--- title: "River crossing: minimum clearance above water" subject: "Transmission and Distribution System Design" chapter: 4 key: "semester_8/transmission_and_distribution_system_design:river-crossing-clearance" source: https://www.bctengineer.com/q/semester_8/transmission_and_distribution_system_design/river-crossing-clearance site: BCT Engineering --- # River crossing: minimum clearance above water Transmission and Distribution System Design · Chapter 4: Conductor and Support Selection - Asked in: 2069 Bhadra (old course), 6 marks - Video: [Problem based on sag, two towers at different heights. Find the clearance between the conductor.](https://www.youtube.com/watch?v=Qpt5FUfdRTE) — BE Technical - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_8/transmission_and_distribution_system_design/river-crossing-clearance - Whole chapter: https://www.bctengineer.com/md/semester_8/transmission_and_distribution_system_design/chapter-4 ## Question A transmission line conductor at a river crossing is supported from two towers at height of 40 m and 60 m above water level. The horizontal distance between the towers is 300 meters. Find the minimum clearance between the conductor and water level. Assume maximum working tension in conductor is 3000 kg and conductor weight = 1200 kg/km. ## Short answer The vertex of the conductor lies 16.7 m outside the span toward the 40 m tower, so the lowest point in the span is at that tower. Minimum clearance to water is therefore 40 m. ## Answer 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 $h$. ```text 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.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For towers at unequal heights, what is the first step in finding the minimum clearance? A. Assume the sag is at mid-span B. Locate the lowest point (vertex) of the conductor and check whether it lies within the span C. Use the higher tower height as the clearance D. Average the two tower heights Answer: B. The vertex position x1 = L/2 - Th/(wL) decides whether the lowest point is inside the span or not. (Key point: Locating the lowest point) 2. Here Th/(wL) = 166.67 m exceeds L/2 = 150 m. What does this mean? A. The vertex is 16.67 m outside the span, beyond the lower tower B. The vertex is at mid-span C. The vertex is 16.67 m inside the span D. The conductor has a sag of 166.67 m Answer: A. x1 = 150 - 166.67 = -16.67 m, negative, so the vertex lies outside the span beyond the 40 m tower. (Key point: Vertex outside span) 3. Where does the minimum clearance occur, and what is its value? A. At mid-span, 50 m B. At the higher tower, 60 m C. At the lower tower, 40 m D. At the vertex, about 39.94 m Answer: C. The conductor rises continuously from the 40 m tower, so the lowest point on the actual conductor is the lower support, 40 m. (Key point: Minimum clearance location) 4. What value of w is used in the calculation for 1200 kg/km? A. 1200 kg/m B. 12 kg/m C. 0.12 kg/m D. 1.2 kg/m Answer: D. 1200 kg/km = 1.2 kg/m, which is used in x1 = L/2 - Th/(wL). 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