Uplift pressure diagram with drainage gallery

Data

Water depth $h=50$ m, tail water $h_t=5$ m, base width $B=30$ m (top width 10 m), $\gamma_w=9.81$ kN/m³. Uplift acts on 60% of the base area, so the intensity is reduced to 60% of the full hydrostatic value (reduction factor 0.6).

Full hydrostatic pressures: heel $p_h=\gamma_wh=490.5$ kN/m²; toe $p_t=\gamma_wh_t=49.05$ kN/m².

(i) Without a drainage gallery

The uplift varies linearly from the heel to the toe, giving a trapezoidal diagram below the base.

 Heel                              Toe
  +--------------------------------+   <- base of dam (30 m)
  |                                |
  |  uplift (trapezoid)        ____|
  |                  _________/    29.43 kN/m2
  |        _________/
  |_______/
  294.3 kN/m2
  (full values: 490.5 at the heel, 49.05 at the toe)

Effective ordinates (60%): heel $=0.6\times490.5=294.3$ kN/m², toe $=0.6\times49.05=29.43$ kN/m².

$$U_1=0.6\times\tfrac12(p_h+p_t)B=0.6\times\tfrac12(490.5+49.05)\times30=4,856\ \text{kN/m}$$

(Full value without the 60% factor: 8,093 kN/m.)

(ii) With a drainage gallery 6 m from the u/s face

The gallery reduces the pressure at its position to the tail water pressure plus one-third of the difference:

$$p_g=p_t+\tfrac13(p_h-p_t)=49.05+\tfrac13(490.5-49.05)=196.2\ \text{kN/m}^2$$

The diagram is then a two-segment line: heel $\to$ gallery $\to$ toe. Effective ordinates (60%): heel 294.3, gallery 117.7, toe 29.43 kN/m².

 Heel      G (6 m)                 Toe
 490.5 --.
          `-.
              196.2 (gallery)
                  `-.______
                           49.05
 0     6                    30 m

Areas:

• Heel to gallery: $\tfrac12(490.5+196.2)\times6=2,060.1$ kN/m
• Gallery to toe: $\tfrac12(196.2+49.05)\times24=2,943.0$ kN/m
• Total (full) $=5,003$ kN/m; with the 60% factor $U_2=0.6\times5,003=3,002$ kN/m

Answer: (i) total uplift ≈ 4,856 kN/m (heel ordinate 294 kN/m², toe 29 kN/m²). (ii) With the gallery the uplift falls to ≈ 3,002 kN/m, a reduction of 38%.