134
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In reference [81], the finite element analysis of inter-strand contact was used to establish the hysteresis loop due to the conductor alternate condition between the stick and the slip regimes.
This was then used to evaluate conductor self-damping for a single-layered conductor. At the contact points, due to the shear force there is the tangential stiffness. These stiffness along the contact point are shown in figure (5.11). The evaluation of the compliance along the contact points are discussed in [89] and these tangential compliances can be used to determine the stiffness along the point’s contacts.
Figure 5.11: Contact points and its rheological representation [89]
Figure (5.11) indicates the number of point contacts between strands and the corresponding stiffness generated. In implementing this model as shown in figure (5.11) numerically can be very challenging and is not worth the effort of implementing as the equivalent line contact can produce relatively good results with very less effort. Therefore, the alternative means is to use an equivalent line contact model and the pressure is assumed to be constant along the line of contact between two strands.
Damping in the conductor is mainly due to hysteresis damping. This form of damping is caused by the axial displacement and the FEM formulation is similar to the finite element formulation for a bar element as shown in figure (5.12) where only the axial strain is considered. This implies that, the damping force is formulated by decoupling the axial parameters from the transverse parameters. Thus, using the bar element for the FEM formulation, taking into account the sinusoidal axial displacement for the two-node bar element. The strain x can be related to the axial displacement u as
d
d
x
ε u
x ………….…………. (5.60)
The axial displacement is interpolated by
2 2 1
1( )u N ( )u N
u ………….…………. (5.61)
136 Where the shape functions are defined as
) 1 2( ) 1
1(
N ; (1 )
2 ) 1
2(
N
………….…………. (5.62) The resultant tensile on the strands can be evaluated as
2 1
2 1
2
1 u u
l EA l EA l T
T
………….…………. (5.63) The above equation can be written in matrix for as follows:
2 1 2 1
T T u u l EA l
EA l EA l
EA
………….…………. (5.64)
Introducing the above equations in form of stiffness matrix for the contact surface is then obtained as:
1 1
1 1 l K EA
………….…………. (5.65) Equation (5.60) provides the stiffness that tends to resist the shearing force between the conductor strands
Figure 5.12: The bar element The axial force acting on the strand can be determined by:
s EA
s q s ds Ts j
i
0
, ………….…………. (5.66)
Hence,
2
3
2 1
cos ( )cos
EA α
u u qEA X α
l
1 cos 1 ) 1 (
1 1
cos 1 3
2
2 1 qEA X
u EA u
………….…………. (5.67) The finite element formulation is expressed on the basis of the finite element equilibrium equation and the force-displacement relation given in compact form as:
K{u}{f} ………….…………. (5.68)
T1 T2
u1 u2
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Where [K] the stiffness matrix, {u} is the vector displacements and {f} is the force vector and displacements {u} are interpolated over the whole conductor strand as {u} = [N] T {d}.
The system equation [K] {d} = {f} was solved for by the system displacements {d}. In reference [16], the author drew the inference that the non-linearity in strand axial movement result in closed hysteresis loops.
To develop the stick-slip contact states, with the generation of hysteresis loop, is to combine the displacement with the developed traction stiffness and incorporating it into a bar model. A hysteresis loop [81] is shown in figure (5.12). and the explanation on how the loop can be use to determermine energy per cylce, see [16] for details. The finite element implementation of the stick- slip models for friction forces usually poses unexpected problems. The difficulty has to do with the implementation of both dynamic and static frictions in the numerical modelling of motion under the effect of friction.
Figure 5.13: The hysteresis loop
This difficulty of modelling along contact arises from the existence of the imposed constraints to determine the state of the strands. This type of constraints can be represented by a set of inequality equations. The major challenge is determining the critical states of the load on the strands that conform the responses accordingly. To generate the hysteresis loop, the force model is implemented with the effect of the friction force fS, determining state of the strands. The strands may be in sliding; still; or transitioning from sliding to still or from still to sliding. If the strands are sliding, the dynamic friction model is implemented. If the strands are still, the static friction model is used as shown in figure (5.10).
To model the inter-strand displacement as a function of friction and then simulates the displacement between strands with the surface stiffness K, the system is assumed to be massless
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as shown in figure (5.14). The strands are pressed together underlying surface with a normal force FN. This initial displacement is assumed to be x0, and oscillates as x0(t) = A sin ωt during the bending. However, there is a need to model the friction force FS, but this depends on whether the strands are sliding or still.
The position of the strand by x (t), its position relative to the bottom strands with the assumption that the strand starts from rest at x (t) = 0 at t = 0. According to Newton’s second law, the force acting in the opposite direction to the surface friction forces, Fs are equal when the strand is in the sticking position and this will occur when the velocity is zero. In this case, the force will be the static friction force, Fs, which is equal to the other forces, Fs = −µFN, so that the net force is zero.
However, this is only true if the static friction force is less than the friction threshold before slip can take place.
s NS k x x t F
F 0
………….…………. (5.69)x = 0 and x0
t Asin
tFigure 5.14: The slip pattern from the outermost towards the innermost layer
Initiation of sliding starts from rest and the strands will start to slip if the force fi exceeds the maximum static friction force. The strands will start moving in a direction given by the force fi, and the strands will then experience a dynamic friction.
Also in this model, it is assumed that the displacement of strands is equal in each layer. This allows the hysteretic behaviour to be treated in a unified manner by a single nonlinear differential equation with no need to distinguish different levels for the various Coulomb friction model but only the
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values for the friction coefficient. The frictional hysteretic behaviour for the conductor in the dynamic state can be expressed as:
N
i i i
i t x k x
F
1
sgn
sin ………….…………. (5.70)
Changing the frictional state from sliding to static in equation (5.70), can only occur if the velocity is equal to zero, and this condition has to be constrain in the equation for this to occur during a simulation. However, when the velocity of the strand changes sign, the strand will actually stop and start sticking to the surface with a static friction force. Therefore, setting the velocity to be exactly zero when the velocity changes sign and enforces the constraint, indicating that strand is in a sliding or a sticking state.
To simulate these equations for the formation of the hysteresis loop, the same principle was used as employed in chapter 4 i.e. simulating the constraint using the Bouc-Wen. Also, the displacement pattern, follows a sequence in which the displacement in a given layer is completed before displacement can be initiated in the layer below and also the displacement starts from the outer layer of the conductor. Unlike, the loop formed in chapter 4 using the moment-curvature relation;
here the loop was formed using the force-displacement relation.