Khosla: percentage pressure and piping, pond 503

Data and assumptions

Reading of the figure: pond level 503 m; d/s floor and bed level 497 m, so $H = 6$ m (no flow, no tail water). U/s pile 1.5 m deep at the u/s end (floor top 500 m, point A; pile bottom 498.5 m); d/s pile 2.0 m deep at the d/s end (floor top 497 m, point D; pile bottom 495 m); floor length $b = 6 + 10 = 16$ m. B is at the foot of the drop ($x = 6$ m) and C at $x = 11$ m. Floor thickness and slope corrections are ignored (as stated), but the mutual interference correction still applies. $G = 2.4$; minimum thickness 0.30 m; permissible exit gradient 0.15.

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 $b = 16$ 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: int = mutual interference at $C_1$ due to the d/s pile, $19\sqrt{D/b'}\,\dfrac{d+D}{b} = 19\sqrt{5/16}\times\dfrac{1.5+5}{16} = 4.31\%$, with $D = 500-495 = 5$ m and $d = 1.5$ m. At $E_2$ the u/s pile bottom (498.5 m) is above the d/s floor (497 m), so there is no interference correction.

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).

All of these exceed the 0.30 m minimum, so they govern.

Exit gradient

$$G_E = \frac{H}{d}\cdot\frac{1}{\pi\sqrt{\lambda}} = \frac{6.00}{2.00}\times\frac{1}{\pi\sqrt{4.531}} = 0.4486\ (\approx 1/2.2)$$

Permissible $= 0.1500$. Not safe against piping; the d/s pile must be deepened or the floor lengthened.

Answer: Corrected pressures: $E_1 = 100\%$, $D_1 = 81.1\%$, $C_1 = 77.2\%$, $E_2 = 31.1\%$, $D_2 = 21.6\%$, $C_2 = 0$. Floor thickness: A 1.17 m, B 2.57 m, C 1.95 m, D 1.33 m. The exit gradient (0.449) is about three times the permissible 0.15, so the floor is not safe against piping.