Uplift force with and without drainage gallery

Data (per metre length)

• Dam height 40 m, free board 3 m, so upstream water depth $h_1 = 37$ m.
• Tailwater depth $h_2 = 5$ m, base width $B = 30$ m, $\gamma_w = 9.81$ kN/m³.
• Full uplift is assumed (C = 1): pressure varies linearly from $\gamma_w h_1$ at the heel to $\gamma_w h_2$ at the toe.

(a) Without drainage gallery

Heel: $9.81 \times 37 = 362.97$ kN/m²; toe: $9.81 \times 5 = 49.05$ kN/m².

$$U = \frac{362.97 + 49.05}{2} \times 30 = 6{,}180.3\ \text{kN/m}$$

(b) With drainage gallery 6 m from upstream face

By the usual rule (IS 6512), the uplift head at the line of drains is the tailwater head plus one-third of the difference:

$$h_d = h_2 + \frac{1}{3}(h_1 - h_2) = 5 + \frac{32}{3} = 15.67\ \text{m} \Rightarrow p_d = 153.69\ \text{kN/m}^2$$

heel  drain                    toe
 |<-6->|<-------- 24 --------->|
 |\    |                       |
 | \   |                       |
 |  \__|______                 |
 |     |      -------______    |
362.97 153.69             49.05

$$\begin{aligned} U_1 &= \frac{362.97 + 153.69}{2} \times 6 = 1{,}550.0\ \text{kN/m} \\ U_2 &= \frac{153.69 + 49.05}{2} \times 24 = 2{,}432.9\ \text{kN/m} \\ U &= 1{,}550.0 + 2{,}432.9 = 3{,}982.9\ \text{kN/m} \end{aligned}$$

Answer: Uplift force = 6,180.3 kN per metre without drains; 3,982.9 kN per metre with the drainage gallery at 6 m. The gallery cuts uplift by about 36%.