5.12 The Formation of Conductor FEM Model
5.12.3 Geometric Mapping for the Conductor
When strung under tension, the overhead transmission lines conductors support its weight thereby sagging and sag/span ratio depends on the axial tensions at both ends. When subjected to transverse vibration, each member of the conductor is subjected to both axial and bending loads.
In the development of the FEM either for the straight or for the conductor structure sagging due to its weight to form a centenary profile, some form of coordinated mapping is necessary. A reference curved beam finite element formulation was done using the natural coordinate system. This was done to take advantage of the iso-parametric interpolation as described by equation (5.42). The geometric equation in terms of x, y and z axis is already defined during the geometric formulation.
For the formulation, the conductor geometry was defined in terms of global coordinate system. To implement the iso-parametric interpolation, the 3D geometric mapping is done. This is implemented using the 3D geometric equations (5.1-5.5). But the relation between the natural coordinate for the normalized curved beam was then transformed into global coordinates, the geometric transformation equations used for this mapping is defined in the next section. It is
142
imperative to emphasize that though the composite structure was done in 3D, the FEM equation implemented for element is that of 2D.
The configuration of the conductor is achieved by taking advantage of the geometric symmetry of the conductor sub-structures. To implement this, all mapping is done with respect to the conductor neutral axis which is assumed to coincide with the centre line of the core.
Figure 5.16: The illustration of iso-parametric mapping
Before mapping of the element is done, the line or the conductor neutral axis is first of all determined for the required stringing configuration i.e. iso-parametric element mapping along the helical path including the core.
Firstly, for the assumed configuration for a specific axial load, along the chord length path defined as the neutral axis, the finite elements for the core are mapped. The core strand coordinates are easily obtained as they are defined along the neutral axis.
The iso-parametric mapping for the strands in each layer, as illustrated (5.16) is done by co- ordinate mapping. It showed the mapping of the reference element into the conductor cross-section as defined by distanceRi, from the neutral axis and with helical angle
i with respect to z-axis in line with coordinate system defined in section (2.4.3) in chapter 2. The discrete functions nodal parameters at each element are expressed in terms of coordinates values that are defined at the centreline of the cross-section of the circular strands. This indicates that the coordinates of the strands in various layer are defined from the neutral axis with the helical radius Riand the helical angle
i with i= 1, 2, 3,……n, i.e. the nodal discretization of the conductor as can be seen in figure (5.15). These values can be transformed into Cartesian coordinates by the conversion of the polar coordinate systems along the curvilinear helical path defined for each strand path.143
To derive the geometric mapping, the process starts by defining each centre cross-section coordinates, for each strand as described in figure (5.15) along both core path and the strand helical path. The implementation of the mapping, i.e. conversion of the polar coordinates into the Cartesian coordinates can be illustrated. For example, the nodes conversion will take the form: R1
and 1 = (x1; y1; z1); R2 and 2 = (x2; y2; z2) and R3 and
3 = (x3; y3; z3) and so on for other successive nodes. Using the interpolation process showed in figure (5.13), the reference strand element is then mapped into the successive finite elements as defined by the nodes points (xi; yi;zi) coordinates along both core and the helical path. The conversion of coordinates can be achieved by using equation (3.27). In the interpolation process only the x and the y coordinates are required.
Finally, the composite structure is formed by the reference 2D curved beam, being geometrically mapped into the global structure of conductor.
The displacements and rotation at the nodes of the 2 D beam element as expressed into the global coordinate system for the conductor and the transformation matrix is defined as:
2 2 2 1
1 1
2 2 1 1
1 0 0 0 0 0
0 0
0 0
0 0
0 0
0 0 0 1 0 0
0 0 0 0
0 0 0 0
y x y x
u u u u
y x
y x y
x y x
v v
v ………….…………. (5.71)
Where L c
x
x x e
2 1 cos
and L s
y
y y e
2 1 sin
Using the beam curved finite element as the reference element, employing the transformation matrix equation to the beam element, the stiffness matrix, mass matrices and load vector for the conductor are obtained as follows:
T
K T T
K TKCe T Be T CA ………….…………. (5.72a)
T
M TKCe T Be ………….…………. (5.72b)
Be e TC T f
f ………….…………. (5.72c) Where T is transformation matrix and is defined as
144
1 0 0 0 0 0
0 0
0 0
0 0
0 0
0 0 0 1 0 0
0 0 0 0
0 0 0 0
s c
s c s
c s c
T
The advantage of using this form of composite formulation to obtain the FEM model for the conductor was that the path of each strand including the core is defined first for the desired configuration catenary curve as a function of its axial loads before mapping the curved beam element along this path. The composite structure is then achieved by the use of the transformation equation to transform the reference element into the conductor structure at the desired configuration.
To implement this FEM model in computer program in terms of the element size was done as a function of the pitch length. Geometrically, the pitch length falls into two quadrants with each half in opposite path. For stress distribution, it is assumed that each quadrant carries equal and opposite stress. Based on this, discrete elements use should have the maximum length or size that is equal or less than half its pitch length. In the aspect of the implementation of global structure, the total length of the conductor used to assemble the finite element along deformed path, the initial conductor points or the point of suspension was made to coincide with the origin of global coordinate. This help defines the initial position of each strand and then the path each strand will follows with the reference to the neutral axis in the final formation of FEM composite model.