Position of maximum sag, unequal supports

When the two supports are at different heights, the lowest point of the conductor is not at mid-span. It shifts towards the lower support. Its position is found from the parabolic approximation.

Derivation

Let:

• $L$ = span (horizontal distance between supports A and B)
• $h$ = difference in level between the supports (B higher than A)
• $w$ = weight of conductor per unit length, and $T$ = horizontal tension
• $x_1$, $x_2$ = horizontal distances of the lowest point O from A (lower) and B (higher), so $x_1 + x_2 = L$

                                B
                              /|
                    S2      /  |
     A                    /    |  h
     |\               /        |
  S1 | \_____ O _____/  .......|
     |<- x1 ->|<----- x2 ---->|

Taking O as the origin, the conductor is the parabola $y = \dfrac{w x^2}{2T}$. So:

$$S_1 = \frac{w x_1^2}{2T}, \qquad S_2 = \frac{w x_2^2}{2T}$$ $$\begin{aligned} h &= S_2 - S_1 = \frac{w}{2T}(x_2^2 - x_1^2) \\ &= \frac{w}{2T}(x_2 + x_1)(x_2 - x_1) = \frac{wL}{2T}(x_2 - x_1) \\ \Rightarrow x_2 - x_1 &= \frac{2Th}{wL} \end{aligned}$$

Solving with $x_1 + x_2 = L$:

$$x_1 = \frac{L}{2} - \frac{Th}{wL}, \qquad x_2 = \frac{L}{2} + \frac{Th}{wL}$$

The lowest point (point of maximum sag) is at $x_1$ from the lower tower and $x_2$ from the higher tower. The sags are $S_1 = w x_1^2/2T$ below A and $S_2 = w x_2^2/2T$ below B. Under wind and ice, $w$ is replaced by the resultant loading $w_r$.

If the supports are at equal height ($h = 0$), $x_1 = x_2 = L/2$ and $S = wL^2/8T$.

Virtual sag

• If $h > \dfrac{wL^2}{2T}$, then $x_1 = \dfrac{L}{2} - \dfrac{Th}{wL}$ becomes negative. The lowest point O of the parabola then lies outside the span, beyond the lower support.
• The conductor rises all the way from A to B and has no real lowest point inside the span. The sag $S_1$ measured to the imaginary point O is called the virtual sag.
• This happens on steep hill slopes, which are common in Nepal. The conductor pulls upward on the lower tower (uplift). That tower must be designed as a tension tower, or guyed, and the string must be a tension string, because a suspension string would be pulled up.