Elementary profile of gravity dam

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).

  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.