--- title: "Khosla: uplift at A, B, thickness, exit gradient 0.2" subject: "Irrigation and Drainage Engineering" chapter: 5 key: "semester_6/irrigation_and_drainage_engineering:khosla-uplift-floor-thickness-fig-103" source: https://www.bctengineer.com/q/semester_6/irrigation_and_drainage_engineering/khosla-uplift-floor-thickness-fig-103 site: BCT Engineering --- # Khosla: uplift at A, B, thickness, exit gradient 0.2 Irrigation and Drainage Engineering · Chapter 5: Diversion Headworks - Asked in: 2070 Chaitra (old course), 16 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/irrigation_and_drainage_engineering/khosla-uplift-floor-thickness-fig-103 - Whole chapter: https://www.bctengineer.com/md/semester_6/irrigation_and_drainage_engineering/chapter-5 ## Question Calculate the uplift pressure at key points of piles in Fig 1. Also check the thickness of the floor at A, B location and exit gradient. The safe exit gradient is 0.2. [Figure: levels 103 m (pond), 100 m, 99.5 m (point C), 97 m, 96 m, 94.5 m (point A), 95 m (point B), 92 m and 92 m (pile bottoms); horizontal distances 15 m and 20 m] ## Short answer Uplift pressures at key points: E1=6.00m, D1=4.36m, C1=3.87m, E2=1.56m, D2=1.37m, C2=0. Floor at B (1.0m provided) requires 2.77m — unsafe. Floor at A (2.5m provided) requires 1.11m — safe. Exit gradient = 0.190 < 0.200 (safe). ## Answer #### 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 $\phi_{C1} = 100-\phi_E$. | Pile | d (m) | α | λ | φE (%) | φD (%) | |---|---|---|---|---|---| | U/s pile | 7.50 | 4.667 | 2.886 | 40.07 | 27.33 | | D/s pile | 5.00 | 7.000 | 4.036 | 33.17 | 22.90 | #### Pressures at the key points Pressure is expressed as a percentage of the head $H = 6.00$ m (no flow). | Point | Simple φ (%) | Corrections (%) | Corrected φ (%) | Pressure head (m) | |---|---|---|---|---| | E1 (top of u/s pile) | 100.00 | 0 | 100.00 | 6.00 | | D1 | 72.67 | - | 72.67 | 4.36 | | C1 | 59.93 | thk +0.85, int +3.77 | 64.55 | 3.87 | | E2 | 33.17 | thk -5.14, int -2.05 | 25.98 | 1.56 | | D2 | 22.90 | - | 22.90 | 1.37 | | C2 (d/s end) | 0 | 0 | 0 | 0 | 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). | Location | x (m) | φ (%) | HG level (m) | Floor top (m) | Unbalanced head (m) | Required t (m) | Provided t (m) | Check | |---|---|---|---|---|---|---|---|---| | B (x = 15 m) | 15 | 48.02 | 99.88 | 96.00 | 3.88 | 2.77 | 1.00 | not safe | | A (d/s end) | 35 | 25.98 | 98.56 | 97.00 | 1.56 | 1.11 | 2.50 | safe | #### 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%): ```text 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. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The head and b adopted are: A. H = 103 - 97 = 6 m, b = 35 m B. H = 3 m, b = 15 m C. H = 11 m, b = 20 m D. H = 6 m, b = 20 m Answer: A. No flow, d/s level at floor 97 m; b = 15 + 20. (Key point: Head and floor length) 2. Unbalanced head at a point is: A. HG level minus floor top level B. Pond - floor C. H x phi only D. Floor thickness Answer: A. The HG level at the section minus the floor top gives the head to be balanced. (Key point: Unbalanced head) 3. At B the required thickness 2.66 m exceeds 1.0 m, so: A. Not safe; thickness must be increased B. Safe C. Pile is deeper D. Exit gradient fails Answer: A. Provided thickness less than t = h/(G-1) means uplift can lift the floor. (Key point: Check at B) 4. At A, 1.07 m required vs 2.5 m provided: A. Safe B. Unsafe C. Equal D. Undefined Answer: A. Provided exceeds required. (Key point: Check at A) 5. G_E = 0.190 vs safe 0.2 means: A. Safe, but only marginally B. Unsafe C. Zero D. Requires larger pile Answer: A. 0.190 < 0.2, so the floor is safe against piping. 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