Economical span, double circuit, A/B/C towers

The most economical span gives the least average tower weight (cost) per km. Here the line has a mix of tower types, so a weighted average tower weight is used.

Tower types assumed (usual IS 802 classification):

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.

Loads and bending moment

For each span $L$ the loads on a tower are found as follows. The wind span is taken equal to the span.

• Each power conductor (6 of them, 2 at each height $h_1, h_2, h_3$):

$$F_c = p\,d_c\,L + 2T_c\sin\frac{\theta}{2} = 100\times0.04\,L + 2\times6000\sin\frac{\theta}{2}\ \text{kg}$$

• Each earth wire (2 of them, at height $H_t$):

$$F_e = p\,d_e\,L + 2T_e\sin\frac{\theta}{2} = 100\times0.02\,L + 2\times2000\sin\frac{\theta}{2}\ \text{kg}$$

• Bending moment at the base:

$$M = 2F_c(h_1+h_2+h_3) + 2F_e H_t$$

Wind on the tower body is neglected because it is about the same for all spans.

The deviation factor $2\sin(\theta/2)$ is 0.0349 for 2°, 0.2611 for 15° and 0.5176 for 30°.

Sample (L = 250 m, tower A, θ = 2°):

$$\begin{aligned} F_c &= 1000 + 6000\times0.0349 = 1209.4\ \text{kg}\\ F_e &= 500 + 2000\times0.0349 = 569.8\ \text{kg}\\ M_A &= 2(1209.4)(44.994) + 2(569.8)(28.444) = 141{,}249\ \text{kg·m}\\ W_A &= 0.0016(93.32)\sqrt{141249\times0.007233\times2} = 6.749\ \text{t} \end{aligned}$$

The weighted weight per tower is $W = 0.8W_A + 0.15W_B + 0.05W_C$, and the weight per km is $W\times1000/L$.

Results

The weight per km is lowest at 250 m (29.819 t/km). 275 m is almost equal (29.834 t/km, only about 0.05% more). From 300 m onwards the weight rises clearly.

Answer: the most economical span is 250 m, with an average tower weight of 7.455 t and 29.82 t of tower steel per km. 275 m is practically as good and could be chosen if tower count or terrain favours it.