Bending moment ratio, 132 kV vs 220 kV towers

The bending moment at the tower base is the sum of each transverse load times its height. Both lines use the same conductor and span, so they have the same sag and the same wind load per wire. The ratio therefore comes from the tower geometry: ground clearance, conductor spacing and earth wire height, all of which grow with voltage.

Data and assumptions

Step 1 – maximum sag (same for both lines)

$$\begin{aligned} T_1 &= 18253/2 = 9126.5\ \text{kg}\\ W_1 &= \sqrt{1.9714^2 + (100\times0.02989)^2} = 3.581\ \text{kg/m}\\ AE &= 5.298\times0.789\times10^6 = 4.180\times10^6\ \text{kg}\\ K_2 &= -9126.5 + 17.73\times10^{-6}\times60\times AE + \frac{W_1^2L^2AE}{24T_1^2}\\ &= -9126.5 + 4446.8 + 2412.8 = -2266.8\\ K_1 &= \frac{1.9714^2\times300^2\times AE}{24} = 6.092\times10^{10} \end{aligned}$$

Solving $T_2^2(T_2 - 2266.8) = 6.092\times10^{10}$ gives $T_2 = 4853$ kg. Then

$$D_{max} = \frac{1.9714\times300^2}{8\times4853} = 4.57\ \text{m}$$

Step 2 – clearances and tower geometry

Conductor heights at the tower:

Step 3 – transverse loads (wind span 300 m)

$$\begin{aligned} F_c &= p\,d\,L = 100\times0.02989\times300 = 896.7\ \text{kg per conductor}\\ F_e &= 100\times0.01098\times300 = 329.4\ \text{kg (earth wire)} \end{aligned}$$

Step 4 – bending moment at the base

$$M = F_c(h_1+h_2+h_3) + F_e H_t$$ $$\begin{aligned} M_{132} &= 896.7(10.666+13.865+17.064) + 329.4(23.086)\\ &= 37{,}298 + 7{,}605 = 44{,}903\ \text{kg·m}\\ M_{220} &= 896.7(11.479+16.361+21.244) + 329.4(29.810)\\ &= 44{,}013 + 9{,}820 = 53{,}833\ \text{kg·m} \end{aligned}$$

Step 5 – ratio

$$\frac{M_{132}}{M_{220}} = \frac{44{,}903}{53{,}833} = 0.834$$

   220 kV               132 kV
     E  29.8 m            E  23.1 m
     |                    |
   --*  21.2 m          --*  17.1 m
     *-- 16.4 m           *-- 13.9 m
   --*  11.5 m          --*  10.7 m
   ===== ground         ===== ground

Answer: the bending moment is about 44.9 t·m for the 132 kV tower and 53.8 t·m for the 220 kV tower, a ratio of M(132) : M(220) ≈ 0.83 : 1. The loads per wire are the same, so the 220 kV tower carries about 20% more moment only because it is taller, through larger clearances and spacing.