River crossing: minimum clearance above water

For supports at unequal heights, the lowest point of the conductor is first located. If it falls outside the span, the minimum clearance occurs at the lower support.

Data: $L = 300$ m, $h = 60-40 = 20$ m, $T = 3000$ kg, $w = 1200$ kg/km $= 1.2$ kg/m.

Position of the lowest point

Let $x_1$ be the horizontal distance from the lowest point to the lower tower, and $x_2$ the distance to the higher tower.

$$x_1 = \frac{L}{2} - \frac{Th}{wL}, \qquad x_2 = \frac{L}{2} + \frac{Th}{wL}$$ $$\frac{Th}{wL} = \frac{3000\times20}{1.2\times300} = 166.67\ \text{m}$$ $$x_1 = 150 - 166.67 = -16.67\ \text{m}, \qquad x_2 = 150 + 166.67 = 316.67\ \text{m}$$

Since $x_1$ is negative, the vertex of the parabola lies 16.67 m outside the span, beyond the 40 m tower.

Check: $\dfrac{w(x_2^2 - x_1^2)}{2T} = \dfrac{1.2(316.67^2 - 16.67^2)}{6000} = 20.0$ m, which equals $h$.

                                  60 m
                              ___--o
                      ___---``     |
   vertex   40 m ___--             |
     .  ___--o``                   |
  <-16.7->   |   <------ 300 m --->|
 ~~~~~~~~~~~~|~~~~~~ water ~~~~~~~~|~~~

Minimum clearance

Within the span, the conductor rises continuously from the 40 m tower to the 60 m tower. The vertex would be only $\dfrac{w x_1^2}{2T} = \dfrac{1.2\times16.67^2}{6000} = 0.056$ m below the lower support level, but it lies outside the span. So the lowest point actually on the conductor is at the lower support.

The tower heights are taken as the heights of the conductor attachment points.

Answer: the minimum clearance between the conductor and the water is 40 m, at the lower tower. There is no sag dip within the span, because the vertex falls 16.7 m outside it.