2.9 Characterization of the Conductor Cross-Sectional Parameters
2.9.2 Tensile Analysis of the Conductor
In every conductor, the axial load generates inner axial stresses, shear, and torsional moments which give rise to coupling effects. The reaction of the conductor to the pure axial loading applied at its ends, assumes that the conductor is in a static condition. The strands are subjected to tensile force producing an axial displacement. This tension-induced elongation on strands generates two components due to coupling; the elongation and rotation. The magnitude of either of these components is likely to be strain rate dependent. It is usually desirable for a conductor to have good torque balance so that it produces little torque when loaded with both ends constrained.
However, an optimum combination of helix angles for the various strength member layers allow each strength member to maintain good stress and torque balance regardless of the magnitude of the tension-induced changes.
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The analysis of the tensile response of the conductors can be done either for only axial displacement of strands neglecting the coupling effect, or for displacement and rotation of strands where the coupling effect is considered.
2.9.2.1 Tensile Analysis of Conductor for only Strands Axial Displacement
The modelling for pure axial load for the helical structure also applicable to power line conductors can be credited to the pioneer work of F.H. Hruska [37, 61, 62], in which he developed the simple axial model for cables. His formulation assumed that the strands are subjected to only tensile forces. He assumed no moments, friction, no radial contraction of strands is not considered but only radial contact is considered. The global strand strain is considered small. This model was used to calculate for the strand interlayer pressure, unwinding torque, and the strand stress and strain.
As the tension is applied to such cables, the strengthened members exert a radial pressure on the strands below. For the case of single layer strand, the strand layer presses against the core. This pressure produces deformations of the core elements and the strands due to both material elongation and the elimination of space within the conductor structure. There may also be slight contact deformations at the interface between the strands and the core. All of these factors contribute to a reduction in overall conductor diameter and a corresponding increase in conductor length. Detailed explanation for this case as the conductors is assumed to experience pure axial displacement can be found in [58], and the extract of this work that are of relevance to this study are reproduced in this section as stated below.
Figure (2.16) shows the strands along the helical path, subjected to loads, and this causes the conductor individual helical strands to experience a tensile force. Due to the helical angle, tensile force, T, gives rise to two components i.e. the tangential and normal forces. These components of the radial tensile force are the axial component, TA, which is parallel to the neutral axis and the tangential component, TT,which is perpendicular to the neutral axis.
Figure 2.16: Axial forces action on the conductor cross-section
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Neglecting bending and torsional effects, the total tensile stress acting across the cross-section of the conductor can be evaluated as:
T
x A
S
.……….……. (2.48)
Where AT is the conductor cross section area and can be calculated by using equation (2.6). The only internal force being considered in the equations is the axial force com p on e nt o r t e ns i l e f o r c e on a strands cross- section. Each individual strand is assumed to be subject to the same tensile force, this can be calculated for using equation (2.28). The tensile stress acting on the individual strands:
j i
j i j
i A
T
……….……. (2.49)
Neglecting the effect of Poisson ratio, the strain acting along the line contact between the strand and the core is given by [15]:
j i i i
j
i E
dX Y
R d 2 ,
2
……….……. (2.50)
The axial strain due to the axial loading along the line contacts as function curve helical path can be determine as
0
( ) ( ) d
s A
i
ε T s ξ q ξ
EA .……….……. (2.51)
Expressing the axial strain along the x-axis for a given strand:
xi c
x
.……….……. (2.52)i xi
i
cos2 ……….……. (2.53a)i i
c
2
cos ……….……. (2.53a) The strain for the core and the strainfor each strand in a given layer, the corresponding axial unit strain is:
C c
c
c AE
T
.……….……. (2.54a)
i i
i i
i AE
T
2
cos .……….……. (2.54b) The total applied force on the conductor in terms of forces acting on each strand including the core can be evaluated as:
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N
i
i i i
c nT
T S
1
cos .……….……. (2.55) Substituting equations (2.54) into (2.55) will results in
N
i
i i
i i c
c c c
c E A nEA
A E S T
1
cos3 ..……….……. (2.56) The component in the bracket in the above equation is known as the axial stiffness of the conductor:
N
i
i i
i i c
cA nEA
E AE
1
cos3 ....……….……. (2.57)
The equation (2.57) can then be used to calculate the conductor’s axial stiffness AE, with the knowledge of conductor geometry and material parameters. Once the axial stiffness is known, strands forces, strain and stresses can be calculated for a given axial load S, on the conductor for each layer, including the core. This can be calculated for as follows:
The forces can be expressed as:
AE S E T A
AE S E T A
i i
i i
c c c
cos . .
2 ..……….……. (2.58)
The corresponding stresses are:
AE S E
AE S E
i i
i
c c
cos . .
2
……….……. (2.59)
2.9.2.2 Tensile Analysis of Conductor for Strands Axial Extension and Rotation In section (2.9.2.1), only the strand extension was considered, neglecting the strand rotation as well as the coupling effect of strand extension and rotation. But actually, in the static condition for the axial loading, axial displacement produces extension and rotation and the coupling effect of extension and rotation. In a conductor, the helical strands generally produce both elongation and rotation of strands simultaneously. The helical strand construction exhibits a geometric symmetric characteristic and due to this helical design of the strands, the overall axial behaviour exhibits a coupling between tensile and torsion strains. This coupling effect has been investigated in [63]
Tangential components of the radial force, TT yield a non-zero moment with respect to the x axis.
When it is not balanced, it leads to a possible rotation of the layer. Such rotation can be minimized
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by having alternate left and right lay layers (reverse-lay). If the balance is not perfect, and depending on the end conditions, there may be a small rotation due to the axial force, S.
The balance of forces and moments in the strands gives the equilibrium equations similar to the helical spring. This governing equation describing the coupling relation between the extension and rotation that is induced on the individual strand can be expressed using the following equation
T A
K K
K K M
F
22 21
12 11
…..………….……. (2.60) Where 𝜀𝐴 denotes the overall axial strain, 𝜀𝑇 the twist angle per unit length, F, the axial force and M, the torque. The four stiffness matrix components (𝐾11 𝐾12 𝐾21 𝐾22) which are for pure tensile (extensional stiffness ), coupling of tensile and torsion (extensional-torsion stiffness), coupling of torsion and tensile (torsion-extensional stiffness) and pure torsion (torsion stiffness) respectively.
For there to be a close-form solution, the stiffness matrix should be symmetric and using Betti’s reciprocal theorem 𝐾12 = 𝐾21. The value for each of the stiffnesses will be determined later in chapter 3.