--- title: "Khosla: two piles, floor at A, B, C, exit 0.2" subject: "Irrigation and Drainage Engineering" chapter: 5 key: "semester_6/irrigation_and_drainage_engineering:khosla-two-piles-thickness-abc-02" source: https://www.bctengineer.com/q/semester_6/irrigation_and_drainage_engineering/khosla-two-piles-thickness-abc-02 site: BCT Engineering --- # Khosla: two piles, floor at A, B, C, exit 0.2 Irrigation and Drainage Engineering · Chapter 5: Diversion Headworks - Asked in: 2063 Asoj (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-two-piles-thickness-abc-02 - Whole chapter: https://www.bctengineer.com/md/semester_6/irrigation_and_drainage_engineering/chapter-5 ## Question Calculate the uplift pressure at key points of two piles in figure below. Also check the thickness of the floor at A, B, C locations and exit gradient. The safe exit gradient is 0.2. [Figure: water level 105 m, crest 102 m; point A at floor level 100 m (pile bottom 94 m, 99 m), point B at 99.5 m, point C at 95 m (93.5 m), downstream pile bottom 90 m; horizontal distances 20 m, 10 m and 20 m] ## Short answer Uplift pressures: E1=10.00 m, C1=7.37 m; E2=2.51 m. Floor thickness: at A need 1.70 m (unsafe), B need 0.66 m (safe), C need 1.79 m (unsafe). Exit gradient 0.271 (>0.2) — not safe against piping. ## Answer #### Data and assumptions Reading of the figure (two pile lines, as the question states): water level 105 m; with no flow the d/s water level is the d/s floor level 95 m, so $H = 10$ m. A = u/s pile junction at $x = 0$: floor top 100 m, floor bottom 99 m (1 m thick), pile bottom 94 m ($d = 6$ m). B = point on the floor at $x = 20$ m: floor top 99.5 m, thickness taken as 1.0 m. C = d/s pile junction at $x = 50$ m ($20 + 10 + 20$): floor top 95 m, bottom 93.5 m (1.5 m thick), pile bottom 90 m ($d = 5$ m). $G = 2.4$, 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 $b = 50$ 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 | 6.00 | 8.333 | 4.697 | 30.53 | 21.16 | | D/s pile | 5.00 | 10.000 | 5.525 | 27.98 | 19.45 | #### Pressures at the key points Pressure is expressed as a percentage of the head $H = 10.00$ m (no flow). | Point | Simple φ (%) | Corrections (%) | Corrected φ (%) | Pressure head (m) | |---|---|---|---|---| | E1 (top of u/s pile) | 100.00 | 0 | 100.00 | 10.00 | | D1 | 78.84 | - | 78.84 | 7.88 | | C1 | 69.47 | thk +1.56, int +2.72 | 73.75 | 7.37 | | E2 | 27.98 | thk -2.56, int -0.32 | 25.10 | 2.51 | | D2 | 19.45 | - | 19.45 | 1.95 | | 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 $b' = 50$ m (at $C_1$: $D = 100-90 = 10$ m, $d = 6$ m; at $E_2$: $D = 95-94 = 1$ 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 | |---|---|---|---|---|---|---|---|---| | A (x = 0) | 0 | 73.75 | 102.37 | 100.00 | 2.37 | 1.70 | 1.00 | not safe | | B (x = 20) | 20 | 54.29 | 100.43 | 99.50 | 0.93 | 0.66 | 1.00 | safe | | C (x = 50) | 50 | 25.10 | 97.51 | 95.00 | 2.51 | 1.79 | 1.50 | not safe | #### Exit gradient $$ G_E = \frac{H}{d}\cdot\frac{1}{\pi\sqrt{\lambda}} = \frac{10.00}{5.00}\times\frac{1}{\pi\sqrt{5.525}} = 0.2708\ (\approx 1/3.7) $$ Permissible $= 0.2000$. Not safe; deepen the d/s pile or lengthen the floor. **Answer:** The floor is not safe at A (1.70 m needed, 1.0 m provided) or at C (1.79 m needed, 1.5 m provided), and is safe at B (0.66 m needed). The exit gradient of 0.271 exceeds 0.2, so the structure is not safe against piping. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Water level 105 m and d/s floor level 95 m give H = A. 3 m B. 5 m C. 12 m D. 10 m Answer: D. H = 105 - 95 = 10 m. (Key point: Head with no flow) 2. At A the required thickness is 1.50 m but 1.00 m is provided. The verdict is A. Safe B. Not safe C. Safe because thickness exceeds 0.65 m D. Not required Answer: B. Provided 1.0 m is less than the required 1.5 m. (Key point: Check at A) 3. At B the requirement is 0.65 m and 1.0 m is provided. Why is B safe? A. Because G is large B. Because B is at the middle C. Because the exit gradient is low D. Provided thickness exceeds the thickness needed to balance uplift Answer: D. t_required = h/(G-1) = 0.65 m, which is less than 1.0 m provided. (Key point: Check at B) 4. G_E = 0.2708 against the safe 0.2. The structure is A. Not safe against piping B. Safe if the floor thickness is 1.5 m C. Safe against piping D. Safe since lambda is above 5 Answer: A. 0.2708 exceeds 0.2, so the exit gradient is unsafe. 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