Most economical span, double circuit line

Method. For each span, the transverse load at the tower comes from two sources:

• wind on the conductors and earth wires: $F = p \times d \times S$, taking the wind span equal to the span $S$ and using the full projected area;
• the resultant of the conductor tensions at a line deviation $\theta$: $F_d = 2T\sin(\theta/2)$.

These loads act at the conductor heights. Their overturning moment at ground level is

$$M = n_c\,(F_c + F_{dc})(h_1 + h_2 + h_3) + n_e\,(F_e + F_{de})\,H_t$$

where $n_c$ is the number of conductors at each level and $n_e$ the number of earth wires. The tower weight is found from Ryle's formula

$$W = K\,H_t\sqrt{M}$$

$K$ is the same for all spans, so it cancels in the comparison. Tower cost is proportional to tower weight, so the span that gives the smallest total tower weight is the most economical span (with the conductor already fixed). Wind on the tower body is neglected.

Data and deviation loads

• Double circuit: 2 power conductors at each of $h_1$, $h_2$, $h_3$, so $n_c = 2$. Two earth wires at $H_t$, so $n_e = 2$.
• Wind pressure $p = 100$ kg/m², $d_c = 0.02597$ m, $d_e = 0.0146$ m.
• All towers are designed for $\theta = 5°$:

$$\begin{aligned} F_{dc} &= 2 \times 6890 \times \sin 2.5° = 601.08\ \text{kg} \\ F_{de} &= 2 \times 3332 \times \sin 2.5° = 290.68\ \text{kg} \end{aligned}$$

Wind loads per span

Overturning moments and tower weight index

Sample calculation (250 m):

$$\begin{aligned} M_c &= 2(649.25 + 601.08)(9.70 + 14.52 + 19.35) = 108{,}953.3\ \text{kg·m} \\ M_e &= 2(365.0 + 290.68)(24.84) = 32{,}574.2\ \text{kg·m} \\ M &= 141{,}527.5\ \text{kg·m}, \quad H_t\sqrt{M} = 24.84\sqrt{141527.5} = 9{,}344.8 \end{aligned}$$

Tower steel per km

Towers per km $= 1000/S$, so the relative cost per km $= (1000/S) \times H_t\sqrt{M}$ (in units of $K$):

Moving from 250 m to 300 m saves more on the number of towers than it adds in tower weight. Beyond 300 m, the heavier and taller towers outweigh the saving.

Answer: the most economical span is 300 m, which has the lowest relative tower cost per km (36,578.1 K). Insulator and foundation costs per km also fall with longer spans, but the spread between 250 m and 350 m is small, so 300 m is a sound choice.