Hydrostatic thrust and uplift with drainage gallery

Data

• Upstream water depth $h_1 = 250 - 200 = 50$ m; tailwater depth $h_2 = 210 - 200 = 10$ m.
• Base width $B = 40$ m; drainage gallery 10 m from the vertical upstream face. $\gamma_w = 9.81$ kN/m³. Values are per metre length of dam.

Hydrostatic thrust

$$\begin{aligned} P_1 &= \frac{1}{2}\gamma_w h_1^2 = \frac{1}{2}\times9.81\times50^2 = 12{,}262.5\ \text{kN} \ (\text{at } 16.67\ \text{m above base}) \\ P_2 &= \frac{1}{2}\gamma_w h_2^2 = \frac{1}{2}\times9.81\times10^2 = 490.5\ \text{kN} \ (\text{at } 3.33\ \text{m above base}) \\ P_{net} &= 12{,}262.5 - 490.5 = 11{,}772\ \text{kN}\ (\text{downstream}) \end{aligned}$$

Uplift pressure

Net seepage head $= h_1 - h_2 = 40$ m. It is reduced by 50% at the gallery:

$$h_g = h_2 + 0.5(h_1 - h_2) = 10 + 20 = 30\ \text{m}$$

 heel      gallery              toe
  |<- 10 ->|<------- 30 -------->|
  |\       |                     |
  | \      |                     |
  |  \_____|_____________        |
  |        |             -----___|
  | 490.5  | 294.3          98.1 |

$$\begin{aligned} U_1 &= \frac{490.5 + 294.3}{2} \times 10 = 3{,}924\ \text{kN} \\ U_2 &= \frac{294.3 + 98.1}{2} \times 30 = 5{,}886\ \text{kN} \\ U &= 3{,}924 + 5{,}886 = 9{,}810\ \text{kN} \end{aligned}$$

Its line of action (moments about heel) is at 15.33 m from the heel.

For comparison, without the gallery: $U = \frac{490.5 + 98.1}{2}\times40 = 11{,}772$ kN, so the gallery reduces uplift by about 17%.

Answer: Net hydrostatic thrust = 11,772 kN/m (upstream 12,262.5 kN/m, tailwater 490.5 kN/m); uplift force = 9,810 kN/m acting 15.33 m from the heel.