Most economical span from sag table

The tower height is built up from the ground clearance, the sag and the conductor spacing. The tower weight is then found from the overturning moment of the wind load with Ryle's formula, and the total tower steel per km is compared.

Tower heights

With 6 m minimum ground clearance and conductors 2 m apart vertically (from the figure), and the ground wire neglected:

Working tension $T = \text{UTS}/\text{FOS} = 15910/2 = 7955$ kg. All towers are straight-line towers, so there is no deviation load. $T$ is needed only to check that the given sags are consistent with it. The 8.7 m horizontal width is the same for every span, so it does not affect the comparison.

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.

Wind load on each conductor ($p = 100$ kg/m², $d = 0.028$ m)

Overturning moment and tower weight index

$M = F_c(h_1 + h_2 + h_3)$. For example, at 250 m: $M = 700 \times (9.24 + 11.24 + 13.24) = 23{,}604$ kg·m and $H_t\sqrt{M} = 13.24\sqrt{23604} = 2{,}034.1$.

Tower steel per km

Here the sag rises quickly with span (3.24 m to 6.12 m), so both tower height and moment grow faster than the number of towers falls.

Answer: the most economical span is 250 m (lowest relative tower cost, 8,136.6 K per km).