Khosla: uplift at A, B, thickness, exit gradient 0.2

Data and assumptions

The figure is partly unclear. Reading adopted: pond level 103 m; two pile lines with floor length $b = 15 + 20 = 35$ m. U/s pile: floor level (point C) 99.5 m, bottom 92 m ($d = 7.5$ m). D/s pile: floor level 97 m, bottom 92 m ($d = 5$ m). With no flow the d/s water level is the d/s floor level 97 m, so $H = 6$ m. Point B is on the depressed floor at 15 m (floor top 96 m, bottom 95 m, thickness 1.0 m); point A is at the d/s end (floor top 97 m, bottom 94.5 m, thickness 2.5 m). Floor thickness 0.5 m at the u/s end. Safe exit gradient 0.2.

Khosla's constants

$\alpha = b/d$, $\lambda = \dfrac{1+\sqrt{1+\alpha^2}}{2}$, $\phi_E = \dfrac{1}{\pi}\cos^{-1}\dfrac{\lambda-2}{\lambda}$, $\phi_D = \dfrac{1}{\pi}\cos^{-1}\dfrac{\lambda-1}{\lambda}$ (in percent), total floor length $b = 35$ m. For the u/s pile, $\phi_{D1} = 100-\phi_D$ and $\phi_{C1} = 100-\phi_E$.

Pressures at the key points

Pressure is expressed as a percentage of the head $H = 6.00$ m (no flow).

Corrections: thk = floor thickness, $+\dfrac{\phi_D-\phi_C}{d}t$ at C and $-\dfrac{\phi_E-\phi_D}{d}t$ at E; int = mutual interference, $19\sqrt{D/b'}\,\dfrac{d+D}{b}$ with $b' = 35$ m (at $C_1$: $D = 99.5-92 = 7.5$ m, $d = 7.5$ m; at $E_2$: $D = 97-92 = 5$ m, $d = 5$ m), positive at $C_1$ and negative at $E_2$. The D points are not corrected.

Floor thickness

Unbalanced head $h$ = hydraulic-gradient level at the floor $-$ floor top level; required thickness $t = \dfrac{h}{G-1}$ with $G = 2.40$ (the HGL is taken as a straight line from $C_1$ to $E_2$ along the floor).

Exit gradient

$$G_E = \frac{H}{d}\cdot\frac{1}{\pi\sqrt{\lambda}} = \frac{6.00}{5.00}\times\frac{1}{\pi\sqrt{4.036}} = 0.1901\ (\approx 1/5.3)$$

Permissible $= 0.2000$. Safe (less than the permissible value).

Hydraulic gradient line

Bars show the corrected pressure (each # is 2.5%):

 E1  100.0% |########################################
 D1   72.7% |#############################
 C1   64.6% |##########################
 E2   26.0% |##########
 D2   22.9% |#########
 C2    0.0% |

The hydraulic gradient line (HGL) drops vertically across each pile (from E to D to C) and runs as a straight line along the floor between the pile junctions ($C_1$ to $E_2$).

Answer: The floor at B (1.0 m provided, 2.77 m needed) is not safe; the floor at A (2.5 m provided, 1.11 m needed) is safe. The exit gradient of 0.190 is just within the safe value of 0.2.