---
title: "Location of maximum sag, supports at different levels"
subject: "Transmission and Distribution System Design"
chapter: 4
key: "semester_8/transmission_and_distribution_system_design:max-sag-location-derivation"
source: https://www.bctengineer.com/q/semester_8/transmission_and_distribution_system_design/max-sag-location-derivation
site: BCT Engineering
---

# Location of maximum sag, supports at different levels

Transmission and Distribution System Design · Chapter 4: Conductor and Support Selection

- Asked in: 2068 Bhadra (old course), 4 marks; 2067 Chaitra (old course), 6 marks (asked 2 times)
- Video: [Lecture#30: Sag Calculation When Supports are at Equal Level and Unequal Levels | Dr. J. A. Laghari](https://www.youtube.com/watch?v=U6FkyPhCpXg) — Dr. J. A. LAGHARI
- 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/max-sag-location-derivation
- Whole chapter: https://www.bctengineer.com/md/semester_8/transmission_and_distribution_system_design/chapter-4

## Question

Starting from the expression for the maximum sag, derive the expression for location of the maximum sag on a transmission conductor supported at different level.

## Short answer

For supports at different levels with height difference h, the lowest point is offset by Th/(wL) from mid-span toward the lower support: x1 = L/2 – Th/(wL), x2 = L/2 + Th/(wL). If h > wL²/(2T), the point lies outside the span.

## Answer

For supports at different levels, the conductor is approximated by a parabola with its lowest point O. The "maximum sag" from each support is measured down to O.

Let $L$ be the span, $h$ the difference in support levels (B higher), $w$ the weight per metre and $T$ the horizontal tension. Let $x_1$ and $x_2$ be the horizontal distances of O from the lower support A and the higher support B.

#### Expressions for maximum sag
Taking O as the origin, $y = \dfrac{w x^2}{2T}$, so

$$
S_1 = \frac{w x_1^2}{2T}\ \text{(from A)}, \qquad S_2 = \frac{w x_2^2}{2T}\ \text{(from B)}
$$

#### Location of maximum sag
The difference of the sags is the difference in support levels:

$$
\begin{aligned} h &= S_2 - S_1 = \frac{w}{2T}\left(x_2^2 - x_1^2\right) \\ &= \frac{w}{2T}(x_2 + x_1)(x_2 - x_1) \end{aligned}
$$

Since $x_1 + x_2 = L$:

$$
x_2 - x_1 = \frac{2Th}{wL}
$$

Adding and subtracting with $x_1 + x_2 = L$:

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

The point of maximum sag therefore lies $\dfrac{Th}{wL}$ from mid-span, towards the lower support.

- With equal supports ($h = 0$): $x_1 = x_2 = L/2$.
- If $h > wL^2/2T$, then $x_1 < 0$ and the lowest point falls outside the span (**virtual sag**). The lower tower then has uplift.
- Under wind and ice, $w$ is replaced by the resultant load per metre.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. With the lowest point O as origin, the sag at horizontal distance x is:
   A. y = wx/T
   B. y = w x^2/(2T)
   C. y = T x^2/(2w)
   D. y = w x^2/T
   Answer: B. Parabolic approximation y = wx^2/(2T). (Key point: Parabola)

2. How is the difference in support levels h expressed?
   A. h = S2 - S1 = (w/2T)(x2^2 - x1^2)
   B. h = S1 + S2
   C. h = w L/T
   D. h = (w/2T)(x1 + x2)
   Answer: A. The two sags differ by the level difference. (Key point: Level difference)

3. Using x1 + x2 = L, the distance x1 of the lowest point from the lower support is:
   A. L/2 + Th/(wL)
   B. L/2 - Th/(wL)
   C. L/2
   D. Th/(wL)
   Answer: B. x2 - x1 = 2Th/(wL), so x1 = L/2 - Th/(wL) (lowest point shifts toward the lower support). (Key point: Result)

4. When does the lowest point fall outside the span?
   A. When h = 0
   B. When h > wL^2/(2T)
   C. When h < wL^2/(2T)
   D. When T is large
   Answer: B. Then x1 < 0, giving virtual sag and uplift at the lower tower. (Key point: Virtual sag)

## Related in Transmission and Distribution System Design

- [Maximum sag with supports at different levels](https://www.bctengineer.com/question.md?key=semester_8%2Ftransmission_and_distribution_system_design%3Amax-sag-derivation-unequal-levels) — Transmission and Distribution System Design, chapter 4
- [Position of maximum sag, unequal supports](https://www.bctengineer.com/question.md?key=semester_8%2Ftransmission_and_distribution_system_design%3Amax-sag-position-unequal-levels) — Transmission and Distribution System Design, chapter 4, asked 4×
- [T/F: sag absent with supports at different heights](https://www.bctengineer.com/question.md?key=semester_8%2Ftransmission_and_distribution_system_design%3Atf-sag-absent-different-heights) — Transmission and Distribution System Design, chapter 4
- [T/F: maximum sag in stringing condition](https://www.bctengineer.com/question.md?key=semester_8%2Ftransmission_and_distribution_system_design%3Atf-max-sag-stringing-condition) — Transmission and Distribution System Design, chapter 4

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