Khosla (analytical): uplift and d/s floor thickness

Data and assumptions

Reading of the figure: u/s water level 291 m; no tail water, so the d/s water level is taken at the d/s floor level 288 m and $H = 3$ m. U/s pile at 0 m: floor 289 m, bottom 285 m ($d = 4$ m). Intermediate pile at 10 m ($7 + 3$): floor 289 m, bottom 282 m ($d = 7$ m); the d/s floor at 288 m starts just downstream of it. D/s pile at 22 m: floor 288 m, bottom 281.5 m ($d = 6.5$ m). Floor length $b = 22$ m. Floor thickness is not known, so the thickness correction is neglected. $G = 2.4$.

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), $b = 22$ 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: 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 below the floor level at the pile being corrected, $b'$ = distance between the two piles; e.g. at $C_2$: $D = 289-281.5 = 7.5$ m, $d = 7$ m, $b' = 12$ m). 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 $G = 2.40$ (on the d/s floor the HGL is taken as a straight line from $C_2$ to $E_3$).

Exit gradient

$$G_E = \frac{H}{d}\cdot\frac{1}{\pi\sqrt{\lambda}} = \frac{3.00}{6.50}\times\frac{1}{\pi\sqrt{2.265}} = 0.0976\ (\approx 1/10.2)$$

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

Answer: Uplift pressures at the key points are in the table above (percent of $H$ = 3 m). The d/s floor of the regulator needs about 0.95 m thickness at its start (just d/s of the intermediate pile), reducing to about 0.9 m at mid-length and 0.83 m at the d/s end; provide at least these values.