The conductor is the piece of hardware through which circulates the electric current. Here, its sizing is essentially to choose/determine: its material, its section area and its structure. This structure reveals how the different strands of conductor have been assembled together, in a smooth or twisted pack, to favour either mechanical resistance or heat dissipation [3]. The most commonly used materials for electrical conductors are copper, aluminium, and alloys produced from one of them. The material met in this study is almelec, and the conductors have the twisted structure. Some useful features of almelec conductors are recorded in Table 1.

After the conductor, follow the line accessories. Among these accessories, we find:

- Insulators (in the form of a seat): these are insulating elements making the connection between the conductors and the supports. They play role both in mechanical (for fixing the conductor to the support, maintaining a suitable tension) and in electrical (for providing insulation between the conductor and the support). Their choice is made essentially according to the level of voltage carried by the conductor. Insulators are used for currents medium and high voltage;

- Suspension and fixing elements (clamps, cross members, uprights, rods, fasteners, plates, fittings…). They are essential to link the conductor to its support;

- Some supports are fitted with anti-cascade devices, to protect the line in case of incident;

- According to the type of conductor, its length, and the wind speed, vibration absorbers may be required on some lines.

It is the hardware element that carries and suspends the conductors at the required height, in order to ensure people and environment safety. It must be able to withstand the applied loads/efforts on it in a sustainable way, without breaking

or significant deformation. Line supports are usually poles, porticos or towers, made of wood, concrete or steel. In this case study, the supports are wooden poles from Eucalyptus saligna wood, which present an advantageous compromise between mechanical characteristics, mass, regrowth period, availability and cost.

The available Eucalyptus wood posts are approximately circular in cross-section, and the trunks are smoothly scrolled, with a shape close to that of a cylinder or a low-angle cone trunk. The sizes to be determined for a pole are its cross-section (or its diameter or its circumference), and its height. The latter is broken down into four parts, as shown in Figure 1 below.

The head height h_head is the pole’s height above the point where the lowest conductor is fixed on the pole. The beam f represents the maximum difference of height between the point where the conductor is fixed on the pole and the lowest point of the suspended conductor. The ground clearance H_ground represents the minimum distance to maintain between the ground and the lowest point of the suspended conductor, for security reasons. The implanted height h_im is the height of the pole to be buried in the ground, so as to ensure stability.

For further sizing computations, it is necessary to define the anchorage height H_a (distance between ground and the point where the conductor is linked to the pole) and the height above ground H_r (the pole’s height measured from the ground level).

The type of wood used is that of the Eucalyptus, whose mechanical characteristics are noted in Table 2.

Eucalyptus wood poles are divided into 7 classes (A, B, C, D, E, F and G). They are classified according to their diameters (measured at the poles base and at the poles head), circumferences, average volume, average weight or admissible load [10]. Regardless, the poles are also declined into 3 categories (see Figure 2),

according to their assembly mode: (a) simple poles, (b) twin poles and (c) strutted poles.

Hence, the sizing to be done for a pole will result in choosing a class, a height and a mode of assembly.

The wooden pole in service undergoes efforts divided into three (3), according to their direction.

· Vertical effort is made of the conductor’s and line accessories weight (Figure 3).

Conductors weight on one side and the side other of the pole. It doesn’t represent the entire weight of the conductors, but only the portion effectively supported by the considered pole.

In the further work, only the resulting force (V) will be considered.

· Longitudinal effort is the result of the conductors’ tensile strength (Figure 4).

The contact action of the conductor on the pole (Ti) has two components: the vertical is nothing else than the conductor’s weight, and the horizontal component (Li) is the tensile strength of the conductor. This is the one counted as longitudinal effort.

In the further work, only (L) will be considered as the resultant of the tensile strengths of the different conductors linked to the pole.

· Transverse effort is due to the action of wind, both on the pole and on the carried conductors (Figure 5).

Wind actions on the conductor (t1) and on the pole itself (t2).

In the further work, only (T) will be considered as the resulting transverse effort applied on top of the pole. This will produce the same effect (bending moment) as the two previous extended forces.

· The balance of the applied efforts on the pole is represented on Figure 6.

The forces are used according to the chosen limits of the dimensioning. The permanent effort is used for dimensioning the pole with regards to its elastic limit. It takes into account the loads that the pole must support at all times, and is determined assuming the minimum wind and the minimum tensile strength of

the conductors.

The nominal effort is used for dimensioning the pole with regards to its breaking limits. A value is determined for each of the possible climatic conditions in the area, and the maximum value is used for dimensioning.

A conductor suspended between two posts takes the form of a chain with the equation:

y = a ⋅ c h ( x / a ) [14]

and its deflection given by:

f = p ⋅ d 2 / 8 T [15] [16] (1)

With f: conductor’s beam (m), p: conductor’s linear weight (N·m⁻¹), T: conductor’s horizontal tensile strength (N), and d: distance between the two poles (m).

The mechanical stress of the conductor is affected by the weather conditions. To take this into account, the following equation of state is to be used:

p 2 ⋅ d 2 24 T 2 − T E ⋅ S − α ⋅ θ = constant [17] [18] (2)

With p: conductor’s linear weight (N·m⁻¹), T conductor’s horizontal tensile strength (N), d: distance between the two poles (m), E: conductor’s Young modulus (MPa), S: conductor’s section area (mm²), α: conductor’s linear expansion coefficient (˚C⁻¹) and θ: temperature (˚C).

In practice, the state Equation (2) is used to determine the value of the mechanical tensile strength T in the conductor in different climatic conditions, and Equation (1) then gives the beam.

The vertical effort corresponds to the amount of the conductors’ weight effectively supported by the pole. It is worth:

p = ρ ⋅ S ⋅ I p ⋅ g (3)

With p: conductor’s weight (N), ρ: conductor’s density (kg·m⁻³), S: conductor’s section area (mm²), I_p: weight rang of conductor (m) and g: gravity (N·kg⁻¹).

The longitudinal effort is the result of mechanical tensile strength of the conductors linked to the pole.

F = 1 2 k ⋅ ρ a ⋅ v 2 ⋅ C x ⋅ D ⋅ l v [19] (4)

With F: wind action on conductor (N), k: dispersive coefficient (no unit), ρ_a: wind density (kg·m⁻³), v: win speed (m·s⁻¹), C_x: aerodynamic coefficient (no unit), D: conductor’s diameter (m) and l_p: weight rang of conductor (m).

The term given by 1 2 k ⋅ ρ a ⋅ v 2 is called wind dynamic pressure, and often

noted q_dyn.

The total height of a support is the sum of: the installation height h_im, the ground clearance H_ground, the maximum conductor’s beam f, and the head height h_head [4]. (Figure 7).

It is given by the formula:

h i m = 0.1 H g r o u n d + 0.5 (5)

With H: pole height (m), h_im: implanted height (m), H_ground: ground clearance (m), f: conductor’s beam (m) and h_head: pole’s head height (m).

Head height depends on the armament of the pole, and is usually between 0 and 0.5 m for the highest placed conductors. Ground clearance is settled by an inter-ministerial order, and depends on the area [20]. Implanted height approximately worth:

h i m = 0.1 H g r o u n d + 0.5 (in meters) (6)

Conductor’s beam is determined by Equation (1) previously given; here, it is its maximum value.

The various efforts are calculated individually, and then combined according to the principle of the resulting moment [12].

1) Longitudinal effort

The moment of the longitudinal effort, calculated regarding the pole’s foot, is given by:

M _ 1 = T . H _ ∝ (7)

With M₁: momentum of the longitudinal effort (N·m), T: conductor’s horizontal tensile strength (N) and H_α: anchorage height of the conductor (m)

And H a = H − h i m − h h e a d (in meters) (8)

2) Transverse effort

The transverse effort is made up of the wind action, both on the pole itself and on the conductors. The related moments are calculated using formulas:

M t 1 = 0.5 C X p ⋅ q d y n ⋅ D ′ ⋅ H r 2 [12] (9)

With M_t1: momentum of the wind action on the pole (N·m), C_xp: pole’s aerodynamic coefficient (no unit), q_dyn: wind dynamic pressure (N·m⁻²), D’: pole average diameter (m) and H_r: height above ground (m).

M t 2 = F ⋅ H a (10)

With M_tα: momentum of the wind on the conductor (N·m), F: wind action on the conductor (N) and H_α: anchorage height of the conductor (m).

Transverse effort is then given by:

F t r a n s v e r s a l = ( M t 1 + M t 2 ) / H r (11)

On another hand, the combined resulting moment worth:

M = M l + M t 1 + M t 2 (12)

The resulting head effort applied to the pole is then given by [4]:

F t e t e = M / H r (13)

This effort, and the height previously determined, is the basis on which the convenient pole is chosen among the common poles listed in the catalogue (see Appendix 1).

For the wood poles of the line that have a particular function, it is preferable to check that the permanent effort remains under the value causing plastic deformation. This value is given by the pole’s catalogue [10].

The aim here is to determine the stresses and strains and to check that they remain below the permissible limit values [11]. We have used the “ideal model with the characteristics of the minor fiber” [12] [21]. It will be the model of a vertical straight cylindrical beam, whose diameter is either the head diameter of the pole or the foot diameter, whichever is the most unfavorable, and whose mechanical characteristics will be taken equal to those of the least resistant fibers found in Saligna eucalyptus trunks.

1) Bending stress

The bending of the pole is due to horizontal loads.

The bending moment M_f at the embedment has the following components:

( H r − h h e a d ) ⋅ F + 0.5 ⋅ q l ⋅ H r 2     et     ( H r − h t e t e ) ⋅ T [22] (14)

The resulting maximum stress is then determined:

σ f = M I G ⋅ D p (15)

With σ_f: maximum flexion stress, M: resulting bending moment (N·m),I_G: quadratic moment of the seciton (m⁴) and D_p: pole’s diameter (m).

This stress must be less than the maximum permissible value (breaking limit for supports in alignment, and elastic limit for the others). Otherwise, a column with a larger diameter is chosen, which will have a greater quadratic moment and a reduced stress.

2) Horizontal beam

Given the horizontal loading of the posts, this maximum deflection has two components, which can be found from the deformation equation:

u ″ ( 1 + u ′ 2 ) 3 / 2 = M E b ⋅ I G [23] (16)

With u pole’s beam (m), M: resulting bending moment (N·m), I_G: quadratic moment of the seciton (m⁴) and D_p: pole’s diameter (m).

To check that it is less than 5% of H_r (the dot is times Hr). If not, a pole with a larger diameter is chosen, which will have an enlarged quadratic moment of the section and a reduced deflection.

3) Buckling

Here the values are calculated to ensure that the pole does not buckle too much under vertical loading. These are on the one hand the pole’s twinge λ and on the other hand the buckling critical load N_c.

Let us recall the formulas of the critical buckling load.

N c = π 2 ⋅ E b ⋅ I G 4 ⋅ H r 2 [21] (17)

With λ: pole’s twinge (no unit), λ_lim: pole’s limit twinge (no unit), N_c buckling critical load (N), I_G: quadratic moment of the section (m⁴), E_b eucalyptus young module (Pa), H_r: height above the ground, (m), S_p pole’s section area (m₂) and R_e pole’s stress limit (Pa).

And critical constraint:

σ c = π 2 ⋅ E b ⋅ I G 4 ⋅ S ⋅ H r 2 = π 2 ⋅ E λ 2 [24] (18)

We determine:

- the limit twinge of Euler of our material, which is around 180:

λ l i m = π ⋅ E R [21] (19)

- the current twinge of our pole is:

λ = 2 ⋅ H r ⋅ S I G [21] (20)

Then, two situations are conceivable.

➢ If λ ≥ λ l i m , Euler’s considerations apply, and it must be verified that:

P: conductor’s linear weight (N·m⁻¹).

(3 is the buckling safety coefficient used in this hypothesis).

If λ l i m > λ ≥ 20 , Rankine’s empirical considerations apply, and it is a question of verifying that:

p ≤ π 2 ⋅ E ⋅ S 2 ⋅ ( λ l i m 2 + λ 2 ) [21] (21)

If 20 ≥ λ , there is safety to buckling, and it is a question of checking that

p ≤ 2 3 ⋅ R e ⋅ S (22)

In any case, if buckling safety is not guaranteed, a pole with a larger diameter should be chosen.

In addition to the scope of data, we must take into consideration: ground clearance, linear weight and minimum mechanical tensile strength of the conductor, at 75˚C [3].

The computation using formula (1) then gives the beam f ≈ 1.08   m .

The attachment methods and the minimum distances to keep between the

NB: The writing “3xalm54” indicates the presence of three conductors (three-phased line), made of almelec, with a section area of 54 mm². The same logic is extended to similar designations.

conductors and the way they are linked to this pole impose a minimum value for the head height h h e a d ≈ 0.8   m .

Therefore, the pole’s height can be extracted from Equation (5): H ≈ 11.75   m .

Appendix 2 gives the ground clearance equal to 8.2 m and Equation (5) then gives the height of the pole: H ≈ 11.75   m . This calculation confirms that a 12 m height is correct for this pole.

In addition to previous data (Table 4), the conductor’s density (Table 1) is to be considered. Formula (3) gives the weight of the conductors effectively supported by the pole: P ≈ 95   N .

The mass of the various line accessories supported by this pole (isolators, fasteners, armament) is evaluated to 37 kg, which implies an additional load of 362.4 N.

The vertical effort is the direct sum of these two loads, giving F v e r t i c a l ≈ 475.4   N .

The total mass of the various accessories supported by the pole (insulators, fixing elements, reinforcement) is evaluated using the data in Appendix 3 is 37 kg, resulting in an additional load of 362.4 N.

The vertical force is the sum of these two loads, i.e., F v e r t i c a l ≈ 475.4   N .

In addition to previous data, must be considered: aerodynamic coefficients for the conductors and the pole [2], conductor’s diameter [2], wind dynamic pressure in the retained climate hypothesis [3], pole’s average diameter [3], anchorage height and height above ground. The data are recorded in Table 5 below, for this pole.

Formulas (4), (9), (10) and (11) enable to find F t r a n s v e r s a l ≈ 4614.1   N .

In addition to previous data, it is needed to know: Young modulus and linear expansion coefficient for the conductor (Table 6), tensile strength imposed at the implementation of the conductors [3], linear masses, and section areas.

A computer-based resolution from the state Equation (2) provides the maximum expectable tensile strength in the conductor, in the hypothesis of considered climate. The longitudinal effort is then deducted as the resultant of these tensile strength: F l o n g i t u n a l ≈ 12921   N .

Considering the previous given data and found results, this effort can be computed from formulas (7), (12) and (13). For this pole, we find F h e a d ≈ 12930   N .

The resulting bending moment applied on the pole and its average diameter are known already. Formula (15) provides the value of the maximum bending stress in the pole σ f ≈ 7.324 × 10 7     Pa .

By applying the same method of calculating forces applied to the first pole of the first canton, and other the poles of canton, the values of the vertical, transverse, longitudinal and nominal forces applied are determined.

Table 7 summarizes the results of these calculations and the nominal allowable stress for each pole.

Poles with particular function in this canton are Pole’s Order 1 (stop pole), Pole’s Order 5 (deviation pole) and Pole’s Order 11 (anchorage pole). For these poles with a special function, it must also be checked that they do not acquire any noticeable plastic deformation. For this checking, the permanent forces have to be calculated (Table 8).

The height found for this first pole is: H ≈ 11.75   m Calculations confirm that a

Green color => the pole suitably withstands the efforts. Yellow color => the pole withstands the efforts, but is close to its limits. Red color => the pole is loaded over its eligible limits.

height of 12 m is well indicated for this pole.

The resulting nominal effort is F t e t e ≈ 12930   N .

This value is compared with the eligible nominal efforts listed in the pole’s catalogue. The lowest pole found capable of supporting this effort is the strutted D12.

The maximum bending stress is σ f ≈ 7.324 × 10 7     Pa The limit value for this pole (given in Table 2) is 5.5 × 10⁷ Pa.

This result leads the need to select a pole with higher bending strength. The closest suitable pole is D12, which is strutted.

The pole in place here (type D12 twin) is no longer suitable for the forces to which it is subjected, in particular the nominal force at the head of the pole and the bending stress. It must be replaced by a D12 type pole that is strutted.

By comparing the head stress of each pole with its nominal allowable stress, we find that: three poles from the first canton (Pole’s Order 1, 2 and 10), are 27.27% of the poles must be replaced, three other poles (Pole’s Order. 4, 5 and 8), are 27.27% of the poles need to be monitored as their head forces are close to their nominal allowable forces five poles are 45.45% of the poles hold the forces perfectly.

By comparing the permanent stress of each pole with its permissible permanent stress from Table 8, we can see that the permanent stress of Pole’s Order 1 is above its yield strength, while that of Pole’s Order 5 is slightly below its yield strength.

Out the 11 poles in the first canton, three poles (27.27%) need to be replaced. Three other poles (27.27%) are considered sensitive, but can still be retained. The types of replacement posts are shown in Table 9.