Elementary profile width by no tension and no sliding

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.

       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.