Khosla: barrage HGL and floor thickness

Data and assumptions

• Pond level 1135.0 m. For the no-flow condition the d/s water level is taken at the d/s floor level 1123.75 m, so the head is $H = 1135.0 - 1123.75 = 11.25$ m.
• The u/s sheet pile (bottom 1120.5 m) is 6.0 m deep below the u/s floor (1126.5 m). The d/s pile (bottom 1118.5 m) is 5.25 m deep below the d/s floor (1123.75 m). Floor length between the piles $b = 40$ m.
• Floor thickness 1.0 m throughout (given); specific gravity of concrete $G = 2.4$.
• Slope correction for a 1:3 slope $= 4.5\%$, multiplied by (slope length / $b$): u/s slope length $1.5$ m (up the slope, subtracted at $C_1$); d/s glacis length $9.75$ m (down the slope, added at $E_2$).

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 $b = 40$ 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 $H = 11.25$ 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, $19\sqrt{D/b'}\,\dfrac{d+D}{b}$ (at $C_1$: $D = 1126.5-1118.5 = 8.0$ m, $d = 6$ m; at $E_2$: $D = 1123.75-1120.5 = 3.25$ m, $d = 5.25$ m; $b' = 40$ m), positive at $C_1$ and negative at $E_2$; slope = slope correction. 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

$$G_E = \frac{H}{d}\cdot\frac{1}{\pi\sqrt{\lambda}} = \frac{11.25}{5.25}\times\frac{1}{\pi\sqrt{4.342}} = 0.3273\ (\approx 1/3.1)$$

Permissible $= 1/5 = 0.2000$. Not safe against piping; deepen the d/s pile or lengthen the floor.

Hydraulic gradient line

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

 E1  100.0% |########################################
 D1   76.6% |###############################
 C1   70.6% |############################
 E2   29.9% |############
 D2   22.0% |#########
 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 $E_2$).

Conclusion: For no flow, the floor needs about 3.7 m thickness at the u/s junction and at the toe of the glacis, reducing to about 2.4 m at the d/s pile. The 1.0 m provided is not enough anywhere, so the floor must be thickened (or the u/s floor lengthened and the u/s pile deepened to reduce the uplift). The exit gradient (1/3.1) exceeds the safe value of 1/5, so the profile is also unsafe against piping; the d/s pile must be deepened.