Khosla: uplift, HGL, thickness, exit gradient 1/5

Data and assumptions

Reading of the figure: water level RL 103.0 m, floor RL 100.0 m, so the head is $H = 3$ m (d/s water at floor level). Pile depths 2 m (u/s, at 0 m), 1.5 m (intermediate, at 10 m) and 3 m (d/s, at 30 m); floor length $b = 10 + 20 = 30$ m. Floor thickness provided: 1.0 m (u/s), 1.25 m (intermediate), 1.0 m (d/s). $G = 2.4$. Safe exit gradient $1/5$.

Khosla's constants

End piles: $\alpha = b/d$, $\lambda = \dfrac{1+\sqrt{1+\alpha^2}}{2}$, $\phi_E = \dfrac{1}{\pi}\cos^{-1}\dfrac{\lambda-2}{\lambda}$, $\phi_D = \dfrac{1}{\pi}\cos^{-1}\dfrac{\lambda-1}{\lambda}$ (in percent), $b = 30$ m; for the u/s pile $\phi_{D1} = 100-\phi_D$, $\phi_{C1} = 100-\phi_E$.

Intermediate pile ($b_1$ from the u/s end, $b_2$ from the d/s end): $\lambda_1 = \dfrac{\sqrt{1+\alpha_1^2}-\sqrt{1+\alpha_2^2}}{2}$, $\lambda = \dfrac{\sqrt{1+\alpha_1^2}+\sqrt{1+\alpha_2^2}}{2}$, $\phi_E = \dfrac{1}{\pi}\cos^{-1}\dfrac{\lambda_1-1}{\lambda}$, $\phi_D = \dfrac{1}{\pi}\cos^{-1}\dfrac{\lambda_1}{\lambda}$, $\phi_C = \dfrac{1}{\pi}\cos^{-1}\dfrac{\lambda_1+1}{\lambda}$.

Pressures at the key points

Pressure is expressed as a percentage of the head $H = 3.00$ 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, $19\sqrt{D/b'}\,\dfrac{d+D}{b}$, positive at C of the u/s pile and negative at E of the d/s pile of each pair ($D$ = depth of the neighbouring pile, $b'$ = distance between the two piles). The D points are not corrected. (Around the shallow intermediate pile the corrected values come out nearly equal, about 60 to 61%.)

Floor thickness

Unbalanced head $h$ = hydraulic-gradient level at the floor $-$ floor top level; required thickness $t = \dfrac{h}{G-1}$ with $G = 2.40$.

Exit gradient

$$G_E = \frac{H}{d}\cdot\frac{1}{\pi\sqrt{\lambda}} = \frac{3.00}{3.00}\times\frac{1}{\pi\sqrt{5.525}} = 0.1354\ (\approx 1/7.4)$$

Permissible $= 1/5 = 0.2000$. Safe (less than the permissible value).

Hydraulic gradient line

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

 E1  100.0% |########################################
 D1   83.9% |##################################
 C1   81.3% |#################################
 E2   60.3% |########################
 D2   60.7% |########################
 C2   61.2% |########################
 E3   24.4% |##########
 D3   19.5% |########
 C3    0.0% |

The hydraulic gradient line (HGL) drops across each pile (from E to D to C) and runs as a straight line along the floor between the pile junctions (C of one pile to E of the next).

Answer: The u/s floor (1.74 m needed, 1.0 m provided) is not safe, the floor at the intermediate pile is marginally short (about 1.3 m needed, 1.25 m provided), and the d/s floor is safe. The exit gradient of 1/7.4 is within the safe value of 1/5.