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0 1 ,
0 r
R and R1r0r1 ..…..……… (2.8)
For the strands arrangement in various layers, parallel to the x-axis, where the distance for the strand centre are defined from the centre of the core. The position of the centre line of each strand in a given layer and the distance along these centres along the curvilinear axis of helix from the conductor neutral axis which is located in the same direction to the x-axis are given as:
ii i i
i r
R,1 , 1sin and
i
i L i
i P
r x x
R 2
sin )
( ……… (2.9)
Where
i incremental helix angle (wrap angle), which is the angular position of strand from the z-axis in the anti-clockwise direction.This form of descriptions given above is applicable to any conductor with strands of circular cross- section.
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The analyses of the strands in terms of stresses generated internally and the energy dissipated requires the knowledge of the inter-strand contacts. The factor of inter-strand contact requires a much more complex theory in order to model and analyse the elastic contact surface and in the presence of friction forces.
2.5.1 The Analysis of Conductor Inter-Strand Contacts
The closely packed arrangement of strands, coupled with the conductor being subjected to tensile force give rise to some form of contacts between strands. The conductor dynamic behaviour and the fatigue characteristics are both dependent on the inter-strand contacts and this is due to the operation of the frictional effects between the strands around these areas of contact. This makes it imperative to have an adequate knowledge of the inter-strand contacts mechanics and forces resulting from such inter-strand contacts. In a conductor, contacts is as a result of the conductor helical geometric arrangement and the axial loading applied at its ends.
In the analysis of the conductor geometry starting from the core strand outward, the first form of contact is the core to strand or the inter-strand contact between strands of adjacent layer. This type of contact occurs due to strands resisting the inwards radial force which tends to lengthen the strands due to the applied axial loads. To model the conductor using this form of contact, it is assumed that the strands in the same layer do not touch each other, and are in contact only with those in adjacent layers either above or below. This form of contact is referred to as the radial contact. The second form of contact within the conductor is due to the closely compact arrangement of the conductor. This results in a strand to strand packed contact, in which strands in the same layer are in contact, and inter-strand contact with adjacent layers are neglected. This form of contact is referred to as lateral or circumficial contact.
The combination of the two forms of contacts above described, is the third type of contact in the conductor. In modelling, this form of contacts is the combination of the radial and the lateral contacts. This form of contact combines the interlayer-strand and alternate layer strand-strand contact i.e. in which lateral contacts occur between strands of the same layer and the radial contacts occurs between strands of different layers. In most models, one form of these contacts has to be selected as the primary aim in order to avoid a statically indeterminate solution, which is mathematically difficult to solve. Under actual loading conditions, a strand may have both contacts.
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Figure 2.9: Inter-strand contacts in helical stands [12]
Various explanations have been given for the phenomena associated with various types of contacts within conductors, as can be found in many literature [11, 18, 19]. A general characteristic of the contacts between the strands depends on the helix angle. The description of the contacts within the cables also applicable to conductors is documented in the paper by T. Hobbs and M. Raoof [12], the authors identified two forms of contacts. They are the line and the point contacts. The diagram in Figure (2.9) shows where these types of contacts occur in a multilayer conductor. As shown in this diagram, the points mark A is used to indicate the areas that experience line contact, while the points mark B, indicates the areas of point or trellis contact.
The line contact occurs between the parallel layered helical strands of the same layer and it also exists between the first layer and the core. The line contact is a form of strand-strand contact i.e., lateral contacts. For the point or trellis contact, this form of contact occurs due to the helical arrangement of strands. This arises due to the opposite lay angle arrangement, strands in alternate layer crosses each other producing a point or trellis contact: inter-layer contact i.e. radial contacts.
In this form of contact, the strands in the one layer touches only those in alternate layers each at a point either below or above depending on the number of layers in the conductor.
The knowledge of any form of deformation and motion in the area contact is very important because of the action of friction, which causes energy dissipation to occur. In [16] it was stated that for stranded conductors, it may be assumed that virtually all of the internal damping energy originates from the mutual dry coulomb friction between the different layers of conductor including the core. In his model, the author used the analyses of the line contacts between the strands under sinusoidal conditions to determine the energy dissipation from the conductor. In a similar process, C. B. Rawlins [19] also used the analysis of line contacts in their models to determine the contact stress, interlayer shear stress, and slip condition under both the axial and
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bending loads. These models use the equivalent of line contacts for the point contacts. Although most authors use the line contact for their analysis, actually the contacts between alternate strands are point contact. To reduce the complexity of numerically when implementing the point contact, in the area of finite element analysis, this study will implement the equivalent of line contact capable of representing the effect of the point contact. The line contact and the equivalent of the line contact for the point contact was used to characterize various contact regions and then used to determine the energy dissipation from a conductor.
2.5.2 Number of Contacting Points
As explained earlier, the helical arrangement of strand gives rise to some form of inter-strand contacts. The shear force that tends to resist the unwinding of strands occurs at inter-strand contacts (line and point) between layers. The shear forces are applied to a strand at the lines or discrete points along its length, where it lay parallel or crosses the strands of the layer above or below. For analysis of the contact areas, it is necessary to determine this array of these line and discrete tractions within a conductor. For point contact, the number of contacts within a lay length determine the number of traction point and compliance function for analysis. For this it becomes imperative to know how many contact points lies along the pitch length.
Consider a point contact between strands, which occurs with the layer above and beneath, and this arrangement produces a number of the contact patches between the layers. The number of contact point can be determined for two contacting layers over a lay length of a strands along the path of the helical strands with opposite lay angles. Reference [57], gives the equation to determine the number of contacts between strands of different layers. The number of contact points between two layers can be defined as the number of contact points between layer, i, and layer, i +1, i.e. contact between a given strand and the strand above it. Hence, the number of contact points per lay length of a strands of layer, i with layer, i+1, as given in [58] is:
1
tan tan
1 1 1
1 ,
i i
i i i i
i R
n R
ncp
.……….……. (2.10)
Conversely, on the other hand, the point contact between a given layer, i, and layer, i-1, i.e. for contact between a given strand and the strand below it.
1
tan tan
1 1 1
1 ,
i i
i i i i
i R
n R
ncp
.……….……. (2.11)
The number of contact points on interface, i, over a conductor unit length is analysed in [58] as:
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i i i
i i i
i
i n R
n R n
ncp
2 tan 2
1 1 tan 1 1 .……….……. (2.12)
Substituting the pitch length as defined by equation (2.1), into equation (2.11):
i L i
L i L i i
i
n n P P P
ncp 1 1 1
1 1
.……….……. (2.13) The number of point contacts in a conductor can be determined by the approximation made by C.B. Rawlins [19], where the number of contacts between layers was given as:
1 1 1
i L
i L i
i P
n P
ncp .……….……. (2.14)
For a given conductor, the actual distance of contact points, measured on the strand centre line can be evaluated [58]:
i i
i i
i i i
i
ci
R R
R R d n
tan tan
2 cos
1 1
1 1
1
1
.……….……. (2.15)To find the normal contact force due to the axial load on conductor using the number of the normal point force at the interface contact points between i-layers and i+1-layer can be found [58].