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*FLEXPROADD

Data format:

A single line with 10 parameters, as follows:

IMAT KTORSION ITYP KINI KLIMIT ACRIT THETA C CTD CBD

Status:

Optional.

Purpose:

To define non-linear flexjoint element properties.

Details :

Parameter	Description
IMAT	Identification number of the flexjoint property.
KTORSION	Flexjoint torsion stiffness; (in Nm/rad)
ITYP	Default value is 0. ITYP = 0: Linear bending moment / angle law. ITYP = 1: Non-linear bending moment / angle law.
KINI	Initial flexjoint stiffness before the critical angle is reached (in Nm/rad)
KLIMIT	Asymptotic flexjoint stiffness (in Nm/rad)
ACRIT	Critical angle from which a plastic curvature appears (in rad)
THETA	First parameter of the hardening rule. Default = 0. (in m)
C	Second parameter of the hardening rule. Default = 0. (in Nm/rad)
CTD	(N.m.s/rad) Torsional damping.
CBD	(N.m.s/rad) Bending damping.

Note

The flexjoint stiffness follows an elasto-plastic law :

\(\vec{M}^{total}_\sigma = \vec{M}_{bin} + \vec{M}_{\sigma} = El_{\iota} \vec{k}^t_X + \vec{M}_{\sigma}\)

The criterion for plasticity defines the state when a limit angle acrit is achieved and the flexjoint stiffness is becoming lower.

The following parameters are introduced :

Mb(M1,M2), the bending moment,

Mo (Mo1,Mo2), the reference in the bending moment space,

Mf, the critical moment.

Then the criterion is rewritten as:

\(f(\vec{M}^{\varsigma}_{b}, \vec{M}^{\varsigma}_{0}) = \| \vec{M}^\varsigma_{b}- \vec{M}^{\varsigma}_{0}\| = M_f\)

At any time, the plastic angle should be adjusted so as to ensure that \(f(\vec{M}^{\varsigma}_{b}, \vec{M}^{\varsigma}_{0}) \leq 0\). This may lead to three possible states :

\(f(\vec{M}^{\varsigma}_{b}, \vec{M}^{\varsigma}_{0}) \leq 0\) defines the elastic domain;

\(f(\vec{M}^{\varsigma}_{b}, \vec{M}^{\varsigma}_{0}) = 0\) and \(f(\vec{M}^{\varsigma}_{b}, \vec{M}^{\varsigma}_{0}) < 0\) represents an elastic evolution;

\(f(\vec{M}^{\varsigma}_{b}, \vec{M}^{\varsigma}_{0}) = 0\) and \(\dot{f}(\vec{M}^{\varsigma}_{b}, \vec{M}^{\varsigma}_{0}) = 0\) is the plastic flow.

Classically, two parameters C and q define the kinematic hardening rule, that is to say, the variation of the plastic criterion center :

\(\vec{\dot{M}} = C\vec{\dot{a}} - \Theta\vec{M}^{\varsigma}_0 \vec{\dot{a}}_p\)

with \(\vec{k}_p^\xi=\int_0^\xi \lVert \dot{\vec{k}_p^\xi} \rVert d \tau\) , hence .

\(\vec{\dot{a}_p} = \| \vec{dot{a}_p} \|\)

Note that C and q can be adjusted for the hardening curve to fit with experimental curves from which three different stiffnesses may be defined.