Most economical EHV span from sag table

The most economical span is the one with the least tower weight (cost) per km. Tower heights come from the ground clearance plus the sag, and the loads are wind only, because all towers are straight-line towers (θ = 0).

Tower heights

From the figure, the vertical spacing is 3 m between the top and middle conductors and 3 m between the middle and bottom conductors. The ground wire is neglected, so the top conductor sets the tower height.

$$\begin{aligned} h_1 &= \text{ground clearance} + D_{max} = 10 + D_{max}\\ h_2 &= h_1 + 3, \qquad h_3 = H_t = h_1 + 6 \end{aligned}$$

Bending moment

The wind load on each conductor, with the wind span equal to the span, is

$$F = p\,d\,L = 100\times0.021\times L = 2.1L\ \text{kg}$$ $$M = F(h_1+h_2+h_3)$$

There is no deviation (straight-line towers), so the conductor tension adds no transverse load. The UTS and area are therefore not needed for this comparison; they only fix the sag values, which are already given.

Tower weight

Tower weight is taken from Ryle's formula (as given with IOE papers):

$$W_t = 0.0016\,H_t\sqrt{M\times FS}\ \ \text{tonne}$$

Here $H_t$ is in ft, $M$ is the bending moment at the base in klb-ft, and FS = 2 (assumed). The conversions are 1 m = 3.281 ft and 1 kg·m = 0.007233 klb-ft. Tower cost is taken as proportional to tower weight, and the number of towers per km is about $1000/L$. So the span with the least tower weight per km is the most economical.

Sample (L = 250 m):

$$\begin{aligned} h_1 &= 14.36,\ h_2 = 17.36,\ h_3 = 20.36\ \text{m}\\ M &= 525\times(14.36+17.36+20.36) = 27{,}342\ \text{kg·m}\\ W_t &= 0.0016\times(20.36\times3.281)\sqrt{27342\times0.007233\times2} = 2.126\ \text{t}\\ \text{Weight/km} &= 2.126\times4 = 8.502\ \text{t/km} \end{aligned}$$

Results

As the span increases, the sag (and so the tower height) and the wind load both rise. The tower weight increases faster than the number of towers per km falls, so the weight per km rises with span.

Answer: the most economical span is 250 m (8.50 t/km of tower steel). In practice this is checked against fixed per-tower costs (foundations, insulators, erection), which favour somewhat longer spans.