---
title: "Elementary profile width by no tension and no sliding"
subject: "Hydro Power"
chapter: 4
key: "semester_6/hydro_power:elementary-profile-no-tension-sliding"
source: https://www.bctengineer.com/q/semester_6/hydro_power/elementary-profile-no-tension-sliding
site: BCT Engineering
---

# Elementary profile width by no tension and no sliding

Hydro Power · Chapter 4: Dam Engineering

- Asked in: 2075 Baisakh, 3+5 marks; 2071 Bhadra, 8 marks (asked 2 times)
- Video: [003_DHS- Elementary Profile of a Gravity Dam](https://www.youtube.com/watch?v=D484z7gorPs) — Athul Mangalath Madhu
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/hydro_power/elementary-profile-no-tension-sliding
- Whole chapter: https://www.bctengineer.com/md/semester_6/hydro_power/chapter-4

## Question

What is meant by elementary profile of a gravity dam? Explain for fixing the width of the elementary profile of concrete gravity dam by considering no tension and no sliding criteria.

## Short answer

The elementary profile is a right-angled triangular shape with vertical upstream face, zero top width, water up to apex, resisting pressure with least material. For no tension, base width B = H / sqrt(Sc - C); for no sliding, B = H / (μ (Sc - C)); the larger value is adopted.

## Answer

#### Elementary profile
The **elementary profile** of a gravity dam is the **right-angled triangle** with a vertical upstream face, zero top width and base width $B$, with water standing up to its apex. It is the theoretical shape that resists water pressure with the least material, because water pressure is zero at the top and increases linearly with depth, exactly like the triangle's width. Under empty reservoir the resultant passes through the inner middle-third point, and under full reservoir (with proper base width) through the outer middle-third point, so there is no tension. A practical profile is made by adding a top width, freeboard and other details to it.

```text
       apex (water level)
        |\
        | \
  P --> |  \     H
        | W \
        |    \
   heel |_____\ toe
        <--B-->
        ^ uplift ^
```

#### Forces (per metre length)
Let $S_c$ = specific gravity of concrete, $\gamma_w$ = unit weight of water, $C$ = uplift intensity factor.
- Self weight: $W = \frac{1}{2}BH S_c \gamma_w$, acting at $B/3$ from the heel.
- Water pressure: $P = \frac{1}{2}\gamma_w H^2$, acting at $H/3$ above the base.
- Uplift: $U = \frac{1}{2}C\gamma_w H B$, acting at $B/3$ from the heel.

#### Base width for no tension
For no tension, the resultant must pass through the **outer middle-third point** (distance $B/3$ from the toe, i.e. $2B/3$ from the heel). Taking moments about this point: $W$ and $U$ both have lever arm $B/3$, and $P$ has lever arm $H/3$.

$$
\begin{aligned}
(W - U)\frac{B}{3} &= P\frac{H}{3} \\
\frac{1}{2}\gamma_w BH(S_c - C)\,B &= \frac{1}{2}\gamma_w H^2 \cdot H \\
B^2(S_c - C) &= H^2 \\
B &= \frac{H}{\sqrt{S_c - C}}
\end{aligned}
$$

Without uplift ($C = 0$): $B = H/\sqrt{S_c}$. For $S_c = 2.4$, $B = 0.645H$; with $C = 1$, $B = 0.845H$.

#### Base width for no sliding
For no sliding, the frictional resistance must at least equal the horizontal force (μ = coefficient of friction):

$$
\begin{aligned}
\mu (W - U) &\geq P \\
\mu \cdot \frac{1}{2}\gamma_w BH(S_c - C) &= \frac{1}{2}\gamma_w H^2 \\
B &= \frac{H}{\mu (S_c - C)}
\end{aligned}
$$

Without uplift: $B = H/(\mu S_c)$.

#### Design base width
The base width adopted is the **larger** of the two values. Example: $S_c = 2.4$, $C = 1$, $\mu = 0.7$ gives $B = 0.845H$ (no tension) and $B = 1.02H$ (no sliding), so $B = 1.02H$ would be adopted. For usual values of $\mu$ (0.65–0.75), the sliding criterion often governs.

The limiting height of the elementary profile (from allowable stress $f$) is $H = f/[\gamma_w(S_c - C + 1)]$; a dam taller than this is a "high" gravity dam.

## Check yourself

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

1. What is the elementary profile?
   A. A trapezoid with wide crest
   B. An arch
   C. A right-angled triangle with vertical upstream face and zero top width, water to the apex
   D. A rectangle
   Answer: C. It is the theoretical minimum-material shape, as pressure increases linearly with depth. (Key point: Elementary profile)

2. Where must the resultant pass for no tension?
   A. Through the heel
   B. Through the outer middle-third point, B/3 from the toe
   C. Through the centre of base
   D. Outside the base
   Answer: B. Taking moments about that point gives B = H/sqrt(Sc - C). (Key point: No-tension condition)

3. What base width follows from no sliding?
   A. B = H/sqrt(Sc - C)
   B. B = mu H/(Sc - C)
   C. B = H/(mu(Sc - C))
   D. B = H (Sc - C)
   Answer: C. From mu(W - U) = P, B = H/[mu(Sc - C)]. (Key point: No sliding width)

4. Which value is adopted for design?
   A. The larger of the no-tension and no-sliding widths
   B. The smaller
   C. The average
   D. The no-tension width always
   Answer: A. The larger satisfies both; e.g. 1.02H governs over 0.845H. (Key point: Design width)

## Related in Hydro Power

- [Elementary profile and 40 m gravity dam stability](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Agravity-dam-elementary-40m) — Hydro Power, chapter 4
- [Elementary profile base width formula proof](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Aelementary-profile-base-width-proof) — Hydro Power, chapter 4
- [Sliding and overturning check of gravity dam](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Agravity-dam-sliding-overturning-figure) — Hydro Power, chapter 4
- [Triangular gravity dam 35 m high](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Atriangular-dam-35m) — Hydro Power, chapter 4
- [Stresses in gravity dam with drain holes](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Agravity-dam-stresses-figure) — Hydro Power, chapter 4
- [Stability of trapezoidal gravity dam (RL 250 m)](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Atrapezoidal-dam-stability-250m) — Hydro Power, chapter 4

## Related in other subjects

- [Elementary profile of gravity dam](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Aelementary-profile-definition-proportioning) — Hydropower Engineering, chapter 4
- [Minimum safe width of 18 m elementary profile](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Aelementary-profile-min-width-18m) — Hydropower Engineering, chapter 4
- [Gravity dam stability: flotation, overturning, sliding](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Agravity-dam-stability-flotation-shear-resistance) — Hydropower Engineering, chapter 4
- [Design elementary profile of masonry dam](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Aelementary-profile-stone-masonry-design) — Hydropower Engineering, chapter 4