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Non linear bending/curvature law

According to experimental results and to the actual understanding of flexible risers internal structure, the bendingp behavior of a flexible pipe may be divided into two main contributions :

1. The contribution of the armor layers for which an elasto-plastic formalism proposed by IFP is applied. Indeed for small curvatures, the armor layers are stuck to each other which results in a high bending stiffness. When the curvature is getting higher, the layers start to slide and the bending stiffness decreases.

2. The contribution of the whole internal structure which is assumed to follow a linear law with an associated bending stiffness which is the asymptotic bending stiffness EIl.

Then the global bending moment writes :

\[ \vec{M}_b^{total} = \vec{M}_{lin} + \vec{M}_b = EI_{l} \vec{C} + \vec{M}_b \]

\(M_b\) follows an elasto-plastic law that is to say that the function bending moment versus curvature is non longer reversible.

Main parameters

  • Initial Bending stiffness : Bending stiffness associated with small curvature variations, named EIrig.

  • Asymptotic bending stiffness : Bending stiffness reached for high curvature variations, named EIl.

  • First critical curvature \(C1\) : For curvature variations lower than \(C1\), the bending stiffness is EIrig.

  • Moment M1 (associated with \(C1\)) :By defintion, \(M1\) = \(EI_{rig}\) * \(C1\)

Elasto-plastic parameters

These are the elasto-plastic parameters which define the non-linear law. They may be either directly specified or derived from experimental results :

  • Specified values : q and C must be entered. q determines the degree of non-linearity of the behaviour law. When q is zero, the behaviour law is a double slopes curve and C is automatically set to \(C = \frac{EI_{rig} EI_l}{EI_{rig} - EI_{l}}\)

  • Derived from experimental results : Three sets of data (Moment, Curvature, Stiffness) must be defined. Provided some conditions are fulfilled, q and C will be automatically calculated when pressing on calculate.

Warning: The requested conditions are the following :

Let define : K1 = EIini Eil,, K2 = EI2 EIl, K3 = EI3 - EIl, B12 = M2 EIl (C2 C1A) M1 and B13 = M3 EIl(C3 C1) M1. Then

  • K1, K2, K3 must be greater than zero otherwise the following warning is displayed : Selected experimental points are not acceptable : negative stiffness found for the armor layers contribution.

  • The stiffness must respect the following order : K1>K2>K3, otherwise the following warning is displayed : Selected experimental points are not acceptable Stiffness should be decreasing.

  • B12 and B13 must be greater than zeros otherwise the following warning is displayed : Selected experimental points are not acceptable : negative moment found for the armor layers contribution.

Note

  • When OK is pressed, the data are stored and the panel is closed,

  • Theta is associated to a length while C is a bending stiffness,

  • Pressing the Graph button will display the curve moment vs curvature for checking.