Uplift at inside corners, 50 m floor weir

Given: floor length $b = 50$ m, cutoffs at both ends, depth $d = 8$ m below the river bed (the floor top is flush with the bed), head $H = 5$ m, floor thickness $t_1 = 1$ m (u/s end) and $t_2 = 2$ m (d/s end). Khosla's theory is used. The inside corners are the junctions of the floor with the u/s pile ($C_1$) and with the d/s pile ($E_2$).

Khosla's constants

$$\alpha = \frac{b}{d} = \frac{50}{8} = 6.25,\qquad \lambda = \frac{1 + \sqrt{1+\alpha^2}}{2} = 3.665$$ $$\begin{aligned} \phi_E &= \frac{1}{\pi}\cos^{-1}\left(\frac{\lambda-2}{\lambda}\right) = 34.99\% \\ \phi_D &= \frac{1}{\pi}\cos^{-1}\left(\frac{\lambda-1}{\lambda}\right) = 24.09\% \end{aligned}$$

Pressures from the simple profile (thin floor)

Correction for mutual interference of piles

Both piles are 8 m deep ($d = D = 8$ m), spaced $b' = 50$ m apart, total floor length $b = 50$ m:

$$C = 19\sqrt{\frac{D}{b'}}\left(\frac{d+D}{b}\right) = 19\sqrt{\frac{8}{50}}\times\frac{16}{50} = 2.43\%$$

It is positive at $C_1$ (upstream of the interfering pile) and negative at $E_2$ (downstream of it).

Correction for floor thickness

The uplift acts on the underside of the floor, a distance $t$ below the key point, so the pressure is interpolated along the pile:

$$\begin{aligned} \text{at } C_1: &\ (\phi_{D_1} - \phi_{C_1})\frac{t_1}{d} = (75.91 - 65.01)\times\frac{1}{8} = +1.36\% \\ \text{at } E_2: &\ (\phi_{E_2} - \phi_{D_2})\frac{t_2}{d} = (34.99 - 24.09)\times\frac{2}{8} = -2.73\% \end{aligned}$$

Corrected pressures and uplift heads

$$\begin{aligned} \phi_{C_1}' &= 65.01 + 2.43 + 1.36 = 68.80\%,\qquad h_{C_1} = 0.6880\times 5 = 3.44\ \text{m} \\ \phi_{E_2}' &= 34.99 - 2.43 - 2.73 = 29.83\%,\qquad h_{E_2} = 0.2983\times 5 = 1.49\ \text{m} \end{aligned}$$

Exit gradient

$$G_E = \frac{H}{d}\cdot\frac{1}{\pi\sqrt{\lambda}} = \frac{5}{8}\times\frac{1}{\pi\sqrt{3.665}} = 0.1039\ (\approx 1/9.6)$$

Answer: Uplift at the u/s inside corner $\approx 68.8\%$ of $H$ (3.44 m); at the d/s inside corner $\approx 29.8\%$ (1.49 m); exit gradient $G_E \approx 0.104$, which is less than the safe gradient of about 1/6 (0.167) for fine sand, so the structure is safe against piping.