The geometric modelling of the conductor with the coordinates system defined in section (5.4) provides a geometric approach for implementing the finite element analysis for the structure when subjected to dynamic loads. Modelling the conductor as composite structure which considered the conductor as the assemblies of either thin beams or rods having distributed mass and elasticity for which the governing equations of motion for such structures are described by partial differential equations. To find the FEM solution in a closed form for this complicated bundle structure is very difficult and almost impossible. This is due to the fact that the integration of these equations as a bundle is generally more complicated than the FEM solution of a single ordinary partial differential equation, as a governing equation for an equivalent single continuous beam. The single beam FEM model is a simplified model which may be used to represent the conductor. Due to the mathematical simplicity, the FEM dynamic analysis of conductor structure as a solid continuous system has limited use in practice. Nevertheless, the analysis of conductor systems with the beam or taut string as a generic model can provide very useful information on the overall dynamic behaviour of conductor structures.
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Contrary to the use of equivalent FEM beam model, in this study because the parameters of interest are related to the conductor internal geometry, the composite model of the assemblies of strands was used. The method of FEM analysis for the conductor is very complex because each conductor strand was subjected to the coupling effect of axial, torsional, and bending. This FEM composite structure concept for the conductor was achieved by the iso-parametric interpolation of the curve 2D beam for each individual strand along the defined geometric path.
5.5.2 The Curved Beams Model
Most FEM formulation for the conductor found in literature were mostly done with the straight beam model. This has restricted the classical beam theory formulation for the conductors to initially straight beams of constant cross-section. This model has been extensively used and has been implemented by several researches investigating conductor dynamics [8, 14]. This simplified model can be accepted in modelling the conductor to some degree of accuracy, but in real problems, where the curvature becomes significant, this model becomes inadequate. In this case for the conductor, the linear variation of strain over the cross section is no longer valid, and the use of curved beam becomes appropriate. Furthermore, curved beams are known to be more efficient in transfer of loads than straight beams, because the load transfer is affected by bending, shear, and axial action. Thus, conductor strands as the form of the sub-structure was modelled by the curve truss beam or the curved frame element.
In this study, the Euler-Bernoulli curved beam theory was used, though there was still the assumption of plane cross sections remaining plane after deformation is valid. The xy- longitudinal plane has been assumed to be the plane of symmetry, with the bending load being applied in this plane. The conductor strands deformations are symmetric; this is why the helical strand can be regarded as an example of a curved beam with body force and restrained sections at the ends. From the conditions of symmetry, the distribution of stress in one quadrant of the helical strands can be balanced by the same continuous strand in the opposite quadrant.
5.5.3 The Beam Constitutive Equation
Before the FEM formulation for the conductor using the curved beam, it is imperative to formulate the constitutive equations for the curved beam. Consider figure (5.1), illustrates the Euler- Bernoulli curved beam, with an axial and transverse displacements, and subjected to rotation. The deflection, y is on the single plane perpendicular to the axis of the beam. The cross-section will have constant density, p, constant modulus, E, constant cross-section, A, and inertia, 1. The
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deflection, y, and the slope, θ, vary along the cross-section, x, and with time, t. The horizontal displacement at the beginning of the beam cross-section is denoted by u.
The forces and moments acting on the beam, these include the vertical forces, V, acting at each end of the segment in the same direction as the vertical deflection, and a moment, M, at each end as shown figure (5.1). There are also the axial forces, T, acting along the beam centreline. Finally, there is the body force per unit length, pgA, which represents the weight of the beam. The equations of motion from the forces and deflection can be determine by summing forces along the x and y directions and taking moments about the left end. This produces equation for tensile, moment and inertia mass respectively:
x gA T
………. (5.7)
gA x V
M
………. (5.8)
x T w x x V t
A w2
2 ………. (5.9)
Figure 5.1: Forces and moments acting on the beam 5.5.4 Curved Beam Strain–Displacement Relationships
Consider figure (5.2), let any point on the cross-section of the beam be displaced from its unstrained position through small components of displacements v and u representing the tangential and radial displacements of points on the cross section, and y represent the transverse displacement in a direction normal to the plane of the beam.
The axial displacement at any point (x, y) may be expressed directly in terms of θ(x) which is the rotation of the normal to the centroidal axis due to flexural effect, and u(x) which is the axial displacement due to the axial effect and the curvature. The axial displacement for the curved beam is given as:
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R
x y u
R x v u y x
u , 0 x …….………. (5.10) The y distance from the centroidal axis of the cross section to the point (x, z) and equals to the
lateral displacement at the centroidal axis. The normal rotation θ(x) is equal to the slope of the centroidal axis minus a rotation, which is due to the transverse shear deformation:
x y x v
x
,
…….………. (5.11)
Figure 5.2: The curved beam The axial strain equation is given as:
x u
………. (5.12) If the curved beam obeys Hook law, the stress-strain relation was obtained by multiplying the
strain by Young's Modulus which yields the stress as:
x E u
………. (5.13) The balance of equation for the curved beam element in terms of forces and moments:
0 0
2 2
x T w dx F d dx
M d
dx F dT
x y
x x x
.………. (5.14a, b)
The axial force and moment can be formulated using the following equations:
The axial tension becomes:
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A A
x s
E u dA
T ………. (5.15) The moment becomes:
2 2
s EI v dA z M
A
x
………. (5.16) WhereI z dAA
2 , is the cross-section moment of inertia, A is the area of the cross-section.
Substituting equations (5.9) and (5.13) into (5.13) to obtain
R v s EA u x t A u2
2 ………. (5.17)
R u T s
x R u s EI v x t
A v 2
2 2
2 2
2 ………. (5.18) Using the Euler-Bernoulli curved beam, the shear strain is neglected. Therefore, since the beam
was assumed to be elastic, the relevant strains in this case are the axial strain and the bending strain, where the bending strain now depends only on the radial displacement, u.
The strain-displacement equations describing the deformation of a curved beam neglecting the shear force is obtained, therefore, the axial and bending strains are given as:
EA A
R v s EA du R
T M
………….…………. (5.19)
R v ds du
A
..……….…………. (5.20)
T B EI
B T
Rds EI u ds T dv Rds
du s
v EI d
M 2 , ,
2
………….…………. (5.21)
Rds
u ds dv Rds
du ds
v d
B 2
2
………….…….……. (5.22)