLINEAR MODEL keyword should be included, and the EI= input set to NONLINEAR. Additionally, the NONLINEAR= input under the same keyword should be set to CURVATURE_SLIPPAGE.

Symmetric and Asymmetric Bending Models

The following section provide some further theory on the distinction between the Symmetric and Asymmetric Bending Models, and its significance in the context of the curvature slippage feature.

Asymmetric Bending

For asymmetric bending with slippage, the stiffness, D, and Nonlinear Material Force, h, for each local y and z axis are calculated independently. For a given local axis, the determination of D and h is illustrated as follows.

Asymmetric Curvature Slippage

where h is calculated as:

                               (1)

and

                       (2)

The stiffness, D, is calculated using either the tangent (default) or secant method (refer to Tangent and Secant Stiffness for further information). If the secant stiffness method is specified using the *NONLINEAR MODEL keyword, then h0 will be zero and hence the non-linear material force will be equal to hs.

Symmetric Bending

For the case of symmetric bending the calculation of D and h is described as follows.

Firstly, the resultant relative curvature, kr, is determined:

               (3)

where:

•ky is the instantaneous local y curvature

•kz is the instantaneous local z curvature

•kys is the slipped local y curvature

•kzs is the slipped local z curvature

A resultant relative bending moment, Mr, is then obtained from the non-linear moment-curvature relationship as follows:

                       (4)

Symmetric Curvature Slippage

The stiffness, D, is calculated using either the tangent (default) or secant method (refer to Tangent and Secant Stiffness for further information), depending on the option specified in the *NONLINEAR MODEL keyword.

Local y and z moments are determined by apportioning the resultant moment, Mr, by the ratio of the change in local curvature (relative to the slipped position), k-ks, to the resultant relative curvature, kr, as follows:

               (5)

and

               (6)

with the addition of Mys and Mzs which are the local y and z slipped moments, respectively.

Once the local moments are determined, the local Non-linear Material Force Terms (hy and hz) are obtained as follows:

                       (7)

and

                       (8)

Relevant Keywords

•*GEOMETRIC SETS is used to assign geometric properties to element sets.

•*MOMENT-CURVATURE is used to define moment-curvature curves for non-linear materials.

•*NONLINEAR MODEL is used to specify a modelling approach for non-linear materials.