Forces on gravity dam and stability requirements

A gravity dam is a solid concrete or masonry dam that resists all external forces by its own weight. The forces acting on it are:

          crest
   ~~~~~ +---+  <- wave pressure Pw
   |     |   |\
 Pi ->   |   | \  <- wind
   |     | W |  \    W = self weight
 P1 ->   |   |   \   (+ EQ: aW horizontal)
   |     |   | W2 \
 Ps ->   |   |     \  ~~~~ tailwater
 (silt)  |   |      \  <- P2
         +---+-------+ toe
  heel   ^  ^  ^  ^  ^
         U (uplift)

Forces acting on a gravity dam

1. Self weight (W): the main stabilising force; $W = \gamma_c \times$ area per metre length, acting at the centroid. Divided into rectangles and triangles for calculation.
2. Water pressure:
   • Upstream horizontal: $P_1 = \frac{1}{2}\gamma_w h_1^2$ at $h_1/3$ above base.
   • Vertical weight of water on a sloping upstream face (if any).
   • Tailwater: $P_2 = \frac{1}{2}\gamma_w h_2^2$ acting upstream, plus weight of water on the downstream slope.
3. Uplift pressure (U): water seeping through the foundation and joints pushes upward. It varies from $C\gamma_w h_1$ at heel to $\gamma_w h_2$ at toe; drainage galleries reduce it (head at drains $= h_2 + \frac{1}{3}(h_1 - h_2)$).
4. Silt pressure (Ps): silt deposited against the upstream face: $P_s = \frac{1}{2}\gamma_{sub} h_s^2 K_a$, $K_a = \frac{1-\sin\phi}{1+\sin\phi}$, at $h_s/3$.
5. Wave pressure (Pw): due to wind waves; wave height $h_w = 0.032\sqrt{VF} + 0.763 - 0.271F^{1/4}$ for fetch $F < 32$ km; $P_w = 2\gamma_w h_w^2$, acting $3h_w/8$ above still water level.
6. Earthquake forces:
   • Inertia force on the dam $= \alpha_h W$ horizontally (and $\alpha_v W$ vertically) at the centroid.
   • Hydrodynamic pressure of water: Westergaard/Zanger, e.g. $P_e = 0.726\,p_e\,h$ with $p_e = C_m\alpha_h\gamma_w h$.
7. Ice pressure: in cold regions, about 250 kN/m² over the ice thickness.
8. Wind pressure: small; usually neglected.

Stability requirements

The dam must be safe for all load combinations (empty reservoir, full reservoir, flood, earthquake):

Also, the foundation must be safe in bearing and the dam must have adequate freeboard to avoid overtopping.