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Non linear flexjoint theory

Elasto-plastic model

The flexjoint stiffness follows an elasto-plastic law :

\[ \vec{M}_b^{total}=\vec{M}_{lin}+\vec{M}_b=EI_l\vec{k}_k^t+\vec{M}_b \]

Where $\varepsilon$ is the strain and $A$, $B$, $C$ are temperature-dependent coefficients. 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 :

$M_b(M_1,M_2)$, the bending moment,
$M_o (M_{o1},M_{o2})$, the reference in the bending moment space,
$M_f$, the critical moment.

Then the criterion is rewritten as:

\[ f(\vec{M_b^\xi},\vec{M_o^\xi})=\lVert \vec{M_b^\xi}-\vec{M_o^\xi} \rVert-M_f \]

At any time, the plastic angle should be adjusted so as to ensure that $f(\vec{M_b^\xi},\vec{M_o^\xi})\le0$. This may lead to three possible states :

$f(\vec{M_b^\xi},\vec{M_o^\xi})\le0$ defines the elastic domain;
$f(\vec{M_b^\xi},\vec{M_o^\xi})=0$ and $\dot f(\vec{M_b^\xi},\vec{M_o^\xi})<0$ represents an elastic evolution;
$f(\vec{M_b^\xi},\vec{M_o^\xi})=0$ and $\dot f(\vec{M_b^\xi},\vec{M_o^\xi})=0$ is the plastic flow.

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

\[ \dot{\vec{M}}_o^\xi=C\dot{\vec{a}}-\theta \vec{M}_o^\xi \dot{\vec{a}}_p \]

with

\[ \vec{k}_p^\xi=\int_0^\xi \lVert \dot{\vec{k}_p^\xi} \rVert d \tau \]

hence

\[ \dot{\vec{a}_p}=\lVert \dot{\vec{a}_p} \rVert \]

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.

Testing and validation

Base Case Description

The base case is a 2 meter long beam connected at one end by a flexjoint and submitted at the other end to an incremental force of 15 kN and to a harmonic force at 10 second of 5.25 kN.

The beam properties are a flexible type with the following characteristics :

Linear mass : 150 $kg.m^{-1}$
Axial stiffness : 1e9 N
Bending stiffness : 1e8 $Nm^2$
Torsion stiffness : 1e9 $Nm^2$
The beam total apparent weight is 2311 N.

Depending on the options used, the flexjoint can be defined by a multi-linear law and a non-linear law.

The multi-linear law is a two slopes curve defining then two stiffness : An initial one of 50 $kNm.deg^{-1}$ until an angle of 0.5 degrees and 20 $kNm.deg^{-1}$ for greater angle.

The non-linear law is defined by an initial stiffness of 70 $kNm.deg^{-1}$ and after a critical angle deviation of 0.1 deg, the asymptotic stiffness is 30 $kNm.deg^{-1}$.

Test 1 : quasi-static multi-linear law & dynamic elasto-plasticity

The following data interpretation is given in the LIS file :

no.   jo    je ipro imat length Multi-linear QS stiffness /  Moment =
fct(dtet)  Torsion stifness (Nm/deg)

3 1 3    3    2 0.0000 |    Moment (N.m)    |
dtet (rad)  |  0.573E+09

\-------------------------------------

  | 0.000E+00    |    0.000E+00 |

  | 0.250E+05    |    0.873E-02 |

  | 0.350E+05    |    0.175E-01 |


Initial bending stiffness   :    7.000E+01 (kNm/deg)

Asymptotic bending stiffness :    3.000E+01 (kNm/deg)

Limit angle variation  :    1.000E-01 (deg)

Non-linear parameters Theta  :    0.000E+00 (-) and C =    5.250E+01 (kNm/deg)

With Podeep, the moment in the flexjoint as well as the vertical angle of the beam axis are post-processed. The following graph shows the moment variation as a function the beam axis.

Quasi-static part :

After checking, the slope of the curve is 49.2 $kNm.deg^{-1}$ at the beginning and 19.9 $kNm.deg^{-1}$ at the end which is quite in-line with the definition of the multi-linear law.

The angle at the end of the QS steps is 0.876 deg which corresponds to the following moment : $$ M_b = 0.5\times50 + (0.876-0.5)\times20 = 32.5~kN.m $$

This value is in-line with the imposed moment (15kNm with a lever arm of 2m) plus the moment due to the beam own weight (2.3 kN).

Dynamic part :

If we focus on the dynamic part of the curve, we can get the following:

This curve exactly follows the introduced behaviour law.

The critical angle is 0.1 deg,
The original stiffness is 70 $kNm.deg^{-1}$,
The asymptotic slope is 30 $kNm.deg^{-1}$.

Test 2 : pure elasto-plasticity (quasi-statics and dynamic)

The following data interpretation is given in the LIS file :

  no.   jo    je ipro length  Non-linear bending stiffness  Torsion stiffness (kNm/deg)


  3 1 3   0 2 0.0000   0.573E+09


  Initial bending stiffness   :    7.000E+01 (kNm/deg)

  Asymptotic bending stiffness :    3.000E+01 (kNm/deg)

  Limit angle variation :    1.000E-01 (deg)

  Non-linear parameters Theta  :    0.000E+00 (-) and C =    5.250E+01 (kNm/deg)

With Podeep, the moment in the flexjoint as well as the vertical angle of the beam axis are post-processed. The following graph shows the moment variation as a function the beam axis.

This graph shows that during the QS steps, the critical angle of 0.1 degree is quickly overtaken and then the slope is 30 $kNm.deg^{-1}$.

If we have a look at the flexjoint moment time series, it turns out that the dynamic variation is first in the same sense as the static part. Indeed, the moment starts to decrease.

That is why there is no change of slope in figure 1 between the static and the dynamic analysis.