---
title: "Elementary profile of gravity dam"
subject: "Hydropower Engineering"
chapter: 4
key: "semester_7/hydropower_engineering:elementary-profile-definition-proportioning"
source: https://www.bctengineer.com/q/semester_7/hydropower_engineering/elementary-profile-definition-proportioning
site: BCT Engineering
---

# Elementary profile of gravity dam

Hydropower Engineering · Chapter 4: Headworks of Storage Plants

- Asked in: 2080 Bhadra, 1+5 marks
- 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_7/hydropower_engineering/elementary-profile-definition-proportioning
- Whole chapter: https://www.bctengineer.com/md/semester_7/hydropower_engineering/chapter-4

## Question

Define the elementary profile of a gravity dam. How do you proportion the dimensions of an elementary profile if the reservoir is full?

## Short answer

The elementary profile is a right-angled triangle with vertical upstream face and apex at maximum water level. Base width is proportioned from no-tension condition B = H/√(S_c−K) and sliding condition B = H/(μ(S_c−K)), taking the larger value, with K=0 for no uplift or 1 for full uplift.

## Answer

#### Elementary profile
The **elementary profile** of a gravity dam is the basic right-angled triangular section with a vertical upstream face, its apex at the maximum water level (zero top width) and the base width $B$ at the foundation. It is the smallest section that is stable under water pressure, self-weight and uplift, and it is the starting point for the practical profile (which adds crest width and freeboard).

```text
  apex (MWL)
     |\
     | \
   H |  \   d/s face
     |   \
     +----\
       B
```

#### Proportioning when the reservoir is full
Let $H$ = height of water, $S_c$ = specific gravity of the dam material, $K$ = uplift coefficient (fraction of full uplift; $K=1$ means full uplift at the heel), $\mu$ = coefficient of friction, $\gamma_w$ = unit weight of water. Per metre length:

- Weight $W=\tfrac12 B H S_c\gamma_w$ at $\tfrac23B$ from the toe (centroid of the triangle)
- Water thrust $P=\tfrac12\gamma_wH^2$ at $H/3$ above the base
- Uplift $U=\tfrac12 K\gamma_w H B$ at $\tfrac23B$ from the toe

**1. No tension (resultant at the lower third point, $e=B/6$):** the resultant should cut the base at $B/3$ from the toe, so moments about the toe give

$$
(W-U)\tfrac{B}{3}=(W-U)\tfrac{2B}{3}-P\tfrac{H}{3}
$$

$$
\tfrac{1}{3}\cdot\tfrac12\gamma_wBH(S_c-K)\,B=\tfrac12\gamma_wH^2\cdot\tfrac{H}{3}\;\Rightarrow\;B=\frac{H}{\sqrt{S_c-K}}
$$

(With no uplift, $K=0$ and $B=H/\sqrt{S_c}$.)

**2. Sliding (friction, $FS=1$):** $\mu(W-U)\ge P$

$$
\mu\cdot\tfrac12\gamma_wBH(S_c-K)\ge\tfrac12\gamma_wH^2\;\Rightarrow\;B=\frac{H}{\mu(S_c-K)}
$$

**3. Stress at the toe:** with the resultant at the kern, $\sigma_{toe}=\dfrac{2\Sigma V}{B}=\gamma_wH(S_c-K)$ and the principal stress is $\sigma_1=\sigma_{toe}(1+\tan^2\phi)=\gamma_wH(S_c-K+1)$ (it must not exceed the permissible stress).

The base width is taken as the **larger** of the widths from the no-tension and sliding conditions, if the stress is within the limit. The apex is then raised by adding a crest width and freeboard to get the practical profile.

## Check yourself

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

1. What is the elementary profile of a gravity dam?
   A. A trapezoid with a wide crest and freeboard
   B. A right-angled triangle with vertical u/s face and apex at the maximum water level, i.e. zero top width
   C. A rectangle with base equal to height
   D. A curved arch section
   Answer: B. It is the minimum triangular section stable under water pressure, weight and uplift; crest width and freeboard are added later for the practical profile. (Key point: Definition of elementary profile)

2. Requiring the resultant at the lower third point (e = B/6) gives which base width?
   A. B = H / (mu (S_c - K))
   B. B = H / sqrt(S_c - K)
   C. B = H sqrt(S_c - K)
   D. B = H (S_c - K)
   Answer: B. Taking moments about the toe with W = 1/2 B H S_c gamma_w and U = 1/2 K gamma_w H B gives B = H / sqrt(S_c - K). (Key point: Width for no tension)

3. What base width results from the sliding condition mu(W - U) >= P?
   A. B = H / sqrt(S_c - K)
   B. B = H / (mu (S_c - K))
   C. B = mu H / (S_c - K)
   D. B = H (S_c - K) / mu
   Answer: B. mu x 1/2 gamma_w B H (S_c - K) = 1/2 gamma_w H^2 gives B = H / (mu (S_c - K)). (Key point: Width for sliding)

4. How is the final base width chosen, and what is the toe stress at the kern?
   A. The smaller of the two widths; sigma_toe = gamma_w H (S_c - K)
   B. The larger of the two widths (if stress is within limit); sigma_toe = 2 SumV / B = gamma_w H (S_c - K)
   C. The average of the two; sigma_toe = SumV / B
   D. The no-tension width always; sigma_toe = 0
   Answer: B. The larger width satisfies both conditions, and with the resultant at the kern the toe stress is 2 SumV / B = gamma_w H (S_c - K), giving sigma_1 = sigma_toe (1 + tan^2 phi). (Key point: Choosing the final width and toe stress)

## Related in Hydropower Engineering

- [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
- [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 equilibrium and overtopping discharge coefficient](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Agravity-dam-reservoir-capacity-curve-hrwl) — 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
- [Gravity dam stability (SG 2.4, RL 0-70)](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Agravity-dam-stability-2082-bhadra) — Hydropower Engineering, chapter 4
- [Gravity dam section: shear stress near toe (2082)](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Agravity-dam-stability-2082-baishakh) — Hydropower Engineering, chapter 4

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